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abstract: 'Interlocus gene conversion (IGC) homogenizes paralogs. Little is known regarding the mutation events that cause IGC and even less is known about the IGC mutations that experience fixation. To disentangle the rates of fixed IGC mutations from the tract lengths of these fixed mutations, we employ a composite likelihood procedure. We characterize the procedure with simulations. We apply the procedure to duplicated primate introns and to protein-coding paralogs from both yeast and primates. Our estimates from protein-coding data concerning the mean length of fixed IGC tracts were unexpectedly low and are associated with high degrees of uncertainty. In contrast, our estimates from the primate intron data had lengths in the general range expected from IGC mutation studies. While it is challenging to separate the rate at which fixed IGC mutations initiate from the average number of nucleotide positions that these IGC events affect, all of our analyses indicate that IGC is responsible for a substantial proportion of evolutionary change in duplicated regions. Our results suggest that IGC should be considered whenever the evolution of multigene families is examined.'
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Article (Methods)\
[**A phylogenetic approach disentangles interlocus gene conversion tract length and initiation rate**]{}
Xiang Ji$^{\ast, 1,2,3}$, Jeffrey L. Thorne$^{\ast, 1,2,4}$\
$^{1}$Bioinformatics Research Center, North Carolina State University, Raleigh, NC, 27695\
$^{2}$Department of Statistics, North Carolina State University, Raleigh, NC, 27695\
$^{3}$Current Address: Department of Biomathematics, University of California, Los Angeles, CA, 90095\
$^{4}$Department of Biological Sciences, North Carolina State University, Raleigh, NC, 27695\
$^{*}$Correspondence: xji3@ucla.edu, thorne@statgen.ncsu.edu
**Keywords:** interlocus gene conversion, multigene family evolution, tract length
[Introduction]{}\[sec:Intro\]
=============================
Interlocus gene conversion (IGC) homogenizes repeats by copying a tract of sequence from one paralog to the equivalent region of another. This means that evidence of nucleotide substitution in one paralog can be erased and that the ancestry of an IGC tract coalesces in two paralogs when IGC events occur. As a result, IGC events partition the sequence sites of a multigene family into regions that have different evolutionary trees. This consequence of IGC has complicated the study of multigene family evolution, especially when the goals are to examine orthology and the history of gene duplication and loss. Incorporating tract length into IGC inference can therefore be helpful for disentangling the local correlation structure of the histories of sequence sites.
In [@Ji2016], we introduced an approach to incorporate IGC into any existing nucleotide substitution model by jointly considering the corresponding nucleotide or codon sites in different paralogs. One limitation of this modeling framework is that it assumes IGC events are independently experienced by sites within a paralog. The framework does not reflect the correlation structure among sites within a paralog that is induced by IGC and we will refer to it as the independent-site (IS) approach. Here, we extend the IS model by incorporating IGC tract information and introduce an accompanying inference procedure. We then illustrate the extensions of our IS approach by applying them to three diverse groups of data sets and by analyzing simulated data.
[New Approach]{} \[sec:NewApproach\]
====================================
The IS model can incorporate IGC by adding one new parameter ($\tau$) to any conventional substitution model that has changes originate with point mutation. The purpose of the additional parameter is to represent the homogenization among paralogs caused by IGC. For example, the HKY model [@Hasegawa1985] describes nucleotide substitutions that originate with point mutation and has substitution rates depend on the type of nucleotide being introduced and whether the substitution is a transition or a transversion. The HKY rate from nucleotide type $i$ to type $i'$ ($i \neq i'$) is $Q_{ii'}$ with $${Q_{ii'}} \propto \left\{ {\begin{array}{*{20}{c}}
{{\pi _{i'}}}&{{\mbox{if transversion}}}\\
{\kappa {\pi _{i'}}}&{{\mbox{if transition}}},
\end{array}} \right.
\label{Eq:HKY}$$ where $\pi_{i'}$ is the stationary probability of nucleotide type $i'$ ($\pi_A+\pi_C+\pi_G+\pi_T =1$) and $\kappa \geq 0$ differentiates transitions and transversions. With two paralogs per genome, the IS extension of the HKY model has $Q_{(i,j),(i',j')}$ be the instantaneous rate at which one paralog changes from state $i$ to $i'$ and the corresponding site of the other paralog changes from state $j$ to $j'$. The resulting rates for possible changes are: $${Q_{(i,j),(i',j')}} \propto \left\{ {\begin{array}{*{20}{c}}
0&{i \neq i',j \neq j'}\\
Q_{ii'} & {i \neq i', j=j', i' \neq j'}\\
Q_{ii'}+ \tau & {i \neq i', j=j', i' = j'}\\
Q_{jj'} & {i = i', j \neq j', i' \neq j'}\\
Q_{jj'} + \tau & {i = i', j \neq j', i'=j'}.
\end{array}} \right.
\label{Eq:HKY+IS-IGC}$$ IGC extensions to other conventional nucleotide or codon substitution models can be made in a similar fashion.
While the IS approach considers dependence due to IGC at corresponding sites in different paralogs, it assumes IGC occurs independently at different sites within the same paralog. As [@Nasrallah2010] noted, the assumption of evolutionary independence among sites can be biologically unreasonable, but is often kept for simplicity and computational tractability. Ideally, dependent evolution among sequence positions would be modeled by treating entire sequences as the state of a system (e.g., @Robinson2003). Having the state space consist of all possible sequences would lead to more realistic Markov models for describing how individual IGC events can affect multiple sites within a paralog. However, the size of the state space becomes large when it matches the number of possible sequences. This large size would not prove computationally tractable if employing the inference strategy outlined in [@Ji2016].
As a compromise between computational feasibility and realism, our new approach extends the IS model by jointly considering pairs of sites in a paralog as well as the corresponding pairs of sites in each other paralog. We refer to the new approach for separately estimating both the distribution of IGC tract lengths and the IGC initiation rate as the pair-site (PS) approach. The PS approach is a composite likelihood procedure that statistically resembles the population genetic technique of [@McVean2002] for estimating homologous recombination rates.
Relaxation of the site-independent assumption {#sec:TractModelSetup}
---------------------------------------------
The parameter $\tau$ of the IS approach (e.g., see Equation \[Eq:HKY+IS-IGC\]) can be interpreted as the average rate at which a site experiences IGC. This average can be further decomposed into two factors. The first is the rate per site of initiation of IGC events that are destined for fixation. The second is the average length of an IGC tract that becomes fixed. We note that the average length of a fixed IGC tract may be less than the average tract length of an IGC mutation because, subsequent to an IGC event, the sequence may experience homologous recombinations that cause the tract lengths of IGC mutations to differ from tract lengths of fixed IGC events. Here, we concentrate on inference regarding fixed IGC tracts. We ignore the possibility of fixation of non-contiguous sequence stretches arising from the tract produced by a single IGC mutation.
We model the rate per site at which fixed IGC events initiate as being $\eta$. We use a geometric distribution with parameter $p$ to model the fixed IGC tract length distribution so that the probability of a fixed IGC event covering $k$ sites is $p{(1 - p)^{k - 1}}$ with $k$ being a positive integer. The mean fixed tract length is therefore $\frac{1}{p}$. Although IGC events cannot initiate at a position that is $5'$ of a duplicated region and cannot extend $3'$ of a duplicated region, we ignore these “edge effects” by having the expected IGC rate be identical at all sites. This treatment can be interpreted as having some IGC tracts initiate $5'$ of the sequence regions being followed and/or terminate $3'$ of those regions. With this treatment, the rate at which a site experiences IGC can be written as $\tau = {\eta }/{p}$. The IS model can be viewed as the special case where $p = 1$.
The PS approach {#sec:PS-IGC}
---------------
Because computational constraints hinder the ability to employ codon-based substitution models, we implemented our PS expansion in conjunction with a modified HKY model [@Hasegawa1985] where the HKY rates are employed to describe nucleotide substitutions that originate with point mutation. Other conventional nucleotide substitution models could be used instead. [@Harpak2017] have independently developed a composite likelihood procedure that extends the IS model of [@Ji2016] to IGC tracts. Their approach is very similar to the one here (see also @Ji2017), but [@Harpak2017] achieve computational feasibility by reducing the $4$ nucleotide types to binary characters. This state space reduction may be especially problematic when sequences being analyzed are not closely related.
To better adapt our modified HKY model to analysis of protein-coding sequences, we add parameters that permit rate heterogeneity among codon positions. Specifically, the parameters denoted $r_2$ and $r_3$ respectively represent the ratios of fixation probabilities at second and third codon positions relative to the first codon position. The resulting model has rates $Q_{ii'}$ of fixed point mutations from nucleotide type $i$ to $i'$ being $${Q_{ii'}} \propto \left\{ {\begin{array}{*{20}{c}}
{r{\pi _{i'}}}&{{\mbox{if transversion}}}\\
{r\kappa {\pi _{i'}}}&{{\mbox{if transition}}},
\end{array}} \right.
\label{Eq:ModifiedHKY}$$ where $r = 1$ for the first codon position, $r = r_2$ for the second codon position and $r = r_3$ for the third codon position. We will refer to this as the independently-evolving paralog model (HKY-IND) in order to contrast it with our models that add dependence among paralogs due to IGC. The IGC treatment that combines the IS parameterization of Equation \[Eq:HKY+IS-IGC\] with the $Q_{ii'}$ rates of Equation \[Eq:ModifiedHKY\] will be referred to as the HKY+IS-IGC model and will be contrasted to the HKY+PS-IGC model that combines HKY-IND with the pair-site (PS) IGC treatment that is described below.
The PS approach jointly considers corresponding sites from all paralogs in the same genome in a pairwise manner. When there are two paralogs, the PS approach jointly considers the states of four nucleotides (two sites from each of the two paralogs). This transforms a 4-state nucleotide substitution model into a $4^4 = 256$-state joint nucleotide substitution model. Consider the site at position $a$ and the site at position $b$ ($a < b$) in paralog $i$. The states of these sites will be denoted $i_a$ and $i_b$. The corresponding states at positions $a$ and $b$ of paralog $j$ will be $j_a$ and $j_b$. We define $Q_{(i_a,i_b,j_a,j_b),(i_a',i_b',j_a',j_b')}$ to be the instantaneous rate at which $i_a$ changes to $i_a'$ and $i_b$ changes to $i_b'$ while $j_a$ changes to $j_a'$ and $j_b$ changes to $j_b'$. An IGC event involving one or both of the sites must yield $i_a' = j_a'$ or $i_b' = j_b'$ or both. This homogenization is reflected in the rates of the HKY+PS-IGC model by considering how often IGC events affect both positions $a$ and $b$ in the two paralogs and how often they affect just one of positions $a$ and $b$. The HKY+PS-IGC model adds these IGC contributions to the rates described by the above HKY-IND model. The rates at which IGC events simultaneously affect both positions $a$ and $b$ depend on the geometric length distribution of fixed IGC tracts. Specifically, assume positions $a$ and $b$ are separated by $n$ sites. The rate of IGC events that affect site $a$ but not site $b$ is $\frac{\eta }{p}\left[ {1 - {{\left( {1 - p} \right)}^n}} \right]$. Likewise, the rate of IGC events that affect site $b$ but not site $a$ is $\frac{\eta }{p}\left[ {1 - {{\left( {1 - p} \right)}^n}} \right]$. The rate of IGC events that simultaneously affect both sites $a$ and $b$ is $\frac{\eta }{p}{\left( {1 - p} \right)^n}$. This means that the evolutionary dependence between sites $a$ and $b$ due to IGC is a function of the separation between $a$ and $b$ along the sequence and becomes weak when ${\left( {1 - p} \right)^n}$ is near $0$.
Changes that can only be caused by point mutations have the rates defined only from the rates of the HKY-IND model (e.g., ${Q_{(A,C,G,T),(C,C,G,T)}} = {Q_{AC}}$). Changes that can be caused by either one point mutation event or by an IGC event that covers only one of the two sites have the rates defined by the sum of the HKY-IND point mutation rate and the IGC rate (e.g., ${Q_{(A,C,G,T),(G,C,G,T)}} = {Q_{AG}} +
\frac{\eta }{p}\left[ {1 - {{\left( {1 - p} \right)}^n}} \right]$). Changes that can be caused by either one point mutation event or by an IGC event that affects either only one or both of the two sites have the rates defined by the sum of the HKY-IND point mutation rate and the two IGC rates (e.g., ${Q_{(A,T,G,T),(G,T,G,T)}} = {Q_{AG}} + \frac{\eta }{p}$). Because the instantaneous rate of point mutations originating at two sites that are both destined for fixation is assumed to be negligible, changes that simultaneously modify both sites are exclusively determined by the IGC contribution (e.g., ${Q_{(A,C,G,T),(G,T,G,T)}} =
\frac{\eta }{p}{\left( {1 - p} \right)^n}$). The off-diagonal entries of this rate matrix that may be non-zero therefore have this structure:
$$Q_{(i_a,i_b,j_a,j_b),(i_a',i_b',j_a',j_b')} = \left\{ {\begin{array}{ll}
Q_{i_a,i_a'} \quad & {\mbox{if } i_a \ne i_a',i_b = i_b',j_a = j_a' \ne i_a',j_b = j_b'}\\
Q_{i_a,i_a'} + \frac{\eta }{p}\left[ {1 - (1 - p)^n} \right] \quad &{\mbox{if }i_a \ne i_a',i_b = i_b',j_a = j_a' = i_a',j_b = j_b' \ne i_b'}\\
Q_{i_a,i_a'} + \frac{\eta }{p} \quad &{\mbox{if }i_a \ne i_a',i_b = i_b', j_a = j_a' = i_a', j_b = j_b' = i_b'}\\
{\frac{\eta }{p}(1 - p)^n} \quad & {\mbox{if }i_a \ne i_a',i_b \ne i_b', j_a = j_a' = i_a', j_b = j_b' = i_b'}\\
{...}&{}
\end{array}} \right.
\label{Eq:HKY+PS-IGC}$$
Although Equation \[Eq:HKY+PS-IGC\] only details rates for changes that alter site $a$ in paralog $i$, other rates can be derived similarly. In summary, instantaneous rates can be positive only if the corresponding change could be caused by a single point mutation or a single IGC event or both. To infer parameter values, we follow [@McVean2002] by forming a composite likelihood that is the product over all possible pairs of sites of pairwise marginal likelihood. Let $s_1, s_2, ..., s_N$ represent the $N$ columns in a multiple sequence alignment. For the situation where there are exactly two paralogs (paralog $i$ and $j$), each alignment column will be assumed to specify the corresponding states of both paralog $i$ and paralog $j$ from all species being considered. Also, a column may have corresponding nucleotides from outgroup taxa that diverged prior to the time when paralogs $i$ and $j$ were formed by the duplication. Parameter values are estimated by maximizing the composite likelihood. Specifically, the branch lengths and rates in the model will be denoted by the vector $\theta$. The maximum composite likelihood estimate (MCLE) of $\theta$ is therefore $${\hat \theta _{MCLE}} = \operatorname*{arg\,max}_{\theta} \prod\limits_{1 \le a < b \le N} {\Pr ({s_a},{s_b}|\theta )},
\label{Eq:Objective}$$ where each pairwise marginal likelihood is calculated with Felsenstein’s pruning algorithm [@Felsenstein1981] and where Equation \[Eq:HKY+PS-IGC\] determines rates for portions of the species phylogeny that have two paralogs present and Equation \[Eq:ModifiedHKY\] specifies rates when only one paralog is present.
Our inference approach treats the complex correlation between multiple sites with a pairwise composite likelihood. As reviewed in [@Varin2011], maximum composite likelihood estimates of pairwise composite likelihood functions are asymptotically unbiased. This means that parameter estimates should approach their true values as paralog length increases. For simple cases where the full likelihood could be calculated in their study of homologous recombination, [@McVean2002] showed that their composite likelihood estimate was close to the maximum (full) likelihood estimate but with variance exceeding that of the full likelihood. Our PS-IGC approach is a pairwise composite likelihood method that is similar in spirit to the one of [@McVean2002] and should have qualitatively similar behavior.
Since it is not a valid likelihood function, conventional asymptotic variance calculations via the Fisher information matrix are not applicable to the PS-IGC approach. The uncertainty of the PS-IGC parameter estimates is approximated in this study through the parametric bootstrap (e.g., see @Goldman1993). A less computationally demanding alternative might be to approximate the uncertainty via the inverse of the Godambe information matrix (e.g., see @Kent1982; @Varin2011).
[Results]{} \[sec:Results\]
============================
We analyzed both actual and simulated data sets. Information regarding the software implementation and the numerical optimization scheme are provided in the Materials and Methods.
Analyses of Simulated Data
--------------------------
Simulations were performed to characterize our pair-site composite likelihood IGC procedure as detailed in Materials and Methods. Figure \[fig:PSIGCSimulationStudy\] summarizes estimates of the average fixed tract length (i.e., $1/p$) from these simulations. Sometimes, simulations yielded inferences for $1/p$ (i.e., estimated average tract lengths) that were more than 10-fold higher than the true value. These extreme values were not included in sample mean calculations but they were included in the reported interquartile ranges. No estimated values were discarded for expected tract lengths of 3, 10, 50 and 100 whereas 3, 7, 6 and 7 estimated values were respectively discarded for the expected tract lengths of 200, 300, 400 and 500.
With the important caveat that not all parameters were estimated from simulated data (see Materials and Methods), the simulations indicate that average estimated mean tract lengths are relatively close to the true values but the variability of expected tract length estimates increases as expected tract length increases. Presumably, this correlation is partially attributable to the fact that all simulation scenarios share the same value for the product of the IGC tract initiation rate and the expected tract length. Therefore, the actual numbers of IGC events will vary more among simulated data sets when tracts are long but the expected number of tracts per simulation is small.
Analyses of Actual Data
-----------------------
We applied the PS approach to analyze three groups of datasets. The first group consists of the $14$ data sets that [@Ji2016] analyzed with their codon-based IS approach for studying IGC. The second group is actually a single primate data set that [@Zhang1998] considered in their pioneering work on the origin of gene function. The third group of data sets are segmentally-duplicated intron regions from primates. [@Harpak2017] studied the effects of IGC on these intron regions with their binary-state treatment and we chose these data to examine how IGC inferences are affected by instead using maximum composite likelihood with a nucleotide-based model.
### Yeast Paralogs {#sec:YeastResults}
Yeast experienced an ancient genome-wide duplication (@Wolfe1997; @Philippsen1997; @Kellis2004; @Dietrich2004; @Dujon2004). [@Ji2016] analyzed 14 data sets of yeast protein-coding genes to characterize IGC that occurred subsequent to the genome-wide duplication. These data sets were the only ones that remained after applying stringent filters that were designed to reduce concerns about sequence alignment and paralogy status (see @Ji2016). While the filters did not require it, all 14 data sets happen to encode ribosomal proteins. In every data set, six yeast species are each represented by two paralogs that stem from the ancient genome-wide duplication. Each data set also includes a sequence from a species ([*L. kluyveri*]{}) that diverged from the other six prior to the genome-wide duplication. The species represented in these data sets are related by the well-established phylogenetic tree topology of Figure \[fig:YeastSpeciesTree\].
Table \[tab:IS-IGCYeastResults\] summarizes some of the results obtained by analyzing the 14 yeast data sets with the HKY+IS-IGC model. As was the case when these data sets were analyzed with the codon-based model of [@Ji2016], Table \[tab:IS-IGCYeastResults\] shows that a substantial proportion of sequence change is attributed to IGC by both the HKY+IS-IGC and HKY+PS-IGC models. Table \[tab:IGCestimates\] contrasts other inferences from the HKY+IS-IGC and HKY+PS-IGC models for these 14 data sets. It shows both models yield very similar estimates of $\tau$. Also, Table \[tab:IGCestimates\] reveals that the expected fixed IGC tract lengths tend to be quite short according to the HKY+PS-IGC estimates. In fact, only 1 of the 14 data sets yields an expected tract length that exceeds 100 nucleotides and 8 of the 14 data sets yield expected tract lengths that are less than 20 nucleotides.
### Primate EDN and ECP
[@Zhang1998] studied primate paralogs that encode eosinophil cationic protein (ECP) and eosinophil-derived neurotoxin (EDN). As described in the Materials and Methods, we used the sequence data and tree topology (see Figure \[fig:PrimateSpeciesTree\]) of [@Zhang1998]. We analyzed these data with both the IS and PS frameworks. Whereas our PS implementation has the advantage of accounting for IGC tracts, our IS implementation is computationally feasible with codon-based substitution models. Using an adaptation of the Muse-Gaut codon model [@Muse1994] that we will denote the MG94+IS-IGC model (see @Ji2016) and considering the changes that occurred subsequent to the EDN/ECP duplication, we estimate that approximately 10.3% of the codon substitutions originated with an IGC event rather than a point mutation.
In [@Ji2016], we introduced a “paralog-swapping” procedure that is intended to investigate whether inferred IGC levels are not actually due to IGC events but are instead attributable to estimation artifacts that arise because of imperfect evolutionary models. The paralog-swapping procedure uses the two paralogs from each of two taxa that are descended from a post-duplication speciation event. It compares the biologically plausible scenario where IGC involves paralogs in the same genome to a biologically implausible scenario that has IGC between paralogs in different genomes (see Figure \[fig:EDN\_ECP\_SwapTest\]). The idea underlying the paralog-swapping procedure is that the inferred IGC levels will be similar between the two scenarios if artifacts due to unrealistic evolutionary models are generating the putative IGC signal. When we apply the paralog-swapping procedure in conjunction with the HKY+IS-IGC model, we infer much more IGC with the biologically plausible scenario. With the biologically plausible scenario of Figure \[fig:EDN\_ECP\_SwapTest\]A, the IGC parameter $\tau$ is estimated to be about $0.28$. With the biologically implausible scenario of Figure \[fig:EDN\_ECP\_SwapTest\]B, the IGC parameter $\tau$ is estimated to be about $0.00004$ (i.e., very close to $0$).
Table \[tab:IS-IGCYeastResults\] includes results obtained by analyzing the EDN/ECP data set with the HKY+IS-IGC model. While the proportion of sequence change attributable to IGC is smaller for the EDN/ECP data set than any of the 14 yeast data sets, Table \[tab:IS-IGCYeastResults\] shows that this inferred proportion is substantial whether the EDN/ECP data are analyzed with the HKY+IS-IGC model or the codon-based IGC treatment of [@Ji2016]. As was the case for most of the yeast data sets, Table \[tab:IGCestimates\] shows that the fixed IGC tract lengths that are inferred from EDN/ECP are very short.
### Primate Introns
[@Harpak2017] examined the effects of IGC on the evolution of segmentally-duplicated primate introns. Here, we analyze a subset of the same data set in order to characterize IGC with our more parameter-rich models. As described in the Materials and Methods, we further filtered the introns of [@Harpak2017] to yield 20 data subsets where all ingroup species have two paralogs and the outgroup species has one paralog. This was done to lessen uncertainty regarding paralogy versus orthology. This filter also serves to make a single branch on the rooted primate tree be where the duplication event likely occurred for all 20 data subsets.
Table \[tab:IS-IGCYeastResults\] includes results obtained by analyzing the primate introns under the HKY+IS-IGC model that has no rate heterogeneity (i.e. $r_2 = r_3 = 1$). It shows that the proportion of sequence change attributable to IGC is smaller for the primate intron data than for EDN/ECP and than for each of the 14 yeast data sets, but this proportion is still relatively large. In contrast to the results from the yeast and EDN/ECP protein-coding data, Table \[tab:IGCestimates\] shows that the fixed IGC tract lengths that are estimated from the intron data have expected values that exceed several hundred nucleotides.
[Discussion]{}\[sec:Discussion\]
=================================
Fixed IGC tracts inferred to be short
-------------------------------------
A distinction can be made between the lengths of IGC mutations and the lengths of IGC tracts that experience fixation. Measurements of the lengths of IGC mutations vary widely among studies and model systems, with some events affecting only about $10$ nucleotides and with more typical lengths involving hundreds of consecutive positions (e.g., see @Chen2007 [@Mansai2011]). Our estimates of the mean lengths of fixed IGC tracts also exhibit substantial variation (see Table \[tab:IGCestimates\]). Whereas our estimates of the mean fixed IGC tract lengths from intron data tend to be not dramatically different than tract length estimates for IGC mutations, our estimates from exon data of the lengths of fixed IGC tracts (see Table \[tab:IGCestimates\]) are substantially smaller than are usually obtained from studies of IGC mutations. This disparity may be partially attributable to homologous recombination events that occur subsequent to IGC mutation and that could thereby make the lengths of fixed IGC tracts shorter than the length of the original IGC mutations. Natural selection may enhance the disparity between lengths of IGC mutation tracts and lengths of fixed tracts. IGC mutations can simultaneously introduce multiple fitness-affecting sequence changes into a paralog. Some of the changes may be advantageous whereas others may be deleterious. Subsequent homologous recombination may separate the advantageous and disadvantageous changes and thereby favor the fixation of IGC tracts that are shorter than the lengths of the original IGC mutations.
Biologically implausible evolutionary models represent a more mundane explanation for the short lengths that we estimate for fixed exonic IGC tracts. One shortcoming of the analyses presented here is that our treatment of changes due to IGC is more unrealistic than the treatment of substitutions that originated with point mutation. For nucleotide substitutions that originated by point mutation, the pair-site approach reflects the difference of fixation rates among the three codon positions by setting the relative rate at the first codon position to $1$ and adding a separate relative rate parameter for the second position (e.g., $r_2$) and another for the third position (e.g., $r_3$). However, the pair-site approach does not have differential fixation rates according to which codon positions are affected by the IGC event. This IGC treatment is convenient for setting up the statistical model but may be biologically unrealistic. Furthermore, we assume that IGC mutations occur and fix at a rate that is independent of the differences between the IGC donor and recipient tracts. This assumption violates evidence that IGC mutations become less likely as paralogs diverge (see @Chen2007).
Whereas the exons analyzed in this study are from genes that contain either no intron or a short one, further investigations with more genes and more sophisticated IGC models may shed light on whether fixed IGC stretches really are shorter in exons than introns. These future investigations might benefit from studying protein-coding genes with both long coding regions and long introns.
IGC inference procedures
------------------------
[@Harpak2017] recently developed two inferential procedures so that they could study IGC in segmentally-duplicated regions of humans and other primates. One of these procedures employed a hidden Markov model (HMM) that relied on the assumption that each sequence site has experienced either $0$ or $1$ IGC events subsequent to the duplication that created the paralogs being studied. Based on analyses of simulated data, we found a closely related but distinct HMM procedure to yield misleading estimates of expected IGC tract lengths, possibly because the IGC levels that were examined caused serious violations of the HMM assumptions [@Ji2017]. HMM approaches have proven useful for diverse DNA and protein sequence analysis tasks, but their value for studying IGC is likely to be particularly sensitive to how closely aligned are the HMM assumptions to the data being analyzed.
The other inferential procedure explored by [@Harpak2017] was a pair-site IGC treatment that resembles the one introduced here (see also @Ji2017). An attractive feature of the pair-site approaches is that they employ maximum composite likelihood for inference rather than relying on HMM assumptions that are most justified when IGC is rare. The pair-site approach of [@Harpak2017] considers two sites and two paralogs jointly but treats all sequence sites as being binary in nature. As a result, their joint state space has a size of $2^4 = 16$ where each site has two possible states representing two allele types with one state being the nucleotide type observed in mouse and with the other binary state collectively representing the other $3$ nucleotide types. Furthermore, @Harpak2017 assume constant generation time and known divergence times on the lineage separating the primates from their most recent common ancestor with mouse. The @Harpak2017 pair-site approach seems most appropriate for evolutionary scales where sequence changes are rare and where species are relatively closely related so that parameters such as generation time have little variation across the tree. It may be less satisfactory for larger timescales, such as the one that encompasses the period from the common ancestor of rodents and primates to the current day. In contrast, our parameterizations have been aimed at timescales of this magnitude.
@Harpak2017 concluded from their intron analyses that the IGC rate per sequence position is an order of magnitude faster than the point mutation rate. Because the rates of the IS-IGC and PS-IGC approaches are normalized so that the expected rate per paralog per site is $1$ for substitutions that originated with a point mutation, our estimated $\tau$ values of approximately $0.5$ from both the IS-IGC and the PS-IGC approaches suggest that IGC happens at roughly the same rate per site of each paralog as point mutation. In addition, we estimate the percentage of nucleotide substitutions that originate with an IGC event rather than a point mutation to be $9.3\%$ for the filtered intron datasets. The estimated average tract length from our PS-IGC approach has the same order of magnitude as estimated by @Harpak2017. Because of the substantial differences between our analyses and those of @Harpak2017, it is unclear how to isolate the cause of the different results.
Abundant substitution due to IGC
--------------------------------
Following [@Ji2016], we used the MG94+IS-IGC model to estimate the proportion of fixed codon changes that were attributable to IGC rather than point mutation. We also employed the HKY+IS-IGC model to infer the proportion of nucleotide changes that were attributable to IGC rather than point mutation. For both the Yeast data sets and the primate EDN/ECP data, the estimated proportions are somewhat higher for the HKY+IS-IGC model than for the MG94+IS-IGC model (see Table \[tab:IS-IGCYeastResults\]). While we expect the proportion for the HKY+IS-IGC model to be somewhat higher than for the MG94+IS-IGC model because individual events that change multiple codon positions are each only counted once in the MG94+IS-IGC calculations, we expect that most of the disparity in proportions is due to the differences between models.
While there is some disparity in IGC estimates among procedures, the more important message from Table \[tab:IS-IGCYeastResults\] is there is a substantial amount of evolutionary change that is due to IGC. This has implications for the evolutionary consequences of gene duplications. Gene duplication is considered an important source of novel gene function. After their formation, duplicated genes may experience neo-functionalization, sub-functionalization or become pseudogenized (e.g., see @Lynch2000 [@Walsh2003]). In the absence of IGC, these events are determined by mutations accumulating independently at each paralog. [@Teshima2008] showed that IGC can slow down the fixation of the neofunctionalized paralog by overwriting the neofunctionalized paralog sequence by that of another paralog. In this way, IGC can oppose natural selection. Furthermore, consideration of IGC could potentially improve the dating of subfunctionalization or neofunctionalization events.
Our conclusion is that fixation of IGC mutations is an important source of evolutionary change in multigene families. While the relative rate of IGC experienced per site (i.e., $\tau$) can be relatively well estimated by our IS and PS techniques, our PS approach yielded estimates of mean tract length from protein-coding data that were unexpectedly short and that were associated with high degrees of uncertainty. This motivates development of additional inference procedures that might be better than the PS procedure at extracting IGC tract information. In addition, there is ample room for improvement of IGC inference procedures. One important future direction will be to have the inference procedures assess how IGC rates decrease with paralog divergence. In addition, many multigene families consist of more than two paralogs and methods are needed for characterizing IGC in these cases. Because our analyses suggest that IGC is responsible for a substantial proportion of molecular evolution in multigene families, we believe that improved IGC inference should be a high priority.
[Materials and Methods]{} \[sec:MaterialsMethods\]
==================================================
Rate Normalization
------------------
For all analyses, substitution rates were normalized so that the expected rate per paralog per nucleotide is $1$ for substitutions that originated with a point mutation and so that the reported values of $\tau$ can then be compared to this normalized value. The normalization leads to one unit of branch length representing one expected substitution arising from point mutation per paralog per site. Because the model has rate heterogeneity among the three codon positions, the normalization to an average rate of $1$ yields respective expected rates of $3/(1+r_2+r_3)$, $3r_2/(1+r_2+r_3)$, and $3r_3/(1+r_2+r_3)$ at the first, second, and third codon positions.
Simulations {#ssec:Simulations}
-----------
The Materials and Methods of [@Ji2016] describe how and why data sets were simulated using parameter values estimated from the YDR418W\_YEL054C data set. With the same simulation procedure and for the same reasons, we again based our simulations on the YDR418W\_YEL054C data set. However, this time we used the inferred values of the HKY+IS-IGC parameters from the YDR418W\_YEL054C data to simulate data sets.
The estimated value of $\tau = \eta / p$ was $5.16$ for the YDR418W\_YEL054C data set. We wanted to explore how the composite likelihood estimates of expected tract length $1/p$ varied for different true values of $1/p$. Because $\eta = \tau / (1/p)$, we used $\tau = 5.16$ and each value of $1/p$ that was explored in the simulations to set the corresponding value of $\eta$. All other parameters were set at the values inferred by maximum composite likelihood from the YDR418W\_YEL054C data set.
To match the YDR418W\_YEL054C data, IGC and point mutation events were simulated according to the yeast species tree for sequences of length 492 nucleotides. While the simulations in [@Ji2016] operated at the codon level, these simulations used nucleotides as the units because the modified HKY model is a nucleotide substitution model that has independent evolution at the three codon positions but has different rates for each codon position. The nucleotides in three alignment columns (i.e., columns 238, 239, 240) were removed from each simulated data set because the actual YDR418W\_YER054C data has a gap in these columns. For each simulation condition, 100 data sets were generated. For each simulation scenario, one hundred data sets were simulated and analyzed with the HKY+PS-IGC model. Rather than finding the combination of parameter values that jointly maximize the composite likelihood, computational concerns and numerical optimization difficulties resulted in our instead optimizing the composite likelihood only over the tract length parameter $p$ with all other free parameters constrained at the values that were used to simulate the data sets. We note that the parameter $\tau = \eta / p$. By constraining $\tau$ at its true value and then inferring $p$, a value for the IGC tract initiation parameter $\eta$ is simultaneously inferred (i.e., $\eta = \frac{\tau }{(1/p)}$). While the computational shortcut of only inferring $p$ from simulated data is not ideal, the simulation results of [@Ji2016] indicated that the value of $\tau$ can be estimated relatively well.
Parametric Bootstrap Analyses
-----------------------------
There were 100 replicates per parametric bootstrap analysis. The maximum composite likelihood procedure was employed to obtain estimates of model parameters from actual data and those estimated values were used to simulate data sets of the same size as the actual data. Each of the 100 simulated data sets was analyzed by the composite likelihood procedure and variability of the resulting estimates was summarized. Paralleling the analyses of simulated data sets described in Section \[ssec:Simulations\], analyses of the 100 bootstrap replicates with the HKY+PS-IGC model were made computationally feasible by estimating the IGC tract length parameter $p$ but constraining all other free parameters to their true values. This imposition of constraints will presumably cause the uncertainty in the tract length parameter $p$ to be underestimated, but we hope that this underestimation is small given that $\tau = \eta /p$ and that [@Ji2016] found that the $\tau$ parameter of the IS model could usually be well estimated.
Yeast Data
----------
In the Materials and Methods of [@Ji2016], we describe how the $14$ data sets of yeast protein-coding genes were selected and prepared. Although the 14 yeast data sets exclude introns, our pair-site analyses of these data attempted to accommodate the fact that an IGC tract may cross exon-intron boundaries. For each pair of [*S. cerevisiae*]{} paralogs that is listed in Table \[tab:IS-IGCYeastResults\] and Table \[tab:IGCestimates\], we identified the exon-intron boundaries and intron lengths of the first of the listed [*S. cerevisiae*]{} paralogs. For each adjacent pair of columns in our alignment of yeast exons, we could therefore determine the separation along the gene sequence of the [*S. cerevisiae*]{} paralog that is listed first for each pair in Table \[tab:IS-IGCYeastResults\] and Table \[tab:IGCestimates\]. We then assumed that the gene sequence separation between adjacent exon columns for this [*S. cerevisiae*]{} paralog was identical to the gene sequence separation for all other paralogs in the exon alignment. Therefore, we did not attempt to account for insertions and deletions or for the possibility that exon-intron boundaries may vary along the evolutionary tree.
Primate EDN and ECP Data
------------------------
We used the EDN and ECP sequence data from the [@Zhang1998] study and the tree topology that was first introduced by [@Rosenberg1995]. Because of the relatively short time separating the common ancestor of humans and chimpanzees from the common ancestor of those two species with gorillas, we excluded human sequence data from our analyses to lessen the variation of gene tree and species tree topologies. The remaining species are gorilla ([*Gorilla gorilla*]{}), chimpanzee ([*Pan troglodytes*]{}), orangutan ([*Pongo pygmaeus*]{}), macaque ([*Macaca fascicularis*]{}), and tamarin ([*Saguinus oedipus*]{}). We obtained the protein-coding DNA sequences via their GenBank accession numbers and aligned them at the amino acid level with version 7.305b of the MAFFT software [@Katoh2013]. The protein sequence alignment was then converted to the corresponding codon-level alignment. Three codon columns that contain gaps were removed from the alignment. The remaining $157$ codon columns were used in the analyses.
Primate Intronic Data
---------------------
We received $550$ sequence alignment files of the intronic data analyzed from [@Harpak2017]. Each alignment file corresponds to one intron of a total of $75$ protein-coding gene pairs. We further filtered the alignment files to retain only those with both paralogs in the five ingroup species: human ([*Homo sapiens*]{}), chimpanzee ([*Pan troglodytes*]{}), gorilla ([*Gorilla gorilla*]{}), orangutan ([*Pongo abelii*]{}) and macaque ([*Macaca mulatta*]{}); and one paralog in the outgroup species: mouse ([*Mus musculus*]{}). This filter is intended to minimize uncertainties of gene duplication and loss histories so that the analyzed data is less likely to violate the gene tree topology that was assumed in our analyses. Twenty sets of paralogous introns and $15370$ sites remained after this filtering of alignment files. Each of these twenty sets of paralogous introns corresponds to one of five pairs of paralogs of protein-coding genes. The ensembl gene ids for the paralog pairs from human are: ENSG00000109272 and ENSG00000163737, ENSG00000136943 and ENSG00000135047, ENSG00000158485 and ENSG00000158477, ENSG00000163564 and ENSG00000163563, and\
ENSG00000187626 and ENSG00000189298.
Because these sequence data represent introns rather than exons, we analyzed the 20 datasets with parameter values under the HKY+PS-IGC model that has no rate heterogeneity (i.e., $r_2 = r_3 = 1$). In all analyses of the segmentally-duplicated intron data, we assumed that IGC tracts affecting one pair of intron paralogs in our data set would not also affect other pairs in our filtered data set. While this treatment of exon-intron boundaries is less sophisticated than that applied to the yeast and EDN/ECP data (see below), it is computationally less demanding than the treatment applied to those data.
We investigated two different sorts of analyses of the intron data. One sort consisted of grouping together all introns associated with the same gene. This results in the 20 intron subsets being divided into 5 groups. We separately analyzed each of the 5 groups. We constrained each intron assigned to a group to share parameter values with the other introns in the group. This treatment caused numerical optimization difficulties for most of the groups. Specifically, substantially different maximum composite likelihood estimates were obtained when numerical optimization was initiated from different sets of parameter values. We believe this difficulty arose because most of the 5 groups of introns contained little information about IGC tract lengths.
Due to this difficulty, we decided to share information across the 5 groups of introns by jointly analyzing them with the assumption that all introns shared the same parameter values. This corresponds to an assumption that the duplications that generated the 5 groups of introns all occurred at the same time. While this assumption neglects some variability in branch lengths among gene trees, it results in sharing information about IGC tract length distributions across the groups. Because this shared treatment resulted in less numerical optimization difficulty, all parameter estimates reported here for the intron data were obtained via the shared treatment.
Exon-Intron Boundaries
----------------------
For yeast and primate EDN/ECP data, we only analyzed the coding sequences of protein-coding genes but we incorporated the possibility that individual IGC events could span entire introns and thereby affect consecutive exons. Consider two neighboring sites in the coding sequence that are at positions $k$ and $k + 1$. Because of the possibility of introns, these two sites may not be neighbors in the gene sequence. We will assume these sites are separated in the gene sequence by $m$ nucleotides where $m \ge 1$ with $m=1$ being the case where the two sites are in the same exon. When the two sites are in different exons, our analyses have the distance $m$ between them be the length of the intron sequence plus one.
Numerical Optimization
----------------------
Model parameters were estimated by numerically optimizing the logarithms of the composite likelihoods. Inferred parameters include branch lengths, parameters related to point mutation, and IGC-related parameters. The “L-BFGS-B” method from the scipy package [@Oliphant2007; @Walt2011] was employed for numerical optimization.
Acknowledgments
===============
We thank Alexander Griffing, Hirohisa Kishino, Eric Stone, Ed Susko, and Marc Suchard for diverse assistance. We also thank Arbel Harpak for sharing his primate intron data. This work was supported by the National Institute of General Medical Sciences at the National Institutes of Health (GM118508) and the National Science Foundation (DEB 1754142). Data sets are available at <https://github.com/xji3>. Software for inferring IGC is available at <https://github.com/xji3/IGCexpansion> and <http://jsonctmctree.readthedocs.org/en/latest/>.
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---------------------------- ------------- --------------- ----------------- --------------- --------------- -------------- --------------
[**HKY**]{} [**MG94**]{}
[**Paralog Pair**]{} [**LnL**]{} [**Diff** ]{} [**$\tau$** ]{} [**$r_2$**]{} [**$r_3$**]{} [**Prop**]{} [**Prop**]{}
YLR406C,YDL075W -1189.81 26.55 5.10 0.43 8.08 0.24 0.20
YER131W,YGL189C -1216.91 31.58 5.27 0.97 19.60 0.27 0.20
YML026C,YDR450W -1368.47 94.72 12.85 0.27 9.67 0.40 0.34
YNL301C,YOL120C -2126.64 129.75 7.94 0.57 7.16 0.32 0.26
YNL069C,YIL133C -2332.61 74.95 3.63 0.43 4.78 0.26 0.22
YMR143W,YDL083C -1217.38 52.80 9.19 0.13 17.72 0.31 0.29
YJL177W,YKL180W -1840.38 63.64 6.45 0.50 8.97 0.28 0.21
YBR191W,YPL079W -1468.95 91.84 13.66 0.15 6.39 0.40 0.32
YER074W,YIL069C -1233.00 131.50 20.90 0.28 6.94 0.39 0.37
YDR418W,YEL054C -1735.40 65.09 5.16 0.54 11.58 0.27 0.21
YBL087C,YER117W -1372.91 79.96 11.05 0.34 10.59 0.38 0.29
YLR333C,YGR027C -1246.67 108.84 9.88 0.60 8.05 0.39 0.29
YMR142C,YDL082W -2033.88 179.32 14.37 1.12 8.42 0.42 0.38
YER102W,YBL072C -2037.26 205.73 14.77 0.82 6.19 0.43 0.36
EDN/ECP -1713.06 12.61 1.79 1.52 1.56 0.16 0.10
Introns -62878.87 93.79 0.44 N.A. N.A. 0.09 N.A.
\[tab:IS-IGCYeastResults\]
---------------------------- ------------- --------------- ----------------- --------------- --------------- -------------- --------------
: HKY+IS-IGC results.
Each row begins with the name of the data set. Yeast datasets are named by the systematic names of the two [*S. cerevisiae*]{} paralogous open reading frames. The “EDN/ECP” row represents the primate EDN/ECP data set. The “Introns” row represents the segmentally-duplicated primate introns. The “lnL IS-IGC” column shows the maximum log-likelihood value of the HKY+IS-IGC model. The “Diff” column specifies the number of log-likelihood units by which the HKY+IS-IGC value exceeds the maximum log-likelihood value of same model when $\tau$ is constrained to $0$. The “$\tau$” column shows the estimated $\tau$ value from the HKY+IS-IGC model. The “$r_2$” and “$r_3$” columns show the estimated relative substitution rates of the second and third codon positions from the HKY+IS-IGC model. The column labelled “HKY Prop” shows the estimated proportions of nucleotide changes attributable to IGC with the HKY+IS-IGC model. The column labelled “MG94 Prop” shows the estimated proportions of codon changes attributable to IGC with the MG94+IS-IGC model. “N.A.” is written in table entries to denote “not applicable”.
---------------------------------------------------------- ------- ---------------- ------- -------- ----------------- -- -- -- -- --
[ **** ]{}
(lr)[2-3]{} (lr)[4-4]{} (lr)[5-6]{} [**Paralog Pair**]{} MCLE Interquartile MCLE MCLE Interquartile
YLR406C,YDL075W 5.10 (4.37, 6.12) 5.10 4.04 (2.97, 4.87)
YER131W,YGL189C 5.27 (3.89, 6.75) 5.27 13.24 (9.46, 16.91)
YML026C,YDR450W 12.85 (10.46, 15.32) 12.84 1.04 (1.00, 1.27)
YNL301C,YOL120C 7.94 (6.88, 8.71) 7.93 103.52 (68.27, 141.12)
YNL069C,YIL133C 3.63 (3.15, 4.11) 3.63 12.37 (8.90, 13.20)
YMR143W,YDL083C 9.19 (7.28, 13.85) 9.20 4.79 (3.55, 6.45)
YJL177W,YKL180W 6.45 (5.13, 7.20) 6.45 8.50 (7.11, 10.53)
YBR191W,YPL079W 13.66 (10.99, 15.41) 13.65 9.08 (7.57, 10.78)
YER074W,YIL069C 20.90 (16.69, 24.06) 20.89 53.05 (38.86, 72.06)
YDR418W,YEL054C 5.16 (4.23, 6.48) 5.16 3.80 (3.13, 4.69)
YBL087C,YER117W 11.05 (8.76, 13.22) 11.03 29.45 (20.14, 38.67)
YLR333C,YGR027C 9.88 (8.11, 11.43) 9.85 36.68 (26.96, 53.72)
YMR142C,YDL082W 14.37 (11.80, 16.77) 14.36 33.61 (29.45, 46.62)
YER102W,YBL072C 14.77 (12.25, 16.36) 14.76 26.34 (20.61, 61.41)
EDN/ECP 1.79 (1.40, 2.21) 1.79 6.90 (4.32, 8.34)
Introns 0.44 (0.41, 0.46) 0.52 370.4 (142.9, 523.0)
\[tab:IGCestimates\]
---------------------------------------------------------- ------- ---------------- ------- -------- ----------------- -- -- -- -- --
: IGC parameter estimates.
Each row begins with the name of the data set, as detailed in the caption to Table \[tab:IS-IGCYeastResults\]. Columns labelled “MCLE” contain maximum composite likelihood estimates and columns labelled “Interquartile” represent the $25{\mbox{th}}$ and $75{\mbox{th}}$ percentiles of the estimated values in $100$ parametric bootstrap samples. The “$\tau$, IS-IGC” columns show the $\tau$ values estimated with the HKY+IS-IGC model. The “$\tau$, PS-IGC” column shows the $\tau$ values estimated with the HKY+PS-IGC model. Because the parametric bootstrap analyses for the HKY+PS-IGC model did not consider uncertainty in the $\tau$ parameter (see Materials and Methods), interquartile ranges are not available for these $\tau$ estimates. The “$1/p$, HKY+PS-IGC” columns show the average tract length in nucleotides of fixed IGC events as estimated with the HKY+PS-IGC model.
|
---
abstract: |
This research is an effort to understand small-scale properties of networks resulting in global structure in larger scales. Networks are modelled by graphs and graph-theoretic conditions are used to determine the structural properties exhibited by the network. Our focus is on *signed networks* which have positive and negative signs as a property on the edges. We analyse networks from the perspective of *balance theory* which predicts *structural balance* as a global structure for signed social networks that represent groups of friends and enemies. The vertex set of *balanced* signed networks can be partitioned into two subsets such that each negative edge joins vertices belonging to different subsets.
The scarcity of balanced networks encouraged us to define the notion of *partial balance* in order to quantify the extent to which a network is balanced. We evaluate several numerical *measures of partial balance* and recommend using the *frustration index*, a measure that satisfies key axiomatic properties and allows us to analyse graphs based on their levels of partial balance.
The exact algorithms used in the literature to compute the frustration index, also called the *line index of balance*, are not scalable and cannot process graphs with a few hundred edges. We formulate computing the frustration index as a graph optimisation problem to find the minimum number of edges whose removal results in a balanced network given binary decision variables associated with graph nodes and edges. We use our first optimisation model to analyse graphs with up to 3000 edges.
Reformulating the optimisation problem, we develop three more efficient *binary linear programming* models. Equipping the models with *valid inequalities* and *prioritised branching* as speed-up techniques allows us to process graphs with 15000 edges on inexpensive hardware. Besides making exact computations possible for large graphs, we show that our models outperform heuristics and approximation algorithms suggested in the literature by orders of magnitude.
We extend the concepts of balance and frustration in signed networks to applications beyond the classic friend-enemy interpretation of balance theory in social context. Using a high-performance computer, we analyse graphs with up to 100000 edges to investigate a range of applications from biology and chemistry to finance, international relations, and physics.
author:
- Samin Aref
bibliography:
- 'thesisrefs.bib'
title: Signed Network Structural Analysis and Applications with a Focus on Balance Theory
---
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to express my very great appreciation to Dr. Mark C. Wilson for supervising this research and motivating me in the past couple of years. The experience of working with him was extremely valuable and I am deeply indebted to him for sharing his knowledge and expertise. It was proven to me numerous times that having him as a supervisor has played a key role in the success of this Ph.D. project.
I also would like to thank Dr. Andrew J. Mason for co-supervising this research. I am very grateful to him not only for his valuable comments, but for sharing his mathematical modelling expertise which strengthened the contributions of this thesis.
I was privileged to have Dr. Serge Gaspers and Prof. Gregory Gutin as examiners of this thesis. I am thankful for their essential comments which helped in revising and improving the thesis.
This research would not be completed without the tremendous support of my partner for whom my heart is filled with gratitude. I cannot thank her enough for her selfless and pure love that has lighten up my life. The challenges we faced could have not possibly been overcome without her devotion and dedication.
I am also indebted to a lifetime of love and support from my parents and my sister. Their presence has always encouraged me to accept new challenges such as a Ph.D. program in New Zealand. I am grateful for having the best father, the best mother, and the best sister I can possibly imagine.
I would like to acknowledge University of Auckland for investing in these ideas. The support provided by Department of Computer Science, Centre for eResearch, and [Te Pūnaha Matatini]{} was greatly appreciated.
In the end, I would like to thank everyone who has taught me something; past teachers and professors as well as authors of the papers I have read and the reviewers who have commented on my works.
Introduction
============
Conclusion and Future Directions
================================
In Chapter \[ch:1\], we discussed quantifying the answer to this simple question: is the enemy of an enemy a friend? We formalised the concept of a measure of partial balance, compared several measures on synthetic and real datasets, and investigated their axiomatic properties. We evaluated measures to be used in future work based on their properties which led to finding key axioms and desirable properties satisfied by a measure known as the frustration index. We recommended its usage in future work, although it requires intensive computation.
The findings of Chapter \[ch:1\] have a number of important implications for future investigation. Although we focused on partial balance, the findings may well have a bearing on link prediction and clustering in signed networks [@Gallier16]. Some other relevant topics of interest in signed networks are network dynamics [@tan_evolutionary_2016] and opinion dynamics [@li_voter_2015]. Effective methods of signed network structural analysis can contribute to these topics as well.
The intensive computations required for obtaining the frustration index encouraged us to focus in Chapter \[ch:2\] on developing computational methods that exactly compute this measure for decent-sized graphs in a reasonable time. Our studies of this graph-theoretic measure revealed that while it has several applications in many fields, it was mostly approximated or estimated using heuristic methods. We also found out that the frustration index was almost never computed exactly in non-trivial examples because of the complexity in its computation which is closely related to classic NP-hard graph problems. We linearised a quadratic programming model to compute this measure exactly. We obtained numerical results on graphs with up to 3000 edges that showed most real-world social networks and some biological networks have small frustration index values which indicate that they are close to a state of structural balance.
In Chapter \[ch:3\], we focused on reformulating the optimisation model we had developed for computing the frustration index. We suggested three new integer linear programming models that were mathematically equivalent, but had major differences in performance. We also took advantage of some structural properties in the networks to develop speed-up techniques. Our algorithms were shown to provide the global optimal solution and outperform all previous methods by orders of magnitude in solve time. We showed that exact values of the frustration index in signed graphs with up to 15000 edges can be efficiently computed using our suggested optimisation models on inexpensive hardware.
Chapters \[ch:2\] – \[ch:3\] have a number of important implications for future investigation. The optimisation models introduced can make network dynamics models more consistent with the theory of structural balance [@antal_dynamics_2005]. Many sign change simulation models that allow one change at a time use the number of balanced triads in the network as a criterion for transitioning towards balance. These models may result in stable states that are not balanced, like *jammed states* and *glassy states* [@marvel_energy_2009]. This contradicts not only the instability of unbalanced states, but the fundamental assumption that networks gradually move towards balance. Deploying decrease in the frustration index as the criterion, the above-mentioned states might be avoided resulting in a more realistic simulation of signed network dynamics that is consistent with structural balance theory and its assumptions.
The efficient computational methods we developed encouraged us to explore the frustration index beyond its classic friend-enemy interpretation in the social context. In Chapter \[ch:4\], we investigated a range of applications from biology and chemistry to finance, international relations, and physics. This helped us unify the concept of signed graph frustration index whose practical applications can be found among mostly unanswered questions in several research areas. We discussed how the frustration index turns out to be a measure of distance to monotonicity in systems biology, a predictor of fullerene chemical stability, a measure of bi-polarisation in international relations, an indicator for well-diversified portfolios in finance, and a proxy for ground-state energy in some models of atomic magnets in physics. We used a high-performance computer to solve a wide range of instances involving graphs with up to 100000 edges concerning applications in several fields.
While Chapter \[ch:4\] provided an overview of the state-of-the-art numerical computations on signed graphs and the vast range of applications to which it can be applied, it is by no means an exhaustive survey on the applications of the frustration index. From a computational perspective, this thesis and some other recent studies [@Giscard2016; @giscard2016general] call for more advanced computational models that put larger networks within the reach of exact analysis. As another future research direction, one may consider formulating edge-based measures of stability for directed signed networks based on theories involving directionality and signed ties [@leskovec_signed_2010; @yap_why_2015].
|
---
abstract: 'Reinforcement learning (RL) has proven its worth in a series of artificial domains, and is beginning to show some successes in real-world scenarios. However, much of the research advances in RL are hard to leverage in real-world systems due to a series of assumptions that are rarely satisfied in practice. In this work, we identify and formalize a series of independent challenges that embody the difficulties that must be addressed for RL to be commonly deployed in real-world systems. For each challenge, we define it formally in the context of a Markov Decision Process, analyze the effects of the challenge on state-of-the-art learning algorithms, and present some existing attempts at tackling it. We believe that an approach that addresses our set of proposed challenges would be readily deployable in a large number of real world problems. Our proposed challenges are implemented in a suite of continuous control environments called [`realworldrl-suite`]{}which we propose an as an open-source benchmark.'
author:
- |
Gabriel Dulac-Arnold$^{1,}$[^1]\
`dulacarnold@google.com`\
Nir Levine$^{3,*}$\
`nirlevine@google.com`\
Daniel J. Mankowitz$^{2,*}$\
`dmankowitz@google.com`\
Jerry Li$^{2}$\
Cosmin Paduraru$^{2}$\
Sven Gowal$^{2}$\
Todd Hester$^{4}$\
bibliography:
- 'references.bib'
title: 'An empirical investigation of the challenges of real-world reinforcement learning'
---
Introduction {#intro}
============
Reinforcement learning (RL) [@sutton2018reinforcement] is a powerful algorithmic paradigm encompassing a wide array of contemporary algorithmic approaches [@mnih2015human; @silver2016mastering; @hafner2018learning]. RL methods have been shown to be effective on a large set of simulated environments [@mnih2015human; @silver2016mastering; @lillicrap2015continuous; @OpenAI_dota; @Tessler2017], but uptake in real-world problems has been much slower. We posit that this is due primarily to a large gap between the casting of current experimental RL setups and the generally poorly defined realities of real-world systems.
We are inspired by a large range of real-world tasks, from control systems grounded in the physical world [@vecerik2019practical] to global-scale software systems interacting with billions of users [@covington2016deep; @ie2019recsim]. Physical systems can range in size from a small drone [@abbeel2010autonomous] to a data center [@DM_Datacenter], in complexity from a one-dimensional thermostat [@hester2018predictively] to a self-driving car, and in cost from a calculator to a spaceship. Software systems range from billion-user recommender systems [@covington2016deep] to on-device controllers for individual smart-phones, they can be scheduling millions of software jobs across the globe or optimizing the battery profile of a single device, and the codebase might be millions of lines of code to a simple kernel module. In all these scenarios, there are recurring themes: the systems have inherent latencies, noise, and non-stationarities that make them hard to predict. They may have large and complicated state & action spaces, safety constraints with significant consequences, and large operational costs both in terms of money and time. This is in contrast to training on a perfect simulated environment where an agent has full visibility of the system, zero latency, no consequences for bad action choices and often deterministic system dynamics.
We posit that these difficulties can be well summarized by a set of nine challenges that are holding back RL from real-world use. At a high level these challenges are:
1. Being able to learn on live systems from limited samples.
2. Dealing with unknown and potentially large delays in the system actuators, sensors, or rewards.
3. Learning and acting in high-dimensional state and action spaces.
4. Reasoning about system constraints that should never or rarely be violated.
5. Interacting with systems that are partially observable, which can alternatively be seen viewed as systems that are non-stationary or stochastic.
6. Learning from multi-objective or poorly specified reward functions.
7. Being able to provide actions in real-time, especially for systems with high control frequencies.
8. Training off-line from the fixed logs of an external behavior policy.
9. Providing system operators with explainable policies.
We hope that by identifying, replicating and solving these challenges, reinforcement learning can be more readily used to solve important real-world problems. Our contributions can be structured into four parts:
- **Identification and definition of real-world challenges:** Our main goal is to more clearly define the issues reinforcement learning is having when dealing with real systems. By making these problems identifiable and well-defined, we hope they can be dealt with more explicitly, and thus solved more rapidly. We structure the difficulties of real-world systems in the aforementioned 9 challenges. For each of the above challenges, we provide some intuition on where it arises anddiscuss potential solutions present in the literature.
- **Experiment design and analysis for each challenge:** For all challenges except explainability, we define a formal definition of the challenge and create a set of environments exhibiting this challenge’s characteristics. We then train common RL agents on the challenge with varying degrees of difficulty and analyze its effects on learning. We provide insights as to which challenges are more difficult and propose calibrated parameters for each challenge.
- **Define and baseline RWRL Combined Challenge Benchmark tasks:** After careful calibration, we combine a subset of our proposed challenges into a single environment and baseline the performance of two state-of-the-art learning agents on this setup in Section \[sec:combined\_challenges\]. We show that state-of-the-art agents fail quickly, even for mild perturbations applied along each challenge dimension. We encourage the community to work on improving upon the combined challenges’ baseline performance. We believe that in doing so, we will take large steps towards developing agents that are implementable on real world systems.
- **Open-source [`realworldrl-suite`]{}codebase:** We present the set of perturbed environments in a parametrizable suite, called [`realworldrl-suite`]{}which extends the DeepMind Control Suite [@tassa2018deepmind] with various perturbations representing the aforementioned challenges. The goal of the suite is to accelerate research in these areas by enabling RL practitioners and researchers to quickly, in a principled and reproducible fashion, test their learning algorithms on challenges that are encountered in many real-world systems and settings. The [`realworldrl-suite`]{}is available for download here: <https://github.com/google-research/realworldrl_suite>. A user manual, found in Appendix \[app:codebase\], explains how to instantiate each challenge and also provides code examples for training an agent.
Analysis of the Real-World Challenges
=====================================
In this section, for each of the challenges presented in the introduction we discuss its importance and present current research directions that attempt to tackle the challenge, providing starting points for practitioners and newcomers to the domain. We then define it more formally, and analyse its effects on state-of-the-art learning algorithms using the [`realworldrl-suite`]{}, to provide insights on how these challenges manifest themselves in isolation. While not all of these challenges are present together in every real system, for many systems they are all present together to some degree. For this reason, in Section \[sec:combined\_challenges\] we also present a set of combined reference challenges, varying in difficulty, that emulate a complete system with all of the introduced challenges. We believe that a learner able to tackle these combined challenges would be a good candidate for many real-world systems.
#### Notation
Environments are formalised as Markov Decision Processes (MDPs). A MDP can be defined as a tuple $\langle \S, \A , p, r, \gamma \rangle$, where an agent is in a state $s_t \in \S$ and takes an action $a_t \in \A$ at timestep $t$. When in state $s_t$ and taking an action $a_t$, an agent will arrive in a new state $s_{t+1}$ with probability $p(s_{t+1} | s_t, a_t)$, and receive a reward $r(s_t, a_t, s_{t+1})$. Our environments are episodic, which is to say that they last a finite number of timesteps, $1 \leq t \leq T$. The value of $\gamma$, the discount factor, reflects the agent’s planning horizon. The full state of the process, $s_t$, respects the Markov property: $p(s_{t+1} | s_t, a_t, \cdots, s_0, a_0) = p(s_{t+1} | s_t, a_t)$, i.e. all necessary information to predict $s_{t+1}$ is contained in $s_t$ and $a_t$. In many of the environments in this paper the *observed* state does not include the full internal state of the [`MuJoCo`]{}physics simulator. It has nevertheless been shown empirically that the observed state is sufficient to control an agent, so we interchange the notion of state and observation unless otherwise specified.
Ultimately, the goal of a RL agent is to find an optimal policy $\pi^*: \S \rightarrow \A$ which maximizes its expected return over a given MDP: $$\pi^* = \operatorname*{arg\,max}_\pi \mathbb{E^\pi}\left[\sum_{t=0}^{\infty} \gamma^t r(s_t, \pi(s_t), s_{t+1} \sim p(s_t, \pi(s_t)))\right]$$
There are many ways to find this policy [@sutton2018reinforcement], and we will use two *model-free* methods described in the following section.
#### Learning algorithms:
For each challenge, we present the results of two state-of-the-art RL learning algorithms: Distributional Maximum a Posteriori Policy Optimization (DMPO) [@Abdolmaleki2018] and Distributed Distributional Deterministic Policy Gradient (D4PG) [@barthmaron2018d4pg].
D4PG is a modified version of Deep Deterministic Policy Gradients (DDPG) [@lillicrap2015continuous], an actor-critic algorithm where state-action values are estimated by a critic network, and the actor network is updated with gradients sampled from the critic network. D4PG makes four changes to improve the critic estimation (and thus the policy): evaluating $n$-step rather than 1-step returns, performing a *distributional* critic update [@BellemareDM17], using prioritized sampling of the replay buffer, and performing distributed training. These improvements give D4PG state of the art results across many DeepMind control suite [@tassa2018deepmind] tasks as well as manipulation and parkour tasks [@heess2017emergence]. The hyperparameters for D4PG can be found in Appendix \[app:algorithms\], Table \[table:d4pg\_hyperparameters\].
MPO [@abbas2018mpo] is an RL method that combines the sample efficiency of off-policy methods with the scalability and hyperparameter robustness of on-policy methods. It is an EM style method, which alternates an E-step that re-weights state-action samples with an M step that updates a deep neural network with supervised training. MPO achieves state of the art results on many continuous control tasks while using an order of magnitude fewer samples when compared with PPO [@Schulman2017ppo]. Distributional MPO (DMPO) is an extension of MPO that uses a distributional value function and achieves superior performance. The hyperparameters for DMPO can be found in Appendix \[app:algorithms\], Table \[table:mpo\_hyperparameters\].
Each algorithm is run for $30$K episodes on [`cartpole:swingup`]{}, [`walker:walk`]{}, [`quadruped:walk`]{}and [`humanoid:walk`]{}tasks from the [`realworldrl-suite`]{}. Unless stated otherwise, the mean value reported in each graph is the mean performance of the last $100$ episodes of training with the corresponding standard deviation. All hyperparameters for all experiments can be found in Table \[app1:hyperparameters\_sweeps\]. To make experiments more easily reproducible we did not use distributed training for either D4PG or DMPO. Additionally, unless otherwise noted, evaluation is performed on the same policy as used for training, to be consistent with the notion that there is no train/eval dichotomy.
Challenge 1: Learning On the Real System from Limited Samples
-------------------------------------------------------------
#### Motivation & Related Work
Almost all real-world systems are either slow-moving, fragile, or expensive enough to operate, that data they produce is costly and therefore learning algorithms must be as data-efficient as possible. Unlike much of the research performed in RL [@mnih2015human; @Espeholt2018; @dqfd; @Tessler2017], real systems do not have separate training and evaluation environments, therefore the agent must quickly learn to act reasonably and safely. In the case where there are off-line logs of the system, these might not contain anywhere near the amount of data or data coverage that current RL algorithms expect. In addition, as all training data comes from the real system, learning agents cannot have an overly aggressive exploration policy during training, as these exploratory actions are rarely without consequence. This results in training data that is low-variance with very little of the state and action space being covered.
Learning iterations on a real system can take a long time, as slower systems’ control frequencies can range from hours in industrial settings, to multiple months in cases with infrequent user interactions such as healthcare or advertisement. Even in the case of higher-frequency control tasks, the learning algorithm needs to learn *quickly* from potential mistakes without having to repeat them multiple times. In addition, since there is often only one instance of the system, approaches that instantiate hundreds or thousands of environments to accelerate training through distributed training [@whoops; @impala; @adamski2018distributed] nevertheless require as much data and are rarely compatible with real systems. For all these reasons, learning on a real system requires an algorithm to be both sample-efficient and quickly performant.
There are a number of related works that deal with RL on real systems and, in particular, focus on sample efficiency. One body of work is Model Agnostic Meta-Learning (MAML) [@finn2017model], which focuses on learning within a task distribution and, with few-shot learning, quickly adapting to solving a new in-distribution task that it has not seen previously. Bootstrap DQN [@osband16] learns an ensemble of Q-networks and uses Thompson Sampling to drive exploration and improve sample efficiency. Another approach to improving sample efficiency is to use expert demonstrations to bootstrap the agent, rather than learning from scratch. This approach has been combined with DQN [@mnih2015human] and demonstrated on Atari [@dqfd], as well as combined with DDPG [@lillicrap2015continuous] for insertion tasks on robots [@insertion]. Recent Model-based deep RL approaches [@hafner2018learning; @chua2018deep; @nagabandi2019deep], where the algorithm plans against a learned transition model of the environment, show a lot of promise for improving sample efficiency. Another common approach is to learn ensembles of transition models and use various sampling strategies from those models to drive exploration and improve sample efficiency [@MLJ12-hester; @chua2018deep; @buckman2018].
#### Experimental Setup & Results
To evaluate this challenge, we measure the normalized regret with respect to the performance of the final policy. If we assume that the final policy has an average return $R_{mean}$ with a $95\%$ confidence interval defined by $[R_{lower}, R_{upper}]$, and the agent first reaches a return $R_{lower}$ in episode $K$, we can consider the agent to be converged at episode $K$. We can then we define the normalized regret as $$\mathcal{L}_{perf}(\pi) =\frac{1}{R_{mean}} \left[ \mathbb{E}_{j > K}[R_j] - \frac{1}{K} \sum_{i=0}^K R_i \right],$$ which can be read as sum of regrets for each episode $i$, i.e., the return that would have been achieved by the final policy, minus the actual return that was achieved. The normalized regret for each of the evaluation domains is shown in Figure \[fig:d4pg\_efficiency\] (D4PG) and \[fig:dmpo\_efficiency\] (DMPO) respectively. The normalized regret can effectively be interpreted as the amount of actual return lost, prior to convergence, due to poor policy performance. As seen in the figure, DMPO has a high normalized regret on [`humanoid:walk`]{}indicating poor performance prior to convergence. The goal of this challenge is to converge as quickly as possible and minimize the regret.
Once converged, we also want to measure the instability of the converged policy during training. To do so, we define the instability regret, which quantifies how much the policy performance drops below the lower confidence interval after convergence. More formally, $$\mathcal{L}_{instability}(\pi)=\frac{1}{M-K}\sum_{i=K}^M \left[ \left(R_{lower}-R_{i}\right)\mathds{1}\left[R_{i}<R_{lower}\right] \right]$$ computes post-convergence regret relative to the lower confidence bound, where $M$ is the episode length.
Large regret indicates instabilities in the converged performance of the algorithm. The average instability regret for each of the domains is shown in Figure \[fig:d4pg\_stability\] (D4PG) and \[fig:dmpo\_stability\] (DMPO) respectively. As can be seen in the figures, both [`cartpole:swingup`]{}and [`quadruped:walk`]{}have low instability regret. In addition, even though [`walker:walk`]{}has lower regret than [`humanoid:walk`]{}in Figure \[fig:dmpo\_efficiency\], it has significantly higher instability regret as seen in Figure \[fig:dmpo\_stability\], indicating that once converged, performance is less stable on [`walker:walk`]{}compared to [`humanoid:walk`]{}. We hope that analysing algorithms in this way will enable a practitioner to (1) develop algorithms that are sample efficient and reduce the regret until convergence; and (2) ensure that, once converged, the algorithm is stable. These two properties are highly desirable in industrial systems such as data center cooling plants.
[.45]{} ![Normalized regret and standard deviation of D4PG (left) and DMPO (right) with respect to final policy performance. Assuming the final policy has an average return of $R_{mean}$ with a 95% confidence interval $[R_{lower}, R_{upper}]$, and that the agent first reaches a return of $R_{lower}$ in episode k, the normalized regret is $\frac{1}{R_{mean}} \sum_{i=0}^K L_i$, where $L_i$ is the regret for episode $i$ with respect to the final policy performance.](figures/D4PG/regret_D4PG.pdf "fig:"){width="\textwidth"}
[.45]{} ![Normalized regret and standard deviation of D4PG (left) and DMPO (right) with respect to final policy performance. Assuming the final policy has an average return of $R_{mean}$ with a 95% confidence interval $[R_{lower}, R_{upper}]$, and that the agent first reaches a return of $R_{lower}$ in episode k, the normalized regret is $\frac{1}{R_{mean}} \sum_{i=0}^K L_i$, where $L_i$ is the regret for episode $i$ with respect to the final policy performance.](figures/DMPO/regret_DMPO.pdf "fig:"){width="\textwidth"}
[.45]{} {width="\textwidth"}
[.45]{} {width="\textwidth"}
Challenge 2: System Delays {#subsec:delays}
--------------------------
#### Motivation & Related Work
Most real systems have delays in either sensing, actuation, or reward feedback. These might occur because of low-frequency sensing and actuation, because of safety checks or other transformations performed on the selected action before it is actually implemented, or because it takes time for an action’s effect to be fully manifested.
[@MLJ12-hester] focus on controlling a robot vehicle with significant delays in the control of the braking system. They incorporate recent history into the state of the agent so that the learning algorithm can learn the delay effects itself. [@delayed-outcomes] look at delays in recommender systems, where the true reward is based on the user’s interaction with the recommended item, which may take weeks to determine. They both present a factored learning approach that is able to take advantage of intermediate reward signals to improve learning in these delayed tasks. [@hung2018optimizing] introduce a method to better assign rewards that arrive significantly after a causative event. They use a memory-based agent, and leverage the memory retrieval system to properly allocate credit to distant past events that are useful in predicting the value function in the current timestep. They show that this mechanism is able to solve previously unsolveable delayed reward tasks. [@arjona2018rudder] introduce the RUDDER algorithm, which uses a backwards-view of a task to generate a return-equivalent MDP where the delayed rewards are re-distributed more evenly throughout time. This return-equivalent MDP is easier to learn and is guaranteed to have the same optimal policy as the original MDP. They improvements using this approach in Atari tasks with long delays.
#### Experimental Setup & Results
The [`realworldrl-suite`]{}implements delays in observation, action and reward with an $n$-step buffer between the environment and the agent. An action delay is defined here as delaying the agent’s action execution for $n$ timesteps, whereas an observation/reward delay is defined as withholding an agent’s observation/reward for $n$ timesteps. We can evaluate the effects of the delay on an agent’s performance by looking at the episodic return upon convergence.
Figures \[fig:d4pg\_all\_delays\] and \[fig:dmpo\_all\_delays\] show the performance of D4PG and DMPO respectively under increasing levels of action, observation and reward delay. As expected, when delays increase, the performance of the algorithm decreases. Both algorithms appear to be less sensitive to reward delay compared to delays in observations or actions. This can be seen in the right-most plot of Figures \[fig:d4pg\_all\_delays\] and \[fig:dmpo\_all\_delays\], where the reward delay (x-axis) has to be increased to $100$ timesteps to see a significant drop in performance. The reason the agent may be more robust to reward delay is that even though the reward is delayed, it can ultimately be credited to an action that led to achieving that reward, even for relatively large delays. However, for more complicated tasks such as [`humanoid:walk`]{}, where action credit assignment is less obvious for large delays, performance degrades quickly. It should also be noted that the performance for observation delays is similar to that of action delays. The subtle difference between these settings is the reward that the agent receives at timestep $t$. In the case of action delays, an agent receives the reward $r(s_t, a_{t-n})$ whereas for observation delays, the reward is $r(s_{t-n}, a_t)$.
[.95]{} {width="\textwidth"}
[.95]{} {width="\textwidth"}
Challenge 3: High-Dimensional Continuous State and Action Spaces
----------------------------------------------------------------
#### Motivation & Related Work
Many practical real-world problems have large and continuous state & action spaces. For example, consider the huge action spaces of recommender systems [@covington2016deep], or the number of sensors and actuators to control cooling in a data center [@DM_Datacenter]. These large state and action spaces can present serious issues for traditional RL algorithms, (e.g., see [@dulac2015deep; @Tessler2019b]).
There are a number of recent works focused on addressing this challenge. @dulac2015deep ([-@dulac2015deep]) present an approach based on generating a vector for a candidate action and then doing nearest neighbor search to find the closest real action available. @zahavy2018learn ([-@zahavy2018learn]) propose an Action Elimination Deep Q Network (AE-DQN) that uses a contextual bandit to eliminate irrelevant actions. @he2015deep ([-@he2015deep]) present the Deep Reinforcement Relevance Network (DRRN) for evaluating continuous action spaces in text-based games. @Tessler2019b ([-@Tessler2019b]) introduce compressed sensing as an approach to reconstruct actions in text-based games with combinatorial action spaces.
#### Experimental Setup & Results
For this particular challenge, we first compared results across all the tasks. The state and action dimensions for each task can be found in Table \[app:state\_dimensions\]. Both stability of the overall system and the dimensionality affect learning progress. For example, as seen in Figures \[fig:d4pg\_regular\_training\_curves\] and \[fig:dmpo\_regular\_training\_curves\] for D4PG and DMPO respectively, `quadruped` is higher dimensional than `walker`, yet converges faster since it is a fundamentally more stable system. On the other hand, dimensionality is also a factor as `cartpole`, which is significantly lower-dimensional than `humanoid`, converges significantly faster.
We subsequently increased the number of state dimensions of each task with dummy state variables sampled from a zero mean, unit variance normal distribution. We then compare the average return for each task as we increase the state dimensionality. Figures \[fig:d4pg\_noise\] and \[fig:dmpo\_noise\] (right) show the converged average performance of the learning algorithm on each task for D4PG and DMPO respectively. Since the added states were effectively injecting noise into the system, the algorithm learns to deal with the noise and converges to the optimal performance for the cases of [`cartpole:swingup`]{}, [`quadruped:walk`]{}and [`walker:walk`]{}. In some cases, e.g. Figures \[fig:d4pg\_efficiency\_learning\] and \[fig:dmpo\_efficiency\_learning\] for [`walker:walk`]{}, the additional dummy dimensions slightly affect convergence speed indicating that the learning algorithm learns to deal with noise efficiently, but it does slow down learning progress.
**Task** **Observation Dimension** **Action Dimension**
---------------------- --------------------------- ----------------------
**Cartpole:Swingup** 5 1
**Walker:Walk** 18 6
**Quadruped:Walk** 78 12
**Humanoid:Walk** 67 21
: The observation and action dimensions for each task.[]{data-label="app:state_dimensions"}
[.45]{} {width="\textwidth"}
[.45]{} {width="\textwidth"}
[.95]{} {width="\textwidth"}
[.95]{} {width="\textwidth"}
[.45]{} {width="\textwidth"}
[.45]{} {width="\textwidth"}
Challenge 4: Satisfying Environmental Constraints {#sec:constraints}
-------------------------------------------------
#### Motivation & Related Work
Almost all physical systems can destroy or degrade themselves and the environment around them if improperly controlled. Software systems can also significantly degrade their performance or crash, as well as provide improper or incorrect interactions with users. As such, considering constraints on their operation is fundamentally necessary to controlling them. Constraints are not only important during system operation, but also during exploratory learning phases as well. Examples of physical constraints include limits on system temperatures or contact forces for safe operation, maintaining minimum battery levels, avoiding dynamic obstacles, or limiting end effector velocities. Software systems might have constraints around types of content to propose to users or system load and throughput limits to respect.
Although system designers may often wrap the learnt controller in a safety watchdog controller, the learnt controller needs to be aware of the constraints to avoid degenerate solutions which lazily rely on the watchdog. We want to emphasize that constraints can be put in place for varying reasons, ranging from monetary costs, to system up-time and longevity, to immediate physical safety of users and operators. Due to the physically grounded nature of our suite, our proposed set of constraints are physically bound and are intended to avoid self-harm, but the suite’s framework provides options for users to define any constraints they wish.
Recent work in RL safety [@galdalal; @AchiamHTA17] has cast safety in the context of Constrained MDPs (CMDPs) [@altman1999constrained], and we will concentrate on pre-defined constraints on the environment in this context. Constrained MDPs define a constrained optimization problem and can be expressed as: $$\vspace{-0.2cm}
\max_{\pi \in \Pi} R(\pi) \mbox{ subject to } C^k(\pi) \leq V_k, k = 1, \ldots,K.$$
Here, $R$ is the cumulative reward of a policy $\pi$ for a given MDP, and $C^k(\pi)$ describes the incurred cumulative cost of a certain policy $\pi$ relative to constraint $k$. The CMDP framework describes multiple ways to consider cumulative cost of a policy $\pi$: the total cost until task completion, the discounted cost, or the average cost. Specific constraints are defined as $c_k(s,a)$.
The CMDP setup allows for arbitrary constraints on state and action to be expressed. In the context of a physical system these can be as simple as box constraints on a specific state variable, or more complex such as dynamic collision-avoidance constraints. One major challenge with addressing these safety concerns in real systems is that safety violations will likely be very rare in logs of the system. In many cases, safety constraints are assumed and are not even specified by the system operator or product manager.
An extension to CMDPs is budgeted MDPs [@bmdp; @Carrara2018AFA]. While for a CMDP, the constraint level $V_k$ is given, for budgeted MDPs, it is unknown. Instead, the policy is learned as a function of constraint level. The user can examine the trade-offs between expected return and constraint level and choose the constraint level that best works for the data. This is a good match for common real-world scenario where the constraints may not be absolute, but small violations may be allowed for a large improvement in expected returns.
Recently, there has a been a lot of work focused on the problem of safety in reinforcement learning. One focus has been the addition of a safety layer to the network [@galdalal; @optlayer]. These approaches focus on safety during training, and have enabled an agent to learn a task with zero safety violations during training. There are other approaches [@AchiamHTA17; @Tessler2018; @bohez2019value] that learn a policy that violates constraints during training but produce a *trained* policy that respects the safety constraints. Additional RL approaches include using Lyapunov functions to learn safe policies [@NIPS2018_chow] and exploration strategies that predict the safety of neighboring states [@TurchettaB016; @WachiSYO18]. A Probabilistic Goal MDP [@Mankowitz2016; @Xu2011] is another type of objective that encourages an agent to achieve a pre-defined reward level irrespective of the time it takes to complete the task. This objective encourages risk-averse behaviour leading to safer and more robust policies.
#### Experimental Setup & Results
To demonstrate the complexity of system constraints, we leverage the CMDP formalism to include a series of binary safety-inspired constraints to our challenge domains. These constraints can be either considered passively, as a measure of an agent’s behavior, or they can be included in the agent’s observation so that the agent may learn to avoid them.
As an example, our `cartpole` environment with variables $x,\theta$ (cart position and pole angle) includes three boolean constraints:
1. `slider_pos`, which restricts the cart’s position on the track: $x_l < x < x_r$.
2. `slider_accel`, which limits cart acceleration: $ \ddot{x} < A_\text{max}$.
3. `balance_velocity`, a slightly more complex constraint, which limits the pole’s angular velocity when it is close to being balanced: $ \left| \theta \right| > \theta_L \vee \dot{\theta} < \dot{\theta}_V$.
The full set of available constraints across all tasks is described in Table \[tab:safe\_envs\_full\]. Each constraint can be tuned by modifying a parameter $\texttt{safety\_coeff} \in [0,1]$ where $0$ is harder and $1$ is easier to satisfy.
To evaluate this challenge, we track the number of constraint violations by the agent, for each constraint, throughout training. We present the effects of `safety_coeff` on all four environments in Figure \[fig:safety\_hyperparameters\_sweeps\]. For each task, we illustrate both the effects of `safety_coeff` as a function of the average number of constraint violations upon convergence (left) as well as the average number of violations throughout an episode of `cartpole_swingup` (right). We can see that `safety_coeff` makes the task more difficult as it tends towards 0, and that constraint violations are non-uniform throughout time e.g. as the cart swings back and forth, the pole, position and acceleration constraints are more frequently violated.
Although the learner presented here ignores the constraints, we also include a multi-objective task which combines the task’s reward function with a constraint violation penalty in Section \[sec:multiobj\].
[| l | l | ]{}\
**Type** & **Constraint**\
`slider_pos` & $x_l < x < x_r$\
`slider_accel` &$ \ddot{x} < A_\text{max}$\
`balance_velocity` & $ \left| \theta \right| > \theta_L \vee \dot{\theta} < \dot{\theta}_V $\
\
[| l | l |]{}\
\
**Type** & **Constraint**\
`joint_angle` & $\bm{\theta}_{L} < \bm{\theta} < \bm{\theta}_{U}$\
`joint_velocity` & $ \max_i \left| \dot{\bm{\theta}_i} \right| < L_{\dot{\theta}} $\
`dangerous_fall` & $0 < (\bm{u}_z \cdot \bm{x})$\
`torso_upright` & $0 < \bm{u}_z$\
\
[| l | l |]{}\
\
**Type** & **Constraint**\
`joint_angle` & $\theta_{L, i} < \bm{\theta}_i < \theta_{U,i}$\
`joint_velocity` & $ \max_i \left| \dot{\bm{\theta}_i} \right| < L_{\dot{\theta}} $\
`upright` & $0 < \bm{u}_z$\
`foot_force` &$ \bm{F}_{\text{EE}} < F_\text{max} $\
\
[| l | l |]{}\
\
**Type** & **Constraint**\
`joint_angle_constraint` & $\theta_{L, i} < \bm{\theta}_i < \theta_{U,i}$\
`joint_velocity_constraint` & $\max_i \left| \dot{\bm{\theta}_i} \right| < L_{\dot{\theta}}$\
`upright_constraint` & $0 < \bm{u}_z$\
& $ \bm{F}_{head} < F_{\text{max},1}$\
& $ \bm{F}_{torso} < F_{\text{max},2}$\
`foot_force_constraint` & $ \bm{F}_{\text{Foot}} < F_\text{max,3} $\
![For each task, the left plot shows the evolution of the number of safety constraints upon convergence for various values of the safety coefficient. The right plot shows, for a safety coefficient of 1, the evolution of safety violations over an episode on average. This is to illustrate how different violations get triggered at different points in an episode.[]{data-label="fig:safety_hyperparameters_sweeps"}](figures/D4PG/d4pg_cartpole_swingup_safety.pdf "fig:"){width="48.00000%"} ![For each task, the left plot shows the evolution of the number of safety constraints upon convergence for various values of the safety coefficient. The right plot shows, for a safety coefficient of 1, the evolution of safety violations over an episode on average. This is to illustrate how different violations get triggered at different points in an episode.[]{data-label="fig:safety_hyperparameters_sweeps"}](figures/D4PG/d4pg_walker_walk_safety.pdf "fig:"){width="48.00000%"} ![For each task, the left plot shows the evolution of the number of safety constraints upon convergence for various values of the safety coefficient. The right plot shows, for a safety coefficient of 1, the evolution of safety violations over an episode on average. This is to illustrate how different violations get triggered at different points in an episode.[]{data-label="fig:safety_hyperparameters_sweeps"}](figures/D4PG/d4pg_quadruped_walk_safety.pdf "fig:"){width="48.00000%"} ![For each task, the left plot shows the evolution of the number of safety constraints upon convergence for various values of the safety coefficient. The right plot shows, for a safety coefficient of 1, the evolution of safety violations over an episode on average. This is to illustrate how different violations get triggered at different points in an episode.[]{data-label="fig:safety_hyperparameters_sweeps"}](figures/D4PG/d4pg_humanoidwalk_safety.pdf "fig:"){width="48.00000%"}
![For each task, the left plot shows the evolution of the number of safety constraints upon convergence for various values of the safety coefficient. The right plot shows, for a safety coefficient of 1, the evolution of safety violations over an episode on average. This is to illustrate how different violations get triggered at different points in an episode.[]{data-label="fig:safety_hyperparameters_sweeps"}](figures/DMPO/dmpo_cartpoleswingup_safety.pdf "fig:"){width="48.00000%"} ![For each task, the left plot shows the evolution of the number of safety constraints upon convergence for various values of the safety coefficient. The right plot shows, for a safety coefficient of 1, the evolution of safety violations over an episode on average. This is to illustrate how different violations get triggered at different points in an episode.[]{data-label="fig:safety_hyperparameters_sweeps"}](figures/DMPO/dmpo_walkerwalk_safety.pdf "fig:"){width="48.00000%"} ![For each task, the left plot shows the evolution of the number of safety constraints upon convergence for various values of the safety coefficient. The right plot shows, for a safety coefficient of 1, the evolution of safety violations over an episode on average. This is to illustrate how different violations get triggered at different points in an episode.[]{data-label="fig:safety_hyperparameters_sweeps"}](figures/DMPO/dmpo_quadrupedwalk_safety.pdf "fig:"){width="48.00000%"} ![For each task, the left plot shows the evolution of the number of safety constraints upon convergence for various values of the safety coefficient. The right plot shows, for a safety coefficient of 1, the evolution of safety violations over an episode on average. This is to illustrate how different violations get triggered at different points in an episode.[]{data-label="fig:safety_hyperparameters_sweeps"}](figures/DMPO/dmpo_humanoidwalk_safety.pdf "fig:"){width="48.00000%"}
Challenge 5: Partial Observability and Non-Stationarity {#sec:partial_obs}
-------------------------------------------------------
#### Motivation & Related Work
Almost all real systems where we would want to deploy RL are partially observable. For example, on a physical system, we likely do not have observations of the wear and tear on motors or joints, or the amount of buildup in pipes or vents. We have no observations on the quality of the sensors and whether they are malfunctioning. On systems that interact with users such as recommender systems, we have no observations of the mental state of the users. Often, these partial observations appear as noise (e.g., sensor wear and tear or uncalibrated/broken sensors), non-stationarity (e.g. as a pump’s efficiency degrades) or as stochasticity (e.g. as each robot being operated behaves differently).
#### Partial observability.
Partially observable problems are typically formulated as a partially observable Markov Decision Process (POMDP) [@cassandra1998survey]. The key difference from the MDP formulation is that the agent’s observation $x \in X$ is now separate from the state, with an observation function $O(x\mid s)$ giving the probability of observing $x$ given the environment state $s$. There are a couple common approaches to handling partial observability in the literature. One is to incorporate history into the observation of the agent: DQN [@mnih2015human] stacks four Atari frames together as the agent’s observation to account for partial observability. Alternatively, an approach is to use recurrent networks within the agent, enabling them to track and recover hidden state. [@HausknechtS15] apply such an approach to DQN, and show that the recurrent version can perform equally well in Atari games when only given a single frame as input. [@nagabandi] propose an approach modeling the system as non-stationary with a time-varying reward function, and use meta-learning to find policies that will adapt to this non-stationarity. Much of the recent work on transferring learned policies from simulation to the real system also focuses on this area, as the underlying differences between the systems are not observable [@andrychowicz2018learning; @peng2018sim].
*Experimental Setup & Results* Many real-world sensor issues can be viewed as a partial observability challenge (unobserved properties describing the functioning of the sensor) that could be helped by recurrent models or other approaches for partial observability. A common issue we see in real-world settings is malfunctioning sensors. On any real task, we can assume that the sensors are noisy, which we reproduce by adding increasing levels of Gaussian noise to the actions and observations. Results of these perturbations can be observed in Figures \[fig:d4pg\_noise\] and \[fig:dmpo\_noise\] (left and middle figures respectively) for D4PG and DMPO. We frequently also see sensors that either get stuck at a certain value for a period of time or drop out entirely, with some default value being sent to the agent. We simulate both of these scenarios by setting both a probability of a sensor being stuck or dropped and varying the length of the malfunction being. Results for these perturbations are presented in Figures \[fig:d4pg\_stuck\], \[fig:dmpo\_stuck\] and Figures \[fig:d4pg\_dropped\], \[fig:dmpo\_dropped\] for stuck and dropped sensors. We see from the figures that both dropped and stuck sensors have a significant effect on degrading the final performance.
#### Non-stationarity.
Real world systems are often stochastic and noisy compared to most simulated environments. In addition, sensor and action noise as well as action delays add to the perturbations an agent may experience in the real-world setting. There are a number of RL approaches that have been utilized to ensure that an agent is robust to different subsets of these factors. We will focus on Robust MDPs, domain randomization and system identification as frameworks for reasoning about noisy, non-stationary systems.
A Robust MDP is defined by a tuple $\langle \S, \A, \mathcal{P}, r, \gamma \rangle$ where $S,A,r$ and $\gamma$ are as previously defined; $\mathcal{P}$ is a set of transition matrices referred to as the uncertainty set [@iyengar2005robust]. The objective that we optimize is the worst-case value function defined as: $$\label{eq:robustness}
J(\pi)=\inf_{p \in \mathcal{P}}\mathbb{E}^p \biggl[\sum_{t=0}^\infty \gamma^t r_t \vert \mathcal{P}, \pi \biggr].$$ At each step, nature chooses a transition function that the agent transitions with so as to minimize the long term value. The agent learns a policy that maximizes this worst case value function. Recently, a number of works have surfaced that have shown this formulation to yield robust policies that are agnostic to a range of perturbations in the environment [@tamar2014scaling; @mankowitz2018learning; @shashua2017deep; @derman2018soft; @Derman2019; @Mankowitz2019b]. The solutions do tend to be overly conservative but some work has been done to yield less conservative, ‘soft-robust’ solutions [@derman2018soft].
In addition to the robust MDP formalism, the practitioner may be interested in both robustness due to domain randomization and system identification. Domain randomization [@peng2018sim] involves explicitly training an agent on various perturbations of the environment and averaging these learning errors together during training. System identification involves training a policy that, once on a new system, can determine the characteristics of the environment it is operating in and modify its policy accordingly [@finn2017model; @nagabandi].
*Experimental Setup & Results* We perform a number of different experiments to determine the effects of non-stationarity. We first want to determine whether perturbations to the environment can have an effect on a converged policy that is trained without any challenges added to the environment. For each of the domains, we perturb each of the supported parameters shown in Table \[tab:perturb\_env\_full\]. The effect of the perturbations on the converged D4PG policy for each domain and supported parameter can be seen in Figure \[fig:nonstationary\_disk\]. It is clear that varying the perturbations does indeed have an effect on the performance of the converged policy; in many instances this causes the converged policy to completely fail. This is consistent with the results in [@Mankowitz2019b]. This hyperparameter sweep also helps determine which parameter settings are more likely to have an effect on the learning capabilities of the agent during training.
The second set of experiments therefore aim to determine the consequences of incorporating non-stationarity effects during training. Every episode, new environment parameters are sampled between a $[perturb_{min}, perturb_{max}]$ where $perturb_{min}$ and $perturb_{max}$ indicate the minimum and maximum perturbation values of a particular parameter that we vary. For example, in [`cartpole:swingup`]{}, the perturbation parameter is pole length and $perturb_{min}=0.5$, $perturb_{max}=3.0$ and the variance used for sampling is $perturb_{std}=0.05$.
Based on the previous set of experiments, for each task, we select domain parameters that we expect may change the optimal policy. We perform four hyperparameter training sweeps on the domain parameters for each domain & each algorithm (D4PG and DMPO). These sweeps are in increasing orders of difficulty and have thus been named $\texttt{diff}_1$, $\texttt{diff}_2$, $\texttt{diff}_3$, $\texttt{diff}_4$ and are shown in Table \[tab:perturb\_parameters\]. We perturb the environment in two different ways: uniform and cyclic perturbations. For uniform perturbations, we sample each episode from a uniform distribution and for the cyclic perturbations, a random positive change was sampled from a normal distribution, and the values were reset to the lower limit once the upper limit had been reached. Additional sampling methods and perturbation parameters are supported in the [`realworldrl-suite`]{}and can also be seen in Table \[tab:perturb\_env\_full\]. Cycle sampling simulates scenarios of equipment degrading over time until being replaced or fixed and returning to peak performance. The slow consistent changes over episodes also enables for the possibility of an algorithm adapting to the changes over time.
Figures \[fig:nonstationary\_uniform\] and \[fig:nonstationary\_cyclic\] show the training performance for D4PG and DMPO when applying uniform and cyclic perturbations per episode respectively. As seen in the figures, increasing the range of the perturbation parameter has the effect of slowing down learning. This seems to be consistent across all of the domains we evaluated.
**Env.** **Supported Parameters**
----------- --------------------------
Cart-Pole Pole length
Pole mass
Joint damping
Slider damping
Walker Thigh length
Torso length
Joint damping
Contact friction
Quadruped Shin length
Torso density
Joint damping
Contact friction
Humanoid Joint Damping
Contact Friction
Head Size
: Supported perturbed parameters for each of the control tasks.[]{data-label="tab:perturb_env_full"}
**Env.** **$Perturb_{min}$** **$Perturb_{max}$** **$Perturb_{std}$** **Default Value**
------------------- --------------------- --------------------- --------------------- -------------------
Cart-Pole
Parameter pole\_length
$\texttt{diff}_1$ 0.9 1.1 0.02 1.0
$\texttt{diff}_2$ 0.7 1.7 0.1 1.0
$\texttt{diff}_3$ 0.5 2.3 0.15 1.0
$\texttt{diff}_4$ 0.3 3.0 0.2 1.0
Walker
Parameter thigh\_length
$\texttt{diff}_1$ 0.225 0.25 0.002 0.225
$\texttt{diff}_2$ 0.225 0.4 0.015 0.225
$\texttt{diff}_3$ 0.15 0.55 0.04 0.225
$\texttt{diff}_4$ 0.1 0.7 0.06 0.225
Quadruped
Parameter shin\_length
$\texttt{diff}_1$ 0.25 0.3 0.005 0.25
$\texttt{diff}_2$ 0.25 0.8 0.05 0.25
$\texttt{diff}_3$ 0.25 1.4 0.1 0.25
$\texttt{diff}_4$ 0.25 2.0 0.15 0.25
Humanoid
Parameter join\_damping
$\texttt{diff}_1$ 0.6 0.8 0.02 0.1
$\texttt{diff}_2$ 0.5 0.9 0.04 0.1
$\texttt{diff}_3$ 0.4 1.0 0.06 0.1
$\texttt{diff}_4$ 0.1 1.2 0.1 0.1
: Perturbed parameters chosen for each control task, with varying levels of difficulty[]{data-label="tab:perturb_parameters"}
[.95]{} {width="\textwidth"}
[.95]{} {width="\textwidth"}
[.95]{} {width="\textwidth"}
[.95]{} {width="\textwidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy0_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy1_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy2_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy3_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy4_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy5_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy6_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy100_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy7_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy8_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy9_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy101_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy10_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy102_DISK.pdf){width="\linewidth"}
![Perturbation effects on a converged D4PG policy due to varying specific environment parameters.[]{data-label="fig:nonstationary_disk"}](figures/D4PG/non-stationarity/converged_policy102_DISK.pdf){width="\linewidth"}
![Uniform perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_uniform"}](figures/D4PG/perturbations/perturb_uniform_cartpole_D4PG.jpg "fig:"){width="48.00000%"} ![Uniform perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_uniform"}](figures/D4PG/perturbations/perturb_uniform_quadruped_D4PG.jpg "fig:"){width="48.00000%"} ![Uniform perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_uniform"}](figures/D4PG/perturbations/perturb_uniform_walker_D4PG.jpg "fig:"){width="48.00000%"} ![Uniform perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_uniform"}](figures/D4PG/perturbations/perturb_uniform_humanoid_D4PG.jpg "fig:"){width="48.00000%"}
![Uniform perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_uniform"}](figures/DMPO/perturbations/perturb_uniform_cartpole_DMPO.jpg "fig:"){width="48.00000%"} ![Uniform perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_uniform"}](figures/DMPO/perturbations/perturb_uniform_quadruped_DMPO.jpg "fig:"){width="48.00000%"} ![Uniform perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_uniform"}](figures/DMPO/perturbations/perturb_uniform_walker_DMPO.jpg "fig:"){width="48.00000%"} ![Uniform perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_uniform"}](figures/DMPO/perturbations/perturb_uniform_humanoid_DMPO.jpg "fig:"){width="48.00000%"}
![Cyclic perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_cyclic"}](figures/D4PG/perturbations/perturb_cyclic_cartpole_D4PG.jpg "fig:"){width="48.00000%"} ![Cyclic perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_cyclic"}](figures/D4PG/perturbations/perturb_cyclic_quadruped_D4PG.jpg "fig:"){width="48.00000%"} ![Cyclic perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_cyclic"}](figures/D4PG/perturbations/perturb_cyclic_walker_D4PG.jpg "fig:"){width="48.00000%"} ![Cyclic perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_cyclic"}](figures/D4PG/perturbations/perturb_cyclic_humanoid_D4PG.jpg "fig:"){width="48.00000%"}
![Cyclic perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_cyclic"}](figures/DMPO/perturbations/perturb_cyclic_cartpole_DMPO.jpg "fig:"){width="48.00000%"} ![Cyclic perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_cyclic"}](figures/DMPO/perturbations/perturb_cyclic_quadruped_DMPO.jpg "fig:"){width="48.00000%"} ![Cyclic perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_cyclic"}](figures/DMPO/perturbations/perturb_cyclic_walker_DMPO.jpg "fig:"){width="48.00000%"} ![Cyclic perturbations applied per episode for each of the four domains for D4PG and DMPO.[]{data-label="fig:nonstationary_cyclic"}](figures/DMPO/perturbations/perturb_cyclic_humanoid_DMPO.jpg "fig:"){width="48.00000%"}
Challenge 6: Multi-Objective Reward Functions {#sec:multiobj}
---------------------------------------------
#### Motivation & Related Work
RL frames policy learning through the lens of optimizing a global reward function, yet most systems have multi-dimensional costs to be minimized. In many cases, system or product owners do not have a clear picture of what they want to optimize. When an agent is trained to optimize one metric, other metrics are often discovered that also need to be maintained or improved. Thus, a lot of the work on deploying RL to real systems is spent figuring out how to trade off between different objectives.
There are many ways of dealing with multi-objective rewards: [@roijers2013survey] provide an overview of various approaches. Various methods exist that deal explicitly with the multi-objective nature of the learning problems, either by predicting a value function for each objective [@vanseijen], or by finding a policy that optimizes each sub-problem [@Li2019], or that fits each Pareto-dominating mixture of objectives [@Moffaert2014]. [@Yang2019] learn a general policy that can behave optimally for any desired mixture of objectives. Multiple trivial objectives have been also used for enriching the reward signal to simply improve learning of the base task [@Jaderberg2016].
In the specific case of dealing with balancing a task reward with negative outcomes, a possible approach is to use a Conditional Value at Risk (CVaR) objective [@Tamar2015], which looks at a given percentile of the reward distribution, rather than expected reward. @Tamar2015 show that by optimizing reward percentiles, the agent is able to improve upon its worst-case performance. Distributional DQN [@dabney18a; @BellemareDM17] explicitly models the distribution over returns, and it would be straight-forward to extend it to use a CVaR objective.
When rewards can’t be functionally specified, there are a number of works devoted to recovering an underlying reward function from demonstrations, such as inverse reinforcement learning [@russell1998learning; @ng2000algorithms; @abbeel2004apprenticeship; @ross2011reduction]. @menell examine how to infer the truly intended reward function from the given reward function and training MDPs, to ensure that the agent performs as intended in new scenarios.
Because the global reward function is generally a balance of multiple sub-goals (e.g., reducing both time-to-target and energy use), a proper evaluation should separate the individual components of the reward function to better understand the policy’s trade-offs. Looking at the Pareto boundaries provides some insights to the relative trade-offs between objectives, but doesn’t scale well beyond 2-3 objectives. We propose a simple multi-objective analysis of return. If we consider that the global reward function is defined as a linear combination of sub-rewards, $r(s,a) = \sum_{j=1}^K \alpha_j r_j(s,a)$, then we can consider the vector of per-component rewards for evaluation: $$\label{eq:multiobj}
\bm{J}^{multi}(\pi) = \left(\sum_{i=1}^{T_n} r_j(s_i,a_i)\right)_{1\leq j \leq K} \in \mathbb{R}^K.$$
When dealing with multi-objective reward functions, it is important to track the different objectives individually when evaluating a policy. This allows for a more clear understanding of the different trade-offs the policy is making and choose which compromises they consider best.
To evaluate the performance of the algorithm across the full distribution of scenarios (e.g. users, tasks, robots, objects,etc.), we suggest independently analyzing the performance of the algorithm on each cohort. This is also important for ensuring fairness of an algorithm when interacting with populations of users. Another approach is to analyze the CVaR return rather than expected returns, or to directly determine whether rare catastrophic rewards are minimized [@Tamar2015; @tamar2015policy]. Another evaluation procedure is to observe behavioural changes when an agent needs to be risk-averse or risk-seeking such as in football [@Mankowitz2016].
#### Experimental Setup & Results
We illustrate the multi-objective challenge by looking at the effects of a multi-objective reward function that encourages both task success and the satisfaction of safety constraints specified in Section \[sec:constraints\]. We use a naive mixture reward: $$\label{eq:mixture}
r_m = (1-\alpha) r_b + \alpha r_c,$$ where $r_b$ is the task’s base reward, $r_c$ is the number of satisfied constraints during that timestep and $\alpha \in[0,1]$ is the multi-objective coefficient that balances between the objectives.
The [`realworldrl-suite`]{}allows multi-objective rewards to be defined, providing the multiple objectives either as observations to the agents, as modifications to the original task’s reward, or both. We use the suite to model the multi-objective problem by letting $\alpha$ correspond to the `multiobj_coeff` in the [`realworldrl-suite`]{}, and changing the task’s reward to correspond to Equation . For each task, we visualize both the per-element reward, as defined in Equation , and the average number of each constraint’s violations upon convergence. Figure \[fig:multiobj\] shows the varying effects of this multi-objective reward on each reward component, $r_b$ and $r_c$, as a function of $\texttt{multiobj\_coeff}$, where we adjust `safety_coeff` to $0.5$ and vary `multiobj_coeff`. We can see the evolution in performance relative to $r_b$ and $r_c$ (left), as well as the resulting effects on constraint satisfaction (right) as `multiobj_coeff` is varied. As $r_c$ becomes more important in the global reward, constraints are quickly taken into account. However, over-emphasis on $r_c$ quickly degrades $r_b$ and therefore base task performance. Although this is a naive way to deal with safety constraints, it illustrates the often contradictory goals that a real-world task might have, and the difficulty in satisfying all of them. We also believe it provides an interesting framework to analyze how different algorithmic approaches better balance the need to satisfy constraints with the ability to maintain adequate system performance.
![Performance vs. constraint satisfaction trade-offs as $\alpha$, the multiobjective coefficient, is varied. The multi-objective coefficient is the reward-mixture coefficient that makes the agent’s perceived reward lean more towards the original task reward or more towards the constraint satisfaction reward. For each task, the left plot shows the evolution of the tasks’ original reward as the reward-mixture mixture coefficient is altered. The right plot shows the average number of constraint violations upon convergence per episode for each individual constraint.[]{data-label="fig:multiobj"}](figures/D4PG/d4pg_cartpoleswingup_multiobj.pdf "fig:"){width="49.00000%"} ![Performance vs. constraint satisfaction trade-offs as $\alpha$, the multiobjective coefficient, is varied. The multi-objective coefficient is the reward-mixture coefficient that makes the agent’s perceived reward lean more towards the original task reward or more towards the constraint satisfaction reward. For each task, the left plot shows the evolution of the tasks’ original reward as the reward-mixture mixture coefficient is altered. The right plot shows the average number of constraint violations upon convergence per episode for each individual constraint.[]{data-label="fig:multiobj"}](figures/D4PG/d4pg_walkerwalk_multiobj.pdf "fig:"){width="49.00000%"} ![Performance vs. constraint satisfaction trade-offs as $\alpha$, the multiobjective coefficient, is varied. The multi-objective coefficient is the reward-mixture coefficient that makes the agent’s perceived reward lean more towards the original task reward or more towards the constraint satisfaction reward. For each task, the left plot shows the evolution of the tasks’ original reward as the reward-mixture mixture coefficient is altered. The right plot shows the average number of constraint violations upon convergence per episode for each individual constraint.[]{data-label="fig:multiobj"}](figures/D4PG/d4pg_quadrupedwalk_multiobj.pdf "fig:"){width="49.00000%"} ![Performance vs. constraint satisfaction trade-offs as $\alpha$, the multiobjective coefficient, is varied. The multi-objective coefficient is the reward-mixture coefficient that makes the agent’s perceived reward lean more towards the original task reward or more towards the constraint satisfaction reward. For each task, the left plot shows the evolution of the tasks’ original reward as the reward-mixture mixture coefficient is altered. The right plot shows the average number of constraint violations upon convergence per episode for each individual constraint.[]{data-label="fig:multiobj"}](figures/D4PG/d4pg_humanoidwalk_multiobj.pdf "fig:"){width="49.00000%"}
![Performance vs. constraint satisfaction trade-offs as $\alpha$, the multiobjective coefficient, is varied. The multi-objective coefficient is the reward-mixture coefficient that makes the agent’s perceived reward lean more towards the original task reward or more towards the constraint satisfaction reward. For each task, the left plot shows the evolution of the tasks’ original reward as the reward-mixture mixture coefficient is altered. The right plot shows the average number of constraint violations upon convergence per episode for each individual constraint.[]{data-label="fig:multiobj"}](figures/DMPO/dmpo_cartpoleswingup_multiobj.pdf "fig:"){width="49.00000%"} ![Performance vs. constraint satisfaction trade-offs as $\alpha$, the multiobjective coefficient, is varied. The multi-objective coefficient is the reward-mixture coefficient that makes the agent’s perceived reward lean more towards the original task reward or more towards the constraint satisfaction reward. For each task, the left plot shows the evolution of the tasks’ original reward as the reward-mixture mixture coefficient is altered. The right plot shows the average number of constraint violations upon convergence per episode for each individual constraint.[]{data-label="fig:multiobj"}](figures/DMPO/dmpo_walkerwalk_multiobj.pdf "fig:"){width="49.00000%"} ![Performance vs. constraint satisfaction trade-offs as $\alpha$, the multiobjective coefficient, is varied. The multi-objective coefficient is the reward-mixture coefficient that makes the agent’s perceived reward lean more towards the original task reward or more towards the constraint satisfaction reward. For each task, the left plot shows the evolution of the tasks’ original reward as the reward-mixture mixture coefficient is altered. The right plot shows the average number of constraint violations upon convergence per episode for each individual constraint.[]{data-label="fig:multiobj"}](figures/DMPO/dmpo_quadrupedwalk_multiobj.pdf "fig:"){width="49.00000%"} ![Performance vs. constraint satisfaction trade-offs as $\alpha$, the multiobjective coefficient, is varied. The multi-objective coefficient is the reward-mixture coefficient that makes the agent’s perceived reward lean more towards the original task reward or more towards the constraint satisfaction reward. For each task, the left plot shows the evolution of the tasks’ original reward as the reward-mixture mixture coefficient is altered. The right plot shows the average number of constraint violations upon convergence per episode for each individual constraint.[]{data-label="fig:multiobj"}](figures/DMPO/dmpo_humanoidwalk_multiobj.pdf "fig:"){width="49.00000%"}
Challenge 7: Real-time Inference Challenge {#sec:real-time-inference}
------------------------------------------
#### Motivation & Related Work
To deploy RL to a production system, policy inference must be done in real-time at the control frequency of the system. This may be on the order of milliseconds for a recommender system responding to a user request [@covington2016deep] or the control of a physical robot, and up to the order of minutes for building control systems [@DM_Datacenter]. This constraint both limits us from running the task faster than real-time to generate massive amounts of data quickly [@silver2016mastering; @impala] and limits us from running slower than real-time to perform more computationally expensive approaches (e.g. some forms of model-based planning [@doya2002multiple; @levine2019prediction; @schrittwieser2019mastering]).
One approach is to take existing algorithms and validate their feasibility to run in real-time [@adam2011experience]. Another approach is to design algorithms with the explicit goal of running in real-time [@cai2017real; @wang2015real]. Recently [@ramstedt2019real] presented a different view on real-time inference and proposed the Real-Time Markov Reward Process, in which the state evolves during an action selection. Anytime inference [@vlasselaer2015anytime; @spirtes2001anytime] is a family of algorithms that can return a valid solution at any time they are being interrupted, and are expected to produce better performing solutions the longer they run.
#### Experimental Setup & Results
The [`realworldrl-suite`]{}offers two ways in which one can measure the effect of real-time inference: *latency* and *throughput*. Latency corresponds to the amount of time it takes an agent to output an action based on an observation. Even if the agent is replicated over multiple machines, allowing it to handle the frequency of the observations arriving from the system, it still may have latency issues due to the time it needs in order to output an action for a single observation. To be able to see how a system will react in the face of latency, we use the action delay mechanism, where at time step $t$ the agent outputs an action $a_t$ based on $s_{t}$, but the system actually responds to $a_{t-n}$, where $n$ is the delay in time steps. Throughput correspond to the frequency of input observations the agent is able to process which depends on the amount of hardware or compute that is available for it as well as the complexity of the agent itself. We modeled the effects of throughput bottlenecks as action repetition: we denote the length of the action repetition by $k$, then at time step $k\cdot t$ the agent outputs an action $a_{k \cdot t}$ based on the observation $s_{k \cdot t}$, however, for the next $k-1$ time steps (i.e., time steps $k \cdot t + 1, k \cdot t + 2, .. (k+1) \cdot t - 1$), the agent repeats the same output $a_{k \cdot t}$. These two perturbations allow us to see how agents that have latency and throughput issues will affect their environment, and additionally can show us how well an agent can learn to plan accordingly to compensate for its computational shortcomings.
Figures \[fig:d4pg\_all\_delays\] and \[fig:dmpo\_all\_delays\] show the performance of D4PG and DMPO, respectively, on the action delay challenge. For discussion on these results we refer the reader to Section \[subsec:delays\]. Figures \[fig:action\_repetition\_d4pg\] and \[fig:action\_repetition\_dmpo\] shows the performance on the action repetition challenge for D4PG and DMPO, respectively. We note that generally, as expected, the performance of the agents deteriorates as the number of repeated actions increases. More interestingly though, we observe that albeit `quadruped` has larger state and action spaces than `cartpole` and `walker`, it still more robust to action repetition. We believe the reason for that lies in the inherit stability of the different tasks, where `humanoid` is the least stable, and `quadruped` is the most stable.
[.45]{} {width="\textwidth"}
[.45]{} {width="\textwidth"}
Challenge 8: Offline Reinforcement Learning - Training from Offline Logs
------------------------------------------------------------------------
#### Motivation & Related Work
For many systems, learning from scratch through online interaction with the environment is too expensive or time-consuming. Therefore, it is important to design algorithms for learning good policies from offline logs of the system’s behavior. In many cases these comes from an existing rule-based, heuristic or myopic policy that we are trying to replace with an RL approach. Reinforcement learning from data logs has traditionally been called batch RL in the literature, but has started to be referred to as offline RL more recently in order to avoid confusion with mini-batch learning. An extension of this setup is the “growing-batch” setting, where a new policy is trained offline at each iteration, with the logs including new data from all the previous policies.
Some of the early examples of offline / batch RL include least squares temporal difference methods [@bradtke1996linear; @lagoudakis2003least] and fitted Q iteration [@ernst2005tree; @riedmiller2005neural]. More works such as @agarwal2019striving, @fujimoto2019off, or @kumar2019stabilizing have shown that naively applying well-known deep RL methods such as DQN [@mnih2015human] in the offline setting can lead to poor performance. This has been attributed to a combination of poor generalization outside the training data’s distribution as well as overly confident Q-function estimates when performing backups with a $\max$ operator. However, distributional deep RL approaches [@dabney18a; @BellemareDM17; @barthmaron2018d4pg] have been shown to produce better performance in the offline setting in both Atari [@agarwal2019striving] and robot manipulation [@cabi2019scaling]. There have also been a number of recent methods explicitly addressing the issues stemming from combining generalization outside the training data along with issues related to the $\max$ operator, which come in two main flavors. The first family of approaches constrain the action choice to the support of the training data [@fujimoto2019off; @kumar2019stabilizing; @siegel2020keep; @jaques2019way; @wu2019behavior]. The second type of approaches start with behavior cloning [BC; @pomerleau1989alvinn], which trains a policy using the objective of predicting the action seen in the offline logs. Works such as @wang2018exponentially, @chen2019bail, or @peng2019advantage then use the advantage function to select the best actions in the dataset for training behavior cloning.
#### Experimental Setup & Results
The [`realworldrl-suite`]{}version of the offline / batch RL challenge is to learn from data logs generated from sub-optimal policies running on the no-challenge setting, where all challenge effects are turned off, and the *combined challenge* setting (see Section \[sec:combined\_challenges\]) where data logs are generated from an environment that includes effects from combining all the challenges (except for safety and multi-objective rewards). The policies were obtained by training three DMPO agents until convergence with different random weight initializations, and then taking snapshots corresponding to roughly $75\%$ of the converged performance. For the *no challenge* setting, we generated three datasets of different sizes for each environment by combining the three snapshots, with the total dataset sizes (in numbers of episodes) provided in Table \[tab:batch\_rl\_data\]. Further, we repeated the procedure with the easy combination of the other challenges (see section \[sec:combined\_challenges\]). We chose to use the “large data” setting for the combined challenge to ensure the task is still solvable. The algorithms used for offline learning were an offline version of D4PG [@barthmaron2018d4pg] that uses the data logs as a fixed experience replay buffer, as well as a variation of the Advantage-weighted Behavioural Model (ABM) [@Siegel2020], which restricts the learned model to mimic the behavior policy when it has a positive advantage. The performance of the ABM algorithm trained on the small, medium and large batch datasets can be found in Figure \[fig:offline\_learning\] (learning curves) for each of the domains. D4PG was also trained on each of the tasks, but failed to learn in each case and therefore the results have been omitted. As seen in the figures, the agent fails to learn properly in the [`humanoid:walk`]{}and [`cartpole:swingup`]{}domain, but manages to reach a decent level of performance in [`walker:walk`]{}and [`quadruped:walk`]{}. In addition, the size of the dataset does not seem to have a significant effect on performance. This may indicate that the dataset sizes are still too large to handicap an agent’s learning capabilities for a state-of-the-art offline RL agent, while being too difficult to solve for D4PG.
For the ‘Easy’ combined challenge offline task, we used DMPO behaviour policies trained on each task. The [`humanoid:walk`]{}DMPO behaviour policy was too poor to generate reasonable data (see Figure \[fig:dmpo\_all\_challenges\_plot\]) and we therefore focused on [`cartpole:swingup`]{}, [`walker:walk`]{}and [`quadruped:walk`]{}for this task. This also motivates why we need to make progress on the combined challenges *online* task (see Section \[sec:combined\_challenges\]) so that we can generate reasonable behaviour policies to generate the datasets for batch RL algorithms to train on.
We subsequently trained ABM and D4PG (offline version) on the data generated from the behaviour policies. The agents failed to achieve any reasonable level of performance on cartpole and walker, and have thus been omitted. The learning curves of ABM trained on quadruped on the combined easy challenge can be found in Figure \[fig:offline\_learning\_combined\]. Although the performance is still sub-optimal, it is encouraging to see that the batch agents can learn something reasonable. The D4PG offline agent failed to learn in each case and the results have therefore been omitted.
cartpole:swingup walker:walk quadruped:walk humanoid:walk
---------------- ------------------ ------------- ---------------- ---------------
Small Dataset 100 1000 100 4000
Medium Dataset 200 2000 200 8000
Large Dataset 500 5000 500 20000
: Amount of data (number of episodes) used for different versions of the offline RL challenge. When we added the combined version of the other challenges as well, we used the “most data” version in order to keep the task solvable. We chose these numbers to be approximately four times the number of epsiodes that it takes for each agent to converge in the online setting.[]{data-label="tab:batch_rl_data"}
![Learning from offline data on small, medium and large datasets in the no challenge setting using Advantage-weighted Behavioural Modelling. For the cartpole domain, the X-axis is extended to show a clearer learning curve.[]{data-label="fig:offline_learning"}](figures/AWM/offline/offline_training0_Advantage-Weighted_Modelling.pdf){width="\linewidth"}
![Learning from offline data on small, medium and large datasets in the no challenge setting using Advantage-weighted Behavioural Modelling. For the cartpole domain, the X-axis is extended to show a clearer learning curve.[]{data-label="fig:offline_learning"}](figures/AWM/offline/offline_training1_Advantage-Weighted_Modelling_first_5000_episodes.pdf){width="\linewidth"}
![Learning from offline data on small, medium and large datasets in the no challenge setting using Advantage-weighted Behavioural Modelling. For the cartpole domain, the X-axis is extended to show a clearer learning curve.[]{data-label="fig:offline_learning"}](figures/AWM/offline/offline_training2_Advantage-Weighted_Modelling_first_5000_episodes.pdf){width="\linewidth"}
![Learning from offline data on small, medium and large datasets in the no challenge setting using Advantage-weighted Behavioural Modelling. For the cartpole domain, the X-axis is extended to show a clearer learning curve.[]{data-label="fig:offline_learning"}](figures/AWM/offline/offline_training3_Advantage-Weighted_Modelling_first_5000_episodes.pdf){width="\linewidth"}
![Learning from offline data on large datasets in the easy combined challenge setting using Advantage-weighted Behavioural Modelling on quadruped.[]{data-label="fig:offline_learning_combined"}](figures/AWM/offline/offline_training_combined_2_Advantage-Weighted_Modelling.pdf){width="\linewidth"}
Combining the Challenges: RWRL Benchmark {#sec:combined_challenges}
----------------------------------------
While each of these challenges present difficulties independently, many real world domains possess all of the challenges together. To demonstrate the difficulty of learning to control a system with multiple dimensions of real-world difficulty, we combine multiple challenges described above into a set of benchmark tasks to evaluate real-world learning algorithms. Our combined challenges include parameter perturbations, additional state dimensions, observation delays, action delays, reward delays, action repetition, observation & action noise, and stuck & dropped sensors. Even taking the relatively easy versions of each challenge (where the algorithm still reached close to the optimal performance individually) and combining them together creates a surprisingly difficult task. Performance on these challenges can be seen in Table \[tab:d4pg\_all\_challenges\] for D4PG and Table \[tab:dmpo\_all\_challenges\] for DMPO, and Figures \[fig:d4pg\_all\_challenges\_plot\] and \[fig:dmpo\_all\_challenges\_plot\] respectively. We can see that both learners’ performance drops drastically, even when applying the smallest perturbations of each challenge.
Due to both the application interest in these combined challenges, as well as their clear difficulty, we believe them to be good benchmark tasks for researchers looking to create RL algorithms for real-world systems. We provide the parameters for each challenge in Table \[tab:challengehyperparams\] (taken from the individual hyperparameters sweeps, see Table \[app1:hyperparameters\_sweeps\] in the Appendix). The [`realworldrl-suite`]{}can load the challenges directly, making it easy to replicate these benchmark environments in any experimental setup. Although the baseline performance we provide is with a naive learner that is not designed to answer these challenges, we believe it provides a good starting point for comparison and look forward to followup work that provides more performant algorithms on these reference challenges.
[.45]{} {width="\textwidth"}
[.45]{} {width="\textwidth"}
----------------------- ------------ -- ------------ -- ------------ --
**Experiment**
**System Delays**
Action
Observation
Rewards
**Action Repetition**
**Gaussian Noise**
Action
Observation
**Stuck/**
**Dropped Noise**
Stuck Sensor
Dropped Sensor
**Perturbation**
**Cartpole**
**Perturbation**
**Quadruped**
**Perturbation**
**Walker**
**Perturbation**
**Humanoid**
**High**
**Dimensionality** *Increase* *Increase* *Increase*
----------------------- ------------ -- ------------ -- ------------ --
: The hyperparameter setting for each combined challenge in increasing levels of difficulty[]{data-label="tab:challengehyperparams"}
cartpole:swingup walker:walk quadruped:walk humanoid:walk
-------- ------------------ ---------------- ----------------- ----------------
859.63 (5.68) 983.24 (9.7) 998.71 (0.32) 934.0 (27.34)
Easy 482.32 (84.56) 514.44 (70.21) 787.73 (86.95) 102.92 (22.47)
Medium 175.47 (51.57) 75.49 (16.94) 268.01 (135.84) 1.28 (0.99)
Hard 108.2 (57.97) 59.85 (17.7) 280.75 (123.21) 1.27 (0.79)
: Mean D4PG performance ($\pm$ standard deviation) when incorporating all challenges into the system.[]{data-label="tab:d4pg_all_challenges"}
cartpole:swingup walker:walk quadruped:walk humanoid:walk
-------- ------------------ ---------------- ----------------- ----------------
859.06 (18.07) 977.71 (14.5) 998.35 (3.71) 788.49 (33.88)
Easy 464.05 (89.11) 474.44 (74.55) 567.53 (210.54) 1.33 (1.14)
Medium 155.63 (35.81) 64.63 (17.03) 180.3 (92.41) 1.27 (0.9)
Hard 138.06 (55.82) 63.05 (18.71) 144.69 (92.85) 1.4 (0.82)
: Mean DMPO performance ($\pm$ standard deviation) when incorporating all challenges into the system.[]{data-label="tab:dmpo_all_challenges"}
Future Iterations
-----------------
In this paper, we have addressed 8 of the 9 challenges originally presented in [@dulacarnold2019challenges]. The remaining challenge is explainability. Objectively evaluating explainability of a policy is not trivial, but we we hope this can be addressed in future iterations of this suite. We provide an overview of this challenge and possible approaches to creating explainable RL agents.
#### Explainability
Another essential aspect of real systems is that they are owned and operated by humans, who need to be reassured about the controllers’ intentions and require insights regarding failure cases. For this reason, policy explainability is important for real-world policies. Especially in cases where the policy might find an alternative and unexpected approach to controlling a system, understanding the longer-term intent of the policy is important for obtaining stakeholder buy-in. In the event of policy errors, being able to understand the error’s origins *a posteriori* is essential. Previous work that is potentially well-suited to this challenge include options [@Sutton1999] that are well-defined hierarchical actions that can be composed together to solve a given task. Previous research in this area includes learning the options from scratch [@Mankowitz2016a; @Mankowitz2016b; @Bacon2017] as well as planning, given a pre-trained set of options [@Schaul2015; @Mankowitz2018]. In addition, research has been done to develop a symbolic planning language that could be useful for explainability [@Konidaris2018; @James2018].
Additional Related Work
=======================
While we covered related work specific to each challenge in the sections above, there are a few other works that relate to ours, either through the goal of practical reinforcement learning or more generally by providing interesting benchmark suites.
In general, the fact that machine learning methods have a tendency to overfit to their evaluation environments is well-recognized. [@wagstaff2012machine] discusses the strong lack of real-world applications in ML conferences and the subsequent impact on research directions this can have. [@henderson2018deep] investigate ways in which RL results can be made to be more reproducible and suggest guidelines for doing so. Their paper ends by asking the question “In what setting would \[a given algorithm\] be useful?”, to which we try to contribute by proposing a specific setting in which well-adapted work should hopefully stand out.
@MLJ12-hester similarly present a list of challenges for real world RL, but specifically for RL on robots. They present four challenges (sample efficiency, high-dimensional state and action spaces, sensor/actuator delays, and real-time inference), all of which we include in our set of challenges. They do not include our other challenges such as satisying constraints, multi-objective, non-stationarity and partial observability (e.g., noisy/stuck sensors). Their approach is to setup a real-time architecture for model-based learning where ensembles of models are learned to improve robustness and sample efficiency. In a spirit similar to ours, the `bsuite` framework [@Osband2019] proposes a set of challenges grounded in fundamental problems in RL such as memory, exploration, credit assignment etc. These problems are equally important and complementary to the more empirically founded challenges proposed in our suite. Recently, other teams have released real-world inspired environments, such as Safety Gym [@Ray2019], which extends a planar world with location-based safety constraints. Our suite proposes a richer and more varied set of constraints, as well as an easy ability to add custom constraints, which we believe provides a more general and difficult challenge for RL algorithms.
The Horizon platform [@gauci2018horizon] and Decision Service [@agarwal2016making] provide software platforms for training, evaluation and deployment of RL agents in real-world systems. In the case of Decision Service, transition probabilities are logged to help make off-policy evaluation easier down the line, and both systems consider different approaches to off-policy evaluation. We believe well-structured frameworks such as these are crucial to productionizing RL systems. [@Kumar_ROBEL] propose a set of simple robot designs with corresponding simulators that have been tuned to be physically realistic, implementing safety constraints and various perturbations.
Challenge Suite Overview
========================
Our open-sourced [`realworldrl-suite`]{}contains:
- Seven real-world challenge wrappers (mentioned above) across $8$ DeepMind Control Suite tasks [@tassa2018deepmind]:\
`cartpole:(swingup and balance)`, `walker:(walk and run)`,\
`quadruped:(walk and run)`, `humanoid:(stand and walk)`
- The flexibility to instantiate different variants of each challenge, as well as the ability to easily combine challenges together using a simple configuration language. See Appendix \[app:codebase\] for more details.
- Examples of how to run RL agents on each challenge environment.
- The ability to instantiate the “Easy”, “Medium” and “Hard” combined challenges.
- A Jupyter notebook enabling an agent to be run on any of the challenges in a browser, as well as accompanying functions to plot the agent’s performance.
#### Evaluation environments.
In this paper, we evaluate RL algorithms on a subset of four tasks from our suite, namely: [`cartpole:swingup`]{}, [`walker:walk`]{}, [`quadruped:walk`]{}and [`humanoid:walk`]{}. We chose these tasks to cover varying levels of task difficulty and dimensionality. It should be noted that [`MuJoCo`]{}possesses an internal dynamics state and that only preprocessed observations are available to the agent [@tassa2018deepmind]. We refer to state in this paper as in the typical MDP setting: the information available to the agent at time $t$. Since we provide all available observations as input to the agent, we use the term observation and state interchangeably in this paper. For each challenge, we have implemented environment wrappers that instantiate the challenge. These wrappers are parameterized such that the challenge can be ramped up from having no effect to being very difficult. For example, the amount of delay added onto the actuators can be set arbitrarily, varying the difficulty from slight to impossible. By implementing the challenges in this way, we can easily adapt them to other tasks and ramp them up and down to measure their effects. Our goal with this task suite is to replicate difficulties seen in complex real systems in a more simplified setup, allowing for methodical and principled research.
Discussion & Conclusion
=======================
To re-iterate from the Introduction, our contributions can be structured into four parts: (1) Identifying and defining a set of challenges; (2) Designing a set of experiments and analysing their effects on common RL agents; (3) Defining and benchmarking RWRL combined challenge tasks for easy algorithmic comparisons; and (4) Open-sourcing an environmental suite, [`realworldrl-suite`]{}, which allows researchers and practicioners to easily replicate and extend the experiments we performed.
#### Identification and definition of real-world challenges
We believe that we provide a set of the most important challenges that RL algorithms need to succeed at before being ready for real-world application. In our own personal experience as well as that of our collaborators, we have been confronted ourselves numerous times with the often difficult task of applying RL to various real-world systems. This set of challenges stems from these experiences, and we are convinced that finding solutions to them will likely provide promissing algorithms that are readily useable in real-world systems. We are particularly interested in results in the off-line domain, as most large systems have a large amount of logs, but little to no tolerance for exploratory actions (datacenter cooling & robotics being good examples of this). We also believe that algorithms able to reason about environmental constraints will allow RL to move onto systems that were previously considered too fragile or expensive for learning-based approaches. Overall, we are excited about the directions that a lot of the cited research is taking and looking forward to interesting results in the near future.
#### Experiment design and analysis for each challenge
Additionally, the design of an experiment for each challenge demonstrates the independent effects of each challenge on an RL agent. This allowed us to show which aspects of real-world tasks present the biggest difficulties for RL agents in a precise and reproducible manner. In the case of *learning on live systems from limited samples*, our proposed efficiency metrics (performance regret and stability) produced interesting findings, showing DMPO to be almost an order of magnitude worse in terms of regret, but significantly more stable once converged. When *dealing with system delays*, we saw that observation and action delays quickly degrade algorithm performance, but reward delays seem to be globally less impactful except on the [`humanoid:walk`]{}task. For *high-dimensional continuous state & action spaces*, we see that additional observation dimensions don’t affect either DMPO or D4PG significantly, and that environments with more action dimensions are not necessarily harder to learn. When *reasoning about system constraints*, we argue that explicit reasoning about constraints is preferable to simply integrating them in the reward, and show that there is no natural way to express constraints in the standard MDP framework. We provide a mechanism in [`realworldrl-suite`]{}that can express constraints in the CMDP setting, and show that constraints can be violated in interesting ways, especially in tasks that have different regimes (e.g. [`cartpole:swingup`]{}’s ‘swing-up’ and ‘balance’ phases). *Partial observability & non-stationarity* are often present in real systems, and can present clear problems for learning algorithms. In small doses stuck sensors pose less of a problem than outright dropped signals however, even though the underlying information is the same. When it comes to non-stationary system dynamics, we can see that the effects depend greatly on the type of element that is varying. Additionally, naive policies clearly degrade more quickly in the face of unstable system dynamics. *Multi-objective rewards* can be difficult to optimize for when they are not well-aligned. By using safety-related constraints that weren’t always compatible with the base task, we showed how naively reasoning about this trad-eoff can quickly degrade system performance, yet that compromising solutions are also possible. We believe that expressing tasks beyond a single reward function is essential in tackling more complex problems and look forward to new methods able to do so. *Real-time policies* are essential for high-frequency control loops present in robotics or low-latency responses necessary in software systems. We showed the effects of both action and state delays on DMPO and D4PG, and showed that these approaches quickly degrade if the system’s control frequency is higher than their response time and actions decorrelate too strongly from observations. Many real-world systems are hard to train on directly, and therefore RL agents need to be able to *train off-line from fixed logs*. It has long been known that this is not a trivial task, as situations that aren’t represented in the data become difficult to respond to. Especially in the case of off-policy td-learning methods, the $\operatorname*{arg\,max}$ over-estimation issue quickly creates divergent value functions. We showed that simply applying D4PG to data from a logged task is not sufficient to find a functional policy, but that offline-specific learning algorithms can deal with even small amounts of data. Finally, *explainable policies* are often desirable (as are explainable machine learning models in general), but not easy to provide or even evaluate. We provide a couple directions of current work in this area, and hope that future work finds clearer approach to this problem.
#### Define and baseline RWRL Combined Challenge Benchmark tasks
By combining a well-tuned set of challenges into a single environment, we were able to generate 12 benchmark tasks (3 levels of difficulty and 4 tasks) which can serve as reference tasks for further research in real-world RL. The choice of challenge parameterizations for each level of difficulty was performed after careful analysis of the combined effects on the learning algorithms we experimented with. We also provided a first round of baselines on our benchmark tasks by running D4PG and DMPO on them: we find that D4PG seems to be slightly more robust for easy perturbations but, aside from the [`quadruped:walk`]{}task, quickly matches DMPO in poor performance. By providing these baseline performance numbers for D4PG and DPMO on these task, we hope that followup work will have a good starting point to understand the quality of their proposed solutions. We encourage the research community to better our current set of RWRL combined challenge baseline results.
#### Open-source the [`realworldrl-suite`]{}codebase
Finally, by implementing all our challenges in the open-sourced [`realworldrl-suite`]{}, we provide a reference implementation for each challenge that allows easy performance comparisons between algorithms hoping to respond to these challenges. By leveraging both the [`realworldrl-suite`]{}and the performance baselines for each challenge presented in this paper, future researchers developing real-world RL algorithms can easily compare their approach against common baselines to provide clear and objective evaluation.
We hope this body of works provides both encouragement to the reinforcement learning community to take up these challenges that are important holdups to bringing RL into real systems, as well as intuition to practicioners who have confronted themselves with attempting to apply RL methods on practical tasks. We strongly believe that robust, dependable, safe, efficient, scaleable RL algorithms are possible, and look forward to seeing the coming years of research in this area.
Appendix {#appendix .unnumbered}
========
Learning Algorithms {#app:algorithms}
===================
Parameters that were used for D4PG and DMPO can be found in Table \[table:d4pg\_hyperparameters\] and Table \[table:mpo\_hyperparameters\], respectively.
Hyperparameters D4PG
------------------------------------- -----------------
Policy net 300-300-200
Number of actions sampled per state 15
Q function net 400-400-300-100
$\sigma$ (exploration noise) 0.1
vmim -150
vmax 150
num atoms 51
n-step 51
Discount factor ($\gamma$) 0.99
Adam learning rate 0.0001
Replay buffer size 2000000
Target network update period 200
Batch size 512
Activation function elu
Layer norm on first layer Yes
Tanh on output of layer norm Yes
: Hyperparameters for D4PG.[]{data-label="table:d4pg_hyperparameters"}
Hyperparameters DMPO
------------------------------------- -----------------
Policy net 300-300-200
Number of actions sampled per state 20
Q function net 400-400-300-100
$\epsilon$ 0.1
$\epsilon_{\mu}$ 0.005
$\epsilon_{\Sigma}$ 0.000001
Discount factor ($\gamma$) 0.99
vmin -150
vmax 150
num atoms 51
Adam learning rate 0.0001
Replay buffer size 1000000
Batch size 256
Activation function elu
Layer norm on first layer Yes
Tanh on output of layer norm Yes
Tanh on Gaussian mean No
Min variance Zero
Max variance unbounded
: Hyperparameters for DMPO.[]{data-label="table:mpo_hyperparameters"}
Parameters {#app:parameters}
==========
The hyperparameters that were used for the individual challenges sweeps can be found in Table \[app1:hyperparameters\_sweeps\].
**Experiment**
---------------------------- ------------------------------- ------------------------------
**System Delays** **Delay (in timesteps)**
Action Delay 0,3,6,9,12,15,18,20
Observation Delay 0,3,6,9,12,15,18,20
Rewards Delay 10,20,40,50,75,100
**Noise** **Std. Deviation**
Gaussian Action Noise 0.0,0.1,0.3,1.0,1.3,2.0,2.3
Gaussian Observation Noise 0.0,0.1,0.3,1.0,1.3,2.0,2.3
Action Repetition Noise 1,2,3,5,7,10,13,16,20
**Stuck/Dropped Probability** **Stuck/Dropped steps**
Stuck Sensor Noise 0.0,0.01,0.05,0.1,0.3,0.5,0.7 0,1,5,10,20,50
Dropped Sensor Noise 0.0,0.01,0.05,0.1,0.3,0.5,0.7 0,1,5,10,20,50
**** **Perturbation Frequency** **Perturbation Schedule**
**Perturbations** 1,2,5,10,50,100 uniform,cyclic\_pos
**** **State Dimension Increase**
**High Dimensionality** 0,10,20,50,100
**Safety Coefficient** **Safety Penalty Weighting**
**Safety** 1.0,0.8,0.5,0.2,0.1 N/A
**Multi-objective** 0.5 1,0.8,0.5,0.2,0.1,0
: Hyperparameter sweeps for each challenge experiment[]{data-label="app1:hyperparameters_sweeps"}
Codebase {#app:codebase}
========
Specifying Challenges {#app:specifying_challenges}
---------------------
Specifying task challenges is done by passing arguments to the `load` method of the environment (see examples in Appendix \[appendix:code\_snippets\]). Comprehensive documentation is available in the codebase itself, however, for completeness we list the different arguments here.
- **Constraints**
- *Description*: Adds a set of constraints on the task. Returns an additional entry in the observations (’constraints’) in the length of the number of the contraints, where each entry is True if the constraint is satisfied and False otherwise. In our implementation we used safety constraints as the constraints. The safety constraints per domain can be found in Table \[tab:safe\_envs\_full\].
- *Input argument*: `safety_spec`, a dictionary that specifies the safety constraints specifications of the task. It may contain the following fields:
- `enable`, a boolean that represents whether safety specifications are enabled.
- `constraints`, a list of class methods returning boolean constraint satisfactions (default ones are provided).
- `limits`, a dictionary of constants used by the functions in ’constraints’ (default ones are provided).
- `safety\_coeff`, a scalar between 1 and 0 that scales safety constraints, 1 producing the base constraints, and 0 likely producing an unsolveable task.
- `observations`, a default-True boolean that toggles the whether a vector of satisfied constraints is added to observations.
- **Delays**
- *Description*: Adds actions, observations and rewards delays. Actions delay is the number of steps between passing the action to the environment when it is actually performed, and observation (reward) delay is the offset of freshness of the returned observation (reward) after performing a step.
- *Input argument*: `delay_spec`, a dictionary that specifies the delay specifications of the task. It may contain the following fields:
- `enable`, a boolean that represents whether delay specifications are enabled.
- `actions`, an integer indicating the number of steps actions are being delayed.
- `observations`, an integer indicating the number of steps observations are being delayed.
- `rewards`, an integer indicating the number of steps rewards are being delayed.
- **Noise**
- *Description*: Adds action or observation noise. Different noise include: white Gaussian actions/observations, dropped actions/observations values, stuck actions/observations values, or repetitive actions.
- *Input argument*: `noise_spec`, a dictionary that specifies the noise specifications of the task. It may contains the following fields:
- `gaussian`, a dictionary that specifies the white Gaussian additive noise. It may contain the following fields:
- `enable`, a boolean that represents whether noise specifications are enabled.
- `actions`, a float indicating the standard deviation of a white Gaussian noise added to each action.
- `observations`, similarly, additive white Gaussian noise to each returned observation.
- `dropped`, a dictionary that specifies the dropped values noise. It may contain the following fields:
- `enable`, a boolean that represents whether dropped values specifications are enabled.
- `observations\_prob`, a float in \[0,1\] indicating the probability of dropping each observation component independently.
- `observations\_steps`, a positive integer indicating the number of time steps of dropping a value (setting to zero) if dropped.
- `actions\_prob`, a float in \[0,1\] indicating the probability of dropping each action component independently.
- `actions\_steps`, a positive integer indicating the number of time steps of dropping a value (setting to zero) if dropped.
- `stuck`, a dictionary that specifies the stuck values noise. It may contain the following fields:
- `enable`, a boolean that represents whether stuck values specifications are enabled.
- `observations\_prob`, a float in \[0,1\] indicating the probability of each observation component becoming stuck.
- `observations\_steps`, a positive integer indicating the number of time steps an observation (or components of) stays stuck.
- `actions\_prob`, a float in \[0,1\] indicating the probability of each action component becoming stuck.
- `actions\_steps`, a positive integer indicating the number of time steps an action (or components of) stays stuck.
- `repetition`, a dictionary that specifies the repetition statistics. It may contain the following fields:
- `enable`, a boolean that represents whether repetition specifications are enabled.
- `actions\_prob`, a float in \[0,1\] indicating the probability of the actions to be repeated in the following steps.
- `actions\_steps`, a positive integer indicating the number of time steps of repeating the same action if it to be repeated.
- **Perturbations**
- *Description*: Perturbs physical quantities of the environment. These perturbations are non-stationary and are governed by a scheduler.
- *Input argument*: `perturb_spec`, a dictionary that specifies the perturbation specifications of the task. It may contain the following fields:
- `enable`, a boolean that represents whether perturbation specifications are enabled.
- `frequency`, an integer, number of episodes between updates perturbation updates.
- `param`, a string indicating which parameter to perturb (supporting multiple parameters, environment-dependent, see Table \[tab:perturb\_env\_full\]).
- `scheduler`, a string indicating the scheduler to apply to the perturbed parameter. Currently supporting:
- constant - constant value determined by the ‘start‘ argument.
- random$\_$walk - random walk governed by a white Gaussian process.
- drift$\_$pos - uni-directional (increasing) random walk which saturates.
- drift$\_$neg - uni-directional (decreasing) random walk which saturates.
- cyclic$\_$pos - uni-directional (increasing) random walk which resets once reaching the maximal value.
- cyclic$\_$neg - uni-directional (decreasing) random walk which resets once reaching the minimal value.
- uniform - uniform sampling process within a bounded support.
- saw$\_$wave - alternating uni-directional random walks between minimal and maximal values.
- `start`, a float indicating the initial value of the perturbed parameter.
- `min`, a float indicating the minimal value the perturbed parameter may be.
- `max`, a float indicating the maximal value the perturbed parameter may be.
- `std`, a float indicating the standard deviation of the white noise for the scheduling process.
- **Dimensionality**
- *Description*: Adds extra dummy features to observations to increase dimensionality of the state space.
- *Input argument*: `dimensionality_spec`, a dictionary that specifies the added dimensions to the state space. It may contain the following fields:
- `num\_random\_state\_observations`, an integer indicating the number of random observations to add (defaults to zero).
- **Multi-Objective Reward**
- *Description*: Provides a reward that gets added onto the base reward and re-normalized to \[0,1\].
- *Input argument*: `multiobj_spec`, a dictionary that sets up the multi-objective challenge. The challenge works by providing an ‘Objective‘ object which describes both numerical objectives and a reward-merging method that allow to both observe the various objectives in the observation and affect the returned reward in a manner defined by the Objective object.
- `objective`, either a string which will load an ‘Objective‘ class from\
utils.multiobj\_objectives.Objective, or an Objective object which subclasses it.
- `reward`, a boolean indicating whether to add the multiobj objective’s reward to the environment’s returned reward.
- `coeff`, a float in \[0,1\] that is passed into the Objective object to change the mix between the original reward and the Objective’s rewards.
- `observed`, a boolean indicating whether the defined objectives should be added to the observation.
Code Snippets {#appendix:code_snippets}
-------------
Below is an example of using the OpenAI PPO baseline with our suite.
``` {.python language="Python"}
from baselines import bench
from baselines.common.vec_env import dummy_vec_env
from baselines.ppo2 import ppo2
import example_helpers as helpers
import realworldrl_suite.environments as rwrl
def _load_env():
"""Loads environment."""
raw_env = rwrl.load(
domain_name='cartpole',
task_name='realworld_swingup',
safety_spec=dict(enable=True),
delay_spec=dict(enable=True, actions=20),
log_output='/tmp/path/to/results.npz',
environment_kwargs=dict(log_safety_vars=True, flat_observation=True))
env = helpers.GymEnv(raw_env)
env = bench.Monitor(env, FLAGS.save_path)
return env
env = dummy_vec_env.DummyVecEnv([_load_env])
ppo2.learn(env=env, network='mlp', lr=1e-3, total_timesteps=1000000,
nsteps=100, gamma=.99)
```
Below is another example running a random policy.
``` {.python language="Python"}
import numpy as np
import realworldrl_suite.environments as rwrl
def random_policy(action_spec):
def _act(timestep):
del timestep
return np.random.uniform(low=action_spec.minimum,
high=action_spec.maximum,
size=action_spec.shape)
return _act
env = rwrl.load(
domain_name='cartpole',
task_name='realworld_swingup',
safety_spec=dict(enable=True),
delay_spec=dict(enable=True, actions=20),
log_output='/tmp/path/to/results.npz',
environment_kwargs=dict(log_safety_vars=True, flat_observation=True))
policy = random_policy(action_spec=env.action_spec())
rewards = []
total_episodes = 100
for _ in range(total_episodes):
timestep = env.reset()
total_reward = 0.
while not timestep.last():
action = policy(timestep)
timestep = env.step(action)
total_reward += timestep.reward
rewards.append(total_reward)
print('Random policy total reward per episode: {:.2f} +- {:.2f}'.format(
np.mean(rewards), np.std(rewards)))
```
Below is an example of instantiating an environment with the ‘easy‘ challenge
``` {.python language="Python"}
import realworldrl_suite.environments as rwrl
env = rwrl.load(
domain_name='cartpole',
task_name='realworld_swingup',
combined_challenge='easy',
log_output='/tmp/path/to/results.npz',
environment_kwargs=dict(log_safety_vars=True, flat_observation=True))
```
[^1]: equally contributed, $^{1}$Google Research, Paris, $^{2}$Deepmind, London, $^{3}$Deepmind, Mountainview, $^{4}$Work done during time at Deepmind
|
---
abstract: 'We investigate the light-matter interaction of a quantum dot with the electromagnetic field in a lossy microcavity and calculate emission spectra for non-zero detuning and dephasing. It is found that dephasing shifts the intensity of the emission peaks for non-zero detuning. We investigate the characteristics of this intensity shifting effect and offer it as an explanation for the non-vanishing emission peaks at the cavity frequency found in recent experimental work.'
author:
- 'A. Naesby'
- 'T. Suhr'
- 'P. T. Kristensen'
- 'J. Mørk'
title: |
Influence of Pure Dephasing\
on Emission Spectra from Single Photon Sources
---
\[!b\]
 Left: Schematic displaying the energy levels of the two-level QD and cavity. [$| e \rangle$]{} and [$| g \rangle$]{} denote excited and ground state of the emitter and [$| 1 \rangle$]{} and [$| 0 \rangle$]{} denote the excited and empty cavity mode. Right: Schematic of a micropillar with a QD in a high-Q cavity. Light escapes from the cavity in the forward direction at a rate $\kappa$, while the QD excitation decays at rate $\gamma$.](Model.eps){width="8.6cm"}
The realization of a solid-state single photon source has been given much attention, because of the many potential applications for such a device. The particularly promising scheme, where a Quantum Dot (QD) is coupled to a high-Q microcavity [@Yamamoto93; @Vahala03], has been investigated both experimentally [@Reithmaier04; @Yoshie04; @Hennessy07] and theoretically [@Carmichael89; @Cui06]. Recent experimental results show a significant emission at the cavity resonance even for strongly detuned systems [@Reithmaier04; @Yoshie04; @Hennessy07], which is not well understood. In order to understand the physics and limitations, it is of significant interest to develop detailed models for such structures, that rely on the coupling between a two-level emitter and a cavity mode resonance. The role of dephasing in QD systems was pointed out by Cui and Raymer [@Cui06], who showed that pure dephasing broadens the emission peaks and softens the features of the emission spectra from a resonantly coupled QD-cavity system. We extend the results of Cui and Raymer to the realistic case of non-zero detuning between cavity and QD resonance and show that detuned systems display a surprisingly large dephasing dependence, which leads to an intensity shift similar to recent experimental observations [@Reithmaier04; @Yoshie04; @Hennessy07].
We consider the model of Cui and Raymer [@Cui06], indicated in fig. \[fig:ModelSystem\], where a QD emitter and a cavity are treated as coupled two-level systems with coupling strength $g_0$. Both QD and cavity couple to output reservoirs, so that photons escape from the cavity at a rate $\kappa$ and the and the excitation of the QD decays nonradiatively and to other modes a a total rate $\gamma$. The resonance frequencies of the QD and cavity mode are denoted $\omega_0$ and $\omega_c$ respectively, and $\Delta = \omega_0-\omega_c$ is the detuning.
The interaction hamiltonian is found for the quantized field in the rotating wave approximation and is given in the interaction picture by [@Cui06; @Scully97] $$\begin{aligned}
H_I &=& \hbar g_0 \sigma_+ae^{i\Delta t} + \hbar\sum_{{\boldsymbol{p}}} A_{{\boldsymbol{p}}}^*\sigma_-d_{{\boldsymbol{p}}}^\dagger e^{i\delta_pt} {\nonumber}\\
&+& \hbar\sum_{{\boldsymbol{k}}} B_{{\boldsymbol{k}}}^*ab_{{\boldsymbol{k}}}^\dagger e^{i\delta_kt} + \mathrm{H.c.} \label{eq:interaction_hamiltonian}\end{aligned}$$ where $\sigma_\pm$ are the raising/lowering operators of the QD and $a^{{\left( \dagger \right)}}$, $b^{{\left( \dagger \right)}}$, $d^{{\left( \dagger \right)}}$ are cavity, cavity reservoir and QD reservoir lowering (raising) operators obeying bosonic statistics. $A_{{\boldsymbol{p}}}^{{\left( * \right)}}$ and $B_{{\boldsymbol{k}}}^{{\left( * \right)}}$ are coupling strengths for the interaction with the $p$’th QD reservoir mode and the $k$’th cavity reservoir mode and $\delta_p = \omega_p-\omega_0$ and $\delta_k = \omega_k-\omega_c$ are detunings for the QD output reservoir and cavity output reservoir, respectively. The last two terms in eqn. (\[eq:interaction\_hamiltonian\]) describe the coupling to emitter and cavity output, respectively.
The system is initiated with an excitation of the emitter and is described by the state vector $${\ensuremath{| \Psi \rangle}} = E{\ensuremath{| e,0 \rangle}} + C{\ensuremath{| g,1 \rangle}} + \sum_{\mathbf{p}} E^r_\mathbf{p} {\ensuremath{| g,\mathbf{p} \rangle}} + \sum_{\mathbf{k}}C^r_\mathbf{k} {\ensuremath{| g,\mathbf{k} \rangle}} \label{eq:State}$$ where $|E(t)|^2$ and $|E^r_{\mathbf{p}}(t)|^2$ ($|C(t)|^2$ and $|C^r_{\mathbf{p}}(t)|^2$) are slowly varying probability amplitudes for the emitter and emitter decay reservoir (cavity and cavity decay reservoir), respectively. By inserting eqns. (\[eq:interaction\_hamiltonian\]) and (\[eq:State\]) into the Schrödinger equation, the envelope functions are extracted by projecting onto the different states of the system and are given by $$\begin{aligned}
\partial_t E{\left( t \right)} &=& -ig_0e^{+i\Delta t}C{\left( t \right)} - \gamma E{\left( t \right)} \label{IV_diff_E1}\\
\partial_t C{\left( t \right)} &=& -ig_0e^{-i\Delta t}E{\left( t \right)} - \kappa C{\left( t \right)} \label{IV_diff_C1}\end{aligned}$$ where the Wigner-Weisskopf approximation has been employed to transform the reservoir coupling terms in the Hamiltonian into the decay terms $\kappa$ and $\gamma$. Dephasing is modeled as a random gaussian process as in [@Cui06; @Mandel97] and included in eqns. (\[IV\_diff\_E1\]) and (\[IV\_diff\_C1\]) by letting $\omega_0 t \rightarrow \omega_0 t + \int_0^t dt f{\left( t \right)}$, where $f{\left( t \right)}$ is a stochastic Langevin noise force with characteristics $\left< f{\left( t \right)} \right> = 0$ and $\left< f{\left( t \right)}f{\left( t' \right)} \right> = 2\gamma_p \delta{\left( t-t' \right)}$, where $\gamma_p$ is the dephasing rate [@Haken04]. By introducing dephasing only in the coupling to the cavity mode, [@Cui06], we neglect dephasing induced broadening of the, assumed weak, emission to other (leaky) modes.
Following [@Cui06; @Wod79], eqns. (\[IV\_diff\_E1\]) and (\[IV\_diff\_C1\]) (with dephasing included) are transformed into simpler equations of motion for $E(t)$ and $C(t)$ and finally solved in order to extract the emission spectra, given by $$\begin{aligned}
S_{E} &=& \frac{2\gamma}{\pi} \Re \left\{\int_0^\infty e^{i{\left( \Omega-\Delta \right)} \tau} {\left< E{\left( t+\tau \right)}E^*{\left( t \right)} \right>} d\tau dt \right\} \label{eq:emitter_emission} \\
S_{C} &=& \frac{2\kappa}{\pi} \Re \left\{\int_0^\infty e^{i\Omega \tau} {\left< C{\left( t+\tau \right)}C^*{\left( t \right)} \right>} d\tau dt \right\} \label{eq:cavity_emission}\end{aligned}$$ where $S_E$ and $S_C$ are the emission spectra for the emitter and cavity, respectively, and $\Omega$ is the frequency of the emitted light. The emission spectra characterize the light that escapes the QD-cavity system through the decay rates $\gamma$ and $\kappa$ and corresponds to what is measured in photoluminescence experiments. In general the measured spectrum is expected to be a combination of $S_E$ and $S_C$ depending on the geometry and exact details of the setup. This is because the emitter can couple to both the cavity mode and to radiation modes outside the cavity and in a photoluminescence experiment, the detector picks up emission from both the cavity and the emitter. For highly directional micropillar type setups, as the one shown in fig. \[fig:ModelSystem\], the cavity emission is expected to dominate the measured light, whereas the emitter spectrum becomes important in photonic crystal QD-cavities, as has been suggested by Auffeves et al. [@auffeves-2007].
\[!tbhp\]
 $S_C$ (full) and $S_E$ (dashed) for (a) zero dephasing and (b) 5 GHz dephasing rate. The emitter ($\omega_0$) and cavity ($\omega_c$) frequencies are indicated with the dashed lines and the total emission intensity $\int d\Omega {\left( S_{C} + S_{E} \right)}$ is constant for all detunings. ](Spectrum_combined_hori.eps){width="8.6cm"}
In fig. \[fig:Spectrum-Comb\] we show $S_C$ (full line) and $S_E$ (dashed line) for a dephasing rate of zero (a) and 5 GHz (b). The parameters are chosen so that the system is in the strong coupling regime with $g_0 = 8$ GHz, $\kappa = 1.6$ GHz and $\gamma =0.32$ GHz, and the Rabi oscillations lead to a splitting of the emission peaks when the emitter and cavity are resonant [@Michler04; @Inoue08]. The anti-crossing characteristic of strong-coupling is clearly seen in fig. \[fig:Spectrum-Comb\]. In this context we define the strong coupling regime as $g_0 > \kappa, \gamma$. The general definition [@Michler04] also contains the detuning $\Delta$ and far detuned systems are thus not necessarily in the strong-coupling regime. For zero dephasing (fig. \[fig:Spectrum-Comb\] (a)) it is noted how the peak at the emitter frequency dominates both $S_E$ and $S_C$ at high detuning, which is a result of the decrease of the coupling as the detuning is increased and of starting the system with an excitation of the emitter.
The inclusion of dephasing (fig. \[fig:Spectrum-Comb\] (b)) considerably changes the emission spectra: First, the peaks are broadened and the splitting originating from Rabi oscillations is blurred, as has already been shown in [@Cui06]. Secondly, and more surprising, the inclusion of dephasing for non-zero detuning leads to a qualitative change in the cavity spectrum $S_C$ as dephasing shifts the emission intensity toward the cavity frequency.
This intensity shifting effect is present both in the strong and weak coupling regime as well as for very large detunings $(|\Delta| \gg g_0)$, but only in the cavity emission spectrum. The intensity shifting effect is illustrated in fig. \[fig:Emission\] (a) where the peak intensity (i.e. the maximum output value in a narrow interval around the peak) of the leftmost peak in $S_C$ (which becomes identical to the cavity emission peak at large detuning) is compared to the sum of both the peak intensities. This is shown as a function of detuning for various dephasing rates. For zero dephasing the emitter peak becomes dominant as the detuning is increased. It can thus be shown that for zero dephasing and in the limit of large detuning the relative cavity peak intensity scales as $\gamma^2/(\gamma^2+\kappa^2)$, which is very small for typical parameters. In contrast, when dephasing is included, the cavity emission peak is seen to become significant and eventually dominant. Close to resonance the inclusion of dephasing merges the peaks into a single peak. The relative peak intensity is not defined for a single peak and thus not included in fig. \[fig:Emission\] for $\gamma_p = 10$ GHz and small detuning values.
\[!tb\]
 (a) Relative left peak (cavity peak) intensity in $S_C$ as a function of detuning $\Delta$. (b) Relative cavity emission intensity as a function of dephasing rate $\gamma_p$. *Parameters:* $g_0=8$ GHz, $\kappa=1.6$ GHz, $\gamma=0.32$ GHz. ](Emission.eps){width="8.6cm"}
The cavity emission intensity compared to the total emission intensity is important for the efficiency of the device and we illustrate this in fig. \[fig:Emission\] (b), where the ratio $\int d\Omega S_C / \int d\Omega {\left( S_C + S_E \right)}$ is shown for varying detuning and dephasing. When $\gamma_p$ is zero and the system is strongly coupled, i.e. when $g_0^2 > {\left( (\kappa-\gamma)/2 \right)}^2$ [@Andreani99; @Rudin99], most of the light is emitted from the cavity, but for increasing detuning the coupling is weakened and the emission directly from the emitter becomes increasingly important. On resonance an increase in dephasing rate leads to a monotonous decrease in $\int d\Omega S_C$ compared to the total output, and for high dephasing the majority of light is emitted from the emitter.
At zero dephasing and when the detuning is increased the emitter emission becomes more significant. For fixed $\left| \Delta \right| > 0$ an intermediate region appears, where the relative cavity emission displays an increase with dephasing before decreasing toward zero.
For high dephasing rates $S_C$ consists of a single peak at the cavity frequency, but the relative cavity emission intensity is smaller compared to zero dephasing. This has to be kept in mind when comparing to measurements, since the distinction between cavity and emitter emission may depend on the experimental set-up and the cavity structure. Before discussing the underlying physics of the intensity shifting effect, let us compare the results of our model to recently published measurements showing a, so far, unexplained detuning dependence. As an example, fig. \[fig:SpectrumCavityReithmaier\] shows emission spectra from the cavity, $S_C$, calculated using parameters comparable to the experiments by Reithmaier et al. [@Reithmaier04] for different detunings. The measured light is expected to be dominated by $S_C$ because of the high directionality of the micropillar setup.
\[tb\]
 Emission spectra calculated using parameters from Reithmaier et al. [@Reithmaier04]. The zero dephasing spectra (red dashed line) are downscaled 5 times compared to the $\gamma_p$ = 20 GHz spectra (black line). *Parameters:* $g_0=38$ GHz, $\kappa=43$ GHz, $\gamma=0.1$ GHz.](Reithmaier.eps){width="4.7cm"}
The spectra including dephasing show much better agreement with the experiment than the spectra calculated in the absence of dephasing. In particular, we note that dephasing favors emission at the cavity frequency although the QD resonance may be far detuned from the cavity resonance. In the experiment [@Reithmaier04] the detuning is varied by changing the temperature. It is well known that the dephasing rate is dependent on temperature [@Bayer02], but we emphasize that the enhancement of the cavity peak is robust with respect to variations in the dephasing rate, which is why a fixed $\gamma_p = 20$ GHz is chosen for all values of detuning. This is also the case for different combinations of the decay rates $\kappa$ and $\gamma$ and the intensity shifting effect is present both in the weak and strong coupling regime as noted above. The model has also been tested against data from Yoshie et al. [@Yoshie04] and Hennessy et al. [@Hennessy07], and in both cases the unexpected, large emission at the cavity frequency can be explained as an effect of intensity shifting. We notice, however, that the model may be less applicable for these structures.
The simplicity of the model makes the results applicable to a range of systems beyond single photon sources, where two-level systems are coupled to microcavities. An example of this is the work by Strauf et al. [@Strauf06] where a few quantum dots were coupled to a nanocavity to realize a photonic-crystal laser. Lasing was witnessed even with the QDs being off resonance with the cavity mode which is surprising and suggests the influence of an effect such as intensity shifting.
In order to get a better physical understanding of the effects responsible for the intensity shifting, we draw upon a mechanical analogue to the QD-cavity system. The differential eqns. (\[IV\_diff\_E1\]) and (\[IV\_diff\_C1\]) are equivalent to the equations describing a system of two masses, each connected by springs to a wall and mutually coupled by another spring. The resonance frequencies of the uncoupled systems are governed by the masses and the spring constants. For identical spring constants the high detuning limit corresponds to one of the masses being much larger than the other and this mass can then be replaced by a driven piston, which makes the system simpler to analyze and understand. This model is illustrated in fig. \[fig:MechSys\]. Dephasing events can be thought of as (instantaneously) moving the piston to a new position while keeping the position of the mass fixed (as well as the total energy of the system). In the case of high detuning the equations reduce to $$\begin{aligned}
\partial_t^2 x{\left( t \right)} + \kappa_c \partial_t x{\left( t \right)} +{\left( k_c+g_c \right)} x{\left( t \right)} = g_c f{\left( t \right)} \label{eq:MechSys}\end{aligned}$$ where $k_c$ and $g_c$ are force constants for the springs, $\kappa_c$ is the damping of the oscillation and $x{\left( t \right)}$ and $f{\left( t \right)}$ are the position of the mass and the piston, respectively. The mass $m_c$ has been set to unity.
\[!bt\]
![\[fig:MechSys\]Schematic of the mechanic model system. The mass $m_c$ is connected to the wall at $x=0$ through a spring with force constant $k_c$ and to the piston through a spring with force constant $g_c$. The position of the piston is given as $f{\left( t \right)}$.](MechSysNew.eps){width="7cm"}
The general solution is the sum of the homogeneous and the inhomogeneous solution, where the former is the damped oscillation of the isolated mass, while the latter is an oscillation at the frequency of the piston. Therefore, the general solution starts out as a combination of the homogeneous and the inhomogeneous oscillation, but over time the transient homogeneous oscillation diminishes and the system oscillates at the frequency of the piston.
Whenever a dephasing event changes the position of the piston, the oscillation of the mass acquires a homogeneous component to compensate for the change. Therefore the mass will acquire a stronger component at its eigenfrequency as the dephasing rate increases, corresponding to a shift in the intensity of the peaks in the Fourier spectrum of the oscillation.
The analogy with the mechanical model demonstrates that the intensity shifting effect is a property of classical as well as quantum mechanical coupled oscillators and the mechanical description of the intensity shifting effect also applies to the quantum mechanical system. At a given time the QD-cavity system is in a superposition of the cavity and emitter state, but the evolution can be changed by a dephasing event, in which case the system must first undergo transient oscillations at the cavity frequency before steady-state oscillation is reestablished.
We note that we have employed the usual assumption that the bare emitter state is excited by a carrier at $t=0$, i.e. $E(0)=1$ [@Carmichael89; @Cui06]. However, in a more detailed approach one should calculate the excitation of the coupled emitter-cavity states based on the physical excitation of carriers in the system, e.g. off-resonant or near resonant.
In summary, we have investigated a coupled system of a two-level emitter and a cavity and found that the frequency of the emitted light shows a surprising dependence of the dephasing rate. Dephasing shifts the emission intensity towards the cavity frequency, which can explain recent experimental results [@Reithmaier04; @Yoshie04; @Hennessy07]. The intensity shifting effect can be qualitatively explained by considering the cumulative effect of many dephasing events at a high dephasing rate. The discontinuous phase change adds transients at the cavity frequency to the oscillation, not unlike the ringing effects seen in classical oscillations, and this gives components at the cavity frequency to the emission spectrum. Other effects may of course contribute to the measured spectra. For example the emitter may not be truly two-level, e.g. due to many-body effects, and a more detailed account of the coupling to leaky electromagnetic modes and their emission pattern may need to be given. In general we believe that the results presented are relavant for a wide range of systems and that the intensity shifting effect due to dephasing is of a generic nature and of general relevance to semiconductor systems, which generally are characterized by high rates of dephasing.
The authors acknowledge helpful discussions with P. Lodahl and P. Kaer Nielsen, Dept. of Photonics Engineering, Technical University of Denmark, and Christian Flindt, Lukin Lab, Department of Physics, Harvard University.
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|
---
author:
- |
Babak Ehteshami Bejnordi, Tijmen Blankevoort & Max Welling [^1]\
Qualcomm AI Research[^2]\
Amsterdam, The Netherlands\
`{behtesha,tijmen,mwelling}@qti.qualcomm.com`\
bibliography:
- 'iclr2019\_conference.bib'
title: |
Batch-Shaping for Learning\
Conditional Channel Gated Networks
---
Introduction {#sec:introduction}
============
Background and related work {#sec:backgroundrelated}
===========================
Batch-shaping {#sec:method}
=============
Channel gated networks {#sec:method}
======================
Experiments {#sec:experiments}
===========
Conclusion {#sec:conclusion}
==========
### Acknowledgments {#acknowledgments .unnumbered}
The authors would like to thank Markus Nagel, Pim de Haan, and jakub tomczak for their valuable discussions and feedback.
Appendix {#sec:appendix}
========
[^1]: Use footnote for providing further information about author (webpage, alternative address)—*not* for acknowledging funding agencies. Funding acknowledgements go at the end of the paper.
[^2]: Qualcomm AI Research is an initiative of Qualcomm Technologies Inc.
|
---
abstract: 'Elastic-scattering phase shifts for four-nucleon systems are studied in an $ab$-$initio$ type cluster model in order to clarify the role of the tensor force and to investigate cluster distortions in low energy $d$+$d$ and $t$+$p$ scattering. In the present method, the description of the cluster wave function is extended from a simple (0$s$) harmonic-oscillator shell model to a few-body model with a realistic interaction, in which the wave function of the subsystems are determined with the Stochastic Variational Method. In order to calculate the matrix elements of the four-body system, we have developed a Triple Global Vector Representation method for the correlated Gaussian basis functions. To compare effects of the cluster distortion with realistic and effective interactions, we employ the AV8$^{\prime}$ potential as a realistic interaction and the Minnesota potential as an effective interaction. Especially for $^1S_0$, the calculated phase shifts show that the $t$+$p$ and $h$+$n$ channels are strongly coupled to the $d$+$d$ channel for the case of the realistic interaction. On the contrary, the coupling of these channels plays a relatively minor role for the case of the effective interaction. This difference between both potentials originates from the tensor term in the realistic interaction. Furthermore, the tensor interaction makes the energy splitting of the negative parity states of $^4$He consistent with experiments. No such splitting is however reproduced with the effective interaction.'
author:
- 'S. Aoyama, K. Arai, Y. Suzuki, P. Descouvemont, D. Baye'
title: ' Four-nucleon scattering with a correlated Gaussian basis method'
---
Introduction {#sect.1}
============
The microscopic cluster model is one of the successful models to study the structure and reactions of light nuclei [@tang77]. In the conventional cluster model, one assumes that the nucleus is composed of several simple clusters with $A \le 4$ which are described by (0$s$) harmonic-oscillator shell model functions, and use an effective $N$-$N$ interaction which is appropriate for such a model space. However, it is well known that the ground states of the typical clusters $d$, $t$, $h$ and $^4$He have non-negligible admixtures of $D$-wave component due to the tensor interaction. Since the conventional cluster model does not directly treat the $D$-wave component, the strong attraction of the nucleon-nucleon interaction due to the tensor term is assumed to be renormalized into the central term of the effective interaction.
Recently, $ab$-$initio$ structure calculations [@benchmark] have been successfully developed: Stochastic Variational Method (SVM) [@varga94; @varga97; @book; @vs95], Global Vector Representation method (GVR) [@DGVR; @GVR], Green’s function Monte Carlo method [@carlson98], no core shell model [@navratil00], correlated hyperspherical harmonics method [@viviani98r], unitary correlation operator method [@ucom], and so on. Although the application of $ab$-$initio$ reaction calculations with a realistic interaction are restricted so much in comparison with structure calculations, it has been intensively applied to the four-nucleon systems $t$+$n$ and $h$+$p$ [@arai10; @hofmann01; @deltuva07; @navratil09; @viviani98; @lazauskas05; @fisher06]. Especially $d$+$d$ scattering states, which couple to $t$+$p$ and $h$+$n$ channels, have attracted much attention, because the $d$+$d$ radiative capture is one of the mechanisms making $^4$He through electro-magnetic transitions [@arriaga91; @sabourov04] and also have posed intriguing puzzles for analyzing powers [@hofmann08; @hofmann97; @deltuva08; @lazauskas04; @ciesielski99], which are motivated by the famous $A_y$ problem in the three-nucleon system.
Furthermore, the $d$+$d$ elastic-scattering phase shifts are interesting because the astrophysical S-factor of the $d$($d$,$\gamma$)$^4$He reaction is not explained by any calculation using an effective interaction that contains no tensor term, and is expected to be contributed by the $D$-wave components of the clusters through $E2$ transitions [@langanke87; @fowler67].
Also, thanks to recent developments of the microscopic cluster model, the simple model using the (0$s$) harmonic-oscillator wave function with an effective interaction is not mandatory any more, at least, in light nuclei. We can use a kind of $ab$-$initio$ cluster model which employs more realistic cluster wave functions with realistic interactions. Therefore, it is interesting to see the difference between the $ab$-$initio$ reaction calculations with a tensor term and the conventional cluster model calculations without a tensor term in few-body systems. The microscopic $R$-matrix method (MRM) with a cluster model (GCM or RGM) has been applied to studies of many nuclei [@baye77; @kanada85; @arai01; @desc10]. It is now used in $ab$ $initio$ descriptions of collisions [@navratil09]. We have also applied the MRM to the $h$+$p$ scattering problem with more realistic cluster wave functions by using a realistic interaction [@arai10]. The Gaussian basis functions for the expansion of the cluster wave functions are chosen by a technique of the SVM [@book]. In the MRM, as will be shown later, the relative wave function between clusters ($a$ and $b$) is connected to the boundary condition at a channel radius. The problem is how to calculate the matrix elements. In this paper, we develop a method called the Triple Global Vector Representation method (TGVR), by which we calculate the matrix elements in a unified way. Although we restricted ourselves to four nucleon systems in the present paper, the formulation of the TGVR itself can be applied to more than four-body systems as in the previous studies of the Global Vector Representation methods (GVR) [@GVR]. Furthermore, for scattering problems, the TGVR can deal with more complicated systems than the double (or single) global vector which was given in the previous papers [@DGVR; @GVR], because we need three representative orbital angular momenta, the total internal orbital momenta of both clusters and the orbital momentum of their relative motion, in order to reasonably describe the scattering states. In other words, the first global vector represents the angular momentum of cluster $a$, the second global vector represents the angular momentum of cluster $b$, and the third global vector represents the relative angular momentum between the clusters.
In this paper, we will investigate the effect of the distortion of clusters on the $d$+$d$ elastic-scattering by comparing the phase shifts calculated with a realistic and an effective interaction. In section 2, we explain the MRM in brief. In section 3, the correlated Gaussian (CG) method with the TGVR, which has newly been developed for the present analysis, will be presented. In section 4, we will explain how to calculate the matrix elements with TGVR basis functions. The typical matrix elements are also given in the appendix. In section 5, we will present and discuss the calculated scattering phase shifts in detail. Finally, summary and conclusions are given in section 6.
Microscopic $R$-matrix method {#sect.2}
=============================
In the present study we calculate $d$+$d$ and $t$+$p$ (and $h$+$n$) elastic scattering phase shifts with the microscopic $R$-matrix method. Though the method is well documented in e.g. Refs. [@baye77; @kanada85; @desc10], we briefly explain it below in order to present definitions and equations needed in the subsequent sections. Since our interest is on low-energy scattering, we consider only two-body channels. A channel $\alpha$ is specified by the two nuclei (clusters) $a,
b$, their angular momenta, $I_a, I_b$, the channel spin $I$ that is a resultant of the coupling of $I_a$ and $I_b$, and the orbital angular momentum $\ell$ for the relative motion of $a$ and $b$. The wave function of channel $\alpha$ with the total angular momentum $J$, its projection $M$, and the parity $\pi$ takes the form $$\begin{aligned}
\Psi^{JM\pi}_{\alpha}=
{\cal A}
\left[\left[\Phi^{a}_{I_a}\Phi^{b}_{I_b}\right]_I
\chi_{\alpha}
(\mbox{\boldmath$\rho$}_{\alpha})\right]_{JM},
\label{wf-ch1}\end{aligned}$$ where $\Phi^{a}_{I_a}$ and $\Phi^{b}_{I_b}$ are respectively antisymmetrized intrinsic wave functions of $a$ and $b$, and ${\cal A}$ is an operator that antisymmetrizes between the clusters. The square bracket $[{I_a} \ {I_b}]_I$ denotes the angular momentum coupling. The coordinate ${{\mbox{\boldmath $\rho$}}}_\alpha$ in the relative motion function $\chi^J_{\alpha}$ is the relative distance vector of the clusters. The channel spin $I$ and the relative angular momentum $\ell$ in $\alpha$ are coupled to give the total angular momentum $J$. The relative-motion functions $\chi_{\alpha}$ also depend on $J$ and $\pi$. For simplicity, this dependence is not displayed explicitly in the notation for $\chi_{\alpha}$ as well as for some other quantities below.
The configuration space is divided into two regions, internal and external, by the channel radius $a$. In the internal region ($\rho_\alpha\le a$), the total wave function may be expressed in terms of a combination of various $\Psi_{\alpha}^{JM\pi}$s $$\begin{aligned}
\Psi^{JM\pi}_{\rm int}&=&\sum_{\alpha} \Psi^{JM\pi}_{\alpha}\nonumber \\
&=&\sum_{\alpha}\sum_{n}f_{\alpha n} {\cal A}
u_{\alpha n}(\rho_\alpha) \phi^{JM\pi}_{\alpha},
\label{wf.int}\end{aligned}$$ with $$\begin{aligned}
\phi^{JM\pi}_{\alpha} = \frac{1}{\sqrt{(1+\delta_{I_aI_b}\delta_{ab})(1+\delta_{ab})}} \left\{
\left[\left[\Phi^{a}_{I_a}\Phi^{b}_{I_b}\right]_I Y_{\ell}(\widehat{{\mbox{\boldmath $\rho$}}}_\alpha)\right]_{JM}
\right. \nonumber \\ \left.
+(-1)^{A_a+I_a+I_b-I+\ell}
\left[\left[\Phi^{b}_{I_b}\Phi^{a}_{I_a}\right]_I Y_{\ell}(\widehat{{\mbox{\boldmath $\rho$}}}_\alpha)\right]_{JM} \delta_{ab}
\right\},
\label{wf.channel}\end{aligned}$$ where $A_a$ is the number of nucleons in cluster $a$, $\delta_{ab}$ is unity if $a$ and $b$ are identical clusters and zero otherwise, and $\delta_{I_aI_b}$ is unity if the clusters are in identical states and zero otherwise. In the second line of Eq. (\[wf.int\]), the relative motion functions of Eq. (\[wf-ch1\]) are expanded in terms of some basis functions as $$\begin{aligned}
\chi_{\alpha m}({{\mbox{\boldmath $\rho$}}}_\alpha) = \sum_n f_{\alpha n}
u_{\alpha n}(\rho_\alpha)Y_{\ell m}(\widehat{{\mbox{\boldmath $\rho$}}}_\alpha).
\label{chi.expansion}\end{aligned}$$ In what follows we take $$\begin{aligned}
u_{\alpha n}(\rho_\alpha)=\rho_\alpha^{\ell}\exp(-\frac{1}{2}\lambda_n \rho_\alpha^2)
\label{u.functions}\end{aligned}$$ with a suitable set of $\lambda_n$s.
In the external region ($\rho_\alpha\ge a$), the total wave function takes the form $$\begin{aligned}
\Psi^{JM\pi}_{\rm ext}=\sum_{\alpha} g_{\alpha}(\rho_\alpha)
\phi^{JM\pi}_{\alpha}
\label{wf.ext}.\end{aligned}$$ Note that the antisymmetrization between the clusters is dropped in the external region under the condition that the channel radius $a$ is large enough. The function $g_{\alpha}(\rho_\alpha)$ of Eq. (\[wf.ext\]) is a solution of the equation $$\begin{aligned}
\left[-\frac{\hbar^2}{2\mu_{\alpha}}\left( \frac{d^2}{d\rho_\alpha^2}
+\frac{2}{\rho_\alpha}\,\frac{d}{d\rho_\alpha}-\frac{\ell(\ell+1)}{\rho_\alpha^2} \right)+\frac{Z_aZ_be^2}{\rho_\alpha}\right] g_{\alpha}(\rho_\alpha) =
E_{\alpha} g_{\alpha}(\rho_\alpha),\end{aligned}$$ where $\mu_{\alpha}$ is the reduced mass for the relative motion in channel $\alpha$, $Z_ae$ and $Z_be$ are the charges of $a$ and $b$, and $E_{\alpha}=E-E_a-E_b$ is the energy for the relative motion, where $E$ is the total energy, and $E_a$ and $E_b$ are the internal energies for the clusters $a$ and $b$, respectively. For the scattering initiated through the channel $\alpha_0$, the asymptotic form of $g_{\alpha}$ for the open channel $\alpha$ $(E_{\alpha} \geq 0)$ is $$\begin{aligned}
g_{\alpha}(\rho_\alpha) = v_\alpha^{-1/2} \rho_\alpha^{-1}
[I_{\alpha}(k_\alpha \rho_\alpha)\delta_{\alpha\,
\alpha_0}-S_{\alpha \, \alpha_0}^{J\pi} O_{\alpha}(k_\alpha \rho_\alpha)],\end{aligned}$$ where $k_\alpha=\sqrt{2\mu_\alpha |E_\alpha |}/\hbar$, $v_\alpha=\hbar k_\alpha/\mu_\alpha$ and $S_{\alpha \, \alpha_0}^{J\pi}$ is an element of the $S$-matrix (or collision matrix) to be determined. Here $I_{\alpha}(k_\alpha \rho_\alpha)$ and $O_{\alpha}(k_\alpha \rho_\alpha)$ are the incoming and outgoing waves defined by $$\begin{aligned}
I_{\alpha}(k_\alpha \rho_\alpha)=O_{\alpha}(k_\alpha \rho_\alpha)^*=
G_\ell(\eta_{\alpha}, k_\alpha \rho_\alpha)-iF_\ell(\eta_{\alpha}, k_\alpha \rho_\alpha),\end{aligned}$$ with the regular and irregular Coulomb functions $F_\ell$ and $G_\ell$. The Sommerfeld parameter $\eta_{\alpha}$ is $\mu_\alpha Z_a Z_b e^2/\hbar^2
k_\alpha$. For a closed channel $\alpha$ $(E_\alpha < 0)$, the asymptotic form of $g_\alpha$ is given by the Whittaker function $$\begin{aligned}
g_{\alpha}(\rho_\alpha) \propto \rho_\alpha^{-1}
W_{-\eta_{\alpha},\ell+1/2}(2k_\alpha\rho_\alpha).\end{aligned}$$
The matrix elements $S_{\alpha \alpha_0}^{J\pi}$ are determined by solving a Schrödinger equation with a microscopic Hamiltonian $H$ involving the $A_a + A_b$ nucleons, $$\begin{aligned}
(H+{\cal L} -E)\Psi_{\rm int}^{JM\pi}={\cal L}\Psi_{\rm ext}^{JM\pi},\end{aligned}$$ with the Bloch operator ${\cal L}$ $$\begin{aligned}
{\cal L}=\sum_{\alpha}\frac{\hbar^2}{2\mu_\alpha a}
\vert \phi_{\alpha}^{JM \pi} \rangle \delta(\rho_\alpha-a)
\left(\frac{\partial}{\partial \rho_\alpha}-\frac{b_\alpha}{\rho_\alpha} \right)\rho_\alpha
\langle \phi_{\alpha}^{JM\pi}\vert,
\label{Bloch}\end{aligned}$$ where the channel radius $a$ is chosen to be the same for all channels, and the $b_\alpha$ are arbitrary constants. Here, we choose $b_\alpha=0$ for the open channels and $b_\alpha=2k_\alpha a W'_{-\eta_\alpha,\ell+1/2}(2k_\alpha a)/W_{-\eta_\alpha,\ell+1/2}(2k_\alpha a)$ for the closed channels. The results do not depend on the choices for $b_\alpha$ but these values simplify the calculations. Notice that the projector on $|\phi_{\alpha}^{JM \pi}\rangle$ in Eq. (\[Bloch\]) is not essential in a microscopic calculation and can be dropped since the various channels are orthogonal at the channel radius.
The Bloch operator ensures that the logarithmic derivative of the wave function is continuous at the channel radius. In addition, $\Psi_{\rm int}^{JM\pi}$ must be equal to $\Psi_{\rm ext}^{JM\pi}$ at $\rho_\alpha=a$. Projecting the Schrödinger equation on a basis state, one obtains $$\begin{aligned}
\sum_{\alpha n} C_{\alpha' n', \alpha n} \, f_{\alpha n}
= \langle \Phi_{\alpha' n'}^{JM\pi} \vert {\cal L} \vert \Psi_{\rm ext}^{JM\pi} \rangle
\label{SEq}\end{aligned}$$ with $$\begin{aligned}
C_{\alpha' n', \alpha n} =
\langle \Phi_{\alpha' n'}^{JM\pi} \vert H+{\cal L} -E
\vert {\cal A}\Phi_{\alpha n}^{JM\pi} \rangle_{\rm int},\end{aligned}$$ and $$\Phi_{\alpha n}^{JM\pi} = u_{\alpha n}(\rho_\alpha)\phi_{\alpha}^{JM\pi}.$$ Here $\langle\vert {\cal O} \vert\rangle_{\rm int}$ indicates that the integration with respect to $\rho_\alpha$ is to be carried out in the internal region. Actually $\langle\vert {\cal O} \vert\rangle_{\rm int}$ is obtained by calculating the matrix element $\langle\vert {\cal O} \vert\rangle$ in the entire space and subtracting the corresponding external matrix element $\langle\vert {\cal O} \vert\rangle_{\rm ext}$ that is easily obtained because no intercluster antisymmetrization is needed. The $R$-matrix and $Z$-matrix are defined by $$\begin{aligned}
&&{\cal R}_{\alpha' \alpha} \equiv \frac{\hbar^2 a}{2}
\left(\frac{k_{\alpha'}}{\mu_{\alpha'}\mu_{\alpha}k_\alpha}\right)^\frac{1}{2}
\sum_{n' n}u_{\alpha' n'}(a)(C^{-1})_{\alpha' n', \alpha n}u_{\alpha n}(a),
\\
&&{\cal Z}_{\alpha' \alpha} \equiv I_{\alpha} (k_\alpha a) \delta_{\alpha' \alpha}-
{\cal R}_{\alpha' \alpha}k_\alpha a I'_{\alpha}(k_\alpha a).\end{aligned}$$ The $S$-matrix is finally obtained as $$\begin{aligned}
S^{J\pi}=({\cal Z}^*)^{-1}{\cal Z}.\end{aligned}$$ In this paper we focus on the elastic phase shifts $\delta_{\alpha}^{J\pi}$ that are defined by the diagonal elements of the $S$-matrix, $$\begin{aligned}
S_{\alpha \alpha}^{J\pi}=\eta_\alpha^{J\pi} e^{2i\delta_\alpha^{J\pi}}.\end{aligned}$$
We study four-nucleon scattering involving the $d$+$d$, $t$+$p$ and $h$+$n$ channels in the energy region around and below the $d$+$d$ threshold. In Table \[chan0\] we list all possible labels $^{2I+1}\ell_J$ of physical channels for $J^{\pi}=0^{\pm}$, $1^{\pm}$, and $2^{\pm}$, assuming $\ell \leq 2$. Here “physical” means that the channels involve the cluster bound states that appear in the external region as well. Non-physical channels involving excited pseudo states will also be included in most calculations. Note that the $d$+$d$ channel must satisfy the condition of $I+\ell$ even (see Eq. (\[wf.channel\])). The channel spin $I=0$ or 2 can couple with only even $\ell$, but $I=1$ with only odd $\ell$. It is noted that the relative motion for the $d$+$d$ scattering can have $\ell=0$ only when $J^{\pi}$ is equal to $0^+$ and $2^+$.
0$^+$ 1$^+$ 2$^+$ 0$^-$ 1$^-$ 2$^-$
------------------------------------------------------------------------------ --------- --------- --------- --------- --------- ---------
$d(1^+)$+$d(1^+)$ $^1S_0$ $^5D_1$ $^5S_2$ $^3P_0$ $^3P_1$ $^3P_2$
$^5D_0$ $^1D_2$
$^5D_2$
$t(\frac{1}{2}^+)$+$p(\frac{1}{2}^+), \;h(\frac{1}{2}^+)$+$n(\frac{1}{2}^+)$ $^1S_0$ $^3S_1$ $^1D_2$ $^3P_0$ $^1P_1$ $^3P_2$
$^3D_1$ $^3D_2$ $^3P_1$
: Channel spins ($^{2I+1}\ell_J$) of physical $d$+$d$, $t$+$p$, and $h$+$n$ channels for $J \le 2$ and $\ell \le 2$.
\[chan0\]
Because one of our purposes in this investigation is to understand the role of the tensor force played in the four-nucleon dynamics, we want to compare the phase shifts obtained with two Hamiltonians that differ in the type of $NN$ interactions. One is a realistic interaction called the AV8$^{\prime}$ potential [@av8p] that includes central, tensor and spin-orbit components. We also add an effective three-nucleon force (TNF) in order to reproduce reasonably the binding energies of $t$, $h$ and $^4$He [@hiyama04], which makes reasonable thresholds. In the present calculation, the TNF is included in all calculations for AV8$^{\prime}$. Another is an effective central interaction called the Minnesota (MN) potential [@thompson77], which reproduce reasonably the binding energies of $t$, $h$ and $^4$He, though it has central terms alone (with an exchange parameter $u=1$). The Coulomb potential is included for both potentials.
The intrinsic wave function $\Phi^{k}_{I_k}$ of cluster $k$ $(k=a,\ b)$ is described with a combination of $N_k$ basis functions with different $L_k$ and $S_k$ values $$\begin{aligned}
\Phi_{I_{k}M_{I_k}}^k=\sum^{N_k} {\cal A} \left[
\psi_{L_k}^{(\rm{space})}\psi_{S_k}^{(\rm{spin})}
\right]_{I_k M_{I_k}}
\psi^{(\rm{isospin})}_{T_k M_{T_k}},
\label{wf.cluster}\end{aligned}$$ where $\psi_{L_k}^{(\rm{space})}$, $\psi_{S_k}^{(\rm{spin})}$ and $\psi_{T_k M_{T_k} }^{(\rm{isospin})}$ denote the space, spin and isospin parts of the cluster wave function. In the case of the AV8$^{\prime}$ potential, the $t$ (or $h$) wave function is approximated with thirty Gaussian basis functions that include $L_k \leq 2$, and $S_k=\frac{1}{2}$ and $\frac{3}{2}$. The deuteron wave function is also approximated with Gaussian basis functions, four terms both in the $S$- and $D$-waves, respectively. The falloff parameters of the Gaussian functions are selected using the SVM [@book] and the expansion coefficients are determined by diagonalizing the intrinsic cluster Hamiltonian. A similar procedure is applied to the case of the MN potential.
The calculated energies $E$, root-mean-square (rms) radii $R^{\rm rms}$ and $D$ state probabilities $P_D$ are given in the fourth to sixth columns in Table \[sub1\]. We use the truncated basis in order to obtain the phase shifts in reasonable computer times, they slightly deviate from more elaborate calculations, which are given in the last three columns. Fortunately, except for the small shift of the threshold energy, the phase shifts are not very sensitive to the details of the cluster wave functions because they are determined by the change of the relative motion function of the clusters. The $N_k$ values in parenthesis for $^4$He are the number of $J^{\pi}=0^+$ basis functions in the major multi-channel calculation. The energy of $^4$He calculated in Table 2 with the multi-channel calculation is thus not optimized but found to be very close to that of the more extensive calculation. It is noted that the calculated $R^{\rm rms}$ value for the deuteron is smaller than in other calculations. This is due to the restricted choice of the length parameters of the basis functions, which permits us to use a relatively small channel radius of $a \sim 15$ fm. We have checked that the phase shifts for a $d+d$ single channel calculation do not change even when more extended deuteron wave functions are employed.
---------------- -------------------- -------- ---------- --------------- -------- ---------- --------------- ---------
potential cluster
$N_k$ $E$ $R^{\rm rms}$ $P_D$ $E$ $R^{\rm rms}$ $P_{D}$
(MeV) (fm) ($\%$) (MeV) (fm) ($\%$)
$d(1^+)$ 8 $-$2.18 1.79 5.9 $-$2.24 1.96 5.8
AV8$^{\prime}$ $t(\frac{1}{2}^+)$ 30 $-$8.22 1.69 8.4 $-$8.41 - -
(with TNF) $h(\frac{1}{2}^+)$ 30 $-$7.55 1.71 8.3 $-$7.74 - -
$^4$He$(0^+)$ (2370) $-$27.99 1.46 13.8 $-$28.44 - 14.1
$d(1^+)$ 4 $-$2.10 1.63 0 $-$2.20 1.95 0
MN $t(\frac{1}{2}^+)$ 15 $-$8.38 1.70 0 $-$8.38 1.71 0
$h(\frac{1}{2}^+)$ 15 $-$7.70 1.72 0 $-$7.71 1.74 0
$^4$He$(0^+)$ (1140) $-$29.94 1.41 0 $-$29.94 1.41 0
---------------- -------------------- -------- ---------- --------------- -------- ---------- --------------- ---------
: Energies $E$, rms radii $R^{\rm rms}$ and $D$-state probabilities $P_D$ of the clusters that appear in four-nucleon scattering and $^4$He with the AV8$^{\prime}$ (with TNF) and MN potentials. $N_k$ is the number of basis functions used to approximate the wave function of cluster $k$. The values in the last three columns for three- and four-body systems are taken from Ref. [@hiyama04] for AV8$^{\prime}$ and Ref. [@DGVR] for MN.
\[sub1\]
Correlated Gaussian function with triple global vectors {#sect.3}
=======================================================
As explained in the previous section, the calculation of the $S$-matrix reduces to that of the Hamiltonian and overlap matrix elements with the functions defined by (\[wf-ch1\]) and (\[wf.cluster\]), and it is conveniently performed by transforming that wave function into an $LS$-coupled form, $$\begin{aligned}
&&{\cal A}
\left[\left[\left[\psi^{(\rm space)}_{L_a}\psi^{(\rm space)}_{L_b}\right]_{L_{ab}}
\chi_\alpha({{\mbox{\boldmath $\rho$}}}_\alpha)\right]_L
\left[\psi^{(\rm spin)}_{S_a} \psi^{(\rm spin)}_{S_b}\right]_{S}
\right]_{JM}.\end{aligned}$$ The transformation can be done as $$\begin{aligned}
& &{\hspace{-10mm}}
{\cal A}\left[ \left[
[\psi_{L_a}^{(\rm{space})}\psi_{S_a}^{(\rm{spin})}]_{I_a}
[\psi_{L_b}^{(\rm{space})}\psi_{S_b}^{(\rm{spin})}]_{I_b}
\right]_I \chi_\alpha({{\mbox{\boldmath $\rho$}}}_\alpha)\right]_{JM}
\psi^{(\rm{isospin})}_{T_a M_{T_a}}\psi^{(\rm{isospin})}_{T_b M_{T_b}}
\nonumber \\
&=&\sum_{L_{ab}LS}
\left[
\begin{array}{ccc}
L_a&S_a&I_a\\
L_b&S_b&I_b\\
L_{ab}&S&I\\
\end{array}\right]
(-1)^{L_{ab}+J-I-L}U(S L_{ab} J \ell; IL)\nonumber\\
&&\times
{\cal A}\left[
\psi_{L_a L_b (L_{ab}) \ell L}^{(\rm space)}
\left[\psi^{(\rm spin)}_{S_a} \psi^{(\rm spin)}_{S_b}\right]_{S}
\right]_{JM}
\psi^{(\rm{isospin})}_{T_a M_{T_a}}\psi^{(\rm{isospin})}_{T_b M_{T_b}}\end{aligned}$$ with $$\begin{aligned}
\psi_{L_a L_b (L_{ab}) \ell L}^{(\rm space)}
=\left[\left[\psi^{(\rm space)}_{L_a}\psi^{(\rm space)}_{L_b}\right]_{L_{ab}}
\chi_\alpha({{\mbox{\boldmath $\rho$}}}_\alpha)\right]_L,
\label{space.part}\end{aligned}$$ where $U$ and $[ \ \ ]$ are Racah and 9$j$ coefficients in unitary form [@book].
The evaluation of the matrix element can be done in the spatial, spin, and isospin parts separately. The spin and isospin parts are obtained straightforwardly. In the following we concentrate on the spatial matrix element. The spatial part (\[space.part\]) of the total wave function is given as a product of the cluster intrinsic parts and their relative motion part. The coordinates used to describe the $2N$+$2N$ channel are depicted in Fig. \[fig.1\](a) with ${{\mbox{\boldmath $\rho$}}}_\alpha={{\mbox{\boldmath $x$}}}_3$, whereas the coordinates suitable for the $t$+$p$ and $h$+$n$ channels are shown in Fig. \[fig.1\](b) with ${{\mbox{\boldmath $\rho$}}}_\alpha={{\mbox{\boldmath $x$}}}_3'$. These coordinate sets are often called H-type and K-type. Therefore the calculation of the spatial matrix element requires a coordinate transformation involving the angular momenta $L_a, L_b,
\ell$ and $L_{ab}$. Moreover the permutation operator in ${\cal A}$ causes a complicated coordinate transformation. All these complexities are treated elegantly by introducing a correlated Gaussian [@boys60; @vs95; @book], provided each part of $\psi_{L_a L_b (L_{ab}) \ell L}^{(\rm space)}$ is given in terms of (a combination of) Gaussian functions as in the present case. In what follows we will demonstrate how it is performed. Because the formulation with the correlated Gaussian is not restricted to four nucleons but can be applied to a system including more particles, the number of nucleons is assumed to be $N$ in this and next sections as well as in Appendices B and C unless otherwise mentioned.
The relative and center of mass coordinates of the $N$ nucleons, ${{\mbox{\boldmath $x$}}}_i\, (i=1,\ldots,N)$, and the single-particle coordinates, ${{\mbox{\boldmath $r$}}}_i\, (i=1,\ldots,N)$, are mutually related by a linear transformation matrix $U$ and its inverse $U^{-1}$ as follows: $${{\mbox{\boldmath $x$}}}_i=\sum_{j=1}^{N}U_{ij}{{\mbox{\boldmath $r$}}}_j,\ \ \ \ \
{{\mbox{\boldmath $r$}}}_i=\sum_{j=1}^{N}(U^{-1})_{ij}{{\mbox{\boldmath $x$}}}_j.
\label{def.matU}$$ We use a matrix notation as much as possible in order to simplify formulas and expressions. Let ${{\mbox{\boldmath $x$}}}$ denote an $(N\!-\!1)$-dimensional column vector comprising all ${{\mbox{\boldmath $x$}}}_i$ but the center of mass coordinate ${{\mbox{\boldmath $x$}}}_N$. Its transpose is a row vector and it is expressed as $$\widetilde{{{\mbox{\boldmath $x$}}}}=({{\mbox{\boldmath $x$}}}_1,{{\mbox{\boldmath $x$}}}_2,...,{{\mbox{\boldmath $x$}}}_{N-1}).$$ The choice for ${{\mbox{\boldmath $x$}}}$ is not unique but a set of Jacobi coordinates is conveniently employed. For the four-body system, the Jacobi set is identical to the K-type coordinate, and the corresponding matrix $U$ is given by $$\begin{aligned}
U_K=
\left(\begin{array}{rrrr}
1&-1&0&0\\
\frac{1}{2}&\frac{1}{2}&-1&0\\
\frac{1}{3}&\frac{1}{3}&\frac{1}{3}&-1\\
\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}\\
\end{array}\right).\end{aligned}$$ The transformation matrix for the H-type coordinate reads $$\begin{aligned}
U_H=
\left(\begin{array}{rrrr}
1&-1&0&0\\
0&0&1&-1\\
\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\\
\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}\\
\end{array}\right).\end{aligned}$$ The K-type coordinate is obtained directly from the H-type coordinate by a transformation matrix $U_K U_H^{-1}$ $$U_KU_H^{-1}=
\left(
\begin{array}{rrrr}
1&0&0&0\\
0&-\frac{1}{2}&1&0\\
0&\frac{2}{3}&\frac{2}{3}&0\\
0&0&0&1\\
\end{array}
\right)
=\left(
\begin{array}{cc}
U_{KH}&0\\
0& 1\\
\end{array}
\right),
\label{HtoK}$$ where $U_{KH}$ is a 3$\times$3 sub-matrix of $U_K U_H^{-1}$.
Each coordinate set emphasizes particular correlations among the nucleons. As mentioned above, the H-coordinate is natural to describe the $d$+$d$ channel, whereas the K-coordinate is suited for a description of the 3$N$+$N$ partition. It is of crucial importance to include both types of motion in order to fully describe the four-nucleon dynamics [@benchmark]. In order to develop a unified method that can incorporate both types of coordinates on an equal footing, we extend the explicitly correlated Gaussian function [@suzuki00; @DGVR] to include triple global vectors $$\begin{aligned}
&&F_{L_1 L_2 (L_{12})L_3 L M}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}})\nonumber\\
&&\quad ={\rm exp}\left(-{\frac{1}{2}}{\widetilde{{{\mbox{\boldmath $x$}}}}} A {{\mbox{\boldmath $x$}}}\right)
\left[[{\cal Y}_{L_1}(\widetilde{u_1}{{\mbox{\boldmath $x$}}})
{\cal Y}_{L_2}(\widetilde{u_2}{{\mbox{\boldmath $x$}}})]_{L_{12}}
{\cal Y}_{L_3}(\widetilde{u_3}{{\mbox{\boldmath $x$}}})\right]_{LM},
\label{cgtgv}\end{aligned}$$ where $${\cal Y}_{L_iM_i}(\widetilde{u_i}{{\mbox{\boldmath $x$}}})=
|\widetilde{u_i}{{\mbox{\boldmath $x$}}}|^{L_i}Y_{L_iM_i}(\widehat{\widetilde{u_i}{{\mbox{\boldmath $x$}}}})$$ is a solid spherical harmonics and its argument, $\widetilde{u_i}{{\mbox{\boldmath $x$}}}$, what we call a global vector, is a vector defined through an $(N-1)$-dimensional column vector $u_i$ and ${{\mbox{\boldmath $x$}}}$ as $$\widetilde{u_i}{{\mbox{\boldmath $x$}}}=\sum_{j=1}^{N-1}(u_i)_j{{\mbox{\boldmath $x$}}}_j,
\label{def.ux}$$ where $(u_i)_j$ is the $j$th element of $u_i$. In Eq. (\[cgtgv\]) $A$ is an $(N-1)\times(N-1)$ real and symmetric matrix, and it must be positive-definite for the function $F$ to have a finite norm, but otherwise may be arbitrary. Non-diagonal elements of $A$ can be nonzero.
The matrix $A$ and the vectors $u_1, u_2, u_3$ are parameters to characterize the “shape” of the correlated Gaussian function. The Gaussian function including $A$ describes a spherical motion of the system, while the global vectors are responsible for a rotational motion. The spatial function (\[space.part\]) is found to reduce to the general form (\[cgtgv\]). Suppose that ${{\mbox{\boldmath $x$}}}$ stands for the H-type coordinate. Then a choice of $\widetilde{u_1}$=(1,0,0), $\widetilde{u_2}$=(0,1,0) and $\widetilde{u_3}$=(0,0,1) together with a diagonal matrix $A$ provides us with the basis function (\[space.part\]) employed to represent the configurations of the $2N$+$2N$ channel. On the other hand, the K-type basis function looks like $${\rm exp}\left(-{\frac{1}{2}}{\widetilde{{{\mbox{\boldmath $x$}}}'}} A_K {{\mbox{\boldmath $x$}}}'\right)
\left[[{\cal Y}_{L_1}({{\mbox{\boldmath $x$}}}'_1)
{\cal Y}_{L_2}({{\mbox{\boldmath $x$}}}'_2)]_{L_{12}}
{\cal Y}_{L_3}({{\mbox{\boldmath $x$}}}'_3)\right]_{LM},
\label{K.basis}$$ where $\widetilde{{{\mbox{\boldmath $x$}}}'}=({{\mbox{\boldmath $x$}}}'_1,{{\mbox{\boldmath $x$}}}'_2,{{\mbox{\boldmath $x$}}}'_3)$ is the K-coordinate set (see Fig. 1(b)) and $A_K$ is a 3$\times$3 diagonal matrix. Noting that ${{\mbox{\boldmath $x$}}}'$ is equal to ${{\mbox{\boldmath $x$}}}'=U_{KH}{{\mbox{\boldmath $x$}}}$, we observe that the basis function (\[K.basis\]) is obtained from Eq. (\[cgtgv\]) by a particular choice of parameters, that is, $\widetilde{u_1}$=(1,0,0), $\widetilde{u_2}$=(0,$-\frac{1}{2}$,$1$) and $\widetilde{u_3}$=(0,$\frac{2}{3}$,$\frac{2}{3}$), and the matrix $A$ is related to $A_K$ by $$\begin{aligned}
A=(u_1 u_2 u_3)A_K \left(
\begin{array}{c}
\widetilde{u_1}\\
\widetilde{u_2}\\
\widetilde{u_3}\\
\end{array}
\right)
=\widetilde{U_{KH}}A_K U_{KH}.\end{aligned}$$ Thus the form of the $F$-function remains unchanged under the transformation of relative coordinates.
Note that $A$ is no longer diagonal. The choice of a different set of coordinates ends up only choosing appropriate parameters for $A$, $u_1$, $u_2$, and $u_3$.
It is also noted that the triple global vectors in Eq. (\[cgtgv\]) are a minimum number of vectors to provide all possible spatial functions with arbitrary $L$ and parity $\pi$. A natural parity state with $\pi=(-1)^L$ can be described by only one global vector, that is, using e.g., $L_1=L$, $L_2=0$, $L_{12}=L$, $L_3=0$ [@vs95; @suzuki98; @GVR]. To describe an unnatural parity state with $\pi=(-1)^{L+1}$ except for $0^-$ case, we need at least two global vectors, say, $L_1=L$, $L_2=1$, $L_{12}=L$, $L_3=0$ [@suzuki00; @DGVR]. The simplest choice for the $0^-$ state is to use three global vectors with $L_1=L_2=L_{12}=L_3=1$ [@suzuki00]. In this way, the basis function (\[cgtgv\]) can be versatile enough to describe bound states of not only four- but also more-particle systems with arbitrary $L$ and $\pi$.
To assure the permutation symmetry of the wave function, we have to operate a permutation $P$ on $F$. Since $P$ induces a linear transformation of the coordinate set, a new set of the permuted coordinates, ${{\mbox{\boldmath $x$}}}_P$, is related to the original coordinate set ${{\mbox{\boldmath $x$}}}$ as ${{\mbox{\boldmath $x$}}}_P\!=\!{\cal P}{{\mbox{\boldmath $x$}}}$ with an $(N-1)\times (N-1)$ matrix ${\cal P}$. As before, this permutation does not change the form of the $F$-function: $$\begin{aligned}
& &PF_{L_1 L_2(L_{12}) L_3 LM}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}})
\nonumber \\
&&\quad =F_{L_1 L_2 (L_{12})L_3 LM}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}}_P)
\nonumber \\
&&\quad =F_{L_1 L_2(L_{12}) L_3 LM}(\widetilde{\cal P}u_1,
\widetilde{\cal P}u_2,\widetilde{\cal P}u_3,\widetilde{\cal P}A{\cal P},
{{\mbox{\boldmath $x$}}}).
\label{trans.cg}\end{aligned}$$ The fact that the functional form of $F$ remains unchanged under the permutation as well as the transformation of coordinates enables one to unify the method of calculating the matrix elements. This unique property is one of the most notable points in the present method.
Calculation of matrix elements {#sect.4}
==============================
Calculations of matrix elements with the correlated Gaussian $F$ are greatly facilitated with the aid of the generating function $g$ [@vs95; @book] $$\begin{aligned}
g({{\mbox{\boldmath $s$}}}; A, {{\mbox{\boldmath $x$}}})=
\exp\Big(-{\frac{1}{2}}{\widetilde{{\mbox{\boldmath $x$}}}}A{{\mbox{\boldmath $x$}}}+
\widetilde{{\mbox{\boldmath $s$}}}{{\mbox{\boldmath $x$}}}\Big),\end{aligned}$$ with $\widetilde{{\mbox{\boldmath $s$}}}=({{\mbox{\boldmath $s$}}}_1,{{\mbox{\boldmath $s$}}}_2,\ldots,{{\mbox{\boldmath $s$}}}_{N-1})$, where ${{\mbox{\boldmath $s$}}}_i=\sum_{j=1}^3\lambda_j(u_j)_i{{\mbox{\boldmath $e$}}}_j$, ${{\mbox{\boldmath $e$}}}_j$ is a 3-dimensional unit vector (${{\mbox{\boldmath $e$}}}_j\cdot{{\mbox{\boldmath $e$}}}_j=1$), and $\lambda_j$ is a scalar parameter. More explicitly $$\widetilde{{\mbox{\boldmath $s$}}}{{\mbox{\boldmath $x$}}}=\sum_{i=1}^{N-1}{{\mbox{\boldmath $s$}}}_i\cdot{{\mbox{\boldmath $x$}}}_i=
\sum_{i=1}^{N-1}\sum_{j=1}^3\lambda_j
(u_{j})_i {{\mbox{\boldmath $e$}}}_j \cdot{{\mbox{\boldmath $x$}}}_i
=\sum_{j=1}^3 \lambda_j{{\mbox{\boldmath $e$}}}_j\cdot (\widetilde{u_j}{{\mbox{\boldmath $x$}}}).$$ The correlated Gaussian $F$ is generated as follows: $$\begin{aligned}
&&F_{L_1 L_2(L_{12}) L_3 LM}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}})\nonumber\\
&&\quad =\left(\prod_{i=1}^3 \frac{B_{L_i}}{L_i !}\int d{{\mbox{\boldmath $e$}}}_i\right)
\left[\left[Y_{L_1}({{\mbox{\boldmath $e$}}}_1)Y_{L_2}({{\mbox{\boldmath $e$}}}_2)\right]_{L_{12}}
Y_{L_3}({{\mbox{\boldmath $e$}}}_3)\right]_{LM}\nonumber\\
&&\quad \times \left(\frac{\partial^{L_1+L_2+L_3}}
{\partial \lambda_1^{L_1} \partial \lambda_2^{L_2} \partial \lambda_3^{L_3} }
\,g({{\mbox{\boldmath $s$}}};A,{{\mbox{\boldmath $x$}}})\right)\Bigg\vert_{\lambda_1=\lambda_2=\lambda_3=0},\label{gfn}\end{aligned}$$ where $$\begin{aligned}
\hspace*{-2cm} B_L&=&\frac{(2L+1)!!}{4\pi}.\end{aligned}$$ When $g({{\mbox{\boldmath $s$}}}; A,{{\mbox{\boldmath $x$}}})$ is expanded in powers of $\lambda_1$, only the term of degree $\lambda_{1}^{L_1}$ contributes in Eq. (\[gfn\]), and this term contains the $L_1$th degree ${{\mbox{\boldmath $e$}}}_1$ because $\lambda_1$ and ${{\mbox{\boldmath $e$}}}_1$ always appear simultaneously. In order for the term to contribute to the integration over ${{\mbox{\boldmath $e$}}}_1$, these $L_1$ vectors ${{\mbox{\boldmath $e$}}}_1$ must couple to the angular momentum $L_1$ because of the orthonormality of the spherical harmonics $Y_{L_1M_1}({{\mbox{\boldmath $e$}}}_1)$, that is, they are uniquely coupled to the maximum possible angular momentum. The same rule applies to $\lambda_2$, ${{\mbox{\boldmath $e$}}}_2$ and $\lambda_3$, ${{\mbox{\boldmath $e$}}}_3$ as well.
We outline a method of calculating the matrix element for an operator ${\cal O}$ $$\langle F_{L_4 L_5(L_{45}) L_6 L'M'}(u_4, u_5, u_6, A',{{\mbox{\boldmath $x$}}})\vert
{\cal O}\vert F_{L_1 L_2(L_{12}) L_3 LM}(u_1, u_2, u_3, A,{{\mbox{\boldmath $x$}}})\rangle.
\label{meofop}$$ In what follows this matrix element is abbreviated as $\langle F'\vert {\cal O}\vert F\rangle $. Using Eq. (\[gfn\]) in Eq. (\[meofop\]) enables one to relate the matrix element to that between the generating functions: $$\begin{aligned}
\left<F'\vert {\cal O} \vert F \right>
&\!=\!&\left(\prod_{i=1}^6 \frac{B_{L_i}}{L_i !}\int d{{\mbox{\boldmath $e$}}}_i\right)
\left[\left[Y_{L_4}({{\mbox{\boldmath $e$}}}_4)Y_{L_5}({{\mbox{\boldmath $e$}}}_5)\right]_{L_{45}}
Y_{L_6}({{\mbox{\boldmath $e$}}}_6)\right]_{L'M'}^* \nonumber\\
&\!\times\!&\left[\left[Y_{L_1}({{\mbox{\boldmath $e$}}}_1)Y_{L_2}({{\mbox{\boldmath $e$}}}_2)\right]_{L_{12}}
Y_{L_3}({{\mbox{\boldmath $e$}}}_3)\right]_{LM}
\nonumber\\
&\!\times\!&\left(\prod_{i=1}^6
\frac{\partial^{L_i}}{\partial \lambda_i^{L_i}}\right)
\left<g({{\mbox{\boldmath $s$}}}',A',{{\mbox{\boldmath $x$}}}')\vert {\cal O} \vert
g({{\mbox{\boldmath $s$}}},A,{{\mbox{\boldmath $x$}}})\right>\Big\vert_{\lambda_i=0},
\label{grandformula}\end{aligned}$$ with $${{\mbox{\boldmath $s$}}}={\lambda}_1u_1{{\mbox{\boldmath $e$}}}_1 \!+\!{\lambda}_2u_2{{\mbox{\boldmath $e$}}}_2
\!+\!{\lambda}_3u_3{{\mbox{\boldmath $e$}}}_3,\hspace*{1cm}
{{\mbox{\boldmath $s$}}}'={\lambda}_4u_4{{\mbox{\boldmath $e$}}}_4 \!+\!{\lambda}_5u_5{{\mbox{\boldmath $e$}}}_5
\!+\!{\lambda}_6u_6{{\mbox{\boldmath $e$}}}_6.$$ The calculation of the matrix element consists of three stages: (1) Evaluate the matrix element between the generating functions, $\left<g({{\mbox{\boldmath $s$}}}',A',{{\mbox{\boldmath $x$}}}')\vert {\cal O} \vert
g({{\mbox{\boldmath $s$}}},A,{{\mbox{\boldmath $x$}}})\right>$. (2) Expand that matrix element in powers of $\lambda_i$ and keep only those terms of degree $L_i$ for each $i$. (3) Recouple the vectors ${{\mbox{\boldmath $e$}}}_i$ and integrate over the angle coordinates. In the second stage the remaining terms should contain ${{\mbox{\boldmath $e$}}}_i$s of degree $L_i$ as well. Hence any term with $\lambda_i^2 {{\mbox{\boldmath $e$}}}_i\!\cdot\!{{\mbox{\boldmath $e$}}}_i\!
=\!\lambda_i^2$ etc. can be omitted because the degree of ${{\mbox{\boldmath $e$}}}_i$ becomes smaller than that of $\lambda_i$.
We will explain the above procedures for the case of an overlap matrix element. The matrix element between the generating functions is $$\begin{aligned}
\left<g({{\mbox{\boldmath $s$}}}',A',{{\mbox{\boldmath $x$}}}')\vert
g({{\mbox{\boldmath $s$}}},A,{{\mbox{\boldmath $x$}}})\right>
=\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}
\exp\left(\frac{1}{2}\widetilde{{{\mbox{\boldmath $z$}}}}B^{-1}{{\mbox{\boldmath $z$}}}\right)\end{aligned}$$ with $$\begin{aligned}
B=A'+A,\qquad {{\mbox{\boldmath $z$}}}={{\mbox{\boldmath $s$}}}+{{\mbox{\boldmath $s$}}}'=\sum_{i=1}^6\lambda_i{{\mbox{\boldmath $e$}}}_iu_i.\end{aligned}$$ To perform the operation in the second stage we note that $$\frac{1}{2}\widetilde{{{\mbox{\boldmath $z$}}}}B^{-1}{{\mbox{\boldmath $z$}}}=\frac{1}{2}\sum_{i,j=1}^6
\rho_{ij}\lambda_i\lambda_j {{\mbox{\boldmath $e$}}}_i\cdot{{\mbox{\boldmath $e$}}}_j$$ with $$\rho_{ij}=\widetilde{u_i}B^{-1}u_j.
\label{def.rho}$$ As mentioned above, here we can drop the diagonal terms, $\lambda_i^2
{{\mbox{\boldmath $e$}}}_i\cdot{{\mbox{\boldmath $e$}}}_i$, and we get $$\begin{aligned}
&&\left(\prod_{i=1}^6\frac{\partial^{L_i}}{\partial \lambda_i^{L_i}}\right)
\left<g({{\mbox{\boldmath $s$}}}',A',{{\mbox{\boldmath $x$}}}')\vert
g({{\mbox{\boldmath $s$}}},A,{{\mbox{\boldmath $x$}}})\right>\Big\vert_{\lambda_i=0} \nonumber\\
&&\quad =\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}\prod_{i=1}^6L_i!
\prod_{i<j}^6\frac{\left(\rho_{ij}{{\mbox{\boldmath $e$}}}_i\cdot{{\mbox{\boldmath $e$}}}_j\right)^{n_{ij}}}{n_{ij}!}.\end{aligned}$$ Here the non-negative integers $n_{ij}$ must satisfy the following equations in order to assure the degree $L_i$ for ${{\mbox{\boldmath $e$}}}_i$ in the different terms, $$\begin{aligned}
&&n_{12}+n_{13}+n_{14}+n_{15}+n_{16}=L_1, \nonumber\\
&&n_{12}+n_{23}+n_{24}+n_{25}+n_{26}=L_2, \nonumber\\
&&n_{13}+n_{23}+n_{34}+n_{35}+n_{36}=L_3, \nonumber\\
&&n_{14}+n_{24}+n_{34}+n_{45}+n_{46}=L_4, \nonumber\\
&&n_{15}+n_{25}+n_{35}+n_{45}+n_{56}=L_5, \nonumber\\
&&n_{16}+n_{26}+n_{36}+n_{46}+n_{56}=L_6.
\label{nij}\end{aligned}$$
The last step is to recouple the angular momenta arising from the various terms. Since we have to couple ${{\mbox{\boldmath $e$}}}_i$s to the angular momentum $L_i$ from the terms of degree $L_i$, we may replace the term $\left(\rho_{ij}{{\mbox{\boldmath $e$}}}_i\cdot{{\mbox{\boldmath $e$}}}_j\right)^{n_{ij}}$ with just one piece $$\frac{ (-\rho_{ij})^{n_{ij}} n_{ij}! \sqrt{2n_{ij}+1}}{B_{n_{ij}}}
\left[Y_{n_{ij}}({{\mbox{\boldmath $e$}}}_i)Y_{n_{ij}}({{\mbox{\boldmath $e$}}}_j)\right]_{00}.$$ Other pieces like $\left[Y_{\kappa}({{\mbox{\boldmath $e$}}}_i)Y_{\kappa}({{\mbox{\boldmath $e$}}}_j)\right]_{00}$ with $\kappa < n_{ij}$ do not contribute to the matrix element. We thus have a product of 15 terms of $\left[Y_{n_{ij}}({{\mbox{\boldmath $e$}}}_i)Y_{n_{ij}}({{\mbox{\boldmath $e$}}}_j)\right]_{00}$. The coupling of these terms is done by defining various coefficients that are all expressed in terms of Clebsch-Gordan, Racah, and 9$j$ coefficients. For example, we make use of the formulas $$\begin{aligned}
& &{\hspace{-1cm}}[Y_a({{\mbox{\boldmath $e$}}}_1)Y_a({{\mbox{\boldmath $e$}}}_2)]_{00}\ [Y_b({{\mbox{\boldmath $e$}}}_1)Y_b({{\mbox{\boldmath $e$}}}_3)]_{00}\
[Y_c({{\mbox{\boldmath $e$}}}_2)Y_c({{\mbox{\boldmath $e$}}}_3)]_{00}
\nonumber \\
& \to& X(abc)\ [[Y_{a+b}({{\mbox{\boldmath $e$}}}_1)Y_{a+c}({{\mbox{\boldmath $e$}}}_2)]_{b+c}Y_{b+c}({{\mbox{\boldmath $e$}}}_3)]_{00},\\
\nonumber\\
& &{\hspace{-1cm}}[Y_a({{\mbox{\boldmath $e$}}}_1)Y_a({{\mbox{\boldmath $e$}}}_4)]_{00}\ [Y_b({{\mbox{\boldmath $e$}}}_1)Y_b({{\mbox{\boldmath $e$}}}_5)]_{00}\
[Y_c({{\mbox{\boldmath $e$}}}_1)Y_c({{\mbox{\boldmath $e$}}}_6)]_{00}
\nonumber \\
&\to& R_3(abc)\ [Y_{a+b+c}({{\mbox{\boldmath $e$}}}_1)\ [[Y_{a}({{\mbox{\boldmath $e$}}}_4)Y_{b}({{\mbox{\boldmath $e$}}}_5)]_{a+b}Y_c({{\mbox{\boldmath $e$}}}_6)]_{a+b+c}]_{00}.\end{aligned}$$ Here the symbol $\to$ indicates that no other terms arising from the left hand side of the equation contribute to the integration over the angles ${{\mbox{\boldmath $e$}}}_i$s, so that only the term on the right hand side has to be retained. Another coefficient is $$\begin{aligned}
&&{\hspace{-1cm}}[[[Y_a({{\mbox{\boldmath $e$}}}_4)Y_b({{\mbox{\boldmath $e$}}}_5)]_qY_c({{\mbox{\boldmath $e$}}}_6)]_Q\
[[Y_{a'}({{\mbox{\boldmath $e$}}}_4)Y_{b'}({{\mbox{\boldmath $e$}}}_5)]_{q'}Y_{c'}({{\mbox{\boldmath $e$}}}_6)]_{Q'}]_{\ell}
\nonumber \\
&\to& \sum_{\ell'}W(abcqQ,a'b'c'q'Q',\ell \ell')
[[Y_{a+a'}({{\mbox{\boldmath $e$}}}_4)Y_{b+b'}({{\mbox{\boldmath $e$}}}_5)]_{\ell'}Y_{c+c'}({{\mbox{\boldmath $e$}}}_6)]_{\ell}.\end{aligned}$$ Expressions for the coefficients, $X, R_3, W$, are given in Appendix A. Performing the integration of the six unit vectors, ${{\mbox{\boldmath $e$}}}_i$s, as prescribed in Eq. (\[grandformula\]) leads to the overlap matrix element $$\begin{aligned}
& &{\hspace{-1cm}}\left<F'\vert F \right>
\nonumber \\
&=&\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}
\left(\prod_{i=1}^6 B_{L_i}\right)
\frac{(-1)^{L_1+L_2+L_3}}{\sqrt{2L+1}}\delta_{LL'}\delta_{MM'}\nonumber\\
&\times & \hspace*{-0.3cm}\sum_{n_{ij}}\left(\prod_{i<j}^6 (-\rho_{ij})^{n_{ij}}
\frac{ \sqrt{2n_{ij}+1}}{B_{n_{ij}}}\right)
O(n_{ij}; L_1L_2L_3L_4L_5L_6,L_{12}L_{45}L)\label{eq6a},
\label{me.overlap}\end{aligned}$$ with $$\begin{aligned}
&&{\hspace{-1cm}}O(n_{ij}; L_1L_2L_3L_4L_5L_6,L_{12}L_{45}L)\nonumber\\
&&{\hspace{-1cm}}=X(n_{12}n_{13}n_{23})R_3(n_{14}n_{15}n_{16})R_3(n_{24}n_{25}n_{26})
R_3(n_{34}n_{35}n_{36})X(n_{45}n_{46}n_{56})\nonumber\\
&&{\hspace{-1cm}}\times Z(n_{12}\!+\!n_{13}\ L_1\!-\!n_{12}\!-\!n_{13})
Z(n_{12}\!+\!n_{23}\ L_2\!-\!n_{12}\!-\!n_{23})
Z(n_{13}\!+\!n_{23}\ L_3\!-\!n_{13}\!-\!n_{23})\nonumber\\
&&{\hspace{-1cm}}\times \sum_{\ell_1 \ell_2 \ell_3}
\left[\begin{array}{ccc}
L_1&L_1\!-\!n_{12}\!-\!n_{13}&n_{12}\!+\!n_{13}\\
L_2&L_2\!-\!n_{12}\!-\!n_{23}&n_{12}\!+\!n_{23}\\
L_{12}&\ell_1&n_{13}\!+\!n_{23}\\
\end{array}\right]
\left[\begin{array}{ccc}
L_{12}&\ell_1&n_{13}\!+\!n_{23}\\
L_3&L_3\!-\!n_{13}\!-\!n_{23}&n_{13}\!+\!n_{23}\\
L&L&0\\
\end{array}\right]
\nonumber\\
&&{\hspace{-1cm}}\times W(n_{14}n_{15}n_{16}\ n_{14}\!+\!n_{15}\ L_1\!-\!n_{12}\!-\!n_{13},
n_{24}n_{25}n_{26}\ n_{24}\!+\!n_{25}\ L_2\!-\!n_{12}\!-\!n_{23}, \ell_1 \ell_2)\nonumber\\
&&{\hspace{-1cm}}\times W(n_{14}\!+\!n_{24}\ n_{15}\!+\!n_{25}\ n_{16}\!+\!n_{26}\ \ell_2\ \ell_1,
n_{34}n_{35}n_{36}\ n_{34}\!+\!n_{35}\ L_3\!-\!n_{13}\!-\!n_{23}, L\ell_3) \nonumber\\
&&{\hspace{-1cm}}\times W(L_4\!-\!n_{45}\!-\!n_{46}\ L_5\!-\!n_{45}\!-\!n_{56}\ L_6\!-\!n_{46}\!-\!n_{56}\ \ell_3\ L, \nonumber\\
&&\hspace*{3cm}
n_{45}\!+\!n_{46}\ n_{45}\!+\!n_{56}\ n_{46}\!+\!n_{56}\ n_{46}\!+\!n_{56}\ 0, L L_{45}),\end{aligned}$$ where $Z$ is the coefficient given in Eq. (\[def.z\]). The summation in Eq. (\[me.overlap\]) extends over all possible sets of $n_{ij}$ that satisfy Eq. (\[nij\]). In most cases the values of $L_i$ are limited up to 2, so that the number of terms to be evaluated is not so large and the calculation of the matrix element is fast.
Expressions for the Hamiltonian matrix elements are collected in Appendix B. One advantage of our method is that the calculation of matrix elements can be done analytically. In addition we do not need to do angular momentum and parity projections because the correlated Gaussian function (\[cgtgv\]) already preserves those quantum numbers.
The Fourier transform of the correlated Gaussian function $F$ is a momentum space function and it becomes a useful tool to calculate various matrix elements that depend on the momentum operators [@DGVR]. For example, the distribution of the relative momentum is obtained by the expectation value of $\delta({{\mbox{\boldmath $p$}}}_i-{{\mbox{\boldmath $p$}}}_j-{{\mbox{\boldmath $p$}}})$, where ${{\mbox{\boldmath $p$}}}_j$ is the momentum of the $j$th particle. It is obviously much easier to calculate the distribution using the momentum space function rather than the coordinate space function. We show in Appendix C that the Fourier transform of $F$ reduces to a linear combination of $F$s in the momentum space.
![ Two-body thresholds calculated with the AV8$^{\prime}$ (left) and MN (middle) potentials. The solid lines are physical channels and the dashed lines are pseudo channels. We also plot experimental two-body thresholds for physical channels (right). The dotted line is the $p$+$p$+$n$+$n$ threshold.[]{data-label="fig:threshold"}](ene_thr.eps){width="9.0" height="8.0"}
Results
=======
$2N$+$2N$ and $3N$+$N$ channels
-------------------------------
In Table \[chan0\], we gave the physical channels, $d$+$d$, $t$+$p$, and $h$+$n$. Fig. \[fig:threshold\] displays two-body decay thresholds in the $d$+$d$ threshold energy region. The three physical channels are the main channels that describe the scattering around the three lowest thresholds ($d$+$d$, $t$+$p$, $h$+$n$). However, the scattering wave function $\Psi^{JM\pi}_{\rm int}$ in the internal region should contain all effects that may occur when all the nucleons come close to each other. It is thus reasonable that $\Psi^{JM\pi}_{\rm int}$ may not be well described in terms of the physical channels alone. Particularly the deuteron can be easily distorted when we use realistic potentials.
We will show that some pseudo $2N$+$2N$ channels are indeed needed to simulate the distortion of the deuteron. These pseudo channels, when they are included in the phase-shift calculation, are expected to take account of the distortion of the clusters of the entrance channel [@kanada85]. Here “pseudo” means that the clusters in the pseudo channels are not physically observable but may play a significant role in the internal region. The wave functions of these $2N$ pseudo clusters are obtained by diagonalizing the intrinsic cluster Hamiltonian similarly to the case of the physical clusters. We take into account the following pseudo clusters: $d^*(1^+, T=0),
\ d^*(0^+, T=1), \ d^*(2^+, T=0), \ d^*(3^+, T=0)$, $2n^*(0^+, T=1)$, and $2p^*(0^+,T=1)$, where the upper suffix \* indicates all the excited state but the ground state of $d$. Among the pseudo clusters, the lowest energy states with 0$^+$ that are related to virtual states would be most important. We especially write them as $\bar{d}(0^+)$, 2$n(0^+)$ (di-neutron) and 2$p(0^+)$ (di-proton). Although they are not bound, they are observed as resonances or quasi-bound states with negative scattering lengths. In fact the scattering lengths are $a_s(nn)=-$16.5 fm and $a_s(pp)=-$17.9 fm, which are comparable to $a_s(np, T=1)=-$23.7 fm. The calculated thresholds of these pseudo channels are also drawn in Fig. \[fig:threshold\].
channel
------ ----------- ----- -----------------------------------------
$2N$+$2N$ I $d(1^+)$+$d(1^+)$
$d(1^+)$+$d^*(1^+)$
$d^*(1^+)$+$d^*(1^+)$
II $\bar{d}(0^+)$+$\bar{d}(0^+)$
$\bar{d}(0^+)$+$d^*(0^+)$
$d^*(0^+)$+$d^*(0^+)$
III $d^*(2^+)$+$d^*(1^+)$
$d^*(2^+)$+$d^*(2^+)$
IV $d^*(3^+)$+$d^*(1^+)$
FULL $d^*(3^+)$+$d^*(2^+)$
$d^*(3^+)$+$d^*(3^+)$
V $2n(0^+)$+$2p(0^+)$
$2n(0^+)$+$2p^*(0^+)$
$2n^*(0^+)$+$2p(0^+)$
$2n^*(0^+)$+$2p^*(0^+)$
$3N$+$N$ 1 $t(\frac{1}{2}^+)$+$p(\frac{1}{2}^+)$
$t^*(\frac{1}{2}^+)$+$p(\frac{1}{2}^+)$
2 $h(\frac{1}{2}^+)$+$n(\frac{1}{2}^+)$
$h^*(\frac{1}{2}^+)$+$n(\frac{1}{2}^+)$
: $2N$+$2N$ and $3N$+$N$ channels. The Roman and Arabic numerals correspond to sets of channels included in the calculations.
\[chan1\]
Though it is expected that the pseudo channels with low threshold energies contribute more strongly to the scattering phase shift, we take into account all of these $2N$+$2N$ channels that include a vanishing total isospin as given in Fig. \[fig:threshold\]. The total isospin of the $3N$+$N$ channel is mixed in the present calculation. Because the $T$=1 component of the scattering wave function only weakly couples to the $d(1^+, T=0)$+$d(1^+, T=0)$ elastic-channel, the channel $d(1^+, T=0)$+$\bar{d}(0^+, T=1)$ is not employed in the calculation.
We also include the excited deuteron channels that comprise the $d^*(2^+, T=0)$ and $d^*(3^+, T=0)$ clusters. The energies of these lowest thresholds are above 10 MeV. These channels are therefore expected not to be very important, but that is not always the case as will be discussed in the case of the $^1$S$_0$ $d$+$d$ phase shift.
Table \[chan1\] summarizes all the channels that are used in our calculation. The 2$N$+2$N$ channels are distinguished by Roman numerals, while the 3$N$+$N$ channels are labeled by Arabic numerals. In the following, we use an abbreviation “2$N$+2$N$” or “3$N$+$N$” to indicate calculations including all 2$N$+2$N$ channels I-V or all 3$N$+$N$ channels (1-2 in Table \[chan1\]), respectively. Here $t^*(\frac{1}{2}^+)$ and $h^*(\frac{1}{2}^+)$ are excited 3$N$ continuum states. A “FULL” calculation indicates that all the channels in the table are included to set up the $S$-matrix. In the case of the MN potential channels III and IV are not included because this potential contains no tensor force.
The relative wave functions $\chi_{\alpha m}$ are expanded with 15 basis functions. We checked the stability of the $S$-matrix against the choice of the channel radius. The channel radius employed in this calculation is about 15 fm.
Positive parity phase shifts
----------------------------
Fig. \[fig:d+d-g3\] displays the $^1S_0$ $d$+$d$ elastic-scattering phase shift obtained with the AV8$^{\prime}$ potential. The dash-dotted line is the phase shift calculated with channel I ($I_d=1^+$), and the dash-dot-dotted line is the phase shift with channels I and II ($I_d=1^+, 0^+$). The phase shifts calculated by including further excited deuterons are also plotted by the dashed and dotted lines that correspond to the channels I-III ($I_d\le2^+$) and I-IV ($I_d\le3^+$), respectively. A naive expectation that the $^1S_0$ $d$+$d$ elastic-scattering phase shift might be well described in channel I ($d(1^+)+d(1^+)$, $d(1^+)+d^*(1^+)$ and $d^*(1^+)+d^*(1^+)$) alone completely breaks down in the case of the AV8$^{\prime}$ potential.
![ $^1S_0$ $d$+$d$ elastic-scattering phase shift calculated with the AV8$^{\prime}$ potential. The phase shifts are all obtained within the $d$+$d$ channels. The set of included channels is successively increased from I to IV. See Table \[chan1\] for the deuteron states included in each channel. []{data-label="fig:d+d-g3"}](fig0+-1.eps){width="7.0" height="6.0"}
Because the deuteron has a virtual state $\bar{d}$ with $0^+$ at low excitation energy, it is reasonable that the inclusion of channel II gives rise to a considerable attractive effect of several tens of degrees on the phase shift, as shown by the dash-dot-dotted line of Fig. \[fig:d+d-g3\]. However, the phase shift exhibits no converging behavior even when the higher spin states such as $d^*$(2$^+$) and $d^*$(3$^+$) are taken into account in the calculation. The additional attractions by these channels are of the same order as that of channel II. One may conclude that the deuteron is strongly distorted even in the low energy $^1S_0$ $d$+$d$ elastic scattering but more physically we have to realize that there exist two observed $0^+$ states below the $d$+$d$ threshold. Obviously the $d$+$d$ scattering wave function is subject to the structure of those states in the internal region.
The second 0$^+$ state of $^4$He lying about 4 MeV below the $d$+$d$ threshold is known to have a $3N$+$N$ cluster structure [@horiuchi08; @hiyama04]. Thus this state together with the ground state of $^4$He cannot be described well in the $2N$+$2N$ model space alone. As seen in Table \[chan0\], the $3N$+$N$ channel contains a $^1S_0$ component, which is the dominant component of the $0^+_2$ state. Since the realistic force strongly couples the $2N$+$2N$ channel to the $3N$+$N$ channel and the $d$+$d$ scattering wave function has to be orthogonal to the main component of the underlying $0^+$ states, we expect that the deuteron in the incoming $d$+$d$ channel never remains in its ground state but has to be distorted largely due to the $3N$+$N$ channel. The phase shift for the channel I-IV (dotted line) shows a resonant pattern. This resonant state is expected to be the second 0$^+$ state because of the restricted model space within the $d$+$d$ channel.
![Comparison of the ground and second 0$^+$ state energies between calculations with the AV8$^{\prime}$ (left) and MN (middle) potentials and experiment (right). The model space for AV8$^{\prime}$ is I-IV, $3N$+$N$ and FULL and the model space for MN is I-II, $3N$+$N$ and FULL. []{data-label="fig:2nd0+"}](2nd0+.eps){width="10.0" height="8.0"}
Fig. \[fig:2nd0+\] displays the calculated ground state energy and the second 0$^+$ energy for the AV8$^{\prime}$ (left) and MN (middle) potentials. The model spaces of the calculations are I-IV, $3N$+$N$ and FULL for AV8$^{\prime}$ and I-II, $3N$+$N$ and FULL for MN. We also plot experimental energies (right) [@tilley92]. For the AV8$^{\prime}$ potential, the energies of the two lowest $0^+$ states do not change very much between the FULL and $3N$+$N$ models. But the second 0$^+$ state with the $d$+$d$ model (channels I-IV) is not bound with respect to the $d$+$d$ threshold as expected before. On the contrary, for the MN potential, the second 0$^+$ state with the $d$+$d$ model (channels I-II) is bound with respect to the $d$+$d$ threshold. We consider that this difference makes the drastic change of the $d$+$d$ phase shifts, between the AV8$^{\prime}$ and MN potentials. It is also interesting to see that the energies of the two lowest $0^+$ states for the MN potential are almost the same between the FULL and $3N$+$N$ models.
Plotted in Fig. \[fig:d+d-3N+N\] are the $^1S_0$ $d$+$d$ elastic-scattering phase shifts obtained with the AV8$^{\prime}$ potential (left) and the MN potential (right). The FULL calculation (solid line) couples all 2$N$+2$N$ and 3$N$+$N$ channels that are listed in Table \[chan1\]. The $R$-matrix analysis (crosses) [@hofmann08] is reproduced well by both the AV8$^{\prime}$ and MN potential with the FULL calculation. Compared to the uncoupled phase shift (dotted line), one clearly sees that the 3$N$+$N$ channel produces a very large effect on the $d$+$d$ elastic phase shift, especially in the case of the AV8$^{\prime}$ potential. We also verified that a calculation excluding the channels III, IV or V from the FULL channel calculation gives only negligible change in the phase shift. The slow convergence seen in Fig. \[fig:d+d-g3\] is thus attributed to the neglect of the $3N$+$N$ channel, indicating that a proper account of the $^1S_0$ $d$+$d$ elastic phase shift at low energy can be possible only when the coupled channels {$d$(1$^+$)+$d$(1$^+$)} +{$d$(0$^+$)+$d$(0$^+$)}+ {$t$(1/2$^+$)+$p(1/2^+)$} +{$h$(1/2$^+$)+$n(1/2^+)$} are considered.
Thus, the slow convergence in Fig. \[fig:d+d-g3\] suggests that the $2N$+$2N$ partition is not an economical way to include the effects of the $3N$+$N$ channel. In the case of the MN potential (right panel in Fig. \[fig:d+d-3N+N\]), the situation is very different from the AV8$^{\prime}$ case. The channel coupling effect is rather modest, and the size of the $^1S_0$ $d$+$d$ elastic phase shift is already accounted for mostly in the $d$+$d$ channel calculation. All these results are very consistent with the $0^+$ spectrum in Fig. \[fig:2nd0+\].
The large distortion effect of the deuteron clusters on the $^1S_0$ $d$+$d$ scattering phase shift is expected to appear in the 3$N$+$N$ phase shift as well because of the coupling between the 3$N$+$N$ and 2$N$+2$N$ channels. We display in Fig. \[fig:3N+N-g3mn\] the $^1S_0$ $t$+$p$ elastic-scattering phase shift at energies below the $d$+$d$ threshold. The 0$^+_2$ state of $^4$He is observed as a sharp resonance with a proton decay width of 0.5 MeV at about 0.4 MeV above the $t$+$p$ threshold. The present energies ($E_r=0.15$ MeV for AV8$^{\prime}$, $E_r=0.12$ MeV for MN) calculated with a bound state approximation are slightly smaller than the experimental value, but they are consistent with a calculation ($E_r=0.105$ MeV and $\Gamma/2=0.129$ MeV for AV18+UIX, $E_r=0.091$ MeV and $\Gamma/2=0.077$ MeV for AV18+UIX+V$^*_3$) with another realistic interaction (AV18) with three nucleon forces by Hofmann and Hale [@hofmann08]. The calculated phase shifts appear slightly larger than that in the $R$-matrix analysis (crosses in Fig. \[fig:3N+N-g3mn\]) [@hofmann08]. It is noted that the phase shift changes so much even for a small change of the 0$^+_2$ resonant pole position ($\sim$0.1 MeV) because it is very near to the threshold. The phase shifts in the FULL calculation, for both AV8$^{\prime}$ and MN cases, show a resonance pattern in a small energy interval and the overall energy dependencies of the phase shifts are similar to each other. However, the phase shifts obtained only in the $3N$+$N$ channel are quite different as indicated by the dotted lines in Fig. \[fig:3N+N-g3mn\]. In the case of the MN potential (right) the phase shift is already close to the FULL phase shift, while in the case of the AV8$^{\prime}$ potential (left) the phase shift is much smaller (by almost 90 degrees) and moreover shows no resonance pattern.
By looking into the wave functions in more detail, we argue that the large distortion effect in the $^1S_0$ $d$+$d$ and $3N$+$N$ coupled channels is really brought about by the tensor force. As shown in Table \[sub1\], the AV8$^{\prime}$ potential with TNF gives 5.8% and 8.4% (8.3%) $D$-state probability for $d$ and $t$ $(h)$, respectively. Thus the $d$+$d$ state in the $^1S_0$ state contains $L=S=0$ components (89%) as well as $L=S=2$ components (11%), where $L$ and $S$ are the total orbital and spin angular momenta of the four-nucleon system. Similarly the $3N$+$N$ state in the $^1S_0$ state contains an $L=S=0$ component (92%) and an $L=S=2$ component (8%). Thus the tensor force couples both states with $\Delta L=2$ and $\Delta S=2$ couplings, which are in fact very large compared to the central matrix element ($\Delta L=0$, $\Delta S=0$). An analysis of this type was performed for some levels of $^4$He in Refs. [@DGVR; @horiuchi08]. The MN potential contains no tensor force, so that the $d$+$d$ and $3N$+$N$ channel coupling is modest.
As listed in Table \[chan0\], there are four channels, $^5S_2$, $^1D_2$, $^3D_2$ and $^5D_2$, for $J^{\pi}=2^+$ at energies around the $d$+$d$ threshold. Among these states, we expect that the effect of the coupling between the 3$N$+$N$ and 2$N$+2$N$ channels occurs most strongly in $^1D_2$ as it appears in all physical channels. However, no sharp resonance is observed in $^4$He up to 28MeV of excitation energy, so that the coupling effect, if any, might be weaker than that observed in the $^1S_0$ case.
Fig. \[fig:2+\] displays the $^1D_2$ elastic-scattering phase shifts obtained in three types of calculations, $3N$+$N$ (dashed line), $2N$+$2N$ (dotted line), and FULL (solid line). The $t$+$p$ and $d$+$d$ phase shifts start from the $t$+$p$ ($E_{\rm c.m.}=0$) and $d$+$d$ thresholds, respectively. The phase shifts of the $3N$+$N$ and $2N$+$2N$ calculations are both slightly positive, indicating a weak attraction in the $t$+$p$ and $d$+$d$ interactions. In the FULL calculation, the $t$+$p$ phase shift becomes more attractive and the $d$+$d$ phase shift turns to be negative (repulsive). The present FULL calculation reproduces the calculation of Ref. [@hofmann08] as expected. Though the effect of the coupling is slightly larger in the AV8$^{\prime}$ potential than in the MN potential, it is much less compared to the case of the $^1S_0$ phase shift. This is understood as follows. In the $^1D_2$ state, the main component of the wave function is given by the $L=2$, $S=0$ state: Its probability is the same as that of $^1S_0$, that is, 92% in $t$+$p$ and 89% in $d$+$d$. However, the probability of finding the state with $L=0$, $S=2$, which causes a strong tensor coupling, is more than one order of magnitude smaller than in the case of $^1S_0$, namely 0.23% in $t$+$p$ and 0.44% in $d$+$d$, respectively. The reason for this small percentage is that, to obtain $L=0$, the incoming $D$-wave in the $^1D_2$ channel must couple with the $D$-components in the clusters, but this coupling leads to several fragmented components with different $L$ values. This relatively weaker coupling of the tensor force explains the phase shift behavior in Fig. \[fig:2+\].
In Fig. \[fig:2+full\] we plot the $t$+$p$ and $d$+$d$ elastic-scattering phase shifts for other channels, $^5S_2$ (solid line), $^3D_2$ (dashed line), and $^5D_2$ (dotted line). We show only the FULL result, because the phase shifts with the truncated basis do not change visibly at the scale of the figure. The obtained phase shifts are not that different between the AV8$^{\prime}$ and MN potentials, and also consistent with the previous calculation [@hofmann08]. Thus, the effect of the distortion of the clusters is very small for 2$^+$ except for $^1D_2$.
We have three channels for $J^{\pi}=1^+$, $^5D_1$, $^3D_1$ and $^3S_1$. No sharp $1^+$ resonance of $^4$He is observed experimentally up to 28 MeV of excitation energy. Another theoretical calculation neither predicts it [@horiuchi08], so that the coupling between the 2$N$+2$N$ and 3$N$+$N$ channels is expected to be weak. Fig. \[fig:1+\] exhibits the $t$+$p$ and $d$+$d$ elastic-scattering phase shifts in the FULL calculation: $^5D_1$ $d$+$d$ (solid line), $^3D_1$ $t$+$p$ (dashed line), and $^3S_1$ $t$+$p$ (dotted line). Only the FULL result is displayed because the phase shift change in other calculations is small. Both AV8$^{\prime}$ and MN potentials produce phase shifts quite similar to each other.
Negative parity phase shifts
----------------------------
As seen from Table \[chan0\], the main components of these negative parity states are considered to be $^3P_J$.
We compare in Fig. \[fig:0-\] the $^3P_0$ elastic-scattering phase shifts calculated with the AV8$^{\prime}$ (left) and MN (right) potentials. The truncated $3N$+$N$ (dashed line) and $2N$+$2N$ (dotted line) calculations are shown together with the FULL result (solid line). The $t$+$p$ phase shift of the $3N$+$N$ calculation is similar with both AV8$^{\prime}$ and MN potentials, while the $d$+$d$ phase shift of the $2N$+$2N$ calculation behaves quite differently between the two potentials: the $d$+$d$ phase shift is weakly attractive with AV8$^{\prime}$ but is very strongly attractive with MN. No typical resonance behavior shows up below the $d$+$d$ threshold, which is in contradiction to experiment. In the FULL model that combines both 3$N$+$N$ and 2$N$+2$N$ configurations, however, the two potentials predict quite different phase shifts especially in the $t$+$p$ channel. The $t$+$p$ phase shift with AV8$^{\prime}$ becomes so attractive that it crosses $\pi/2$, indicating a resonance at about 1 MeV above the $t$+$p$ threshold. The $d$+$d$ phase shift changes sign from attractive to repulsive. The result based on the AV8$^{\prime}$ potential is thus consistent with experiment. Furthermore, we reproduce the flat structure of the $^3P_0$ phase shift around several MeV above the $t$+$p$ threshold which was discussed as the coupling to the $h$+$n$ channel [@hofmann08]. On the other hand, the MN potential changes the $t$+$p$ phase shift only mildly and produces no sharp resonance behavior. The $d$+$d$ phase shift changes drastically to the repulsive side.
As seen in the above figure, the sharp 0$^-$ state appears provided a full model space with a realistic potential is employed. The mechanism to produce this resonance is unambiguously attributed to the tensor force as discussed in Ref. [@DGVR] for the realistic interaction G3RS [@tamagaki68]. According to it, the $0^-$ state consists of only two components, $L=S=1$ (95.5%) and $L=S=2$ (4.5%), ignoring a tiny component with $L=S=0$. The $L=S=2$ component arises from the coupling of the incoming $P$-wave with the $D$-states contained in the $3N$ and $d$ clusters. All the pieces of the Hamiltonian but the tensor force have no coupling matrix element between the two components. The uncoupled Hamiltonian thus gives a too high energy to accommodate a resonance. The tensor force, however, couples the two components very strongly, bringing down its energy to a right position.
The second lowest negative parity state has spin-parity 2$^-$. The physical channel for this state is only $^3P_2$ as seen in Table \[chan0\]. Fig. \[fig:2-\] compares the $^3P_2$ elastic-scattering phase shifts in a manner similar to Fig. \[fig:0-\]. The phase shift obtained with the MN potential is almost the same as the $^3P_0$ phase shift, which is consistent with the previous result [@horiuchi08] that the energies of the negative parity states calculated with the MN potential are found to be degenerate. In the case of the AV8$^{\prime}$ potential, the $^3P_2$ phase shifts grows significantly in the FULL calculation, indicating a resonant behavior. The coupling effect between the $3N$+$N$ and $2N$+$2N$ channels is however much less compared to the $0^-$ state. This is because the incoming $P$-wave coupled to the $D$-states in the clusters gives rise to several $L$ values to produce the $2^-$ state and therefore the tensor coupling does not concentrate sufficiently to produce a sharp resonance.
Fig. \[fig:1-\] displays the $^3P_1$ and $^1P_1$ elastic-scattering phase shifts calculated with the AV8$^{\prime}$ (left) and MN (right) potentials. Note that no physical $d$+$d$ channel exists in the case of the $^1P_1$ state. Because both FULL and $3N$+$N$ calculations give almost the same phase shifts, only the FULL result is shown in the figure. The $^3P_1$ phase shift calculated with the MN potential is again almost the same as those of the $^3P_0$ and $^3P_2$ cases, supporting that the three negative parity states become almost degenerate. The $^3P_1$ elastic-scattering phase shift calculated with the AV8$^{\prime}$ potential is qualitatively similar to that of $^3P_2$. The attractive nature of the $t$+$p$ phase shift becomes further weaker, and to identify a resonance appears to be very hard. Even though it is possible in some way, its width would be a few MeV, which is not in contradiction to experiment. The $^1P_1$ phase shifts are very small in both AV8$^{\prime}$ and MN cases.
For the negative parity states, the FULL model with the AV8$^{\prime}$ potential gives results that are consistent with both experiment and the theoretical calculation of Ref. [@horiuchi08]. We have pointed out that the phase shift behavior reveals the importance of the tensor force particularly in the case of $0^-$. Its effect is often masked however by the coupling between the $D$ states in the clusters and the incoming partial wave.
In this subsection, we investigate the phase shifts of the negative parity states which have dominant $T=0$ components. In Fig. \[fig.13\], we represent three experimental negative parity $T=0$ energies (left). The states are observed at $-7.29$ ($0^-$), $-6.46$ ($2^-$) and $-4.05$ ($1^-$)MeV below the four-nucleon threshold [@tilley92]. The former two are located below the $d$+$d$ threshold and their widths are 0.84 and 2.01MeV, respectively, whereas the last one is above the $d$+$d$ threshold and its width is fairly broad (6.1 MeV). Here, we calculate these energies as $-7.57$ ($0^-$), $-6.82$ ($2^-$) and $-5.95$ ($1^-$)MeV, which are approximated by the half-value position from the maximum phase shifts. The present calculation is not projected out to $T=0$, but the dominant configurations of $t$+$p$, $h$+$n$ and $d$+$d$ elastic scattering are $T=0$. Our calculated energies with AV8$^{\prime}$ reproduce the ordering of $0^-$, $2^-$ and $1^-$ (middle in Fig. \[fig.13\]). The splitting between the two lower states $0^-$ and $2-$ is reproduced, but the experimental $1^-$ energy is higher than the calculation. However, the determination of the energy for such a high energy state with large decay width (6.1 MeV) is very difficult from both experimental and theoretical side, and it usually has a large ambiguity.
This type of analysis was done by Horiuchi and Suzuki, who applied the correlated Gaussian basis with two global vectors to study the energy spectrum of $^4$He [@horiuchi08]. Because the results of Ref. [@horiuchi08] are based on approximate solutions that impose no proper resonance boundary condition, it is interesting to see how the tensor force changes the phase shifts in the negative parity states as shown in this subsection. These authors also found that the negative parity states with $T=0$ turn out to be almost degenerate when the MN potential that contains no tensor force is employed. In the present calculation, three states ($0^-$, $1^-$, $2^-$) completely degenerate at the same energy, $E=-6.64$ MeV (right), and the same phase shift pattern (solid lines in Figs. \[fig:0-\], \[fig:2-\], \[fig:1-\]). Thus we can expect to see a clear evidence for the tensor force in the scattering involving the negative parity states.
Summary and conclusion
======================
We have investigated the distortion of clusters appearing in the low-energy $d$+$d$ and $t$+$p$ elastic scattering using a microscopic cluster model with the triple global vector method. We showed that the tensor interaction changes the phase shifts very much by comparing a realistic interaction and an effective interaction. In the present $ab$-$initio$ type cluster model, the description of the cluster wave functions is extended from a simple (0$s$) harmonic-oscillator shell model to a few-body model. To compare distortion effects of the clusters with realistic and effective interactions, we employed the AV8$^{\prime}$ potential as a realistic interaction and the MN potential as an effective interaction.
For the realistic interaction, the calculated $^1S_0$ phase shift shows that the $t$+$p$ and $h$+$n$ channels strongly couple with the $d$+$d$ channel. These channels are coupled because of the tensor interaction. On the contrary, the coupling of these $3N$+$N$ channels plays a relatively minor role for the case of the effective interaction because of the absence of tensor term. In other words, the $3N$+$N$ channels strongly affect the $d$+$d$ elastic phase shift with the realistic interaction, but not with the effective interaction.
For the 2$^+$ phase shifts, there is a $^1D_2$ component in all physical channels ($d$+$d$, $t$+$p$ and $h$+$n$). The coupling of the $2N$+$2N$ and $3N$+$N$ channels in $^5D_2$ is weaker than in $^1S_2$ because of a weaker tensor coupling as discussed in section \[results\], and the calculated phase shifts are very similar for the realistic and effective potentials. For other positive parity cases, the phase shift behavior of the realistic and effective potentials are very similar, and the coupling between the $2N$+$2N$ and $3N$+$N$ channels can be neglected or is very small. Furthermore, the tensor interaction makes the energy splitting of the $0^-$, $2^-$ and $1^-$ negative parity states of $^4$He consistent with experiment. No such splitting is however reproduced with the effective interaction.
We believe that the physical picture obtained in the large model space with the realistic interaction should be close to the real physical situation. It is needless to say that $ab$-$initio$ reaction calculations are very important to understand the underlying reaction dynamics involving continuum states. Simpler calculations using effective interactions in the same framework, as carried out in the present paper, are also meaningful because we can understand more clearly the effect of the tensor force by comparing both calculations. The reaction calculations with the microscopic cluster model, whose model space and interactions are restricted, have been successfully applied to many heavier nuclei. Therefore, it is instructive to see the difference from the realistic interaction by employing a simple conventional effective interaction as MN in the few-nucleon systems.
It will be quite interesting to see the importance of the tensor force in reaction observables of four nucleons. As a direct application of the present study the radiative capture reaction $d(d,\gamma)^4$He at energies of astrophysical interest is of prime importance. It is expected to take place predominantly via $E2$ transitions [@santos85; @langanke87; @wachter88; @arriaga91; @carlson98]. As is seen from Table \[chan0\], the two deuterons can approach each other in the $S$-wave only when $J^{\pi}$ is either 0$^+$ ($^1S_0$) or 2$^+$ ($^5S_2$). The former case is excluded because a radiative capture reaction of $0^+ \to 0^+$ is forbidden in the lowest-order electromagnetic interaction, and hence the $E2$ transition should be predominant. If there were no tensor force present, the radiative capture would be suppressed near $E=0$ because neither $d$ nor $^4$He would have a $D$-wave component in contradiction with the flat behavior of the astrophysical $S$-factor [@angulo99]. The tensor force strongly changes this story because it can couple $S$- and $D$-waves, bringing a significant amount of $D$-state probability in both $^4$He and $d$. Details of this analysis will be reported elsewhere.\
\
[**Acknowledgment**]{}\
We thank Dr. R. Kamouni for helpful discussions based on his PhD thesis (in French). This work presents research results of Bilateral Joint Research Projects of the JSPS (Japan) and the FNRS (Belgium). Y. S. is supported by a Grant-in-Aid for Scientific Research (No. 21540261). This text presents research results of the Belgian program P6/23 on interuniversity attraction poles initiated by the Belgian-state Federal Services for Scientific, Technical and Cultural Affairs (FSTC). D. B. and P. D. also acknowledge travel support of the Fonds de la Recherche Scientifique Collective (FRSC). The part of computational calculations were carried out in T2K-Tsukuba.
Definitions of recoupling coefficients {#app.A}
======================================
We define an auxiliary coefficient $Z$ that appears in the coupling $$[[Y_a({{\mbox{\boldmath $e$}}}_1)[Y_b({{\mbox{\boldmath $e$}}}_1)Y_b({{\mbox{\boldmath $e$}}}_2)]_0]_a
\to Z(ab)[Y_{a+b}({{\mbox{\boldmath $e$}}}_1)Y_b({{\mbox{\boldmath $e$}}}_2)]_a.$$ By introducing a coefficient $$C(a b, c)=\sqrt{\frac{(2a+1)(2b+1)}
{4\pi(2c+1)}}\langle a\ 0\ b\ 0 \vert c \ 0\rangle \\$$ for the coupling $[Y_a({{\mbox{\boldmath $e$}}}_1)Y_b({{\mbox{\boldmath $e$}}}_1)]_{c}=C(ab, c)Y_{c}({{\mbox{\boldmath $e$}}}_1)$, we can express $Z$ as $$Z(ab)=\sqrt{\frac{2(a+b)+1}{(2a+1)(2b+1)}}C(ab,a+b)=
\frac{1}{\sqrt{4\pi}}\langle a\ 0\ b\ 0 \vert a+b \ 0\rangle.
\label{def.z}$$ Note that $C(ab,c)$ vanishes unless $a+b+c$ is even.
The coefficients that appear in Sect. \[sect.4\] are given as follows: $$\begin{aligned}
&&{\hspace{-5mm}}X(a\ b\ c)=Z(ab)Z(ac)C(bc, b+c)U(a\!+\!c\ c\ a\!+\!b\ b;\ a\ b\!+\!c),\\
&&{\hspace{-5mm}}R_3(a\ b\ c)=Z(ab)Z(a\!+\!b \ c),\\
&&{\hspace{-5mm}}W(a\ b\ c\ q\ Q,\ a'\ b'\ c'\ q'\ Q',\ \ell \ \ell')\nonumber\\
&&{\hspace{-5mm}}=
\left[\begin{array}{ccc}
q &c & Q \\
q'&c' & Q'\\
\ell'&c\!+\!c' &\ell\\
\end{array}\right]
\left[\begin{array}{ccc}
a &b & q\\
a' &b' & q'\\
a\!+\!a'&b\!+\!b' &\ell'\\
\end{array}\right]C(aa', a\!+\!a')C(bb', b\!+\!b')C(cc', c\!+\!c').\end{aligned}$$
Matrix elements for various operators
=====================================
The purpose of this appendix is to collect formulas for various matrix elements. The main procedure to derive the formulas is sketched in Sect. \[sect.4\]. More details for the case of two global vectors are given in Ref. [@DGVR].
\[app.B\]
Kinetic energy
--------------
Let ${{\mbox{\boldmath $\pi$}}}_j$ denote the momentum operator conjugate to ${{\mbox{\boldmath $x$}}}_j$, ${{\mbox{\boldmath $\pi$}}}_j=-i\hbar \frac{\partial}{\partial {{\mbox{\boldmath $x$}}}_j}$. The total kinetic energy operator for the $N$-nucleon system with its center of mass kinetic energy being subtracted takes the form $$\sum_{i=1}^N \frac{{{\mbox{\boldmath $p$}}}_i^2}{2m}-\frac{{{\mbox{\boldmath $\pi$}}}_N^2}{2Nm}
=\frac{1}{2}\widetilde{{\mbox{\boldmath $\pi$}}}\Lambda{{\mbox{\boldmath $\pi$}}},$$ where ${{\mbox{\boldmath $\pi$}}}_N=\sum_{i=1}^N{{\mbox{\boldmath $p$}}}_i$ is the total momentum, $\widetilde{{\mbox{\boldmath $\pi$}}}=({{\mbox{\boldmath $\pi$}}}_1,{{\mbox{\boldmath $\pi$}}}_2,\ldots,{{\mbox{\boldmath $\pi$}}}_{N-1})$, and $\Lambda$ is an $(N-1)\times(N-1)$ symmetric mass matrix. Defining $N-1$-dimensional column vectors $\Gamma_i$ as $$\begin{aligned}
& &\Gamma_i=A'B^{-1}u_i\ \ \ \ \ \ (i=1,2,3),\nonumber \\
& &\Gamma_i=-AB^{-1}u_i\ \ \ \ \ (i=4,5,6)
\label{def.gamma}\end{aligned}$$ and an $(N-1)\times(N-1)$ matrix $Q$ $$Q_{ij}=2\widetilde{\Gamma_i}\Lambda \Gamma_j,$$ we can calculate the matrix element for the kinetic energy through the overlap matrix element $$\begin{aligned}
\langle F'\vert \frac{1}{2}\widetilde{{\mbox{\boldmath $\pi$}}}\Lambda{{\mbox{\boldmath $\pi$}}} \vert F \rangle
= \frac{\hbar^2}{2}\left(R-\sum_{i<j}Q_{ij}\frac{\partial}{\partial \rho_{ij}}\right)
\left<F'\vert F \right>,\end{aligned}$$ where $$R=3{\rm Tr}(B^{-1}A'\Lambda A).$$ The $\rho_{ij}$ values are defined in Eq. (\[def.rho\]).
$\delta$-function
-----------------
A two-body interaction $V({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j)$ can be expressed as $$V({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j)=\int d{{\mbox{\boldmath $r$}}} V({{\mbox{\boldmath $r$}}})\ \delta({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j-{{\mbox{\boldmath $r$}}}).$$ Once the matrix element of $\delta({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j-{{\mbox{\boldmath $r$}}})$ is obtained, the matrix element of the interaction is calculated by integrating over ${{\mbox{\boldmath $r$}}}$ the $\delta$-function matrix element weighted with the form factor $V({{\mbox{\boldmath $r$}}})$. Similarly, for a one-body operator $$D({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $x$}}}_N)=\int d{{\mbox{\boldmath $r$}}} D({{\mbox{\boldmath $r$}}})\ \delta({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $x$}}}_N-{{\mbox{\boldmath $r$}}}),$$ its matrix element can be obtained from that of the $\delta$-function. Because both ${{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j$ and ${{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $x$}}}_N$ can be expressed in terms of a linear combination of the relative coordinate ${{\mbox{\boldmath $x$}}}_i$, it is enough to calculate the matrix element of $\delta(\widetilde{w}{{\mbox{\boldmath $x$}}}-{{\mbox{\boldmath $r$}}})$, where $\widetilde{w}=(w_1,w_2,\ldots,
w_{N-1})$ is a combination constant to express ${{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j$ or ${{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $x$}}}_N$.
The matrix element of the $\delta$-function is given by $$\begin{aligned}
& &{\hspace{-1cm}}\left<F'\vert \delta(\widetilde{w}{{\mbox{\boldmath $x$}}}-{{\mbox{\boldmath $r$}}}) \vert F \right>
\nonumber \\
&=&\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}
\left(\prod_{i=1}^6 B_{L_i}\right)
\frac{(-1)^{L_1+L_2+L_3}\sqrt{2L+1}}{\sqrt{2L'+1}}
\ \left(\frac{c}{2\pi}\right)^{3/2}{\rm e}^{-\frac{1}{2}cr^2} \nonumber\\
&\times & \sum_{\kappa \mu} \langle LM\kappa \mu| L'M'\rangle
Y_{\kappa \mu}^*(\widehat{{\mbox{\boldmath $r$}}})
\sum_{p_i} \left(\prod_{i=1}^6 (-c\gamma_i r)^{p_{i}}
\frac{ \sqrt{2p_{i}+1}}{B_{p_{i}}} \right)
\nonumber \\
&\times& \sum_{\ell_{12}\ell_{45}\ell \ell' \overline{L}_{12}
\overline{L}_{45} \overline{L}}
\frac{(-1)^{\ell+\ell'}}{\sqrt{(2\ell+1)(2\overline{L}+1)}}U(L\overline{L}\kappa
\ell';\ell L')\
\overline{O}(p_i ; \ell_{12} \ell_{45} \ell \ell' \kappa )
\nonumber \\
&\times&
W(p_1 p_2 p_3 \ell_{12} \ell, L_1\!-\!p_1\ L_2\!-\!p_2\ L_3\!-\!p_3\
\overline{L}_{12}\overline{L}, L L_{12})\nonumber \\
&\times&
W(p_4 p_5 p_6 \ell_{45} \ell', L_4\!-\!p_4\ L_5\!-\!p_5\ L_6\!-\!p_6\
\overline{L}_{45}\overline{L}, L' L_{45})\nonumber \\
&\times&
\sum_{n_{ij}}\left(\prod_{i<j}^6 (-\overline{\rho}_{ij})^{n_{ij}}
\frac{ \sqrt{2n_{ij}+1}}{B_{n_{ij}}}\right)
\nonumber \\
&\times&
O(n_{ij}; L_1\!-\!p_1\ L_2\!-\!p_2\ L_3\!-\!p_3\ L_4\!-\!p_4,
L_5\!-\!p_5\ L_6\!-\!p_6,\overline{L}_{12}\overline{L}_{45}\overline{L}),
\label{me.del}\end{aligned}$$ with $$\begin{aligned}
& &c=(\widetilde{w}B^{-1}w)^{-1},\ \ \ \ \ \gamma_i=\widetilde{w}B^{-1}u_i,
\ \ \ \ \ \overline{\rho}_{ij}=\rho_{ij}-c\gamma_i\gamma_j.\end{aligned}$$ The summation over non-negative integers $n_{ij}$ and $p_i$ is restricted by the following equations $$\begin{aligned}
&&n_{12}+n_{13}+n_{14}+n_{15}+n_{16}+p_1=L_1, \nonumber\\
&&n_{12}+n_{23}+n_{24}+n_{25}+n_{26}+p_2=L_2, \nonumber\\
&&n_{13}+n_{23}+n_{34}+n_{35}+n_{36}+p_3=L_3, \nonumber\\
&&n_{14}+n_{24}+n_{34}+n_{45}+n_{46}+p_4=L_4, \nonumber\\
&&n_{15}+n_{25}+n_{35}+n_{45}+n_{56}+p_5=L_5, \nonumber\\
&&n_{16}+n_{26}+n_{36}+n_{46}+n_{56}+p_6=L_6. \end{aligned}$$ Here $\overline{O}(p_i ; \ell_{12} \ell_{45} \ell \ell' \kappa )$ is defined as a coefficient that appears in the coupling of a product of six terms $$\begin{aligned}
& &{\hspace{-1cm}}\prod_{i=1}^6 [Y_{p_i}({{\mbox{\boldmath $e$}}}_i)Y_{p_i}(\widehat{{\mbox{\boldmath $r$}}})]_{00}
= \sum_{\ell_{12}\ell_{45}\ell \ell' \kappa}\overline{O}(p_i ; \ell_{12} \ell_{45} \ell \ell' \kappa)
\nonumber \\
&\times&
[[[[Y_{p_1}({{\mbox{\boldmath $e$}}}_1)Y_{p_2}({{\mbox{\boldmath $e$}}}_2)]_{\ell_{12}}Y_{p_3}({{\mbox{\boldmath $e$}}}_3)]_{\ell}\
[[Y_{p_4}({{\mbox{\boldmath $e$}}}_4)Y_{p_5}({{\mbox{\boldmath $e$}}}_5)]_{\ell_{45}}Y_{p_6}({{\mbox{\boldmath $e$}}}_6)]_{\ell'}]_{\kappa}\
Y_{\kappa}(\widehat{{\mbox{\boldmath $r$}}})]_{00},\end{aligned}$$ and it is given by $$\begin{aligned}
& &\overline{O}(p_i ; \ell_{12} \ell_{45} \ell \ell' \kappa )
\nonumber \\
& &\quad=\sqrt{\frac{2\kappa+1}{\prod_{i=1}^6(2p_i+1)}}
C(p_1 p_2,\ell_{12})C(\ell_{12} p_3,\ell)C(p_4 p_5,\ell_{45})C(\ell_{45} p_6,\ell')
C(\ell \ell', \kappa).\end{aligned}$$
The ${{\mbox{\boldmath $r$}}}$-dependence of the matrix element (\[me.del\]) is $${\rm e}^{-\frac{1}{2}cr^2} r^{p_1+p_2+p_3+p_4+p_5+p_6} Y_{\kappa \mu}^*(\widehat{{\mbox{\boldmath $r$}}}).$$ For a central interaction, $V({{\mbox{\boldmath $r$}}})$ is a scalar function, and the sum over $\kappa$ in Eq. (\[me.del\]) is limited to 0. For a tensor interaction, the angular dependence of $V({{\mbox{\boldmath $r$}}})$ is proportional to $Y_2(\hat{{\mbox{\boldmath $r$}}})$, and $\kappa$ is limited to 2. The electric multipole operator is a special case of one-body operator, so that one can make use of the formula (\[me.del\]) to calculate its matrix element. More explicitly, we give the matrix element of $V(|\widetilde{w}{{\mbox{\boldmath $x$}}}|)
Y_{\kappa \mu}(\widehat{\widetilde{w}{{\mbox{\boldmath $x$}}}})$ that includes all the cases mentioned above: $$\begin{aligned}
& &{\hspace{-1cm}}\left<F'\vert V(|\widetilde{w}{{\mbox{\boldmath $x$}}}|)
Y_{\kappa \mu}(\widehat{\widetilde{w}{{\mbox{\boldmath $x$}}}}) \vert F \right>
\nonumber \\
&=&\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}
\left(\prod_{i=1}^6 B_{L_i}\right)
\frac{(-1)^{L_1+L_2+L_3}\sqrt{2L+1}}{\sqrt{2L'+1}} \nonumber\\
&\times & \langle LM\kappa \mu| L'M'\rangle
\sum_{p_i} \left(\prod_{i=1}^6 (-\gamma_i )^{p_{i}}
\frac{ \sqrt{2p_{i}+1}}{B_{p_{i}}} \right){\cal I}^{(2)}_{p_1+p_2+p_3+p_4+p_5+p_6}(c)
\nonumber \\
&\times& \sum_{\ell_{12}\ell_{45}\ell \ell' \overline{L}_{12}
\overline{L}_{45} \overline{L}}
\frac{(-1)^{\ell+\ell'}}{\sqrt{(2\ell+1)(2\overline{L}+1)}}U(L\overline{L}\kappa
\ell';\ell L')\
\overline{O}(p_i ; \ell_{12} \ell_{45} \ell \ell' \kappa )
\nonumber \\
&\times&
W(p_1 p_2 p_3 \ell_{12} \ell, L_1\!-\!p_1\ L_2\!-\!p_2\ L_3\!-\!p_3\
\overline{L}_{12}\overline{L}, L L_{12})\nonumber \\
&\times&
W(p_4 p_5 p_6 \ell_{45} \ell', L_4\!-\!p_4\ L_5\!-\!p_5\ L_6\!-\!p_6\
\overline{L}_{45}\overline{L}, L' L_{45})\nonumber \\
&\times&
\sum_{n_{ij}}\left(\prod_{i<j}^6 (-\overline{\rho}_{ij})^{n_{ij}}
\frac{ \sqrt{2n_{ij}+1}}{B_{n_{ij}}}\right)
\nonumber \\
&\times&
O(n_{ij}; L_1\!-\!p_1\ L_2\!-\!p_2\ L_3\!-\!p_3\ L_4\!-\!p_4,
L_5\!-\!p_5\ L_6\!-\!p_6,\overline{L}_{12}\overline{L}_{45}\overline{L}),\end{aligned}$$ with the integral of the potential form factor $$\begin{aligned}
{\cal I}^{(m)}_n(c)=\left(\frac{c}{2\pi}\right)^{3/2} c^n\int_0^{\infty} dr \,r^{n+m} V(r) {\rm e}^{-\frac{1}{2}cr^2}. \end{aligned}$$ In case $V(r)$ takes the form of $r^q {\rm e}^{-\rho r^2-\rho'r}$ $(q
\geq -m)$, the integral ${\cal I}^{(m)}_n(c)$ can be obtained analytically, giving a closed form for the matrix element.
It should be noted that the matrix element for a special class of a three-body force can be evaluated with ease. For example, if the radial part of the three-body force has a form $$V_{TNF}=\exp(-\rho_1({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j)^2-
\rho_2({{\mbox{\boldmath $r$}}}_j-{{\mbox{\boldmath $r$}}}_k)^2
-\rho_3({{\mbox{\boldmath $r$}}}_k-{{\mbox{\boldmath $r$}}}_i)^2),$$ the exponent can be rewritten as $-\widetilde{{\mbox{\boldmath $x$}}}\Omega {{\mbox{\boldmath $x$}}}$ with an $(N-1)\times(N-1)$ symmetric matrix $\Omega=\rho_1 w_{ij}\widetilde{w_{ij}}+\rho_2
w_{jk}\widetilde{w_{jk}}+\rho_3 w_{ki}\widetilde{w_{ki}}$, where $w_{ij}$, $w_{jk}$ and $w_{ki}$ are defined by ${{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j=\widetilde{w_{ij}}{{\mbox{\boldmath $x$}}}$, ${{\mbox{\boldmath $r$}}}_j-{{\mbox{\boldmath $r$}}}_k=\widetilde{w_{jk}}{{\mbox{\boldmath $x$}}}$ and ${{\mbox{\boldmath $r$}}}_k-{{\mbox{\boldmath $r$}}}_i=\widetilde{w_{ki}}{{\mbox{\boldmath $x$}}}$. Thus the matrix element reduces to that of the overlap with $A$ being replaced with $A+2\Omega$ $$\langle F'\vert V_{TNF} \vert F \rangle
= \langle F'\vert F_{L_1 L_2(L_{12}) L_3 LM}(u_1, u_2, u_3,
A+2\Omega,{{\mbox{\boldmath $x$}}}) \rangle.$$
Spin-orbit potential
--------------------
The spatial form of a spin-orbit interaction reads $$V(|{{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j|)(({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j)\times \frac{1}{2}({{\mbox{\boldmath $p$}}}_i-{{\mbox{\boldmath $p$}}}_j))_{\mu},$$ where $({{\mbox{\boldmath $a$}}}\times{{\mbox{\boldmath $b$}}})_{\mu}$ $(\mu=0, \pm 1)$ stands for the $\mu$th component of a vector product of ${{\mbox{\boldmath $a$}}}$ and ${{\mbox{\boldmath $b$}}}$. As in the $\delta$-function matrix element, the spin-orbit potential is written as $$V(|\widetilde{w}{{\mbox{\boldmath $x$}}}|) (\widetilde{w}{{\mbox{\boldmath $x$}}}\times \widetilde{\zeta}{{\mbox{\boldmath $\pi$}}})_{\mu},$$ where $\frac{1}{2}({{\mbox{\boldmath $p$}}}_i-{{\mbox{\boldmath $p$}}}_j)$ is expressed in terms of the momentum operators ${{\mbox{\boldmath $\pi$}}}$, $\widetilde{\zeta}{{\mbox{\boldmath $\pi$}}}=\sum_{i=1}^{N-1} \zeta_i {{\mbox{\boldmath $\pi$}}}_i$.
The matrix element of the spin-orbit potential is given by $$\begin{aligned}
& &{\hspace{-1cm}}\left<F'\vert V(|\widetilde{w}{{\mbox{\boldmath $x$}}}|) (\widetilde{w}{{\mbox{\boldmath $x$}}}\times \widetilde{\zeta}{{\mbox{\boldmath $\pi$}}})_{\mu} \vert F \right>
\nonumber \\
&=&\frac{4\pi \sqrt{2} \hbar}{3}\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}
\left(\prod_{i=1}^6 B_{L_i}\right)
\frac{(-1)^{L_1+L_2+L_3}\sqrt{2L+1}}{\sqrt{2L'+1}}
\nonumber\\
&\times & \langle LM 1 \mu| L'M'\rangle
\sum_{p_i} \left(\prod_{i=1}^6 (-\gamma_i )^{p_{i}}
\frac{ \sqrt{2p_{i}+1}}{B_{p_{i}}} \right){\cal I}^{(3)}_{p_1+p_2+p_3+p_4+p_5+p_6}(c)
\nonumber \\
&\times& \sum_{\ell_{12}\ell_{45}\ell \ell' \overline{\ell}_{12}
\overline{\ell}_{45} \overline{\ell}\ \overline{\ell}'
\overline{L}_{12} \overline{L}_{45} \overline{L} }
\frac{(-1)^{\overline{\ell}+\overline{\ell}'}}
{\sqrt{(2\overline{\ell}+1)(2\overline{L}+1)}}
U(L\overline{L}1 \overline{\ell}'; \overline{\ell} L')\
\overline{O}(p_i ; \ell_{12} \ell_{45} \ell \ell' 1)
\nonumber \\
&\times&
\sum_{k=1}^6 (\widetilde{\zeta}\Gamma)_k \ T_k(p_i,\ell_{12}\ell_{45}\ell \ell', \overline{\ell}_{12} \overline{\ell}_{45} \overline{\ell}\ \overline{\ell}')
\sum_{n_{ij}}\left(\prod_{i<j}^6 (-\overline{\rho}_{ij})^{n_{ij}}
\frac{ \sqrt{2n_{ij}+1}}{B_{n_{ij}}}\right)
\nonumber \\
&\times&
O(n_{ij}; L_1\!-\!p_1^k\ L_2\!-\!p_2^k\ L_3\!-\!p_3^k\ L_4\!-\!p_4^k\
L_5\!-\!p_5^k\ L_6\!-\!p_6^k,\overline{L}_{12}\overline{L}_{45}\overline{L}),\end{aligned}$$ where ${p}_i^k$ $(k=1,2,\ldots,6)$ is $${p}_i^k=p_i+\delta_{ik},$$ and where the non-negative integers $n_{ij}$ and $p_i$ are constrained to satisfy the equations $$\begin{aligned}
&&n_{12}+n_{13}+n_{14}+n_{15}+n_{16}+p_1^k=L_1, \nonumber\\
&&n_{12}+n_{23}+n_{24}+n_{25}+n_{26}+p_2^k=L_2, \nonumber\\
&&n_{13}+n_{23}+n_{34}+n_{35}+n_{36}+p_3^k=L_3, \nonumber\\
&&n_{14}+n_{24}+n_{34}+n_{45}+n_{46}+p_4^k=L_4, \nonumber\\
&&n_{15}+n_{25}+n_{35}+n_{45}+n_{56}+p_5^k=L_5, \nonumber\\
&&n_{16}+n_{26}+n_{36}+n_{46}+n_{56}+p_6^k=L_6. \end{aligned}$$ The symbol $(\widetilde{\zeta}\Gamma)_k$ stands for the factor $$(\widetilde{\zeta}\Gamma)_k = \sum_{i=1}^6 \zeta_i (\Gamma_k)_i,$$ where $(\Gamma_k)_i$ is the $i$th element of the column vector $\Gamma_k$ defined in Eq. (\[def.gamma\]). The coefficient $T_k$ appears in the coupling $$\begin{aligned}
& &[Y_1({{\mbox{\boldmath $e$}}}_k)\ [[[Y_{p_1}({{\mbox{\boldmath $e$}}}_1)Y_{p_2}({{\mbox{\boldmath $e$}}}_2)]_{\ell_{12}}Y_{p_3}({{\mbox{\boldmath $e$}}}_3)]_{\ell}\
[[Y_{p_4}({{\mbox{\boldmath $e$}}}_4)Y_{p_5}({{\mbox{\boldmath $e$}}}_5)]_{\ell_{45}}Y_{p_6}({{\mbox{\boldmath $e$}}}_6)]_{\ell'}]_{1}]_{1\mu}
\nonumber \\
& &\to \sum_{\overline{\ell}_{12} \overline{\ell}_{45} \overline{\ell}\ \overline{\ell}'}T_k(p_i,\ell_{12}\ell_{45}\ell \ell', \overline{\ell}_{12} \overline{\ell}_{45} \overline{\ell}\ \overline{\ell}')
\nonumber \\
& &\qquad \times
[[[Y_{{p}_1^k}({{\mbox{\boldmath $e$}}}_1)Y_{{p}_2^k}({{\mbox{\boldmath $e$}}}_2)]_{\overline{\ell}_{12}}Y_{{p}_3^k}({{\mbox{\boldmath $e$}}}_3)]_{\overline{\ell}}\
[[Y_{{p}_4^k}({{\mbox{\boldmath $e$}}}_4)Y_{{p}_5^k}({{\mbox{\boldmath $e$}}}_5)]_{\overline{\ell}_{45}}Y_{{p}_6^k}({{\mbox{\boldmath $e$}}}_6)]_{\overline{\ell}'}]_{1\mu}.\end{aligned}$$ The coefficients $T_k(p_i,\ell_{12}\ell_{45}\ell \ell', \overline{\ell}_{12} \overline{\ell}_{45} \overline{\ell}\ \overline{\ell}')$ are given below: $$\begin{aligned}
& &T_1=U(1\ell 1 \ell'; \overline{\ell}1)
U(1 \ell_{12} \overline{\ell}p_3; \overline{\ell}_{12} \ell)
U(1 p_1 \overline{\ell}_{12} p_2; p_1\!+\!1\ \ell_{12})
C(1 p_1; p_1\!+\!1)
\nonumber \\
& &T_2=-(-1)^{\ell_{12}+\overline{\ell}_{12}}
U(1\ell 1 \ell'; \overline{\ell}1)
U(1\ell_{12}\overline{\ell}p_3; \overline{\ell}_{12} \ell)
U(1 p_2 \overline{\ell}_{12} p_1; p_2\!+\!1\ \ell_{12})
C(1 p_2; p_2\!+\!1)
\nonumber \\
& &T_3=-(-1)^{\ell+\overline{\ell}}
U(1\ell 1 \ell'; \overline{\ell}1)
U(1p_3 \overline{\ell} \ell_{12} ; p_3\!+\!1 \ell)
C(1 p_3; p_3\!+\!1)
\nonumber \\
& &T_4=(-1)^{\ell'+\overline{\ell}'}
U(1\ell' 1 \ell; \overline{\ell}'1)
U(1 \ell_{45} \overline{\ell}' p_4; \overline{\ell}_{45} \ell')
U(1 p_4 \overline{\ell}_{45} p_5; p_4\!+\!1 \ \ell_{45})
C(1 p_4; p_4\!+\!1)
\nonumber \\
& &T_5=-(-1)^{\ell'+\overline{\ell}'+\ell_{45}+\overline{\ell}_{45}}
U(1\ell' 1 \ell; \overline{\ell}'1)
U(1\ell_{45} \overline{\ell}' p_6; \overline{\ell}_{45}\ell')
U(1 p_5 \overline{\ell}_{45} p_4; p_5\!+\!1\ \ell_{45})
C(1 p_5; p_5\!+\!1)
\nonumber \\
& &T_6=-U(1\ell' 1 \ell; \overline{\ell}'1)
U(1 p_6 \overline{\ell}' \ell_{45} ; p_6\!+\!1\ \ell')
C(1 p_6; p_6\!+\!1).\end{aligned}$$
Momentum representation of correlated Gaussian basis {#app.C}
====================================================
The Fourier transform of the correlated Gaussian function (\[cgtgv\]) defines the corresponding basis function in momentum space. The momentum space function is useful to evaluate those matrix elements which depend on the momentum operator [@DGVR]. Suppose that we want to evaluate the matrix element of a two-body operator $V({{\mbox{\boldmath $p$}}}_i-{{\mbox{\boldmath $p$}}}_j)$ or a one-body operator $D({{\mbox{\boldmath $p$}}}_i-\frac{1}{N}{{\mbox{\boldmath $\pi$}}}_N)$. Obviously evaluating the matrix element can be done more easily in momentum space. For this purpose we need to obtain the Fourier transform of the coordinate space function. A great advantage in the correlated Gaussian function $F$ is that its Fourier transform is a linear combination of the correlated Gaussian functions in the momentum space. Thus by expressing ${{\mbox{\boldmath $p$}}}_i-{{\mbox{\boldmath $p$}}}_j$ or ${{\mbox{\boldmath $p$}}}_i-\frac{1}{N}{{\mbox{\boldmath $\pi$}}}_N$ as $\widetilde{\zeta}{{\mbox{\boldmath $\pi$}}}$, we can calculate the matrix element of the momentum-dependent operators in exactly the same way as in the coordinate space.
As in the case with two global vectors [@DGVR], the transformation from the coordinate to momentum space is achieved by a function $$\Phi({{\mbox{\boldmath $k$}}},{{\mbox{\boldmath $x$}}})=\frac{1}{(2\pi)^{\frac{3}{2}(N-1)}}\,
\exp\,(i\tilde{{\mbox{\boldmath $k$}}}{{\mbox{\boldmath $x$}}}),$$ where ${{\mbox{\boldmath $k$}}}$ is an $(N\!-\!1)$-dimensional column vector whose $i$th element is ${{\mbox{\boldmath $k$}}}_i$. With a straightforward integration together with the recoupling of angular momenta, we can show that $$\begin{aligned}
& &\langle \Phi({{\mbox{\boldmath $k$}}},{{\mbox{\boldmath $x$}}})\vert
F_{L_1 L_2 (L_{12})L_3 L M}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}})\rangle
\nonumber \\
& &\quad =\frac{(-i)^{L_1+L_2+L_3}} {({\rm det}A)^{3/2}}
\sum_{\ell_1 \ell_2 \ell_3 \ell_{12}}
{\cal K}(L_1L_2(L_{12})L_3 L; \ell_1 \ell_2 \ell_3 \ell_{12})
\nonumber \\
& &\quad \times F_{L_1\!-\!\ell_1\!-\!\ell_2\ L_2\!-\!\ell_1\!-\!\ell_3\
(\ell_{12})\ L_3\!-\!\ell_2\!-\!\ell_3\ L M}
(A^{-1}u_1, A^{-1}u_2, A^{-1}u_3, A^{-1},{{\mbox{\boldmath $k$}}}),\end{aligned}$$ where the coefficient ${\cal K}$ is given by $$\begin{aligned}
& &{\cal K}(L_1L_2(L_{12})L_3 L; \ell_1 \ell_2 \ell_3 \ell_{12})
\nonumber \\
& &\ =\frac{(-1)^{L-L_3+\ell_2+\ell_3-\ell_{12}}}
{\sqrt{2L\!+\!1}}
\frac{B_{L_1}B_{L_2}B_{L_3}}
{B_{\ell_1}B_{\ell_2}B_{\ell_3}B_{L_1\!-\!\ell_1\!-\!\ell_2}
B_{L_2\!-\!\ell_1\!-\!\ell_3}B_{L_3\!-\!\ell_2\!-\!\ell_3}}
\nonumber \\
& &\quad \times
\sqrt{(2\ell_1\!+\!1)(2\ell_2\!+\!1)(2\ell_3\!+\!1)
(2(L_1\!-\!\ell_1\!-\!\ell_2)+1)(2(L_2\!-\!\ell_1\!-\!\ell_3)+1)(2(L_3\!-\!\ell_2\!-\!\ell_3)+1)}
\nonumber \\
& &\quad \times X(\ell_1 \ell_2 \ell_3)
Z(L_1\!-\!\ell_1\!-\!\ell_2\ \ell_1\!+\!\ell_2)
Z(L_2\!-\!\ell_1\!-\!\ell_3\ \ell_1\!+\!\ell_3)
Z(L_3\!-\!\ell_2\!-\!\ell_3\ \ell_2\!+\!\ell_3)
\nonumber \\
& &\quad \times U(\ell_{12}\ L_3\!-\!\ell_2\!-\!\ell_3\ L_{12}\ L_3; L\
\ell_2\!+\!\ell_3)
\left[
\begin{array}{ccc}
L_1\!-\!\ell_1\!-\!\ell_2 & L_1 & \ell_1\!+\!\ell_2\\
L_2\!-\!\ell_1\!-\!\ell_3 & L_2 & \ell_1\!+\!\ell_3\\
\ell_{12} & L_{12} & \ell_2\!+\!\ell_3 \\
\end{array}
\right]
\nonumber \\
& &\quad \times
(\widetilde{u_1}A^{-1}u_2)^{\ell_1}\
(\widetilde{u_1}A^{-1}u_3)^{\ell_2}\
(\widetilde{u_2}A^{-1}u_3)^{\ell_3},\end{aligned}$$ where $Z$ and $X$ are defined in Appendix A. Non-negative integers $\ell_i$ run over all possible values that satisfy $\ell_1\!+\!\ell_2 \leq L_1,\ \ell_1\!+\!\ell_3 \leq L_2,\
\ell_2\!+\!\ell_3 \leq L_3$. The value of ${\ell}_{12}$ is restricted by the triangular relations among ($\ell_{12}, L_1\!-\!\ell_1\!-\!\ell_2, L_2\!-\!\ell_1\!-\!\ell_3$) and ($\ell_{12}, L_{12}, \ell_2\!+\!\ell_3 $).
[99]{} Wildermuth K, Tang Y C (1977) A Unified Theory of the Nucleus (Vieweg, Braunschweig). Kamada H, Nogga A, Gl[ö]{}ckle W, Hiyama E, Kamimura M [*et al.*]{} (2001) Phys Rev 64:044001 Varga K, Suzuki Y, R. G. Lovas (1994) Nucl Phys A 571:447 Varga K, Ohbayasi K, Suzuki Y (1997) Phys Lett B 396:1; Varga K, Usukura J, Suzuki Y (1998) Phys Rev Lett 80:1876; Usukura J, Varga K, Suzuki Y (1998) Phys Rev A58:1918 Suzuki Y, Varga K (1998) Stochastic variational approach to quantum-mechanical few-body problems (Lecture notes in physics, Vol. 54). Springer, Berlin Heidelberg New York Varga K, Suzuki Y (1995) Phys Rev C 52:2885 Suzuki Y, Horiuchi W, Orabi M, Arai K (2008) Few-Body Syst 42:33 Varga K, Suzuki Y, Usukura J (1998) Few-Body Syst 24:81 Carlson J, Schiavilla R (2008) Rev Mod Phys 70:743; Pudliner B.S, Pandharipande V.R, Carlson J, Pieper S.C, Wiringa R.B (1997) Phys Rev C 56:1720 Navratil P, Kamuntavicius G.P, Barrett B.R (2000) Phys Rev C 61:044001 Viviani M (1998) Few-Body Syst 25:197 Feldmeier H, Neff T, Roth R, Schnack J (1998) Nucl Phys A632:61; Neff T, Feldmeier H (2003) Nucl Phys A 713:311 Arai K, Aoyama S, Suzuki Y (2010) Phys Rev C 81:037301 Phitzinger B, Hofmann M, Hale G.M (2001) Phys Rev C 64:044003 Deltuva A, Fonseca A.C (2007) Phys Rev C 75:014005; Deltuva A, Fonseca A.C (2007) Phys Rev Lett 98:162502 Quaglioni S, Navratil P (2009) Phys Rev C 79:044606; Quaglioni S, Navratil P (2008) Phys Rev Lett 08:092501 Viviani M, Rosati S, Kievsky A (1998) Phys Rev Lett 81:1580; Viviani M, Kievsky A, Rosati S, George E.A, Knulson L.D (2001) Phys Rev Lett 86:3739;Viviani M, Kievsky A,Girlanda L, Marcucci L.E, Rosati S (2009) Few-Body Syst 45:119 Lazauskas R, Carbonell J, Fonseca A.C, Viviani M, Kievsky A, Rosati S (2005) Phys Rev C 71:034004 Fisher B.M $et$ $al.$ (2006) Phys Rev C 74:034001 Arriaga A, Pandharipande V.R, Schiavilla (1991) Phys Rev C 43:983 Sabourov K $et$ $al.$ (2004) Phys Rev C 70:064601 Hofmann H.M, Hale G.M (2008) Phys Rev C 77:044002 Hofmann H.M, Hale G.M (1997) Nucl Phys A 613:69; Hofmann H.M, Hale G.M (2003) Phys Rev C 68:021002 Deltuva A, Fonseca A.C (2007) Phys Rev C 76:021001; Deltuva A, Fonseca A.C, Sauer P.U (2008) Phys Lett B 660:471 Lazauskas R, Carbonell J (2004) Few-Body Syst 34:105 Ciesielski F, Carbonell J, Gignoux C (1999) Phys Lett B 447:199 Assenbaum H.J, Langanke K (1987) Phys Rev C 36:17 Fowler W.A, Caughlan, Zimmenrman (1967) Annu Rev Astron Astrophys 5:525 Baye D, Heenen P H, Libert-Heinemann M (1977) Nucl Phys A 291:230 Kanada H, Kaneko T, Saito S, Tang Y C (1985) Nucl Phys A 444:209 Arai K, Descouvemont P , Baye D, (2001) Phys Rev C 63:044611 Descouvemont P, Baye D (2010) Rep Prog Phys 73:036301 Pudliner B S, Pandharipande V R, Carlson J, Pieper S C, Wiringa R B (1997): Phys Rev C 56, 1720 Hiyama E, Gibson B F, Kamimura M (2003) Phys Rev C 70:031001 Thompson D R, LeMere M, Tang Y C (1977) Nucl Phys A 286:53
Boys S F (1960) Proc R Soc London Ser A 258:402 ; Singer K (1960) [*ibid.*]{} 258:412 Suzuki Y, Usukura J (2000) Nucl Inst Method B 171:67 Suzuki Y, Usukura J, Varga K (1998) J Phys B 31:31 Horiuchi W, Suzuki Y (2008) Phys Rev C 78:034305 Tilley D R, Weller H R, Hale G M (1992), Nucl Phys A 541:1
Tamagaki R (1968) Prog Theor Phys 39:91
Santos F D, Arriaga A, Eiró A M, Tostevin J A (1985) Phys Rev C 31:707 Wachter B, Mertelmeier T, Hofmann H M (1988) Phys Lett B 200:246 Angulo C, Arnould M, Rayet M, Descouvemont P, Baye D [*et al.*]{} (1999) Nucl Phys A 656:3
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abstract: 'An expression for the spin–orbit interaction coupling between different levels, which was shown to be aberrant more than thirty years ago persists in the literature without clear indication of what is used. It leads to expressions quite simpler than they should be. After an attempt to warn the community of the nuclear physicists on this strange situation (nucl-th/0312038), the authors of the publication in which the “aberrant” interaction is described and used, try to justify their work (nucl-th/0401055), by a very strange “symmetrization” of something already symmetric. They claim also that their method allows to solve some problem related to the Pauli principle and give some references, among which a book which reports the solution of such problem almost forty years ago, with a very small effect. An examination of their own results shows that their optimism is not completely justified. Nevertheless, any user of [ECIS]{}, sensitive to their arguments, is requested to ask their opinion to these five coauthors before publishing. .5truecm PACS numbers 24.10.-i,25.40.Dn,25.40.Ny,28.20.Cz'
author:
- Jacques Raynal
date:
title: 'Reply to K. Amos et al nucl-th/0401055.'
---
epsf 0.2cm 0.1cm 0.1cm 0.5cm 0.1cm
*4 rue du Bief, 91380 Chilly–Mazarin, France*
After the publication of some article [@RAP1] in Nuclear Physic A, I wanted to publish a comment [@RAP2] warning the nuclear physicist community that two different deformed spin–orbit are used in the literature for nucleon–nucleus inelastic scattering without notifying which one is employed. Because everybody uses the same expression in nuclear structure studies (for this reason, I qualified it as “normal”), for inelastic scattering, I called “normal” the one which has the same behavior between partial waves and “aberrant” the other one. As I am not of native of English language, I was thinking to be allowed to use this word, because I found on page 3 of [@RAP3] : -1.cm
> [**ab–er–rant**]{} ($\dots $),[*adj.*]{} $[\dots ]$, deviating from what is true, correct, normal, or typical.
and on page 3 of the first volume of [@RAP4] :
> [**ab–er–rant**]{} $\backslash \dots \backslash $ [*adj*]{} $[\dots ]$, [**1 :**]{} straying from the right or normal way : deviating from truth, rectitude, propriety [**2 :**]{} deviating from the usual or natural type [**:**]{}
($\dots $ denoting pronunciation and etymology). My problem to publish such an advertisement is that I have only [@RAP1] to cite.
On the 14th of June, I was visiting the office of the Nuclear Energy Agency, at Issy–les–Moulineaux, which send nuclear codes to who wants them all over the world (except in USA). We found on the web the answer [@RAP5] of which the title is quite terrifying for some body who sent almost 300 [ECIS]{} to more than 50 countries. I find this answer largely out of the subject. I have to answer to the spin–orbit question, but also on the antisymmetrization with occupied states, another subject on which I have comments to do but which I did not want to publish. .5truecm
In [@RAP2], Eq. (1) is not the spin–orbit interaction but its coefficient; the spin–orbit interaction involves the two following lines. The six–parameter spin–orbit interaction of [ECIS]{} can be used to compare results with the two different deformed spin–orbit interactions and many other expressions by who wants it; anyway, the parameters are read only if a special logical is set true. This is like that since [ECIS67]{}. Instead of laughing at, why the authors of [@RAP1] do not do the same?
As the authors of [@RAP1] say that the “normal” is [*derived in some extremely hard–to–find publications”*]{}, let us resume it. First of all, the two-body spin–orbit interaction as described in [@RAP6; @RAP7; @RAP8; @RAP9] [[^1]]{} is “normal”; therefore, the intermediate step of its use as a one-body interaction, described by the sums of two terms of Eq. (84) of [@RAP6], the Eqs. (21-22) of [@RAP7], the Eqs. (51-52) of [@RAP8], and Eqs. (4,49-50) of [@RAP9] are also “normal”. With a finite range, these expressions are a kind of folding potential for the direct term, to which must be added similar expressions for the exchange term. At the zero–range limit (not $\delta ({\mbox{\boldmath $r$}} - {\mbox{\boldmath $r$}}')$ but $\delta
''({\mbox{\boldmath $r$}} - {\mbox{\boldmath $r$}}')$), direct and exchange terms for the two–body interaction are identical : they are given by Eq. (30) of [@RAP7] and Eq. (55) of [@RAP8]. Assuming zero for the eigenvalues of $({\mbox{\boldmath $l$}} \ . \ {\mbox{\boldmath $\sigma$}} )$ of one particle (which means complete shells for $l=j+1/2$ and $j=l-1/2$ with same radial function), one get the same expression as with the “full Thomas term”. The result must be made hermitian, but not in the sense used by the authors of [@RAP1; @RAP5] by dropping the derivative term acting on the right side but by replacing it by a derivation on the left side with opposite sign (that is : acting also on the form–factor).
As the “normal” coupled–channel spin–orbit potential is [*derived in some extremely hard–to–find publications*]{} [@RAP5] but cannot been published anywhere because it is not new, let us give it here in details. First of all, the spin–orbit obtained when the Dirac equation is changed into its Schrödinger equivalent (“full Thomas form”) can be written as : $$V^{LS} = \sum_{\lambda ,\mu }\Big( \nabla V_{\lambda }(r)
Y^{\mu }_{\lambda } ({\hat r})\Big) \times {\nabla \over i} \ . \
{{\mbox{\boldmath $\sigma$}} } \label{E1}$$ which avoids to deal with more equations than necessary : the zeroth order term of [@RAP1] is in $V_0(r)$, the first order term for some $\beta _2$ is in $V_2(r)$, the second order is $V_4(r)$ and also in $V_0(r)$ and in $V_2(r)$ (for the $n^{th}$ order, it is in all even $V$, from $V_0(r)$ to $V_{2n}(r)$). Using the following identities : $$\begin{aligned}
\nabla ={{\mbox{\boldmath $r$}} \over r} \ {d \over {dr}} - i {{{\mbox{\boldmath $r$}} \times
{\mbox{\boldmath $\ell$}} } \over {r^2}}, \qquad \qquad i {\mbox{\boldmath $\sigma$}} .({\mbox{\boldmath $A$}}
\times {\mbox{\boldmath $B$}} ) = ({\mbox{\boldmath $\sigma$}} .{\mbox{\boldmath $A$}}) ({\mbox{\boldmath $\sigma$}} . {\mbox{\boldmath $B$}})
- ({\mbox{\boldmath $A$}} .{\mbox{\boldmath $B$}}),&& \nonumber \\ ({\mbox{\boldmath $\sigma$}} .{\mbox{\boldmath $\nabla$}})=
{({\mbox{\boldmath $\sigma$}} .{\mbox{\boldmath $r$}}) \over r} \Big({d \over {dr}}
- {{({\mbox{\boldmath $\ell$}} .{\mbox{\boldmath $\sigma$}} )} \over r} \Big), \
({\mbox{\boldmath $\sigma$}} .{\mbox{\boldmath $\ell$}}) ({\mbox{\boldmath $\sigma$}} .{\mbox{\boldmath $r$}}) = - ({\mbox{\boldmath $\sigma$}} .
{\mbox{\boldmath $r$}}) ({\mbox{\boldmath $\sigma$}} .{\mbox{\boldmath $\ell$}}), \ ({\mbox{\boldmath $\sigma$}} .{\mbox{\boldmath $r$}})^2
= r^2,&& \label{E2}\end{aligned}$$ $V^{LS}(r)$ can be written as : $$\begin{aligned}
V^{LS} &=& \sum_{\lambda ,\mu } - \Big( \Big[{d \over {dr}} +
{{({\mbox{\boldmath $\ell$}} \ . \ {\mbox{\boldmath $\sigma$}} )}\over r} \Big] V_{\lambda }(r)
Y^{\mu }_{\lambda } ({\hat r}) \Big) \Big[{d \over {dr}} -
{{({\mbox{\boldmath $\ell$}} \ . \ {\mbox{\boldmath $\sigma$}} )}\over r} \Big] \nonumber \\
&+& \Big({d \over {dr}} \ V_{\lambda }(r)
Y^{\mu }_{\lambda } \ ({\hat r}) \Big) \ {d \over {dr}} \ - \
\Big( {{{\mbox{\boldmath $r$}} \times {\mbox{\boldmath $\ell$}} } \over {r^2}} V_{\lambda }(r)
Y^{\mu }_{\lambda } ({\hat r}) \Big) \ {{{\mbox{\boldmath $r$}} \times {\mbox{\boldmath $\ell$}} }
\over {r^2}} \label{E3}\end{aligned}$$ The terms with two derivatives cancel one another. Noting by $\ell _i$ and $\gamma _i$ the angular momentum and the eigenvalue of $({\mbox{\boldmath $\ell$}} .{\mbox{\boldmath $\sigma$}} )$ of the right side, $\ell _f$ and $\gamma _f$ for the left side, $({\mbox{\boldmath $\ell$}} .{\mbox{\boldmath $\sigma$}} )$ acting on $Y^{\mu }_{\lambda } ({\hat r})$ can be replaced by $(\gamma _f -\gamma _i)$ because ${\mbox{\boldmath $\ell$}} _f = {\mbox{\boldmath $\ell$}} _i +
{\mbox{\boldmath $\lambda$}} $. The last term can be simplified, using the relation : $$({\mbox{\boldmath $A$}} \times {\mbox{\boldmath $B$}}) . ({\mbox{\boldmath $C$}} \times {\mbox{\boldmath $D$}}) =
({\mbox{\boldmath $A$}} . {\mbox{\boldmath $C$}}) ({\mbox{\boldmath $B$}} . {\mbox{\boldmath $D$}})
- ({\mbox{\boldmath $B$}} . {\mbox{\boldmath $C$}}) ({\mbox{\boldmath $A$}} . {\mbox{\boldmath $D$}}) \label{E4}$$ which replaces the two cross products by $r^2 ({\mbox{\boldmath $\lambda$}} .
{\mbox{\boldmath $\ell$}} _i)$. But, as ${\mbox{\boldmath $\ell$}} _f = {\mbox{\boldmath $\ell$}} _i + \lambda $ and $({\mbox{\boldmath $\ell$}} . {\mbox{\boldmath $\ell$}}) = ({\mbox{\boldmath $\ell$}} . {\mbox{\boldmath $\sigma$}} )^2 +
({\mbox{\boldmath $\ell$}} . {\mbox{\boldmath $\sigma$}} )$ : $$- 2 ({\mbox{\boldmath $\lambda$}} . {\mbox{\boldmath $\ell$}}_i) = \lambda (\lambda +1) +
(\gamma _i - \gamma _f)(\gamma _i + \gamma _f +1) \label{E5}$$ With these quite simple manipulations, the result is obtained as : $$\begin{aligned}
V^{LS} &=& \sum_{\lambda ,\mu } Y^{\mu }_{\lambda }(\hat r) \Big[
{{d V_{\lambda }(r)}\over {dr}} \ \gamma _i \ + \ {{V_{\lambda }(r)}
\over r} \ (\gamma _i - \gamma _f) {d \over {dr}} \nonumber \\ &+&
{{V_{\lambda }(r)}\over {2 r^2}} \ \Big\{ \lambda (\lambda +1) -
(\gamma _f - \gamma _i )(\gamma _f - \gamma _i \pm 1) \Big\} \Big]
\label{E6}\end{aligned}$$ where $\pm 1$ is $+ 1$ in this tri–dimensional derivation and is $-1$ if the wave functions are multiplied by $r$ as usual. This derivation should not be a problem to people used to angular momenta, ${\mbox{\boldmath $\sigma$}} $–matrices, scalar and vector products.
To say that the first term of Eq. (\[E6\]) is [*fully consistent*]{} with the whole is quite strange. The fact that the two last terms can be replaced [@RAP1; @RAP5; @RBP1] by a $({\mbox{\boldmath $\ell$}} .{\mbox{\boldmath $\ell$}} )$ and a $({\mbox{\boldmath $s$}} .{\mbox{\boldmath $s$}})$ interactions as yet to be proven. Note that the first publications which used the “full Thomas term” [@RBP2; @RBP3] did not notice the behavior $(\gamma _i - \gamma _f)$ of this term because they ignored the (quite simple) derivation presented above. This interaction is expected to play a role primarily for the asymmetry of the inelastic scattering, but less than the deformed central interaction : it should be so in the relation of the amplitude of this asymmetry with the sign of the deformation in the rotational model [@RBP4]. Anyway, If the deformed spin–orbit interaction plays no role in their problem [@RAP1], why they use it.
The symmetrization of an operator including $d/dr$ acting on the right side is its replacement by $-d/dr$ acting on the left side, that is on the form–factor as well as the function. The use of the deformed spin–orbit [@RAP1] is equivalent to the use of : $$V^{LS}_{ijkl}(r)=V_{ijkl}(r) \{[\ell .{\mbox{\boldmath $s$}}]_i+[\ell .{\mbox{\boldmath $s$}}]_j
+[\ell .{\mbox{\boldmath $s$}}]_k+[\ell .{\mbox{\boldmath $s$}}]_l\} \label{E7}$$ for the two–body spin–orbit interaction, as can be seen after one integration. .5truecm
In [@RAP5], there are many comments and references related to the [*Pauli principle*]{} of which it was not question in [@RAP2]. I was allowed by the Service de Physique Theorique to photocopy all the reference \[2\] of [@RAP5] in its library (including the second one in Saclay’s central library) and also the 3 references of the article in Nuclear Physics related to Pauli principle (130 pages for all that) and to borrow [@RBP5]. It seems that they never opened this book; I did not remember of its content. There is a very good table of references by which I found myself cited 8 times as RA 67b, once as RA 68 and also twice as GI 67 and once as ME 66. The first [@RBP6] of their references \[2\], of G. Pisent, one of the coauthors, is also cited twice as PI 67c in this book : a footnote on page 103 ([*In the papers concerned with ${}^{13}C$ as a compound nucleus the exclusion principle could not be exactly satisfied $\vert $ ... , ... , PI 67c, ... $\vert $*]{}.), and the last paragraph of page 113 ([*... and Pisent and Saruis $\vert $PI 67c$\vert $. This various works suffer from the drawback that the Pauli exclusion principle is violated at some stage of the calculation.*]{}).
In Spring 1965, I was theoretically at USC, practically at UCLA, in Los Angeles. With M. A. Melkanoff and T. Sawada, we decided to do some calculations on the giant resonance of ${}^{16}O$ using the shell model with a continuum theory of C. Bloch and V. Gillet [@RBP7]. This work has been published in Nuclear Physics A [@RBP8] and as a seminar at a Summer School in Varenna [@RBP9]; in the same book, there is another seminar from me on the “Stretch scheme” and a seminar entitled [*Results of Hartree–Fock calculations with non–local and hard–core potentials*]{}, by J. P. Svenne, Canadian of Copenhagen University, who I think to be one of the coauthors of Ken Amos. This work is partly reported in [@RBP5]. The space used is described in the book on page 23 in the text together with figure 3.2 and its legend. C. Bloch and V. Gillet could obtain values only at points which they choose for the grid, but we managed to obtain continuous results (footnote on page 77 :[*Care must be taken because the integrands involve $a^c(E'';c)$ which is singular at $E''=E$ $\vert $RA 67b*]{}$\vert $.) with a minimum number of points and did computation from 16 to 30 MeV (this is scattering on ${}^{15}O$ for which 16 MeV is the threshold with respect to ${}^{16}O$). Then, we decided to do the same calculation in the $r$–space instead of the $E$–space. We got different results; looking why, we orthogonalized with the $1s_{1/2}$ occupied bound-state, thinking that a small mixture of this state give very important effects for ${}^{16}O(\gamma ,n)$, more than for the elastic scattering because this result is the integral of the solution multiplied by $r$ and the hole function. We obtained the same result as in $E$-representation. That was the proof that these two approaches, mathematically equivalent are numerically equivalent (discarding error or imprecision on one of them). This is presented pages 106-110 with the results in figure 6.1 . On page 103, [*6.3a. Coupled channels approach*]{} the first paragraph includes two citations prior to [@RBP8] as not taking into account antisymmetrization and quote RA 67b as showing this effect. The second paragraph is : [*The most complete coupled channels calculation of the reactions ${}^{16}O(\gamma ,n)$ and ${}^{16}O(\gamma ,n)$ (for $E1$ transitions) was carried through by Raynal, Melkanoff and Sawada $\vert $RA 67b$\vert $. These authors treat antisymmetrization correctly.*]{} In fact, I never saw the [*Pauli principle*]{} expressed more clearly than by Eqs. (19-20) of [@RBP9] or Eqs. (55-58) of [@RBP8] (Eqs. (59-61) for a zero–range interaction). The reference [@RCP1] given in [@RAP1] uses only a $1s (\alpha )$ state with no generalization.
More details can be found in [@RBP8; @RBP9] : figures 12 and 13 in the first reproduced by figures 4 and 3 in the second show the results obtained respectively for ${}^{16}O(\gamma ,n)$ and ${}^{16}O(\gamma ,p)$ with five channels and coupling the ten channels. In neutron figures, there are :
[$\bullet $]{} the five channels result,
[$\bullet $]{} the ten channels one,
[$\bullet $]{} the experimental results known at that time,
[$\bullet $]{} and also results obtained with five channels without taking into account the occupation of the $1s_{1/2}$ state.
Unhappily, this last curve is not given in figure 6.1 of [@RBP5] which shows only five channels results and experimental data for neutrons and not this fourth curve which is essential to clarify the point in discussion: there is no noticeable effect up to 20 MeV (4.5 MeV above threshold) but a shift of the maximum around 22 MeV, about the same as between five and ten channels calculations. In the same two publications, we showed that the difficulty to deal with a resonance $d_{3/2}$ in the continuum in E–representation can be overcome by using a bound–state and taking into account the difference of the Saxon–Woods wells in r–representation or E–representation. In [@RBP8; @RCP2], we studied the effect of a 2p–2h state as quoted GI 67 by [@RBP5] on pages 108 and 225.
I foresee Ken Amos’ answer: it is not the same problem, you deal with ${}^{15}O$ and ${}^{15}N$ and not ${}^{12}C$, you use some two–body interaction instead of a pure one–body, and so on, and so on ... But look to their own results, table I, page 86 of [@RAP1] the three lines where there are experimental data and results with and without OPP : for the first, the energy is shifted by 33% of what is needed, in the good direction but the width is increased of 20% only instead of 148%; in the second one the energy is shifted of only 4% of what is needed and the width unchanged; in the third one, the energy is shifted of 82% of what is needed, in the good direction (great success) but the width is increased 15% instead of being divided by 4,33; in a fourth case, there is no effect. With these values, they claim that this antisymmetrization is absolutely necessary; with the same values, I feel that it disturbs the results.
On the 25th of June 1975, I participated to the jury for the thesis of J.-M. Normand [@RCP3] at Orsay with V. Gillet, P. Benoist–Gueutal, M. Goldman and R. Arvieu as president. He showed the effect of the Pauli principle at threshold energies [@RCP4]. He studied scattering lengths and effective ranges of neutrons, which I think quite sensitive to these effects, on ${}^{12}C$, ${}^{13}C$, ${}^{16}O$, ${}^{17}O$, ${}^{19}F$ and ${}^{40}Ca$. Only the scattering lengths were known at that time. In table 5, using 4 different interactions with different strengths (in all 14 calculations), he found for ${}^{12}C$ a decrease of 8% to 21% for the scattering length, of 4% to 7% for the effective range (but, among these 14 calculations, the smallest value is 54% of the largest one for the scattering length and 30% for the effective range, 30% and 20% discarding the largest value). In table 6, for ${}^{16}C$, also with 4 interactions and 13 calculations, these figures reduce to 5% to 7% for the scattering length, 2% to 4% for the effective range and values spread by 12% for both. In Table 7 for ${}^{40}Ca$, with 2 interactions and 6 calculations, there is no effect of the $1s_{1/2}$ state (less than 0.3%) but a large effect of the $2s_{1/2}$, 21% to 42% for the scattering length, 15% to 31% for effective range, leading to almost identical results; the variation of this last results are 0.6% and 1.1% for a variation of 4.2% and 3.3% of the strength of the interaction (very special case for which the Pauli principle is more important than the model, id est, than the strength of the interaction). In table 8 are given 6 results for ${}^{13}C$ and 2 for ${}^{17}O$ ; in this table, there are results for two values of $J$. In all these cases, one can see that the corrected results are quite near the uncorrected ones obtained with an increase of the interaction by about 3%. Results for ${}^{19}F$ show the importance of the choice of the space of configuration.
Even if the effects are more important for ${}^{12}C$ than ${}^{16}O$, I do not see in this very sensitive calculation a justification of the assertion of Ken Amos that the Pauli principle affects strongly results at low energy and it is not their publication which can convince looking their table 1. Anyway, such phenomena are weaker with a complex potential (because the wave function is damped inside the nucleus) and [ECIS]{} was not written for such problems. As said in the title, any user of these codes who has the smallest doubt about this subject should ask their opinion to K. Amos et al. Anyway, it is a lot easier to add that to [ECIS]{} than to introduce a quadratic spin-orbit which was never seriously used in [DWBA90]{}.
In [@RAP5], these is a reference to page 426 of Hodgson’s book [@RCP5]; in the following pages, there is a presentation of [@RBP1] and of some publications of G. Pisent, before or after [@RBP6] with no allusion to the “Pauli exclusion principle”. In the subject index, seven pages are indicated for this topic :
[$\bullet $]{} page 90 on semiclassical optical model,
[$\bullet $]{} page 113 for application in nuclear medium,
[$\bullet $]{} pages 130, 131, 132 for the calculation of the imaginary part of the optical potential,
[$\bullet $]{} page 162 related to consequences for nuclear medium,
[$\bullet $]{} page 581 consider the effect of the excess of neutrons on the difference between the number of reactions $(p,n)$ and $(n,p)$.
The “Pauli exclusion principle” applied to the scattering wave is completely ignored in this book where this effect is only applied to nucleon–nucleon scattering in nuclear medium as needed in [@RCP6] : the critics of [@RAP5] on the conception of [ECIS]{} are also valid for it. There is no question of spin–orbit deformation : [@RBP2; @RBP3] are not cited. My own thesis is cited for different points (an error on page 148, not reproduced on pages 231, 235 and 244) including figure 10.6 on elastic deuteron scattering.
The third book [@RCP7] cited in [@RAP5] is not available at Saclay. I cannot afford to buy it and analyze it as I did above for the two first ones.
I hope that every user of [ECIS]{} will make his own opinion on [@RAP5] and my answer, even those who use it for heavy–ion scattering because the title does not exclude this subject. If they have any doubt, they should communicate their results to the five coauthors of which they can find the e–mail address in [@RAP1]. .5truecm
[**Conclusions**]{}
[$\bullet $]{} When they say : [*the spin–orbit expression we use is fully consistent with the ${\mbox{\boldmath $S$}} . {\mbox{\boldmath $L$}}$ term that comes from the full–fledged Thomas term*]{}, they forget to add that they pluck.
[$\bullet $]{} If they open the books which they give as reference, they can see that the effects which they claim to be at low energy are seen only at higher energy. Even if they are some effects [@RCP3; @RCP4], they can disappear if there is a search on parameters as in [@RAP1].
[$\bullet $]{} If they look at the table which they publish, they have to agree that the Pauli principle is inefficient to give good results : readers can conclude that their method is bad or that the Pauli principle has no notable effects but cannot agree with their optimistic comments.
Anyway, the Pauli principle was not the subject of [@RAP2] but the fact to see in the literature an expression which I believed forgotten since a long time and was certainly used many times without being quoted. The “normal” expression is easy to derive as shown by equations (\[E2\]) to (\[E6\]) there in. The allusion to a mosquito and an elephant at the end of [@RAP5] reminds a tale of La Fontaine about a frog and an ox. [*Errare humanum est*]{} (see footnote); [*perseverare diabolicum*]{}.
[999]{} K. Amos, L. Canton, G.Pisent, J. P. Svenne and D. van der Knijff, Nucl. Phys. [**A728**]{}, 65 (2003). J. Raynal, ArXiv:nucl-th/0312038 vl 15 Dec 2003 Webster’s New World Dictionary of the American Language, College Edition, The World Publishing Company (1962) Webster’s Third New International Dictionary, Encyclop[æ]{}dia Britannica, inc – William Benton, editor (1966) K. Amos, L. Canton, G.Pisent, J. P. Svenne and D. van der Knijff, ArXiv:nucl-th/0401055 vl 27 Jan 2004 J. Raynal, Nucl. Phys. [**A97**]{}, 572 (1967) (IAEA, 1972), 281. J. Raynal, Symposium sur les Mécanismes des Réaction Nucléaire et les Phénoménes de Polarisation, Québec, Canada, Sept.1-2, 1969, Les Presses de l’Université Laval, Québec, 75 (1970). J. Raynal, The Structure of Nuclei, International Course on Nuclear Theory, Trieste, Italie, Jan.13 - March 12, 1971, (IAEA,1972) 75. K. A. Amos, P. J. Dortmans, H. V. von Geramb, S. Karataglidis, J. Raynal, Adv. Nucl. Phys. [**25**]{}, 275 (2000) O. Mikoshiba, T. Terasawa, and Tanifugi, Nucl. Phys. [**A168**]{}, 417 (1971). H. Sherif, J. S. Blair, Phys. Let. [**26B**]{}, 489 (1968). H. Sherif, Spin–dependent effects in proton inelastic scattering, Thesis University of Washington (1968). A.B. Kurepin, R.M. Lombard, J. Raynal, Phys. Lett. [**45B**]{}, 184 (1973) C. Mahaux, H.A. Weidenmuller, Shell model approach to nuclear reactions, North–Holland, Amsterdam, 1969 G. Pisent, A.M. Saruis, Nucl.Phys. [**91**]{}, 561 (1967) C. Bloch, V. Gillet, Phys. Letters [**16**]{}, 62 (1965) J. Raynal, M. A. Melkanoff, T. Sawada, Nucl. Phys. [**A101**]{}, 369 (1967) Raynal J. Nuclear Structure and Nuclear Reactions: Proceedings, International School of “Physics”, Enrico Fermi, course XL, Varenna, Italy, June 25 - July 15 1967, Academic Press, , 683 (1969) S. Saito, Prog. Theor. Phys. [**41**]{}, 705 (1969) V. Gillet, M.A. Melkanoff, J. Raynal, Nucl. Phys. [**A97**]{}, 631 (1967) J.-M. Normand, Description microscopique de la diffusion de neutrons de basse énergie par les noyaux, Thesis, Orsay (1975) J.-M. Normand, Nucl. Phys. [**A291**]{}, 126 (1977) P.E. Hodgson, Nuclear Reactions and Nuclear Structure, Clarendon Press (1961) 426 P.J. Dortmans, K. Amos, Phys. Rev. C, [**49**]{}, 1309 (1994) W. Greiner, J.A. Marhun, Nuclear Models, Springer–Verlag, Berlin, 1996.
[^1]: In [@RAP6] there was an error for $a^J(1,2)$ as said in [@RAP7] : the coefficients of $V_{J-1}$ and $V_{J+1}$ are $J(J-1)$ and $(J+1)(J+2)$ instead of $J(J+3)$ and $(J+1)(J-2)$ respectively. There is also a factor 4 in the three first publications, coming from an error in writing the derivative with respect to $({\mbox{\boldmath $r$}}_1 - {\mbox{\boldmath $r$}}_2)$ and assimilation of ${\mbox{\boldmath $\sigma$}} $ to ${\mbox{\boldmath $s$}}$. In [@RAP9], $-(\alpha ^J_{j_2j'_2})^2$ and a factor $1/4$ are missing in the expression of $d^J(1,2)$.
|
---
abstract: |
The reductive perturbation method has been employed to derive the Korteweg-de Vries (KdV) equation for small but finite amplitude electrostatic waves. The Lagrangian of the time fractional KdV equation is used in similar form to the Lagrangian of the regular KdV equation. The variation of the functional of this Lagrangian leads to the Euler-Lagrange equation that leads to the time fractional KdV equation. The Riemann-Liouvulle definition of the fractional derivative is used to describe the time fractional operator in the fractional KdV equation. The variational-iteration method given by He is used to solve the derived time fractional KdV equation. The calculations of the solution with initial condition $A_{0}\sec h(cx)^{2}$ are carried out. Numerical studies have been made using plasma parameters close to those values corresponding to the dayside auroral zone. The effects of the time fractional parameter on the electrostatic solitary structures are presented.
Keywords: Euler-Lagrange equation, Riemann-Liouvulle fractional derivative, fractional KdV equation, He’s variational-iteration method.
PACS: 05.45.Df, 05.30.Pr
author:
- 'El-Said A. El-Wakil, Essam M. Abulwafa,'
- |
Emad K. Elshewy and Aber A. Mahmoud\
Theoretical Physics Research Group, Physics Department,\
Faculty of Science, Mansoura University, Mansoura 35516, Egypt
title: 'Time-Fractional KdV Equation for the plasma in auroral zone using Variational Methods'
---
Introduction
============
Because most classical processes observed in the physical world are nonconservative, it is important to be able to apply the power of variational methods to such cases. A method used a Lagrangian that leads to an Euler-Lagrange equation that is, in some sense, equivalent to the desired equation of motion. Hamilton’s equations are derived from the Lagrangian and are equivalent to the Euler-Lagrange equation. If a Lagrangian is constructed using noninteger-order derivatives, then the resulting equation of motion can be nonconservative. It was shown that such fractional derivatives in the Lagrangian describe nonconservative forces \[1, 2\]. Further study of the fractional Euler-Lagrange can be found in the work of Agrawal \[3, 4\], Baleanu and coworkers \[5, 6\] and Tarasov and Zaslavsky \[7, 8\]. During the last decades, Fractional Calculus has been applied to almost every field of science, engineering and mathematics. Some of the areas where Fractional Calculus has been applied include viscoelasticity and rheology, electrical engineering, electrochemistry, biology, biophysics and bioengineering, signal and image processing, mechanics, mechatronics, physics, and control theory \[9\].
On the other hand, electron acoustic waves (EAWs) propagation in plasmas has received a great deal attention because of its vital role in understanding different types of collective processes in laboratory devices \[10, 11\] as well as in space environments \[12, 13\]. It has been argued that when the hot to cold electron temperature ratio is greater than 10, the electron-acoustic mode may be the principal mode of the plasma in which the restoring force comes from the pressure of the hot electrons, while the inertia comes from the mass of the cold electron component \[14\]. The ions play the role of a neutralizing background, i.e., the ion dynamics does not influence the EAWs because its frequency is much larger than the ion plasma frequency. Several theoretical attempts have been made to explain nonlinear EAWs in plasma systems \[15-17\].
To the author’s knowledge, the problem of time fractional KdV equation in collisionless plasma has not been addressed in the literature before. So, our motive here is to study the effects of time fractional parameter on the electrostatic structures for a system of collisionless plasma consisting of a cold electron fluid and non-thermal hot electrons obeying a non-thermal distribution and stationary ions. Our choice of non-thermal distribution of electrons is prompted by its convenience rather than as precise fitting of the observations. We expect that the inclusion of the non-thermal electrons, time fractional parameter will change the properties as well as the regime of existence of solitons.
Several methods have been used to solve fractional differential equations such as: the Laplace transform method, the Fourier transform method, the iteration method and the operational method \[18, 19\]. Recently, there are some papers deal with the existence and multiplicity of solution of nonlinear fractional differential equation by the use of techniques of nonlinear analysis \[20-23\]. In this paper, the resultant fractional KdV equation will be solved using a variational-iteration method (VIM) firstly used by He \[24, 25\].
This paper is organized as follows: Section 2 is devoted to describe the formulation of the time-fractional KdV (FKdV) equation using the variational Euler-Lagrange method. In section 3, the resultant time-FKdV equation is solved approximately using VIM. Section 4 contains the results of calculations and discussion of these results.
Basic Equations and Time-fractional KdV equation
================================================
We consider a homogeneous system of unmagnetized collisionless plasma consisting of cold electrons fluid, non-thermal hot electrons obeying a non-thermal distribution and stationary ions. Such system is governed by the following normalized equations in one dimension \[26\]:
$$\frac{\partial }{\partial t}u_{c}(x,~t)+\frac{\partial }{\partial x}[u_{c}(x,~t)~n_{c}(x,~t)]=0\text{,} \tag{1a}$$
$$\frac{\partial }{\partial t}u_{c}(x,~t)+u_{c}(x,~t)\frac{\partial }{\partial
t}u_{c}(x,~t)-\gamma \phi (x,~t)=0\text{,} \tag{1b}$$
with Poisson’s equation
$$\frac{\partial ^{2}}{\partial x^{2}}\phi (x,~t)-\frac{1}{\gamma }n_{c}(x,~t)-n_{h}(x,~t)+(1+\frac{1}{\gamma })=0\text{.} \tag{1c}$$
The non-thermal hot electrons density $n_{h}(x,~t)$ is given by:
$$n_{h}(x,~t)=[1-\beta ~\phi (x,~t)+\beta ~\phi (x,~t)^{2}]~\exp [\phi (x,~t)]\text{, }\beta =4\delta /(1+3\delta )\text{.} \tag{2}$$
In these equations, $n_{c}(x,~t)$\[$n_{h}(x,~t)$\] is the cold (non-thermal hot) electrons density normalized by equilibrium value $n_{c0}$\[$n_{h0}$\], $u_{c}(x,~t)$ is the cold electrons fluid velocity normalized by hot electron acoustic speed $C_{e}=\sqrt{\frac{k_{B}T_{h}}{\gamma m_{e}}}$, $\phi (x,~t)$ is the electric potential normalized by $\frac{k_{B}T_{h}}{e}$, $\gamma =\frac{n_{h0}}{n_{c0}}$ is the hot to cold electron equilibrium densities ratio, $m_{e}$ is the electron mass, $\delta $ is a parameter which determines the population of energetic non-thermal hot electrons, $e$ is the electron charge, $x$ is the space co-ordinate and $t$ is the time variable. The distance is normalized to the hot electron Debye length $\lambda _{Dh}$, the time is normalized by the inverse of the cold electron plasma frequency $\omega _{ce}^{-1}$ and $k_{B}$ is the Boltzmann’s constant. Equations (1a) and (1b) represent the inertia of cold electron and (1c) is the Poisson’s equation needed to make the self consistent. The hot electrons are described by non-thermal distribution given by (2).
According to the general method of reductive perturbation theory, we introduce the slow stretched co-ordinates \[27\]:
$$\tau =\epsilon ^{2/3}t\text{ and }\xi =\epsilon ^{1/2}(x-\lambda t)\text{,}
\tag{3}$$
where $\epsilon $ is a small dimensionless expansion parameter and $\lambda $ is the wave speed normalized by $C_{e}$. All physical quantities appearing in (1) are expanded as power series in $\epsilon $ about their equilibrium values as:
$$n_{c}(\xi ,~\tau )=1+\epsilon n_{1}(\xi ,~\tau )+\epsilon ^{2}n_{2}(\xi
,~\tau )+\epsilon ^{3}n_{3}(\xi ,~\tau )+...\text{,} \tag{4a}$$
$$u_{c}(\xi ,~\tau )=\epsilon u_{1}(\xi ,~\tau )+\epsilon ^{2}u_{2}(\xi ,~\tau
)+\epsilon ^{3}u_{3}(\xi ,~\tau )+...\text{,} \tag{4b}$$
$$\phi (\xi ,~\tau )=\epsilon \phi _{1}(\xi ,~\tau )+\epsilon _{{}}^{2}\phi
_{2}(\xi ,~\tau )+\epsilon _{{}}^{3}\phi _{3}(\xi ,~\tau )+...\text{.}
\tag{4c}$$
We impose the boundary conditions as $\xi \rightarrow \infty $, $n_{c}=n_{h}=1$, $u_{c}=0$ and $\phi =0$.
Substituting (3) and (4) into (1) and equating coefficients of like powers of $\epsilon $, the lowest-order equations in $\epsilon $ lead to the following results:
$$n_{1}(\xi ,\tau )=\frac{\gamma }{\lambda ^{2}}\phi _{1}(\xi ,\tau )\text{
and }u_{1}(\xi ,\tau )=\frac{\gamma }{\lambda }\phi _{1}(\xi ,\tau )\text{.}
\tag{5}$$
Poisson’s equation gives the linear dispersion relation
$$\lambda =\sqrt{\frac{1}{1-\beta }}=\sqrt{\frac{1+3\delta }{1-\delta }}\text{.} \tag{6}$$
Considering the coefficients of $O(\epsilon ^{2})$ and eliminating the second order-perturbed quantities $n_{2}$, $u_{2}$ and $\phi _{2}$ lead to the following KdV equation for the first-order perturbed potential:
$$\frac{\partial }{\partial \tau }\phi _{1}(\xi ,~\tau )+A~\phi _{1}(\xi
,~\tau )\frac{\partial }{\partial \xi }\phi _{1}(\xi ,~\tau )+B~\frac{\partial ^{3}}{\partial \xi ^{3}}\phi _{1}(\xi ,~\tau )=0\text{,} \tag{7a}$$
where
$$A=-\frac{3\gamma +\lambda ^{4}}{2\lambda }\text{, }B=\frac{\lambda ^{3}}{2}\text{,} \tag{7b}$$
$\phi _{1}(\xi ,~\tau )$ is a field variable, $\xi $ is a space coordinate in the propagation direction of the field and $\tau \in T$($=[0,T_{0}]$) is the time. The resultant KdV equation (7a) can be converted into time-fractional KdV equation as follows:
Using a potential function $v(\xi ,~\tau )$ where $\phi _{1}(\xi ,~\tau
)=v_{\xi }(\xi ,~\tau )$ gives the potential equation of the regular KdV equation (1) in the form
$$v_{\xi \tau }(\xi ,~\tau )+A~v_{\xi }(\xi ,~\tau )v_{\xi \xi }(\xi ,~\tau
)+B~v_{\xi \xi \xi \xi }(\xi ,~\tau )=0\text{,} \tag{8}$$
where the subscripts denote the partial differentiation of the function with respect to the parameter. The Lagrangian of this regular KdV equation (7) can be defined using the semi-inverse method \[28, 29\] as follows.
The functional of the potential equation (8) can be represented by
$$J(v)=\dint\limits_{R}d\xi \dint\limits_{T}d\tau \{v(\xi ,\tau )[c_{1}v_{\xi
\tau }(\xi ,\tau )+c_{2}Av_{\xi }(\xi ,\tau )v_{\xi \xi }(\xi ,\tau
)+c_{3}Bv_{\xi \xi \xi \xi }(\xi ,\tau )]\}\text{,} \tag{9}$$
where $c_{1}$, $c_{2}$ and $c_{3}$ are constants to be determined. Integrating by parts and taking $v_{\tau }|_{R}=v_{\xi }|_{R}=v_{\xi
}|_{T}=0 $ lead to
$$J(v)=\dint\limits_{R}d\xi \dint\limits_{T}d\tau \{v(\xi ,\tau )[-c_{1}v_{\xi
}(\xi ,\tau )v_{\tau }(\xi ,\tau )-\frac{1}{2}c_{2}Av_{\xi }^{3}(\xi ,\tau
)+c_{3}Bv_{\xi \xi }^{2}(\xi ,\tau )]\}\text{.} \tag{10}$$
The unknown constants $c_{i}$ ($i=$ $1$, $2$, $3$) can be determined by taking the variation of the functional (10) to make it optimal. Taking the variation of this functional, integrating each term by parts and make the variation optimum give the following relation
$$2c_{1}v_{\xi \tau }(\xi ,\tau )+3c_{2}Av_{\xi }(\xi ,\tau )v_{\xi \xi }(\xi
,\tau )+2c_{3}Bv_{\xi \xi \xi \xi }(\xi ,\tau )=0\text{.} \tag{11}$$
As this equation must be equal to equation (8), the unknown constants are given as
$$c_{1}=1/2\text{, }c_{2}=1/3\text{ and }c_{3}=1/2\text{.} \tag{12}$$
Therefore, the functional given by (10) gives the Lagrangian of the regular KdV equation as
$$L(v_{\tau },~v_{\xi },v_{\xi \xi })=-\frac{1}{2}v_{\xi }(\xi ,\tau )v_{\tau
}(\xi ,\tau )-\frac{1}{6}Av_{\xi }^{3}(\xi ,\tau )+\frac{1}{2}Bv_{\xi \xi
}^{2}(\xi ,\tau )\text{.} \tag{13}$$
Similar to this form, the Lagrangian of the time-fractional version of the KdV equation can be written in the form
$$\begin{aligned}
F(_{0}D_{\tau }^{\alpha }v,~v_{\xi },v_{\xi \xi }) &=&-\frac{1}{2}[_{0}D_{\tau }^{\alpha }v(\xi ,\tau )]v_{\xi }(\xi ,\tau )-\frac{1}{6}Av_{\xi }^{3}(\xi ,\tau )+\frac{1}{2}Bv_{\xi \xi }^{2}(\xi ,\tau )\text{, }
\notag \\
0 &\leq &\alpha <1\text{,} \TCItag{14}\end{aligned}$$
where the fractional derivative is represented, using the left Riemann-Liouville fractional derivative definition as \[18, 19\]
$$_{a}D_{t}^{\alpha }f(t)=\frac{1}{\Gamma (k-\alpha )}\frac{d^{k}}{dt^{k}}[\int_{a}^{t}d\tau (t-\tau )^{k-\alpha -1}f(\tau )]\text{, }k-1\leq \alpha
\leq 1\text{, }t\in \lbrack a,b]\text{.} \tag{15}$$
The functional of the time-FKdV equation can be represented in the form
$$J(v)=\dint\limits_{R}d\xi \dint\limits_{T}d\tau F(_{0}D_{\tau }^{\alpha
}v,~v_{\xi },v_{\xi \xi })\text{,} \tag{16}$$
where the time-fractional Lagrangian $F(_{0}D_{\tau }^{\alpha }v,~v_{\xi
},v_{\xi \xi })$ is defined by (14).
Following Agrawal’s method \[3, 4\], the variation of functional (16) with respect to $v(\xi ,\tau )$ leads to
$$\delta J(v)=\dint\limits_{R}d\xi \dint\limits_{T}d\tau \{\frac{\partial F}{\partial _{0}D_{\tau }^{\alpha }v}\delta _{0}D_{\tau }^{\alpha }v+\frac{\partial F}{\partial v_{\xi }}\delta v_{\xi }+\frac{\partial F}{\partial
v_{\xi \xi }}\delta v_{\xi \xi }\}\text{.} \tag{17}$$
The formula for fractional integration by parts reads \[3, 18, 19\]
$$\int_{a}^{b}dtf(t)_{a}D_{t}^{\alpha
}g(t)=\int_{a}^{t}dtg(t)_{t}D_{b}^{\alpha }f(t)\text{, \ \ \ }f(t)\text{, }g(t)\text{ }\in \lbrack a,~b]\text{.} \tag{18}$$
where $_{t}D_{b}^{\alpha }$, the right Riemann-Liouville fractional derivative, is defined by \[18, 19\]
$$_{t}D_{b}^{\alpha }f(t)=\frac{(-1)^{k}}{\Gamma (k-\alpha )}\frac{d^{k}}{dt^{k}}[\int_{t}^{b}d\tau (\tau -t)^{k-\alpha -1}f(\tau )]\text{, }k-1\leq
\alpha \leq 1\text{, }t\in \lbrack a,b]\text{.} \tag{19}$$
Integrating the right-hand side of (17) by parts using formula (18) leads to
$$\delta J(v)=\dint\limits_{R}d\xi \dint\limits_{T}d\tau \lbrack _{\tau
}D_{T_{0}}^{\alpha }(\frac{\partial F}{\partial _{0}D_{\tau }^{\alpha }v})-\frac{\partial }{\partial \xi }(\frac{\partial F}{\partial v_{\xi }})+\frac{\partial ^{2}}{\partial \xi ^{2}}(\frac{\partial F}{\partial v_{\xi \xi }})]\delta v\text{,} \tag{20}$$
where it is assumed that $\delta v|_{T}=\delta v|_{R}=\delta v_{\xi }|_{R}=0$.
Optimizing this variation of the functional $J(v)$, i. e; $\delta J(v)=0$, gives the Euler-Lagrange equation for the time-FKdV equation in the form
$$_{\tau }D_{T_{0}}^{\alpha }(\frac{\partial F}{\partial _{0}D_{\tau }^{\alpha
}v})-\frac{\partial }{\partial \xi }(\frac{\partial F}{\partial v_{\xi }})+\frac{\partial ^{2}}{\partial \xi ^{2}}(\frac{\partial F}{\partial v_{\xi
\xi }})=0\text{.} \tag{21}$$
Substituting the Lagrangian of the time-FKdV equation (14) into this Euler-Lagrange formula (21) gives
$$-\frac{1}{2}~_{\tau }D_{T_{0}}^{\alpha }v_{\xi }(\xi ,\tau )+\frac{1}{2}~_{0}D_{\tau }^{\alpha }v_{\xi }(\xi ,\tau )+Av_{\xi }(\xi ,\tau )v_{\xi \xi
}(\xi ,\tau )+Bv_{\xi \xi \xi \xi }(\xi ,\tau )=0\text{.} \tag{22}$$
Substituting for the potential function, $v_{\xi }(\xi ,\tau )=\phi _{1}(\xi
,\tau )=\Phi (\xi ,\tau )$, gives the time-FKdV equation for the state function $\Phi (\xi ,\tau )$ in the form
$$\frac{1}{2}[_{0}D_{\tau }^{\alpha }\Phi (\xi ,\tau )-_{\tau
}D_{T_{0}}^{\alpha }\Phi (\xi ,\tau )]+A\Phi (\xi ,\tau )\Phi _{\xi }(\xi
,\tau )+B\Phi _{\xi \xi \xi }(\xi ,\tau )=0\text{,} \tag{23}$$
where the fractional derivatives $_{0}D_{\tau }^{\alpha }$ and $_{\tau
}D_{T_{0}}^{\alpha }$ are, respectively the left and right Riemann-Liouville fractional derivatives and are defined by (15) and (19).
The time-FKdV equation represented in (14) can be rewritten by the formula
$$\frac{1}{2}~_{0}^{R}D_{\tau }^{\alpha }\Phi (\xi ,\tau )+A~\Phi (\xi ,\tau
)\Phi _{\xi }(\xi ,\tau )+B~\Phi _{\xi \xi \xi }(\xi ,\tau )=0\text{,}
\tag{24}$$
where the fractional operator $_{0}^{R}D_{\tau }^{\alpha }$ is called Riesz fractional derivative and can be represented by \[4, 18, 19\]
$$\begin{aligned}
~_{0}^{R}D_{t}^{\alpha }f(t) &=&\frac{1}{2}[_{0}D_{t}^{\alpha
}f(t)+~(-1)^{k}{}_{t}D_{T_{0}}^{\alpha }f(t)] \notag \\
&=&\frac{1}{2}\frac{1}{\Gamma (k-\alpha )}\frac{d^{k}}{dt^{k}}[\int_{a}^{t}d\tau |t-\tau |^{k-\alpha -1}f(\tau )]\text{, } \notag \\
k-1 &\leq &\alpha \leq 1\text{, }t\in \lbrack a,b]\text{.} \TCItag{25}\end{aligned}$$
The nonlinear fractional differential equations have been solved using different techniques \[18-23\]. In this paper, a variational-iteration method (VIM) \[24, 25\] has been used to solve the time-FKdV equation that formulated using Euler-Lagrange variational technique.
Variational-Iteration Method
============================
Variational-iteration method (VIM) \[24, 25\] has been used successfully to solve different types of integer nonlinear differential equations \[30, 31\]. Also, VIM is used to solve linear and nonlinear fractional differential equations \[32, 33\]. This VIM has been used in this paper to solve the formulated time-FKdV equation.
A general Lagrange multiplier method is constructed to solve non-linear problems, which was first proposed to solve problems in quantum mechanics \[24\]. The VIM is a modification of this Lagrange multiplier method \[25\]. The basic features of the VIM are as follows. The solution of a linear mathematical problem or the initial (boundary) condition of the nonlinear problem is used as initial approximation or trail function. A more highly precise approximation can be obtained using iteration correction functional. Considering a nonlinear partial differential equation consists of a linear part $\overset{\symbol{94}}{L}U(x,t)$, nonlinear part $\overset{\symbol{94}}{N}U(x,t)$ and a free term $f(x,t)$ represented as
$$\overset{\symbol{94}}{L}U(x,t)+\overset{\symbol{94}}{N}U(x,t)=f(x,t)\text{,}
\tag{26}$$
where $\overset{\symbol{94}}{L}$ is the linear operator and $\overset{\symbol{94}}{N}$ is the nonlinear operator. According to the VIM, the ($n+1$) approximation solution of (26) can be given by the iteration correction functional as \[24, 25\]
$$U_{n+1}(x,t)=U_{n}(x,t)+\int_{0}^{t}d\tau \lambda (\tau )[\overset{\symbol{94}}{L}U_{n}(x,\tau )+\overset{\symbol{94}}{N}\hat{U}_{n}(x,\tau )-f(x,\tau )]\text{, }n\geq 0\text{,} \tag{27}$$
where $\lambda (\tau )$ is a Lagrangian multiplier and $\hat{U}_{n}(x,\tau )$ is considered as a restricted variation function, i. e; $\delta \hat{U}_{n}(x,\tau )=0$. Extreme the variation of the correction functional (27) leads to the Lagrangian multiplier $\lambda (\tau )$. The initial iteration can be used as the solution of the linear part of (26) or the initial value $U(x,0)$. As n tends to infinity, the iteration leads to the exact solution of (26), i. e;
$$U(x,t)=\underset{n\rightarrow \infty }{\lim }U_{n}(x,t)\text{.} \tag{28}$$
For linear problems, the exact solution can be given using this method in only one step where its Lagrangian multiplier can be exactly identified.
Time-fractional KdV equation Solution
=====================================
The time-FKdV equation represented by (24) can be solved using the VIM by the iteration correction functional (27) as follows:
Affecting from left by the fractional operator on (24) leads to
$$\begin{aligned}
\frac{\partial }{\partial \tau }\Phi (\xi ,\tau ) &=&~_{0}^{R}D_{\tau }^{\
\alpha -1}\Phi (\xi ,\tau )|_{\tau =0}\frac{\tau ^{\alpha -2}}{\Gamma
(\alpha -1)} \notag \\
&&-\ _{0}^{R}D_{\tau }^{\ 1-\alpha }[A~\Phi (\xi ,\tau )\frac{\partial }{\partial \xi }\Phi (\xi ,\tau )+B~\frac{\partial ^{3}}{\partial \xi ^{3}}\Phi (\xi ,\tau )]\text{, } \notag \\
0 &\leq &\alpha \leq 1\text{, }\tau \in \lbrack 0,T_{0}]\text{,} \TCItag{29}\end{aligned}$$
where the following fractional derivative property is used \[18, 19\]
$$\begin{aligned}
\ _{a}^{R}D_{b}^{\ \alpha }[\ _{a}^{R}D_{b}^{\ \beta }f(t)]
&=&~_{a}^{R}D_{b}^{\ \alpha +\beta }f(t)-\overset{k}{\underset{j=1}{\sum }}\
_{a}^{R}D_{b}^{\ \beta -j}f(t)|_{t=a}~\frac{(t-a)^{-\alpha -j}}{\Gamma
(1-\alpha -j)}\text{, } \notag \\
k-1 &\leq &\beta <k\text{.} \TCItag{30}\end{aligned}$$
As $\alpha <1$, the Riesz fractional derivative $_{0}^{R}D_{\tau }^{\ \alpha
-1}$ is considered as Riesz fractional integral $_{0}^{R}I_{\tau }^{1-\alpha
}$ that is defined by \[18, 19\]
$$\ _{0}^{R}I_{\tau }^{\ \alpha }f(t)=\frac{1}{2}[_{0}I_{\tau }^{\ \alpha
}f(t)\ +~_{\tau }I_{b}^{\ \alpha }f(t)]=\frac{1}{2}\frac{1}{\Gamma (\alpha )}\int_{a}^{b}d\tau |t-\tau |^{\alpha -1}f(\tau )\text{, }\alpha >0\text{.}
\tag{31}$$
where $_{0}I_{\tau }^{\ \alpha }f(t)$ and $_{\tau }I_{b}^{\ \alpha }f(t)$ are the left and right Riemann-Liouvulle fractional integrals, respectively \[18, 19\].
The iterative correction functional of equation (29) is given as
$$\begin{aligned}
\Phi _{n+1}(\xi ,\tau ) &=&\Phi _{n}(\xi ,\tau )+\int_{0}^{\tau }d\tau
^{\prime }\lambda (\tau ^{\prime })\{\frac{\partial }{\partial \tau ^{\prime
}}\Phi _{n}(\xi ,\tau ^{\prime }) \notag \\
&&-~_{0}^{R}I_{\tau ^{\prime }}^{1-\alpha }\Phi _{n}(\xi ,\tau ^{\prime
})|_{\tau ^{\prime }=0}\frac{\tau ^{\prime \alpha -2}}{\Gamma (\alpha -1)}
\notag \\
&&+\ _{0}^{R}D_{\tau ^{\prime }}^{\ 1-\alpha }[A~\overset{\symbol{126}}{\Phi
_{n}}(\xi ,\tau ^{\prime })\frac{\partial }{\partial \xi }\overset{\symbol{126}}{\Phi _{n}}(\xi ,\tau ^{\prime })+B~\frac{\partial ^{3}}{\partial \xi
^{3}}\overset{\symbol{126}}{\Phi _{n}}(\xi ,\tau ^{\prime })]\}\text{,}
\TCItag{32}\end{aligned}$$
where $n\geq 0$ and the function $\overset{\symbol{126}}{\Phi _{n}}(\xi
,\tau )$ is considered as a restricted variation function, i. e; $\delta
\overset{\symbol{126}}{\Phi _{n}}(\xi ,\tau )=0$. The extreme of the variation of (32) using the restricted variation function leads to
$$\begin{aligned}
\delta \Phi _{n+1}(\xi ,\tau ) &=&\delta \Phi _{n}(\xi ,\tau
)+\int_{0}^{\tau }d\tau ^{\prime }\lambda (\tau ^{\prime })~\delta \frac{\partial }{\partial \tau ^{\prime }}\Phi _{n}(\xi ,\tau ^{\prime }) \\
&=&\delta \Phi _{n}(\xi ,\tau )+\lambda (\tau )~\delta \Phi _{n}(\xi ,\tau
)-\int_{0}^{\tau }d\tau ^{\prime }\frac{\partial }{\partial \tau ^{\prime }}\lambda (\tau ^{\prime })~\delta \Phi _{n}(\xi ,\tau ^{\prime })=0\text{.}\end{aligned}$$
This relation leads to the stationary conditions $1+\lambda (\tau )=0$ and $\frac{\partial }{\partial \tau ^{\prime }}\lambda (\tau ^{\prime })=0$, which leads to the Lagrangian multiplier as $\lambda (\tau ^{\prime })=-1$.
Therefore, the correction functional (32) is given by the form
$$\begin{aligned}
\Phi _{n+1}(\xi ,\tau ) &=&\Phi _{n}(\xi ,\tau )-\int_{0}^{\tau }d\tau
^{\prime }\{\frac{\partial }{\partial \tau ^{\prime }}\Phi _{n}(\xi ,\tau
^{\prime }) \notag \\
&&-~_{0}^{R}I_{\tau ^{\prime }}^{1-\alpha }\Phi _{n}(\xi ,\tau ^{\prime
})|_{\tau ^{\prime }=0}\frac{\tau ^{\prime \alpha -2}}{\Gamma (\alpha -1)}
\notag \\
&&+\ _{0}^{R}D_{\tau ^{\prime }}^{\ 1-\alpha }[A~\Phi _{n}(\xi ,\tau
^{\prime })\frac{\partial }{\partial \xi }\Phi _{n}(\xi ,\tau ^{\prime })+B~\frac{\partial ^{3}}{\partial \xi ^{3}}\Phi _{n}(\xi ,\tau ^{\prime })]\}\text{,} \TCItag{33}\end{aligned}$$
where $n\geq 0$.
In Physics, if $\tau $ denotes the time-variable, the right Riemann-Liouville fractional derivative is interpreted as a future state of the process. For this reason, the right-derivative is usually neglected in applications, when the present state of the process does not depend on the results of the future development \[3\]. Therefore, the right-derivative is used equal to zero in the following calculations.
The zero order correction of the solution can be taken as the initial value of the state variable, which is taken in this case as
$$\Phi _{0}(\xi ,\tau )=\Phi (\xi ,0)=A_{0}\sec \text{h}^{2}(c\xi )\text{.}
\tag{34}$$
where $A_{0}$ and $c$ are constants.
Substituting this zero order approximation into (33) and using the definition of the fractional derivative (25) lead to the first order approximation as
$$\begin{aligned}
\Phi _{1}(\xi ,\tau ) &=&A_{0}\sec \text{h}^{2}(c\xi ) \notag \\
&&+2A_{0}c~\sinh (c\xi )~\sec \text{h}^{3}(c\xi )[4c^{2}B \notag \\
&&+(A_{0}A-12c^{2}B)\sec h^{2}(c\xi )]\frac{\tau ^{\alpha }}{\Gamma (\alpha
+1)}\text{.} \TCItag{35}\end{aligned}$$
Substituting this equation into (33), using the definition (25) and the Maple package lead to the second order approximation in the form
$$\begin{aligned}
\Phi _{2}(\xi ,\tau ) &=&A_{0}\sec \text{h}^{2}(c\xi ) \notag \\
&&+2A_{0}c~\sinh (c\xi )~\sec \text{h}^{3}(c\xi
)[4c^{2}B+(A_{0}A-12c^{2}B)\sec h^{2}(c\xi )]\frac{\tau ^{\alpha }}{\Gamma
(\alpha +1)} \notag \\
&& \TCItag{36}\end{aligned}$$
The higher order approximations can be calculated using the Maple or the Mathematica package to the appropriate order where the infinite approximation leads to the exact solution.
Results and calculations
========================
For small amplitude electron-acoustic waves, the time fractional Korteweg-de Vries equation has been derived. The Riemann-Liouvulle fractional derivative \[18, 19\] is used to describe the time fractional operator in the FKdV equation. He’s variational-iteration method \[24, 25\] is used to solve the derived time-FKdV equation.
To make our result physically relevant, numerical studies have been made using plasma parameters close to those values corresponding to the dayside auroral zone \[15, 34\].
However, since one of our motivations was to study effects of time fractional order ($\alpha $), the energetic population parameter ($\delta $) and the hot to cold electron equilibrium densities ratio ($\gamma $) on the existence of solitary waves. The electrostatic potential as a function of space variable $\xi $ and time variable $\tau $ is represented in Figure (1) that has a single rarefactive soliton shape. In Figure (2), the relation between the hot to cold electron equilibrium densities ratio ($\gamma $) and the amplitude of the electrostatic potential solitary wave $|\Phi (0,\tau )|$ is obtained at different values of the time variable $\tau $. It is seen that the soliton amplitude decreases with the increase of ($\gamma $). In addition the soliton amplitude $|\Phi (0,\tau )|$ against the fractional order ($\alpha $) is represented in Figure (3) at different time variable values. It is seen that $|\Phi (0,\tau )|$ increases with the increase of ($\alpha $). Figure (4) shows the relation between $|\Phi (0,\tau )|$ and the energetic population parameter ($\delta $). The effect of the fractional order ($\alpha $) on the electrostatic soliton amplitude $|\Phi (0,\tau )|$ for different values of the energetic population parameter ($\delta $) is given in Figure (5). To compare our result (the amplitude of the electrostatic potential $|\Phi (0,\tau )|$) with that observed in the auroral zone, we choose a set of available parameters corresponding to the dayside auroral zone where an electric field amplitude $E_{0}=100$ $\unit{mV}/\unit{m}$ has been observed \[15, 34\] with $T_{c}=5\unit{eV}$, $T_{h}=250\unit{eV}$, $n_{c0}=0.5\unit{m}^{-3}$ and $n_{h0}=2.5\unit{m}^{-3}$. These parameters correspond to $\lambda _{Dh}\approx 7430\unit{cm}$ and the normalized electrostatic wave potential amplitude $\Phi _{0}=\frac{E_{0}\lambda _{Dh}e}{k_{B}T_{h}}\approx 0.03\unit{V}$, which is obtained for different values of ($\alpha $) and ($\delta $) \[cf. Figure (5)\]. This value of the electrostatic potential amplitude is given at values of ($\alpha $) and ($\delta $) as: ($\alpha =0.61$, $\delta =0.01$), ($\alpha =0.69$, $\delta =0.05$), ($\alpha =0.78$, $\delta =0.1$) and ($\alpha =0.9$, $\delta
=0.15$) at velocity $v=0.04$, $\gamma =5$ and time $\tau =10$.
In summery, it has been found that the amplitude of the electron acoustic solitary waves as well as the parametric regime where the solitons can exist are sensitive to the time fractional order.
We have stressed out that it is necessary to include the space fractional parameter. This is beyond the scope of the present paper and it will be include in a further work in electron-acoustic solitary wave. The application of our model might be particularly interesting in the auroral region.
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**Figure Captions:**
Fig. (1): The electrostatic potential $\Phi (\xi ,\tau )$ against $\xi $ and $\tau $ at $\gamma =5$, $v=0.04$, $\alpha =0.5$ and $\delta =0.1$.
Fig. (2): The amplitude of the electrostatic potential $|\Phi (0,\tau )|$ against $\gamma $ at different values of time for $v=0.04$, $\alpha =0.5$ and $\delta =0.1$.
Fig. (3): The amplitude of the electrostatic potential $|\Phi (0,\tau )|$ against $\alpha $ at different values of time for $\gamma =5$, $v=0.04$ and $\delta =0.1$.
Fig. (4): The amplitude of the electrostatic potential $|\Phi (0,\tau )|$ against $\delta $ at different values of time for $\gamma =5$, $v=0.04$ and $\alpha =0.5$.
Fig. (5): The amplitude of the electrostatic potential $|\Phi (0,\tau )|$ against $\alpha $ at different values of $\delta $ for $v=0.04$, $\tau =10$ and $\gamma =5$.
|
---
abstract: 'Using a method of Korobenko, Maldonado and Rios we show a new characterization of doubling metric-measure spaces supporting Poincaré inequalities without assuming a priori that the measure is doubling.'
address:
- 'Ryan Alvarado,Department of Mathematics and Statistics, Amherst College, 502 Seeley Mudd, Amherst, Massachusetts 01002'
- 'Piotr Hajłasz,Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260'
author:
- Ryan Alvarado
- Piotr Hajłasz
title: 'A note on metric-measure spaces supporting Poincaré inequalities'
---
[*Dedicated to Professor Vladimir Maz’ya on the occasion of his 80th birthday.*]{}
Non-smooth functions have played a key role in analysis since the nineteenth century. One fundamental development in this vein came with the introduction of Sobolev spaces, which turned out to be a key tool in studying nonlinear partial differential equations and calculus of variations. Although classically Sobolev functions themselves were not smooth, they were defined on smooth objects such as domains in the Euclidean space or, more generally, Riemannian manifolds. By the late 1970s it became well recognized that several results in real analysis required little structure from the underlying ambient, and could be generalized to non-smooth settings, such as to the so-called spaces of homogeneous type. The latter spaces are metric spaces equipped with a doubling Borel measure (see [@CoWe71; @CoWe77]). In fact, maximal functions, Hardy spaces, functions of bounded mean oscillation, and singular integrals of Calderón-Zygmund-type all continue to have a fruitful theory in context of spaces of homogeneous type. However, this rich theory was, in a sense, only zeroth-order analysis given that no derivatives were involved. The study of first-order analysis with suitable generalizations of derivatives, a fundamental theorem of calculus, and Sobolev spaces, in the setting of spaces of homogeneous type, was initiated in the 1990s. This area, known as analysis on metric spaces, has since grown into a multifaceted theory which continues to play an important role in many areas of contemporary mathematics. For an introduction to the subject we recommend [@AlMi; @AGS; @AT; @BB; @cheeger; @hajlasz; @SMP; @HKST; @HeinonenK; @shanmugalingam].
One of the main objects of study in analysis on metric spaces are so called spaces supporting Poincaré inequalities introduced in [@HeinonenK]. To define this notion, recall that a [*metric-measure space*]{} $(X,d,\mu)$ is a metric space $(X,d)$ with a Borel measure $\mu$ such that $0<\mu\big(B(x,r)\big)<\infty$ for all $x\in X$ and all $r\in(0,\infty)$. If the measure $\mu$ is [*doubling*]{}, i.e., there exists a constant $C\in(0,\infty)$ such that $\mu(2B)\leq C\mu(B)$ for all balls$B\subseteq X$, then we call $(X,d,\mu)$ a [*doubling metric-measure space*]{}. The notation $\tau B$ stands for the dilation of the ball $B$ by the factor $\tau\in(0,\infty)$, i.e., $\tau B:=B(x,\tau r)$, $x\in X$, $r\in(0,\infty)$. A Borel function $g:X\to[0,\infty]$ is said to be an [*upper gradient*]{} of another Borel function $u:X\to\mathbb{R}$ if $$|u(x)-u(y)|\leq\int_{\gamma_{xy}} g\,ds,$$ holds for each $x,y\in X$ and all rectifiable curves $\gamma_{xy}$ joining $x,y$. Finally, a metric-measure space $(X,d,\mu)$ is said to [*support a $p$-Poincaré inequality*]{}, $p\in[1,\infty)$, if there exist constants $C\in(0,\infty)$ and $\sigma\in[1,\infty)$ such that $$\label{ppoin}
\mvint_{B} |u-u_{B}|\,d\mu\leq Cr
\left(\,\, \mvint_{\sigma B}g^{p}\, d\mu\right)^{1/p},$$ whenever $B$ is a ball of radius $r\in(0,\infty)$, $u\in L^1_{\rm loc}(X,\mu)$, and $g:X\to[0,\infty]$ is an upper gradient of $u$. Here and in what follows the barred integral and $f_E$ stand for the integral average: $$f_E=\mvint_Ef\, d\mu =\frac{1}{\mu(E)}\int_E f\, d\mu,$$ where $E$ is a $\mu$-measurable set of positive measure. To be consistent with the definition of the upper gradient, in what follows we will always assume that functions $u\in L^1_{\rm loc}(X,\mu)$ are everywhere finite Borel representatives. The above definitions of the upper gradient and spaces supporting Poincaré inequalites are due to Heinonen and Koskela in [@HeinonenK] (see also [@HKST] for a more detailed exposition).
It was proved in [@SMP2] and [@SMP Theorem 5.1] that if a doubling metric-measure space supports a Poincaré inequality, then the $p$-Poincaré inequality self-improves in the sense that for some $q\in(p,\infty)$ and $C'\in(0,\infty)$, there holds $$\label{REW-1}
\left(\, \mvint_{B} |u-u_{B}|^{q}\, d\mu\right)^{1/q}\leq
C'r
\left(\,\, \mvint_{5\sigma B}g^{p}\, d\mu\right)^{1/p},$$ whenever $B$ is a ball of radius $r\in(0,\infty)$, $u\in L^1_{\rm loc}(X,\mu)$, and $g:X\to[0,\infty]$ is an upper gradient of $u$. The purpose of this note is to show that the family of inequalities in on a metric measure space imply that the underlying measure is doubling, and thus providing a characterization of doubling metric-measure spaces supporting Poincaré inequalities without assuming a priori that the measure is doubling, see Theorem \[EPequiv\], below. This result is a minor refinement of a beautiful result in [@korobenkomr], where it was proved that in a related context, a family of weak Sobolev inequalities imply that the measure is doubling. However, the authors considered Sobolev inequalities where the balls had the same radius on both sides, and such condition is stronger than the one in . Moreover, they did not address the important applications to Sobolev spaces supporting Poincaré inequalities.
While the proof presented below is almost the same as the one in [@korobenkomr], it is important to provide details: the proof employs an infinite iteration of Sobolev inequalities and since now we have balls of different size on both sides, it is not obvious without checking details that this will not cause estimates to blow up. This paper should be regarded as a supplement to the work of [@korobenkomr] and an advertisement of their work. Different, but related iterative arguments to the one presented below were used in [@carron; @gorka; @hajlaszkt1; @hajlaszkt2; @hebey] in the proofs that a Sobolev inequality implies a measure density condition. Another application of a method developed in [@korobenkomr] is given in a forthcoming paper [@AGH].
We now state the main result of this note.
\[EPequiv\] Let $(X,d,\mu)$ be a metric-measure space and fix $p\in[1,\infty)$. Then the following two statements are equivalent.
1. The measure $\mu$ is doubling and the space $(X,d,\mu)$ supports a $p$-Poincaré inequality. .08in
2. There exist $q\in(p,\infty)$, $C_P\in [1,\infty)$, and $\sigma\in[1,\infty)$ such that $$\label{eq20}
\left(\, \mvint_{B} |u-u_{B}|^{q}\, d\mu\right)^{1/q}\leq
C_Pr
\left(\,\, \mvint_{\sigma B}g^{p}\, d\mu\right)^{1/p},$$ whenever $B$ is a ball of radius $r\in(0,\infty)$, $u\in L^1_{\rm loc}(X,\mu)$, and $g:X\to[0,\infty]$ is an upper gradient of $u$.
We could assume that holds with $C_P\in (0,\infty)$, but the estimates presented below are more elegant if $C_P\geq 1$. Clearly, if holds with a constant strictly greater than zero, then we can increase it to a constant greater than or equal to $1$.
A positive locally integrable function $0<w\in L^1_{\rm loc}({\mathbb R}^n)$ defines an absolutely continuous measure $d\mu=w(x)\, dx$ with the weight $w$. A class of the so called [*$p$-admissible*]{} weights plays a fundamental role in the nonlinear potential theory [@HKM]. To make the presentation brief, we will not recall the definition of a $p$-admissible weight, but we refer the reader to [@HKM] for details. As an immediate consequence of Theorem \[EPequiv\] and [@SMP2 Theorem 2] we obtain a new characterization of $p$-admissible weights. A variant of this result has also been proved in [@korobenkomr].
A function $0<w\in L^1_{\rm loc}({\mathbb R}^n)$ is a $p$-admissible weight for some $1<p<\infty$, if and only if there exist $q\in (p,\infty)$, $C\in [1,\infty)$ and $\sigma\in [1,\infty)$ such that $$\left(\,\mvint_B |u-u_B|^q\, d\mu\right)^{1/q}\leq
Cr\left(\,\mvint_{\sigma B} |\nabla u|^p\, d\mu\right)^{1/p}$$ whenever $B\subset{\mathbb R}^n$ is a ball of radius $r\in (0,\infty)$, $u\in C^\infty(\sigma B)$ and $d\mu=w\, dx$.
The implication [*(a)*]{} $\Rightarrow$ [*(b)*]{} follows immediately from [@SMP2 Theorem 1]. Note however, that the constant $\sigma$ in might be larger than that in the $p$-Poincaré inequality (see ). Thus we will focus on proving that [*(a)*]{} follows from [*(b)*]{}. To this end, suppose that $X$ satisfies the condition displayed in . Making use of Hölder’s inequality and the fact that $1\leq p<q$, we may conclude that the $(q,p)$-Poincaré inequality in implies that the space $(X,d,\mu)$ supports a $p$-Poincaré inequality (see ).
There remains to show that the condition in forces the measure $\mu$ to be doubling. Fix a ball $B:=B(x,r)$, $x\in X$, $r\in(0,\infty)$, and observe that specializing to the case when $B$ is replaced by $2\sigma B$ yields $$\label{eq-JK2}
\left(\,\, \mvint_{2\sigma B} |u-u_{2\sigma B}|^q\, d\mu\right)^{1/q}
\leq 2C_P\sigma r\left(\,\, \mvint_{2\sigma^2 B} g^p\, d\mu\right)^{1/p},$$ whenever $u\in L^{1}_{\rm loc}(X,\mu)$ and $g:X\to[0,\infty]$ is an upper gradient of $u$. Since $p\geq1$, it follows from and Hölder’s inequality that, $$\begin{aligned}
\label{eq-JK3}
\left(\,\, \mvint_{2\sigma B} |u|^q\, d\mu\right)^{1/q}
&\leq
\left(\,\, \mvint_{2\sigma B} |u-u_{2\sigma B}|^q\, d\mu\right)^{1/q}
+|u_{2\sigma B}|
\nonumber\\[4pt]
&\leq 2\sigma rC_P \left(\,\, \mvint_{2\sigma^2 B} g^p\, d\mu\right)^{1/p}+
\left(\,\, \mvint_{2\sigma B} |u|^p\, d\mu\right)^{1/p}.\end{aligned}$$ We now define a collection of functions $\{u_j\}_{j\in\mathbb{N}}$ as follows: for each fixed $j\in\mathbb{N}$, let $r_j:=(2^{-j-1}+2^{-1})r$ and set $B_j:=B(x,r_j)$. Then $$\label{JG-1}
\frac{1}{2}r<r_{j+1}<r_j\leq\frac{3}{4}r,
\quad\forall\,j\in\mathbb{N}.$$ For each $j\in\mathbb{N}$, let $u_j:X\to\mathbb{R}$ be the function defined by setting for each $y\in X$, $$\begin{aligned}
\label{udef}
u_j(y):=
\left\{
\begin{array}{ll}
\,\qquad 1\quad &\mbox{if $y\in B_{j+1}$,}
\\[6pt]
\displaystyle\frac{r_j-d(x,y)}{r_j-r_{j+1}}
&\mbox{if $y\in B_j\setminus B_{j+1}$,}
\\[15pt]
\,\qquad 0 &\mbox{if $y\in X\setminus B_j$.}
\end{array}
\right.\end{aligned}$$ Noting that $\displaystyle (r_j-r_{j+1})^{-1}=2^{j+2}r^{-1}$, a straightforward computation will show that $u_j$ is $2^{j+2}r^{-1}$-Lipschitz on $X$ and that the function $g_j:=2^{j+2}r^{-1}\chi_{B_j}$ is an upper gradient of $u$, where $\chi_{B_j}$ denotes the characteristic function of the set $B_j$. In particular, we have that $u_j\in L^{1}_{\rm loc}(X,\mu)$ and that the functions $u_j$ and $g_j$ satisfy . Observe that for each fixed $j\in\mathbb{N}$, we have (keeping in mind $\sigma\geq1$) $$\begin{aligned}
\label{HD-1}
2\sigma rC_P\left(\,\, \mvint_{2\sigma^2 B} g_j^p\, d\mu\right)^{1/p}
= \sigma C_P2^{j+3}\left(\frac{\mu(B_j)}{\mu(2\sigma^2 B)}\right)^{1/p}
\leq \sigma C_P 2^{j+3}\left(\frac{\mu(B_j)}{\mu(2\sigma B)}\right)^{1/p}\end{aligned}$$ and $$\begin{aligned}
\label{HD-2}
\left(\,\, \mvint_{2\sigma B}|u_j|^p\, d\mu\right)^{1/p}
\leq \left(\frac{\mu(B_j)}{\mu(2\sigma B)}\right)^{1/p}.\end{aligned}$$ Moreover, $$\label{HD-3}
\left(\,\, \mvint_{2\sigma B} |u_j|^{q}\, d\mu\right)^{1/q}\geq\left(\frac{\mu(B_{j+1})}{\mu(2\sigma B)}\right)^{1/q}.$$ In concert, - and the extreme most sides of the inequality in , give $$\begin{aligned}
\label{HD-4}
\left(\frac{\mu(B_{j+1})}{\mu(2\sigma B)}\right)^{1/q}
&\leq\sigma C_P 2^{j+4}
\left(\frac{\mu(B_j)}{\mu(2\sigma B)}\right)^{1/p},
\quad\forall\,j\in\mathbb{N}.\end{aligned}$$ Therefore $$\begin{aligned}
\label{HD-42}
\mu(B_{j+1})^{1/q}
\leq\sigma C_P 2^{j+4}\frac{\mu(B_j)^{1/p}}{\mu(2\sigma B)^{(q-p)/pq}},
\quad\forall\,j\in\mathbb{N}.\end{aligned}$$ With $\alpha:=q/p\in(1,\infty)$ we raise both sides of the inequality in to the power $p/\alpha^{j-1}$ in order to obtain $$\begin{aligned}
\label{HD-5}
\mu(B_{j+1})^{1/\alpha^{j}}\leq
2^{p(j+4)/\alpha^{j-1}}
\bigg(\frac{\sigma C_P}{\mu(2\sigma B)^{(q-p)/pq}}\bigg)^{p/\alpha^{j-1}}\mu(B_j)^{1/\alpha^{j-1}},
\quad\forall\,j\in\mathbb{N}.\end{aligned}$$ If we let $P_j:=\mu(B_j)^{1/\alpha^{j-1}}$, then the inequality in becomes $$\begin{aligned}
\label{HD-6}
P_{j+1}\leq
2^{p(j+4)/\alpha^{j-1}}
\bigg(\frac{\sigma C_P}{\mu(2\sigma B)^{(q-p)/pq}}\bigg)^{p/\alpha^{j-1}}P_j,
\quad\forall\,j\in\mathbb{N},\end{aligned}$$ which, together with an inductive argument and the fact that $P_1\leq\mu(B)$, implies $$\begin{aligned}
\label{HD-7}
P_{j+1}&\leq P_1\prod_{k=1}^j
\left[2^{p(k+4)/\alpha^{k-1}}
\bigg(\frac{\sigma C_P}{\mu(2\sigma B)^{(q-p)/pq}}\bigg)^{p/\alpha^{k-1}}\right]
\nonumber\\[4pt]
&\leq \mu(B)\prod_{k=1}^j
\left[2^{p(k+4)/\alpha^{k-1}}
\bigg(\frac{\sigma C_P}{\mu(2\sigma B)^{(q-p)/pq}}\bigg)^{p/\alpha^{k-1}}\right],
\quad\forall\,j\in\mathbb{N}.\end{aligned}$$
We claim that the product in converges as $j\to\infty$. Indeed, observe that $$\begin{aligned}
\label{QW-1}
\prod_{k=1}^\infty \bigg(\frac{\sigma C_P}{\mu(2\sigma B)^{(q-p)/pq}}\bigg)^{p/\alpha^{k-1}}
&=\bigg(\frac{\sigma C_P}{\mu(2\sigma B)^{(q-p)/pq}}\bigg)^{p\sum_{k=1}^\infty\alpha^{1-k}}
\nonumber\\[4pt]
&=
\bigg(\frac{\sigma C_P}{\mu(2\sigma B)^{(q-p)/pq}}\bigg)^{\frac{p\alpha}{\alpha-1}}
=\bigg(\frac{\sigma C_P}{\mu(2\sigma B)^{(q-p)/pq}}\bigg)^{\frac{pq}{q-p}},\end{aligned}$$ and $$\label{QW-3}
\prod_{k=1}^\infty\big(2^{p(k+4)}\big)^{1/\alpha^{k-1}}
=2^{\sum_{k=1}^\infty p(k+4)\alpha^{1-k}}=:A(p,q)\in(0,\infty).$$ On the other hand, it follows from that $$0<\mu(2^{-1}B)^{1/\alpha^{j-1}}\leq
P_j=\mu(B_j)^{1/\alpha^{j-1}}\leq \mu(B)^{1/\alpha^{j-1}}<\infty,$$ which, in turn, further implies $\displaystyle\lim\limits_{j\to\infty}P_j=1$. Consequently, passing to the limit in yields $$\begin{aligned}
\label{ME12}
1&\leq \mu(B)\frac{\big(\sigma C_P\big)^{pq/(q-p)}}{\mu(2\sigma B)}A(p,q).\end{aligned}$$ Hence, $$\begin{aligned}
\label{ME13}
\mu(2\sigma B)\leq \big(\sigma C_P\big)^{pq/(q-p)}A(p,q)\,
\mu(B).\end{aligned}$$ Since $\sigma\geq1$, it follows that $\mu$ is doubling. This finishes the proof of the second implication and, in turn, the proof of the theorem.
In the proof of the [*(b)*]{} $\Rightarrow$ [*(a)*]{} in Theorem \[EPequiv\], one can compute the constant $A(p,q)$ appearing in by observing that (keeping in mind $\alpha=q/p$), $$\begin{aligned}
{\sum_{k=1}^\infty p(k+4)\alpha^{1-k}}
&=p\sum_{k=1}^\infty \frac{k}{\alpha^{k-1}}+4p\sum_{k=1}^\infty \frac{1}{\alpha^{k-1}}
\nonumber\\[4pt]
&=\frac{p}{(1-1/\alpha)^2}+\frac{4p\alpha}{\alpha-1}
=\frac{pq^2}{(q-p)^2}+\frac{4pq}{q-p}.\end{aligned}$$ Therefore, $$A(p,q)=2^{\frac{pq^2}{(q-p)^2}+\frac{4pq}{q-p}}.$$ Hence, condition implies that measure $\mu$ satisfies the following doubling condition: $$\mu(2B)\leq\Big(\sigma C_P2^{\frac{q}{(q-p)}+4}\Big)^{pq/(q-p)}\mu(B)\quad
\mbox{for all balls\,\,$B\subseteq X$.}$$
[99]{} Sobolev embedding for $M^{1,p}$ spaces is equivalent to a lower bound of the measure. [*Preprint.*]{} Hardy spaces on Ahlfors-regular quasi metric spaces. A sharp theory. Lecture Notes in Mathematics, 2142. Springer, Cham, 2015. Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. Oxford Lecture Series in Mathematics and its Applications, 25. Oxford University Press, Oxford, 2004. EMS Tracts in Mathematics, 17. European Mathematical Society (EMS), Zürich, 2011. Inégalités isopérimétriques et inégalités de Faber-Krahn. Sémin. Théor. Spectr. Géom., 13, Année 1994–1995, pp. 63–66, Univ. Grenoble I, Saint-Martin-d’Hères, 1995. Differentiability of Lipschitz functions on metric measure spaces. [*Geom. Funct. Anal.*]{} 9 (1999), 428–517. Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes, Lecture Notes in Mathematics, Vol.242, Springer-Verlag, 1971. Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), no.4, 569–645. In metric-measure spaces Sobolev embedding is equivalent to a lower bound for the measure. [*Potential Anal.*]{} 47 (2017), 13–19. Sobolev spaces on metric-measure spaces. In: [*Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002)*]{}, pp. 173–218, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003. Sobolev met Poincaré. [*Mem. Amer. Math. Soc.*]{} 145 (2000), no. 688. Sobolev meets Poincaré. [*C. R. Acad. Sci. Paris Sér. I Math.*]{} 320 (1995), 1211–1215. Sobolev embeddings, extensions and measure density condition. [*J. Funct. Anal.*]{} 254 (2008), 1217–1234. Measure density and extendability of Sobolev functions. [*Rev. Mat. Iberoam*]{} 24 (2008), 645–669. Lecture Notes in Mathematics, 1635. Springer-Verlag, Berlin, 1996. New Mathematical Monographs, 27. Cambridge University Press, Cambridge, 2015. Quasiconformal maps in metric spaces with controlled geometry. [*Acta Math.*]{} 181 (1998), 1–61. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. From Sobolev inequality to doubling. [*Proc. Amer. Math. Soc.*]{} 143 (2015), 4017–4028. Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. [*Rev. Mat. Iberoamericana*]{} 16 (2000), 243–279.
|
---
abstract: 'We derive an analytic formula using the Mueller matrix formalism that parameterizes the non-idealities of a half-wave plate (HWP) made from dielectric AR-coated birefringent slabs. This model accounts for frequency-dependent effects at normal incidence, including effects driven by the reflections at dielectric boundaries. The model also may be used to guide the characterization of an instrument that uses a HWP. We discuss the coupling of a HWP to different source spectra, and the potential impact of that effect on foreground removal for the SPIDER CMB experiment. We also describe a way to use this model in a map-making algorithm that fully corrects for HWP non-idealities.'
author:
- 'Sean A. Bryan'
- 'Thomas E. Montroy'
- 'John E. Ruhl'
title: 'Modeling dielectric half-wave plates for CMB polarimetry using a Mueller matrix formalism'
---
Introduction
============
CMB polarization encodes information about Inflation [@baumann09], re-ionization [@zaldarriaga08], and the large-scale structure of the universe [@smith08]. Many upcoming experiments including SPIDER [@crill08], EBEX [@sagiv10], POLARBEAR [@lee08], Keck [@sheehy10], ABS [@essinger-hileman09] and others will use a half-wave plate (HWP) to modulate the polarization state of light from the sky before measurement by the detectors. Such HWPs may be periodically stepped (eg between maps of the same part of the sky) to reduce the effect of beam asymmetries and instrumental polarization of optical elements skyward of the HWP. Alternatively, they can be continuously rotated to modulate the signal and to also reject atmospheric variations and 1/f noise [@johnson06].
An ideal HWP with one of its crystal axes oriented at an angle $\theta_{hwp}$ to the plane of polarization of incident light rotates that polarization plane by $2\theta_{hwp}$ as the light passes through it. Real HWPs made from birefringent materials have several important non-idealities. Since the phase delay is only a half-wave at a single frequency, the exact angle through which a HWP rotates the polarization state is frequency-dependent. Additionally, reflections from the material interfaces reduce transmission and induce non-ideal rotation; to minimize these effects anti-reflection (AR) coatings are typically used on the surfaces of the HWP. However, AR coatings are frequency dependent and do not fully eliminate these non-ideal behaviors.
These effects have been treated in a variety of ways by other authors. O’Dea et. al. [@odea07] and Brown et. al. [@brown09] used a Mueller matrix formalism to parameterize a HWP and polarized detector, but did not connect their parameterization with a physical model of a HWP. Savini et. al. [@savini06] described in detail a physical model for stacks of dielectric birefringent materials. Their model works directly with the electric fields, which allows it to handle the input and output polarization state of the light, multiple reflections from dielectric interfaces, and the finite bandwidth over which the stack is a half-wave retarder. Matsumura [@matsumura06] modeled a multiple-layer HWP using a similar approach. We employ the methods of Savini et. al. [@savini06] in this paper, using Jones and Mueller matrix methods to derive exact couplings from sources (of arbitrary but known spectra) on the sky to a detector, through a non-ideal HWP made of any number of dielectric layers. Our resulting model can be evaluated quickly in observing simulations such as those done for SPIDER [@mactavish08], an important property for upcoming CMB experiments. Additionally, we show how to use this model to correct for known HWP non-idealities during the mapmaking process (using an input data timestream to make a map of intensity and polarization on the sky), greatly reducing the potential impact of systematic effects induced by such non-idealities.
The HWP Mueller Matrix
======================
Our goal is to calculate the combined Mueller matrix of a HWP plus polarized-detector system for arbitrary orientations of the instrument and HWP relative to the coordinates defining the Stokes parameters of the incoming radiation, including the effects of reflections at the various dielectric interfaces and the effect of band averaging. A partially polarized detector can be modeled as a partial linear polarizer with Mueller matrix $\mathsf{M}_{pol}$ followed by a total power detector that is sensitive only to the $I$ Stokes parameter. The total Mueller matrix $\mathsf{M}$ is found by combining the Mueller matrices of the partial polarizer and the HWP, taking rotational alignments into account. The total detected power $d$ for incident light with a Stokes vector $S = [I, Q, U, V]$ is given by summing over the top row the product of $\mathsf{M}$ and $S$ [@jones08], $$\label{eqn_one}
d = I M_{II} + Q M_{IQ} + U M_{IU} + V M_{IV} ,$$ where the $M_{XX}$ are elements of the top row of the total Mueller matrix.
Since physical models of the action of a birefringent HWP are constructed using electric fields, we start the calculation in the Jones formalism. A Jones matrix is a 2-by-2 matrix of complex numbers that that describes the action of an optical system on the $x$- and $y$-components of the electric field of an incident plane wave.
We start by considering the Jones matrix of a general retarder, $$\label{j_hwp}
\mathsf{J}_{ret}(f) = \left[ \begin{array}{cc} a(f) & \epsilon_1(f) \\ \epsilon_2(f) & b(f) e^{i \phi(f)} \end{array} \right],$$ where $a(f)$, $b(f)$, and $\phi(f)$ are real, and $\epsilon_1(f)$ and $\epsilon_2(f)$ are small and complex [@odea07]. To convert this to a Mueller matrix, we follow Jones et. al. [@jones08] and use $$\label{jones2mueller}
M_{ij} = \frac{1}{2} \mathrm{trace} ( \sigma_i \mathsf{J} \sigma_j \mathsf{J}^\dagger )$$ from Born and Wolf [@born], where $\sigma_i$ are the Pauli matrices, $$\begin{aligned}
\begin{array}{cc}
\sigma_1 = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] &
\sigma_2 = \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right] \\
& \\
\sigma_3 = \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right] &
\sigma_4 = \left[ \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right]. \\
\end{array}\end{aligned}$$ This will yield the Mueller matrix as a function of frequency $\mathsf{M}_{ret}(f)$ of the HWP. Since Mueller matrices can be band-averaged, we integrate this single-frequency Mueller matrix against a CMB or foreground spectrum $S(f)$, as well as the detector passband $F(f)$. This gives the band-averaged Mueller matrix $$\label{band_average}
\mathsf{M}_{HWP} = \frac{\int df \mathsf{M}_{ret}(f) S(f) F(f)}{\int df S(f) F(f)}.$$
Single-Plate HWP
----------------
For a HWP made from a single layer of birefringent dielectric material, $x$- and $y$-polarized states defined in the crystal axes cannot couple into each other. This means that in the HWP Jones matrix of Eq. \[j\_hwp\], $\epsilon_1(f)~=~\epsilon_2(f)~=~0$. This leaves only three parameters $a(f)$, $b(f)$ and $\phi(f)$ that are necessary to completely characterize the HWP. The transmission coefficients $a(f)$ and $b(f)$ vary with frequency because of the passband of the AR coating and the interference of multiple reflections inside the birefringent layer. The relative phase delay $\phi(f)$ also varies with frequency because the path length difference for polarization states traveling along the slow and fast crystal axes is $(n_s - n_f) d = \frac{\phi(f)}{2 \pi} \frac{c}{f} $.
Applying Eq. \[jones2mueller\] to the Jones matrix of a single-plate HWP $$\label{j_single_plate_hwp}
\mathsf{J}_{ret}(f) = \left[ \begin{array}{cc} a(f) & 0 \\ 0 & b(f) e^{i \phi(f)} \end{array} \right]$$ gives the corresponding Mueller matrix $\mathsf{M}_{ret}(f)$ $$\begin{aligned}
\label{m_ret}
\left[ \begin{array}{cccc} \frac{1}{2}(a^2 + b^2) & \frac{1}{2}(a^2 - b^2) & 0 & 0 \\ \frac{1}{2}(a^2 - b^2) & \frac{1}{2}(a^2 + b^2) & 0 & 0 \\ 0 & 0 & a b \cos(\phi) & - a b \sin(\phi) \\ 0 & 0 & a b \sin(\phi) & a b \cos(\phi) \end{array} \right],\end{aligned}$$ where $a$, $b$, and $\phi$ are all functions of frequency. This reduces to the result given in Tinbergen [@tinbergen96] for an ideal retarder ($a = b = 1$).
In general, the band-averaged HWP Mueller matrix $\mathsf{M}_{HWP}$ calculated using Eq. \[band\_average\] will not be of the same form as $\mathsf{M}_{ret}(f)$ and will have four (rather than three) independent non-zero elements, $$\label{m_hwp}
\mathsf{M}_{HWP} \equiv \left[ \begin{array}{cccc} T & \rho & 0 & 0 \\ \rho & T & 0 & 0 \\ 0 & 0 & c & -s \\ 0 & 0 & s & c \end{array} \right].$$ For the specific case of the SPIDER 145 GHz HWP, we have calculated this Mueller matrix as a function of frequency, and plotted it in Fig. \[hwp\_mueller\]. Since the CMB is not expected to be circularly-polarized, in the case where the subsequent detector and optical system does not induce sensitivity to circular polarization the $s$ parameter will not be relevant for CMB polarimetry.
{width="100.00000%"}
Multiple-Plate HWP
------------------
For a multiple-layer HWP, we cannot simply multiply together the Mueller matrices of several single-layer HWPs, because this would not account for multiple reflections among the interfaces between each layer. We instead return to the original Jones matrix of a general retarder in Eq. \[j\_hwp\]. For a multiple-layer HWP, $\epsilon_1$ and $\epsilon_2$ will be small, but still non-zero. This means that all seven of the parameters of the Jones matrix, $a(f)$, $b(f)$, $\phi(f)$, ${\operatorname{Re}}[\epsilon_1(f)]$, ${\operatorname{Im}}[\epsilon_1(f)]$, ${\operatorname{Re}}[\epsilon_2(f)]$, and ${\operatorname{Im}}[\epsilon_2(f)]$ are required to characterize the HWP. After applying Eq. \[jones2mueller\], we obtain the corresponding Mueller matrix
$$\begin{aligned}
\mathsf{M}_{ret}(f) = \left[ \begin{array}{cccc}
\frac{1}{2} \left( a^{2} + b^{2} + |\epsilon_{1}|^{2} + |\epsilon_{2}|^{2} \right) & \frac{1}{2} \left( a^{2} - b^{2} - |\epsilon_{1}|^{2} + |\epsilon_{2}|^{2} \right) & \dots \\
\frac{1}{2} \left( a^{2} - b^{2} + |\epsilon_{1}|^{2} - |\epsilon_{2}|^{2} \right) & \frac{1}{2} \left( a^{2} + b^{2} - |\epsilon_{1}|^{2} - |\epsilon_{2}|^{2} \right) & \dots \\
a \cdot {\operatorname{Re}}[\epsilon_{2}] + b \cdot \left( {\operatorname{Re}}[\epsilon_{1}] \cos(\phi) + {\operatorname{Im}}[\epsilon_{1}] \sin(\phi) \right) & a \cdot {\operatorname{Re}}[\epsilon_{2}] - b \cdot \left( {\operatorname{Re}}[\epsilon_{1}] \cos(\phi) + {\operatorname{Im}}[\epsilon_{1}] \sin(\phi) \right) & \dots \\
a \cdot {\operatorname{Im}}[\epsilon_{2}] + b \cdot \left( {\operatorname{Re}}[\epsilon_{1}] \sin(\phi) - {\operatorname{Im}}[\epsilon_{1}] \cos(\phi) \right) & a \cdot {\operatorname{Im}}[\epsilon_{2}] - b \cdot \left( {\operatorname{Re}}[\epsilon_{1}] \sin(\phi) - {\operatorname{Im}}[\epsilon_{1}] \cos(\phi) \right) & \dots
\end{array} \right. \nonumber \\
\left. \begin{array}{cccc}
\textrm{Row~1~cont.} & a \cdot {\operatorname{Re}}[\epsilon_{1}] + b \cdot \left( {\operatorname{Re}}[\epsilon_{2}] \cos(\phi) + {\operatorname{Im}}[\epsilon_{2}] \sin(\phi) \right) & - a \cdot {\operatorname{Im}}[\epsilon_{1}] - b \cdot \left( {\operatorname{Re}}[\epsilon_{2}] \sin(\phi) - {\operatorname{Im}}[\epsilon_{2}] \cos(\phi) \right) \\
\textrm{Row~2~cont.} & a \cdot {\operatorname{Re}}[\epsilon_{1}] - b \cdot \left( {\operatorname{Re}}[\epsilon_{2}] \cos(\phi) + {\operatorname{Im}}[\epsilon_{2}] \sin(\phi) \right) & - a \cdot {\operatorname{Im}}[\epsilon_{1}] + b \cdot \left( {\operatorname{Re}}[\epsilon_{2}] \sin(\phi) - {\operatorname{Im}}[\epsilon_{2}] \cos(\phi) \right) \\
\textrm{Row~3~cont.} & {\operatorname{Re}}[\epsilon_{1}] \cdot {\operatorname{Re}}[\epsilon_{2}] + {\operatorname{Im}}[\epsilon_{1}] \cdot {\operatorname{Im}}[\epsilon_{2}] + a b \cos(\phi)& \textcolor{white}{-}{\operatorname{Re}}[\epsilon_{1}] \cdot {\operatorname{Im}}[\epsilon_{2}] - {\operatorname{Im}}[\epsilon_{1}] \cdot {\operatorname{Re}}[\epsilon_{2}] - a b \sin(\phi) \\
\textrm{Row~4~cont.} & {\operatorname{Re}}[\epsilon_{1}] \cdot {\operatorname{Im}}[\epsilon_{2}] - {\operatorname{Im}}[\epsilon_{1}] \cdot {\operatorname{Re}}[\epsilon_{2}] + a b \sin(\phi) & -{\operatorname{Re}}[\epsilon_{1}] \cdot {\operatorname{Re}}[\epsilon_{2}] - {\operatorname{Im}}[\epsilon_{1}] \cdot {\operatorname{Im}}[\epsilon_{2}] + a b \cos(\phi)
\end{array} \right].\end{aligned}$$
This reduces to the single plate result of Eq. \[m\_ret\] if $\epsilon_{1} = \epsilon_{2} = 0$. Note that none of the elements of this matrix have the same functional form. This means that after applying Eq. \[band\_average\] to band-average against the source and detector spectra, all 16 elements of the resulting Mueller matrix will be independent. Since the CMB is not expected to be circularly-polarized and the subsequent optical and detector system should have no sensitivity to circular polarization, only the top-left 9 elements of the Mueller matrix will be relevant for CMB polarimetry.
Rotating the Instrument and HWP
===============================
Jones et. al. [@jones08] modeled a polarization-sensitive detector as a rotatable instrument with a partial-polarizer followed by a total power detector. The Jones matrix of a vertical partial polarizer is $$\label{j_pol_a}
\mathsf{J}_{pol} = \left[ \begin{array}{cc} \eta & 0 \\ 0 & \delta \end{array} \right],$$ which can be turned into a corresponding Mueller matrix $$\begin{aligned}
\label{m_pol}
\mathsf{M}_{pol} = \left[ \begin{array}{cccc} \frac{1}{2}(\eta^2 + \delta^2) & \frac{1}{2}(\eta^2 - \delta^2) & 0 & 0 \\ \frac{1}{2}(\eta^2 - \delta^2) & \frac{1}{2}(\eta^2 + \delta^2) & 0 & 0 \\ 0 & 0 & \eta \delta & 0 \\ 0 & 0 & 0 & \eta \delta \end{array} \right].\end{aligned}$$ Given the instrument rotation matrix [@tinbergen96] $$\mathsf{M}_{\psi} = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \textcolor{white}{-}\cos{2 \psi_{inst}} & \sin{2 \psi_{inst}} & 0 \\ 0 & -\sin{2 \psi_{inst}} & \cos{2 \psi_{inst}} & 0 \\ 0 & 0 & 0 & 1 \end{array} \right],$$ the detected radiation is given by the coupling to $I$ in the product $\mathsf{M}_{pol}~\mathsf{M}_{\psi}$; for an ideal polarizer with $\eta = 1$ and $\delta = 0$ and an arbitrary instrument angle $\psi_{inst}$, the detector signal is $$d = \frac{1}{2} \left(I + Q \cos(2 \psi_{inst}) + U \sin(2 \psi_{inst}) \right).$$
The addition of a HWP to the instrument can be modeled with the Mueller matrix product $$\label{mat_multiply}
\mathsf{M} = \mathsf{M}_{pol}~\mathsf{M}_\xi~\mathsf{M}_{-\theta}~\mathsf{M}_{HWP}~\mathsf{M}_{\theta}~\mathsf{M}_{\psi},$$ where $\mathsf{M}_{\theta}$ is the rotation matrix by the HWP angle $\theta_{hwp}$, and $\mathsf{M}_{\xi}$ is the rotation matrix by the detector orientation angle $\xi_{det}$. Fig. \[instrument\_and\_hwp\_angles\] illustrates the definition of these angles; while only two angles are needed to define the single-detector problem, we use three here to make the problem more straightforward to visualize, and to easily accommodate calculations of focal planes with detectors at multiple orientation angles. Here we consider only a single detector with $\xi_{det} = 0$.
The top row of the resulting Mueller matrix for a single-plate HWP is $$\begin{aligned}
\label{hwp_matrix_elements}
M_{II} &=& \frac{1}{2} \left[ T (\eta^2 + \delta^2) + \rho \cos(2 \theta_{hwp}) (\eta^2 - \delta^2) \right] \nonumber \label{start_of_big_equation} \\
M_{IQ} &=& \textcolor{white}{-}\mathcal{F} \sin(2 \psi_{inst}) + \mathcal{G} \cos(2 \psi_{inst}) \nonumber \\
M_{IU} &=& - \mathcal{F} \cos(2 \psi_{inst}) + \mathcal{G} \sin(2 \psi_{inst}) \nonumber \\
M_{IV} &=& \frac{1}{2} s \sin(2 \theta_{hwp}) (\eta^2 - \delta^2),\end{aligned}$$ where $\mathcal{F}$ and $\mathcal{G}$ are $$\begin{aligned}
\mathcal{F} &\equiv& - \frac{1}{4} (T-c) \sin(4 \theta_{hwp}) (\eta^2 - \delta^2) \nonumber \\
&-& \frac{1}{2} \rho \sin(2 \theta_{hwp})(\eta^2 + \delta^2) \\
\mathcal{G} &\equiv& \frac{1}{4} \left[T + c + (T-c) \cos(4 \theta_{hwp}) \right] (\eta^2 - \delta^2) \nonumber \\
&+& \frac{1}{2} \rho \cos(2 \theta_{hwp}) (\eta^2 + \delta^2). \label{end_of_big_equation}\end{aligned}$$ This reduces to the result given in Jones et. al. [@jones08] in the limit of no HWP ($T = c = 1,~\rho = s = 0$), and in the case of an ideal HWP ($T=1,~c=-1,~\rho = s = 0$) reduces to $$\begin{aligned}
M^{ideal}_{II} &=& \frac{1}{2} (\eta^2 + \delta^2) \nonumber \\
M^{ideal}_{IQ} &=& \frac{1}{2} \cos(2 ( \psi_{inst} + 2 \theta_{hwp})) (\eta^2 - \delta^2) \nonumber \\
M^{ideal}_{IU} &=& \frac{1}{2} \sin(2 ( \psi_{inst} + 2 \theta_{hwp})) (\eta^2 - \delta^2) \nonumber \\
M^{ideal}_{IV} &=& 0.\end{aligned}$$
Armed with the band-averaged Mueller matrix representations of the HWP plus detector system given by Eq. \[mat\_multiply\], we can use Eq. \[eqn\_one\] to calculate the detector output as a function of input Stokes parameters.
![Our definitions of the detector, HWP, and instrument angles. The instrument angle is defined relative to a fixed reference on the sky that determines the absolute orientations of $Q$ and $U$, and the detector and HWP angles are defined relative to the instrument. \[instrument\_and\_hwp\_angles\]](hwp_inst_and_det_angles){width="48.00000%"}
An example of the output from our model, along with the percent-level differences between that and a naiive treatment, is shown in Fig. \[cmb\_sim\] for the specific case of the SPIDER experiment, which uses a single birefringent sapphire HWP with quarter-wave AR coats.
Detector Pair Summing and Differencing
======================================
Many CMB polarization experiments use pairs of Polarization-Sensitive Bolometers (PSBs) located at the same point on the focal plane [@jones02]. Both detectors in a pair view the same patch of the sky, but detector B is oriented at $90^\circ$ with respect to detector A. Detector differencing within a pair reduces common-mode noise while retaining sensitivity to linear polarization. The sum of a pair measures the $I$ Stokes parameter. We take Eq. \[eqn\_one\] with $\mathsf{M}$ given by Eq. \[mat\_multiply\] with $\xi_{det} = 0$ as a model for the A detector timestream $d_i^A$. The B detector timestream $d_i^B$ can be similarly calculated by setting $\xi_{det} = 90^\circ$; we note however that the Jones matrix of a horizontal polarizer, $$\label{j_pol_b}
\mathsf{J}_{pol}^B = \left[ \begin{array}{cc} \delta & 0 \\ 0 & \eta \end{array} \right],$$ is related to that of a vertical polarizer (Eq. \[j\_pol\_a\]) via the substitutions $\eta \rightarrow \delta$ and $\delta \rightarrow \eta$. We can thus just make those substitutions in Equations \[start\_of\_big\_equation\] through \[end\_of\_big\_equation\] to find a model for the B detector timestream. We then construct the sum and difference timestreams $$\begin{aligned}
d_i^{sum} &\equiv& d^A_i + d^B_i , \nonumber\\
d_i^{diff} &\equiv& d^A_i - d^B_i.\end{aligned}$$ Only $(\delta^2 + \eta^2)$ terms will remain in the sum timestream and only $(\delta^2 - \eta^2)$ terms will remain in the difference timestream. The matrix elements for the sum timestream with a single-plate HWP are therefore $$\begin{aligned}
M_{II}^{sum} &=& T (\eta^2 + \delta^2) \nonumber \\
M_{IQ}^{sum} &=& \rho \cos(2 (\psi_{inst} + \theta_{hwp}) ) (\eta^2 + \delta^2) \nonumber \\
M_{IU}^{sum} &=& \rho \sin(2 (\psi_{inst} + \theta_{hwp}) ) (\eta^2 + \delta^2) \nonumber \\
M_{IV}^{sum} &=& 0.\end{aligned}$$ Even with the non-idealities of the HWP, the sum timestream coupling to intensity is independent of HWP angle. There is also a small coupling to linear polarization. The matrix elements for the difference timestream are $$\begin{aligned}
M_{II}^{diff} &=& \rho \cos(2 \theta_{hwp}) (\eta^2 - \delta^2) \nonumber \\
M_{IQ}^{diff} &=& \textcolor{white}{-}\left[ \mathcal{F}' \sin(2 \psi_{inst}) + \mathcal{G}' \cos(2 \psi_{inst}) \right] (\eta^2 - \delta^2) \nonumber \\
M_{IU}^{diff} &=& \left[-\mathcal{F}' \cos(2 \psi_{inst}) + \mathcal{G}'\sin(2 \psi_{inst}) \right] (\eta^2 - \delta^2) \nonumber \\
M_{IV}^{diff} &=& s \sin(2 \theta_{hwp}) (\eta^2 - \delta^2) ,\end{aligned}$$ where $\mathcal{F}'$ and $\mathcal{G}'$ are $$\begin{aligned}
\mathcal{F}' &\equiv& -\frac{1}{2} (T-c) \sin(4 \theta_{hwp}) \\
\mathcal{G}' &\equiv& \frac{1}{2} \left[T + c + (T-c) \cos(4 \theta_{hwp}) \right] .\end{aligned}$$ The difference timestream unfortunately has small couplings to intensity and circular polarization. Note that for both the sum and difference timestreams, detector cross-polarization only shows up as an overall factor of $(\eta^2~\pm~\delta^2)$, and will not result in leakage between the estimates of $Q$ and $U$.
Application to a Specific Instrument
====================================
The model we derived requires input values for the HWP parameters $a(f)$, $b(f)$, $\phi(f)$, $\epsilon_{1}(f)$, and $\epsilon_{2}(f)$. We can calculate these parameters from first principles, or measure them using a polarized Fourier transform spectrometer (FTS) with the assembled instrument.
Physical Modeling
-----------------
To calculate the HWP parameters from first principles, we use a physical optics model similar to the one in Savini et al. [@savini06]. The model extends the 2-by-2 matrix formalism described by Hecht and Zajac [@hecht74] for modeling multiple dielectric layers of isotropic materials to a 4-by-4 matrix formalism for multiple layers of potentially birefringent materials. The model uses the electromagnetic boundary conditions at the interface between each layer of material to map the incident electric and magnetic fields onto the transmitted fields. This fully treats multiple reflections and interference effects, and can also handle lossy materials. We use this model to calculate the elements of the HWP Jones matrix shown in Eq. \[j\_hwp\] by calculating the transmitted electric field amplitude $\vec{E}_{out}$ at a frequency $f$ with an incident electric field amplitude $\vec{E}_{in}$, for both $x$- and $y$-polarized incident waves. We assemble the transmitted amplitudes into the Jones matrix for that HWP, $$\begin{aligned}
\label{hwp_jones_calculation}
\left[ \begin{array}{c} a(f) \\ \epsilon_2(f) \end{array} \right] = \vec{E}_{out} \left(\left[ \begin{array}{c} 1 \\ 0 \end{array} \right],f \right), \nonumber \\
\left[ \begin{array}{c} \epsilon_1(f) \\ b(f) e^{i \phi(f)} \end{array} \right] = \vec{E}_{out} \left(\left[ \begin{array}{c} 0 \\ 1 \end{array} \right],f \right).\end{aligned}$$
Instrument Calibration
----------------------
The parameterization we developed can also be used to characterize the HWP, with the aim of better understanding the coupling to intensity and linear polarization signals from the CMB and foregrounds with other spectra. This characterization process depends on the manner in which the HWP will be used, ie whether it will be continuously rotated or occasionally stepped during the observation process. For both the stepped and continuously rotating cases, a polarized FTS can be used to measure the frequency and polarization response of the instrument and HWP. Taking the Fourier transform of the detector timestream as the FTS mirror moves at a constant speed will yield the spectral response of the combined instrument [@lesurf90].
### Stepped HWP
As an illustration of how the calibration process of an instrument with a stepped single-plate HWP might work, consider the case where we align the instrument with the FTS such that $\xi_{det}=\psi_{inst} = 0^\circ$. The FTS spectrum $\mathcal{S}(\theta_{hwp},f)$ at a particular HWP angle can be modeled using Eq. \[eqn\_one\] with the Mueller matrix elements given in Eq. \[mat\_multiply\]. Before proceeding, it is useful to change variables from $\eta$ and $\delta$ to the optical efficiency and the polarization efficiency $\gamma$. Here we define the optical efficiency as $\varepsilon \equiv \frac{1}{2}(\eta^2 + \delta^2)$ and follow Jones et. al. [@jones08] and take the polarization efficiency $\gamma \equiv \frac{\eta^2 - \delta^2}{\eta^2 + \delta^2}$. Setting $\psi_{inst} = 0$ and collecting terms in Eq. \[eqn\_one\] gives
$$\begin{aligned}
\label{detout_cal}
\mathcal{S}(\theta_{hwp},f) &=& \left(I + \frac{1}{2} Q \cos(4 \theta_{hwp})\right) T(f) \varepsilon(f) + \left(\frac{1}{2} Q + \frac{1}{2} U \sin(4 \theta_{hwp})\right) T(f) \varepsilon(f) \gamma(f) \nonumber \\
&+& \left(\frac{1}{2} Q \cos(2 \theta_{hwp}) + \frac{1}{2} U \sin(2 \theta_{hwp})\right) \rho(f) \varepsilon(f) + I \cos(2 \theta_{hwp}) \rho(f) \varepsilon(f) \gamma(f) \nonumber \\
&+& \frac{1}{2}\left[Q (1 -\cos(4 \theta_{hwp})) - U \sin(4 \theta_{hwp}) \right] c(f) \varepsilon(f) \gamma(f),\end{aligned}$$
where $I$, $Q$, and $U$ are the Stokes parameters of the light coming out of the FTS. Placing a polarizing grid at the output of the FTS produces light with $I = 1$ and $Q = 1$. We call spectra taken in this configuration $\mathcal{S}_{0}$. Rotating the grid by $45^{\circ}$ produces light with $I=1$ and $U=1$; spectra taken with this configuration we call $\mathcal{S}_{45}$.
We have five parameters to constrain (for a single-plate system; a multi-plate HWP has more), so we need at least five FTS spectra with different instrument and HWP angle combinations. Considering noiseless data, we need spectra with the polarizing grid at both 0 and $45^{\circ}$ to independently estimate all of the parameter combinations. With this data, we can set up a system of linear equations to estimate the HWP and detector parameters,
$$\left[ \begin{array}{c} \mathcal{S}_{0}(\theta_{1},f) \\ \mathcal{S}_{0}(\theta_{2},f) \\ \mathcal{S}_{0}(\theta_{3},f) \\ \vdots \\ \mathcal{S}_{45}(\theta_{N+1},f) \\ \mathcal{S}_{45}(\theta_{N+2},f) \\ \vdots \end{array} \right] =
\frac{1}{2} \left[ \begin{array}{ccccc}
2+\cos(4 \theta_{1}) & 1 & \cos(2 \theta_{1}) & 2 \cos(2 \theta_{1}) & 1 - \cos(4 \theta_{1}) \\
2+\cos(4 \theta_{2}) & 1 & \cos(2 \theta_{2}) & 2 \cos(2 \theta_{2}) & 1 - \cos(4 \theta_{2}) \\
2+\cos(4 \theta_{3}) & 1 & \cos(2 \theta_{3}) & 2 \cos(2 \theta_{3}) & 1 - \cos(4 \theta_{3}) \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
2 & \sin(4 \theta_{N+1}) & \sin(2 \theta_{N+1}) & 2\cos(2 \theta_{N+1}) & - \sin(4 \theta_{N+1}) \\
2 & \sin(4 \theta_{N+2}) & \sin(2 \theta_{N+2}) & 2\cos(2 \theta_{N+2}) & - \sin(4 \theta_{N+2}) \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\end{array} \right]
\left[ \begin{array}{c} T(f) \varepsilon(f) \textcolor{white}{\gamma(f)}\\ T(f) \varepsilon(f) \gamma(f) \\ \rho(f) \varepsilon(f)\textcolor{white}{\gamma(f)} \\ \rho(f) \varepsilon(f) \gamma(f) \\ c(f) \varepsilon(f) \gamma(f) \\ \end{array} \right] .$$
Operationally, estimating this combination of the parameters is sufficient to characterize the full instrument (detectors + HWP) for polarized CMB mapmaking. However, since the detector optical efficiency $\varepsilon(f)$ appears as an overall factor in these equations, we cannot independently estimate the HWP-only parameters $T(f)$, $\rho(f)$, and $c(f)$. To check the HWP physical optics model, we would need separate FTS spectra without the HWP in place to estimate the detector optical efficiency.
In this illustration, we do not consider possible parameter covariances or biases induced by detector noise and non-optimal instrument and HWP angles. One possible way to handle this would be to use FTS spectra taken at many HWP and instrument angles to obtain least-squares estimates of the calibration parameters and errors as a function of frequency. This would also allow a test in the $\chi^{2}$ sense on whether the performance of the HWP and instrument is well-described by our formalism.
### Rapidly-Rotating HWP
If an instrument designed for a rapidly-rotating single-plate HWP has a mode in which the HWP can be stepped, then obviously the characterization procedure in the previous section may be used to estimate the HWP parameters.
As a substitute or complement, the HWP can be characterized in the instrument as it is rapidly rotating. For a single-plate HWP, taking $\psi_{inst} = 0^\circ$ and collecting the constant, $2 f_{hwp}$ and $4 f_{hwp}$ terms in Eq. \[eqn\_one\] gives $$\begin{aligned}
\label{detout_psi_zero}
&&d(\theta_{hwp}) = \left[ T \varepsilon I + \frac{1}{2}(T+c) \varepsilon \gamma Q \right] \\
&+& \left[ \frac{1}{2} (T - c) \varepsilon \gamma U \right] \sin{4 \theta_{hwp}} + \left[ \frac{1}{2} (T - c) \varepsilon \gamma Q \right] \cos{4 \theta_{hwp}} \nonumber \\
&+& \left[ \rho \varepsilon \gamma I + \frac{1}{2} \rho \varepsilon Q \right] \cos{2 \theta_{hwp}} + \left[ s \varepsilon \gamma V + \frac{1}{2} \rho \varepsilon U \right] \sin{2 \theta_{hwp}} \nonumber.\end{aligned}$$ This shows that only polarization information is contained in the $4f_{HWP}$ component of the detector timestream. A lock-in at $4 f_{HWP}$ on the timestream $d_{i}$ will produce companion $x_i$ and $y_i$ timestreams, where $x_{i}$ is the sine wave lock-in output timestream, and $y_{i}$ is the cosine timestream. The $Q$ and $U$ timestreams in instrument coordinates (since $\xi_{det} = \psi_{inst} = 0^\circ$) may be estimated using $$\begin{aligned}
Q_i = \frac{2}{(T-c) \varepsilon \gamma} y_i \nonumber \\
U_i =\frac{2}{(T-c) \varepsilon \gamma} x_i \end{aligned}$$ This means that only one calibration factor is necessary to make polarization maps with a rapidly-rotating single-plate HWP. Doing this as a function of frequency with an FTS allows for the estimation of this calibration factor for different source spectral shapes.
Polarized Mapmaking {#mapmaker}
===================
In many CMB polarization experiments, the instrument continuously scans the sky, and the detector timestreams are used to estimate maps of the $I$, $Q$, and $U$ Stokes parameters. Since the CMB is not expected to be circularly-polarized, we take $V=0$. To estimate the maps, a mapmaking algorithm such as the one in Jones et. al. [@jones08] assumes that without cross-polarization, the detector timestream $d_{i}$ can be modeled as $$\label{det_out_no_hwp}
d_i = \frac{1}{2} \left(I + Q \cos(2 \psi_i) + U \sin(2 \psi_i) \right) + n_i,$$ where $\psi_{i}$ is the instrument angle timestream, and $n_i$ is atmospheric and detector noise. The algorithm then combines this detector timestream with the pointing timestream of the telescope, and uses an iterative matrix method to compute maximum likelihood maps of $I$, $Q$, and $U$, as well as an estimate of the noise timestream. The algorithm described in Jones et. al. [@jones08] also can treat cross polarization in the detector.
As an illustration of the principle of the method applied to mapmaking with a HWP, we consider $N$ noiseless detector samples taken at different instrument angles $\psi_{i}$ and HWP angles $\theta_{i}$ when the telescope was pointed at the same spot on the sky. To estimate the Stokes parameters at that point on the sky, we set up a system of linear equations using the detector timestream model, $$\vec{d} = \mathsf{A} \left[ \begin{array}{c} I \\ Q \\ U \end{array} \right]$$ Here $\vec{d}$ is a $N$-element column vector of detector samples, and $\mathsf{A}$ is a $N$-by-$3$ matrix containing the detector and HWP model. If we take Eq. \[det\_out\_no\_hwp\] and model an ideal polarized detector with an ideal HWP, the $\mathsf{A}$ matrix is $$\mathsf{A}_{ideal} = \frac{1}{2} \left[ \begin{array}{ccc} 1 & \cos{(2 (\psi_1 + 2 \theta_1))} & \sin{(2 (\psi_1 + 2 \theta_1))} \\ 1 & \cos{(2 (\psi_2 + 2 \theta_2))} & \sin{(2 (\psi_2 + 2 \theta_2))} \\ \vdots & \vdots & \vdots \\ 1 & \cos{(2 (\psi_N + 2 \theta_N))} & \sin{(2 (\psi_N + 2 \theta_N))} \\ \end{array} \right].$$ To estimate the polarization state of the incident light, we invert the $\mathsf{A}$ matrix, $$\label{pol_est}
\left[ \begin{array}{c} I \\ Q \\ U \end{array} \right] = \mathsf{A}^{-1} \vec{d}.$$
To treat the effects of the HWP non-idealities, we can use the timestream model of Eq. \[eqn\_one\] and the Mueller matrix elements from Eq. \[mat\_multiply\] to set up the relevant system of linear equations. In this case the $\mathsf{A}$ matrix becomes $$\mathsf{A}_{real} = \left[ \begin{array}{ccc} M_{II}(\theta_1,\psi_1) & M_{IQ}(\theta_1,\psi_1) & M_{IU}(\theta_1,\psi_1) \\ M_{II}(\theta_2,\psi_2) & M_{IQ}(\theta_2,\psi_2) & M_{IU}(\theta_2,\psi_2) \\ \vdots & \vdots & \vdots \\ M_{II}(\theta_N,\psi_N) & M_{IQ}(\theta_N,\psi_N) & M_{IU}(\theta_N,\psi_N) \\ \end{array} \right].$$ Even though this matrix has more complicated elements, Eq. \[pol\_est\] still holds and we can still estimate the polarization state of the incident light with a matrix inversion. Thus, simply changing the functional form of the $\mathsf{A}$ matrix in the mapmaking algorithm allows it to exactly compensate for the HWP non-idealities in our model.
Application to the SPIDER HWP
=============================
Mueller Matrix
--------------
The HWPs in the SPIDER instrument will consist of a single 330 mm diameter single-crystal sapphire plate for each telescope with a quarter-wave quartz layer attached to each sides. Each HWP will be cooled to $\sim 4$ K. Here we consider the prototype HWPs for the SPIDER 145 GHz band.
We have measured the indices of refraction of sapphire at 5 K to be $n_s~=~3.336~\pm~.003$ and $n_f~=~3.019~\pm~.003$, from 100 GHz to 240 GHz [@bryan10a]. Our fused quartz AR coats have a room temperature index of refraction of $n_{ar} = 1.951$ at 245 GHz [@lamb95]. We use Eq. \[hwp\_jones\_calculation\] to calculate the Mueller matrix of a thickness-optimized SPIDER 145 GHz HWP as a function of frequency; the results are shown in Fig. \[hwp\_mueller\]. We assumed all materials were lossless, though that assumption can easily be relaxed and the output used in the Mueller formalism described above without modification.
We then use Eq. \[band\_average\] to calculate the band-averaged matrix elements of the SPIDER HWP, assuming a top-hat detector spectrum from 130 GHz to 160 GHz. For the source spectra we use $$\begin{aligned}
S(f) \propto
\begin{cases}
1 & \mathrm{Flat} \\
\frac{dB}{dT} (f,2.725~\mathrm{K}) & \textrm{CMB~\cite{fixsen03}} \\
f^{1.67} B(f,9.6~\mathrm{K}) \\
~~~+ 0.0935 f^{2.7} B(f,16.2~\mathrm{K}) & \textrm{Dust~\cite{finkbeiner99}} \\
f^{-1} & \textrm{Synchrotron~\cite{bennett03}} \\
f^{-.14} & \textrm{Free-free~\cite{oster61}},
\end{cases}\end{aligned}$$ where $B(f,T)$ is the blackbody function, as estimates of the CMB and astrophysical foregrounds. The results are shown in Table \[hwp\_params\]. The $c$ parameter deviates from ideality by almost $5\%$, which is a relatively large effect. This pushes us towards handling the non-idealities through calibration and a modified mapmaker as discussed in Section \[mapmaker\].
$T$ $\mathbf{\rho}$ $c$ $s$
----------------- --------- ----------------- ---------- ----------
**Flat** 0.97389 0.01069 -0.95578 0.00170
**CMB** 0.97396 0.01069 -0.95591 -0.00952
**Dust** 0.97391 0.01080 -0.95563 -0.03598
**Synchrotron** 0.97382 0.01070 -0.95565 0.01280
**Free-free** 0.97388 0.01069 -0.95577 0.00325
1 0 -1 0
: Calculated Mueller matrix elements for an optimized cryogenic sapphire HWP with a quartz AR-coat for the *Spider* 145 GHz HWP. We optimized the HWP thickness based on our measured cold indices, and the AR-coat thickness based on the index in Lamb [@lamb95]. The first row shows the HWP parameters averaged within the *Spider* 145 GHz passband. The CMB, Dust, Synchrotron, and Free-free rows all are band-averaged against the source spectra within the passband. The last row shows the parameter values of an ideal HWP for comparison. \[hwp\_params\]
In addition to being large amplitude, the $s$ parameter depends significantly on which source spectrum is used; this is not surprising given the form of $s$ across the band, shown in Fig. \[hwp\_params\], which shows that tailoring $s$ to be near zero requires near symmetric placement and weighting across the band, since $s$ is large but asymmetric. The variations of $T$, $\rho$, and $c$ between the CMB and foreground sources are not as worrisome. The parameters vary among the different sources at less than the $10^{-4}$ level.
For comparison, SPIDER is targeting $B$-modes at the $r\sim.03$ level, and we will remove dust foregrounds by combining maps from several frequencies. To reach our target $B$-mode sensitivity, our goal for dust foreground subtraction is to reduce it by $\sim 95\%$. The HWP response varies among the different sources at the $10^{-4}$ level, which suggests that residuals from foreground subtraction should be dominated by incomplete dust removal, not source-varying HWP systematics. Next-generation experiments targeting $r<.01$ will require better foreground removal, and may be impacted by this effect.
Mapmaking
---------
A full time-domain simulation of a given instrument’s scan strategy is necessary to precisely quantify the effects of using a conventional mapmaking algorithm with a non-ideal HWP, and to see if the modified mapmaker described above improves the results. To get a sense for what the results might be, we performed a simpler simulation. We input a Stokes vector of $[100~\mu \mathrm{K},1~\mu \mathrm{K},0,0]$ to the detector model in Eq. \[eqn\_one\]. This Stokes vector is representative of the amplitudes of temperature and $E$-mode CMB fluctuations at the angular scales relevant for SPIDER. We fixed $\xi_{det} = \psi_{inst} = 0^\circ$ and generated one detector sample at each of the HWP angles $0^\circ, 1\times \frac{\theta_{max}}{10-1}, 2\times \frac{\theta_{max}}{10-1}, \dots, \theta_{max}$ to make a total of 10 samples evenly spaced in angle from $0^{\circ}$ to $\theta_{max}$. We used these simulated detector samples to estimate the $Q$ and $U$ Stokes parameters using Eq. \[pol\_est\]. We simulated the effect of using a conventional mapmaker by using the matrix $A_{ideal}$, and simulated a modified mapmaker by using $A_{real}$. We repeated this process with varying $\theta_{max}$ to generate Fig. \[hwp\_mapmaker\_comparison\], which is a plot of reconstructed $Q$ and $U$ as a function of $\theta_{max}$. We generated the second pair of plots shown in Fig. \[hwp\_mapmaker\_comparison\] with an input Stokes vector of $[100~\mu \mathrm{K},0,1~\mu \mathrm{K},0]$.
{width="\textwidth"}
Not shown in Fig. \[hwp\_mapmaker\_comparison\] is the fact that the conventional mapmaker also mis-estimates the $I$ Stokes parameter when using samples from a detector with a non-ideal HWP. However, most CMB instruments (including SPIDER) will calibrate by cross-correlating the intensity maps with the WMAP [@jarosik10] or upcoming Planck satellite results, which would change the overall calibration to compensate for the $I$ mis-estimation. To approximate the effect of this calibration process, we scaled our estimates of $I$, $Q$ and $U$ made with the conventional mapmaker by a factor to force the $I$ estimate to be the same as the input value. This was not necessary for the modified mapmaker since it accurately estimates the Stokes $I$ term. The reason the conventional mapmaker generates the large residuals shown in Fig. \[hwp\_mapmaker\_comparison\] is that it does not handle the temperature-polarization leakage generated by the HWP.
Quantifying the systematics from using a non-ideal HWP with a conventional mapmaker will require full time-domain instrument simulations. These will be presented for the case of SPIDER in an upcoming paper [@odea_future].
Conclusions
===========
We have derived an analytic model for birefringent, AR-coated HWPs that simulates the response of an instrument to different source spectra on the sky at arbitrary instrument and HWP angles. We also modeled the sum and difference timestreams of a PSB pair looking through a HWP, and showed that it should be possible to characterize the HWP and detectors with lab testing of a completed instrument. Motivated by the presence of potentially significant non-idealities in the HWP, we presented a mapmaking algorithm that accounts for these known non-idealities of the HWP. This model will be integrated into an end-to-end simulation of the SPIDER instrument to translate these non-idealities into detailed limits on their possible contamination of power spectral and parameter estimates.
After the initial calculation of band-averaged parameters, our Mueller matrix method is analytic and does not require repetitive matrix multiplications or repetitive integration over frequency. Therefore, modeling detector timestreams is far faster than repeated use of a Jones-formalism code. As an example, in 10 seconds of computer time in Matlab on a laptop, the direct Jones method can simulate 300 SPIDER detector samples, while the Mueller matrix method can simulate 24,000, a speedup by a factor of $\sim75$. A version of the Mueller matrix code written in C for use in the instrument simulation code for SPIDER can simulate 10 million detector samples in 10 seconds on the same machine.
Acknowledgements
================
The authors are supported by NASA grant number NNX07AL64G. We would like to thank the anonymous referee for the helpful suggestions.
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abstract: 'An overview is given of recent progress on a variety of fronts in the global QCD analysis of the parton structure of the nucleon and its implication for collider phenomenology, carried out by various subgroups of the CTEQ collaboration.'
author:
- |
Wu-Ki Tung$^{1,2}$, H.L. Lai$^{1,2,3}$, J. Pumplin$^1$, P. Nadolsky$%
^4$, and C.-P. Yuan$^1$.\
$^1$ Michigan State University, East Lansing, MI - USA\
$^2$ University of Washington, Seattle, Washington - USA\
$^3$ Taipei Municipal University of Education, Taipei, Taiwan\
$^4$ Argonne National Laboratory, Argonne, IL, USA\
title: 'Global QCD Analysis and Collider Phenomenology—CTEQ '
---
Introduction
============
Parton distribution functions (PDFs) are essential input to all calculations on high energy cross sections with initial state hadrons. PDFs are extracted from comprehensive *global analysis* of available hard scattering data within the framework of *perturbative QCD.* This report covers recent progress on global QCD analysis made by members of the CTEQ collaboration on a variety of fronts. [@url]
The basis of most of the recent progress is a new implementation of the general mass (GM) formulation for perturbative QCD that systematically includes heavy quark mass effects, both in kinematics and in the order-by-order factorization formula. [@Tung:2006tb] The next section describes the main implications of the new global QCD analysis on collider phenomenology at the Tevatron and the LHC. [@YuanEw]
This is followed by the first in-depth study of the strangeness sector of the parton parameter space, based on the most up-to-date global analysis. [@Lai:2007dq] We found that current data imply a symmetric component of the strange parton distribution, $s(x)+\bar{s}(x)$, that has a shape independent of that of the isospin singlet non-strange sea; and a strangeness asymmetry function $s(x)-%
\bar{s}(x)$ that has a slightly positive first moment.
The same formalism has been applied to investigate the possibility of a non-perturbative (intrinsic) charm component in the nucleon. [Pumplin:2007wg]{} This study is discussed in a separate talk in this workshop [@TungHQ]. In a significant expansion of global QCD analysis, we have succeeded in combining the traditional fixed-order global fits with $%
p_{t}$ resummation calculations. Combined conventional and $p_{t}$-resummed global fits can now be made to pin down parton degrees of freedom that are most pertinent for precision $W$-mass measurement and Higgs particle phenomenology. [@YuanPt] Another subgroup of CTEQ has performed a detailed investigation of the role of recent neutrino scattering experiments (NuTeV, Chorus) and fixed-target Drell-Yan cross section measurement (E866) on global analysis, particularly pertaining to the large-$x$ behavior of parton distributions. The results are reported in [@Owens:2007kp].
Due to space limitation, it is impossible to include in this short written report the figures that illustrate the results discussed in the corresponding talk, as summarized above. However, since the slides for the talk have been made available at the official conference URL [@url], we shall make use of these, and refer the reader to the actual figures by the slide numbers where they appear in the posted talk [@url]. The same space limitation restricts citations to only the papers and talks on which this report is directly based.
New Generation of PDFs and Their Implications for Collider Phenomenology
========================================================================
The base parton distribution set from the new generation of global analysis incorporating the GM formalism for heavy quark mass effects is the CTEQ6.5M PDF set [@Tung:2006tb]. The main improvements over the previous generation of PDFs—CTEQ6.0 and CTEQ6.1—are the mass treatment and the incorporation of the full HERA Run 1 cross section measurements, with their correlated systematic errors.
The most noticeable change in the output parton distributions is a sizable increase in the $u$- and $d$-quark distributions in the region $x\sim%
10^{-3}$ for a wide range of $Q$. The three figures on slides 4/5 of [url]{} show the ratio of the CTEQ6.1 $u,d$-quark and the gluon distribution to that of CTEQ6.5 at $Q=2$ GeV. The differences are moderated by QCD evolution, but still persist to a high energy scale such as the W/Z masses. The origin of these differences can be traced to the treatment of the heavy quark mass, as explained in slide 6. This change has immediate phenomenological consequences. The figure on slide 7 shows ratios of predicted cross sections at the LHC for the standard model (SM) processes $W^{\pm }/Z$ production, Higgs production $gg\rightarrow H^{0}$, and associated production of $W^{\pm }H$; as well as some representative beyond standard model (BSM) processes, e.g. charged Higgs production $\bar{s}c\rightarrow H^{+}\rightarrow \bar{b}t$.
Of immediate interest is the 7% increase in the predicted W and Z production cross sections at LHC (which are sensitive to PDFs in the $x\sim%
10^{-3} $ range) compared to previous estimates. The plot on slide 10 shows the predicted Z vs. the W cross sections for several commonly available PDF sets. The predictions seem to fall into two groups, with no obvious pattern. The results on slide 11 represent an attempt to see whether the difference between Zeus and H1 predictions can be reproduced in the CTEQ framework. We do not see a substantial difference between the two experimental inputs, but do see a clear dependence on how mass effects are treated. Further mysteries are (i) why are the Zeus predictions independent of their mass treatment; and (ii) why are their predictions with mass effects so different from that of MRST, even though they use the MRST formalism for mass treatment. The resolution of these apparent puzzles concerning the W and Z cross sections at the LHC is clearly of great importance to its physics program.
To see the impact of the new PDFs on collider phenomenology in general, it is convenient to examine the luminosity curves. These are shown in slides 8-9 over the range 10 GeV $<\hat{s}<$ 5 TeV for LHC (normalized to that of CTEQ6.1), including bands representing the estimated uncertainties due to experimental input to the global analysis. The quark-quark ($q\text{-}q$) luminosity curves show the largest change between the two generations of PDFs; the $g\text{-}g$ luminosity is shifted only slightly, and the $g\text{-}q$ luminosity shift lies in between.
The cross sections shown in slide 7 also include some typical BSM processes. Notice in particular the very large predicted cross section for the last process due to a new PDF set CTEQ6.5C that permits a non-perturbative (intrinsic) charm component of the nucleon [@Pumplin:2007wg].
In the base PDF set CTEQ6.5M, we adopted the conventional assumptions that the strange distributions $s(x)$ and $\bar{s}(x)$ are of the same shape as the isospin symmetric non-strange sea at the initial scale $\mu =Q_{0}$ for QCD evolution, and that the charm distribution $c(x)$ is zero at the scale $\mu =m_{c}$. There are of no independent degrees of freedom for strange and charm. The improved theoretical and experimental inputs to the new generation of global analysis now permit us to relax these ad hoc assumptions, and hence to study where the truth lies. The results on strange PDFs obtained by [@Lai:2007dq] will be summarized in the following section. The subject of charm PDF is covered in [TungHQ]{}.
Systematic Study of the Strangeness PDFs
========================================
Within the global QCD analysis framework, the only currently measurable process that is directly sensitive to the strange distributions $s(x)$ and $%
\bar{s}(x)$ is dimuon (charm) production in neutrino (and anti-neutrino) scattering off nucleons, via the partonic process $W^{+}(W^{-})+s\,(\bar{s}%
)\rightarrow c\,(\bar{c})$. The final data of the NuTev experiment [Mason:2006qa]{} is thus crucial for this analysis. The constraining power of these data can be realized, however, *only within the framework of a comprehensive global analysis*, since the same final state is produced also by the down quarks, and since the strange sea is intricately coupled to the gluon and the non-strange partons by QCD interactions. Also, because of the presence of the charm particle in the final state of the dimuon signal, a consistent theoretical treatment of heavy quark mass effects for both charged-current and neutral-current DIS processes [@Tung:2006tb] is essential to obtain reliable results.
For convenience, we define the symmetric strange sea** **$%
s_{+}(x)\equiv s(x)+\bar{s}(x)$ and the strangeness asymmetry $%
s_{-}(x)\equiv s(x)-\bar{s}(x)$. All these functions refer to the initial distributions at $\mu =Q_{0}$; QCD evolution then dictates their $\mu $ dependence at higher energy scales. We address the following three issues in turn.
**Is the shape of the symmetric strange sea independent of that of the non-strange sea?** The answer appears to be yes, according to the up-to-date global analysis [@Lai:2007dq]. The evidence is shown on slide 14. The table gives the changes in the goodness-of-fit for the full set of 3542 data points used in the global analysis, as well as the subset of 149 points for neutrino dimuon data sets, that we found in performing a series of global analysis, using 2/3/4 independent strangeness shape parameters, compared to the CTEQ6.5M reference fit that tied the shape of $s(x)$ and $\bar{s}(x)$ to that of the non-strange sea. We see that there is a substantial improvement in the quality of the fit to the dimuon data with $s_{+}(x)$ different from that of the non-strange sea. We also see that current data cannot discriminate between 2-, 3-, or 4-parameter forms for $s_{+}(x)$. Thus, a 2-parameter form will serve as a practical working hypothesis.
**What is the size of the symmetric strange sea, and what are the allowed ranges for its size and shape**? Slide 15 presents results of our study on these issues. The upper figure shows the goodness-of-fit in terms of $\chi ^{2}/$point for the dimuon data (deep parabola) and for the global data (shallow parabola) as a function of the momentum fraction carried by the strange sea, $\langle x\rangle _{s+}=\int xs_{+}(x)dx$. The dimuon data clearly favor a central value of $\langle x\rangle _{s}\sim 0.027$. The range of allowed size is obtained by adopting a 90% confidence level criterion. In terms of the ratio of the first moments (fractional momentum) of the strange to non-strange sea, this range corresponds to ($0.27,$ $0.67$), as indicated on the slide. The range of possible shape of $s_{+}(x)$ is a little more elusive to quantify. The lower figure presents a range of possible candidates, within the 90% C.L. criterion, when both the size and shape parameters are allowed to vary. These representative PDF sets are labeled CTEQ6.5S$n$, $n=0,1,...,4$, with $n=0$ being the central fit.
**Current status of the strangeness asymmetry:** Non-perturbative models of nucleon structure suggest a possible non-vanishing strangeness asymmetry. Within the PQCD framework, QCD evolution beyond the first two leading orders causes a non-vanishing $s_{-}(x,\mu ),$ even if one starts with a symmetric strange sea. Historically, $s_{-}(x)$ was first studied phenomenologically in 2003 as a possible explanation for the NuTeV anomaly associated with the Weinberg angle measurement. Therefore, it is natural to ask: what can we say about $s_{-}(x)$ currently, now that both the theory and experimental situation have improved? The results of our study, [@Lai:2007dq], are summarized in slide 17: (i) current global analysis still does not require a non-zero $%
s_{-}(x)$, although it is consistent with one; (ii) the best fit corresponds to a positive asymmetry $\langle x\rangle _{s-}=\int xs_{-}(x)dx\sim 0.002$; and (iii) the 90% C.L. range for $\langle x\rangle _{s-}$ is ($-0.001,~0.005
$). These results are consistent with both the 2003 CTEQ study and the most recent NuTeV analysis [@Mason:2006qa]. The figures on slide 17 show the shape of $s_{-}(x)$ and the momentum distribution $xs_{-}(x)$ for a variety of possible candidate PDFs within the 90% C.L. criterion.
New neutrino DIS and Drell-Yan data and large-x PDFs
====================================================
It has been known for some time that the relatively recent NuTeV total cross section and E866 Drell-Yan cross section data sets pose puzzling dilemmas for quantitative global QCD analysis of PDFs, as indicated in slides 20 and 21. Attempts to incorporate these data in global analysis by Owens *et.al.* [@Owens:2007kp] led to the following key observations: (i) the NuTeV data set pulls against several of the other data sets, notably the E-866 and the BCDMS and NMC data. Nuclear corrections (heavy target) do not improve the situation. (In fact, assuming no nuclear correction lessens, but does not remove, the problem.); (ii) the conflicts are most pronounced when one examines the $d/u$ ratio. Adding NuTeV and E-866 simultaneously in the global analysis causes the $d/u$ ratio to flatten out substantially, resulting in worsened fits to other precision DIS data; and (iii) the E866 $pp$ data is more comparable with precision DIS data sets than the $pd$ data. Slides 23 - 26 show the figures that support these observations.
[**Conclusion:**]{} Results presented here, in conjunction with those covered in [@YuanEw; @TungHQ; @YuanPt], represent significant evolutionary advancement of global QCD analysis, as well as some ground-breaking development (such as the incorporation of $p_t$ resummation [@YuanPt]). There are, however, also open problems that require further study and resolution [@Owens:2007kp]. Much remains to be done.
[9]{}
D. A. Mason, Proceedings of 14th International Workshop on Deep Inelastic Scattering (DIS 2006), Tsukuba, Japan; and FERMILAB-THESIS-2006-01.
|
---
abstract: 'In the present paper the conditions for the validity of the Tsallis’ Statistics are analyzed. The same has been done following the analogy with the traditional case: starting from the microcanonical description of the systems and taking into account their self-similarity scaling properties in the thermodynamic limit, it is analyzed the necessary conditions for the equivalence of microcanonical ensemble with the Tsallis’ generalization of the canonical ensemble. It is shown that the Tsallis’ Statistics is appropriate for the macroscopic description of systems with potential scaling laws of the asymptotic accessible states density of the microcanonical ensemble. Our analysis shows many details of the Tsallis’ formalism: the q-expectation values, the generalized Legendre’s transformations between the thermodynamic potentials, as well as the conditions for its validity, having a priori the possibility to estimate the value of the entropic index without the necessity of appealing to the computational simulations or the experiment. On the other hand, the definition of physical temperature received a modification which differs from the Toral’s result. For the case of finite systems, we have generalized the microcanonical thermostatistics of D. H. E. Gross with the generalization of the curvature tensor for this kind of description.'
address:
- |
Departamento de Física, Universidad de Pinar del Río\
Martí 270, esq. 27 de Noviembre, Pinar del Río, Cuba.
- |
Departamento de Física Nuclear\
Instituto Superior de Ciencias y Tecnologías Nucleares\
Quinta de los Molinos. Ave Carlos III y Luaces, Plaza\
Ciudad de La Habana, Cuba.
author:
- 'L. Velazquez[^1]'
- 'F. Guzmán[^2]'
title: 'Some remarks about the Tsallis’ Formalism'
---
Introduction
============
In the last years many researchers have been working in the justification of the Tsallis’ Formalism. Many of them have pretended to do it in the context of the Information Theory [@Cura; @Sumi; @Plst] without appealing to the microscopic properties of the systems. Through the years many functional forms of the information entropy similar to the Shanonn-Boltzmann-Gibbs’ have been proposed in order to generalize the traditional Thermodynamics (see for example in refs.[@renyi; @shamit; @abes; @lans]). This way to derive the Thermostatistics is very atractive, since it allows us to obtain directly the probabilistic distribution function of the generalized canonical ensemble at the thermodynamic equilibrium, as well as to develop the dynamical study of systems in non-equilibrium processes.
The main dificulty for this kind of description is to determine the necesary conditions for the application of each specific entropic form. For example, in the Tsallis’ Statistics, the theory is not be able to determine univocally the value of the entropy index, $q$, so that, it is needed the experiment or the computational simulation in order to precise it (see for example in the refs. [@reis; @Pla2; @tir3]). Similar arguments can be applied for other formulations of the Thermodynamics based on a parametric information entropic form. That is the reason why we consider that the statistical description of nonextensive systems should start from the microscopic characteristics of them.
Following the traditional analysis, the derivation of the Thermostatistics from the microscopic properties of the systems could be performed by considerating the microcanonical ensemble. For the case of the Tsallis’ Statistics this is not a new idea.
In 1994 A. Plastino and A. R. Plastino [@Plt2] had proposed one way to justify the q-generalized canonical ensemble with similar arguments employed by Gibbs himself in deriving his canonical ensemble. It is based on the consideration of a closed system composed by a subsystem weakly interacting with a [*finite thermal bath*]{}. They showed that the macroscopic characteristics of the subsystem are described by the Tsallis’ potential distribution, relating the [*entropy index,* ]{}$q$[*,*]{} with the finiteness of the last one. In this approach the Tsallis [*ad-hoc*]{} cut-off condition comes in a natural fashion.
Another attempt was made by S. Abe and A. K. Rajagopal [@Raj1]: a closed system composed by a subsystem weakly interacting with a very large thermal bath, this time analyzing the behavior of the systems around the equilibrium, considering this as a state in which the most probable configurations are given. They showed that the Tsallis’ canonical ensemble can be obtained if the entropy counting rule is modified, introducing the Tsallis’ generalization of the logarithmic function for arbitrary entropic index [@qln], showing in this way the possibility of the nonuniqueness of the canonical ensemble theory.
So far it has been said that the Tsallis’ Statistics allows to extend the Thermodynamics to the study of systems that are anomalous from the traditional point of view, systems with long-range correlations due to the presence of long-range interactions, with a dynamics of non markovian stochastic processes, where the [*entropic index gives a measure of the non extensivity degree of a system, an intrinsic characteristic of the same* ]{} [@tsal]. The identification of this parameter with the finiteness of a thermal bath is limited, since this argument is non-applicable on many other contexts in which the Tsallis’ Statistics is expected to work: astrophysical systems [@pla1; @pla2], turbulent fluids and non-screened plasma [@bog], etc.
The Abe-Rajogopal’s analysis suggests that there is an arbitrariness in the selection of the entropy counting rule, which determines the form of the distribution. In their works they do not establish a criterium that allows to define the selection of the entropy counting rule univocally.
In the Boltzmann-Gibbs’ Statistics the entropy counting rule is supported by means of the scaling behavior of the microcanonical states density and the fundamental macroscopic observables, the integrals of motion and external parameters, with the increasing of the system degrees of freedom, and its Thermodynamic Formalism, based on the Legendre’s Transformations between the Thermodynamic potentials,[* by the equivalence between the microcanonical and the canonical ensembles in the Thermodynamic Limit*]{} (ThL).
In the ref. [@vel1] it was addressed the problem of generalizing the extensive postulates in order to extend the Thermostatistics for some Hamiltonian non-extensive systems. Our proposition was that this derivation could be carried out taking into consideration the self-similarity scaling properties of the systems with the increasing of their degrees of freedom and analyzing the conditions for the equivalence of the microcanonical ensemble with the generalized canonical ensemble in the ThL. The last argument has a most general character than the Gibbs’, since it does not demand the separability of one subsystem from the whole system. The Gibbs’ argument is in disagreement with the long-range correlations of the nonextensive systems. The consideration of the self-similarity scaling properties of the systems allows us to precise the counting rule for the generalized Boltzmann’s entropy [@vel1], as well as the equivalence of the microcanonical ensemble with the generalized canonical one determines the necesary conditions for the applicability of the generalized canonical description in the ThL.
The Legendre’s Formalism
========================
In this section the analysis of the necessary conditions for the equivalence of the microcanonical ensemble with the Tsallis’ canonical one will be performed in analogy with our previous work [@vel2], which was motived by the methodology used by D. H. E. Gross in deriving his[*Microcanonical Thermostatistics*]{} [@gro1] through the technique of the steepest descend method.
In the ref. [@vel2] was shown that the Boltzmann-Gibbs’ Statistics can be applied to the macroscopic study of the pseudoextensive systems, those with [*exponential self-similarity scaling laws*]{} [@vel1; @vel2] in the ThL, using an adequate selection of the representation of the motion integrals space [@vel1], $\Im _{N}$. The previous analysis suggests that a possible application of Tsallis’ formalism could be found for those systems with an scaling behavior weaker than the exponential.
In this analysis the following [*potential self-similarity scaling*]{} laws will be considered:
$$\left.
\begin{array}{c}
N\rightarrow N\left( \alpha \right) =\alpha N \\
I\rightarrow I\left( \alpha \right) =\alpha ^{\chi }I \\
a\rightarrow a\left( \alpha \right) =\alpha ^{\pi _{a}}a
\end{array}
\right\} \Rightarrow W_{asym}\left( \alpha \right) =\alpha ^{\kappa
}W_{asym}\left( 1\right) \text{,} \label{ps}$$
where $W_{asym}$ is the accessible volume of the microcanonical ensemble in the system configurational space in the ThL, $I$ are the system integrals of motion of the macroscopic description in a specific representation ${\cal R}%
_{I}$ of $\Im _{N}$, $a$ is a certain set of parameters, $\alpha $ is the scaling parameter, $\chi $ , $\pi _{a}$ and $\kappa $ are real constants characterizing the scaling transformations. The nomenclature $W_{asym}\left(
\alpha \right) $ represents:
$$W_{asym}\left( \alpha \right) =W_{asym}\left[ I\left( \alpha \right)
,N\left( \alpha \right) ,a\left( \alpha \right) \right] \text{.}$$
This kind of self-similarity scaling laws demands an entropy counting rule different from the logarithmic. It is supposed that the Tsallis’ generalization of exponential and logarithmic functions [@qln]:
$$e_{q}\left( x\right) =\left[ 1+\left( 1-q\right) x\right] ^{\frac{1}{1-q}}%
\text{ \ \ \ }\ln _{q}\left( x\right) =\frac{x^{1-q}-1}{1-q}\equiv
e_{q}^{-1}\left( x\right)$$
are more convenient to deal with it.
Let us consider a finite Hamiltonian system with this kind of scaling behavior in the ThL. We postulate that the [**Generalized Boltzmann’s Principle**]{} [@vel1] adopts the following form:
$$\left( S_{B}\right) _{q}=\ln _{q}W\text{.} \label{cr}$$
The accessible volume of the microcanonical ensemble in the system configurational space, $W$, is given by:
$$W\left( I,N,a\right) =\Omega \left( I,N,a\right) \delta I_{o}=\delta
I_{o}\int \delta \left[ I-I_{N}\left( X;a\right) \right] dX\text{,}$$
where $\delta I_{o}$ is a [*suitable*]{} constant volume element which makes $W$ dimensionless. The corresponding information entropy for the q-generalized Boltzmann’s entropy, the Eq.(\[cr\]), is the Tsallis’ nonextensive entropy (TNE) [@Tsal1]:
$$S_{q}=-%
%TCIMACRO{\underset{k}{\sum }}%
%BeginExpansion
\mathrel{\mathop{\sum }\limits_{k}}%
%EndExpansion
p_{k}^{q}\ln _{q}p_{k}\text{.}$$
In the thermodynamic equilibrium the TNE leads to the q-exponential generalization of the Boltzmann-Gibbs’ Distributions: $$\omega _{q}\left( X;\beta ,N,a\right) =\frac{1}{Z_{q}\left( \beta
,N,a\right) }e_{q}\left[ -\beta \cdot I_{N}\left( X;a\right) \right] \text{,}$$ where $Z_{q}\left( \beta ,a,N\right) $ is the partition function [@Tsal2] For this ensemble, the [*q-Generalized Laplace’s Transformation*]{} is given by: $$Z_{q}\left( \beta ,N,a\right) =\int e_{q}\left( -\beta \cdot I\right)
W\left( I,N,a\right) \frac{dI}{\delta I_{o}}\text{.}$$
The Laplace’s Transformation establishes the connection between the fundamental potentials of both ensembles, the q-generalized [*Planck’spotential*]{}:
$$P_{q}\left( \beta ,N,a\right) =-\ln _{q}\left[ Z_{q}\left( \beta ,N,a\right) %
\right] \text{,}$$
and the generalized Boltzmann’s entropy defined by the Eq.(\[cr\]):
$$e_{q}\left[ -P_{q}\left( \beta ,N,a\right) \right] =\int e_{q}\left( -\beta
\cdot I\right) e_{q}\left[ \left( S_{B}\right) _{q}\left( I,N,a\right) %
\right] \frac{dI}{\delta I_{o}}\text{.} \label{e1}$$
The q-logarithmic function satisfies the [*subadditivity relation*]{}:
$$\ln _{q}\left( xy\right) =\ln _{q}\left( x\right) +\ln _{q}\left( y\right)
+\left( 1-q\right) \ln _{q}\left( x\right) \ln _{q}\left( y\right) \text{,}$$
and therefore:
$$e_{q}\left( x\right) e_{q}\left( y\right) =e_{q}\left[ x+y+\left( 1-q\right)
xy\right] \text{.}$$
The last identity allows us to rewrite the Eq.(\[e1\]) as:
$$e_{q}\left[ -P_{q}\left( \beta ,N,a\right) \right] =\int e_{q}\left[
-c_{q}\beta \cdot I+\left( S_{B}\right) _{q}\left( I,N,a\right) \right]
\frac{dI}{\delta I_{o}}\text{,} \label{e3}$$
where:
$$c_{q}=1+\left( 1-q\right) \left( S_{B}\right) _{q}\text{.}$$
In the Tsallis’ case, the linear form of the Legendre’s Transformation [*is violated*]{} and therefore, the[* ordinary Legendre’s Formalism does not establish the correspondence between the two ensembles*]{}. In order to preserve the homogeneous scaling in the q-exponential function argument, it must be demanded the scaling invariance of the set of admissible representations of the integrals of motion space [@vel1], ${\cal M}_{c}$, that is, the set ${\cal M}_{c}$ is composed by those representations $%
{\cal R}_{I}$ satisfying the restriction:
$$\chi \equiv 0\text{,}$$
in the scaling transformation given in Eq.(\[ps\]). In these cases, when the ThL is invoked, the main contribution to the integral of the Eq.(\[e3\]) will come from the maxima of the q-exponential function argument. The equivalence between the microcanonical and the canonical ensemble will only take place when there is only one sharp peak. Thus, the argument of the q-exponential function leads to assume the nonlinear generalization of the Legendre’s Formalism [@Abe1; @Fran] given by:
$$\widetilde{P}_{q}\left( \beta ,N,a\right) =Max\left[ c_{q}\beta \cdot
I-\left( S_{B}\right) _{q}\left( I,N,a\right) \right] \text{.} \label{ltp}$$
We recognized immediately the formalism of the [*normalized q-expectations values* ]{}[@Abe1]. The maximization leads to the relation:
$$\beta =\frac{\nabla \left( S_{B}\right) _{q}}{1+\left( 1-q\right) \left(
S_{B}\right) _{q}}\left( 1-\left( 1-q\right) \beta \cdot I\right) \text{.}
\label{r1}$$
Using the identity:
$$\nabla S_{B}=\frac{\nabla \left( S_{B}\right) _{q}}{1+\left( 1-q\right)
\left( S_{B}\right) _{q}}\text{,}$$
where $S_{B}$ is the usual Boltzmann’s entropy, the Eq.(\[r1\]) can be rewritten as:
$$\beta =\nabla S_{B}\left( 1-\left( 1-q\right) \beta \cdot I\right) .$$
Finally it is arrived to the relation:
$$\beta =\frac{\nabla S_{B}}{\left( 1+\left( 1-q\right) I\cdot \nabla
S_{B}\right) }\text{.} \label{gzl}$$
This is a very interesting result because it allows to limit the values of the entropy index. If this formalism is arbitrarily applied to a pseudoextensive system (see in ref.[@vel2]), then $I\cdot \nabla S_{B}$ will not bound in the ThL and $\beta $ will vanish trivially. The only possibility in this case is to impose the restriction $q\equiv 1$, that is, [*the Tsallis’ Statistics with an arbitrary entropy index can not be applied to the pseudoextensive systems*]{}. There are many examples in the literature in which the Tsallis’ Statistics has been applied indiscriminately without minding if the systems are extensive or not, i.e., [* gases* ]{} [@Fran; @Abe2][*, blackbody radiation* ]{}[@Tir1][*, and others*]{}.
In some cases, the authors of these works have introduced some artificial modifications to the original Tsallis’ formalism in order to obtain the same results as those of the classical Thermodynamics, i.e., the [*q-dependent Boltzmann’s constant*]{} (see for example in ref.[@Abe3]). The above results indicate the non applicability of the Tsallis’ Statistics for these kind of systems. It must be pointed out that this conclusion is supported with a great accuracy by direct experimental mensurements trying to find nonextensive effects in some ordinary extensive systems (cosmic background blackbody radiation [@Pla2], fermion systems [@Tsal5; @Pla3], gases [@Pla4]).
The Tsallis’ formalism introduces a correlation to the canonical intensive parameters of the Boltzmann-Gibbs’ Probabilistic Distribution Function. This result differs from the one obtained by Toral [@toral], who applied to the microcanonical ensemble the physical definitions of temperature and pressure introduced by S. Abe in the ref.[@Abe2]:
$$\begin{tabular}{l}
$\frac{1}{kT_{phys}}=\frac{1}{1+\left( 1-q\right) S_{q}}\frac{\partial }{%
\partial E}S_{q}$, \\
$\frac{P_{phys}}{kT_{phys}}=\frac{1}{1+\left( 1-q\right) S_{q}}\frac{%
\partial }{\partial V}S_{q}$.
\end{tabular}$$
When these definitions are applied to the microcanonical ensemble asuming the generalized Boltzmann’s Principle, the Eq.(\[cr\]), the physical temperature coincides with the usual Boltzmann’s relation:
$$\frac{1}{kT_{phys}}=\frac{\partial }{\partial E}S_{B}\text{.}$$
It is easy to show, that this result does not depend on the entropy counting rule of the generalized Boltzmann’s Principle [@vel1], but on separability of a closed system in subsystems weakly correlated among them, and the additivity of the integrals of motion and the macroscopic parameters. It must be recalled that these exigencies are only valid for the extensive systems, but, it is not the case that we are studying here. Our result comes in fashion as consequence of the system scaling laws in the thermodynamic limit.
An important second condition must be satisfied for the validity of the Legendre’s transformation, [*the stability of the maximum*]{}. This condition leads to the q-generalization of the [*Microcanonical Thermostatistics*]{} of D. H. E. Gross [@gro1]. In this approach, the stability of the Legendre’s formalism is supported by the concavity of the entropy, the negative definition of the quadratic forms of the curvature tensor [@gro1; @vel2]. In the Tsallis’ case, the curvature tensor must be modified as:
$$\left( K_{q}\right) _{\mu \nu }=\frac{1}{1-\left( 1-q\right) \widetilde{P}%
_{q}}\left[ \left( 2-q\right) \frac{\partial }{\partial I^{\mu }}\frac{%
\partial }{\partial I^{\nu }}\left( S_{B}\right) _{q}+\right.$$
$$\left. +\left( 1-q\right) \left( \beta _{\mu }\frac{\partial }{\partial
I^{\nu }}\left( S_{B}\right) _{q}+\beta _{\nu }\frac{\partial }{\partial
I^{\mu }}\left( S_{B}\right) _{q}\right) \right] \text{.}$$
Taking into consideration that the scaling behavior of the functions $\left(
S_{B}\right) _{q}$ and $\widetilde{P}_{q}$ are identical, which is derived from the Eq.(\[ltp\]), it is easy to see that the curvature tensor is scaling invariant. Using the above definition and developing the Taylor’s power expansion up to the second order term in the q-exponential argument, we can approximate the Eq.(\[e3\]) as:
$e_{q}\left[ -P_{q}\left( \beta ,N,a\right) \right] \simeq \int e_{q}\left[ -%
\widetilde{P}_{q}\left( \beta ,N,a\right) \right] \times $$$\times e_{q}\left[ -\frac{1}{2}\left( I-I_{M}\right) ^{\mu }\cdot \left.
\left( -K_{q}\right) _{\mu \nu }\right| _{I=I_{M}}\cdot \left(
I-I_{M}\right) ^{\nu }\right] \frac{dI}{\delta I_{o}}\text{.} \label{e4}$$
The maximum will be stable if all the eingenvalues of the q-curvature tensor are [*negative*]{} and [*very large*]{}. In this case, in the q-generalized canonical ensemble there will be [*small fluctuations of the integrals of motion around its q-expectation values*]{}. The integration of Eq.(\[e4\]) yields:
$e_{q}\left[ -P_{q}\left( \beta ,N,a\right) \right] \simeq e_{q}\left[ -%
\widetilde{P}_{q}\left( \beta ,N,a\right) \right] \times $$$\times \frac{1}{\delta I_{o}\det^{\frac{1}{2}}\left( \frac{1-q}{2\pi }\left.
\left( -K_{q}\right) _{\mu \nu }\right| _{I=I_{M}}\right) }\frac{\Gamma
\left( \frac{2-q}{1-q}\right) }{\Gamma \left( \frac{2-q}{1-q}+\frac{1}{2}%
n\right) }\text{.} \label{e5}$$
Denoting $K_{q}^{-1}$ by:
$$K_{q}^{-1}=\frac{1}{\delta I_{o}\det^{\frac{1}{2}}\left( \frac{1-q}{2\pi }%
\left. \left( -K_{q}\right) _{\mu \nu }\right| _{I=I_{M}}\right) }\frac{%
\Gamma \left( \frac{2-q}{1-q}\right) }{\Gamma \left( \frac{2-q}{1-q}+\frac{1%
}{2}n\right) }\text{,}$$
and rewriting Eq.(\[e5\]) again:
$e_{q}\left[ -P_{q}\left( \beta ,N,a\right) \right] \simeq e_{q}\left[ -%
\widetilde{P}_{q}\left( \beta ,N,a\right) +\right. $$$\left. \ln _{q}\left( K_{q}^{-1}\right) -\left( 1-q\right) \ln _{q}\left(
K_{q}^{-1}\right) \widetilde{P}_{q}\left( \beta ,N,a\right) \right] \text{,}$$ it is finally arrived to the condition:
$$R\left( q;\beta ,N,a\right) =\left| \left( 1-q\right) \ln _{q}\left(
K_{q}^{-1}\right) \right| \ll 1\text{.}$$
The last condition could be considered as an [*optimization problem*]{}, since the entropic index is an independent variable in the functional dependency of the physical quantities. The specific value of $q$ could be chosen in order to minimize the function $R\left( q;\beta ,N,a\right) $ for all the possible values of the integrals of motion . In this way, the problem of the determination of the entropic index could be solved in the frame of the microcanonical theory without appealing to the computational simulation or the experiment.
Thus, the q-generalized Planck’s potential could be obtained by means of the generalized Legendre’s transformation:
$$P_{q}\left( \beta ,N,a\right) \simeq c_{q}\beta \cdot I-\left( S_{B}\right)
_{q}\left( I,N,a\right) \text{,}$$
where the canonical parameters $\beta $ hold the Eq.(\[gzl\]). Thus, the q-generalization of the Boltzmann’s entropy will be equivalent with the Tsallis’ entropy in the ThL:
$$\left( S_{B}\right) _{q}\simeq S_{q}\text{.}$$
If the uniquenees of the maximum is not guarantized, that is, any of the eingenvalues of the q-curvature tensor is non negative in a specific region of the integrals of motion space, there will be a catastrophe in the generalized Legendre’s Transformation. In analogy with the traditional analysis, this peculiarity can be related with the occurrence of a phenomenon similar to the phase transition in the ordinary extensive systems.
Conclusions
===========
We have analyzed the conditions for the validity of the Tsallis’ generalization of the Boltzmann-Gibbs’ Statistics. Starting from the microcanonical ensemble, we have shown that the same one can be valid for those Hamiltonian systems with potential self-similarity scaling laws in the asymptotic states density. Systems with this kind of scaling laws must be composed by strongly correlated particles, and therefore, these systems must exhibit an anomalous dynamical behavior. There are some computational evidences that suggest that the Tsallis’ Statistics could be applied for dissipative dynamical systems at the edge of chaos (see in the ref.[@Tsal4; @Lato; @Lato2; @s1; @s2; @s3; @s4; @s5]) and Hamiltonian systems with long-range interactions [@hs1; @hs2; @hs3; @hs4].
In this context we have shown an entire series of details of Tsallis’ formalism that in this approach appear in a natural way: the [*q*]{}-expectation values, the generalized Legendre’s transformations between the thermodynamic potentials, as well as the conditions for the validity of the same one, having a priori the possibility to estimate the value of the entropic index without the necessity of appealing to the computational simulations or the experiment.
For the case of finite systems satisfying this kind of scaling laws in the thermodynamic limit, we have generalized the Microcanonical Thermostatistics of D. H. E. Gross assuming the Tsallis’ generalization of the Boltzmann’s entropy. This assumption leads to the generalization of the curvature tensor, which is the central object in the thermodynamic formalism of this theory, since it allows us to access to the ordering information of a finite system.
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[^1]: luisberis@geo.upr.edu.cu
[^2]: guzman@info.isctn.edu.cu
|
---
abstract: 'We investigate the newtonian stationary accretion of a polytropic perfect fluid onto a central body with a hard surface. The selfgravitation of the fluid and its interaction with luminosity is included in the model. We find that for a given luminosity, asymptotic mass and temperature of the fluid there exist two solutions with different cores.'
address: |
M. Smoluchowski Institute of Physics, Jagiellonian University,\
Reymonta 4, 30-059 Kraków, Poland
author:
- Edward Malec and Krzysztof Roszkowski
title: Gravastars and bifurcation in quasistationary accretion
---
Introduction
============
The question we want to address in this paper is the following *inverse problem*: having a complete set of data describing a compact body immersed in a spherically symmetric accreting fluid, find the mass of the central body. We assume that we know the total mass, luminosity, asymptotic temperature, the equation of state of the accreting gas and the gravitational potential at the surface of the core.
The fundamental question is whether observers can distinguish between gravastars [@MM] versus black holes as engines of luminous accreting systems (see a controversy in [@AKL; @Narayan]). While we do not address this problem here, we show a related ambiguity in a simple newtonian model.
The Shakura model
=================
The first investigation of stationary accretion of spherically symmetric fluids, including luminosity close to the Eddington limit, was provided by Shakura [@Shakura]. It was later extended to models including the gas pressure, its selfgravity and relativistic effects [@OS; @Thorne; @RM; @Park].
In the following we will denote the areal velocity by $U(r,t)=\partial_t R$ (where $t$ is comoving time and $R$ the areal radius), the local, Eddington and total luminosities by $L(R)$, $L_E$ and $L_0$, quasilocal mass by $m(R)$ and total by $M$, pressure by $p$, the baryonic mass density by ${\varrho}$ (the polytropic equation of state will be $p = K {\varrho}^\Gamma$, $1< \Gamma \leq 5/3$) and the gravitational potential by $\phi(R)$. The radius of the central body is $R_0$ while its “modified radius” is defined by ${\tilde{R}_0}= GM/|\phi(R_0)|$. Under the assumption that at the outer boundary of the fluid the following holds true: $$U^2_\infty \ll \frac{G m(R_\infty)}{R_\infty} \ll a^2_\infty,$$ we have the following set of equations: $$\dot M = - 4 \pi R^2 {\varrho}U,
\label{roszkowski:eq1}$$ $$U \partial_R U = -\frac{G m(R)}{R^2} - \frac{1}{{\varrho}} \partial_R p
+ \alpha \frac{L(R)}{R^2},$$ $$\partial_R \dot M = 0,$$ $$L_0 - L(R) = \dot M \left( \frac{a^2_\infty}{\Gamma - 1}
- \frac{a^2}{\Gamma -1} - \frac{U^2}{2} - \phi(R) \right).
\label{roszkowski:eq4}$$ $\alpha$ is a dimensional constant $\alpha = \sigma_T / 4 \pi m_p c$. The details of solving the system are provided elsewhere [@MRK; @MalecR] and we will present only the main results here.
We assume that the accretion is critical, [*i.e.*]{}, there exists a sonic point, where the speed of accreting gas $U$ is equal to the speed of sound $a$. All values measured at that point will be denoted with an asterisk. We define: $$x = \frac{L_0}{L_E}, \qquad y = \frac{M_\ast}{M}, \qquad
\gamma = \frac{{\tilde{R}_0}}{R_\ast},$$ and obtain the total luminosity: $$L_0 = {\phi_0 \chi_\infty G^{2} \pi^2 \frac{M^{3}}{a_\infty^3}} {(1-y)[y-x\exp(-x\gamma)]^2}{\left( \frac{2}{5-3\Gamma}\right) ^{(5-3\Gamma)/2(\Gamma-1)}}.
\label{roszkowski:luminosity}$$ $\chi_\infty$ is approximately the inverse of the volume of the gas located outside the sonic point. In sake of brevity we will use $$\beta = \alpha {\phi_0 \chi_\infty G^{} \pi^2 \frac{M^{2}}{a_\infty^3}} {\left( \frac{2}{5-3\Gamma}\right) ^{(5-3\Gamma)/2(\Gamma-1)}}$$ to obtain in a form using the relative luminosity: $$x = \beta {(1-y)[y-x\exp(-x\gamma)]^2}.
\label{roszkowski:relative}$$
Bifurcation
===========
For the relative luminosity, fulfilling we proved the following theorem:
For the functional $F(x, y) = x - \beta {(1-y)[y-x\exp(-x\gamma)]^2}$ there exists a critical point $x =a$, $y=b$ such that $F(a, b) = 0$ and $\partial_y F(a, b) = 0$ with $0 < a < b < 1$ and $a = 4 \beta (1-b)^3$, $b = [2+a \exp(-a \gamma)]/3$.
For any $0 < x < a$ there exist two solutions $y(x)^+_-$ bifurcating from $(a,b)$. They are locally approximated by: $$y^+_- = b \pm \left(\frac{(a-x)[b+a\exp(-a\gamma)(1-2a\gamma)]}{\beta[b-a\exp(-a\gamma)][1-a\exp(-a\gamma)]} \right)^{1/2} .$$
The relative luminosity $x$ is extremized at the critical point $(a,b)$.
Discussion
==========
In the paper we have assumed the existence of an accreting system which satisfies certain conditions. Under those assumptions the complicated set of integro-differential nonlinear equations (\[roszkowski:eq1\]–\[roszkowski:eq4\]) can be simplified to an algebraic one (\[roszkowski:relative\]). We checked numerically that the performed simplification causes errors of the order of $10^{-3}$ (see [@MRK] for details).
The analysis of shows that there exist two different solutions, having the same total luminosity and total mass, but different masses of the core objects. One can also conclude that for sufficiently large $\beta$ the maximal relative luminosity $a$ can get close to $1$, [*i.e.*]{}, the total luminosity approaches the Eddington limit.
As the two solution branches bifurcate from the point $(a, b)$, there is no much difference between the central masses of bright objects (see [@MalecR; @MRK] for plots). However, when luminosity is small ($L_0 \ll L_E$), this difference can become arbitrarily large. This can be understood intuitively, because the radiation is small for test fluids (since the layer of gas is thin), or when the central object is light (therefore weakly attracting surrounding gas).
The results obtained here are consistent with relativistic analysis neglecting interaction between the gas and the radiation [@KKMMS; @KMM].
Acknowledgments {#acknowledgments .unnumbered}
===============
This paper has been partially supported by the MNII grant 1P03B 01229.
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---
abstract: 'We define a class of quadratic differential algebras which are generated as differential graded algebras by the elements of an Euclidean space. Such a differential algebra is a differential calculus over the quadratic algebra of its elements of differential degree zero. This generalizes for arbitrary quadratic algebras the differential graded algebra of exterior polynomial differential forms for the algebra of polynomial functions on $\mathbb R^n$. We investigate the structure of these differential algebras and their connection with the Koszul complexes of quadratic algebras.'
---
[**QUADRATIC DIFFERENTIAL ALGEBRAS**]{}\
Michel DUBOIS-VIOLETTE [^1] and Giovanni LANDI [^2]
Introduction
============
In this paper we shall be concerned with quadratic algebras $\cala$ generated by $n$ elements $x^\lambda$ $(\lambda\in \{1,\dots, n\})$ with relations $$x^\lambda x^\mu=R^{\lambda\mu}_{\nu\rho} x^\nu x^\rho$$ for $\lambda, \mu\in \{1,\dots,n\}$ where the $R^{\lambda\mu}_{\nu\rho}$ are real numbers for $\lambda, \mu,\nu, \rho \in \{1,\dots, n\}$.\
Let us show that for any real quadratic algebra generated by $n$ elements $x^\lambda$, one may write its relations as above. Indeed let us introduce on the real vector space $E=\sum_\lambda \mathbb R x^\lambda$ of its generators the unique Euclidean structure such that $(x^\lambda)$ is an orthonormal basis and let $\calr\subset E\otimes E$ be the space of its relations. Then $E\otimes E$ is also canonically an Euclidean space and let $P$ be the orthogonal projection onto its subspace $\calr$. One has $$P=\frac{1}{2}(\bbbone -R)$$ where $R$ is an involutive symmetric endomorphism of $E\otimes E$ with matrix element $R^{\lambda\mu}_{\nu\rho}$. Then the space of relations $\calr$ is spanned by the $$x^\lambda x^\mu-R^{\lambda\mu}_{\nu\rho} x^\nu x^\rho$$ with $\lambda,\mu\in \{1,\dots, n\}$ so one may write the relations of $\cala$ as expected above with furthermore the properties $$R^{\lambda\mu}_{\alpha \beta} R^{\alpha\beta}_{\nu\rho}= \delta^\lambda_\nu \delta^\mu_\nu,\>\>\>\> R^{\lambda\mu}_{\nu\rho}=R^{\nu\rho}_{\lambda\mu}$$ for $\lambda, \mu, \nu, \rho\in \{1,\dots, n\}$. Since all these properties will be useful, one will start in the following by real quadratic algebras generated by an Euclidean space $E$ which is therefore not a restriction.\
Assuming $\cala$ given as above, one can define a differential calculus over $\cala$ by setting $$\label{rdiff}
\left\{
\begin{array}{lll}
x^\lambda x^\mu & = & R^{\lambda\mu}_{\nu\rho} x^\nu x^\rho \\
\\
x^\lambda dx^\mu & = & R^{\lambda\mu}_{\nu\rho} dx^\nu x^\rho \\
\\
dx^\lambda x^\mu & = & R^{\lambda\mu}_{\nu\rho} x^\nu dx^\rho \\
\\
dx^\lambda dx^\mu & = & - R^{\lambda\mu}_{\nu\rho} dx^\nu dx^\rho \\
\end{array}
\right.$$ which define a quadratic algebra (generated by the $x^\lambda, dx^\mu$) on which $x^\lambda \mapsto dx^\lambda$ extends uniquely as an antiderivation $d$ of square 0 $(d^2=0)$, i.e. as a differential, for the graduation given by the degree in the $dx^\lambda$. The corresponding differential graded algebra $\Omega(\cala)$ is a differential calculus over $\cala$ in the sense of [@wor:1989] (see also in [@mdv:2001]).\
It is worth noticing here that in the case where the $R^{\lambda\mu}_{\nu\rho}$ are given by $\delta^\lambda_\rho \delta^\mu_\nu$ then the algebra $\cala$ is the polynomial algebra $\mathbb R[x^\lambda]=S(\mathbb R^n)$ and $\Omega(\cala)$ is the algebra $S(\mathbb R^n)\otimes \wedge(\mathbb R^n)$ of polynomial differential forms on $\mathbb R^n$ while $d$ is the usual exterior differential. In the following, this case will be called [*the classical case*]{}.\
Our aim in this paper is to investigate the properties of such differential graded algebras.\
The above construction has been introduced and used in [@lan-pag:2018] within a specific framework in which the symmetry condition for the $R^{\lambda\mu}_{\nu\rho}$ is not satisfied but is replaced by conditions which are adapted for a complex $\ast$-algebraic variant of the previous real formulation (see Section \[StarAlg\]).\
. We use throughout the Einstein convention of summation on the repeated up-down indices. Given a finite-dimensional vector space $E$, we denote its tensor algebra by $T(E)$, its symmetric algebra by $S(E)$ and its exterior algebra by $\wedge(E)$. All the graded algebras involved in this paper are $\mathbb N$-graded algebras but it is convenient to consider these graded algebras as $\mathbb Z$-graded algebras such that their components of negative degree vanish.
Canonical form of a quadratic relation
======================================
Throughout this paper $E$ is a finite-dimensional Euclidean space with orthonormal coordinate basis $(x^\lambda)$ $\lambda\in \{1,\dots, n\}$, $n=\dim(E)$.\
Let $\calr$ be a linear subspace of $E\otimes E$ and let $\cala$ be the quadratic algebra defined by $$\label{A}
\cala=T(E)/(\calr)$$ where $(\calr)$ denotes the two-sided ideal of the tensor algebra $T(E)$ of $E$ generated by $\calr$. Since the relations of $\cala$ are homogeneous (quadratic) of degree strictly greater than one, $\cala$ is a graded algebra $$\cala=\oplus_{n\in \mathbb N} \cala_n$$ for the degree induced by the one of $T(E)$ which is unital and connected, that is $$\cala_0=\mathbb R\cdot \bbbone$$ where $\bbbone$ is the unit of $\cala$ induced from the one of $T(E)$, and one has $\cala_1=E$ while $\cala_2=E\otimes E/\calr$.\
The space $E\otimes E$ is canonically an Euclidean space and we denote by $P$ the orthogonal projection onto the subspace $\calr$ of $E\otimes E$. One has $$\label{R}
P=\frac{1}{2}(\bbbone-R)$$ where $R$ is a symmetric involution of $E\otimes E$. In terms of components in the orthonormal system $(x^\lambda\otimes x^\mu)$ one has $$\label{mR}
R=(R^{\lambda\mu}_{\nu\rho})$$ with $$\label{invol}
R^{\lambda\mu}_{\alpha\beta}\>\> R^{\alpha\beta}_{\nu\rho}=\delta^\lambda_\nu\>\> \delta^\mu_\rho$$ and $$\label{sym}
R^{\lambda\mu}_{\nu\rho}=R^{\nu\rho}_{\lambda\mu}$$ for the real numbers $R^{\lambda\mu}_{\nu\rho}$. The quadratic algebra $\cala$ is then the real unital associative algebra generated by the $x^\lambda$ $(\lambda\in \{1,\dots, n\})$ with relations $$\label{cqr}
x^\lambda x^\mu - R^{\lambda\mu}_{\nu\rho}\>\> x^\nu x^\rho=0$$ for $\lambda,\mu \in \{1,\dots,n\}$.\
As pointed out in the introduction in the classical case, i.e. when $R^{\lambda\mu}_{\nu\rho}$ coincides with $\delta^\lambda_\rho \delta^\mu_\nu$, $\cala$ is the polynomial algebra $\mathbb R[x^\lambda]=S(\mathbb R^n)=S(E)$.
The Koszul dual $\cala^!$ of $\cala$
====================================
Let $E^\ast$ be the dual vector space of $E$ and $(\theta_\lambda)$ be the dual basis of $(x^\lambda)$. The Koszul dual $\cala^!$ of $\cala$ is the quadratic algebra [@pri:1970], [@man:1988] $$\label{Akd}
\cala^!=T(E^\ast)/(\calr^!)$$ where $\calr^!$ is the orthogonal in $E^\ast\otimes E^\ast=(E\otimes E)^\ast$ of the subspace $\calr$ of $E\otimes E$. In terms of components, $\cala^!$ is the real unital associative algebra generated by the $\theta_\lambda$ with relations $$\label{cqrd}
\theta_\lambda \theta_\mu + R^{\nu\rho}_{\lambda\mu} \>\> \theta_\nu \theta_\rho=0$$ for $\lambda,\mu\in \{1,\dots,n\}$.\
In view of the symmetry of the $R^{\lambda\mu}_{\nu\rho}$, that is of relations (\[sym\]), $\cala^!$ is isomorphic to the real unital algebra $\cala'$ generated by elements $y^\lambda$ for $\lambda\in\{1,\dots,n\}$ with relations $$\label{cqrprim}
y^\lambda y^\mu + R^{\lambda \mu}_{\nu\rho}\>\> y^\nu y^\rho=0$$ for $\lambda,\mu\in \{1,\dots, n\}$. This latter algebra is the quadratic algebra $$\label{Aprim}
\cala'=T(E)/(\calr^\perp)$$ where $\calr^\perp$ is the orthogonal subspace to $\calr$ in the Euclidean space $E\otimes E$. Note that the orthogonal projection onto $\calr^\perp$ is $$\label{-R}
P=\frac{1}{2}(\bbbone +R)$$ thus one passes from $\cala$ to $\cala'$ by changing $R$ into $-R$ and the $x^\lambda$ into the $y^\lambda$.\
Notice that with the identification (\[Aprim\]), $(y^\lambda)$ is the orthonormal basis of $E$.\
It is not hard to see that in the classical case $\cala'$ coincides with the exterior algebra $\wedge(\mathbb R^n)=\wedge(E)$.
The bigraded algebra $A$
========================
Let $A$ be the quadratic algebra generated by the $2n$ elements $x^\lambda,y^\mu$ with relations $$\label{rA}
\left\{
\begin{array}{llll}
x^\lambda x^\mu & - & R^{\lambda\mu}_{\nu\rho}\>\> x^\nu x^\rho &= 0 \\
\\
x^\lambda y^\mu & - & R^{\lambda\mu}_{\nu\rho}\>\> y^\nu x^\rho & = 0 \\
\\
y^\lambda x^\mu & - & R^{\lambda\mu}_{\nu\rho}\>\> x^\nu y^\rho & = 0\\
\\
y^\lambda y^\mu & + & R^{\lambda\mu}_{\nu\rho}\>\> y^\nu y^\rho & = 0
\end{array}
\right.$$ for $\lambda,\mu \in \{ 1,\dots, n\}$. Since these relations are separately homogeneous in the $x^\lambda$ and in the $y^\mu$, the algebra $A$ is a bigraded algebra, that is one has $$\label{bigrA}
\begin{array}{l}
A = \oplus_{r,s\geq 0}\>\> A^{(r,s)}\\
\\
A^{(p,q)} A^{(r,s)} \subset A^{(p+r, q+s)}
\end{array}$$ where in $(r,s)$, $r$ denotes the degree in the $x^\lambda$ while $s$ denotes the degree in the $y^\mu$.\
The subalgebra of $A$ given by $\oplus_r A^{(r,0)}$ coincides with $\cala$ while the subalgebra given by $\oplus_s A^{(0,s)}$ coincides with $\cala' (\simeq \cala^!)$ that is $$\begin{array}{lll}
\cala & = & \oplus_r A^{(r,0)} \subset A\\
\\
\cala' & = & \oplus_s A^{(0,s)} \subset A
\end{array}$$ by definition.\
\[pprod\] In $A$, one has $$A^{(r,s)}=A^{(r,0)} A^{(0,s)}=A^{(0,s)}A^{(r,0)}$$ for any $r,s\in \mathbb N$
. By using the second and the third equalities of \[rdiff\], one can write any product of $(r)$ $s's$ and $(s)y's$ as a linear combination of products where the $s's$ are on the left-hand side (resp. right-hand side) of the $y's$. This implies $A^{(r,s)}\subset A^{(r,0)} A^{(0,s)}\subset A^{(r,s)}$ and $A^{(r,0)}\subset A^{(r,s)}$ and therefore $A^{r,s)}=A^{(r,0)} A^{(0,s)}=A^{(0,s)} A^{(r,0)}.\>\square$
In other words one has $$\label{phr}
A=\cala \cala'=\cala'\cala$$ in the bigraded algebra $A$ and $$\label{strgr}
A^{(p,q)} A^{(r,s)}=A^{(p+r,q+s)}$$ follows easily for any $p,q,r,s\in \mathbb N$.\
In the classical case one has $A^{(r,s)}=\cala_r\otimes \cala'_{ s}$ since then the $x^\lambda$ and the $y^\mu$ commute, and therefore in this case $A^{(r,s)}$ coincides with $S^r(E)\otimes \wedge^s(E)=S^r(\mathbb R^n)\otimes \wedge^s(\mathbb R^n)$.
The differential calculus $\Omega(\cala)$ on $\cala$
====================================================
Let us define the graded algebra $\Omega(\cala)$ by setting $$\label{omega}
\Omega(\cala)=\oplus_{k\in\mathbb N} \Omega^k(\cala)$$ with $$\label{omegak}
\Omega^k(\cala)=\oplus_{r\in \mathbb N} A^{(r,k)}$$ then the linear mapping $d$ of $A^{(1,0)}$ into $A^{(0,1)}$ and of $A^{(0,1)}$ into $A^{(0,2)}$ defined by $$\label{d}
\left\{
\begin{array}{lllll}
x^\lambda &\mapsto & dx^\lambda & = & y^\lambda \\
y^\lambda & \mapsto & dy^\lambda & = & 0
\end{array}
\right.$$ extends uniquely as an antiderivation of degree 1 of the graded algebra $\Omega(\cala)$ in view of the relations (\[rA\]). It follows that one has $$\label{diffo}
d^2=0$$ for the derivation $d^2$ since it vanishes on the generators of $\Omega(\cala)$ (=$A$ as algebra). Thus $\Omega(\cala)$ endowed with the differential $d$ is a differential graded algebra which is generated by $E$ as graded differential algebra with the quadratic relation (\[rdiff\]) of the introduction. Notice that the first equality of (\[phr\]) reads $$\label{prd}
\Omega^k(\cala)=\omega_{\lambda_1\cdots \lambda_k}\>\> dx^{\lambda_1}\dots dx^{\lambda_k}$$ for any $k\in \mathbb N$ with $\omega_{\lambda_1\cdots \lambda_k}\in \cala$ in view of (\[omegak\]).\
In the classical case the differential graded algebra $\Omega(\cala)$, that is $\Omega^\bullet(\cala)=S(\mathbb R^n)\otimes \wedge^\bullet(\mathbb R^n)$ endowed with $d$, is the differential graded algebra of exterior polynomial differential forms on $\mathbb R^n$.
The codifferential $\delta$ of $\Omega(\cala)$
==============================================
By using again the relations (\[rA\]), it follows that the linear mapping $\delta$ of $A^{(1,0)}$ into $A^{(2,-1)}=0$ and of $A^{(0,1)}$ into $A^{(1,0)}$ defined by $$\label{delta}
\left\{
\begin{array}{llll}
x^\lambda & \mapsto &\delta x^\lambda & = 0 \\
y^\lambda & \mapsto &\delta y^\lambda & = x^\lambda
\end{array}
\right.$$ extends uniquely as an antiderivation of degree -1 of the graded algebra $\Omega(\cala)$. Then since the derivation $\delta^2$ is vanishing on the generators of $\Omega(\cala)$ ($=A$ as algebra), one has $$\label{codiffo}
\delta^2=0$$ that is $\delta$ is a differential of degree -1 of $\Omega(\cala)$ which will be referred to as the [*codifferential of*]{} $\Omega(\cala)$.\
Since by definition $\Omega(\cala)=A$ as algebra $\Omega(\cala)$ is bigraded and its [*differential degree*]{} for which $d$ and $\delta$ are antiderivations is the second degree of this bigraduation, one has $$\Omega^p(\cala)=\sum_r\Omega^{(r,p)}(\cala)=\sum_r A^{(r,p)}$$ for the differential degree $p$. For this bigraduation $d$ is of bidegree $(-1,1)$ while $\delta$ is of bidegree $(1,-1)$.\
In the classical case the complex $(\Omega^\bullet(\cala),\delta)$, that is $$\Omega^\bullet(\cala)=S(\mathbb R^n)\otimes \wedge^\bullet(\mathbb R^n)$$ endowed with $\delta$, identifies canonically with the Koszul complex of the quadratic algebra $\cala=S(\mathbb R^n)$.\
A generalization of the Poincaré lemma
======================================
The derivation $d\delta+\delta d$ is of bidegree $(0,0)$ and one has the following lemma.
\[todeg\] The derivation $d\delta +\delta d$ is the total degree that is one has $$\label{derdeg}
(d\delta +\delta d) a = (r+s) a$$ for $a\in A^{(r,s)}$ and $r,s \in \mathbb N$.
. This follows immediately from $$\left\{
\begin{array}{lll}
(d\delta + \delta d) x^\lambda & = & x^\lambda\\
(d\delta + \delta d) y^\mu & = & y^\mu
\end{array}
\right.$$ for the generators $x^\lambda, y^\mu$, ($\lambda, \mu\in \{1,\dots, n\}$).$\>\>\square$
\[triv\] The homology $H^\bullet(d)=\oplus_{k\in \mathbb N} H^k(d)$ of $(\Omega(\cala),d)$ is given by $$H^k(d)=0\>\> \text{for}\>\> k\geq 1\>\> \text{and}\>\> H^0(d)=\mathbb R$$ while the homology $H_\bullet(\delta)=\oplus_{k\in \mathbb N}H_k(\delta)$ of $(\Omega(\cala),\delta)$ is given by $$H_k(\delta)=0\>\> \text{for}\>\> k\geq 1\>\> \text{and}\>\> H_0(\delta)=\mathbb R.$$
. In view of (\[derdeg\]), if $r+s>0$ then $da=0$ is equivalent to $a=d\delta \left(\frac{a}{r+s}\right)$ while $\delta a=0$ is equivalent to $a=\delta d\left(\frac{a}{r+s}\right)$. On the other hand $r+s=0$ means that $a=\lambda\bbbone$ for some $\lambda\in \mathbb R$. $\square$\
Notice that $(\Omega(\cala),d)$ is a cochain complex while $(\Omega(\cala),\delta)$ is a chain complex, it is why the homology $H^\bullet(d)$ of $d$ is noted with the degree up while the homology $H_\bullet(\delta)$ is noted with the degree down. Both $H^\bullet(d)$ and $H_\bullet(\delta)$ are real graded algebras which here are trivial that is reduced to the ground field $\mathbb R$ (in view of above corollary).\
Corollary (\[triv\]) means in particular that the sequence of $\cala$-modules $$\label{ResR}
\dots \stackrel{\delta}{\rightarrow}\Omega^k(\cala)\stackrel{\delta}{\rightarrow} \Omega^{k-1}(\cala)\stackrel{\delta}{\rightarrow}\dots \stackrel{\delta}{\rightarrow} \Omega^1(\cala)\stackrel{\delta}{\rightarrow}\cala\stackrel{\varepsilon}{\rightarrow}\mathbb R\rightarrow 0$$ is a resolution of the trivial module $\mathbb R$, $\varepsilon$ being the projection on degree 0 of the graded algebra $\cala$. A priori, this is not a sequence of free (i.e. projective here see in [@car:1958]) $\cala$-modules. However in the classical case $\cala=S(\mathbb R^n)$, (\[ResR\]) is the Koszul resolution of $\mathbb R$ by the free $S(\mathbb R^n)$-modules $S(\mathbb R^n)\otimes \wedge^\bullet(\mathbb R^n)$.\
It is our aim in the following to discuss this point and the relation of $(\Omega(\cala),\delta)$ with the Koszul complex of the quadratic algebra $\cala$, [@man:1988].
The Koszul complex $K(\cala)$
=============================
By the very definition of the Koszul dual algebra $\cala^!$ of $\cala$, the dual vector spaces $\cala^{!\ast}_p$ of the homogeneous components $\cala^!_p$ of $\cala^!$ are given by $$\label{kn}
\cala^{!\ast}_p=\cap_{0\leq s\leq n-2} E^{\otimes s}\otimes \calr \otimes E^{\otimes p-s-2}$$ for $p\geq 2$ and $\cala^{!\ast}_1=E$, $\cala^{!\ast}_0=\mathbb R\bbbone$, (as for any quadratic algebra with vector space of generators $E$ and vector space of relations $\calr \subset E\otimes E)$.\
Notice that one has $$\label{aen}
\cala^{!\ast}_p\subset E^{\otimes p}$$ for any $p\in \mathbb N$ and that furthermore $$\label{aepn}
\cala^{!\ast}_p\subset \calr \otimes E^{\otimes p-2}$$ for $p\geq 2$. Therefore the linear mappings of $\cala\otimes E^{\otimes p}$ into $\cala\otimes E^{p-1}$ defined by $$a \otimes (x_1\otimes \dots \otimes x_p) \mapsto ax_1 \otimes (x_2\otimes \dots \otimes x_p)$$ induces linear mappings $$\label{bord}
b: \cala\otimes \cala^{!\ast}_p \rightarrow \cala \otimes \cala^{!\ast}_{p-1}$$ satisfying $b^2=0$ (with the convention that $\cala^{!\ast}_p=0$ for $p<0$). This defines the Koszul complex $K(\cala)$ of $\cala$ as $$\label{K(cala)}
\dots \stackrel{b}{\rightarrow} \cala \otimes \cala^{!\ast}_p \stackrel{b}{\rightarrow} \cala\otimes \cala^{!\ast}_{p-1} \stackrel{b}{\rightarrow} \cala \otimes \cala^{!\ast}_{p-2} \stackrel{b}{\rightarrow}\dots$$ which ends as $$\label{K0}
\dots \stackrel{b}{\rightarrow} \cala \otimes \calr \stackrel{b}{\rightarrow} \cala \otimes E \stackrel{b}\rightarrow \cala$$ since $\cala^{!\ast}_2 = \calr$ and $\cala^{!\ast}_1=E$. As well known and easy to show the exactness of the following sequence $$\label{pres}
\cala\otimes \calr \stackrel {b}{\rightarrow} \cala \otimes E\stackrel{b}{\rightarrow}\cala \stackrel{\varepsilon}{\rightarrow} \mathbb R \rightarrow 0$$ is equivalent to the presentation of $\cala$ by generators and relations. Thus if the Koszul complex $K(\cala)$ is acyclic in positive degrees, (i.e. in degrees $\geq 2$ since by (\[pres\]) it is acyclic in degree =1), it gives a free resolution $$\label{kreso}
K(\cala) \stackrel{\varepsilon}{\rightarrow} \mathbb R \rightarrow 0$$ of the trivial module $\mathbb R$. A quadratic algebra $\cala$ is said to be a [*Koszul algebra*]{} [@pri:1970], [@man:1988] whenever its Koszul complex $K(\cala)$ is acyclic in positive degrees and then the resolution (\[kreso\]) is refered to as the [*Koszul resolution*]{} of $\mathbb R$. In this case, the Koszul resolution is a minimal free resolution.
Comparison of $K(\cala)$ with $(\Omega(\cala),\delta)$
======================================================
There are surjective homomorphisms $$\label{hp}
h_p:\cala\otimes \cala^{!\ast}_p\rightarrow \Omega^p(\cala)=\cala \cdot \cala'_p$$ of (left) $\cala$-modules which are induced by the isomorphisms $\cala^{!\ast}_p\simeq \cala'_p$ of vector spaces. However the corresponding diagram of homomorphisms of $\cala$-modules $$\label{DKO}
\begin{diagram}
\node{\dots} \arrow{e,t}{b} \node{K_p(\cala)}\arrow{s,r}{h_p} \arrow{e,t}{b}\node{K_{p-1}(\cala)}\arrow{s,r}{h_{p-1}}
\arrow{e,t}{b} \node{K_{p-2}(\cala)}\arrow{s,r}{h_{p-2}}
\arrow{e,t}{b}\node{\dots}\\
\node{\dots} \arrow{e,t}{\delta} \node{\Omega^p(\cala)}\arrow{e,t}{\delta}\node{\Omega^{p-1}(\cala)}\arrow{e,t}{\delta} \node{\Omega^{p-2}(\cala)}\arrow{e,t}{\delta}\node{\dots}
\end{diagram}$$ is generally not commutative and, in contrast to the $K_p(\cala)=\cala\otimes \cala^{!\ast}_p$ the $\Omega^p(\cala)=\cala \cdot \cala'_p$ are generally not free modules.\
However it is worth noticing here that the exact sequence (\[pres\]) corresponding to the presentation of $\cala$ coincides canonically with the end $$\Omega^2(\cala) \stackrel{\delta}{\rightarrow} \Omega^1(\cala) \stackrel{\delta}{\rightarrow} \cala \stackrel{\varepsilon}{\rightarrow} \mathbb R \rightarrow 0$$ of the exact sequence (\[ResR\]) since one has $\Omega^1(\cala)=\cala\otimes E$ and $\Omega^2(\cala)= \cala\otimes \cala'_2\simeq \cala\otimes \calr$ canonically in view of $\cala'_2=E\otimes E/\calr^\perp$ and $E\otimes E=\calr\oplus \calr^\perp$ and since furthermore with these identifications $\delta$ coincide with $b$ up to an irrelevant factor. Thus the end of (\[DKO\]) $$\begin{diagram}
\node{K_2(\cala)}\arrow{s,r}{h_2} \arrow{e,t}{b} \node{K_1(\cala)}\arrow{s,r}{h_1} \arrow{e,t}{b}\node{\cala}\arrow{s,r}{h_0}
\arrow{e,t}{\varepsilon } \node{\mathbb R}\arrow{s}\arrow{s,r}{=}
\arrow{e}\node{0}\\
\node{\Omega^2(\cala)} \arrow{e,t}{\delta} \node{\Omega^1(\cala)}\arrow{e,t}{\delta}\node{\cala}\arrow{e,t}{\varepsilon} \node{\mathbb R}\arrow{e}\node{0}
\end{diagram}$$ is a commutative diagram of $\cala$-modules with vertical isomorphisms. It is also worth noticing that in the classical case where $\cala =S(\mathbb R^n)$ and $R^{\lambda\mu}_{\nu\rho}=\delta^\lambda_\rho \delta^\mu_\nu$, the diagram (\[DKO\]) is commutative with vertical isomorphisms. It is therefore natural to ask for conditions on $(\cala, R)$ for which this occurs. In view of the acyclicity of (\[ResR\]), this can occur only if $\cala$ is Koszul and if the $\Omega^p(\cala)$ are free modules. A very natural condition under which this is satisfied is when $R:E\otimes E \rightarrow E\otimes E$ satisfies the so called quantum Yang-Baxter equation $$\label{YB}
(R\otimes I)(I\otimes R)(R\otimes I)=(I\otimes R)(R\otimes I)(I\otimes R)$$ as endomorphim of $E\otimes E \otimes E$, where $I$ denotes the identity mapping of $E=\mathbb R^n$ into itself. Indeed it is well known that then $\cala$ is a Koszul algebra [@gur:1990], [@wam:1993] and that by using the fact that the $I^{\otimes^r} \otimes R\otimes I^{\otimes^s}$ generate a representation of the symmetric group $\frak S(r+s+2)$ in $E^{\otimes r+s+ 2}$ one can show that the product $\cala \cdot \cala'$ in $A$ is isomorphic to $\cala\otimes \cala'$ and that the $b's$ coincides then to the $\delta's$ up to unimportant normalizations.
The complex quadratic $\ast$-algebra variant {#StarAlg}
============================================
The variant described in this section is useful and natural for a quantum theoretical formalism and for discussion of reality conditions in a noncommutative geometrical framework. It consists here in considering complex quadratic algebras which are complex $\ast$-algebras generated by hermitian elements which belong to an Euclidean subspace generating a Hilbertian space.\
So let again $E$ be a $n$-dimensional Euclidean space with orthonormal coordinate basis $(x^\lambda)\>\> \lambda\in \{1,\dots, n\}$ and let $E_c$ be the complexified of $E$ endowed with the corresponding sesquilinear product. Thus $E_c$ is a Hilbertian space endowed with an antilinear involution $x\mapsto x^\ast$ with $E=\{x\in E_c\vert x^\ast=x\}$ and $(x^\lambda)$ is an orthonormal basis of $E_c$ which consists of hermitian elements. Then there is a unique extension of the involution of $E_c$ to the complex tensor algebra $T(E_c)$ for which $T(E_c)$ is a complex $\ast$-algebra.\
Let $\calr$ be a $\ast$-invariant subspace of $E_c\otimes E_c$, (i.e. invariant by $x\otimes y\mapsto y^\ast\otimes x^\ast)$, the complex quadratic algebra $$\cala=T(E_c)/(\calr)$$ is a $\ast$-algebra which is generated by the hermitian elements $x^\lambda,\>\> \lambda\in \{1,\dots,n\}$. The space $E_c\otimes E_c$ is canonically a Hilbertian space and let $P$ be the hermitian projector onto $\calr\subset E_c \otimes E_c$. One has $$P=\frac{1}{2}(\bbbone -R)$$ where $R$ is a hermitian involution. In terms of components in the $x^\lambda\otimes x^\mu$ one has $R=(R^{\lambda\mu}_{\nu\rho})$ with $R^{\lambda\mu}_{\nu\rho}\in \mathbb C$ satisfying $$R^{\lambda\mu}_{\alpha\beta} R^{\alpha\beta}_{\nu\rho} = \delta^\lambda_\nu \delta^\mu_\rho$$ as before and $$\label{astR}
R^{\lambda\mu}_{\nu\rho} = \bar R^{\nu\rho}_{\lambda\mu} = \bar R^{\mu\lambda}_{\rho\nu}$$ instead of relation (\[sym\]) of Section 2. The relations of $\cala$ reads $$x^\lambda x^\mu= R^{\lambda\mu}_{\nu\rho} x^\nu x^\rho$$ for $\lambda, \mu \in \{1,\dots, n\}$ and by the same construction as in Sections 4, 5, 6, 7 using a complex $R$ as above instead of the real one there, one constructs a differential calculus $\Omega(\cala)$ for $\cala$ which is a quadratic complex differential graded $\ast$-algebra generated by $E$ and one defines the codifferential $\delta$ on $\Omega(\cala)$, etc. This variant of the construction has been used in a particular 4-dimensional setting in [@lan-pag:2018].
Generalizations
===============
Finally it is worth noticing that Relations (\[rdiff\]) define a quadratic differential graded algebra for any $R=(R^{\lambda\mu}_{\nu\rho})$ which is involutive $(R^2=\bbbone)$ or more generally which is invertible and satisfies an equation of the form $$\label{pHe}
R^2=\alpha R + \beta \bbbone$$ where $\alpha$ and $\beta$ are two scalar (with $\beta\not = 0$), in order that the equations with $R$ replaced by $R^n$ with $n\geq 2$ do not imply other constraints.\
Then all the above analysis applies, even the one of Section 9 with some slight modifications in view of the work of [@gur:1990] and [@wam:1993] implying the Koszulity of $\cala$ under the assumption (\[YB\]), (Hecke case).
Acknowlegements {#acknowlegements .unnumbered}
===============
This work was partially supported by INDAM-GNSAGA and by the INDAM-CNRS Project LIA-LYSM.
[1]{}
H. Cartan. Homologie et cohomologie d’une alg[è]{}bre gradu[é]{}e. , 11(2):1–20, 1958.
M. Dubois-Violette. Lectures on graded differential algebras and noncommutative geometry. In Y. Maeda and al., editors, [*Noncommutative Differential Geometry and Its Applications to Physics*]{}, pages 245–306. Shonan, Japan, 1999, Kluwer Academic Publishers, 2001.
D.I. Gurevich. Algebraic aspects of the quantum [Y]{}ang-[B]{}axter equation. , 2:119–148, 1990 ; (Transl. in Leningrad Math. J. 2 (1991) 801-828).
G. Landi and C. Pagani. A class of differential quadratic algebras and their symmetries. , 12:1469–1501, 2018.
Yu. I. Manin. . CRM Universit[é]{} de Montr[é]{}al, 1988.
S.B. Priddy. Koszul resolutions. , 152:39–60, 1970.
M. Wambst. Complexes de [Koszul]{} quantiques. , 43:1083–1156, 1993.
S.L. Woronowicz. Differential calculus on compact matrix pseudogroups (quantum groups). , 122:129–170, 1989.
[^1]: Laboratoire de Physique Théorique, UMR 8627, CNRS et Université Paris-Sud 11, Bâtiment 210, F-91 405 Orsay Cedex, France\
Michel.Dubois-Violette$@$u-psud.fr
[^2]: Matematica, Università di Trieste, Via A. Valerio, 12/1, 34127 Trieste, Italy\
Institute for Geometry and Physics (IGAP) Trieste, Italy\
and INFN, Trieste, Italy\
landi$@$units.it
|
---
abstract: 'Networks of nonlinear resonators offer intriguing perspectives as quantum simulators for non-equilibrium many-body phases of driven-dissipative systems. Here, we employ photon correlation measurements to study the radiation fields emitted from a system of two superconducting resonators, coupled nonlinearly by a superconducting quantum interference device (SQUID). We apply a parametrically modulated magnetic flux to control the linear photon hopping rate between the two resonators and its ratio with the cross-Kerr rate. When increasing the hopping rate, we observe a crossover from an ordered to a delocalized state of photons. The presented coupling scheme is intrinsically robust to frequency disorder and may therefore prove useful for realizing larger-scale resonator arrays.'
author:
- 'Michele C. Collodo'
- Anton Potočnik
- Simone Gasparinetti
- 'Jean-Claude Besse'
- Marek Pechal
- Mahdi Sameti
- 'Michael J. Hartmann'
- Andreas Wallraff
- Christopher Eichler
title: |
[Observation of the Crossover from Photon Ordering to Delocalization\
in Tunably Coupled Resonators]{}
---
Engineering optical nonlinearities that are appreciable on the single photon level and lead to nonclassical light fields has been a central objective for the study of light-matter interaction in quantum optics [@haroche2013exploring; @raimond_manipulating_2001; @chang_quantum_2014]. While such nonlinearities have first been realized in individual optical cavities [@thompson_observation_1992; @birnbaum_photon_2005] and with Rydberg atoms [@brune_quantum_1996; @peyronel_quantum_2012], more recently superconducting circuit quantum electrodynamics (QED) [@wallraff_strong_2004] has proven to be a powerful platform for the study of nonclassical light fields. Circuit QED systems facilitate strong effective interactions between individual photons [@houck_generating_2007; @lang_observation_2011], long coherence times [@koch_charge-insensitive_2007] as well as precise control of drive fields [@motzoi_simple_2009; @heeres_implementing_2017] within a large variety of possible design implementations. Particularly, *in-situ* tunable or nonlinear couplers have been explored more recently for superconducting elements [@bertet_parametric_2006; @baust_tunable_2015; @mckay_universal_2016; @chen_qubit_2014; @lu_universal_2017; @kounalakis_tuneable_2018; @eichler_realizing_2018].
Well-controllable engineered quantum systems, in which strong optical nonlinearities occur in extended volumes [@carusotto_quantum_2013] or networks of multiple nonlinear resonators offer interesting perspectives to study interacting many-body systems with photons [@hartmann_quantum_2016; @noh_quantum_2017] and to mimic the dynamics of otherwise less accessible systems [@georgescu_quantum_2014; @schmidt_circuit_2013; @houck_-chip_2012], such as supersolids [@leonard_supersolid_2017] or topological quantum matter [@gross_quantum_2017]. Photons are trapped in resonators only for a limited time, even in high quality devices. Interacting photons are thus typically explored in a non-equilibrium regime, in which continuous driving compensates for excitation loss and yields stationary states of light fields [@noh_out--equilibrium_2017]. It has been predicted that these non-equilibrium systems offer rich phase diagrams of novel exotic states which have no analogue in equilibrium systems [@prosen_quantum_2008], featuring *e.g.* synchronization [@leib_synchronized_2014] or bistability [@le_boite_steady-state_2013; @kessler_dissipative_2012].
Non-equilibrium coupled resonator systems have more recently also been investigated experimentally, both in a semiclassical and in a quantum regime. Macroscopic self trapping of exciton polaritons has been observed in a dimer of coupled Bragg stack microcavities [@abbarchi_macroscopic_2013], vacuum squeezing was demonstrated in a dimer of superconducting resonators [@eichler_quantum-limited_2014], the unconventional photon blockade has been observed in the optical and the microwave domain [@vaneph_observation_2018; @snijders_observation_2018], and signatures of bistability have been found in a chain of superconducting resonators [@fitzpatrick_observation_2017]. Moreover, a transition from a classical to a quantum regime has been observed in the decay dynamics of a resonator dimer [@raftery_observation_2014], chiral currents of one or two photons have been generated in a three qubit ring [@roushan_chiral_2017], and spectral signatures of many-body localization [@roushan_spectroscopic_2017] as well as a Mott insulator of photons [@ma_dissipatively_2018] have been observed in a qubit chain.
In this Letter, we explore the interaction between individual photons in a driven-dissipative system of two nonlinearly coupled superconducting resonators (see Fig. \[fig1\]a). The nonlinear coupler mediates a cross-Kerr interaction $V$, on-site Kerr interactions $U_a$ and $U_b$, and an effective linear hopping interaction [with]{} *in-situ* tunable rate $J{\ensuremath{_{\text{ac}}}}$. We measure the on-site $g^{(2)}_{aa} := g^{(2)}_{aa}(\tau = 0)$ and cross correlations at zero time delay $g^{(2)}_{ab} := g^{(2)}_{ab}(\tau = 0)$ between the emitted field from both resonators. In the limit of small $J{\ensuremath{_{\text{ac}}}}$/$V$, a photon trapped in one resonator blocks the excitation of the neighboring resonator and vice versa, leading to a spontaneous self-ordering of microwave photons [@chang_crystallization_2008; @hartmann_polariton_2010]. Such an inter-site photon blockade regime has been predicted for resonator arrays with nonlinear couplers [@jin_photon_2013; @jin_steady-state_2014]. When increasing $J{\ensuremath{_{\text{ac}}}}/V$, however, a delocalization of photons and a simultaneous occupation of both resonators becomes favorable, leading to a change in the photon statistics.
For this experiment we utilize an on-chip superconducting circuit consisting of two lumped element resonators with characteristic impedance $Z = 80 {\ensuremath{\,\mathrm{\Omega}}}$ (see Fig. \[fig1\]b,c). The aforementioned nonlinear coupling circuit, interconnecting the two resonators, consists of a capacitively shunted superconducting quantum interference device (SQUID) with capacitance $C_J = 95 {\ensuremath{\,\mathrm{fF}}}$ and Josephson energy $E{\ensuremath{_{\text{J}}}}{\ensuremath{^{\text{max}}}}/h = 80 {\ensuremath{\,\mathrm{GHz}}}$, with the Planck constant $h$. We use a superconducting coil and an on-chip flux drive line (port 5) to ensure full dc and ac control of the magnetic flux threading the SQUID loop. Each resonator is weakly coupled to an input port (3 and 4), through which we drive the system, and to an output port (1 and 2) into which approximately $50\%$ of the intra-cavity field is emitted and measured using a linear detection chain. The total decay rates are measured to be $(\kappa_a, \kappa_b)/2\pi = (2.8, 2.4) {\ensuremath{\,\mathrm{MHz}}}$.
First, we characterize the sample by measuring the transmitted amplitude $|S_{21}|$ as a function of external magnetic flux $\Phi{\ensuremath{_{\text{dc}}}}$. At each flux bias point we observe two resonances corresponding to the two eigenmodes of the system, see Fig. \[supp\_cal1\]a. The flux dependence of the measured eigenfrequencies is well explained by a linear circuit impedance model comprising a tunable effective Josephson energy, which allows us to determine the aforementioned circuit parameters. From a normal mode model we extract the tuning range of the corresponding linear hopping rate $J{\ensuremath{_{\text{dc}}}}/2\pi = -0.8\ldots0.8{\ensuremath{\,\mathrm{GHz}}}$ (Fig. \[supp\_cal1\]b). The tunability of $J{\ensuremath{_{\text{dc}}}}$ results from an interplay between the capacitive and the flux-dependent inductive coupling between the two resonators. As these carry opposite signs, we are able to cancel both contributions achieving approximately zero net static linear coupling $J{\ensuremath{_{\text{dc}}}} \approx 0$ at a dc flux bias point of $\Phi{\ensuremath{_{\text{dc}}}} \approx -0.37{\ensuremath{\,\mathrm{\Phi_0}}}$, where $\Phi_0 = \frac{h}{2 e}$ is the magnetic flux quantum. At this bias point the two measured resonances $(\omega_a, \omega_b)/2\pi = (6.802, 7.164) {\ensuremath{\,\mathrm{GHz}}}$ are separated by the bare detuning $\Delta/2\pi \equiv (\omega_b- \omega_a)/2\pi = 362 {\ensuremath{\,\mathrm{MHz}}}$ and correspond to good approximation to the local modes of the system (Fig. \[supp\_cal1\]c,d). As a result, the radiation of each mode ($a$, $b$) is collected in its respective output line at port (1, 2). Notably, the finite detuning $\Delta$ between the bare cavity modes suppresses undesired nonlinear interactions, which would otherwise give rise to pair hopping and correlated hopping (see supplementary material).
In order to recover a well controllable linear hopping rate despite the finite cavity detuning, we implement a parametric coupling scheme [@bertet_parametric_2006; @tian_parametric_2008; @lu_universal_2017]. Here, we apply an ac modulated flux drive to the SQUID with a variable amplitude $\Phi{\ensuremath{_{\text{ac}}}}$ and a modulation frequency $\omega{\ensuremath{_{\text{ac}}}}$, which equals the resonator detuning $\omega{\ensuremath{_{\text{ac}}}}=\Delta$. For $\Phi{\ensuremath{_{\text{ac}}}}=0$ we recover the uncoupled resonator modes when probing the transmission spectra $|S_{13}|$ and $|S_{24}|$ (see Fig. \[fig2\]a). However, as we increase $\Phi{\ensuremath{_{\text{ac}}}}$, we observe a simultaneous frequency splitting of both modes, which scales linearly with $\Phi{\ensuremath{_{\text{ac}}}}$, and which we interpret as the result of a parametrically induced photon hopping with rate $J{\ensuremath{_{\text{ac}}}}/2\pi = 0\ldots40 {\ensuremath{\,\mathrm{MHz}}}$ (see supplementary material).
In an appropriate doubly rotating frame, where each mode rotates at its resonance frequency, our system is well described by an effective Hamiltonian $$\begin{aligned}
\frac{1}{\hbar}\mathcal{H}_\Delta &= \delta_a {\color{black}a{^{\dagger}}a} + \delta_b {\color{black}b{^{\dagger}}b} + J{\ensuremath{_{\text{ac}}}}\left( {\color{black}a{^{\dagger}}b}+ {\color{black}b{^{\dagger}}a}\right)\\
&+ \frac{1}{2} U_a \, {\color{black}a{^{\dagger 2}}a^2} + \frac{1}{2} U_b \,{\color{black}b{^{\dagger 2}}b^2} + V\, {\color{black}a{^{\dagger}}a b{^{\dagger}}b}\\
&+ \Omega_a (a{^{\dagger}}+ a) + \Omega_b (b{^{\dagger}}+ b)
$$
with the drive detuning $\delta_i = \omega{\ensuremath{_{\text{drive,\textit{i}}}}} - \omega_i$ ($i \in \{a,b\}$) and the drive rates $\Omega_i$. The on-site and the cross-Kerr interaction rates at zero coupling bias are $(U_a, U_b, V)/2\pi = -(3.1\pm0.3, 2.7\pm0.2, 7.0\pm0.3) {\ensuremath{\,\mathrm{MHz}}}$, which have been extracted from a spectroscopic measurement (see supplementary material). In the absence of a parametric modulation the eigenstates of this Hamiltonian correspond to the photon number states $|n_a n_b\rangle$ in the local basis (compare Fig. \[fig2\]b). The second order transitions are red shifted by the corresponding Kerr rates. For finite $J{\ensuremath{_{\text{ac}}}}$ the eigenstates hybridize in both the one- and two-photon manifold.
We focus on a parameter regime in which $|V|$ and $J{\ensuremath{_{\text{ac}}}}$, as well as $\kappa_i$ and $\Omega_i$, are comparable in magnitude, featuring a competition between nonlinear interaction and linear hopping, as well as between drive and dissipation. In our system we additionally have $|U_i| \approx \kappa_i$. Both $\Omega = \Omega_a = \Omega_b$, setting the average number of excitations in the system, and $J{\ensuremath{_{\text{ac}}}}$, setting the rate at which the resonators exchange excitations, are utilized as tunable control parameters, while $V$, $U_i$ and $\kappa_i$ are constant. In the experiment we keep the drive frequencies, and thus $\delta_i = 0$, fixed. We eliminate influences of the phase of $J{\ensuremath{_{\text{ac}}}}$ on the measured results by averaging over multiple randomized phase configurations.
We characterize the quantum states of the uniformly and continuously driven two-resonator system by measuring the second order cross $g^{(2)}_{ab}$ and on-site correlation $g^{(2)}_{aa}$ of the emitted radiation as a function of $J{\ensuremath{_{\text{ac}}}}$ and $\Omega$ (see Fig. \[fig3\]a,c). To this aim, we linearly amplify and digitize the radiation fields at both output ports in order to obtain the second order photon correlations [@bozyigit_antibunching_2011; @eichler_quantum-limited_2014; @eichler_exploring_2015]. To enhance the signal-to-noise ratio, we use a quantum-limited Josephson parametric amplifier [@eichler_quantum-limited_2014] operated in a phase-sensitive mode (see supplementary material for details about the detection process). The measured $g^{(2)}$ correlations are compared with the results of a numerical master equation simulation [@qutip] (see Fig. \[fig3\]b,d). As confirmed by this simulation, the average resonator occupations remain at or below the single photon level for all the data presented in Fig. \[fig3\].
In the regime of small $J{\ensuremath{_{\text{ac}}}}$ and low $\Omega$ we measure the radiation to be anti-bunched, see Fig. \[fig3\]. In this limit, the cross-Kerr interaction effectively shifts the transition frequency of one cavity when a photon is present in the other and thus detunes the ($|01\rangle, |10\rangle \leftrightsquigarrow |11\rangle$) transition from the drive tones. This inhibits simultaneous occupation of both cavities, leading to a dynamic self-ordered photon state manifested as anti-bunching in the photon cross statistics. Equivalently, the on-site Kerr interaction prevents each mode from being doubly excited, leading to anti-bunched on-site correlations.
Increasing the hopping rate $J{\ensuremath{_{\text{ac}}}}$ results in a hybridization of the modes in both the one- and two-excitation manifold, see Fig. \[fig2\]c for a level diagram as a function of $J{\ensuremath{_{\text{ac}}}}$. When $J{\ensuremath{_{\text{ac}}}}$ becomes comparable to the Kerr rate $U_i$, the transitions of the single photon manifold become detuned from the drive frequency, while the two-photon-transition into the symmetric $|02+20+11\rangle$ branch becomes resonant with the drive. This leads to a more efficient drive into the second excitation manifold compared to the originally dominating single photon states and to an admixture of simultaneous cavity occupations ($|11\rangle$, $|20\rangle$, and $|20\rangle$). This causes a crossover from anti-bunched to bunched statistics in both the measured $g^{(2)}_{ab}$ and $g^{(2)}_{aa}$, see Fig. \[fig3\]e. Interestingly, we find a regime in which the on-site correlation $g^{(2)}_{aa}$ is already close to unity, while the cross correlation $g^{(2)}_{ab}$ is still anti-bunched. We attribute this effect to $V$ being larger than $U_i$.
Studying the dependence on the drive rate $\Omega$, we find that $g^{(2)}_{aa}$ approaches unity when $\Omega$ exceeds $U_i$ (Fig. \[fig3\]f), which we explain by the breakdown of the photon blockade. This effect is found to be largely independent of $J{\ensuremath{_{\text{ac}}}}$. We observe a similar behavior for the cross correlations. In this case, however, the measured $g^{(2)}_{ab}$ approaches one half in the limit of large drive rate $\Omega$, which is in good agreement with the result obtained from the numerical simulations.
In conclusion, we have realized a coupled cavity system, featuring a tunable ratio between linear hopping and cross-Kerr interaction rate and observed the crossover from photon ordering to delocalization. Inspired by the proposals by Jin *et al.* [@jin_photon_2013; @jin_steady-state_2014], we interpret the measured cross correlations as an order parameter in a ($J{\ensuremath{_{\text{ac}}}}$, $\Omega$)-dependent phase diagram of the system. The observed crossover closely resembles the onset of a driven-dissipative photon ordering phase transition, from a fully ordered crystalline phase dominated by spontaneous symmetry breaking towards a uniform delocalized steady-state phase [@brown_localization_2018; @fink_signatures_2018].
We expect the demonstrated coupling mechanism to be well extendable towards larger resonator arrays. Resilience to disorder in electrical parameters [@underwood_low-disorder_2012] can be achieved by frequency staggering of neighboring cavities along with the adjustability of the parametric modulation frequencies. Additionally, the employed lumped element structures excel in this scenario thanks to a compact footprint and high design versatility.
The presented system and variations thereof could be used to explore regimes, in which inter-site interactions exceed on-site interactions [@elliott_designing_2018; @busche_contactless_2017]. Additionally, the controllability of the phase of the hopping rate could be employed to create artificial gauge fields in plaquette systems and to study non-reciprocal dynamics with photons [@roushan_chiral_2017]. Furthermore, the variability of flux modulation frequencies could enable the controllable activation of additional interaction terms such as a parametric coupling between neighboring resonators [@tangpanitanon_hidden_2018] or pair hopping [@peropadre_tunable_2013], *e.g.* for the study of supersolid phases [@huang_extended_2016].
This work is supported by the National Centre of Competence in Research “Quantum Science and Technology” (NCCR QSIT), a research instrument of the Swiss National Science Foundation (SNSF) and by ETH Zurich. M. J. H. acknowledges support by the EPSRC under grant No. EP/N009428/1.
******
**
System Engineering
==================
Linear Impedance Model
----------------------
We construct a linear impedance model based on the equivalent circuit of our sample (see Fig. \[fig1\]b) using an ABCD matrix formalism. The model is then fitted to the measured eigenfrequencies, which we extract from measurements of the dc flux dependent transmission amplitude spectrum $|S_{21}|$ (see Fig. \[supp\_cal1\]a). This allows us to determine the electrical parameters, see Tab. \[tab:1\].
--------------------------------------------------------------------- ------------------------------------
$C_a$ $260 {\ensuremath{\,\mathrm{fF}}}$
$C_b$ $300 {\ensuremath{\,\mathrm{fF}}}$
$C_J$ $95 {\ensuremath{\,\mathrm{fF}}}$
$E{\ensuremath{_{\text{J}}}}{\ensuremath{^{\text{max}}}}/h$$\qquad$ $80 {\ensuremath{\,\mathrm{GHz}}}$
$L_a$ $1.9 {\ensuremath{\,\mathrm{nH}}}$
$L_b$ $1.9 {\ensuremath{\,\mathrm{nH}}}$
$L{\ensuremath{_{\text{s}}}}$ $0.3 {\ensuremath{\,\mathrm{nH}}}$
--------------------------------------------------------------------- ------------------------------------
: \[tab:1\] List of electrical parameters of the microwave circuit presented in the main text.
We neglect the corrections due to the coupling to the environment. However, we take into account the contribution of a spurious inductance $L{\ensuremath{_{\text{s}}}}$ caused by the lead wires to the SQUID. This will modify the dc contribution of the effective inductance of the SQUID as $L{\ensuremath{_{\text{J}}}} = L{\ensuremath{_{\text{s}}}} + \Phi_0/E{\ensuremath{_{\text{J}}}}(\Phi{\ensuremath{_{\text{dc}}}})$ with $E{\ensuremath{_{\text{J}}}}(\Phi{\ensuremath{_{\text{dc}}}}) = E{\ensuremath{_{\text{J}}}}{\ensuremath{^{\text{max}}}} |\cos(\pi \Phi{\ensuremath{_{\text{dc}}}} / \Phi_0)|$.
In order to prevent the creation of a large closed ground loop through the SQUID, which could alter the circuit’s dynamics in the presence of magnetic flux, we opt for a floating resonator configuration via large shunt capacitors, designed to be $C{\ensuremath{_{\text{s}}}} = 800 {\ensuremath{\,\mathrm{fF}}}$.
Effective Hamiltonian under Parametric Modulation
-------------------------------------------------
In order to be able to study the competition of linear hopping interaction and nonlinear cross-Kerr interaction between adjacent resonators, it is crucial to construct a system with a Hamiltonian featuring exclusively these two coupling mechanisms.
Starting from the Lagrangian of the implemented nonlinear coupling circuit [@jin_photon_2013] discussed in the main text (see Fig. \[fig1\]), we find the full local mode Hamiltonian $$\begin{aligned}
\frac{1}{\hbar} \mathcal{H{\ensuremath{^{\text{full}}}}} &=
\omega_a \, \uline{{\color{{black}} a^{\dagger}a}} + \omega_b\, \uline{{\color{{black}}b^{\dagger} b}} \\
&- (J{\ensuremath{_{\text{c}}}} - J_\ell) \left({\color{{black}}a{^{\dagger 2}}} + {\color{{black}}b{^{\dagger 2}}} + {\color{{black}}a^2} + {\color{{black}}b^2} \right)\\
&+ (J{\ensuremath{_{\text{c}}}} - J_\ell) \left( \dotuline{{\color{{black}}a{^{\dagger}}b}} + \dotuline{{\color{{black}}b{^{\dagger}}a}} \right)\\
&- (J{\ensuremath{_{\text{c}}}} + J_\ell) \left( {\color{{black}}a{^{\dagger}}b{^{\dagger}}} + {\color{{black}}b a} \right)\\
&+\frac{\tilde V}{24} \underbrace{\left( a + a{^{\dagger}}- b - b{^{\dagger}}\right)^4}_{
\large{
\begin{subarray}{l}
= a{^{\dagger 4}}+ 4\,{\color{{black}}a{^{\dagger 3}}a}+ 6\, \uline{{\color{{black}}a{^{\dagger 2}}a^2}} + 4\, {\color{{black}}a{^{\dagger}}a^3} + a^4\\[7pt]
- 4\, b{^{\dagger}}a{^{\dagger 3}}- 4\,{\color{{black}}a{^{\dagger 3}}b} - 12 \,{\color{{black}}b{^{\dagger}}a{^{\dagger 2}}a}- 12\, \dotuline{{\color{{black}}a{^{\dagger 2}}b a}}\\
- 12\, \dotuline{{\color{{black}} b{^{\dagger}}a{^{\dagger}}a^2}}- 12\,{\color{{black}} a{^{\dagger}}b a^2} - 4\, {\color{{black}}b{^{\dagger}}a^3} - 4\, b a^3 \\[7pt]
+ 6\, b{^{\dagger 2}}a{^{\dagger 2}}+ 12\, {\color{{black}}b{^{\dagger}}a{^{\dagger 2}}b} + 6\, \dashuline{{\color{{black}}a{^{\dagger 2}}b^2}}\\
+ 12\, {\color{{black}}b{^{\dagger 2}}a{^{\dagger}}a} + 24\, \uline{{\color{{black}}b{^{\dagger}}a{^{\dagger}}b a}}+ 12\,{\color{{black}} a{^{\dagger}}b^2 a}\\
+ 6\,\dashuline{{\color{{black}} b{^{\dagger 2}}a^2}} + 12\, {\color{{black}}b{^{\dagger}}b a^2} + 6\, b^2 a^2 \\[7pt]
- 4\,b{^{\dagger 3}}a{^{\dagger}}- 4\,{\color{{black}}b{^{\dagger 3}}a} - 12\, {\color{{black}}b{^{\dagger 2}}a{^{\dagger}}b} - 12\, \dotuline{{\color{{black}}b{^{\dagger 2}}b a}}\\
- 12\, \dotuline{{\color{{black}}b{^{\dagger}}a{^{\dagger}}b^2}} - 12\, {\color{{black}}b{^{\dagger}}b^2 a} - 4\, {\color{{black}}a{^{\dagger}}b^3 } - 4\, b^3 a \\[7pt]
+b{^{\dagger 4}}+ 4\,{\color{{black}}b{^{\dagger 3}}b} +6\, \uline{{\color{{black}} b{^{\dagger 2}}b^2}} + 4\, {\color{{black}}b{^{\dagger}}b^3} + b^4\end{subarray}
}
} + \mathcal{O}(a^6)\\\end{aligned}$$
expressed in terms of ladder operators $a^{\dagger}$, $b^{\dagger}$ with the bare resonator frequencies $\omega_a$, $\omega_b$, the capacitively (inductively) mediated linear hopping rate $J{\ensuremath{_{\text{c}}}}$ ($J_\ell$) and the cross-Kerr rate $\tilde V$. For simplicity we assume comparable characteristic impedances $Z_a \approx Z_b$ resulting in Kerr rates $\tilde V \approx 2 \tilde U_a \approx 2 \tilde U_b$ for the two resonators. Corrections from normal ordering are omitted. The selected highlighted terms rotate at a frequency of $
\{
\dashuline{{\color{{black}} 2\omega_a-2\omega_b}},
\dotuline{{\color{{black}}\omega_a - \omega_b}},
\uline{{\color{{black}}0}} \}
$ with respect to a doubly rotating frame, which is locked to the resonance frequencies of both modes $\omega_a$ and $\omega_b$. In particular, this shows the importance of a substantial detuning $\Delta = \omega_b - \omega_a \neq 0$ in order to suppress the detrimental pair hopping ($\sim\dashuline{{\color{{black}}a^{\dagger 2} b^{2}}}$) and correlated hopping ($\sim\dotuline{{\color{{black}}a^{\dagger} \left(a^{\dagger}a + b^{\dagger}b\right) b}}$) terms while keeping the desired Kerr terms ($\sim\uline{{\color{{black}}a^{\dagger}a b^{\dagger}b}}$) resonant.
In order to preserve a linear hopping interaction (${\sim\dotuline{{\color{{black}}a^{\dagger} b}}}$) despite this detuning $\Delta$ we employ a parametric modulation scheme by driving the nonlinear coupling circuit with the ac modulated flux $$\varphi(t)~=~\varphi{\ensuremath{_{\text{dc}}}} + \varphi{\ensuremath{_{\text{ac}}}} \cos\left(\omega{\ensuremath{_{\text{ac}}}}t \right),$$ where $\varphi_{i} = 2\pi \Phi_i / \Phi_0$ are the magnetic flux amplitudes normalized to the flux quantum. The applied flux drive tone at frequency $\omega{\ensuremath{_{\text{ac}}}}$ is responsible for selecting and activating specific interactions from $\mathcal{H{\ensuremath{^{\text{full}}}}}$ within a subsequent rotating wave approximation. The interaction rates are effectively altered as $ J_\ell\mapsto J_\ell \cos \frac{\varphi(t)}{2} \text{ and } \tilde V \mapsto \tilde V \cos \frac{\varphi(t)}{2} $, an expansion to first order leads to $$\begin{aligned}
\cos \left(\frac{\varphi(t)}{2}\right) &= \cos \left(\frac{\varphi{\ensuremath{_{\text{dc}}}}}{2} + \frac{\varphi{\ensuremath{_{\text{ac}}}}}{4} \left({\ensuremath{\mathrm e^{{\mathrm i}\Delta t}}} + {\ensuremath{\mathrm e^{-{\mathrm i}\Delta t}}}\right) \right)\\
&\approx \cos \left(\frac{\varphi{\ensuremath{_{\text{dc}}}}}{2}\right) - \sin \left(\frac{\varphi{\ensuremath{_{\text{dc}}}}}{2}\right) \frac{\varphi{\ensuremath{_{\text{ac}}}}}{4} \left({\ensuremath{\mathrm e^{{\mathrm i}\Delta t}}} + {\ensuremath{\mathrm e^{-{\mathrm i}\Delta t}}}\right)\end{aligned}$$ for a modulation frequency $\omega{\ensuremath{_{\text{ac}}}} = \Delta$.
We choose individual rotating frames for each mode, locked to their respective resonance frequency $\omega_{a}$, $\omega_{b}$. After a rotating wave approximation for sufficiently large resonator-resonator detuning $\Delta$ (*i.e.* keeping solely resonant terms) and operating with balanced capacitive and inductive hopping rates $J{\ensuremath{_{\text{c}}}} = J_\ell \cos (\varphi{\ensuremath{_{\text{dc}}}}/2)$, we are left with the effective Hamiltonian
$$\begin{aligned}
\frac{1}{\hbar}\mathcal{H{\ensuremath{^{\textsf{i}}}}{\ensuremath{_{\textsf{$\Delta$}}}}} &= \delta_a \uline{{\color{{black}}a{^{\dagger}}a}} + \delta_b \uline{{\color{{black}}b{^{\dagger}}b}}\\
&+ J_\ell \sin \left(\frac{\varphi{\ensuremath{_{\text{dc}}}}}{2}\right) \frac{\varphi{\ensuremath{_{\text{ac}}}}}{4} \left( \dotuline{{\color{{black}}a{^{\dagger}}b}}+ \dotuline{{\color{{black}}b{^{\dagger}}a}}\right)\\
&+\frac{1}{2} \tilde U_a \cos \left(\frac{\varphi{\ensuremath{_{\text{dc}}}}}{2}\right)\, \uline{{\color{{black}}a{^{\dagger 2}}a^2}}\\
&+ \frac{1}{2} \tilde U_b \cos \left(\frac{\varphi{\ensuremath{_{\text{dc}}}}}{2}\right)\,\uline{{\color{{black}}b{^{\dagger 2}}b^2}} \\
&+ \tilde V \cos \left(\frac{\varphi{\ensuremath{_{\text{dc}}}}}{2}\right) \, \uline{{\color{{black}}a{^{\dagger}}a b{^{\dagger}}b}}\\
&- \tilde J_n \sin \left(\frac{\varphi{\ensuremath{_{\text{dc}}}}}{2}\right) \frac{\varphi{\ensuremath{_{\text{ac}}}}}{4} \left( \dotuline{{\color{{black}}a{^{\dagger}}\left(a{^{\dagger}}a + b{^{\dagger}}b \right) b}} + \text{h.c.}\right)\\
&+ \Omega_a (a{^{\dagger}}+ a) + \Omega_b (b{^{\dagger}}+ b)\end{aligned}$$
In this form, the linear scaling of the effective linear hopping rate $J{\ensuremath{_{\text{ac}}}} = J_\ell \sin(\varphi{\ensuremath{_{\text{dc}}}}/2) \, \varphi{\ensuremath{_{\text{ac}}}}/4$ with the flux modulation amplitude $\Phi{\ensuremath{_{\text{ac}}}}$ is evident. For our device we find $J_\ell/2\pi = 1.6 {\ensuremath{\,\mathrm{GHz}}}$. The nonlinear Kerr rates $(U_a, U_b, V) = (\tilde U_a, \tilde U_b, \tilde V) \cos (\varphi{\ensuremath{_{\text{dc}}}}/2)$ are dependent on the flux bias point, but are inherently robust against detuning $\Delta$ or flux modulation. We infer the correlated hopping rate $\tilde J_n/2\pi = 8 {\ensuremath{\,\mathrm{MHz}}} $ from the measured cross-Kerr rate. For sufficiently weak flux modulation ($\varphi{\ensuremath{_{\text{ac}}}}/2\pi \lesssim 3\times10^{-3}$ in the experiment), the contribution of the pair (correlated) hopping terms are quadratically (linearly) suppressed, allowing us to neglect their influence on the system dynamics and to finally reconstruct the Hamiltonian as presented in the main text.
Experimental Setup and Data Collection
======================================
Sample Fabrication
------------------
All linear elements of the device presented in Fig. \[fig1\] are fabricated by patterning a sputtered $150 {\ensuremath{\,\mathrm{nm}}}$ thin niobium film on a sapphire substrate with photolithography and reactive ion etching. In a subsequent step Josephson junctions are added using electron-beam lithography and double-angle shadow evaporation of Aluminum. We operate the sample in a dilution refrigerator at a base temperature of $20 {\ensuremath{\,\mathrm{mK}}}$. The full microwave wiring diagram is shown in Fig. $\ref{supp2}$.
Linear Amplification Chain and Driving Scheme
---------------------------------------------
The resonators $a$, $b$ are driven at their respective bare resonance frequencies $\omega_a$, $\omega_b$ with a set of symmetric drive lines via ports 3 and 4. Scattered radiation is collected at ports 1 and 2 and routed to a set of symmetric linear amplification chains. The itinerant signal is amplified by near-quantum limited Josephson parametric amplifiers (JPA) [@eichler_quantum-limited_2014] at base temperature, operated in phase-sensitive mode in a dual pump tone configuration [@kamal_signal--pump_2009]. The pump tones are symmetrically detuned from the signal frequency by $\pm 250 {\ensuremath{\,\mathrm{MHz}}}$ ($\pm 300 {\ensuremath{\,\mathrm{MHz}}}$) for the amplifier at mode $a$ ($b$). The pump tones are thus far detuned from the measurement band, alleviating the need for pump tone cancellation. We determine a bandwidth of $13 {\ensuremath{\,\mathrm{MHz}}}$ ($20 {\ensuremath{\,\mathrm{MHz}}}$) at a gain of $18.8 {\ensuremath{\,\mathrm{dB}}}$ ($19.6 {\ensuremath{\,\mathrm{dB}}}$) of the parametric amplifiers.
Subsequently, the signals are amplified by high-electron-mobility-transistor (HEMT) amplifiers, thermally anchored at $4{\ensuremath{\,\mathrm{K}}}$, and low noise room temperature amplifiers. The signals are then down converted by mixing with individual local oscillator tones to an intermediate frequency of $25 {\ensuremath{\,\mathrm{MHz}}}$, filtered to avoid backfolding of noise and amplified before being digitized by an analog-to-digital converter with a sampling rate of $100 {\ensuremath{\,\mathrm{MHz}}}$. A field programmable gate array (Xilinx Virtex-4) digitally down converts the digitized signals and extracts the $I$, $Q$ quadratures for each channel.
The JPA pump field and local oscillator phases are locked to each other and chosen such that $I$ corresponds to the amplified quadrature. In order to uniformly sample the entire phase space distribution of the field we slowly cycle the relative phase of the individual local oscillators with respect to the corresponding input drive field. Consequently, we are able to reconstruct the second order correlations from a measurement of a single quadrature $I$ per channel.
Correlation Function Measurements
---------------------------------
We collect the resulting $I_a$, $I_b$ quadrature values of several million repetitions of the experiment with the drive fields turned on in a two dimensional histogram (“on”) and extract the normally ordered statistical moments $\langle (I_a)^n (I_b)^m\rangle{\ensuremath{_{\text{on}}}}$ (with $n+m \leq 4$) of this distribution [@lang_correlations_2013]. It is necessary to mitigate the influence of added thermal noise on the on-histogram in order to reconstruct the statistics of the radiation field emitted from the sample at base temperature. To this aim, we repeat the measurement without applying any drive field and extract the statistical moments $\langle (I_a)^n (I_b)^m\rangle{\ensuremath{_{\text{off}}}}$ from the corresponding histogram (“off”). Assuming a linear amplification chain, the moments of the on-histogram $$\begin{aligned}
\langle (I_a)^n (I_b)^m\rangle{\ensuremath{_{\text{on}}}} = \sum^{n,m}_{k,l = 0} \binom{n}{k}\binom{m}{l} &\langle (I_a)^{n-k} (I_b)^{m-l}\rangle{\ensuremath{_{\text{off}}}} \\
\times&\langle (I_a)^{k} (I_b)^{l}\rangle{\ensuremath{_{\text{diff}}}}\end{aligned}$$ are composed of the added noise captured in the moments of the off-histogram and the moments of the signal at the output of the sample $\langle (I_a)^{k} (I_b)^{l}\rangle{\ensuremath{_{\text{diff}}}}$. This set of linear equations is solved in order to obtain the latter [@eichler_characterizing_2012].
The reconstructed moments of the quadratures can be expressed as normally ordered moments of the mode operators $\langle (I_a)^k (I_a)^l\rangle{\ensuremath{_{\text{diff}}}} \propto \langle:\left(a + a{^{\dagger}}\right)^k \left(b + b{^{\dagger}}\right)^l :\rangle$, where $I_a = \frac{1}{2} \left(a{^{\dagger}}+ a\right)$, $I_{b} = \frac{1}{2}\left(b{^{\dagger}}+ b\right)$. This allows us to calculate the correlations at zero time delay $g^{(2)}_{aa} := g^{(2)}_{aa}(0) = \frac{\langle a{^{\dagger}}a{^{\dagger}}a a \rangle}{\langle a{^{\dagger}}a\rangle^2}$ and $g^{(2)}_{ab} := g^{(2)}_{ab}(0) = \frac{\langle a{^{\dagger}}a b{^{\dagger}}b \rangle}{\langle a{^{\dagger}}a\rangle \langle b{^{\dagger}}b\rangle}$ as $$g^{(2)}_{aa} = \frac{2}{3} \frac{\langle (I_a)^4 \rangle{\ensuremath{_{\text{diff}}}}}{\langle (I_a)^2\rangle^2{\ensuremath{_{\text{diff}}}}}
\text{, }
g^{(2)}_{ab} = \frac{\langle (I_a)^2 (I_b)^2 \rangle{\ensuremath{_{\text{diff}}}}}{\langle (I_a)^2\rangle{\ensuremath{_{\text{diff}}}} \langle (I_b)^2\rangle{\ensuremath{_{\text{diff}}}}}$$ Error bars shown in Fig. \[fig3\] of the main text are extracted from the standard deviation of the mean of repeated measurements. The validity of the analysis is verified by measuring the statistics of a coherent tone $g^{(2)}_{aa} = 1.01 \pm 0.01$ and $g^{(2)}_{bb} = 1.0 \pm 0.02$.
Measured Kerr Rates
===================
We measure the nonlinear interaction rates at the dc flux bias point $\Phi{\ensuremath{_{\text{dc}}}} \approx -0.37{\ensuremath{\,\mathrm{\Phi_0}}}$, *i.e.* at vanishing linear hopping $J{\ensuremath{_{\text{dc}}}}/2\pi \approx 0 {\ensuremath{\,\mathrm{MHz}}}$ via spectroscopic characterization of two-photon transitions (Fig. \[supp\_kerr\]). The on-site Kerr rates are measured with a strong probe tone, we find $(U_a, U_b)/2\pi = -(3.1\pm0.3, 2.7\pm0.2) {\ensuremath{\,\mathrm{MHz}}}$. The cross-Kerr rates are measured using a weak probe tone while simultaneously pumping the respective single photon transitions strongly. Fitting two Lorentzian curves to the measured spectra allows us to extract the detuning between the transitions of the one- and two-photon manifolds, which directly corresponds to the Kerr rates in question. We find the cross-Kerr rate $(V{\ensuremath{_{\text{a}}}}, V{\ensuremath{_{\text{b}}}})/2\pi = - (7.2, 6.7){\ensuremath{\,\mathrm{MHz}}}$ to be dependent on the port from which the transition is pumped. We attribute this to imprecisions in the frequency extraction and combine the findings to the value reported in the main text $V/2\pi = - (7.0\pm0.3) {\ensuremath{\,\mathrm{MHz}}}$.
|
---
author:
- |
M. De Pasquale[^1], L. Piro, B. Gendre, L. Amati, L.A. Antonelli, E. Costa, M. Feroci,\
F. Frontera, L. Nicastro, P. Soffitta,
- 'J. in’t Zand'
date: 'Received –; accepted –'
title: 'The BeppoSAX catalog of GRB X-ray afterglow observations'
---
Introduction
============
Discovered in the early 70’s [@kle73], Gamma-Ray Bursts (GRBs) have been a mysterious phenomenon for 25 years. The lack of any optical counterpart prevented observers from determining the distance - galactic or extragalactic - and therefore the amount of energy involved, which was uncertain within 10 orders of magnitude. A lot of different models were at that time able to explain the observed prompt gamma-ray emission.
The situation changed dramatically with the first fast and precise localization of GRB, that was obtained by the BeppoSAX satellite [@piro95; @boe97] in 1997. This satellite was combining a gamma-ray burst monitor (that provided the burst trigger) with X-ray cameras (that were able to asses a precise position and to carry out follow-up observations). This observational strategy led to the discovery of the X-ray [@cos97], optical [@van97] and radio [@fra97] afterglows. The spectroscopy of the optical counterpart of the burst also allowed the distance of these events to be firmly established as cosmologic [@met97]. With the end of the BeppoSAX mission (April 2002) and its reentry, a page of the GRB afterglow study was turned, but the observations remain within the archives. To prepare the future, we have initiated a complete re-analysis of all X-ray observations done. In this first paper, we present the legacy of BeppoSAX : its X-ray afterglow catalog, focusing on the continuum properties. We will also compare our results with those of previous studies on GRB X-ray afterglows [@fro05; @piro04]. In two forthcoming papers [@gen05 Gendre et al. in preparation], we will discuss GRB afterglow observations made by XMM-Newton and Chandra, and a systematic study of line emission in the X-ray afterglow spectra. This article is organized as follows. In Sec. \[sec\_analyse\] we present the data analysis and the results. We discuss these results in Sec. \[sec\_discu\] in the light of the fireball model. We investigate the so called [*Dark Burst*]{} phenomenon in Sec. \[sec\_dark\], before concluding.
Data reduction and analysis {#sec_analyse}
=============================
BeppoSAX detected and localized simultaneously in the Gamma Ray Burst Monitor [GRBM, @fro97] and Wide Field Cameras [WFC, @jag97] 51 GRBs within its six year long lifetime [@fro04]. These bursts have been included in our analysis sample. We note that this set is biased against X-ray rich GRBs and especially X-ray flashes [@hei03], i.e. bursts with weak or absent signal in the GRBM and normal counterpart in the WFC. In our sample, we also included GRB991106, GRB020410 and GRB020427, although they gave no detection in the GRBM[^2], due to the fact that a subsequent observation with BeppoSAX was performed after the localization with the WFC. Data on these bursts are reported in Tables \[table1\] and \[table2\]. We have not included the bursts discovered after an archive re-analysis. Overall, it was possible to follow up 36 burst with the narrow field instruments. One other afterglow observation () was carried out following external triggers. Finally, in the case of , BeppoSAX detected the burst while it was outside failed the WFC field of view, and the follow up observation was performed on the basis of a localization by the RXTE All Sky Monitor. In this paper we present the data gathered by the Narrow Field Instruments (NFI) Low Energy Concentrator Spectrometer [LECS, 0.1 - 10 keV, @par97] and Medium Energy Concentrator Spectrometer [MECS, 1.6 - 10 keV, @boe97]. The first of this sample () was followed up late, while 38 had fast (within 1 day[^3]) follow up observations. We analyzed 37 of these fast follow-up, excluding due to its high contamination of a nearby X-ray source.
A typical observation starts $\sim 8-9$ hours after the burst and its duration is about $1\times 10^{5}$ seconds for MECS and $7
\times 10^{4} $ for LECS. The net exposure lasts $\sim 1/2$ of the observation for MECS and $1/4$ for LECS.
Afterglow identification and temporal analysis
----------------------------------------------
The first step of data analysis is source detection, in order to find the afterglow. For this purpose, we used the MECS data, because this instrument has a sensitivity higher than that of the LECS. We extracted the image, ran the detection tool [*Ximage*]{} 4.3[^4] on this image and selected all the sources with at least a $3\sigma$ significance located inside the WFC error box. In the special cases of and we used the IPN error box [@hur00] and ASM error box [@lev98] respectively as these bursts were outside the WFC field of view. The afterglow was recognized by its fading behavior. The light curves were generated from counts extracted within a circle area centered on the source with a radius of 4 arcminutes. We chose this value because $\gtrsim90\%$ of the source energy is within this region [@fio99]. We also selected counts between 1.6 and 10 keV interval, which is the optimal range of work for the MECS.
The associated background was extracted using an annulus centered at the same position than the source extraction region, with inner and outer radii of 4.5 and 10 arcminutes respectively. To take into account the effects of effective area variation and the MECS support, we renormalized the counts extracted in the annuli by a factor determined by comparing the counts in the same regions of the library background fields.
We used the local background rather than the library background for light curves in order to take into account any possible time fluctuation. We developed an IDL script to construct and fit light curves. This algorithm can calculate adequate errors even in the case of few counts per bin, by using a Poissonian statistics. However, if possible, the width of temporal bins was chosen wide enough to have at least 15-20 counts/bin (background subtracted) at least, in order to apply a proper Gaussian fit (see below). When available, subsequent TOOs were also used to better constraint the light curve behavior.
The light curves were fitted using a simple power law, using the Levenberg-Marquardt method to minimize the $\chi^2$ statistic. We detected 31 sources with a positive decaying index (in the following, we used the convention $F_{X} \varpropto t^{-\delta}$, thus a decaying source has a positive decay index) at the $90\%$ confidence level. These sources were identified as the X-ray afterglow of each burst[^5]. For three of these sources (, and ) the value of the decay index is greater than zero but not well constrained. We report in Table \[table3\] the decay index we obtained for all these 31 sources (henceforth, all errors reported are at $1\sigma$, while upper limits are quoted at the 90% confidence level, unless otherwise specified).
In three cases (, and ), we detected within the WFC error box only one source that did not display any significant fading behavior. We refer to these as [*candidate*]{} afterglows. We have calculated the probability to observe a serendipitous source at the observed flux level within the WFC error box for these 3 bursts, adopting the Log N - Log S distribution for BeppoSAX released by @gio00. The probability are $\cong 0.027$ for and $\cong0.05$ for and . The probability that all of these 3 sources are not afterglows is $\sim
10^{-4}$. We note, however, that these probabilities have been calculated for extragalactic sources; for low Galactic latitude events, like GRB991106 ($b\simeq-3\degr$), the value may differ significantly. @cor02 indicated that [GRB 991106]{} could in fact be a Galactic type-I X-ray burster.
In two cases ( and ) we did not detect any source with $3\sigma$ significance within the WFC error box. We report in Table \[table3\] the $3\sigma$ detection upper limits.
Some observations deserve special comments. was observed for $\sim1000$ seconds only and no decaying behavior can be detected within the light curve of the source found inside the WFC error box. However, given the high flux of this source ($\sim10^{-12}$ erg cm$^{-2}$ sec$^{-1}$ in the 1.6-10 keV band), the probability to have observed a serendipitous source was $\sim10^{-3}$. We have thus assumed that this source was indeed the X-ray afterglow of . In the case of , we analyzed the source S1 coincident with [@pia99]. We do not include it in the following discussion as the detected X-ray emission could be strongly affected by .
We present the light curves in Fig. \[fig1\].
Spectral analysis
-----------------
The X-ray afterglow spectra have been accumulated from the LECS and MECS during the first TOO only, for those afterglows with more than 150 photons in the MECS (background subtracted). 15 GRBs passed this criterion; their spectra are presented in Fig. \[fig2\].
We have generally collected LECS counts within a circle centered on the source with radius $r=8$ arcminutes, which again encircles $>90\%$ of source energy. We operated with LECS data in the range 0.1-4.0 keV, where the response matrix is more accurate. As for MECS, we collected counts with the same criteria we applied for the time analysis. For spectral analysis, we used the library spectral backgrounds for both LECS and MECS as they have a very good signal-to-noise ratio, due to long exposition[^6]. However, the library backgrounds have been taken at high Galactic latitudes, with an average Galactic absorption around 2-3 $\times10^{20}$. Several afterglows in our sample have been observed in fields with an absorption much higher than this value. For these bursts, the local background would differ from the library one at low energy (e.g. below 0.3 keV). The use of a library background from 0.1 keV would result in an underestimate of the low-energy signal and consequently a too high estimate of the intrinsic absorbing column of the burst. Therefore, to evade this problem, we have taken the minimum energy for LECS to be 0.4 keV if the Galactic column density was $N_{H}\ge 5\times 10^{20}$ cm$^{-2}$. Similarly to the time analysis, the spectral analysis was performed by requiring at least $15-20$ counts/bin. The standard model to fit the spectral data consists of a constant, a Galactic absorption, an extragalactic absorption (i.e. [*in situ*]{}) and a power law. The constant has been included because of the differences in the LECS and MECS instruments. Its value is obtained in each case by fitting LECS and MECS data in the 1.6 - 4 keV interval (to avoid absorption effects) with a simple power law model.
In our work, we have calculated the 1.6 - 10 keV flux of X-ray afterglows 11 hours after the burst trigger. We have chosen this time to avoid effects of changes in the decaying slope. The average count rate in the MECS has been associated with the average flux given by the spectrum. Successively, we have taken the count rate at 11 hours, which is given by the light curves, to compute the flux at that time. In most cases, observations include it. In a few cases (e.g. ) the flux has been extrapolated.
For those afterglows with $<150$ counts, we used a canonical model with an power law energy index of $\alpha=1.2$ (which is typical of X-ray afterglow spectra) to convert the count rate 11 hours after the trigger to the corresponding flux.
---------- ------------------------- ------------------------ ------------------------ ------------------------ --
GRB name $1.6 - 10$ keV Flux Decay Spectral Density
($10^{-13}$ index index column
erg cm$^{-2}$ s$^{-1}$) $\delta$ $\alpha$ ($10^{22}$ cm$^{-2}$)
$0.75 \pm 0.47$ $2.8^{-3.7}$ — —
$20.8 \pm 2.7$ $1.32^{+0.15}_{-0.20}$ $1.04^{+0.21}_{-0.27}$ $<1.12 $
$1.35 \pm 0.73$ $1.11^{+1.5}_{-0.76}$ — —
$5.72 \pm 0.90$ $0.80^{+0.18}_{-0.15}$ $1.40^{+0.32}_{-0.27}$ $2.63^{+2.5}_{-1.37}$
$6.36 \pm 0.91$ $1.00 \pm 0.22$ $1.08^{+0.40}_{-0.23}$ $<53$
— $>0.4$ — —
$6.00 \pm 0.56$ $1.42^{+0.62}_{-0.48}$ $1.44^{+0.32}_{-0.26}$ $<3.07$
$2.82 \pm 0.59$ $0.10\pm0.06$ — —
$5.6 \pm 2.2 $ $>0.51$ — —
$3.9^{+1.2}_{-1.1}$ $2.18^{+0.89}_{-0.65}$ $2.43^{+0.97}_{-0.65}$ $5.1^{+6.0}_{-3.8}$
$2.6^{+1.2}_{-1.1}$ $1.49^{+1.9}_{-0.86}$ — —
$14.0^{+7.0}_{-3.2}$ $1.10^{+0.36}_{-0.28}$ $1.71 \pm0.29 $ $2.6^{+2.0}_{-1.3}$
$2.8^{2.1}_{1.3}$ $0.66^{+0.68}_{-0.44}$ — —
$54.2 \pm 1.7$ $1.45\pm0.06$ $0.99\pm0.05$ $0.10^{+0.08}_{-0.06}$
$2.8^{+5.1}_{-1.4}$ $>0$ — —
$34.7 \pm 2.1 $ $1.4\pm0.1$ $1.17\pm 0.09$ $<0.93$
$3.3^{+1.6}_{-1.5}$ $1.32^{+1.7}_{-0.92}$ — —
$5.87 \pm 0.84$ $0.88^{+0.28}_{-0.20}$ $1.68^{+0.45}_{-0.38}$ $4.1^{+3.4}_{-2.3}$
$3.20 \pm 0.87$ $0.9^{+0.47}_{-0.42}$ $1.31^{+0.57}_{-0.43}$ $<13.15$
$10.6 \pm 4.0 $ — — —
$5.4^{+1.9}_{-1.5}$ $1.10^{+0.50}_{-0.32}$ — —
$1.26 \pm 0.47 $ $1.1^{+2.5}_{-2.1}$ — —
$3.10^{+0.90}_{-0.96}$ $1.41^{+0.98}_{-0.77}$ $1.54^{+0.31}_{-4}$ $2.1^{+2.0}_{-1.3}$
$6.2^{+2.1}_{-1.8}$ $0.68\pm0.41$ $1.04\pm0.27$ $<0.36$
$3.0^{+4.1}_{-1.4}$ $0.8^{+0.5}_{-1.5}$ — —
$1.6 \pm 1.2$ $>0$ — —
$1.28 \pm 0.38 $ $-0.23^{+1.4}_{-0.94}$ — —
$32.6^{+15.7}_{-8.7}$ $1.79^{+0.21}_{-0.16}$ — —
$23.2^{+5.8}_{-4.5}$ $1.47^{+0.22}_{-0.27}$ $1.29^{+0.27}_{-0.26}$ $3.4^{+2.3}_{-1.7}$
$3.06^{+0.71}_{-0.64}$ $1.90^{+0.90}_{-0.53}$ — —
$<1.43$ — — —
$70.6 \pm 3.4$ $1.35\pm0.06$ $1\pm0.06$ $1.27^{+0.33}_{-0.31}$
$13.6 \pm 1.5$ $1.30\pm0.03$ — —
$<3.4$ — — —
$3.8 \pm 0.8$ $0.84^{+0.46}_{-0.35}$ — —
$77.8^{+6.3}_{-6.9}$ $0.92\pm0.12$ $1.3\pm0.19$ $<4.8$
$4.8 \pm 1.7$ $1.3^{+0.10}_{-0.12}$ — —
---------- ------------------------- ------------------------ ------------------------ ------------------------ --
All the results of our X-ray afterglow analysis are summarized in Table \[table3\]. In Table \[table3b\], we report results of the previous analysis on single *BeppoSAX* GRBs, mostly taken by a review of @fro04. We can see a general agreement of the previous results with ours. In order to increase the statistical significance of the sample of X-ray afterglows with known redshift, we included in our successive analysis GRB011211. For this burst, which was observed by , we assumed a flux of $1.7\pm0.04$ 10$^{-13}$ erg cm$^{-2}$ s$^{-1}$, a spectral and decay index of $\alpha=1.3\pm0.06$ and $\delta=2.1\pm0.2$ respectively (Gendre et al. 2005).
------ --------------------------- ----------------------- --------------------------------------- ------------------------- -------------------- --
GRB Temporal Energy n$_H$ /n$_H^G$ 2–10 keV flux Ref.
name index$^a$ index at 10$^5$ s $^{a}$
$\delta$ $\alpha$ ($\times10^{21}$ cm$^{-1}$) (erg cm$^{-2}$s$^{-1}$)
$>$1.5 — — $<1.0\times10^{-13}$ @fer98
1.3 $\pm$ 0.2 1.1$\pm$0.3 3.5$_{-2.3}^{+3.3}$ / 1.6 $\sim6.8\times10^{-13}$ @cos97 [@fro98]
$1.45 \pm 0.15$ 1.7$\pm$0.6 $<$20 / 2.0 $\sim4.5\times10^{-14}$ @nic98v
1.1$\pm$0.1$^b$ 1.5$\pm$0.55 6.0$_{-3.3}^{+7.9}$ / 0.5 $3.5\times10^{-13}$ @piro98b [@piro99]
$\sim1.2$ 0.6$\pm$0.2 1.0$_{-1.0}^{+2.3}$ / 0.6 — @dal00
1.12$^{+0.08}_{-0.05}$(W) \[1.1\] \[0.13\] / 0.13 $\sim1.4\times10^{-13}$ @ant99
1.3$\pm$0.03 (W) 1.4$\pm$0.4 10$\pm$4 / 0.9 $2.0\times10^{-13}$ @zan98
0.16$\pm$0.04 1.0$\pm$0.18 \[0.39\] / 0.39 $\sim4.0\times10^{-13}$ @pia00
1.83$\pm$0.30 1.8$^{+0.6}_{-0.5}$ 3–20 / 1.73 $8.0\times10^{-14}$ @nic98
1.19$\pm$0.17(W) —- — $\sim2.3\times10^{-13}$ @sof02
$>$0.91 1.51$\pm$0.32 36$_{-13}^{+22}$$^{\mathrm c}$ / 0.34 $4.5\times10^{-13}$ @vre99
1.3$^{+0.5}_{-0.4}$ 0.92$\pm$0.47 \[0.18\] / 0.18 $\sim2.0\times10^{-13}$ @fro00b
$^d$ 1.46$\pm$0.04 0.94$\pm$0.08 0.9$^{+15} _{0.9}$ / 0.21 $1.25\times10^{-12}$ @mai05
1.42$\pm$0.07 1.03$\pm$0.08 2.1$\pm$0.6/0.94 $9.6\times10^{-13}$ @kuu00
0.83$\pm$0.16 0.7$^{+0.4}_{-0.2}$ \[0.3\] / 0.3 $\sim3.3\times10^{-13}$ @fer01
1.58$\pm$0.06 — - $<1.2\times10^{-13}$ @fro05
1.15$\pm$0.03(W) 1.16$^{+0.3}_{-0.37}$ \[0.35\] / 0.35 $\sim2.0\times10^{-13}$ @mon01
$>$0.4 0.53$\pm$0.25 \[2.5\]/ 2.5 $\sim3.0\times10^{-13}$ @zan00b
$1.38\pm0.03$(W) $0.75\pm0.3$ $<4\times10^{21}$ $\sim2\times 10^{-13}$ @piro02
0.8$\pm$0.3 1.0$\pm$0.18 0.7$_{-0.7}^{+7.5}$/ 0.55 $\sim3.5\times10^{-13}$ @ant00
1.89$^{+0.16 e}_{-0.19}$ $0.9\pm0.42$ 4/0.27$^e,^f$ $9.0\times10^{-13}$ @piro01
1.18$\pm$0.05 1.4$\pm$0.3 8.7$\pm$0.4$ /0.42$ $\sim8.0\times10^{-13}$ @ama02b
2.1$^{+0.6}_{-1.0}$ 0.3$^{+0.8}_{-0.6}$ \[0.27\] / 0.27 — @gui03
1.33$\pm$0.04 0.97$\pm$0.05 1.5$\pm$0.3/ 0.16 $2.4\times10^{-12}$ @zan01
$1.29\pm0.03$(W) $1.6\pm0.5$ $<100/$ $\sim 10^{-13}$ @piro05
— — — $<3\times10^{-13}$ @zand03
$0.81\pm0.07$ $1.05\pm0.08$ — $\sim3.5\times10^{-12}$ @nic04
$1.3^{+0.13} _{0.09}$ $1^{+2.2} _{-1.1}$ 0.29/0.29 $\sim10^{-13}$ @ama04
------ --------------------------- ----------------------- --------------------------------------- ------------------------- -------------------- --
$^a$ All upper limits are 3$\sigma$ except for GRB990705 which are 2$\sigma$.\
$^b$ from 6$\times$10$^{4}$ s to 5.8$\times$10$^{5}$ s\
$^c$ n$_H$ value corrected for redshift.\
$^d$ Spectral data of the first 20,000 s. The time decaying index includes the whole NFI TOO.\
$^e$ *SAX* plus *CHANDRA* data [@piro01].\
$^f$ Corrected for redshift [@piro01]. This n$_H^z$ value was added to the Galactic column density n$_H^G$.\
Results and Discussion {#sec_discu}
======================
General properties of X-ray afterglows {#section31}
--------------------------------------
![The distribution of 1.6-10 keV fluxes in the BeppoSAX GRB afterglow sample. All fluxes are indicated 11 hours after the burst. Upper limits have been set to $10^{-13}$ for clarity. \[fig3\]](Flux_X_11hr_all_in_one_10.ps){width="8cm"}
{width="8cm"} {width="8cm"}
We detect an X-ray afterglow in 31 of 36 cases. This constitutes $86\%$ of the sample. If all doubtful sources are considered as afterglows, then the fraction of X-ray afterglows increases up to $94\%$.
In Fig. \[fig3\] we present the distribution of the X-ray afterglow flux F$_{X}$ in the 1.6-10 keV band. It spans approximately 2 orders of magnitude. GRB 020410 afterglow is the object with the highest flux, $\sim8\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$, while the weakest is $970402$, $\sim10^{-13}$ erg cm$^{-2}$ s$^{-1}$. The fit of this distribution with a Gaussian provides a logarithmic mean and width of $<F_{X}>-12.2\pm 0.1$ and $\sigma_{Fx}
= 0.5$ respectively. One may wonder if some faint afterglows could be missed due to the detection limit (either due to a low luminosity or to a large distance). In this case, the true distribution could be broader than that we measure. However, the fact that we detect X-ray afterglows in $\sim90\%$ of follow-up observations indicates that this is not the case.
We have also estimated the distribution of the spectral and decay indexes (Fig.\[fig:alphadelta\]). The values we have obtained for those parameters are the result of the convolution of the intrinsic distribution with the measurement error. Under the assumption that both are Gaussian, it is possible to deconvolve the two distributions. We have adopted a maximum likelihood method [see @dep03; @mac88] to gather the best estimates of the parent distribution in the BeppoSAX sample. We have obtained from the spectral index distribution a mean value of $\alpha=1.2\pm0.1$ with a width of $0.13_{-0.05}^{+0.11}$, and from the decay index distribution a mean value of $\delta=1.3\pm0.1$ with a width of $0.3 \pm 0.1$. These values depend on the value of $p$, the energy power law index of the electrons which radiate by synchrotron emission within the fireball, and the state of the fireball itself (fast/slow cooling, position of the cooling frequency, beaming, surrounding medium). In section 3.4 we will show that the average properties of the afterglow are consistent with a cooling frequency below the X-ray range. In this case, following Sari et al. 1998, we can determine an average value for $p=2.4\pm0.2$.
---------- ------------------------ --------------------- ---------- ------------------------ --
GRB name $L_{X} ^{iso}$ $E_{\gamma} ^{iso}$ $\theta$ $L_{X} ^{corr}$
$10^{44}$ erg s$^{-1}$ $10^{51}$ erg rad $10^{44}$ erg s$^{-1}$
28.6 9.9 $>0.32$ $>1.46$
16.1 3.5 0.391 1.23
147 125 $>0.1$ $>0.74$
7.21 4.26 $>0.226$ $>0.2$
37.4 74.1 0.2 0.75
373 692 0.089 1.48
269.7 144.5 0.054 0.39
79.4 0.096
3.32 $>0.777$
6.96 130 $>0.139$ $>0.07$
3.4 3.17 $>0.115$ $>0.023$
335 155 0.140 2.14
377 375 0.08 13.1
5.1 3.74 0.145 0.05
20 68.8 0.115 0.12
---------- ------------------------ --------------------- ---------- ------------------------ --
: \[table4\]X-ray luminosity (assuming isotropy, $L_{X} ^{iso}$, and after beaming correction, $L_{X}
^{corr}$), Energy emitted during the prompt $\gamma$-ray event (assuming isotropy, $E_{\gamma} ^{iso}$) in units of $10^{51}$ erg, and beaming angle for BeppoSAX GRBs with a measured beaming angle (extracted from literature).
General properties of the prompt emission and selection effects
---------------------------------------------------------------
We list in Table \[table2\] the properties of the prompt emission of GRB detected by BeppoSAX, extracted from the literature. Figure \[fig3bis\] displays the distribution of the $\gamma$-ray fluence of the BeppoSAX sample. The fit with a Gaussian provides a mean logarithmic fluence of $S_{\gamma}=-5.31$ and a width of distribution $\sigma_{S\gamma}=0.77$[^7].
![The 40-700 keV fluence distribution of the BeppoSAX GRB sample. Data are extracted from the literature.\[fig3bis\]](Fluence_Gamma_all_in_one_7.ps){width="8cm"}
An important question regards the possible selection effects on the flux of the prompt phase. In Fig. \[fig:selection\] we present the isotropic gamma-ray energy and X-ray energy for events of known redshift, emitted in the 40-700 and 2-10 keV band respectively in the GRB cosmological rest frames. They have been calculated by using the k-correction of Bloom et al. 2001, with cosmological parameters $H_{0}$=65 km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\Lambda}$=0.7, $\Omega_{M}$=0.3.
The continuous lines indicate the detection thresholds as function of the redshift, *for a typical GRB.* Note that these are indicative values because the sensitivity depends on the exposed area as function of the off-set angle and the duration of the event. The minimum energy required for a detection have been calculated taking the fluence detection thresholds of the two instruments, around $S=10^{-7}$ erg cm$^{-2}$ for the GRBM and $S=8\times
10^{-8}$ erg cm$^{-2}$ for the WFC. In the case of the WFC this corresponds to about 200 mCrab in 20 seconds. From the figures it is evident that the gamma-ray energies are well above the GRBM threshold. On the contrary the sample is limited by the WFC detection threshold, roughly corresponding to a isotropic energy in the 2-10 keV range of $\sim 10^{50}$ erg at z=1 and $\sim 10^{51}$ erg at z=4.
We note, however, that this may not be true for X-ray rich GRBs and X-ray Flashes [@hei03]: the $\gamma$-ray emission of these objects is weak or absent. In these cases, only the WFC could detect distant events.
Correlation between Afterglow Luminosity and Gamma-Ray Energy. {#sec_energetic}
--------------------------------------------------------------
We note that the width of the $\gamma$-ray fluence distribution is not very different from that of the X-ray afterglow flux distribution (see Fig. \[fig3\] and Fig. \[fig3bis\]) A few authors, e.g.@kup00, have proposed that the energy emission from the fireball surface need not be isotropic, but that large spatial variations of $dE_{\gamma}/d\Omega$ in the fireball could exist. During the prompt emission phase, the radiation is highly beamed, due to very high Lorentz factor of the ejecta. These circumstances would lead to a large spread of $\gamma$-ray fluences. In the afterglow phase, X-rays are less beamed due to the lower Lorentz factor, and hence the fluctuations are averaged over a larger region. Therefore, X-ray flux afterglow distribution would be less broad than the $\gamma$-ray fluence. As we do not observe such a difference in the two distribution widths, we cannot support the hypothesis of @kup00.
{width="8cm"} {width="8cm"}
The distribution of S$_{\gamma}$ - F$_{X}$ ratio is not very broad ($\sigma =0.71$), suggesting a correlation between the X-ray afterglow luminosity and the gamma-ray energy (see Fig. \[fig4\]). For the sample of burst with known redshift we have then derived L$_{X}$ by the formula [@lam00] :
$$F(\nu,t) = \frac{L_{\nu}(\nu,t)} {4\pi D^2(z) (1+z)^{1+\alpha-\delta}}$$
Luminosity is obtained in the 1.6-10 keV energy band and at 11 hours after the burst in the rest frame. We have adopted the average values of $\alpha$, $\delta$ reported in the previous section. The cosmological parameters used are the same as for the computation of the emitted energy (see Sec. \[section31\]) [^8].
In Fig.\[fig6\] we plot L$_{X}$ vs E$_{\gamma}$. The correlation coefficient is r=0.74 and the probability of chance correlation is 0.008. It is worth noting that some indication of correlation between prompt and afterglow luminosity is also found in a small set of *Swift* bursts [@chi05].
![Distribution of the logarithmic ratios of the prompt $\gamma$-ray fluence versus the X-ray afterglow flux for the BeppoSAX GRB sample. \[fig4\]](Ratio_Flunce_vs_Flux_ALL_4.ps){width="9cm"}
![\[fig6\] 1.6-10 keV Afterglow Luminosity vs 40-700 keV Energy of the prompt emission. The fit between these two quantities, discussed in the text, is also shown together with its confidence interval (dot-dashed box). The correlation coefficient is $r=0.74$. The dotted lines represent the Eqn. \[eqn\_max\] is case of $\epsilon_{e}^{1.3}/\epsilon_{\gamma} = 10$ (upper line), 1 and 0.1 (middle lines) and 0.01 (lower line)](fig_7_new_new.ps){width="9cm"}
Assuming that the observed X-ray frequency $\nu_{X}$ is above the cooling frequency $\nu_{c}$, the measurement of X-ray luminosity at a fixed time after the burst gives an estimate of isotropic kinetic energy of the fireball $E_{K,A}$ [@fre01] :
$$\label{equation_2}
E_{K,A} = C \epsilon_{e} ^{\frac{-4p+4}{p+2}} \nu t L_{X}$$
In that equation, $C$ is a parameter which depends very weakly on the fraction of energy carried by the magnetic field $\epsilon_{B}$, the luminosity distance, the flux density, the time $t$ and the frequency of observation $\nu$. $C$ has a stronger dependence on the value of $p$, however henceforth we will make the simplifying assumption that the value of this parameter is the same for all bursts examined. For our purposes, the value of $C$ can thus be considered constant. We also note that Eqn. \[equation\_2\] does not depend on the value of the density of the circumbust medium, so it holds either in the case of expansion in interstellar medium, with constant density, or in the case of medium affected by wind of the progenitor star, with a typical density profile decreasing from the center of the explosion.
Using $p=2.4$, the value determined from the data, a luminosity distance of $3\times10^{28}$ cm, time and frequency of observation of $40000$ sec and $2.4\times 10^{17}$ Hz, a flux density of $0.3 \mu$Jy, $\epsilon_{B}$=0.01, Eqn. \[equation\_2\] becomes :
$$E_{K,A} = 5.8\times 10^{6} \epsilon _{e} ^{-1.3} L_{X}$$
In the case of gamma-ray emission, we have to consider an unknown coefficient of conversion of relativistic energy of the fireball into gamma-ray energy [@pira01].
$$E_{\gamma} = \epsilon_{\gamma} E_{K,P}$$
where $E_{K,P}$ is the isotropic relativistic energy of the fireball in the prompt phase. We may suppose $E_{K,P}\backsimeq E_{K,A}$, because $\epsilon_{\gamma}$ cannot be too close to unity otherwise there will not be an afterglow [@kob97; @pira01]. We assume that radiative losses also are negligible. From the previous equations we derive:
$$\label{eqn_max}
L_{X} = 1.73 \times 10^{-7} \epsilon_{e}^{1.3} \epsilon_{\gamma}^{-1} E_{\gamma}$$
We plot in Fig. \[fig6\] this relationship (dotted lines), assuming $\epsilon_{e}^{1.3}/\epsilon_{\gamma}$ equal to 0.01, 0.1, 1 and 10 respectively. As one can see, the correlation we have found implies that the ratio $\epsilon_{e}^{1.3}/\epsilon_{\gamma}$ does not strongly vary from burst to burst. Assuming that $\epsilon_e$ is not too close to zero [a common value observed is $\sim$ 0.3 @fre01], this implies that $\epsilon_{e}$ is approximately proportional to $\epsilon_{\gamma}$. Thus, the fraction of fireball energy carried by relativistic electrons in the external shock and emitted in the afterglow is roughly proportional to the fraction of the fireball relativistic energy converted into $\gamma$-rays during the prompt phase.
Jet collimation {#sec_beaming}
---------------
According to @sar98 [@che99; @rho97], the decay index and the spectral index values are linked together by closure relationships that depend on the burst geometry and environment. We present the closure relationships for each burst in Fig. \[fig7\], and focus first on the burst geometry (shown in the top panel of Fig. \[fig7\]).
As one can see, the jet signature is ruled out in most of the cases from our analysis. This is also evident when we calculate the mean value for the closure relationship. For a jet signature, this is :
$$\begin{aligned}
\label{eq1}
\delta - 2 \alpha - 1 = -2.1 \pm 0.22 & & \nu_x < \nu_c \\
\label{eq1bis}
\delta - 2 \alpha \phantom{-1} = -1.1\pm 0.22 & & \nu_x > \nu_c\end{aligned}$$
In Eq. \[eq1\] and \[eq1bis\] we should expect a value of 0, clearly excluded by the data. This implies that the beaming angle may be large. We can set a lower limit on its value ($\theta$). According to @sar99, we have :
$$\label{eq2}
\theta = 0.057 \left(\frac{n_{-1}}{E_{\gamma,i,53}} \right)^{1/8}
t_{\theta,day}^{3/8} \left(\frac{\epsilon_{\gamma}}{0.2}\right)^{1/8}
\left(\frac{1+z}{2}\right)^{-3/8}$$
In Eqn. \[eq2\], E$_{\gamma,i,53}$ represents the isotropic energy emitted in $\gamma$-rays by the fireball in units of $10^{53}$ erg, $n_{-1}$ is the density in 0.1 particle cm$^{-3}$ unit, $\epsilon_{\gamma}$ is the efficiency of conversion of explosion energy into $\gamma$-rays, and $t_{\theta,day}$ the date when the break of light curve, due to the beamed emission, appears (expressed in days).
BeppoSAX TOOs are mostly carried out within 2 days after the GRB. Because decay and spectral slopes are not consistent with a collimated outflow, we can derive $t_{\theta,day} > 2$. Assuming a typical E$_{\gamma,i,53}$=$1$, $\epsilon_{\gamma}=0.2$, $n_{-1}=100$ [@ber03] we obtain a limit of $\theta \gtrsim 0.1 $ rad, which in turn give us a lower limit on the beaming factor $f_{b} \approxeq 0.005$. This result is of the same order of magnitude of that claimed by @fra01. We note that the majority of beaming angles, mostly inferred by breaks in optical light curves, are consistent with this result. Only GRB 990510 and GRB010222 seem to represent exceptions (see table \[table4\]).
A density of $n_{-1}=100$ is typical of the interstellar medium. On the other hand, several authors proposed that GRBs are originated by massive stars [e.g. @woo93]. In such a case, these stars should produce the GRB within their original forming region, which are usually very dense. If we assume $n_{-1}=10^{4}$, which is typical of Giant Molecular Clouds, the beaming angle limit increase to $\theta \gtrsim 0.24$ rad, which corresponds to a beaming factor limit of $f_{b} \approxeq 0.03$.
![Afterglow Luminosity of BeppoSAX GRBs with known redshift. Solid line : before correction for beaming. Dashed line : after correction for beaming.\[fig8\]](luminosity_distrib.ps){width="8cm"}
@ber03 claimed that the distribution of X-ray afterglow luminosity appears to converge significantly toward a common value after beaming correction. We have tested this hypothesis with our sample, using the beaming angle values reported in the literature [see Table \[table4\]; most of them are extracted from the article by @ber03]. The isotropic luminosity is corrected for beaming by applying a multiplicative factor depending on the beaming angle [see @ber03 for details]. Before beaming correction, the luminosity distribution displays a logarithmic width of 0.8 (see Fig. \[fig8\]), with a mean value of $7.2 \times 10^{45}$ erg s$^{-1}$. After the beaming correction, the distribution width shrinks to a value of 0.4, very similar to the 0.3 value @ber03 obtained. The mean luminosity decreases to $9.5 \times
10^{43}$ erg s$^{-1}$ (Fig. \[fig8\]).
One may note that the beaming angle was calculated assuming a density of 10 cm$^{-3}$ when it was unknown. This may have strong consequences. As an example, @zan01 has reported a density value of $10^6$ cm$^{-3}$ for . When using this value, rather than that reported by @ber03, the beaming angle increases up to 0.26 rad. This leads the beaming corrected luminosity distribution width to increase to a value of 0.7, clearly not supporting anymore the hypothesis of a standard energy release in the afterglow. Thus, such claims should be accepted with caution, depending on the assumptions made on the density values.
The density profile of the environment {#sec_environ}
--------------------------------------
Figure \[fig7\] displays also the closure relationships for an expansion into a wind environment (the WIND case, middle panel) and a constant density medium (the ISM case, bottom panel). These closure relationships present a degeneration when $\nu_{c}<\nu_{X}$, which prevents us from drawing any conclusion. One can see from Fig. \[fig7\] that most of the bursts are in that situation. The uncertainties of other bursts do not allow us to draw any conclusion for most of them using only the X-ray data. This is also shown by the mean closure relationships reported in Table \[table6\]: the two medium cases can fit the mean value if the cooling frequency is below the X-rays, while none of them can fit the mean value in the opposite case.
----------------------------------------------------------------------------------------
ISM Wind
------------------- --------------------------------------- ----------------------------
$\nu_{X}<\nu_{c}$ $\delta -1.5\alpha = -0.5\pm0.2 $ $\delta -1.5\alpha -0.5 =
-1\pm0.2$
$\nu_{c}<\nu_{X}$ $\delta -1.5\alpha + 0.5 = 0 \pm0.2 $ $\delta - 1.5\alpha +0.5 =
0\pm 0.2 $
----------------------------------------------------------------------------------------
: \[table6\] Mean closure relationship from our sample. We indicate the wind and ISM closure relationships, depending of the cooling frequency position.
To get rid of this degeneration, we need to use also the optical observations. From the fireball model, the X-ray decay index is larger than the optical one if the cooling frequency is between the optical and X-ray bands and if the fireball is expanding into a constant density medium [@sar98]. The difference between the optical and X-ray decay index is $-0.25$. If the fireball expands into a wind environment (also assuming the cooling frequency to be between the optical and X-ray bands), then it is the optical decay index which is larger than the X-ray decay index. The difference between the optical and X-ray decay index is now 0.25. Assuming that the cooling frequency is indeed between the optical and the X-ray bands, we can remove the degeneration.
GRB $\delta_{O}$ $\delta_{X} - \delta_{O}$ Reference
----- --------------- --------------------------- -----------
$1.21\pm0.02$ 0.11 1
$0.15\pm0.02$ 0.65 2
$1.20\pm0.02$ -0.20 3
$1.28\pm0.19$ 0.14 4
$0.8\pm0.5$ 0.69 5
$1.22\pm0.35$ -0.12 6
$1.10\pm0.35$ 0.34 7
$0.8\pm0.2$ 0.64 8
$1.32\pm0.03$ 0.03 9
$1.63\pm0.61$ -0.33 10
$0.95\pm0.2$ 1.15 11
$0.5\pm0.25$ 0.34 12, 13
: \[table5\]Optical decay indexes and comparison with the X-ray band decay indexes.
References : 1: @mas98 2: @gal98 [the index shown is relative to the BeppoSAX observation interval] 3: @die98 4: @rei99 5: @hjo02 6: @blo99 7: @kul99 8: @har99 9 : @mas01 10: @pri02 11: @jac03 12: @blo02 13: @gre02
![\[fig9\]Difference of the X-ray and optical decay indexes of BeppoSAX sample. Right line: $\delta_{X}=\delta{o}+0.25$ (as expected for an ISM environment). Left line: $\delta_{X}=\delta{o}-0.25$ (as expected for a wind environment). ](fig10_nnn.ps){width="8cm"}
In Table \[table5\] we show the optical vs X-ray band decay indexes (results taken from the literature). We excluded GRB 980519 and GRB000926 from our set because in their case the jet phase started slightly after the beginning of BeppoSAX observations (Jaunsen et al. 2001, Fynbo et al. 2001), therefore we may have their decaying behavior largely affected by the change of slope.
For the remaining GRBs with both X-ray and optical afterglows detected, the average value of the decay index is $\delta_{O}=1\pm0.2$ in the optical and $\delta_{X}=1.3\pm0.2$ in the X-rays. The difference between these two values is $0.3 \pm 0.3$. A constant density medium surrounding the burst is thus favored, but a wind environment is not ruled out. This is also visible in Fig. \[fig9\], where we plot the $\delta_{X} - \delta_{O}$ value for each single burst. For a majority of them, the value 0.25 is preferred, thus implying also that we observe a constant density medium surrounding the burst, for some others, we observe indeed a wind medium. This is tricky, as one should expect, if the long GRB progenitor is indeed a massive star [as the GRB-supernova association claimed for several GRBs implies, see e.g. @sta03; @hjo03], the surrounding medium to be the wind arising from the star for all bursts [@che99]. @ram01 suggested the existence of a termination shock that could maintain the wind close to the star [see also @che04]. This would explain our observations. In such a case, this implies that the termination shock has been crossed before the observations (thus early after the burst), which should then imply a dense surrounding medium. This is supported by the large absorption observed around the bursts (see Table \[table3\]): such a high density column may be due to a compact and dense layer around the burst. This is also supported by the observation of . For this burst, the surrounding medium is indeed the interstellar medium (see Fig. \[fig7\]). @zan01 has proposed this burst to be surrounded by a very dense ($10^6$ cm$^{-3}$) medium or affected by a jet effect. We can discard the jet effects (see Fig. \[fig7\]), and thus confirm the proposed explanation. Such a medium, with a large density, would be very efficient to maintain the termination shock nearby the GRB progenitor.
Finally, we would like to underline the fact that inferences drawn from our afterglow analysis are in general agreement with those of the reviews of @fro05 and @piro04. This is not very surprising, however, because of the wide consistency of @fro05 results with ours, while @piro04 used a large part of the same GRB X-ray afterglow set and basically the same data analysis to derive his conclusions.
Dark GRBs {#sec_dark}
=========
About 90 % of the GRBs detected by BeppoSAX present an X-ray afterglow. On the other hand, only 16 GRBs present an optical afterglow. Taking into account the late follow up of and the absence of optical observations of and , this implies that only 42 % of the GRBs detected by BeppoSAX have an identified optical afterglow. This led to the definition of the so called [*Dark*]{} bursts [@dep03]. Several authors [e.g. @fyn01; @fox03; @rol05] pointed out that this definition can in fact hide an instrumental bias (as this does not take into account the date of the optical follow up and the decay rate of the optical afterglow). In fact, the non detection of the optical afterglow can be due to several reasons: a late follow up, a steep decay, an intrinsic faintness, a large dust extinction and a distant burst. While the first two possibilities are instrumental bias, the last three give information about the burst.
For those bursts with a rapid optical follow up and a non detection of the optical follow up, it has been shown that on average the optical flux should be $2$ magnitude lower than bursts with an optical afterglow in order to explain the non detection of the optical source [@laz02]. Another study made with a sample of 31 BeppoSAX GRB afterglows indicated that the X-ray afterglow fluxes of dark GRBs are, on average, $4.8$ times weaker than those of normal bursts [@dep03]. The probability that this flux distribution comes from a single population of burst is 0.002, i.e.a $3\sigma$ rejection. Using the whole BeppoSAX sample, this probability does not change significantly. The results exposed in Sec. \[sec\_energetic\] imply that this X-ray faintness should extend to the prompt phase, and thus that dark GRBs should present a fainter $\gamma$-ray fluence. We have tested this hypothesis and present the result in Fig. \[fig10\]. As one can see, there is indeed a trend for the dark burst (dotted line) to have a low $\gamma$-ray fluence compared to GRBs with optical transient (OT GRBs). The ratio between the average dark GRB fluence and OT GRBs fluence is 5.7, similar to the value of the ratio of X-ray fluxes and the expected value derived from the correlation observed in Sec. \[sec\_energetic\]. The probability that optically bright GRBs and dark GRBs fluence distributions derive from an unique population of burst is 0.01. It thus seems that faintness is an intrinsic property of dark GRBs at all wavelengths.
The above statements can explain the non detection of the optical afterglow. But they imply that [*the whole afterglow*]{} is affected by this effect (i.e. the faintness is observed in all the observation bands). On the contrary, extinct optical afterglow and distant bursts should also feature a faintness that is wavelength dependent (due to dust-to gas laws in the first case and due to the Lymann-$\alpha$ forest redshifted in the optical band in the second case). To discriminate all these effects and to validate their interpretation, @dep03 also carried out a comparison of the X-ray and optical fluxes. They found that 75 % of dark bursts were compatible with a global faintness, and thus that these bursts were dark because searches were not fast or deep enough. For the remaining GRBs, the optical-to-X-ray flux ratio is at least a factor 5-10 lower than the average value observed in normal GRBs. In terms of spectral index, these events have optical to X-ray spectral index $\alpha_{OX}\lesssim0.6$, whereas for OT GRBs the average value is $\simeq0.75$. These facts strongly suggest that for these bursts the spectrum is depleted in the optical band. @jac04, using a similar method and comparing their results with the fireball model expectations, indicated that at least 10 % of their sample was not compatible with the fireball model and thus were [*truly dark*]{} GRBs. It is worth noting that the *Swift* mission [@geh05], recently begun, has already confirmed that a considerable fraction of GRBs has tight upper limits for the optical emission (Roming et al. 2005, in preparation) We can thus indicate that about 10-20 % of GRBs is characterized by an optical afterglow emission fainter than that expected from the X-ray afterglow flux. These bursts could be distant (z$>5$) or extinct bursts.
Two dark bursts have been associated with host galaxies at z $<5$ [@djo01; @piro02]. We also note (see Table \[table3\]) that the X-ray absorption around some bursts is important and could be responsible of an important optical extinction [see e.g. @str04]. Thus, for some of these events, the likely explanation of the darkness is an optical depletion by dust in star forming region. This in turn supports the massive star progenitor hypothesis for long GRBs, as these massive stars are likely to explode in their original star forming region. On the other hand, this does not rule out the distance explanation for some dark bursts with no known host. In fact, it is likely that the dark burst population is the sum of these three (faint, distant and extinct) populations. In principle, these cases could be disentangled by other measurements such as column density, prompt E$_{peak}$, X-ray flux. However, it is important to be cautious, because a few X-ray flashes (see Heise et al. 2001) could have the values of these parameters consistent with those of very high redshift GRBs, even if they are not actually placed at $z>5$.
![\[fig10\]Comparison of the $\gamma$-ray fluences of dark (dotted line) and optically bright (dashed line) GRBs.](Fluence_Gamma_all_6.ps){width="8cm"}
Conclusions {#sec_conclu}
===========
We have presented the BeppoSAX X-ray afterglow catalog. Thirty-nine BeppoSAX afterglow observations were carried out on a sample of 52 detected GRBs. Thirty-one X-ray afterglows were securely identified due to their fading behavior. Three other observations led to the detection of only one source within the prompt positional error box. Thus, X-ray afterglows are present in $\sim90$% of the observations.
We derived the main properties - flux, decay index, spectral index, absorption - for 15 afterglows, and give constraints on decay slope and flux for the remaining. The width of the prompt fluence and X-ray afterglow flux distributions are similar, suggesting no strong spatial variation of the energy emission within the beamed fireball. We pointed out a likely correlation between the X-ray afterglow luminosity and the energy emitted during the prompt $\gamma$-ray event. Such a correlation suggests that the fraction of fireball energy carried by relativistic electrons in the external shock and emitted in the afterglow is roughly proportional to the fraction of the fireball relativistic energy converted into $\gamma$-ray during the prompt phase.
We do not detect significant jet signature within the afterglow observations, implying a lower limit on the beaming angle of $\sim0.1$. Moreover, we note that the hypothesis of standard energy release in the afterglow as claimed by @ber03 may be consistent with our sample, but it strongly depends on the assumptions made about the density of the surrounding medium.
The average value of the spectral index of the electron energy distribution, inferred by our time and spectral analysis, is $p=2.4\pm0.2$.
Our data support the fact that GRBs should be typically surrounded by a medium with a constant density rather than a wind environment, and that this medium should be dense. This may be explained by a termination shock located near the burst progenitor. We finally pointed out that some bursts without optical counterpart may be explained by an intrinsic faintness of the event, while others can be strongly absorbed.
A first comparison with the bursts observed by XMM-Newton and Chandra are presented in @gen05. In a forthcoming paper (Gendre et al., in preparation), we will search the spectra for metal lines and other deviations from the continuum properties.
The BeppoSAX satellite was a joint program of Italian (ASI) and Dutch (NIVR) space agencies. BG acknowledges a support by the EU FP5 RTN ’Gamma ray bursts: an enigma and a tool’.
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[lcccccl]{}
\
GRB name & Position & Localization & First TOO & Sum of & Other TOOs & Optical afterglow\
& (Right Ascention, & & start-end & GTI$^{a}$ & start-end & detection (redshift)\
& Declination) & & (hours) & (ksec) & (hours) &\
\
GRB name & Position & Localization & First TOO & Sum of & Other TOOs & Optical afterglow\
& (Right Ascension, & & start-end & GTI$^{a}$ & start-end & detection (redshift)\
& Declination) & & (hours) & (ksec) & (hours) &\
& $17^h30^m37^s \phantom{00} +49\degr05\arcmin48\arcsec$ & WFC & 3715-3765.2 & 49.1 & — & N\
& $15^h28^m10^s \phantom{00} +19\degr36\arcmin17\arcsec$ & NFI & 16-46.5 & 56 & — & N\
& $05^h01^m47^s \phantom{00} +11\degr46\arcmin41\arcsec$ & NFI & 8-16.7 & 14.3 & 89.6 - 98.8 & Y (z=0.695)\
& $14^h50^m03^s \phantom{00} -69\degr20\arcmin06\arcsec$ & NFI & 8-19 & 23.6 & 40.9-58.5 & N\
& $06^h53^m49^s \phantom{00} +79\degr16\arcmin20\arcsec$ & NFI & 6-21.6 & 35.5 & 66-74 & Y (z=0.835)\
& & & & & 136.3-160 &\
& $11^h56^m25^s \phantom{00} +65\degr12\arcmin43\arcsec$ & NFI & 6.5-60.7 & 101 & — & Y (z=3.42)\
& $12^h57^m15^s \phantom{00} +59\degr23\arcmin26\arcsec$ & NFI & 12-31.2 & 37 & — & N\
& $00^h25^m56^s \phantom{00} -63\degr01\arcmin24\arcsec$ & WFC & — & — & — & N\
& $08^h36^m26^s \phantom{00} -18\degr53\arcmin00\arcsec$ & WFC & — & — & — & Y\
& $07^h02^m37^s \phantom{00} +38\degr50\arcmin46\arcsec$ & NFI & 7-48.6 & 63.8 & — & Y\
& $19^h35^m02^s \phantom{00} -52\degr50\arcmin16\arcsec$ & NFI & 10.2-52.4 & 52.1 & 161-185 & SN (z=0.0085)\
& & & & & Nov 10.75-12&\
& $21^h16^m44^s \phantom{00} -67\degr13\arcmin05\arcsec$ & NFI & 10-47.2 & 49.1 & 218-265 & No study\
& $23^h22^m17^s \phantom{00} +77\degr15\arcmin53\arcsec$ & NFI & 9.7-35.2 & 78 & — & Y\
& $10^h18^m04^s \phantom{00} +71\degr33\arcmin58\arcsec$ & NFI & 8.6-35.3 & 61.5 & — & Y (z=1.1)\
& $23^h59^m07^s \phantom{00} +08\degr35\arcmin06\arcsec$ &(RXTE) & 22.3-45.6 & 39.2 & 110.3-132.6 & Y (z=0.97)\
& $23^h29^m37^s \phantom{00} -23\degr55\arcmin45\arcsec$ & NFI & 6.5-61 & 89 & 172-191 & N\
& $15^h25^m31^s \phantom{00} +44\degr45\arcmin52\arcsec$ & NFI & 5.8-53.9 & 81.9 & — & Y (z=1.62)\
& $03^h02^m45^s \phantom{00} -53\degr06\arcmin11\arcsec$ & NFI & 6-44 & 56.4 & — & N\
& $13^h38^m03^s \phantom{00} -80\degr29\arcmin44\arcsec$ & NFI & 8-44.4 & 67.9 & — & Y (z=1.6)\
& $00^h26^m34^s \phantom{00} -32\degr12\arcmin00\arcsec$ & WFC & — & — & — & No study\
& $01^h48^m23^s \phantom{00} -77\degr05\arcmin22\arcsec$ & NFI & 8-39.7 & 30 & — & N\
& $12^h19^m28^s \phantom{00} -03\degr50\arcmin00\arcsec$ & NFI & 7.5-29.5 & 37 & 169.8-195 & N\
& $05^h09^m52^s \phantom{00} -72\degr08\arcmin02\arcsec$ & WFC & 11-33.8 & 77.8 & — & Y (z=0.86)\
& $22^h31^m49^s \phantom{00} -73\degr24\arcmin24\arcsec$ & WFC & — & — & — & Y (z=0.43)\
& $03^h10^m36^s \phantom{00} -68\degr07\arcmin13\arcsec$ & NFI & 8-48.9 & 77.9 & — & N\
& $07^h31^m07^s \phantom{00} -69\degr27\arcmin24\arcsec$ & NFI & 11-11.4 & 1.1 & — & N\
& $06^h52^m53^s \phantom{00} -74\degr59\arcmin17\arcsec$ & WFC & — & — & — & N\
& $06^h51^m02^s \phantom{00} +11\degr35\arcmin37\arcsec$ & NFI & 13-33.9 & 36.1 & 258-285.8 & N\
& $12^h03^m29^s \phantom{00} -67\degr45\arcmin25\arcsec$ & WFC & — & — & — & N\
& $22^h24^m43^s \phantom{00} +54\degr23\arcmin22\arcsec$ & NFI & 8-26.8 & 31.6 & — & N\
& $01^h59^m17^s \phantom{00} -40\degr39\arcmin17\arcsec$ & NFI & 7.2-40.2 & 44.4 & — & N (z=0.835)\
& $18^h54^m28^s \phantom{00} -66\degr27\arcmin59\arcsec$ & NFI & 12-41.5 & 50.8 & — &N (z=0.37-0.47)\
& $10^h45^m09^s \phantom{00} -33\degr59\arcmin01\arcsec$ & NFI & 12-27.3 & 26.6 & 78.8-99 & N\
& $00^h09^m27^s \phantom{00} -61\degr31\arcmin43\arcsec$ & NFI & 7.4-50.5 & 34.8 & — & N\
& $15^h32^m42^s \phantom{00} +73\degr47\arcmin23\arcsec$ & NFI & 10-41.6 & 44.6 & — & N\
& $07^h35^m29^s \phantom{00} +69\degr11\arcmin56\arcsec$ & WFC & — & — & — & N\
& $17^h04^m06^s \phantom{00} +51\degr47\arcmin37\arcsec$ & (IPN) & 48.9-61 & 19.6 & — & Y (z=2.066)\
& $18^h23^m04^s \phantom{00} +50\degr53\arcmin56\arcsec$ & WFC & — & — & — & N\
& $18^h30^m08^s \phantom{00} +55\degr18\arcmin14\arcsec$ & NFI & 16-37.8 & 33.2 & 70-106 & N\
& $17^h09^m22^s \phantom{00} +39\degr15\arcmin36\arcsec$ & WFC & — & — & — & no study\
& $17^h40^m56^s \phantom{00} +48\degr34\arcmin52\arcsec$ & NFI & 6-51.8 & 83 & — & N\
& $02^h36^m59^s \phantom{00} +61\degr45\arcmin57\arcsec$ & WFC & 15-36 & 17.2 & — & N\
& $14^h52^m12^s \phantom{00} +43\degr01\arcmin00\arcsec$ & NFI & 8-64 & 88.3 & — & Y (z=1.48)\
& $21^h06^m22^s \phantom{00} +53\degr12\arcmin36\arcsec$ & WFC & — & — & — & no study\
& $19^h39^m39^s \phantom{00} +13\degr37\arcmin05\arcsec$ & WFC & — & — & — & N\
& $19^h06^m50^s \phantom{00} -70\degr10\arcmin48\arcsec$ & WFC & — & — & — & no study\
& $10^h46^m43^s \phantom{00} -57\degr47\arcmin37\arcsec$ & WFC & — & — & — & no study\
& $11^h34^m29^s \phantom{00} -76\degr01\arcmin52\arcsec$ & NFI & 21.9-65 & 32.5 & 86.7-120 & Y (z=0.36)\
& $11^h15^m16^s \phantom{00} -21\degr55\arcmin44\arcsec$ & WFC & — & — & — & Y (z=2.14)\
& $16^h13^m05^s \phantom{00} -83\degr42\arcmin35\arcsec$ & WFC & 6-10.8 & 6.1 & — & N\
& $18^h00^m58^s \phantom{00} +81\degr06\arcmin41\arcsec$ & NFI & 6-12.4 & 12.3 & 26.8-33.2 & Y\
& $08^h45^m14^s \phantom{00} +66\degr41\arcmin16\arcsec$ & WFC & — & — & — & N\
& $22^h06^m27^s \phantom{00} -83\degr49\arcmin28\arcsec$ & NFI & 20-27.5 & 22.8 & 54.3-59.6 & Y\
& $22^h09^m21^s \phantom{00} -65\degr19\arcmin42\arcsec$ & NFI & 11-14.3 & 6.8 & 60.2-66 & N\
$a$ First TOO.
[lccccc]{}
\
GRB name & $\gamma$-ray & X-ray & $\gamma$-ray & X-ray & Ref.\
& duration & duration & fluence & fluence &\
& (T, s) & (T, s) & $10^{-7}$ erg cm$^{-2}$ & $10^{-7}$ erg cm$^{-2}$\
\
GRB name & $\gamma$-ray & X-ray & $\gamma$-ray & X-ray & Ref.\
& duration & duration & fluence & fluence &\
& (T, s) & (T, s) & $10^{-7}$ erg cm$^{-2}$ & $10^{-7}$ erg cm$^{-2}$\
& 8 & 17 & $26\pm3$ & $0.8\pm0.2$ & 1, 2, 3\
& 43 & 60 & $430\pm30$ & $16\pm1$ & 4, 2, 3\
& 80 & 80 & $64.5$ & $15.4$ & 5\
& 150 & 150 & $82\pm9$ & $4.7\pm1.5$ & 2\
& 15 & 29 & $14.5$ & $5.3$ & 5\
& 35 & 35 & $64.9$ & $2.34$ & 5,3\
& 7 & 7 & $6.6\pm0.7$ & 1 & 6,3\
& 20 & 20 & $32.3\pm3$ & — & 3,7\
& 9 & 9 & $7.5\pm1.5$ & $2\pm0.3$ & 5, 3\
& 58 & 68 & $650\pm50$ & $9.7\pm0.7$ & 5,\
& 31 & 40 & $28.5\pm5$ & $7.8\pm0.2$ & 2, 3\
& 15 & 20 & $23\pm3$ & - & 7, 3\
& 30 & 190 & $81\pm5$ & 18 & 8,9, 3\
& 50 & 50 & $9.9$ & $2.3$ & 5, 3\
& 20 & 260 & $4\pm1$ & $5.7\pm1$ & 10,3\
& 90 & — & $300\pm100$ & — & 11\
& 100 & 100 & $1790$ & $22.9$ & 5, 3\
& 25 & 25 & $12.7\pm1.5$ & — & 7, 3\
& 75 & 80 & $181$ & $17.9$ & 3, 5\
& 11 & 11 & — & — & 3\
& 28 & 60 & — & $\sim15$ & 3,12\
& 23 & 40 & $10\pm1$ & $15\pm0.8$ & 13, 3\
& 42 & 45 & $423$ & $22.5$ & 5, 3\
& 30 & 30 & $65\pm3$ & $28.6$ & 5, 3\
& 30 & 30 & $\sim42$ & $\sim2.5$ & 14, 3\
& 1 & 220 & — & — & 3\
& 50 & 130 & — & — & 3\
& 3 & 10 & $9\pm1$ & 1 & 15,16, 3\
& 13 & 40 & — & — & 3\
$^{\mathrm a}$ & — & 5 & $<1.2$ $^{\mathrm b}$ & & 17\
& 10 & 115 & $610\pm20$ & $\sim15$ & 18, 3\
& 115 & 100 & $61.7$ & $11.6$ & 5, 3\
& 80 & 120 & $14.4\pm0.4$ & — & 19, 20\
& 14 & 30 & — & — & 3\
& 12 & 120 & $9.8\pm0.9$ & $17\pm1$ & 21, 3\
& 15 & 20 & — & — & 3\
& 31 & 60 & — & — & 3\
& 60 & 65 & $49.7\pm1.9$ & $6.4\pm0.33$ & 22, 3\
& 23 & 25 & — & — & 3\
& 15 & 30 & $45\pm0.8$ & $2\pm0.3$ & 23\
& 40 & 150 & — & — & 3\
& 170 & 280 & $753$ & $95$ & 5, 3\
& 15 & 24 & — & — & 3\
& 37 & 41 & — & — & 3\
& 74 & 90 & — & — & 3\
& 25 & 30 & — & — & 3\
& 105 & 100 & $1000\pm20$ & $140\pm3$ & 24, 3\
& 400 & 400 & $37\pm4$ & $11\pm1$ & 24, 3\
& 70 & 90 & 30 & 0.9 & 25, 3\
& 15 & 50 & — & — & 3\
& 40 & 60 & — & — & 3\
& 1800 &$>$1290& $\sim290$ & $>47$ & 26, 3\
& — & 60 & $<2.9$ & $3.7\pm0.3$ & 27, 3\
$a$ Perhaps not a GRB. See Cornelisse et al. 2000.
$b$ Conservative $3\sigma$ upper limit based on GCN 448
References : 1: @piro98, 2: @fro00a, 3: @fro04, 4: @fer98, 5: @ama02a, 6. @ant99, 7: @ama99, 8 : @nic98, 9 : @zan99, 10: @fro00b, 11: @ama98, 12: @mul99b, 13: @fer01, 14: @mon01, 15: @tas99, 16: @zan00b, 17: @gan99, 18: @piro02, 19: @gui00, 20: @zan00a, 21: @nic01, 22: @gui03, 23: @gui03, 24: @piro05, 25: @zand03, 26: @nic04, 27: @ama04. Note 1: When not available, values of 2-10 keV fluences have been calculated from the 2-26 keV fluences and assuming the spectral parameters reported in the references.
Note 2: The X-ray and $\gamma$ fluences reported by @ama02a have been obtained by reporting at z$=0$ the parameters of the WFC and GRBM spectra fit (see table 2 of the same article).
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[^1]: Present address : Mullard Space Science Laboratory, Holmbury St. Mary, Dorking, Surrey RH5 6NT, United Kingdom
[^2]: In the case of GRB020410, GRBM was actually switched off at the time of the burst. A $\gamma$-ray signal was detected by Konus[@nic04]
[^3]: 2 days for GRB000926
[^4]: see *http://heasarc.gsfc.nasa.gov/docs/xanadu/ximage/ximage.html*
[^5]: In the cases of GRB000926 and GRB020427, we have used data gathered by the [Chandra]{} X-ray observatory to constrain the decay index [see @piro01; @gen05]. For GRB011121, we have used the last WFC data points [see @piro05]
[^6]: In the case of GRB970111 and GRB970402, better results were obtained by using local background
[^7]: GRB 980425 has not been included in this calculation and in the successive ones for its peculiarity.
[^8]: As for GRB000214, $z=0.44$ was adopted.
|
---
abstract: 'Unordered feature sets are a nonstandard data structure that traditional neural networks are incapable of addressing in a principled manner. Providing a concatenation of features in an arbitrary order may lead to the learning of spurious patterns or biases that do not actually exist. Another complication is introduced if the number of features varies between each set. We propose convolutional deep averaging networks (CDANs) for classifying and learning representations of datasets whose instances comprise variable-size, unordered feature sets. CDANs are efficient, permutation-invariant, and capable of accepting sets of arbitrary size. We emphasize the importance of nonlinear feature embeddings for obtaining effective CDAN classifiers and illustrate their advantages in experiments versus linear embeddings and alternative permutation-invariant and -equivariant architectures.'
author:
- 'Andrew Gardner$^{1}$, Jinko Kanno$^{1}$, Christian A. Duncan$^{2}$ and Rastko R. Selmic$^{3}$[^1][^2][^3]'
bibliography:
- 'references.bib'
title: |
**Classifying Unordered Feature Sets\
with Convolutional Deep Averaging Networks**
---
Introduction
============
We propose for classifying and learning feature representations of datasets containing instances with unordered features, where each feature is considered a tuple composed of one or more values. accept variable-size input and are invariant to permutations of the input’s order. In addition, as a side-effect of the training process, learn discriminative, nonlinear embeddings of individual input elements into a space of chosen dimensionality. Contrary to their name, which is inspired by the work of Iyyer [@imbgd-ducrsmtc], could perhaps be more accurately termed convolutional deep pooling networks as we also consider the effects of functions other than averaging such as taking element-wise maximums or sums.
Contributions
-------------
We propose for classifying unordered feature sets. We show that a with nonlinear embeddings is competitive with and perhaps even superior to and known permutation-invariant architectures for classifying instances containing variable-size sets of unordered features. We also find that the type of pooling plays a significant role in determining the efficacy of the network with `sum`-pooling clearly outperforming `max`- and `average`-pooling.
Related Research
----------------
Sets, particularly those without an inherent ordering, comprise a class of data for which an obvious deep learning [@lbh-dl] treatment is somewhat elusive. A simple feed-forward neural network such as a [@jmm-annat] is insufficient without enormous amounts of data and even more so if the sets are not of constant size. In addition, are generally insufficient since the order of the elements may be unreliable or bias the network toward certain spurious or transient patterns. Recently, the deep learning community has begun to explicitly consider architectures specifically made to address the unique challenges proposed by sets and other unusually structured data such as graphs [@ccm-udhsg] and ordered sequences [@vbk-omsss]. These architectures usually work by exploiting or preserving symmetries in the data (see @gd-dsn or [@cw-gecn] for general frameworks). In the remainder of the section, we focus on work more directly related to our own.
@imbgd-ducrsmtc proposed for classifying text from an unordered list of words and showed that this rivaled more complex network architectures for the same task. A is essentially a traditional feed-forward neural network whose main distinguishing feature lies in the nature of its input: the element-wise average of word embeddings in a vector space. @imbgd-ducrsmtc did not consider learning word embeddings as part of the architecture, instead opting to use a set of predefined embeddings. In addition, only averaging was considered as a means of aggregating the word embeddings.
@hck-ldrsud considered learning linear embeddings as part of the network architecture and summing instead of averaging the embeddings. The resulting network was cast as an with identity weight matrices and served as a baseline against the article’s primary architectures. We show that linear embeddings are not sufficient for all tasks and indeed are unnecessary with certain pooling operations including averaging and summing.
@rg-bwernnar develop a neural bag-of-words model that is equivalent to a single-layer-embedding with average pooling. Each dimension of the embedding is interpreted as the probability of a Gaussian-distributed visual word given the embedded element. Consequently, the embedding is constrained by a softmax output. @rg-bwernnar do not appear to explicitly treat instances as sets rather than sequences, but their architecture is nevertheless permutation invariant. A specialized layer representing a with certain types of nonlinear kernels is incorporated after pooling.
Permutation equivariance is closely related to the concept of invariance. Whereas invariance prescribes that the output of a function is unchanged when the input is permuted, equivariance indicates that the output (presumed to be a sequence or set of the same cardinality as the input) is permuted in the same manner as the input. In other words, equivariance dictates that when a function $f:X^n \to Y^n$ is given $x \in X^n$ permuted by any $\pi \in \mathcal{S}_n$, where $\mathcal{S}_n$ is the symmetric group on $n$ symbols, then $$f(\pi x) = \pi f(x).$$ Note that invariance means that $$f(\pi x) = f(x).$$ @rsp-dlspc propose a computationally efficient permutation equivariant layer accomplished via a precise pattern of weight sharing. The following equation computes the output $\vec{y}$ of a recommended version of this layer given an $n$ element $d$-dimensional input set represented as a matrix $\vec{x} \in \mathbb{R}^{n\times d}$, $$\vec{y} = \vec{\sigma}(\vec{1}_n\transpose{\beta} + (\vec{x}-\vec{1}_n\transpose{\vec{x}_{\max}})\Gamma),
\label{eq:pmeqLayer}$$ where $\vec{\sigma}$ is some nonlinear activation function, $\vec{x}_{\max} \in \mathbb{R}^d$ is a vector of the column-wise maximum values of $\vec{x}$, $\Gamma \in \mathbb{R}^{d \times m}$ is a weight matrix, $\beta \in \mathbb{R}^m$ is a bias, and $\vec{1}_n$ is a vector of $n$ ones. @gvwak-pennadp also propose a permutation-equivariant layer for dynamics prediction but base their version on applying an arbitrary function to all pairwise combinations of input elements and averaging (pooling) the output, given $n$ inputs $\vec{x}_i \in U$, $i\in [1,n]$ and a function $f:U \times U \to \mathbb{R}$, the $j$-th index of the output $\vec{y}$ is given by $$y_j = \frac{1}{n}\sum_{i =1}^n f(\vec{x}_i, \vec{x}_j)
\label{eq:permutationalLayer}$$ As noted by @rsp-dlspc, permutation invariance can be obtained from a permutation equivariant function by pooling over its output.
@es-tns propose a variational autoencoder [@kw-aevb] for learning statistics of independent and identically distributed data. This work is perhaps the most similar to our own in that the proposed statistical network implicitly contains a as part of its structure. The application of the implicit is distinguished from ours in that it is applied at the instance level rather than the feature level. Whereas we are embedding individual features, @es-tns embed instances. In addition, @es-tns appear to focus solely on average pooling.
Convolutional Deep Averaging Networks
=====================================
Suppose we have a dataset $\mathbb{X}$ composed of $l$ subsets $X_i$, $i \in [1,l]$ of some set $U$ (theoretically, each $X_i$ may in fact be a multiset). Let us assume $U \subset \mathbb{R}^d$ so that a given subset $X_i$ contains $n_i$ arbitrarily indexed vectors $\vec{x}^{(i)}_j$, $j \in [1,n_i]$. Our objective is to design a neural network architecture capable of converting each of these variable-size subsets into a fixed-size representation that is useful for machine learning tasks such as classification.
One could certainly use an by treating each $X_i$ as a sequence. However, if there is no inherent ordering to the elements, then an possesses some significant disadvantages. The may learn or be biased towards spurious patterns that are a result of the chosen ordering scheme. In addition, the removal of an element in the middle of the sequence could lead to unexpected results.
We reason that the ideal architecture for this problem is invariant to the order of the input, and we propose augmenting the architecture by directly incorporating the embedding function $f: U \to \mathbb{R}^m$ into the structure of the network, where $m$ is the chosen size of the embedding. We call the resulting architecture a due to its similarity to a , which will become apparent shortly.
In theory, we place no restriction on the form that $f$ may take except that it be parameterized in a manner compatible with backpropagation-based training. For the sake of simplicity, we assume that $f$ can be represented by an , although an is also conceivable if elements of $U$ are sequences or time series. When given a set $X_i$, the embedding function is applied separately to each $\vec{x} \in X_i$. One could informally interpret the embedding layer as a sort of convolution of $f$ with the elements of $X_i$. The embeddings are then combined in a manner that does not depend on their order, through a binary, commutative, and associative operator. To borrow familiar language from , the embeddings are pooled. Let $\rho: 2^{\mathbb{R}^m} \to \mathbb{R}^p$ denote the pooling function and note that usually $p=m$ as is the case for typical pooling operations such as summation. A is then defined by the function $X \mapsto g(\rho(\mathcal{X}))$, where $\mathcal{X} = \{f(x) \suchthat x \in X\}$ and $g$ represents a neural network with arbitrary structure. A with single-layer $f$ can be cast as a special type of by considering each set $X_i$ as an $n_i \times d$ image where $f$ is a bank of $m$ $1 \times d$ filters. Alternatively, simply removing $\vec{1}_n\transpose{\vec{x}_{\max}}$ from yields an equivalent layer. with embeddings may also be considered with multiple convolutional layers. See for an illustration of the proposed architecture.
In an alternative interpretation of the embedding, we posit that $f$ effectively performs a type of bin or bucket sort of the set elements by allocating them to $m$ bins. Each dimension of the embedding is thus associated with a certain region within the input space $U$. Unlike an actual bucket sort, we do not require the bins to be disjoint. By constraining the output of $f$ with a softmax function, however, one could produce a probability distribution over the bins. This interpretation generalizes the neural bag-of-words model of Richard and Gall [@rg-bwernnar] by allowing the distribution of each visual word to be learned rather than constrained to be Gaussian. In a sense, such a network computes a probabilistic $k$-means with non-linear clusters. Depending on the dimensionality of $U$, this interpretation provides us one way to visualize and examine the embedded feature space by plotting the activation of a bin or the distribution of the visual word in the input space.
The form of the embedding function plays a significant role in the performance of the network. In the following subsections, we show that nonlinear embeddings are generally preferable to linear.
Disadvantage of Linear Embeddings
---------------------------------
Consider a linear embedding $f_\text{lin}: U \to \mathbb{R}^m$ defined by $$f_\text{lin}(x) = \transpose{W}\vec{x}+\vec{b},$$ where $W$ is an $d \times m$ weight matrix and $\vec{b} \in \mathbb{R}^m$ is a bias vector. Assume that the pooling layer consists of an `average` operation. The output of the pooling layer and input to the deep portion of the network given $X_i$ is then $$\begin{split}
\frac{1}{n_i}\sum_{j=1}^{n_i}f_\text{lin}\left(\vec{x}^{(i)}_j\right) &= \frac{1}{n_i}\sum_{j=1}^{n_i} \left[\transpose{W}\vec{x}^{(i)}_j + \vec{b}\right]\\
&= \transpose{W}\sum_{j=1}^{n_i}\frac{\vec{x}^{(i)}_j}{n_i} + \vec{b} \\
&= f_\text{lin}\left(\frac{1}{n_i}\sum_{j=1}^{n_i}\vec{x}^{(i)}_j\right).
\end{split}$$ We see that we could have simply pooled the input elements directly. In addition, if $V$ and $\vec{c}$ are the weights and bias of the first post-pooling layer, then $f_\text{lin}$ could be merged into the layer by substituting $V$ and $\vec{c}$ with $WV$ and $\transpose{V}\vec{b}+\vec{c}$. In other words, the linear embedding is computationally unnecessary and can be eliminated. A similar conclusion may be reached if `sum`-pooling is used instead (or any linear operation). `Max`-pooling is an exception as it introduces a nonlinearity. However, `max`-pooling with linear embeddings still has potential issues with ambiguity.
Nonlinear Embeddings Mitigate Ambiguity
---------------------------------------
Based on the previous subsection’s result, one may consider simply skipping a learned embedding and working directly with the input points as the plain of @imbgd-ducrsmtc suggests. In general, though, this course of action may be unwise. In particular, suppose there are two sets $X_i$, $X_j$ such that $$\sum_{\vec{x}_i \in X_i} \vec{x}_i = \sum_{\vec{x}_j \in X_j} \vec{x}_j.$$ One could even construct a situation wherein both sets also have the same element-wise maximums by choosing $X_i$ and $X_j$ to have the same convex hull. In such an event, $X_i$ and $X_j$ are indistinguishable under linear embeddings with `max`-pooling since the maximum (and minimum) of a linear function will always lie on the boundary (vertices) of a convex set. Regardless of the cause of the ambiguity, the consequence is that instances with potentially significant differences are functionally identical from the network’s perspective. The primary issue, though, is the fact that these ambiguities are not caused by particularly exotic circumstances.
A nonlinear embedding allows the network to learn functions that can differentiate sets that are ambiguous under linear pooling. Note that ambiguity is still possible with a nonlinear embedding. However, since the embedding is learned to satisfy some objective, one can expect these ambiguities to either be benign or to indicate some inherent similarity between the ambiguous instances. For example, consider the sets of black and white points in (a) that are ambiguous under `sum`- and `max`-pooling. Using a pair of sigmoidal activation functions each defined by $$\sigma(a) = \frac{1}{1+e^{-a}},$$ with inputs $a_1 = x \pm \epsilon_1$, $a_2 = y \pm \epsilon_2$, where $\epsilon_1$, $\epsilon_2$ are each small and positive, we can compute nonlinear embeddings that are unambiguous under `sum`-pooling.
The nonlinear embedding of the entire set is the key point; linear point-wise embeddings followed by `max`-pooling may be sufficient when equivalent convex hulls are rare. However, we hypothesize that nonlinear embeddings are inherently more powerful and thus more useful since they have greater representational capacity.
![An example of simultaneous `sum`-, `average`-, and `max`-pooling ambiguity and its partial resolution via a nonlinear embedding. (a) The set of black points and the set of white points shown have the same coordinate-wise sums and maximums. The shading shows the activation of two sigmoidal functions that can be used to construct nonlinear 2D embeddings (b and c) that distinguish the two sets under `sum`- and `average`-pooling. (b) The embedding of the black points. (c) The embedding of the white points. Note that two points share nearly the same embedding.[]{data-label="fig:ambiguityExample"}](AmbiguityExample.png){width="\linewidth"}
Experiments
===========
We conduct experiments to evaluate the performance of against alternative architectures as well as examine the effects of different pooling operations. Our experiments focus especially on variable-size sets, which do not seem to have many existing results in the literature. All models were implemented and tested using the Keras [@keras] deep learning framework with the Theano [@theano-short] backend.
Posture Recognition from Point Sets
-----------------------------------
A motion capture dataset of hand postures provides the primary basis for our evaluation.[^4] The dataset consists of variable-size point sets representing five hand postures captured from 12 users. The size of each point set ranges from 3 to 12, although it should be noted that only 11 markers were physically present. Each point set shares the same coordinate system, so no rotations or translations should be required to process the data. Regardless, we center each point set to have zero mean in each dimension. The goal is to classify each point set as one of the five postures.
In order to make the problem more challenging, we do a leave-one-user-out evaluation where all but one user contribute to the training and validation sets and the test set is drawn exclusively from the left-out user. Each user is iteratively left out, and the resulting test accuracies are averaged to obtain a reasonable evaluation of the tested classifier’s generalization error. Training, validation, and test sets are disjoint and each consist of 75 uniformly randomly selected instances per class per user without replacement. This process is repeated five times in order to obtain some measure of confidence in the results.
Model Specification
-------------------
We compare a variety of architectures for this task, including linear embeddings with `max`-pooling, linear embedding with `sum`-pooling (no embedding), and nonlinear embeddings with `average`-, `sum`-, and `max`-pooling. These models are compared against an with [@cgcb-eegrnnsm] as well as an experimental variant of the architecture with recurrent connections between the embeddings, which we call a . In an , we trade permutation invariance and independent embeddings for increased functional capacity. Note that an is effectively just an that is pooled over the entire time axis. Finally, we implement the permutation-equivariant layers of @gvwak-pennadp (defined by ) and @rsp-dlspc (defined by ) for an external, contemporaneous comparison. One or more permutational layers enable one to obtain dependent nonlinear embeddings that are permutation invariant (after pooling) as opposed to the . From this point forth, we refer to the permutational layer of @gvwak-pennadp as a pairwise layer due to its structure and to distinguish it from the permutation-equivariant layer of @rsp-dlspc. We refer to the respective types of model as and for brevity. Despite the fact that the nonlinear embeddings of a are not always technically convolutions, we will sometimes refer to them as convolutional layers when compared against the recurrent, pairwise, and permutational layers of competing architectures. Given the incredibly diverse array of architectural and training options available in the literature, we tried to make our architectures as uniform as possible in order to enable fair comparison. Since the recurrent architectures depend on the order of the input, points in each set were lexicographically sorted by their $x$-, $y$- and $z$-coordinates. Gaussian noise with a standard deviation of 20 (millimeters, which is the scale of the input) was applied to the input as a form of regularization for each network. Dropout [@shkss-dswpnno] of 10% was applied to the hidden layers of each network, and $l_2$ regularization with a magnitude of 0.001 was applied to the weights of each layer. We did not apply dropout to the input, but we did adopt the simultaneous dropout suggested by @rsp-dlspc for the , which consists of dropping a feature simultaneously in all elements of an input set rather than independently. A default embedding size of 11 was chosen for computational expedience as well as to let each embedded dimension hypothetically represent one of the physical markers. Two special (convolutional, recurrent, ) layers with 11 neurons each were used in each architecture. Each tested recurrent network was bidirectional [@sp-brnn] with the forward and backward outputs concatenated at each timestep (22 dimensional output). To clarify, the final timesteps in each direction were concatenated in the case of the plain without pooling. Except in the case of linear embeddings, maxout activations [@gwfmcb-mn] with 2 pieces were used in each layer except for the network output, which incorporated a softmax activation. Each model used the same post-embedding architecture, which consisted of one 11-neuron layer with a residual connection [@hzrs-drlir] followed by the 5-neuron (one per class) softmax layer.
offer significant computational and practical advantages over the other architectures that arise primarily from the fact that the embeddings are independent. Unlike an or , the embeddings can be computed in parallel rather than sequentially. In addition, only $O(n)$ embedding function evaluations are required as opposed to $O(n^2)$ function evaluations for a . The fact that the embeddings are independent also enables their re-use in intersecting sets whereas recurrent or permutational architectures must re-evaluate each point. are of similar complexity, although they lack any advantages derived from independent embeddings. For these reasons we were able to experiment with and with embedding sizes that are an order of magnitude higher than the other models (100 to be precise) yet still require less computation.
The RMSProp [@th-rmsprop] implementation provided by Keras [@keras] was used with a learning rate of 0.001 and minibatches of size 64. Training was terminated for a model if the validation loss did not improve after 40 epochs.
Results and Discussion
----------------------
Results are presented in , where we can see that the highest average accuracy was achieved by a with `sum`-pooling and 100-dimensional nonlinear embeddings. We also immediately notice a significant difference between types of pooling for permutation invariant architectures. , on the other hand, appear to be robust to changes in the pooling mode and just as effective if not marginally better than the .
[|l|| c|c| c| c|]{} **Type** & **Embedding** & **Embedding** & **Pooling** & **Accuracy**\
**** & **** & **Size** & **** &\
& None & N/A & `sum` & $\underset{\pm 0.10}{20.05}$\
& Linear & 11 & `max` & $\underset{\pm 4.30}{70.00}$\
& Linear & 100 & `max` & $\underset{\pm 0.96}{75.94}$\
& Nonlinear & 11 & `average` & $\underset{\pm 1.00}{71.61}$\
& Nonlinear & 11 & `max` & $\underset{\pm 1.85}{65.44}$\
& Nonlinear & 11 & `sum` & $\underset{\pm 1.37}{89.14}$\
& Nonlinear & 100 & `average` & $\underset{\pm 1.81}{77.00}$\
& Nonlinear & 100 & `max` & $\underset{\pm 1.60}{75.81}$\
& Nonlinear & 100 & `sum` & $\underset{\pm 1.72}{92.24}$\
& Recurrent & 11 & `average` & $\underset{\pm 1.15}{90.25}$\
& Recurrent & 11 & `max` & $\underset{\pm 0.98}{90.31}$\
& Recurrent & 11 & `sum` & $\underset{\pm 1.25}{90.38}$\
& Recurrent & 11 & N/A & $\underset{\pm 1.59}{89.28}$\
& Pairwise & 11 & `average` & $\underset{\pm 1.91}{85.86}$\
& Pairwise & 11 & `max` & $\underset{\pm 0.59}{81.94}$\
& Pairwise & 11 & `sum` & $\underset{\pm 1.45}{89.67}$\
& Permutational & 11 & `average` & $\underset{\pm 1.41}{70.64}$\
& Permutational & 11 & `max` & $\underset{\pm 0.59}{61.31}$\
& Permutational & 11 & `sum` & $\underset{\pm 2.17}{76.59}$\
& Permutational & 100 & `average` & $\underset{\pm 2.86}{80.84}$\
& Permutational & 100 & `max` & $\underset{\pm 0.67}{77.65}$\
& Permutational & 100 & `sum` & $\underset{\pm 1.53}{87.31}$\
In general, we note that the highest accuracies achieved for each type are clustered around 90% accuracy. Our results do not provide enough confidence to say that the best-performing models are significantly different (in a statistical sense) than one another, but they do suggest a potential advantage to certain and disadvantage to . The slightly inferior performance may be explained by the fact that their permutation-equivariant layers are slightly more constrained than the competition. Furthermore, we tested only a portion of the possible architectures proposed by the framework of @rsp-dlspc. Regardless of whether the best does achieve significantly higher accuracy than the competition, the computational advantages of a over and certainly warrants their utility. In particular, reducing the embedding size to 11 renders a significantly more efficient classifier (with relatively few parameters) with only a marginal drop in accuracy.
We can hypothesize potential reasons for the pattern of results induced by different pooling modes. Note first that the difference between `average`- and `sum`-pooling must arise from the fact that the input sets are not of constant size. If the size was constant, then both pooling modes would be the same but for a constant factor. A potential cause for their difference here may thus arise from the fact that `average`-pooling effectively removes information (the implicitly encoded size of the set) and introduces ambiguity between certain set embeddings. On the other hand, `max`-pooling’s relatively poor performance may be partially due to the choice of the maxout activation function. We noted in some exploratory trials that its accuracy significantly improved when paired with activations. Some theoretical basis for `sum`-pooling’s apparent advantage may be given by a probabilistic interpretation. Though embeddings were not constrained by a softmax output, we may interpret them as the logarithm of unscaled posterior probabilities as indicated by a neural bag-of-words model [@rg-bwernnar]. The sum of the embeddings then gives the log-likelihood (shifted by some amount) for the parameters associated with each visual word given the point set.
We also show that nonlinear embeddings can yield significant gains over linear or identity (no) embeddings. Indeed, for this problem the identity embedding yields a network no better than guessing. The linear embedding with `max`-pooling, on the other hand, is competitive with its counterpart nonlinear embedding. However, whereas the nonlinear embedding could potentially be improved by adding more layers or changing its activation functions, the linear embedding is already exhausting its functional capacity.
Conclusion
==========
We introduced the , a class of neural networks designed for classifying instances containing unordered, variable-size feature sets. The proposed architecture works by directly incorporating a function into the network’s structure that embeds the features in a high-dimensional space and pooling the subsequent embeddings. As the name implies, an equivalence can be drawn between the convolution operation in and the application of the embedding function. Experiments show that in terms of accuracy, are competitive with competing recurrent and permutation-equivariant architectures. are also computationally efficient compared to alternative architectures, favoring parallel implementations and re-use of prior results since feature embeddings are set-invariant. In addition, the learned feature embeddings are a useful by-product that can potentially solve related problems such as nonlinear clustering.
Future work may include further exploration of properties and optimal architectures when applied to other problems or datasets. One should note that networks incorporating convolutional, recurrent, or permutation-equivariant layers need not be mutually exclusive. Architectures that perform a convolutional embedding prior to a permutation-equivariant layer (or vice-versa) may be worth exploring and could be capable of achieving results superior to either method when used alone.
[^1]: $^{1}$Andrew Gardner and Jinko Kanno are with the College of Engineering and Science, Louisiana Tech University, Ruston, LA 71272, USA [andrew.gardner1@ieee.org]{}, [jkanno@latech.edu]{}
[^2]: $^{2}$Christian A. Duncan is with the Department of Engineering, Quinnipiac University, Hamden, CT 06518, USA [christian.duncan@quinnipiac.edu]{}
[^3]: $^{3}$Rastko R. Selmic is with the Department of Electrical and Computer Engineering, Concordia University, Montreal, QC H4B 1R6, Canada [rastko.selmic@concordia.ca]{}
[^4]: The dataset along with further documentation is available at <http://www.latech.edu/~jkanno/collaborative.htm>.
|
---
author:
- |
[^1]\
University of Graz\
E-mail:
- |
Christof Gattringer\
University of Graz\
E-mail:
title: 'Dualization of non-abelian lattice gauge theory with Abelian Color Cycles (ACC)'
---
Introduction
============
Exactly rewriting lattice field theories in terms of new, so-called “dual variables” is a strategy that has been developed and used in recent years to overcome complex action problems for lattice field theories at finite density. The Boltzmann factor is decomposed into local factors which are then expanded, such that subsequently the original degrees of freedom can be integrated out in closed form. The partition function turns into a sum over configurations of the expansion indices which constitute the dual variables. For several models it was found that this strategy leads to a representation of the partition function with only real and positive weights, such that a Monte Carlo simulation in terms of the dual variables solves the complex action problem.
However, mapping lattice field theories to a dual representation is interesting beyond a possible application for finite density simulations. The dual variables have to obey constraints which give rise to an interesting geometrical interpretation of the dual variables: The gauge field degrees of freedom are described by surfaces that can either be closed surfaces or are bounded by loops that represent the matter fields (see, e.g., the reviews [@Chandrasekharan:2008gp; @deForcrand:2010ys; @Gattringer:2014nxa]). The structure of the constraints and thus the geometrical structure of the dual degrees of freedom is of course determined by the symmetries of the theory in the conventional representation. While for U(1) gauge fields the geometrical structure is simple and well understood (see, e.g., [@Mercado:2013ola]), for non-abelian gauge theories no clear picture has emerged yet (for different non-abelian dualization strategies see the references in [@Gattringer:2016lml]).
In this contribution we discuss a new approach for the dualization of non-abelian lattice gauge theories, where the traces in the Wilson plaquette action are decomposed into color sums over colored loops around plaquettes, so-called Abelian color cycles (ACC). The ACCs commute such that the same dualization strategy as in the U(1) case can be applied. We present the ACC approach for the gauge groups SU(2) and SU(3) and discuss the corresponding constraints.
ACC dualization for SU(2) lattice gauge theory
==============================================
The Wilson action for SU(2) lattice gauge theory reads: $$S_G[U] \; = \; -\dfrac{\beta}{2} \sum_{x,\mu < \nu} \!\!
\operatorname{Tr}\; U_{x,\mu} \, U_{x+\hat{\mu},\nu} \, U_{x+\hat{\nu},\mu}^{\dagger} \, U_{x,\nu}^\dagger \; ,
\label{eq:actionsu2}$$ where $U_{x,\mu} \in$ SU(2), living on the links of a 4-dimensional lattice, are the dynamical degrees of freedom of the theory. We decompose the action into a sum over ACCs, by explicitly writing the color sums for trace and the matrix products, $$\label{eq:abelianactionsu2}
S_G[U] \; = \; -\dfrac{\beta}{2} \sum_{x,\mu < \nu} \; \sum_{a,b,c,d=1}^{2}
U_{x,\mu}^{ab} \, U_{x+\hat{\mu},\nu}^{bc} \, U_{x+\hat{\nu},\mu}^{dc \ \star} \, U_{x,\nu}^{ad \ \star} \; .$$ The ACCs are the products $U_{x,\mu}^{ab} U_{x+\hat{\mu},\nu}^{bc} U_{x+\hat{\nu},\mu}^{dc \ \star} U_{x,\nu}^{ad \ \star}$ of the matrix elements of the four link elements and are labelled by 4 color indices $a,b,c$ and $d$. Since each color index has 2 possible values we have $2^4 = 16$ different ACCs, which are complex numbers and therefore commute with each other.
It is convenient to introduce a geometrical representation for the link elements as arrows on a 4-dimensional lattice with 2 layers representing the two possible values of the color indices. More specifically, the element $U_{x,\mu}^{ab}$ is represented by an arrow connecting the layer $a$ at site $x$ to the layer $b$ at $x + \hat{\mu}$. Complex conjugation corresponds to reversing the arrow. With this convention the ACCs $U_{x,\mu}^{ab} U_{x+\hat{\mu},\nu}^{bc} U_{x+\hat{\nu},\mu}^{dc \ \star} U_{x,\nu}^{ad \ \star}$ correspond to paths in color space closing around plaquettes. In Fig. \[fig:allcycles\] we show all 16 cycles that are generated when varying the color labels $a,b,c,d$.
![The 16 possible abelian color cycles attached to a given plaquette. In the dual representation their occupation is given by the corresponding cycle occupation number $p_{x,\mu\nu}^{abcd} \in \mathds{N}_0$.[]{data-label="fig:allcycles"}](cycles.pdf)
Using the ACCs we can now rewrite the partition sum as follows: $$\begin{aligned}
Z & = \int \! D[U] \, e^{-S_G[U]} \; = \; \int \! D[U] \prod_{x,\mu<\nu} \prod_{a,b,c,d = 1}^{2}
e^{\frac{\beta}{2} U_{x,\mu}^{ab} U_{x+\hat{\mu},\nu}^{bc} U_{x+\hat{\nu},\mu}^{dc \ \star} U_{x,\nu}^{ad \ \star}}
\nonumber \\
& = \int \! D[U] \prod_{x,\mu<\nu} \prod_{a,b,c,d = 1}^{2} \sum_{p_{x,\mu\nu}^{abcd} = 0}^{\infty}
\dfrac{ \left( \beta/2 \right)^{p_{x,\mu\nu}^{abcd}}}{p_{x,\mu\nu}^{abcd}\, !}
\left( U_{x,\mu}^{ab} U_{x+\hat{\mu},\nu}^{bc} U_{x+\hat{\nu},\mu}^{dc \ \star}
U_{x,\nu}^{ad \ \star} \right)^{p_{x,\mu\nu}^{abcd}}
\nonumber \\
& = \sum_{\{p\}} \left[ \prod_{x,\mu<\nu} \prod_{a,b,c,d}
\dfrac{ \left(\beta/2 \right)^{p_{x,\mu\nu}^{abcd}}}{p_{x,\mu\nu}^{abcd}\, !} \right]
\prod_{x,\mu} \int \! \! dU_{x,\mu} \; \prod_{a,b} \left( U_{x,\mu}^{ab} \right) ^{N_{x,\mu}^{ab}}
\left( U_{x,\mu}^{ab \ \star} \right) ^{\overline{N}_{x,\mu}^{ab}} \; .
\label{eq:partitionsumsu2}\end{aligned}$$ In the first line we rewrite the exponential of the action, which is a sum over plaquettes and over color indices, into a product over plaquettes and color indices, such that we obtain individual expontentials for all ACCs. In the second step we expand each of these exponentials in a Taylor series, introducing individual expansion coefficients $p_{x,\mu\nu}^{abcd} \in \mathds{N}_0$ for each ACC, which we refer to as *cycle occupation numbers*. Finally, in the last line we reorganize the factors and introduce the sum over all configurations of cycle occupation numbers, $\sum_{\{p\}} = \prod_{x,\mu<\nu} \prod_{a,b,c,d = 1}^{2} \sum_{p_{x,\mu\nu}^{abcd} = 0}^{\infty}$. After reordering the factors of link elements it is convenient to introduce the exponents for $U_{x,\mu}^{ab}$ and $U_{x,\mu}^{ab\, \star}$ as $N_{x,\mu}^{ab} \; = \; \sum_{\nu:\mu<\nu}p_{x,\mu\nu}^{abss} + \sum_{\rho:\mu>\rho}p_{x-\hat{\rho},\rho\mu}^{sabs}$ and $\overline{N}_{x,\mu}^{abcd} \; = \; \sum_{\nu:\mu<\nu}p_{x-\hat{\nu},\mu\nu}^{ssba} + \sum_{\rho:\mu>\rho}p_{x,\rho\mu}^{assb} \; ,$ where the label $s$ stands for the independent summation of the color indices replaced by it, e.g., $p_{x,\mu\nu}^{abss} = \sum_{c,d =1}^{2} p_{x,\mu\nu}^{abcd}$.
In order to compute the remaining integrals over the gauge links in the last line of , we choose an explicit parametrization of the SU(2) link variables $$\label{eq:parametrizationsu2}
U_{x,\mu} =\left(
\begin{array}{cc}
\cos\theta_{x,\mu} \, e^{i\alpha_{x,\mu}} & \sin\theta_{x,\mu} \, e^{i\beta_{x,\mu}}\\
-\sin\theta_{x,\mu} \, e^{-i\beta_{x,\mu}} & \cos\theta_{x,\mu} \, e^{-i\alpha_{x,\mu}}
\end{array} \right) , \ \theta_{x,\mu} \in [0,2\pi], \ \ \alpha_{x,\mu}, \, \beta_{x,\mu} \in [0,\pi/2] \; .$$ The corresponding Haar measure reads $dU_{x,\mu} = (2 \pi^2)^{-1} \, d\theta_{x,\mu}\sin\theta_{x,\mu} \cos\theta_{x,\mu} \,
d\alpha_{x,\mu} \, d\beta_{x,\mu}$. All gauge integrals can now be computed in closed form and one finds for the partition sum, $$Z \; = \; \sum_{\{p\}} W_{\beta}[p] \; W_H[p] \; (-1)^{\sum_{x,\mu}J_{x,\mu}^{21}} \;
\prod_{x,\mu} \delta(J_{x,\mu}^{11}-J_{x,\mu}^{22}) \; \delta(J_{x,\mu}^{12}-J_{x,\mu}^{21}) \; ,
\label{eq:partitionsum4}$$ where $W_{\beta}[p]$ is the weight factor collecting the coefficients of the Taylor expansion (the term inside the square brackets in (\[eq:partitionsumsu2\])). Evaluating the gauge link integrals in (\[eq:partitionsumsu2\]) gives additional weight factors $W_H[p]$ from integrating the $\theta_{x,\mu}$, which are related to beta-functions (see [@Gattringer:2016lml] for their explicit form). Both, $W_{\beta}[p]$ and $W_H[p]$ are real and positive. However, note that the partition sum also contains the explicit sign factor $(-1)^{\sum_{x,\mu}J_{x,\mu}^{21}}$ which origins from the minus sign in the 2,1 matrix element in the parametrization (\[eq:parametrizationsu2\]) of our SU(2) link variables. The two Kronecker deltas come from the integration over the phases $\alpha_{x,\mu}$ and $\beta_{x,\mu}$ and give rise to two constraints on each link. These constraints link together components of the currents $J_{x,\mu}^{ab}$ defined as $$\label{eq:Jfluxes}
J_{x,\mu}^{ab} \; = \; N_{x,\mu}^{ab} - \overline{N}_{x,\mu}^{ab} \; = \;
\sum_{\nu:\mu<\nu}[\, p_{x,\mu\nu}^{abss} - p_{x-\hat{\nu},\mu\nu}^{ssba} \, ] -
\sum_{\rho:\mu>\rho}[\, p_{x,\rho\mu}^{assb} - p_{x-\hat{\rho},\rho\mu}^{sabs} \, ] \; .$$
![Lhs.: Graphical illustration of the contributions from the cycle occupation numbers to the $J$-flux using the example of the $J_{x,\mu}^{12}$ element. For a description of the plot see the text. Rhs.: Geometrical illustration of the two constraints in Eq. for the fluxes $J_{x,\mu}^{ab}$ on all links $(x,\mu)$. The first constraint (top) requires the sum over all 1-1 fluxes to equal the sum over all 2-2 fluxes. The second constraint (bottom) requires the sum over 1-2 fluxes to equal the sum over 2-1 fluxes.[]{data-label="fig:J"}](constraints.pdf)
The $J_{x,\mu}^{ab}$ sum over all cycle occupation numbers that contribute to the flux from color $a$ on site $x$ to color $b$ on site $x+\hat{\mu}$. In lhs. plot of Fig. \[fig:J\] we show four of the plaquettes attached to the link $(x,\mu)$ and illustrate how they contribute to $J_{x,\mu}^{12}$ as an example. On the link $(x,\mu)$ the flux from color 1 to 2 is kept fixed and represented with solid arrows. For every plaquette attached to the link this flux gets contributions from four different cycle occupation numbers, which are summed over in the definition , and illustrated with dotted lines. Thus $J_{x,\mu}^{ab}$ is the total flux from color $a$ on site $x$ to color $b$ on site $x+\hat{\mu}$.
With this interpretation of the $J_{x,\mu}^{ab}$ it is now clear how to interpret the constraints given by the two Kronecker deltas in : For every link of the lattice, the fluxes on the two color layers have to be equal, and the fluxes between the two layers have to match, as represented in the rhs. plot of Fig. \[fig:J\]. Moreover, the constraints allow for a simple interpretation of the sign factor: Since by the constraints the $J_{x,\mu}^{21}$ flux equals the $J_{x,\mu}^{12}$ flux, configurations that have an odd number of flux crossings contribute to the partition function with a negative sign.
In its dual form (\[eq:partitionsum4\]) the partition function is a sum over configurations of cycle occupation numbers $p_{x,\mu\nu}^{abcd} \in \mathbb{N}_{0}$ attached to the plaquettes $(x, \mu < \nu)$. At each link $(x,\mu)$ the $p_{x,\mu\nu}^{abcd}$ have to obey constraints which are expressed in terms of the two Kronecker deltas that relate components of the fluxes $J_{x,\mu}^{ab}$ at each link. It is easy to see that a large class of admissible dual pure gauge configurations are closed surfaces made of cycle occupation numbers such that at each link the fluxes compensate to 0, or are such that nontrivial 1-2 fluxes cancel with 2-1 fluxes and 1-1 fluxes with 2-2 fluxes. The latter possibility also allows for non-orientable surfaces that are absent in the case of U(1) gauge fields. All these surface configurations have positive signs.
However, configurations with negative sign are not excluded completely by the constraints. We were able [@Gattringer:2016lml] to construct such configurations by stacks of 4 occupied ACCs on a single plaquette, i.e., these configurations appear at ${\cal O}(\beta^4)$. So far we did not find any other genuine configurations with negative sign that could not be decomposed into factors with the negative sign 4-stacks among them. This local nature of the negative sign contributions hints at a possible resummation.
The ACC construction for SU(3)
==============================
For SU(3), we follow the same procedure as in SU(2). Starting from the Wilson action: $$S_G[U] \; = \; -\dfrac{\beta}{3} \sum_{x,\mu < \nu}
\operatorname{Re}\operatorname{Tr}\, U_{x,\mu} \; U_{x+\hat{\mu},\nu} \, U_{x+\hat{\nu},\mu}^{\dagger} \, U_{x,\nu}^\dagger \; ,
\label{eq:actionsu3}$$ we explicitly write the trace and matrix multiplications as color sums, $$\label{eq:abelianactionsu3}
S_G[U] \; = \; -\dfrac{\beta}{6} \sum_{x,\mu < \nu} \; \sum_{a,b,c,d=1}^{3} \left[
U_{x,\mu}^{ab} U_{x+\hat{\mu},\nu}^{bc} U_{x+\hat{\nu},\mu}^{dc \ \star} U_{x,\nu}^{ad \ \star} +
U_{x,\mu}^{ab \ \star} U_{x+\hat{\mu},\nu}^{bc \ \star} U_{x+\hat{\nu},\mu}^{dc} U_{x,\nu}^{ad} \right] \; .$$ As in the SU(2) case we refer to the products of link matrix elements as ACCs. Note that for SU(3) we explicitly have an ACC and its complex conjugate, while in SU(2) there is no such pairing due to the pseudo-reality of SU(2). Again we write the Boltzmann factor $e^{-S_G[U]}$ as a product over plaquette coordinates $(x,\mu < \nu)$ and color indices $(a,b,c,d)$ and for each combination of indices obtain two Boltzmann factors for the ACC and its complex conjugate. Both are expanded, giving rise to two sets of expansion indices $n_{x,\mu\nu}^{abcd} \in \mathds{N}_0$ and $\bar{n}_{x,\mu\nu}^{abcd} \in \mathds{N}_0$. After reorganizing the products over link matrix elements we find the representation that corresponds to (\[eq:partitionsumsu2\]) in the SU(2) case, $$Z = \; \sum_{\{n, \bar{n}\}} \left[ \prod_{x,\mu<\nu} \prod_{a,b,c,d}
\dfrac{ \left( \beta/6 \right)^{n_{x,\mu\nu}^{abcd} + \bar{n}_{x,\mu\nu}^{abcd}}}{n_{x,\mu\nu}^{abcd} ! \; \;
\bar{n}_{x,\mu\nu}^{abcd} !} \right]
\prod_{x,\mu} \int \! \! dU_{x,\mu} \; \prod_{a,b} \left( U_{x,\mu}^{ab} \right) ^{N_{x,\mu}^{ab}}
\left( U_{x,\mu}^{ab \ \star} \right) ^{\overline{N}_{x,\mu}^{ab}} \; ,
\label{eq:partitionsumsu3}$$ where $$\begin{gathered}
N_{x,\mu}^{abcd} \; = \;
\sum_{\nu:\mu<\nu}n_{x,\mu\nu}^{abss} + \bar{n}_{x-\hat{\nu},\mu\nu}^{ssba} +
\sum_{\rho:\mu>\rho}\bar{n}_{x,\rho\mu}^{assb} + n_{x-\hat{\rho},\rho\mu}^{sabs} \; \; ,
\\
\overline{N}_{x,\mu}^{abcd} \; = \;
\sum_{\nu:\mu<\nu}\bar{n}_{x,\mu\nu}^{abss} + n_{x-\hat{\nu},\mu\nu}^{ssba} +
\sum_{\rho:\mu>\rho}n_{x,\rho\mu}^{assb} + \bar{n}_{x-\hat{\rho},\rho\mu}^{sabs} \; \; .\end{gathered}$$
For integrating out the SU(3) gauge links $U_{x,\mu}$ we choose the parametrization [@brozan]: $$\label{eq:parametrizationsu3}
U_{x,\mu} =\left(
\begin{array}{ccc}
c_1 c_2 \, e^{i\phi_1} \; & s_1 \, e^{i\phi_3} \; & c_1 s_2 \, e^{i\phi_4}\\
s_2 s_3 \, e^{-i\phi_4 -i\phi_5} - s_1 c_2 c_3 \, e^{i\phi_1 +i\phi_2 -i\phi_3} \; & c_1 c_3 \, e^{i\phi_2} \; & - c_2 s_3 \, e^{-i\phi_1 -i\phi_5} - s_1 s_2 c_3 \, e^{i\phi_2 -i\phi_3 +i\phi_4}\\
- s_2 c_3 \, e^{-i\phi_2 -i\phi_4} - s_1 c_2 s_3 \, e^{i\phi_1 -i\phi_2 +i\phi_5} \; & c_1 s_3 \, e^{i\phi_5} \; & c_2 c_3 \, e^{-i\phi_1 -i\phi_2} - s_1 s_2 s_3 \, e^{-i\phi_3 +i\phi_4 +i\phi_5}
\end{array} \right) \; ,$$ where $c_i = \cos \theta^{(i)}_{x,\mu}$, $s_i = \sin \theta^{(i)}_{x,\mu}$, with $\theta^{(i)}_{x,\mu} \in [0,\pi/2]$, and $\phi_i = \phi^{(i)}_{x,\mu}$, with $\phi^{(i)}_{x,\mu} \in [0,2\pi]$, and Haar measure $dU_{x,\mu} = (2\pi^5)^{-1} \; d\theta_1 c_1^3 s_1 \; d\theta_2 c_2 s_2 \; d\theta_3 c_3 s_3 \;
d\phi_1 \; d\phi_2 \; d\phi_3 \; d\phi_4 \; d\phi_5$. In the following, it will prove convenient to perform the change of variables: $$\begin{aligned}
&n_{x,\mu\nu}^{abcd} - \bar{n}_{x,\mu\nu}^{abcd} = p_{x,\mu\nu}^{abcd} \quad , \quad p_{x,\mu\nu}^{abcd} \in \mathbb{Z} \, , \\
&n_{x,\mu\nu}^{abcd} + \bar{n}_{x,\mu\nu}^{abcd} = |p_{x,\mu\nu}^{abcd}| + 2 \, l_{x,\mu\nu}^{abcd} \quad , \quad l_{x,\mu\nu}^{abcd} \in \mathbb{N}_0 \, ,\end{aligned}$$ and to introduce the fluxes $J_{x,\mu}^{ab} \; = \; N_{x,\mu}^{ab} - \overline{N}_{x,\mu}^{ab}$ and $S_{x,\mu}^{ab} \; = \; N_{x,\mu}^{ab} + \overline{N}_{x,\mu}^{ab}$ given explicitly by $$\begin{gathered}
\label{eq:Jfluxessu3}
J_{x,\mu}^{ab} \; = \!
\sum_{\nu:\mu<\nu} [\, p_{x,\mu\nu}^{abss} - p_{x-\hat{\nu},\mu\nu}^{ssba} \, ] - \!\!
\sum_{\rho:\mu>\rho}[\, p_{x,\rho\mu}^{assb} - p_{x-\hat{\rho},\rho\mu}^{sabs} \, ] \; ,\\ \nonumber
S_{x,\mu}^{ab} = \!\!\!\!
\sum_{\nu:\mu<\nu} \!\!\! [ |p_{x,\mu\nu}^{abss}| + |p_{x-\hat{\nu},\mu\nu}^{ssba}| +
2(l_{x,\mu\nu}^{abss} + l_{x-\hat{\nu},\mu\nu}^{ssba}) ]
+ \!\!\!\!
\sum_{\rho:\mu>\rho} \!\!\! [ |p_{x,\rho\mu}^{assb}| - |p_{x-\hat{\rho},\rho\mu}^{sabs}| +
2(l_{x,\rho\mu}^{assb} - l_{x-\hat{\rho},\rho\mu}^{sabs}) ] \, .\end{gathered}$$ The geometrical interpretation of the $J_{x,\mu}^{ab}$ is the same as for SU(2), i.e., they represent the total flux from color $a$ on site $x$ to color $b$ on site $x+\hat{\mu}$, where now the color indices can be $1,2$ or $3$.
To obtain the final result for the partition sum we substitute the parametrization and the Haar measure in . An additional step is still required in order to be able to perform the Haar measure integration, because some of the elements $U_{x,\mu}^{ab}$ of the matrix are not in the simple form $U_{x,\mu}^{ab} = r_{x,\mu}^{ab} e^{i \varphi_{x,\mu}^{ab}}$, but are sums $U_{x,\mu}^{ab} = \rho_{x,\mu}^{ab} e^{i \alpha_{x,\mu}^{ab}} + \omega_{x,\mu}^{ab} e^{i \beta_{x,\mu}^{ab}}$. For the latter we make use of the binomial theorem $(x + y)^{n} = \sum_{k = 0}^{n} \binom{n}{k}x^{k} y^{n-k}$ and rewrite the integrand in as $$\left( U_{x,\mu}^{ab} \right)^{\!N_{x,\mu}^{ab}} \!
\left( U_{x,\mu}^{ab \ \star} \right) ^{\!\overline{N}_{x,\mu}^{ab}} \; = \!\!
\sum_{m_{x,\mu}^{ab} = 0}^{N_{x,\mu}^{ab}} \,
\sum_{\overline{m}_{x,\mu}^{ab} = 0}^{\overline{N}_{x,\mu}^{ab}} \!\!
\! \binom{N_{x,\mu}^{ab}}{m_{x,\mu}^{ab}} \! \binom{\overline{N}_{x,\mu}^{ab}}{\overline{m}_{x,\mu}^{ab}} \!
\left( \! \rho_{x,\mu}^{ab} \! \right)^{\!s_{x,\mu}^{ab}} \! \left( \! \omega_{x,\mu}^{ab} \! \right)^{\!S_{x,\mu}^{ab} - s_{x,\mu}^{ab}}
e^{i\alpha_{x,\mu}^{ab} j_{x,\mu}^{ab}} \, e^{i \beta_{x,\mu}^{ab} \left(J_{x,\mu}^{ab} - j_{x,\mu}^{ab}\right)}.$$ This procedure introduces new sets of dual variables, $m_{x,\mu}^{ab}$ and $\overline{m}_{x,\mu}^{ab}$. For the sums and differences of these we use the shorthand notation $j_{x,\mu}^{ab} = m_{x,\mu}^{ab} - \overline{m}_{x,\mu}^{ab}$, $s_{x,\mu}^{ab} = m_{x,\mu}^{ab} + \overline{m}_{x,\mu}^{ab}$.
Inserting the matrix elements from (\[eq:parametrizationsu3\]) and performing the gauge field integration one finds $$\begin{aligned}
&Z \; = \; \sum_{\{p,l\}} \sum_{\{m,\overline{m}\}} W_{\beta}[p,l] \; W_H[p,l,m,\overline{m}] \;
(-1)^{\sum_{x,\mu}J_{x,\mu}^{12} + J_{x,\mu}^{23} + J_{x,\mu}^{31} - j_{x,\mu}^{23} - j_{x,\mu}^{31}} \; \prod_{x,\mu}
\delta(J_{x,\mu}^{11} + J_{x,\mu}^{12} - J_{x,\mu}^{33} - J_{x,\mu}^{23})
\nonumber \\
& \hspace{5mm} \times \;
\delta(J_{x,\mu}^{22} + J_{x,\mu}^{12} - J_{x,\mu}^{33} - J_{x,\mu}^{31}) \;
\delta(J_{x,\mu}^{13} + J_{x,\mu}^{12} - J_{x,\mu}^{31} - J_{x,\mu}^{21}) \;
\delta(J_{x,\mu}^{32} + J_{x,\mu}^{12} - J_{x,\mu}^{23} - J_{x,\mu}^{21}) \; .
\label{eq:partitionsumsu32}\end{aligned}$$ The partition function is a sum over configurations of the cycle occupation numbers $p_{x,\mu\nu}^{abcd} \in \mathbb{Z}$ and the dual variables $l_{x,\mu\nu}^{abcd} \in \mathbb{N}_{0}$, $m_{x,\mu\nu}^{abcd}$ and $\overline{m}_{x,\mu\nu}^{abcd}$. Each configuration comes with the real and positive weight factors $W_{\beta}[p,l]$ and $W_{H}[p,l,m,\overline{m}]$ which collect the coefficients of the Taylor expansion and the combinatorial factors from the Haar measure integral. Again we find a sign factor $(-1)^{\sum_{x,\mu}J_{x,\mu}^{12} + J_{x,\mu}^{23} + J_{x,\mu}^{31} - j_{x,\mu}^{23} - j_{x,\mu}^{31}}$, which comes from the explicit minus signs in (\[eq:parametrizationsu3\]). The $p_{x,\mu\nu}^{abcd}$ have to obey constraints which are expressed in terms of the four Kronecker deltas in that relate components of the fluxes $J_{x,\mu}^{ab}$ at each link. The geometrical interpretation of the constraints is illustrated in Fig. \[fig:fluxconservationsu3\] using a straightforward generalization of the SU(2) graphical representation.
![Geometrical illustration of the constraints in Eq. (3.11) for the fluxes $J_{x,\mu}^{ab}$ on all links $(x,\mu)$. The constraints in the top row imply that the flux out of a color has to equal the flux into that color. The bottom row of constraints governs the exchange of flux between two colors. In the absence of exchange all three colors must have the same flux. Note that the diagrams are overcomplete and only four of them are independent, corresponding to the four constraints in (3.11). []{data-label="fig:fluxconservationsu3"}](constraints_su3.pdf)
Concluding remarks
==================
In this paper we have presented a new method for finding a dual representation for non-abelian lattice gauge theories, based on strong coupling expansion. The key ingredient for the success of the dualization is a decomposition of the gauge action in terms of abelian color cycles (ACC) which are loops in color space around plaquettes. The ACCs are abelian in nature, i.e., they commute, and the dualization proceeds as in the abelian case. The link integration can be performed explicitly and all expansion coefficients are known in closed form – they are simple combinatorial factors.
For the case of SU(2), in [@Gattringer:2016lml] we presented an extension of the dualization with ACCs by including staggered fermions. A remarkable fact is that in the leading terms of the coupled hopping/strong coupling expansion the minus signs cancel such that in this limit also a dual simulation is possible without the aforementioned resummation. The exploratory results presented here for SU(3) aim at a first assessment of the structure of constraints to be expected for that group.
An interesting open question is whether the dualization strategy based on ACCs allows for a full dualization in the sense that new gauge fields are introduced on the dual lattice such that the constraints are automatically fulfilled. While for U(1) lattice field theory such a dualization is well known (see, e.g., the review [@Savit]), the non-abelian case is less understood. Maybe the ACC strategy, which is patterned after the abelian approach, leads to progress towards such a full dualization.
[99]{}
S. Chandrasekharan, PoS LATTICE [**2008**]{} (2008) 003 \[arXiv:0810.2419 \[hep-lat\]\]. P. de Forcrand, PoS LAT [**2009**]{} (2009) 010 \[arXiv:1005.0539 \[hep-lat\]\]. C. Gattringer, PoS LATTICE [**2013**]{} (2014) 002 \[arXiv:1401.7788 \[hep-lat\]\]. Y. Delgado Mercado, C. Gattringer and A. Schmidt, Phys. Rev. Lett. [**111**]{} (2013) no.14, 141601 \[arXiv:1307.6120 \[hep-lat\]\]. C. Gattringer and C. Marchis, arXiv:1609.00124 \[hep-lat\]. J.B. Brozan, Phys. Rev. [**D38**]{} (1988) 1994.
R. Savit, Rev. Mod. Phys. [**52**]{} (1980) 453.
[^1]: This work is supported by the FWF DK W1203 [*”Hadrons in Vacuum, Nuclei and Stars”*]{}, and partly also by the FWF Grant. Nr. I 1452-N27, as well as the DFG TR55, [*”Hadron Properties from Lattice QCD”*]{}.
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abstract: 'Possible anisotropic spin singlet pairings in Bi$_2$X$_3$ (X is Se or Te) are studied. Among six pairings compatible with the crystal symmetry, two novel pairings show nontrivial surface Andreev bound states, which form flat bands and could produce zero bias conductance peak in measurements like point contact spectroscopy. By considering purely repulsive short range Coulomb interaction as the pairing mechanism, the dominant superexchange terms are all antiferromagnetic, which would usually favor spin singlet pairing in Bi$_2$X$_3$. Mean field analyses show that the interorbital pairing interaction favors a mixed spatial-parity anisotropic pairing state, and one pairing channel with zero energy surface states has a sizable component. The results provide important new information for future experiments.'
address: |
$^1$Department of Physics, Southeast University, Nanjing 210096, China\
$^2$Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan\
$^3$Department of Physics, National Sun Yat-sen University, Kaohsiung 804, Taiwan
author:
- 'Lei Hao$^{1}$, Guiling Wang$^{1}$, Ting-Kuo Lee$^{2}$, Jun Wang$^{1}$, Wei-Feng Tsai$^{3}$, and Yong-Hong Yang$^1$'
title: 'Novel anisotropic spin singlet pairings in Cu$_x$Bi$_2$Se$_3$ and Bi$_2$Te$_3$'
---
\[sec:Introduction\]Introduction
================================
Cu$_x$Bi$_2$Se$_3$ is the first superconductor emerging from a three dimensional topological insulator (TI).[@hor10; @wray10] As a bran-new superconducting (SC) material with topologically nontrivial normal state, it has been suspected to be a time reversal invariant (TRI) topological superconductor (TSC), which supports Andreev bound states (ABS) on the surface.[@fu10; @hao11; @sasaki11; @hsieh12; @yamakage12] The surface ABS in TSCs are massless Majorana fermions, which are novel particles identical to their antiparticles and are under intensive search also because of their prospect of application in topological quantum computing.[@wilczeketal]
Based on a phenomenological on-site attractive interaction, Fu and Berg indeed find some triplet pairings that support surface ABS in Cu$_x$Bi$_2$Se$_3$.[@fu10] Subsequent point contact spectroscopy experiments observed zero bias conductance peaks (ZBCPs), providing smoking-gun evidence for the existence of surface ABS.[@sasaki11] Though recently there are reports advocating conventional $s$ wave pairing[@levy12], the majority of experiments are indicating the unconventional nature of the pairing in Cu$_x$Bi$_2$Se$_3$.[@sasaki11; @koren11; @kirzhner12; @chen12; @kriener11; @das11; @bay12] For examples, a specific heat measurement on one hand suggests a fully gapped pairing while on the other hand is not in full agreement with BCS prediction.[@kriener11] Meissner effect measurements show an unusual field dependence of the magnetization, which is argued to be consistent with odd-parity spin triplet pairing.[@das11] Upper critical field measurements show the absence of Pauli limiting effect, contrary to conventional isotropic s wave pairings, pointing to a very likely triplet pairing.[@bay12]
There are definitely more works needed to identify the genuine pairing symmetries of Cu$_x$Bi$_2$Se$_3$ and the pressure induced SC state of Bi$_{2}$Te$_{3}$ and Bi$_{2}$Se$_{3}$.[@zhang11pa; @zhang11pb; @kirshenbaum13] One important question, which is by and large disregarded up to now, is the possible relevance of anisotropic spin singlet pairings to these new superconductors.[@hao11; @bay12] Since the anisotropic spin singlet parings are more commonly realized than triplet pairings among all unconventional superconductors, including the well-known cuprates and iron pnictides, it is highly desirable to explore this possibility. In this direction, one interesting open question is whether a possible anisotropic spin singlet paring can support surface ABS that could give a ZBCP in point contact spectra and scanning tunneling microscopy (STM) experiments. Since the observation of ZBCP is considered to be a clear indication of possible unconventional pairing, it is very important to identify theoretically this possibility.
The above topic is interesting also in the symmetry classification of superconductors. Since spin-orbit interaction is important for Bi$_2$X$_3$ (X is Se or Te), TRI pairings all belong to the DIII symmetry class.[@schnyder08] In this symmetry class, while known topologically nontrivial pairings are all spin triplet[@schnyder08; @kitaev09; @sato09; @fu10], it is unclear if spin singlet pairings can also have nontrivial topological properties. The present system of Bi$_2$X$_3$ provides a good candidate to explore this possibility. In the work presented below we show that, when anisotropic pairing is considered, two interorbital singlet pairings give zero energy surface states and hence ZBCP in point contact spectra and STM experiments.
The remaining part of the paper is organized as follows. In Sec. II we construct a tight binding model by using group theory and mapping to an existing minimal model defined close to the Brillouin zone (BZ) center. Then we analyze the possible anisotropic spin singlet pairings from purely symmetry considerations, with an antiferromagnetic exchange term in the mind as a phenomenological mechanism for giving these pairings. The gap structures and spectral properties of the various pairings are then analyzed in detail. In Sec. III, we start from a purely repulsive short range Coulomb interaction and derive the dominant superexchange channels relevant to Bi$_{2}$Se$_{3}$ and Bi$_{2}$Te$_{3}$. Then a self-consistent mean field calculation in terms of a $t$$-$$U$$-$$V$$-$$J$ model is performed, which shows that the actual pairing is always a mixture of several components of the anisotropic singlet pairings identified in Sec. II. In Sec. IV, we discuss the implications of our work for experiments and then give a summary of the results. More technical details related to the results are provided in the Appendices.
\[sec:result1\]singlet pairings emerging from a phenomenological mechanism and symmetry analysis
================================================================================================
To analyze the possible anisotropic spin singlet pairings for Bi$_2$X$_3$, we first identify all the possible anisotropic spin singlet pairing channels in a general manner from symmetry considerations and then analyze their properties. The actual relevance and the most probable channel of the anisotropic spin singlet pairings are to be presented in the next section. So, we consider in this section a realistic model for the band structure and a phenomenological correlation term that supports only spin singlet pairings. The model thus consists of two terms $H=H_{0}+H_{ex}$, in which $H_{0}$ is a tight binding term giving rise to the normal state band structure and $H_{ex}$ is an exchange term that could give rise to the desired spin singlet pairings.
Bi$_2$X$_3$ materials belong to the $D_{3d}^{5}$ space group, which consists of Bi$_2$X$_3$ quintuple layers stacked along the out of plane direction. Former theoretical studies on this material system are mostly based on two orbital $\mathbf{k}\cdot\mathbf{p}$ models defined close to the $\boldsymbol{\Gamma}$ point.[@zhang09; @liu10; @fu09] To consider the SC phase transition, we construct a tight binding model with the correct symmetry in the full BZ.[@dresselhaus08] Instead of working in the original BZ for a $D_{3d}^{5}$ space group[@zhang09; @liu10], we consider a hexagonal BZ corresponding to an equivalent hexagonal lattice with two orbitals per unit cell, for its simplicity and capability to respect the low energy symmetries and to account for physical properties of the system, as verified in previous studies.[@wang10; @hao11; @sasaki11] Take the basis vector as $\phi_{\mathbf{k}}^{\dagger}=[a_{\mathbf{k}\uparrow}^{\dagger},
b_{\mathbf{k}\uparrow}^{\dagger},a_{\mathbf{k}\downarrow}^{\dagger},b_{\mathbf{k}\downarrow}^{\dagger}]$, in which $a$ and $b$ operators correspond to the two orbitals, the tight binding model is (see Appendix A for details) $$\begin{aligned}
H_{0}(\mathbf{k})&=&\epsilon(\mathbf{k})I_{4}+M(\mathbf{k})\Gamma_5+B_{0}c_{z}(\mathbf{k})\Gamma_{4}
+A_{0}[c_{y}(\mathbf{k})\Gamma_{1} \notag \\
&&-c_{x}(\mathbf{k})\Gamma_{2}]+R_{1}d_{1}(\mathbf{k})\Gamma_{3}+R_{2}d_{2}(\mathbf{k})\Gamma_{4},\end{aligned}$$ in which $\epsilon(\mathbf{k})=C_{0}+2C_{1}[1-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{4})]
+\frac{4}{3}C_{2}[3-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})
-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{3})]$. $M(\mathbf{k})$ is obtained from $\epsilon(\mathbf{k})$ by making the substitutions $C_{i}\rightarrow M_{i} (i=0,1,2)$. $c_{x}(\mathbf{k})=\frac{1}{\sqrt{3}}
[\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})]$, $c_{y}(\mathbf{k})=\frac{1}{3}[\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})
-2\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{3})]$, and $c_{z}(\mathbf{k})=\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{4})$. Finally, $d_{1}(\mathbf{k})=-\frac{8}{3\sqrt{3}}[\sin(\mathbf{k}\cdot\mathbf{a}_{1})+\sin(\mathbf{k}\cdot\mathbf{a}_{2})
+\sin(\mathbf{k}\cdot\mathbf{a}_{3})]$ and $d_{2}(\mathbf{k})=-8[\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})
+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{3})]$. Here, the four independent nearest neighbor (NN) bond vectors of the effective hexagonal lattice are $\boldsymbol{\delta}_{1}=(\frac{\sqrt{3}}{2}a, \frac{1}{2}a, 0)$, $\boldsymbol{\delta}_{2}=(-\frac{\sqrt{3}}{2}a$, $\frac{1}{2}a, 0)$, $\boldsymbol{\delta}_{3}=(0, -a, 0)$, and $\boldsymbol{\delta}_{4}=(0, 0, c)$, with $a$ and $c$ denoting in plane and out-of-plane lattice parameters.[@acparameters] The three in plane second nearest neighbor (2NN) bond vectors in $d_{1}(\mathbf{k})$ are $\mathbf{a}_{1}=\boldsymbol{\delta}_{1}-\boldsymbol{\delta}_{2}$, $\mathbf{a}_{2}=\boldsymbol{\delta}_{2}-\boldsymbol{\delta}_{3}$, and $\mathbf{a}_{3}=\boldsymbol{\delta}_{3}-\boldsymbol{\delta}_{1}$. Expanding close to the $\boldsymbol{\Gamma}$ point, the above model is easily shown to reduce to the same form as Eqs. (16) and (17) in Liu *et al*.[@liu10] Demanding that the expanded model is the same as that in Liu *et al*, the parameters are determined as shown in Table I. In actual calculations, we change the value of $M_{1}$ to the bracketed value of 0.62 eV (0.102 eV) for Bi$_2$Se$_3$ (Bi$_2$Te$_3$) which yields a band gap of approximately 0.26 eV (0.06 eV).[@zhang09] All other parameters will be kept as given in Table I.
------------------------ --------- --------- --------- --------- ---------------
$$ $C_{0}$ $C_{1}$ $C_{2}$ $M_{0}$ $M_{1}$
\[0.2ex\] Bi$_2$Se$_3$ -0.0083 0.063 1.774 -0.28 0.0753 (0.62)
Bi$_2$Te$_3$ -0.18 0.0634 2.59 -0.3 0.027 (0.102)
$$ $M_{2}$ $A_{0}$ $B_{0}$ $R_{1}$ $R_{2}$
\[0.2ex\] Bi$_2$Se$_3$ 2.596 0.804 0.237 0.713 -1.597
Bi$_2$Te$_3$ 2.991 0.655 0.0295 0.536 -1.064
------------------------ --------- --------- --------- --------- ---------------
: Parameters in the tight binding model obtained by comparing with a $\mathbf{k}\cdot\mathbf{p}$ model[@liu10], in units of electron volts. Bracketed values of $M_{1}$ are those actually used.
In Eq. (1), $I_4$ is the $4\times4$ unit matrix. The form of the $\Gamma$ matrices depends on the choice of bases for the two orbitals. In the original $\mathbf{k}\cdot\mathbf{p}$ model[@liu10; @zhang09], the two orbitals are chosen to have definite parity. Here, we choose a basis set in which the two orbitals have the physical meaning of local $p_z$ orbitals residing on the top and bottom Se (or Te) layers of a quintuple unit hybridized with $p_z$ orbitals in neighboring Bi layers.[@liu10; @wang10; @hao11; @fu09; @fu10; @sasaki11] The two basis sets are related by a simple unitary transformation. We thus have $\Gamma_1=s_{1}\otimes\sigma_{3}$, $\Gamma_2=s_{2}\otimes\sigma_{3}$, $\Gamma_3=s_{3}\otimes\sigma_{3}$, $\Gamma_{4}=-s_{0}\otimes\sigma_{2}$, and $\Gamma_{5}=s_{0}\otimes\sigma_{1}$.[@liu10; @zhang09] $s_{i}$ and $\sigma_{i}$ are Pauli matrices for the spin and orbital degrees of freedom.
To study possible spin singlet pairings, $H_{ex}$ is restricted to contain only antiferromagnetic (AF) terms up to NN in plane bonds. The AF exchange terms are known to be able to give rise to spin singlet pairings.[@seo08; @kotliar88; @goswami10] Since there are two orbitals, we have both intraorbital and interorbital terms, thus $H_{ex}=H_{intra}+H_{inter}$. They are written generally as $$H_{intra}=\sum\limits_{\mathbf{i}\boldsymbol{\delta}\alpha}
J^{\alpha}_{\mathbf{i},\mathbf{i}+\boldsymbol{\delta}}(\mathbf{S}_{\mathbf{i}\alpha}\cdot\mathbf{S}_{\mathbf{i}+\boldsymbol{\delta},\alpha}
-\frac{1}{4}\hat{n}_{\mathbf{i}}^{\alpha}\hat{n}_{\mathbf{i}+\boldsymbol{\delta}}^{\alpha}),$$ and $$H_{inter}=\sum\limits_{\mathbf{i}\boldsymbol{\delta}}
J^{ab}_{\mathbf{i},\mathbf{i}+\boldsymbol{\delta}}
(\mathbf{S}_{\mathbf{i}a}\cdot\mathbf{S}_{\mathbf{i}+\boldsymbol{\delta},b}
-\frac{1}{4}\hat{n}_{\mathbf{i}}^{a}\hat{n}_{\mathbf{i}+\boldsymbol{\delta}}^{b}),$$ where $\mathbf{i}$ runs over unit cells, $\boldsymbol{\delta}$ runs over the six NN in plane bonds $\pm\boldsymbol{\delta}_{j}$ ($j$=1, 2, 3), and the $\alpha$ summation in $H_{intra}$ runs over the two orbitals. $\hat{n}_{\mathbf{i}}^{\alpha}=\hat{n}_{\mathbf{i}\alpha\uparrow}+\hat{n}_{\mathbf{i}\alpha\downarrow}$ is the electron number operator for $\alpha$ orbital.
Out of all the possible spin singlet pairing channels contained in $H_{ex}$, we focus on those pairings compatible with the crystal symmetry of the Bi$_{2}$X$_{3}$ materials. Taking advantage of the various irreducible representations identified earlier by Liu *et al*[@liu10], we find six TRI $\mathbf{k}$-dependent pairings which are compatible with the crystal symmetry and are spin singlets (See Appendix C). The six pairings identified include two intraorbital channels belonging to the $\tilde{\Gamma}_{3}^{+}$ representation, which are $\Delta_{j}\phi_{\mathbf{k}}^{\dagger}i\Gamma_{31}(\phi_{-\mathbf{k}}^{\dagger})^{\text{T}}\varphi_{j}(\mathbf{k})$ ($j$=1, 2). $\Delta_{j}$ are the magnitudes of the pairing terms. $\Gamma_{31}=s_{2}\otimes\sigma_{0}$. $\varphi_{1}(\mathbf{k})=\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2}) =-2\sin(\frac{\sqrt{3}}{2}k_{x}a)\sin(\frac{1}{2}k_{y}a)$ and $\varphi_{2}(\mathbf{k})=2\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{3})-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})
-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})=2[\cos(k_{y}a)-\cos(\frac{\sqrt{3}}{2}k_{x}a)\cos(\frac{1}{2}k_{y}a)]$. There are also two interorbital pairings belonging to the $\tilde{\Gamma}_{3}^{+}$ representation, which could be obtained from the above two pairing terms by substituting $\Gamma_{31}$ by $\Gamma_{24}=s_{2}\otimes\sigma_{1}$ and identifying $\varphi_{3(4)}(\mathbf{k})=\varphi_{1(2)}(\mathbf{k})$. The other two pairing channels belong to $\tilde{\Gamma}_{3}^{-}$ and are $\Delta_{j}\phi_{\mathbf{k}}^{\dagger}i\Gamma_{25}(\phi_{-\mathbf{k}}^{\dagger})^{\text{T}}\varphi_{j}(\mathbf{k})$ ($j$=5, 6), in which $\Gamma_{25}=s_{2}\otimes\sigma_{2}$, $\varphi_{5}(\mathbf{k})=\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2}) =2\sin(\frac{\sqrt{3}}{2}k_{x}a)\cos(\frac{1}{2}k_{y}a)$ and $\varphi_{6}(\mathbf{k})=\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})
-2\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{3})=2\sin(\frac{1}{2}k_{y}a)[\cos(\frac{\sqrt{3}}{2}k_{x}a)+2\cos(\frac{1}{2}k_{y}a)]$. In the following discussions, we use $\Delta_j$ to refer to the $j$-th pairing defined above, in places where confusion is not incurred.
The wave vector dependencies of the various pairing components determine their gap structures. For experimentally relevant chemical potentials, states on the Fermi surface are all close to the $\boldsymbol{\Gamma}=(0,0,0)$ point of the BZ, it is thus enough to focus on small wave vectors. For the first and the third pairings, $\varphi_{1}(\mathbf{k})=0$ gives two line nodes of the gap along $k_{x}=0$ and $k_{y}=0$. The other nodes determined by $k_{x}=\frac{2\pi}{\sqrt{3}a}$ or $k_{y}=\frac{2\pi}{a}$ are unlikely to occur since they lie on the BZ boundary and are far away from the Fermi surface. For the second and the fourth pairings, the line nodes are determined by $\varphi_{2}(\mathbf{k})=0$ and satisfy $4\cos(\frac{1}{2}k_{y})=\cos(\frac{\sqrt{3}}{2}k_{x})+\alpha\sqrt{\cos^{2}(\frac{\sqrt{3}}{2}k_{x})+8}$, $\alpha=\pm$. Note that, for each $k_{x}$, there are two solutions for $k_{y}$ for each $\alpha$. Usually, only the two solutions related to $\alpha=+$ lie on the Fermi surface, so the second and fourth pairings in general also have two lines nodes. Similarly, from $\varphi_{5}(\mathbf{k})=0$ we know that the fifth pairing usually has only one line node determined by states on the Fermi surface with $k_{x}=0$. And finally the sixth pairing also has only one set of line nodes determined by $\varphi_{6}(\mathbf{k})=0$, consisting of states on the Fermi surface with $k_{y}=0$. Among the six pairings identified, the fifth ($\Delta_{5}$) and the sixth ($\Delta_{6}$) pairings are peculiar in that, though spin singlet, they have *$p$-wave like odd $\mathbf{k}$ dependencies* for small wave vectors, which together with their odd orbital-parity is consistent with their spin singlet nature. Here, we define spatial-parity and orbital-parity as the parities of the SC order parameter related with reversal of the wave vector and exchange of the two orbitals, respectively.
To check properties of the pairings identified above, we first calculate their surface local density of states (SLDOS), which are directly observable via point contact spectra.[@sasaki11; @koren11; @kirzhner12] The SLDOS are defined as the surface spectral function averaged over the surface wave vectors, that is $\rho_{s}(\omega)=N_{s}^{-1}\sum_{\mathbf{k}_{xy}}A(\mathbf{k}_{xy},\omega)$, in which $N_{s}$ is the number of wave vectors $\mathbf{k}_{xy}$ in the surface BZ.[@sasaki11] $A(\mathbf{k}_{xy},\omega)$ is the surface spectral function defined as imaginary part of the electronic surface Green’s functions (GF) obtained in terms of the iterative GF method (or, transfer matrix method).[@hao11; @wang10] See Appendix D for a brief explanation of our usage of the iterative GF method. To distinguish possible surface-state contributions, we calculate simultaneously the bulk local density of states (BLDOS), which are obtained easily from the bulk GF. In these calculations, we first add directly a certain pairing term to $H_{0}$ without regarding its origin, to focus on a single pairing channel.
{width="8.8cm" height="12cm"}
As shown in Figs. 1(a) to 1(f) are the LDOS for three typical pairings, for Bi$_{2}$Se$_{3}$ (Bi$_{2}$Te$_{3}$) system at electron (hole) fillings specified by the chemical potentials as indicated. For the same pairing, qualitatively similar results are obtained for the two systems. BLDOS for the $\Delta_{1}$ to $\Delta_{4}$ pairings all show a V-shape demonstrating the linear density of states at the chemical potential. The SLDOS for these pairings are all similar to the BLDOS, indicating that there are no topological nontrivial surface ABS. Very interestingly, in Figs. 1(e) and 1(f) which are typical results for both $\Delta_{5}$ and $\Delta_{6}$ pairings, while the BLDOS still shows a linear density of states, a sharp zero energy peak appears in the SLDOS. This peak structure is reminiscent of the ZBCP observed in some point contact spectra measurements in superconducting Cu$_{x}$Bi$_{2}$Se$_{3}$.[@sasaki11; @kirzhner12] While these zero energy peaks imply sharp ZBCPs in point contact spectra and STM experiments, they would broaden at finite temperature and thus within the experimental resolution.[@sasaki11; @hsieh12] We also find that, a TRI pairing consisting of a mixture of $\Delta_{5}$ or $\Delta_{6}$ and other pairings preserves the novel zero energy peak in the SLDOS, see Figs. 1(g) and 1(h) for typical examples. Note that, because all the six pairings share the gap node at $k_{x}=k_{y}=0$, the composite pairing consisting of several pairing components is still gapless in the bulk. This explains the V-shape BLDOS shown in Figs.1(g) and 1(h).
The surface ABS are more clearly seen from the surface spectral functions, as shown in Fig. 2 for $\Delta_{6}$. Along ($k_{x}$, 0) direction of the surface BZ, the superconductor has a line node. Along other directions such as (0, $k_{y}$) in Fig. 2, a SC gap opens, the surface ABS are clearly present and form a flat band. This is similar to the surface ABS in some nodal spin triplet pairings.[@sasaki11; @hao11] Though the topological numbers for $\Delta_5$ and $\Delta_6$ with nodal lines are difficult to calculate directly [@sato10; @sasaki11; @matsuura13], the number counting of zero energy surface ABS are also indicative of important conclusions. By explicitly calculating the eigenstates for a thin film of the $\Delta_{6}$ pairing, we confirmed that a *single* Kramers’ pair of ABS exists on both the top and the bottom surfaces. Since an odd number of Kramers’ pairs of surface states are generally protected from TRI perturbations, the gapless surface ABS for $\Delta_5$ and $\Delta_6$ should also be topologically stable. In this sense, we may say that $\Delta_{5}$ and $\Delta_{6}$ are topologically nontrivial spin singlet pairings. To our knowledge, they are the first reported spin singlet pairings in DIII class that might be topologically nontrivial.
{width="8.5cm" height="12.5cm"}
\[sec:result2\]pairings emerging from purely repulsive Coulomb interactions
===========================================================================
Before studying further properties of the novel spin singlet pairings identified above, it is important to ask if the spin singlet pairings are actually relevant to the superconducting state of Bi$_2$X$_3$ (X is Se or Te). That is, whether or not spin singlet pairing is the dominant pairing channel for realistic pairing mechanisms. Since evidences have appeared that Bi$_2$Se$_3$ is a sizable correlated electron system [@wang11; @craco12], whereas the electron-phonon coupling (EPC) in the present system is generally considered to be smaller than BCS superconductors[@giraud11; @hatch11; @zhu12; @pan12], we would here take the *purely repulsive* short range Coulomb interaction as the pairing mechanism for Bi$_2$X$_3$. Since EPC usually favors spin singlet pairings, the following study stands as a more stringent test for the relevance of spin singlet pairings.
Compatible with our orbital convention, the Coulomb repulsion terms are conveniently added to $H_{0}$ as $$H_{1}=U\sum\limits_{\mathbf{i}\alpha}\hat{n}_{\mathbf{i}\alpha\uparrow}\hat{n}_{\mathbf{i}\alpha\downarrow}
+V\sum\limits_{\mathbf{i}}\hat{n}_{\mathbf{i}}^{a}\hat{n}_{\mathbf{i}}^{b},$$ in which $\alpha$ runs over the two orbitals. Since the two orbitals in the present model reside on different sites within a unit cell, we do not include the Hund’s coupling term between them.[@craco12] In addition, we expect the on-site intraorbital correlation stronger than the inter-site interorbital correlation, that is $U>V>0$.
Starting from $H_{0}+H_{1}$, the dominant superexchange couplings are derived using standard projection operator method (Schrieffer-Wolf transformation).[@schrieffer66] Here, we retain the lowest order terms up to two site correlations. The superexchange terms are derived first at half filling. The relation $U>V$ imposes a local intraorbital no-double-occupancy condition. In principle, each hopping parameter and every pair of two different hopping parameters listed in Table I could mediate a superexchange term. However, since the magnitudes of the various parameters differ greatly, we expect that the superexchange terms mediated by the largest several parameters in Table I dominate the actual pairing instability. We thus select $C_2$, $M_2$ and $R_2$, which are apparently larger than the other parameters. We find that $C_2$ mediates the term $H_{intra}$, with $J^{\alpha}_{\mathbf{i},\mathbf{i}+\boldsymbol{\delta}}=\frac{8C^{2}_{2}}{9U}$ independent of $\boldsymbol{\delta}$. $M_2$, $R_2$ and their crossing mediate three interorbital superexchange terms, which combine to give $H_{inter}$, with $J^{ab}_{\mathbf{i},\mathbf{i}+\boldsymbol{\delta}}=
\frac{(8R_{2})^2}{U}[1+(-1)^{\eta(\boldsymbol{\delta})}\frac{M_{2}}{6R_{2}}]^{2}$. $\eta(\boldsymbol{\delta})$ arises from the crossing term between $M_{2}$ and $R_{2}$ and is defined as $0$ ($1$) for $\boldsymbol{\delta}=\boldsymbol{\delta}_{j}$ ($\boldsymbol{\delta}=-\boldsymbol{\delta}_{j}$), with $j$=1, 2, 3. Thus, the AF exchange terms conceived in Eqs. (2)-(3) emerge naturally in Bi$_2$X$_3$ if we consider short range repulsive Coulomb interaction as the pairing mechanism. Other parameters, such as $A_{0}$ in Table I which could mediate superexchange correlations favoring triplet pairings, are too small to be competitive with the identified terms, for both Bi$_{2}$Se$_{3}$ and Bi$_{2}$Te$_{3}$. More details about the derivation of the dominant superexchange terms can be found in Appendix B. In conclusion, pairings mediated by purely repulsive short range Coulomb interactions are dominantly spin singlet in Bi$_2$X$_3$.
Besides confirming the hypothetical form of $H_{ex}$, an interesting new consequence of the above derivation is that $H_{inter}$ only has $C_3$ symmetry with respect to $c$-axis, which is clear from the $\boldsymbol{\delta}$-dependency of $\eta(\boldsymbol{\delta})$. This feature inherits directly from the $R_{2}$ term, because the crossing term between $R_{2}$ and $M_{2}$ is linear in $R_{2}$. An immediate consequence of this real space anisotropic correlation is that it explicitly *breaks the in plane inversion symmetry* of $H_{ex}$, which implies that the *in plane spatial-parity* is not a good quantum number and the resulting pairing would be a mixture of even and odd spatial-parity states. The spatial-parity mixing effect is more clearly seen from the Fourier transformation of the interorbital pairing potential in $H_{inter}$, which is $$\begin{aligned}
J^{ab}_{\mathbf{k},\mathbf{k'}}&=&\frac{(8R_{2})^2}{NU}\sum\limits_{j=1}^{3}\{[1+\frac{M_{2}^{2}}{(6R_{2})^2}]
\cos(\mathbf{k}-\mathbf{k'})\cdot\boldsymbol{\delta}_{j} \notag \\
&&+i\frac{M_{2}}{3R_{2}}
\sin(\mathbf{k}-\mathbf{k'})\cdot\boldsymbol{\delta}_{j}\},\end{aligned}$$ with $N$ the number of unit cells in the whole lattice. This dynamical generation of an inversion symmetry breaking *correlation* term is an essential feature of the present model and is intrinsic to Bi$_{2}$X$_{3}$ materials. We also mention that, though the $R_2$ term in $H_{0}$ already breaks the in-plane inversion, the present correlation term is different because firstly it depends also on $M_2$ and emerges as a crossing term between the $M_2$ and $R_2$ terms, secondly the mixing of pairings with even and odd spatial-parity is now explicit and their relative phase and weight are determined by the correlation term itself.
To see the relative importance of the six pairings identified formerly by symmetry, we have performed mean field calculations at zero temperature, in terms of a $t$$-$$U$$-$$V$$-$$J$ type full model $H=H_{0}+H_{1}+H_{ex}$ (see Appendix C for details). First, we decouple $H_{ex}$ by introducing nonequivalent bond pairing terms.[@seo08; @goswami10; @kotliar88] Six intraorbital pairings for the $a$ orbital are introduced as $\chi_{a}^{\nu\pm}=\langle
a_{\mathbf{i}\pm\boldsymbol{\delta}_{\nu},\downarrow}a_{\mathbf{i}\uparrow}
-a_{\mathbf{i}\pm\boldsymbol{\delta}_{\nu},\uparrow}a_{\mathbf{i}\downarrow}\rangle$ ($\nu=1, 2, 3$). Another six intraorbital pairings $\chi_{b}^{\nu\pm}$ are similarly defined for the $b$ orbital. Six interorbital pairings are introduced as $\chi_{ba}^{\nu\pm}=\langle
b_{\mathbf{i}\pm\boldsymbol{\delta}_{\nu},\downarrow}a_{\mathbf{i}\uparrow}
-b_{\mathbf{i}\pm\boldsymbol{\delta}_{\nu},\uparrow}a_{\mathbf{i}\downarrow}\rangle$ ($\nu=1, 2, 3$). A translational invariant pairing phase is assumed, so that the $18$ pairing terms are independent of $\mathbf{i}$. Since we focus on the SC phase, the decoupling of $H_{ex}$ to the normal phase is disregarded. Terms in $H_{1}$ are decoupled in the simplest manner as $\hat{n}_{1}\hat{n}_{2}\rightarrow\langle\hat{n}_{1}\rangle\hat{n}_{2}+\hat{n}_{1}\langle\hat{n}_{2}\rangle
-\langle\hat{n}_{1}\rangle\langle\hat{n}_{2}\rangle$. For each set of parameters ($U$, $V$ and doping or chemical potential), we then get the $18$ mean field pairing order parameters self-consistently from many different initial values. Finally, the amplitudes of the six pairing terms are projected out of the solution.[@seo08; @goswami10; @kotliar88] We find that, for parameters typical for Bi$_{2}$Se$_{3}$ and Bi$_{2}$Te$_{3}$, the dominant pairing channel is a mixture of $\Delta_{4}$ and $\Delta_{6}$ pairings with a tiny admixture of the $\Delta_{2}$ pairing. All other pairing components are identically zero. In addition, though there are usually several coexisting pairing components, the pairing is *time reversal invariant* up to a global $U(1)$ phase, same as the conclusion of a similar calculation for iron pnictides.[@seo08] We emphasize that the two main features of the results, coexistence of several pairing components and the time reversal invariance of the full pairing, are true for all model parameters that we have tried, which are spanned by $U\sim[8$ eV, 20 eV\], $V\sim[5$ eV, 8 eV\] and $x\sim[-0.08, 0.16]$ ($x$ is the number of excess electrons in each unit cell). The robustness of the two features are consequences of the interorbital superexchange correlation term, as shown in Eq.(5), from which the pairings of even spatial-parity and of odd spatial-parity always appear together in a time reversal invariant combination. We should confess that the present mean field studies underestimate the fluctuation effect and the competition from possible magnetic normal states, so that the pairing instability is overestimated. However, the dominant pairing channel should still be spin singlet even if these corrections are taken into account, which are left to future works.
\[sec:Summary\]Experimental implications and Summary
====================================================
The novel spin singlet pairings, $\Delta_{5}$ and $\Delta_{6}$, could be distinguished experimentally from other candidate pairings. Firstly, since they give quite different surface spectral functions[@hao11; @sasaki11], the correct pairing symmetry could be read from ARPES if the precision of measurement can reach the order of $\sim0.1$ meV.[@wray10] Secondly, the SLDOS which could be probed by point contact spectroscopy or STM can also be used to discriminate among the candidate pairings. For example, the fully gapped interorbital triplet pairing gives an in-gap nonzero energy double-peak structure in SLDOS at low temperature and for good contact.[@sasaki11; @hsieh12] However, the $\Delta_{5}$ or $\Delta_{6}$ pairing always gives a single ZBCP. While our proposal is in better agreement with existing experiments[@sasaki11; @koren11; @kirzhner12; @chen12], more measurements are desired to get a definite conclusion. Thirdly, the static spin susceptibilities show clear differences for different candidate pairings and thus could be used to discriminate some of them. Fourthly, the thermal conductivity which depends sensitively on the anisotropy of the pairings was proposed to discriminate two triplet pairings.[@nagai12] It should also be able to tell $\Delta_5$ or $\Delta_6$ pairing from the other candidate (triplet) pairings. Details that lead to the above conclusions are to be published elsewhere. Besides the above proposals, our spin singlet pairing state is not in direct contradiction with existing experiments. Not only for experiments pointing to polar or anisotropic pairings, it could also be in agreement with a specific heat experiment which shows that the pairing has a fully gapped component.[@kriener11] Though the novel singlet pairings $\Delta_5$ or $\Delta_6$ are both gapless, a fully gapped $\mathbf{k}$-independent interorbital spin singlet component could be readily added into our mixed pairing state to give the experimental feature (see Appendix B).
To summarize, we have studied the possible anisotropic spin singlet pairings in Bi$_2$X$_3$ (X is Se or Te). Two novel interorbital spin singlet pairings with odd spatial-parity and odd orbital-parity support surface ABS, which form zero energy flat bands. The presence of only one Kramers’ pair of ABS on each surface implies that they should be topologically stable against TRI perturbations. Considering purely repulsive short range Coulomb interaction as the pairing mechanism, the low energy effective model turns out indeed to be dominated by spin-singlet-favoring AF correlations. Besides, the interorbital AF correlation favors a pairing state with mixed spatial-parity. It would be interesting to see if this prediction can be verified by future experiments.
L.H. and G.L.W. are supported by NSFC.11204035 and SRFDP.20120092120040. T.K.L. acknowledges the support of NSC in Taiwan under Grant No.103-2120-M-001-009. J.W. is supported by NSFC.11274059 and NSF of Jiangsu Province BK20131284. W.F.T. is supported by the NSC in Taiwan under Grant No.102-2112-M-110-009. Part of the calculations was performed in the National Center for High-Performance Computing in Taiwan.
\[sec:TBModel\]Tight Binding Model
==================================
Here, we construct a tight binding model for the bulk electronic structures of Bi$_2$Se$_3$, Cu$_x$Bi$_{2}$Se$_3$ and Bi$_2$Te$_3$ materials from symmetry considerations. As illustrated in the main text, we replace the actual lattice with $D_{3d}^{5}$ space group symmetry by a hexagonal lattice with two orbitals per unit cell. Take in-plane (labeled as the $xy$ plane) and out-of-plane (labeled as the $z$ direction) lattice parameters as $a$ and $c$, the four independent nearest-neighbor (NN) bond vectors of the effective hexagonal lattice are $\boldsymbol{\delta}_{1}=(\frac{\sqrt{3}}{2}a, \frac{1}{2}a, 0)$, $\boldsymbol{\delta}_{2}=(-\frac{\sqrt{3}}{2}a$, $\frac{1}{2}a, 0)$, $\boldsymbol{\delta}_{3}=(0, -a, 0)$, and $\boldsymbol{\delta}_{4}=(0, 0, c)$. We take the lattice parameters as $a$=4.14 Å(4.38 Å) and $3c$=28.64 Å(30.487 Å) for Bi$_2$Se$_3$ (Bi$_2$Te$_3$).[@acparameters] Small changes in $a$ and $c$ for the SC state of Bi$_2$X$_3$ are neglected.[@hor10; @wray10; @zhang11pa; @zhang11pb] For Bi$_2$Te$_3$, we consider the SC transition under ambient pressure without structural transition and so the symmetry keeps as $D_{3d}^{5}$.[@zhang11pb]
Since spin-orbit interaction is important in Bi$_2$X$_3$, we have to consider the double group of the $D_{3d}^{5}$ space group to get a proper tight binding model. Following the notations of Liu *et al* [@liu10], we write the generators of the point group for $D_{3d}^{5}$ as $R_{3}$ (threefold rotation, about the symmetry line parallel to $z$ axis), $R_{2}$ (twofold rotation, about the symmetry line parallel to $x$ axis) and $P$ (inversion). For the double group, introduce the operator $\mathcal{C}$ to represent $2\pi$ rotation. The characters for the various irreducible representations are then as shown in Table I.[@dresselhaus08; @liu10]
[c c c c c c c c c c c c c]{}\
\[-1.5ex\] $D_{3d}(\bar{3}m)$ & $E$ & 2$R_{3}$ & 3$R_{2}$ & $P$ & 2$P$$R_{3}$ & 3$P$$R_{2}$ & $\mathcal{C}$ & 2$\mathcal{C}$$R_{3}$ & 3$\mathcal{C}$$R_{2}$ & $\mathcal{C}$$P$ & 2$\mathcal{C}$$P$$R_{3}$ & 3$\mathcal{C}$$P$$R_{2}$\
\[0.2ex\]\
\[-2ex\] $\tilde{\Gamma}_{1}^{+}$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\
\
\[-2ex\] $\tilde{\Gamma}_{2}^{+}$ & 1 & 1 & -1 & 1 & 1 & -1 & 1 & 1 & -1 & 1 & 1 & -1\
\
\[-2ex\] $\tilde{\Gamma}_{3}^{+}$ & 2 & -1 & 0 & 2 & -1 & 0 & 2 & -1 & 0 & 2 & -1 & 0\
\
\[-2ex\] $\tilde{\Gamma}_{4}^{+}$ & 1 & -1 & $i$ & 1 & -1 & $i$ & -1 & 1 & $-i$ & -1 & 1 & $-i$\
\
\[-2ex\] $\tilde{\Gamma}_{5}^{+}$ & 1 & -1 & $-i$ & 1 & -1 & $-i$ & -1 & 1 & $i$ & -1 & 1 & $i$\
\
\[-2ex\] $\tilde{\Gamma}_{6}^{+}$ & 2 & 1 & 0 & 2 & 1 & 0 & -2 & -1 & 0 & -2 & -1 & 0\
\
\[-2ex\] $\tilde{\Gamma}_{1}^{-}$ & 1 & 1 & 1 & -1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & -1\
\
\[-2ex\] $\tilde{\Gamma}_{2}^{-}$ & 1 & 1 & -1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & -1 & 1\
\
\[-2ex\] $\tilde{\Gamma}_{3}^{-}$ & 2 & -1 & 0 & -2 & 1 & 0 & 2 & -1 & 0 & -2 & 1 & 0\
\
\[-2ex\] $\tilde{\Gamma}_{4}^{-}$ & 1 & -1 & $i$ & -1 & 1 & $-i$ & -1 & 1 & $-i$ & 1 & -1 & $i$\
\
\[-2ex\] $\tilde{\Gamma}_{5}^{-}$ & 1 & -1 & $-i$ & -1 & 1 & $i$ & -1 & 1 & $i$ & 1 & -1 & $-i$\
\
\[-2ex\] $\tilde{\Gamma}_{6}^{-}$ & 2 & 1 & 0 & -2 & -1 & 0 & -2 & -1 & 0 & 2 & 1 & 0\
Same as in the main text, we take the basis vector as $\phi_{\mathbf{k}}^{\dagger}=[a_{\mathbf{k}\uparrow}^{\dagger},
b_{\mathbf{k}\uparrow}^{\dagger},a_{\mathbf{k}\downarrow}^{\dagger},b_{\mathbf{k}\downarrow}^{\dagger}]$, in which the two orbitals represented by the $a$ (not to be confused with the in-plane lattice parameter) and $b$ operators denote local $p_z$ orbitals residing on the top and bottom Se (Te) layers of a Bi$_2$Se$_3$ (Bi$_2$Te$_3$) quintuple unit hybridized with $p_z$ orbitals in neighboring Bi layers. The fact that a minimal model consisting of the above two hybridized $p_z$ orbitals is enough for the low energy physics of topological insulators like Bi$_2$Se$_3$ and Bi$_2$Te$_3$ has been established in previous works.[@liu10; @zhang09; @fu09] With this basis at hand, and define $s_{i}$ and $\sigma_{i}$ as Pauli matrices for the spin and orbital degrees of freedom, we can write the various symmetry operations in matrix form.[@liu10] The time reversal operator is $T=is_{2}\otimes\sigma_{0}K$, where $K$ denotes the complex conjugation and $\sigma_{0}$ is the $2\times2$ unit matrix in orbital subspace. Matrix for the threefold rotation is $R_{3}=e^{i(s_{3}\otimes\sigma_{0}/2)\theta}=\cos{\frac{\theta}{2}}+is_{3}\otimes\sigma_{0}\sin{\frac{\theta}{2}}$, with $\theta=2\pi/3$. The twofold rotation is $R_{2}=is_{1}\otimes\sigma_{1}$. The matrix for inversion is $P=s_{0}\otimes\sigma_{1}$, with $s_{0}$ the $2\times2$ unit matrix in spin subspace. Finally, we may also write out the matrix for $\mathcal{C}$, the $2\pi$ rotation, which should be written as $-s_{0}\otimes\sigma_{0}$.
Denote by $H_{0}(\mathbf{k})$ the 4$\times$4 Hamiltonian matrix for wave vector $\mathbf{k}$ in the basis of $\phi_{\mathbf{k}}^{\dagger}$. The translational invariance (or, periodicity) of the material in real space implies that $H_{0}(\mathbf{k})$ is a periodic function of the reciprocal lattice vectors in the extended zone scheme.[@dresselhaus08] According to Bloch’s theorem, $H_{0}(\mathbf{k})$ could be Fourier expanded in terms of the real space lattice. Since the reciprocal lattice have the same symmetry as the real space lattice, $H_{0}(\mathbf{k})$ should be an invariant under the action of the $D_{3d}^{5}$ double group. So, the general form of $H_{0}(\mathbf{k})$ conforming to symmetry is [@dresselhaus08] $$H_{0}(\mathbf{k})=\sum\limits_{j,\alpha\nu}a_{j\alpha\nu}g_{j\alpha}(\mathbf{k},\mathbf{d}_{\nu})O^{j\alpha},$$ where $\mathbf{d}_{\nu}$ represent lattice vectors connecting sites at the $\nu$-th nearest-neighbor, $g_{j\alpha}(\mathbf{k},\mathbf{d}_{\nu})$ is symmetrized combination of Fourier functions of the form $e^{i\mathbf{k}\cdot\mathbf{d}_{\nu}}$ which transforms as the $\alpha$-th component of the $j$-th irreducible representation of $D_{3d}^{5}$ double group, $O^{j\alpha}$ is a basis matrix function also transforming as the $\alpha$-th component of the $j$-th irreducible representation of the symmetry group, and $a_{j\alpha\nu}$ is a constant indicating the contribution of this term to the band structure. The above form is uniquely determined by group theory. Once the energy bands at high symmetry points of the BZ is known from experiments or first principle calculations, the coefficients $\{a_{j\alpha\nu}\}$ could be fixed by a minimization procedure. The model determined by the above Slater-Koster method is naturally a tight binding model if the number of relevant $\mathbf{d}_{\nu}$ is finite and $a_{j\alpha\nu}$ for further neighbor contributions are negligible.
For the materials of interest to us here, the most reliable and detailed data about the low energy band structure is around the BZ center, that is the $\boldsymbol{\Gamma}$ point.[@liu10] Since the properties close to the $\boldsymbol{\Gamma}$ point is of most interest to us, this information though insufficient to obtain a model which could produce the correct complete band structure should still be able to provide us a model of the correct global symmetry with correct behavior close to the $\boldsymbol{\Gamma}$ point. On the other hand, we suppose that a tight binding model (TBM) is sufficient to give a good description of the electronic band structure, in terms of $\mathbf{d}_{\nu}$ as short ranged as possible.
The basis matrix functions are formerly constructed by Liu *et al*.[@liu10] They are cited here as the second column of Table II. Their time reversal properties are cited as the third column. In our orbital convention, the $\Gamma$ matrices are defined as $\Gamma_1=s_{1}\otimes\sigma_{3}$, $\Gamma_2=s_{2}\otimes\sigma_{3}$, $\Gamma_3=s_{3}\otimes\sigma_{3}$, $\Gamma_{4}=-s_{0}\otimes\sigma_{2}$, $\Gamma_{5}=s_{0}\otimes\sigma_{1}$, and $\Gamma_{ij}=[\Gamma_{i}, \Gamma_{i}]/2i$ are commutators of corresponding $\Gamma_{i}$.[@liu10; @zhang09] In addition, $I_{4}$ is the 4$\times$4 unit matrix.
[c c c c c]{}\
\[-1.5ex\] Representation & Basis matrices & $T$ & Basis Fourier Functions & $T'$\
\[0.2ex\]\
\[-2ex\] $\tilde{\Gamma}_{1}^{+}$ & $I_{4}$ & $+$ & $1, \frac{1}{3}[\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})+\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})+\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{3})], \cos(\mathbf{k}\cdot\boldsymbol{\delta}_{4})$ & $+$\
\
\[-2ex\] $\tilde{\Gamma}_{1}^{+}$ & $\Gamma_{5}$ & $+$ & $1, \frac{1}{3}[\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})+\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})+\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{3})], \cos(\mathbf{k}\cdot\boldsymbol{\delta}_{4})$ & $+$\
\
\[-2ex\] $\tilde{\Gamma}_{2}^{+}$ & $\Gamma_{12}$ & $-$ & none & none\
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\[-2ex\] $\tilde{\Gamma}_{2}^{+}$ & $\Gamma_{34}$ & $-$ & none & none\
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\[-2ex\] $\tilde{\Gamma}_{3}^{+}$ & {$\Gamma_{13}$, $\Gamma_{23}$} & $-$ & $\{\frac{1}{2}[\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})], \frac{1}{2\sqrt{3}}[\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})+\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})-2\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{3})]\}$ & $+$\
\
\[-2ex\] $\tilde{\Gamma}_{3}^{+}$ & {$\Gamma_{14}$, $\Gamma_{24}$} & $-$ & $\{-\frac{1}{2\sqrt{3}}[\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})+\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})-2\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{3})], \frac{1}{2}[\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})]\}$ & $+$\
\
\[-2ex\] $\tilde{\Gamma}_{1}^{-}$ & $\Gamma_{3}$ & $-$ & $\frac{1}{3}[\sin(\mathbf{k}\cdot\mathbf{a}_{1})+\sin(\mathbf{k}\cdot\mathbf{a}_{2})+\sin(\mathbf{k}\cdot\mathbf{a}_{3})]$ & $-$\
\
\[-2ex\] $\tilde{\Gamma}_{1}^{-}$ & $\Gamma_{35}$ & $+$ & $\frac{1}{3}[\sin(\mathbf{k}\cdot\mathbf{a}_{1})+\sin(\mathbf{k}\cdot\mathbf{a}_{2})+\sin(\mathbf{k}\cdot\mathbf{a}_{3})]$ & $-$\
\
\[-2ex\] $\tilde{\Gamma}_{2}^{-}$ & $\Gamma_{4}$ & $-$ & $\frac{1}{3}[\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})
+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{3})], \sin(\mathbf{k}\cdot\boldsymbol{\delta}_{4})$ & $-$\
\
\[-2ex\] $\tilde{\Gamma}_{2}^{-}$ & $\Gamma_{45}$ & $+$ & $\frac{1}{3}[\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})
+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{3})], \sin(\mathbf{k}\cdot\boldsymbol{\delta}_{4})$ & $-$\
\
\[-2ex\] $\tilde{\Gamma}_{3}^{-}$ & {$\Gamma_{1}$, $\Gamma_{2}$} & $-$ & $\{-\frac{1}{2\sqrt{3}}[\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})-2\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{3})], \frac{1}{2}[\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})]\}$ & $-$\
\
\[-2ex\] $\tilde{\Gamma}_{3}^{-}$ & {$\Gamma_{15}$, $\Gamma_{25}$} & $+$ & $\{-\frac{1}{2\sqrt{3}}[\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})-2\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{3})], \frac{1}{2}[\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})]\}$ & $-$\
The $g_{j\alpha}(\mathbf{k},\mathbf{d}_{\nu})$ function is constructed by projection operators pertaining to various irreducible representations.[@dresselhaus08] Since only the character table is available, we could only construct the character projection operator which generate symmetrized Fourier functions that are linear combinations of the various components of an irreducible representation. The *character projection operator* is defined as $$\hat{P}^{(j)}=\frac{n_{j}}{h}\sum\limits_{R}\chi^{(j)}(R)^{\ast}\hat{P}_{R},$$ where $j$ labels a certain irreducible representation, $n_{j}$ is the dimension of this representation, $h$ is total number of symmetry elements $R$ in the group, $\chi^{(j)}(R)$ is the character of $R$ in the $j$-th irreducible representation, and $\hat{P}_{R}$ is the symmetry operator for the symmetry element $R$. To get a basis pertaining to wave vector $\mathbf{k}$ and the $j$-th irreducible representation in terms of $\mathbf{d}_{\nu}$, we act $\hat{P}^{(j)}$ on $e^{i\mathbf{k}\cdot\mathbf{d}_{\nu}}$. First of all, since $\hat{P}_{\mathcal{C}}$ keeps $e^{i\mathbf{k}\cdot\mathbf{d}_{\nu}}$ invariant, no basis of the prescribed form could be constructed for the six irreducible representations $\tilde{\Gamma}_{4}^{\pm}$, $\tilde{\Gamma}_{5}^{\pm}$, and $\tilde{\Gamma}_{6}^{\pm}$. For $\tilde{\Gamma}_{1}^{+}$, first consider $\mathbf{d}_{0}=(0,0,0)$, we get $\hat{P}^{(\tilde{\Gamma}_{1}^{+})}1=1$. So, a constant could be taken as a basis for (and only for) $\tilde{\Gamma}_{1}^{+}$. Now consider a NN bond $\boldsymbol{\delta}_{1}=(\frac{\sqrt{3}}{2}a, \frac{1}{2}a, 0)$, we have $\hat{P}^{(\tilde{\Gamma}_{1}^{+})}e^{i\mathbf{k}\cdot\boldsymbol{\delta}_{1}}=\frac{1}{3}[\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})
+\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})+\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{3})]$. The result do not change if we replace $\boldsymbol{\delta}_{1}$ by $\boldsymbol{\delta}_{2}$ or $\boldsymbol{\delta}_{3}$. For $\boldsymbol{\delta}_{4}$, we have $\hat{P}^{(\tilde{\Gamma}_{1}^{+})}e^{i\mathbf{k}\cdot\boldsymbol{\delta}_{4}}=\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{4})$. We define the in plane 2NN lattice vectors as $\mathbf{a}_{1}=\boldsymbol{\delta}_{1}-\boldsymbol{\delta}_{2}$, $\mathbf{a}_{2}=\boldsymbol{\delta}_{2}-\boldsymbol{\delta}_{3}$, and $\mathbf{a}_{3}=\boldsymbol{\delta}_{3}-\boldsymbol{\delta}_{1}$. A basis for $\tilde{\Gamma}_{1}^{+}$ could also be constructed in terms of these 2NN bonds, which turns out to be $\frac{1}{3}[\cos(\mathbf{k}\cdot\mathbf{a}_{1})
+\cos(\mathbf{k}\cdot\mathbf{a}_{2})+\cos(\mathbf{k}\cdot\mathbf{a}_{3})]$. However, we restrict to NN bonds if it could be used to construct a basis set, to keep the model minimal.
Basis functions for other representations up to 2NN in plane bonds are similarly constructed. For $\tilde{\Gamma}_{1}^{-}$ representation, a direct calculation shows that $\hat{P}^{(\tilde{\Gamma}_{1}^{-})}e^{i\mathbf{k}\cdot\boldsymbol{\delta}_{l}}=0$ for $l=1,2,3,4$. So no basis could be constructed in terms of the NN bonds. Consider the 2NN bond $\mathbf{a}_{1}$, we have $\hat{P}^{(\tilde{\Gamma}_{1}^{-})}e^{i\mathbf{k}\cdot\mathbf{a}_{1}}=\frac{i}{3}[\sin(\mathbf{k}\cdot\mathbf{a}_{1})
+\sin(\mathbf{k}\cdot\mathbf{a}_{2})+\sin(\mathbf{k}\cdot\mathbf{a}_{3})]$. For $\tilde{\Gamma}_{2}^{+}$ representation, calculation shows that no basis could be constructed up to 2NN bonds. But since the $\tilde{\Gamma}_{2}^{+}$ representation does not appear in the $\mathbf{k}$$\cdot$$\mathbf{p}$ model[@liu10], we would still restrict our model within 2NN bonds. For $\tilde{\Gamma}_{2}^{-}$ representation, we get $\hat{P}^{(\tilde{\Gamma}_{2}^{-})}e^{i\mathbf{k}\cdot\boldsymbol{\delta}_{1}}=\frac{i}{3}[\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})
+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{3})]$, $\hat{P}^{(\tilde{\Gamma}_{2}^{-})}e^{i\mathbf{k}\cdot\boldsymbol{\delta}_{4}}=i\sin(\mathbf{k}\cdot\boldsymbol{\delta_{4}})$, and $\hat{P}^{(\tilde{\Gamma}_{2}^{-})}e^{i\mathbf{k}\cdot\mathbf{a}_{l}}=0$ ($l=1,2,3$).
$\tilde{\Gamma}_{3}^{\pm}$ are two dimensional representations. For them, the character projection operators would in general generate a linear combination of two basis functions when operating it on an arbitrary Fourier exponential. If an arbitrary set of basis functions are required, we could take this as one basis and generate another basis which is orthogonal to it to form a basis set. However, since we would form invariants in terms of these symmetrized Fourier functions and the basis matrix functions, the two sets of basis functions should transform identically under the group operation. Enforcing this requirement, we could get the proper sets of basis Fourier functions which have the same group transformation properties as the corresponding basis matrix functions. The symmetrized Fourier functions are thus as shown in the fourth column of Table II. The time reversal property of the basis Fourier functions are as shown in the fifth column under the title of $T'$. In the fourth column of Table II, two functions in a single brace form a basis set for the corresponding two dimensional representation. The non-braced functions are optional bases for the corresponding one dimensional representation.
Having the basis matrix functions and the symmetrized Fourier functions at hand, the tight binding model is constructed by multiplying the corresponding components of the two together to form invariants of the symmetry group. Since the material and hence the model for it preserves time reversal symmetry, the terms to be multiplied together to form invariants should also have the same $T$ value. We are thus lead by the requirement of group symmetry, time reversal invariance, and Hermiticity to write $H_{0}(\mathbf{k})$ in the most general form (with the restriction of keeping only short-range hoppings) as $$\begin{aligned}
H_{0}(\mathbf{k})&=&\epsilon(\mathbf{k})I_{4}+M(\mathbf{k})\Gamma_5+B_{0}c_{z}(\mathbf{k})\Gamma_{4}
+A_{0}[c_{y}(\mathbf{k})\Gamma_{1} \notag \\
&&-c_{x}(\mathbf{k})\Gamma_{2}]+R_{1}d_{1}(\mathbf{k})\Gamma_{3}+R_{2}d_{2}(\mathbf{k})\Gamma_{4}.\end{aligned}$$ Definitions of the various terms are the same as in the main text.
To determine the parameters in the model, we compare the model with the $\mathbf{k}\cdot\mathbf{p}$ model defined close to $\boldsymbol{\Gamma}=(0,0,0)$. So, we expand the various terms close to $\boldsymbol{\Gamma}$ as $1-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{4})\simeq\frac{1}{2}k_{z}^{2}c^{2}$, $3-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{3})\simeq\frac{1}{2}(k_{x}^{2}+k_{y}^{2})a^{2}$, $\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{4})\simeq k_{z}c$, $\frac{1}{3}[\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})-2\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{3})]
\simeq k_{y}a$, $\frac{1}{\sqrt{3}}[\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})]\simeq k_{x}a$, $\sin(\mathbf{k}\cdot\mathbf{a}_{1})+\sin(\mathbf{k}\cdot\mathbf{a}_{2})+\sin(\mathbf{k}\cdot\mathbf{a}_{3})\simeq \frac{3\sqrt{3}}{8}(3k_{x}k_{y}^{2}-k_{x}^{3})a^{3}$, and $\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{3})
\simeq\frac{1}{8}(k_{y}^{3}-3k_{x}^{2}k_{y})a^{3}$. Substituting these approximations into the above model, it clearly has a form identical to the $\mathbf{k}$$\cdot$$\mathbf{p}$ model proposed by Liu *et al*.[@liu10] Demanding that our tight binding model reduce to the same model as that used by Liu *et al*, we could derive the values of the various parameters as shown in Table I of the main text. A calculation of the bulk density of states shows that the bulk energy gaps of Bi$_{2}$Se$_{3}$ and Bi$_{2}$Te$_{3}$ corresponding to the above parameters are both too small to be comparable to experiment. Test calculations show that changing $M_{1}$ to 0.62 eV for Bi$_2$Se$_3$ and 0.102 eV for Bi$_2$Te$_3$ and keeping other parameters unchanged, the bulk energy gaps of Bi$_2$Se$_3$ and Bi$_2$Te$_3$ are approximately 0.26 eV and 0.06 eV, which are close to known experimental and first principle theoretical results.[@zhang09]
\[sec:exchange\]Superexchange Coupling Terms
============================================
We derive the dominant superexchange couplings that could arise from the model proposed in the main text. The model combines the tight binding model and the short range correlation terms and is written as $H=H_{0}+H_{1}$, with $$H_{1}=U\sum\limits_{\mathbf{i}\alpha}\hat{n}_{\mathbf{i}\alpha\uparrow}\hat{n}_{\mathbf{i}\alpha\downarrow}
+V\sum\limits_{\mathbf{i}}\hat{n}_{\mathbf{i}}^{a}\hat{n}_{\mathbf{i}}^{b},$$ in which $\alpha$ runs over the two orbitals $a$ and $b$. While the actual strength of the on-site correlations for Bi$_{2}$X$_{3}$ (X is Se or Te) materials are presently unknown, evidences have appeared that they should be sizeable to explain the experimental findings.[@craco12; @giraud11] In light of this information, we could consider a strongly correlated system when deriving the low energy effective model. Working with the two orbital model $H_{0}$, the materials are doped semiconductors close to half filling. We thus generalize the routine procedure of deriving the $t-J$ model from the one orbital Hubbard model to the present two-orbital model [@schrieffer66], first get the effective model at half filling and then dope the model to approximately represent the actual materials.
Since the two orbitals within a single unit cell reside on different sites, we expect that $U>V>0$ holds. At half filling and for sufficiently large $U$ and $V$, the system would in most of the time be restricted within the subspace of local intraorbital single occupation. Projecting out the subspace with doubly occupied orbitals, we have[@schrieffer66] $$\tilde{H}=PHP-\frac{1}{U}PHQHP,$$ where $P$ and $Q$ are projection operators which project into the subspace of intraorbital no-double-occupancy and the subspace with intraorbital double occupancy (and thus with empty orbital in the case of half filling of interest), respectively. $P^2=P$ and $Q^2=Q=1-P$. In the present two orbital systems, virtual hoppings could be intraorbital or interorbital, the excitation energy of which are both $U$. The second term in $\tilde{H}$ is the exchange term arising from mixing of the two subspaces and would be denoted as $H_{ex}$. Since $H_{0}$ has many terms, we expect that a lot of different terms would appear for $H_{ex}$ when the multiplication is carried out. However, we note that the parameters in Table I of the main text show very large differences. The different superexchange terms thus derived would also show very large differences. Since we are interested in finding out the dominant pairing channel, it is sufficient to retain only the largest several superexchange terms mediated by the hopping terms corresponding to the largest several parameters in Table I (the main text). We thus retain the superexchange terms mediated by $C_2$, $M_2$ and $R_2$, which are apparently larger than other parameters in Table I (the main text). As a further approximation, we retain only two-site terms. Since before the virtual hop or after two complementary virtual hops the system only has singly occupied orbitals, the two complementary hop must both be intra-orbital or inter-orbital.
Straightforward deductions show that $C_{2}$ mediates an intraorbital superexchange $$H_{intra}=\frac{8C_{2}^{2}}{9U}\sum\limits_{\mathbf{i}\boldsymbol{\delta}\alpha}
(\mathbf{S}_{\mathbf{i}\alpha}\cdot\mathbf{S}_{\mathbf{i}+\boldsymbol{\delta},\alpha}
-\frac{1}{4}\hat{n}_{\mathbf{i}}^{\alpha}\hat{n}_{\mathbf{i}+\boldsymbol{\delta}}^{\alpha}),$$ where $\mathbf{i}$ runs over unit cells, $\boldsymbol{\delta}$ runs over the six NN in plane bonds $\pm\boldsymbol{\delta}_{j}$ ($j$=1, 2, 3), and the $\alpha$ summation runs over the two orbitals. $\hat{n}_{\mathbf{i}}^{\alpha}=\hat{n}_{\mathbf{i}\alpha\uparrow}+\hat{n}_{\mathbf{i}\alpha\downarrow}$ is the electron number operator for $\alpha$ orbital. Written as above, the single occupation condition per unit cell and per orbital is imposed. That is, we use $\hat{n}_{\mathbf{i}\alpha\uparrow}+\hat{n}_{\mathbf{i}\alpha\downarrow}=1$ to simplify terms. For example, $(1-\hat{n}_{\mathbf{i}\alpha\bar{\sigma}})a_{\mathbf{i}\sigma}^{\dagger}\simeq \hat{n}_{\mathbf{i}\alpha\sigma}a_{\mathbf{i}\sigma}^{\dagger}=a_{\mathbf{i}\sigma}^{\dagger}$ ($\bar{\sigma}$ is the opposite spin of $\sigma$).
Following the same convention, the superexchange terms mediated by $M_2$ and $R_2$ are obtained $$H_{M_{2}}=\frac{(4M_{2})^2}{9U}\sum\limits_{\mathbf{i}\boldsymbol{\delta}}
(\mathbf{S}_{\mathbf{i}a}\cdot\mathbf{S}_{\mathbf{i}+\boldsymbol{\delta},b}
-\frac{1}{4}\hat{n}_{\mathbf{i}}^{a}\hat{n}_{\mathbf{i}+\boldsymbol{\delta}}^{b}),$$ $$H_{R_{2}}=\frac{(8R_{2})^2}{U}\sum\limits_{\mathbf{i}\boldsymbol{\delta}}
(\mathbf{S}_{\mathbf{i}a}\cdot\mathbf{S}_{\mathbf{i}+\boldsymbol{\delta},b}
-\frac{1}{4}\hat{n}_{\mathbf{i}}^{a}\hat{n}_{\mathbf{i}+\boldsymbol{\delta}}^{b}).$$ Besides the the above two terms, the crossing terms between $M_2$ and $R_2$ also mediate a superexchange term in the same interorbital channel, which turns out to be $$H_{M_2R_2}=\frac{64M_{2}R_{2}}{3U}\sum\limits_{\mathbf{i}\boldsymbol{\delta}}
(-1)^{\eta(\boldsymbol{\delta})}(\mathbf{S}_{\mathbf{i}a}\cdot\mathbf{S}_{\mathbf{i}+\boldsymbol{\delta},b}
-\frac{1}{4}\hat{n}_{\mathbf{i}}^{a}\hat{n}_{\mathbf{i}+\boldsymbol{\delta}}^{b}).$$ $\eta(\boldsymbol{\delta})$ is defined as $0$ ($1$) for $\boldsymbol{\delta}=\boldsymbol{\delta}_{j}$ ($\boldsymbol{\delta}=-\boldsymbol{\delta}_{j}$), with $j$=1, 2, 3. Arising from the crossing term between $M_{2}$ and $R_{2}$, $\eta(\boldsymbol{\delta})$ inherits the bond-wise sign change character from $R_{2}$. Whereas the sign change in $R_{2}$ hopping is canceled out in the $R_{2}^{2}$ term. It is clear that the coefficients of the above three terms combine nicely into a square term, written as $J^{ab}_{\mathbf{i},\mathbf{i}+\boldsymbol{\delta}}=
\frac{(8R_{2})^2}{U}[1+(-1)^{\eta(\boldsymbol{\delta})}\frac{M_{2}}{6R_{2}}]^{2}$. Thus the interorbital superexchange term is written as $$H_{inter}=\sum\limits_{\mathbf{i}\boldsymbol{\delta}}
J^{ab}_{\mathbf{i},\mathbf{i}+\boldsymbol{\delta}}
(\mathbf{S}_{\mathbf{i}a}\cdot\mathbf{S}_{\mathbf{i}+\boldsymbol{\delta},b}
-\frac{1}{4}\hat{n}_{\mathbf{i}}^{a}\hat{n}_{\mathbf{i}+\boldsymbol{\delta}}^{b}).$$ Since the coupling constants in both $H_{intra}$ and $H_{inter}$ are positive, the above superexchange terms favor only spin singlet pairings. The other hopping terms in $H_{0}$, such as the $A_{0}$ term, might favor triplet pairings. A similar derivation shows that it is indeed the case. However, since it is much smaller than the above two terms, the pairing favored by it can not survive the competition. On the other hand, though the spin-orbit interaction in $H_{0}$ mixes spin singlet and spin triplet components, the induced spin triplet component is always much smaller than the spin singlet component.
Finally, we mention that the interorbital constant term in $H_{0}(\mathbf{k})$, that is $(M_{0}+2M_{1}+4M_{2})\Gamma_{5}$, also mediates an intra-unit-cell interorbital superexchange term. This term is not negligible compared to the terms retained in the main text. However, this term favors a $\mathbf{k}$-independent interorbital spin singlet pairing, which is equivalent to the third or fourth pairing defined in the main text by setting $\varphi_{3(4)}(\mathbf{k})=1$ and was already studied in previous works [@fu10; @hao11]. So, we have ignored this superexchange term in this study which focuses on anisotropic spin singlet pairings. But, as mentioned in the main text, this fully gapped interorbital spin singlet pairing component could readily be incorporated into our framework to account for some experimental features, such as the specific heat experiments which indicates the presence of a fully gapped pairing component.[@kriener11]
\[sec:MFA\] Pairing Symmetry and Mean Field Calculations
========================================================
With the above model at hand, the simplest and fastest way to evaluate the most probable pairing symmetry is to perform a mean field calculation.[@seo08; @kotliar88; @goswami10] Here, we show some details of the mean field calculations and analysis of the results. In order to derive proper superexchange type of terms that favor pairing, we have imposed a condition of strong correlation. However, no firm evidences showing that the system belongs to strong correlation *limit* have appeared. So, we do not impose the projection of $P$ on $H_{0}$. Instead, we keep the on-site correlation terms together with $H_{0}$ unchanged to qualitatively reflect the constraint on intraorbital double occupancy. Thus, we use a $t-U-V-J$ model, that is $H=H_{0}+H_{1}+H_{ex}$, as a starting point to perform the mean field analyses.
At the mean filed level, pairing comes only from $H_{ex}$. In a full mean field decoupling over $H_{ex}$, we would get not only pairing terms but also terms for magnetic orders and various bond orders. However, since on one hand we are here interested only in the superconducting phase and on the other hand no evidences for the existence of other phases have appeared, we would keep only the decoupling channel that leads to pairing. In terms of the identity $\boldsymbol{\sigma}_{\alpha\beta}\cdot\boldsymbol{\sigma}_{\alpha'\beta'}=2\delta_{\alpha\beta'}\delta_{\alpha'\beta}
-\delta_{\alpha\beta}\delta_{\alpha'\beta'}$ for the scalar product of two Pauli vectors, we get $$\begin{aligned}
&&\mathbf{S}_{\mathbf{i}a}\cdot\mathbf{S}_{\mathbf{i}+\boldsymbol{\delta},a}
-\frac{1}{4}\hat{n}_{\mathbf{i}}^{a}\hat{n}_{\mathbf{i}+\boldsymbol{\delta}}^{a} \notag \\
&&=-\frac{1}{2}(a_{\mathbf{i}\uparrow}^{\dagger}a_{\mathbf{i}+\boldsymbol{\delta},\downarrow}^{\dagger}
-a_{\mathbf{i}\downarrow}^{\dagger}a_{\mathbf{i}+\boldsymbol{\delta},\uparrow}^{\dagger})
(a_{\mathbf{i}+\boldsymbol{\delta},\downarrow}a_{\mathbf{i}\uparrow}
-a_{\mathbf{i}+\boldsymbol{\delta},\uparrow}a_{\mathbf{i}\downarrow}), \notag \\
&&\mathbf{S}_{\mathbf{i}b}\cdot\mathbf{S}_{\mathbf{i}+\boldsymbol{\delta},b}
-\frac{1}{4}\hat{n}_{\mathbf{i}}^{b}\hat{n}_{\mathbf{i}+\boldsymbol{\delta}}^{b} \notag \\
&&=-\frac{1}{2}(b_{\mathbf{i}\uparrow}^{\dagger}b_{\mathbf{i}+\boldsymbol{\delta},\downarrow}^{\dagger}
-b_{\mathbf{i}\downarrow}^{\dagger}b_{\mathbf{i}+\boldsymbol{\delta},\uparrow}^{\dagger})
(b_{\mathbf{i}+\boldsymbol{\delta},\downarrow}b_{\mathbf{i}\uparrow}
-b_{\mathbf{i}+\boldsymbol{\delta},\uparrow}b_{\mathbf{i}\downarrow}), \notag \\
&&\mathbf{S}_{\mathbf{i}a}\cdot\mathbf{S}_{\mathbf{i}+\boldsymbol{\delta},b}
-\frac{1}{4}\hat{n}_{\mathbf{i}}^{a}\hat{n}_{\mathbf{i}+\boldsymbol{\delta}}^{b} \notag \\
&&=-\frac{1}{2}(a_{\mathbf{i}\uparrow}^{\dagger}b_{\mathbf{i}+\boldsymbol{\delta},\downarrow}^{\dagger}
-a_{\mathbf{i}\downarrow}^{\dagger}b_{\mathbf{i}+\boldsymbol{\delta},\uparrow}^{\dagger})
(b_{\mathbf{i}+\boldsymbol{\delta},\downarrow}a_{\mathbf{i}\uparrow}
-b_{\mathbf{i}+\boldsymbol{\delta},\uparrow}a_{\mathbf{i}\downarrow}), \notag \\\end{aligned}$$ where $\boldsymbol{\delta}=\pm\boldsymbol{\delta}_{1}$, $\pm\boldsymbol{\delta}_{2}$, and $\pm\boldsymbol{\delta}_{3}$. Since the coefficients for the intraorbital and interorbital superexchange couplings are both positive, the above decomposition makes it clear that $H_{ex}$ favors and only favors spin singlet pairings. Take advantage of the above expressions, eighteen independent mean field parameters are introduced to describe the superconducting pairing. Firstly, we define twelve operators $\hat{\chi}_{a}^{\nu\pm}=
a_{\mathbf{i}\pm\boldsymbol{\delta}_{\nu},\downarrow}a_{\mathbf{i}\uparrow}
-a_{\mathbf{i}\pm\boldsymbol{\delta}_{\nu},\uparrow}a_{\mathbf{i}\downarrow}$ ($\nu=1, 2, 3$) and $\hat{\chi}_{b}^{\nu\pm}=
b_{\mathbf{i}\pm\boldsymbol{\delta}_{\nu},\downarrow}b_{\mathbf{i}\uparrow}
-b_{\mathbf{i}\pm\boldsymbol{\delta}_{\nu},\uparrow}b_{\mathbf{i}\downarrow}$ ($\nu=1, 2, 3$). Their expectation values, $\chi_{a}^{\nu\pm}=\langle\hat{\chi}_{a}^{\nu\pm}\rangle$ and $\chi_{b}^{\nu\pm}=\langle\hat{\chi}_{b}^{\nu\pm}\rangle$, define the twelve mean field parameters for the intraorbital spin singlet pairing. Then we define another six operators $\hat{\chi}_{ba}^{\nu\pm}=
b_{\mathbf{i}\pm\boldsymbol{\delta}_{\nu},\downarrow}a_{\mathbf{i}\uparrow}
-b_{\mathbf{i}\pm\boldsymbol{\delta}_{\nu},\uparrow}a_{\mathbf{i}\downarrow}$ ($\nu=1, 2, 3$). Their expectation values, $\chi_{ba}^{\nu\pm}=\langle\hat{\chi}_{ba}^{\nu\pm}\rangle$, define the six mean field parameters for the interorbital spin singlet pairing.
Retaining only the decoupling to the superconducting channel, we make the mean field approximation to $H_{ex}$ in terms of $(\hat{\chi}_{\alpha}^{\nu\pm})^{\dagger}\hat{\chi}_{\alpha}^{\nu\pm}\simeq(\chi_{\alpha}^{\nu\pm})^{\ast}\hat{\chi}_{\alpha}^{\nu\pm}
+(\hat{\chi}_{\alpha}^{\nu\pm})^{\dagger}\chi_{\alpha}^{\nu\pm}-(\chi_{\alpha}^{\nu\pm})^{\ast}\chi_{\alpha}^{\nu\pm}$ ($\alpha$ is $a$ or $b$), and $(\hat{\chi}_{ba}^{\nu\pm})^{\dagger}\hat{\chi}_{ba}^{\nu\pm}\simeq(\chi_{ba}^{\nu\pm})^{\ast}\hat{\chi}_{ba}^{\nu\pm}
+(\hat{\chi}_{ba}^{\nu\pm})^{\dagger}\chi_{ba}^{\nu\pm}-(\chi_{ba}^{\nu\pm})^{\ast}\chi_{ba}^{\nu\pm}$.[@seo08; @kotliar88; @goswami10] For $H_{1}$, it is easy to see that it does not favor superconducting state at the mean field level. We make the mean field decoupling to $H_{1}$ in the simplest way as $\hat{n}_{1}\hat{n}_{2}\rightarrow\langle\hat{n}_{1}\rangle\hat{n}_{2}+\hat{n}_{1}\langle\hat{n}_{2}\rangle
-\langle\hat{n}_{1}\rangle\langle\hat{n}_{2}\rangle$, which introduces four mean field parameters $n_{\alpha\sigma}$ ($\alpha$ is for the $a$ or $b$ orbital, $\sigma$ is for the $\uparrow$ or $\downarrow$ spin).
After the above mean field decoupling, the Hamiltonian is now a bilinear of electron operators and is easily transformed into the reciprocal space. Then the mean field calculation is performed in a self-consistent manner starting from an arbitrary set of initial values for the $18$ spin singlet pairing amplitudes and $4$ on-site occupation numbers. The mean field calculation turns out to converge very well. When convergence is arrived at for a certain set of parameters, we analyze the pairings that are contained in the results.
For the spin singlet solution obtained by the above self-consistent mean field calculation, we are interested in those pairing components contained in it which hold symmetry compatible with the crystal symmetry. In addition, we focus on time reversal invariant pairings. At first sight, the spin singlet pairing terms in the superconducting Hamiltonian are to be constructed completely parallel to the construction of $H_{0}(\mathbf{k})$. However, since the 4$\times$4 pairing term which appears in the off-diagonal position of the Bogoliubov-de Gennes (BdG) Hamiltonian needs *not* be Hermitian, the number of possible time reversal invariant combinations for it is increased. A direct survey over Table II shows that the four one dimensional representations $\tilde{\Gamma}_{1}^{\pm}$ and $\tilde{\Gamma}_{2}^{\pm}$ all describes spin triplet pairings, so drop out of our present analysis. For the remaining two dimensional representations $\tilde{\Gamma}_{3}^{\pm}$, it is interesting that for each of the four realizations of them, one basis matrix function corresponds to spin singlet pairing while the other basis matrix function corresponds to spin triplet pairing. For example, for the $\{\Gamma_{13},\Gamma_{23}\}$ realization of $\tilde{\Gamma}_{3}^{+}$, $\Gamma_{13}=-s_{2}\otimes\sigma_{0}$ is in the spin singlet channel while $\Gamma_{23}=s_{1}\otimes\sigma_{0}$ describes spin triplet pairing. In addition, $\Gamma_{24}=s_{2}\otimes\sigma_{1}$ for $\tilde{\Gamma}_{3}^{+}$, $\Gamma_{2}=s_{2}\otimes\sigma_{3}$ and $\Gamma_{25}=s_{2}\otimes\sigma_{2}$ for $\tilde{\Gamma}_{3}^{-}$ all belong to the spin singlet channel.
The coexistence of spin singlet and spin triplet pairing components in a single irreducible representation might be a direct result of the presence of spin-orbit interaction, which makes the pairing with a definite spin state not well defined. That is, even though the correlation term favors only spin singlet pairing, some spin triplet pairing component would be induced from the spin singlet pairing by the spin-orbit interaction. In the present study, we do not analyze the spin triplet pairings induced by the spin-orbit interaction and concentrate on the dominant spin singlet pairing components. So, we have four basis matrices pertaining to two irreducible representations that are possibly of interest. The wave vector dependence of the pairings are taken from the basis Fourier functions in Table II. Since now only one component of the basis matrix functions exists for a certain set of the two dimensional representation, we consider both of the two basis Fourier functions as possible candidates, since a symmetry transformation would mix the two basis Fourier functions and the absence of spin triplet pairing then leaves a product of a linear combination of the two basis Fourier functions with the spin singlet basis matrix function.
In the above convention, we would get eight independent spin singlet pairings. The symmetry factors ($\mathbf{k}$-dependency) of the pairings are defined as $\varphi_{1}(\mathbf{k})=\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})$ and $\varphi_{2}(\mathbf{k})=2\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{3})-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})
-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})$, $\varphi_{3(4)}(\mathbf{k})=\varphi_{1(2)}(\mathbf{k})$, $\varphi_{5}(\mathbf{k})=\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})$ and $\varphi_{6}(\mathbf{k})=\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})+\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})
-2\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{3})$, $\varphi_{7(8)}(\mathbf{k})=\varphi_{5(6)}(\mathbf{k})$. The first two pairings correspond to $\Gamma_{31}=-\Gamma_{13}$ and are $\Delta_{j}\phi_{\mathbf{k}}^{\dagger}i\Gamma_{31}(\phi_{-\mathbf{k}}^{\dagger})^{\text{T}}\varphi_{j}(\mathbf{k})$ ($j$=1, 2). The next two pairings correspond to $\Gamma_{24}$ and are $\Delta_{j}\phi_{\mathbf{k}}^{\dagger}i\Gamma_{24}(\phi_{-\mathbf{k}}^{\dagger})^{\text{T}}\varphi_{j}(\mathbf{k})$ ($j$=3, 4). The fifth and sixth pairings corresponds to $\Gamma_{25}$ and are $\Delta_{j}\phi_{\mathbf{k}}^{\dagger}i\Gamma_{25}(\phi_{-\mathbf{k}}^{\dagger})^{\text{T}}\varphi_{j}(\mathbf{k})$ ($j$=5, 6). The last two pairings correspond to $\Gamma_{2}$, and are written as $\Delta_{j}\phi_{\mathbf{k}}^{\dagger}\Gamma_{2}(\phi_{-\mathbf{k}}^{\dagger})^{\text{T}}\varphi_{j}(\mathbf{k})$ ($j$=7, 8). $\Delta_{j}$ ($j=1,\ldots,8$) are the amplitudes of the corresponding pairing components. However, since the summation of the last two pairings over wave vectors in the BZ vanishes, that is $\sum_{\mathbf{k}}\phi_{\mathbf{k}}^{\dagger}\Gamma_{2}(\phi_{-\mathbf{k}}^{\dagger})^{\text{T}}\varphi_{j}(\mathbf{k})=0$ ($j$=7, 8), we only have six spin singlet time reversal invariant parings that are compatible with crystal symmetry and are anisotropic.
For a set of parameters such as a specific $U$ and $V$ and doping level, after obtaining the $18$ pairing order parameters ($\chi_{a}^{\nu\pm}$, $\chi_{b}^{\nu\pm}$, and $\chi_{ba}^{\nu\pm}$, $\nu$=$1$, $2$, $3$) through self-consistent calculations, we could extract the amplitudes for the six symmetry channels defined above.[@seo08; @kotliar88; @goswami10] That is, we could express the value of $\Delta_{i}$ ($i$=$1$, $\ldots$, $6$) in terms of the $18$ mean field pairing order parameters. Since the six pairings are mutually independent, we could get the representation of their amplitudes in terms of all the self-consistent pairing fields. This is as much as to say, when extracting the pairing amplitude of a certain channel among the six possibilities, we regard it as the only pairing that is contained in the self-consistent solution. First consider $\Delta_{1}$. The upper-right $4\times4$ block of the BdG Hamiltonian for this pairing is written explicitly as $$\begin{aligned}
&&\Delta_{1}\sum\limits_{\mathbf{k}}\phi^{\dagger}_{\mathbf{k}}i\Gamma_{13}(\phi^{\dagger}_{-\mathbf{k}})^{\text{T}}\varphi_{1}(\mathbf{k}) \notag \\
&&=\Delta_{1}\sum\limits_{\mathbf{k}}[-a^{\dagger}_{\mathbf{k}\uparrow}a^{\dagger}_{-\mathbf{k}\downarrow}+a^{\dagger}_{\mathbf{k}\downarrow}a^{\dagger}_{-\mathbf{k}\uparrow}
-b^{\dagger}_{\mathbf{k}\uparrow}b^{\dagger}_{-\mathbf{k}\downarrow}+b^{\dagger}_{\mathbf{k}\downarrow}b^{\dagger}_{-\mathbf{k}\uparrow}] \notag \\
&& \cdot[\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})-\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})] \notag \\
&&=\frac{1}{2}\Delta_{1}\sum\limits_{\mathbf{i}}[(a^{\dagger}_{\mathbf{i}\uparrow}a^{\dagger}_{\mathbf{i}+\boldsymbol{\delta}_{2},\downarrow}
-a^{\dagger}_{\mathbf{i}\downarrow}a^{\dagger}_{\mathbf{i}+\boldsymbol{\delta}_{2},\uparrow}
+a^{\dagger}_{\mathbf{i}\uparrow}a^{\dagger}_{\mathbf{i}-\boldsymbol{\delta}_{2},\downarrow} \notag \\
&&-a^{\dagger}_{\mathbf{i}\downarrow}a^{\dagger}_{\mathbf{i}-\boldsymbol{\delta}_{2},\uparrow}
-a^{\dagger}_{\mathbf{i}\uparrow}a^{\dagger}_{\mathbf{i}+\boldsymbol{\delta}_{1},\downarrow}
+a^{\dagger}_{\mathbf{i}\downarrow}a^{\dagger}_{\mathbf{i}+\boldsymbol{\delta}_{1},\uparrow} \notag \\
&&-a^{\dagger}_{\mathbf{i}\uparrow}a^{\dagger}_{\mathbf{i}-\boldsymbol{\delta}_{1},\downarrow}
+a^{\dagger}_{\mathbf{i}\downarrow}a^{\dagger}_{\mathbf{i}-\boldsymbol{\delta}_{1},\uparrow})+(a\rightarrow b)].\end{aligned}$$ The above pairing term is then compared with the mean field decoupling to $H_{ex}$. Requiring the identical terms to be equal to each other, we obtain the representation of $\Delta_{1}$ in terms of the mean field pairing amplitudes as $$\begin{aligned}
\Delta_{1}&=&\frac{1}{8}J_{1}[\chi_{a}^{1+}+\chi_{a}^{1-}-\chi_{a}^{2+}-\chi_{a}^{2-} \notag \\
&&+\chi_{b}^{1+}+\chi_{b}^{1-}-\chi_{b}^{2+}-\chi_{b}^{2-}],\end{aligned}$$ where $J_{1}=\frac{8C_{2}^{2}}{9U}$. Furthermore, since we consider a uniform solution, the pairing fields are independent of lattice sites. So the above value for $\Delta_{1}$ could be expressed in the wave vector representation as $$\Delta_{1}=\frac{J_{1}}{2N}\sum\limits_{\mathbf{k}}d(\mathbf{k})\varphi_{1}(\mathbf{k}),$$ where $N$ is number of unit cells in the lattice, $d(\mathbf{k})=\langle a_{-\mathbf{k}\downarrow}a_{\mathbf{k}\uparrow}+b_{-\mathbf{k}\downarrow}b_{\mathbf{k}\uparrow}\rangle$.
Amplitudes for the other five pairings could similarly be extracted from a solution of the $18$ mean field pairings. They are written in the wave vector representation as follows. $$\Delta_{2}=\frac{J_{1}}{6N}\sum\limits_{\mathbf{k}}d(\mathbf{k})\varphi_{2}^{'}(\mathbf{k}),$$ where $\varphi_{2}^{'}(\mathbf{k})=\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{3})-2\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{1})
-2\cos(\mathbf{k}\cdot\boldsymbol{\delta}_{2})$. $$\Delta_{3}=-\frac{(2M_{2})^2}{9NU}\sum\limits_{\mathbf{k}}d^{'}(\mathbf{k})[(1+\frac{36R_{2}^{2}}{M^{2}})\varphi_{3}(\mathbf{k})
-i\frac{12R_{2}}{M_{2}}\varphi_{5}(\mathbf{k})],$$ where $d^{'}(\mathbf{k})=\langle b_{-\mathbf{k}\downarrow}a_{\mathbf{k}\uparrow}-b_{-\mathbf{k}\uparrow}a_{\mathbf{k}\downarrow}\rangle$. $$\Delta_{4}=-\frac{(2M_{2})^2}{27NU}\sum\limits_{\mathbf{k}}d^{'}(\mathbf{k})[(1+\frac{36R_{2}^{2}}{M_{2}^{2}})\varphi_{4}^{'}(\mathbf{k})
+i\frac{24R_{2}}{M_{2}}\varphi_{6}^{'}(\mathbf{k})],$$ where $\varphi_{4}^{'}(\mathbf{k})=\varphi_{2}^{'}(\mathbf{k})$ and $\varphi_{6}^{'}(\mathbf{k})=2\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{1})+2\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{2})
-\sin(\mathbf{k}\cdot\boldsymbol{\delta}_{3})$. $$\Delta_{5}=-\frac{(2M_{2})^2}{9NU}\sum\limits_{\mathbf{k}}d^{'}(\mathbf{k})[i(1+\frac{36R_{2}^{2}}{M_{2}^{2}})\varphi_{5}(\mathbf{k})
-\frac{12R_{2}}{M_{2}}\varphi_{3}(\mathbf{k})].$$ $$\Delta_{6}=-\frac{(2M_{2})^{2}}{27NU}\sum\limits_{\mathbf{k}}d^{'}(\mathbf{k})[i(1+\frac{36R_{2}^{2}}{M_{2}^{2}})\varphi_{6}^{'}(\mathbf{k})
+\frac{12R_{2}}{M_{2}}\varphi_{4}^{'}(\mathbf{k})].$$
Two interesting features are clear in the above equations that determine $\Delta_1$ to $\Delta_6$. The first is that $\varphi_{i}^{'}(\mathbf{k})\neq\varphi_{i}(\mathbf{k})$ for $i$=2, 4, 6. From the above procedure of determining these pairing amplitudes we know that this is because the pairings with symmetry factors $\varphi_{2}(\mathbf{k})$, $\varphi_{4}(\mathbf{k})$ and $\varphi_{6}(\mathbf{k})$ are unequal amplitude superposition of $e^{i\mathbf{k}\cdot\boldsymbol{\delta}_{l}}$ ($l$=$\pm1$, $\pm2$, $\pm3$) factors. That is, the pairing fields on the $\pm\boldsymbol{\delta}_{3}$ bonds are stronger than those on the $\pm\boldsymbol{\delta}_{1}$ and $\pm\boldsymbol{\delta}_{2}$ bonds. The second feature is also discussed in the main text, that is the pairings $\Delta_{3}$ is explicitly mixed with $\Delta_{5}$ while $\Delta_{4}$ is explicitly mixed with $\Delta_{6}$. We have defined the concept of spatial-parity and orbital-parity in the main text. The above mixing of two kinds of pairings with opposite spatial-parity is a consequence of the interorbital superexchange term $H_{inter}$, which breaks explicitly the in-plane inversion symmetry of the correlation term. Another consequence of $H_{inter}$ is that, since the pairing potential is a time reversal invariant combination of even spatial-parity and odd spatial-parity components (see Eq.(5) of the main text), the self-consistent mean field superconducting solution consisting of several pairing components are time reversal invariant up to a global $U(1)$ phase. A time reversal invariant multi-component superconducting state was also found for iron pnictides.[@seo08]
\[sec:iGF\] The iterative Green’s function method
=================================================
Here, we explain how we get the surface Green’s functions (GFs) in terms of the iterative GF method, which produce Figures 1 and 2 of the main text. First, we add a pairing term $\underline{\Delta}(\mathbf{k})$ to the normal state Hamiltonian $H_{0}(\mathbf{k})$. $\underline{\Delta}(\mathbf{k})$ can be one of the six spin singlet pairings defined in Sec.III (or, in the main text) or a specific linear combination of several pairing components. Introducing the Nambu basis $\psi^{\dagger}_{\mathbf{k}}=[\phi^{\dagger}_{\mathbf{k}},(\phi_{-\mathbf{k}})^{\textbf{T}}]$, we get the Bogoliubov-de Gennes (BdG) Hamiltonian as $$H(\mathbf{k})=
\begin{pmatrix} H_{0}(\mathbf{k})-\mu I_{4} & \underline{\Delta}(\mathbf{k}) \\
-\underline{\Delta}^{\ast}(-\mathbf{k}) & \mu I_{4}-H^{\ast}_{0}(-\mathbf{k})
\end{pmatrix},$$ where $\mu$ is the chemical potential and $I_{4}$ is the fourth-order unit matrix. The bulk GF is defined simply as $G_{b}(\mathbf{k},\omega)=[(\omega+i\eta)I_{8}-H(\mathbf{k})]^{-1}$, where $\eta$ is the positive infinitesimal and $I_{8}$ is the eighth-order unit matrix. In actual calculations, $\eta$ is taken as a small finite positive number (e.g., 10$^{-5}$ eV is used in this work).
To study the surface states living on the $xy$ surface, we consider a sample of the Bi$_{2}$X$_{3}$ (X is Se or Te) superconductor occupying the lower half space ($z<0$). The corresponding model is obtained by transforming the $z$ direction of the bulk model, Eq. (1) in the main text, from wave vector space to real space. Introducing an integral label $n$ to represent the various quintuple layers, with bigger $n$ indicating a larger $z$ coordinate, we can write the model as $\hat{H}=\hat{H}_{xy}+\hat{H}_{z}$, where $\hat{H}_{xy}$ contains the intra-quintuple-layer terms and $\hat{H}_{z}$ consists of the inter-quintuple-layer hopping terms. Denoting the Nambu basis in terms of the layer label $n$ and the two dimensional wave vectors $\tilde{\mathbf{k}}$ defined on the $k_{x}k_{y}$ plane, we have $$\begin{aligned}
&&\hat{H}_{xy}=
\frac{1}{2}\sum\limits_{n\tilde{\mathbf{k}}}\psi^{\dagger}_{n\tilde{\mathbf{k}}}\begin{pmatrix} H^{'}_{0}(\tilde{\mathbf{k}})-\mu I_{4} & \underline{\Delta}(\tilde{\mathbf{k}}) \\
-\underline{\Delta}^{\ast}(-\tilde{\mathbf{k}}) & \mu I_{4}-H^{'\ast}_{0}(-\tilde{\mathbf{k}})
\end{pmatrix}\psi_{n\tilde{\mathbf{k}}} \notag \\
&&=\frac{1}{2}\sum\limits_{n\tilde{\mathbf{k}}}\psi^{\dagger}_{n\tilde{\mathbf{k}}}h_{xy}(\tilde{\mathbf{k}})\psi_{n\tilde{\mathbf{k}}},\end{aligned}$$ where $$\begin{aligned}
H^{'}_{0}(\tilde{\mathbf{k}})&=&\epsilon^{'}(\tilde{\mathbf{k}})I_{4}+M^{'}(\tilde{\mathbf{k}})\Gamma_5
+A_{0}[c_{y}(\tilde{\mathbf{k}})\Gamma_{1} \notag \\
&&-c_{x}(\tilde{\mathbf{k}})\Gamma_{2}]+R_{1}d_{1}(\tilde{\mathbf{k}})\Gamma_{3}+R_{2}d_{2}(\tilde{\mathbf{k}})\Gamma_{4}.\end{aligned}$$ While the dependencies of $c_{x}$, $c_{y}$, $d_{1}$ and $d_{2}$ on the wave vectors keep unchanged, $\epsilon^{'}(\tilde{\mathbf{k}})=C_{0}+2C_{1}
+\frac{4}{3}C_{2}[3-\cos(\tilde{\mathbf{k}}\cdot\boldsymbol{\delta}_{1})-\cos(\tilde{\mathbf{k}}\cdot\boldsymbol{\delta}_{2})
-\cos(\tilde{\mathbf{k}}\cdot\boldsymbol{\delta}_{3})]$ and $M^{'}(\tilde{\mathbf{k}})=M_{0}+2M_{1}
+\frac{4}{3}M_{2}[3-\cos(\tilde{\mathbf{k}}\cdot\boldsymbol{\delta}_{1})-\cos(\tilde{\mathbf{k}}\cdot\boldsymbol{\delta}_{2})
-\cos(\tilde{\mathbf{k}}\cdot\boldsymbol{\delta}_{3})]$. The inter-quintuple-layer terms are $$\begin{aligned}
&&\hat{H}_{z}=
\frac{1}{2}\sum\limits_{n\tilde{\mathbf{k}}}\psi^{\dagger}_{n\tilde{\mathbf{k}}}\begin{pmatrix} H_{0z} & 0 \\
0 & -H^{\ast}_{0z}
\end{pmatrix}\psi_{n+1,\tilde{\mathbf{k}}}+\text{H.c.} \notag \\
&&=\frac{1}{2}\sum\limits_{n\tilde{\mathbf{k}}}\psi^{\dagger}_{n\tilde{\mathbf{k}}}h_{z}\psi_{n+1,\tilde{\mathbf{k}}}+\text{H.c.},\end{aligned}$$ where $\text{H.c.}$ means taking the Hermite conjugation of the terms explicitly written out, and $$H_{0z}=-M_{1}\Gamma_{5}-\frac{i}{2}B_{0}\Gamma_{4}.$$
With the above model at hand, the surface GF for the surface layer of the semi-infinite sample occupying the $z<0$ half space is obtained iteratively by[@wang10; @hao11] $$G^{(m)}_{s}(\tilde{\mathbf{k}},\omega)=[g^{-1}-h^{\dagger}_{z}G^{(m-1)}_{s}h_{z}]^{-1},$$ where $g=[(\omega+i\eta)I_{8}-h_{xy}(\tilde{\mathbf{k}})]^{-1}$ is the GF for an isolated quintuple layer. The superscripts $m$ and $m-1$ label the iteration steps. To begin, we set $G^{(0)}_{s}=g$ and get $G^{(1)}_{s}$. Then $G^{(1)}_{s}$ is put into the right side of Eq.(D6) to give $G^{(2)}_{s}$. The iteration is repeated until the difference between every corresponding component of $G^{(m)}_{s}$ and $G^{(m-1)}_{s}$ is smaller than a certain small positive number, which is set by hand to control the precision.
Once we have obtained the surface GF $G_{s}(\tilde{\mathbf{k}},\omega)$ in terms of the above iterative GF method (or, transfer matrix method), we can get the surface spectral function by summing up the imaginary parts of the four particle surface GFs as (note that, the wave vectors in the surface BZ are denoted as $\mathbf{k}_{xy}$ in the main text) $$A(\tilde{\mathbf{k}},\omega)=-\frac{1}{\pi}\sum\limits_{i=1}^{4}\text{Im}G_{s,ii}(\tilde{\mathbf{k}},\omega).$$
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---
abstract: 'Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being [compact open subgroups of reductive groups over non-archimedean local fields]{}. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a test case, we examine in detail the induction and restriction of representations from and to the Siegel Levi subgroup of the symplectic group $\operatorname{Sp}_4$ over a finite local principal ideal ring of length two. We obtain in this case a Mackey-type formula for the composition of these induction and restriction functors which is a perfect analogue of the well-known formula for the composition of Harish-Chandra functors. In a different direction, we study representations of the Iwahori subgroup $I_n$ of $\operatorname{GL}_n(F)$, where $F$ is a non-archimedean local field. We establish a bijection between the set of irreducible representations of $I_n$ and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.'
address:
- 'Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany'
- 'Department of Mathematics, University of Hamburg, Bundesstr. 55, 20146 Hamburg, Germany'
- 'Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel'
author:
- Tyrone Crisp
- Ehud Meir
- Uri Onn
bibliography:
- 'CMO.bib'
title: 'A variant of Harish-Chandra functors'
---
Introduction {#sec:intro}
============
Overview
--------
Harish-Chandra (or parabolic) induction and restriction are fundamental operations in the representation theory of reductive groups over finite fields, allowing efficient transport of representations between such groups and establishing a close connection to the representation theory of finite Coxeter groups; see [@Zelevinsky; @vanLeeuwen] for a particularly elegant development of this connection for finite classical groups. Recall that Harish-Chandra induction is an instance of the following general construction. Given a finite group $G$, and subgroups $L$ and $U$ such that $L$ normalises $U$, one obtains a functor $\operatorname{i}_L^G$ from the complex representations of $L$ to the complex representations of $G$ by tensor product with the ${\mathcal H}(G)$-${\mathcal H}(L)$ bimodule ${\mathcal H}(G)e_U$, where ${\mathcal H}(G)$ is the complex group algebra of $G$ and $e_U$ is the idempotent associated with the trivial representation of $U$. Dually, tensoring with the bimodule $e_U{\mathcal H}(G)$ gives a functor $\operatorname{r}^G_L$ that is adjoint to $\operatorname{i}_L^G$. A variant of Mackey’s double-coset formula applies to the composite functor $\operatorname{r}^G_L\operatorname{i}_L^G$, yielding a decomposition of the endomorphism algebra of an induced representation $\operatorname{i}_L^G(M)$ into a direct sum indexed by the double-coset space $LU\backslash G/LU$; cf. [@vanLeeuwen Theorem 2.3.1]. In the case of Harish-Chandra induction, $G$ is the group of rational points of a connected reductive group defined over a finite field, and $L$ and $U$ are the respective groups of rational points of a Levi factor of and the unipotent radical of a rational parabolic subgroup $P$ of $G$. The Bruhat decomposition gives a parametrisation of the $P$-double cosets in $G$ by the double cosets of the Weyl group of $L$ in the Weyl group $G$, and the Mackey formula becomes $$\label{eq:Mackey_intro}
\operatorname{r}^G_{L} \operatorname{i}_{L}^G \cong \bigoplus_{g\in W_{L}\backslash W_G / W_{L}} \operatorname{i}_{L\cap gLg^{-1}}^{L} \, \operatorname{Ad}_g \, \operatorname{r}^{L}_{g^{-1}Lg\cap L}.$$ See [@DM] for the precise general formulation and proof, and for a sampling of the applications of this formula; and [see]{} [@Harish-Chandra; @Springer_cusp] for the original work of Harish-Chandra.
In this paper we study induction and restriction functors which generalise the Harish-Chandra functors to a rich family of profinite groups, to which the family of reductive groups over finite fields is only a partial first approximation. Our motivating examples are classical groups over compact discrete valuation rings, but our framework covers many other cases, including arbitrary open compact subgroups of reductive groups over local fields. Certain representations of such open compact subgroups play an important role in the construction and classification of smooth representations of the reductive groups via the theory of types. However, the representation theory of these compact subgroups per se is not so well understood.
Before we introduce the functors that are at the heart of the present paper we remark that the most obvious generalisation of the Harish-Chandra functors to the setting considered here tends to produce representations that are far from irreducible, and in this sense lacks the efficiency of the classical Harish-Chandra functors. For a concrete example, let ${\mathfrak o}$ be the ring of integers in a non-archimedean local field $F$ (so $F$ is either the field of Laurent series over a finite field, or a finite extension of the $p$-adic numbers). Let ${\mathfrak}p$ denote the maximal ideal of ${\mathfrak o}$, and for every $\ell \in {\mathbb{N}}$ set ${\mathfrak o}_\ell={\mathfrak o}/{\mathfrak}p^\ell$. Let $\operatorname{T}_n\subset \operatorname{B}_n \subset \operatorname{GL}_n$ denote the standard diagonal torus and the standard upper-triangular Borel subgroup in the general linear group, and let $\operatorname{U}_n$ denote the unipotent radical of $\operatorname{B}_n$. The subgroup $\operatorname{T}_n({\mathfrak o}_\ell)$ normalises $\operatorname{U}_n({\mathfrak o}_\ell)$, and so the construction described in the first paragraph gives a functor from representations of $\operatorname{T}_n({\mathfrak o}_\ell)$ to representations of $\operatorname{GL}_n({\mathfrak o}_\ell)$, which in particular sends the trivial representation of $\operatorname{T}_n({\mathfrak o}_\ell)$ to the permutation representation of $\operatorname{GL}_n({\mathfrak o}_\ell)$ given by $$\label{eqn.HC.for.GLn}
{\mathcal H}(\operatorname{GL}_n({\mathfrak o}_\ell))e_{\operatorname{U}_n({\mathfrak o}_\ell)} \otimes_{{\mathcal H}(\operatorname{T}_n({\mathfrak o}_\ell))} 1 \cong {\mathcal H}(\operatorname{GL}_n({\mathfrak o}_\ell)/\operatorname{B}_n({\mathfrak o}_\ell)).$$
When $\ell=1$ the Mackey formula gives a decomposition of according to the regular representation of the symmetric group on $n$ letters. For $\ell > 1$ the decomposition of into irreducibles gets very quickly out of control, owing to the complicated nature of the double-coset space $\operatorname{B}_n({\mathfrak o}_\ell)\backslash \operatorname{GL}_n({\mathfrak o}_\ell)/ \operatorname{B}_n({\mathfrak o}_\ell)$. Misleadingly simple is the case $n=2$, where the induced representation has $\ell+1$ irreducible components (see [@Casselman]); already for $n=3$ the decomposition of the induced representation is rather complicated and, in particular, depends on the degree of the residue field ${\mathfrak o}_1={\mathfrak o}/{\mathfrak}p$, see [@OnnSingla].
Our proposed variant of Harish-Chandra induction, in this $\operatorname{GL}_n$ example, sends the trivial representation of $\operatorname{T}_n({\mathfrak o}_\ell)$ to the image of the intertwining operator $${\mathcal H}\left(\operatorname{GL}_n({\mathfrak o}_\ell)/\operatorname{B}_n({\mathfrak o}_\ell)\right) \to {\mathcal H}\left(\operatorname{GL}_n({\mathfrak o}_\ell)/\operatorname{B}_n^{{t}}({\mathfrak o}_\ell)\right)$$ which averages right $\operatorname{B}_n({\mathfrak o}_\ell)$-invariant functions on $\operatorname{GL}_n({\mathfrak o}_\ell)$ by the right action of $\operatorname{U}_n^{t}({\mathfrak o}_\ell)$, where ${t}$ means transpose, to obtain right $\operatorname{B}_n^{t}({\mathfrak o}_\ell)$-invariant functions. This image is isomorphic—regardless of $\ell$—to the module ${\mathcal H}(\operatorname{GL}_n({\mathfrak o}_1)/\operatorname{B}_n({\mathfrak o}_1))$, on which $\operatorname{GL}_n({\mathfrak o}_\ell)$ acts through the quotient map $\operatorname{GL}_n({\mathfrak o}_\ell)\to \operatorname{GL}_n({\mathfrak o}_1)$.
This process of passing to the image of a canonical intertwining operator between two induced representations fits into a rather general setting, which we shall now describe. In the main body of the paper we study representations of profinite groups, such as groups of matrices over compact discrete valuation rings; but our results also apply to (and are interesting for) finite groups, such as matrix groups over the finite rings ${\mathfrak o}_{\ell}$, and in order to minimise the technicalities in this introduction we shall restrict our attention here to the finite case.
Let $G$ be a finite group, and suppose that $U$, $L$ and $V$ are subgroups of $G$ such that $L$ normalises $U$ and $V$, and such that the map $$U\times L\times V \hookrightarrow G$$ given by multiplication in $G$ is injective. We let $e_U$ and $e_V$ denote the idempotents in the complex group algebra ${\mathcal H}(G)$ associated to the trivial representations of $U$ and $V$, and we consider the ${\mathcal H}(G)$-${\mathcal H}(L)$ bimodule ${\mathcal H}(G)e_U e_V$. Let $\operatorname{i}_{U,V}$ be the functor from the category $\operatorname{\mathcal{R}}(L)$ of complex representations of $L$ to the category of complex representations of $G$ defined by tensoring with this bimodule: $$\operatorname{i}_{U,V}:\operatorname{\mathcal{R}}(L) \to \operatorname{\mathcal{R}}(G), \qquad M \mapsto {\mathcal H}(G)e_U e_V \otimes_{{\mathcal H}(L)}M.$$ Similarly, define $$\operatorname{r}_{U,V}:\operatorname{\mathcal{R}}(G) \to \operatorname{\mathcal{R}}(L), \qquad N \mapsto e_Ue_VH(G) \otimes_{{\mathcal H}(G)}N.$$
This definition is closely related to, and directly inspired by, a construction of Dat [@Dat_parahoric]. Note, though, that we consider only complex representations, whereas Dat studied representations over more general commutative rings. The relationship between our definition and Dat’s, for complex coefficients, is discussed further in Remark \[rem:CMO-vs-Dat\] and in Section \[parahoric\_section\]. The main novelty of the above definition relative to Dat’s is that we do not require the product $ULV$ to be a group, so that for instance we could as above take $G$ to be $\operatorname{GL}_n({\mathfrak o}_\ell)$, and let $L=\operatorname{T}_n({\mathfrak o}_\ell)$, $U=\operatorname{U}_n({\mathfrak o}_\ell)$ and $V=\operatorname{U}_n^{t}({\mathfrak o}_\ell)$. When $\ell=1$, a theorem of Howlett and Lehrer ([@Howlett-Lehrer_HC Theorem 2.4]; cf. Example \[ex:pind-field\]) implies that the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$ in this $\operatorname{GL}_n$ example are isomorphic to the functors of Harish-Chandra induction and restriction. When $\ell>1$ these functors are proper subfunctors of the more obvious generalisations of the Harish-Chandra functors mentioned above.
Description of the main results
-------------------------------
Basic properties of the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$, in the abstract setting [for profinite groups]{}, are presented in Section \[sec:definition\]. For instance, these functors are adjoints on both sides; they do not depend on the order of $U$ and $V$, up to natural isomorphism; they preserve finite-dimensionality; and they satisfy a version of induction in stages.
The analysis of the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$ becomes considerably less complicated in cases where the product map $U\times L\times V\to G$ is a bijection. In many examples, such as the $\operatorname{GL}_n$ example considered above, this is not the case, but there is a normal subgroup $G_0 \lhd G$ such that the product map $U_0\times L_0\times V_0\to G_0$ is a bijection, where $H_0$ means $H\cap G_0$. In the $\operatorname{GL}_n$ example we can take $G_0$ to be the principal congruence subgroup $G_0=\{g\in \operatorname{GL}_n({\mathfrak o}_\ell)\ |\ g\equiv 1 \textrm{ modulo }{\mathfrak}p\}$.
Suppose that $G$ admits such a normal subgroup $G_0$. The representation categories $\operatorname{\mathcal{R}}(L)$ and $\operatorname{\mathcal{R}}(G)$ decompose according to $L_0$- and $G_0$-isotypic components, and the individual components can be described using Clifford theory. In Section \[sec:Clifford\] we prove that the Clifford analysis is compatible with the induction and restriction functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$.
More precisely, let $\psi$ be an irreducible representation of $L_0$, and let ${\varphi}=\operatorname{i}_{U_0,V_0}(\psi)$ be the corresponding (irreducible) induced representation of $G_0$. Let $L(\psi)$ and $G({\varphi})$ denote the inertia groups of $\psi$ and ${\varphi}$. We prove in Theorems \[thm:C1\], \[thm:C2\] and \[thm:C3\] that there is a commutative diagram for induction (and a similar diagram for restriction): $$\xymatrix@C=80pt{ \operatorname{\mathcal{R}}(L)_\psi \ar[r]^-{\operatorname{i}_{U,V}} & \operatorname{\mathcal{R}}(G)_{\varphi}\\
\operatorname{\mathcal{R}}(L(\psi))_\psi \ar[r]^-{\operatorname{i}_{U({\varphi}),V({\varphi})}} \ar[u]^-{\cong} & \operatorname{\mathcal{R}}(G({\varphi}))_{\varphi}\ar[u]_-{\cong} \\
\operatorname{\mathcal{R}}^{\gamma } (L(\psi)/L_0) \ar[r]^-{\operatorname{i}_{U({\varphi})/U_0, V({\varphi})/V_0}} \ar[u]^-{\cong} & \operatorname{\mathcal{R}}^{\gamma }(G({\varphi})/G_0) \ar[u]_-{\cong}
}$$ where $\operatorname{\mathcal{R}}(H)_\theta$ stands for the representations of $H$ whose restriction to $H_0\lhd H$ contains the irreducible representation $\theta$; and $\operatorname{\mathcal{R}}^{\gamma}(H(\theta)/H_0)$ stands for projective representations (for a certain cocycle $\gamma$) of the quotient $H(\theta)/H_0$.
As for the groups $L_0$ and $G_0$, in many of our motivating examples they are amenable to the orbit method: their irreducible representations correspond bijectively to coadjoint orbits in the Pontryagin duals of certain Lie algebras ${\mathfrak}l_0$ and ${\mathfrak}g_0$. This situation is studied in Section \[sec:orbit\], where we show that under appropriate assumptions the induction functor $\operatorname{i}_{U_0,V_0}:\operatorname{\mathcal{R}}(L_0) \to \operatorname{\mathcal{R}}(G_0)$ corresponds to a natural inclusion of coadjoint orbits $\Lambda^*:L_0\backslash \widehat{{\mathfrak}l_0} {\hookrightarrow}G_0\backslash \widehat{{\mathfrak}g_0}$. That is, the diagram $$\xymatrix@R=30pt@C=50pt{ \operatorname{Irr}(L_0) \ar[r]^-{\operatorname{i}_{U_0,V_0}} \ar[d]_-{\cong} & \operatorname{Irr}(G_0) \ar[d]^-{\cong}\\
{L_0}\backslash \widehat{{\mathfrak}l_0} \ar[r]^-{\Lambda^*} & G_0\backslash \widehat{{\mathfrak}g_0}
}$$ commutes.
Returning from the abstract setting to our motivating examples, the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$ provide a new approach to the representation theory of classical groups over compact discrete valuation rings, and the results of Sections \[sec:Clifford\] and \[sec:orbit\] provide tools to analyse these functors. In Section \[sec:sp\] we illustrate the method for the symplectic group $\operatorname{Sp}_4({\mathfrak o}_2)$. The main result is a Mackey-type formula for the composition of restriction and induction to/from the Siegel Levi subgroup. The formula is the same as the usual formula for the composition of Harish-Chandra induction and restriction for the corresponding group $\operatorname{Sp}_4({\mathfrak o}_1)$ over the residue field of ${\mathfrak o}$, which lends some support to the analogy between our functors and the Harish-Chandra functors. This analogy is further supported by an analysis for the groups $\operatorname{GL}_n$, which will be presented in a sequel to this paper.
The general methods developed in Sections \[sec:Clifford\] and \[sec:orbit\] and used in Section \[sec:sp\] apply equally well to Dat’s parahoric induction and restriction functors. In Section \[parahoric\_section\] we prove that Dat’s parahoric induction and restriction functors are not isomorphic to ours, in the example of the Siegel Levi in $\operatorname{Sp}_4({\mathfrak o}_2)$; this gives a negative answer to Dat’s question [@Dat_parahoric Question 2.15]. We also prove that the parahoric induction and restriction functors do not satisfy the analogue of in this example.
While our primary motivation for studying the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$ is their application to classical groups, these functors are defined in much broader generality, and we believe that they have a useful role to play in the representation theory of more general matrix groups. In Section \[sec:Iwahori\] we use these functors to study one such example, the representation theory of the Iwahori subgroup $I_n$ of $\operatorname{GL}_n({\mathfrak o})$. The Iwahori in $\operatorname{SL}_2$ was previously studied from a similar point of view in [@Dat_parahoric] and [@Crisp_parahoric]. The main result of this section, Theorem \[thm:Iw\_primitive\], states that the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$ in this context give a bijection $$\label{eq:intro_prim}
\operatorname{Irr}(I_n) \longleftrightarrow \bigsqcup_{n_1+\cdots+n_k=n} \operatorname{Prim}(I_{n_1})\times \cdots \times \operatorname{Prim}(I_{n_k})$$ between the irreducible representations of $I_n$, and tuples of [*primitive*]{} irreducible representations of smaller Iwahori subgroups (where [primitivity]{} is defined by the vanishing of the functors $\operatorname{r}_{U,V}$). The problem of classifying the irreducible representations of $I_n$ remains a very difficult one—it contains the problem of counting the conjugacy classes in the group of upper-triangular matrices over the residue field ${\mathfrak o}_1$—but the bijection shows that part of this classification is very simple and combinatorial in nature.
Related constructions
---------------------
Representations of open compact subgroups of reductive groups over local fields have received much attention in the past two decades. One approach, taken by Lusztig and Stasinski, is to generalise Deligne-Lusztig theory [@Deligne-Lusztig] (which is itself a generalisation of the Harish-Chandra theory) to such groups; see [@Lusztig1; @Stasinski], and also [@Chen-stasinski] and [@Lusztig2]. Another approach, taken by Hill [@Hill_Jord; @Hill_nilp; @Hill_Reg; @Hill_SSandCusp], consists of a direct Clifford-theoretic analysis of representations according to their restrictions to congruence kernels. In particular, in [@Hill_Jord] Hill establishes a Jordan decomposition for characters of general linear groups over [rings of integers in $p$-adic fields]{}, analogous to the Jordan decomposition of irreducible characters of finite reductive groups established by Lusztig, cf. [@Lusztig84]. Hill’s work relies on an analysis of certain Hecke algebras building on the work of Howe and Moy [@Howe-Moy]. Another approach was proposed by the third author in [@Onn] using a different variant of Harish-Chandra induction that allows one to import representations from automorphism groups of finite modules over discrete valuation rings, yielding a complete and characteristic-independent treatment in rank two. The work of Dat [@Dat_parahoric], in which representations of parahoric subgroups of $p$-adic reductive groups are studied using methods closely related to those of the present paper, has already been mentioned above.
It would be of great interest to understand how all these approaches align with the one taken in this paper. The relationship between our work and that of Dat is addressed in Remark \[rem:CMO-vs-Dat\] and Section \[parahoric\_section\]. As for the other works cited above, let us make a couple of general observations.
The first point to note is that the natural filtration on the valuation ring ${\mathfrak o}$ does not enter a priori into the definition of our induction/restriction functors, and in this sense our approach is more elementary than those of the above-cited works. It is consequently more general—applying for instance to $\operatorname{GL}_n(R)$ for an arbitrary (pro)finite commutative ring $R$—although the usefulness of our methods beyond the setting of discrete valuation rings remains to be tested.
A second difference is one of scope. Our functors are defined with a view to making the induced representations as small as possible, and the set of representations which cannot be obtained by induction in our sense (the cuspidal representations) will be accordingly large—certainly larger than the corresponding sets for the approaches listed above. Our goal is to develop an analogue of Harish-Chandra theory which mirrors as closely as possible the theory for reductive groups over a finite field, yielding a description of arbitrary representations in terms of cuspidal ones and of Weyl group combinatorics. We leave untouched for now the problem of constructing (let alone classifying) the cuspidal representations.
Acknowledgments
---------------
We thank George Willis and Helge Glöckner for helpful discussions on tidy subgroups. The first two authors were partly supported by the Danish National Research Council through the Centre for Symmetry and Deformation (DNRF92). The third author acknowledges the support of the Israel Science Foundation and of the Australian Research Council.
Notation, definitions, and basic properties {#sec:definition}
===========================================
In this section we define and develop basic properties of the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$ in an abstract setting. The pivotal point in this section is Proposition \[prop:z\], which allows us to generalise many of Dat’s results from [@Dat_parahoric Section 2] to the situation considered in this paper. We begin by setting up the notation that will be used throughout the paper.
Notation {#subsec:notation}
--------
For a profinite group $G$ we let $\operatorname{\mathcal{R}}(G)$ denote the category of smooth, complex representations of $G$, that is, linear representations ${\varphi}:G\to \operatorname{GL}_{{\mathbb{C}}}(M)$ in which each vector in $M$ is fixed by some open subgroup of $G$. We will denote such a representation either by the map ${\varphi}$ or by the space $M$, as convenient. If $M$ is any representation of $G$ (not necessarily smooth), we let $M^\infty$ denote the $G$-subspace of vectors fixed by some open subgroup of $G$.
Let ${\mathcal H}(G)$ denote the algebra of locally constant, complex-valued functions on $G$, with product given by convolution with respect to some Haar measure on $G$. Different choices of Haar measure give isomorphic algebras, the isomorphism being multiplication by the ratio $\operatorname{vol}_1(G)/\operatorname{vol}_2(G)$ of the total volumes of the two measures. The category $\operatorname{\mathcal{R}}(G)$ is equivalent to the category of nondegenerate [left]{} ${\mathcal H}(G)$-modules, i.e. those modules $M$ which satisfy $M={\mathcal H}(G)M$. If $G$ is finite then we will usually use counting measure as the Haar measure on $G$, in which case the map sending $g\in G$ to the $\delta$-function $\delta_g$ at $g$ extends to an isomorphism from the complex group ring ${\mathbb{C}}(G)$ to ${\mathcal H}(G)$.
Let $\operatorname{Irr}(G)$ denote the set of isomorphism classes of irreducible smooth representations of $G$. When chances for confusion are slim we will also write $\rho\in \operatorname{Irr}(G)$ for an actual irreducible representation. For each $\rho \in \operatorname{Irr}(G)$ let $\operatorname{ch}_\rho\in {\mathcal H}(G)$ be the character $g\mapsto \operatorname{tr}(\rho(g))$ of $\rho$, and let $e_\rho \in {\mathcal H}(G)$ be the idempotent defined by $$e_\rho : g \mapsto \frac{\dim_{\mathbb{C}}(\rho)}{\operatorname{vol}(G)} \operatorname{ch}_\rho(g^{-1}).$$ If $M$ is a smooth representation of $G$ then $e_\rho$ acts on $M$ by projecting $M$ onto its $\rho$-isotypical submodule. For the special case of the trivial representation we write $e_G$ for the corresponding idempotent, namely, the function on $G$ with constant value $1/\operatorname{vol}(G)$. The element $e_G$ acts on each smooth representation $M$ by projecting onto the submodule $M^G$ of $G$-fixed vectors.
If $M$ is a nondegenerate left ${\mathcal H}(G)$-module, and $N$ is a nondegenerate right ${\mathcal H}(G)$-module, then by definition $$N\otimes_{{\mathcal H}(G)} M = N\otimes_{{\mathbb{C}}} M \big / \operatorname{span}\{nf\otimes m - n\otimes fm\ |\ n\in N,\ m\in M,\ f\in {\mathcal H}(G)\}.$$ Equivalently, viewing $N$ and $M$ as smooth representations of $G$, $N\otimes_{{\mathcal H}(G)} M$ is the space of coinvariants for the action $g:n\otimes m\to ng^{-1}\otimes gm$ of $G$ on $N\otimes_{{\mathbb{C}}} M$.
If $H$ is a closed subgroup of $G$, then ${\mathcal H}(G)$ is a smooth representation of $H$ under both left- and right-translation, and consequently ${\mathcal H}(G)$ is an ${\mathcal H}(H)$-bimodule. Given a smooth representation $M$ of $H$ we write $$\operatorname{ind}_{H}^G M = \left.\left\{ f:G\xrightarrow[\text{constant}]{\text{locally}} M\ \right|\ f(h g)=h\cdot f(g), \forall h \in H,g\in G \right\}$$ for the induced representation, on which $G$ acts by right-translation. This is isomorphic to the tensor product ${\mathcal H}(G)\otimes_{{\mathcal H}(H)} M$. If the subgroup $H$ is a semidirect product $U\rtimes L$, then representations may be induced from $L$ to $G$ by first inflating to $H$ (i.e. pulling back along the quotient map $H\to L$), and then applying the functor $\operatorname{ind}_H^G$. The resulting functor from $\operatorname{\mathcal{R}}(L)$ to $\operatorname{\mathcal{R}}(G)$ is isomorphic to the functor of tensor product with the ${\mathcal H}(G)$-${\mathcal H}(L)$ bimodule ${\mathcal H}(G)e_U \cong {\mathcal H}(G/U)$, where ${\mathcal H}(G/U)$ denotes the space of locally constant functions on $G/U$.
Whenever a group $G$ acts on a set $X$ we write $G(x)$ for the stabiliser in $G$ of $x \in X$.
The first three chapters of [@Renard] are a convenient reference for all of the above. Many of the examples considered here will be groups of matrices over compact subrings of non-archimedean local fields; see [@Renard Chapter V], for instance, for more background on these.
[Virtual Iwahori decompositions]{}
----------------------------------
Let us begin by describing the kind of groups that we shall be interested in, and giving several examples.
\[def:vI\] Let $G$ be a profinite group. A *virtual Iwahori decomposition* of $G$ is a triple of closed subgroups $(U,L,V)$ of $G$, where $L$ normalises $U$ and $V$, such that
(1) The multiplication map $U\times L \times V \to G$ is an open embedding (and therefore a homeomorphism onto its image).
(2) $G$ contains arbitrarily small open, normal subgroups $K$ for which the multiplication map $$(U\cap K) \times (L\cap K) \times (V\cap K) \to K$$ is a homeomorphism.
An *Iwahori decomposition* of $G$ is a virtual Iwahori decomposition for which the multiplication map in (1) is surjective (and therefore, a homeomorphism).
The following immediate observation shows that the notion of virtual Iwahori decomposition is inherited by subgroups and quotients.
\[ex:vI-subquotients\] Let $(U,L,V)$ be a virtual Iwahori decomposition of $G$.
1. If $J$ is a closed subgroup of $G$, then $(U\cap J, L\cap J, V\cap J)$ is a virtual Iwahori decomposition of $J$.
2. If $(X,H,Y)$ is a virtual Iwahori decomposition of $L$, then $(U\rtimes X, H, Y\ltimes V)$ is a virtual Iwahori decomposition of $G$.
3. If $K$ is an open normal subgroup of $G$ with an Iwahori decomposition as in part (2) of Definition \[def:vI\], then $(U/(U\cap K), L/(L\cap K), V/(V\cap K))$ is a virtual Iwahori decomposition of $G/K$.
[The concept of Iwahori decomposition first appeared in the work of Iwahori and Matsumoto on $p$-adic Chevalley groups [@IwMat]. The virtual version defined above is likewise motivated by examples occurring naturally in the study of reductive groups:]{}
\[ex:vI-p-adic\] Let $\mathbf G$ be a connected reductive group over a non-archimedean local field $F$, and let $G$ be any compact open subgroup of $\mathbf G(F)$. There is a maximal $F$-split torus $\mathbf T\subset \mathbf G$ (depending on $G$) with the property that if $\mathbf L$ is an $F$-rational Levi subgroup of $\mathbf G$ containing $\mathbf T$, and $\mathbf U$ and $\mathbf V$ are the unipotent radicals of an opposite pair of $F$-rational parabolic subgroups of $\mathbf G$ with common Levi factor $\mathbf L$, then the triple of subgroups $(G\cap \mathbf U(F), G\cap \mathbf L(F), G\cap \mathbf V(F))$ is a virtual Iwahori decomposition of $G$. [This follows from the Bruhat-Tits theory: one can take $\mathbf T$ to be any torus whose associated apartment in the affine building of $\mathbf{G}(F)$ contains a point fixed by $G$. An explicit filtration of $G$ by open normal subgroups admitting Iwahori decompositions is constructed in [@Schneider-Stuhler Section 1.2]; cf. [@Dat_parahoric 2.11].]{}
\[ex:vI-GLnO\] For a specific instance of the previous example, let $G=\operatorname{GL}_n({\mathfrak o})$, where ${\mathfrak o}$ is the ring of integers in a non-archimedean local field. Given an ordered partition $n=n_1+\cdots+ n_m$ of $n$ as a sum of positive integers, let $L\cong \operatorname{GL}_{n_1}({\mathfrak o})\times \cdots \times \operatorname{GL}_{n_m}({\mathfrak o})$ be the corresponding subgroup of block-diagonal matrices in $G$. Let $U$ be the group of upper-triangular matrices in $G$ with diagonal blocks $1_{n_1\times n_1}\times \cdots \times 1_{n_m\times n_m}$, and let $V$ be the transpose of $U$. Then the triple $(U,L,V)$ is a virtual Iwahori decomposition of $G$; the principal congruence subgroups $$K_\ell \coloneq \ker\left( \operatorname{GL}_n({\mathfrak o})\to \operatorname{GL}_n({\mathfrak o}_\ell)\right)$$ (where ${\mathfrak o}_\ell = {\mathfrak o}/{\mathfrak}p^{\ell}$, ${\mathfrak}p$ being the maximal ideal of ${\mathfrak o}$) all admit Iwahori decompositions. Passing to quotients by the $K_\ell$ yields virtual Iwahori decompositions of the finite groups $\operatorname{GL}_n({\mathfrak o}_\ell)$.
\[ex:vI-parahoric\] A second virtual Iwahori decomposition of $G=\operatorname{GL}_n({\mathfrak o})$ is given by $(U,L, V_1)$, where $U$ and $L$ are as in Example \[ex:vI-GLnO\], and $V_1= V\cap K_1$. In this case the image $ULV_1$ of the product mapping $U\times L\times V_1\to G$ is a subgroup of $G$. For instance, if the partition is $n=1+\cdots+1$, then $ULV_1$ is the standard *Iwahori subgroup* of $\operatorname{GL}_n({\mathfrak o})$, comprising those matrices which are upper-triangular modulo ${\mathfrak}p$.
If $G$ is finite then the condition (2) in Definition \[def:vI\] is always satisfied, e.g. by the trivial subgroup $K=\{1\}$. Since the smooth representation theory of a profinite group $G$ is determined in a very simple way by the representations of the finite quotients of $G$, the condition (2) is therefore not essential to much of the sequel. On the other hand, this condition is convenient in places for shortening some proofs, and it is satisfied by all of our motivating examples. Nevertheless, let us note the following [quite]{} general construction of examples which satisfy condition (1) without—at least a priori—satisfying (2).
\[ex:vI-tidy\] Let $\mathcal G$ be a totally disconnected locally compact group and let $\alpha:\mathcal G\to\mathcal G$ be a topological group automorphism. Suppose that the contraction subgroups $$\mathcal U_\alpha=\{g\in \mathcal G\ |\ \alpha^n(g)\to 1\textrm{ as }n\to \infty\}\quad\text{and}\quad \mathcal V_\alpha = \mathcal U_{\alpha^{-1}}$$ are closed in $\mathcal G$. This is [always]{} the case, for example, if $\mathcal G$ is a $p$-adic Lie group.
These contraction subgroups are both normalised by the closed subgroup $$\mathcal L_\alpha = \{g\in \mathcal G\ |\ \{\alpha^n(g)\ |\ n\in {\mathbb{Z}}\}\textrm{ is precompact in $\mathcal G$}\},$$ and the multiplication map $$\mathcal U_\alpha \times \mathcal L_\alpha \times \mathcal V_\alpha \to \mathcal G$$ is an open embedding. So if $G$ is any compact open subgroup of $\mathcal G$, then the triple $(\mathcal U_\alpha\cap G, \mathcal L_\alpha\cap G, \mathcal V_\alpha\cap G)$ satisfies condition (1) of Definition \[def:vI\]. Moreover, $G$ contains arbitrarily small open subgroups $K$ for which the multiplication map $$(\mathcal U_\alpha\cap K) \times (\mathcal L_\alpha\cap K)\times (\mathcal V_\alpha\cap K)\to K$$ is a [homeomorphism]{} (the so-called *tidy* subgroups for $\alpha$). It is not clear to us whether $G$ contains arbitrarily small open *normal* subgroups [$K$]{} with this property. If $\mathcal G$ is an analytic Lie group over a local field and the automorphism $\alpha$ is analytic (keeping the assumption that the contraction groups are closed), then it is at least true that [$\mathcal G$ contains arbitrarily small open subgroups $K$ with Iwahori decomposition $(\mathcal U_\alpha\cap K, \mathcal L_\alpha\cap K, \mathcal V_\alpha\cap K)$; cf. Example \[ex:orbit\_tidy\] for the characteristic $0$ case.]{} We thank George Willis and Helge Glöckner for a discussion of this example. See [@Baumgartner-Willis] and [@Gloeckner] for details.
Definition and basic properties of the functors $\boldsymbol\operatorname{i}$ and $\boldsymbol\operatorname{r}$
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We now come to the main definition of the paper. Whenever $H$ is a closed subgroup of a profinite group $G$, [the space]{} ${\mathcal H}(G)$ is a bimodule over ${\mathcal H}(H)$. If $L$, $U$ and $V$ are closed subgroups of $G$, and $L$ normalises $U$ and $V$, then the action of ${\mathcal H}(L)$ on ${\mathcal H}(G)$ commutes with the idempotents $e_U\in {\mathcal H}(U)$ and $e_V\in {\mathcal H}(V)$. Thus ${\mathcal H}(G)e_U e_V$ is an ${\mathcal H}(G)$-${\mathcal H}(L)$ bimodule, and $e_U e_V {\mathcal H}(G)$ is an ${\mathcal H}(L)$-${\mathcal H}(G)$ bimodule.
\[def:pind\] Let $(U,L,V)$ be a virtual Iwahori decomposition of a profinite group $G$. Define the following functors: $$\operatorname{i}_{U,V}: \operatorname{\mathcal{R}}(L)\to \operatorname{\mathcal{R}}(G), \qquad \operatorname{i}_{U,V}:M\mapsto {\mathcal H}(G)e_U e_V \otimes_{{\mathcal H}(L)} M$$ $$\operatorname{r}_{U,V}: \operatorname{\mathcal{R}}(G) \to \operatorname{\mathcal{R}}(L), \qquad \operatorname{r}_{U,V}: N \mapsto e_U e_V {\mathcal H}(G) \otimes_{{\mathcal H}(G)} N.$$
\[rem:CMO-vs-Dat\] The definition in the case where $ULV$ is a subgroup of $G$ is due to Dat, who considered situations like Example \[ex:vI-parahoric\], see [@Dat_parahoric 2.6, 2.11]. The novelty of Definition \[def:pind\] is that we relax the requirement that $ULV$ be a group, so as to cover cases like Example \[ex:vI-GLnO\]. See Section \[parahoric\_section\] for an example of the difference between our definition and Dat’s definition of [*parahoric induction*]{}. Also note that Dat makes a further assumption in [@Dat_parahoric], namely that the group $L$ should contain an open normal subgroup $L^\dagger$ such that the set $UL^\dagger V$ is a pro-$p$ subgroup of $G$. This assumption, which is needed to ensure the integrality of certain constructions in [@Dat_parahoric], plays no role here, where all representations are over ${\mathbb{C}}$.
Let us make a few further remarks on Definition \[def:pind\]. Firstly, since ${\mathcal H}(G)e_U e_V$ is the image of the bimodule map $f\mapsto fe_V$ from ${\mathcal H}(G)e_U$ to ${\mathcal H}(G)e_V$, and since every $M\in \operatorname{\mathcal{R}}(L)$ is a direct sum of representations of finite quotients of $L$, and hence flat as a module over ${\mathcal H}(L)$, the module $\operatorname{i}_{U,V}(M)$ is isomorphic to the image of the map $$\label{eq:intertwiner}
J_V:{\mathcal H}(G)e_U\otimes_{{\mathcal H}(L)} M \xrightarrow{\ f\otimes m\mapsto fe_V\otimes m\ } {\mathcal H}(G)e_V\otimes_{{\mathcal H}(L)} M.$$ The module ${\mathcal H}(G)e_U\otimes_{{\mathcal H}(L)} M$ is isomorphic as a representation of $G$ to the induced representation $\operatorname{ind}_{LU}^G(M)$, where $M$ is inflated to a representation of $LU$ by letting $U$ act trivially (cf. Section \[subsec:notation\]). We similarly have ${\mathcal H}(G)e_V\otimes_{{\mathcal H}(L)} M\cong \operatorname{ind}_{LV}^G(M)$, and the map $J_V$ corresponds in this picture to the standard intertwining operator $$J_V:\operatorname{ind}_{LU}^G(M) \to \operatorname{ind}_{LV}^G(M), \qquad J_V(f): g\mapsto \int_V f(vg)\,{d}v .$$ Similarly, $\operatorname{r}_{U,V}(N)$ is isomorphic to the image of the canonical projection $$e_U:N^V \to N^U, \qquad n\mapsto \int_U un\,{d}u$$ from the $V$-invariants to the $U$-invariants of $N$.
[As a final remark on Definition \[def:pind\], we note that the definition makes sense if we assume only that $L$, $U$ and $V$ are closed subgroups of $G$ such that $L$ normalises $U$ and $V$. Some of the properties of the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$ that we shall establish below remain valid in this degree of generality: e.g., parts , , and of Theorem \[thm:pind-properties\]. For the applications we have in mind, the assumption that $(U,L,V)$ is a virtual Iwahori decomposition is both a natural and a useful one.]{}
\[ex:pind-field\] Let $G$ be a [reductive group over a finite field]{}, and let $LU$ and $LV$ be an opposite pair of parabolic subgroups of $G$. A theorem of Howlett and Lehrer (see [@Howlett-Lehrer_HC Theorem 2.4]) asserts that in this case the map is an isomorphism for every $M\in\operatorname{\mathcal{R}}(L)$, and this implies that the functor $\operatorname{i}_{U,V}$ is equal to the Harish-Chandra induction functor $M\mapsto {\mathcal H}(G)e_V \otimes_{{\mathcal H}(L)} M$ (and isomorphic to the analogous functor with $U$ in place of $V$). Similarly, $\operatorname{r}_{U,V}$ is isomorphic to the functor of [Harish-Chandra restriction]{}. See [@DM Chapter 4] for background on Harish-Chandra functors for [finite reductive groups]{}.
Example \[ex:pind-field\] notwithstanding, the map is usually far from being an isomorphism. For instance, if $G$ is a compact open subgroup of a reductive group $\mathbf{G}(F)$ as in Example \[ex:vI-p-adic\], and $(U,L,V)$ is the virtual Iwahori decomposition of $G$ corresponding to an opposite pair of proper parabolic subgroups of $\mathbf{G}$, then the subgroups $LU$ and $LV$ have infinite index in $G$, and hence the representations $\operatorname{ind}_{LU}^G(M)$ and $\operatorname{ind}_{LV}^G(M)$ are infinite-dimensional for every $M\in \operatorname{\mathcal{R}}(L)$. By contrast, the representation $\operatorname{i}_{U,V}(M)$ is finite-dimensional whenever $M$ is: see Theorem \[thm:pind-properties\].
\[ex:pind-commuting\] Suppose that $(U,L,V)$ is a virtual Iwahori decomposition of $G$ such that the subgroups $U$ and $V$ commute with one another. Then the product $H\coloneq ULV$ is an open subgroup of $G$, isomorphic to $(U\times V)\rtimes L$. We have $e_U e_V = e_{U\times V}$, and there are isomorphisms of ${\mathcal H}(G)$-${\mathcal H}(L)$ bimodules $${\mathcal H}(G)e_U e_V = {\mathcal H}(G)e_{U\times V} \cong {\mathcal H}(G/(U\times V)).$$ Consequently the functor $\operatorname{i}_{U,V}$ is of the form $\operatorname{\mathcal{R}}(L) \xrightarrow{\operatorname{inf}} \operatorname{\mathcal{R}}(H) \xrightarrow{\operatorname{ind}} \operatorname{\mathcal{R}}(G)$ discussed in Section \[subsec:notation\].
We shall now establish some basic properties of the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$. Many of these properties were established in [@Dat_parahoric] for the case where $(U,L,V)$ is an actual, as opposed to a virtual, Iwahori decomposition of $G$. The proofs in [@Dat_parahoric] mostly carry over with only minor changes to the case of a virtual Iwahori decomposition, thanks to the following analogue of [@Dat_parahoric Proposition 2.2]. The proofs of these propositions, though, are quite different.
\[prop:z\] Let $G$ be a profinite group and let $L$, $U$ and $V$ be closed subgroups of G such that $L$ normalises $U$ and $V$. For every $M \in \operatorname{\mathcal{R}}(G)$ there is a linear automorphism $z_M\in \operatorname{GL}(M)$, commuting with the actions of $L$, $e_U$ and $e_V$, such that $z_M^{-1} e_U e_V$ is an idempotent in $\operatorname{End}(M)$.
Each smooth representation $M\in \operatorname{\mathcal{R}}(G)$ may be regarded as a representation of the infinite dihedral group $\Gamma=\langle s,t\ |\ s^2=t^2=1\rangle$, by sending $s\mapsto 2e_U-1$ and $t\mapsto 2e_V-1$. Since $G$ is profinite, every $M\in \operatorname{\mathcal{R}}(G)$ is isomorphic to a direct sum of finite-dimensional unitary representations of $G$, which restrict to finite-dimensional unitary representations of $\Gamma$ (unitary because the idempotents $e_U$ and $e_V$ are self-adjoint in ${\mathcal H}(G)$). It follows that every $M\in \operatorname{\mathcal{R}}(G)$ is semisimple as a representation of $\Gamma$, and so $M$ decomposes (uniquely) as the direct sum of its $\Gamma$-isotypic components.
We claim that in each irreducible representation $W$ of $\Gamma$ there is a nonzero $z_W\in {\mathbb{C}}$ such that $z_W^{-1} pq$ is an idempotent in $\operatorname{GL}(W)$, where $p=\frac{1}{2}(s+1)$ and $q=\frac{1}{2}(t+1)$. Indeed, since the dihedral group has an abelian normal subgroup of index two, every irreducible representation of $\Gamma$ is either one- or two-dimensional. In the one-dimensional case $p$ and $q$ commute and so we may take $z_W=1$. In the two-dimensional case, $pq$ and $(pq)^2$ are two nonzero maps between the one-dimensional subspaces $qW$ and $pW$, and so there is a (unique) nonzero scalar $z_W$ such that $pq=z_W^{-1}(pq)^2$.
Having established the claim, we let $z_M\in \operatorname{GL}(M)$ to be the automorphism of $M$ which acts as the scalar $z_W$ on the $W$-isotypical component of $M$. It is clear from the construction that $z_M$ commutes with $e_U$ and $e_V$, and that $z_M^{-1} e_U e_V$ is an idempotent. If $T\in \operatorname{End}(M)$ commutes with $e_U$ and $e_V$ then $T$ preserves the $\Gamma$-isotypic components, and so commutes with $z_M$. In particular, $z_M$ commutes with the $L$-action on $M$.
\[rem:z\] If $W$ is a two-dimensional irreducible unitary representation of the infinite dihedral group, then $z_W=\cos ^2 (\alpha_W)$, where $\alpha_W$ is the angle between the images of $p$ and $q$ in the Hilbert space $W$. Thus the eigenvalues of $z_M$ all lie in the interval $(0,1]$. [If the multiplication map $U\times L\times V\to G$ is a homeomorphism]{}, and $M$ is an irreducible representation of $G$, then $z_M$ is the scalar operator $$z_M = \begin{cases} \dim \operatorname{r}_{U,V}(M) / \dim M & \text{if }\operatorname{r}_{U,V} (M) \neq 0, \\ 1 & \text{if }\operatorname{r}_{U,V}(M)=0; \end{cases}$$ see [@Crisp_parahoric Proposition 1.11]. Moreover, Dat has shown that if $L$ contains an open normal subgroup $L^\dagger$ such that $UL^\dagger V$ is a pro-$p$ subgroup of $G$, then the eigenvalues lie in ${\mathbb{Z}}[1/p]$; see [@Dat_parahoric Proposition 2.2].
With the automorphisms $z_M$ in hand, many of the arguments from [@Dat_parahoric Section 2] carry over to our setting, and establish the following properties of the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$.
\[thm:pind-properties\] Let $(U, L,V)$ be a virtual Iwahori decomposition of a profinite group $G$, and consider the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$. Then:
1. \[item:pind-UV-VU\] There are natural isomorphisms $\operatorname{i}_{U,V}\cong \operatorname{i}_{V,U}$ and $\operatorname{r}_{U,V}\cong \operatorname{r}_{V,U}$.
2. \[item:pind-Hom\] $\operatorname{i}_{U,V}$ is naturally isomorphic to the functor $$\operatorname{i}'_{U,V} : M\mapsto \operatorname{Hom}_{{\mathcal H}(L)}(e_V e_U {\mathcal H}(G), M)^{\infty} ,$$ and is therefore right-adjoint to $\operatorname{r}_{U,V}$.
3. \[item:pres-Hom\] $\operatorname{r}_{U,V}$ is naturally isomorphic to the functor $$\operatorname{r}'_{U,V} : N\mapsto \operatorname{Hom}_{{\mathcal H}(G)} ({\mathcal H}(G)e_V e_U, N)^\infty,$$ and is therefore right-adjoint to $\operatorname{i}_{U,V}$.
4. \[item:pind-compat\] Let $(U',L,V')$ be a second virtual Iwahori decomposition of $G$, such that $$U= (U\cap U') (U\cap V'),\quad V=(V\cap U') (V\cap V'),$$ $$U'=(U'\cap U)(U'\cap V),\quad \text{and}\quad V'=(V'\cap U)(V'\cap V).$$ Then $\operatorname{i}_{U,V}\cong \operatorname{i}_{U',V'}$ and $\operatorname{r}_{U,V}\cong \operatorname{r}_{U',V'}$.
5. \[item:pind-finite\] Let $K$ be an open normal subgroup of $G$ with an Iwahori decomposition $(U_K, L_K, V_K)\coloneq (U\cap K, L\cap K, V\cap K)$. The diagrams $$\xymatrix@C=60pt{
\operatorname{\mathcal{R}}(L) \ar[r]^-{\operatorname{i}_{U,V}} & \operatorname{\mathcal{R}}(G) \\
\operatorname{\mathcal{R}}(L/L_K) \ar[u]^-{\operatorname{inf}} \ar[r]^-{\operatorname{i}_{U/U_K,V/V_K}} & \operatorname{\mathcal{R}}(G/ K) \ar[u]_-{\operatorname{inf}}
}
\qquad\text{and}\qquad
\xymatrix@C=60pt{
\operatorname{\mathcal{R}}(G) \ar[r]^-{\operatorname{r}_{U,V}} & \operatorname{\mathcal{R}}(L) \\
\operatorname{\mathcal{R}}(G/K) \ar[u]^-{\operatorname{inf}} \ar[r]^-{\operatorname{r}_{U/U_K,V/V_K}} & \operatorname{\mathcal{R}}(L/L_K)\ar[u]_-{\operatorname{inf}}
}$$ commute up to natural isomorphism. (Here $\operatorname{inf}$ denotes inflation.)
6. \[item:pind-nonzero\] $\operatorname{i}_{U,V}(M)$ is nonzero whenever $M$ is nonzero, and $\operatorname{i}_{U,V}(M)$ is finite-dimensional whenever $M$ is finite-dimensional.
7. \[item:pind-stages\] If $(X,H,Y)$ is a virtual Iwahori decomposition of $L$, then $$\operatorname{i}_{U,V} \circ \operatorname{i}_{X,Y} \cong \operatorname{i}_{U\rtimes X,Y\ltimes V}$$ as functors $\operatorname{\mathcal{R}}(H)\to \operatorname{\mathcal{R}}(G)$.
Parts and are instances of the following general fact, whose proof generalises the argument of [@Dat_parahoric Corollaire 2.7]:
\[lem:ind\_coind\] Let $H$ and $K$ be closed subgroups of a profinite group $G$. Let $X\subseteq {\mathcal H}(G)$ be an ${\mathcal H}(H)$-${\mathcal H}(K)$ subbimodule, and denote by $X^*$ the image of $X$ under the involution $f^*(g)=\overline{f(g^{-1})}$ on ${\mathcal H}(G)$; note that $X^*$ is an ${\mathcal H}(K)$-${\mathcal H}(H)$ bimodule. Suppose that for every open normal subgroup $H_1\subseteq H$ there is an open normal subgroup $G_1\subseteq G$ satisfying $$\label{ind_coind_eq}
e_{H_1}X \subseteq e_{G_1}{\mathcal H}(G).$$ Then the functors $\operatorname{\mathcal{R}}(K)\to \operatorname{\mathcal{R}}(H)$ defined by $$M \mapsto X\otimes_{{\mathcal H}(K)} M \qquad \text{and} \qquad M \mapsto \operatorname{Hom}_{{\mathcal H}(K)} (X^*, M)^{\infty}$$ are naturally isomorphic.
Consider the natural transformation $$\Phi : X\otimes_{{\mathcal H}(K)} M \to \operatorname{Hom}_{{\mathcal H}(K)}(X^*, M), \qquad
\Phi (x_1\otimes m) : x_2^* \mapsto (x_2^* x_1){\big{\vert}}_K \cdot m$$ where $(x_2^* x_1){\big{\vert}}_K$ means the restriction of the convolution product $x_2^* x_1\in {\mathcal H}(G)$ to the subgroup $K$. Fix an open normal subgroup $H_1\subseteq H$. We will show that the map $\Phi$ restricts to an isomorphism between the respective subspaces of $H_1$-fixed vectors.
Let $G_1$ be an open normal subgroup of $G$ satisfying condition , [so that]{} the subspace ${e_{H_1}X\subset {\mathcal H}(G)}$ consists exclusively of $G_1$-invariant functions. We may then replace $G$ by $G/G_1$, and assume for the rest of the proof that $G$ is a finite group. Furthermore, the module $M$ decomposes as a direct sum of finite-dimensional modules, and the natural map $\Phi$ commutes with direct sums, so we may assume that $M$ is finite-dimensional.
Now, the pairing $$X^*\times X \to {\mathbb{C}}\qquad (x_2^*, x_1) \mapsto (x_2^*x_1)(1)$$ is nondegenerate, since it is the restriction to $X$ of the natural $L^2$-inner product on ${\mathcal H}(G)$. It follows from this, and from the standard duality theory of finite-dimensional vector spaces, that the map $$\Psi: X\otimes_{{\mathbb{C}}} M \to \operatorname{Hom}_{{\mathbb{C}}}(X^*, M) \qquad \Psi(x_1\otimes m): x_2^* \mapsto (x_2^*x_1)(1)\cdot m$$ is an isomorphism. The map $\Psi$ descends to an isomorphism of $K$-coinvariants $$\label{ind_coind_proof1}
X\otimes_{{\mathcal H}(K)} M \xrightarrow{\cong} \left(\operatorname{Hom}_{{\mathbb{C}}}(X^*, M)\right)_K,$$ where the $K$-action on $\operatorname{Hom}_{{\mathbb{C}}}(X^*,M)$ is by conjugation. Averaging over $K$ gives an isomorphism of $K$-coinvariants with $K$-invariants: $$\label{ind_coind_proof2}
\left(\operatorname{Hom}_{{\mathbb{C}}}(X^*, M)\right)_K \xrightarrow[\cong]{T \mapsto \int_K k T k^{-1}\, {d}k} \operatorname{Hom}_{{\mathcal H}(K)}(X^*,M),$$ and the map $\Phi$ is the composition of the isomorphisms and .
To prove part , let $z_{{\mathcal H}(G)}$ be the automorphism of ${\mathcal H}(G)$ obtained by applying Proposition \[prop:z\] to the left-translation action of $G$. Then the maps $$e_U e_V {\mathcal H}(G) \xrightarrow{f\mapsto e_V f} e_V e_U {\mathcal H}(G) \quad \text{and} \quad e_V e_U {\mathcal H}(G) \xrightarrow{f\mapsto z_{{\mathcal H}(G)}^{-1} e_U f} e_U e_V {\mathcal H}(G)$$ are mutually inverse isomorphisms of ${\mathcal H}(L)$-${\mathcal H}(G)$ bimodules, giving rise to a natural isomorphism of functors ${\operatorname{r}_{U,V}\cong \operatorname{r}_{V,U}}$. A similar argument, using the right action of $G$ on ${\mathcal H}(G)$, gives $\operatorname{i}_{U,V}\cong \operatorname{i}_{V,U}$.
To prove part we apply Lemma \[lem:ind\_coind\] with $H=G$, $K=L$, and $X={\mathcal H}(G)e_U e_V$. The hypothesis is trivially satisfied and we conclude that the functor $\operatorname{i}_{U,V}$ is naturally isomorphic to $\operatorname{i}'_{U,V}$. The standard $\otimes$-$\operatorname{Hom}$ adjunction implies that $\operatorname{i}'_{U,V}$ is right-adjoint to $\operatorname{r}_{V,U}$, and we have $\operatorname{r}_{V,U} \cong \operatorname{r}_{U,V}$ by part (1); see [@Renard I.2.2 (Corollaire)] for a formulation and proof of the adjunction in the present context.
To prove part we apply Lemma \[lem:ind\_coind\] again, this time with $H=L$, $K=G$ and ${X=e_U e_V {\mathcal H}(G)}$. To verify the hypothesis , fix an open normal subgroup $H_1\subseteq L$. Then there is an open normal subgroup $G_{0}\subseteq G$ having an Iwahori decomposition $(U_{0}, L_{0}, V_{0})$, where $L_{0}$ is contained in $H_1$. Here $Y_{0}$ means $Y\cap G_{0}$ for every subset $Y \subset G$. We then have $$e_{H_1}e_U e_V{\mathcal H}(G) \subseteq e_{L_{0}}e_U e_V {\mathcal H}(G) = e_U(e_{U_{0}}e_{L_{0}}e_{V_{0}})e_V {\mathcal H}(G) \subseteq e_{G_{0}}{\mathcal H}(G),$$ so is satisfied [by $G_1=G_0$]{}. Now Lemma \[lem:ind\_coind\] implies that $\operatorname{r}_{U,V}$ is isomorphic to $\operatorname{r}'_{U,V}$, which is right-adjoint to $\operatorname{i}_{U,V}$ [by the argument of part ]{}.
Part follows from Proposition \[prop:z\], as in [@Dat_parahoric Lemme 2.9].
Part follows from the equality $e_K = e_{U_K}e_{L_K} e_{V_K}$, as was remarked in [@Dat_parahoric p.272]. [It is also a consequence of Theorem \[thm:C1\], below.]{}
The finite-dimensionality assertion in part follows from part : every finite-dimensional smooth representation of $L$ is inflated from a representation of some finite quotient $L/L_K$, and the functor $\operatorname{i}_{U/U_K,V/V_K}$ obviously preserves finite-dimensionality. To prove that $\operatorname{i}_{U,V}(M)\neq 0$ as long as $M\neq 0$, fix a nonzero $m\in M$ and let $f\in \operatorname{ind}_{LU}^G (M)$ be the function supported on the open set $ULV\subset G$, and given there by $f(ulv)=l\cdot m$. The image of $f$ under the intertwiner $J_V:\operatorname{ind}_{LU}^G (M) \to \operatorname{ind}_{LV}^G (M)$ (see ) is nonzero, because $$(J_V f)(1) = \int_V f(v)\, {d}v =m,$$ and so $\operatorname{i}_{U,V}(M)\cong \operatorname{Im}(J_V)$ is nonzero.
For part , convolution over $L$ gives an isomorphism of ${\mathcal H}(G)$-${\mathcal H}(H)$ bimodules $${\mathcal H}(G)e_U e_V \otimes_{{\mathcal H}(L)} {\mathcal H}(L)e_X e_Y \xrightarrow{\cong} {\mathcal H}(G)e_U e_V e_X e_Y.$$ Now $e_X$ commutes with $e_V$ since $X$ normalises $V$, and we have $e_U e_X = e_{U\rtimes X}$ and $e_V e_Y = e_{Y\ltimes V}$, and so we have produced an isomorphism between the bimodules representing the functors $\operatorname{i}_{U,V}\circ \operatorname{i}_{X,Y}$ and $\operatorname{i}_{U\rtimes X, Y\ltimes V}$.
If the triple $(U,L,V)$ is an actual Iwahori decomposition of $G$, then the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$ enjoy the following additional properties, which we recall from [@Dat_parahoric] and [@Crisp_parahoric] for the reader’s convenience:
\[thm:pind-Iw\] Suppose that $(U,L,V)$ is an Iwahori decomposition of a profinite group $G$. Then:
1. \[item:pind-Iw-irr\] $\operatorname{i}_{U,V}$ sends $\operatorname{Irr}(L)$ to $\operatorname{Irr}(G)$, while $\operatorname{r}_{U,V}$ sends $\operatorname{Irr}(G)$ to $\operatorname{Irr}(L)\sqcup \{0\}$.
2. \[item:pind-Iw-prespind\] $\operatorname{r}_{U,V}\circ\operatorname{i}_{U,V}\cong {\mathrm{id}}_{\operatorname{\mathcal{R}}(L)}$.
3. \[item:pind-Iw-pindpres\] If $M\in \operatorname{Irr}(G)$ and $\operatorname{r}_{U,V}(M)\neq 0$, then $\operatorname{i}_{U,V} \operatorname{r}_{U,V} (M)\cong M$.
4. \[item:pind-Iw-Hom\] If $M\in \operatorname{Irr}(G)$, then either $\operatorname{Hom}_L(M^U, M^V)$ is zero, in which case $\operatorname{r}_{U,V}(M)=0$; or $\operatorname{Hom}_L(M^U, M^V)$ is one-dimensional, in which case it is spanned by the operator $e_Ve_U$, which is an isomorphism, and $\operatorname{r}_{U,V}(M)\cong M^U\cong M^V$.
5. \[item:pind-Iw-UV\] Given $M\in \operatorname{Irr}(G)$ and $N\in \operatorname{Irr}(L)$, one has $M \cong \operatorname{i}_{U,V}(N)$ if and only if $N$ is a common subrepresentation of $M^U$ and $M^V$.
6. \[item:pind-Iw-character\] For each $({\varphi},M) \in \operatorname{Irr}(G)$ there is a nonzero scalar $c$ such that $$e_U e_{\varphi}e_V = c e_U e_{\operatorname{r}_{U,V}({\varphi})} e_V$$ as operators on ${\mathcal H}(G)$.
Parts and are proved in [@Dat_parahoric Corollaire 2.10]. Part is proved in [@Crisp_parahoric Lemma 1.10], and part follows from parts and .
To prove part : if $\operatorname{r}_{U,V}(M)\neq 0$ then the adjunction in part of Theorem \[thm:pind-properties\] gives a nonzero intertwiner $M\to \operatorname{i}_{U,V} \operatorname{r}_{U,V}(M)$. Both of these representations are irreducible (by part ), and so they are isomorphic.
Part follows from the character formula for $\operatorname{r}_{U,V}$ proved in [@Crisp_parahoric Proposition 1.11]. That formula implies that there is a nonzero $s\in {\mathbb{C}}$ such that $$\operatorname{ch}_{\operatorname{r}_{U,V}({\varphi})}(l) = s \int_U \int_V \operatorname{ch}_{\varphi}(vlu)\, {d}u \, {d}v$$ for all $l\in L$. Writing $\sim$ to indicate equality up to a nonzero scalar multiple, the operator $e_U e_{\operatorname{r}_{U,V}({\varphi})} e_V$ is thus given by $$\begin{aligned}
e_U e_{\operatorname{r}_{U,V}({\varphi})} e_V & \sim \int_U \int_L \int _V \operatorname{ch}_{\operatorname{r}_{U,V}({\varphi})}(l^{-1}) ulv\, {d}u\, {d}l \, {d}v \\
&\sim \int_U \int_L \int _V \left( \int_U \int_V \operatorname{ch}_{\varphi}(v_1^{-1} l^{-1} u_1^{-1})\, {d}u_1\,\ {d}v_1\right) ulv\, {d}u\, {d}l \, {d}v \\
&\sim \int_U \int_L \int _V \left( \int_U \int_V \operatorname{ch}_{\varphi}(v_1^{-1} l^{-1} u_1^{-1})\, {d}u_1\,\ {d}v_1\right) u(u_1lv_1)v\, {d}u\, {d}l \, {d}v \\
&\sim \int_U \int_G \int_V \operatorname{ch}_{\varphi}(g^{-1}) ugv\, {d}u\, {d}g\, {d}v \\
&\sim e_U e_{\varphi}e_V
\end{aligned}$$ where in the third step we used invariance of the Haar measures, and in fourth we used the fact that the product of the Haar measures on $U$, $L$ and $V$ is a Haar measure on $G=ULV$.
Relations of the functors $\operatorname{i}$ and $\operatorname{r}$ with Clifford theory {#sec:Clifford}
========================================================================================
The functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$ can be difficult to work with, since the bimodule ${\mathcal H}(G)e_U e_V$ is not obviously the space of functions on any nice $G\times L$-space. The situation where $(U,L,V)$ is an actual, rather than a virtual, Iwahori decomposition of $G$ is significantly easier to deal with; see Sections \[sec:orbit\] and \[sec:Iwahori\], for instance. If $G$ admits only a virtual Iwahori decomposition, then $G$ contains an open normal subgroup $G_0$ which admits an actual Iwahori decomposition (this is part of the definition). In this section we will firstly recall how Clifford theory reduces the study of representations of $G$ to that of projective representations of certain subgroups of $G/G_0$; and then we will show how the induction and restriction functors $\operatorname{i}$ and $\operatorname{r}$ are compatible with this reduction.
Review of Clifford theory {#subsec:Clifford_review}
-------------------------
Let us recall the basic assertions of Clifford theory. Details can be found in [@Karpilovsky], for example.
Let $G_0$ be an open normal subgroup of a profinite group $G$. Then $G$ acts by conjugation on the set $\operatorname{Irr}(G_0)$ of isomorphism classes of irreducible representations of $G_0$. For each ${\varphi}\in \operatorname{Irr}(G_0)$ we let $\operatorname{\mathcal{R}}(G)_{\varphi}$ denote the category of smooth representations of $G$ whose restriction to $G_0$ contains only representations in the $G$-orbit $G\cdot{\varphi}$. The first assertion of Clifford theory is: $$\tag{C1}\label{C1}
\text{$\operatorname{\mathcal{R}}(G)$ is equivalent to the product }\prod_{G\cdot {\varphi}\in G\backslash \operatorname{Irr}(G_0)} \operatorname{\mathcal{R}}(G)_{\varphi}.$$
Fix an irreducible representation ${\varphi}:G_0\to \operatorname{GL}(W)$ of $G_0$, and let $G({\varphi})$ denote the stabiliser of ${\varphi}$ in $G$. Since ${\varphi}$ is smooth, it is trivial on some open normal subgroup $G_{00}$ of $G$, and replacing $G$ by $G/G_{00}$ we might as well assume—as we shall, for the rest of Section \[subsec:Clifford\_review\]—that $G$ is finite. We use the counting measure on $G$ to define the convolution on ${\mathcal H}(G)$, so that the $\delta$-functions $\delta_g$ satisfy $\delta_g\delta_h=\delta_{gh}$.
Representations may be induced from $G({\varphi})$ to $G$ in the usual way (see Section \[subsec:notation\]). The second assertion of Clifford theory is: $$\tag{C2}\label{C2}
\text{The functor \ $\operatorname{ind}:\operatorname{\mathcal{R}}(G({\varphi}))_{\varphi}\to \operatorname{\mathcal{R}}(G)_{{\varphi}}$ \ is an equivalence of categories.}$$ An inverse is given by the functor which sends a representation $M \in \operatorname{\mathcal{R}}(G)_{{\varphi}}$ to its [$G({\varphi})$-]{}subspace $e_{\varphi}M$, where $e_{\varphi}\in {\mathcal H}(G_0)$ is the central idempotent associated to ${\varphi}$. Note that the category $\operatorname{\mathcal{R}}(G({\varphi}))_{\varphi}$ is equivalent, in an obvious way, to the category of modules over the direct-summand $e_{\varphi}{\mathcal H}(G({\varphi}))$ of the algebra ${\mathcal H}(G({\varphi}))$.
Let ${\overline}{G}$ denote the quotient $G/G_0$, and let $\theta:G\to {\overline}{G}$ be the quotient map. Schur’s lemma implies that ${\varphi}$ admits a *projective extension* [to $G({\varphi})$]{}, i.e. a map ${\varphi}':G({\varphi})\to \operatorname{GL}(W)$ which becomes a group homomorphism upon passing to the quotient $\operatorname{PGL}(W)$, and which satisfies $${\varphi}'(g_0 g) = {\varphi}(g_0){\varphi}'(g) \quad \text{and}\quad {\varphi}'(g g_0) = {\varphi}'(g){\varphi}(g_0)$$ for all $g\in G({\varphi})$ and all $g_0\in G_0$. These two properties imply that there is a two-cocycle $\gamma:{\overline}{G}({\varphi})\times {\overline}{G}({\varphi})\to {\mathbb{C}}^\times$ whose inflation to a cocycle on $G({\varphi})$, which we also denote by $\gamma$, satisfies $$\label{eqn:def.of.gamma}
{\varphi}'(g_1){\varphi}'(g_2)= \gamma(g_1,g_2)^{-1}{\varphi}'(g_1 g_2).$$
We call $\gamma^{-1}$ the two-cocycle associated to ${\varphi}'$; the cocycles associated to different choices of projective extensions of ${\varphi}$ are cohomologous. The projective representation ${\varphi}'$ may be regarded as a module over the twisted group algebra ${\mathcal H}^{\gamma^{-1}}(G({\varphi}))$, a construction that we now recall. To a two-cocycle $\alpha$ on a finite group $\Gamma$, one may associate the [*twisted group algebra*]{} ${\mathcal H}^\alpha(\Gamma)$, that is, the algebra of complex-valued functions on $G$, with twisted convolution multiplication $\cdot_\alpha$ defined on the basis $\left\{ \delta_g \ |\ g \in \Gamma \right \}$ of $\delta$-functions by $$\delta_{g_1}\cdot_\alpha \delta_{g_2} \coloneq \alpha(g_1,g_2) \delta_{g_1 g_2}.$$
We let $\operatorname{\mathcal{R}}^\alpha(\Gamma)$ denote the category of ${\mathcal H}^\alpha(\Gamma)$-modules. An immediate consequence of the definition is that if $\alpha,\beta: \Gamma \times \Gamma \to {\mathbb{C}}^\times$ are two-cocycles, and $M$ and $N$ are modules over ${\mathcal H}^\alpha(\Gamma)$ and ${\mathcal H}^\beta(\Gamma)$ respectively, then $M \otimes N$ is naturally an ${\mathcal H}^{\alpha \beta}(\Gamma)$-module with respect to the diagonal action.
Returning to our setup, if $M$ is an ${\mathcal H}^\gamma({\overline}{G}({\varphi}))$-module, then by inflation $M$ is also an ${\mathcal H}^\gamma(G({\varphi}))$-module. As $W \in \operatorname{\mathcal{R}}^{\gamma^{-1}}(G({\varphi}))$ we get that $M \otimes W$ is an ${\mathcal H}^{\gamma \cdot \gamma^{-1}}({\overline}{G}({\varphi}))$-module, i.e. an ordinary (as opposed to a projective) representation of $G({\varphi})$, and the third assertion of Clifford theory is: $$\tag{C3}\label{C3}
\text{The functor \ $\otimes{\varphi}':\operatorname{\mathcal{R}}^{\gamma}({\overline}{G}({\varphi}))\xrightarrow{M \mapsto M\otimes W} \operatorname{\mathcal{R}}(G({\varphi}))_{{\varphi}}$ \ is an equivalence of categories.}$$ An equivalent formulation of , which we shall use below, is that the map $$\theta\otimes{\varphi}' : {\mathcal H}(G({\varphi}))e_{\varphi}\to {\mathcal H}^\gamma({\overline}{G}({\varphi}))\otimes_{{\mathbb{C}}} \operatorname{End}(W) \qquad \delta_g e_{\varphi}\mapsto \delta_{\theta(g)}\otimes {\varphi}'(g)$$ is an isomorphism of algebras. Then the equivalence decomposes as $$\label{eq:C3_decomposition}
\operatorname{\mathcal{R}}^\gamma({\overline}{G}({\varphi})) \xrightarrow[\cong]{M\mapsto M\otimes W} \operatorname{Mod}\left({\mathcal H}^\gamma({\overline}{G}({\varphi}))\otimes \operatorname{End}(W)\right) \xrightarrow[\cong]{ (\theta\otimes{\varphi}')^* } \operatorname{\mathcal{R}}(G({\varphi}))_{\varphi}.$$ [Here $\operatorname{Mod}(R)$ denotes the category of left $R$-modules.]{}
This ends our review of Clifford theory. We shall now explain the compatibility with the functors $\operatorname{i}$ and $\operatorname{r}$.
Induction and Clifford theory
-----------------------------
In this section we let $G$ be a profinite group, with a virtual Iwahori decomposition $(U,L,V)$ as in Definition \[def:vI\]. We also fix one open normal subgroup $G_0\subset G$ for which the product mapping $$U_0 \times L_0 \times V_0 \to G_0$$ is a homeomorphism, where $H_0\coloneq H\cap G_0$ for every subgroup $H\subseteq G$. We consider the induction functors $$\operatorname{i}=\operatorname{i}_{U,V}:\operatorname{\mathcal{R}}(L)\to \operatorname{\mathcal{R}}(G) \qquad \text{and}\qquad \operatorname{i}_0 = \operatorname{i}_{U_0,V_0}:\operatorname{\mathcal{R}}(L_0)\to \operatorname{\mathcal{R}}(G_0),$$ along with their adjoint restriction functors $\operatorname{r}$ and $\operatorname{r}_0$, as in Definition \[def:pind\].
It follows from part of Theorem \[thm:pind-Iw\] that the functor $\operatorname{i}_0$ sends irreducible representations of $L_0$ to irreducible representations of $G_0$, and thus produces a map from $\operatorname{Irr}(L_0)$ to $\operatorname{Irr}(G_0)$.
\[L\_eq\_lemma\] The map $\operatorname{i}_0:\operatorname{Irr}(L_0)\to \operatorname{Irr}(G_0)$ is $L$-equivariant and injective.
$L$ normalises $U_0$ and $V_0$, and so commutes with $e_{U_0}$ and $e_{V_0}$. The injectivity follows from part of Theorem \[thm:pind-Iw\].
The functors $\operatorname{i}$ and $\operatorname{r}$ are compatible with the decomposition , in the following sense.
\[thm:C1\] With the above notation one has $\operatorname{i}(\operatorname{\mathcal{R}}(L)_\psi)\subseteq \operatorname{\mathcal{R}}(G)_{\operatorname{i}_0(\psi)}$, for every $\psi\in \operatorname{Irr}(L_0)$.
We first claim that ${\mathcal H}(G)e_U e_V$ is isomorphic, as an ${\mathcal H}(G)$-${\mathcal H}(L)$ bimodule, to some submodule of ${\mathcal H}(G)e_{U_0}e_{V_0}$. This is because $${\mathcal H}(G)e_U e_V \subseteq {\mathcal H}(G) e_{U_0} e_V \cong {\mathcal H}(G)e_V e_{U_0} \subseteq {\mathcal H}(G)e_{V_0} e_{U_0} \cong {\mathcal H}(G)e_{U_0} e_{V_0} ,$$ where the inclusions hold because $U_0$ and $V_0$ are subgroups of $U$ and $V$, respectively, and the isomorphisms hold by part of Theorem \[thm:pind-properties\].
For each $N\in \operatorname{\mathcal{R}}(L)_\psi$ we now have (up to $G$-equivariant isomorphism) $$\label{eqn:submodules.induction}
\operatorname{i}(N) \subseteq {\mathcal H}(G)e_{U_0} e_{V_0} \otimes_{{\mathcal H}(L)} N \subseteq {\mathcal H}(G)e_{U_0}e_{V_0}\otimes_{{\mathcal H}(L_0)} \operatorname{res}^L_{L_0}(N)$$ where the first inclusion holds because of the inclusion of bimodules established above, and the second inclusion holds because the tensor product over ${\mathcal H}(L)$ is a quotient—and therefore also a submodule—of the tensor product over the subalgebra ${\mathcal H}(L_0)$. The restriction to $G_0$ of the right-hand side in is a direct sum of $G$-conjugates of $\operatorname{i}_0 (\operatorname{res}^L_{L_0} (N))$, where $\operatorname{res}^L_{L_0}(N)$ is a direct sum of $L$-conjugates of $\psi$. Since $\operatorname{i}_0$ is $L$-equivariant this implies that the restriction of $\operatorname{i}(N)$ to $G_0$ is a direct sum of $G$-conjugates of $\operatorname{i}_0(\psi)$, as claimed.
For the rest of this section we shall fix an irreducible representation $\psi:L_0\to \operatorname{GL}(W)$ of $L_0$, and study the functor $\operatorname{i}$ on the subcategory $\operatorname{\mathcal{R}}(L)_\psi$. There is an open normal subgroup $G_{00}\subset G$ with an Iwahori decomposition $G_{00}= U_{00} L_{00} V_{00}$, such that $\psi$ is trivial on $L_{00}$. Part of Theorem \[thm:pind-properties\] allows us to replace $G$ by the quotient $G/G_{00}$, and so we may assume without loss of generality for the rest of this section that **$\boldsymbol G$ is a finite group**. We consequently take all Haar measures to be counting measures, so that the $\delta$ functions on elements of $G$ satisfy $\delta_g\delta_h=\delta_{gh}$ inside ${\mathcal H}(G)$.
To simplify the notation let us write ${\varphi}$ for $\operatorname{i}_0(\psi)$. Lemma \[L\_eq\_lemma\] implies that $G({\varphi})\cap L = L(\psi)$. Let $U({\varphi})$ and $V({\varphi})$ denote the inertia groups of ${\varphi}$ in $U$ and $V$, respectively, and consider the functor $$\operatorname{i}_\psi\coloneq \operatorname{i}_{U({\varphi}),V({\varphi})}:\operatorname{\mathcal{R}}(L(\psi))_{\psi} \to \operatorname{\mathcal{R}}(G({\varphi}))_{{\varphi}}$$ given by tensor product with the $e_{\varphi}{\mathcal H}(G({\varphi}))$-$e_\psi{\mathcal H}(L(\psi))$ bimodule $e_{\varphi}{\mathcal H}(G({\varphi}))e_{U({\varphi})}e_{V({\varphi})}e_\psi$.
The functors $\operatorname{i}$ and $\operatorname{r}$ are compatible with the equivalence as follows:
\[thm:C2\] The diagram $$\xymatrix@C=50pt{
\operatorname{\mathcal{R}}(L)_{\psi} \ar[r]^-{\operatorname{i}} & \operatorname{\mathcal{R}}(G)_{{\varphi}} \\
\operatorname{\mathcal{R}}(L(\psi))_\psi \ar[u]^-{\operatorname{ind}}_-{\cong} \ar[r]^-{\operatorname{i}_\psi} & \operatorname{\mathcal{R}}(G({\varphi}))_{{\varphi}} \ar[u]_-{\operatorname{ind}}^-{\cong}
}$$ commutes up to a natural isomorphism.
We will replace the right-hand vertical arrow in the diagram by its inverse, and prove that the diagram $$\label{Clifford_diagram2}
\xymatrix@C=50pt{
\operatorname{\mathcal{R}}(L)_{\psi} \ar[r]^-{\operatorname{i}} & \operatorname{\mathcal{R}}(G)_{{\varphi}}\ar[d]^-{e_{\varphi}} \\
\operatorname{\mathcal{R}}(L(\psi))_\psi \ar[u]^-{\operatorname{ind}} \ar[r]^-{\operatorname{i}_\psi} & \operatorname{\mathcal{R}}(G({\varphi}))_{{\varphi}}
}$$ commutes up to natural isomorphism. This amounts to producing an isomorphism of $G({\varphi})$-$L(\psi)$ bimodules $$\label{Clifford_equation}
e_{{\varphi}}{\mathcal H}(G) e_{U} e_{V} e_\psi \cong
e_{{\varphi}} {\mathcal H}(G({\varphi}))e_{U({\varphi})}e_{V({\varphi})} e_\psi.$$
We first claim that $$\label{Clifford_step1}
e_{{\varphi}}{\mathcal H}(G) e_{U} e_{V}e_\psi = e_{{\varphi}}{\mathcal H}(G({\varphi}))e_{U({\varphi})} e_{V} e_\psi,$$ the equality holding inside ${\mathcal H}(G)$. For vectors $x$ and $y$ we write $x \sim y$ if they differ by a non-zero scalar multiple. To prove we compute for every $g\in G$: $$\delta_g e_{U}e_{V} e_\psi {=} \delta_g e_{U} ( e_{U_0} e_\psi e_{V_0} )e_{V} {e_\psi}
\sim \delta_g e_{U} (e_{U_0}e_{\varphi}e_{V_0})e_{V} {e_\psi}
{=} \delta_g e_U e_{{\varphi}} e_V {e_\psi}
\sim \sum_{u\in U/U({\varphi})} {\delta_{gu}} e_{\varphi}e_{U({\varphi})} e_{V} {e_\psi},$$ where [in the first step we have used that $e_\psi$ is an idempotent which commutes with $e_V$, and in the second step we have used part of Theorem \[thm:pind-Iw\]]{}. Orthogonality of characters then implies $$\begin{aligned}
e_{\varphi}\delta_g e_{U}e_{V}e_\psi & \sim
\sum_{u\in U/U({\varphi})} e_{\varphi}e_{(gu)\cdot{\varphi}} {\delta_{gu}} e_{U({\varphi})} e_{V} {e_\psi}\\
& \sim
\begin{cases} e_{\varphi}{\delta_{gu}} e_{U ({\varphi})} e_{V}e_\psi & \text{if $\exists u\in U$ with $gu\in G({\varphi})$,}\\ 0 &\text{otherwise.}
\end{cases}\end{aligned}$$ This shows that every element of the left-hand side of can be written as an element of the right-hand side, and vice versa.
Now, the $G({\varphi})\times L(\psi)$-equivariant map $$\label{Clifford_step2}
e_{\varphi}{\mathcal H}(G({\varphi})) e_{U({\varphi})}e_{V({\varphi})}e_\psi \xrightarrow{f\mapsto fe_{V}}
e_{\varphi}{\mathcal H}(G({\varphi})) e_{U({\varphi})}e_{V}e_\psi$$ is obviously surjective. It is injective as well, for if $f\in {\mathcal H}(G({\varphi}))e_{V({\varphi})}$ then $$fe_V \sim \sum_{v\in V({\varphi})\backslash V} fe_{V({\varphi})} \delta_v = \sum_{v\in V({\varphi})\backslash V} f\delta_v,$$ where the functions $f\delta_v$ (as $v$ varies over $V({\varphi})\backslash V$) are supported on the disjoint cosets $G({\varphi})v$ and are therefore linearly independent. Thus the map is an isomorphism. Composing with the equality gives the desired isomorphism .
We are still fixing an irreducible representation $\psi:L_0\to \operatorname{GL}(W)$ and [letting]{} ${\varphi}$ [denote the induced representation]{} $\operatorname{i}_0(\psi):G_0\to GL(\operatorname{i}_0 (W))$. Consider the quotients $${\overline}{G}({\varphi})= G({\varphi})/G_0,\quad {\overline}{L}(\psi) = L(\psi)/L_0,\quad {\overline}{U}({\varphi})= U({\varphi})/U_0,\quad {\overline}{V}({\varphi}) =V({\varphi})/V_0.$$
Choose a projective extension ${\varphi}'$ of ${\varphi}$ to $G({\varphi})$, and let $\gamma^{-1}$ be the associated two-cocycle as in . Part of Theorem \[thm:pind-Iw\] implies that there is an $L_0$-equivariant isomorphism $$\Theta:W\xrightarrow{\cong} {\varphi}(e_{U_0}){\varphi}(e_{V_0})\operatorname{i}_0 (W),$$ unique up to a nonzero scalar [multiple]{}. Since the subgroup $L(\psi)$ normalizes the subgroups $U_0$ and $V_0$ it follows that ${\varphi}'(l)$ commutes with ${\varphi}(e_{U_0}){\varphi}(e_{V_0})$ for every $l\in L(\psi)$, and therefore ${\varphi}'(l)$ stabilises the subspace ${\varphi}(e_{U_0}){\varphi}(e_{V_0})\operatorname{i}_0 (W) \subset \operatorname{i}_0 (W)$. The map $\psi':L(\psi)\to \operatorname{GL}(W)$ defined by $$\label{eq:psiprime_def}
\psi'(l) = \Theta^{-1}{\varphi}'(l)\Theta$$ is then a projective extension of $\psi$ to $L(\psi)$, independent of the choice of $\Theta$. (It does depend, however, on the choice of ${\varphi}'$.) An easy argument shows that the resulting two-cocycle on ${\overline}{L}(\psi)$ is just the restriction of the two-cocycle $\gamma$ to ${{\overline}{L}(\psi)}$, and we shall therefore denote both two-cocycles by the same letter.
\[lem:C3\_eX\] Given a projective extension ${\varphi}'$ of ${\varphi}$ as above, there are unique scalars $a_x,b_y\in {\mathbb{C}}^\times$ for $x\in {\overline}{U}({\varphi})$ and $y\in {\overline}{V}({\varphi})$ such that the elements $$e^{{\varphi}'}_{{\overline}{U}({\varphi})} \coloneq \frac{1}{|{\overline}{U}({\varphi})|} \sum_{x\in {\overline}{U}({\varphi})} a_x \delta_x
\qquad\text{and}\qquad e_{{\overline}{V}({\varphi})}^{{\varphi}'} \coloneq \frac{1}{|{\overline}{V}({\varphi})|}\sum_{y\in {\overline}{V}({\varphi})} b_y \delta_y$$ are idempotents in ${\mathcal H}^\gamma({\overline}{G}({\varphi}))$, commute with the subalgebra ${\mathcal H}^{\gamma}({\overline}{L}(\psi))$, and such that the image of the elements $e_{U({\varphi})}e_{\varphi}$ and $e_{V({\varphi})}e_{\varphi}$ under the isomorphism of algebras $$\theta{\otimes}{\varphi}':{\mathcal H}(G({\varphi}))e_{{\varphi}} \to {\mathcal H}^{\gamma}({\overline}{G}({\varphi})){\otimes}\operatorname{End}(\operatorname{i}_0 (W))$$ are $e_{{\overline}{U}({\varphi})}^{{\varphi}'}{\otimes}{\varphi}(e_{U_0})$ and $e_{{\overline}{V}({\varphi})}^{{\varphi}'}{\otimes}{\varphi}(e_{V_0})$, respectively.
We will prove the lemma for the $U$-subgroups. The proof for the $V$-subgroups is identical.
We know, by Parts and of Theorem \[thm:pind-Iw\], that ${\varphi}(e_{U_0})\operatorname{i}_0 (W)\cong W$ as representations of $L_0$, and in particular that ${\varphi}(e_{U_0})\operatorname{i}_0(W)$ is irreducible over $L_0$.
Let $u\in U({\varphi})$. Since $U_0$ is a normal subgroup of $U({\varphi})$, we know that $u$ commutes with $e_{U_0}$ in ${\mathcal H}(G({\varphi}))$, and so ${\varphi}'(u)$ induces a linear automorphism of the subspace ${\varphi}(e_{U_0})\operatorname{i}_0 (W)$. Moreover, for $l\in L_0$ we have that ${ulu^{-1}l^{-1}\in U\cap G_0 = U_0}$. Writing $ul=luu_0$ with $u_0\in U_0$, we observe that on ${\varphi}(e_{U_0})\operatorname{i}_0 (W)$ the operator ${\varphi}'(u)$ commutes with ${\varphi}(l)$ for every $l\in L_0$. Hence, by Schur’s lemma, the operator ${\varphi}'(u)$ acts on ${\varphi}(e_{U_0})\operatorname{i}_0(W)$ as a non-zero scalar $a_u \in {\mathbb{C}}^\times$. The scalar $a_u$ depends only on the class of $u$ in the quotient ${\overline}{U}({\varphi})=U({\varphi})/U_0$, and so for each $x\in {{\overline}{U}({\varphi})}$ we may define $a_x\coloneq a_{\tilde{x}}$, where $\tilde{x}\in U({\varphi})$ is any lift of $x$.
The image of the idempotent $e_{U({\varphi})}e_{\varphi}$ under $\theta{\otimes}{\varphi}'$ is therefore $$\begin{split}
\theta{\otimes}{\varphi}' (e_{U({\varphi})}e_{\varphi}) &= \frac{1}{|U({\varphi})|} \sum_{u\in U({\varphi})} \theta(\delta_u)\otimes {\varphi}'(u)\\
&= \frac{1}{|{{\overline}{U}({\varphi})}|\cdot |U_0|} \sum_{\substack{x\in {{\overline}{U}({\varphi})}\\ u_0\in U_0}} \delta_x \otimes {\varphi}'(\tilde{x}){\varphi}(u_0) \\
&= \frac{1}{|{{\overline}{U}({\varphi})}|} \sum_{x\in {{\overline}{U}({\varphi})}} \delta_x \otimes {\varphi}'(\tilde{x}){\varphi}(e_{U_0}) = e^{{\varphi}'}_{{\overline}{U}({\varphi})} \otimes {\varphi}(e_{U_0}).
\end{split}$$
From the fact that the element $e_{U({\varphi})}e_{\varphi}$ is an idempotent which commutes with the elements of $L(\psi)$ in ${\mathcal H}(G({\varphi}))$, and the fact that $\theta{\otimes}{\varphi}'$ is an algebra homomorphism, it follows immediately that $e_{{\overline}{U}({\varphi})}^{{\varphi}'}$ is an idempotent which commutes with the subalgebra ${\mathcal H}^{\gamma}({{\overline}{L}(\psi)})$.
Finally, the uniqueness of the scalars $a_x$ follows from the linear independence of the elements $\delta_x\otimes {\varphi}(e_{U_0})$ in ${\mathcal H}^\gamma({\overline}{G}({\varphi}))\otimes \operatorname{End}(\operatorname{i}_0(W))$.
[From the proof of [Lemma \[lem:C3\_eX\]]{} it follows that]{} the restriction of $\gamma$ to ${{\overline}{U}({\varphi})}$ and to ${{\overline}{V}({\varphi})}$ is cohomologous to the trivial two-cocycle. Indeed, following the proof of the lemma, we see that for ${x_1,x_2\in {{\overline}{U}({\varphi})}}$ [one has]{} $a_{x_1}a_{x_2}\gamma(x_1,x_2) = a_{x_1x_2}$. In other words, $a:{{\overline}{U}({\varphi})}\to {\mathbb{C}}^\times$ is a coboundary which provides a trivialisation of the restriction of $\gamma$ to ${{\overline}{U}({\varphi})}$. The same is true for ${{\overline}{V}({\varphi})}$ and $b$. By changing the choice of ${\varphi}'$ by a suitable coboundary, [one]{} can therefore arrange that all the numbers $a_x$ and $b_y$ are 1. [We shall continue to work with an arbitrary choice of ${\varphi}'$ in what follows.]{}
Lemma \[lem:C3\_eX\] implies that ${\mathcal H}^\gamma({\overline}{G}({\varphi}))e_{{\overline}{U}({\varphi})}^{{\varphi}'} e_{{\overline}{V}({\varphi})}^{{\varphi}'}$ is an ${\mathcal H}^\gamma({\overline}{G}({\varphi}))$-${\mathcal H}^\gamma({{\overline}{L}(\psi)})$ bimodule. We denote by ${\operatorname{i}_{{{\overline}{U}({\varphi})},{{\overline}{V}({\varphi})}}^{{\varphi}'}:\operatorname{\mathcal{R}}^\gamma({{\overline}{L}(\psi)})\to \operatorname{\mathcal{R}}^\gamma({\overline}{G}({\varphi}))}$ the functor of tensor product with this bimodule. Likewise, we denote by $\operatorname{r}_{{{\overline}{U}({\varphi})},{{\overline}{V}({\varphi})}}^{{\varphi}'}$ the functor of tensor product with the ${\mathcal H}^\gamma({{\overline}{L}(\psi)})$-${\mathcal H}^\gamma({\overline}{G}({\varphi}))$ bimodule $e_{{\overline}{U}({\varphi})}^{{\varphi}'} e_{{\overline}{V}({\varphi})}^{{\varphi}'}{\mathcal H}^\gamma({\overline}{G}({\varphi}))$. The arguments of Theorem \[thm:pind-properties\] carry over to this twisted setting, and show that the functors $\operatorname{i}_{{{\overline}{U}({\varphi})},{{\overline}{V}({\varphi})}}^{{\varphi}'}$ and $\operatorname{r}_{{{\overline}{U}({\varphi})},{{\overline}{V}({\varphi})}}^{{\varphi}'}$ are two-sided adjoints, and that up to natural isomorphism they do not depend on the order of ${{\overline}{U}({\varphi})}$ and ${{\overline}{V}({\varphi})}$.
Our induction functors are compatible with the final assertion of Clifford theory, as follows:
\[thm:C3\] Let ${\varphi}'$ be a projective extension of ${\varphi}$, with corresponding cocycle $\gamma^{-1}$, and let $\psi'$ be the projective extension of $\psi$ defined by . The diagram $$\xymatrix@C=50pt{
\operatorname{\mathcal{R}}(L(\psi))_{\psi} \ar[r]^-{\operatorname{i}_{\psi}} & \operatorname{\mathcal{R}}(G({\varphi}))_{{\varphi}} \\
\operatorname{\mathcal{R}}^{\gamma }({{\overline}{L}(\psi)}) \ar[u]_-{\cong}^-{\otimes\psi'} \ar[r]^-{\operatorname{i}_{{{\overline}{U}({\varphi})},{{\overline}{V}({\varphi})}}^{{\varphi}'}} & \operatorname{\mathcal{R}}^{\gamma }({\overline}{G}({\varphi})) \ar[u]_-{\otimes {\varphi}'}^-{\cong}
}$$ is commutative up to natural isomorphism.
\[independence\_example\] Suppose that the irreducible representation $\psi$ of $L_0$ satisfies $G(\operatorname{i}_0\psi)=L(\psi)G_0$. We then have ${\overline}{G}({\varphi})={{\overline}{L}(\psi)}$ in Theorem \[thm:C3\], and $\operatorname{i}_{{{\overline}{U}({\varphi})},{{\overline}{V}({\varphi})}}^{{\varphi}'}$ is the identity functor, and so we conclude from Theorems \[thm:C2\] and \[thm:C3\] that in this case $$\operatorname{i}_{U,V}:\operatorname{\mathcal{R}}(L)_\psi \to \operatorname{\mathcal{R}}(G)_{\operatorname{i}_0(\psi)}$$ is an equivalence of categories. Specific examples of this kind arise in Section \[sec:sp\].
The proof of Theorem \[thm:C3\] uses the following lemma, whose proof is a matter of straightforward linear algebra:
\[lem:lin\_alg\] Let $E$ be a finite dimensional vector space over ${\mathbb{C}}$, and let $S:E\to E$ be a linear endomorphism. Let $A\subseteq \operatorname{End}_{{\mathbb{C}}}(E)$ be the centraliser of $S$ in $\operatorname{End}_{{\mathbb{C}}}(E)$. Then [we have isomorphisms $$E\otimes \left(\operatorname{Im}(S)\right)^* \xrightarrow{e\otimes f\longmapsto [e'\mapsto ef(e')]} \operatorname{Hom}\left(\operatorname{Im}(S), E\right) \xrightarrow{T\mapsto T\circ S} \operatorname{End}(E)S$$ of $\operatorname{End}(E)$-$A$-bimodules.]{}
In our application the space $E$ will be $\operatorname{i}_0(W)$, and the endomorphism $S$ will be the action of $e_{U_0}e_{V_0}e_\psi$.
Let $$T:\operatorname{Mod}\left( {\mathcal H}^\gamma({{\overline}{L}(\psi)})\otimes \operatorname{End}(W)\right) \to \operatorname{Mod}\left( {\mathcal H}^\gamma({\overline}{G}({\varphi}))\otimes \operatorname{End}(\operatorname{i}_0(W)\right)$$ be the functor of tensor product with the bimodule ${\mathcal H}^\gamma({\overline}{G}({\varphi}))e_{{\overline}{U}({\varphi})}^{{\varphi}'} e_{{\overline}{V}({\varphi})}^{{\varphi}'} \otimes_{{\mathbb{C}}} \left(\operatorname{i}_0 (W)\otimes_{{\mathbb{C}}} W^*\right)$, where $\operatorname{i}_0(W)$ is viewed as a left $\operatorname{End}(\operatorname{i}_0(W))$ module and $W^*$ is viewed as a right $\operatorname{End}(W)$-module in the obvious way.
We shall decompose the equivalences $\otimes\psi'$ and $\otimes{\varphi}'$ into compositions of two equivalences, as in , and show both squares in the diagram $$\label{eq:C3_proof_diagram}
\xymatrix@C=60pt@R=40pt{
\operatorname{\mathcal{R}}(L(\psi))_\psi \ar[r]^-{\operatorname{i}_\psi} & \operatorname{\mathcal{R}}(G({\varphi}))_{\varphi}\\
\operatorname{Mod}\left( {\mathcal H}^\beta({{\overline}{L}(\psi)})\otimes \operatorname{End}(W)\right) \ar[u]_-{\cong}^-{(\theta\otimes\psi')^*} \ar[r]^-{T} &
\operatorname{Mod}\left( {\mathcal H}^\gamma({\overline}{G}({\varphi})) \otimes \operatorname{End}(\operatorname{i}_0(W))\right) \ar[u]^-{\cong}_-{(\theta\otimes{\varphi}')^*} \\
\operatorname{\mathcal{R}}^\beta({{\overline}{L}(\psi)}) \ar[u]_-{\cong}^-{\otimes W} \ar[r]^-{\operatorname{i}_{{{\overline}{U}({\varphi})},{{\overline}{V}({\varphi})}}^{{\varphi}'}} & \operatorname{\mathcal{R}}^\gamma({\overline}{G}({\varphi})) \ar[u]^-{\cong}_-{\otimes \operatorname{i}_0(W)}
}$$ commute.
To show that the bottom square of commutes, let $M$ be an ${\mathcal H}^{\gamma}({{\overline}{L}(\psi)})$ module. We have natural isomorphisms of ${\mathcal H}^\gamma({\overline}{G}({\varphi}))\otimes \operatorname{End}(\operatorname{i}_0(W))$ modules $$\begin{aligned}
T(M\otimes W)&= \left({\mathcal H}^\gamma({\overline}{G}({\varphi}))e_{{{\overline}{U}({\varphi})}}^{{\varphi}'}e_{{{\overline}{V}({\varphi})}}^{{\varphi}'} \otimes (\operatorname{i}_0(W)\otimes W^*) \right)\otimes_{{\mathcal H}^\gamma({{\overline}{L}(\psi)})\otimes \operatorname{End}(W)} (M\otimes W) \\
& \cong \left({\mathcal H}^\gamma({\overline}{G}({\varphi}))e_{{\overline}{U}({\varphi})}^{{\varphi}'} e_{{{\overline}{V}({\varphi})}}^{{\varphi}'}\otimes_{{\mathcal H}^\gamma({{\overline}{L}(\psi)})}M\right) \otimes \left( \operatorname{i}_0(W)\otimes (W^*\otimes_{\operatorname{End}(W)} W)\right) \\
& \cong \operatorname{i}_{{{\overline}{U}({\varphi})},{{\overline}{V}({\varphi})}}^{{\varphi}'}(M) \otimes \operatorname{i}_0(W),\end{aligned}$$ because $W^*\otimes_{\operatorname{End}(W)} W\cong {\mathbb{C}}$, as $W$ is finite-dimensional. Thus the bottom square of the diagram commutes.
To show that the top square of commutes, it is enough to construct a linear isomorphism $$F: e_{\varphi}{\mathcal H}(G({\varphi}))e_{U({\varphi})}e_{V({\varphi})}e_\psi \longrightarrow {\mathcal H}^\gamma({\overline}{G}({\varphi}))e_{{\overline}{U}({\varphi})}^{{\varphi}'} e_{{\overline}{V}({\varphi})}^{{\varphi}'} \otimes_{{\mathbb{C}}} \left(\operatorname{i}_0 (W)\otimes_{{\mathbb{C}}} W^*\right)$$ between the bimodules associated to the functors $\operatorname{i}_\psi$ and $T$, satisfying $$\label{eq:C3_proof_F}
F( f \cdot h \cdot k) = (\theta\otimes {\varphi}')(f) \cdot F(h) \cdot (\theta\otimes\psi')(k)$$ for all $f\in e_{\varphi}{\mathcal H}(G({\varphi}))$, $h\in e_{\varphi}{\mathcal H}(G({\varphi}))e_{U({\varphi})}e_{V({\varphi})}e_\psi$ and $k\in e_\psi{\mathcal H}(L(\psi))$.
We shall construct $F$ as a composition $F=F_3 F_2 F_1$: $$\begin{aligned}
e_{\varphi}{\mathcal H}(G({\varphi}))e_{U({\varphi})}e_{V({\varphi})}e_\psi & \xrightarrow[\cong]{F_1}
{\mathcal H}^\gamma({\overline}{G}({\varphi}))e_{{\overline}{U}({\varphi})}^{{\varphi}'}e_{{{\overline}{V}({\varphi})}}^{{\varphi}'} \otimes \operatorname{End}(\operatorname{i}_0(W)){\varphi}(e_{U_0}e_{V_0}e_\psi) \\
& \xrightarrow[\cong]{F_2} {\mathcal H}^\gamma({\overline}{G}({\varphi}))e_{{\overline}{U}({\varphi})}^{{\varphi}'}e_{{\overline}{V}({\varphi})}^{{\varphi}'} \otimes \operatorname{i}_0(W)\otimes \left( {\varphi}(e_\psi e_{U_0}e_{V_0})\operatorname{i}_0(W)\right)^* \\
& \xrightarrow[\cong]{F_3} {\mathcal H}^\gamma({\overline}{G}({\varphi})) e_{{\overline}{U}({\varphi})}^{{\varphi}'} e_{{\overline}{V}({\varphi})}^{{\varphi}'} \otimes \operatorname{i}_0(W)\otimes W^* ,\end{aligned}$$ where the isomorphisms $F_1$, $F_2$ and $F_3$ are defined below.
The map $F_1$ is the restriction of the algebra isomorphism $\theta\otimes {\varphi}'$ to the bimodule $e_{\varphi}{\mathcal H}(G({\varphi}))e_{U({\varphi})}e_{V({\varphi})}e_\psi$. The image is as claimed because of Lemma \[lem:C3\_eX\]. By definition, $F_1$ satisfies $$F_1(f\cdot h \cdot k) = (\theta\otimes{\varphi}')(f) \cdot F_1(h) \cdot (\theta\otimes{\varphi}')(k)$$ (where $f$, $h$ and $k$ are as in ).
The map $F_2$ is the identity on the first tensor factor, while on the second factor it is the isomorphism given by Lemma \[lem:lin\_alg\], with $E=\operatorname{i}_0(W)$ and $S={\varphi}(e_{U_0} e_{V_0} e_\psi)$. Clearly $F_2$ is a map of left ${\mathcal H}^\gamma({\overline}{G}({\varphi}))\otimes \operatorname{End}(\operatorname{i}_0(W))$ modules. Turning to the right module structure, fix $l\in L(\psi)$. The operator ${\varphi}'(e_\psi\delta_l)\in \operatorname{End}(\operatorname{i}_0(W))$ commutes with ${\varphi}(e_\psi e_{U_0}e_{V_0})$, because $L(\psi)$ centralises $\psi$ and normalises $U_0$ and $V_0$. Therefore ${\varphi}'(e_\psi\delta_l)$ lies in the algebra $A$ of Lemma \[lem:lin\_alg\], and so the isomorphism $F_2$ satisfies $$F_2 \left( F_1(h)\cdot \theta(l)\otimes{\varphi}'(e_\psi\delta_l)\right) = F_2 F_1(h) \cdot \left(\theta(l)\otimes {\varphi}'(e_\psi\delta_l)\right).$$
The map $F_3$ is the identity on the first two tensor factors, while on the third factor it is the linear dual $\Theta^*$ of the isomorphism $\Theta: W\to {\varphi}(e_{U_0}e_{V_0})\operatorname{i}_0(W)$. (Note that $e_\psi$ acts as the identity on $W$.) Clearly $F_3$ is a map of left ${\mathcal H}^\gamma({\overline}{G}({\varphi}))\otimes \operatorname{End}(\operatorname{i}_0(W))$ modules. The isomorphism $\Theta$ satisfies $
{\varphi}'(e_\psi\delta_l)\circ \Theta = \Theta\circ \psi'(e_\psi \delta_l),
$ by the definition of $\psi'$, and we therefore have $$F_3\left[ F_2 F_1(h) \cdot \left(\theta(l)\otimes{\varphi}'(e_\psi\delta_l)\right) \right] = F(h)\cdot \left(\theta(l)\otimes\psi'(e_\psi\delta_l)\right).$$ We have now shown that the isomorphism $F$ satisfies , and this completes the proof of Theorem \[thm:C3\].
The functor $\operatorname{i}$ and the orbit method {#sec:orbit}
===================================================
In this section we examine the induction functor $\operatorname{i}_{U,V}$ in situations to which the orbit method applies, and show that it corresponds to a natural inclusion map on coadjoint orbits. We begin with an abstract formulation and then discuss a natural family of groups to which it applies, namely uniform [pro-$p$ groups and finite $p$-groups of nilpotency class less than $p$]{}. In particular, this family includes many compact open subgroups in reductive groups over $p$-adic fields.
An abstract formulation {#subsec:abstract}
-----------------------
The orbit method in the context of profinite groups goes back to the work of Howe [@Howe]. An abstract formulation was given by Boyarchenko and Sabitova in [@Boy-Sab], and it is this latter point of view that we shall adopt here.
Let $G$ be a profinite group, let ${\mathfrak}{g}$ be an abelian profinite group, and let $\exp:{\mathfrak}{g}\to G$ be a homeomorphism satisfying
(A) The formula $\operatorname{Ad}_g(x)\coloneq \log(g\exp(x)g^{-1})$ for $g\in G$, $x\in {\mathfrak}{g}$, and $\log=\exp^{-1}$ defines an action of $G$ on ${\mathfrak}{g}$ by group automorphisms.
(B) The pullback map $\exp^*:{\mathcal H}(G)^{G}\to {\mathcal H}({\mathfrak}{g})^{G}$, from the $\operatorname{Ad}_G$-invariant locally constant functions on $G$ to those on ${\mathfrak}{g}$, is an isomorphism of convolution algebras.
The adjoint action of $G$ on ${\mathfrak}{g}$ induces a coadjoint action on the Pontryagin dual group $\widehat{{\mathfrak}{g}}$. It is shown in [@Boy-Sab Theorem 1.1] that for each irreducible smooth representation $\tau$ of $G$, with character $\operatorname{ch}_\tau\in {\mathcal H}(G)^G$, there is an $\operatorname{Ad}_G$-orbit $\Omega\subset\widehat{{\mathfrak}{g}}$ such that $$\label{orbit_equation}
\exp^*(\operatorname{ch}_\tau) = |\Omega|^{-1/2}\sum_{\psi\in \Omega} \psi,$$ and the map $\operatorname{ch}_\tau \mapsto \Omega$ sets up a bijection $\mathcal O_G:\operatorname{Irr}(G) \to G\backslash \widehat{{\mathfrak}{g}}$ from the set of isomorphism classes of irreducible representations of $G$ to the set of coadjoint orbits in $\widehat{{\mathfrak}g}$.
\[orbit\_theorem\] Let $G$, ${\mathfrak}{g}$ and $\exp$ be as above. Let $U$, $L$ and $V$ be closed subgroups of $G$ such that:
1. $(U,L,V)$ is an Iwahori decomposition of $G$.
2. The preimages ${\mathfrak}{u},{\mathfrak}{l},{\mathfrak}{v}$ of $U,L,V$ under $\exp$ are subgroups of ${\mathfrak}{g}$, and ${\mathfrak}g = {\mathfrak}u \oplus {\mathfrak}l \oplus {\mathfrak}v$ as abelian groups.
3. The map $\exp:{\mathfrak}{g}\to G$ restricts to homeomorphisms $${\mathfrak}{l}\to L,\quad {\mathfrak}{l}\oplus{\mathfrak}{u} \to LU\quad \text{and}\quad {\mathfrak}{l}\oplus{\mathfrak}{v}\to LV,$$ each of which satisfies the conditions (A) and (B).
[Then ]{}the projection $\Lambda$ of ${\mathfrak}{g}$ onto its summand ${\mathfrak}{l}$ induces an injective map $$\Lambda^*:L\backslash \widehat{{\mathfrak}{l}} \to G\backslash \widehat{{\mathfrak}{g}}, \qquad \Omega \mapsto \operatorname{Ad}_G^*(\Omega\circ \Lambda)$$ which makes the diagram $$\xymatrix@C=40pt{ \operatorname{Irr}(L) \ar[r]^-{\operatorname{i}_{U,V}} \ar[d]_-{{\mathcal O_L}} & \operatorname{Irr}(G) \ar[d]^-{{\mathcal O_G}}\\
L\backslash \widehat{{\mathfrak}l} \ar[r]^-{\Lambda^*} & G\backslash \widehat{{\mathfrak}g}
}$$ commutative.
We require the following lemma:
\[orbit\_U\_lemma\] Let $\Omega$ be an orbit in $LU\backslash\widehat{{\mathfrak}{l}{\mathfrak}{u}}$, corresponding via the orbit method to an irreducible representation $\tau$ of $LU$. Then $\tau$ is trivial on $U$ if and only if every $\psi\in \Omega$ is trivial on ${\mathfrak}{u}$.
If every $\psi\in \Omega$ is trivial on ${\mathfrak}{u}$, then the character formula ensures that the character of $\tau$ is constant on $U$, and therefore that $\tau$ is trivial on $U$. Conversely, if $\tau$ is trivial on $U$ then its character is constant on $U$, with constant value $\dim(\tau)=|\Omega|^{1/2}$, and so implies that $\sum_{\psi\in \Omega}\psi(y)=|\Omega|$ for every $y\in {\mathfrak}{u}$. Since each $\psi(y)$ is a complex number of modulus one, this equality forces $\psi(y)=1$ for every $\psi$ and every $y$.
Note that by Theorem \[thm:pind-Iw\] the functor $\operatorname{i}= \operatorname{i}_{U,V}:\operatorname{\mathcal{R}}(L)\to \operatorname{\mathcal{R}}(G)$ preserves irreducibility, and therefore induces a map $\operatorname{i}: \operatorname{Irr}(L)\to \operatorname{Irr}(G)$. We recall the following characterisation of the map $\operatorname{i}$ from Theorem \[thm:pind-Iw\]: given irreducible representations $\tau \in \operatorname{\mathcal{R}}(G)$ and $\sigma\in \operatorname{\mathcal{R}}(L)$, one has $\tau\cong \operatorname{i}(\sigma)$ if and only if $\sigma$ is a common subrepresentation of $\tau^U$ and $\tau^V$.
Fix an orbit $\Omega\in L\backslash \widehat{{\mathfrak}{l}}$ and let $\sigma={\mathcal O_L^{-1}(\Omega)\in \operatorname{Irr}(L)}$ be [the corresponding]{} irreducible representation of $L$. Let $\tau={\mathcal O_G^{-1}(\Lambda^*\Omega)\in \operatorname{Irr}(G)}$ be [the corresponding]{} irreducible representation [of $G$]{}. We will show that $\sigma$ is isomorphic to a subrepresentation of $\tau^U$. The same argument shows that $\sigma$ is isomorphic to a subrepresentation of $\tau^V$, and then Theorem \[thm:pind-Iw\] gives $\tau \cong \operatorname{i}(\sigma)$ as required.
The characters $\operatorname{ch}_\rho$, where $\rho$ ranges over $\operatorname{Irr}(LU)$, constitute a linear basis for the space ${\mathcal H}(LU)^{LU}$. We let $P:{\mathcal H}(LU)^{LU}\to {\mathcal H}(LU)^{LU}$ be the projection $$P(\operatorname{ch}_\rho) = \begin{cases} \operatorname{ch}_\rho &\text{if $\rho$ is trivial on $U$} \\ 0 & \text{otherwise.}\end{cases}$$ On the other hand, the functions $$\chi_\Psi \coloneq |\Psi|^{-1/2}\sum_{\psi\in \Psi} \psi,$$ as $\Psi$ ranges over $LU\backslash\widehat{{\mathfrak}{l}{\mathfrak}{u}}$, constitute a basis for ${\mathcal H}({\mathfrak}{l}{\mathfrak}{u})^{LU}$, and we let $Q$ be the idempotent operator on ${\mathcal H}({\mathfrak}{l}{\mathfrak}{u})^{LU}$ defined by $$Q(\chi_{\Psi}) = \begin{cases} \chi_{\Psi} & \text{if every $\psi\in \Psi$ is trivial on ${\mathfrak}{u}$} \\ 0 & \text{otherwise}.\end{cases}$$
Lemma \[orbit\_U\_lemma\] implies the commutativity of the middle square in the diagram $$\label{orbit_proof_eq}
\xymatrix@R=30pt@C=50pt{
{\mathcal H}(G)^{G} \ar[r]^-{\text{restrict}} \ar[d]^-{\exp^*} & {\mathcal H}(LU)^{LU} \ar[r]^-{P} \ar[d]^-{\exp^*} & {\mathcal H}(LU)^{LU} \ar[r]^-{\text{restrict}} \ar[d]^-{\exp^*} & {\mathcal H}(L)^{L} \ar[d]^-{\exp^*} \\
{\mathcal H}({\mathfrak}{g})^{G} \ar[r]^-{\text{restrict}} & {\mathcal H}({\mathfrak}{l}{\mathfrak}{u})^{LU} \ar[r]^-{Q} & {\mathcal H}({\mathfrak}{l}{\mathfrak}{u})^{LU} \ar[r]^-{\text{restrict}} & {\mathcal H}({\mathfrak}{l})^L
}$$ where restrict means restriction of functions. The two outer squares in the diagram obviously commute. For each irreducible representation $\rho$ of $G$, the composition along the top row of sends the character of $\rho$ to the character of $\rho^U$.
Choose a point $\psi$ in the orbit $\Omega \subset \widehat{{\mathfrak}{l}}$, and write $\Lambda^*\Omega=\{{\psi\circ \Lambda},{\varphi}_1,\ldots,{\varphi}_n\}$. Since the character $\psi\circ\Lambda\in \widehat{{\mathfrak}{g}}$ is trivial on ${\mathfrak}{u}$, and has $\psi\circ\Lambda{\big{\vert}}_{{\mathfrak}{l}}=\psi$, we find that the composition along the bottom row of sends $\chi_{\Lambda^*\Omega}=\exp^*(\operatorname{ch}_\tau)$ to the function $$|\Lambda^*\Omega|^{-1/2} \left( \psi + \sum_{{\varphi}_i\equiv 1\text{ on } {\mathfrak}{u}} {\varphi}_i{\big{\vert}}_{{\mathfrak}{l}}\right).$$ Since this sum contains $\psi$—and hence $\chi_\Omega=\exp^*(\operatorname{ch}_\sigma)$—with a positive coefficient, we conclude from the commutativity of that $\tau^U$ contains a copy of $\sigma$.
Application to (pro-) $\boldsymbol p$-groups
--------------------------------------------
The results of the previous subsection apply to a rich and well-behaved family of (pro-) $p$-groups which we now discuss. Roughly speaking these groups admit good linearisations, that is, to each such group one may associate a Lie algebra that carries complete information on the group.
### Uniform pro-$p$-groups {#uniform-pro-p-groups .unnumbered}
A finite $p$-group is called [*powerful*]{} if $[G,G] \subset G^p$ when $p$ is odd (and $[G,G] \subset G^4$ when $p=2$). Here $G^m$ is the group generated by $m$-powers. A pro-$p$ group is called powerful if it is the inverse limit of finite powerful groups. A pro-$p$ group is called [*uniform*]{} if it is powerful, finitely generated (as a pro-$p$ group), and torsion-free. To each uniform pro-$p$ group $G$ one may associate a [*uniform*]{} [${\mathbb{Z}}_p$-Lie algebra]{} ${\mathfrak}g = \operatorname{Lie}(G)$, that is, a ${\mathbb{Z}}_p$-Lie algebra which is free of finite rank as a ${\mathbb{Z}}_p$-module, and which satisfies $[{\mathfrak}g, {\mathfrak}g]_{\operatorname{Lie}} \subset p{\mathfrak}g$ for $p$ odd (and $[{\mathfrak}g, {\mathfrak}g]_{\operatorname{Lie}} \subset 4{\mathfrak}g$ for $p=2$); see [@DDMS] for a comprehensive treatment. This association defines an equivalence of categories between the category of uniform pro-$p$ groups and uniform ${\mathbb{Z}}_p$-Lie algebras. Starting with a uniform Lie algebra ${\mathfrak}g$ this association is made concrete using the Campbell-Hausdorff series $$H(u,v)=\log\left(\exp(u)\exp(v)\right) = u+v+(\textrm{Lie brackets}) \in {\mathbb{Q}}\langle \!\langle u,v \rangle \!\rangle,$$ which is expressible in terms of $u,v\in {\mathfrak}g$ by means of the Lie bracket, and which allows one to define a uniform pro-$p$ group $G$ having the same underlying set as ${\mathfrak}g$ and group operation $u \cdot v = H(u,v)$. Let $\exp$ denote the identity map on ${\mathfrak}g$, thought of as a map from the Lie algebra ${\mathfrak}g$ to the group $G$. This map is well-behaved with respect to passage to subgroups or quotients: Lie subalgebras of ${\mathfrak}g$ correspond bijectively to closed uniform subgroups of $G$, and ideals in ${\mathfrak}g$ correspond to normal subgroups in $G$; see [@DDMS Section 4.5]. Moreover, it is shown in [@Boy-Sab Theorem 2.6] that for $p\neq 2$ this map $\exp$ satisfies the conditions (A) and (B) from Section \[subsec:abstract\], meaning that the orbit method applies and gives a bijection $\mathcal O_G:\operatorname{Irr}(G)\to G\backslash \widehat{g}$. [This generalises an earlier result of Howe [@Howe Theorem 1.1].]{}
### Finite $p$-groups of nilpotency class less than $p$ {#finite-p-groups-of-nilpotency-class-less-than-p .unnumbered}
There is a similar Lie-type correspondence for finite $p$-groups of nilpotency class less than $p$. To each group $G$ of this type one may associate a finite ${\mathbb{Z}}$-Lie algebra ${\mathfrak}g=\operatorname{Lie}(G)$ which is nilpotent of class less than $p$, and whose additive group is a $p$-group, such that $G$ is isomorphic to the group $\exp({\mathfrak}g)$ whose underlying set is ${\mathfrak}g$ and whose multiplication is given by the Campbell-Hausdorff series (which is finite, in this case); see [@Khukhro Section 10.2]. If $p$ is odd then the orbit method applies to the map $\exp:{\mathfrak}g\to G$; see [@Boy-Sab Theorem 2.6].
### Application of Theorem \[orbit\_theorem\] {#application-of-theorem-orbit_theorem .unnumbered}
For the rest of this section let $G=\exp({\mathfrak}g)$ be either a uniform pro-$p$ group or a finite $p$-group of nilpotency class less than $p$, with corresponding Lie algebra ${\mathfrak}g$. For each subalgebra ${\mathfrak}h$ of ${\mathfrak}g$ we write $H$ for the corresponding subgroup $\exp({\mathfrak}h)$ of $G$.
\[def:Lie\_Iwahori\] An *Iwahori decomposition* of ${\mathfrak}g$ is a triple of Lie subalgebras $({\mathfrak}u, {\mathfrak}l, {\mathfrak}v)$ of ${\mathfrak}g$ such that $[{\mathfrak}l,{\mathfrak}u]\subseteq {\mathfrak}u$, $[{\mathfrak}l, {\mathfrak}v]\subseteq {\mathfrak}v$, and such that ${\mathfrak}g = {\mathfrak}u \oplus {\mathfrak}l \oplus v$ as ${\mathbb{Z}}_p$-modules (in the uniform pro-$p$ case) or as ${\mathbb{Z}}$-modules (in the finite $p$-group case).
\[lem:Lie\_Iwahori\] If $({\mathfrak}u, {\mathfrak}l, {\mathfrak}v)$ is an Iwahori decomposition of ${\mathfrak}g$, then $(U,L,V)$ is an Iwahori decomposition of $G$.
The Lie correspondence ensures that $U$, $L$ and $V$ are closed subgroups of $G$ such that $L$ normalises $U$ and $V$. The subgroups $V$ and $B\coloneq UL$ have trivial intersection in $G$, because the subalgebras ${\mathfrak}v$ and ${\mathfrak}u\oplus {\mathfrak}l$ have trivial intersection in ${\mathfrak}g$, and so the product map $U\times L \times V\to G$ is injective. We shall now show that this map is surjective.
We must show that for each $x\in {\mathfrak}b\coloneq {\mathfrak}u\oplus {\mathfrak}l$ and each $y\in {\mathfrak}v$ one has $\exp(x+y)\in BV$. The Campbell-Hausdorff formula implies that $\exp(x+y) = \exp(x)\exp(z_1)\exp(y)$ for some $z_1\in {\mathfrak}g_1=[{\mathfrak}g,{\mathfrak}g]$. Writing $z_1=x_1+y_1$, where $x_1\in {\mathfrak}b$ and $y_1\in {\mathfrak}v$, another application of Campbell-Hausdorff gives $\exp(z_1)=\exp(x_1)\exp(z_2)\exp(y_1)$ for some $z_2\in {\mathfrak}g_2=[{\mathfrak}g,{\mathfrak}g_1]$. Continuing in this way we find $z_n \in {{\mathfrak}g}_n = [{\mathfrak}g,{\mathfrak}g_{n-1}]$, $x_{n-1} \in {\mathfrak}b$ and $y_{n-1} \in {\mathfrak}v$, for every $n \in {\mathbb{N}}$, such that $\exp(z_{n-1})=\exp(x_{n-1})\exp(z_n)\exp(y_{n-1})$, and we deduce that $$\exp(x+ y ) \in \bigcap_{n\geq 0} B\exp({\mathfrak}g_n) V = B\left(\bigcap_{n\geq 0} \exp({\mathfrak}g_n)\right) V = BV,$$ where the first equality holds because the groups $\exp({\mathfrak}g_n)$ form a descending chain and $G$ is compact, and the second holds because ${\mathfrak}g$ is either uniform or nilpotent.
We are left to verify condition (2) of Definition \[def:vI\]. If $G$ is finite this condition is trivially satisfied, so suppose that $G$ is a uniform pro-$p$ group. For each $n\geq 0$ the triple $(p^n{\mathfrak}u, p^n{\mathfrak}l, p^n{\mathfrak}v)$ is an Iwahori decomposition of the [ideal]{} $p^n{\mathfrak}g$ of ${\mathfrak}g$, and so the above argument shows that the open [normal]{} subgroups $K_n = \exp(p^n{\mathfrak}g)$ of $G$ satisfy condition (2).
We now have the following corollary of Theorem \[orbit\_theorem\]:
\[cor:Lie\_Iwahori\] Let $p$ be an odd prime. Let $G$ be either a uniform pro-$p$ group, or a finite $p$-group of nilpotency class less than $p$. Let $({\mathfrak}u, {\mathfrak}l, {\mathfrak}v)$ be an Iwahori decomposition of the Lie algebra ${\mathfrak}g$ of $G$, and let $(U,L,V)$ be the corresponding Iwahori decomposition of $G$. The diagram $$\xymatrix@R=30pt@C=50pt{ \operatorname{Irr}(L) \ar[r]^-{\operatorname{i}_{U,V}} \ar[d]_-{{\mathcal O_L}} & \operatorname{Irr}(G) \ar[d]^-{{\mathcal O_G}}\\
L\backslash \widehat{{\mathfrak}l} \ar[r]^-{\Lambda^*} & G\backslash \widehat{{\mathfrak}g}
}$$ is commutative.
This follows from Theorem \[orbit\_theorem\]. The hypothesis (1) of that theorem is satisfied because of Lemma \[lem:Lie\_Iwahori\]; hypothesis (2) is satisfied by assumption; and the hypothesis (3) is satisfied because of [@Boy-Sab Theorem 2.6].
We remark that for uniform pro-$2$ groups the orbit method does not fully apply, though one has weaker versions; see [@Jaikin; @Boy-Sab].
\[ex:orbit\_tidy\] In real life one may find a rich supply of groups to which the corollary may be applied. [Let $\mathcal G$ be a $p$-adic Lie group, let $\alpha$ be an automorphism of $\mathcal G$, and denote by $\alpha_*$ the derived automorphism of the Lie algebra ${\mathfrak}g$ [of $G$]{}. Then $${\mathfrak}u_\alpha \coloneq \{x\in {\mathfrak}g\ |\ \alpha_*^n(x)\to 0\text{ as }n\to\infty\}\quad \text{and}\quad {\mathfrak}v_\alpha\coloneq {\mathfrak}u_{\alpha^{-1}}$$ are nilpotent Lie subalgebras of ${\mathfrak}g$, normalised by the subalgebra $${\mathfrak}l_\alpha \coloneq \{ x\in {\mathfrak}g\ |\ \{\alpha_*^n(x)\ |\ n\in {\mathbb{Z}}\}\text{ is precompact in }{\mathfrak}g\},$$ and we have ${\mathfrak}g = {\mathfrak}u_\alpha\oplus {\mathfrak}l_\alpha\oplus {\mathfrak}v_\alpha$ as ${\mathbb{Q}}_p$-vector spaces. Moreover, ${\mathfrak}u_\alpha$, ${\mathfrak}l_\alpha$ and ${\mathfrak}v_\alpha$ are the respective Lie algebras of the closed subgroups $\mathcal U_\alpha$, $\mathcal L_\alpha$ and $\mathcal V_\alpha$ of $\mathcal G$, where we are using the notation of Example \[ex:vI-tidy\]. These assertions are proved in [@Wang Theorem 3.5]. It is shown in [@Gloeckner_manuscripta Lemma 3.3] that ${\mathfrak}g$ contains arbitrary small open uniform ${\mathbb{Z}}_p$-Lie subalgebras ${\mathfrak}k$ having ${\mathfrak}k = ({\mathfrak}u_\alpha\cap {\mathfrak}k)\oplus ({\mathfrak}l_\alpha \cap {\mathfrak}k) \oplus ({\mathfrak}v_\alpha \cap {\mathfrak}k)$. The compact open subgroups $K= \exp({\mathfrak}k)$ of $\mathcal G$ then have Iwahori decompositions $(\mathcal U_\alpha\cap K, \mathcal L_\alpha\cap K, \mathcal V_\alpha\cap K)$. If $p$ is odd, Corollary \[cor:Lie\_Iwahori\] describes the induction functor $\operatorname{i}_{\mathcal U_\alpha\cap K, \mathcal V_\alpha\cap K}:\operatorname{\mathcal{R}}(\mathcal L_\alpha\cap K)\to \operatorname{\mathcal{R}}(K)$ in terms of the orbit method and of the projection ${\mathfrak}k\to {\mathfrak}l_\alpha\cap {\mathfrak}k$. ]{}
For a finite example, let ${\mathfrak o}$ be a compact discrete valuation ring with maximal ideal ${\mathfrak}p$ and residue characteristic $p$. Let $K=K_1$ be the first principal congruence subgroup in $\operatorname{GL}_n({\mathfrak o}/{\mathfrak}p^{\ell})$, for $\ell>1$. Then $K$ is a finite $p$-group of nilpotency class $\ell-1$, with Lie algebra ${\mathfrak}k=M_n({\mathfrak}p/{\mathfrak}p^\ell)$. As explained in Example \[ex:vI-GLnO\], each partition $n=n_1+\cdots+n_m$ gives an Iwahori decomposition $(U\cap K, L\cap K, V\cap K)$ of $K$, corresponding to the decomposition of ${\mathfrak}k$ into block-upper-triangular, block-diagonal and block-lower-triangular matrices. If ${p>\ell-1}$ and odd, Corollary \[cor:Lie\_Iwahori\] gives a description of the resulting induction functor $\operatorname{i}_{U\cap K, V\cap K}:\operatorname{\mathcal{R}}(L\cap K)\to \operatorname{\mathcal{R}}(K)$ in terms of the orbit method and the projection of ${\mathfrak}k$ onto its subalgebra of block-diagonal matrices.
We have taken the point of view of the theory of uniform groups due to its fairly concrete and algebraic formulation. Historically, the Lie correspondence [for (pro-)p-groups]{} goes back to the seminal work of Lazard [@Lazard-finite], [@Lazard]. [The technique of obtaining Iwahori decompositions of groups from decompositions of Lie algebras is well known in the setting of $p$-adic reductive groups: see [@WorksInProgress] and [@Deligne_support], for example.]{}
Case study: Siegel Levi subgroup in $\operatorname{Sp}_4({\mathfrak o}_2)$ {#sec:sp}
==========================================================================
Let ${\mathfrak o}$ be a compact discrete valuation ring with maximal ideal $\mathfrak{p}$, a fixed uniformiser $\pi$ and finite residue field ${\Bbbk}$ of odd characteristic. Let ${\mathfrak o}_\ell:={\mathfrak o}/\mathfrak{p}^\ell$. In this section we illustrate how the results of the previous sections may be applied to study the representations of the symplectic group $\operatorname{Sp}_4({\mathfrak o}_2)$ that are induced, in the sense of Definition \[def:pind\], from the Siegel Levi subgroup of $2\times 2$ block-diagonal matrices. [Note that this is equivalent to studying [those]{} induced representations of $\operatorname{Sp}_4({\mathfrak o})$ which factor through $\operatorname{Sp}_4({\mathfrak o}_2)$: see Theorem \[thm:pind-properties\].]{} The main results in this section are a double-coset formula, à la Mackey, for the composition of induction and restriction for these groups (Theorem \[thm:sp\_Mackey\]); and an answer to a question of Dat regarding parahoric induction (Corollary \[Dat\_corollary\]).
Let us introduce the notation used to state the Mackey formula. Let $$G= \operatorname{Sp}_4({\mathfrak o}_2) = \{g\in \operatorname{GL}_4({\mathfrak o}_2) \ |\ g^{{t}} j g = j\}, \quad\text{where}\quad j=\left[\begin{smallmatrix} & & -1 & \\ & & & -1 \\ 1 & & & \\ & 1 & &\end{smallmatrix}\right].$$ This group admits a virtual Iwahori decomposition $(U,L,V)$, with $$\begin{split}
L &= \left. \left\{ \begin{bmatrix} a & 0 \\ 0 & a^{-{t}} \end{bmatrix} \ \right|\ a\in \operatorname{GL}_2({\mathfrak o}_2)\right\}, \quad \\
U &= \left. \left\{ \begin{bmatrix} 1 & m \\ & 1 \end{bmatrix} \ \right|\ m\in M_2({\mathfrak o}_2),\ m=m^{{t}} \right\},\quad \text{and}\quad V =U^{t},
\end{split}$$ where $(\cdot)^{{t}}$ means transpose and $(\cdot)^{-{t}}$ means transpose inverse. We consider the associated functors $$\operatorname{i}_L^G\coloneq \operatorname{i}_{U,V}:\operatorname{\mathcal{R}}(L)\to \operatorname{\mathcal{R}}(G) \qquad \text{and}\qquad \operatorname{r}^G_L\coloneq \operatorname{r}_{U,V}:\operatorname{\mathcal{R}}(G)\to \operatorname{\mathcal{R}}(L).$$
The subgroup $L\cong\operatorname{GL}_2({\mathfrak o}_2)$ has a virtual Iwahori decomposition $(U',D,V')$, where $$\begin{split}
D &= \{ \operatorname{diag}(\alpha,\delta,\alpha^{-1},\delta^{-1})\ |\ \alpha,\delta\in {\mathfrak o}_2^\times\},\quad \\
U'&= \left.\left\{ \operatorname{diag}\left( \left[\begin{smallmatrix} 1 & \beta \\ & 1 \end{smallmatrix}\right], \left[\begin{smallmatrix} 1 & \\ -\beta& 1\end{smallmatrix}\right]\right) \ \right|\ \beta \in {\mathfrak o}_2\right\} \quad\text{and}\quad V'= (U')^{t}.
\end{split}$$ We consider the associated functors $$\operatorname{i}_D^L\coloneq \operatorname{i}_{U',V'}:\operatorname{\mathcal{R}}(D)\to \operatorname{\mathcal{R}}(L) \qquad \text{and}\qquad \operatorname{r}^L_D\coloneq \operatorname{r}_{U',V'}:\operatorname{\mathcal{R}}(L)\to \operatorname{\mathcal{R}}(D).$$
[We let $$W_G \coloneq N_G(D)/D \qquad \text{and}\qquad W_L\coloneq N_L(D)/D$$ denote the Weyl groups of $D$ in $G$ and in $L$, respectively]{}. We write $\operatorname{Ad}_g$ for the conjugation action of a group on itself and subsets thereof, and with a slight abuse of notation also for the corresponding action on representations.
\[thm:sp\_Mackey\] There is a natural isomorphism of functors $\operatorname{\mathcal{R}}(L)\to \operatorname{\mathcal{R}}(L)$, $$\operatorname{r}^G_L \operatorname{i}_L^G \cong \bigoplus_{g\in W_L\backslash W_G/W_L} \operatorname{i}_{gLg^{-1}\cap L}^L \, \operatorname{Ad}_g \, \operatorname{r}^L_{L\cap g^{-1}Lg} .$$
\[rem:sp\_Mackey\] Let us unpack Theorem \[thm:sp\_Mackey\] a [little]{}.
(1) The right-hand side of the formula in Theorem \[thm:sp\_Mackey\] is a sum over a set of representatives $g\in N_G(D)$ for the double cosets of $W_L$ in $W_G$; the resulting functor does not depend on the choices made, up to natural isomorphism.
(2) For each $g\in N_G(D)$, the intersection $gLg^{-1}\cap L$ is either $L$ or $D$. The functors $\operatorname{i}_D^L$ and $\operatorname{r}^L_D$ were defined above; the functors $\operatorname{i}_L^L$ and $\operatorname{r}^L_L$ are, by definition, the identity functors on $\operatorname{\mathcal{R}}(L)$.
(3) The group $W_L$ is the two-element group generated (modulo $D$) by the matrix $$t{=} \operatorname{diag}\left( \sigma,\sigma \right) \in L,\qquad \text{where}\qquad \sigma=\left[\begin{smallmatrix} & -1 \\ 1& \end{smallmatrix}\right]\in \operatorname{GL}_2({\mathfrak o}_2).$$ The eight-element dihedral group $W_G$ is generated (modulo $D$) by $W_L$ together with the matrix $$w\coloneq \begin{bmatrix} {0} & & -1 & \\ & 1 & & \\ 1 & & {0} & \\ & & & 1 \end{bmatrix}.$$ Defining $$s\coloneq \begin{bmatrix} & \sigma \\ \sigma^{-1} & \end{bmatrix} \in G,$$ we have the double-coset decomposition $$W_G = W_L \sqcup W_L s W_L \sqcup W_L w W_L.$$ The element $s$ normalises $L$, while $wLw^{-1}\cap L = D$. Putting all of this together, the formula in Theorem \[thm:sp\_Mackey\] takes the following more explicit form: $$\operatorname{r}^G_L \operatorname{i}_L^G \cong {\mathrm{id}}\oplus \operatorname{Ad}_s \oplus \operatorname{i}_D^L \operatorname{Ad}_w \operatorname{r}^L_D.$$
(4) Note that the definition of the functors $\operatorname{i}_L^G$ and $\operatorname{r}^G_L$, and the statement of Theorem \[thm:sp\_Mackey\], continue to make sense when ${\mathfrak o}_2$ is replaced by ${\mathfrak o}_\ell$, or indeed by any finite (or profinite) commutative ring. Over ${\mathfrak o}_1$, the formula is valid: as explained in Example \[ex:pind-field\], the functors $\operatorname{i}_L^G$, $\operatorname{r}^G_L$, $\operatorname{i}_D^L$ and $\operatorname{r}^L_D$ are isomorphic in that case to Harish-Chandra functors, and the formula in Theorem \[thm:sp\_Mackey\] is an instance of the well-known formula for the composition of these functors (cf. [@DM Theorem 5.1]). We do not know whether the formula in Theorem \[thm:sp\_Mackey\] is valid for $\operatorname{Sp}_4$ over more general rings; the proof presented below relies on some very special features of ${\mathfrak o}_2$.
Our strategy for proving Theorem \[thm:sp\_Mackey\] is as follows. Reduction modulo $\pi$ gives rise to a surjective group homomorphism $G= \operatorname{Sp}_4({\mathfrak o}_2) \to \operatorname{Sp}_4({\Bbbk})$, whose kernel is an abelian group isomorphic to the Lie algebra ${\mathfrak}{sp}_4({\Bbbk})$. In Sections \[subsec:Sp4\_congruence\] and \[subsec:Sp4\_Clifford\] we apply the orbit method and Clifford theory to reduce Theorem \[thm:sp\_Mackey\] to a statement about orbits and representations of stabilisers for the adjoint action of $\operatorname{Sp}_4({\Bbbk})$ on ${\mathfrak}{sp}_4({\Bbbk})$. In Sections \[subsec:Sp4\_centralisers\] and \[subsec:Sp4\_Mackey\_proof\] we verify the theorem through a case-by-case analysis of the orbits (with some details postponed to Appendix \[appendix\]).
For the semisimple orbits our induction and restriction functors correspond to Harish-Chandra induction and restriction for (reductive) subgroups of $\operatorname{Sp}_4({\Bbbk})$, and our Mackey formula follows from the well-known Mackey formula for Harish-Chandra functors. The computation for the non-semisimple orbits—and in particular, for the one nilpotent orbit that is relevant here—is more subtle. In Corollary \[Dat\_corollary\] we shall see that it is precisely this nilpotent orbit that witnesses the difference between our induction functor and Dat’s parahoric induction.
The congruence subgroup {#subsec:Sp4_congruence}
-----------------------
Let $G_0$ denote the kernel of the reduction map $\operatorname{Sp}_4({\mathfrak o}_2) \to \operatorname{Sp}_4({\Bbbk})$, and let $${\mathfrak}{g}={\mathfrak}{sp}_4({\Bbbk}) = \{ y\in M_4({\Bbbk})\ |\ jy + y^{{t}} j =0\},$$ viewed as an additive abelian group on which $G$ acts via the adjoint action of its quotient $\operatorname{Sp}_4({\Bbbk})$. To [reduce]{} the notational load we shall write $$g \cdot y = \operatorname{Ad}_g(y)=gyg^{-1}~~\text{(modulo $\pi$)}, \quad g \in G, y \in {\mathfrak}g.$$
\[lem:Sp4\_exp\] The map $\exp:{\mathfrak}{g}\to G_0$ defined as the composition $${\mathfrak}{g} \xrightarrow{y\mapsto \pi y} \pi {\mathfrak}{sp}_4({\mathfrak o}_2) \xrightarrow{z \mapsto 1+z} G_0$$ is a $G$-equivariant group isomorphism.
Clear.
For every subgroup $H$ of $G$ we set $$H_0 \coloneq H \cap G_0,\quad \overline{H}_{\phantom{0}} \coloneq HG_0/G_0 \cong H/H_0,\quad \text{and}\quad
{\mathfrak}{h} \coloneq \log(H_0),$$ where $\log:G_0\to {\mathfrak}{g}$ denotes the inverse to $\exp$. In particular, ${\mathfrak}l$ is the additive subgroup of $M_4({\Bbbk})$ consisting of the block-diagonal matrices $\operatorname{diag}(x,-x^{{t}})$, for $x\in M_2({\Bbbk})$.
It is easily checked that the triple of subgroups $({\mathfrak}u, {\mathfrak}l, {\mathfrak}v)$ forms an Iwahori decomposition of ${\mathfrak}g$, and it follows that the triple $(U_0, L_0, V_0)$ is an Iwahori decomposition of $G_0$. Similarly, $({\mathfrak}u', {\mathfrak}d, {\mathfrak}v')$ is an Iwahori decomposition of ${\mathfrak}l$, and so $(U'_0, D_0, V_0')$ is an Iwahori decomposition of $L_0$.
\[lem:Sp4\_pairing\] Choose and fix a nontrivial character $\zeta:{\Bbbk}\to {\mathbb{C}}^\times$. For each $y\in {\mathfrak}g$, denote by ${\varphi}_y:G_0\to {\mathbb{C}}^\times$ the character $${\varphi}_y :g \mapsto \zeta\circ \operatorname{tr}\left( \log (g ) y\right).$$ The mapping $y\mapsto {\varphi}_y$ is a $G$-equivariant bijection ${\mathfrak}g\xrightarrow{\cong} \operatorname{Irr}(G_0)$, which restricts to an $L$-equivariant bijection ${\mathfrak}l \xrightarrow{\cong} \operatorname{Irr}(L_0)$, and to a $D$-equivariant bijection ${\mathfrak}d\xrightarrow{\cong} \operatorname{Irr}(D_0)$.
Let $\langle z,y\rangle\coloneq \zeta\circ\operatorname{tr}(zy)$ for $z,y\in M_4({\Bbbk})$. It is well-known that the map $M_4({\Bbbk})\to \widehat{M_4({\Bbbk})}$ sending $y$ to $\langle \cdot, y\rangle$ is an isomorphism. By Pontryagin duality this map restricts to an isomorphism between ${\mathfrak}g$ and the dual of $M_4({\Bbbk})/{\mathfrak}{g}^\perp$, where ${\mathfrak}{g}^\perp=\{z\in M_4({\Bbbk})\ |\ \langle z,{\mathfrak}g\rangle =1\}$. Let ${\mathfrak}{g}'=\{z\in M_4({\Bbbk})\ |\ jz-z^{t}j=0\}$. For each $z\in {\mathfrak}{g}'$ and $y\in {\mathfrak}g$ we have $\operatorname{tr}(zy)=\operatorname{tr}(\operatorname{Ad}_j(z)\operatorname{Ad}_j(y)) = -\operatorname{tr}(zy)$, showing that ${\mathfrak}{g}'\subseteq {\mathfrak}{g}^\perp$. We also have $M_4({\Bbbk})={\mathfrak}{g}\oplus {\mathfrak}{g}'$ (this is the eigenspace decomposition for the involution $y\mapsto \operatorname{Ad}_j(y^{t})$), and since ${\mathfrak}{g}$ and its dual $M_4({\Bbbk})/{\mathfrak}{g}^\perp$ have the same cardinality we must have ${\mathfrak}{g}'={\mathfrak}{g}^\perp$. Thus the pairing $\langle\cdot,\cdot\rangle$ restricts to an isomorphism ${\mathfrak}g\to \widehat{{\mathfrak}g}$. Composing with the isomorphism $\widehat{\log}:\widehat{{\mathfrak}{g}}\to \widehat{G_0}=\operatorname{Irr}(G_0)$ shows that $y\mapsto {\varphi}_y$ is an isomorphism ${\mathfrak}g\to \operatorname{Irr}(G_0)$. The $G$-equivariance of this map follows from the invariance of the trace. Similar arguments apply to ${\mathfrak}l$ and ${\mathfrak}d$.
Theorem \[orbit\_theorem\], applied to this particularly simple setting, gives the following identification of the induction maps $$\operatorname{i}_0 \coloneq \operatorname{i}_{U_0,V_0}:\operatorname{Irr}(L_0)\to \operatorname{Irr}(G_0)\quad \text{and}\quad \operatorname{i}_0'\coloneq \operatorname{i}_{U_0',V_0'}:\operatorname{Irr}(D_0)\to \operatorname{Irr}(L_0).$$
\[lem:Sp4\_orbit\] The diagram $$\xymatrix@R=30pt@C=50pt{
\operatorname{Irr}(D_0) \ar[r]^-{\operatorname{i}_0'}
& \operatorname{Irr}(L_0) \ar[r]^-{\operatorname{i}_0} & \operatorname{Irr}(G_0) \\
{\mathfrak}d \ar[r]^-{\mathrm{inclusion}} \ar[u]^-{y\mapsto {\varphi}_y}_-{\cong} & {\mathfrak}l \ar[r]^-{\mathrm{inclusion}} \ar[u]^-{y\mapsto {\varphi}_y}_-{\cong} & {\mathfrak}g \ar[u]^-{y\mapsto {\varphi}_y}_-{\cong}
}$$ is commutative.
In view of Theorem \[orbit\_theorem\] and Lemma \[lem:Sp4\_exp\], it is enough to observe that the diagram $$\xymatrix@R=30pt@C=60pt{
\widehat{{\mathfrak}d} \ar[r]^-{(\Lambda')^*} & \widehat{{\mathfrak}l} \ar[r]^-{\Lambda^*} & \widehat{{\mathfrak}g} \\
{\mathfrak}d \ar[u]^-{y\mapsto \langle \cdot, y\rangle}_-{\cong}\ar[r]^-{\mathrm{inclusion}} &
{\mathfrak}l \ar[u]^-{y\mapsto \langle\cdot, y\rangle}_-{\cong} \ar[r]^-{\mathrm{inclusion}} & {\mathfrak}g \ar[u]^-{y\mapsto \langle \cdot, y\rangle } _-{\cong}
}$$ commutes, where $\Lambda$ is the projection of ${\mathfrak}g = {\mathfrak}u \oplus {\mathfrak}l \oplus {\mathfrak}v$ onto its summand ${\mathfrak}l$, and $\Lambda'$ is the projection of ${\mathfrak}l$ onto its summand ${\mathfrak}d$.
Application of Clifford theory {#subsec:Sp4_Clifford}
------------------------------
We shall use Theorems \[thm:C1\], \[thm:C2\] and \[thm:C3\] to transport the functors $\operatorname{i}_L^G$ and $\operatorname{r}_L^G$ to the setting of (projective) representations of the centralisers $\overline{L}(y)\subseteq \operatorname{GL}_2({\Bbbk})$ and $\overline{G}(y)\subseteq \operatorname{Sp}_4({\Bbbk})$ associated to the characters ${\varphi}_y$.
The first assertion of Clifford theory decomposes the categories $\operatorname{\mathcal{R}}(D)$, $\operatorname{\mathcal{R}}(L)$ and $\operatorname{\mathcal{R}}(G)$ as products over the sets $D\backslash \operatorname{Irr}(D_0)$, $L\backslash \operatorname{Irr}(L_0)$ and $G\backslash \operatorname{Irr}(G_0)$, respectively. For each $y\in {\mathfrak}g$, let ${\varphi}_y$ be the character in $\operatorname{Irr}(G_0)$ defined in Lemma \[lem:Sp4\_pairing\]. We denote by $$E^G_{y}: \operatorname{\mathcal{R}}(G)\to \operatorname{\mathcal{R}}(G)_{{\varphi}_y}$$ the projection onto the subcategory associated to (the $G$-orbit of) the character ${\varphi}_y$. We similarly define $E^L_y$ and $E^D_y$, for $y\in {\mathfrak}l$ and $y\in{\mathfrak}d$ respectively.
For each $y\in {\mathfrak}l$ we write $$G(y,{\mathfrak}l) \coloneq \{g\in G\ |\ g\cdot y\in {\mathfrak}l\}$$ for the set of elements in $G$ which conjugate $y$ back into ${\mathfrak}l$. Notice that $G(y,{\mathfrak}l)$ is stable under left multiplication by $L$, and under right multiplication by $G(y)$.
The first step is to show that we may deal with ordinary, as opposed to projective, representations of the centralisers.
\[lem:Sp4\_extension\] There is a family of maps $({\varphi}'_y)_{y\in {\mathfrak}l}$ with the following properties:
1. ${\varphi}'_y$ is a one-dimensional (ordinary) representation of the centraliser $G(y)$ that extends ${\varphi}_y$.
2. For each $g\in G(y,{\mathfrak}l)$ one has $\operatorname{Ad}_g({\varphi}'_y)={\varphi}'_{g\cdot y}$.
3. ${\varphi}'_y(g)=1$ for all $g\in U(y)\cup V(y)$.
4. If $y\in {\mathfrak}d$ then ${\varphi}'_y(g)=1$ for all $g\in U'(y)\cup V'(y)$.
For each $y\in {\mathfrak}l \subset M_4({\Bbbk})$, let $H(y)$ denote the centraliser of $y$ in the group $\operatorname{GL}_4({\mathfrak o}_2)$ (which acts on $M_4({\Bbbk})$ through the adjoint action of its quotient $\operatorname{GL}_4({\Bbbk})$). Singla showed in [@Singla_GLn Proposition 2.2] that the character ${\varphi}_y$ extends to a linear character of $H(y)$. If ${\varphi}'_{y}$ is such an extension, then for each $g\in G(y,{\mathfrak}l)$ the character $\operatorname{Ad}_g({\varphi}'(y))$ is an extension of ${\varphi}_{g\cdot y}$ to $H(g\cdot y)$. Moreover, if $g\in G(y)$ then $\operatorname{Ad}_g({\varphi}'_y)={\varphi}'_y$. We may thus choose a family of characters ${\varphi}'_y$ satisfying (1) and (2) by fixing one $y$ in each $G$-orbit, choosing an extension ${\varphi}'_y$ as above, and then defining ${\varphi}'_{g\cdot y}\coloneq \operatorname{Ad}_g({\varphi}'_y)$ for each $g\in G(y,{\mathfrak}l)$.
We will prove that the characters ${\varphi}'_y$ constructed above are trivial on $U(y)$ and $V(y)$ by showing that these two groups belong to the commutator subgroup of $H(y)$. Indeed, let $m\in M_2({\mathfrak o}_2)$ be any matrix such that the $4\times 4$ matrix $
u= \left[\begin{smallmatrix} 1 & m \\ & 1\end{smallmatrix} \right]
$ lies in $H(y)$. Then the matrices $
u' = \left[\begin{smallmatrix} 1 & m/2 \\ & 1\end{smallmatrix}\right]$ and $z = \left[\begin{smallmatrix} 1 & \\ & -1\end{smallmatrix}\right]
$ also lie in $H(y)$, and we have $u=[u',z]$. This shows that $U(y)$ lies in $[H(y),H(y)]$, and a similar argument applies to $V(y)$. Thus the family ${\varphi}'_y$ constructed above satisfies condition (3).
Finally, if $y\in {\mathfrak}d$, then a similar argument to the above shows that $U'(y)$ and $V'(y)$ belong to the commutator subgroup of the centraliser of $y$ inside the block-diagonal subgroup $\operatorname{diag}(\operatorname{GL}_2({\mathfrak o}_2),\operatorname{GL}_2({\mathfrak o}_2))\subset \operatorname{GL}_4({\mathfrak o}_2)$, and so property (4) is also satisfied.
For the rest of [Section \[sec:sp\]]{} we fix a family of characters ${\varphi}'_y$ as in Lemma \[lem:Sp4\_extension\]. [As explained in Section \[subsec:Clifford\_review\],]{} Clifford theory gives equivalences of categories $$F^L_y : \operatorname{\mathcal{R}}({\overline}{L}(y))\xrightarrow{\otimes{\varphi}'_y} \operatorname{\mathcal{R}}(L(y))_{{\varphi}_y} \xrightarrow{\operatorname{ind}_{L(y)}^L} \operatorname{\mathcal{R}}(L)_{{\varphi}_y}$$ and $$F^G_y : \operatorname{\mathcal{R}}({\overline}{G}(y))\xrightarrow{\otimes{\varphi}'_y} \operatorname{\mathcal{R}}(G(y))_{{\varphi}_y} \xrightarrow{\operatorname{ind}_{G(y)}^G} \operatorname{\mathcal{R}}(G)_{{\varphi}_y} .$$
\[lem:Sp4\_FG\] For each $y\in {\mathfrak}l$, each $g\in G(y,{\mathfrak}l)$ and each $h\in N_G(L)$, the diagrams $$\xymatrix@R=30pt@C=50pt{
\operatorname{\mathcal{R}}(G)_{{\varphi}_y} \ar[r]^-{{\mathrm{id}}} & \operatorname{\mathcal{R}}(G)_{{\varphi}_{g\cdot y}} \\
\operatorname{\mathcal{R}}({\overline}{G}(y)) \ar[u]^-{F^G_y} \ar[r]^-{\operatorname{Ad}_{ {g}}} & \operatorname{\mathcal{R}}({\overline}{G}(g\cdot y))\ar[u]_-{F^G_{g\cdot y}}
} \qquad \text{and}\qquad
\xymatrix@R=30pt@C=50pt{
\operatorname{\mathcal{R}}(L)_{{\varphi}_y} \ar[r]^-{\operatorname{Ad}_h} & \operatorname{\mathcal{R}}(L)_{{\varphi}_{h\cdot y}} \\
\operatorname{\mathcal{R}}({\overline}{L}(y)) \ar[u]^-{F^L_y} \ar[r]^-{\operatorname{Ad}_{ {h}}} & \operatorname{\mathcal{R}}({\overline}{L}(h\cdot y))\ar[u]_-{F^L_{h\cdot y}}
}$$ commute up to natural isomorphism.
The commutativity of the second diagram follows from property (2) of Lemma \[lem:Sp4\_extension\], and from the well-known fact that $\operatorname{Ad}_h\circ \operatorname{ind}_{L(y)}^L \cong \operatorname{ind}_{L(h\cdot y)}^L \circ \operatorname{Ad}_h$. The commutativity of the first diagram follows from the same argument, plus the fact that $\operatorname{Ad}_g$ is isomorphic to the identity functor on $\operatorname{\mathcal{R}}(G)$ for every $g\in G$.
For each $y\in {\mathfrak}l$ we consider the functors $$\operatorname{i}_{\overline{L}(y)}^{{\overline}{G}(y)} \coloneq \operatorname{i}_{{\overline}{U}(y),{\overline}{V}(y)}:\operatorname{\mathcal{R}}({\overline}{L}(y))\to \operatorname{\mathcal{R}}({\overline}{G}(y)) \quad \text{and}\quad \operatorname{r}^{{\overline}{G}(y)}_{{\overline}{L}(y)}: \operatorname{\mathcal{R}}({\overline}{G}(y))\to \operatorname{\mathcal{R}}({\overline}{L}(y)).$$
\[lem:Sp4\_C3\_LG\] For each $y\in {\mathfrak}l$ the diagrams $$\xymatrix@R=30pt@C=50pt{
\operatorname{\mathcal{R}}(L)_{{\varphi}_y} \ar[r]^-{\operatorname{i}_L^G E^L_y} & \operatorname{\mathcal{R}}(G)_{{\varphi}_y} \\
\operatorname{\mathcal{R}}(\overline{L}(y)) \ar[r]^-{\operatorname{i}_{{\overline}{L}(y)}^{{\overline}{G}(y)}} \ar[u]^-{F^L_y}_-{\cong} & \operatorname{\mathcal{R}}({\overline}{G}(y))\ar[u]_-{F^G_y}^-{\cong}
}
\qquad \text{and}\qquad
\xymatrix@R=30pt@C=50pt{
\operatorname{\mathcal{R}}(G)_{{\varphi}_y} \ar[r]^-{E^L_y\operatorname{r}_L^G} & \operatorname{\mathcal{R}}(L)_{{\varphi}_y} \\
\operatorname{\mathcal{R}}(\overline{G}(y)) \ar[r]^-{\operatorname{r}_{{\overline}{L}(y)}^{{\overline}{G}(y)}} \ar[u]^-{F^G_y}_-{\cong} & \operatorname{\mathcal{R}}({\overline}{L}(y))\ar[u]_-{F^L_y}^-{\cong}
}$$ commute up to natural isomorphism.
The fact that ${\varphi}'_y$ is trivial on the subgroups $U(y)$ and $V(y)$ ensures that the functions $a$ and $b$ of Lemma \[lem:C3\_eX\] are [identically]{} equal to $1$, and thus that the functor $\operatorname{i}_{{\overline}{U}(y),{\overline}{V}(y)}^{{\varphi}'_y}$ appearing in Theorem \[thm:C3\] is equal to $\operatorname{i}_{{\overline}{L}(y)}^{{\overline}{G}(y)}$. This proves the commutativity of the $\operatorname{i}$-diagram; taking adjoints proves the commutativity of the $\operatorname{r}$-diagram.
Combining Lemmas \[lem:Sp4\_FG\] and \[lem:Sp4\_C3\_LG\] gives immediately:
\[lem:Sp4\_C3\_yz\] For each $y\in {\mathfrak}l$ and each $g\in G(y,{\mathfrak}l)$ the diagram $$\xymatrix@R=30pt@C=50pt{
\operatorname{\mathcal{R}}(L)_{{\varphi}_y} \ar[r]^-{\operatorname{i}_L^G E^L_y} & \operatorname{\mathcal{R}}(G)_{{\varphi}_y} \ar[r]^-{{\mathrm{id}}} & \operatorname{\mathcal{R}}(G)_{{\varphi}_{g\cdot y}} \ar[r]^{E^L_{g\cdot y}\operatorname{r}^G_L} & \operatorname{\mathcal{R}}(L)_{{\varphi}_{g\cdot y}} \\
\operatorname{\mathcal{R}}({\overline}{L}(y)) \ar[u]^-{F^L_y} \ar[r]^-{\operatorname{i}_{{\overline}{L}(y)}^{{\overline}{G}(y)}} & \operatorname{\mathcal{R}}({\overline}{G}(y)) \ar[u]^-{F^G_y} \ar[r]^-{\operatorname{Ad}_{ {g}}} & \operatorname{\mathcal{R}}({\overline}{G}(g\cdot y)) \ar[u]^-{F^G_{g\cdot y}} \ar[r]^-{\operatorname{r}_{{\overline}{L}(g\cdot y)}^{{\overline}{G}(g\cdot y)}} & \operatorname{\mathcal{R}}({\overline}{L}(g\cdot y)) \ar[u]^-{F^L_{g\cdot y}}
}$$ commutes up to natural isomorphism.
Now we use Clifford theory to analyse the right-hand side ${\mathrm{id}}\oplus \operatorname{Ad}_s \oplus \operatorname{i}_{D}^L\operatorname{Ad}_w\operatorname{r}^L_D$ of the Mackey formula (cf. Remarks \[rem:sp\_Mackey\](3)). For each pair of elements $y,z\in {\mathfrak}l$, define a functor $$\Delta(z,y):\operatorname{\mathcal{R}}(\overline{L}(y)) \to \operatorname{\mathcal{R}}(\overline{L}(z)),\qquad \Delta(z,y)=\begin{cases} \operatorname{Ad}_{ {l}} & \text{if }z=l\cdot y \\ 0 & \text{if }z\not\in L\cdot y.\end{cases}$$ Note that $\Delta(z,l)$ is well-defined up to natural isomorphism, because $\operatorname{Ad}_l\cong {\mathrm{id}}$ on $\operatorname{\mathcal{R}}({\overline}{L}(y))$ for every $l\in L(y)$.
\[lem:Sp4\_RHS\_1\] For each $y,z\in {\mathfrak}l$ the diagrams $$\xymatrix@R=30pt@C=70pt{
\operatorname{\mathcal{R}}(L)_{{\varphi}_y} \ar[r]^-{E^L_z E^L_y} & \operatorname{\mathcal{R}}(L)_{{\varphi}_z} \\
\operatorname{\mathcal{R}}({\overline}{L}(y)) \ar[r]^-{\Delta(z,y)} \ar[u]^-{F^L_y} & \operatorname{\mathcal{R}}({\overline}{L}(z))\ar[u]_-{F^L_z}
}\qquad \text{and}\qquad
\xymatrix@R=30pt@C=70pt{
\operatorname{\mathcal{R}}(L)_{{\varphi}_y} \ar[r]^-{E^L_z \operatorname{Ad}_s E^L_y} & \operatorname{\mathcal{R}}(L)_{{\varphi}_z} \\
\operatorname{\mathcal{R}}({\overline}{L}(y)) \ar[r]^-{\Delta(z,s\cdot y)\operatorname{Ad}_s} \ar[u]^-{F^L_y} & \operatorname{\mathcal{R}}({\overline}{L}(z))\ar[u]_-{F^L_z}
}$$ commute up to natural isomorphism.
The commutativity of the first diagram follows from Lemma \[lem:Sp4\_FG\], and from the fact that $E_z^L E_y^L=0$ unless $y$ and $z$ are $L$-conjugate. The commutativity of the second diagram follows from a similar argument, plus the equality $\operatorname{Ad}_s E_y^L = E_{s\cdot y}^L \operatorname{Ad}_s$.
The analysis of the functors $\operatorname{i}_D^L$ and $\operatorname{r}^L_D$ follows the above analysis of $\operatorname{i}_L^G$ and $\operatorname{r}^G_L$. Because $D$ is abelian we have $D(y)=D$ for each $y\in {\mathfrak}d$. Clifford theory gives an equivalence of categories $$F^D_y: \operatorname{\mathcal{R}}({\overline}{D}) \xrightarrow{\otimes{\varphi}'_y} \operatorname{\mathcal{R}}(D)_{{\varphi}_y},$$ such that for each $h\in N_G(D)$ the diagram $$\xymatrix@R=30pt@C=50pt{
\operatorname{\mathcal{R}}(D)_{{\varphi}_y} \ar[r]^-{\operatorname{Ad}_h} & \operatorname{\mathcal{R}}(D)_{{\varphi}_{h\cdot y}} \\
\operatorname{\mathcal{R}}({\overline}{D}) \ar[r]^-{\operatorname{Ad}_{ {h}}} \ar[u]^-{F^D_y} & \operatorname{\mathcal{R}}({\overline}{D}) \ar[u]_-{F^D_{h\cdot y}}
}$$ commutes up to natural isomorphism.
We consider the functors $$\operatorname{i}_{{\overline}{D}}^{{\overline}{L}(y)} \coloneq \operatorname{i}_{{\overline}{U_0'}(y),{\overline}{V_0'}(y)}:\operatorname{\mathcal{R}}({\overline}{D})\to \operatorname{\mathcal{R}}({\overline}{L}(y)) \qquad \text{and}\qquad \operatorname{r}^{{\overline}{L}(y)}_{{\overline}{D}}\coloneq \operatorname{r}_{{\overline}{U_0'}(y),{\overline}{V_0'}(y)}:\operatorname{\mathcal{R}}({\overline}{L}(y)) \to \operatorname{\mathcal{R}}({\overline}{D}).$$
For each $y,z\in {\mathfrak}l$, we define the functor $$\Xi(z, y): \operatorname{\mathcal{R}}({\overline}{L}(y))\to \operatorname{\mathcal{R}}({\overline}{L}(z))$$ as the direct sum, over $d\in {\mathfrak}d$, of the compositions $$\operatorname{\mathcal{R}}({\overline}{L}(y)) \xrightarrow{\Delta(d,y)} \operatorname{\mathcal{R}}({\overline}{L}(d)) \xrightarrow{\operatorname{r}^{{\overline}{L}(d)}_{{\overline}{D}}} \operatorname{\mathcal{R}}({\overline}{D}) \xrightarrow{\operatorname{Ad}_w} \operatorname{\mathcal{R}}({\overline}{D}) \xrightarrow{\operatorname{i}_{{\overline}{D}}^{{\overline}{L}(w\cdot d)}} \operatorname{\mathcal{R}}({\overline}{L}(w\cdot d)) \xrightarrow{\Delta(z, w\cdot d)} \operatorname{\mathcal{R}}({\overline}{L}(z)).$$
\[lem:Sp4\_RHS\_2\] For each $y,z\in {\mathfrak}l$ the diagram $$\xymatrix@R=30pt@C=80pt{
\operatorname{\mathcal{R}}(L)_{{\varphi}_y}\ar[r]^-{E_z^L \left( \operatorname{i}_D^L \operatorname{Ad}_w \operatorname{r}^L_D\right) E_y^L} & \operatorname{\mathcal{R}}(L)_{{\varphi}_z} \\
\operatorname{\mathcal{R}}({\overline}{L}(y)) \ar[r]^-{ \Xi(z, y)} \ar[u]^-{F^L_y} &
\operatorname{\mathcal{R}}({\overline}{L}(z)) \ar[u]_-{F^L_z}
}$$ commutes up to natural isomorphism.
Decomposing the category $\operatorname{\mathcal{R}}(D)$ over $\operatorname{Irr}(D_0)\cong {\mathfrak}d$, we have $$\operatorname{i}_D^L \operatorname{Ad}_w \operatorname{r}^L_D = \bigoplus_{d\in {\mathfrak}d} \operatorname{i}_D^L \operatorname{Ad}_w E_d^D \operatorname{r}^L_D = \bigoplus_{d\in {\mathfrak}d} \operatorname{i}_D^L E_{w\cdot d}^D \operatorname{Ad}_w E_d^D \operatorname{r}^L_D = \bigoplus_{d\in {\mathfrak}d} E_{w\cdot d}^L \operatorname{i}_D^L E_{w\cdot d}^D \operatorname{Ad}_w E_d^D \operatorname{r}^L_D E_d^L$$ where in the last equality we have used Theorem \[thm:C1\]. After applying Lemma \[lem:Sp4\_RHS\_1\] to the functors $E_y^L E_{w\cdot d}^L$ and $E_d^L E_z^L$, we are left to prove the commutativity of $$\xymatrix@R=30pt@C=50pt{
\operatorname{\mathcal{R}}(L)_{{\varphi}_d} \ar[r]^-{E^D_d \operatorname{r}^L_D} & \operatorname{\mathcal{R}}(D)_{{\varphi}_d} \ar[r]^-{\operatorname{Ad}_w} & \operatorname{\mathcal{R}}(D)_{{\varphi}_{w\cdot d}} \ar[r]^-{\operatorname{i}_D^L} & \operatorname{\mathcal{R}}(L)_{{\varphi}_{w\cdot d}} \\
\operatorname{\mathcal{R}}({\overline}{L}(d)) \ar[r]^-{\operatorname{r}^{{\overline}{L}(d)}_{{\overline}{D}}} \ar[u]^-{F^L_d} & \operatorname{\mathcal{R}}({\overline}{D}) \ar[r]^-{\operatorname{Ad}_w} \ar[u]^-{F^D_d} & \operatorname{\mathcal{R}}({\overline}{D})\ar[r]^-{\operatorname{i}_{{\overline}{D}}^{{\overline}{L}(w\cdot d)}} \ar[u]^-{F^D_{w\cdot d}} & \operatorname{\mathcal{R}}({\overline}{L}(w\cdot d)) \ar[u]^-{F^L_{w\cdot d}}
}$$ for each $d\in {\mathfrak}d$. This follows from Theorem \[thm:C3\], just as in Lemma \[lem:Sp4\_C3\_yz\].
The end result of our Clifford analysis is as follows:
\[cor:Sp4\_Clifford\] Theorem \[thm:sp\_Mackey\] is equivalent to the assertion that for every $y\in {\mathfrak}l$ and every $g\in G(y,{\mathfrak}l)$, there is a natural isomorphism $$\operatorname{r}^{{\overline}{G}(g\cdot y)}_{{\overline}{L}(g\cdot y)}\, \operatorname{Ad}_{ {g}} \, \operatorname{i}_{{\overline}{L}(y)}^{{\overline}{G}(y)} \cong \Delta(g\cdot y,y) \bigoplus \Delta(g\cdot y, s\cdot y)\operatorname{Ad}_s \bigoplus \Xi(g\cdot y, y)$$ of functors $\operatorname{\mathcal{R}}({\overline}{L}(y))\to \operatorname{\mathcal{R}}({\overline}{L}(g\cdot y))$.
By Lemma \[lem:Sp4\_pairing\] and we have $$\operatorname{r}^G_L \operatorname{i}_L^G = \bigoplus_{L\cdot y,\ L\cdot z\, \in L\backslash {\mathfrak}l} E^L_z \operatorname{r}^G_L \operatorname{i}_L^G E^L_y.$$ Theorem \[thm:C1\] implies that $$E^L_z \operatorname{r}^G_L \operatorname{i}_L^G E^L_y = E^L_z \operatorname{r}^G_L E^G_z E^G_y \operatorname{i}_L^G E^L_y,$$ and $E^G_z E^G_y=0$ unless $z$ and $y$ lie in the same $G$-orbit. This proves that $$\label{eq:Sp4_Clifford_cor1}
\operatorname{r}^G_L\operatorname{i}_L^G = \bigoplus_{\substack{L\cdot y\in L\backslash {\mathfrak}l, \\ g\in L\backslash G(y,{\mathfrak}l)/G(y)}} E^L_{g\cdot y} \operatorname{r}^G_L \operatorname{i}_L^G E^L_y.$$
Clifford theory likewise gives a decomposition $${\mathrm{id}}\oplus \operatorname{Ad}_s \oplus \operatorname{i}_D^L \operatorname{Ad}_w \operatorname{r}^L_D = \bigoplus_{L\cdot y,\ L\cdot z\, \in L\backslash {\mathfrak}l} \left( E^L_{z}E^L_y \oplus E^L_z\operatorname{Ad}_s E^L_y \oplus E^L_z\operatorname{i}_D^L \operatorname{Ad}_w \operatorname{r}^L_D E^L_y\right),$$ and Lemmas \[lem:Sp4\_RHS\_1\] and \[lem:Sp4\_RHS\_2\] imply that each term in the sum vanishes if $z$ and $y$ are not $G$-conjugate. This proves that $$\label{eq:Sp4_Clifford_cor2}
{\mathrm{id}}\oplus \operatorname{Ad}_s \oplus \operatorname{i}_D^L \operatorname{Ad}_w \operatorname{r}^L_D = \bigoplus_{\substack{L\cdot y\in L\backslash {\mathfrak}l, \\ g\in L\backslash G(y,{\mathfrak}l)/G(y)}} \left(E^L_{g\cdot y}E^L_y \oplus E^L_{g\cdot y} \operatorname{Ad}_s E^L_y \oplus E^L_{g\cdot y} \operatorname{i}_D^L \operatorname{Ad}_w \operatorname{r}^L_D E^L_y\right) .$$
Thus the reformulation of Theorem \[thm:sp\_Mackey\] given in Remarks \[rem:sp\_Mackey\](3) is equivalent to the existence of a natural isomorphism, for each $y\in {\mathfrak}l$ and each $g\in G(y,{\mathfrak}l)$, between the $(y,g)$ terms on the right-hand sides of and . Conjugating each of these terms by the equivalences $F^L_{y}$ and $F^L_{g\cdot y}$, and applying Lemmas \[lem:Sp4\_C3\_yz\], \[lem:Sp4\_RHS\_1\] and \[lem:Sp4\_RHS\_2\], we bring Theorem \[thm:sp\_Mackey\] into the asserted form.
Centralisers {#subsec:Sp4_centralisers}
------------
We now present the facts about the centralisers $\overline{G}(y)$ and $\overline{L}(y)$ and about the orbit spaces $L\backslash G(y,{\mathfrak}l)/G(y)$ that will be needed for the proof of Theorem \[thm:sp\_Mackey\]. More details, and the proofs of the assertions made here, are given in Appendix \[appendix\].
Fix $x\in M_2({\Bbbk})$, and let $y=\operatorname{diag}(x,-x^{t})$ be the corresponding element of ${\mathfrak}l$. We divide our analysis according to the Jordan normal form of $x$. Up to conjugacy by ${\overline}{L}\cong \operatorname{GL}_2({\Bbbk})$, the following nine cases exhaust all of the possibilities. In the following $s$, $t$, $w$ and $\sigma$ are as in Remarks \[rem:sp\_Mackey\]. We shall write $L\backslash G(y,{\mathfrak}l)/G(y)=\{g,h,k\}$ to mean that $G(y,{\mathfrak}l) = L g G(y) \sqcup L h G(y) \sqcup Lk G(y)$.
[**Case 1:**]{} $x=\operatorname{diag}(\mu,\mu)$, $\mu\in {\Bbbk}$. In this case $\overline{L}(y)=\overline{L} \cong \operatorname{GL}_2({\Bbbk})$. There are two subcases:
[**1A:**]{} $\mu \ne 0$. Here $\overline{G}(y)=\overline{L}(y)$, and $L\backslash G(y,{\mathfrak}l)/G(y)=\{1,s,w\}$.
[**1B:**]{} ${\mu=0}$. Here ${\overline}{G}(y)={\overline}{G}$ and $L\backslash G(y,{\mathfrak}l)/G(y)=\{1\}$.
[**Case 2:**]{} $x=\operatorname{diag}(\mu,\nu)$, $\mu \ne \nu$. In this case $\overline{L}(y)=\overline{D}$. There are three subcases:
[**2A:**]{} [$\mu \neq \pm \nu$, $\mu\neq 0\neq \nu$.]{} Here $\overline{G}(y)=\overline{L}(y)$, and $L\backslash G(y,{\mathfrak}l)/G(y) = \{1,s,w,wt\}$.
[**2A$\!^\star$:**]{} [$\nu= 0$.]{} Here $\overline{G}(y)$ is a reductive group over ${\Bbbk}$; ${\overline}{L}(y)={\overline}{D}$ is a rational maximal torus, whose Weyl group in ${\overline}{G}(y)$ is generated (modulo ${\overline}{D}$) by the involution $t^{-1}w t$; and ${\overline}{U}(y)$ and ${\overline}{V}(y)$ are the unipotent radicals of an opposite pair of rational Borel subgroups of ${\overline}{G}(y)$ containing ${\overline}{L}(y)$. We have $L\backslash {\overline}{G}(y)/G(y)=\{1,w\}$.
[**2B:**]{} [${\mu =-\nu}$.]{} In this case we have ${\overline}{G}(y)=\operatorname{Ad}_w(\overline{L})$, ${\overline}{U}(y) = \operatorname{Ad}_w\left( {\overline}{V'} \right)$ and ${\overline}{V}(y) = \operatorname{Ad}_w\left( {\overline}{U'} \right)$, while $L\backslash G(y,{\mathfrak}l)/G(y)=\{1,w,wt\}$.
[**Case 3:**]{} $x=\big[\begin{smallmatrix} \alpha & \beta \\ \mu\beta & \alpha\end{smallmatrix}\big], \mu \in {\Bbbk}$ non-square, $\alpha\in {\Bbbk}$, $\beta\in {\Bbbk}^\times$. In this case ${\Bbbk}_2\coloneq M_2({\Bbbk})(x)$ is a quadratic field extension of ${\Bbbk}$, and $\overline{L}(y)=\{\operatorname{diag}(a,a^{-{t}})\ | \ a\in {\Bbbk}_2^\times \} \cong {\Bbbk}_2^\times$. There are two subcases.
[**3A:**]{} [${\alpha\neq 0}$.]{} We have ${\overline}{G}(y)=\overline{L}(y)$ and $L\backslash G(y,{\mathfrak}l)/G(y) = \{1,s\}$.
[**3B:**]{} [$\alpha=0$.]{} Here ${\overline}{G}(y)$ is a reductive group over ${\Bbbk}$; ${\overline}{L}(y)$ is a rational maximal torus of ${\overline}{G}(y)$ whose Weyl group is generated by $s$; and ${\overline}{U}(y)$ and ${\overline}{V}(y)$ are the unipotent radicals of an opposite pair of rational Borel subgroups of ${\overline}{G}(y)$ containing ${\overline}{L}(y)$. In this case $L\backslash G(y,{\mathfrak}l)/G(y)=\{1\}$.
[**Case 4:**]{} ${x=\left[\begin{smallmatrix} \mu & 1 \\ & \mu \end{smallmatrix}\right]}$. In this case $\overline{L}(y)=\{\operatorname{diag}(a,a^{-{t}})\ |\ a\in {\Bbbk}[x]\} \cong \operatorname{GL}_1({\Bbbk}[{\varepsilon}]/({\varepsilon}^2))$. There are two subcases.
[**4A:**]{} [${\mu\neq 0}$.]{} Here $\overline{G}(y)=\overline{L}(y)$ and $L\backslash G(y,{\mathfrak}l)/G(y) = \{1,s\}$.
[**4B:**]{} [$\mu =0$.]{} The subgroups ${\overline}{U}(y)$ and ${\overline}{V}(y)$ commute with one another in ${\overline}{G}(y)$, and we have $${\overline}{G}(y) = \left( {\overline}{U}(y)\times {\overline}{V}(y) \right) \rtimes \left( {\overline}{L}(y)\rtimes S\right)$$ where $S$ is the two-element group generated by $s$. We have $L\backslash G(y,{\mathfrak}l)/G(y)=\{1\}$.
Proof of Theorem \[thm:sp\_Mackey\] {#subsec:Sp4_Mackey_proof}
-----------------------------------
In this section we shall use the results of the previous section to prove that for each $y\in {\mathfrak}l$ and each $g\in G(y,{\mathfrak}l)$, there is a natural isomorphism $$\label{eq:sp_Mackey_proof}
\operatorname{r}^{{\overline}{G}(g\cdot y)}_{{\overline}{L}(g\cdot y)}\, \operatorname{Ad}_{ {g}} \, \operatorname{i}_{{\overline}{L}(y)}^{{\overline}{G}(y)} \cong \Delta(g\cdot y,y) \bigoplus \Delta(g\cdot y, s\cdot y)\operatorname{Ad}_s \bigoplus \Xi(g\cdot y, y)$$ of functors $\operatorname{\mathcal{R}}({\overline}{L}(y))\to \operatorname{\mathcal{R}}({\overline}{L}(g\cdot y)$. By Corollary \[cor:Sp4\_Clifford\], this constitutes a proof of Theorem \[thm:sp\_Mackey\]. We recall from Section \[subsec:Sp4\_Clifford\] that $$\Delta(z,y)=\begin{cases} \operatorname{Ad}_{ {l}} & \text{if }z=l\cdot y \\ 0 & \text{if }z\not\in L\cdot y \end{cases} \qquad \text{and}\qquad
\Xi(z,y) = \bigoplus_{d\in {\mathfrak}d} \left(\Delta(z,w\cdot d) \operatorname{i}_{{\overline}{D}}^{{\overline}{L}(w\cdot d)} \operatorname{Ad}_w \operatorname{r}^{{\overline}{L}(d)}_{{\overline}{D}} \Delta(d,y)\right)$$ for all $y,z\in {\mathfrak}l$.
The proof of goes through a case-by-case analysis of the various possibilities for $y=\operatorname{diag}(x,-x^{t})$. The cases are labelled as in Section \[subsec:Sp4\_centralisers\]. The reader who is more interested in ideas than in details might like to focus on cases 1A , 2A$\!^\star$ and 4B, which together contain all of the techniques used in the other cases.
[**Case 1A:**]{} Take $y=\operatorname{diag}(\mu,\mu,-\mu,-\mu)$, $\mu\neq 0$. We must consider $g=1$, $g=s$ and $g=w$.
For $g=1$, the left-hand side of is the identity on $\operatorname{\mathcal{R}}({\overline}{L})$, because all of the centralisers are equal to ${\overline}{L}$. We have $\Delta(y,s\cdot y)=0$ because $y$ and $s\cdot y=-y$ are not $L$-conjugate. The only diagonal matrix $d\in {\mathfrak}d$ that is $L$-conjugate to $y$ is $d=y$ itself, and we have $\Delta(y,w\cdot y)=0$, and so $\Xi(y,y)=0$. Thus the only nonzero term on the right-hand side of is $\Delta(y,y)$, which is the identity on $\operatorname{\mathcal{R}}({\overline}{L})$. Thus the two sides of are isomorphic.
For $g=s$ all of the centralisers are again equal to $\overline{L}$, and so the left-hand side of equals $\operatorname{Ad}_s$. One finds as above that the only nonzero term on the right-hand side is $\Delta(s\cdot y, s\cdot y)\operatorname{Ad}_s$, which equals $\operatorname{Ad}_s$.
For $g=w$ we have $w\cdot y = \operatorname{diag}(-\mu,\mu,\mu,-\mu)$, and so the centralisers of $g\cdot y$ are as in case 2B. The left-hand side of is thus equal to $\operatorname{r}^{{\overline}{G}(w\cdot y)}_{{\overline}{D}} \operatorname{Ad}_w$. Since $\Delta(w\cdot y, y)$ and $\Delta(w\cdot y, s\cdot y)$ are both zero, the only potentially nonzero term on the right-hand side of is $\Xi(w\cdot y, y)$. Since the only diagonal matrix that is $L$-conjugate to $y$ is $y$ itself, we have $$\Xi(w\cdot y, y) = \Delta(w\cdot y, w\cdot y) \operatorname{i}_{{\overline}{D}}^{{\overline}{L}(w\cdot y)} \operatorname{Ad}_w \operatorname{r}^{{\overline}{L}(y)}_{{\overline}{D}} \Delta(y,y) = \operatorname{Ad}_w \operatorname{r}^{{\overline}{L}}_{{\overline}{D}} .$$ Now, we have ${\overline}{G}(w\cdot y)=\operatorname{Ad}_w({\overline}{L})$, ${\overline}{U}(w\cdot y)=\operatorname{Ad}_w({\overline}{V'})$, and ${\overline}{V}(w\cdot y) = \operatorname{Ad}_w({\overline}{U'})$ (see case 2B in Section \[subsec:Sp4\_centralisers\]), and therefore $$\operatorname{r}^{{\overline}{G}(w\cdot y)}_{{\overline}{D}} \operatorname{Ad}_w = \operatorname{r}_{{\overline}{U}(w\cdot y),{\overline}{V}(w\cdot y)} \operatorname{Ad}_w \cong \operatorname{Ad}_w \operatorname{r}_{{\overline}{V'},{\overline}{U'}} \cong \operatorname{Ad}_w\operatorname{r}_{{\overline}{U'},{\overline}{V'}} = \operatorname{Ad}_w \operatorname{r}^{{\overline}{L}}_{{\overline}{D}}$$ where we used Theorem \[thm:pind-properties\] to switch ${\overline}{U'}$ and ${\overline}{V'}$. This completes the proof of in case 1A.
[**Case 2A:**]{} Take $y=\operatorname{diag}(\mu ,\nu,-\mu ,-\nu)$, where $\mu$ and $\nu$ are nonzero and $\mu \neq \pm \nu$. We must consider $g=1$, $g=s$, $g=w$ and $g=wt$. For each of these $g$ the matrix $g\cdot y$ is again of the form 2A, and so all of the centralisers appearing in are equal to $\overline{D}$, and the left-hand side of is equal to the functor $\operatorname{Ad}_g$ on $\operatorname{\mathcal{R}}(\overline{D})$.
For $g=1$ the functor $\Delta( y, y)$ equals the identity, while $\Delta(y, s\cdot y)=0$ (because $y$ and $s\cdot y$ are not $L$-conjugate) and $\Xi(y,y)=0$ (because the only diagonal matrices that are $L$-conjugate to $y$ are $y$ and $t\cdot y$, and neither of these is $L$-conjugate to $w\cdot y$). So both sides of equal the identity.
For $g=s$ the functor $\Delta(s\cdot y, s\cdot y)$ is the identity, while $\Delta(s\cdot y,y)$ and $\Xi(s\cdot y, s\cdot y)$ are both zero. So both sides of equal $\operatorname{Ad}_s$.
For $g=w$, the only potentially nonzero term on the right-hand side of is $\Xi(w\cdot y, y)$. There are two diagonal matrices $d\in {\mathfrak}d$ that are $L$-conjugate to $y$, namely $y$ itself and $t\cdot y$. Since $w\cdot y=\operatorname{diag}(-\mu,\nu,\mu,\-\nu)$ and $wt\cdot y=\operatorname{diag}(-\nu,\mu,\nu,-\mu)$ are not $L$-conjugate, we have $\Delta(w\cdot y, wt\cdot y)=0$, and so the summand in $\Xi(w\cdot y, w\cdot y)$ corresponding to $d=t\cdot y$ is equal to zero. Therefore $$\Xi(w\cdot y, y) = \Delta(w\cdot y, w\cdot y) \operatorname{i}_{{\overline}{D}}^{{\overline}{L}(w\cdot y)} \operatorname{Ad}_w \operatorname{r}^{{\overline}{L}(y)}_{{\overline}{D}} \Delta(y,y) = \operatorname{Ad}_w$$ as required.
For $g=wt$ the argument of the previous paragraph shows that the right-hand side of is equal to $\Xi(wt\cdot y, y)$, and that only the $d=t\cdot y$ summand in the latter is nonzero. We have $$\Xi(wt\cdot y, y) = \Delta(wt\cdot y,wt\cdot y) \operatorname{i}_{{\overline}{D}}^{{\overline}{L}(wt\cdot y)} \operatorname{Ad}_w \operatorname{r}^{{\overline}{L}(t\cdot y)}_{{\overline}{D}} \Delta(t\cdot y,y)
= \operatorname{Ad}_w \operatorname{Ad}_t$$ because $\Delta(t\cdot y,y)=\operatorname{Ad}_t$ and all of the centralisers equal ${\overline}{D}$. This completes the proof of in case 2A.
[**Case 2A$\!^\star$:**]{} Let $y=\operatorname{diag}(\mu,0,-\mu,0)$ where $\mu\neq 0$. We must consider $g=1$ and $g=w$.
For $g=1$, the left-hand side of equals $\operatorname{r}^{{\overline}{G}(y)}_{{\overline}{D}} \operatorname{i}_{{\overline}{D}}^{{\overline}{G}(y)}$. We are in the situation of Example \[ex:pind-field\], and so $\operatorname{r}^{{\overline}{G}(y)}_{{\overline}{D}}$ and $\operatorname{i}_{{\overline}{D}}^{{\overline}{G}(y)}$ are isomorphic to the functors of Harish-Chandra restriction and induction (respectively) for the maximal torus ${\overline}{D}\subset {\overline}{G}(y)$. Since the Weyl group of ${\overline}{D}$ in ${\overline}{G}(y)$ is equal to $\{1,t^{-1}wt\}$, the usual Mackey formula (cf. [@DM Theorem 5.1]) for the composition of Harish-Chandra functors gives $$\operatorname{r}^{{\overline}{G}(y)}_{{\overline}{D}} \operatorname{i}_{{\overline}{D}}^{{\overline}{G}(y)} \cong {\mathrm{id}}\oplus \operatorname{Ad}_{t^{-1}w t}.$$
Still taking $g=1$, we have $\Delta(g\cdot y,s\cdot y)=0$, and so the right-hand side of equals ${\mathrm{id}}\oplus \Xi(y,y)$. The only diagonal matrices that are $L$-conjugate to $y$ are $d=y$ and $d=t\cdot y$. For $d=y$ we have $\Delta(y,w\cdot y)=0$, and so the only potentially nonzero summand in $\Xi(y,y)$ is the one corresponding to $d=t\cdot y$. Computing this summand, we find $$\Xi(y,y)=\Delta(y,wt\cdot y) \operatorname{i}_{{\overline}{D}}^{{\overline}{L}(wt\cdot y)} \operatorname{Ad}_w \operatorname{r}^{{\overline}{L}(t\cdot y)}_{{\overline}{D}} \Delta(t\cdot y, y) = \operatorname{Ad}_{t^{-1}} \operatorname{Ad}_w \operatorname{Ad}_t,$$ because ${\overline}{L}(wt\cdot y)={\overline}{L}(t\cdot y)={\overline}{D}$. Thus the right-hand side of is, like the left-hand side, isomorphic to ${\mathrm{id}}\oplus \operatorname{Ad}_{t^{-1}wt}$.
Now take $g=w$. Notice that $w\cdot y = -y$. The left-hand side of is $$\operatorname{r}^{{\overline}{G}(y)}_{{\overline}{D}} \operatorname{Ad}_w \operatorname{i}_{{\overline}{D}}^{{\overline}{G}(y)} \cong \operatorname{Ad}_w \operatorname{r}^{{\overline}{G}(y)}_{{\overline}{D}} \operatorname{i}_{{\overline}{D}}^{{\overline}{G}(y)} \cong \operatorname{Ad}_w \oplus \operatorname{Ad}_{ts},$$ where for the first isomorphism we have used Theorem \[thm:pind-properties\], and for the second we have used the Mackey formula for Harish-Chandra induction together with the equality $wt^{-1}wt=ts$ in $G$.
Keeping $g=w$ and turning to the right-hand side of , the term $\Delta(w\cdot y,y)$ vanishes, while the fact that $w\cdot y = ts\cdot y$ implies that $\Delta(w\cdot y, s\cdot y)\operatorname{Ad}_s=\operatorname{Ad}_{ts}$. So we are left to show that $\Xi(w\cdot y,y)=\operatorname{Ad}_w$. The $d=y$ term in $\Xi(w\cdot y, y)$ is equal to $$\Delta(w\cdot y, w\cdot y) \operatorname{i}_{{\overline}{D}}^{{\overline}{D}} \operatorname{Ad}_w \operatorname{r}^{{\overline}{D}}_{{\overline}{D}} \Delta(y,y) = \operatorname{Ad}_w,$$ while the $d=t\cdot y$ term vanishes because $\Delta(w\cdot y, t\cdot y)=0$. Thus both sides of are isomorphic to $\operatorname{Ad}_w\oplus \operatorname{Ad}_{ts}$ in this case.
[**Case 3A:**]{} Take $x=\left[\begin{smallmatrix} \alpha & \beta \\ \beta\mu & \alpha \end{smallmatrix}\right]$, where ${\mu\in {\Bbbk}}$ is a non-square and $\alpha,\beta\in {\Bbbk}^\times$, and let $y=\operatorname{diag}(x,-x^{t})$. We must consider $g=1$ and $g=s$. We have ${\overline}{G}(y)={\overline}{G}(s\cdot y) = {\overline}{L}(s\cdot y)={\overline}{L}(y)$, so that the left-hand side of is equal to $\operatorname{Ad}_g$ for each $g$. Note that since $y$ is not $L$-conjugate to a diagonal matrix we have $\Xi(z,y)=0$ for every $z$.
For $g=1$ we have $\Delta(y,y)={\mathrm{id}}$ while $\Delta(y,s\cdot y)=0$, so both sides of equal the identity.
For $g=s$ we have $\Delta(s\cdot y,y)=0$ while $\Delta(s\cdot y,s\cdot y)={\mathrm{id}}$ and so both sides of equal $\operatorname{Ad}_s$. So holds in case 3A .
[**Case 4A:**]{} Take ${x=\left[\begin{smallmatrix} \mu & 1 \\ & \mu \end{smallmatrix}\right]}$, where ${\mu\neq 0}$, and let $y=\operatorname{diag}(x,-x^{t})$. The argument is the same as in case 3A.
[**Case 1B:**]{} Take $y=0$. We need only consider $g=1$. Then becomes the assertion that $$\operatorname{r}^{{\overline}{G}}_{{\overline}{L}} \operatorname{i}_{{\overline}{L}}^{{\overline}{G}} \cong {\mathrm{id}}\oplus \operatorname{Ad}_s \oplus \operatorname{i}_{{\overline}{D}}^{{\overline}{L}}\operatorname{Ad}_w \operatorname{r}^{{\overline}{L}}_{{\overline}{D}}.$$ This is true: the functors $\operatorname{i}_{{\overline}{L}}^{{\overline}{G}}$ and $\operatorname{r}^{{\overline}{G}}_{{\overline}{L}}$ identify, as in Example \[ex:pind-field\], with the functors of Harish-Chandra induction and restriction for the Siegel Levi subgroup in ${\overline}{G}=\operatorname{Sp}_4({\Bbbk})$, and the above formula is just the standard Mackey formula for the composition of these functors.
[**Case 2B:**]{} Let $y=\operatorname{diag}(\mu,-\mu,-\mu,\mu)$, $\mu\neq 0$. We must consider $g=1$, $g=w$ and $g=wt$.
For $g=1$ the left-hand side of is equal to $$\operatorname{r}^{\operatorname{Ad}_w({\overline}{L})}_{{\overline}{D}} \operatorname{i}_{{\overline}{D}}^{\operatorname{Ad}_w({\overline}{L})} = \operatorname{Ad}_w \operatorname{r}^{{\overline}{L}}_{{\overline}{D}} \operatorname{i}_{{\overline}{D}}^{{\overline}{L}} \operatorname{Ad}_{w^{-1}} \cong \operatorname{Ad}_w({\mathrm{id}}\oplus \operatorname{Ad}_t) \operatorname{Ad}_{w^{-1}} \cong {\mathrm{id}}\oplus \operatorname{Ad}_{s}$$ where we have identified $\operatorname{i}_{{\overline}{D}}^{{\overline}{L}}$ and $\operatorname{r}^{{\overline}{L}}_{{\overline}{D}}$ with Harish-Chandra functors and applied the usual Mackey formula for the group ${\overline}{L}\cong \operatorname{GL}_2({\Bbbk})$ and its diagonal torus ${\overline}{D}$. On the right-hand side of we have $\Delta(y,y)={\mathrm{id}}$ and $\Delta(y,s\cdot y)={\mathrm{id}}$, so we are left to show that $\Xi(y,y)=0$. The only $d\in {\mathfrak}d$ with $\Delta(d,y)\neq 0$ are $d=y$ and $d=t\cdot y$. In both of these cases we have $\Delta(y, w\cdot d)=0$, and so $\Xi(y,y)=0$ as required.
The $g=w$ and $g=wt$ cases follow the argument for the $g=w$ component of case 1A. The left-hand side of is isomorphic to $\operatorname{Ad}_g \operatorname{i}_{{\overline}{D}}^{\operatorname{Ad}_g({\overline}{L})}$, while the right-hand side is isomorphic to $\operatorname{i}_{{\overline}{D}}^{{\overline}{L}} \operatorname{Ad}_g$, and the two sides are isomorphic to each other by Theorem \[thm:pind-properties\].
[**Case 3B:**]{} Let ${x=\left[\begin{smallmatrix} & \beta \\ \beta\mu & \end{smallmatrix}\right]}$, ${\mu\in {\Bbbk}}$ a non-square, ${\beta\in {\Bbbk}^\times}$, and take $y=\operatorname{diag}(x,-x^{t})$. We need consider only $g=1$. We have on the one hand $\Delta(y,s\cdot y)={\mathrm{id}}$, while on the other hand $\Xi(y,y)=0$ (since $y$ is not $L$-conjugate to any $d\in {\mathfrak}d$), and so reads $$\operatorname{r}^{{\overline}{G}(y)}_{{\overline}{L}(y)} \operatorname{i}_{{\overline}{L}(y)}^{{\overline}{G}(y)} \cong {\mathrm{id}}\oplus \operatorname{Ad}_s.$$ This is true: the functors on the right-hand side are isomorphic to Harish-Chandra functors as in Example \[ex:pind-field\], and the above formula is the usual Mackey formula for these functors.
[**Case 4B:**]{} Take $x=\left[\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix}\right]$ and $y=\operatorname{diag}(x,-x^{t})$. We need only consider $g=1$. As in case 3B, the right-hand side of is ${\mathrm{id}}\oplus \operatorname{Ad}_s$, while the left-hand side is $\operatorname{r}^{{\overline}{G}(y)}_{{\overline}{L}(y)} \operatorname{i}_{{\overline}{L}(y)}^{{\overline}{G}(y)}$. Since ${\overline}{U}(y)$ and ${\overline}{V}(y)$ commute, the latter functor is isomorphic as in Example \[ex:pind-commuting\] to the tensor product with the ${\mathcal H}({\overline}{L}(y))$-bimodule $$e_{{\overline}{U}(y)} e_{{\overline}{V}(y)} {\mathcal H}({\overline}{G}(y)) e_{{\overline}{U}(y)} e_{{\overline}{V}(y)} \cong {\mathcal H}\left( ({\overline}{U}(y)\times {\overline}{V}(y))\backslash {\overline}{G}(y) / ({\overline}{U}(y)\times {\overline}{V}(y)\right) = {\mathcal H}\left( ({\overline}{U}(y)\times {\overline}{V}(y))\backslash {\overline}{G}(y)\right),$$ with the last equality holding because ${\overline}{U}(y)\times {\overline}{V}(y)$ is normal in ${\overline}{G}(y)$. The semidirect product decomposition of ${\overline}{G}(y)$ given in Section \[subsec:Sp4\_centralisers\] for this case implies that $${\mathcal H}\left( ({\overline}{U}(y)\times {\overline}{V}(y))\backslash {\overline}{G}(y)\right) \cong {\mathcal H}({\overline}{L}(y)\rtimes S)\cong {\mathcal H}({\overline}{L}(y))\oplus {\mathcal H}({\overline}{L}(y))s$$ as ${\mathcal H}({\overline}{L}(y))$-bimodules, and so the corresponding tensor product functor $\operatorname{r}^{{\overline}{G}(y)}_{{\overline}{L}(y)}\operatorname{i}_{{\overline}{L}(y)}^{{\overline}{G}(y)}$ is isomorphic to ${\mathrm{id}}\oplus \operatorname{Ad}_s$ as required.
This completes the proof of and hence, by Corollary \[cor:Sp4\_Clifford\], of Theorem \[thm:sp\_Mackey\].
Comparison with parahoric induction {#parahoric_section}
-----------------------------------
We now come to the second corollary of the analysis of Sections \[subsec:Sp4\_congruence\]–\[subsec:Sp4\_centralisers\]. In addition to the functor $$\operatorname{i}_{U,V}:\operatorname{\mathcal{R}}(L)\to \operatorname{\mathcal{R}}(G)$$ that we have been considering until now, we also have the functor $$\operatorname{i}_{U_0,V} :\operatorname{\mathcal{R}}(L)\to \operatorname{\mathcal{R}}(G),$$ which is an example of *parahoric induction* as defined by Dat in [@Dat_parahoric]. More precisely, $\operatorname{i}_{U_0,V}$ is the restriction of a parahoric induction functor for $\operatorname{Sp}_4({\mathfrak o})$ to the subcategory of representations inflated from $\operatorname{Sp}_4({\mathfrak o}_2)$, where ${\mathfrak o}$ is the ring of integers in a non-archimedean local field. It follows immediately from the definitions that we have a natural inclusion $ \operatorname{i}_{U,V} \subseteq \operatorname{i}_{U_0, V}$. We shall show that this inclusion is proper. This gives a negative answer to [@Dat_parahoric Question 2.15] in this case.
\[Dat\_corollary\] Let $x\in M_2({\Bbbk})$ and consider $y=\operatorname{diag}(x,-x^{t})\in {\mathfrak}l$. The restrictions of the functors $$\operatorname{i}_{U,V} ,\ \operatorname{i}_{U_0,V} :\operatorname{\mathcal{R}}(L) \to \operatorname{\mathcal{R}}(G)$$ to the subcategory $\operatorname{\mathcal{R}}(L)_{{\varphi}_y}$ are mutually nonisomorphic if $x$ is nonzero and nilpotent; and these restrictions are mutually isomorphic if $x$ is zero or non-nilpotent.
The computations of Section \[subsec:Sp4\_Clifford\] show that there are commutative (up to natural isomorphism) diagrams $$\xymatrix@R=30pt@C=50pt{
\operatorname{\mathcal{R}}(L)_{{\varphi}_y} \ar[r]^-{\operatorname{i}_{U,V}} & \operatorname{\mathcal{R}}(G)_{{\varphi}_y} \\
\operatorname{\mathcal{R}}({\overline}{L}(y)) \ar[r]^-{\operatorname{i}_{{\overline}{U}(y),{\overline}{V}(y)}} \ar[u]^-{F^L_y}_-{\cong} & \operatorname{\mathcal{R}}({\overline}{G}(y)) \ar[u]_-{F^G_y}^-{\cong}
} \qquad \text{and}\qquad
\xymatrix@R=30pt@C=50pt{
\operatorname{\mathcal{R}}(L)_{{\varphi}_y} \ar[r]^-{\operatorname{i}_{U_0,V}} & \operatorname{\mathcal{R}}(G)_{{\varphi}_y} \\
\operatorname{\mathcal{R}}({\overline}{L}(y)) \ar[r]^-{\operatorname{i}_{{\overline}{V}(y)}} \ar[u]^-{F^L_y}_-{\cong} & \operatorname{\mathcal{R}}({\overline}{G}(y)) \ar[u]_-{F^G_y}^-{\cong}
}$$ If $x$ is semisimple then ${\overline}{G}(y)$ is a finite reductive group, and ${\overline}{U}(y)$ and ${\overline}{V}(y)$ are the unipotent radicals of an opposite pair of rational parabolic subgroups with common Levi subgroup ${\overline}{L}(y)$. The functor $\operatorname{i}_{{\overline}{V}(y)}$is the Harish-Chandra induction functor associated to the parabolic subgroup ${\overline}{L}(y){\overline}{V}(y)$ of ${\overline}{G}(y)$, and as in Example \[ex:pind-field\] the natural inclusion $\operatorname{i}_{{\overline}{U}(y),{\overline}{V}(y)}\subseteq \operatorname{i}_{{\overline}{V}(y)}$ is an isomorphism.
If $x$ is neither semisimple nor nilpotent, as in Case 4A, then the groups ${\overline}{U}(y)$ and ${\overline}{V}(y)$ are both trivial, ${\overline}{G}(y)={\overline}{L}(y)$, and the functors $\operatorname{i}_{{\overline}{U}(y),{\overline}{V}(y)}$ and $\operatorname{i}_{{\overline}{V}(y)}$ are both isomorphic to the identity.
We are left to consider the case where $x$ is nilpotent; say $x=\left[\begin{smallmatrix} 0 & 1\\ 0 & 0 \end{smallmatrix}\right]$. In this case we have $${\overline}{G}(y)= \left( {\overline}{U}(y)\times {\overline}{V}(y)\right) \rtimes \left({\overline}{L}(y)\rtimes S\right),$$ from which it follows (cf. Example \[ex:pind-commuting\]) that the functors $\operatorname{i}_{{\overline}{U}(y),{\overline}{V}(y)}$ and $\operatorname{i}_{{\overline}{V}(y)}$ are isomorphic, respectively, to the compositions $$\begin{aligned}
& \operatorname{i}_{{\overline}{U}(y),{\overline}{V}(y)}:\operatorname{\mathcal{R}}({\overline}{L}(y)) \xrightarrow{\operatorname{inf}} \operatorname{\mathcal{R}}\left( \left({\overline}{U}(y)\times {\overline}{V}(y) \right)\rtimes {\overline}{L}(y)\right) \xrightarrow{\operatorname{ind}} \operatorname{\mathcal{R}}({\overline}{G}(y)), \\
& \operatorname{i}_{{\overline}{V}(y)}: \operatorname{\mathcal{R}}({\overline}{L}(y)) \xrightarrow{\operatorname{inf}} \operatorname{\mathcal{R}}({\overline}{V}(y)\rtimes {\overline}{L}(y)) \xrightarrow{\operatorname{ind}} \operatorname{\mathcal{R}}({\overline}{G}(y)).
\end{aligned}$$ The functor $\operatorname{i}_{{\overline}{U}(y),{\overline}{V}(y)}$ thus scales the ${\mathbb{C}}$-dimension of representations by a factor of $$\left[{\overline}{G}(y) : ({\overline}{U}(y)\times {\overline}{V}(y))\rtimes {\overline}{L}(y) \right] = |S| = 2,$$ while $\operatorname{i}_{{\overline}{V}(y)}$ scales the dimension by $$\left[{\overline}{G}(y) :{\overline}{V}(y)\rtimes {\overline}{L}(y) \right] = |S|\cdot |{\overline}{U}(y)| = 2 |{\Bbbk}|.$$ Thus $\operatorname{i}_{U,V}$ is not isomorphic to $\operatorname{i}_{U_0,V}$ as functors on $\operatorname{\mathcal{R}}(L)_{{\varphi}_y}$.
The above proof also shows that the parahoric induction and restriction functors do not satisfy the analogue of Theorem \[thm:sp\_Mackey\]:
Let $x\in M_2({\Bbbk})$ be nonzero and nilpotent, and let $y=\operatorname{diag}(x,-x^{t})$. The restriction of the functor $$\operatorname{r}_{U_0,V}\operatorname{i}_{U_0,V} : \operatorname{\mathcal{R}}(L) \to \operatorname{\mathcal{R}}(L)$$ to the subcategory $\operatorname{\mathcal{R}}(L)_{{\varphi}_y}$ is not isomorphic to ${\mathrm{id}}\oplus \operatorname{Ad}_s$.
The proof of Corollary \[Dat\_corollary\] showed that for each nonzero $M\in \operatorname{\mathcal{R}}(L)_{{\varphi}_y}$ there is a proper inclusion $\operatorname{i}_{U,V}(M)\subsetneq \operatorname{i}_{U_0,V}(M)$, and hence a proper inclusion $$\operatorname{Hom}_L(M,M\oplus \operatorname{Ad}_s(M)) \cong \operatorname{End}_G(\operatorname{i}_{U,V}(M)) \subsetneq \operatorname{End}_G(\operatorname{i}_{U_0,V}(M)) \cong \operatorname{Hom}_L(M, \operatorname{r}_{U_0,V}\operatorname{i}_{U_0,V}(M)).$$ Thus $\operatorname{r}_{U_0,V}\operatorname{i}_{U_0,V}(M)$ is not isomorphic to $M\oplus \operatorname{Ad}_s(M)$.
(1) A straightforward computation with the functors $\operatorname{i}_{{\overline}{V}(y)}$ and $\operatorname{r}_{{\overline}{V}(y)}$, using the semidirect product decomposition of ${\overline}{G}(y)$, shows that for each irreducible $M\in \operatorname{\mathcal{R}}(L)_{{\varphi}_y}$ one has $$\dim_{{\mathbb{C}}} \operatorname{End}_G(\operatorname{i}_{U_0,V}(M)) = \begin{cases} |{\Bbbk}|+1 & \text{if }M\cong \operatorname{Ad}_s(M),\\
|{\Bbbk}| & \text{if }M\not\cong \operatorname{Ad}_s(M).\end{cases}$$
(2) The nilpotent orbit $L\cdot y$ is the only one on which the Mackey formula fails to hold for the functors $\operatorname{i}_{U_0,V}$ and $\operatorname{r}_{U_0,V}$: on all of the other orbits our proof of Theorem \[thm:sp\_Mackey\] carries over to the parahoric functors, thanks to Corollary \[Dat\_corollary\].
Representations of the Iwahori subgroup of the general linear group {#sec:Iwahori}
===================================================================
Let ${\mathfrak o}$ be a [compact]{} discrete valuation ring with maximal ideal ${\mathfrak}p$. In this section we shall present a simple application of the functors $\operatorname{i}_{U,V}$ and $\operatorname{r}_{U,V}$ to the representation theory of the *Iwahori subgroups* $$I_n = I_n({\mathfrak o}) = \{ g\in \operatorname{GL}_n({\mathfrak o})\ |\ g\textrm{ is upper-triangular modulo }{\mathfrak}p\}$$ We shall relate the representations of $I_n$ to representations of its block-diagonal subgroups. Before stating the main result let us establish some notation (borrowed from [@Bernstein-Zelevinsky]) for these subgroups.
Let $\mathcal P_n$ denote the set of compositions (also called ordered partitions) of $n$: an element $\alpha\in\mathcal P_n$ is thus an ordered tuple of positive integers $(\alpha_1,\alpha_2,\ldots,\alpha_m)$ having $\sum \alpha_i = n$. The *blocks* of $\alpha$ are the subsets $$b_1(\alpha) = \{1,\ldots,\alpha_1\},\quad b_2(\alpha)=\{\alpha_1+1,\ldots,\alpha_1+\alpha_2\},\quad \textrm{etc.}$$ of $\{1,\ldots,n\}$. We shall usually write $n$, instead of $(n)$, for the composition with one block.
The set $\mathcal P_n$ is partially ordered by refinement: $\alpha\leq \beta$ if each block of $\beta$ is a union of blocks of $\alpha$. This partial order makes $\mathcal P_n$ into a lattice, the greatest lower bound $\alpha\wedge\beta$ of two compositions being the composition whose blocks are the nonempty intersections $b_i(\alpha)\cap b_j(\beta)$ of the blocks of $\alpha$ and $\beta$. We also have an associative order-preserving product $$\mathcal P_n \times \mathcal P_m \to \mathcal P_{n+m}, \qquad (\alpha,\beta)\mapsto \alpha\cdot \beta$$ given by concatenation.
Given a composition $\alpha\in \mathcal P_n$ we denote by $$I_\alpha = \{g\in I_n\ |\ g_{ij}=0 \textrm{ unless $i$ and $j$ lie in the same block of $\alpha$}\}$$ the closed subgroup of $\alpha$-block-diagonal matrices in $I_n$. These groups are compatible with the concatenation product: $$\label{eq:Iw_concat}
I_{\alpha\cdot \beta} \cong I_{\alpha}\times I_{\beta}$$ in an obvious way, and this gives an equivalence on smooth representations, $$\operatorname{\mathcal{R}}(I_\alpha)\times \operatorname{\mathcal{R}}(I_\beta) \xrightarrow[\cong]{(M_\alpha,M_\beta)\mapsto M_\alpha\otimes M_\beta} \operatorname{\mathcal{R}}(I_{\alpha\cdot\beta}).$$
We also consider the groups $$\begin{split}
U_\alpha &= \left\{g\in I_n\ \left|\ \ \begin{aligned}&g\textrm{ is upper-triangular};\ g_{ii}=1 \textrm{ for every $i$}; \textrm{ and}\\ &g_{ij}=0\textrm{ if $i\neq j$ and $i$ and $j$ lie in the same block of $\alpha$} \end{aligned}\ \right. \right\}, ~\text{and}\ \\ V_\alpha &= U_\alpha^{t}\cap I_n.
\end{split}$$
If $\beta$ is a second composition with $\alpha\leq \beta$, we define $$U_\alpha^\beta = U_\alpha \cap I_\beta \quad \text{and}\quad V_\alpha^\beta= V_\alpha \cap I_\beta.$$ If $\alpha\leq \beta\in \mathcal P_n$ and $\gamma\leq \delta\in \mathcal P_m$, then the isomorphism $I_{\beta\cdot\delta}\cong I_\beta\times I_\delta$ of restricts to isomorphisms $$\label{eq:Iw_concat2}
U_{\alpha\cdot\gamma}^{\beta\cdot\delta} \cong U_\alpha^\beta\times U_\gamma^\delta \qquad \text{and}\qquad V_{\alpha\cdot\gamma}^{\beta\cdot\delta} \cong V_\alpha^\beta\times V_\gamma^\delta.$$
If $\alpha=(2,1)$ and $\beta=(3)$, then $$I_\alpha = \left[\begin{smallmatrix}{\mathfrak o}^\times & {\mathfrak o}\\ {\mathfrak}p & {\mathfrak o}^\times & \\ & & {\mathfrak o}^\times \end{smallmatrix}\right], \quad
U_\alpha^\beta = \left[\begin{smallmatrix} 1 & & {\mathfrak o}\\ & 1 & {\mathfrak o}\\ & & 1 \end{smallmatrix}\right],\quad
V_\alpha^\beta = \left[\begin{smallmatrix} 1 & & \\ & 1 & \\ {\mathfrak}p & {\mathfrak}p & 1 \end{smallmatrix}\right]$$ where the blanks indicate zeros.
\[lem:Iw-linalg\]
1. For each pair of compositions $\alpha \leq \beta$ in $\mathcal P_n$, the triple $(U_\alpha^\beta, I_\alpha, V_\alpha^\beta)$ is an Iwahori decomposition of $I_\beta$.
2. For each triple of compositions $\alpha \leq \beta \leq \gamma$ one has $$U_{\alpha}^\gamma = U_{\alpha}^\beta \ltimes U_{\beta}^\gamma \qquad \text{and} \qquad V_{\alpha}^\gamma = V_{\alpha}^\beta \ltimes V_{\beta}^\gamma.$$
3. For each pair of compositions $\alpha,\beta\in \mathcal P_n$ one has $$U_{\alpha\wedge\beta}^\alpha = U_{\beta} \cap I_\alpha \qquad \text{and}\qquad V_{\alpha\wedge\beta}^\alpha = V_\beta \cap I_\alpha.$$
Part (1) is well-known, and can be established by elementary linear algebra as in [@Bernstein-Zelevinsky 3.11]. Part (2) follows immediately from the Iwahori decompositions. Part (3) boils down to the (manifestly true) assertion that for integers $i$ and $j$ lying in the same block of $\alpha$, $i$ and $j$ lie in the same block of $\alpha\wedge\beta$ if and only if they lie in the same block of $\beta$.
For each pair of compositions $\alpha \leq \beta $ in $\mathcal P_n$, consider the functors $$\operatorname{i}_{\alpha}^\beta = \operatorname{i}_{U_{\alpha}^\beta, V_{\alpha}^\beta} : \operatorname{\mathcal{R}}(I_\alpha) \to \operatorname{\mathcal{R}}(I_\beta) \qquad \text{and}\qquad
\operatorname{r}^\beta_\alpha = \operatorname{r}_{U_{\alpha}^\beta, V_{\alpha}^\beta} : \operatorname{\mathcal{R}}(I_\beta) \to \operatorname{\mathcal{R}}(I_\alpha).$$
[The functors $\operatorname{i}_{\alpha}^\beta$ and $\operatorname{r}^\beta_\alpha$ are examples of parahoric induction as defined in [@Dat_parahoric].]{} Theorems \[thm:pind-properties\] and \[thm:pind-Iw\] give some basic properties of these functors. Let us mention two that will be used below:
\[lem:pind-Iw-props\]
1. If $\alpha\leq \beta\leq \gamma$ are compositions of $n$, then $$\operatorname{i}_{\alpha}^\gamma \cong \operatorname{i}_{\beta}^\gamma \operatorname{i}_{\alpha}^\beta \qquad \text{and}\qquad \operatorname{r}^\gamma_\alpha \cong \operatorname{r}^\beta_\alpha \operatorname{r}^\gamma_\beta.$$
2. If $\alpha\leq \beta\in \mathcal P_n$ and $\gamma\leq \delta \in \mathcal P_m$, then the diagram $$\xymatrix@C=50pt{
\operatorname{\mathcal{R}}(I_\alpha)\times \operatorname{\mathcal{R}}(I_{\gamma}) \ar[r]^-{\operatorname{i}_\alpha^\beta \times \operatorname{i}_\gamma^\delta} \ar[d]_-{\otimes} & \operatorname{\mathcal{R}}(I_\beta)\times \operatorname{\mathcal{R}}(I_\delta) \ar[d]^-{\otimes} \\
\operatorname{\mathcal{R}}(I_{\alpha\cdot\gamma}) \ar[r]^-{\operatorname{i}_{\alpha\cdot\gamma}^{\beta\cdot\delta}} & \operatorname{\mathcal{R}}(I_{\beta\cdot\delta})
}$$ commutes up to natural isomorphism, as does the corresponding diagram of adjoint functors $\operatorname{r}$.
Part (1) follows from part (2) of Lemma \[lem:Iw-linalg\] and part of Theorem \[thm:pind-properties\]. Part (2) follows from the compatibility of the decompositions and .
An irreducible representation $M$ of $I_n$ will be called *primitive* if $\operatorname{r}^n_\alpha(M) =0$ for every composition $\alpha\in \mathcal P_n$ except for $\alpha=n$. We denote the set of isomorphism classes of primitive irreducible representations by $\operatorname{Prim}(I_n)$.
The following lemma is key to our analysis of the functors $\operatorname{i}$ and $\operatorname{r}$.
\[lem:Iw\_pres\] Let $\alpha,\beta\in \mathcal P_n$ be compositions of $n$, and let $M$ be an irreducible representation of $I_n$. If $\operatorname{r}^n_\alpha (M)$ and $\operatorname{r}^n_\beta (M)$ are both nonzero, then so is $\operatorname{r}^n_{\alpha\wedge\beta}(M) $.
Since the representation $M$ is irreducible and smooth, it factors through the quotient map $I_n({\mathfrak o}) \to I_n({\mathfrak o}/{\mathfrak}p^{\ell})$ for some $\ell$. The functors $\operatorname{i}$ and $\operatorname{r}$ commute with inflation (Theorem \[thm:pind-properties\]), and so we may replace ${\mathfrak o}$ by ${\mathfrak o}/{\mathfrak}p^{\ell}$ and assume throughout the proof that $I_n$ is a finite group.
We know that $N\coloneq \operatorname{r}^n_{\alpha} (M)$ is nonzero. Therefore, up to isomorphism, we can write $$M = \operatorname{i}^n_{\alpha} (N) = {\mathcal H}(I_n) e_{U_{\alpha}}e_{V_{\alpha}}{\otimes}_{{\mathcal H}(I_{\alpha})} N= {\mathcal H}(I_n)e_{U_{\alpha}} e_{V_{\alpha}} e_{U_{\alpha}} e_{V_{\alpha}}{\otimes}_{{\mathcal H}(I_{\alpha})} N.$$ (In the last equality we used Proposition \[prop:z\].)
We know that the subspace $$\label{eqn:res.n.beta.M}
\operatorname{r}^n_{\beta}(M) = e_{U_{\beta}}e_{V_{\beta}}{\mathcal H}(I_n) e_{U_{\alpha}}e_{V_{\alpha}}e_{U_{\alpha}}e_{V_{\alpha}}{\otimes}_{{\mathcal H}(I_{\alpha})} N$$ of $M$ is nonzero. By part (2) of Lemma \[lem:Iw-linalg\] we know that each element of $V_{\alpha}\subseteq V_{\alpha\wedge\beta}$ can be written as the product of an element of $V_{\beta}$ with an element of $V^{\beta}_{\alpha\wedge\beta}= V_{\alpha}\cap I_{\beta}$. Therefore, using the Iwahori decomposition of $I_n$ with respect to $\alpha$, we get that $I_n=V_\alpha I_\alpha U_\alpha=V_\beta V^{\beta}_{\alpha\wedge\beta} I_\alpha U_\alpha$, which allows us to replace ${\mathcal H}(I_n)$ by ${\mathcal H}(V^{\beta}_{\alpha\wedge\beta})$ in and write $$\begin{split}
\operatorname{r}^n_{\beta}(M)&=e_{U_{\beta}}e_{V_{\beta}}{\mathcal H}(V^{\beta}_{\alpha\wedge\beta}) e_{U_{\alpha}}e_{V_{\alpha}} e_{U_{\alpha}} e_{V_{\alpha}}{\otimes}_{{\mathcal H}(I_{\alpha})} N \\
&= {\mathcal H}(V^{\beta}_{\alpha\wedge\beta}) e_{U_{\beta}} e_{V_{\beta}} e_{U_{\alpha}} e_{V_{\alpha}} e_{U_{\alpha}} e_{V_{\alpha}} {\otimes}_{{\mathcal H}(I_{\alpha})} N,
\end{split}$$ where the second equality holds because the elements of ${\mathcal H}(V^{\beta}_{\alpha\wedge\beta})\subset {\mathcal H}(I_{\beta})$ commute with $e_{U_{\beta}}$ and $e_{V_{\beta}}$. So we see that $\operatorname{r}^n_\beta(M)$ is generated as a representation of $I_\beta$ by its subspace $$\label{eq:Iw_pres_pf_subspace}
e_{U_{\beta}} e_{V_{\beta}} e_{U_{\alpha}} e_{V_{\alpha}} e_{U_{\alpha}} e_{V_{\alpha}} {\otimes}_{{\mathcal H}(I_{\alpha})} N$$ and hence that this subspace is nonzero.
We now write $e_{V_{\beta}} = e_{V_{\beta}}e_{V_{\alpha\wedge\beta}^{\alpha}}$. We use the fact that elements of ${\mathcal H}(V_{\alpha\wedge\beta}^{\alpha})\subset {\mathcal H}(I_{\alpha})$ commute with $e_{U_{\alpha}}$ and that $e_{V_{\alpha\wedge\beta}^\alpha}e_{V_{\alpha}} = e_{V_{\alpha\wedge\beta}}$ (by part (2) of Lemma \[lem:Iw-linalg\]), to obtain $$e_{V_\beta} e_{U_\alpha} e_{V_\alpha} = e_{V_\beta}e_{V_{\alpha\wedge\beta}^{\alpha}}e_{U_\alpha} e_{V_\alpha} =
e_{V_\beta} e_{U_\alpha} e_{V_{\alpha\wedge\beta}^{\alpha}} e_{V_\alpha} =
e_{V_\beta} e_{U_\alpha} e_{V_{\alpha\wedge\beta}}.$$ A similar argument shows that $e_{U_\beta} e_{V_\beta} e_{U_\alpha} = e_{U_{\alpha\wedge\beta}} e_{V_\beta} e_{U_\alpha}$, and so the subspace is equal to $$e_{U_{\alpha\wedge\beta}} e_{V_\beta} e_{U_\alpha} e_{V_{\alpha\wedge\beta}} e_{U_{\alpha}} e_{V_{\alpha}} {\otimes}_{{\mathcal H}(I_{\alpha})} N.$$ This non-zero subspace of $M$ is contained in the subspace $$\begin{aligned}
e_{U_{\alpha\wedge\beta}} {\mathcal H}(I_n) e_{V_{\alpha\wedge\beta}} e_{U_{\alpha}} e_{V_{\alpha}} {\otimes}_{{\mathcal H}(I_{\alpha})} N & =
e_{U_{\alpha\wedge\beta}} {\mathcal H}(I_{\alpha\wedge\beta}) e_{V_{\alpha\wedge\beta}} e_{U_{\alpha}} e_{V_{\alpha}} {\otimes}_{{\mathcal H}(I_{\alpha})} N \\
& = e_{U_{\alpha\wedge\beta}} e_{V_{\alpha\wedge\beta}} e_{U_{\alpha}} e_{V_{\alpha}} {\otimes}_{{\mathcal H}(I_{\alpha})} N,
\end{aligned}$$ where we have used the Iwahori decomposition of $I_n$ with respect to $\alpha\wedge\beta$, and the inclusion $I_{\alpha\wedge\beta}\subseteq I_\alpha$. But this last nonzero subspace of $M$ is exactly $\operatorname{r}^n_{\alpha\wedge\beta}(M)$, so we are done.
Let us now present the main results of this section:
\[thm:Iw\_primitive\] Let $ M $ be an irreducible representation of the Iwahori subgroup $I_n\subset \operatorname{GL}_n({\mathfrak o})$. There is a unique [composition]{} $\alpha=(\alpha_1,\ldots,\alpha_m)$ of $n$, and unique primitive irreducible representations $ M _i\in \operatorname{Prim}(I_{\alpha_i})$, such that $$M \cong \operatorname{i}_\alpha^n ( M _1\otimes\cdots\otimes M _m).$$
First note the following consequence of part (2) of Lemma \[lem:pind-Iw-props\]: if $ M _1,\ldots, M _m$ are irreducible representations of $I_{\alpha_1},\ldots,I_{\alpha_m}$, then $$\label{eq:Iw_prim}
M _i\textrm{ is primitive for all } i \quad \Longleftrightarrow \quad \operatorname{r}^\alpha_\gamma( M _1\otimes\cdots\otimes M _m)=0\textrm{ for all } \gamma \lneq \alpha.$$
Consider the set $$\mathcal Q = \{\alpha\in \mathcal P_n\ |\ \operatorname{r}^n_\alpha (M) \neq 0\},$$ which is nonempty since it contains the [composition]{} $n$. Let $\alpha=(\alpha_1,\ldots,\alpha_m)$ be the greatest lower bound of $\mathcal Q$ in the lattice $\mathcal P_n$; Lemma \[lem:Iw\_pres\] implies that $\alpha\in \mathcal Q$. The (nonzero) irreducible representation $\operatorname{r}^n_\alpha (M)$ of the group $I_\alpha$ decomposes uniquely as a tensor product $$\operatorname{r}^n_\alpha(M) \cong \bigotimes_{i=1}^m M _i$$ of irreducible representations of the factors $I_{\alpha_i}$ of $I_\alpha$ (cf. ). If $\gamma\lneq\alpha$ then $$\operatorname{r}^\alpha_\gamma\left( \bigotimes M _i\right) \cong \operatorname{r}^n_\gamma (M) =0$$ by Lemma \[lem:pind-Iw-props\] part (1) and the minimality of $\alpha$, and so all of the $ M _i$’s are primitive by . Since by part of Theorem \[thm:pind-Iw\] we have ${M \cong \operatorname{i}_\alpha^n\operatorname{r}^n_\alpha (M)}$, we are done with the existence part of the proof.
The uniqueness follows from : if $\operatorname{r}^n_\beta (M) \cong N_1\otimes\cdots\otimes N_\ell$, where the $N_i$ are all primitive, then we must have $\beta=\alpha$ by minimality, and then $N_i\cong M_i$ for each $i$ by the uniqueness of the tensor product decomposition.
Lemma \[lem:Iw\_pres\] also implies the following simple formula for the composition of induction and restriction:
\[prop:Iw\_ri\] For all $\alpha,\beta \in \mathcal P_n$ and all $ M \in \operatorname{Irr}(I_\alpha)$ one has $$\operatorname{r}^n_\beta \operatorname{i}_{\alpha}^n (M) \cong \operatorname{i}_{\alpha\wedge\beta}^\beta \operatorname{r}^\alpha_{\alpha\wedge\beta} (M) .$$
If $\operatorname{r}^n_\beta(\operatorname{i}_\alpha^n (M) )$ is nonzero, then—since $\operatorname{r}^n_\alpha(\operatorname{i}_\alpha^n M )\cong M $ is also nonzero—Lemma \[lem:Iw\_pres\] implies that $$\operatorname{r}^\alpha_{\alpha\wedge\beta} (M) \cong \operatorname{r}^n_{\alpha\wedge\beta}(\operatorname{i}_\alpha^n (M) ) \neq 0.$$ In other words, if $\operatorname{r}^\alpha_{\alpha\wedge\beta} (M) =0$, then $\operatorname{r}^n_\beta\operatorname{i}_\alpha^n (M) =0$ too.
If $\operatorname{r}^\alpha_{\alpha\wedge\beta} (M) \neq 0$, then we can use Theorem \[thm:pind-Iw\] and Lemma \[lem:pind-Iw-props\](1) to compute $$\operatorname{r}^n_\beta \operatorname{i}_\alpha^n (M) \cong
\operatorname{r}^n_\beta \operatorname{i}_\alpha^n (\operatorname{i}_{\alpha\wedge\beta}^\alpha\operatorname{r}^\alpha_{\alpha\wedge\beta} (M) ) \cong
\operatorname{r}^n_\beta \operatorname{i}_{\alpha\wedge\beta}^n \operatorname{r}^\alpha_{\alpha\wedge\beta} (M) \cong
\operatorname{r}^n_\beta \operatorname{i}_\beta^n (\operatorname{i}_{\alpha\wedge\beta}^\beta \operatorname{r}^\alpha_{\alpha\wedge\beta} (M) ) \cong
\operatorname{i}_{\alpha\wedge\beta}^\beta \operatorname{r}^\alpha_{\alpha\wedge\beta} (M)$$ as claimed.
Theorem \[thm:Iw\_primitive\] has the following corollary, which gives a neat description of the way the representations of all the groups $I_n$ (for $n\geq 0$) fit in together. Namely, let $\mathcal{K}\coloneq \bigoplus_{n\geq 0} K_0\left({\operatorname{\mathcal{R}}_f}(I_n)\right)$ denote the direct sum of the Grothendieck groups of the categories of [finite-dimensional smooth representations of the groups $I_n$]{}, with the convention that $I_0$ is the trivial group . The maps induced [on Grothendieck groups]{} by the functors $$\operatorname{\mathcal{R}}(I_n)\times \operatorname{\mathcal{R}}(I_m) \to \operatorname{\mathcal{R}}(I_{n+m}),\qquad (M_1,M_2)\mapsto \operatorname{i}_{(n,m)}^{n+m}(M_1\otimes M_2)$$ equip $\mathcal{K}$ with a graded multiplication structure. It follows from Lemma \[lem:pind-Iw-props\] that this multiplication is associative. Since the irreducible representations of $I_n$ constitute a ${\mathbb{Z}}$-basis for $K_0({\operatorname{\mathcal{R}}_f}(I_n))$, Theorem \[thm:Iw\_primitive\] implies the following result:
The ring $\mathcal{K}$ is isomorphic to $
{\mathbb{Z}}\left\langle \bigsqcup_{n\geq 0} \operatorname{Prim}(I_n) \right\rangle
$, the non-commutative polynomial algebra with indeterminates the primitive irreducible representations.
Centralisers for the adjoint action of $\operatorname{Sp}_4({\Bbbk})$ {#appendix}
=====================================================================
In this section we give proofs of the assertions in Section \[subsec:Sp4\_centralisers\] regarding the centralisers $\overline{G}(y)$ and $\overline{L}(y)$ and the spaces $L\backslash G(y,{\mathfrak}l)/G(y)$. Part of the computations here can be deduced from [@Srinivasan], where the cardinalities of the centralisers of elements of $\operatorname{Sp}_4({\Bbbk})$ are computed, by using the Cayley map. As we require the precise structure of the centralisers we give a detailed computation below.
Fix $x\in M_2({\Bbbk})$, and let $y=\operatorname{diag}(x,-x^{t})$ be the corresponding element of ${\mathfrak}l$. Clearly we have $${\overline}{L}(y) = \{ \operatorname{diag}(a,a^{-{t}})\ |\ a\in M_2({\Bbbk})(x)\},$$ where $M_2({\Bbbk})(x)$ denotes the centraliser of $x$ in the algebra $M_2({\Bbbk})$. Elements of $M_2({\Bbbk})$ are either scalar or regular (in the sense of admitting a cyclic vector in ${\Bbbk}^2$). We therefore have $$\label{eqn:dichotomy}
M_2({\Bbbk})(x) =
\begin{cases}
M_2({\Bbbk})& \text{if $x$ is a scalar matrix}, \\
{\Bbbk}[x] & \text{if $x$ is non-scalar}.
\end{cases}$$
Turning to the centralisers in $\overline{G}=\operatorname{Sp}_4({\Bbbk})$, let us first note that the matrices $x$ and $-x^{t}$ give rise to two ${\Bbbk}[T]$-module structures on ${\Bbbk}^2$, and that the centraliser of $y$ in $\operatorname{GL}_4({\Bbbk})$ is isomorphic, in an obvious way, to the automorphism group of the direct sum ${\Bbbk}^2_x\oplus {\Bbbk}^2_{-x^{t}}$ of these modules.
\[lem:Sp\_trace0\] For each $x\in M_2({\Bbbk})$ with $\operatorname{tr}(x)=0$, the centraliser $\operatorname{GL}_4({\Bbbk})(y)$ of $y=\operatorname{diag}(x,-x^{t})\in M_4({\Bbbk})$ inside $\operatorname{GL}_4({\Bbbk})$ is given by $$\operatorname{GL}_4({\Bbbk})(y) =
\Sigma\cdot \operatorname{GL}_2\left( M_2({\Bbbk})(x) \right)\cdot \Sigma^{-1}$$ where $\sigma = \left[\begin{smallmatrix} & -1 \\ 1 & \end{smallmatrix}\right]\in \operatorname{GL}_2({\Bbbk})$ and $\Sigma=\left[\begin{smallmatrix} 1& \\ & \sigma \end{smallmatrix}\right]\in \operatorname{GL}_4({\Bbbk})$.
If $\operatorname{tr}(x)=0$ then $\sigma x \sigma^{-1} = -x^{t}$, and so ${\mathrm{id}}\oplus\sigma:{\Bbbk}^2_x\oplus {\Bbbk}^2_x\to {\Bbbk}^2_x\oplus {\Bbbk}^2_{-x^{t}}$ is a ${\Bbbk}[T]$-module isomorphism. Conjugating $\operatorname{GL}_2(M_2({\Bbbk})(x))=\operatorname{Aut}({\Bbbk}^2_x\oplus {\Bbbk}^2_x)$ by this isomorphism gives the asserted description of $\operatorname{GL}_4({\Bbbk})(y)$.
We now proceed to the computation of ${\overline}{L}(y)$, ${\overline}{G}(y)$ and $L\backslash G(y,{\mathfrak}l)/G(y)$ in each of the cases listed in Section \[subsec:Sp4\_centralisers\]. Note that $\operatorname{tr}(x)\neq 0$ in the A cases, while $\operatorname{tr}(x)=0$ in the B cases.
[**Case 1:**]{} $x=\operatorname{diag}(\mu,\mu)$.
We have $M_2({\Bbbk})(x)=M_2({\Bbbk})$, so ${\overline}{L}(y)={\overline}{L}$. For ${\overline}{G}(y)$ and $L\backslash G(y,{\mathfrak}l)/G(y)$ there are two subcases to consider:
[**1A:**]{} $\mu\neq 0$. Since $x$ and $-x^{t}$ share no eigenvalue, there are no nonzero morphisms between the $k[T]$ modules ${\Bbbk}^2_x$ and ${\Bbbk}^2_{-x^{t}}$, and consequently we have $\overline{G}(y)=\overline{L}(y)=\overline{L}$.
We claim that $L\backslash G(y,{\mathfrak}l)/G(y)=\{1,s,w\}$. This is equivalent to the claim that there are, up to conjugacy by $L$, three $G$-conjugates of $y$ lying in ${\mathfrak}l$: namely $y$ itself, $s\cdot y$, and $w\cdot y$. Indeed, any $G$-conjugate of $y$ in ${\mathfrak}{l}$ must be split and semisimple, and must therefore be $L$-conjugate to a diagonal matrix $z$ whose entries form a permutation of the entries of $y$. Since $z$ lies in ${\mathfrak}l$, and hence is of the form $\operatorname{diag}(z_1,z_2,-z_1,-z_2)$, the only possibilities for $z$ are $$\operatorname{diag}(\mu,\mu,-\mu,-\mu),\ \operatorname{diag}(-\mu,-\mu,\mu,\mu),\ \operatorname{diag}(-\mu,\mu,\mu,-\mu),\text{ or } \operatorname{diag}(\mu,-\mu,-\mu,\mu).$$ The first three are equal to $y$, $s\cdot y$ and $w\cdot y$ respectively, while the last is $L$-conjugate to $w\cdot y$.
[**1B:**]{} [$\mu=0$.]{} Obviously $\overline{H}(y)=\overline{H}$ for all $H\subseteq G$, and $G(y,{\mathfrak}l)=G(y)$.
[**Case 2:**]{} $x=\operatorname{diag}(\mu,\nu)$, $\mu \neq \nu$.
We have $M_2({\Bbbk})(x) = \left\{\left. \left[\begin{smallmatrix} \alpha & \\ & \beta \end{smallmatrix}\right]\ \right|\ \alpha,\beta\in {\Bbbk}\right\}\cong {\Bbbk}\oplus {\Bbbk}$, and so ${\overline}{L}(y)={\overline}{D}$ is the group of diagonal matrices in ${\overline}{G}$. There are three subcases to consider:
[**2A:**]{} $\nu\neq \pm\mu$, $\mu\neq 0\neq \nu$. Similar arguments to those of Case 1A show that $\overline{G}(y)=\overline{L}(y)$, and that $L\backslash G(y,{\mathfrak}l)/G(y) = \{1,s,w,wt\}$.
[**2A$\!^\star$:**]{} $\nu= 0$. The space $\operatorname{Hom}_{{\Bbbk}[T]}({\Bbbk}^2_x,{\Bbbk}^2_{-x^{t}})$ is one-dimensional, spanned by $p = \left[\begin{smallmatrix} 0 & \\ & 1\end{smallmatrix}\right]$, and so we have $$\operatorname{GL}_4({\Bbbk})(y) = \left\{\left. \begin{bmatrix} a & b \\ c& d\end{bmatrix}\ \right|\ a,d\in \overline{D},\ b,c\in {\Bbbk}p\right\}.$$ Applying the condition $j^{-1}g^t j=g^{-1}$ defining $\operatorname{Sp}_4({\Bbbk})$ to a matrix of the above form, we find that $${\overline}{G}(y) = \left\{\left. \begin{bmatrix} \alpha_1 & & & \\ & \alpha_2 & &\beta \\ & & \delta_1 & \\ & \gamma & & \delta_2 \end{bmatrix}\in \operatorname{GL}_4({\Bbbk})\ \right|\ \alpha_1\delta_1=1=\alpha_2\delta_2-\beta\gamma \right\}\cong GL_1({\Bbbk}) \times \operatorname{SL}_2({\Bbbk}).$$ The Weyl group of $\operatorname{SL}_2({\Bbbk})$ with respect to its diagonal torus is generated by the matrix $\sigma$, and so the Weyl group of ${\overline}{G}(y)$ with respect to ${\overline}{D}$ is generated by the matrix $$\begin{bmatrix} 1 & & & \\
& 0 & & -1 \\
& & 1 & \\ & 1 & & 0 \end{bmatrix} = t^{-1} w t.$$ Up to $L$-conjugacy, the $G$-conjugates of $y$ lying in ${\mathfrak}l$ are $y$ and $-y=w\cdot y$, and so $L\backslash G(y,{\mathfrak}l)/G(y)=\{1,w\}$.
[**2B:**]{} $\nu=-\mu$. Let $z=\operatorname{diag}(-\mu,-\mu, \mu, \mu)$, so that $y=w\cdot z$. Then ${\overline}{G}(y)=\operatorname{Ad}_w({\overline}{G}(z))$, and ${\overline}{G}(z)={\overline}{L}$ as in Case 1A. Since $\operatorname{Ad}_{w}^{-1}({\overline}{U})\cap {\overline}{L}={\overline}{V'}$, and $\operatorname{Ad}_w^{-1}({\overline}{V}) \cap {\overline}{L} = {\overline}{U'}$, we have ${\overline}{U}(y) = \operatorname{Ad}_w\left( {\overline}{V'} \right)$ and ${\overline}{V}(y) = \operatorname{Ad}_w\left( {\overline}{U'} \right)$. The argument of Case 1A gives $L\backslash G(y,{\mathfrak}l)/G(y)=\{1,w,wt\}$.
[**Case 3:**]{} $x=\big[\begin{smallmatrix} \alpha & \beta \\ \mu\beta & \alpha\end{smallmatrix}\big], \mu \in {\Bbbk}$ non-square, $\alpha\in {\Bbbk}$, $\beta\in {\Bbbk}^\times$.
In this case $M_2({\Bbbk})(x)=\left\{\big[\begin{smallmatrix} \alpha_1 & \beta_1 \\ \mu \beta_1 & \alpha_1 \end{smallmatrix}\big] \mid \alpha_1,\beta_1 \in {\Bbbk}\right\}$ is a quadratic field extension of ${\Bbbk}$, which we shall denote by ${\Bbbk}_2$. There are two subcases to consider.
[**3A:**]{} $\alpha \neq 0$. Similar arguments to those of case 1A (considering the eigenvalues in ${\Bbbk}_2$) show that $\overline{G}(y)=\overline{L}(y)$, while $L\backslash G(y,{\mathfrak}l)/G(y)=\{1,s\}$.
[**3B:**]{} $\alpha=0$. Since $\operatorname{tr}(x)= 0$, Lemma \[lem:Sp\_trace0\] implies that $\operatorname{Ad}_{\Sigma^{-1}}: \operatorname{GL}_4({\Bbbk})(y) \to \operatorname{GL}_2({\Bbbk}_2)$ is an isomorphism. Observing that $\operatorname{Ad}_{\Sigma^{-1}}(j)= \left[\begin{smallmatrix} & \sigma^{-1} \\ \sigma^{-1} & \end{smallmatrix}\right]$, and that $\Sigma^{t}= \Sigma^{-1}$, we find that the isomorphism $\operatorname{Ad}_{\Sigma^{-1}}$ sends $\overline{G}(y)$ to $$\operatorname{Ad}_{\Sigma^{-1}}(\overline{G}(y)) = \{ g\in \operatorname{GL}_2({\Bbbk}_2)\ |\ g^* g = 1\},$$ where $$\begin{bmatrix} a & b \\ c & d\end{bmatrix}^* = \begin{bmatrix} \sigma d^{t}\sigma^{-1} & \sigma b^{t}\sigma^{-1} \\ \sigma c^{t}\sigma^{-1} & \sigma a^{t}\sigma^{-1} \end{bmatrix}.$$ The map $a\mapsto \sigma a^{t}\sigma^{-1}$ is a nontrivial ${\Bbbk}$-algebra automorphism of ${\Bbbk}_2$, and so is equal to the nontrivial element $a\mapsto a^{|{\Bbbk}|}$ in $\operatorname{Gal}({\Bbbk}_2/{\Bbbk})$.
Let $\overline{{\Bbbk}}$ denote an algebraic closure of ${\Bbbk}_2$. The above computations show that $\operatorname{Ad}_{\Sigma^{-1}}$ restricts to an isomorphism from $\overline{G}(y)$ to the (unitary) group $\operatorname{GU}_2({\Bbbk})$ of fixed points of the automorphism $$\operatorname{GL}_2(\overline{{\Bbbk}})\to \operatorname{GL}_2(\overline{{\Bbbk}}),\qquad \begin{bmatrix} a & b \\ c & d \end{bmatrix} \mapsto \begin{bmatrix} d^{|{\Bbbk}|} & b^{|{\Bbbk}|} \\ c^{|{\Bbbk}|} & a^{|{\Bbbk}|} \end{bmatrix}^{-1}.$$ The subgroup $\overline{L}(y)$ corresponds under this isomorphism to the [non-split rational]{} maximal torus of diagonal matrices [$\{\operatorname{diag}(a,a^{-|{\Bbbk}|}) \mid a \in {\Bbbk}_2^\times \}$]{} in $\operatorname{GU}_2({\Bbbk})$, while $\overline{U}(y)$ and $\overline{V}(y)$ correspond to the unipotent radicals of [rational]{} Borel subgroups of upper / lower triangular matrices. The Weyl group of $\operatorname{GU}_2({\Bbbk})$ with respect to its diagonal torus is generated by the matrix $\left[\begin{smallmatrix} & -1 \\ -1 & \end{smallmatrix}\right]=\operatorname{Ad}_{\Sigma^{-1}}(s)$.
The argument of case 1A shows that all of the $G$-conjugates of $y$ lying in $L$ are already $L$-conjugate, and so we have $L\backslash G(y,{\mathfrak}l)/G(y)=\{1\}$.
[**Case 4:**]{} ${x=\left[\begin{smallmatrix} \mu & 1 \\ & \mu \end{smallmatrix}\right]}$.
We have $M_2({\Bbbk})(x)= \left\{\left[\begin{smallmatrix} \alpha & \beta \\ & \alpha \end{smallmatrix}\right] \mid \alpha,\beta \in {\Bbbk}\right\} \cong {\Bbbk}[{\varepsilon}]/({\varepsilon}^2)$. There are two subcases to consider.
[**4A:**]{} $\mu\neq 0$. Arguing as in case 1A once again, we find that $\overline{G}(y)=\overline{L}(y)$, while $L\backslash G(y,{\mathfrak}l)/G(y)= \{1,s\}$.
[**4B:**]{} $\mu=0$. Arguing as in case 3B, we find that the isomorphism $$\operatorname{Ad}_{\Sigma^{-1}}: \operatorname{GL}_4({\Bbbk})(y) \to \operatorname{GL}_2({\Bbbk}[x])$$ of Lemma \[lem:Sp\_trace0\] restricts to an isomorphism between $\overline{G}(y)$ and the group $Q\subset \operatorname{GL}_2({\Bbbk}[x])$ of fixed points of the involution $$\operatorname{GL}_2({\Bbbk}[x])\to \operatorname{GL}_2({\Bbbk}[x]), \qquad \begin{bmatrix} a & b \\ c & d \end{bmatrix} \mapsto \begin{bmatrix} d^\# & b^\# \\ c^\# & a^\#\end{bmatrix}^{-1}$$ where $\#$ denotes the ${\Bbbk}$-automorphism $x\mapsto -x$ of ${\Bbbk}[x]$. We have furthermore $$\begin{aligned}
\operatorname{Ad}_{\Sigma^{-1}}(\overline{L}(y))& =\left\{\left. \begin{bmatrix} a & \\ & a^{-\#}\end{bmatrix}\in \operatorname{GL}_2({\Bbbk}[x]) \ \right|\ a\in {\Bbbk}[x]^\times \right\}\eqcolon H, \\
\operatorname{Ad}_{\Sigma^{-1}}(\overline{U}(y)) &=\left\{ \left. \begin{bmatrix} 1 & b \\ & 1 \end{bmatrix}\in \operatorname{GL}_2({\Bbbk}[x]) \ \right| \ b\in x{\Bbbk}[x] \right\}\eqcolon X, \quad \text{and} \\
\operatorname{Ad}_{\Sigma^{-1}}(\overline{V}(y))& = \left\{ \left. \begin{bmatrix} 1 & \\ c & 1 \end{bmatrix}\in \operatorname{GL}_2({\Bbbk}[x]) \ \right| \ c\in x{\Bbbk}[x] \right\}\eqcolon Y.
\end{aligned}$$ Let $S$ denote the two-element subgroup of ${\overline}{G}(y)$ generated by $s$, and let $R$ denote the subgroup $\operatorname{Ad}_{\Sigma^{-1}}(S)$ of $Q$; thus $R$ is the two-element group generated by $r=\operatorname{Ad}_{\Sigma^{-1}}(s)=\left[\begin{smallmatrix} & -1 \\ -1 & \end{smallmatrix}\right]$.
The subgroups $X$ and $Y$ commute in $Q$, because $x^2=0$. Since $H$ normalises $X$ and $Y$, this implies that the product $XHY$ is a subgroup of $Q$, equal to $(X\times Y)\rtimes H$. Explicitly, $$XHY = \left\{ \left. \begin{bmatrix} a & b \\ c & a^{-\#} \end{bmatrix}\ \right| \ a\in {\Bbbk}[x]^\times,\ b,c\in x{\Bbbk}[x]\right\},$$ i.e. the group of $q\in Q$ such that $q$ is diagonal modulo $x$.
Now, for each $q\in Q$, the reduction of $q$ modulo $x$ is a fixed point of the involution $$\operatorname{GL}_2({\Bbbk})\to \operatorname{GL}_2({\Bbbk}), \qquad \begin{bmatrix} a & b \\ c & d \end{bmatrix} \mapsto \begin{bmatrix} d & b \\ c & a \end{bmatrix}^{-1}$$ and so $q$ modulo $x$ is either of the form $\left[\begin{smallmatrix} a & \\ & a^{-1}\end{smallmatrix}\right]$ or $\left[\begin{smallmatrix} & b \\ b^{-1} & \end{smallmatrix}\right]$. Thus the homomorphism $$Q \to \{\pm 1\}, \qquad q\mapsto \det(q\text{ modulo } x)$$ has kernel $XHY$, and is split by the homomorphism $$\{\pm 1\}\to Q,\qquad -1\mapsto r.$$ This gives a decomposition $Q = \left( (X\times Y)\rtimes H \right) \rtimes R$. Since conjugation by $r$ preserves $H$ and permutes $X$ and $Y$, we may rewrite this decomposition as $Q= \left(X\times Y\right)\rtimes \left( H\rtimes R\right)$. Applying $\operatorname{Ad}_{\Sigma}$ gives $${\overline}{G}(y) = \left( {\overline}{U}(y)\times {\overline}{V}(y)\right) \rtimes \left({\overline}{L}(y)\rtimes S\right).$$
As in Case 3B we have $G\cdot y\cap {\mathfrak}l = L\cdot y$, and so $L\backslash G(y,{\mathfrak}l)/G(y)=\{1\}$.
|
---
abstract: |
This paper is focused on the language modelling for task-oriented domains and presents an accurate analysis of the utterances acquired by the Dialogos spoken dialogue system. Dialogos allows access to the Italian Railways timetable by using the telephone over the public network.
The language modelling aspects of specificity and behaviour to rare events are studied. A technique for getting a language model more robust, based on sentences generated by grammars, is presented. Experimental results show the benefit of the proposed technique. The increment of performance between language models created using grammars and usual ones, is higher when the amount of training material is limited. Therefore this technique can give an advantage especially for the development of language models in a new domain.
address:
title: 'Language Modelling For Task-Oriented Domains'
---
INTRODUCTION
============
Statistical language modelling (LM) is currently used for two different classes of applications: dictation systems and task-oriented spoken dialogue systems (SDS).
The first kind of systems are tested with a very large vocabulary (60-20,000 words) and they need the availability of a huge amount of training data, for instance WSJ-NAB has a 45 million word text corpora [@cit8].
SDSs are used in specific task-oriented domains, and they need special training material, which can be obtained either by expensive simulations [@cit6] or by using the SDS itself. The use of a general task-independent corpus for LM of a SDS could increase, in comparison to LM that use a task-dependent one, the perplexity by an order of magnitude [@cit9]. This is due to the mismatch between the general corpus and the specific application domain. In any case the acquired material is very limited, for instance the LM in the Air Travel Information System (ATIS) is based on a training-set of only 250,000 words [@cit10].
This paper is focused on the language modelling for task-oriented domains. The tests made uses the utterances acquired by the Dialogos, the SDS which allows access to the Italian Railways timetable by using the telephone over the public network [@cit1]. Other similar systems are described in [@cit2; @cit5; @cit7]. The vocabulary of Dialogos contains 3,471 words, clustered in 358 classes. The semantically important words are grouped into classes, such as city names (2,983 words), numbers (76 words), and so on. During the recognition, a class-based bigram LM is used, and the 25-best sequences are rescored using a trigram LM.
Section \[sec2\] shows how well a LM captures the specificity of the domain, while Section \[sec3\] studies the behaviour of the LM to rare events. Finally Section \[sec4\] illustrates a technique for generalising a LM by adding n-grams generated by a grammar.
SPECIFICITY OF A LANGUAGE MODEL {#sec2}
===============================
A relevant characteristic of a task-oriented domain is the distribution of the user utterances in a corpus. Using the Dialogos SDS, a corpus of 1,363 spoken dialogues has been acquired, from 493 unexperienced subjects, that called the system from all over Italy [@cit1; @cit3].
For the present study, the collected material was divided into two parts: a training-set of 20,511 utterances and a test-set of 2,040 utterances. Each utterance was transformed in a normalised form (NU), by changing each city name, month name and number into a class tag. For instance the user utterance:
[*“I want to leave from Naples to Rome Monday at five (o’clock)”*]{}
becomes the following NU:
[*“I want to leave from CITY-NAME to CITY-NAME WEEK-DAY at HOUR-NUMBER”.*]{}
For the sake of the language modelling, the NU is equivalent to the original utterance [^1].
It is worth noticing that even a small number of very frequent NUs cover a great part of the acquired data (see Figure \[fig1\]). The 7-th most frequent NUs cover 58% of the training-set, and 54% of test-set, and the first 191-st cover nearly 80% of test-set and over 85% of training- set. On the other hand the NUs with just one occurrence are 2,060, and more then 56% of them contain some spontaneous speech phenomena. This result shows that a few frequent NUs can already give a quite sensible picture of the user utterance distribution.
Moreover some partial training-sets were selected, which include the first n utterances in the whole training set, for n ranging from 100 to 20,511 utterances. For each partial training-set a LM was created and the recognition (WA) and understanding (SU) rates are given in Figure \[fig2\]. The performances of the LMs created on a partial training-set were compared with an experiment without any LM, which is even reported in Figure \[fig2\] as 0-utterance training-set. A LM trained on only 100 utterances achieves a remarkable error rate reduction of 30% of SU and 23% of WA, especially if it is compared with the error reduction when the whole 20,511 training-set is used, that is of 43% of SU and 39% of WA.
A coherent behaviour is also confirmed by perplexity values (PP) depicted in Figure \[fig3\], where the utterances were classified according to the kind of prompt generated by the system. Three representative points have been selected, which are the request of: departure and arrival city (City), time of departure (Time), and date of departure (Date). For these categories the PP of a 100-utterance LM is two times higher than a 1,000-utterance one and three times the LM trained on the whole training-set. The fact that, the PP values for the City requests are the highest, can be explained by the large number of city-names in the vocabulary (2,983, near 85% of the whole vocabulary).
ROBUSTNESS TO RARE EVENTS {#sec3}
=========================
In this Section the behaviour of the LM with respect to rare events is studied. The test-set of 2,040 utterances was split into two parts: The first part contains 362 utterances, whose 351 NUs do not appear in any of the partial training-sets. This is referred below as the unseen part of the test-set. The second part includes the rest of the test-set (1,678 utterances, but only 257 NUs). The NUs in the partial training-sets cover progressively the utterances of the second part. For instance, the 100-utterance training-set contains only 29 NUs, which cover 1,317 of these 1,678 utterances.
Both recognition, and overall understanding results show quite similar values for the 1,678 utterances (82-85% of SU), but they are very different for the unseen part (33-46% of SU), see Figure \[fig4\]. The performance on the unseen part is an indicator of the robustness of the model. In the following the reason for the low performance on the unseen part is further analysed.
The NUs with more then three occurrences in the global training-set, and different one to each other, were selected. Table \[tab1\] shows the number of this NUs, that exists in each one of the partial training-sets. They were divided into groups according to the different kind system request. The growth of NUs for City and Date is fast until 5,000 utterances are reached, then it becomes very slow. This indicates that there is a kind of saturation. While Time NUs increase nearly proportionally.
-------------- ----- ----- ------- ------- -------- --------
100 500 1,000 2,000 10,000 20,511
[**City**]{} 11 17 26 43 46 47
[**Date**]{} 8 17 25 43 48 51
[**Time**]{} 6 12 15 36 42 49
-------------- ----- ----- ------- ------- -------- --------
: \[tab1\] Number of frequent NUs in partial training-sets.
Moreover, the NUs, whose frequency in the training-set is greater than 0.1%, were compared with the ones in the test-set. We observed that the selected NUs of the training-set covers more then 90% of the test-set NUs, in case of City and Date, but only 55% in case of Time. Therefore, the City and the Date groups are considered much more robust than the Time group, because the frequent NUs do not indicate a saturation, and because there is a lack of the training-set NUs in the test-set. This is due to the high variability of the time expressions.
INCREASING ROBUSTNESS BY ADDING N-GRAMS GENERATED BY GRAMMARS {#sec4}
=============================================================
Another coverage test was made using grammars. A grammar was created (explained in Section \[sec4-1\]) on the basis of the NUs in the 500-utterance training-set. The sentences generated by the grammar showed a coverage of 85% of the NUs in the 20,511 training-set. This suggests that, the robustness of a LM may be increased by the use of a simple grammar derived from the common NUs in the training material.
At first the sentences generated by grammar were added to the training material. The obtained LMs, did not improve results, because the addition of the grammar generated sentences, greatly changes the frequency distribution of the n-grams, and reduces the specificity of the training-set.
The adopted solution was to create the LM starting from a data-base that contains n-grams, and not from a data-base of generated sentences. This made possible to add only the not-existing n-grams which do not highly affect the specificity. Therefore the tool used for training the LMs was changed, in order to be able to process both sentences and n-grams. Commonly when the n-grams are extracted from a sentence, they get automatically all their contexts (the (n-1)-gram that precedes the n-th word of the n-gram). On the other hand, if an n-gram is artificially added, it is necessary to incorporate even the missing contexts for this n-gram.
Grammar creation {#sec4-1}
----------------
The grammars used in the following tests were manually created, and they started from a set of correct NUs selected from a training-set. For each NU, semantic concepts were identified, then for each of these concepts a non-terminal was introduced, and, finally, each non-terminal was generalised. For instance, in the case of a [**Time**]{} NU:
[*“in the morning after seven o’clock”*]{},
the following non-terminal sequence could be identified:
[*Part\_of\_Day Time\_Specifier Time\_Identifier*]{}.
[*Part\_of\_Day*]{} can become also [*“in the afternoon”*]{}, [*“in the evening”*]{} or [*“at lunch time”*]{}, can be expressed as: [*“before”*]{}, [*“not earlier than”*]{}, while for [*Time\_Identifier*]{} other forms are: [*“a quarter to seven”*]{}, [*“twenty minutes past seven”*]{}.
At this point both the 1,000-utterance training-set (SPTS-1,000) and the global one (STS) were split according to the system request. Concentrating the analysis on the [**City**]{}, [**Date**]{}, and [**Time**]{} requests, for the syntactically and semantically correct NUs in SPTS-1,000 a grammar was created. For instance, there are 107 NUs in the SPTS-1,000 [**Date**]{} requests, and 2,483 NUs in STS.
For [**Date**]{} and [**Time**]{} requests group one grammar was created (Gr\_D, and Gr\_T respectively), whereas two for the [**City**]{} requests: Gr\_C which generalises only NUs about departure and arrival location, and Gr\_Cdt which also generalises data and time, because the answers to the [**City**]{} requests could also contain that information.
-------------- --------- ------ -------- ------ ----- ------- ------- ------ -----
part all part all part all part all
[**City**]{} Gr\_C 534 9,861 159 859 166 43 1 1
[**City**]{} Gr\_Cdt 568 10,088 125 632 1,326 1,156 4 2
[**Date**]{} Gr\_D 316 6,921 96 849 150 82 1 1
[**Time**]{} Gr\_T 276 6,431 36 616 1,748 1,488 10 1
-------------- --------- ------ -------- ------ ----- ------- ------- ------ -----
: \[tab2\] Event composition of the training-sets.
Creation of generalised LMs {#sec4-2}
---------------------------
The merge between the n-grams extracted from a training set and from sentences generated by the grammar was done using the following technique. At first, both the training-set and the sentences generated by a grammar were transformed in n-grams (n=3), then three type of events were considered: n-grams which are present both in the training-set and in the generated sentences (called [*usual events*]{}), n-grams which exist only in the training-set (called [*rare events*]{}), and n-grams which exist only in the generated sentences (called [*unknown events*]{}).
Into the new LM, the unknown events were added only once, while the rare events maintained their frequencies (which is quite low). In many cases the number of unknown events is much more higher than the number of usual events. For instance in the case of time there are 276 usual events obtained from SPTS-1,000, 36 rare events and 1,748 unknown events. Therefore the quantities of usual and unknown events are weighted, by multiplying them with a balance-factor. At this point, a language model is created, then the best value for the balance-factor (BaFa) is empirically determined by the minimisation of the PP on the test-set.
Using Table \[tab2\], the event composition of each one of the studied LMs can be computed. For each request group many LMs were created by the generalisation of SPTS-1,000 and STS, respectively [*part*]{} and [*all*]{} in the Table. It is worth noticing that in a baseline LM only the usual and rare events are considered.
Experimental Results {#sec40}
--------------------
In this Section, the performances of the LMs that include n-grams generated by a grammar were compared with baseline LMs which does not make use of grammar n-grams. These baseline LMs are reported in the Tables \[tab3\]-\[tab6\], with the tag [*unused*]{} in the grammar column.
-------------- -------------- ------ ------ ------ ------
WA SU WA SU
[**City**]{} [*unused*]{} 77.5 68.5 82.3 71.4
[**City**]{} Gr\_C 78.8 69.3 82.5 71.4
[**City**]{} Gr\_Cdt 80.0 70.1 82.3 72.6
[**Date**]{} [*unused*]{} 82.0 80.9 82.8 80.9
[**Date**]{} Gr\_D 82.7 81.3 82.9 80.9
[**Time**]{} [*unused*]{} 79.7 85.5 83.6 86.7
[**Time**]{} Gr\_T 82.5 86.1 83.6 86.7
-------------- -------------- ------ ------ ------ ------
: \[tab3\]Recognition and understanding results.
Table \[tab3\] shows that the LMs created using the grammars, obtain better results for the SPTS-1,000 LMs, while for the STS LMs the increment is rather limited. In particular, for [**Time**]{} and [**City**]{} the improvement of WA is significant. The reasons are: the high variability of time expressions and the fact that sometimes the [**City**]{} requests even include information about [**Date**]{} and [**Time**]{}, especially in the first utterance to the system. This fact is evident from the improvement obtained by the use of the Gr\_Cdt grammar, which even increases the performance of the STS LM.
Moreover the merge of with SPTS-1,000 with grammars improve the results, but they could not reach the performances of the baseline STS LMs. An explanation is that the used grammars do not model the highly frequent extra-linguistic phenomena.
In addition the perplexity of these LMs has been studied. For each group the analyses of the PP has been performed on the test-set and even on the sentences generated by the grammar. Table \[tab4\] shows PP results for all the LMs tested on the specific part of the test-set. The generalisation of the LMs by using grammar n-grams does not significantly affect the PP.
SPTS-1000 STS
-------------- -------------- ----------- -----
PP PP
[**City**]{} [*unused*]{} 117 79
[**City**]{} Gr\_C 118 78
[**City**]{} Gr\_Cdt 122 96
[**Date**]{} [*unused*]{} 33 24
[**Date**]{} Gr\_D 32 24
[**Time**]{} [*unused*]{} 20 14
[**Time**]{} Gr\_T 19 15
: \[tab4\]Perplexity results on the test-set.
The use of a test-set of sentences generated by the grammars, even if it does not give a correct insight of the behaviour of the system on a test-set acquired from real users, because the sentence distribution is artificial, it can show the degree of generalisation. These PP results have been reported in Table \[tab5\] and Table \[tab6\] according to the number of unknown events reported in Table \[tab2\]. In the former are shown the results for small grammars (G\_C, and G\_D), while in the latter the results for large ones (G\_Cdt, and G\_T).
SPTS-1000 STS
-------------- -------------- ----------- -----
PP PP
[**City**]{} [*unused*]{} 72 36
[**City**]{} Gr\_C 24 24
[**Date**]{} [*unused*]{} 52 25
[**Date**]{} Gr\_D 17 16
: \[tab5\]Perplexity results on the grammar sentences.
SPTS-1000 STS
-------------- -------------- ----------- -----
PP PP
[**City**]{} [*unused*]{} 117 207
[**City**]{} Gr\_Cdt 36 42
[**Time**]{} [*unused*]{} 231 53
[**Time**]{} Gr\_T 12 11
: \[tab6\]Perplexity values for Gr\_C and Gr\_T.
In Table \[tab5\], a clear reduction of the PP could be observed for the LMs which includes grammar n-grams. This reduction is higher for the LMs trained over SPTS- 1,000 (66%), but it is relevant even for the LMs trained on STS (33%).
Making a similar comparison of the PP results, presented in Table \[tab6\], for the large sets of unknown evens, as expected, a more significant reduction was obtained, that goes from a minimum of 77% to a maximum of 94%.
\[sec5\]CONCLUSIONS
===================
This papers shows that, in a task-oriented domain, a LM trained out with a small amount of training material (1,000 utterances) acquired form naive users, allows to obtain rather good results, especially in the case of the more common NUs. This is because common NUs are a few, but very frequent.
Secondly, in a task-oriented domain with a very limited training-set, the robustness of a language modelling can be increased by the use of a simple grammar derived from the common NUs in the training material.
A technique for the generalisation of a language model adding n-grams generated by a grammar is described. The advantage of this technique is shown by experimental results. The improvements obtained by using this technique, are especially good for language models trained on a small amount of training material, and therefore the technique can be used in the first phases of the development of a LM for a new domain. Even if the generalised LMs do not increase the performance of a model trained on a large training-set, the perplexity indicates a better behaviour of the models in the case of rare events.
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Aust, H., M. Oerder, F. Seide, V. Steinbiss, “The Philips Automatic Train Timetable Information System”, in [*Speech Communications*]{}, 1995, vol. 17, pp. 249–262.
Baggia, P., E. Gerbino, E. Giachin, C. Rullent, “Experiences of Spontaneous Speech Interaction with a Dialogue System”, in [*Proc. of CRIM/FORWISS Workshop*]{}, München, 1994, pp. 241–248.
Besling, S., H.-G. Meier, “Language Model Speaker Adaptation”, in [*Proc. of EUROSPEECH’95*]{}, Madrid, 1995, pp. 1755–1758.
Eckert, W., T. Kuhn, H. Niemann, S. Rieck, A. Scheuer, E. G. Schukat-Talamazzini, “A Spoken Dialogue System for German Intercity Train Timetable Inquiries”, in [*Proc. of EUROSPEECH’93*]{}, Berlin, 1993, vol. 3, pp. 1871–1874.
Fraser, N., G. N. Gilbert, “Simulating Speech Systems”, in [*Computer Speech and Language*]{}, 1991, vol. 5, pp. 81–99.
Goddeau, D., E. Brill, J. Glass, C. Pao, M. Phillips, J. Polifroni, S. Seneff, V. Zue, “GALAXY: A Human Language Interface to On-line Travel Information”, in [*Proc. of ICSLP’94*]{}, Yokoama, 1994, pp. 707–710.
Gauvain, J. L., L. Lamel, G. Adda, D. Matrouf, “Developments in Continuous Speech Dictation using the 1995 ARPA NAB New Task”, in [*Proc. of ICASSP’96*]{}, Atlanta, 1996, vol. 1, pp. 73–76.
Placeway, P., R. Schwartz, P. Fung, L. Nguyen, “The Estimation of Powerful Language Models from Small and Large Corpora”, in [*Proc. of ICASSP’93*]{}, Minneapolis, 1993, vol. 2, pp. 33–36.
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[^1]: This is because these classes are being used by the class-based LM and each word in a class has been considered with equal probability.
|
---
author:
- 'Filippo Callegaro & Ivan Marin'
bibliography:
- 'biblio.bib'
date: 'November 18, 2010'
title: '[**Homology computations for complex braid groups**]{}'
---
Scuola Normale Superiore\
Piazza dei Cavalieri, 7\
56126 Pisa\
Italy\
—\
Institut de Mathématiques de Jussieu\
Université Paris 7\
175 rue du Chevaleret\
75013 Paris\
France
[**Abstract.**]{} Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincaré polynomial with coefficients in a finite field for one large series of such groups, and compute the second integral cohomology group for all of them. As a consequence we get non-isomorphism results for these groups.
[**MSC 2010 : 20J06, 20F36, 20F55**]{}
Complexes from Garside theory
=============================
We recall a few homological constructions from the theory of Garside monoids and groups. Recall that a Garside group $G$ is the group of fractions of a Garside monoid $M$, where Garside means that $M$ satisfy several conditions for which we refer to [@dehpar]. In particular, $M$ admits (left) lcm’s, and contains a special element, called the Garside element. We denote $\mathcal{X}$ the set of atoms in $M$, assumed to be finite. The homology of $G$ coincides with the homology of $M$. Garside theory provides two useful resolutions of $\Z$ by free $\Z M$-modules.
The first one was defined in [@CMW]. Another one, with more complicated differential but a smaller number of cells, has been defined in [@DL].
The Dehornoy-Lafont complex
---------------------------
Let $M$ be a Garside monoid with a finite set of atoms $\mathcal{X}$. We choose an arbitrary linear order $<$ on $\mathcal{X}$. For $m \in M$, denote $\md(m)$ denote the smaller element in $\mathcal{X}$ which divides $m$ on the right ($m = a\md(m)$ for some $a \in M$). Recall that $\lcm(x,y)$ for $x,y \in M$ denotes the least common multiple on the left, that is $v = gx = h y$ implies $v = j \lcm(x,y)$ for some $j \in M$. If $A = (x,B)$ is a list of elements in $M$ we define inductively $\lcm(A) = \lcm(x, \lcm(B))$.
A $n$-cell is a $n$-tuple $[x_1,\dots,x_n]$ of elements in $\mathcal{X}$ such that $x_1 < \dots < x_n$ and $x_i
= \md (\lcm(x_i,x_{i+1},\dots,x_n))$. Let $\mathcal{X}_n$ denote the set of all such $n$-cells. By convention $\mathcal{X}_0
= \{ [\emptyset] \}$. The set $C_n$ of $n$-chains is the free $\Z M$-module with basis $\mathcal{X}_n$. A differential $\partial_n : C_n \to C_{n-1}$ is defined recursively through two auxiliary $\Z$-module homomorphisms $s_n : C_n \to C_{n+1}$ and $r_n : C_n \to C_n$. Let $[\alpha,A]$ be a $(n+1)$-cell, with $\alpha \in \mathcal{X}$ and $A$ a $n$-cell. We let $\alpha_{/A}$ denote the unique element in $M$ such that $(\alpha_{/A}) lcm(A) = lcm(\alpha,A)$. The defining equations for $\partial$ and $r$ are the following ones. $$\partial_{n+1}[\alpha,A] = \alpha_{/A} [A] - r_n(\alpha_{/A} [A]), \ \ r_{n+1} = s_n \circ
\partial_{n+1}, \ \ r_0(m[\emptyset]) = [\emptyset].$$ In order to define $s_n$, we say that $x [A]$ for $x \in M$ and $A$ a $n$-cell is *irreducible* if $x = 1$ and $A = \emptyset$, or if $\alpha = \md(x \lcm(A))$ coincides with the first coefficient in $A$. In that case, we let $s_n(x [A]) = 0$, and otherwise $$s_n(x[A]) = y[\alpha,A] + s_n(y r_n(\alpha_{/A} [A]))$$ with $x = y \alpha_{/A}$.
The Charney-Meyer-Wittlesey complex
-----------------------------------
Let again $G$ denote the group of fractions of a Garside monoid $M$, with Garside element $\Delta$. Let $\mathcal{D}$ denote the set of simple elements in $M$, namely the (finite) set of proper divisors of $\Delta$. We let $\mathcal{D}_n$ denote the set of $n$-tuples $[\mu_1|\dots|\mu_n]$ such that each $\mu_i$ as well as the product $\mu_1 \dots \mu_n$ lie in $\mathcal{D}$. The differential from the free $\Z M$-modules $\Z M \mathcal{D}_n$ to $\Z M \mathcal{D}_{n-1}$ is given by $$\partial_n [\mu_1|\dots|\mu_n] = \mu_1 [\mu_2|\dots|\mu_n]
+ \sum_{i=1}^{n-1} (-1)^i [\mu_1,\dots,\mu_i \mu_{i+1},\dots,\mu_n]
+ (-1)^n [\mu_1|\dots|\mu_{n-1}]$$ This complex in general has larger cells than the previous one. Its main advantage for us is that the definition of the differential is simpler, and does not involve many recursion levels anymore.
Application to the exceptional groups
-------------------------------------
When $W$ is well-generated, meaning that it can be generated by $n$ reflections, where $n$ denotes the rank of $W$, then $B$ is the group fractions of (usually) several Garside monoids that generalize the Birman-Ko-Lee monoid of the usual braid groups. These monoids have been introduced by D. Bessis in [@BESSISKPI1] and call there dual braid monoids. They are determined by the choice of a so-called Coxeter element $c$. Such an element is regular, meaning that it admits only one eigenvalue different from 1 with the corresponding eigenvector outside the reflection hyperplanes. A Coxeter element is a regular element with eigenvalue $\exp(2 \ii \pi/h)$, where $h$ denotes the (generalized) Coxeter number for $W$, namely its highest degree as a reflection group.
The corresponding Garside monoid $M_c$ is then generated by some set $R_c$ of braided reflections with relations of the form $r r' = r' r''$ (see [@BESSISKPI1] for more details). The above complexes for these monoids have been implemented by Jean Michel and the second author, using the (development version of) the CHEVIE package for GAP3. The chosen Coxeter element are indicated in Table \[coxeterdualmonoids\], in terms of the usual presentations of these groups (see [@BMR] for an explanations of the diagrams).
= lcircle10 scaled 1950 = lcircle10 scaled 3 = lcircle10 scaled 2 = lcircle10 \#1\#2[[-0.4pt\_[\#2]{}-8.6pt[\#1]{}2.3pt]{}]{} \#1pt[[ width\#1pt height4pt depth-3pt]{}]{} \#1[[-0.6pt\_[\#1]{}-1pt]{}]{} |\#1pt[[width\#1pt height3pt depth-2pt]{}]{} \#1[[-0.4pt-7pt[\#1]{}2.6pt]{}]{} \#1pt[[ width\#1pt height5pt depth-4pt]{}]{} \#1\#2[-1.5pt\^[\#1]{}-2pt]{} \#1\#2\#3[ \#1|14pt-13pt7.5pt \#2 -29pt9.5pt -2pt 17.5pt -1pt9.5pt ]{} \#1\#2[ 19.4pt]{} \#1\#2 \#1\#2\#3
$$\begin{array}{c||c|c|c|c|c}
\mbox{Group} & G_{24} & G_{27} & G_{29} & G_{33} & G_{34} \\
\hline
\mbox{Diagram} & \nnode s\rlap{\kern 2.5pt{\raise 6pt\hbox{$\triangle$}}}
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\kern-26.2pt\raise8.5pt\hbox{$\diagup$}
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\raise16.7pt\hbox{$\node$\rlap{\raise 2pt\hbox{$\kern 1pt\scriptstyle u$}}}
\kern -3pt\raise8.3pt\hbox{$\diagdown$}
\kern -7.6pt\raise9pt\hbox{$\diagdown$}
\kern 4pt & \nnode s\rlap{\kern 2.5pt{\raise 6pt\hbox{$\triangle$}}}
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\rlap{\kern-6pt\raise11pt\hbox{$\scriptstyle 5$}}
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\raise16.5pt\hbox{$\node$\rlap{\raise 2pt\hbox{$\kern 1pt\scriptstyle u$}}}
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\kern 4pt & \nnode s\bar10pt
\nnode t\rlap{\kern 1pt\raise9pt\hbox{$\underleftarrow{}$}}
\rlap{\kern 6.5pt\vrule width1pt height16pt depth-11pt
\kern 1pt \vrule width1pt height16pt depth-11pt}
\dbar14pt\nnode u
\kern-27pt\raise9.5pt\hbox{$\diagup$}
\kern -3pt\raise18.5pt
\hbox{$\node$\rlap{\raise 2pt\hbox{$\kern 1pt\scriptstyle v$}}}
\kern -2pt\raise9.5pt\hbox{$\diagdown$} & \nnode s\bar10pt\trianglerel tvu\bar10pt\nnode w &
\nnode s\bar10pt\trianglerel tvu\bar10pt\nnode w\bar10pt\nnode x \\
\hline
\mbox{Coxeter\ element} & stu & uts & stvu & wvtsu & xwuvts \\
\hline
\end{array}$$
Using the HAP package for GAP4 we then obtained the homologies described in Table \[tableexc\] (we recall in Table \[tableexcCox\] the ones obtained earlier by Salvetti for the Coxeter groups) except for the groups $G_{12}, G_{13}, G_{22}, G_{31}$, which are not well-generated, as well as the $H_3(B,\Z)$ of type $G_{33}$. When $W$ has type $G_{13}$, the group $B$ is the same as when $W$ has Coxeter type $I_2(6)$, and the result is known. For $G_{12}$ and $G_{22}$ one can use Garside monoids introduced by M. Picantin in [@THESEPIC].
A complex for $G_{31}$ can be obtained from the theory of Garside categories by considering it as the the centralizer of some regular element in the Coxeter group $E_8$. This viewpoint was used in [@BESSISKPI1] in order to prove that the corresponding spaces $X$ and $X/W$ are $K(\pi,1)$. More precisely, a simplicial complex (reminiscent from the Charney-Meyer-Wittlesey complex) is constructed in [@BESSISKPI1], which is homotopically equivalent to $X/W$. From this construction, we got a complex from an implementation by Jean Michel in CHEVIE.
However, for $G_{31}$, $G_{33}$ and $G_{34}$, the complexes obtained are too large to be dealt with completely through usual computers and software. The one missing for $G_{31}$ and $G_{33}$ are the middle homology $H_2(B,\Z)$ for $G_{31}$ and $H_3(B,\Z)$ for $G_{33}$. For $G_{33}$ the Dehornoy-Lafont complex for $G_{33}$ is however computable in reasonable time, and its small size enables to compute the whole homology by standard methods. For $G_{31}$, for which there is so far no construction analogous to the Dehornoy-Lafont complex, we used the following method for computing $H_2(B,\Z)$.
We first get $H_2(B,\Q) = 0$ by computing the second Betti number from the lattice. Indeed, recall from [@OT] (cor. 6.17, p.223) that the Betti numbers of $X/W$ can be in principle computed from the lattice of the arrangement. Precisely, the second Betti number of $X/W$ is given by $\sum_{Z \in T_2} |\mathcal{H}_Z/W_Z| -1$ where $T_i$ is a system of representatives modulo $W$ of codimension $i$ subspaces in the arrangement lattice ; for $Z$ such a subspace, $\mathcal{H}_Z = \{ H \in \mathcal{A} \ | \
H \supset Z \}$, $W_Z = \{ w \in W \ | \ w(Z) = Z \}$. More generally, the $i$-th Betti number is given by $$(-1)^i \sum_{Z \in T_i} \sum_{\sigma \in U_Z} (-1)^{d(\sigma)}$$ where $U_Z$ is the set of classes modulo $W$ of the set of simplices of the augmented Folkman complex of the lattice $\mathcal{A}_z$, and $d(\sigma)$ denotes the dimension of a cell. The Folkman complex of a lattice is defined (see [@OT]) as the complex of poset obtained by removing the minimal and maximal elements of the lattice ; when the maximal codimension of the lattice is 1, then the Folkman complex is empty. The augmented Folkman complex is defined by adding to the Folkman complex one $G$-invariant simplex of dimension $-1$. In the case of $G_{31}$ the computation of this formula is doable and we get 0 for the second Betti number.
We then reduce our original complex mod $p^r$, for $p^r$ small enough so that we can encode each matrix entry inside one byte. Then we wrote a C program to compute $H_2(B,\Z_4) = H_2(B,\Z_2) = \Z_2$, $H_2(B,\Z_9) = H_2(B,\Z_3) = \Z_3$ and $H_2(B,\Z_5) = 0$ (the matrix of $d_3$ has size $11065 \times 15300$). Since $G_{31}$ has order $2^{10} .3^2. 5$ and $H_*(P)$ is torsion-free, for $p \not\in \{ 2, 3, 5 \}$ we have $H_2(B,\Z_p) = H_2(P,\Z_p)^W = (H_2(P,\Z)^W) \otimes \Z_p$. But $0=H_2(B,\Q) = H_2(P,\Q)^W = H_2(P,\Z)^W \otimes \Q$ whence $H_2(P,\Z)^W = 0$ and $H_2(B,\Z_p) = 0$. Now $H_1(B,\Z) = \Z$ is torsion-free, hence $H_2(B,\Z_n) \simeq
H_2(B,\Z) \otimes \Z_n$ for any $n$ by the universal coefficients theorem. Since $H_2(B,\Z)$ is a $\Z$-module of finite type this yields $H_2(B,\Z) = \Z_6$ and completes the computation for $G_{31}$.
Embeddings between Artin-like monoids {#sectembed}
-------------------------------------
We end this section by proving a few lemmas concerning submonoids, which will be helpful in computing differentials in concrete cases.
We consider Garside monoids with set of generators $S$ and endowed with a length function, namely a monoid morphism $\ell : M \to \N = \Z_{\geq 0}$ such that $\ell(x) = 0 \Leftrightarrow
x = 1$ and $\ell(s) = 1$ for all $s \in S$. We consider the divisibility relation on the left (that is $U | V$ means $\exists m \ V = U m$) and recall that such a monoid admit lcm’s (on the left).
Let $M,N$ be two such monoids, and $\varphi : M \to N$ a monoid morphism such that
1. $\forall s \in S \ | \ \varphi(s) \neq 1$
2. $\forall s,t \in S \ \lcm(\varphi(s),\varphi(t)) = \varphi(\lcm(s,t))$
The following results on such morphisms are basically due to J. Crisp, who proved them in [@crisp] in the case of finite-type Artin groups.
Let $U,V \in M$. If $\varphi(U) | \varphi(V)$ then $U | V$.
By induction on $\ell(V)$. Since $\forall s \in S \ \ell(\varphi(s)) \geq 1 = \ell(s)$, we have $\ell(\varphi(U)) \geq \ell(U)$. Since $\varphi(U) | \varphi(V)$, we have $\ell(\varphi(U)) \leq \ell(\varphi(V))$ hence $\ell(U) \leq \ell(\varphi(V))$. Hence $\varphi(V) = 1$ implies $\ell(U) = 0$ and $U = 1$, which settles the case $\ell(V) = 0$.
We thus assume $\ell(V) \geq 1$. The case $U=1$ being clear, we can assume $U \neq 1$. Then there exists $s,t \in S$ with $s | U$ and $t|V$. It follows that $\varphi(t) | \varphi(V)$ and $\varphi(s) | \varphi(U) | \varphi(V)$, hence $\lcm(\varphi(s),\varphi(t)) | \varphi(V)$.
Now $\lcm(s,t) = t m$ for some $m \in M$ and $V = t V'$ for some $V' \in M$, hence $\varphi(t) \varphi(m) | \varphi(V) = \varphi(t) \varphi(V')$ and this implies $\varphi(m) | \varphi(V')$ by cancellability in $M$. Since $\ell(V') < \ell(V)$, from the induction assumption follows that $m | V'$ hence $tm | V$ that is $\lcm(s,t) | V$. In particular we get $s | V$. Writing $V = s V''$ and $U = s U'$ for some $V'',U' \in M$, the assumption $\varphi(U) | \varphi(V)$ implies $\varphi(U') | \varphi(V'')$ by cancellability, and then $U' | V''$ by the induction assumption. It follows that $U|V$ which proves the claim.
The lemma has the following consequence.
The morphism $\varphi : M \to N$ is injective. If $G_M$, $G_N$ denotes the group of fractions of $M,N$, then $\varphi$ can be extended to $\widetilde{\varphi} : G_M \into G_N$.
Let $U,V \in M$ with $\varphi(U) = \varphi(V)$. By the lemma we get $U|V$ and $V | U$. This implies $\ell(U) = \ell(V)$ hence $U = V$. Composing $\varphi : M \to N$ with the natural morphism $N \into G_N$ yields a monoid morphism $M \to G_N$. Since $G_N$ is a group this morphism factors through the morphism $M \to G_M$ and this provides $\widetilde{\varphi} : G_M \to G_N$. Let $g \in \Ker \widetilde{\varphi}$. Since $g \in G_M$ there exists $a,b \in M$ with $g = ab^{-1}$ hence $\varphi(a) = \varphi(b)$, $a=b$ and $g = 1$.
We consider the following extra assumption on $\varphi$. We assume that, for all $m \in M$ and $n \in N$, $n | m$ implies
We can now identify in this $M,N,G_M$ to subsets of $G_N$. We consider the following extra assumption. We assume that, for all $m \in M, n \in N$, if $n$ divides $m$ in $N$ then $n \in M$.
Under this assumption, $U,V$ in $M$ have the same lcm in $M$ and in $N$. Moreover, $M = N \cap G_M$.
Since $\lcm_M(U,V)$ divides $U,V$ in $N$, we have that $\lcm_N(U,V))$ divides $\lcm_M(U,V)$ in $N$. Conversely, since $\lcm_N(U,V)$ divides $U$ in $N$ and $U \in M$, by the assumption we get $\lcm_N(U,V) \in M$. From the lemma we thus get that $\lcm_N(U,V)$ divides $U$ and $V$ in $M$ hence $\lcm_M(U,V)$ divides $\lcm_N(U,V)$ in $N$. It follows that $\lcm_M(U,V) = \lcm_N(U,V)$.
We have $M \subset N \cap G_M$. Let $n \in N \cap G_M$. Since $n \in G_M$ there exists $a,b \in M$ with $n = ab^{-1}$, hence $nb = a \in M$. Hence $n \in N$ divides $a \in M$ in $M$. By the assumption we get $n \in M$ and the conclusion.
The groups ${{\mathrm{B}}}(e,e,r)$ {#s:beer}
==================================
The Corran-Picantin monoid
--------------------------
We denote ${{\mathrm{B}}}(e,e,r)$ for $e \geq 1$ and $r \geq 2$ the braid group associated to the complex reflection group $G(e,e,r)$. The ${{\mathrm{B}}}(e,e,r)$ are the group of fractions of a Garside monoid introduced by R. Corran and M. Picantin (see [@corpic]). This monoid, that we denote $M(e,e,r)$, has generators (atoms) $t_0,t_1,\dots,t_{e-1}, s_3, s_4, \dots, s_r$ and relations
1. $t_{i+1} t_i = t_{j+1} t_j$, with the convention $t_e = t_0$,
2. $s_3 t_i s_3 = t_i s_3 t_i$
3. $s_k t_i = t_i s_k$ for $k \geq 4$
4. $s_k s_{k+1} s_k = s_{k+1} s_k s_{k+1} $ for $k \geq 3$
5. $s_k s_l = s_l s_k$ when $| l-k | \geq 2$.
Link with the topological definition
------------------------------------
The connection between this monoid and the group ${{\mathrm{B}}}(e,e,r)$ defined as a fundamental group is quite indirect. In [@BMR] a first presentation is obtained by combining embeddings into usual braid groups, fibrations and coverings. The presentation used here is deduced from this one in a purely algebraic matter, by adding generators in order to get a Garside presentation. Although it is folklore the description of all generators as braided reflection does not appear in the literature (see however [@BESSISCORRAN] for a statement without proof in a related context).
In order to provide this connection, we need to recall the way these generators are constructed. For clarity, we stick to the notations of [@BMR] ; in this paper, the authors introduce 4 different spaces, $\mathcal{M}(r+1) = \{ (z_0,\dots,z_r) \in \C^{r+1} \ | \ z_i \neq z_j \}$, $\mathcal{M}^{\#}(m,r) = \{ (z_1,\dots,z_r) \in \C^r \ | \ z_i \neq 0,
z_i /z_j \not\in \mu_m \}$, $\mathcal{M}(e,r) = \{ (z_1,\dots,z_r) \in \C^r \
| \ z_i \not\in \mu_e z_j \}$, and $\mathcal{M}^{\#}(r) = \{ (z_1,\dots,z_r) \in \C^r \ | \ z_i \neq 0 \}$, where $\mu_n$ denotes the set of $n$-th roots of 1 in $\C$. We have a Galois covering $r : \mathcal{M}^{\#}(m,r)
\to \mathcal{M}^{\#}(r) = \mathcal{M}^{\#}(m,r) / (\mu_m)^r$, a locally trivial fibration $p : \mathcal{M}(r+1) \to \mathcal{M}^{\#}(r)$ with fiber $\C$ given by $(z_0,\dots,z_r) \mapsto (z_0 - z_1,\dots,
z_0-z_r)$, and a natural action of $\mathfrak{S}_r$ on $\mathcal{M}(r+1)$ that leaves the $(r+1)$-st coordinate fixed. We choose a fixed point $x \in \mathcal{M}(r+1)/\mathfrak{S}_r$, and a lift $\widetilde{p(x)}$ of $p(x) \in \mathcal{M}^{\#}(r)/\mathfrak{S}_r$ in $\mathcal{M}^{\#}(d,r)/G(d,1,r) = (\mathcal{M}(d,r)/(\mu_m)^r)/\mathfrak{S}_r$. We get an isomorphism $\psi : \pi_1(\mathcal{M}^{\#}(d,r)/G(d,1,r),
\widetilde{p(x)}) \to \pi_1(\mathcal{M}(r+1)/\mathfrak{S}_r,x)$ by composing the isomorphisms induced by $r$ and $p$. $$\xymatrix{
\pi_1(\mathcal{M}^{\#}(d,r)/G(d,1,r), \widetilde{p(x)}) \ar[dr]_{r}^{\simeq} \ar[rr]^{\psi} & &
\ar[dl]^{p}_{\simeq} \pi_1(\mathcal{M}(r+1)/\mathfrak{S}_r, x) \\
& \pi_1(\mathcal{M}^{\#}(r)/\mathfrak{S}_r, p(x)) & \\
}$$ Since $\pi_1(\mathcal{M}^{\#}(d,r)/G(d,1,r)) = {{\mathrm{B}}}(d,1,r)$, $\psi$ identifies the latter group with $\pi_1(\mathcal{M}(r+1)
/\mathfrak{S}_r)$. The generators of ${{\mathrm{B}}}(d,1,r)$ are then obtained in [@BMR] by taking the preimages under $\psi$ and the covering of $\mathcal{M}(r+1)/\mathfrak{S}_r \to \mathcal{M}(r+1)/\mathfrak{S}_{r+1}$. Note that this covering provides an injection between fundamental groups, hence an embedding $\tilde{\psi} : {{\mathrm{B}}}(d,1,r) \into {{\mathrm{Br}}}(r+1)$, where ${{\mathrm{Br}}}(r+1)$ denotes the usual braid group on $r+1$ strands. We choose for base point in $\mathcal{M}(r+1)$ the point $x = (0,x_1,\dots,x_r)$ with the $x_i \in \R$ and $x_{i+1} \ll x_i$, and for generators of the usual braid group $\mathcal{M}(r+1)/\mathfrak{S}_{r+1}$ the elements $\xi_0, \xi_1,\dots,\xi_{r-1}$ as described below :
Then (see [@BMR]), the group $\pi_1(\mathcal{M}(r+1)/\mathfrak{S}_r)$ is generated by $\xi_0^2,\xi_1,\dots,\xi_{r-1}$. The element $\xi_0^2$ is the class in $\mathcal{M}(r+1)$ of the loop $(\frac{x_1}{2}(1-e^{2 \ii \pi t}), \frac{x_1}{2}( e^{2 \ii \pi t} +1),x_2,
\dots,x_r )$. Taking its image by $p$ provides a loop based at $(-x_1,-x_2,\dots,-x_r)$ described by $(-x_1 e^{2 \ii \pi t},
\frac{x_1}{2} (1 - e^{2 \ii \pi t}) -x_2,\dots,\frac{x_1}{2} (1 - e^{2 \ii \pi t}) - x_r)$. Since $|x_i | \ll |x_{i+1}|$, this path is homotopic to $(-x_1 e^{2 \ii \pi t}, -x_2, \dots, -x_r)$, both in $\mathcal{M}^{\#}(r)$ and in $\mathcal{M}^{\#}(r)/\mathfrak{S}_r$. Letting $a_i = -x_i$, we have $0 < a_1 < a_2 < \dots < a_r$, and we choose $y = \widetilde{p(x)}$ to be $y = (a_1^{\frac{1}{d}},\dots,
a_r^{\frac{1}{d}})$. The above loop thus lifts under $r$ to the path $(a_1 e^{2 \ii \pi t/d},a_2,\dots,a_r)$ in $\mathcal{M}^{\#}(d,r)$. By definition of $\psi$, the class of this path $\sigma = \psi^{-1}(\xi_0^2)$. Similarly, we can determine $\psi^{-1}(\xi_i)$ when $i \geq 1$ : the image of $\xi_i$ under $p$ is a path in $\mathcal{M}^{\#}(r)$ homotopic to
On the open cone described in the picture, the map $z \mapsto z^d$ is a positive homeomorphism, and this enables one to lift this path to
meaning that $\tau_i = \psi^{-1}(\xi_i)$ is the class of this path, from $(a_1^{1/d}, \dots, a_i^{1/d}, a_{i+1}^{1/d}, \dots, a_r^{1/d})$ to $(a_1^{1/d}, \dots, a_{i+1}^{1/d}, a_{i}^{1/d}, \dots, a_r^{1/d})$. We recall that ${{\mathrm{B}}}(de,e,r)$ is defined by $\pi_1( \mathcal{M}^{\#}(de,r)/G(de,e,r))$ when $d > 1$. If moreover $e=1$, then ${{\mathrm{B}}}(d,1,r)$ is generated by $\sigma,
\tau_1,\dots,\tau_{r-1}$ ; in general, it is generated by $\sigma^e,
\tau_1,\dots,\tau_{r-1}$. Now, the morphism $\tilde{\psi} : {{\mathrm{B}}}(d,1,r) \into
{{\mathrm{Br}}}(r+1) = \pi_1(\mathcal{M}(r+1)/\mathfrak{S}_{r+1})$ commutes with the natural morphisms $$\xymatrix{ {{\mathrm{B}}}(d,1,r) \ar[rr] \ar[d] & & {{\mathrm{Br}}}(r+1) \ar[d] \\
G(d,1,r) \ar[rr] \ar[dr] & & \mathfrak{S}_{r+1} \\
& \mathfrak{S}_r \ar[ur] & }$$ Letting as in [@BMR] $\xi'_1 = \xi_0^2 \xi_1 \xi_0^{-2} \in \pi_1(
\mathcal{M}(r+1)/\mathfrak{S}_r, x)$, we have $\tau'_1 = \psi^{-1}(\xi'_1)
\in \pi_1(\mathcal{M}^{\#}(d,r) / G(d,1,r),y)$. As before we let $\zeta = \exp(2 \ii \pi/d)$ and $g_i \in G(d,1,r)$ being defined by $g_i.(z_1,\dots,z_r) = (z_1,
z_2,\dots,\zeta z_i, \dots, z_r)$. We let $b_i = a_i^{1/d}$. $$\xymatrix{
(b_1,\dots,b_r) \ar@/^1pc/[d]^{\xi_0^{-2}} \ar@/_5pc/@{-->}[dd]_{g_1^{-1} s_1}
\ar@/_10pc/@{-->}[ddd]_{g_1^{-1}s_1g_1}\ar@/^10pc/[ddd]_{\xi'_1}\\
(\zeta^{-1} b_1,b_2,\dots,b_r) \ar@/^1pc/[d]^{g_1^{-1}.\xi_1}\\
(\zeta^{-1} b_2, b_1,\dots,b_r) \ar@/^1pc/[d]^{g_1^{-1} s_1 . \xi_0^2}\\
(\zeta^{-1} b_2, \zeta b_1,\dots,b_r) }$$ In order to generate ${{\mathrm{B}}}(e,e,r) = \pi_1(\mathcal{M}(e,r)/\mathfrak{S}_r, x)$, and letting $e=d$, we only need to take the image of $\tau'_1,\tau_1,\dots,\tau_{r-1}$ under $i^*$ where $i : \mathcal{M}^{\#}(e,r) \to \mathcal{M}(e,r)$ is the natural inclusion. We will use the following definition.
Let $X$ be the complement of an hyperplane arrangement $\mathcal{A}$ in $\C^l$, and $v,v' \in X$. A line segment from $v$ to $v'$ is $t \mapsto (1-t) v + t v'$ for $t \in [0,1]$, If this line segment crosses exactly one hyperplane of $\mathcal{A}$ at one point, a positive *detour* from $v$ to $v'$ is a path of the form $\gamma(t) = (1-t) v + t v' + \ii t (1-t) (v-v') \eps$ for $\eps > 0$ small enough so that it and the similar paths $\gamma'$ for $0 < \eps' < \eps$ do not cross any hyperplane in $\mathcal{A}$. All such detours are clearly homotopic to each other. A negative detour is defined similarly with $\ii$ replaced by $- \ii$.
Note that, for $v \in \mathcal{M}(e,r)$ and $s$ a reflection in $G(e,e,r)$, if there exists a positive detour from the base point $\underline{b} = (b_1,\dots,b_r)$ to $w.\underline{b}$, then it provides a braided reflection around the hyperplane attached to $s$.
The elements $i^*(\tau_k)$ are now easy-to-describe braided reflections, as the positive detours from $\underline{b}$ to their images by the corresponding reflections. In case $e = 2$, the given monoid is then clearly the classical Artin monoid of type $D_r$, so we can assume $e \geq 3$. The paths corresponding to $\xi_0^{2}$ and to its translates are homotopic to a line segment in $\mathcal{M}(e,r)$. The fact that $\tau'_1$ is a braided reflection essentially amounts to the fact that $i^*(g_0^{-1} .\tau_1)$ is a braided reflection in $\pi_1(\mathcal{M}(e,r)/G(e,e,r), g_0^{-1} . y)$, and this holds true because $\tau_1$ is a braided reflection in $\mathcal{M}^{\#}(e,r)/G(e,e,r)$.
We consider the plane $P$ defined by the equations $z_i = b_i$ for $i = 3,\dots,r$, and identify it to $\C^2$ through $(z_1,z_2)$. We let $P^0 = \C^2 \setminus \bigcup \{ z_2 = z_1 \eta \ | \ \eta \in \mu_e \}
= P \cap \mathcal{M}(e,r)$. Then $\tau_1,\tau'_1$ lie in the plane $P$, and $\tau'_1$ is homotopic in $P^0$ to
where the half-circle represents the positive detour from $(\zeta^{-1} b_1,b_2)$ to $(\zeta^{-1} b_2,b_1)$. We let now $t_0 = \tau_1$, $t_1 = \tau'_1$, $t_{i+1} = t_i^{-1} t_{i-1}t_i$ for $1 \leq i \leq e-2$. A way to understand paths in $P \simeq \C^2$ is to use the projection $\C^2 \to \mathbbm{P}^1(\C)$ given by $(z_1,z_2) \mapsto
z_2/z_1$. Note for example that two paths $\gamma_1,\gamma_2$ in $P$ with the same endpoints whose images are homotopic in $\mathbbm{P}^1(\C) \setminus
\mu_e$ are homotopic in $P^0$ as soon as, writing $\gamma_i(t)
=(x_i(t),y_i(t))$, the set $x_1([0,1]) \cup x_2([0,1])$ is contained in some simply connected subspace of $\C \setminus \{ 0 \}$. We let $\alpha = b_2/b_1 \gg 1$. Then the positive detour $t_0$ is mapped to a path from $\alpha$ to $\alpha^{-1}$ close to the line segment, with image in the positive half-plane. The line segments of the form $\gamma(t) = (z_1,z_2(t)$ are mapped to line segments, and lines form $\gamma(t) = (z_1(t),z_2)$ are mapped to images of a line under $z \mapsto 1/z$, which is the composite of the complex conjugation with the geometric inversion with respect to the unit circle ; they are thus mapped to a line if the original line passes through 0, and otherwise to a circle passing through the origin. The induced action of $G(e,e,r)$ is given by $s_1 : z \mapsto \frac{1}{z}$, $g_1 : z \mapsto \zeta^{-1} z$, $g_2 : z \mapsto \zeta z$. The images of $t_1$ and $t_2$ are depicted in figure \[figt1t2\]. The images of $t_2$ and of the positive detour from $(b_1,b_2)$ to $(\zeta^{-2} b_2,\zeta^2 b_1)$ are then clearly homotopic (see figure \[figt2detour\]), and the first coordinate of both paths is easily checked to remain in a simply connected region of $\C \setminus \{ 0 \}$. With the same argument, using the relation $t_{i+1} = t_i^{-1} t_0 t_1$ and possibly using $(z_1,z_2) \mapsto z_1/z_2$ instead of $(z_1,z_2)
\mapsto z_2/z_1$, we get that each $t_i$ is (homotopic to) the positive detour from $(b_1,b_2)$ to $(\zeta^{-i}b_2,\zeta^i b_1)$. We thus got the following
Let $\underline{b} = (b_1,\dots,b_r) \in \mathcal{M}(e,r)$ with $0 < b_1 \ll b_2 \ll \dots \ll b_r$. Then ${{\mathrm{B}}}(e,e,r) = \pi_1( \mathcal{M}(e,r)/G(e,e,r),\underline{b})$ is generated by braided reflections $t_0,\dots,t_{e-1},s_3,\dots,s_r$ which are positive detours from $\underline{b}$ to their images under the corresponding reflection. Under ${{\mathrm{B}}}(e,e,r) \onto
G(e,e,r)$, $t_i$ is mapped to $(z_1,z_2,\dots,z_r) \mapsto
(\zeta^{-i} z_2,\zeta^i z_1,\dots,z_r)$, and $t_0,s_3,\dots,
s_r$ are mapped to the successive transpositions of $\mathfrak{S}_r$ in that order. These generators provide a presentation of ${{\mathrm{B}}}(e,e,r)$ with the relations (2)-(5) of page , and with (1) replaced by $t_i t_{i+1} = t_j t_{j+1}$.
We notice that the slight change in the presentation is meaningless in monoid-theoretic terms, as both monoids are isomorphic under $t_i \mapsto t_{- i}$, but it is not in topological terms, as $t_1 t_0 t_1^{-1}$ is *not* homotopic to a detour from $\underline{b}$ to its image (see figure \[figt2plus\]).
Let $S_0 = \{ t_i,s_3,\dots,s_r \}$. The subgroup of ${{\mathrm{B}}}(e,e,r) = \pi_1(\mathcal{M}(e,r)/G(e,e,r)$ generated by $S_0$ is a parabolic subgroup in the sense of [@BMR], and can be naturally identified with the braid group on $r$ strands as the fundamental group of $\{ (z_1,\dots,z_r) \ | \ z_i \neq z_j, z_1+\dots+z_r = 0 \}/\mathfrak{S}_r$, with base point $(-\zeta^{-i}(b_1+b_2+\dots + b_r),b_2,\dots,b_r)$, in such a way that the elements of $S_0$ are identified with positive detours.
The parabolic subgroup of $G(e,e,r)$ defined as the fixer of $(\zeta^i, 1,1 , \dots, 1)$ is obviously conjugated to the one fixing $(1,1,\dots,1)$, the latter being the natural $\mathfrak{S}_r \subset G(e,e,r)$. We thus need only consider the case $i = 0$. Let $\alpha = -(b_1+\dots+b_r) \ll 0$, $\underline{b}_0
= (\alpha,b_2,\dots,b_r)$, $X = \mathcal{M}(e,r)$ and $X_0 = \{ (z_1,\dots,z_r) \ | \ z_i \neq z_j, z_1+\dots+z_r = 0 \}$. By [@BMR] we get an embedding $\pi_1(X_0/\mathfrak{S}_r,
\underline{b}_0) \into \pi_1(X/G(e,e,r),\underline{b})$, natural only up to the choice of a path from $\underline{b}$ to $\underline{b}_0$ in $Y$. The line segment $\gamma$ from $\underline{b}$ to $\underline{b}_0$ provides such a natural choice.
We now need to prove that composing the positive detours from $\underline{b}$ with this path provides the positive detours from $\underline{b}_0$, up to homotopy in $Y = \{ (z_1,\dots,z_r) \ | \ z_i \neq z_j \}$. For $s_3,\dots,s_r$ this is true because the first component of the first path can be homotoped to the second one in $(\R_{\leq b_1}, b_1)$. For $t_0$ we let $\sigma_0$ and $\sigma$ denote the positive detours in $P^0$ from $(b_1,b_2)$ to $(b_2,b_1)$ and from $(\alpha,b_2)$ to $(b_2,\alpha)$, respectively. Let $\gamma, \gamma'$ denote the line segments $(b_1,b_2) \to (\alpha,b_2)$ and $(b_2,\alpha) \to (b_2,b_1)$. We need to prove that $\sigma_0$ is homotopic to $\gamma' \sigma \gamma$ in $\{ (z_1,z_2) \in \C^2 \ | \
z_1 \neq z_2 \}$, the other coordinates $z_3,\dots,z_r$ being the same for both paths. Since $b_1 - b_2$ and $\alpha -b_2$ have the same (negative) sign, we can homotope $\gamma' \sigma \gamma$ to a path with the same real part (for both coordinates), and with imaginary part the same as $\sigma$, up to possibly diminishing the chosen factor $\eps$ in the definition of the detours. Choosing then an homotopy in $\R_{\leq b_2}^2$ between the real parts of these two paths, provides an homotopy between them in $Y$.
Parabolic submonoids
--------------------
We apply the results of section \[sectembed\] on submonoids to the monoid $N = M(e,e,r)$, with generators $S = \{ t_0,\dots,t_{e-1}, s_3,\dots,s_r \}$. Let $\mathcal{C} = \{ t_0 , \dots ,t_{e-1} \}$. For $S_0 \subset
S$, let $M(S_0)$ be the monoid generated by $S_0$ with the defining relations of $M(e,e,r)$ which involve only elements of $S_0$. We get a natural morphism $\varphi : M(S_0) \to M(S) = M(e,e,r)$. We ask for the following extra assumption on $S_0$ : $$S_0 \cap \mathcal{C} \in \{ \emptyset, \mathcal{C}, \{ t_i \} \} \mbox{ for some } i \in \{0,1,\dots,e-1 \} .$$ In other terms, $S_0$ contains none, all or exactly one of the $t_i$’s. Note that all the corresponding monoids are known to be Garside and are endowed with a suitable length function.
This condition implies the extra condition on $\varphi$ in section \[sectembed\], namely that, if $n \in M(S)$ divides $\varphi(m)$ for some $m \in M(S_0)$, then $n \in \varphi(M(S_0))$. Indeed, if we have such $n,m$,then $n \in \varphi(M(S_0))$ unless $n$ can be written as a word containing some $x \in S \setminus S_0$. But in that case $\varphi(m)$ can also be written as a word in $S$ containing $x$. Now note that the defining relations involving such a $x$ cannot make it disappear, except when $x \in \mathcal{C}$. By contradiction this settles the cases $S_0 \cap \mathcal{C} = \emptyset$ and $S_0
\supset \mathcal{C}$. In case $S_0 \cap \mathcal{C} = \{ t_{i} \}$, we can assume $x = t_j$ for $j \neq i$, and would get equality in $M(S)$ of two words on $S$, one involving $t_i$ and no other element of $\mathcal{C}$, and the other involving $t_j$. But we check on the defining relations that all relations involving $t_i$ either involve only $t_i$ and no other elements of $\mathcal{C}$ in which case they preserve that property and do not make the $t_i$’s disappear, or they involve several elements of $\mathcal{C}$ and cannot be applied to the first word. This leads to a contradiction, which proves this property.
This condition also implies the property (2) for $\varphi$. For this we need to compute the lcm’s between two elements $x,y$ of $S$. We need to prove the following in $M(S_0)$, for any $S_0 \subset S$ satisfying the above condition that contains $x$ and $y$.
- $\lcm(s_i,s_j) = s_i s_j = s_j s_i$ if $|j-i| \geq 2$
- $\lcm(s_i,s_{i+1}) = s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}$
- $\lcm(t_i,t_j) = t_1 t_0 = t_i t_{i-1} = t_jt_{j-1}$
- $\lcm(t_i ,s_3) = t_i s_3 t_i = s_3 t_i s_3 $
- $\lcm(t_i, s_j) = t_is_j = s_j t_i $ if $j \geq 4$.
The identities with length two are clear, as the lcm exist and cannot have length 1. For the ones of length 3, namely $\{ x, y \} = \{ s_i,s_{i+1} \}$ and $\{ x, y \} = \{ t_i ,s_3 \}$, we use that $\{ x , y \} \subset S_0 \subset S$ satisfies our condition. Since the lcm of $x,y$ in $M(S_0)$ should divide $xyx=yxy$, it should then come from $M(\{ x, y \})$, meaning that is should be a word in $x$ and $y$, of length at most $2$. Thus only few possibilities remain, all of them easily excluded.
Using the previous section, we thus get injective monoid morphisms $M(S_0) \to M(S) = M(e,e,r)$. Let $B(S_0)$ the group of fractions of $M(S_0)$. It is proved in [@corpic] that $B(S) = {{\mathrm{B}}}(e,e,r)$. We call the $B(S_0)$ the parabolic submonoids of ${{\mathrm{B}}}(e,e,r)$. Crucial examples of such submonoids are described below.
### Second homology group
We choose on the atoms the ordering $s_r < s_{r-1} < \dots< s_3 < t_0 < t_1 < \dots < t_{e-1}$. By the above construction, the parabolic submonoid $M(e,e,r-1) = M(\{ s_{r-1},\dots,s_3,t_0,\dots,t_{e-1} \})$ is indeed a submonoid of $M(e,e,r)$, and the lcm of a family of elements in $M(e,e,r-1)$ is also its lcm in $M(e,e,r)$. The same holds true for the following submonoids :
- the ones generated by $s_3,t_i$, which is an Artin monoid of type $A_2$ ;
- the ones generated by $s_k,t_i$, $k \geq 4$, which is an Artin monoid of type $A_1 \times A_1$ ;
- the ones generated by $s_4,s_3,t_i$, which is an Artin monoid of type $A_3$ ;
- the ones generated by $s_k,s_3,t_i$, $k \geq 5$ which is an Artin monoid of type $A_1 \times A_2$ ;
- the ones generated by $s_k,s_l,t_i$, $k \geq l+2$, $l \geq 4$, which is an Artin monoid of type $A_1 \times A_1 \times A_1$ ;
- the ones generated by $s_k,s_l,s_r$, which is an Artin monoid of type given by the obvious subdiagram (of type $A_1 \times A_1 \times A_1$, $A_2 \times A_1$, $A_1 \times A_2$ or $A_3$).
We first compute the differentials of the top cell for the corresponding Artin monoids (see Table \[typesA\]), and then use this remark for computing the differentials of the 2-cells and 3-cells. We let $d_n = \partial_n \otimes_{\Z M} \Z : C_n \otimes_{\Z M} \Z \to C_{n-1} \otimes_{\Z M} \Z$ denote the differential with trivial coefficients.
The 2-cells are the following : $[t_0,t_{i}]$ for $1<i < e$, $[s_3,t_i]$ , $[s_k,t_i]$ for $k \geq 4$ and $[s_k,s_l]$ for $k<l$. From Table \[typesA\] we get $$\begin{array}{lcll}
d_2 [t_0,t_i] &=& [t_i] + [t_{i+1}] - [t_0] - [t_1] \\
d_2 [s_3, t_i] &=& [t_i] - [s] \\
d_2 [s_k, t_i] &=& 0 \mbox{ if } k \geq 4 \\
d_2 [s_k,s_l] &=& 0 \mbox{ if } l> k+2 \geq 4 \\
d_2 [s_k,s_{k+1}] &=& [s_{k+1}] - [s_k]
\end{array}$$ We let $\delta_2 = t_1 t_0$ denote the Garside element of $M(e,e,2)$ and we assume $e > 1$. For the 3-cells, we also need to compute
$$\begin{array}{lcl}
\partial_3 [s,t_0,t_j] &=& (s\delta_2 s - t_{j+2} t_{j+1} s +t_{j+2} s)[t_0,t_j]
-t_{j+2}st_{j+1}[s,t_j] \\ & & +(t_{j+2}-st_{j+2})[s,t_{j+1}] + (s-t_{j+2}s-1)[t_0,t_{j+1}]
+(st_2-t_2)[s,t_1] \\ & & + (t_2 s +1 -s)[t_0,t_1]
+[s,t_{j+2}] +t_2 s t_1[s,t_0]
-[s,t_2]
\end{array}$$ when $j \not\equiv -1 \mod e$, and $$\begin{array}{lcl}
\partial_3 [s,t_0,t_{-1}] &=& (s\delta_2 s -t_1t_0s +t_1s)[t_0,t_{-1}]
-t_{1}st_{0}[s,t_{-1}] + (1-t_2+st_2)[s,t_1] \\ & & + (1+t_2 s - s)[t_0,t_1]
+(t_1 - st_1)[s,t_0]
+t_2 s t_1[s,t_0]
-[s,t_2]
\end{array}$$ This means $d_3 [s,t_0,t_{-1}] = [t_0,t_{-1}]
-[s,t_{-1}] + [s,t_1] + [t_0,t_1]
+[s,t_0]
-[s,t_2]
$ and $d_3 [s,t_0,t_j] = [t_0,t_j]
-[s,t_j] -[t_0,t_{j+1}]
+ [t_0,t_1]
+[s,t_{j+2}] +[s,t_0]
-[s,t_2]
$ for $j \not\equiv 1 \mod e$.
$$\begin{array}{|c|c|c|lcl|}
\hline
\mathrm{Type} & \mathrm{Atoms} & \mathrm{Relations} & & & \mathrm{Differential\ of\ top\ cell} \\
\hline
M(e,e,2) & t_0 < \dots < t_r & t_i t_{i+1} = t_j t_{j+1} & \partial_2 [t_0,t_i] & = & t_{i+1} [t_i] + [t_{i+1}] - t_1[t_0] - [t_1] \\
\hline
M(e,e,2) \times A_1 & s < t_0 < \dots < t_r & t_i t_{i+1} = t_j t_{j+1} & \partial_2 [s,t_0,t_i,] & = &(s-1)[t_0,t_i] - t_{i+1} [s,t_i] \\
& & t_i s = st_i & & + & t_1 [s,t_0] - [s,t_{i+1}] + [s,t_1] \\
\hline
A_2 & s < t & sts = tst & \partial_2 [s,t] & =& (ts+1-s)[t] + (t-st-1)[s] \\
\hline
A_1 \times A_1 & s<u & su=us & \partial_2 [s,u] &=& (s-1)[u] - (u-1)[s] \\
\hline
A_3 & s<t<u & sts = tst & \partial_3 [s,t,u] &=& (u+stu-tu-1)[s,t] - [s,u] \\
& & su=us & & + & (su-u-s+1-tsu)t[s,u] \\
& & tut = utu & & + & (s-1-ts+uts)[t,u] \\
\hline
A_2 \times A_1 & s<t<u & tu = ut & \partial_3 [s,t,u] &= & (1-s+ts)[t,u] \\
& & su = us & & + & (t-1-st)[s,u] \\
& & sts = tst & & + & (u-1)[s,t]\\
\hline
A_1 \times A_2 & s < t < u & st=ts & \partial_3 [s,t,u] &=& (1+tu-u)[s,t]\\
& & su = us & & + &(t-1-ut)[s,u] \\
& & tut=utu & & + & (s-1)[t,u] \\
\hline
A_1 \times A_1 \times A_1 & s<t<u & su = us & \partial_3[s,t,u] &=& (1-t)[s,u] \\
& & st = ts & & + & (u-1) [s,t]\\
& & tu = ut & & + & (s-1)[t,u] \\
\hline
\end{array}$$
We now compute the second homology group, starting with $\Ker d_2$. Let $$v_i = [t_0,t_i] + [s,t_0] + [s,t_1] - [s,t_i] - [s,t_{i+1}] \in \Ker d_2$$ for $1 \leq i \leq e-1$. Let $K_1$ denote the submodule of $\Ker d_2$ spanned by the $v_i$. It is easy to show that $K_1$ is free on the $v_i$, and $K_1 = \Ker d_2$ for $r = 3$ ; if $r > 3$ we have $\Ker d_2 = K_1 \oplus K_2$ where $K_2$ is the free $\Z$-module with basis the $[s_k,t_i]$ for $k \geq 4$ and, if $r \geq 5$, the $[s_l,s_k]$ for $l \geq k+2$, $k \geq 3$.
Now decompose $\Z \mathcal{X}_3 = C_1 \oplus C_2$ where $C_1$ has for basis the $[s_3,t_0,t_i]$ and $C_2$ has for basis the other 3-cells. By the above computations we get $d_3(C_1) \subset K_1$, and $d_3(C_2) \subset K_2$. Thus $H_2(B,\Z) = (K_1/d_3(C_1)) \oplus (K_2 / d_3(C_2))$. We first compute $K_1/d_3(C_1)$. We have $$\begin{array}{lclr}
d_3[s_3,t_0,t_j] & = & v_j - v_{j+1} + v_1 & \mbox{ if } 0<j<e-1 \\
d_3[s_3,t_0,t_{e-1}] & = & v_{e-1} + v_{1} + v_1
\end{array}$$ We denote $u_{i} = [s_3,t_0,t_i]$ for $1 \leq i \leq e-1$, and let $w_i = u_i + u_{i+1} + \dots + u_{e-1}$. Then $d_3 w_i = v_i + (e-i) v_1$. Written on the $\Z$-basis $(w_i)$ and $(v_i)$, $d_3$ is in triangular form, and the only diagonal coefficient that differs from $1$ is $e$, since $d_3 w_1 = e v_1$. It follows that $K_1/d_3(C_1) \simeq \Z_e$. Since $H_2(B,\Z) = K_1/d_3(C_1)$ for $r = 3$, we can now assume $r \geq 4$.
First assume $r=4$. In $K_2/d_3(C_2)$ we have $2[s_4,t_i] \equiv 0$, because $d_3[s_4,s_3,t_i] = -2[s_4,t_i]$. Since $d_3[s_4,t_0,t_i] = -[s_4,t_i]+[s_4,t_0]-[s_4,t_{i+1}]+[s_4,t_1]$. we get $[s_4,t_i] + [s_4,t_{i+1}] \equiv [s_4,t_0]+[s_4,t_1]$ when $i>0$. In particular, $[s_4,t_i] + [s_4,t_{i+1}] \equiv [s_4,t_{i+1}] + [s_4,t_{i+2}]$ that is $[s_4,t_i] \equiv [s_4,t_{i+2}]$, at least if $0<i<e-1$. From $d_3[s_4,t_0,t_1]\equiv 0$ we deduce $[s_4,t_2] \equiv [s_4,t_0]$, and from $d_3[s_4,t_0,t_{e-1}] \equiv 0$ we deduce $[s_4,t_{e-1}] \equiv [s_4,t_1]$. Thus $[s_4,t_i] \equiv [s_4,t_{i+2}]$ for every $i$. When $e$ is odd, $K_2/d_3(C_3)$ is then spanned by the class of $[s_4,t_0]$. From the other relations one easily gets that this class is nonzero, and since $2[s_4,t_i] \equiv 0$ we get $K_2/d_3(C_2) \simeq \Z_2$. When $e$ is even, this quotient is spanned by the classes of $[s_4,t_0]$ and $[s_4,t_1]$, and we get similarly $K_2/d_3(C_2) \simeq \Z_2^2$.
We now assume $r \geq 5$. Then $d_3([s_5,s_3,t_i]) = [s_5,s_3]-[s_5,t_i]$ whence $a := [s_5,t_0] \equiv [s_5,t_i]$ for all $i$, regardless whether $e$ is even or odd. From $d_3[s_5,s_4,t_i] = [s_4,t_i] - [s_5,t_i]$ we get $[s_4,t_i] \equiv a$ and from $[s_{k+1},s_k,t_i] = [s_k,t_i] - [s_{k+1},t_i]$ we deduce by induction $[s_k,t_i] \equiv a$. The only remaining relation involving $a$ is then as before $2a \equiv 0$.
On the other hand, we have $[s_5,s_3] \equiv a$. Assume we have $[s_l,s_k] \equiv a$ for some $l,k$ with $l \geq k+2$. From $d_3[s_l,s_{k+1},s_k] = [s_l,s_{k+1}] - [s_l,s_k]$ for $l \geq k+3$ we get $[s_l,s_{k'}] \equiv a$ for all $l-2 \geq k' \geq 3$, and then that $[s_{l'},s_{k'}] \equiv a$ for all $l'-2 \geq k' \geq 3$. We thus get $K_2/d_3 C_2 \simeq \Z/2\Z$.
As a consequence, we get the following result.
\[H2BEER\] Let $B = {{\mathrm{B}}}(e,e,r)$ with $r \geq 3$ and $e \geq 2$.
- When $r = 3$, $H_2(B,\Z) \simeq \Z_e$
- When $r = 4$ and $e$ is odd, $H_2(B,\Z) \simeq \Z_e \times \Z_2 \simeq \Z_{2e}$
- When $r = 4$ and $e$ is even, $H_2(B,\Z) \simeq \Z_e \times \Z_2^2$
- When $r \geq 5$, $H_2(B,\Z) \simeq \Z_e \times \Z_2$
The case $r=2$ is when $W$ is a dihedral group, and this case is known by [@SALVETTI] : we have $H_2(B,\Z) = 0$ if $e$ is odd, $H_2(B,\Z) = \Z$ if $e$ is even.
Low-dimensional homology {#s:low}
========================
The second homology group {#ss:H2}
-------------------------
The computations above provide the second integral homology group $H_2(B,\Z)$. In the case of the finite group $W$, the group $H_2(W,\Z)$ can be identified with the Schur multiplier $H^2(W,\C^{\times})$, which is relevant for dealing with projective representations. We use the determination of the $H_2(B,\Z)$ to show a direct connection between the two groups $H^2(B,\C^{\times})$ and $H^2(W,\C^{\times})$. We first start with a lemma.
Let $W$ be an irreducible finite complex 2-reflection group, and $B$ the associated braid group. The inflation morphism $H^2(W,\C^{\times}) \to H^2(B,\C^{\times})$ is into.
The Hochschild-Serre exact sequence associated to $1 \to P \to B \to W \to 1$ is $$0 \to H^1(W,\C^{\times}) \to H^1(B,\C^{\times}) \to
H^1(P,\C^{\times})^W \to H^2(W,\C^{\times}) \to H^2(B,\C^{\times}).$$ Now $H^1(P,\C^{\times}) = \operatorname{\mathrm{Hom}}(P^{ab},\C^{\times})^W
= \operatorname{\mathrm{Hom}}((P^{ab})^W ,\C^{\times})$ and $H^1(B,\C^{\times})
= \operatorname{\mathrm{Hom}}(B^{ab},\C^{\times})$. Now $P^{ab} = H_1(P,\Z)$ and $B^{ab}$ (see [@BMR] thm. 2.17) are torsion-free, with $B^{ab}
\simeq \Z^r$ where $r$ denotes the number of hyperplane orbits, and $(P^{ab})^W$ can be identified with $(2 \Z)^r$. The induced map $\operatorname{\mathrm{Hom}}(\Z^r,\C^{\times})\to \operatorname{\mathrm{Hom}}((2 \Z)^r,\C^{\times}) $ is then onto, since $\C$ is algebraically closed. By the Hochschild-Serre exact sequence above the conclusion follows.
[**Remark.**]{} Another proof of the lemma can be given using projective representations instead of the Hochschild-Serre exact sequence. Let $\alpha \in Z^2(W,\C^{\times})$ with zero image in $H^2(B,\C^{\times})$, choose some projective representation $R$ of $W$ with 2-cocycle $\alpha$, and consider its lift $\tilde{R}$ to $B$. By assumption, it is linearizable into some linear representation $\tilde{S}$. Choosing one generator of the monodromy $\sigma_i$ in $X/W$ for each hyperplane orbit (see [@BMR] appendix A) we find that $\tilde{S}(\sigma_i^2) = {\lambda}_i \in
\C^{\times}$. By [@BMR] Theorem 2.17 there exists a morphism $\varphi : B \to \C^{\times}$ with $\varphi(\sigma_i) = 1/{\lambda}_i$, and then $\tilde{T} = \tilde{S} \circ \varphi$ is a linear representation of $B$ that factors through $W$ and linearizes $R$, thus proving that $\alpha$ has zero image in $H^2(W,\C^{\times})$.
It is known by work of Read [@READ] and van der Hout [@HOUT] that $H^2(W,\C^{\times}) \simeq H_2(W,\Z)$ is a free $\Z_2$-module in all cases. A nice property that follows from our computation is that the part of $H^2(B,\C^{\times})$ that comes from $H^2(W,\C^{\times})$ is exactly the 2-torsion (except for 2 exceptional cases). Indeed, since $H_1(B,\Z)$ is torsion-free and $\C^{\times}$ is divisible, by the Universal Coefficients Theorem we get $H^2(B,\C^{\times}) \simeq \operatorname{\mathrm{Hom}}(H_2 B, \C^{\times})$ and the proposition below is a consequence of our computation of $H_2(B,\Z)$ (see Table \[tableH2\] for the exceptional groups, Theorems \[H2B2EER\] and \[H2BEER\] for the $G(2e,e,r)$ and the $G(e,e,r)$) and of the works of Read and van der Hout on $W$. We recall their computation of $H_2(W,\Z)$ in Table \[tableH2\] for the exceptional groups, and the rank over $\Z_2$ for the other ones in Table \[schurgen\].
$$\begin{array}{|c|l|c|c|}
\hline
r & e& G(e,e,r) & G(2e,e,r) \\
\hline
2 & \mbox{odd} & 0 & 1 \\
& \mbox{even} & 1 & 2 \\
\hline
3 & \mbox{odd} & 0 & 2 \\
& \mbox{even} & 1 & 2 \\
\hline
4 & \mbox{odd} & 1 & 3 \\
& \mbox{even} & 3 & 4 \\
\hline
5 & \mbox{odd} & 1 & 3 \\
& \mbox{even} & 2 & 3 \\
\hline
\end{array}$$
Except for $W = G_{33}$ or $W = G_{34}$, $H^2(W,\C^{\times})$ coincides with the 2-torsion of $H^2(B,\C^{\times})$.
$$\begin{array}{|c|c|c||c|c|c|}
\hline
W & H_2 W & H_2 B & W & H_2 W & H_2 B \\
\hline
G_{12} & 0 & 0 & G_{30} & \Z_2 & \Z_2 \\
G_{13} & \Z_2 & \Z & G_{31} & \Z_2 & \Z_6 \\
G_{22} & 0 & 0 & G_{33} & 0 & \Z_6 \\
G_{23} & \Z_2 & \Z & G_{34} & 0 & \Z_6 \\
G_{24} & \Z_2 & \Z & G_{35} & \Z_2 & \Z_2 \\
G_{27} & \Z_2 & \Z_3 \times \Z & G_{36} & \Z_2 & \Z_2 \\
G_{28} & (\Z_2)^2 & \Z^2 & G_{37} & \Z_2 & \Z_2 \\
G_{29} & (\Z_2)^2 & \Z_2 \times \Z_4 & & & \\
\hline
\end{array}$$
First homology in the sign representation {#ss:sign}
-----------------------------------------
If $r = |\mathcal{A}/W|$ denotes the number of hyperplane classes, the abelianization $B_{ab}$ is isomorphic to $\Z^r$. There are thus $2^r-1$ nonzero morphisms $B \onto \Z_2$, which define $2^r-1$ subgroups of even braids. When $r = 1$, there is only one such morphism $\eps : B \to \Z_2$ and group $B^{(2)} = \Ker \eps$. We investigate here two abelian invariants of $B$ which are naturally attached to this group : the abelianization $B^{(2)}_{ab}$ of $B^{(2)}$ and $H_1(B,\Z_{\eps})$.
Let $u \in B \setminus B^{(2)}$. The group $H_1(B,\Z_{\eps})$ is isomorphic to the quotient of $B^{(2)}_{ab}$ by the relations $[u^2] \equiv 0$ and $[h^u] \equiv
-[h]$ for $h \in B^{(2)}_{ab}$, where $h^u = u^{-1} h u$.
We start from the bar resolution $C_2 \to C_1 \to C_0$, where $C_i$ is a free $\Z B$-module with basis the $[g_1,\dots,g_i]$ for $g_i \in B$, we have $d_1([g]) = (g-1)[\emptyset]$, $d_2([g_1,g_2]) = g_1 [g_2] - [g_1 g_2] + [g_1]$. Denoting $d_i^{\eps}$ the differential with coefficients in $\Z_{\eps}$ and $C_i^{\eps} = C_i \otimes_{\Z B} \Z_{\eps}$ with $\Z$-basis the $[g_1,\dots,g_i]$, we get that $\Ker d_1^{\eps}$ is the direct sum $\Z B^{(2)} \oplus I$ where $I = \{ \sum_{g \not\in B^{(2)}} x_g [g] \ | \ x_g \in \Z, \sum x_g = 0 \}$. Choose some $u \in B \setminus B^{(2)}$. The image of $d_2^{\eps}$ is spanned by the $[g_1g_2] - \eps(g_1)
[g_2] - [g_1]$. Among them we find
1. $[u^2] + [u] - [u] = [u^2]$
2. $[h_1 h_2] - [h_1] - [h_2]$, for $h_1,h_2$ in $B^{(2)}$
3. $[uh] - [u] + [h]$ for $h \in B^{(2)}$
4. $[h^u] + [h]$ for $h \in B^{(2)}$.
Indeed, the element (4) is the difference of two elements clearly in $\mathrm{Im} d_2^{\eps}$, $[hu] - [u] - [h]$ and $[u h^u] + [h^u] - [u]$, where $h^u = u^{-1} h u$, since $u h^u = hu$. By (3), and since $I$ is spanned by the $[hu] - [u]$ for $h \in B^{(2)}$, we see that $H_1(B,\Z_{\eps})$ is generated by the images of the $[h]$ for $h \in B^{(2)}$. It is easy to check that the relations of the form $d_2^{\eps}([g_1,g_2]) \equiv 0$ are consequences of (1-4), hence $H_1(B,\Z_{\eps})$ is the quotient of $ B^{(2)}_{ab}$ by the relations $(1)$ and $(4)$.
The computation of $B^{(2)}$ can be done for exceptional groups by using the Reidemeister-Schreier method (see [@MKS]) and the presentations of [@BMR] and [@bessismichel]. Note that they are known to provide presentations of $B$ for all groups but $G_{31}$, for which our results as well will be conjectural. We start from one of these standard presentation of $B$ by braided reflections $\sigma_1,\dots,\sigma_{n}$ and use $\{ 1, \sigma_1\}$ for Schreier transversal. Then generators for $B^{(2)}$ are given by $\sigma_1^2,\sigma_1 \sigma_2,
\sigma_1 \sigma_3, \dots,\sigma_1 \sigma_n$ and $\sigma_2 \sigma_1^{-1}$, $\sigma_3 \sigma_1^{-1},\dots ,\sigma_n \sigma_1^{-1}$. We then apply the Reidemeister-Schreier process and find a presentation of $B^{(2)}$ from the relations $R$, $\sigma_1 R \sigma_1^{-1}$ where $R$ runs among the relations for $B$. The presentations obtained for exceptional groups are tabulated in figure \[figpreseven\] (the column ‘ST’ refers to the Shephard-Todd number of the group). It is then easy to abelianize these relations. We choose $u = \sigma_1$.
In order to get $H_1(B,\Z_{\eps})$ from $B^{(2)}_{ab}$ we start by adding the relation $[\sigma_1^2] \equiv 0$. Note that $\sigma_1 (\sigma_i \sigma_1^{-1}) \sigma_1^{-1}
= (\sigma_1 \sigma_i) (\sigma_1^{-2})$ hence $-[\sigma_i \sigma_1^{-1}]
\equiv [\sigma_1 \sigma_i] - [\sigma_1^2] \equiv [\sigma_1 \sigma_i]$, and that $\sigma_1 (\sigma_1 \sigma_i)\sigma_1^{-1} = \sigma_1^2.
\sigma_i \sigma_1^{-1}$ hence $-[\sigma_1 \sigma_i] \equiv [\sigma_i \sigma_1^{-1}]$. The relations defining $H_1(B,\Z_{\eps})$ from $B^{(2)}_{ab}$ thus boil down to $-[\sigma_1 \sigma_i] \equiv [\sigma_i \sigma_1^{-1}]$ and $[\sigma_1^2] \equiv 0$.
ß
In order to get $H_1(B,\Z_{\eps})$ for the groups $G(*e,e,r)$, instead of using the complicated presentations of $B$ afforded by [@BMR], we use the semidirect product decomposition described in section \[s:iso\]. Recall that $B = \Z \ltimes \tilde{A}$ where we denote by $A$ the affine Artin group of type $\tilde{A}_{r-1}$. Then $A$ has Artin generators $\ss_1,\dots,\ss_r$ and the semidirect product is defined by $\tau \ss_i \tau^{-1} = \ss_{i+e}$ where addition is considered modulo $r$. From the split exact sequence $1 \to A \to B \to \Z \to 1$ we get the Hochschild-Serre short exact sequence $$0 = H_2(\Z,H_0(A,\Z_{\eps})) \to H_0(\Z,H_1(A,\Z_{\eps})) \to H_1(B,\Z_{\eps}) \to H_1(\Z,H_0(A,\Z_{\eps})) \to 0$$ with $H_2(\Z,H_0(A,\Z_{\eps})) = 0$ since $\Z$ has homological dimension 1. Since $A$ acts on $\Z_{\eps}$ through $\ss_i \mapsto -1$ we have $H_0(A,\Z_{\eps}) = \Z/2\Z = \Z_2$ ; since $\tau$ acts trivially on $H_0(A,\Z_{\eps})$ we thus get $H_1(\Z,H_0(A,\Z_{\eps})) \simeq H_1(\Z,\Z_2) \simeq \Z_2$. The short exact sequence thus boils down to $0 \to H_0(\Z,H_1(A,\Z_{\eps})) \to
H_1(B,\Z_{\eps}) \to \Z_2 \to 0$ and our task is reduced to computing $H_1(A,\Z_{\eps})$ while keeping track of the action of $\tau$.
In order to compute $H_1(A,\Z_{\eps})$ we apply the above process. Generators for $A^{(2)}$ are given by $u = \ss_1^2$, $x_i = \ss_1 \ss_i$ and $y_i = \ss_i \ss_1^{-1}$ for $2 \leq i \leq r$, and relations are given by rewriting $R$ and $\ss_1 R \ss_1^{-1}$ with $R$ running along the braid relations for $A$. These braid relations are the following (where $|j-i| \geq 2$ actually mean that $j,i$ are not connected in the braid diagram) $$\begin{array}{lllcl}
(R) & 1 \not\in \{ i, i+1\} & \ss_i \ss_{i+1} \ss_i \ss_{i+1}^{-1} \ss_i^{-1} \ss_{i+1}^{-1} & \rightsquigarrow &
y_i x_{i+1} y_i y_{i+1}^{-1} x_i^{-1} y_{i+1}^{-1}
\\
& |j-i| \geq 2, 1 \not\in \{ i,j \} & \ss_i \ss_j \ss_i^{-1} \ss_j^{-1} & \rightsquigarrow &
y_i x_j x_i^{-1}y_j^{-1}\\
& & \ss_1 \ss_2 \ss_1 \ss_2^{-1} \ss_1^{-1} \ss_2^{-1} & \rightsquigarrow &
x_2 y_2^{-1}u^{-1} y_2^{-1} \\
& & \ss_1 \ss_r \ss_1 \ss_r^{-1} \ss_1^{-1} \ss_r^{-1} & \rightsquigarrow &
x_r y_r^{-1} u^{-1}y_r^{-1}\\
& i \not\in \{2,r \} & \ss_1 \ss_i \ss_1^{-1} \ss_i^{-1} & \rightsquigarrow & x_i u^{-1} y_i^{-1}\\
(\ss_1 R \ss_1^{-1} ) & 1 \not\in \{ i, i+1\} & \ss_1\ss_i \ss_{i+1} \ss_i \ss_{i+1}^{-1} \ss_i^{-1} \ss_{i+1}^{-1}\ss_1^{-1} & \rightsquigarrow &
x_i y_{i+1}x_i x_{i+1}^{-1} y_i^{-1} x_{i+1}^{-1} \\
& |j-i| \geq 2, 1 \not\in \{ i,j \} & \ss_1\ss_i \ss_j \ss_i^{-1} \ss_j^{-1} \ss_1^{-1}& \rightsquigarrow &
x_i y_j y_i^{-1}x_j^{-1}
\\
& & \ss_1\ss_1 \ss_2 \ss_1 \ss_2^{-1} \ss_1^{-1} \ss_2^{-1}\ss_1^{-1} & \rightsquigarrow &
u y_2 u x_2^{-1} x_2^{-1}
\\
& & \ss_1\ss_1 \ss_r \ss_1 \ss_r^{-1} \ss_1^{-1} \ss_r^{-1}\ss_1^{-1} & \rightsquigarrow &
u y_ru x_r^{-1} x_r^{-1}
\\
& i \not\in \{2,r \} &\ss_1 \ss_1 \ss_i \ss_1^{-1} \ss_i^{-1}\ss_1^{-1} & \rightsquigarrow &
u y_i x_i^{-1}
\\
\end{array}$$ Abelianizing and dividing out by the relations $y_i = -x_i$ yields an abelian presentation for $H_1(A,\Z_{\eps})$ by generators $u, x_i$ for $2 \leq i \leq r$ and relations $$\begin{array}{lll}
1 \not\in \{ i, i+1\} & 3 x_{i+1} =3 x_i
\\
|j-i| \geq 2, 1 \not\in \{ i,j \} &
2 x_j =2 x_i \\
&
3x_2 =0 \\
&
3x_r =0\\
i \not\in \{2,r \} & 2 x_i = 0\\
\end{array}$$ Thus, for $r = 3$, $H_1(A,\Z_{\eps}) = <x_2,x_3 | 3x_2 = 3x_3 = 0> = \Z_3 x_2 \oplus \Z_3 x_3 \simeq \Z_3^2$, for $r = 4$, $$H_1(A,\Z_{\eps}) = < x_2,x_3,x_4 | 3x_2 = 3x_4 = 0, 2x_3 = 0 , 2x_2 =2x_4, 3x_3 = 3x_2=3x_4 >$$ hence $H_1(A,\Z_{\eps} ) = < x_2, x_4 | 3x_2 = 3 x_4 = 0 , x_2 = x_4 > = \Z_3 x_2 \simeq \Z_3$. When $r \geq 5$ , $H_1(A,\Z_{\eps})$ is generated by $x_2,\dots,x_r$, and we have $3 x_2 = 3 x_r= 0$. We have $2 < 3 < r-1 < r$. Then $2x_3 = 2x_{r-1} = 0$ but $0= 3x_2 = 3x_3$ and $0 = 3x_r = 3x_{r-1}$. It follows that $x_3 = 0$ and $x_{r-1} = 0$. Since $2x_3 = 2x_r$ and $2x_2 = 2x_{r-1}$ we get $x_2 = x_{r-1}$ and $x_3 = x_{r}$ hence $x_i = 0$ for all $i$ and $H_1(A,\Z_{\eps}) = 0$.
For $r \in \{ 3 , 4 \}$ it remains to compute the action of $\tau$ on $H_1(A,\Z_{\eps})$. We have $\tau.\ss_i = \ss_{i+e}$ hence $\tau.(\ss_1 \ss_i) = \ss_{1+e} \ss_{i+e}
= \ss_{1+e} \ss_1^{-1} \ss_1 \ss_{i+e}$. For $e \equiv 0 \mod r$ we have $\tau.x_i = x_i$ and $H_0(\Z,H_1(A,\Z_{\eps})) \simeq H_1(A,\Z_{\eps})$. For $r = 3$, $e \equiv 1 \mod 3$, $\tau.x_2 = \ss_2 \ss_3
\equiv y_2 + x_3 \equiv -x_2 + x_3 $ and $\tau.x_3 = \ss_2\ss_1
= \ss_2 \ss_1^{-1} \ss_1^2 \equiv -x_2$. It follows that $H_0(\Z,H_1(A,\Z_{\eps})) \simeq \Z_3$. For $r=4$, $e \equiv 1 \mod 4$, $\tau.x_2 = \ss_2 \ss_3 = \ss_2 \ss_1^{-1} \ss_1 \ss_3
\equiv y_2 + x_3 \equiv -x_2 + x_3 \equiv -x_2$ hence $H_0(\Z,H_1(A,\Z_{\eps}))=0$. For $r=4$, $e \equiv 2 \mod 4$, $\tau.x_2 = \ss_3 \ss_4 = \ss_3 \ss_1^{-1} \ss_1 \ss_4 \equiv x_2$ hence $H_0(\Z,H_1(A,\Z_{\eps}))= H_1(A,\Z_{\eps})$. Altogether, this yields
For $B = {{\mathrm{B}}}(*e,e,r)$, and $r \geq 3$, $$\begin{array}{ll}
H_1(B,\Z_{\eps}) \simeq \Z_2 & \mbox{ for } r \geq 5 \\
H_1(B,\Z_{\eps}) \simeq \Z_6 & \mbox{ for } r=4, e \equiv 0,2 \mod 4 \\
H_1(B,\Z_{\eps}) \simeq \Z_2 & \mbox{ for } r=4, e \equiv 1 \mod 4 \\
H_1(B,\Z_{\eps}) \simeq \Z_3 \oplus \Z_3 \oplus \Z_2 & \mbox{ for } r=3, e \equiv 0 \mod 4 \\
H_1(B,\Z_{\eps}) \simeq \Z_6 & \mbox{ for } r=3, e \equiv 1 \mod 4 \\
\end{array}$$
Finally, for groups of type $G(e,e,r)$, we use the Dehornoy-Lafont complex associated to the Corran-Picantin monoid. The 1-cells $[s]$ are mapped to $(\eps(s) - 1)[\emptyset]
= -2 [\emptyset]$, hence the 1-cycles are spanned by the $[s]-[t]$ for $s,t$ two atoms. We have $d_{\eps}[s_j,s_i] = 2(s_j-s_i)$ when $|j-i| \geq 2$, $d_{\eps}[t_0,t_i] = - t_i + t_{i+1} + t_0 - t_1$, $d_{\eps}[s_3,t_i] = 3t_i - 3 s_3$, $d_{\eps}[s_i,t_0] = 2(s_i-t_0)$ for $i \geq 4$, and $d_{\eps} [s_{i+1},s_i] = 3 (s_{i+1}-s_i)$. Since a basis of the 1-cycles is given by the $t_i - t_0$, $t_0-s_3$, $s_3-s_4$, …, $s_{r-1}-s_r$, $H_1(B,\Z_{\eps})$ is spanned by $t_1-t_0, t_0-s_3,
\dots,s_{r-1}-s_r$, each of these elements being annihilated by 3. From $d_{\eps}[s_i,t_0] = 2(s_i-t_0)$ for $i \geq 4$ we get that $s_4 - s_3 \equiv
s_3 - t_0$, from $d_{\eps}[s_5,s_3] = 2(s_5-s_3)$ we get $s_5 - s_4 \equiv s_4 - s_3$, and so on. Finally, from $$d_{\eps}[t_1,s_4] = 2(t_1-s_4) = 2(t_1 - t_0) + 2(t_0-s_3) + 2(s_3-s_4)
\equiv 2(t-1-t_0) +(t_0-s_3)$$ we get that $t_1 - t_0 \equiv t_0-s_3$. It follows that, for $r \geq 4$, $H_1(B,\Z_{\eps})$ is generated by $t_1-t_0$ hence $H_1(B,\Z_{\eps}) \simeq \Z_3$ ; for $r = 3$, it is generated by $t_1 - t_0$ and $t_0 - s_3$ and $H_1(B,\Z_{\eps}) \simeq \Z_3\oplus \Z_3$ ; it is generated by $t_1-t_0$ for $r =2$.
The case $e=1$ (that is, of the usual braid group) follows the same pattern. On the whole, we get the following.
For the groups ${{\mathrm{B}}}(e,e,r)$ with $e \geq 2$, $H_1(B,\Z_{\eps}) \simeq \Z_3$ if $r \geq 4$. If $r = 3$ then $H_1(B,\Z_{\eps}) \simeq \Z_3 \oplus \Z_3$. If $r=2$ then $H_1(B,\Z_{\eps}) \simeq \Z$. When $e=1$, we have $H_1(B,\Z_{\eps}) = 0$ for $r = 2$ or $r \geq 5$, and $H_1(B,\Z_{\eps})=
\Z_3$ if $r=3$ or $r = 4$.
$$\begin{array}{|c|l|}
ST & \mbox{Presentations for the group of even braids} \\
\hline
12 & vbu=awv,uaw=vbua,vbu=buaw,uaw=wvb\\
13 & buaw=awv,wvb=vbua,vbua=buaw,uawv=wvbu\\
22 & vbua=awvb,uawv=vbuaw,vbua=buaw,uawv=wvbu\\
23 & auaua=vv,vvv=uauau,bu=w,w=ub,bvb=awa,waw=vbv\\
24 & vv=auau,uaua=vv,awaw=bvbv,vbvb=wawa,w=bub,ubu=ww, \\
& auawvb=vbuaw,vvbuaw=uawvbu\\
27 & w=bub,ubu=ww,vv=auau,uaua=vv,awawa=bvbvb,vbvbv=wawaw, \\ &
bvwaua=waubv,waubvv=ubvwau\\
28 & aua=v,vv=uau,bu=w,w=ub,bvbv=awaw,wawa=vbvb,cu=x,x=uc,cv=ax, \\ &
cv=ax, xa=vc,cwc=bxb,xbx=wcw\\
29 & v=aua,uau=vv,axa=cvc,vcv=xax,bxb=cwc,wcw=xbx,awaw=bvbv, \\ &
vbvb=wawa,w=bu,ub=w,x=cu,uc=x,cwaxbv=bvcwax,xbvcwa=waxbvc\\
30 & auaua=vv,vvv=uauau,bu=w,w=ub,bvb=awa,waw=vbv,cu=x,x=uc, \\ &
cv=ax, xa=vc,cwc=bxb,xbx=wcw\\
31^* & x=cuc,ucu=xx,axa=cvc,vcv=xax,dwd=byb,yby=wdw,aya=dvd,vdv=yay, \\ &
vb=aw,uaw=vbu,aw=bua,vbu=wv,y=du,ud=y,bx=cw,wc=xb,dx=cy,\\
& yc=xd \\
33 & v=aua,uau=vv,bvb=awa,waw=vbv,cvc=axa,xax=vcv,cwc=bxb,xbx=wcw,\\ &
cyc=dxd,xdx=ycy,w=bu,ub=w,x=cu,uc=x,y=du,ud=y,ay=dv,vd=ya, \\ &
by=dw,wd=yb,cvbxaw=bxawcv,xawcvb=wcvbxa\\
34 & v=aua,uau=vv,bvb=awa,waw=vbv,cvc=axa,xax=vcv,cwc=bxb,xbx=wcw, \\ &
cyc=dxd,xdx=ycy,w=bu,ub=w,x=cu,uc=x,y=du,ud=y,ay=dv,vd=ya, \\ &
by=dw,wd=yb,dzd=eye,yey=zdz,z=eu,ue=z,az=ev,ve=za,bz=ew,\\ &
we=zb,cz=ex,xe=zc,cvbxaw=bxawcv,xawcvb=wcvbxa\\
35 & au=v,v=ua,bub=w,ww=ubu,bv=aw,wa=vb,cu=x,x=uc,cvc=axa, \\ &
xax=vcv,cwc=bxb,xbx=wcw,du=y,y=ud,dv=ay,ya=vd,dw=by,yb=wd, \\ &
dxd=cyc,ycy=xdx,eu=z,z=ue,ev=az,za=ve,ew=bz,zb=we,\\
& ex=cz,zc=xe,eye=dzd,zdz=yey \\
36 & au=v,v=ua,bub=w,ww=ubu,bv=aw,wa=vb,cu=x,x=uc,cvc=axa,\\ &
xax=vcv,cwc=bxb,xbx=wcw,du=y,y=ud,dv=ay,ya=vd,dw=by, \\ &
yb=wd,dxd=cyc,ycy=xdx,eu=z,z=ue,ev=az,za=ve,ew=bz,zb=we, \\ &
ex=cz,zc=xe,eye=dzd,zdz=yey,fu=x_2,x_2=uf,fv=ax_2,x_2a=vf,\\ &
fw=bx_2,x_2b=wf,fx=cx_2,x_2c=xf,fy=dx_2,x_2d=yf,fzf=ex_2e,x_2ex_2=zfz\\
37 & au=v,v=ua,bub=w,ww=ubu,bv=aw,wa=vb,cu=x,x=uc,cvc=axa, \\ &
xax=vcv,cwc=bxb,xbx=wcw,du=y,y=ud,dv=ay,ya=vd,dw=by,\\ &
yb=wd,dxd=cyc,ycy=xdx,eu=z,z=ue,ev=az,za=ve,ew=bz,zb=we,\\ &
ex=cz,zc=xe,eye=dzd,zdz=yey,fu=x_2, x_2=uf,fv=ax_2,x_2a=vf, \\ &
fw=bx_2,x_2b=wf,fx=cx_2,x_2c=xf,fy=dx_2,x_2d=yf,fzf=ex_2e,\\ &
x_2ex_2=zfz,gu=y_2,y_2=ug,gv=ay_2,y_2a=vg,gw=by_2,y_2b=wg,gx=cy_2,\\
& y_2c=xg,gy=dy_2,y_2d=yg,gz=ey_2,y_2e=zg,gx_2g=fy_2f,y_2fy_2=x_2gx_2 \\
\end{array}$$ $^*$ Provided that the presentation of $B$ suggested in [@BMR] for $G_{31}$ is correct.
$$\begin{array}{|c|ccccccc|}
& H_0 & H_1 & H_2 & H_3 & H_4 & H_5 & H_6 \\
\hline
G_{12} & \Z & \Z & 0 & & & & \\
G_{13} & \Z & \Z^2 & \Z & & & & \\
G_{22} & \Z & \Z & 0 & & & & \\
G_{24} & \Z & \Z & \Z & \Z & & & \\
G_{27} & \Z & \Z & \Z_3 \times \Z & \Z & & & \\
G_{29} & \Z & \Z & \Z_2 \times \Z_4 & \Z_2 \times \Z & \Z & & \\
G_{31} & \Z & \Z & \Z_6 & \Z & \Z & & \\
G_{33} & \Z & \Z & \Z_6 & \Z_6 & \Z & \Z & \\
G_{34} & \Z & \Z & \Z_6 & ? & ? & ? & ? \\
\end{array}$$
$$\begin{array}{|l|ccccccccc|}
& H_0 & H_1 & H_2 & H_3 & H_4 & H_5 & H_6 & H_7 & H_8\\
\hline
I_2(2m) & \Z & \Z^2 & \Z & & & & & & \\
I_2(2m+1) & \Z & \Z & 0 & & & & & & \\
H_3 = G_{23} & \Z & \Z & \Z & \Z & & & & & \\
H_4 = G_{30} & \Z & \Z & \Z_2 & \Z & \Z & & & & \\
F_4 = G_{28} & \Z & \Z^2 & \Z^2 & \Z^2 & \Z & & & & \\
E_6=G_{35} & \Z & \Z & \Z_2 & \Z_2 & \Z_6 & \Z_3 & 0 & & \\
E_7 = G_{36} & \Z & \Z & \Z_2 & \Z_2^2 & \Z_6^2 & \Z_3 \times \Z_6 & \Z & \Z & \\
E_8 = G_{37} & \Z & \Z & \Z_2 & \Z_2 & \Z_2 \times \Z_6 & \Z_3 \times \Z_6 & \Z_2 \times \Z_6 & \Z & \Z \\
\end{array}$$
$$\begin{array}{|c|c|c|}
ST & B^{(2)}_{ab} & H_1(B,\Z_{\eps}) \\
\hline
12 & \Z_3\times \Z & \Z_3 \\ 13 & \Z\times \Z & \Z_2 \\ 22 & \Z & 0 \\
23 & \Z & 0 \\ 24 & \Z & 0 \\ 27 & \Z & 0 \\ 28 & \Z\times \Z & \Z_2 \\
29 & \Z & 0 \\ 30 & \Z & 0 \\ 31^* & \Z & 0 \\ 33 & \Z & 0 \\ 34 & \Z & 0 \\
35 & \Z & 0 \\ 36 & \Z & 0 \\ 37 & \Z & 0 \\
\end{array}$$
|
---
author:
- |
*Xiao Xu and Qing Zhao*\
*Cornell University, Ithaca, NY.* *Email: {xx243, qz16}@cornell.edu* [^1]
bibliography:
- 'References.bib'
title: |
[**Distributed No-Regret Learning in\
Multi-Agent Systems**]{}\
---
Introduction
============
Game theory is a well-established tool for studying interactions among self-interested players. Under the assumption of complete information on the game composition at each player, the focal point of game-theoretic studies has been on the *Nash equilibrium* (NE) in analyzing game outcomes and predicting strategic behaviors of rational players.
The difficulty in obtaining complete information in real-world applications gives rise to the formulation of *repeated unknown games*, where each player has access to only local information such as his own actions and utilities, but is otherwise unaware of the game composition or even the existence of opponents. In such a setting, a rational player improves his decision-making through real-time interactions with the system and learns from past experiences [@cesa2006prediction]. The problem can be viewed through the lens of distributed online learning, where the central question is whether learning dynamics of distributed players lead to a system-level equilibrium in some sense. Studies in the past few decades have revealed intriguing connections between various notions of *no-regret* learning at each player and certain relaxed versions of NE at the system level [@cesa2006prediction; @young2004strategic].
While one-step closer to real-world systems, repeated unknown games, in their canonical forms, often adopt idealistic assumptions in terms of the stationarity of the player population and their utilities, availability of complete and perfect feedback, full rationality of players with unbounded cognition and computation capacity, and homogeneity among players in their knowledge of the game. Many emerging multi-agent systems, however, are inherently dynamic and heterogeneous, and inevitably limited in terms of available information and the cognition and computation capacity of the players. We give below two examples.
*Example: adversarial machine learning.* Security issues are at the forefront of machine learning and deep learning research, especially in safety-critical and risk-sensitive applications. The interaction between the defender and the attacker can be modeled as a two-player game. While the player population may be small, the game is highly complex in terms of the action space, utilities, feedback models, and the available knowledge each player has about the other. In particular, the attacker is characterized by its knowledge—how much information it has for designing attacks—and power—how often a successful attack can be launched. Both can be dynamically changing and adaptive to the strategies of the defender. A full spectrum of attacker profiles has been considered, ranging from the so-called black-box model to the white-box model (i.e., an omniscient attacker). The attack process is also dynamic, often exhibiting bursty behaviors following a successful intrusion or a system malfunction. The action space of the attacker can be equally diverse, including poisoning attacks and perturbation attacks. The former targets the training phase by injecting corrupted labels and examples for the purpose of embedding wrong decision rules into the machine learning algorithm. The latter targets the blind spots of a fully trained artificial intelligence using strategically perturbed instances that trigger wrong outputs, even when the perturbation is so minute as being indiscernible to humans. In terms of utilities, the attacker’s goal may be to compromise the integrity of the system (i.e., to evade detection by causing false negatives) or the availability of the system (to flood the system with false positives). See a comprehensive taxonomy of attacks against machine learning systems in [@barreno2010security].
*Example: transportation systems.* Route selection in urban transportation is a typical example of a non-cooperative game repeated over time. The game is characterized by a large population of players that is both dynamic and heterogeneous, with vehicles leaving and joining the system and utilities varying across players and over time. The envisioned large-scale adoption of autonomous vehicles will further diversify the traffic composition. Autonomous vehicles are significantly different from human drivers in terms of decision-making rationality, access to and usage of system-level knowledge, and memory and computation power. Bounded rationality is more evident in human drivers: they are likely to select a familiar route and inclined to settle for sufficing yet suboptimal options.
Complex multi-agent systems as in the above examples call for new game models, new concepts of regret, new design of distributed learning algorithms, and new techniques for analyzing game outcomes. We present in this article representative results on distributed no-regret learning in multi-agent systems. We start in Sec. \[sec:classic\] with a brief review of background knowledge on classical repeated unknown games. In the subsequent four sections, we explore four game characteristics—dynamicity, incomplete and imperfect feedback, bounded rationality, and heterogeneity—that challenge the classical game models. For each characteristic, we illuminate its implications and ramifications in game modeling, notions of regret, feasible game outcomes, and the design and analysis of distributed learning algorithms. Limited by our understanding of this expansive research field and constrained by the page limit, the coverage is inevitably incomplete. We hope the article nevertheless provides an informative glimpse of the current landscape of this field and stimulates future research interests.
Distributed Learning in Repeated Unknown Games {#sec:classic}
==============================================
In this section, we review key concepts in game theory and highlight classical results on distributed learning in repeated unknown games.
Static Games and Equilibria
---------------------------
An $N$-player *static game* is represented by a tuple $\mathcal{G}(\mathcal{N},\mathcal{A},u)$, where $\mathcal{N}=\{1,..., N\}$ is the set of players, $\mathcal{A}=\mathcal{A}_1\times\cdots\times\mathcal{A}_N$ the Cartesian product of each player’s action space $\mathcal{A}_i$, and $u=(u_1,...,u_N)$ the utility functions that capture the interaction among players. Specifically, the utility function $u_i$ of player $i$ encodes his preference towards an action. It is a mapping from the action profile ${\bf a}=(a_1,...,a_N)$ of all players to player $i$’s reward $u_i({\bf a})$.
A *Nash equilibrium* (NE) is an action profile ${\bf{a}}^*=(a^*_1,...,a^*_N)$ under which no player can increase his reward via a unilateral deviation. Specifically, $u_i({\bf a}^*)\ge u_i(a_i',{\bf a}^*_{-i})$ for all $i$ and all $a_i'\neq a_i^*$, where ${\bf a}^*_{-i}$ denotes the action profile after excluding player $i$. Due to the focus on deterministic actions (also called *pure strategies*), the resulting equilibrium is a *pure Nash equilibrium*. A player may also adopt a *mixed strategy*, which is a probability distribution $s_i$ over the action space. Correspondingly, a *mixed Nash equilibrium* is a product distribution ${\bf{s}}^*=s^*_1\times\cdots\times s_N^*$ under which the expected utility $\mathbb{E}_{{\bf a}^*\sim{\bf s}^*}[u_i({\bf a}^*)]$ for every player $i$ is no smaller than that under a unilateral deviation $s_i'\neq s_i^*$ in player $i$’s strategy. A game with a finite population and a finite action space has at least one mixed NE but may not have any pure NE [@nisan2007algorithmic].
NE is defined under the assumption that players adopt independent strategies (note the product form of ${\bf{s}}^*$). A more general equilibrium—*correlated equilibrium* (CE)—allows correlation across players’ strategies. We note that for equilibrium definitions introduced here, we focus on games with a finite action space. Specifically, a CE is a *joint* probability distribution $\bf s$ (not necessarily in a product form) satisfying $\mathbb{E}_{{\bf a}\sim{\bf s}}[u_i(a_i, {\bf a}_{-i})|a_i]\ge\mathbb{E}_{{\bf a}\sim{\bf s}}[u_i(a_i',{\bf a}_{-i})|a_i]$ for all $i$, $a_i$, and $a_i'$, where the expectation is over the joint strategy $\bf{s}$ conditioned on that the realized action of player $i$ is $a_i$. The concept of CE can be interpreted by introducing a mediator, who draws an outcome $\bf a$ from $\bf s$ and privately recommends action $a_i$ to player $i$. The equilibrium condition states that no player has the incentive to deviate from the outcome of the correlated draw from $\bf s$ after his part is revealed. CE can be further relaxed to the so-called *coarse correlated equilibrium* (CCE), which is a joint distribution $\bf s$ satisfying $\mathbb{E}_{{\bf a}\sim{\bf s}}[u_i({\bf a})]\ge\mathbb{E}_{{\bf a}\sim{\bf s}}[u_i(a_i',{\bf a}_{-i})]$ for all $i$ and all $a_i'\neq a_i$. Different from CE, CCE imposes an equilibrium condition that is realization independent.
The four types of equilibria exhibit a sequential inclusion relation as illustrated in Fig. \[equilibria\]. The more general set of strategy profiles (i.e., allowing correlated strategies across players) in CE and CCE may lead to higher expected utilities summed over all players. CE and CCE can also be computed via linear programming, while pure NE and mixed NE are hard to compute [@nisan2007algorithmic]. More importantly, CE and CCE are learnable through certain learning dynamics of players when a game is played repeatedly as discussed next. A caveat is that the set of CCE may contain highly non-rational strategies that choose only strictly dominated actions (actions that are suboptimal responses to all action profiles of the other players). See [@viossat2013no] for specific examples.
Repeated Unknown Games and No-Regret Learning
---------------------------------------------
A *repeated game* consists of $T$ repetitions of a static game (referred to as the stage game in this context)[^2]. In a repeated unknown game, after taking an action $a_i^t$ (potentially randomized according to a mixed strategy) in the $t$-th stage, player $i$ accrues a utility $u_i({\bf a}^t)$ and observes the entire utility vector $(u_i(a_i',{\bf a}^t_{-i}))_{a_i'\in\mathcal{A}_i}$ for all actions $a_i'$ in his action space (we focus on a finite action space here) against the action profile ${\bf a}_{-i}^t$ of the other players. The actions and utilities of the other players, however, are unknown and unobservable.
From a single player’s perspective, a repeated unknown game can be viewed as an online learning problem where the player chooses actions sequentially in time by learning from past experiences. A commonly adopted performance measure in online learning is *regret*, defined as the cumulative reward loss against a properly defined benchmark policy with hindsight vision and/or certain clairvoyant knowledge about the game. In other words, the benchmark policy defines the learning objective that an online algorithm aims to achieve over time. Different benchmark policies lead to different regret measures. Two classical regret notions are the *external regret* and the *internal regret* as detailed below.
Let $\pi_i$ denote the online learning algorithm adopted by player $i$. For a fixed action sequence $\{{\bf a}_{-i}^t\}_{t=1}^{T}$ of the other players, the external regret of $\pi_i$ is defined as: $$\max_{a'\in\mathcal{A}_i}\mathbb{E}_{\pi_i}\left[ \sum_{t=1}^{T}(u_i(a',{\bf a}_{-i}^t)-u_i({\bf a}^t))\right],$$ where $\mathbb{E}_{\pi_i}$ denotes the expectation over the random action process $\{a_i^t\}_{t=1}^{T}$ induced by $\pi_i$. In other words, the benchmark policy in the external regret chooses the best fixed response to the other players’ actions in hindsight. The internal regret of $\pi_i$ is defined as: $$\label{Rin}
\max_{a,a'\in\mathcal{A}_i}\mathbb{E}_{\pi_i}\left[\sum_{t=1}^{T}\mathbb{I}\{a_i^t=a\}(u_i(a',{\bf a}_{-i}^t)-u_i({\bf a}^t))\right],$$ where $\mathbb{I}\{\cdot\}$ is the indicator function. In this definition, the benchmark policy is the best hindsight *modification* of $\pi_i$ by swapping a single action with another throughout all stages.
An online learning algorithm $\pi_i$ is said to achieve the *no-regret* condition if against all action sequences $\{{\bf a}_{-i}^t\}_{t=1}^{T}$ of the other players, the cumulative regret has a sublinear growth rate with the time horizon $T$. In other words, $\pi_i$ offers, asymptotically as $T\to\infty$, the same average reward per stage as the specific benchmark policy adopted in the corresponding regret measure. No-regret learning is also referred to as Hannan consistency due to the original work [@hannan1957approximation] as well as [@blackwell1956analog].
It is clear that the significance of no-regret learning depends on the adopted benchmark policy which the learning algorithm is measured against. A benchmark policy with stronger performance leads to a stronger notion of regret. In particular, the internal regret is a stronger notion than the external regret: no-regret learning under the former implies no-regret learning under the latter, but not vice versa [@stoltz2005internal].
A number of no-regret learning algorithms exist in the literature. Representative algorithms achieving no-external-regret learning include *Multiplicative Weights* (MW) (also known as the Hedge algorithm) and Follow the Perturbed Leader [@cesa2006prediction]. Both are randomized policies, as randomization is necessary for achieving no-regret learning in an adversarial setting with general reward functions [@cesa2006prediction]. In particular, under the MW algorithm, each player maintains a weight $W_a(t)$ of each action $a$ at every stage $t$ based on past rewards: $W_a(t)=e^{\epsilon \sum_{\tau=1}^{t}r_a(\tau)}=W_a(t-1)e^{\epsilon r_a(t)}$, where $r_a(\tau)$ is the reward received under $a$ at stage $\tau$ and $\epsilon>0$ is the learning rate. The probability of choosing $a$ in the next stage is proportional to its weight given by $\frac{W_a(t)}{\sum_{a'}W_{a'}(t)}$.
For no-internal-regret learning, a representative algorithm is *Regret Matching* [@hart2000simple]. Let $R^{a\to a'}(t)=\frac{1}{t}\sum_{\tau=1}^{t}\mathbb{I}\{a_i^{\tau}=a\}(u_i(a',{\bf a}_{-i}^{\tau})-u_i({\bf a}^{\tau}))$ denote the average gain per play by switching from action $a$ to an alternative $a'$ in the past $t$ plays. In the ($t+1$)-th stage, the probability of switching from the previous action $a_t$ to an alternative $a'$ is given by $\frac{1}{\epsilon}R^{a_{t}\to a'}(t)$, where $\epsilon>0$ is a normalization parameter chosen to ensure a positive probability of staying with action $a_t$. Regret Matching also offers no-external-regret learning by setting the probability of selecting an action $a$ at the ($t+1$)-th stage to the normalized average gain per play from playing action $a$ throughout the past $t$ plays, i.e., $\frac{R^a(t)}{\sum_{a'}R^{a'}(t)}$, where $R^a(t)=\frac{1}{t}\sum_{\tau=1}^{t}(u_i(a,{\bf a}_{-i}^{\tau})-u_i({\bf a}^{\tau}))$ [@hart2000simple].
System-Level Performance under No-Regret Learning
-------------------------------------------------
Regret captures the learning objective of an individual player. At the system level, it is desirable to know whether the dynamical behaviors of distributed players converge to an equilibrium in some sense and whether the self-interested regret minimization promises a certain level of optimality in terms of social welfare.
For the first question, it has been shown that if every player adopts a no-external-regret learning algorithm, the empirical distribution of the sequence of actions taken by all players converges to the set of CCE of the stage game [@roughgarden2015intrinsic]. No-regret learning under the internal regret measure guarantees convergence to the more restrictive set of CE [@hart2000simple]. Such convergence results are, however, in terms of the empirical frequency of the players’ actions rather than the actual sequence of plays. The convergence is also only to the set of equilibria, rather than to an equilibrium in the corresponding set. In fact, by treating learning in games as a dynamical system, recent studies have shown that in the continuous-time setting, the actual plays under no-regret learning algorithms (such as Follow the Regularized Leader) may exhibit cycles rather than convergence [@mertikopoulos2018cycles]. In the discrete-time setting, it has been shown that in zero-sum games, the actual plays under the MW algorithm (starting from a non-equilibrium initial strategy) diverges from every fully mixed NE [@bailey2018multiplicative]. For games with special structures (e.g., potential games [@heliou2017learning] with a finite action space and bilinear smooth games [@gidel2019negative] with a continuum of actions), however, stronger results on the convergence of the actual plays to the more restrictive set of (mixed) NE have been established.
In addition to the convergence of learning dynamics, the social welfare resulting from the self-interested learning of individual players is of great interest in many applications. In (known) static games, the loss in social welfare $W({\bf s})=\mathbb{E}_{{\bf{a}}\sim{\bf s}}\left[\sum_{i=1}^{N}u_i({\bf a})\right]$ (i.e., the system-level utility under a strategy profile $\bf s$) due to the self-interested behaviors of players is quantified by the *price of anarchy* (POA). It is defined as the ratio of the optimal social welfare $\textrm{OPT}=\max_{{\bf s}}W({\bf s})$ among all strategies to the smallest social welfare in the set of mixed NE. For repeated unknown games, a corresponding concept, *price of total anarchy* (POTA), is defined as: $$\label{POTA}
\frac{\textrm{OPT}}{\min_{{\bf s}^1,...,{\bf s}^T}\frac{1}{T}\sum_{t=1}^{T}{W({\bf s}^t)}},$$ where ${\bf s}^1,...,{\bf s}^T$ is the sequence of strategy profiles in the no-regret dynamics of all players. It has been shown that in games with special structures (e.g., valid games and congestion games), no-regret learning guarantees a POTA that converges to the POA of the stage game even though the sequence of actual plays may not converge to a (mixed) NE [@blum2008regret]. The convergence of the POTA to the POA of the stage game implies that no-regret learning can fully negate the impact of the unknown nature of the game on social welfare. The result was later extended in [@roughgarden2015intrinsic] to a general class of games referred to as *smooth games* (which includes valid games and congestion games as special cases). To achieve higher social welfare, cooperation among players is necessary. For example, if every player agrees to follow a learning algorithm designed specifically for optimizing the system-level performance, the optimal action profile will be selected a high percentage of time [@marden2014achieving].
Dynamicity {#IS1}
==========
In a dynamic repeated game, the stage game is time-varying. The dynamicity may be in any of the three elements of the game composition: the set of players, the action space, and the utility functions[^3].
Notions of Regret
-----------------
Dynamic unknown games call for new notions of regret to provide meaningful performance measures for distributed online learning algorithms. Specifically, the benchmark policy of a fixed single best action used in the external regret and that of a fixed single best action modification used in the internal regret can be highly suboptimal in dynamic games. As a result, achieving no-regret learning under thus-defined regret measures can no longer serve as a stamp for good performance.
A rather immediate extension of the external regret is to consider every interval of the learning horizon and measure the cumulative loss against a single best action in hindsight that is specific to each interval. This leads to the notion of *adaptive regret*, under which no-regret learning requires a sublinear growth of the cumulative reward loss in every interval as the interval length tends to infinity. The adaptive regret is particularly suitable for piecewise stationary systems where changes can be abrupt but infrequent. Classical learning algorithms such as MW can be extended to achieve no-adaptive-regret [@luo2015achieving]. The key issue in algorithm design is a mechanism to discount experiences from the distant past.
Another extension of the external regret is the so-called *dynamic regret*, in which the benchmark policy can be an arbitrary sequence of actions, as opposed to a fixed action throughout an interval of growing length. Achieving diminishing reward loss against all sequences of actions is, however, unattainable. Constraints on either the benchmark action sequence or the reward functions are necessary for defining a meaningful measure. On the variation of the benchmark action sequence, a commonly adopted constraint in the setting with finite actions is that the benchmark sequence is piecewise-stationary with at most $K$ changes (the thus-defined regret is also referred to as the *K-shifting regret*). In this case, the no-adaptive-regret condition directly implies no-dynamic-regret [@luo2015achieving]. With a continuum of actions, the constraint is often imposed on the cumulative distance between every two consecutive actions in the sequence, i.e., $V_T(\{a^{t}\}_{t=1}^{T})=\sum_{t=1}^{T-1}||a^{t+1}-a^t||$. It has been shown that if the benchmark sequence is slow-varying, i.e., $V_T=o(T)$, no-dynamic-regret is achievable through well-designed restart procedures [@duvocelle2018learning]. The variation constraint can also be applied to the reward functions. A typical example with a continuum of actions is the sublinear “variation budget” assumption. Specifically, the cumulative variation between the reward functions in two consecutive stage games grows sublinearly in $T$, i.e., $\sum_{t=1}^{T-1}\sup_a|u_{t+1}(a)-u_{t}(a)|=o(T)$. Similar constraints can be imposed on the gradient $\nabla u_t(a)$ of the utility function and with the variation measured by the $L_p$-norm. See [@mokhtari2016online] and references therein for details and corresponding no-regret learning algorithms.
The external regret and its extensions are measured against an alternative strategy of a single player. A new notion of regret—*Nash equilibrium regret*—considers a benchmark policy that is jointly determined by the strategies of all players [@pmlr-v97-cardoso19a]. Consider a repeated game with time-varying utility functions $\{u_i^t\}_{t=1}^{T}$ for each player $i$. Let $\bar{u}_i=\frac{1}{T}\sum_{t=1}^{T}u_i^t$ be the average utility function and ${\bf s}^*$ the mixed NE of the static game defined by the average utility functions $\bar{u}=(\bar{u}_1,...,\bar{u}_N)$. The NE regret of player $i$ following a policy $\pi_i$ is then given by $\mathbb{E}_{\pi}[\sum_{t=1}^{T}u_i^t({\bf{a}}^t)]-T\mathbb{E}_{{\bf a}^*\sim{\bf s}^*}[\bar{u}_i({\bf{a}}^*)]$, where ${\bf a}^t$ is the action profile selected by the policies $\pi=(\pi_1,...,\pi_N)$ of all players at stage $t$. No-regret learning under the NE regret ensures that each player’s average reward asymptotically matches that promised by the mixed NE under the average utility functions. A centralized learning algorithm achieving no-NE-regret was developed in [@pmlr-v97-cardoso19a] for repeated two-player zero-sum games with arbitrarily varying utility functions. Achieving no-regret learning under the measure of NE regret in a distributed setting, however, remains open.
System-Level Performance
------------------------
The two key measures—convergence to equilibria and POTA—for system-level performance also need to be modified to take into account game dynamics. The time-varying sequence $\{\mathcal{G}^t\}_{t=1}^{T}$ of stage games defines a sequence of equilibria and a sequence $\{\textrm{OPT}^t\}_{t=1}^{T}$ of optimal social welfare. The desired relation between no-regret learning dynamics at individual players and the system-level equilibria is thus in terms of tracking rather than converging. For the definition of POTA, the optimal social welfare in the numerator in (\[POTA\]) needs to be replaced with the *average* optimal social welfare $\frac{1}{T}\sum_{t=1}^{T}\textrm{OPT}^t$.
An online learning algorithm is said to successfully track the sequence of (mixed) NE in a dynamic game if the average distance between the sequence of (mixed) action profiles resulting from the algorithm and the sequence of (mixed) NE vanishes as $T$ tends to infinity. A representative study in [@duvocelle2018learning] considers a game with a continuum of actions and dynamicity manifesting only in the utility functions. Under the assumptions that the sequence of NE is slow-varying and the utility functions are monotonic, it was shown that learning algorithms with sublinear dynamic regret successfully track the sequence of NE. The monotonicity of the utility functions plays a key role in the analysis: it translates the closeness between the learning dynamics and the NE in terms of the cumulative reward (as in the regret measure) to the closeness in terms of their distance in the action space (the concern of the tracking outcome).
The performance of no-regret learning in terms of social welfare was studied in [@lykouris2016learning] for games with a dynamic population of players. Specifically, in each stage, each player may independently exit with a fixed probability and is subsequently replaced with a new player with a potentially different utility function (the population size is therefore fixed and the player set is a stationary process over time). For structural games such as first-price auctions, bandwidth allocation, and congestion games, the relation between no-adaptive-regret learning and the average optimal social welfare was examined.
Game dynamics can be in diverse forms. There lacks a holistic understanding on the matching between regret notions and the underlying dynamics of the game. Different forms of game dynamics demand different benchmark policies in order to arrive at a meaningful regret measure that lends significance to the stamp of “no-regret learning” yet at the same time is attainable. Viewing from a different angle, one may pose the fundamental question on what kinds of game dynamics are tamable through distributed online learning and make no-regret learning and approximately optimal social welfare feasible.
Incomplete and Imperfect Feedback {#IS2}
=================================
Learning and adaptation rely on feedback. Quality of the feedback in terms of completeness and accuracy thus has significant implications in no-regret learning. We explore this issue in this section.
Incomplete Feedback
-------------------
Incomplete feedback stands in contrast to full-information feedback where utilities of all actions a player could have taken are observed in each stage. Incompleteness can be spatial across the action space or temporal across decision stages. In the former case, a commonly studied model is the so-called *bandit feedback*, where only the utility of the chosen action is revealed. In the latter, the feedback model is referred to as *lossy feedback* where there are decision stages with no feedback [@zhou2018learning]. One can easily envision a more general model compounding bandit feedback with lossy feedback. Studies on this general model are lacking in the literature.
The term “bandit feedback” has its roots in the classical problem of *multi-armed bandit* [@zhao2019multi]. The name of the problem comes from likening an archetypical single-player online learning problem to playing a multi-armed slot machine (known as a bandit for its ability of emptying the player’s pocket). Each arm, when pulled, generates rewards according to an unknown stochastic model or in an adversarial fashion. Only the reward of the chosen arm is revealed after each play. Due to the incomplete feedback, the player faces the tradeoff between exploration (to gather information from less explored arms) and exploitation (to maximize immediate reward by favoring arms with a good reward history).
In a multi-player game setting with bandit feedback, no-regret learning from an individual player’s perspective can be cast as a single-player *non-stochastic*/*adversarial* bandit model where the payoff of each arm/action is adversarially chosen and aggregates the interaction with the other players in the game. The concept of external regret in the game setting corresponds to the weak regret in the adversarial bandit model [@auer2002nonstochastic], which adopts the best single-arm policy in hindsight as the benchmark. The MW algorithm was modified in [@auer2002nonstochastic] to handle the change of the feedback model from full-information to bandit. Specifically, the weight $W_a(t)$ of action $a$ at time $t$ is updated as $W_{a}(t)=W_{a}(t-1)e^{\epsilon r_{a}(t)/p_{a}(t)}$ where $p_{a}(t)$ is the probability of selecting action $a$ at time $t$ and $r_a(t)=0$ if $a$ is unselected. Dividing the observed reward by the corresponding probability of the chosen action ensures the unbiasedness of the observation. Quite intuitively, the price for not observing the rewards of all actions is the degradation of the regret order in the size of the action space, i.e., from $\Theta(\sqrt{\log(|\mathcal{A}|)T})$ in the full-information setting [@cesa2006prediction], to $\Theta(\sqrt{|\mathcal{A}|T})$ in the bandit setting [@audibert2009minimax].
The multi-player bandit problem explicitly models the existence of $N$ players competing for $M$ ($M>N$) arms [@liu2010distributed]. Originally motivated by applications in wireless communication networks where distributed users compete for access to multiple channels, this specific game model is characterized by a special form of interaction among players: a collision occurs when multiple players select the same arm, which results in utility loss. The objective of this distributed learning problem is to minimize the *system-level regret* over all players against the optimal centralized (hence collision-free) allocation of the players to the best set of arms [@liu2010distributed]. In addition to the exploration-exploitation tradeoff in the single-player setting, this distributed learning problem under a system-level objective also faces the tradeoff between selecting a good arm and avoiding colliding with competing players. A number of distributed learning algorithms have been developed to achieve a sublinear system-level regret with respect to $T$ [@liu2010distributed]. Recent extensions of the multi-player bandit problem further consider the setting where each arm offers different payoffs across players [@bistritz2018distributed].
The multi-player bandit problem is a special game model in that the players have identical action space and their interaction is only in the form of “collisions” when choosing the same action. In a general game setting, the impact of incomplete feedback on no-regret learning and system-level performance is largely open. One quantitative measure of the impact is the regret order with respect to the size of the action space. As mentioned above, bandit feedback results in an additional $\sqrt{|\mathcal{A}|}$ term in the regret order, which can be significant when the action space is large. Recent work [@cesa2019delay; @bar2019individual] has shown that local communications among neighboring players in a network setting can mitigate the negative impact of bandit feedback on the regret order in $|\mathcal{A}|$. In terms of the impact on the system-level performance, it has been shown under a game model with a continuum of actions that bandit feedback degrades the convergence rate of the learning dynamics to equilibria [@bravo2018bandit].
Imperfect Feedback
------------------
Imperfect feedback refers to the inaccuracy of the observed utilities in revealing the quality of the selected actions. Recall that mixed strategies are necessary for achieving no-regret learning in the adversarial setting. The quality of a mixed strategy is characterized by the expected utility where the expectation is taken over the randomness of strategies of all players. Referred to as *expected feedback*, the feedback model assuming observations on the expected utility, however, can be unrealistic. A more commonly adopted feedback model is the *realized feedback* where only the utility of the realized action profile is revealed. The realized feedback can be viewed as a noisy unbiased estimate of the expected feedback where the noise is due to the randomness of players’ strategies.
The so-called *noisy feedback* assumes a different source of noise: it comes from the external environment and is additive to either the observed utility vectors in the so-called semi-bandit feedback [@heliou2017learning] with a finite action space, or the gradient of the utility functions in the first-order feedback [@mertikopoulos2019learning] with a continuum of actions. Under the assumptions of unbiasedness and bounded variance, the issue of the additive noise can be addressed by rather standard estimation techniques and analysis. A more challenging setting is to consider non-stochastic noise due to adversarial attacks, especially in applications such as adversarial machine learning. This problem was recently studied in the single-player setting [@jun2018adversarial]. Studies in the multi-agent setting are still lacking.
Bounded Rationality {#IS3}
===================
The concept of *bounded rationality* was first introduced in economics [@simon1955behavioral] to provide more realistic models than the often adopted perfect rationality that assumes the decision-making of players is the result of a full optimization of their utilities. In reality, players often take reasoning shortcuts that may lead to suboptimal decisions. Such reasoning shortcuts may be a result of limited cognition of human minds or necessitated by the available computation time and power relative to the complexity of action optimization.
Cognitive limitations include the limited ability in anticipating other decision-makers’ strategic responses and certain psychological factors that interfere with the valuation of options. Various models exist for capturing the limitations in the players’ valuation of options. For example, a player may be myopic, focusing only on the short-term reward [@gabaix2005bounded]. Even with forward-thinking, a player may settle for suboptimal actions perceived as acceptable by the player [@simon1955behavioral]. The limitation in a player’s ability to anticipate other players’ strategies can be modeled through a *cognitive hierarchy* by grouping players according to their cognitive abilities and characterizing them in an iterative fashion. Specifically, players with the lowest level of cognitive ability are grouped as the level-$0$ players who make decisions randomly. Level-$k$ ($k>0$) players are then defined iteratively as those who assume they are playing against lower-level players and anticipate the opponents’ strategies accordingly. Recent work draws an interesting connection between the cognitive hierarchy model and the *Optimistic Mirror Descent* (OMD) algorithm for solving the saddle point problem with applications in generative adversarial networks [@daskalakis2018training]. The saddle-point problem can be viewed as a two-player zero-sum game with a continuum of actions. The solutions to the problem correspond to the set of NE. It has been shown that the OMD algorithm guarantees a converging system dynamic to an NE in terms of the actual plays while Gradient Descent (GD) may lead to cycles [@daskalakis2018training]. In the language of cognitive hierarchy, players adopting GD can be regarded as level-0 thinkers in the sense that they do not anticipate the strategies of their opponents. Players adopting OMD are level-1 thinkers since they take advantage of the fact that their opponents are taking similar gradient methods, which will not lead to abrupt gradient changes between two consecutive stages [@daskalakis2018training]. Consequently, an extra gradient update is applied in OMD to accelerate learning.
Besides cognitive limitations, players are also constrained in terms of physical resources such as memory and computation power. Acquiring, storing, and processing all relevant information for decision-making may be infeasible, especially in complex systems with a large action space. For example, players may only choose from strategies with bounded complexity [@scarsini2012repeated], or use only recent observations in decision-making due to memory constraints [@chen2017k].
While models for bounded rationality abound in economics, political science, and other related disciplines, incorporating such models into distributed online learning is still in its infancy. A holistic understanding on the implications of bounded rationality in distributed online learning is yet to be gained. An intriguing aspect of the problem is that bounded rationality may not necessarily imply degraded performance. For example, in dynamic games, bounded memory of past experiences may have little effect since no-regret learning dictates that the distant past be forgotten (see discussions in Sec. 3).
Heterogeneity {#IS4}
=============
The heterogeneity of complex multi-agent systems characterizes the asymmetry across players in three aspects: the available information and knowledge about the system, available actions, and the level of adaptivity to opponents’ strategies. In the example of mixed traffic in urban transportation, autonomous vehicles, while likely to have greater computation power for solving complex decision problems, may have to obey an additional set of regulations on available actions.
In adversarial machine learning, in addition to the asymmetry on the knowledge and power, the attacker and the defender may also have different levels of real-time adaptivity to the other player’s strategy. Classical regret notions such as the external regret that assumes fixed actions of the other players, while applicable to *oblivious* attackers, are no longer valid under *adaptive* attacks. A partial solution is to adopt a new notion of *policy regret* defined against an adaptive adversary who assigns reward vectors based on previous actions of the player [@arora2012online]. Specifically, let $u_t(\cdot;a_{1:t-1})$ denote the player’s reward function determined by the adversary at time $t$, given the sequence of actions $a_{1:t-1}$ taken by the player in the past. The policy regret with reward functions $\{u_t\}_{t=1}^{T}$ is defined as $$\max_{a\in\mathcal{A}}\mathbb{E}\left[\sum_{t=1}^{T}u_t(a; \{a,...,a\})-\sum_{t=1}^{T}u_t(a_t; a_{1:t-1})\right],$$ where $u_t(\cdot;\{a,...,a\})$ denotes the reward function determined by the adversary if the player took actions $\{a,...,a\}$ in the past. The $m$-memory policy regret is defined by assuming that the reward function depends only on the past $m$ actions of the player.
The difference between the external regret and the policy regret may not be crucial if the adversary and the player have homogeneous objectives (e.g., mixed traffic in transportation systems). It has been shown that there exists a wide class of algorithms that can ensure no-regret learning under both regret definitions, as long as the adversary is also using such an algorithm [@arora2018policy]. In applications such as adversarial machine learning where the adversary may be a malicious opponent, the two notions of regret are incompatible: there exists an $m$-memory adaptive adversary that can make any action sequence of the player with sublinear regret in one notion suffer from linear regret in the other [@arora2018policy]. A general technique for developing no-policy-regret algorithms in the single-player setting was proposed in [@arora2012online]. In terms of the system-level performance, it was shown in two-player games that no-policy-regret learning guarantees convergence of the system dynamic to a new notion of equilibrium called *policy equilibrium* [@arora2018policy]. However, the understanding of policy equilibrium is limited. In games with more than two players, even the definition of policy equilibrium is unclear.
[^1]: This work was supported by the National Science Foundation under Grant CCF-1815559.
[^2]: In a general definition of a repeated game [@laraki2015advances], the stage game is parameterized by a state, which affects the utility function. Two basic settings exist in the literature: (i) the state evolves over time following a Markov transition rule (the state in the next stage depends on the state and actions in the current stage); (ii) the state is fixed throughout all stages. We focus on the second setting in discussing classical results on repeated games.
[^3]: Note that the general definition of repeated games in [@laraki2015advances] includes dynamicity in the utility function, as the state parameter may evolve over time following a Markov transition rule. The dynamic repeated game discussed in this section differs from the general repeated game in two aspects: (i) the set of players and the action space can also be time-varying; (ii) the utility functions are in general independent across stages.
|
---
abstract: 'Coecke, Sadrzadeh, and Clark [@LambekFest] developed a compositional model of meaning for distributional semantics, in which each word in a sentence has a meaning vector and the distributional meaning of the sentence is a function of the tensor products of the word vectors. Abstractly speaking, this function is the morphism corresponding to the grammatical structure of the sentence in the category of finite dimensional vector spaces. In this paper, we provide a concrete method for implementing this linear meaning map, by constructing a corpus-based vector space for the type of sentence. Our construction method is based on structured vector spaces whereby meaning vectors of all sentences, regardless of their grammatical structure, live in the same vector space. Our proposed sentence space is the tensor product of two noun spaces, in which the basis vectors are pairs of words each augmented with a grammatical role. This enables us to compare meanings of sentences by simply taking the inner product of their vectors.'
author:
- |
Edward Grefenstette$^\ast$, Mehrnoosh Sadrzadeh$^\ast$, Stephen Clark$^\dagger$, Bob Coecke$^\ast$, Stephen Pulman$^\ast$\
$^\ast$Oxford University Computing Laboratory, $^\dagger$University of Cambridge Computer Laboratory\
`firstname.lastname@comlab.ox.ac.uk`, `stephen.clark@cl.cam.ac.uk`
bibliography:
- 'iWCS.bib'
title: '**Concrete Sentence Spaces for Compositional Distributional Models of Meaning**'
---
Background
==========
Coecke, Sadrzadeh, and Clark [@LambekFest] develop a mathematical framework for a compositional distributional model of meaning, based on the intuition that *syntactic analysis guides the semantic vector composition*. The setting consists of two parts: a formalism for a type-logical syntax and a formalism for vector space semantics. Each word is assigned a grammatical type and a meaning vector in the space corresponding to its type. The meaning of a sentence is obtained by applying the function corresponding to the grammatical structure of the sentence to the tensor product of the meanings of the words in the sentence. Based on the type-logic used, some words will have atomic types and some compound function types. The compound types live in a tensor space where the vectors are weighted sums (i.e. superpositions) of the pairs of bases from each space. Compound types are “applied” to their arguments by taking inner products, in a similar manner to how predicates are applied to their arguments in Montague semantics.
For the type-logic we use Lambek’s Pregroup grammars [@LambekBook]. The use of pregoups is not essential, but leads to a more elegant formalism, given its proximity to the categorical structure of vector spaces (see [@LambekFest]). A Pregroup is a partially ordered monoid where each element has a right and left cancelling element, referred to as an *adjoint*. It can be seen as the algebraic counterpart of the cancellation calculus of Harris [@Harris]. The operational difference between a Pregroup and Lambek’s Syntactic Calculus is that, in the latter, the monoid multiplication of the algebra (used to model juxtaposition of the types of the words) has a right and a left adjoint, whereas in the pregroup it is the elements themselves which have adjoints. The adjoint types are used to denote functions, e.g. that of a transitive verb with a subject and object as input and a sentence as output. In the Pregroup setting, these function types are still denoted by adjoints, but this time the adjoints of the elements themselves.
As an example, consider the sentence “dogs chase cats”. We assign the type $n$ (for noun phrase) to “dog” and “cat”, and $n^rsn^l$ to “chase”, where $n^r$ and $n^l$ are the right and left adjoints of $n$ and $s$ is the type of a (declarative) sentence. The type $n^rsn^l$ expresses the fact that the verb is a predicate that takes two arguments of type $n$ as input, on its right and left, and outputs the type $s$ of a sentence. The parsing of the sentence is the following reduction: $$n (n^r s n^l) n\leq 1s1 = s$$ This parse is based on the cancellation of $n$ and $n^r$, and also $n^l$ and $n$; i.e. $n n^r \leq 1$ and $n^l n \leq 1$ for 1 the unit of juxtaposition. The reduction expresses the fact that the juxtapositions of the types of the words reduce to the type of a sentence.
On the semantic side, we assign the vector space $N$ to the type $n$, and the tensor space $N \otimes S \otimes N$ to the type $n^r s n^l$. Very briefly, and in order to introduce some notation, recall that the tensor space $A \otimes B$ has as a basis the cartesian product of a basis of $A$ with a basis of $B$. Recall also that any vector can be expressed as a weighted sum of basis vectors; e.g. if $(\overrightarrow{v_1},\ldots,\overrightarrow{v_n})$ is a basis of $A$ then any vector $\overrightarrow{a}\in A$ can be written as $\overrightarrow{a} = \sum_i{C_i \overrightarrow{v_i}}$ where each $C_i \in \mathbb{R}$ is a weighting factor. Now for $(\overrightarrow{v_1},\ldots,\overrightarrow{v_n})$ a basis of $A$ and $(\overrightarrow{v'_1},\ldots,\overrightarrow{v'_n})$ a basis of $B$, a vector $\overrightarrow{c}$ in the tensor space $A \otimes
B$ can be expressed as follows: $$\sum_{ij}{C_{ij} \,
(\overrightarrow{v_i}\otimes\overrightarrow{v'_j})}$$ where the tensor of basis vectors $\overrightarrow{v_i}\otimes\overrightarrow{v'_j}$ stands for their pair $(\overrightarrow{v_i},\overrightarrow{v'_j})$. In general $\overrightarrow{c}$ is not separable into the tensor of two vectors, except for the case when $\overrightarrow{c}$ is not [*entangled*]{}. For non-entangled vectors we can write $\overrightarrow{c} = \overrightarrow{a} \otimes \overrightarrow{b}$ for $\overrightarrow{a} = \sum_i{C_i \overrightarrow{v_i}}$ and $\overrightarrow{b} = \sum_j{C'_j \overrightarrow{v'_j}}$; hence the weighting factor of $\overrightarrow{c}$ can be obtained by simply multiplying the weights of its tensored counterparts, i.e. $C_{ij} =
C_i \times C'_j$. In the entangled case these weights cannot be determined as such and range over all the possibilities. We take advantage of this fact to encode meanings of verbs, and in general all words that have compound types and are interpreted as predicates, relations, or functions. For a brief discussion see the last paragraph of this section. Finally, we use the Dirac notation to denote the dot or inner product of two vectors $\langle \overrightarrow{a} \mid \overrightarrow{b} \rangle \in \mathbb{R}$ defined by $\sum_{i} C_i \times C'_i$.
Returning to our example, for the meanings of nouns we have $\overrightarrow{\text{dogs}}, \overrightarrow{\text{cats}} \in N$, and for the meanings of verbs we have $\overrightarrow{\text{chase}} \in
N \otimes S \otimes N$, i.e. the following superposition: $$\sum_{ijk} C_{ijk} \,(\overrightarrow{n_i} \otimes
\overrightarrow{s_j} \otimes \overrightarrow{n_k})$$ Here $\overrightarrow{n_i}$ and $\overrightarrow{n_k}$ are basis vectors of $N$ and $\overrightarrow{s_j}$ is a basis vector of $S$. From the categorical translation method presented in [@LambekFest] and the grammatical reduction $n (n^r s n^l)
n\leq s$, we obtain the following linear map as the categorical morphism corresponding to the reduction: $$\epsilon_N \otimes 1_s
\otimes \epsilon_N : N \otimes (N \otimes S \otimes N) \otimes N \to
S$$ Using this map, the meaning of the sentence is computed as follows: $$\begin{aligned}
\overrightarrow{\text{dogs} \ \text{chase} \ \text{cats}} &\quad = \quad \left(\epsilon_N \otimes 1_s \otimes \epsilon_N\right) \left(\overrightarrow{\text{dogs}} \otimes \overrightarrow{\text{chase}} \otimes \overrightarrow{\text{cats}}\right)\\
& \quad = \quad \left(\epsilon_N \otimes 1_s \otimes \epsilon_N\right) \left(\overrightarrow{\text{dogs}} \otimes \left(\sum_{ijk} C_{ijk} (\overrightarrow{n_i} \otimes \overrightarrow{s_j} \otimes \overrightarrow{n_k})\right) \otimes \overrightarrow{\text{cats}} \right)\\
&\quad = \quad \sum_{ijk} C_{ijk} \langle \overrightarrow{\text{dogs}} \mid \overrightarrow{n_i}\rangle \overrightarrow{s_j} \langle \overrightarrow{n_k}\mid \overrightarrow{\text{cats}}\rangle\vspace{-2mm}\end{aligned}$$
The key features of this operation are, first, that the inner-products reduce dimensionality by ‘consuming’ tensored vectors and by virtue of the following component function: $$\epsilon_N : N \otimes N \to
\mathbb{R} :: \overrightarrow{a} \otimes \overrightarrow{b} \mapsto
\langle \overrightarrow{a} \mid \overrightarrow{b} \rangle$$ Thus the tensored word vectors $\overrightarrow{\text{dogs}} \otimes
\overrightarrow{\text{chase}} \otimes \overrightarrow{\text{cats}}$ are mapped into a sentence space $S$ which is common to all sentences regardless of their grammatical structure or complexity. Second, note that the tensor product $\overrightarrow{\text{dogs}} \otimes \overrightarrow{\text{chase}}
\otimes \overrightarrow{\text{cats}}$ does not need to be calculated, since all that is required for computation of the sentence vector are the noun vectors and the $C_{ijk}$ weights for the verb. Note also that the inner product operations are simply picking out basis vectors in the noun space, an operation that can be performed in constant time. Hence this formalism avoids two problems faced by approaches in the vein of [@Smolensky; @ClarkPulman], which use the tensor product as a composition operation: first, that the sentence meaning space is high dimensional and grammatically different sentences have representations with different dimensionalities, preventing them from being compared directly using inner products; and second, that the space complexity of the tensored representation grows exponentially with the length and grammatical complexity of the sentence. In constrast, the model we propose does not require the tensored vectors being combined to be represented explicitly.
Note that we have taken the vector of the transitive verb, e.g. $\overrightarrow{\text{chase}}$, to be an entangled vector in the tensor space $N \otimes S \otimes N$. But why can this not be a separable vector, in which case the meaning of the verb would be as follows: $$\overrightarrow{\text{chase}} \quad = \quad \sum_{i}{C_i \overrightarrow{n_i}} \quad \otimes \quad \sum_{j}{C'_j \overrightarrow{s_j}} \quad \otimes \quad \sum_{k}{C''_k \overrightarrow{n_k}}$$ The meaning of the sentence would then become $\sigma_1 \sigma_2 \sum_{j}{C'_j \overrightarrow{s_j}}$ for $\sigma _1 = \sum_{i}{C_i \langle \overrightarrow{\text{dogs}} \mid\overrightarrow{n_i}\rangle}$ and $\sigma_2 = \sum_{k}{C''_k \langle \overrightarrow{\text{cats}}\mid\overrightarrow{n_k}\rangle}$. The problem is that this meaning only depends on the meaning of the verb and is independent of the meanings of the subject and object, whereas the meaning from the entangled case, i.e. $\sigma_1 \sigma_2\sum_{ijk}{C_{ijk} \overrightarrow{s_j}}$, depends on the meanings of subject and object as well as the verb.
From Truth-Theoretic to Corpus-based Meaning
============================================
The model presented above is compositional and distributional, but still abstract. To make it concrete, $N$ and $S$ have to be constructed by providing a method for determining the $C_{ijk}$ weightings. Coecke, Sadrzadeh, and Clark [@LambekFest] show how a truth-theoretic meaning can be derived in the compositional framework. For example, assume that $N$ is spanned by all animals and $S$ is the two-dimensional space spanned by $\overrightarrow{\text{true}}$ and $\overrightarrow{\text{false}}$. We use the weighting factor to define a model-theoretic meaning for the verb as follows: $$C_{ijk} \overrightarrow{s_j} = \begin{cases} \overrightarrow{\text{true}} &
{chase} (\overrightarrow{n_i}, \overrightarrow{n_k}) = \text{true}\,,\\ \overrightarrow{\text{false}} & o.w.\end{cases}$$ The definition of our meaning map ensures that this value propagates to the meaning of the whole sentence. So $chase(\overrightarrow{dogs}, \overrightarrow{cats})$ becomes true whenever “dogs chase cats” is true and false otherwise. This is exactly how meaning is computed in the model-theoretic view on semantics. One way to generalise this truth-theoretic meaning is to assume that ${chase}
(\overrightarrow{n_i}, \overrightarrow{n_k}) $ has degrees of truth, for instance by defining $chase$ as a combination of $run$ and $catch$, such as: $$chase = {2 \over 3} run + {1 \over 3} catch$$ Again, the meaning map ensures that these degrees propagate to the meaning of the whole sentence. For a worked out example see [@LambekFest]. But neither of these examples provide a [*distributional*]{} sentence meaning.
Here we take a first step towards a corpus-based distributional model, by attempting to recover a meaning for a sentence based on the meanings of the words derived from a corpus. But crucially this meaning goes beyond just composing the meanings of words using a vector operator, such as tensor product, summation or multiplication [@Lapata]. Our computation of sentence meaning treats some vectors as functions and others as function arguments, according to how the words in the sentence are typed, and uses the syntactic structure as a guide to determine how the functions are applied to their arguments. The intuition behind this approach is that *syntactic analysis guides semantic vector composition*.
The contribution of this paper is to introduce some concrete constructions for a compositional distributional model of meaning. These constructions demonstrate how the mathematical model of [@LambekFest] can be implemented in a concrete setting which introduces a richer, not necessarily truth-theoretic, notion of natural language semantics which is closer to the ideas underlying standard distributional models of word meaning. We leave full evaluation to future work, in order to determine whether the following method in conjunction with word vectors built from large corpora leads to improved results on language processing tasks, such as computing sentence similarity and paraphrase evaluation.
[**Nouns and Transitive Verbs.**]{} We take $N$ to be a *structured vector space*, as in [@ErkPado; @Greffen]. The bases of $N$ are annotated by ‘properties’ obtained by combining dependency relations with nouns, verbs and adjectives. For example, basis vectors might be associated with properties such as “arg-fluffy”, denoting the argument of the adjective fluffy, “subj-chase” denoting the subject of the verb chase, “obj-buy” denoting the object of the verb buy, and so on. We construct the vector for a noun by counting how many times in the corpus a word has been the argument of ‘fluffy’, the subject of ‘chase’, the object of ‘buy’, and so on.
The framework in [@LambekFest] offers no guidance as to what the sentence space should consist of. Here we take the sentence space $S$ to be $N \otimes N$, so its bases are of the form $\overrightarrow{s_j} =
{(\overrightarrow{n_i}, \overrightarrow {n_k})}$. The intuition is that, for a transitive verb, the meaning of a sentence is determined by the meaning of the verb together with its subject and object.[^1] The verb vectors $C_{ijk}{(\overrightarrow{n_i}, \overrightarrow
{n_k})}$ are built by counting how many times a word that is $n_i$ (e.g. has the property of being fluffy) has been subject of the verb and a word that is $n_k$ (e.g. has the property that it’s bought) has been its object, where the counts are moderated by the extent to which the subject and object exemplify each property (e.g. *how fluffy* the subject is). To give a rough paraphrase of the intuition behind this approach, the meaning of “dog chases cat" is given by: the extent to which a dog is fluffy and a cat is something that is bought (for the $N \otimes N$ property pair “arg-fluffy" and “obj-buy"), and the extent to which fluffy things [*chase*]{} things that are bought (accounting for the meaning of the verb for this particular property pair); plus the extent to which a dog is something that runs and a cat is something that is cute (for the $N \otimes N$ pair “subj-run" and “arg-cute"), and the extent to which things that run [*chase*]{} things that are cute (accounting for the meaning of the verb for this particular property pair); and so on for all noun property pairs.
[**Adjective Phrases.**]{} Adjectives are dealt with in a similar way. We give them the syntactic type $nn^l$ and build their vectors in $N \otimes N$. The syntactic reduction $nn^l n
\to n$ associated with applying an adjective to a noun gives us the map $1_N \otimes \epsilon_N$ by which we semantically compose an adjective with a noun, as follows: $$\overrightarrow{\text{red fox}} = (1_N \otimes \epsilon_N)(\overrightarrow{\text{red}} \otimes \overrightarrow{\text{fox}}) = \sum_{ij}{C_{ij}\overrightarrow{n_i} \langle \overrightarrow{n_j} \mid \overrightarrow{\text{fox}} \rangle}$$ We can view the $C_{ij}$ counts as determining what sorts of properties the arguments of a particular adjective typically have (e.g. arg-red, arg-colourful for the adjective “red”).
[**Prepositional Phrases.**]{} We assign the type $n^r n$ to the whole prepositional phrase (when it modifies a noun), for example to “in the forest” in the sentence “dogs chase cats in the forest”. The pregroup parsing is as follows: $$n (n^rsn^l) n (n^r n) \leq 1sn^l 1 n \leq sn^l n \leq s1 = s$$ The vector space corresponding to the prepositional phrase will thus be the tensor space $N \otimes N$ and the categorification of the parse will be the composition of two morphisms: $(1_S \otimes \epsilon^l_N) \circ (\epsilon^r_N \otimes 1_S \otimes 1_N \otimes \epsilon^r_N \otimes 1_N)$. The substitution specific to the prepositional phrase happens when computing the vector for “cats in the forest” as follows: $$\begin{aligned}
\overrightarrow{\text{cats \ in \ the \ forest}} &=
(\epsilon^r_N \otimes 1_N) \left (\overrightarrow{\text{cats}} \otimes \overrightarrow{\text{in \ the \ forest}}\right)\\
&= (\epsilon^r_N \otimes 1_N) \left (\overrightarrow{\text{cats}} \otimes \sum_{lw} C_{lw} \overrightarrow{n_l} \otimes \overrightarrow{n_k}\right)\\
& = \sum_{lw} C_{lw} \langle \overrightarrow{\text{cats}} \mid \overrightarrow{n_l} \rangle \overrightarrow{n_w}\end{aligned}$$ Here we set the weights $C_{lw}$ in a similar manner to the cases of adjective phrases and verbs with the counts determining what sorts of properties the noun modified by the prepositional phrase has, e.g. the number of times something that has attribute $n_l$ has been in the forest.
[**Adverbs.**]{} We assign the type $s^r s$ to the adverb, for example to “quickly” in the sentence “Dogs chase cats quickly”. The pregroup parsing is as follows: $$n (n^rsn^l) n (s^rs) \leq 1s1s^rs = ss^rs\leq 1s = s$$ Its categorification will be a composition of two morphisms $(\epsilon^r_S \otimes 1_S) \circ (\epsilon^r_N \otimes 1_S \otimes \epsilon^l_N \otimes 1_S \otimes 1_S)$. The substitution specific to the adverb happens after computing the meaning of the sentence without it, i.e. that of “Dogs chase cats”, and is as follows: $$\begin{aligned}
\overrightarrow{\text{Dogs \ chase \ cats \ quickly}} &=
(\epsilon^r_S \otimes 1_S) \circ (\epsilon^r_N \otimes 1_S \otimes \epsilon^l_N \otimes 1_S \otimes 1_S)
\left (\overrightarrow{\text{Dogs}} \otimes \overrightarrow{\text{chase}} \otimes \overrightarrow{\text{cats}} \otimes \overrightarrow{\text{quickly}}\right)\\
&=
(\epsilon^r_S \otimes 1_S) \left (\sum_{ijk} C_{ijk} \langle \overrightarrow{\text{dogs}} \mid \overrightarrow{n_i}\rangle \overrightarrow{s_j} \langle \overrightarrow{n_k}\mid \overrightarrow{\text{cats}}\rangle \otimes \overrightarrow{\text{quickly}}\right)\\
&= (\epsilon^r_S \otimes 1_S) \left(\sum_{ijk} C_{ijk} \langle \overrightarrow{\text{dogs}} \mid \overrightarrow{n_i}\rangle \overrightarrow{s_j} \langle \overrightarrow{n_k}\mid \overrightarrow{\text{cats}}\rangle \otimes \sum_{lw} C_{lw} \overrightarrow{s_l} \otimes \overrightarrow{s_w}\right)\\
&= \sum_{lw} C_{lw} \left \langle \sum_{ijk} C_{ijk} \langle \overrightarrow{\text{dogs}} \mid \overrightarrow{n_i}\rangle \overrightarrow{s_j} \langle \overrightarrow{n_k}\mid \overrightarrow{\text{cats}}\rangle \mid \overrightarrow{s_l}\right \rangle
\overrightarrow{s_k}\end{aligned}$$ The $C_{lw}$ weights are defined in a similar manner to the above cases, i.e. according to the properties the adverb has, e.g. which verbs it has modified. Note that now the basis vectors $\overrightarrow{s_l}$ and $\overrightarrow{s_w}$ are themselves pairs of basis vectors from the noun space, $(\overrightarrow{n_i}, \overrightarrow{n_j})$. Hence, $C_{lw}(\overrightarrow{n_i}, \overrightarrow{n_j})$ can be set only for the case when $l=i$ and $w=j$; these counts determine what sorts of properties the verbs that happen quickly have (or more specifically what properties the subjects and objects of such verbs have). By taking the whole sentence into account in the interpretation of the adverb, we are in a better position to semantically distinguish between the meaning of adverbs such as “slowly” and “quickly”, for instance in terms of the properties that the verb’s subjects have. For example, it is possible that elephants are more likely to be the subject of a verb which is happening slowly, e.g. run slowly, and cheetahs are more likely to be the subject of a verb which is happening quickly.
Concrete Computations
=====================
In this section we first describe how to obtain the relevant counts from a parsed corpus, and then give some similarity calculations for some example sentence pairs.
Let $\mathcal{C}_l$ be the set of grammatical relations (GRs) for sentence $s_l$ in the corpus. Define $\mathit{verbs}(\mathcal{C}_l)$ to be the function which returns all instances of verbs in $\mathcal{C}_l$, and $\mathit{subj}$ (and similarly $\mathit{obj}$) to be the function which returns the subject of an instance $V_{\textit{instance}}$ of a verb $V$, for a particular set of GRs for a sentence:
$$subj(V_{\textit{instance}}) = \begin{cases} noun & \text{if $V_{\textit{instance}}$ is a verb with subject $noun$}\\
\varepsilon_n & o.w.\end{cases}$$
where $\varepsilon_n$ is the empty string. We express $C_{ijk}$ for a verb $V$ as follows: $$C_{ijk} = \begin{cases} \sum_l\sum_{v \in \mathit{verbs}(\mathcal{C}_l)}{\delta(v,V) \langle \overrightarrow{subj(v)} \mid \overrightarrow{n_i} \rangle \langle \overrightarrow{obj(v)} \mid \overrightarrow{n_k} \rangle} & \text{if}\ \overrightarrow{s_j} ={(\overrightarrow{n_i} , \overrightarrow{n_k})} \\
0 & o.w.\end{cases}\vspace{-3mm}$$ where $\delta(v,V) = 1$ if $v = V$ and 0 otherwise. Thus we construct $C_{ijk}$ for verb $V$ only for cases where the subject property $n_i$ and the object property $n_k$ are paired in the basis $\overrightarrow{s_j}$. This is done by counting the number of times the subject of $V$ has property $n_i$ and the object of $V$ has property $n_k$, then multiplying them, as prescribed by the inner products (which simply pick out the properties $n_i$ and $n_k$ from the noun vectors for the subjects and objects).
The procedure for calculating the verb vectors, based on the formulation above, is as follows:
1. For each GR in a sentence, if the relation is $subject$ and the head is a verb, then find the complementary GR with $object$ as a relation and the same head verb. If none, set the object to $\varepsilon_n$.
2. Retrieve the noun vectors $\overrightarrow{subject}, \overrightarrow{object}$ for the subject dependent and object dependent from previously constructed noun vectors.
3. For each $(n_i,n_k) \in basis(N) \times basis(N)$ compute the inner-product of $\overrightarrow{n_i}$ with $\overrightarrow{subject}$ and $\overrightarrow{n_k}$ with $\overrightarrow{object}$ (which involves simply picking out the relevant basis vectors from the noun vectors). Multiply the inner-products and add this to $C_{ijk}$ for the verb, with $j$ such that $\overrightarrow{s_j} = (\overrightarrow{n_i},\overrightarrow{n_k})$.
The procedure for other grammatical types is similar, based on the definitions of $C$ weights for the semantics of these types.
We now give a number of example calculations. We first manually define the distributions for nouns, which in practice would be obtained from a corpus:
bankers cats dogs stock kittens
------------------- --------- ------ ------ ------- ---------
1\. arg-fluffy 0 7 3 0 2
2\. arg-ferocious 4 1 6 0 0
3\. obj-buys 0 4 2 7 0
4\. arg-shrewd 6 3 1 0 1
5\. arg-valuable 0 1 2 8 0
We aim to make these counts match our intuitions, in that bankers are shrewd and a little ferocious but not furry, cats are furry but not typically valuable, and so on.
We also define the distributions for the transitive verbs ‘chase’, ‘pursue’ and ‘sell’, again manually specified according to our intuitions about how these verbs are used. Since in the formalism proposed above, $C_{ijk}=0$ if $\overrightarrow{s_j} \neq (\overrightarrow{n_i},\overrightarrow{n_k})$, we can simplify the weight matrices for transitive verbs to two dimensional $C_{ik}$ matrices as shown below, where $C_{ik}$ corresponds to the number of times the verb has a subject with attribute $n_i$ and an object with attribute $n_k$. For example, the matrix below encodes the fact that something ferocious ($i=2$) chases something fluffy ($k=1$) seven times in the hypothetical corpus from which we might have obtained these distributions. $$C^{\textrm{chase}} = \left[
\begin{tabular}{ccccc}
1 & 0 & 0 & 0 & 0 \\
7 & 1 & 2 & 3 & 1 \\
0 & 0 & 0 & 0 & 0 \\
2 & 0 & 1 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 \\
\end{tabular}
\right]
\quad
C^{\textrm{pursue}} = \left[
\begin{tabular}{ccccc}
0 & 0 & 0 & 0 & 0 \\
4 & 2 & 2 & 2 & 4 \\
0 & 0 & 0 & 0 & 0 \\
3 & 0 & 2 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 \\
\end{tabular}
\right]
\quad
C^{\textrm{sell}} = \left[
\begin{tabular}{ccccc}
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 3 & 0 & 4 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 5 & 0 & 8 \\
0 & 0 & 1 & 0 & 1 \\
\end{tabular}
\right]$$ These matrices can be used to perform sentence comparisons: $$\begin{aligned}
\langle \overrightarrow{\text{dogs chase cats}} & \mid \overrightarrow{\text{dogs pursue kittens}} \rangle = \\
& = \left\langle \left(\sum_{ijk}{C^{\text{chase}}_{ijk} \langle \overrightarrow{\text{dogs}} \mid \overrightarrow{n_i}\rangle \overrightarrow{s_j} \langle \overrightarrow{n_k}\mid \overrightarrow{\text{cats}}\rangle}\right) \right| \left. \left(\sum_{ijk}{C^{\text{pursue}}_{ijk} \langle \overrightarrow{\text{dogs}} \mid \overrightarrow{n_i}\rangle \overrightarrow{s_j} \langle \overrightarrow{n_k}\mid \overrightarrow{\text{kittens}}\rangle} \right) \right\rangle \\
&\\
& = \sum_{ijk}{C^{\text{chase}}_{ijk}C^{\text{pursue}}_{ijk} \langle \overrightarrow{\text{dogs}} \mid \overrightarrow{n_i}\rangle \langle \overrightarrow{\text{dogs}} \mid \overrightarrow{n_i}\rangle \langle \overrightarrow{n_k}\mid \overrightarrow{\text{cats}}\rangle \langle \overrightarrow{n_k}\mid \overrightarrow{\text{kittens}}\rangle}\end{aligned}$$ The raw number obtained from the above calculation is 14844. Normalising it by the product of the length of both sentence vectors gives the cosine value of $0.979$.
Consider now the sentence comparison $\langle \overrightarrow{\text{dogs chase cats}} \mid \overrightarrow{\text{cats chase dogs}} \rangle $. The sentences in this pair contain the same words but the different word orders give the sentences very different meanings. The raw number calculated from this inner product is 7341, and its normalised cosine measure is $0.656$, which demonstrates the sharp drop in similarity obtained from changing sentence structure. We expect some similarity since there is some non-trivial overlap between the properties identifying cats and those identifying dogs (namely those salient to the act of chasing).
Our final example for transitive sentences is $\langle \overrightarrow{\text{dogs chase cats}} \mid \overrightarrow{\text{bankers sell stock}} \rangle $, as two sentences that diverge in meaning completely. The raw number for this inner product is 6024, and its cosine measure is $0.042$, demonstrating the very low semantic similarity between these two sentences.
Next we consider some examples involving adjective-noun modification. The $C_{ij}$ counts for an adjective $A$ are obtained in a similar manner to transitive or intransitive verbs: $$C_{ij} = \begin{cases} \sum_l\sum_{a\in\mathit{adjs}(\mathcal{C}_l)}{\delta(a,A) \langle \overrightarrow{arg\mhyphen{}of(a)} \mid \overrightarrow{n_i} \rangle } & \text{if}\ \overrightarrow{n_i} = \overrightarrow{n_j} \\
0 & o.w.\end{cases}$$ where $\mathit{adjs}(\mathcal{C}_l)$ returns all instances of adjectives in $\mathcal{C}_l$; $\delta(a,A) = 1$ if $a = A$ and $0$ otherwise; and $arg\mhyphen{}of(a) = noun$ if $a$ is an adjective with argument $noun$, and $\varepsilon_n$ otherwise.
As before, we stipulate the $C_{ij}$ matrices by hand (and we eliminate all cases where $i \neq j$ since $C_{ij} = 0$ by definition in such cases): $$C^{\text{fluffy}} = [9\ 3\ 4\ 2\ 2] \qquad
C^{\text{shrewd}} = [0\ 3\ 1\ 9\ 1]\qquad
C^{\text{valuable}} = [3\ 0\ 8\ 1\ 8]$$ We compute vectors for “fluffy dog” and “shrewd banker” as follows: [$$\begin{aligned}
\overrightarrow{\text{fluffy \ dog}} &= (3 \cdot 9)\, \overrightarrow{\text{arg-fluffy}} + (6 \cdot 3)\, \overrightarrow{\text{arg-ferocious}} + (2 \cdot 4)\, \overrightarrow{\text{obj-buys}} + (5 \cdot 2)\, \overrightarrow{\text{arg-shrewd}} + (2 \cdot 2)\, \overrightarrow{\text{arg-valuable}}\\
\overrightarrow{\text{shrewd \ banker}} &=( 0 \cdot 0) \, \overrightarrow{\text{arg-fluffy}} + (4 \cdot 3)\, \overrightarrow{\text{arg-ferocious}} + (0 \cdot 0)\, \overrightarrow{\text{obj-buys}} + (6 \cdot 9)\, \overrightarrow{\text{arg-shrewd}} + (0 \cdot 1)\, \overrightarrow{\text{arg-valuable}}\end{aligned}$$]{} Vectors for $\overrightarrow{\text{fluffy \ cat}}$ and $\overrightarrow{\text{valuable \ stock}}$ are computed similarly. We obtain the following similarity measures: $$cosine ( \overrightarrow{\text{fluffy \ dog}} , \overrightarrow{\text{shrewd \ banker}} ) = 0.389\qquad
cosine ( \overrightarrow{\text{fluffy \ cat}} , \overrightarrow{\text{valuable \ stock}}) = 0.184$$ These calculations carry over to sentences which contain the adjective-noun pairings compositionally and we obtain an even lower similarity measure between sentences: $$cosine( \overrightarrow{\text{fluffy dogs chase fluffy cats}} , \overrightarrow{\text{shrewd bankers sell valuable stock}} ) = 0.016$$
To summarise, our example vectors provide us with the following similarity measures:
[**Sentence 1**]{} [**Sentence 2**]{} [**Degree of similarity**]{}
------------------------------- ------------------------------------ ------------------------------
dogs chase cats dogs pursue kittens $0.979$
dogs chase cats cats chase dogs $0.656$
dogs chase cats bankers sell stock $0.042$
fluffy dogs chase fluffy cats shrewd bankers sell valuable stock $0.016$
Different Grammatical Structures {#ssec:different_grammatical_structures}
================================
So far we have only presented the treatment of sentences with transitive verbs. For sentences with intransitive verbs, the sentence space suffices to be just $N$. To compare the meaning of a transitive sentence with an intransitive one, we embed the meaning of the latter from $N$ into the former $N \otimes N$, by taking $\overrightarrow{\varepsilon_n}$ (the ‘object’ of an intransitive verb) to be $\sum_i{\overrightarrow{n_i}}$, i.e. the superposition of all basis vectors of $N$.
Following the method for the transitive verb, we calculate $C_{ijk}$ for an instransitive verb $V$ and basis pair $\overrightarrow{s_j} =
{(\overrightarrow{n_i} , \overrightarrow{n_k})}$ as follows, where $l$ ranges over the sentences in the corpus: $$\sum_l\sum_{v \in \mathit{verbs}(\mathcal{C}_l)}{\delta(v,V)
\langle \overrightarrow{subj(v)} \mid
\overrightarrow{n_i} \rangle \langle
\overrightarrow{obj(v)} \mid \overrightarrow{n_k}
\rangle}
= \sum_l\sum_{v \in \mathit{verbs}(\mathcal{C}_l)}{\delta(v,V) \langle \overrightarrow{subj(v)} \mid \overrightarrow{n_i} \rangle \langle \overrightarrow{\varepsilon_n} \mid \overrightarrow{n_k} \rangle}$$ and $\langle
\overrightarrow{\varepsilon_n} \mid \overrightarrow{n_i} \rangle = 1$ for any basis vector $n_i$.
We can now compare the meanings of transitive and intransitive sentences by taking the inner product of their meanings (despite the different arities of the verbs) and then normalising it by vector length to obtain the cosine measure. For example:
$$\begin{aligned}
\langle \overrightarrow{\text{dogs chase cats}} \mid \overrightarrow{\text{dogs chase}} \rangle & = \left\langle \left(\sum_{ijk}{C_{ijk} \langle \overrightarrow{\text{dogs}} \mid \overrightarrow{n_i}\rangle \overrightarrow{s_j} \langle \overrightarrow{n_k}\mid \overrightarrow{\text{cats
}}\rangle}\right) \right| \left. \left(\sum_{ijk}{C'_{ijk} \langle \overrightarrow{\text{dogs}} \mid \overrightarrow{n_i}\rangle \overrightarrow{s_j}} \right) \right\rangle\\
&\\
& = \sum_{ijk}{C_{ijk}C'_{ijk} \langle \overrightarrow{\text{dogs}}
\mid \overrightarrow{n_i}\rangle \langle \overrightarrow{\text{dogs}} \mid
\overrightarrow{n_i}\rangle \langle \overrightarrow{n_k}\mid \overrightarrow{\text{cats}}\rangle} \end{aligned}$$
The raw number for the inner product is 14092 and its normalised cosine measure is 0.961, indicating high similarity (but some difference) between a sentence with a transitive verb and one where the subject remains the same, but the verb is used intransitively.
Comparing sentences containing nouns modified by adjectives to sentences with unmodified nouns is straightforward: $$\begin{aligned}
\langle \overrightarrow{\textrm{fluffy dogs chase fluffy cats}} \mid \overrightarrow{\textrm{dogs chase cats}}\rangle &=\\
&\\
\sum_{ij} C^{\text{fluffy}}_i C^{\text{fluffy}}_j C^{\text{chase}}_{ij} C^{\text{chase}}_{ij} \langle \overrightarrow{dogs} \mid \overrightarrow{n_i} \rangle^2 \langle \overrightarrow{n_j} \mid \overrightarrow{cats} \rangle^2 &= 2437005\\ \end{aligned}$$ From the above we obtain the following similarity measure: $$cosine(\overrightarrow{\textrm{fluffy dogs chase fluffy cats}},\overrightarrow{\textrm{dogs chase cats}}) = 0.971$$ For sentences with ditransitive verbs, the sentence space changes to $N \otimes N \otimes N$, on the basis of the verb needing two objects; hence its grammatical type changes to $n^r s n^l n^l$. The transitive and intransitive verbs are embedded in this larger space in a similar manner to that described above; hence comparison of their meanings becomes possible.
Ambiguous Words
===============
The two different meanings of a word can be distinguished by the different properties that they have. These properties are reflected in the corpus, by the different contexts in which the words appear. Consider the following example from [@ErkPado]: the verb “catch” has two different meanings, “grab” and “contract”. They are reflected in the two sentences “catch a ball” and “catch a disease”. The compositional feature of our meaning computation enables us to realise the different properties of the context words via the grammatical roles they take in the corpus. For instance, the word ‘ball’ occurs as argument of ‘round’, and so has a high weight for the base ‘arg-round’, whereas the word ‘disease’ has a high weight for the base ‘arg-contagious’ and as ‘mod-of-heart’. We extend our example corpus from previously to reflect these differences as follows:
ball disease
-------------------- ------ ---------
1\. arg-fluffy 1 0
2\. arg-ferocious 0 0
3\. obj-buys 5 0
4\. arg-shrewd 0 0
5\. arg-valuable 1 0
6\. arg-round 8 0
7\. arg-contagious 0 7
8\. mod-of-heart 0 6
In a similar way, we build a matrix for the verb ‘catch’ as follows: $$C^{\textrm{catch}} = \left[
\begin{tabular}{cccccccc}
3 & 2 & 3 & 3 & 3 & 8 & 6 & 2\\
3 & 2 & 3 & 0 & 1 & 4 & 7 & 4\\
2 & 4 & 7 & 1 & 1 & 6 & 2 & 2\\
3 & 1 & 2 & 0 & 0 & 3 & 6 & 2\\
1 & 1 & 1 & 0 & 0 & 2 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\end{tabular}
\right]$$ The last three rows are zero because we have assumed that the words that can take these roles are mostly objects and hence cannot catch anything. Given these values, we compute the similarity measure between the two sentences “dogs catch a ball” and “dogs catch a disease” as follows: $$\langle \overrightarrow{\text{dogs catch a ball}} \mid \overrightarrow{\text{dogs catch a disease}}\rangle \quad = \quad 0$$ In an idealised case like this where there is very little (or no) overlap between the properties of the objects associated with one sense of “catch” (e.g. a disease), and those properties of the objects associated with another sense (e.g. a ball), disambiguation is perfect in that there is no similarity between the resulting phrases. In practice, in richer vector spaces, we would expect even diseases and balls to share some properties. However, as long as those shared properties are not those typically held by the object of catch, and as long as the usages of catch play to distinctive properties of diseases and balls, disambiguation will occur by the same mechanism as the idealised case above, and we can expect low similarity measures between such sentences.
Related Work {#sec:relatedwork}
============
Mitchell and Lapata introduce and evaluate a multiplicative model for vector composition [@Lapata]. The particular concrete construction of this paper differs from that of [@Lapata] in that our framework subsumes truth-theoretic as well as corpus-based meaning, and our meaning construction relies on and is guided by the grammatical structure of the sentence. The approach of [@ErkPado] is more in the spirit of ours, in that extra information about syntax is used to compose meaning. Similar to us, they use a structured vector space to integrate lexical information with selectional preferences. Finally, Baroni and Zamparelli model adjective-noun combinations by treating an adjective as a function from noun space to noun space, represented using a matrix, as we do in this paper [@BaroniEMNLP10].
[^1]: Intransitive and ditransitive verbs are interpreted in an analagous fashion; see $\S$\[ssec:different\_grammatical\_structures\].
|
---
author:
- |
, Gernot Münster, Owe Philipsen\
Westfälische Wilhelms-Universität Münster, Institut für Theoretische Physik, Wilhelm-Klemm-Str.9, 48149 Münster, Germany\
E-mail: jens.langelage@uni-muenster.de\
E-mail: munsteg@uni-muenster.de\
E-mail: ophil@uni-muenster.de
title: |
[ ]{}\
Strong coupling expansion for Yang-Mills theory at finite temperature
---
Introduction
============
Contrary to weak coupling expansions, strong coupling expansions are known to be convergent series with a finite radius of convergence. In the early days of lattice gauge theory they were used to get analytical results for some physical quantities of interest, such as glueball masses or the energy density of lattice Yang-Mills theories. These calculations were done at zero temperature, i.e. at infinite volume $N_s^3$ and temporal extent $N_t$ of the lattice.
Here we calculate such series expansions for the free energy density and screening masses with an infinite spatial volume and a compactified temporal extension $N_t$ of the lattice. In this way finite temperature effects are generated, giving us the opportunity to study the physical, temperature dependent free energy density in the confined phase. The physical deconfinement phase transition then corresponds to a finite convergence radius of the series, which one may try to estimate from the behaviour of the coefficients.
Free energy density
===================
Cluster expansion
-----------------
The partition function of the lattice Yang-Mills theory is given by a functional integration of the exponentiated Wilson action over the corresponding SU(N) group space, $$\begin{aligned}
Z&=&\int \,DU\,\exp\left[\sum_p\frac{\beta}{2N}\left( \mathrm{Tr}\, U+\mathrm{Tr}\,U^{\dagger}-2N\right) \right], \\
\beta&=&\frac{2N}{g^2}.\nonumber\end{aligned}$$ An expansion in the lattice coupling $\beta$ by group characters $\chi_r(U)$ and a cluster expansion yields the free energy density [[@Montvay:1994cy]]{} $$\tilde{f}\equiv-\frac{1}{\Omega}\ln Z=-6\ln\,c_0(\beta)-\frac{1}{\Omega}
\sum_{C=(X_i^{n_i})}\,a(C)\prod_i\Phi(X_i)^{n_i}.
\label{free}$$ where $\Omega=V\cdot N_t$ is the lattice volume and $c_0$ is the expansion coefficient of the trivial representation, which has been factored out. The combinatorial factor $a(C)$ is introduced via a moment-cumulant-formalism, and equals $1$ for clusters $C$ which consist of only one so-called polymer $X_i$. The quantity in equation (\[free\]) is customarily called a free energy, even at zero physical temperature, because the path integral corresponds to a partition function if one formally identifies $\beta$ with $1/T$. Here we are interested in a physical temperature $T=1/(aN_t)$, realized by compactifying the temporal extension of the lattice. The physical free energy is then obtained by subtracting the formal ($N_t=\infty$) free energy, which is analogous to a subtraction of the divergent vacuum energy in the continuum. Thus the physical free energy density reads $$f(N_t,u)=\tilde{f}(N_t,u)-\tilde{f}(\infty,u).$$ The contributing polymers $X_i$ have to be objects with a closed surface, since $$\begin{aligned}
\int dU \chi_r(U)=\delta_{r,0}.\end{aligned}$$ This means the group integration projects out the trivial representation at each link. To calculate the group integrals one uses the integration formula $$\int dU \chi_r(UV)\chi_r(WU^{-1})=\chi_r(VW).$$ For a more detailed introduction to strong coupling calculations we refer to [[@Montvay:1994cy]]{}.
Results
-------
The graph contributing to the lowest order of the free energy density is a tube of length $N_t$ with a cross-section of one single plaquette. The contribution of these tubes together with inner plaquettes is $$\begin{aligned}
\mbox{SU(2):}\qquad f_1(N_t,u)&=&-\frac{3}{N_t}u^{4N_t}a^{N_t},\\
\mbox{SU(3):}\qquad f_1(N_t,u)&=&-\frac{3}{N_t}u^{4N_t}\left[b^{N_t}+c^{N_t}\right],\end{aligned}$$ where $u$, $v$ and $w$ are the expansion parameters of the lowest dimensional representations of the corresponding gauge groups, $$\begin{aligned}
\mbox{SU(2):}\qquad u&=&\frac{\beta}{4}+{{\cal{O}}(\beta^{2})}\qquad v=\frac{\beta^2}{24}+{{\cal{O}}(\beta^{4})},\qquad \nonumber\\
\mbox{SU(3):}\qquad u&=&\frac{\beta}{18}+{{\cal{O}}(\beta^{2})}\qquad v=\frac{\beta^2}{432}+{{\cal{O}}(\beta^{4})}\qquad w=\frac{\beta^2}{288}+{{\cal{O}}(\beta^{4})},\end{aligned}$$ and we have used the abbreviations $$\begin{aligned}
a&=&1+3v-4u^2,\nonumber\\
b&=&1-3u-6v+8w,\nonumber\\
c&=&1+3u+6v+8w-18u^2.\end{aligned}$$
Higher order contributions consist of such tubes with local decorations of additional plaquettes either in the fundamental or in higher representations. For the interesting cases SU(2) and SU(3), these contributions up to the calculated orders are $$\begin{aligned}
\mbox{SU(2):}\quad f(N_t,u)=&-&\frac{3}{N_t}u^{4N_t}a^{N_t}\left[ 1+12N_tu^4-\frac{1556}{81}N_tu^6+\left(86N_t^2+\frac{35828}{243}N_t \right)u^8 \right], \\
\nonumber\\
\mbox{SU(3):}\quad f(N_t,u)=&-&\frac{3}{N_t}u^{4N_t}c^{N_t}\Big[1+12N_tu^4+42N_tu^5-\frac{115343}{2048}N_tu^6-\frac{1095327}{2048}N_tu^7\Big]\nonumber\\
\nonumber\\
&-&\frac{3}{N_t}u^{4N_t}b^{N_t}\Big[1+12N_tu^4+30N_tu^5-\frac{17191}{256}N_tu^6+63N_tu^7\Big],\end{aligned}$$ which are valid only for $N_t\geq5$. For smaller $N_t$ there are modifications of these formulae coming from polymers with cross-sections larger than one plaquette. The complete results for $N_t=2$ and $3$ in SU(2) are $$\begin{aligned}
N_t=2:\quad f(2,u)=&-&\frac{3}{2}u^{8}\left[ 1-4u^2+\frac{110}{3}u^4-\frac{58472}{405}u^6+\frac{64897681}{65610}u^8 \right],\\
N_t=3:\quad f(3,u)=&-&u^{12}\left[ 1-6u^2+50u^4-\frac{37966}{135}u^6+\frac{843898}{405}u^8 \right],\end{aligned}$$
Free energy density as a glueball gas
-------------------------------------
Recognizing the first orders of the corresponding glueball masses (see [@Munster:1981es] and [@Seo:1982jh]) for SU(2) $$\begin{aligned}
m(A_1^{++})&=&-4\ln\,u+2u^2-\frac{98}{3}u^4-\frac{20984}{405}u^6-\frac{151496}{243}u^8,\\
m(E^{++})&=&-4\ln\,u+2u^2-\frac{26}{3}u^4+\frac{13036}{405}u^6-\frac{28052}{243}u^8,\end{aligned}$$ and SU(3) $$\begin{aligned}
m(A_1^{++})&=&-4\ln\,u-3u+9u^2-\frac{27}{2}u^3-7u^4-\frac{297}{2}u^5+\frac{858827}{10240}u^6+\frac{47641149}{71680}u^7,\\
m(E^{++})&=&-4\ln\,u-3u+9u^2-\frac{27}{2}u^3+17u^4-\frac{153}{2}u^5+\frac{1104587}{10240}u^6+\frac{29577789}{71680}u^7,\\
m(T_1^{+-})&=&-4\ln\,u+3u+\frac{9}{2}u^3-\frac{98}{4}u^4+\frac{33}{4}u^5-\frac{36771}{1280}u^6+\frac{117897}{448}u^7,\end{aligned}$$ one can write $$\begin{aligned}
SU(2):\qquad f(N_t,u)&=&-\frac{1}{N_t}\left[e^{-m(A_1^{++})N_t} + 2e^{-m(E^{++})N_t} + {\cal{O}}(u^4)\right], \\
SU(3):\qquad f(N_t,u)&=&-\frac{1}{N_t}\left[e^{-m(A_1^{++})N_t} + 2e^{-m(E^{++})N_t} + 3e^{-m(T_1^{+-})N_t}+ {\cal{O}}(u^4)\right],\end{aligned}$$ corresponding to a gas of non-interacting glueballs in a hadron-resonance-gas model [@Karsch:2003zq], where $$\begin{aligned}
f\simeq-T\sum_i e^{-\frac{E_i}{T}}.\end{aligned}$$ This is a rather remarkable result. It allows to see from first principles that the pressure $p=-f$ is exponentially small in the confined phase, and it explains the success of the hadron-resonance-gas model in reproducing the confined phase equation of state. Since the partition function is not directly measurable in Monte-Carlo simulations, the pressure is usually obtained by the integral method [@Boyd:1996bx], where the expectation values of derivatives are computed and then integrated numerically, $$\begin{aligned}
\frac{p}{T^4}\,\bigg\vert_{\beta_0}^{\beta}=N_t^4\int_{\beta_0}^{\beta}d\beta' \left[ S_0-S_T\right], \end{aligned}$$ with $S_0=6P_0$ and $S_T=3(P_t+P_s)$, where $P_0$ denotes the plaquette expectation value on symmetric lattices and $P_{t,s}$ are those of space-time and space-space plaquettes with $N_t<N_s$. The lower integration limit is usually set to zero by hand, arguing with an exponentially small pressure in the low temperature regime. Our results now justify this assumption from first principles.
Phase transition
----------------
Physical phase transitions limit the radius of convergence on the real $\beta$-axis, signalled by a singularity in the full free energy. We model the full function from the series coefficients by Padé approximants $[L,M]$ with $$\begin{aligned}
[L,M](u)\equiv \frac{1+a_1u+\dots +a_Lu^L}{b_0+b_1u+\dots+b_Mu^M},\end{aligned}$$ and search for the zeroes of the denominator. The resulting $L+M=2,3,4$ Padé tables for $N_t=2,3$ with the nearest real singularities are shown in table (\[pade\]).
[**[SU(2): $N_t=2$]{}**]{}\
$[L,M]$ $u_c$ $\beta_c$ $|u_c-u_0|$
--------- ----------- ----------- -------------
$[1,2]$ $0.4033 $ 1.8227 0.0899
$[0,3]$ $0.4675 $ 2.2201
$[2,2]$ $0.5201 $ 2.5981
$[1,3]$ $0.4684 $ 2.2262
$[0,4]$ $0.4684 $ 2.2261
: Zeroes of the denominator ($u_c$) and the numerator ($u_0$) of the $[L,M]$ Padé approximants and the corresponding value of $\beta_c$.[]{data-label="pade"}
[**[SU(2): $N_t=3$]{}**]{}\
$[L,M]$ $u_c$ $\beta_c$ $|u_c-u_0|$
--------- ----------- ----------- -------------
$[1,2]$ $0.3467 $ 1.5133 0.0219
$[0,3]$ $0.5009 $ 2.4538
$[2,2]$ $0.4623 $ 2.1853 0.2388
$[1,3]$ $0.4347 $ 2.0098 0.1373
$[0,4]$ $0.4617 $ 2.1820
: Zeroes of the denominator ($u_c$) and the numerator ($u_0$) of the $[L,M]$ Padé approximants and the corresponding value of $\beta_c$.[]{data-label="pade"}
[![Left: Plot of the pressure density $p$ vs. $\beta$ ($L+M=4$ Padés for SU(2) and $N_t=3$). The plot range corresponds to the confined phase up to the critical coupling. Right: Plot of $p$ vs. $T/T_c$ for SU(3) from Monte Carlo data [@Boyd:1996bx].[]{data-label="fig_pressure"}](gnuplot_mapledat2.ps "fig:"){width="7cm"}]{}
![Left: Plot of the pressure density $p$ vs. $\beta$ ($L+M=4$ Padés for SU(2) and $N_t=3$). The plot range corresponds to the confined phase up to the critical coupling. Right: Plot of $p$ vs. $T/T_c$ for SU(3) from Monte Carlo data [@Boyd:1996bx].[]{data-label="fig_pressure"}](fige4.ps){width="5cm"}
Zeroes $u_0$ of the Padé approximant which are very close to a singularity often indicate that the singularity is superfluous and disappears as the full funtion is approached. Hence, removing the singularities with a nearby zero, we obtain estimates for the critical couplings, which are not far from the Monte Carlo results $\beta_c=1.8800(30)$ for $N_t=2$ and $\beta_c=2.1768(30)$ for $N_t=3$ [@Fingberg:1992ri].
The $L+M=4$ Padé’s for $N_t=3$, SU(2), are shown in figure (\[fig\_pressure\]). The spread in the curves gives an estimate of the systematic error of the approximants at that order. The exponential suppression in the confined phase as well as the onset of the pressure upon approaching $T_c$ is reproduced by the strong coupling series.
Screening masses
================
Zero temperature
----------------
Screening masses are defined by the exponential decay of the spatial correlation of suitable operators. We used plaquette operators in our calculations. Temporarily assigning separate gauge couplings to all plaquettes, the correlator can be defined as [@Munster:1981es] $$C(z)=\langle\mathrm{Tr}\,U_{p_1}(0)\,\,\mathrm{Tr}\,U_{p_2}(z)
\rangle=N^2\frac{\partial^2}{\partial\beta_1\partial\beta_2}\ln\,
Z(\beta,\beta_1\beta_2)\bigg\vert_{\beta_{1,2}=\beta}.$$ At zero temperature the exponential decay is the same as for correlations in the time direction, and thus determined by the gluball masses, the lowest of which may be extracted as $$m=-\lim_{z\rightarrow\infty}\frac{1}{z}\ln\,C(z).\\$$ The leading order graphs for the strong coupling series are shown in figure (\[mass\]). This leads to the lowest order contribution: $$C(z)=A\,u^{4z}=A\mathrm{e}^{-m_sz}.$$ Thus the leading order for the screening mass is given by $$m_s=-4\ln\,u(\beta).$$
Finite temperature
------------------
The graph contributing to the lowest order of the difference between the screening masses at zero and finite temperature is shown in figure (\[fig\_scrmass\]). To lowest order the mass difference is $$\begin{aligned}
\Delta m_s(N_t)&=&m_s(N_t)-m_s(\infty)\\
\\
&=&-\frac{2}{3}N_tu^{4N_t-6}\end{aligned}$$ Thus one can see that the finite temperature effect on the screening mass is very small below $T_c$, as is also observed in Monte Carlo simulations (for references, see [@Laermann:2003cv]).
Conclusions
===========
We performed explorative studies of strong coupling expansions at finite temperature. Our series for the free energy density is to the lowest orders consistent with a free glueball gas. This result justifies the neglect of the lower integration constant in numerical calculations of the equation of state by the integral method from first principles. Moreover, it gives an explanation for the success of the hadron-resonance-gas model in reproducing lattice data in the confined phase. By extrapolating the power series via Padé approximants and looking for the zeroes of the denominator, it is possible to get estimates for the critical value $\beta_c$ of the deconfining phase transition, although higher order terms seem necessary in order to obtain some accuracy here. Finally, glueball screening masses show a weak temperature dependence in the confined phase, consistent with what is found in numerical simulations.
\[mass\]
[99]{} I. Montvay and G. Münster, *Quantum fields on a lattice,* Cambridge University Press, UK (1994).
G. Münster, *Strong Coupling Expansions For The Mass Gap In Lattice Gauge Theories,* *Nucl. Phys.* [**B 190**]{} (1981) 439. K. Seo, *Glueball Mass Estimate By Strong Coupling Expansion In Lattice Gauge Theories,* *Nucl. Phys.* [**B 209**]{} (1982) 200. F. Karsch, K. Redlich and A. Tawfik, *Thermodynamics at non-zero baryon number density: A comparison of lattice and hadron resonance gas model calculations,* *Phys. Lett.* [**B 571**]{} (2003) 67 \[[arXiv:hep-ph/0306208]{}\]. G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lütgemeier and B. Petersson, *Thermodynamics of SU(3) Lattice Gauge Theory,* *Nucl. Phys.* [**B 469**]{} (1996) 419 \[[arXiv:hep-lat/9602007]{}\]. J. Fingberg, F. Karsch and U. M. Heller, *Scaling and asymptotic scaling in the SU(2) gauge theory,* *Nucl. Phys. Proc. Suppl.* [**30**]{} (1993) 343 \[[arXiv:hep-lat/9208012]{}\]. E. Laermann and O. Philipsen, *Status of lattice QCD at finite temperature,* *Ann. Rev. Nucl. Part. Sci.* [**53**]{} (2003) 163 \[[arXiv:hep-ph/0303042]{}\].
|
---
abstract: 'The elementary excitation spectrum of the spin-$\frac{1}{2}$ antiferromagnetic (AFM) Heisenberg chain is described in terms of a pair of freely propagating spinons. In the case of the Ising-like Heisenberg Hamiltonian spinons can be interpreted as domain walls (DWs) separating degenerate ground states. In dimension $d>1$, the issue of spinons as elementary excitations is still unsettled. In this paper, we study two spin-$\frac{1}{2}$ AFM ladder models in which the individual chains are described by the Ising-like Heisenberg Hamiltonian. The rung exchange interactions are assumed to be pure Ising-type in one case and Ising-like Heisenberg in the other. Using the low-energy effective Hamiltonian approach in a perturbative formulation, we show that the spinons are coupled in bound pairs. In the first model, the bound pairs are delocalized due to a four-spin ring exchange term in the effective Hamiltonian. The appropriate dynamic structure factor is calculated and the associated lineshape is found to be almost symmetric in contrast to the 1d case. In the case of the second model, the bound pair of spinons lowers its kinetic energy by propagating between chains. The results obtained are consistent with recent theoretical studies and experimental observations on ladder-like materials.'
author:
- 'Indrani Bose [^1] and Amit Kumar Pal'
title: Motion of Bound Domain Walls in a Spin Ladder
---
Department of Physics Bose Institute 93/1 A. P. C. Road, Kolkata - 700009
Introduction
============
Low-dimensional quantum antiferromagnets exhibit a rich variety of phenomena indicative of novel ground and excited state properties [@key-1; @key-2; @key-3]. In one dimension (1d), the ground and low-lying excited states of the spin chain, in which nearest neighbour spins of magnitude $\frac{1}{2}$ interact via the antiferromagnetic (AFM) Heisenberg exchange interaction, can be determined exactly using the Bethe Ansatz [@key-4; @key-5]. The ground state has no long range order and the spin-spin correlations are characterized by a power-law decay. The elementary excitation is not the conventional spin-1 magnon but a pair of spin-$\frac{1}{2}$ excitations termed spinons. The physical origin of spinons can be best understood in the Ising limit of the exchange interaction Hamiltonian given by
where $S_{i}^{\pm}$ are the spin raising and lowering operators and $N$ is the total number of spins. In the Ising limit $(J_{xy}=0)$, the ground states of the Hamiltonian are the doubly degenerate Néel states, one of which is shown in figure 1(a). An excited state is created by flipping a spin from its ground state arrangement, e.g., a down spin is flipped into an up spin in figure 1(b). The flip gives rise to two domain walls (DWs) consisting of parallel spins and shown by dotted lines in the figure [@key-1; @key-6]. The transverse exchange interaction term in (1) interchanges the spins in an antiparallel spin pair and has the effect of making the DWs propagate independently (figure 1(c)). Since the original spin flip carries spin one, each of the DWs or spinons has spin-$\frac{1}{2}$ associated with it. Spinons are thus examples of fractional excitations in an interacting spin system.
Spinons can be detected through inelastic neutron scattering in which neutrons scatter against spins to create spin flips. Due to energy and momentum conservation, the energy absorption spectrum for spin flips at different wave vectors can be measured. One observes a peak at a well-defined energy if the spin flip creates a single particle excitation. In the case of a pair of spinons, the total energy $\epsilon$ and momentum $k$ of the spin excitation are given by $\epsilon(k)=\epsilon_{1}(k_{1})+\epsilon_{2}(k_{2})$ and $k=k_{1}+k_{2}$ where $\epsilon_{i}$, $k_{i}$ $(i=1,2)$ denote individual spinon energy and momentum [@key-1; @key-6]. The total momentum $k$ of the spin flip can be distributed in a continuum of ways among the spinons giving rise to a continuous absorption spectrum. For a single particle excitation, the energy versus momentum relation defines a single branch of excitations whereas for spinons a continuum of excitations with well-defined lower and upper boundaries is obtained. The compounds $CsCoCl_{3}$ and $CsCoBr_{3}$ are good examples of Ising-like Heisenberg antiferromagnets in 1d above the Néel temperature and provide evidence of the two-spinon continuum in neutron scattering experiments [@key-6; @key-7; @key-8]. In the case of the isotropic Heisenberg Hamiltonian ($J_{z}=J_{xy}$ in equation (1)), the spinon spectrum has been clearly observed in the linear chain compound $KCuF_{3}$ [@key-9] though a physical interpretation of spinons is not as straightforward as in the Ising-like case.
The existence of spinons, with fractional quantum number spin-$\frac{1}{2}$, is well established in 1d Heisenberg-type antiferromagnets. In higher dimensions, the spin-1 magnons are the elementary excitations in magnetically ordered ground states. There are theoretical suggestions that quantum antiferromagnets with spin-liquid (no magnetic long range order and without broken symmetry) ground states may support elementary spinon-like excitations with fractional quantum numbers [@key-10; @key-11; @key-12]. A well-known example is that of a resonating-valance-bond (RVB) state, a linear superposition of VB states, in which the spins are paired in singlet (VB) configurations. A broken VB gives rise to a pair of free spins which may propagate independently to give rise to spinon excitations. If the energetic cost of deconfinement is high, the spinons propagate as a bound pair (confinement) so that the elementary excitation has spin-1.
Despite considerable effort, there are few experimental evidences of spinon-like excitations in $d>1$ [@key-13; @key-14]. One strong candidate is $Cs_{2}CuCl_{4}$, a spin-$\frac{1}{2}$ Heisenberg antiferromagnet defined on a spatially anisotropic triangular lattice [@key-13]. The dynamical structure factor $S(k,\omega)$ for $Cs_{2}CuCl_{4}$, where $k$ and $\omega$ are the momentum and energy transfers in the neutron scattering experiment, is dominated by a broad continuum which has been cited as evidence for fractionalized excitations. Kohno et al [@key-15] reanalyzed the neutron scattering data to show that the spinons are not characteristic of some exotic 2d state but are descendants of the weakly-coupled excitations of individual chains in the material. The spectrum also has a sharp dispersing peak attributed to ‘triplon’ bound states of the spinons. The bound pair lowers its kinetic energy through propagation between chains. The issue of fractional versus integer excitations has been extensively investigated in AFM spin-$\frac{1}{2}$ ladders [@key-16; @key-17; @key-18; @key-19; @key-20]. A two-chain ladder consists of two AFM chains coupled by rung exchange interactions. The spinons of individual chains are confined even if the rung exchange interaction strength is infinitesimal. Ladders with strong rung exchange couplings suppress spinon excitations at all energy scales. Recently, Lake et al [@key-21] have carried out neutron scattering experiments on a weakly-coupled ladder material, $CaCu_{2}O_{3}$, and shown that deconfined spinons at high energies evolve into $S=1$ excitations at lower energies. The spinons are associated with individual chains whereas the $S=1$ excitations are the triplon excitations, i.e., bound states of spinons. Two approaches are usually adopted in probing the nature of excitations in spin ladders: (i) the rung coupling strength dominates and (ii) the rung coupling strength is weaker than the intra-chain coupling strengths [@key-21]. In this paper, we consider a different case, not studied earlier, in which two $S=\frac{1}{2}$ Ising-like Heisenberg AFM chains, each of which is described by a Hamiltonian of the type shown in equation (1), are coupled by Ising or Ising-like Heisenberg AFM exchange interactions. In section II, we investigate the nature of the low-lying excitations with Ising rung exchange interactions. In section III, the rung exchange interactions are considered to be Ising-like Heisenberg AFM in nature. Section IV contains concluding remarks.
Ising Rung Exchange Interactions
================================
We consider an antiferromagnetic two-chain spin ladder with the spins of magnitude $\frac{1}{2}$. The individual chains of the ladder are described by the Ising-Heisenberg Hamiltonian (equation(1)). The chains are coupled by rungs with the corresponding exchange interactions being of the Ising-type. The ladder Hamiltonian $H_{L}$ is given by where the index $\alpha=1(2)$ refers to the top (bottom) chain of the ladder, $i$ denotes the site index and $N$ is the total number of rungs. We also assume that the anisotropy constant $\epsilon=\frac{J_{XY}}{J_{Z}}$ is $<<1$. Hence, the Ising part of the Hamiltonian, $H_{Z}$, can be considered to be the unperturbed Hamiltonian with $H_{XY}$, containing the transverse exchange interactions, providing the perturbation. Since $H_{Z}$ is AFM in nature, the lowest energy states are the Néel states with n.n. spin pairs antiparallel. In the spirit of Villain [@key-6], we first consider a ladder with an odd number of rungs, i. e., $N$ = odd and periodic boundary conditions (PBCs). Energies are measured w. r. t. that of a Néel configuration of spins. Since there are $3N$ n.n. spin pairs, the energy of a configuration in which all such pairs are antiparallel is $E_{N\acute{e}el}=-3N\;\frac{J_{Z}}{4}$. Since $N$ is odd, a perfect Néel configuration is not possible and the lowest energy states of $H_{Z}$ contain a pair of parallel spin pairs which define the DWs or spinons (figure 2 (a)). The DWs form a bound pair to ensure minimal energy loss. Any other arrangement of DWs in the individual chains gives rise to higher energy states. The lowest energy states are $N$-fold degenerate as there are $N$ possibilities for the location of the bound pair which is an $S_{z}=0$ object.
We next consider the effect of the perturbing Hamiltonian on the minimum energy states. The transverse exchange interaction can give rise to independent DW motion in the chains which, however, costs energy as a propagating DW leaves in its wake ferromagnetically aligned rung spins (figure 2(b)). The energy cost increases as the distance between the DWs increases resulting in confinement of the DW pair. We investigate the dynamics of the DW pair perturbatively using a low-energy effective Hamiltonian (LEH) [@key-22; @key-23]. The $N$-fold degenerate DW pair states $|p_{i}\rangle,\; i=1,...,N$ (figure 2(a)) constitute the low-energy manifold and have energy $E_{0}=-\frac{3NJ_{Z}}{4}+J_{Z}$. The higher energy states of $H_{Z}$ are denoted by $|q_{\alpha}\rangle$ with energy $E_{\alpha}$. The perturbing Hamiltonian $H_{XY}$ connects the low-energy manifold to the manifold of higher-energy states. In general, perturbation lifts the degeneracy of the low-energy manifold leading to an effective Hamiltonian operating in the space of states associated with the low-energy manifold. Diagonalization of the $n$-th order $(n=1,2,...)$ effective Hamiltonian in the low-energy subspace of states reproduces the $n$-th order energy corrections to the low-energy unperturbed states. Using degenerate perturbation theory, the first order LEH is given, up to an overall constant, by [@key-23] The second-order LEH has the form Since the matrix element $\langle p_{i}|H_{XY}|p_{j}\rangle=0$, the LEH is determined using the second-order expression in equation (4). Two types of processes contribute to $H_{eff}^{(2)}$. In “diagonal” processes, the spins in an antiparallel pair exchange and then reexchange back to the original configuration $(|p_{i}\rangle=|p_{j}\rangle)$. Such processes do not lift the degeneracy and give rise to a constant energy shift. We neglect this contribution in deriving the effective Hamiltonian. In the off-diagonal processes, the spins in two antiparallel pairs belonging to the four-spin plaquettes bordering the bound DW pair are interchanged. For the DW pair state shown in figure 2(a), the intermediate states $|q_{i}\rangle,\; i=1,..,4$ are given by $$\begin{aligned}
|q_{1}\rangle & = & \begin{array}{ccccccc}
\uparrow & \Uparrow & \Downarrow & \uparrow & \downarrow & \uparrow & \downarrow\\
\downarrow & \uparrow & \downarrow & \downarrow & \uparrow & \downarrow & \uparrow\end{array}\\ \nonumber
|q_{2}\rangle & = & \begin{array}{ccccccc}
\uparrow & \downarrow & \uparrow & \uparrow & \downarrow & \uparrow & \downarrow\\
\downarrow & \Downarrow & \Uparrow & \downarrow & \uparrow & \downarrow & \uparrow\end{array}\\ \nonumber
|q_{3}\rangle & = & \begin{array}{ccccccc}
\uparrow & \downarrow & \uparrow & \Downarrow & \Uparrow & \uparrow & \downarrow\\
\downarrow & \uparrow & \downarrow & \downarrow & \uparrow & \downarrow & \downarrow\end{array}\\ \nonumber
|q_{4}\rangle & = & \begin{array}{ccccccc}
\uparrow & \downarrow & \uparrow & \uparrow & \downarrow & \uparrow & \downarrow\\
\downarrow & \uparrow & \downarrow & \Uparrow & \Downarrow & \downarrow & \uparrow\end{array}\end{aligned}$$
The spin pairs deviated from the arrangement shown in figure 2 (a) are marked by double arrows. The perturbing Hamiltonian, $H_{XY}$, acting on the intermediate $|q_{\alpha}\rangle$ states shifts the location of the bound DW pair by two lattice constants either towards the left or the right. This is illustrated in figure 3 for the state $|q_{1}\rangle$. The energy of the states $|q_{\alpha}\rangle\;(\alpha=1,...,4)$ is $E_{\alpha}=-\frac{3NJ_{Z}}{4}+2J_{Z}$. The second-order LEH is thus given by where $\epsilon=\frac{J_{XY}}{J_{Z}}$. The off-diagonal processes are equivalent to “ring” exchanges involving four spins. The full second-order Hamiltonian, defined in the low-energy manifold, is thus given by where with $J_{ring}=-\frac{\epsilon^{2}J_{Z}}{2}$ and the sum over all elementary plaquettes of the ladder. Ring or cyclic exchange interactions (equation (8)) also appear in the perturbative effective Hamiltonian theories developed for the XXZ Heisenberg model on the checkerboard lattice [@key-24] and in the case of an easy-axis Kagomé antiferromagnet [@key-25]. In the ladder model, the ring exchange interaction has the effect of deconfining the bound DW pair. In the low-energy subspace, the dispersion relation of the bound pair can be determined in a straightforward manner. The eigenstate $\psi(k)$ of the pair can be written as a linear combination of staes $|j\rangle(j=1,2,...,N)$ where $j$ denotes the location of the bound DW pair. $H_{eff}$ (equation (7)) operating on $|\psi(k)\rangle$ yields the eigenvalue The subscript ‘$b$’ in $\omega_{b}$ denotes that the dispersion relation is that of a bound DW pair.
We next consider a two-chain spin ladder with an even number $N$ of rungs and described by the Hamiltonian in equation (7) satisfying PBCs. The low-lying excitation spectrum is obtained in the subspace of degenerate eigenstates of $H_{Z}$ which are generated by flipping all the spins in a block of $\mu$ ($\mu$ may be odd/even) adjacent rungs in the ground state (Néel state) of $H_{Z}$. Each of the states contains two bound DW pairs (figure 4) and has energy
The $z$-component of the total spin of each state, $S_{z}^{tot}=0$. We consider $\mu$ to be odd with the degenerate eigenstates of $H_{Z}$ given by where $\mathcal{S}_{k}=S_{k,1}^{+}S_{k,2}^{-}$ and $\mathcal{S}^{\prime}_{k}=S_{k,1}^{-}S_{k,2}^{+}$. $\psi_{N\acute{e}el1}$ is the ground state of $H_{Z}$ shown in figure 4(c). The Hamiltonian $H_{ring}$ in (7) has the following matrix elements between the Ising eigenstates : where $J_{ring}=-\frac{\epsilon^{2}J_{Z}}{2}$. The low-lying excited state of the Hamiltonian $H_{eff}$ (equation (7)) is given by From the eigenvalue equation $H_{eff}\psi_{DW}(k)=\lambda\psi_{DW}(k)$, one obtains where The diagonal matrix element is $2J_{Z}$ as energies are measured with respect to the Néel state energy $-\frac{3NJ_{Z}}{4}$ (see equation (11)). We choose the coefficients $C_{\nu}$’s to be $C_{\nu}=e^{-i\phi\nu}$. The eigenvalues constitute an excitation continuum given by where $-\pi<\phi\leq\pi$.
Figure 5 shows the excitation continuum with upper and lower bounds given by $2J_{Z}(1\pm\epsilon^{2}\cos k)$. The degenerate eigenstates of $H_{Z}$ defined in (12) correspond to the top chain of the spin ladder being in as $S_{tot,1}^{z}=+1$ and the bottom chain being in an $S_{tot,2}^{z}=-1$ state. One can construct a set of degenerate eigenstates with the situation reversed. Also, $\mu$, the number of adjacent rungs constituting the block of flipped spins can be even $(\mu=2,4,6,...etc.)$. There are two distinct sets of such states [@key-8; @key-26]. All these subspaces of states give rise to the same excitation continuum (figure 5). The excitation continuum arises due to the motion of two walls each of which consists of a bound pair of DWs.
One notes that the effects of $H_{ring}$, for the two-chain ladder, and $H_{XY}$, for the 1d chain, on the DW states are similar. In the first case, a bound pair of DWs shifts by two lattice constants and in the second case a single DW shifts by the same distance. In the case of a single chain, the eigenvalues $\lambda_{1d}$ constituting the excitation continuum are Comparing equations (17) and (18), one finds that the spread of the continuum around the unperturbed level is less in the case of the spin ladder.
The dynamic form factor, $S_{11}(q,\omega)$, associated with the bound DW pair is defined at $T=0$ by where In (19), $|g\rangle$ and $|f\rangle$ are the ground and excited states of $H_{eff}$ connected by $A(q)$, with energies $E_{g}$ and $E_{f}$ respectively, $\omega$ and $q$ are the frequency and wave number of the excitation. $S_{11}(q,\omega)$, involving a pair of spin deviations, could be probed by the light-scattering techniques [@key-27; @key-28]. Upto the first order of $J_{ring}$ ($J_{ring}<<J_{Z}$ in (7)), where $E_{0}$ is the energy of $\psi_{N\acute{e}el1}$. Since $H_{ring}$ acting on $\psi_{N\acute{e}el1}$ creates two bound DW pairs, $\frac{1}{E_{0}-H_{Z}}=-\frac{1}{2J_{Z}}.$Thus, where $\psi_{1}(q)$ and $\psi_{3}(q)$ are as defined in equation (12) and $v^{*}=-\frac{\epsilon^{2}J_{Z}}{2}\left(1+e^{2iq}\right)$. Using equation (22) and (19) and the expression (14) for $|f\rangle=\psi_{DW}(k)$, one gets following the procedures described in [@key-7; @key-26] with $\Omega=\omega-2J_{Z}$.
The expression (23) is similar to that for the dynamic structure factor $S_{xx}(q,\omega)$ of the $1d$ chain obtained in first order perturbation theory [@key-7] ($A_{l}=S_{l}^{x}$ in equation (20)) except that in the latter case, $\Omega=\omega-J_{Z}$ and the contribution of the anisotropy term is to first order in $\epsilon$. Figure 6 shows the plots of $S_{11}(q,\omega)\times2J_{Z}|\cos q|$ versus $\frac{\omega}{J_{Z}}$ for $\epsilon=0.15$ and for various values of the wave number $q$. The lineshape is almost symmetric in contrast to the prominent asymmetry found in the 1d case [@key-7].
   
Ising-Heisenberg Rung Exchange Interactions
===========================================
We now consider the case in which the rung exchange interactions of the two-chain spin ladder are Ising-like Heisenberg-type. The Hamiltonian is given by The Hamiltonian (24) differs from $H_{L}$ is equation (2) by the addition of the last term. The ground states of $H_{Z}$ are the doubly degenerate Néel states. We consider the Néel state $\psi_{N\acute{e}el1}$ shown in figure 4(c). The ladder can be divided into two sublattices $A$ and $B$ such that in $\psi_{N\acute{e}el1}$ the $A(B)$ sublattice spins are pointing up (down). The ground state energy $E_{0}=-\frac{3NJ_{Z}}{4}$.
The lowest energy excitation of the unperturbed Hamiltonian is obtained by flipping a single spin in either the $A\;(S_{tot}^{z}=-1)$ or the $B\;(S_{tot}^{z}=+1)$ sublattice. We consider the latter case with $|i\rangle$ denoting the state in which the flipped spin is located in the $i$th rung (figure 7(a)). These excited states are $N$-fold degenerate with the energy The perturbing Hamiltonian acting on the state $|i\rangle$ generates the following states where $\epsilon=\frac{J_{XY}}{J_{Z}}$. The states $|m\rangle(m=1,...,4)$ are shown in figure 7(b) with energy The other states which are generated when $H_{XY}$ acts on the state $|i\rangle$ have higher energies and are hence not considered. $H_{XY}$ acting on the states $|m\rangle$ gives
In first order perturbation theory there is no energy correction. A finite energy correction is obtained in the second order perturbation theory. The effective LEH (equation (4)) with $|p_{i}\rangle=|i\rangle,\;|p_{j}\rangle=|j\rangle,\;|q_{\alpha}\rangle=|m\rangle$ and $E_{\alpha}=E_{m}$ in the same order is given by The effect of this Hamiltonian on the low-energy excited state $|i\rangle$ (figure 7(a)) is to shift the flipped spin from the $i$th to the $(i+1)$th or the $(i-1)$th rungs. Since the flipped spins are located in the $B$ sublattice, the shift is in the diagonal direction. The flipped spin is associated with a bound pair of DWs in a chain. The bound pair lowers its kinetic energy by propagating between chains. The full second-order Hamiltonian defined in the low-energy manifold of states with single spin flips, is As before, we have not included the terms arising from the “diagonal” processes in equation (4) as they give rise to a constant energy shift. The low energy excited state with $S_{tot}^{z}=+1$ can be constructed as where ‘$B$’ denotes the $B$ sublattice. The dispersion relation for the propagation of the flipped spin or equivalently the bound DW pair is given by where the energy is measured w. r. t the Néel state energy. The bound DW pair moves diagonally across the spin ladder. The unperturbed Hamiltonian, $H_{Z}$, is the same irrespective of whether the rung exchange interactions are Ising-type or Ising-Heisenberg-type. Thus, the lowest unperturbed excited state is the single flip state in both the cases. The excitation has a localized character when the rung exchange interactions are of the Ising-type. The bound DW pair associated with the single spin-flip can not propagate between the chains as the inter-chain interactions are Ising-like and propagation of the DWs in a single chain is, as pointed out before, energetically prohibitive. The energy, $-\frac{3J_{Z}}{2}$, of the localized excitation is lower than that of the propagating excitations involving two bound DW pairs when the rung exchange interactions are of the Ising-type (section 2). In this case, propagating excitations with the lowest energy involve two bound DW pairs rather than one.
Concluding Remarks
==================
AFM spin models in which the existence of spinons is well-established include the spin-$\frac{1}{2}$ Heisenberg AFM chain [@key-1], the Majumdar-Ghosh model [@key-29] and the Haldane-Shastry model [@key-30; @key-31]. The physical picture of a spinon as a DW between two degenerate ground states emerges in the Ising-Heisenberg limit of the AFM Hamiltonian [@key-1; @key-7; @key-8]. In the case of the MG model, the spin-$\frac{1}{2}$ excitation acts as a DW between the two dimerized ground states of the model [@key-32; @key-33]. In a closed chain, the DWs occur in pairs so that the lowest-lying excitation is given by the two-spinon continuum. The spinons are deconfined in this case and can move away from each other. There is no energy cost in moving the spinons far apart. This is not so when two AFM chains are coupled in the form of a spin ladder. Let $J_{\bot}$and $J_{||}$ be the strengths of the rung and intra-chain n.n. exchange interactions respectively. The spinon excitations of individual chains are confined by even an infinitesimal coupling strength $J_{\bot}$[@key-18; @key-21]. The two $S=\frac{1}{2}$ spinons form a bound state giving rise to singlet and triplet excitation branches. In the case of the strongly coupled ladder $(J_{\bot}>>J_{||})$, the elementary excitation is a triplet. Lake et al. [@key-21] carried out neutron scattering experiments on the weakly-coupled $(J_{\bot}<<J_{||})$ ladder material $CaCu_{2}O_{3}$ and obtained evidence of the singlet excitation mode. The spinon continuum was observed at high energies for which the chains are effectively decoupled. The spinons in a chain evolve into an $S=1$ excitation at lower energies thereby confirming that the $S=1$ “triplon” excitation is a bound state of two spinons and not a conventional magnon.
In this paper, we study a two-chain $S=\frac{1}{2}$ AFM spin ladder in which the individual chains are described by the Ising-like Heisenberg Hamiltonian and the rung couplings are of the Ising-type. Using a low-energy effective Hamiltonian approach, we establish that in a ladder with an odd number of rungs the spinons (DWs) form a bound pair. A four-spin ring exchange interaction in the effective Hamiltonian is responsible for the delocalization of the bound pair. In the case of a ladder with an even number of rungs, the low-lying propagating excitation involves two bound pairs of DWs which can move away from each other giving rise to a continuum of excitations. The physical origin of the excitation continuum is similar to that in the case of the Ising-Heisenberg AFM chain in 1d except that in the former case the spinons form bound pairs and the dispersion of the excitation spectrum is a higher-order effect in perturbation theory. This results in an almost symmetric lineshape in the case of the dynamic structure factor $S_{11}(q,\omega)$ (Figure 6) in contrast to the asymmetry observed in the structure factor $S_{xx}(q,\omega)$ in 1d [@key-7]. The delocalization of a bound spinon pair is brought about via a ring or a diagonal exchange interaction term in the effective Hamiltonian.
We further consider a second model in which the rung exchange interactions are described by the Ising-like Heisenberg Hamiltonian. In this case also, the spinon pair in a single chain is bound and the bound pair lowers its kinetic energy by hopping between chains. Kohno et al. [@key-15] have studied a $S=\frac{1}{2}$ spatially anisotropic frustrated Heisenberg antiferromagnet in 2d in the weak interchain coupling regime. The model provides a good quantitative fit to the inelastic inelastic neutron scattering data of the triangular antiferromagnet $Cs_{2}CuCl_{4}.$ The spectrum consists of a continuum arising from the deconfinement of spinons in individual chains and a sharp dispersing peak associated with the coherent propagation of a triplon bound state of two spinons between neighbouring chains. In the case of our model, the bound pair has $S^{z}=+1$. One can similarly construct an $S^{z}=-1$ excitation. In summary, we have studied ladder models with the Ising-like Heisenberg Hamiltonian describing the interactions in individual chains. The rung interactions may be pure Ising or Ising-like Heisenberg. The ladder models studied in this paper share some common features with models in which the exchange interactions are isotropic. In the latter case, two types of models have generally been considered : ladder models in which the rung exchange interactions are the most dominant and models which describe spin chains coupled by weak exchange interactions. The models considered in this paper belong to a category not studied earlier and provide considerable physical insight on the origin of spinon confinement and how the bound spinon pairs delocalize. The isotropic models have, however, a richer dynamics with interactions generating cascades of virtual particles so that the two-body confinement problem becomes a many-body one [@key-21]. The major common feature emerging from the study of ladder models with both Ising-like and isotropic exchange interactions appears to be the confinement of spinons in the form of bound states. The origin of excitation continua in specific cases lies in multi-triplon excitations rather than in the fractionalization of excitations [@key-18].
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[^1]: indrani@bosemain.boseinst.ac.in
|
---
abstract: 'A structural model for intermediate sized silicon clusters is proposed that is able to generate unique structures without any dangling bonds. This structural model consists of bulk-like core of five atoms surrounded by fullerene-like surface. Reconstruction of the ideal fullerene geometry results in the formation of crown atoms surrounded by $\pi$-bonded dimer pairs. This model yields unique structures for , , and clusters without any dangling bonds and hence explains why these clusters are least reactive towards chemisorption of ammonia, methanol, ethylene, and water. This model is also consistent with the experimental finding that silicon clusters undergo a transition from prolate to spherical shapes at . Finally, reagent specific chemisorption reactivities observed experimentally is explained based on the electronic structures of the reagents.'
address: 'The Department of Chemistry, New York University, New York, NY 10003-6621.'
author:
- 'Jun Pan[^1] and Mushti V. Ramakrishna'
date: 'Submitted to Phys. Rev. Lett., '
title: Magic Numbers of Silicon Clusters
---
= 10000
Smalley and co-workers carried out extensive studies on chemisorption of various reagents on intermediate sized silicon clusters [@Elkind:87]. These studies revealed that the reactivity rates for ammonia (NH$_3$) chemisorption vary by over three orders of magnitude as a function of cluster size, with 21, 25, 33, 39, and 45 atom clusters being particularly unreactive. Clusters containing more than forty seven atoms are highly reactive and they do not display strong oscillations in reactivities as a function of the cluster size. Similar results were obtained with methanol (CH$_3$OH), ethylene (C$_2$H$_4$), and water (H$_2$O). On the other hand, nitric oxide (NO) and oxygen (O$_2$) were found to react readily with all clusters in this size range.
Several structural models have been proposed to explain these experimental data [@Phillips:88; @Jelski:88; @Kaxiras:89; @Patterson:90; @Swift:91]. However, these models do not yield unique and consistent structures for different cluster sizes and each cluster structure has to be obtained independently. Furthermore, these models do not satisfy the essential criterion that the structures of the unreactive clusters should not have any dangling bonds.
In this Letter we propose a consistent model that generates the structures of the intermediate sized unreactive silicon clusters in a systematic way. The structures thus generated are unique for , , and clusters. Furthermore, these clusters do not have any dangling bonds and hence explains why these clusters are unreactive for chemisorption. Finally, our model is consistent with the experimental finding of Jarrold and Constant that silicon clusters undergo a shape transition from prolate to spherical shapes at [@Jarrold:91].
Our structural model for silicon clusters consists of bulk-like core of five atoms surrounded by fullerene-like surface. The core atoms bind to the twelve surface atoms, thus rendering them bulk-like with four-fold coordination. The surface then relaxes from its ideal fullerene geometry to give rise to crown atoms and dimer pairs. The crown atoms are called adatoms in surface science literature [@crown]. These crown atoms are formally three-fold coordinated and possess one dangling bond each. The dimer pairs are also formally three-fold coordinated, but they eliminate their dangling bonds through local $\pi$ bonding. This model yields structures with seventeen four-fold coordinated atoms, four crown atoms, and the rest dimer pairs. The essential feature of this construction is that the bulk-like core and fullerene-like surface make these structures stable. This model is applicable to clusters containing more than twenty atoms.
Unlike carbon, silicon does not form strong delocalized $\pi$ bonds. Consequently, fullerene cage structures [@Curl:91; @Boo:92], which require strong delocalized $\pi$ bonds, are energetically unfavorable for silicon. For this reason, silicon clusters prefer as few surface atoms as possible. Nonetheless, the fullerene geometry for the surface, consisting of interlocking pentagons and hexagons, gives special stability to the surface atoms. Furthermore, since delocalized $\pi$ bonding is not favorable in silicon, we expect the surface atoms to relax from their ideal fullerene geometry to allow for dimer formation through strong local $\pi$ bonding. Our model accounts for all these facts.
The 5-atom core in our model has the exact structure of bulk silicon with one atom in the center bonded to four atoms arranged in a perfect tetrahedral symmetry. There are numerous ways to orient the 5-atom core inside the fullerene cage and bind it to the surface atoms. Furthermore, structural isomers may exist for fullerenes of any size [@Boo:92; @Fowler:92]. Thus it is possible to use this model to generate all possible structural isomers for any odd numbered intermediate sized cluster. However, a particular orientation of the 5-atom core inside the fullerene cage will yield structures with maximum number of dimer pairs and least number of dangling bonds. Such isomers are likely to be most stable.
The 28- and 40-atom fullerenes are the only ones belonging to the symmetry group $T_d$ in the 20-60 atom size range [@Boo:92]. We generate the and structures by inserting the 5-atom core inside the 28- and 40-atom fullerene, respectively. We orient the 5-atom pyramid in such a way that the central atom (A, violet), the apex atom (B, blue) of the pyramid, and the crown atom (C, red) on the surface lie on a line. The C atom is the central atom shared by three fused pentagons. This atom is surrounded by three other surface atoms (D, green), which relax inwards to bind to the four core atoms B. This relaxation motion is identical to that necessary to form the 2 $\times$ 1 reconstruction on the bulk Si(111) surface [@Cohen:84; @Lannoo:91]. With an activation barrier of only $\approx$ 0.01 eV [@Cohen:84; @Northrup:82] and gain of 2.3 eV/bond due to $\sigma$ bond formation between B and D [@Lannoo:91; @Brenner:91], such a relaxation of fullerene surface is feasible even at 100 K. Finally, the remaining surface atoms (E, orange) readjust to form as many dimers as possible. This relaxation is similar to that on Si(100) surface yielding dimer pairs [@Lannoo:91; @Chadi:79]. The dimers are $\sigma$-bonded pair of atoms whose dangling bonds are saturated through strong local $\pi$ bonds. Because of the tetrahedral symmetry of the core as well as of the reconstructed fullerene cage, the final and structures also have tetrahedral symmetry. The structure is also generated in this way, starting from the 34-atom fullerene cage and stuffing five atoms inside it. However, the structure thus generated has only C$_{3v}$ symmetry, because of the lower symmetry of the fullerene cage [@Boo:92].
The , , and cluster structures thus generated are displayed in Fig. 1. These structures have seventeen four-fold coordinated atoms, four crown atoms, and variable number of dimer pairs. The structure has six dimer pairs, whereas has twelve dimer pairs forming four hexagons. On the other hand, the surface of the cluster consists of nine dimer pairs, three of which form a hexagon.
We constructed and clusters also using the proposed model, but found that these structures do not possess the characteristic crown-atom-dimer pattern found in unreactive clusters. Consequently, these structures possessed large number of dangling bonds. This explains the highly reactive nature of these clusters.
In principle, the proposed model can be used to construct structure also. However, cage is too small to accommodate five atoms inside it without undue strain. Consequently, will prefer a different geometry. Indeed, there is experimental evidence that clusters smaller than do not possess spherical shapes characteristic of larger clusters; instead they seem to prefer prolate shapes [@Jarrold:91]. Our inability to generate a compact structure for explains this experimental observation also.
We verified our model by constructing structures of all the clusters discussed here using the molecular modelling kits [@Jones:94]. These structures unambiguously demonstrated that only 33-, 39-, and 45-atom clusters possess the crown-atom-dimer pattern on their surfaces. We then calculated the atomic coordinates of these clusters from the tetrahedral geometry of the 5-atom core and the known geometries of the fullerene structures [@Boo:92], assuming that all bond lengths are equal to the bulk value of 2.35 Å. The structures thus generated are displayed in Fig. 1. We then relaxed these structures by carrying out molecular dynamics at 100 K using the Kaxiras-Pandey potential [@Kaxiras:88]. The final structures obtained from these simulations are nearly identical to the initial ones. This indicates that the proposed structures are locally stable.
The computer generated structures displayed in Fig. 1 revealed that the crown atoms are able to form a fourth bond to the core atoms B, thus rendering the B atoms formally five-fold coordinated. The B-C bond arises from the back donation of the electrons from C to B and it weakens the neighboring bonds through electronic repulsion. The dangling bond on the crown atom is thus eliminated and these magic number clusters become unreactive.
The classical potentials available at present [@Kaxiras:88; @Stillinger:85] are most appropriate for bulk silicon and related structures. Such potentials may not be suitable for describing unusual bonding patterns, such as the five-fold coordinated silicon atoms found in these clusters. Consequently, we cannot use the available classical potentials to prove that the proposed structures correspond to the lowest energy structures. Finally, structural determination based on the first principles electronic structure calculations are extremely demanding computationally at present for such large clusters, especially if adequate basis sets and grid sizes are employed and the calculations are converged to better than 0.01 eV accuracy. Furthermore, such complicated calculations do not provide the conceptual framework for understanding cluster properties. On the other hand, our simple physically motivated stuffed fullerene model yields insights into the nature of bonding in silicon clusters and explains the experimental trends in reactivities.
There exist several competing structural models [@Phillips:88; @Jelski:88; @Kaxiras:89; @Patterson:90; @Swift:91] of silicon clusters that attempt to explain the experimental reactivity data of Smalley and co-workers [@Elkind:87]. For example, Kaxiras has proposed structures of Si$_{33}$ and Si$_{45}$ clusters based on the reconstructed 7 $\times$ 7 and 2 $\times$ 1 surfaces of bulk Si(111), respectively [@Kaxiras:89]. However, this model does not explain the reactivity data since bulk surfaces are highly reactive. Furthermore, it is inconsistent that the surface of should be the metastable 2 $\times$ 1 surface rather than the more stable 7 $\times$ 7 surface [@Lannoo:91]. Finally, the structure of Kaxiras has forty dangling bonds, which make it highly reactive, contrary to the experiments. To overcome this discrepancy between experiment and theory, Kaxiras has proposed that the dangling bonds on form $\pi$-bonded chains, analogous to the 2 $\times$ 1 reconstruction of the bulk Si(111) surface [@Pandey:81]. However, silicon favors strong local $\pi$ bonds over delocalized $\pi$-bonded chains. This is the reason why the 2 $\times$ 1 reconstruction, involving $\pi$-bonded chains [@Pandey:81], is metastable with respect to the locally $\pi$-bonded 7 $\times$ 7 reconstruction on the bulk Si(111) surface [@Lannoo:91]. For the same reason, the structure proposed by Kaxiras is a metastable and highly reactive structure. Indeed, Jelski and co-workers disputed the Kaxiras model of by constructing alternative structures for Si$_{45}$ that are lower in energy, but do not possess any of the features of the reconstructed bulk surfaces [@Swift:91].
The bonding characteristics of silicon differ in subtle ways from that of carbon. In carbon, delocalized $\pi$ bonds are favored over local $\pi$ bonds, whereas the opposite is true in silicon. For this reason, graphite is the most stable form of carbon at room temperature and atmospheric pressure, but not the graphite form of silicon. Likewise, the bulk (111) surface of the diamond form of carbon exhibits the 2 $\times$ 1 reconstruction, but not the 7 $\times$ 7 reconstruction [@Pate:86; @Bokor:86]. These examples, illustrate how subtle differences in bonding characteristics determine possible crystal structures and surface reconstructions. The same is true of clusters and the models of cluster structures should account for these characteristics. Our model of silicon clusters accounts for these facts by focussing on structures that are able to form maximum number of $\sigma$ bonds and eliminate their surface dangling bonds through local $\pi$ bonding.
Our structure for is identical to that proposed by Kaxiras [@Kaxiras:89] and Patterson and Messmer [@Patterson:90]. This structure has been shown to be locally stable [@Feldman:91]. But our structure is different from that of Kaxiras [@Kaxiras:89]. However, we can generate the structure of Kaxiras by stuffing one atom inside a 44-atom fullerene cage and allowing for the reconstruction of the fullerene surface. Thus our model is very general, subsuming the Kaxiras model within it.
The reactivity patterns of NO and O$_2$ are different from those found for NH$_3$, CH$_3$OH, C$_2$H$_4$, and H$_2$O [@Elkind:87]. This may be explained based on the ground state electronic structures of these reagents. NH$_3$, CH$_3$OH, C$_2$H$_4$, and H$_2$O in their ground states have closed shell electronic structure with all electrons paired. On the other hand, NO and O$_2$ in their ground states are $^2\Pi_g$ and $^3\Sigma_g^-$, possessing one and two unpaired electrons, respectively [@Herzberg:50]. Consequently, NH$_3$, CH$_3$OH, C$_2$H$_4$, and H$_2$O can chemisorb only at those sites where excess electron density is present due to dangling bonds. Such a selectivity gives rise to highly oscillatory pattern in the reactivities, because the number of dangling bonds varies as a function of cluster size. The magic number clusters are unreactive because they do not possess any dangling bonds. On the other hand, NO $(^2\Pi_g)$ and O$_2$ $(^3\Sigma_g^-)$ can chemisorb anywhere, because these reagents carry the necessary dangling bonds for instigating the reaction anywhere on the cluster surface. Hence, NO and O$_2$ readily react with all clusters and do not display the oscillatory pattern in their chemical reactivities. This explains the reagent specific chemisorption reactivities observed experimentally [@Elkind:87].
The magic number clusters are not completely inert towards the closed shell reagents [@Elkind:87]. These clusters are more reactive towards NH$_3$, CH$_3$OH, and H$_2$O than towards C$_2$H$_4$. This is because NH$_3$, CH$_3$OH, and H$_2$O have lone pairs on either nitrogen or oxygen and these lone pairs have a small probability of instigating reaction on the cluster surface. A lone pair is a pair of electrons that is not part of a bond. C$_2$H$_4$ does not have any lone pairs and hence the magic number clusters are quite unreactive towards this molecule. The electronic the structure of reagents thus explains even subtle variations in the reactivities of magic number clusters towards a group of related reagents.
In summary, we propose a structural model for the unreactive silicon clusters containing more than twenty atoms. This model consists of bulk-like core of five atoms surrounded by reconstructed fullerene surface. The resulting structures for , , and are unique, have maximum number of four-fold coordinated atoms, minimum number of surface atoms, and zero dangling bonds. Such unique structures cannot be built for other intermediate sized clusters and hence they will have larger number of dangling bonds. This explains why , , and clusters are least reactive towards closed shell reagents ammonia, methanol, ethylene, and water [@Elkind:87]. Our model also indicates that cluster cannot be formed in a spherical shape. This result is consistent with the experimental finding that silicon clusters undergo a shape transition from prolate to spherical shapes at [@Jarrold:91]. Finally, two distinct patterns of chemisorption reactivities observed experimentally are explained based on the electronic structures of the reagents. The reactivities of closed shell reagents depend on the available number of dangling bond sites, whereas the reactivities of free radical reagents are not so dependent. Consequently, only the closed shell reagents are sensitive to the cluster structure and hence exhibit the highly oscillatory pattern in reactivities as a function of the cluster size.
This research is supported by the New York University and the Donors of The Petroleum Research Fund (ACS-PRF \# 26488-G), administered by the American Chemical Society.
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[^1]: The Department of Physics, New York University, New York, NY 10003-6621.
|
---
author:
- |
Saiful A. Mojumder$^1$, Yifan Sun$^2$, Leila Delshadtehrani$^1$, Yenai Ma$^1$, Trinayan Baruah$^2$,\
José L. Abellán$^3$, John Kim$^4$, David Kaeli$^2$, Ajay Joshi$^1$\
\
\
\
bibliography:
- 'tsm1.bib'
- 'kaeli.bib'
- 'trinayan.bib'
- 'ajay.bib'
title: |
: A Hardware-Level Timestamp-based\
Cache Coherence Scheme for Multi-GPU systems
---
|
---
abstract: 'We introduce a quantum Minkowski space-time based on the quantum group $SU(2)_q$ extended by a degree operator and formulate a quantum version of the anti-self-dual Yang-Mills equation. We construct solutions of the [*quantum*]{} equations using the [*classical*]{} ADHM linear data, and conjecture that, up to gauge transformations, our construction yields all the solutions. We also find a deformation of Penrose’s twistor diagram, giving a correspondence between the quantum Minkowski space-time and the classical projective space ${\mathbb{P}}^3$.'
author:
- |
Igor Frenkel\
Yale University\
Department of Mathematics\
10 Hillhouse Avenue\
New Haven, CT 06520-8283 USA\
\
Marcos Jardim\
University of Massachusetts at Amherst\
Department of Mathematics and Statistics\
Amherst, MA 01003-9305 USA
title: |
Quantum instantons\
with classical moduli spaces
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction {#intro .unnumbered}
============
Our present view of the mathematical structure of space-time was first formulated by H. Minkowski in [@Mi], based on Einstein’s discovery of Special Relativity. Since then, many mathematicians and physicists have tried to develop various generalizations of Minkowski’s concept of space-time, in different directions.
One such generalization was strongly advocated by R. Penrose [@P]. He studied the conformal compactification of the complexified Minkowski space-time, denoted by ${\mathbb{M}}$, and the associated space of null straight lines, denoted by ${\mathbb{P}}$, showing that various equations from Mathematical Physics (e.g. Maxwell and Dirac equations) over ${\mathbb{M}}$ can be transformed into natural holomorphic objects over ${\mathbb{P}}$.
One of the most celebrated examples of the Penrose programme was the solution of the anti-self-dual Yang-Mills (ASDYM) equations by M. Atiyah, V. Drinfeld, N. Hitchin and Yu. Manin. These authors described explicitly the moduli space ${\cal M}^{\rm reg}(n,c)$ of finite action solutions of the ASDYM equation over ${\mathbb{R}}^4$, usually called [*instantons*]{}, of fixed rank $n$ and charge $c$ in terms of some linear data [@ADHM]. In particular, they constructed an instanton associated to any linear data, and proved, conversely, that every instanton can be obtained in this way, up to gauge transformations.
The Penrose approach has also been successfully applied to massless linear equations, the full Yang-Mills equation, self-dual Einstein equation, etc. (see [@WW] and the references therein).
Another generalization of the Minkowski space-time based on noncommutative geometry was proposed by A. Connes [@C]. The first step in this approach is the replacement of geometric objects by their algebras of functions and the reformulation of various geometric concepts in algebraic language. The next step is the deformation of the algebraic structures and the introduction of noncommutativity. The [*algebraic*]{} structures so obtained are no longer associated with the original geometry, being regarded as [*geometric*]{} structures on a “noncommutative space”.
The simplest example of noncommutative space-time can be obtained by replacing the commutative algebra ${\mathbb{R}}^4$ by the Heisenberg algebra. Recently, N. Nekrasov and A. Schwarz [@NS] have defined instantons on such noncommutative ${\mathbb{R}}^4$ and argued that the corresponding moduli space can be parametrized by a modification of the Atiyah, Drinfeld, Hitchin and Manin (ADHM) linear data.
Another source for the deformation of the Minkowski space-time emerged with the discovery of [*quantum groups*]{} by V. Drinfeld [@Dr] and M. Jimbo [@J]. It quickly led to the notion of a quantum Minkowski space-time and various related structures; some of the early papers are [@CSSW; @Mj; @SWZ]. In particular, a quantum version of the ASDYM equation was studied in [@Z].
One expects that the linear data of Atiyah, Drinfeld, Hitchin and Manin should also be deformed in order to yield solutions of the quantum ASDYM equation. Thus the quantum Minkowski space-time and related structures, though mathematically sensible, seem rather dubious in terms of physical applications. In fact, the quantum deformation destroys the classical symmetry groups and the possible reconstruction of these symmetries is far from apparent.
In the present paper we observe a new phenomenon, which goes against the typical intuition related to quantum deformations of the Minkowski space-time and various equations on it. We show that there exists a natural quantum deformation of the Minkowski space-time (in fact, of the whole compactified complexified Minkowski space ${\mathbb{M}}$, along with its real structures) and the ASDYM equation, such that [*the moduli space of quantum instantons is naturally parameterized by the classical, non-deformed ADHM data*]{}.
Starting from the classical ADHM data, we will explicitly construct solutions of the quantum ASDYM equation. Furthermore, we conjecture that our construction yields all the solutions up to gauge equivalence. We hope to prove this conjecture using a generalization of the Penrose twistor transform, which takes the [*quantum*]{} Minkowski space-time to the [*classical*]{} space of straight lines. Such procedure thus realizes Penrose’s dream, who regarded light rays as more fundamental than points in space-time: in our construction, space-time is being deformed while the space of light rays is kept fixed.
Moreover, this phenomenon of quantum equations with classical solutions does not seem to be restricted only to the ASDYM equation, but it is also steadfast for massless linear equations, full Yang-Mills equations, etc. This opens new venues for physical applications of the quantum Minkowski space-time here proposed: the classical symmetry groups can be restored if one considers [*all*]{} quantum deformations of the classical Minkowski space-time, since the spaces of solutions of the quantum equations admit natural identifications.
Our constructions are based on the theory of the quantum group $SU(2)_q$ extended by a natural degree operator, which makes possible for a surprising construction of solutions of the quantum ASDYM equation from the classical ADHM data. The extended quantum group $\widetilde{SU}(2)_q$ has functional dimension $4$, the same as the quantum space-time, and we show that the relation between them extends to deep structural levels.
To formulate the ASDYM equation we need a theory of exterior forms on quantum Minkowski space-time. This is derived from the differential calculus on $SU(2)_q$ first developed in [@WZ; @W], with the addition of the differential of the degree operator. The $R$-matrix formulation of the quantum group $SU(2)_q$ [@FRT] and its exterior algebra [@S] allows us to present the construction of the quantum connections even more compactly than in the classical case, and efficiently verify the quantum ASDYM equation.
Our results bring us to the conclusion that [*the correct notion for a quantum Minkowski space-time is precisely the extended quantum group $\widetilde{SU}(2)_q$*]{}. This relation, which might seem artificial at the classical level ($q=1$), is imposed on us by the mathematical structure itself. We believe that future research will reveal the full potential of this new incarnation of the Minkowski space-time.
Quantum Minkowski space-time from $SU(2)_q$ {#mink}
===========================================
Algebraic structures on Minkowski space-time
--------------------------------------------
We begin with some well known facts regarding Penrose’s approach to the Minkowski space-time for the convenience of the reader; further details can be found in [@Ma; @WW].
Let ${\mathbb{T}}$ be a 4 dimensional complex vector space and consider ${\mathbb{M}}= \mathbf{G}_2({\mathbb{T}})$, the Grassmannian of planes in ${\mathbb{T}}$. As usual in the literature, we will often refer to ${\mathbb{M}}$ as the [*compactified complexified Minkowski space*]{}, since ${\mathbb{M}}$ can be obtained via a conformal compactification of $M^4\otimes{\mathbb{C}}$, where $M^4$ denotes the usual Minkowski space.
The Grassmannian ${\mathbb{M}}$ can be realized as a quadric in ${\mathbb{P}}^5$ via the [*Plücker embedding*]{}. More precisely, note that $\mathbf{P}(\Lambda^2{\mathbb{T}})\simeq{\mathbb{P}}^5$, and take homogeneous coordinates $[z_{rs}]$ for $r,s=1,2,3,4$ (where $z_{rs}=-z_{sr}$ is the coefficient of $dz_r\wedge dz_s$). Then ${\mathbb{M}}$ becomes the subvariety of $\mathbf{P}(\Lambda^2{\mathbb{T}})$ given by the quadric: $$\label{quadric}
z_{12}z_{34} - z_{13}z_{24} + z_{14}z_{23} = 0$$
Let us now fix a direct sum decomposition of ${\mathbb{T}}$ into two 2-dimensional subspaces: $$\label{decomp}
{\mathbb{T}}= {\mathbb{L}}\oplus {\mathbb{L}}'$$ Such choice induces a decomposition of the second exterior power as follows: $$\Lambda^2{\mathbb{T}}= \Lambda^2{\mathbb{L}}\oplus \Lambda^2{\mathbb{L}}' \oplus {\mathbb{L}}\wedge{\mathbb{L}}'$$ Now fix basis $\{e_1,e_2\}$ and $\{e_{1'},e_{2'}\}$ in ${\mathbb{L}}$ and ${\mathbb{L}}'$, respectively. To help us keep track of the choices made, we will use indexes $\{1,2,1',2'\}$ instead of $\{1,2,3,4\}$. Since the variables $z_{12}$ and $z_{1'2'}$ will play a very special role in our discussion, we will introduce the notation $D=z_{12}$ and $D'=z_{34}=z_{1'2'}$. The quadric (\[quadric\]) is then rewritten in the following way: $$\label{quad}
z_{11'}z_{22'}-z_{12'}z_{21'} = DD'$$
The decomposition (\[decomp\]) also induces the choice of a [*point at infinity*]{} in the compactified complexified Minkowski space ${\mathbb{M}}$. Let ${\cal S}(x)$ denote the plane in ${\mathbb{T}}$ corresponding to the point $x\in{\mathbb{M}}$ and $\ell$ denote the point in ${\mathbb{M}}$ corresponding to the plane ${\mathbb{L}}$. Consider the sets: $$\begin{aligned}
{\mathbb{M}}^{\rm I} = \left\{ x\in{\mathbb{M}}\ |\ {\cal S}(x)\cap{\mathbb{L}}=\{0\} \right\} & \ \ \ &
{\rm complexified\ Minkowski\ space} \nonumber\\
C(\ell) = \left\{ x\in{\mathbb{M}}\ |\ \dim\left({\cal S}(x)\cap{\mathbb{L}}\right)=1 \right\} & \ \ \ &
{\rm light\ cone\ at\ infinity} \nonumber\end{aligned}$$ Then clearly ${\mathbb{M}}={\mathbb{M}}^{\rm I} \cup C(\ell) \cup \{\ell\}$, and ${\mathbb{M}}^{\rm I}$ is an affine space, being isomorphic to ${\mathbb{C}}^4=M^4\otimes{\mathbb{C}}$. Moreover, we note that the light cone at infinity $C(\ell)$ has complex codimension one in ${\mathbb{M}}$.
We will denote the local coordinates on ${\mathbb{M}}^{\rm I}$ by $x_{rs'}=z_{rs'}/D$, where $r,s=1,2$. They are related to the Euclidean coordinates $x^k$ on ${\mathbb{M}}^{\rm I}$ in the following way: $$\label{coords1} \begin{array}{ccc}
x_{11'} = x^1 - ix^4 & \ \ \ &
x_{12'} = -ix^2 - x^3 \\
x_{21'} = -ix^2 + x^3 & \ \ \ &
x_{22'} = x^1 + ix^4
\end{array}$$ We will denote by $E^4$ the real Euclidean space spanned by $x^k$.
Similarly, let $\ell'$ be the point in ${\mathbb{M}}$ corresponding to the plane ${\mathbb{L}}'$. It can be regarded as the [*origin*]{} in ${\mathbb{C}}^4$. We define: $${\mathbb{M}}^{\rm J} = \left\{ x\in{\mathbb{M}}\ |\ {\cal S}(x)\cap{\mathbb{L}}'=\{0\} \right\}$$ $$C(\ell') = \left\{ x\in{\mathbb{M}}\ |\ \dim\left({\cal S}(x)\cap{\mathbb{L}}'\right)=1 \right\}$$ so that ${\mathbb{M}}^{\rm J}$ is also a 4-dimensional affine space and ${\mathbb{M}}={\mathbb{M}}^{\rm J} \cup C(\ell') \cup \{\ell'\}$. We will denote the local coordinates on ${\mathbb{M}}^{\rm J}$ by $y_{rs'}=z_{rs'}/D'$, where $r,s=1,2$.
It is important to note that even though the affine spaces ${\mathbb{M}}^{\rm I}$ and ${\mathbb{M}}^{\rm J}$ do not cover the entire compactified complexified Minkowski space ${\mathbb{M}}$, only a codimension two submanifold is left out, since ${\mathbb{M}}\setminus \left( {\mathbb{M}}^{\rm I} \cup {\mathbb{M}}^{\rm J} \right) = C(\ell) \cap C(\ell')$.
The intersection ${\mathbb{M}}^{\rm IJ}={\mathbb{M}}^{\rm I} \cap {\mathbb{M}}^{\rm J}$ is given by the set of all $x\in{\mathbb{M}}^{\rm I}$ such that $\det(X)=x_{11'}x_{22'}-x_{12'}x_{21'}\neq0$. Equivalently, this is also the set of all $y\in{\mathbb{M}}^{\rm J}$ such that $\det(Y)=y_{11'}y_{22'}-y_{12'}y_{21'}\neq0$. The gluing map $\tau:{\mathbb{M}}^{\rm IJ}{\rightarrow}{\mathbb{M}}^{\rm IJ}$ relates the local coordinates on ${\mathbb{M}}^{\rm I}$ and ${\mathbb{M}}^{\rm J}$ in the following way: $$\begin{aligned}
x_{11'} = \frac{y_{22'}}{\det(Y)} & \ \ \ &
x_{12'} = -\frac{y_{12'}}{\det(Y)} \\
x_{21'} = -\frac{y_{21'}}{\det(Y)} & \ \ \ &
x_{22'} = \frac{y_{11'}}{\det(Y)}\end{aligned}$$
Since ${\mathbb{M}}$ is an algebraic variety, it can also be characterized via its homogeneous coordinate algebra: $${\mathfrak{M}}= {\mathbb{C}}[D,D',z_{11'},z_{12'},z_{21'},z_{22'}]_h/{\cal I}_m$$ where ${\cal I}_m$ is the ideal generated by the quadric (\[quad\]). The subscript “$h$" means that ${\mathfrak{M}}$ consists only of the homogeneous polynomials.
In this picture, the coordinate algebras of the affine varieties ${\mathbb{M}}^{\rm I}$ and ${\mathbb{M}}^{\rm J}$ introduced above can be interpreted as certain localizations of the quadratic algebra ${\mathfrak{M}}$. Indeed, the coordinate rings for ${\mathbb{M}}^{\rm I}$ and ${\mathbb{M}}^{\rm J}$ are respectively given by: $$\begin{aligned}
{\mathfrak{M}}^{\rm I} = {\mathfrak{M}}[D^{-1}]_0 =
{\mathbb{C}}\left[ x_{11'},x_{12'},x_{21'},x_{22'} \right] \\
{\mathfrak{M}}^{\rm J} = {\mathfrak{M}}[D'^{-1}]_0 =
{\mathbb{C}}\left[ y_{11'},y_{12'},y_{21'},y_{22'} \right]\end{aligned}$$ where the subscript “$0$" means that we take only the degree zero part of the localized graded algebra.
Finally, we note that ${\mathfrak{M}}^{\rm I}$ and ${\mathfrak{M}}^{\rm J}$ can be made isomorphic by adjoining the inverses of the determinants $\det(X)$ and $\det(Y)$, respectively. Indeed, define the matrices of generators: $$\label{XY} \begin{array}{ccc}
X = \left( \begin{array}{cc}
x_{11'} & x_{12'} \\ x_{21'} & x_{22'}
\end{array} \right) & \ \ &
Y = \left( \begin{array}{cc}
y_{22'} & -y_{12'} \\ -y_{21'} & y_{11'}
\end{array} \right)
\end{array}$$ The map $\eta:{\mathfrak{M}}^{\rm I}[\det(X)^{-1}] \rightarrow {\mathfrak{M}}^{\rm J}[\det(Y)^{-1}]$ given by: $$\eta(X) = \frac{Y}{\det(Y)}$$ is an isomorphism. It is the algebraic analogue of the gluing map $\tau$ described above in geometric context, while ${\mathfrak{M}}^{\rm I}[\det(X)^{-1}] \simeq {\mathfrak{M}}^{\rm J}[\det(Y)^{-1}]$ plays the role of the intersection ${\mathbb{M}}^{\rm IJ}={\mathbb{M}}^{\rm I}\cap{\mathbb{M}}^{\rm J}$.
The quantum group $SU(2)_q$ and its extension
---------------------------------------------
Let $q$ be a formal parameter. Recall that the quantum group $GL(2)_q$ is the bialgebra over ${\mathbb{C}}$ generated by $g_{11'},g_{12'},g_{21'},g_{22'}$ subject to the following commutation relations, see e.g. [@CP]: $$\begin{aligned}
g_{11'}g_{12'}=q^{-1}g_{12'}g_{11'} & \ \ \
& g_{11'}g_{21'}=q^{-1}g_{21'}g_{11'} \nonumber\\
g_{12'}g_{22'}=q^{-1} g_{22'}g_{12'} & \ \ \
& g_{21'}g_{22'}=q^{-1}g_{22'}g_{21'} \label{gl2q1}\end{aligned}$$ $$g_{12'}g_{21'}=g_{21'}g_{12'}$$ $$\label{gl2q2}
g_{11'}g_{22'} - q^{-1}g_{12'}g_{21'} = g_{22'}g_{11'} - qg_{21'}g_{12'}$$ The comultiplication $\Delta$ and the counit $\varepsilon$ are given by: $$\label{comult}
\Delta(g_{rs'}) = \sum_{k=1,2} g_{rk'} \otimes g_{ks'}, \ \ \ r,s=1,2$$ $$\label{conunit}
\varepsilon(g_{rs'}) = \delta_{rs'}, \ \ \ r,s=1,2$$
The expression (\[gl2q2\]) is called the [*quantum determinant*]{} and it is denoted by $\det_q(g)$. One easily checks that: $$\label{qdet.commutes}
g_{rs'}{\rm det}_q(g) = {\rm det}_q(g)g_{rs'}, \ \ \ r,s=1,2$$ $$\label{qdet.comult}
\Delta({\rm det}_q(g)) = {\rm det}_q(g)\otimes {\rm det}_q(g)$$
In order to define a Hopf algebra structure on $GL(2)_q$ we adjoin the inverse of the quantum determinant $\det_q(g)^{-1}$ to the generators $g_{rs'}$ with the obvious relations. Then the antipode exists, and it is given by: $$\begin{array} {ccc}
\gamma(g_{11'}) = g_{22'}{\rm det}_q(g)^{-1} & &
\gamma(g_{12'}) = -qg_{12'}{\rm det}_q(g)^{-1} \\
\gamma(g_{21'}) = -q^{-1}g_{21'}{\rm det}_q(g)^{-1} & &
\gamma(g_{22'}) = g_{11'}{\rm det}_q(g)^{-1}
\end{array}$$ We will assume that the quantum group $GL(2)_q$ contains $\det_q(g)^{-1}$ and is therefore a Hopf algebra.
The quantum group $SL(2)_q$ is defined by the quotient: $$SL(2)_q = GL(2)_q/\langle {\rm det}_q(g)=1 \rangle$$ It is a well defined Hopf algebra by equations (\[qdet.commutes\]) and (\[qdet.comult\]).
It is also useful to re-express the commutation relations for the quantum group $GL(2)_q$ by means of the $R$-matrix; let $$\label{rmatrix}
R_{12} = \left( \begin{array}{cccc}
p^{-1} & 0 & 0 & 0 \\ 0 & 1 & p^{-1}-q & 0 \\
0 & p^{-1}-q^{-1} & 1 & 0 \\ 0 & 0 & 0 & p^{-1}
\end{array} \right), \ \ p=q^{\pm 1}$$ and let $T$ be the matrix of generators, i.e.: $$T = \left( \begin{array}{cc}
g_{11'} & g_{12'} \\ g_{21'} & g_{22'}
\end{array} \right)$$ Then the relations (\[gl2q1\]) and (\[gl2q2\]) can then be put in the following compact form: $$R_{12} T_1T_2 = T_2T_1 R_{12}$$ where $T_1=T\otimes{\mathbf{1}}$ and $T_2={\mathbf{1}}\otimes T$ [@FRT]. Note that the commutation relations (\[gl2q1\]) and (\[gl2q2\]) do not depend on the parameter $p$. The $R$-matrix (\[rmatrix\]) also satisfies the Hecke relation: $$\label{hecke1}
R_{12} - (R_{21})^{-1} = (p^{-1} - p) P$$ where $R_{21}=R_{12}^{\rm t}$ (with the superscript “t” meaning transposition) and $P$ is the permutation matrix. Equivalently, we also have for $\hat{R}_{12}=PR_{12}$: $$\label{hecke2}
(\hat{R}_{12})^2 = (p^{-1} - p) \hat{R}_{12} + {\mathbf{1}}$$
Next, we will extend the quantum groups $GL(2)_q$ and $SL(2)_q$ by introducing a new generator $\delta$ and its inverse, satisfying the following commutation relations with the quantum group generators: $$\label{deltacommut}
\begin{array} {ccc}
\delta g_{11'} = g_{11'} \delta & \ \ \ &
\delta g_{12'} = qg_{12'} \delta \\
\delta g_{21'} = q^{-1}g_{21'} \delta & \ \ \ &
\delta g_{22'} = g_{22'} \delta
\end{array}$$ In matrix form, the above relations become: $$\delta T Q^2 = Q^2 T \delta$$ where $Q$ is the following matrix: $$\label{matrixq}
Q = \left( \begin{array}{cc}
q^{\frac{1}{4}} & 0 \\ 0 & q^{-\frac{1}{4}}
\end{array} \right)$$ Strictly speaking, one should consider $q^{\frac{1}{4}}$ to be the formal parameter, instead of $q$. However, fractional powers of $q$ will rarely appear in this paper.
In other words, we define $$\widetilde{GL}(2)_q = GL(2)_q[\delta,\delta^{-1}]/(\ref{deltacommut})
\ \ \ {\rm and}\ \ \
\widetilde{SL}(2)_q = SL(2)_q[\delta,\delta^{-1}]/(\ref{deltacommut})$$ The comultiplication, counit and antipode in $\widetilde{GL}(2)_q$ and $\widetilde{SL}(2)_q$ are given by: $$\Delta(\delta) = \delta\otimes\delta, \ \ \
\varepsilon(\delta) = 1, \ \ \
\gamma(\delta)=\delta^{-1}$$ It is easy to check that $\widetilde{GL}(2)_q$ and $\widetilde{SL}(2)_q$ satisfy the axioms of a Hopf algebra, and can thus be thought as quantum groups. Besides, we have the identity: $$\gamma^2(g)=\delta^2 g \delta^{-2}, \ \ \
{\rm for\ all}\ g\in GL(2)_q, \ SL(2)_q$$ Thus conjugation by $\delta$ can be viewed as a square root of the antipode squared.
In this paper, we will be primarily interested in the quantum group $SL(2)_q$ and its extension $\widetilde{SL}(2)_q$. For these quantum groups we define an involution $\dagger$ which fixes the formal parameter (i.e. $q^\dagger = q$) and acts on the generators as follows: $$\label{involution}
\begin{array} {ccc}
g_{11'}^\dagger = g_{22'}, & & g_{12'}^\dagger = -g_{21'} \\
g_{21'}^\dagger = -g_{12'}, & & g_{22'}^\dagger = g_{11'} \\
\delta^\dagger = \delta, & & (\delta^{-1})^\dagger = \delta^{-1}
\\
\end{array}$$ We extend the $\dagger$-involution to the quantum group $\widetilde{SL}(2)_q$ by requiring it to be a conjugate linear anti-homomorphism, that is: $$\label{ext.star} \begin{array}{l}
(xy)^\dagger = y^\dagger x^\dagger ,\ \ {\rm where}\ \ x,y\in\widetilde{SL}(2)_q;\\
a^\dagger=\overline{a},\ \ \ {\rm for\ all}\ \ a\in{\mathbb{C}}.
\end{array}$$
We define $SU(2)_q$ and $\widetilde{SU}(2)_q$ as the quantum groups $SL(2)_q$ and $\widetilde{SL}(2)_q$, respectively, equipped with the $\dagger$-involution: $$SU(2)_q = (SL(2)_q,\dagger)
\ \ \ {\rm and}\ \ \
\widetilde{SU}(2)_q=(\widetilde{SL}(2)_q,\dagger)$$
Note that instead of the formal parameter $q$ in the definitions of the quantum groups, we could have used a positive real number, which is automaticaly fixed by the involution $\dagger$. However, we prefer to consider various specializations of formal $q$ later on, the most interesting one being the specialization to a root of unity (see also Section \[ru\] below).
Quantum Minkowski space-time
----------------------------
Let us introduce two new sets of variables on the extended quantum group $\widetilde{SL}(2)_q$: $$\label{xvar} \begin{array} {lcr}
x_{11'} = \delta g_{11'} = g_{11'}\delta, & \ \ \ &
x_{12'} = q^{-1/2}\delta g_{12'} = q^{1/2}g_{12'}\delta, \\
x_{21'} = q^{1/2}\delta g_{21'} = q^{-1/2}g_{21'}\delta,
&\ \ \ & x_{22'} = \delta g_{22'} = g_{22'}\delta
\end{array}$$ and $$\label{yvar} \begin{array} {lcr}
y_{11'} = \delta^{-1} g_{11'} = g_{11'}\delta^{-1}, & \ \ \ &
y_{12'} = q^{1/2}\delta^{-1} g_{12'} = q^{-1/2}g_{12'}\delta^{-1}, \\
y_{21'} = q^{-1/2}\delta^{-1} g_{21'} = q^{1/2}g_{21'}\delta^{-1},
& \ \ \ & y_{22'} = \delta^{-1} g_{22'} = g_{22'}\delta^{-1}
\end{array}$$ It is easy to determine their commutation relations: $$\label{xcommut1}
x_{11'}x_{12'} = x_{12'}x_{11'}, \ \ \ x_{21'}x_{22'} = x_{22'}x_{21'},$$ $$\label{xcommut3}
\left[ x_{11'} ,x_{22'} \right] + [x_{21'},x_{12'}] = 0$$ $$\label{xcommut2} \begin{array} {c}
x_{11'}x_{21'} = q^{-2}x_{21'}x_{11'}, \ \ \ x_{12'}x_{22'} = q^{-2}x_{22'}x_{12'}, \\
x_{21'}x_{12'} = q^2 x_{12'} x_{21'}
\end{array}$$ and $$\label{ycommut1}
y_{11'}y_{21'} = y_{21'}y_{11'}, \ \ \ y_{12'}y_{22'} = y_{22'}y_{12'}$$ $$\label{ycommut3}
\left[ y_{11'},y_{22'} \right] + [y_{12'},y_{21'}] = 0$$ $$\label{ycommut2} \begin{array} {c}
y_{11'}y_{12'} = q^{-2}y_{12'}y_{11'}, \ \ \ y_{21'}y_{22'} = q^{-2}y_{22'}y_{21'}, \\
y_{12'}y_{21'} = q^2 y_{21'} y_{12'}
\end{array}$$ The determinant condition $\det_q(g)=1$ in the new variables becomes: $$\label{xdtx}
x_{11'}x_{22'} - x_{12'}x_{21'} =
x_{22'}x_{11'} - x_{21'}x_{12'} = \delta^2$$ and $$\label{ydty}
y_{11'}y_{22'} - y_{21'}y_{12'} =
y_{22'}y_{11'} - y_{12'}y_{21'} = \delta^{-2}$$ Furthermore, the $\dagger$ involution acts on these new variables in the following way: $$\label{star.xy}
\begin{array}{ccccccc}
x_{11'}^\dagger = x_{22'} & \ \ & x_{12'}^\dagger =-x_{21'} & \ \ &
x_{21'}^\dagger =-x_{12'} & \ \ & x_{22'}^\dagger = x_{11'} \\
y_{11'}^\dagger = y_{22'} & \ \ & y_{12'}^\dagger =-y_{21'} & \ \ &
y_{21'}^\dagger =-y_{12'} & \ \ & y_{22'}^\dagger = y_{11'}
\end{array}$$
We will consider the following subalgebras of $\widetilde{SL}(2)_q$: $${\mathfrak{M}}^{\rm I}_q = {\mathbb{C}}[x_{11'},x_{12'},x_{21'},x_{22'}]/(\ref{xcommut1}-\ref{xcommut2})$$ and $${\mathfrak{M}}^{\rm J}_q = {\mathbb{C}}[y_{11'},y_{12'},y_{21'},y_{22'}]/(\ref{ycommut1}-\ref{ycommut2})$$ regarding them as $q$-deformations of ${\mathfrak{M}}^{\rm I}$ and ${\mathfrak{M}}^{\rm J}$, the coordinate algebras corresponding to the affine subspaces ${\mathbb{M}}^{\rm I}$ and ${\mathbb{M}}^{\rm J}$ of the compactified complexified Minkowski space-time ${\mathbb{M}}$. Moreover, we introduce the $\dagger$-algebras: $$S^{\rm I}_q=({\mathfrak{M}}^{\rm I}_q,\dagger)
\ \ \ {\rm and}\ \ \
S^{\rm J}_q=({\mathfrak{M}}^{\rm J}_q,\dagger)$$ regarding them as $q$-deformations of $S^4\setminus\{\infty\}$ and $S^4\setminus\{0\}$, respectively.
Note also that ${\mathfrak{M}}^{\rm I}_q$ and ${\mathfrak{M}}^{\rm J}_q$ are isomorphic (as Hopf algebras) once the generators $\delta$ and $\delta^{-1}$ are adjoined. Let $X$ and $Y$ be again the matrices of generators as defined in (\[XY\]); the map: $$\begin{aligned}
\nonumber \eta: {\mathfrak{M}}^{\rm I}_q[\delta^{-1}] & \longrightarrow & {\mathfrak{M}}^{\rm J}_q[\delta] \\
\label{nc.glueing} X & \mapsto & \delta^2 Y \end{aligned}$$ is an isomorphism. More explicitly, in coordinates: $$\begin{array}{ccc}
x_{11'} \mapsto \delta^2 y_{22'} & \ \ \ &
x_{12'} \mapsto -\delta^2 y_{12'} \\
x_{21'} \mapsto -\delta^2 y_{21'} & \ \ \ &
x_{22'} \mapsto \delta^2 y_{11'}
\end{array}$$
The inverse map $\eta^{-1}$ is given by $Y\mapsto X \delta^{-2}$. Moreover, $\eta(X^\dagger) = \eta(X)^\dagger$, so that $\eta(f^\dagger)=\eta(f)^\dagger$ for all $f\in{\mathfrak{M}}^{\rm I}_q[\delta^{-1}]$.
Geometrically, the algebras ${\mathfrak{M}}^{\rm I}_q[\delta^{-1}]$ and ${\mathfrak{M}}^{\rm J}_q[\delta]$ play the role of the intersection ${\mathbb{M}}^{\rm IJ}={\mathbb{M}}^{\rm I}\cap{\mathbb{M}}^{\rm J}$, while $\eta$ plays the role of the gluing map $\tau$.
Furthermore, as in the case of the quantum group $GL(2)_q$, the above commutation relations can also be put in a compact form by means of an $R$-matrix. It is now convenient to consider the following matrix of generators: $$\label{XY2} \begin{array}{ccc}
X = \left( \begin{array}{cc}
x_{11'} & x_{12'} \\ x_{21'} & x_{22'}
\end{array} \right)
& \ \ \ &
Y = \left( \begin{array}{cc}
y_{11'} & y_{12'} \\ y_{21'} & y_{22'}
\end{array} \right)
\end{array}$$ Notice that: $$\begin{array}{ccc}
X = Q^{-1} \delta T Q = Q T \delta Q^{-1}
& \ \ \ {\rm and}\ \ \ &
Y = Q \delta^{-1} T Q^{-1} = Q^{-1} T \delta^{-1} Q
\end{array}$$ where $Q$ was defined in (\[matrixq\]). In order to write down the above commutation relations in matrix form, define $X_1=X\otimes{\mathbf{1}}$, $X_2={\mathbf{1}}\otimes X$ and similarly $Q_1=Q\otimes{\mathbf{1}}$, $Q_2 = {\mathbf{1}}\otimes Q$ . Therefore: $$\begin{aligned}
X_1X_2 & = &
\left( Q_1^{-1} \delta T_1 Q_1 \right) \left( Q_2 T_2 \delta Q_2^{-1} \right) =
\left( Q_1^{-1} Q_2 \right) \delta T_1T_2 \delta \left( Q_1Q_2^{-1} \right) = \\
& = & \left( Q_1^{-1} Q_2 R^{-1}_{12} Q_2 Q_1^{-1} \right) X_2X_1
\left( Q_2^{-1} Q_1 R_{12} Q_1 Q_2^{-1} \right)\end{aligned}$$ Thus we define $R^{\rm I}_{12} = Q_2^{-1} Q_1 R_{12} Q_1 Q_2^{-1}$; more precisely: $$\label{rmatrixI}
R^{\rm I}_{12} = \left( \begin{array}{cccc}
p^{-1} & 0 & 0 & 0 \\ 0 & q^{-1} & p^{-1}-q & 0 \\
0 & p^{-1}-q^{-1} & q & 0 \\ 0 & 0 & 0 & p^{-1}
\end{array} \right), \ \ p=q^{\pm1}$$ The commutation relations (\[xcommut1\]-\[xcommut2\]) can then be presented in matrix form: $$R^{\rm I}_{12} X_1X_2 = X_2X_1 R^{\rm I}_{12}$$
Performing a similar calculation for the $y_{rs'}$ variables, we obtain: $$\label{rmatrixJ}
R^{\rm J}_{12} = Q_2 Q_1^{-1} R_{12} Q_1^{-1} Q_2 = \left( \begin{array}{cccc}
p^{-1} & 0 & 0 & 0 \\ 0 & q & p^{-1}-q & 0 \\
0 & p^{-1}-q^{-1} & q^{-1} & 0 \\ 0 & 0 & 0 & p^{-1}
\end{array} \right), \ \ p=q^{\pm1}$$ with the commutation relations (\[ycommut1\]-\[ycommut2\]) being given by: $$R^{\rm J}_{12} Y_1Y_2 = Y_2Y_1 R^{\rm J}_{12}$$
Differential forms on quantum Minkowski space-time {#forms}
==================================================
Differential forms on Minkowski space-time
------------------------------------------
Since ${\mathbb{M}}^{\rm I}$ and ${\mathbb{M}}^{\rm J}$ are affine spaces, their modules of differential forms are very simple to describe. Indeed, recall that: $${\mathfrak{M}}^{\rm I}={\mathbb{C}}[x_{11'},x_{12'},x_{21'},x_{22'}]$$ Therefore, the module of differential 1-forms is given by the free ${\mathfrak{M}}^{\rm I}$-module generated by $dx_{rs'}$: $$\Omega^1_{{\mathfrak{M}}^{\rm I}} = {\mathfrak{M}}^{\rm I} \langle dx_{rs'} \rangle$$ while the module of differential 2-forms is given by the free ${\mathfrak{M}}^{\rm I}$-module generated by $dx_{rs'}\wedge dx_{kl'}$: $$\Omega^2_{{\mathfrak{M}}^{\rm I}} = \Lambda^2(\Omega^1_{{\mathfrak{M}}^{\rm I}}) =
{\mathfrak{M}}^{\rm I} \langle dx_{rs'}\wedge dx_{kl'} \rangle$$ with $r,s,k,l=1,2$.
The action of the de Rham operator $d: {\mathfrak{M}}^{\rm I}{\rightarrow}\Omega^1_{{\mathfrak{M}}^{\rm I}}$ is given on the generators as $x_{rs'}\mapsto dx_{rs'}$, and it is then extended to the whole ${\mathfrak{M}}^{\rm I}$ by ${\mathbb{C}}$-linearity and the Leibnitz rule: $$\label{l}
d(fg) = g df + f dg$$ where $f,g \in {\mathfrak{M}}^{\rm I}$. One also defines the de Rham operator $d : \Omega^1_{{\mathfrak{M}}^{\rm I}}{\rightarrow}\Omega^2_{{\mathfrak{M}}^{\rm I}}$ on the generators as $f dx_{rs'}\mapsto df \wedge dx_{rs'}$, also extending it by ${\mathbb{C}}$-linearity and the Leibnitz rule (\[l\]).
The modules of differential forms and de Rham operators over ${\mathfrak{M}}^{\rm J}$ are similarly described.
Now let $\Omega^2_{E^4}$ denote the bundle of 2-forms on Euclidean space $E^4$ with coordinates $\{x^1,x^2,x^3,x^4\}$. Recall that the Hodge operator $*:\Omega^2_{E^4}{\rightarrow}\Omega^2_{E^4}$ is defined as follows: $$\begin{array}{ccc}
*dx^1\wedge dx^2 = dx^3\wedge dx^4 & \ \ \ \ &
*dx^3\wedge dx^4 = dx^1\wedge dx^2
\end{array}$$ $$\begin{array}{ccc}
*dx^1\wedge dx^3 = -dx^2\wedge dx^4 & \ \ \ \ &
*dx^2\wedge dx^4 = -dx^1\wedge dx^3
\end{array}$$ $$\begin{array}{ccc}
*dx^1\wedge dx^4 = dx^2\wedge dx^3 & \ \ \ \ &
*dx^2\wedge dx^3 = dx^1\wedge dx^4
\end{array}$$ We can then use the relation between Euclidean and twistor coordinates on ${\mathbb{M}}^{\rm I}=E^4\otimes{\mathbb{C}}$ given by (\[coords1\]) to express the action of the Hodge operator on $\Omega^2_{{\mathfrak{M}}^{\rm I}}$. One obtains: $$\begin{array}{ccc}
*dx_{11'}\wedge dx_{12'} = dx_{11'}\wedge dx_{12'} & \ \ \ \ &
*dx_{11'}\wedge dx_{21'} = - dx_{11'}\wedge dx_{21'}
\end{array}$$ $$\begin{array}{ccc}
*dx_{11'}\wedge dx_{22'} = - dx_{12'}\wedge dx_{21'} & \ \ \ \ &
*dx_{12'}\wedge dx_{21'} = - dx_{11'}\wedge dx_{22'}
\end{array}$$ $$\begin{array}{ccc}
*dx_{12'}\wedge dx_{22'} = - dx_{12'}\wedge dx_{22'} & \ \ \ \ &
*dx_{21'}\wedge dx_{22'} = dx_{21'}\wedge dx_{22'}
\end{array}$$
Clearly $*^2=1$, thus the complexified Hodge operator $*:\Omega^2_{{\mathfrak{M}}^{\rm I}}{\rightarrow}\Omega^2_{{\mathfrak{M}}^{\rm I}}$ induces a splitting of $\Omega^2_{{\mathfrak{M}}^{\rm I}}$ into two submodules corresponding to eigenvalues $\pm1$. More explicitly, we have: $$\Omega^{2}_{{\mathfrak{M}}^{\rm I}}=\Omega^{2,+}_{{\mathfrak{M}}^{\rm I}}\oplus
\Omega^{2,-}_{{\mathfrak{M}}^{\rm I}}$$ where $$\Omega^{2,+}_{{\mathfrak{M}}^{\rm I}} = {\mathfrak{M}}^{\rm I} \langle dx_{11'}\wedge dx_{12'},
dx_{21'}\wedge dx_{22'}, dx_{11'}\wedge dx_{22'} - dx_{12'}\wedge dx_{21'}
\rangle$$ $$\Omega^{2,-}_{{\mathfrak{M}}^{\rm I}} = {\mathfrak{M}}^{\rm I} \langle dx_{11'}\wedge dx_{21'},
dx_{12'}\wedge dx_{22'}, dx_{11'}\wedge dx_{22'} + dx_{12'}\wedge dx_{21'}
\rangle$$
#### Connection and curvature
Let $E$ be a ${\mathfrak{M}}^{\rm I}$-module; a [*connection*]{} on $E$ is a ${\mathbb{C}}$-linear map: $$\nabla: E {\rightarrow}E\otimes_{{\mathfrak{M}}^{\rm I}}\Omega^1_{{\mathfrak{M}}^{\rm I}}$$ satisfying the Leibnitz rule: $$\label{leib1}
\nabla(f\sigma)=\sigma\otimes df + f\nabla\sigma$$ where $f\in{\mathfrak{M}}^{\rm I}$ and $\sigma\in E$. The connection $\nabla$ also acts on 1-differentials, being defined as the additive map: $$\nabla: E\otimes_{{\mathfrak{M}}^{\rm I}}\Omega^1_{{\mathfrak{M}}^{\rm I}} {\rightarrow}E\otimes_{{\mathfrak{M}}^{\rm I}}\Omega^2_{{\mathfrak{M}}^{\rm I}}$$ satisfying: $$\label{leib2}
\nabla(\sigma\otimes\omega)=\sigma\otimes d\omega +
\omega\wedge\nabla\sigma$$ where $\omega\in\Omega^1_{{\mathfrak{M}}^{\rm I}}$.
Moreover, two connections $\nabla$ and $\nabla'$ are said to be gauge equivalent if there is $g\in{\rm Aut}_{{\mathfrak{M}}^{\rm I}}(E)$ such that $\nabla = g^{-1} \nabla' g$.
The curvature $F_\nabla$ is defined by the composition: $$E \stackrel{\nabla}{\longrightarrow}
E\otimes_{{\mathfrak{M}}^{\rm I}}\Omega^1_{{\mathfrak{M}}^{\rm I}}
\stackrel{\nabla}{\longrightarrow} E\otimes_{{\mathfrak{M}}^{\rm I}}
\Omega^2_{{\mathfrak{M}}^{\rm I}}$$ and it is easy to check that it is actually ${\mathfrak{M}}^{\rm I}$-linear. Therefore, $F_\nabla$ can be regarded as an element of ${\rm End}_{{\mathfrak{M}}^{\rm I}}(E)\otimes_{{\mathfrak{M}}^{\rm I}}\Omega^2_{{\mathfrak{M}}^{\rm I}}$. Furthermore, if $\nabla$ and $\nabla'$ are gauge equivalent, then there is $g\in{\rm Aut}_{{\mathfrak{M}}^{\rm I}}(E)$ such that $F_\nabla = g^{-1} F_{\nabla'} g$.
If $E$ is projective (hence free), any connection $\nabla$ can be encoded into a matrix $A\in{\rm End}_{{\mathfrak{M}}^{\rm I}}(E)\otimes_{{\mathfrak{M}}^{\rm I}}\Omega^1_{{\mathfrak{M}}^{\rm I}}$. Simply choose a basis $\{\sigma_k\}$ for $E$, and let $s=\sum a^k\sigma_k$, so that: $$\nabla s = \sum_k \left( \sigma_k\otimes da^k + a^k\nabla\sigma_k \right)$$ Thus it is enough to know how $\nabla$ acts on the basis $\{\sigma_k\}$: $$\nabla\sigma_k = \sum_{l,\alpha} A_k^{l,\alpha}\sigma_l\otimes dx_\alpha$$ and we define $A$ as the matrix with entries given by the 1-forms $\sum_\alpha A_k^{l,\alpha}dx_\alpha$.
Conversely, given $A\in{\rm End}_{{\mathfrak{M}}^{\rm I}}(E)\otimes_{{\mathfrak{M}}^{\rm I}}\Omega^1_{{\mathfrak{M}}^{\rm I}}$, we define the connection: $$\nabla_A s = \sum_{k,l,\alpha}
\left( \sigma_k\otimes da^k + A_k^{l,\alpha}a^k\sigma_l\otimes dx_\alpha \right)$$
Differential forms on the quantum group $SU(2)_q$
-------------------------------------------------
Let us now recall a few facts regarding the exterior algebra over the relevant quantum groups [@S; @WZ; @W]. The module of 1-forms over the quantum group $GL(2)_q$, which we shall denote by $\Omega^1_{GL}$, is the $GL(2)_q$-bimodule generated by $dg_{rs'}$ satisfying the following relations (written in matrix form): $$\label{nc1forms}
R_{12} T_1 dT_2 = dT_2 T_1 (R_{21})^{-1}$$ where $R_{12}$ is again the matrix (\[rmatrix\]) and $dT_2 = {\mathbf{1}}\otimes dT$, with: $$dT = \left( \begin{array}{cc}
dg_{11'} & dg_{12'} \\ dg_{21'} & dg_{22'}
\end{array} \right)$$ Similarly, the module of 2-forms $\Omega^2_{GL}$, is the $GL(2)_q$-bimodule generated by $dg_{rs'}\wedge dg_{kl'}$, which satisfy the relations (written in matrix form): $$\label{nc2forms}
R_{12} dT_1 \wedge dT_2 = - dT_2 \wedge dT_1 (R_{21})^{-1}$$ where $dT_1 = dT \otimes{\mathbf{1}}$. Furthermore, the commutation relations between $g_{mn'}$ and $dg_{rs'}\wedge dg_{kl'}$ can be deduced from (\[nc1forms\]) and (\[nc2forms\]) as follows. Let $dT_3 = {\mathbf{1}}\otimes{\mathbf{1}}\otimes dT$. Denoting $R_{ba}=(R_{ab})^{\rm t}$, we have $R_{12} T_1 dT_2 R_{21} = dT_2 T_1$ and $R_{13} T_1 dT_3 R_{31} = dT_3 T_1$. Therefore, $$\begin{aligned}
dT_3 \wedge dT_2 T_1 & = &
dT_3 \wedge (R_{12} T_1 dT_2 R_{21}) = R_{12} dT_3 T_1 \wedge
dT_2 R_{21} = \\
& = & R_{12}R_{13} T_1 dT_3 R_{31} \wedge dT_2 R_{21} = \left( R_{12}R_{13} \right) T_1
dT_3 \wedge dT_2 \left( R_{12}R_{13} \right)^{\rm t}\end{aligned}$$
The noncommutative de Rham operators are given by their action on the generators as follows: $$\begin{array}{ccc}
d : GL(2)_q {\rightarrow}\Omega^1_{GL} & \ \ \ \ &
d : \Omega^1_{GL} {\rightarrow}\Omega^2_{GL} \\
g_{rs'} \mapsto dg_{rs'} & \ \ \ \ &
g_{kl'}dg_{rs'} \mapsto dg_{kl'} \wedge dg_{rs'}
\end{array}$$ This is then extended to the whole $GL(2)_q$ and $\Omega^1_{GL}$ by ${\mathbb{C}}$-linearity and the Leibnitz rule: $$d(f_1f_2) = df_1 f_2 + f_1 df_2$$ for all $f_1,f_2\in GL(2)_q$.
The modules $\Omega^1_{GL}$ and $\Omega^2_{GL}$ also have natural involutions, extended from $\dagger$ in a natural way, namely: $$(f dg_{rs'})^\dagger = dg_{rs'}^\dagger f^\dagger$$ $$(f dg_{rs'} \wedge dg_{kl'})^\dagger = -
dg_{kl'}^\dagger\wedge dg_{rs'}^\dagger f^\dagger$$
To get the modules of forms on $SL(2)_q$ it is enough to take the quotient by the appropriate relations: $$\Omega^1_{SL} = \Omega^1_{GL} \left/
\begin{array}{c} \det_q T = 1 \\ d (\det_q T) = 0 \end{array}
\right. \ \ \ {\rm and} \ \ \ \Omega^2_{SL}=\Lambda^2(\Omega^1_{SL})$$ Finally, modules of forms on $SU(2)_q$ are then defined as the pairs: $$\Omega^k_{SU} = ( \Omega^k_{SL} , \dagger ), \ \ \ k=1,2$$
It is also important for our purposes to describe the modules of 1- and 2-forms on the extended quantum group $\widetilde{SL}(2)_q$. To do that, we add the generators $\delta$ and $d\delta$ satisfying the relations (written in matrix form): $$\begin{array}{ccc}
\delta dT Q^2 = Q^2 dT \delta
& \ \ {\rm and} \ \ &
d\delta dT Q^2 = - Q^2 dT d\delta
\end{array}$$
We extend the involution $\dagger$ to $\Omega^k_{\widetilde{SL}}$ by declaring that $(d\delta)^\dagger = d\delta$; thus we have: $$\Omega^k_{\widetilde{SU}} = ( \Omega^k_{\widetilde{SL}} , \dagger ), \ \ \ k=1,2$$
Let us now introduce the noncommutative analogue of the Hodge operator on $\Omega^2_{GL}$. This will serve as a model on our Definition of self-dual and anti-self-dual 2-forms on quantum space-time.
Recall that $R_{12}$ denotes the $R$-matrix (\[rmatrix\]) for $GL(2)_q$. Define: $$\label{p+p-}
P^+ = \frac{\hat{R}_{12} + p {\mathbf{1}}}{p+p^{-1}}
\ \ \ {\rm and} \ \ \
P^- = \frac{-\hat{R}_{12} + p^{-1} {\mathbf{1}}}{p+p^{-1}}$$ Clearly, $P^+ + P^- = {\mathbf{1}}$. Moreover, using the Hecke relations (\[hecke2\]), one easily checks that $(P^+)^2 = P^+$ and $(P^-)^2 = P^-$. Now the Hodge $*$-operator is defined on the generators of $\Omega^2_{GL}$ by: $$\label{hodgeGL}
* dT_1 \wedge dT_2 = \left( P^+ - P^- \right) dT_1 \wedge dT_2$$
Differential forms on quantum Minkowski space-time {#differential-forms-on-quantum-minkowski-space-time}
--------------------------------------------------
In analogy with the classical case, we define the module of 1-forms over the algebra ${\mathfrak{M}}^{\rm I}_q$, which we shall denote by $\Omega^1_{{\mathfrak{M}}^{\rm I}_q}$, as the ${\mathfrak{M}}^{\rm I}_q$-bimodule generated by: $$dX =
\left( \begin{array}{cc}
dx_{11'} & dx_{12'} \\ dx_{21'} & dx_{22'}
\end{array} \right) = Q^{-1} \delta dT Q = Q dT \delta Q^{-1}$$
Moreover, the generators $dx_{rs'}$ satisfy the following relations (written in matrix form): $$\begin{aligned}
X_1 dX_2 & = &
\left( Q_1^{-1}Q_2 \right) \delta T_1 dT_2 \delta \left( Q_1Q_2^{-1} \right) =
\left( Q_1^{-1}Q_2 \right) \delta R_{12}^{-1} dT_2 T_1 R_{21}^{-1} \delta \left( Q_1 Q_2^{-1} \right) = \\
& = & \left( Q_1^{-1}Q_2 R_{12}^{-1} Q_2 Q_1^{-1} \right) dX_2 X_1 \left( Q_2^{-1} Q_1 R_{21}^{-1} Q_1 Q_2^{-1} \right) \end{aligned}$$ where $R_{12}$ is again the matrix (\[rmatrix\]) and $dX_2 = {\mathbf{1}}\otimes dX$. Thus using the $R$-matrix for ${\mathfrak{M}}^{\rm I}_q$ (\[rmatrixI\]) and defining $R^{\rm I}_{21}=Q_1^{-1}Q_2R_{21}Q_1^{-1}Q^{2}$, we obtain: $$\label{xnc1forms}
R^{\rm I}_{12} X_1 dX_2 = dX_2 X_1 (R^{\rm I}_{21})^{-1}$$
Similarly, the module of 2-forms $\Omega^2_{{\mathfrak{M}}^{\rm I}_q}$, is the ${\mathfrak{M}}^{\rm I}_q$-bimodule generated by $dx_{rs'}\wedge dx_{kl'}$ satisfying the relations below (written in matrix form), which can be deduced from (\[xnc1forms\]): $$\label{xnc2forms}
R^{\rm I}_{12} dX_1 \wedge dX_2 = - dX_2 \wedge dX_1 (R^{\rm I}_{21})^{-1}$$ where $dX_1 = dX \otimes {\mathbf{1}}$.
Performing the same calculations for ${\mathfrak{M}}^{\rm J}_q$, we conclude that $\Omega^1_{{\mathfrak{M}}^{\rm J}_q}$ and $\Omega^2_{{\mathfrak{M}}^{\rm J}_q}$ are the ${\mathfrak{M}}^{\rm J}_q$-bimodules generated by $dy_{rs'}$ and $dy_{rs'}\wedge dy_{kl'}$, respectively, satisfying the following relations (written in matrix form): $$\label{yncforms} \begin{array}{ccc}
R^{\rm J}_{12} Y_1 dY_2 = dY_2 Y_1 (R^{\rm J}_{21})^{-1}
& \ \ {\rm and}\ \ &
R^{\rm J}_{12} dY_1 \wedge dY_2 = - dY_2 \wedge dY_1 (R^{\rm J}_{21})^{-1}
\end{array}$$ where $R^{\rm J}_{12}$ is the $R$-matrix for ${\mathfrak{M}}^{\rm J}_q$ (\[rmatrixJ\]), $R^{\rm J}_{21}=Q_2^{-1}Q_1R_{21}q_1Q_2^{-1}$ and $$dY = \left( \begin{array}{cc}
dy_{11'} & dy_{12'} \\ dy_{21'} & dy_{22'}
\end{array} \right) = Q\delta^{-1} dT Q^{-1} = Q^{-1} dT \delta^{-1} Q$$ with $dY_1 = dY \otimes{\mathbf{1}}$ and $dY_2 = {\mathbf{1}}\otimes dY$.
We now introduce the concept of anti-self-duality of quantum 2-forms over the quantum Minkowski space-time. Notice that the matrices $R^{\rm I}_{12}$, $R^{\rm I}_{21}$ and $R^{\rm J}_{12}$, $R^{\rm J}_{21}$ also satisfy Hecke relations (recall that $P$ denotes the permutation matrix): $$R^{\rm I}_{12} - (R^{\rm I}_{21})^{-1} = (p^{-1}-p) P
\ \ \ {\rm and}\ \ \
R^{\rm J}_{12} - (R^{\rm J}_{21})^{-1} = (p^{-1}-p) P$$ Moreover, defining $\hat{R}^{\rm I}_{12}=PR^{\rm I}_{12}$ and $\hat{R}^{\rm J}_{12}=PR^{\rm J}_{12}$, we obtain: $$\label{heckeIJ}
(\hat{R}^{\rm I}_{12})^2 = {\mathbf{1}}+ (p^{-1}-p) \hat{R}^{\rm I}_{12}
\ \ \ {\rm and}\ \ \
(\hat{R}^{\rm J}_{12})^2 = {\mathbf{1}}+ (p^{-1}-p) \hat{R}^{\rm J}_{12}$$ in analogy with (\[hecke2\]). Therefore we can proceed as in the case of $\widetilde{GL}(2)_q$ discussed above and define the projectors: $$P^{{\rm I}+} = \frac{\hat{R}^{\rm I}_{12} + p {\mathbf{1}}}{p+p^{-1}}
\ \ \ {\rm and} \ \ \
P^{{\rm I}-} = \frac{-\hat{R}^{\rm I}_{12} + p^{-1} {\mathbf{1}}}{p+p^{-1}}$$ $$P^{{\rm J}+} = \frac{\hat{R}^{\rm J}_{12} + p {\mathbf{1}}}{p+p^{-1}}
\ \ \ {\rm and} \ \ \
P^{{\rm J}-} = \frac{-\hat{R}^{\rm J}_{12} + p^{-1} {\mathbf{1}}}{p+p^{-1}}$$ Now we define the Hodge operator on $\Omega^2_{{\mathfrak{M}}^{\rm I}_q}$ (in matrix form): $$* dX_1 \wedge dX_2 = \left( P^{{\rm I}+} - P^{{\rm I}-} \right) dX_1 \wedge dX_2$$ Since $*^2={\mathbf{1}}$, the module $\Omega^2_{{\mathfrak{M}}_q^{\rm I}}$ can be decomposed into two submodules corresponding to eigenvalues $\pm1$. Denote such submodules by $\Omega^{2,+}_{{\mathfrak{M}}_q^{\rm I}}$ and $\Omega^{2,-}_{{\mathfrak{M}}_q^{\rm I}}$.
In order to compare with the commutative case, it is instructive to write down their bases, which are given by the entries of the matrices $P^{{\rm I}+} dX_1 \wedge dX_2$ and $P^{{\rm I}-} dX_1 \wedge dX_2$. After applying the commutation relations (\[nc2forms\]), we conclude that: $$\label{nc-sd2f}
\Omega^{2,+}_{{\mathfrak{M}}^{\rm I}_q} = {\mathfrak{M}}^{\rm I}_q \langle dx_{11'}\wedge dx_{12'},
dx_{21'}\wedge dx_{22'}, dx_{11'}\wedge dx_{22'} - dx_{12'}\wedge dx_{21'}
\rangle$$ $$\label{nc-asd2f}
\Omega^{2,-}_{{\mathfrak{M}}^{\rm I}_q} = {\mathfrak{M}}^{\rm I}_q \langle dx_{11'}\wedge dx_{21'},
dx_{12'}\wedge dx_{22'}, dx_{11'}\wedge dx_{22'} + dx_{12'}\wedge dx_{21'}
\rangle$$ in complete analogy with the commutative case.
Finally, replacing all the I’s by J’s, we define the Hodge operator on $\Omega^2_{{\mathfrak{M}}_q^{\rm J}}$ as well: $$* dY_1 \wedge dY_2 =
\left( P^{{\rm J}+} - P^{{\rm J}-} \right) dY_1 \wedge dY_2$$
#### Connection and curvature
Let $E$ be a right ${\mathfrak{M}}^{\rm I}_q$-module. In analogy with the commutative case, a connection on $E$ is a ${\mathbb{C}}$-linear map: $$\nabla: E {\rightarrow}E\otimes_{{\mathfrak{M}}^{\rm I}_q}\Omega^1_{{\mathfrak{M}}^{\rm I}_q}$$ satisfying the Leibnitz rule: $$\nabla(\sigma f)=\sigma\otimes df + \nabla(\sigma) f$$ where $f\in{\mathfrak{M}}^{\rm I}_q$ and $\sigma\in E$. The connection $\nabla$ also acts on 1-forms, being defined as the ${\mathbb{C}}$-linear map: $$\nabla: E\otimes_{{\mathfrak{M}}^{\rm I}_q}\Omega^1_{{\mathfrak{M}}^{\rm I}_q} {\rightarrow}E\otimes_{{\mathfrak{M}}^{\rm I}_q}\Omega^2_{{\mathfrak{M}}^{\rm I}_q}$$ satisfying: $$\nabla(\sigma\otimes\omega)=\sigma\otimes d\omega +
\nabla\sigma \wedge \omega$$ where $\omega\in\Omega^1_{{\mathfrak{M}}^{\rm I}_q}$.
Moreover, two connections $\nabla$ and $\nabla'$ are said to be gauge equivalent if there is $g\in{\rm Aut}_{{\mathfrak{M}}^{\rm I}_q}(E)$ such that $\nabla = g^{-1} \nabla' g$.
The curvature $F_\nabla$ is defined by the composition: $$E \stackrel{\nabla}{\longrightarrow}
E\otimes_{{\mathfrak{M}}^{\rm I}_q}\Omega^1_{{\mathfrak{M}}^{\rm I}_q}
\stackrel{\nabla}{\longrightarrow} E\otimes_{{\mathfrak{M}}^{\rm I}_q}
\Omega^2_{{\mathfrak{M}}^{\rm I}_q}$$ and it is easy to check that $F_\nabla$ is actually right ${\mathfrak{M}}^{\rm I}_q$-linear. Therefore, $F_\nabla$ can be regarded as an element of ${\rm End}_{{\mathfrak{M}}^{\rm I}_q}(E)\otimes_{{\mathfrak{M}}^{\rm I}_q}\Omega^2_{{\mathfrak{M}}^{\rm I}_q}$. Furthermore, if $\nabla$ and $\nabla'$ are gauge equivalent, then there is $g\in{\rm Aut}_{{\mathfrak{M}}^{\rm I}}(E)$ such that $F_\nabla = g^{-1} F_{\nabla'} g$. A connection $\nabla$ is said to be anti-self-dual if $F_\nabla\in
{\rm End}_{{\mathfrak{M}}^{\rm I}_q}(E)\otimes_{{\mathfrak{M}}^{\rm I}_q}\Omega^{2,-}_{{\mathfrak{M}}_q^{\rm I}}$.
Finally, if $E$ is projective (though not necessarily free in this case), any connection $\nabla$ can be encoded into the [*connection matrix*]{} $A\in{\rm End}_{{\mathfrak{M}}^{\rm I}_q}( \oplus^n{\mathfrak{M}}^{\rm I}_q )\otimes_{{\mathfrak{M}}^{\rm I}_q}\Omega^1_{{\mathfrak{M}}^{\rm I}_q}$ (where $n$ is the rank of $E$) in the following way. First, we recall the following basic result from algebra (see e.g. [@Lm]):
A finitely generated right $R$-module $M$ is projective of rank $n$ if and only if there are elements $\sigma_k\in M$ and $\rho^k\in M^\vee={\rm Hom}_R(M,R)$ for $k=1,...,n$ such that $m=\sum_k \sigma_k \rho^k(m)$, for any $m\in M$.
Let $\{\sigma_k,\rho^k\}$ be a dual basis for $E$, so that any $s\in E$ can be written as $s=\sum_k \sigma_k \rho^k(s)$; applying the Leibnitz rule, we get: $$\nabla s =
\sum_k \left( \sigma_k\otimes d(\rho^k(s)) + \nabla(\sigma_k) \rho^k(s) \right)$$ Thus, as in the commutative case, it is enough to know how $\nabla$ acts on $\{\sigma_k\}$; we set: $$\nabla\sigma_k = \sum_{l} \sigma_l A_k^{l},
\ \ \ {\rm with}\ \ \ A_k^l = \rho^l\otimes{\mathbf{1}}(\nabla\sigma_k) \in \Omega^1_{{\mathfrak{M}}^{\rm I}_q}$$ and we define $A$ as the matrix with entries given by the 1-forms $A_k^l$. Clearly, $A$ depends on the choice of dual basis; changing the dual basis amounts to a change of gauge for $A$.
Conversely given a matrix $A\in{\rm End}_{{\mathfrak{M}}^{\rm I}_q}(\oplus^n{\mathfrak{M}}^{\rm I}_q)\otimes_{{\mathfrak{M}}^{\rm I}_q}\Omega^1_{{\mathfrak{M}}^{\rm I}_q}$, we define the connection: $$\nabla_A s = \sum_{l,k} \sigma_k\otimes d(\rho^k(s)) +
\sigma_l A_k^l \rho^k(s)$$
Construction of quantum instantons {#ward}
==================================
Classical instantons and the ADHM data
--------------------------------------
As we mentioned in Introduction, solutions of the classical ASDYM equations can be constructed from certain linear data, so-called [*ADHM data*]{}. We will now briefly review some relevant facts regarding this correspondence.
Recall that the ASDYM equation can be defined for a complex vector bundle $E$ over any four dimensional Riemannian manifold $X$, provided with a connection $\nabla$. This connection is said to be [*anti-self-dual*]{} if the corresponding curvature 2-form $F_\nabla$ satisfies the equation ($*$ is the Hodge operator on 2-forms): $$*F_\nabla = - F_\nabla$$ i.e., if $F_\nabla\in{\rm End}(E)\otimes\Omega^{2,-}_X$. An ASD connection is usually called an [*instanton*]{} if the integral: $$c = \frac{1}{8\pi^2} \int_X {\rm Tr}(F_\nabla \wedge F_\nabla)$$ converges. If $X$ is a compact manifold, $c$ is actually an integer, so-called [*instanton number*]{} or [*charge*]{}, and coincides with the second Chern class of $E$. In view of the symmetry between self-dual and anti-self-dual connections (they only differ by the choice of orientation on $X$), one can assume without loss of generality that $c>0$. Furthermore, a framing for an instanton on $X$ at a point $p\in X$ is the choice of an isomorphism $E_p\simeq{\mathbb{C}}^n$, where $E_p$ denotes the fiber of $E$ at $p$.
In the celebrated paper [@ADHM], Atiyah, Drinfeld, Hitchin and Manin constructed a class of ASD connections on the simplest compact four dimensional manifold, namely the four dimensional sphere $S^4$. They also proved that their construction is complete in the sense that any ASD connection on $S^4$ is gauge equivalent to a connection constructed by them from a certain algebraic data that depends on the rank $n$ of the vector bundle $E$ and the instanton number $c$.
More precisely, let $V$ and $W$ be Hermitian vector spaces of dimensions $c$ and $n$, respectively. The algebraic data of Atiyah, Drinfeld, Hitchin and Manin consists of four linear operators: $$\label{adhm.ops}
B_1, B_2 \in {\rm End}(V), \ \ \ \ \ \
i \in {\rm Hom}(W,V), \ \ \ \ \ \ j \in {\rm Hom}(V,W)$$ satisfying the following linear relations ($\dagger$ denotes Hermitian conjugation): $$\begin{aligned}
\label{adhm.eqn1} [ B_1 , B_2 ] + ij & = & 0 \\
\label{adhm.eqn2} [ B_1 , B_1^\dagger ] + [ B_2 , B_2^\dagger ] + ii^\dagger -
j^\dagger j & = & 0\end{aligned}$$ plus a regularity condition which we describe below.
The ADHM data admits a natural action of the unitary group $U(V)$: $$\label{action}
g(B_1,B_2,i,j) = (gB_1g^{-1},gB_2g^{-1},gi,jg^{-1}), \ \ \ g\in U(V)$$ We say that the ADHM datum $(B_1,B_2,i,j)$ is [*regular*]{} if its stabilizer subgroup is trivial. Equivalently, $(B_1,B_2,i,j)$ is regular if and only if it satisfies the following two conditions:
- [*stability*]{}: there is no proper subspace $S\subset V$ such that $B_k(S)\subset S$ ($k=1,2$) and $i(W)\subset S$;
- [*costability*]{}: there is no proper subspace $S\subset V$ such that $B_k(S)\subset S$ ($k=1,2$) and $S\subset \ker j$.
The following lemma is probably well known to the experts:
\[st=cost\] Suppose that $(B_1,B_2,i,j)$ satisfies the ADHM equations (\[adhm.eqn1\]) and (\[adhm.eqn2\]). Then $(B_1,B_2,i,j)$ is stable if and only if it is also costable.
If $(B_1,B_2,i,j)$ is not stable, then by duality on $V$ there is a proper subspace $S^\perp\subset V$ such that $B_k^\dagger(S^\perp)\subset S^\perp$ and $S^\perp\subset\ker i^\dagger$. So restricting (\[adhm.eqn2\]) to $S^\perp$ and taking the trace, we conclude that ${\rm Tr}(j^\dagger j|_{S^\perp}) = 0$. Hence $S^\perp \subset \ker j$, and $(B_1,B_2,i,j)$ is not costable.
Conversely, $(B_1,B_2,i,j)$ is not costable, take $S^\perp\subset V$ nonempty such that $B_k^\dagger(S^\perp)\subset S^\perp$ and $S^\perp\subset \ker j$. Restricting (\[adhm.eqn2\]) to $S^\perp$ and taking the trace we conclude that ${\rm Tr}(i i^\dagger|_{S^\perp}) = 0$. Hence $S^\perp \subset \ker i^\dagger$, and dualizing equation (\[adhm.eqn1\]) we see that $[B_1^\dagger,B_2^\dagger]|_{S^\perp}=0$. Thus $S$ is a proper subspace of $V$ such that $B_k(S)\subset S$ and $i(W)\subset S$, contradicting stability. $\Box$
We denote the space of regular orbits by: $$\label{moduli}
{\cal M}^{\rm reg}(n,c) = \{ (B_1,B_2,i,j) \ | \
(\ref{adhm.eqn1}) , (\ref{adhm.eqn2}) \} / U(V)$$ The main result of [@ADHM] is the following:
The space ${\cal M}^{\rm reg}(n,c)$ is the moduli space of framed instantons of rank $n$ and charge $c$ on $S^4$. In other words, there is a bijection between points of ${\cal M}^{\rm reg}(n,c)$ and gauge equivalence classes of framed ASD connections of rank $n$ and charge $c$.
Quantum Haar measure and duality
--------------------------------
For later reference, we now explain the notion of duality on free ${\mathfrak{M}}^{\rm I}_q$- and ${\mathfrak{M}}^{\rm J}_q$-modules. We also now specialize the formal parameter $q$ to a positive real number for the remainder of this section.
Recall that a Haar functional on a Hopf algebra $\cal A$ is a linear functional $H:{\cal A}{\rightarrow}{\mathbb{C}}$ satisfying the following conditions:
- bi-invariance: $(H\otimes{\mathbf{1}})\circ\Delta(a) = ({\mathbf{1}}\otimes H)\circ\Delta(a) = H(a)$;
- antipode invariance: $H(\gamma(a))=H(a)$;
- normalization: $H(1)=1$.
\[thm-h\] There is a unique Haar functional $H$ on the quantum group $SL(2)_q$, which induces a positive definite Hermitian form on $SU(2)_q$, namely: $$\label{h}
( g_1 , g_2 ) = H(f^\dagger g), \ \ \ g_1,g_2\in SL(2)_q$$
The existence and uniqueness of the Haar functional was established in [@K; @MU; @W0]. The verification of the fact that $H$ induces a Hermitian form also can be found in these papers. $\Box$
Now let $U$ be a finite dimensional Hermitian vector space, and let $(\cdot,\cdot)$ denote its Hermitian inner product, which is chosen to be conjugate linear in the first argument. We define the pairing: $$\begin{aligned}
(\cdot,\cdot) :
U\otimes{\mathfrak{M}}^{\rm I}_q \times U\otimes{\mathfrak{M}}^{\rm I}_q & {\rightarrow}&
{\mathfrak{M}}^{\rm I}_q \nonumber \\
( v_1\otimes f_1,v_2\otimes f_2 ) & = &
(v_1,v_2) f_1^\dagger f_2
\label{innerprod}\end{aligned}$$ From a geometrical point of view, the pairing above plays the role of a Hermitian metric on a (trivial) vector bundle over the Euclidean ${\mathbb{R}}^4$: the pairing of two sections of the bundle gives a function on ${\mathbb{R}}^4$.
\[n-deg2\] The pairing (\[innerprod\]) is non-degenerate, i.e.: $$( \sigma , \sigma ) = 0 \ \ \Leftrightarrow \ \ \sigma=0$$
We extend the Haar functional to $\widetilde{SL}(2)_q$ as the homomorphism $\tilde{H}:\widetilde{SL}(2)_q\to {\mathbb{C}}[\delta,\delta^{-1}]$ such that ($n\in{\mathbb{Z}}$): $$\tilde{H} (\delta^n g) = \delta^n H(g)$$
Now let $\{v_k\}$ be an orthonormal basis for $U$, and take $\sigma=\sum_k v_k\otimes f_k$, where $f_k\in{\mathfrak{M}}^{\rm I}_q$; we can assume that $f_k=\sum_{\alpha\geq
0}\delta^\alpha f_{k\alpha}$, with $f_{k\alpha}\in SL_q(2)$.
Then $0=\tilde{H}((\sigma,\sigma))$ forces, by the non degeneracy of $H$, $f_{k0}=0$, [*i.e.*]{} $\sigma=\sigma_1\delta$, for some $\sigma_1$. Then $0=(\sigma,\sigma)=\delta(\sigma_1,\sigma_1)\delta$, and $(\sigma_1,\sigma_1)=0$ by the invertibility of $\delta$ in $\widetilde{SL_q(2)}$. By iterating this procedure we get that $\sigma = 0$.
$\Box$
Now consider $F_l=U_l\otimes{\mathfrak{M}}^{\rm I}_q$ as right ${\mathfrak{M}}^{\rm I}_q$-modules, where $U_l$ are Hermitian vector spaces, $l=1,2$. Let $F_l^\dagger$ denote the set of all maps $\mu:F_l\to{\mathfrak{M}}^{\rm I}_q$ such that $\mu(\sigma x)=\mu(\sigma)x$, for all $\sigma\in F_l$ and $x\in{\mathfrak{M}}^{\rm I}_q$, with the structure of a right ${\mathfrak{M}}^{\rm I}_q$-module defined by: $$(\mu x) (\sigma) = x^\dagger\mu(\sigma)
\ \ \ \forall \sigma\in F_l$$
By Proposition \[n-deg2\], the map: $$\begin{aligned}
{\cal I}_l: F_l & \longrightarrow & F_l^\dagger \label{idn} \\
\sigma & \mapsto & ( \sigma , \cdot ) \nonumber \end{aligned}$$ is injective, and it is easy to see that $\cal I$ must also be surjective. Moreover, notice that $${\cal I}(\sigma x) = x^\dagger {\cal I}(\sigma)$$ In other words, the map $\cal I$ provides an identification between a free ${\mathfrak{M}}^{\rm I}_q$-module and its dual. In particular, one can also regard $\dagger$ as an involution on $F_l$, defined as follows: $$\label{modinv}
(v\otimes f)^\dagger = \overline{v}\otimes f^\dagger$$ and it is easy to see that $(\sigma x)^\dagger = x^\dagger\sigma^\dagger$ (with this equality being understood in terms of the bimodule structure of the free modules $F_l$).
Any map ${\cal L}:F_1\to F_2$ satisfying ${\cal L}(\sigma x)={\cal L}(\sigma)x$ induces a dual map ${\cal L}^\vee:F_2^\dagger \to F_1^\dagger$ in the usual way: $${\cal L}^\vee (\varphi) = \varphi\circ {\cal L}$$ Finally, the identification (\[idn\]) yields a map ${\cal L}^\dagger=
{\cal I}_1^{-1}{\cal L}^\vee{\cal I}_2:F_2\to F_1$ with the property: $$( {\cal L}^\dagger \sigma_1 , \sigma_2 ) =
( \sigma_1 , {\cal L} \sigma_2 ) \ \ \ \ \sigma_l\in F_l$$ In particular, let us consider ${\cal L}=L\otimes x$, where $L\in{\rm Hom}(U_1,U_2)$ and $x$ means multiplication by $x\in{\mathfrak{M}}^{\rm I}_q$ in the left. Then the definition (\[innerprod\]) immediately implies that ${\cal L}^\dagger = L^\dagger\otimes {x^\dagger}$.
\[decomposition\] If ${\cal L}:F_1\to F_2$ is injective, then $$F_2 = {\rm Im}{\cal L} \oplus \ker {\cal L}^\dagger$$
Since ${\cal L}$ is injective, it is easy to see that ${\rm Im}{\cal L}$ is a free submodule of $F_2$. Let $N\subset F_2$ be such that $F_2={\rm Im}{\cal L} \oplus N$. Define: $$({\rm Im}{\cal L})^0 =
\{ \psi\in F_2^\dagger \ | \ \psi(\nu)=0 \ \forall \nu\in {\rm Im}{\cal L} \}$$ One can then show that $N^\dagger \simeq ({\rm Im}{\cal L})^0 \simeq \ker{\cal L}^\vee$. The desired decomposition now follows from the identification $N\simeq N^\dagger$. $\Box$
Construction of quantum instantons {#construction-of-quantum-instantons}
----------------------------------
First of all, we must explain precisely what we mean by a quantum instanton on the quantum affine Minkowski space.
\[nc.instanton\] A quantum instanton on $S^{\rm I}_q$ is a triple $(E,\nabla,\dagger)$ consisting of:
- a finitely generated, projective right ${\mathfrak{M}}^{\rm I}_q$-module $E$ equipped with an involution $\dagger: E \rightarrow E$ satisfying $(\sigma x)^\dagger = x^\dagger \sigma^\dagger$ for all $x\in{\mathfrak{M}}^{\rm I}_q$ and $\sigma\in E$;
- anti-self-dual connection $\nabla:E{\rightarrow}E\otimes\Omega^1_{{\mathfrak{M}}^{\rm I}_q}$ which is compatible with the involution $\dagger$, i.e. $\nabla \dagger = \dagger \nabla$.
Quantum instantons on $S^{\rm J}_q$ are similarly defined. Moreover, we also define the notion of [*consistency*]{} between quantum instantons on $S^{\rm I}_q$ and $S^{\rm J}_q$:
\[consistency\] Quantum instantons $(E_{\rm I},\nabla_{\rm I},\dagger_{\rm I})$ on $S^{\rm I}_q$ and $(E_{\rm J},\nabla_{\rm J},\dagger_{\rm J})$ on $S^{\rm J}_q$ are said to be [*consistent*]{} if:
- there is an isomorphism $$\label{nc.mod.glue.map}
\Gamma: E_{\rm I} [\delta^{-1}] {\rightarrow}E_{\rm J}[ \delta ]$$ such that $\Gamma(\sigma f)=\Gamma(\sigma)\eta(f)$, for all $\sigma\in E_{\rm I}[\delta^{-1}]$ and $f\in {\mathfrak{M}}^{\rm I}_q[\delta^{-1}]$;
- $\nabla_{\rm J} \Gamma = \Gamma \nabla_{\rm I}$;
- $\dagger_{\rm J} \Gamma = \Gamma \dagger_{\rm I}$.
Recall that $\eta$ is the isomorphism ${\mathfrak{M}}^{\rm I}_q[\delta^{-1}] {\rightarrow}{\mathfrak{M}}^{\rm J}_q[\delta]$ described in (\[nc.glueing\]). Geometrically, the consistency condition means that the quantum instantons $\nabla_{\rm I}$ and $\nabla_{\rm J}$ coincide in the “intersection” variety ${\mathfrak{M}}^{\rm I}_q[\delta^{-1}]\simeq{\mathfrak{M}}^{\rm J}_q[\delta]$, up to a gauge transformation.
The goal of this Section is to convert ADHM data into a consistent pair of quantum instantons, in close analogy with the classical case.
#### Quantum Instantons on quantum Minkowski space-time
As before, let $V$, $W$ denote Hermitian vector spaces of dimension $c$ and $n$, respectively. Let $\tilde{W}=V\oplus V\oplus W$.
Let $(B_1,B_2,i,j)$ be an ADHM datum, as in (\[adhm.ops\]). We start by considering the following sequence of free ${\mathfrak{M}}_q^{\rm I}$-modules: $$V\otimes{\mathfrak{M}}_q^{\rm I} \stackrel{\alpha_{\rm I}}{\longrightarrow}
\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I} \stackrel{\beta_{\rm I}}{\longrightarrow}
V\otimes{\mathfrak{M}}_q^{\rm I}$$ where the maps $\alpha_{\rm I}$ and $\beta_{\rm I}$ are given by: $$\label{alpha}
\alpha_{\rm I} = \left( \begin{array}{c}
B_1\otimes{\mathbf{1}}- {\mathbf{1}}\otimes x_{21'} \\
B_2\otimes{\mathbf{1}}- {\mathbf{1}}\otimes x_{22'} \\
j\otimes{\mathbf{1}}\end{array} \right)$$ and $$\label{beta}
\beta_{\rm I} = \left( \begin{array}{ccc}
-B_2\otimes{\mathbf{1}}+ {\mathbf{1}}\otimes x_{22'} \ \ &
B_1\otimes{\mathbf{1}}- {\mathbf{1}}\otimes x_{21'} \ \ &
i\otimes{\mathbf{1}}\end{array} \right)$$
The induced dual maps $\beta_{\rm I}^\dagger:V\otimes{\mathfrak{M}}_q^{\rm I}\to\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I}$ and $\alpha_{\rm I}^\dagger:\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I}\to V\otimes{\mathfrak{M}}_q^{\rm I}$ are given by: $$\label{betadual}
\beta_{\rm I}^\dagger = \left( \begin{array}{c}
-B_2^\dagger\otimes{\mathbf{1}}+ {\mathbf{1}}\otimes x_{11'} \\
B_1^\dagger\otimes{\mathbf{1}}+ {\mathbf{1}}\otimes x_{12'} \\
i^\dagger\otimes{\mathbf{1}}\end{array} \right)$$ and $$\label{alphadual}
\alpha_{\rm I}^\dagger = \left( \begin{array}{lcr}
B_1^\dagger\otimes{\mathbf{1}}+ {\mathbf{1}}\otimes x_{12'} \ \ &
B_2^\dagger\otimes{\mathbf{1}}- {\mathbf{1}}\otimes x_{11'} \ \ &
j^\dagger\otimes{\mathbf{1}}\end{array} \right)$$
\[compo\]
1. $\beta_{\rm I}\alpha_{\rm I}=0$ if and only if $[B_1,B_2]+ij=0$.
2. $\beta_{\rm I}\beta_{\rm I}^\dagger=
\alpha_{\rm I}^\dagger\alpha_{\rm I}$ if and only if $[B_1,B_1^\dagger]+[B_2,B_2^\dagger]+ii^\dagger-j^\dagger j=0$.
It is easy to check that: $$\beta_{\rm I}\alpha_{\rm I} =
\left( [B_1,B_2]+ij \right) \otimes{\mathbf{1}}+
{\mathbf{1}}\otimes [x_{22'},x_{21'}]$$ Since $[x_{22'},x_{21'}]=0$ (see equation (\[xcommut1\])), the first statement follows.
Another straightforward calculation reveals that: $$\beta_{\rm I}\beta_{\rm I}^\dagger -\alpha_{\rm I}^\dagger\alpha_{\rm I} =
\left( [B_1,B_1^\dagger]+[B_2,B_2^\dagger]+ii^\dagger-j^\dagger j \right) \otimes {\mathbf{1}}+ {\mathbf{1}}\otimes ([x_{11'},x_{22'}]-[x_{12'},x_{21'}])$$ and the second statement follows from equation (\[xcommut3\]). $\Box$
\[inj-surj\]
1. $\alpha_{\rm I}$ is injective.
2. $\beta_{\rm I}$ is surjective if and only if $(B_1,B_2,i,j)$ is stable.
Let ${\cal X}_1={\mathbb{C}}[x_{11'},x_{12'}]$ and ${\cal X}_2={\mathbb{C}}[x_{21'},x_{22'}]$. Then ${\mathfrak{M}}^{\rm I}_q$ can be represented in the following way: $${\mathfrak{M}}^{\rm I}_q = {\cal X}_1 \otimes {\cal X}_2 /
(\ref{xcommut3}),(\ref{xcommut2})$$ Clearly, $\alpha_{\rm I}$ and $\beta_{\rm I}$ can be restricted to maps $$V\otimes{\cal X}_2 \stackrel{\alpha_{\rm I}}{\longrightarrow}
\tilde{W}\otimes{\cal X}_2 \stackrel{\beta_{\rm I}}{\longrightarrow}
V\otimes{\cal X}_2$$ It then follows from Nakajima (see [@N2], lemma 2.7) that $\alpha_{\rm I}|_{{\cal X}_2}$ is injective and that $\beta_{\rm I}|_{{\cal X}_2}$ is surjective if and only if $(B_1,B_2,i,j)$ is stable. Since $\alpha_{\rm I}={\mathbf{1}}_{{\cal X}_1}\otimes\alpha_{\rm I}|_{{\cal X}_2}$ and $\beta_{\rm I}={\mathbf{1}}_{{\cal X}_1}\otimes\beta_{\rm I}|_{{\cal X}_2}$, we conclude that $\alpha_{\rm I}$ is also injective, and that $\beta_{\rm I}$ is surjective if and only if $(B_1,B_2,i,j)$ is stable. $\Box$
Thus we have the short sequence of free ${\mathfrak{M}}^{\rm I}_q$-modules: $$\label{nc.monad2}
0 {\rightarrow}V\otimes{\mathfrak{M}}_q^{\rm I} \stackrel{\alpha_{\rm I}}{\longrightarrow}
\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I} \stackrel{\beta_{\rm I}}{\longrightarrow}
V\otimes{\mathfrak{M}}_q^{\rm I} {\rightarrow}0$$ which is exact on the first and last terms. Its middle cohomology $E_{\rm I}={\rm ker}\beta_{\rm I}/{\rm Im}\alpha_{\rm I}$ is then a well defined right ${\mathfrak{M}}^{\rm I}_q$-module since both $\beta_{\rm I}$ and $\alpha_{\rm I}$ are right linear. Clearly $E_{\rm I}$ is torsion-free, since it is a submodule of a free module; we argue that it is projective.
Furthermore, notice that the pairing (\[innerprod\]) also induces a pairing $E_{\rm I}\times E_{\rm I}\to {\mathfrak{M}}^{\rm I}_q$.
\[inj-surj2\]
1. $\beta_{\rm I}^\dagger$ is injective.
2. $\alpha_{\rm I}^\dagger$ is surjective if and only if $(B_1,B_2,i,j)$ is costable.
Same argument as Proposition \[inj-surj\], just replacing ${\cal X}_2$ by ${\cal X}_1$ and vice-versa. $\Box$
\[iso\] $\xi_{\rm I}=\beta_{\rm I}\beta_{\rm I}^\dagger=\alpha_{\rm I}^\dagger\alpha_{\rm I}$ is an isomorphism if and only if $(B_1,B_2,i,j)$ is stable.
Applying Proposition \[decomposition\] to the map $\beta_{\rm I}^\dagger$, we get that: $$\label{decomp-beta}
\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I} =
{\rm Im}\beta_{\rm I}^\dagger \oplus \ker\beta_{\rm I}$$ In particular, ${\rm Im}\beta_{\rm I}^\dagger \cap \ker \beta_{\rm I} =\{0\}$ so that the injectivity of $\xi_{\rm I}$ follows from the injectivity of $\beta_{\rm I}^\dagger$.
Now if $(B_1,B_2,i,j)$ is stable then $\beta_{\rm I}$ is surjective, so given $\nu\in V\otimes{\mathfrak{M}}_q^{\rm I}$, there is $\mu\in\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I}$ such that $\beta_{\rm I}(\mu)=\nu$. By the decomposition (\[decomp-beta\]), we know that $\mu=\mu'+\mu''$ for some $\mu'\in {\rm Im}\beta_{\rm I}^\dagger$ and $\mu''\in \ker\beta_{\rm I}$. Let $\mu'=\beta_{\rm I}^\dagger(\nu')$. Then $\nu=\beta_{\rm I}(\mu)=\beta_{\rm I}(\mu')=
\beta_{\rm I}(\beta_{\rm I}^\dagger(\nu'))$, hence $\xi_{\rm I}$ is also surjective.
Conversely, if $(B_1,B_2,i,j)$ is not stable, then $\beta_{\rm I}$ is not surjective, so $\beta_{\rm I}\beta_{\rm I}^\dagger$ is not surjective either, and $\xi_{\rm I}$ is not an isomorphism. $\Box$
Now consider the [*Dirac operator*]{}: $$\begin{aligned}
{\cal D}_{\rm I} :
\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I} & {\rightarrow}& (V\oplus V)\otimes{\mathfrak{M}}_q^{\rm I} \nonumber \\
\label{dirac} {\cal D}_{\rm I} & = &
\left( \begin{array}{c} \beta_{\rm I} \\ \alpha_{\rm I}^\dagger \end{array} \right) \end{aligned}$$ Moreover, we define the [*Laplacian*]{}: $$\Xi_{\rm I} : (V\oplus V)\otimes{\mathfrak{M}}_q^{\rm I} {\rightarrow}(V\oplus V)\otimes{\mathfrak{M}}_q^{\rm I}$$ $$\label{nc.laplacian}
\Xi_{\rm I} = {\cal D}_{\rm I}{\cal D}_{\rm I}^\dagger =
\left( \begin{array}{cc} \beta_{\rm I}\beta_{\rm I}^\dagger & 0 \\
0 & \alpha_{\rm I}^\dagger\alpha_{\rm I} \end{array} \right)$$ Proposition \[iso\] implies that $\Xi_{\rm I}=\xi_{\rm I}{\mathbf{1}}_{V\oplus V}$ and $\Xi_{\rm I}$ is an isomorphism too. We can then define the projection map: $$P_{\rm I} : \tilde{W}\otimes{\mathfrak{M}}_q^{\rm I} {\rightarrow}E_{\rm I}$$ $$\label{proj}
P_{\rm I} = {\mathbf{1}}- {\cal D}_{\rm I}^\dagger\Xi_{\rm I}^{-1}{\cal D}_{\rm I}$$ For the next two Propositions, we assume that $(B_1,B_2,i,j)$ is stable. We have:
\[proj2\] $E_{\rm I} \simeq {\rm ker}{\cal D}_{\rm I}$. In particular, $E_{\rm I}$ is a projective right ${\mathfrak{M}}_q^{\rm I}$-module.
Given $\psi\in{\rm ker}\beta_{\rm I}$, we show that there is a unique $\nu\in V\otimes{\mathfrak{M}}_q^{\rm I}$ such that $\psi'=\psi+\alpha_{\rm I}(\nu)\in\ker{\cal D}_{\rm I}$, i.e. $\beta_{\rm I}(\psi')=\alpha_{\rm I}^\dagger(\psi')=0$. Indeed: $$\beta_{\rm I}(\psi')=\beta_{\rm I}\alpha_{\rm I}(\nu)=0$$ $$\alpha_{\rm I}^\dagger(\psi')=0 \Leftrightarrow
\alpha_{\rm I}^\dagger(\psi)= -\alpha_{\rm I}^\dagger\alpha_{{\rm I}}(\nu)$$ but $\xi_{\rm I}=\alpha_{\rm I}^\dagger\alpha_{\rm I}$ is an isomorphism, thus $\nu=\xi_{\rm I}^{-1}\alpha_{\rm I}^\dagger(\psi)$, as desired.
Finally, it is easy to see that given $\nu\in\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I}$, there are unique $\psi\in\ker{\cal D}_{\rm I}$ and $\varphi\in{\rm Im}{\cal D}_{\rm I}^\dagger$ such that $\nu=\psi+\varphi$. Indeed, just take $\psi=P_{\rm I}\nu$ and $\varphi={\cal D}_{\rm I}^\dagger\Xi_{\rm I}^{-1}{\cal D}_{\rm I}\nu$.
In other words, we conclude that $E_{\rm I}\oplus{\rm Im}{\cal D}_{\rm I}^\dagger=
\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I}$, which implies that $E_{\rm I}$ is projective as a ${\mathfrak{M}}_q^{\rm I}$-module. $\Box$
Note also that $E_{\rm I}$ is finitely generated and has rank $n=\dim W$.
To define the connection, let $\iota_{\rm I}:E_{\rm I}{\rightarrow}\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I}$ denote the natural inclusion and $d:{\mathfrak{M}}_q^{\rm I}{\rightarrow}\Omega^1_{{\mathfrak{M}}_q^{\rm I}}$ denote the quantum de Rham operator. We define $\nabla_{\rm I}$ via the composition: $$\xymatrix{
E_{\rm I} \ar[r]^{\iota_{\rm I}} &
\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I} \ar[r]^{{\mathbf{1}}\otimes d} &
\tilde{W}\otimes\Omega^1_{{\mathfrak{M}}_q^{\rm I}} \ar[r]^{P_{\rm I}\otimes{\mathbf{1}}} &
E_{\rm I}\otimes_{{\mathfrak{M}}_q^{\rm I}}\Omega^1_{{\mathfrak{M}}_q^{\rm I}}
}$$
\[asd\] $F_{\nabla_{\rm I}}$ is anti-self-dual.
Note that $F_{\nabla_{\rm I}}=\nabla_{\rm I}\nabla_{\rm I}=P_{\rm I}dP_{\rm I}d$; therefore given $e\in E_{\rm I}$ we have: $$\begin{aligned}
F_{\nabla_{\rm I}} e & = &
P_{\rm I} \left( d ( {\mathbf{1}}- {\cal D}_{\rm I}^\dagger\Xi_{\rm I}^{-1}{\cal D}_{\rm I} ) de \right) =
P_{\rm I} \left( d {\cal D}_{\rm I}^\dagger\Xi_{\rm I}^{-1} (d{\cal D}_{\rm I}) e \right) = \\
& = & P_{\rm I} \left( (d{\cal D}_{\rm I}^\dagger)\Xi_{\rm I}^{-1} (d{\cal D}_{\rm I}) e +
{\cal D}_{\rm I}^\dagger d(\Xi_{\rm I}^{-1} (d{\cal D}_{\rm I}) e) \right) = \\
& = & P_{\rm I} \left( (d{\cal D}_{\rm I}^\dagger)\Xi_{\rm I}^{-1} (d{\cal D}_{\rm I}) e \right)\end{aligned}$$ for $P_{\rm I} \left({\cal D}_{\rm I}^\dagger d(\Xi_{\rm I}^{-1} (d{\cal D}_{\rm I}) e) \right) = 0$. Since $\Xi_{\rm I}^{-1}=\xi_{\rm I}^{-1}{\mathbf{1}}$, we conclude that $F_{\nabla_{\rm I}}$ is proportional to $d{\cal D}_{\rm I}^\dagger\wedge d{\cal D}_{\rm I}$, as a 2-form.
It is then a straightforward calculation to show that each entry of $d{\cal D}_{\rm I}^\dagger\wedge d{\cal D}_{\rm I}$ belongs to $\Omega^{2,-}_{{\mathfrak{M}}_q^{\rm I}}$; indeed: $$d{\cal D}_{\rm I}^\dagger\wedge d{\cal D}_{\rm I} =
\left( \begin{array}{lr}
dx_{11'} \ \ & \ \ -dx_{21'} \\
dx_{12'} \ \ & \ \ -dx_{22'} \\ 0 & 0
\end{array} \right) \wedge
\left( \begin{array}{lrc}
dx_{22'} \ \ & \ \ -dx_{21'} & \ 0 \\
dx_{12'} \ \ & \ \ -dx_{11'} & \ 0
\end{array} \right) =$$ $$= \left( \begin{array}{ccc}
dx_{11'}dx_{22'}-dx_{21'}dx_{12'} \ \ &
\ \ -dx_{11'}dx_{21'}+dx_{21'}dx_{11'} & \ 0 \\
dx_{12'}dx_{22'}-dx_{22'}dx_{12'} \ \ &
\ \ -dx_{12'}dx_{21'}+dx_{22'}dx_{11'} & \ 0 \\
0 & 0 & \ 0
\end{array} \right)$$ Applying the commutation relations (\[xnc2forms\]), we obtain: $$d{\cal D}_{\rm I}^\dagger\wedge d{\cal D}_{\rm I} =
\left( \begin{array}{ccc}
dx_{11'}dx_{22'}+dx_{12'}dx_{21'} &
-2dx_{11'}dx_{21'} & \ 0 \\
2dx_{12'}dx_{22'} &
-(dx_{11'}dx_{22'}+dx_{12'}dx_{21'})& \ 0 \\
0 & 0 & \ 0
\end{array} \right)$$ Comparing with (\[nc-asd2f\]), we have proved our claim. $\Box$
#### Gauge equivalence
We show that if $(B_1,B_2,i,j)$ and $(B_1',B_2',i',j')$ are equivalent ADHM data, then the respective pairs $(E_{\rm I},\nabla_{\rm I})$ and $(E_{\rm I}',\nabla'_{\rm I})$ are gauge equivalent, in the sense that there is a ${\mathfrak{M}}^{\rm I}_q$-isomorphism $G:E_{\rm I}'{\rightarrow}E_{\rm I}$ such that $\nabla_{\rm I}'=G^{-1}\nabla_{\rm I}G$.
To do that, recall that $(B_1,B_2,i,j)$ and $(B_1',B_2',i',j')$ are equivalent if there exists $g\in{\rm U}(V)$ such that: $$\begin{array}{ccccc}
B_k' = g B_k g^{-1},\ k=1,2 & \ \ \ &
i' = g i & \ \ \ & j' = j g^{-1}
\end{array}$$ Let $G\in U(\tilde{W})$ be given by $g\times g\times{\mathbf{1}}_{W}$. It is then easy to check that the following diagram is commutative: $$\label{geqv}
\xymatrix{
0 \ar[r] & V\otimes{\mathfrak{M}}^{\rm I}_q \ar[r]^{\alpha_{\rm I}'}
\ar[d]^{g\otimes{\mathbf{1}}} &
\tilde{W}\otimes{\mathfrak{M}}^{\rm I}_q \ar[r]^{\beta_{\rm I}'}
\ar[d]^{G\otimes{\mathbf{1}}} &
V\otimes{\mathfrak{M}}^{\rm I}_q \ar[r] \ar[d]^{g\otimes{\mathbf{1}}} & 0 \\
0 \ar[r] & V\otimes{\mathfrak{M}}^{\rm I}_q \ar[r]^{\alpha_{\rm I}} &
\tilde{W}\otimes{\mathfrak{M}}^{\rm I}_q \ar[r]^{\beta_{\rm I}}&
V\otimes{\mathfrak{M}}^{\rm I}_q \ar[r] & 0
}$$ Therefore the modules $E_{\rm I}={\rm ker}\beta_{\rm I}/{\rm Im}\alpha_{\rm I}$ and $E_{\rm I}'={\rm ker}\beta_{\rm I}'/{\rm Im}\alpha_{\rm I}'$ are isomorphic; indeed, it is easy to see that $G$ maps $E_{\rm I}'$ onto $E_{\rm I}$ (regarded as submodules of $\tilde{W}\otimes{\mathfrak{M}}^{\rm I}$). We shall also denote by $G$ the induced isomorphism $E_{\rm I}'{\rightarrow}E_{\rm I}$).
Now denote by $\iota_{\rm I}':E_{\rm I}'{\rightarrow}\tilde{W}\otimes{\mathfrak{M}}^{\rm I}_q$ the inclusion and by $P_{\rm I}':\tilde{W}\otimes{\mathfrak{M}}^{\rm I}_q{\rightarrow}E_{\rm I}'$ the projection (\[proj\]). Clearly, $\iota_{\rm I}'=G^{-1}\iota_{\rm I} G$ and $P_{\rm I}'=G^{-1} P_{\rm I} G$. In addition, we have: $$\begin{aligned}
\nabla_{\rm I}' & = & P_{\rm I}' d \iota_{\rm I}' =
G^{-1} P_{\rm I} G d G^{-1}\iota_{\rm I} G = \\
& = & G^{-1} P_{\rm I} \left( GdG^{-1}(\iota_{\rm I} G) +
d \iota_{\rm I} G \right) = \\
& = & G^{-1} P_{\rm I} d \iota_{\rm I} G = G^{-1}\nabla_{\rm I}G \end{aligned}$$ since $G$ acts as the identity on ${\mathfrak{M}}^{\rm I}_q$, so that $dG^{-1}=0$.
#### Real structure
Dualizing the monad (\[nc.monad2\]) and using the identifications $(V\otimes{\mathfrak{M}}_q^{\rm I})^\dagger\simeq V\otimes{\mathfrak{M}}_q^{\rm I}$ and $(\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I})^\dagger\simeq\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I}$ one obtains: $$0 {\rightarrow}V\otimes{\mathfrak{M}}_q^{\rm I} \stackrel{\beta_{\rm I}^\dagger}{{\rightarrow}}
\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I}
\stackrel{\alpha_{\rm I}^\dagger}{{\rightarrow}}
V\otimes{\mathfrak{M}}_q^{\rm I} {\rightarrow}0$$ From Proposition \[inj-surj2\], this monad is again exact at the first and last terms. Let us denote its cohomology by $E_{\rm I}^\dagger$, which also has the structure of a right ${\mathfrak{M}}_q^{\rm I}$-module. Moreover, via the procedure in the proof of Proposition \[proj2\], $E_{\rm I}^\dagger$ can be identified with the kernel of the map: $$\left( \begin{array}{c}
\alpha_{\rm I}^\dagger \\ \beta_{\rm I}
\end{array} \right) : \tilde{W}\otimes{\mathfrak{M}}_q^{\rm I}
{\rightarrow}(V\oplus V)\otimes{\mathfrak{M}}_q^{\rm I}$$ which is clearly isomorphic to ${\rm ker}{\cal D}_{\rm I}\simeq E_{\rm I}$. Therefore, the involution $\dagger: \tilde{W}\otimes{\mathfrak{M}}_q^{\rm I}\to\tilde{W}\otimes{\mathfrak{M}}_q^{\rm I}$ induces a map $\dagger:E_{\rm I}{\rightarrow}E_{\rm I}$; the desired property follows easily from (\[modinv\]). To check the compatibility of $\dagger$ with the connection $\nabla_{\rm I}$, note that: $$\nabla_{\rm I}(e^\dagger) = P_{\rm I}d(e^\dagger) =
P_{\rm I}(de)^\dagger = (P_{\rm I}de)^\dagger = (\nabla_{\rm I}e)^\dagger$$
Summing up the work done so far, we have constructed a well-defined map from the set of equivalence classes of [*classical*]{} ADHM data to the set of gauge equivalence classes of [*quantum*]{} instantons on quantum Minkowski space-time ${\mathfrak{M}}_q^{\rm I}$, in the sense of Definition \[nc.instanton\].
#### Connection matrix
Finally, let us describe the connection matrix associated with the connection $\nabla_{\rm I}$ given above. To do that, let $\{\sigma_k,\rho^k\}_{k=1}^{r}$ be a dual basis for $E_{\rm I}$. Let also $\{w_k\}_{k=1}^{r}$ be an orthonormal basis for $W$. These choices induce a natural map: $$\begin{aligned}
\Psi : W\otimes{\mathfrak{M}}^{\rm I}_q & {\rightarrow}&
\tilde{W}\otimes{\mathfrak{M}}^{\rm I}_q \\
\Psi(w_k\otimes f) & = & \iota_{\rm I}(\sigma_k)f\end{aligned}$$ extended by linearity. Then $\Psi$ has the property ${\cal D}_{\rm I}\circ\Psi=0$: $${\cal D}_{\rm I}\circ\Psi (\sum_k w_k\otimes f_k) =
{\cal D}_{\rm I} (\sum_k f_k \iota(\sigma_k)) =
\sum f_k {\cal D}_{\rm I}(\iota(\sigma_k)) = 0$$ The map $\Psi$ is clearly injective, so that $\Psi^\dagger\Psi:W\otimes{\mathfrak{M}}^{\rm I}_q{\rightarrow}W\otimes{\mathfrak{M}}^{\rm I}_q$ is an isomorphism. Moreover, the basis $\{w_k\}_{k=1}^{r}$ can be chosen such that $\Psi^\dagger\Psi={\mathbf{1}}$.
On the other hand, consider the map: $$\begin{aligned}
\rho : E_{\rm I} & {\rightarrow}&
W\otimes{\mathfrak{M}}^{\rm I}_q \\
\rho(e) & = & \sum_k w_k\otimes \rho^k(e)\end{aligned}$$ Then $\rho P_{\rm I} \Psi = {\mathbf{1}}$, thus $\rho P_{\rm I} = \Psi^\dagger$.
Recalling that the entries of the connection matrix are given by $A_k^l = \rho^l\otimes{\mathbf{1}}(\nabla_{\rm I} \sigma_k)$, we have: $$A_k^l = \rho^l \left( P _{\rm I} d\Psi (w_k\otimes 1) \right)$$ But the right hand side are just the matrix coefficients of $\Psi^\dagger d\Psi$, so that $A=\Psi^\dagger d\Psi$, as in the classical ADHM construction.
#### Quantum instantons on $S_q^{\rm J}$
Consider the monad $$\label{nc.monad3}
0 {\rightarrow}V\otimes{\mathfrak{M}}_q^{\rm J} \stackrel{\alpha_{\rm J}}{{\rightarrow}}
\tilde{W}\otimes{\mathfrak{M}}_q^{\rm J} \stackrel{\beta_{\rm J}}{{\rightarrow}}
V\otimes{\mathfrak{M}}_q^{\rm J} {\rightarrow}0$$ with the maps: $$\alpha_{\rm J} = \left( \begin{array}{c}
B_1\otimes{\mathbf{1}}- {\mathbf{1}}\otimes y_{12'} \\
B_2\otimes{\mathbf{1}}- {\mathbf{1}}\otimes y_{22'} \\
j\otimes{\mathbf{1}}\end{array} \right)$$ and $$\beta_{\rm J} = \left( \begin{array}{ccc}
-B_2\otimes{\mathbf{1}}+ {\mathbf{1}}\otimes y_{22'} &
B_1\otimes{\mathbf{1}}- {\mathbf{1}}\otimes y_{12'} &
i\otimes{\mathbf{1}}\end{array} \right)$$ so that $\alpha_{\rm J} = ({\mathbf{1}}\otimes\eta) \alpha_{\rm I} ({\mathbf{1}}\otimes\eta^{-1})$ and $\beta_{\rm J} = ({\mathbf{1}}\otimes\eta) \beta_{\rm I} ({\mathbf{1}}\otimes\eta^{-1})$
It is again easy to check that $\beta_{\rm J}\alpha_{\rm J}=0$, that $\alpha_{\rm J}$ is injective, and that $\beta_{\rm J}$ is surjective if and only if $(B_1,B_2,i,j)$ is stable. Thus, the cohomology of (\[nc.monad3\]), denoted by $E_{\rm J}$, is a projective right ${\mathfrak{M}}_q^{\rm J}$-module. The connection $\nabla_{\rm J}$ and real structure $\dagger_{\rm J}:E_{\rm J}{\rightarrow}E_{\rm J}$ can be similarly defined, and we obtain a quantum instanton on ${\mathfrak{M}}^{\rm J}_q$.
#### Consistency
If the modules $E_{\rm I}$ and $E_{\rm J}$ are constructed as above, the consistency map (\[nc.mod.glue.map\]) arises in the following way. Consider the diagram: $$\label{nc.diag1} \xymatrix{
0 \ar[r] &
V\otimes{\mathfrak{M}}^{\rm I}_q[\delta^{-1}] \ar[r]^{\alpha_{\rm I}}
\ar[d]^{{\mathbf{1}}_V\otimes\eta} &
\tilde{W}\otimes{\mathfrak{M}}^{\rm I}_q[\delta^{-1}] \ar[r]^{\beta_{\rm I}}
\ar[d]^{{\mathbf{1}}_{\tilde{W}}\otimes\eta} &
V\otimes{\mathfrak{M}}^{\rm I}_q[\delta^{-1}] \ar[r] \ar[d]^{{\mathbf{1}}_V\otimes\eta} & 0 \\
0 \ar[r] &
V\otimes{\mathfrak{M}}^{\rm J}_q[\delta] \ar[r]^{\alpha_{\rm J}} &
\tilde{W}\otimes{\mathfrak{M}}^{\rm J}_q[\delta] \ar[r]^{\beta_{\rm J}}&
V\otimes{\mathfrak{M}}^{\rm J}_q[\delta] \ar[r] & 0 }$$ Clearly, the cohomology of the first row is $E_{\rm I}[\delta^{-1}]$, while the cohomology of the second row is $E_{\rm J}[\delta]$. Moreover, the diagram is commutative. Therefore, the isomorphism ${\mathbf{1}}_{\tilde{W}}\otimes\eta$ induces an isomorphism $\Gamma:E_{\rm I}[\delta^{-1}]{\rightarrow}E_{\rm J}[\delta]$, as required in Definition \[consistency\].
To establish the consistency between the connections $\nabla_{\rm I}$ and $\nabla_{\rm J}$, it is enough to show that the connection matrices $A_{\rm I}$ and $A_{\rm J}$ are related via a gauge transformation on the “intersection algebra” ${\mathfrak{M}}^{\rm I}_q[\delta^{-1}]\stackrel{\eta}{{\rightarrow}}{\mathfrak{M}}^{\rm J}_q[\delta]$. Indeed, fix a trivialization: $\Psi_{\rm I}$ of $E_{\rm I}$ such that $\Psi_{\rm I}^\dagger \Psi_{\rm I}={\mathbf{1}}$; consider the diagram: $$\xymatrix{
0 \ar[r] &
W\otimes{\mathfrak{M}}^{\rm I}_q[\delta^{-1}] \ar[r]^{\Psi_{\rm I}}
\ar[d]^{{\mathbf{1}}_V\otimes\eta} &
\tilde{W}\otimes{\mathfrak{M}}^{\rm I}_q[\delta^{-1}] \ar[r]^{{\cal D}_{\rm I}}
\ar[d]^{{\mathbf{1}}_{\tilde{W}}\otimes\eta} &
V\otimes{\mathfrak{M}}^{\rm I}_q[\delta^{-1}] \ar[r] \ar[d]^{{\mathbf{1}}_V\otimes\eta} & 0 \\
0 \ar[r] &
V\otimes{\mathfrak{M}}^{\rm J}_q[\delta] \ar@{-->}[r]&
\tilde{W}\otimes{\mathfrak{M}}^{\rm J}_q[\delta] \ar[r]^{{\cal D}_{\rm J}}&
V\otimes{\mathfrak{M}}^{\rm J}_q[\delta] \ar[r] & 0 }$$ The commutativity of the second square on the above follows from the commutativity of the diagram (\[nc.diag1\]). This means that $\Psi_{\rm J}=({\mathbf{1}}_{\tilde{W}}\otimes\eta)\Psi_{\rm I}({\mathbf{1}}_V\otimes\eta^{-1})$ is a trivialization of $E_{\rm J}$. To simplify notation, let us simply use $\eta$ to denote ${\mathbf{1}}\otimes\eta$. Hence: $$\begin{aligned}
A_{\rm J} & = &
(\eta\Psi_{\rm I}\eta^{-1})^\dagger d_{\rm J} (\eta\Psi_{\rm I}\eta^{-1}) =
\eta\Psi_{\rm I}^\dagger\eta^{-1} \eta d_{\rm I} (\Psi_{\rm I}\eta^{-1}) = \\
& = & \eta (\Psi_{\rm I}^\dagger d_{\rm I} \Psi_{\rm I})\eta^{-1} +
\eta \Psi_{\rm I}^\dagger \Psi_{\rm I} d_{\rm I} (\eta^{-1}) =
\eta A_{\rm I} \eta^{-1} + \eta d_{\rm I} \eta^{-1}\end{aligned}$$ where $d_{\rm I}$ and $d_{\rm J}$ denote the de Rham operators on ${\mathfrak{M}}_{\rm I}$ and ${\mathfrak{M}}_{\rm J}$, respectively.
In other words, $\nabla_{\rm J}=\Gamma\nabla_{\rm I}\Gamma^{-1}$. We also used the fact that $(\eta^{-1})^\dagger=\eta$ and that $d_{\rm J} \eta = \eta d_{\rm I}$.
Finally, recall that $\dagger\eta=\eta\dagger$. Since $\Gamma$ is induced from $\eta$ and $\dagger$ is induced from $\dagger$, we conclude that indeed: $\dagger_{\rm J}\Gamma = \Gamma\dagger_{\rm I}$.
We sum up the work done in this Section in the following statement, which motivated the title of this paper:
\[oneway\] There exists a well-defined map from the set of equivalence classes of regular ADHM data to the moduli space of gauge equivalence classes of consistent pairs of quantum instantons.
Quantum Penrose transform and further perspectives {#conc}
==================================================
Completeness conjecture and the quantum Penrose transform
---------------------------------------------------------
As we mentioned in Introduction, we conjecture that all anti-self-dual connections on ${\mathfrak{M}}^{\rm I}_q$ are gauge equivalent to the ones produced above. In other words, the map given by Theorem \[oneway\] is invertible: given a consistent pair of quantum instantons, there is an ADHM datum $(B_1,B_2,i,j)$ such that the consistent pair can be reconstructed from $(B_1,B_2,i,j)$ via the procedure above.
As a consequence of this conjecture, we are able to conclude that the moduli space of [*quantum*]{} instantons actually coincides with the moduli space of [*classical*]{} instantons, therefore fully justifying the title of this paper.
The key ingredient in the proof of the classical version of this conjecture is Penrose’s twistor diagram (also termed the [*flat self-duality diagram*]{} by Manin [@Ma]): $$\xymatrix{
& \mathbf{F}_{1,2}({\mathbb{T}}) \ar[dl]^\mu \ar[dr]_\nu & \\
\mathbf{P}({\mathbb{T}}) & & \mathbf{G}_2({\mathbb{T}})={\mathbb{M}}}$$ where $\mathbf{F}_{1,2}({\mathbb{T}})$ denotes the flag manifold of lines within planes in ${\mathbb{T}}$, a 4-dimensional complex vector space.
Recall also that the four sphere $S^4$ is naturally embedded into ${\mathbb{M}}$. It can be realized as the fixed point set of the following real structure $\sigma$ on ${\mathbb{M}}$: $$\label{m.invol}
\begin{array}{ccc}
\sigma(z_{11'}) = \overline{z_{22'}} & \ \ \ & \sigma(z_{12'}) = -\overline{z_{21'}} \\
\sigma(z_{21'}) = -\overline{z_{12'}}& \ \ \ & \sigma(z_{22'}) = \overline{z_{11'}}\\
\sigma(D) = \overline{D} & \ \ \ & \sigma(D') = \overline{D'}
\end{array}$$ Moreover, we observe that even though the affine pieces ${\mathbb{M}}^{\rm I}$ and ${\mathbb{M}}^{\rm J}$ do not cover ${\mathbb{M}}$, all of its real points lie within their union, that is $S^4\hookrightarrow{\mathbb{M}}^{\rm I}\cup{\mathbb{M}}^{\rm J}$.
We claim that there are $q$-deformations of $\mathbf{G}_2({\mathbb{T}})$ and $\mathbf{F}_{1,2}({\mathbb{T}})$ which provide a correspondence between a $q$-deformed Grassmannian and the [*classical*]{} twistor space $\mathbf{P}({\mathbb{T}})$. Furthermore, our quantum Minkowski space-time ${\mathfrak{M}}^{\rm I}_q$ is an [*affine patch*]{} of the $q$-deformed Grassmannian. Moreover, all relevant noncommutative varieties can also be obtained from the quantum group $GL(4)_q$ extended by appropriate derivations. However, in the construction below, we will only use the quantum group $SL(2)_q$ enlarged by its corepresentations of functional dimension 2 and by certain degree operators.
Once these noncommutative varieties are constructed, we hope that our conjecture will be proved in complete parallel with the classical version.
#### Quantum Grassmannian
Let us now describe the noncommutative variety from which the quantum Minkowski space-time ${\mathfrak{M}}^{\rm I}_q$ is obtained via localization. Recall that $q$ is a formal parameter.
\[mpq\] The quantum compactified, complexified Minkowski space ${\mathfrak{M}}_{p,q}$ is the associative graded ${\mathbb{C}}$-algebra generated by $z_{11'},z_{12'},z_{21'},z_{22'},D,D'$ satisfying the relations (\[relations1\]) to (\[relations5\]) below $(p=q^{\pm 1})$:
$$\label{relations1}
\begin{array}{lcr}
z_{11'}z_{12'}=z_{12'}z_{11'} & \ \ \ & z_{11'}z_{21'}=z_{21'}z_{11'} \\
z_{12'}z_{22'}=z_{22'}z_{12'} & \ \ \ & z_{21'}z_{22'}=z_{22'}z_{21'} \\
& z_{12'}z_{21'}=z_{21'}z_{12'} &
\end{array}$$
$$\label{relations2}
q^{-1} (z_{11'}z_{22'}-z_{12'}z_{21'})=q (z_{22'}z_{11'}-z_{12'}z_{21'})$$
$$\label{relations3}
\begin{array}{lcr}
Dz_{11'}=pq^{-1} z_{11'}D & \ \ \ & D'z_{11'}=p^{-1}q^{-1} z_{11'}D' \\
Dz_{12'}=pq^{-1} z_{12'}D & \ \ \ & D'z_{12'}=p^{-1}q z_{12'}D' \\
Dz_{21'}=pq z_{21'}D & \ \ \ & D'z_{21'}=p^{-1}q^{-1} z_{21'}D' \\
Dz_{22'}=pq z_{22'}D & \ \ \ & D'z_{22'}=p^{-1}q z_{22'}D' \\
\end{array}$$
$$\label{relations4}
p^{-1} DD'=p D'D$$
$$\label{relations5}
q^{-1} (z_{11'}z_{22'}-z_{12'}z_{21'}) = p^{-1} DD'$$
The relations (\[relations1\])-(\[relations4\]) are simply commutation relations, while (\[relations5\]) plays the role of the quadric (\[quad\]) that defines ${\mathbb{M}}$ as a subvariety of ${\mathbb{P}}^5$. In other words, the algebra ${\mathfrak{M}}_{p,q}$ can be regarded as a [*quantum Grassmannian*]{}. Note also that the relations (\[relations1\])-(\[relations5\]) can be expressed in $R$-matrix form.
Furthermore, in analogy with the classical case, it is not difficult to establish the following:
\[local\] The algebras ${\mathfrak{M}}^{\rm I}_q$ and ${\mathfrak{M}}^{\rm J}_q$ are localizations of ${\mathfrak{M}}_{p,q}$ with respect to $D$ and $D'$, respectively, independently on $p=q^{\pm1}$. In particular, $$x_{rs'} = \frac{z_{rs'}}{D} \ \ \ {\rm and} \ \ \ y_{rs'} =
\frac{z_{rs'}}{D'}$$
The proof is a straightforward calculation left to the reader. Note only that the notation $\frac{z_{rs'}}{D}$ means $c_{rs'}^{-1/2}D^{-1}z_{rs'} = c_{rs'}^{1/2}z_{rs'}D^{-1}$ whenever $D^{-1}z_{rs'}=c_{rs'}z_{ij'}D^{-1}$ for some constant $c_{rs'}$, and similarly for $\frac{z_{rs'}}{D'}$. $\Box$
It is also important to note that it follows from the comparison of equations (\[relations3\]) with equations (\[xvar\]) and (\[yvar\]) that $\delta^2=\frac{D'}{D}$.
In geometric terms, our quantum Minkowski space-time ${\mathfrak{M}}^{\rm I}_q$ appears as an affine patch of the quantum Grassmannian ${\mathfrak{M}}_{p,q}$, thus justifying our Definition.
#### The quantum 4-sphere
Let us define the action of the $\dagger$-involution on the generators of ${\mathfrak{M}}_{p,q}$ as follows: $$\begin{array}{lcl}
z_{11'}^\dagger = z_{22'} & \ \ \ & z_{12'}^\dagger = - z_{21'} \\
z_{21'}^\dagger = - z_{12'} & \ \ \ & z_{22'}^\dagger = z_{11'} \\
D^\dagger = D & \ \ \ & D'^\dagger = D'
\\
q^\dagger = q & \ \ \ & p^\dagger = p
\end{array}$$ We then extend $\dagger$ to ${\mathfrak{M}}_{p,q}$ by requiring it to be a conjugate linear anti-homomor-phism. Note that $\dagger$ interchanges the quantum Grassmannians, i.e. $\dagger:{\mathfrak{M}}_{p,q}\to{\mathfrak{M}}_{p^{-1},q}$. One easily checks that it is consistent with the $\dagger$-involutions previously defined on ${\mathfrak{M}}^{\rm I}_q$ and ${\mathfrak{M}}^{\rm J}_q$. Also, comparing with (\[m.invol\]), we see that the map $\dagger$ just defined is the analogue of the real structure $\sigma$ on ${\mathbb{M}}$. Therefore, we propose the following definition.
\[spq\] The quantum 4-dimensional sphere $S_{q}$ is the pair of algebras ${\mathfrak{M}}_{q^{\pm1},q}$ equipped with the map $\dagger:{\mathfrak{M}}_{p,q}\to{\mathfrak{M}}_{p^{-1},q}$: $$S_q = ({\mathfrak{M}}_{q^{\pm1},q},\dagger)$$
Alternative definitions of quantum 4-dimensional spheres have been proposed by several other authors, see in particular [@BCT; @CL; @DLM]. Note however that ${\mathfrak{M}}_{p,q}$ is a deformation of the algebra of homogeneous coordinates, not of the algebra of polynomial functions as in the examples discussed in the references mentioned above. Our definition is justified by the analogy with the commutative case and by the construction of quantum instantons on the two “affine patches” $S^{\rm I}_q$ and $S^{\rm J}_q$ of $S_q$ done in Section 3.
#### Quantum flag variety
Let ${\mathbb{P}}$ denote the projective twistor space $\mathbf{P}({\mathbb{T}})$; recall that the flag variety ${\mathbb{F}}=\mathbf{F}_{1,2}({\mathbb{T}})$ can be naturally embedded in the product ${\mathbb{P}}\times{\mathbb{M}}$ by sending the flag \[line$\subset$plane\]$\in{\mathbb{F}}$ to the pair (line,plane)$\in{\mathbb{P}}\times{\mathbb{M}}$. More precisely, let $[z_k]$ ($k=1,2,1',2'$, consistently with the splitting (\[decomp\])) be homogeneous coordinates in ${\mathbb{P}}$; then ${\mathbb{F}}$ can be described as an intersection of the following quadrics in ${\mathbb{P}}\times{\mathbb{M}}$: $$\begin{aligned}
Dz_{1'} + z_1z_{21'} - z_2z_{11'} = 0 & \ \ \ &
Dz_{2'} + z_1z_{22'} - z_2z_{12'} = 0\label{q1} \\
D'z_1 + z_{2'}z_{11'} - z_{1'}z_{12'} = 0 & \ \ \ &
D'z_2 + z_{2'}z_{21'} - z_{1'}z_{22'} = 0 \label{q2}\end{aligned}$$
Let us now introduce the quantum flag variety that provides a correspondence between the quantum Grassmannian described above and the classical twistor space. Notice that the coordinate algebra of the projective twistor space ${\mathbb{P}}$ is simply given by: $${\mathfrak{P}}= {\mathbb{C}}[z_1,z_2,z_{1'},z_{2'}]_h$$ where the subscript “$h$" means that ${\mathfrak{P}}$ consists only of the homogeneous polynomials.
Again, the starting point lies in a natural extension of the quantum group $SU(2)_q$, this time by left and right corepresentations of functional dimension 2. We define left and right corepresentations of $GL(2)_q$ as the vector spaces $L_q$ and $R_q$ of polynomials on two noncommutative variables $g_1,g_2$ and $g_{1'},g_{2'}$, respectively, satisfying the commutation relations: $$\label{left}
g_1g_2 = q^{-1}g_2g_1$$ $$\label{right}
g_{1'}g_{2'} = q^{-1}g_{2'}g_{1'}$$ The coaction is given by, respectively: $$\label{lcoation}
\Delta_L(g_r) = \sum_{k=1,2} g_{rk'}\otimes g_k$$ $$\label{rcoation}
\Delta_R(g_{s'}) = \sum_{k=1,2} g_{k'}\otimes g_{ks'}$$ The commutation relations (\[gl2q1\]) and (\[gl2q2\]) for the quantum group $GL(2)_q$ imply that $\Delta_L$ and $\Delta_R$ are indeed corepresentations (in fact, they are also necessary conditions).
We can also regard $L_q$ and $R_q$ as corepresentations of the quantum groups $SL(2)_q$, $\widetilde{GL}(2)_q$ and $\widetilde{SL}(2)_q$. Again, we will be primarily interested in $\widetilde{SL}(2)_q$.
We define the $\dagger$-involution on the generators of $L_q$ and $R_q$ as follows: $$\label{co.star}
\begin{array} {ccc}
g_1^\dagger = g_2, & & g_2^\dagger = -g_1 \\
g_{1'}^\dagger = g_{2'}, & & g_{2'}^\dagger = -g_{1'}
\end{array}$$ As before, we extend $\dagger$ to a conjugate linear anti-homomorphisms of $L_q$ and $R_q$; see page . We also get: $$\label{corep.star}
\Delta_L(x^\dagger) = (\Delta_L(x))^\dagger\otimes(\Delta_L(x))^\dagger,
\ \ \ {\rm and}\ \ \
\Delta_R(y^\dagger) = (\Delta_R(y))^\dagger\otimes(\Delta_R(y))^\dagger$$ for $x\in L_q$ and $y\in R_q$, which means that $\Delta_L$ and $\Delta_R$ are well defined as corepresentations of the quantum groups $SU(2)_q$ and $\widetilde{SU}(2)_q$.
Next, we define the semi-direct product quantum groups $(SL(2)_q\ltimes L_q)_{p}$ and $(R_q\rtimes SL(2)_q)_{p}$ as the algebras with the combined generators of $SL(2)_q$ and $L_q$ and $SL(2)_q$ and $R_q$, respectively, satisfying the relations (in $R$-matrix form): $$\begin{array}{ccc}
p^{1/2} R T_1 S_2 = S_2 T_1 &
\ \ \ {\rm and}\ \ \ &
T_1 S_2^\prime = S_2^\prime T_1 R p^{1/2}
\end{array}$$ where $R$ is the $R$-matrix given in (\[rmatrix\]) and $$\begin{array}{ccc}
S = \left( \begin{array}{c} g_1 \\ g_2 \end{array} \right) &
\ \ \ {\rm and}\ \ \ &
S' = \left( \begin{array}{c} g_{1'} \\ g_{2'} \end{array} \right)
\end{array}$$ are the generating matrices for $L_q$ and $R_q$, respectively, with $S_2={\mathbf{1}}\otimes S$, $S_2'={\mathbf{1}}\otimes S'$. We define a comultiplication on $(SL(2)_q\ltimes L_q)_{p}$ and $(R_q\rtimes SL(2)_q)_{p}$ by the formulas (\[comult\]), (\[lcoation\]) and (\[rcoation\]). Remark that:
\[cons\] The multiplication and comultiplication in $(SL(2)_q\ltimes L_q)_{p}$ and $(R_q\rtimes SL(2)_q)_{p}$ are consistent, and they yield a bialgebra structure.
The bialgebras $(SL(2)_q\ltimes L_q)_{p}$ and $(R_q\rtimes SL(2)_q)_{p}$ are quantum versions of the classical groups $SL(2)\ltimes L$ and $R\rtimes SL(2)$, respectively, where $L$ and $R$ denote left and right two-dimensional representations of $SL(2)$. It is well known that these two groups are isomorphic, where the isomorphism is given by the identification $L\simeq R$. It turns out that this isomorphism has a quantum analogue, and the bialgebras $(SL(2)_q\ltimes L_q)_{p}$ and $(R_q\rtimes SL(2)_q)_{p}$ are also isomorphic. The quantum isomorphism $\kappa:(SL(2)_q\ltimes L_q)_{p}{\rightarrow}(R_q\rtimes SL(2)_q)_{p}$ is the identity on $SL(2)_q$ and acts in the generators of $L_q$ as follows: $$\begin{aligned}
\kappa(g_1) & = &
p^{-3/4} \left( -q^{1/2}g_{11'}g_{2'} + q^{-1/2}g_{12'}g_{1'} \right) \\
\kappa(g_2) & = &
p^{-3/4} \left( -q^{1/2}g_{21'}g_{2'} + q^{-1/2}g_{22'}g_{1'} \right)\end{aligned}$$ Therefore we define $F_{p,q}$ as $R_q\rtimes SL(2)_q\ltimes L_q$ modulo the equivalence given by the isomorphism $\kappa$. One can compute the following commutation relations: $$\label{g.commut} \begin{array}{ccc}
g_1g_{1'} = p^{1/2} g_{1'}g_1 & \ \ \ & g_2g_{1'} = p^{1/2} g_{1'}g_2 \\
g_1g_{2'} = p^{1/2} g_{2'}g_1 & \ \ \ & g_2g_{2'} = p^{1/2} g_{2'}g_2
\end{array}$$
We proceed by adjoining two new generators $\partial$ and $\partial'$ to $F_{p,q}$, and postulating the following relations (keeping in mind that $p=q^{\pm1}$): $$\label{dd'grs} \begin{array}{lcl}
\partial g_{11'} = q^{-1/2} g_{11'} \partial & \ \ \ &
\partial' g_{11'} = q^{-1/2} g_{11'} \partial' \\
\partial g_{12'} = q^{-1/2} g_{12'} \partial & \ \ \ &
\partial' g_{12'} = q^{1/2} g_{12'} \partial' \\
\partial g_{21'} = q^{1/2} g_{21'} \partial & \ \ \ &
\partial' g_{21'} = q^{-1/2} g_{21'} \partial' \\
\partial g_{22'} = q^{1/2} g_{22'} \partial & \ \ \ &
\partial' g_{22'} = q^{1/2} g_{22'} \partial' \\
\end{array}$$ $$\label{dd'gk} \begin{array}{lcl}
\partial g_1 = p^{1/2}q^{-1/2} g_1 \partial & \ \ \ &
\partial' g_1 = p^{1/2} g_1 \partial' \\
\partial g_2 = p^{-1/2}q^{1/2} g_2 \partial & \ \ \ &
\partial' g_2 = p^{1/2} g_2 \partial' \\
\partial g_{1'} = p^{-1/2} g_{1'} \partial & \ \ \ &
\partial' g_{1'} = p^{1/2}q^{-1/2} g_{1'} \partial' \\
\partial g_{2'} = p^{-1/2} g_{2'} \partial & \ \ \ &
\partial' g_{2'} = p^{1/2}q^{1/2} g_{2'} \partial' \\
\end{array}$$ $$\label{dd'}
\partial \partial' = p^{1/2} \partial' \partial$$ We denote the extended algebra $F_{p,q}[\partial,\partial']$ by $\widetilde{F}_{p,q}$; this is the prototype of our quantum flag variety.
Furthermore, we also define the $\dagger$-involution on $\widetilde{F}_{p,q}$ by extending the $\dagger$-involutions on $\widetilde{SL}(2)_q$, $L_q$ and $R_q$ defined on (\[involution\]) and (\[co.star\]) and declaring: $$\label{dd'star}
\partial^\dagger =\partial
\ \ \ {\rm and} \ \ \
\partial'^\dagger = \partial'$$
As in the construction of the quantum Minkowski space-time, we introduce a new set of variables $z_{rs'},z_k,z_{k'}$ in the bialgebra $\widetilde{F}_{p,q}=F_{p,q}[\partial,\partial']$, related to the original variables $g_{rs'},g_k,g_{k'}$ via: $$\frac{z_{rs'}}{\Delta} = g_{rs'} \ \ \ \ \
\frac{z_k}{\partial} = g_k \ \ \ \ \
\frac{z_{k'}}{\partial'} = g_{k'}$$ where $\Delta=p^{1/4} \partial' \partial=p^{-1/4} \partial \partial'$. The fractions above have the same meaning as in Proposition \[local\]. We also set: $$D = \partial^2 \ \ \ \ \ D' = \partial'^2$$
Using relations (\[dd’grs\]) and (\[dd’\]), one re-obtains the relations (\[relations1\]-\[relations5\]), which define ${\mathfrak{M}}_{p,q}$. In other words, the quantum compactified, complexified Minkowski space ${\mathfrak{M}}_{p,q}$ can be regarded as a subalgebra of the extended quantum group $SL(2)_q[\partial,\partial']$. Notice in particular that the quantum quadric (\[relations5\]) is just the relation $\det_q(T)=1$ expressed in the new variables $z_{rs'}$.
Now using relations (\[dd’gk\]) and (\[dd’\]), it is easy to check that: $$\label{relations6.5}
z_1,z_2,z_{1'}z_{2'}\ \text{commute with one another}$$ Furthermore, we also obtain: $$\label{relations7}
\begin{array}{lcr}
z_1z_{11'}=z_{11'}z_1 & \ \ \ & z_1z_{12'}=z_{12'}z_1 \\
z_2z_{21'}=z_{21'}z_2 & \ \ \ & z_2z_{22'}=z_{22'}z_2 \\
z_{1'}z_{11'}=z_{11'}z_{1'} & \ \ \ & z_{1'}z_{21'}=z_{21'}z_{1'} \\
z_{2'}z_{12'}=z_{12'}z_{2'} & \ \ \ & z_{2'}z_{22'}=z_{22'}z_{2'}
\end{array}$$ and, keeping in mind that $p=q^{\pm1}$: $$\label{relations8}
\begin{array}{c}
z_1z_{21'} = pq z_{21'}z_1 + (1-pq) z_{11'}z_2 \\
z_1z_{22'} = pq z_{22'}z_1 + (1-pq) z_{12'}z_2 \\
z_2z_{11'} = pq^{-1} z_{11'}z_{2} + (1-pq^{-1}) z_{21'}z_1 \\
z_2z_{12'} = pq^{-1} z_{22'}z_{2} + (1-pq^{-1}) z_{22'}z_1 \\
z_{1'}z_{12'} = p^{-1}q z_{12'}z_{1'} + (1-p^{-1}q) z_{11'}z_{2'} \\
z_{1'}z_{12'} = p^{-1}q z_{22'}z_{1'}+ (1-p^{-1}q) z_{21'}z_{2'} \\
z_{2'}z_{11'} = p^{-1}q^{-1} z_{11'}z_{2'} + (1-p^{-1}q^{-1}) z_{12'}z_{1'} \\
z_{2'}z_{21'} = p^{-1}q^{-1} z_{21'}z_{2'}+ (1-p^{-1}q^{-1}) z_{22'}z_{1'}
\end{array}$$ The commutation relations between the degree operators $D$ and $D'$ and the commuting generators $z_1,z_2,z_{1'},z_{2'}$ are given by: $$\label{relations9}
\begin{array}{lcl}
Dz_1= p^{-1}q^{-1} z_1D & \ \ \ & D'z_1=z_1D' \\
Dz_2= p^{-1}q z_2D & \ \ \ & D'z_2=z_2D' \\
Dz_{1'}=z_{1'}D & \ \ \ & D'z_{1'}=pq^{-1} z_{1'}D'\\
Dz_{2'}=z_{2'}D & \ \ \ & D'z_{2'}=pq z_{2'}D'
\end{array}$$ The last set of relations below follows from the identifications induced by the isomorphism $\kappa$ and its inverse, and yields the quantum analogues of the quadrics (\[q1\]) and (\[q2\]) defining ${\mathbb{F}}$ as a subvariety of ${\mathbb{P}}\times{\mathbb{M}}$: $$\label{relations10}
\begin{array}{c}
Dz_{1'} = p ( z_{11'}z_2 - z_{21'}z_{1} ) \\
Dz_{2'} = p ( z_{12'}z_2 - z_{22'}z_{1} ) \\
D'z_1 = p^{-1} (-z_{11'}z_{2'} + z_{12'}z_{1'}) \\
D'z_2 = p^{-1} (-z_{21'}z_{2'} + z_{22'}z_{1'})
\end{array}$$ This observation motivates our next definition:
\[qflag\] The quantum flag variety ${\mathfrak{F}}_{p,q}$ is the associative graded ${\mathbb{C}}$-algebra with generators $z_{11'},z_{12'},z_{21'},z_{22'},D,D',z_1,z_2,z_{1'}z_{2'}$ satisfying relations (\[relations1\])-(\[relations5\]) and (\[relations7\])-(\[relations10\]) above.
In particular, note that ${\mathfrak{F}}_{p,q}$ can be regarded as a subalgebra of $\widetilde{F}_{p,q}$. We also remark that: $$\label{qflag2}
{\mathfrak{F}}_{p,q}={\mathfrak{M}}_{p,q}\otimes_{{\mathbb{C}}}{\mathfrak{P}}/(\ref{relations7})-(\ref{relations10})$$ in close analogy with the classical case. All the relations (\[relations7\]-\[relations10\]) can also be expressed in $R$-matrix form.
Now let $${\mathfrak{F}}_q^{\rm I} = {\mathfrak{F}}_{p,q}[D^{-1}]_0 \simeq
{\mathfrak{M}}_q^{\rm I}\otimes{\mathbb{C}}[z_1,z_2]_h$$ $${\mathfrak{F}}_q^{\rm J}={\mathfrak{F}}_{p,q}[D'^{-1}]_0 \simeq
{\mathfrak{M}}_q^{\rm J}\otimes{\mathbb{C}}[z_1,z_2]_h$$ where ${\mathfrak{F}}_{p,q}[D^{-1}]$ and ${\mathfrak{F}}_{p,q}[D'^{-1}]$ denote the localization of ${\mathfrak{F}}_{p,q}$ as a ${\mathfrak{M}}_{p,q}$-bimodule and the subscript “$0$" means that we take only the degree zero part of the localized graded algebra, and the subscript “$h$" means that only the homogeneous polynomials are considered. Geometrically, notice that these algebras are playing the roles of $q$-deformations of the “affine” flag varieties ${\mathbb{F}}^{\rm I}=\nu^{-1}({\mathbb{M}}^{\rm I})={\mathbb{M}}^{\rm I}\times{\mathbb{P}}^1$ and ${\mathbb{F}}^{\rm J}=\nu^{-1}({\mathbb{M}}^{\rm J})={\mathbb{M}}^{\rm J}\times{\mathbb{P}}^1$ [@WW]. To further justify Definition \[qflag\] we prove:
\[nc.diagram\] The maps ${\mathsf{m}}_{p,q}:{\mathfrak{P}}{\rightarrow}{\mathfrak{F}}_{p,q}$ and ${\mathsf{n}}_{p,q}:{\mathfrak{M}}_{p,q}{\rightarrow}{\mathfrak{F}}_{p,q}$ defined as identities on the generators are injective homomorphisms. Furthermore, the multiplication in the associative algebras ${\mathfrak{M}}_{p,q}$, ${\mathfrak{M}}_{p,q}^{\rm I}$, ${\mathfrak{M}}_{p,q}^{\rm J}$ and ${\mathfrak{F}}_{p,q}$, ${\mathfrak{F}}_{p,q}^{\rm I}$, ${\mathfrak{F}}_{p,q}^{\rm J}$ described above is consistent, and they have the same Hilbert polynomials as their commutative counterparts ${\mathbb{M}}$, ${\mathbb{M}}^{\rm I}$, ${\mathbb{M}}^{\rm J}$ and ${\mathbb{F}}$, ${\mathbb{F}}^{\rm I}$, ${\mathbb{F}}^{\rm J}$, respectively.
The statement regarding the maps ${\mathsf{m}}_{p,q}$ and ${\mathsf{n}}_{p,q}$ is clear from our construction, see (\[qflag2\]).
For the second part of the Theorem, we must first check the consistency of multiplication in the algebra ${\mathfrak{M}}_{p,q}$ by embedding it into its localization ${\mathfrak{M}}_{p,q}[D^{-1}]$. The commutation relations (\[relations1\]), (\[relations2\]) and the first column of (\[relations3\]) allow us to choose a basis in ${\mathbb{C}}[z_{rs'},D^{\pm1}]$ of the form: $$\label{basis}
z_{11'}^{n_{11'}}z_{12'}^{n_{12'}}z_{21'}^{n_{21'}}z_{22'}^{n_{22'}}D^n,
\ \ \ {\rm with} \ \ \ n_{rs'}\in\mathbb{Z}_+,\ \ n\in\mathbb{Z}$$ Inverting $D$ in (\[relations5\]) we obtain an expression for $D'$, and we can check directly the second column of (\[relations3\]) and relation (\[relations4\]). Thus the basis (\[basis\]) is in fact a basis of ${\mathfrak{M}}_{p,q}[D^{-1}]$, and its Hilbert polynomial coincides with the classical one.
Next we find the Hilbert polynomial of the algebra ${\mathfrak{M}}_{p,q}$. Using (\[relations5\]), we can present an arbitrary element from ${\mathfrak{M}}_{p,q}$ uniquely in the following form: $$\label{poly}
P_0(z_{rs'}) + P_1(z_{rs'},D)D + P_2(z_{rs'},D')D'$$ where $P_0,P_1,P_2$ are polynomials and $z_{rs'}$ are ordered as in (\[basis\]). This presentation coincides with the classical one, and allows to compute explicitly the Hilbert polynomial of ${\mathfrak{M}}_{p,q}$.
Similarly, we check the consistency of multiplication in the algebra ${\mathfrak{F}}_{p,q}$ by embedding it into ${\mathfrak{F}}_{p,q}[D^{-1}]$. In addition to the relations used above, we use the the first half of the commutation relations (\[relations7\]), (\[relations8\]) and the first column of (\[relations9\]) to choose a basis in ${\mathbb{C}}[z_{rs'},z_{k},D^{\pm1}]$ of the form: $$\label{basis2}
z_{11'}^{n_{11'}}z_{12'}^{n_{12'}}z_{21'}^{n_{21'}}z_{22'}^{n_{22'}}
z_1^{n_1}z_2^{n_2}D^n,
\ \ \ {\rm with} \ \ \ n_{rs'},n_k\in\mathbb{Z}_+,\ \ n\in\mathbb{Z}$$ Inverting $D$ in (\[relations5\]) and also in the first half of (\[relations10\]), we obtain expressions for $D'$, $z_{1'}$ and $z_{2'}$ and check directly all the other relations involving these three generators, namely the second half of the commutation relations (\[relations7\]), (\[relations8\]) and the second column of (\[relations9\]). We conclude that the multiplication in ${\mathfrak{F}}_{p,q}[D^{-1}]$ is consistent and its basis is given by (\[basis2\]).
Finally, we obtain the Hilbert polynomial of ${\mathfrak{F}}_{p,q}$ by noticing that using (\[relations5\]) and (\[relations10\]) we can present an arbitrary element of ${\mathfrak{F}}_{p,q}$ uniquely in the form $$\label{poly2}
P_0(z_{rs'},z_k,z_{k'}) + P_1(z_{rs'},z_k,D)D + P_2(z_{rs'},z_{k'},D')D'$$ where $P_0,P_1,P_2$ are polynomials and $z_{rs'},z_k,z_{k'}$ are ordered as in (\[basis2\]). Again this presentation coincides with the classical one, and yields an explicit formula for the Hilbert polynomial of ${\mathfrak{F}}_{p,q}$.
Alternatively, one could also check the consistency of multiplication more efficiently using the $R$-matrix formulation. $\Box$
Summing up, we have constructed noncommutative varieties ${\mathfrak{M}}_{p,q}$ and ${\mathfrak{F}}_{p,q}$, thought as quantum deformations of ${\mathbb{M}}$ and ${\mathbb{F}}$, and injective morphisms ${\mathsf{m}}_{p,q}$ and ${\mathsf{n}}_{p,q}$ fitting into the following diagram: $$\label{nc-dblfib}
\xymatrix{ & {\mathfrak{F}}_{p,q} & \\
{\mathfrak{P}}\ar[ur]_{{\mathsf{m}}_{p,q}} & & {\mathfrak{M}}_{p,q} \ar[ul]^{{\mathsf{n}}_{p,q}} }$$ where ${\mathfrak{P}}$ is just the commutative projective 3-space. In analogy with the classical case, diagram (\[nc-dblfib\]) will be called the [*quantum twistor diagram*]{}.
Furthermore, we also have the [*quantum local twistor diagrams*]{}: $$\label{loc.nc-dblfib} \begin{array}{lr}
\xymatrix{ & {\mathfrak{F}}_q^{\rm I} & \\
{\mathfrak{P}}^{\rm I} \ar[ur] & & {\mathfrak{M}}_q^{\rm I} \ar[ul] } &
\xymatrix{ & {\mathfrak{F}}_q^{\rm J} & \\
{\mathfrak{P}}^{\rm J} \ar[ur] & & {\mathfrak{M}}_q^{\rm J} \ar[ul] }
\end{array}$$ where $${\mathfrak{P}}^{\rm I} = {\mathbb{C}}[z_1 , z_2 ,
x_{11'}z_2-x_{21'}z_1 , x_{12'}z_2-x_{22'}z_1]_h$$ $${\mathfrak{P}}^{\rm J} = {\mathbb{C}}[ y_{12'}z_{1'}-y_{11'}z_{2'} ,
y_{22'}z_{1'}-y_{21'}z_{2'} , z_{1'} , z_{2'}]_h$$ as commutative subalgebras of ${\mathfrak{F}}^{\rm I}_q$ and ${\mathfrak{F}}^{\rm J}_q$, respectively. Notice that ${\mathfrak{P}}^{\rm I}$ and ${\mathfrak{P}}^{\rm J}$ are exactly the coordinate algebras of $\mu(\nu^{-1}({\mathbb{M}}^{\rm I}))$ and $\mu(\nu^{-1}({\mathbb{M}}^{\rm J}))$, the “affine” twistor spaces [@WW].
It is also clear from our construction that the quantum twistor diagram (\[nc-dblfib\]) comes with a [*real structure*]{} defined by the $\dagger$ involutions on ${\mathfrak{F}}_{p,q}$ and ${\mathfrak{M}}_{p,q}$ defined above, acting on the generators of ${\mathfrak{P}}$ as follows: $$z_1^\dagger = z_2 \ \ \ z_2^\dagger = - z_1
\ \ \ z_{1'}^\dagger = z_{2'} \ \ \ z_{2'}^\dagger = - z_{1'}$$
After rewriting the classical Penrose transform in algebraic terms, we can use the quantum twistor diagrams (\[nc-dblfib\]) and (\[loc.nc-dblfib\]) to transform [*quantum*]{} objects (solutions of the quantum ASDYM equation) into [*classical*]{} ones (holomorphic bundles with a real structure). This is the strategy for the proof of our completeness conjecture, and a topic for a future paper.
Remarks on roots of unity and representation theory {#ru}
---------------------------------------------------
In our construction of quantum instantons, we have specialized the formal parameter $q$ to a positive real number. Thus it is natural to ask what happens if we consider other specializations. Since our construction is deeply related with the behaviour of the Haar functional on the quantum group $SL(2)_q$, we expect that our results remain valid for all specializations of $q$ for which the Haar functional has a similar structure, namely for any complex $q$ that is not a root of unity.
It is an interesting problem to obtain a modified ADHM data for any given root of unity, and then describe the corresponding moduli space of quantum instantons in terms of this data. We expect that such description will be given by certain quiver varieties.
Moreover, equivariant anti-self-dual connections on ${\mathbb{R}}^4$ has long been a topic of intense research. In particular, instantons on the quotient spaces ${\mathbb{C}}^2/\Gamma$, where $\Gamma$ is a finite subgroup of $SU(2)$, and on their desingularizations (known as ALE spaces) have been studied by Kronheimer and Nakajima in [@KN; @N4]. The moduli spaces of instantons on ALE spaces play a fundamental role on Nakajima’s construction of representations of affine Lie algebras [@N3].
To obtain a quantum analogue of this construction, we first need to describe “finite subgroups” of $SU(2)_q$. This turns out to be a deep problem, recently solved by Ostrik [@O], though only at the level of representation categories. The resulting classification, as in the classical case, is given by Dynkin diagrams of ADE type. This suggests that the corresponding moduli spaces of quantum instantons might shed a new light at Nakajima’s quiver varieties and his results on representation theory.
We would also like to notice the similarities between the deformation of the representation theory for $SU(2)$ (as well as other compact simple Lie groups) and the deformation of instanton theory presented in this paper. As it is well known, the classification of irreducible corepresentations of the quantum group $SU(2)_q$, the behaviour of the Haar functional, the decomposition of their tensor products, etc, is the same as in the classical, non-deformed case if $q$ is not a root of unity, see [@CP].
The anti-self-duality condition can be viewed as [*half*]{} of the relations necessary to determine a representation of the quantum group $SU(2)_q$, and we also obtain a classification identical to the non-deformed case. Tensor products of representations of quantum groups correspond in our context to the composition of instantons of different ranks, and again we expect it not to depend on the quantization parameter, if it is not a root of unity.
Further perspectives
--------------------
It is well known that the Penrose transform can be used to obtain solutions not only of the ASDYM equation, but also of a variety of other differential equations on Minkowski space-time [@WW]. One can then expect to define the quantum analogue of such equations in a way that our proposed quantum Penrose transform still yields the same result for generic $q$.
The most basic class of examples are the massless free field equations, which includes the wave, Dirac and Maxwell equations. In particular, the polynomial solutions to the wave equation on ${\mathbb{M}^{\rm I}}$ are given by the matrix coefficients of irreducible representations of $SU(2)$. Thus the appropriate quantum wave equation should have solutions on ${\mathfrak{M}}^{\rm I}_q$ given by the matrix coefficients of representations of the quantum group $SU(2)_q$, and a similar relation should also be true for the higher spin equations.
Moreover, the wave equation for the anti-self-dual connection plays a prominent role in the proof of completeness of instantons [@Ma] and its solutions correspond, via Penrose transform, to certain sheaf cohomology classes defined over $\mathbf{P}({\mathbb{T}})$. We expect that all this structure will be preserved under our quantum deformation of the twistor diagram, leading to a proof of our completeness conjecture.
Finally, we cannot avoid to mention another remarkable noncommutative deformation of instantons discovered by Nekrasov and Schwarz [@NS]. The moduli spaces of such noncommutative instantons can be regarded as a smooth compactification of the classical ADHM data, though the relation with the latter is not as straightforward as in our case [@N1].
A version of the Penrose transform for the Nekrasov-Schwarz noncommutative instantons has been discussed in [@KKO]. Also, it has been observed in [@L] that the factorization of the noncommutative Minkowski space by a finite subgroup $\Gamma\hookrightarrow SU(2)$ in the Nekrasov-Schwarz setting yields the smooth varieties previously introduced by Nakajima [@N3].
It is an interesting and challenging problem to obtain a natural compactification of the moduli spaces of quantum instantons by considering a more general type of ${\mathfrak{M}}^{\rm I}_q$-modules than those considered here, and then compare with the smooth compactification appearing in the Nekrasov-Schwarz approach. This might lead to a further extension of our understanding of the mathematical structure of space-time.
#### Acknowledgement.
We thank Yu. Berest for a discussion. We also thank the referee for his comments on the first version of this paper.
I.F. is supported by the NSF grant DMS-0070551, and would like to thank the MSRI for its hospitality at the later stages of this project. M.J. thanks the Departments of Mathematics at Yale University and at the University of Pennsylvania.
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---
abstract: 'We report on infrared spectroscopy of bilayer graphene integrated in gated structures. We observed a significant asymmetry in the optical conductivity upon electrostatic doping of electrons and holes. We show that this finding arises from a marked asymmetry between the valence and conduction bands, which is mainly due to the inequivalence of the two sublattices within the graphene layer. From the conductivity data, the energy difference of the two sublattices is determined.'
author:
- 'Z. Q. Li'
- 'E. A. Henriksen'
- 'Z. Jiang'
- 'Z. Hao'
- 'M. C. Martin'
- 'P. Kim'
- 'H. L. Stormer'
- 'D. N. Basov'
title: Band structure asymmetry of bilayer graphene revealed by infrared spectroscopy
---
Recently there has been unprecedented interest in carbon-based materials due to the discovery of graphene [@PheneNature]. Among all carbon systems, bilayer graphene stands out due to its remarkable properties such as the formation of a tunable band gap between the valence and conduction bands [@Castro][@Ohta][@Gap][@McCann]: a property not attainable in common semiconductors. The vast majority of previous experimental and theoretical studies of bilayer graphene assumed a symmetric band structure that is governed by the interlayer coupling energy $\gamma
_{1}$. This is in contrast with a significant electron-hole asymmetry observed in cyclotron resonance [@Erik] and cyclotron mass experiments [@Castro]. Several theoretical proposals have been put forward to explain these results [@Castro][@Kusminskiy]. Yet, the microscopic origin of the observed effects remains unknown.
Here we present the first investigation of the optical conductivity of bilayer graphene via infrared spectroscopy. We observed dramatic differences in the evolution of the conductivity for electron and hole polarities of the gate voltage. We show that small band parameters other than $%
\gamma _{1}$ give rise to an asymmetry between the valence and conduction bands, in contrast to the commonly assumed symmetric band structure. The systematic character of our IR data enables us to extract an energy difference between the A and B sublattices within the same graphene layer (Fig 1(b)) of $\delta _{AB}\approx $18meV. We analyze some of the implications of these findings for other properties of bilayer graphene.
Infrared (IR) reflectance R($\omega $) and transmission T($\omega $) measurements were performed on bilayer graphene samples on SiO$_{2}$/Si substrate [@Erik] as a function of gate voltage V$_{g}$ at 45K employing synchrotron radiation, as described in [@Li08]. We find that both R($%
\omega $) [@Wang; @Kuzmenko-bilayer] and T($\omega $) spectra of the bilayer graphene device can be strongly modified by a gate voltage. Figure 1 shows the transmission ratio data at several voltages normalized by data at the charge neutrality voltage V$_{CN}$: T(V)/T(V$_{CN}$), where V$_{CN}$ is the voltage corresponding to the minimum DC conductivity, and V= V$_{g}-$V$%
_{CN}$. The T(V)/T(V$_{CN}$) spectra are dominated by a dip at around 3000 cm$^{-1}$, the magnitude of which increases systematically with voltage. Apart from the main dip, a peak was observed in the T(V)/T(V$_{CN}$) data below 2500 cm$^{-1}$, which shifts systematically with voltage. This latter feature is similar to the T(V)/T(V$_{CN}$) data for single layer graphene [@Li08]. The gate-induced enhancement in transmission (T(V)/T(V$_{CN}$)$>
$1) below 2500 cm$^{-1}$ and above 3500 cm$^{-1}$ implies a decrease of the absorption with voltage in these frequency ranges.
The most informative quantity for exploring the quasiparticle dynamics in bilayer graphene is the two dimensional (2D) optical conductivity $\sigma
_{1}\left( \omega \right) +i\sigma _{2}\left( \omega \right) $ [@Li08][@LiPRL]. First, we extracted the optical conductivity at V$_{CN}$ from the reflectance data (not shown) employing a multilayer analysis of the device [@Li08][@LiPRL]. We find that $\sigma _{1}\left( \omega
,V_{CN}\right) $ has a value of $2\ast (\pi e^{2}/2h)$ at high energies, with a pronounced peak at 3250 cm$^{-1}$ (inset of Fig 2(b)). This observation is in agreement with theoretical analysis on undoped bilayer graphene[@BilayerIR][@Nilsson][@Nicol]. Our high energy data agree with recent experiments in the visible region [@GeimIR]. The peak around 3250 cm$^{-1}$ can be assigned to the interband transition in undoped bilayer near the interlayer coupling energy $\gamma _{1}$.
An applied gate voltage shifts the Fermi energy E$_{F}$ to finite values leading to significant modifications of the optical conductivity. The $%
\sigma _{1}\left( \omega ,V\right) $ and $\sigma _{2}\left( \omega ,V\right)
$ spectra extracted from voltage-dependent reflectance and transmission data [@Li08] are shown in Fig 2. At frequencies below 2500 cm$^{-1}$, we observe a suppression of $\sigma _{1}\left( \omega ,V\right) $ below $2\ast
(\pi e^{2}/2h)$ and a well-defined threshold structure, the energy of which systematically increases with voltage. Significant conductivity was observed at frequencies below the threshold feature. These observations are similar to the data in single layer graphene [@Li08]. The threshold feature below 2500 cm$^{-1}$ can be attributed to the onset of interband transitions at 2E$_{F}$, as shown by the arrow labeled e$_{1}$ in the inset of Fig 2(a) and (b). The observed residual conductivity below 2E$_{F}$ is in contrast to the theoretical absorption for ideal bilayer graphene [@Nilsson][@Nicol] that shows nearly zero conductivity up to 2E$_{F}$. Similar to single layer graphene, the residual conductivity may originate from disorder effects [Nilsson]{} or many body intertactions [@Li08]. Apart from the above similarities, the optical conductivity of bilayer graphene is significantly different from the single layer conductivity. First, the energy range where the conductivity $\sigma _{1}\left( \omega ,V\right) $ is impacted by the gate voltage extends well beyond the 2E$_{F}$ threshold. Furthermore, we find a pronounced peak near 3000 cm$^{-1}$, the oscillator strength of which shows a strong voltage dependence. This peak originates from the interband transition between the two conduction bands or two valence bands (inset of Fig.2a) [@Nilsson][@Nicol].
The voltage dependence of the Fermi energy in bilayer graphene can be extracted from $\sigma _{2}\left( \omega ,V\right) $ using a similar procedure as in [@Li08]. In order to isolate the 2E$_{F}$ feature, we fit the main resonance near 3000 cm$^{-1}$ with Lorentzian oscillators and then subtracted them from the experimental $\sigma _{2}\left( \omega
,V\right) $ spectra to obtain $\sigma _{2}^{diff}\left( \omega ,V\right) $. The latter spectra reveal a sharp minimum at $\omega $=2E$_{F}$ (Fig 2(c)) in agreement with single layer graphene [@Li08]. Figure 3a depicts the experimental 2E$_{F} $ values along with the theoretical result in [McCann]{}. Assuming the Fermi velocity v$_{F}$ in bilayer graphene is similar to that in single layer graphene (v$_{F}$=1.1$\times $10$^{6}$ m/s), we find that our data can be fitted with $\gamma _{1}$=450$\pm $80meV. Equally successful fits can be obtained assuming the Fermi velocity and interlayer coupling in the following parameter space: v$_{F}$=1.0-1.1$\times $10$^{6}$ m/s and $\gamma _{1}$=360-450 meV. Previous theoretical and experimental studies showed that an applied gate voltage opens a gap $\Delta $ between the valence and conduction bands [@Castro][@Ohta][@Gap][McCann]{}. Because $\Delta (V)$ is much smaller than 2E$_{F}$(V) for any applied bias in bottom-gate devices [@McCann], it has negligible effects on the experimentally observed 2E$_{F}$(V) behavior.
The central result of our study is an observation of a pronounced asymmetry in evolution of the optical conductivity upon injection of electrons or holes in bilayer graphene. Specifically, the frequencies of the main peak $%
\omega _{peak}$ in $\sigma _{1}\left( \omega ,V\right) $ are very distinct for E$_{F}$ on the electron and hole sides, as shown in Fig 3(b). In addition, $\omega _{peak}$ on the electron side shows a much stronger voltage dependence compared to that on the hole side. All these features are evident in the raw data in Fig.1, where the resonance leads to a dip in T(V)/T(V$_{CN}$) spectra. These behaviors are reproducible in multiple gated samples. Such an electron-hole asymmetry is beyond a simple band structure only taking $\gamma _{1}$ into account, which predicts symmetric properties between electron and hole sides.
We propose that the electron-hole asymmetry in our $\sigma _{1}\left( \omega
,V\right) $ data reflects an asymmetry between valence and conduction bands. Such an asymmetric band structure arises from finite band parameters $\delta
_{AB}$ and v$_{4}$, where $\delta _{AB}$ (denoted as $\Delta $ in [Graphite]{}[@Nilsson]) is the energy difference between A and B sublattices within the same graphene layer, and v$_{4}$=$\gamma _{4}/\gamma
_{0}$. $\gamma _{4}$ and $\gamma _{0}$ are defined as interlayer next-nearest-neighbor coupling energy and in-plane nearest-neighbor coupling energy, respectively [@Graphite][@Nilsson]. We first illustrate the effects of $\delta _{AB}$ and v$_{4}$ on the energy bands of bilayer graphene E$_{i}$(k) (i=1,2,3,4), which can be obtained from solving the tight binding Hamiltonian Eq (6) in Ref. [@Nilsson]. We find that finite values of $\delta _{AB}$ and v$_{4}\ $break the symmetry between valence and conduction bands, as schematically shown in the inset of Fig 2(a). Specifically, $\delta _{AB}$ induces an asymmetry in E$_{1}$ and E$_{4}$ bands such that E$_{1}>$-E$_{4}$ at k=0, whereas v$_{4}$ induces an electron-hole asymmetry in the slope of the valence and conduction bands. With finite v$_{4}$, the bands E$_{1}$ and E$_{2}$ are closer and E$_{3}$ and E$_{4}$ are further apart at high k compared to those with zero v$_{4}$ value.
Next we examine the effects of $\delta _{AB}$ and v$_{4}$ on $\sigma
_{1}\left( \omega ,V\right) $. It was predicted theoretically [@Nicol] that the main peak in $\sigma _{1}\left( \omega ,V\right) $ occurs in the frequency range between two transitions labeled e$_{2}$ and e$_{3}$ as shown in the inset of Fig 2(a) and (b). Here e$_{2}$=$-$E$_{4}$(k=0)$-\Delta /2$ and e$_{3}$=E$_{3}$(k=k$_{F}$)$-$E$_{4}$(k=k$_{F}$) for the hole side, and e$%
_{2}$=E$_{1}$(k=0)$-\Delta /2$ and e$_{3}$=E$_{1}$(k=k$_{F}$)$-$E$_{2}$(k=k$%
_{F}$) for the electron side[@Nicol], with $\Delta $ defined as the gap at k=0. For zero values of $\delta _{AB}$ and v$_{4}$, e$_{2}$ and e$_{3}$ transitions are identical on the electron and hole sides. The finite values of $\delta _{AB}$ and v$_{4}$ induce a significant inequality between e$_{2}$ and e$_{3}$ on the electron and hole sides. We first focus on the low voltage regime, where $\omega _{peak}$=e$_{2}$=e$_{3}$. Because v$_{F}$ and v$_{4}$ always enter the Hamiltonian in the form of v$_{F}$k and v$_{4}$k products[@Nilsson], these terms give vanishing contributions at low V, where k goes to zero. Consequently, $\omega _{peak}$ value at low bias is solely determined by $\gamma _{1}$ and $\delta _{AB}$, with $\omega _{peak}$=$\gamma _{1}+\delta _{AB}$ and $\omega _{peak}$=$\gamma _{1}-\delta _{AB}$ for the electron and hole sides, respectively. At V$_{CN}$ (0V), interband transitions between the two conduction bands and the two valence bands are both allowed, which leads to a broad peak centered between $\gamma
_{1}+\delta _{AB}$ and $\gamma _{1}-\delta _{AB}$ (Fig 3(b)). From the two distinct low voltage $\omega _{peak}$ values on the electron and hole sides shown in Fig 3(b), the values of $\gamma _{1}$ and $\delta _{AB}$ can be determined with great accuracy: $\gamma _{1}$=404$\pm $10meV and $\delta
_{AB}$=18$\pm $2meV. Therefore, the $\sigma _{1}\left( \omega ,V\right) $ data at low biases clearly indicates an asymmetry between valence and conduction bands in bilayer graphene due to finite energy difference of A and B sublattices.
In order to explore the V dependence of $\omega _{peak}$ and the width of the main peak in $\sigma _{1}\left( \omega ,V\right) ,$ $\Gamma _{peak}$, we plot the e$_{2}$ and e$_{3}$ transition energies [@Nicol] as a function of V (Fig. 3b), using the gap formula $\Delta $(V) in [@McCann][note1]{} and our calculated asymmetric dispersion E$_{i}$(k) (i=1,2,3,4) [note2]{}, with v$_{F}$=1.1$\times $10$^{6}$m/s, $\gamma _{1}$=404meV, $\delta
_{AB}$=18meV, and for both v$_{4}=0$ and v$_{4}=0.04$. We find that e$_{2}$ does not depend on v$_{4}$ [@note1], whereas e$_{3}$ is strongly affected by v$_{4}$. With a finite value of v$_{4}$ ($\approx $0.04), an assignment of $\omega _{peak}$ to (e$_{2}$+e$_{3}$)/2 appears to fit our data well on both electron and hole sides. Nevertheless, larger separation of e$_{2}$ and e$_{3}$ on the hole side is inconsistent with the relatively narrow peak in $\sigma_{1}\left( \omega ,V\right) $ for both electron and hole injection with nearly identical width. Yet the finite value of the v$%
_{4} $ parameter is essential to qualitatively account for the voltage dependence of $\omega _{peak},$ because with v$_{4}$$\approx$0 $\omega
_{peak}$ follows e$_{2}$ and e$_{3}$ on the electron and hole sides (Fig 3b), respectively, eluding a consistent description. A quantitative understanding of the V dependence of $\omega _{peak}$ and $\Gamma _{peak}$ is lacking at this stage. Our results highlight a need for further experimental and theoretical investigation of v$_{4}$ including its possible V dependence.
We stress that $\gamma _{1}$ and $\delta _{AB}$ are determined from the low bias (low k$_F$) data. Therefore the values of $\gamma _{1}$ and $\delta
_{AB}$ reported here do not suffer from the currently incomplete understanding of V dependence of $\omega _{peak}$ and $\Gamma_{peak}$ discussed above. The $\gamma _{1}$ value (404$\pm $10meV) is directly determined from measurements of transitions between the two conduction bands or valence bands. It has been predicted theoretically that the band structure of bilayer graphene as well as the parameters $\gamma _{1}$ and v$%
_{F}$ can be strongly renormalized by electron-electron interactions [Kusminskiy]{}. The $\gamma _{1}$ value inferred from our data is close to theoretical estimates of the renormalized $\gamma _{1}$ [@Kusminskiy].
IR measurements reported here have enabled accurate extraction of $\delta
_{AB}$ in bilayer graphene free from ambiguities of alternative experimental methods. Interestingly, the energy difference between A and B sublattices $%
\delta _{AB}$ in bilayer graphene (18meV) is much greater than that in graphite ($\delta _{AB}\approx $8meV) [@Graphite]. Such a large value of $\delta _{AB}$ in bilayer is very surprising. We propose that this observation stems from different interlayer coupling between the B sublattices in bilayer graphene and graphite. In bilayer, the direct interlayer coupling between A$_{1}$ and A$_{2}$ (Fig 1(b)) considerably enhances the energy of A sublattices due to Coulomb repulsion between the $%
\pi $ orbits. However, the sublattices B$_{1}$ and B$_{2}$ are not on top of each other as shown in Fig 1(b) and thus are more weakly coupled. Therefore, the energy of the B sites is lower than that of the A sites within the same layer, leading to a large $\delta _{AB}$ in bilayer graphene. On the other hand, in graphite the B sublattice in the third layer B$_{3}$ is on top of B$%
_{1}$, and that in the fourth layer B$_{4}$ is right above B$_{2}$. The coupling between the B sites in the next nearest neighbor layers (B$_{1}$ and B$_{3}$, B$_{2}$ and B$_{4}$, etc) increases the energy of the B sites compared to that in bilayer, giving rise to a smaller $\delta _{AB}$ value.
The asymmetry between valence and conduction bands uncovered by our study has broad implications on the fundamental understanding of bilayer graphene. An electron-hole asymmetry was observed in the cyclotron resonance [Erik]{} and cyclotron mass experiments [@Castro] in bilayer, both of which have eluded a complete understanding so far. Our accurate determination of finite values of $\delta _{AB}$ and v$_{4}$ calls for explicit account of the asymmetric band structure in the interpretation of the cyclotron data. Moreover, the different $\delta _{AB}$ values in bilayer graphene and graphite reveal the importance of interlayer coupling in defining the electronic properties and band structure of graphitic systems.
During the preparation of this paper, we became aware of another infrared study of bilayer graphene by A.B. Kuzmenko et al [@Kuzmenko-bilayer]. We thank M.L. Zhang and M.M. Fogler for fruitful discussions. Work at UCSD is supported by DOE (No. DE-FG02-00ER45799). Research at Columbia University is supported by the DOE (No. DE-AIO2-04ER46133 and No. DE-FG02-05ER46215), NSF (No. DMR-03-52738 and No. CHE-0117752), NYSTAR, the Keck Foundation and Microsoft, Project Q. The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
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E. V. Castro, et al, Phys. Rev. Lett. 99, 216802 (2007).
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E. McCann, Phys. Rev. B 74, 161403(R) (2006).
E. Henriksen et al, Phys. Rev. Lett. 100, 087403 (2008).
S. V. Kusminskiy, D. K. Campbell, A. H. Castro Neto, arXiv:cond-mat/0805.0305; private communication.
Z.Q. Li et al, Nature Physics 4, 532 (2008).
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Z.Q. Li et al, Phys. Rev. Lett. 99, 016403 (2007).
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The gap formula $\Delta $(V) in [@McCann] did not take into account $\delta _{AB}$ and v$_{4}$. However, we find that finite $%
\delta _{AB}$ and v$_{4}$ values have no effect on the gap $\Delta $. Specifically, $\delta _{AB}$ primarily modifies the E$_{1}$ and E$_{4}$ bands, while leaving unchanged the gap $\Delta $ between the E$_{2}$ and E$%
_{3}$ bands at k=0, as shown in Fig 2(a). In addition, v$_{4}$ always appears in a term v$_{4}$k in the Hamiltonian[@Nilsson], therefore it has zero effect on $\Delta $.
In the calculation of the energy bands E$_{i}$(k) (i=1,2,3,4), we used the approximation $\Delta $=0, which can be justified for the purpose of estimating e$_{2}$ and e$_{3}$. The gap $\Delta $ is very small ($<$80 meV) in the voltage range studied in our work [@McCann], and does not affect the higher energy bands E$_{1}$(k=0) or E$_{4}$(k=0) and therefore the value of e$_{2}$. Note that $\Delta $=0 is only assumed when calculating E$_{i}$(k), but not in the $\Delta $ term in the expression of e$%
_{2}$. Moreover, E$_{2}$ and E$_{3}$ bands are modified by the gap only at energies below $\Delta $/2. Because E$_{F}$ is much larger than $\Delta $/2 under applied voltage [@McCann], E$_{2}$(k=k$_{F}$)and E$_{3}$(k=k$_{F}$) are not affected by $\Delta$. Therefore, a finite gap does not modify the value of e$_{3}$ compared to that with $\Delta $=0.
![(color online) T(V)/T(V$_{CN}$) spectra of bilayer graphene. (a) and (b): data for E$_{F}$ on the hole side and electron side. Inset of (a): a schematic of the device and infrared measurements. Inset of (b): a schematic of bilayer graphene. The solid lines indicate bonds in the top layer A$_{1}$B$_{1}$, whereas the dashed lines indicate bonds in the bottom layer A$_{2}$B$_{2}$. The sublattice A$_{1}$ is right on top of the sublattice A$_{2}$.[]{data-label="Fig.1"}](fig1.eps){width="6cm" height="8.3cm"}
![(color online) The optical conductivity of bilayer graphene. (a) and (b): $\protect\sigma _{1}\left( \protect\omega ,V\right) $ data for E$%
_{F}$ on the hole side and electron side. (c): $\protect\sigma %
_{2}^{diff}\left( \protect\omega ,V\right) $ spectra in the low frequency range, after subtracting the Lorentzian oscillators describing the main resonacne around 3000 cm$^{-1}$ from the whole $\protect\sigma _{2}\left(
\protect\omega ,V\right) $ spectra. Inset of (a): Schematics of the band structure of bilayer with zero values of $\protect\delta _{AB}$ and v$_{4}$ (red) and finite values of $\protect\delta _{AB}$ and v$_{4}$ (black), together with allowed interband transitions. Insets of (b): $\protect\sigma %
_{1}\left( \protect\omega ,V\right) $ at 0V (V$_{CN}$) and 40V on the hole side with assignments of the features. []{data-label="Fig.2"}](fig2.eps){width="6cm" height="12.5cm"}
![(color online) (a) Symbols: the 2E$_{F}$ values extracted from the optical conductivity detailed in the text. The error bars are estimates of the uncertainties of $\protect\sigma _{2}^{diff}\left( \protect\omega %
,V\right) $ spectra in Fig 2(c). Solid lines: the theoretical 2E$_{F}$ values using v$_{F}$=1.1$\times $10$^{6}$ m/s and $\protect\gamma _{1}$=450meV. (b) Solid symbols, the energy of the main peak $\protect\omega %
_{peak}$ in the $\protect\sigma _{1}\left( \protect\omega ,V\right) $ spectrum. Open symbols: the energy of the dip feature $\protect\omega _{dip}$ in the T(V)/T(V$_{CN}$) spectra. Note that $\protect\omega _{peak}$ in $%
\protect\sigma _{1}\left( \protect\omega ,V\right) $ is shifted from $%
\protect\omega _{dip}$ in the raw T(V)/T(V$_{CN}$) data with an almost constant offset, which is due to the presence of the substrate. Solid lines: theoretical values of the transitions at e$_{2}$, e$_{3}$ and (e$_{2}$+e$_{3}
$)/2 with v$_{F}$=1.1$\times $10$^{6}$m/s, $\protect\gamma _{1}$=404meV and $%
\protect\delta _{AB}$=18meV and v$_{4}$=0.04. Red dashed lines: e$_{3}$ with similar parameters except v$_{4}$=0.[]{data-label="Fig.3"}](fig3.eps){width="6cm" height="7.18cm"}
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abstract: 'Fermat’s principle and variational analysis is used to analyze the trajectories of light propagating in a radially inhomogeneous medium with a singularity in the center. It is found that the light trajectories are similar to those around a black hole, in the sense that there exists a critical radius within which the light cannot escape, but spirals into the singularity.'
author:
- |
M. Marklund, D. Anderson, F. Cattani, M. Lisak and L. Lundgren\
*Department of Electromagnetics, Chalmers University of Technology,*\
*SE–412 96 Göteborg, Sweden*
date: '()'
title: 'Fermat’s principle and variational analysis of an optical model for light propagation exhibiting a critical radius'
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Introduction
============
There has recently, within the scientific community, been much interest focused on the extra-ordinary properties associated with Bose-Einstein condensates - clouds of atoms cooled down to nano-Kelvin temperatures where all atoms are in the same quantum state and macroscopic quantum conditions prevail [@Leonhardt-Piwnicki-99; @Leonhardt-Piwnicki-00; @Hau-etal-99].
One of the most dramatic experiments which has been performed in Bose-Einstein condensates is the demonstration of optical light pulses traveling at extremely small group velocities, e.g. velocities as small as 17 m/s have been reported [@Hau-etal-99].
In a physically suggestive application of the optical properties of Bose-Einstein condensates, it was recently suggested that they could be used to create, in the laboratory, dielectric analogues of relativistic astronomical phenomena like those associated with a black hole [@Leonhardt-Piwnicki-00].
The astrophysical concept of a black hole is one of the most fascinating and intriguing phenomena related to the interaction between light and matter through the curvature of space-time within the framework of general relativity. Black holes are believed to form when compact stars undergo complete gravitational collapse due to, e.g., accretion of matter from its surroundings (in general relativity, the pressure contributes to the gravitational mass of a fluid, and an increased pressure will, after a certain point, therefore only help to accelerate the collapse phase).
In ordinary dielectric media, unrealistic physical conditions would be required in order to demonstrate any of the spectacular effects of general relativity. However, in dielectric media characterized by very small light velocities, new possibilities appear. In particular, it was recently suggested that by creating a vortex structure in such a dielectric medium it is possible to mimic the properties of an optical black hole [@Leonhardt-Piwnicki-00]. What this study in fact suggested was that one could construct, by using the aforementioned vortex, an unstable photon orbit, much like the orbit at a radius $r = 3M$ in the Schwarschild geometry [@Misner-Thorne-Wheeler]. In order to construct an event horizon, one would need to supplement the vortex flow by a radial motion of the fluid [@Visser-00; @Leonhardt-Piwnicki-00b].
The vortex, which involves a rotating cylindrical velocity field, tends to attract light propagating perpendicular to its axis of rotation and to make it deviate from its straight path. In the model considered in Refs.[@Leonhardt-Piwnicki-99; @Leonhardt-Piwnicki-00], the increasing, in fact diverging, rotational velocity field of the vortex core attracts light by the optical Aharanov–Bohm effect and causes a bending of the light ray similar to that of the gravitational field in general relativity. It is also shown that there exist a critical radius, $r_{crit}$, from the vortex core with the properties that if light rays come closer to the core than $r_{crit}$,the light will fall towards the singularity of the vortex core. In this sense, the critical radius, $r_{crit}$, plays the role of an optical unstable photon orbit, analogous to the unstable orbit in the Schwarzschild geometry.
The basic physical effect involved in the analysis of the ray propagation in Refs.[@Leonhardt-Piwnicki-99; @Leonhardt-Piwnicki-00] is the fact that the refractive index, $ n$, of a medium changes when the medium is moving. In Refs.[@Leonhardt-Piwnicki-99; @Leonhardt-Piwnicki-00], the medium is assumed to have a cylindrical vortex velocity field $ \bf V= V(\bf r)$ given by $${\bf V(r)} = \frac{W}{r}{\bf \widehat{\varphi}}$$ where $r$ is the radius from the vortex core centre, $\widehat{\varphi }$ is the azimuthal unit vector and $2\pi W$ is the vorticity.
Advanced physical concepts like Bose-Einstein condensates, vortex velocity fields and black holes are fascinating physical effects, which however may be difficult to present in a simplified manner for undergraduate students. Nevertheless, it is important to try to convey inspiration for and arouse curiosity in such phenomena. An interesting example of such an effort was recently made in Ref. [@McDonald-00], where the basic physical mechanism, in the form of a classical two level system was used to demonstrate the possibility of the very low light velocities observed in Bose- Einstein condensates. In a later work by the same author, Ref. [@McDonald-01], an inspiring investigation is given of a simple mechanical model that exhibits a gravitational critical radius. The purpose of the present work is similar to that of Refs. [@McDonald-00; @McDonald-01]. We will try to discuss a simple classical example of a medium where the refractive index has a divergence at $r = 0$ and which exhibits some of the dramatic properties of the light behaviour around a black hole as discussed in Refs.. In addition we want to illustrate the power and beauty of the classical principle of Fermat and of variational methods in connection with this new fascinating concepts.
Fermat’s principle {#sec:Fermat}
==================
It is well known that light tends to be deflected towards regions with higher refractive index. Consider light propagating in a cylindrically symmetric medium where the refractive index increases towards the centre. In such a situation we expect the light path to look qualitatively as in Fig. 1. The actual light path is determined by Fermat’s principle i.e.$$\delta\int n({\bf r})ds=0$$ where$ ds$ is an infinitesimal element along the light ray. We will consider light propagation in a medium where the refractive index is of the qualitative form, cf Eq. (1) $$n(r)\simeq \left\{
\begin{array} {r@{\quad as\quad}l}1 & r\gg r_{0}\\
r_0/r & r\ll r_{0}\end{array} \right.$$
One possible realization of $n(r)$ with the desired properties is given by $$n^{2}(r) = 1 + \left(\frac{r_{0}}{r}\right)^2$$
Using Fermat’s principle and expressing the light path as the relation $\theta=\theta(r)$ where $\theta$ is the polar angle, Eq. (2) implies $$\delta\int{n(r)\sqrt{1+r^{2}\left(\frac{d\theta}{dr}\right)^2}}dr
= 0 \ .$$
The Euler-Lagrange variational equation corresponding to Eq. (5) reads $$\frac{d}{dr}\left[ n(r)\frac{r^2 \, d\theta/dr}%
{\sqrt{1 + r^2(d\theta/dr)^2}}\right]=0$$ which determines the trajectory of the light.
Solution of the light trajectory
================================
Equation (6) directly implies that $$\frac{r^2n(r) \, d\theta/dr }%
{\sqrt{1 + r^2 (d\theta/dr)^2}} = r_{i}$$ where the constant $r_{i}$ is determined by initial conditions, i.e. the properties of the incident light.
Equation (7) is easily inverted to read $$\frac{d\theta}{dr}=\pm\frac{r_{i}}{\sqrt{r^4n^2(r) - r_{i}^2r^2}}$$
For a light ray incident as shown in Fig. 1, we clearly have $d\theta/dr <0$ (at least up to some minimum radius $ r=r_{min}$. It is illustrative to first consider the trivial case of a homogeneous medium with $n(r)\equiv1$. In this case Eq. (9) becomes $$\frac{d\theta}{dr}=-\frac{r_{i}}{\sqrt{r^4 - r_{i}^2r^2}}$$ which can easily be interpreted to yield the light path in the form $$\theta=\mathrm{arccot}\sqrt{\frac{r^2}{r_{i}^2} - 1}$$ or simpler $$r=\frac{r_{i}}{\sin \theta}$$ Clearly this is the straight line solution $$y=r_{i}$$ where the parameter $r_{i}$ plays the role of “impact parameter” or minimum distance from the centre.
Let us now consider the model variation for $n(r)$ as given by Eq. (5). Within this model Eq. (10) becomes $$\frac{d\theta}{dr} =
- \frac{r_i}{\sqrt{r^4 - (r_i^2 - r_0^2)r^2}}$$ Clearly the solution of Eq. (13) will depend crucially on the relative magnitude of $r_i$ and $r_0$, i.e. the impact parameter relative to the characteristic radial extension of the inhomogeneity. Consider first the case when $r_0 < r_i$. Equation (13) can then be rewritten as $$\begin{aligned}
\frac{b_1}{r_i}\frac{d\theta}{dr}
&=& -\frac{b_1}{\sqrt{r^4 -b_1^2r^2}} \ , \\
b_1 &\equiv& \sqrt{r_{i}^{2}-r_{0}^{2}} \ .\end{aligned}$$ Equation (15) is of the same form as Eq. (9) and we directly infer the following solution $$r = \frac{\sqrt{r_{i}^{2} - r_{0}^{2}}}%
{\sin\left[\theta\sqrt{1-r_{0}^{2}/r_{i}^{2}}\,\right]}$$
As $\theta\rightarrow0$, we still have asymptotically $$r\simeq\frac{r_{i}}{\sin \theta}$$ However, the trajectory is now bending towards the origin and the minimum distance occurs at the polar angle $\theta=\theta_{m}$ given by $$\theta_m = \frac{\pi}{2}\frac{1}{\sqrt{1-r_0^2/r_i^2}}$$
The corresponding minimum distance, $r_m$ is $$r_m \equiv r{\theta_m} = \sqrt{r_i^2 - r_0^2}$$
We also note that the trajectory is symmetrical around the angle $\theta_{m}$ and that the asymptotic angle of the outgoing light ray is $$\theta_{\infty}\equiv \lim_{r\rightarrow\infty} \theta(r)
=\frac{\pi}{\sqrt{1 - r_0^2/r_i^2}} = 2 \theta_m$$
The solution given by eq.(10) describes a trajectory which is bent towards the centre of attraction at $ r=0$. Depending on the ratio $r_{0}/r_{i}$, the trajectory is either more or less bent or may even perform a number of spirals towards the centre before again turning outwards and escaping, cf. Fig. 2. The number of turns, $N$,which the light ray does around the origin before escaping is simply $$N = \left\lfloor \frac{2\theta_{m}}{2\pi} \right\rfloor
= \left\lfloor \frac{1}{2\sqrt{1-r_{0}^{2}/r_{i}^{2}}} \right\rfloor$$ where $\left\lfloor x \right\rfloor$ denotes the largest integer less than $x$.
Let us now consider the special case when the impact parameter equals the characteristic width of the refractive index core i.e.$r_{i}=r_{0}$. The equation for the trajectory now simplifies to $$\frac{d\theta}{dr} = -\frac{r_{i}}{r^{2}}$$ with the simple solution $$\theta=\frac{r_{i}}{r}$$ i.e. the trajectory describes a path in the form of Arkimede’s spiral as the light falls towards the origin, cf. Fig. 3. The form of the light trajectory in the situation when $ r_{i}< r_{0}$ is now obvious, it will spiral into the singularity more or less directly depending on the magnitude of the ratio $r_{0}/r_{i}>1$.
The actual trajectory in this case is given by a slight generalization of Eq. (16) viz $$r=\frac{\sqrt{r_{0}^{2}-r_{i}^{2}}}{\sinh
[\theta\sqrt{1-r_{i}^{2}/r_{0}^{2}}]}$$
This solution does indeed convey the expected behaviour, the trajectory spirals monotonously into the singularity of the refractive index.
Final comments
==============
The present analysis is inspired by recent discoveries and discussions about light propagation in Bose-Einstein condensates where extremely low light velocities can be obtained. This has triggered speculations about possible laboratory demonstrations of effects, which normally are associated with general relativistic conditions. In particular, it has been suggested that it may be possible to create the analogue of a black hole using a divergent in-spiral of a Bose-Einstein condensate.
In the present work we have analyzed a simple classical example of light propagation as determined by Fermat’s principle in a medium characterized by a radially symmetric refractive index. In analogy with the variation of the vortex velocity field suggested in [@Leonhardt-Piwnicki-00], the refractive index is here assumed to diverge towards the centre. This classical example exhibit some of the characteristic properties of light propagating around a black hole where the gravitational attraction deflects the light and where, under certain conditions, the light may be “swallowed” by the black hole.
In the example analyzed here, the unstable photon orbit of the Schwarzschild black hole is similar to the characteristic radius, $r_0$, of the refractive index variation, which together with the impact parameter, $r_i$, of the incident light completely determines the light trajectory. If $r_0 < r_i$, the light is more or less deflected, but ultimately escapes. However, if $r_0 \geq r_i$, the light spirals into the singularity.It should be cautioned that this result depends crucially on the presence of the singularity in the refractive index (4). If this is removed the in-spiraling photon orbit will eventually turn and start spiraling outwards. In the recent discussion about the possibility of generating analogues of optical black holes in Bose-Einstein condensates, it has been suggested that the vortex motion must also have a velocity component in the radial direction.
In the classical model considered here, a number of additional physical effects will obviously affect the light path when it comes close to the axis and will in fact remove the mathematical singularity. Nevertheless, the model provides a simple example of light dynamics, which resembles some of the properties of light propagation around a black hole. Furthermore,the investigation is based on Fermat’s principle and variational analysis, in this way illustrating the use of classical methods in connection with very new and fascinating concepts at the front line of modern research.
Figure Captions. {#figure-captions. .unnumbered}
================
Fig.1 Qualitative plot of a light ray trajectory in a cylindrically symmetric medium with a refractive index, which increases towards the centre.\
Fig.2 Light trajectories for the refractive index model of eq.() for different impact radii $r_{i}>r_{0}$\
Fig.3 Light trajectory in the case of $r_{i}=r_{0}$, the spiral of Arkimede.\
[99]{}
U. Leonhardt and P. Piwnicki, Phys. Rev. A **60**, 4301 (1999)
U. Leonhardt and P. Piwnicki, Phys. Rev. Letters **84**, 822 (2000)
L. V. Hau, S. E. Harris, Z. Dutton and C. H. Behroozi, Nature (London) **397**, 594 (1999)
C. Misner, K. S. Thorne and J. A. Wheeler *Gravitation* (Freeman, 1973) M. Visser, Phys. Rev. Letters **85**, 5252 (2000)
U. Leonhardt and P. Piwnicki, Phys. Rev. Letters **85**, 5253 (2000)
K. T. Mc Donald, Am. J. Phys. **68**, 293 (2000) K. T. Mc Donald, Am. J. Phys. **69**, 617 (2001)
|
---
author:
- 'Houman Owhadi[^1] and Lei Zhang[^2]'
date: 'December 21, 2005'
title: 'Homogenization of parabolic equations with a continuum of space and time scales. '
---
Introduction and main results
=============================
Let $\Omega$ be a bounded and convex domain of class $C^2$ of $\R^n$. Let $T>0$. Consider the following parabolic PDE $$\label{ghjh52}
\begin{cases}
\partial_t u=\diiv\big(a(x,t)\nabla u(x,t)\big)+g\quad \text{in}\quad \Omega\times (0,T) \\
u(x,t)=0 \quad \text{for}\quad (x,t) \in \big(\partial \Omega \times
(0,T) \big) \cup \big(\Omega \cup \{t=0\}\big).
\end{cases}$$ Write $\Omega_T:=\Omega \times (0,T)$. $g$ is a function in $L^{2}(\Omega_T)$. $(x,t) \rightarrow a(x,t)$ is a mapping from $\Omega_T$ into the space of symmetric positive definite matrices with entries in $L^\infty(\Omega_T)$. Assume $a$ to be uniformly elliptic on the closure of $\Omega_T$. This paper addresses the issue of the homogenization of in space and time in situations where scale separation and ergodicity at small scales are not available (see [@BeLiPa78], [@JiKoOl91] and [@Al01] for an introduction to classical homogenization theory). For that purpose, we will introduce in subsection \[kjskjsjshj871\] theorems establishing under Cordes type conditions the increase of regularity of solutions of when derivatives are taken with respect to harmonic coordinates instead of Euclidean coordinates. In subsections \[sub2\], \[jksjhs89b\] these results will be used to homogenize in space and in time. More precisely, assume $a$ to be written on a fine tessellation with $N$ degrees of freedom. If $a$ is time independent, then by solving $n$-times an elliptic boundary value-problem associated to (at a cost of $O(N (\ln
N)^{n+3})$ operations using the Hierarchical matrix method [@Beben05]) it is possible approximate the solutions of by solving an homogenized operator with $N^\alpha$ degrees of freedom ($\alpha <1$, $\alpha=0.2$ for instance) or with a fixed number $M$ of degrees of freedom (numerical experiments given at the end of this paper have been conducted with $N=16641$ and $M=9$), this problem is of practical importance for oil extraction and reservoir modeling in geophysics. If $a$ is also characterized by a continuum of time scales, then the method presented here does not reduce the number of operation counts necessary to solve only one time. However if one needs to solve $K$ ($K>n$) times (with different right hand sides) then by solving $n$ times it is possible to obtain an approximation of the solutions of by solving an homogenized (in space and time) parabolic equation written on a coarse tessellation with coarse time steps.
Compensation phenomenon {#kjskjsjshj871}
-----------------------
Let $F$ be the solution of the following parabolic equation $$\label{ghjagsash52}
\begin{cases}
\partial_t F=\diiv\big(a(x,t)\nabla F(x,t)\big)\quad \text{in}\quad \Omega_T \\
F(x,t)=x \quad \text{for}\quad (x,t) \in \big(\partial \Omega \times
(0,T) \big)\\
\diiv\big(a(x,0)\nabla F(x,0)\big)=0\quad \text{in}\quad \Omega.
\end{cases}$$ By we mean that $F:=(F_1,\ldots, F_n)$ is a $n$-dimensional vector field such that each of its entries satisfies $$\label{ghjasgsash52}
\begin{cases}
\partial_t F_i=\diiv\big(a(x,t)\nabla F_i(x,t)\big)\quad \text{in}\quad \Omega_T \\
F_i(x,t)=x_i \quad \text{for}\quad (x,t) \in \big(\partial \Omega
\times (0,T) \big) \\
\diiv\big(a(x,0)\nabla F_i(x,0)\big)=0\quad \text{in}\quad \Omega.
\end{cases}$$ Observe that if $a$ is time independent then $F$ is the solution of an elliptic boundary value problem.
Write $$\label{jhscczxhjd}
\sigma:={^t\nabla F}a\nabla F.$$
Write $\beta_\sigma$ the Cordes parameter associated to $\sigma$ defined by $$\label{sshgdd7641}
\beta_{\sigma}:=\esssup_{(x,t)\in \Omega_T}\Big(
n-\frac{\big(\Tr[\sigma]\big)^2}{\Tr[^t\sigma \sigma] }\Big).$$ Observe that since $$\label{sshgdmmmd7641}
\beta_{\sigma}=\esssup_{(x,t)\in \Omega_T}\Big(
n-\frac{\big(\sum_{i=1}^n
\lambda_{i,\sigma(x,t)}\big)^2}{\sum_{i=1}^n
\lambda_{i,\sigma(x,t)}^2}\Big).$$ where $(\lambda_{i,M})$ denotes the eigenvalues of $M$, $\beta_{\sigma}$ is a measure of the anisotropy of $\sigma$.
### Time independent medium.
In this subsection we assume that $a$ does not depend on time $t$. Write for $p\geq 2$, $W^{2,p}_{D}$ ($D$ for Dirichlet boundary condition) the Banach space $W^{2,p}_D(\Omega)\cap
W^{1,p}_0(\Omega)$. Equip $W^{2,p}_D(\Omega)$ with the norm $$\|v\|_{W^{2,p}_D(\Omega)}^2:=\int_{\Omega}\big(\sum_{i,j}(\partial_i
\partial_j v)^2\big)^\frac{p}{2}.$$ Equip the space $L^p(0,T,W^{2,p}_D(\Omega))$ with the norm $$\|v\|_{L^p(0,T,W^{2,p}_D(\Omega))}^p=\int_0^T\int_{\Omega}
\big(\sum_{i,j} (\partial_i
\partial_j v)^2\big)^\frac{p}{2} \,dx\,dt.$$
\[ksjshjsxxsd8721\] Assume that $\partial_t a\equiv 0$, $g\in L^2(\Omega_T)$, $\Omega$ is convex, $\beta_\sigma<1$ and $(\Tr[\sigma])^{\frac{n}{4}-1}\in
L^\infty(\Omega)$ then $u\circ F^{-1}\in L^2(0,T,W^{2,2}_D(\Omega))$ and $$\|u\circ F^{-1}\|_{L^2(0,T,W^{2,2}_D(\Omega))}\leq \frac{C
}{1-\beta_\sigma^\frac{1}{2}} \|g\|_{L^2(\Omega_T)}.$$
The constant $C$ can be written $$C=\frac{C_n}{(\lambda_{\min}(a))^\frac{n}{4}}
\big\|(\Tr[\sigma])^{\frac{n}{4}-1}\big\|_{L^\infty(\Omega)}.$$ Through this paper, we write $$\lambda_{\min}(a):=\inf_{(x,t)\in\Omega_T}\inf_{l\in \R^n,
|l|=1}{^tl. a(x,t).l}.$$
According to theorem \[ksjshjsxxsd8721\] although the second order derivatives of $u$ with respect to Euclidean coordinates are only in $L^2(0,T,H^{-1}(\Omega))$, they are in $L^2(\Omega_T)$ with respect to harmonic coordinates.
Observe that if $a$ is time independent then $F$ and $\sigma$ are time independent and $F$ is the solution of the following elliptic problem: $$\label{dgdgfsghsxzf62}
\begin{cases}
\diiv a \nabla F=0 \quad \text{in}\quad \Omega\\
F(x)=x \quad \text{on}\quad \partial \Omega.
\end{cases}$$
In dimension one $F$ is trivially an homeomorphism. In dimension $2$ this property follows from topological constraints [@MR2001070] (even with $a_{i,j}\in
L^\infty(\Omega)$), [@MR1892102] (one can also deduce from [@MR1892102] that for $n=2$, if $a$ is smooth then the conditions $\beta_\sigma<1$ and $(\Tr[\sigma])^{-1}\in
L^\infty(\Omega)$ are satisfied). In dimension three and higher $F$ can be non-bijective even if $a$ is smooth, we refer to [@MR1892102] and [@MR2073507], however in dimension $3$ the assumption $(\Tr[\sigma])^{\frac{n}{4}-1}\in L^\infty(\Omega_T)$ implies that $F$ is an homeomorphism. If $n\geq 4$ we need to assume that $F$ is an homeomorphism to prove the theorem.
In fact the condition $(\Tr(\sigma))^{-1}\in L^p(\Omega_T)$ for $p<\infty$ depending on $n$ is sufficient to obtain theorem \[ksjshjsxxsd8721\] and the following compensation theorems. For the sake of clarity this paper has been restricted to $(\Tr(\sigma))^{-1} \in L^\infty(\Omega_T)$.
Write $$\label{sjjdhd27}
\mu_\sigma:=\esssup_{\Omega_T}\frac{\lambda_{\max}(\sigma)}{\lambda_{\min}(\sigma)}.$$ It is easy to check that $\mu_{\sigma}$ is bounded by an increasing function of $(1-\beta_{\sigma})^{-1}$ and in dimension two $\beta_{\sigma}<1$ is equivalent to $\mu_{\sigma}<\infty$.
Theorem \[ksjshjsxxsd8721\] has been called compensation phenomenon because the composition by $F^{-1}$ increases the regularity of $u \in L^2(0,T, H^1_0(\Omega))$. The choice of this name has been motivated by F. Murat and L. Tartar’s work on H-convergence [@MR1493039] which is also based on on a regularization property called compensated compactness or div-curl lemma introduced in the 70’s by Murat and Tartar [@MR506997], [@MR584398] (we also refer to [@MR1225511] for refinements of the div-curl lemma).
The compensation phenomena presented in this subsection can be observed numerically. In figure \[ap7\], the value of $a$ is set to be equal to $1$ or $100$ with probability $1/2$ on each triangle of a fine mesh characterized by $16641$ nodes and $32768$ triangles. has been solved numerically on that mesh with $g=1$. $u$, $u\circ F^{-1}$, $\partial_x u$ and $\partial x (u\circ F^{-1})$ have been plotted at time $t=1$ in figure \[unut100tip7\].
[ ]{}\
[ ]{}\
In situations where $g\in L^\infty(0,T,L^2(\Omega))$, $\partial_t
g\in L^2(0,T, H^{-1}(\Omega))$ or $g\in L^p(\Omega_T)$ with $p>2$, one can obtain a higher regularity for $u\circ F^{-1}$. This is the object of the following theorems.
\[ksjshszjscxsxsdded8721\] Assume that $\Omega$ is convex, $g\in L^\infty(0,T,L^2(\Omega))$,\
$\partial_t g\in L^2(0,T, H^{-1}(\Omega))$, $\partial_t a\equiv 0$, $\beta_\sigma<1$ and $(\Tr[\sigma])^{\frac{n}{4}-1}\in
L^\infty(\Omega_T)$ then for all $t\in [0,T]$, $u\circ F^{-1}(.,t)
\in W^{2,2}_D(\Omega)$ and $$\label{equzalundi}
\|u\circ F^{-1}(.,t)\|_{W^{2,2}_D(\Omega)}\leq \frac{C
}{1-\beta_\sigma^\frac{1}{2}}
\Big(\big\|g\big\|_{L^\infty(0,T,L^2(\Omega))}+\|\partial_t
g\|_{L^2(0,T,H^{-1}(\Omega))}\Big).$$
The constant $C$ can be written $$C=\frac{C_{n,\Omega}}{(\lambda_{\min}(a))^\frac{n}{4}}
\big\|(\Tr[\sigma])^{\frac{n}{4}-1}\big\|_{L^\infty(\Omega_T)}(1+\frac{1}{\lambda_{\min}(a)})^\frac{1}{2}.$$
\[ksjshjsxgghxsd8721\] Assume that $\Omega$ is convex, $g(.,0)\in L^2(\Omega)$,\
$\partial_t g\in L^2(0,T, H^{-1}(\Omega))$, $g\in L^p(\Omega_T)$, $\partial_t a\equiv 0$, $\beta_\sigma<1$ and $(\Tr[\sigma])^{\frac{n}{4}-1}\in L^\infty(\Omega_T)$ then there exists a real number $p_0>2$ depending only on $n,\Omega$ and $\beta_\sigma$ such that for each $p$ such that $2\leq p<p_0$ one has $$\begin{split}
\|u\circ F^{-1}\|_{L^p(0,T,W^{2,p}_D(\Omega))} \leq &\frac{C
}{1-\beta_\sigma^\frac{1}{2}}
\big(\|g\|_{L^p(\Omega_T)}\\&+\big\|g(.,0)\big\|_{L^2(\Omega)}+\|\partial_t
g\|_{L^2(0,T,H^{-1}(\Omega))}\big).
\end{split}$$
The constant $C$ can written $$C=\frac{C_{n,\Omega,p}}{(\lambda_{\min}(a))^\frac{n}{4}}
\big\|(\Tr[\sigma])^{\frac{n}{4}-1}\big\|_{L^\infty(\Omega_T)}\big(1+\frac{1}{\lambda_{\min}(a)}\big)^\frac{1}{2}.$$
Write $$\label{hdaassz7}
\begin{split}
\|v\|_{C^{\gamma}(\Omega)}:=\sup_{x,y\in \Omega,
x\not=y}\frac{|v(x)-v(y)|}{|x-y|^\gamma}.
\end{split}$$
\[hdgwsssawwsd7\] Assume that $n\leq 2$, $\Omega$ is convex, $g(.,0)\in L^2(\Omega)$, $\partial_t g\in L^2(0,T, H^{-1}(\Omega))$, $g\in L^p(\Omega_T)$, $\partial_t a\equiv 0$, $\beta_\sigma<1$, $(\Tr[\sigma])^{-1}\in
L^\infty(\Omega_T)$ and $g\in L^2[0,T;L^{p^*}(\Omega)]$ with $2<p^*$. Then there exists $p\in (2,p^*]$ and $\gamma(p)>0$ such that $$\label{hdhsxdzssaqssgc7}
\begin{split}
\big(\int_0^T \big\|\nabla (u\circ
F^{-1})(.,t)\big\|_{C^{\gamma}(\Omega)}^2\,dt\big)^\frac{1}{2} \leq
&\frac{C }{1-\beta_\sigma^\frac{1}{2}}
\big(\|g\|_{L^p(\Omega_T)}\\&+\big\|g(.,0)\big\|_{L^2(\Omega)}+\|\partial_t
g\|_{L^2(0,T,H^{-1}(\Omega))}\big).
\end{split}$$
The constant $C$ in depends on $n$, $p$, $\Omega$, $\lambda_{\min}(a)$ and $\big\|(\Tr(\sigma))^{-1}\big\|_{L^\infty(\Omega_T)}$. It is easy to check that if $n=1$ then the theorem is valid with $\gamma=1/2$.
In the following theorems $\Omega$ is not assumed to be convex.
\[hdgwsswdswsd7\] Assume $n\geq 2$ and $\partial_t a\equiv 0$. Let $p>2$. There exist a constant $C^*=C^*(n,\partial \Omega)>0$ a real number $\gamma>0$ depending only on $n,\Omega$ and $p$ such that if $\beta_{\sigma}<C^*$ then $$\label{hdhsxdznnnnsgxc7}
\begin{split}
\big(\int_0^T \big\|\nabla (u\circ
F^{-1})(.,t)\big\|_{C^{\gamma}(\Omega)}^2\,dt\big)^\frac{1}{2} \leq
C
\big(&\|g\|_{L^p(\Omega_T)}+\big\|g(.,0)\big\|_{L^2(\Omega)}\\&+\|\partial_t
g\|_{L^2(0,T,H^{-1}(\Omega))}\big).
\end{split}$$
The constant $C$ in depends on $n$, $\gamma$, $\Omega$, $C^*$, $\lambda_{\min}(a)$ and $\big\|(\Tr(\sigma))^{\frac{n}{2p}-1}\big\|_{L^\infty(\Omega_T)}$.
It is easy to check that if $a=e(x)S(x,t)$ where $e$ is a time independent symmetric uniformly elliptic matrix with $L^\infty(\Omega)$ entries and $S$ is a regular uniformly positive function then the results given in this sub-section and the homogenization schemes of sub-section \[sub2\] remain valid with the time independent harmonic coordinates associated to $e$, i.e. solution of $-\diiv e \nabla F=0$.
### Medium with a continuum of time scales.
In this subsection the entries of $a$ are merely in $L^\infty(\Omega_T)$. We need to introduce the following Cordes type condition.
\[slkssjk88271\] We say that condition \[slkssjk88271\] is satisfied if and only if there exists $\delta\in (0,\infty)$ and $\epsilon>0$ such that $$\esssup_{\Omega_T}
\frac{\delta^2\Tr[{^t\sigma\sigma}]+1}{\big(\delta\Tr[\sigma]+1\big)^2}\leq
\frac{1}{n+\epsilon}.$$
Write $$z_\sigma:=\esssup_{\Omega_T} n \frac{\Tr[{^t\sigma
\sigma}]}{(\Tr[\sigma])^2}.$$ Observe that $z_\sigma$ is a measure of anisotropy of $\sigma$, in particular $1\leq z_\sigma \leq n$ and $z_\sigma=1$ if $\sigma$ is isotropic. Write $$y_\sigma:=\|\Tr[\sigma]\|_{L^\infty(\Omega_T)}\big\|(\Tr[\sigma])^{-1}\big\|_{L^\infty(\Omega_T)}.$$
\[kwjdjwj7\] If $\|\Tr[\sigma]\|_{L^\infty(\Omega_T)}<\infty$ and $\big\|(\Tr[\sigma])^{-1}\big\|_{L^\infty(\Omega_T)}<\infty$ then condition \[slkssjk88271\] is satisfied with $$\delta:=n \big\|(\Tr[\sigma])^{-1}\big\|_{L^\infty(\Omega_T)}$$ and with $\epsilon:=\frac{2ny_\sigma-n}{2n y^2_\sigma}$ provided that $z_\sigma \leq 1+\frac{\epsilon}{n}$.
Observe that in dimension one $z_\sigma=1$, thus for $n=1$ condition \[slkssjk88271\] is satisfied is $\Tr[\sigma]\in
L^\infty(\Omega_T)$ and $(\Tr[\sigma])^{-1}\in L^\infty(\Omega_T)$.
We have the following theorems
\[skjhsj823\] Assume that $\Omega$ is convex, and condition \[slkssjk88271\] is satisfied then $u\circ F^{-1}\in
L^2\big(0,T,W^{2,2}_D(\Omega)\big)$, $\partial_t (u\circ F^{-1})\in
L^2(\Omega_T)$ and $$\|u\circ F^{-1}\|_{L^2(0,T,W^{2,2}_D(\Omega))}+\|\partial_t(u\circ
F^{-1})\|_{L^2(\Omega_T)}\leq C \|g\|_{L^2(\Omega_T)}$$ where $C$ depends on $\Omega$, $n$, $\delta$ and $\epsilon$.
According to theorem \[skjhsj823\] although the second order space derivatives and first order time derivatives of $u$ with respect to Euclidean coordinates are only in $L^2(0,T,H^{-1}(\Omega))$, they are in $L^2(\Omega_T)$ with respect to harmonic coordinates.
Similarly we obtain the following theorems in situations where $g\in
L^p(\Omega_T)$ with $p>2$.
\[skjhsjd823\] Assume that $\Omega$ is convex, and condition \[slkssjk88271\] is satisfied then there exists a number $p_0>2$ depending on $n,\Omega,\epsilon$ such that for $p\in (2,p_0)$, $u\circ F^{-1}\in
L^p\big(0,T,W^{2,p}_D(\Omega)\big)$, $\partial_t (u\circ F^{-1})\in
L^p(\Omega_T)$ and $$\|u\circ F^{-1}\|_{L^p(0,T,W^{2,p}_D(\Omega))}+\|\partial_t(u\circ
F^{-1})\|_{L^p(\Omega_T)}\leq C \|g\|_{L^p(\Omega_T)}$$ where $C$ depends on $\Omega$, $n$, $\delta$ and $\epsilon$.
\[skjhsssej823\] Assume that $\Omega$ is convex, and condition \[slkssjk88271\] is satisfied then there exists a number $\alpha_0>2$ depending on $n,\Omega,\epsilon$ such that for $\alpha \in (0,\alpha_0)$, $\nabla
(u\circ F^{-1})\in L^2(0,T,C^\alpha(\Omega))$ and $$\|\nabla (u\circ F^{-1})(.,t)\|_{L^2(0,T,C^\alpha(\Omega))}\leq C
\|g\|_{L^p(\Omega_T)}$$ where $C$ depends on $\Omega$, $\delta$, $n$, and $\epsilon$.
These compensation phenomena can be observed numerically. We consider in dimension $n=2$, $$\label{ksjhskjhdhkdjh872y2}
\begin{split}
a(x,y,t)=\frac{1}{6}(\sum_{i=1}^5 \frac{1.1+\sin(2\pi
x'/\epsilon_{i})}{1.1+\sin(2\pi
y'/\epsilon_{i})}+\sin(4x'^{2}y'^{2})+1)
\end{split}$$ with $x'=x+\sqrt{2}t$, $y'=y-\sqrt{2}t$, $\epsilon_{1}=\frac{1}{5}$, $\epsilon_{2}=\frac{1}{13}$, $\epsilon_{3}=\frac{1}{17}$, $\epsilon_{4}=\frac{1}{31}$ and $\epsilon_{5}=\frac{1}{65}$. This medium has been plotted in figure \[mediap4\] at time $0$ (observe that $\lambda_{\max}(a)/\lambda_{\min}(a)\sim 100$).
has been solved numerically on that mesh with $g\equiv
1$ on the fine mesh characterized by $16641$ nodes and $32768$ triangles. Figure \[unut03p4\] shows $\partial_x u$ and $\partial_x (u\circ F^{-1})$ at time $0.3$.
[ ]{}\
In figure \[fixpointu00\] and \[fixpointu\], the value of $x_0$ is set to $x_0:=(0.75,-0.25)$ and the curves $t\rightarrow u(x_0,t),
u\circ F^{-1}(x_0,t), \nabla u(x_0,t),\nabla u\circ F^{-1}(x_0,t)$ are plotted from $t=0$ to $t=0.3$
[ ]{}\
Homogenization in space. {#sub2}
------------------------
Let $X_h$ be a finite dimensional subspace of $H^1_0(\Omega)\cap
W^{1,\infty}(\Omega)$[^3] with the following approximation property: there exists a constant $C_X$ such that for all $f\in W^{2,2}_D(\Omega)$
$$\label{approp}
\inf_{v\in X_h} \|f-v\|_{H^1_0(\Omega)}\leq C_X h
\|f\|_{W^{2,2}_D(\Omega)}.$$
It is known and easy to check that the set of piecewise linear functions on a triangulation of $\Omega$ satisfies condition provided that the length of the edges of the triangles are bounded by $h$ ($C_X$ in being given by the aspect ratio of the triangles).
For media characterized by a continuum of time scales we will consider twice differentiable elements satisfying the following usual inverse inequalities (see section 1.7 of [@ErGu04]): for $v\in
X_h$, $$\label{appsrop3}
\|v\|_{W^{2,2}_D(\Omega)}\leq
C_X h^{-1} \|v\|_{H^1_0(\Omega)}.$$ and $$\label{appssswerop3}
\| v\|_{H^1_0(\Omega)}\leq
C_X h^{-1} \|v\|_{L^2(\Omega)}.$$ In this paper we will use splines to ensure that condition is satisfied (observe that it requires the quasi-uniformity of the (coarse) mesh, i.e. a bound on the aspect ratio of the (coarse) triangles).
For $t\in (0,T)$ let us define $$V_h(t):=\big\{\varphi \circ F(x,t)\, :\, \varphi \in X_h\big\}.$$ Write $L^2\big(0,T;H^1_0(\Omega)\big)$ the usual Sobolev space associated to the norm $$\|v\|_{L^2(0,T;H^1_0(\Omega))}^2:=\int_0^T
\big\|v(.,t)\big\|_{H^1_0(\Omega)}^2\,dt.$$ Write $Y_T$ the subspace of $L^2\big(0,T;H^1_0(\Omega)\big)$ such that for each $v \in Y_T$ and $t\in [0,T]$, $x\rightarrow v(x,t)$ belongs to $V_h(t)$.\
Write $u_h$ the solution in $Y_T$ of the following system of ordinary differential equations:
$$\label{ghjbbfh52}
\begin{cases}
(\psi,\partial_t u_h)_{L^2(\Omega)}+a[\psi,u_h]=(\psi,g)_{L^2(\Omega)}\quad \text{for all $t\in (0,T)$ and $\psi \in V_h(t)$} \\
u_h(x,0)=0.
\end{cases}$$
Write $$\label{gaz52}
a[v,w]:=\int_\Omega {^t\nabla v(x,t)}a(x,t)\nabla w(x,t)\,dx.$$
### Time independent domain
We have the following theorem
\[slsakdeswj8kksjd23\] Assume that $\partial_t a \equiv 0$, $\Omega$ is convex, $\beta_{\sigma}<1$ and $(\Tr[\sigma])^{-1}\in L^\infty(\Omega_T)$ then $$\label{sjswhsawdskjdkjxwsdlswe}
\big\|(u-u_h)(.,T)\big\|_{L^2(\Omega)}+\big\|u-u_h\big\|_{L^2(0,T;H^1_0(\Omega))}
\leq C h \|g\|_{L^2(\Omega_T)}.$$
The constant $C$ depends on $C_X$, $n$, $\Omega$, $\lambda_{\min}(a)$ and\
$\big\|(\Tr[\sigma])^{-1}\big\|_{L^\infty(\Omega_T)}$. If $n\geq 5$ it also depends on $\big\|\Tr[\sigma]\big\|_{L^\infty(\Omega_T)}$ and if $n=1$ it also depends on $\lambda_{\max}(a)$.
### Medium with a continuum of time scales.
Assume that $\Omega$ is convex, and condition \[slkssjk88271\] is satisfied then $$\label{ghhjaawssjncssbbhfhs52az}
\begin{split}
\big\|(u-u_h)(T)\big\|_{L^2(\Omega)}+\big\|u-u_h\big\|_{L^2(0,T,H^1_0(\Omega))}\leq
C h \|g\|_{L^2(\Omega_T)}.
\end{split}$$
The constant $C$ depends on $C_X$, $n$, $\Omega$, $\delta$ and $\epsilon$, $\lambda_{\min}(a)$ and $\lambda_{\max}(a)$.
The system of ordinary differential equations is still characterized by a continuum of time scales in situations where the entries of $a$ merely belong to $L^\infty(\Omega_T)$. They need to be discretized (homogenized) in time in order to be solved numerically. This will be the object of the next subsection. Loosely speaking, although is associated to a fine tessellation and fine time steps, it is possible to approximate its operator on a coarse tessellation with coarse time steps.
Homogenization in space and time. {#jksjhs89b}
---------------------------------
Let $M\in \N^*$. Let $(t_n=n \frac{T}{M})_{0\leq n\leq M}$ be a discretization of $[0,T]$. Let $(\varphi_i)$ be a basis of $X_h$. Write $Z_T$ the subspace of $Y_T$ such that $w\in Z_T$ if and only if $w$ can be written $$w(x,t)=\sum_{i} c_i(t) \varphi_i(F(x,t)).$$ and the functions $t\rightarrow c_i(t)$ are constants on each intervals $(t_n,t_{n+1}]$. Write $V$ the subspace of $Y_T$ such that its elements $\psi$ can be written $$\psi(x,t)=\sum_{i} d_i \varphi_i(F(x,t)).$$ where the parameters $d_i$ are constants (on $[0,T]$). For $w\in
Y_T$, define $w_n \in V$ by $$w_n(x,t):=\sum_{i} c_i(t_n) \varphi_i(F(x,t)).$$ Write $v$ the solution in $Z_T$ of the following system of implicit ordinary differential equations (such that $v(x,0)\equiv 0$): for $n\in \{0,\ldots,M-1\}$ and $\psi\in V$, $$\label{ghjddwsdcszdbbsfh52}
\begin{split}
\big(\psi(t_{n+1}),
v_{n+1}(t_{n+1})\big)_{L^2(\Omega)}=&\big(\psi(t_{n}),
v_{n}(t_{n})\big)_{L^2(\Omega)}\\&+\int_{t_n}^{t_{n+1}}\Big(\big(\partial_t\psi(t),
v_{n+1}(t)\big)_{L^2(\Omega)}\\&-a\big[\psi(t),v_{n+1}(t)]+\big(\psi(t),g(t)\big)_{L^2(\Omega)}
\Big)\,dt.
\end{split}$$
The following theorem shows the stability of the implicit scheme .
\[sshjhsjjs823\] Let $v\in Z_T$ be the solution of . We have $$\label{ghnnjddsdmmsswsdcszddsbbsfh52}
\begin{split}
\big\|v(T)\big\|_{L^2(\Omega)}+\|v\|_{L^2(0,T,H^1_0(\Omega))}
\leq C \|g\|_{L^2(\Omega_T)}.
\end{split}$$
The constant $C$ depends on $n$, $\Omega$ and $\lambda_{\min}(a)$.
The following theorem gives an error bound on the accuracy of time discretization scheme when $a$ does not depend on time.
\[jskjdhdcxjdh723\] Let $v\in Z_T$ be the solution of and $u_h$ be the solution of . Assume that $\partial_t
a\equiv 0$. We have $$\label{gazoszwscxbbfmbh52}
\begin{split}
\big\| (u_h-v)(T)\big\|_{L^2(\Omega)}+
&\|u_h-v\|_{L^2(0,T,H^1_0(\Omega))} \leq C |\Delta t|
\\&\Big(\|\partial_t
g\|_{L^2(0,T,H^{-1}(\Omega))}+\big\|g(.,0)\big\|_{L^2(\Omega)}\Big).
\end{split}$$
The constant $C$ depends on $n$, $\Omega$ and $\lambda_{\min}(a)$.
The following theorem gives an error bound on the accuracy of the time discretization scheme when $a$ has no bounded time derivatives.
\[skjhseeeddfnmnddsf23\] Assume that $\Omega$ is convex, and condition \[slkssjk88271\] is satisfied. Let $v\in Z_T$ be the solution of and $u_h$ be the solution of , we have $$\label{ghjdzmmssasdssabfh52}
\begin{split}
\big\| (u_h-v)(T)\big\|_{L^2(\Omega)}+
\|u_h-v\|_{L^2(0,T,H^1_0(\Omega))} \leq C \frac{|\Delta t|}{h}
\|g\|_{L^2(\Omega_T)}
\end{split}$$ where $C$ depends on $\Omega$, $n$, $\delta$, $\epsilon$, $\lambda_{\min}(a)$ and $\lambda_{\max}(a)$.
Observe that the accuracy of the time discretization scheme requires that $|\Delta t|<<h$ when $a$ has no bounded time derivatives.
We refer to section \[ksjsskhs821\] for numerical experiments.
Literature and further remarks.
-------------------------------
For early works on homogenization with random mixing coefficients we refer to [@PaVa83], [@KiVa86], [@PaVa79], [@MR0383530], [@ZhKoOl79], [@Ko85], [@Ko87], [@MR1429379]. Papanicolaou and Varadhan [@MR0461684] have considered a two-component Markov process $(x(t), y(t))$ where $y(t)$ is rapidly varying (and is not assumed to be ergodic in dimension one) and enters in the coefficients of the stochastic process driving $x(t)$. They have studied the convergence properties of $x(t)$ as the fluctuations of $y(t)$ becomes more rapid using the martingale approach to diffusion, developed by Stroock and Varadhan [@MR0359025], [@MR0359024], [@MR532498], [@MR0410912].
The numerical homogenization method implemented in this paper is a finite element method. The idea of using oscillating tests functions can be back tracked to the work of Murat and Tartar on homogenization and H-convergence, we refer in particular to [@MR557520] and [@MR1493039]. Those papers also contain convergence proofs for the finite element method in an abstract setting for a sequence of $H$-converging elliptic operators (recall that the framework of H-convergence is independent from ergodicity or scale separation assumptions and are based on the compactness of any sequence of solutions of $-\diiv a_\epsilon \nabla u_\epsilon=g$ with uniformly bounded and elliptic conductivities $a_\epsilon$, we also refer to the initial work of Spagnolo [@MR0240443] for G-convergence).
The numerical implementation and practical application of oscillating test functions in numerical finite element homogenization have been called multi-scale finite element methods and have been studied by several authors [@MR1286212], [@MR1740386], [@MR1613757], [@MR1455261], [@MR2123115], [@MR1232956], [@MR1194543], [@AlBr05]. The work of Hou and Wu [@MR1455261] has been a large source of inspiration in numerical applications (particularly for reservoir modeling in geophysics, we refer to [@MR1898136], [@MR1956022], [@MR2111701] and [@HouEf05] for recent developments) since it was leading to a coarse scale operator while keeping the fine scale structures of the solutions. With the method introduced by Hou and Wu, the construction of the base functions is decoupled from element to element leading to a scheme adapted to parallel computers. A proof of the convergence of the method is given in periodic settings when the size of the heterogeneities is smaller than the grid size and an ‘oversampling technique’ is proposed to remove the so called cell resonance error [@MR2119937] when the size of the heterogeneities is comparable to the grid size.
Allaire and Brizzi [@AlBr04] have observed that multiscale finite element method with splines would have a higher accuracy and have introduced the composition rule (we also refer to [@MR1286212]). In [@OwZh05], it has been observed that if $u$ is the solution of the divergence form elliptic equation $$\label{ghjh5nsx2}
\begin{cases}
-\diiv\big(a(x)\nabla u(x)\big)=g\quad \text{in}\quad \Omega \\
u=0 \quad \text{in}\quad \partial \Omega.
\end{cases}$$ and $F$ are harmonic coordinates defined by $$\label{dgdgfsghsf6bg2}
\begin{cases}
\diiv a \nabla F=0 \quad \text{in}\quad \Omega\\
F(x)=x \quad \text{on}\quad \partial \Omega.
\end{cases}$$ then under the Cordes type condition $\beta_{\sigma}<1$ on $\sigma$ given by , one has for some $p>2$. $$\|u\circ F^{-1}\|_{W^{2,p}(\Omega)}\leq C \|g\|_{L^p(\Omega)}.$$ It has been deduced from this compensation phenomenon that numerical homogenization methods based on oscillating finite elements can converge in the presence of a continuum of scales if one uses global harmonic coordinates to obtain the test functions instead of solutions of a local cell problem [@OwZh05]. In dimension three and higher it has been known since the work of Fenchenko and Khruslov [@MR615994], [@MR1145750] that the homogenization of divergence form elliptic operators $-\diiv a_\epsilon \nabla
u_\epsilon=g$ can lead to a non local homogenized operator if the sequence of matrices $a_\epsilon$ is uniformly elliptic but with entries uniformly bounded only in $L^1(\Omega)$. From a numerical point of view this non-local effects imply that a nonlocal numerical homogenization method cannot be avoided to obtain accuracy. Hence in [@OwZh05], it is shown that the accuracy of local methods depend on the aspect ratio of the triangles of the tessellation with respect to harmonic coordinates (which is not the case if one uses non local finite elements, we refer to [@OwZh05] for further discussions on the apparition of non local effects in numerical homogenization). Recently Briane has shown [@Bri05] that this non-local effect is absent in dimension two in the H-convergence setting.
The phenomenon is similar here, however observe that if one has solved the initial parabolic equation at least $n$ times and those solutions are (locally) linearly independent it is also possible to use them as new coordinates for numerical homogenization. Observe that in dimension higher than three the harmonic coordinates are not always invertible, an idea to bypass this difficulty could be either to choose the change of coordinates locally and adaptively or to enrich the coordinates by writing down the initial equations as degenerate equations in a space of higher dimension [@Var05], these points have not been explored. For divergence form elliptic equations, recall that fast methods based on hierarchical matrices[^4] are available [@MR1993936; @BeCh05; @Beb05; @Bebe05; @Beben05] for solving and in $O\big(N (\ln
N)^{n+3}\big)$ operations ($N$ being the number of interior nodes of the fine mesh).
The issue of numerical homogenization partial differential equations with heterogeneous coefficients has received a great deal of attention and many methods have been proposed. A few of them are cited below.
- Multi-scale finite volume methods [@JLP03].
- Heterogeneous Multi-scale Methods [@EV04], [@MR2164241].
- Wavelet based homogenization [@MR1492791], [@MR1618846], [@LC04], [@MR1614457], [@MR1614980], [@MR1354913].
- Residual free bubbles methods [@MR2006324].
- Discontinuous enrichment methods [@MR2007030], [@MR1870426].
- Partition of Unity Methods [@FY05].
- Energy Minimizing Multi-grid Methods [@MR1756048].
Following the methods of [@OwZh05], it is possible to implement a finite-volume method based on the compensation theorems given in this paper. The elements given in this paper contain the fine scale structure of $F$, as it has been done in [@OwZh05], it is possible to approximate the initial parabolic operator by a homogenized parabolic operator associated to the coarse mesh (the test functions in this case would be piecewise linear on the coarse mesh and the approximation error associated to the homogenized operator would depend on the aspect ratio of the triangles of the coarse mesh in the metric induced by $F$).
Finally, in this paper $a$ has been assumed to be bounded and uniformly elliptic. Without these assumptions the diffusion associated to homogenized operator can be anomalously slow [@BeOw00b], [@Ow00a] or fast (super-diffusive) [@Ow04]. If $a$ has an unbounded skew symmetric component, the homogenization of can give rise to a degenerate operator [@Ow04].
Proofs
======
Compensation.
-------------
### Time independent medium.
We will need the following lemmas. Let $\A_T$ be the bilinear form on $L^2\big(0,T;H^1_0(\Omega)\big)$ defined by $$\A_T[v,w]:=\int_0^T a[v,w](t) \,dt$$ where $$a[v,u](t):=\int_\Omega {^t\nabla v}(x,t)a(x,t)\nabla u(x,t)\,dx.$$ We write $\A_T[u]:=\A_T[u,u]$.
\[sjhsgh8\] We have $$\label{eqra1}
\big\|u(.,T)\big\|_{L^2(\Omega)}^2+ \A_T[u]\leq
\frac{C_{n,\Omega}}{\lambda_{\min}(a)} \|g\|_{L^2(\Omega_T)}^2.$$
Multiplying by $u$ and integrating with respect to time we obtain that $$\label{gsesfdssw2}
\frac{1}{2}\big\|u(.,T)\big\|_{L^2(\Omega)}^2+
\A_T[u]=(u,g)_{L^2(\Omega_T)}.$$ Using Poincaré and Minkowksi inequalities leads us to .
\[jshsszskjsh61RED\] Assume $\partial_t a\equiv 0$. We have $$\label{sxdsgxssssxsesw2RED}
\begin{split}
\big\|\partial_t u\big\|_{L^2(\Omega_T)}^2+a\big[u(.,T)\big]\leq
\big\|g\big\|_{L^2(\Omega_T)}^2.
\end{split}$$
Multiplying by $\partial_t u$ and integrating by parts we obtain that $$\label{ghjbsanjxzassxxsxch52RED}
\begin{split}
\big\|\partial_t u(.,t)\big\|_{L^2(\Omega)}^2+a[\partial_t
u,u]=(\partial_t u, g)_{L^2(\Omega)}.
\end{split}$$ Observing that $$\label{ghjbsanjxzasswsxxsxch52RED}
\begin{split}
a[\partial_t u,u]=\frac{1}{2}\partial_t
\big(a[u]\big)-\frac{1}{2}\int_{\Omega}{^t\nabla u \partial_t a
\nabla u}.
\end{split}$$ we conclude by integration with respect to time and using Minkowski inequality.
\[jshsszskjsh61\] Assume $\partial_t a \equiv 0$. We have $$\label{sxdsgxssssxsesw2}
\begin{split}
\big\|\partial_t u(.,T)\big\|_{L^2(\Omega)}^2+\A_T[\partial_t u]\leq
& \frac{C_{n,\Omega}}{\lambda_{\min}(a)} \|\partial_t
g\|_{L^2(0,T,H^{-1}(\Omega))}^2+\big\|g(.,0)\big\|_{L^2(\Omega)}^2.
\end{split}$$
We obtain from that $$\label{gchudyhjh52}
\begin{split}
\partial_t^2 u=\diiv\big(a(x,t)\nabla \partial_t u(x,t)\big)+\diiv\big(\partial_t a(x,t)\nabla u(x,t)\big) +\partial_t
g.
\end{split}$$ Multiplying by $\partial_t u$ and integrating with respect to time we obtain that $$\label{ghjbsanjxzassxxsxch52p}
\begin{split}
\frac{1}{2}\big\|\partial_t
u(.,T)\big\|_{L^2(\Omega)}^2+\A_T[\partial_t u]=&\int_0^T
(\partial_t u,\partial_t g)_{L^2(\Omega)}dt-\int_0^T (\partial_t
a)\big[\partial_t u, u\big]\,dt\\&+\frac{1}{2}\big\|\partial_t
u(.,0)\big\|_{L^2(\Omega)}^2.
\end{split}$$ We conclude by the $H^{-1}$-duality inequality and Minkowski inequality.
We now need a variation of Campanato’s result [@CM5] on non-divergence form elliptic operators. Let us write for a symmetric matrix $M$, $$\nu_M:=\frac{\Tr(M)}{\Tr({^tM M})}.$$ We consider the following Dirichlet problem: $$\label{dcaqssaslkwaq21}
L_M v=f$$ with $L_M:=\sum_{i,j=1}^n M_{ij}(x) \partial_i \partial_j$. The following theorems \[hdgjhsdswasgd7\] and \[hdgjhdgd7\] are straightforward adaptations of theorem 1.2.1 of [@MPG00]. They are proven in [@MPG00] under the assumption that $M$ is bounded and elliptic. It is easy to check that the conditions $\beta_{M}<1$ and $\nu_M\in L^\infty(\Omega)$ are sufficient for the validity of those theorems. We refer to [@OwZh05] for that adaptation.
\[hdgjhsdswasgd7\] Assume that $\beta_M<1$, $\nu_M\in L^\infty(\Omega)$ and $\Omega$ is convex. Then if $f\in L^2(\Omega)$ the Dirichlet problem has a unique solution satisfying $$\label{hdhazdgc7}
\|v\|_{W^{2,2}_D(\Omega)}\leq \frac{C }{1-\beta_M^\frac{1}{2}}
\|\nu_M f\|_{L^2(\Omega)}.$$
$\beta_M$ is the Cordes parameter associated to $M$.
\[hdgjhdgd7\] Assume that $\beta_M<1$, $\nu_M\in L^\infty(\Omega)$ and $\Omega$ is convex. Then, there exists a real number $p_0>2$ depending only on $n,\Omega$ and $\beta_M$ such that for each $f\in L^p(\Omega)$, $2\leq p<p_0$ the Dirichlet problem has a unique solution satisfying $$\label{hdhdgc7}
\|v\|_{W^{2,p}_D(\Omega)}\leq \frac{C_{n,\Omega,p}
}{1-\beta_M^\frac{1}{2}} \|\nu_M f\|_{L^p(\Omega)}.$$
Let us now prove the compensation theorems. Choose $$\label{ksjshsj7622}
M:=\frac{\sigma}{|\det(\nabla F)|^\frac{1}{2}}\circ F^{-1}.$$ It is easy to check that $\beta_\sigma<1$ implies that $F$ is an homeomorphism from $\Omega$ onto $\Omega$, thus is well defined. Moreover observe that $\beta_M=\beta_\sigma$ and $$\label{hdhxcszswc7}
\|\nu_M\|_{L^\infty(\Omega_T)}^2 \leq
\frac{C_n}{(\lambda_{\min}(a))^\frac{n}{2}}
\big\|(\Tr[\sigma])^{\frac{n}{4}-1}\big\|^2_{L^\infty(\Omega_T)}.$$ Fix $t\in [0,T]$. Choose $$\label{ghjhssalzz52}
\begin{split}
f:=\frac{(\partial_t u - g)}{|\det (\nabla F)|^\frac{1}{2}}\circ
F^{-1}.
\end{split}$$ Observe that by the change of variable $y=F(x)$ one obtains that if $\partial_t a\equiv 0$ (which implies that $F$ is time independent), $\partial_t u \in L^2(\Omega)$ and $g(.,t)\in L^2(\Omega)$ that $f\in L^2(\Omega)$ and $$\label{ghjhssadsfalzz52}
\begin{split}
\|f\|_{L^2(\Omega)}\leq \|\partial_t
u\|_{L^2(\Omega)}+\|g\|_{L^2(\Omega)}.
\end{split}$$ It follows from theorem \[hdgjhsdswasgd7\] that there exists a unique $v\in W^{2,2}_D(\Omega)$ satisfying $$\label{hdhsszdgc7}
\|v\|_{W^{2,2}_D(\Omega)}^2\leq
\frac{C\|\nu_M\|_{L^\infty(\Omega_T)}^2
}{(1-\beta_\sigma^\frac{1}{2})^2} \big(\|\partial_t
u\|_{L^2(\Omega)}^2+\|g\|_{L^2(\Omega)}^2\big).$$ and the following equation $$\label{ghjhsxfazz52}
\begin{split}
\partial_t \hat{u}(y,t)=\sum_{i,j}\big(\sigma(F^{-1}(y,t),t)\big)_{i,j}
\partial_i\partial_j v(y,t)+\hat{g}(y,t).
\end{split}$$ We use the notation $\hat{g}:=g\circ F^{-1}$ and $\hat{u}:=u\circ
F^{-1}$. Using the change of variable $y=F(x)$ and using the property $\diiv a \nabla F=0$ when $\partial_t a\equiv 0$ we obtain that can be written $$\label{ghjhsxfsaazsdz52j}
\begin{split}
\partial_t u=\diiv \big(a \nabla (v\circ F)\big)+g.
\end{split}$$ If $\partial_t u \in L^2(\Omega)$ and $g(.,t)\in L^2(\Omega)$ we can use the uniqueness property of the solution of the divergence form elliptic Dirichlet problem $$\label{ghjhsxfsaazsdz52}
\begin{split}
\diiv \big(a \nabla w\big)=\partial_t u -g.
\end{split}$$ to obtain that $v\circ F=u$. Thus using lemma \[jshsszskjsh61\] we have proven theorem \[ksjshszjscxsxsdded8721\]. Moreover assume that $g\in L^2(\Omega_T)$ and $\partial_t u\in L^2(\Omega_T)$. It follows that for $t\in [0,T]-B$, $g(.,t)\in L^2(\Omega)$ and $\partial_t u(.,t)\in L^2(\Omega)$ where $B$ is a subset of $[0,T]$ of $0$-Lebesgue measure. It follows from the previous arguments that on $[0,T]-B$, $u\circ F^{-1}(.,t) \in W^{2,2}_D(\Omega)$ and satisfies $$\label{hdhsszwweeddgc7}
\|u\circ F^{-1}(.,t)\|_{W^{2,2}_D(\Omega)}^2\leq
\frac{C\|\nu_M\|_{L^\infty(\Omega_T)}^2
}{(1-\beta_\sigma^\frac{1}{2})^2} \big(\|\partial_t
u(.,t)\|_{L^2(\Omega)}^2+\|g(.,t)\|_{L^2(\Omega)}^2\big).$$ Integrating with respect to time we obtain that\
$u\circ F^{-1}\in L^2(0,T, W^{2,2}_D(\Omega))$ and $$\label{hdhsszwxweeddgc7}
\|u\circ F^{-1}\|_{L^2(0,T, W^{2,2}_D(\Omega))}^2\leq
\frac{C\|\nu_M\|_{L^\infty(\Omega_T)}^2
}{(1-\beta_\sigma^\frac{1}{2})^2} \big(\|\partial_t
u\|_{L^2(\Omega_T)}^2+\|g\|_{L^2(\Omega_T)}^2\big).$$ Thus using lemma \[jshsszskjsh61RED\] we have obtained theorem \[ksjshjsxxsd8721\].
Let us now prove theorem \[ksjshjsxgghxsd8721\]. Assume that there exists $q_0>2$ such that for $2\leq p<q_0$, $\partial_t u \in
L^p(\Omega_T)$ and $g\in L^p(\Omega_T)$. Let us now apply theorem \[hdgjhdgd7\] with $p<\min(p_0,q_0)$, $M$ given by and $f$ given by . It follows that for $t\in [0,T]-B$ (where $B$ is a subset of $[0,T]$ of $0$-Lebesgue measure), $g(.,t)\in L^p(\Omega)$ and $\partial_t
u(.,t)\in L^p(\Omega)$. We deduce from theorem \[hdgjhdgd7\] and the argumentation related to equation that on $[0,T]-B$, $u\circ F^{-1}(.,t)\in W^{2,p}_D(\Omega)$ and $$\label{hdhssdwedeesewddgc7}
\|u\circ F^{-1}(.,t)\|_{W^{2,p}_D(\Omega)}^p\leq
\frac{C_{n,p,\Omega}\|\nu_M\|_{L^\infty(\Omega_T)}^p
}{(1-\beta_\sigma^\frac{1}{2})^p} \big(\|\partial_t
u(.,t)\|_{L^p(\Omega)}^p+\|g(.,t)\|_{L^p(\Omega)}^p\big).$$ Integrating with respect to time we obtain that\
$u\circ F^{-1}\in L^p(0,T,W^{2,p}_D(\Omega))$ and
$$\label{hdhssssesewddgc7}
\|u\circ F^{-1}\|_{L^p(0,T,W^{2,p}_D(\Omega))}\leq
\frac{C_{n,p,\Omega}\|\nu_M\|_{L^\infty(\Omega_T)}
}{1-\beta_\sigma^\frac{1}{2}} \big(\|\partial_t
u\|_{L^p(\Omega_T)}+\|g\|_{L^p(\Omega_T)}\big).$$
It remains to show that under the assumptions of theorem \[ksjshjsxgghxsd8721\], $\partial_t u \in L^p(\Omega_T)$.
In order to bound $\big\|\partial_t u(.,t)\big\|_{L^p(\Omega)}$ we use general Sobolev inequalities (chapter 5.6 of [@Evans97]).
- If $n\geq 3$, write $p^*=2n/(n-2)$. We have for $2<p\leq p^*$, $$\label{jskjsdh83}
\big(\int_{\Omega}(\partial_t u )^{p} \,dx\big)^\frac{2}{p} \leq
C_{n,\Omega} \big(\int_{\Omega}(\partial_t u )^{p^*}
\,dx\big)^\frac{2}{p^*}$$ thus, using Gagliardo-Nirenberg-Sobolev inequality $$\label{hjsshsghs61}
\big(\int_{\Omega}(\partial_t u )^{p} \,dx\big)^\frac{2}{p} \leq
C_{n,p,\Omega} \frac{1}{\lambda_{\min}(a)} a[\partial_t u].$$
- If $n=2$, we write for $(x_1,x_2,x_3)\in \Omega\times (0,1)$, $v(x_1,x_2,x_3):=\partial_t
u(x_1,x_2)$. Using Gagliardo-Nirenberg-Sobolev inequality in dimension three we obtain that for $2<p\leq 6$ $$\big(\int_{\Omega}(\partial_t u )^{p} \,dx\big)^\frac{2}{p} \leq
C_{n,p,\Omega} \int_{\Omega}(\nabla \partial_t u )^{2} \,dx.$$ Which leads us to .
- If $n=1$ then using Morrey’s inequality we obtain that with $\gamma:=1/2$, $$\|\partial_t u\|_{C^{0,\gamma}(\Omega)}^2 \leq C_{\Omega}
\frac{1}{\lambda_{\min}(a)} a[\partial_t u].$$
We conclude the proof of theorem \[ksjshjsxgghxsd8721\] by using lemma \[jshsszskjsh61\].
We deduce theorem \[hdgwsssawwsd7\] from Morrey’s inequality and theorem \[ksjshjsxgghxsd8721\].
#### Hölder continuity for $n\geq 3$ or non-convexity of $\Omega$.
In this paragraph we will not assume $\Omega$ to be convex. Let $N^{p,\lambda}(\Omega)$ $(1<p<\infty,\, 0<\lambda<n$) be the weighted Morrey space formed by the functions $v:\Omega\rightarrow \R$ such that $\|v\|_{N^{p,\lambda}(\Omega)}<\infty$ with $$\|v\|_{N^{p,\lambda}(\Omega)}:=\sup_{x_0\in
\Omega}\Big(\int_{\Omega}|x-x_0|^{-\lambda} |v(x)|^p
\Big)^\frac{1}{p}.$$ To obtain the Hölder continuity of $u\circ F^{-1}$ in dimension $n\geq 3$ we use corollary 4.1 of [@MR1903306]. We give the result of S. Leonardi below in a form adapted to our context. Consider the Dirichlet problem . We do not assume $\Omega$ to be bounded. We write $W^{2,p,\lambda}(\Omega)$ the functions in $W^{2,p}_D(\Omega)$ such that their second order derivatives belong to $N^{p,\lambda}(\Omega)$.
\[ksjhs721\] There exist a constant $C^*=C^*(n,p,\lambda,\partial \Omega)>0$ such that if $\beta_M <C^*$ and $f\in N^{p,\lambda}(\Omega)$ then the Dirichlet problem has a unique solution in $W^{2,p,\lambda}\cap W^{1,p}_0(\Omega)$. Moreover, if $0<\lambda
<n<p$ then $\nabla v \in C^{\alpha}(\Omega)$ with $\alpha=1-n/p$ and $$\|\nabla v\|_{C^\alpha(\Omega)}\leq
\frac{C}{\lambda_{\min}(M)}\|f\|_{N^{p,\lambda}(\Omega)}$$ where $C=C(n,p,\lambda,\partial \Omega)$.
The proof of theorem \[hdgwsswdswsd7\] is an application of theorem \[ksjhs721\]. We just need to observe that from Hölder inequality we have for $0<\epsilon<0.5$ $$\|f\|_{N^{p,\epsilon}(\Omega)} \leq C_{n,p,\Omega,\epsilon}
\|f\|_{L^{p (1+\epsilon)}(\Omega)}.$$ From this point the proof is similar to the proof of theorem \[ksjshjsxgghxsd8721\].
### Medium with a continuum of time scales.
We will need theorems 1.6.2 and 1.6.3 of [@MPG00]. For the sake of completeness we will remind those theorems below in version adapted to our framework. Consider the following parabolic problem: $$\label{dcaqssssswaslkwaq21}
\partial_t v=\sum_{i,j=1}^n M_{ij}(x) \partial_i \partial_j v+ f.$$ We assume $M$ to be symmetric bounded and elliptic and $v=0$ at $t=0$ and on the boundary $\partial \Omega$. Write $$\eta_M:=\sup_{x\in \Omega_T}
\frac{\Tr[{^tMM}]+1}{\big(\Tr[M]+1\big)^2}.$$ and $$\alpha_M:=\sup_{x\in \Omega_T} \frac{\Tr[M]+1}{\Tr[{^tMM}]+1}.$$ Write for $p\geq 2$ $$S_p(\Omega_T):=\Big\{v\in L^p\big(0,T,W^{2,p}_D(\Omega)\big);
\partial_t v \in L^p(\Omega_T); v(.,0)\equiv 0\Big\}$$ and $$\|v\|_{S_p(\Omega_T)}^p:=\int_{\Omega_T}\big(\sum_{i,j}(\partial_i
\partial_j v)^2+(\partial_t v)^2\big)^\frac{p}{2}\,dy\,dt.$$
\[hgft5411\] Assume $\Omega$ to be convex and that there exists $\epsilon
\in(0,1)$ such that $\eta_M \leq 1/(n+\epsilon)$, then for each $f\in L^2(\Omega_T)$ the Cauchy-Dirichlet problem admits a unique solution in $S_2(\Omega_T)$ which satisfies the bound $$\|v\|_{S_2(\Omega_T)}\leq
\frac{\alpha_M}{1-\sqrt{1-\epsilon}}\|f\|_{L^2(\Omega_T)}.$$
\[hghjg6665542212\] Assume $\Omega$ to be convex and that there exists $\epsilon
\in(0,1)$ such that $\eta_M \leq 1/(n+\epsilon)$, then there exists a number $p_0>2$ depending on $\Omega,n,\epsilon$ such that for each $f\in L^p(\Omega_T)$ the Cauchy-Dirichlet problem admits a unique solution in $S_p(\Omega_T)$ which satisfies the bound $$\label{jshshsg651}
\|v\|_{S_p(\Omega_T)}\leq C_p
\frac{\alpha_M}{1-\sqrt{1-\epsilon}}\|f\|_{L^p(\Omega_T)}.$$
In fact theorem 1.6.3 of [@MPG00] is written with $1-C(p)\sqrt{1-\epsilon}$ in the denominator of but it is easy to modify it to obtain by lowering the value of $p_0$ changing the value of $C_p$.
Let $\delta >0$. Let us now apply theorem \[hgft5411\] on $[0,T/\delta ]$ with $$M:=\delta \sigma \circ F^{-1}(y,\delta t)$$ and $$f:=\delta (g\circ F^{-1})(y,\delta t).$$ Observe that if condition \[slkssjk88271\] is satisfied then $F$ is an homeomorphism and $M$ is well defined, bounded and elliptic. Moreover $\eta_M<\infty$ and $\alpha_M<\infty$ since $$\esssup_{\Omega_{\frac{T}{\delta}}}\frac{\Tr[{^tMM}]+1}{\big(\Tr[M]+1\big)^2}=\esssup_{\Omega_T}
\frac{\delta^2\Tr[{^t\sigma\sigma}]+1}{\big(\delta\Tr[\sigma]+1\big)^2}\,.$$ It follows that the following equation admits a unique solution in $S_2(\Omega_{\frac{T}{\delta}})$. $$\begin{split}
\partial_t w(y,t)=\sum_{i,j}M_{i,j}(y,t)
\partial_i\partial_j w(y,t)+k(y,t)
\end{split}$$ with $k(y,t)=\delta \hat{g}(y,\delta t)$. And we have
$$\int_0^\frac{T}{\delta}\int_{\Omega} \big((\partial_t
w)^2+\sum_{i,j}(\partial_i
\partial_j w)^2\big)\,dy\,dt\leq
\frac{C}{(1-\sqrt{1-\epsilon})^2}\|f\|_{L^2(\Omega_{\frac{T}{\delta}})}.$$
Using the change of variables $t\rightarrow \delta t$ and writing $$w(y,t):=v(y,\delta t).$$ we obtain that $v$ satisfies the following equation on $\Omega_T$ $$\label{ghjhdrsazz52}
\begin{split}
\partial_t v(y,t)=\sum_{i,j}\big(\sigma(F^{-1}(y,t),t)\big)_{i,j}
\partial_i\partial_j v(y,t)+\hat{g}(y,t).
\end{split}$$ Using the change of variable $y=F(x)$ and observing that $\partial_t
F=\diiv a\nabla F$ we obtain that $v\circ F$ satisfies $$\label{ghjhdrsdwazdz52}
\begin{split}
\partial_t (v\circ F)=\diiv \big(a\nabla (v\circ F)\big)+g.
\end{split}$$ It follows from the uniqueness of the solution of that $u=v\circ F$. In resume we have obtained theorem \[skjhsj823\] (we use lemma \[dkdjjddh\] to control the constants). The proof of \[skjhsjd823\] is similar and based on theorem \[hghjg6665542212\]. The proof of \[skjhsssej823\] follows from \[skjhsjd823\] and Morrey’s inequality.
Let us now prove proposition \[kwjdjwj7\]. Write $x=\Tr[\sigma]$ and $z=n \frac{\Tr[{^t\sigma \sigma}]}{(\Tr[\sigma])^2}$ (observe that $1\leq z\leq n$). It is easy to check that condition \[slkssjk88271\] can be written $$\label{kjshs71}
-\delta^2 x^2 (\frac{\epsilon +n}{n}z-1)+2x\delta-(n+\epsilon-1)\geq
0.$$ Choose $\delta=n
\big\|(\Tr[\sigma])^{-1}\big\|_{L^\infty(\Omega_T)}$. Observing that $\delta x \geq n$ and $\delta x \leq n y_\sigma$ it is easy to conclude the proof of proposition \[kwjdjwj7\]. Similarly obtains the following lemma by straightforward computation from equation .
\[dkdjjddh\] Assume that condition \[slkssjk88271\] is satisfied then $\mu_\sigma<C(n,\epsilon,\delta)$ $$\big\|(\Tr[\sigma])^{-1}\big\|_{L^\infty(\Omega_T)}\leq
C(n,\epsilon,\delta)$$ and $$\big\|\Tr[\sigma]\big\|_{L^\infty(\Omega_T)}\leq
C(n,\epsilon,\delta).$$
Convergence of the finite element method. {#PreLem}
-----------------------------------------
Write $\Rh$ the projection operator mapping $L^2\big(0,T;H^1_0(\Omega)\big)$ onto $Y_T$ defined by: for all $v\in Y_T$ $$\A_T[v,u-\Rh u]=0.$$ Write $\rho:=u-\Rh u$ and $\theta:=\Rh u-u_h$.
\[jkshssde732\] $$\label{ghhjaawjszsbbhfh52az}
\frac{1}{2}\big\|(u-u_h)(T)\big\|^2_{L^2(\Omega)}+\A_T[u-u_h]=\int_{\Omega_T}\rho\partial_t(u-u_h)+\A_T[\rho,u-u_h].$$
Subtracting (integrated against $\psi$) and we obtain that $$\label{ghjjjbaabbfh52}
\big(\psi,\partial_t (u-u_h)\big)+a[\psi,u-u_h]=0\quad \text{for all
$\psi \in V_h(t)$}.$$ Integrating by parts with respect to time we deduce that $$\label{ghjjjbssabfh52}
\big(\psi, (u-u_h)(.,t)\big)+a[\psi,u-u_h]=\int_{\Omega_t}
\partial_t \psi (u-u_h).$$ Taking $\psi=\theta$ in we deduce that $$\label{ghjjjbssabsasafh52}
\begin{split}
\big\|(u-u_h)(.,t)\big\|^2_{L^2(\Omega)}+\A_t[u-u_h]=&\int_{\Omega_t}
\partial_t \theta (u-u_h)+\big(\rho, (u-u_h)(.,t)\big)\\&+\A_t[\rho,u-u_h].
\end{split}$$ Observing that $$\label{ghjjjsbssabsasdafh52}
\int_0^t \big(\partial_t \theta, u-u_h\big)+\big(\rho,
(u-u_h)(.,t)\big)=\frac{1}{2}\big\|(u-u_h)(.,t)\big\|^2_{L^2(\Omega)}+\int_0^t(
\rho, \partial_t (u-u_h)).$$ we deduce the lemma.
### Time independent medium.
\[jkshssssde7a32\] $$\label{ghhjaawjbbhfhs52az}
\begin{split}
\big\|(u-u_h)(T)\big\|^2_{L^2(\Omega)}+\A_T[u-u_h]\leq 2\Big(&
\|\rho\|_{L^2(\Omega_T)}\|\partial_t u-\partial_ t
u_h\|_{L^2(\Omega_T)}\\&+ \A_T[\rho]\Big).
\end{split}$$
Lemma \[jkshssssde7a32\] is a straightforward consequence of lemma \[jkshssde732\] and Cauchy-Schwartz and Minkowski inequalities.
\[sjhsgh7\] We have $$\label{gsessnbx6fdssw2}
\big\|u_h(.,T)\big\|_{L^2(\Omega)}^2+ \A_T[u_h]\leq
\frac{C_{n,\Omega}}{\lambda_{\min}(a)} \|g\|_{L^2(\Omega_T)}^2.$$
Taking $\psi=u_h$ in and integrating with respect to time we obtain that $$\label{gsesfdssmmw2}
\frac{1}{2}\big\|u_h(.,T)\big\|_{L^2(\Omega)}^2+
\A_T[u_h]=(u_h,g)_{L^2(\Omega_T)}.$$ Using Poincaré and Minkowksi inequalities leads us to .
\[jshsszskssjsbh61RED\] Assume $\partial_t a \equiv 0$. We have $$\label{sxdsgxssssbxsesw2RED}
\begin{split}
\big\|\partial_t u_h\big\|_{L^2(\Omega_T)}^2+a\big[u_h(.,T)\big]\leq
\big\|g\big\|_{L^2(\Omega_T)}^2.
\end{split}$$
The proof is similar to lemma \[jshsszskjsh61RED\]. We need to take $\psi=\partial_t u_h$ in .
Let $t\in [0,T]$ and $v\in H^1_0(\Omega)$, we will write $\Rht
v(.,t)$ the solution of: $$\int_\Omega {^t\nabla \psi}a(x,t)(\psi,v-\Rht v)\,dx=0 \quad
\text{for all $\psi\in V_h(t)$}.$$
We will need the following lemma,
\[lemrho\] Assume the mapping $x\rightarrow F(x,t)$ to be invertible, then for $v\in H^1_0(\Omega)$ we have
- For $n=1$, $$\label{ghhjsdaswwwfh52}
\big(a[v-\Rht v]\big)^\frac{1}{2} \leq C_{X} h
\|v\circ F^{-1}(.,t)\|_{W^{2,2}_D} \|a\nabla
F\|_{L^\infty(\Omega_T)}^\frac{1}{2}.$$
- For $n\geq 2$, $$\label{ghhjsdaswwszwfh52}
\begin{split}
\big(a[v-\Rht v]\big)^\frac{1}{2} \leq & C_{X} h
\|v\circ F^{-1}(.,t)\|_{W^{2,2}_D} \\& \times C_n \mu_\sigma^{\frac{n-1}{4}}
\big\|(\Tr[\sigma])^{-1}\big\|_{L^\infty(\Omega_T)}^{\frac{n-2}{4}}.
\end{split}$$
Recall that $\mu_{\sigma}$ is given by equation and it is easy to check that $\mu_{\sigma}$ is bounded by an increasing function of $(1-\beta_{\sigma})^{-1}$.
Using the change of coordinates $y=F(x,t)$ we obtain that (we write $\hat{v}:=v \circ F^{-1}$) $$\label{ghhjasabbhfh52}
a[v]=Q[\hat{v}]$$ with $$\label{ghhssabbhfh52}
\Q[w]:=\int_\Omega {^t\nabla w(y,t)}Q(y,t)\nabla w(y,t)\,dy$$ and $$\label{ghhsssh52}
Q(y,t):=\frac{\sigma}{\det(\nabla F)}\circ F^{-1}.$$ Using the definition of $\Rht v$ we obtain that $$\label{ghhjasszbbhfh52}
\Q[\hat{v}-\widehat{\Rht v}]=\inf_{\varphi \in X_h
}Q[\hat{v}-\varphi].$$ Using property we obtain that $$\label{ghhjawjbbhfh52}
\Q[\hat{v}-\widehat{\Rht v}]\leq \lambda_{\max}(Q) C_{X}^2 h^2
\|\hat{v}\|_{W^{2,2}_D (T)}^2.$$ It is easy to obtain that
- $n=1$. $$\lambda_{\max}(Q)\leq \|a\nabla F\|_{L^\infty(\Omega_T)}.$$
- $n\geq 2$. $$\lambda_{\max}(Q)\leq C_n \mu_\sigma^{\frac{n-1}{2}}
\big\|(\Tr[\sigma])^{-1}\big\|_{L^\infty(\Omega_T)}^{\frac{n}{2}-1}.$$
\[slskjdddd823\] Assume that $\partial_t a \equiv 0$, $\Omega$ is convex, $\beta_{\sigma}<1$ and $(\Tr[\sigma])^{-1}\in L^\infty(\Omega_T)$ then $$\label{sjswhdawdsdlswe}
\A_T[\rho] \leq C h^2 \|g\|_{L^2(\Omega_T)}^2.$$
The constant $C$ depends on $C_X$, $n$, $\Omega$, $\lambda_{\min}(a)$ and\
$\big\|(\Tr[\sigma])^{-1}\big\|_{L^\infty(\Omega_T)}$. If $n\geq 5$ it also depends on $\big\|\Tr[\sigma]\big\|_{L^\infty(\Omega_T)}$ and if $n=1$ it also depends on $\lambda_{\max}(a)$.
The proof is a straightforward application of lemma \[lemrho\] and theorem \[ksjshjsxxsd8721\]. Observe that in dimension one $a\nabla F=(\int_\Omega a^{-1})^{-1}$
\[slskj823\] Assume that $\partial_t a \equiv 0$, $\Omega$ is convex, $\beta_{\sigma}<1$ and $(\Tr[\sigma])^{-1}\in L^\infty(\Omega_T)$ then $$\label{sjswhawdsdlswe}
\|\rho\|_{L^2(\Omega_T)} \leq C h^2 \|g\|_{L^2(\Omega_T)}.$$
The constant $C$ depends on $C_X$, $n$, $\Omega$, $\lambda_{\min}(a)$ and\
$\big\|(\Tr[\sigma])^{-1}\big\|_{L^\infty(\Omega_T)}$. If $n\geq 5$ it also depends on $\big\|\Tr[\sigma]\big\|_{L^\infty(\Omega_T)}$ and if $n=1$ it also depends on $\lambda_{\max}(a)$.
The proof follows from standard duality techniques (see for instance theorem 5.7.6 of [@BreSco02]). We choose $v\in
L^2(0,T,H^1_0(\Omega))$ to be the solution of the following linear problem: for all $w\in L^2(0,T,H^1_0(\Omega))$ $$\label{dhjdssh61}
A_T[w,v]=(w,\rho)_{L^2(\Omega_T)}.$$ Choosing $w=\rho$ in equation we deduce that $$\|\rho\|_{L^2(\Omega_T))}^2 =\A_T[\rho,v-\Rht v].$$ Using Cauchy Schwartz inequality we deduce that $$\label{sjhdlssswe}
\|\rho\|_{L^2(\Omega_T)}^2 \leq \big(\A_T[\rho]\big)^\frac{1}{2}
\big(\A_T[v-\Rht v]\big)^\frac{1}{2}.$$ Using theorem \[ksjshjsxxsd8721\] we obtain that $$\label{ksjsjh61}
\|\hat{v}\|_{L^2(0,T,W^{2,2}_D(\Omega))}\leq C
\|\rho\|_{L^2(\Omega_T)}.$$ Using lemma \[lemrho\] we obtain that $$\label{sjhddsdlswe}
\big(\A_T[v-\Rht v]\big)^\frac{1}{2} \leq C h
\|\rho\|_{L^2(\Omega_T)}.$$ It follows that $$\label{sjswhawdsdlsmxwe}
\|\rho\|_{L^2(\Omega_T)} \leq C h \big(\A_T[\rho]\big)^\frac{1}{2}.$$ We deduce the lemma by applying lemma \[slskjdddd823\] to bound $A_T[\rho]$.
\[slsakdeswj823\] Assume that $\partial_t a \equiv 0$, $\Omega$ is convex, $\beta_{\sigma}<1$ and $(\Tr[\sigma])^{-1}\in L^\infty(\Omega_T)$ then $$\label{sjswhsawdswsdlswe}
\big\|(u-u_h)(.,T)\big\|_{L^2(\Omega)}^2+\big\|u-u_h\big\|_{L^2(0,T;H^1_0(\Omega))}^2
\leq C h^2 \|g\|_{L^2(\Omega_T)}^2.$$
The constant $C$ depends on $C_X$, $n$, $\Omega$, $\lambda_{\min}(a)$ and\
$\big\|(\Tr[\sigma])^{-1}\big\|_{L^\infty(\Omega_T)}$. If $n\geq 5$ it also depends on $\big\|\Tr[\sigma]\big\|_{L^\infty(\Omega_T)}$ and if $n=1$ it also depends on $\lambda_{\max}(a)$.
The proof is a straightforward application of lemmas \[slskj823\], \[slskjdddd823\], \[jshsszskssjsbh61RED\], \[jkshssssde7a32\] and \[jshsszskjsh61RED\].
### Medium with a continuum of time scales.
In this subsection we will assume that the finite elements are in $H^2(\Omega)\cap H^1_0(\Omega)$ and satisfy inverse inequality .
\[jkshssssdess7a32\] $$\label{ghhjaawjssbbhfhs52az}
\begin{split}
\frac{1}{2}\big\|(u-u_h)(t)\big\|^2_{L^2(\Omega)}+\A_t[u-u_h]=&\int_{\Omega_t}\frac{\hat{\rho}}{|\det
\nabla F|\circ F^{-1}}\\& \big(\hat{g}+\sum_{i,j=1}^n
\sigma_{i,j}\circ F^{-1}\partial_i \partial_j \hat{u}_h-\partial_t
\hat{u}_h\big).
\end{split}$$
Consider equation . We have $$\int_{\Omega_t}\rho\partial_t(u-u_h)=\int_{\Omega_t}\frac{\hat{\rho}}{|\det
\nabla F|\circ F^{-1}}\partial_t (\hat{u}-\hat{u}_h)+
\int_{\Omega_t}\rho \partial_t F (\nabla F)^{-1}\nabla (u-u_h).$$ Using equation we obtain that $$\int_{\Omega_t}\rho \partial_t F (\nabla F)^{-1}\nabla
(u-u_h)=-\A_t[\rho,u-u_h]-\sum_{i,j=1}^n \int_{\Omega_t}\hat{\rho}
Q_{i,j}\partial_i \partial_j (\hat{u}-\hat{u}_h).$$
\[jskjshsj73\] $$\big\|\frac{\partial_t \hat{u}_h}{|\det(\nabla
F)|^\frac{1}{2}\circ F^{-1} }\big\|_{L^2(\Omega_T)}\leq 2
\|g\|_{L^2(\Omega_T)}+ C \|\hat{u}_h\|_{L^2(0,T,W^{2,2}_D(\Omega))}.$$ where the constant $C$ depends on $n$, $\lambda_{\max}(a)$, $\|\Tr[\sigma]\|_{L^\infty(\Omega_T)}$, $\mu_\sigma$.
Using the change of variable $y=F(x,t)$ in we obtain that for all $\varphi\in X_h$ $$\label{ghjbbfh533e2}
\begin{cases}
(\varphi,\frac{\partial_t \hat{u}_h}{|\det(\nabla F)|\circ F^{-1} }
)_{L^2(\Omega)}=&\sum_{i,j=1}^n \int_{\Omega}
(\varphi,Q_{i,j}\partial_i\partial_j
\hat{u}_h)_{L^2(\Omega)}\\&+(\varphi,\frac{\hat{g}}{|\det(\nabla
F)|\circ F^{-1} })_{L^2(\Omega)}\\
\hat{u}_h(x,0)=0.
\end{cases}$$ Recall that $Q$ is given by . We choose $\varphi=\partial_t \hat{u}$ and observe that $$\frac{\sigma}{|\det \nabla F|^\frac{1}{2}}=\frac{\sigma}{|\det
\sigma|^\frac{1}{4}}|\det a|^\frac{1}{4}.$$ Thus $$\big\|\frac{\sigma}{|\det \nabla
F|^\frac{1}{2}}\big\|\leq
C\big(n,\lambda_{\max}(a),\|\Tr[\sigma]\|_{L^\infty(\Omega_T)},\mu_\sigma\big).$$ We deduce the lemma by Minkowski inequality.
Combining lemma \[jkshssssdess7a32\] and lemma \[jskjshsj73\] we obtain the following lemma
\[jkshsssssdess7a32\] $$\label{ghhjaawssjssbcbhfhs52az}
\begin{split}
\frac{1}{2}\big\|(u-u_h)(T)\big\|^2_{L^2(\Omega)}+\A_T[u-u_h]\leq
&\|\rho\|_{L^2(\Omega_T)}\Big(\|g\|_{L^2(\Omega_T)}\\&+ C
\|\hat{u}_h\|_{L^2(0,T,W^{2,2}_D(\Omega))} \Big).
\end{split}$$ where the constant $C$ depends on $n$, $\lambda_{\max}(a)$, $\|\Tr[\sigma]\|_{L^\infty(\Omega_T)}$, $\mu_\sigma$.
\[slskjsseee823l\] Assume that $\Omega$ is convex, and condition \[slkssjk88271\] is satisfied then $$\label{sjswhfawssdsdlswe}
\|\rho\|_{L^2(\Omega_T)} \leq C h^2 \|g\|_{L^2(\Omega_T)}.$$
The constant $C$ depends on $C_X$, $n$, $\Omega$, $\delta$ and $\epsilon$, $\lambda_{\min}(a)$ and $\lambda_{\max}(a)$.
The proof is similar to the proof of lemma \[slskj823\]. As in \[slskj823\] we choose $v\in L^2(0,T,H^1_0(\Omega))$ to be the solution of the following linear problem: for all $w\in
L^2(0,T,H^1_0(\Omega))$ $$\label{dhjdssh61m}
A_T[w,v]=(w,\rho)_{L^2(\Omega_T)}.$$ Choosing $w=\rho$ in equation we deduce that $$\|\rho\|_{L^2(\Omega_T))}^2 =\A_T[\rho,v-\Rht v].$$ Using Cauchy Schwartz inequality we deduce that $$\label{sjhdlssswem}
\|\rho\|_{L^2(\Omega_T)}^2 \leq \big(\A_T[\rho]\big)^\frac{1}{2}
\big(\A_T[v-\Rht v]\big)^\frac{1}{2}.$$ Using theorem \[skjhsj823\] we obtain that $$\label{ksjsjh61m}
\|\hat{v}\|_{L^2(0,T,W^{2,2}_D(\Omega))}\leq C
\|\rho\|_{L^2(\Omega_T)}.$$ Using lemma \[lemrho\] we obtain that $$\label{sjhddsdlswem}
\big(\A_T[v-\Rht v]\big)^\frac{1}{2} \leq C h
\|\rho\|_{L^2(\Omega_T)}.$$ It follows that $$\label{sjswhawdsdlsmxwem}
\|\rho\|_{L^2(\Omega)} \leq C h \big(\A_T[\rho]\big)^\frac{1}{2}.$$ We deduce the lemma by applying lemma \[lemrho\] and theorem \[skjhsj823\] to bound $A_T[\rho]$.
\[slskjsseee823\] Assume that $\Omega$ is convex, and that $\Tr[\sigma]\in
L^\infty(\Omega_T)$. $$\label{sjswhawssdsdlswe}
\|\hat{u}_h\|_{L^2(0,T,W^{2,2}_D(\Omega))}\leq \frac{C}{h}
\|g\|_{L^2(\Omega_T)}.$$
The constant $C$ depends on $C_X$, $n$, $\Omega$, $\lambda_{\min}(a)$ and $\lambda_{\max}(a)$ and $\|\Tr[\sigma]\|_{L^\infty(\Omega_T)}$.
Using the inverse inequality of the finite elements we obtain that $$\label{sjswhawssdsdflswe}
\|\hat{u}_h\|_{L^2(0,T,W^{2,2}_D(\Omega))}\leq \frac{C_X}{h}
\|\nabla \hat{u}_h\|_{L^2(0,T,W^{2,2}_D(\Omega))}.$$ Using the change of variables $y=F(x)$ we obtain that $$\label{sjswhawssdsdfflswe}
\|\nabla
\hat{u}_h\|_{L^2(0,T,W^{2,2}_D(\Omega))}^2 \leq C \A_T[u_h]$$ where $C$ depends on $n$, $\lambda_{\min}(a)$ and $\|\Tr[\sigma]\|_{L^\infty(\Omega_T)}$. We deduce the lemma by using lemma \[sjhsgh7\].
Assume that $\Omega$ is convex, and condition \[slkssjk88271\] is satisfied then $$\label{ghhjaawssjssbbhfhsnb52az}
\begin{split}
\frac{1}{2}\big\|(u-u_h)(T)\big\|^2_{L^2(\Omega)}+\A_T[u-u_h]\leq C
h \|g\|_{L^2(\Omega_T)}^2.
\end{split}$$
The constant $C$ depends on $C_X$, $n$, $\Omega$, $\delta$ and $\epsilon$, $\lambda_{\min}(a)$ and $\lambda_{\max}(a)$.
The proof is a straightforward application of lemma \[jkshsssssdess7a32\], lemma \[slskjsseee823l\] and lemma \[slskjsseee823\].
### Homogenization and discretization in time.
We use the notation of subsection \[jksjhs89b\]. First let us observe that the numerical scheme associated to is stable. Indeed choosing $\psi=v_{n+1}$ one gets $$\label{ghjddsdsswsdcszdsbbsfh52}
\begin{split}
\big|v_{n+1}(t_{n+1})\big|^2_{L^2(\Omega)}=&\big(v_{n+1}(t_{n}),
v_{n}(t_{n})\big)_{L^2(\Omega)}\\+\frac{1}{2}\big( \big|
v_{n+1}(t_{n+1})&\big|^2_{L^2(\Omega)}- \big|
v_{n+1}(t_{n})\big|^2_{L^2(\Omega)}\big)
\\-\int_{t_n}^{t_{n+1}}\Big(a\big[v_{n+1}(t)]&+\big(v_{n+1}(t),g(t)\big)_{L^2(\Omega)}\Big)\,dt.
\end{split}$$ It follows by Cauchy-Schwartz inequality that $$\label{ghjddsdscswsdcszddsbbsfh52}
\begin{split}
\frac{1}{2}\big|v_{n+1}(t_{n+1})\big|^2_{L^2(\Omega)}\leq & \frac{1}{2}\big|v_{n}(t_{n})\big|^2_{L^2(\Omega)}
\\-\int_{t_n}^{t_{n+1}}\Big(a\big[v_{n+1}(t)]&+\big(v_{n+1}(t),g(t)\big)_{L^2(\Omega)}\Big)\,dt.
\end{split}$$ Hence using Poincaré and Minkowski inequalities one obtains that $$\label{ghjddsdsswsdcszddsbbsfh52}
\begin{split}
\big|v_{n+1}(t_{n+1})\big|^2_{L^2(\Omega)}+\int_{t_n}^{t_{n+1}}a\big[v_{n+1}(t)]\,dt \leq& \big|v_{n}(t_{n})\big|^2_{L^2(\Omega)}
\\+\frac{C_{n,\Omega}}{\lambda_{\min}(a)}&\int_{t_n}^{t_{n+1}}\big|g(t)\big|^2_{L^2(\Omega)}\,dt.
\end{split}$$ which implies theorem \[sshjhsjjs823\] and the stability of the scheme. Integrating with respect to time we obtain that for $\psi\in V$, $$\label{ghjddwszdbbfh52}
\begin{split}
\big(\psi(t_{n+1}),
u_h(t_{n+1})\big)_{L^2(\Omega)}=&\big(\psi(t_{n}),
u_h(t_{n})\big)_{L^2(\Omega)}+\int_{t_n}^{t_{n+1}}\Big(\big(\partial_t\psi(t),
u_h(t)\big)_{L^2(\Omega)}\\&-a\big[\psi(t),u_h(t)]+\big(\psi(t),g(t)\big)_{L^2(\Omega)}
\Big)\,dt.
\end{split}$$ Let us write $(u^i)$ the coordinates of $u_h$ associated to the basis $(\varphi_i \circ F)$, i.e. $$u_h(x,t):=\sum_{i} u^i(t) \varphi_i(F(x,t)).$$ Let us define $$u_n(x,t):=\sum_{i} u^i(t_n) \varphi_i(F(x,t)).$$ Subtracting and we obtain that for $\psi \in Z_T$, $$\label{ghjdddxxwszdssdbbfh52}
\begin{split}
\big(\psi(t_{n+1}),
(u_{n+1}-v_{n+1})(t_{n+1})\big)_{L^2(\Omega)}=&\big(\psi(t_{n}),
(u_n-v_{n})(t_{n})\big)_{L^2(\Omega)}\\&+\int_{t_n}^{t_{n+1}}\Big(\big(\partial_t\psi(t),
(u_h-v_{n+1})(t)\big)_{L^2(\Omega)}\\&-a\big[\psi(t),(u_h-v_{n+1})(t)]
\Big)\,dt.
\end{split}$$ Choosing $\psi=u_{n+1}-v_{n+1}$ we deduce using Cauchy-Schwartz inequality that $$\label{ghjdzddxxwszsssssdbbfh52}
\begin{split}
\frac{1}{2} & \big| (u_{n+1}-v_{n+1})(t_{n+1})\big|^2_{L^2(\Omega)}+
\int_{t_n}^{t_{n+1}}a\big[(u_{n+1}-v_{n+1})(t)] \,dt \leq\\&
\frac{1}{2}\big|
(u_n-v_{n})(t_{n})\big|^2_{L^2(\Omega)}+\int_{t_n}^{t_{n+1}}\Big(\big(\partial_t(u_{n+1}-v_{n+1})(t),
(u_h-u_{n+1})(t)\big)_{L^2(\Omega)}\\&-a\big[(u_{n+1}-v_{n+1})(t),(u_{h}-u_{n+1})(t)]
\Big)\,dt.
\end{split}$$
#### Time independent medium.
Observe that if the medium is time independent then can be written $$\label{gazoszsssssdbbfh52}
\begin{split}
\frac{1}{2} & \big| (u_{n+1}-v_{n+1})(t_{n+1})\big|^2_{L^2(\Omega)}+
\int_{t_n}^{t_{n+1}}a\big[(u_{n+1}-v_{n+1})(t)] \,dt \leq\\&
\frac{1}{2}\big|
(u_n-v_{n})(t_{n})\big|^2_{L^2(\Omega)}-\int_{t_n}^{t_{n+1}}
a\big[(u_{n+1}-v_{n+1})(t),(u_{h}-u_{n+1})(t)] \,dt
\end{split}$$ which leads us to $$\label{gazoszwsbbfh52}
\begin{split}
\frac{1}{2} & \big| (u_{n+1}-v_{n+1})(t_{n+1})\big|^2_{L^2(\Omega)}+
\int_{t_n}^{t_{n+1}}a\big[(u_{n+1}-v_{n+1})(t)] \,dt \leq\\&
\frac{1}{2}\big|
(u_n-v_{n})(t_{n})\big|^2_{L^2(\Omega)}\\&+\int_{t_n}^{t_{n+1}}
\int_{t_n}^{t_{n+1}} 1(t<s)\,a\big[(u_{n+1}-v_{n+1})(t),\partial_s
u_h(s)] \,ds\,dt.
\end{split}$$ Write $\Delta t:=t_{n+1}-t_n$. Using Minkowski inequality we obtain that $$\label{gazoszwsbbfh52ppp}
\begin{split}
a\big[(u_{n+1}-v_{n+1})(t),\partial_s u_h(s)]\leq &\frac{1}{2 \Delta
t}a\big[(u_{n+1}-v_{n+1})(t)]\\&+ \frac{1}{2}\Delta t
a\big[\partial_s u_h(s)].
\end{split}$$
It follows from that $$\label{gazoszwsbbfh52c}
\begin{split}
& \big| (u_{n+1}-v_{n+1})(t_{n+1})\big|^2_{L^2(\Omega)}+
\int_{t_n}^{t_{n+1}}a\big[(u_{n+1}-v_{n+1})(t)] \,dt \leq\\& \big|
(u_n-v_{n})(t_{n})\big|^2_{L^2(\Omega)}+ |\Delta t|^2
\int_{t_n}^{t_{n+1}} \,a\big[\partial_s u_h(s)] \,ds.
\end{split}$$ Observing that $$\label{gazoszwsbbfh52cc}
\begin{split}
\int_{t_n}^{t_{n+1}}a\big[(u_{n+1}-v_{n+1})(t)] \,dt\geq& 0.5
\int_{t_n}^{t_{n+1}}a\big[(u_{h}-v_{n+1})(t)]
\,dt\\&-\int_{t_n}^{t_{n+1}}a\big[(u_{h}-u_{n+1})(t)] \,dt
\end{split}$$ and $$\label{gazoszwsbbfh52vcb}
\begin{split}
\int_{t_n}^{t_{n+1}}a\big[(u_{h}-u_{n+1})(t)] \,dt \leq |\Delta t|^2
\int_{t_n}^{t_{n+1}} \,a\big[\partial_s u_h(s)] \,ds
\end{split}$$ we obtain that $$\label{gazoszwsbbfh52vcdf}
\begin{split}
& \big| (u_{n+1}-v_{n+1})(t_{n+1})\big|^2_{L^2(\Omega)}+ 0.5
\int_{t_n}^{t_{n+1}}a\big[(u_{h}-v_{n+1})(t)] \,dt \leq\\& \big|
(u_n-v_{n})(t_{n})\big|^2_{L^2(\Omega)}+\frac{3}{2} |\Delta t|^2
\int_{t_n}^{t_{n+1}} \,a\big[\partial_s u_h(s)] \,ds.
\end{split}$$ In conclusion we have obtained the following following lemma
\[jskjdhdjdh723\] Let $v\in Z_T$ be the solution of . We have $$\label{gazoszwsbbfhgf52}
\begin{split}
\big\| (u_h-v)(T)\big\|^2_{L^2(\Omega)}+
\int_{0}^{T}a\big[(u_h-v)(t)] \,dt \leq 3 |\Delta t|^2 \int_{0}^{T}
\,a\big[\partial_s u_h(s)] \,ds.
\end{split}$$
Combining lemma \[jshsszskjsh61\] with lemma \[jskjdhdjdh723\] we obtain the following theorem:
\[jskjdhdjdhxzs723\] Let $v\in Z_T$ be the solution of . We have $$\label{gazoszwsbdsbfh52}
\begin{split}
\big\| (u_h-v)(T)\big\|^2_{L^2(\Omega)}+
&\int_{0}^{T}a\big[(u_h-v)(t)] \,dt \leq 3 |\Delta t|^2
\\&\Big(\frac{4}{\lambda_{\min}(a)} \|\partial_t
g\|_{L^2(0,T,H^{-1}(\Omega))}^2+\big\|g(.,0)\big\|_{L^2(\Omega)}^2\Big).
\end{split}$$
#### Time dependent medium.
Observe that $$\partial_t(u_{n+1}-v_{n+1})=\partial_t F (\nabla
F)^{-1}\nabla (u_{n+1}-v_{n+1}).$$ It follows after writing $\partial_t F =\diiv a \nabla F$, integration by parts and using the change of variables $y=F(x,t)$ in that $$\label{ghjdzddxxwszsskksssdddbbfh52}
\begin{split}
\frac{1}{2} & \big| (u_{n+1}-v_{n+1})(t_{n+1})\big|^2_{L^2(\Omega)}+
\int_{t_n}^{t_{n+1}}a\big[(u_{n+1}-v_{n+1})(t)] \,dt \leq\\&
\frac{1}{2}\big|
(u_n-v_{n})(t_{n})\big|^2_{L^2(\Omega)}-2\int_{t_n}^{t_{n+1}}
a\big[(u_{n+1}-v_{n+1})(t),(u_{h}-u_{n+1})(t)\big] \,dt\\&
-\sum_{i,j} \int_{t_n}^{t_{n+1}} \int_{\Omega}
(\hat{u}_{h}-\hat{u}_{n+1})Q_{i,j}\partial_i \partial_j
(\hat{u}_{n+1}-\hat{v}_{n+1}) \,dt\,dy.
\end{split}$$ Hence using Minkowski inequality we obtain that $$\label{ghjdzddxxwszsskksssdddbbfcih52}
\begin{split}
& \big| (u_{n+1}-v_{n+1})(t_{n+1})\big|^2_{L^2(\Omega)}+
\int_{t_n}^{t_{n+1}}a\big[(u_{n+1}-v_{n+1})(t)] \,dt \leq\\&
\big| (u_n-v_{n})(t_{n})\big|^2_{L^2(\Omega)}+4 \int_{t_n}^{t_{n+1}}
a\big[(u_{h}-u_{n+1})(t)\big] \,dt\\& -2\sum_{i,j}
\int_{t_n}^{t_{n+1}} \int_{\Omega}
(\hat{u}_{h}-\hat{u}_{n+1})Q_{i,j}\partial_i \partial_j
(\hat{u}_{n+1}-\hat{v}_{n+1}) \,dt\,dy.
\end{split}$$ Using Minkowski inequality we obtain that $$\label{ghjdzddxxwszsskksssdddiibbfh52}
\begin{split}
\Big|\sum_{i,j} \int_{t_n}^{t_{n+1}}& \int_{\Omega}
(\hat{u}_{h}-\hat{u}_{n+1})Q_{i,j}\partial_i \partial_j
(\hat{u}_{n+1}-\hat{v}_{n+1}) \,dt\,dy\Big|\leq\\& C_A n^2
\int_{t_n}^{t_{n+1}} \int_{\Omega}
|\hat{u}_{h}-\hat{u}_{n+1}|^2\,dt\,dy
\\&+\frac{\lambda_{\max}(Q)}{C_A} \int_{t_n}^{t_{n+1}} \sum_{i,j}\int_{\Omega}
|\partial_i \partial_j (\hat{u}_{n+1}-\hat{v}_{n+1})|^2
\,dt\,dy.
\end{split}$$ Using the inverse inequality and the change of variable $y=F(x)$ we obtain that $$\label{ghjdzddxxwsiizsskksssdddbbfh52}
\begin{split}
\int_{t_n}^{t_{n+1}}
\sum_{i,j}\int_{\Omega} |\partial_i \partial_j
(\hat{u}_{n+1}-\hat{v}_{n+1})&|^2 \,dt\,dy\leq \frac{C_X}{h^2
\lambda_{\min}(Q)}
\\&\int_{t_n}^{t_{n+1}}a\big[(u_{n+1}-v_{n+1})(t)] \,dt
.\end{split}$$ In resume, choosing $C_A= \frac{4 C_X \lambda_{\max}(Q)}{h^2
\lambda_{\min}(Q)}$ we have obtained that $$\label{ghjdzddxxwsdbfh52}
\begin{split}
& \big| (u_{n+1}-v_{n+1})(t_{n+1})\big|^2_{L^2(\Omega)}+0.5
\int_{t_n}^{t_{n+1}}a\big[(u_{n+1}-v_{n+1})(t)] \,dt \leq\\&
\big| (u_n-v_{n})(t_{n})\big|^2_{L^2(\Omega)}+8 \int_{t_n}^{t_{n+1}}
a\big[(u_{h}-u_{n+1})(t)\big] \,dt\\& + \frac{8 C_X
\lambda_{\max}(Q)}{h^2 \lambda_{\min}(Q)} n^2 \int_{t_n}^{t_{n+1}}
\int_{\Omega} |\hat{u}_{h}-\hat{u}_{n+1}|^2\,dt\,dy .\end{split}$$ And a computation similar to the one leading to gives us $$\label{ghjdzsxxwsdbfh52o}
\begin{split}
& \big| (u_{n+1}-v_{n+1})(t_{n+1})\big|^2_{L^2(\Omega)}+\frac{1}{4}
\int_{t_n}^{t_{n+1}}a\big[(u_h-v_{n+1})(t)] \,dt \leq\\&
\big| (u_n-v_{n})(t_{n})\big|^2_{L^2(\Omega)}+9 \int_{t_n}^{t_{n+1}}
a\big[(u_{h}-u_{n+1})(t)\big] \,dt\\& + \frac{8 C_X
\lambda_{\max}(Q)}{h^2 \lambda_{\min}(Q)} n^2 \int_{t_n}^{t_{n+1}}
\int_{\Omega} |\hat{u}_{h}-\hat{u}_{n+1}|^2\,dt\,dy .\end{split}$$ Moreover using the change of variables $F(x)=y$ and the inverse inequality we obtain that $$\label{ghjdzsxxwsdbfh52}
\begin{split}
\int_{t_n}^{t_{n+1}}
a\big[(u_{h}-u_{n+1})(t)\big] \,dt \leq \frac{C_X
\lambda_{\max}(Q)}{h^2} \int_{t_n}^{t_{n+1}} \int_{\Omega}
|\hat{u}_{h}-\hat{u}_{n+1}|^2\,dt\,dy.
\end{split}$$ Let us also observe that $$\label{ghjdzsswaqwsdbfh52}
\begin{split}
\int_{t_n}^{t_{n+1}} \int_{\Omega}
|\hat{u}_{h}-\hat{u}_{n+1}|^2\,dt\,dy\leq |\Delta t|^2
\int_{t_n}^{t_{n+1}} \int_{\Omega} |\partial_t
\hat{u}_{h}|^2\,dt\,dy.
\end{split}$$ It follows that $$\label{ghjdzsxxwsdbfh52oo}
\begin{split}
& \big| (u_{n+1}-v_{n+1})(t_{n+1})\big|^2_{L^2(\Omega)}+\frac{1}{4}
\int_{t_n}^{t_{n+1}}a\big[(u_h-v_{n+1})(t)] \,dt \leq\\&
\big| (u_n-v_{n})(t_{n})\big|^2_{L^2(\Omega)}+ C_B \frac{|\Delta
t|^2}{h^2} \int_{t_n}^{t_{n+1}} \int_{\Omega} |\partial_t
\hat{u}_{h}|^2\,dt\,dy.
\end{split}$$ with $$\label{ghssawsdbfh52}
\begin{split}
C_B=C_n C_X \lambda_{\max}(Q) (1+\frac{1}{\lambda_{\min}(Q)})
.\end{split}$$ We deduce that $$\label{ghjdzsxxwsdssabfh52}
\begin{split}
\big\| (u_h-v)(T)\big\|^2_{L^2(\Omega)}+\frac{1}{4}
\int_{0}^{T}a\big[(u_h-v)(t)] \,dt \leq C_B \frac{|\Delta t|^2}{h^2} \int_{0}^{T} \int_{\Omega}
|\partial_t \hat{u}_{h}|^2\,dt\,dy.
\end{split}$$ Using lemma \[dkdjjddh\] to control $C_B$ and combining with theorem \[skjhsj823\] we obtain the following theorem.
\[skjhseeeddfddsf23\] Assume that $\Omega$ is convex, and condition \[slkssjk88271\] is satisfied. Let $v\in Z_T$ be the solution of , we have $$\label{ghjdzssasdssabfh52}
\begin{split}
\big\| (u_h-v)(T)\big\|^2_{L^2(\Omega)}+\frac{1}{4}
\int_{0}^{T}a\big[(u_h-v)(t)] \,dt \leq C \frac{|\Delta t|^2}{h^2}
\|g\|_{L^2(\Omega_T)}^2
.\end{split}$$ where $C$ depends on $\Omega$, $n$, $\delta$, $\epsilon$, $\lambda_{\min}(a)$ and $\lambda_{\max}(a)$.
Numerical Experiments {#ksjsskhs821}
=====================
The purpose of this section is to give several illustrations of the implementation of this method. The domain is the unit square in dimension two. Equation is solved on a fine tessellation characterized by $16129$ interior nodes (degree of freedoms).
Three different coarse tessellations are considered, one with $9$ degrees of freedoms (noted *dof* in the tables), one with $49$ and the last one with $225$.
The parabolic operator associated to equation has been homogenized onto these coarse meshes using the method the method presented in this paper. We have chosen splines to span the space $X_h$ introduced in subsection \[sub2\].
Time independent examples.
--------------------------
\[exa:siteprco\]Time independent site percolation.
In this example we consider the site percolating medium associated to figure \[ap7\]. The fine mesh is characterized by $16641$ nodes. has been homogenized to three different coarse meshes with $9$, $49$ and $225$ interior nodes using the method described here and splines for the space $X_h$. has been solved with the fine mesh operator and the coarse mesh operators with $g=1$ and $g=\sin(2.4 x-1.8 y+2 \pi t)$. The fine mesh and coarse mesh errors are given in tables \[cerrtindepp7\], \[ferrtindepp7\], \[cerrtindeptdrhsp7\], \[ferrtindeptdrhsp7\]. Figure \[eruuhzaa4\] shows $u$ computed on $16641$ interior nodes and $u_h$ computed on $9$ interior nodes in the case $g=1$ at time $1$.
[ ]{}\
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0142 0.0389 0.0168 0.0366
49 0.0077 0.0450 0.0101 0.0482
225 0.0035 0.0228 0.0060 0.0293
: Coarse Mesh Error. Time Independent Site Percolation with $g=1$.[]{data-label="cerrtindepp7"}
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0196 0.0843 0.0251 0.1193
49 0.0136 0.0698 0.0184 0.1028
225 0.0040 0.0243 0.0070 0.0485
: Fine Mesh Error. Time Independent Site Percolation with $g=1$.[]{data-label="ferrtindepp7"}
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0236 0.0569 0.0262 0.0477
49 0.0181 0.0571 0.0215 0.0558
225 0.0119 0.0774 0.0167 0.0939
: Coarse Mesh Error. Time Independent Percolation Case with $g=\sin(2.4 x-1.8 y+2 \pi t)$.[]{data-label="cerrtindeptdrhsp7"}
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0424 0.1099 0.0512 0.1712
49 0.0277 0.0985 0.0348 0.1451
225 0.0174 0.0886 0.0242 0.1192
: Fine Mesh Error. Time Independent Percolation with $g=\sin(2.4 x-1.8 y+2 \pi t)$.[]{data-label="ferrtindeptdrhsp7"}
\[exa:chan\]Time independent high conductivity channel.
In this example $a$ is random and characterized by a fine and long ranged high conductivity channel. We choose $a(x)=100$, if $x$ is in the channel, and $a(x)=O(1)$ and random, if $x$ is not in the channel. The media is illustrated in figure \[ap6\]
Tables \[cerrp6tindepsp\] and \[ferrp6tindepsp\] give the coarse and fine meshes errors.
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0159 0.0496 0.0207 0.0477
49 0.0067 0.0389 0.0102 0.0345
225 0.0035 0.0228 0.0060 0.0293
: Coarse Mesh Error, high conductivity channel.[]{data-label="cerrp6tindepsp"}
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0178 0.0564 0.0257 0.0947
49 0.0079 0.0388 0.0129 0.0660
225 0.0040 0.0243 0.0070 0.0485
: Fine Mesh Error, high conductivity channel.[]{data-label="ferrp6tindepsp"}
Time dependent examples.
------------------------
In the following examples we consider media characterized by a continuum of time scales. In the following examples the ODE obtained on the coarse mesh from the homogenization of have also been homogenized in time according to the method described in subsection \[jksjhs89b\].
\[exa:tdep\_trignometric\]Time Dependent Multiscale trigonometric.
In this example $a$ is given by equation . Although the number fine time steps to solve is $2663$, only $134$ coarse time steps have been used to solve the homogenized equation. Hence if one also takes into account homogenization in space, the compression factor is of the order of $35000$ for the coarse mesh with $9$ interior nodes.
Figure \[fixpointF\] shows the curves of $t\rightarrow a(x_0,t)$ and $t\rightarrow F(x_0,t)$ for a given $x_0\in \Omega$.
[ ]{}\
The coarse and fine mesh relative $L^1$, $L^2$, $L^\infty$, and $H^1$ errors with respect to time have been plotted in figures \[errL1tdepp4\], \[errL2tdepp4\], \[errLitdepp4\] and \[errH1tdepp4\]. The initial increase of the relative error has its origin in the initial value $u\equiv 0$ at time $0$.
[ ]{}\
[ ]{}\
[ ]{}\
[ ]{}\
The coarse and fine meshes errors are given in tables \[cerrtdp4sp\] and \[ferrtdp4sp\] for $g=1$ at $t=0.1$, those errors are given in tables \[cerrtdrhsp4\] and \[ferrtdrhsp4\] for $g=\sin(2.4 x-1.8y+2 \pi t)$ at $t=0.1$
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0018 0.0045 0.0019 0.0039
49 0.0012 0.0054 0.0015 0.0060
: Coarse Mesh Error. Multiscale trigonometric time dependent Medium. $g=1$.[]{data-label="cerrtdp4sp"}
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0031 0.0096 0.0034 0.0242
49 0.0014 0.0059 0.0016 0.0166
: Fine Mesh Error. Multiscale trigonometric time dependent Medium. $g=1$.[]{data-label="ferrtdp4sp"}
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0043 0.0087 0.0044 0.0085
49 0.0033 0.0079 0.0035 0.0084
: Coarse mesh error. Multiscale trigonometric time dependent Medium. $g=\sin(2.4 x-1.8y+2 \pi t)$. []{data-label="cerrtdrhsp4"}
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0082 0.0199 0.0087 0.0379
49 0.0038 0.0104 0.0040 0.0244
: Fine mesh error. Multiscale trigonometric time dependent medium. $g=\sin(2.4 x-1.8y+2 \pi t)$. []{data-label="ferrtdrhsp4"}
\[exa:tdep\_fourier\]Time Dependent Random Fourier Modes.
In this example $a(x,y,t)=e^{h(x,y,t)}$ where $h$ is given by the following equation $$h(x,y,t)=\sum_{|k|\leq R}(a_{k}\sin(2\pi
k.x')+b_{k}\cos(2\pi k.x'))$$ where $R=4$, $x'=x+\sqrt{2}t$, $y'=y-\sqrt{2}t$, $a_{k}$ and $b_{k}$ are independent identically distributed random variables on $[-0.2,0.2]$. In this example, one can compute that $\frac{\lambda_{\max}(a)}{\lambda_{\min}(a)}=95.7$.
Figure \[mediap5\] is a plot of $a$ at time $0$. Although the number fine time steps to solve is $1332$ only $67$ coarse time steps have been used to solve the homogenized equation. Hence if one also takes into account homogenization in space, the compression factor is of the order of $35000$ for the coarse mesh with $9$ interior nodes.
The coarse and fine mesh relative $L^1$, $L^2$, $L^\infty$, and $H^1$ errors with respect to time (up to time $t=0.1$) have been plotted in figures \[errL1tdepp5\], \[errL2tdepp5\], \[errLitdepp5\] and \[errH1tdepp5\]. Those errors are also given up to $t=1$ in figures \[errL1t100p5\], \[errL2t100p5\], \[errLit100p5\], \[errH1t100p5\].
The coarse and fine meshes errors are given in tables \[cerrtdp5sp\] and \[ferrtdp5sp\] for $g=1$ at $t=0.1$.
[ ]{}\
[ ]{}\
[ ]{}\
[ ]{}\
[ ]{}\
[ ]{}\
[ ]{}\
[ ]{}\
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0025 0.0028 0.0064 0.0052
49 0.0032 0.0098 0.0034 0.0100
: Coarse mesh error at $t=0.1$. Time dependent random Fourier modes.[]{data-label="cerrtdp5sp"}
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0063 0.0344 0.0079 0.0481
49 0.0042 0.0207 0.0049 0.0337
: Fine mesh error at $t=0.1$. Time dependent random Fourier modes.[]{data-label="ferrtdp5sp"}
\[exa:tdep\_randomfractal\]Time Dependent Random Fractal
In this example, $a$ is given by a product of discontinuous functions oscillating randomly at different space and time scales. Namely $a(x,t):=a_{1}(x,t)a_{2}(x,t)\cdots a_{n}(x,t)$ with $n=6$ and $a_{i}(x,t)=c_{pq}^i(t)$ for $x\in[\frac{p}{2^{i}},\frac{p+1}{2^{i}})\times[\frac{q}{2^{i}},\frac{q+1}{2^{i}})$. The coefficients $c_{pq}^i(t)$ are chosen at random with uniform law in $[\frac{1}{\gamma},\gamma]$ with $\gamma=0.7$ and independently in subdivision in space and in time, thus they are assumed to be constant in each time interval $0.1\times[\frac{k}{4^i},\frac{k+1}{4^i})$. In this example we have $\frac{\lambda_{\max}(a)}{\lambda_{\min}(a)}=160.3295$. Although the number of fine time steps to solve is $3482$, only $175$ coarse time steps have been used to solve the homogenized equation which corresponds to a reduction of the complexity of the scheme by a factor of $35000$ in the case of the coarse tessellation with $9$ interior nodes. $a$ and the map $(F_1,F_2)$ are drawn in figure \[mediatdp6\]. $L^1$, $L^2$, $L_{\infty}$ and $H_1$ errors are given in figure \[errL1t10tdp6\] to \[errH1t10tdp6\]. Coarse and fine mesh errors are given in table \[cerrtdp6sp\] and \[ferrtdp6sp\] at time $t=0.1$. We have chosen $g=1$ in this numerical experiment, one obtains similar results by choosing $g=\sin(2.4 x-1.8y+2 \pi t)$.
[ ]{}\
[ ]{}\
[ ]{}\
[ ]{}\
[ ]{}\
[ ]{}\
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0046 0.0074 0.0052 0.0065
49 0.0036 0.0046 0.0036 0.0059
: Coarse Mesh Error for the time dependent random fractal medium with spline element[]{data-label="cerrtdp6sp"}
dof $L^{1}$ $L^{\infty}$ $L^{2}$ $H^{1}$
----- --------- -------------- --------- ---------
9 0.0039 0.0082 0.0043 0.0222
49 0.0033 0.0054 0.0034 0.0168
: Fine Mesh Error for the time dependent random fractal medium with spline element[]{data-label="ferrtdp6sp"}
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[^1]: California Institute of Technology Applied & Computational Mathematics, Control & Dynamical systems, MC 217-50 Pasadena , CA 91125, owhadi@caltech.edu
[^2]: California Institute of Technology Applied & Computational Mathematics MC 217-50 Pasadena , CA 91125, zhanglei@acm.caltech.edu
[^3]: $W^{1,\infty}$ is the usual space of uniformly Lipschitz continuous functions.
[^4]: As for the fast multipole method and the hierarchical multipole method designed by L. Greengard and V. Rokhlin [@MR918448], these methods are based on the singular value decomposition of operators Green’s function.
|
---
abstract: 'FeSe is a fascinating superconducting material at the frontier of research in condensed matter physics. Here we provide an overview on the current understanding of the electronic structure of FeSe, focusing in particular on its low energy electronic structure as determined from angular resolved photoemission spectroscopy, quantum oscillations and magnetotransport measurements of single crystal samples. We discuss the unique place of FeSe amongst iron-based superconductors, being a multi-band system exhibiting strong orbitally-dependent electronic correlations and unusually small Fermi surfaces, prone to different electronic instabilities. We pay particular attention to the evolution of the electronic structure which accompanies the tetragonal-orthorhombic structural distortion of the lattice around 90 K, which stabilizes a unique nematic electronic state. Finally, we discuss how the multi-band multi-orbital nematic electronic structure has an impact on the understanding of the superconductivity, and show that the tunability of the nematic state with chemical and physical pressure will help to disentangle the role of different competing interactions relevant for enhancing superconductivity.'
author:
- 'Amalia I. Coldea$^1$ and Matthew D. Watson$^2$'
title: The key ingredients of the electronic structure of FeSe
---
FeSe, electronic structure, quantum oscillations, ARPES, Fe-based superconductors
INTRODUCTION
============
FeSe is structurally the simplest of the iron-based superconductors, but is host to some of the richest physics. Shortly after high-$T_c$ superconductivity was found in iron-pnictide systems by Kamihara *et. al.* [@Kamihara2008], superconductivity in the Fe-chalcogenide compound FeSe below 8 K was discovered by Hsu *et. al.* [@Hsu2008]. All these materials share a common structural motif of a square lattice of Fe, bonded to pnictogen or chalcogen ions which sit alternately above and below the plane. The appearance of unconventional superconductivity in the Fe-based systems is commonly thought to arise from a spin fluctuation pairing mechanism, linked with the suppression of spin-density wave ordering observed in the parent compounds. However the unique properties of non-magnetic FeSe provides a challenging test-case for this view, and has generated intense detailed experimental and theoretical investigation of its electronic and magnetic properties.
The superconductivity of FeSe is remarkably tunable. Under applied pressure, $T_c$ reaches 36.7 K at 8.9 GPa [@Medvedev2009]. Furthermore, the FeSe layers are held by weak van der Waals bonds that make them susceptible to mechanical exfoliation and chemical intercalation. Ionic gating using a field-effect layer transistor in thin flakes of FeSe induces dramatic changes in the carrier density and enhances superconductivity towards 43 K [@Lei2016]. Intercalation has the effect of separating the layers and also effectively dopes them with electrons, and a similarly enhanced $T_c$ can be achieved [@Sun2015; @BurrardLucas2013].
The recent renaissance in research on FeSe has been mainly driven by significant advances in materials development. Firstly, the discovery of high temperature superconductivity in a monolayer FeSe grown by molecular beam epitaxy on SrTiO$_3$ [@Wang2012monolayer; @Wang2017review] provided a exciting new route towards strongly enhanced superconductivity over 65 K in an two-dimensional iron-based superconductor. Secondly, it was found that the chemical vapor transport method [@Bohmer2013; @Chareev2013; @Bohmer2016] could yield mm-sized plate-like single crystals free of impurity phases and close to stoichiometry, enabling detailed study of the intrinsic physics of bulk FeSe.
In this review we will discuss the electronic behaviour of FeSe in high quality single crystals in order to reveal the key ingredients behind its unusual nematic state and superconducting properties. We will focus on the experimental insights provided by high resolution ARPES studies over a large range of temperatures, combined with low temperature quantum oscillations and magnetotransport experiments, and discuss the interplay of order parameters in this intriguing system.
Basic physical properties of FeSe
---------------------------------
![Temperature-dependent resistivity measurement of a single crystal of FeSe. Insets show thermal expansion data adapted from Böhmer *et. al.* [@Bohmer2013] and nematic susceptibility data from Ref. [@Watson2015a]. Schematic Fermi surfaces are based on ARPES measurements [@Watson2016; @Rhodes2017_arxiv].[]{data-label="fig:r-t-schematic"}](Figure1){width="0.7\linewidth"}
[**Figure \[fig:r-t-schematic\]**]{} show some of the basic properties of FeSe at ambient pressure. At room temperature the resistivity starts to saturate around 0.4 m$\Omega$cm, similar to other Fe-based superconductors, which may indicate that the system approaches the Mott-Ioffe limit [@Gunnarsson2003], when the mean-free path of electrons has become smaller than the in-plane lattice constant. In the low temperature regime the resistivity display almost linear resistivity, before becoming superconducting at $T_c \sim 8.7$ K.
FeSe exhibits a structural transition, from a tetragonal $P4/nmm$ to a weakly orthorhombic $Cmma$ unit cell at $T_s \sim$ 90 K [@Mcqueen2009a; @Khasanov2010]. The magnitude of the structural distortion, as well as the preceding softening of the lattice, mimics other parent compounds of iron-based superconductors, like BaFe$_2$As$_2$ [@Bohmer2013]. Other parent and underdoped compounds undergo a structural transition at high temperature followed closely by a magnetic transition, however FeSe is unusual in that no long-range magnetic order has been detected.
The driving force of this symmetry-breaking transition is hotly debated [@Chubukov2015; @Fernandes2014; @Onari2016; @Wang2015]. The transition at $T_s$ is unlikely to be driven by the lattice degrees of freedom due to the very small changes in in-plane lattice parameters [@Bohmer2013], shown in [**Figure \[fig:r-t-schematic\]**]{}. Instead, the breaking of fourfold rotational symmetry at $T_s$, is driven by the development of a [*nematic*]{} electronic state, characterised by strong in-plane anisotropy observed in resistivity [@Tanatar2016] and quasiparticle interference [@Kasahara2014; @Sprau2016_arxiv]. Important evidence for an electronically-driven transition is the divergence of the [*nematic susceptibility*]{} approaching $T_s$ [@Watson2015a; @Hosoi2016], measured as the induced resistivity anisotropy in response to an external strain, shown in [**Figure \[fig:r-t-schematic\]**]{}. Thus we expect signatures of this electronic nematic state to manifest either in strongly anisotropy effects of its Fermi surface, often referred as a [*Pomeranchuk instability*]{} with $d$-wave symmetry [@Pomeranchuk1959; @Chubukov2015] or charge nematic fluctuations [@Massat2016], and anisotropic scattering below $T_s$.
The lack of long-range magnetic order, despite the presence of significant spin-fluctuations detected by neutron diffraction [@Wang2016], is a significant puzzle. Amongst several possible explanations, it has been suggested that FeSe may be close to several competing magnetic instabilities [@Glasbrenner2015], or may be a strongly fluctuating quantum paramagnet [@Wang2015]. On the other hand, it has been proposed that the structural transition may be non-magnetic in origin, instead being a manifestation of orbital ordering [@Onari2016; @Chubukov2016]. The origin of the structural transition at $T_s$ is closely linked to the debate concerning the mechanism of superconductivity in FeSe, with wide implications on iron-based superconductivity in general. The nematic order is likely to significantly influence the superconductivity, with recent evidence of orbitally-selective pairing and strongly anisotropic and twofold-symmetric superconducting gap structure [@Sprau2016_arxiv; @Xu2016]. Thus, the understanding of these intriguing physical properties requires detailed knowledge about the experimental electronic structure in both the tetragonal and nematic phases.
ARPES STUDIES OF FESE
=====================
What does ARPES actually measure in FeSe?
=========================================
The $P4/nmm$ unit cell of tetragonal FeSe includes two Fe sites which are related by a glide symmetry. It has been argued that the essential physics can be captured in an effective 1-Fe unit cell with half the number of bands, constructed by [*unfolding*]{} the 2-Fe Brillouin zone [@YWang2015; @Tomic2014]. This raises questions about what structure will be observed in ARPES measurements. Experimentally, ARPES measurements can detect the entire Fermi surface expected for the 2-Fe unit cell in FeSe at 100 K [@Watson2016; @Fedorov2016], and also in e.g. LiFeAs [@Borisenko2015]. However the underlying symmetry of the 1-Fe unit cell modulates the observations [@Brouet2012; @Moreschini2014] such that the spectral intensity on some branches may be suppressed ([**Figure \[fig:arpesschematic\]c)**]{}. On top of this, the intensity of features always depends strongly on the polarisation of the incident beam due to the different orbital characters and parities of the bands ([**Figure \[fig:arpesschematic\]b**]{}). The effective $k_z$ depends on the incident photon energy, which also affects the relative intensity of features.
Both thin films and high quality vapor-grown single crystals of FeSe are very suitable and popular systems for ARPES investigations, having large, flat and non-polar (001) surfaces. Here, we discuss mainly ARPES measurements of single crystals of FeSe with a specific focus on understanding of the nematic ordering, rather the case of the high $T_c$ monolayer FeSe on different substrates reviewed elsewhere [@Pustovit2016; @Liu2015a; @Wang2017review]. The many other ARPES studies on related iron selenides, including FeTe$_{1-x}$Se$_x$, alkali-metal dosed FeSe, electron-doped (K,Rb)$_x$Fe$_{2-y}$Se$_2$, and (Li$_{0.8}$Fe$_{0.2}$)OHFeSe are reviewed elsewhere [@Pustovit2016; @Liu2015a; @Wang2017review; @Huang2017_arxiv], and general overviews on ARPES studies of iron-based superconductors may be found in Refs. [@Richard2011; @Kordyuk2012; @Richard2015; @vanRoekeghem2016].
![a) DFT calculation of the Fermi surface of FeSe, from Ref. [@Watson2015a]. b) Schematic diagram of the matrix elements considerations based on parities of atomic orbitals, in this case showing the $d_{xz}$ orbital, which would conventionally appear in LH polarisation only. c) The effect on the glide symmetry on the spectral weight as probed by ARPES, based on Brouet *et. al.* [@Brouet2012]. Dashed lines indicate dispersions which are expected to have a suppressed spectral weight.[]{data-label="fig:arpesschematic"}](Figure2){width="0.95\linewidth"}
The hole pockets
----------------
![a) High and b) low temperature spectra at the Z point (23 eV) , and c) low temperature Fermi surface map at Z. d,e) Extracted band positions, overlaid with guides to the eye indicating the band dispersions. f) Schematic diagram of the hole pockets in the tetragonal and orthorhombic phases. The anisotropic distortion of the Fermi surface is a signature of a nematic electronic phase. Figures adapted from Ref. [@Watson2015a].[]{data-label="fig:holepockets"}](Figure3){width="0.9\linewidth"}
According to DFT calculations ([**Figure \[fig:arpesschematic\]a**]{}), FeSe should exhibit three quasi-2D hole pockets at the zone centre, which occupy a sizeable fraction of the Brillouin zone, but the experimental picture varies substantially. The observed dispersions are significantly renormalised compared to the DFT dispersions, and shifted down such that the pockets are much smaller [@Watson2015a]. In particular the $d_{xy}$ band is much flatter than expected, with a renormalisation factor of 8 [@Maletz2014; @Watson2015a], and is found at $\sim$50 meV below the Fermi level. The hole pockets are quasi-2D, with rather small $k_F$ values at the $\Gamma$ point, becoming larger at the Z point. When measured in the tetragonal phase above 90 K ([**Figure \[fig:holepockets\]a**]{}), the Fermi surface of FeSe consists of an outer quasi-2D hole pocket and a small 3D inner hole pocket which just crosses the Fermi level around the Z point [@Watson2015a]. These Fermi surface are essentially circular, and come from the $d_{xz/yz}$ bands which are split at the $\Gamma$/Z point by spin-orbit coupling only [@Watson2015a; @Fernandes2014b], with a band separation $\Delta_{SO}\sim$ 20 meV [@Watson2015a; @Watson2017c_arxiv; @Borisenko2015]. In the nematic phase, the Fermi surface distorts substantially into an elliptical shape, and in addition the inner hole pocket moves completely below the Fermi level. Thus the low temperature Fermi surface of FeSe around the zone centre consists of a single elliptical quasi-2D band ([**Figure \[fig:holepockets\]f**]{}). Since samples will naturally form twin domains below 90 K, two superposed ellipses are observed in most ARPES measurements ([**Figure \[fig:holepockets\]e**]{}), though a single ellipse may be observed in detwinned measurements [@Shimojima2014; @Suzuki2015].
The electron pockets
--------------------
![a) ARPES measurements in the high-symmetry direction through the M point, above $T_s$, showing curvature data at 37 eV. b) ARPES measurements at 56 eV in the same orientation. The MDC at the Fermi level in the 56 eV data consists of four peaks. c) Schematic electronic structure in the tetragonal phase. d) Fermi surface map at 100 K, above $T_s$. e-g) Equivalent measurements at low temperatures.[]{data-label="fig:electronpockets"}](Figure4){width="1\linewidth"}
The electron pockets around the M point are also quasi-2D and much smaller than expected in the DFT calculation, also showing renormalisation by a factor of $\sim$ 4 [@Watson2015a] for the $d_{yz}$ band in the tetragonal phase. [**Figure \[fig:electronpockets\]c**]{} shows that at low temperature, the high-symmetry cut through the M point contain four bands, with two features in the Energy Dispersion Curve (EDC) at the M point separated by a famously large energy scale of $\Delta_M\sim$50 meV ($\sim$600 K).
The understanding of the ARPES spectra at the M point has been a contentious issue, and some historical context is useful. Pioneering work on multilayer thin film samples of FeSe [@Tan2013] obtained spectra qualitatively similar to [**Figure \[fig:electronpockets\]c**]{}. Due to the analogy with similar features seen on an equally large energy scale in magnetic parent compounds NaFeAs and BaFe$_2$As$_2$ around the M point, the 50 meV energy scale was first associated with spin density wave order in thin films [@Tan2013], but later on in single crystals was linked to a nematic ordering [@Nakayama2014], due to the lack of long-range magnetic order in FeSe. It was suggested that in the nematic phase the $d_{xz/yz}$ band degeneracy at the M point is dramatically lost, and the 50 meV energy scale is the magnitude of the $d_{yz}-d_{xz}$ orbital ordering (at M), with the onset of band shifts occurring at $T_s \sim 90$ K [@Shimojima2014; @Watson2015a].
Recent improvements in the data quality, particularly at temperatures above 90 K, clearly point towards a different interpretation of the band dispersions at the M point. The crucial evidence is the observation of distinct $d_{xy}$ dispersions in data obtained above 90 K [@Watson2016; @Fedorov2016; @Fanfarillo2016; @Zhang2016]. In [**Figure \[fig:electronpockets\]a**]{} the curvature analysis reveals clear evidence for the lower branch of the $d_{xy}$ dispersion at M in addition to the $d_{xz},d_{yz}$ dispersions, while the branch forming the outer $d_{xy}$ section of the Fermi surface is clearly seen in data obtained at 56 eV in [**Figure \[fig:electronpockets\]b**]{} and [**d**]{}. The fact that $d_{xy}$ dispersions also contribute brightly to the observed spectra rules out the assignment of the 50 meV splitting to only the $d_{xz}$ and $d_{yz}$ bands. Instead it is the separation of $d_{xy}$ and $d_{xz/yz}$ bands at the M point. This band separation does appear to increase in magnitude below 90 K and is therefore linked with nematic order, but is not intrinsically a measure of any orbital symmetry-breaking.
![a) Temperature-dependence of EDCs at the $\Gamma$ point, lower panel showing extracted band positions from a fitting analysis. b) Similar analysis at the M point. c) Analysis of MDCs at the A point. Figure adapted from Refs. [@Watson2016] and [@Watson2017c_arxiv].[]{data-label="fig:reviewfigarpestdepdata"}](Figure5){width="0.95\linewidth"}
Temperature-dependence of the position of the bands
---------------------------------------------------
We now focus on the details of the band shifts in the nematic phase, with the aim of determining the magnitude and momentum-dependence of the nematic order parameter. We start with the hole pockets at high temperatures in the tetragonal phase, where at the $\Gamma$ point the $d_{xz}$/$d_{yz}$ bands are separated by spin-orbit coupling only [@Fernandes2014b], with the magnitude of this separation being $\sim$20 meV [@Watson2015a; @Borisenko2015; @Watson2017c_arxiv]. While Zhang *et. al.* found no significant temperature-dependence at the $\Gamma$ point, most studies agree that there is an increased separation of the $d_{xz}/d_{yz}$ hole bands in the nematic phase [@Watson2015a; @Fedorov2016; @Watson2016], which was studied in detail in Ref. [@Watson2017c_arxiv], reproduced in [**Figure \[fig:reviewfigarpestdepdata\]a**]{}. As the spin-orbit term is expected to be temperature-independent, this increase in splitting is the signature of a symmetry-breaking orbital order parameter. It is important to note that the band separation due to spin-orbit coupling will add in quadrature, not linearly, with the extra splitting associated with the nematic order at the $\Gamma$ point [@Fedorov2016; @Fernandes2014; @Watson2017c_arxiv]. From measurements of twinned samples, it is impossible to determine whether it is the $d_{xz}$ or $d_{yz}$ orbital which is raised/lowered in energy at the $\Gamma$ point. This information is obtained by [*detwinning*]{} the samples using a mechanical strain [@Shimojima2014; @Suzuki2015], from which it may be deduced that the $d_{xz}$ orbital is raised in energy while the $d_{yz}$ orbital is lowered in energy ([**Figure \[fig:orderparameters\]a**]{}). Extracting the position of the $d_{xz}$ band in the unoccupied states is a difficult task, but in Ref. [@Watson2017c_arxiv] a value of 37.5 meV was estimated, implying that the symmetry-breaking component $\Delta_\Gamma \approx$ 29 meV after the spin-orbit coupling is accounted for. The downward shift of the $d_{yz}$ band brings the small 3D hole pocket seen at the Z point completely below the Fermi level ([**Figure \[fig:holepockets\]b**]{}). Therefore the low temperature Fermi surface at the zone centre consists of a single elliptical quasi-2D hole pocket, with the longer direction of the hole pocket oriented along the shorter $b$ axis of the orthorhombic structure ([**Figure \[fig:holepockets\]f**]{}).
[**Figure \[fig:reviewfigarpestdepdata\]b**]{} shows the EDCs at the M point as a function of temperature. In Ref. [@Watson2016] it was shown based on fitting analysis that above 90 K there are actually two features in the spectra, which are separated by $\sim$ 20 meV. These must be associated with the separate $d_{xz/yz}$ and $d_{xy}$ bands which are expected at M ([**Figure \[fig:electronpockets\]c**]{}). Below 90 K, in the orthorhombic phase the band degeneracies at the M point are no longer protected by fourfold symmetry, thus it would be natural to expect that the $d_{xz}$ and $d_{yz}$ dispersions should separate, which would give extra features in the EDC at the M point in a twinned sample. Such a separation was also observed in Ref. [@Fedorov2016], on the order of 10 meV. However most data sets at the M point show only two features even at the lowest temperatures [@Shimojima2014; @Watson2015a; @Watson2016]. If there are only two features, this would imply an unexpected situation: the $d_{xz/yz}$ and $d_{xy}$ bands have an increased separation below 90 K, reaching $\sim$50 meV by low temperatures, but no $d_{xz/yz}$ splitting is resolved. It is helpful to additionally analyse the MDCs as a function of temperature. This analysis is best performed with an incident photon energy of 56 eV, ([**Figure \[fig:reviewfigarpestdepdata\]c**]{}), where the outer branch of the electron pockets is easily distinguished at temperatures above 90 K. By following the features in the MDC by fitting with Lorentzian peak profiles, it can be easily deduced that the $d_{yz}$ $k_F$ shrinks substantially, while the outer $d_{xy}$ section increases below 90 K, so the pocket evolves from the two crossed ellipses seen at high temperature into the crossed peanut-shaped (or [*bow-tie shaped*]{} [@Sprau2016_arxiv]) bands at low temperature.
![a) Tight binding simulations of the Fermi surface in the tetragonal phase, closely based on the experimental dispersions. b) Calculated Fermi surface including the unidirectional nematic bond ordering term. c,d) Experimental Fermi surfaces at high and low temperatures. e,f) Other proposed orbital ordering scenarios, the ferro-orbital ordering and $d$-wave bond nematic orders. Figures adapted from Ref. [@Watson2016]. []{data-label="fig:orderparameters"}](Figure6){width="0.95\linewidth"}
The orbital order in FeSe
-------------------------
We now consider the implications of these observations on the determination of the form of the orbital order parameter. The simplest order parameter would be a momentum-independent orbital polarization, or ferro-orbital ordering, $\frac{\Delta}{2}(n_{yz}-n_{xz})$, shown in [**Figure \[fig:orderparameters\]e**]{}. Early on, based primarily on the M point data several groups proposed a 50 meV ferro-orbital ordering [@Shimojima2014; @Nakayama2014; @Watson2015a; @Watson2015b; @Kreisel2015]. However, this interpretation has been revised, as this effect is of much smaller magnitude at the $\Gamma$ point [@Watson2015a]. Moreover, ARPES studies under strain have established that the $d_{yz}$ band moves in opposite directions at the $\Gamma$ and at the M points [@Suzuki2015], which clearly points towards a momentum-dependent order parameter. If ferro-orbital ordering is excluded, or at least it is known to be not the primary order parameter of the transition, one must start to consider bond-type orderings [@Jiang2016], such as a $d$-wave bond nematic order [@Jiang2016; @Yi2015; @Zhang2015], plotted in [**Figure \[fig:orderparameters\]f**]{}. However since a pure $d$-wave symmetry order parameter would give no extra splitting at the $\Gamma$ point [@Jiang2016], this is excluded. A composite order parameter involving both ferro-orbital and $d$-wave bond nematic order parameter has been recently discussed [@Sprau2016_arxiv; @Scherer2017], which could perhaps account for the $\Gamma$ point data with some fine-tuning, but would give a $d_{xz/yz}$ splitting at the M point.
Another candidate is the [*unidirectional nematic bond order*]{}, introduced in Ref. [@Watson2016], which provides a simple and accurate description of the band shifts and splitting in the nematic phase. This order parameter, written as ${\Delta_u}(n'_{yz}-n'_{xz})\cos(k_x)$, does not affect the on-site orbital energies, but instead breaks fourfold symmetry in the inter-site hopping terms (i.e. $n'$ indicates a hopping term) [@Watson2016]. It has the effect of giving an extra splitting of $d_{xz}$-$d_{yz}$ bands at the $\Gamma$ point, giving a total separation of $\sqrt{\Delta_{SO}^2+(2\Delta_u)^2}$, but shifting the $d_{yz}$ and $d_{xz}$ dispersions up together by $\Delta_u$ at the M point, without breaking the degeneracy there. The experimental data at the $\Gamma$ [@Watson2017c_arxiv] and M [@Watson2016] points would indicate values of $\Delta_{u,\Gamma} \approx$14.5 and $\Delta_{u,M} \approx$ 20 meV, consistent within the experimental uncertainty. The Fermi surface and band dispersions with and without this order parameter ([**Figures \[fig:orderparameters\]a,b**]{}), which show an good correspondence with the experimental data in both phases. However at the present time a microscopic motivation for this order parameter is lacking. Moreover high resolution detwinned ARPES measurements will help to eliminate uncertainty over the role of different domains, particularly at the M point. Finally we note that with all these scenarios, a small contribution from other symmetry-allowed order parameters is to be expected, for example a small ferro-orbital ordering contribution cannot be ruled out in addition to the unidirectional nematic bond order.
FeSe as a strongly correlated system
------------------------------------
So far we have focused on the $\sim$20 meV band shifts associated with nematic ordering, but a complete picture of the physics of the system includes an understanding of the few-eV energy scale. FeSe is a strongly correlated material, exhibiting an enhancement of the quasiparticle effective masses in quantum oscillations [@Shimojima2014; @Watson2015a; @Watson2015b] and the Sommerfeld coefficient [@Bohmer2014]. However the electron-electron interactions which are primarily responsible for these effects around the Fermi level also manifest on energy scales comparable with the Fe $3d$ bandwidth, which can be probed by measuring photoemission spectra across a wide range of binding energies.
![a,b) ARPES spectra measured through the $\Gamma$ point, plotted on a wide energy scale. b) Schematic understanding of features seen in panel a). QP and LHB refer to quasiparticle dispersions and the Lower Hubbard Band. Hints of the Se $4p$ dispersions are also seen. c) Comparison of total ARPES intensity with DFT and DFT+DMFT calculations. Figures adapted from Ref. [@Watson2017a][]{data-label="fig:reviewfigarpeshighenergy"}](Figure7){width="0.95\linewidth"}
In this context, it is worth emphasizing that the quantity probed by ARPES is the one-particle spectral function, mulitplied by the relevant photoemission matrix element and the Fermi function. The spectral function contains very rich information about many-body effects in the system. Around the Fermi level, the ARPES spectra show sharp quasiparticle bands, which have primarily $d_{xz}$,$d_{yz}$ and $d_{xy}$ orbital character. [**Figure \[fig:reviewfigarpeshighenergy\]a**]{} shows an ARPES measurement obtained in LV polarisation at the $\Gamma$ point, which highlights the $d_{yz}$ spectral weight. It can be seen that the quasiparticle bands give way to a featureless region giving a dip in the total spectral intensity around $\sim$0.5 eV binding energy, after which a dispersive but much broader feature is seen, around 1-2.5 eV binding energy. With a width of $\sim$1 eV, this can hardly be described as a quasiparticle band, but rather should be considered as incoherent spectral weight, or a [*Hubbard band*]{} [@Watson2017a; @Evtushinsky2016_arxiv]. At $\sim$3-6 eV binding energies, hints of the Se $4p$ band dispersions are seen in some measurement geometries [@Watson2017a; @Evtushinsky2016_arxiv], very close to their expected locations according to DFT calculations.
As shown in [**Figure \[fig:reviewfigarpeshighenergy\]c**]{}, this structure in the Fe $3d$ bandwidth is captured well within the DFT+DMFT technique, which accounts for the large on-site Coulomb repulsion term $U$=4 eV and the Hund’s rule interaction $J_H$=0.8 eV which are not adequately treated within DFT alone. The calculated total spectral function exhibits a similar peak-dip-hump structure to the experimental data. Not every detail can be accounted for in the DFT+DMFT calculations, and in particular both DFT and DFT+DMFT predict much larger sizes of the Fermi surfaces than are seen experimentally. The general picture is that FeSe is a strongly correlated material, in which the on-site $U$ and $J_H$ play an important role in shaping the overall structure of the $3d$ bandwidth, but other effects such as the interatomic Coulomb interactions $V$ [@Jiang2016; @Scherer2017] may also be relevant for understanding finer details of the electronic structure.
QUANTUM OSCILLATIONS IN FESE
============================
The observation of quantum oscillations in single crystals of FeSe grown using chemical vapor transport have been reported by different groups [@Terashima2014; @Audouard2014; @Watson2015a; @Watson2015b], using magnetotransport and tunnel diode oscillator techniques. [**Figure \[fig:QOs\]**]{} shows the transverse magnetoresistivity as a function of magnetic field up to 45 T at 0.4 K for a single crystal of FeSe with a residual resistivity ratio larger than 25. Shubnikov-de Haas oscillations are detected in the normal state above 18 T on top of the strongly magnetoresistive background and a complex oscillatory signal is revealed after the background subtraction in the inset of [**Figure \[fig:QOs\]a**]{}. The fast Fourier transform (FFT) spectra show a large number of closely spaced peaks corresponding to quantum oscillation frequencies below 700 T ([**Figure \[fig:QOs\]b**]{}). These frequencies are significantly smaller than those seen in other highly crystalline iron-based superconductors such as LiFeAs, LiFeP and BaFe$_2$(As$_{1-x}$P$_x$)$_2$, suggesting that the low temperature Fermi surface of FeSe has small pockets, a factor five smaller that the largest frequency predicted by DFT calculations of FeSe [@Watson2015a], and occupying less than 2.3% of the first Brillouin zone [@Terashima2014].
What can be learned from quantum oscillations?
==============================================
Quantum oscillations is a well-established and powerful technique for the experimental characterisation of the Fermi surface at low temperatures. Due to the Landau quantisation of electronic states in an applied magnetic field $B$, oscillations of various physical properties periodic in $1/B$ are observed ([**Figure \[fig:QOs\]a**]{}). The frequency of oscillation relates to extremal areas of the Fermi surface, and the temperature-dependence of the amplitude of oscillations reveals the orbitally-averaged quasiparticle masses.
The first major constraint for the observation of quantum oscillations is that the cyclotron energy which separates Landau levels needs to be larger than the broadening of the levels $\hbar/\tau$ due to scattering Thus only very high quality single crystals with long mean free paths show quantum oscillations. Secondly, the amplitude of the quantum oscillations is significantly damped for heavier quasiparticle masses as a result of the smearing of the Landau levels by the Fermi-Dirac distribution. Lastly, in order to observe quantum oscillations in a superconducting system, the superconductivity needs to be suppressed by a magnetic field which exceeds the upper critical field, since the gapping of quasiparticle states very strongly suppresses any oscillatory signal.
While quantum oscillations give values of the extremal cross sectional areas with a greater precision than ARPES, the assignment of quantum oscillation frequencies to different pockets in $k$-space can only be done by comparison with theoretical modelling or other techniques, and can be a complex problem in the presence of several Fermi surface pockets. However the shape of the Fermi surface can be determined from the angular dependence of these frequencies [@Terashima2014; @Watson2015a]. For quasi-2D pockets it is instructive to plot $F\cos{\theta}$ as a function of $\theta$, the angle between the applied field and the sample normal ([**Figure \[fig:kz-dispersion\]c**]{}). On such a plot, a flat line would correspond to a perfect cylinder, while minimal (maximal) extremal areas of a warped quasi-2D band would have upward (downward) curvature.
In ultra-high magnetic fields, magnetotransport and Hall effect data provided an interesting insight into the origin of the largest-amplitude $\beta$,$\gamma$ and $\delta$ peaks [@Watson2015b]. By considering the relative amplitudes of the quantum oscillations of the $\rho_{xx} $ and $\rho_{xy} $ components, together with the positive sign of the high-field Hall signal at very low temperatures, it was revealed that the single hole band is associated with the $\beta$ and $\delta$ extremal areas in [**Figure \[fig:QOs\]b**]{} and [**Figure \[fig:kz-dispersion\]**]{}. These bands also have similar effective masses of around 4 $m_e$ [@Terashima2014; @Audouard2014; @Watson2015a]. This assignment is in good agreement with the areas estimated from ARPES, with $k_F$ values varying from 0.1-0.15 Å$^{-1}$ around the hole pocket at Z [@Watson2015a]. Thus the hole band gives small carrier density of 3.58$\times 10^{20}$ cm$^{-3}$ and contributes 3.1(4) mJ/mol K$^2$ to the electronic specific heat.
![Three-dimensional representation of the low temperature quasi-2D Fermi surfaces of FeSe, from photon-energy dependent ARPES measurements for the hole band in a) and electron bands in e) . The schematic representation of the Fermi surface in (b,d) to illustrate the position of the maximum and minima extremal orbits determined in quantum oscillations. c) Angular dependence of the quantum oscillation frequencies; intensity map represents the amplitude of oscillations.[]{data-label="fig:kz-dispersion"}](Figure9){width="1\linewidth"}
![a) Low temperature Fermi surface map of FeSe [@Watson2016]. b) Possible orbits on the Fermi surface responsible for the observed quantum oscillation frequencies in FeSe and FeSe$_{1-x}$S$_{x}$ [@Watson2015a; @Coldea2016]. c) An alternative scenario of the electron pockets which could give tiny electron pockets with a Dirac-like dispersion. d) Schematic Fermi surface of FeSe at high temperatures in the tetragonal phase and e) for a FeSe$_{1-x}$S$_{x}$ sample where nematicity is fully suppressed. f) Low temperature tetragonal Fermi surface of FeSe$_{0.82}$S$_{0.18}$ [@Reiss2017].[]{data-label="fig:FS_evolution_nematicity"}](Figure10){width="1\linewidth"}
The $\gamma$ frequency, the maximum of a quasi-2D pocket associated with the outer electron orbit, has a notably heavier effective mass of $\sim 7(1)$ $m_e$ [@Watson2015a; @Terashima2014]. The minimum orbit of this pocket has been assigned either to $\alpha_1$ [@Terashima2014], suggesting that only one electron pocket is present, or alternatively associated to the weak $\epsilon$ peak [@Coldea2016], which also account for the presence of an additional very small electron pocket ($\alpha_1$ and $\alpha_2$) ( [**Figure \[fig:kz-dispersion\]**]{}). The presence of a large electron pocket, having an almost compensated carrier density to the hole bands, as well as that of a second tiny electron pocket was also determined from magnetotransport studies in FeSe [@Watson2015b].
We now consider the different possible scenarios for the origin of the quantum oscillations associated with the electron pockets. In general, the orbits that are detected by quantum oscillations for the electron bands in non-magnetic iron-based superconductors originate from the inner and outer electron bands, separated by a sizable spin-orbit hybridisation ([**Figure \[fig:FS\_evolution\_nematicity\]d,e**]{}), as seen in LaFePO, LiFeAs and LiFeP [@Coldea2008; @Putzke2012]. However, in FeSe the impact of nematic ordering, and the sensitivity of the Fermi surface to very small changes of the chemical potential, complicates the interpretation of the data. From the ARPES perspective, $\gamma$ frequency could be assigned to the four-leaf clover shaped orbit at the A point, shown in [**Figure \[fig:FS\_evolution\_nematicity\]b**]{}. Alternatively, due to the strongly distorted electron Fermi surface, this frequency could originate from other trajectories either caused by breakdown orbits or other effects induced by the strong magnetic fields and at very low temperatures. Small changes in the band positions relative to the chemical potential (1–2 meV) could push the inner electron bands above the Fermi level at the M point ([**Figure \[fig:reviewfigarpestdepdata\]**]{}), and lead to single elliptical orbit and two tiny, [*Dirac-like*]{}, electron pockets ([**Figure \[fig:FS\_evolution\_nematicity\]c**]{}). Thus, the exact details of the electron bands in FeSe are very sensitive to small changes in the band structure parameters and there can be topologically significantly different Fermi surfaces arising from variations of the band positions of only a few meV in the presence of spin-orbit coupling as well as in high magnetic fields.
Magnetotransport: small and tiny electron pockets
-------------------------------------------------
At temperatures above 100 K, the magnetoresistance and Hall effect can be well-described as a compensated two carrier system [@Watson2015b]. The Hall effect is linear, and the hole and electron pockets have rather similar mobilities leading to a small overall value of the Hall coefficient which changes sign more than once as a function of temperature. However at temperatures below $\sim$80-90 K the Hall effect becomes noticeable non-linear, changing sign between the low and high magnetic field regimes ([**Figure \[fig:QOs\]c)**]{}. This behavior can be described by going beyond the two-band model, and considering the presence of an additional tiny electron band with higher mobility than the either the hole band or the larger electron band [@Watson2015b; @Huynh2016; @Sun2016mob].
The origin of a tiny electron pocket with a carrier density of about 0.7$\times 10^{20}$ cm$^{-3}$, a factor 5 smaller than the hole band, could originate as an inner electron band [**Figure \[fig:FS\_evolution\_nematicity\]d**]{} and give rise to the lowest frequencies in the quantum oscillations spectra ($\alpha_1$ and/or $\alpha_2$ in [**Figure \[fig:QOs\]b**]{}). Another scenario is that the presence of the small number of highly mobile carrier is linked to the [*Dirac-like*]{} dispersions in the nematic phase [@Tan2016] on some sections of the electron pockets [@Huynh2016]. As a result of the band shifts in the nematic phase, the $d_{xz}$ and $d_{xy}$ bands cross away from the M point near the ends of the peanut-shapes ([**Figure \[fig:FS\_evolution\_nematicity\]c**]{}). However these band crossings are gapped due to spin-orbit coupling [@Watson2016], (although this has been hard to resolve with ARPES), and near-linear dispersions exist only in a very limited regime. High mobility carriers have been detected across the whole nematic phase of FeSe$_{1-x}$S$_x$ [@Sun2016mob; @Ovchenkov2017].
However, the understanding of magnetotransport in FeSe is a subtle problem, and the complex large crossed-peanut shaped electron bands will have a strongly varying effective mass, and the Fermi surface curvature effects could play a role [@Ong1991]. Moreover, the scattering rate around the pocket could also vary strongly, being related to the orbital character on various sections. Anisotropic scattering rates were attributed to spin fluctuation scattering in Ref. [@Tanatar2016], and anisotropy in the relaxation time of excited electrons was detected in pump-probe measurements [@Luo2016_arxiv]. A large distribution of electron mobilities was also suggested by the mobility spectrum analysis in Ref. [@Huynh2016].
Orbitally-dependent electronic correlations
-------------------------------------------
The Fe-chalcogenides are widely considered to be more strongly correlated than their Fe-pnictide cousins [@Yin2011; @vanRoekeghem2016; @Lanata2013; @Yi2015_natcomm]. The effect of electronic correlations can be estimated by comparing DFT band structure calculations with the measured quasiparticle band dispersion in ARPES in the tetragonal phase of FeSe. It was found that the band renormalisation factors vary significantly between $\sim$ 3.2, 2.1 for the outer hole bands ($\alpha$,$\beta$) with $d_{xz/yz}$ orbital character, and a factor $\sim$8-9 for the $\gamma$ pocket, with $d_{xy}$ orbital character [@Maletz2014; @Watson2015a]. However substantially larger band renormalisations have been observed in FeSe$_x$Te$_{1-x}$, where the band-selective renormalization can reach 17 [@Tamai2010; @ZKLiu2015], while correlations are expected to be weaker in FeS [@Miao2017]. The effective masses observed by quantum oscillations are also renormalised, with a particularly large mass enhancement on the outer electron pocket with largely $d_{xy}$ character [@Watson2015a; @Coldea2016]. In electron-doped FeSe-based systems it has been shown that the $d_{xy}$ spectral weight depletes with increasing temperature due to particularly strong correlations effects [@Yi2015_natcomm].
The orbital selectivity of the observed renormalisations has been interpreted as evidence that the proximity to an orbital-selective Mott transition is a key to the understanding of the Fe-based superconductors [@Lanata2013]. In this picture, FeSe is a system in which the $d_{xy}$ orbital is closer to localisation than the other orbitals, due to its occupation being closer to half-filling. Calculations using DMFT on FeSe give band renormalisations of $\sim$2.8 for the $d_{xz/yz}$ orbitals, comparable to the measured values, although the predicted value of $\sim$3.5 for the $d_{xy}$ band underestimates the experimental value [@Yin2011]. Moreover the incoherent [*Hubbard bands*]{} observed by ARPES at high binding energies are qualitatively captured by DMFT [@Watson2017a; @Evtushinsky2016_arxiv]. However a narrowing of the bandwidth is also predicted from the inter-site Coulomb interaction $V$ [@Jiang2016], and non-local correlations are likely to be important.
FERMI SURFACE SHRINKING
=======================
Quantum oscillations, magnetotransport and ARPES agree on an important and unusual aspect of the electronic structure of FeSe: the small size of the quasi-2D electron and hole pockets, which are much smaller than the expectations from DFT. In a compensated system, the total charge is conserved when the holes and electron pockets shrink simultaneously, but the origin of this effect is not yet established. Similar Fermi surface shrinking effects have been observed with quantum oscillations in e.g. LaFePO [@Coldea2008] and BaFe$_2$(As$_{1-x}$P$_x$)$_2$ [@Shishido2010], which were sometimes quantified in terms of the typically 50-100 meV rigid shifts of the unrenrormalised DFT bands that would be required to account for the experimental Fermi surface area.
In FeSe, in order to bring the calculated Fermi surface close to the those determined experimentally, large shifts of up to 200 meV [@Watson2015a] would be needed. Momentum-dependent band shifts with opposite sign at the $\Gamma$ and M points have also been described from the ARPES perspective [@Borisenko2015]. A magnetic reconstruction of the Fermi surface can lead to small quantum oscillation frequencies, as seen in BaFe$_2$As$_2$ [@Terashima2011]. However in FeSe the nematic order gives in-plane distortions but does not dramatically affect the in-plane areas; the pockets are small even in the tetragonal phase [@Watson2015a; @Watson2016].
This shrinking effect has been suggested to be a natural consequence of the strong particle-hole asymmetry of electronic bands, providing an indirect experimental evidence of strong interband scattering, as a direct consequence of the coupling to a bosonic mode [@Ortenzi2009]. Recently, it has been suggested that in FeSe, due to the strong orbital-depending correlation effects, spin fluctuations between hole and electron pockets are responsible for an orbital-dependent shrinking of the Fermi surface that affects mainly the $xz/yz$ parts of the Fermi surface [@Fanfarillo2016]. An alternative perspective to explain the small and strongly renormalized low-energy band structure of FeSe is related to the consideration nearest neighbor Coulomb interaction, $V$. This interaction generates hopping corrections to the band dispersions and pushes the van Hove singularity at M point up towards the Fermi energy while pulling down the top of the hole band at the $\Gamma$ point. The interatomic $V$ has also been discussed as a possible microscopic origin of the nematic ordering, where it is thought to favor a $d$-wave nematic bond order [@Jiang2016; @Scherer2017].
Chemical potential effects in the tetragonal phase
--------------------------------------------------
Recent temperature-dependent ARPES studies of FeSe up to room temperature report continuous temperature-dependent band shifts up to $\sim $ 25 meV even in the tetragonal phase [@Rhodes2017_arxiv; @AbdelHafiez2016; @Kushnirenko2017_arxiv]. Similar effects have been observed other iron-based superconductors, such as Ba(Fe,Co)$_2$As$_2$ [@Brouet2013] and Ba(Fe,Ru)$_2$As$_2$ [@Dhaka2013], being assigned to temperature-induced chemical potential shifts. In semimetals, where the top of the hole bands and bottom of the electron bands are very close to the chemical potential within the energy range of thermal broadening ($\mu \sim k_B T$), significant chemical potential shifts can occur due to balance the thermal population of carriers. This effect has interesting implications on the understanding of transport and magnetic measurements in this temperature range [@Rhodes2017_arxiv]. The chemical potential shift may also have influence at the nematic transition: a natural consequence of the nematic order parameter would be a shift of the chemical potential to preserve charge, which might account for the $\sim$ 10 meV momentum-independent downward shift of the $d_{xy}$ bands ([**Figure \[fig:reviewfigarpestdepdata\]**]{}).
TUNING NEMATIC ORDER WITH PHYSICAL AND CHEMICAL PRESSURE
========================================================
The nematic state of bulk FeSe can be tuned and suppressed either by using applied hydrostatic pressures, isoelectronic substitution or chemical doping. Applied hydrostatic pressure ([**Figure \[fig:fesesx-schematicphasediagram\]b**]{}) initially suppresses nematicity [@Medvedev2009; @Terashima2015; @Sun2016; @Kothapalli2016] and then stabilizes a new magnetic state [@Bendele2012; @Kothapalli2016], likely to be a stripe magnetic ordering. From this point onwards, the phase diagram resembles other Fe-based superconductors, with a high-$T_c$ phase being found once this magnetic order is suppressed at very high pressures [@Sun2016].
Another tuning parameter of nematicity is isoelectronic substitution achieved by replacing selenium ions for isoelectronic but smaller sulfur atoms in FeSe$_{1-x}$S$_x$, which acts as a chemical pressure effect [@Watson2015c; @Mizuguchi2009]. Surprisingly, this tuning parameter does not stabilize any long range magnetic order outside the nematic phase [@Watson2015c; @Coldea2016]. Thus FeSe$_{1-x}$S$_x$ is an interesting and unique phase diagram in which the nematic transition temperature can be tuned to zero temperature without any magnetic order ([**Figure \[fig:fesesx-schematicphasediagram\]a)**]{}; perhaps surprisingly the superconducting transition temperature change very little in response.
![a) Schematic phase diagram of FeSe$_{1-x}$S$_x$, based on Refs. [@Watson2015c; @Coldea2016]. Nematicity is suppressed both as a function of temperature and sulphur substitution [@Watson2015c; @Reiss2017]. b) Schematic phase diagram of FeSe under pressure, based on Ref. [@Sun2015]. Hatched area could indicate a regime of coexisting superconductivity and antiferromagnetism [@Bendele2012], while SC2 refers to the second dome of superconductivity with much higher $T_c$.[]{data-label="fig:fesesx-schematicphasediagram"}](Figure11){width="0.8\linewidth"}
ARPES studies of FeSe$_{1-x}$S$_x$ showed a clear reduction in Fermi surface anisotropy and orbital ordering effects for both electron and hole Fermi surfaces [@Watson2015c]. For compositions beyond the nematic phase boundary, no anisotropies remain ([**Figure \[fig:fesesx-schematicphasediagram\]a**]{}) [@Reiss2017]. Recent quantum oscillations studies of FeSe$_{1-x}$S$_x$ suggest a continuous expansion of the outer electron and hole bands with chemical pressure even outside the nematic state [@Coldea2016]. This is in contrast to the observation of only small Fermi surfaces in FeSe under pressure, suggested to result from magnetic reconstruction [@Terashima2015]. Nematic susceptibility studies as a function of chemical pressure have revealed the possible presence of a nematic critical point [@Hosoi2016]. However further evidence for quantum criticality, such as a divergent quasiparticle effective masses in quantum oscillations [@Coldea2016] or thermodynamic probes [@Abdel-Hafiez2015], remains to be found.
Another interesting aspect of the electronic structure of FeSe$_{1-x}$S$_x$ are the possible changes of the Fermi surface topology, resulting from the suppression of nematic ordering. The inner hole pocket which is pushed below the chemical potential by nematic ordering in FeSe emerges at low temperatures in ARPES data for substitutions higher than 11% [@Watson2015c], while the electron pockets lose their in-plane distortions. Quantum oscillations measurements in very high magnetic fields detect a prominent low frequency oscillation around $x=0.12$ [@Coldea2016] that could tentatively be linked with the re-emerging inner hole pocket ([**Figure \[fig:FS\_evolution\_nematicity\]e**]{}). However, this large amplitude low frequency oscillation is not detected at higher sulphur substitution beyond the nematic phase [@Coldea2016]. This implies that there could be other topological changes in the Fermi surface, a possible Liftshitz transition, as a function of chemical pressure, involving other small electron bands or breakdown orbits [@Coldea2016]. Hall effect measurements in low fields suggest that the high mobility electrons seem to disappear outside the nematic state in FeSe$_{1-x}$S$_x$ [@Ovchenkov2016].
THE INTERPLAY OF NEMATIC AND SUPERCONDUCTING ORDERS
===================================================
The normal state nematic electronic structure of FeSe with highly anisotropic Fermi surfaces has profound implications on the superconducting pairing and the symmetry of the superconducting order parameter. A highly twofold anisotropic superconducting gap has been recently found by ARPES in FeSe$_{0.93}$S$_{0.07}$ [@Xu2016]. Moreover recent Bogoliubov quasiparticle scattering interference imaging at very low temperatures also found an extremely anisotropic superconducting gap in FeSe. The gap structure is nodeless, though the gap reaches rather small values on the major axis of the elliptical hole pocket, and the gap has opposite sign between the hole and the small electron pockets [@Sprau2016_arxiv]. Although there has been some reports suggesting the presence of nodes in the superconducting gap of FeSe [@Song2011; @Moore2015; @Kasahara2014], most of the thermodynamic and thermal conductivity studies of bulk FeSe in the superconducting phase can be explained if at least two different nodeless superconducting gaps are present [@Lin2011; @Bourgeois-Hope2016].
The strong anisotropy of the superconducting gap is an important indication of unconventional pairing in FeSe. It has been suggested that the anisotropic gap is due to an orbital-dependent pairing mechanism, with the maximum gap on sections with $d_{yz}$ character, and a small gap on sections with $d_{xz}$ or $d_{xy}$ character [@Kreisel2016_arxiv; @Sprau2016_arxiv]. In this picture, superconductivity results from the nesting of $d_{yz}$ sections of the hole and electron bands which couples strongly to ($\pi,0$) magnetic fluctuations, whereas the $d_{xy}$ character does not participate.
Due to the presence of a small electron band with small Fermi energy and comparable to the superconducting gap is suggested that FeSe could be placed into BCS-BEC cross-over regime [@Kasahara2014]. Furthermore, as the Zeeman energy for this small electron band also becomes comparable to its Fermi energy and gap energy, a highly spin-polarized, pairing superconductivity is inferred to form in high magnetic fields at low temperatures [@Watashige2017].
As a function of chemical pressure in FeSe$_{1-x}$S$_x$, the presence of multi-gap superconductivity is preserved [@Abdel-Hafiez2015] whereas tunnelling experiments found that the vortex core anisotropy is strongly suppressed [@Moore2015]. In FeSe, high-resolution thermal expansion showed a lack of coupling between the orthorhombicity and superconductivity [@Bohmer2013]. However in samples with up to 15% sulphur substitution and correspondingly reduced tetragonal-to-orthorhombic transition temperatures, an enhancement of the orthorhombic distortion in the superconducting state [@Wang2016Meingast] was observed. This may indeed indicate that superconductivity favors the nematic state.
The interplay of nematic, superconducting and magnetic orders has attracted much interest. There is a large body of recent theoretical work on FeSe addressing the competing instabilities of the nematic order in relation to spin-density wave and superconductivity, as detailed in Refs. [@Xing2017; @Kreisel2015; @Yamakawa2016], as well as on the role played by strong inter-site Coulomb interactions [@Jiang2016; @Scherer2017].
CONCLUSION
==========
FeSe has opened up new avenues towards understanding unconventional superconductivity in Fe-based superconductors. Experimental efforts on this simple system have started to reveal the key ingredients responsible for its still mysterious nematic behavior and its the highly tunable superconductivity, while exposing its complexities. Insights have been gained from various experimental probes, which are now pointing towards a common picture: FeSe is a system where strong correlations, orbital-selectivity, spin-orbit coupling, nematic bond ordering and fluctuating magnetism are all important. FeSe is perhaps the cleanest example of a nematic phase in condensed matter, with the small size of the Fermi surface pockets and small Fermi energies of the bands being distinguishing features of its multi-band, multi-orbital electronic structure.
1. FeSe is a fascinating quantum material which displays a unique nematic electronic state, from which a highly anisotropic superconductivity emerges.
2. The nematic state of FeSe is characterised by the development of an unusual momentum-dependent orbital ordering, described by a bond ordering and not on-site occupations.
3. FeSe is a strongly correlated system, belonging to the class of iron chalcogenides, in which orbital-dependent quasiparticle mass renormalizations are important, with the $d_{xy}$ orbitals being the most correlated.
4. The Fermi surface of FeSe at low temperature is that of a compensated semimetal, with one small elliptical hole band and two electron bands, one of the electron bands having an unusually high mobility.
5. The sizes of the quasi-2D Fermi surfaces are much smaller than predicted by *ab-initio* calculations, likely linked to non-local interactions such as the inter-site Coulomb repulsion.
6. The low energy electronic structure of FeSe reveals many small comparable energy scales: spin-orbit coupling, nematic order, the effective Fermi energies of bands, and superconductivity.
7. Tuning the system with chemical and physical pressure can lead to quite dramatic effects: induced magnetism and high-$T_c$ are found under pressure, while the suppression of nematic order does not lead to any strongly enhanced $T_c$ in FeSe$_{1-x}$S$_x$.
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1. Is the origin of the anisotropy of the superconducting gap in FeSe a simple consequence of its nematic electronic structure or is a manifestation of a highly anisotropic orbitally-dependent pairing mechanism ?
2. Are nematic fluctuations important for enhancing superconductivity in FeSe under pressure or the spin fluctuations are always in the driving seat ?
3. What is the role played by the small electronic bands with high mobility for superconductivity in FeSe ?
4. Can chemical and applied pressure in FeSe help to disentangle the role of nematicity, magnetism, orbital order and electronic correlations in FeSe and identify the direct route towards high-$T_c$ superconductivity ?
5. Can high-resolution detwinned ARPES measurements of FeSe finally settle the open questions regarding the orbital order parameter in FeSe?
6. The large 50 meV energy scale seen in ARPES in FeSe is very similar to previous results in NaFeAs and BaFe$_2$As$_2$. Is this a universal feature of the nematic phase, independent of magnetic order?
DISCLOSURE STATEMENT {#disclosure-statement .unnumbered}
====================
The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
We are very grateful to all our numerous collaborators for their important scientific contribution to the understanding of the electronic structure of FeSe. Special thanks go to Timur Kim and Moritz Hoesch, for the development of I05 and technical support which was instrumental in obtaining the high-resolution ARPES data on FeSe. We are also very thankful to Amir Haghighirad, Thomas Wolf and Shigeru Kasahara for the growth of high quality crystals, Andy Schofield, Roser Valenti, Steffen Backes for theoretical support, Andrey Chubukov, Peter Hirschfeld, Brian Anderson, Rafael Fernandes and Oskar Vafek, A. Kreisel, Ilya Eremin and Takasada Shibauchi for numerous insightful discussions. We thank Alix McCollam, David Graf, Eung San, William Knafo, Pascal Reiss, Samuel Blake, Mara Bruma for their contributions to high magnetic field studies as well as Puning Zhao for technical support. We acknowledge the financial support provided by EPSRC (EP/I004475/1, EP/I017836/1) and Diamond Light Source for experimental access to the I05 Beamline. Amalia Coldea is grateful for an EPSRC Career Acceleration Fellowship (EP/I004475/1).
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|
---
author:
- Sébastien Boucksom
title: Higher dimensional Zariski decompositions
---
\[section\] \[theo\][Corollary]{} \[theo\][Definition]{} \[theo\][Proposition]{} \[theo\][Lemma]{}
$Author's$ $address$: Institut Fourier, 100 rue des Maths, BP74, 38402 Saint-Martin d’Hères Cedex, France.\
$e-mail$: sbouckso@ujf-grenoble.fr\
$Abstract$: using currents with minimal singularities, we construct minimal multiplicities for a real pseudo-effective $(1,1)$-class $\alpha$ on a compact complex $n$-fold $X$, which are the local obstructions to the numerical effectivity of $\alpha$. The negative part of $\alpha$ is then defined as the real effective divisor $N(\alpha)$ whose multiplicity along a prime divisor $D$ is just the generic multiplicity of $\alpha$ along $D$, and we get in that way a divisorial Zariski decomposition of $\alpha$ into the sum of a class $Z(\alpha)$ which is nef in codimension 1 and the class of its negative part $N(\alpha)$, which is exceptional in the sense that it is very rigidly embedded in $X$. The positive parts $Z(\alpha)$ generate a modified nef cone, and the pseudo-effective cone is shown to be locally polyhedral away from the modified nef cone, with extremal rays generated by exceptional divisors. We then treat the case of a surface and a hyper-Kähler manifold in some detail: under the intersection form (resp. the Beauville-Bogomolov form), we characterize the modified nef cone and the exceptional divisors; our divisorial Zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a Zariski decomposition on a projective surface, which is thus extended to the hyper-Kähler case. Finally, we explain how the divisorial Zariski decomposition of (the first Chern class of) a big line bundle on a projective manifold can be characterized in terms of the asymptotics of the linear series $|kL|$ as $k\to\infty$.\
2000 Mathematics Subject Classification: 32J25
Introduction
============
It is known since the pioneering work of O.Zariski \[Zar62\] that the study of the linear series $|kL|$ where $L$ is a line bundle on a projective surface can be reduced to the case where $L$ is numerically effective (nef). The more precise result obtained by Zariski is that any effective ${\mathbf{Q}}$-divisor $D$ on a projective surface $X$ can be uniquely decomposed into a sum $D=P+N$ where $P$ is a nef ${\mathbf{Q}}$-divisor (the positive part), $N=\sum a_j D_j$ is an effective ${\mathbf{Q}}$-divisor (the negative part) such that the Gram matrix $(D_i\cdot D_j)$ is negative definite, and $P$ is orthogonal to $N$ with respect to the intersection form. Zariski shows that the natural inclusion $H^0(kP)\to H^0(kL)$ is necessarily an isomorphism in that case, relating the decomposition to the original problem.\
The proof of the uniqueness in this decomposition shows that the negative part $N$ only depends on the class $\{D\}$ of $D$ in the Néron-Severi group $NS(X)$, so that $\{D\}\mapsto\{P\}$ yields a map from part of the pseudo-effective cone to the nef cone, which we want to study geometrically.\
Building upon the construction by J.-P.Demailly of metrics with minimal singularities on a pseudo-effective line bundle $L$ over a compact complex $n$-fold, we define the minimal multiplicity $\nu(\alpha,x)$ of an arbitrary real pseudo-effective $(1,1)$-class $\alpha$ on a compact complex $n$-fold $X$ at some point $x\in X$. This multiplicity $\nu(\alpha,x)$ is the local obstruction at $x$ to the numerical effectivity of $\alpha$, and we then get the negative part of such a class $\alpha$ by setting $N(\alpha)=\sum\nu(\alpha,D)D$, where $D$ ranges over the prime divisors of $X$ and $\nu(\alpha,D)$ is the generic multiplicity of $\alpha$ along $D$ (cf. section 3). This negative part $N(\alpha)$ is an effective ${\mathbf{R}}$-divisor which is exceptional in the sense that it is very rigidly imbedded in $X$. When $X$ is a surface, the divisors we obtain in that way are exactly the effective ${\mathbf{R}}$-divisors whose support $D_1,...,D_r$ have negative definite Gram matrix $(D_i\cdot D_j)$.\
The difference $Z(\alpha):=\alpha-\{N(\alpha)\}$ is a real $(1,1)$-class on $X$ which we call the Zariski projection of $\alpha$. It is not a nef class, but is somehow nef in codimension 1. More precisely, we define the modified nef cone of a Kähler $n$-fold to be the closed convex cone generated by the classes in ${H^{1,1}(X,{\mathbf{R}})}$ which can be written as the push-forward of a Kähler class by a modification. We then show that the Zariski projection $Z(\alpha)$ of a pseudo-effective class $\alpha$ belongs to this modified nef cone. The decomposition $\alpha=Z(\alpha)+\{N(\alpha)\}$ we call the divisorial Zariski decomposition, and it is just induced by the Siu decomposition of a positive current with minimal singularities in $\alpha$ when the latter is big. For such a big class, we give a criterion to recognize a decomposition $\alpha=p+\{N\}$ into a modified nef and big class and the class of an effective real divisor as the divisorial Zariski decomposition of $\alpha$, in terms of the non-Kähler locus of $p$ (cf. section 3.5)\
The geometric picture is now as follows: the pseudo-effective cone of a compact complex $n$-fold $X$ is locally polyhedral away from the modified nef cone, with extremal rays that write ${\mathbf{R}}_+\{D\}$ for some exceptional prime $D$ of $X$. The Zariski projection $Z$ yields a projection from the pseudo-effective cone to the modified nef cone parallel to these exceptional rays, which map is concave (in some sense) and homogeneous, but not continuous up to the boundary of the pseudo-effective cone in general. The fibre $Z^{-1}(p)$ of $Z$ above a modified nef class $p$ is a countable union of simplicial cones generated by exceptional families of primes.\
When $X$ is a surface, a modified nef class is just a nef class; when $\alpha$ is the class of an effective ${\mathbf{Q}}$-divisor $D$ on a projective surface, the divisorial Zariski decomposition of $\alpha$ is just the original Zariski decomposition of $D$. More generally, we show that the divisorial decomposition of a pseudo-effective class $\alpha$ on a Kähler surface is the unique orthogonal decomposition of $\alpha$ into the sum of a modified nef class and the class of an exceptional (in some sense) effective ${\mathbf{R}}$-divisor. This fact accounts for the rationality of the Zariski decomposition on a surface, meaning that the negative part $N$ is rational when $D$ is.\
An interesting fact is that much of the well-known case of a surface carries on to the case where $X$ is a compact hyper-Kähler manifold. Using the quadratic Beauville-Bogomolov form on ${H^{1,1}(X,{\mathbf{R}})}$ and deep results due to D.Huybrechts, we can prove the following facts: a family of primes is exceptional in our sense iff their Gram matrix is negative definite. In particular, a prime is exceptional iff it has negative square, and this forces it to be uniruled. The modified nef cone of a hyper-Kähler manifold is just the dual cone to the pseudoeffective cone, which is also the closure of the so-called birational (or bimeromorphic) Kähler cone. Finally, the divisorial Zariski decomposition is the unique orthogonal decomposition into a modified nef class and an exceptional divisor. In particular, the divisorial Zariski decomposition is also rational in that case.\
In a last part, we explain how to tackle the above constructions in a more algebraic fashion. When $L$ is a big divisor on a projective manifold, we prove that the divisorial Zariski projection of $L$ is the only decomposition $L=P+N$ into real divisors with $P$ modified nef and $H^0(\lfloor kP\rfloor)=H^0(kL)$ for every $k$. The minimal multiplicities of $\{L\}$ (and thus its negative part) can be recovered from the asymptotic behaviour of the sections of $kL$. The case of a general pseudo-effective line bundle $L$ is then handled by approximating it by $L+{\varepsilon}A$, where $A$ is ample.\
The methods used in this paper are mostly “transcendental”, since we heavily rely on the theory of currents, but the results we aim at definitely belong to algebraic geometry, and we have thus tried to make the paper legible for more algebraically inclined readers by providing in the first section a rather detailed account of the tools we need afterwards.
Technical preliminaries
=======================
${\partial\overline{\partial}}$-cohomology
------------------------------------------
When $X$ is an arbitrary complex manifold, the ${\partial\overline{\partial}}$-lemma of Kähler geometry does not hold, and it is thus better to work with ${\partial\overline{\partial}}$-cohomology. We will just need the $(1,1)$-cohomology space $H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$, which is defined as the quotient of the space of $d$-closed smooth $(1,1)$-forms modulo the ${\partial\overline{\partial}}$-exact ones. The real structure on the space of forms induces a real structure on $H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$, and we denote by ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ the space of real points.\
The canonical map from $H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$ to the quotient of the space of $d$-closed $(1,1)$-currents modulo the ${\partial\overline{\partial}}$-exact ones is injective (because, for any degree $0$ current $f$, ${\partial\overline{\partial}}f$ is smooth iff $f$ is), and is also surjective: given a closed $(1,1)$-current $T$, one can find a locally finite open covering $U_j$ of $X$ such that $T={\partial\overline{\partial}}f_j$ is ${\partial\overline{\partial}}$-exact on $U_j$. If $\rho_j$ is a partition of unity associated to $U_j$ and $f:=\sum\rho_jf_j$, then $T-{\partial\overline{\partial}}f$ is smooth. Indeed, on $U_i$, it is just ${\partial\overline{\partial}}(\sum_j\rho_j(f_i-f_j)$, and each $f_i-f_j$ is smooth since it is even pluri-harmonic. As a consequence, a class $\alpha\in H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$ can be seen as an affine space of closed $(1,1)$-currents. We denote by $\{T\}\in H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$ the class of the current $T$. Remark that $i{\partial\overline{\partial}}$ is a real operator (on forms or currents), so that if $T$ is a real closed $(1,1)$-current, its class $\{T\}$ lies in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ and consists in all the closed currents $T+i{\partial\overline{\partial}}\varphi$ where $\varphi$ is a real current of degree $0$.\
When $X$ is furthermore compact, it can be shown that $H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$ is finite dimensional. The operator ${\partial\overline{\partial}}$ from smooth functions to smooth closed (1,1)-forms is thus an operator between Fréchet spaces with finite codimensional range; it therefore has closed range, and the quotient map $\theta\mapsto\{\theta\}$ from smooth closed $(1,1)$-forms to $H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$ endowed with its unique finite-dimensional complex vector space Hausdorff topology is thus continuous and open.
General facts about currents
----------------------------
### Siu decomposition
Let $T$ be a closed positive current of bidegree $(p,p)$ on a complex $n$-fold $X$. We denote by $\nu(T,x)$ its Lelong number at a point $x\in X$. The Lelong super-level sets are defined by $E_c(T):=\{x\in
X,\nu(T,x)\geq c\}$ for $c>0$, and a well known result of Y.T.Siu \[Siu74\] asserts that $E_c(T)$ is an analytic subset of $X$, of codimension at least $p$. As a consequence, for any analytic subset $Y$ of $X$, the generic Lelong number of $T$ along $Y$, defined by $$\nu(T,Y):=\inf\{\nu(T,x),x\in Y\},$$ is also equal to $\nu(T,x)$ for a very general $x\in Y$. It is also true that, for any irreducible analytic subset $Y$ of codimension $p$ in $X$, the current\
$T-\nu(T,Y)[Y]$ is positive. The symbol $[Y]$ denotes the integration current on $Y$, which is defined by integrating test forms on the smooth locus of $Y$. Since $E_+(T):=\cup_{c>0}E_c(T)$ is a countable union of $p$-codimensional analytic subsets, it contains an at most countable family $Y_k$ of $p$-codimensional irreducible analytic subsets. By what we have said, $T-\nu(T,Y_1)[Y_1]-...-\nu(T,Y_k)[Y_k]$ is a positive current for all $k$, thus the series $\sum_{k\geq 0}\nu(T,Y_k)[Y_k]$ converges, and we have $$T=R+\sum_k\nu(T,Y_k)[Y_k]$$ for some closed positive $(p,p)$-current $R$ such that each $E_c(R)$ has codimension $>p$. The decomposition above is called the Siu decomposition of the closed positive $(p,p)$-current $T$. Since $\nu(T,Y)=0$ if $Y$ is a $p$-codimensional subvariety not contained in $E_+(T)$, it makes sense to write $\sum_k\nu(T,Y_k)[Y_k]=\sum\nu(T,Y)[Y]$, where the sum is implicitely extended over all $p$-codimensional irreducible analytic subsets $Y\subset X$.\
### Almost positive currents
A real $(1,1)$-current $T$ on a complex manifold $X$ is said to be almost positive if $T\geq\gamma$ holds for some smooth real $(1,1)$-form $\gamma$. Let $T\geq\gamma$ be a closed almost positive $(1,1)$-current. On a small enough open set $U$ with coordinates $z=(z_1,...,z_n)$, we write $T={\partial\overline{\partial}}\varphi$ where $\varphi$ is a degree $0$ current. Since $\gamma+Ci{\partial\overline{\partial}}|z|^2$ is a positive $(1,1)$-form on $U$ for $C>0$ big enough, we get that $i{\partial\overline{\partial}}(\varphi+C|z|^2)$ is positive, which means that $\varphi+C|z|^2$ is (the current associated to) a (unique) pluri-subharmonic function on $U$. A locally integrable function $\varphi$ on $X$ such that $i{\partial\overline{\partial}}\varphi$ is almost positive is called an almost pluri-subharmonic function, and is thus locally equal to a pluri-subharmonic function modulo a smooth function.\
The Lelong number $\nu(T,x)$ of a closed almost positive $(1,1)$-current $T$ can be defined as $\nu(T+Ci{\partial\overline{\partial}}|z|^2,x)$ as above, since this does not depend on the smooth function $C|z|^2$. Consequently, the Siu decomposition of $T$ can also be constructed, and writes $T=R+\sum\nu(T,D)[D]$, where $D$ ranges over the prime divisors of $X$, and $R$ is a closed almost positive $(1,1)$-current. In fact, we have $R\geq\gamma$ as soon as $T\geq\gamma$ for a smooth form $\gamma$.\
### Pull-back of a current
When $f:Y\to X$ is a $surjective$ holomorphic map between compact complex manifolds and $T$ is a closed almost positive $(1,1)$-current on $X$, it is possible to define its pull back $f^{\star}T$ by $f$ using the analogue of local equations for divisors: write $T=\theta+i{\partial\overline{\partial}}\varphi$ for some smooth form $\theta\in\{T\}$. $\varphi$ is then an almost pluri-subharmonic function, thus locally a pluri-subharmonic function modulo $\mathcal{C}^{\infty}$. One defines $f^{\star}T$ to be $f^{\star}\theta+i{\partial\overline{\partial}}f^{\star}\varphi$, as this is easily seen to be independent of the choices made. Of course, we then have $\{f^{\star}T\}=f^{\star}\{T\}$.
### Gauduchon metrics and compactness
On any compact complex $n$-fold $X$, there exists a Hermitian metric $\omega$ such that $\omega^{n-1}$ is ${\partial\overline{\partial}}$-closed. Such a metric is called a Gauduchon metric. As a consequence, for every smooth real $(1,1)$-form $\gamma$, the quotient map $T\mapsto\{T\}$ from the set of closed $(1,1)$-currents $T$ with $T\geq\gamma$ to ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is proper. Indeed, the mass of the positive current $T-\gamma$ is controled by $\int(T-\gamma)\wedge\omega^{n-1}$, and $\int T\wedge\omega^{n-1}=\{T\}\cdot\{\omega\}$ only depends on the class of $T$. The result follows by the weak compactness of positive currents with bounded mass. Another consequence is that the kernel of $T\mapsto\{T\}$ meets the cone of closed positive $(1,1)$-currents at the origin only.\
### Cycles as currents
One can associate to any effective $p$-codimensional ${\mathbf{R}}$-cycle $Y=a_1Y_1+...+a_rY_r$ a closed positive $(p,p)$-current $[Y]=a_1[Y_1]+...+a_r[Y_r]$, called the integration current on $Y$. The map $Y\mapsto[Y]$ so defined is injective, and a result of Thie says that the Lelong number $\nu([Y],x)$ is just the multiplicity of $Y$ at $x$. Consequently, we shall drop the brackets in $[Y]$ when no confusion is to be feared, and write for instance $T=R+\sum\nu(T,D)D$ for a Siu decomposition, because this is more in the spirit of this work.
Cones in the ${\partial\overline{\partial}}$-cohomology
-------------------------------------------------------
We now assume that $X$ is compact, and fix some reference Hermitian form $\omega$ (i.e. a smooth positive definite $(1,1)$-form). A cohomology class\
$\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is said to be pseudo-effective iff it contains a positive current;\
$\alpha$ is nef (numerically effective) iff, for each ${\varepsilon}>0$, $\alpha$ contains a smooth form $\theta_{{\varepsilon}}$ with $\theta_{{\varepsilon}}\geq-{\varepsilon}\omega$;\
$\alpha$ is big iff it contains a Kähler current, i.e. a closed $(1,1)$-current $T$ such that $T\geq{\varepsilon}\omega$ for ${\varepsilon}>0$ small enough. Finally, $\alpha$ is a Kähler class iff it contains a Kähler form (note that a smooth Kähler current is the same thing as a Kähler form).\
Since any two Hermitian forms $\omega_1$, $\omega_2$ are commensurable (i.e. $C^{-1}\omega_2\leq\omega_1\leq C\omega_2$ for some $C>0$), these definitions do not depend on the choice of $\omega$.\
The set of pseudo-effective classes is a closed convex cone ${\mathcal{E}}\subset{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$, called the pseudo-effective cone. It has compact base, because so is the case of the cone of closed positive $(1,1)$-currents. Similarly, one defines the nef cone ${\mathcal{N}}$ (a closed convex cone), the big cone ${\mathcal{B}}$ (an open convex cone), and the Kähler cone ${\mathcal{K}}$ (an open convex cone). We obviously have the inclusions $${\mathcal{K}}\subset{\mathcal{B}}\subset{\mathcal{E}}$$ and $${\mathcal{K}}\subset{\mathcal{N}}\subset{\mathcal{E}}.$$ By definition, $X$ is a Kähler manifold iff its Kähler cone ${\mathcal{K}}$ is non-empty. Similarly (but this is a theorem, cf. \[DP01\]) $X$ is a Fujiki manifold (i.e. bimeromorphic to a Kähler manifold) iff its big cone ${\mathcal{B}}$ is non-empty (see also the proof of proposition 2.3 below). If $X$ is Kähler, ${\mathcal{K}}$ is trivially the interior ${\mathcal{N}}^0$ of the nef cone. Similarly, if $X$ is Fujiki, ${\mathcal{B}}$ is trivially the interior ${\mathcal{E}}^0$ of the pseudo-effective cone.\
We will now and then denote by $\geq$ the partial order relation on ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ induced by the convex cone ${\mathcal{E}}$.
The Néron-Severi space
----------------------
Given a line bundle $L$ on $X$, each smooth Hermitian metric $h$ on $L$ locally writes as $h(x,v)=|v|^2e^{-2\varphi(x)}$ for some smooth local weight $\varphi$; the curvature form $\Theta_h(L):=\frac{i}{\pi}{\partial\overline{\partial}}\varphi$ is a globally defined real $(1,1)$-form, whose class in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ we denote by $c_1(L)$, the first Chern class of $L$. We write $dd^c=\frac{i}{\pi}{\partial\overline{\partial}}$ for short. A singular Hermitian metric $h$ on $L$ is by definition a metric $h=h_{\infty}e^{-2\varphi}$, where $h_{\infty}$ is a smooth Hermitian metric on $L$ and the weight $\varphi$ is a locally integrable function. The curvature current of $h$ is defined as $\Theta_h(L):=\Theta_{h_{\infty}}(L)+dd^c\varphi$; it also lies in $c_1(L)$. Conversely, given a smooth Hermitian metric $h_{\infty}$ on $L$, any closed real $(1,1)$-current $T$ in $c_1(L)$ can be written (by definition) as $T=\Theta_{h_{\infty}}(L)+dd^c\varphi$. But $\varphi$ is just a degree $0$ current $a$ $priori$. However, $\varphi$ is automatically $L^1_{loc}$ in case $T$ is almost positive (cf. section 2.2.2), thus each almost positive current $T$ in $c_1(L)$ is the curvature form of a singular Hermitian metric on $L$.\
The image of the homomorphism Pic$(X)\to{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ $L\mapsto c_1(L)$ is called the Néron-Severi group, denoted by $NS(X)$. It is a free ${\mathbf{Z}}$-module, whose rank is denoted by $\rho(X)$, and called the Picard number of $X$. The real Néron-Severi space ${NS(X)_{{\mathbf{R}}}}$ is just the real subspace of dimension $\rho(X)$ in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ generated by $NS(X)$. Kodaira’s embedding theorem can be formulated as follows: $X$ is a projective manifold iff the intersection of the Kähler cone ${\mathcal{K}}$ with ${NS(X)_{{\mathbf{R}}}}$ is non-empty. Similarly, $X$ is a Moishezon manifold (i.e. bimeromorphic to a projective manifold) iff the intersection of the big cone ${\mathcal{B}}$ with ${NS(X)_{{\mathbf{R}}}}$ is non-empty (cf. \[DP01\]).
Currents with analytic singularities
------------------------------------
### Definition
A closed almost positive $(1,1)$-current $T$ on a compact complex $n$-fold $X$ is said to have analytic singularities (along a subscheme $V({\mathcal{I}})$ defined by a coherent ideal sheaf ${\mathcal{I}}$) if there exists some $c>0$ such that $T$ is locally congruent to $\frac{c}{2}dd^c\log(|f_1|^2+...+|f_k|^2)$ modulo smooth forms, where $f_1,...,f_k$ are local generators of ${\mathcal{I}}$. $T$ is thus smooth outside the support of $V({\mathcal{I}})$, and it is an immediate consequence of the Lelong-Poincaré formula that $\sum\nu(T,D)D$ is just $c$ times the divisor part of the scheme $V({\mathcal{I}})$. If we first blow-up $X$ along $V({\mathcal{I}})$ and then resolve the singularities, we get a modification $\mu:{\widetilde}{X}\to X$, where ${\widetilde}{X}$ is a compact complex manifold, such that $\mu^{-1}{\mathcal{I}}$ is just ${\mathcal{O}}(-D)$ for some effective divisor $D$ upstairs. The pull-back $\mu^{\star}T$ clearly has analytic singularities along $V(\mu^{-1}{\mathcal{I}})=D$, thus its Siu decomposition writes $$\mu^{\star}T=\theta+cD$$ where $\theta$ is a smooth $(1,1)$-form. If $T\geq\gamma$ for some smooth form $\gamma$, then $\mu^{\star}T\geq\mu^{\star}\gamma$, and thus $\theta\geq\mu^{\star}\gamma$. This operation we call a resolution of the singularities of $T$.
### Regularization(s) of currents
We will need two basic types of regularizations (inside a fixed cohomology class) for closed $(1,1)$-currents, both due to J.-P.Demailly.
Let $T$ be a closed almost positive $(1,1)$-current on a compact complex manifold $X$, and fix a Hermitian form $\omega$. Suppose that $T\geq\gamma$ for some smooth real $(1,1)$-form $\gamma$ on $X$. Then:\
(i) There exists a sequence of smooth forms $\theta_k$ in $\{T\}$ which converges weakly to $T$, and such that $\theta_k\geq\gamma-C\lambda_k\omega$ where $C>0$ is a constant depending on the curvature of $(T_X,\omega)$ only, and $\lambda_k$ is a decreasing sequence of continuous functions such that $\lambda_k(x)\to\nu(T,x)$ for every $x\in X$.\
(ii) There exists a sequence $T_k$ of currents with analytic singularities in $\{T\}$ which converges weakly to $T$, such that $T_k\geq\gamma-{\varepsilon}_k\omega$ for some sequence ${\varepsilon}_k>0$ decreasing to $0$, and such that $\nu(T_k,x)$ increases to $\nu(T,x)$ uniformly with respect to $x\in X$.
Point (ii) enables us in particular to approximate a Kähler current $T$ inside its cohomology class by Kähler currents $T_k$ with analytic singularities, with a very good control of the singularities. A big class therefore contains plenty of Kähler currents with analytic singularities.
Intersection of currents
------------------------
Just as cycles, currents can be intersected provided their singular sets are in an acceptable mutual position. Specifically, let $T$ be a closed positive $(1,1)$-current on a complex manifold $X$. Locally, we have $T=dd^c\varphi$ with $\varphi$ a pluri-subharmonic function, which is well defined modulo a pluri-harmonic (hence smooth) function. We therefore get a globally well-defined unbounded locus $L(T)$, which is the complement of the open set of points near which $\varphi$ is locally bounded. Assume now that $T_1$, $T_2$ are two closed positive $(1,1)$-currents such that $L(T_j)$ is contained in an analytic set $A_j$ (which may be $X$); locally, we write $T_j=dd^c\varphi_j$ with $\varphi_j$ a pluri-subharmonic function. If $A_1\cap A_2$ has codimension at least $2$, then it is shown in \[Dem92\] that $\varphi_1dd^c\varphi_2$ has locally finite mass, and that $dd^c\varphi_1\wedge dd^c\varphi_2:=dd^c(\varphi_1
dd^c\varphi_2)$ yields a globally defined closed positive $(2,2)$-current, denoted by $T_1\wedge T_2$. It is also true that $T_1\wedge T_2$ lies in the product cohomology class $\{T_1\}\cdot\{T_2\}\in H^{2,2}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})$.\
We will only need the following two special cases: if $T_1$ is a closed positive $(1,1)$-current with analytic singularities along a subscheme of codimension at least $2$, then $T_1\wedge T_2$ exists for every closed positive $(1,1)$-current $T_2$.\
If $D_1$ and $D_2$ are two distinct prime divisors, then $[D_1]\wedge
[D_2]$ is a well defined closed positive $(2,2)$-current. Since its support is clearly contained in the set-theoretic intersection $D_1\cap D_2$ (whose codimension is at least $2$), we have $[D_1]\wedge [D_2]=\sum a_j[Y_j]$, where the $Y_j$’s are the components of $D_1\cap D_2$. In fact, it can be shown that $\sum
a_jY_j$ is just the $2$-cycle associated to the scheme-theoretic intersection $D_1\cap D_2$, thus $[D_1]\wedge[D_2]$ is just the integration current associated to the cycle $D_1\cdot D_2$.
The modified nef cone
---------------------
For our purposes, we need to introduce a new cone in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$, which is somehow the cone of classes that are nef in codimension 1. Let $X$ be a compact complex $n$-fold, and $\omega$ be some reference Hermitian form.
Let $\alpha$ be a class in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$.
\(i) $\alpha$ is said to be a modified Kähler class iff it contains a Kähler current $T$ with $\nu(T,D)=0$ for all prime divisors $D$ in $X$.
\(ii) $\alpha$ is said to be a modified nef class iff, for every ${\varepsilon}>0$, there exists a closed $(1,1)$-current $T_{{\varepsilon}}$ in $\alpha$ with $T_{{\varepsilon}}\geq-{\varepsilon}\omega$ and $\nu(T_{{\varepsilon}},D)=0$ for every prime $D$.
This is again independent of the choice of $\omega$ by commensurability of the Hermitian forms. The set of modified Kähler classes is an open convex cone called the modified Kähler cone and denoted by ${\mathcal{MK}}$. Similarly, we get a closed convex cone ${\mathcal{MN}}$, the modified nef cone. Using the Siu decomposition, we immediately see that ${\mathcal{MK}}$ is non-empty iff the big cone ${\mathcal{B}}$ is non-empty, in which case ${\mathcal{MK}}$ is just the interior of ${\mathcal{MN}}$.\
[**Remark 1**]{}: upon regularizing the currents using (ii) of theorem 2.1, we can always assume that the currents involved in the definition have analytic singularities along a subcheme of codimension at least 2.\
[**Remark 2**]{}: the modified nef cone of a compact complex surface is just its nef cone (cf. section 4.2.1).\
[**Remark 3**]{}: just as for nef classes, one cannot simply take ${\varepsilon}=0$ in the definition of a modified nef class. We recall the example given in \[DPS94\]: there exists a ruled surface $X$ over an elliptic curve such that $X$ contains an irreducible curve $C$ with the following property: the class $\{C\}\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is nef, but contains only one positive current, which is of course the integration current $[C]$.
The following proposition gives a more “algebraic” characterization of ${\mathcal{MK}}$, which also explains the (seemingly dumb) terminology.
A class $\alpha$ lies in ${\mathcal{MK}}$ iff there exists a modification\
$\mu:{\widetilde}{X}\to X$ and a Kähler class ${\widetilde}{\alpha}$ on ${\widetilde}{X}$ such that $\alpha=\mu_{\star}{\widetilde}{\alpha}$.
$Proof$: the argument is adapted from \[DP01\], theorem 3.4. If ${\widetilde}{\omega}$ is a Kähler form on ${\widetilde}{X}$ and $\omega$ is our reference Hermitian form on $X$, then $\mu^{\star}\omega\leq C{\widetilde}{\omega}$ for some $C>0$, since ${\widetilde}{X}$ is compact. Since $\mu$ is a modification, we have $\mu_{\star}\mu^{\star}\omega=\omega$, so we get $T:=\mu_{\star}{\widetilde}{\omega}\geq C^{-1}\omega$, and $T$ is thus a Kähler current. Since the singular values of $\mu$ are in codimension at least 2, we immediately see that $\nu(T,D)=0$ for every prime divisor $D$ in $X$, and $\{T\}=\mu_{\star}\{\omega\}$ lies in ${\mathcal{MK}}$ as desired. Conversely, if $\alpha\in{\mathcal{MK}}$ is represented by a Kähler current $T$ with $\nu(T,D)=0$ for all $D$, there exists by (ii) of theorem 2.1 a Kähler current $T_k$ in $\alpha$ with analytic singularities along a subscheme $V_k$ with $\nu(T_k,D)\leq\nu(T,D)$, so that $V_k$ has no divisor component. We select a resolution of the singularities of $T_k$ $\mu:{\widetilde}{X}\to X$, and write $\mu^{\star}T_k=\theta+F$, where $\theta$ is a smooth form and $F$ is an effective ${\mathbf{R}}$-divisor. Since $T_k\geq{\varepsilon}\omega$ for ${\varepsilon}>0$ small enough, we get that $\theta\geq\mu^{\star}{\varepsilon}\omega$. Denoting by $E_1,...,E_r$ the $\mu$-exceptional prime divisors on ${\widetilde}{X}$, it is shown in \[DP01\], lemma 3.5, that one can find ${\delta}_1,...,{\delta}_r>0$ small enough and a closed smooth $(1,1)$-form $\tau$ in $\{{\delta}_1E_1+...+{\delta}_rE_r\}$ such that $\mu^{\star}{\varepsilon}\omega-\tau$ is positive definite everywhere. It follows that $\theta-\tau$ is a Kähler form upstairs. Now, we have $$\alpha=\mu_{\star}\{T_k\}=\mu_{\star}\{\theta-({\delta}_1E_1+...+{\delta}_rE_r)\}=\mu_{\star}\{\theta-\tau\},$$ since $E_j$ is $\mu$-exceptional and so is $F$ because $\mu_{\star}F$ is an effective divisor contained in the scheme $V_k$; this concludes the proof of proposition 2.3.
That a modified nef classe is somehow nef in codimension 1 is reflected in the following
If $\alpha$ is a modified Kähler (resp. nef) class, then $\alpha_{|D}$ is big (resp. pseudo-effective) for every prime divisor $D\subset X$.
$Proof$: if $\alpha$ is a modified nef class and ${\varepsilon}>0$ is given, choose a current $T_{{\varepsilon}}\geq-{\varepsilon}\omega$ in $\alpha$ with analytic singularities in codimension at least 2. Locally, we have $\omega\leq Cdd^c|z|^2$ for some $C>0$, thus $T_{{\varepsilon}}+{\varepsilon}Cdd^c|z|^2$ writes as $dd^c\varphi_{{\varepsilon}}$, where $\varphi_{{\varepsilon}}$ is pluri-subharmonic and is not identically $-\infty$ on $D$. Thus the restriction $(\varphi_{{\varepsilon}})_{|D}$ is pluri-subharmonic, and $(T_{{\varepsilon}}+{\varepsilon}Cdd^c|z|^2)_{|D}$ is a well defined closed positive current. It follows that $(T_{{\varepsilon}})_{|D}$ is a well defined almost positive current on $D$, with $(T_{{\varepsilon}})_{|D}\geq-{\varepsilon}C\omega_{|D}$. This certainly implies that $\alpha_{|D}$ is pseudo-effective. The case $\alpha\in{\mathcal{MK}}$ is treated similarly.
Currents with minimal singularities
-----------------------------------
Let $\varphi_1$, $\varphi_2$ be two almost pluri-subharmonic functions on a compact complex manifold $X$. Then, following \[DPS00\], we say that $\varphi_1$ is less singular than $\varphi_2$ (and write $\varphi_1\preceq\varphi_2$) if we have $\varphi_2\leq\varphi_1+C$ for some constant $C$. We denote by $\varphi_1\approx\varphi_2$ the equivalence relation generated by the pre-order relation $\preceq$. Note that $\varphi_1\approx\varphi_2$ exactly means that $\varphi_1=\varphi_2$ mod $L^{\infty}$.\
When $T_1$ and $T_2$ are two closed almost positive $(1,1)$-currents on $X$, we can also compare their singularities in the following fashion: write $T_i=\theta_i+dd^c\varphi_i$ for $\theta_i\in\{T_j\}$ a smooth form and $\varphi_i$ an almost pluri-subharmonic function. Since any $L^1_{loc}$ function $f$ with $dd^cf$ smooth is itself smooth, it is easy to check that $\varphi_i$ does not depend on the choices made up to equivalence of singularities, and we compare the singularities of the $T_i$’s by comparing those of the $\varphi_i$’s.\
Let now $\alpha$ be a class in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ and $\gamma$ be a smooth real $(1,1)$-form, and denote by $\alpha[\gamma]$ the set of closed almost positive $(1,1)$-currents $T$ lying in $\alpha$ with $T\geq\gamma$. It is a (weakly) compact and convex subset of the space of $(1,1)$-currents. We endow it with the pre-order relation $\preceq$ defined above. For any family $T_j$, $j\in J$ of elements of $\alpha[\gamma]$, we claim that there exists an infimum $T=\inf_{j\in J}T_j$ in $(\alpha[\gamma],\preceq)$, which is therefore unique up to equivalence of singularities. The proof is pretty straightforward: fix a smooth form $\theta$ in $\alpha$, and write $T_j=\theta+dd^c\varphi_j$ for some quasi pluri-subharmonic functions $\varphi_j$. Since $X$ is compact, $\varphi_j$ is bounded from above; therefore, upon changing $\varphi_j$ into $\varphi_j-C_j$, we may assume that $\varphi_j\leq 0$ for all $ j\in J$. We then take $\varphi$ to be the upper semi-continuous upper enveloppe of the $\varphi_j$’s, $j\in J$, and set $T:=\theta+dd^c\varphi$. It is immediate to check that $T\preceq T_j$ for all $ j$, and that for every $S\in\alpha[\gamma]$, $S\preceq T_j$ for all $ j$ implies that $S\preceq T$. We should maybe explain why $T\geq\gamma$: locally, we can choose coordinates $z=(z_1,...,z_n)$ and a form $q(z)=\sum\lambda_j|z_j|^2$ such that $dd^cq\leq\gamma$ and $dd^cq$ is arbitrarily close to $\gamma$. Writing $\theta=dd^c\psi$ for some smooth local potential $\psi$, the condition $\theta+dd^c\varphi_j\geq\gamma$ implies that $\psi+\varphi_j-q$ is pluri-subharmonic. The upper enveloppe $\psi+\varphi-q$ is thus also pluri-subharmonic, which means that $T=\theta+dd^c\varphi\geq dd^cq$; letting $dd^cq$ tend to $\gamma$, we get $T\geq\gamma$, as desired.\
Since any two closed almost positive currents with equivalent singularities have the same Lelong numbers, the Lelong numbers of $\inf T_j$ do not depend on the specific choice of the current. In fact, it is immediate to check from the definitions that $$\nu(\inf_{j\in J} T_j,x)=\inf_{j\in J}\nu(T_j,x).$$\
As a particular case of the above construction, there exists a closed almost positive $(1,1)$-current $T_{\min,\gamma}\in\alpha[\gamma]$ which is a least element in $(\alpha[\gamma],\preceq)$. $T_{\min,\gamma}$ is well defined modulo $dd^cL^{\infty}$, and we call it a current with minimal singularities in $\alpha$, for the given lower bound $\gamma$. When $\gamma=0$ and $\alpha$ is pseudo-effective, we just write $T_{\min}=T_{\min,0}$, and call it a positive current with minimal singularities in $\alpha$. It must be noticed that, even for a big class $\alpha$, $T_{\min}$ will be a Kähler current only in the trivial case:
A pseudo-effective class $\alpha$ contains a positive current with minimal singularities $T_{\min}$ which is a Kähler current iff $\alpha$ is a Kähler class.
$Proof$: we can write $T_{\min}=\theta+dd^c\varphi$ with $\theta$ a smooth form. If $T_{\min}$ is Kähler, then so is ${\varepsilon}\theta+(1-{\varepsilon})T_{\min}=\theta+dd^c(1-{\varepsilon})\varphi$ for ${\varepsilon}>0$ small enough. We therefore get $\varphi\preceq(1-{\varepsilon})\varphi$ by minimality, that is: $(1-{\varepsilon})\varphi\leq\varphi+C$ for some constant $C$. But this shows that $\varphi$ is bounded, and thus $T_{\min}$ is a Kähler current with identically zero Lelong numbers. Using (i) of theorem 2.1, we can therefore regularize it into a Kähler form inside its cohomology class, qed.\
Finally, we remark that a positive current with minimal singularities in a pseudo-effective class is generally non-unique (as a current), as the example of a Kähler class already shows.
The divisorial Zariski decomposition
====================================
In this section $X$ denotes a compact complex $n$-fold, and $\omega$ is a reference Hermitian form, unless otherwise specified.
Minimal multiplicities and non-nef locus
----------------------------------------
When $\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is a pseudo-effective class, we want to introduce minimal multiplicities $\nu(\alpha,x)$, which measure the obstruction to the numerical effectivity of $\alpha$. For each ${\varepsilon}>0$, let $T_{\min,{\varepsilon}}=T_{\min,{\varepsilon}}(\alpha)$ be a current with minimal singularities in $\alpha[-{\varepsilon}\omega]$ (cf. section 2.8 for the notation). We then introduce the following
The minimal multiplicity at $x\in X$ of the pseudo-effective class $\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is defined as $$\nu(\alpha,x):=\sup_{{\varepsilon}>0}\nu(T_{\min,{\varepsilon}},x).$$
The commensurability of any two Hermitian forms shows that the definition does not depend on $\omega$. When $D$ is a prime divisor, we define the generic minimal multiplicity of $\alpha$ along $D$ as $$\nu(\alpha,D):=\inf\{\nu(\alpha,x),x\in D\}.$$ We then have $\nu(\alpha,D)=\sup_{{\varepsilon}>0}\nu(T_{\min,{\varepsilon}},D)$, and $\nu(\alpha,D)=\nu(\alpha,x)$ for the very general $x\in D$.
Let $\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ be a pseudo-effective class.
\(i) $\alpha$ is nef iff $\nu(\alpha,x)=0$ for every $x\in X$.
\(ii) $\alpha$ is modified nef iff $\nu(\alpha,D)=0$ for every prime $D$.
$Proof$: if $\alpha$ is nef (resp. modified nef), $\alpha[-{\varepsilon}\omega]$ contains by definition a smooth form (resp. a current $T_{{\varepsilon}}$ with $\nu(T_{{\varepsilon}},D)=0$ for every prime $D$). We thus have $\nu(T_{\min,{\varepsilon}},x)=0$ (resp. $\nu(T_{\min,{\varepsilon}},D)=0$) for every ${\varepsilon}>0$, and thus $\nu(\alpha,x)=0$ (resp. $\nu(\alpha,D)=0$). Conversely, if $\nu(\alpha,x)=0$ for every $x\in X$, applying (i) of theorem 2.1 to $T_{\min,{\varepsilon}}$, we see that $\nu(T_{\min,{\varepsilon}},x)=0$ for every $x\in X$ implies that $\alpha[-{\varepsilon}'\omega]$ contains a smooth form for every ${\varepsilon}'>{\varepsilon}$, and $\alpha$ is thus nef. Finally, if $\nu(\alpha,D)=0$ for every prime $D$, we have $\nu(T_{\min,{\varepsilon}},D)=0$ for every prime $D$. Since $T_{\min,{\varepsilon}}$ lies in $\alpha[-{\varepsilon}\omega]$, $\alpha$ is modified nef by the very definition.\
In view of proposition 3.2, we propose the
The non-nef locus of a pseudo-effective class $\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is defined by $$E_{nn}(\alpha):=\{x\in X,\nu(\alpha,x)>0\}.$$
Recall that the set $E_+(T):=\{x\in X,\nu(T,x)>0\}$ is a countable union of closed analytic subsets for every closed almost positive $(1,1)$-current $T$. Since $E_{nn}(\alpha)=\cup_{{\varepsilon}>0}E_+(T_{\min,{\varepsilon}})$, the non-nef locus is also a countable union of closed analytic subsets. We do not claim however that each super-level set $\{x\in X,\nu(\alpha,x)\geq c\}$ $(c>0)$ is an analytic subset (this is most certainly not true in general). Using results of M.Paun, proposition 3.2 generalizes as follows:
A pseudo-effective class $\alpha$ is nef iff $\alpha_{|Y}$ is nef for every irreducible analytic subset $Y\subset E_{nn}(\alpha)$.
$Proof$: since the restriction of a nef class to any analytic subset is nef, one direction is clear. To prove the converse, we cannot directly apply the results of M.Paun \[Pau98\], since we allow slightly negative currents, so we sketch the proof to show that it immediately carries on to our situation. We quote without proof the following results:
Let $\alpha$ be any class in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$, and $Y_1$, $Y_2$ two analytic subsets of $X$. If $\alpha_{|Y_i}$ is nef $(i=1,2)$, then $\alpha_{|Y_1\cup Y_2}$ is nef.
Let $\theta$ be a closed smooth $(1,1)$-form and $\gamma$ be any smooth $(1,1)$-form. Let $Y\subset X$ be an analytic subset, and assume that $$\theta_{|Y}+dd^c\varphi\geq\gamma_{|Y}$$ for some smooth function $\varphi$ on $Y$. Then, for every ${\varepsilon}>0$, there exists a neighbourhood $V$ of $Y$ and a smooth function $\varphi_{{\varepsilon}}$ on $V$ such that $$\theta+dd^c\varphi_{{\varepsilon}}\geq\gamma-{\varepsilon}\omega$$ on $V$.
We now select once for all a smooth form $\theta$ in $\alpha$. By applying (ii) of theorem 2.1 to $T_{\min,{\varepsilon}}\geq-{\varepsilon}\omega$, we can select a closed $(1,1)$-current with analytic singularities $T^{(1)}_{{\varepsilon}}=\theta+dd^c\varphi^{(1)}_{{\varepsilon}}$ such that $T^{(1)}_{{\varepsilon}}\geq-2{\varepsilon}\omega$ and $\nu(T^{(1)}_{{\varepsilon}},x)\leq\nu(T_{\min,{\varepsilon}},x)$. Let $Y_{{\varepsilon}}$ be the analytic subset along which $T^{(1)}_{{\varepsilon}}$ is singular; we have $Y_{{\varepsilon}}\subset E_+(T_{\min,{\varepsilon}})\subset E_{nn}(\alpha)$. Since $\alpha$ is nef by assumption on every component of $Y_{{\varepsilon}}$, using the above two lemma, we can find a neighbourhood $V_{{\varepsilon}}$ of $Y_{{\varepsilon}}$ and a smooth function $\varphi^{(2)}_{{\varepsilon}}$ on $V_{{\varepsilon}}$ such that $\theta+dd^c\varphi^{(2)}_{{\varepsilon}}\geq-{\varepsilon}\omega$ on $V_{{\varepsilon}}$. We choose a smaller neighbourhood $W_{{\varepsilon}}$ of $Y_{{\varepsilon}}$ with $\overline{W_{{\varepsilon}}}\subset V_{{\varepsilon}}$, and we then set $$\varphi^{(3)}_{{\varepsilon}}:=\cases{\varphi^{(1)}_{{\varepsilon}}& on $X-W_{{\varepsilon}}$,\cr
\max_{\eta}(\varphi_{{\varepsilon}}^{(2)}-C_{{\varepsilon}},\varphi^{(1)}_{{\varepsilon}})& on $\overline{W_{{\varepsilon}}}$\cr}$$ where $\max_{\eta}(x,y):=\max\star\rho_{\eta}$ denotes a regularized maximum function obtained by convolution with a regularizing kernel $\rho_{\eta}$ ($\eta$ is chosen so small that $\max_{\eta}(x,y)=x$ when $y<x-1/2$), and $C_{{\varepsilon}}$ is a positive constant, large enough to achieve $\varphi^{(1)}_{{\varepsilon}}\geq\varphi_{{\varepsilon}}^{(2)}-C_{{\varepsilon}}+1$ near $\partial W_{{\varepsilon}}$ (we use that $\varphi^{(1)}_{{\varepsilon}}$ is smooth away from $Y_{{\varepsilon}}$, hence locally bounded near $\partial W_{{\varepsilon}}$). The two parts to be glued then coincide near $\partial W_{{\varepsilon}}$, thus $\varphi^{(3)}_{{\varepsilon}}$ is smooth. Since both $\theta+dd^c\varphi^{(1)}_{{\varepsilon}}$ and $\theta+dd^c\varphi_{{\varepsilon}}^{(2)}-C_{{\varepsilon}}$ are greater than $-2{\varepsilon}\omega$, the gluing property of pluri-subharmonic functions yields that $\theta+dd^c\varphi^{(3)}_{{\varepsilon}}\geq-2{\varepsilon}\omega$. Since this is true for every ${\varepsilon}>0$, this shows that $\alpha$ is indeed nef.
We now investigate the continuity of $\alpha\mapsto\nu(\alpha,x)$ and $\nu(\alpha,D)$:
For every $x\in X$ and every prime $D$, the maps ${\mathcal{E}}\to{\mathbf{R}}$ $\alpha\mapsto\nu(\alpha,x)$ and $\nu(\alpha,D)$ are convex, homogeneous. They are continuous on the interior ${\mathcal{E}}^0$, and lower semi-continuous on the whole of ${\mathcal{E}}$.
$Proof$: let $\alpha$, $\beta$ be two pseudo-effective classes. If $T_{\min,{\varepsilon}}(\alpha)$ and $T_{\min,{\varepsilon}}(\beta)$ are currents with minimal singularities in $\alpha[-{\varepsilon}\omega]$ and $\beta[-{\varepsilon}\omega]$ respectively, then\
$T_{\min,{\varepsilon}}(\alpha)+T_{\min,{\varepsilon}}(\alpha)$ belongs to $(\alpha+\beta)[-2{\varepsilon}\omega]$, thus $$\nu(T_{\min,2{\varepsilon}}(\alpha+\beta),x)\leq\nu(T_{\min,{\varepsilon}}(\alpha),x)+\nu(T_{\min,{\varepsilon}}(\beta),x)\leq\nu(\alpha,x)+\nu(\beta,x).$$ We infer from this $\nu(\alpha+\beta,x)\leq\nu(\alpha,x)+\nu(\beta,x)$, and a similar sub-additivity property for $\nu(\cdot,D)$ is obtained along the same lines. Since the homogeneity of our two maps is obvious, the convexity also follows.\
The quotient map $\theta\mapsto\{\theta\}$ from the Fréchet space of closed smooth real $(1,1)$-forms to ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is surjective, thus open. If $\alpha_k\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is a sequence of pseudo-effective classes converging to $\alpha$ and ${\varepsilon}>0$ is given, we can thus find a smooth form $\theta_k\in\alpha-\alpha_k$ for each $k$ big enough such that\
$-{\varepsilon}\omega\leq\theta_k\leq{\varepsilon}\omega$. The current $T_{\min,{\varepsilon}}(\alpha_k)+\theta_k$ then lies in $\alpha[-2{\varepsilon}\omega]$, and thus $\nu(T_{\min,2{\varepsilon}}(\alpha),x)\leq\nu(T_{\min,{\varepsilon}}(\alpha_k),x)\leq\nu(\alpha_k,x)$, for each $k$ big enough. We infer from this that $\nu(T_{\min,2{\varepsilon}}(\alpha),x)\leq\liminf_{k\to\infty}\nu(\alpha_k,x)$ for each ${\varepsilon}>0$, hence $\nu(\alpha,x)\leq\liminf_{k\to\infty}\nu(\alpha_k,x)$, by taking the supremum of the left hand-side for ${\varepsilon}>0$. This means that $\alpha\mapsto\nu(\alpha,x)$ is lower semi-continuous, and similarly for $\nu(\alpha,D)$, just replacing $x$ by $D$ in the above proof.\
Finally, the restrictions of our maps to ${\mathcal{E}}^0$ are continuous as any convex map on an open convex subset of a finite dimensional vector space is.
Let $\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ be a pseudo-effective class, and $T_{\min}$ be a positive current with minimal singularities in $\alpha$.
\(i) We always have $\nu(\alpha,x)\leq\nu(T_{\min},x)$ and $\nu(\alpha,D)\leq\nu(T_{\min},D)$.
\(ii) When $\alpha$ is furthermore big, we have $\nu(\alpha,x)=\nu(T_{\min},x)$ and $\nu(\alpha,D)=\nu(T_{\min},D)$.
$Proof$: since $T_{\min}$ belongs to $\alpha[-{\varepsilon}\omega]$ for every ${\varepsilon}>0$, $\nu(\alpha,x)\leq\nu(T_{\min},x)$ follows for every $x\in
X$, for any pseudo-effective class $\alpha$. If $\alpha$ is furthermore big, we can choose a Kähler current $T$ in $\alpha$ with $T\geq\omega$ for some Hermitian form $\omega$. If $T_{\min,{\varepsilon}}$ is a current with minimal singularities in $\alpha[-{\varepsilon}\omega]$, then $(1-{\varepsilon})T_{\min,{\varepsilon}}+{\varepsilon}T$ is a positive current in $\alpha$, and thus $\nu((1-{\varepsilon})T_{\min,{\varepsilon}}+{\varepsilon}T,x)\geq\nu(T_{\min}
,x)$ by minimality of $T_{\min}$, from which we infer $$(1-{\varepsilon})\nu(\alpha,x)+{\varepsilon}\nu(T,x)\geq\nu(T_{\min},x).$$ We thus get the converse inequality $\nu(\alpha,x)\geq\nu(T_{\min},x)$ by letting ${\varepsilon}\to 0$. The case of $\nu(\alpha,D)$ is similar.
Definition of the divisorial Zariski decomposition
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Let $\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ be again a pseudo-effective class, and choose a positive current with minimal singularities $T_{\min}$ in $\alpha$. Since $\nu(\alpha,D)\leq\nu(T_{\min},D)$ for every prime $D$ by proposition 3.8, the series of currents $\sum\nu(\alpha,D)[D]$ is convergent, since it is dominated by $\sum\nu(T_{\min},D)[D]$.
The negative part of a pseudo-effective class $\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is defined as $N(\alpha):=\sum\nu(\alpha,D)[D]$. The Zariski projection of $\alpha$ is $Z(\alpha):=\alpha-\{N(\alpha)\}$. We call the decomposition $\alpha=Z(\alpha)+\{N(\alpha)\}$ the divisorial Zariski decomposition of $\alpha$.
It is certainly highly desirable that the negative part $N(\alpha)$ of a pseudo-effective class be a divisor, i.e. that $\nu(\alpha,D)=0$ for almost every prime $D$. We will see in section 3.3 that it is indeed the case. For the time being, we concentrate on the Zariski projection, which we see as a map $Z:{\mathcal{E}}\to{\mathcal{E}}$.
Let $\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ be a pseudo-effective class. Then:
\(i) Its Zariski projection $Z(\alpha)$ is a modified nef class.
\(ii) We have $Z(\alpha)=\alpha$ iff $\alpha$ is modified nef.
\(iii) $Z(\alpha)$ is big iff $\alpha$ is.
\(iv) If $\alpha$ is not modified nef, then $Z(\alpha)$ belongs to the boundary $\partial{\mathcal{MN}}$ of the modified nef cone.
$Proof$:(i) Let $T_{\min,{\varepsilon}}$ be as before a current with minimal singularities in $\alpha[-{\varepsilon}\omega]$, and consider its Siu decomposition $T_{\min,{\varepsilon}}=R_{{\varepsilon}}+\sum\nu(T_{\min,{\varepsilon}},D)[D]$. First, we claim that $N_{{\varepsilon}}:=\sum\nu(T_{\min,{\varepsilon}},D)[D]$ converges weakly to $N(\alpha)$ as ${\varepsilon}$ goes to $0$. For any smooth form $\theta$ of bidimension $(1,1)$, $\theta+C\omega^{n-1}$ is a positive form for $C>0$ big enough. Every such $\theta$ is thus the difference of two positive forms, and it is enough to show that $\int
N_{{\varepsilon}}\wedge\theta\to\int N(\alpha)\wedge\theta$ for every smooth positive form $\theta$. But $\int
N_{{\varepsilon}}\wedge\theta=\sum\nu(T_{\min,{\varepsilon}},D)\int[D]\wedge\theta$ is a convergent series whose general term $\nu(T_{\min,{\varepsilon}},D)\int[D]\wedge\theta$ converges to $\nu(\alpha,D)\int[D]\wedge\theta$ as ${\varepsilon}\to 0$ and is dominated by $\nu(T_{\min},D)\int[D]\wedge\theta$; since $\sum\nu(T_{\min},D)\int[D]\wedge\theta\leq\int T_{\min}\wedge\theta$ converges, our claim follows by dominated convergence.\
In particular, the class $\{N_{{\varepsilon}}-N(\alpha)\}$ converges to zero. Since the map $\theta\mapsto\{\theta\}$ is open on the space of smooth closed $(1,1)$-form, we can find a sequence $\theta_k\geq-{\delta}_k\omega$ of smooth forms with $\theta_k\in\{N_{{\varepsilon}_k}-N(\alpha)\}$ for some sequences ${\varepsilon}_k<<{\delta}_k$ going to zero. It remains to notice that $T_k:=R_{{\varepsilon}_k}+\theta_k$ is a current in $Z(\alpha)$ with $T_k\geq-({\varepsilon}_k+{\delta}_k)\omega$ and $\nu(T_k,D)=0$ for every prime $D$. Since ${\varepsilon}_k+{\delta}_k$ converges to zero, $Z(\alpha)$ is modified nef by definition.
\(ii) Since $N(\alpha)=\sum\nu(\alpha,D)[D]$ is a closed positive $(1,1)$-current, it is zero iff its class $\{N(\alpha)\}\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is. The assertion is thus just a reformulation of (ii) in proposition 3.2.
\(iii) If $Z(\alpha)$ is big, then of course $\alpha=Z(\alpha)+\{N(\alpha)\}$ is also big, as the sum of a big class and a pseudo-effective one. If conversely $\alpha$ is big, it contains a Kähler current $T$, whose Siu decomposition we write $T=R+\sum\nu(T,D)[D]$. Note that $R$ is a Kähler current since $T$ is; since $T$ belongs to $\alpha[-{\varepsilon}\omega]$ for every ${\varepsilon}>0$, we have $\nu(T,D)\geq\nu(\alpha,D)$, and $R+\sum(\nu(T,D)-\nu(\alpha,D))[D]$ is thus a Kähler current in $Z(\alpha)$ as desired.
\(iv) Assume that $Z(\alpha)$ belongs to the interior ${\mathcal{MN}}^0$ of the modified nef cone. By proposition 3.2, we have to see that $\nu(\alpha,D)=0$ for every prime $D$. Suppose therefore that $\nu(\alpha,D_0)>0$ for some prime $D_0$. The class $Z(\alpha)+{\varepsilon}\{D_0\}$ has to lie in the open cone ${\mathcal{MN}}^0$ for ${\varepsilon}$ small enough, thus we can write for $0<{\varepsilon}<\nu(\alpha,D_0)$: $$\alpha=(Z(\alpha)+{\varepsilon}\{D_0\})+(\nu(\alpha,D_0)-{\varepsilon})\{D_0\}+\{\sum_{D\neq
D_0}\nu(\alpha,D)D\}.$$ We deduce that $\nu(\alpha,D_0)\leq\nu(Z(\alpha)+{\varepsilon}\{D_0\},D_0)+(\nu(\alpha,D_0)-{\varepsilon})$. Indeed, the class $\{D_0\}$ (resp. $\{\sum_{D\neq D_0}\nu(\alpha,D)D\}$) has minimal multiplicity $\leq 1$ (resp. 0) along $D_0$, because so is the generic Lelong numbers of the positive current $[D_0]$ (resp. $\sum_{D\neq D_0}\nu(\alpha,D)[D]$) along $D_0$. Now, we also have $\nu(Z(\alpha)+{\varepsilon}\{D_0\},D_0)=0$ since $Z(\alpha)+{\varepsilon}\{D_0\}$ is modified nef by assumption, hence the contradiction $\nu(\alpha,D_0)\leq\nu(\alpha,D_0)-{\varepsilon}$.
\(i) The map $\alpha\mapsto N(\alpha)$ is convex and homogeneous on ${\mathcal{E}}$. It is continuous on the interior of the pseudo-effective cone.
\(ii) The Zariski projection $Z:{\mathcal{E}}\to{\mathcal{MN}}$ is concave and homogeneous. It is continuous on the interior of ${\mathcal{E}}$.
$Proof$: we have already noticed that $\nu(\alpha+\beta,D)\leq\nu(\alpha,D)+\nu(\beta,D)$ for every prime $D$ and every two pseudo-effective classes $\alpha$, $\beta$. This implies that $N(\alpha+\beta)\leq N(\alpha)+N(\beta)$. Homogeneity is obvious, and the first assertion follows. To show continuity, it is enough as above to show that $\alpha\mapsto\int N(\alpha)\wedge\theta$ is continuous on ${\mathcal{E}}^0$ for every positive form $\theta$. But the latter map is convex, and thus continuous on ${\mathcal{E}}^0$ as any convex map on an open convex subset of a finite dimensional vector space is. (ii) is now an obvious consequence of (i) and the relation $Z(\alpha)=\alpha-\{N(\alpha)\}$.
Negative part and exceptional divisors
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If $A=D_1,...,D_r$ is a finite family of prime divisors, we denote by $V_+(A)\subset{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ the closed convex cone generated by the classes $\{D_1\},...,\{D_r\}$. Every element of $V_+(A)$ writes $\alpha=\{E\}$ for some effective ${\mathbf{R}}$-divisor supported by the $D_j$’s. Since $[E]$ is a positive current in $\alpha$, we have $N(\alpha)\leq E$, and thus $Z(\alpha)$ can be represented by the effective ${\mathbf{R}}$-divisor $E-N(\alpha)$, which is also supported by the $D_j$’s. We conclude: $V_+(A)$ is stable under the Zariski projection $Z$. In particular, we have $Z(V_+(A))=0$ iff $V_+(A)$ meets ${\mathcal{MN}}$ at $0$ only.
\(i) A family $D_1,...,D_q$ of prime divisors is said to be an exceptional family iff the convex cone generated by their cohomology classes meets the modified nef cone ${\mathcal{MN}}$ at $0$ only.
\(ii) An effective ${\mathbf{R}}$-divisor $E$ is said to be exceptional iff its prime components constitute an exceptional family.
We have the following
\(i) An effective ${\mathbf{R}}$-divisor $E$ is exceptional iff $Z(\{E\})=0$.
\(ii) If $E$ is an exceptional effective ${\mathbf{R}}$-divisor, we have $E=N(\{E\})$.
\(iii) If $D_1,...,D_q$ is an exceptional family of primes, then their classes $\{D_1\},...,\{D_q\}$ are linearly independent in ${NS(X)_{{\mathbf{R}}}}\subset{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$. In particular, the length of the exceptional families of primes is uniformly bounded by the Picard number $\rho(X)$.
$Proof$: (i) let $A=D_1,...,D_r$ denote the family of primes supporting $E$, and choose a Gauduchon metric $\omega$ (cf. section 2.2.4). Since $\omega^{n-1}$ is ${\partial\overline{\partial}}$-closed, $\int
Z(\alpha)\wedge\omega^{n-1}$ is well defined, and defines a map ${\mathcal{E}}\to{\mathbf{R}}$ $\alpha\mapsto\int Z(\alpha)\wedge\omega^{n-1}$, which is concave and homogeneous (by proposition 3.11), and everywhere non-negative. The restriction of this map to $V_+(A)$ shares the same properties, and the class $\alpha:=\{E\}$ is a point in the relative interior of the convex cone $V_+(A)$ at which $\int
Z(\alpha)\wedge\omega^{n-1}=0$. By concavity, we thus get $\int
Z(\alpha)\wedge\omega^{n-1}=0$ for every $\alpha\in V_+(A)$, and thus $Z(\alpha)=0$ for every such $\alpha\in V_+(A)$, qed.\
(ii) When $E$ is exceptional, we have both $E\geq N(\{E\})$ (because the positive current $[E]$ lies in the class $\{E\}$) and $\{E\}=\{N(\{E\})\}$ (because $Z(\{E\})=0$). Since a closed positive current which yields zero in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is itself zero, we get the result.\
(iii) Since $D_1,...,D_q$ are linearly independent in Div$(X)\otimes{\mathbf{R}}$, the assertion is equivalent to the fact that the quotient map $D\mapsto\{D\}$ is injective on the ${\mathbf{R}}$-vector space of divisors generated by the $D_j$’s. But this is easy: if $E=\sum a_j D_j$ lies in the kernel, we can write $E=E_+-E_-$ with $E_+$ and $E_-$ effective such that $\{E_+\}=\{E_-\}$. By (iii), we get $E_+=E_-$, whence $E=0$, qed.
We state as a theorem the following important consequences of (iii):
\(i) For every pseudo-effective class $\alpha\in{\mathcal{E}}$, the negative part $N(\alpha)$ is an exceptional effective ${\mathbf{R}}$-divisor supported by at most $\rho(X)$ primes.
\(ii) $X$ carries at most countably many exceptional primes.
\(iii) The exceptional fiber $Z^{-1}(0)$ is contained in ${NS(X)_{{\mathbf{R}}}}$, and is a union of at most countably many simplicial cones over exceptional families of primes.
$Proof$: (i) We have $Z(\alpha)\geq Z(Z(\alpha))+Z(\{N(\alpha)\})$, and $Z(Z(\alpha))=Z(\alpha)$ by proposition 3.10, thus $Z(\{N(\alpha)\})=0$. We immediately deduce from this that any family of primes $D_1,...,D_r$ such that $\nu(\alpha,D_j)>0$ for every $j$ is an exceptional family, and the assertion follows from (iii) of proposition 3.13.\
(ii) We just have to notice that $D\mapsto\{D\}$ is injective on the set of exceptional primes, and maps into the lattice $NS(X)\subset{NS(X)_{{\mathbf{R}}}}$.\
(iii) Since $\{A\}$ is a linearly independent set for every exceptional family of primes $A$, we see that $V_+(A)=\sum_{D\in A}{\mathbf{R}}_+\{D\}$ is a simplicial cone. It remains to observe that $\alpha$ lies in the exceptional fiber $Z^{-1}(0)$ iff $\alpha=\{N(\alpha)\}$, thus $Z^{-1}(0)$ is covered by the simplicial cones $V_+(A)$.\
We will see in section 4.3 that a family $D_1,...,D_q$ of primes on a surface is exceptional iff the Gram matrix $(D_i\cdot D_j)$ is negative definite, i.e. iff $D_1,...,D_q$ can all be blown down to points by a modification towards an analytic surface (singular in general). On a general compact complex $n$-fold $X$, an exceptional divisor is still very rigidly embedded in $X$:
If $E$ is an exceptional effective ${\mathbf{R}}$-divisor, then its class $\{E\}$ contains but one positive current, which is $[E]$. In particular, when $E$ is rational, its Kodaira-Iitaka dimension $\kappa(X,E)$ is zero.
$Proof$: if $T$ is a positive current in $\{E\}$, we have $\nu(T,D)\geq\nu(\{E\},D)$ for every prime $D$. Using the Siu decomposition of $T$, we thus see that $T\geq\sum\nu(\{E\},D)D=N(\{E\})=E$, since $E$ is exceptional. But we also have $\{T\}=\{E\}$, hence $T=E$, as was to be shown. To get the last point, let $D$ be an element of the linear system $|kE|$ for some integer $k>0$ such that $kE$ is Cartier. The positive current $\frac{1}{k}[D]$ then lies in $\{E\}$, thus we have $[D]=k[E]$ as currents, hence $D=kE$ as divisors. This shows that $h^0(kE)=1$ for each $k>0$, qed.
Discontinuities of the Zariski projection
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It is remarkable that the Zariski projection $Z$ is not continuous in general up to the boundary $\partial{\mathcal{E}}$.
If $X$ carries infinitely many exceptional primes, then the Zariski projection $Z:{\mathcal{E}}\to{\mathcal{MN}}$ is not continuous.
$Proof$: we use the following
If $D_k$ is an infinite sequence of divisors, the rays ${\mathbf{R}}_+\{D_k\}\subset{\mathcal{E}}$ can accumulate on ${\mathcal{MN}}$ only.
$Proof$: suppose that $t_k\{D_k\}$ converges to some non-zero $\alpha\in{\mathcal{E}}$ (for $t_k>0$). For each prime $D$, we then have $D_k\neq D$ and thus $\nu(t_k\{D_k\},D)=t_k\nu(\{D_k\},D)=0$ for infinitely many $k$, because the family $D_k$ is infinite. By lower semi-continuity (proposition 3.7) we deduce $\nu(\alpha,D)=0$ for every prime $D$, i.e. $\alpha$ is modified nef (by proposition 3.2).
Assume now that an infinite sequence of exceptional prime divisors $D_k$ exists. Since ${\mathcal{E}}$ has compact base, upon extracting a subsequence, we can assume that $t_k\{D_k\}$ converges to some non-zero $\alpha\in{\mathcal{E}}$ (with $t_k>0$ an appropriate sequence). Since $D_k$ is exceptional, we have $Z(t_k\{D_k\})=0$ for every $k$, but $Z(\alpha)=\alpha$ since $\alpha$ is modified nef by the above lemma. Consequently, $Z^{-1}(0)$ is not closed, and $Z$ is not continuous.\
To get an example of discontinuous Zariski projection, just take $X$ to be the blow-up of ${\mathbf{P}}^2$ in at least 9 general points. Such a rational surface is known to carry countably many exceptional curves of the first kind (cf. \[Har77\], p.409). Since a prime divisor $C$ on a surface is exceptional iff $C^2<0$ (cf. section 4.3), the set of exceptional primes on $X$ is infinite, and we have our example.
When is a decomposition the Zariski decomposition?
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Suppose that we have a decomposition $\alpha=p+\{N\}$ of a pseudo-effective class $\alpha$ into the sum of a modified nef class $p$ and the class of an effective ${\mathbf{R}}$-divisor $N$. We want a criterion that tells us when it is the Zariski decomposition of $\alpha$. We have $N(\alpha)\leq N(p)+N$, and $N(p)=0$ since $p$ is modified nef, thus $N(\alpha)=N$ happens iff $Z(\alpha)=p$, and our question is equivalent to the study of the fibers $Z^{-1}(p)$, with $p\in{\mathcal{MN}}$.\
We will need the following
If $\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is a big class, we define its non-Kähler locus as $E_{nK}(\alpha):=\cap_TE_+(T)$ for $T$ ranging among the Kähler currents in $\alpha$.
Let us explain the terminology:
Let $\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ be a big class. Then:
\(i) The non-nef locus $E_{nn}(\alpha)$ is contained in the non-Kähler locus $E_{nK}(\alpha)$.
\(ii) There exists a Kähler current with analytic singularities $T$ in $\alpha$ such that $E_+(T)=E_{nK}(\alpha)$. In particular, the non-Kähler locus $E_{nK}(\alpha)$ is an analytic subset of $X$.
\(iii) $\alpha$ is a Kähler class iff $E_{nK}(\alpha)$ is empty. More generally, $\alpha$ is a Kähler class iff $\alpha_{|Y}$ is a Kähler class for every irreducible component $Y$ of the analytic set $E_{nK}(\alpha)$.
$Proof$:(i) Since $\alpha$ is big, its non-nef locus $E_{nn}(\alpha)$ is just the set $\{x\in X,\nu(T_{\min},x)>0\}$, since we have $\nu(\alpha,x)=\nu(T_{\min},x)$ in that case (cf. proposition 3.8). For every Kähler current $T$ in $\alpha$, we have $\nu(T,x)\geq\nu(T_{\min},x)$ by minimality, and the inclusion $E_{nn}(\alpha)\subset E_{nK}(\alpha)$ ensues.
\(ii) First, we claim that given two Kähler currents $T_1$, $T_2$ in $\alpha$, there exists a Kähler current with analytic singularities $T$ such that $E_+(T)\subset E_+(T_1)\cap E_+(T_2)$. Indeed, we can find ${\varepsilon}>0$ small enough such that $T_j\geq{\varepsilon}\omega$. Our currents $T_1$ and $T_2$ thus belong to $\alpha[{\varepsilon}\omega]$, and admit an infimum $T_3$ in that set with respect to $\preceq$ (cf. section 2.8). In particular, $T_3$ is a current in $\alpha$ with $T_3\geq{\varepsilon}\omega$ and $\nu(T_3,x)=\min\{\nu(T_1,x),\nu(T_2,x)\}$ for every $x\in X$. By (ii) of theorem 2.1, there exists a Kähler current with analytic singularities $T$ in $\alpha$ such that $\nu(T,x)\leq\nu(T_3,x)$ for every $x\in X$, hence $E_+(T)\subset E_+(T_1)\cap E_+(T_2)$, and this proves the claim.
Using the claim and (ii) of theorem 2.1, it is easy to construct a sequence $T_k$ of Kähler currents with analytic singularities such that $E_+(T_k)$ is a decreasing sequence with $E_{nK}(\alpha)=\cap_kE_+(T_k)$. Since $T_k$ has analytic singularities, $E_+(T_k)$ is an analytic subset, thus the decreasing sequence $E_+(T_k)$ has to be stationary (by the strong Nötherian property), and we eventually get $E_{nK}(\alpha)=E_+(T_k)$ for some $k$, as desired.
\(iii) If $\alpha$ is a Kähler class, $E_+(\omega)$ is empty for every Kähler form $\omega$ in $\alpha$, and thus so is $E_{nK}(\alpha)$. Conversely, assume that $\alpha_{|Y}$ is a Kähler class for every component $Y$ of $E_+(\alpha)$, and let $T$ be a Kähler current with analytic singularities such that $E_+(T)=E_{nK}(\alpha)$. $\alpha$ is then a Kähler class by proposition 3.3 of \[DP01\], qed.\
We can now state the following
Let $p$ be a big and modified nef class. Then the primes $D_1,...,D_r$ contained in the non-Kähler locus $E_{nK}(p)$ form an exceptional family $A$, and the fiber of $Z$ above $p$ is the simplicial cone $Z^{-1}(p)=p+V_+(A)$. When $p$ is an arbitrary modified nef class, $Z^{-1}(p)$ is an at most countable union of simplicial cones $p+V_+(A)$, where $A$ is an exceptional family of primes.
$Proof$: note that, by the very definitions, for every pseudo-effective class $\alpha$, the prime components of its negative part $N(\alpha)$ are exactly the set $A$ of primes $D$ contained in the non-nef locus $E_{nn}(\alpha)$. Furthermore, $Z(\alpha)+V_+(A)$ is entirely contained in the fiber $Z^{-1}Z(\alpha)$. Indeed, the restriction of $Z$ to this simplicial cone is a concave map above the affine constant map $Z(\alpha)$, and both coincide at the relative interior point $\alpha$, thus they are equal on the whole of $Z(\alpha)+V_+(A)$. This already proves the last assertion.\
Assume now that $p$ is modified nef and big, and suppose first that $\alpha$ lies in $Z^{-1}(p)$. To see that $\alpha$ lies in $p+V_+(A)$, we have to prove that every prime $D_0$ with $\nu(\alpha,D_0)>0$ lies in $E_{nK}(p)$, that is: $\nu(T,D_0)>0$ for every Kähler current $T$ in $p$. If not, choose a smooth form $\theta$ in $\{D_0\}$. Since $T$ is a Kähler current, so is $T+{\varepsilon}\theta$ for ${\varepsilon}$ small enough. For $0<{\varepsilon}<\nu(\alpha,D_0)$ small enough, $T_{{\varepsilon}}:=T+{\varepsilon}\theta+(\nu(\alpha,D_0)-{\varepsilon})[D_0]+\sum_{D\neq
D_0}\nu(\alpha,D)[D]$ is then a positive current in $\alpha$ with $\nu(T_{{\varepsilon}},D_0)=\nu(\alpha,D_0)-{\varepsilon}<\nu(\alpha,D)=\nu(T_{\min},D_0)$ (the last equality holds by proposition 3.8 because $\alpha$ is big since $p$ is); this is a contradiction which proves the inclusion $Z^{-1}(p)\subset p+V_+(A)$.\
In the other direction, let $T$ be a Kähler current in $p$, and let $T=R+\sum\nu(T,D)D$ be its Siu decomposition. $R$ is then a Kähler current with $\nu(R,D)=0$ for every prime $D$, thus its class $\beta:=\{R\}$ is a modified Kähler class. We first claim that we have $D_j\subset E_{nn}(p-{\varepsilon}\beta)$ for every ${\varepsilon}>0$ small enough and every prime component $D_j$ of the non-Kähler locus $E_{nK}(p)$ of $p$. Indeed, since $p-{\varepsilon}\beta$ is big for ${\varepsilon}>0$ small enough, we have $\nu(p-{\varepsilon}\beta,D_j)=\nu(T,D_j)$ if $T$ is a positive current with minimal singularities in $p-{\varepsilon}\beta$, and we have to see that $\nu(T,D_j)>0$. But $T+{\varepsilon}R$ is a Kähler current in $p$, thus $D_j\subset E_{nK}(p)\subset E_+(T+{\varepsilon}R)$ by definition, which exactly means that $\nu(T+{\varepsilon}R,D_j)>0$. The claim follows since $\nu(R,D_j)=0$ by construction of $R$.\
As a consequence of this claim, each prime $D_1,...,D_r$ of our family $A$ occurs in the negative part $N(p-{\varepsilon}\beta)$ for ${\varepsilon}>0$ small enough. Consequently, by the first part of the proof, the Zariski projection of $Z(p-{\varepsilon}\beta)+\{E\}$ is just $Z(p-{\varepsilon}\beta)$ for every effective ${\mathbf{R}}$-divisor $E$ supported by the $D_j$’s and every ${\varepsilon}>0$ small enough. Since $p$ is big, $Z$ is continuous at $p$, thus $Z(p-{\varepsilon}\beta)$ converges to $Z(p)$, which is just $p$ because the latter is also modified nef. Finally, $Z$ is also continuous at the big class $p+\{E\}$, thus the Zariski projection of $Z(p-{\varepsilon}\beta)+\{E\}$ converges to that of $p+\{E\}$, and thus $Z(p+\{E\})=p$ holds. This means that $p+V_+(A)\subset Z^{-1}(p)$, and concludes the proof of theorem 3.20.
Structure of the pseudo-effective cone
--------------------------------------
Using our constructions, we will prove the
The boundary of the pseudo-effective cone is locally polyhedral away from the modified nef cone, with extremal rays generated by (the classes of) exceptional prime divisors.
$Proof$: this is in fact rather straightforward by now: for each prime $D$, the set ${\mathcal{E}}_D:=\{\alpha\in{\mathcal{E}},\nu(\alpha,D)=0\}$ is a closed convex subcone fo ${\mathcal{E}}$. This follows from the fact that $\alpha\mapsto\nu(\alpha,D)$ is convex, homogeneous, lower semi-continuous and everywhere non-negative. If $\alpha\in\partial{\mathcal{E}}$ does not belong to ${\mathcal{MN}}$, it does not belong to ${\mathcal{E}}_D$ for some prime $D$ by proposition 3.2. For every $\beta\in{\mathcal{E}}$, we have either $\beta\in{\mathcal{E}}_D$, or $D$ occurs in the negative part $N(\beta)$. Therefore, ${\mathcal{E}}$ is generated by ${\mathbf{R}}_+\{D\}$ and ${\mathcal{E}}_D$, and the latter does not contain $\alpha$. This means that $\partial{\mathcal{E}}$ is locally polyhedral near $\alpha$. Since $\nu(\alpha,D)>0$, we also see that $D$ is exceptional. Finally, the extremal rays of ${\mathcal{E}}$ not contained in ${\mathcal{MN}}=\cap_D{\mathcal{E}}_D$ have to lie outside ${\mathcal{E}}_D$ for some exceptional prime $D$, and since ${\mathcal{E}}={\mathcal{E}}_D+{\mathbf{R}}_+\{D\}$, each such extremal ray is generated by $\{D\}$ for some $D$, qed.
Volumes
-------
Recall that the volume of a pseudo-effective class $\alpha$ on a compact Kähler $n$-fold is defined to be the supremum $v(\alpha)$ of $\int_XT_{ac}^n$ for $T$ a closed positive $(1,1)$-current in $\alpha$ (cf. \[Bou02\]). A class $\alpha$ is big iff $v(\alpha)>0$, and the volume is a quantitative measure of its bigness. We have already noticed that $Z(\alpha)$ is big iff $\alpha$ is; we have the following quantitative version:
Let $\alpha$ be a pseudo-effective class on $X$ compact Kähler. Then $v(Z(\alpha))=v(\alpha)$.
The proof is in fact immediate: if $T$ is a positive current in $\alpha$, then we have $T\geq N(\alpha)$ since $T$ belongs to $\alpha[-{\varepsilon}\omega]$ for each ${\varepsilon}>0$, and we deduce that $T\to T-N(\alpha)$ is a bijection between the positive currents in $\alpha$ and those in $Z(\alpha)$. It remains to notice that $(T-N(\alpha))_{ac}=T_{ac}$ to conclude the proof.
Zariski decomposition on a surface and a hyper-Kähler manifold
==============================================================
It is known since the pioneering work of Zariski \[Zar62\] that any effective divisor $D$ on a projective surface admits a unique Zariski decomposition $D=P+N$, i.e. a decomposition into a sum of ${\mathbf{Q}}$-divisors $P$ and $N$ with the following properties:
\(i) $P$ is nef, $N=\sum a_jN_j$ is effective,
\(ii) $P\cdot N=0$,
\(iii) The Gram matrix $(N_i\cdot N_j)$ is negative definite.
We want to show that our divisorial Zariski decomposition indeed is a generalization of such a Zariski decomposition on a surface.
Notations
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$X$ will stand for a compact Kähler surface, or a compact hyper-Kähler manifold. For such an $X$, we denote by $q$ the quadratic form on ${H^{1,1}(X,{\mathbf{R}})}$ defined as follows: when $X$ is a surface, we set $q(\alpha):=\int\alpha^2$, and when $X$ is hyper-Kähler, we choose a symplectic holomorphic form $\sigma$, and let $q(\alpha):=\int\alpha^2(\sigma\overline{\sigma})^{m-1}$ be the usual Beauville-Bogomolov quadratic form, with $\sigma$ normalized so as to achieve $q(\alpha)^m=\int_X\alpha^{2m}$ (with $\dim X=n=2m$). In both cases $({H^{1,1}(X,{\mathbf{R}})},q)$ is Lorentzian, i.e. it has signature $(1,h^{1,1}(X)-1)$; the open cone $\{\alpha\in{H^{1,1}(X,{\mathbf{R}})},q(\alpha)>0\}$ has thus two connected components which are convex cones, and we denote by ${\mathcal{P}}$ the component containing the Kähler cone ${\mathcal{K}}$. We call ${\mathcal{P}}$ the positive cone (attached to the quadratic form $q$). In general, given a linear form $\lambda$ on ${H^{1,1}(X,{\mathbf{R}})}$, we will denote its kernel by $\lambda^{\perp}$ and the two open half-spaces it defines by $\lambda_{>0}$ and $\lambda_{<0}$. The dual $\mathcal{C}^{\star}$ of a convex cone $\mathcal{C}$ in ${H^{1,1}(X,{\mathbf{R}})}$ is seen as a cone in ${H^{1,1}(X,{\mathbf{R}})}$, using the duality induced by $q$.
The dual pseudo-effective cone
------------------------------
In both cases, we shall prove that the modifies nef cone is the dual cone to the pseudo-effective cone.
### The case of a surface
We suppose that $X$ is a surface. We prove the following essentially well-known
When $X$ is surface, the Kähler cone and the modified Kähler cone coincide. The dual pseudo-effective cone is just the nef cone.
$Proof$: if $\alpha\in{\mathcal{MK}}$, it can be represented by a Kähler current with analytic singularities in codimension 2, that is at some points $x_1,...,x_r$. Therefore we see that the non-Kähler locus $E_{nK}(\alpha)$ is a discrete set. Since the restriction of any class to a point is (by convention) a Kähler class, theorem 3.19 shows that $\alpha$ lies in fact in ${\mathcal{K}}$.\
Since $\int_X\omega\wedge T$ is positive for every Kähler form $\omega$ and every positive current $T$, we of course have ${\mathcal{K}}\subset{{\mathcal{E}}^{\star}}$, and thus also ${\mathcal{N}}=\overline{K}\subset{{\mathcal{E}}^{\star}}$. The other inclusion is much deeper, since it is a consequence of the Nakai-Moishezon criterion for Kähler classes on a surface, as given in \[Lam99\]. Indeed, this criterion implies that a real $(1,1)$-class $\alpha$ on a Kähler surface is a nef class iff $\alpha\cdot\omega\geq 0$ for every $\omega\in{\mathcal{K}}$ and $\alpha\cdot C\geq 0$ for every irreducible curve $C$. Since a class in ${{\mathcal{E}}^{\star}}$ clearly satisfies these conditions, we get ${{\mathcal{E}}^{\star}}\subset{\mathcal{N}}$, and the proof of theorem 4.1 is over.\
As a consequence, since ${\mathcal{K}}$ is contained in ${\mathcal{P}}$ and since $\overline{{\mathcal{P}}}$ is self dual (just because $q$ is Lorentzian), we get dually that $\overline{{\mathcal{P}}}\subset{\mathcal{E}}$ and thus that ${\mathcal{P}}\subset{\mathcal{E}}^0={\mathcal{B}}$, which means the following: if $\alpha$ is a real $(1,1)$-class with $\alpha^2>0$, then $\alpha$ or $-\alpha$ is big. This generalizes the well known case where $\alpha$ is (the first Chern class of) a line bundle (whose proof is based on Riemann-Roch).
### The hyper-Kähler case
In that case, the dual peudo-effective cone is also equal to the modified nef cone, but the proof uses another description, due to D.Huybrechts, of the dual pseudo-effective cone. In the easy direction, we have:
\(i) The modified nef cone ${\mathcal{MN}}$ is contained in both the dual pseudo-effective cone ${{\mathcal{E}}^{\star}}$ and the closure of the positive cone $\overline{{\mathcal{P}}}$
\(ii) We have $q(D,D')\geq 0$ for any two distinct prime divisors $D\neq D'$.
$Proof$: to prove (i), we only have to prove that ${\mathcal{MK}}\subset{{\mathcal{E}}^{\star}}$. Indeed, ${\mathcal{MK}}\cap{{\mathcal{E}}^{\star}}\subset{\mathcal{E}}\cap{{\mathcal{E}}^{\star}}$ is trivially contained in $\overline{{\mathcal{P}}}$. We pick a modified Kähler class $\alpha$ and a pseudo-effective class $\beta\in{\mathcal{E}}$, and choose a Kähler current $T$ in $\alpha$ with analytic singularities in codimension at least 2, and a positive current $S$ in $\beta$. By section 2.6, the wedge product $T\wedge S$ is well defined as a closed positive $(2,2)$-current, and lies in the class $\alpha\cdot\beta$. Since $(\sigma\overline{\sigma})^{m-1}$ is a smooth positive form of bidimension $(2,2)$, the integral $\int_XT\wedge S\wedge(\sigma\overline{\sigma})^{m-1}$ is positive. But $(\sigma\overline{\sigma})^{m-1}$ is also closed, thus we have $$\int_XT\wedge S\wedge(\sigma\overline{\sigma})^{m-1}=\alpha\cdot\beta\cdot\{(\sigma\overline{\sigma})^{m-1}\}=q(\alpha,\beta),$$ so we have proven that $q(\alpha,\beta)\geq 0$ as desired.\
The second contention is obtained similarly, noting that $\{D\}\cdot\{D'\}$ contains a closed positive $(2,2)$-current, which is $[D\cdot D']$, where $D\cdot D'$ is the effective intersection cycle.\
The other direction ${{\mathcal{E}}^{\star}}\subset{\mathcal{MN}}$ is much deeper. The effective $1$-dimensional cycles $C$ and the effective divisors $D$ define linear forms on ${H^{1,1}(X,{\mathbf{R}})}$ via the intersection form and the Beauville-Bogomolov form $q$ respectively, and we define a rational (resp. uniruled) chamber of the positive cone ${\mathcal{P}}$ to be a connected component of ${\mathcal{P}}-\cup C^{\perp}$ (resp. ${\mathcal{P}}-\cup
D^{\perp}$), where $C$ (resp. $D$) runs over the rational curves (resp. the uniruled divisors). By a rational curve (resp. a uniruled divisor) we mean an effective $1$-dimensional cycle all of whose components are irreducible rational curves (resp. an effective divisor all of whose components are uniruled prime divisors). The rational chamber of ${\mathcal{P}}$ cut out by all the $C_{>0}$’s (resp. $D_{>0}$)’s will be called the fundamental rational chamber (resp. the fundamental uniruled chamber). When $X$ is a $K3$ surface, the rational and uniruled chambers are the same thing and coincide with the traditional chambers in that situation. We can now state the following fundamental result:
\(i) The positive cone ${\mathcal{P}}$ is contained in ${\mathcal{E}}$.
\(ii) If $\alpha\in{\mathcal{P}}$ belongs to one of the rational chambers, then there exists a bimeromorphic map $f:X-\to X'$ to a hyper-Kähler $X'$ such that $$f_{\star}\alpha=\omega'+\{D'\},$$ where $\omega'\in{\mathcal{K}}_{X'}$ is a Kähler class and $D'$ is an uniruled ${\mathbf{R}}$-divisor.\
(iii) When $\alpha\in{\mathcal{P}}$ lies in both the fundamental uniruled chamber and one of the rational chambers, then no uniruled divisor $D'$ occurs in (ii).\
(iv) The fundamental rational chamber coincides with the Kähler cone of $X$.
In fact, \[Huy99\] states this only for a very general element $\alpha\in{\mathcal{P}}$, but we have noticed in \[Bou01\] that the elements of the rational chambers are already very general in that respect.\
In the situation (iii), $\alpha$ lies in $f^{\star}{\mathcal{K}}_{X'}$ for some bimeromorphic $f:X-\to X'$ towards a hyper-Kähler $X'$. The union of such open convex cones ${\mathcal{K}}_f:=f^{\star}{\mathcal{K}}_{X'}$ is called the bimeromorphic Kähler cone, and is denoted by ${\mathcal{BK}}$. The union in question yields in fact a partition of ${\mathcal{BK}}$ into open convex cones ${\mathcal{K}}_f$ (since a bimeromorphic map between minimal manifolds which sends one Kähler class to a Kähler class is an isomorphism by a result of A.Fujiki); ${\mathcal{BK}}$ is an open cone, but definitely not convex in general. (iii) tells us that each intersection of a rational chamber with the fundamental uniruled chamber is contained in ${\mathcal{BK}}$, and thus in one of the ${\mathcal{K}}_f$’s.\
We can now describe the dual pseudo-effective cone:
The dual pseudo-effective ${{\mathcal{E}}^{\star}}$ of a hyper-Kähler manifold coincides with the modified nef cone ${\mathcal{MN}}$.
$Proof$: by proposition 4.2, it remains to see that ${{\mathcal{E}}^{\star}}$ is contained in the modified nef cone ${\mathcal{MN}}$. By (i) of theorem 4.3, we have ${{\mathcal{E}}^{\star}}\subset\overline{{\mathcal{P}}}$, and it will thus be enough to show that an element of the interior of ${{\mathcal{E}}^{\star}}$ which belongs to one of the rational chambers lies in ${\mathcal{MN}}$. But an element $\alpha$ of the interior of ${{\mathcal{E}}^{\star}}$ has $q(\alpha,D)>0$ for every prime $D$, thus it certainly lies in the fundamental uniruled chamber. If $\alpha$ lies in both the interior of ${{\mathcal{E}}^{\star}}$ and one of the rational chambers, it therefore lies in ${\mathcal{K}}_f=f^{\star}{\mathcal{K}}_{X'}$ for some bimeromorphic $f:X-\to X'$, and it remains to see that ${\mathcal{K}}_f\subset{\mathcal{MN}}$. But if $\omega$ is a Kähler form on $X'$, its pull-back $T:=f^{\star}\omega$ can be defined using a resolution of $f$, and it is easy to check that $T$ is a Kähler current with $\nu(T,D)=0$ for every prime $D$, since $f$ induces an isomorphism $X-A\to X'-A'$ for $A$, $A'$ analytic subsets of codimension at least 2 (this is because $X$ and $X'$ are minimal). Therefore, $\{T\}=f^{\star}\{\omega\}$ belongs to ${\mathcal{MK}}\subset{\mathcal{MN}}$, qed.
Exceptional divisors
--------------------
When $X$ is a surface or a hyper-Kähler manifold, the fact that a family $D_1,...,D_r$ of prime divisors is exceptional can be read off its Gram matrix.
A family $D_1,...,D_r$ of prime divisors is exceptional iff its Gram matrix $(q(D_i,D_j))$ is negative definite.
$Proof$: let $V$ (resp. $V_+$) be the real vector space of ${\mathbf{R}}$-divisors (resp. effective ${\mathbf{R}}$-divisors) supported by the $D_j$’s. We begin with a lemma of quadratic algebra:
Assume that $(V,q)$ is negative definite. Then every $E\in V$ such that $q(E,D_j)\leq 0$ for all $j$ belongs to $V_+$.
$Proof$: if $E\in V$ is non-positive against each $D_j$, we write $E=E_+-E_-$ where $E_+$ and $E_-$ are effective with disjoint supports. We have to prove that $E_-=0$, and this is equivalent by assumption to $q(E_-)\geq 0$. But $q(E_-)=q(E_-,E_+)-q(E_-,E)$. The first term is positive because $E_+$ and $E_-$ have disjoint supports, using (ii) of proposition 4.2, whereas the second is positive by assumption on $E$.\
Let $D_1,...,D_r$ be primes with negative definite Gram matrix. In particular, we then have that $\{V_+\}\subset{H^{1,1}(X,{\mathbf{R}})}$ meets $\overline{{\mathcal{P}}}$ at $0$ only. Since the modified nef cone ${\mathcal{MN}}$ is contained in $\overline{{\mathcal{P}}}$ by proposition 4.2, $\{V_+\}$ $a$ $fortiori$ meets the modified nef cone at $0$ only, which means by definition that $D_1,...,D_r$ is an exceptional family, and this proves necessity in theorem 4.5. In the other direction, assume that $D_1,...,D_r$ is an exceptional family of primes. We first prove that the matrix $(q(D_i,D_j))$ is semi-negative. If not, we find an ${\mathbf{R}}$-divisor $E$ in $V$ with $q(E)>0$. Writing again $E=E_+-E_-$, with $E_+$ and $E_-$ two effective divisors in $V_+$ with disjoint supports, we have again $q(E_+,E_-)\geq 0$ by (ii) of proposition 4.2, and thus $q(E_+)+q(E_-)\geq q(E)>0$. We may therefore assume that $E$ lies in $V_+$, with $q(E)>0$. But then $E$ or $-E$ is big, and it has to be $E$ because it is already effective. Its Zariski projection $Z(\{E\})$ is then non-zero since it is also big (by proposition 3.10), and it lies in both $\{V_+\}$ and ${\mathcal{MN}}$, a contradiction.\
To conclude the proof of theorem 4.5, we may assume (by induction) that the Gram matrix of $D_1,...,D_{r-1}$ is negative definite. If $(V,q)$ is degenerate, the span $V'$ of $D_1,...,D_{r-1}$ is such that its orthogonal space $V'^{\perp}$ in $V$ is equal to the null-space of $V$. We then decompose $D_r=E+F$ in the direct sum $V=V'\oplus V'^{\perp}$. Since $q(E,D_j)=q(D_r,D_j)\geq 0$ for $j<r$, lemma 4.6 yields that $E\leq 0$. Therefore, $F=D_r-E$ lies in $V_+$, and is certainly non-zero. We claim that $\{F\}$ is also modified nef, which will yield the expected contradiction. But $F$ lies in the null-space of $V$, and is therefore non-negative against every prime divisor $D$. If $\alpha$ is a pseudo-effective class, we have $q(\{F\},\alpha)=q(\{F\},Z(\alpha))+q(F,N(\alpha))$. The first term is positive since $Z(\alpha)\in{\mathcal{MN}}={{\mathcal{E}}^{\star}}$, and the the second one is positive because $F$ is positive against every effective divisor. We infer from all this that $\{F\}$ lies in ${{\mathcal{E}}^{\star}}={\mathcal{MN}}$, and the claim follows.\
The theorem says in particular that a prime divisor $D$ is negative iff $q(D)<0$. On a K3 surface, an easy and well-known argument using the adjunction formula shows that the prime divisors with negative square are necessarily smooth rational curves with square $-2$. In higher dimension, we have:
On a hyper-Kähler manifold $X$, the exceptional prime divisors are uniruled.
$Proof$: since $D$ is exceptional, it lies outside $\overline{{\mathcal{P}}}={\mathcal{P}}^{\star}$, and we thus find a class $\alpha\in{\mathcal{P}}$ lying in one of the rational chambers such that $q(\alpha,D)<0$. By (ii) of theorem 4.3, there exists a bimeromorphic map between hyper-Kähler manifolds $f:X-\to X'$ such that $f_{\star}\alpha=\omega'+\sum a_j D_j'$ with $\omega'$ a Kähler class, $a_j\geq 0$ and $D_j'$ a uniruled prime divisor. Since the quadratic form is preserved by $f$, we have $0>q(\alpha,D)=q(\omega',f_{\star}D)+\sum a_j q(D_j',f_{\star}D)$, and $q(D_j',f_{\star}D)$ has to be negative for some $j$. But this implies that the two primes $D_j'$ and $f_{\star}D$ coincide, and thus $D=f^{\star}D_j'$ is uniruled since $D_j'$ is.
Rationality of the Zariski decomposition
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We want to prove that the divisorial Zariski decomposition is rational (when $X$ is a surface or a hyper-Kähler manifold) in the sense that $N(\alpha)$ is a rational divisor when $\alpha$ is a rational class. We first show the following characterization of the divisorial Zariski decomposition:
If $\alpha\in{H^{1,1}(X,{\mathbf{R}})}$ is a pseudo-effective class, its divisorial Zariski decomposition $\alpha=Z(\alpha)+\{N(\alpha)\}$ is the unique orthogonal decomposition of $\alpha$ into the sum of a modified nef class and the class of an exceptional effective ${\mathbf{R}}$-divisor.
$Proof$: we first prove uniqueness: assume that $\alpha=p+\{N\}$ is an orthogonal decomposition with $p$ a modified nef class and $N$ an effective exceptional ${\mathbf{R}}$-divisor. We claim that $N(\alpha)=N$. To see this, let $D_1,...,D_r$ be the support of $N$; the Gram matrix $(q(D_i,D_j))$ is negative definite by theorem 4.5, and $p$ is orthogonal to each $D_j$ because $q(p,N)=0$ and $q(p,D_j)\geq 0$ for all $j$ since $p$ is a modified nef class. We have $N(\alpha)\leq
N(p)+N$ and $N(p)=0$ since $p$ is modified nef, thus $N(\alpha)\leq
N$. But $N(\alpha)-N$ is supported by primes $D_1,...,D_r$ whose Gram matrix is negative definite, and $q(N(\alpha)-N,D_j)=q(p,D_j)-q(Z(\alpha),D_j)$ is non-positive since $p$ is orthogonal to $D_j$ and $Z(\alpha)$ belongs to ${\mathcal{MN}}={{\mathcal{E}}^{\star}}$. Lemma 4.6 thus yields $N(\alpha)\geq N$, and the claim follows. To prove theorem 4.8, we will show the existence of an orthogonal decomposition $\alpha=p+\{N\}$ with $p$ a modified nef class and $N$ an exceptional ${\mathbf{R}}$-divisor. When this is done, we must have $N=N(\alpha)$ by the claim, so that that $\alpha=Z(\alpha)+\{N(\alpha)\}$ is itself an orthogonal decomposition.
A pseudo-effective class $\alpha$ lies in ${{\mathcal{E}}^{\star}}$ iff $q(\alpha,D)\geq 0$ for every prime $D$.
$Proof$: if $\beta$ is a pseudo-effective class, we write $q(\alpha,\beta)=q(\alpha,Z(\beta))+q(\alpha,N(\beta))$. The first term is positive because $Z(\beta)$ lies in ${{\mathcal{E}}^{\star}}$, and the second one is positive if $q(\alpha,D)\geq 0$ for each prime $D$.
Let $\alpha$ be a pseudo-effective class and let $D_1,...,D_r$, $E_1,...,E_p$ be two families of primes such that:
\(i) $q(\alpha,D_j)<0$ and $q(\alpha,E_i)\leq 0$ for every $j$ and $i$.
\(ii) $E_1,...,E_r$ is an exceptional family.
Then the union of these two families is exceptional.
$Proof$: let $F$ be an effective divisor supported by $D_j$’s and $E_i$’s, and assume that $\{F\}$ is a modified nef class. We have to see that $F=0$. But $q(\alpha,F)$ is positive since $F$ is modified nef, thus we see using (i) that $F$ is in fact supported by $E_i$’s, and then (ii) enables us to conclude that $F=0$ as desired.\
At this point, the argument is similar to \[Fuj79\]. If the pseudo-effective class $\alpha$ is already in ${{\mathcal{E}}^{\star}}$, we trivially have our decomposition. Otherwise, consider the family $A$ of primes $D$ such that $q(\alpha,D)<0$. That family is exceptional by lemma 4.10 with $E_1,...,E_p$ an empty family, thus $A$ is finite with negative definite Gram matrix, and is non-empty by lemma 4.9. Let $$\alpha=\alpha_1+\{N_1\}$$ be the decomposition in the direct sum $V^{\perp}\oplus V$, where $V\subset{H^{1,1}(X,{\mathbf{R}})}$ is spanned by $A$. We claim that $N_1$ is effective and that $\alpha_1$ is pseudo-effective. Since $q(N_1,D)=q(\alpha,D)<0$ for every $D\in A$, lemma 4.6 yields that $N_1$ is effective. We can also write $N(\alpha)=E+F$ where $E$ and $F$ are effective with disjoint supports and $F$ is supported by elements of $A$. Then for every $D\in A$ we have $q(F-N_1,D)\leq q(N(\alpha)-N_1,D)$ since $E$ and $D$ are disjoint, and $q(N(\alpha)-N_1,D)=q(\alpha_1,D)-q(Z(\alpha),D)$ is non-positive because $\alpha_1$ and $D$ are orthogonal and $Z(\alpha)$ lies in ${{\mathcal{E}}^{\star}}$. We infer from this that $N(\alpha)\geq N_1$ using lemma 4.6, and $\alpha_1=Z(\alpha)+\{N(\alpha)-N_1\}$ is thus pseudo-effective, and this proves our claim.\
If $\alpha_1$ lies in ${{\mathcal{E}}^{\star}}$, we have our decomposition by construction; otherwise, we iterate the construction: let $B$ be the non-empty exceptional family of primes $D$ such that $q(\alpha_1,D)<0$. Since $A$ is already exceptional and $q(\alpha_1,D)=0$ for $D\in A$, we infer from lemma 4.10 that the union $A_1$ of $A$ and $B$ is again an exceptional family. We decompose $$\alpha_1=\alpha_2+\{N_2\}$$ in the direct sum $V_1^{\perp}\oplus V_1$, where $V_1\subset{H^{1,1}(X,{\mathbf{R}})}$ is spanned by $A_1$. The same arguments as above show in that case also that $\alpha_2$ is pseudo-effective, and also that $N_2$ is effective (since $q(N_2,D)=q(\alpha_1,D)\leq 0$ for each $D\in A_1$). But since $B$ is non-empty, $A_1$ is an exceptional family strictly bigger than $A$. Since the length of the exceptional families is uniformly bounded by the Picard number $\rho(X)$ by theorem 3.14, the iteration of the construction has to stop after $l$ steps, for which we get a class $\alpha_l$ which is modified nef. The desired decomposition is then obtained by setting $p:=\alpha_l$ and $N:=N_1+...+N_l$, which is exceptional since it is supported by elements of $A\cup A_1\cup...\cup A_l=A_l$ (since $A\subset A_1\subset ...\subset A_1$ by construction). This concludes the proof of theorem 4.8.
The divisorial Zariski decomposition is rational in case $X$ is a surface or a hyper-Kähler manifold. In particular, when $D$ is a pseudo-effective divisor on $X$, the modified nef ${\mathbf{R}}$-divisor $P:=D-N(\{D\})$ is rational and such that the canonical inclusion of $H^0(X,{\mathcal{O}}(kP))$ in $H^0(X,{\mathcal{O}}(kD))$ is surjective for every $k$ such that $kP$ is Cartier.
$Proof$: if $\alpha\in NS(X)\otimes{\mathbf{Q}}$ is a rational class, $N(\alpha)$ is necessarily the image of $\alpha$ by the orthogonal projection $NS(X)\otimes{\mathbf{Q}}\to V_{{\mathbf{Q}}}(\alpha)$, where $V_{{\mathbf{Q}}}(\alpha)$ is the ${\mathbf{Q}}$-vector space generated by the cohomology classes of the components of $N(\alpha)$. The latter is therefore rational. As to the second part, let $E$ be an element of the linear system $|kD|$. Since the integration current $\frac{1}{k}[E]$ is positive and lies in $\{D\}$, we have $E\geq kN(\{D\})$. But this exactly means that $kN(\{D\})$ is contained in the base scheme of $|kD|$, as was to be shown.
If $p\in{H^{1,1}(X,{\mathbf{R}})}$ is a modified nef class on $X$, its volume is equal to $$v(p)=q(p)^m=\int p^{\dim X}.$$ In general, we have $v(\alpha)=\int Z(\alpha)^{\dim X}$; in particular, the volume of a rational class is rational.
$Proof$: we have already proven in proposition 3.22 that $v(\alpha)=v(Z(\alpha))$, so only the first assertion needs a proof. We have shown in \[Bou02\] that the equality $v(p)=\int p^{\dim
X}$ is always true when $p$ is a nef class, so the contended equality holds on a surface. In the hyper-Kähler case, since we have chosen the symplectic form $\sigma$ so that $q(\alpha)^m=\alpha^{2m}$ for any class $\alpha$, we just have to prove $v(p)=q(p)^m$ for $p\in{\mathcal{MN}}$. The latter cone is also the closure of the bimeromorphic Kähler cone ${\mathcal{BK}}$, so we may assume that $p$ lies in $f^{\star}{\mathcal{K}}_{X'}$ for some bimeromorphic map $f:X-\to X'$ between hyper-Kähler manifolds (because both $q$ and the volume are continuous). But since $f$ is an isomorphism in codimension 1, the volume is invariant under $f$, and so is the quadratic form $q$, so we are reduced to the case where $p$ is a Kähler class, for which the equality is always true as we’ve said above.
The algebraic approach
======================
In this section, we would like to show what the constructions we have made become when $\alpha=c_1(L)$ is the first Chern class of a line bundle on a projective complex manifold $X$. The general philosophy is that the divisorial Zariski decomposition of a big line bundle can be defined algebraically in terms of the asymptotic linear series $|kL|$. When $L$ is just pseudo-effective, sections are of course not sufficient, but we are led back to the big case by approximating. For those who are reluctant to assume projectivity too quickly, we remark that a compact Kähler manifold carrying a big line bundle is automatically projective.
From sections to currents and back
----------------------------------
Let $L\to X$ be a line bundle over the projective manifold $X$. Each time $L$ has sections $\sigma_1,...,\sigma_l\in H^0(X,L)$, there is a canonical way to construct a closed positive current $T\in c_1(L)$ with analytic singularities as follows: choose some smooth Hermitian metric $h$ on $L$, and consider $$\varphi(x):=\frac{1}{2}\log\sum_jh(\sigma_j(x)).$$ Then we define $T=\Theta_h(L)+dd^c\varphi$, where $\Theta_h(L)$ is the first Chern form of $h$. One immediately checks that $T$ is positive and independent of the choice of $h$, and thus depends on the sections $\sigma_j$ only. $T$ has analytic singularities exactly along the common zero-scheme $A$ of the $\sigma_j$’s, and its Siu decomposition therefore writes $T=R+D$, where $D$ is the divisor part of $A$. When $(\sigma_j)$ is a basis of $H^0(X,L)$, we set $T_{|L|}:=T$. Another way to see $T_{|L|}$ is as the pull-back of the Fubiny-Study form on ${\mathbf{P}}H^0(X,L)^{\star}={\mathbf{P}}^N$ (the identification is determined by the choice of the basis of $H^0(L)$) by the rational map\
$\phi_{|L|}:X-\to{\mathbf{P}}H^0(X,L)^{\star}$. $T_{|L|}$ is independent of the choice of the basis up to equivalence of singularities, and carries a great deal of information about the linear system $|L|$: the singular scheme $A$ of $T_{|L|}$ is the base scheme $B_{|L|}$ of the linear system $|L|$, the Lelong number $\nu(T_{|L|},x)$ at $x$ is just the so-called multiplicity of the linear system at $x$, which is defined by $$\nu(|L|,x):=\min\{\nu(E,x),E\in|L|\}.$$ If a modification $\mu:{\widetilde}{X}\to X$ is chosen such that $\mu^{\star}|L|=|M|+F$, where $M$ has non base-point and $F$ is an effective divisor, then $\mu^{\star}T_{|L|}=T_{\mu^{\star}|L|}=T_{|M|}+F$ where $T_{|M|}$ is smooth since $|M|$ is generated by global sections. The so-called moving self-intersection of $L$, which is by definition $L^{[n]}:=M^n$, is thus also equal to $\int_X(T_{|L|})_{ac}^n$.\
When $L$ is a big line bundle, we get for each big enough $k>0$ a positive current $T_k:=\frac{1}{k}T_{|kL|}$ in $c_1(L)$. A result of Fujita (cf. \[DEL00\]) claims that the volume $v(L)$ is the limit of $\frac{1}{k^n}(kL)^{[n]}$, thus we have $v(L)=\lim_{k\to+\infty}\int_XT_{k,ac}^n$.\
Finally, if $T_{\min}$ is a positive current with minimal singularities in $c_1(L)$, we can choose a singular Hermitian metric $h_{\min}$ on $L$ whose curvature current is $T_{\min}$ (by section 2.4). If $L$ is still big and if for each $k$ we choose the basis of $H^0(kL)$ to be orthonormal with respect to $h_{\min}^{\otimes k}$, then it can be shown that $T_k\to T_{\min}$, and we will see in 5.2 that $\nu(T_k,x)=\frac{1}{k}\nu(|kL|,x)$ converges to $\nu(T_{\min},x)=\nu(c_1(L),x)$. In some sense, the family $T_k$ deriving from $|kL|$ is cofinite $(c_1(L)^+,\preceq)$.\
It should however be stressed that $T_{|kL|}$ will in general $not$ be a Kähler current, even if $L$ is big. Indeed, consider the pull-back $L=\mu^{\star}A$ of some ample line bundle $A$ by a blow-up $\mu$. Then $kL$ will be generated by global sections for $k$ big enough, and $T_{|kL|}$ is thus smooth for such a $k$, but not a Kähler current, since $L$ is not ample and a smooth Kähler current is just a Kähler form.\
Conversely, to go from currents to sections is the job of the $L^2$ estimates for the $\overline{\partial}$ operator, e.g. in the form of Nadel’s vanishing theorem. Recall that the multiplier ideal sheaf ${\mathcal{I}}(T)$ of a closed almost positive $(1,1)$-current $T$ is defined locally as follows: write $T=dd^c\varphi$ locally at some $x$. Then the stalk ${\mathcal{I}}(T)_x$ is the set of germs of holomorphic functions at $x$ such that $|f|^2e^{-2\varphi}$ is locally integrable at $x$. Then Nadel’s vanishing states that if $T$ is a Kähler current in the first Chern class $c_1(L)$ of a line bundle $L$, then $H^q(X,{\mathcal{O}}(K_X+L)\otimes{\mathcal{I}}(T))=0$ for every $q>0$. In particular, if $V(T)$ denotes the scheme $V({\mathcal{I}}(T))$, then the restriction map $$H^0(X,{\mathcal{O}}_X(K_X+L))\to H^0(V(T),{\mathcal{O}}_{V(T)}(K_X+L))$$ is surjective. This affords a tool to prove generation of jets at some points, using the following lemma (cf. \[DEL00\]):
If $\nu(T,x)<1$, then ${\mathcal{I}}(T)_x={\mathcal{O}}_x$. If $\nu(T,x)\geq n+s$, we have ${\mathcal{I}}(T)_x\subset\mathcal{M}_x^{s+1}$.
To illustrate how this works, let us prove the following algebraic characterization of the non-Kähler locus:
If $L$ is a big line bundle, then the non-Kähler locus\
$E_{nK}(c_1(L))$ is the intersection of the non-finite loci $\Sigma_k$ of the rational maps $\phi_{|kL|}$, defined as the union of the reduced base locus $B_{|kL|}$ and the set of $x\in X-B_{|kL|}$ such that the fiber through $x$ $\phi_{|kL|}^{-1}(\phi_{|kL|}(x))$ is positive dimensional somewhere.
$Proof$: If $x_1,...,x_r\in X$ lie outside $E_{nK}(c_1(L))$, then we can find a Kähler current $T\in c_1(L)$ with analytic singularities such that each $x_j$ lies outside the singular locus of $T$. The latter being closed, there exists a neighbourhood $U_j$ of $x_j$ such that $\nu(T,z)=0$ for every $z\in U_j$. We artificially force an isolated pole at each $x_j$ by setting ${\widetilde}{T}=T+\sum_{1\leq j\leq
r}dd^c({\varepsilon}\theta_j(z)\log|z-x_j|)$, where $\theta_j$ is a smooth cut-off function near $x_j$, and ${\varepsilon}>0$ is so small that ${\widetilde}{T}$ is still Kähler. We have $\nu({\widetilde}{T},x_j)={\varepsilon}$, whereas $\nu({\widetilde}{T},z)$ is still zero for every $z\neq x_j$ in $U_j$. We now choose a some smooth form $\tau$ in $c_1(K_X)$, and consider the current $T_k:=k{\widetilde}{T}-\tau$. It lies in the first Chern class of $L_k:=kL-K_X$, and is certainly still Kähler for $k$ big enough. We also have $\nu(T_k,z)=0$ for every $z\neq x_j$ close to $x_j$, and $\nu(T_k,x_j)=k{\varepsilon}$. Given $s_1,...,s_r$, we see that, for $k$ big enough, each $x_j$ will be isolated in $E_1(T_k)$, whereas ${\mathcal{I}}(T_k)_{x_j}\subset\mathcal{M}_{x_j}^{s_j+1}$, using Skoda’s lemma. Nadel’s vanishing then implies that the global sections of $kL$ generate $s_j$-jets at $x_j$ for every $j$. This implies that the non-finite locus $\Sigma_k$ is contained in $E_{nK}(c_1(L))$.\
To prove the converse inclusion, we have to find for each $m$ a Kähler current $T_m$ in $c_1(L)$ with $E_+(T_m)\subset\Sigma_m$. To do this, we copy the proof of proposition 7.2 in \[Dem97\].
If $L$ is any line bundle such that the non-finite locus $\Sigma_m$ of $mL$ is distinct from $X$ for some $m$, then, for every line bundle $G$, the base locus of $|kL-G|$ is contained in $\Sigma_m$ for $k$ big enough.
We then take $G$ to be ample, and set $T_m:=\frac{1}{k}(T_{|kL-G|}+\omega)$ with $k$ big enough so that $B_{|kL-G|}\subset\Sigma_m$ and $\omega$ a Kähler form in $c_1(G)$.\
To prove lemma 5.3, note that $|mL|$ is not empty, so we can select a modification $\mu:{\widetilde}{X}\to X$ such that $\mu^{\star}|mL|=|{\widetilde}{L}|+F$, where $|{\widetilde}{L}|$ is base point free. It is immediate to check that it is enough to prove the lemma for ${\widetilde}{L}$, so we can assume from the beginning that $L$ is base-point free, with $m=1$. We set $\phi:=\phi_{|L|}:X\to{\mathbf{P}}^N$ and $\Sigma:=\Sigma_1$. Upon adding a sufficiently ample line bundle to $G$, it is also clear that we may assume $G$ to be very ample. If $x\in X$ lies outside $\Sigma$, the fiber $\phi^{-1}(\phi(x))$ is a finite set, so we can find a divisor $D\in|G|$ which doesn’t meet it. Therefore we have $\phi(x)\in{\mathbf{P}}^N-\phi(D)$, so that for $k$ big enough there exists $H\in|{\mathcal{O}}_{{\mathbf{P}}^N}(k)|$ with $H\geq\phi_{\star}D$ which doesn’t pass through $\phi(x)$. The effective divisor $\phi^{\star}H-D$ is then an element of $|kL-G|$ which doesn’t pass through $x$. The upshot is: for every $x\in X$ outside $\Sigma$, we have $x\in X-B_{|kL-G|}$ for $k$ big enough. By Nötherian induction, we therefore find $k$ big enough such that $B_{|kL-G|}$ is contained in $\Sigma$, as was to be shown.
Minimal Lelong numbers
----------------------
When $L$ is a big ${\mathbf{R}}$-divisor, we denote by $L_k:=\lfloor kL\rfloor$ the round-down of $kL$, and by $R_k:=kL-L_k$ the fractional part of $kL$. We then consider the sequence $\frac{1}{k}\nu(|L_k|,x)$. It is easily seen to be subadditive, and therefore $\nu(||L||,x):=\lim_{k\to+\infty}\frac{1}{k}\nu(|kL|,x)$ exists. We then prove the following
If $L$ is a big ${\mathbf{R}}$-divisor on $X$ and $\alpha:=\{L\}\in NS(X)_{{\mathbf{R}}}$, then $$\nu(\alpha,x)=\nu(||L||,x)$$ for every $x\in X$.
$Proof$: let $L=\sum a_jD_j$ be the decomposition of $L$ into its prime components. We choose arbitrary smooth forms $\eta_j$ in $\{D_j\}$, and denote by $\tau_k:=\sum(ka_j-\lfloor ka_j\rfloor)\eta_j$ the corresponding smooth form in $\{R_k\}$. Since $\tau_k$ has bounded coefficients, we can choose a fixed Kähler form $\omega$ such that $-\omega\leq\tau_k\leq\omega$ for every $k$. If $E$ is an effective divisor in $|L_k|$, then $1/k([E]+\tau_k)$ is a current in $\alpha[-1/k\omega]$, therefore $\frac{1}{k}\nu(E,x)\geq\nu(T_{\min,1/k},x)$, where $T_{\min,1/k}$ is a current with minimal singularities in $\alpha[-1/k\omega]$, and this yields $\lim_{k\to\infty}\frac{1}{k}\nu(|L_k|,x)\geq\lim_{k\to\infty}\nu(T_{\min,1/k},x)=\nu(\alpha,x)$. In the other direction, we use a related argument in \[DEL00\], Theorem 1.11. The Ohsawa-Takegoshi-Manivel $L^2$ extension theorem says in particular that if we are given a Hermitian line bundle $(A,h_A)$ with sufficiently positive curvature form, then for every pseudo-effective line bundle $G$ and every singular Hermitian metric $h$ on $G$ with positive curvature current $T\in c_1(G)$ and every $x\in X$, the evaluation map $$H^0(X,\mathcal{O}(G+A)\otimes\mathcal{I}(T))\to\mathcal{O}_x(G+A)\otimes\mathcal{I}(T)_x$$ is surjective, with an $L^2$ estimate independent of $(G,h)$ and $x\in X$.\
We now fix a Hermitian line bundle $(A,h_A)$ with a sufficiently positive curvature form $\omega_A$ to satisfy the Ohsawa-Takegoshi theorem. We select a positive current with minimal singularities $T_{\min}$ in $\alpha$, and also a Kähler current $T$ in $\alpha$, which is big by assumption; we can then find almost pluri-subharmonic functions $\varphi_{\min}$ and $\varphi$ on $X$ such that $T_{\min}-dd^c\varphi_{\min}$ and $T-dd^c\varphi$ are smooth. We set $G_k:=L_k-A=kL-R_k-A=(k-k_0)L+(k_0L-R_k-A)$, and fix $k_0$ big enough so that $k_0T-\omega-\omega_A$ is a Kähler current. For $k\geq k_0$, the current $T_k:=(k-k_0)T_{\min}+(k_0T-\tau_k-\omega_A)$ is then a positive current in $c_1(G_k)$, thus we can choose for each $k$ a smooth Hermitian metric $h_k$ on $G_k$ such that $T_k$ is the curvature current of the singular Hermitian metric $\exp(-2(k-k_0)\varphi_{\min}-2k_0\varphi)h_k$. Applying the Ohsawa-Takegoshi to $G_k$ equipped with this singular Hermitian metric, we thus get a section $\sigma\in H^0(X,L_k)$ such that $$h_k(\sigma(x))\exp(-2(k-k_0)\varphi_{\min}(x)-2\varphi(x))=1$$ and $$\int_Xh_k(\sigma)\exp(-2(k-k_0)\varphi_{\min}-2\varphi)dV\leq C_1,$$ where $C_1$ does not depend on $k$ and $x$. If we choose a basis $\sigma_1,...,\sigma_l$ of $H^0(X,L_k)$, we infer from this that $$\varphi_{\min}(x)+\frac{1}{k-k_0}\varphi(x)=\frac{1}{2(k-k_0)}\log h_k(\sigma(x))$$ $$\leq\frac{1}{2(k-k_0)}\log\sum h_k(\sigma_j(x))+C_2,$$ where $C_2$ does not depend on $x$. The latter inequality comes from the bound on the $L^2$ norm of $\sigma$, since the $L^2$ norm dominates the $L^{\infty}$ norm. Therefore $$\frac{1}{k-k_0}\nu(|L_k|,x)\leq\nu(\varphi_{\min},x)+\frac{C_3}{k-k_0},$$ where $C_3$ is a bound on the Lelong numbers of $T$. If we let $k\to\infty$ in the last inequality, we get $\nu(||L||,x)\leq\nu(\alpha,x)$ as desired.
Zariski decompositions of a divisor
-----------------------------------
The usual setting for the problem of Zariski decompositions is the following: let $X$ be a projective manifold, and $L$ a divisor on it. One asks when it is possible to find two ${\mathbf{R}}$-divisors $P$ and $N$ such that:
\(i) $L=P+N$
\(ii) $P$ is nef,
\(iii) $N$ is effective,
\(iv) $H^0(X,kL)=H^0(X,\lfloor kP\rfloor)$ for all $ k>0$, where the round-down $\lfloor F\rfloor$ of an ${\mathbf{R}}$-divisor $F$ is defined coefficient-wise.
This can of course happen only if $L$ is already pseudo-effective. When this is possible, one says that $L$ admits a Zariski decomposition (over ${\mathbf{R}}$ or ${\mathbf{Q}}$, depending whether the divisors are real or rational). We want to show that, for a big divisor $L$, this can be read off the negative part $N(\{L\})$.
Let $L$ be a big divisor on $X$, and let $N(L):=N(\{L\})$ and $P(L):=L-N(L)$. Then $L=P(L)+N(L)$ is the unique decomposition $L=P+N$ into a modified nef ${\mathbf{R}}$-divisor $P$ and an effective ${\mathbf{R}}$-divisor $N$ such that the canonical inclusion $H^0(\lfloor kP\rfloor)\to H^0(kL)$ is an isomorphism for each $k>0$.
$Proof$: first, we have to check that $H^0(X,kL)=H^0(X,\lfloor kP(L)\rfloor)$. If $E$ is an effective divisor in the linear system $|kL|$, we have to see that $E\geq\lceil kN(L)\rceil$. But $\frac{1}{k}[E]$ is a positive current in $\{L\}$, thus $E\geq kN(L)$, and so $E\geq\lceil kN(L)\rceil$ since $E$ has integer coefficients.\
Conversely, assume that $L=P+N$ is a decomposition as in theorem 5.5. We have to show that $N=N(L)$, i.e. $\nu(\{L\},D)=\nu(N,D)$ for every prime $D$. In view of theorem 5.4, this will be a consequence of the following
Suppose that a big divisor $L$ writes $L=P+N$, where $P$ is an ${\mathbf{R}}$-divisor and $N$ is an effective ${\mathbf{R}}$-divisor such that $H^0(X,kL)=H^0(X,\lfloor kP\rfloor)$ for every $k>0$. Then we have:
\(i) If $P$ is nef, then $\nu(||L||,x)=\nu(N,x)$ for every $x\in X$.
\(ii) If $P$ is modified nef, then $\nu(||L||,D)=\nu(N,D)$ for every prime $D$.
$Proof$: the assumption $H^0(X,kL)=H^0(X,\lfloor kP\rfloor)$ means precisely that for every $E\in|kL|$ we have $E\geq\lceil kN\rceil$, thus $\nu(|kL|,x)\geq\sum\frac{\lceil ka_j\rceil}{k}\nu(D_j,x)$ if we write $N=\sum
a_jD_j$. We deduce from this the inequality $\lim_{k\to\infty}\frac{1}{k}\nu(|kL|,x)\geq\sum a_j\nu(D_j,x)=\nu(N,x)$. To get the converse inequalities, notice that $$\nu(|kL|,x)\leq\nu(|P_k|,x)+\nu(kN,x)$$ with $P_k:=\lfloor kP\rfloor$ as before; dividing this out by $k$ and letting $k\to+\infty$, we deduce $$\lim_{k\to\infty}\frac{1}{k}\nu(|kL|,x)\leq\lim_{k\to\infty}\frac{1}{k}\nu(kN,x)=\nu(N,x)$$ when $P$ is nef, since $\nu(\{P\},x)=\lim_{k\to\infty}\frac{1}{k}\nu(P_k,x)$ is then always zero, and similarly with $D$ in place of $x$ when $P$ is modified nef (remark that $P$ is big because $L$ is). This concludes the proof of theorem 5.5.
Let $L$ be a big divisor on $X$, and assume that $\nu(\{L\},D)$ is irrational for some irreducible divisor $D$. Then there cannot exists a modification $\mu:{\widetilde}{X}\to X$ such that $\mu^{\star}L$ admits a Zariski decomposition over ${\mathbf{Q}}$.
$Proof$: if a modification $\mu$ as stated exists, then the negative part $N(\mu^{\star}L)$ has to be rational by theorem 5.5, and we get a contradiction using the following easy
Let $\alpha$ be a pseudo-effective class on $X$, and let $\mu:{\widetilde}{X}\to X$ be a modification. Then we have $$N(\alpha)=\mu_{\star}N(\mu^{\star}\alpha).$$
$Proof$: very easily checked using that a modification is an isomorphism in codimension 1.
### An example of Cutkosky.
We propose to analyze in our setting an example due to S.D.Cutkosky \[Cut86\] of a big line bundle $L$ on a 3-fold $X$ whose divisorial Zariski decomposition is not rational, but whose Zariski projection $Z(\{L\})$ is nef. We start from any projective manifold $Y$ for which ${\mathcal{N}}_Y={\mathcal{E}}_Y$. Thus $Y$ might be a smooth curve or any manifold with nef tangent bundle (cf. \[DPS94\]). We pick two very ample divisors $D$ and $H$ on $Y$, and consider $X:={\mathbf{P}}({\mathcal{O}}(D)\oplus{\mathcal{O}}(-H))$, with its canonical projection $\pi:X\to Y$. If we denote by $L:={\mathcal{O}}(1)$ the canonical relatively ample line bundle on $X$, then it is well known that $${H^{1,1}(X,{\mathbf{R}})}=\pi^{\star}H^{1,1}(Y,{\mathbf{R}})\oplus{\mathbf{R}}L.$$ Since $D$ is ample, $L$ is big, but it won’t be nef since $-H$ is not. We are first interested in the divisorial Zariski decomposition of $L$. We have a hypersurface $E:={\mathbf{P}}({\mathcal{O}}(-H))\subset X$, and since $D$ has a section, we see that $E+\pi^{\star}D\in |L|$. Therefore we get $N(L)\leq N(\pi^{\star}D)+E$; but $\pi^{\star}D$ is nef, so has $N(\pi^{\star}D)=0$, and we deduce $N(L)\leq E$. Consequently, $N(L)=\mu_LE$ for some $0\leq\mu_L\leq 1$, and $L=Z(L)+\mu_L E$. We claim that $$\mu_L=\min\{t>0, (L-tE)_{|E}\in{\mathcal{N}}_E\}.$$ First, we have $L-tE=\pi^{\star}D+(1-t)E$, and since $\pi^{\star}D$ is nef, we get that the non-nef locus $E_{nn}(L-tE)$ is contained in $E$ for $0<t<1$. Therefore $L-tE\in{\mathcal{N}}_X$ iff $(L-tE)_{|E}\in{\mathcal{N}}_E$. If this is the case, we have $N(L)\leq N(L-tE)+tE=tE$, and thus $t\geq\mu_L$. Conversely, since $L-\mu_L E=Z(L)$ lies in ${\mathcal{MN}}$, we get that $Z(L)_{|E}\in{\mathcal{E}}_E={\mathcal{N}}_E$ by proposition 2.4 (since $E$ is isomorphic to $Y$ via $\pi$), and we deduce the equality. Now, notice that the projection $\pi$ induces an isomorphism $E\to Y$ such that $L$ becomes $-H$ and thus $E_{|E}$ becomes $-D-H$. The condition $(L-tE)_{|E}\in{\mathcal{N}}_E$ is turned into $-H+t(D+H)\in{\mathcal{N}}_Y$, and we get in the end $$\mu_L=\min\{t>0, -H+t(D+H)\in{\mathcal{N}}_Y\}$$ The picture can be made more precise:
\(i) The nef cone ${\mathcal{N}}_X$ is generated by $\pi^{\star}{\mathcal{N}}_Y$ and $L+\pi^{\star}H$.
\(ii) The pseudo-effective cone ${\mathcal{E}}_X$ is generated by $\pi^{\star}{\mathcal{N}}_Y$ and by $E$.
\(iii) The only exceptional divisor on $X$ is $E$, and the modified Kähler cone coincides with the Kähler cone. The Zariski projection $Z(\alpha)$ of a pseudo-effective class $\alpha$ is thus the projection of $\alpha$ on ${\mathcal{N}}_X$ parallel to ${\mathbf{R}}_+ E$.
$Proof$: given line bundles $L_1,...,L_r$ on a compact Kähler manifold $Y$, a class $\alpha=\pi^{\star}\beta$ over $X:={\mathbf{P}}(L_1\oplus...\oplus L_r)$ is nef (resp. pseudo-effective) iff $\beta$ is. A class $\alpha={\mathcal{O}}(1)+\pi^{\star}\beta$ is nef iff $\beta+L_j$ is nef forall $ j$, and $\alpha$ is big iff the convex cone generated by $\beta+L_1,...,\beta+L_r$ meets the big cone of $Y$, which condition is equivalent (by homogeneity) to: $\beta+$conv$(L_1,...,L_r)$ meets the big cone; finally $\alpha$ is pseudo-effective iff $\beta+$conv$(L_1,...,L_r)$ meets ${\mathcal{E}}_Y$. In our case $\alpha=\pi^{\star}\beta+L$ is thus nef iff $\beta-H$ is nef, and $\alpha$ is pseudo-effective iff $\alpha+[-H,D]$ meets ${\mathcal{N}}_Y$. The latter condition is clearly equivalent to $\alpha-D\in{\mathcal{N}}_Y$. Now an arbitrary class $\alpha$ on $X$ uniquely writes $\alpha=tL+\pi^{\star}\beta$. If $\alpha$ is pseudo-effective, then $t\geq 0$ (since $L$ is relatively ample); if $t=0$, then $\alpha\in\pi^{\star}{\mathcal{N}}_Y$. Otherwise, we may assume by homogeneity that $t=1$, and thus (i) and (ii) follow from the above discussion.\
By (ii), a pseudo-effective class $\alpha$ writes $\pi^{\star}\beta+tE$ with $\beta$ nef. Therefore we get $N(\alpha)\leq tE$, and $E$ is thus the only exceptional divisor on $X$. In fact, we even have $E_{nn}(\alpha)\subset E$, and thus $\alpha$ is nef iff $\alpha_{|E}$ is nef. In particular, we see that ${\mathcal{MK}}={\mathcal{K}}$ as desired (use proposition 2.4 again).\
We now assume that $Y$ is a surface. The assumption ${\mathcal{N}}_Y={\mathcal{E}}_Y$ implies that ${\mathcal{N}}_Y=\overline{{\mathcal{P}}}_Y={\mathcal{E}}_Y$, and $\mu_L$ is none but the least of the two roots of the quadratic polynomial in $t$ $(-H+t(D+H))^2$; it will thus be irrational for most choices of $H$ and $D$ (on, say, an abelian surface). This already yields that the divisorial Zariski decomposition of the rational class $c_1(L)$ will not be rational in general, that is, the analogue of corollary 4.11 is not true in general on a 3-fold.\
Since $Z(L)$ is nef, the volume of $L$ is just $v(Z(L))=Z(L)^3$, with $Z(L)=(1-\mu_L)L+\mu_L\pi^{\star}D$. The cubic intersection form is explicit on ${H^{1,1}(X,{\mathbf{R}})}$ from the relations $$L^3-\pi^{\star}(D-H)\cdot L^2-D\cdot H\cdot L=0$$ and $\pi_{\star}L=1$, $\pi_{\star}L^2=D-H$, thus we can check that $v(L)$ is an explicit polynomial of degree 3 in $\mu_L$ which is also irrational for most choices of $D$ and $H$. We conclude: there exists a big line bundle on a projective 3-fold with an irrational volume, by contrast with proposition 4.12.
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|
---
author:
- 'Matija Bucić[^1]'
- 'Benny Sudakov[^2]'
title: Counting odd cycle free orientations of graphs
---
Introduction
============
Given a fixed graph $H$, over all $n$-vertex graphs $G$ what is the maximum number of $2$-edge colourings of $G$ which contain no monochromatic copy of $H$? We denote the answer by $F(n,H)$. This very natural question was first asked by Erdős and Rothschild [@E-R] in 1984 for the special case of $H=K_3$. This case was resolved by Yuster [@yuster] for large $n$ who in turn raised the problem of determining $F(n,K_k)$. This problem, again for large $n$, was solved by Alon, Balogh, Keevash and Sudakov [@ABKS] who in addition solve it for $H$ being any edge-colour critical graph (defined as graphs in which removal of some edge results in decrease in the chromatic number). This question has attracted a lot of attention over the years and has been generalised in a number of ways, we point the interested reader to the numerous papers citing [@ABKS].
In 2006 Alon and Yuster [@alon-yuster] raised a closely related problem of maximising, over all $n$-vertex graphs $G$, the number of orientations of $G$ which contain no copy of some fixed tournament $T$. We denote the answer by $D(n,T)$. An immediate lower bound comes from taking $G$ to be a $K_{k}$-free graph with maximum number of edges. Since any orientation of such a $G$ is free of any tournaments on $k$ vertices this gives $D(n,T) \ge 2^{|E(G)|}=2^{t_{k-1}(n)}$ for any $k$-vertex tournament $T$, where $t_{k-1}(n)$ denotes the Turán number. Alon and Yuster [@alon-yuster] show that, for large $n$, this easy lower is in fact the answer. Their general argument, which follows the approach used in [@ABKS], relies on a regularity lemma and hence results in a requirement for $n$ to be extremely large. For the special case of $3$ vertex tournaments they give a different approach which solves the problem for the transitive tournament on $3$ vertices for all $n$ and only requires $n$ to be larger than about $10^4$ for $C_3$. Recently Araujo, Botler and Mota [@mota] determine the answer for $C_3$ for all values of $n$.
Araujo, Botler and Mota [@mota] raise a very natural question of what happens if instead of tournaments we are interested in $D(n,H)$ for an arbitrary oriented graph. In particular, they single out the question of what happens if $H$ is a strongly connected directed cycle $C_{k}$, even if we are only interested in the case of large $n$. In this short remark we answer their question for odd cycles.
\[thm:main\] For any $k \ge 1$ there exists $n_0=n_0(k)$ such that if $n \ge n_0$ $$D(n,C_{2k+1})=2^{{
\lfloor n^2/4 \rfloor
}}.$$
In fact, our argument applies for any $H$ which is an orientation of an edge-colour critical graph. Since our argument follows closely the ideas of both [@ABKS; @alon-yuster] we will only give a short sketch and only for the case of odd cycles. Another benefit of this approach is that we are able to present the key ideas behind all of these arguments without burying them under the details as tends to happen when making regularity based proofs formal.
Proof sketch
============
We refer the reader to [@alon-yuster] for how to fill in various details and computations since their results are in the same setting as we are working in. We will assume some familiarity with the basic directed regularity lemma, the specific details needed are given in Section 2 of [@alon-yuster].
The following lemma says that if there are many orientations of $G$ which are $C_{2k+1}$-free then $G$ is not far from being bipartite. It is analogous to Lemma 2.1 in [@alon-yuster] which replaces $C_{2k+1}$ with an arbitrary tournament (and adjusts the numbers accordingly).
\[lem:stability\] Let $k \ge 1$ and $\delta>0$ there exists $n_0=n_0(\delta,k)$ such that if $G$ is a graph of order $n \ge n_0$ which has at least $2^{{
\lfloor n^2/4 \rfloor
}}$ distinct $C_{2k+1}$-free orientations then there is a bipartition of $V(G)$ with at most $\delta n^2$ edges inside parts.
Let $\overrightarrow{G}$ be a $C_{2k+1}$-free orientation of $G$. We apply directed regularity lemma to $\overrightarrow{G}$ to obtain an ${\varepsilon}$-regular partition $V(\overrightarrow{G})=V_1 \cup \ldots \cup V_m$ (all $V_i$’s should have sizes as equal as possible, and all but ${\varepsilon}m^2$ pairs $(V_i, V_j)$ should satisfy that linear sized subsets have about the same density of edges in both directions as the density between $V_i,V_j$). We then consider a cluster graph of density $\eta$ (so vertices being parts of our partition and two parts joined by a directed edge if they are ${\varepsilon}$-regular and the density of edges in the corresponding direction is at least $\eta$).
We first want to show that there must exist some orientation $\overrightarrow{G}$ for which the resulting cluster graph has at least $m^2/4-\beta m^2$ edges directed both ways. If this is not the case there would be too few (less than $2^{n^2/4}$) orientations possible. Indeed, we will fix a partition $\mathcal{P}$ and a cluster graph $C$ and count how many orientations could result with this partition and the cluster graph. Since there are relatively few (recall that regularity lemma gives us a constant number of parts) possible partitions and cluster graphs and every orientation gives rise to some partition and cluster graph this will result in too few orientations in total. There are few edges inside parts of our fixed $\mathcal{P}$ and between non ${\varepsilon}$-regular pairs (at most ${\varepsilon}n^2$ in both cases) and each edge may be oriented in two ways so total contribution of such edges is at most a factor of $2^{2{\varepsilon}n^2}$ to the number of orientations. Similarly, for any ${\varepsilon}$-regular pair $(V_i,V_j)$ which is not an edge of $C$ in some direction there must be at most about $\eta n^2/m^2$ directed edges, so a large proportion of the edges are oriented the same way. An easy estimate tells us that edges can be oriented this way in at most $2^{c_{\eta}n^2/m^2}$ many ways where $c_{\eta}$ is a small constant depending on $\eta$. Since there are at most $m^2$ such pairs, orienting edges between them contributes at most a factor of $2^{c_{\eta}n^2}$ to the total number of orientations. Finally, for any edge of $C$ there are at most $2^{(n/m)^2}$ orientations of edges between the corresponding pair, but since we are assuming $C$ has at most $m^2/4-\beta m^2$ edges they contribute at most a factor of $2^{n^2/4-\beta n^2}$ to the total number of orientations. Choosing $\eta$ to be small enough compared to $\beta$ gives us a contradiction to having at least $2^{{
\lfloor n^2/4 \rfloor
}}$ orientations.
Let now $\overrightarrow{G}$ be an orientation for which the resulting cluster graph $C$ has at least $m^2/4-\beta m^2$ edges directed both ways. We claim $C$ can not contain a bidirected triangle missing only a single directed edge as otherwise $\overrightarrow{G}$ would contain a $C_{2k+1}$. This is a consequence of a standard embedding lemma along the lines of Lemma 2.5 of [@alon-yuster] and can be deduced from it by refining the partition (splitting each part into $k$ parts, while preserving the regularity and density with somewhat worse constants) and then using their lemma to embed $C_{2k+1}$ one vertex per new part.
In particular, this tells us that the graph consisting only of bidirected edges of $C$ is both triangle free and has at least $m^2/4-\beta m^2$ edges. The stability theorem of Simonovits [@simonovits-stability] tells us that there is a bipartition of $V(C)=W_1 \cup W_2$ with at most $\alpha m^2$ bidirected edges within a part (for any $\alpha$, provided $\alpha \gg \beta$). If we consider a bipartite subgraph consisting of bidirected edges of $C$ between $W_1$ and $W_2$ it has at least $m^2/4-(\beta+\alpha) m^2$ edges. In particular, if $C$ had in addition $9(\alpha+\beta)m^2$ directed edges we would find a bidirected triangle with one directed edge removed in $C$. This follows since at least $8(\alpha+\beta)m^2$ of these additional edges must be inside parts so at least $4(\alpha+\beta)m^2$ inside a single part, say $W_1$, and we can pass to a bipartite subgraph of size $2(\alpha+\beta)m^2$ within $W_1$. Taking into account these edges might also be bidirected there are $(\alpha+\beta)m^2$ distinct pairs spanning a directed edge. These edges together with the bidirected edges across make a subgraph with at least $m^2/4$ edges so by Mantel’s theorem make a triangle. This triangle needs to have at most one vertex inside the part (since edges inside the part make a bipartite graph) so they make a desired triangle in $C$.
This tells us that after removing edges within $V_i$’s, between non-${\varepsilon}$-regular pairs, between pairs having density less than $\eta$ in some direction (since we know there are at most $9(\alpha+\beta)m^2$ such pairs) and edges inside $W_1$ or $W_2$ above (at most $\alpha m^2$ such pairs) we removed at most a small constant proportion of edges and are left with a bipartite graph, as desired.
The following lemma replaces the embedding Lemma 3.1 of [@alon-yuster]. Let us introduce some notation for convenience. Given a directed graph $G$ and an integer $k$ we say a pair of disjoint subsets $W_1,W_2\subseteq V(G), |W_i|\ge 2k$ are $k$-regular if for any $X_i \subseteq W_i, |X_i|\ge |W_i|/20$ for $i=1,2$ we have at least $1/10$ proportion of edges of $G$ directed from $X_1$ to $X_2$ as well as from $X_2$ to $X_1.$
\[lem:embedding\] Let $G$ be a directed graph and $W_1,W_2\subseteq V(G)$ a $k$-regular pair. Then one can find a directed path of length $2k$ starting in either of $W_i$.
We iteratively find our directed path. Assume that at stage $2i-1$ we found a path $v_1,\ldots, v_{2i-1}$ and a subset $V_{2i-1}\subseteq W_2 \setminus \{v_2,v_4,\ldots,v_{2i-2}\}$ of at least $|W_2|/20$ out-neighbours of $v_{2i-1}$. Then since there is a 1/10 proportion of edges oriented from $V_{2i-1}$ to $W_1 \setminus \{v_1,v_3,\ldots,v_{2i-1}\}$ there must be a vertex $v_{2i}$ in $V_{2i}$ with a set $V_{2i}$ of at least $(|W_1|-k)/10\ge |W_1|/20$ out-neighbours in $W_1 \setminus\{v_1,v_3,\ldots,v_{2i-1}\}$. Repeating from the other side completes the iteration. After $2k$ iterations we find the desired path.
We now turn to the proof of our main result.
Let $n_0$ be given by with sufficiently small $\delta$.
Let us take a graph $G$ on $n>n_0^2+n_0$ vertices which has at least $2^{{
\lfloor n^2/4 \rfloor
}+m}$ $C_{2k+1}$-free orientations for some $m \ge 0$. We will show that if $G$ is not the Turán graph then we can find a vertex $v$ such that $G\setminus v$ has at least $2^{{
\lfloor (n-1)^2/4 \rfloor
}+m+1}$ $C_{2k+1}$-free orientations. We then iterate (note that no subgraph we consider can any longer be a Turán graph since it has too many orientations, so also edges) as long as our graph has at least $n_0$ vertices. When we stop we obtain a graph with less than $n_0$ vertices which has at least $2^{n_0^2}$ orientations which is impossible since it has at most $n_0^2/2$ edges. Let us assume $G$ is not the Turán graph on $n \ge n_0$ vertices and proceed to find such a vertex $v$.
Every vertex needs to have degree at least ${
\lfloor n/2 \rfloor
}$ as otherwise its edges contribute at most a factor of $2^{{
\lfloor n/2 \rfloor
}-1}$ to the number of orientations so it would work as our vertex $v$ above.
Let $V_1,V_2$ make a bipartition of $V(G)$ which minimises the number of edges within parts. Since $n \ge n_0$ by we have few (in particular at most $\delta n^2$) edges within parts. This implies $|V_1|,|V_2| \le (1/2+\delta^{1/2})n$ as otherwise there would be less than $n^2/4$ edges, so too few orientations. Similarly, there can be at most $\delta n^2$ edges missing between parts.
We first claim that there can be only few orientations for which there exists a pair of subsets $X_1\subseteq V_1, X_2\subseteq V_2$, both of size at least $2\delta n$, which have at most 1/10 proportion of edges directed from $X_1$ to $X_2$. The number of such orientations of edges between $X_1,X_2$ is at most $\binom{e(X_1,X_2)}{\le e(X_1,X_2)/10}\le 2^{0.5e(X_1,X_2)}$. Since the total number of edges is at most $n^2/4+\delta n^2$ there are at most $2^{n^2/4+\delta n^2-0.5e(X_1,X_2)}$ such orientations of the whole graph. Since we are missing at most $\delta n^2$ edges between $V_1,V_2$ we have $e(X_1,X_2)\ge |X_1||X_2|-\delta n^2 \ge 3 \delta n^2$ the number of such orientations is at most $2^{n^2/4-0.5\delta n^2}$. Since we can choose possible locations of $X_1$ and $X_2$ in at most $2^{2n}$ many ways there can be at most $2^{2n}\cdot 2^{n^2/4-0.5\delta n^2}\le 2^{{
\lfloor n^2/4 \rfloor
}}/2$ orientations for which such a pair $X_1,X_2$ exists.
Let us now consider only $C_{2k+1}$-free orientations for which any pair of subsets $X_1,X_2$ of size at least $2\delta n$ have at least $1/10$ proportion of edges oriented in both ways. In particular, any pair of subsets both of size at least $40\delta n$ is $k$-regular. We call such an orientation relevant and by above counting there are at least $2^{{
\lfloor n^2/4 \rfloor
}+m}-2^{{
\lfloor n^2/4 \rfloor
}}/2\ge 2^{{
\lfloor n^2/4 \rfloor
}+m-1}$ relevant orientations.
**Case 1.** Some vertex $v$ has at least $800\delta n$ neighbours in its own part, say $V_1$.
Note that $v$ must have at least $800 \delta n$ neighbours in $V_2$ as well (by us taking the max-cut). If in a relevant orientation $v$ has $40\delta n$ out-neighbours and $40\delta n$ in-neighbours belonging to different parts then since these sets make a $k$-regular pair we can find a path of length $2k-1$ and join it with $v$ to find a $C_{2k+1}$, a contradiction. This implies that $v$ must have at most $80\delta n$ in neighbours or at most $80\delta n$ out-neighbours. In particular, its edges may be oriented in such a way in at most $2\cdot \binom{d(v)}{\le 80 \delta n}\le 2\binom{d(v)}{\le d(v)/10} \le 2^{0.49d(v)}$ many ways.
In other words $G \setminus \{v\}$ must have at least $2^{{
\lfloor n^2/4 \rfloor
}+m-1-0.49d(v)}\ge 2^{{
\lfloor (n-1)^2/4 \rfloor
}+m+1}$ $C_{2k+1}$-free orientations (since $d(v) \le n$ and $n$ is large) as desired.
**Case 2.** Every vertex of $G$ has at most $800\delta n$ neighbours in its own part.
Since $G$ is not the Turán graph there must exist an edge $uv$ inside a part. Both $u$ and $v$ have at least ${
\lfloor n/2 \rfloor
}-800\delta n \ge n/3$ neighbours in the other part, in particular they have $d(u,v) \ge n/8$ common neighbours since parts have size at most $n/2+\delta^{1/2}n$. If in a relevant orientation $uv$ is an edge then there can be at most $40 \delta n$ out-neighbours of $v$ which are also in-neighbours of $u$ in the other part, as otherwise provides us with $C_{2k+1}$. This will severely reduce the number of possible orientations of edges incident to $u,v.$ In particular, the edges from $u$ and $v$ to their common neighbours can be oriented in at most $\binom{d(u,v)}{40 \delta n} \cdot 4^{40\delta n} \cdot 3^{d(u,v)-40\delta n}\le 4^{0.99d(u,v)}.$ The same bound analogously holds if $vu$ is the edge instead. In particular, there are at most $2^{d(u)+d(v)-0.02d(u,v)}\le 2^{n-n/1000}$ possible orientations of edges incident to $u$ and $v$ (we are using that both $u$ and $v$ have degree at most $n/2+ \delta^{1/2}n+800\delta n$). In particular, the total number of orientations of $G \setminus \{u,v\}$ is going to be at least $2^{{
\lfloor (n-2)^2/4 \rfloor
}+m+2}$ so we made two steps of our argument at once and are done.
[1]{}
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[^1]: Department of Mathematics, ETH, Zürich, Switzerland. Email: [](mailto:matija.bucic@math.ethz.ch).
[^2]: Department of Mathematics, ETH, Zürich, Switzerland. Email: [](mailto:benjamin.sudakov@math.ethz.ch). Research supported in part by SNSF grant 200021-175573.
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abstract: 'We derive a new microscopic spin Hamiltonian for Rashba-coupled double exchange metals. The Hamiltonian consists of anisotropic interactions of the Dzyaloshinskii-Moriya (DM) and Kitaev form, in addition to the standard isotropic term. We validate the spin Hamiltonian by comparing results with those on the exact spin-fermion model, and present its phase diagram using large scale Monte Carlo simulations. In addition to ferromagnetic, planar spiral and flux states, the model hosts skyrmion crystal and classical spin-liquid states characterized, respectively, by multiple peaks and a diffuse ring pattern in the spin structure factor. The filamentary domain wall structures in the spin-liquid state are in remarkable agreement with experimental data on thin films of MnSi-type B20 metals and transition metals and their alloys.'
address: |
Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S.A.S. Nagar, Manauli PO 140306, India\
[*sanjeev@iisermohali.ac.in*]{}
author:
- 'Deepak S. Kathyat'
- Arnob Mukherjee
- Sanjeev Kumar
title: New Microscopic Magnetic Hamiltonian for Exotic Spin Textures in Metals
---
Search for magnetic materials supporting unusual spin textures has become an important theme of research in recent years [@Woo2016; @Yu2018; @Yu2010a; @Hoffmann2017; @Nayak2017; @Kurumaji2017; @Kanazawa2012]. Presence of such textures in insulators and metals holds promise for technological applications [@Fert2017; @Laurita2017; @Wiesendanger2016]. In particular, topologically protected magnetic textures such as skyrmions, are considered building blocks of race-track memory devices [@Fert2013; @Nagaosa2013b; @Gobel2019; @Karube2018]. Presence of such spin textures in metals allows for their control using ultra-low currents. Furthermore, noncoplanar magnetic states in metals are known to dramatically influence the spin-polarized charge transport – a feature that can be utilized in spintronics applications [@Zhou2019; @Kindervater2019; @Zang2011; @Sorn2019; @Gao2018; @Barcza2010; @Woo2018]. There are various metallic magnets, [*e.g.*]{} MnSi, FeGe, Co-Zn-Mn alloys, etc., that support exotic spin textures not only in the ground state but also at higher temperatures [@Stishov2007; @Nayak2017; @Yu2011; @Zhao2016; @Pfleiderer2004]. Similar spin textures are also observed in thin films as well as multilayers involving transition metals [@Dupe2014; @Pollard2017; @Soumyanarayanan2017; @Meyer2019].
The key step towards designing or discovering materials with unconventional spin textures is to understand the physics of minimal microscopic models incorporating essential elementary mechanisms [@Farrell2014a; @Chen2016; @Rossler2010]. Spin Hamiltonians naturally emerge in insulators as the charge degrees of freedom become inactive and the low energy physics is determined by the spin degrees of freedom. In contrast, spin Hamiltonians in metals are phenomenologically motivated. Exceptions exist in metals that consist of a subsystem of localized magnetic moments interacting with conduction band. The RKKY model is a famous example in this category [@Ruderman1954; @Kasuya1956a; @Yosida1957a; @Hayami2018; @Bezvershenko2018]. Explanation of skyrmion-like spin textures relies on the presence of DM interactions [@Seki2012b; @Seki2012c; @Dzyaloshinsky1958; @Moriya1960; @Rossler2006]. However, such anisotropic terms are derived by invoking the effect of spin-orbit coupling (SOC) on Mott insulators [@Farrell2014a], and should not be used for metals. Therefore, a consistent microscopic description of exotic spin textures in metallic magnets is currently missing.
In this work, we present a closed form expression for a spin Hamiltonian for Rashba coupled double-exchange (DE) magnets. The resulting model consists of anisotropic terms resembling DM and Kitaev interactions, and it is the first example of a frustrated spin Hamiltonian for metals with nearest neighbor (nn) interactions. After presenting the derivation, we explicitly test the validity of the pure spin model by comparing results against exact diagonalization based simulations on the starting electronic model. The magnetic phase diagram of the new spin model is obtained via large-scale Monte Carlo simulations. The model supports, in addition to a ferromagnetic (FM) phase, (i) single-Q (SQ) spiral states, (ii) diagonally-oriented flux (d-Flux) state, (iii) multiple-Q (MQ) states with noncoplanar skyrmion crystal (SkX) patterns, and (iv) a classical spin liquid (CSL) state characterized by diffuse ring patterns in the spin structure factor (SSF). The CSL state shows filamentary domain wall structure of remarkable similarity to the experimental data on thin films and multilayers of B20 compounds and transition metals [@Soumyanarayanan2017; @Pollard2017; @Woo2018]. The new spin model introduced here has wide range of applicability as it originates from the FM Kondo lattice model (FKLM) – a generic model for metals with local moments. Some of the well known families of materials where FKLM is realized are, manganites, doped magnetic semiconductors and Heusler compounds [@Dagotto2002; @Alvarez2002; @Berciu2001; @Pradhan2017; @Yaouanc2020; @Bombor2013; @Felser2015; @Sasoglu2008].
Our starting point is the FKLM in the presence of Rashba SOC on a square lattice, described by the Hamiltonian,
$$\begin{aligned}
H & = & - t \sum_{\langle ij \rangle,\sigma} (c^\dagger_{i\sigma} c^{}_{j\sigma} + {\textrm H.c.})
+ \lambda \sum_{i} [(c^{\dagger}_{i \downarrow} c^{}_{i+x\uparrow} - c^{\dagger}_{i\uparrow} c^{}_{i+x\downarrow}) \nonumber \\
& & + \textrm{i} (c^{\dagger}_{i\downarrow} c^{}_{i+y\uparrow} + c^{\dagger}_{i\uparrow} c^{}_{i+y\downarrow}) + {\textrm H.c.}] - J_H \sum_{i} {\bf S}_i \cdot {\bf s}_i.
\label{Ham}\end{aligned}$$
Here, $c_{i\sigma} (c_{i\sigma}^\dagger$) annihilates (creates) an electron at site ${i}$ with spin $\sigma$, $\langle ij \rangle$ implies that $i$ and $j$ are nn sites. $\lambda$ and $J_H$ denote the strengths of Rashba coupling and ferromagnetic Kondo (or Hund’s) coupling, respectively. $\bf{s}_i$ is the electronic spin operator at site $i$, and ${\bf S}_i$, with $|{\bf S}_i| = 1$, denotes the localized spin at that site. We parameterize $t = (1-\alpha) t_0$ and $\lambda = \alpha t_0$ in order to connect the weak and the strong Rashba limits, $\alpha = 0$ and $\alpha = 1$, respectively. $t_0=1$ sets the reference energy scale.
Note that coupling between localized spins ${\bf S}_i$ is mediated via the conduction electrons. In the limit of weak Kondo coupling, this leads to a modified RKKY Hamiltonian which is discussed in a recent work [@Okada2018]. To clarify the physics of the above Hamiltonian in the large $J_H$ limit, also known as the DE limit, we rewrite the Hamiltonian in a basis where the spin-quantization axes are site dependent and align with the direction of the local magnetic moment [@SM]. Since antiparallel orientations are strongly suppressed for large $J_H$, the low energy physics is determined by effectively spinless fermions with the spin quantization axis parallel to the local moments. Projecting onto the parallel subspace, we obtain the Rashba DE (RDE) Hamiltonian,
$$\begin{aligned}
H_{{\rm RDE}} &=& \sum_{\langle ij \rangle, \gamma} [g^{\gamma}_{ij} d^{\dagger}_{ip} d^{}_{jp} + {\textrm H.c.}],
\label{Ham-DE}\end{aligned}$$
where, $d^{}_{ip} (d^{\dagger}_{ip})$ annihilates (creates) an electron at site ${i}$ with spin parallel to the localized spin. Site $j = i + \gamma$ is the nn of site $i$ along spatial direction $\gamma = x,y$. The projected hopping $g^{\gamma}_{ij} = t^{\gamma}_{ij} + \lambda^{\gamma}_{ij}$ have contributions from the standard hopping integral $t$ and the Rashba coupling $\lambda$, and depend on the orientations of the local moments. The two contributions to $g^{\gamma}_{ij}$ are given by, $$\begin{aligned}
t^{\gamma}_{ij} & = & -t \big[\cos(\frac{\theta_i}{2}) \cos(\frac{\theta_j}{2})
+ \sin(\frac{\theta_i}{2}) \sin(\frac{\theta_j}{2})e^{-\textrm{i} (\phi_i-\phi_j)} \big],
\nonumber \\
\lambda_{{ij}}^{x} & = & \lambda \big[\sin(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2})e^{-\textrm{i} \phi_i} - \cos(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})e^{\textrm{i} \phi_j}\big],
\nonumber \\
\lambda_{{ij}}^y & = & \textrm{i} \lambda \big[\sin(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2})e^{-\textrm{i} \phi_i} + \cos(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})e^{\textrm{i} \phi_j}\big].\end{aligned}$$ Writing $g^{\gamma}_{ij}$ in the polar form, $g^{\gamma}_{ij} = f^{\gamma}_{ij} e^{{\rm i} h^{\gamma}_{ij}}$, and defining the ground state expectation values $ D^{\gamma}_{ij} = \langle [e^{{\rm i} h^{\gamma}_{ij}} d^{\dagger}_{ip} d^{}_{jp} + {\textrm H. c.}] \rangle_{gs}$ as coupling constants, we obtain the low-energy spin Hamiltonian,
$$\begin{aligned}
H_{{\rm S}} &=& -\sum_{\langle ij \rangle, \gamma} D^{\gamma}_{ij}f^{\gamma}_{ij}, \nonumber \\
\sqrt{2} f^{x}_{ij} & = & \big[ t^2(1+{\bf S}_i \cdot {\bf S}_j)+\lambda^2(1-{\bf S}_i \cdot {\bf S}_j+2S_i^{y}S_j^{y}) \nonumber \\
& & + 2t\lambda \hat{y} \cdot ({\bf S}_i \times{\bf S}_j) \big]^{1/2}, \nonumber \\
\sqrt{2} f^{y}_{ij} & = & \big[ t^2(1+{\bf S}_i \cdot {\bf S}_j)+\lambda^2(1-{\bf S}_i \cdot {\bf S}_j+2S_i^{x}S_j^{x}) \nonumber \\
& & -2t\lambda \hat{x} \cdot ({\bf S}_i \times{\bf S}_j) \big]^{1/2}.
\label{Ham-eff}\end{aligned}$$
The key question is, how well does $H_{{\rm S}}$ Eq. (\[Ham-eff\]) describe the low energy magnetic states of the spin-fermion model $H_{{\rm RDE}}$? We directly address this by comparing energetics of the two models in the low temperature regime. Hybrid simulations combining exact diagonalization and Monte Carlo (EDMC) are carried out for $H_{{\rm RDE}}$ at electronic filling fraction of $n = 0.3$ [@Yunoki1998b; @Dagotto2002]. Results are compared with simulations on $H_{{\rm S}}$ using $D^{\gamma}_{ij}$ as coupling constants. Energy per site $E$ is defined as statistical average $ \overline{H_{{\rm S}}}/N$ for the pure spin model, and as quantum statistical average $\overline{\langle H_{{\rm RDE}} \rangle}/N$ for the spin-fermion model, where the bar denotes the averaging over Monte Carlo steps and $N$ is the number of lattice sites. Comparison of energy per site with varying temperature is shown for representative values of $\alpha$ (see Fig. \[fig1\] ($a$)-($b$)).
Ground states are correctly captured by $H_{{\rm S}}$ for all choices of $\alpha$, and the energies between $H_{{\rm RDE}}$ and $H_{{\rm S}}$ match very well in the low temperature regime. The quantitative agreement can be further improved by using simulation techniques already known for DE systems [@Kumar2005a; @Calderon1998a]. More importantly, we find that most of the ground states obtained in EDMC on $H_{{\rm RDE}}$ lead to values of $D^{\gamma}_{ij}$ that are independent of $ij$ [@SM]. This leads to a simplified effective spin Hamiltonian with $D^{\gamma}_{ij} \equiv D_0$ in Eq. (\[Ham-eff\]). We will now describe the magnetic properties of this effective model using large scale Monte Carlo simulations.
In order to investigate the magnetic phase diagram of the spin Hamiltonian Eq. (\[Ham-eff\]) with $D^{\gamma}_{ij} \equiv D_0 = 1$, we use classical Monte Carlo simulations with the standard Metropolis algorithm. The simulations are carried out on lattice sizes varying from $N = 40^2$ to $N = 200^2$, and $\sim 5\times 10^4$ Monte Carlo steps are used for equilibration and averaging at each temperature point. The phases are characterized with the help of component resolved SSF,
$$\begin{aligned}
S^{\mu}_{f}({\bf q}) &=& \frac{1}{N^2}\sum_{ij} \overline{S^{\mu}_i S^{\mu}_j}~ e^{-{\rm i}{\bf q} \cdot ({\bf r}_i - {\bf r}_j)},
\label{SSF}\end{aligned}$$
where, $\mu = x, y, z$ denotes the component of the spin vector and ${\bf r}_i$ is the position vector for spin ${\bf S}_i$. The total structure factor can be computed as, $S_f({\bf q}) = \sum_{\mu} S^{\mu}_{f}({\bf q})$. Fig. \[fig2\] shows the temperature variations of characteristic features in the SSF for different values of $\alpha$. In the small $\alpha$ regime, the ground state is FM (characterized by $S_f({\bf q})$ at ${\bf q} = (0,0)$ in Fig. \[fig2\]($a$)) and the Curie temperature reduces with increasing $\alpha$. In the large $\alpha$ limit, d-Flux state characterized by simultaneous appearance of peaks at ${\bf q}=(\pi,0)$ and ${\bf q} = (0, \pi)$ in SSF is stabilized. The corresponding ordering temperature increases with increasing $\alpha$ (see Fig. \[fig2\]($d$)). We find two other ordered states at intermediate values of $\alpha$: SQ spiral states with SSF peaks either at ${\bf q} = (q,0)$ or at ${\bf q} = (0,q)$ (see Fig. \[fig2\]($b$)), and noncoplanar MQ states with all three components, $\mu = x,y,z$, contributing to total SSF at different ${\bf q}$. For $0.06 \leq \alpha \leq 0.34$, the SSF displays a circular pattern without any prominent peaks, suggestive of a liquid-like magnetic state [@Tokiwa2014; @Okabe2019; @Nakatsuji2006]. The detailed form of SSF for these unusual phases is discussed below.
We summarize the simulation results in the form of a phase diagram in Fig. \[fig3\]($g$). The ground state changes from a FM at small $\alpha$ to a d-Flux at large $\alpha$, via three non-trivial phases for intermediate values of $\alpha$. The evolution of the ground state SSF is displayed in Fig. \[fig3\]($a$)-($f$). As the FM state is destabilized upon increasing $\alpha$, we do not find any ordered phase. Instead, the SSF shows a diffuse circular pattern (see Fig. \[fig3\]($b$)) characteristic of a disordered liquid-like state. The radius of the ring increases upon increasing $\alpha$, and the intensity near the axial points, $(\pm q, 0)$ and $(0, \pm q)$, becomes relatively large (see Fig. \[fig3\]($c$)). For $0.34 < \alpha < 0.58$, we find SQ spiral states with either horizontal or vertical FM stripes (see Fig. \[fig3\]($d$) and Fig. \[fig4\]($c$)). In a narrow window, $0.58 < \alpha < 0.66$, MQ noncoplanar states are stabilized. Finally the planar d-Flux state is obtained as the ground state for $\alpha > 0.66$. Inflexion point in the temperature dependence of relevant components of SSF are used to identify the boundaries between the paramagnet (PM) and ordered phases. Note that, in case of CSL state a well defined order parameter does not exist, and dashed line indicates the temperature at which the diffuse ring pattern appears in the SSF.
We provide a clear understanding of the ground state evolution in terms of typical low temperature spin configurations in Fig. \[fig4\]. Upon increasing $\alpha$, the FM state is destabilized and typical configurations consist of filamentary structures of domain walls (see Fig. \[fig4\]($a$)-($b$)). The stability of the filamentary structures is related to an unusual degeneracy of spiral states that originates from the presence of mutually orthogonal directions of the two DM vectors in our spin model [@SM]. The fact that domain walls can turn in arbitrary direction with negligible energy cost is responsible for the presence of the diffuse circular pattern in the SSF (see Fig. \[fig3\] ($b$)). For larger values of $\alpha$, the width of domain walls decreases and a preference for horizontal or vertical orientations of the domain walls is found (see Fig. \[fig4\] ($b$)). This is reflected in the appearance of arc features in SSF near the axial points (see Fig. \[fig3\] ($c$)). For $\alpha > 0.58$ we obtain long-range ordered SQ and MQ states. The MQ states can be non-coplanar (see Fig. \[fig4\] ($d$)-($e$)) or coplanar (see Fig. \[fig4\] ($f$)). The noncoplanar patterns in the MQ states are identical to lattices of smallest skyrmions [@McKeever2019].
We have derived a new spin Hamiltonian for DE metals in the presence of Rashba SOC. Anisotropic interactions, similar to those required for stabilizing exotic spin textures, naturally arise in our model. We explicitly compare the energetics in the low temperature regime between the exact Hamiltonian and our spin model in order to prove the validity of the latter. Increasing the relative strength of Rashba term w.r.t. the hopping generates CSL, SQ spiral and MQ SkX states, starting from the trivial FM phase. An elegant description of this evolution emerges from the ground state degeneracy analysis. Our spin model provides a consistent description of spin textures in itinerant magnets. In particular, the filamentary domain wall structures obtained in our simulations are in excellent agreement with the experimental observations in thin films and multilayers of transition metals [@Soumyanarayanan2017; @Pollard2017; @Woo2017; @Woo2018]. Our results predict that inducing Rashba SOC in DE metals is a robust approach to generate exotic noncoplanar spin textures.
The weak coupling approach to understand magnetism in spin orbit coupled itinerant magnets is via RKKY type effective models [@Okada2018]. Such models are long ranged and strongly depend on the filling fraction of the conduction band. In contrast, the form of our spin Hamiltonian is independent of the electronic filling fraction. Therefore, in our description, the exotic magnetic states do not originate from Fermi surface nesting features. Consequently, such states are expected without fine-tuning of electron density. This is consistent with the fact that such spin textures are experimentally observed in a variety of thin films and multilayers of transition metals. While the model is derived starting from the FKLM, at the mean-field level similar physics should hold for the Hubbard model where localized and itinerant electrons are associated with the same band [@Martin2008a; @Pasrija2016]. Furthermore, short-range interactions and a closed form expression are two highly desirable features of any model Hamiltonian. Therefore, in addition to its applicability in understanding magnetism of Rashba coupled itinerant systems, the new spin model should attract attention from pure statistical mechanics viewpoint.
We acknowledge the use of computing facility at IISER Mohali.
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Derivation of the new spin Hamiltonian
======================================
The ferromagnetic Kondo lattice model (FKLM) in the presence of Rashba coupling on a square lattice is described by the Hamiltonian, $$\begin{aligned}
H & = & - t \sum_{\langle ij \rangle,\sigma} (c^\dagger_{i\sigma} c^{}_{j\sigma} + {\textrm H.c.})
+ \lambda \sum_{i} [(c^{\dagger}_{i \downarrow} c^{}_{i+x\uparrow} - c^{\dagger}_{i\uparrow} c^{}_{i+x\downarrow}) \nonumber \\
& & + \textrm{i} (c^{\dagger}_{i\downarrow} c^{}_{i+y\uparrow} + c^{\dagger}_{i\uparrow} c^{}_{i+y\downarrow}) + {\textrm H.c.}] - J_H \sum_{i} {\bf S}_i \cdot {\bf s}_i.
\label{eq:Ham}\end{aligned}$$ The notations remain identical to that used in the main text. In order to handle the large $J_H$ limit, we perform a site dependent rotation of the spin-$\frac{1}{2}$ basis given by the canonical $SU(2)$ transformation,
$\begin{bmatrix}
c_{i\uparrow} \\
c_{i\downarrow}
\end{bmatrix}
=
\begin{bmatrix}
\cos(\frac {\theta_i}{2}) & -\sin(\frac {\theta_i}{2}) e^{-\textrm{i} \phi_i} \\
\sin(\frac {\theta_i}{2}) e^{\textrm{i} \phi_i} & \cos(\frac {\theta_i}{2})
\end{bmatrix} \begin{bmatrix}
d_{ip} \\
d_{ia}
\end{bmatrix}$.\
\
Here, $d_{ip} (d_{ia})$ annihilates an electron at site ${i}$ with spin parallel (antiparallel) to the localized spin and $\theta_i$, $\phi_i$ are the polar and azimuthal angles describing the direction of the local spin ${\bf S}_i$. In the large $J_H$ limit, the low energy physics is retained in parallel subspace, leading to the Rashba double-exchange (RDE) Hamiltonian,\
$$\begin{aligned}
H_{{\rm RDE}} & = & \sum_{\langle ij \rangle,\gamma} [g^{\gamma}_{ij} d^\dagger_{ip} d^{}_{jp} + {\textrm H.c.}],
\label{eq:Ham2}\end{aligned}$$\
where, site $j = i + \gamma$ is the nn of site $i$ along spatial direction $\gamma \in \{x,y\}$. The projected hopping parameters, $g^{\gamma}_{ij}= t^{\gamma}_{ij} + \lambda^{\gamma}_{ij}$ , are given by,
$$\begin{aligned}
t^{\gamma}_{ij} & = & -t \big[\cos(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2})
+ \sin(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})e^{-\textrm{i} (\phi_i-\phi_j)} \big] ,
\nonumber \\
\lambda_{{ij}}^x & = & \lambda \big[\sin(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2})e^{-\textrm{i} \phi_i} - \cos(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})e^{\textrm{i} \phi_j}\big] ,
\nonumber \\
\lambda_{{ij}}^y & = & \textrm{i} \lambda \big[\sin(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2})e^{-\textrm{i} \phi_i} + \cos(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})e^{\textrm{i} \phi_j}\big] .\end{aligned}$$
$$\begin{rcases}
t^{\gamma}_{ij} = -t \big[\cos(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2})
+ \sin(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})e^{\textrm{i} (\phi_i-\phi_j)} \big] ,
\\
\lambda_{{ij}}^x = \lambda \big[\sin(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2})e^{\textrm{i} \phi_i} - \cos(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})e^{-\textrm{i} \phi_j}\big] ,
\\
\lambda_{{ij}}^y = \textrm{i} \lambda \big[\sin(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2})e^{\textrm{i} \phi_i} + \cos(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})e^{-\textrm{i} \phi_j}\big] .
\end{rcases}$$
Writing $g^{\gamma}_{ij}$ in polar form, $g^{\gamma}_{ij}= f^{\gamma}_{ij} e^{\textrm{i} h^{\gamma}_{ij}}$, we obtain the following closed form expressions for $f^{x}_{ij}$ and $f^{y}_{ij}$ :
$$\begin{aligned}
f^{x}_{ij} & = & \sqrt{\frac {1}{2}[t^2(1+S_i^{x}S_j^{x}+S_i^{y}S_j^{y}+S_i^{z}S_j^{z})+\lambda^2(1-S_i^{x}S_j^{x}+S_i^{y}S_j^{y}-S_i^{z}S_j^{z})-2t\lambda(S_i^{x}S_j^{z}-S_i^{z}S_j^{x})]} \nonumber \\
& = & \sqrt{\frac {1}{2}[t^2(1+{\bf S}_i \cdot {\bf S}_j)+\lambda^2(1-{\bf S}_i \cdot {\bf S}_j+2S_i^{y}S_j^{y})+2t\lambda \hat{\bf y} \cdot ({\bf S}_i \times{\bf S}_j)]} ,\end{aligned}$$
$$\begin{aligned}
\label{eq:ESH}
f^{y}_{ij} & = & \sqrt{\frac {1}{2}[t^2(1+S_i^{x}S_j^{x}+S_i^{y}S_j^{y}+S_i^{z}S_j^{z})+\lambda^2(1+S_i^{x}S_j^{x}-S_i^{y}S_j^{y}-S_i^{z}S_j^{z})+2t\lambda(S_i^{z}S_j^{y}-S_i^{y}S_j^{z})]} \nonumber \\
& = & \sqrt{\frac {1}{2}[t^2(1+{\bf S}_i \cdot {\bf S}_j)+\lambda^2(1-{\bf S}_i \cdot {\bf S}_j+2S_i^{x}S_j^{x})-2t\lambda \hat{\bf x} \cdot ({\bf S}_i \times{\bf S}_j)]} .\end{aligned}$$
\
The phase angles, $h^{\gamma}_{ij}$, are easily obtained via, $$\begin{aligned}
h^{\gamma}_{ij} & = &\arctan \left(\frac{\text{Im}(g^{\gamma}_{ij})}{\text{Re}(g^{\gamma}_{ij})}\right),\end{aligned}$$
where, real and imaginary parts of $g^{\gamma}_{ij}$ are given by,
$$\begin{aligned}
\text{Re}(g^{x}_{ij}) &=
\begin{aligned}[t]
&-t(\cos(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2}) + \sin(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})\cos(\phi_i-\phi_j)) \\
&+\lambda(\sin(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2}) \cos\phi_i- \cos(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})\cos\phi_j ),
\end{aligned} \\
\nonumber
\text{Im}(g^{x}_{ij}) &=
\begin{aligned}[t]
&t(\sin(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})\sin(\phi_i-\phi_j))-\lambda(\sin(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2}) \sin\phi_i + \cos(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})\sin\phi_j ),
\end{aligned} \\
\nonumber
\text{Re}(g^{y}_{ij}) &=
\begin{aligned}[t]
& -t(\cos(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2}) + \sin(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})\cos(\phi_i-\phi_j)) \\
& - \lambda(\cos(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2}) \sin\phi_j- \sin(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2})\sin\phi_i ),
\end{aligned} \\
\nonumber
\text{Im}(g^{y}_{ij}) &=
\begin{aligned}
& t(\sin(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})\sin(\phi_i-\phi_j))+\lambda(\sin(\frac {\theta_i}{2}) \cos(\frac {\theta_j}{2}) \cos\phi_i + \cos(\frac {\theta_i}{2}) \sin(\frac {\theta_j}{2})\cos\phi_j).
\end{aligned}\end{aligned}$$
The ground state expectation values of the Hamiltonian Eq. (\[eq:Ham2\]) is identical to the expression, $-\sum_{\langle ij \rangle,\gamma} D^{\gamma}_{ij} f^{\gamma}_{ij}$ , where $ D^{\gamma}_{ij} = \langle [e^{\textrm{i} h^{\gamma}_{ij}} d^\dagger_{ip} d^{}_{jp} + {\textrm H.c.}]\rangle_{gs} $. Following the strategy used in double exchange models, we promote the above expression to a spin Hamiltonian,
$$\begin{aligned}
H_{{\rm S}} & = & -\sum_{\langle ij \rangle,\gamma} D^{\gamma}_{ij} f^{\gamma}_{ij} .
\label{eq:Heff}\end{aligned}$$
We emphasize that, by construction, the magnetic ground states of $H_{{\rm S}}$ Eq. (\[eq:Heff\]) and $H_{{\rm RDE}}$ Eq. (\[eq:Ham2\]) are identical.
Distribution of coupling constants
==================================
The coupling constants of the effective Hamiltonian Eq. (\[eq:Heff\]) are determined as the expectation values in the ground states obtained via EDMC on $H_{{\rm RDE}}$. We calculate the distributions of $D^{\gamma}_{ij}$ for the pairs of nearest neighbor sites in different ground states obtained via EDMC. The density of $D^{\gamma}_{ij}$ is defined as,
$$\begin{aligned}
\mathcal{N}(D) = 1/N \sum_{\langle ij \rangle} \delta(D - D^{\gamma}_{ij}) \nonumber
\approx 1/N \sum_{\langle ij \rangle} \frac{\eta/\pi}{\eta^2 + (D - D^{\gamma}_{ij})^2}, \end{aligned}$$
where, $\eta$ is Lorentzian broadening parameters which is set to $0.001$ for calculations.
The density of $D^{\gamma}_{ij}$s is shown in Fig. (\[fig:DOD\]) for different values of $\alpha$. We find that $D^{\gamma}_{ij}$ is independent of $ij$ for most of the ground states. This justifies the use of a single coupling constant in the effective spin Hamiltonian. For the spiral state with wave vector $(0,q)$ we find a slight separation of scales between $D^{x}_{ij}$ and $ D^{y}_{ij}$. This difference is expected to further re-enforce the stability of the $(0,q)$ spiral states.
Origin of classical spin liquid (CSL) behavior
==============================================
In this section we provide a simple description of CSL states observed in the region $0.15 \leq \alpha \leq 0.34$. A careful look at the form of the Hamiltonian Eq. (\[eq:ESH\]) suggests that for small values of $\alpha$, terms proportional to $\lambda^2$ may be ignored. The only non-trivial effect then comes from terms proportional to $t\lambda$. These terms prefer spiral states with competing orientations of the spiral planes. Along $x$-direction, a spiral in $xz$ plane is preferred and along $y$-direction a spiral in $yz$ plane is preferred. This motivates us to construct the following variational ansatz where the plane of the spiral is one of the variational parameters:
$$\begin{aligned}
S_i^x &=& S_0 \sin(\boldsymbol{q}.\boldsymbol{r}_i)\cos(\Phi_p), \nonumber \\
S_i^y &=& S_0 \sin(\boldsymbol{q}.\boldsymbol{r}_i)\sin(\Phi_p),\nonumber \\
S_i^z &=& S_0 \cos(\boldsymbol{q}.\boldsymbol{r}_i).
\label{eq:vari_ansatz}\end{aligned}$$
In the above, $S_0$ is the unit magnitude of the classical spin vectors, $\Phi_p$ is the orientation of the spiral plane ($\Phi_p = 0$ for $xz$ plane and $\Phi_p = \frac{\pi}{2}$ for $yz$ plane) and $\boldsymbol{q} = q(\cos\beta,\sin\beta)$ is the spiral wave-vector. In the CSL state, we find that the energy of a spiral is independent of the spiral plane angle $\Phi_p$, provided the wave-vector angle $\beta$ is related to $\Phi_p$ via $\beta - \Phi_p = \pi$. This explains the stability of filamentary domain wall structure in the CSL regime: the domain walls can freely reorient as long as the spiral plane also undergoes a reorientation in such a way that the spiral plane is oriented perpendicular to the local orientation of the domain wall.
In order to quantify this degeneracy of spiral states, we define $\Delta E = \max[E_{min}(\Phi_p)] - \min[E_{min}(\Phi_p)]$. $E_{min}(\Phi_p)$ represents the minimum energy obtained for a given orientation of the spiral plane, marked by a square symbol Fig. \[fig:vari\_E vs beta\]. Exact degeneracy is characterized by $\Delta E = 0$. We show the variation of $\Delta E$ with the coupling constant $\alpha$ as an inset in Fig. \[fig:vari\_E vs beta\] ($b$). The degree of degeneracy clearly reduces near $\alpha=0.35$, which coincides with the crossover point between CSL and SQ spiral states.
|
---
abstract: |
This paper investigates large fluctuations of Locational Marginal Prices (LMPs) in wholesale energy markets caused by volatile renewable generation profiles. Specifically, we study events of the form
$$\Prob \Big ( \bLMP \notin \prod_{i=1}^n [\alpha_i^-, \alpha_i^+] \Big),$$
where $\bLMP$ is the vector of LMPs at the $n$ power grid nodes, and $\balpha^-,\balpha^+\in\Real^n$ are vectors of price thresholds specifying undesirable price occurrences. By exploiting the structure of the supply-demand matching mechanism in power grids, we look at LMPs as deterministic piecewise affine, possibly discontinuous functions of the stochastic input process, modeling uncontrollable renewable generation. We utilize techniques from large deviations theory to identify the most likely ways for extreme price spikes to happen, and to rank the nodes of the power grid in terms of their likelihood of experiencing a price spike. Our results are derived in the case of Gaussian fluctuations, and are validated numerically on the IEEE 14-bus test case.
author:
- 'T. Nesti'
- 'J. Moriarty'
- 'A. Zocca'
- 'B. Zwart'
bibliography:
- 'BiblioThesisCameraReadyCompletePaper.bib'
title: Large Fluctuations in Locational Marginal Prices
---
Introduction
============
Modern-day power grids are undergoing a massive transformation, a prominent reason being the increase of intermittent renewable generation registered in the first two decades of the 21st century [@Ren212019]. The inherently uncertain nature of renewable energy sources like wind and solar photovoltaics is responsible for significant amounts of variability in power output, with important consequences for energy markets operations. In particular, energy prices can exhibit significant volatility throughout different hours of the day, and are usually negatively correlated with the amount of renewable generation in the grid [@Paraschiv2014]. In this paper, we focus on the Locational Marginal Pricing (LMP) mechanism [@Ferc2003], a market architecture adopted by many US energy markets. Under the LMP market architecture prices are location-dependent, and the presence of congested transmission lines causes them to vary wildly across different locations, contributing to their erratic behavior. On the other hand, under the zonal pricing mechanism [@Neuhoff2011] (as in most European markets) a single price is calculated for each zone in the market. The topic of energy price forecasting has received a lot of attention in the forecasting community in the last 20 years, since the restructuring of energy markets from a government-controlled system to a deregulated market [@Weron2014]. Thanks to the particularly rich mathematical structure of the LMP mechanism, prediction models for LMPs are not limited to traditional statistical analysis and stochastic model-based techniques, but include structural methods exploiting the mathematical properties of the supply-demand matching process performed by grid operators, known as the Optimal Power Flow (OPF) problem [@Hunueault1991].
The relevant literature on structural prediction models can be categorized based on whether it takes an operator-centric [@Bo2009; @Ji2017] or a participant-centric [@Zhou2011; @Geng2016; @Radovanovic2019holistic] point of view.
In the former case, it is assumed that the modeler has full knowledge of all the parameters defining the OPF formulation, such as generation cost functions, grid topology, and physical properties of the network. In [@Bo2009], the authors analyze the uncertainty in LMPs with respect to total load in the grid, relying on the structural property that changes in LMPs occur at the so-called critical load levels. In [@Ji2017], both load and generation uncertainty is considered, and a multiparametric programming approach is proposed. The market participant-centric approach, conversely, relies only on publicly available data, usually limited to grid-level, aggregated demand and generation, and nodal price data, without assuming knowledge of the network parameters. In [@Zhou2011], the authors utilizes the structure of the OPF formulation to infer the congestion status of transmission lines based only on zonal load levels, while in [@Geng2016] a semi-decentralized data-driven approach, based on learning nodal prices as a function of nodal loads using support vector machines, is proposed. In [@Radovanovic2019holistic], a fully decentralized forecasting algorithm combining machine learning techniques with structural properties of the OPF is presented, and validation on the Southwest Power Pool market data results in accurate day-ahead predictions of real-time prices. The methodologies described above have varying levels of performance in predicting expected intra-day variations, but they all have limitations when predicting extreme *price spike* values. Even when assuming a a fully centralized perspective, forecasting price spikes is a notoriously difficult problem [@Lu2005], and is mostly undertaken within the framework of zonal electricity markets [@Lu2005; @Hagfors2016; @Paraschiv2016; @Veraart2016], while the corresponding problem for LMP-based markets has received less attention. At the same time, the connection between locational marginal pricing and congestion status of the grid makes this problem particularly relevant for the discussion on financial transmission rights [@Bushnell1999], while a deeper understanding of the occurrence of high price events can inform network upgrades aimed at mitigating them [@Wu2006]. In this paper, we study the problem of predicting large price fluctuations in LMP-based energy markets from a centralized perspective, proposing a novel approach combining multiparametric programming techniques [@Tondel2003] with large deviations theory [@Dembo1998]. Large deviations techniques have been successfully used in fields such as queueing theory, telecommunication engineering, and finance [@Bucklew1990]. In the recent years, they also have been applied in the context of power systems in order to study transmission line failures [@Nesti2019temperature; @Nesti2018emergent] and statistical properties of blackouts [@Nesti2019blackout].
In the present work we study the probability of nodal price spikes occurrences of the form $\Prob\bigl( \bLMP \notin \prod_{i=1}^n [\alpha_i^-, \alpha_i^+] \bigr)$, where $\bLMP$ is the vector of Locational Marginal Prices at the $n$ grid nodes and $\balpha^-,\balpha^+\in\Real^n$ are vectors of price thresholds specifying undesirable price occurrences. Assuming full knowledge of the power grid parameters, we first derive the deterministic function linking the stochastic input process, modeling renewable generation, to the $\bLMP$ vector. This, in turn, allows us to use large deviations theory to identify the most likely ways for extreme LMP spikes to happen as a result of unusual volatile renewable generation profiles. The large deviations approach offers a powerful and flexible framework that holistically combines the network structure and operation paradigm (the OPF) with a stochastic model for renewable generation. This approach enables us to: i) approximate the probability of price spikes by means of solving a deterministic convex optimization problem, ii) rank the nodes of the power grids according to their likelihood of experiencing price spike events, iii) handle the *multimodal* nature of the LMP’s probability distribution, and iv) relax the LICQ regularity condition, an assumption that is usually required in the relevant literature [@Zhou2011; @Bo2009; @Li2009; @Bo2012].
The rest of this paper is organized as follows. A rigorous formulation of the problem under consideration is provided in Section \[s:model\], while a connection to the field of multiparametric programming is established in Section \[s:mpt\]. In Section \[s:ldp\], we derive our main large deviations result relating the event of a rare price spike to the solution of a deterministic optimization problem, which is further analyzed in Section \[s:optimization\]. We illustrate the potential of the proposed methodology in Section \[s:numerics\] with a case study on the IEEE 14-bus test case and draw our conclusions in Section \[s:conclusion\].
System model and problem formulation {#s:model}
====================================
The power grid is modeled as a connected graph $\Graph=\Graph(\Nodes,\Edges)$, where the set of nodes $\Nodes$ represents the $n$ buses in the system, and the set of edges $\Edges$ model the $m$ transmission lines. We assume that $\Nodes = \Nodes_g \sqcup \Nodes_{\theta}$, with $|\Nodes_g|=n_g, |\Nodes_{\theta}|=n_{\theta}$, $n_g + n_{\theta} = n$, and where $\sqcup$ denotes a disjoint union. Each bus $i\in\Nodes_g$ houses a traditional controllable generator $\gen_i$, while each bus $i\in\Nodes_{\theta}$ houses a stochastic uncontrollable generating unit $\theta_i$. Finally, we assume that a subset of nodes $\Nodes_d\subseteq \Nodes$ houses loads, with $|\Nodes_d|=n_d$. We denote the vectors of conventional generation, renewable generation, and demand, as the vectors $\bgen\in\Real_{+}^{\Nodes_g}, \bw\in\Real_{+}^{\Nodes_{\theta}}$, and $\bdem\in\Real_{+}^{\Nodes_d}$, respectively. [^1] To simplify notation, we extend the vectors $\bgen,\bw,\bdem$ to $n$-dimensional vectors $\bgenext,\bwext,\bdemext\in\Real^n$ by setting $\tilde{g}_i=0$ whenever $i\notin \Nodes_g$, and similarly for $\bwext$ and $\bdemext$. The vectors of *net power injections* and *power flows* are denoted by $\bpow := \bgenext + \bwext -\bdemext\in\Real^n$ and $\bflow\in\Real^m$, respectively. To optimally match power demand and supply while satisfying the power grid operating constraints, the Independent System Operator (ISO) solves the Optimal Power Flow (OPF [@Hunueault1991]) problem and calculates the optimal energy dispatch vector $\bg^*\in\Real^{n_g}$, as well as the vector of *nodal prices* $\bLMP\in\Real^{n}$, as we will describe in Section \[s:model\]\[ss:lmp\].In its full generality, the OPF problem is a nonlinear, nonconvex optimization problem, which is difficult to solve [@Bienstockbook]. In this paper we focus on the widely used approximation of the latter known as DC-OPF, which is based on the *DC approximation* [@Purchala2005]. The DC approximation relates any zero-sum vector $\bpow$ of net power injections and the corresponding power flows $\bflow$ via the linear relationship $\bflow=\bPTDF \bpow$, where the matrix $\bPTDF\in\Real^{m\times n}$, known as the power transfer distribution factor (PTDF) matrix, encodes information on the grid topology and parameters, cf. Section \[ss:PTDF\_derivation\]. The DC-OPF can be formulated as the following quadratic optimization problem: $$\begin{aligned}
{5}
& \underset{\bgen\in\Real^{n_g}}{\min} & & \,\,\sum_{i=1}^{n_g} J_i(g_i)= \frac{1}{2}\bgen^{\tr} \bH\bgen + \bh^\tr \bg \label{eq:obj}\\
& \, \, \, \, \, \text{s.t.} & & \ones^{\tr} (\bgenext+\bwext - \bdemext)=0 & &\qquad :\lmpen \label{eq:balance} \\
& & &\bfminuslimit \le \bPTDF(\bgenext +\bwext - \bdemext)\le \bfpluslimit & &\qquad :\bmum,\bmup \label{eq:lines}\\
& & & \bgminuslimit\le \bgenext \le \bgpluslimit & &\qquad :\btaum,\btaup \label{eq:gen}\end{aligned}$$ where the variables are defined as follows:\
[lp]{} $\bH\in\Real^{n_g\times n_g}$ & diagonal positive definite matrix appearing in the quadratic term of ;\
$\bh\in\Real^{n_g}$ & vector appearing in the linear term of the objective function ;\
$\bPTDF \in\Real^{m\times n}$ & PTDF matrix (see definition later, in Eq. );\
$\bfminuslimit,\bfpluslimit \in\Real^m$ & vector of lower/upper transmission line limits;\
$\bgminuslimit,\bgpluslimit \in \Real^{n_g}$ & vector of lower/upper generation constraints;\
$\lmpen\in\Real$ & dual variable of the energy balance constraint;\
$\bmum, \bmup \in \Real^m_+$ & dual variables of the transmission line constraints ,\
$\btaum, \btaup \in\Real^{n_g}_+$ & dual variables of the generation constraints ;\
$\ones\in\Real^n$ & $n$-dimensional vector of ones.\
Following standard practice [@Sun2010], we model the generation cost function $J_i(\cdot)$ at $i\in\Nodes_g$ as an increasing quadratic function of $g_i$, and denote by $\bJ(\bgen) := \sum_{i=1}^n J_i(g_i)$ the aggregated cost.
Locational marginal prices {#ss:lmp}
--------------------------
In this paper, we focus on energy markets adopting the concept of Locational Marginal Prices (LMPs) as electricity prices at the grid nodes. Under this market architecture, the LMP at a specific node is defined as the marginal cost of optimally supplying the next increment of load at that particular node while satisfying all power grid operational constraints, and can be calculated by solving the OPF problem in Eqs. -. More precisely, let $\bg^*$ and $J^*=\bJ(\bg^*)$ denote, respectively, the optimal dispatch and the optimal value of the objective function of the OPF problem in Eqs. -, and let $\mathcal{L}$ be the Lagrangian function. The LMP at bus $i$ is defined as the partial derivative of $J^*$ with respect to the demand $d_i$, and is equal to the partial derivative of the Lagrangian with respect to demand $d_i$ evaluated at the optimal solution: $$\label{eq:LMP_def}
\LMP_i=\frac{\partial{J^*}}{\partial d_i}=\frac{\partial{\mathcal{L}}}{\partial d_i}\Bigr\rvert_{\bgen^*}.$$ Following the derivation in [@Radovanovic2019holistic], the LMP vector can be represented as $$\label{eq:LMP_dec}
\bLMP=\lmpen\ones+\bPTDF^\top\bmu\in\Real^n,$$ where $\bmu=\bmum-\bmup$. Note that $\mu_{\ell}=0$ if and only if line $\ell$ is not congested, that is, if and only if $\fminuslimit_{\ell}<f_{\ell}<\fpluslimit_{\ell}$. In particular, $\mu^+_{\ell}>0$ if $f_{\ell}=\fpluslimit_{\ell}$, and $\mu^-_{\ell}<0$ if $f_{\ell}=\fminuslimit_{\ell}$.[^2] As a consequence, if there are no congested lines, the LMPs at all nodes are equal, i.e., $\LMP_i=\lmpen$ for every $i=1\,\ldots,n,$ and the common value $\lmpen$ in is known as the marginal *energy component*. The energy component $\lmpen$ reflects the marginal cost of energy at the reference bus. If instead at least one line is congested, the LMPs are not all equal anymore and the term $\tilde{\bpi}:=\bPTDF^\top\bmu$ in Eq. is called the marginal *congestion component*. When ISOs calculate the LMPs, they also include a *loss component*, which is related to the heat dissipated on transmission lines and is not accounted for by the DC-OPF model. The loss component is typically negligible compared to the other price components [@SPPreport2016], and its inclusion goes beyond the scope of this paper.
Problem statement {#ss:prob_statement}
-----------------
In this paper, we adopt a *functional* perspective, i.e., we view the uncontrollable generator as a variable parameter, or input, of the OPF. In particular, we are interested in a setting where the objective function, PTDF matrix, nodal demand $\bdem$, line limits and generation constraints are assumed to be known and fixed. Conversely, the uncontrollable generation $\bw\in\Real^{\Nodes_{\theta}}$ corresponds to a *variable parameter* of the problem, upon which the solution of the OPF problem in Eqs. - (to which we will henceforth refer as $\OPF(\btheta)$), and thus the LMP vector, depend.
In other words, the LMP vector is a deterministic function of $\btheta$ $$\label{eq:mapping}
\Real^{n_{\theta}} \supseteq \bTheta \ni \btheta \rightarrow \bLMP(\btheta)\in\Real^n,$$ where $\bTheta \subseteq \Real^{n_{\theta}}$ is the *feasible parameter space* of the OPF, i.e., the set of parameters $\btheta$ such that $\OPF(\btheta)$ is feasible. In particular, we model $\btheta$ as a non-degenerate multivariate Gaussian vector $\btheta_{\eps} \sim \mathcal{N}_{n_{\theta}}(\bmuth,\eps\bSigmath)$, where the parameter $\eps>0$ quantifies the magnitude of the noise. The mean $\bmuth\in \mathring{\bTheta}$ (where $\mathring{A}$ denotes the interior of the set $A$) of the random vector $\btheta$ is interpreted as the expected, or nominal, realization of renewable generation for the considered time interval. Furthermore, we assume that $\bSigmath$ is a known positive definite matrix, and consider the regime where $\eps\to 0$. In view of the mapping , $\bLMP$ is a $n$-dimensional random vector whose distribution depends on that of $\btheta$, and on the deterministic mapping $\btheta \to \bLMP(\btheta)$. We assume that the $\bLMP$ vector corresponding to the expected renewable generation $\bmuth$ is such that $$\label{eq:condition_mu}
\bLMP(\bmuth) \in \mathring{\Pi},$$ where $\Pi:=\prod_{i=1}^n [\alpha_i^-, \alpha_i^+]$, and $\balpha^{-},\balpha^{+}\in\Real^n$ are vectors of price thresholds. We are interested in the event $Y=Y(\balpha^{-},\balpha^{+})$ of *anomalous price fluctuations* (or *price spikes*) defined as $$\begin{aligned}
Y(\balpha^{-},\balpha^{+}) &= \bigl\{ \btheta\in \bTheta \,:\, \bLMP (\btheta) \notin \prod_{i=1}^n [\alpha_i^-, \alpha_i^+] \bigr\} \label{eq:PriceSpike}\\
&=\bigcup_{i=1}^n \{ \btheta\in \bTheta \,:\, \LMP_i(\btheta) < \alpha_i^{-}
\text{ or } \LMP_i(\btheta) > \alpha_i^{+}\},\label{eq:PriceSpike_2}\end{aligned}$$ which, in view of Eq. and the regime $\eps \to 0$, is a *rare event*. Without loss of generality, we only consider thresholds $\balpha^{-},\balpha^{+}$ such that the event $Y(\balpha^{-},\balpha^{+})$ has a non-empty interior in $\Real^{n_{\theta}}$. Otherwise, the fact that $\bmuth$ is non degenerate would imply $\Prob \bigl(Y(\balpha^{-},\balpha^{+})\bigr) = 0$.
We observe that the above formulation of a price spike event is quite general, and can cover different application scenarios, as we now outline. For example, if $\alpha = \alpha^{+}_i=-\alpha^{-}_i > 0$ for all $i$, then the price spike event becomes $$Y(\alpha) =\{\btheta\in\Theta\, : \, \norm{\bLMP}=\max_{i=1,\ldots,n} |\LMP_i| > \alpha\},$$ and models the occurrence of a price spike larger than a prescribed value $\alpha$. On the other hand, if we define $\balpha^{-}= \bLMP(\bmuth) -\bbeta$ and $\balpha^{+}= \bLMP(\bmuth) + \bbeta$, for $\bbeta\in\Real_{+}^{n}$, the spike event $$Y(\bbeta) = \bigcup_{i=1}^n\{\btheta\in\bTheta\, : \, |\LMP_i -\LMP_i(\bmuth)|> \beta_i\},$$ models the event of any $\LMP_i$ deviating from its nominal value $\LMP_i(\bmuth)$ more than $\beta_i>0$. Moreover, by setting $\balpha^{-}= \bLMP(\bmuth) -\bbeta^{-}$ and $\balpha^{+}= \bLMP(\bmuth) + \bbeta^{+}$, $\bbeta^{-},\bbeta^{+}\in\Real^{m,+}$ and $\bbeta^{-}\neq \bbeta^{+}$, we can weigh differently negative and positive deviations from the nominal values.
We remark that *negative* price spikes are also of interest [@Gerster2016; @Gonzales2017] and can be covered in our framework, by choosing the threshold vectors $\balpha^{-},\balpha^{+}$ accordingly. Finally, note that we can study price spikes at a more granular level by restricting the union in Eq. to a particular subset of nodes $\widetilde{\Nodes}\subseteq \Nodes$.
Derivation of the PTDF matrix $\bPTDF$ {#ss:PTDF_derivation}
--------------------------------------
Choosing an arbitrary but fixed orientation of the transmission lines, the network topology is described by the *edge-vertex incidence matrix* $\binc\in\Real^{m\times n}$ defined as $\inc_{\ell, i}=1$ if $\ell=(i,j)$, $\inc_{\ell, i}=-1$ if $\ell=(j,i)$, and $\inc_{\ell, i} = 0$ otherwise. We associate to every line $\ell \in \Edges$ a *weight* $\weight_{\ell}$, which we take to be equal to the inverse of the reactance $x_{\ell}>0$ of that line, i.e., $\weight_{\ell}=x_{\ell}^{-1}$ [@Bienstockbook]. Let $\bdiagdc\in\Real^{m\times m}$ be the diagonal matrix containing the line weights $\bdiagdc=\diag(\weight_{1}^,\dots, \weight_{m})$. The network topology and weights are simultaneously encoded in the *weighted Laplacian matrix* of the graph $\Graph$, defined as $\blap = \binc^\tr \bdiagdc \binc$. Finally, by setting node $1$ as the reference node, the PTDF matrix is given by $$\label{eq:PTDF_der}
\bPTDF :=[\zero\,\, \bdiagdc\bincsub\blapsub^{-1}],$$where $\bincsub\in\Real^{m\times(n-1)}$ is the matrix obtained by deleting the first columns of $\binc$, $\blapsub^{(n-1)\times (n-1)}$ by deleting the first row and column of $\blap$.
Multiparametric programming {#s:mpt}
===========================
As discussed in Section \[s:model\]\[ss:prob\_statement\], LMPs can be thought as deterministic functions of the parameter $\btheta$. Therefore, in order to study the distribution of the random vector $\bLMP$, we need to investigate the structure of the mapping $\btheta \to \bLMP(\btheta)$. We do this using the language of Multiparametric Programming Theory (MPT) [@Tondel2003], which is concerned with the study of optimization problems which depend on a vector of parameters, and aims at analyzing the impact of such parameters on the outcome of the problem, both in terms of primal and dual solutions. In our setting, we stress that the parameter $\btheta $ models the uncontrollable renewable generation. Hence, the problem $\OPF(\btheta)$ can be formulated as a standard Multiparametric Quadratic Program (MPQ) as follows: $$\begin{aligned}
\underset{\bgen \in \Real^{n_g}}{\min} &\quad \frac{1}{2}\bgen^\tr \bH^\tr \bgen + \bgen^\tr \bh \qquad \label{eq:OPF_MQP_1} \\
\text{s.t.} &\quad \bA\bgen\le \bb+\bE\btheta, \label{eq:OPF_MQP_2} \end{aligned}$$ where $\bA\in\Real^{(2+2m+2n_g)\times n_g},\bE\in\Real^{(2+2m+2n_g)\times n_{\theta}},\bb\in\Real^{(2+2m+2n_g)}$ are defined as $$\label{eq:MQP}
\bA=\begin{bmatrix}
\phantom{-}\mathbf{1}_{n_g}^\tr\\
-\mathbf{1}_{n_g}^\tr\\
\phantom{-}\bPTDF_{\Nodes_g}\\
-\bPTDF_{\Nodes_g}\\
\phantom{-}\bI_{n_g} \\
-\bI_{n_g}\\
\end{bmatrix},\quad
\bb=\begin{bmatrix}
\phantom{-}\ones^\tr \bdem\\
-\ones^\tr\bdem\\
\phantom{-}\bPTDF_{\Nodes_d} \bdem +\bfpluslimit\\
-\bPTDF_{\Nodes_d} \bdem -\bfminuslimit\\
\phantom{-}\bgpluslimit\\
-\bgminuslimit\\
\end{bmatrix},\quad
\bE=\begin{bmatrix}
-\ones_{n_{\theta}}^\tr \\
\phantom{-}\ones_{n_{\theta}}^\tr \\
-\bPTDF_{\Nodes_{\theta}} \\
\phantom{-}\bPTDF_{\Nodes_{\theta}} \\
\phantom{-}\zeros_{n_{\theta}} \\
\phantom{-}\zeros_{n_{\theta}} \\
\end{bmatrix}.
$$ For $k\in\Nat$, denote by $\ones_k, \zeros_k\in\Real^k$ and $\bI_{k}\in\Real^{k\times k}$ the vector of ones, zeros, and the identity matrix of dimension $k$, respectively. Moreover, $\bPTDF_{\Nodes_g}\in\Real^{m\times n_g}$ and $\bPTDF_{\Nodes_{\theta}}\in\Real^{m\times n_{\theta}}$ denote the submatrices of $\bPTDF$ obtained by selecting only the columns corresponding to nodes in $\Nodes_g$ and $\Nodes_{\theta}$, respectively.
A key result in MPT [@Tondel2003] is that the feasible parameter space $\bTheta\subseteq \Real^{n_{\theta}}$ of the problem Eqs. - can be partitioned into a finite number of convex polytopes, each corresponding to a different *optimal partition*, i.e., a grid-wide state vector that indicates the saturated status of generators and congestion status of transmission lines.
\[def:partition\] Given a parameter vector $\btheta\in\bTheta$, let $\bg^*=\bg^*(\btheta)$ denote the optimal generation vector obtained by solving the problem defined by Eqs. -. Let $\mathcal{J}$ denote the index set of constraints in Eq. , with $|\mathcal{J}|=2+2m+2n$. The *optimal partition* of $\mathcal{J}$ associated with $\btheta$ is the partition $\mathcal{J} = \B(\btheta)\sqcup \B^{\complement}(\btheta)$, with $\mathcal{B}(\btheta)= \{i\in\mathcal{J}\,\,|\, \bA_i\bgen^*=\bb+\bE_i\theta\}$ and $
\mathcal{B}^\complement(\btheta)=\{i\in\mathcal{J}\,|\, \bA_i\bg^*<\bb+\bE_i\btheta\}$.
The sets $\mathcal{B}$ and $\mathcal{B}^\complement$, respectively, correspond to binding and non-binding constraints of the OPF and, hence, identify congested lines and nonmarginal generators. With a minor abuse of notation, we identify the optimal partition $(\B,\B^\complement)$ with the corresponding set of binding constraints $\B$. Given an optimal partition $\B$, let $\bA_{\B},\bE_{\B}$ denote the submatrices of $\bA$ and $\bE$ containing the rows $\bA_i,\bE_i$ indexed by $i\in\B$, respectively.
\[rm:redundant\] The energy balance equality constraint in the original OPF formulation is rewritten as two inequalities indexed by $i=1,2$ in Eq. , which are always binding and read $\bA_i\bg^*=\bb_i+\bE_i\btheta$, $i=1,2$. Looking at Eq. , we see that the two equations $\bA_i\bg^*=\bb_i+\bE_i\btheta, \, i=1,2,$ are identical, and thus one of them is redundant. In the rest of this paper, we eliminate one of the redundant constraints from the set $\B$, namely the one corresponding to $i=2$. Therefore, we write $\B = \{1\} \sqcup \Bcong \sqcup \Bsat$, where $\Bcong\subseteq\{3,\ldots,2+2m\}$ describes the congestion status of transmission lines, and $\Bsat\subseteq \{2+2m+1,2+2m+2n\}$ describe the saturated status of generators.
\[def:licq\] Given an optimal partition $\B$, we say that the *linear independent constraint qualification* (LICQ) holds if the matrix $\bA_{\B}\in\Real^{|\B|\times n}$ has full row rank.
Since there is always at least one binding constraint, namely $i=1$ (corresponding to the power balance constraint), we can write $|\B| = 1 + |\B\pr|$, where $\B\pr =
\Bcong \sqcup \Bsat$ contains the indexes of binding constraints corresponding to line and generator limits. Since line and generation limits cannot be binding both on the positive and negative sides, we have that $|\B\pr| = |\Bsat| + |\Bcong|\le n_g + m$. Moreover, it is observed in [@Zhou2011] that the row rank of $\bA_{\B}$ is equal to $ \min(1+|\Bsat|+|\Bcong|, n_g)$, implying that the LICQ condition is equivalent to $$\label{eq:LICQ_eq}
1+|\Bsat|+|\Bcong|\le n_g.$$ The following theorem is a standard result in MPT, see [@Tondel2003 Theorem 1]. It states that there exist $M$ affine maps defined in the interiors of the critical regions $$\mathring{\bTheta_k} \ni \btheta \to \bLMP_{\rvert \mathring{\bTheta_k}} (\btheta) = \bCtildek \btheta + \bctildek,\, k=1\ldots,M$$ where $\bCtildek,\bctildek$ are suitably defined matrices and vectors. Moreover, if LICQ holds for every $\btheta\in\bTheta$, then the maps agree on the intersections between the regions $\bTheta_k$’s, resulting in an overall continuous map $
\bTheta \ni \btheta \to \bLMP(\btheta) \in \Real^n.
$
\[th:mpt1\] Assume that $\bH$ is positive definite, $\bTheta$ a fully dimensional compact set in $\Real^{n_{\theta}}$, and that the LICQ regularity condition is satisfied for every $\btheta\in\bTheta$. Then, $\bTheta$ can be covered by the union of a finite number $M$ of fully-dimensional compact convex polytopes $\bTheta_1,\ldots,\bTheta_M$, referred to as *critical regions*, such that: (i) their interiors are pairwise disjoint $\mathring{\bTheta}_k \cap \mathring{\bTheta}_h = \emptyset$ for every $k\neq h$, and each interior $\mathring{\bTheta}_k $ corresponds to the largest set of parameters yielding the same optimal partition; (ii) within the interior of each critical region $\mathring{\bTheta}_k$, the optimal generation $\bgen^*$ and the associated $\bLMP$ vector are affine functions of $\btheta$, (iii) the map $\bTheta\ni\btheta \to \bLMP(\btheta)$ defined over the entire parameter space is piecewise affine and continuous.
Relaxing the LICQ assumption
----------------------------
One of the assumptions of Theorem \[th:mpt1\], which is standard in the literature [@Zhou2011; @Bo2009; @Li2009; @Bo2012], is that the LICQ condition holds for every $\btheta\in\bTheta$. In particular, this means that LICQ holds in the interior of two neighboring regions, which we denote as $\mathring{\bTheta}_i$ and $\mathring{\bTheta}_j$. Let $\Hyp$ be the hyperplance separating $\mathring{\bTheta}_i$ and $\mathring{\bTheta}_j$. The fact that LICQ holds at $\mathring{\bTheta}_i$ implies that, if $\{i_1,\ldots,i_q\}$ are the binding constraints at optimality in the OPF for $\btheta\in\mathring{\bTheta}_i$, then in view of Eq. we have $q\le n_g$, where we recall that $n_g$ is the number of decision variables in the OPF (i.e., the number of controllable generators).
Requiring LICQ to hold everywhere means that, in particular, it must hold in the common facet between regions. As we move from $\mathring{\bTheta}_k$ on to the common facet $\F = \bTheta_i \cap \Hyp$ between regions $\bTheta_i$ and $\bTheta_j$, which has dimension $n_{\theta}-1$, there could be an additional constraint becoming active (coming from the neighboring region $\bTheta_j$), and therefore the LICQ condition implies $q+1\le n_g$. In general, critical regions can intersect in faces of dimensions $1,\ldots,n_{\theta}-1$, and enforcing LICQ to hold on all these faces could imply the overly-conservative assumption $q+n_{\theta}-1\le n_g$.
In what follows, we relax the assumptions of Theorem \[th:mpt1\] by allowing LICQ to be violated on the union of these lower-dimensional faces $$\label{eq:Theta0}
\bTheta_{\circ} :=\bTheta \setminus \bigcup_{k=1}^M \mathring{\bTheta}_k.$$ Since this union has zero $n_{\theta}$-dimensional Lebesgue measure, the event $\btheta\in\bTheta_{\circ}$ rarely happens in practice, and thus is usually ignored in the literature, but it does cause a technical issue that we now address. If LICQ is violated on $\btheta \in \bTheta_{\circ}$, the Lagrange multipliers of the OPF, and thus the LMP, need not be unique. Therefore, the map $\btheta \to \bLMP(\btheta)$ is not properly defined on $\bTheta_{\circ}$. In order to extend the map from $\bigcup_{k=1}^M \mathring{\bTheta}_k$ to the full feasible parameter space $\bTheta$, we incorporate a tie-breaking rule to consistently choose between the possible LMPs. Following [@Tang2013nash], we break ties by using the lexicographic order. This choice defines the LMP function over the whole feasible parameter space $\bTheta$, but may introduce jump discontinuities on the zero-measure set $\bTheta_{\circ}$. In the next section, we address this technicality and formally derive our main large deviations result.
Large deviations results {#s:ldp}
========================
\[prop:ldp\] Let $\btheta_{\eps} \sim \mathcal{N}_{n_{\theta}}(\bmuth,\eps\bSigmath)$ be a family of nondegenerate $n_{\theta}$-dimensional Gaussian r.v.’s indexed by $\eps>0$. Assume that the LICQ condition is satisfied for all $\btheta\in\bTheta\setminus \bTheta_{\circ}$. Consider the event $$Y=Y(\balpha^-,\balpha^+) = \bigcup_{i=1}^n \{ \btheta\in \bTheta \,:\, \LMP_i(\btheta) \notin [\alpha_i^{-},\alpha_i^{+}]\},$$ defined in Eq. , assume that the interior of $Y$ is not empty [^3] and that $$\bLMP(\bmuth) \in \mathring{\Pi},\quad \Pi:=\prod_{i=1}^n [\alpha_i^-, \alpha_i^+].$$ Then, the family of random vectors $\{\btheta_{\eps}\}_{\eps>0}$ satisfies $$\label{eq:ldp_th}
\lim_{\eps \to 0} \eps \log \Prob (\btheta_{\eps} \in Y) = -\inf_{\btheta \in Y} I(\btheta),$$ where $\I(\btheta) = \frac{1}{2}(\btheta -\bmuth)^\top \bSigmath^{-1} (\btheta -\bmuth).$.
For notational compactness, in the rest of the proof we will write $Y$ without making explicit its dependence on $(\balpha^-,\balpha^+)$. Defining $Z := \bigcup_{i=1}^n \{\btheta \in\Real^{n_{\theta}} \,:\ \LMP_i(\btheta) \notin [\alpha_i^{-},\alpha_i^{+}]\}$, the event $Y$ can be decomposed as the disjoint union $Y = Y_*\cup Y_{\circ}$, where $$\begin{aligned}
Y_*=
\bigcup_{k=1}^M \mathring{\bTheta}_k \cap Z,\quad
Y_{\circ} = \bTheta_{\circ} \cap Z,\end{aligned}$$ and $Y_{\circ} \subseteq \bTheta_{\circ} = \bTheta \setminus \bigcup_{k=1}^M \mathring{\bTheta}_k$ is a zero-measure set. As $\btheta_{\eps}$ is non nondegenerate, it has a density $f$ with respect to the $n_{\theta}$-dimensional Lebesgue measure in $\Real^n_{\theta}$. Since the $n_{\theta}$-dimensional Lebesgue measure of $ Y_{\circ}$ is zero, we have $$\Prob (\btheta_{\eps} \in Y_{\circ})=\int_{\bx\in Y_{\circ}} f(\bx) d\bx =0$$ and $\Prob (\btheta_{\eps} \in Y) = \Prob (\btheta_{\eps} \in Y_*)$. As a consequence, we can restrict our analysis to the event $Y_*$. Thanks to Cramer’s theorem in $\Real^{n_{\theta}}$ [@Dembo1998], we have $$\begin{aligned}
\label{eq:LDP-general}
-\inf _{\btheta \in \mathring{Y}_* }I(\btheta)
\leq &\liminf_{\eps\to 0}\eps\log {\big (}{\Prob}(\btheta_{\eps} \in Y_*){\big )}\\
\leq &\limsup _{\eps\to 0}\eps\log {\big (}{\Prob}(\btheta_{\eps} \in Y_*){\big )}\leq -\inf_{\btheta \in \overline{Y_*}}I(\btheta),\end{aligned}$$ where $I(\btheta)$ is the Legendre transform of the log-moment generating function of $\btheta_{\eps}$. It is well-known (see, for example, [@Touchette2009]) that when $\btheta_{\eps}$ is Gaussian then $\I(\btheta) = (\btheta -\bmuth)^\tr \bSigmath^{-1} (\btheta -\bmuth)$. In order to prove , it remains to be shown that $$\label{eq:two_infs}
\inf _{\btheta \in \mathring{Y}_*}I(\btheta) = \inf_{\btheta \in \overline{Y_*}}I(\btheta).$$ Thanks to the continuity of the maps $\bLMP\rvert_{\mathring{\bTheta}_k}$, the set $Y_*$ is open, since $$\begin{aligned}
\label{eq:union1}
Y_* = &\bigcup_{k=1}^M \mathring{\bTheta_k} \cap Z
=\bigcup_{k=1}^M \bigl(\mathring{\bTheta_k} \cap \bigcup_{i=1}^n\{\ \LMP_i\rvert_{\mathring{\bTheta}_k}(\btheta) \notin [\alpha_i^{-},\alpha_i^{+}] \}\bigr)\\
=&\bigcup_{k=1}^M \Bigr(\mathring{\bTheta_k} \cap \bigcup_{i=1}^n
\{\bCtildek_i \btheta + \bctildek_i \notin [\alpha_i^{-},\alpha_i^{+}]\}\Bigl)\\
=&\bigcup_{k=1}^M \bigcup_{i=1}^n \Bigr(\mathring{\bTheta_k} \cap
(\{\bCtildek_i \btheta + \bctildek_i < \alpha_i^-\} \cup \{\bCtildek \btheta + \bctildek > \alpha_i^+\})\Bigl).
$$ Therefore, $\mathring{Y_*} = Y_*$, $\overline{\mathring{Y_*}} = \overline{Y_*}\supseteq Y_*$ and Eq. follows from the continuity of $I(\btheta)$.
Proposition \[prop:ldp\] allows us approximate the probability of a price spike, for small $\eps$, as $$\label{eq:ldp_approx}
\Prob (\btheta_{\eps} \in Y) \approx \exp\,\Bigl(\frac{ -\inf_{\btheta \in Y} I(\btheta)}{\eps}\Bigr),$$ as it is done in [@Nesti2019temperature; @Nesti2018emergent] in the context of studying the event of transmission line failures. Moreover, the minimizer of the optimization problem corresponds to the most likely realization of uncontrollable generation that leads to the rare event. Furthermore, the structure of the problem allows us to efficiently rank nodes in terms of their likelihood to experience a price spike, as we illustrate in Section \[s:numerics\].
Solving the optimization problem {#s:optimization}
================================
In view of Proposition \[prop:ldp\], in order to study $\lim_{\eps \to 0} \eps \log \Prob (\btheta_{\eps} \in Y(\balpha^-,\balpha^+))$ we need to solve the deterministic optimization problem $\inf_{\btheta \in Y_{*}} I(\btheta).$ The latter, in view of Theorem \[th:mpt1\] and the definition of $Y_*$, the latter is equivalent to $$\begin{aligned}
\inf_{\btheta \in Y_{*}} I(\btheta) =
&\min_{k=1,\ldots,M} \inf_{\btheta \in \mathring{\Theta}_k \cap Z} I(\btheta)
=\min_{i=1,\ldots,n }\min_{k=1,\ldots,M} \inf_{\btheta \in \mathring{\Theta}_k, \bCtildek_i \btheta + \bctildek_i\notin [\alpha_i^-,\alpha_i^+]} I(\btheta).\end{aligned}$$ This amounts to solving at most $nM$ quadratic optimization problems of the form $\inf_{\btheta \in T_{i,k}} I(\btheta)$ for $i=1,\ldots,n$, $k=1,\ldots,M$, where $$T_{i,k} = T^-_{i,k} \sqcup T^+_{i,k},\quad
T^-_{i,k} = \mathring{\Theta}_k \cap \ \{ \bCtildek_i \btheta + \bctildek_i< \alpha_i^- \},\quad
T^+_{i,k} =\mathring{\Theta}_k \cap \{ \bCtildek_i \btheta + \bctildek_i> \alpha_i^+ \}.$$
In the rest of this section, we show how we can significantly reduce the number of optimization problems that need to be solved by exploiting the geometric structure of the problem. First, since $$\begin{aligned}
\inf_{\btheta \in Y_{*}} I(\btheta)
=&\min_{i=1,\ldots,n }\inf_{\btheta \in \bigcup_{k=1}^M T_{i,k}} I(\btheta),\end{aligned}$$ we fix $i=1\ldots, n$ and consider the sub-problems $$\label{eq:problem_fixed_i}
\inf_{\btheta \in \bigcup_{k=1}^M T_{i,k}} I(\btheta) =
\min\,\Bigl\{
\inf_{\btheta \in \bigcup_{k=1}^M T^-_{i,k}} I(\btheta),
\inf_{\btheta \in \bigcup_{k=1}^M T^+_{i,k}} I(\btheta)
\Bigr\}.$$ The reason why we want to solve the problems in Eq. individually for every $i$ is because we are not only interested in studying the overall event $Y$, but also in the more granular events of node-specific price spikes. For example, this would allow us to rank the nodes in terms of their likelihood of experiencing a price spike (see Section \[s:numerics\]). Define $$\begin{aligned}
&L^-_{(i,k)} := \bTheta_k \cap (T^-_{i,k})^{\complement}=
\bTheta_k \cap \{\bCtildek_i \btheta + \bctildek_i \ge \alpha_i^{-}\},\quad
L_i^- := \bigcup_{k=1}^M L^-_{i,k} = \bTheta \cap \{\LMP_i \ge \alpha_i^-\},\\
& L^+_{(i,k)} := \bTheta_k \cap (T^+_{i,k})^{\complement}=
\bTheta_k \cap \{\bCtildek_i \btheta + \bctildek_i \le\alpha_i^{+}\},\quad
L_i^+ := \bigcup_{k=1}^M L^+_{i,k} = \bTheta \cap \{\LMP_i \le \alpha_i^+\},\end{aligned}$$ and consider the partition of the sets $L_i^+$ and $L_i^-$ into disjoint closed connected components, i.e., $$L_i^-= \bigsqcup_{\ell \in \text{conn. comp. of } L_i^- } W^{(i,-)}_{\ell},\,
L_i^+= \bigsqcup_{\ell \in \text{conn. comp. of } L_i^+ } W^{(i,+)}_{\ell},$$ and let $W^{(i,-)}_{\ell^{-*}},W^{(i,+)}_{\ell^{+*}}$ be the components containing $\bmuth$. Since $\partial(A \cup B) = \partial A \cup \partial B$ if $\overline{A}\cap B = A \cap \overline{B} = \emptyset$, the boundary $\partial L_i^+ = \bigsqcup_{\ell \in \F^+_{i}} \partial W^{(i,+)}_{\ell}$ is the union of the set of parameters $\btheta\in\bTheta$ such that $\bLMP(\btheta)=\alpha_i^+$ with, possibly, a subset of the boundary of $\bTheta$ (and similarly for $\partial L_i^-$). As stated by Proposition \[prop:opt\_1\], we show that, in order to solve the two problems in the right hand side of Eq. we need to look only at the boundaries $\partial W^{(i,-)}_{\ell^*},\partial W^{(i,+)}_{\ell^*}$.
\[prop:opt\_1\] Under the same assumptions of Theorem \[th:mpt1\], we have $$\begin{aligned}
&\inf_{\btheta \in \bigcup_{k=1}^M T^{+}_{i,k}} I(\btheta) =
\inf_{\btheta \in \partial \bigcup_{k=1}^M T^{+}_{i,k}} I(\btheta), \quad
\inf_{\btheta \in \bigcup_{k=1}^M T^{-}_{i,k}} I(\btheta) =
\inf_{\btheta \in \partial \bigcup_{k=1}^M T^{-}_{i,k}} I(\btheta).
\label{eq:opt_1}\end{aligned}$$ Moreover, $$\begin{aligned}
&\inf_{\btheta \in \bigcup_{k=1}^M T^{+}_{i,k}} I(\btheta) =
\inf_{\btheta \in \partial W^{(i,+)}_{\ell^{*+}}} I(\btheta),
\quad
\inf_{\btheta \in \bigcup_{k=1}^M T^{-}_{i,k}} I(\btheta) =
\inf_{\btheta \in \partial W^{(i,-)}_{\ell^{*-}}} I(\btheta).\label{eq:lake_minus}\end{aligned}$$
First note that the rate function $I(\btheta)$ is a (strictly) convex function, since $\bSigmath$ is positive definite. Since $\bigcup_{k=1}^M T^{+}_{i,k}$ is open and $I(\btheta)$ is a continuous function, it holds that $$\inf_{\btheta \in \bigcup_{k=1}^M T^{+}_{i,k}} I(\btheta) =
\inf_{\btheta \in \overline{\bigcup_{k=1}^M T^{+}_{i,k}}} I(\btheta) .$$ Moreover, since $I(\btheta)$ is continuous and $\overline{\bigcup_{k=1}^M T^{+}_{i,k}}$ compact, the infimum is attained. The fact that $\overline{\bigcup_{k=1}^M T^{+}_{i,k}}\supseteq \partial \bigcup_{k=1}^M T^{+}_{i,k}$ immediately implies that $$\inf_{\btheta \in \bigcup_{k=1}^M T^{+}_{i,k}} I(\btheta) \le
\inf_{\btheta \in \partial \bigcup_{k=1}^M T^{+}_{i,k}} I(\btheta).$$ On the other hand, assume by contradiction that $$\inf_{\btheta \in \bigcup_{k=1}^M T^{+}_{i,k}} I(\btheta) <
\inf_{\btheta \in \partial \bigcup_{k=1}^M T^{+}_{i,k}} I(\btheta).$$ In particular, there exists a point $\btheta_0$ in the interior of $\bigcup_{k=1}^M T^{+}_{i,k}$ such that $I(\btheta_0)<I(\btheta)$ for all $\btheta \in \bigcup_{k=1}^M T^{+}_{i,k}$. Define, for $t\in[0,1]$, the line segment joining $\bmuth$ and $\btheta_0$, i.e. $\btheta_{t} = (1-t)\bmuth + t \btheta_0$. Since $\btheta_0$ lies in the interior of $\bigcup_{k=1}^M T^{+}_{i,k}$, and $\bmuth\notin \bigcup_{k=1}^M T^{+}_{i,k}$, there exist a $0< t_*<1$ such that $\btheta_t \in \bigcup_{k=1}^M T^{+}_{i,k}$ for all $t\in [t_*,1]$. Due to the convexity of $I(\btheta)$, and the fact that $I(\bmuth)=0$, we have $$I(\btheta_{t_*}) < (1-{t_*}) I(\bmuth) + {t_*} I(\btheta_0) = {t_*} I(\btheta_0) < I(\btheta_0),$$ thus reaching a contradiction. Hence, $$\inf_{\btheta \in \bigcup_{k=1}^M T^{+}_{i,k}} I(\btheta) =
\inf_{\btheta \in \partial \bigcup_{k=1}^M T^{+}_{i,k}} I(\btheta),$$ and the minimum is achieved on $ \partial \bigcup_{k=1}^M T^{+}_{i,k}$, proving Eq. .
In view of Eq. , in order to prove Eq. it is enough to show that $$\inf_{\btheta \in \partial \bigcup_{k=1}^M T^{+}_{i,k}} I(\btheta)=
\inf_{\btheta \in \partial W^{(i,+)}_{\ell^{*+}}} I(\btheta).$$
Given that the sets $T^{+}_{i,k}=\mathring{\bTheta}_k \cup \{\LMP_i(\btheta)>\alpha_i^+\}$, for $k=1,\ldots,M$, are disjoint, the boundary of the union is equal to the union of the boundaries, i.e., $\partial \bigcup_{k=1}^M T^{+}_{i,k} = \bigcup_{k=1}^M \partial T^{+}_{i,k}.$ Each term $\partial T^{+}_{i,k}$ is the boundary of the polytope $\overline{T^{+}}_{i,k} = \bTheta_k \cap \{\LMP_i \ge \alpha_i^+\}$, and thus consists of the union of a subset of $\bigcup_{k=1}^M \partial \bTheta_k$ (a subset of the union of the facets of the polytope $\bTheta_k$) with the segment $\bTheta_k \cap \{\LMP_i = \alpha_i^+\}.$ As a result, $\partial \bigcup_{k=1}^M T^{+}_{i,k} I(\btheta)$ intersects $\partial W^{(i,+)}_{\ell^{*+}}$ in $\bTheta \cap \{\LMP_i=\alpha_i^+\}$.
We now show that (i) the minimum is attained at a point $\btheta_0$ such that $\LMP_i(\btheta_0) = \alpha_i^+$, so that $\btheta_0 \in \bigsqcup_{\ell \in \text{conn. comp. of } L_i^+ } \partial W^{(i,+)}_{\ell}$, and (ii) $\btheta_0\in \partial W^{(i,+)}_{\ell^{*+}}$. Assume by contradiction that $\LMP_i(\btheta_0)>\alpha_i^+$, and consider the line segment joining $\bmuth$ and $\btheta_0$, $\btheta_{t} = (1-t)\bmuth + t \btheta_0, \, t\in[0,1]$. The function $$[0,1] \ni t \to g(t) := \LMP_i (\btheta_t) = \LMP_i ((1-t)\bmuth + t \btheta_0) \in \Real,$$ is continuous and such that $g(0) = \bLMP(\bmuth)< \alpha$ and $g(1)=\bLMP(\btheta_0)>\alpha_i^*$. Thanks to the intermediate value theorem, there exists a $0<t_*<1$ such that $g(t_*)=\LMP_i(\btheta_{t_*})=\alpha_i^*$, and $$I(\btheta_{t_*}) < (1-t_*) I(\bmuth) + t_* I(\btheta_0) = t_* I(\btheta_0) < I(\btheta_0),$$ which is a contradiction, since $\btheta_0$ is the minimum. Therefore, $\btheta_0\in \bigsqcup_{\ell \in \text{conn. comp. of } L_i^+ } \partial W^{(i,+)}_{\ell}$. The same argument, based on the convexity of the rate function and the fact that $I(\bmuth)=0$, shows that $\btheta\in \partial W^{(i,+)}_{\ell^{*,+}}$. Lastly, Eq. can be derived in the same way.
Proposition \[prop:opt\_1\] shows that in order to solve the problem in Eq. we only need to look at the boundaries $\partial W^{(i,+)}_{\ell^*}, \partial W^{(i,-)}_{\ell^*}$. Determining such boundaries is a non-trivial problem, for which dedicated algorithms exist. However, such algorithms are beyond the scope of this paper and we refer the interested reader to the contour tracing literature and, in particular, to [@Dobkin1990contour].
Numerics {#s:numerics}
========
In this section, we illustrate the potential of our large deviations approach using the standard IEEE 14-bus test case from MATPOWER [@Zimmerman2011]. This network consists of 14 nodes housing loads, 6 controllable generators, and 20 lines. As line limits are not included in the test case, we set them as $\bfpluslimit = \lambda \bfpluslimit^{(\text{planning)}}$, where $ \bfpluslimit^{(\text{planning})} := \gamma_{\text{line}} |\bflow|$, $\bflow$ is the solution of a DC-OPF using the data in the test file, and $\gamma_{\text{line}}\ge 1$. We interpret $\bfpluslimit^{(\text{planning})}$ as the maximum allowable power flow before the line trips, while $\lambda$ is a safety tuning parameter satisfying $1/\gamma_{\text{line}}\le\lambda\le 1$. In the rest of this section, we set $\gamma_{\text{line}}=2$ and $\lambda=0.6$.
We add two uncontrollable renewable generators at nodes $4$ and $5$, so that $n_d=14,n_g=6$ and $n_{\theta}=2$. All the calculations related to multiparametric programming are performed using the MPT3 toolbox [@MPT3]. We model the renewable generation as a 2-dimensional Gaussian random vector $\btheta\sim \mathcal{N}_2(\bmuth,\bSigmath)$, where $\bmuth$ is interpreted as the nominal, or forecast, renewable generation. The covariance matrix $\bSigmath$ is calculated as in [@Hoffmann2019consistency] to model positive correlations between neighboring (thus geographically “close”) nodes. More specifically, we consider normalized symmetric graph Laplacian $L_{\text{sym}} = \Delta ^{-1/2} L_{\text{sym}} \Delta^{-1/2},$ where $\Delta\in\Real^{n\times n}$ is the diagonal matrix with entries equal to $\Delta_{i,i}=\sum_{j\neq i} \weight_{i,j}$. We then compute the matrix $$\label{eq:cov_IEEE14}
\bC = \tau^{2\kappa} (L_{\text{sym}} + \tau^2 I)^{-\kappa}\in\Real^{n\times n},$$ for $\kappa=2$ and $\tau^2=1$ as in [@Hoffmann2019consistency], and consider the $n_{\theta}\times n_{\theta}$ submatrix $\btildeSigmath$ obtained from of $\bC$ by choosing rows and columns indexed by $\Nodes_{n_{\theta}}=\{4,5\}$, and we define $\bSigmath$ as $$\bSigmath := \diag(\{\delta_i\}_{i=1}^{n_{\theta}}) \,\btildeSigmath\, \diag(\{\delta_i\}_{i=1}^{n_{\theta}})\,\in\Real^{n_{\theta}\times n_{\theta}},$$ where the parameters $\delta_i$’s control the magnitudes of the standard deviations $\sigma_{i}:=\sqrt{\bSigmath(i,i)}$, $i=1,2$. In particular, the $\delta_i$’s are chosen in such a way that the standard deviations $\sigma_{i}$’s match realistic values for wind power forecasting error expressed as a fraction of the corresponding installed capacity, over different time windows $T$, namely $\sigma_{i} = q(T) \times \mu_i^{(\text{installed})},\, i=1,2,$ where $q=[0.01,0.018,0.04]$, corresponding to time windows of $5,15$ and $60$ minutes, respectively (see [@Nesti2019temperature], Section V.B). Finally, the installed capacity of the renewable generators are chosen based on the boundary of the $2$-dimensional feasible space $\bTheta$, namely $\mu_1^{(\text{installed})} = \max\{x\,:\, (x,y) \in \bTheta\},
\mu_2^{(\text{installed})} = \max\{y\,:\, (x,y) \in \bTheta\}.$ Although $\btheta\sim \mathcal{N}_2(\bmuth,\bSigmath)$ is in principle unbounded, we choose the relevant parameters in such a way that, in practice, $\btheta$ never exceeds the boundary of the feasible space $\bTheta$. Since $\bSigmath$ is obtained from realistic values for wind power forecasting error, the question is whether the matrix $\bSigmath$ used in the numerics is close enough to the small-noise regime to make the large deviations results meaningful. As we show, the answer to this question is affirmative, validating the use of the large deviations methodology.
Multimodality and sensitivity with respect to $\vect{\mu}_{\vect{\theta}}$
--------------------------------------------------------------------------
Given $\bmuth \in \bTheta$, we set the price thresholds defining the spike event as $\balpha^{-}_i= \LMP_i(\bmuth) - \errrel |\LMP_i(\bmuth)| ,\,\balpha^{+}_i= \LMP_i(\bmuth) + \errrel |\LMP_i(\bmuth)|,$ where $\errrel>0$. In other words, we are interested in studying the event of a *relative price deviation* of magnitude greater than $\errrel>0$: [$$Y=\bigcup_{i=1}^n Y_i, \qquad Y_i = \{\btheta\in\bTheta\,:\,
|\LMP_i(\btheta) -\LMP_i(\bmuth)| > \errrel |\LMP_i(\bmuth)|\}.$$ ]{}
Next, we consider two scenarios, corresponding to low and high expected wind generation, i.e. $\bmuth^{(\text{low})} = 0.1 \times\bmu^{(\text{installed})}$, $\bmuth^{(\text{high})} = 0.5 \times\bmu^{(\text{installed})}$,and variance corresponding to the 15-minute window, i.e. $q^{(\text{medium})} = 0.018$. Fig. \[fig:panel\] shows the location of $\bmuth$, together with $10^6$ samples from $\btheta$, the corresponding empirical density of the random variable $\LMP_{10}$ obtained through Monte Carlo simulation, and a visualization of the piecewise affine mapping $\btheta\to \bLMP_{10}(\btheta)$.
We observe that the results are extremely sensitive to the standard deviation and location of the forecast renewable generation $\bmuth$ relative to the geometry of the critical regions, as this affects whether the samples of $\btheta$ will cross the boundary between adjacent regions or not. In turn, the crossing of a boundary can result in the distribution of the LMPs being *multimodal* (see Fig. \[fig:panel\], left panels), due to the piecewise affine nature of the map $\btheta\to \bLMP$. This observation shows how the problem of studying LMPs fluctuations is intrinsically harder than that of emergent line failures, as in [@Nesti2017line; @Nesti2018emergent]. The phenomenon is more pronounced in the presence of steep gradient changes at the boundary between regions (or in the case of discontinuities), as can be observed in the right panels of Fig. \[fig:panel\], which show the piecewise affine map $\btheta\to \bLMP_{10}(\btheta)$ for the two different choices of $\bmuth$. In particular, the expected $\bLMP$ can differ greatly from $\bLMP(\bmuth)$.
![forecast generation $\bmuth$ and empirical distribution of renewable generation $\btheta$ (left); Empirical density of the random variable $\LMP_{10}$, with the thresholds $\alpha^-_i$ and $\alpha^+_i$ represented as red and blue vertical bars, respectively (middle); Piecewise affine map $\btheta\to \bLMP_{10}(\btheta)$ with price thresholds $\alpha^{\pm}_{10}= \LMP_{10}(\bmuth) \pm \errrel |\LMP_{10}(\bmuth)|$, for two different choices of $\bmuth$ (right). []{data-label="fig:panel"}](cloud_lambda06_increment025_boundary_q0018_mu01cap.png "fig:"){width="32.00000%"} ![forecast generation $\bmuth$ and empirical distribution of renewable generation $\btheta$ (left); Empirical density of the random variable $\LMP_{10}$, with the thresholds $\alpha^-_i$ and $\alpha^+_i$ represented as red and blue vertical bars, respectively (middle); Piecewise affine map $\btheta\to \bLMP_{10}(\btheta)$ with price thresholds $\alpha^{\pm}_{10}= \LMP_{10}(\bmuth) \pm \errrel |\LMP_{10}(\bmuth)|$, for two different choices of $\bmuth$ (right). []{data-label="fig:panel"}](density_10_lambda06_increment025_boundary_q0018_mu01cap_cropped "fig:"){width="32.00000%"} ![forecast generation $\bmuth$ and empirical distribution of renewable generation $\btheta$ (left); Empirical density of the random variable $\LMP_{10}$, with the thresholds $\alpha^-_i$ and $\alpha^+_i$ represented as red and blue vertical bars, respectively (middle); Piecewise affine map $\btheta\to \bLMP_{10}(\btheta)$ with price thresholds $\alpha^{\pm}_{10}= \LMP_{10}(\bmuth) \pm \errrel |\LMP_{10}(\bmuth)|$, for two different choices of $\bmuth$ (right). []{data-label="fig:panel"}](landscape_lambda06_increment025_cap=boundary_q0018_mu=01_cap_node10_new_PROVA.png "fig:"){width="32.00000%"}
![forecast generation $\bmuth$ and empirical distribution of renewable generation $\btheta$ (left); Empirical density of the random variable $\LMP_{10}$, with the thresholds $\alpha^-_i$ and $\alpha^+_i$ represented as red and blue vertical bars, respectively (middle); Piecewise affine map $\btheta\to \bLMP_{10}(\btheta)$ with price thresholds $\alpha^{\pm}_{10}= \LMP_{10}(\bmuth) \pm \errrel |\LMP_{10}(\bmuth)|$, for two different choices of $\bmuth$ (right). []{data-label="fig:panel"}](%cloud_lambda06_increment025_cap2_q0018_mu05cap.png
cloud_lambda06_increment025_boundary_q0018_mu05cap.png "fig:"){width="32.00000%"} ![forecast generation $\bmuth$ and empirical distribution of renewable generation $\btheta$ (left); Empirical density of the random variable $\LMP_{10}$, with the thresholds $\alpha^-_i$ and $\alpha^+_i$ represented as red and blue vertical bars, respectively (middle); Piecewise affine map $\btheta\to \bLMP_{10}(\btheta)$ with price thresholds $\alpha^{\pm}_{10}= \LMP_{10}(\bmuth) \pm \errrel |\LMP_{10}(\bmuth)|$, for two different choices of $\bmuth$ (right). []{data-label="fig:panel"}](density_10_lambda06_increment025_boundary_q0018_mu05cap_cropped "fig:"){width="32.00000%"} ![forecast generation $\bmuth$ and empirical distribution of renewable generation $\btheta$ (left); Empirical density of the random variable $\LMP_{10}$, with the thresholds $\alpha^-_i$ and $\alpha^+_i$ represented as red and blue vertical bars, respectively (middle); Piecewise affine map $\btheta\to \bLMP_{10}(\btheta)$ with price thresholds $\alpha^{\pm}_{10}= \LMP_{10}(\bmuth) \pm \errrel |\LMP_{10}(\bmuth)|$, for two different choices of $\bmuth$ (right). []{data-label="fig:panel"}](landscape_lambda06_increment025_cap=boundary_q0018_mu=05_cap_node10_new.png "fig:"){width="32.00000%"}
[0.23]{} ![Comparison between empirical probabilities $\widehat{\mathbb{P}}(Y_i)$ based on Monte Carlo simulation and normalized decay rates $-\min_{i} I^*_i/I^*_i$ for various level of $\errrel.$[]{data-label="fig:_bars"}](bar_lambda06_increment025_cap=boundary_q0018_mu=05cap.png "fig:"){width="\textwidth"}
[0.23]{} ![Comparison between empirical probabilities $\widehat{\mathbb{P}}(Y_i)$ based on Monte Carlo simulation and normalized decay rates $-\min_{i} I^*_i/I^*_i$ for various level of $\errrel.$[]{data-label="fig:_bars"}](bar_lambda06_increment05_cap=boundary_q0018_mu=05cap.png "fig:"){width="\textwidth"}
[0.23]{} ![Comparison between empirical probabilities $\widehat{\mathbb{P}}(Y_i)$ based on Monte Carlo simulation and normalized decay rates $-\min_{i} I^*_i/I^*_i$ for various level of $\errrel.$[]{data-label="fig:_bars"}](bar_lambda06_increment1_cap=boundary_q0018_mu=05cap.png "fig:"){width="\textwidth"}
[0.23]{} ![Comparison between empirical probabilities $\widehat{\mathbb{P}}(Y_i)$ based on Monte Carlo simulation and normalized decay rates $-\min_{i} I^*_i/I^*_i$ for various level of $\errrel.$[]{data-label="fig:_bars"}](bar_lambda06_increment10_cap=boundary_q0018_mu=05cap.png "fig:"){width="\textwidth"}
Ranking of nodes based on their likelihood of having a price spike
------------------------------------------------------------------
As illustrated by Eq. , large deviations theory predicts the most likely node to be $\argmin_{i=1,\ldots,n} I^*_i,$ where $I^*_i := \inf_{\btheta \in \bigcup_{k=1}^M T_{i,k}} I(\btheta)$. Indirectly, this approach produces also a *ranking* of nodes according to their likelihood of having a price spike. The use of large deviations theory to rank power grid components according to their likelihood of experiencing anomalous deviations from a nominal state has been validated in [@Nesti2018emergent] in the context of transmission line failures. In order to validate the accuracy of the LDP methodology also for ranking nodes according to the likelihood of their price spikes, we compare the LD-based ranking with the one obtained via crude Monte Carlo simulation, as described in Table \[tab:\_ranking\_1\]. We observe that the LD-based approach is able to recover the exact ranking of nodes, for various levels of relative error $\errrel$. Table \[tab:\_ranking\_1\] reports the values of the probability $\widehat{\mathbb{P}}(Y_i)$ of a price spike in node $i$, calculated using Monte Carlo simulation, together with the corresponding decay rates $I^*_i = \inf_{\btheta \in \bigcup_{k=1}^M T_{i,k}} I(\btheta)$, showing that the LD-based approach correctly identifies the ranking. This property is validated more extensively in Fig. \[fig:\_bars\], which depicts the values of $\widehat{\mathbb{P}}(Y_i)$ against $-\min_{k} I^*_k/I^*_i$ across a wider range of price thresholds $\errrel$, corresponding to decreasing probability of the price spike event.
$i$ $ \widehat{\mathbb{P}}(Y_i)$ $ I^*_i $
----- ------------------------------ ------------ ----
9 8.6371e-01 8.1160e-04 1
8 6.8984e-01 8.5572e-04 2
7 6.8984e-01 8.5572e-04 3
10 4.8713e-01 1.0786e-03 4
11 4.8690e-01 1.1123e-03 5
6 4.8613e-01 1.2438e-03 6
12 4.8586e-01 1.2849e-03 7
13 4.8586e-01 1.3296e-03 8
14 4.7559e-01 1.6548e-03 9
4 2.1282e-02 4.0854e+00 10
5 0 6.8384e+01 11
1 0 1.1584e+02 12
2 0 1.2971e+02 13
3 0 2.6984e+03 14
: Ranking of nodes based on the likelihood of having a price spike, according to both Monte Carlo simulation (in terms of probabilities $\widehat{\mathbb{P}}(Y_i)$) and large deviations results (in terms of decay rates $ I^*_i $), for the case $\bmuth^{(\text{high})} = 0.5 \times\bmu^{(\text{installed})}$, $\errrel=0.25$, $q=0.018$. The values $\widehat{\mathbb{P}}(Y_i)$, for $i = 1,2,3,5$, are not reported as the Monte Carlo simulation is not sufficiently accurate for such small probabilities. []{data-label="tab:_ranking_1"}
Concluding remarks and future work {#s:conclusion}
==================================
In this paper, we illustrate the potential of concepts from large deviations theory to study the events of rare price spikes caused by fluctuations of renewable generation. By assuming a centralized perspective, we are able to use large deviations theory to approximate the probabilities of such events, and to rank the nodes of the power grids according to their likelihood of experiencing a price spike. Our technical approach is able to handle the multimodality of LMP’s distributions, as well as violations of the LICQ regularity condition.
Future research directions include extending the present framework to non-Gaussian fluctuations, as well as incorporating a source of discrete noise in the form of line outages. Moreover, it would be of interest to study the sensitivity of the approximation in Eq. with respect to the tuning parameter $\lambda$, which quantifies the conservatism in the choice of the line limits. This would allow us to extend the notion of safe capacity regions [@Nesti2019temperature; @Nesti2017line] in the context of energy prices. An alternative approach to deal with non-Gaussian fluctuations and more involved price spike structures could be to efficiently sample *conditionally* on a price spike to have occurred, a problem for which specific Markov chain Monte Carlo (MCMC) methods have been developed, e.g., the Skipping Sampler [@Moriarty2019]. In the case of a complicated multi-modal conditional distribution, the large deviations results derived in this paper can be of extreme help in identifying all the relevant price spikes modes, thus speeding up the MCMC procedure.
#### Acknowledgements
This research is supported by NWO VICI grant 639.033.413, NWO Rubicon grant 680.50.1529 and EPSRC grant EP/P002625/1. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “The mathematics of energy systems” when work on this paper was undertaken. This work was supported by EPSRC grant number EP/R014604/1.
[^1]: The notation $x\in\Real^{A}$ indicates that the entries in the $|A|$-dimensional vector $x$ are indexed by the set $A$.
[^2]: $\mu^-,\mu^+$ cannot be both strictly positive, since lower and upper line flow constraints cannot be simultaneously binding.
[^3]: If $\mathring{Y}=\emptyset$, then trivially $\Prob (\btheta_{\eps} \in Y )= 0$.
|
---
abstract: 'Short distance structure of spacetime may show up in the form of high frequency dispersion. Although such dispersion is not locally Lorentz invariant, we show in a scalar field model how it can nevertheless be incorporated into a generally covariant metric theory of gravity provided the locally preferred frame is dynamical. We evaluate the resulting energy-momentum tensor and compute its expectation value for a quantum field in a thermal state. The equation of state differs at high temperatures from the usual one, but not by enough to impact the problems of a hot big bang cosmology. We show that a superluminal dispersion relation can solve the horizon problem via superluminal equilibration, however it cannot do so while remaining outside the Planck regime unless the dispersion relation is artificially chosen to have a rather steep dependence on wavevector.'
address: 'Department of Physics, University of Maryland, College Park, MD 20742-4111, USA'
author:
- 'Ted Jacobson[^1] and David Mattingly[^2]'
title: |
Generally covariant model of a scalar field with high frequency dispersion\
and the cosmological horizon problem
---
ł ø Ł[[L]{}]{} §
Introduction
============
Recent studies of the role of trans-Planckian physics in the Hawking effect and in inflationary cosmology have exploited scalar fields with high frequency dispersion as a model of how short distance physics might affect the behavior of quantum fields in such settings. These models involve a preferred frame in which the distinction between high and low frequency is made. If such a matter field is to be coupled to the spacetime metric in a generally covariant theory of gravity, the preferred frame must be treated as a dynamical quantity rather than as a fixed background structure. One way of doing this was formulated by us in a previous paper[@aether], in which the preferred frame is determined by a dynamical unit timelike vector field $u^a$. Using this formulation we obtained an expression for the stress tensor of a scalar field with dispersion.
In the present paper we use this stress tensor to evaluate the equation of state in flat spacetime for a thermal state of the field. This equation of state is then used in a fluid description of the matter field in a cosmological model, and the implications for a hot big bang cosmology are examined. We also investigate under what circumstances superluminal equilibration associated with high frequency dispersion can solve the cosmological horizon problem.
In considering only the flat space thermal state of the matter field we are adopting an adiabatic approximation which precludes effects related to out of equilibrium phenomena including particle creation in the dynamical background of a cosmological metric such as were considered recently for fields with high frequency dispersion in the context of inflation[@Martin; @Niemeyer]. This is a good approximation for frequencies larger than the expansion rate $H$. It can therefore be used to study the effects of dispersion for frequencies of order $k_0$ provided $H\ll k_0$.
Model field theory
==================
Various possibilities exist for the kinetic terms in the action for $u^a$. For the purposes of illustration in this paper we choose the “minimal theory" of Ref. [@aether], S\_[min]{}\[&&g\_[ab]{},u\^a,ł\]= d\^4x\
&& ( -a\_1 R - b\_1 F\^[ab]{}F\_[ab]{} +ł(g\_[ab]{}u\^a u\^b -1) ), \[Smin\] where F\_[ab]{}:= 2\_[\[a]{} u\_[b\]]{}. The field $\l$ is just a Lagrange multiplier whose variation enforces the constraint that $u^a$ be a unit vector.
For the matter content we are interested in a scalar field with high frequency dispersion $\o^2=|\vec{k}|^2[1+g(|\vec{k}|/k_0)]$, where $g$ is a function that vanishes at zero, and $k_0$ is a constant with the dimensions of inverse length which sets the scale for deviations from Lorentz invariance. It has been suggested that such a modified dispersion relation might arise in loop quantum gravity[@GP; @MT], or in string theory or other approaches to quantum gravity, or more generally from an unspecified modification of the short distance structure of spacetime (see for example [@Amelino; @river]). Possible observational consequences have been the subject of recent study (see for example [@Amelino; @Bertolami; @Bear; @Carmona] and references therein), and the role of such dispersion in the Hawking process[@river] and in the generation of inflationary primordial density fluctuations[@Martin; @Niemeyer] have been examined.
Absent a reliable theory of such modifications, it makes sense simply to expand in $k/k_0$. We consider here the lowest order modification for a scalar field that is invariant under rotations and analytic in $k$, which is given by ø(k)\^2=||\^2 -||\^4/k\_0\^2, \[dr\] where $k_0$ is a dimensionful parameter. With $k_0^2>0$ this yields a subluminal group velocity, and with $k_0^2<0$ it is superluminal. The pathology at $|\vec{k}|=k_0$ in the subluminal case is irrelevant since (\[dr\]) is only regarded as the first terms in an expansion. When we evaluate the equation of state, we shall impose a cutoff that avoids this pathology.
The dispersion relation (\[dr\]) can be produced by adding a term to the action with four spatial derivatives. Previously this has been done in 1+1 dimensional models (for a review see [@river]), and recently such models have been generalized to field theory in a background 3+1 dimensional Robertson-Walker spacetime[@Martin; @Niemeyer]. Here we extend these models to a general 3+1 dimensional spacetime. This can be accomplished in a generally covariant manner, consistently with spatial rotation invariance, with the action $S_\phi=\int d^4x\, \sqrt{-g}\,{\cal L}_\phi$, where \_=(\^a\_a+k\_0\^[-2]{}(D\^2)\^2). \[Lmod\] Here $D^2$ is the covariant spatial Laplacian, i.e., D\^2=-D\^aD\_a=-q\^[ac]{}\_a(q\_c\^b\_b ), where $D_a$ is the spatial covariant derivative operator[@wald] and $q_{ab}$ is the (positive definite) spatial metric orthogonal to the dynamical unit vector $u^a$, q\_[ab]{}:=-g\_[ab]{}+u\_au\_b. \[q\]
The equation of motion for the metric takes the Einstein form G\_[ab]{}= 8G(T\^[(u)]{}\_[ab]{} + T\^[()]{}\_[ab]{}), \[geq\] where $G=1/16\pi a_1$, and with T\^[(u)]{}\_[ab]{}= -4b\_1 (F\_[am]{}F\_b\^m-F\^2 g\_[ab]{}) +2łu\_au\_b, \[tu\] &&T\^[()]{}\_[ab]{}= \_a\_b- [L]{}\_ g\_[ab]{}\
&&- k\_0\^[-2]{}. \[tdisp\] (The constraint equation $g_{ab}u^au^b=1$ has been used to drop the contribution to (\[tu\]) that would have come from the variation of $\sqrt{-g}$ in the constraint term of the action (\[Smin\]).)
The equation of motion for the field $u^a$ takes the form \^b F\_[ba]{}=- (łu\_a + ). \[ueqmatter\] with &&= 2k\_0\^[-2]{}u\^b\
&&. \[dsdu\] Contracting (\[ueqmatter\]) with $u^a$ we obtain an expression for $\l$ in terms of the other fields: ł=2b\_1 u\^a \^b F\_[ab]{}-u\^a, \[l\] where, from (\[dsdu\]), u\^a= 4k\_0\^[-2]{}u\^au\^b D\^2 \_[\[m]{} \_[a\]]{}q\_b\^m. \[udsdu\]
Thermal state in a Robertson-Walker cosmology
=============================================
Let us now specialize to a Robertson-Walker (RW) spacetime, in the semiclassical framework where the metric and $u^a$ are treated as classical fields and the scalar field is a quantum field (which therefore has a well-defined thermal equilibrium state). In the field equations for the metric and for $u^a$ we take the expectation value of the $\phi$-terms. Assuming $u^a$ shares the RW symmetry it must be the cosmological rest frame. The tensor $F_{ab}$ then vanishes, so $\l$ is just determined by the matter term in (\[l\]). (If there are further terms involving $u^a$ in the action then there are additional contributions to $\l$.)
With RW symmetry we have $\nab_a u_b = H q_{ab}$, where $H=\dot{a}/a$ is the usual Hubble “constant". Iterating this identity we find $\nabla_a q_{mn} = H(q_{am}\, u_n + q_{an}\, u_m)$, from which it follows that the contraction (\[udsdu\]) is given by u\^a= 6k\_0\^[-2]{} H D\^2, \[udsduRW\] where $\dot{\phi}=u^m\nab_m\phi$.
Suppose now that the scalar field is well approximated by an adiabatically evolving thermal state. This would be the case if (i) there are interactions that produce an equilibration rate which is large compared to the expansion rate, and (ii) the thermal frequency is also large compared to the expansion rate. In this case the expectation value of (the Hermitian part of) the operator $\dot{\phi}\, D^2\phi$ vanishes, since the operator is odd under time reversal while a thermal state is invariant. Using (\[l\]) and (\[udsduRW\]) this implies that, in the minimal model, $\la \l\ra=0$. (If other terms are included in the action for $u^a$ then $\la\l\ra$ would not vanish, although it would still not receive contributions from this matter field.) Therefore in this model $\la T^{(u)}_{ab}\ra=0$, so the only contribution to the cosmological stress tensor comes from the scalar field.
Thermal equation of state
-------------------------
Consider now the expectation value $\la T^{(\phi)}_{ab}\ra$ of the stress tensor (\[tdisp\])—or more precisely of its Hermitian part—in a thermal state. Note first that time reversal invariance of the thermal state requires an even number of time derivatives of $\phi$, and spatial isotropy requires an even number of spatial derivatives of $\phi$, in order for the thermal expectation value not to vanish. Thus terms with an odd number of derivatives of $\phi$ do not contribute. Let us call this the “odd derivative rule". We can use this rule to see that the expectation value $\la{\cal L}_\phi\ra$ vanishes. Integrating by parts, ${\cal L}_\phi$ can be expressed as a term that vanishes since $\phi$ satisfies its equation of motion, plus the total derivative of an expression involving only terms with an odd number of derivatives of $\phi$. The total derivative can be taken out of the expectation value, hence that term vanishes.
The part of (\[tdisp\]) multipled by $k_0^{-2}$ has three terms. The expectation value of the first term has a single time derivative, hence vanishes by time reversal symmetry. The expectation value of the third term is the gradient of an expression with three spatial derivatives which vanishes by the odd derivative rule. The second term can be integrated by parts, and the resulting total derivative piece has vanishing expectation value by the odd derivative rule, which leaves only $2q_{(a}{}^m\la D^2\phi\,\nab_m\nab_{b)}\phi\ra$. In a thermal state this must have the form $Au_au_b + Bq_{ab}$, where $u^a$ is the rest frame defined by the thermal bath. To find $A$ we contract with $u^au^b$, which yields $A=0$ due to the factor $q_{a}{}^m$. To find $B$ we contract with $q^{ab}$ and divide by 3, hence this term contributes $-\frac{2}{3}\la D^2\phi\, D^2\phi\ra$. The expectation value of the stress tensor thus takes the form T\_[ab]{}= u\_a u\_b + P q\_[ab]{}, with energy density $\rho$ and pressure $P$ given by &=& \^2 \[rho\]\
P &=& (D)\^2 -k\_0\^[-2]{} (D\^2)\^2.\[P\]
To evaluate the density and pressure we expand the field in Fourier components using the dispersion relation (\[dr\]) and sum the contributions from the modes, weighting each by the thermal expectation value of the number operator $e^{\o(\vec{k})/T} -1$, which yields = , \[rint\] P= . \[pint\] For low temperatures $T\ll k_0$, only modes with $k\ll k_0$ contribute significantly, hence we recover the standard result for massless radiation, $P=\frac{1}{3}\rho$, and the energy density scales as $\rho\propto T^4$.
Before looking at the exact temperature dependence, let us consider the high temperature limit. The nature of this limit depends on whether the dispersion is sub- or super-luminal. In the superluminal case $k_0^2<0$, we can sensibly use the dispersion relation (\[dr\]) out to arbitrarily large $k$, so in the high temperature limit only wave vectors $|\vec{k}|\gg |k_0|$ are relevant. For such wave vectors we have $\o(\vec{k})^2\simeq |\vec{k}|^4/|k_0^2|$, hence the energy density scales as $\rho\propto T^{5/2}k_0^{3/2}$, and $P\simeq \frac{2}{3}\rho$. This should be taken only as a qualitative indication of what might occur, since one would expect that for $|\vec{k}|\gsim k_0$ further terms in the $k$-expansion of the dispersion relation become important.
In the subluminal case $k_0^2>0$ we must impose a cutoff so as not to enter the unphysical region where the dispersion relation would yield $\o(\vec{k})^2<0$. We choose to impose the cutoff at $|\vec{k}|=k_0/\sqrt{2}$, where $\o(\vec{k})$ attains its maximum value. This is similar to a lattice cutoff for which, in one dimension, the dispersion relation is $\o(k)=(2/\d)\sin k\d/2$, and for which $|k|\le \pi/\d$ exhausts all the independent modes. With this cutoff in place, $\o(\vec{k})/T\ll1$ for all modes in the high temperature limit, hence the expectation value of the number operator tends toward $T/\o(\vec{k})$. Thus the energy density scales as $\rho\propto T k_0^3$ in this limit. As for the equation of state, the ratio of the integrals yields $P/\rho\simeq 0.174$.
The interpolation between the low and high temperature limits can be determined by a numerical calculation of the integrals. In Fig. \[eos\] we plot $P/\rho$ vs. $\log_{10}(T/k_0)$. It is seen that the equation of state smoothly connects the low and high temperature limits. For the subluminal case most of the interpolation takes place over the range of temperatures $10^{-1.5}<T/k_0<10^{-0.5}$, while in the superluminal case most of the interpolation takes place over the somewhat higher range $10^{-1}<T/k_0<10$.
Cosmological implications
-------------------------
The modified equation of state derived here can be used in a cosmological model where thermal matter with non-Lorentz-invariant dispersion acts as a source of gravity. The Bianchi identity implies the energy-momentum tensor is divergence-free, which for a perfect fluid implies $\dot{\r}=-3H(\r+p)$. Together with the equation of state this determines the evolution of the temperature. In the superluminal case local energy-momentum conservation is also independently implied by the matter field equations as usual, however in the subluminal case the dynamics is not self-contained because of the high wave vector cutoff. If the cutoff is imposed at a fixed proper wave vector, new modes are added as the universe expands, and the field dynamics does not specify into which state these modes are born. The assumptions of a thermal stress tensor and of energy-momentum conservation require that, whatever their birth state, they rapidly equilibrate with the rest of the modes. This is reasonable under the assumption that the system is in the adiabatic regime.
It is interesting to ask whether high frequency dispersion could have any impact on the cosmological problems that led to the invention of the inflationary scenario. In particular, could it provide a solution of these problems not requiring inflation? The modification of the equation of state caused by the dispersion would affect the quantitative details of the evolution of the scale factor at times when the typical wavevectors are of order $k_0$ or greater, which is presumably only in the very early universe. However, as we have seen, the equation of state only changes from $P=\r/3$ to $P=(2/3)\r$ in the superluminal case and to $P=0.17\r$ in the subluminal case. Neither are significant enough to qualitatively change the dynamics or horizon size.
Superluminal equilibration and the horizon problem
--------------------------------------------------
In the superluminal case, the fact that influences travel faster than light at wavevectors $k\gsim k_0$ opens up the possibility of solving the horizon problem via superluminal equilibration. The coordinate distance covered by a wavepacket with proper group velocity $v_g$ is $\Delta x=\int v_g dt/a$. For a dispersion relation that goes as $\o\sim k^n$ at large wave vectors the group velocity goes as $v_g\sim k^{n-1}$. It is easy to show that the typical wavevector at the peak of a thermal distribution scales as $a^{-1}$ (as long as the dispersion relation is homogeneous), hence the typical group velocity scales as $a^{1-n}$, so we have $\Delta x\sim\int dt/a^n$. If this diverges then the horizon problem is solved assuming the framework of this model.
With the above dispersion relation, the equation of state is $P=(n/3)\r$, which according to the Einstein equation yields the evolution of the scale factor $a(t) \propto t^{2/(n+3)}$. Thus $\Delta x\sim\int dt\, t^{-2n/(n+3)}$, which diverges at the lower limit for any $n\ge3$.
Unfortunately it does not make much sense to view this as a solution to the horizon problem, because most of the $\Delta x$ is traversed during a regime in which the typical wavevector and energy density are so much larger than the Planck scale that we have no reason to trust the semiclassical model at all. In particular, we have checked that in order for $\Delta x$ to surpass the horizon size, the evolution must be extrapolated all the way back to a time at which the typical wavevector is roughly $k_0$ times $[10^{57}(k_0/k_{Planck})^2]^{1/(n-3)}$. Presuming that $k_0$ is within a few orders of magnitude of the Planck scale, this typical wave vector exceeds the Planck scale unless $n\gsim50$. Since we have no theory that would determine the maximum exponent $n$ appearing in a superluminal dispersion relation, it would seem artificial at this stage to adjust $n$ in order to achieve a sub-Planckian resolution of the horizon problem.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by the National Science Foundation under grant No. PHY98-00967.
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T. Jacobson and D. Mattingly, “Gravity with a dynamical preferred frame", gr-qc/0007031.
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J. C. Niemeyer, “Inflation with a high frequency cutoff,” astro-ph/0005533.
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[^1]: jacobson@physics.umd.edu
[^2]: davemm@physics.umd.edu
|
=1
Introduction
============
The “bispectral problem” seeks to identify linear operators $L$ acting on functions of the variable $x$ and $\Lambda$ acting on functions of the variable $z$ such that there exists an eigenfunction $\psi(x,z)$ satisfying the equations $$\begin{gathered}
L\psi=p(z)\psi\qquad\mbox{and}\qquad \Lambda\psi=\pi(x)\psi.\end{gathered}$$ In other words, the components of the *bispectral triple* $(L,\Lambda,\psi(x,z))$ satisfy two different eigenvalue equations, but with the roles of the spacial and spectral variables exchanged.
The search for bispectral triples was originally formulated and investigated by Duistermaat and Grünbaum [@DG] in a paper which completely resolved the question in the special case in which the operators were scalar differential operators with one being a Schrödinger operator. Since then, the bispectrality of many different sorts of operators have been considered and many connections to different areas of math and physics have also been discovered. (See [@BispBook] and the articles referenced therein.)
The present paper will be considering a type of bispectrality in which both the operators and eigenvalues behave differently when acting from the left than from the right. It is necessary to introduce some notation and reorder the terms in the eigenvalue equations in order to properly describe the main results.
Throughout the paper, $M$ and $N$ should be considered to be fixed (but arbitrary) natural numbers. In addition, $H$ is a fixed (but arbitrary) invertible constant $N\times N$ matrix. Let $\Delta_{n,\lambda}$ denote the linear functional acting from the right on functions of $z$ by differentiating $n$ times and evaluating at $z=\lambda$: $$\begin{gathered}
(f(z))\Delta_{n,\lambda}=f^{(n)}(\lambda).
\end{gathered}$$ The set of linear combinations of these finitely-supported distributions with coefficients from $\CN$ will be denoted $\Dist$: $$\begin{gathered}
\Dist=\left\{\sum_{j=1}^{m}\Delta_{n_j,\lambda_j}C_j\colon
m,(n_j+1)\in\N,\ \lambda_j\in\C,\ C_j\in\CN\right\}. $$ Finally, let $\bdelta\subset\Dist$ be an $MN$-dimensional space of distributions.
The goal of this paper is to produce from the selection of $H$ and $\bdelta$ a $(L,\Lambda,\psi)$ such that
- $\psi(x,z)=(I+O(z^{-1}))e^{xzH}$ is an $N\times N$ matrix function of $x$ and $z$ with the specified asymptotics in $z$,
- $\psi(x,z)(zH)^M$ is holomorphic in $z$ and in the kernel of every element of $\bdelta$ (i.e., $\psi$ “satisfies the conditions”),
- $L\psi(x,z)=\psi(x,z) p(z)$ for the matrix differential operator in $x$ acting from the left and some matrix function $p(z)$,
- and $\psi(x,z)\Lambda=\pi(x)\psi(x,z)$ for a matrix differential-translation operator in $z$ acting from the right and some matrix function $\pi(x)$.
In the case $N=1$, this goal is already achieved constructively for *any* choice of distributions [@BispSol]. One interesting result of the present paper is that for $N>1$, lack of commutativity with $H$ may impose an obstacle to finding such a bispectral triple in that no such triple exists for certain choices of $\bdelta$. The three subsections below each offer some motivation for interest in the existence of such triples.
New bispectral triples
----------------------
One source of interest in the present paper is simply the fact that it generates examples of bispectral triples that have not previously been studied. The construction outlined below produces many bispectral triples $(L,\Lambda,\psi)$, where $L$ is a matrix coefficient differential operator in $x$, $\Lambda$ is an operator in $z$ which acts by both differentiation and *translation* in $z$, and $\psi(x,z)$ is a matrix function which asymptotically approaches $e^{xzH}$ for a chosen invertible matrix $H$.
This can be seen either as a matrix generalization of the paper [@BispSol] which considered exactly this sort of bispectrality in the scalar case or as a generalization of the matrix bispectrality in [@BGK; @BL; @WilsonNotes] to the a more general class of eigenfunctions and operators.
Bispectral duality of integrable particle systems
-------------------------------------------------
Among the applications found for bispectrality is its surprising role in the duality of integrable particle systems. Two integrable Hamiltonian systems are said to be “dual” in the sense of Ruijsenaars if the action-angle maps linearizing one system is simply the inverse of the action-angle map of the other [@Ruijsenaars]. When the Hamiltonians of the systems are quantized, the Hamiltonian operators themselves share a common eigenfunction and form a bispectral triple [@FGNR].
Moreover, the duality of the classical particle systems can also be manifested through bispectrality in that the dynamics of the two operators in a bispectral triple under some integrable hierarchy can be seen to display the particle motion of the two dual systems respectively. This classical bispectral duality was observed first in the case of the self-duality of the Calogero–Moser system [@cmbis; @Wilson2]. In [@BispSol], it was conjectured that certain bispectral triples involving scalar operators that translate in $z$ would similarly be related to the duality of the rational Ruijsenaars–Schneider and hyperbolic Calogero–Moser particle systems. This was later confirmed by Haine [@Haine].
The spin generalization of the Calogero–Moser system similarly exhibits classical bispectral duality [@BGK; @WilsonNotes]. Achieving this result essentially involved generalizing the scalar case [@cmbis; @Wilson2] to bispectrality for matrix coefficient differential operators. It is hoped that the construction presented in this paper which generalize that in [@BispSol] will similarly find application to classical bispectral duality in some future matrix generalization of the results in [@Haine].
Non-commutative bispectral Darboux transformations
--------------------------------------------------
In the original context of operators on scalar functions, the eigenvalue equations defining bispectrality were originally written with all operators and eigenvalues acting from the left. However, research into non-commutative versions of the discrete-continuous version of the bispectral problem in non-commutative contexts found it necessary to have operators in different variables acting from opposite sides in order to ensure that they commute [@matrix1; @matrix3; @GPT1; @GPT2; @GPT3; @GPT4] (cf. [@Duran]). Continuous-continuous bispectrality in the non-commutative context is also a subject of interest [@matrix2; @matrix3b; @matrix4; @matrix5]. As in the discrete case, it was found that the generalizing the results from the scalar case to the matrix case required letting the operators act from opposite sites [@BGK; @BL; @WilsonNotes]. Building on this observation, the present paper seeks to further consider the influence of non-commutativity on constructions and results already known for scalar bispectral triples.
In this regard, the wave functions of the $N$-component KP hierarchy are of interest since they asymptotically look like matrices of the form $e^{xzH}$, where $H$ is an $N\times N$ matrix [@BtK; @DJKM]. Consequently, unlike the scalar case or the case $H=I$ considered in [@BGK; @WilsonNotes], the vacuum eigenfunction itself may not commute with the coefficients of the operators. In fact, for the purpose of more fully investigating the consequences of non-commutativity for bispectral Darboux transformations, this paper will go beyond the standard formalism for the $N$-component KP hierarchy by considering the case in which $H$ is not even diagonalizable and therefore has a centralizer with more interesting structure. Furthermore, following the suggestion of Grünbaum [@matrix3b], the present paper will consider the case in which both of the *eigenvalues* are matrix-valued.
By generalizing the construction from [@BispSol] to the context in which the vacuum eigenfunction, eigenvalues, and operator coefficients all generally fail to commute with each other, this investigation has identified some results that are surprisingly different than in the commutative case. For example, it is shown that in this context there exist rational Darboux transformations that do *not* preserve the bispectrality of the eigenfunction (see Section \[nogo\]) and that bispectral triples do not always exhibit ad-nilpotency (see Remark \[rem:ad-examp\]). These will be summarized in the last section of the paper.
Additional notation
===================
Distributions and matrices
--------------------------
Let $M$, $N$, $H$ and $\bdelta$ be as in the Introduction. The set of constant $N$-component column vectors will be denoted by $\CN$ and $\MN$ is the set of $N\times N$ constant matrices. Associated to the selection of $H$ one has $$\begin{gathered}
\Cent= \{Q\in\MN\colon [Q,H]=0 \},\end{gathered}$$ the centralizer of $H$ in $\MN$.
Let $\{\delta_1,\ldots,\delta_{MN}\}$ be a basis for $\bdelta\subset\Dist$. Unlike the selection of $N$, $M$ and $H$ which were indeed entirely arbitrary, two additional assumptions regarding the choice of $\bdelta$ will have to be made so that a bispectral triple may be produced from it. However, rather than making those assumptions here at the start, the additional assumptions will be introduced only when they become necessary. This should help to clarify which results are independent of and which rely on the assumptions.
Nearly all of the objects and constructions below depend on the choice of the number $N$, the matrix $H$ and the distributions $\bdelta$ that have been selected and fixed above, but to avoid complicating the notation the dependence on these selections will not be written explicitly. (For instance, the matrix $\Phi$ in could be called $\Phi_{N,M,H,\bdelta}$ because it does depend on these selections, but it will simply be called $\Phi$.)
Operators and eigenvalues
-------------------------
The operators in $x$ to be considered in this paper will all be differential operators in the variable $x$ (also sometimes called $t_1$) which are polynomials in $\partial=\frac{\partial}{\partial x}$ having coefficients that are $N\times N$ matrix functions of $x$. The operators in $z$ will be written in terms of $\partial_z=\frac{\partial}{\partial z}$ or the translation operator $\Trans{\alpha}\colon f(z)\mapsto f(z+\alpha)$. More generally, they will be polynomials in these having coefficients that are $N\times N$ matrix rational functions of $z$.
Because operator coefficients and eigenvalues will be matrix-valued, the action of an operator will depend on whether its coefficients multiply from the right or the left. It also matters whether the eigenvalue acts by multiplication on the right or the left of the eigenfunction. This paper will adopt the convention that all operators in $x$ that are independent of $z$ (whether they are differential operators or simply functions acting by multiplication) act from the left and that all operators in $z$ that are independent of $x$ (including functions, translation operators, finitely-supported distributions and differential operators) act from the right. The action of an operator in $z$ will be denoted simply by writing the operator to the right of the function it is acting on. So, for instance, the function $e^{xzH}$ satisfies the eigenvalue equations $$\begin{gathered}
\partial e^{xzH}=e^{xzH}
(zH)\qquad\text{and}\qquad e^{xzH}\partial_z=(xH)e^{xzH}.\end{gathered}$$
The decision to have operators in $x$ and $z$ acting from different sides is not merely a matter of notation. The need for such an assumption for the differential operators in $x$ and $z$ respectively was already noted in prior work on matrix bispectrality [@BGK; @BL; @matrix3; @GPT1; @GPT2; @GPT3; @GPT4; @WilsonNotes]. The present work extends this convention to the eigenvalues and finitely-supported distributions as well, and does so because the theorems fail to be true otherwise.
\[rem:eigenchange\] Note that one needs to be cautious about applying intuition about eigenfunctions in a commutative setting without considering how non-commutativity may affect it. For example, although a non-zero multiple of an eigenfunction in the commutative setting always remains an eigenfunction with the same eigenvalue, here there are two other possibilities. Suppose $L\psi(x,z)=\psi(x,z)p(z)$, so that $\psi$ is an eigenfunction for $L$ with eigenvalue $p$, and that $g$ is an invertible constant matrix. Then $g\psi$ may not be an eigenfunction for $L$ if $[L,g]\not=0$ and more surprisingly even though $\psi g$ is an eigenfunction for $L$, the corresponding eigenvalue changes to $g^{-1}pg$.
Dual construction for $\boldsymbol{N}$-component KP
===================================================
The purpose of this section is to produce a matrix coefficient pseudo-differential operator satisfying the Lax equations of the multicomponent KP hierarchy introduced by Date–Jimbo–Kashiwara–Miwa [@DJKM] and mostly follows the approach of Segal–Wilson [@SW]. The proof methods utilized here are rather standard in the field of integrable systems. However, one of the main points of this paper is that some of the novel features of this situation pose unexpected obstacles to the standard methods used to study bispectrality. So, although it is not surprising that the operators produced in this way satisfy these Lax equations, the proofs are presented with sufficient detail to ensure that they work despite the non-diagonalizability of $H$ and the fact that the distributions here are acting from the right.
The $\boldsymbol{N}$-component Sato Grassmannian
------------------------------------------------
Let $\HN$ denote the Hilbert space of square-integrable vector-valued functions $S^{1}\to(\C^N)^{\top}$ where $S^{1}\subset\C$ is the unit circle $|z|=1$ and $(\C^N)^{\top}$ is the set of complex valued row[^1] $N$-vectors. Denote by $e_i$ for $0\leq i\leq N-1$ the $1\times N$ matrix which has the value $1$ in column $i+1$ and zero in the others. This extends to a basis $\{e_i\colon i\in\Z\}$ of $\HN$ for which $e_i=z^ae_b$ when $i=aN+b$ for $0\leq b\leq N-1$. The Hilbert space has the decomposition $$\begin{gathered}
\HN=\HN_+\oplus \HN_-,\end{gathered}$$ where $\HN_+$ is the Hilbert closure of the subspace spanned by $e_i$ for $0\leq i$ and $\HN_-$ is the Hilbert closure of the subspace spanned by $e_i$ for $i<0$.
The Grassmannian ${\rm Gr}^{(N)}$ is set of all closed subspaces $V\subset \HN$ such that the orthogonal projections $V\to
\HN_-$ is a compact operator and such that the orthogonal projection $V\to \HN_+$ is Fredholm of index zero [@BtK; @Sato; @SW].
The notion of $N$-component KP hierarchy to be considered in this paper is compatible with, but somewhat different from that addressed by previous authors as the following remark explains.
For the $N$-component KP hierarchy, the construction of solutions from a point in the Grassmannian usually involves a collection of diagonal constant matrices $H_{\alpha}$ ($1\leq \alpha\leq N$) such that powers of $zH_{\alpha}$ infinitesimally generate the continuous flows and $z$-dependent matrices $T_{\beta}$ ($1\leq \beta\leq N-1$) that generate discrete flows (sometimes called “Schlesinger transformations”) of the hierarchy [@BtK; @DJKM]. In the present paper, however, only the continuous flows generated infinitesimally by powers of $z$ times the (not necessarily diagonal) matrix $H$ selected earlier will be considered.
A point of $\boldsymbol{{\rm Gr}^{(N)}}$ associated to the selection of distributions
-------------------------------------------------------------------------------------
As usual, one associates a subspace of $\HN$ to the choice of $\bdelta$ by taking its dual in $\HN_+$ and multiplying on the right by the inverse of a matrix polynomial in $z$ whose degree depends on the dimension of $\bdelta$ (cf. [@cmbis; @nKdV2KP; @SW; @Wilson] where the analogous procedure involved dividing by a scalar polynomial):
\[def:Wd\] Let $\Wd\subset \HN$ be defined by $$\begin{gathered}
\Wd=\big\{p(z)(zH)^{-M}\colon p(z)\in \HN_+,\ (p)\delta=0\ \text{for} \ \delta\in\bdelta\big\}.\end{gathered}$$
$\Wd\in {\rm Gr}^{(N)}$.
The image of $\Wd$ under the projection map onto $\HN_-$ is contained in the finite-dimensional subspace spanned by the basis elements $e_i$ for $-MN\leq i\leq -1$. This is sufficient to conclude that the projection map is compact. The map $w\mapsto w(zH)^M$ from $W$ to $\HN_+$ has Fredholm index $MN$ because it has no kernel and the image is the common solution set of $MN$-linearly independent conditions. The map from $\HN_+$ which first right multiplies by $(zH)^{-M}$ and then projects onto $\HN_+$ has index $-MN$ since its kernel is spanned by the basis vectors $e_i$ with $0\leq i\leq MN-1$ but the image is all of $\HN_+$. The composition of these maps is the projection from $\Wd$ to $\HN_+$ and so its index is the sum of the indices which is zero.
$\boldsymbol{N}$-component KP wave function
-------------------------------------------
Let $\psi_0=\exp\big(\sum\limits_{i=1}^{\infty} t_iz^iH^i\big)$ where $\t=(t_1,t_2,\ldots)$ are the continuous KP time variables, with the variables $t_1$ and $x$ considered to be identical. Let $\phi_i(\t)=(\psi_0)\delta_i$ ($1\leq i\leq MN$) be the $\CN$-valued functions obtained by applying each element of the basis of $\bdelta$ to $\psi_0$. Combine them as blocks into the $N\times MN$ matrix $\bphi=(\phi_1\ \cdots\ \phi_{MN})$ and define the matrix $\Phi$ as the $MN\times MN$ block Wronskian matrix $$\begin{gathered}
\Phi(\t)=\left(\begin{matrix}\bphi\\ \dfrac{\partial}{\partial x}\bphi\\
\vdots\\ \dfrac{\partial^{M-1}}{\partial x^{M-1}}\bphi\end{matrix}\right)
= \left(\begin{matrix}\phi_1&\phi_2&\cdots&\phi_{MN}\\
\phi_1'&\phi_2'&\cdots&\phi_{MN}'\\ \vdots & \vdots&\ddots&\vdots&\\
\phi_1^{(M-1)}&\phi_2^{(M-1)}&\cdots&\phi_{MN}^{(M-1)}\end{matrix}\right
).\label{eqn:Phi}\end{gathered}$$
\[assumpA\] Henceforth, assume that $\bdelta$ was chosen so that the matrix $\Phi$ in is invertible for some values of $x=t_1$ (i.e., so that $\det\Phi\not\equiv0$).
The following remarks offer two different interpretations of the fact that Assumption \[assumpA\] is necessary here but not for the analogous result in the scalar case [@BispSol].
When $N=1$, the requirement that $\det(\Phi)\not=0$ is equivalent to the requirement that $\{\phi_1,\ldots,\phi_M\}$ is a linearly independent set of functions. Then, the independence of the basis of distributions would already ensure that Assumption \[assumpA\] is satisfied. However, when $N>1$ the Wronskian matrix $\Phi$ can be singular even if the functions $\phi_i$ are linearly independent as functions of $x$. (For example, consider the case $M=1$, $N=2$, $\phi_1=(1\ 1)^{\top}$, $\phi_2=(x\ x)^{\top}$).
The determinant of $\Phi$ can be interpreted as the determinant of the projection map from $\psi_0^{-1}(\t)\Wd$ to $\HN_+$. In other words, it is the $\tau$-function of $\Wd$. (The proof of this claim is essentially the same as the proof of Theorem 7.5 in [@nKdV2KP].) The $\tau$-function is non-zero when $\psi_0^{-1}(\t)\Wd$ is in the “big cell” of the Grassmannian. In the case $N=1$, the orbit of any point $W$ in the Grassmannian under the action of $\psi_0^{-1}$ intersects the big cell [@SW]. In contrast, for the $N$-component KP hierarchy it is known that there are points in the Grassmannian whose orbit under the continuous flows never intersect the big cell [@BtK].
Due to Assumption \[assumpA\], we may define the differential operator $K$ as $$K=\partial^MI-\left(\phi_1^{(M)}\ \cdots\
\phi_{MN}^{(M)}\right)\Phi^{-1}\left(\begin{matrix} I\\ \partial I\\
\vdots\\ \partial^{M-1}I\end{matrix}\right).\label{eqn:K}$$
\[Klemma\]
1. The operator $K$ defined in is the unique monic $N\times N$ differential operator of order $M$ such that[^2] $K\bphi=0$.
2. If $L$ is any $N\times N$ matrix differential operator satisfying $L\bphi=0$ then $L=Q\circ K$ for some differential operator $Q$.
It is easy to check that $K\bphi=0$. Alternatively, Lemma \[Klemma\](a) follows from results of Etingof, Gelfand and Retakh on quasi-determinants [@EGR]. However, Lemma \[Klemma\](b) is apparently a new result. Although the lemma was originally formulated for this paper[^3], a self-contained proof is being published separately as [@MDO-Note].
The theory of quasi-determinants is not explicitly being used here but the operator $K$ defined in could alternatively be computed as a quasi-determinant of a Wronskian matrix with $N\times N$ matrix entries [@EGR]. The method of quasi-determinants was applied to the bispectral problem for matrix coefficient operators in [@BL]. So, in this sense, the use of this operator $K$ here is a continuation of the approach adopted there.
Let $ \psi(\t,z)=K(\psi_0)(zH)^{-M}. $ This function will play an important role as the eigenfunction for the operators in $x$ and (given one additional assumption) $z$ to be introduced below.
The function $\psi$ defined above has the following properties:
- $\psi(\t,z)=(I+O(z^{-1}))\psi_0$ where $I$ is the $N\times N$ identity matrix,
- $\psi(\t,z)\in W$ for all $\t$ in the domain of $\psi$.
Consequently, $\psi=\psi_{\Wd}$ is the $N$-component KP wave function of the point $\Wd\in {\rm Gr}^{(N)}$.
Because $\partial(\psi_0)=\psi_0zH$, applying the monic differential operator $K=\partial^M+\cdots$ to $\psi_0$ produces a function of the form $K\psi_0=P(\t,z)\psi_0$ where $P$ is a polynomial of degree $M$ in $z$ with leading coefficient $H^M$. Then $K\psi_0H^{-M}z^{-M}=(I+O(z^{-1}))\psi_0$ as claimed. It remains to be shown that $\psi$ is an element of $\Wd$. By Definition \[def:Wd\] it is sufficient to note that for each $1\leq i\leq MN$, $\psi
z^MH^M=K\psi_0$ satisfies $$\begin{gathered}
(K\psi_0)\delta_i=K ((\psi_0)\delta_i )=K\phi_i=0.\tag*{\qed}\end{gathered}$$
Lax equations of the $\boldsymbol{N}$-component KP hierarchy
------------------------------------------------------------
Let $K^{-1}$ denote the unique multiplicative inverse of the monic differential operator $K$ in the ring of matrix-coefficient pseudo-differential operators and let $\L= K \circ \partial \circ K^{-1}$ be the pseudo-differential operator obtained by conjugating $\partial$ by $K$.
\[KPLax\] $\L$ satisfies the Lax equations $$\begin{gathered}
\frac{\partial}{\partial t_i}\L=[(\L^i)_+,\L]\end{gathered}$$ for each $i\in\N$, where $\big(\sum\limits_{i=-\infty}^n \alpha_i(\t)\partial^i\big)_+=\sum\limits_{i=0}^n
\alpha_i(\t)\partial^i$.
Let $\phi=(\psi_0)\delta$ for some $\delta\in\bdelta$. A key observation is that for each $i\in\N$: $$\begin{gathered}
\frac{\partial}{\partial t_i}\phi = \frac{\partial}{\partial t_i}
(\psi_0)\delta = \left(\frac{\partial}{\partial t_i}
\psi_0\right)\delta = \left((zH)^i \psi_0\right)\delta =
\left(\frac{\partial^i}{\partial x^i} \psi_0\right)\delta =
\partial^i (\psi_0)\delta=\frac{\partial^i}{\partial x^i}( \phi).\end{gathered}$$ Using the fact that $\phi$ satisfies these “dispersion relations” and the intertwining relationship $\L^i\circ K= K\circ
\partial^i$ we differentiate the identity $ K(\phi)=0$ (which follows from Lemma \[Klemma\]) by $t_i$ to get $$\begin{gathered}
0 =
K_{t_i}(\phi)+ K(\phi_{t_i}) = K_{t_i}(\phi)+ K(\partial^i \phi)
= K_{t_i}(\phi)+\L^i\circ K(\phi)\\
\hphantom{0} = K_{t_i}(\phi)+(\L^i)_-\circ
K(\phi)+(\L^i)_+\circ K(\phi).
\end{gathered}$$ Since both $K$ and $(\L^i)_+$ are ordinary differential operators (as “$+$” denotes the projection onto the subring of ordinary differential operators), $(\L^i)_+\circ K$ is an ordinary differential operator with a right factor of $K$. Therefore, $\phi$ is in its kernel and the last term in the sum above is zero. We may therefore conclude that $$\begin{gathered}
K_{t_i}(\phi)+(\L^i)_-\circ
K(\phi)=0.\label{eqstar}\end{gathered}$$ (Note that the second term in this sum need not be zero since $(\L^i)_-$ is not an ordinary differential operator.)
Using $\L^i\circ K = K\circ \partial^i$ we can split $\L^i$ into its positive and negative parts to get $$\begin{gathered}
(\L^i)_-\circ K = K\circ
\partial^i - (\L^i)_+\circ K.
\end{gathered}$$ Since the object on the right is just a difference of differential operators we know that $(\L^i)_-\circ K$ is a differential operator.
According to , $\Gamma(\bphi)=0$ where $\Gamma$ is the ordinary differential operator $$\begin{gathered}
\Gamma= K_{t_i}+(\L^i)_-\circ K.\end{gathered}$$ Then by Lemma \[Klemma\], there exists a differential operator $Q$ so that $\Gamma=Q\circ K$. However, $\Gamma$ has order strictly less than $M$ since the coefficient of the $M^{\rm th}$ order term of $ K$ is constant by construction and since multiplying by $(\L^i)_-$ will necessarily lower the order. This is only possible if $Q=0$ and $\Gamma$ is the zero operator. Hence, $K_{t_i} =-(\L^i)_-\circ K$. The Lax equation follows because $$\begin{gathered}
\L_{t_i} = K_{t_i}\circ \partial \circ
K^{-1}- K\circ \partial \circ K^{-1}\circ K_{t_i}\circ K^{-1}\\
\hphantom{\L_{t_i} }{} =
-(\L^i)_-\circ K\circ \partial \circ K^{-1}+ K\circ \partial \circ
K^{-1}\circ (\L^i)_-\circ K\circ K^{-1}\\
\hphantom{\L_{t_i} }{}=
[\L,(\L^i)_-]=[(\L^i)_+,\L]. \tag*{{}}\end{gathered}$$
Operators in $\boldsymbol{x}$ having $\boldsymbol{\psi}$ as eigenfunction
=========================================================================
From this point onwards, the goal is to determine whether the wave function $\psi(\t,z)$ is an eigenfunction for an operator in $x=t_1$ with $z$-dependent eigenvalue and vice versa. The higher indexed time variables will only complicate the notation. So, it will henceforth be assumed that $t_i=0$ for $i\geq 2$. Then, $\bphi$ and the coefficients of $K$ can be considered to be functions of only the variable $x$, and $\psi_0(x,z)=e^{xzH}$ and $\psi(x,z)$ (sometimes called the “stationary wave function”) are functions of $x$ and $z$. Unlike the numbered assumptions, this one is made for notational simplicity only. Dependence on the KP time variables can be added to the objects to be discussed below so that all claims remain valid.
A distribution $\delta\in\Dist$ can be composed with $p(z)\in\MN[z]$ by defining $p(z)\circ \delta$ to be the distribution whose value when applied to $f(z)$ is the same as that of $\delta$ applied to the product $f(z)p(z)$ for any $f(z)\in\MN[z]$. Associate to the choice $\bdelta$ of distributions the ring of polynomials *with coefficients in $\Cent$* which turn elements of $\bdelta$ into elements of $\bdelta$: $$\begin{gathered}
\A=\big\{p(z)\in\Cent[z]\colon p(z)\circ
\delta\in\bdelta\ \forall\, \delta\in\bdelta \big\}.\end{gathered}$$ In particular, for each $p\in\A$ and each basis element $\delta_i$ there exist numbers $c_j$ such that $p\circ\delta_i=\sum\limits_{i=1}^{MN} \delta_j c_j$.
As in the scalar case, elements of $\A$ are stabilizers of the point in the Grassmannian: if $p\in \A$ then $\Wd p\subset \Wd$. The main significance of the ring $\A$ is that there is a differential operator $L_p$ of positive order satisfying $L_p\psi=\psi p(z)$ for every non-constant $p\in \A$. Interestingly, unlike the scalar case, it will be shown that the question of which *constant* matrices are in $\A$ is also of interest in that whether $\psi$ is part of a bispectral triple is related to whether $H$ is an element of $\A$. Before those facts are established, however, the following definitions and results show that $\A$ contains polynomials of every sufficiently high degree.
Let $\Supp\subset\C$ denote the support of the distributions in $\bdelta$. That is, $\lambda\in\Supp$ if an only if $\Delta_{n,\lambda}$ appears with non-zero coefficient for some $n$ in at least one $\delta_i$. For each $\lambda\in\Supp$ let $m_{\lambda}$ denote the *largest* number $n$ such that $\Delta_{n,\lambda}$ appears with non-zero coefficient in at least one element of $\bdelta$. The scalar polynomial $$\begin{gathered}
p_0(z)=\prod_{\lambda\in\Supp}(z-\lambda)^{m_{\lambda}+1}\end{gathered}$$ will be used in the next lemma, in Definition \[def:L0\] and also in Theorem \[mainresult\] below.
For any $p\in\Cent[z]$, the product $p_0(z)p(z)$ is in $\A$. So, $p_0(z)\Cent[z]\subset\A.$
Let $p\in\Cent[z]$ and $\delta\in\bdelta$. Then applying the distribution $p_0p\circ\delta$ to any polynomial $q$ is equal to $(qp_0p)\delta$. The distribution $\delta$ will act by differentiating and evaluating at $z=\lambda$ for each $\lambda\in\Supp$. However, $p_0$ was constructed so that it has zeroes of sufficiently high multiplicity at each $\lambda$ to ensure that this will be equal to zero. Hence $p_0p\circ
\delta$ is the zero distribution, which trivially satisfies the criterion in the definition of $\A$.
\[bispinvdef\] For $p\in\Cent[z]$ where $p=\sum\limits_{i=0}^n C_i z^i$ for $C_i\in\Cent$ define $$\begin{gathered}
\bisp^{-1}(p)=\sum_{i=0}^n C_i H^{-i} \partial^i.\end{gathered}$$
For any $p\in\Cent[z]$, the constant coefficient differential operator $\bisp^{-1}(p)$ has $\psi_0=e^{xzH}$ as an eigenfunction with eigenvalue $p(z)$.
If $p$ is the polynomial with coefficients $C_i$ as in Definition \[bispinvdef\] then $$\begin{gathered}
\bisp^{-1}(p)\psi_0 = \left(\sum_{i=0}^n C_i
H^{-i}\partial^i\right)\psi_0 =\sum_{i=0}^n C_i H^{-i}\partial^i\psi_0
= \sum_{i=0}^n C_i H^{-i}\psi_0(zH)^i\\
\hphantom{\bisp^{-1}(p)\psi_0}{} =\psi_0\left(\sum_{i=0}^n C_i
H^{-i}(zH)^i\right) = \psi_0\left(\sum_{i=0}^n C_iz^i\right)
=\psi_0p(z). \tag*{{}}\end{gathered}$$
\[claimLp\] For any $p\in \A$ there is an $N\times N$ ordinary differential operator $L_p$ in $x$ satisfying the intertwining relationship $$\begin{gathered}
L_p\circ K= K\circ \bisp^{-1}(p)
\end{gathered}$$ and the eigenvalue equation $L_p\psi(x,z)=\psi p(z)$.
Let $p\in \A$. Observe that $$\begin{gathered}
K\circ
\bisp^{-1}(p)(\phi_i) = K\circ \bisp^{-1}(p)((\psi_0)\delta_i) =K\big(\big(
\bisp^{-1}(p)\psi_0\big)\delta_i\big) = K(( \psi_0)p(z)\circ\delta_i)\\
\hphantom{K\circ\bisp^{-1}(p)(\phi_i)}{}
=K\left(\sum_{j=1}^{MN} (\psi_0)\delta_jc_j \right) = \sum_{j=1}^{MN}
K(\phi_j)c_j =\sum_{j=1}^{MN} 0\times c_j =0.
\end{gathered}$$ But, that means that each function $\phi_i$ is in the kernel of $K\circ
\bisp^{-1}(p)$ and hence by Lemma \[Klemma\] there is a differential operator $L_p$ such that $ K\circ \bisp^{-1}(p)=L_p\circ K$. This establishes the intertwining relationship.
Applying $L_p$ to $\psi= K \psi_0(zH)^{-M}$ one finds $$\begin{gathered}
L_p\psi = L_p\big( K\psi_0(zH)^{-M}\big) = (L_p\circ K)\psi_0(zH)^{-M} = \big(
K\circ \bisp^{-1}(p)\big)\psi_0(zH)^{-M} \\
\hphantom{L_p\psi}{} = K \psi_0 p(z)(zH)^{-M} = K
\psi_0 (zH)^{-M}p(z)=\psi p(z). \tag*{{}}\end{gathered}$$
Operators in $\boldsymbol{z}$ having $\boldsymbol{\psi}$ as eigenfunction
=========================================================================
For any $\alpha\in\C$ let the translation operator $\Trans{\alpha}$ act on functions of $z$ according to the definition $$\begin{gathered}
(f(z))\Trans{\alpha}=f(z+\alpha).\end{gathered}$$ Further let $\Trans{\alpha}^H=
\sum_{i,j} \Trans{\alpha \gamma_i}(\alpha \partial_z)^jC_{ij}$ where the matrices $C_{ij}$ and constants $\gamma_i$ are defined by the formula $$\begin{gathered}
\exp\big({xH^{-1}}\big)=\sum_{i,j}
\exp({\gamma_i x})x^j C_{ij}\qquad
\text{(with $\gamma_i\not=\gamma_{i'}$ if $i\not=i'$).}
\label{eqn:gammadef}\end{gathered}$$
The differential-translation operator $\Trans{\alpha}^H$ has $\psi_0=e^{xzH}$ as an eigenfunction with eigenvalue $e^{\alpha x}$: $$\begin{gathered}
\psi_0\Trans{\alpha}^H=e^{\alpha x}\psi_0.\end{gathered}$$
By definition, $$\begin{gathered}
e^{xzH}\Trans{\alpha}^H=e^{xzH}\sum_{i,j} T_{\alpha \gamma_i}(\alpha
\partial_z^j)C_{ij}=e^{xzH}\sum_{i,j} e^{x\alpha \gamma_i H}(\alpha x
H)^jC_{ij}.\end{gathered}$$ Then the claim follows if it can be shown that the sum in the last expression is equal to $e^{\alpha x}$.
However, holds not only for any scalar $x$ but also when it is replaced by some matrix which commutes with the matrices $H^{-1}$ and $C_{ij}$ which appear in it. In particular, since $\alpha x H$ commutes with $H^{-1}$, its commutator with the expression on the left side of equation is zero. From this one can determine that its commutator with each coefficient matrix $C_{ij}$ on the right is zero as well. However, replacing $x$ with $\alpha x H$ yields the formula $$\begin{gathered}
\exp(\alpha x)=\sum_{i,j}
\exp({\gamma_i \alpha x H})(\alpha x H)^j C_{ij}\end{gathered}$$ as needed.
The goal of this section is to produce operators in $z$ which are matrix differential-translation operators in $z$ having rational coefficients that share the eigenfunction $\psi$ with the differential operators $L_p$ in $x$ constructed in the previous section. In order for the construction to work, an additional assumption is required:
\[assumpB\] Henceforth, it is assumed that $H\in\A$. Equivalently, assume that for each $1\leq i\leq MN$ there exist numbers $c_j$ such that $$\begin{gathered}
H\delta_i=\sum_{j=1}^{MN}c_j \delta_j.\end{gathered}$$
The anti-isomorphism {#sec:antiiso}
--------------------
This section will introduce an anti-isomorphism between rings of operators in $x$ and $z$ respectively such that an operator and its image have the same action on $\psi_0$. The use of such an anti-isomorphism as a method for studying bispectrality was pioneered in special cases in [@Wilson] and [@KR] and extended to a very general commutative context in [@BHY]. (Additionally, three months after the present paper was posted and submitted, a new preprint by two of the same authors as [@BHY] appeared which seeks to further generalize those results to the non-commutative context [@GHY].)
The two rings of operators of interest to this construction are $$\begin{gathered}
\Weyl=\bigoplus_{\alpha\in\C}e^{\alpha
x}\Cent[x,\partial] \qquad \hbox{and} \qquad
\WeylFlat=\bigoplus_{\alpha\in\C}\Trans{\alpha}^H\Cent[z,\partial] .\end{gathered}$$
Note that operators from both rings involve polynomial coefficient matrix differential operators which commute with the constant matrix $H$, but elements of $\Weyl$ may also include a finite number of factors of the form $e^{\alpha x}$ while elements of $\WeylFlat$ may similarly include factors of $\Trans{\alpha}^H$ for a finite number of complex numbers $\alpha$.
Let $\bisp\colon \Weyl\to\WeylFlat$ be defined by $$\begin{gathered}
\bisp\left(\sum_{i=0}^l\sum_{j=0}^m\sum_{k=1}^n
C_{ijk}e^{\alpha_kx}x^i\partial^j\right)
=\sum_{i=0}^l\sum_{j=0}^m\sum_{k=1}^n\partial_z^i \Trans{\alpha_k}^H
z^jC_{ijk}H^{j-i},
\label{eqn:bisp}\end{gathered}$$ where $C_{ijk}\in\Cent$ are the coefficient matrices.
\[lem:bisp\] For any $L_0\in\Weyl$, the operators $L_0$ and $\bisp(L_0)$ have the same action on $\psi_0=e^{xzH}$: $$\begin{gathered}
L_0\psi_0=\psi_0\bisp(L_0).\end{gathered}$$ Moreover, $\bisp$ is an anti-isomorphism.
Since $$\begin{gathered}
\sum_{i=0}^m\sum_{j=0}^n \sum_k C_{ij}x^ie^{\alpha_k
x}\partial^j(\psi_0) = \sum_{i=0}^m\sum_{j=0}^n \sum_k
C_{ij}e^{\alpha_k x}x^i(zH)^j(\psi_0)\\
\qquad{}
= \sum_{i=0}^m\sum_{j=0}^n\sum_k C_{ij}e^{\alpha_k
x}x^i(\psi_0)(zH)^j
= \sum_{i=0}^m\sum_{j=0}^n\sum_k
C_{ij}e^{\alpha_k x}(\psi_0)\big(H^{-1}\partial_z\big)^i(zH)^j\\
\qquad{}
= \sum_{i=0}^m\sum_{j=0}^n\sum_k
C_{ij}(\psi_0)\Trans{\alpha_k}^H\big(H^{-1}\partial_z\big)^i(zH)^j
= \sum_{i=0}^m\sum_{j=0}^n \sum_k
(\psi_0)\big(H^{-1}\partial_z\big)^i\Trans{\alpha_k}^H(zH)^jC_{ij}\\
\qquad{}
= (\psi_0)\sum_{i=0}^m\sum_{j=0}^n\sum_k
\big(H^{-1}\partial_z\big)^i\Trans{\alpha_k}^H(zH)^jC_{ij}
= \psi_0\bisp\left(\sum_{i=0}^m\sum_{j=0}^n \sum_k C_{ij}x^ie^{\alpha_k
x}\partial^j\right)\end{gathered}$$ it follows that $L_0$ and $\bisp(L_0)$ have the same action on $\psi_0$.
The inverse map $\bisp^{-1}$ is found by simply moving the factor of $H^{j-i}$ to the other side of equation , which confirms that $\bisp$ is a bijection. If $K_1,K_2\in \Weyl$ we have $$\begin{gathered}
(K_2\circ K_1)\psi_0=K_2(K_1(\psi_0))=K_2(\psi_0\bisp(K_1))=(K_2(\psi_0))\bisp(K_1
)\\
\hphantom{(K_2\circ K_1)\psi_0}{}
=(\psi_0\bisp(K_2))\bisp(K_1)=\psi_0(\bisp(K_2)\circ
\bisp(K_1)).\end{gathered}$$ However, we also know that $(K_2\circ
K_1)\psi_0=\psi_0\bisp(K_2\circ K_1)$. Then, $\bisp(K_2\circ
K_1)-\bisp(K_2)\circ\bisp(K_1)$ has the $\psi_0$ in its kernel. The only operator in $\WeylFlat$ having $\psi_0$ in its kernel is the zero operator. Therefore, $$\begin{gathered}
\bisp(K_2\circ
K_1)=\bisp(K_2)\circ\bisp(K_1)\label{eqn:anti-iso}\end{gathered}$$ and the map is an anti-isomorphism.
\[rem:antiq\] The reader may be surprised to see “anti-isomorphism” being used to describe a map satisfying equation . Generally, a map having this property is called an *isomorphism*. There are two reasons this terminology is being used here. First, that is the terminology that was used for the analogous map in previous papers in the scalar setting [@BHY; @BispSol; @KR]. More importantly, the fact that the order of operators here is preserved is merely a consequence of the fact that the operators in $z$ act from the right while the operators in $x$ act from the left. Note that when $K_2\circ K_1$ acts on a function it is $K_1$ that acts first while when $\bisp(K_2)\circ \bisp(K_1)$ acts it is $\bisp(K_2)$ that acts first and it is in this sense that it is an *anti*-isomorphism. Nevertheless, it is interesting to note that in this context the map does preserve the order of operators in a product. It may be that had the operators in different variables been written as acting from opposite sides even in the commutative case from the beginning, the map would have been described as an isomorphism instead.
Eigenvalue equations for operators in $\boldsymbol{z}$
------------------------------------------------------
\[def:L0\] Let $L_0$ be the constant coefficient matrix differential operator $$\begin{gathered}
L_0=\bisp^{-1}(p_0(z)I).\end{gathered}$$
\[lem:L0kills\] $L_0\bphi=0$ and so $L_0=Q\circ K$ for some $Q$.
Because differentiation in $x$ and multiplication on the left commute with the application of the distributions $\delta_i$ we have $$\begin{gathered}
L_0\phi_i=L_0(\psi_0)\delta_i=(L_0\psi_0)\delta_i.\end{gathered}$$ By Lemma \[lem:bisp\], $L_0\psi_0=\psi_0p_0(z)$. The polynomial $p_0$ was chosen so that it has a zero of high enough multiplicity at each point in the support of $\delta_i$ to guarantee that $$\begin{gathered}
(\psi_0
p_0(z))\delta_i=0.\end{gathered}$$ It then follows from Lemma \[Klemma\] that $L_0=Q\circ K$.
In the remainder of this construction, $Q$ will denote the differential operator such that $L_0=Q\circ K$.
\[KcommuteswithH\] Given Assumption [\[assumpB\]]{}, the operators $L_0$, $K$ and $ Q$ commute with $H$.
Applying the operator $H^{-1}KH$ to $\phi_i$ we see that $$\begin{gathered}
H^{-1}KH\phi_i = H^{-1}KH\big(e^{xzH}\big)\delta_i=H^{-1}K\big(e^{xzH}\big)H\delta_i
=H^{-1}K\left(\sum_{j=1}^{MN} c_j\phi_j\right) \\
\hphantom{H^{-1}KH\phi_i}{} = H^{-1}\left(\sum_{j=1}^{MN}
c_jK(\phi_j)\right) =0.\end{gathered}$$ However, by Lemma \[Klemma\], $K$ is the unique monic operator of order $M$ having all the basis functions $\phi_i$ in its kernel so $H^{-1}KH=K$.
We also know that $L_0$ commutes with $H$ because $p_0$ is a scalar multiple of the identity and $\bisp^{-1}$ which turns $p_0$ into $L_0$ only introduces additional powers of $H$. Then $$\begin{gathered}
HQ\circ
K = H L_0=L_0H = Q \circ KH=Q \circ HK = QH \circ K.\end{gathered}$$ Multiplying this by $K^{-1}$ on the right yields $HQ=QH$.
Note that $K$ and $Q$ are probably not in $\Weyl$, as they may be *rational* functions in $x$ and a finite number of functions of the form $e^{\alpha x}$. However, it is possible to clear their denominators either by multiplying by a function on the left or by composing with a function on the right, which motivates the following definition:
Let $\Aflat$ be the set of $x$-dependent $N\times N$ matrix functions defined as follows: $$\begin{gathered}
\Aflat=\left\{ \piQK\in
\bigoplus_{\alpha\in\C}\Cent[x]e^{\alpha x}\colon \piQK=\pi_Q\pi_K,\
\pi_K(x)K\in\Weyl, Q\circ \pi_K(x)\in\Weyl\right\}.\end{gathered}$$ In other words, it is the set of zero order elements of $\Weyl$ which factor as a product such the right factor times $K$ and $Q$ composed with the left factor are both elements of $\Weyl$.
$\Aflat$ is non-empty and contains matrix functions that are non-constant in $x$.
Note that $K$ and $Q$ are operators that are rational in $x$ and a finite number of terms of the form $e^x$. If we let $\pi_K$ be the least common multiple of the denominators of the coefficients in $K$ then $\pi_K K$ is in $\Weyl$ (since we have simply cleared the denominator by multiplication). Similarly, if we let $\pi_Q$ be a high enough power of the least common multiple of the denominators of $Q$ then $Q\circ \pi_Q\in\Weyl$. Then, $\piQK=\pi_Q\pi_K$ is by construction an element of $\Aflat$. Since, $\pi_Q f \pi_K$ is also an element of $\Aflat$ for any order zero element of $\Weyl$, $\Aflat$ contains non-constant matrix functions.
The main result is the construction of an operator $\Lambda$ in $z$ with eigenvalue $\piQK$ when applied to the wave function $\psi$ for every $\piQK\in\Aflat$:
\[mainresult\] Let $\piQK=\pi_Q\pi_K\in\Aflat$ where $\bar K=\pi_K K\in\Weyl$ and $\bar Q=Q\circ \pi_Q\in\Weyl$. Define $\Lambda:=(zH)^M\circ (p_0(z))^{-1} \circ \bisp(\bar Q)\circ \bisp(\bar
K)\circ (zH)^{-M}$, then $$\begin{gathered}
\psi\Lambda=\piQK(x)\psi.\end{gathered}$$
Write $L_0$ as $L_0=\bar Q\circ
(\pi_K(x)\pi_Q(x))^{-1}\circ \bar K$. Applying this operator to $\psi_0$ and multiplying each side by $p_0^{-1}(z)$ gives $$\begin{gathered}
(\bar Q\circ (\pi_K(x)\pi_Q(x))^{-1}\circ \bar K)\psi_0p_0^{-1}(z)=\psi_0.\end{gathered}$$ Moving $\bar K$ to the other side using the anti-isomorphism and applying $\flat(\bar Q)$ to both sides this becomes $$\begin{gathered}
\big(\bar Q\circ (\pi_K(x)\pi_Q(x))^{-1}\big)\psi_0(\flat (\bar K)\circ p_0^{-1}(z)\circ \flat(\bar Q))=\psi_0\flat (\bar Q).\end{gathered}$$ Moving the last expression to the other side of the equality and moving $\flat(\bar Q)$ in it to the other side, we finally get $$\begin{gathered}
\bar Q((\pi_K\pi_Q)^{-1}\psi_0\big(\flat (\bar K)\circ p_0^{-1}\circ \flat (\bar Q)\big)-\psi_0)=0.\end{gathered}$$ Note that $\bar Q$ is a differential operator in $x$ with a non-singular leading coefficient (since it is a factor of the monic differential operator $L_0$). Hence, its kernel is finite-dimensional. The only way the expression to which it is applied, a polynomial in $z$ multiplied by $e^{xzH}$, could be in its kernel for all $z$ is if it is equal to zero. From this, we conclude that $$\begin{gathered}
\flat(\bar K)\circ p_0^{-1}\circ\flat( \bar Q) = \flat(\pi_K\pi_Q).\end{gathered}$$
Using this one finds that the action of $\Lambda$ on $\psi$ is $$\begin{gathered}
(\psi)\Lambda = \big(K\psi_0(zH)^{-M}\big)\Lambda=\big(\pi_K^{-1}(x)\bar K
\psi_0(zH)^{-M}\big)\Lambda\\
\hphantom{(\psi)\Lambda}{}
= \big(\pi_K^{-1}(x)(\psi_0)\bisp{\bar
K}(zH)^{-M}\big)\Lambda
= \big(\pi_K^{-1}(x)\psi_0\big)\bisp{\bar K}\circ
(zH)^{-M}\circ\Lambda \\
\hphantom{(\psi)\Lambda}{}
= \big(\pi_K^{-1}(x)\psi_0\big)\bisp{\bar K}\circ
(p_0(z))^{-1} \circ \bisp(\bar Q)\circ \bisp(\bar K)\circ (zH)^{-M}\\
\hphantom{(\psi)\Lambda}{}
= \big(\pi_K^{-1}(x)\psi_0\big)\Lambda_0\circ \bisp(\bar K)\circ (zH)^{-M}
= \big(\pi_K^{-1}(x)\pi_K(x)\pi_Q(x)\psi_0\big)\circ \bisp(\bar K)\circ
(zH)^{-M}
\\
\hphantom{(\psi)\Lambda}{}
= (\pi_Q(x)\psi_0)\circ \bisp(\bar K)\circ (zH)^{-M}
= \big(\pi_Q(x)\pi_K(x)\pi_K^{-1}(x)\psi_0\big)\circ \bisp(\bar K)\circ
(zH)^{-M}
\\
\hphantom{(\psi)\Lambda}{}
= \piQK(x)(\pi_K^{-1}(x)\psi_0)\bisp(\bar K)\circ
(zH)^{-M}
= \piQK(x)\big(\pi_K^{-1}(x)\bar K\psi_0\big)
(zH)^{-M}
\\
\hphantom{(\psi)\Lambda}{}
=\piQK(x)(K\psi_0)(zH)^{-M}=\piQK(x)\psi. \tag*{{}}\end{gathered}$$
Ad-nilpotency {#sec:ad}
-------------
We now have eigenvalue equations $ L_p\psi=\psi p(z)$ and $\psi\Lambda=\piQK(x)\psi $. In the commutative case, Duistermaat and Grünbaum noticed that these operators and eigenvalues satisfied an interesting relationship that goes by the name of “ad-nilpotency” [@DG]. As usual, for operators $R$ and $P$ we recursively define $\ad_R^nP$ by $$\begin{gathered}
\ad_R^1P=R\circ P-P\circ R\ \hbox{and}\ \ad_R^nP=R\circ
\big(\ad_R^{n-1}P\big)-\big(\ad_R^{n-1}P\big)\circ R,\qquad n\geq 2.\end{gathered}$$ When $R$ is a differential operator of order greater than one, then one generally expects the order of $\ad_R^nP$ to get large as $n$ goes to infinity, but they found that when $R$ is a scalar bispectral differential operator and $P$ is the eigenvalue of a corresponding differential operator in the spectral variable, then surprisingly $\ad_R^nP$ is the zero operator for large enough $n$. In fact, in that scalar Schrödinger operator case they considered, Duistermaat and Grünbaum found that ad-nilpotency was both a necessary and sufficient condition for bispectrality [@DG].
As it turns out, in the this new context even the easier of these two statements fails to hold. In particular, the ad-nilpotency in the commutative case is a consequence of the fact that the leading coefficients of $R\circ P$ and $P\circ R$ are equal, but this is not generally true for differential operators with matrix coefficients. Consequently, it is *not* the case that ad-nilpotency holds for all of the bispectral operators produced by the procedure described above. The following results (and an example in Remark \[rem:ad-examp\]) show how and to what extent the old result generalizes.
\[ad\] Given $L_p$, $\Lambda$, $\piQK(x)$ and $p(z)$ as above, the results of applying $\ad_{\piQK}^nL_p$ and $\ad_{\Lambda}^n p$ on $\psi$ are equal for every $n\in\N$.
The case $n=1$ is obtained by subtracting the equations $$\begin{gathered}
\piQK(x)L_p\psi=\piQK(x)\psi p(z)=\psi\Lambda p(z)
\end{gathered}$$ from the equations with the orders of the operators reversed $$\begin{gathered}
L_p\piQK(x)\psi=L_p\psi \Lambda=\psi p(z)\Lambda.\end{gathered}$$ If one assumes the claim is true for $n=k$ then the case $n=k+1$ is proved similarly and the general case follows by induction.
So, the equivalence of the actions of the two operators generated by iterating “ad” remains true regardless of non-commutativity. One cannot conclude from this alone that either of them is zero without making further assumptions, but if $\ad_{\piQK}^nL=0$ (which would be the case, for instance, if one could be certain that $\piQK$ would commute with the coefficients) then because $\psi$ is not in the kernel of any non-zero operator in $z$ alone, the one could conclude the same is true for the corresponding operator written in terms of $\Lambda$ and $p$:
\[cor\] If $\ad_{\piQK}^nL=0$ then $\ad_{\Lambda}^n
p=0$.
This may be the first time that ad-nilpotency has been considered for translation operators as well as for matrix operators. It is therefore relevant to note that Corollary \[cor\] is valid even when the operator $\Lambda$ involves shift operators of the form $\Trans{\alpha}$ (see Remark \[rem:ad-examp\]).
Examples
========
A wave function that is not part of a bispectral triple {#nogo}
-------------------------------------------------------
Consider the case $\bdelta=\operatorname{span}\left\{\delta_1,\delta_2\right\}$, $$\begin{gathered}
H=\left(\begin{matrix}1&0\\0&-1\end{matrix}\right), \qquad
\delta_1=\left(\begin{matrix}\Delta_{1,0}\\\Delta_{0,0}\end{matrix}\right), \qquad
\text{and} \qquad
\delta_2=\left(\begin{matrix}0\\\Delta_{1,0}\end{matrix}\right).\end{gathered}$$ Then Assumption \[assumpA\] is met and $$\begin{gathered}
\psi(x,z)=\left(\begin{matrix}\dfrac{xz-1}{xz}e^{xz}&0\vspace{1mm}\\
\dfrac{1}{x^2z}e^{xz}&\dfrac{xz+1}{xz}e^{-xz}\end{matrix}\right).\end{gathered}$$
The matrix polynomial $ p(z)=z^2I$ satisfies $p\circ \delta_i=0$ for $i=1$ and $i=2$ and so $p\in\A$ and as predicted by Theorem \[claimLp\], the corresponding operator $$\begin{gathered}
L_p=\left(\begin{matrix}\partial^2-\dfrac{2}{x^2}&0\vspace{1mm}\\
\dfrac{4}{x^3}&\partial^2-\dfrac{2}{x^2}\end{matrix}\right)\end{gathered}$$ satisfies $L_p\psi=\psi p$.
Note that this is a rational Darboux transformation of the first operator and eigenfunction from the bispectral triple $(\partial^2,\partial_z^2,e^{xzH})$. Since in the scalar case it has been found that rational Darboux transformations preserve bispectrality, one might expect $L_p$ and $\psi$ to be part of a bispectral triple. However, $H\delta_i\not\in \bdelta$ and so Assumption \[assumpB\] is not satisfied. Thus, Theorem \[mainresult\] does not guarantee the existence of a differential-translation operator $\Lambda$ in $z$ having $\psi$ as an eigenfunction. In fact, in this simple case we can see that no such operator exists.
Briefly, the argument is as follows. Consider a differential-translation operator $\Lambda$ acting on $\psi$ from the right and suppose there is a function $\pi(x)$ such that $\psi\Lambda=\pi(x)\psi$. By noting the coefficients of $e^{xz}$ and $e^{-xz}$ in the top right entry of each side of this equality, we see immediately that $\Lambda_{12}=\pi_{12}=0$ (i.e., $\Lambda$ and $\pi$ are both lower triangular.). Then,we have the scalar eigenvalue equation $\psi_{11}\Lambda_{11}=\pi_{11}\psi_{11}$ and also $\psi_{21}\Lambda_{11}=\pi_{21}\psi_{11}+\pi_{22}\psi_{21}$. Combining these with $\psi_{11}+x\psi_{21}=e^{xz}$ one concludes that $e^{xz}\Lambda_{11}$ can be written in the form $(f(x)+g(x)/z)e^{xz}$. It follows that $\Lambda_{11}$ as an operator in $z$ has only coefficients that are constant or are a constant multiplied by $1/z$. In fact, all of the operators in $z$ having $\psi_{11}$ as eigenfunction are known; they are the operators that intertwine by $\partial-\frac{1}{z}$ with a constant coefficient operator. The only ones that meet both the criteria of the previous two sentences are the order zero operators. In other words, $\Lambda_{11}$ would have to be a number. Then the equation for the action of $\Lambda_{11}$ on $\psi_{21}$ tells us that $\pi_{21}=0$ and $\pi_{22}=\pi_{11}$ is also a number. Since the eigenvalue of this operator $\Lambda$ does not depend on $x$, the operator does not form a bispectral triple with $L$ and $\psi$.
A rational example with non-diagonalizable $\boldsymbol{H}$
-----------------------------------------------------------
Consider $H=I+U$, where $$\begin{gathered}
U=\left(\begin{matrix}0&1\\0&0\end{matrix}\right), \qquad \text{so
that}\qquad \psi_0=e^{xzH}=\left(\begin{matrix}e^{xz}&ze^{xz}\\
0&e^{xz}\end{matrix}\right).\end{gathered}$$ The distributions chosen are $\bdelta=\operatorname{span}\{\delta_1,\dots,\delta_4\}$ with $$\begin{gathered}
\delta_1=\left(\begin{matrix}\Delta_{2,1}\\ 0\end{matrix}\right), \qquad
\delta_2=\left(\begin{matrix}0\\ \Delta_{2,1}\end{matrix}\right), \qquad
\delta_3=\left(\begin{matrix}\Delta_{0,1}\\ 0\end{matrix}\right), \qquad
\text{and}\qquad
\delta_4=\left(\begin{matrix}0\\\Delta_{0,1}\end{matrix}\right).\end{gathered}$$ Assumption \[assumpA\] is met and the unique monic operator of order $2$ satisfying $K(\phi_i)=0$ where $\phi_i=(\psi_0)\delta_i$ for $1\leq
i\leq 4$ is $$\begin{gathered}
K=\partial^2 I +\left(\begin{matrix}\dfrac{-2 x-1}{x} & -2
\\ 0 & \dfrac{-2 x-1}{x} \end{matrix}\right)\partial
+\left(\begin{matrix} \dfrac{x+1}{x} & \dfrac{2 x^2+x}{x^2} \vspace{1mm}\\ 0 &
\dfrac{x+1}{x} \end{matrix}\right).\end{gathered}$$ Now define $$\begin{gathered}
\psi=K(\psi_0)(zH)^{-2}=\left(\begin{matrix} \dfrac{x z^2-2 x z-z+x+1}{x
z^2} & \dfrac{z-1}{x z^2} \vspace{1mm}\\ 0 & \dfrac{x z^2-2 x z-z+x+1}{x z^2}
\end{matrix}\right)e^{xzH}.\end{gathered}$$
Assumption \[assumpB\] is met because $H\delta_i=\delta_i$, $H\delta_{i+1}=\delta_i+\delta_{i+1}$ for $i=1,3$. Consequently, it will be possible to produce operators in $z$ sharing the eigenfunction $\psi$. The matrix polynomial $$\begin{gathered}
p(z)=\left(\begin{matrix}(z-1)^2&(z-1)^3\\
0&(z-1)^2\end{matrix}\right)=(z-1)^2I+(z-1)^3U\end{gathered}$$ is in $\A$ because $p\circ\delta_i=2\delta_{i+2}$ and $p\circ\delta_{i+2}=0$ for $i=1,2$. So, $$\begin{gathered}
\bisp^{-1}(p(z))=\left(\begin{matrix}\partial^2-2\partial+1&\partial^3-5
\partial^2+5\partial-1\\ 0&\partial^3-2\partial+1\end{matrix}\right)\end{gathered}$$ satisfies the intertwining relationship $$\begin{gathered}
K\circ
\bisp^{-1}(p(z))=L_p\circ K\end{gathered}$$ with $$\begin{gathered}
L_p=\left(\begin{matrix}0&1\cr0&0\end{matrix}\right)\partial^3+
\left(\begin{matrix}1 & -5 \\ 0 & 1 \end{matrix}\right)\partial^2+
\left(\begin{matrix} -2 & 5-\dfrac{3}{x^2} \vspace{1mm}\\ 0 & -2
\end{matrix}\right)\partial+ \left(\begin{matrix}1-\dfrac{2}{x^2} &
-1+\dfrac{7}{x^2}+\dfrac{3}{x^3} \vspace{1mm}\\ 0 & 1-\dfrac{2}{x^2}
\end{matrix}\right).\end{gathered}$$ This operator has eigenfunction $\psi$ with eigenvalue $p$: $$\begin{gathered}
L_p\psi=L_p\circ K \psi_0=K\circ
L_0\psi_0=K\psi_0p(z)=\psi p(z).\end{gathered}$$ (As mentioned in Remark \[rem:eigenchange\], multiplying $\psi$ by a matrix on the right has the effect of conjugating the eigenvalue. It is therefore worth noting that the eigenvalue $p$ here is not only non-diagonal but in fact non-diagonalizable.)
To produce an operator in $z$ sharing $\psi$ as eigenfunction, we consider the polynomial $ p_0(z)=(z-1)^3 $ (which has this form because $\lambda=1$ is the only point in the support of the distributions in $\bdelta$ and because the highest derivative they take there is $m_{\lambda}=2$). Then $$\begin{gathered}
L_0=\bisp^{-1}(p_0(z)I)=\big(\partial^3-3\partial^2+3\partial-1\big)I+\big({-}3\partial^3+6\partial^2-3\partial\big)U\end{gathered}$$ is the operator satisfying $L_0\psi_0=\psi_0p_0(z)$ and $L_0$ factors as $L_0=Q\circ K$ with $$\begin{gathered}
Q=\left(\begin{matrix}\partial+\dfrac{1-x}{x}&-3\partial+\dfrac{2x-3}{x}\vspace{1mm}\\
0&\partial+\dfrac{1-x}{x}\end{matrix}\right).\end{gathered}$$
The next step is to choose two functions from $\Cent[x]$, so that $\bar
K=\pi_K K$ and $\bar Q=Q\circ \pi_Q$ are in $\C[x,\partial]$. It turns out that the selection $$\begin{gathered}
\pi_K=\left(\begin{matrix}x&x^2\\0&x\end{matrix}\right)= \pi_Q\end{gathered}$$ works and we get that $$\begin{gathered}
\Lambda=(zH)^2\circ(p_0(z))^{-1}\circ \bisp(\bar
Q)\circ \bisp(\bar K)\circ (zH)^{-2} = \partial_z^3 I +
\partial_z^2\left(\begin{matrix} 1 & -\dfrac{2
\left(z^2-z+6\right)}{(z-1) z} \\ 0 & 1 \end{matrix}\right) \\
\hphantom{\Lambda=}{} +
\partial_z\left(\begin{matrix} \dfrac{4}{z-z^2} & \dfrac{2 \left(z^2+8
z+6\right)}{(z-1)^2 z^2} \vspace{1mm}\\ 0 & \dfrac{4}{z-z^2} \end{matrix}\right) +
\left(\begin{matrix}\dfrac{-2 z^2+4 z+2}{(z-1)^2 z^2} & \dfrac{4 z^3-6
z^2-8 z-20}{(z-1)^3 z^2} \vspace{1mm}\\ 0 & \dfrac{-2 z^2+4 z+2}{(z-1)^2 z^2}
\end{matrix}\right)\end{gathered}$$ does indeed satisfy $\psi\Lambda=\pi_Q(x)\pi_K(x)\psi$.
Computing $\boldsymbol{\Trans{\alpha}^H}$ for non-diagonal $\boldsymbol{H}$ {#TransHexamp}
---------------------------------------------------------------------------
Suppose $ H=\lambda I +U $ where $U$ is still the upper-triangular matrix from the previous example. Since $$\begin{gathered}
\hbox{exp}\big(xH^{-1}\big)=e^{x/\lambda}I-\frac{x}{\lambda^2}e^{x/\lambda}U,\end{gathered}$$ the corresponding operator formed by replacing $e^{ x /\lambda}$ with $\Trans{\alpha /\lambda}$ and other occurrences of $x$ with $\alpha\partial_z$ would be $$\begin{gathered}
\Trans{\alpha}^H=\Trans{\alpha/\lambda}I-\frac{1}{\lambda^2}
\Trans{\alpha/\lambda}\alpha\partial_z U.\end{gathered}$$ This operator in $z$ that combines differentiation and translation has the property that $(e^{xzH})\Trans{\alpha}^H$ $=e^{\alpha x}e^{xzH}$.
An exponential example with $\boldsymbol{H=I}$
----------------------------------------------
Finally, consider $H=I$ and $$\begin{gathered}
\bdelta=\operatorname{span}\left\{
\left(\begin{matrix} \Delta_{0,0}+\Delta_{0,1}\\\Delta_{0,1}
\end{matrix}\right), \left(\begin{matrix}0\\ -\Delta_{0,0}+\Delta_{0,1}
\end{matrix}\right)\right\} .\end{gathered}$$ Then $$\begin{gathered}
K=\left(\begin{matrix}\partial-\dfrac{e^x}{1+e^x}&0\vspace{1mm}\\
\dfrac{e^{x}}{e^{2x}-1}&\partial+\dfrac{e^x}{1-e^x}\end{matrix}\right)
\qquad\text{and}\\
\psi(x,z)=\left(\begin{matrix} \dfrac{e^{x z}
\left(e^x (z-1)+z\right)}{\left(1+e^x\right) z} & 0 \vspace{1mm}\\ \dfrac{e^{z
x+x}}{\left(-1+e^{2 x}\right) z} & \dfrac{e^{x z} \left(e^x
(z-1)-z\right)}{\left(-1+e^x\right) z} \end{matrix}\right).\end{gathered}$$
Because of the support of the distributions, $p_0(z)=z^2-z$ and we may choose $p\in\A$ to be a multiple of that by a constant matrix $$\begin{gathered}
p(z)=\left(\begin{matrix}0&z^2-z\\ z^2-z&0\end{matrix}\right).\end{gathered}$$ The constant operator $\bisp^{-1}(p)=p(\partial)$, satisfies the intertwining relationship $K\circ p(\partial)=L_p\circ K$ with $$\begin{gathered}
L_p = \left( \begin{matrix} -\dfrac{e^x \left(-1+e^x-2
e^{2 x}\right)}{\left(-1+e^{2 x}\right)^2} & -\dfrac{2 e^{2
x}}{\left(-1+e^x\right)^2 \left(1+e^x\right)} \vspace{1mm}\\ \dfrac{e^{2 x}
\left(-3+2 e^x\right)}{\left(-1+e^{2 x}\right)^2} &
\dfrac{e^x}{\left(-1+e^x\right)^2 \left(1+e^x\right)} \end{matrix}
\right)\\
\hphantom{L_p =}{}
+ \left( \begin{matrix} -\dfrac{e^x}{-1+e^{2 x}} &
\dfrac{-1-e^x+3 e^{2 x}-e^{3 x}}{\left(-1+e^x\right)^2
\left(1+e^x\right)} \vspace{1mm}\\ \dfrac{-1+2 e^x+2 e^{2 x}-2 e^{3 x}-e^{4
x}}{\left(-1+e^{2 x}\right)^2} & \dfrac{e^x}{\left(-1+e^x\right)
\left(1+e^x\right)} \end{matrix} \right)\partial+ \left(
\begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix}\right)\partial_x^2.\end{gathered}$$ Hence $ L_p\psi=\psi p(z). $ (Since $p(z)$ and $\psi$ do not commute, it is necessary to write the eigenvalue on the right rather than the left: $L\psi\not=p(z)\psi$.)
Now, to compute an operator in $z$ having the same function $\psi$ as an eigenfunction, we factor $p_0(\partial)I$ as $Q\circ K$ to obtain $$\begin{gathered}
Q=\partial I + \left( \begin{matrix} -\frac{1}{1+e^x} & 0 \\
\frac{e^x}{1-e^{2 x}} & \frac{1}{-1+e^x} \end{matrix} \right).
\end{gathered}$$ Letting $\pi_K$ and $\pi_Q$ be $$\begin{gathered}
\pi_K=\left( \begin{matrix} e^{2 x}-1
& 0 \\ 0 & 1-e^{2 x} \end{matrix} \right), \qquad \pi_Q=\left(
\begin{matrix} e^{2 x}-1 & 0 \\ 0 & e^{2 x} -1 \end{matrix} \right)\end{gathered}$$ we get $$\begin{gathered}
\bar K = \pi_KK=\left( \begin{matrix} -e^x
\left(-1+e^x\right) & 0 \\ -e^x & e^x+e^{2 x} \end{matrix} \right)+
\left( \begin{matrix} \left(-1+e^x\right) \left(1+e^x\right) & 0 \\ 0 &
1-e^{2 x} \end{matrix} \right)\partial \in\Weyl
\end{gathered}$$ and $$\begin{gathered}
\bar Q =
Q\circ \pi_Q = \left( \begin{matrix} 1-e^x+2 e^{2 x} & 0 \\ -e^x &
1+e^x+2 e^{2 x} \end{matrix} \right)+ \left( \begin{matrix} -1+e^{2
x} & 0 \\ 0 & -1+e^{2 x} \end{matrix} \right)\partial\in\Weyl.
\end{gathered}$$
Replacing $e^{\alpha x}$ by $\Trans{\alpha}^H=\Trans{\alpha}$ and $\partial$ by $z$ we get the corresponding translation operators $$\begin{gathered}
\bisp(\bar K)=\left( \begin{matrix} -z & 0 \\ 0 & z \end{matrix}
\right)+\Trans1 \left( \begin{matrix} 1 & 0 \\ -1 & 1 \end{matrix}
\right)+\Trans2 \left( \begin{matrix} z-1 & 0 \\ 0 & 1-z \end{matrix}
\right)\end{gathered}$$ and $$\begin{gathered}
\bisp(\bar Q)=\left( \begin{matrix} 1-z & 0 \\ 0 & 1-z
\end{matrix} \right)+\Trans1 \left( \begin{matrix} -1 & 0 \\ -1 & 1
\end{matrix} \right)+\Trans2 \left( \begin{matrix} z+2 & 0 \\ 0 & z+2
\end{matrix} \right).
\end{gathered}$$ As predicted by Theorem \[mainresult\], $$\begin{gathered}
\Lambda = \frac{z}{p_0(z)}\circ \bisp(\bar Q)\circ
\bisp(\bar K)\circ \frac{1}{z} = \left( \begin{matrix} \dfrac{z^3+5
z^2+6 z}{z (z+2) (z+3)} & 0 \vspace{1mm}\\ 0 & \dfrac{-z^3-5 z^2-6 z}{z (z+2) (z+3)}
\end{matrix} \right)\\
\hphantom{\Lambda =}{}
+\Trans1 \left( \begin{matrix} 0 & 0 \vspace{1mm}\\ \dfrac{2
z^2+6 z+4}{z (z+1) (z+2)} & 0 \end{matrix} \right)
+\Trans2 \left(
\begin{matrix} \dfrac{-2 z^3-10 z^2-12 z}{z (z+2) (z+3)} & 0 \vspace{1mm}\\ \dfrac{-2
z-4}{z (z+1) (z+2)} & \dfrac{2 z^3+10 z^2+12 z}{z (z+2) (z+3)}
\end{matrix} \right)\\
\hphantom{\Lambda =}{}
+\Trans3 \left( \begin{matrix} \dfrac{4 z+12}{z (z+2)
(z+3)} & 0 \vspace{1mm}\\ \dfrac{-2 z^2-4 z-2}{z (z+1) (z+2)} & \dfrac{4 z+12}{z (z+2)
(z+3)} \\ \end{matrix} \right)\\
\hphantom{\Lambda =}{}
+\Trans4 \left( \begin{matrix}
\dfrac{z^3+5 z^2+2 z-8}{z (z+2) (z+3)} & 0 \vspace{1mm}\\ 0 & \dfrac{-z^3-5 z^2-2
z+8}{z (z+2) (z+3)} \end{matrix} \right)\end{gathered}$$ satisfies $\psi\Lambda=\piQK\psi$ ($\piQK=\pi_Q\pi_K$).
\[rem:ad-examp\] This is a good example for demonstrating ad-nilpotency and how it has changed in this non-commutative context. Using $L$, $\Lambda$, $p(z)$ and $\pi_K$, $\pi_Q$ as above, it is indeed true that $$\begin{gathered}
\ad_{\piQK}^nL\psi=\psi\ad_{\Lambda}^np\end{gathered}$$ (with the operator in $z$ acting from the right as usual). However, contrary to our expectation from the commutative case in which the order of the operator on the left decreases to zero when $n$ gets large, $\ad_{\piQK}^nL$ is an operator of order $2$ for every $n$. This happens because the action of iterating $\ad_{\piQK}$ on the leading coefficient of $L$ itself is not nilpotent. On the other hand, if instead of $\pi_K$ and $\pi_Q$ we had chosen $$\begin{gathered}
\pi_K^*=\pi_Q^*=\big(e^{2x}-1\big)I,\qquad \piQK^*=\pi_K^*\pi_Q^*=\big(e^{2x}-1\big)^2I\end{gathered}$$ (intentionally selected so as to commute with the coefficients of any operator) then Theorem \[mainresult\] would have produced a different operator $\Lambda^*$ satisfying $\piQK^*\psi=\psi\Lambda^*$. In this case, because the order is lowered at each iteration, $\ad_{\piQK^*}^3L=0$. This is not particularly surprising or special as the same could be said if $\piQK^*$ were replaced by any function of the form $f(x)I$. However, because of the correspondence in Theorem \[ad\], we can conclude from this that $\ad_{\Lambda^*}^3p(z)=0$ also. That is more interesting because in general repeatedly taking the commutator of the operator $\Lambda^*$ with some function will not produce the zero operator, even if all coefficients are assumed to commute. Nevertheless, with these specific choices everything cancels out leaving exactly zero when $\ad_{\Lambda^*}$ is applied three times to the function $p(z)$ that is the eigenvalue above.
Conclusions and remarks
=======================
Given a choice of an invertible $N\times N$ matrix $H$ and $MN$-dimensional space $\bdelta$ of vector-valued finitely supported distributions, this paper sought to produce a bispectral triple $(L,\Lambda,\psi)$ where $L$ is a differential operator acting from the left, $\Lambda$ is a differential-translation operator acting from the right and $\psi$ is a common eigenfunction that is asymptotically of the form $e^{xzH}$ satisfying the “conditions” generated by the distributions. In the scalar case, this was achieved in [@BispSol] for *any* choice of $\bdelta$. In the matrix generalization above, however, the construction only works given Assumptions \[assumpA\] and \[assumpB\]. The bispectral triples produced given those two assumptions include many new examples both in the form of the eigenfunction (asymptotically equal to $e^{xzH}$) and the fact that the matrix-coefficient operator $\Lambda$ may involve translation in $z$ as well as differentiation in $z$. More importantly, this investigation yielded some observations that may be useful in future studies of bispectrality in a non-commutative context.
This paper sought to develop a general construction of bispectral triples with matrix-valued eigenvalues and also to understand what obstructions there might be to generalizing the construction from [@BispSol] to the matrix case. It is interesting to note that these seemingly separate goals both turn out to depend on the non-commutativity of the ring $\A$ of functions that stabilize the point in the Grassmannian. Since $\A$ is also the ring of eigenvalues for the operators in $x$, it is not a surprise that by letting its elements be matrix-valued gives us a non-commutative ring $\{L_p\colon p\in\A\}$ of operators sharing the eigenfunction $\psi$. It was not clear at first that the ring $\A$ would also be the source of the obstruction to producing bispectral triples. However, the construction of operators in $z$ sharing $\psi$ as eigenfunction uses the assumption that $H$ is an element of $\A$. (Specifically, this is used in the proof of Lemma \[KcommuteswithH\].) It is interesting to note that this *also* depends on the fact that we considered a matrix-valued stabilizer ring.
Section \[sec:ad\] explored the extent to which the property of ad-nilpotency, which has been a feature of papers on the bispectral problem since it was first noted in [@DG], continues to apply in the case of matrix coefficient operators. It is still the case that the operator formed by iterating the adjoint of one of the operators in a bispectral triple on the eigenvalue of the other operator has the same action on the eigenfunction as the operator formed by iterating the adjoint action of its eigenvalue on the other operator. However, unlike the scalar case, only if additional assumptions about the coefficients of the operators are met will either of those be zero for a large enough iterations.
It was previously observed [@BGK; @BL; @WilsonNotes] (see also [@matrix3; @GPT1; @GPT2; @GPT3; @GPT4]) that requiring the operators in $x$ and $z$ to act on the eigenfunctions from opposite side resulted in a form of bispectrality whose structure and applications more closely resembled that in the scalar case. Because generalizing the scalar results of [@BispSol] necessitated also requiring that the eigenvalues and finitely-supported distributions act from the same side as the operators acting in the same variables, this previous observation can now be extended to those other objects as well.
This is the first time that the bispectral anti-isomorphism was used to construct bispectral operators in a context involving operators with matrix coefficients, see section \[sec:antiiso\] (cf. [@GHY]). Some modifications were necessary since the rather general construction in [@BHY] assumed that there were no zero-divisors so that formal inverses could be introduced. In addition, the use of the method here depended on the convention of considering operators in $x$ and $z$ to be acting from opposite sides and required that the coefficients of the operators on which it acted were taken from the centralizer of $H$. The most interesting difference may have been that because the operators in $z$ are acting from the right rather than the left, the map actually *preserves* the order of a product.
Unlike the scalar case, not every choice of distributions $\bdelta$ corresponds to a bispectral triple. For instance, if Assumption \[assumpA\] fails to be met then there simply is no wave function $\psi$ satisfying the conditions. More interestingly, if Assumption \[assumpA\] is met but Assumption \[assumpB\] is not then there is a wave function that is an eigenfunction for a ring of differential operators in $x$, but Theorem \[mainresult\] does not produce a corresponding operator in $z$. In fact, as Section \[nogo\] demonstrates, in at least some cases there actually is no bispectral triple of the form considered above which includes that wave function. This is very different than the scalar case. Unfortunately, this paper does not entirely answer the question of which choices of matrix $H$ and distributions $\bdelta$ produce a wave function $\psi$ that is part of a bispectral triple of the type considered here. In particular, this paper does not show or claim that there *cannot be* a differential-difference operator $\Lambda$ in $z$ having $\psi$ as an eigenfunction with $x$-dependent eigenvalue when $H\not\in\A$.
Arguably, some of the non-commutativity involved in the construction above was “artificially inserted” in the form of the choice of the matrix $H$. If one chooses to consider only the case $H=I$, then Assumption \[assumpB\] is automatically met and Theorems \[claimLp\] and \[mainresult\] produce a bispectral triple for any $\bdelta$ satisfying Assumption \[assumpA\]. Since any operator $L_p$ produced by the construction above for some choice of $H$ *can* be produced using a different choice of distributions but with $H=I$, one may conclude that each of these operators is part of a bispectral triple. (In other words, the obstruction to bispectrality that is visible when one seeks a bispectral triple for a given wave function $\psi$ disappears if one instead focuses on the operator $L_p$ and seeks a corresponding bispectral triple.) However, it seems plausible that some examples of bispectrality to be considered in the future will involve vacuum eigenfunctions that are necessarily non-commutative (unlike these examples in which the non-commutativity of the vacuum eigenfunctions can always be eliminated through a change of variables), and the observations and results above will prove useful in those contexts.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author thanks the College of Charleston for the sabbatical during which this work was completed, Maarten Bergvelt and Michael Gekhtman for mathematical assistance as well as serving as gracious hosts, Chunxia Li for carefully reading and commenting on early drafts, and the referees for their advices.
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[^1]: In previous papers the elements of $\HN$ have been written as column vectors. However, because the construction of bispectral operators below is most easily described in terms of matrix finitely-supported distributions in $z$ acting from the *right*, they will be written here as $1\times N$ matrices.
[^2]: The expression $D\bphi=0$ is a convenient way to write that applying the $N\times N$ differential operator $D$ to each function $\phi_i$ ($1\leq i\leq MN$) results in the zero vector of $\CN$.
[^3]: Lemma \[Klemma\](b) is used in the proofs of Theorem \[KPLax\], Lemma \[KcommuteswithH\] and Theorem \[claimLp\].
|
---
abstract: 'We provide a new class of indecomposable entanglement witnesses. In $4 \times 4$ case it reproduces the well know Breuer-Hall witness. We prove that these new witnesses are optimal and atomic, i.e. they are able to detect the “weakest" quantum entanglement encoded into states with positive partial transposition (PPT). Equivalently, we provide a new construction of indecomposable atomic maps in the algebra of $2k \times 2k$ complex matrices. It is shown that their structural physical approximations give rise to entanglement breaking channels. This result supports recent conjecture by Korbicz [*et. al.*]{}'
author:
- 'Dariusz Chruściński, Justyna Pytel and Gniewomir Sarbicki[^1]'
title: '**Constructing new optimal entanglement witnesses**'
---
Introduction
============
The interest on quantum entanglement has dramatically increased during the last two decades due to the emerging field of quantum information theory [@QIT]. It turns out that quantum entanglement may be used as basic resources in quantum information processing and communication. The prominent examples are quantum cryptography, quantum teleportation, quantum error correction codes and quantum computation.
Since the quantum entanglement is the basic resource for the new quantum information technologies it is therefore clear that there is a considerable interest in efficient theoretical and experimental methods of entanglement detection (see [@HHHH] for the review).
The most general approach to characterize quantum entanglement uses a notion of an entanglement witness (EW) [@EW1; @EW2]. A Hermitian operator $W$ defined on a tensor product $\mathcal{H}=\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$ is called an EW iff 1) $\mbox{Tr}(W\sigma_{\rm sep})\geq 0$ for all separable states $\sigma_{\rm sep}$, and 2) there exists an entangled state $\rho$ such that $\mbox{Tr}(W\rho)<0$ (one says that $\rho$ is detected by $W$). It turns out that a state is entangled if and only if it is detected by some EW [@EW1]. There was a considerable effort in constructing and analyzing the structure of EWs [@Terhal2; @O; @Lew1; @Lew2; @Lew3; @Bruss; @Toth; @Bertlmann; @Brandao; @Gniewko; @how]. In fact, entanglement witnesses have been measured in several experiments [@EX; @Wu]. Moreover, several procedures for optimizing EWs for arbitrary states were proposed [@O; @O1; @O2; @O3]. It should be stressed that there is no universal $W$, i.e. there is no entanglement witness which detects all entangled states. Each entangled state $\rho$ may be detected by a specific choice of $W$. It is clear that each EW provides a new separability test and it may be interpreted as a new type of Bell inequality [@W-Bell]. There is, however, no general procedure for constructing EWs.
Due to the Choi-Jamio[ł]{}kowski isomorphism [@Choi; @Jam] any EW corresponds to a linear positive map $\Lambda :
\mathcal{B}(\mathcal{H}_A) \rightarrow \mathcal{B}(\mathcal{H}_B)$, where by $\mathcal{B}(\mathcal{H})$ we denote the space of bounded operators on the Hilbert space $\mathcal{H}$. Recall that a linear map $\Lambda$ is said to be positive if it sends a positive operator on $\mathcal{H}_A$ into a positive operator on $\mathcal{H}_B$. It turns out [@EW1] that a state $\rho$ in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$ is separable iff $(\oper_A {{\,\otimes\,}}\Lambda)\rho$ is positive definite for all positive maps $\Lambda :
\mathcal{B}(\mathcal{H}_B) \rightarrow \mathcal{B}(\mathcal{H}_A)$ (actually this result is based on [@Woronowicz1]). Unfortunately, in spite of the considerable effort, the structure of positive maps is rather poorly understood [@Woronowicz1; @Stormer1; @Choi1; @Woronowicz2; @Robertson; @Tang; @TT; @Osaka; @Benatti; @Kye; @Ha; @Kossak1; @Hall; @Breuer; @OSID-W; @atomic; @CMP; @kule].
In the present paper we provide a construction of a new class of positive maps in $\mathcal{B}(\mathbb{C}^{2k})$ with $k\geq 2$. Our construction uses the well-known reduction map as a building block. It turns out that for $k=2$ our construction reproduces Breuer-Hall maps [@Breuer; @Hall] but for $k>2$ it gives completely new family of maps. It is shown that proposed maps are indecomposable (i.e. they are able to detect entangled PPT states) and atomic (i.e. they are able to detect “weakly” entangled PPT states). As a byproduct we construct new families of PPT entangled states detected by our maps.
The paper is organized as follows: for pedagogical reason we collect basic definitions and introduce the most important properties of positive maps and entanglement witnesses in Section \[DEF\]. Section \[RED\] provides basic construction. Then in Section \[PROP\] we study basic properties of our maps/witnesses (indecomposability, atomicity, optimality). Section \[SPA\] discusses structural physical approximation (SPA) [@SPA1; @SPA2; @SPA3] of our maps. It is shown that the corresponding SPA gives rise to entanglement breaking channels and hence it supports recent conjecture by Korbicz [*et. al.*]{} [@SPA3]. Final conclusions are collected in the last Section.
Positive maps, entanglement witnesses and all that {#DEF}
==================================================
For the reader convenience we recall basic definitions and properties which are important throughout this paper.
Positive maps
-------------
Let $\Lambda : \mathcal{B}(\mathcal{H}_A) \rightarrow
\mathcal{B}(\mathcal{H}_B)$ be a positive linear map. In what follows we shall consider only finite dimensional Hilbert spaces such that ${\rm dim}\mathcal{H}_A =d_A$ and ${\rm dim}\mathcal{H}_B
=d_B$. One calls $\Lambda$ $k$-positive if $$\label{}
\oper_k {{\,\otimes\,}}\Lambda : M_k {{\,\otimes\,}}\mathcal{B}(\mathcal{H}_A) \longrightarrow
M_k {{\,\otimes\,}}\mathcal{B}(\mathcal{H}_B)\ ,$$ is positive. In the above formula $M_k$ denotes a linear space of $k
\times k$ complex matrices and $\oper_k : M_k \rightarrow M_k$ is an identity map, i.e. $\oper_k(A) = A$ for each $A \in M_k$. A positive map which is $k$-positive for each $k$ is called completely positive (CP). Actually, if $d_A,d_B < \infty$ one shows [@Choi] that $\Lambda$ is CP iff it is $d$-positive with $d = \min\{d_A,d_B\}$.
A positive map $\Lambda$ is decomposable if $$\label{}
\Lambda = \Lambda_1 + \Lambda_2 \circ T\ ,$$ where $\Lambda_1$ and $\Lambda_2$ are CP and $T$ denotes transposition in a given basis. Maps which are not decomposable are called indecomposable (or nondecomposable).
A positive map $\Lambda$ is atomic if it cannot be represented as $$\label{}
\Lambda = \Lambda_1 + \Lambda_2 \circ T\ ,$$ where $\Lambda_1$ and $\Lambda_2$ are 2-positive.
A positive map $\Lambda$ is optimal if and only if for any CP map $\Phi$, the map $\Lambda - \Phi$ is no longer positive.
Entanglement witnesses
----------------------
Using Choi-Jamio[ł]{}kowski isomorphism [@Choi; @Jam] each positive map $\Lambda$ gives rise to entanglement witness $W$ $$\label{CJ}
W = d_A (\oper_A {{\,\otimes\,}}\Lambda)P^+_A\ ,$$ where $P^+_A$ denotes maximally entangled state in $\mathbb{C}^{d_A}
{{\,\otimes\,}}\mathbb{C}^{d_A}$ and $\oper_A$ denotes an identity map acting on $\mathcal{B}(\mathcal{H}_A)$. One has an obvious
An entanglement witness $W$ defined by (\[CJ\]) is decomposable/indecomposable (atomic) \[optimal\] $\{$ $k$–EW $\}$ if and only if the corresponding positive map $\Lambda$ is decomposable/indecomposable (atomic) \[optimal\] $\{$ $k$–positive $\}$.
It is clear that $W \in \mathcal{B}(\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B)$ is a decomposable EW iff $$\label{Wd}
W = A + B^\Gamma\ ,$$ where $A,B \geq 0$ and $B^\Gamma = (\oper_A {{\,\otimes\,}}T)B$ denotes partial transposition. Witnesses which cannot be represented as in (\[Wd\]) are indecomposable
Let $\psi$ be a normalize vector in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$. Denote by ${\rm SR}(\psi)$ the number of nonvanishig Schmidt coefficients of $\psi$. One has $$\label{}
1 \leq {\rm SR}(\psi) \leq d \ .$$ Now, $W$ is $k$–EW iff $$\label{}
\< \psi |W|\psi\> \geq 0 \ ,$$ for each $\psi$ such that ${\rm SR}(\psi) \leq k$. Evidently, $W\geq
0$ iff $W$ is $d$–EW. Now, $W$ is atomic if it cannot be represented as $$\label{}
W = W_1 + W_2^\Gamma\ ,$$ where $W_1$ and $W_2$ are 2–EWs. Finally, $W$ is optimal EW iff for any $P\geq 0$, $W-P$ is no longer EW. Following [@O] one has the following criterion for the optimality of $W$: if the set of product vectors $\psi {{\,\otimes\,}}\phi \in \mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$ satisfying $$\label{}
\< \psi {{\,\otimes\,}}\phi|W|\psi{{\,\otimes\,}}\phi\>=0\ ,$$ span the total Hilbert space $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$, then $W$ is optimal.
Detecting quantum entanglement
------------------------------
Positive maps and EWs are basic tools in detecting quantum entanglement. A state $\rho$ in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$ is separable if and only if for all positive maps $\Lambda :
\mathcal{B}(\mathcal{H}_B) \rightarrow \mathcal{B}(\mathcal{H}_A)$ one has $$\label{}
(\oper_A {{\,\otimes\,}}\Lambda)\rho \geq 0 \ .$$ Equivalently, iff for each entanglement witness $W$ $$\label{}
{\rm Tr}(\rho W) \geq 0\ .$$ Note that entangled PPT states can be detected by indecomposable maps/witnesses only. Let $\sigma$ be a density operator in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$. Following [@SN] one introduces its Schmidt number $$\label{SN-rho}
\mbox{SN}(\sigma) = \min_{p_k,\psi_k}\, \left\{ \,
\max_{k}\, \mbox{SR}(\psi_k)\, \right\} \ ,$$ where the minimum is taken over all possible pure states decompositions $$\label{}
\sigma = \sum_k \, p_k\, |\psi_k\>\<\psi_k|\ ,$$ with $p_k\geq 0$, $\sum_k\, p_k =1$ and $\psi_k$ are normalized vectors in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$. Note, that if $\sigma
= |\psi\>\<\psi|$, then ${\rm SN}(\sigma) = {\rm SR}(\psi)$. Again one has $ 1 \leq {\rm SN}(\sigma) \leq d$. Suppose now that $\sigma$ is PPT but entangled. Intuitively, the ‘weakest’’ quantum entangled encoded in $\sigma$ corresponds to the situation when ${\rm
SR}(\sigma) = {\rm SR}(\sigma^\Gamma) = 2$. Such “weakly” entangled PPT states can be detected by atomic maps/witnesses only.
Reduction map as a building block {#RED}
=================================
Let us start with an elementary positive map in $\mathcal{B}(\mathbb{C}^n)$ called reduction map [@Horodecki] $$\label{Rn}
R_n(X)=\mathbb{I}_{n}\mathrm{Tr}X-X\ ,$$ for $X \in \mathcal{B}(\mathbb{C}^n)$. It is well known that $R_n$ is completely co-positive (i.e. $R_n \circ T$ is CP) and hence optimal. Recently, this map was generalized by Breuer and Hall [@Breuer; @Hall] to the following family of positive maps $$\label{BH}
\Phi^U_{2k}(X) = \frac{1}{2(k-1)}\Big( R_{2k}(X) - UX^TU^\dagger\Big) \ ,$$ where $U$ is an arbitrary antisymmetric unitary $2k \times 2k$ matrix. It was shown that these map are indecomposable [@Breuer; @Hall] and optimal [@Breuer]. Such antisymmetric unitary matrix may be easily construct as follows $$\label{}
U = V U_0 V^\dagger\ ,$$ where $V$ stands for real orthogonal matrix $(VV^\dagger =
VV^T=\mathbb{I}_{2k})$ and $$\label{}
U_0 = \mathbb{I}_k {{\,\otimes\,}}J\ ,$$ with $J$ being $2 \times 2$ symplectic matrix $$\label{J}
J=\left( \begin{array}{c c} 0 & 1 \\ -1 & 0\end{array}
\right)\ .$$ It is therefore clear that in this case one has $$\label{BH1}
\Phi^U_{2k}(X) = V \Phi^0_{2k}(V^\dagger X V) V^\dagger \ ,$$ where $\Phi^0_{2k}$ corresponds to $\Phi^U_{2k}$ with $U=U_0$. Actually, one can always find a basis in $\mathbb{C}^{2k}$ such that $U$ takes the “canonical form” $U_0$. Interestingly for $k=2$ the Breuer-Hall map $\Phi^0_4$ reproduces well known Robertson map [@Robertson] who provided it as an example of an extremal (and hence optimal) indecomposable positive map. Moreover, Robertson construction may be nicely described in terms of $R_2$ as follows [@atomic]
$$\label{R4}
\Phi^0_4\left( \begin{array}{c|c} X_{11} & X_{12} \\ \hline X_{21} & X_{22} \end{array}
\right) = \frac 12 \left( \begin{array}{c|c} \mathbb{I}_2\,
\mbox{Tr} X_{22} & -[X_{12} + R_2(X_{21})] \\ \hline -[X_{21} +
R_2(X_{12})] & \mathbb{I}_2\, \mbox{Tr} X_{11}
\end{array} \right) \ ,$$
where $X_{kl} \in \mathcal{B}(\mathbb{C}^2)$. This pattern is reproduced for arbitrary $k$. It is easy to show that the action of $\Phi^0_{2k}$ may be represented as follows:
$$\begin{aligned}
\label{Phi-2k}
\Phi^0_{2k}\left( \begin{array}{c|c|c|c} X_{11} & X_{12} & \cdots &
X_{1k}
\\ \hline X_{21} & X_{22} & \cdots & X_{2k} \\ \hline \vdots & \vdots & \ddots & \vdots \\
\hline X_{k1} & X_{k2} & \cdots & X_{kk}\end{array} \right) = \frac{1}{2(k-1)}\left( \begin{array}{c|c|c|c}
\mathbb{I}_{2}(\mathrm{Tr}X-\mathrm{Tr}X_{11}) &
-(X_{12}+R_2(X_{21})) & \cdots & -(X_{1k}+R_2(X_{k1})) \\ \hline -(X_{21}+R_2(X_{12})) &
\mathbb{I}_{2}(\mathrm{Tr}X-\mathrm{Tr}X_{22}) & \cdots & -(X_{2k}+R_2(X_{k2})) \\
\hline \vdots & \vdots & \ddots & \vdots \\ \hline
-(X_{k1}+R_2(X_{1k})) & -(X_{k2}+R_2(X_{2k})) & \cdots & \mathbb{I}_{2}(\mathrm{Tr}X-\mathrm{Tr}X_{kk}) \end{array}
\right)\ ,
\nonumber\end{aligned}$$
where again $X_{ij}$ are $2 \times 2$ blocks. Hence $\Phi^0_{2k}$ is defined in (\[BH\]) by $R_{2k}$ but the above pattern shows that it basically uses reduction map $R_2$ only. We stress that $R_2$ is exceptional: it is not only optimal but also extremal. Indeed, the corresponding entanglement witness $W_2 = 2(\oper {{\,\otimes\,}}R_2)P^+_2$ reads as follows $$\label{}
W_2 = \mathbb{I}_2 {{\,\otimes\,}}\mathbb{I}_2 - P^+_2 =
\left( \begin{array}{c c|c c} \cdot & \cdot & \cdot & -1 \\ \cdot & 1 & \cdot & \cdot \\
\hline \cdot & \cdot & 1 & \cdot \\ -1 & \cdot & \cdot & \cdot \end{array}
\right)\ .$$ and $W_2 = P^\Gamma$, where $P=|\psi\>\<\psi|$ with $$\label{}
|\psi\> = |01\> - |10\>\ .$$ Note, that $R_2$ may be nicely represented as follows $$\label{R-J}
R_2(X)=J X^T J^\dagger \ ,$$ with $J$ defined in (\[J\]). Formula (\[R-J\]) provides Kraus representation for $R_2 \circ T$ and shows that $R_2$ is completely co-positive.
In the present paper we propose another construction of maps in $\mathcal{B}(\mathbb{C}^{2k})$. Now, instead of treating a $2k
\times 2k$ matrix $X$ as a $k \times k$ matrix with $2\times 2$ blocks $X_{ij}$ we consider alternative possibility, i.e. we consider $X$ as a $2 \times 2$ with $k \times k$ blocks and define
$$\label{Psi-2k}
\Psi^0_{2k}\left( \begin{array}{c|c} X_{11} & X_{12} \\ \hline X_{21} & X_{22} \end{array}
\right) = \frac 1k \left( \begin{array}{c|c} \mathbb{I}_k\,
\mbox{Tr} X_{22} & -[X_{12} + R_k(X_{21})] \\ \hline -[X_{21} +
R_k(X_{12})] & \mathbb{I}_k\, \mbox{Tr} X_{11}
\end{array} \right) \ .$$
Again, normalization factor guaranties that the map is unital, i.e. $\Psi^0_{2k}(\mathbb{I}_2 {{\,\otimes\,}}\mathbb{I}_k) = \mathbb{I}_2 {{\,\otimes\,}}\mathbb{I}_k$. It is clear that for $k=2$ one has $$\label{}
\Phi^0_4 = \Psi^0_4 \ .$$ We stress that our new construction is much simpler than $\Phi^0_{2k}$ and it uses as a building block the true reduction map in $\mathcal{B}(\mathbb{C}^k)$. Moreover, it is clear that it provides a natural generalization of the original Robertson map in $\mathcal{B}(\mathbb{C}^4)$.
Now, our task is to prove that $\Psi^0_{2k}$ defines a positive map. It is enough to show that each rank-1 projector $P$ is mapped via $\Psi^0_{2k}$ into a positive element in $\mathcal{B}(\mathbb{C}^{2k})$, that is, $\Psi^0_{2k}(P)\geq 0$. Let $P = |\psi\>\<\psi|$ with arbitrary $\psi$ from $\mathbb{C}^{2k}$. Now, due to $\mathbb{C}^{2k} = \mathbb{C}^k \oplus \mathbb{C}^k$ one has $$\label{}
\psi = \psi_1 \oplus \psi_2\ ,$$ with $\psi_1,\psi_2 \in\mathbb{C}^k$ and hence $$\label{}
P = \left( \begin{array}{c|c} X_{11} & X_{12} \\ \hline X_{21} & X_{22} \end{array}
\right) = \left( \begin{array}{c|c} |\psi_1\>\<\psi_1| & |\psi_1\>\<\psi_2| \\
\hline |\psi_2\>\<\psi_1| & |\psi_2\>\<\psi_2| \end{array}
\right) \ .$$ One has therefore $$\label{}
\Psi^0_{2k}(P) = \frac 1k \left( \begin{array}{c|c} \mathbb{I}_k\, \mbox{Tr}
X_{22} & - A \\ \hline - A^\dagger & \mathbb{I}_k\, \mbox{Tr}
X_{11} \end{array} \right) \ ,$$ where the linear operator $A : \mathbb{C}^k \rightarrow
\mathbb{C}^k$ reads as follows $$\label{}
A = |\psi_1\>\<\psi_2| - |\psi_2\>\<\psi_1| + \<
\psi_1|\psi_2\>\, \mathbb{I}_k \ .$$ Let $\<\psi_j|\psi_j\> = a_j^2 > 0$ (if one of $a_j$ vanishes then evidently one has $\Psi^0_{2k}(P)\geq 0$). Defining $$\label{}
L = \sqrt{k} \left( \begin{array}{c|c} \mathbb{I}_k\, a_2^{-1}
& \mathbb{O}_k \\ \hline \mathbb{O}_k & \mathbb{I}_k\, a_1^{-1}
\end{array} \right) \ ,$$ one finds $$\label{}
L \Psi^0_{2k}(P) L^\dagger = \left( \begin{array}{c|c} \mathbb{I}_k & - \widetilde{A} \\ \hline
- \widetilde{A}^\dagger & \mathbb{I}_k \end{array} \right) \ ,$$ with $$\label{}
\widetilde{A} = |\widetilde{\psi}_1\>\<\widetilde{\psi}_2| - |\widetilde{\psi}_2\>\<\widetilde{\psi}_1| + \<
\widetilde{\psi}_1|\widetilde{\psi}_2\>\, \mathbb{I}_k \ ,$$ and normalized $\widetilde{\psi}_j = \psi_j/a_j$. Hence, to show that $\Psi^0_{2k}(P)\geq 0$ one needs to prove $$\label{*}
\left( \begin{array}{c|c} \mathbb{I}_k & - \widetilde{A} \\ \hline
- \widetilde{A}^\dagger & \mathbb{I}_k \end{array} \right) \geq 0
\ ,$$ for arbitrary $\psi_j \neq 0$. Now, the above condition is equivalent to $$\label{AAI}
\widetilde{A}\widetilde{A}^\dagger \leq \mathbb{I}_k \ .$$ Vectors $\{\psi_1,\psi_2\}$ span 2-dimensional subspace in $\mathbb{C}^k$ and let $\{e_1,e_2\}$ be a 2-dim. orthonormal basis such that $\psi_1 = e_1$ and $$\label{}
\psi_2 = e^{i\lambda} s e_1 + c e_2\ ,$$ with $s = \sin\alpha$, $c=\cos\alpha$ for some angle $\alpha$. Now, completing the basis $\{e_1,e_2,e_3,\ldots,e_k\}$ in $\mathbb{C}^k$ one easily finds that the matrix elements of $\widetilde{A}$ has a form of the following direct sum $$\label{}
\widetilde{A} = \left( \begin{array}{c|c} e^{-i\lambda} s & c \\ \hline
- c & e^{i\lambda}s \end{array} \right)\, \oplus\, e^{-i\lambda}s
\mathbb{I}_{k-2} \ .$$ Hence $$\label{}
\widetilde{A}\widetilde{A}^\dagger = \mathbb{I}_2 \, \oplus \, s^2
\mathbb{I}_{k-2}\ ,$$ which proves (\[AAI\]) since all eigenvalues of $\widetilde{A}\widetilde{A}^\dagger$ – $\{1,1,s^2,\ldots,s^2\}$ – are bounded by 1.
Now, our new positive maps can be useful in detecting entanglement only if they are not completely positive. It is easy to check that the corresponding Choi matrix $$\label{W2k}
W_{2k} = \sum_{i,j=1}^{2k} e_{ij} {{\,\otimes\,}}\Psi^0_{2k}(e_{ij})\ ,$$ possesses 2 negative eigenvalues $\{-1,(2-k)/k\}$ (unless $k=2$). Hence, (\[W2k\]) defines true entanglement witness in $\mathbb{C}^{2k} {{\,\otimes\,}}\mathbb{C}^{2k}$. As usual using Dirac notation we define $e_{kl} := |e_k\>\<e_l|$. Note, that the corresponding Brauer-Hall witness possesses only one negative eigenvalue “$-1$". Hence these two classes are different (unless $k=2$).
Properties of new entanglement witnesses {#PROP}
========================================
In this section we study basis properties of $W_{2k}$.
$W_{2k}$ are indecomposable
---------------------------
To show that $W_{2k}$ is indecomposable one needs to define a PPT state $\rho$ in $\mathbb{C}^{2k} {{\,\otimes\,}}\mathbb{C}^{2k}$ such that ${\rm Tr}(W_{2k} \rho_{2k}) < 0$. Consider the following operator $$\label{our-rho}
\rho_{2k} = \sum_{i,j=1}^{2k} e_{ij} {{\,\otimes\,}}\rho^{(2k)}_{ij}\ ,$$ where the $2k \times 2k$ blocks are defined as follows: diagonal blocks $$\label{}
\rho^{(2k)}_{ii} = N_k \left( \begin{array}{c|c} k\mathbb{I}_k
& \mathbb{O}_k \\ \hline \mathbb{O}_k & \mathbb{I}_k
\end{array} \right)\ ,$$ for $i=1,\ldots,k$, and $$\label{}
\rho^{(2k)}_{ii} = N_k \left( \begin{array}{c|c} \mathbb{I}_k
& \mathbb{O}_k \\ \hline \mathbb{O}_k & k\mathbb{I}_k
\end{array} \right)\ ,$$ for $i=k+1,\ldots,2k$. The off-diagonal blocks are form: $$\label{}
\rho^{(2k)}_{i,i+k} = - N_k W^{(2k)}_{i,i+k} \ ,$$ for $i=1,\ldots,k$, $$\label{}
\rho^{(2k)}_{ij} = N_k e_{ij} \ ,$$ for $i=1,\ldots,k$, $j=k+1,\ldots,2k$ and $j\neq i+k$ and $$\label{}
\rho^{(2k)}_{ij} = \mathbb{O}_k\ ,$$ otherwise. The normalization factor $N_k$ is given by $$1/N_k = 2k^2(k+1)\ .$$ Direct calculation shows that $$\rho \geq 0\ , \ \ \ \rho^\Gamma \geq 0 \ , \ \ \ {\rm Tr}\rho =
1\ .$$ For example for $k=2$ one obtains the following density operator
$$\label{}
\rho_{4} = \frac{1}{24} \left( \begin{array}{cccc|cccc|cccc|cccc}
2& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& 1\\
\cdot& 2& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\
\cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\
\cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \hline
\cdot& \cdot& \cdot& \cdot& 2 & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\
\cdot& \cdot& \cdot& \cdot& \cdot& 2 & \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& 1 \\
\cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot \\
\cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \hline
\cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\
\cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\
1& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& 2 & \cdot& \cdot& \cdot& \cdot& \cdot \\
\cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 2 & \cdot& \cdot& \cdot& \cdot \\ \hline
\cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot \\
\cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot \\
\cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 2& \cdot \\
1& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 2 \end{array}
\right)\ .$$
One easily finds for the trace $$\label{}
{\rm Tr}(W_{2k} \rho_{2k} ) = - \frac{k-1}{k^2(k+1)}\ ,$$ which proves indecomposability of $W_{2k}$.
$W_{2k}$ are atomic
-------------------
In order to prove that $W_{2k}$ is atomic one has to define a PPT state $D_{2k}$ such that Schmidt rank of $D_{2k}$ and of its partial transposition $D_{2k}^\Gamma$ is bounded by 2 and show that ${\rm
Tr}(W_{2k} D_{2k})< 0$. It is clear that atomicity implies indecomposability but for clarity of presentation we treat these two notions independently. Let us introduce the following family of product vectors $$\begin{aligned}
\phi_1 &=& e_1 {{\,\otimes\,}}e_1 \ ,\\
\phi_2 &=& e_1 {{\,\otimes\,}}e_{k+1} \ , \\
\phi_3 &=& e_k {{\,\otimes\,}}e_1 \ , \\
\phi_4 &=& e_k {{\,\otimes\,}}e_{2k} \\
\phi_5 &=& e_{k+1} {{\,\otimes\,}}e_1\ , \\
\phi_6 &=& e_{k+1} {{\,\otimes\,}}e_{k+1}\ , \\
\phi_7 &=& e_{k+1} {{\,\otimes\,}}e_{2k}\ .\end{aligned}$$ Define now the following positive operator $$\begin{aligned}
\label{D}
D_{2k} &=& \frac 17 \Big( |\phi_1 + \phi_6\>\<\phi_1 + \phi_6| + |\phi_5 - \phi_4\>\<\phi_5 -
\phi_4| \nonumber \\ &+& |\phi_2\>\<\phi_2| + |\phi_3\>\<\phi_3| +
|\phi_7\>\<\phi_7| \Big)\ .\end{aligned}$$ One easily finds for its partial transposition $$\begin{aligned}
\label{DG}
D_{2k}^\Gamma &=& \frac 17 \Big( |\phi_2 + \phi_5\>\<\phi_2 + \phi_5| + |\phi_3 - \phi_7\>\<\phi_3 -
\phi_7| \nonumber \\ &+& |\phi_1\>\<\phi_1| + |\phi_4\>\<\phi_4| +
|\phi_6\>\<\phi_6| \Big)\ .\end{aligned}$$ Now, it is clear from that both $D_{2k}$ and $D_{2k}^\Gamma$ are constructed out of rank-1 projectors and Schmidt rank of each projector is 1 or 2. Therefore $${\rm SN}(D_{2k}) \leq 2 \ , \ \ \ {\rm SN}(D^\Gamma_{2k}) \leq 2
\ .$$ Finally, one finds for the trace $$\label{}
{\rm Tr}(W_{2k}D_{2k}) = - \frac{1}{7k} \ ,$$ which shows that $W_{2k}$ defines atomic entanglement witness.
$W_{2k}$ are optimal
--------------------
To show that $W_{2k}$ is optimal we use the following result Lewenstein et. al. [@Lew1]: if the family of product vectors $\psi {{\,\otimes\,}}\phi \in \mathbb{C}^{2k} {{\,\otimes\,}}\mathbb{C}^{2k}$ satisfying $$\label{}
\< \psi {{\,\otimes\,}}\phi| W|\psi {{\,\otimes\,}}\phi \> = 0 \ ,$$ span the total Hilbert space $\mathbb{C}^{2k} {{\,\otimes\,}}\mathbb{C}^{2k}$, then $W$ is optimal. Let us introduce the following sets of vectors: $$\begin{aligned}
f_{mn} = e_m + e_n \ ,\end{aligned}$$ and $$\begin{aligned}
g_{mn} = e_m + i e_n \ ,\end{aligned}$$ for each $1\leq m < n \leq 2k$. It is easy to check that $(2k)^2$ vectors $\psi_\alpha {{\,\otimes\,}}\psi^*_\alpha$ with $\psi_\alpha$ belonging to the set $$\{\, e_l\, , f_{mn}\, , g_{mn} \, \}\ ,$$ are linearly independent and hence they do span $\mathbb{C}^{2k} {{\,\otimes\,}}\mathbb{C}^{2k}$. Direct calculation shows that $$\label{}
\< \psi_\alpha {{\,\otimes\,}}\psi^*_\alpha| W|\psi_\alpha {{\,\otimes\,}}\psi^*_\alpha \> = 0 \ ,$$ which proves that $W_{2k}$ is an optimal EW.
$W_{2k}$ have circulant structure
---------------------------------
Finally, let us note that $W_{2k}$ displays so called circulant structure [@PPT-nasza; @CIRCULANT]. Let $\{e_1,\ldots,e_d\}$ be an orthonormal basis in $\mathbb{C}^d$ and let $S : \mathbb{C}^d
\rightarrow \mathbb{C}^d$ be a shift operator (elementary permutation) defined by $$\label{}
S\, e_j = e_{j+1} \ , \ \ \ j=1,\ldots,d\ \ \ ({\rm mod} \ d) \ .$$ Now, introducing $$\label{}
\Sigma_0 = {\rm span} \{ e_1{{\,\otimes\,}}e_1, \ldots, e_d {{\,\otimes\,}}e_d\} \ ,$$ define $$\label{}
\Sigma_\alpha = (\oper {{\,\otimes\,}}S^\alpha)\Sigma_1 \ , \ \ \
\alpha=0,\ldots,d-1\ .$$ A bipartite operator $X : \mathbb{C}^d {{\,\otimes\,}}\mathbb{C}^d \rightarrow
\mathbb{C}^d {{\,\otimes\,}}\mathbb{C}^d$ displays circulant structure if $$\label{}
X = X_0 \oplus \ldots \oplus X_{d-1}\ ,$$ such that each $X_\alpha$ is supported on $\Sigma_\alpha$. It is therefore clear that $$\label{}
X_\alpha = \sum_{i,j=1}^d x^{(\alpha)}_{ij} e_{ij} {{\,\otimes\,}}S^\alpha
e_{ij} S^{\alpha\,\dagger} \ ,$$ where $[x^{(\alpha)}_{ij}]$ is $d {{\,\otimes\,}}d$ complex matrix for each $\alpha=0,\ldots,d-1$, i.e. a circulant bipartite operator $X$ is uniquely defined by the collection of $d$ complex matrices $[x^{(\alpha)}_{ij}]$.
Structural physical approximation {#SPA}
=================================
It is well know that positive maps cannot be directly implemented in the laboratory. The idea of [*structural physical approximation*]{} (SPA) [@SPA1; @SPA2] is to mix a positive map $\Lambda$ with some completely positive map making the mixture $\widetilde{\Lambda}$ completely positive. In the recent paper [@SPA3] the authors analyze SPA to a positive map $\Lambda : \mathcal{\mathcal{H}_A}
\rightarrow \mathcal{\mathcal{H}_B}$ obtained through minimal admixing of white noise $$\label{}
\widetilde{\Lambda}(\rho) = p \frac{\mathbb{I}_B}{d_B} \, {\rm
Tr}(\rho) + (1-p) \Lambda(\rho)\ .$$ The minimal means that the positive mixing parameter $0 < p < 1$ is the smallest one for which the resulting map $\widetilde{\Lambda}$ is completely positive, i.e. it defines a quantum channel. Equivalently, one may introduce SPA of an entanglement witness $W$: $$\label{}
\widetilde{W} = \frac{p}{d_A d_B} \mathbb{I}_A {{\,\otimes\,}}\mathbb{I}_B
+ (1-p) W \ ,$$ where $p$ is the smallest parameter for which $\widetilde{W}$ is a positive operator in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$, i.e. it defines (possibly unnormalized) state.
It was conjectured [@SPA3] that SPA to optimal positive maps correspond to entanglement breaking maps (channels). Equivalently, SPA to optimal entanglement witnesses correspond to separable (unnormalized) states. It turns out that the family of optimal maps/witnesses constructed in this paper does support this conjecture.
The corresponding SPA of $W_{2k}$ is given by $$\label{}
\widetilde{W}_{2k} = \frac{p}{(2k)^2} \mathbb{I}_{2k} {{\,\otimes\,}}\mathbb{I}_{2k} + (1-p) W_{2k} \ .$$ Using the fact that the maximal negative eigenvalue of $W_{2k}$ equals “$-1$” one easily finds the following condition for the positivity of $\widetilde{W}_{2k}$ $$\label{p}
p \geq \frac{d}{d+1}\ ,$$ with $d = 2k$. Surprisingly, one obtains the same bound for $p$ as in Eqs. (26), (33) and (65) in [@SPA3].
Now, to show that SPA of $\Psi^0_{2k}$ (or eqivalently $W_{2k}$) is entanglement breaking (equivalently separable) we use the following
[@AA] Let $\Lambda : \mathcal{B}(\mathbb{C}^d) \rightarrow
\mathcal{B}(\mathbb{C}^d)$ be a positive unital map. Then SPA of $\Lambda$ is entanglement breaking if $\Lambda$ detects all entangled isotropic states in $\mathbb{C}^d {{\,\otimes\,}}\mathbb{C}^d$.
Indeed, let $$\label{}
\rho_p = \frac{p}{d^2}\, \mathbb{I}_d {{\,\otimes\,}}\mathbb{I}_d + (1-p)
P^+_d\ ,$$ be an isotropic state which is known to be entangled iff $$\label{p-iso}
p < \frac{1}{d+1}\ .$$ Now, assume that $(\oper {{\,\otimes\,}}\Lambda) \rho_p$ is not positive if $\rho_p$ is entangled. Using $\Lambda(\mathbb{I}_d) =
\mathbb{I}_d$ one obtains $$\label{}
(\oper {{\,\otimes\,}}\Lambda) \rho_p = \frac{p}{d^2}\, \mathbb{I}_d {{\,\otimes\,}}\mathbb{I}_d + (1-p) W\ ,$$ that is, $(\oper {{\,\otimes\,}}\Lambda) \rho_p = \widetilde{W}$. Now, if $\widetilde{W}$ is positive, then $\rho_p$ has to be separable (otherwise it would be detected by $\Lambda$). But since $\oper {{\,\otimes\,}}\Lambda$ sends separable states into separable states one concludes that $\widetilde{W}$ is separable (or equivalently $\widetilde{\Lambda}$ is entanglement breaking).
If in addition $\Lambda$ is self-dual, i.e. $$\label{self}
{\rm Tr}(A \cdot \Lambda(B)) = {\rm Tr}(\Lambda(A) \cdot B)\ ,$$ for all $A,B \in \mathcal{B}(\mathbb{C}^d)$, then it is enough to check whether all entangled isotropic states are detected by the witness $W$, i.e. ${\rm Tr}(W \rho_p) < 0$ for all $p$ satisfying (\[p-iso\]).
Again, the proof is very easy. One has $$\begin{aligned}
\label{}
{\rm Tr}(\rho_p \cdot W) &=& {\rm Tr}(\rho_p \cdot (\oper {{\,\otimes\,}}\Lambda)P^+_d)\nonumber \\ &=& {\rm Tr}((\oper {{\,\otimes\,}}\Lambda)\rho_p \cdot P^+_d)\ ,\end{aligned}$$ where in the last equality we used the self-duality of $\Lambda$. Now, if ${\rm Tr}(\rho_p \cdot W) < 0$ for $p$ satisfying (\[p-iso\]), then $(\oper {{\,\otimes\,}}\Lambda)\rho_p$ is not positive (otherwise its trace with the projector $P^+_d$ would be positive). Hence by Lemma 1 SPA $\widetilde{\Lambda}$ is entanglement breaking.
We are prepared to show that SPA for $\Psi^0_{2k}$ is entanglement breaking.
$\Psi^0_{2k}$ is self-dual.
One checks by direct calculations that $$\label{}
{\rm Tr}(e_{kl} \cdot \Psi^0_{2k}(e_{mn})) = {\rm Tr}(\Psi^0_{2k}(e_{kl}) \cdot e_{mn})\ ,$$ for all $k,l,m,n=1,\ldots,2k$. Hence, due to the Lemma 2, to show that SPA for $\Psi^0_{2k}$ is entanglement breaking is it enough to prove
${\rm Tr}(W_{2k} \rho_p) < 0$ for all $p$ satisfying (\[p-iso\]).
To prove it let us note that $$\begin{aligned}
\label{}
{\rm Tr}(W_{2k} \rho_p) = \frac{p}{(2k)^2} {\rm Tr}\,W_{2k} + (1-p)
{\rm Tr}(W_{2k} P^+_{2k})\ .\end{aligned}$$ Now, ${\rm Tr}\,W_{2k}=2k$, and $$\label{}
{\rm Tr}(W_{2k} P^+_{2k}) = \frac{1}{2k} \sum_{m,n=1}^{2k} \<
m|\Psi^0_{2k}(e_{mn})|n\> \ .$$ Finally, using definition of $\Psi^0_{2k}$ one gets $$\label{}
\sum_{m,n=1}^{2k} \< m|\Psi^0_{2k}(e_{mn})|n\> = - 2k\ ,$$ and hence $$\label{}
{\rm Tr}(W_{2k} \rho_p) = \frac{p(d+1) -1}{d}\ ,$$ with $d=2k$. Now, if $\rho_p$ is entangled, i.e. $p < 1/(d+1)$, then ${\rm Tr}(W_{2k} \rho_p) < 0$ which shows that $W_{2k}$ detects all entangled isotropic states.
Conclusions
===========
We have provided a new construction of EWs in $\mathbb{C}^d {{\,\otimes\,}}\mathbb{C}^d$ with $d=2k$. It was shown that these EWs are indecomposable, i.e. they are able to detect PPT entangled state. Moreover, they are so called atomic EWs, i.e. they able to detect PPT entangled states $\rho$ such that both $\rho$ and $\rho^\Gamma$ possess Schmidt number 2. The crucial property of our witnesses is they optimality, i.e. they are no other witnesses which can detect more entangled states.
Equivalently, our construction gives rise to the new class of positive maps in algebras of $d \times d$ complex matrices. For $k=2$ this construction reproduces old example of Robertson map [@Robertson] and hence [@atomic] defines the special case of Brauer-Hall maps [@Breuer; @Hall].
Let us observe that if $\Lambda : \mathcal{B}(\mathbb{C}^d)
\rightarrow \mathcal{B}(\mathbb{C}^d)$ is a positive indecomposable map then for any unitaries $U_1,U_2 : \mathbb{C}^d \rightarrow
\mathbb{C}^d$ a new map $$\label{}
\Lambda^{U_1U_2}(A) := U_1 \Lambda(U_2^\dagger A
U_2)U_1^\dagger\ ,$$ is again positive and indecomposable [@atomic]. This observation enables one to generalize Robertson map $\Phi^0_{2k}$ and our new map $\Psi^0_{2k}$ to $\Phi^{U_1U_2}_{2k}$ and $\Psi^{U_1U_2}_{2k}$. Note, that if $U_1 = U_2 = U$ given by (\[BH\]), then $\Phi^{U}_{2k} := \Phi^{UU}_{2k}$ defines Breuer-Hall map. Therefore, $\Psi^{U}_{2k} := \Psi^{UU}_{2k}$ may be regarded as a Breuer-Hall-like generalization of our primary map $\Psi^0_{2k}$.
It should be stressed , an EW defined by the Breuer-Hall map and EW $W_{2k}$ introduces in this paper are different, i.e. they do detect different classes of PPT entangled states. Direct calculation shows that the PPT entangled state (\[our-rho\]) is not detected by the Breuer-Hall witness. On the other hand consider the family of PPT entangled state introduced in [@Breuer] $$\label{lambda}
\rho(\lambda) = \lambda P^+_d + (1-\lambda) \rho_0\ ,$$ with $$\label{}
\rho_0 = \frac{2}{d(d+1)}\, U_0 P_S U_0^\dagger\ ,$$ and $P_S$ being the projector onto the subspace of states symmetric under the swap operation. It was shown that $\rho(\lambda)$ is PPT for $0 \leq \lambda \leq 1(d+2)$. Moreover, Breuer-Hall witness detects all entangled states within $\lambda$-family (\[lambda\]) (both PPT and NPT). Direct calculation shows that our witness $W_{2k}$ does not detect PPT entangled states in (\[lambda\]).
Interestingly, the partial transposition $W_{2k}^\Gamma$ defines an EW with $k(k-1)/2$ negative eigenvalues (all equal to ‘$-1$’). For $k=2$ it gives exactly one negative eigenvalue (the fact well known from the family of Breuer-Hall maps in 4 dimensions). Therefore, this example provide an EW with multiple negative eigenvalues. However, contrary to the Breuer-Hall maps we were not able to show that $W_{2k}^\Gamma$ is optimal.
We have shown that structural physical approximation for our new class of positive maps gives rise to entanglement breaking channels. This result supports recent conjecture by Korbicz et. al. [@SPA3].
Finally, let us mention some open questions. In this paper we have used reduction map as a building block to construct other optimal maps. Can we use other positive maps as building blocks? Is it true that properties of building blocks (like optimality and/or atomicity) are shared by the map which is built out of them?
Acknowledgement {#acknowledgement .unnumbered}
===============
We thank Antonio Acin and Andrzej Kossakowski for valuable discussions. This work was partially supported by the Polish Ministry of Science and Higher Education Grant No 3004/B/H03/2007/33 and by the Polish Research Network [*Laboratory of Physical Foundations of Information Processing*]{}.
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[^1]: email: darch@phys.uni.torun.pl
|
---
abstract: 'Growing observational evidence now indicates that nebular line emission has a significant impact on the rest-frame optical fluxes of $z\sim5-7$ galaxies observed with Spitzer. This line emission makes $z\sim5-7$ galaxies appear more massive, with lower specific star formation rates. However, corrections for this line emission have been very difficult to perform reliably due to huge uncertainties on the overall strength of such emission at $z\gtrsim5.5$. Here, we present the most direct observational evidence yet for ubiquitous high-equivalent width (EW) \[OIII\]+H$\beta$ line emission in Lyman-break galaxies at $z\sim7$, while also presenting a strategy for an improved measurement of the sSFR at $z\sim7$. We accomplish this through the selection of bright galaxies in the narrow redshift window $z\sim6.6-7.0$ where the IRAC 4.5 micron flux provides a clean measurement of the stellar continuum light. Observed 4.5 micron fluxes in this window contrast with the 3.6 micron fluxes which are contaminated by the prominent \[OIII\]+H$\beta$ lines. To ensure a high S/N for our IRAC flux measurements, we consider only the brightest ($H_{160}<26$ mag) magnified galaxies we have identified in CLASH and other programs targeting galaxy clusters. Remarkably, the mean rest-frame optical color for our bright seven-source sample is very blue, $[3.6]-[4.5]=-0.9\pm0.3$. Such blue colors cannot be explained by the stellar continuum light and require that the rest-frame EW of \[OIII\]+H$\beta$ be greater than 637[Å]{} for the average source. The bluest four sources from our seven-source sample require an even more extreme EW of 1582[Å]{}. Our derived lower limit for the mean \[OIII\]+H$\beta$ EW could underestimate the true EW by $\sim2\times$ based on a simple modeling of the redshift distribution of our sources. We can also set a robust lower limit of $\gtrsim\rm 4\,Gyr^{-1}$ on the specific star formation rates based on the mean SED for our seven-source sample. Planned follow-up spectroscopy of our sample and deeper IRAC imaging with the SURF’S Up program will further improve these results.'
author:
- 'R. Smit, R. J. Bouwens, I. Labb[é]{}, W. Zheng, L. Bradley, M. Donahue, D. Lemze, J. Moustakas, K. Umetsu, A. Zitrin, D. Coe, M. Postman, V. Gonzalez, M. Bartelmann, N. Ben[í]{}tez, T. Broadhurst, H. Ford, C. Grillo, L. Infante, Y. Jimenez-Teja, S. Jouvel, D.D. Kelson, O. Lahav, D. Maoz, E. Medezinski, P. Melchior, M. Meneghetti, J. Merten, A. Molino, L. Moustakas, M. Nonino, P. Rosati, S. Seitz'
title: 'EVIDENCE FOR UBIQUITOUS HIGH-EW NEBULAR EMISSION in $\lowercase{z}\sim7$ GALAXIES: TOWARDS A CLEAN MEASUREMENT of the SPECIFIC STAR FORMATION RATE USING A SAMPLE OF BRIGHT, MAGNIFIED GALAXIES'
---
Introduction {#sec:intro}
============
In the last decade the evolution of galaxies at the earliest times has been predominantly mapped out by studying the rest-frame UV light in galaxies across cosmic time [e.g. @Stanway2003; @Bouwens2007; @Bouwens2011; @Lorenzoni2011; @Oesch2012; @Oesch2013; @Bradley2012b; @Bowler2012; @Schenker2013b]. Despite great progress in this area, an equally important part of the story regards the build-up of mass in galaxies and the specific star formation rate (sSFR, i.e. the star formation rate divided by the stellar mass), which provide direct constraints on the growth time scale of individual sources [@Stark2009; @Gonzalez2010]. Typical sSFRs of star-forming galaxies at $z\sim2$ ($M_\ast\sim 5\times10^9M_\odot$) are $\rm\sim2\,Gyr^{-1}$, equivalent to a doubling time of $\sim500\rm\,Myr$.
Over the last few years, there has been a substantial improvement in our characterization of the sSFR in high-redshift galaxies and how it evolves. Initial observational studies found little evolution in the sSFR from $z\sim2$ to higher redshift in apparent disagreement with theories of star formation fueled by cold accretion (Stark et al. 2009; Gonzalez et al. 2010; Labbe et al. 2010a,b). However, the effect of nebular emission lines (e.g., \[OIII\], \[OII\], H$\alpha$) that can contaminate the IRAC measurements of the stellar continuum light had not been taken into account [e.g. @Schaerer2009; @Schaerer2010].
The effect of this emission on broadband IRAC measurements can be quite considerable. Extrapolating the H$\alpha$ EWs measured by @Fumagalli2012 and @Erb2006 to higher redshifts suggests H$\alpha$ EWs as large as 1000[Å]{} at $z\gtrsim6$. This would indicate that $\sim$45% of the flux in \[4.5\] is due to H$\alpha$ for galaxies at $z\sim6-7$, while \[OIII\]+H$\beta$ can contribute $\sim$55% of the flux in \[3.6\]. Correcting for the effects of nebular emission, one can derive sSFRs which are plausibly consistent with theoretical expectations [@Stark2013; @Gonzalez2012; @deBarros2012].
As the previous discussion indicates, it is essential in quantifying the sSFR at $z>5$ to characterize the EWs of nebular emission lines and their impact on the IRAC photometry. Pioneering studies in the last two years have quantified the strength of nebular emission lines at $z\gtrsim4$, through the measured flux offsets to the Spitzer/IRAC \[3.6\] and \[4.5\] bands. @Shim2011 compare the \[3.6\] and \[4.5\] fluxes at $z\sim4$ and show that the $[3.6]-[4.5]$ color correlates with the star formation rate (SFR), implying that the source of the offset is likely due to the presence of H$\alpha$ emission lines. @Stark2013 estimate the influence of H$\alpha$ on the \[3.6\] flux at $z\sim3-4$ by comparing the color distribution of contaminated and uncontaminated spectroscopic confirmed galaxies [see also @Schenker2013b] and extrapolating the observed emission line contamination to $z\sim5-7$.
The first attempt to derive nebular line EWs for a large sample of Lyman-break galaxies at $z\gtrsim5$ is presented in @Labbe2012, based on a comparison of a stacked \[3.6\] and \[4.5\] flux measurement at $z\sim8$ from the IRAC Ultra Deep Field (IUDF) program with similar flux measurements from a stacked sample at $z\sim7$ (see also @Gonzalez2012a who make inferences about the EWs of nebular emission lines from the stacks of $z\sim4-6$ galaxies). Estimates of the nebular-line EWs have also been made from direct fits to large number of spectroscopic ally-confirmed $z\sim4-7$ galaxies [@deBarros2012; @Ono2012; @Tilvi2013; @Curtis2013]
Even making use of the above methods, the sSFR in $z\sim6-8$ galaxies is still very uncertain. While one can certainly estimate the sSFR in this redshift range by utilizing an extrapolation of the H$\alpha$ EWs found at $z\sim4$ to higher redshift, extrapolations are inherently uncertain. Results on the sSFR at $z\sim8$ [@Labbe2012], though providing good leverage to constrain the redshift evolution, are limited by the extreme faintness of the individual galaxies whose redshift distribution is only approximately known. Finally, the typical H$\alpha$ EW in $z\sim4$ galaxies used for sSFR estimates has been established primarily through sources which show Ly$\alpha$ in emission; however, it is unclear if those sources are representative of the broader $z\sim4$ population (for more discussion see @Schenker2013b).
To overcome these issues, here we make use of a new strategy for measuring the sSFRs and stellar masses for galaxies at very high redshifts, while simultaneously obtaining very good constraints on the EWs of \[OIII\]+H$\beta$ line emission. Our plan is to take advantage of the considerable quantity of deep, wide-area observations over the 524-orbit, 25-cluster Cluster Lensing And Supernova survey with Hubble (CLASH) program [@Postman2012] and other programs observing strong lensing clusters with deep multiband HST data. We select a small sample of bright, magnified galaxies for which we can obtain a clean measurement of the stellar continuum light from the deep IRAC observations over these clusters. One particularly fruitful redshift window in which we can obtain such clean measurements is the redshift window $z\sim6.6-7.0$, where \[4.5\] is completely free of any emission lines. This should allow us to place much more robust constraints on the sSFR and the EW of nebular emission of star-forming galaxies at $z\sim7$.
This paper is organized as follows. In §\[sec:Observations\] we discuss our data set, our photometric procedure, and source selection. In §\[sec:Results\] we present the properties of our selected $z\sim7$ sample. We discuss the constraints we put on the EWs of H$\alpha$, H$\beta$ and \[OIII\] and the sSFR. We present a summary and discussion of our results in §\[sec:Summary\].
Throughout this paper we adopt a Salpeter IMF with limits 0.1-100$\,M_\odot$ [@Salpeter]. For ease of comparison with previous studies we take $H_0=70\,\rm km\,s^{-1}\,Mpc^{-1},\,\Omega_{\rm{m}}=0.3\,$and$\,\Omega_\Lambda=0.7$. Magnitudes are quoted in the AB system [@OkeGun]
Observations {#sec:Observations}
============
Data
----
In selecting our small sample of bright, magnified $z\sim7$ galaxies, we make use of the deep HST observations available over the first 23 clusters in the CLASH multi-cycle treasury program (GO \#12101: PI Postman), Abell 1689 and Abell 1703 (GO \#11802: PI Ford), the Bullet cluster (GO \#11099: PI Bradac), and 9 clusters from the Kneib et al. (GO \#11591) program. The CLASH cluster fields are each covered with 20-orbit HST observations spread over 16 bands using the Advanced Camera for Surveys (ACS: $B_{435},\,g_{475},\,V_{606},\,r_{625},\,i_{775},\,I_{814},$ and $z_{850}$), Wide Field Camera WFC3/UVIS ($UV_{225},\,UV_{275},\,U_{336}$ and $U_{390}$) and WFC3/IR instrument ($Y_{105},\,J_{110},\,J_{125},\, JH_{140}$ and $H_{160}$). Abell 1703 was covered with 22 orbits of ACS and WFC3/IR ($B_{435},\,g_{475},\,V_{606},\,r_{625},\,i_{775},\,z_{850},\,J_{125},\,H_{160}$) while clusters in the Kneib et al. program were covered with 6 orbits ($I_{814},\,J_{110},\,H_{160}$). HST mosaics were produced using the Mosaicdrizzle pipeline (see @Koekemoer2011 for further details), and individual bands in the deep imaging data reach 5$\sigma$ depths of 26.4-27.7 mag (0.4"-diameter aperture). Deep Spitzer/IRAC observations of our fields in the \[3.6\] and \[4.5\] bands were provided for by the ICLASH (GO \#80168: @Bouwens2011b) and Spitzer IRAC Lensing Survey program (GO \#60034: PI Egami). The typical exposure time per cluster was 3.5 to 5 hours per band, allowing us to reach 26.5 mag at 1 sigma. Reductions of the IRAC observations used in this paper were performed with MOPEX [@Makovoz2005].
![The impact of emission lines on the \[3.6\] and \[4.5\] band fluxes and our strategy for deriving sSFRs and \[OIII\]+H$\beta$ EWs from our $z\sim7$ sample. *Top panel:* The redshift range over which strong nebular emission lines, H$\alpha$, H$\beta$, \[OIII\] and \[OII\], will contaminate the \[3.6\] and \[4.5\] flux of galaxies. *Bottom panel:* The expected $[3.6]-[4.5]$ colors as a function of redshift due to nebular emission lines. The solid and dotted lines show the expected color assuming relatively low EWs, i.e., EW$_0$(\[OIII\]+H$\beta)\sim140$Å, and assuming strong evolution, i.e., EW$_0$(\[OIII\]+H$\beta)\propto(1+z)^{1.8}$[Å]{} [@Fumagalli2012], respectively, similar to the models considered in @Gonzalez2012 and @Stark2013. We select sources in the redshift range $z_{phot}=6.6-7.0$, where \[OIII\]$\lambda\lambda$4959,5007 and H$\beta$ are present in \[3.6\], while \[4.5\] receives no significant contamination from nebular emission lines, falling exactly in between the H$\alpha$ and \[OIII\] lines. The red solid circles and 1 sigma upper limit show the observed colors in our sample. We find that most sources show blue $[3.6]-[4.5]$ colors, falling in the range between our two models. Four sources from our sample exhibit extremely blue rest-frame optical colors, with $[3.6]-[4.5]\lesssim-0.8$, indicating contamination of \[OIII\]+H$\beta$ with a mean EW of $\gtrsim1582$[Å]{} (see §\[sec:stack\]), even higher than using the @Fumagalli2012 extrapolation indicated by the dotted line. Two sources at $z\sim6.75$ have been offset by $\Delta z=0.05$ for clarity. []{data-label="fig:emlines"}](fig1.ps){width="0.9\columnwidth"}
Photometry and Selection {#sec:phot}
------------------------
The photometry we obtain for sources in our cluster fields follows a similar procedure as described in @Bouwens2012. In short, we run the SExtractor software [@Bertin1996] in dual-image mode. The detection images are constructed from all bands redwards of the Lyman break (i.e. $Y_{105},\,J_{110},\,J_{125},\, JH_{140}$ and $H_{160}$). After PSF-matching the observations to the $H_{160}$-band PSF, colors are measured in Kron-like apertures and total magnitudes derived from 0.6"-diameter circular apertures.
Our initial source selection is based on the Lyman-break technique [@Steidel1999], with the requirement that the source drops out in the $I_{814}$ band. Specifically, our requirements for $z\sim6-7$ sources are $$(I_{814}-J_{110}>0.7)\,\wedge\,(J_{110}-JH_{140}<0.45).$$ For sources in the CLASH program we require $H_{160}<26$ AB, while we select sources to the brighter magnitude limit $H_{160}<25$ AB in all other fields to ensure good photometric redshift constraints for all our sources. We also require sources to have either a non-detection in the $V_{606}$ band ($<2\sigma$) or to have a very strong Lyman break, i.e. $V_{606}-J_{125}>2.5$. We require sources to be undetected in the optical $\chi^2$ image [@Bouwens2011] we construct from the observations bluewards of the $r_{625}$ band. Finally we require the SExtractor stellarity parameter (equal to 0 and 1 for extended and point sources, respectively) in the $ J_{110}$ band be less than 0.92 to ensure that our selection is free of contamination by stars.
To identify those sources where we can obtain clean rest-frame optical stellar continuum, we also require that sources have a best-fit photometric redshift between $z=6.6$ and $7.0$, as determined by the photometric redshift software EAZY [@Brammer2008]. All available HST photometry (i.e. 16 bands for CLASH clusters) is used in the redshift determinations. No use of the Spitzer/IRAC photometry is made in the photometric redshift determination to avoid coupling the selection of our sources to the $[3.6]-[4.5]$ colors we will later measure. We use templates of young stellar populations with no Ly$\alpha$ emission.
Strong Ly$\alpha$ emission can systematically influence the photometric redshift estimate. However, we emphasize that any potential sources from outside our desired redshift interval that could be in our sample due to uncertainties in the photometric redshift estimate would only serve to increase the flux in the \[4.5\] band and redden the $[3.6]-[4.5]$ color (i.e. due to contamination in the \[4.5\] band of H$\alpha$ at $z<6.6$ and \[OIII\]+H$\beta$ at $z>7.0$). Correcting for this possible source of interlopers would result in higher EWs and sSFRs than in the case of no contamination. This reinforces the point we will make in §\[sec:Results\] that the EWs we derive for the \[OIII\]+H$\beta$ emission and the sSFRs are strong lower limits on the actual values.
Figure \[fig:emlines\] shows the redshift range where we would expect the strongest emission lines, H$\alpha$, H$\beta$, \[OIII\]$\lambda\lambda4959,5007$ and \[OII\]$\lambda3727$, to impact the \[3.6\] and \[4.5\] fluxes. The top panel indicates which lines fall in specific IRAC filters at a given redshift, while the bottom panel indicates the estimated $[3.6]-[4.5]$ color offset due to the various emission lines. We select sources in the redshift range $z_{phot}=6.6-7.0$, where we know that both \[OIII\] and H$\beta$ fall in \[3.6\], while \[4.5\] falls exactly between \[OIII\] and H$\alpha$ where no significant emission lines are present (see for example Figure \[fig:emlines\_example\]).
![HST $H_{160}$, Spitzer/IRAC \[3.6\], and \[4.5\] postage stamp images (6.5“ $\times$ 6.5”) of our sample of bright, magnified $z\sim6.6-7.0$ galaxies behind clusters. The IRAC postage stamps have already been cleaned for contamination from neighboring sources (§\[sec:iracphot\]). It is obvious that a large fraction of the sources in our selection are much brighter at 3.6$\mu$m than at 4.5$\mu$m.[]{data-label="fig:exampleclean"}](figure_stamps_z7.eps){width="1.\columnwidth"}
{width="60.00000%"}
\[fig:results\_sed\]
IRAC Photometry {#sec:iracphot}
---------------
Photometry of sources in the available Spitzer/IRAC data over our fields is challenging, due to blending with nearby sources from the broad PSF. We therefore use the automated cleaning procedure described in @Labbe2010a [@Labbe2010b]. In short, we use the high-spatial resolution HST images as a template with which to model the positions and flux profiles of the foreground sources. The flux profiles of individual sources are convolved to match the IRAC PSF and then simultaneously fit to all sources within a region of $\sim$13“ around the source. Flux from all the foreground galaxies is subtracted and photometry is performed in 2.5”-diameter circular apertures. We apply a factor of $\sim2.0\times$ correction to account for the flux outside of the aperture, based on the radial light profile of the PSF. Figure \[fig:exampleclean\] shows the cleaned IRAC images of our sample. Our photometric procedure fails when contaminating sources are either too close or bright. Sources with badly subtracted neighbors are excluded. In total, clean photometry is obtained for 78% of the sources, resulting in 7 sources in our final selection (excluding only one source behind RXJ1347 and one source behind MACS1206 from our sample).
[lrrccccccc]{} MACS0429Z-9372034910 & $04{:}29{:}37.20$ & $-2{:}53{:}49.10$ & $6.9 \pm 0.2$ & $24.3 \pm 0.1$ & $0.7 \pm 0.1$ & $-0.3 \pm 0.1$ & $-1.4 \pm 0.4$ & $2.5 \pm 0.2$ & $-21.6 \pm 0.1$\
A611Z-0532603348 & $08{:}00{:}53.26$ & $36{:}03{:}34.8$ & $6.7 \pm 0.2$ & $25.7 \pm 0.1$ & $0.3 \pm 0.4$ & $-1.5 \pm 0.4$ & $-1.5 \pm 0.5$ & $1.8 \pm 0.1$ & $-20.5 \pm 0.1$\
RXJ1347Z-7362045151 & $13{:}47{:}36.20$ & $-11{:}45{:}15.1$ & $6.7 \pm 0.2$ & $25.7 \pm 0.1$ & $0.2 \pm 0.4$ & $-1.3 \pm 0.4$ & $-2.2 \pm 0.5$ & $2.7 \pm 0.2$ & $-20.1 \pm 0.1$\
A209Z-1545136005 & $01{:}31{:}54.51$ & $-13{:}36{:}00.5$ & $6.9 \pm 0.2$ & $25.3 \pm 0.1$ & $1.0 \pm 0.3$ & $-0.3 \pm 0.3$ & $-2.7 \pm 0.6$ & $1.2 \pm 0.0$ & $-21.4 \pm 0.1$\
A2261Z-2269808378$^{\rm c}$ & $17{:}22{:}26.99$ & $32{:}08{:}37.8$ & $6.9 \pm 0.1$ & $25.2 \pm 0.1$ & $<$0.0 & $<-$0.8 & $-2.0 \pm 0.3$ & $5.6 \pm 1.7$ & $-19.9 \pm 0.3$\
MACS1423Z-3469204207 & $14{:}23{:}46.92$ & $24{:}04{:}20.7$ & $6.7 \pm 0.3$ & $25.6 \pm 0.1$ & $1.0 \pm 0.3$ & $0.2 \pm 0.5$ & $-1.2 \pm 0.8$ & $4.7 \pm 1.0$ & $-19.6 \pm 0.3$\
A1703-zD1$^{\rm d}$ & $13{:}14{:}59.41 $ & $ 51{:}50{:}00.8 $ & $6.8 \pm 0.1$ & $23.9 \pm 0.1$ & $0.2 \pm 0.2$ & $-1.3 \pm 0.5$ & $-1.4 \pm 0.3$ & $9.0 \pm 4.5$ & $-20.6 \pm 0.5$\
Mean stack & & & $6.8\pm0.2$ & $25.5\pm0.1$ & $0.2\pm0.2$ & $-0.9\pm0.3$ & $-1.9\pm0.3$ & &
Results {#sec:Results}
=======
Our search for bright ($H_{160}\lesssim26$) LBGs in the redshift range $z\sim6.6-7.0$ behind strong lensing clusters results in 9 candidates. One of the sources in our $z\sim7$ sample was previously reported by @Bradley2012a based on a study of Abell 1703. For seven sources we obtain reasonably clean IRAC photometry, as shown in the postage stamps in Figure \[fig:exampleclean\]. The properties of the sources are summarized in Table \[tab:clusters\] and they range in $H_{160}$ band magnitude from 24.3 to 25.7. Typical magnification factors, $\mu$, for our sources are $\sim2-9$, using the lensing models of @Zitrin2010 [@Zitrin2011] and Zitrin et al. (in prep). Though the magnification of the sources improves the S/N of our measurements, we stress that measurements of emission line EWs and sSFRs only depend on the colors of the SED and therefore are not impacted by uncertainties in the model magnification factors.
![ The constraint on the evolution of \[OIII\]+H$\beta$ EWs (and equivalent H$\alpha$ EWs) from our stacking analysis and references from the literature [@Erb2006; @Shim2011; @Fumagalli2012; @Stark2013; @Labbe2012]. The robust lower limit (red arrow) assumes all sources are at $z=6.76$ where the \[3.6\]-\[4.5\] color is expected to be the most extreme for a given set of EWs and that the underlying stellar continuum has a $[3.6]-[4.5]$ color of $\sim-0.4$ (which would only be the case if all galaxies have an age of $\sim3\times10^{6}\rm yr$). Any bluer $[3.6]-[4.5]$ color would therefore arise from the impact of the \[OIII\]+H$\beta$ emission lines on the \[3.6\] flux. For the model estimate (red open circle) we model the effects of a broader redshift distribution as described in §\[sec:stack\]. For comparison with lower redshift estimates we have converted our EWs to EW(H$\alpha$+\[NII\]) using the conversion factors from @Anders2003. The high EW inferred from our mean stacked sample indicates significantly stronger emission lines than observed at redshift $z\sim0-2$, possibly consistent with an extrapolation of the trends with redshift and mass found by @Fumagalli2012 [indicated by the dashed black lines]. ](EW.ps){width="0.9\columnwidth"}
\[fig:results\_EW\]
$[3.6]-[4.5]$ color distribution and nebular emission lines {#sec:EmLines}
-----------------------------------------------------------
Our selection of sources in the redshift range $z\sim6.6-7.0$ provides us with the valuable opportunity to establish the typical EW of the nebular emission lines in $z\gtrsim6$ sources through a comparison of the flux in \[3.6\] and \[4.5\]. LBGs at high redshift are expected to exhibit flat optical stellar continuum, based on stellar population synthesis models. In these models young galaxies with typical ages between 50-200Myr and low dust extinction, e.g. E(B-V)$\sim0.1$, will have a (\[3.6\]-4.5\])$_{\rm continuum}$ color of $\sim0\pm0.1$ mag. However, extremely young (i.e. $\sim3\times10^6\rm yr$), dust-free galaxies can exhibit (\[3.6\]-4.5\])$_{\rm continuum}$ colors as blue as $\sim-0.4$. To be conservative, we will adopt this for the color of the underlying stellar continuum, and assume that any bluer $[3.6]-[4.5]$ color arises from the impact of emission lines to establish robust lower limits.
In the bottom panel of Figure \[fig:emlines\] the dotted line shows a prediction of the observed optical color due to emission lines for a model of strongly increasing rest-frame emission line EWs as a function of redshift (dotted line), with EW$_0$(\[OIII\]+H$\beta)\propto(1+z)^{1.8}$[Å]{}, based on the evolution in EW$_0$(H$\alpha$) found by @Fumagalli2012 for star forming galaxies over the redshift range $0\lesssim z\lesssim2$. The red points show the observed colors for our sample. Most of our sources show quite blue $[3.6]-[4.5]$ colors and essentially all of them are bluer than that expected based on a conservative model of constant rest-frame EW (solid black line: i.e. assuming no evolution from $z\sim2$ where EW$_0$(\[OIII\]+H$\beta$)$\sim$140[Å]{}, derived from the H$\alpha$ EWs found by @Erb2006). Interestingly enough, three of the sources from our sample have $[3.6]-[4.5]$ colors even bluer than expected at $z\sim6.7-6.8$ for the model from @Fumagalli2012 with EW$_0$(H$\alpha)\propto(1+z)^{1.8}$[Å]{}. Four of the sources have $[3.6]-[4.5]$ colors bluer than $-0.8$. Since we would only expect galaxies to show such sources extreme $[3.6]-[4.5]$ colors in the narrow redshift range $z\sim6.6-7.0$, this provides us with additional confidence that our selection is effective at identifying sources in the desired redshift range.
Inferred \[OIII\]+H$\beta$ EWs of $z\sim7$ galaxies from the mean SED {#sec:stack}
---------------------------------------------------------------------
To obtain our best measurement of the \[4.5\] flux and hence stellar continuum light from $z\sim7$ galaxies, we use a mean stack of the clean \[3.6\] and \[4.5\] images after dividing by the observed rest-frame UV luminosity (the geometric mean of the $J_{125},\,JH_{140}$ and $H_{160}$ luminosities). We measure the flux in a 2.5" diameter aperture on the stacked image and apply an aperture correction measured from the PSF images ($\sim2.0\times$). The mean SED of our stacked $z\sim6.8$ sample is shown in Figure \[fig:results\_sed\]. Errors are obtained through bootstrap resampling.
We use the stacked detections in the IRAC bands to evaluate the mean contribution of the emission lines. From the mean $[3.6]-[4.5]$ color we estimate the \[OIII\]+H$\beta$ EW by assuming that our entire sample is at $z=6.76$, where we expect the most extreme colors because \[4.5\] is completely free of emission lines, while \[3.6\] is contaminated by both the \[OIII\] doublet and H$\beta$. In practice, this results in an underestimate of the intrinsic line strength, since we know that the \[OIII\] lines start to drop out of \[3.6\] at $z\sim6.9-7.0$. Therefore we expect a less extreme $[3.6]-[4.5]$ color for a given mean EW at $z\sim6.9-7.0$ than at $z\sim6.7-6.8$. It is also possible that due to uncertainties in the photometric redshifts, sources outside of our target redshift range have been included in our selection and therefore the measurement of the \[4.5\] flux is contaminated by either H$\alpha$ ($z<6.6$) or \[OIII\] ($z>7$). This would also make the mean $[3.6]-[4.5]$ color redder and accordingly make the emission lines appear to be less extreme.
The mean observed $[3.6]-[4.5]$ color for our sample is $-0.9\pm0.3$ (error obtained through bootstrap resampling). In the most conservative estimate, we assume that the underlying stellar continuum exhibits a $[3.6]-[4.5]$ color of $\sim-0.4$ and therefore the \[OIII\] and H$\beta$ are responsible for a color of $[3.6]-[4.5]\sim-0.5$, which would give a robust lower limit of EW$_0$(\[OIII\]+H$\beta)\gtrsim637$[Å]{} for the mean $z\sim7$ galaxy distribution. The \[OIII\]+H$\beta$ EW we estimate here is equivalent to EW$_0$(H$\alpha$+\[NII\])$\gtrsim495$[Å]{} adopting the tabulated values from @Anders2003 for 0.2Z$_\odot$ metallicity and assuming case B recombination.
The mean observed $[3.6]-[4.5]$ color for our four bluest sources is $-1.4\pm0.4$. If we assume again a very blue underlying continuum (i.e., $-0.4$), line emission would be responsible for a color of $[3.6]-[4.5]\sim-1.0$, consistent with a robust lower limit of EW$_0$(\[OIII\]+H$\beta)\gtrsim1582$[Å]{} for these sources. While the four bluest sources in our sample are, given their extreme colors, almost certainly at a redshift $z\sim6.7-6.8$, it is unclear if the three other sources are less extreme due to a lower \[OIII\]+H$\beta$ EW or simply because they lie in a different redshift range (i.e. $z\gtrsim6.9$ or close to $z\sim6.6$), where for a given EW we expect somewhat redder colors. To obtain a good estimate of the EW, detailed knowledge of the redshift distribution and the underlying stellar continuum color is required. Given our lack of deep spectroscopy for our sample we will make some reasonable assumptions to obtain a model estimate of the rest-frame EW of our sample. We assume that the redshift probability distribution of our sources is given by the sum of the probability distributions obtained from our photometric redshift code, corrected for the fact that galaxies are more difficult to observe at higher redshift scaling roughly as $\frac{d\log{\phi}}{dz}=0.3$ [@Bouwens2012b] and assuming no sources are outside our desired redshift range $z\sim6.6-7.0$. We use this redshift probability distribution in our sample to estimate the expected $[3.6]-[4.5]$ color distribution, for a given mean EW$_0$(\[OIII\]+H$\beta)$ and 0.3 dex scatter around the mean. We assume the underlying continuum color is $-0.25$, as would be expected for a galaxy age of $\sim$10Myr. We randomly draw sources from the distribution and calculate the 68% likelihood of finding a mean $<[3.6]-[4.5]>$ color given a total of seven observed sources and the observed photometric errors. Based on this modeling, the observed $[3.6]-[4.5]\sim-$0.9 mag color is consistent with a possible EW$_0$(\[OIII\]+H$\beta)$ EW of $\sim 1806_{-863}^{+1826}$[Å]{}. This is equivalent to EW$_0$(H$\alpha+\rm [NII])\sim 1323^{+1338}_{-632}$[Å]{}, adopting the same conversion factor from @Anders2003 assumed above.
The modeling we perform above indicates that the true EW$_0$(\[OIII\]+H$\beta)$ may be $\sim 2-3\times$ larger than our robust lower limit of 637[Å]{}. Figure \[fig:results\_EW\] compares our results with other determinations from the literature [@Erb2006; @Shim2011; @Fumagalli2012; @Stark2013; @Labbe2012]. The solid and dashed lines in Figure \[fig:results\_EW\] show the expected evolution of the H$\alpha$+\[NII\] EWs extrapolating the evolution found in @Fumagalli2012 at $z\sim0-2$. We note however that a direct comparison is difficult to make since the @Fumagalli2012 relation was derived for galaxies in the mass range $M_\ast=10^{10}-10^{10.5} M_\odot$, while we are probing galaxies in the mass range $10^{9}-10^{9.5} M_\odot$. For reference we show the possible evolution of galaxies $M_\ast=10^{9}-10^{9.5} M_\odot$ (top dashed line), using the same scaling with redshift EW$_0$(H$\alpha$+\[NII\]$)\propto(1+z)^{1.8}$[Å]{} but extrapolating the normalization to lower masses, based on the mass trend in the SDSS-DR7 data derived in @Fumagalli2012.
In general, the EWs we infer are in good agreement with extrapolations from previous results at lower redshift. However our results are based on a UV-selected sample, which could yield different results from a mass complete sample. It is clear nonetheless, that our EWs estimates strongly support the high EWs used by @Stark2013 and @Gonzalez2012 in correcting the SEDs of $z\sim5-7$ samples to derive higher values of the sSFRs.
![ The evolution of the sSFR as a function of redshift, based on the fitted SED to our stacked spectrum, excluding the \[3.6\] flux from the fit. Error bars are the 68% confidence interval, based on the photometric uncertainties. The red filled circle indicates the best fit using the assumptions described in §\[sec:ssfr\], leaving dust as a free parameter. The red open circle indicates a fit where the dust content is fixed at $A_V=0.38\pm0.16$. We also show a lower limit (red arrow) on the sSFR we derive from the EW of \[OIII\]+H$\beta$, by converting to H$\alpha$ assuming the line ratios from @Anders2003. For context, we also include many previous sSFR results from the literature [@Noeske2007; @Daddi2007; @Stark2009; @Stark2013; @Gonzalez2010; @Gonzalez2012; @Reddy2012; @Labbe2012]. Our results indicate possible strong evolution in the sSFRs from redshift $z\sim2$ to $z\sim7$, consistent with other recent results based on an extrapolation of the $z\sim4$ H$\alpha$ EW distribution [@Stark2013; @Gonzalez2012] or for single $z\sim7$ galaxies [e.g. @Ono2012; @Tilvi2013; @Ouchi2013]. Our results are also in agreement with theoretical predictions (e.g., @Neistein2008 \[dark grey line\] and @Dave2011 \[light grey line\]).](ssfr.ps){width="0.9\columnwidth"}
\[fig:results\_ssfr\]
Specific Star Formation Rates {#sec:ssfr}
-----------------------------
The redshift range where we select galaxies is the only redshift window at $z\gtrsim5$ where we can probe the rest-frame stellar continuum light in an uncontaminated fashion, using the \[4.5\] IRAC band (see figure \[fig:emlines\]). This allows us to estimate the sSFR, with minimal contamination from emission lines.
We obtain the mean sSFR through stellar population modeling of our stacked photometry, leaving out the \[3.6\] measurement. The modeling was performed with FAST [@Kriek2009], using the @Bruzual2003 [hereafter BC03] stellar populations synthesis models. We use a @Salpeter IMF with limits 0.1-100$\,M_\odot$ and a @Calzetti2000 dust-law. We consider ages between 10Myr and the age of the universe at $z\sim6.8$ and dust extinction between $A_V=0-2$. A constant star formation history and subsolar metallicity (0.2Z$_\odot$) is assumed. We fix the redshift to the median of the photometric redshifts, at $z=6.77$.
Given this freedom of parameters the mean SED is best described by a fairly young galaxy (age $\lesssim100$Myr) and reasonable dust ($A_V\sim0.7$) in order to fit both the small Balmer break ($H_{160}-[4.5]\sim0.2$) and moderately red UV-continuum slope ($\beta\sim-1.9$), resulting in a notably high sSFR of $\rm 52^{+50}_{-41}\,Gyr^{-1}$ (see the SED in Figure \[fig:results\_sed\]). However, the interpretation of this result is not straightforward. First of all, we have only the \[4.5\] band in the rest-frame optical to break the age-dust degeneracy. Given our modest sample there is still a range of models that can fit the data well. A possibly more insightful answer is obtained when we fix the dust to the expected value derived from the typical spread of UV-continuum slopes and the @Meurer1999 law [e.g. @Bouwens2012] similar to the assumptions made in @Bouwens2012, @Stark2013, @Gonzalez2012 and @Labbe2012. This results in a dust content of $A_V=0.38\pm0.16$ using the latest numbers from @Bouwens2013. The fit is shown with the thin red line in Figure \[fig:results\_sed\]. We obtain a sSFR of $\rm 7^{+7}_{-3}\,Gyr^{-1}$.
Alternatively, we can estimate the sSFR of our $z\sim7$ sample from the \[OIII\]+H$\beta$ EWs we infer, by converting to H$\alpha$ EW assuming same line ratios from @Anders2003 as described in §\[sec:stack\]. We use the @Kennicutt1998 relation to convert H$\alpha$ luminosity to star formation rate and we use BC03 models (assuming no dust) to convert the rest-frame optical continuum light to stellar mass. Using these assumption we obtain a lower limit on the sSFR for our seven source sample and the bluest four sources of $\sim\rm14\,Gyr^{-1}$ and $\sim\rm130\,Gyr^{-1}$ respectively, based on the robust lower limits on the \[OIII\]+H$\beta$ EW derived in §\[sec:stack\].
Comparing with direct constraints at $z\sim2$ we estimate $\gtrsim 2\times$ evolution in the sSFR over this redshift range, in good agreement with estimates at $z\sim7$ based on a few spectroscopically confirmed sources [@Ono2012; @Tilvi2013] and extrapolating the H$\alpha$ EWs from lower redshifts [@Stark2013; @Gonzalez2012]. Our derived constraint is also in agreement with theoretical models that predict the sSFR to follow the specific infall rate of baryonic matter [e.g. @Neistein2008].
Summary and Discussion {#sec:Summary}
======================
In this paper, we present the cleanest evidence yet for very high \[OIII\]+H$\alpha$ EWs in the $z\sim7$ galaxy population. We also simultaneously explore a strategy for obtaining a clean measurement of the sSFR at $z\sim7$ based on the stellar continuum flux measured in the \[4.5\] micron band – which is largely free of contamination from the strongest nebular lines. Nebular emission lines (\[OIII\], H$\alpha$, H$\beta$) and the extreme faintness of $z\gtrsim5.5$ galaxies make it extremely challenging to establish the stellar masses and sSFRs of the typical galaxy at high redshift.
To overcome these issues, we have isolated a small sample of nine bright ($H_{160} < 26$ mag), magnified galaxies in the redshift range $z\sim6.6-7.0$ from CLASH and other programs, seven of which we can perform high-quality IRAC photometry. Galaxies with photometric redshifts in the range $z\sim6.6-7.0$ are useful, since there the \[4.5\] band from Spitzer/IRAC provides us with a clean measurement of the stellar continuum flux from galaxies in the rest-frame optical, free of contamination from dominant nebular emission lines (Figure \[fig:emlines\] and Figure \[fig:emlines\_example\]).
For the mean source in our sample, we find that we can set a robust lower limit on the rest-frame EW of \[OIII\]+H$\beta$ of 637[Å]{}. For this lower limit, we adopt the bluest conceivable $[3.6]-[4.5]$ colors for the stellar continuum and assume that all sources in our sample are at $z=6.76$ where a given \[OIII\]+H$\beta$ EW would produce the most extreme $[3.6]-[4.5]$ color. Use of a more realistic redshift distribution for our sample, i.e., consistent with the photometric redshift estimates and not assuming that all sources are at $z=6.76$, suggest that these lower limits may underestimate the true EWs by a factors of $\sim2\times$.
The four bluest sources in our selection (58% of our sample) show evidence for even more extreme line emission, with $[3.6]-[4.5]\lesssim-0.8$. For these 4 sources, we can set a robust lower limit of 1582[Å]{} on the rest-frame EW in \[OIII\]+H$\beta$.
Extreme line emission with EWs greater 1000[Å]{} has been found at lower redshift in low mass galaxies [@vanderwel2011; @Atek2011]. Our results are consistent with the idea that extreme line emission may be present in the typical star-forming galaxy at $z\sim7$.
Furthermore, our \[4.5\] stack results imply a firm lower limit on the sSFR of $\sim\rm 4\,Gyr^{-1}$ for star-forming galaxies at $z\sim7$. If any sources from our $z\sim6.6-7.0$ photometric redshift sample lie at lower or higher redshifts than this, it would imply even lower \[4.5\] micron fluxes for the stack and hence higher sSFRs. Compared with sSFRs measurements at $z\sim2$ [@Daddi2007; @Reddy2012], this implies at least a $\gtrsim 2\times$ evolution in the sSFR over the redshift range $z\sim2$ to $z\sim7$. Similar to a few other spectroscopically confirmed $z\sim7$ galaxies in the literature [@Ono2012; @Tilvi2013], this provides strong evidence that the sSFRs at $z\sim7$ are high.
We expect improvement in these results through the measurement of spectroscopic redshifts for our sample from deep spectroscopy. This should allow us to obtain an even cleaner selection of $z\sim6.6-7.0$ galaxies from which to quantify the emission line contamination and sSFRs. Follow-up observations of our sample are facilitated by the fact that these candidates are typically $\sim$1 magnitude brighter than similar candidates found in the field, making these efforts quite feasible in terms of the telescope time required.
Moreover, our bright $z\sim7$ sample is small and the S/N we have per source is still modest. Increases in sample size can come from shallow surveys over a larger numbers of clusters, such as those available from recent snapshot programs. S/N increases will come from very deep HST+Spitzer observations being taken by the Frontier Fields program and the SURF’S Up program [@Bradac2012].
We thank Jeff Cooke, Rob Crain, Eichii Egami, Andrea Ferrara, Marijn Franx, Max Pettini, Norbert Pirzkal and Vivienne Wild for interesting conversations. Eichii Egami independently discovered the same extreme $[3.6]-[4.5]$ colors in at least one of the sources from the present sample. We thank Pascal Oesch for useful feedback on our manuscript. We acknowledge support from ERC grant HIGHZ \#227749, an NWO vrij competitie grant, and the NASA grant for the CLASH MCT program. AZ is supported by contract research “Internationale Spitzenforschung II/2-6” of the Baden Württemberg Stiftung.
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abstract: |
In the present article, we discuss jackknife empirical likelihood (JEL) and adjusted jackknife empirical likelihood (AJEL) based inference for finding confidence intervals for probability weighted moment (PWM). We obtain the asymptotic distribution of the JEL ratio and AJEL ratio statistics. We compare the performance of the proposed confidence intervals with recently developed methods in terms of coverage probability and average length. We also develop JEL and AJEL based test for PWM and study it properties. Finally we illustrate our method using rainfall data of Indian states.\
[Keywords:]{} Empirical Likelihood; JEL; Probability weighted moment; $U$-statistics.
author:
- |
D B$^*$, S K K$^{**,\dag}$ S N$^{***}$\
$^{*}$D S C U R, I\
$^{**}$I S I, C, I\
$^{***}$ I I T, C, I
title: Jackknife empirical likelihood based inference for Probability weighted moments
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[^1]
Introduction
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Probability weighted moments (PWM) generalize the concept of moments of a probability distribution.It is generally used to estimate the parameters of extreme distributions of natural phenomena. Particularly, in the fields of Hydrology and Climatology, researchers use PWM for the estimation of parameters of the distributions related to water discharges and maxima of temperatures. Greenwood et al. (1979) proposed the concept of probability weighted moments to estimate the parameters involved in the models of extremes of natural phenomena. The PWM of a random variable $X$ with distribution function $F(.)$ is defined as $$\mathcal{M}_{p,r,s}=\mathbb{E}\left \lbrace X^pF^r(X)(1-F(X))^s\right\rbrace,$$ where $p$, $r$ and $s$ are any real numbers. Hosking et al. (1985) studied PWM of the form given by $$\label{pwm}
\beta_r=\mathbb{E}\left \lbrace XF^r(X)\right\rbrace,$$ to characterize various distributional properties such as assessment of scale parameter, skewness of the distribution and L-moments. In this article we discuss the empirical likelihood inference of $\beta_r$.
Given a random sample $X_1,X_2,\cdots, X_n$ of size $n$ from $F$, let $X_{(i)},i=1,2,\ldots,n$ be the $i$-$th$ order statistic. David and Nagaraja (2003) proposed an estimator (D-N estimator) for $\beta_r$ by replacing $F$ in (\[pwm\]) with its empirical version, $\hat{F}_n(x)=\frac{1}{n}\sum\limits_{i=1}^{n}\mathbb{I}_{\lbrace X_i\le x\rbrace}$, where $\mathbb{I}$ denotes the indicator function. Their estimator is given by $$\label{1}
\overline{\beta}_r=\frac{1}{n}\sum\limits_{i=1}^{n}\left(\frac{i}{n}\right)^r X_{(i)}.$$ To develop an empirical likelihood inference for $\beta_r$, Vexler et al. (2017) proposed an estimator (Vexler’s estimator) given by $$\label{2}
\tilde{\beta}_r= \frac{1}{r+1}\sum\limits_{i=1}^{n} X_{(i)} \left\lbrace \left(\frac{i}{n} \right)^{r+1} -\left(\frac{i-1}{n} \right)^{r+1} \right\rbrace$$ and showed that asymptotic behaviour of both $\tilde{\beta}_r$ and $\overline{\beta}_r$ are same.
Empirical likelihood is a non-parametric inference tool which make use of likelihood principle. This inference procedure is firstly used by Thomas and Grunkemeier (1975) to obtain the confidence interval for survival probability when data contain censored observations. Pioneering papers by Owen (1988, 1990) for finding the confidence interval of regression parameters take the empirical likelihood method into a general methodology and have wide applications in many statistical areas. This approach enjoys the wide acceptance among the researchers as it combines the effectiveness of the likelihood approach with the reliability of non-parametric procedure.
Empirical likelihood finds applications in regression, survival analysis and inference for income inequality measures \[Shi and Lau (2000), Whang (2006), Qin et al. (2010), Peng (2011), Qin et al. (2013), Zhou (2015), Wang et al.(2016), Wang and Zhao (2016)\]. Recently, Vexler et al. (2017) proposed an empirical likelihood based inference for PWM and showed that the limiting distribution of log empirical likelihood ratio statistic is $\chi^{2}$ distribution with one degree of freedom and obtained confidence intervals and likelihood ratio tests for PWM.
In empirical likelihood approach, we need to maximize the non-parametric likelihood function subject to some constraints. When the constraints are linear, the maximization of the likelihood is not difficult. However, when the constraints are based on nonlinear statistics such as $U$-statistics with higher degree $(\ge 2)$ kernel the implementation of empirical likelihood becomes challenging. To overcome this difficulty, Jing et al. (2009) introduced the jackknife empirical likelihood (JEL) inference, which combines two of the popular non-parametric approaches namely, the jackknife and the empirical likelihood approach. Chen et al. (2008) proposed the concept of adjusted empirical likelihood which preserves the asymptotic properties of empirical likelihood. Even though it is an adjustment given to empirical likelihood, adjusted empirical likelihood procedure yields better coverage probability than bootstrap calibration and Bartlett correction methods. Recently Zhao et al. (2015) introduced adjusted jackknife empirical likelihood (AJEL) inference so that restriction on parameter values on the convex hull of estimating equation is relaxed.
Motivated by these recent works, in this article, we develop JEL and AJEL based inference to construct confidence intervals and likelihood ratio tests for PWM. The present article is structured as follows. In Section 2, we derive the jackknife empirical log likelihood ratio for $\beta_r$ and obtained its limiting distribution. Using this result, we construct jackknife empirical likelihood based confidence interval and likelihood ratio test for $\beta_r$. We further derive AJEL based confidence interval and likelihood ratio test for $\beta_r$. In Section 3, a comparison of the proposed methods with empirical likelihood method are given using Monte Carlo simulation. Finally an illustration of our methods using a real data is given in Section 4. Major findings of the study are given in Section 5.
Jackknife empirical likelihood inference for PWM
================================================
In this section, first we discuss JEL based inference for confidence interval and likelihood ratio test for $\beta_r$. Later we discuss same problems using AJEL based inference. The implementation of these methods require jackknife pseudo values obtained using an estimator of $\beta_r$. For this purpose, we introduce an estimator of $\beta_r$ using theory of $U$-statistics.
To obtain a $U$-statistic based estimator for $\beta_r$ we rewrite (\[pwm\]) as $$\begin{aligned}
\beta_{r}=& \frac{1}{r+1}\int\limits_{-\infty}^{\infty} (r+1)xF^r(x) dF(x) \\
=& \frac{1}{r+1}\cdotp \mathbb{E}(\max(X_1,X_2,\cdots,X_{r+1})),\end{aligned}$$provided $r$ is a positive integer. Therefore an unbiased estimator of $\beta_r$ is given by $$\label{est}
\widehat{\beta}_{r}=\frac{1}{r+1} \frac{1}{C_{m,n}} \sum\limits_{C_{m,n}}h{(X_{i_1},X_{i_2},\cdots,X_{i_{r+1}})},$$ where $h{(X_{i_1},X_{i_2},\cdots,X_{i_{r+1}})}=\max(X_{i_1},X_{i_2},\cdots,X_{i_{r+1}})$ and the summations is over the set $C_{m,n}$ of all combinations of $(r+1)$ distinct elements $\lbrace i_1,i_2,\cdots,i_{r+1}\rbrace$ chosen from $\lbrace 1,2,\cdots,n\rbrace$. Clearly $\widehat{\beta}_{r}$ is a consistent estimator of ${\beta}_{r}$ (Lehmann, 1951). Use of consistent and unbiased estimator to construct an empirical likelihood based confidence interval give better coverage probability and average length. However, $\beta_r$ has a $U$-statistics based estimator with kernel of degree greater than one which results in non-linear constraints in the optimization problem associated with empirical likelihood. This makes the implementation of empirical likelihood theory very difficult. This leads us to construct JEL based confidence interval and test for $\beta_r$.
Next we discuss how to derive the jackknife empirical likelihood ratio for $\beta_r$. The jackknife pseudo-values for $\beta_r$ is given by $$\label{jsv}
\widehat{V}_{k}= n \widehat{\beta}_{r}-(n-1)\widehat{\beta}_{r,k}; \qquad k=1,2,\cdots,n,$$ where $\widehat{\beta}_{r,k}$ is the estimator of $\beta_r$ obtained from (\[est\]) by using $(n-1)$ observations $X_1,X_2,...,X_{k-1},X_{k+1},...,X_n$. The jackknife estimator $\widehat{\beta}_{r,jack}$ of $\beta_r$ is the average of the jackknife pseudo-values, that is $$\widehat{\beta}_{r,jack}=\frac{1}{n}\sum\limits_{k=1}^{n}\widehat{V}_{k}.$$ As the pseudo-values are constructed using a $U$-statistic, the two estimators $\widehat{\beta}_{r}$ and $\widehat{\beta}_{r,jack}$ coincide. We use the jackknife pseudo-values $\widehat{V}_{k}$ defined in equation (\[jsv\]) to construct JEL for $\beta_r$. The jackknife empirical likelihood of $\beta_{r}$ is defined as $$\label{6}
J(\beta_{r})=\sup_{\bf p} \left(\prod_{k=1}^{n}{p_k};\,\, p_k \ge 0;\, \, \sum_{k=1}^{n}{p_k}=1;\,\,\sum_{k=1}^{n}{p_k (\widehat{V}_k}-\beta_r)=0\right),$$ where ${\bf p}=(p_1,p_2,...,p_n)$ is a probability vector. The maximum of (\[6\]) occurs at $$p_k=\frac{1}{n}\left(1+\lambda(\widehat{V}_{k}-\beta_r)\right)^{-1}, k=1,2,...,n,$$ where $\lambda$ is the solution of $$\label{eq7}
\frac{1}{n}\sum_{k=1}^{n}{\frac{\widehat{V}_{k}-\beta_r}{1+\lambda (\widehat{V}_{k}-\beta_r)}}=0,$$provided $$\min_{{1\le k\le n}}\widehat{V}_{k}<\widehat \beta_r< \max_{1\le k\le n}\widehat{V}_{k}.$$ Also note that, $\prod\limits_{k=1}^{n}p_i$, subject to $\sum\limits_{i=1}^{n}p_i=1$, attains its maximum $n^{-n}$ at $p_i=n^{-1}$. Hence, the jackknife empirical log-likelihood ratio for $\beta_r$ is given by $$\label{jelrat}
l(\beta_r)=-\sum_{i=1}^{n}\log\left[1+\lambda (\hat{V}_{k}-\beta_r)\right].$$
Next theorem explains the limiting distribution of $l(\beta_r)$ which can be used to construct the JEL based confidence interval and test for $\beta_r$.
Suppose that $E\left(h^2(X_{1}, X_{2},..., X_{r+1})\right)<\infty$ and $\sigma^2=Var(g(X))>0$, where $g(x)=E\left(h(X_1, X_{2},..., X_{r+1})|X_1=x\right)-\beta_{r}$. Then, as $n\rightarrow\infty$, the distribution of $-2l(\beta_r)$ is $\chi^{2}(1)$.
Let $S=\frac{1}{n}\sum_{k=1}^{n}(\widehat{V}_{k}-\beta_r)^{2}$. Since $\widehat\beta_r=\frac{1}{n}\sum_{k=1}^{n}\widehat{V}_k$, by strong law of large numbers we obtain $$\label{eq8}
S= \sigma^{2}+o(1).$$ Using Lemma A.4 of Jing et al. (2009) we have $$\label{eq9}
\max_{1\le k\le n} |\widehat{V}_{k}-\beta_{r}|=o(\sqrt{n}).$$ Hence using (\[eq8\]) and (\[eq9\]) we have $$\label{eq111}
\frac{1}{n}\sum_{k=1}^{n}|\widehat{V}_{k}-\beta_{r}|^{3}\le |\widehat{V}_{k}-\beta_{r}|\frac{1}{n}\sum_{k=1}^{n}(\widehat{V}_{k}-\beta_{r})^{2}=o(\sqrt{n}).$$ The $\lambda$ satisfying the equation (\[eq7\]) has the property (Jing et al. 2009) $$\label{lo}
|\lambda|=O_{p}(n^{-\frac{1}{2}}).$$ Hence using (\[eq9\]) we obtain $$\label{eq10}
\max_{1\le k\le n} \lambda|\widehat{V}_{k}-\beta_{r}|=o({1}).$$Therefore, using equations (\[eq111\]), (\[eq10\]) and (\[lo\]) we have $$\frac{1}{n}\sum_{k=1}^{n}(\widehat{V}_{k}-\beta_{r})^{3}\lambda^2|1+\lambda(\widehat V_k-\beta_r)|^{-1}=o_p(\sqrt{n})O_p({1/n})O_p(1)=o_p(1/\sqrt{n}).$$Hence from (\[eq7\]) we obtain $$\label{eq11}
\lambda=\frac{(\widehat{\beta}_{r}-\beta_{r})}{S}+o_p(1/\sqrt n).$$ Using Taylor’s theorem, we can express $l(\beta_r)$ given in (\[jelrat\]) as $$\label{asylike}
-2l(\beta_r)=2n\lambda(\widehat{\beta}_{r}-\beta_{r})-nS\lambda^2+R(\beta_r),$$where $R(\beta_r)$ is the reminder term. Using $|\lambda|=O_{p}(n^{-\frac{1}{2}})$ and (\[eq111\]) we obtained the reminder term $R(\beta_r)=o_p(1)$. Hence using (\[eq11\]), we can express (\[asylike\]) as $$\label{eq15}
-2l(\beta_r)=\frac{n(\widehat{\beta}_{r}-\beta_{r})^2}{S}+o_p(1).$$ Using the central limit theorem for $U$-statistics, as $n\rightarrow \infty$, the asymptotic distribution of $\sqrt{n}(r+1)\left(\widehat{\beta}_{r}-{\beta}_{r}\right)$ is Gaussian with mean zero and variance $(r+1)^2\sigma^2$, where $$\begin{aligned}
\sigma^{2}&=&Var \left(E(max(X_1,X_2,\cdots,X_{r+1})|X_1=x)\right)\\&=&Var\left(XF^{r}(X)+r\int_{X}^{\infty}yF^{r-1}(y)dF(y)\right).
\end{aligned}$$ Hence, as $n \to \infty$, $\sqrt{n}\left(\widehat{\beta}_{r}-{\beta}_{r}\right)$ converges in distribution to normal with mean zero and variance $\sigma^2$. Accordingly $\frac{n(\widehat{\beta}_{r}-{\beta}_{r})^2}{\sigma^{2}}$ converges in distribution to $ \chi^{2}$ with one degree of freedom. In view of (\[eq8\]), by Slutsky’s theorem, from (\[eq15\]) we have the result.\
Using Theorem 1, JEL based confidence interval for $\beta_r$ at $100(1-\alpha)\%$ is given by $$CI_1=\left\{\beta_r|-2l(\beta_r)\le \chi^2_{1,1-\alpha}\right\},$$where $\chi^2_{1,1-\alpha}$ is the $(1-\alpha)$th percentile of chi-square distribution with one degree of freedom. The performance of these confidence intervals in terms of coverage probabilities and average lengths were evaluated via a Monte Carlo simulation and the results are reported in Section 3.
Using the asymptotic distribution of jackknife empirical log likelihood ratio we can develop JEL based test for testing the hypothesis $\beta_r=\beta_r^0$, where $\beta_r^0$ is a specific value of $\beta_r$. We reject the null hypothesis at significance level $\alpha$ if $$-2l(\beta_r)> \chi^2_{1,1-\alpha}.$$ Simulation study shows that the type 1 error rate of the test converges to desired significance level and has good power. The results of the simulation study are also reported in Section 3.
Next we discuss construction of AJEL based confidence interval for $\beta_r$. Define $$\widehat{V}_{n+1}=-\frac{a_n}{n}\sum\limits_{k=1}^{n} \widehat{V}_{k},$$ for some positive $a_n$. Chen et al. (2008) suggested to take $a_n=\max\lbrace 1,\log_e (n/2)\rbrace$. The adjusted jackknife estimator of $\beta_r$ is defined as $$\widehat{\beta}_{r,adjjack}=\frac{1}{n+1}\sum\limits_{k=1}^{n+1}\widehat{V}_{k}.$$ The adjusted jackknife empirical likelihood of $\beta_r$ is given by $$\label{jel}
R(\beta_{r})=\sup_{\bf p} \left(\prod_{k=1}^{n+1}{(n+1) p_k};\, \, p_k \ge 0; \,\, \sum_{k=1}^{n+1}{p_k}=1;\,\,\sum_{k=1}^{n+1}{p_k (\widehat{V}_k}-\beta_r)=0\right).$$ The maximum of (\[6\]) occurs at $$p_k=\frac{1}{n+1}\left(1+\lambda_1(\widehat{V}_{k}-\beta_r)\right)^{-1}, k=1,2,...,n+1,$$ where $\lambda_1$ is the solution of $$\frac{1}{n+1}\sum_{k=1}^{n+1}{\frac{\widehat{V}_{k}-\beta_r}{1+\lambda_1 (\widehat{V}_{k}-\beta_r)}}=0.$$ Hence, the adjusted jackknife empirical log likelihood ratio for $\beta_r$ is given by $$l_1(\beta_r)=-\sum_{k=1}^{n+1}\log\left[1+\lambda_1 (\hat{V}_{k}-\beta_r)\right].$$
Under the assumption of Theorem 1, and for $a_n = o_p(n^{2/3})$, as $n\rightarrow \infty$, $-2l_{1}(\beta_{r})$, is distributed as $\chi^2(1)$.
The proof follows on similar lines of proof of Theorem 1. Note that as long as $a_n = o_p(n)$, we have $|\lambda_1| = O_p(1/\sqrt n)$. Consider $$\begin{aligned}
\label{ajel}
-2l_1(\beta_r)&=&2\sum_{k=1}^{n+1} \log(1+\lambda \widehat V_k)
\\&=&2\sum_{k=1}^{n+1}\left(\lambda\widehat V_k- \lambda^2\widehat V_{k}^{2}/2\right)+ o_p(1)\\&=& 2n\lambda(\widehat{\beta}_{r}-\beta_{r})-nS\lambda^2+o_p(1)\\&=&
\frac{n(\widehat{\beta}_{r}-\beta_{r})^2}{S}+o_p(1),\end{aligned}$$ where the second last identity follows from the fact that the $(n + 1)$th term of the summation is $a_n O_p(n^{-3/2})=o_p(n)O_p(n^{-3/2})=o_p(1)$. Hence by Slutsky’s theorem we have the result.
Using the asymptotic distribution of adjusted jackknife empirical log likelihood ratio we can construct a confidence interval and likelihood ratio test for $\beta_r$. A $100(1-\alpha) \%$ AJEL based confidence interval for $\beta_r$ is given by $$CI_2=\left\{\beta_r|l_{1}(\beta_{r})\le \chi^2_{1,1-\alpha}\right\}.$$ In AJEL based ratio test for testing the hypothesis $\beta_r=\beta_r^0$, we reject the hypothesis when $ -2l_{1}(\beta_{r})> \chi^2_{1,1-\alpha}$. The performance of these confidence intervals and tests are also evaluated in Section 3.
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The box plots of four estimators $\overline{\beta}_1$, $\tilde{\beta}_1$, $\widehat{\beta}_{1,jack}$ and $\widehat{\beta}_{1,adjjack}$ obtained for different sample sizes $n=25,50, 150, 300$ are shown in Figures 1-3. In Figure 1, we generate observations from standard exponential distribution and computed the four estimators for $n=25,50, 150, 300$. The Monte-Carlo simulation procedure is repeated 5000 times. The red horizontal line across the box plot represents the actual value of $\beta_1$ when $X$ follows standard exponential. Similarly in Figure 2 and 3 show box plots of the estimators when the observations are generated from standard normal and standard lognormal distributions respectively.
Monte carlo study
=================
In this section, we perform a Monte Carlo simulation study to compare the performance of JEL and AJEL based confidence intervals with the confidence intervals obtained by empirical likelihood method constructed using D-N estimator (DNEL) as well as Vexler’s estimator (VXL). The simulation is done using R. We find empirical type 1 error as well as power of the proposed JEL and AJEL based tests. In our study, the observations are simulated from standard exponential, standard normal and standard log normal distributions with different sample sizes, $n=25, 50,100, 200$ and $300$. The simulation procedure is repeated for five thousand times. Note that we need to find the values of $\lambda$ and $\lambda_1$ to construct the confidence intervals and to obtain the critical regions of the proposed tests. We used the R functions $uniroot$ and $optimize$ to find the values of $\lambda$ and $\lambda_1$.
The Table 1 gives the Monte Carlo variances of the four estimators $\overline{\beta}_{r}$, $\tilde{\beta}_{r}$, $\widehat{\beta}_{r,jack}$ and $\widehat{\beta}_{r,adjjack}$ calculated based on the observations generated from standard exponential, standard normal and standard log normal distributions, respectively. We can see that $\widehat{\beta}_{r,adjjack}$ provides the least variance in almost all cases.
In Table 2 we report the coverage probability obtained for the four confidence intervals mentioned above. From Table 2 it is clear that in all three cases, the coverage probabilities of all four confidence intervals converge to the actual target value $(0.95)$. Hence it is worth to compare the average length of these intervals. In Table 3 we report the average length of the confidence intervals obtained for the confidence level $0.95$. When $X$ has standard exponential and standard normal distribution JEL based confidence intervals performs better than the other confidence intervals in terms of average length. In log normal case, in most of the cases JEL and AJEL based confidence intervals have smaller length compared to other two confidence intervals.
The empirical type 1 error of the proposed tests are listed in Table 4. We find the type 1 error rate for $\beta_r$ for $r=1,2,3,4$ when the samples are generated from above listed three distributions. For small sample sizes $n=25,50$, AJEL based test has well controlled type 1 error rates for normally distributed data. From Table 4, it is clear that for all the cases, empirical type 1 error reaches the nominal value $\alpha=0.05$ as the sample size increases. Tables 5, 6 and 7 provide the power comparison for the JEL, AJEL, DNEL and VXL based tests when the null hypothesis is related to the following cases.
1. $\beta_1=0.68 $, $\beta_2= 0.55$, $\beta_3=0.47 $, $\beta_4= 0.41$ (or $X \sim Exp(0.9)$)
2. $\beta_1= 0.6$, $\beta_2=0.49 $, $\beta_3= 0.42$, $\beta_4=0.37 $ (or $X \sim Exp(0.8)$)
3. $\beta_1=0.56 $, $\beta_2=0.56 $, $\beta_3= 0.51$, $\beta_4=0.47 $ (or $X \sim Normal(0,4)$)
4. $\beta_1=0.84 $, $\beta_2=0.84 $, $\beta_3=0.77 $, $\beta_4=0.70 $ (or $X \sim Normal(0,9)$)
5. $\beta_1=2.63 $, $\beta_2=2.36 $, $\beta_3=2.16 $, $\beta_4=2.01 $ (or $X \sim Lognormal(0,1.5)$)
6. $\beta_1=6.81 $, $\beta_2=6.39 $, $\beta_3=6.08 $, $\beta_4=5.82 $ (or $X \sim Lognormal(0,2)$)
From Tables 5, 6 and 7, we can see that empirical power of the JEL based test higher than other tests.
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$n$ $W_1$ $W_2$ $W_3$ $W_4$ $W_1$ $W_2$ $W_3$ $W_4$ $W_1$ $W_2$ $W_3$ $W_4$
25 0.5965 0.5665 0.5831 0.4719 0.3136 0.2929 0.2963 0.2398 3.6880 3.7375 3.6053 3.5759
50 0.5885 0.5736 0.5819 0.5164 0.2935 0.2835 0.2850 0.2529 3.6245 3.6538 3.5490 3.5917
150 0.5709 0.5661 0.5688 0.5427 0.2980 0.2946 0.2952 0.2816 3.6509 3.7183 3.6696 3.6020
300 0.5994 0.5969 0.5983 0.5831 0.2933 0.2916 0.2919 0.2844 3.5998 3.5548 3.4058 3.5104
25 0.4266 0.3884 0.4091 0.3311 0.1690 0.1519 0.1556 0.1259 2.8852 2.6441 2.8250 2.2864
50 0.4119 0.3929 0.4031 0.3577 0.1618 0.1534 0.1553 0.1378 3.0319 2.9030 3.0012 2.6634
150 0.4255 0.4188 0.4224 0.4030 0.1583 0.1555 0.1561 0.1490 3.0024 2.9592 2.9922 2.8549
300 0.4155 0.4123 0.4140 0.4035 0.1582 0.1568 0.1571 0.1531 3.0001 2.9785 2.9949 2.9188
25 0.3433 0.3000 0.3227 0.2612 0.1128 0.0976 0.1014 0.0821 2.8900 2.5455 2.8099 2.2742
50 0.3306 0.3090 0.3203 0.2842 0.1068 0.0993 0.1012 0.0898 2.6063 2.4449 2.5654 2.2766
150 0.3130 0.3059 0.3096 0.2954 0.1008 0.0983 0.0989 0.0944 2.6165 2.5615 2.6034 2.4840
300 0.3263 0.3226 0.3246 0.3163 0.1054 0.1042 0.1045 0.1018 2.7465 2.7175 2.7397 2.6701
25 0.2880 0.2419 0.2659 0.2152 0.0865 0.0720 0.0759 0.0614 2.7197 2.3014 2.6157 2.1170
50 0.2700 0.2473 0.2590 0.2298 0.0800 0.0729 0.0747 0.0663 2.6946 2.4782 2.6426 2.3452
150 0.2560 0.2486 0.2524 0.2408 0.0741 0.0718 0.0725 0.0691 2.3813 2.3156 2.3653 2.2567
300 0.2672 0.2633 0.2653 0.2586 0.0722 0.0711 0.0714 0.0696 2.3917 2.3584 2.3834 2.3228
-- ----- -------- -------- -------- -------- -- -------- -------- -------- -------- -- -------- -------- -------- --------
: Monte-Carlo Variance times the sample size for each estimator: $W_1=n Var(\overline{\beta}_r)$, $W_2=n Var(\widehat{\beta}_r)$, $W_3=n Var(\widehat{\beta}_{r,jack})$ and $W_4=n Var(\widehat{\beta}_{r,adjjack})$.[]{data-label=""}
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$n$
DNEL VLX JEL AJEL DNEL VLX JEL AJEL DNEL VLX JEL AJEL
25 0.913 0.908 0.902 0.905 0.919 0.928 0.912 0.953 0.869 0.853 0.851 0.839
50 0.939 0.933 0.932 0.934 0.943 0.948 0.940 0.960 0.903 0.891 0.889 0.890
100 0.941 0.938 0.936 0.947 0.950 0.951 0.947 0.958 0.925 0.916 0.918 0.917
200 0.943 0.942 0.942 0.946 0.949 0.951 0.950 0.957 0.933 0.925 0.925 0.927
300 0.946 0.946 0.946 0.950 0.952 0.950 0.948 0.953 0.935 0.933 0.932 0.930
25 0.903 0.951 0.881 0.891 0.912 0.967 0.906 0.947 0.862 0.948 0.839 0.826
50 0.935 0.924 0.924 0.932 0.94 0.948 0.932 0.961 0.898 0.886 0.881 0.876
100 0.945 0.939 0.938 0.948 0.945 0.949 0.941 0.958 0.929 0.912 0.912 0.912
200 0.949 0.948 0.945 0.954 0.950 0.951 0.949 0.956 0.935 0.924 0.923 0.922
300 0.951 0.948 0.949 0.954 0.944 0.941 0.943 0.947 0.933 0.924 0.923 0.927
25 0.903 0.888 0.878 0.892 0.903 0.927 0.897 0.941 0.855 0.831 0.829 0.819
50 0.930 0.874 0.911 0.926 0.936 0.916 0.926 0.957 0.899 0.878 0.873 0.876
100 0.934 0.915 0.925 0.935 0.946 0.900 0.937 0.957 0.917 0.885 0.899 0.902
200 0.950 0.951 0.944 0.950 0.947 0.908 0.944 0.955 0.936 0.915 0.922 0.920
300 0.947 0.942 0.940 0.947 0.949 0.913 0.948 0.957 0.931 0.915 0.917 0.921
25 0.837 0.881 0.858 0.884 0.897 0.929 0.887 0.932 0.846 0.866 0.804 0.808
50 0.926 0.843 0.903 0.922 0.918 0.808 0.909 0.945 0.888 0.821 0.86 0.864
100 0.935 0.925 0.927 0.938 0.937 0.894 0.931 0.952 0.914 0.878 0.892 0.890
200 0.947 0.944 0.940 0.950 0.947 0.952 0.942 0.954 0.926 0.922 0.914 0.913
300 0.942 0.944 0.934 0.941 0.947 0.948 0.945 0.956 0.937 0.925 0.927 0.926
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: [Coverage probability of likelihood ratio tests for different $r$ and different distributions]{}[]{data-label="my-label"}
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$n$
DNEL VLX JEL AJEL DNEL VLX JEL AJEL DNEL VLX JEL AJEL
25 0.9776 0.8696 0.8388 0.8743 0.6463 0.6178 0.5473 0.6403 3.5341 3.5113 3.4718 3.7529
50 0.6737 0.5592 0.5475 0.5816 0.3700 0.3618 0.3556 0.3557 2.6065 2.5382 2.5306 2.3840
100 0.4448 0.3504 0.3478 0.3837 0.2523 0.2420 0.2358 0.2413 1.3274 1.2837 1.2805 1.2168
200 0.3060 0.2643 0.2549 0.2530 0.1751 0.1748 0.1711 0.1626 0.9948 0.8870 0.8821 0.9110
300 0.2446 0.1957 0.1933 0.2063 0.1422 0.1322 0.1306 0.1313 0.6882 0.6394 0.6303 0.7153
25 0.9031 0.6514 0.6409 0.6859 0.4788 0.4745 0.4173 0.4419 3.6222 3.5692 3.5686 3.1831
50 0.6508 0.4793 0.4733 0.4908 0.3280 0.2651 0.2524 0.2871 2.0383 1.7987 1.4620 1.9150
100 0.4338 0.3209 0.3053 0.3390 0.2247 0.1854 0.1782 0.1946 1.3018 1.1537 1.1531 1.1154
200 0.3141 0.2205 0.2193 0.2176 0.1616 0.1221 0.1203 0.1214 0.9575 0.8599 0.8539 0.8384
300 0.2383 0.1822 0.1806 0.1813 0.1254 0.0978 0.0956 0.0994 0.7941 0.7565 0.7508 0.7294
25 0.9111 0.7413 0.7188 0.6450 0.4431 0.3817 0.3778 0.3769 2.8677 3.2130 2.6444 2.9005
50 0.6432 0.5342 0.4397 0.4273 0.3292 0.2688 0.2416 0.2579 2.0338 1.9462 1.9358 2.3499
100 0.4622 0.2875 0.2748 0.3007 0.2256 0.2220 0.1519 0.1530 1.4327 1.3802 1.3180 1.2739
200 0.3025 0.2023 0.1943 0.1956 0.1519 0.1178 0.0999 0.1024 0.9472 0.8419 0.8336 0.8184
300 0.2457 0.1625 0.1499 0.1606 0.1207 0.0900 0.0804 0.0816 0.6521 0.5749 0.5667 0.5585
25 0.8023 0.5576 0.5485 0.6734 0.4546 0.2925 0.2922 0.3064 3.3343 2.6987 2.3671 2.7833
50 0.5911 0.3720 0.3659 0.4000 0.3216 0.1965 0.1918 0.2022 1.6487 1.5862 1.5822 2.0474
100 0.4308 0.2940 0.2923 0.3067 0.2104 0.1360 0.1346 0.1303 1.5489 1.2440 1.2403 1.3687
200 0.2997 0.1719 0.1691 0.1796 0.1426 0.1174 0.0900 0.0922 0.9711 0.9209 0.8783 0.8510
300 0.2333 0.1352 0.1302 0.1422 0.1145 0.0806 0.0760 0.0797 0.7077 0.5738 0.5716 0.5934
-- ----- -------- -------- -------- -------- -- -------- -------- -------- -------- -- -------- -------- -------- --------
: Average length[]{data-label="my-label"}
-- ----- ------- ------- ------- ------- -- ------- ------- ------- ------- -- ------- ------- ------- -------
$n$
DNEL VLX JEL AJEL DNEL VLX JEL AJEL DNEL VLX JEL AJEL
25 0.095 0.092 0.098 0.087 0.081 0.072 0.088 0.047 0.131 0.147 0.149 0.161
50 0.061 0.067 0.068 0.066 0.057 0.052 0.060 0.040 0.097 0.109 0.111 0.110
100 0.059 0.062 0.064 0.053 0.050 0.049 0.053 0.042 0.075 0.084 0.082 0.083
200 0.057 0.058 0.058 0.054 0.051 0.049 0.050 0.043 0.067 0.075 0.075 0.073
300 0.054 0.054 0.054 0.050 0.048 0.047 0.052 0.050 0.065 0.067 0.068 0.070
25 0.097 0.049 0.119 0.109 0.088 0.063 0.094 0.053 0.138 0.052 0.161 0.174
50 0.065 0.076 0.076 0.065 0.060 0.052 0.068 0.039 0.102 0.114 0.119 0.124
100 0.055 0.061 0.062 0.052 0.055 0.051 0.059 0.042 0.071 0.088 0.088 0.088
200 0.051 0.052 0.055 0.046 0.050 0.049 0.051 0.044 0.065 0.076 0.077 0.078
300 0.049 0.052 0.051 0.046 0.056 0.059 0.057 0.053 0.067 0.076 0.077 0.073
25 0.100 0.112 0.122 0.108 0.097 0.073 0.103 0.059 0.145 0.169 0.171 0.181
50 0.070 0.126 0.089 0.074 0.064 0.084 0.074 0.043 0.101 0.122 0.127 0.124
100 0.066 0.085 0.075 0.065 0.054 0.100 0.063 0.043 0.083 0.115 0.101 0.098
200 0.054 0.058 0.060 0.053 0.053 0.092 0.056 0.045 0.064 0.085 0.078 0.080
300 0.051 0.049 0.056 0.050 0.051 0.087 0.052 0.043 0.069 0.085 0.083 0.079
25 0.116 0.119 0.142 0.121 0.103 0.071 0.113 0.068 0.154 0.134 0.196 0.192
50 0.074 0.157 0.097 0.078 0.082 0.192 0.091 0.055 0.112 0.179 0.140 0.136
100 0.062 0.075 0.073 0.065 0.063 0.106 0.069 0.048 0.086 0.122 0.108 0.110
200 0.058 0.056 0.060 0.059 0.053 0.048 0.058 0.046 0.074 0.078 0.086 0.087
300 0.050 0.056 0.056 0.053 0.053 0.052 0.055 0.044 0.063 0.075 0.073 0.074
-- ----- ------- ------- ------- ------- -- ------- ------- ------- ------- -- ------- ------- ------- -------
: Size of the likelihood ratio tests for different $r$ []{data-label="my-label"}
-- ----- ------- ------- ------- ------- -- ------- ------- ------- -------
$n$ DNEL VLX JEL AJEL DNEL VLX JEL AJEL
25 0.088 0.099 0.126 0.041 0.147 0.215 0.280 0.018
50 0.097 0.124 0.152 0.028 0.233 0.359 0.409 0.092
100 0.124 0.184 0.210 0.069 0.386 0.608 0.642 0.386
200 0.202 0.328 0.353 0.204 0.643 0.868 0.881 0.784
300 0.275 0.440 0.462 0.337 0.808 0.965 0.968 0.940
25 0.092 0.072 0.152 0.047 0.131 0.161 0.301 0.023
50 0.091 0.127 0.180 0.033 0.178 0.340 0.449 0.100
100 0.107 0.190 0.235 0.080 0.304 0.600 0.670 0.407
200 0.159 0.307 0.347 0.206 0.512 0.866 0.885 0.791
300 0.215 0.443 0.476 0.343 0.674 0.965 0.971 0.942
25 0.103 0.152 0.186 0.057 0.123 0.267 0.328 0.027
50 0.082 0.171 0.176 0.035 0.153 0.409 0.442 0.100
100 0.095 0.204 0.226 0.081 0.235 0.611 0.645 0.383
200 0.128 0.300 0.333 0.201 0.407 0.864 0.885 0.780
300 0.175 0.429 0.461 0.338 0.553 0.958 0.965 0.936
25 0.102 0.178 0.188 0.061 0.121 0.351 0.353 0.030
50 0.088 0.233 0.295 0.042 0.131 0.453 0.455 0.106
100 0.087 0.238 0.322 0.076 0.202 0.626 0.652 0.389
200 0.123 0.354 0.386 0.215 0.344 0.856 0.879 0.776
300 0.156 0.440 0.462 0.337 0.472 0.949 0.959 0.924
-- ----- ------- ------- ------- ------- -- ------- ------- ------- -------
: Power of the likelihood ratio tests for exponential distribution[]{data-label="my-label"}
-- ----- ------- ------- ------- ------- -- ------- ------- ------- -------
$n$ DNEL VLX JEL AJEL DNEL VLX JEL AJEL
25 0.291 0.261 0.380 0.371 0.509 0.434 0.554 0.421
50 0.495 0.462 0.561 0.545 0.781 0.717 0.783 0.725
100 0.782 0.764 0.829 0.808 0.972 0.952 0.965 0.955
200 0.970 0.965 0.984 0.973 1.000 0.999 1.000 1.000
300 0.997 0.997 0.998 0.997 1.000 1.000 1.000 1.000
25 0.403 0.523 0.648 0.645 0.680 0.692 0.838 0.690
50 0.649 0.700 0.854 0.847 0.921 0.944 0.974 0.955
100 0.898 0.948 0.989 0.978 0.998 1.000 1.000 1.000
200 0.994 1.000 1.000 1.000 1.000 1.000 1.000 1.000
300 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
25 0.421 0.479 0.686 0.683 0.684 0.810 0.932 0.805
50 0.641 0.861 0.931 0.927 0.914 0.985 0.995 0.989
100 0.889 0.991 0.998 0.996 0.996 1.000 1.000 1.000
200 0.993 1.000 1.000 1.000 1.000 1.000 1.000 1.000
300 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
25 0.392 0.636 0.794 0.756 0.653 0.911 0.960 0.860
50 0.602 0.941 0.997 0.965 0.892 0.996 0.999 0.996
100 0.862 0.998 1.000 0.999 0.992 1.000 1.000 1.000
200 0.990 1.000 1.000 1.000 0.993 1.000 1.000 1.000
300 0.996 1.000 1.000 1.000 1.000 1.000 1.000 1.000
-- ----- ------- ------- ------- ------- -- ------- ------- ------- -------
: Power of the likelihood ratio tests for Normal distribution[]{data-label=""}
-- ----- ------- ------- ------- ------- -- ------- ------- ------- -------
$n$ DNEL VLX JEL AJEL DNEL VLX JEL AJEL
25 0.378 0.458 0.496 0.159 0.742 0.822 0.842 0.553
50 0.592 0.704 0.730 0.525 0.940 0.972 0.974 0.942
100 0.856 0.928 0.935 0.885 0.997 0.999 0.999 0.999
200 0.984 0.996 0.997 0.994 1.000 1.000 1.000 1.000
300 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000
25 0.359 0.409 0.553 0.198 0.701 0.679 0.866 0.592
50 0.552 0.702 0.766 0.580 0.916 0.872 0.979 0.957
100 0.828 0.940 0.949 0.913 0.997 0.940 1.000 0.999
200 0.979 0.997 0.998 0.996 1.000 0.972 1.000 1.000
300 0.998 1.000 1.000 1.000 1.000 0.984 1.000 1.000
25 0.341 0.432 0.580 0.228 0.696 0.507 0.894 0.649
50 0.532 0.721 0.798 0.622 0.912 0.630 0.986 0.968
100 0.794 0.932 0.955 0.924 0.994 0.715 1.000 1.000
200 0.969 0.994 0.999 0.998 1.000 0.806 1.000 1.000
300 0.996 0.998 1.000 1.000 1.000 0.853 1.000 1.000
25 0.335 0.530 0.606 0.246 0.663 0.503 0.908 0.661
50 0.505 0.780 0.806 0.644 0.884 0.639 0.990 0.972
100 0.759 0.943 0.962 0.931 0.990 0.739 1.000 1.000
200 0.956 0.995 1.000 0.999 1.000 0.823 1.000 1.000
300 0.973 0.998 1.000 1.000 1.000 0.962 1.000 1.000
-- ----- ------- ------- ------- ------- -- ------- ------- ------- -------
: Power of the likelihood ratio tests for log normal distribution[]{data-label=""}
Application
===========
Modelling rain fall frequency values is an important area of research for planning various projects such as construction of bridges and spillways of dams. Many probability distributions are used for this modelling purpose and this includes using method of PWM for the estimation of the parameters of interest. Here we look into 95 % interval estimates of $\beta_r$ for rain fall data set of Indian States.
Government of India releases data set related to various sectors and the same can be accessible through the web page [www.data.gov.in](www.data.gov.in). These sectors contain agriculture, education, health and family welfare, labour and employment, travel and tourism etc. We use weighted average of monthly rain fall (in mm) data for the whole country starting from 1901 to 2014 released by Meteorological Department, Ministry of Earth Sciences, India. This data set is based on more than 2000 rain guage readings spread over the entire country. During the span of 1901 to 2014, rain fall reached maximum for the year 1917 and recorded minimum frequency for the year 1972. India had witnessed severe drought in the year 1972-73 in drought prone areas especially large parts of Maharashtra, a state of India.
Table 8 gives 95 % interval estimates of $\beta_1$ and corresponding average lengths for the rain fall data. From the data, it is clear that for all most all months JEL interval has shorter average length. Also performances of JEL interval is comparable with VLX interval except for the month September where VLX interval is less stable. For AJEL based approach, the intervals are more wide for the months June to September.
[|l|l|l|l|l|]{} Month & DNEL & VLX & JEL & AJEL\
January & &
-------------------
(11.0018,13.3237)
2.3219
-------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(11.0817, 13.3838)
[2.3021]{}
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(10.2383, 13.0992)
2.8609
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
\
February &
--------------------
(12.4763, 17.2933)
4.8170
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(13.4675, 15.9107)
2.4433
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(13.5546, 15.9952)
[2.4406]{}
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(12.5195, 15.6855)
3.1661
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
\
March &
--------------------
(14.5653, 20.1041)
5.5387
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(15.7437, 18.5202)
2.7765
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(15.8361, 18.6097)
[2.7736]{}
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(14.5958, 18.2250)
3.6291
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
\
April &
--------------------
(18.9144, 24.9451)
6.0308
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(20.5914, 22.8184)
2.2270
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(20.6697, 22.8892)
[2.2195]{}
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(18.7482, 22.4825)
3.7343
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
\
May &
--------------------
(31.1654, 41.1201)
9.9546
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(33.9171, 37.6333)
[3.7192]{}
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(34.0352, 37.7536)
3.7184
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(30.8670, 37.1203)
6.2533
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
\
June &
---------------------
(82.7641, 107.3357)
24.5717
---------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(90.5382, 97.7330)
7.1948
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(90.8288, 97.9655)
[7.1367]{}
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(81.7213, 96.3148)
14.5935
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
\
July &
--------------------
(138.682, 175.931)
37.2482
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
----------------------
(152.4140, 159.2123)
6.7983
----------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(152.696, 159.460)
[6.7641]{}
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
----------------------
(135.8374, 156.8447)
21.0063
----------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
\
August &
----------------------
(122.7836, 156.2862)
33.5026
----------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
----------------------
(134.8720, 141.3698)
6.4978
----------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
----------------------
(135.1352, 141.6145)
[6.4793]{}
----------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
----------------------
(120.4020, 139.2415)
18.8395
----------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
\
September &
---------------------
(84.7445, 109.9898)
25.2453
---------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
---------------------
(92.6231, 108.6973)
[16.0743]{}
---------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
---------------------
(92.9210, 100.4291)
[7.5081]{}
---------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(83.6334, 98.5444)
14.9110
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
\
October &
--------------------
(39.6969, 53.8395)
14.1426
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(42.9838, 49.4825)
6.4987
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
---------------------
(43.20343, 49.6888)
[6.4854]{}
---------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
----------------------
(39.58949, 48.73922)
9.1497
----------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
\
November &
--------------------
(16.3331, 23.2432)
6.9101
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(17.5455, 21.4974)
[3.9620]{}
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(17.6720, 21.6304)
3.9584
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(16.4306, 21.1816)
4.7510
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
\
December &
-------------------
(8.2039, 11.9599)
3.7561
-------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
---------------------
(8.776057, 11.1920)
2.4160
---------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
-------------------
(8.8470, 11.2497)
2.4028
-------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
&
--------------------
(8.21451, 11.0223)
2.8078
--------------------
: 95% confidence interval estimators of $\beta_1$ and average lengths for rainfall data[]{data-label="my-label"}
\
Conclusions
===========
Probability weighted moments generalize the concept of moments of a probability distribution and are used to estimates parameters involved in model extremes of the natural phenomena. In this article we proposed JEL and AJEL based inference for $\beta_r$. We showed that the log JEL and AJEL ratio statistics is asymptotically distributed as chi-square distribution with one degree of freedom and constructed JEL and AJEL based confidence interval for $\beta_r$. We also developed JEL and AJEL based tests for testing the hypothesis $\beta_r=\beta_r^0$, where $\beta_r^0$ is a specific value of $\beta_r$. Simulation results shows that proposed JEL and AJEL test perform better than other recently developed tests in terms of empirical power. We also showed that the JEL based confidence interval has minimum length compared to competitor and has coverage probability of desired confidence level. The method is illustrated by using a rainfall data of Indian states.
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[^1]: [$^{\dag}$]{} [Corresponding E-mail: skkattu@isichennai.res.in. ]{} D B would like to thanks Indian Statistical Institute, Chennai for the support during his visit to ISI Chennai.
|
---
abstract: 'In this work we develop the theory of solution-regions with a constructive approach. We also extend the theory to the case of general linear conditions and provide various sets of sufficient hypotheses for existence and multiplicity results.'
author:
- |
F. Adrián F. Tojo[^1]\
e-mail: fernandoadrian.fernandez@usc.es\
*Instituto de Matemáticas, Facultade de Matemáticas,*\
*Universidade de Santiago de Compostela, Spain.*\
title: 'A constructive approach towards the Method of Solution-Regions'
---
**Keywords:** System of First Order Differential Equations, Multiple Solutions, Fixed Point Index, General Boundary Conditions, Solution-Regions, Upper and Lower Solutions
**MSC:** Primary 34B15; secondary 34A34, 34A12
Introduction
============
In a recent paper of Frigon [@Frigon] the author developed a theory, named *method of solution-regions*, in order to obtain results concerning the existence and multiplicity of solutions of the equation $$\label{eq1}u'(t)=f(t,u(t))\text{ for a.e. } t\in I:=[a,b],\ u\in B,$$ where $I$ is a non-degenerate interval and $B$ is the family of functions satisfying initial conditions of the form $$\label{ci}
u(a)=r$$ or the periodic boundary conditions $$\label{cp}
u(a)=u(b).$$
The method of solution-regions is an outstanding generalization of various methods of obtaining uniqueness and multiplicity of solutions of differential problems, namely the methods of upper and lower solutions [@Pouso2001; @Frigon1995], strict upper and lower solutions [@mawhin1987; @graefkong; @el2015], solution-tubes [@frigon2007; @frigono] and strict solution-tubes [@frigon2007; @frigonlotf]. Furthermore, the method is closely related to that of Gaines and Mahwin concerning what they called *bound sets* [@GM1; @GM2]. The definition of bound set extends that of solution region in the way presented in Remark \[rembs\], but the theory concerning them is developed in a non-comparable way as we will point out.
In this paper we take a constructive approach towards admissible regions. It is in Section 2 that we prove that, indeed, admissible regions, such as are defined in Definition \[defar\], always have an admissible pair, something which formed part of the assumptions before [@Frigon Definition 3.1]. By unlinking the topological and analytical aspects of admissible regions we are able to reach many interesting conclusions regarding their nature (see Remarks \[br\] and \[remint\]).
In what concerns solution regions (Section 3) we state the refined definition of *$C$-solution region*. This parameter dependent definition relaxes the restrictions imposed on the admissible pair (condition (H5)) while maintaining a simple proof. We also provide some alternative or complementary hypotheses (see (H4’), (H5’) and (H6)) in order to derive, in the next section, existence results.
In Section 4 we generalize the problems studied in [@Frigon] by allowing more general boundary conditions. First we deal with general linear conditions of the kind $$\label{cg}
\Gamma (u-u(a))=r,$$ where $r\in{\mathbb R}$ and $\Gamma:{\mathcal C}([a,b],{\mathbb R})\to {\mathbb R}$ is a linear functional (which can be thought as $\Gamma:{\mathcal C}([a,b],{\mathbb R}^n)\to {\mathbb R}^n$ by action on each component) such that $M:=\Gamma(1)\ne0$ ($1$ understood as the constant function $1$ on $[a,b]$). Observe that, by Riesz–Markov–Kakutani Representation Theorem [@Ben Theorem 7.2.4], there is a unique regular Borel (signed) measure $\mu$ on $[a,b]$ such that $\Gamma u=\int u\operatorname{d} \mu$ for every $u\in{\mathcal C}([a,b],{\mathbb R})$. Also, condition generalizes condition . Just take $r=0$ and $\Gamma u=u(b)$ (observe that $\Gamma(1)=1\ne0$).
We will also work with the boundary condition $$\label{cg2}
\Gamma u=r,$$ It is clear that that condition generalizes condition by defining $\Gamma$ as before. Also, observe that conditions and overlap in some cases, but neither of them covers all of the cases of the other. At the end of Section 4 we provide an example to which the theory is applied.
In Section 5 we deal with multiplicity results in the usual way through Fixed Point Index Theory. This scenario requires refined hypotheses (such as (H0’), (H4”) and (H5”)).
Finally we present our conclusions in Section 6, where we talk about the prowess and limitations of this approach and present some guidelines to overcome the occurring difficulties.
Throughout this paper we will work with the spaces ${\mathbb R}^n$ with the euclidean norm $\|\cdot\|$ and ${\mathcal C}(X,{\mathbb R}^n)$ the space of continuous functions with the supremum norm $\|\cdot\|_0$ where $X$ is some set. ${\mathbb R}^+$ will denote the interval $(0,+\infty)$.
Admissible Regions
==================
Let us first state some basic definitions for the theory ahead.
Let $A\subset{\mathbb R}\times{\mathbb R}^n$. Given $t\in {\mathbb R}$ and $x\in{\mathbb R}^n$ we write $$A_t:=\{x\in{\mathbb R}^n\ :\ (t,x)\in A\},$$ and $$A^x:=\{t\in {\mathbb R}\ :\ (t,x)\in A\}.$$ If we consider the continuous natural inclusions $i_t:{\mathbb R}^n\to{\mathbb R}\times{\mathbb R}^n$ and $i^x:{\mathbb R}\to{\mathbb R}\times{\mathbb R}^n$ such that $i_t(x)=i^x(t)=(t,x)$, then $A_t=(i_t)^{-1}(A)$ and $A^x=(i^x)^{-1}(A)$. Similarly, we consider the natural projections $\pi_1:{\mathbb R}\times{\mathbb R}^n\to{\mathbb R}$ and $\pi_2:{\mathbb R}\times{\mathbb R}^n\to{\mathbb R}^n$ such that $\pi_1(t,x)=t$ and $\pi_2(t,x)=x$.
Let $D\subset I\times{\mathbb R}^n$. A map $f:D\to{\mathbb R}^m$ is a *Carathéodory function* if
1. $f(t,\cdot)$ is continuous on $D_t$ for almost every $t\in I$,
2. $f(\cdot,x)$ is measurable for all $x\in\pi_2(R)$,
3. for all $k\in{\mathbb R}^+$, there exists $\psi_k\in L^1(I,{\mathbb R})$ such that $\|f(t,x)\|\le\psi_k(t)$ for a.e. $t$ and every $x$ such that $\|x\|\le k$ and $(t,x)\in D$, that is, the set $$\{t\in I\ :\ \|f(t,x)\|>\psi_k(t)\text{ for some }x\in R_t,\ \|x\|\le k\}$$ has measure zero.
\[defar\] Let $n\in{\mathbb N}$. A set $R\subset I\times{\mathbb R}^n$, is called an *admissible region* if $R$ is compact and
1. $R_t\ne\emptyset$ for every $t\in I$.
Let $R\subset I\times{\mathbb R}^n$ be an admissible region. A pair of continuous functions $(h,p)$ where $h:I\times{\mathbb R}^n\to{\mathbb R}$ and $p=(p_1,p_2):I\times{\mathbb R}^n\to I\times{\mathbb R}^n$ is called an *admissible pair associated to $R$* if
- $R=h^{-1}{((-\infty,0])}$.
- The map $h$ has partial derivatives at $(t,x)$ for almost every $t$ and every $x$ with $(t,x)\in (I\times {\mathbb R}^n)\backslash R$ and $\frac{\partial h}{\partial t}$ and $\nabla_x h$ are locally Carathéodory maps on $(I\times {\mathbb R}^n)\backslash R$.
- $p$ is bounded and such that $p(t,x)=(t,x)$ for every $(t,x)\in R$ and $$\label{trcon}\left\langle\nabla_xh(t,x),p_2(t,x)-x\right\rangle\le0\text{ for a.\,e. $t$ and every $x$ with $(t,x)\in (I\times {\mathbb R}^n)\backslash R$.}$$
\[br\] There are several things to take into account in this definition.
1. The definition of admissible region in [@Frigon] is more stringent in condition , where the inequality is taken in the strict sense. We will show that this is not necessary in order to get the same results, provided that we add another condition, (H6), later on, or that we alter condition (H4) ahead (see Remark \[remalt4\]).
2. The definition of admissible region in [@Frigon] imposes the existence of an associated admissible pair (that is, it includes axioms (H1)-(H3) in the definition), something which we prove is always the case.
3. The definition of admissible region in [@Frigon] does not require explicitly $R$ to be compact, although that is a direct consequence of it.
4. How do we interpret the set on which condition holds? In the proof of [@Frigon Theorem 5.1] they use that, *for any function $u\in W^{1,1}(I,{\mathbb R}^n)$ such that $(t,u(t))\in R$ and for every $t\in I$ almost everywhere on $\{t\ :\ h(t,u(t))>0\}$, $$\left\langle\nabla_xh(t,u(t)),p_2(t,u(t))-u(t)\right\rangle<0.$$* In order to ensure this inequality holds in terms of a variable $x$ instead of $u(t)$ it is enough to have that the set $$\{t\in I\ :\ \left\langle\nabla_xh(t,x),p_2(t,x)-x\right\rangle\ge0\text{ for some }x\in R_t\}$$ has measure zero.
5. In [@Frigon] the author highlighted, by providing several examples, the versatility of admissible regions, showing that they may have corners, their boundary may not be smooth and that they may not be proximate retracts. With the definition presented here it is obvious that those are possibilities for admissible regions.
6. If $(h,p)$ is an admissible pair and $\beta\in{\mathcal C}^1(I,{\mathbb R}^+)$, then $(\widetilde h,p)$ is also an admissible pair where $\widetilde h(t,x)=\beta(t)h(t,x)$. Hence, if $h(t,R_t)$ is bounded for every $t\in I$, taking $\beta$ such that $\beta(t)<1/\sup \|h(t,R_t)\|$, we have that $\widetilde h$, defined as before, is bounded.
7. If $(h_k,p)$ are an admissible pairs for $k=1,2$; then $(h_1+h_2,p)$ is an admissible pair.
Now we are ready to state some definitions and results in order to construct an admissible pair for a given region function.
Let $C,D\subset{\mathbb R}^n$. We define the *distance of a point $x$ to the set* $C$ as $$d_C(x):=\inf\{||x-y||\ :\ y\in C\}.$$ Analogously, we define the distance between $C$ and $D$ as $$d(C,D):=\inf\{||x-y||\ :\ x\in C,\ y\in D\}.$$ Furthermore, if $C$ is convex, we denote by $P_C:{\mathbb R}^n\to C$ the *projection* onto $C$. Remember that $P_C$ is a continuous function [@Holmes p. 108].
\[thmci\] Let $C$ be a closed convex subset of a Hilbert space $H$, and let $\varphi(x): =\frac{1}{2} d_C(x)^2$ for every $x\in H$. Then $\varphi$ is a $C^1$ convex function on $H$ and $\nabla \varphi(x)=x-P_C(x)$.
\[prowhit\] Let $R\subset{\mathbb R}^n$ be a closed set. Then there is\
$\varphi\in{\mathcal C}^\infty({\mathbb R}^n,[0,+\infty))$ such that $\varphi^{-1}(0)=R$.
\[remmin\]Observe that, in Proposition \[prowhit\], since $\varphi$ attains an absolute minimum at every point of $R$, $\nabla \varphi(x)=0$ for every $x\in R$.
\[thmar\] Let $R\subset I\times{\mathbb R}^n$ be an admissible region. Then $R$ has an associated admissible pair $(h,p)$.
Let $r\in{\mathbb R}^+$ such that $\pi_2(R)\subset C:=B_{{\mathbb R}^n}[0,r]$. Let $D:=B_{{\mathbb R}^n}(0,r+1)$ and $\varphi:{\mathbb R}^{n+1}\to{\mathbb R}$ be the function provided by Proposition \[prowhit\] for $R\cup(I\times({\mathbb R}^n\backslash D))$. Define $$h(t,x):=\begin{dcases}\varphi(t,x), & (t, x)\in I\times C,\\ [1- d_C(x)^2]\varphi(t,x)+\frac{1}{2} d_C(x)^2, & (t, x)\in I\times ( D\backslash C),\\ \frac{1}{2} d_C(x)^2, & (t, x)\in I\times ({\mathbb R}^n\backslash D).\end{dcases}$$ $h$ is of class ${\mathcal C}^1$ and $R=h^{-1}((-\infty,0])$, so (H1) and (H2) are satisfied. In fact, $$\nabla_xh(t,x):=\begin{dcases}\nabla_x\varphi(t,x), & (t, x)\in I\times C,\\ -2[x-P_C(x)]\varphi(t,x)+[1-d_C(x)^2]\nabla_x\varphi(t,x)+x-P_C(x), & (t, x)\in I\times ( D\backslash C),\\ x-P_C(x), & (t, x)\in I\times ({\mathbb R}^n\backslash D).\end{dcases}$$ Furthermore, $\nabla_x h(t,x)=0$ for every $(t,x)\in R$ (see Remark \[remmin\]). Let $p=(p_1,p_2):I\times{\mathbb R}^n\to I\times{\mathbb R}^n$ with $p_1(t,x)=t$ and $p_2(t,x):=x-\nabla_xh(t,x)=$ $$\begin{dcases} x-\nabla_x\varphi(t,x), & (t, x)\in I\times C,\\ 2[x-P_C(x)]\varphi(t,x)-[1-d_C(x)^2]\nabla_x\varphi(t,x)+P_C(x), & (t, x)\in I\times ( D\backslash C),\\ P_C(x), & (t, x)\in I\times ({\mathbb R}^n\backslash D).\end{dcases}$$ $p$ is continuous, bounded, and $p(t,x)=(t,x)$ for every $(t,x)\in R$. Furthermore, $$\left\langle\nabla_xh(t,x),p_2(t,x)- x\right\rangle=-\|\nabla_x h(t,x)\|^2\le0,$$ so (H3) holds as well.
Solution Regions
================
In the previous section we have presented the basic aspects of admissible regions, including the construction of admissible pairs. Unfortunately, the properties of these regions are independent of the problem of study (in this case problem ). It is for this reason that we need the concept of solutions regions.
Let $C\in{\mathbb R}^+$. A triple $(R,(h,p))$ where $R\subset I\times{\mathbb R}^n$ is an admissible region and $(h,p)$ is an admissible pair for $R$ is called a *$C$-solution region of problem *, if the following hold:
1. $\frac{\partial h}{\partial t}(t,x)+\left\langle\nabla_xh(t,x),f(p(t,x))\right\rangle\le0$ for a.e. $t$ and every $x$ with $(t,x)\not\in R$.
2. $h(a,u(a))\le0$ or $h(a,u(a))\le h(b,u(b))$ for any $u\in B$ such that $\|u\|_0\le C$.
In [@Frigon] they talk about solution regions and not $C$-solution regions. We will use this extra information given by the parameter $C$ for the more general boundary conditions and we deal with in this work.
\[remint\]We can rewrite condition (H4) as $\left\langle\nabla h(t,x),\widetilde f(p(t,x))\right\rangle\le0$ for a.e. $t$ and every $x$ with $(t,x)\not\in R$ where $\widetilde f=(1,f)$. Written in this way, this expression is reminiscent of the *transversality condition* [@Cid Equation (2.1)], necessary for some generalized Lipschitz uniqueness results, where $\widetilde f$ is the function defined to transform a non-autonomous problem into an autonomous one.
Thus expressed, (H4) has a geometric interpretation similar to the transversality condition. The function $\nabla h(t,x)$ can be thought as a normal field to the local foliation in $I\times {\mathbb R}^n$ where the leaves are defined by $h^{-1}(c)$ for each value $c\in{\mathbb R}$. Hence, (H4) implies that the function $\widetilde f$ points, at each point in $h^{-1}(c)$, in the direction of decreasing value of $c$, that is, against the flow $\nabla h(t,x)$ that takes one leave to another.
In this article we will only use the statement of (H5) in the case conditions $B$ refer to conditions or . The reader can check that, for the cases and (studied in [@Frigon]), condition (H5) can be rewritten as
- $h(a,r)\le0$ or $h(a,u(a))\le h(b,x)$ for any $x\in{\mathbb R}^n$, $\|x\|\le C$ in the case of condition and
- $h(a,x)\le0$ or $h(a,x)\le h(b,x)$ for any $x\in{\mathbb R}^n$, $\|x\|\le C$.
They are used with a similar statement in [@Frigon].
\[rembs\] The concept of solution region is closely related to that of bound set in [@GM1; @GM2].
We say that a set $A\subset I\times{\mathbb R}^n$ is a *bound set relative to problem * if
1. $A$ is open in the relative topology of $I\times{\mathbb R}^n$.
2. For any $(t_0,x_0)\in\partial A$ with $t_0\in(a,b)$ there exists $V\in{\mathcal C}^1(I\times{\mathbb R}^n,{\mathbb R})$ such that
1. $A\subset V^{-1}(-\infty,0)$,
2. $V(t_0,x_0)=0$,
3. $\frac{\partial V}{\partial t}(t_0,x_0)+\left\langle\nabla_xV(t_0,x_0),f(t_0,x_0)\right\rangle\ne 0$.
Let $(R,(h,p))$ be a solution region of problem . and consider $A:=\mathring R$. $A$ is open in the relative topology of $I\times {\mathbb R}^n$. Fix $(t_0,x_0)\in \partial A$. Using Proposition \[prowhit\], let $\psi\in{\mathcal C}^\infty(I\times{\mathbb R}^n,{\mathbb R}^n)$ such that $\psi\ge 0$ and $\psi^{-1}(0)={\mathbb R}^n\backslash A$. Let $V=h-\psi$. Taking the functional $V$ for every $(t_0,x_0)\in\partial A$ we can show that $A$ is a bound set relative to problem . Clearly, in the relative topology of $I\times {\mathbb R}^n$, not all compact sets are the closure of an open set and not all open sets have compact closures but, in practice, bound sets generalize solution regions.
The results concerning bound sets in [@GM2] have a different flavor than those presented here. For instance they require bound sets to be *autonomous*, that is, with $V$ independent of $t_0$ and $t$ –cf. [@GM2 Definition 3.2] and the $f$ occurring in problem has to be continuous –see [@GM2 Theorem 3.1].
As said before, in [@Frigon], the inequality in condition is taken in the strict sense. We will need the following condition to make up for it.
- Let $(R,(h,p))$ be a solution region of problem . Then there exists $\widehat h$ such that $(R,(\widehat h,p))$ is a solution region of problem and there exist $t_1,t_2\in I$ such that $\widehat h(t_1,x)= h(t_1,x)$ for every $x\in{\mathbb R}^n$ and $\widehat h(t_2,x)\ne h(t_2,x)$ for every $x\in\ {\mathbb R}^n\backslash R_{t_2}$.
In the light of Remark \[br\], points 6 and 7, condition (H6) is not as stringent as it may seem. In this line, the following lemma shows a sufficient condition for (H6) to hold.
Let $(R,(h,p))$ be a solution region for . Assume that $h$ is bounded and $(h,p)$ satisfies
1. There exist $\varepsilon\in{\mathbb R}^+$, $\delta\in{\mathbb R}^+$ and $t_0\in \mathring I$ such that $t\in[t_0-\delta,t_0+\delta]\subset \mathring I$ and$$\frac{\partial h}{\partial t}(t,x)+\left\langle\nabla_xh(t,x),f(p(t,x))\right\rangle\le-\varepsilon,$$ for a.e. $t\in[t_0-\delta,t_0+\delta]$ and every $x$ with $(t,x)\not\in R$.
Then (H6) holds.
Let $\beta\in{\mathcal C}^1(I,{\mathbb R}^+)$ such that $\beta|_{I\backslash[t_0-\delta,t_0+\delta]}=1$ and $\beta(t)>1$ for $t\in(t_0-\delta,t_0+\delta)$, $\|\beta'\|_0<\varepsilon/\|h\|_0$ for $t\in I$. Define $\widehat h(t,x)=\beta(t)h(t,x)$ for every $(t,x)\in I\times{\mathbb R}^n$. Clearly, $(\widehat h,p)$ is an admissible pair and $\widehat h(a,x)=h(a,x)$ and $\widehat h(b,x)=h(b,x)$, so $\widehat h$ satisfies (H5). Also $\widehat h(t_0,x)\ne h(t_0,x)$ for every $x\in R_{t_0}$.
Furthermore,
$$\begin{aligned}
& \frac{\partial \widehat h}{\partial t}(t,x)+\left\langle\nabla_x\widehat h(t,x),f(p(t,x))\right\rangle\\ = & \beta'(t)h(t,x)+\beta(t)\frac{\partial h}{\partial t}(t,x)+\left\langle\nabla_x(\beta h)(t,x),f(p(t,x))\right\rangle \\= & \beta'(t)h(t,x)+\beta(t)\left[\frac{\partial h}{\partial t}(t,x)+\left\langle\nabla_xh(t,x),f(p(t,x))\right\rangle\right] \\ \le &
\beta'(t)h(t,x)-\beta(t)\varepsilon\le \|\beta'\|_0\|h\|_0-\varepsilon\le0.\end{aligned}$$
Existence results
=================
Now we recall the following lemma which will be the key to proving various results ahead. We present it in a slightly modified way, although the proof is the same as in [@Frigon].
\[lembar\] Let $z\in{\mathbb R}$, $w\in{\mathcal C}(I,{\mathbb R})$ and $J=w^{-1}((z,+\infty))$. Assume that
1. $w\in W^{1,1}_{\operatorname{loc}}(J,{\mathbb R})$,
2. $w'(t)\le0$ for a.e. $t\in J$,
3. one of the following conditions holds:
1. $w(a)\le z$,
2. $w(a)\le w(b)$.
Then $w(t)\le z$ for all $t\in I$ or there exists $k>0$ such that $w(t)=k$ for all $t\in I$.
\[remlembar\] In Lemma \[lembar\], if $w'(t)<0$ for a.e. $t\in J$, then $w$ cannot be constant so, in that case, $w(t)\le z$.
Let $(h,p)$ be an admissible pair associated to an admissible region $R$. Consider again $r\in{\mathbb R}^n$ and a positive linear functional $\Gamma:{\mathcal C}(I,{\mathbb R})\to {\mathbb R}$. Let $M:=\Gamma(1)$. In what follows we will assume $M>0$. In the case we would like to work with a functional $\Gamma$ such that $M<0$, it is enough to do the change of variables $v=-u$ and study the problem $v'(t)=\widetilde f(t,v(t))$, $\widetilde\Gamma (v-v(a))=r$ ( $\widetilde\Gamma v=r$ respectively), where $\widetilde f(t,x)=-f(t,-x)$ and $\widetilde\Gamma=-\Gamma$.
Define $g(s):=\Gamma(\chi_{[a,\cdot]}(s))$ for every $s\in I$ where $\chi_{[a,t]}(s)$ denotes the characteristic function of of the interval $[a,t]$. Observe that $g$ is nonnegative. Let $c\in L^1(I,{\mathbb R})$ be such that $c(t)>\|f(p(t,x))\|$ for a.e. $t\in I$ and every $x\in {\mathbb R}^n$. Now, define $$f_R(t,x):=\begin{dcases} f(t,x), & (t,x)\in R,\\
f(p(t,x))+c(t)(p_2(t,x)- x), & (t,x)\in (I\times{\mathbb R}^n)\backslash R,\end{dcases}$$
and $$\Theta u:= \Gamma\int_a^tf_R(s,u(s))\operatorname{d} s.$$ Since $\Gamma u=\int u\operatorname{d} \mu$ for some unique regular Borel measure $\mu$ on $I$, we can apply Fubini’s Theorem [@Ben Theorem 3.7.5], and so $$\Theta u= \Gamma\int_a^bf_R(s,u(s))\chi_{[a,t]}(s)\operatorname{d} s=\int_a^{b} f_R(s,u(s))g(s)\operatorname{d} s .$$
Let $m:=\max\{\|x\|\ :\ x\in\pi_2(R)\}$, $C:=\max\left\{m,1+\|p_2\|_0\right\}$, $K>C$, $D:=B_{{\mathbb R}^n}[0,C]$, $E:=B_{{\mathbb R}^n}(0,K)$ and ${\mathcal U}:=\{u\in{\mathcal C}(I,{\mathbb R}^n)\ :\ \|u\|_0<K\}$.
With the previous ingredients we can consider the problem $$\begin{aligned}
\label{eql} u'(t)= &\lambda f_R(t,u(t)),\\ \label{mbc}
u(a)= & P_D\left( \frac{\Gamma u- r}{M}\right) .\end{aligned}$$
Observe that condition is equivalent to $\Gamma(u-u(a))= r$ when $u\in D$. The reason for the projection is that, due to the nature of the boundary condition , we cannot, in general, bound $\|u(a)\|$ (unlike in the cases of and , cf. [@Frigon]) which is necessary in order for condition 3.(a) in Lemma \[lembar\] to be satisfied. A similar approach using a projection in the case of upper and lower solutions was used in [@CaPo Section 3], where the nonlinear condition $L_1(u(a),u(b),u'(a),u'(b),u)=0$ occurred.
We could have presented ondition as $$\label{mbc2}Mu(a)= P_{MD}\left( \Gamma u- r\right) ,$$ where $MD:=\{Mx\in {\mathbb R}^n\ :\ x\in D\}$ thanks to the following lemma. Observe that, although would allow to write condition for the case $M=0$, doing this would be fruitless: for $M=0$ condition reads $0=0$.
For any $\alpha\in[0,+\infty)$, $x\in{\mathbb R}^n$, we have that $\alpha P_D(x)= P_{\alpha D}\left( \alpha x\right) $.
We have that $$\alpha P_D(x)=\begin{dcases} \alpha x, & \|x\|\le C, \\ \alpha C\frac{x}{\|x\|}, & \|\alpha x\|
> C,\end{dcases}\quad P_{\alpha D}\left( \alpha x\right) =\begin{dcases} \alpha x, & \|\alpha x\|\le \alpha C, \\ \alpha C\frac{\alpha x}{\|\alpha x\|}, & \|\alpha x\|> \alpha C.\end{dcases}$$ Written this way, the result is clear.
Let, $\displaystyle\Phi u:=\Gamma u- P_{MD}\left( \Gamma u- r\right) $ and consider the operator ${\mathcal J}:[0,1]\times{\mathcal C}(I,{\mathbb R}^n)\to{\mathcal C}(I,{\mathbb R}^n)$ defined by $${\mathcal J}(\lambda,u):=u(a)+\lambda \int_a^tf_R(s,u(s))\operatorname{d} s-(1+\lambda(t-a))(\lambda\Theta u- \Phi u).$$
Inspired in [@Frigon], we present the following results concerning the existence of solutions.
\[proM0\] Let $f:I\times{\mathbb R}^n\to{\mathbb R}^n$ be a Carathéodory function and $(h,p)$ an admissible pair associated to an admissible region $R$. Assume (H6) holds and $M>0$. Then, for every $\lambda\in[0,1]$, problem , has at least one solution. Moreover, $\operatorname{ind}({\mathcal J}(\lambda,\cdot),{\mathcal U})=(-1)^n$ for every $\lambda\in[0,1]$.
${\mathcal J}$ is a continuous and completely continuous operator (cf. [@Frigon Lemma 2.2]). Also, the fixed points of ${\mathcal I}$ are solutions of ,. Indeed, if $u={\mathcal J}(\lambda,u)$ then, for every $t\in I$, $$u(t)=u(a)+\lambda \int_a^tf_R(s,u(s))\operatorname{d} s-(1+\lambda(t-a))(\lambda\Theta u- \Phi u).$$ In particular, for $t=a$, we get that $\lambda\Theta u=\Phi u$, so $$\label{eqpfb}u(t)=u(a)+\lambda \int_a^tf_R(s,u(s))\operatorname{d} s.$$ Applying $\Gamma$ to both sides in , we obtain $\Gamma u=Mu(a)+ \lambda\Theta u=Mu(a)+ \Phi u$. Hence, $$u(a)=\frac{\Gamma u-\Phi u}{M}=P_D\left( \frac{\Gamma u- r}{M}\right) ,$$ and condition is satisfied. Also, for a.e. $t\in I$, $$u'(t)=\lambda f_R(t,u(t)),$$ so $u$ is a solution of , for every $\lambda\in[0,1]$.
Now we prove that the solutions of , are bounded by $C$. Assume $u$ is a solution of equation , for some $\lambda\in[0,1]$, that is, $$u(t)=u(a)+\lambda \int_a^tf_R(s,u(s))\operatorname{d} s.$$ Consider first the case $\lambda=0$. In that case $u\equiv x$ for some constant $x\in{\mathbb R}^n$, so $$\|u\|_0=\|x\|=\|u(a)\|=\left\| P_D\left( \frac{\Gamma u- r}{M}\right) \right\|\le C.$$
Now we study the case $\lambda>0$.
Then, almost everywhere on $\{t\in I\ :\ \|u(t)\|>C\}$, $$\begin{aligned}
\|u(t)\|'= & \left\langle\frac{u(t)}{\|u(t)\|},u'(t)\right\rangle=\left\langle\frac{u(t)}{\|u(t)\|},\lambda f_R(s,u(t))\right\rangle \\= & \left\langle\frac{u(t)}{\|u(t)\|},\lambda [f(p(t,u(t)))+c(t)(p_2(t,u(t))- u(t))]\right\rangle
\\= & -\lambda c(t)\|u(t)\|+\left\langle\frac{u(t)}{\|u(t)\|},\lambda [f(p(t,u(t)))+c(t)p_2(t,u(t))]\right\rangle
\\ \le & -\lambda c(t)C+\left|\left\langle\frac{u(t)}{\|u(t)\|},\lambda [f(p(t,u(t)))+c(t)p_2(t,u(t))]\right\rangle\right|
\\ \le & \lambda c(t)\left(1+\|p_2\|_0-C\right)<0,\end{aligned}$$ where the last inequality is a consequence of the definition of $c$. Then, by Lemma \[lembar\] and Remark \[remlembar\], $\|u\|_0\le C$.
By the homotopy property of the fixed point index, $\operatorname{ind}({\mathcal J}(\lambda,\cdot),{\mathcal U})=\operatorname{ind}({\mathcal J}(0,\cdot),{\mathcal U})$ for every $\lambda\in[0,1]$.
Observe that $${\mathcal J}(0,u)(t)=u(a)+ \Phi u.$$ is constant for $t\in I$. By the contraction property of the index (see [@Granas]), $$\operatorname{ind}({\mathcal J}(0,\cdot),{\mathcal U})=\operatorname{ind}({\mathcal J}(0,\cdot)|_{E},E).$$
Now let ${\mathcal P}(\lambda,x)= 2x+ \lambda\Phi x$ for $x\in E$, $\lambda\in[0,1]$. Observe that ${\mathcal P}(1,\cdot)={\mathcal J}(0,\cdot)$. Clearly, for $\lambda=0$, there is no $x\in\partial E$ such that $x={\mathcal P}(0,x)$. Assume there exists $x\in\partial E$ and $\lambda\in(0,1]$ such that $x={\mathcal P}(\lambda,x)$. We have that $$\begin{aligned}
0= & {\mathcal P}(\lambda,x)-x= x+\lambda\left( \Gamma x- P_{MD}\left( Mx- r\right) \right) =
x+\lambda\left( Mx- P_{MD}\left( Mx- r\right) \right) .\end{aligned}$$ Thus, $$\begin{aligned}
& (1+\lambda M)x=\lambda P_{MD}\left( Mx- r\right) .\end{aligned}$$ Taking norms, $$\begin{aligned}
& (1+\lambda M)K=\lambda\left\|P_{MD}\left( Mx- r\right) \right\|\le \lambda M C<\lambda MK ,\end{aligned}$$ which is a contradiction.
Therefore, by the homotopy property of the index, $$\begin{aligned}
\operatorname{ind}({\mathcal J}(\gamma,\cdot),{\mathcal U}) = &\operatorname{ind}({\mathcal J}(0,\cdot),E)= \operatorname{ind}({\mathcal P}(1,\cdot),E)=\operatorname{ind}({\mathcal P}(0,\cdot),E) = \operatorname{ind}(\operatorname{Id},E)=(-1)^n.\end{aligned}$$ Thus, ${\mathcal P}(\lambda,\cdot)$ has a fixed point, and hence problem , a solution, for every $\lambda\in[0,1]$.
\[thmprin\]Let $f:I\times{\mathbb R}^n\to{\mathbb R}^n$ be a Carathéodory function and $(h,p)$ an admissible pair associated to an admissible region $R$ such that $(R,(h,p))$ is a $C$-solution region of . Assume (H6) holds and $M> 0$. Then problem , has a solution $u\in W^{1,1}(I,{\mathbb R}^n)$ such that $(t,u(t))\in R$ for every $t\in I$.
By Proposition \[proM0\] there is a solution $u$ of problem , for $\lambda=1$ such that $\|u\|_0\le C$. Then, a.e. on $\{t\in I\ :\ h(t,u(t))>0\}$, we have that, by (H3) and (H4), $$\label{pasoint}\begin{aligned}\frac{\operatorname{d} h}{\operatorname{d} t}(t,u(t))= & \frac{\partial h}{\partial t}(t,u(t))+\left\langle\nabla_xh(t,u(t)),u'(t)\right\rangle \\ = & \frac{\partial h}{\partial t}(t,u(t))+\left\langle\nabla_xh(t,u(t)),f(p(t,x))+c(t)(p_2(t,x)-x)\right\rangle\le 0.
\end{aligned}$$
Hence, by (H5) and Lemma \[lembar\], either $h(t,u(t))\le 0$, and thus $(t,u(t))\in R$. As a consequence, $u$ is a solution of ,, or $h(t,u(t))=k$ for every $t\in I$.
Assume we are in the last case. Then, by (H6), there exists another admissible pair $(\widehat h,p_2)$ and two points $t_1,t_2\in I$ such that (H5) holds and $\widehat h(t_1,x)= h(t_1,x)$ for every $x\in{\mathbb R}^n$ and $\widehat h(t_2,x)\ne h(t_2,x)$ for every $x\in\ {\mathbb R}^n\backslash R_{t_2}$. Then, either $\widehat h(t,u(t))$ is not constant, and we are done, or it is constant, and hence $$k=h(t_1,u(t_1))=\widehat h(t_1,u(t_1))=\widehat h(t_2,u(t_2))\ne h(t_2,u(t_2))=k,$$ a contradiction.
\[remalt4\]Observe that, in the previous theorem, we could have dispensed with (H6) if the inequality in (H4) were strict for, in that case, the last inequality in would be strict (see Remark \[remlembar\]).
Observe that in the proof of Theorem hypothesis (H5) is used, exclusively, on solutions of problem , for $\lambda\in[0,1]$. Hence, we could provide the following weaker form of (H5) taking this into account.
1. $h(a,u(a))\le0$ or $h(a,u(a))\le h(b,u(b))$ for any solution $u$ of problem , such that $\|u\|_0\le C$ for any $\lambda\in[0,1]$.
Observe that Proposition was established for condition . Now we will develop a result for the case in an analogous fashion.
Consider the problem $$\label{eql3}\begin{aligned} u'(t)= &\lambda f_R(t,u(t)),\\
u(a)= & \widetilde\Phi u:=P_D\left( \frac{\Gamma u+Mu(a)- r}{M}\right) .
\end{aligned}$$
Consider now the operator ${\mathcal K}:[0,1]\times{\mathcal C}(I,{\mathbb R}^n)\to{\mathcal C}(I,{\mathbb R}^n)$ defined by $${\mathcal K}(\lambda,u):=M^{-1}(\widetilde\Phi u-\lambda \Theta u)+\lambda \int_a^tf_R(s,u(s))\operatorname{d} s.$$
Proposition \[proM\] has a proof very similar to that of Proposition \[proM0\]. We provide a sketch of it.
\[proM\] Let $f:I\times{\mathbb R}^n\to{\mathbb R}^n$ be a Carathéodory function and $(h,p)$ an admissible pair associated to an admissible region $R$. Assume (H6) holds. Then, for every $\lambda\in[0,1]$, problem has at least one solution. Moreover, there exists $$C>\max\{\|x\|\ :\ x\in\pi_2(R)\}$$ such that $$\operatorname{ind}({\mathcal K}(\lambda,\cdot),{\mathcal U})=(-1)^n$$ for every $\lambda\in[0,1]$ with ${\mathcal U}:=\{u\in{\mathcal C}(I,{\mathbb R}^n)\ :\ \|u\|<C\}$.
${\mathcal K}$ is a continuous and completely continuous operator and the fixed points of ${\mathcal K}$ are solutions of . Indeed, if $u={\mathcal K}(\lambda,u)$ then, for every $t\in I$, $$\label{eqpf3}u(t)=M^{-1}(\widetilde\Phi u-\lambda \Theta u)+\lambda \int_a^tf_R(s,u(s))\operatorname{d} s.$$
Applying $\Gamma$ to both sides in we get $\Gamma u= \widetilde\Phi u$. Also, for a.e. $t\in I$, $u'(t)=\lambda f_R(t,u(t))$, so $u$ is a solution of for every $\lambda\in[0,1]$. The proof continues as in Proposition \[proM\].
Theorem \[thmprin2\] is analogous to Theorem \[thmprin\].
\[thmprin2\]Let $f:I\times{\mathbb R}^n\to{\mathbb R}^n$ be a Carathéodory function and $(h,p)$ an admissible pair associated to an admissible region $R$ such that $(R,(h,p))$ is a solution region of . Assume (H6) holds and $M>0$. Then problem , has a solution $u\in W^{1,1}(I,{\mathbb R}^n)$ such that $(t,u(t))\in R$ for every $t\in I$.
An example
----------
Let $I:=[0,1]$ and consider the problem $$\label{eqeje}\begin{aligned} x'(t)= & -2x(t)e^{y(t)}, & \int_{0}^1 x(s)\operatorname{d} s=1,\\ y'(t)= & -y(t)e^{x(t)}, & \int_{0}^1 y(s)\operatorname{d} s= 1.\end{aligned}$$ for a.e. $t\in I$. Let $R:=B_{{\mathbb R}^3}[0,2]$, $f(t,x,y):=(-2xe^{y},-ye^x)$, $h(t,x,y)=\frac{1}{2}d_{R}^2(t,x,y)$ and $$p(t,x,y):=(t,x,y)-\nabla h(t,x,y)=P_{R}(t,x,y),$$ for $(t,x,y)\in I\times{\mathbb R}^2$. Now we show that $(R,(h,p))$ is a $2$-solution region of .
(H1) $R=h^{-1}{((-\infty,0])}$.
(H2) The map $h$ es continuously differentiable (Theorem \[thmci\]).
(H3) $p$ is bounded and such that $p(t,x,y)=(t,x,y)$ for every $(t,x,y)\in R$ and $$\begin{aligned}
& \left\langle\pi_2(\nabla h(t,x,y)),\pi_2(p(t,x,y)-(t,x,y))\right\rangle= -\|\pi_2(\nabla h(t,x,y))\| \\ = &
\begin{dcases}
0, & (t,x,y)\in R, \\
-\frac{\left(x^2+y^2\right) \left(\sqrt{t^2+x^2+y^2}-2\right)^2}{t^2+x^2+y^2}, & (t,x,y)\not\in R, \\
\end{dcases}\end{aligned}$$ which is non-positive for $(t,x,y)\in (I\times {\mathbb R}^2)\backslash R$.
(H4) Since $$f(p(t,x,y))=
\begin{dcases}
\left( -2 x e^{-y},-ye^{-x} \right) , & (t,x,y)\in R, \\
\left( -\frac{4 x e^{-\frac{2 y}{\sqrt{t^2+x^2+y^2}}}}{\sqrt{t^2+x^2+y^2}},-\frac{2 y e^{-\frac{2 x}{\sqrt{t^2+x^2+y^2}}}}{\sqrt{t^2+x^2+y^2}}\right) , & (t,x,y)\not\in R,\\
\end{dcases}$$ we have that $$\begin{aligned}
& \frac{\partial h}{\partial t}(t,x,y)+\left\langle\nabla_{(x,y)}h(t,x,y),f(p(t,x,y))\right\rangle \\ = & -\frac{e^{-\frac{2 (x+y)}{\sqrt{t^2+x^2+y^2}}} \left(\sqrt{t^2+x^2+y^2}-2\right) }{\left(t^2+x^2+y^2\right)^{3/2}}\\ & \cdot\left( 4 x^2 e^{\frac{2 x}{\sqrt{t^2+x^2+y^2}}} \sqrt{t^2+x^2+y^2}-t e^{\frac{2 (x+y)}{\sqrt{t^2+x^2+y^2}}} \left(t^2+x^2+y^2\right)+2 y^2 e^{\frac{2 y}{\sqrt{t^2+x^2+y^2}}} \sqrt{t^2+x^2+y^2}\right)
< 0,\end{aligned}$$ for a.e. $t$ and every $(x,y)$ with $(t,x,y)\not\in R$. This fact is illustrated in Figure \[fig:cont\].
![The inner surface is the boundary of the region $R$ (where $h=0$), the outer surface is the level set $h=1/2$ and the vector field the map $\widetilde f=(1,f)$. Observe that, as stated in Remark \[remint\], for $c>0$, $\widetilde f$ points, at each point in $h^{-1}(c)$, in the direction of decreasing value of $c$.[]{data-label="fig:cont"}](cont){width="0.5\linewidth"}
Observe that $M=1$. Also, we have that $m=\max\{\|x\|\ :\ x\in\pi_2(R)\}=2$, $\|p_2\|_0=2$ and $C:=\max\left\{m,1+\|p_2\|_0\right\}=2$.
(H5’) Let us check that, for any solution $u$ of problem such that $\int_I u= r=\left( 1,1\right) $ and $\|u\|_0\le 2$, we have that $h(0,u(0))\le0$ or $h(0,u(0))\le h(1,u(1))$. In other words, that $d_{R}(0,u(0))\le 0$ or $d_{R}(0,u(0))\le d_{R}(1,u(1))$, which is equivalent to $\|u(0)\|\le 2$ or $$\max\{\|u(0)\|-2,0\}\le \max\{\sqrt{1+\|u(0)\|^2}-2,0\}.$$
Problem , in this context, can be expressed as $$\begin{aligned} u'(t)= &\lambda \begin{dcases} \left( -2 x e^{-y},-ye^{-x} \right) , & (t,x,y)\in R,\\
f(p(t,x,y))+c(t)(p_2(t,x,y)- (x,y)), & (t,x,y)\in (I\times{\mathbb R}^n)\backslash R,\end{dcases}\\
u(-1)= & \begin{dcases} \int_{-1}^0 u+u(-1)- r, & \left\|\int_{-1}^0 u+u(-1)- r\right\|\le 2, \\ 2\frac{\int_{-1}^0 u+u(-1)- r}{\left\|\int_{-1}^0 u+u(-1)- r\right\|}, & \left\|\int_{-1}^0 u+u(-1)- r\right\|>2.\end{dcases}
\end{aligned}$$
Taking norms in the boundary conditions it is clear that $\|u(-1)\|\le 2$ and, hence, (H5’) is satisfied.
We can conclude that there is a solution of problem in $R=B_{{\mathbb R}^3}[0,2]$. A numerical approximation of this solution is shown in Figure \[fig:plot\].
![Representation of the numerical solution of problem . The continuous line represents $x(t)$, the dashed line $y(t)$ and the dotted line $\|(t,x(t),u(t))\|$. Observe that the dotted line is always below two.[]{data-label="fig:plot"}](plot){width="0.7\linewidth"}
Multiplicity results
====================
As we did in the case of admissible regions and solution regions, we present here a slightly modified definition of strict solution region [@Frigon].
A solution region $(R,(h,p))$ is a *strict $C$-solution region* of problem if the following hold:
1. $\mathring R_t\ne\emptyset$ for every $t\in I$.
2. There exists $\varepsilon\in{\mathbb R}^+$ such that $\frac{\partial h}{\partial t}(t,x)+\left\langle\nabla_xh(t,x),f(p(t,x))\right\rangle\le0$ for a.e. $t$ and every $x$ with $(t,x)\in h^{-1}((-\varepsilon,0))$. Also, $\frac{\partial h}{\partial t}$ and $\nabla_x h$ are locally Carathéodory in $h^{-1}((-\varepsilon,0))$.
3. $h(a,u(a))<0$ or $h(a,u(a))< h(b,u(b))$ for any $u\in B$ such that $\|u\|_0\le C$.
(H0’) is a stronger condition than (H0) and (H5”) than (H5). On the other hand, (H4”) complements (H4) by extending it to part of the interior of $R$. Clearly, if $h\ge0$, such as would be the case if it had been obtained by the construction in Theorem \[thmar\] or in the previous example, this condition would be vacuous.
Now we extend [@Frigon Proposition 6.3] to the case of conditions or . The proof is basically the same, but we include it for completeness.
\[ssl\] Let $f:I\times{\mathbb R}^n\to{\mathbb R}^n$ be a Carathéodory function and $(R,(h,p))$ a strict solution region of problem . Assume $u$ is a solution of problem such that $(t,u(t))\in R$ for every $t\in I$. Then $(t,u(t))\in \mathring R$ for every $t\in I$.
We do the proof in the case of the case of conditions or . Assume $J:=\{t\in I\ :\ h(t,u(t))=0\}\ne\emptyset$. Observe that $a\not\in J$. If that were the case, $0=h(a,u(a))<h(b,u(b))\le0$, which is a contradiction. Let $r=\min J\in(a,b]$. We have that $h(t,u(t))<0$ for $t\in [0,r)$. Let $\varepsilon\in{\mathbb R}^+$ as in (H4”). Since $u$ and $h$ are continuous, there exist $t_1,t_2\in[0,r)$, $t_1<t_2$ such that $h(t_1,u(t_1))<h(t_2,u(t_2))$ and $h(t,u(t))\in(-\varepsilon,0)$ for all $t\in[t_1,t_2]$. Using (H4”) we get $$\begin{aligned}
0< & h(t_2,u(t_2))-h(t_1,u(t_1))=\int_{t_1}^{t_2}\frac{\operatorname{d} }{\operatorname{d} t}h(t,u(t))\operatorname{d} t\\ = & \int_{t_1}^{t_2}\left( \frac{\partial h}{\partial t}(t,x)+\left\langle\nabla_xh(t,u(t)),f(p(t,u(t)))\right\rangle\right) \operatorname{d} t\le 0,\end{aligned}$$ which is a contradiction and, hence, $J=\emptyset$ and $h(t,u(t))<0$ for every $t\in I$ Since $h$ is continuous and $R=h^{-1}((-\infty,0])$, $\partial R=h^{-1}(0)$, so $(t,u(t))\in \mathring R$ for every $t\in I$.
The statement *‘$(t,u(t))\in \mathring R$ for every $t\in I$’* in Proposition \[ssl\] is slightly stronger than the statement *‘$u(t)\in \mathring R_t$ for every $t\in I$’* in [@Frigon Proposition 6.3], but the proof in [@Frigon] also covers this case. It can be checked that $$\mathring R\subset\bigcup_{t\in I}(\{t\}\times\mathring R_t),$$ but the reverse content is not in general true. In fact, $\pi_1\left( \bigcup_{t\in I}(\{t\}\times\mathring R_t)\right) $ might be connected while $\pi_1(\mathring R)$ might be not.
Proposition \[ssl\] allows us to prove a variety of multiplicity results in a standard way through Fixed Point Index Theory (cf. [@Frigon; @infante2016non]).
Conclusions
===========
As it was pointed out before, we have substituted the original condition
- $p$ is bounded and such that $p(t,x)=(t,x)$ for every $(t,x)\in R$ and $$\left\langle\nabla_xh(t,x),p_2(t,x)-x\right\rangle<0\text{ for a.\,e. $t$ and every $x$ with $(t,x)\in (I\times {\mathbb R}^n)\backslash R$.}$$
in the definition of admissible region [@Frigon] by the weaker version (H3). As a result we had to ask for (H6) to be satisfied in order to prove Propositions \[proM0\] and \[proM\] and Theorems \[thmprin\] and \[thmprin2\]. The reason for this change was the need to prove de existence of admissible pairs associated to a given admissible function. Theorem \[thmar\] provides such a result and it may be possible to improve it by obtaining an admissible pair satisfying (H3’) instead of (H3).
Such a proof would not be devoid of difficulties. There are several ways to construct a ${\mathcal C}^\infty$ function $f$ that vanishes only on $R$, for instance convolving the indicator function with a bump function, using a Whitney’s cover (see [@Krantz Proposition 3.3.6], [@krantz2 Section 5.3] or [@Whitney]), etc. Furthermore, it seems possible to obtain such a function further satisfying that $(\nabla_xf)^{-1}(\{0\})$ is countable (in the case of a closed subset of ${\mathbb R}$ this is straightforward) although the topology of $R$ may play an important role in the construction.
Various approaches may be taken at this point. Analytic functions on ${\mathbb R}$ satisfy that, if non-constant, they cannot vanish on a set with accumulation points, a fact that could be relevant when we observe that, Whitney’s Extension Theorem [@Whitney Theorem I] provides an extension that is analytic outside of the original domain of the function.
As said before, the topology of $R$ plays an important role. This is due to the following result.
Let $f:{\mathbb R}^n\to[0,+\infty)$ be differentiable. Let $R:=f^{-1}(\{0\})=R$. Let $C$ be a bounded connected component of ${\mathbb R}^n\backslash R$. Then there exists $x\in C$ such that $\nabla f(x)=0$.
Since $C$ is bounded, $\overline C$ is compact and, hence, being $f$ continuous, it reaches a maximum value in $\overline C$. Since $f>0$ in $C$ and $f$ is $0$ in $\partial C$, the maximum is attained at some point $x\in C$. Since $f$ differentiable in the open set $C$ we have that $f'(x)=0$.
Furthermore, any open set in ${\mathbb R}^n$ may have at most a countable number of connected components. This is due to the fact that ${\mathbb R}^n$ is a second countable topological space. This means that we have a lower bound for the number of zeros the gradient of a function $f:{\mathbb R}^n\to[0,+\infty)$ may have in $f^{-1}({\mathbb R}^+)$.
The relation of critical points of a function with the connected components of the domain is also consequence of a basic result in Morse Theory known as the Morse inequalities [@Milnor Theorem 5.2]. To be precise, it relates the topological invariants known as Betti numbers (the first of them being the number of connected components) to the number of the different types of non-degenerate singularities of the function. Thus, Morse Theory might shed light on our problem. We have the following well known results.
A non-degenerate critical point is isolated.
Let $M$ be a closed $m$-manifold and $g\in\mathcal C^\infty(\mathbb R^n,\mathbb R)$. Then there exists a Morse function $f\in\mathcal C^\infty(\mathbb R^n,\mathbb R)$ arbitrarily close to $g$.
Since Morse functions have only non-degenerate critical points, they have at most a countable number of such points, making them another suitable approach to the problem.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to acknowledge his gratitude towards Professor Santiago Codesido for his help with the code for the computation of the numerical solution in the example, to Professor Marlène Frigon for her explanatory comments concerning her paper [@Frigon] and to Professors Rodrigo López Pouso and Ignacio Márquez Albés for their insightful advice regarding linear functionals defined by Lebesgue-Stieltjes integration.
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[^1]: The author was partially supported by Ministerio de Economía y Competitividad, Spain, and FEDER, project MTM2013-43014-P, and by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDER.
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abstract: |
We present a proof for the existence and uniqueness of weak solutions for a cut-off and non cut-off model of non-linear diffusion equation in finite-dimensional space $R^D$ useful for modelling flows on porous medium with saturation, turbulent advection, etc. - and subject to deterministic or stochastic (white noise) stirrings. In order to achieve such goal, we use the powerful results of compacity on functional $L^p$ spaces (the Aubin-Lion Theorem). We use such results to write a path-integral solution for this problem.
Additionally, we present the rigourous functional integral solutions for the Linear Diffussion equation defined in Infinite-Dimensional Spaces (Separable Hilbert Spaces). These further results are presented in order to be useful to understand Polymer cylindrical surfaces probability distributions and functionals on String theory.
---
**Non-linear diffusion in $R^D$ and**
**in Hilbert Spaces, a Cylindrical/Functional**
**Integral Study**
.3in
[Luiz C.L. Botelho]{}
Departamento de Matemática Aplicada,
Instituto de Matemática, Universidade Federal Fluminense,
Rua Mario Santos Braga
24220-140, Niterói, Rio de Janeiro, Brazil
e-mail: botelho.luiz@ig.com.br
.3in
Introduction
============
The deterministic non-linear diffusion equation is one of the most important topics in the Mathematical-Physics of the non-linear evolution equation theory \[1-3\]. An important class of initial-value problems in turbulence has been modeled by non-linear diffusion stirred by random sources \[4\].
The purpose of this paper in Mathematical methods for Physics is to provide a model of non-linear diffusion were one can use and understand the compacity functional analytic arguments to produce theorems of existence and uniqueness on weak solutions for deterministic sitirring in $L^\infty ([0,T] \times L^2 ({{\Omega}}))$. We use these results to give a first step “proof" for the famous Rosen path integral representation for the Hopf charactheristic functional associated to the white-noise stirred non-linear diffusion model. These studies are presented on section II.
In section III we present a study of a Linear diffusion equation in a Hilbert Space, which is the basis of the famous Loop Wave Equations in String and Polymer surface theory (\[3\], \[4\]).
The Non-Linear Diffusion
========================
Let us start our paper by considering the following non-linear diffusion equation in some strip $\Omega \times[0,T]$ with ${\overline}{\Omega} $ denoting a $C^\infty$-compact domain of $R^D$. [^1] $$\frac{\partial U (x,t)}{\partial t}=(+ \Delta U) (x,t) + \Delta^{(\wedge)} (F(U (x,t))+f(x,t))$$ with initial and Dirichlet boundary conditions as given below. $$U(x,0)=g(x)\in L^2(\Omega)$$ $$U(x,t)\mid_{\partial\Omega} \equiv 0\quad \text{(for $t > 0$)}$$
We note that the non-linearitity of the diffusion-spatial term of the parabolic problem eq(1) takes into account the physical properties of non-linear porous medium’s diffusion saturation physical situation where this model is supposed to be applied \[1\] - by means of the hypothesis that the regularized Laplacean operator $\Delta^{(\wedge)}$ in the non-linear term of the governing diffusion eq.(1) has a cut-off in its spectral range. Additionaly we make the hypothesis that the non-linear function $F(x)$ is a compact support real continuously differentiable function on the extended interval $(- \infty,\infty)$. The external source $f(x,t)$ is supposed to belong to the space $L^{\infty} ([0,T] \times L^2(\Omega))$ or to be a white-noise external stirring of the form (\[2\] - pp. 61) when in the random case $$F(\cdot,t) = \frac{d}{dt} \left\{{\sum_{n \in Z} \sqrt{\lambda_n} \beta_n}(t) \varphi_n (\cdot)\right\} = \frac{d}{dt} w(t).$$
Here $\{\varphi_n\}$ denotes a complete orthonormal set on $L^2 (\Omega)$ and $\beta_{n}(t)$, $n \in Z$ are independent Wiener processes.
Let us show the existence and uniqueness of weak solutions for the diffusion problem above stated by means of Galerking Method for the case of deterministic $ f(x,t) \in
L^\infty([0,T] \times L^2(\Omega))$.
Let $\{\varphi_n (x)\}$ be spectral eigen-functions associated to the Laplacean $\Delta.$ Note that each $\varphi_n (x) \in H^2 (\Omega) \cap H_{0}^{1} (\Omega)$ \[3\]. We introduce now the (finite-dimensional) Galerkin approximants $$\begin{aligned}
U^{(n)} (x,t) &= \sum_{i=1}^{n} U_{i}^{(n)} (t) \varphi_i (x) \notag \\
f^{(n)} (x,t) &= \sum_{i=1}^{n}(f(x,t),\varphi_i (x))_{L^{2} (\Omega)} \varphi_i (x)\end{aligned}$$ subject to the initial-conditions $$U^{(n)} (x,0) = \sum_{i=1}^{n} (g(x),\varphi_i)_{L^{2} (\Omega)} \varphi_i(x)$$ here $(\,\, ,\,\, )_{L^{2}(\Omega)}$ denotes the usual inner product on $L^{2} (\Omega).$
After substituting eqs.(5), (6) in eq.(1), one gets the weak form of the non-linear diffusion equation in the finite-dimension approximation as a mathematical well-defined systems of ordinary non-linear differential equations, as a result of an application of the Peano existence-solution theorem. $$\begin{aligned}
& \left( \frac{\partial U^{(n)} (x,t)}{\partial t} ,\varphi_j(x)\right)_{L^2 (\Omega)} + \left(-\Delta U^{(n)} (x,t) , \varphi_ j(x)\right)_ {L^2 (\Omega)} \notag \\
&= \left(\nabla^{(\wedge)} \cdot [(F'(U^{(n)}(x,t)) \nabla^{(\wedge)} U^{(n)}(x,t)], \varphi_ j (x)
\right)_{L^2(\Omega)} + (f^{(n)} (x,t), \varphi_ j (x))_{L^2(\Omega)} \end{aligned}$$
By multiplying the associated system eq.(7) by $U^{(n)}$ we get the diffusion equation in the finite dimensional Galerking sub-space in the integral form: $$\begin{aligned}
& \frac12 \frac d{dt} \Vert U^{(n)} \Vert^2_{L^2 (\Omega)} \notag \\
& +(-\Delta U^{(n)}, U^{(n)} )_{L^2 (\Omega)} \notag \\
&+\int_\Omega d^3x (F'(U ^{(n)} ) (\nabla^{(\wedge)} U^{(n)} \cdot \overline{\nabla^{(\wedge)} U^{(n)}} ) (x,t) = (f, U^{(n)})_{L^2 (\Omega)}\end{aligned}$$
This result, by its turn, yields a prior estimate for any positive integer $p$: $$\begin{aligned}
& \frac12\frac{d}{dt} \left( \Vert U^{(n)} \Vert^2_{L^2 (\Omega)} \right) + \gamma (\Omega) \Vert U^{(n)} \Vert^2_{L^2 (\Omega)} + \Vert (F'(U^{(n)}))^{\frac{1}{2}} (\nabla U^{(n)}) \Vert^2_{L^2 (\Omega)}\notag \\
& \leq \frac{1}{2} \left\{p \Vert f(x,t) \Vert^2_{L^2 (\Omega)} + \frac{1}{p} \Vert U^{(n)} \Vert^2_{L^2 (\Omega)} \right\} \end{aligned}$$ Here $\gamma (\Omega)$ is the Garding-Poincaré constant on the inequalite of the quadratic form associated to the Laplacean operator defined on the domain $H^2 ({{\Omega}}) \cap H_0^1 ({{\Omega}}).$ $$\Vert U^{(n)} \Vert^2_{H^1 (\Omega)} = \bigl( -\Delta U^{(n)} , U^{(n)} \bigr)_{L^2(\Omega)} \geq \gamma (\Omega) \Vert U^{(n)}||^2_{L^2 (\Omega)} .$$
By chosing the integer $p$ big enough and applying the Gronwall lema, we obtain that the set of function $ \{U^{(n)} (x,t) \}$ forms a bounded set in $L^\infty ([0,T], L^2 (\Omega)) \cap L^\infty ([0,T], H^1_0 (\Omega) )$ and in $L^2 ([0,T],L^2(\Omega) ).$ As a consequence of this boundeness property of the function set $ \{U^{(n)} \}$, there is a sub-sequence weak-star convergent to a function ${\overline}U (t,x) \in L^\infty ([0,T], L^2 (\Omega))$, which is the candidate for our “weak" solution of eq.(1).
Another important estimate is to consider again eq.(9), but now considering the Sobolev space $H^1_0 (\Omega)$ on this estimate eq.(9), namely: $$\begin{aligned}
& \frac{1}{2} \bigl( \Vert U^{(n)} (T) \Vert^2_{L^2(\Omega)} - \Vert U^{(n)} (0) \Vert^2 \bigr) + {\overline}C_0 \int_0^T dt||U^{(n)}||^2_{H_0^1({{\Omega}})} \notag \\
& \le \frac12 p\left( \int_0^T ||f||^2_{L^2 ({{\Omega}})}dt\right)+\frac1{2p} \left(\int_0^T
\Vert U^{(n)} \Vert^2_{L^2(\Omega)}dt \right) < {\overline}M<\infty\end{aligned}$$ since we have the coerciviness condition for the Laplacean operator $$(- \Delta U^{(n)}, U^{(n)} )_ {L^{2}(\Omega)} \geq {\overline}C_0 (U^{(n)} , U^{(n)} )_{H^1_0(\Omega)}.$$ Note that $\Vert U^{(n)} (0)\Vert^2 \leq 2 \Vert g(x) \Vert^2_{L^2 (\Omega)}$ (see eq.(8)) and $ \{ \Vert U^{(n)} (T) \Vert^2_{L^2(\Omega)} \}$ is a bounded set of real positive numbers.
As a consequence of a prior estimate of eq.(11), one obtains that the previous sequence of functions $\{ U^{(n)} \} \in L^\infty ([0,T], H^1_0 (\Omega) \cap H^{2} (\Omega))$ forms a bounded set on the vector valued Hilbert space $L^2 ([0,T], H^1_0 (\Omega))$ either.
Finally, one still has another a prior estimate after multiplying the Galerkin system eq.(7) by the time-derivatives $\dot U^{(n)}$, namely (with $f(x,t)\equiv0$ for simplicity) $$\begin{aligned}
& \int^T_0 dt \left\Vert \frac{dU_n (t)}{dt} \right\Vert^2_{L^2(\Omega)} \notag\\
&\leq \overbrace{\text{ Real} (AU_n (T), U_n (T) )}^{(<0)} - \overbrace{(AU_n (0), U_n (0) )}^{(\ge {{\gamma}}({{\Omega}})(U_n(0),U_n(0)))} \notag\\
& \quad +\int^T_0 dt \left \Vert \Delta^{(\wedge)} F (U_n (t)) \frac{dU_n}{dt} \right \Vert_{L^2(\Omega)} \notag\\
& \leq \frac{1}{2} p \left( \int^T_0 \left\Vert \Delta^{(\wedge)} F (U_n (t) ) \right\Vert^2_{L^2(\Omega)} dt \right) \notag\\
& \quad + \frac{1}{2p} \left( \int^T_0 dt \left\Vert \frac{dU_n}{dt} \right\Vert^2_{L^2(\Omega)} \right) + \overbrace{{{\gamma}}({{\Omega}})||U_n(0)||^2}^{\equiv 0}\end{aligned}$$
By noting that $$\begin{aligned}
\, & \int^T_0 \Vert \Delta^{(\wedge)} F (U_n (t) ) \Vert^2_{L^2(\Omega)} dt \notag\\
&\leq \Vert \Delta^{(\wedge)} \Vert^2_{op} \times \begin{pmatrix}
\sup & \{ F(x) \}\\
x \in [-\infty,\infty] &
\end{pmatrix}^2 \notag \\
& \times \int^T_0 dt \Vert U_n (t) \Vert^2_{L^2(\Omega)} < \infty\end{aligned}$$ one obtains as a further result that the set of the derivatives $\{ \frac{dU_n}{dt}\}$ is bounded in $L^2 ([0,T], L^2 (\Omega) )$ (so in $L^2 ([0,T], H^{-1} (\Omega) )$.
At this point we apply the famous Aubin-Lion theorem \[3\] to obtain the strong convergence on $L^2(\Omega)$ of the set of the Galerkin approximants $\{U_n (x,t)\}$ to our candidate ${\overline}U(x,t),$ since this set is a compact set in $ L^2 ([0,T],L^2(\Omega) )$ (see appendix A).
By collecting all the above results we are lead to the strong convergence of the $L^2(\Omega)$-sequence of functions $F(U_n (x,t) )$ to the $L^2(\Omega)$ function $F ({\overline}U (x,t) ).$
We now assemble the above obtained rigorous mathematical results to obtain ${\overline}U (x,t) $ as a weak solution of eq.(1) for any test function $v (x,t) \in C^\infty_0 ([0,T]), H^2 (\Omega) \cap H^1_0 (\Omega) )$ $$\begin{aligned}
\, & \lim_{n \rightarrow \infty} \int^T_0 dt \Bigl [ \Bigl ( U^{(n)} , - \frac{dv}{dt} \Bigr )_{L^2(\Omega)} + (-\Delta U^{(n)} , v)_{L^2(\Omega)} \notag \\
&\big (F (U^{(n)}) , - \Delta^{(\wedge)} v)_{L^2(\Omega)} \Bigr] \notag \\
&= \lim_{n\to\infty} \int^T_0 dt (f^{(n)} , v)\end{aligned}$$ or in the weak-generalized sense above mentioned $$\begin{aligned}
\, & \int^T_0 dt \Bigl ( {\overline}U (x,t), - \frac{dv (x,t)}{dt} \Bigr )_{L^2(\Omega)} \notag \\
&+ \bigl ( {\overline}U(x,t), (- \Delta v) (x,t) \bigr )_{L^2(\Omega)} \notag \\
&+ \bigl ( F ({\overline}U (x,t),- ( \Delta^{(\wedge)} v (x,t) \bigr )_{L^2(\Omega)} \notag\\
&= \int^T_0 dt (f(x,t), v(x,t) )_{L^2(\Omega)},\end{aligned}$$ since $v(0,x)= v (T,x)\equiv 0$ by our proposed space of time-dependent test functions as $C^\infty_0 ([0,t], \linebreak H^2({{\Omega}}) \cap H^1_0 (\Omega) ),$ suitable to be used on the Rosens path integrals representations for stochastic systems (see equations (22a)-(22b) in what follows).
The uniqueness of our solution ${\overline}U (x,t)$, comes from the following lemma \[4\], if $F(x)$ is an injective function.
[**Lemma 1.**]{} If ${\overline}U_{(1)}$ and ${\overline}U_{(2)}$ in $L^\infty ([0,T] \times L^2(\Omega))$ are two functions satisfying the weak relationship below $$\begin{aligned}
\, & \int^T_0 dt\left \{ \left( {\overline}U_{(1)} - {\overline}U_{(2)}, - \frac{\partial v}{\partial t}
\right )_{L^2(\Omega)} \right. \notag \\
& + ( {\overline}U_{(1)} - {\overline}U_{(2)} , + \Delta v )_{L^2(\Omega)} \notag \\
& \bigl ( F({\overline}U_{(1)} - F ( {\overline}U_{(2)}); + \Delta v\bigr)_{L^2(\Omega)} \Big\}\equiv 0
\end{aligned}$$ then ${\overline}U_{(1)}= {\overline}U_{(2)}$ a.e in $L^\infty ([0,T] \times L^2 ({{\Omega}}))$. The proof of eq.(17) is easily obtained by considering the family of test functions on eq.(16) of the following form $v_{n} (x,t)=g_{(\varepsilon)} (t) e^{+ \alpha_n t} \varphi_n (x)$ with $- \Delta\varphi_n (x)= {{\alpha}}_n{{\varphi}}_n(x)$ and $g(t)= 1$ for $(\varepsilon, T- \varepsilon)$ with $\varepsilon > 0$ arbitrary. We can see that it reduces to the obvious identity $(\alpha_{n}> 0)$. $$\int^{T -\varepsilon}_{\varepsilon} dt \exp(\alpha_n t) \left(F( {\overline}U_{(1)}) - F ( {\overline}U_{(2)} ), \varphi_n\right )_{L^2 ({{\Omega}})} \equiv 0,$$ which means that $F({\overline}U_{(1)}) = F ( {\overline}U_{(2)})$ a.e on $(0,T) \times \Omega$ since $\varepsilon$ is an arbitrary number. We have thus $$\overline U_1 = \overline U_2\quad \text{ almost everywhere}$$
Let us now consider a path-integral solution of eq.(1) (with $g(x) = 0$) for $f(x,t)$ denoting the white-noise stirring \[4\]. $$E (f(x,t) f(x',t')) = \lambda \delta^{(D)} (x-x') \delta (t-t')$$ where $\lambda$ is the noise-strenght.
The first step is to write the generating process stochastic functional through the Rosen-Feynman path integral identities \[4\] $$Z[J(x,t)] = E_{f} \left[ \exp \left\{i \int_0^T dt \int_\Omega d^D x U (x,t, [f]) J (x,t)\right\}\right]\tag{21-a}$$ $$= E_f \left[ \int D^F [U] \delta^{(F)} (\partial_t U - \Delta U - \Delta^{(\wedge)} (F(U) - f)) \right]
\times \exp \left\{ i \int^T_0 dt \int_\Omega d^D x U (x,t) J (x,t) \right\} \tag{21-b}$$ $$= E_f \left[ \int D^F [U] D^F [\lambda ] \exp \left\{ i \int_0^T dt \int_\Omega d^D x \lambda (x,t) \right.\right.
\times \big({{\partial}}_t U - \Delta U - \Delta^{(\wedge)} (F (U) - f )\big) \Big\} \Big]$$ $$\times \exp \left\{ i \int^T_0 dt \int_\Omega d^D x U (x,t) J (x,t) \right\} \tag{21-c}$$ $$= \int D^F [U] \exp \left\{ - \frac{1}{2 \lambda} \int_0^T dt \int_\Omega d^D x \right.
\times ({{\partial}}_t U - \Delta U - \Delta^{(\wedge)} (F (U))^{2} (x,t) ) \Big\}$$ $$\times \exp \left\{ i \int^T_0 dt \int_\Omega d^D x U (x,t) J (x,t) \right\}\tag{21-d}$$
The important step made rigorous mathematically possible on the above written (still formal) Rosen’s path integral representation by our previous rigorous mathematical analysis is the use of the delta functional identity on eq.(21-b) which is true only in the case of the existence and uniqueness of the solution of the diffusion equation in the weak sense at least for multiplier Lagrange fields $\lambda (x,t) \in C_{0}^{\infty} ([0,T], H^2 (\Omega) \cap H_0^1 (\Omega) ).$
As an important mathematical result to be pointed out is that in general case of a non-porous medium \[4\] in $R^3$, where one should model the diffusion non linearity by a complete Laplacean $\Delta F(U (x,t) )$, one should observes that the set of (cut-off) solutions $\{{\overline}U^{(\wedge)} (x,t) \}$ of eq.(1) still remains a bounded set on $L^\infty ([0,T], L^{2} (\Omega)$. Since we have the a priori estimate uniform bound for the $U^{(n)}$-derivatives below in $D=3$ (with $G' (x)=F(x)$ and $F(0)=0$). Namely: $$\begin{aligned}
&\left| \int^T_0 dt \left\Vert \frac{dU^{(n)}}{dt}\right\Vert^2_{L^2(\Omega)}\right|\leq \left| \int^T_0 dt\left(\int_\Omega d^3 x (\Delta F ( U^{(n)}(t)))\cdot \left(\frac{{\overline}{dU^{(n)}(t)}}{dt}\right) \right)\right|\notag\\
&\quad + \left| \int_\Omega d^3 x f(x,t) \frac{d(U^{(n)}(x,t))}{dt}\right|\notag\\
&\leq \left|\int^T_0 dt \text{ Real}\left\{\frac{d}{dt}\int_\Omega d^3 x \Delta G(U^{(n)} (t)\right\}\right| + \frac12 \left\{\,\sup_{0 \leq t \leq T}\, p\Vert f(x,t)\Vert^2_{L^2 (\Omega)} + \frac{1}{p}\Vert \overset{\cdot}{U}^{(n)} \Vert^2_{L^2 (\Omega)}\right\}\notag\\
& \leq \left| \text{Real} \left(\int_\Omega d^3 x (\Delta G(U^{(n)} (T,x) )-\Delta G (U^{(n)} (0,x))\right) \right|\notag \\
& \quad +\frac{1}{2}\, \sup_{0 \leq t \leq T}\, \left\{ p \Vert f(x,t)\Vert^2_{L^2 (\Omega)} + \frac{1}{p}\Vert \overset{\cdot}{U}^{(n)} \Vert _{L^2 (\Omega)} \right\} \notag \\
& \leq \frac{1}{2} p \Vert f \Vert^2_{L^\infty ((0,t), L^2 (\Omega))} + \frac{1}{2p} \Vert \overset{\cdot}{U}^{(n)} (t) \Vert_{L^\infty( (0,t), L^2 (\Omega))} < \infty \tag{22}\end{aligned}$$
Where $ U^{(n)} (T,x)\Big|_{\partial \Omega} =U^{(n)} (0,x)\Big|_{\partial \Omega} = 0$ (see eq.(3). The uniform bound for the derivatives is achieved by choosing $\frac{1}{2p}<1$.
As another point worth to call the attention for we note that the above considered function space is the dual of the Banach space $L^1 ([0,T], L^{2} (\Omega))$. So, one can extract from the above set of cut-off solutions a candidate ${\overline}U^{(\infty)} (x,t) $, in the weak-star topology of $L^\infty ([0,T], L^{2} (\Omega))$ for the above cited case of cut-off removing $\wedge = + \infty$ \[6\]. However, we will not proceed throughly in this straightforward technical question of cut-off removing in our model of non-linear diffusion in this paper for general spaces $R^D$.
Finally, we remark that in the one-dimensional case $\Omega \in R^1$, one can further show by using the same compacity methods the existence and uniqueness of the diffusion equation added with the hydrodynamic advective term $\frac{1}{2}\frac{{{\partial}}}{dx} (U (x,t) )^2$, which turns the diffusion eq.(1) as a kind of non-linear Burger equation on a porous medium.
It appears very important to remark that Galerking methods applied directly to the finite-dimensional stochastic eq.(7) (see eq.(4)) may be saving-time computer simulation candidates for the “turbulent" path-integral eq.(22a)-eq.(22d) evaluations by approximate numerical methods (\[2\]-second reference).
The Linear Diffusion in the space $L^2 (\Omega)$
=================================================
Let us now present some mathematical results for the diffusion problem in Hilbert Spaces formed by square-integrable functions $L^2 (\Omega)$ \[5\], with the domain $\Omega$ denoting a compact set of $R^D$.
The diffusion equation in the infinite-dimensional space $L^2 (\Omega)$ is given by the following functional differential equation (see first reference of \[5\] for the mathematical notation). $$\frac{\partial \psi [f (x); t]}{\partial t} = \frac{1}{2} Tr_{L^2(\Omega)}\left( [QD^2_f \psi [f (x, t)]\right) \qquad
\psi [f(x),t \rightarrow 0^{+}] = \Omega [f(x)],\tag{23}$$
Here $\psi [f(x) , \cdot]$ is a time-dependent functional to be determined through the governing eq.(23) and belonging to the space $L^2(L^2(\Omega), d_{Q}{\mu} (f))$ with $d_{Q}{\mu} (f)$ denoting the Gaussian measure on $L^2(\Omega)$ associated to $Q $ – a fixed positive self-adjoint trace class operator $ \oint_{1} (L^2(\Omega))$[^2] – and $D^2_f$ is the second – Frechet derivative of the functional$\psi [f(x),t]$ which is given by a $f(x)$-dependent linear operator on $L^2 (\Omega)$ with associated quadratic form $(D^{2}_{f} \psi [f(x), t] \cdot g(x), h(x))_{L^2(\Omega)}$.
By considering explicitly the spectral base of the operator $Q$ on $L^2 (\Omega)$ $$Q \varphi_n = \lambda_n \varphi_n,\tag{24}$$ The $L^2(\Omega)$-infinite – dimensional diffusion equation takes the usual form: $$\Psi [\sum_{n}f_n \varphi_n , t] = \psi^{(\infty)} [(f_n),t] \tag{25a}$$ $$\Omega [\sum_{n}f_n \varphi_n ] = \Omega^{(\infty)} [(f_n)] \tag{25b}$$ $$\frac{\partial \psi^{(\infty)}[(f_n),t]}{\partial t}= \sum_{n}[(\lambda_n \Delta_{f_n}) \psi^{(\infty)}[(f_n),t]]\tag{25c}$$ $$\psi^{(\infty)}[(f_n),0] = \Omega^{(\infty)} [(f_n)] \tag{25d}$$
or in the Physicist’ functional derivative form (see ref. \[5\]). $$\frac{\partial}{\partial t} \psi [f(x),t] = \int_\Omega d^{D} x \int_\Omega d^{D} x' Q (x, x') \frac{\delta^{2}}{\delta f(x') \delta f(x)} \psi [f(x),t] \tag{26a}$$ $$\psi [f(x),0]= \Omega[f(x)] \tag{26b}$$
Here the integral operator Kernel of the trace class operator is explicitly given by $$Q (x,x') = \sum_{n} (\lambda_{n} \varphi_{n}(x) \varphi_{n}(x')) \tag{26c}$$
A solution of eq.(26a) is easily written in terms of Gaussian path-integrals \[5\] which reads on the physicist’s notations $$\begin{aligned}
& \psi [f(x),t] = \int_{L^{2}(\Omega)} D^F [g(x)] \Omega [f(x) + g(x)] \times \displaystyle\det^{+\frac{1}{2}} \left[\frac{1}{t} Q^{-1} \right] \notag\\
& \times \exp \left\{-\frac{1}{2t} \int_{\Omega}d^{D}x \int_\Omega d^D x' g(x) \cdot Q^{-1} (x,x') g(x')\right\}\tag{27}\end{aligned}$$
Rigorously, the correct functional measure on eq.(27) is the normalized Gaussian measure with the following Generating functional $$\begin{aligned}
Z [j(x)] &= \int_{L^2 (\Omega)} d_{tQ} \mu [g(x)] \exp \left\{ i \int_\Omega j(x) g(x) d^D x\right\} \notag \\
&= \exp \left\{ -\frac{t}{2} \int_{\Omega}d^{D}x \int_\Omega d^D x' j(x) Q^{+1} (x,x') j(x')\right\}\tag{28}\end{aligned}$$
At this point, it becomes important remark that when writting the solution as a Gaussian-path integral average as done in eq.(27), all the $L^2 (\Omega)$ functions in the functional domain of our diffusion functional field $\psi[f(x),t]$ belongs to the functional domain of the quadratic form associated to the classe trace operator $Q$ the so-called reproducing kernel of the operator $Q$ which is not the whole Hilbert Space $L^2 (\Omega)$ as naively indicated on eq.(27), but the following subset of it: $$\text{Dom} (\psi [\cdot,t]) = \{f(x) \in L^2 (\Omega) \vert Q^{- \frac{1}{2}}f \in L^{2} (\Omega)\} \overset{\subset}{\neq} L^2 (\Omega) \tag{29}$$
The above written result gives a new generalization of the famous Cameron-Martin theorem that the usual Wienner measure (defined by the one-dimensional Laplacean with Dirichlet conditions on the interval end-points) is translation invariant, i.e $d^{\text{Wien}} \mu[f+g]= d^{\text{Wien}} \mu[f] \times \left(\frac{d^{\text{Wien}} \mu [f+g]}{d^{\text{Wien}} \mu [f]}\right)$, if and only if the shift function $g(x)$ is absolutely continuous with derivative on $L^2 ([a,b])$. In other words $g \in H_0^1([a,b])=\text{Dom} \left\{ \sqrt{-\frac{d^2}{d^2 x}}\right\}$.
Another point important to call the reader attention is that one can writte eq.(27) in the usual form of Diffusion in finite dimensional case (see appendix B) $$\begin{aligned}
&\psi [f(x), t] = \int_{L^2(\Omega)} D^F [g(x)] \Omega [f(x) + \sqrt tg(x)] \times \det[Q^{-1}] \notag \\
&\exp \left\{ - \frac{1}{2t}\int_\Omega d^{D} x \int_\Omega d^D x'g(x) Q^{-1} (x,x') g(x) \right\}, \tag{30}\end{aligned}$$
At this point is worth call the reader attention that $d_{t Q} \mu$ and $d_{Q} \mu$ Gaussian measures are singular to each other by a direct application of Kakutani theorem for Gaussian infinite dimensional measures for any time $t>0$. $$d_{t Q} \mu [g(x)] \big/ d_{Q} \mu [g(x)] = +\infty \tag{31}$$
Let us apply the above results for the Physical diffusion of Polymer Rings (closed strings) described by Periodic Loops $\vec X (\sigma) \in R^D, 0 \leq \sigma \leq T, \vec X (\sigma + T) = \vec X(\sigma)$ with a non-local diffusion coeficient $Q (\sigma, \sigma')$ (such that $\int_0^T d\sigma \int_0^T d\sigma'Q (\sigma, \sigma') = Tr [Q] < \infty$). The funcional governing equation in Loop Space (formed by Polymer rings) is given by $$\frac{\partial \psi^{(\varepsilon)}[\vec X (\sigma); A]}{\partial A} = \int_0^T d \sigma \int_0^T d\sigma' Q_{ij}^{(\varepsilon)} (\sigma, \sigma')\frac{\delta^2}{\delta \vec X_i(\sigma)\delta \vec X_j(\sigma)} \psi^{(\varepsilon)} [\vec X(\sigma), A]\tag{32a}$$ $$\psi^{(\varepsilon)} [\vec X (\sigma); 0] = \exp \left\{- \frac \lambda {2} \int_0^T d \sigma \int_0^T d\sigma' \vec X_i (\sigma) M_{ij} (\sigma, \sigma')\vec X_j (\sigma')\right\}.\tag{32b}$$
Here the ring polymer surface probability distribuition $\psi^{(\varepsilon)} [\vec X(\sigma),A]$ depends on the area parameter $A$, the area of the cylindrical polymer surface of our surface-polymer chain. Note the presence of a parameter $\varepsilon$ on the above written objects takes into account the local (the integral operator kernel) case $Q(\sigma, \sigma')=\delta (\sigma - \sigma')$ as a limiting case of the rigorously mathematical well-defined (class trace) situation on the end of the observable evaluations $$Q_{ij}^{(\varepsilon)}(\sigma,\sigma')=\frac{1}{\sqrt {\pi \varepsilon }}\left[\exp \left(-\frac{(\sigma -\sigma')^2}{\varepsilon^2}\right)\right] \times \delta_{ij} \tag{33}$$
The solution of eq.(32a) is straightforwardly written in the case of a self-adjoint kernel $M$ on $L^2 (\Omega \times \Omega)$ (with $[M,Q]=0$). $$\begin{aligned}
&\exp \left\{ -\frac{1}{2}\int_0^T d\sigma \int_0^T d\sigma'\vec X_i (\sigma) M_{ij} (\sigma, \sigma')\vec X_j (\sigma')\right\}\notag \\
&\times \text{det}^{-\frac{1}{2}} [1+ A \lambda M (Q^{(\varepsilon)})^{-1}] \notag \\
&\exp \left\{ +\frac{1}{2}\int_0^T d\sigma \int_0^T d\sigma' \vec{(MX)}_i (\sigma) \left( (\lambda M+ (Q^{(\varepsilon)})^{-1} \cdot \frac{1}{A} \right) \vec{(MX)}_j(\sigma') \right\} \tag{34} \end{aligned}$$
The functional determinant can be reduced to the evaluation of an integral equation $$\begin{aligned}
&\text{det}^{\frac{1}{2}} [1+ A \lambda M (Q^{(\varepsilon)})^{-1}] \notag \\
&=\exp \left\{ -\frac{1}{2} Tr_{L^2(\Omega)}\text{lg} (1+\lambda AM(Q^{(\varepsilon)})^{-1}\right\} \notag \\
&=\exp \left\{ -\frac{A}{2}Tr_{L^2(\Omega)} \int_0^\lambda d\lambda'[(Q^{(\varepsilon)})^{-1} M)(1+ \lambda' A(Q^{(\varepsilon)})^{-1} M)^{ -1}] \right\} \notag \\
& =\exp \left\{ -\frac{A}{2}Tr_{L^2(\Omega)} \int_0^\lambda d \lambda' R(\lambda')\right\} \tag{35}\end{aligned}$$
Here the kernel operator $R(\lambda')$ satisfies the integral equation (accesible for numerical analysis) $$R(\lambda')(1+ \lambda'A(Q^{(\varepsilon)})^{-1} M=(Q^{(\varepsilon)})^{-1} M\tag{36}$$
Which in the local case of $\varepsilon \rightarrow 0^+ ,$ when considered in the final result eq.(34) - eq.(35), produces the explicitly candidate solutions for our Polymer-surface probalility distribuition with $M$ a class trace operator on the Loop space: $L^2([0,T])$. $$\begin{aligned}
&\psi [\vec X (\sigma), A] = \exp \left\{ -\frac{1}{2}Tr_{L^2(\Omega)} \int_0^\lambda d\lambda'[M(Q^{(\varepsilon)}+ \lambda' AM)^{-1}]\right\}\notag\\
&\times \exp \left\{ -\frac{\lambda}{2}\int_0^T d\sigma \int_0^T d\sigma'X_i (\sigma) \cdot M_{ij} (\sigma, \sigma') \vec X_j (\sigma')\right\} \notag \\
&\times \exp \left\{ +\frac{1}{2}\int_0^T d\sigma \int_0^T d \sigma' \vec{(MX)}_i (\sigma) \left( \lambda M + (Q^{(\varepsilon)})^{-1} \cdot \frac{1}{A}\right) (\sigma, \sigma') \vec{(MX)}_j (\sigma')\right\}\tag{37}\end{aligned}$$
It is worth call the reader attention that if $A \in \oint_1$ and $B$ is a bounded operator - so $A\cdot B$ is a class trace operator-, the functional determinant $\text{det}[1+AB]$ is a well-defined object as a direct result of the obvious estimate, result which was used to arrive at eq.(37). $$\displaystyle\lim_{N\rightarrow \infty}\displaystyle\prod_{n=0}^{N} (1+ \lambda_n )\leq \exp \left( \displaystyle\sum_{n=0}^{N}\lambda_n \right) = \exp(Tr AB)$$
As a last comment on the linear infinite-dimensional diffusion problem eq.(23), let us sketchy a (rigorous) proof that eq.(27) is the unique solution of eq.(23). Firstly, let us consider the initial condition on eq.(23) as belonging to the space of all mappings $G\vdots L^2 (\Omega) \rightarrow R$ that are twice Fréchet differentiable on $L^2 (\Omega)$ with uniformly continuous and bounded second derivative $D_f^2 G$ (a bounded operator of $\mathcal L( L^2 (\Omega))$ with norm ${\overline}C$). This set of mappings will be denoted by $uC^2 [L^2 (\Omega), R].$ It is, thus, straightforward to see through an application of the mean value theorem that the following estimate holds true $$\begin{aligned}
&\displaystyle\sup_{f(x)\in Q^\frac{1}{2}L^2 (\Omega)} \vert \psi [f(x),t] -G[f(x)]\vert \notag\\
&\leq \int_{L^2 (\Omega)} \vert G(f(x) + g(x)) - G (g(x)) \vert d_{t Q} \mu [g(x)] \notag\\
&\leq \int_{L^2(\Omega)}\left[ \vert DG (f(x), g(x))_{L^2(\Omega)} + \int_0^1 d \sigma (1-\sigma) (D^2 G [f(x) + \sigma g(x)] g(x), g(x)_{L^2(\Omega)}\vert \right] \notag\\
&\times \quad d_{t Q} \mu[g(x)] \notag\\
&\leq 0+ {\overline}C \int_0^1 d \sigma (1-\sigma) \int_{L^2(\Omega)} \Vert g(x)\Vert^2_{L^2(\Omega)} d_{t Q} \mu [g(x)] \notag\\
&\leq {\overline}C \left( \int_{L^2(\Omega)} \Vert g(x)\Vert^2_{L^2(\Omega)} d_{t Q} \mu [g(x)] \right) \notag\\
&\leq {\overline}CTr (tQ) = ({\overline}CTr (Q)t \rightarrow 0 \quad as \quad t \rightarrow 0^+ . \tag{38}\end{aligned}$$
We have thus defined a strongly continuous semi-group on the Banach Space $UC^2[L^2(\Omega),R]$ with infinitesimal generator given by the infinite-dimensional Laplacean $Tr [QD^2]$ acting on the space $L^2 (Q^{\frac{1}{2}}(L^2 (\Omega)),R)$. By the general theory of semi-groups on Banach spaces we obtain that eq.(27) satisfies the infinite-dimensional diffusion initial value problem eq.(23), at least for initial conditions on the space $uC^2 [L^2 (\Omega),R].$ Since purely Gaussian functionals belong to $uC^2 [L^2 (\Omega),R]$ and they form a dense set on the space $L^2 (L^2(\Omega), d_Q \mu)$, we get the proof of our result for general initial condition on $L^2 (L^2(\Omega), d_Q \mu)$.
Finally, we point out that the general solution of the diffusion problem on Hilbert Space with sources and sinks, namely $$\frac{\partial}{dt} \psi [f(x),t] = \frac{1}{2} Tr_{L^2(\Omega)} [QD^2_f \psi [f(x),t]] -V[f(x)]\psi[f(x),t]\tag{39}$$ with $$\psi [f(x),t \rightarrow 0^+]=\Omega [f(x)],\tag{40}$$ posseses a generalized Feynman-Wiener-Kac Hilbert $L^2(\Omega)$ space valued path integral representation, which in the Feynman Physicist formal notation reads as $$\begin{aligned}
&\psi [h(x),T]= \int_{C([0,T],L^2(\Omega))} D^F [X(\sigma)] \notag\\
&\times \exp \left\{ - \frac{1}{2}\int_0^T d \sigma \left(\frac{dX}{d\sigma}, Q^{-1} \frac{dX}{d\sigma}\right)_{L^2(\Omega)} (\sigma)\right\} \notag\\
&\times \Omega \left[ \left(\int_0^T X (\sigma) d\sigma\right) + X(0)\right] \notag\\
&\times \exp \left\{ - \int_0^T d\sigma V \left[ \left(\int_0^T X (\sigma') d\sigma'\right) + X(0)\right ]\right\}\tag{41}\end{aligned}$$
Where the paths satisfy the end-point constraint $X(T)= h(x) \in L^2(\Omega); X(0)=f(x)\in L^2(\Omega)$.
[[**Appendix A:**]{} [**The Aubin-Lion Theorem**]{}]{}
.2in
Just for completenesse in this mathematical appendix for our mathematical oriented readers, we intend to give a detailed proof of the basic result on compacity of sets in function spaces of the form $L^2(\Omega)$ and througout used on section 2. We have, thus, the Aubin-Lion Theorem\[3\] in the Gelfand triplet $H_0^1 (\Omega) \hookrightarrow L^2(\Omega) \hookrightarrow H^{-1}(\Omega) = (H_0^1 (\Omega))^*$
“[**Aubin-Lion**]{} - If $ \{U_n (x,t)\}$ is a sequence of time-differentiable functions in a bounded set of $L^2 ([0,T],H^1_0 (\Omega))$ such that its time derivatives forms a bounded set of $L^2 ([0,T],H^{-1}_0 (\Omega))$, we have that $ \{U_n (x,t)\}$ is a compact set on $L^2([0,T],L^2(\Omega))$".
[**Proof:**]{} the basic fact we are going to use to give a mathematical proof of this theorem is the following identity (Ehrling’s lemma): For any given $\varepsilon>0$, there is a constant $C(\varepsilon)$ such that $$\Vert U_n\Vert_{L^2(\Omega)} \leq \varepsilon\Vert U_n \Vert_{H_0^1(\Omega)}+C(\varepsilon)\Vert U_n \Vert^2_{H^{-1}(\Omega)}\tag{A-1}$$
As a consequence, we have the following estimate $$\begin{aligned}
&\int_0^T\Vert U_n -U_m \Vert ^2_{L^2 ([0,T]), L^2 (\Omega)} \notag\\
&\leq \int_0^T dt (\varepsilon \Vert U_n - U_m \Vert_{H_0^1(\Omega)} + C(\varepsilon) \Vert U_n - U_m \Vert_{H^{-1}(\Omega)})^2 \notag\\
&\leq \varepsilon^2 \left(\int_0^T dt \Vert U_n - U_m \Vert^2_{ H_0^1(\Omega)} \right) + (C (\varepsilon)^2 \left( \int_0^T dt \Vert U_n - U_m \Vert^2_{H^{-1}(\Omega)}\right) \notag\\
&\quad +2 \varepsilon C(\varepsilon) \left(\int_0^T dt (\Vert U_n - U_m \Vert_{H_0^1(\Omega)} \times \Vert U_n - U_m \Vert_{H_0^{-1}(\Omega)})\right) \notag\\
&\leq \varepsilon^2 \left(\int_0^T dt \Vert U_n - U_m \Vert^2_{H_0^1(\Omega)}\right) + C(\varepsilon)^2 \left(\int_0^T dt \Vert U_n - U_m \Vert^2_{H^{-1}(\Omega)}\right) \notag\\
&\quad +2 \varepsilon C(\varepsilon) \left( \int_0^T dt \Vert U_n - U_m \Vert^2_{H_0^1(\Omega)}\right)^\frac{1}{2} + \left(\int_0^T dt \Vert U_n - U_m \Vert^2_{H^{-1}(\Omega)}\right)^\frac{1}{2}\notag\\
&\leq 2 \varepsilon^2 M+2 \varepsilon C (\varepsilon) M^\frac{1}{2} \left(\int_0^T dt \Vert U_n - U_m \Vert^2_{H^{-1}(\Omega)} \right)^\frac{1}{2} \notag\\
&+ (C(\varepsilon)^2) \left(\int_0^T dt \Vert U_n - U_m \Vert^2_{ H^{-1}(\Omega)} \right)\tag{A2}\end{aligned}$$
At this point, we use the Arzela-Ascoli theorem to see that $\{U_n (x,t)\}$ is a compact set on the space $C ([0,T],H^{-1} (\Omega))$ since we have the set equicontinuity: $$\begin{aligned}
\Vert U_n (t) - U_m (s) \Vert_{H^{-1}(\Omega)} &\leq \int_s^t \Vert U'_n (\tau) \Vert_{H_0^{-1}(\Omega)} d \tau \notag \\
&\leq \vert t - s \vert^{(1 - \frac{1}{2})} \times \left( \int_0^T \Vert U'_n (\tau) \Vert^2_{H^{-1}(\Omega)} d \tau \right)^{\frac{1}{2}}\notag\\
&\leq {\overline}M \vert t - s \vert^{\frac{1}{2}}\tag{A3}\end{aligned}$$
It is a crucial step now by remarking that $H_0^1(\Omega)$ is compactly immerse in $L^2 (\Omega)$ (Rellich Theorem). Let us not that for each $t$ (almost everywhere in $[0,T]$), $U_n (x,t)$ is a bounded set on $H_0^1 (\Omega)$ since $U_n (x,t)$ belongs to a bounded set $L^2 ([0,T]$, $H_0^1 (\Omega))$ by hypothesis. As a consequence, $\{U_{n_k},(x,t)\}$ is a compact set on $L^2 (\Omega)$ (Rellich Theorem) and so in $H^{-1} (\Omega)$ almost everywhere in $[0,T]$ since $L^2 (\Omega) \hookrightarrow H^{-1} (\Omega)$. By an application of the Arzela-Ascoli theorem, there is a sub-sequence$\{U_{nk} (x,t)\}$ of $\{U_n (x,t)\}$ (and still denoted by $\{U_n (x,t)\}$ such that it converges uniformly to a given function ${\overline}U (x,t) \in C ([0,T], H^{-1} (\Omega))$. As a direct result of this fact we, have that (for $T< \infty !)$ for $(n,m) \rightarrow \infty$. $$\left( \int_0^T \Vert U_n - U_m \Vert^2_{H^{-1}(\Omega)}\right)^\frac{1}{2} \leq (\sup \vert U_n - U_m \vert_{C ([0,T], H^{-1} (\Omega)}) \times \left( \int_0^T 1 \cdot dt \right)^\frac{1}{2} \rightarrow 0 \tag{A4}$$
Returning to our estimate eq.(A2), we see that this sub-sequence is a Cauchy sequence in $L^2 ([0,T], L^2(\Omega))$. As a consequence, for each fixed $t \in [0,T]$ (almost everywhere), $U_n (x,t)$ converges to ${\overline}U (x,t)$ in $L^2(\Omega)$.
[[**Appendix B:**]{} [**The Linear Diffusion Equation**]{}]{}
.2in
Let us show mathematically the basic functional integral representation eq.(30) for the $L^2 (\Omega)$- Space Diffusion Equation eq.(23) .
As a first step for such proof, let us call the reader attention that one should consider the second order (Laplacean) $ D^2 U (x,t)$ as a bounded operator in $L^2 (\Omega)$ in order to the operatorial composition with the positive definete class trace operator $Q$ still be a class trace operator as it is explicitly supposed in the right-hand side of eq.(23).
We thus impose as the sub-space of initial condition the Diffusion Equation eq.(23) for the (dense) vector sub-space of $ C(L^2 (\Omega), R)$ composed of all functionals of the form. $$f(x)= \int_{L^2 (\Omega)} d_{Q} \mu (p) F(p) \exp \left(i\langle p,x\rangle_{L^2 (\Omega)} \right) \tag{B1}$$ with $F(p)\in L^2 (L^2 (\Omega),d_Q \mu).$
By substituting the inital condition eq.(B1) into the integral representation eq.(30) and by using the Fubbini-Toneli Theorem to exchange the needed integrations order in the estimate below, we get: $$\begin{aligned}
U(x,t) &= \int_{L^2 (\Omega)} f(x+ \sqrt{t} \xi ) d_{Q} \mu (\xi)\notag \\
&= \int_{L^2} d_{Q} \mu (\xi) \left\{ \int_{L^2} d_{Q} \mu (p) F(p) e^{i \langle p,x+ \sqrt{t} \xi \rangle_{L^2}}\right\}\notag \\
& =\int_{L^2} d_{Q} \mu (p)F(p)\cdot e^{i \langle p,x \rangle_{L^2}} e^{- \frac{1}{2}t \langle p,Q p \rangle_{L^2}} . \tag{B2}\end{aligned}$$
Note that we have already proved that $U(x,t)$ is a bounded functional of $ C\left( L^2 (\Omega) \times [0,\infty]; R \right)$ on the basis of our hypothesis on the initial functional date eq.(B1).
At this point we observe that the second order Frechet derivatives of the Functional $\exp i\langle p,x\rangle_{L^2}$ are easily (explicitly) evaluated as \[(7)\] $$\begin{aligned}
& Q D^2 \left( e^{i \langle p,x \rangle_{L^2}} \right) = \left(\sum_{\ell=1}^{\infty} \lambda_\ell \frac{\partial^2}{\partial^2 x_\ell} \right) \left[e^{i \left(\sum_{n=1}^{\infty} p_n x_n \right)} \right] = - \left(\langle p, Q p\rangle_{L^2}\right) e^{i \langle p,x \rangle_{L^2}} \tag{B3}\end{aligned}$$
We have thus a straightforward proof of our claim above cited on the basis again of the chosen initial date sub-space $$\begin{aligned}
&Tr[Q D^2 U(x,t)]\notag \\
&\leq \int_{L^2(\Omega)} d_Q \mu (p) |F(p)|\langle p, Q p \rangle_{L^2}\notag\\
&\leq \left( \int_{L^2(\Omega)} d_Q \mu (p) |F(p)|^2\right)^{\frac{1}{2}} \left( \int_{L^2(\Omega)} d_Q \mu (p) |\langle p, Q p\rangle_{L^2(\Omega)}|^2\right)^{\frac{1}{2}} \notag \\
&\leq (Tr Q) \Vert F\Vert_{L^2 (L^2 (\Omega), d_Q \mu)} < \infty . \tag{B4}\end{aligned}$$
Now, it is a simply application to verify that eq.(B2) satisfies the Diffusion Equation in $L^2 (\Omega)$ (or in any other Separable Hilbert Space). Namely: $$\begin{aligned}
&\frac{\partial U(x,t)}{\partial t}=\int_{L^2(\Omega)} d_Q \mu (p) F(p)e^{i \langle p, x \rangle_{L^2}} \left\{-\frac{1}{2}\langle p,Qp \rangle_{L^2 (\Omega)} \right\}\notag \\
& \times e^{-\frac{t}{2} \langle p, Q p \rangle_{L^2(\Omega)}}\tag{B5} \end{aligned}$$ $$\begin{aligned}
T r_{L^2(\Omega)} [QD^2 U(x,t)] &= \int_{L^2(\Omega)} d_Q \mu (p) F(p)\left\{D^2 e^{i \langle p, x \rangle}\right\} e^{-\frac t2 \langle p, Qp \rangle_{L^2({{\Omega}})}} \notag \\
&= \int_{L^2(\Omega)}d_Q \mu (p) F(p) \left\{ - \langle p, Qp \rangle_{L^2} \right\} e^{i \langle p, x \rangle} e^{-\frac t2 \langle p, Qp \rangle_{L^2({{\Omega}})}} \tag{B6}\end{aligned}$$ with $$U(x,0) = \int_{L^2(\Omega)}d_Q \mu (p) F(p) e^{i \langle p, x \rangle} \left\{ \lim_{t \to 0^+} e^{-\frac{t}{2} \langle p, Qp \rangle_{L^2 (\Omega)}}\right\} = f(x) . \tag{B7}$$
.5in
[**References**]{}
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: Ian Sneddon - Fourier Transforms, McGraw-Hill Book Company, INC, USA, 1951; Patrick A. Domenico and Franklin W. Schwartz, “Physical and Chemical Hydrogeology", second edition, John Wiley and Sons, Inc., 1998.P.L. Schdev, “Nonlinear Diffusive Waves", Cambridge University Press, (1987).Werner E. Kohler; B.S. White (Editors), “Mathematics of Random Media", Lectures in Applied Mathematics, vol. 27, American Math. Soc., Providence, Rhode Island, (1980).R.Z. Sagdeev (Editor), “Non-Linear and Turbulent Processes in Physics", vol.1, vol.2, vol.3, Harwood Academic Publishers. (1987)Geoffrey Grimmett, “Percolation", Springer-Verlag, (1989).
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: G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional System", London Math. Soc. Lect. Notes, vol.229, Cambridge University Press, (1996).R. Teman, C. Foias, O. Manley and Y. Treve, Physica 6D, 157 (1983).
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: Robert Dortray and Jacques - Louis Lion “Analyse Mathématique et Calcul Numérique", Masson, Paris, vol.1-vol.9., Avner Friedman, “Variational Principles and Free-Boundary Problems - Pure and Applied, Math." John-Wiley and Sons (1988).
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: Luiz C.L. Botelho, Journal of Physics A: Math. Gen.[**34**]{}, L31-L37 (2001). Luiz C. L. Botelho, II Nuovo Cimento [**117**]{} B1, 15 (2002); J. Math. Phys. [**42**]{}, 1682 (2001)Luiz C. L. Botelho, Int. J. Mod. Physics B[**13**]{} (13), 1663 (1999); Int. J. Mod. Physics. [**12**]{}; Mod. Phys. B[**14**]{} (3), 331 (2002)A.S. Monin and A. M. Yaglon, “Statistical Fluid Mechanics" - Mit Press, Cambridge, vol.2, 1971G. Rosen, Journ. Math. Phys. [**12**]{}, 812 (1971).Luiz C. L. Botelho, Mod. Phys. Lett [**13**]{}B, 317 (1999).V. Gurarie and A. Migdal, Phys. Rev. E[**54**]{}, 4908 (1996).U.Frisch, “Turbulence", Cambridge Press, cambridge, (1996)W.D. Mc-Comb, “The Physics of Fluid Turbulence", Oxford University, Oxford, (1990).Luiz C. L. Botelho, Mod. Phys. Lett B[**13**]{}, 363 (1999).S.I. Denisov, W. Horsthemke, Phys. Lett. A[**282**]{} (6), 367 (2001). Luiz C. L. Botelho, Int. J. Mod. Phys. B [**13**]{}, 1663 (1999).Luiz C. L. Botelho, Mod. Phys. Lett B[**12**]{}, 301 (1998).Luiz C. L. Botelho, Mod. Phys. Lett B[**12**]{}, 569 (1998).Luiz C. L. Botelho, Mod. Phys. Lett B[**12**]{}, 1191 (1998).Luiz C. L. Botelho, II Nuovo Cimento [**117**]{}B(1), 15 (2002); J. Math. Phys. [**42**]{}, 1682 (201).
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[^1]: See example 9.2-2 and 9.3 in “Transport Phenomena´´ by R. Byran Bird, Warren E. Stewart, Edwin N. Lightfoot; John Wiley & Sons, 1960, pages 272–276, 304–309.
[^2]: For instance $$Q^{+1}(x,y)=I_\Omega(x)((-\Delta)^\alpha+m^2)^{-1}(x,y)I_\Omega(y) \text{ in } R^n$$ with $I_\Omega(z)=\begin{cases}
1 \text{ if } z\in \Omega \\
0 \text{ if } z\in \Omega' \end{cases}$ and $\alpha>\frac n2$. Note that $Tr_{L^2(\Omega)}(Q) = (\text{vol}(\Omega)) \Big[\int \frac{d^n p}{p^{2\alpha}+m^2}\Big]$
|
[**Short closed geodesics with self-intersections**]{}
[Viveka Erlandsson and Hugo Parlier ]{}
[**Abstract.**]{} Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in $k$ (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like $k$ for growing $k$.
Introduction {#s:introduction}
============
Closed geodesics play an important part in describing the geometry and dynamics of hyperbolic surfaces and their moduli. In particular, the length spectrum of a hyperbolic surface is closely related to analytic problems on surfaces as it determines the spectrum of the Laplacian. Among the closed curves, the simple ones play a particular role and are related to geometric and topological problems on moduli spaces including the study of homeomorphism groups and metrics on Teichmüller space.
Among all closed geodesics, the shortest one is somewhat special and is called the systole of the surface. Unless a hyperbolic surface $X$ (with non-trivial fundamental group of finite type) is homeomorphic to a thrice punctured sphere, its systole is a simple closed geodesic. With this in mind, we are interested in the following problem introduced and studied by Basmajian and Buser. Given a fixed integer $k$, we consider the set of closed geodesics of $X$ that self-intersect at least $k$ times. Since the length spectrum is discrete, among them there is one of minimal length, say $\gamma$. By definition, $\gamma$ self-intersects at least $k$ times. The question is to find an upper bound on the number of self-intersection points of $\gamma$.
As mentioned before, for $k=0$, this is asking for the number of self-intersections of the systole of $X$ and so unless $X$ contains no simple closed geodesics, the answer is $0$. For $k=1$, Buser [@BuserBook Theorem 4.2.4] solved the problem by showing that among all non-simple closed geodesics of $X$, the shortest one has a single intersection point (it is a so-called [*figure eight geodesic*]{}). The proof is an involved cut and paste type argument based on the observation that a non-simple closed geodesic contains a simple loop as a subset. Perhaps surprisingly, as far as exact values go, there are no further results known.
A general result, due to Basmajian [@BasmajianStable], provides a first answer to the question in the case where $X$ is complete, finite area and finite type. He shows that there exists a constant (that can be made explicit) which depends on $k$ and the topology of $X$ (but not its geometry) such that the number of self-intersections of $\gamma$ is upper bounded by this constant. If one works out the explicit bound, the dependence on $k$ is exponential. The bound on the topology is used to bound the lengths of curves in a pair of pants decomposition via a theorem of Bers [@Bers], quantified by Buser and others [@BalacheffParlier; @BuserBook; @ParlierShort]. For general surfaces (those not necessarily of finite area), the methods proposed by Basmajian provide a bound which this time depends on the geometry of the surface, and in particular on a bound on the length of curves in a pants decomposition.
Let $I_k(X)$ denote the maximum number of self-intersections of a shortest geodesic on $X$ with at least $k$ self-intersections. We prove the following:
\[thm:universal\] Let $X$ be an orientable complete hyperbolic surface with non-abelian fundamental group. Then $$I_k(X)\leq31\sqrt{k+\frac{1}{4}}\left(16\sqrt{k+\frac{1}{4}}+1\right)$$
The two main features of our result is that the growth is linear in $k$ (for instance the upper bound is less that $600\, k$ for all $k\geq 2$) and that there is no dependence on the geometry or the topology of the surface. In particular, it holds for [*any*]{} hyperbolic surface where the question makes sense (meaning with non-abelian fundamental group, including infinite area or infinite type surfaces, although this is not our focus point). While the final result does not depend on the geometry of the surface, one of the main ideas of our proof is to use the specific geometry of the surface to find appropriate decompositions of candidate curves.
Although the proof is mostly self-contained, it is certainly inspired by a flurry of recent results [@AGPS; @ChasRelations; @ChasPhillips; @ErlandssonSouto; @SapirBounds; @SapirLower] focused on understanding the relationship between self-intersection and the length of closed geodesics. One of the tools we do use is the upper bounds of Basmajian [@BasmajianUniversal; @BasmajianShort] on the length of the shortest curve with at least $k$ self-intersections. We note that these length bounds can be used directly to find a linear upper bound on $I_k(X)$ but the bound depends on the geometry of $X$ (see Section \[sec:setup\] for more details).
Basmajian also shows that there is a considerable difference in the length growth depending on whether surfaces have a cusp or not: the growth rate for closed surfaces is roughly $\sqrt{k}$ whereas it is $\log(k)$ if the surface has cusps. We are able to exploit that growth difference to prove an asymptopically optimal result for cusped surfaces.
\[thm:cuspcase\] Let $X$ be an orientable complete finite type hyperbolic surface with at least one cusp. Then there exists constants $D(X),K(X)$, depending on $X$, such that $$I_k(X)\leq k+D(X) \log (k)$$ for all $k>K(X)$.
Exactly where the constants $D(X)$ and $K(X)$ come from can be found in Section \[sec:cusp\]. Unlike in the previous theorem, the bounds here depends on the geometry of $X$. Although we do not want to dwell on it here, the condition on $X$ being of finite type can be relaxed to there being a positive lower bound on the systole length of $X$.
Note that Theorem \[thm:cuspcase\] implies that $$\lim_{k\to\infty}\frac{I_k(X)}{k}=1$$ when $X$ has a cusp. We conjecture that the above limit is always equal to $1$, regardless of whether $X$ has a cusp or not, but our methods do not seem to extend easily to more general surfaces.
Our proof of Theorem \[thm:cuspcase\] requires a generalization of Basmajian’s lower bounds on lengths [@BasmajianUniversal]. In particular, we need to be able to control the relationship between length and intersection in the $\varepsilon$-thick part of a surface (which we denote $X_T$). As our result may be of independent interest, we state it here.
\[thm:thick\] For $\varepsilon \leq \frac{1}{2}$, the intersection $\gamma_T= \gamma \cap X_T$ satisfies $$\ell(\gamma_T) >\frac{\varepsilon}{12} {\sqrt{i(\gamma_T,\gamma_T)}}$$
Note that a closed surface is $\varepsilon$-thick for sufficiently small $\varepsilon$, so we recuperate Theorem 1.1 from [@BasmajianUniversal] with a somewhat different proof.
We end the introduction by addressing the very natural question of lower bounds on $I_k(X)$. By definition, $I_k(X)\geq k$ with equality for infinitely many $k$. In fact, it is not a priori obvious that equality does not hold for [*all*]{} $k\geq 1$. However, there is a heuristic argument, inspired by results from [@BasmajianUniversal], for why this should not always be the case. We illustrate it with a pair of pants $P$, say with three cuff lengths of length $1$. The local behavior of a closed geodesic is to either loop around one of the three boundary curves, or to follow some trajectory in the middle portion of the pair of pants, for instance that of a figure eight geodesic. If a closed geodesic closely follows a figure eight geodesic $n$ times, this creates roughly $n^2$ self-intersection points. On the other hand, a curve that loops $n$ times around a cuff creates roughly $n$ self-intersection points. Now assume there is a minimal length curve realizing $I_k(P)$ that has exactly $k$ self-intersections. Suppose you want to modify it to get a candidate for $I_{k+k_0}(P)$ for some $k_0$ relatively small compared to $k$. Each loop around a boundary costs you roughly $1$ in length, but although this is less than taking an extra copy of a figure eight curve, you are only getting one extra intersection point per loop. Thus, in terms of length, it would be more efficient to take (quasi) copies of a figure eight to generate self-intersection points than by looping around a boundary. Making the above argument rigorous would require a more delicate analysis of curves in pairs of pants, very different in nature from the methods used in this paper, but nonetheless, we expect that $$\limsup_{k\to \infty} (I_k(X) -k) = \infty$$ for any compact $X$.
Closed curves and their lengths {#sec:setup}
===============================
Setup and known results
-----------------------
Let $X$ be an orientable complete hyperbolic surface with non-abelian fundamental group. Said differently, we ask that $X$ is not the hyperbolic plane and is not topologically a cylinder. We want $X$ to have an interesting set of closed geodesics.
We will denote by ${{\mathcal G}}(X)$ the set of closed geodesics, by ${{\mathcal G}}_k(X)$ the subset of those that self-intersect exactly $k$ times, and by ${{\mathcal G}}_{\geq k}(X)$ those that intersect at least $k$ times. Basmajian studied the following quantity [@BasmajianUniversal; @BasmajianShort]: $$s_k(X):= \inf \{ \ell(\gamma) : \gamma \in {{\mathcal G}}_k(X)\}$$ showing that $$s_k(X) \leq 2 C_8(X) \sqrt{k+\frac{1}{4}}$$ where $C_8(X)$ is the length of the shortest figure eight closed geodesic on $X$. (As mentioned above, Buser showed that $C_8(X)$ is also the length of the shortest non-simple closed geodesic of $X$.) The general gist of the proof of the above inequality is to construct a closed geodesic which follows the figure eight curve multiple times. The number of self-intersections of such a curve is roughly the square of the number of copies of the figure eight curve. To create a primitive closed curve, and to get the correct intersection number on the nose, require more delicate arguments. We remark that the above bound, from [@BasmajianShort], is an improvement on previous bounds in [@BasmajianUniversal] where lower bounds on $s_k(X)$ are also explored. A fact about $s_k(X)$ that we will use in the sequel is the discrepancy between the growths when $X$ has cusps or not. The growth is logarithmic in $k$ when $X$ has a cusp.
By discreteness of the length spectrum (for finite type surfaces), the value $s_k(X)$ is realized by the length of at least one closed geodesic. In particular, for $k=0$ this is the systole which, unless $X$ is a three holed sphere, is realized by a simple closed curve since the shortest non-trivial curve is always simple. If $X$ is a three holed sphere, the systole is a figure eight geodesic.
A related quantity is the following: $$s_{\geq k}(X):= \inf \{ \ell(\gamma) : \gamma \in {{\mathcal G}}_{\geq k}(X)\}$$ and again it must be realized by the length of certain closed geodesics which may or may not have $k$ self-intersections. The actual number of self-intersections is our main concern in this article, and we will denote this number by $I_k(X)$. As $s_{\geq k}(X) \leq s_k(X)$, the inequality stated above for $s_k(X)$ also holds for $s_{\geq k} (X)$.
When $X$ is compact the upper bounds on $s_{\geq k} (X)$ are matched by lower bounds [@BasmajianUniversal] of the form $C(X)\sqrt{k}$. Here the constant depends on the geometry of $X$ in such a way that $C(X)$ tends to $0$ when $X$ approaches the boundary of moduli space. These bounds, when appropriately put together, give a linear upper bound on $I_k(X)$ of type $U(X) k$ but where $U(X)$ this time goes to infinity as $X$ approaches the boundary of moduli space. In constrast, Basmajian’s upper bounds [@BasmajianStable] on $I_k(X)$, when $X$ is complete and of finite area, only depend on the topology of $X$: $$I_k(X) \leq F(g, n, k)$$ Here $g$ is the genus of $X$, $n$ the number of cusps and $F$ an explicit function. The proof is based on a generalization of the classical collar lemma for simple closed geodesics to closed geodesics. This generalized collar lemma implies that (self-)intersection points must create length, and as there is a bound on the length of the shortest curves with given lower bound on number of self-intersections, there cannot be arbitrarily many self-intersection points.
Intersections and length
------------------------
We begin with the following lemma which relates lengths of simple closed geodesics and lengths of figure eight geodesics.
\[lem:simple8\] Let $\alpha,\beta$ be simple closed geodesics on $X$ with $i(\alpha,\beta) =1$ and $\ell(\alpha), \ell(\beta) \leq L$. Then $$C_8(X) < 4 L$$
We think of $\alpha$ and $\beta$ as oriented loops based in their intersection point. The geodesic in the homotopy class of the closed curve obtained by the following concatenations $$\alpha * \beta * \alpha^{-1} * \beta$$ is a figure eight geodesic whose length is strictly less than $2 \ell(\alpha) + 2\ell( \beta)$ which is at most $4L$.
As a corollary we have the following.
\[cor:balls\] For any $p \in X$ and for all $r_0\leq \frac{C_8(X)}{8}$, the set $B_{r_0} (p)$ is topologically either a disk or a cylinder.
If not, then there is a point $p$ which is the base point of at least two distinct (and thus non-homotopic) simple geodesic loops $\alpha$ and $\beta$ of length at most $2 r_0$. These two loops could generate a pair of pants in which case the geodesic in the homotopy class of $\alpha*\beta$ is a figure eight geodesic of length at most $4 r_0 \leq \frac{C_8}{2}$ which is impossible. Otherwise they generate a one-holed torus in which case we refer to the previous lemma to conclude that $C_8(X) < 8 \frac{C_8}{8}$, again a contradiction.
The above observation will be crucial in the sequel.
Bounding intersection numbers
=============================
We can now turn our attention to the problem at hand, namely the proof of Theorem \[thm:universal\]. For clarity of exposition, we suppose that $X$ is of finite type. What we really use is the discreteness of the length spectrum which may fail if $X$ is of infinite type. In Remark \[rem:infinite\] below, we discuss how to adapt the argument to when $X$ has a non-discrete length spectrum. However, we insist on the fact that this is not our focus point and the remark can be ignored by the reader only interested in finite type surfaces.
Let $\gamma \in {{\mathcal G}}_{\geq k} (X)$ be of minimal length. We seek to find an upper bound on $i(\gamma,\gamma)$. Once and for all, set $r_0$ to be the quantity $$r_0:= \frac{C_8(X)}{8}$$ We cut $\gamma$ into segments $c_1, c_2, \hdots, c_m$, all of length $r_0$ except possibly $c_m$ which may be shorter. Note that by Basmajian’s inequality $$\ell(\gamma) < 2 C_8(X) \sqrt{k+\frac{1}{4}} = 16 r_0 \sqrt{k+\frac{1}{4}}$$ and as such $$m \leq \ceil[\Bigg]{16 \sqrt{k+\frac{1}{4}}} < 16 \sqrt{k+\frac{1}{4}} + 1$$
\[rem:infinite\] When the length spectrum of $X$ is not discrete, we cannot guarantee that $\gamma$ of minimal length exists (see [@BasmajianKim] for results about infinite type surfaces with non-discrete length spectra). However, Basmajian’s inequality above continues to hold as we will briefly explain. The inequality depends only on $C_8(X)$, which may or may not be realized by a figure eight geodesic on $X$. Suppose it is not. Then there is a sequence of figure eight geodesics whose lengths $L_i$ tend to $C_8(X)$. Thus, for each $i\in {{\mathbb N}}$, there is a geodesic $\gamma_i$ with self-intersection at least $k$ satisfying the inequality $$\ell(\gamma_i) < 2 L_i \sqrt{k+\frac{1}{4}}$$ From this we can deduce the existence of a $\gamma$ with self-intersection $k$ such that $$\ell(\gamma) \leq 2 C_8(X) \sqrt{k+\frac{1}{4}} = 16 r_0 \sqrt{k+\frac{1}{4}}$$ The arguments presented in what follows can all be adapted to the the non-discrete case by suitably replacing a minimal length $\gamma$ by a curve $\gamma$ of length arbitrarily close to the infimum of lengths. However, for clarity, we will not continually refer to how to adapt the arguments in this more general setting in the sequel.
Note that due to our choice of $r_0$ and Corollary \[cor:balls\], any pair of intersecting segments $c_i, c_j$ (not necessarily distinct) all live in either disks or cylinders. If they live in a disk, then they are simple and can pairwise intersect at most once. We observe therefore that if all pairs of segments lived in disks, there would be an immediate upper bound on self-intersection given by $$\frac{m^2}{2} - m$$ Replacing $m$ with the upper bound in terms of $k$ proves the main theorem in this case, but of course we cannot a priori suppose this to be the case.
We use the word *strand* for a segment in a cylinder which has both its endpoints on the boundary of the cylinder. In general this is not always the case for our segments $c_i$, however, we will often extend segments to strands. By abuse of notation, we denote the strand also by $c_i$.
If a segment $c_i$ lives in a cylinder ${{\mathcal C}}$, it can be one of two types. Consider $\delta_{+}$ and $\delta_{-}$ the two boundary curves of ${{\mathcal C}}$. If the strand $c_i$ intersects both $\delta_{+}$ and $\delta_{-}$ in its endpoints, it is a simple geodesic segment as there is no topology to create self-intersection. We refer to this type as a *crossing* strand (an example is the leftmost strand in Figure \[fig:strands\]).
The other type, which we will call a *returning* strand, has both its endpoints on the same boundary curve, say $\delta_{-}$. In this case, it may have self intersection points which appear as a result of it wrapping around the core curve of the cylinder. In Figure \[fig:strands\], the middle and right strands have $1$ and $2$ self-intersection points.
If the cylinder ${{\mathcal C}}$ has core curve $\delta$ we define the *winding number* $\omega(c_i)$ of a strand $c_i$ in ${{\mathcal C}}$ (with respect to ${{\mathcal C}}$) in the following way. Every point of $c_i$ projects to a well-defined point of $\delta$. The winding number of $c_i$ is given by the length of the projection of $c_i$ (thought of as a parameterized segment) divided by the length of $\delta$.
Understanding the behavior of segments lying in embedded cylinders will be crucial. Here we record a fact about the intersection numbers of segments lying in the same cylinder.
\[lem:int\] Let $s_1, s_2$ be two distinct crossing strands, $r_1, r_2$ two distinct returning strands, all lying in the same cylinder, with $\omega(s_1)\leq\omega(s_2)$ and $\omega(r_1)\leq\omega(r_2)$. Then:
1. $i(s_1,r_1)\leq \lceil \omega(r_1) \rceil$
2. $i(r_1,r_1)\leq\lceil \omega(r_1) \rceil$
3. $i(r_1,r_2)\leq2\lceil\omega(r_1)\rceil$
4. $i(s_1,s_2)\leq\lceil\omega(s_1)\rceil$
Suppose the cylinder ${{\mathcal C}}$ has boundary curves $\delta_-$ and $\delta_+$ and core curve $\delta$. For each strand $c_i$ in ${{\mathcal C}}$ we will construct a representative $c_i'$ homotopic to $c_i$ (relative its endpoints on $\delta_-$ and $\delta_+$) and use it to get an upper bound on the intersection numbers. Suppose $c_i$ has endpoints $p$ and $q$ on $\delta_-$ or $\delta_+$. Note that if $r_1$ has both its endpoints on $\delta_-$ and $r_2$ has both its endpoints on $\delta_+$ then $i(r_1,r_2)=0$. Hence we can assume with out loss of generality that $c_i$ has at least one endpoint on $\delta_-$. We construct $c_i'$ the following way. Choose a simple loop $\delta_{c_i}$ in the interior of ${{\mathcal C}}$ such that every point is on it is equidistant to $\delta$. Let $c_i'$ be the curve consisting of the perpendicular segment between $p$ and $\delta_{c_i}$, a segment winding around $\delta_{c_i}$ according to $\omega(c_i)$, and finally the perpendicular segment between $\delta_{c_i}$ and $q$. Moreover, if $c_i$ is a returning strand, chose $\delta_{c_i}$ to be closer to $\delta_-$ than $\delta_+$, and if its a crossing strand chose it closer to $\delta_+$. Finally, if $c_i$ and $c_j$ are of the same type and $\omega(c_i)<\omega(c_j)$ choose $\delta_{c_i}$ to be closer to the boundary of ${{\mathcal C}}$ than $\delta_{c_j}$ is (and when they have the same winding number, make an arbitrary choice). Clearly $c_i'$ is homotopic to $c_i$.
For $i=1,2$, let $s_i'$ and $r_i'$ be the representatives of $s_i$ and $r_i$ obtained as above. It is clear that $\vert s_1'\cap r_1'\vert\leq\lceil\omega(r_1)\rceil$ and since $i(s_1,r_1)\leq\vert s_1'\cap r_1'\vert$ we have proved the first part of the lemma. The remaining parts follow similarly.
Unwinding curves
----------------
We begin by finding a bound on $i(\gamma,\gamma)$ in the case where a segment $c_i$ self-intersects more than $2$ times. Note that if this happens it necessarily lives inside a cylinder and is a returning strand.
\[lem:unwind1\] If there exists $c_i$ with $i(c_i,c_i) \geq 2$, then $$i(\gamma,\gamma) \leq k -1 + 16 \sqrt{k+\frac{1}{4}}$$
The segment $c_i$ contains a point of self-intersection $p$ and a geodesic simple loop based in $p$ as a subset. This loop generates a cylinder ${{\mathcal C}}$ of core geodesic $\delta$ (or possibly a cusp - in this case we set $\delta$ to be a small horocyclic neighborhood of the cusp disjoint and very far away from $c_i$). We observe that the parallel line $h_p$ to $\delta$ passing through $p$ is embedded in $X$ and moreover, the line parallel to $h_p$ consisting of points distance $r_0$ from $h$ is also embedded and is the boundary of an embedded cylinder. This is because otherwise there would be a point $p'$ with two geodesic loops of length at most $2 r_0$. As before, this would imply the existence of a figure eight geodesics of length strictly less than $C_8(X)$ which is not possible.
We extend this cylinder maximally by boundary lines parallel to $\delta$ (both ’up’ and ’down’) and so that it remains embedded. The resulting cylinder we denote ${{\mathcal C}}$ and we extend (if necessary) the segment $c_i$ so that both its endpoints lie on the other boundary curve of ${{\mathcal C}}$ which we’ll denote $\delta_{-}$. Note that $c_i$ is entirely contained in the half cylinder with boundary curves $\delta$ and $\delta_{-}$.
Ł(.525\*.33) $h_p$\
Ł(.465\*.1) $c_i$\
Ł(.425\*.35) $p$\
Ł(.398\*.27) $r_0$\
Ł(.595\*.97) $\delta$\
Ł(.49\*-0.08) $\delta_{-}$\
An important feature of this cylinder, which we will need below, is the following: Any geodesic arc $a$ which essentially crosses ${{\mathcal C}}$ and has endpoints on $\partial {{\mathcal C}}$, has length at least $2 r_0$.
To see this consider a point $q$ which is the base point of a simple geodesic loop of length at most $2 r_0$ ($p$ is such a point). By repeating the argument above, the parallel line $h_q$ to $\delta$ at the level of $q$ is embedded in ${{\mathcal C}}$, as is the cylinder consisting of all points at distance at most $r_0$ from $h_q$. In particular, the width of ${{\mathcal C}}$ is at least $2 r_0$.
Now consider an essential arc $a$ on ${{\mathcal C}}$. If it is simple and goes across the cylinder it has length at least the width of the cylinder, thus at least $2r_0$. If it is non-simple with both endpoints on $\delta_{-}$, then it must have a point at distance at least $r_0$ from $\delta_{-}$ and so it must be of length at least $2r_0$.
Because $i(c_i,c_i) \geq 2$, we have $w(c_i) \geq 2$. It will be convenient to think of ${{\mathcal C}}$ as the quotient of its universal cover $\tilde{{{\mathcal C}}}$ by the standard action of ${{\mathbb Z}}$ and look at copies of $c_i$ in this “unwrapped” version of ${{\mathcal C}}$ (see Figure \[fig:lift\]).
Ł(.62\*.28) $\tilde{c}_i$\
Ł(.62\*.88) $\tilde{\delta}$\
Ł(.49\*-0.1) $\tilde{\delta}_{-}$\
Let $c_i(t)$, $t\in [0,1]$ be a parametrization of $c_i$ and note that by standard hyperbolic geometry, the distance function $d_{{\mathcal C}}(c_i(t), \delta)$ is strictly convex. (The function $d_{{\mathcal C}}$ is the intrinsic distance function of ${{\mathcal C}}$.)
Let $p$ be the closest self-intersection point of $c_i$ to $\delta$. It is the base point of a geodesic simple loop $\alpha$, which is a subset of $c_i$. We consider the closed geodesic $\gamma'$ in the homotopy class of the curve obtained from $\gamma$ by removing the loop $\alpha$ from $\gamma$. Note that necessarily $\ell(\gamma') < \ell(\gamma)$ and because of our choice of loop removal, $\gamma'$ is not only non-trivial, we will be able to lower bound its self-intersection number. We begin by noting however that $$i(\gamma',\gamma') \leq k-1$$ otherwise $\gamma$ would not be minimal among elements of ${{\mathcal G}}_{\geq k}(X)$.
To get a lower bound we will construct a representative of $\gamma$ from the geodesic $\gamma'$. Begin by observing that there is an arc of $\gamma'$ which lives on ${{\mathcal C}}$ and which corresponds to the truncated strand $c_i$.
Ł(.45\*.45) $c'$\
Ł(.49\*.65) $p'$\
Ł(.528\*.505) $\alpha'$\
We’ll denote it $c'$ and assume that it is oriented following some orientation of $\gamma'$. Consider its closest point $p'$ to $\delta$ and the loop $\alpha'$ formed by all points of ${{\mathcal C}}$ of equal distance to $\delta$. Note that $\alpha'$ is freely homotopic to $\delta$ and thus to the loop $\alpha$ previously considered. We orient $\alpha'$ following the same orientation as $c'$. We consider the arc $c''$ obtained by following $c'$ from its orientation point until $p'$, then following $\alpha'$ and then continuing along $c'$. The important observation is that by replacing $c'$ with $c''$, we’ve recuperated the homotopy class of $\gamma$.
The number of self-intersection points of this representative of $\gamma$ is at least $i(\gamma,\gamma)$, but we’ll be able to find an upper bound on this intersection number as well, which in turn will give us a bound on $i(\gamma,\gamma)$.
We consider all the arcs of $\gamma'$ which are contained in the connected components of $\gamma' \cap {{\mathcal C}}$ that might possibly intersect $\alpha'$. They must of course be essential strands that intersect ${{\mathcal C}}$, and as observed above, must hence be of length at least $2r_0$. We can thus bound their number using our upper bound on the length of $\gamma'$. As $$\ell(\gamma') \leq 16 r_0 \sqrt{k+\frac{1}{4}}$$ we have that the number of strands is at most $$8 \sqrt{k+\frac{1}{4}}$$ Because distance from points in ${{\mathcal C}}$ to $\delta$ is strictly convex along parametrized geodesics, each strand can intersect $\alpha'$ at most twice. We thus have that $$i(\alpha', \gamma') \leq 16 \sqrt{k+\frac{1}{4}}$$ Therefore $$i(\gamma,\gamma) \leq i(\gamma',\gamma') + \i(\alpha',\gamma') < k-1 + 16 \sqrt{k+\frac{1}{4}}$$ as desired.
Observe that we can thus suppose in what follows that all of our segments are either simple or satisfy $i(c_i,c_i)=1$. A segment of the latter type we will call of $\alpha$-type, for obvious reasons.
The same “unwinding" technique from the proof of Lemma \[lem:unwind1\] can be used to bound $i(\gamma,\gamma)$ when we have two simple arcs $c_i,c_j$ that intersect at least twice. First we need the following fact.
Suppose there exists a crossing strand $c_i$ lying in a cylinder ${{\mathcal C}}$ with $\omega(c_i)>\frac{1}{2}$. Then $$i(\gamma,\gamma)\leq k-1+ 16 \sqrt{k+\frac{1}{4}}$$
Suppose $c_i$ lies in the cylinder ${{\mathcal C}}$ with core curve $\delta$. We extend the cylinder maximally in parallel directions so that it remains embedded to obtain cylinder $C'$, still with core curve $\delta$. Note that the winding number of the corresponding strand $c_i$ still satisfies $\omega(c_i)>\frac{1}{2}$ with respect to $C'$. Also, by the same argument as in the proof of Lemma \[lem:unwind1\] any geodesic arc that essentially crosses $C$ and has endpoints on $\partial C$ has length at least $2r_0$ and hence there are at most $$8\sqrt{k+\frac{1}{4}}$$ such strands.
We now unwind $c_i$ once (by applying a single Dehn twist around $\delta$ to $c_i$, in such a way that the winding number of $c_i$ decreases). Let $\gamma'$ be the geodesic representative in the homotopy class of the resulting curve. Since $\omega(c_i)>1/2$ it follows that $\ell(\gamma')<\ell(\gamma)$ and hence, by the definition of $\gamma$, $i(\gamma',\gamma')\leq k-1$.
Ł(.33\*.78) $c_i$\
Ł(.747\*.78) $c'$\
We proceed in a manner similar to the proof of Lemma \[lem:unwind1\]: we will reconstruct a representative of $\gamma$ from $\gamma'$ and use it to bound the self-intersection number of $\gamma$. Note that there is a strand $c'$ in a component of $\gamma'\cap{{\mathcal C}}$ corresponding to $c_i$. Let $p$ be the intersection point between $c'$ and $\delta$. We choose some orientation of $\gamma'$ and orient $\delta$ in the ’winding’ direction. Consider the arc $c''$ obtained by following $c'$ from one of its endpoints until $p$, then the loop $\delta$, and then continuing along $c'$ to its other endpoint. Let $\gamma''$ be the curve obtained from $\gamma'$ by replacing $c'$ with $c''$. Clearly $\gamma''$ is homotopic to $\gamma$ and hence $i(\gamma,\gamma)\leq i(\gamma'',\gamma'')$. By the exact same argument as in Lemma \[lem:unwind1\] we have $$i(\gamma,\gamma)\leq i(\gamma',\gamma')+i(\alpha,\gamma')\leq k-1+16\sqrt{k+\frac{1}{4}}$$ as desired.
If $c_i$ is a crossing strand in some cylinder $C$ with $\omega(c_i)\leq1/2$, it follows from Lemma \[lem:int\] that it can intersect any other simple segment at most once. Hence we have:
\[cor:unwind2\] If there exists crossing strands $c_i,c_j$ with $i(c_i,c_j) \geq 2$, then $$i(\gamma,\gamma) \leq k-1 + 16 \sqrt{k+\frac{1}{4}}$$
$\alpha$-type segments and final estimates
------------------------------------------
We now place ourself in the situation where all of our segments are either simple or of $\alpha$-type. Furthermore, by Corollary \[cor:unwind2\], we can suppose that any two simple segments intersect at most once.
We begin with a lemma about how an $\alpha$-type segment can intersect another segment:
Let $c_i,c_j$ be two of our segments and suppose that $c_i$ is of $\alpha$-type. Then $$i(c_i,c_j) \leq 4$$
Since $c_i$ is $\alpha$-type we must have $\omega(c_i)\leq2$ (with respect to the cylinder for which it is $\alpha$-type). It follows from Lemma \[lem:int\] that $i(c_i,c_j)\leq2$ if $c_j$ is simple and $i(c_i,c_j)\leq4$ if $c_j$ is $\alpha$-type.
We can now bound the intersection number of $\gamma$. Recall that the only cases left to consider are when $i(c_i,c_i)\leq 1$ and $i(c_i,c_j) \leq 4$ for all $i, j$. Hence we have: $$i(\gamma,\gamma) \leq \frac{1}{2} \sum_{i,j=1}^m i(c_i,c_j) - \sum_{l=1}^m i(c_l,c_l) \leq 2 m^2 - m \leq 32 \sqrt{k+\frac{1}{4}} ( 16 \sqrt{k+\frac{1}{4}} +1)$$ which proves the theorem.
Intersections in the thick part and surfaces with cusps {#sec:cusp}
=======================================================
The main goal of this section is to prove Theorem \[thm:cuspcase\] but before doing so we study thick-thin decompositions of surfaces.
Thick parts of closed curves
----------------------------
Given a hyperbolic surface $X$ and fixed $\varepsilon > 0$, we define the $\varepsilon$-thick part $X_T$ of $X$ to be the subset of $X$ consisting of points with injectivity radius at least $\varepsilon$. The $\varepsilon$-thin part $X_t$ is the subset of $X$ with injectivity radius at most $\varepsilon$. Now given a curve $\gamma \subset X$, we can decompose it into $\gamma_T := X_T \cap \gamma$ and $\gamma_t := X_t \cap \gamma$.
Note that $\gamma$ might go in and out of the thick part, so $\gamma_T$ is not necessarily the continuous image of an interval. Nonetheless $\gamma_T$ can be broken into arcs that are continuous images of intervals with endpoints lying on the boundary of the thick part and we will denote these components by $\gamma_1, \hdots, \gamma_r$. Our first observation is that, provided $\varepsilon$ is small enough, each $\gamma_i$ has a certain length.
If $\varepsilon \leq \frac{1}{2}$, then $$\ell(\gamma_i) \geq \frac{3}{4}$$ for $i= 1,\hdots,r$
The boundary of $X_T$ consists of points of injectivity radius exactly $\varepsilon$ and, in particular, for any point of the boundary there is a simple geodesic loop of length $2 \varepsilon$ based in that point. Suppose that $\gamma_i$ joins points $p,q$ on the boundary of $X_T$ and denote by $\alpha$ and $\beta$ the simple loops of length $2 \varepsilon$ based in $p$ and $q$, respectively. Note that $\alpha$ and $\beta$ are either disjoint or freely homotopic. We orient $\gamma_i$, $\alpha$ and $\beta$ such that $\alpha$ and $\beta$ have opposite orientations. We now obtain a homotopy class of curve given by the concatenation $$\alpha * \gamma_i *\beta *\gamma_i$$ The main observation is that the geodesic $\delta$ in the homotopy class of the above concatenation is a non-simple closed geodesic and thus has length at least $4 \log(1+\sqrt{2})$ (see for instance [@BuserBook]). Now as $\ell(\alpha) + \ell(\beta) + 2 \ell(\gamma_i)$ is a strict upper bound for $\ell(\delta)$, we have the inequality $$2 \ell(\gamma_i) > 4 \log(1+\sqrt{2}) - 2 > \frac{3}{2}$$ and the result follows.
The constants in the above proof are clearly not optimal, and the choice of $\varepsilon \leq \frac{1}{2}$ is somewhat arbitrary.
We now turn our attention to finding a lower bound on $\ell(\gamma_T)$ in terms of $i(\gamma_T, \gamma_T)$, proving Theorem \[thm:thick\] of the introduction which gives a lower bound on length in terms of intersection number.
We begin by considering a set of points $\{p_j\}_{j\in I}$ which form an $\varepsilon$-net for $X_T$ ($I$ is just an index set). Specifically, the points all belong to $X_T$, are pairwise at least distance $\varepsilon$ apart and are maximal for inclusion. In particular, any $x\in X_T$ is distance at most $\varepsilon$ from at least one $p_j$. As such we can consider the Voronoi cells $\{V_j \}_{j\in I}$ around each of the $p_i$. As $\varepsilon \leq \frac{1}{2}$, each of the Voronoi cells are (topological) disks.
The intersection between $\gamma_T$ and any Voronoi cell $V_j$ is a collection of simple geodesic segments each of length at most $2 \varepsilon$. As $\gamma_T$ is of finite length, we can decompose $\gamma_T$ into these simple geodesic segments that traverse Voronoi cells. Denote them by $c_1, \hdots, c_m$. We note that an immediate upper bound on $i(\gamma_T, \gamma_T)$ is given by $$\frac{m(m-1)}{2}$$ as any two of these segments can intersect at most once. We’ll now proceed to bound $m$ in terms of $\ell(\gamma_T)$.
Recall our notation of $\gamma_1, \hdots, \gamma_r$ for the components of $\gamma_T$. By the previous lemma, we have $\ell(\gamma_i) \geq \frac{3}{4}$.
Consider the arc $\gamma_i$ consisting of multiple $c_j$’s, the intersections with the Voronoi cells. We suppose the number of them is $m_i$ and we have $$\sum_{i=1}^{r} m_i = m$$ We will now bound $m_i$ in terms of $\ell(\gamma_i)$. To do so we lift to the universal cover and consider the set of lifts of centers of Voronoi cells encountered by $\gamma_i$. We denote by $\tilde{\gamma}_i$ the lift of $\gamma_i$ and by $q_1,\hdots, q_{m_i}$ the lifts of the centers of the Voronoi cells. Note that the (open) balls of radius $\frac{\varepsilon}{2}$ around each $q_j$ are all pairwise disjoint. These balls are also all contained in the $\frac{3}{2} \varepsilon$ neighborhood of $\tilde{\gamma}_i$. The area of this neighborhood is obtained by computing the area of a strip of width $\frac{3}{2} \varepsilon$ around $\tilde{\gamma}_i$ and by adding the area of a ball of radius $ \frac{3}{2} \varepsilon$ for each of the two endpoints of $\tilde\gamma_i$. The resulting area is $$A:= 2\left(\ell(\gamma_i) \sinh\sfrac{3\varepsilon}{2} + \pi(\cosh \sfrac{3\varepsilon}{2}-1)\right)$$ In comparison, the total area of the balls of radius $\frac{\varepsilon}{2}$ around each $q_j$ is $$B:=m_i 2\pi (\cosh \sfrac{\varepsilon}{2}-1)$$ and as $B < A$ we can deduce that $$m_i < \frac{\ell(\gamma_i) \sinh\sfrac{3\varepsilon}{2} + \pi(\cosh \sfrac{3\varepsilon}{2}-1)}{\pi (\cosh \sfrac{\varepsilon}{2}-1)}$$ We are not trying to optimize the constants we obtain, so we will simplify the above expression somewhat. Seen as a linear function in $\ell(\gamma_i)$, the leading coefficient can be bounded by $$\frac{\sinh\sfrac{3\varepsilon}{2} }{\pi (\cosh \sfrac{\varepsilon}{2}-1)}< \frac{5}{\varepsilon}$$ as $\varepsilon \leq \frac{1}{2}$. The second coefficient is strictly increasing in $\varepsilon$ so, again using $\varepsilon \leq \frac{1}{2}$, we bound it by $10$. We thus have $$m_i < \frac{5}{\varepsilon} \ell(\gamma_i) + 10 < \frac{5}{\varepsilon} (\ell(\gamma_i)+1)$$ Using the fact that $\ell(\gamma_i) > \frac{3}{4}$, this implies the following (highly non-optimal) inequality: $$m_i < \frac{12}{\varepsilon} \ell(\gamma_i)$$ We now return to $\gamma_T$ and $m$.
We have $$\begin{aligned}
i(\gamma_T,\gamma_T) & \leq &
\frac{1}{2} m (m-1)\\
& =& \frac{1}{2} \sum_{i=1}^r m_i \left(\sum_{i=1}^r m_i -1\right)\\
& < &\left( \frac{12}{\varepsilon}\right)^2 \left(\sum_{i=1}^r \ell(\gamma_i)\right)^2 = \left( \frac{12}{\varepsilon}\right)^2 \left(\ell(\gamma_T)\right)^2\end{aligned}$$ and thus $$\ell(\gamma_T) > \frac{\varepsilon \sqrt{i(\gamma_T,\gamma_T)}}{12}$$ as desired.
Note that if $X$ is closed, setting $\varepsilon := \min \{\frac{1}{2}, \frac{{{\rm sys}}(X)}{2}\}$ where ${{\rm sys}}(X)$ is the systole length of $X$, then $X=X_T$. In particular $\gamma$ is entirely contained in the thick part of $X$ and we have a lower bound on its length that grows like the root of its intersection. This is exactly the statement of Theorem 1.1 in [@BasmajianUniversal]. In what follows, we will need to apply our estimate to surfaces with cusps.
Surfaces with cusps
-------------------
Armed with Theorem \[thm:thick\] and using Basmajian’s upper bounds on length for surfaces with cusps [@BasmajianUniversal], we can now prove Theorem \[thm:cuspcase\].
Let $X$ be a complete hyperbolic surface with at least one cusp. If $\gamma$ is a closed geodesic on $X$ with at least $k\geq2$ self-intersections, it is a result by Basmajian [@BasmajianUniversal Corollary 1.3] that there exists a constant $C=C(k,X)$ such that $\ell(\gamma)<C$. In fact, $C=2\sinh^{-1}{(k)}+d_X+1$ where $d_X$ is the shortest orthogonal distance from the length 1 horosphere boundary of a cusp to itself. Note that $\sinh^{-1}(k)$ is comparable to $\log{(k)}$, and therefore so is $C(k,X)$.
Let $\varepsilon'=\frac{1}{4}$ and let $s$ be the systole length of the $\varepsilon'$-thick part of $X$. Note that $\frac{1}{4}<\cosh^{-1}\left({\frac{\sqrt{11}}{3}}\right)$ which is the injectivity radius of a cusp with boundary horosphere of length $\frac{2}{3}$.
Now, let $\varepsilon=\min\left\{\frac{1}{4}, \frac{s}{2}\right\}$. Choose $K\geq2$ such that $C(k,X)<\frac{\varepsilon}{12}\sqrt{k}$ for all $k>K$. Let $k>K$ and $\gamma$ a shortest geodesic on $X$ with at least $k$ self-intersections. By Theorem \[thm:thick\] $\gamma$ must intersect $X_t$, the $\varepsilon$-thin part of $X$. By the choice of $\varepsilon$, $\gamma$ must enter a cusp of $X$, and in fact a cusp neighborhood with boundary horosphere $\delta$ of length $\frac{2}{3}$. This implies that $\gamma_t$ contains a strand $c$ (a continuous image of an interval with endpoints lying on the boundary horosphere) that intersects itself at least 3 times. We use a similar unwinding argument as in Lemma \[lem:unwind1\] to get a bound on the intersection number of $\gamma$. Let $p$ be the self-intersection point of $c$ furthest away from $\delta$. It is the base point of a geodesic loop $\alpha$. Remove this loop from $\gamma$ and consider the resulting geodesic $\gamma'$. Clearly $\ell(\gamma')<\ell(\gamma)$ and hence, by definition of $\gamma$, $i(\gamma',\gamma')\leq k-1$.
Let $c'$ be the strand of $\gamma'$ corresponding to the truncated strand $c$. Note that $c'$ self-intersects at least twice, and hence enters the cusp neighborhood (of the same cusp as $c$) with boundary horosphere $\delta'$ of length 1. Pick a point $p'$ on $c'$ in this cusp neighborhood and consider the simple loop $\alpha'$ based at this point consisting of all points equidistant from $\delta'$. As in Lemma \[lem:unwind1\], let $c''$ be the arc obtained by concatenating $c'$ and $\alpha'$ and let $\gamma''$ be the curve obtained by replacing $c'$ with $c''$ in $\gamma'$, and note that $\gamma''$ is homotopic to $\gamma$. Hence $$i(\gamma,\gamma)\leq i(\gamma'',\gamma'')=i(\gamma',\gamma')+i(\alpha',\gamma').$$
To estimate $i(\alpha',\gamma')$ note that it is bounded from above by twice the number of strands of $\gamma'$ that enters the cusp neighborhood with boundary horosphere of length 1 (since each such strand can intersect $\alpha'$ at most twice). Each such strand has to pass through the cylinder of width $\log (2)$ in the cusp bounded by the horospheres of length 2 and 1, and then return. Hence each strand has length at least $2\log (2)$ and since $\ell(\gamma')<C(k,X)$ there are less than $C(k,X)/(2\log (2))$ such strands, and $i(\alpha',\gamma')<C(k,X)/\log (2)$. Therefore, $$i(\gamma,\gamma)<k-1+ \frac{C(k,X)}{\log (2)}$$ and, as noted above, $C(k,X)$ is comparable to $\log (k)$, proving the theorem.
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---
abstract: 'We demonstrate that, in a many-particle system, particles can be strongly confined to their sites. The localization is obtained by constructing a sequence of on-site energies that efficiently suppresses resonant hopping. The time during which a many-particle state remains strongly localized in an infinite chain can exceed the reciprocal hopping frequency by $\agt 10^5$ already for a moderate energy bandwidth. The results show viability of quantum computers with time-independent qubit coupling.'
author:
- 'M.I. Dykman$^{1}$, F.M. Izrailev$^{3}$, L.F. Santos$^{1}$, and M. Shapiro$^{2}$'
title: ' Many-particle localization by constructed disorder: enabling quantum computing with perpetually coupled qubits'
---
Disorder-induced localization has been one of the central problems of condensed-matter physics [@Anderson]. The problem is particularly challenging for many-body systems, where only a limited number of results has been obtained [@Altshuler03]. Recently it has also attracted much interest in the context of quantum computing. In many proposed physical implementations of a quantum computer (QC) the qubit-qubit interaction is not turned off [@liquid_NMR; @Makhlin01; @mark; @mooij99; @Yamamoto02; @van_der_Wiel03]. Generally, the interaction leads to hopping of excitations between the qubits. Preventing hopping is a prerequisite for quantum computation. To control a QC and measure its state excitations must remain localized between operations. Several approaches to quantum computation with interacting qubits were proposed recently [@Zhou02; @Benjamin03; @Berman-01].
In this paper we study strong on-site many-particle localization. It implies that each particle (or excitation) is nearly completely confined to one site. This is a stronger condition than just exponential decay of the wave function, and it is this condition that must be met in a QC.
Strong on-site localization does not arise in a disordered many-particle system with bounded random on-site energies [@Shepelyanskiy]. Indeed, consider a state where particles occupy $N$ sites. For short-range hopping it is directly coupled to $\sim N$ other $N$-particle states. With probability $\propto N$ one of them will be in resonance with the initial state, provided the on-site energies are uniformly distributed over a finite-width band. For large $N$ this leads to state hybridization over time $\sim
J^{-1}$, where $J$ is the intersite hopping integral (we set $\hbar =
1$).
Here we show that many-particle localization can be obtained in an [*infinite*]{} chain by [*constructing*]{} a sequence of on-site energies. For the proposed narrow-band sequence, all many-particle states remain confined for a time that largely exceeds $J^{-1}$. We find that stationary many-particle states in moderately long chains are also strongly localized.
A QC allows both studying and using strong localization. Here, on-site excitation energies are interlevel distances of the qubits. They can often be individually controlled, which makes it possible to construct an arbitrary energy sequence. However, since the qubit tuning range is limited, so should be the energy bandwidth. A smaller bandwidth is also desirable for a higher speed of quantum gate operations.
To localize one particle, the difference between excitation energies on neighboring sites should be much larger than $J$. In addition, even for nearest neighbor coupling, the energies of remote sites should also differ to prevent virtual transitions via intermediate sites. However, the further away the sites are, the smaller their energy difference can be. We use this to obtain strong on-site single-particle confinement for a bounded energy bandwidth.
For many-particle localization one has to suppress not only single-particle, but also combined resonances, where several interacting excitations make a transition simultaneously. There is no known way to eliminate all such resonances. However, the problem can be approached from a different point of view by looking at a lifetime of a localized state. The effective many-particle hopping integral quickly falls off with the increasing number of involved excitations and intermediate nonresonant sites, which gives the effective “order” of a transition. To obtain a desired lifetime it is sufficient to eliminate resonances up to a certain order. As we show, this can be done for an infinite system.
We will consider a one-dimensional chain of $S=1/2$ spins in a magnetic field. This model also describes an array of qubits. The excitation energy of a qubit is the Zeeman energy of a spin. The qubit-qubit interaction is the exchange spin coupling. For many proposed realizations of QC’s [@liquid_NMR; @Makhlin01; @mark; @mooij99; @Yamamoto02; @van_der_Wiel03] it has a form ${1\over 2}\sum^{\prime}
J_{nm}^{\mu\mu}S_n^{\mu}S_m^{\mu}$, where $n,m$ are spin sites, $\mu=x,y,z$ are spin projections, and $J_{nm}^{xx}=J_{nm}^{yy}$ for the effective magnetic field in the $z$-direction. The 1D spin system can be mapped, via Jordan-Wigner transformation, onto a system of fermions. For nearest neighbor coupling, the fermion Hamiltonian is $$\begin{aligned}
\label{hamiltonian_fermions}
H=&&\sum\nolimits_n\varepsilon_na_n^{\dagger}a_n+ {1\over
2}J\sum\nolimits_n\bigl( a_n^{\dagger}a_{n+1}+a_{n+1}^{\dagger}a_n\bigr)
\nonumber\\ &&+ J\Delta\sum\nolimits_n
a_n^{\dagger}a_{n+1}^{\dagger}a_{n+1}a_n.\end{aligned}$$ Here, $a^{\dagger}_n, a_n$ are the fermion creation and annihilation operators. Presence of a fermion on site $n$ corresponds to the $n$th spin (qubit) being in the excited state. The on-site energies $\varepsilon_n$ in Eq. (\[hamiltonian\_fermions\]) are the Zeeman energies counted off from the characteristic central energy, $J\equiv
J_{n\,n+1}^{xx}$ is the hopping integral, and $J\Delta \equiv
J_{n\,n+1}^{zz}$ is the fermion interaction energy; we set $J,\Delta>
0$.
Localization of stationary states can be conveniently characterized by the inverse participation ratio (IPR), which shows over how many sites the wave function spreads. For an $N$-particle eigenstate $|\psi_{N\lambda}\rangle$ ($\lambda$ enumerates the eigenstates) it is given by $$\label{IPR}
I_{N\lambda}=\left(\sum\nolimits_{n_1<\ldots<n_N}\bigl\vert\langle
\Phi_{n_1\ldots n_N}|\psi_{N\lambda}\rangle\bigr\vert^4\right)^{-1},$$ where $|\Phi_{n_1\ldots n_N}\rangle = a^{\dagger}_{n_1}\ldots
a^{\dagger}_{n_N}|0\rangle$ is an on-site $N$-particle wave function (quantum register).
For fully localized stationary states $I_{N\lambda}=1$. For delocalized states $I_{N\lambda}\gg 1$ \[for an $L$-site chain $I_{N\lambda}\lesssim L!/N!(L-N)!$\]. Strong localization corresponds to $I_{N\lambda}$ being close to $1$ for all states.
Localization requires that the on-site energies $\varepsilon_n$ be tuned away from each other. For nearest neighbor coupling a natural first step is to separate $\varepsilon_n$’s into two subbands, for even and odd $n$, with the inter-subband distance $h$ that significantly exceeds $J$. Then we further split each subband into two subbands to detune next nearest neighbors. Here the splitting can be smaller, because next-nearest-neighbor hopping occurs via a nonresonant site, and the effective hopping integral is $\sim J^2/h$. The procedure of band splitting is continued, with higher-order splitting being smaller and smaller.
A simple sequence of $\varepsilon_n$ that implements the above idea has the form $$\label{sequence}
\varepsilon_n={1\over 2}h\left[(-1)^n -\sum\nolimits_{k=2}^{n+1}(-1)^{\lfloor
n/k\rfloor}\alpha^{k-1}\right], \quad n\geq 1$$ ($\lfloor \cdot\rfloor$ is the integer part). The energies (\[sequence\]) are illustrated in Fig. \[fig:one\_excitation\](a). Besides the scaling factor $h$, they are characterized by one dimensionless parameter $\alpha <1$. For small $\alpha$, the two major subbands have width $\approx \alpha h$ and are separated by a gap of width $\approx h$. The splitting of higher-order subbands are proportional to higher powers of $\alpha$. For $\alpha \agt 0.4$ all subbands overlap and the subband structure disappears.
One can see from Eq. (\[sequence\]) and Fig. \[fig:one\_excitation\](a) that sites with close energies are indeed spatially separated. Analytical estimates of the energy difference can be obtained for small $\alpha$. We have $|\varepsilon_{n+m}-\varepsilon_n| \sim h$ for odd $m$ and $\sim \alpha h$ for odd $m/2$. In general, the larger is $m$ the higher may be the order in $\alpha$ of the leading term in $|\varepsilon_{n+m}-\varepsilon_n|$.
It is important for localization that the sequence (\[sequence\]) has no simple symmetry. It is neither self-similar nor quasi-periodic (which is another example of “constructed” disorder [@Sokoloff85]). For analytical estimates it is essential that the coefficients at any given power $\alpha^q$ are repeated with period $2(q+1)$ [@elsewhere].
A convenient characteristic of the on-site energy sequence is the amplitude of a particle transition from site $n$ to site $n+m$. To the lowest order in $J$ it has the form $$\label{amplitude}
K_n(m)= \prod\nolimits_{k=1}^{m}
J/\left[2(\varepsilon_n - \varepsilon_{n+k})\right].$$ It can be shown using some results from number theory that $K_n(m)$ decays with $m$ nearly exponentially [@elsewhere]. For small $\alpha$ and large $|m|$ we have $$\label{result}
K_n(m)= \alpha^{-\nu|m|}\,(J/2h)^{|m|}.$$ The decrement $\nu$ depends on $n,m$. However, it is limited to a narrow region around $\nu= 1$ with $0.89 < \nu < 1.19$, cf. Fig. \[fig:one\_excitation\](b). For estimates one can use $\nu=1$, i.e., set $K_n(m)=K^m, K=J/2\alpha h$.
![(color online). Single-particle localization for the on-site energy sequence (\[sequence\]). (a) The energies $\varepsilon_n/h$ for $\alpha=0.3$. (b) The decrement $\nu$ of the $\alpha$-dependence of the transition amplitude $K_n(m)$ (\[result\]) for $m=1000$ as function of $n$. The dashed lines show the analytical limits on $\nu$. (c) The mean single-particle inverse participation ratio $\langle I_1\rangle$ vs. $\alpha$ for $h/J=20$ and for different chain lengths $L$. The vertical dashed line shows the analytical estimate for the threshold of strong localization. The inset shows the maximal IPR, $I_{1\max}=\max_{\lambda} I_{1\lambda}$, demonstrating strong localization.[]{data-label="fig:one_excitation"}](fig1new_loc.eps){width="2.8in"}
Equation (\[result\]) describes the tail of the transition amplitude for $J/2h\alpha\ll 1$. Strong single-particle localization occurs for $\alpha \gg \alpha_{\rm th}$, where the threshold value of $\alpha$ is $\alpha_{\rm th} \approx J/2h$. The condition $\alpha_{\rm th}\ll
\alpha < 0.4$ can be satisfied for a moderately large ratio of the energy bandwidth $h$ to the hopping integral $J$.
Strong single-particle localization for $h/J=20$, as evidenced by $I_{1\lambda}$ being very close to 1, is seen from Fig. \[fig:one\_excitation\](c). The data are obtained by diagonalizing the Hamiltonian (\[hamiltonian\_fermions\]) for open chains with different numbers of sites $L$.
In the limit $\alpha\to 0$ the stationary single-particle states are sinusoidal, which gives $\langle I_1\rangle \approx L/3$, cf. Fig. \[fig:one\_excitation\](c) ($\langle\cdot\rangle$ means averaging over the eigenstates). As $\alpha$ increases, the bands are split into more and more subbands, and $\langle I_1\rangle$ decreases. It sharply drops to $\approx 1$ in a narrow region, which can be conditionally associated with a smeared transition to strong localization. The center of the transition region gives $\alpha_{\rm
th}$. It appears to be independent of the chain length $L$. The estimate $\alpha_{\rm th}=J/2h$ is in good agreement with the numerical data for different $h/J$.
When $\alpha \gg \alpha_{\rm th}$, all states are localized. The wave function tails are small and limited mostly to nearest neighbors. At its minimum over $\alpha$ for given $h/J$, for all states $I_{1\lambda} - 1 \approx J^2/h^2$, see inset in Fig. \[fig:one\_excitation\](c). The agreement with the above estimate becomes better with increasing $h/J$. For $\alpha \agt 0.4$, when the bands of $\varepsilon_n$ start overlapping, the IPR increases with $\alpha$.
The difference in the localization problems for many-particle and single-particle systems stems from the interaction term $\propto
J\Delta$ in the Hamiltonian (\[hamiltonian\_fermions\]). If $\Delta =
0$ (the interaction between the underlying spins or qubits is of the $XY$-type), the above single-particle results apply to the many-particle system. For nonzero $\Delta$, on the other hand, (i) the energy levels are shifted depending on the occupation of neighboring sites, potentially leading to many-particle resonances, and (ii) there occur interaction-induced many-particle transitions.
To analyze many-particle effects, it is convenient to change from $a_n^{\dagger},a_n$ to new creation and annihilation operators $b_n^{\dagger},b_n$ that diagonalize the single-particle part of the Hamiltonian (\[hamiltonian\_fermions\]), $a_n=\sum\nolimits_kU_{nk}b_k$. The interaction part of the Hamiltonian becomes $$\label{H_int}
H_i=J\Delta\sum
V_{k_1k_2k_3k_4}b_{k_1}^{\dagger}b_{k_2}^{\dagger}b_{k_3}b_{k_4},$$ where the sum runs over $k_{1,2,3,4}$, and $V_{k_1k_2k_3k_4} =
\sum\nolimits_pU^*_{pk_1}U^*_{p+1\,k_2}U_{p+1\,k_3}U_{pk_4}$.
If all single-particle stationary states are strongly localized, the off-diagonal matrix elements $U_{nk}$ are small. They are determined by the decay of the wave functions and fall off exponentially, $U_{nk}\sim K^{|k-n|}$ for $|k-n| \gg 1$. At the same time, the diagonal matrix element is $U_{nn} \approx 1$. Therefore the major terms in the matrix $V_{k_1k_2k_3k_4}$ are those with $\varkappa=0$, where $$\varkappa=\min_p(|k_1-p|+|k_2-p-1|+|k_3-p-1|+|k_4-p|).$$ They lead to an energy shift $\propto J\Delta$ for the many-particle states with occupied neighboring sites.
The terms $V_{k_1k_2k_3k_4}$ with $\varkappa> 0$ lead to two-particle intersite transitions $(k_3,k_4)\to (k_1,k_2)$, and $VJ\Delta $ plays the role of a two-particle hopping integral. The parameter $\varkappa$ gives the number of intermediate steps involved in a transition. The steps are counted off from the configuration where the particles occupy neighboring sites [@elsewhere]. For large $\varkappa$ and $\alpha\gg \alpha_{\rm th}$ we have $V\propto
(J/2h\alpha^{\nu})^{\varkappa}\ll 1$, i.e., many-particle hopping integrals are small and rapidly decrease with the number of involved virtual steps.
Numerical results on the many-particle IPR are shown in Fig. \[fig:many\_loc\]. We have studied chains of length $L=10, 12$, and 14 with $L/2$ excitations, which have the largest number of states for given $L$ ($\propto 2^L$ for large $L$). The results were similar, and we present the data for $L=12$, in which case the total number of states is 924.
For small $\alpha$, the IPR is independent of $\alpha$ and is large because of the large number of resonating on-site states $|\Phi_{n_1\ldots n_6}\rangle$. It is reduced by the interaction $\propto J\Delta$ that splits the energy spectrum into subbands depending on the number of occupied neighboring sites. On the whole, the IPR decreases with increasing $\alpha$ as long as $\alpha \alt
0.4$. In the region $0.2\alt \alpha \alt 0.4$ we have $\langle
I_6\rangle \approx 1.01$ except for narrow peaks. This indicates that away from the peaks all stationary states are strongly localized, as confirmed by the data on $I_{6\max}=\max_{\lambda} I_{6\lambda}$.
![(color) Many-particle localization for a chain of length $L=12$ with $6$ excitations. The data refer to the first 12 sites of the chain (\[sequence\]), the reduced bandwidth is $h/J=20$. The purple, red, green, and blue curves give IPR for the coupling parameter $\Delta=0,0.3,1$, and 3, respectively. The inset shows the maximal $I_6$. Sharp isolated peaks for $\Delta\neq 0$ result from the hybridization of many-particle on-site states that are in resonance for the corresponding $\alpha$. The peaks for $\Delta=0$ are due to the boundaries.[]{data-label="fig:many_loc"}](fig2new_loc.eps){width="2.8in"}
A distinctive feature of the many-particle IPR as function of $\alpha$ are multiple resonant peaks, a part of which is resolved in Fig. \[fig:many\_loc\]. They occur when two on-site states resonate, which gives $I_{6\max}\lesssim 2$. The strongest peaks come from two-particle resonances. They happen when the two-particle energy difference $$\label{energy_difference}
\delta\varepsilon
=|\varepsilon_{k_1}+\varepsilon_{k_2}-
\varepsilon_{k_3}-\varepsilon_{k_4}|$$ is close to $MJ\Delta$ with $M=0,1,2$.
As we increase $\alpha$ starting from $\alpha=0$, pronounced peaks of $\langle I_6\rangle$ appear first for $\delta\varepsilon \approx
s\alpha h \approx J\Delta$ with $s=1,2$. They are due to hybridization of pairs on sites $(n,n+1)$ and $(n,n+3)$ for $s=1$, and $(n,n+1)$ and $(n-1,n+2)$ for $s=2$, for example ($\varkappa = 2$-transitions).
For larger $\alpha$, resonances occur when $s\alpha^nh\approx
MJ\Delta$ with $n\geq 2$. Such resonances require more intermediate steps, with $\varkappa \geq 4$. The widths of the IPR peaks are small and are in good agreement with simple estimates based on Eq. (\[H\_int\]) [@elsewhere]. In between the peaks $I_{6\max} = 1.02$ for $0.2\lesssim \alpha\lesssim 0.4$ and $h/J=20$.
A special role is played by two-particle resonances where $\delta
\varepsilon \ll J$ for all $\alpha < 0.4$. They emerge already for $\varkappa = 2$-transitions $(n,n+1) \leftrightarrow (n-1,n+2)$. Here, if $n$ and $n+2$ are prime numbers, $\delta \varepsilon\sim
\alpha^{n-1}h$ is extremely small for large $n$. Strong resonance occurs for all $n=6k-1$, in which case $\delta \varepsilon/h\propto
\alpha^{\xi}$ with $\xi \geq 4$. Such $\delta \varepsilon$ is unusually small for $\varkappa =2$. More many-particle resonances happen for higher $\varkappa$. For different sections of the chain (\[sequence\]) we found that they could increase $\langle
I_6\rangle$ up to 1.15 between the peaks, for $h/J=20,0.2<\alpha<0.4$, and $\Delta=1$. These resonances can be eliminated by modifying the sequence (\[sequence\]): $\varepsilon_n \to \varepsilon_n+\alpha^{\prime} h/2$ for $n=6k$. For appropriate $\alpha^{\prime} \sim 0.1$, this modification brings $\langle I_6\rangle$ and $I_{6\max}$ back to $\approx 1.01$ and $\approx 1.02$, respectively [@elsewhere; @footnote2].
We now outline an [*alternative approach*]{} to the problem of strong localization, which is particularly relevant for quantum computing. Qubit states have finite coherence time, estimated as $\lesssim
10^5 - 10^6J^{-1}$ for most models. It is sufficient to show that the states remain localized for a time that exceeds the coherence time. This will enable both gate operations, that take time $\sim
J^{-1}$, and measurement, that often requires more time.
Delocalization occurs through a transition to a resonating on-site state. All resonant two-particle transitions up to a given number of steps $\varkappa_0$ will be eliminated if, for $\varkappa <
\varkappa_0$, the on-site energy difference $\delta\varepsilon$ exceeds the maximum change of the interaction energy $\propto
J\Delta$. For $\varkappa_0=4$ this requires $J\Delta/h <
\alpha^2,\alpha^{\prime}/2, |\alpha-\alpha^{\prime}/2|$. From Eq. (\[H\_int\]), the time needed for a transition with $\varkappa_0$ steps is $\agt K^{-\varkappa_{_0}}/J\Delta$, it scales as $h^{\varkappa_{_0}}$. Then if $\varkappa_0=4$, the lifetime of [*all*]{} on-site states exceeds $J^{-1}$ by a factor $10^5$ for $h=30$, $\alpha=0.3$, $\alpha^{\prime}\approx 0.2$, and $\Delta \alt 1$.
![(color online) The low-energy part of the two-particle energy differences $\delta \varepsilon _n /h$ (\[energy\_difference\]) for all transitions with $\varkappa\leq 5$ in which one of the involved particles is on the $n$th site ($n> 2$). The data refer to $\alpha =0.25$. The left panel corresponds to the sequence (\[sequence\]). The right panel refers to the modified sequence with $\alpha^{\prime} = 0.22$ and shows the zero-energy gap.[]{data-label="fig:correction"}](fig3_6k.eps){width="2.8in"}
The transition time is further dramatically increased if $\varkappa=4$ resonances are eliminated. The low-energy gap in $\delta\varepsilon$ for the corresponding transitions is $\sim\alpha^3h$ when $\alpha^{\prime} \gg
\alpha^3$. For example, for $\alpha = 0.25$ and $\alpha^{\prime} = 0.22$ this gap is $\delta \varepsilon/h > 0.01$ for [*all*]{} sites $n>4$, cf. Fig. \[fig:correction\]. This means that all states will remain localized on their sites for a time $\agt K^6(J\Delta)^{-1}$, if $2J\Delta/h < 0.01$.
In terms of the operation of a quantum computer, the on-site energy sequence (\[sequence\]) is advantageous, since one radiation frequency can be used to resonantly excite different qubits. Selective tuning to this frequency can be done without bringing neighboring qubits in resonance with each other. In our approach, localization does not require refocusing [@liquid_NMR], which is not always easy to implement. We avoid delocalization due to indirect resonant $n\to n+2$ transitions, which undermines the approach [@Benjamin03]. Compared to Ref. [@Zhou02], in our approach the interaction does not have to be ever turned off, and no multi-qubit encoding is necessary. In addition, our results are not limited to systems with nearest neighbor coupling.
In conclusion, we have proposed a sequence of on-site energies that leads to strong localization of single- and many-particle stationary states of interacting spins or qubits. For an infinite chain, we eliminate resonances between on-site states with the effective interstate hopping integral up to $(J/h)^5$. This leads to a long lifetime of localized many-particle states already for a comparatively narrow energy bandwidth. The results show viability of scalable quantum computers where the interqubit interaction is not turned off.
This research was supported in part by the Institute for Quantum Sciences at Michigan State University and by the NSF through grant No. ITR-0085922.
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The obtained $\langle I_6\rangle-1$ were much smaller than for a chain with random $\varepsilon_n$ uniformly distributed over a band of the same width as (\[sequence\]). On the other hand, $\langle I_6\rangle$ for the sequence (\[sequence\]) did not increase when random terms up to $\sim h\alpha^4$ were added to $\varepsilon_n$, which indicates that the sequence (\[sequence\]) is robust with respect to errors.
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abstract: 'A classic theorem of Dirac from 1952 states that every graph with minimum degree at least $n/2$ contains a Hamiltonian cycle. In 1963, Pósa conjectured that every graph with minimum degree at least $2n/3$ contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac’s theorem by proving that every graph with $\deg(u) + \deg(v) \geq n$ for every $uv \notin E(G)$ contains a Hamiltonian cycle. Recently, Châu proved an Ore-type version of Pósa’s conjecture for graphs on $n\geq n_0$ vertices using the regularity–blow-up method; consequently the $n_0$ is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller $n_0$, we believe that our method of proof will be of independent interest.'
author:
- 'Louis DeBiasio[^1] Safi Faizullah[^2] Imdadullah Khan[^3]'
title: 'Ore-degree threshold for the square of a Hamiltonian cycle'
---
Introduction
============
Notation and Definitions
------------------------
Given a graph $G$, we denote the vertex set and edge set by $V(G)$ and $E(G)$ respectively, when the graph $G$ is clear by the context we refer to them as $V$ and $E$ respectively. When $uv\in E(G)$ we denote it by $u\sim v$ otherwise $u\not\sim v$. For a vertex $v\in V$, $N(v)$ is the set of neighbors of $v$ in $V$ and the degree of $v$ is $|N(v)|$ and we denote it by $\deg(v)=\deg_G(v)$. For $A\subseteq V(G)$, $N(v,A)$ is the set of neighbors of $v$ in $A$ and $\deg(v,A)$ is $|N(v,A)|$. We denote by $\delta(G)$ the minimum degree over all vertices in $G$ and by $\Delta(G)$ the maximum degree over all vertices in $G$. We write $N(v_1,v_2,\dots, v_l)=\bigcap_{i=1}^{l}N(v_i)$ for the set of common neighbors of $v_1, v_2, \ldots, v_l$. Similarly, $N(v_1,v_2,\dots, v_l, A):=\bigcap_{i=1}^{l}N(v_i,A)$ and $deg(v_1,v_2,\ldots,v_l, A) := |N(v_1,v_2,\dots, v_l, A)|$. We denote a cycle on $t$ vertices by $C_t$ and a path on $t$ vertices by $P_t$. When $G$ is a graph on $n$ vertices and $C_n\subseteq G$, we call $C_n$ a Hamiltonian cycle. A bipartite graph $G=(V,E)$, where $V = A \cup B$, $A\cap B=\emptyset$ will be denoted by $G(A,B)$. The balanced complete $r$-partite graph with color classes of size $t$ is denoted by $K_r(t)$. For $A\subseteq V(G)$, $G[A]$ is the restriction of $G$ to $A$. When $A$ and $B$ are subsets of $V(G)$, we denote by $e(A,B)$ the number of edges of $G$ with one endpoint in $A$ and the other in $B$, and by $e(A)=|E(G[A])|$ the number of edges with both endpoints in $A$. Let $\delta(A,B)=\min_{v\in A} \deg(v, B)$. For non-empty $A$ and $B$, $$d(A,B)=\frac{e(A,B)}{|A||B|}$$ is the *density* of the graph between $A$ and $B$. We write $d(A)=2e(A)/|A|^2$. A graph $G$ on $n$ vertices is $\gamma$-*dense* if it has at least $\gamma \binom{n}{2}$ edges. A bipartite graph $G(A,B)$ is $\gamma$-*dense* if it contains at least $\gamma |A||B|$ edges. Throughout the paper $\log$ denotes the base 2 logarithm.
Powers of Cycles
----------------
A classical result of Dirac [@DI] asserts that if $G$ is a graph on $n\geq 3$ vertices with $\delta(G)\geq n/2$, then $G$ contains a Hamiltonian cycle. Note that when $n=2t$, Dirac’s theorem implies that $G$ contains $t$ vertex disjoint copies of $K_2$. In 1963, Corrádi and Hajnal [@CH] proved that if $G$ is a graph on $n=3t$ vertices with $\delta(G)\geq \frac{2n}{3}$, then $G$ contains $t$ vertex disjoint triangles. Generalizing the Corrádi-Hajnal theorem, Erdős conjectured [@E] and Hajnal and Szemerédi later proved [@HS] the following:
\[HSthm\] Let $G$ be a graph on $n=t(k+1)$ vertices. If $\delta(G)\geq \frac{kn}{k+1}$, then $G$ contains $t$ vertex disjoint copies of $K_{k+1}$.
Finally in 1976, Bollobas and Eldridge [@BE], and independently Catlin [@Cat], made a conjecture which would generalize the Hajnal-Szemerédi theorem: If $G$ and $H$ are graphs on $n$ vertices with $\Delta(H)\leq k$ and $\delta(G)\geq \frac{kn-1}{k+1}$, then $H\subseteq G$. While this conjecture is still open in general, we will only be interested in the $k=2$ case which was proved by Aigner and Brandt in 1993 [@AB].
\[ABthm\] Let $G$ and $H$ be graphs on $n$ vertices. If $\Delta(H)\leq 2$ and $\delta(G)\geq \frac{2n-1}{3}$, then $H\subseteq G$.
Note that all of these degree conditions are easily seen to be best possible.
Let $H$ be a graph with vertex set $V$. The $k^{th}$ power of $H$, denoted $H^k$, is defined as follows: $V(H^k) = V$ and $uv\in E(H^k)$ if and only if the distance between $u$ and $v$ in $H$ is at most $k$. When $k=2$ we call $H^2$ the *square of $H$*. Notice that $C_n^{k-1}$ contains ${\left\lfloor\frac{n}{k}\right\rfloor}$ vertex disjoint copies of $K_k$. Furthermore, notice that $C_n^2$ contains every graph $H$ on $n$ vertices with $\Delta(H)\leq 2$ (actually $P_n^2$ also has this property). In 1963, Pósa made a conjecture (see [@E]) that would significantly strengthen the Corrádi-Hajnal theorem (and retroactively Theorem \[ABthm\], see [@FK3]).
\[posa\] Let $G$ be a graph on $n$ vertices. If $\delta(G)\geq\frac{2n}{3}$, then $C_n^2\subseteq G$.
After Erdős’ conjecture became the Hajnal-Szemerédi theorem, Seymour made a conjecture in 1974 [@SE] which generalizes Pósa’s conjecture to handle all values of $k$ (note that for $k\geq 4$, this does not generalize the Bollobás-Eldridge, Catlin conjecture).
\[sey\] Let $G$ be a graph on $n$ vertices. If $\delta(G)\geq\frac{kn}{k+1}$, then $C_n^{k}\subseteq G$.
Starting in the 90’s a substantial amount of progress was made on these conjectures. Jacobson (unpublished) first established that the square of a Hamiltonian cycle can be found in any graph $G$ given that $\delta(G)\geq 5n/6$. Later Faudree, Gould, Jacobson and Schelp [@F2] improved the result, showing that the square of a Hamiltonian cycle can be found if $\delta(G)\geq(3/4+\eps)n$. The same authors further relaxed the degree condition to $\delta(G)\geq3n/4$. Fan and Häggkvist lowered the bound first in [@FH] to $\delta(G) \geq 5n/7$ and then in [@FK1] to $\delta(G)\geq(17n+9)/24$. Faudree, Gould and Jacobson [@F1] further lowered the minimum degree condition to $\delta(G)\geq7n/10$. Then Fan and Kierstead [@FK2] achieved the almost optimal $\delta(G)\geq\left(\frac23+\eps\right)n$. They also proved in [@FK3] that $\delta(G)\geq(2n-1)/3$ is sufficient for the existence of the square of a Hamiltonian [*path*]{}. Finally, they proved in [@FK4] that if $\delta(G)\geq2n/3$ and $G$ contains the square of a cycle with length greater than $2n/3$, then $G$ contains square of a Hamiltonian cycle.
Regarding Conjecture \[sey\], Faudree [*et al*]{} [@F2] proved that for any $\eps > 0$ and positive integer $k$ there is a $C$ such that if $G$ is a graph on $n\geq C$ vertices with $\delta(G)\geq\left(\frac{2k-1}{2k}+\eps\right)n,$ then $G$ contains the $k^{th}$ power of a Hamiltonian cycle.
Using the regularity–blow-up method first in [@KSS4] Komlós, Sárközy and Szemerédi proved Conjecture \[sey\] in asymptotic form, then in [@KSS2] and [@KSS5] they proved both conjectures for $n\geq n_0$. The proofs used the regularity lemma [@SZ1], the blow-up lemma [@KSS3; @KSS6], and the Hajnal-Szemerédi theorem [@HS]. Since the proofs used the regularity lemma the resulting $n_0$ is very large (it involves a tower function). A new proof of Pósa’s conjecture was given by Levitt, Sárközy and Szemerédi [@LSS] which avoided the use of the regularity lemma and thus significantly decreased the value of $n_0$. An explicit bound on $n_0$ was determined by Châu, DeBiasio, and Kierstead in [@CDK]; however, for small $n_0$ the conjecture is still open. Finally, Jamshed and Szemerédi [@JSz] gave a new proof of Seymour’s conjecture that avoided the use of the regularity lemma.
Ore-type generalizations of Dirac-type results
----------------------------------------------
For a pair of non-adjacent vertices $(u,v)$, the value of $\deg(u)+\deg(v)$ is called the Ore-degree of $(u,v)$. We denote by $\delta_2(G)$ the minimum Ore-degree over all non-adjacent pairs of vertices in $G$. In 1960, Ore [@O] proved that if $G$ is graph on $n\geq 3$ vertices with $\delta_2(G) \geq n$, then $G$ contains a Hamiltonian cycle. Since any graph with $\delta(G)\geq \frac{n}{2}$ satisfies $\delta_2(G)\geq n$, Ore’s theorem strengthens Dirac’s theorem. Inspired by this, researchers have sought to generalize minimum degree (“Dirac-type") conditions to Ore-type degree conditions; for a survey of such results see [@KKY].
Two important examples of Ore-type results are the following generalizations of Theorem \[HSthm\] and \[ABthm\].
\[KKthm\] Let $G$ be a graph on $n=t(k+1)$ vertices. If $\delta_2(G)\geq \frac{2kn}{k+1}-1$, then $G$ contains $t$ vertex disjoint copies of $K_{k+1}$.
\[KYthm\] Let $G$ and $H$ be graphs on $n$ vertices. If $\Delta(H)\leq 2$ and $\delta_2(G)\geq \frac{4n}{3}-1$, then $H\subseteq G$.
A natural Ore-type generalization of Pósa’s conjecture suggests that if $\delta_2(G) \geq \frac{4n}{3}$, then $C_n^2\subseteq G$. It turns out that this natural generalization is not quite true as Châu [@C] gave a construction of a graph $G$ for which $\delta_2(G)=\frac{4n}{3}$, but $G$ does not contain the square of a Hamiltonian cycle. However, in the same paper, Châu uses the regularity–blow-up method to prove that if $G$ is a graph on $n\geq n_0$ vertices with $\delta_2(G)>\frac{4n}{3}$, then $C_n^2\subseteq G$. In fact, he is able to give an even more refined degree condition:
\[Cthm\] Let $G$ be a graph on $n$ vertices. If $\delta_2(G)\geq \frac{4n-1}{3}$ and
1. $\delta(G)\leq \frac{n}{3}+2$, then $P_n^2\subseteq G$.
2. $\delta(G)>\frac{n}{3}+2$, then there exists $n_0$ such that if $n\geq n_0$, then $C_n^2\subseteq G$.
(See [@C], Proposition 9.1 for an explanation of why this result actually implies Theorem \[KYthm\] and the $k=2$ case of Theorem \[KKthm\] for sufficiently large $n$ despite the fact that $\frac{4n-1}{3}>\frac{4n}{3}-1$.)
One of the purposes of this paper is to present another proof of Theorem \[Cthm\].(ii) which avoids the use of the regularity lemma, thus resulting in a much smaller value of $n_0$.
\[main\] There exists $n_0$ such that if $G$ is a graph on $n\geq n_0$ vertices with $$\label{minDeg}
\delta_2(G) \geq \frac{4n-1}{3} \mbox{ and } \delta(G) > \frac{n}{3} + 2,$$ then $C_n^2\subseteq G$.
Aside from lowering the bound on $n_0$, we believe that the techniques used in this paper are of independent interest and can have more applications. In particular, our proof provides a simpler template for approaching the following Ore-type version of Conjecture \[sey\].
\[oreseymour\] Let $G$ be a graph on $n$ vertices. If $\delta_2(G)\geq \frac{2kn-1}{k+1}$ and $\delta(G)>\frac{(k-1)n}{k+1}+2$, then $C_n^k\subseteq G$.
Outline of the Proof
--------------------
As is common in these types of problems, our proof is divided into extremal and non-extremal cases. The extremal conditions will resemble the properties found in Figure \[Examples\]; either there is a vertex close to smallest possible degree, or there is a set of size approximately $n/3$ with very few edges. We formally define the extremal conditions below.
Let $0<\alpha\ll\frac{1}{3}$ and let $G$ be a graph on $n$ vertices.
1. We say that $G$ satisfies extremal condition 1 with parameter $\alpha$ if there exists $v\in V(G)$ such that $\deg(v) < (\frac{1}{3}+\alpha)n$.
2. We say that $G$ satisfies extremal condition 2 with parameter $\alpha$ if there exist disjoint sets $A_1, A_2$ such that for $i=1,2$, $|A_i|\geq (1/3-\alpha)n$ and $d(A_i)<\alpha$.
3. We say that $G$ satisfies extremal condition 3 with parameter $\alpha$ if there exists a set $A_1$ such that $|A_1|\geq (1/3-\alpha)n$, $d(A_1)<\alpha$, and for all $A_2\subseteq V(G)\setminus A_1$ with $|A_2|\geq (1/3-\alpha)n$, $d(A_2)\geq \alpha$.
Let $0<\alpha\ll\frac{1}{3}$. If $G$ does not satisfy extremal condition 1,2, and 3 with parameter $\alpha$, then we say $G$ is not $\alpha$-extremal. Specifically, this implies that $\delta(G)\geq (1/3+\alpha)n$ and for all $A\subseteq V(G)$ with $|A|\geq (1/3-\alpha)n$, $d(A)\geq \alpha$.
These extremal cases are dealt with in [@C] without the use of the regularity lemma; however, the blow-up lemma is used in multiple cases. In Section \[EXsection\] we provide an alternate argument which can be used in [@C] to replace each use of the blow-up lemma.
The non-extremal case is where our proof differs most significantly from [@C] and is the main focus of our paper. We avoid the use of the regularity lemma, the blow-up lemma, and Theorem \[KKthm\] by instead using Erdős-Stone type results to cover all but a small fraction of the vertex set with disjoint balanced complete tripartite graphs of size about $\log n$. Then we prove a new connecting lemma which allows us to connect the complete tripartite graphs by square paths. Aside from any leftover vertices, we have a nearly spanning structure which contains a square cycle and is quite robust in the sense that most of the vertices are in complete tripartite graphs of size about $\log n$. Finally, we take advantage of the robustness of our structure by inserting the leftover vertices in such a way that the resulting structure contains the square of a Hamiltonian cycle. All of this will be made precise in Section \[nonex\_section\].
Extremal case {#EXsection}
=============
In [@C], the extremal cases are handled with very detailed, yet elementary arguments – with one exception. In certain cases of [@C], the problem is reduced to finding the square of a Hamiltonian cycle in a balanced tripartite graph where each pair is nearly complete, with the exception of a small number of vertices which still satisfy some minimum degree condition. Here Châu uses the fact that these very dense pairs are $({\varepsilon}, \delta)$-super regular so the blow-up lemma can be applied to show that the desired square cycle exists. However, these dense pairs have a property which is far stronger than the property of being $({\varepsilon}, \delta)$-super regular. Thus, our goal in this section is simply to provide an elementary argument which could be used to replace all of the uses of the blow-up lemma in the extremal cases of [@C]. Note that we will not reproduce the proof found in [@C], as we are only providing a minor diversion to the conclusion of certain cases of the argument.
\[cleanExtCaseLemma1\] Let $0<\alpha' \ll 1$ and let $H$ be a balanced tripartite graph on $3m= n \geq n_0$ vertices with $V(H)$ partitioned as $A_1,A_2,A_3$. If for all $i\neq j$, $\delta(A_i ,A_j) \geq (1-\alpha')m$, then we can cover $V(H)$ by disjoint triangles.
We first find a perfect matching $M_1$ between $A_1$ and $A_2$ by an application of the König-Hall theorem. Then we find a perfect matching between $M_1$ and $A_3$, such that $e=xy\in M_1$ is matched with a vertex $z \in N(x,y,A_3)$. For any edge $e=xy \in M_1$ we have $\deg(x,y,A_3) \geq (1-2\alpha')m$, therefore, by König-Hall theorem there exists a perfect matching between $M_1$ and $A_3$ as desired.
\[cleanExtCaseLemma2\] Let $0<\alpha' \ll 1$ and let $H$ be a balanced tripartite graph on $3m= n \geq n_0$ vertices with $V(H)$ partitioned as $A_1,A_2,A_3$. If $T = \{t_1,t_2,\ldots,t_m\}$ is a triangle cover of $V(H)$ and if for all $i\neq j$, $\delta(A_i, A_j) \geq (1-\alpha')m$, then $H$ contains the square of a Hamiltonian cycle. Furthermore, $H$ contains the square of a Hamiltonian path which starts with $t_1$ and ends with $t_m$.
Let $t=(x_1,x_2,x_3)$ and $t' = (y_1,y_2,y_3)$ be any two triangles in $T$ such that $x_i,y_i \in A_i$. We say that $t$ [*precedes*]{} $t'$, if $x_i$ is adjacent to $y_1,\ldots,y_{i-1}$ for $2 \leq i \leq 3$ (if $t$ [*precedes*]{} $t'$, then $x_1x_2x_3y_1y_2y_3$ is a square-path). We say that $\{t, t'\}$ is a [*good*]{} pair, if $t$ precedes $t'$ and $t'$ precedes $t$. By the degree conditions above, any $t_i\in T$ makes a [*good*]{} pair with at least $(1-\sqrt{\alpha'}) m$ other triangles in $T$.
Make an auxiliary graph $H'$ over $T$ such that each triangle $t_i\in T$ is adjacent to the triangle $t_j$ if and only if $\{t_i, t_j\}$ is a good pair. By the above observation we clearly have $\delta(H') > m/2$, hence by the Dirac’s theorem there is a Hamiltonian cycle in $H'$. Also since $\delta(H') > m/2$, $H'$ is Hamiltonian connected and thus there is a Hamiltonian path in $H'$ which starts with $t_1$ and ends with $t_m$. It is easy to see that this Hamiltonian cycle (path) in $H'$ corresponds to the square of a Hamiltonian cycle (path) in $H$.
Finally we arrive at the main lemma which can be used to replace the use of the blow-up lemma in the extremal cases of [@C].
Let $0<\alpha' \ll \beta\ll \gamma\ll 1$ and let $H$ be a balanced tripartite graph on $3m= n \geq n_0$ vertices with $V(H)$ partitioned as $A_1,A_2,A_3$. If for all $i\neq j$, there are at least $(1-\beta)m$ vertices in $A_i$ with at least $(1-\alpha')m$ neighbors in $A_j$ and $\delta(A_i, A_j)\geq \gamma m$, then $H$ contains the square of a Hamiltonian cycle. Furthermore, if we specify two edges $u_1u_2$ and $u_{3m-1}u_{3m}$ such that for all $u\in \{u_1, u_2, u_{3m-1}, u_{3m}\}$, $\deg(u, A_j)\geq (1-\alpha')m$, then $H$ contains the square of a Hamiltonian path $P=u_1u_2\dots u_{3m-1}u_{3m}$.
Call a vertex $u$ in $A_i$ [*bad*]{} if $u$ has less than $(1-\alpha')m$ neighbors in $A_j$ for some $j\neq i$. By the hypothesis, there are at most $2\beta m$ bad vertices in each $A_i$. Now with a simple greedy procedure, for each bad vertex $u\in A_1$ we find a triangle $t_2 = (b_1,b_2,b_3)$, such that $b_1=u$ and $b_2$ and $b_3$ are typical (not bad) vertices in $A_2$ and $A_3$. We find two more similar triangles $t_1 = (a_1,a_2,a_3)$ and $t_3 = (c_1,c_2,c_3)$, such that $a_2\in N(u)$, $a_3\in N(u,b_2)$ and $c_1\in N(b_2,b_3)$, $c_2\in N(b_3)$. Clearly $a_1a_2a_3b_1b_2b_3c_1c_2c_3$ is a square path. We replace these three triangles with an exceptional triangle $(d_1,d_2,d_3)$ with one vertex each in $A_1$, $A_2$ and $A_3$, such that for $1\leq i\leq 3$, $d_i$ is connected to common neighbors of $a_i$ and $c_i$. By the fact that $a_i$ and $c_i$ are not bad vertices every $d_i$ has at least $(1-3\alpha')m$ neighbors in both of the other two sets. We similarly make an exceptional triangle for each of the remaining bad vertices. Since the total number of bad vertices is at most $6\beta m$ and the minimum degree is $\gamma m\gg 6\beta m$, this greedy procedure can be easily carried out. In the remaining parts of $A_1$, $A_2$, and $A_3$ by Lemma \[cleanExtCaseLemma1\] we find a triangle cover and add all the exceptional triangles to the cover. Then by Lemma \[cleanExtCaseLemma2\], we find the square of a Hamiltonian cycle.
Now suppose $u_1u_2$ and $u_{3m-1}u_{3m}$ are given edges such that for all $u\in \{u_1, u_2, u_{3m-1}, u_{3m}\}$, $\deg(u, A_j)\geq (1-\alpha')m$ for all $A_j$ such that $u\notin A_j$. We make $t_1 = (u_1,u_2,u_3)$ and $t_m = (u_{3m-2},u_{3m-1},u_{3m})$ such that $u_3$ is a typical vertex in $N(u_1,u_2)$ and $u_{3m-2}$ is a typical vertex in $N(u_{3m-1},u_{3m})$. Now by applying Lemma \[cleanExtCaseLemma1\] we find a triangle cover and add all the exceptional triangles to the cover. Then by Lemma \[cleanExtCaseLemma2\], we find the square of a Hamiltonian path which starts with $t_1$ and ends with $t_m$.
Non-extremal case {#nonex_section}
=================
Before we give an overview of the non-extremal case, it would be helpful to have some idea of how the non-extremal case is proved in [@C] (which is a generalization of the arguments in [@KSS2], [@KSS4], [@KSS5]). Suppose $G$ is a non-extremal graph on $n$ vertices ($n$ sufficiently large) with $\delta_2(G)\geq \frac{4n}{3}$. Using the regularity lemma and Theorem \[KKthm\], one can show that $G$ contains a set of disjoint balanced $4$-partite and $3$-partite graphs spanning almost all of $G$ each having size $\Omega(n)$. Each of these multi-partite graphs $H$ has the property that every pair of color classes forms a suitably dense psuedorandom bipartite graph, so by applying the blow-up lemma, one obtains an almost spanning square path in $H$. If we connect these multi-partite graphs together with square paths before applying the blow-up lemma, one will obtain an almost spanning square path of $G$. Finally the remaining vertices need to somehow be inserted, which is an elementary, but detailed argument.
We are able to avoid the regularity–blow-up method by showing that for sufficiently large $n$ (but nowhere near as big as needed for the regularity lemma), $G$ can be partitioned into disjoint balanced complete tripartite graphs spanning almost all of $G$, each having size $\Omega(\log n)$; we call this “the cover" and it is built in Section \[sec:cover\]. Since the tripartite graphs are complete, we do not have to apply the blow-up lemma; if we go around a complete tripartite graph picking vertices from each of the color classes sequentially we get a square-path. Next we must prove a Connecting Lemma which allows us to connect the tripartite graphs by short square-paths giving us a “cycle of cliques"; this is done in Section \[sec:connecting\]. At the end of this process there will be a few leftover vertices which need to be inserted; this is done in Section \[sec:inserting\].
Here is the statement of the non-extremal case (notice that in the non-extremal case we are able to slightly relax the Ore-degree condition).
\[thm:nonextremal\] For all $0<{\varepsilon}\ll \alpha\ll 1$ there exists $n_0$ such that if $G$ is a graph on $n\geq n_0$ vertices with \[minDegree\_nonext\] \_2(G)(-2)n and $G$ is not $\alpha$-extremal, then $C_n^2\subseteq G$.
Given a graph $G$ with $\delta_2(G)\geq (\frac{4}{3}-2{\varepsilon})n$, let $L$ be the set of vertices in $G$ with degree less than $(2/3-{\varepsilon})n$. We say that the vertices in $L$ are *low-degree vertices* and the vertices in $V(G)\setminus L$ are *high-degree vertices*. Note that by the definition of $\delta_2(G)$, $G[L]$ is a clique.
The Cover {#sec:cover}
---------
In order to cover most of the vertices in $G$ with complete tripartite graphs as mentioned above, we will need quantitative versions of some classical results in extremal graph theory.
### Lemmas {#sec:lemmas}
\[denseSubgraph\] Let $0<d,\gamma<1$. If $G(A,B)$ is a $(d+2\gamma)$-dense bipartite graph, then there must be at least $\gamma|B|$ vertices in $B$ for which the degree in $A$ is at least $(d+\gamma)|A|$.
Indeed, otherwise the total number of edges would be less than $$(d+\gamma) |A|\cdot|B| + \gamma |B|\cdot|A| = (d+2\gamma) |A||B|$$ a contradiction to the fact that $G(A,B)$ is $(d+2\gamma)$-dense.
\[rpartVolArg\] Let $0<c,\gamma<1/3$, $s={\left\lfloorc\log n\right\rfloor}$, and let $G$ be a graph on $n\geq n_0$ vertices with $K:=K_3(s)=(A_1,A_2,A_3)\subseteq G$. If $B\subseteq V(G)\setminus V(K)$ with $|B|\geq \gamma n$ and $d(B,K)\geq \frac{2}{3} + 2\gamma$, then $G[B\cup V(K)]$ contains a $K':=K_4(\gamma s) = (A_1',A_2',A_3',B')$, where $A_i'\subset A_i$ and $B'\subset B$.
Let $B_1=\{v\in B: \deg(v, K)\geq (\frac{2}{3} + \gamma)|V(K)|\}$. By Fact \[denseSubgraph\] we have $|B_1|\geq \gamma |B|\geq \gamma^2 n$. By the degree condition, each vertex in $B_1$ has at least $\gamma s$ neighbors in each $A_i$. There are at most $2^{|V(K)|}=2^{3s} \le n^{3c}$ different possible neighborhoods, so by averaging there must be a neighborhood that appears for a set $B_2$ of at least $\frac{|B_1|}{n^{3c}} \geq \frac{\gamma^2n }{n^{3c}}=\gamma^2 n^{(1-3c)}$ vertices of $B_1$. Selecting an appropriate subset $B'$ of $B_2$, we get the desired complete $K_4(\gamma s)$.
We need a version of the Erdős-Stone theorem where we have control of the parameters. While there are a sequence of improvements by Bollobás-Erdős, Bollobás-Erdős-Simonovits, and Bollobás-Kohayakawa (to name a few), we will state a version due to Nikiforov [@Nik1] which gives an explicit lower bound on $n$.
\[BES\] Let $c$ and $n$ be such that $0<c<1$ and $n\geq \exp(64/c^3)$, and let $G$ be a graph on $n$ vertices. If $e(G)\geq (\frac{1}{2}+c)\frac{n^2}{2}$, then $G$ contains $K_{3}(s)$ where $s={\left\lfloor\frac{c^3}{64}\log n\right\rfloor}$.
Finally, we need a simple fact which allows us to translate our Ore-degree condition into an appropriate edge density condition.
\[OreEdges\] Let $0<d<1$ and let $G$ be a graph on $n\geq 2$ vertices. If $\delta_2(G) \geq 2dn$, then $e(G) \geq d\frac{n^2}{2}$.
Define $\gamma$ so that $e(G)=\gamma\binom{n}{2}$ and suppose $\delta_2(G) \geq 2d(n-1)$. We have $$\begin{aligned}
2d(n-1)(1-\gamma)\binom{n}{2}\leq \sum_{\{u,v\}\not\in E(G)}(\deg(u)+\deg(v))&=\sum_{v\in V(G)}\deg(v)(n-1-\deg(v))\\
&=2\gamma\binom{n}{2}(n-1)-\sum_{v\in V(G)}(\deg(v))^2\\
&\leq 2\gamma\binom{n}{2}(n-1)-\frac{1}{n}\left(\sum_{v\in V(G)}\deg(v)\right)^2\\
&= 2\gamma\binom{n}{2}(n-1)-\frac{4\gamma^2\binom{n}{2}^2}{n}.\end{aligned}$$ Dividing both sides by $2(n-1)(1-\gamma)\binom{n}{2}$ gives $\gamma\geq d$, and thus $\delta_2(G) \geq 2d(n-1)$ implies $e(G)\geq d\binom{n}{2}$. Thus if $\delta_2(G)\geq 2dn=2\frac{dn}{n-1}(n-1)$, then $e(G)\geq \frac{dn}{n-1}\binom{n}{2}=d\frac{n^2}{2}$ as stated.
### Building the cover
Let $s, n'\in \mathbb{R}^+$. A $(s, n')$ tripartite cover is a collection ${{\mathcal T}}$ of vertex disjoint copies of $K_3(t_i)=:T_i$ with ${\left\lfloors\right\rfloor}\leq t_i\leq {\left\lceil2s\right\rceil}$ such that $|\bigcup_{T_i\in {{\mathcal T}}}V(T_i)|\geq n'$.
Note that in the following lemma we do not assume that $G$ is non-extremal.
\[lem:cover\] For all $0<{\varepsilon}\ll \eta\ll 1$, there exists $n_0$ and $c>0$ such that if $G$ is a graph on $n\geq n_0$ vertices with $\delta_2(G)\geq (\frac{4}{3}-2{\varepsilon})n$, then $G$ contains a $(c\log n, (1-\eta)n)$ tripartite cover.
Set $c_0=\frac{\eta^6}{64}$ and $t_0={\left\lfloorc_0\log n\right\rfloor}$. By (\[minDegree\_nonext\]) and Fact \[OreEdges\] we have $e(G) \geq (\frac{2}{3}-{\varepsilon})\frac{n^2}{2}$. We repeatedly apply Lemma \[BES\] (with $c=\eta^2$) to find complete tripartite graphs with each color class of size $t_0$ until the remaining graph contains no copy of $K_3(t_0)$. Let ${{\mathcal T}}$ be the collection of tripartite graphs obtained in this way, and let $U=V(G)\setminus V({{\mathcal T}})$, where $V({{\mathcal T}})=\bigcup_{T\in {{\mathcal T}}}V(T)$. If $|U| < \eta n$, then we are done, so suppose $|U|\geq \eta n$.
Set $U_0=U$, ${{\mathcal T}}_0={{\mathcal T}}$, and for $i\geq 0$ set $c_i=\eta^{2i}c_0=\eta^{2i+6}/64$ and $t_i={\left\lfloorc_i\log n\right\rfloor}$.
\[enlargecover\] If $|U_i|\geq \eta n$, then either
1. $G[U_i]$ contains $K:=K_3(t_i)$, in which case we reset $U_i:=U_i\setminus V(K)$ and ${{\mathcal T}}_{i}:={{\mathcal T}}_i \cup K$ or
2. $G[U_i]$ does not contain a copy of $K_3(t_i)$, in which case there exists a cover ${{\mathcal T}}_{i+1}$ such that $|V({{\mathcal T}}_{i+1})|\geq |V({{\mathcal T}}_i)|+\eta^4n$ and every color class in the cover has size between $t_{i+1}$ and $2t_{i+1}$.
If Claim \[enlargecover\] holds, then for some $j\leq \frac{1}{\eta^4}$, we must have $|U_j|<\eta n$ (as at least $\eta^4 n$ vertices are added to the cover before we increase the index). Note that $c_j\geq \eta^{2j}\eta^6/64\geq \eta^{\frac{2}{\eta^4}+6}/64=:c$ and thus the balanced complete tripartite graphs in ${{\mathcal T}}_j$ have parts of size between ${\left\lfloorc\log n\right\rfloor}$ and ${\left\lceil2c\log n\right\rceil}$ as required. We now finish the proof of the cover lemma by proving Claim \[enlargecover\].
Let $0\leq i\leq \frac{1}{\eta^4}$ and suppose $G[U_i]$ does not contain a copy of $K_3(t_i)$. In this case by Lemma \[BES\] \[Udensity\] d(U\_i)<(+\^[(2i+6)/3]{})(+\^2).
Start by setting $Z=\emptyset$. We will consider each $T\in {{\mathcal T}}_i$ one by one. If $d(U_i,T) < (\frac{2}{3}+6\eta^2)$, then consider the next element of ${{\mathcal T}}_i$. If $d(U_i,T) \geq (\frac{2}{3}+6\eta^2)$, then by Lemma \[rpartVolArg\] there exists $K_{4}(3\eta^2 t_i)$ in $G[U_i\cup V(T)]$, which can be split into four copies of $K_3(\eta^2 t_i)=K_3(t_{i+1})$. Move the used vertices from $U_i$ into $Z$ and reset $U_i:=U_i\setminus Z$. Let ${{\mathcal T}}_i'$ be the set of $3$-partite graphs in ${{\mathcal T}}_i$ for which the procedure succeeded. If $|{{\mathcal T}}_i'|\geq \eta^2\frac{n}{3t_i}$, then we will have increased the cover by at least $3\eta^2 t_i\cdot\eta^2\frac{n}{3t_i} =\eta^4 n$. If $|U_i|<\eta n$ or we have increased the cover by $\eta^4n$, we partition each color class into parts of size at least $t_{i+1}$ (which implies that all parts have size at most $2t_{i+1}$).
So suppose we have increased the cover by less than $\eta^4n$ and we still have $|U_i|\geq \eta n$. In this case we have $|{{\mathcal T}}_i'|<\eta^2\frac{n}{3t_i}$ which implies \[T’\] |V([[T]{}]{}\_i’)Z|= |[[T]{}]{}\_i’|(3t\_i+3\^2t\_i)<3(1+\^2)t\_i\^2<2\^2n.
For every $T\in {{\mathcal T}}_i\setminus {{\mathcal T}}_i'$, we have $$\begin{aligned}
e(U_i, T) \leq
(\frac{2}{3}+6\eta^2)|V(T)||U_i| \label{UtoT}.\end{aligned}$$
Now by and we have $$\begin{aligned}
e(U_i, V({{\mathcal T}}_i)\cup Z)=e(U_i,V({{\mathcal T}}_i')\cup Z) + e(U_i, V({{\mathcal T}}_i\setminus {{\mathcal T}}_i'))
&\leq 2\eta^2n |U_i| + (\frac{2}{3}+6\eta^2)|V({{\mathcal T}}_i)||U_i|\notag \\
&\leq (\frac{2}{3}|V({{\mathcal T}}_i)| + 8\eta^2 n)|U_i|\label{UtoOutside}\end{aligned}$$
Recall that $G[L]$ (the graph induced by the low-degree vertices) induces a clique and since $G[U_i]$ contains no $K_3(t_i)$, we have $|L\cap U_i|< 3t_i<{\varepsilon}|U_i|<{\varepsilon}n$. Also note that $n=|U_i|+|Z|+|V({{\mathcal T}}_i)|\geq |U_i|+|V({{\mathcal T}}_i)|$. Now we get $$\begin{aligned}
e(U_i, V({{\mathcal T}}_i)\cup Z)=\sum_{v\in U_i} \deg(v) - 2 e(U_i) &\stackrel{\eqref{Udensity}}{\geq} \sum_{v\in U_i\setminus L} (\frac{2}{3}-\eps)n - 2(\frac{1}{2}+\eta^2)\frac{|U_i|^2}{2} \notag\\
&\geq ((1-{\varepsilon})(\frac{2}{3}-\eps)n - (\frac{1}{2}+\eta^2)|U_i|)|U_i|\notag\\
&\geq (\frac{2}{3}|V({{\mathcal T}}_i)| + \frac{2}{3}|U_i|- 2\eps n - (\frac{1}{2}+\eta^2)|U_i|)|U_i|\notag\\
&\geq (\frac{2}{3}|V({{\mathcal T}}_i)| + \frac{1}{6}|U_i|- 2\eps n -\eta^2|U_i|)|U_i|\notag\\
&\geq (\frac{2}{3}|V({{\mathcal T}}_i)| + \frac{1}{6}\eta n- 2\eps n -\eta^2|U_i|)|U_i|
\label{UtoOutside_lower}\end{aligned}$$
By and , we have $(\frac{2}{3}|V({{\mathcal T}}_i)|+\frac{1}{7}\eta n)|U_i|\leq e(U_i, V({{\mathcal T}}_i)\cup Z)\leq (\frac{2}{3}|V({{\mathcal T}}_i)| + 8\eta^2 n)|U_i|$, a contradiction.
Connecting {#sec:connecting}
----------
In this section we will prove that if $G$ is non-extremal, then we can find a short square path between any two disjoint ordered edges provided that each edge consists of high degree vertices or has a triangle in the common neighborhood of the endpoints. This Lemma will then be used to connect the tripartite graphs coming from Lemma \[lem:cover\].
### Connecting ordered edges
First note the following simple fact.
\[4or5edges\] Given disjoint triangles $T$ and $T'$, either there exists an ordering of the vertices of $T=x_3x_1x_2$ and $T'=y_1y_2y_3$ such that $x_3x_1x_2y_1y_2y_3$ is a square path, or there exist vertices $x_1,x_2\in T$ and $y_1,y_2\in T'$ such that $x_1\not\sim y_1$ and $x_2\not\sim y_2$.
One very special case of a result of Faudree and Schelp [@FS] says that in every $2$-coloring of $K_{3,3}$ there is either a red path on $4$ vertices or a blue path on $4$ vertices (this special case is easily checked). Applying this to the induced bipartite graph between $T$ and $T'$ implies that (with the appropriate labeling of the vertices) $x_1y_1$, $y_1x_2$, and $x_2y_2$ are either all edges or all non-edges; the former implies that $x_3x_1x_2y_1y_2y_3$ is a square path and the latter implies that $x_1\not\sim y_1$ and $x_2\not\sim y_2$.
\[generalconnecting\] Let $\frac{1}{n_0}\ll {\varepsilon}\ll \alpha\ll 1$ and let $G$ be a graph on $n\geq n_0$ vertices with $\delta_2(G)\geq (\frac{4}{3}-4{\varepsilon})n$ such that $G$ is not $\alpha$-extremal. For all distinct $u_1,u_2,v_1,v_2\in V(G)$ with $u_1u_2, v_1v_2\in E(G)$, if
1. $\deg(u_1), \deg(u_2) \geq (\frac{2}{3}-2{\varepsilon})n$ or there exists a triangle $T$ in $G[N(u_1,u_2)\setminus \{v_1,v_2\}]$, and
2. $\deg(v_1), \deg(v_2) \geq (\frac{2}{3}-2{\varepsilon})n$ or there exists a triangle $T'$ in $G[N(v_1,v_2)\setminus \{u_1,u_2\}]$,
then there exists $Q\subseteq G-\{u_1,u_2,v_1,v_2\}$ such that $P=u_1u_2Qv_1v_2$ is a square path with $|V(Q)|\leq 18$.
Suppose first that there exists a triangle $T$ in $G[N(u_1,u_2)\setminus \{v_1,v_2\}]$ and there exists a triangle $T'$ in $G[N(v_1,v_2)\setminus \{u_1,u_2\}]$; let $G'=G-T- T'-\{u_1,u_2,v_1,v_2\}$. If $T$ and $T'$ are vertex disjoint, then by Fact \[4or5edges\], we either immediately find a square path from $T$ to $T'$ or there exist two disjoint non-adjacent pairs of vertices in $T\times T'$. Let $(x_i,y_i)$ and $(x_j,y_j)$ be two such pairs and define $C_{i,j}=\{v\in V(G'): \deg(v, \{x_i,y_i,x_j,y_j\})=4\}$. Consider two disjoint non-edges $(x_i,y_i)$, $(x_j,y_j)$ such that $|C_{i,j}|$ is maximum. We may label the vertices of $T$ as $x_1,x_2,x_3$ and the vertices of $T'$ as $y_1,y_2,y_3$ such that the disjoint non-edges which maximize $|C_{i,j}|$ are $(x_1,y_1)$ and $(x_2,y_2)$; i.e., $C_{i,j}=C_{1,2}$. Let $A=\{v\in V(G): \deg(v,\{x_1,x_2\})=2$, $\deg(v, \{y_1, y_2\})=1\}$ and $B=\{v\in V(G): \deg(v,\{x_1,x_2\})=1$, $\deg(v, \{y_1, y_2\})=2\}$. Set $C:=C_{1,2}$ and note that $A, B$ and $C$ are disjoint. Since $x_1\not\sim y_1$ and $x_2\not\sim y_2$, we have $$\label{x1x2y1y2}
\deg(\{x_1,y_1,x_2,y_2\})\geq 2\left(\frac{4}{3}-4{\varepsilon}\right)n= \left(\frac{8}{3}-8{\varepsilon}\right)n.$$ Also we have $$\deg(\{x_1,y_1,x_2,y_2\})\leq 4|C|+3(|A|+|B|)+2(n-|A|-|B|-|C|).$$ Together this gives $$\label{AB2C}
|A|+|B|+2|C|\geq \left(\frac{2}{3}-8{\varepsilon}\right)n.$$ If $T$ and $T'$ have an edge $e$ in common, then set $Q=e$ and note that $u_1u_2Qv_1v_2$ is the desired square path with $|Q|= 2$. If $T$ and $T'$ have exactly one vertex in common, call it $z$. Now, if there exist vertices $x\in V(T)$ and $y\in V(T')$, both distinct from $z$, such that $xy\in E(G)$, then set $Q=xzy$ and note that $u_1u_2Qv_1v_2$ is the desired square path with $|Q|= 3$. So if $T$ and $T'$ have exactly one vertex in common, we can label the vertices of $T$ as $x_1,x_2,x_3$ and the vertices of $T'$ as $y_1,y_2,y_3$ such that $x_3=y_3$ and $x_1\not\sim y_1$ and $x_2\not\sim y_2$. Letting $G'=G-\{u_1,u_2,v_1,v_2\}-T-T'$, we may define $A$, $B$, and $C$ as above, with and consequently holding.
Now suppose, without loss of generality, that there is no triangle in $G[N(u_1,u_2)\setminus \{v_1v_2\}]$ but there is a triangle $T'$ in $G[N(v_1,v_2)\setminus \{u_1,u_2\}]$. We have $|N(u_1,u_2)\setminus L|\geq (\frac{1}{3}-8{\varepsilon})n-|L|>(\frac{1}{3}-\alpha)n$ and since we are not in the extremal case, we have an edge $x_1'x_2'\in G[N(u_1,u_2)\setminus L]-T'$. If there is a triangle in $G[N(x_1',x_2')\setminus \{u_1,u_2,v_1,v_2\}]$, then we call it $T$ and proceed as in the first paragraph, noting that in this case $Q$ must begin with $x_1'x_2'$. So suppose there is no triangle in $G[N(x_1',x_2')\setminus \{u_1,u_2,v_1,v_2\}]$. This implies $|N(x_1',x_2')\setminus L|\geq (\frac{1}{3}-8{\varepsilon})n-|L|>(\frac{1}{3}-\alpha)n$ and since we are not in the extremal case, we have an edge $x_1''x_2''\in G[N(x_1',x_2')\setminus L]$. Again, if there is a triangle in $G[N(x_1'',x_2'')\setminus \{u_1,u_2,x_1',x_2',v_1,v_2\}]$, then we call it $T$ and proceed as in the first paragraph, noting that in this case $Q$ must begin with $x_1'x_2'x_1''x_2''$. So suppose there is no triangle in $G[N(x_1'',x_2'')\setminus \{u_1,u_2,v_1,v_2,x_1',x_2'\}]$. Suppose $x_1''$ or $x_2''$, say $x_1''$, has $3$ neighbors in $T\rq{}$. If there exists $i\in [2]$ such that $x_i'$ has at least one neighbor in $T'$, call it $y_1$ and let $y_2$ be a distinct vertex in $T'$, we may set $Q=x_{3-i}'x_i'x_1''y_1y_2$ and note that $u_1u_2Qv_1v_2$ is the desired square path with $|Q|=5$. On the other hand, if $x_1''$ has $3$ neighbors in $T'$, but $x_1'$ and $x_2'$ have no neighbors in $T'$, then we set $x_1:=x_1'$ and $x_2:=x_2'$ and with $A$, $B$, $C$ defined as before, and consequently holds. So suppose $x_1''$ and $x_2''$ each have less than $3$ neighbors in $T'$. Either $x_1''$ and $x_2''$ have the same two neighbors in $T'$, say $y_1',y_2'$, in which case we set $Q=x_1'x_2'x_1''x_2''y_1'y_2'$, giving us the desired square path $u_1u_2Qv_1v_2$ with $|Q|=6$, or else there exists $y_1,y_2\in T'$ such that with $x_1:=x_1''$ and $x_2:=x_2''$, we have $x_1\not\sim y_1$ and $x_2\not\sim y_2$. So with $A$, $B$, $C$ defined as before, and consequently holds.
Finally suppose that there is no triangle in $G[N(u_1,u_2)\setminus \{v_1v_2\}]$ and no triangle in $G[N(v_1,v_2)\setminus \{u_1,u_2\}]$; this implies that $|N(u_1,u_2)\cap L|< 3$ and $|N(v_1,v_2)\cap L|< 3$ . So we have $|N(u_1,u_2)\setminus L|,|N(v_1,v_2)\setminus L|\geq (\frac{1}{3}-8{\varepsilon})n-|L|>(\frac{1}{3}-\alpha)n$ and since we are not in the extremal case, we have an edge $x_1'''x_2'''\in G[N(u_1,u_2)\setminus L]$ and a disjoint edge $y_1'''y_2'''\in G[N(v_1,v_2)\setminus L]$. If either $x_1''',x_2'''$ has a triangle in their common neighborhood or $y_1''',y_2'''$ has a triangle in their common neighborhood, we proceed as in one of the previous cases, noting that we append $x_1'''x_2'''$ or $y_1'''y_2'''$ to the beginning or end of $Q$ respectively. So suppose not; in this case we set $x_i:=x_i'''$ and $y_i:=y_i'''$ for all $i\in [2]$. Letting $G'=G-\{u_1,u_2,v_1,v_2,x_1,x_2,y_1,y_2\}$, we may define $A$, $B$, and $C$ as before and since $x_1,x_2,y_1,y_2\not\in L$, and consequently holds.
With the initial segments of the square path now in place, suppose there exists a square path $Q'$ having at most 10 vertices starting with either direction of $x_1x_2$ and ending with either direction of $y_1y_2$. From the cases above, we may have to append at most 6 vertices to the beginning of $Q'$ and at most $2$ vertices to the end of $Q'$, giving us the desired square path $P=u_1u_2Qv_1v_2$ with $|Q|\leq 18$.
\[ABCempty\] If following conditions do not hold, then we have the desired square path $Q'$.
1. $d(C)=0$, $d(B,C)=0$, $d(A,C)=0$, and $d(A,B)=0$; and
2. for all $S\in \{A,B,C\}$, if $d(S)=0$, then $|S|\leq 1$. In particular, $|C|\leq 1$.
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1. From the definition of $A$, $B$, $C$, if any of the given edge sets were non-empty, we would immediately have a square path $Q'$.
2. First suppose $|A|+|B|\leq \alpha n$. Then by , we have $$|C|\geq \frac{1}{2}\left(\frac{2}{3}-8{\varepsilon}-\alpha\right)n\geq \left(\frac{1}{3}-\alpha\right)n.$$ Since we are not in the extremal case, we have $d(C)>0$ contradicting Claim \[ABCempty\](i). So we may suppose $$\label{A+B}
|A|+|B|>\alpha n$$
Let $S\in \{A,B,C\}$ with $d(S)=0$ and suppose $|S|\geq 2$. From Claim \[ABCempty\](i), we have $\deg(v, A\cup B\cup C)=0$ for all $v\in S$. Let $v_1,v_2\in S$ and since $v_1\not\sim v_2$ we have $\deg(v_1)+\deg(v_2)\geq \left(\frac{4}{3}-4{\varepsilon}\right)n$. But now we have the following contradiction $$\begin{aligned}
\left(\frac{4}{3}-4{\varepsilon}\right)n\leq \deg(v_1)+\deg(v_2)&\leq 2(n-|A|-|B|-|C|)\\
&\leq 2n-(|A|+|B|+2|C|)-(|A|+|B|)\\
&\leq 2n-\left(\frac{2}{3}-8{\varepsilon}\right)n-\alpha n ~~~\text{(by \eqref{AB2C} and \eqref{A+B})}\\
&<\left(\frac{4}{3}-4{\varepsilon}\right)n.\end{aligned}$$
Note that since $d(C)=0$ by Claim \[ABCempty\](i), we have $|C|\leq 1$ by Claim \[ABCempty\](ii).
From this point we assume that the conditions of Claim \[ABCempty\] hold, as otherwise we would be done. By Claim \[ABCempty\](ii) and , we have $$\label{ABbig}
|A|+|B|\geq \left(\frac{2}{3}-8{\varepsilon}\right)n-2|C|\geq \left(\frac{2}{3}-9{\varepsilon}\right)n.$$
We consider two cases based on the edge density of $A$ and $B$.
**Case 1:** $d(A)>0$ and $d(B)>0$
Let $a_1a_2\in G[A]$ and $b_1b_2\in G[B]$. By Claim \[ABCempty\](i), $a_1\not\sim b_1$ and $a_2\not\sim b_2$. Thus $|N(a_1, b_1)|,|N(a_2, b_2)|\geq \left(\frac{1}{3}-8{\varepsilon}\right)n$. Furthermore, by Claim \[ABCempty\](i), $|N(a, b)\cap (A\cup B\cup C)|=0$ for all $a\in A$, $b\in B$. Combining this with gives $$\begin{aligned}
|N(a_1,b_1,a_2,b_2)|\geq 2\left(\frac{1}{3}-8{\varepsilon}\right)n-(n-|A\cup B\cup C|)&\geq \left(\frac{2}{3}-16{\varepsilon}\right)n-\left(\frac{1}{3}+9{\varepsilon}\right)n \\
&>\left(\frac{1}{3}-\alpha\right)n.\end{aligned}$$ Since we are not in the extremal case, we have an edge $d_1d_2\in G[N(a_1,b_1,a_2,b_2)]$ which gives the desired square path $Q'=x_1x_2a_1a_2d_1d_2b_1b_2y_1y_2$ with $|Q'|\leq 10$.
**Case 2:** $d(A)=0$ or $d(B)=0$.
If $d(B)=0$, then by Claim \[ABCempty\](ii), $|B|\leq 1$ and since $|C|\leq 1$, implies \[Abig\] |A|(-8)n-|B|-2|C|(-9)n. Likewise if $d(A)=0$, then \[Bbig\] |B|(-8)n-|A|-2|C|(-9)n.
Recall that so far we have constructed square paths $u_1u_2\dots \{x_1,x_2\}$ and $\{y_1,y_2\}\dots v_1v_2$ and we are attempting to construct $Q'$ to complete the square path from some ordering of $x_1x_2$ to some ordering of $y_1y_2$. If $x_1x_2$ is in a triangle $T$ in the common neighborhood of the two vertices preceding $x_1x_2$ on $u_1u_2\dots x_1x_2$, we say that $x_1x_2$ is *supported by $T$*. Likewise, if $y_1y_2$ is in a triangle $T'$ in the common neighborhood of the two vertices following $y_1y_2$ on $y_1y_2\dots v_1v_2$, we say that $y_1y_2$ is *supported by $T'$*. This distinction is important, because if say $x_1x_2$ is not supported by $T$, then based on the initial construction this implies that $x_1,x_2$ are high degree vertices and have no triangle in their common neighborhood.
**Case 2.1:** $d(A)=0$ and $x_1x_2$ is supported by $T$, or $d(B)=0$ and $y_1y_2$ is supported by $T'$.
Without loss of generality, suppose $d(B)=0$ and $y_1y_2$ is supported by $T'$. By and $\delta(G)\geq (1/3+\alpha)n$, we have $$|A\cap N(y_3)|\geq (\alpha - 9{\varepsilon})n\geq 3.$$ First suppose there exists $i\in \{1,2\}$ such that $y_3\sim x_i$. Let $a\in N(y_3)\cap A$ and let $j\in \{1,2\}$ such that $a\sim y_j$ (by definition of $A$). Thus $Q'=x_{3-i}x_iay_3y_jy_{3-j}$ is the desired square path with $|Q'|\leq 6$. So suppose that $y_3\not\sim x_1$ and $y_3\not\sim x_2$ (if $x_1x_2$ is supported by $T$, this of course implies that we are in the case where $T$ and $T'$ do not have a vertex in common). Since $|N(y_3)\cap A|\geq 3$, there exists $j\in \{1,2\}$ such that $y_j\sim a$ and $y_j\sim a'$ for distinct $a,a'\in N(y_3)\cap A$. Furthermore, since $a,a'\in A$, we have $\{x_1,x_2,y_3,y_j\}\subseteq N(a)\cap N(a')$, but since $|C|\leq 1$ the non-adjacent pairs $(x_j, y_j)$ and $(x_{3-j}, y_3)$ along with $a,a'$ contradict the maximality of $|C|$.
**Case 2.2:** $d(A)=0$ and $x_1x_2$ is not supported by $T$, or $d(B)=0$ and $y_1y_2$ is not supported by $T'$.
Without loss of generality, suppose $d(A)=0$ and $x_1x_2$ is not supported by $T$. By , we have $|B|\geq (\frac{2}{3}-9{\varepsilon})n$. By the case we have $|N(x_1,x_2)|\geq (\frac{1}{3}-4{\varepsilon})n$ and there exists $z\in N(x_1,x_2)\setminus L$. Since $\deg(z)\geq (\frac{2}{3}-2{\varepsilon})n$ and $|B|\geq (\frac{2}{3}-9{\varepsilon})n$, we have $|N(z)\cap B|\geq (\frac{1}{3}-11{\varepsilon})n>(\frac{1}{3}-\alpha)n$. As $G$ is not $\alpha$-extremal, there exists $b_1b_2\in E(G[N(z)\cap B])$. Since $b_1\in B$, there exists $i\in\{1,2\}$ such that $b_1\sim x_i$. Thus $Q'=x_{3-i}x_izb_1b_2y_1y_2$ is the desired square path with $|Q'|\leq 7$.
### Connecting complete tripartite graphs
Let $q, s ,n'\in \mathbb{R}^+$. A $(q, s, n')$ connected tripartite cover is a $(s, n')$ tripartite cover $\{K^0, \dots, K^{m-1}\}$ together with a collection of $m$ vertex disjoint square paths $\{P_0, \dots, P_{m-1}\}$ such that for all $0\leq i\leq m-1$, $P_i=u_1u_2u_3Qv_1v_2v_3$ is a square path from $K^i=(U_1,U_2,U_3)$ to $K^{i+1}=(V_1,V_2,V_3)$ (addition modulo $m$), such that $u_i\in U_i$, $v_i\in V_i$ for all $i\in [3]$ with $|Q|\leq q$ and the vertices of $Q$ are disjoint from the vertices of $\bigcup V(K^i)$.
Note that a $(q, s, n')$ connected tripartite cover contains a square cycle on at least $n'$ vertices (by “winding around" each balanced complete tripartite graph and using each $P_i$ to get to the next tripartite graph).
\[coverconnecting\] For all $0<{\varepsilon}\ll \alpha\ll 1$ there exists $n_0$ such that if $G$ is a graph on $n\geq n_0$ vertices with $\delta_2(G)\geq (\frac{4}{3}-4{\varepsilon})n$ and $G$ is not $\alpha$-extremal, then the following statement holds. Given disjoint balanced complete tripartite subgraphs $K=(U_1,U_2,U_3)$ and $K'=(V_1,V_2,V_3)$ of $G$ with color classes of size at least 19, there exists a square path $P=u_1u_2u_3Qv_1v_2v_3$ where $u_i\in U_i$, $v_i\in V_i$ for all $i\in [3]$, such that $|Q|\leq 18$.
First let $u_1\in U_1$ and $v_3\in V_3$. If possible, we choose $u_2\in U_2\setminus L$ and $u_3\in U_3\setminus L$. If this is not possible, then $U_i\subseteq L$ for some $i=2,3$, in which case we let $T$ be a triangle in $U_i$ and note that every vertex of $T$ is a neighbor of both $u_2$ and $u_3$. Likewise, if possible, we choose $v_1\in V_1\setminus L$ and $v_2\in V_2\setminus L$. If this is not possible, then $V_j\subseteq L$ for some $j=1,2$, in which case we let $T'$ be a triangle in $V_j$ and note that every vertex of $T'$ is a neighbor of both $v_1$ and $v_2$. Now applying Lemma \[generalconnecting\] gives a square path $u_1u_2u_3Qv_1v_2v_3$ with $|Q|\leq 18$. Note that since $Q$ uses at most 18 vertices, in the worst case, we would use at most $19$ vertices from a single color class.
\[connectedcover\] For all $0<{\varepsilon}\ll \eta\ll \alpha\ll 1$ there exists $n_0$ and $c>0$ such that if $G$ is a graph on $n\geq n_0$ vertices with $\delta_2(G)\geq (\frac{4}{3}-2{\varepsilon})n$ and $G$ is not $\alpha$-extremal, then $G$ contains a $(18, c\log n, (1-2\eta)n)$ connected triangle cover.
First apply Lemma \[lem:cover\] to get a $(c'\log n, (1-\eta)n)$ tripartite cover ${{\mathcal T}}= \{K^0,\ldots,K^{m-1}\}$ (where $c'$ is the constant coming from Lemma \[lem:cover\]). Fix an orientation for each tripartite graph in ${{\mathcal T}}$ and applying Lemma \[coverconnecting\] to connect $K^i=(U_1,U_2,U_3)$ to $K^{i+1}=(V_1,V_2,V_3)$ by a square path $P_i =u_1u_2u_3Qv_1v_2v_3$ where $v_h\in V_h$ and $v'_h\in V'_h$ for all $h\in [3]$ and $|Q|\leq 18$. We make all of the vertices of $P_i$ forbidden to be used for any later connection. If at some point in this process some $K\in {{\mathcal T}}$ has more than $\frac{c'}{2}\log n$ forbidden vertices, we make all the vertices in $K$ forbidden. Note that there are at most $\frac{n}{c'\log n}$ connections to be made, each one causing $24$ vertices to become forbidded – with the additional rule that once half of the vertices from a color class are used, all vertices in that tripartite graph are made forbidden – we have that the total number of forbidden vertices at any point in the process is at most $2\cdot 24 \frac{n}{c'\log n} < \eps n$. Thus, after every step the remaining graph still satisfies $\deg(u)+\deg(v) \geq (\frac{4}{3} - 4{\varepsilon})n$, hence we can continue to apply Lemma \[coverconnecting\] until we construct $P_{m-1}$ from $K^{m-1}$ to $K^0$. Finally we remove all vertices from the tripartite graphs that are part of some $P_i$ except the starting and ending triangles and rebalance the tripartite graphs by discarding arbitrary subset of vertices from larger color classes, noting that at most $18$ vertices could be removed from each color class; so at most $18\cdot \frac{n}{c'\log n} < \eps n$ in total. We now have the desired $(18, c\log n, (1-2\eta)n)$ connected tripartite cover (with $c=c'/2$).
Inserting the remaining vertices {#sec:inserting}
--------------------------------
Finally we show that if we are given a connected tripartite cover, we can assign the remaining vertices to the tripartite graphs in such a way that they can be incorporated into a square cycle.
\[lemma:inserting\] Let $0<{\varepsilon},c \ll \eta\ll \alpha\ll 1$ and $G$ be a graph on $n$ vertices containing a $(18, c\log n, (1-2\eta)n)$ connected tripartite cover $\mathcal{K}$ with square paths ${{\mathcal P}}= \{P_1,\dots, P_{m}\}$. If $n$ is sufficiently large, $\delta_2(G)\geq (\frac{4}{3}-2{\varepsilon})n$, and $G$ is not $\alpha$-extremal, then $G$ contains a collection ${{\mathcal T}}$ of disjoint (not necessarily balanced) complete tripartite graphs $\{T^1, \dots, T^m\}$, such that $|V(T^i)|\geq (1-\eta^{1/3})c\log n$ and $|V({{\mathcal T}})|\geq (1-\sqrt{\eta})n$ together with a function $f$ from $V(G)-V({{\mathcal T}})-V({{\mathcal P}})$ to $\{T^1, \dots, T^m\}$ having the property that $|f^{-1}(T^i)|\leq \frac{1}{\eta^{1/3}}\frac{2\eta n}{m}$ and $V(T_i)\cup f^{-1}(T_i)$ contains a square path starting with the last edge of $P_{i-1}$ and ending with the first edge of $P_{i}$.
Let $U = V(G) - V(\mathcal{K}) - V({{\mathcal P}})$ and note that $|U|\leq 2\eta n$. We will try to assign the vertices of $U$ to the complete tripartite graphs, but in the process we will end up having to modify the original cover. For convenience, we let the original cover consist of complete tripartite graphs $\{T^1,\dots, T^m\}$ and square paths $\{P_1,\dots, P_m\}$ where $\frac{1}{6c}\frac{n}{\log n}\leq m\leq \frac{1}{3c}\frac{n}{\log n}$ and throughout the process, we will refer to the tripartite graphs by these same names even if they are modified. We assume that size of a color class in $T^i$ is $t$. However, we will maintain a set ${{\mathcal T}}^*$ of triangles which cannot be modified as they are being used to insert vertices into some $T^i$.
Let $w\in U$. We will prove that we can assign $w$ to some $T^i$ while only adding at most $8$ triangles to ${{\mathcal T}}^*$. Once $\eta^{1/3}c\log n$ vertices have been assigned to $T^i$, then we make all of the vertices of $T^i$ forbidden. Since there are at most $2\eta n$ vertices to be assigned, this will make at most $\frac{2\eta n}{\eta^{1/3}c\log n}$ tripartite graphs forbidden and a total of at most $\frac{2\eta n}{\eta^{1/3}c\log n}\cdot 6c\log n \leq 12\eta^{2/3} n$ forbidden vertices $Z$. For any vertex we only consider its neighborhood in $V(G)-V({{\mathcal T}}^*)-V({{\mathcal P}})-Z$ so for the rest of the proof we will assume that $$\delta_2(G) \geq (\frac{4}{3} - 2{\varepsilon})n- 2(|V({{\mathcal P}})|-|V({{\mathcal T}}^*)|-|Z|)>(\frac{4}{3} - 2\sqrt{\eta})n.$$
First, if $w$ has at least $2$ neighbors in every color class of $T^i = (T_1^i,T_2^i,T_3^i)$, then there are two triangles $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ in $N(w)$, such that $x_j,y_j \in T^i_j$. Clearly we can assign $w$ to $T^i$. We add the two triangles $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ to ${{\mathcal T}}^*$; we say that these triangles are blocked by $w$.
Suppose this is not the case; without loss of generality, for all $i\in [m]$ assume that $w$ has at most one neighbor in $T^i_3$. Let $R(w)=\{v\in T^i_3\setminus N(w): |N(w)\cap T^i_1|, |N(w)\cap T^i_2|\geq 2\}$. Since $\delta(G)\geq (\frac{1}{3}+\alpha)n>mt+\sqrt{\eta} n$, $R(w)$ is non-empty. If $\deg(w)\leq (\frac{2}{3}-\sqrt{\eta})n$, then let $w'\in R(w)$. By the degree condition, $\deg(w')\geq (\frac{2}{3}-\sqrt{\eta})n$. So we may insert $w$ into $T^i$ adding two triangles to ${{\mathcal T}}^*$ and try to insert $w'$ instead. So we assume $\deg(w)\geq (\frac{2}{3}-\sqrt{\eta})n$ and we will try to insert $w$ by adding at most six triangles to ${{\mathcal T}}^*$. We may also assume that for all $v\in \{w\}\cup R(w)$, $\deg(v)<(\frac{2}{3}+\sqrt{\eta})n$, otherwise $v$ will have at least two neighbors in each color class of some $T^i$, in which case we can move $v$ to $T^i$ and replace $v$ with $w$. This implies that for all $v\in V(G)$, if there exists $u\in \{w\}\cup R(w)$ such that $v\not\sim u$, then $$\label{relaxedlower}
\deg(v)\geq (\frac{4}{3}-2\sqrt{\eta})n-(\frac{2}{3}+\sqrt{\eta})n\geq (\frac{2}{3}-3\sqrt{\eta})n.$$ Since $\deg(w)\geq (\frac{2}{3}-\sqrt{\eta})n$ and $\deg(w, T^i_3)\leq 1$ for all $i$, it is the case that $w$ has at least $(1-2\sqrt[4]{\eta})t$ neighbors in two color classes of at least $(1-2\sqrt[4]{\eta})m$ tripartite graphs in ${{\mathcal T}}$, as otherwise we would have $$(\frac{2}{3}-\sqrt{\eta})n\leq \deg(w)\leq 2t(1-2\sqrt[4]{\eta})m+2\sqrt[4]{\eta}m(2-2\sqrt[4]{\eta})t\leq (2-4\sqrt{\eta})tm\leq (2-4\sqrt{\eta})\frac{n}{3}$$ a contradiction.
Let $I=\{i\in [m]: |N(w)\cap T^i_1|, |N(w)\cap T^i_2|\geq (1-2\sqrt[4]{\eta})t\}$ and $R'(w)=\bigcup_{i\in I} T^i_3\setminus N(w)$. Note that $|R'(w)|\geq (1-2\sqrt[4]{\eta})tm\geq (\frac{1}{3}-\alpha)n$ and since $G$ is not $\alpha$-extremal, $e(R'(w))\geq \alpha n^2$. At least $(\alpha-\sqrt{\eta})n^2$ of these edges are not incident with a triangle in ${{\mathcal T}}^*$. If any of these edges lie inside some $T^i_3$, then we can insert $w$ into $T^i$ (adding $w$ to $T^i_3$ unbalances $T^i_3$, but we can use the edge inside $T^i_3$ to rebalance). Let $\alpha'=\alpha-\sqrt{\eta}$, so there are at least $\alpha' m^2$ pairs $\{i,j\}$ such that $e(T^i_3, T^j_3)\geq \alpha' t^2$ and $i,j\in I$; let $I_2$ be the set of such pairs.
Either we can insert $w$ according to the rules above or for all $\{i,j\}\in I_2$, there exists $h\in [3]$ such that (i) $e(T^i_h, T^j)<(2-\alpha'/8)t^2$, (ii) $e(T^i_h, V(G))\geq (\frac{2}{3}-3\sqrt{\eta})nt$, and (iii) $e(T^i_h, T)\leq 2t^2$ for all $T\in {{\mathcal T}}$.
Suppose the claim holds and we are unable to insert $w$. For each pair in $I_2$, there is some color class $T^i_h$ having the stated property. Since there are at least $\alpha' m^2$ pairs in $I_2$ and at most $3m$ color classes, some color class $T^i_h$ has the property for at least $\alpha' m/3$ pairs. This implies that $$\left(\frac{2}{3}-3\sqrt{\eta}\right)nt\leq e(T^i_h, V(G))\leq 2t^2m-\frac{\alpha' m}{3}\cdot\frac{\alpha' t^2}{8}\leq \left(\frac{2}{3}-\frac{\alpha'^2}{72}\right)nt,$$ a contradiction since $\eta <\alpha'$. We now finish the proof by proving the claim.
Let $\{i,j\}\in I_2$ and let $X_i=\{x\in T^i_3: \deg(x, T^j_3)\geq \alpha' t\}$ and $X_j=\{x\in T^j_3: \deg(x, T^i_3)\geq \alpha' t\}$. Since $e(T^i_3, T^j_3)\geq \alpha' t^2$, we have $|X_i|, |X_j|\geq \alpha' t$. If any vertex $v\in T^i_3\cup T^j_3$ has at least $2$ neighbors in every color class of some $T\in {{\mathcal T}}$ ($T\neq T^i$ and $T\neq T^j$), then we may move $v$ to $T$ without unbalancing and replace $v$ with $w$. So in particular for all $v\in T^i_3$, we have $$\label{smallone}
\deg(v, T^j_1)\leq 1 ~\text{ or }~ \deg(v, T^j_2)\leq 1$$
Let $X_i''=\{x\in X_i: \deg(x, T^j_1),\deg(x,T^j_2)\leq \alpha' t\}$. If $|X_i''|\geq |X_i|/2\geq \alpha' t/2$, then $$e(T^i_3,T^j)\leq (1-\alpha'/2)t\cdot 2t+\alpha' t/2\cdot(1+2\alpha')t\leq(2-\alpha'/2)t^2$$ (clearly conditions (i) and (ii) are satisfied since $T^i_3\subseteq R(w)$); so suppose $|X_i''|<|X_i|/2$. Since every vertex in $X_i\setminus X_i''$ has at least $\alpha' t$ neighbors in either $T^j_1$ or $T^j_2$, we can set $X_i'$ to be those vertices with at least $\alpha' t$ neighbors in $T^j_2$ and without loss of generality we may suppose $|X_i'|\geq |X_i\setminus X_i''|/2\geq \alpha' t/4$. We will now show that $T^j_1$ satisfies the conditions of the Claim.
Note that every vertex in $X_i'$ has at most one neighbor in $T^j_1$ by , and thus $$\label{incidentXi}
e(T^j_1,T^i_3)\leq t(t-|X_i'|)+|X_i'|\leq (1-\alpha'/6)t^2$$
Also, if some vertex $w'$ in $T^j_1\cap N(w)$ has at least $2$ neighbors in each of $N(w)\cap T^i_1$ and $N(w)\cap T^i_2$, then we may move $w$ and $w'$ into $T^i_3$ and replace $w'$ with some vertex $x\in X_i'$; so suppose not. This implies $$e(T^j_1, T^i_1\cup T^i_2)\leq (1-2\sqrt[4]{\eta})t\cdot t+2\sqrt[4]{\eta}t\cdot 2t=(1+2\sqrt[4]{\eta})t^2.$$ Combining this with gives $$e(T^j_1, T_i)\leq (1-\alpha'/6)t^2+(1+2\sqrt[4]{\eta})t^2\leq (2-\alpha'/8)t^2$$
If there were more than one vertex in $T^j_1$ which is adjacent to every vertex in $X_i'$, then we violate ; so suppose not. Thus by , $e(T^j_1, V(G))\geq (\frac{2}{3}-3\sqrt{\eta})nt$. Finally if some vertex $v\in T^j_1$ has at least two neighbors in every color class of some $T\in {{\mathcal T}}$, then we could move $v$ to $T$, replace it with a vertex from $X_i'$ (which has at least $\alpha' t$ neighbors in $T^j_2$ and $T^j_3$) and replace the vertex from $X_i'$ with $w$; thus $e(T^j_1, T)\leq 2t^2$ for all $T\neq T^i$.
Given $G$ we first apply Lemma \[connectedcover\] to get a $(18, c\log n, (1-2\eta)n)$ connected cover in $G$. We insert the remaining vertices into the cover using Lemma \[lemma:inserting\] and get a set ${{\mathcal T}}$ of $m$ disjoint complete tripartite graphs, a set of square paths ${{\mathcal P}}$, and the function $f:V(G)-V({{\mathcal T}})-V({{\mathcal P}}) \rightarrow {{\mathcal T}}$. Note that for any $w$ such that $f(w)=T^i$, there are two triangles blocked by $w$. Let $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ be the triangles blocked by $w$, notice that by construction $x_i,y_i \in N(w)$. Create an auxiliary triangle $(z_1,z_2,z_3)$ in $T^i$ to replace these two triangles and connect $z_i$ to the common neighbors of $x_i$ and $y_i$. Note that the modified $T^i$ is still a complete tripartite graph. We similarly introduce such an auxiliary triangle for each vertex $w \in V(G)-V({{\mathcal T}})-V({{\mathcal P}})$. Find a triangle cover in the remaining part of $T^i$ except for the two triangles that are part of $P_{i-1}$ and $P_{i}$ by an application of Lemma \[cleanExtCaseLemma1\]. Combining these triangles with the auxiliary triangles, we find a Hamiltonian square path by applying Lemma \[cleanExtCaseLemma2\] that starts with the last triangle $t_{i-1}$ of $P_{i-1}$ and ends with the first triangle $t_i$ of $P_i$.
**Acknowledgements:** We would like to thank the anonymous referees for suggesting a nicer presentation of Lemma \[generalconnecting\].
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[^1]: Department of Mathematics, Miami University, Oxford, OH 45056 USA. E-mail address: debiasld@miamioh.edu. Research supported in part by Simons Foundation Collaboration Grant \#402337.
[^2]: Hewlett-Packard and Department of Computer Science, Rutgers University, Piscataway, NJ 08854, USA. E-mail address: safi@hp.com
[^3]: Department of Computer Science, School of Science and Engineering, Lahore University of Management Sciences, Lahore 54792, Pakistan. E-mail address: imdad.khan@lums.edu.pk. Research supported in part by LUMS Faculty Startup Grant. Part of the work was done while the author was at Umm Al-Qura Univeristy, KSA.
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abstract: 'A synchronized molecular dynamics simulation via macroscopic heat and momentum transfer is proposed to model the non-isothermal flow behaviors of complex fluids. In this method, the molecular dynamics simulations are assigned to small fluid elements to calculate the local stresses and temperatures and are synchronized at certain time intervals to satisfy the macroscopic heat- and momentum-transport equations. This method is applied to the lubrication of a polymeric liquid composed of short chains of ten beads between parallel plates. The rheological properties and conformation of the polymer chains coupled with local viscous heating are investigated with a non-dimensional parameter, the Nahme-Griffith number, which is defined as the ratio of the viscous heating to the thermal conduction at the characteristic temperature required to sufficiently change the viscosity. The present simulation demonstrates that strong shear thinning and a transitional behavior of the conformation of the polymer chains are exhibited with a rapid temperature rise when the Nahme-Griffith number exceeds unity. The results also clarify that the reentrant transition of the linear stress-optical relation occurs for large shear stresses due to the coupling of the conformation of polymer chains and due to the heat generation under shear flows.'
author:
- 'Shugo Yasuda$^1$ [^1] and Ryoichi Yamamoto$^2$ [^2]'
title: 'Synchronized molecular dynamics simulation via macroscopic heat and momentum transfer: an application to polymer lubrication'
---
Introduction
============
To predict the transport phenomena of complex fluids caused by the coupled heat and momentum transfer processes is challenging from both a scientific and engineering point of view. Molecular dynamics (MD) simulations are often used to predict material properties (e.g., the rheological, thermal, and electrical properties), in which the simulation is performed for a very small piece of the material under a certain ideal environment.[@book:89AT; @book:08EM] However, in actual engineering and biological systems, the macroscopic features of complex fluids are highly affected by the spatial heterogeneity caused by the macroscopic transport phenomenon involved in the boundary conditions. A typical example is the generation of heat in lubrication systems.[@book:87BAH] To predict such complicated behavior in complex fluids, the entire system, including the boundary conditions, must be considered on the basis of an appropriate molecular model. In principle, full MD simulations of the entire system can meet these requirements. However, it is difficult in practice to perform full MD simulations on the macroscopic scale, which is common in actual engineering systems and is far beyond the molecular size. Multiscale modeling is a promising candidate to address this type of problem.
The multiscale simulation for the flow behaviors of complex fluids was first advanced in the CONNFFESSIT approach for polymeric liquids by Laso and Öttinger[@art:93LO; @art:95FLO; @art:97LPO], where the local stress in the fluid solver is calculated using the microscopic simulation instead of using any constitutive relations. The GENERIC approach is also presented for the non-isothermal polymeric flows by making important corrections and clarifications to the CONNFFESSIT scheme.[@art:99DEO] The strategy exploited in the CONNFFESSIT approach is also introduced into the heterogeneous multiscale modeling (HMM), which was proposed by E and Enquist,[@art:03EE] as a general methodology for the efficient numerical computation of problems with multiscale characteristics. HMM has been applied to various problems, such as the simple polymeric flow[@art:05RE], coarsening of a binary polymer blend[@art:11MD] and the channel flow of a simple Lennard-Jones liquid[@art:13BLR]. The equation-free multiscale computation proposed by Kevrekids et al. is also based on a similar idea and has been applied to various problems.[@art:03KGHKRT; @art:09KS] De et al. proposed the scale-bridging method, which can correctly reproduce the memory effect of a polymeric liquid, and demonstrated the non-linear viscoelastic behavior of a polymeric liquid in slab and cylindrical geometries.[@art:06DFSKK; @art:13D] The multiscale simulation for polymeric flows with the advection of memory in two and three dimensions was developed by Murashima and Taniguchi.[@MT2010; @MT2011; @MT2012] We have also developed a multiscale simulation of MD and computational fluid dynamics (CFD). The multiscale method was first developed for simple fluids[@art:08YY] and subsequently extended to polymeric liquids with the memory effect.[@art:09YY; @art:10YY; @art:11YY; @art:13MYTY]
However, a multiscale simulation for the coupled heat and momentum transfer of complex fluids has yet to be proposed. For complex fluids, it is usually difficult to describe the heat generation coupled with the momentum transport using only macroscopic quantities. The spatial variation in the temperature also becomes notable at the macroscopic scale due to local viscous heating under shear flow. Thus, multiscale modeling is important for the coupled heat and momentum transfer of complex fluids. In the present paper, we propose a multiscale simulation, termed the synchronized molecular dynamics simulation (SMD), for the coupled heat and momentum transfer in complex fluids by extending the multiscale simulation for momentum transport and apply it to the polymer lubrication. Using this method, we investigate the rheological properties and conformation of polymer chains for the thermohydrodynamic lubrication of a polymeric liquid composed of short chains in a gap between parallel plates, in which the width of the gap is sufficiently large compared to the characteristic length of the flow behaviors, e.g., the length of the viscous boundary layer, such that the macroscopic quantities, e.g., the velocity, stress and temperature, become spatially heterogeneous.
The full MD simulations for confined short-chain molecules in slab geometry were previously studied in Refs. [@art:99JAT] and [@art:01KPY]. The viscous heating of simple liquids in the same geometry was also studied using a full MD simulation in Refs. [@art:97KPY] and [@art:10KBC]. However, these results are exclusively for the molecularly thin films, i.e., the slab width is approximately ten times the molecular size, whereas in the present study, a width of over a thousand times the molecular size is considered in the SMD simulation.
In the following, we first describe the problem considered in the present paper. The simulation method is explained after the presentation of the problem. The SMD simulation of polymer lubrication is performed, and the results are discussed; these results are mainly the rheological properties and the coupling of the conformations of the polymer chains with heat generation under shear flows. We also present a critical analysis of the conceptual and technical issues of the SMD method. Finally, a short summary and a perspective on the future of SMD are given.
Problem
=======
We consider a polymeric liquid contained in a gap of width $H$ between parallel plates with a constant temperature $T_0$ (see Fig. \[fig\_problem\](a)). The polymeric liquid is composed of short Kremer-Grest chains[@art:90KG] of ten beads, in which all of the bead particles interact with a truncated Lennard-Jones potential defined by $$U_{\rm LJ}(r)=\left\{
\begin{array}{l r}
4\epsilon\left[
\left(\frac{\sigma}{r}\right)^{12}
-\left(\frac{\sigma}{r}\right)^{6}
\right]
+\epsilon,
&\quad (r\le 2^{1/6}\sigma),\\
0,
&\quad (r\ge 2^{1/6}\sigma),
\end{array}
\right.$$ and consecutive beads on each chain are connected by an anharmonic spring potential, $$U_{\rm F}(r)=-\frac{1}{2}k_c R_0^2 \ln
\left[
1-\left(\frac{r}{R_0}\right)^2
\right],$$ with $k_c$=30$\epsilon/\sigma^2$ and $R_0$=$1.5\sigma$. The polymeric liquid is in a quiescent state with a uniform density $\rho_0$ and a uniform temperature $T_0$ before a time $t=0$. Hereafter, the $y$-axis is perpendicular to the parallel plates, and the boundaries between the upper and lower plates and the polymeric liquid are located at $y=H$ and 0, respectively. The upper plate starts to move in the x direction with a constant shear stress $\sigma_0$ at a time $t$=0, while the lower plate is at rest.
The macroscopic behavior of the polymeric liquid is described by the following transport equations:
\[eq\_macro\] $$\begin{aligned}
\rho_0\frac{\partial v_x}{\partial t}&=\frac{\partial \sigma_{xy}}{\partial y},
\label{eq_flow}\\
%
\rho_0\frac{\partial e}{\partial t}&=\sigma_{xy}\dot\gamma - \lambda \frac{\partial^2 T}{\partial y^2},
\label{eq_ene}\end{aligned}$$
where $v_\alpha$ is the velocity, $\sigma_{\alpha\beta}$ is the stress tensor, $e$ is the internal energy per unit mass, and $\dot \gamma$ is the shear rate, i.e., $\dot \gamma=\partial v_x/\partial y$. Hereafter, the subscripts $\alpha$, $\beta$, and $\gamma$ represent the index in Cartesian coordinates, i.e., {$\alpha$,$\beta$,$\gamma$}$\in${$x$,$y$,$z$}. Here, we assume that the macroscopic quantities are uniform in the $x$ and $z$ directions, $\partial /\partial x$=$\partial /\partial z$=0, and the density of the polymeric liquid is constant. Fourier’s law for a heat flux with a constant and uniform thermal conductivity $\lambda$ is also considered in Eq. (\[eq\_ene\]). Note that the thermal conductivity of polymeric liquids is anisotropic under shear flows in general[@art:90BB; @art:96OP; @art:96BC; @art:97BCB], and some experimental studies have reported that the linear stress-thermal relation between the stress tensor and thermal conductivity holds.[@art:01VSIGB; @art:04SVBBS; @art:12SVG; @art:13GSV] However, in the present study, we only consider the isotropic thermal conductivity as the first step because the effect of shear thinning of the viscosity is thought to be more crucial to viscous heating under strong shear flows than that of the anisotropy of the thermal conductivity. Involving the anisotropic thermal conductivity in the SMD simulation is an important future work. We also assume that the velocity and temperature of the polymeric liquid are the same as those of the plates at the boundaries, i.e., the non-slip and non-temperature-jump boundary conditions.
The effect of viscous heating is estimated using the ratio of the first and second terms in Eq. (\[eq\_ene\]) to be $\sigma_0\dot\Gamma H^2/\lambda \Delta T_0$. Here, $\dot\Gamma$ is the gross shear rate of the system, which is defined by the ratio of the velocity of the upper plate $v_w$ to the width of the gap $H$, $\dot \Gamma=v_w/H$, and $\Delta T_0$ is a characteristic temperature rise for the polymeric liquid. In the present problem, we consider a characteristic temperature necessary to substantially change the viscosity of the polymeric liquid, i.e., $\Delta T_0=|\eta_0/(\partial \eta_0/\partial T_0)|$, where $\eta_0$ is the characteristic viscosity of the polymeric liquid at a temperature of $T_0$. Thus, the Nahme-Griffith number $Na$, defined as $$\label{eq_nahme-griffith}
Na=\frac{\sigma_0\dot\Gamma H^2}{\lambda|\partial \log(\eta_0)/\partial T_0|^{-1}}$$ represents the effect of viscous heating on the changes in the rheological properties.[@book:87BAH; @art:08PMM] Usually, in lubrication systems and in high-speed processing operations with polymeric liquids, the Nahme-Griffith number is not negligibly small because of the large viscosity and the small thermal conductivity of the polymeric liquid.[@art:08PMM] For example, when a lubrication oil in a gap with a width of 1 $\mu$m is subjected to shear deformation with a strain rate of $1\times 10^6$ $\rm s^{-1}$, the Nahme-Griffith number is estimated to be $Na\gtrsim 0.1$. Thus, the rheological properties of the lubricant in such micro devices must be significantly affected not only by the large velocity gradient but also by the temperature increase caused by local viscous heating. To predict the rheological properties of the polymeric liquid in these systems, one must consider the temperature variation in Eq. (\[eq\_ene\]) coupled with Eq. (\[eq\_flow\]).
Simulation Method
=================
In the present simulation, the gap between the parallel plates is divided into $M$ mesh intervals with a uniform width of $\Delta y=H/M$, and the local velocities are calculated at each mesh node through the typical finite volume scheme shown in Eq. (\[eq\_flow\]). The local shear stresses $\sigma_{xy}(y)$ are calculated from the local shear rates in the MD cells associated with each mesh interval using the NEMD simulation with the SLLOD algorithm. The MD simulations are performed in a time interval $\Delta t$, and the time integrals of the instantaneous shear stresses ${\cal P}_{xy}$ in each MD cell are used to update the local velocities at the next time step in accordance with the macroscopic momentum transport Eq. (\[eq\_flow\]), $$v_x^n(y)=v_x^{n-1}(y)+\frac{\partial}{\partial y}\int_{(n-1)\Delta t}^{n\Delta t}{\cal P}_{xy}(\tau;\dot \gamma^{n-1}(y))d\tau.$$ Here, the superscript $n$ represents the time step number, ${\cal P}_{xy}(\tau;\dot \gamma^{n-1}(y))$ is the instantaneous shear stress in the NEMD simulation with a shear rate of $\dot \gamma^{n-1}(y)$, and $\tau$ is the temporal progress of the NEMD simulation. Note that the time-step size of the MD simulation is different from $\Delta t$. The local viscous heating caused by the shear flow, i.e., the first term of Eq. (\[eq\_ene\]), is calculated in the NEMD simulations without the use of a thermostat algorithm in each MD cell; however, at each time interval $\Delta t$, the instantaneous kinetic energies of the molecules per unit mass $\cal K$ in each MD cell are corrected according to the heat fluxes between neighboring MD cells. Figure \[fig\_calc\_temp\] is a schematic of the calculation of the temperature in the SMD method. The instantaneous temperatures [$\cal T$]{} at each MD cell and their integrals over the duration of each MD run, i.e., $\int_{(n-1)\Delta t}^{n\Delta t}{\cal T}(\tau)d\tau$, are calculated at each MD cell. The heat fluxes between neighboring MD cells $\delta {\cal K}$ are calculated on the global mesh system (depicted on the upper side in Fig. \[fig\_calc\_temp\]) as $$\delta {\cal K}=-\frac{\lambda}{\rho_0} \frac{\partial^2}{\partial y^2}\int_{(n-1)\Delta t}^{n\Delta t}{\cal T}(\tau)d\tau,$$ and the instantaneous kinetic energies ${\cal K}$ at each MD cell are corrected by rescaling the molecular velocities according to the corrected temperature ${\cal T'}$ (depicted on the lower side in Fig. \[fig\_calc\_temp\]) via $${\cal T'}^n={\cal T}^n+\frac{2}{3}\delta {\cal K}.$$
![A schematic for the calculation of the temperature in the SMD method. The upper side represents the progress on the global mesh system, and the lower side represents the progress at each MD cell. []{data-label="fig_calc_temp"}](figure2.eps)
Thus, the MD simulations assigned to each fluid element are synchronized at time intervals of $\Delta t$ to satisfy the macroscopic heat and momentum transport equations (see Fig. \[fig\_problem\](b)). Note that it is difficult to rewrite Eq. (\[eq\_ene\]) as the time evolution of temperature in general because the internal energy depends on not only the macroscopic variables but also on the conformations of the polymer chains. In the present SMD simulation, the temperature increase caused by local viscous heating is calculated autonomously in the MD simulation and satisfies the macroscopic energy balance of Eq. (\[eq\_ene\]).
Hereafter, we measure the length, time, temperature and density in units of $\sigma$, $\tau_0=\sqrt{m\sigma^2/\epsilon}$, $\epsilon/k_B$, and $m/\sigma^3$, respectively. Here, $k_B$ is the Boltzmann constant, and $m$ is the mass of the LJ particle. In the following simulations, the density and thermal conductivity of the polymeric liquid are fixed to be $\rho_0=1$ and $\lambda$=150, respectively, and the temperature of the plates and the width of the gap between the plates are fixed to be $T_0=0.2$ and $H=2500$, respectively, whereas the shear stress applied to the upper plate $\sigma_0$ varies. At this number density $\rho_0$ and this temperature $T_0$, the conformation of the bead particles becomes severely jammed and results in complicated rheological properties.[@art:11YY; @art:02YO]
We have carried out the numerical tests of the present method for various calculation conditions by varying the number of mesh intervals $M$, time interval $\Delta t$, and number of polymer chains in each MD cell $N_p$. The results and a critical analysis of the present method are given in Sec. V. In the present paper, unless otherwise stated, the following calculation parameters are used: The number of mesh intervals $M$ and the time interval $\Delta t$ are $M$=32 and $\Delta t$=1, respectively. A total of 100 polymer chains, i.e., $N_p=100$, of ten beads, i.e., 1000 bead particles, are contained in each MD cell. Thus, the mesh width and side length of the MD cell are $\Delta y$=78.125 and $l_{\rm MD}$=10, respectively. The time-step size in the MD simulation $\Delta \tau$ is set to be $\Delta \tau=0.001$. Thus, MD simulations are performed for 1000 time steps in the time interval $\Delta t$, i.e., $\Delta t=1000\Delta \tau$.
Results
=======
We performed SMD simulations with various values for the shear stress applied to the plate $\sigma_0$, i.e., $\sigma_0=$0.002, 0.01, 0.03, 0.05, 0.055, 0.06, 0.07, 0.08, and 0.09, and investigated the behaviors of a polymeric liquid at steady state. In the following, we present quantities averaged for a long period of time at steady state, in which the shear stress is spatially uniform and the time derivative of the local internal energy, i.e., the left-hand side of Eq. (\[eq\_ene\]), is negligible.
![The spatial variation in the shear stress $\sigma_{xy}(y)$ (a), temperature $T(y)$ (b), shear rate $\dot \gamma(y)$ (c), and $xy$ component of the bond orientation tensor in Eq. (\[bond\_orientation\]) (d) for various values of $\sigma_0$. []{data-label="fig_local"}](figure3.eps)
Figure \[fig\_local\] shows the spatial variations in local quantities for various values of the shear stress applied to the plate, i.e., for $\sigma_0=$0.01, 0.05, 0.06, and 0.08. The shear stress $\sigma_{xy}$ is found to be spatially uniform, and this fact also indicates that the condition necessary for the polymeric liquid to be at steady state is satisfied in the present simulation. The spatial variation in the temperature $T$ is small when the applied shear stress $\sigma_0$ is smaller than 0.05, i.e., $\sigma_0 \le 0.05$, whereas for $\sigma_0 \ge 0.06$, the spatial variation in $T$ becomes notable. The spatial variation in the shear rate $\dot \gamma$ also increases rapidly when the applied shear stress $\sigma_0$ is larger than 0.05. However, the behaviors of the spatial variations in the temperature and in the shear rate are different. The spatial variation in the shear rate is only enhanced in the vicinity of the plate and is rather moderate, except in the vicinity of the plate, whereas the spatial variation in the temperature is a parabolic curve throughout the region. Note that in Fig. \[fig\_local\](c), we omit the result for $\sigma_0=$0.01 because the amplitude of the shear rate is very small and occasionally exhibits negative values as a result of the fluctuations; the spatial average of the shear rate for $\sigma_0=$0.01 is $7.2\times 10^{-6}$. In Fig. \[fig\_local\](d), we show the $xy$ component of the bond orientation tensor $Q_{xy}$, which is defined as $$\label{bond_orientation}
Q_{\alpha\beta}=\frac{1}{N_{\rm p}}\sum_{\rm chain}\frac{1}{N_{\rm b}-1}\sum_{j=1}^{N_{\rm b}-1}\frac{b_{j\alpha}}{b_{\rm min}}\frac{b_{j\beta}}{b_{\rm min}},$$ where $N_{\rm p}$ is the number of polymer chains in each MD cell, $N_{\rm b}$ is the number of bead particles in a polymer chain, ${\bm b}_j$ for $1\le j\le N_{\rm b}-1$ is the bond vector between consecutive beads in the same chain, and $b_{\rm min}$ is the distance at which the sum $U_{\rm LJ}(r)+U_{\rm F}(r)$ has a minimum and is calculated to be $b_{\rm min}\simeq 0.97$. The spatial variation in $Q_{xy}$ is also small for small applied shear stresses, i.e., $\sigma_0 \le 0.05$, but this spatial variation becomes notable for $\sigma_0 \ge 0.06$. In the vicinity of the plate, $Q_{xy}$ increases monotonically with increasing applied shear stress $\sigma_0$, whereas in the middle of the plates, $Q_{xy}$ varies non-monotonically with the applied shear stress. This complicated behavior arises from the temperature variation; the conformation of the polymer chains in the vicinity of the plate is mainly influenced by the shear rate because the variation in the temperature from the reference value $T_0$ is small there, but at the middle of the plates, the conformation of the polymer chains is greatly influenced by the temperature because the variation in the temperature from the reference value $T_0$ is large there. The relations among the temperature, shear rate, and conformation of the polymer chains are also discussed in detail later.
![The apparent viscosity $\eta$, defined as $\eta=\sigma_0/\dot \Gamma$, (a) and the spatial averages of the shear stress and temperature, $\bar \sigma_{xy}$ and $\bar T$, (b) as functions of the gross shear rate $\dot \Gamma$. In figure (a), the asterisks “$*$” show the results of the viscosity for a uniform temperature $T=0.2$, i.e., $Na=0$, obtained by the NEMD simulations. The downward arrows in both figures represent the shear rate at which the Nahme-Griffith number $Na$ defined in Eq. (\[eq\_nahme-griffith\]) equals unity. []{data-label="fig_gross2"}](figure4.eps)
![The spatial average of the bond-orientation tensor of the polymer chains $\bar Q_{\alpha\beta}$ as a function of the gross shear rate $\dot \Gamma$. In addition, see the caption in Fig. \[fig\_gross2\] []{data-label="fig_qab"}](figure5.eps)
In the following, we investigate the rheological properties of the system. Hereafter, the spatial average of a quantity $a(y)$ is represented as $\bar a$, i.e., $\bar a=\frac{1}{H}\int_0^Ha(y)dy$. Figure \[fig\_gross2\] shows the apparent viscosity of the system and the spatial averages of the shear stress and temperature as a function of the gross shear rate. The downward arrows in Fig. \[fig\_gross2\] indicate the reference shear rate at which the Nahme-Griffith number $Na$ defined in Eq. (\[eq\_nahme-griffith\]) equals unity, where the reference temperature is calculated from the viscosities of the model polymeric liquid that were obtained by the NEMD simulation at the uniform and constant temperatures of $T=0.2$ and 0.4 in Ref. as $\Delta T_0=0.15$. In Fig. \[fig\_gross2\](a), the apparent viscosity is compared to the viscosity of the polymeric liquid with the uniform and constant temperature $T_0$ obtained from the NEMD simulation; this case corresponds to a negligible Nahme-Griffith number, i.e., $Na=0$. Strong shear thinning occurs when the Nahme-Griffith number exceeds unity, i.e., $Na>1$. The slope in the power law, i.e., the index $\nu$ in the approximate relation $\eta(\dot\Gamma)\propto \dot \Gamma^{-\nu}$, is almost unity (but never exceeds unity), i.e., $\nu \simeq 1$, for $\dot \Gamma=1\times 10^{-4}\sim 1\times 10^{-3}$. A discrepancy between the apparent viscosity and that for $Na=0$ is thought to be caused by the fact that the temperature slightly increases in the present simulation even at the smallest gross shear rate (the average temperature $\bar T$ increases by 2.5 percent of the wall temperature $T_0$), and the temporal-spatial fluctuations of the temperature and shear rate are also induced in the present simulation, whereas the temperature and the shear rate are kept constant and uniform in the NEMD simulation. Figure \[fig\_gross2\](b) shows the spatial averages of the local shear stress and temperature as functions of the gross shear rate. We note that the spatial average of the shear stress $\bar \sigma_{xy}$ coincides with the applied shear stress $\sigma_0$ on the plate because the local shear stress is almost spatially uniform at steady state, as shown in Fig. \[fig\_local\](b). The average shear stress $\bar \sigma_{xy}$ is observed to increase monotonically with the gross shear rate. The plateau region of the curve corresponds to the strongly shear-thinning regime with the index $\nu\simeq 1$ in Fig. \[fig\_local\] (a). The average temperature increases rapidly with increasing gross shear rate when the Nahme-Griffith number exceeds unity. The rate of increase of the average temperature is lower than that of the average shear stress at small gross shear rates, i.e., $\dot\Gamma \lesssim 1\times 10^{-4}$, but reverses at large gross shear rates, i.e., $\dot\Gamma \gtrsim 1\times 10^{-4}$.
Figure \[fig\_qab\] shows the conformation changes in the polymer chains as a function of the gross shear rate. A non-monotonic dependence of the conformation of the polymer chains on the gross shear rate is observed. A transition occurs at the entrance of the strong shear-thinning regime, with $\nu\simeq 1$ in Fig. \[fig\_gross2\](a), i.e., $\dot \Gamma\simeq 1\times 10^{-4}$. The polymer chains are stretched in the $x$ direction, and the $xy$ component of the bond-orientation tensor increases as $\dot \Gamma$ for small gross shear rates, i.e., $\dot \Gamma \lesssim 1\times 10^{-4}$. However, for large gross shear rates, i.e., $\dot\Gamma \gtrsim 1\times 10^{-4}$, the alignment of the polymer chains is disturbed, and the conformation of the polymer chains becomes more isotropic as the gross shear rate increases. This transitional behavior is caused by the temperature variation; when the temperature increases sufficiently for the large gross shear rates, the coherent structure becomes disturbed by the thermal motion of the bead particles.
![The stress-optical relation $\bar \sigma_{xy}/\bar T$ vs. $\bar Q_{xy}$. The squares $\square$ indicate the present results, and the symbols $+$, $\times$, and $*$ indicate the results obtained in Ref. at uniform temperatures. []{data-label="fig_stress_optic"}](figure6.eps)
Figure \[fig\_stress\_optic\] shows the stress-optical relation for the present problem.[@book:83J; @book:86DE; @book:07S] For the present model polymeric liquid, the NEMD simulations for the isothermal shear flows, i.e., $Na$=0, showed that a universal curve in the stress-optical relation holds in both the linear ($\bar Q_{xy} \lesssim 0.05$) and nonlinear ($\bar Q_{xy} \lesssim 0.05$) regimes.[@art:02YO] In the figure, the present results are compared with those for the case $Na=0$. The leftmost square symbol in the present study represents the result for the smallest gross shear rate, i.e., $\dot \Gamma\simeq 1\times 10^{-6}$. It is observed that $\bar \sigma_{xy}/\bar T$ increases with $\bar Q_{xy}$, while the gross shear rate increases up to $\dot\Gamma \lesssim 1\times 10^{-4}$ because the shear stress increases more rapidly with the gross shear rate than does the temperature (see Fig. \[fig\_gross2\](b)); moreover, $\bar Q_{xy}$ also increases with the gross shear rate in this regime (see Fig. \[fig\_qab\]). However, for large gross shear rates, i.e., $\dot \Gamma \gtrsim 1\times 10^{-4}$, the temperature increases more rapidly than does the shear stress, and the $xy$ component of the bond-orientation tensor decreases with the gross shear rate. Interestingly, the linear stress-optical relation is recovered for shear stresses larger than that for the transitional behavior of the conformation tensor, although the temperature, shear stress, and conformation of the polymer chains exhibit very complicated nonlinear behavior. This reentrant transition of the linear stress-optical relation for large shear stresses can never be reproduced in the NEMD simulations using a thermostat because both the shear stress $\bar \sigma_{xy}$ and the $xy$ component of the bond orientation tensor $\bar Q_{xy}$ monotonically increase with the shear rate at a constant temperature $\bar T$, but $\bar Q_{xy}$ saturates to a limiting value $\bar Q_{xy}\sim 0.1$; therefore, the nonlinear stress-optical relation forms, as shown in Fig. \[fig\_stress\_optic\], for the case $Na=0$.
Critical analysis of the method
===============================
In this section, numerical tests for various calculation conditions are implemented by varying the number of mesh intervals $M$ (i.e., $\Delta x$=$H/M$), time interval $\Delta t$, and number of polymer chains in each MD cell $N_p$, which are given in Table \[t1\]. The comparisons between the results of the present method and the analytic solutions given by Gavis and Laurence are also presented.[@art:68GL]
$M$ $\Delta t$ $N_p$
---- ----- ------------ -------
C1 32 1.0 100
C2 64 1.0 100
C3 16 1.0 100
C4 32 0.5 100
C5 32 5.0 100
C6 32 1.0 1000
: Calculation Conditions[]{data-label="t1"}
![The comparisons of the velocity (a) and temperature (b) profiles under the various calculation conditions for the applied shear stress $\sigma_0=0.01$. []{data-label="fig_compari_pw001"}](figure7.eps)
![The comparisons of the velocity (a) and temperature (b) profiles under the various calculation conditions for the applied shear stress $\sigma_0=0.08$. []{data-label="fig_compari_pw008"}](figure8.eps)
Figures \[fig\_compari\_pw001\] and \[fig\_compari\_pw008\] show the comparisons of the velocity and temperature profiles obtained under the different calculation conditions for the applied share stresses $\sigma_0=0.01$ and $\sigma_0=0.08$, respectively. The comparisons between C1, C2, and C3 show the effect of changing the mesh interval $\Delta x$, and the comparisons between C1, C4, and C5 show the effect of changing the time interval $\Delta t$. Note that the time-step size of the MD simulation $\Delta \tau$ is fixed to be $\Delta \tau=0.001$. Thus, the MD simulations are performed for 500 and 5000 time steps for C4 and C5, respectively, in each time interval $\Delta t$. The comparison between C1 and C6 shows the effect of changing the number of particles in each MD cell, i.e., the effect of the noise intensity arising from the MD simulations.
It is observed in Fig. \[fig\_compari\_pw001\] and in \[fig\_compari\_pw008\] that for $\sigma_0=0.01$, the velocity profiles for C2 and C5 significantly deviate from those under other calculation conditions, whereas for $\sigma_0=0.08$, the deviations under different calculation conditions are not notable. The deviations of C2 and C5 for $\sigma_0=0.01$ are considered to be a result of the numerical stability condition for the momentum transport equation Eq. (\[eq\_flow\]), which is written as $\Delta t < \Delta x^2/2\eta$ for a Newtonian fluid with a constant viscosity $\eta$. The exact stability condition for the SMD method is unknown because in the SMD simulations, the fluctuations are involved in local stresses, and the local viscosities are also autonomously modulated according to the local flow velocities. In the present numerical tests, only the calculation conditions C2 and C5 do not satisfy the condition $\Delta t < \Delta x^2/2\eta$, where $\eta$ is the proper viscosity obtained under other calculation conditions. A similar situation is also found in the previous study in Ref. [@art:10YY], in which numerical tests for the creep motion of the model polymer melt with a uniform temperature are performed. In the previous study, we found that the multiscale simulation in which the time-step size $\Delta t$ is larger than the viscous diffusion time $\Delta x^2/\eta$, i.e., $\Delta t > \Delta x^2/\eta$, reproduces the velocity profile with a pseudo viscosity that is smaller than the proper viscosity. These facts indicate that the time interval $\Delta t$ must be smaller than the viscous diffusion time $\Delta x^2/\eta$, within which the viscous force propagates for the mesh interval $\Delta x$, to obtain the proper solutions in the SMD simulations. This also explains why the deviations of the solutions between different calculation conditions for $\sigma_0=0.08$ are small because for $\sigma_0=0.08$, the local viscosities decrease as a result of both the shear thinning and the temperature increase so that the relation $\eta \ll \Delta x^2/\Delta t$ is satisfied everywhere, except at the vicinity of the plate, where the temperature is close to the reference temperature. Incidentally, a similar condition is also required for the energy transport equation, but it is usually satisfied as long as that for the momentum transport equation is satisfied because the thermal conductivity is usually small compared to the viscosity (i.e., the Prandtl number is small) for polymeric liquids.
The temperature profiles for $\sigma_0=0.01$ do not coincide with each other, although the absolute differences are small, i.e., at most one percent of the wall temperature $T_0$, while for $\sigma_0=0.08$, the deviations between different calculation conditions are very small. This may be caused by the fluctuations of local shear rates and stresses because the signal-to-noise ratio is smaller for $\sigma_0=0.01$ than for $\sigma_0=0.08$. However, by increasing the particle numbers in each MD cell, the fluctuations due to the noise arising from the MD simulations can be reduced.
![The comparisons of temperature profiles between the SMD simulations and the analytic solutions for various shear stresses.[]{data-label="fig_compari_analy"}](figure9.eps)
Figure \[fig\_compari\_analy\] shows the comparisons between the SMD simulations and the analytic solutions for different applied shear stresses. Here, the analytic solution is obtained for a Newtonian liquid with an exponential dependence of the viscosity on temperature[@art:68GL; @art:00BM], i.e., $\eta(T)=\eta_0\exp[-(T-T_0)/\Delta T_0]$, where $\eta_0$ is a characteristic viscosity at a uniform temperature $T_0$ and $\Delta T_0$ is defined above Eq. (\[eq\_nahme-griffith\]). The analytic solution for the temperature is written as[@art:68GL; @art:00BM] $$\begin{aligned}
T(\hat y)&=T_0+\Delta T_0\times \nonumber \\
&\log\left\{\left(1+\frac{\tilde {Na}}{8}\right){\rm sech}^2\left[\left({\rm arcsinh}\sqrt{\frac{\tilde {Na}}{8}}\right)(2\hat y-1)\right]\right\},\end{aligned}$$ where $\hat y=y/H$, and $\tilde {Na}$ is obtained by replacing $\sigma_0$ with $\eta_0\dot \Gamma$ in Eq. (\[eq\_nahme-griffith\]). Here, the characteristic viscosity $\eta_0$ is estimated using the NEMD simulation with a uniform temperature $T_0$ at a shear rate $\dot \gamma=1\times 10^{-6}$ as $\eta_0=1930$.
In Fig. \[fig\_compari\_analy\], it can be observed that the temperature profile for $\sigma_0=0.01$ under the calculation condition C1 deviates considerably from the analytic solution; however, under the calculation condition C6, where the number of polymer chains in the MD cell is increased tenfold, the temperature profile becomes much closer to the analytic solution. Note that for $\sigma_0=0.01$, the effect of the shear thinning of the viscosity is very small, as observed in Fig. \[fig\_gross2\]. Thus, the analytic solution is thought to represent an accurate temperature profile for which the noise effects in the SMD simulation are completely neglected. This fact indicates that the temperature profile obtained by the SMD simulation for small applied shear stresses is greatly affected by the noise even after the long-time average is taken. Thus, to obtain an accurate temperature profile for a small applied shear stress, one needs a large number of molecules in each MD cell, although the cell size $l_{\rm MD}$ should be smaller than the mesh interval $\Delta x$ for an efficient computation. The temperature profile obtained using the SMD simulation for $\sigma_0=0.05$ also deviates from the analytic solution; this is caused by shear thinning, which is not considered for the analytic solution.
In addition to the overall technical issues, there are also concerns about the conceptual issues of the SMD method. The synchronous scheme via the macroscopic transport equations imposes ignoring the molecular transports of constituents across the mesh interval. Thus, the SMD method is not applicable for the dilute polydisperse fluids but is rather designed for dense fluids such as the polymer melt. In the concept of locality of the SMD method, the viscous diffusion and the vortex structures are resolved in the global mesh system. Thus, the mesh interval and the size of the MD cell must be sufficiently small such that the fluid inertia and convection can be ignored at those scales, i.e., the local Reynolds number at the mesh interval $\Delta x$ must be very small. This is also related to the technical aspects of the SMD method because this condition warrants exploiting the SLLOD algorithm and the homogeneous rescaling of the kinetic energies in each MD cell.
Summary
=======
We have proposed a synchronized molecular dynamics simulation via macroscopic heat and momentum transfer and applied this method to the analysis of the lubrication of a polymeric liquid, coupled with viscous heating. The rheological properties and the conformations of the polymer chains are investigated using a non-dimensional parameter, i.e., the Nahme-Griffith number. The SMD simulation demonstrates that strong shear thinning, which is almost inversely proportional to the shear rate, and the transitional behavior for the conformation of the polymer chains occur with a rapid temperature increase when the Nahme-Griffith number exceeds unity. The results show that the linear stress-optical relation holds despite the complicated behaviors of the temperature, shear rate, and conformation of the polymer chains.
We have also carried out numerical tests under various calculation conditions by varying the number of mesh intervals $M$ (i.e., $\Delta x$=$H/M$), time interval $\Delta t$, and number of polymer chains in each MD cell $N_p$ and found the following critical issues in the implementation of the SMD simulations:
1. The time interval $\Delta t$ must be smaller than the viscous diffusion time $\Delta x^2/\eta$, within which the viscous force propagates for the mesh interval $\Delta x$; otherwise, the SMD simulations reproduce the solutions with a pseudo viscosity that is smaller than the true viscosity.
2. The temperature profile at a small applied shear stress is strongly affected by the noise arising from the local MD cells. To obtain an accurate solution, taking the long-time averages as well as increasing the number of particles in each MD cell is required.
3. Concerning the concept of locality in the SMD method, the flow behaviors involving the viscous dissipation and the vortex structures are resolved in the global mesh system. Thus, the mesh interval $\Delta x$ and the size of the MD cell $l_{\rm MD}$ must be sufficiently small such that the fluid inertia and convection can be ignored at those scales, i.e., the local Reynolds numbers at those scales must be very small.\
The first issue concerns the numerical stability condition for the macroscopic transport equations Eq. (\[eq\_macro\]), although the exact stability condition is unknown for the SMD method. The second issue is the consequence of the comparisons between the SMD simulations and the analytic solutions. For the present problem, an SMD simulation using 10,000 particles in each MD cell can successfully reproduce an accurate temperature profile that is described by the analytic solution. The third issue is concerned with the locality concept of the SMD method, and this also involves the technical aspects of the SMD method because the use of the small local Reynolds number warrants exploiting the SLLOD algorithm and the homogeneous rescaling of the kinetic energies at each time interval $\Delta t$, which are described in Sec. III.
Although these issues must be considered, the SMD simulation has two distinctive advantages over the full MD simulations: First, the SMD simulation can reduce the computational effort (i.e., the number of molecules) compared to that of the full MD simulation by a factor of $(l_{\rm MD}/\Delta x)^d$, where $d$ is the dimension number of the macroscopic transport equation. In the present simulation, the factor is 0.128 because $d$ is one. However, the extension to the two- and three-dimensional cases is also possible by incorporating the algorithms developed in, for example, Refs. . Second, almost perfect parallelization efficiency (i.e., using $N$ CPUs in the parallel computation speeds up the calculation compared to using a single CPU by $N$ times) is achieved in a parallel computation with as many CPUs as there are MD cells by assigning each CPU to an MD cell.[@art:10YY] This efficiency is obtained because each MD simulation is performed independently in the time interval $\Delta t$. The advantage in parallel computation holds not only for the short-range interaction molecular models but also for any complicated molecular models for which the parallelization of MD simulations is difficult. These advantages enable us to analyze the complicated flow behaviors of complex liquids at the macroscopic scales found in actual engineering and biological systems on the basis of the appropriate molecular model.
This study was financially supported by the Hyogo Science and Technology Association and by the Grant for Basic Science Research Projects from The Sumitomo Foundation. The computations have been performed using the facilities of the Supercomputer Center at the Institute for Solid State Physics, the University of Tokyo.
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[^1]: Electronic mail: yasuda@sim.u-hyogo.ac.jp
[^2]: Electronic mail: ryoichi@cheme.kyoto-u.ac.jp
|
---
abstract: |
While semidefinite programming (SDP) problems are polynomially solvable in theory, it is often difficult to solve large SDP instances in practice. One technique to address this issue is to relax the global positive-semidefiniteness (PSD) constraint and only enforce PSD-ness on smaller $k\times k$ principal submatrices — we call this the *sparse SDP relaxation*. Surprisingly, it has been observed empirically that in some cases this approach appears to produce bounds that are close to the optimal objective function value of the original SDP. In this paper, we formally attempt to compare the strength of the sparse SDP relaxation vis-à-vis the original SDP from a theoretical perspective.
In order to simplify the question, we arrive at a data independent version of it, where we compare the sizes of SDP cone and the [$k$-PSD closure]{}, which is the cone of matrices where PSD-ness is enforced on all $k\times k$ principal submatrices. In particular, we investigate the question of how far a matrix of unit Frobenius norm in the [$k$-PSD closure]{}can be from the SDP cone. We provide two incomparable upper bounds on this farthest distance as a function of $k$ and $n$. We also provide matching lower bounds, which show that the upper bounds are tight within a constant in different regimes of $k$ and $n$. Other than linear algebra techniques, we extensively use probabilistic methods to arrive at these bounds. One of the lower bounds is obtained by observing a connection between matrices in the [$k$-PSD closure]{}and matrices satisfying the restricted isometry property (RIP).
author:
- 'Grigoriy Blekherman, Santanu S. Dey, Marco Molinaro, Shengding Sun'
bibliography:
- 'QCQP.bib'
title: Sparse PSD approximation of the PSD cone
---
Introduction
============
Motivation
----------
Semidefinite programming (SDP) relaxations are an important tool to provide dual bounds for many discrete and continuous non-convex optimization problems [@wolkowicz2012handbook]. These SDP relaxations have the form $$\begin{aligned}
\label{eq:SDP}
\begin{array}{rcl}
&\textup{min} & \langle C, X\rangle\\
&\textup{s.t.} & \langle A^i, X\rangle \leq b_i ~~\ \forall i \in \{1, \dots, m\}\\
&& X \in {\mathcal{S}^n_+},
\end{array}
\end{aligned}$$ where $C$ and the $A^i$’s are $n \times n$ matrices, ${\langle M, N\rangle} := \sum_{i,j} M_{ij} N_{ij}$, and ${\mathcal{S}^n_+}$ denotes the cone of $n\times n$ symmetric positive semidefinite (PSD) matrices:
$${\mathcal{S}^n_+}=\{ X\in{\mathbb{R}}^{n\times n}\,|\,X=X^T, ~x^{\top} Xx\geq 0, ~\forall x\in{\mathbb{R}}^n\}.$$
In practice, it is often computationally challenging to solve large-scale instances of SDPs due to the global PSD constraint $X \in {\mathcal{S}^n_+}$. One technique to address this issue is to consider a further relaxation that replaces the PSD cone by a larger one ${\mathcal{S}}\supseteq {\mathcal{S}^n_+}$. In particular, one can enforce PSD-ness on (some or all) smaller $k\times k$ principal submatrices of $X$, i.e., we consider the problem $$\begin{aligned}
\label{eq:sparseSDP}
\begin{array}{rcl}
&\textup{min} & \langle C, X\rangle\\
&\textup{s.t.} & \langle A^i, X\rangle \leq b_i \ \forall i \in \{1, \dots, m\}\\
&& \textup{selected } k \times k \textup{ principal submatrices of }X \in \mathcal{S}^k_+.
\end{array}
\end{aligned}$$ We call such a relaxation the *sparse SDP relaxation*.
One reason why these relaxations may be solved more efficiently in practice is that we can enforce PSD constraints by iteratively separating linear constraints. Enforcing PSD-ness on smaller $k\times k$ principal submatrices leads to linear constraints that are sparser, an important property leveraged by linear programming solvers that greatly improves their efficiency [@bixby2002solving; @walter2014sparsity; @amaldi2014coordinated; @coleman1990large; @reid1982sparsity]. This is an important motivation for using [sparse SDP]{} relaxations [@qualizza2012linear; @baltean2018selecting; @DeyWorking]. (This is also the motivation for studying approximations of polytopes [@dey2015approximating], convex hulls of integer linear programs [@dey2017analysis; @walter2014sparsity; @dey2018theoretical], and integer programming formulations [@hojny2017size] by sparse linear inequalities.) This is our reason for calling the relaxation obtained by enforcing the SDP constraints on smaller $k\times k$ principal submatrices of $X$ as the sparse SDP relaxation.
It has been observed that sparse SDP relaxations not only can be solved much more efficiently in practice, but in some cases they produce bounds that are close to the optimal value of the original SDP. See [@qualizza2012linear; @baltean2018selecting; @DeyWorking] for successful applications of this technique for solving box quadratic programming instances, and [@sojoudi2014exactness; @kocuk2016strong] for solving the optimal power flow problem in power systems.
Despite their computational success, theoretical understanding of sparse SDP relaxations remains quite limited. In this paper, we initiate such theoretical investigation. Ideally we would like to compare the objective function values of (\[eq:SDP\]) and (\[eq:sparseSDP\]), but this appears to be a very challenging problem. Therefore, we consider a simpler data-independent question, where we ignore the data of the SDP and the particular selected principal submatrices, to arrive at the following:
\
To formalize this question, we begin by defining the *[$k$-PSD closure]{}*, namely matrices that satisfy all $k \times k$ principal submatrices PSD constraints.
Given positive integers $n$ and $k$ where $2 \leq k \le n$, the [$k$-PSD closure]{}$\mathcal{S}^{n,k}$ is the set of all $n \times n$ symmetric real matrices where all $k \times k$ principal submatrices are PSD.
It is clear that the [$k$-PSD closure]{}is a relaxation of the PSD cone (i.e., ${\mathcal{S}}^{n,k} \supseteq {\mathcal{S}^n_+}$ for all $2 \leq k \le n$) and is an increasingly better approximation as the parameter $k$ increases, i.e., we enforce that larger chunks of the matrix are PSD (in particular ${\mathcal{S}}^{n,n} = {\mathcal{S}^n_+}$). The SOCP relaxation formulated in [@sojoudi2014exactness] is equivalent to using the [$k$-PSD closure]{}with $k=2$ to approximate the PSD cone. Our definition is a generalization of this construction.
It is worth noting that the dual cone of $\mathcal{S}^{n,k}$ is the set of symmetric matrices with factor width $k$, defined and studied in [@boman2005factor]. In particular, the set of symmetric matrices with factor width 2 is the set of scaled diagonally dominant matrices [@wang2019polyhedral], i.e., symmetric matrices $A$ such that $DAD$ is diagonally dominant for some positive diagonal matrix $D$. Note that [@ahmadi2019dsos] uses scaled diagonally dominant for constructing inner approximation of the SDP cones for use in solving polynomial optimization problems.
Problem setup
-------------
We are interested in understanding how well the [$k$-PSD closure]{}approximates the PSD cone for the different values of $k$ and $n$. To measure this approximation we would like to consider the matrix in the [$k$-PSD closure]{}that is farthest from the PSD cone. We need to make two choices here: the norm to measure this distance and a normalization method (since otherwise there is no upper bound on the distance between matrices in the PSD cone and the [$k$-PSD closure]{}).
We will use the Frobenius norm $\|\cdot\|_F$ for both purposes. That is, the distance between a matrix $M$ and the PSD cone is measured as ${\textup{dist}}_F(M,{\mathcal{S}^n_+}) = \textup{inf}_{N \in {\mathcal{S}}^n_{+}}\|M - N\|_F$, and we restrict our attention to matrices in [$k$-PSD closure]{}with Frobenius norm equal to $1$. Thus we arrive at the *(normalized) Frobenius distance* between the [$k$-PSD closure]{}and the PSD cone, namely the largest distance between a unit-norm matrix $M$ in ${\mathcal{S}}^{n,k}$ and the cone ${\mathcal{S}^n_+}$: $$\begin{aligned}
{\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},{\mathcal{S}^n_+})&=\sup_{M\in\mathcal{S}^{n,k},\,\|M\|_F=1}{\textup{dist}}_F(M,{\mathcal{S}^n_+})\\
& = \sup_{M\in\mathcal{S}^{n,k},\,\|M\|_F=1} \inf_{N \in {\mathcal{S}^n_+}} \|M - N\|_F.\end{aligned}$$ Note that since the origin belongs to ${\mathcal{S}^n_+}$ this distance is at most $1$. The rest of the paper is organized as follows: Section \[sec:res\] presents all our results and Section \[sec:con\] concludes with some open questions. Then Section \[sec:pre\] presents additional notation and background results needed for proving the main results. The remaining sections present the proofs of the main results.
Our results {#sec:res}
===========
In order to understand how well the [$k$-PSD closure]{}approximates the PSD cone we present:
- Matching upper and lower bounds on ${\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},{\mathcal{S}^n_+})$ for different regimes of $k$.
- Show that [a polynomial number]{} of $k \times k$ PSD constraints are sufficient to provide a good approximation (in Frobenius distance) to the full [$k$-PSD closure]{}(which has ${n \choose k} \approx \big(\frac{en}{k}\big)^k$ such constraints).
We present these result in more details in the following subsections.
Upper bounds {#main:upper}
------------
First we show that the distance between the [$k$-PSD closure]{}and the SDP cone is at most roughly $\approx \frac{n-k}{n}$. In particular, this bound approximately goes from $1$ to $0$ as the parameter $k$ goes from $2$ to $n$, as expected.
\[thm:upper1\] For all $2\leq k<n$ we have $$\label{upper1}
{\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},\mathcal{S}^n_+)\leq \frac{n -k}{ n + k - 2}.$$
The idea for obtaining this upper bound is the following: given any matrix $M$ in the [$k$-PSD closure]{}$\mathcal{S}^{n,k}$, we construct a PSD matrix ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$ by taking the average of the (PSD) matrices obtained by zeroing out all entries of $M$ but those in a $k \times k$ principal submatrix; the distance between $M$ and ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$ provides an upper bound on ${\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},\mathcal{S}^n_+)$. The proof of Theorem \[thm:upper1\] is provided in Section \[sec:upper1\].
It appears that for $k$ close to $n$ this upper bound is not tight. In particular, our next upper bound is of the form $(\frac{n-k}{n})^{3/2}$, showing that the gap between the [$k$-PSD closure]{}and the PSD cone goes to 0 as $n-k \rightarrow n$ at a faster rate than that prescribed by the previous theorem. In particular, for $k = n - c$ for a constant $c$, Theorem \[thm:upper1\] gives an upper bound of $O\left(\frac{1}{n}\right)$ whereas the next Theorem gives an improved upper bound of $O\left(\frac{1}{n^{3/2}}\right)$.
\[thm:upper2\] Assume $n \ge 97$ and $k \ge \frac{3n}{4}$. Then $$\label{upper2} {\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},\mathcal{S}^n_+) \le 96 \, \bigg(\frac{n-k}{n}\bigg)^{3/2}.$$
It is easy to verify that for sufficiently large $r$ if $k > rn$, then the upper bound given by Theorem \[thm:upper2\] dominates the upper bound given by Theorem \[thm:upper1\].
The proof of Theorem \[thm:upper2\] is more involved than that of Theorem \[thm:upper1\]. The high-level idea is the following: Using Cauchy’s Interlace Theorem for eigenvalues of hermitian matrices, we first verify that every matrix in $\mathcal{S}^{n,k}$ has at most $n - k$ negative eigenvalues. Since the PSD cone consists of symmetric matrices with non-negative eigenvalues, it is now straightforward to see that the distance from a unit-norm matrix $M \in \mathcal{S}^{n,k}$ to ${\mathcal{S}^n_+}$ is upper bounded by the absolute value of the most negative eigenvalue of $M$ times $\sqrt{n -k}$. To bound a negative eigenvalue $-\lambda$ of $M$ (where $\lambda \geq 0$), we consider an associated eigenvector $v \in {\mathbb{R}}^n$ and randomly sparsify it to obtain a random vector $V$ that has at most $k$ non-zero entries. By construction we ensure that $V \approx v$, and that $V$ remains almost orthogonal to all other eigenvectors of $M$. This guarantees that $V^{\top} M V \approx -\lambda + \textrm{``small error''}$. On the other hand, since only $k$ entries of $V$ are non-zero, it guarantees that $V^\top M V$ only depends on a $k \times k$ submatrix of $M$, which is PSD by the definition of the [$k$-PSD closure]{}; thus, we have $V^\top M V \ge 0$. Combining these observations we get that $\lambda \leq \textrm{``small error''}$. This eigenvalue bound is used to upper bound the distance from $M$ to the PSD cone. A proof of Theorem \[thm:upper2\] is provided in Section \[Sec:proofThm2\].
Lower bounds
------------
We next provide lower bounds on ${\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},\mathcal{S}^n_+)$ that show that the upper bounds presented in Section \[main:upper\] are tight for various regimes of $k$. The first lower bound, presented in the next theorem, is obtained by a simple construction of an explicit matrix in the [$k$-PSD closure]{}that is far from being PSD. Its proof is provided in Section \[sec:proofThmlower1\].
\[thm:lower1\] For all $2\leq k<n$, we have $$\label{lower1}
{\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},\mathcal{S}^n_+)\geq\frac{n - k}{\sqrt{ (k-1)^2\, n + n(n-1)} }.$$
Notice that for small values of $k$ the above lower bound is approximately $\approx \frac{n - k}{n}$ which matches the upper bound from Theorem \[thm:upper1\]. For very large values of $k$. i.e. $k = n - c$ for a constant $c$, the above lower bound is approximately $\approx \frac{c}{n^{3/2}}$ which matches the upper bound by Theorem \[thm:upper2\].
Now consider the regime where $k$ is a constant fraction of $n$. While our upper bounds give ${\overline{\textup{dist}}}_F({\mathcal{S}^{n,k}},\mathcal{S}^n_+) = O(1)$, Theorem \[thm:lower1\] only shows that this distance is at least $\Omega(\frac{1}{\sqrt{n}})$, leaving open the possibility that the [$k$-PSD closure]{}provides a sublinear approximation of the PSD cone in this regime. Unfortunately, our next lower bound shows that this is not that case: the upper bounds are tight (up to a constant) in this regime.
\[thm:lower2\] Fix a constant $r < \frac{1}{93}$ and let $k = rn$. Then for all $ k\geq 2$, $${\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},\mathcal{S}^{n}_+)>\frac{\sqrt{r-93r^2}}{\sqrt{162r+3}},$$ which is independent of $n$.
For this construction we establish a connection with the *Restricted Isometry Property* (RIP) [@CandesTao1; @candes2006stable], a very important notion in signal processing and recovery [@Candes2008survey; @Chartrand2008RIP]. Roughly speaking, these are matrices that approximately preserve the $\ell_2$ norm of sparse vectors. The details of this connection and the proof of Theorem \[thm:lower2\] are provided in Section \[sec:proofThm3\].
Achieving the strength of $\mathcal{S}^{n,k}$ by a polynomial number of PSD constraints
---------------------------------------------------------------------------------------
In practice one is unlikely to use the full [$k$-PSD closure]{}, since it involves enforcing the PSD-ness for all ${n \choose k} \approx \left(\frac{en}{k}\right)^k$ principal submatrices. Is it possible to achieve the upper bounds mentioned above while enforcing PSD-ness on fewer principal submatrices? We show that the upper bound given by can also be achieved with factor $1+\epsilon$ and probability at least $1-\delta$ by randomly sampling $O\left(\frac{n^2}{{\varepsilon}^2}\ln \frac{n}{\delta}\right)$ of the $k\times k$ principal submatrices.
\[prop:sampling\] Let $2 \leq k \leq n -1$. Consider ${\varepsilon},\delta>0$ and let $$m :=\frac{12n(n-1)^2}{{\varepsilon}^2 (n-k)^2 k}\ln\frac{2n^2}{\delta} \in O\left(\frac{n^2}{{\varepsilon}^2}\ln \frac{n}{\delta}\right).$$ Let ${\mathcal{I}}= (I_1,\ldots,I_m)$ be a sequence of random $k$-sets independently uniformly sampled from ${[n] \choose k}$, and define ${\mathcal{S}}_{{\mathcal{I}}}$ as the set of matrices satisfying the PSD constraints for the principal submatrices indexed by the $I_i$’s, namely $$\begin{aligned}
{\mathcal{S}}_{{\mathcal{I}}} := \{M \in {\mathbb{R}}^{n \times n} : M_{I_i} \succeq 0,~\forall i \in [m]\}.
\end{aligned}$$ Then with probability at least $1 - \delta$ we have $${\overline{\textup{dist}}}_F({\mathcal{S}}_{{\mathcal{I}}},{\mathcal{S}^n_+}) \leq (1+{\varepsilon})\frac{n -k}{ n + k - 2}.$$
Since the zero matrix is PSD, by definition we always have ${\overline{\textup{dist}}}_F({\mathcal{S}}_{{\mathcal{I}}},{\mathcal{S}^n_+}) \leq 1$. So in order for the bound given by Theorem \[prop:sampling\] to be of interest, we need $(1+{\varepsilon})\frac{n -k}{ n + k - 2} \leq 1$, which means ${\varepsilon}\leq \frac{2k-2}{n-k}$. Plugging this into $m$, we see that we need [at least $\frac{3n(n-1)^2}{k(k-1)^2}\ln \frac{2n^2}{\delta}=\tilde{O}(\frac{n^3}{k^3})$ samples to obtain a nontrivial upper bound on the distance]{}.
Recall that a collection $\mathcal{D}$ of $k$-sets of $[n]$ (called blocks) is called a $2$-design (also called a balanced incomplete block design or BIBD) if every pair of elements in $[n]$ belongs to the same number of blocks, denoted $\lambda$. It follows that every element of $[n]$ belongs to the same number of blocks, denoted $r$. Let $b$ be the total number of blocks. The following relation is easily shown by double-counting: $$\frac{\lambda}{r}=\frac{k-1}{n-1}.$$ For background on block designs we refer to [@MR2029249 Chapters 1 and 2]. It immediately follows from the discussion in Sections \[sec:upper1\] and \[sec:propsampling\] that the strength of the bound in can be achieved by the blocks of a $2$-design, instead of using all $k \times k$ submatrices.
It is known from the work of Wilson [@MR366695 Corollary A and B] that, a $2$-design with $b=n(n-1)$ exists for all sufficiently large values of $n$, although to the best of our knowledge no explicit construction is known. (Wilson’s theorem gives a much more general statement for existence of $2$-designs). Therefore, for almost all $n$ we can achieve the strength of bound while only using $n(n-1)$ submatrices.
Fisher’s inequality states that $b\geq n$, so we need to enforce PSD-ness of at least $n$ minors if we use a $2$-design. A $2$-design is called *symmetric* if $b=n$. Bruck-Ryser-Chowla Theorem gives necessary conditions on $b$, $k$ and $\lambda$, for which a symmetric $2$-designs exist, and this is certainly a limited set of parameters. Nevertheless, symmetric $2$-designs may be of use in practice, as they give us the full strength of while enforcing PSD-ness of only $n$ $k\times k$ minors. Some important examples of symmetric $2$-designs are finite projective planes (symmetric $2$-designs with $\lambda=1$), biplanes ($\lambda=2$) and Hadamard $2$-designs.
Conclusion and open questions {#sec:con}
=============================
In this paper, we have been able to provide various upper and lower bounds on ${\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},\mathcal{S}^n_+)$. In two regimes our bounds on ${\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},\mathcal{S}^n_+)$ are quite tight. These are: (i) $k$ is small, i.e., $2 \leq k \leq \sqrt{n}$ and (ii) $k$ is quite large, i.e., $k = n - c$ where $c$ is a constant. These are shown in the first two rows of Table \[table:one\]. When $k/n$ is a constant, we have also established upper and lower bounds on ${\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},\mathcal{S}^n_+)$ that are independent of $n$. However, our upper and lower bounds are not quite close when viewed as a function of the ratio $k/n$. Improving these bounds as a function of this ratio is an important open question.
Regime Upper bound Lower bound
------------------------------------- --------------------------------------------------------------------------- -----------------------------------------------------------------------------
(small k) $ 2 \leq k \leq \sqrt{n}$ $\frac{n -k}{n }$ (Simplified from Thm \[thm:upper1\]) $\frac{1}{\sqrt{2}}\frac{n -k}{n}$ (Simplified from Thm \[thm:lower1\])
(large k) $k \geq n-c$ $96 \left(\frac{c}{n} \right)^{3/2}$ (Simplified from Thm \[thm:upper2\]) $\frac{1}{\sqrt{2}} \frac{c}{n^{3/2}}$ (Simplified from Thm \[thm:lower1\])
$(n \geq 97, k \geq 0.75n)$
($k/n$ is a constant) $k = rn$ Constant, independent of $n$ Constant, independent of $n$
($r < \frac{1}{93}$) $1 - r$ (Simplified from Thm \[thm:upper1\]) $ \sqrt{\frac{r - 93r^2}{5}} $ (Simplified from Thm \[thm:lower2\])
: Bounds on ${\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},\mathcal{S}^n_+)$ for some regimes[]{data-label="table:one"}
We also showed that instead of selecting all minors, only a polynomial number of randomly selected minors realizes upper bound (\[upper1\]) within factor $1+{\varepsilon}$ with high probability. An important question in this direction is to deterministically and strategically determine principal submatrices to impose PSD-ness, so as to obtain the best possible bound for (\[eq:sparseSDP\]) As discussed earlier, such questions are related to exploring 2-designs and perhaps further generalizations of results presented in [@kim2011exploiting].
Notation and Preliminaries {#sec:pre}
==========================
The *support* of a vector is the set of its non-zero coordinates, and we call a vector *$k$-sparse* if its support has size at most $k$. We will use $[n]$ to denote the set $\{1,...,n\}$. A *$k$-set* of a set $A$ is a subset $B\subset A$ with $|B|=k$. Given any vector $x\in{\mathbb{R}}^n$ and a $k$-set $J\subset [n]$ we define $x_J\in{\mathbb{R}}^k$ as the vector where we remove the coordinates whose indices are not in $J$. Similarly, for a matrix $M \in \mathbb{R}^{n \times n}$ and a $k$-set $J\subset [n]$, we denote the principal submatrix of $M$ corresponding to the rows and columns in $J$ by $M_J$.
Linear algebra
--------------
Given any $n\times n$ matrix $A=[a_{ij}]$ its trace (the sum of its diagonal entries) is denoted as $\operatorname{Tr}(A)$. Recall that $\operatorname{Tr}(A)$ is also equal to the sum of all eigenvalues of $A$, counting multiplicities. Given a symmetric matrix $A$, we use $\lambda_1(A) \geq \lambda_2(A) \ge \dots$ to denote its eigenvalues in non-increasing order.
We remind the reader that, a real symmetric $n\times n$ matrix $M$ is said to be PSD if $x^{\top} Mx\geq 0$ for all $x\in{\mathbb{R}}^n$, or equivalently all of its eigenvalues are non-negative. We also use the notation that $A\succeq B$ if $A-B$ is PSD.
We next present the famous Cauchy’s Interlace Theorem which will be important for obtaining an upper bound on the number of negative eigenvalues of matrices in $\mathcal{S}^{n,k}$. A proof can be found in [@Horn1985matrix].
\[thm:Cauchy\]
Consider an $n \times n$ symmetric matrix $A$ and let $A_J$ be any of its $k\times k$ principal submatrix. Then for all $1\leq i\leq k$, $$\lambda_{n-k+i}(A)\leq \lambda_i (A_J)\leq \lambda_i (A).$$
Probability
-----------
These following concentration inequalities will be used throughout, and can be found in [@concentration].
Let $X$ be a non-negative random variable. Then for all $a \ge 1$,
$$\Pr(X\ge a\mathbb{E}(X))\le \frac{1}{a}.$$
Let $X$ be a random variable with finite mean and variance. Then for all $a>0$, $$\Pr(|X - \mathbb{E}(X)| \geq a) \leq \frac{\operatorname{Var}(X)}{a^2}.$$
Let $X_1,...,X_n$ be i.i.d. Bernoulli random variables, with $\Pr(X_i=1)=\mathbb{E}(X_i)=p$ for all $i$. Let $X=\sum_{i=1}^n X_i$ and $\mu=\mathbb{E}(X)=np$. Then for any $0<\delta<1$,
$$\Pr\big(|X-\mu|>\delta \mu\big)\leq 2\exp\bigg(-\frac{\mu\delta^2}{3}\bigg).$$
Proof of Theorem \[thm:upper1\]: Averaging operator {#sec:upper1}
===================================================
Consider a matrix $M$ in the [$k$-PSD closure]{}${\mathcal{S}}^{n,k}$ with $\|M\|_F = 1$. To upper bound its distance to the PSD cone we transform $M$ into a “close by” PSD matrix ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$.
The idea is clear: since all $k \times k$ principal submatrices of $M$ are PSD, we define ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$ as the average of these minors. More precisely, for a set $I \subseteq [n]$ of $k$ indices, let $M^I$ be the matrix where we zero out all the rows and columns of $M$ except those indexed by indices in $I$; then ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$ is the average of all such matrices: $$\begin{aligned}
{\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} := \frac{1}{{n \choose k}} \sum_{I \in {[n] \choose k}} M^I.
\end{aligned}$$ Notice that indeed since the principal submatrix $M_I$ is PSD, $M^I$ is PSD as well: for all vectors $x \in {\mathbb{R}}^n$, $x^\top M^I x = x_I M_I x_I \ge 0$. Since the average of PSD matrices is also PSD, we have that ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$ is PSD, as desired.
Moreover, notice that the entries of ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$ are just scalings of the entries of $M$, depending on how many terms of the average it is not zeroed out:
1. **Diagonal terms:** These are scaled by the factor $$\frac{{n \choose k} - {n -1 \choose k} }{{n \choose k}} = \frac{k}{n},$$ that is, ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}_{ii} = \frac{k}{n}M_{ii}$ for all $i \in [n]$.
2. **Off-diagonal terms:** These are scaled by the factor $$\frac{{n \choose k} - (2{n - 1 \choose k} - {n - 2 \choose k}) }{{n \choose k}} = \frac{k(k-1)}{n(n-1)},$$ that is, ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}_{ij} = \frac{k(k-1)}{n(n-1)}M_{ij}$ for all $i \neq j$.
To even out these factors, we define the scaling $\alpha := \frac{2n(n-1)}{k (n + k -2)}$ and consider $\alpha {\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$. Now we have that the difference between $M$ and $\alpha {\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$ is a uniform scaling (up to sign) of $M$ itself: $(M - \alpha {\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu})_{ii} = (1 - \alpha \frac{k}{n})\, M_{ii} = -\frac{n -k}{ n + k - 2}\, M_{ii}$, and $(M - \alpha {\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu})_{ij} = (1-\alpha \frac{k(k-1)}{n(n -1)})\,M_{ij} = \frac{n -k}{ n + k - 2}\, M_{ij}$ for $i\neq j$. Therefore, we have $$\begin{aligned}
\textup{dist}_{F}(M, {\mathcal{S}^n_+}) ~\leq~ \|M - \alpha {\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}\|_F ~=~ \frac{n -k}{ n + k - 2}\,\|M\|_F ~=~ \frac{n -k}{ n + k - 2}.
\end{aligned}$$ Since this holds for all unit-norm matrix $M \in {\mathcal{S}}^{n,k}$, this upper bound also holds for ${\overline{\textup{dist}}}_F({\mathcal{S}}^{n,k},{\mathcal{S}^n_+})$. This concludes the proof of Theorem \[thm:upper1\].
Proof of Theorem \[thm:upper2\]: Randomized sparsification {#Sec:proofThm2}
==========================================================
Let $M \in {\mathcal{S}}^{n,k}$ be a matrix in the [$k$-PSD closure]{}with $\|M\|_F = 1$. To prove Theorem \[thm:upper2\], we show that the Frobenius distance from $M$ to the PSD cone is at most $O\big((\frac{n-k}{n})^{3/2}\big)$. We assume that $M$ is not PSD, otherwise we are done, and hence it has a negative eigenvalue. We write $M$ in terms of its eigendecomposition: Let $-\lambda_1 \le -\lambda_2 \le \ldots \le -\lambda_\ell$ and $\mu_1,\ldots,\mu_{n-\ell}$ be the negative and non-negative eigenvalues of $M$, and let $v^1,\ldots,v^{\ell} \in {\mathbb{R}}^n$ and $w^1,\ldots,w^{n-\ell} \in {\mathbb{R}}^n$ be orthonormal eigenvectors relative to these eigenvalues. Thus $$\label{eq:decomp}
M=- \sum_{i \le \ell} \lambda_i v^i (v^i)^\top + \sum_{i \le n-\ell} \mu_i w^i (w^i)^\top.$$ Notice that since $\|M\|_F = 1$ we have $$\begin{aligned}
\sum_{i \le \ell} \lambda_i^2 + \sum_{i \le n-\ell} \mu_i^2 = 1. \label{eq:normEigen}
\end{aligned}$$
We first relate the distance from $M$ to the PSD cone to its negative eigenvalues.
Distance to PSD cone and negative eigenvalues.
----------------------------------------------
We start with the following general observation.
\[prop:neval\] Suppose $M$ is a symmetric $n\times n$ matrix with $\ell\leq n$ negative eigenvalues. Let $-\lambda_1\leq -\lambda_2\leq ...\leq -\lambda_\ell < 0$ and $\mu_1,...,\mu_{n-l}\geq 0$ be the negative and non-negative eigenvalues of $M$. Then $${\textup{dist}}_F(M,{\mathcal{S}^n_+})=\sqrt{\sum_{i=1}^\ell \lambda_i^2}.$$
Let $V$ be the orthonormal matrix that diagonalizes $M$, i.e., $$V^{\top} MV=D :={\textup{diag}}(-\lambda_1,...,-\lambda_\ell,\mu_1,...,\mu_{n-\ell}).$$ It is well-known that the Frobenius norm is invariant under orthonormal transformation. Therefore, for any $N
\in{\mathcal{S}^n_+}$ we have $${\textup{dist}}_F(M,N)~=~\|M-N\|_F~=~\|V^{\top }(M-N)V\|_F ~=~ {\textup{dist}}_F(D,\,V^{\top}NV).$$ Since $N\in {\mathcal{S}^n_+}$ iff $V^{\top}NV\in {\mathcal{S}^n_+}$, we see that ${\textup{dist}}_F(M,{\mathcal{S}^n_+})={\textup{dist}}_F(D,{\mathcal{S}^n_+})$. So we only need to show that the latter is $\sqrt{\sum_{i=1}^\ell \lambda_i^2}$.
Let $D_+={\textup{diag}}(0,...,0,\mu_1,...,\mu_{n-\ell})$ be obtained from $D$ by making all negative eigenvalues zero. Then $$\|D-D_+\|_F=\sqrt{\sum_{i=1}^n\sum_{i=1}^n (D-D_+)_{ij}^2}=\sqrt{\sum_{i=1}^\ell \lambda_i^2}.$$ It then suffices to show that $D^+$ is the PSD matrix closest to $D$. For that, let $N$ be any PSD matrix. Then $N_{ii}=e_i^{\top} N e_i\geq 0$ for all $i$, where $e_i$ is the standard unit vector on $i^{th}$ coordinate. Thus we have $$\begin{aligned}
\|D-N\|_F&~=~\sqrt{\sum_{i=1}^\ell (N_{ii}+\lambda_{i})^2+\sum_{i=\ell+1}^{n} (\mu_{i-\ell}-N_{ii})^2+\sum_{i=1}^n\sum_{j\neq i}N_{ij}^2} ~\geq~ \sqrt{\sum_{i=1}^\ell \lambda_i^2}.
\end{aligned}$$ This concludes the proof.
In addition, Cauchy’s Interlace Theorem gives an upper bound on the number of negative eigenvalues of matrices in ${\mathcal{S}^{n,k}}$.
\[cor:nevals\] Any $A\in \mathcal{S}^{n,k}$ has at most $n-k$ negative eigenvalues.
Let $J$ be any $k$-subset of $[n]$. Since $A\in\mathcal{S}^{n,k}$ we have that $A_J$ is PSD, so in particular $\lambda_k({A_J})\geq 0$. Thus, by Theorem \[thm:Cauchy\] the original matrix $A$ also has $\lambda_k(A)\geq 0$, and so the first $k$ eigenvalues of $A$ are nonnegative.
Using Proposition \[prop:neval\] and Proposition \[cor:nevals\], given any symmetric matrix $M\in{\mathcal{S}^{n,k}}$ we can get an upper bound on ${\textup{dist}}_F(M,{\mathcal{S}^n_+})$ using its smallest eigenvalue.
\[prop:ub:smallev\] Consider a matrix $M\in{\mathcal{S}^{n,k}}$ with smallest eigenvalue $-\lambda_1 < 0$. Then $${\textup{dist}}_F(M,{\mathcal{S}^n_+})\leq\sqrt{n-k} \cdot\lambda_1.$$
Letting $-\lambda_1,\ldots,-\lambda_\ell$ be the negative eigenvalues of $M$, we have from Proposition \[prop:neval\] that ${\textup{dist}}_F(M,{\mathcal{S}^n_+})=\sqrt{\sum_{i=1}^\ell \lambda_i^2} \le \sqrt{\ell}\cdot \lambda_1$, since $-\lambda_1$ is the smallest eigenvalue. Since $\ell \le n-k$, because of Proposition \[cor:nevals\], we obtain the result.
Upper bounding $\lambda_1$
--------------------------
Given the previous proposition, fix throughout this section a (non PSD) matrix $M \in {\mathcal{S}^{n,k}}$ with smallest eigenvalue $- \lambda_1 < 0$. Our goal is to upper bound $\lambda_1$.
The first observation is the following: Consider a symmetric matrix $A$ and a set of coordinates $I \subseteq [n]$, and notice that for every vector $x \in {\mathbb{R}}^n$ supported in $I$ we have $x^\top A x = x_I^\top A_I x_I$. Thus, the principal submatrix $A_I$ is PSD iff for all vectors $x \in {\mathbb{R}}^n$ supported in $I$ we have $x^\top A x \ge 0$. Applying this to all principal submatrices gives a characterization of the [$k$-PSD closure]{}via $k$-sparse test vectors.
\[obs:sparse\] A symmetric real matrix $A$ belongs to ${\mathcal{S}}^{n,k}$ iff for all $k$-sparse vectors $x \in {\mathbb{R}}^n$ we have $x^\top A x \ge 0$.
Using this characterization, and the fact that $M \in {\mathcal{S}}^{n,k}$, the idea to upper bound $\lambda_1$ is to find a vector $\bar{v}$ with the following properties (informally):
1. $\bar{v}$ is $k$-sparse
2. $\bar{v}$ is similar to the eigenvector $v^1$ relative to $\lambda_1$
3. $\bar{v}$ is almost orthogonal to the eigenvectors of $M$ relative to its non-negative eigenvalues.
Such vector gives a bound on $\lambda_1$ because using the eigendecomposition $$\begin{aligned}
0 \stackrel{\textrm{Obs \ref{obs:sparse}}}{\le} \bar{v}^\top M \bar{v} = - \sum_{i \le \ell} \lambda_i \, {\langle v^i, \bar{v}\rangle}^2 + \sum_{i \le n-\ell} \mu_i \, {\langle w^i, \bar{v}\rangle}^2 \lesssim -\lambda_1 + \textrm{``small error''},
\end{aligned}$$ and hence $\lambda_1 \lesssim \textrm{``small error''}$.
We show the existence of such $k$-sparse vector $\bar{v}$ via the probabilistic method by considering a random sparsification of $v^1$. More precisely, define the random vector $V \in {\mathbb{R}}^n$ as follows: in hindsight set $p:= 1 - \frac{2 (n-k)}{n}$, and let $V$ have independent entries satisfying $$V_i=\begin{cases}
v^1_i & \text{ if }(v^1_i)^2>2/n, \\
\frac{v^1_i}{p} \text{ with probability } p & \text{ if } (v^1_i)^2\leq \frac{2}{n}, \\
0 \text{ with probability } 1-p & \text{ if } (v^1_i)^2\leq \frac{2}{n}.
\end{cases}$$
The choice of $p$ guarantees that $V$ is $k$-sparse with good probability.
\[lemma:sparse\] $V$ is $k$-sparse with probability at least $\frac{1}{2}$.
Let $m$ be the number of entries in $v^1$ with $(v^1_i)^2\leq \frac{2}{n}$. Since $\|v^1\|_2=1$ we have $m\geq \frac{n}{2}$. By the randomized construction, the number of coordinates of value 0 in $V$ is lower bounded by a binomial random variable $B$ with $m$ trials and success probability $1-p$. Using the definition of $p$ we have the expectation $${\mathbb{E}}B = m (1-p) \ge \frac{n}{2} \cdot \frac{2(n-k)}{n} = n-k;$$ since $n-k$ is integer we have $\lfloor {\mathbb{E}}B \rfloor \ge n-k$. Moreover, it is known that the median of a binomial distribution is at least the expectation rounded down to the nearest integer [@kaas1980mean], hence $\Pr(B \ge \lfloor {\mathbb{E}}B \rfloor) \ge \frac{1}{2}$. Chaining these observations we have $$\begin{aligned}
\Pr\big(\text{\# of coordinates of value 0 in $V$} \ge n-k\big) \ge \Pr\big(B \ge n-k\big) \ge \Pr\big(B \ge \lfloor {\mathbb{E}}B \rfloor\big) \ge \frac{1}{2}.
\end{aligned}$$ In other words, our randomized vector $V$ is $k$-sparse with probability at least $\frac{1}{2}$.
Next, we show that with good probability $V$ and $v_1$ are in a “similar direction”.
\[lemma:parallel\] With probability $> 1 - \frac{1}{6}$ we have ${\langle V, v^1\rangle} \ge \frac{1}{2}$.
To simplify the notation we use $v$ to denote $v^1$. By definition of $V$, for each coordinate we have ${\mathbb{E}}[V_i v_i] = v_i^2$, and hence ${\mathbb{E}}{\langle V, v\rangle} = \|v\|_2^2 = 1$.
In addition, let $I$ be the set of coordinates $i$ where $v_i^2 \le \frac{2}{n}$. Then for $i \notin I$ we have $\operatorname{Var}(V_i v_i) = 0$, and for $i \in I$ we have $\operatorname{Var}(V_i v_i) = v_i^2 \operatorname{Var}(V_i) \le \frac{2}{n} \operatorname{Var}(V_i)$. Moreover, since $p \ge \frac{1}{2}$ (implied by the assumption $k \ge \frac{3n}{4}$) we have by construction $V_i \le \frac{v_i}{p} \le 2 v_i$, and hence $$\operatorname{Var}(V_i) \le {\mathbb{E}}V_i^2 \le 2 v_i {\mathbb{E}}V_i = 2 v_i^2.$$ So using the independence of the coordinates of $V$ we have $$\operatorname{Var}{\langle V, v\rangle} = \sum_{i \in I} \operatorname{Var}(V_i v_i) \le \frac{4}{n}\, \sum_i v_i^2 = \frac{4}{n}.$$ Then by Chebyshev’s inequality we obtain that $$\Pr\left(\langle V, v\rangle \leq \frac{1}{2}\right) \leq \Pr \left( |\langle V, v\rangle - 1| \geq \frac{1}{2}\right) \leq \frac{16}{n}.$$ Since $n \geq 97$, this proves the lemma.
Finally, we show that $V$ is almost orthogonal to the eigenvectors of $M$ relative to non-negative eigenvalues.
\[lemma:ortho\] With probability $\ge 1 -\frac{1}{3}$ we have $\sum_{i \le n - \ell} \mu_i\,{\langle V, w^i\rangle}^2 \le \frac{24(n-k)}{n^{3/2}}$.
Again we use $v$ to denote $v^1$. Define the matrix ${\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} := \sum_{i \le n - \ell} \mu_i w_i w_i^\top$, so we want to upper bound $V^\top {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} V$. Moreover, let $\Delta=V-v$; since $v$ and the $w_i$’s are orthogonal we have ${\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} v = 0$ and hence $$\begin{aligned}
V^\top {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} V = v {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} v + 2 \Delta^\top {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} v + \Delta^\top {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} \Delta = \Delta^\top {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} \Delta, \label{eq:delta}
\end{aligned}$$ so it suffices to upper bound the right-hand side.
For that, notice that $\Delta$ has independent entries with the form $$\Delta_i=\begin{cases}
0 & \text{ if }v_i^2>\frac{2}{n}, \\
\frac{v_i(1-p)}{p} \text{ with probability } p & \text{ if } v_i^2\leq \frac{2}{n}, \\
-v_i \text{ with probability } 1-p & \text{ if } v_i^2\leq \frac{2}{n}.
\end{cases}$$ So ${\mathbb{E}}[\Delta_i \Delta_j] = {\mathbb{E}}\Delta_i {\mathbb{E}}\Delta_j = 0$ for all $i\neq j$. In addition ${\mathbb{E}}\Delta_i^2 = 0$ for indices where $v^2_i > \frac{2}{n}$, and $$\begin{aligned}
{\mathbb{E}}\Delta_i^2 \le \frac{v_i^2 (1-p)^2}{p} + v_i^2 (1-p) = v_i^2 \frac{1-p}{p} \le \frac{2(1-p)}{np}.
\end{aligned}$$ Using these we can expand ${\mathbb{E}}[\Delta^T {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}\Delta]$ as $$\begin{aligned}
{\mathbb{E}}[\Delta^T {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}\Delta] = {\mathbb{E}}\bigg[\sum_{i,j} {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}_{ij}\Delta_i \Delta_j\bigg] = \sum_{i,j} {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}_{ij} \,{\mathbb{E}}[\Delta_i\Delta_j] &= \sum_{i=1}^{n} {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}_{ii}\,{\mathbb{E}}\Delta_i^2 \notag\\
&\leq \frac{2(1-p)}{np}\operatorname{Tr}({\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}) \notag\\
&= \frac{4 (n-k)}{n^2 p}\operatorname{Tr}({\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}), \label{eq:delta2}
\end{aligned}$$ where the last equation uses the definition of $p$.
Since the $\mu_i$’s are the eigenvalues of of ${\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$, we can therefore bound the trace as $$\operatorname{Tr}({\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu})=\sum_{i \le n-\ell} \mu_i \leq \sqrt{n - \ell} \cdot \sqrt{\sum_{i \le n - \ell} \mu_i^2} \le \sqrt{n - \ell} \leq \sqrt{n},$$ where the first inequality follows from the well-known inequality that $\| u\|_1 \leq \sqrt{n}\|u\|_2$ for all $u \in \mathbb{R}^n$ and the second inequality uses $1 = \|M\|_F = \sqrt{\sum_{i \le \ell} \lambda_i^2 + \sum_{i \le n-\ell} \mu_i^2}$. Further using the assumption that $p \ge \frac{1}{2}$, we get from that $${\mathbb{E}}[\Delta^T {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}\Delta] \le \frac{8 (n-k)}{n^{3/2}}.$$
Finally, since all the eigenvalues $\mu_i$ of ${\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$ are non-negative, this matrix is PSD and hence the random variable $\Delta^\top {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} \Delta$ is non-negative. Markov’s inequality then gives that $$\begin{aligned}
\Pr\bigg(\Delta^\top {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} \Delta \ge \frac{24 (n-k)}{n^{3/2}} \bigg) \le \Pr\bigg(\Delta^\top {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} \Delta \ge 3\, {\mathbb{E}}[\Delta^\top {\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} \Delta] \bigg) \le \frac{1}{3}.
\end{aligned}$$ This concludes the proof of the lemma.
With these properties of $V$ we can finally upper bound the modulus $\lambda_1$ of the most negative eigenvalue of $M$.
\[lemma:lambda\] $\lambda_1 \le \frac{96(n-k)}{n^{3/2}}.$
We take the union bound over Lemmas \[lemma:sparse\] to \[lemma:ortho\]. In other words, the probability that $V$ fails at least one of the properties in above three lemmas is strictly less than $\frac{1}{2}+\frac{1}{6}+\frac{1}{3}=1$. Therefore, with strictly positive probability $V$ satisfies all these properties. That is, there is a vector $\bar{v} \in {\mathbb{R}}^n$ that is $k$-sparse, has ${\langle \bar{v}, v^1\rangle} \ge \frac{1}{2}$ and $\sum_{i \le n - \ell} \mu_i\,{\langle \bar{v}, w^i\rangle}^2 \le \frac{24(n-k)}{n^{3/2}}$. Then using Observation \[obs:sparse\] and the eigendecomposition $$\begin{aligned}
0 \stackrel{\textrm{Obs \ref{obs:sparse}}}{\le} \bar{v}^\top M \bar{v} = - \sum_{i \le \ell} \lambda_i \, {\langle \bar{v}, v^i\rangle}^2 + \sum_{i \le n-\ell} \mu_i \, {\langle \bar{v}, w^i\rangle}^2 \le -\frac{\lambda_1}{4} + \frac{24(n-k)}{n^{3/2}}.
\end{aligned}$$ Reorganizing the terms proves the lemma.
Concluding the proof of Theorem \[thm:upper2\]
----------------------------------------------
Plugging the upper bound on $\lambda_1$ from Lemma \[lemma:lambda\] into Proposition \[prop:ub:smallev\] we obtain that $${\textup{dist}}_F(M, {\mathcal{S}^n_+}) \le 96\, \bigg(\frac{n-k}{n}\bigg)^{3/2}.$$ Since this holds for all unit-norm $M \in {\mathcal{S}}^{n,k}$, we have that ${\overline{\textup{dist}}}_F({\mathcal{S}}^{n,k},{\mathcal{S}^n_+})$ also satisfies the same upper bound. This concludes the proof.
Proof of Theorem \[thm:lower1\]: A specific family of matrices in ${\mathcal{S}}^{n,k}$ {#sec:proofThmlower1}
=======================================================================================
To prove the lower bounds on ${\overline{\textup{dist}}}_F(\mathcal{S}^{n,k},\mathcal{S}^n_+)$ we construct specific families of matrices in $\mathcal{S}^{n,k}$ with Frobenius norm 1, and then lower bound their distance to the PSD cone.
For the first lower bound in Theorem \[thm:lower1\], we consider the construction where all diagonal entries are the same, and all off-diagonal ones are also the same. More precisely, given scalars $a,b \ge 0$ we define the matrix $$\begin{aligned}
G(a,b, n):= (a + b) I_n -a \textbf{1} \textbf{1}^{\top},
\end{aligned}$$ where $I_n$ is the $n \times n$ identity matrix, and $\textbf{1}$ is the column vector with all entries equal to $1$. In other words, all diagonal entries of $G(a,b,n)$ are $b$, and all off-diagonal ones are $-a$.
The parameter $a$ will control how far this matrix is from PSD: for $a=0$ it is PSD, and if $a$ is much bigger than $b$ it should be “far” from the PSD cone. We then directly compute its eigenvalues, as well as its Frobenius distance to the PSD cone.
\[Gprop:evals\] The eigenvalues of $G(a,b,n)$ are $b-(n-1)a$ with multiplicity 1, and $b+a$ with multiplicity $n-1$.
Let $\{v^1,...,v^n\}$ be an orthonormal basis of ${\mathbb{R}}^n$ such that $\sqrt{n}v^1=\textbf{1}$. Then we can rewrite $G(a,b,n)$ as $$\begin{aligned}
G(a,b,n)& = (a+b)\sum_{i=1}^n v^i (v^i)^{\top}-nav^1 (v^1)^{\top}\\
& = \big(b-(n-1)a\big)v^1 (v^1)^{\top}+(a+b)\sum_{i=2}^n v^i (v^i)^{\top}.
\end{aligned}$$ This gives a spectral decomposition of $G(a,b,n)$, so it has the aforementioned set of eigenvalues.
The next two corollaries immediately follow from Proposition \[Gprop:evals\].
\[cor:Gprop1\] If $a,b \geq 0$, then $G(a, b, n) \in \mathcal{S}^{n,k}$ iff $b \geq (k -1)a$. In particular, since ${\mathcal{S}}^{n,n}={\mathcal{S}^n_+}$, $G(a,b,n)\in{\mathcal{S}^n_+}$ iff $b\geq (n-1)a$.
Note that every $k \times k$ principal submatrix of $G(a, b, n)$ is just the matrix $G(a, b, k)$, which belongs to ${\mathcal{S}^n_+}$ iff $b-(k-1)a\geq 0$, since $a,b\geq 0$.
\[cor:Gprop2\] If $a,b \geq 0$, then ${\textup{dist}}_F(G(a,b,n), {\mathcal{S}^n_+}) = \max\{(n -1) a - b, 0\}$.
If $b\geq (n-1)a$, then $G(a,b,n)\in{\mathcal{S}^n_+}$ from first corollary, so ${\textup{dist}}_F(G(a,b,n), {\mathcal{S}^n_+})=0$ by definition.
If $b<(n-1)a$, then $G(a,b,n)$ has only one negative eigenvalue $b-(n-1)a$. Thus using Proposition \[prop:neval\] we get ${\textup{dist}}_F(G(a,b,n), {\mathcal{S}^n_+})=(n-1)a-b$.
To conclude the proof of Theorem \[thm:lower1\], let $\bar{a} = \frac{1}{\sqrt{ ( k - 1)^2n + n(n -1)} }$ and $\bar{b} = (k-1) \bar{a}$. From Corollary \[cor:Gprop1\] we know that $G(\bar{a}, \bar{b}, n)$ belongs to the [$k$-PSD closure]{}$\mathcal{S}^{n,k}$, and it is easy to check that it has Frobenius norm 1. Then using Corollary \[cor:Gprop2\] we get $$\begin{aligned}
{\overline{\textup{dist}}}_F({\mathcal{S}}^{n,k},{\mathcal{S}^n_+}) \ge {\textup{dist}}_F(G(\bar{a} , \bar{b}, n), {\mathcal{S}^n_+}) = (k-1) \bar{a} = \frac{n - k}{\sqrt{ ( k - 1)^2n + n(n -1)}}.
\end{aligned}$$ This concludes the proof.
Proof of Theorem \[thm:lower2\]: RIP construction when $k= O(n)$ {#sec:proofThm3}
================================================================
Again, to prove the lower bound ${\overline{\textup{dist}}}_F({\mathcal{S}}^{n,k}, {\mathcal{S}^n_+}) \ge cst$ for a constant $cst$ we will construct (randomly) a unit-norm matrix $M$ in the [$k$-PSD closure]{}${\mathcal{S}}^{n,k}$ that has distance at least $cst$ from the PSD cone ${\mathcal{S}^n_+}$; we will use its negative eigenvalues to assess this distance, via Proposition \[prop:neval\].
#### Motivation for connection with RIP property.
Before presenting the actual construction, we give the high-level idea of how the RIP property (Definition \[def:RIP\] below) fits into the picture. For simplicity, assume $k = n/2$. (The actual proof will not have this value of $k$). The idea is to construct a matrix $M$ where about half of its eigenvalues take the negative value $-\frac{1}{\sqrt{n}}$, with orthonormal eigenvectors $v^1,v^2,\ldots, v^{n/2}$, and rest take a positive value $\frac{1}{\sqrt{n}}$, with orthonormal eigenvectors $w^1,w^2,\ldots, w^{n/2}$). This normalization makes $\|M\|_F = \Theta(1)$, so the reader can just think of $M$ being unit-norm, as desired. In addition, from Proposition \[prop:neval\] this matrix is far from the PSD cone: ${\textup{dist}}_F(M,{\mathcal{S}^n_+}) \gtrsim \sqrt{\left(\frac{1}{\sqrt{n}}\right)^2 \cdot \frac{n}{2}} = cst$. So we only need to guarantee that $M$ belongs to the [$k$-PSD closure]{}; for that we need to carefully choose its positive eigenspace, namely the eigenvectors $w^1,w^2,\ldots, w^{n/2}$.
Recall that from Observation \[obs:sparse\], $M$ belongs to the [$k$-PSD closure]{}iff $x^\top M x$ for all $k$-sparse vectors $x \in {\mathbb{R}}^n$. Letting $V$ be the matrix with rows $v^1,v^2,\ldots,$ and $W$ the matrix with rows $w^1,w^2,\ldots$, the quadratic form $x^\top M x$ is
$$\begin{aligned}
x^\top M x = - \frac{1}{\sqrt{n}} \sum_i {\langle v^i, x\rangle}^2 + \frac{1}{\sqrt{n}} \sum_i {\langle w^i, x\rangle}^2 = -\frac{1}{\sqrt{n}} \|Vx\|_2^2 + \frac{1}{\sqrt{n}} \|Wx\|_2^2.
\end{aligned}$$
Since the rows of $V$ are orthonormal we have $\|Vx\|_2^2 \le \|x\|_2^2$. Therefore, if we *could construct the matrix $W$ so that for all $k$-sparse vectors $x \in {\mathbb{R}}^n$ we had $\|Wx\|_2^2 \approx \|x\|_2^2$*, we would be in good shape, since we would have $$\begin{aligned}
x^\top M x \gtrsim - \frac{1}{\sqrt{n}} \|x\|_2^2 + \frac{1}{\sqrt{n}} \|x\|_2^2 \gtrsim 0 \qquad\qquad\textrm{for all $k$-sparse vectors $x$}, \label{eq:preRIP}
\end{aligned}$$ thus $M$ would be (approximately) in the [$k$-PSD closure]{}. This approximate preservation of norms of sparse vectors is precisely the notion of the *Restricted Isometry Property* (RIP) [@CandesTao1; @candes2006stable].
\[def:RIP\] Given $k<m<n$, an $m\times n$ matrix $A$ is said to be $(k,\delta)$-RIP if for all $k$-sparse vectors $x \in {\mathbb{R}}^n$, we have $$(1-\delta)\|x\|_2^2\leq \|Ax\|_2^2\leq (1+\delta)\|x\|_2^2.$$
This definition is very important in signal processing and recovery [@Candes2008survey; @Chartrand2008RIP; @CandesTao1; @candes2006stable], and there has been much effort trying to construct deterministic [@Calderbank2010RIPdet; @Bandeira2013RIPdet] or randomized [@baraniuk2008simple] matrices satisfying given RIP guarantees.
The following theorem in [@baraniuk2008simple] provides a probabilistic guarantee for a random Bernoulli matrix to have the RIP.
\[thm:RIP\] Let $A$ be an $m\times n$ matrix where each entry is independently $\pm 1/\sqrt{m}$ with probability $1/2$. Then $A$ is $(k,\delta)$-RIP with probability at least
$$\label{rip}
1-2\left(\frac{12}{\delta}\right)^k e^{-\left(\delta^2/16-\delta^3/48\right)m}.$$
#### Proof of Theorem \[thm:lower2\]
After we have observed the above connection between matrices in $\mathcal{S}^{n,k}$ and RIP matrices, in the actual proof we adopt a strategy that does not “flow” exactly as described above but is easier to analyze. We will: 1) select $W$, a RIP matrix by selecting parameters $m$ and $\delta$ and applying Theorem \[thm:RIP\]; 2) use it to construct a matrix $M \in \mathcal{S}^{n, k}$; 3) rescale the resulting matrix so that its Frobenius norm is $1$, and; 4) finally compute its distance from ${\mathcal{S}^n_+}$ and show that this is a constant independent of $n$.
#### Actual construction of $M$.
Set $m = 93 k$ and $\delta = 0.9$. Then we can numerically verify that whenever $k\geq 2$, the probability is at least $0.51>\frac{1}{2}$. Then let $W$ be a random $m \times n$ matrix as in Theorem \[thm:RIP\], and define the matrix $$M := - (1-\delta) I + W^\top W.$$ First observe that $M$ has a large relative distance to the PSD cone and with good probability belongs to the [$k$-PSD closure]{}.
\[lemma:RIP1\] The matrix $M$ satisfies the following:
1. With probability at least $0.51$, $M \in {\mathcal{S}}^{n,k}$
2. ${\textup{dist}}_F(M,{\mathcal{S}^n_+}) \ge \sqrt{n-m}\,(1-\delta) $.
Whenever $W$ is $(k,\delta)$-RIP, by definition, for all $k$-sparse $x$ we have $x^{\top} W^{\top} Wx=\|Wx\|^2\geq (1-\delta)x^{\top }x$. Therefore $x^{\top} Mx\geq 0$ for all $k$-sparse $x$, and hence $M\in{\mathcal{S}^{n,k}}$ by Observation \[obs:sparse\]. This gives the first item of the lemma.
For the second item, notice that all vectors in the kernel of $W$, which has dimension $n-m$, are eigenvectors of $M$ with eigenvalue $-(1-\delta)$. So the negative eigenvalues of $M$ include at least $n-m$ copies of $-(1-\delta)$, and the result follows from Proposition \[prop:neval\].
Now we need to normalize $M$, and for that we need to control its Frobenius norm.
\[lemma:RIPnorm\] With probability at least $\frac{1}{2}$, $\|M\|_F^2 \le 2n\delta^2+\frac{2n(n-1)}{m}$.
Notice that the diagonal entries of $W^\top W$ equal $1$, so $$\|M\|_F^2 = \sum_{i = 1}^n M_{ii}^2 + \sum_{i , j \in [n], i \neq j} M_{ij}^2 = n \delta^2 + \sum_{i , j \in [n], i \neq j} (W^{\top}W)_{ij}^2.$$ We upper bound the last sum. Let the columns of $W$ be $C^1,...,C^n$, and denote by $X_{ij} = {\langle C^i, C^j\rangle}$ the $ij$-th entry of $W^{\top} W$. Notice that when $i\neq j$, $X_{ij}$ is the sum of $m$ independent random variables $C^i_\ell C^j_\ell$ that take values $\{-\frac{1}{m},\frac{1}{m}\}$ with equal probability, where $\ell$ ranges from $1$ to $m$. Therefore, $${\mathbb{E}}X_{ij}^2 = \operatorname{Var}(X_{ij}) = \sum_{\ell \in [m]} \operatorname{Var}(C^i_\ell C^j_\ell) = m \,\frac{1}{m^2} = \frac{1}{m}.$$ This gives that $${\mathbb{E}}\, \|M\|_F^2 = n\delta^2+\frac{n(n-1)}{m}.$$ Since $\|M\|_F^2$ is non-negative, from Markov’s inequality $\|M\|_F^2 \le 2 {\mathbb{E}}\, \|M\|_F^2$ with probability at least $1/2$. This gives the desired bound, concluding the proof.
Taking a union bound over Lemmas \[lemma:RIP1\] and \[lemma:RIPnorm\], with strictly positive probability the normalized matrix $\frac{M}{\|M\|_F}$ belongs to ${\mathcal{S}}^{n,k}$ and has $${\textup{dist}}_F\left(\frac{M}{\|M\|_F}, {\mathcal{S}^n_+}\right) \ge \frac{\sqrt{n-m}\,(1-\delta)}{\sqrt{2n(n-1)/m+2n\delta^2}} \geq \frac{\sqrt{n-m}\,(1-\delta)}{\sqrt{2n^2/m+2n\delta^2}}.$$ Thus, there is a matrix with such properties.
Now plugging in $k=rn, m=93k, \delta=0.9$, the right hand side is at least $\frac{\sqrt{r-93r^2}}{\sqrt{162r+3}}$. This concludes the proof of Theorem \[thm:lower2\].
Proof of Theorem \[prop:sampling\] {#sec:propsampling}
==================================
The idea of the proof is similar to that of Theorem \[thm:upper1\] (in Section \[sec:upper1\]), with the following difference: Given a unit-norm matrix $M \in {\mathcal{S}}_{{\mathcal{I}}}$, we construct a matrix ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$ by averaging over the principal submatrices indexed by *only the $k$-sets in ${\mathcal{I}}$* instead of considering all $k$-sets, and upper bound the distance from $M$ to the PSD cone by ${\textup{dist}}_F(M, \alpha{\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu})$. Then we need to provide a uniform upper bound on ${\textup{dist}}_F(M, {\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu})$ that holds *for all $M$’s simultaneously with good probability* (with respect to the samples ${\mathcal{I}}$). This will then give an upper bound on ${\overline{\textup{dist}}}_F({\mathcal{S}}_{{\mathcal{I}}},{\mathcal{S}^n_+})$. Recall that ${\mathcal{I}}= (I_1,\ldots,I_m)$ is a sequence of independent uniform samples from the $k$-sets of $[n]$. As defined in Section \[sec:upper1\], let $M^I$ be the matrix where we zero out all the rows and columns of $M$ except those indexed by indices in $I$. Let $T_{{\mathcal{I}}}$ be the (random) partial averaging operator, namely for every matrix $M \in {\mathbb{R}}^{n \times n}$ $$\begin{aligned}
T_{{\mathcal{I}}}(M) := \frac{1}{|{\mathcal{I}}|} \sum_{I \in {\mathcal{I}}} M^I.
\end{aligned}$$
As we showed in Section \[sec:upper1\] for the full average ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} := T_{{[n] \choose k}}(M)$, the first observation is that if $M \in {\mathcal{S}}_{{\mathcal{I}}}$, that is, all principal submatrices $\{M_I\}_{I \in {\mathcal{I}}}$ are PSD, then the partial average $T_{{\mathcal{I}}}(M)$ is also PSD.
\[lemma:partialPSD\] If $M \in {\mathcal{S}}_{{\mathcal{I}}}$, then $T_{{\mathcal{I}}}(M)$ is PSD.
This is straightforward, since each $M^I$ is PSD.
Consider a unit-norm matrix $M$. Now we need to upper bound ${\textup{dist}}_F(M, \alpha\,T_{{\mathcal{I}}}(M))$, for a scaling $\alpha$, in a way that is “independent” of $M$. In order to achieve this goal, notice that $(T_{\mathcal{I}}(M))_{ij} = f_{ij} M_{ij}$, where $f_{ij}$ is the fraction of sets in ${\mathcal{I}}$ that contain $\{i,j\}$. Then it is not difficult to see that the Frobenius distance between $M$ and $T_{{\mathcal{I}}}(M)$ can be controlled using only these fractions $\{f_{ij}\}$, since the Frobenius norm of $M$ is fixed to be 1. The next lemma makes this observation formal. Since the fractions $\{f_{ij}\}$ are random (they depend on ${\mathcal{I}}$), the lemma focuses on the typical scenarios where they are close to their expectations.
Notice that the probability that a fixed index $i$ belongs to $I_\ell$ is $\frac{k}{n}$, so the fraction $f_{ii}$ is $\frac{k}{n}$ in expectation. Similarly, the expected value of $f_{ij}$ is $\frac{k(k-1)}{n(n-1)}$ when $i\neq j$. In other words, the expectation of $T_{{\mathcal{I}}}(M)$ is ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}$.
\[lemma:typical\] Consider ${\varepsilon}\in [0,1)$ and let $\gamma := \frac{k(n-k)}{2n(n-1)}$. Consider a scenario where ${\mathcal{I}}$ satisfies the following for some ${\varepsilon}\in [0,1)$:
1. For every $i \in [n]$, the fraction of the sets in ${\mathcal{I}}$ containing $i$ is in the interval $\left[\frac{k}{n} - {\varepsilon}\gamma, \frac{k}{n} + {\varepsilon}\gamma \right]$.
2. For every pair of distinct indices $i,j \in [n]$, the fraction of the sets in ${\mathcal{I}}$ containing both $i$ and $j$ is in the interval $\left[\frac{k(k-1)}{n(n-1)} - {\varepsilon}\gamma, \frac{k(k-1)}{n(n-1)} + {\varepsilon}\gamma \right]$.
Then there is a scaling $\alpha > 0$ such that for all matrices $M \in {\mathbb{R}}^{n \times n}$ we have $${\textup{dist}}_F(M, \alpha\, T_{{\mathcal{I}}}(M)) \le (1+{\varepsilon}) \frac{n-k}{n+k-2}\,\|M\|_F.$$
As in Section \[sec:upper1\], let ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu} = T_{{[n] \choose k}}(M)$ be the full average matrix. Recall that ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}_{ii} = {\mkern 1.5mu\widetilde{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu}_{ii} M_{ii}$ for ${\mkern 1.5mu\widetilde{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu}_{ii} = \frac{k}{n}$, and ${\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}_{ij} = {\mkern 1.5mu\widetilde{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu}_{ij} M_{ij}$ for ${\mkern 1.5mu\widetilde{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu}_{ij} = \frac{k (k-1)}{n (n-1)}$ when $i \neq j$. Also let $\alpha := \frac{2n(n-1)}{k (n + k -2)}$. Finally, define $\Delta := \tilde{M} - T_{{\mathcal{I}}}(M)$ as the error between the full and partial averages.
From triangle inequality we have $$\begin{aligned}
\|M - \alpha\, T_{{\mathcal{I}}}(M)\|_F \le \|M - \alpha\, {\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}\|_F + \alpha\,\|\Delta\|_F.
\end{aligned}$$ Moreover, in Section \[sec:upper1\] we proved the full average bound $\|M - \alpha\, {\mkern 1.5mu\widetilde{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu}\|_F \le \frac{n-k}{n+k-2} \|M\|_F$. Moreover, from our assumptions we have $f_{ij} \in [{\mkern 1.5mu\widetilde{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu}_{ij} - {\varepsilon}\gamma, {\mkern 1.5mu\widetilde{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu}_{ij} + {\varepsilon}\gamma]$ for all $i,j$, and hence $|\Delta_{ij}| \le {\varepsilon}\gamma\, |M_{ij}|$; this implies the norm bound $\|\Delta\|_F \le {\varepsilon}\gamma \|M\|_F$. Putting these bounds together in the previous displayed inequality gives $$\begin{aligned}
\|M - \alpha\, T_{{\mathcal{I}}}(M)\|_F \le \bigg(\frac{n-k}{n+k-2} + {\varepsilon}\alpha \gamma\bigg)\,\|M\|_F = (1+{\varepsilon}) \frac{n-k}{n+k-2}\,\|M\|_F.
\end{aligned}$$ This concludes the proof.
Finally, we use concentration inequalities to show that the “typical” scenario assumed in the previous lemma holds with good probability.
With probability at least $1 - \delta$ and the parameter $m$ given in Theorem \[prop:sampling\], the sequence ${\mathcal{I}}$ is in a scenario satisfying the assumptions of Lemma \[lemma:typical\].
As stated in Lemma \[lemma:typical\], we only need that for all entries $i,j$ the fraction $f_{ij}$ deviates from its expectation by at most $+{\varepsilon}\gamma$, with failure probability at most $\delta$. From union bound, this can be achieved if for each entry, the probability that the deviation of its fraction $f_{ij}$ fails to be within $[-{\varepsilon}\gamma, \epsilon\gamma]$ is at most $\frac{\delta}{n^2}$. Now we consider both diagonal and off-diagonal terms:
1. Diagonal terms $f_{ii}$: For each $k-$set sample $I$, let $X_I$ be the indicator variable that is 0 if $i\notin I$, and 1 if $i\in I$. Notice that they are independent, with expectation $\frac{k}{n}$. Let $X=\sum_{i\in{\mathcal{I}}} X_I$ be the sum of these variables.
From definition of $f_{ii}$ we have $X=f_{ii}m$, where $m$ is the total number of samples. From Chernoff bound, have that $$\begin{aligned}
\Pr\bigg(\left|f_{ii}-\frac{k}{n}\right|> {\varepsilon}\frac{(n-k)k}{2n(n-1)}\bigg) &= \Pr\bigg(\left|X-\frac{mk}{n}\right|> {\varepsilon}m\frac{(n-k)k}{2n(n-1)}\bigg)\\
&\le 2\exp\bigg(-\frac{{\varepsilon}^2 (n-k)^2k m}{12n(n-1)^2 }\bigg)\\
&\le \frac{\delta}{n^2}
\end{aligned}$$ as long as $$m\geq \frac{12n(n-1)^2}{{\varepsilon}^2 (n-k)^2 k}\ln\frac{2n^2}{\delta}.$$
2. Off-diagonal terms $f_{ij}$: Similar to first case, now for each $k-$set sample $I$, let $X_I$ be the indicator variable that is 1 if $\{i,j\}\subset I$, and 0 otherwise. Now the expectation of each $X_I$ becomes $\frac{k(k-1)}{n(n-1)}$. Again let $X=\sum_{i\in{\mathcal{I}}} X_I$.
Using same argument as above, $X=f_{ij}m$. From Chernoff bound we get $$\begin{aligned}
\Pr\bigg(\left|f_{ij}-\frac{k(k-1)}{n(n-1)}\right|> {\varepsilon}\frac{(n-k)k}{2n(n-1)}\bigg) &= \Pr\bigg(\left|X-\frac{mk(k-1)}{n(n-1)}\right|> {\varepsilon}m\frac{(n-k)k}{2n(n-1)}\bigg)\\
&\le 2\exp\bigg(-\frac{{\varepsilon}^2 (n-k)^2k m}{12n(n-1)(k-1) }\bigg)\\
&\le \frac{\delta}{n^2}
\end{aligned}$$ as long as $$m\geq \frac{12n(n-1)(k-1)}{{\varepsilon}^2 (n-k)^2 k}\ln\frac{2n^2}{\delta}.$$
Since we chose $m$ large enough so it satisfies both of these cases, taking a union bound over all $i,j$’s we get that the probability that any of the $f_{ij}$’s is $+{\varepsilon}\gamma$ more than their expectations is at most $\delta$. This concludes the proof.
Combining this with Lemma \[lemma:typical\], we conclude the proof of Theorem \[prop:sampling\].\
**Acknowledgements.** Grigoriy Blekherman was partially supported by NSF grant DMS-1901950.
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abstract: 'We show that the light curve of the double GeV+optical flash in GRB 130427A is consistent with radiation from the blast wave in a wind-type medium with density parameter $A=\rho r^2\sim 5\times 10^{10}$ g cm$^{-1}$. The peak of the flash is emitted by copious $e^\pm$ pairs created and heated in the blast wave; our first-principle calculation determines the pair-loading factor and temperature of the shocked plasma. Using detailed radiative transfer simulations we reconstruct the observed double flash. The optical flash is dominated by synchrotron emission from the thermal plasma behind the forward shock, and the GeV flash is produced via inverse Compton (IC) scattering by the same plasma. The seed photons for IC scattering are dominated by the prompt MeV radiation during the first tens of seconds, and by the optical to X-ray afterglow thereafter. IC cooling of the thermal plasma behind the forward shock reproduces all GeV data from a few seconds to $\sim 1$ day. We find that the blast wave Lorentz factor at the peak of the flash is $\Gamma\approx 200$, and the forward shock magnetization is $\epsB\sim 2\times 10^{-4}$. An additional source is required by the data in the optical and X-ray bands at times $>10^2$ s; we speculate that this additional source may be a long-lived reverse shock in the explosion ejecta.'
author:
- 'Indrek Vurm, Romain Hascoët and Andrei M. Beloborodov'
title: 'Pair-dominated GeV-optical flash in GRB 130427A'
---
\[firstpage\]
Introduction
============
GRB 130427A was an exceptionally bright gamma-ray burst due to its relative proximity (cosmological redshift $z=0.34$, @Levan13) and high luminosity reaching $L_{\rm MeV}\sim 3\times 10^{53}~\mbox{erg s}^{-1}$ in the MeV band (@Ackermann14, hereafter A14; @Golenetskii13). The burst was accompanied by a GeV flash with peak luminosity $L_{\rm GeV} \sim 10^{51}~\mbox{erg s}^{-1}$ (A14) and an optical flash with peak luminosity $L_{\rm O}\sim 10^{49}~\mbox{erg s}^{-1}$ [@Vestrand14]. It is the first gamma-ray burst (GRB) observed at early times $\tobs<100$ s by both optical and GeV telescopes.
Remarkably, the optical and GeV flashes peaked at approximately the same time $\tobs\sim 15$ s, and both showed a smooth decay after the peak; the optical flux decay $F_\nu\propto t^{-1.67}$ was steeper than that in the GeV band. Such double (optical+GeV) flashes were predicted to result from copious $e^\pm$ pair creation in the blast wave of the GRB explosion (@BHV13, hereafter B13). In this Letter, we apply this model to GRB 130427A.
In our model, the GeV emission is produced by inverse Compton (IC) cooling of the blast wave [see also @B05a; @Fan05]. The observed spectrum extends to at least $\sim 100$ GeV, with a 95 GeV photon detected at $243$ s and a $32~\mbox{GeV}$ photon at 34 ks. Such high-energy photons cannot be produced by synchrotron emission (@deJager92 [@PiranNakar10]; A14; @Fan13), which makes a strong case for their IC origin.
We calculate the synchrotron and IC cooling of the plasma heated in the forward shock of the explosion using the Monte Carlo radiative transfer code developed in B13. The code self-consistently solves the coupled problem of radiative transfer, pair creation, and blast wave dynamics. The original version of the code included only the prompt radiation as a source of target photons for IC scatterings; here we also include the optical to X-ray afterglow radiation, which dominates seed photons for IC scattering at late times. The prompt and afterglow radiation densities used in our calculations are taken from observations.
GeV flash
=========
Pair-dominated peak
-------------------
The external medium ahead of the blast wave is exposed to the prompt GRB radiation, which pre-accelerates the medium and loads it with copious $e^\pm$ pairs [@ThompsonMadau00; @B02a]. Bright bursts $e^\pm$-enrich the external medium by a factor $Z_\pm\gg 1$ at radii $R < 10^{17}$ cm. B13 showed that this effect leads to a bright GeV+optical flash. The forward shock heats the pair-enriched medium to the thermal Lorentz factor given by $$\begin{aligned}
\ginj \approx \frac{\Gbw}{\gpre(1+\betapre)}\left( 1 + \epse\frac{\mue\mprot}{\Zpm\me} \right),
\label{eq:ginj}\end{aligned}$$ where $\Gbw$ is the blast wave Lorentz factor, $\gpre$ is the pre-acceleration Lorentz factor of the $e^\pm$-loaded medium ahead of the blast wave, $\betapre = (1-1/\gpre^2)^{1/2}$, $\mue$ is the ion mass per proton in units of $\mprot$ ($\mue=1$ for hydrogen and $2$ for heavier elements), and $\epse$ is the fraction of shocked ion energy transferred to leptons; B13 showed that at early times $\epse\approx 1$. In our numerical model presented below we assume $\epse=1$ as long as $\Zpm>500$; at later times we take $\epse=0.3$, as suggested by plasma shock simulations [@SironiSpitkovsky2011].
The pair-loading factor $Z_\pm$ steeply decreases at $R\simgt 10^{16}$ cm and hence $\ginj$ grows (Figure \[fig:dyn\]). This implies a steep rise in the energy of the IC photons, $\EIC\sim \ginj^2 E_t$, where $E_t$ are the energies of the seed/target photons. As long as the blast wave overlaps with the prompt radiation, the seed radiation is dominated by the prompt photons with $E_t\simlt 1$ MeV. The onset (and peak) of the GeV flash marks the moment when $\EIC$ reaches the GeV band. This occurs when $\ginj$ exceeds $\sim 30$.
The condition $\ginj\sim 30$ together with the observed peak time $T_p$ determines the radius and Lorentz factor of the blast wave (B13). For GRB 130427A we find, $$\begin{aligned}
R_p= 1.6\times 10^{16} \, \left( \frac{\EGRB}{8\times 10^{53} \, {\rm erg}}\right)^{1/2}
{\rm cm},\end{aligned}$$
$$\begin{aligned}
\Gbw(R_p) \approx 150 \left( \frac{\EGRB}{8\times 10^{53} \, {\rm erg}}\right)^{1/4}
\left( \frac{\tGeV}{15 \, {\rm s}}\right)^{-1/2}.
\label{eq:gamma}\end{aligned}$$
Here $\EGRB$ is normalized to the energy of the main prompt MeV episode [@Golenetskii13] and we have used $z=0.34$ [@Levan13].
Assuming that the external medium is a wind from the massive progenitor of the burst, the expected number of GeV photons in the peak of the flash is (B13), $$\begin{aligned}
\NGeV \sim 8\times 10^{51} \, \Zpm\,\frac{A_{11} R_{16}}{\mu_e} \, \M,
\label{eq:Nem}\end{aligned}$$ where $A = \rho R^2=\mbox{const}$ is the wind density parameter, and $\M\sim 5-10$ is the multiplicity of photons emitted above $100$ MeV by a single fast-cooling electron. The pair-loading factor $\Zpm$ steeply drops from $10^3$ to $10^2$ at $R\approx R_p$ (Figure \[fig:dyn\]). Comparing Equation (\[eq:Nem\]) with the observed $\NGeV \sim 5\times 10^{54}$ (A14, @Fan13), we conclude that $A=10^{10}-10^{11}$ g cm$^{-1}$ is required. Our detailed transfer simulations show that $A\sim 5\times 10^{10}$ g cm$^{-1}$ gives a GeV flash that is close to the observed one.
The simulated GeV light curve is shown in the upper panel of Figure \[fig:LC\]; the corresponding high-energy spectra at five time intervals are plotted in Figure \[fig:spec\]. The emission above $100$ MeV is initially soft, but quickly hardens as $\ginj$ exceeds 30 and then the spectrum remains roughly flat in $\nu F_{\nu}$. The maximal photon energy, $E_{\rm IC,max} = \me c^2 \Gamma\ginj (1+z)^{-1}$, evolves to the TeV range within a few dynamical times as $\Zpm$ drops.
Blast wave deceleration
-----------------------
Our model for the GeV flash gives the parameters $A$, $R_p$, and $\Gamma_p$, and implies the explosion energy $$\Ekin \approx 2.5\times 10^{53} \,\mbox{erg}.$$ It is consistent with a high radiative efficiency of the prompt emission, $\EMeV/(\Ekin+\EMeV)\approx 0.8$. If the prompt emission is considered as a proxy for the ejecta power, one infers that most of the ejecta kinetic energy is contained in a shell of material about 15 light-seconds thick. The (formal) deceleration radius of the blast wave is $$\begin{aligned}
\Rdec &= 5 \times 10^{15} \, \frac{{\cal E}_{\rm kin}}{2.5\times 10^{53}\, {\rm erg}} \nonumber \\
&\times \left(\frac{A}{5\times 10^{10} \, {\rm g}\, {\rm cm}^{-1}}\right)^{-1} \left(\frac{\Gamma}{200}\right)^{-2} \, {\rm cm}.\end{aligned}$$ The corresponding timescale $\Rdec/(2c\Gamma^2)$ is shorter than the duration of the prompt emission. Therefore the reverse shock in this explosion must be relativistic; it crosses the shell in approximately the same time as it takes the main prompt episode to completely overtake the forward shock. The reverse shock crossing marks the time when the bulk of the jet kinetic energy has been transferred to the blast wave. At this point the blast wave is still radiatively efficient (as the pair loading factor $\Zpm$ is still high); the explosion loses a substantial fraction of its initial energy during the first $20$-$100$ s. This results in the steep decline of $\Gbw(R)$ at $R\sim (2-5)\times 10^{16}$ cm (Figure \[fig:dyn\]). At $t\sim 100$ s the blast wave approaches the adiabatic self-similar regime with $\Gamma \propto [\Ekin/(A t)]^{1/4}$ and $R \propto (\Ekin t/A)^{1/2}$.
Transition to synchrotron-self-Compton cooling {#sec:SSC}
----------------------------------------------
The prompt radiation decouples from the forward shock at $\sim 30$ s and does not contribute to IC cooling at later times. Then the blast wave is mainly cooled by IC scattering of the afterglow radiation, which is produced by the blast wave itself via synchrotron emission [see also @Liu13; @Tam13]. Remarkably, the transition to the “synchrotron-self-Compton” (SSC) phase is smooth, with no easily recognizable feature in the GeV light curve (Figure \[fig:LC\]). The main reason for this is that the electrons at this stage are still in the fast-cooling regime, which renders their IC emission insensitive to the target photon luminosity.
At the beginning of the SSC phase, $\ginj$ is already high and the IC scattering is dominated by low-energy photons, below the Klein-Nishina energy, $$\begin{aligned}
\EKN
\approx \frac{\Gamma\, \me c^2}{\ginj (1+z)} =
1\, \varepsilon_{{\rm e}, -1}^{-1} \, {\rm keV},
\label{eq:E_KN}\end{aligned}$$ where we have used Equation (\[eq:ginj\]) with $\Zpm=1$ and $\gpre=1$, as pair creation is weak at late times. Equation (\[eq:E\_KN\]) along with $\EIC \propto E_t$ implies that the energies of target photons upscattered to the LAT band range from optical to soft X-rays. We approximate the spectral luminosity of the target (afterglow) radiation as $$\begin{aligned}
L_E=L_E^0\,
\left(\frac{E_t}{1 \,\mbox{keV}}\right)^{-\alpha} t_3^{-\beta},
\label{eq:AG}\end{aligned}$$ where $t=\tobs/(1+z)$. We use $L_E^0=3\times 10^{56}~\mbox{s}^{-1}$, $\alpha = 0.55$ for the optical to X-ray spectral index, and $\beta=1.1$ for the temporal index [e.g. @Perley13].
The IC cooling time of the thermal plasma behind the forward shock becomes longer than the dynamical time at $\sim 10^4$ s. Our numerical calculations show that the transition to the slow-cooling regime is very gradual with no easily identifiable spectral or temporal signature in the GeV emission (Figure \[fig:LC\]).
The decay slope of the high-energy light curve cannot be described by a simple analytical model. Naively, in the fast-cooling stage one would expect $\EIC\LE(\EIC) \propto \epse L_{\rm diss} ( \EIC/E_{\rm IC,max})^{-\alpha+1}$, where $L_{\rm diss}=8\pi c^3 A\Gamma^4$ is the total luminosity dissipated at the shock, yielding $\EIC \LE (\EIC)
\propto \varepsilon_{\rm e}^{0.55} \Ekin^{0.78} A^{0.22} \EIC^{0.45} t^{-0.78}$. Similarly, in the slow-cooling phase $\EIC\LE(\EIC) \propto \tauT \ginj^2 E_t
L_E(E_t) \propto
\varepsilon_{\rm e}^{1.1} \Ekin^{-0.23} A^{1.2} \EIC^{0.45} t^{-1.9}$. The simulated light curve is inconsistent with either regime and decays approximately as $t^{-1.2}$ up to $\sim 10^4~\mbox{s}$.
This behavior results from a few effects. In the fast-cooling phase the temporal decay is steeper than the naive prediction due to the contribution from secondary pairs produced by the partial absorption of the GeV flash; this effect declines with time and becomes negligible at a few $100~\mbox{s}$. The decreasing pair loading (up to $\sim 100$ s) also somewhat steepens the light-curve. Furthermore, the large-angle GeV radiation from the main peak affects the observed light curve after the peak. The gradual transition to the slow cooling regime around $\sim 10^4~\mbox{s}$ results in a broad bump in the light curve as electrons start accumulating at $\ginj$; the asymptotic slow cooling regime is only approached at $t\gtrsim 1$ d.
The maximum energy of IC photons produced by the thermal electron population, $E_{\rm IC,max} = 270 \, \varepsilon_{{\rm e},-1} {\cal E}_{{\rm kin},54}^{1/2} A_{11}^{-1/2} t_3^{-1/2} \mbox{GeV}$, can accommodate the observed multi-GeV photons at late times, in particular the $32~\mbox{GeV}$ photon observed at 34 ks.
TeV emission
------------
The relative proximity and high luminosity of GRB 130427A makes it an interesting target for very high energy (VHE) observations. Our model predicts emission of photons of energies $\sim 1$ TeV. The simulated VHE light curve above 100 GeV is shown in the top panel of Figure \[fig:LC\] (magenta line). The luminosity above $100$ GeV reaches the peak of $\sim 8\times 10^{49}~\mbox{erg s}^{-1}$ during the first minute, and most of the VHE fluence should be received in $\sim 1000$ s. Such flashes are detectable with current Cerenkov telescopes. To our knowledge no rapid VHE follow-up was performed for GRB 130427A by presently operating observatories. VERITAS obtained an upper limit at $\sim 1$ d, which indicates a (temporal or spectral) break when compared with the extrapolation of the earlier LAT observation below 100 GeV (J. McEnery, private communication). This is consistent with our model, as the predicted VHE emission from the thermal electrons behind the shock cuts off at about $50$ ks, when the characteristic IC photon energy falls below 100 GeV.
Optical flash
=============
The optical flash is produced by synchrotron emission from the same thermal electrons injected at the forward shock that give rise to the GeV emission (B13). The mechanism of the delayed onset, peak and early decay is also analogous. The bright optical flash occurs when the synchrotron emission reaches the optical band as the electron injection Lorentz factor increases, i.e. when $\ginj=\gopt$, where $$\begin{aligned}
\gopt = \left( \frac{5 E}{\Gamma h\nu_{B}} \right)^{1/2}
&\approx 500
\, (\varepsilon_{{\rm B}, -3} A_{11})^{-1/4} \, R_{16}^{1/2} \, \Gamma_{2}^{-1} \nonumber \\
&\times\left[ \gpre(1+\betapre) \right]^{1/4} \, \left( \frac{E}{2 \, {\rm eV}} \right)^{1/2},\end{aligned}$$ and $\nu_{B}=eB/(2\pi \me c)$ is the cyclotron frequency.
The energetic pairs behind the shock are in the fast-cooling regime at the peak of the flash. As the electron/positron cools, most of the optical radiation is emitted when its Lorentz factor $\gamma\sim \gopt$. The approximate optical luminosity is given by (B13) $$\begin{aligned}
L_{\rm O}\approx \Zpm \frac{dN_{\rm p}}{dR} \frac{dR}{dt} \frac{\me c^2 \gopt \Gamma}{2} \fsyn,
\label{eq:Lopt}\end{aligned}$$ where the factor $\fsyn \approx \UB/\Urad$ accounts for the fraction of energy radiated as synchrotron emission. The observed optical luminosity near the peak, $\sim 10^{49}$ erg/s, requires $\epsB\sim 10^{-4}-10^{-3}$.
The theoretical optical light curve at $2~\mbox{eV}$ is plotted in the lower panel of Figure \[fig:LC\]. Compared to the GeV flash, the onset is slightly delayed, because the threshold $\ginj$ for producing synchrotron optical radiation is somewhat higher than that for producing IC GeV radiation. The decay of the optical flash is controlled by the declining pair loading factor $\Zpm$ and is consistent with the observed light curve up to $\sim 100$ s. At later times synchrotron emission from nonthermal electrons must take over, which is not included in the model shown in Figure 2.
Discussion
==========
The observed GeV flash in GRB 130427A can be explained as IC emission from the thermal plasma behind the blast wave in a wind medium, once the pair loading of the blast wave is correctly taken into account. The same model reproduced the GeV flash in GRB 080916C (B13). The exceptional LAT data for GRB 130427A, which extends to $\sim 1$ d, made it possible to test the model at longer times, when the seed photons for IC scattering change from the prompt radiation to the afterglow. We found that this transition leaves no sharp features and is consistent with the entire observed light curve of GeV emission.
The hot $e^\pm$ plasma in the blast wave must also emit synchrotron radiation, in particular in the optical band. The predicted optical light curve is very close to the optical flash observed during the first 100 s (Figure \[fig:LC\]). This provides further support to the proposed model.
Figure 2, the main result of this paper, shows only emission from the [*thermal*]{} plasma behind the forward shock, which is a robust consequence of shock heating and is straightforward to model from first principles. We also performed a simulation including a nonthermal population of leptons in the forward shock, with an injection spectrum $dN_{\rm inj}/d\gamma\propto \gamma^{-p}$ (with $p=2.2$) carrying a fraction $\epsnth=0.1$ of the shock energy. We found that the additional synchrotron and IC radiation produced by this nonthermal component weakly affects the predicted GeV+optical flash. We conclude that the thermal postshock plasma dominates the flash, at least in the region of parameter space explored by our simulations. A higher $\epsnth$ and a flat electron spectrum $p\approx 2$ would make the contribution from nonthermal particles more significant, especially before the peak of the flash, making the rise toward the peak less sharp. Detailed models with thermal+nonthermal shocked plasma are deferred to a future paper.
After the peak, the synchrotron frequency of the thermal electrons heated by the forward shock $\nu_{\rm syn,th}$ remains above the optical band until $\sim 10^4$ s. In this situation, the addition of nonthermal electrons with $\gamma>\ginj$ does not significantly increase the optical emission from the forward shock. The additional (nonthermal) contribution to the optical afterglow observed at $10^2$-$10^4$ s should be produced by a different source, most likely a long-lived reverse shock [@Uhm07; @Genet07]. This agrees with the suggestion of previous works on GRB 130427A [@Panaitescu13; @Laskar13; @Perley13].
Our flash model requires the wind density parameter $A\sim 5\times 10^{10}$ g cm$^{-1}$. It is much higher (and more typical of Wolf-Rayet stars) than previously inferred from nonthermal afterglow modeling at $t>10$ min [@Panaitescu13; @Laskar13; @Perley13]. A constraint on $A$ from the late afterglow comes from the following consideration. When the characteristic synchrotron frequency of the forward shock crosses the optical band (which happens at $t\sim 10^4$ s in our model) its predicted optical flux is $F_\nu\sim 2 \, A_{11}(\varepsilon_e/0.3)^{-2}(\mu_e/2)^{-3}\, t_4$ mJy. It should not exceed the observed flux of 2 mJy, a condition satisfied by our model. Models assuming $\mu_e=1$ (hydrogen) and $\epsilon_e=0.1$ require smaller $A$, in agreement with Panaitescu et al. (2013). One should also keep in mind that the wind density profile may deviate from $R^{-2}$, i.e. the effective $A$ may change in the late afterglow. Yet we find no conflict between $A\sim 5\times 10^{10}$ g cm$^{-1}$ and radio data at $t>1$ d; the blast wave can produce radio emission without significant self-absorption in the forward or reverse shock.
Our model implies a high radiative efficiency of the blast wave at early times, when pair loading is strong. During the first few $100$ s the blast wave energy $\Ekin$ drops from $2.5\times 10^{53}$ erg to $\sim 10^{53}$ erg. We expect that in a more detailed model a long-lived reverse shock will add energy to the blast wave and keep $\Ekin$ from falling to such low values. A few lines of evidence suggest this energy injection. First, this would help to explain the high X-ray luminosity. Without additional energy the power dissipated in the forward shock is low, $$\begin{aligned}
L_{\rm diss} = \Ekin/4t = 2.5\times 10^{49} \, E_{{\rm kin}, 53} \, t^{-1}_3 \,\mbox{erg s}^{-1}.\end{aligned}$$ It is only a factor of 9 higher than the observed $0.3$-$10$ keV luminosity at $t\gtrsim 10^{3}$ s, which would require a very high efficiency of X-ray emission. Secondly, the observed X-ray spectral index indicates that the (nonthermal) electrons are radiating X-rays in the slow cooling regime already at $\sim 1000$ s. At these early times, electrons are mainly cooled by IC scattering (not synchrotron) and the cooling frequency $\nu_{\rm syn,c}$ is very sensitive to the blast wave energy, $\nu_{\rm syn,c}\propto \epsB^{1/2}\Ekin^p t^q$, where $p=(4+\alpha)/2\alpha\approx 4.1$, $q = (4\beta-3\alpha)/2\alpha\approx 2.5$, and $\alpha$, $\beta$ are the afterglow spectral and temporal indices defined in Equation (\[eq:AG\]). Energy injection via the reverse shock would help to keep $\nu_{\rm syn,c}$ above the X-ray band. We find that supplying $\Ekin\sim 10^{54}$ erg by $t\sim 1000$ s may be sufficient to explain the slow-cooling regime in the X-ray band. This can be accomplished by a tail of the GRB jet with $\Gamma_{\rm tail}\approx 50-100$ carrying energy comparable to the jet head.
The increased $\Ekin$ will boost the optical luminosity, which can overshoot the observed afterglow, in particular when $\nu_{\rm syn,th}$ crosses the optical band at $t\sim 10^4$ s. This problem could be resolved if $\epsB$ is reduced by a factor of $\sim30$ by that time. It is not unreasonable to assume that $\epsB$ evolves, as physical conditions change in the expanding blast wave; e.g., the pair loading is quickly decreasing. Another factor that can reduce $\epsB$ is the increasing cooling length of the shock-heated plasma. Note that $\epsB$ describes the [*average*]{} value of the magnetic field in the emission region and depends on how quickly the field decays downstream of the shock [e.g. @Lemoine13].
The reduction of $\epsB$ in the late afterglow phase is also suggested by the high value of the cooling frequency, $\nu_{\rm syn,c}$, inferred from observations by NuSTAR at $t\sim 1$ d. NuSTAR identified a break at $\sim 100$ keV in the afterglow spectrum, which was interpreted as a cooling break [@Kouveliotou13]. With no evolution of $\epsB$, our model would predict the break at a few keV while a reduction of $\epsB$ by a factor of $\sim 10$ between $10^2$ and $10^5$ s would move the cooling break to $\sim 100$ keV (note that cooling at 1 d is dominated by synchrotron emission, not by IC scattering, and therefore $\nu_{\rm syn,c}\propto \epsB^{-3/2} \Ekin^{1/2} t^{1/2}$).
Detailed modeling of the nonthermal optical and X-ray emission from the forward and long-lived reverse shocks is an involved problem, which we defer to a future work. It should not, however, change the results of the present paper. We emphasize that both the optical flash and the [*entire*]{} GeV light curve are insensitive to the details of energy injection and the evolution of $\epsB$. The same is true for our estimate of the wind density parameter $A$.
This work was supported by NSF grant AST-1008334 and NASA Fermi Cycle 6 grant NNX 13AP246.
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|
---
abstract: |
Let $\Gamma$ be a finitely generated, torsion-free, two step nilpotent group. Let $C^*(\Gamma)$ be the universal $C^*$-algebra of $\Gamma$. We show that $acsr( C^*(\Gamma)) =
acsr( C( (\widehat{ \Gamma })_1)$, where for a unital $C^*$-algebra ${\mathcal{A}}$, $acsr({\mathcal{A}})$ is the absolute connected stable rank of ${\mathcal{A}}$, and where $(\widehat{\Gamma})_1$ is the space of one-dimensional representations of $\Gamma$. For the case of stable rank, we have close results. In the process, we give a stable rank estimate for maximal full algebras of operator fields over metric spaces.
address:
- |
Department of Mathematics\
University of Toronto\
100 St. George St., Room 4072\
Toronto, Ontario\
M5S 3G3\
Canada
- |
Department of Mathematical Sciences\
Faculty of Science\
University of the Ryukyus\
Nishihara-cho, Okinawa\
903-0213\
Japan
author:
- Ping Wong Ng
- Takahiro Sudo
title: On the stable rank of algebras of operator fields over metric spaces
---
Introduction
============
Rieffel \[**17**\] introduced the notion of $\emph{stable rank}$ for $C^*$-algebras as the noncommutative version of complex dimension of ordinary topological spaces. It turns out that the stable rank of a unital $C^*$-algebra is the same as its Bass stable rank (see \[**8**\]).
There has been much work in computing the stable ranks of the universal $C^*$-algebras of various connected Lie groups. The greatest progress has been made in the case of type I solvable Lie groups (see \[**20**\], \[**22**\] and \[**23**\]). Roughly speaking, it has been shown that the stable rank of the universal $C^*$-algebra of a type I solvable Lie group $G$ is controlled by the ordinary topological dimension of the space of one-dimensional representations of $G$.
Recently, the stable ranks of the universal $C^*$-algebras of a class of non-type I solvable Lie groups (which include the Mautner group) have been computed (see \[**21**\]).
In this paper, we compute the stable ranks of the universal $C^*$-algebras of a class of non-type I amenable discrete groups. Specifically, our main result is
Let $\Gamma$ be a finitely generated, torsion-free, two-step nilpotent group. Let $C^*(\Gamma)$ be the universal $C^*$-algebra of $\Gamma$. Then
1. $acsr( C^*( \Gamma )) = acsr(C( (\widehat{\Gamma})_1))$,
2. $sr( C( (\widehat{\Gamma})_1)) \leq sr( C^*( \Gamma )) \leq
sr( C( (\widehat{\Gamma})_1)) ) + 1$,
3. if the topological dimension $dim( (\widehat{\Gamma})_1)$ is even, then $sr(C^*(\Gamma)) = sr( C( (\widehat{\Gamma})_1))$.
Here $(\widehat{\Gamma})_1$ is the space of one-dimensional representations of $\Gamma$. Also, for a unital $C^*$-algebra ${\mathcal{A}}$, $sr( {\mathcal{A}})$ is the stable rank of ${\mathcal{A}}$, and $acsr({\mathcal{A}})$ is the absolute connected stable rank of ${\mathcal{A}}$.
We note that for a unital $C^*$-algebra ${\mathcal{A}}$, the absolute connected stable rank of ${\mathcal{A}}$ is numerically the same as the stable rank of the tensor product $C[0,1] \otimes {\mathcal{A}}$.
A key step in our proof of Theorem 1.1, is the following stable rank estimate for algebras of operator fields over metric spaces, which is of independent interest:
Suppose that $X$ is a $\sigma$-compact, locally compact, $k$-dimensional metric space. Suppose that ${\mathcal{A}}$ is a maximal full algebra of operator fields over $X$ with fibre algebras, say, $\{ {\mathcal{A}}_t \}_{ t \in X}$ such that ${\mathcal{A}}_t$ is unital for all $t \in X$. Then the stable rank of ${\mathcal{A}}$ satisfies the inequality $$sr( {\mathcal{A}}) \leq sup_{ t \in X } sr( C([0,1]^k ) \otimes {\mathcal{A}}_t ).$$
We note that Theorems 1.1 and 1.2 generalize results from \[**13**\], where we compute the stable ranks of the universal $C^*$-algebras of the (possibly higher rank) discrete Heisenberg groups.
General references for stable rank are \[**14**\] and \[**17**\]. General references for full algebras of operator fields are \[**6**\], \[**11**\] and \[**24**\]. General references for the representation theory of finitely generated, two-step nilpotent groups are \[**2**\], \[**15**\] and \[**16**\] (also see \[**9**\]).
In what follows, for a $C^*$-algebra ${\mathcal{A}}$, “$sr({\mathcal{A}})$” and “$acsr({\mathcal{A}})$" will denote the stable rank and absolute connected stable rank of ${\mathcal{A}}$ respectively. If, in addition, ${\mathcal{A}}$ is unital, then for every positive integer $M>0$, $Lg_M( {\mathcal{A}})$ will be the set of all $M$-tuples $(a_1, a_2, ..., a_M)$ in ${\mathcal{A}}^M$ such that $\sum_{j=1}^M (a_j)^* a_j$ is an invertible element of ${\mathcal{A}}$. Also, for a metric space $X$, for a point $x \in X$ and real number $r > 0$, “$B(x,r)$" will denote the open ball of radius r (with respect to the metric on $X$) about $x$.
Main results
============
In \[**13**\], we proved the following result:
Suppose that ${\mathcal{A}}$ is a unital maximal full algebra of operator fields with base space the $k$-cube $[0,1]^k$ and fibre algebras, say, $\{ {\mathcal{A}}_t \}_{ t \in [0,1]^k}$. Then the stable rank of ${\mathcal{A}}$ satisfies the inequality $$sr( {\mathcal{A}}) \leq sup_{t \in [0,1]^k } sr( C([0,1]^k) \otimes {\mathcal{A}}_t).$$
The key technique within the proof of Theorem 2.1, was the following technical result, which we state as a lemma:
Suppose that ${\mathcal{A}}$ is a unital maximal full algebra of operator fields with base space $[0,1]$ and fibre algebras, say, $\{ {\mathcal{A}}_t \}_{t \in [0,1]}$. Suppose that $M =_{df} sup_{t \in [0,1]} sr( C[0,1] \otimes {\mathcal{A}}_t )$ is a finite number. Let $q$, $r$ be real numbers, with $0 < q < r < 1$, and let ${\mathcal{A}}([0,r])$ and ${\mathcal{A}}([q,1])$ be the restrictions of the operator fields in ${\mathcal{A}}$ to $[0,r]$ and $[q,1]$ respectively. Now let $\epsilon>0$ be given and suppose that for $j=1,2,...,M$, $\{ a_j(t) \}_{ t \in [0,r]}$ is an operator field in ${\mathcal{A}}([0,r])$ and $\{ b_j(t) \}_{ t \in [q,1]}$ is an operator field in ${\mathcal{A}}([q,1])$ such that
1. $\| a_j(t) - b_j(t) \| < \epsilon$ for all $t \in [q,r]$ and for $j=1,2,..,M$,
2. $\sum_{j=1}^M (a_j(t))^* a_j(t)$ is an invertible element of ${\mathcal{A}}_t$ for all $t \in [0,r]$, and
3. $\sum_{j=1}^M (b_j(t))^* b_j(t)$ is an invertible element of ${\mathcal{A}}_t$ for all $t \in [q,1]$.
Then there are operator fields $\{ c_j(t) \}_{t \in [0,1]}$ in ${\mathcal{A}}$, $j=1,2,...,M$, such that
1. $\| c_j(t) -a_j(t) \| < \epsilon$ for all $t \in [0,r]$ and for $j=1,2,..., M$,
2. $\| c_j(t) - b_j(t) \| < \epsilon$ for all $t \in [q,1]$ and for $j=1,2,...,M$, and
3. $\sum_{j=1}^M (c_j(t))^* c_j(t)$ is invertible in ${\mathcal{A}}_t$ for all $t \in [0,1]$.
By \[**12**\] IV.7 page 85 last paragraph and \[**19**\] Theorem 1.1, we may assume that we have a metric on $X$ such that for every point $x \in X$ and for every real number $r>0$, the boundary of the open ball $B(x,r)$ (with respect to this new metric) is at most $k-1$-dimensional. Henceforth, we will be working with this metric.
Suppose that $X$ is noncompact. Then let $X_{\infty}$ be the one-point compactification of $X$, with point at infinity $\infty$. We may view ${\mathcal{A}}$ as a maximal full algebra of operator fields with base space $X_{\infty}$ and fibre algebras $\{ {\mathcal{A}}_t \}_{t \in X_{\infty} }$, where ${\mathcal{A}}_t$ is the same as before when $t \neq \infty$, and ${\mathcal{A}}_{\infty} = \{ 0 \}$ the zero $C^*$-algebra. Let ${\mathcal{A}}^{+}$ be the unitization of ${\mathcal{A}}$. By \[**10**\] Theorem 1 and Corollary 1, ${\mathcal{A}}^{+}$ is a unital maximal full algebra of operator fields with base space $X_{\infty}$ and fibre algebras $\{ ({\mathcal{A}}_t)_1 \}_{t \in X_{\infty}}$ where $({\mathcal{A}}_t )_1 = {\mathcal{A}}_t$ for $t \neq \infty$ and $({\mathcal{A}}_{\infty})_1 = \mathbb{C}$ (the complex numbers). A continuity structure $\widetilde{{\mathcal{F}}}$ is the set of all operator fields of the form $a + \alpha 1$, where $a$ is in ${\mathcal{A}}$ and $\alpha$ is a complex number. If $X$ is compact, then ${\mathcal{A}}$ will automatically be unital, and (in the arguments that follow) we let $X_{\infty} = X$ and ${\mathcal{A}}^{+} = {\mathcal{A}}$, and we need not consider the point $\infty$ at all.
Now suppose that $M=_{df} sup_{t \in X} sr( C([0,1]^k) \otimes {\mathcal{A}}_t )$ is a finite number. Let $(a_1, a_2, ..., a_M)$ be an $M$-tuple in $({\mathcal{A}}^+)^M$ and let $\epsilon > 0$ be given. By adding a small scalar multiple of the unit if necessary, we may assume that $(a_1, a_2, ..., a_M)$ is nonzero at $\infty$ (noncompact) case. We may also assume that $\epsilon$ is small enough so that for any other $M$-tuple $(c_1, c_2, ..., c_M)$, if $c_j (\infty)$ is within $\epsilon$ of $a_j (\infty)$ for all $j$, then the $M$-tuple $(c_1, c_2, ..., c_M)$ is also not the zero vector at $\infty$.
Now we can choose a sequence of nonempty open balls $\{ B(x_i, r_i) \}_{i=1}^{\infty}$ in $X$ and a sequence of $M$-tuples $\{ (f_{i,1}, f_{i,2},..., f_{i,M} ) \}_{i = 1}^{\infty}$ in $({\mathcal{A}}^+)^M$ such that
1. $X$ is covered by the union of all the open balls $B(x_i, r_i)$, $r = 1,2
......$,
2. for every $i$, $i=1,2,3...$, there is a strictly positive number $\delta_i$ such that $\sum_{j=1}^M f_{i,j}(t)^* f_{i,j}(t)$ is an invertible element of ${\mathcal{A}}_t$ for all $t \in B( x_i, r_i + \delta_i)$, and
3. there is an increasing sequence of integers $\{ N_n \}_{n = 1}^{\infty}$ such that $f_{i,j}$ is within $\epsilon/{2^n}$ of $a_j$ for $i \geq N_n$, $i=1,2,3,....$ and $1 \leq j
\leq M$.
Condition (a) uses the $\sigma$-compactness of $X$. Condition (b) requires the use of existence and continuity of operator fields in full algebras of operator fields (see the definition of full algebras of operator fields in \[**6**\], \[**10**\] or \[**24**\]). One also needs to use the fact that in a unital $C^*$-algebra, any element close enough to the unit is invertible. Condition (c) requires the maximality of the full algebra of operator fields (see \[**10**\] Proposition 1 and \[**24**\] Theorem 1.1) as well as the $\sigma$-compactness of $X$. Henceforth, we let “(+)" denote properties (a) - (c) collectively.
By the $\sigma$-compactness of $X$, we may additionally assume that such that for every $n$, if $i \leq N_n$ and $j > N_n$ then $r_j < (1/8)r_i$, for all $i,j$, i.e., the size of the open balls $B(x_i, r_i)$ are, approximately, “decreasing uniformly" with rate $1/8$. We may also assume that for each $i$, $i=1,2,3,...$, the closure of $B(x_i, r_i + \delta_i)$ is a compact subset of $X$.
Our procedure for constructing an $M$-tuple in $Lg_M( {\mathcal{A}}^+ )$ which will approximate $(a_1, a_2, ..., a_M)$ to within $\epsilon$ is to construct $M$ sequences of operator fields $\{ \alpha_{j}^n \}_{n=1}^{\infty}$ $j=1,2,..., M$, which satisfy the following conditions:
1. $\alpha_{j}^n$ is an operator field over $\bigcup_{i=1}^{N_n} \overline{B(x_i, r_i)}$ for $n=1,2,3,...$ and for $1 \leq j \leq M$,
2. $\alpha_{j}^n$ is within $\epsilon/2$ of $a_j$ (over $\bigcup_{i=1}^{N_n} \overline{B(x_i, r_i)}$), for $n=1,2,3,...$ and for $1 \leq j \leq M$,
3. $\alpha_{j}^n$ is within $\epsilon/{2^n}$ of $a_j$ over $\overline{B(x_{N_n}, r_{N_n})}$,
4. $\sum_{j=1}^M (\alpha_{j}^n (t) )^* \alpha_{j}^n (t)$ is invertible in ${\mathcal{A}}_t$ for all $t \in \overline{B(x_i, r_i)}$ and for $i \leq N_n$, and
5. for $i \leq m \leq n$, $(\alpha_{1}^m, \alpha_{2}^m,..., \alpha_{M}^m)
= (\alpha_{1}^n, \alpha_{2}^n, ..., \alpha_{M}^n)$ over the ball $B(x_i, r_i / 2)$.
We let “(\*)" denote conditions (1) - (4) collectively.
For simplicity, let us assume that for every integer $n$, $N_n = n$. We now construct the operator fields $\{ \alpha_{j}^n \}_{n=1}^{\infty}$, $1 \leq j \leq M$, recursively on $n$ (for all $j$ at each step $n$). For $n=1$, just let $( \alpha_{1}^1, \alpha_{2}^1, ..., \alpha_{M}^1 ) = ( f_{1,1}, f_{1,2}, ...,
f_{1,M})$. Now suppose that $(\alpha_{1}^{n}, \alpha_{2}^{n}, ..., \alpha_{M}^n)$ has been constructed. To construct $(\alpha_{1}^{n+1}, \alpha_{2}^{n+1}, ..., \alpha_{M}^{n+1})$, we need to “connect" $(f_{n+1, 1}, f_{n+1, 2},..., f_{n+1, M})$ with $(\alpha_{1}^{n}, \alpha_{2}^{n}, ..., \alpha_{M}^n)$ over an appropriate subset of $X$. We may assume that $\bigcup_{i=1}^{n} \overline{B(x_i, r_i)}$ is nonempty (for otherwise, it would be immediate).
Let $d$ be the positive real number which is the minimum of the quantities $\delta_{n+1}$ and $r_{n+1}$. Let $F$ be the set of all points $x$ in $\bigcup_{i=1}^n
\overline{B(x_i, r_i)}$ whose distance from $x_{n+1}$ is between (and including) $r_{n+1}$ and $r_{n+1} + d$. For $s \in [0,1]$, let $F_s$ be the set of points in $F$ which have distance $(1-s)r_{n+1} + s(r_{n+1} + d)$ from $x_{n+1}$. Let ${\mathcal{A}}(F)$ be the $C^*$-algebra gotten by taking the restriction of (the operator fields in) ${\mathcal{A}}$ to $F$. Then ${\mathcal{A}}(F)$ can be realized as a unital maximal full algebra of operator fields with base space $[0,1]$ and fibre algebras, say, $\{ B_s \}_{s \in [0,1]}$. For each $s \in [0,1]$, the fibre algebra ${\mathcal{B}}_s$ is the restriction of ${\mathcal{A}}$ to $F_s$, and for each element $a \in {\mathcal{A}}(F)$, its fibre at $s \in [0,1]$ (with respect to this continuous field representation) is the restriction of $a$ to $F_s$. Continuity and maximality follows from the continuity and maximality of the algebra of operator fields ${\mathcal{A}}$.
Therefore, $C[0,1] \otimes {\mathcal{A}}(F)$ can be realized as a unital maximal full algebra of operator fields with base space $[0,1]$ and fibre algebras $\{ C[0,1] \otimes {\mathcal{B}}_s \}_{s \in [0,1]}$. The continuity structure consists of all operator fields of the form $s \mapsto \sum_{i=1}^N f_i \otimes b_i (s)$, where the $f_i$s are in $C[0,1]$ and the $b_i$s are continuous operator fields in ${\mathcal{A}}(F)$ (with respect to the continuous field decomposition of ${\mathcal{A}}(F)$ in the previous paragraph). Hence, by Theorem 2.1, the stable rank of $C[0,1] \otimes {\mathcal{A}}(F)$ satisfies $sr( C[0,1] \otimes {\mathcal{A}}(F)) \leq sup_{s \in [0,1]} sr( C[0,1] \otimes
{\mathcal{B}}_s )$.
But for $s \in [0,1]$, ${\mathcal{B}}_s$ can be realized as a unital maximal full algebra of operator fields with base space $F_s$ (a compact metric space) and fibre algebras $\{ {\mathcal{A}}_t \}_{ t \in F_s}$ (since ${\mathcal{B}}_s$ is the restriction of ${\mathcal{A}}$ to $F_s$). Hence, for $s \in [0,1]$, $C[0,1] \otimes {\mathcal{B}}_s$ can be realized as a unital maximal full algebra of operator fields with base space $F_s$ and fibre algebras $\{ C[0,1] \otimes {\mathcal{A}}_t \}_{t \in F_s }$. But by our assumption on the metric in the first paragraph of this proof, $F_s$ is a metric space with dimension less than or equal to $k-1$. Hence, we have, by induction, that $sr( C[0,1] \otimes {\mathcal{B}}_s ) \leq sup_{t \in F_s} sr( C([0,1]^{k-1}
\otimes C[0,1] \otimes {\mathcal{A}}_t )$ (The induction is on the dimension of the base space. Note that when $F_s$ is zero-dimensional, the stable rank estimate will be immediate, since we can choose a *finite, clopen* covering for $F_s$, which satisfies the properties in (+). Hence the base case is immediate). From this and the previous paragraph, $sr( C[0,1] \otimes {\mathcal{A}}(F) ) \leq M$.
By Lemma 2.2, it follows that there is an $M$-tuple of operator fields $(\alpha_{1}^{n+1}, \alpha_{2}^{n+1}, ..., \alpha_{M}^{n+1})$ on $\overline{B(x_{n+1} , r_{n+1})} \cup \bigcup_{i=1}^n \overline{B(x_i, r_i)}$ such that $\alpha_{j}^{n+1} = f_{n+1, j}$ on $B(x_{n+1}, r_{n+1})$, $\alpha_{j}^{n+1} = \alpha_{j}^n$ on $\bigcup_{i=1}^M B(x_i, r_i) -
F$, and the $\alpha_{j}^{n+1}$s satisfy (1), (2) and (4) in (\*). Condition (5) in (\*) is satisfied, since $d$ was chosen to be less than or equal to $r_{n+1}$, and the latter is strictly less than $(1/8)r_n$. Finally, condition (3) in (\*) is satisfied since $f_{n+1,j}$ is within $\epsilon/{2^{n+1}}$ of $a_j$.
In the general case where $N_{n+1}$ is not necessarily equal to $n+1$, we need to repeat the preceding procedure a finite number of times, in the natural way, in order to go from $(\alpha_{1}^n, \alpha_{2}^n, ...,
\alpha_{M}^n)$ to $(\alpha_{1}^{n+1},
\alpha_{2}^{n+1}, ..., \alpha_{M}^{n+1})$.
Also, when $X$ is compact, the preceding procedure will stop at finitely many steps and the sequences $\{ \alpha_{j}^n \}_{n =1}^{\infty}$, $1 \leq j \leq M$, will all be finite. We leave to the reader the obvious modifications that need to be made.
Now suppose that we have constructed sequences of operator fields $\{ \alpha_{j}^n \}_{n=1}^{\infty}$, $j = 1, 2, ..., M$ as in (\*). We then construct an $M$-tuple of continuous operator fields in $({\mathcal{A}}^+)^M$ as follows: let $\alpha_j (t) = \alpha_{j}^n (t)$ for $t \in B(x_n, {r_n}/2)$ and (in the noncompact case) let $\alpha_j (\infty) = a_j (\infty)$. Continuity at the point $\infty$ is ensured by condition (3) in (\*). Then $\alpha_j$ is within $\epsilon /2$ of $a_j$ for $j=1,2,...,M$. Moreover, that $ \sum_{i=1}^M (\alpha_j(\infty))^* \alpha_j (\infty)$ is invertible follows from the invertibility of $\sum_{i=1}^M (a_j (\infty))^* a_j (\infty)$ and the smallness of the the $\epsilon$, both of which were assumed at the beginning. Hence, $(\alpha_1, \alpha_2, ..., \alpha_M) \in Lg_M ( {\mathcal{A}}^+)$ and $\alpha_j$ is within $\epsilon$ of $a_j$ for $j=1,2,..., M$.
The result of the next computation is surely known (See \[**18**\], the comments after the proof of Proposition 3.10).
If ${\mathcal{A}}_{\Theta}$ is a simple noncommutative torus and ${\mathbb T}^k$ the ordinary $k$-torus, then $sr( C({\mathbb T}^k) \otimes {\mathcal{A}}_{\Theta}) = 2$.
By \[**3**\] Theorem 1.5 and \[**17**\] the proof of Corollary 7.2, $sr( C({\mathbb T}^k) \otimes {\mathcal{A}}_{\Theta})$ is a finite number. Hence by \[**17**\] Theorem 6.1, let $l$ be a positive integer such that both $sr( \mathbb{M}_{2^l}({\mathbb C}) \otimes C({\mathbb T}^k) \otimes {\mathcal{A}}_{\Theta})$ and $sr( \mathbb{M}_{3^l}({\mathbb C}) \otimes C({\mathbb T}^k) \otimes {\mathcal{A}}_{\Theta})$ are less than or equal to $2$. Let ${\mathcal{A}}_{\Theta} =
\overline{\bigcup_{n=1}^{\infty} {\mathcal{A}}_n}$ be the inductive limit decomposition of ${\mathcal{A}}_{\Theta}$ given in \[**3**\] Corollary 2.10.
Now let a positive real number $\epsilon > 0$ and a positive integer $m > 0$ be given. Let $a_1$ and $a_2$ be arbitrary elements of $C({\mathbb T}^k)
\otimes {\mathcal{A}}_m$. Choose an integer $n > m$ such that there are $(b_1, b_2)
\in Lg_2 ( \mathbb{M}_{2^l}({\mathbb C}) \otimes C({\mathbb T}^k) \otimes {\mathcal{A}}_n)$ and $(c_1, c_2) \in Lg_2 ( \mathbb{M}_{3^l}({\mathbb C}) \otimes C({\mathbb T}^k)
\otimes {\mathcal{A}}_n)$, with $b_j$ within $\epsilon$ of $a_j \otimes 1_{{\mathcal M}_{2^l}}$ and $c_j$ within $\epsilon$ of $a_j \otimes 1_{{\mathcal M}_{3^l}}$, $j = 1,2$. Then it follows from the proof of \[**3**\] Corollary 2.10, that we can choose an integer $N > n$ and choose a finite dimensional subalgebra ${\mathcal{B}}\subseteq {\mathcal{A}}_N$ such that there exists $(d_1, d_2) \in Lg_2 (C({\mathbb T}^k) \otimes C^*({\mathcal{A}}_n, {\mathcal{B}}))$ with $d_j$ being within $\epsilon$ of $a_j$ for $j=1,2$. But $\epsilon$, $m$ and $a_j$ were arbitrary. Hence, $sr( C({\mathbb T}^k) \otimes
{\mathcal{A}}_{\Theta}) \leq 2$.
Now $K_1 ({\mathcal{A}}_{\Theta}) = \mathbb{Z}^{2^{p-1}} \neq 0$ where $p$ is the dimension of the noncommutative torus ${\mathcal{A}}_{\Theta}$ (i.e., ${\mathcal{A}}_{\Theta}$ is a noncommutative $p$-torus). So we can find a positive integer $n$ such that $GL_n({\mathcal{A}}_{\Theta}) \neq GL_n ({\mathcal{A}}_{\Theta})_0$, where $GL_n({\mathcal{A}}_{\Theta})$ is the group of invertibles in ${\mathcal M}_n({\mathcal{A}}_{\Theta})$ and $GL_n ({\mathcal{A}}_{\Theta})_0$ is the connected component of the identity in $GL_n({\mathcal{A}}_{\Theta})$. Therefore, the connected stable rank $csr({\mathcal M}_n({\mathcal{A}}_{\Theta}) \geq 2$. But $csr({\mathcal M}_n({\mathcal{A}}_{\Theta}) \leq
sr(\mathbb{M}_{n}({\mathbb C}) \otimes C({\mathbb T}^k) \otimes {\mathcal{A}}_{\Theta})
\leq sr( C({\mathbb T}^k) \otimes {\mathcal{A}}_{\Theta})$. Hence, $sr( C({\mathbb T}^k) \otimes {\mathcal{A}}_{\Theta}) \geq 2$.
Since $C((\widehat{\Gamma})_1)$ is naturally a quotient of $C^*(\Gamma)$, we must have that $sr(C^*(\Gamma)) \geq sr(((\widehat{\Gamma})_1))$ and $acsr(C^*(\Gamma)) \geq acsr(((\widehat{\Gamma})_1))$.
By \[**2**\] page 390 last paragraph and page 391 first paragraph, and by \[**15**\] Theorem 1.2, $C^*(\Gamma)$ can be realized as a unital maximal full algebra of operator fields with base space $\widehat{Z(\Gamma)}$ and fibre algebras, say, $\{ {\mathcal{A}}_{\lambda} \}_{\lambda \in \widehat{ Z(\Gamma )} }$. Here, $Z(\Gamma)$ is the centre of $\Gamma$, and $\widehat{ Z(\Gamma)}$ is the Pontryagin dual of the centre of $\Gamma$. Moreover, the continuous open surjection corresponding to this continuous field decomposition of $C^*(\Gamma)$ is the map $p: Prim(C^* (
\Gamma )) \rightarrow \widehat{ Z(\Gamma)}$ which brings a primitive ideal of $C^*(\Gamma)$ to its restriction to $Z(\Gamma)$.
Also, by \[**2**\] page 390 last paragraph and page 391 first paragraph, by \[**15**\] Theorem 1.2, and by \[**16**\] Theorem 1, for fixed $\lambda \in \widehat{Z(\Gamma)}$, ${\mathcal{A}}_{\lambda}$ (as in the previous paragraph) can in turn be realized as a unital maximal full algebra of operator fields with base space of the form ${\mathbb T}^g \times T$ for some commutative $g$-torus ${\mathbb T}^g$ and finite set $T$. The integer $g$ is less than or equal to the rank of $\Gamma/ Z(\Gamma)$. The fibre algebras are all isomorphic. Let ${\mathcal{B}}_{\lambda}$ be the unique $C^*$-algebra which all the fibre algebras are isomorphic to. Then ${\mathcal{B}}_{\lambda}$ will be either of the form ${\mathcal M}_n( {\mathbb C})$ (a full matrix algebra) or ${\mathcal M}_n ({\mathbb C}) \otimes {\mathcal{A}}_{\Theta}$ where ${\mathcal{A}}_{\Theta}$ is a simple noncommutative torus (the former case will occur if $p^{-1}(\lambda)$ consists of $n$-dimensional representations and the latter will occur if $p^{-1}(\lambda)$ consists of infinite dimensional representations). We note that $g$, $T$ and ${\mathcal{B}}_{\lambda}$ will all depend on $\lambda$.
Now let $\Gamma^{(2)}$ be the commutator subgroup of $\Gamma$ (i.e., the subgroup of $\Gamma$ generated by elements of the form $xy x^{-1} y^{-1}$ where $x,y \in \Gamma$). Let $\Gamma^{(2)}_s$ be the saturation of $\Gamma^{(2)}$ (i.e., the smallest sugroup $H$ of $\Gamma$ containing $\Gamma^{(2)}$ such that for every $x \in \Gamma$, if $x^n \in H$ for some strictly positive integer $n$ then $x \in H$). Since $Z(\Gamma)$ (the centre of $\Gamma$) is a saturated subgroup of $\Gamma$ (i.e., $x \in \Gamma$ and $x^n \in Z(\Gamma)$ for some strictly positive integer $n$ implies that $x \in Z(\Gamma)$), $\Gamma^{(2)}_s$ is a saturated subgroup of $Z(\Gamma)$. Hence, we have a decomposition $Z(\Gamma) = \Gamma^{(2)}_s \oplus F$, where $F$ is a saturated free abelian subgroup of $Z(\Gamma)$. This in turn gives a decomposition $\widehat{Z(\Gamma)} = \widehat{\Gamma^{(2)}_s} \times
\widehat{F}$.
Now let $N$ be a positive integer such that for all $n \geq N$, $sr( {\mathcal M}_n ({\mathbb C}) \otimes C( {\mathbb T}^{h+1} ) ) \leq 2$, where $h$ is the rank of $\Gamma$. Let $S$ be the set of all $\lambda \in \widehat{Z(\Gamma)}$ such that $p^{-1}(\lambda)$ consists of $m$-dimensional representations with $m \leq N$. With respect to the decomposition of $\widehat{Z(\Gamma)}$ given in the previous paragraph, $S$ must have the form $\{ \lambda_1, \lambda_2,...,
\lambda_k \} \times \widehat{F}$, for a finite set of points $\lambda_i \in
\widehat{\Gamma^{(2)}_s}$. (Suppose that $x_1, x_2, ..., x_q$ are elements of $\Gamma$ so that $x_1 / Z(\Gamma), x_1 / Z(\Gamma), ..., x_q / Z(\Gamma)$ give a basis for $\Gamma /Z(\Gamma)$. Suppose that $\pi$ is an $m$-dimensional representation of $\Gamma$. Then the scalar values $\pi (x_i x_j (x_i)^{-1} (x_j)^{-1})$, for $ 1 \leq i, j \leq q$, must all be rational numbers which can be placed in the form $r/q$ where $q \leq m$. These scalar values determine the values of $\pi$ on $\Gamma^{(2)}$ and there are only finitely many possibilities for them. And since $\Gamma^{(2)}$ has finite index in $\Gamma^{(2)}_s$, there are only finitely many possibilities for the restriction of $\pi$ to $\Gamma^{(2)}_s$.)
Let $J$ be the ideal of $C^*(\Gamma)$ consisting of all operator fields which vanish on $S$. Then $J$ is a maximal full algebra of operator fields with base space $\widehat{Z(\Gamma)} - S$. The quotient $C^*(\Gamma) / J$ is a unital maximal full algebra of operator fields with base space $S$. Indeed, $C^*(\Gamma) / J$ is the restriction, to $S$, of the operator fields in $C^*(\Gamma)$. Now from the exact sequence $
0 \rightarrow C[0,1] \otimes J \rightarrow
C[0,1] \otimes C^*(\Gamma) \rightarrow C[0,1] \otimes
C^*(\Gamma)/ J \rightarrow
0,
$ and by \[**7**\] Corollary 2.22, we get that $sr(C[0,1] \otimes C^*(\Gamma) ) = max \{
sr( C[0,1] \otimes J ) , sr ( C[0,1] \otimes C^*(\Gamma)/J ) \}$. By Theorem 1.2, and our definitions of $N$ and $S$, $sr( C[0,1] \otimes J ) \leq
sup_{ \lambda \in \widehat{Z(\Gamma)} - S} sr( C[0,1] \otimes {\mathcal{A}}_{\lambda})$. But for $\lambda \in \widehat{Z(\Gamma)} - S$, we have, by the definitions of $N$ and $S$, by Lemma 2.3, by \[**17**\] Proposition 1.7 and Theorem 6.1, and by our discussion of the continuous field decomposition of ${\mathcal{A}}_{\lambda}$, that $sr( C[0,1] \otimes {\mathcal{A}}_{\lambda} ) \leq 2$. Hence, $sr( C[0,1] \otimes J ) \leq 2$.
Now by Theorem 1.2, $sr( C[0,1] \otimes C^*(\Gamma) / J ) \leq
sup_{\lambda \in S } sr( C[0,1] \otimes {\mathcal{A}}_{\lambda} )$. But for $\lambda \in S$, we have that the continuous field decomposition of ${\mathcal{A}}_{\lambda}$ has base space ${\mathbb T}^g \times T$ where $g$ is less than the rank of $\Gamma / Z(\Gamma)$ and $T$ is a finite set. The fibre algebras are all isomorphic to ${\mathcal M}_m({\mathbb C})$, for some integer $m$. Hence, by Theorem 1.2, for $\lambda \in S, sr( C[0,1] \otimes C^*(\Gamma)) \leq
sr( C({\mathbb T}^{g + 1} ) ) = sr( C[0,1] \otimes C( (\widehat{ \Gamma })_1 )$. From this and the previous paragraph, $acsr( C^*( \Gamma ) ) = acsr( C( ( \widehat{ \Gamma })_1 ))$.
The proofs of the other statements of the theorem now follow from our computation for absoluted connected stable rank.
By \[**17**\] Theorem 4.3 and our result for absolute connected stable rank, we have that $sr( C^*( \Gamma ) ) \leq
sr( C[0,1] \otimes C^*(\Gamma))
= sr( C[0,1] \otimes C((\widehat{\Gamma})_1)))$
But by \[**17**\] Corollary 7.2, $sr( C[0,1] \otimes C^*(\Gamma) )
\leq sr( C^*(\Gamma)) + 1$. Hence, $sr( C( (\widehat{\Gamma})_1))) \leq
sr( C^*( \Gamma )) \leq sr( C( (\widehat{\Gamma})_1))) + 1$.
Also, by \[**17**\] Proposition 1.7, if $dim( (\widehat{\Gamma})_1 )$ is even, then $sr( C[0,1] \otimes C( (\widehat{\Gamma})_1) ) = sr(C((\widehat{\Gamma})_1))$. Hence, if $dim( (\widehat{\Gamma})_1 )$ is even, then $sr(C^*(\Gamma)) =
sr( C((\widehat{\Gamma})_1))$.
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abstract: 'Simulating quantum contextuality with classical systems requires memory. A fundamental yet open question is what is the minimum memory needed and, therefore, the precise sense in which quantum systems outperform classical ones. Here, we make rigorous the notion of classically simulating quantum state-independent contextuality (QSIC) in the case of a single quantum system submitted to an infinite sequence of measurements randomly chosen from a finite QSIC set. We obtain the minimum memory needed to simulate arbitrary QSIC sets via classical systems under the assumption that the simulation should not contain any oracular information. In particular, we show that, while classically simulating two qubits tested with the Peres-Mermin set requires $\log_2 24 \approx 4.585$ bits, simulating a single qutrit tested with the Yu-Oh set requires, at least, $5.740$ bits.'
author:
- Adán Cabello
- Mile Gu
- Otfried Gühne
- 'Zhen-Peng Xu'
title: 'Optimal Classical Simulation of State-Independent Quantum Contextuality'
---
[*Introduction.—*]{}Quantifying the resources needed to simulate quantum phenomena with classical systems is crucial to making precise the sense in which quantum systems provide an advantage over classical ones. While the extra resources needed for simulating entanglement and quantum nonlocality (i.e., the quantum violation of Bell inequalities [@Bell64]) have been studied extensively [@BCT99; @Steiner00; @CGM00; @BT03; @TB03; @Pironio03; @CGMP05], the resources needed to simulate quantum contextuality [@Bell66; @KS67], a natural generalization of quantum nonlocality to the case of nonspacelike separated systems and witnessed by the quantum violation of noncontextuality inequalities [@KCBS08; @Cabello08; @BBCP09; @YO12; @KBLGC12], have been less explored [@K11; @Cabello12; @FK16]. In a nutshell, while simulating quantum nonlocality with classical systems requires superluminal communication [@BCT99; @BT03; @TB03; @Pironio03; @CGMP05], simulating quantum contextuality requires memory [@K11; @Cabello12; @FK16] or, more precisely, the ability of storing and recovering a certain amount of classical information. It is known that, in some cases, the required memory is larger than the information-carrying capacity of the corresponding quantum system [@K11]. The problem is that only lower bounds to the minimum memory are known for some particular scenarios [@K11; @FK16]. In addition, it is not known how the minimum memory scales with, e.g., the size of the set of possible measurements.
A particularly interesting case is that of quantum state-independent contextuality (QSIC) in experiments with sequential measurements [@Cabello08; @BBCP09; @YO12; @KBLGC12] on a single recycled quantum system [@K11; @WLK16; @LMZNACH17]. In this case, a single quantum system is submitted to an unlimited sequence of measurements, randomly chosen from a finite set of measurements, as illustrated in Fig. \[Fig1\]. After each measurement, the outcome is observed and recorded. The set of measurements has the peculiarity of being able to produce contextuality no matter what the initial quantum state of the system is. These sets are called QSIC sets [@CKB15; @CKP16] and, for each of them, there are optimal combinations of correlations for detecting contextuality [@KBLGC12]. The interest of this case comes from the fact that unbounded strings of data with contextual correlations can be produced using a [*single*]{} system initially prepared in an arbitrary state [@LMZNACH17], a situation that strongly contrasts with the case of nonlocality generated through the violation of a Bell inequality, where thousands of spacelike separated pairs of quantum systems in an entangled quantum state are needed. The question we want to address in this Letter is what is the minimal amount of memory a classical system would need to simulate the predictions of quantum theory for QSIC experiments with unlimited sequential measurements. Contrary to the previous approaches [@K11; @FK16], we aim at simulating all statistics arising in quantum theory and not only the perfect correlations leading to a violation of a contextuality inequality. We consider the most general simulation under the restriction that the classical model used for simulation should not contain oracular information, as explained below.
[*Scenario.—*]{}We consider ideal experiments in which successive measurements are performed on a single quantum system at times $t_1<t_2<\cdots$ At each $t_i$, a measurement belonging to a QSIC set is randomly chosen and performed. We assume that the quantum state after the measurement at $t_i$ is the quantum state before the measurement at $t_{i+1}$. The process is repeated infinitely many times. Our aim is to extract conclusions valid for any QSIC set. However, the sake of clarity, we will present our results using two famous QSIC sets.
![Observables in the Peres-Mermin set. $z_1$ denotes the quantum observable represented by the operator $\sigma_z^{(1)} \otimes \openone^{(2)}$. Similarly, $zx$ denotes $\sigma_z^{(1)} \otimes \sigma_x^{(2)}$. Observables in each row or column are mutually compatible and their corresponding operators have four common eigenstates. In the figure these eigenstates are represented by straight lines numbered from $1$ to $24$. For example, quantum state $|2\rangle$ is the one satisfying $\sigma_z^{(1)} \otimes \openone^{(2)} |2\rangle=|2\rangle$, $\openone^{(1)} \otimes \sigma_z^{(2)} |2\rangle=-|2\rangle$, and $\sigma_z^{(1)} \otimes \sigma_z^{(2)} |2\rangle=-|2\rangle$. \[Fig2\]](Fig2.pdf){width="5.6cm"}
[*The Peres-Mermin set.—*]{}The QSIC set with the smallest number of observables known has nine 2-qubit observables and it is shown in Fig. \[Fig2\]. It was introduced by Peres [@Peres90] and Mermin [@Mermin90b] and first implemented in experiments with sequential measurements by Kirchmair [*et al.*]{} [@KZG09] on trapped ions and by Amselem [*et al.*]{} [@ARBC09] on single photons. In addition, it has been recently implemented on entangled photons by Liu [*et al.*]{} [@LHC16].
When one uses this set for unlimited sequential measurements on a single system, from the moment two different observables that are in the same row or column in Fig. \[Fig2\] are measured consecutively, the system remains in one of the 24 quantum states defined in Fig. \[Fig2\]. After that, any other subsequent measurement leaves the system in one of these 24 quantum states and each of them occurs with the same probability.
[*The Yu-Oh set.—*]{}As proven in Ref. [@CKP16], the QSIC set with the smallest number of observables represented by rank-one projectors has 13 single-qutrit observables. It was introduced by Yu and Oh [@YO12] and is a subset of a QSIC set previously considered by Peres [@Peres91]. Its associated optimal noncontextuality inequality was found by Kleinmann [*et al.*]{} [@KBLGC12]. It inspired a photonic experiment by Zu [*et al.*]{} [@ZWDCLHYD12] (see also Amselem [*et al.*]{} [@ABB13]) and was implemented as an experiment with sequential measurements on a single ion by Zhang [*et al.*]{} [@ZUZ13], and, recently, it was used to implement the scheme in Fig. \[Fig1\] by Leupold [*et al.*]{} [@LMZNACH17].
When one uses the Yu-Oh set for unlimited sequential measurements on a single system, if at any point the system is in one of the 13 pure states of the Yu-Oh set and one measures one randomly chosen projector onto one of these 13 states, then the number of possible postmeasurement states does not remain constant but grows with the number of sequential measurements. In fact, some states are more probable than others (see Fig. \[fig:dis\]). This contrasts with the case of the Peres-Mermin set, where the number of possible postmeasurement states is constant and all states are equally probable.
[cccc]{} &\
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[*The notion of simulation and the relation to previous works.—*]{}When talking about a classical simulation of a temporal process, it is important to specify what precisely shall be simulated and which conditions a simulation apparatus should meet. A general strategy for simulating temporal correlations is to use hidden Markov models (HMMs) [@RJ86] or, when deterministic effects are considered, Mealy machines [@Mealy55]. There, the simulation apparatus is always in a definite internal state $k$, and for each internal state $k$, there is an output mechanism (e.g., a table $R_k$ containing all the results of the potential measurements) and an update mechanism (e.g., a table $U_k$ that describes the change of the internal state depending on the measurement). In such a model, however, it can easily happen that the simulation apparatus contains information about the future that cannot be derived from the past. By this we mean the following: consider two persons, where the first one only knows the current internal state of the machine and the second one only knows the past observation of measurements and results. Clearly, if the simulation apparatus simulates all the correlations properly, the first person can predict the future as well as the second person. For many processes, however, it can happen that the first person can predict the future better, e.g., if the given internal state $k$ predicts a deterministic outcome for the next measurement, which cannot be deduced from the past. In this way, a simulation apparatus can contain oracular information (i.e., information that cannot be obtained from the past) [@CEJM10].
For our simulation, we restrict our attention to a simulation without oracular information. This leads to the notion of causal models and, more specifically, to $\varepsilon$ transducers, as explained below. These are also so-called unifilar processes, meaning that they are special HMMs, where the output derived from the internal state $k$ determines the update of the internal state. We note that with more general HMMs the memory required for the simulation can sometimes be reduced [@CEJM10; @LA08] and that such models have been used to simulate the Peres-Mermin set [@K11; @FK16]. Our restriction to causal models, however, is physically motivated by the demand that only the past observations should be used for simulating the future.
[*Tools.—*]{}To calculate the minimum memory that a classical system must have, a key observation is that our ideal experiments are examples of stochastic input-output processes that can be analyzed in information-theoretic terms. A stochastic process $\overleftrightarrow{{\cal Y}}$ is a one-dimensional chain $\ldots,Y_{-2},Y_{-1},Y_0,Y_1,Y_2,\ldots$ of discrete random variables $\{Y_t\}_{t \in \mathbb{Z}}$ that take values $\{y_t\}_{t \in \mathbb{Z}}$ over a finite or countably infinite alphabet ${\cal Y}$. An input-output process $\overleftrightarrow{Y}|\overleftrightarrow{X}$ with input alphabet ${\cal X}$ and output alphabet ${\cal Y}$ is a collection of stochastic processes $\overleftrightarrow{Y}|\overleftrightarrow{X} \equiv \{ \overleftrightarrow{Y} | \overleftrightarrow{x} \}_{\overleftrightarrow{x} \in \overleftrightarrow{{\cal X}}}$, where each such process $\overleftrightarrow{Y}|\overleftrightarrow{x}$ corresponds to all possible output sequences $\overleftrightarrow{Y}$ given a particular bi-infinite input sequence $\overleftrightarrow{x}$. It can be represented as a finite-state automaton or, equivalently, as a hidden Markov process. In our experiment, $x_t$ is the observable measured at time $t$ and $y_t$ is the corresponding outcome. By $\overleftarrow{X}$ we denote the chain of previous measurements, $\ldots,X_{t-2},X_{t-1}$, by $\overrightarrow{X}$ we denote $X_t,X_{t+1},\ldots$, and by $\overleftrightarrow{X}$ we denote the chain $\ldots,X_{t-1},X_{t},X_{t+1},\ldots$. Similarly, $\overleftarrow{Y}$, $\overrightarrow{Y}$, and $\overleftrightarrow{Y}$ denote the past, future, and all outcomes, respectively, while $\overleftarrow{Z}$, $\overrightarrow{Z}$, and $\overleftrightarrow{Z}$ denote the past, future, and all pairs of measurements and outcomes. For deriving physical consequences, we have to consider the minimal and optimal representation of this process.
As proven in Ref. [@BC14], the fact that each of our experiments is an input-output process implies that for each of them there exists a unique minimal and optimal predictor of the process, i.e., a unique finite-state machine with minimal entropy over the state probability distribution and maximal mutual information with the process’s future output given the process’s input-output past and the process’s future input. This machine is called the process’s $\varepsilon$ transducer [@BC14] and is the extension of the so-called $\varepsilon$ machines [@CY89; @SC01]. An $\varepsilon$ transducer of an input-output process is a tuple $({\cal X}, {\cal Y}, {\cal S}, {\cal T})$ consisting of the process’s input and output alphabets ${\cal X}$ and ${\cal Y}$, the set of causal states ${\cal S}$, and the set of corresponding conditional transition probabilities ${\cal T}$. The causal states $s_{t-1} \in {\cal S}$ are the equivalence classes in which the set of input-output pasts $\overleftarrow{{\cal Z}}$ can be partitioned in such a way that two input-output pasts $\overleftarrow{z}$ and $\overleftarrow{z}'$ are equivalent if and only if the probabilities $P(\overrightarrow{Y}|\overrightarrow{X},\overleftarrow{Z}=\overleftarrow{z})$ and $P(\overrightarrow{Y}|\overrightarrow{X},\overleftarrow{Z}=\overleftarrow{z}')$ are equal. The causal states are a so-called sufficient statistic of the process. They store all the information about the past needed to predict the output and as little as possible of the remaining information overhead contained in the past. The Shannon entropy over the stationary distribution of the causal states $H({\cal S})$ is the so-called statistical complexity and represents the minimum internal entropy needed to be stored to optimally compute future measurement outcomes (this quantity generally depends on how our measurements $\overleftrightarrow{X}$ are selected; here, we assume each $X_t$ is selected from a uniform probability distribution). The set of conditional transition probabilities ${\cal T} \equiv \{P(S_{t+1} = s_j,Y_t=y|S_t=s_i,X_t=x)\}$ governs the evolution.
![Classical memory in bits needed to simulate a sequential Yu-Oh and Peres-Mermin experiments, as given by Eq. (\[main\]), as a function of the number of steps, as defined in the caption of Fig. \[fig:dis\] for the case of Yu-Oh (and similarly for the case of Peres-Mermin). Values are obtained from considering all possible measurement sequences of a given number of steps and then analytically calculating the corresponding results and postmeasurement states. \[Fig4\][]{data-label="fig:step10entropy"}](Fig4.pdf){width="7.2cm"}
[*Minimum memory needed to simulate QSIC.—*]{}The $\varepsilon$ transducers of associated with the QSIC experiments have a particular property, namely, that there is a one-to-one correspondence between causal states $s_t$ and quantum states $|\psi_{t}\rangle \in \Phi$, where $\Phi$ is the set of possible states occurring after a measurement (for completeness, a proof is presented in the Supplemental Material). Therefore, the minimum number of bits a finite-state classical machine must have to simulate the predictions of quantum theory for a QSIC experiment with unlimited sequential measurements chosen uniformly at random is given by the Shannon entropy $$H=-\sum_i p_i \log_2 p_i.
\label{main}$$ In , $p_i$ is the probability of each quantum state achievable during the experiment’s occurrence and, in general, depends on the distribution in which different measurements are chosen.
For the Peres-Mermin set, there are 24 causal states, each occurring with equal probability (see Fig. \[Fig2\]). Hence, a simulation with an $\varepsilon$ transducer requires $\log_2(24)=4.585$ bits to imitate a quantum system of 2 qubits. This classical memory is significantly higher than the classical information-carrying-capacity of the quantum system that produces these correlations.
For the Yu-Oh set, the calculations are more involved. The reason is that the longer the measurement sequence is, the more possible quantum states can occur as postmeasurement states. In addition, the quantum states do not occur with the same probability; see Fig. \[fig:dis\]. For small sequences up to length ten, however, all the states and probabilities can be analytically computed. The results imply that if only the last ten measurements and results are included, at least $5.740$ bits are required for the simulation (see Fig. \[Fig4\]).
A proper comparison with the amount of memory required to simulate noncontextual sets is obtained by noticing that the memory required to reproduce the predictions of quantum mechanics when we restrict the measurements to subsets (of the QSIC sets) that cannot produce contextuality is $2$ bits for the Peres-Mermin set and $\log_2 3 \approx 1.585$ bits for the Yu-Oh set. These values are obtained as follows. Contextuality is an impossibility of a joint probability distribution over a single probability space. For sequences of projective measurements, incompatibility implies the nonexistence of a joint probability distribution. Therefore, the memory needed to simulate noncontextual sets is the one required to reproduce the predictions of quantum mechanics for subsets of mutually compatible measurements of the QSIC set, which is $\log_2 d$ bits for any QSIC set of dimension $d$. Notice that contextuality requires incompatibility, but also that measurements can be grouped into mutually compatible subsets so that each measurement belongs to at least two of them. Therefore, simulating a set of incompatible measurements not restricted by these rules may require more memory.
One might conjecture that the minimal memory necessary to classically simulate QSIC must be related to the degree of contextuality. However, the relation is difficult to trace. For example, while the minimal memory necessary to simulate classically the Peres-Mermin and Yu-Oh sets is larger for Yu-Oh, the degree of contextuality that can be measured by, e.g., the ratio between the violation and the noncontextual bound for the optimal noncontextuality inequalities [@KBLGC12] is $1.5$ for Peres-Mermin and $1.107$ for Yu-Oh, showing that contextuality is higher for Peres-Mermin. The same conclusion can be reached by adopting other measures of contextuality [@AB11; @GHHHJKW14]. Therefore, understanding the connection between memory and the degree of contextuality is an interesting open problem that should be addressed in the future. Here, also the effects of noise and imperfections should be considered.
[*Conclusions.—*]{}The question of which classical resources are needed for simulating quantum effects is central for the connection of the foundations of quantum theory with quantum information. By applying the tools of complexity science, we have shown how to calculate the amount of memory a classical system would need to simulate quantum state-independent contextuality in the case of a single quantum system submitted to an infinite sequence of measurements randomly chosen from any finite set. Our result precisely quantifies the quantum vs classical advantage of a phenomenon, quantum state-independent contextuality, discovered 50 years ago and shows how profitable may be combining previously unrelated disciplines, such as complexity and quantum information.
Our result opens a way to test systems for their quantumness. Suppose we have a system whose internal functioning is unknown and that is submitted to sequential measurements for which a classical simulation requires more memory than the one allowed by the Bekenstein bound. Here, the Bekenstein bound refers to the limit on the entropy that can be contained in a physical system with given size and energy [@Bekenstein72]. We may assume that no system can store and process information beyond the Bekenstein bound and can test whether the system is not emitting heat due to Landauer’s principle (which states that the erasure of classical information implies some heat emission [@Landauer61]). If this heat is not found, then our result allows us to certify that the system is in fact quantum and not a classical simulation. Therefore, we can use its quantum features for information processing.
On the other hand, our result could also inspire new techniques in complexity science, where there is a growing interest in the value of quantum theory for simulating otherwise difficult to simulate classical processes (e.g., [@TGVG17; @PGHWP17]). In this respect, our result could pinpoint the properties of classical processes that make them particularly amenable to improved modeling using quantum systems and thus also further catalyze the use of quantum methods in complexity science.
We thank Matthias Kleinmann, Jan-[Å]{}ke Larsson, and Karoline Wiesner for discussions, and Jayne Thompson for help in the Supplemental Material. This work was supported by Project No. FIS2014-60843-P, “Advanced Quantum Information” (MINECO, Spain) with FEDER funds, the project “Photonic Quantum Information” (Knut and Alice Wallenberg Foundation, Sweden), and the Singapore National Research Foundation Fellowship No. NRF-NRFF2016-02. A.C. is also supported by the FQXi Large Grant “The Observer Observed: A Bayesian Route to the Reconstruction of Quantum Theory.” M.G. is also supported by the John Templeton Foundation Grant No. 53914 “Occam’s Quantum Mechanical Razor: Can Quantum Theory Admit the Simplest Understanding of Reality?” and the FQXi Large Grant “Observer-Dependent Complexity: The Quantum-Classical Divergence over ‘What is Complex?’” O.G. is also supported by the DFG and the ERC (Consolidator Grant No. 683107/TempoQ). Z.-P.X. is also supported by the Natural Science Foundation of China (Grant No. 11475089) and the China Scholarship Council.
Supplemental Material {#supplemental-material .unnumbered}
=====================
The $\varepsilon$-transducers associated to quantum state-independent contextuality (QSIC) experiments have a particular property, namely, that there is a one-to-one correspondence between causal states $s_t$ and quantum states $|\psi_{t}\rangle \in \Phi$, where $\Phi$ is the set of possible states occurring after a measurement. This can be seen as follows.
Consider a quantum system measured at discrete time steps $t$ by some chosen rank-$1$ projector $M_t \in \mathcal{X}$ with measurement outcome $y_t \in \{0,1\}$. Let the input-output process $\overleftrightarrow{Y}|\overleftrightarrow{X}$, with input alphabet $\mathcal{X}$ and output alphabet $\mathcal{Y} = \{0,1\}$, be the input-output process associated to the experiment. We define the experiment to be causally complete if it satisfies that: (a) Each measurement choice $M_t$ is made independently, such that $P(M_t = x)$ is finite for each $x \in \mathcal{X}$. (b) The input-output past of the system is sufficient to ascertain exactly the state the system is in. (c) Any two different quantum states of the system attained during the experiment can be distinguished statistically by measurements in $\mathcal{X}$.
Note that (a) is always true when measurements are chosen at random. For finite QSIC sets (b) is generically true, i.e., the probability for a past that does not determine the actual state decreases exponentially with the number of steps. (c) holds for the Yu-Oh and Peres-Mermin sets and also for any QSIC set made of rank-$1$ projectors, and any QSIC set can always be implemented using a QSIC set with only rank-$1$ projectors. The fact that (c) holds for any QSIC set made of rank-$1$ projectors follows from the fact that any state of a QSIC set belongs to, al least, two different basis, and from the fact the attained states are orthogonal to some state of the QSIC set (see, e.g., Fig. 3 in the main text). Then, to distinguish between two attained states $|\psi_{a}\rangle$ and $|\psi_{b}\rangle$ that are not orthogonal to the same state of the QSIC set, one can measure the projection on one state of the QSIC set orthogonal to $|\psi_{a}\rangle$ but not to $|\psi_{b}\rangle$. To distinguish between two attained states that are both orthogonal to the same subspace span by states of the QSIC set, one can measure the projection on one state of the QSIC set orthogonal to that subspace. We can now establish the following:
Let the input-output process $\overleftrightarrow{Y}|\overleftrightarrow{X}$ with input alphabet ${\cal X}$ and output alphabet ${\cal Y}$ be one that describes the input-output behavior of a causally complete measurement procedure. Let $\Phi = \{{|\phi_i\rangle}\}$ be the set of possible states that the quantum system can take after the measurements. Then, $\Phi$ is in one-to-one correspondence with the causal states of $\overleftrightarrow{Y}|\overleftrightarrow{X}$.
Observe that conditions (a) and (b) guarantee that observation of each possible past $\overleftarrow{z} = (\overleftarrow{x}, \overleftarrow{y})$ implies that the actual state can be deduced from the past. Thus, the state of the quantum system in the present does not contain any oracular information – it is possible to determine the state at $t = 0$ entirely by looking at past input-output behavior. Thus the encoding function $s(\overleftarrow{z}) = \phi_{\overleftarrow{z}} \in \Phi$ that maps each possible past to the resulting state of the system in the present exists. Therefore, to establish $\Phi$ is in one-to-one correspondence with the causal states, we need only to show that $s(\overleftarrow{z}) = s(\overleftarrow{z}')$ if and only if they are of coinciding future input-output behavior, i.e., $P(\overrightarrow{Y} | \overrightarrow{X}, \overleftarrow{Z} = \overleftarrow{z})
= P(\overrightarrow{Y} | \overrightarrow{X}, \overleftarrow{Z} = \overleftarrow{z}')$. In both directions we will use proof by contrapositive.
To prove the forward direction, note that the state of the quantum system uniquely determines the distribution over future outcomes $\overrightarrow{Y}$, for any potential measurement sequence $\overrightarrow{x}$. Thus whenever $P(\overrightarrow{Y} | \overrightarrow{x}, \overleftarrow{z}) \neq P(\overrightarrow{Y} | \overrightarrow{x}, \overleftarrow{z}')$ for some $\overrightarrow{x}$, we must have $\phi_{\overleftarrow{z}} \neq \phi_{\overleftarrow{z}'}$. For the reverse direction note that by condition (c), the set of measurements $M_x \in \mathcal{X}$ allows us to distinguish statistically any state. Thus, if $\phi_{\overleftarrow{z}} \neq \phi_{\overleftarrow{z}'}$, then there exists $M \in \mathcal{M}$ such that $\rm{tr} (\rho_{\overleftarrow{z}} M ) \neq \rm{tr} (\phi_{\overleftarrow{z}'}M)$. Thus $\rho_{\overleftarrow{z}} \neq \rho_{\overleftarrow{z}'}$ implies $P(Y_0 | X_0 = M, \overleftarrow{z}) \neq P(Y_0 | X_0 = M, \overleftarrow{z}')$.
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abstract: 'We study the emergence of cooperation in structured populations with any arrangement of cooperators and defectors on the evolutionary graph. Using structure coefficients defined for configurations describing such arrangements of any number of mutants, we provide results for weak selection to favor cooperation over defection on any regular graph with $N \leq 14$ vertices. Furthermore, the properties of graphs that particularly promote cooperation are analyzed. It is shown that the number of graph cycles of certain length is a good predictor for the values of the structure coefficient, and thus a tendency to favor cooperation. Another property of particularly cooperation–promoting regular graphs with a low degree is that they are structured to have blocks with clusters of mutants that are connected by cut vertices and/or hinge vertices.'
author:
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Hendrik Richter\
HTWK Leipzig University of Applied Sciences\
Faculty of Engineering\
Postfach 301166, D–04251 Leipzig, Germany.\
Email: hendrik.richter@htwk-leipzig.de
title: |
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**Evolution of Cooperation for Multiple Mutant Configurations on All Regular Graphs with $N \leq 14$ players**
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Introduction
============
Describing conditions for the emergence of cooperation in structured populations is a fundamental problem in evolutionary game theory [@broom13; @new18; @nowak06; @wu13]. In structured populations the network describing which players interact with each other may be crucial for the fixation of a strategy. Recently, several attempts have been made to explore the universe of interaction graphs in order to link graph properties to fixation. For a single cooperator this question has been studied intensively and recently relationships have been mapped for a large variety of different interaction graphs connecting which strategy is favored with the fixation probabilities and the fixation times [@allen17; @moell19; @pav18; @tka19]. These results clarify for a single mutant the relationships between the graph structure, on the one hand, and fixation probability and fixation time, on the other. The main findings are that generally fixation probability and fixation time is correlated such that a higher fixation probability comes with a higher fixation time. Within this general rule, it has further been shown that generalized stars maximize fixation probability while minimizing fixation time, while comet–kites minimize fixation probability while maximizing fixation time [@moell19]. Furthermore, if we allow self loops and weighted links, we may construct arbitrarily strong amplifiers of selection [@pav18]. Compared with these findings, the problem of multiple cooperators (or more than one mutant) is far less studied. One approach uses configurations and structure coefficients [@chen16] and has shown that cooperation is favored over defection under conditions which can be linked to spectral graphs measures and cooperator path length [@rich19a; @rich19b].
This study deals with strategy selection for multiple mutants on evolutionary graphs and addresses two central questions. The first is to find out which interaction network modeled as a regular graph yields the largest structure coefficient and therefore is most suited to promote the evolution of cooperation. This is reported for all regular graphs with $N \leq 14$ vertices ($=$ players). This question is studied subject to three parameters, the number of players, coplayers and cooperators. Answering this question may inform designing interaction networks with prescribed abilities to promote or suppress cooperation. As there are some trends over varying these three parameters, it appears possible to conjecture for beyond the considered parameters. The second question studied takes up the observation that there are differences in the values of the structure coefficients over regular interaction graphs and asks what makes some graphs different from others in terms of promoting the evolution of cooperation. Our main interest is what these differences are from a graph–theoretical point of view. This goes along with identifying certain properties of regular cooperation–promoting graphs. The main result is that the number of graph cycles of certain length is a good predictor of a large value of the structure coefficient. Especially for a smaller number of coplayers graphs that particularly promote cooperation have rather cycles with small length. Furthermore, these graphs are structured to have blocks that are connected by cut vertices and/or hinge vertices. Cooperators cluster on these blocks and serve as a mutant family that may invade the remaining graph. The study presented here uses structure coefficients, which have been derived for birth–death and death–birth processes [@chen16]. However, as the structure coefficients solely depend on the distribution of cooperators and defectors on the evolutionary graph, they could be, at least in principle, also calculated for other strategy updating processes as long as these processes are not completely random. Thus, the methodology reported here is also applicable for other types of non–imitative dynamics.
The paper is structured as follows. In Sec. 2 the main results are given. In particular, upper and lower bounds on the structure coefficients are presented for all interaction networks modelled as regular graphs with $N\leq 14$ players. Furthermore, it is shown that between maximal structure coefficients (and thus conditions favoring the prevalence of cooperation) and the count of cycles with certain length, there is an approximately linear relationship. The results are discussed in Sec. 3, while the Appendices review the methodological framework of configurations, regular graphs and structure coefficients, discuss graph isomorphism, and give a collection of graphs with maximal structure coefficients.
Evolution of Cooperation
========================
Upper and lower bounds on the structure coefficients
----------------------------------------------------
The structure coefficient $\sigma(\pi,\mathcal{G})$ introduced by Chen et al. [@chen16] (see [@rich19a; @rich19b] for further analysis) is a measure of whether or not cooperation is favored over defection in games with any arrangement of cooperators and defectors on regular evolutionary graphs. More strictly speaking, in an evolutionary game with weak selection and a payoff matrix (\[eq:payoff\]), the fixation probability of cooperation is larger than the fixation probability of defection if $\sigma(\pi,\mathcal{G}) (a-d)> (c-b)$, see also Appendix 1. This condition connects the values of the payoff matrix, the structure of the evolutionary graph $\mathcal{G}$ and the arrangement of cooperators and defectors on this graph expressed by the configuration $\pi$ with long–term prevalence of cooperation. The structure coefficient $\sigma(\pi,\mathcal{G})$ generalizes the structure coefficient $\sigma$ introduced by Tarnita et al. [@tarnita09] which yields the same condition for favoring cooperation, $\sigma (a-d)> (c-b)$, but applies to a single cooperator (or a single mutant). By contrast, $\sigma(\pi,\mathcal{G})$ is valid for any arrangement of cooperators and defectors on the evolutionary graph and specifically for several cooperators (or multiple mutants).
As the structure coefficient varies over configurations $\pi$ and graphs $\mathcal{G}$, it is natural to ask about upper and lower bounds of $\sigma(\pi,\mathcal{G})$. In this paper, we approach this question by checking all $\sigma(\pi,\mathcal{G})$, which appears feasible for a small number of players $N \leq 14$ and all regular graphs with up to 14 vertices. We classify the structure coefficients and graphs with respect to the number of players $N$. Furthermore, the configurations $\pi$ are also grouped according to the number of cooperators $c(\pi)$, $2 \leq c(\pi) \leq N-2$, while the graphs $\mathcal{G}$ are sorted according to the number of coplayers $k$ (which equals the degree of the graph). As the structure coefficients $\sigma(\pi,\mathcal{G})$ vary over configurations *and* graphs $\mathcal{G}$, we may define two bounds. A first is over all $2^N-2$ non–absorbing configurations, which we call $\sigma_{max_i}$. Thus, we obtain for each graph $\mathcal{G}_i$, $i=1,2,\ldots \mathcal{L}_k(N)$, the quantity $\sigma_{max_i}=\underset{\pi}{\max} \: \sigma(\pi,\mathcal{G}_i)$. A second bound, called $\sigma_{max}$, is derived from the first bound and additionally collects over all $\mathcal{L}_k(N)$ regular graphs with a given $N$ and $k$ according to Tab. \[tab:graphs\]. Thus, we get $\sigma_{max}=\underset{i}{\max} \: \sigma_{max_i}$. For the minimum, the bounds are defined like–wise.
Fig. \[fig:sigmax1\] shows the maximal structure coefficient $\sigma_{max}$ and the maximal difference $\Delta \sigma=\sigma_{max}-\sigma_{min}$ over players $N$ and coplayers $k$. As discussed in Appendix 2 these results apply to any instance of a regular graph, for example to random regular graphs. It can be seen that the maximal structure coefficient $\sigma_{max}$ is largest for $k=3$, which is cubic graphs. For $k>3$, the values of $\sigma_{max}$ get gradually smaller. In other words, the more coplayers there are, the smaller is $\sigma_{max}$. Also, for a constant number of coplayers, $\sigma_{max}$ increases with $N$, which is the number of players. The increase, however, gets gradually smaller and converges for $N \rightarrow \infty$ to a constant, which is $\sigma(\pi,\mathcal{G}) \rightarrow \sigma=(k+1)/(k-1)$ [@chen16; @ohts06]. For instance, for $k=3$, the structure coefficients converge to $\sigma(\pi,\mathcal{G}) \rightarrow \sigma=2$.
\(a) (b)
\(a) $N=12$ (b) $N=12$
\(c) $N=14$ (d) $N=14$
In other words, for the thermodynamic limit with an infinite population, prevalence of cooperation only depends on the number of coplayers $k$ of a regular graph, but not on the graph structure or the number and arrangement of cooperators on the graph. The largest difference between maximal and minimal structure coefficient $\Delta \sigma=\sigma_{max}-\sigma_{min}$ we also get for $k=3$. Here, $\Delta \sigma$ increase to a largest values (for instance for $k=3$ this happens for $N=10$) before falling for $N$ getting even larger, converging to $\Delta \sigma=0$ for $N \rightarrow \infty$.
We next analyze the maximal structure coefficients depending on the number of cooperators $c(\pi)$. Thus, the maximum is over all $\#_{c(\pi)}=\left(\begin{smallmatrix} N \\ c(\pi) \end{smallmatrix} \right)$ configurations with the same number of cooperators $2 \leq c(\pi) \leq N-2$ and all regular graphs according to Tab. \[tab:graphs\]. The maximal values of $\sigma_{max}$ and $\Delta \sigma$ are obtained for $c(\pi)=N/2$ for $N$ even and for both $(N+1)/2$ and $(N-1)/2$ for $N$ odd. An exception is $N=12$ and $k=3$, where $\sigma_{max}$ is obtained for $c(\pi)=5$ and $c(\pi)=7$. Furthermore, we get the following results, see Fig. \[fig:sigmax2\] as examples for $N=12$ and $N=14$. The value $\sigma_{max}$ and $\Delta \sigma$ are symmetric with the number of cooperators $c(\pi)$ and generally higher for the number of cooperators and defectors exactly or approximately the same than for a small number of cooperators or a small number of defectors. For the number of coplayers $k$ getting larger, the differences over the number of cooperators $c(\pi)$ for both $\sigma_{max}$ and $\Delta \sigma$ are levelled.
\(a) (b) $N=14$
Apart from the numerical values of the maximal structure coefficients $\sigma_{max}$ and their relations to the number of players $N$, coplayers $k$ and cooperators $c(\pi)$, it is also interesting to know for which of the $\mathcal{L}_k(N)$ graphs the maximal values occurs. We call the graphs for which this happens the $\sigma_{max}$–graphs. Their number is $\#_{\sigma_{max}}$. Tab. \[tab:graphs1\] give the number of $\sigma_{max}$–graphs, $\#_{\sigma_{max}}$, for all $N$ and $k$ considered here, see also Appendix 3 for some examples of $\sigma_{max}$–graphs. If we compare these numbers with the total number $\mathcal{L}_k(N)$ of $k$–regular graphs on $N$ vertices, see Tab. \[tab:graphs\], we observe that $\mathcal{L}_k(N)$ grows much faster than $\#_{\sigma_{max}}$. In other words, the $\sigma_{max}$–graphs become rare as $N$ increases. Fig. \[fig:sig\_log1\] shows the quantity $\#_{log}=- \frac{1}{N^2} \log \left(\frac{\#_{\sigma_{max}}}{(4k-1/4k^2)\mathcal{L}_k(N)} \right)$ over $N$ and $k$ (Fig. \[fig:sig\_log1\]a), and over $c(\pi)$ and $k$ for $N=14$ (Fig. \[fig:sig\_log1\]b). We may conclude that as a rough approximation the ratio $\frac{\#_{\sigma_{max}}}{\mathcal{L}_k(N)}$ falls exponentially in $N$ and polynomially in $k$ for $k \approx N/2$ and $N$ getting larger. Furthermore, observe from Fig. \[fig:sig\_log1\]b that for small and large values of the number of cooperators $c(\pi)$ there is a larger number of graphs that are $\sigma_{max}$–graphs. The $\sigma_{max}$–graphs become rarer for $c(\pi) \approx N/2$, for which but $\sigma_{max}$ is largest.
$\: {}_k \: \backslash \: {}^N$ 6 7 8 9 10 11 12 13 14
--------------------------------- --- --- --- --- ---- ---- ---- ---- ---- --
3 1 0 1 0 1 0 4 0 10
4 0 2 1 1 1 1 2 10 14
5 0 0 2 0 1 0 1 0 1
6 0 0 0 3 2 5 1 2 1
7 0 0 0 0 2 0 4 0 1
8 0 0 0 0 0 5 6 49 4
9 0 0 0 0 0 0 4 0 14
10 0 0 0 0 0 0 0 7 14
11 0 0 0 0 0 0 0 0 4
: The numbers $\#_{\sigma_{max}}$ of graphs with maximal $\sigma_{max}$ for all regular graphs with $\mathcal{L}_k(N)>1$ and $6 \leq N \leq 14$. []{data-label="tab:graphs1"}
(a)
\(b) $N=12$ (c) $N=14$
Relationships between structure coefficients and graph cycles
--------------------------------------------------------------
Recently, Giscard et al. [@gis19] proposed an algorithm to count efficiently the number of cycles with length $\ell$ in a graph: $\mathcal{C}_\ell(N,k)$ with $3 \leq \ell \leq N$. Thus, it is feasible to count $\mathcal{C}_\ell(N,k)$ for all $\mathcal{L}_k(N)$ regular graphs with $N\leq 14$, as given in Tab. \[tab:graphs\]. As an example see Fig. \[fig:graph\_6\_3\] with the count $\mathcal{C}_\ell(6,3)$, $\ell=\{3,4,5,6\}$, for the $\mathcal{L}_3(6)=2$ graphs with $N=6$ and $k=3$. The following discussion is based on taking into account these numerical results.
\(a) $N=12$, $k=3$ (b) $N=12$, $k=8$
\(c) $N=14$, $k=3$ (d) $N=14$, $k=10$
In the previous section, it was shown that the maximal structure coefficients vary over interaction networks modelled as regular graphs, even if the number of players, coplayers and cooperators is constant. Thus, it appears reasonable to assume that some features of the graphs may be associated with these differences. In the following, results are presented in support for an approximately linear relationship between the number of graph cycles with certain length and the maximal structure coefficients. Two previous results can be interpreted as to point at the validity of such a relationship between the number of graph cycles and fixation properties. A first is from evolutionary games on lattice grids [@hau01; @hau04; @lang08; @page00]. For these games, it has been shown that clusters of cooperators have a higher fixation probability than cooperators that are widely distributed on the grid. The location of the cluster on the grid does not matter. As lattice grids can be described by regular graphs (a Von Neumann neighborhood is a 4–regular graph, a Moore neighborhood a 8–regular graph) clusters imply short and closed paths between the nodes of the grid. Furthermore, the grid means an abundance of cycles with even cycle length. A second result is that between the structure coefficients and the path length between the cooperators there is a strong negative correlation [@rich19b]. Cooperator path length is defined as the path length averaged over all pairs of cooperators on the evolutionary graph. If there are more than two cooperators, the cooperator path length has particularly small values if the cooperators cluster next to each other and are linked by loops. Thus, small values of the cooperator path length correspond with the abundance of cycles of certain length.
\(a) $N=12$, $k=3$ (b) $N=12$, $k=8$
\(c) $N=14$, $k=3$ (d) $N=14$, $k=10$
As there are $\mathcal{L}_k(N)$ regular graphs for a given $N$ and $k$, we obtain $\mathcal{L}_k(N)$ maximal structure coefficients $\sigma_{max_i}$, $i=1,2,\ldots\mathcal{L}_k(N)$ together with the same count of cycle length $\mathcal{C}_{\ell_i}(N,k)$. Thus, we may assume for each $i$ a linear relationship $\sigma_{max_i}=\mathcal{C}_{\ell_i}(N,k) x +\epsilon_i$ for some variables $x$ with an error term $\epsilon_i$. To test the validity of this linear relationship, we calculate the residual error $$\text{res}=\frac{1}{\mathcal{L}_k(N)} \| \mathbfcal{C}_\ell x^* - \boldsymbol\sigma_{max} \|, \label{eq:res_error}$$ where $\mathbfcal{C}_\ell$ comprises of all $\mathcal{L}_k(N)$ cycle length $\mathcal{C}_{\ell_i}(N,k)$ and $\boldsymbol\sigma_{max}$ contains all $\mathcal{L}_k(N)$ structure coefficients $\sigma_{max_i}$ for a given $N$ and $k$. The variable $x^*$ is the solution of the non–negative least square problem $$x^*=\arg \min_x \| \mathbfcal{C}_\ell x - \boldsymbol\sigma_{max} \|.$$ As the length of $x^*$ varies with varying $\mathcal{L}_k(N)$, the residual error in (\[eq:res\_error\]) is weighted by $\mathcal{L}_k(N)$ to make it comparable over all $N$ and $k$. Note that the residual error (\[eq:res\_error\]) gives equivalent results to the root–mean–square deviation, which is also sometimes used to measure the accuracy of a (linear) model. The results are given in Fig. \[fig:sig\_res\]. We see that the residual error $\text{res}$ is small for all $6 \leq N \leq 14$, $3 \leq k \leq N-3$ and gets even smaller for $N$ getting larger. Generally, the error $\text{res}$ is slightly larger for $k=3$ and $k=N-3$ than for intermediate values of $k$. This is also true for calculating $\text{res}$ for each number of cooperators $c(\pi)$, see Figs. \[fig:sig\_res\]b and \[fig:sig\_res\]c, which show the results for $N=12$ and $N=14$. For $N=14$ the values of $\text{res}$ are generally smaller than for $N=12$ and the largest values of $\text{res}$ are obtained for small and large $k$ for all $c(\pi)$. To conclude we can observe that the results for the residual error $\text{res}$ are generally very small, which is equivalent to saying that the error term $\epsilon_i$ in the assumed linear relationship $\sigma_{max_i}=\mathcal{C}_{\ell_i}(N,k) x +\epsilon_i$ has an expected value $\mathbb{E}(\epsilon_i) \approx 0$. Thus, there is some justification to observe that between the maximal structure coefficients $\sigma_{max_i}$ and the cycle count $\mathcal{C}_{\ell_i}(N,k)$ there is an approximately linear relationship.
Finally, another aspect of the interplay between graph structure and fixation properties should be highlighted. To begin with, we analyze the cycle count $\mathcal{C}_{\ell}(N,k)$ of $\sigma_{max}$–graphs, which are those graphs among the $\mathcal{L}_k(N)$ regular graphs that have maximal structure coefficients. Consider the example $N=12$ and $k=3$. There are $\mathcal{L}_3(12)=85$ graphs of which $\#_{\sigma_{max}}=4$ are $\sigma_{max}$–graphs, compare Tab. \[tab:graphs1\] with Tab. \[tab:graphs\]. For these 4 graphs we analyze how the count $\mathcal{C}_{\ell}(12,3)$ is distributed over $\ell=3,4,\ldots,12$. A possible way to visualize such an analysis is based on schemaballs [@krz04; @rich19a], see Fig. \[fig:balls\]a. In such a schemaball we draw Bezier curves connecting the count $\mathcal{C}_\ell(N,k)$ in the upper half of the ball with the associated cycle length $\ell$ in the lower half. The actual values of both $\ell$ and $\mathcal{C}_\ell(N,k)$ are written on the ball. The curves are colored in such a way that equal values of the cycle length $\ell$ have the same (and specific) color, no matter to which cycle count $\mathcal{C}_\ell(N,k)$ they are belonging. The colors are selected equidistant from a RGB color wheel. If there are several $\sigma_{max}$–graphs, as there are $\#_{\sigma_{max}}=4$ for $N=12$, $k=3$ in Fig. \[fig:balls\]a, each graph has its own set of curves between $\ell$ and $\mathcal{C}_\ell$. The schemaball thus contains all of them, which means there may be curves between the same value of $\ell$ and several $\mathcal{C}_\ell$ (and vice versa). For instance, in Fig. \[fig:balls\]a showing the schemaball for $N=12$ and $k=3$, we see that for $\ell=3$, which is cycles of length 3, also known as triangle, we find connection to $\mathcal{C}_3(13,3)=(2,3,4,5)$. This means each of the $\#_{\sigma_{max}}=4$ graphs has triangle, one has 2 of them, another one has 3, still another one has 4 and the final one has 5 triangle.
From the visualization using a schemaball it can be immediately seen that for $N=12$ and $k=3$ small cycles lengths, that is $\ell=\{3,4,\ldots,7\}$, have generally a count $\mathcal{C}_\ell(12,3)>0$. For large cycle lengths, that is $\ell=\{8,9,\ldots,12\}$, we have $\mathcal{C}_\ell(12,3)=0$. For $N=14$ and $k=3$, see Fig. \[fig:balls\]c, we get very similar results. By contrast, for larger $k$, not only the cycle count $\mathcal{C}_\ell(N,k)$ is much higher than for lower $k$, but also the distribution over cycles lengths $\ell$ is quite different, see the examples $N=12$, $k=8$, Fig. \[fig:balls\]b and $N=14$, $k=10$, Fig. \[fig:balls\]d. Here, small as well as large cycle lengths $\ell$ have a substantial count $\mathcal{C}_\ell(N,k)$. Moreover, every cycles length $\ell$ is connected to a distinct interval of $\mathcal{C}_\ell(N,k)$. This means that the $\sigma_{max}$–graphs have very similar counts $\mathcal{C}_\ell(N,k)$ for each $\ell$. These properties becomes even more clear if we additionally consider the schemaballs for $\sigma_{min}$–graphs, which are the graphs with minimal structure coefficients see Fig. \[fig:balls1\] for the same examples as Fig. \[fig:balls\]. Not only there are more $\sigma_{min}$–graphs than $\sigma_{max}$–graphs, (for instance 77 vs. 4 for $N=12$, $k=3$, or 359 vs. 6 for $N=12$, $k=8$), the balls for small $k$ look very different, compare Figs. \[fig:balls1\]a and \[fig:balls1\]c with Figs. \[fig:balls\]a and \[fig:balls\]c. For the $\sigma_{min}$–graphs and small $k$ even large cycle length $\ell$ have a substantial count $\mathcal{C}_\ell(N,k)$. The count is actually much higher, which means that $\sigma_{min}$–graphs have generally more cycles of a given length than $\sigma_{max}$–graphs. On the other hand, for large $k$ the differences are rather marginal. The only difference is that the schemaballs are more dense, which means that $\sigma_{min}$–graphs have more different counts for a given cycle length than $\sigma_{max}$–graphs. For the other tested number of players $N \leq 14$ similar results are obtained as shown in Figs. \[fig:balls\] and \[fig:balls1\]. We next discuss some implications of these results on the evolution of cooperation on regular evolutionary graphs.
Discussion and Conclusions
==========================
In this paper structure coefficients $\sigma(\pi,\mathcal{G})$ introduced by Chen et al. [@chen16] (see [@rich19a; @rich19b] for further analysis) are studied for all regular interaction graphs with $N\leq 14$ players and $3 \leq k \leq N-3$ coplayers. These structure coefficients provide a simple condition connecting long–term prevalence of cooperation with the values of the payoff matrix (\[eq:payoff\]), the structure of the evolutionary graph $\mathcal{G}$ and the arrangement of any number of cooperators and defectors on this graph, which is expressed by the configuration $\pi$. Cooperation is favored for weak selection and a configuration $\pi$ on a graph $\mathcal{G}$ if $$\sigma(\pi,\mathcal{G})>\frac{c-b}{a-d}. \label{eq:cond1}$$ For $\sigma(\pi,\mathcal{G})<1$, the game favors the evolution of spite, which can be seen as a sharp opposite to cooperation. For $\sigma(\pi,\mathcal{G})=1$, the condition (\[eq:cond1\]) matches the standard condition of risk–dominance. For $\sigma(\pi,\mathcal{G})>1$, the diagonal elements of the payoff matrix (\[eq:payoff\]), $a$ and $d$, are more critical than the off–diagonal elements, $b$ and $c$, for determining which strategy is favored. For instance, cooperation can be favored in the Prisoner’s Dilemma game, which is specified by $c>a>d>b$. The condition (\[eq:cond1\]) implies that a larger value of $\sigma(\pi,\mathcal{G})$ still allows cooperation to emerge if $a-d$ is small (or $c-b$ is large). For the Stag Hunt game (Coordination game), characterized by $a>c \geq d>b$, the condition $\sigma(\pi,\mathcal{G})>1$ means to favor a Pareto–efficient strategy ($a>d$) over a risk–dominant strategy ($a+b<c+d$). Again, a larger value of $\sigma(\pi,\mathcal{G})$ tolerates a smaller Pareto–efficiency $a-d$. Put differently, cooperation is favored even if the difference between reward and punishment is rather low. A generalization of these discussions can be achieved by the universal scaling approach for payoff matrices that facilitates studying a continuum of social dilemmas [@wang15]. According to this approach a larger value of $\sigma(\pi,\mathcal{G})$ implies a larger section of the parameter space spanned by gamble–intending and risk–averting dilemma strength [@rich19c]. Based on this interpretation of the structure coefficient $\sigma(\pi,\mathcal{G})$, we review the following major results of the numerical experiments presented in Sec. 2.
- There is an approximately linear relationship between maximal structure coefficients and the count of cycles of the interaction graph with certain length. Moreover, the number of $\sigma_{max}$–graphs grows much slower for a rising number of players than the number of $k$–regular graphs on $N$ vertices. Thus, graphs with maximal structure coefficients get rare for the number of players $N$ getting large.
- The values of the structure coefficients are larger for a small number of coplayers, that is for graphs with a small degree, and maximal for $k=3$, which is cubic graphs, than for larger numbers of coplayers. This is also the case for the largest differance between maximal and minimal structure coefficients. Thus, for regular evolutionary graphs describing the interactions between players, the results for $N \leq 14$ players suggest that a smaller number of coplayers is particularly prone to promote cooperation if a favorable graph is selected. The selection of graphs does matter less for a larger number of coplayers. The $\sigma_{max}$–graphs with small numbers of coplayers $k$ not only have largest maximal structure coefficients, they are also characterized by the absence of cycles with a length above a certain limit, see examples in the collection of $\sigma_{max}$–graphs in Appendix 3.
- There are not only no long cycles in $\sigma_{max}$–graphs with small $k$. The graphs are also structured into blocks that are connected by cut vertices and/or hinge vertices. A cut vertex is a vertex whose removal disconnects the graph, while a hinge vertex is a vertex whose removal makes the distance longer between at least two other vertices of the graphs [@chang97; @ho96]. For instance, for $N=12$ and $k=3$, the vertices occupied by the players $\mathcal{I}_3$ and $\mathcal{I}_9$, see Fig. \[fig:graph\_12\_3\], are cut vertices, while for $N=10$ and $k=4$, see Fig. \[fig:graph\_10\_x\]b, the vertices occupied by the players $\mathcal{I}_5$ and $\mathcal{I}_6$ are hinge vertices as their removal would make the distance between $\mathcal{I}_4$ and $\mathcal{I}_7$ longer. The blocks are occupied by clusters of cooperators. The clusters can be seen as to serve as a mutant family that invades the remaining graph. As vertices with players of opposing strategies are connected by cut and/or hinge vertices there is only a small number of (or even just a single) migration path for the cooperators and/or defectors. A similar observation has been reported for evolutionary games on lattices grids [@hau01; @lang08], see also the discussion in Sec. 2.2. To summarize: the results suggest that $\sigma_{max}$–graphs for small numbers of coplayers have some distinct graph–theoretical properties. Searching for these properties in a given graph may inform the design of interactions graphs that are either particularly prone to cooperation or particularly opposed to it.
- The property of missing long cycles is also a possible explanation as to why regular graphs with small degree differ substantially from graphs with larger degree in terms of promoting cooperation in evolutionary games. A larger degree makes it impossible to have blocks that are connected by only a few edges. As the number of edges increases linearly with the degree by $kN/2$ and each vertex has the same number of edges, there is an ample supply on connections. These results imply that connectivity properties of the interaction graph play an important role in the emergence of cooperation. It may be interesting to see if these connectivity issues may possibly also show in algebraic graph measures, for instance algebraic connectivity expressed by the Fiedler vector.
The results given above show a clear dependency between the long–term prevalence of cooperation in evolutionary games on regular graphs and some of their graph–theoretical properties, which generally confirm previous findings on clusters of cooperators in games on lattice grids [@hau01; @hau04; @lang08; @page00], on pairs of mutants on a circle graph ($k=2$) [@xiao19], and on short cooperator path lengths on some selected regular graphs with $N=12$ and $k=3$, among them the Frucht, the Tietze and the Franklin graph [@rich19b]. However, apart from statements about the prevalence of cooperation there are also other quantifiers of evolutionary dynamics that are highly relevant. In other words, some of the difficulty in the given approach for evaluating the emergence of cooperation in evolutionary games on graphs arises from structure coefficients merely treating a comparison of fixation probabilities. The condition indicates that the fixation probability of cooperation is higher than the fixation probability of defection. This, however, does not entail the values of these probabilities. However, structure coefficients can be calculated with polynomial time complexity [@chen16], while computing fixation probabilities is generally intractable due to an exponential time complexity [@hinder16; @ibs15; @vor13]. In other words, by using the approach involving structure coefficients, we exchange computational tractability by obtaining just a comparison of fixation probabilities instead of their exact values. Moreover, apart from the difference in the information obtained, the variety in the descriptive power of the structure coefficients as compared to the fixation probabilities is salient in another way. Most likely, there is a rather complex relationship between structure coefficients and fixation probability, which is illustrated by the example of a single cooperator for which the structure coefficient does precisely not imply unique values of the fixation probability of cooperation. For a single cooperator we get a single value of the structure coefficient, but fixation probabilities vary over initial configurations as shown for the Frucht and for the Tietze graph [@mcavoy15].
All these considerations show that calculating fixation probabilities and fixation times for multiple mutant configurations is not only computationally expensive, but also has a huge number of possible setups, for instance, which one of the considerable number of graphs to analyze, or where to place cooperators on the evolutionary graph and how many. There are various experimental parameters to be taken into account, which might be why so far systematically conducted numerical studies are sparse. In this sense, another contribution of this paper might be seen in pointing at settings for numerical experiments calculating fixation probabilities and fixation times. The results given in this paper show that among all the regular interaction graphs with $N\leq 14$ players and $3 \leq k \leq N-3$ coplayers, there is a comparably small number of graphs (as given in Tab. \[tab:graphs1\]) which favor cooperation more than others. It may be interesting to see if these graphs also stand out in terms of fixation probability and fixation time as compared to a graph randomly drawn from the other ones.
Acknowledgments: {#acknowledgments .unnumbered}
================
I wish to thank Markus Meringer for making available the `genreg` software [@mer99] used for generating the regular graphs according to Tab. \[tab:graphs\] and for helpful discussions.
Appendix A Configurations, regular graphs and structure coefficients {#appendix-a-configurations-regular-graphs-and-structure-coefficients .unnumbered}
====================================================================
The co–evolutionary games we consider here have $N$ players $\mathcal{I}= \{ \mathcal{I}_i \}$, $i=1,2,\ldots,N$, that each uses either of two strategies $\pi_i \in \{C,D \}$, which we may interpret as cooperating or defecting. Each player $\mathcal{I}_i$, which interacts with a coplayer $\mathcal{I}_j$, receives payoff according to the $2 \times 2$ payoff matrix $$\bordermatrix{\: {}_i \: \backslash \: {}^j & C & D \cr
C & a & b \cr
D & c & d \cr}. \label{eq:payoff}$$ Which player interacts with whom is described by the interaction graph $\mathcal{G}=(V,E)$, where the vertices $v_i \in V$ represent the players and the edges $e_{ij} \in E$ indicate that the players $\mathcal{I}_i$ and $\mathcal{I}_j$ interact as mutual coplayers [@lieb05; @ohts07; @rich17]. Which strategy is used by which player at a given point of time is specified by a configuration $\pi=(\pi_1,\pi_2,\ldots,\pi_N)$ with $\pi_i \in \{C,D \}$. If we represent the two strategies by a binary code $\{C,D \} \rightarrow \{1,0 \}$, a configuration appears as a binary string the Hamming weight of which denotes the number of cooperators $c(\pi)$. For games with $N$ players, there are $2^N$ configurations with 2 configurations ($\pi=(00\ldots 0)$ and $\pi=(11\ldots1)$) absorbing. Players may update their strategies in an updating process, for instance death–birth (DB) or birth–death (BD) updating [@allen14; @patt15]. Recently, it was shown by Chen et al. [@chen16] that strategy $\pi_i=1=C$ is favored over $\pi_i=0=D$ if $$\sigma(\pi,\mathcal{G}) (a-d)> (c-b). \label{eq:cond}$$ This results applies to weak selection and $2 \times 2$ games with $N$ players, payoff matrix (\[eq:payoff\]), any configuration $\pi$ of cooperators and defectors and for any interaction network modeled by a simple, connected, $k$–regular graph.
The quantity $\sigma(\pi,\mathcal{G})$ in Eq. (\[eq:cond\]) is the structure coefficient of the configuration $\pi$ and the graph $\mathcal{G}$. It may not have the same value for different arrangements of cooperators and defectors described by the configuration $\pi$ and also for different interaction networks modeled by a regular graph $\mathcal{G}$. In particular, it was shown that for weak selection and the graph $\mathcal{G}$ describing interaction as well as replacement graph, the structure coefficient $\sigma(\pi,\mathcal{G})$ can be calculated with time complexity $\mathcal{O}(k^2N)$ for DB and BD updating [@chen16]. For DB updating there is $$\sigma(\pi,\mathcal{G})=\frac{N\left(1+1/k \right) \overline{\omega^1} \cdot \overline{\omega^0}-2\overline{\omega^{10}}-\overline{\omega^1 \omega^0} }{N\left(1-1/k \right) \overline{\omega^1} \cdot \overline{\omega^0}+\overline{\omega^1 \omega^0}}, \label{eq:sigma}$$ with 4 local frequencies ($\overline{\omega^1}$, $\overline{\omega^0}$, $\overline{\omega^{10}}$ and $\overline{\omega^1 \omega^0}$), which depend on $\pi$ and $\mathcal{G}$, see [@chen16; @rich19a; @rich19b] for a probabilistic interpretation of these frequencies. Our focus here is on DB updating as it has been shown that BD updating never favors cooperation [@chen16].
$\: {}_k \: \backslash \: {}^N$ 6 7 8 9 10 11 12 13 14
--------------------------------- --- --- --- ---- ---- ----- ------- --------- ------------ --
3 2 0 5 0 19 0 85 0 509
4 1 2 6 16 59 265 1.544 10.778 88.168
5 1 0 3 0 60 0 7.848 0 3.459.383
6 0 1 1 4 21 266 7.849 367.860 21.609.300
7 0 0 1 0 5 0 1.547 0 21.609.301
8 0 0 0 1 1 6 94 10.786 3.459.386
9 0 0 0 0 1 0 9 0 88.193
10 0 0 0 0 0 1 1 10 540
11 0 0 0 0 0 0 1 0 13
12 0 0 0 0 0 0 0 1 1
13 0 0 0 0 0 0 0 0 1
: The numbers $\mathcal{L}_k(N)$ of simple connected $k$–regular graphs on $N$ vertices, [@mer99], which corresponds to the number of regular interaction graphs with $N$ players and $k$ coplayers for $6 \leq N \leq 14$ and $3 \leq k \leq N-1$. Note that there is more than one graph, $\mathcal{L}_k(N)>1$, only for $k \leq N-3$. []{data-label="tab:graphs"}
Appendix B Isomorphic graphs, isomorphic configurations and cycle counts {#appendix-b-isomorphic-graphs-isomorphic-configurations-and-cycle-counts .unnumbered}
========================================================================
The structure coefficient $\sigma(\pi,\mathcal{G})$, as for instance defined for DB updating by Eq. (\[eq:sigma\]), may vary over configurations $\pi$ and graphs $\mathcal{G}$. This suggests the question of upper and lower bounds of $\sigma(\pi,\mathcal{G})$. For a rather low number of players it appears feasible to check all $\sigma(\pi,\mathcal{G})$, as demonstrated in the paper for $N \leq 14$ and all regular graphs with up to 14 vertices. For a $2 \times 2$ game with $N$ players, there are $2^N-2$ non–absorbing configurations $\pi$. These configurations can be grouped according to the number of cooperators $c(\pi)$, $2 \leq c(\pi) \leq N-2$. The number of simple, connected regular graphs is known for small numbers of vertices, e.g. [@mer99], see Tab. \[tab:graphs\]. Note that these numbers apply to graphs that are all not isomorphic with each isomorphism class being represented by exactly one graph. In other words, Tab. \[tab:graphs\] also gives the number of isomorphism classes for all $6 \leq N \leq 14$ and $3 \leq k \leq N-1$. Isomorphism refers to the property that two graphs are structurally alike and merely differ in how the vertices and edges are named. More precisely, two graphs are isomorphic if there is a bijective mapping $\theta$ between their vertices which preserves adjacency [@bon08], pp. 12–14.
Consider, for example, the $\mathcal{L}_3(6)=2$ interaction graphs with $N=6$ players, each with $k=3$ coplayers, see Fig. \[fig:graph\_6\_3\]. For the graph in Fig. \[fig:graph\_6\_3\]a we get the maximal structure coefficient $\sigma_{max}=1.1818$ for 2 configurations, $\pi=(111000)$ as shown in Fig. \[fig:graph\_6\_3\]a and $\pi=(000111)$. By the isomorphism $\theta=\left(\begin{smallmatrix}v_1 &v_2&v_3&v_4&v_5&v_6 \\ 1 &2&3&4&5&6 \\ 6 & 1 &2 &3&4&5 \end{smallmatrix} \right)$, we obtain an isomorphic graph as shown in Fig. \[fig:graph\_6\_3\]b. For this graph, the configuration $\pi=(111000)$ has $\sigma=1.0000$, but $\pi=(110001)$ and $\pi=(001110)$ have $\sigma_{max}=1.1818$. Note that between the configurations with $\sigma_{max}$ the same isomorphic mapping $\theta$ applies. In other words, the structure coefficients are invariant under isomorphic mappings. For each pair of isomorphic graphs, there are isomorphic configurations that have the same value of the structure coefficient. For the graph in Fig. \[fig:graph\_6\_3\]c, we obtain the result that the structure coefficient is constant over all configurations (except the absorbing configurations). Thus, isomorphic transformations do not alter the values of $\sigma(\pi,\mathcal{G})$.
\(a) (b) $ \hspace{2.7cm} (c) $
These results apply generally to structure coefficients $\sigma(\pi,\mathcal{G})$ of regular graphs. The local frequencies in Eq. (\[eq:sigma\]) solely depend on counting two types of paths on the interaction graph [@chen16; @rich19a; @rich19b]. The quantities $\overline{\omega^1}$, $\overline{\omega^0}$ and $\overline{\omega^1 \omega^0}$ relate to the number of paths with length 1 that connect any vertex with adjacent vertices that hold a cooperator (or defector). The quantity $\overline{\omega^{10}}$ relates to the number of paths with length 2 from any vertex to adjacent vertices on which the first vertex of the path holds a cooperator and the second vertex holds a defector. As an isomorphic reshuffling of vertices preserves adjacency, these numbers stay the same if the isomorphism acts on both the vertices and the configurations. Thus, suppose two graphs $\mathcal{G}_i$ and $\mathcal{G}_j$ are isomorphic with isomorphism $\theta$. Then, it follows $\sigma(\pi,\mathcal{G}_i)=\sigma(\theta(\pi),\mathcal{G}_j)$. Furthermore, the maximal structure coefficient is invariant as well, that is for isomorphic graphs $\mathcal{G}_i$ and $\mathcal{G}_j$ there is $\sigma_{max_i}=\underset{\pi}{\max} \: \sigma(\pi,\mathcal{G}_i)=\sigma_{max_j}=\underset{\pi}{\max} \: \sigma(\pi,\mathcal{G}_j)$. Any regular graph belongs to one of the isomorphism classes and can be obtained by isomorphic transformations by any member of this class. Regular interaction graphs that are isomorphic have the same distribution of structure coefficients $\sigma(\pi,\mathcal{G})$ over the number of cooperators $c(\pi)$. Thus, by considering one representative of each isomorphism class, we can make statements about structure coefficients for all regular graphs.
For each graph, there is a specific count $\mathcal{C}_\ell(N,k)$ of cycles with length $\ell$, $3 \leq \ell \leq N$. There are efficient algorithms to count these cycles [@gis19]. Consider again the $\mathcal{L}_3(6)=2$ graphs with $N=6$ players and $k=3$ coplayers, see Fig. \[fig:graph\_6\_3\]. We find the graph in Fig. \[fig:graph\_6\_3\]a and Fig. \[fig:graph\_6\_3\]b has $\mathcal{C}_{\ell_1}(6,3)=(2,3,6,2)$ with $\ell=\{3,4,5,6\}$ (there are 2 cycles of length $\ell=3$, 3 cycles of length $\ell=4$, 6 cycles of length $\ell=5$ and so on), while the graph in Fig. \[fig:graph\_6\_3\]c has $\mathcal{C}_{\ell_2}(6,3)=(0,9,0,6)$. It generally applies that isomorphic graphs have the same $\mathcal{C}_\ell(N,k)$. Graphs that are not isomorphic have frequently a distinct count $\mathcal{C}_\ell(N,k)$, but there are also cases, particularly for $N$ getting larger, where 2 not isomorphic graphs have the same count $\mathcal{C}_\ell(N,k)$.
Appendix C Collection of $\sigma_{max}$–graphs with $N \leq 14$ {#appendix-c-collection-of-sigma_maxgraphs-with-n-leq-14 .unnumbered}
===============================================================
We here give a collection of selected $\sigma_{max}$–graphs with $N \leq 14$. The graphs are shown to illustrate some graph–theoretical properties associated with prevalence of cooperation. The single $\sigma_{max}$–graph with $N =6$ is already shown in Fig. \[fig:graph\_6\_3\]a. For $N=7$, there are $\mathcal{L}_4(7)=2$ regular graph, which both have the same maximal structure coefficients. In other words, the count of graphs equals the count of $\sigma_{max}$–graph, which is why they are not included in the collection.
\(a) $k=3$ (b) $k=4$ (c) $k=5$ (d) $k=5$
\(a) $k=4$ (b) $k=6$ (c) $k=6$ (d) $k=6$
\(a) $k=3$ (b) $k=4$ (c) $k=5$
\(d) $k=6$ (e) $k=6$ (f) $k=7$ (g) $k=7$
\(a) (b) (c) (d)
\(a) $k=4$ (b) $k=4$ (c) $k=6$
\(a) (b) (c) (d)
\(e) (f) (g) (h)
Figs. \[fig:graph\_8\_x\]–\[fig:graph\_10\_x\] shown all $\sigma_{max}$–graphs for $N=8,9,10$ and $3 \leq k \leq N-3$ together with $\sigma_{max}$ and the associated configurations. For $N=12$ and $N=14$, only some examples of $\sigma_{max}$–graphs are given in Figs. \[fig:graph\_12\_3\]–\[fig:graph\_14\_3\] due to brevity. A full list of all $\sigma_{max}$–graphs for $11 \leq N \leq 14$ and $3 \leq k \leq N-3$ is made available here [@rich20]. It is particularly noticeable that the $\sigma_{max}$–graphs are structured to have blocks with clusters of mutants. For instance, we see such a block with $(\mathcal{I}_1,\mathcal{I}_2,\mathcal{I}_3,\mathcal{I}_4)$ for the graph with $N=8$ and $k=3$ in Fig. \[fig:graph\_8\_x\]a and for $N=9$ and $k=4$ in Fig. \[fig:graph\_9\_x\]a, or for $N=10$ and $k=3$, Fig. \[fig:graph\_10\_x\]a and for the cubic graphs ($k=3$) with $N=12$ and $N=14$ as well, see Figs. \[fig:graph\_12\_3\] and \[fig:graph\_14\_3\]. The $\sigma_{max}$–graphs with larger degree ($=$ coplayers) still somewhat retains such a “blockish” appearance (for instance $(\mathcal{I}_1,\mathcal{I}_2,\mathcal{I}_3,\mathcal{I}_4,\mathcal{I}_5)$ in Fig. \[fig:graph\_10\_x\]c) but to a far lesser degree. In addition, $\sigma_{max}$–graphs with larger degree are frequently vertex–transitive (for instance Figs. \[fig:graph\_9\_x\]d, \[fig:graph\_10\_x\]e and \[fig:graph\_10\_x\]g) which is not the case for cubic ($k=3$) and quartic ($k=4$) $\sigma_{max}$–graphs with $N \leq 14$, with the exception of $N=6$ and $k=3$, see Fig. \[fig:graph\_6\_3\]a. Furthermore, it can be observed that the blocks are occupied by clusters of cooperators which are frequently connected by cut vertices and/or hinge vertices. For instance, for $N=12$ and $k=3$, the vertices occupied by the players $\mathcal{I}_3$ and $\mathcal{I}_9$, see Fig. \[fig:graph\_12\_3\], are cut vertices, while for $N=10$ and $k=4$, see Fig. \[fig:graph\_10\_x\]b, the vertices occupied by the players $\mathcal{I}_5$ and $\mathcal{I}_6$ are hinge vertices as their removal would make the distance between $\mathcal{I}_4$ and $\mathcal{I}_7$ longer. As discussed above, the clusters can be seen as to serve as a mutant family that invades the remaining graph. As vertices with players of opposing strategies are connected by cut and/or hinge vertices there is only a small number of (or even just a single) migration path for the cooperators and/or defectors.
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|
---
author:
- 'G. Israelian'
- 'N. Shchukina'
- 'R. Rebolo'
- 'G. Basri'
- 'J. I. González Hernández'
- 'T. Kajino'
date: 'Received; accepted'
title: 'Oxygen and Magnesium Abundance in the Ultra-Metal-Poor Giants CS22949-037 and CS29498-043: Challenges in Models of Atmospheres.'
---
Introduction
============
Surveys of metal-poor stars (Beers et al. 1992) and their abundance studies (Ryan, Norris & Beers 1996; Norris, Ryan & Beers1999) are aimed at investigating the chemical evolution of the Galaxy and the nucleosynthetic yields of supernovae. Despite the numerous studies in this field, the current situation with abundance trends of various $\alpha$-elements is very confusing. Observations of Stephens & Boesgaard (2002) demonstrate that the \[$\alpha$/Fe\] ratios for Ca, Si, Ti and Mg do not show a flat [*plateau*]{} at \[Fe/H\] $< -1$, as was claimed in previous studies. The results obtained by Idiart & Thevenin(1999) and Stephens & Boesgaard (2002) cannot be called [*consistent*]{} with the analysis presented by Carretta et al.(2002) and many others (see McWilliam 1997). The situation with oxygen is far from being resolved (Israelian, García López & Rebolo 1998; Israelian et al.2001; Nissen et al. 2002; Takeda 2003; Fulbright & Johnson 2003), while a new debate over the sulfur abundance in metal-poor stars has already emerged (Israelian & Rebolo 2001; Takada-Hidai et al. 2002; Nissen et al. 2003). These and many other studies clearly show that \[$\alpha$/Fe\] $>$ 0 for the great majority of metal-poor stars in the Galaxy. However, it is hard to speak of any trend when the abundance ratios in many stars computed by different authors disagree by more than 0.3–0.4 dex. There are many aspects to this serious problem, and we shall not discuss them further here. Abundance analysis of ultra-metal-poor stars with \[Fe/H\] $< -$3 gives rise to even more enigmas into this field.
It is well known that the chemical composition of the atmospheres of halo dwarfs is not altered by any internal mixing and therefore provides a good opportunity to constrain Galactic chemical evolution models. Unfortunately, most of the known ultra-metal-poor stars are not dwarfs but giants (Beers et al. 1992), which pose two serious problems. First, the atmospheric parameters of giants are more uncertain and second, their surfaces can be polluted by enriched material that has been either dredged from the stellar interior or transferred from a companion star. Oxygen is a key element in this scheme as it can help to distinguish between pristine and pollution origins of other elements and also show which of the aforementioned processes was dominant. There have been intensive investigations of the oxygen abundances in halo stars over the last five years. Abundances derived from near-UV OH lines in metal-poor dwarf stars (Israelian et al.1998; Boesgaard et al. 1999; Israelian et al.2001) show that the \[O/Fe\] (\[O/Fe\] = log(O/Fe)$_\star$–log (O/Fe)$_\odot$) ratio increases from 0 to 1 between \[Fe/H\] = 0 and $-3$. The abundances derived from low-excitation OH lines agreed well with those derived from high-excitation lines of the O[i]{} triplet at 7771–5 Å (Israelian et al. 1998, 2001; Boesgaard et al. 1999 ; Nissen et al.2002). It seems that even the \[O[i]{}\] forbidden line at 6300 Å supports the “quasi-linear” trend of \[O/Fe\] (Nissen et al.2002) when standard 1D atmospheric models are employed. While some authors claim a good agreement between the forbidden line \[O[i]{}\] 6300 Å and the near-IR triplet (Mishenina et al. 2000; Nissen et al.2002), others suggest the opposite (Carretta, Gratton, & Sneden 2000). In a recent study Takeda (2003) found that the disagreement between the triplet and the forbidden line tends to be larger for cool giants. Fulbright & Johnson (2003) support this conclusion in a detailed study of 55 subgiants and giants. These authors conclude that it is impossible to resolve the disagreement in the two indicators without adopting an ad hoc temperature scale that is incompatible with standard temperature scales such as IRFM and H$\alpha$. In fact, the \[O/Fe\] trend obtained based on the ad hoc scale (see Fig. 13 of Fulbright & Johnson 2003) is identical to the trends presented by Israelian et al. (2001) and Nissen et al. (2002). There is no [plateau]{} at \[O/Fe\] = 0.5. Despite considerable observational effort the trend of the \[O/Fe\] ratio in the halo is still unclear. However, the latest studies (Israelian et al. 2001; Nissen et al. 2002; Takeda 2003; Fulbright & Johnson 2003) suggest that dwarfs provide more reliable and consistent abundances than giants. It is not clear how the 3D effects will resolve this conflict since the latter do not predict an agreement between different oxygen abundance indicators in dwarfs (Asplund & García Pérez 2002).
McWilliam et al.(1995) were the first to carry out a detailed spectroscopic analysis of CS22949-037 and to confirm that the star is very metal-poor with an $\alpha$-element excess. Furthermore, Depagne et al. (2002) performed a more detailed investigation of this object and found a large excess of oxygen (\[O/Fe\] = 2.0) and sodium (\[Na/Fe\] = 2.1). Zero-heavy-element supernovae models with fall back have been invoked in order to interpret the elemental abundance ratios in this star. Aoki et al. (2002) have presented a detailed analysis of another ultra-metal-poor giant CS29498–043 with a very high abundance excess of \[Mg/Fe\] = 1.81. Both, CS22949–037 and CS29498–043 exhibit a large overabundance of N and C but show no significant enhancement of neutron-capture elements. It is possible that the surfaces of these stars have been polluted by enriched material, either dredged from the star’s inner core or transferred from a companion star. The abundances of $\alpha$-elements could be used to discriminate in favour of one of these hypotheses or to confirm a pristine origin. The detailed comparison of elemental abundances may provide important constraints on the properties of the first supernova progenitors.
In this article we present observations of the oxygen triplet in CS22949–037 and CS29498-043, as well as the detection of the forbidden line \[O[i]{}\] 6300 Å in CS29498–043. The stellar parameters and the abundances of iron, oxygen and magnesium were derived in non-LTE. We report a significant discrepancy between the abundances derived from the oxygen triplet and the forbidden line. The conflict cannot be resolved under any circumstances, at least for CS22949-037. A similar conflict was found for Mg. We question the validity of standard plane–parallel models of atmospheres employed in the present analysis.
Observations
============
The observations of CS29498–043 and CS22949–037 were performed on 2002 October 31 and 2002 May 20 (only CS22949–037) at the KeckI using the high-resolution spectrograph HIRES and the TEK 2048 $\times$ 2048 pixel$^{2}$ CCD. A 1.1 arcsec entrance slit provided a resolving power $R \sim$60000. A red wavelength setting was used to observe oxygen triplet lines at 7771–5 Å. The average signal-to-noise (S/N) ratio of the combined spectrum near 7770 Å was S/N $>$ 100. Five spectra of CS29498–043 with a total exposure time 13500 s were obtained on 2002 September 26 and 27 with the configuration CD4 at TNG/SARG (La Palma). A resolving power of 29000 was achieved with a 1.6 arcsec slit. The \[O[i]{}\] 6300 Å was not blended with any telluric features in this spectral window and the sky emission at 6300 Å was carefully subtracted using the off-slit spectra. The forbidden line with an equivalent width EW = 60 $\pm$ 10 mÅ was observed in every exposure, providing an independent confirmation of the detection. Rotational velocity of CS29498–043 derived from the triplet (HIRES) and the forbidden line (SARG) was 10 $\pm$ 3 and 8 $\pm$ 3 kms$^{-1}$, respectively. The S/N ratio 30 was reached near 6300 Å in the unbind spectrum of CS29498–043. However, given the rotational velocity of the star we could apply a factor of two bining providing S/N $\sim$ 40 to the final spectrum. All the spectra were reduced using standard [iraf]{}[^1] procedures (bias subtraction, flat-field correction, and extraction of one-dimensional spectra). Different spectra for each object were co-added before wavelength calibration and continuum normalization.
[lccccl]{} Ion & $\lambda$ (Å) & $\log gf$ & $\chi$ & EW\
O[i]{} & 7771.960 & 0.324 & 9.11 & 41.0\
O[i]{} & 7774.180 & 0.174 & 9.11 & 30.0\
O[i]{} & 7775.400 & $-$0.046 & 9.11 & 19.0\
O[i]{} & 6300.230 & $-$9.759 & 0.00 & 5.0\
Fe[i]{} & 3899.709 & $-$1.531 & 0.09 & 102.2\
Fe[i]{} & 3920.260 & $-$1.746 & 0.12 & 95.5\
Fe[i]{} & 3922.914 & $-$1.651 & 0.05 & 102.5\
Fe[i]{} & 4005.246 & $-$0.610 & 1.55 & 61.5\
Fe[i]{} & 4045.815 & 0.280 & 1.48 & 99.6\
Fe[i]{} & 4063.597 & 0.070 & 1.55 & 89.8\
Fe[i]{} & 4071.740 & $-$0.022 & 1.60 & 84.1\
Fe[i]{} & 4076.636 & $-$0.360 & 3.20 & 6.3\
Fe[i]{} & 4132.060 & $-$0.648 & 1.60 & 58.2\
Fe[i]{} & 4143.871 & $-$0.450 & 1.55 & 66.2\
Fe[i]{} & 4147.673 & $-$2.104 & 1.48 & 7.7\
Fe[i]{} & 4181.758 & $-$0.180 & 2.82 & 10.5\
Fe[i]{} & 4187.044 & $-$0.548 & 2.44 & 18.6\
Fe[i]{} & 4199.098 & 0.250 & 3.03 & 23.9\
Fe[i]{} & 4202.031 & $-$0.708 & 1.48 & 60.3\
Fe[i]{} & 4222.219 & $-$0.967 & 2.44 & 8.3\
Fe[i]{} & 4227.434 & 0.272 & 3.32 & 15.1\
Fe[i]{} & 4250.125 & $-$0.405 & 2.46 & 22.6\
Fe[i]{} & 4260.479 & $-$0.020 & 2.39 & 44.2\
Fe[i]{} & 4271.159 & $-$0.349 & 2.44 & 32.7\
Fe[i]{} & 4282.406 & $-$0.810 & 2.17 & 16.0\
Fe[i]{} & 4325.765 & $-$0.010 & 1.60 & 91.2\
Fe[i]{} & 4404.752 & $-$0.142 & 1.55 & 85.1\
Fe[i]{} & 4415.125 & $-$0.615 & 1.60 & 61.8\
Fe[i]{} & 4447.722 & $-$1.342 & 2.21 & 6.0\
Fe[i]{} & 4461.654 & $-$3.210 & 0.09 & 33.0\
Fe[i]{} & 4528.619 & $-$0.822 & 2.17 & 21.3\
Fe[i]{} & 4871.323 & $-$0.410 & 2.85 & 10.7\
Fe[i]{} & 4872.144 & $-$0.600 & 2.87 & 7.8\
Fe[i]{} & 4891.496 & $-$0.140 & 2.84 & 19.8\
Fe[i]{} & 4920.509 & 0.060 & 2.82 & 23.5\
Fe[i]{} & 4994.133 & $-$3.080 & 0.91 & 7.5\
Fe[i]{} & 5001.871 & $-$0.010 & 3.87 & 3.1\
Fe[i]{} & 5049.825 & $-$1.355 & 2.27 & 5.5\
Fe[i]{} & 5051.636 & $-$2.795 & 0.91 & 11.2\
Fe[i]{} & 5068.774 & $-$1.230 & 2.93 & 3.9\
Fe[i]{} & 5110.414 & $-$3.760 & 0.00 & 15.9\
Fe[i]{} & 5123.723 & $-$3.068 & 1.01 & 4.8\
Fe[i]{} & 5127.363 & $-$3.307 & 0.91 & 3.0\
Fe[i]{} & 5166.286 & $-$4.195 & 0.00 & 7.8\
Fe[i]{} & 5171.599 & $-$1.793 & 1.48 & 19.5\
Fe[i]{} & 5194.943 & $-$2.090 & 1.55 & 6.9\
Fe[i]{} & 5232.946 & $-$0.057 & 2.93 & 15.6\
Fe[i]{} & 5266.562 & $-$0.490 & 2.99 & 6.2\
Fe[i]{} & 5324.185 & $-$0.100 & 3.20 & 7.2\
Fe[i]{} & 5339.935 & $-$0.680 & 3.25 & 3.2\
Fe[i]{} & 5371.493 & $-$1.645 & 0.95 & 62.6\
Fe[i]{} & 5383.374 & 0.500 & 4.29 & 2.3\
Fe[i]{} & 5397.131 & $-$1.993 & 0.91 & 46.8\
Fe[i]{} & 5405.778 & $-$1.844 & 0.99 & 44.8\
Fe[i]{} & 5429.699 & $-$1.879 & 0.95 & 48.5\
Fe[i]{} & 5434.527 & $-$2.122 & 1.01 & 31.5\
Fe[i]{} & 5446.920 & $-$1.930 & 0.99 & 43.8\
Fe[i]{} & 5506.782 & $-$2.797 & 0.99 & 9.4\
Fe[ii]{} & 4178.85 & $-$2.480 & 2.57 & 5.0\
Fe[ii]{} & 4233.16 & $-$2.000 & 2.57 & 20.2\
Fe[ii]{} & 4416.81 & $-$2.600 & 2.77 & 4.9\
Fe[ii]{} & 4515.33 & $-$2.480 & 2.83 & 3.4\
Fe[ii]{} & 4520.22 & $-$2.600 & 2.79 & 2.8\
Fe[ii]{} & 4555.89 & $-$2.290 & 2.82 & 7.3\
\[tab2\]
Stellar parameters from the non-LTE computations of iron
========================================================
It is well known that a strong over-ionization of neutral iron in metal-poor stars leads to the systematic difference in abundances determined from the Fe[i]{} and Fe[ii]{} lines. This difference increases with decreasing metallicity and may reach 0.4 dex in very metal-poor dwarfs (Thévenin & Idiart 1999). In addition, the non-LTE modeling predicts a dependence of the non-LTE abundance corrections of Fe[i]{} lines on the lower excitation potential ($\chi$). The corrections are particularly large for the low-excitation Fe[i]{} lines, while the non-LTE effects are not important for the Fe[ii]{} lines. The non-LTE abundance corrections in the Sun are in the range 0.02–0.1 dex (Shchukina & Trujillo Bueno 2001). In general, non-LTE effects play a significant role in metal-poor stars because of decrease in electron density when collisions with free electrons no longer dominate the kinetic equilibrium. Another consequence of the metal deficiency is an appreciable weakening in UV blanketing. Non-LTE over-ionization of Fe[i]{} substantially reduces the UV line opacity and allows more flux to escape. These effects lead us to suspect that the gravities of metal-poor giants derived from the LTE Fe analysis are underestimated because of the neglect of non-LTE effects. This effect has been studied in metal-poor dwarfs (Thévenin & Idiart 1999) and in the subgiant BD +231330 (Israelian et al.2001). In fact, Thévenin & Idiart (1999) derived gravity corrections of up to 0.5 dex with respect to LTE values, for the case of stars with \[Fe/H\] $\sim -3.0$. They have shown that non-LTE effects are important in determining stellar parameters from the iron ionization balance.
The stellar parameters of our targets were obtained using the ionization equilibrium of Fe. The Fe model atom used in our study provides very consistent results for a 3D atmospheric model of the Sun (Shchukina & Trujillo Bueno 2001). The microturbulent velocity was fixed at 2 . Non-LTE analysis of Fe based on plane–parallel atmosphere models of Kurucz (1992) was carried out with the code NATAJA (Shchukina & Trujillo Bueno 2001), and the atmospheric parameters were derived using the same method as in Israelian et al.(2001). The equivalent widths of 26 (CS29498–043) and 60 (CS22949–037) Fe lines listed in Tables 1 and 2 were taken from the articles of Aoki et al.(2002) and Depagne et al.(2002), respectively. In the present analysis we used solar abundances from Grevesse and Sauval (1998). The only exception was oxygen, for which we used the solar abundance $\log \epsilon(O)$=8.74 from Nissen et al. (2002).
Figures 1 to 6 show our computations for a grid of atmospheric models for CS22949-037 and CS29498-043. The non-LTE corrections to \[Fe/H\] are around 0.3–0.4 dex (Figs 3 and 6) and abundance scatter obtained for the best parameter sets are less than 0.2 dex (Figs 2 and 5). After many iterations with different input parameters, our final results were $T_{\rm eff}$ = 4900$\pm$125 K, $\log g$ = 2.5$\pm$0.3 and \[Fe/H \] = $-$3.5$\pm$0.2 for CS22949–037 and $T_{\rm eff}$ = 4300$\pm$160 K, $\log g$ = 1.5$\pm$0.35 and \[Fe/H\] = $-3.5\pm0.24$ for CS29498–043. The errors were computed following McWilliam et al. (1995) and assuming those authors 1-$\sigma$ uncertainties of $\sim$0.03 and $\sim$0.07 dex for the oscillator strengths of the and lines, respectively. We also note that the oscillator strengths of the lines used in our calculations (Fuhr et al. 1988) are very similar to those compiled by McWilliam et al. (1995). The gravities that we obtained are about 1dex larger compared with those reported by Aoki et al. (2002) and Depagne et al.(2002).
[lccccl]{} Ion & $\lambda$ (Å) & $\log gf$ & $\chi$ & EW\
O[i]{} & 7771.960 & 0.324 & 9.11 & 18.0\
O[i]{} & 7774.180 & 0.174 & 9.11 & 15.0\
O[i]{} & 7775.400 & $-$0.046 & 9.11 & 10.0\
O[i]{} & 6300.230 & $-$9.759 & 0.00 & 60.\
Fe[i]{} & 4005.246 & $-$0.610 & 1.55 & 87.7\
Fe[i]{} & 4459.121 & $-$1.279 & 2.17 & 33.1\
Fe[i]{} & 4461.654 & $-$3.210 & 0.09 & 83.2\
Fe[i]{} & 4489.741 & $-$3.966 & 0.12 & 43.3\
Fe[i]{} & 4528.619 & $-$0.822 & 2.17 & 52.1\
Fe[i]{} & 4890.762 & $-$0.430 & 2.86 & 27.6\
Fe[i]{} & 4891.496 & $-$0.140 & 2.84 & 46.0\
Fe[i]{} & 4918.999 & $-$0.370 & 2.85 & 35.4\
Fe[i]{} & 4920.509 & 0.060 & 2.82 & 56.7\
Fe[i]{} & 4939.690 & $-$3.340 & 0.86 & 36.0\
Fe[i]{} & 4957.603 & 0.043 & 2.80 & 67.4\
Fe[i]{} & 4994.133 & $-$3.080 & 0.92 & 27.0\
Fe[i]{} & 5012.071 & $-$2.642 & 0.86 & 55.8\
Fe[i]{} & 5049.825 & $-$1.355 & 2.28 & 27.5\
Fe[i]{} & 5051.636 & $-$2.795 & 0.91 & 43.6\
Fe[i]{} & 5083.342 & $-$2.958 & 0.96 & 36.8\
Fe[i]{} & 5110.414 & $-$3.760 & 0.00 & 65.9\
Fe[i]{} & 5123.723 & $-$3.068 & 1.01 & 26.6\
Fe[i]{} & 5127.363 & $-$3.307 & 0.91 & 21.3\
Fe[i]{} & 5166.286 & $-$4.195 & 0.00 & 36.5\
Fe[i]{} & 5171.599 & $-$1.793 & 1.48 & 34.4\
Fe[i]{} & 5194.943 & $-$2.090 & 1.55 & 39.9\
Fe[ii]{} & 4522.634 & $-$2.030 & 2.83 & 28.0\
Fe[ii]{} & 4583.829 & $-$2.020 & 2.79 & 25.9\
Fe[ii]{} & 4923.921 & $-$1.320 & 2.88 & 50.1\
Fe[ii]{} & 5018.434 & $-$1.220 & 2.88 & 60.3\
\[tab2\]
Oxygen
======
We have detected all three lines of the oxygen triplet in CS22949–037 and CS29498–043 (Fig. 7) and the \[O[i]{}\] 6300 Å line in CS29498-043 (Fig. 8). The equivalent width of the forbidden line in CS29498-043 EW$\sim$35 mÅ measured by Aoki et al.(2003) using SUBARU/HDS is smaller than the value obtained from our TNG/SARG. Our TNG/SARG measurement agrees within 3-$\sigma$. This modest agreement is not surprising since the line detected by Aoki et al.(2003) is strongly blended with a telluric feature and our line was affected by sky emission. The relatively large difference may then be related to uncertainties in the corrections for telluric absorption and/or sky emission. The non-LTE computations of the oxygen atom were carried out using the atomic model with 23 levels of O[i]{} and one level of O[ii]{} described by Carlsson & Judge (1993). While only 31 bound–bound and 23 bound–free radiative transitions were considered in our computations, we note that the consideration of additional levels and transitions does not affect our results (Shchukina 1987; Takeda 2003). Our computations for the Sun and metal-poor dwarfs predict non-LTE corrections that are very close to those reported recently by Takeda (2003) and Nissen et al. (2002) for 1D models.
It is well known that inelastic collisions with hydrogen atoms tend to cancel out the non-LTE effects. The forbidden line is not affected by these collisions since it is formed under LTE. The effect on the triplet may reach 0.1 dex in hot and metal-poor subdwarfs such as LP815-43 (e.g. Nissen et al. 2002). However, it is often stated that Drawin’s formalism (Drawin 1968) gives very uncertain results for hydrogen collision rates (e.g. Belyaev et al. 1999). Obviously, we are not concerned with these problems in our targets since the density of H atoms (and therefore the collision rates) in the atmosphere of K giants is about two orders of magnitude lower than in dwarfs. Moreover, inelastic collisions with hydrogen atoms bring the oxygen abundance derived from the 7771-5 Å triplet close to its LTE value. This will make the “oxygen conflict” even more severe as the discrepancy between the triplet and the forbidden line will be greater. Thus, collisions with H atoms were not taken into account in our computations. We also note that the effects of triplet–quintet system coupling, CO-binding and $L_{\beta}$ pumping are negligible in the atmospheres of our targets.
Our non-LTE oxygen abundances from the near-IR triplet yield \[O/Fe\] = 3.13$\pm$0.21 and \[O/Fe\] = 3.02$\pm$0.27 in CS22949–037 and CS29498–043, respectively. The non-LTE corrections ($\Delta \epsilon=\epsilon({\rm non-LTE})-\epsilon({\rm LTE})$) on the triplet and 6300 Å are listed in Table 5. Assuming our stellar parameters for CS22949–037 and the equivalent width of the forbidden line from Depagne et al.(2002) we find \[O/Fe\] = 1.95. The large difference between the abundances derived from the triplet and the forbidden line cannot be explained by the non-LTE abundance corrections for the triplet (Table 4) and/or by a noise/telluric correction for the forbidden line. In fact, one needs \[O[i]{}\] 6300 Å line with an EW $>$ 50 mÅ to provide \[O/Fe\] $>$ 3. This is clearly ruled out by observations of Depagne et al. (2002). Even if we assume for CS22949–037 the stellar parameters from Depagne et al. (2002), the difference between 7771–5 and 6300 Å is as high as 1.55 dex. The forbidden line measured in the SARG spectra of CS29498–043 has EW = 60$\pm$10mÅ, providing \[O/Fe\] = 2.49$\pm$0.13 (Fig. 8).
It is hard to resolve the conflict between the triplet and the forbidden line by playing with the stellar parameters since the errors in $T_{\rm eff}$ and $\log g$ are of the order of 150 K and 0.3 dex, respectively. We have to understand the formation mechanisms of these lines in the atmospheres of our targets. The intensity contribution functions (CFI) (see Gray 1976) of the triplet and the forbidden lines decrease rapidly with atmospheric height (Fig. 9), and one can assume, as a working hypothesis, that these lines are optically thin. In this case their equivalent widths will be proportional to the ratio $K^{\rm line}$ / $K^{\rm cont}$, where $K^{\rm line}$ and $K^{\rm cont}$ are the absorption coefficients in the line and in the continuum, respectively. Many authors state that the strength of the forbidden line is sensitive to the structure of the upper atmosphere. Our analysis (Fig. 9) shows that this is not the case since the forbidden line has a very small oscillator strength and is therefore formed deep in the atmosphere. The triplet lines are formed at this depth because they are excited from high excitation levels which are populated only in the hot layers of the atmosphere. The absorption coefficient in the line will be very sensitive to $T_{e}$ since the population of the lower level is proportional to exp($- \chi_{l}$/$kT_{e}$) and therefore EW$_{7771-5}$ $\sim$ $T{_e}$. The strength of the forbidden line will be controlled by $K^{\rm cont}$ because $K^{line} \ll K^{\rm cont}$. Since $K^{\rm cont} \sim N{_e}$ (where $N{_e}$ is the electron density), EW$_{6300}$ $\sim$ 1/$N{_e}$. Thus, the EW$_{6300}$ will decrease (while EW$_{7771-5}$ will increase) with an increasing $T{_e}$ since $N{_e}$ $\sim$ $T{_e}$ in stellar atmospheres. Given the EWs of the triplet and the forbidden line, one may find the effective temperature of the model where these two indicators provide the same abundance. This test has been carried out for CS22949–037 and the results are shown in Fig.10. While the $T_{\rm eff}$ of the star derived from the Fe lines is 4900 K, consistent oxygen abundance can only be obtained when $T_{\rm eff}$ = 5600 K.
The temperature distributions in two stars with $T_{\rm eff}$ = 4900 and 5600 K and the location of the maximum CFI in the line centre for each model are shown in the Figs.11 and 12. These figures clearly demonstrate that the formation depth of the forbidden line moves toward the surface and to much higher temperatures in the model with a higher $T_{\rm eff}$. In fact, the maximum contribution to the line strength comes from the hot layers (Fig.11). The same plot for the near-IR triplet shows (Fig.12) that the maximum contribution to the equivalent width comes from the layers that are closer to the surface compared with the forbidden line. In fact, our analysis demonstrates that the formation layer of the forbidden line is optically thinner and therefore less sensitive to the temperature gradient (i.e. $T_{\rm eff}$). Our analysis shows that the failure of the Kurucz (1992) models to provide consistent abundances from the near-IR triplet and the forbidden line, comes from the temperature and density gradients near the continuum-forming region. This problem has the same roots as that related to the discrepancy between the temperatures obtained from colours (continuum) and the Fe ionization balance. It is clear that we cannot resolve the conflict between the triplet and the forbidden line by modifying the temperature distribution in the upper atmosphere. The key to this problem is hidden deep in the atmosphere. There must be a certain combination of the electron density and the temperature gradients close to the continuum formation region to provide consistent abundances from the triplet and the forbidden line. It is easy to show that the forbidden line is formed deep in the atmosphere of our targets. We have repeated calculations of this line for CS22949-037 by gradually removing the upper parts of the atmosphere until we reached those layers where the 6300 Å line is actually “formed”. These simple tests have demonstrated that more than 80% of the EW of the forbidden line is formed at $\log {\it mass} >$ 1.4 (e.g. Fig.9). The discrepancy between the near-IR triplet and the 6300 Å line also exists in moderately metal-poor giants. These problems have been addressed in detail by Takeda (2003) and Fulbright & Johnson (2003). Our targets provide the most extreme cases of the “oxygen conflict” at very low metallicities.
We have found that the discrepancy between the near-IR triplet and the forbidden line cannot be explained by non-LTE effects, uncertainties in the stellar parameters or the quality of observations. The oxygen puzzle arises from the failure of standard plane-parallel atmosphere to describe physical conditions in the atmospheres of very metal-poor giants. Spherical effects cannot be the cause of this discrepancy since these lines are formed in very deep and narrow layers where the plane–parallel approximation is certainly applicable. The oxygen puzzle is not unique to CS22949–037 and CS29498-043. On a smaller scale it exists in other halo giants (Takeda 2003; Fulbright & Johnson 2003). We suspect that a similar discrepancy exists for the most metal-poor star in our Galaxy, HE0107-5240.
[lcccc]{} Line & EW & (4400/0.5/$-$3.5) & (4400/1.5/$-$3.5) & (4300/1.5/$-$3.5)\
& mÅ & \[Mg/Fe\] = 1.8 & \[Mg/Fe\] = 1.0 & \[Mg/Fe\] = 0\
Mg[i]{} 4571 & 104 & 7.286 & 6.233 & 5.626\
Mg[i]{} 5172 & 218 & 5.031 & 4.905 & 5.147\
Mg[i]{} 5183 & 228 & 4.937 & 4.785 & 5.012\
O[i]{} 6300 & 60 & 7.626 & 7.732 & 7.694\
O[i]{} 7772 & 18 & 7.984 & 8.232 & 8.296\
O[i]{} 7774 & 15 & 8.019 & 8.274 & 8.343\
O[i]{} 7775 & 10 & 8.004 & 8.272 & 8.345\
[lccc]{} Line & EW (mÅ) & log $\epsilon$ & $\Delta \epsilon({\rm NLTE-LTE})$\
Mg[i]{} 3829 & 156.1 & 5.224 & 0.187\
Mg[i]{} 3832 & 185.2 & 4.986 & 0.158\
Mg[i]{} 3838 & 202.7 & 4.874 & 0.160\
Mg[i]{} 4571 & 52.9 & 5.810 & 0.670\
Mg[i]{} 5172 & 176.1 & 5.347 & 0.141\
Mg[i]{} 5183 & 199.4 & 5.353 & 0.124\
O[i]{} 6300 & 5. & 7.184 & $-$0.013\
O[i]{} 7772 & 41. & 8.430 & $-$0.238\
O[i]{} 7774 & 30. & 8.362 & $-$0.222\
O[i]{} 7775 & 19. & 8.303 & $-$0.200\
[lcccccc]{} Target & $\Delta_{\rm [OI]}$ & $\Delta_{\rm Triplet}$ & $\Delta_{\rm Mg 4571}$ & $\Delta_{\rm Mg(Sum)}$ & $\Delta A_{\rm O}$ & $\Delta A_{\rm Mg}$\
CS29498–043 & $-$0.012 & $-$0.188 & 0.179 & $-$0.071 & 0.53 & 0.546\
CS22949–037 & $-$0.013 & $-$0.22 & 0.670 & 0.154 & 1.18 & 0.653\
Magnesium
=========
The non-LTE computations of the magnesium atom were carried out using the 24 levels simplified version of the model atom described by Carlsson, Rutten & Shchukina (1992). Addition of new levels and transitions does not influence on the solution for the lines considered in this paper. The abundance of Mg in CS29498–043 was derived from the spectral lines at 4571, 5172 and 5183 Å using the equivalent width measurements of Aoki et al. (2002). These lines provide very different abundances just as in the case of oxygen (Table 3). The forbidden resonance line Mg[i]{} 4571 Å in CS29498–043 provides \[Mg/Fe\] = 1.626 with a non-LTE correction of $\sim$0.18 dex while Mg[i]{} lines at 5172 and 5183 Å provided consistent abundance with a mean \[Mg/Fe\] = 1.08. The non-LTE corrections in 4571 Å are 0.18 dex while in two other lines they even do not reach $-$0.1 dex (Table 5). The difference between the 4571 and 5172 + 5183 Å is larger in the non-LTE than in the LTE case. The mean abundance from the three lines is \[Mg/Fe\] = 1.26 which, of course, does not makes any sense given the huge discrepancy between the 4571 and 5172 + 5183 Å lines. As for the CS22949–037, there are six Mg[i]{} lines available (Table 4) in our model atom from the article of Depagne et al.(2002). From the five lines listed in Depagne et al.(2002) we obtained a mean \[Mg/Fe\] = 1.156 while the 4571 Å line again yields a much larger abundance \[Mg/Fe\] = 1.81.
The CFIs of the 4571 and 5183 Å lines in CS22949–037 have very different shapes (Fig. 9). The 5183 Å line is formed in an extended region of the upper atmosphere while the forbidden resonance line is produced by the same deep layers where the 6300 Å forbidden line of the neutral oxygen is formed. The 5183 Å line is very strong and less sensitive to the effective temperature compared with the 4571 Å line. The agreement between abundances provided by 4571 Å and 5183 Å lines can be achieved if we increase the temperature in the inner layers of the atmosphere. This will increase the strength of the 4571 Å line (and the abundance obtained from this line will decrease) while the 5183 Å will remain almost unchanged. Given the negligible sensitivity of the 5183 Å line to the effective temperature one may suggest that this line provides a more reliable abundance. This is similar to the oxygen atom, where the forbidden line is believed to be the best abundance indicator since it is less sensitive to $T_{\rm eff}$.
Mg is an important source of free electrons and its overabundance may have a non-negligible effect on the stellar surface gravity derived from the ionization balance of Fe. According Aoki et al. (2002), the effect of Mg overabundance on the gravity of CS29498–043 is 0.4 dex. We have repeated the non-LTE Fe and Mg analysis of CS29498–043 for the cases when \[Mg/Fe\] = 1.0 and \[Mg/Fe\] = 1.8 (see Figs 13 and 14). It appears that the gravity of the star is not changed when the Mg abundance is increased by a factor of 10 (i.e. from \[Mg/Fe\] = 0 to \[Mg/Fe\] = 1). However, the gravity drops to $\log g$ = 0.5 if we set \[Mg/Fe\] = 1.8. Thus, the effect is indeed very large if Mg is as abundant as oxygen. However, as we have already stated, the Mg abundance derived from the 4571 Å line (\[Mg/Fe\]=1.8) is most probably overestimated and therefore we do not find it necessary to revise the stellar parameters obtained for \[Mg/Fe\] = 0 or \[Mg/Fe\] = 1. The final parameters for CS29498–043 are assumed $T_{\rm eff}$ = 4400 K, $\log g$ = 1.5 and \[Fe/H\] = $-3.5$ when \[Mg/Fe\] = 1.0.
Discussion
==========
Our results may have an interesting impact on the physics of the supernova progenitor which gave birth to these ultra-metal-poor stars. Should we use the abundance from the triplet or the forbidden line to set constrains on the supernova models ? The situation with our targets clearly demonstrates that non-LTE effects on the near-IR oxygen triplet are not responsible for this conflict, and that therefore this abundance indicator is as reliable (or unreliable) as the forbidden line.
Both CS22949–037 and CS29498–043 are distinguished from other carbon-rich metal-poor stars with large excesses of neutron capture s-process elements. The overabundance of s-elements is usually attributed to nucleosynthesis in the thermally pulsing AGB stars. These ultra-metal-poor giants have moderately high abundances of Mg, Si and Al, while C and N are unusually overabundant. In this paper we find that both stars are also rich in oxygen. It appears that the class of objects with high \[O/Fe\] is not limited by CS22949–037 and CS29498–043. Aoki et al. (2002) proposed the existence of a new class of very metal-poor stars which originate from supernova in which most of the matter were absorbed by the iron core. The most metal-poor star in the Galaxy, HE0107–5240, also belongs to this class of objects and has a large \[O/Fe\]=2.4 derived from the UV OH lines (Bessell et al. 2004). The abundance pattern in this star is consistent with a model (Umeda & Nomoto 2003) in which the supernova undergoes some mixing followed by a fallback into a massive black hole. However, the oxygen abundance of HE0107–5240 is smaller than the prediction of Umeda & Nomoto (2003) by as much as 1 dex. The prototype of this type of supernova is SN1997D, which was very underluminous because of the small amount (2$\times10^{-3}$M$_{\sun}$) of $^{56}$Ni ejected during explosion. Four new faint supernova have been reported recently (Pastorello et al. 2003). While such supernovae are not observed frequently, their real number is expected to be much higher because of the faintness of the supernovae. These faint supernovae are expected to be more frequent in the early Galaxy (Umeda & Nomoto 2003), while their ejecta are characterized by very high \[O/Fe\] ratios. In fact, the monotonically rising trend of \[O/Fe\] can possibly be explained if we assume that the iron yield decreases with stellar mass (i.e. most massive stars form massive black holes). It is clear that the formation rate of massive black holes in the early Galaxy may affect the observed \[O/Fe\] trend. A massive black hole in the low mass X-ray binary system Nova Sco 1994 with \[O/Fe\] = 1.0 (Israelian et al. 1999) may serve as a prototype for such “failed” supernovae, where almost all the Fe has been accreted by the black hole. Apparently CS22949-037 is not an “exceptional" star as noted by Depagne et al.(2002). There are another four high \[O/Fe\] stars: CS29498-043, HE0107-5240, CS29497-030 (Sivarani et al.2003) and LP625-44 (Aoki et al. 2002b). Abundances in stars such as LP625-44 and CS29497-030 are assumed to result from mass transfer from an AGB star across a binary on to the observed companion star. However, it is not clear whether this explains the \[O/Fe\] excess in these s-process-enhanced stars. Other s-process-rich stars such as LP706-7 (Norris et al. 1997) do not show any radial velocity variations. There is much work to be done before we will be able to understand why some s-process stars are also oxygen rich. Displaying these stars on the \[O/Fe\] versus \[Fe/H\] diagram with a representative sample of halo dwarfs and giants from the literature (e.g. Nissen et al.2002, Cayrel et al.2003, Israelian et al.2001), we can see the general trend of oxygen in the galaxy and the relative position of these “extreme" stars with respect measurements of other “normal" stars (Fig. 15). This diagram gives a broader perspective on the evolution of oxygen in the Galaxy. We propose the existence of an upper envelope of the \[O/Fe\] ratio represented by the dashed line which suggests a monotonically increasing trend toward lower metallicities (Fig. 15). Very high \[O/Fe\] ratios are possibly indicating that most of the Fe nuclei synthesized in the inner core are actually held by a massive compact object, i.e a black hole. Smaller ratios possibly indicate that a significant fraction of the Fe nuclei is incorporated into the supernova ejecta, therefore a smaller mass cut is required and the likely formation of less massive compact objects, i.e neutron stars. Assuming that below metallicity $-$3, we are seeing direct yields from the first generation of supernovae, the range in \[O/Fe\] encompassed by the dashed lines in our plot may be populated by stars contaminated by supernovae that led to the formation of compact objects with different masses.
The fact that the atmospheres of these stars do not contain large amounts of Ca, Ti, Si and Mg supports the idea of massive black holes left from the supernova explosions of the first stars. It has been suggested that the bulk of light r-nuclei (with $A$ $<$ 130) appear to have different sources from those for heavy r-nuclei (Wasserburg, Busso & Gallino 1996). The meteoritic data require at least two distinct types of SN r-process events: the high-frequency events, $H$, producing heavy nuclei with $A$ $>$ 130, including $^{182}$Hf, and the low-frequency $L$ events producing light nuclei with $A$ $>$ 130, including $^{129}$I. The r-process production in the SN environments associated with the $H$ and $L$ events has been discussed in some detail by Wasserburg & Qian (2000). The abundance analysis of CS22949–037 and CS29498–043 extended to heavy neutron capture elements may directly test the speculation by Wasserburg and Qian (2000) that $H$ events are associated with supernovae producing black holes, whereas $L$ events are associated with supernovae producing neutron stars. According to this model, the parent supernovae of our targets come from the $H$-events.
The large disagreements found from different abundance indicators of Mg and O reveal that the atmospheric models used in this study are not reliable. The conflict is so severe that we cannot question the quality of the data and/or the model atoms used in our non-LTE. The problem with the oxygen and magnesium abundances may have the same roots as that discussed by Dalle Ore (1993). This author found large discrepancies among the temperatures obtained from the excitation and ionization equilibrium of several cool giants. In fact, the systematic disagreement between different temperature scales found from the continuum energy distribution, H$\alpha$ and Fe lines is the best indication that the models employed in these studies are to some extent unreliable.
Our analysis suggests that the gravities of very metal-poor giants derived from the LTE Fe analysis are strongly underestimated because non-LTE effects are neglected. The oxygen abundances in CS22949–037 and CS29498–043 derived from the triplet and the forbidden line differ by a large factor. It is interesting that the oxygen forbidden line at 6300 Å and the near-IR triplet are formed in the same layers deep in the atmosphere (Fig. 9). This suggests that one needs a different atmospheric structure in order to achieve consistency for O and Mg. It is clear that the standard 1D models of Kurucz (1992) are unreliable for ultra-metal-poor giants. As a final check of these results, we used models without convective overshooting (Castelli, Gratton & Kurucz 1997) and found that they do not resolve the discrepancy either.
Conclusions
===========
Observations with Keck I/HIRES have revealed strong lines of the oxygen near-IR triplet in the spectra of the ultra-metal-poor giants CS22949–037 and CS29498–043. The forbidden line of oxygen with EW = 60$\pm$10mÅ was observed with TNG/SARG. A detailed non-LTE analysis of Fe has been carried out and a new set of the atmospheric parameters have been obtained. Our analysis suggests that the gravities of metal-poor giants derived from the LTE Fe analysis are strongly underestimated because of the neglect of non-LTE effects.
The oxygen abundance in CS22949–037 and CS29498–043 derived from the triplet and the forbidden line differ by 1.18 and 0.53 dex, respectively. This disagreement cannot be explained by a non-LTE effects, quality of the data and/or uncertainties in stellar parameters. Other mechanisms must be invoked in order to explain this puzzle. A similar discrepancy was found for Mg when comparing the abundances obtained from the resonance line 4571 Å and several strong subordinate lines. Based on the present analysis we propose that the Kurucz (1992) models are not reliable for these ultra-metal-poor giants.
Acknowledgments
===============
The data presented here were obtained with 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. We are grateful to Wako Aoki and Martin Asplund for several helpful discussions and Piercarlo Bonifacio for providing interpolated models of Castelli et al. without overshooting. We thank the anonymous referee for useful suggestions and comments. N. S. would like to thank R. Kostik for several discussions and Irina Vasiljeva for a help with computations. This research was partially supported by the Spanish DGES under project AYA2001-1657 and by INTAS grant 00-00084.
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[^1]: [iraf]{} is distributed by the National Optical Astronomical Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation, USA.
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abstract: 'We study the structure of generalized Baumslag–Solitar groups from the point of view of their (usually non-unique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of smallest complexity (*fully reduced* decompositions) and give a simplified proof of the existence of deformations. We also prove a finiteness theorem and solve the isomorphism problem for generalized Baumslag–Solitar groups with no non-trivial integral moduli.'
address: 'Mathematics Department, University of Oklahoma, Norman OK 73019, USA'
author:
- Max Forester
title: 'Splittings of generalized Baumslag–Solitar groups'
---
Introduction {#introduction .unnumbered}
============
This paper explores the structure of generalized Baumslag–Solitar groups from the point of view of their (usually non-unique) splittings as fundamental groups of graphs of groups. By definition, a generalized Baumslag–Solitar group is the fundamental group of a graph of infinite cyclic groups. Equivalently, it is a group that acts on a simplicial tree with infinite cyclic vertex and edge stabilizers. We call such tree actions generalized Baumslag–Solitar trees. These groups have arisen in the study of splittings of groups, both in the work of Kropholler [@krop:torus; @krop:gbsgroups] and as useful examples of JSJ decompositions [@forester:jsj]. They were classified up to quasi-isometry in [@whyte:gbsgroups; @farbmosher:bs1], but their group-theoretic classification is still unknown.
Our approach to understanding generalized Baumslag–Solitar groups is to study the space of all generalized Baumslag–Solitar trees for a given group. In most cases this is a *deformation space*, consisting of $G$-trees related to a given one by a deformation (a sequence of collapse and expansion moves [@herrlich; @bass:remarks]). Equivalently this is the set of $G$-trees having the same elliptic subgroups (subgroups fixing a vertex) as the given one [@forester:trees]. It is important to note that $G$-trees having the same elliptic subgroups need not have the same vertex stabilizers. This is one of the main issues arising in this paper.
Two notions of complexity for $G$-trees are the number of edge orbits and the number of vertex orbits. Within a deformation space, the local minima for both notions occur at the reduced trees: those for which no collapse moves are possible. In the first part of this paper we study *fully reduced* $G$-trees. These are $G$-trees in which no vertex stabilizer contains the stabilizer of a vertex from a different orbit. Fully reduced trees, when they exist, globally minimize complexity in a deformation space. They are somewhat canonical (cf. Proposition \[VAbijection\]) but are not always unique. The slide-inequivalent trees given in [@forester:jsj] are both fully reduced, for example.
One of our main results is Theorem \[fullreducibility\] which states that every generalized Baumslag–Solitar tree can be made fully reduced by a deformation. After developing properties of fully reduced trees in Section \[sec4\] we use these results to give a simplified proof of the fact (originally proved in [@forester:trees]) that all non-elementary generalized Baumslag–Solitar trees with the same group lie in a single deformation space. These results also lay the groundwork for further study on the classification of generalized Baumslag–Solitar groups.
In the second part of the paper we focus on generalized Baumslag–Solitar groups having no non-trivial integral moduli. It turns out that this class of groups can be understood reasonably well. One key property is given in Theorem \[slidethm\]: for such groups, deformations between reduced trees can be converted into sequences of slide moves, which do not change complexity. We then prove a finiteness theorem for such trees (Theorem \[finiteness\]), and these two results together yield a solution to the isomorphism problem (Corollary \[isomproblem\]). This is our second main result.
This paper is based on my PhD dissertation, prepared under the direction of Peter Scott. I wish to express my gratitude to Peter Scott for his support and for many valuable discussions and suggestions. I also thank Gilbert Levitt and Noel Brady for helpful discussions and comments.
Preliminaries {#sec1}
=============
We will use Serre’s notation for graphs and trees [@serre:trees]. Thus a graph $A$ is a pair of sets $(V(A)$, $E(A))$ with maps $\partial_0, \partial_1 \co E(A) \to V(A)$ and an involution $e
\mapsto \overline{e}$ (for $e \in E(A)$), such that $\partial_i\overline{e} = \partial_{1-i} e$ and $e \not= \overline{e}$ for all $e$. An element $e \in E(A)$ is to be thought of as an oriented edge with initial vertex $\partial_0 e$ and terminal vertex $\partial_1
e$. We denote by $E_0(v)$ the set of all edges having initial vertex $v$. An edge $e$ is a *loop* if $\partial_0 e = \partial_1 e$.
Let $G$ be a group. A *$G$-tree* is a tree with a $G$-action by automorphisms, without inversions. A $G$-tree is *proper* if every edge stabilizer is strictly smaller that its neighboring vertex stabilizers. It is *minimal* if there is no proper $G$-invariant subtree, and it is *cocompact* if the quotient graph is finite.
Given a $G$-tree $X$, an element $g\in G$ is *elliptic* if it fixes a vertex of $X$ and is *hyperbolic* otherwise. If $g$ is hyperbolic then there is a unique $G$-invariant line in $X$, called the *axis* of $g$, on which $g$ acts as a translation. A subgroup $H$ of $G$ is *elliptic* if it fixes a vertex.
Suppose a graph of groups has an edge $e$ which is a loop. Let $A$ be the vertex group and $C$ the edge group, with inclusion maps $i_0, i_1 \co C
\hookrightarrow A$. If one of these maps, say $i_0$, is an isomorphism, then $e$ is an *ascending loop*. The *monodromy* is the composition $i_1 \circ i_0^{-1} \co A \hookrightarrow A$.
In a *collapse move*, an edge in a graph of groups carrying an amalgamation of the form $A \ast_C C$ is collapsed to a vertex with group $A$. Every inclusion map having target group $C$ is reinterpreted as a map into $A$, via the injective map of vertex groups $C \hookrightarrow A$.
[90]{}[10]{}
An *expansion move* is the reverse of a collapse move. Both of these moves are called *elementary moves*. A *deformation* (also called an *elementary deformation* in [@forester:trees; @forester:jsj]) is a finite sequence of such moves.
A graph of groups is *reduced* if it admits no collapse moves. This means that if an inclusion map from an edge group to a vertex group is an isomorphism, then the edge is a loop. Correspondingly, a $G$-tree is reduced if, whenever an edge stabilizer is equal to the stabilizer of one of its endpoints, both endpoints are in the same orbit. Note that reduced $G$-trees are minimal.
The deformation shown below (cf. [@herrlich]) is called a *slide move*. In order to perform the move it is required that $D \subseteq C$ (regarded as subgroups of $A$).
[120]{}[12]{}
It is permitted for the edge carrying $C$ to be a loop; in this case the only change to the graph of groups is in the inclusion map $D \hookrightarrow A$. See Proposition \[gbsmoves\] for an example.
An *induction move* is an expansion and collapse along an ascending loop. In the diagram below the ascending loop has vertex group $A$ and monodromy $\phi \co A \to A$, and $B$ is a subgroup such that $\phi(A)
\subseteq B \subseteq A$. The map $\iota\co B \to A$ is inclusion. The lower edge is expanded and the upper edge is collapsed, resulting in an ascending loop with monodromy the induced map $\phi\vert_{B}\co B \to B$.
[104]{}[12]{}
The reverse of this move is also considered an induction move. Notice that the vertex group changes, in contrast with slide moves.
A *fold* is most easily described directly in terms of $G$-trees. The graph of groups description includes many different cases which are explained in [@bestvina:accessibility]. To perform a fold in a $G$-tree one chooses edges $e$ and $f$ with $\partial_0 e =
\partial_0 f$, and identifies $e$ and $f$ to a single edge. One also identifies $g e$ with $g f$ for every $g \in G$, so that the resulting quotient graph has a $G$-action. It is not difficult to show that the new graph is a tree.
The following basic result is proved in [@forester:trees Proposition 3.16].
\[foldfac\] Suppose a fold between $G$-trees preserves hyperbolicity of elements of $G$. Then the fold can be represented by a deformation.
Generalized Baumslag–Solitar groups {#sec2}
===================================
A *generalized Baumslag–Solitar tree* is a $G$-tree whose vertex and edge stabilizers are all infinite cyclic. The groups $G$ that arise are called *generalized Baumslag–Solitar groups*. Basic examples include Baumslag–Solitar groups [@baumslagsolitar], torus knot and link groups, and finite index subgroups of these groups.
The quotient graphs of groups have all vertex and edge groups isomorphic to ${{\mathbb Z}}$, and the inclusion maps are multiplication by various non-zero integers. Thus any example is specified by a graph $A$ and a function $i\co E(A) \to ({{\mathbb Z}}- \{0\})$. The corresponding graph of groups will be denoted by $(A,i)_{{{\mathbb Z}}}$. If $X$ is the $G$-tree above $(A,i)_{{{\mathbb Z}}}$ then the induced function $i \co E(X) \to ({{\mathbb Z}}- \{0\})$ satisfies $$\label{absindex}
{\left\lverti(e)\right\rvert} \ = \ [G_{\partial_0 e} : G_e]$$ for all $e\in E(X)$.
\[signs\] There is generally some choice involved in writing down a quotient graph of groups of a $G$-tree. This issue is explored fully in [@bass:covering Section 4]. Without changing the $G$-tree it describes, a graph of groups may be modified by twisting an inclusion map by an inner automorphism of the target vertex group. Any two quotient graphs of groups of a $G$-tree are related by modifications of this type.
In the case of generalized Baumslag–Solitar trees there are no such inner automorphisms and the quotient graph of groups is uniquely determined by the $G$-tree. The associated edge-indexed graph is then very nearly uniquely determined; the only ambiguity arises from the choice of generators of edge and vertex groups. One may simultaneously change the signs of all indices at a vertex, or change the signs of $i(e)$ and $i(\overline{e})$ together for any $e$, with no change in the graph of groups or the $G$-tree it encodes.
Elementary moves and deformations can be described directly in terms of edge-indexed graphs, as follows. The verifications are left to the reader. In the diagrams below, each index $i(e)$ is shown next to the endpoint $\partial_0 e$. Note in particular that any deformation performed on a generalized Baumslag–Solitar tree results again in a generalized Baumslag–Solitar tree.
\[gbsmoves\] If an elementary move is performed on a generalized Baumslag–Solitar tree, then the quotient graph of groups changes locally as follows:
[90]{}[10]{}
A slide move has the following description:
[100]{}[11]{}
or
[100]{}[10]{}
An induction move is as follows (cf. Lemma \[essentialdivides\]):
[80]{}[10]{}
A $G$-tree is *elementary* if there is a $G$-invariant point or line, and *non-elementary* otherwise. In [@forester:jsj Lemma 2.6] it is shown that a generalized Baumslag–Solitar tree is elementary if and only if the group is isomorphic to ${{\mathbb Z}}$, ${{\mathbb Z}}\times {{\mathbb Z}}$, or the Klein bottle group. Thus we may speak of generalized Baumslag–Solitar groups as being elementary or non-elementary.
A fundamental property of generalized Baumslag–Solitar groups is that the elliptic subgroups are canonical, except in the elementary case. Recall that two subgroups $H, K$ of $G$ are *commensurable* if $H \cap K$ has finite index in both $H$ and $K$. The following lemma is proved in [@forester:trees Corollary 6.10] and [@forester:jsj Lemma 2.5].
\[elliptic\] Let $X$ be a non-elementary generalized Baumslag–Solitar tree with group $G$. A nontrivial subgroup $H \subseteq G$ is elliptic if and only if it is infinite cyclic and is commensurable with all of its conjugates.
The property of $H$ given in the lemma is rather special. Kropholler showed in [@krop:gbsgroups] that among finitely generated groups of cohomological dimension $2$, the existence of such a subgroup exactly characterizes the generalized Baumslag–Solitar groups.
Full reducibility {#sec3}
=================
A graph of groups is *fully reduced* if no vertex group can be conjugated into another vertex group. Correspondingly, a $G$-tree is fully reduced if, whenever one vertex stabilizer contains another vertex stabilizer, they are conjugate. Notice that a fully reduced graph of groups is minimal and reduced. Two basic examples of fully reduced trees are proper trees and trees having a single vertex orbit.
We shall see that for generalized Baumslag–Solitar groups, fully reduced decompositions exist and have underlying graphs of smallest complexity (Theorems \[fullreducibility\] and \[complexity\] below).
\[bs530\] The $G$-tree shown on the left is reduced but not fully reduced. The valence three vertex group can be conjugated into the other vertex group (by conjugating around the loop). After performing an induction move and a collapse one finds that $G$ is the Baumslag–Solitar group $BS(5,30)$, which was perhaps not obvious initially.
[100]{}[10]{}
The following result generalizes this procedure to arbitrary generalized Baumslag–Solitar trees.
\[fullreducibility\] Every cocompact generalized Baumslag–Solitar tree is related by a deformation to a fully reduced (generalized Baumslag–Solitar) tree.
The proof relies strongly on the fact that stabilizers are infinite cyclic, and therefore contain a unique subgroup of any given index. Before proving the theorem we establish some preliminary facts concerning generalized Baumslag–Solitar trees. Our first objective (Corollary \[fixpathcor\]) is to characterize the paths that are fixed by vertex stabilizers.
\[fixpath\] Let $X$ be a generalized Baumslag–Solitar tree with group $G$. Suppose $G_x
\subseteq n G_{x'}$ for vertices $x \not= x'$. Let $(e_1, \ldots, e_k)$ be the path from $x$ to $x'$, with vertices $x_0 = x$, $x_i = \partial_1
e_i$ for $1 {\leqslant}i {\leqslant}k$. Define $m_i = i(\overline{e}_i)$, $n_i =
i(e_{i+1})$, and $n_k = n$. Then $i(e_1) = \pm 1$ and
1. $G_x = \left(\Pi_{i=1}^{k} \, m_i \, / \, \Pi_{i=1}^{k-1}
\, n_i\right) G_{x_k}$
2. $\Pi_{i=1}^k \ n_i$ divides $\Pi_{i=1}^k \ m_i$.
The statement $i(e_1)=\pm 1$ is clear because $G_x$ fixes $e_1$. The other two statements are proved together by induction on $k$.
If $k=1$ then (i) says that $G_x = m_1 G_{x'}$, which holds because $i(e_1) = \pm 1$. Then the assumption $G_x \subseteq n_1 G_{x'}$ implies that $n_1$ divides $m_1$, because $G_x = G_{e_1} = m_1 G_{x'}$.
Now let $k > 1$ be arbitrary. Since $G_x$ fixes $e_k$ we have $G_x
\subseteq n_{k-1} G_{x_{k-1}}$, and the induction hypothesis gives that $\Pi_{i=1}^{k-1} n_i$ divides $\Pi_{i=1}^{k-1} m_i$. We also have $G_x = \left(\Pi_{i=1}^{k-1} \, m_i \, / \, \Pi_{i=1}^{k-2}
\, n_i\right)
G_{x_{k-1}}$ and $n_{k-1} G_{x_{k-1}} = G_{e_k} = m_k
G_{x_k}$. Therefore $$G_x \ = \ \left(\Pi_{i=1}^{k-1} \, m_i \, / \,
\Pi_{i=1}^{k-1} \, n_i\right) n_{k-1} \, G_{x_{k-1}} \ = \
\left(\Pi_{i=1}^{k-1} \, m_i \, / \,
\Pi_{i=1}^{k-1} \, n_i\right) m_k \, G_{x_k},$$ proving (i).
Next, the assumption $G_x \subseteq n_k G_{x_k}$ becomes $\left(\Pi_{i=1}^{k} \, m_i \, / \, \Pi_{i=1}^{k-1} \, n_i\right)
G_{x_k} \subseteq n_k G_{x_k}$ by (i), establishing (ii).
The lemma is valid in any locally finite $G$-tree, provided one interprets statements such as $G_x \subseteq n G_{x'}$ correctly. For example this statement would mean that $G_x \subseteq G_{x'}$ and $n$ divides $[G_{x'}:G_x]$. The following corollary, however, is specific to generalized Baumslag–Solitar trees.
\[fixpathcor\] Let $(e_1, \ldots, e_k)$ be a path in $X$ and define $m_i$, $n_i$ as in the previous lemma. Then ${G}_{\partial_0 e_1}$ fixes the path $(e_1,
\ldots, e_k)$ if and only if $i(e_1) = \pm 1$ and for every $l{\leqslant}(k-1)$ $$\label{fixpathineq2}
\Pi_{i=1}^l \ n_i \quad \text{divides} \quad \Pi_{i=1}^l \ m_i.$$
The forward implication is given by Lemma \[fixpath\](ii). The converse is proved by induction on $k$. Suppose holds for each $l$ and that ${G}_{\partial_0 e_1}$ fixes the path $(e_1,
\ldots, e_{k-1})$. Then $G_{\partial_0 e_1}$ is the subgroup of $G_{\partial_0 e_k}$ of index $\Pi_{i=1}^{k-1} m_i \, / \,
\Pi_{i=1}^{k-2}n_i$ by Lemma \[fixpath\](i). Property for $l= k-1$ implies that $n_{k-1}$ divides this index, and so ${G}_{\partial_0 e_1} \subseteq G_{e_k}$.
Next we describe the steps needed to construct the deformation of Theorem \[fullreducibility\]. The kind of example one should have in mind is one similar to Example \[bs530\], but with several loops incident to the left-hand vertex.
Throughout the rest of this section $X$ denotes a generalized Baumslag–Solitar tree with group $G$ and quotient graph of groups $(A,i)_{{{\mathbb Z}}}$.
Let $f\in E(A)$ be an edge with $\partial_0 f \not= \partial_1 f$. Let $\rho = (e_1, \ldots, e_k, f)$ be a path in $A$ such that each $e_i$ is a loop at $\partial_0 f$. We say that $\rho$ is an *admissible path for $f$* if, for some lift $\tilde{\rho} =
(\tilde{e}_1, \ldots, \tilde{e}_k, \tilde{f}) \subset X$, $${G}_{\partial_0 \tilde{e}_1} \ \ \subseteq \ \
{G}_{\tilde{f}}.$$ This condition depends only on the indices along the path $\rho$ by Corollary \[fixpathcor\], so it is independent of the choice of $\tilde{\rho}$. When dealing with admissible paths we will use the notation $m_i = i(\overline{e}_i)$, $n_i = i(e_{i+1})$, and $n_k =
i(f)$; then the path is admissible if and only if $i(e_1) = \pm 1$ and holds for each $l$.
An edge $e\in E(A)$ is *essential* if $i(e) \not= \pm 1$, and *inessential* otherwise. The *length* of $\rho$ is $k$, and the *essential length* of $\rho$ is the number of essential edges occurring in $\rho$.
\[inessentialfirst\] If $\rho = (e_1, \ldots, e_k, f)$ is an admissible path then there is a permutation $\sigma$ such that the path $\rho_{\sigma} = (e_{\sigma(1)},
\ldots, e_{\sigma(k)}, f)$ is admissible and all of the essential edges of $\rho_{\sigma}$ occur after the inessential edges.
We show first that if $e_j$ is inessential for some $j > 1$ then $$\rho' = \rho_{((j-1) \ j)} = (e_1, \ldots, e_{j-2}, e_j, e_{j-1}, e_{j+1},
\ldots, e_k, f)$$ is an admissible path for $f$. Letting $m'_i$ and $n'_i$ be the indices along $\rho'$, we have $m'_{j-1} = m_j$, $m'_j =
m_{j-1}$, and $n'_{j-2} = n_{j-1}$, $n'_{j-1} = n_{j-2}$, with all other indices unchanged. Clearly still holds for $l \not=
j-2, j-1$. One easily verifies for these other two cases as well, using the fact that $n_{j-1} = \pm 1$ (because $e_j$ is inessential). Hence $\rho'$ is admissible for $f$. Next, by using transpositions of this type, one can move all of the inessential edges in $\rho$ to the front of the path.
\[inessentialsame\] Let $\rho = (e_1, \ldots, e_k, f)$ be an admissible path such that $e_1,
\ldots, e_j$ are inessential and $e_{j+1}, \ldots, e_k$ are essential. Then there is a sequence of slide moves, after which the path $\rho' = (e_1, \ldots, e_1, e_{j+1}, \ldots, e_k, f)$ is admissible. The inessential part $(e_1, \ldots, e_1)$ of $\rho'$ may have length greater than $j$, though $\rho$ and $\rho'$ have the same essential length ($k-j$).
First we slide $\overline{e}_1$ over each edge of $(A,i)_{{{\mathbb Z}}}$ (other than $e_1$) that appears in $(e_2, \ldots, e_j)$. Since these edges are all inessential loops, these slides can be performed. The index $i(e_1)$ is unchanged so occurrences of $e_1$ in the path are still inessential.
Each slide of $\overline{e}_1$ over $e_i$ multiplies $i(\overline{e}_1)$ by $\pm i(\overline{e}_i)$. The end result is that $m_1$ gets multiplied by a product $\pm \Pi_{\nu=1}^r n_{i_{\nu}}$. By itself this change does not violate the conditions . However if some $e_i$ is equal to $\overline{e}_1$ (where $i > j$), then $n_{i-1}$ is also multiplied by $\pm \Pi_{\nu=1}^r n_{i_{\nu}}$, and may fail. To remedy this we adjoin several copies of $e_1$ to the front of the path, one for each occurrence of $\overline{e}_1$ in $(e_{j+1}, \ldots, e_k)$. Then the products $\Pi_{i=1}^l m_i$ acquire enough additional factors $\pm \Pi_{\nu=1}^r n_{i_{\nu}}$ to remain divisible by $\Pi_{i=1}^l n_i$. This extended path is therefore admissible. As a result of the slide moves, $m_i$ now divides $m_1$ for each $i {\leqslant}j$.
We now replace $e_i$ by $e_1$ for $i {\leqslant}j$. The quantities $\Pi_{i=1}^l m_i$ increase and each $\Pi_{i=1}^l n_i$ remains unchanged (up to sign), so property still holds for every $l$.
\[essentialdivides\] Let $(e_1, \ldots, e_1, e_{j+1}, \ldots, e_k, f)$ be an admissible path such that $e_1$ is inessential and $e_{j+1}, \ldots, e_k$ are essential. Then there is a sequence of induction moves after which a path of the same form is admissible, has essential length at most ($k -
j$), and satisfies $i(e_{j+1}) = \pm i(\overline{e}_1)^r$ for some $r$.
If $e_{j+1} = \overline{e}_1$ then we can discard it from the path without affecting admissibility. Thus we can assume that $e_{j+1} \not=
\overline{e}_1$. Admissibility implies that $i(e_{j+1})$ divides $i(\overline{e}_1)^j$. Let $r$ be minimal so that $i(e_{j+1})$ divides $i(\overline{e}_1)^r$ and let $l$ be any factor of $i(\overline{e}_1)^r / i(e_{j+1})$ that divides $i(\overline{e}_1)$. We show how to make $i(e_{j+1})$ become $l \cdot i(e_{j+1})$. By repeating this procedure the desired result can be achieved.
Writing $i(\overline{e}_1)$ as $lm$, we perform an induction move along $e_1$ as follows:
[100]{}[10]{}
The index of every edge incident to $\partial_0 e_1$ is multiplied by $l$, except for $i(e_1)$ and $i(\overline{e}_1)$, which remain the same. As a result the indices $m_i$ and $n_{i-1}$ are multiplied by $l$ whenever $e_i$ is not equal to $e_1$ or $\overline{e}_1$. For every such $i$ we adjoin a copy of $e_1$ to the front of the path, making it admissible as in the proof of the preceding lemma.
\[esslengthone\] The previous argument is still valid when $j=k$. That is, if the path $(e_1,
\ldots, e_1, f)$ is admissible, then there is a sequence of induction moves after which $i(f) = i(\overline{e}_1)^r$ for some $r$.
We show that if $(A,i)_{{{\mathbb Z}}}$ is not fully reduced then there is a deformation to a decomposition having fewer edges. Repeating the procedure will eventually produce a fully reduced decomposition.
If $(A,i)_{{{\mathbb Z}}}$ is not fully reduced then there exist vertices $v$, $w$ of $X$ such that ${G}_v \subseteq {G}_w$ and $v \not\in G w$. The path $(\tilde{e}_1, \ldots, \tilde{e}_r)$ from $v$ to $w$ contains an edge that does not map to a loop in $A$. Let $\tilde{f} = \tilde{e}_{k+1}$ be the first such edge. Since ${G}_v$ stabilizes $(\tilde{e}_1, \ldots, \tilde{e}_k, \tilde{f})$, the image $\rho = (e_1, \ldots, e_k, f)$ of this path in $A$ is an admissible path for $f$.
Next we show how to produce an admissible path for $f$ having essential length smaller than that of $\rho$, assuming this length is positive. Suppose $\rho$ has essential length $s$. Applying Lemmas \[inessentialfirst\], \[inessentialsame\], and \[essentialdivides\] in succession to the path $\rho$ we can arrange that there is an admissible path $\rho' = (e'_1, \ldots, e'_1, e'_{k'-s+1}, \ldots, e'_{k'}, f)$ such that $e'_1$ is inessential, $e'_{k'-s+1}, \ldots, e'_{k'}$ are essential, and $i(e'_{k'-s+1}) = i(\overline{e}'_1)^r$ for some $r$. (The essential length $s$ may have decreased, but then we are done for the moment.) In applying these lemmas the decomposition $(A,i)_{{{\mathbb Z}}}$ changes by a deformation to $(A', i')_{{{\mathbb Z}}}$ where $A'$ has the same number of edges as $A$. Now we can slide $e'_{k'-s+1}$ over $\overline{e}'_1$ $r$ times to make $i(e'_{k'-s+1}) = \pm 1$.
These slide moves affect the indices of the edges $e'_i$ that are equal to $e'_{k'-s+1}$ or $\overline{e}'_{k'-s+1}$. If $e'_i = e'_{k'-s+1}$ then $n_{i-1}$ is divided by $(m_1)^r = i(\overline{e}'_1)^r$ and this change does not affect the admissibility of $\rho'$. If $e'_i =
\overline{e}'_{k'-s+1}$ then $m_i$ is divided by $(m_1)^r$. In order to keep $\rho'$ admissible we adjoin $r$ copies of $e'_1$ to the front of the path for each such $e'_i$. This done, we have produced an admissible path for $f$ of smaller essential length because $e'_{k'-s+1}$ is now inessential.
By repeating this process we can obtain a decomposition $(A',i')_{{{\mathbb Z}}}$ related by a deformation to $(A,i)_{{{\mathbb Z}}}$ (with ${\left\lvertE(A')\right\rvert} =
{\left\lvertE(A)\right\rvert}$), and an admissible path for $f$ having essential length zero. Applying Lemmas \[inessentialsame\] and \[essentialdivides\] once more, this path has the form $(e'_1, \ldots, e'_1, f)$ where $i(f) =
i(\overline{e}'_1)^r$. Now we slide $f$ over $\overline{e}'_1$ $r$ times to make $i(f) = \pm 1$, and collapse $f$. The resulting decomposition has fewer edges than $(A,i)_{{{\mathbb Z}}}$.
Vertical subgroups {#sec4}
==================
In this section we link the structure of a fully reduced $G$-tree to that of the group $G$, using the notion of a vertical subgroup. We are concerned with the difference between elliptic subgroups (which may be uniquely determined) and vertex stabilizers (which often are not). It turns out that vertical subgroups are a useful intermediate notion. See in particular Example \[verticaleg\].
Let $X$ be a $G$-tree. A subgroup $H \subseteq G$ is *vertical* if it is elliptic and every elliptic subgroup containing $H$ is conjugate to a subgroup of $H$.
\[vertical\] If $X$ is fully reduced then an elliptic subgroup is vertical if and only if it contains a vertex stabilizer.
Suppose $H$ contains $G_v$, and let $H'$ be an elliptic subgroup containing $H$. Since $H'$ is elliptic, it is contained in $G_w$ for some $w$, and hence $G_v \subseteq G_w$. Full reducibility implies that $(G_w)^g
= G_v$ for some $g$, and therefore $(H')^g \subseteq H$.
Conversely suppose $H$ is vertical. Then $H\subseteq G_v$ for some $v$, and so $(G_v)^g \subseteq H$ for some $g\in G$. Hence $H$ contains $G_{gv}$.
\[verticaleg\] Let $G$ be the Baumslag–Solitar group $BS(1,6)$ with its standard decomposition $G = {{\mathbb Z}}\,\ast_{\, {{\mathbb Z}}}$ and presentation $\langle x, t
\mid t x t^{-1} = x^6 \rangle$. The vertex stabilizers of the Bass–Serre tree $X$ are the conjugates of the subgroup $\langle x \rangle$. Among the subgroups of the form $\langle x^n \rangle$, notice that all are elliptic, and only those where $n$ is a power of $6$ are vertex stabilizers. According to Lemma \[vertical\], $\langle x^n \rangle$ is vertical if and only if $n$ divides a power of $6$.
Now consider the automorphism $\phi\co {G} \rightarrow {G}$ defined by $\phi(x) = x^3$, $\phi(t) = t$ (with inverse $x \mapsto t^{-1} x^2 t$, $t\mapsto t$). If we twist the action of $G$ on $X$ by $\phi$ then the vertex stabilizers will be the conjugates of $\langle x^2
\rangle$ rather than $\langle x \rangle$. Thus there is no hope of characterizing vertex stabilizers from the structure of $G$ alone. On the other hand, the vertical subgroups are uniquely determined (because the elliptic subgroups are).
In this particular example, the set of vertical subgroups is the smallest $\operatorname{Aut}(G)$-invariant set of elliptic subgroups containing a vertex stabilizer. Every vertical subgroup can be realized as a vertex stabilizer by twisting by an automorphism.
\[eqreln1\] Now we define an equivalence relation on the set of vertical subgroups of $G$. Set $H \sim K$ if $H$ is conjugate to a subgroup of $K$. This relation is symmetric: suppose $H \sim K$, so that $H^g \subseteq K$ for some $g\in G$. Then $H \subseteq K^{g^{-1}}$ and so $K^{g^{-1}}$ is conjugate to a subgroup of $H$, as $H$ is vertical. Therefore $K \sim
H$. Reflexivity and transitivity are clear.
\[VAbijection\] If $X$ is fully reduced then the vertex orbits correspond bijectively with the $\sim$-equivalence classes of vertical subgroups of $G$. The bijection is induced by the natural map $v \mapsto G_v$.
The induced map is well defined since $G_v \sim (G_v)^g = G_{gv}$ for any $g$. For injectivity, suppose that $G_v \sim G_w$. Then $G_{gv} \subseteq
G_w$ for some $g\in G$. Full reducibility implies that $gv$ and $w$ are in the same orbit, hence $v$ and $w$ are as well.
For surjectivity, suppose $H$ is vertical. It contains a stabilizer $G_v$ by Lemma \[vertical\], and $G_v \subseteq H$ implies $G_v \sim
H$.
The following application of Proposition \[VAbijection\] explains the choice of the term *fully reduced*.
\[complexity\] A non-elementary cocompact generalized Baumslag–Solitar tree is fully reduced if and only if it has the smallest number of edge orbits among all generalized Baumslag–Solitar trees having the same group.
The proof of Theorem \[fullreducibility\] shows that any generalized Baumslag–Solitar tree with the smallest number of edge orbits is fully reduced. For the converse we show that no tree with more edge orbits can also be fully reduced.
Suppose the given tree is fully reduced. Let $N\subseteq G$ be the normal closure of the set of elliptic elements. This subgroup is uniquely determined since the tree is non-elementary. Note that $G/N$ is the fundamental group of the quotient graph. Hence the homotopy type of this graph is uniquely determined. Proposition \[VAbijection\] implies that the number of vertices is also uniquely determined, and so the number of edges is as well. Thus any two fully reduced trees have the same number of edge orbits.
Existence of deformations {#sec5}
=========================
We now know that generalized Baumslag–Solitar trees can be made fully reduced (Theorem \[fullreducibility\]) and that for such trees, the structure of the tree is partially encoded in the set of elliptic subgroups (Proposition \[VAbijection\]). Using these facts we may now give a quick proof of the existence of deformations between generalized Baumslag–Solitar trees. This result is a special case of Theorem 1.1 of [@forester:trees].
\[defthm\] Let $X$ and $Y$ be non-elementary cocompact generalized Baumslag–Solitar trees with isomorphic groups. Then $X$ and $Y$ are related by a deformation.
A map between trees is a *morphism* if it sends vertices to vertices and edges to edges (and respects the maps $\partial_0$, $\partial_1$, $e
\mapsto \overline{e}$). Geometrically it is a simplicial map which does not send any edge into a vertex.
The following result is taken from [@bestvina:accessibility Section 2].
\[bfprop\] Let $G$ be a finitely generated group and suppose that $\phi\co X
\rightarrow Y$ is an equivariant morphism of $G$-trees. Assume further that $X$ is cocompact, $Y$ is minimal, and the edge stabilizers of $Y$ are finitely generated. Then $\phi$ is a finite composition of folds.
Let $G$ be the common group acting on $X$ and $Y$. Note that both trees define the same elliptic and vertical subgroups. By Theorem \[fullreducibility\] we can assume that both trees are fully reduced (and minimal). Applying Propositions \[bfprop\] and \[foldfac\], it now suffices to construct a morphism from $X$ to $Y$. In fact we shall construct such a map from $X'$ to $Y$, where $X'$ is obtained from $X$ by subdivision (a special case of a deformation).
Let $x_1, \ldots, x_n \in V(X)$ be representatives of the vertex orbits of $X$. Then there are vertices $y_1, \ldots, y_n \in
V(Y)$ such that $G_{x_i} \subseteq G_{y_i}$, since each $G_{x_i}$ is elliptic. We define a map $\phi\co V(X) \to V(Y)$ by setting $\phi(x_i)
= y_i$ and extending equivariantly. We then extend $\phi$ to a topological map $X \to Y$ by sending an edge $e$ to the unique reduced path in $Y$ from $\phi(\partial_0 e)$ to $\phi(\partial_1
e)$. Subdividing where necessary, we obtain an equivariant simplicial map $\phi' \co X' \to Y$.
Now we verify that $\phi'$ is a morphism. It suffices to check that $\phi(x) \not= \phi(x')$ whenever $x$ and $x'$ are vertices of $X$ that bound an edge. There are two cases. If $x' = gx$ for some $g\in G$ then $g$ is hyperbolic, since it has translation length one in $X$, and equivariance implies that $\phi(x) \not= \phi(x')$. Otherwise, if $x$ and $x'$ are in different orbits, then $G_x \not\sim G_{x'}$ by Proposition \[VAbijection\]. Here we are using the fact that $X$ is fully reduced. Equivariance yields $G_{x} \subseteq G_{\phi(x)}$ and $G_{x'} \subseteq
G_{\phi(x')}$, and since these are all vertical subgroups (by Lemma \[vertical\]) we now have $G_{x} \sim G_{\phi(x)}$ and $G_{x'} \sim
G_{\phi(x')}$. Hence $G_{\phi(x)} \not\sim G_{\phi(x')}$, and in particular $\phi(x) \not= \phi(x')$.
The modular homomorphism {#sec6}
========================
Let ${{{\mathbb Q}}}^{\times}_{>0}$ denote the positive rationals considered as a group under multiplication. The following notion was first defined by Bass and Kulkarni [@bass:treelat].
The *modular homomorphism* $q\co G \rightarrow {{{\mathbb Q}}}^{\times}_{>0}$ of a locally finite $G$-tree is given by $$q(g) \ = \ [V:V\cap V^{g}] \ / \ [V^{g}: V \cap
V^{g}]$$ where $V$ is any subgroup of $G$ commensurable with a vertex stabilizer. In this definition we are using the fact that in locally finite $G$-trees, vertex stabilizers are commensurable with all of their conjugates. One can easily check that $q$ is independent of the choice of $V$.
In the case of generalized Baumslag–Solitar trees the modular homomorphism may be defined directly in terms of the graph of groups $(A,i)_{{{\mathbb Z}}}$, as in [@bass:treelat]. First note that $q$ factors through $H_1(A)$ because it is trivial on elliptic subgroups and ${{\mathbb Q}}^{\times}_{>0}$ is abelian. Writing $q$ as a composition $G \to
H_1(A) \to {{\mathbb Q}}^{\times}_{>0}$, the latter map is then given by $$\label{modhom}
(e_1, \ldots, e_k) \ \mapsto \ \Pi_{j=1}^k \, {\left\lverti(e_j) /
i(\overline{e}_j)\right\rvert} .$$ To verify note that given $g\in G$, the corresponding $1$-cycle in $H_1(A)$ is obtained by projecting any (oriented) segment of the form $[v,gv]$ to $A$. One then uses $V = G_v$ to evaluate $q(g)$, by applying to the edges of $[v,gv]$.
The next definition is not actually needed in this paper. We mention it for completeness, with the expectation that it will be useful in future work.
\[signedmodhom\] The *signed modular homomorphism* $\hat{q} \co G \to {{\mathbb Q}}^{\times}$ of a generalized Baumslag–Solitar tree with quotient graph of groups $(A,i)_{{{\mathbb Z}}}$ is defined via the map $H_1(A) \to {{\mathbb Q}}^{\times}$ given by $$\label{smodhom}
(e_1, \ldots, e_k) \ \mapsto \ \Pi_{j=1}^k \, i(e_j) /
i(\overline{e}_j) .$$ One should verify that this is well defined in light of Remark \[signs\]. Clearly, changing the signs of $i(e)$ and $i(\overline{e})$ together, for any $e$, has no effect. Similarly, since $(e_1, \ldots,
e_k)$ is a cycle, changing all signs at a vertex will introduce an even number of sign changes in .
There is also an *orientation homomorphism* $G \to \{\pm 1\}$ defined by $g \mapsto \hat{q}(g)/q(g)$.
\[modhominvariant\] The modular homomorphisms are invariant under deformations. For the unsigned case, note that that during an elementary move there is a vertex stabilizer that remains unchanged. Taking $V$ to be this stabilizer, one obtains invariance of $q$. Alternatively, one may verify directly that the homomorphisms defined by and are invariant, using Proposition \[gbsmoves\].
Deformations and slide moves {#sec7}
============================
In this section we show how to rearrange elementary moves between generalized Baumslag–Solitar trees. Our goal is to replace deformations by sequences of slide moves, which are considerably easier to work with. It should be noted that in general, reduced generalized Baumslag–Solitar trees with the same group $G$ need not be related by slide moves; see [@forester:jsj]. Nevertheless this does occur in a special case, given in Theorem \[slidethm\] below.
Suppose $(A,i)_{{{\mathbb Z}}}$ has a loop $e$ with $(i(e), i(\overline{e})) =
(m,n)$. If $m$ divides $n$ then $e$ is a *virtually ascending loop*. It is *strict* if $n \not= \pm m$. Similarly, a *strict ascending loop* is one with indices of the form $(\pm 1, n)$, $n \not= \pm 1$.
The next two propositions are valid for sequences of moves between generalized Baumslag–Solitar trees.
\[expslide\] Suppose an expansion is followed by a slide move. Either
the moves remove a strict virtually ascending loop and create a strict ascending loop, or
the moves may be replaced by a (possibly empty) sequence of slides, followed by an expansion.
Suppose the expansion creates $e$ and the second move slides $e_0$ over $e_1$ (from $\partial_0 e_1$ to $\partial_1 e_1$). If $e$ is not $e_i$ or $\overline{e}_i$ ($i=0,1$) then the moves may be performed in reverse order as they do not interfere with each other.
If $e = e_1$ or $\overline{e}_1$ then there is no need to perform the slide at all. When performing an expansion at a vertex, the incident edges are partitioned into two sets, which are then separated by a new edge. Sliding $e_0$ over the newly created edge is equivalent to including $e_0$ in the other side of the partition before expanding.
If $e=e_0$ or $\overline{e}_0$ then there are several cases to consider. Orient $e$ so that $i(e)=1$, and $e_1$ so that $e$ slides over $e_1$ from $\partial_0(e_1)$ to $\partial_1(e_1)$. The cases depend on which of the vertices $\partial_0 e$, $\partial_1 e$, $\partial_0 e_1$, $\partial_1 e_1$ coincide (after the expansion and before the slide).
After expanding $e$, $e_1$ has endpoints $\partial_0(e)$ and $\partial_1(e)$. If $\partial_1(e) = \partial_0(e_1)$ and $\partial_0(e)
= \partial_1(e_1)$ then $i(e)$ is still $1$ after the slide and $e$ has become an ascending loop. In addition, since the slide takes place we must have $i(e_1) \mid
i(\overline{e})$. Writing $i(e_1) = k$ and $i(\overline{e}) = kl$, we must have had $kl \mid i(\overline{e}_1)$ and $i(e_1) = k$ before the expansion, so $e_1$ was a virtually ascending loop. Writing $i(\overline{e}_1)
= klm$ (before the expansion) the modulus of the loop $e_1$ is $lm$. If $lm \not= \pm 1$ then alternative (i) holds. If $lm = \pm 1$ then after the expansion, $i(\overline{e}_1) = \pm 1$. The expansion and slide may then be replaced by a single expansion.
Otherwise $\partial_1(e) = \partial_1(e_1)$ and $\partial_0(e) =
\partial_0(e_1)$. The two moves have the form:
[111]{}[16]{}
The same $G$-tree results if we first perform slides and then expand, including $\partial_0(e_1)$ in the same side as $\partial_1(e_1)$, and then exchange the names of $e$ and $e_1$.
The edge $e_1$ has distinct endpoints and is incident to only one endpoint of $e$. Let $\{f_i\}$ be the edges with $\partial_0(f_i) = \partial_0(e)$ just before the slide move (not including $e_1$). We replace the expansion and slide by slides and an expansion as follows: first slide each $f_i$ over $e_1$, then expand at $\partial_1(e_1)$ so that the new expansion edge $e$ separates $\{f_i\}$ from the rest of the edges at $\partial_1(e_1)$. For example:
[100]{}[20]{}
becomes
[100]{}[20]{}
As in Case 1, the names of $e$ and $e_1$ must be exchanged after the new moves. This procedure works whether $e_1$ is incident to $\partial_0(e)$ or to $\partial_1(e)$.
The edge $e_1$ is a loop incident to $\partial_1(e)$. Then the procedure from Case 2 works. The two moves are replaced by a sequence of slides (around the loop $e_1$) followed by an expansion.
The edge $e_1$ is a loop incident to $\partial_0(e)$. Let $l = i(\overline{e})$. Since $\partial_0(e)$ is the end of $e$ that slides over $e_1$ and $i(e)=1$, the loop $e_1$ must be an ascending loop (before and after the slide). Note that before the expansion of $e$, the indices of $e_1$ were $l$ times their current values; hence $e_1$ was originally a virtually ascending loop. Let $k$ be the modulus of $e_1$. If $k \not= \pm 1$ then alternative (i) holds. If $k=1$ then the slide move may simply be omitted. If $k=-1$ then first perform the slide moves as described in Case 2, and then expand the edge $e$ as before, but with $i(\overline{e}) = l$ and $i(e) = -1$.
\[expcollapse\] Suppose an expansion creating the edge $e$ is followed by the collapse of an edge $e'$. Then either
$e'$ is a strict ascending loop before the expansion and $e$ is a strict ascending loop after the collapse,
both moves may be deleted,
both moves may be replaced by a sequence of slides, or
the collapse may be performed before the expansion move.
If $e = e'$ or $e = \overline{e}'$ then clearly (ii) holds. Otherwise $e$ and $e'$ are distinct, proper edges just after the expansion and before the collapse. If they do not meet then conclusion (iv) holds.
Now assume that $e$ and $e'$ meet in one or two vertices. Orient both edges so that $i(e) = i(e') = 1$.
The edges $e$ and $e'$ have two vertices in common. If $\partial_0(e) = \partial_1(e')$ and $\partial_1(e) = \partial_0(e')$ then alternative (i) holds. Otherwise, if $\partial_0(e) =
\partial_0(e')$ and $\partial_1(e) = \partial_1(e')$ then set $k =
i(\overline{e})$ and $l = (\overline{e}')$. The moves have the form:
[102]{}[16]{}
Evidently the moves may be replaced by slides around the loop $e'$.
The edges $e$ and $e'$ meet in one vertex. Again let $k =
i(\overline{e})$ and $l = (\overline{e}')$. There are four configurations. If $\partial_0(e) = \partial_0(e')$ then we see:
[110]{}[10]{}
We may replace the two moves by slides over the edge $e'$.
In the other three configurations the collapse may be performed before the expansion. To illustrate, the case $\partial_1(e) = \partial_1(e')$ has the following configuration:
[111]{}[10]{}
and it is easy to see that the collapse may be performed first. The remaining two cases are entirely similar.
\[slidethm\] Let $X$ and $Y$ be reduced non-elementary cocompact generalized Baumslag–Solitar trees with group $G$, and suppose that $q({G}) \cap {{\mathbb Z}}= 1$. Then $X$ and $Y$ are related by slide moves.
The property $q({G}) \cap {{\mathbb Z}}= 1$ guarantees that no generalized Baumslag–Solitar decomposition of ${G}$ contains strict virtually ascending loops. Starting with a sequence of moves from $X$ to $Y$ (given by Theorem \[defthm\]) we claim that Propositions \[expslide\] and \[expcollapse\] can be applied to obtain a new sequence consisting of collapses, followed by slides, followed by expansions. To see this, note that case (i) of either proposition cannot occur. Therefore expansions can be pushed forward past slides (by \[expslide\]) and past collapses (by \[expcollapse\]), and collapses can be pulled back before slides (by \[expslide\] applied to the reverse of the sequence of moves). That is, we have the replacement rules $ES \to S^* E$, $EC \to (S^* \mbox{ or } CE)$, and $SC \to CS^*$, where $E$ and $C$ denote expansion and collapse moves respectively and $S^*$ denotes a (possibly empty) sequence of slide moves.
The algorithm for simplifying a sequence of moves is to repeatedly perform either of the following two steps, until neither applies. The first step is to find the first collapse move that is preceded by an expansion or slide, and apply the replacement $EC \to (S^* \mbox{ or }
CE)$ or $SC \to CS^*$ accordingly. The second step is to find the last expansion move that is followed by a collapse or slide and apply the replacement $EC \to (S^* \mbox{ or } CE)$ or $ES \to S^* E$. This procedure terminates, in a sequence of the form $C^* S^* E^*$. Then since $X$ and $Y$ are reduced, the new sequence of moves has no collapses or expansions.
The isomorphism problem {#sec8}
=======================
Next we approach the problem of classifying generalized Baumslag–Solitar groups. At the minimum, a classification should include an algorithm for determining when two indexed graphs define the same group. This is the problem considered here.
For certain generalized Baumslag–Solitar groups the isomorphism problem is trivial. This occurs when the deformation space contains only one reduced tree (such a tree is called *rigid*). The basic rigidity theorem for generalized Baumslag–Solitar trees was proved independently in [@pettet; @gilbertetal; @forester:trees] and it states that $(A,i)_{{{\mathbb Z}}}$ is rigid if there are no divisibility relations at any vertex. Levitt [@levitt:char] has extended this result by giving a complete characterization of trees that are rigid. Then to solve the isomorphism problem for such groups one simply makes the trees reduced and compares them directly (cf. Remark \[signs\]).
In this section we solve the isomorphism problem for the case of generalized Baumslag–Solitar groups having no non-trivial integral moduli. The general case is still open.
\[indexbound\] Let $Q \subset {{\mathbb Q}}^{\times}_{>0}$ be a finitely generated subgroup such that $Q \cap {{\mathbb Z}}= 1$. Then for any $r \in {{\mathbb Q}}$ the set $rQ \cap {{\mathbb Z}}$ is finite.
We consider ${{\mathbb Q}}^{\times}_{>0}$ as a free ${{\mathbb Z}}$-module with basis the prime numbers, via prime decompositions. Note that a positive rational number is an integer if and only if it has nonnegative coordinates in ${{\mathbb Q}}^{\times}_{>0} = {{\mathbb Z}}\oplus {{\mathbb Z}}\oplus \cdots$, and so the positive integers comprise the first “orthant” of ${{\mathbb Q}}^{\times}_{>0}$.
We are given that $Q$ meets the first orthant only at the origin. By taking tensor products with ${{\mathbb R}}$ we may think of ${{\mathbb Q}}^{\times}_{>0}$ as a vector space and $Q$ a finite dimensional subspace. Since multiplication by $r$ is a translation in ${{\mathbb Z}}\oplus {{\mathbb Z}}\oplus \cdots$, we have that $rQ$ is an affine subspace parallel to $Q$. It suffices to show that this affine subspace meets the first orthant in a compact set.
This is clear if $Q$ is a codimension $1$ subspace of a coordinate subspace ${{\mathbb R}}\oplus \cdots \oplus {{\mathbb R}}$, because the subspace would have a strictly positive normal vector, and then $rQ$ would meet the first orthant of ${{\mathbb R}}\oplus \cdots \oplus {{\mathbb R}}$ in a simplex or a point (or not at all). Otherwise we can choose a coordinate subspace ${{\mathbb R}}\oplus \cdots
\oplus {{\mathbb R}}$ containing $Q$ and then enlarge $Q$ to make it codimension $1$, preserving the property that it meets the first orthant only at the origin. The result then follows easily.
\[finiteness\] Let $G$ be a finitely generated generalized Baumslag–Solitar group. If $q(G) \cap {{\mathbb Z}}= 1$ then there are only finitely many reduced graphs of groups $(A,i)_{{{\mathbb Z}}}$ with fundamental group $G$.
If $G$ is elementary then there are only four reduced graphs of groups whose universal covering trees have at most two ends and the result is clear. These are: a single vertex, a loop with indices $\pm 1$ (two cases: equal signs or opposite signs), and an interval with indices $\pm
2$.
If $G$ is non-elementary then any two reduced trees are related by slide moves, by Theorem \[slidethm\]. In particular there are only finitely many possible quotient graphs. Thus we may consider sequences of slide moves in which every edge returns to its original position in the quotient graph. To prove the theorem it then suffices to show that after such a sequence, there are only finitely many possible values for each edge index $i(e)$.
Suppose $X$ is a generalized Baumslag–Solitar tree with group $G$ and $e$ is an edge of $X$ with initial vertex $v$. Consider a sequence of slide moves after which $e$ has initial vertex $gv$ for some $g \in G$. Then we have $G_e \subset (G_v \cap G_{gv})$. This implies that $$\begin{split}
[G_v:G_e] \ &= \ [G_v:(G_v \cap
G_{gv})] \, [(G_v \cap G_{gv}):G_e] \\
&= \ q(g)\, [G_{gv}:G_e],
\end{split}$$ or equivalently $[G_{gv}:G_e] = [G_v:G_e] \, q(g^{-1})$. Therefore, since $i(e) = \pm [G_v:G_e]$ before the slide moves, the new index after the moves is an element of the set $\pm i(e) q(G) \cap
{{\mathbb Z}}$. This set is finite by Lemma \[indexbound\].
\[isomproblem\] There is an algorithm which, given finite graphs of groups $(A,i)_{{{\mathbb Z}}}$ and $(B,j)_{{{\mathbb Z}}}$ such that $(A,i)_{{{\mathbb Z}}}$ has no non-trivial integral moduli, determines whether the associated generalized Baumslag–Solitar groups are isomorphic.
First make $(A,i)_{{{\mathbb Z}}}$ and $(B,j)_{{{\mathbb Z}}}$ reduced. If one or both is elementary then it is a simple matter to check for isomorphism. Among the reduced elementary graphs of groups, a single vertex has group ${{\mathbb Z}}$, a loop with both indices equal to $1$ has group ${{\mathbb Z}}\times {{\mathbb Z}}$, and the remaining two cases yield the Klein bottle group.
Otherwise, by Theorem \[slidethm\], the groups are isomorphic if and only if there is a sequence of slide moves taking $(A,i)_{{{\mathbb Z}}}$ to $(B,j)_{{{\mathbb Z}}}$. Now consider the set of graphs of groups related to $(A,i)_{{{\mathbb Z}}}$ by slide moves. This is the vertex set of a connected graph ${{\mathscr G}}$ whose edges correspond to slide moves. We claim that every vertex of ${{\mathscr G}}$ is a reduced graph of groups. This then implies that ${{\mathscr G}}$ is finite by Theorem \[finiteness\]. To prove the claim one observes, using Proposition \[gbsmoves\], that an edge in a reduced graph of groups cannot be made collapsible during a slide move unless it slides over a strict ascending loop, but there are no such loops because the group has no non-trivial integral moduli.
Now search ${{\mathscr G}}$ by performing all possible sequences of slide moves of length $n$, for increasing $n$ until no new graphs of groups are obtained. Then the two generalized Baumslag–Solitar groups are isomorphic if and only if $(B,j)_{{{\mathbb Z}}}$ has been found by this point.
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|
---
abstract: 'The boolean satisfiability problem is a famous NP-complete problem in computer science. An effective way for this problem is the stochastic local search (SLS). However, in this method, the initialization is assigned in a random manner, which impacts the effectiveness of SLS solvers. To address this problem, we propose [*NLocalSAT*]{}. [*NLocalSAT*]{}combines SLS with a solution prediction model, which boosts SLS by changing initialization assignments with a neural network. We evaluated [*NLocalSAT*]{}on five SLS solvers (CCAnr, Sparrow, CPSparrow, YalSAT, and probSAT) with problems in the random track of SAT Competition 2018. The experimental results show that solvers with [*NLocalSAT*]{}achieve 27% $\sim$ 62% improvement over the original SLS solvers.'
author:
-
- Wenjie Zhang$^1$
- Zeyu Sun$^1$
- Qihao Zhu$^1$
- Ge Li$^1$
- |
\
Shaowei Cai$^2$
- |
Yingfei Xiong$^1$Lu Zhang$^1$\
$^1$Key Laboratory of High Confidence Software Technologies (Peking University), MoE;\
Software Institute, Department of Computer Science and Technology, Peking University;\
$^2$State Key Laboratory of Computer Science, Institute of Software;Chinese Academy of Sciences;\
School of Computer Science and Technology, University of Chinese Academy of Sciences;\
{zhang\_wen\_jie,szy\_,zhuqh,lige,xiongyf,zhanglucs}@pku.edu.cn, caisw@ios.ac.cn
bibliography:
- 'ijcai20.bib'
title: 'NLocalSAT: Boosting Local Search with Solution Prediction'
---
Introduction
============
Boolean satisfiability (also referred to as propositional satisfiability and abbreviated as SAT) is the problem to determine whether there exists a set of assignments for a given boolean formula to make the formula evaluate to true. SAT is widely used in solving combinatorial problems, which are generated from various applications, such as program analysis [@DBLP:conf/popl/HarrisSIG10], program verification [@DBLP:conf/lpar/Leino10], and scheduling [@DBLP:conf/icse/KasiS13]. These applications first reduce the target problem into a SAT formula and then find a solution using a SAT solver. However, the SAT problem has proven to be NP-complete [@DBLP:conf/stoc/Cook71], which means that algorithms for solving SAT problems may need exponential time in the worst case. Therefore, many techniques have been proposed to increase the efficiency of the search process of SAT solvers.
The state-of-the-art SAT solvers can be divided into two categories, CDCL (Conflict Driven Clause Learning) solvers and SLS (Stochastic Local Search) solvers. CDCL solvers are based on the deep backtracking search, which assigns one variable each time and backtracks when a conflict occurs. On the other hand, SLS solvers initialize an assignment for all variables and then find a solution by constantly flipping the assignment of variables to optimize some score.
Over the last few years, artificial neural networks have been widely used in many problems [@DBLP:journals/corr/EdwardsX16; @DBLP:conf/sat/SelsamB19]. A neural network is a machine learning model with a large number of parameters. Neural networks have been used on many data structures, such as sequences [@DBLP:conf/interspeech/MikolovKBCK10], images [@DBLP:journals/corr/SimonyanZ14a], and graphs [@DBLP:journals/corr/EdwardsX16]. The graph convolutional network (GCN) [@DBLP:journals/corr/EdwardsX16] is a neural network model on graph structures, which extracts both structural information and information on nodes in a graph. GCN performs well on many tasks on graphs.
There have been some studies on solving SAT problems with neural networks. Some of them use end-to-end neural networks to solve SAT problems directly as the outputs of the neural networks, while others use neural network predictions to boost CDCL solvers. Selsam et al. proposed an end-to-end neural network model to predict whether a SAT problem is satisfiable [@DBLP:conf/iclr/SelsamLBLMD19] in 2019. Later, Selsam et al. modified NeuroSAT to NeuroCore [@DBLP:conf/sat/SelsamB19]. NeuroCore guides CDCL solvers with unsat-core predictions which are computed every certain interval in the neural network on GPUs. CDCL solvers with NeuroCore solve 6%-11% more problems than the original.
In this paper, we propose [*NLocalSAT*]{}, which is the first method that uses a neural network to boost SLS solvers and the first off-line method to boost SAT solvers with neural networks. Different from NeuroCore which induces large overhead to CDCL by an on-line prediction[^1], [*NLocalSAT*]{}uses the prediction in an off-line way. In this method, the neural network is computed only once for each SAT problem.
In our proposed method, we first train a neural network to predict the solution space of a SAT problem. Then, we combine SLS solvers with the neural network by modifying the solvers’ initialization assignments under the guidance of the output of the neural network. Such combination induces limited overhead, and it can easily be applied to SLS solvers. Furthermore, we evaluated SLS solvers, and [*NLocalSAT*]{}solves 27% $\sim$ 62% more problems than the original SLS solvers. Such experimental results show the effectiveness of [*NLocalSAT*]{}.
#### Contributions
\(1) We train a neural network to predict the solution of a SAT problem. (2) We propose a method to boost SLS solvers by modifying its initialization of assignments with the guidance of predictions of the neural network. (3) To the best of our knowledge, we are the first to combine the SLS with a neural network model [[and we are the first to propose an off-line method to boost SAT solvers with a neural network]{}]{}.
Approach
========
![The overview of our model, [*NLocalSAT*]{}.[]{data-label="fig:overview"}](imgs/Overview_SAT.pdf){width="\linewidth"}
Figure \[fig:overview\] shows an overview of our model, [*NLocalSAT*]{}. Our model combines a neural network and an SLS solver. [[Given a satisfiable input formula, the neural network is to output a candidate solution and the solver is to find a final solution with the guidance of the neural network.]{}]{} For an input formula, we first transfer it into a formula graph, which is further fed to a graph-based neural network for extracting features. [[Then, different from previous works, where the neural network is to predict the satisfiability of the whole problem in NeuroSAT and to predict whether the variable is in the unsat-core in NeuroCore, the neural network in [*NLocalSAT*]{}outputs a candidate solution for the formula by a multi-layer perceptron after the neural network.]{}]{} We use this candidate solution to initialize the assignments of an SLS SAT solver to guide the search process.
Formula Graph
-------------
To take the structural information of an input formula into consideration, we first transfer it into a formula graph. A general boolean formula can be any expressions consisting of variables, conjunctions, disjunctions, negations, and constants. All boolean formulas can be reduced into an equisatisfiable conjunctive normal form (CNF) with linear length in linear time [@tseitin1968complexity]. In a CNF, a SAT problem is a conjunction of clauses $C_1 \wedge C_2 \wedge \cdots \wedge C_n.$ Each clause is a disjunction of literals (i.e., variables and negated variables) $C_i = L_{i1} \vee L_{i2} \vee \cdots \vee L_{in},$ where $L_{ij} = x_k$ or $L_{ij} = \neg x_k$. In this paper, we assume that all SAT problems are in CNFs. [[A SAT problem $S$ in the CNF can be seen as a bipartite graph $G = (C, L, E)$, where $C$ (clause set of $S$) and $L$ (literal set of $S$) are the node sets of $G$ and $E$ is the edge set of $G$. $(c, l)$ is in $E$ if and only if the literal $l$ is in the clause $c$.]{}]{} $A$ is the adjacent matrix of the bipartile graph $G$. The element $A_{ij}$ of the adjacent matrix equals to one when there is an edge between node $i$ and node $j$, otherwise 0. For example, $\left(x_1 \lor x_2\right) \land \neg \left(x_1 \land x_3\right)$ is a boolean formula. This boolean formula can be converted into $\left(x_1 \lor x_2\right) \land \left(\neg x_1 \lor \neg x_3\right)$ in the conjunctive normal form. The bipartite graph for this problem is shown in Figure \[fig:bipartite\]. The adjacency matrix for this graph is $$A = \left(\begin{matrix}
1 & 0 & 1 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 1
\end{matrix}\right).$$
(c1) at (-1,1) [$c_1$]{}; (c2) at (1,1) [$c_2$]{}; (x1) at (-2.5,0) [$x_1$]{}; (nx1) at (-1.5,0) [$\neg x_1$]{}; (x2) at (-0.5,0) [$x_2$]{}; (nx2) at (0.5,0) [$\neg x_2$]{}; (x3) at (1.5,0) [$x_3$]{}; (nx3) at (2.5,0) [$\neg x_3$]{}; (c1) edge (x1) (c1) edge (x2) (c2) edge (nx1) (c2) edge (nx3);
Graph-Based Neural Network
--------------------------
The graph-based neural network aims to predict the candidate solution for a SAT problem. The network consists of a gated graph convolutional network to extract structural information about the graph and a two-layer perceptron to predict the solution.
### Gated Graph Convolutional Network
[[Inspired by NeuroSAT [@DBLP:conf/iclr/SelsamLBLMD19], we use a similar gated graph convolutional network (GGCN) to extract features of variables. The gated graph convolutional network (GGCN) takes the adjacency matrix as the input and outputs the features of each variable extracted from the graph.]{}]{}
In a SAT problem, the satisfiability is not influenced by the names of clauses and literals (e.g., the satisfiability of two formulas $\left(x_1 \lor x_2\right)$, $\left(x_3 \lor x_4\right)$). To use this property, for an input formula graph $G$, we initialize each clause $c_i \in G$ as a vector $\bm{c}^\text{(init)} \in \mathbb{R}^d$, each literal $l_i \in G$ as another vector $\bm{l}^\text{(init)} \in \mathbb{R}^d$, where $d$ is the embedding size and $d$ is set to 64 in this paper. These vectors are further fed to GGCN to extract structural information in the graph.
Each iteration of GGCN is an update for the vectors of these nodes where each node updates its vector by taking its neighbors’ information (vectors). Formally, at the $t$-th iteration, the detailed computations for clause $c$ and literal $l$ are represented by $$\bm c_{t} = \mathrm{LSTMCell}(\bm c_{t-1}, \sum_{l' \in G} \tilde{A}_{cl'} \bm l_{t-1}')$$ $$\bm l_{t} = \mathrm{LSTMCell}(\bm l_{t-1}, \sum_{c' \in G} \tilde{A}_{c'l} \bm c_t' + \neg \bm l_{t-1})$$ Here, $\tilde{A}$ is a normalized adjacency matrix for the graph (the detailed computation is presented in Equation \[eq:norm\]). $\bm l'_t$ and $\bm c'_t$ denote the vector of the literal $l'$ and clause $c'$ at the $t$-th iteration. $\neg \bm l_{t}$ is the vector of negated literal of $l$ at the $t$-th iteration. $\bm c_0$ = $\bm c'_0$ = $\bm{c}^\text{(init)}$, $\bm l_0$ = $\bm l_0'$ = $\bm{l}^\text{(init)}$. The $\mathrm{LSTMCell}$ is a long short-term memory (LSTM) unit with layer normalization.
We use normalization in convolution to improve scalability. Because in a SAT problem, there can be any number of nodes adjacent to one, so if we simply add all vectors together, the summation is unbounded. We use symmetrical normalization $$\tilde{A} = S_1^{-1/2} A S_2^{-1/2},
\label{eq:norm}$$ where $S_1$ and $S_2$ are the diagonal matrices with summation of $A$ in columns and rows. In the implementation, a small number ($10^{-6}$) is added to the diagonal elements of $S_1$ and $S_2$ to avoid division by zero.
We apply the GGCN layer of 16 iterations on the initial value and get a vector containing structural information about each literal. Then, the two vectors for a literal and its negation are concatenated for each variable.
### Two-layer Perceptron
After applying GGCN, vectors of nodes contain structural information of literals. A two-layer perceptron $\mathrm{MLP}$ with hidden 512 size is applied on the vector for each variable to extract classification information from the structural information. Through a softmax function, we get the probability for the variable to be true. $$\begin{aligned}
&P(v=\mathrm{FALSE}), P(v=\mathrm{TRUE})\\
= &\mathrm{softmax}\left\{\mathbf{W}_2~\mathrm{ReLU}\left[\mathbf{W}_1(\mathrm{\mathbf l_{v}: \mathbf l_{\neg v}})+\mathbf{b}_1\right]+\mathbf{b}_2\right\},
\end{aligned}$$ [[where $\mathbf W_1, \mathbf W_2, \mathbf b_1, \mathbf b_2$ are weights and biases for the two perceptrons and the colon indicates the connection of two vectors.]{}]{}
### Loss Function
Our model is trained by minimizing the cross-entropy loss against the ground truth. For the predicted variables $<v_1,v_2\cdots,v_n>$, where $n$ denotes the number of variables, the cross-entropy loss is computed as $$\begin{aligned}Loss = - \sum_{i=1}^{n}&[g(v_i)\log(P(v=\mathrm{TRUE}))\\
+ &\left(1-g(v_i)\right)\log(P(v=\mathrm{FALSE}))],\end{aligned}$$ where $g(v_i)$ denotes the ground truth of $v_i$.
Training Data
-------------
The goal of our network is to predict the solution to the SAT problem, so we should generate training data for SAT problems with solution labeling. Due to the scarcity of SAT Competition data, using additionally generated small SAT instances could help provide thorough training. We generate two training datasets and train our model in order. Our model is first pretrained with a large number of generated small SAT instances. We generate tens of millions of small SAT instances with solution labeling because these small problems are easy to solve. Such large amounts of data can make the training process more effective and avoid overfitting. These small instances can help our network better learn structural information. Then, our model is fine-tuned on a dataset generated from SAT Competitions. By finely tuning on the dataset from SAT Competition instances, our model can learn specific information in the field and learn to predict the solution on large instances.
### Small Generated Instances
We generate small instances containing 10 to 20 variables by a random generator. The number of clauses is 2 to 6 times the number of variables. Each clause contains 3 variables with the probability of 0.8 and 2 variables with the probability of 0.2. Each clause is generated randomly by randomly selecting variables or negated variables from the problem.
After the generation of problems, we solve these problems with a complete backtracking solver. We drop those unsatisfied problems because our model only learns to predict the solution of a satisfiable SAT problem. However, there can be more than one solution for a specific SAT instance, which can confuse the neural network. We depend on the solution space to tackle this problem. The solver maintains a counter for each variable and each negated variable. In the solving process, the solver explores all solutions instead of stopping after one specific solution. When the solver finds a solution, the counters of literals in the solution will increase by 1. When the whole search process finishes, the solver reports for each variable whether the variable itself or its negated form appears more in the solutions. If the variable itself appears more, we label this variable as 1. If the negated variable appears more, we label this variable as 0. We use this value as training labeling for the variable.
After data generation, we pretrain [*NLocalSAT*]{}on these small instances. In the implementation, small data instances have higher overhead during model training, so we combine several instances into one larger batch. Since our model is independent of the order of variables, we can simply combine two or more instances via concatenating edges and labels of these instances.
### SAT Competition Instances
To train our model on larger instances, we generate training data from SAT problems in random tracks of SAT Competitions. We use a SAT solver to get one solution for each problem. Problems that are not solved by the solver will be removed. We then use the solution as labeling in the training data.
[[Training on GPU requires more memory than evaluation. So, if the problem is too large to fit in the memory of our GPU during training, we will cut these problems into smaller ones.]{}]{} Let us denote the largest number of variables that can fit into the memory of our GPU as $N_L$. For every large instance, we first get a solution $S_0$ with a SAT solver. Then, we sample $N_L$ variables $X_0$ from all variables. For one clause $c$ in the original SAT problem, if $c$ contains no literals from $X_0$, the clause is removed. If $c$ is not satisfied on $X_0$ after removing literals that are not from $X_0$, the clause is also removed. Clauses with only one literal are also removed to prevent the problem from being too easy. Otherwise, the clause $c$ remains in the problem. If the sampling generates a problem with too few clauses, the problem is removed because this will lead to too many solutions.
Combination with Local Search
-----------------------------
Stochastic local search algorithms can be considered as an optimization process. The solver flips the assignments of variables to maximize some total score. For example, we can use the number of clauses that evaluate to true as the score. When the score reaches the number of clauses, the problem is solved. Algorithm \[alg:sls\] shows a general algorithm in a local search solver.
**Data**: SAT problem $P$
$S \leftarrow$ initialize assignment randomly;\
Return $S$; $l \leftarrow$ Select a variable by some heuristics;\
Flip($S$, $l$);
In SLS solvers, it is not counter-intuitive that the initial assignment has a great impact on whether it can quickly find a solution, because there can be many local minima in the problem and badly initialized assignments near a local optimum can cause the SLS solvers to get stuck. Intuitively, the closer to the solution of the problem, the less likely it is to encounter local minima. In order to avoid falling into the local optimum, these solvers restart the searching process by reassigning new random values to variables after a period of time without a score increase. However, most of the existing SLS solvers initialize assignments in a random manner. Random generation of initial values can explore more space for the SAT problem. However, if the distance between initial values and solution values is too large, the solver is more likely to fall into a local optimum.
We propose a new initialization method using the output of our neural network. Before starting the solver, we run our neural network to predict a solution. We replace the initialization function with our neural initialization, where, instead of randomly generating 0 or 1, the function assigns the predicted values to a variable with a probability of $p_0$ and assigns the negation of the predicted value with a probability of $1-p_0$. This neural-initialization function keeps the assignment with a probability of $p_0$ to explore near the candidate solution and explores new solution space with a probability of $1-p_0$ in case that the neural networks’ prediction is wrong. The initialization process is shown in Algorithm \[algo:init\]. Algorithm \[alg:nlocalsat\] shows the architecture of our modified SLS solvers. Note that our neural network model is executed only once for one SAT problem. Though the cost of computing a neural network is high, the cost of calling a neural network only once is acceptable, which consumes 0.1 seconds to tens of seconds depending on the size of the problem.
**Data**: Probability $p_0$, Assignment of variables $\mathrm{assignment}$, Neural network predictions $N$
$\mathrm{assignment}[i] \leftarrow N[i]$ $\mathrm{assignment}[i] \leftarrow \neg N[i]$
**Data**: SAT problem $P$
Return $S$; $l \leftarrow$ Select a variable by some heuristics; Flip($S$, $l$);
Experiments
===========
Datasets
--------
Our model was trained on a dataset with generated problems with small SAT instances (denoted as $Dataset_{small}$) and a dataset with problems in random tracks of SAT Competitions in 2012, 2013, 2014, 2016, 2017 (denoted as $Dataset_{comp}$) (The competition of SAT in 2015 was called SAT Race 2015. There was no random track in SAT Race 2015). Our model was evaluated on problems in the random track of SAT Competition in 2018 (denoted as $Dataset_{eval}$) with 255 SAT instances in total. We found that there are several duplicate problems in 2018 and previous years, so we removed them from the training and validation datasets [[to ensure problems in the $Dataset_{eval}$ are generated with different random seeds with those in $Dataset_{comp}$. So, it’s almost impossible to have isomorphic problems between these two datasets. However, there will be some similar substructures between the training set and the test set, so that neural networks can predict by learning these substructures.]{}]{}
[[The $Dataset_{comp}$ and the $Dataset_{eval}$ both contain two categories of problems, i.e., uniformly generated random SAT problems (denoted as *Uniform*) [@belov2014generating] and hard SAT problems generated with a predefined solution (denoted as *Predefined*) [@DBLP:conf/socs/BalyoC18].]{}]{}
Pretraining
-----------
In $Dataset_{small}$, we generated about $2.5 \times 10^7$ small problems and combined them into about $4 \times 10^5$ batches with about ten thousand variables each as our pretraining dataset. We generated 200 batches in the same approach with different random seeds as validation data during pretraining.
We trained our model to converge using the ADAM [@DBLP:journals/corr/KingmaB14] optimizer with its default parameters by minimizing the cross-entropy of predicted values and label values on $Dataset_{small}$. After pretraining, the precision on the validation dataset is 98%, i.e., the network can predict correctly 98% of variables whose solution space is larger on these small SAT problems.
Training
--------
We used $Dataset_{comp}$ as the training dataset and the validation dataset. We loaded the pretrained model and continued to train with the same optimizer and loss function. After training, the precision on the validation dataset is 95%.
Evaluation
----------
We tested our proposed method on five recent SLS solvers, i.e., CCAnr [@DBLP:conf/sat/CaiLS15], Sparrow [@DBLP:conf/sat/BalintF10], CPSparrow, YalSAT[@biere2016splatz], probSAT[@balint2018probsat]. These solvers have performed very well among SLS solvers on random tracks of SAT Competitions in recent years. CCAnr is an SLS solver proposed in 2015 to capture structural information on SAT problems. CCAnr is a variant of CCASat [@DBLP:journals/ai/CaiS13]. CCAnr performs better on all tracks of SAT Competitions than CCASat. Sparrow is a clause weighting SLS solver. CPSparrow is a combination of Sparrow and a preprocessor Coprocessor [@DBLP:conf/sat/BalintM13]. CPSparrow is the best pure SLS solver in the random track of SAT Competition 2018. [[YalSAT is the champion of the random track of SAT Competition 2017.]{}]{}
[[Due to the strong randomness of SLS solvers, the experiments for SLS solvers were performed three times with three different random seeds and then aggregated the results.]{}]{}
We evaluated these original solvers and those modified with [*NLocalSAT*]{}on $Dataset_{eval}$. We also evaluated three other solvers MapleLCMDistChronoBT [@ryvchin2018maple_lcm_dist_chronobt], gluHack, and Sparrow2Riss [@balint2018sparrowtoriss] under the same set. MapleLCMDistChronoBT and Sparrow2Riss are the champions of SAT Competition 2018 in the main track and the random track. gluHack is the best CDCL solver of SAT Competition 2018 in the random track. MapleLCMDistChronoBT is a CDCL solver with recently proposed techniques to improve performance such as chronological backtracking [@DBLP:conf/sat/NadelR18], learned clause minimization [@DBLP:conf/ijcai/LuoLXML17], and so on. Sparrow2Riss is a combination of Coprocessor, Sparrow, and a CDCL solver Riss.
We set up a timeout limit to 1000 seconds. Solvers failed to find a solution within the time limit will be killed immediately. Our experiments were performed on a work station with an Intel Xeon E5-2620 CPU and a TITAN RTX GPU. During our experiment, the time for initialization of the GPU environment was ignored but the time of the GPU computation was included in the total time.
Results
=======
Number of Problems Solved
-------------------------
Table \[tab:num-problems\] shows the number of problems solved in 1000 seconds time limit. [[Each row represents a tested solver. The experiments of SLS solvers are performed three times to reduce the randomness of results. Each number in the rows of SLS solvers is the average and the standard deviation of results in the three experiments.]{}]{} Each column in the table represents a category of problems in the dataset. The number in parentheses indicates the total number of problems in the category. [[$Dataset_{eval}$ contains unsatisfiable problems [@belov2014generating]. However, no solvers reported unsatisfiability on any problems within the time limit. So, the solved problems in Table \[tab:num-problems\] are all satisfiable ones.]{}]{}
The experimental result shows that solvers with [*NLocalSAT*]{}solve more problems than the original ones. CCAnr, Sparrow, CPSparrow, YalSAT, and probSAT with [*NLocalSAT*]{}solve respectively 41%, 30%, and 27%, 62%, and 62% more problems than the original solvers. This improvement has been shown in *Predefined* problems and in *Uniform* problems on Sparrow and CPSparrow. Sparrow with [*NLocalSAT*]{}and CPSparrow with [*NLocalSAT*]{}solve more problems than all other solvers including the champions on SAT Competition 2018. Sparrow2Riss is a combination of a preprocessor, an SLS solver, and a CDCL solver, thus showing good performance, but the SLS solvers with [*NLocalSAT*]{}still outperforms Sparrow2Riss. CDCL solvers perform well on *Predefined* problems and [*NLocalSAT*]{}can help to improve performance particularly on this category, from which we can conclude that [*NLocalSAT*]{}can improve particularly on those problems, on which CDCL solvers perform well. [*NLocalSAT*]{}may capture some structural information to boost SLS solvers on these problems.
Time of Solving Problems
------------------------
Figure \[img:cactus\] shows the relationship between the number of problems solved and time consumption comparing solvers with [*NLocalSAT*]{}and without [*NLocalSAT*]{}. In this figure, we can see that some simple problems which are solved within 1 second with the original solver need more solving time with [*NLocalSAT*]{}than the original solver. This is because the neural network computation takes a certain amount of time before the solver starts and this time is especially noticeable for simple problems. But on hard problems, our modifications can improve the solver significantly.
Table \[tab:speedup-or-slowdown\] compares the average running time with the timeout penalty (PAR-2) between different solvers. The PAR-2 running time of problems that are not solved by the solver in the time limit is twice of the time limit (i.e., 2000 seconds in our experiments). The PAR-2 score was used in previous SAT Competitions. Values in this table show that solvers with [*NLocalSAT*]{}are slightly slower on easy problems but much faster on hard problems, particularly, on *Predefined*. Solvers with [*NLocalSAT*]{}can find a solution faster than those without [*NLocalSAT*]{}on most problems, i.e., solvers with [*NLocalSAT*]{}are more effective than the original ones.
Related Work
============
Recently, several studies have investigated how to make use of neural networks in solving NP-complete constraint problems. There are two categories of methods to solve NP-complete problems using neural networks. The first category of methods is end-to-end approaches using end-to-end neural networks to solve problems, i.e., given the problem as an input, the neural network outputs the solution directly. In these methods, the neural network can learn to solve the problem itself [@DBLP:conf/iclr/AmizadehMW19; @DBLP:conf/cpaior/Galassi0MM18; @DBLP:conf/iclr/SelsamLBLMD19; @DBLP:conf/cp/0003KK18; @DBLP:conf/aaai/PratesALLV19]. However, due to the accuracy and structural limitations of neural networks, the end-to-end methods can only solve small problems. The other category of methods is heuristic methods that treat neural networks as heuristics [@DBLP:conf/nips/BalunovicBV18; @DBLP:conf/nips/LiCK18; @DBLP:conf/sat/SelsamB19]. Among these methods, traditional solvers work with neural networks together. Neural networks generate some predictions, and the solvers will use these predictions as heuristics to solve the problem. Constraints in the problems can be maintained in the solver, so these methods can solve large-scale problems. Our proposed method (i.e., [*NLocalSAT*]{}) belongs to the second category, heuristic methods, so [*NLocalSAT*]{}can be used for larger instances than those end-to-end methods. Balunovic et al. [@DBLP:conf/nips/BalunovicBV18] proposed a method to learn a strategy for Z3, which greatly improves the efficiency of Z3. Li et al. [@DBLP:conf/nips/LiCK18] proposed a model on solving maximal independent set problems with a tree search algorithm. NeuroCore [@DBLP:conf/sat/SelsamB19] is a method to improve CDCL solvers with predictions of unsat-cores. [[None of these methods is used to boost stochastic local search solvers with solution predictions and none of these is an off-line method to boost SAT solvers. The training data used in [*NLocalSAT*]{}are solutions of problems or solution space distribution of problems, which is also different from previous works, where NeuroSAT uses the satisfiability of problems and NeuroCore uses unsat-core predictions.]{}]{}
Conclusion and Future Work
==========================
This paper explores a novel perspective of combining SLS with a solution prediction model. We propose [*NLocalSAT*]{}to boost SLS solvers. Experimental results show that [*NLocalSAT*]{}significantly increases the number of problems solved and decreases the solving time for hard problems. In particular, Sparrow and CPSparrow with our proposed [*NLocalSAT*]{}perform better than state-of-the-art CDCL, SLS, and hybrid solvers on the random track of SAT Competition 2018.
[[[*NLocalSAT*]{}can boost SLS SAT solvers effectively. With this learning-based method, we may build a domain-specific SAT solver without expertise in the domain by training [*NLocalSAT*]{}with SAT instances from the domain. It is also interesting to further explore building domain-specific solvers with [*NLocalSAT*]{}.]{}]{}
[^1]: The on-line prediction means to predict every certain interval in CDCL.
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---
abstract: 'The X2.2-class solar flare of February 15, 2011, produced a powerful ‘sunquake’ event, representing a helioseismic response to the flare impact in the solar photosphere, which was observed with the HMI instrument on the Solar Dynamics Observatory (SDO). The impulsively excited acoustic waves formed a compact wavepacket traveling through the solar interior and appearing on the surface as expanding wave ripples. The initial flare impacts were observed in the form of compact and rapid variations of the Doppler velocity, line-of-sight magnetic field and continuum intensity. These variations formed a typical two-ribbon flare structure, and are believed to be associated with thermal and hydrodynamic effects of high-energy particles heating the lower atmosphere. The analysis of the SDO/HMI and X-ray data from the Ramaty High Energy Solar Spectroscopic Imager (RHESSI) shows that the helioseismic waves were initiated by the photospheric impact in the early impulsive phase, observed prior to the hard X-ray ($50-100$ keV) impulse, and were probably associated with atmospheric heating by relatively low-energy electrons ($\sim 6-50$ keV) and heat flux transport. The impact caused a short motion in the sunspot penumbra prior to the appearance of the helioseismic wave. It is found that the helioseismic wave front traveling through a sunspot had a lower amplitude and was significantly delayed relative to the front traveling outside the spot. These observations open new perspectives for studying the flare photospheric impacts and for using the flare-excited waves for sunspot seismology.'
author:
- 'A.G. Kosovichev'
title: 'Helioseismic response to X2.2 solar flare of February 15, 2011'
---
Introduction
============
‘Sunquakes’, the helioseismic response to solar flares, are caused by strong localized hydrodynamic impacts in the photosphere during the flare impulsive phase. The helioseismic waves are observed directly as expanding circular-shaped ripples on the solar surface, which can be detected in Dopplergram movies and as characteristic ridges in time-distance diagrams, [@Kosovichev1998; @Kosovichev2006a; @Kosovichev2006b], or indirectly by calculating the distribution of an integrated acoustic emission [@Donea1999; @Donea2005].
Solar flares are sources of high-temperature plasma, strong hydrodynamic motions and heat flux in the solar atmosphere. Perhaps, in all flares such perturbations generate acoustic waves traveling through the interior. However, only in some flares is the impact sufficiently localized and strong to produce the seismic waves with the amplitude above the convection noise level. The sunquake events with expanding ripples are relatively rare, and [ have been]{} observed only in some high-M and X-class flares. The last previous observation of the seismic waves was reported for X1.2 flare of January 15, 2005.
It has been found in the initial July 9, 1996, flare observations [@Kosovichev1998] that the hydrodynamic impact in the photosphere (‘sunquake source’) followed the hard X-ray flux impulse, and hence, the impact of high-energy electrons. [ They]{} suggested that the mechanism of sunquakes can be explained by a hydrodynamic ‘thick-target’ model of solar flares [@Kostiuk1975]. Several other mechanisms [ of the helioseismic response]{}, including impact by high-energy protons and back-warming radiation heating [e.g. @Donea2005; @Zharkova2007], and also due to magnetic field variations [@Hudson2008]. However, the mechanism, which converts a part of the flare energy and momentum into the helioseismic acoustic waves, is currently unknown. It is also unknown why only some flares generate [ large-amplitude waves observed as ripples or enhanced acoustic emission]{} [see a review of @Hudson2011 for a recent discussion and references].
Most of the previous observations of sunquakes were obtained with the Michelson Doppler Imager instrument on SOHO. However, the full-disk observations with the full 2 arcsec/pixel resolution suitable for flare studies were obtained uninterruptedly only for 2 months a year. Thus, many flares were not observed, and the statistics of sunquakes and their relation to the flare properties were not established. Except short eclipse periods in March and September, the Solar Dynamics Observatory launched in February 2010 provides uninterrupted observations of the Sun. The Helioseismic and Magnetic Imager (HMI) on SDO observes variations of intensity, magnetic field and plasma velocity (Dopplergrams) on the surface of Sun almost uninterruptedly with high spatial resolution (0.5 arcsec/pixel) and high cadence (45 sec) [@Schou2010]. The flare of February 15, 2011, was the first X-class flare of the new solar cycle, 24, and the first observed by HMI. This paper presents results of the initial analysis, which revealed the sunquake event. This event shows some curious properties, which make it different from the previously observed ‘sunquakes’.
Results
=======
The X2.2 flare of February 15, 2011, occurred in the central sunspot of active region NOAA 1158, which had a $\delta$-type magnetic configuration (Fig. \[fig1\]). According to the GOES-15 soft X-ray measurements, the flare started at 01:44 UT, reached maximum at 01:56 UT and ended at 02:06 UT. The flare signals are clearly detected in all HMI observables, and show that the flare had a typical two-ribbon structure with the ribbons located on both sides of the magnetic neutral line. This is well seen in the magnetograms (Fig. \[fig1\]a). [ We consider two places of strong localized photospheric impacts as ‘Impact 1’ and ‘Impact 2’ for further analysis. The sunquake originated from ‘Impact 1’.]{}
The sunquake event was initially revealed in the running difference movie of the raw Doppler velocity data [@Kosovichev2011a]. However, the wave structure is better seen after applying to the data a Gaussian frequency filter with a central frequency of 6 mHz and a characteristic width of 2 mHz. This filter enhances the high-frequency sunquake signal relative to the lower-frequency background solar granulation noise. In addition, the images are remapped onto the heliographic Carrington coordinates [ using the Postel’s azimuthal equidistant projection]{} and tracked with the differential rotation rate. Figure \[fig2\] shows two frames of the frequency-filtered Doppler-velocity movie [ (available as supplementary online-only material)]{}. The sunquake wave appears about 20 minutes after the initial flare impact of the photosphere. The wave front has a circular shape, but it is not isotropic. The wavefront traveling outside the magnetic region in the North-East direction (‘Wave 1’) has the highest amplitude, and is most clearly visible. In the opposite direction the wave travels through a sunspot (‘Wave 2’), and its amplitude is suppressed. Also, the wave front traveling through the sunspot is visibly delayed relative to ‘Wave 1’ traveling outside. Figure \[fig3\] shows positions of the two wave fronts at 02:08 UT in the corresponding magnetogram and white-light images.
Figure \[fig4\] shows the time-distance diagrams obtained by remapping the frequency-filtered Dopplergrams onto the polar coordinates with the center at ‘Impact 1’, and averaging over the range of angles corresponding to the two parts of the wave fronts in Fig. \[fig2\]. In these diagrams, the helioseismic acoustic waves form characteristic ridges, the slope of which corresponds to the local group-speed of the wave packets traveling between two surface points through the interior. The speed increases with the distance because for larger distances the waves travel through the deeper interior where the sound-speed is higher [e.g. @Kosovichev2011b]. For comparison, the theoretical travel times calculated in the ray approximation are shown by dashed curves. The starting points of these curves are chosen to approximately match the position of the ridges. Evidently, the ridge of ‘Wave 2’ is much weaker and shorter than the ridge of ‘Wave 1’. In these diagrams, the wave source (‘Impact 1’) produces strong variations at zero distance at about 01:50 UT. Also, the ‘Wave 2’ front is delayed by $\sim 100$ sec with respect to ‘Wave 1’.
The time-distance diagrams show two interesting features. During approximately the first three minutes the wave source is moving in the direction of ‘Wave 1’ (Fig. \[fig4\]) with a speed of about 15-17 km/s, which is higher than the local sound speed but may correspond to the magneto-acoustic speed of the sunspot penumbra in the vicinity of the source. [ The source motions, which can be supersonic,]{} have been observed for other ‘sunquake’ events [@Kosovichev2006b; @Kosovichev2006c; @Kosovichev2007]. Similar to this case, the source motions are at least partly responsible for the anisotropy of the wave amplitude. In the previous cases the source motion was associated with apparent motions of the point-like photospheric impacts in the flare ribbons. In this case, the source motion may be associated with MHD waves excited by the flare momentum impact in the almost horizontal field on the penumbra. However, this process requires a special separate investigation.
The sunquake source is associated with one of the impacts located along the flare ribbons. The flare ribbons consist of individual patches representing impacts of flare impulses. In these data it is easy to find that the location of the sunquake source was in the penumbra area near the edge of the active region. This area is identified as ‘Impact 1’ in Figures \[fig1\]–\[fig3\]. It is characterized by strong and rapid variations of the Doppler velocity and magnetic field, and also by an impulsive increase of the continuum intensity (Fig. \[fig5\]a). There were strong photospheric impacts in several other locations. However, these impacts did not provide clearly visible seismic waves. The reason for this is not clear. For comparison, in Figure \[fig5\]b we show the variations in one of the strongest compact impacts identified as ‘Impact 2’. This impact was located in a region of strong magnetic field in the sunspot outer umbra near the magnetic neutral line. During the impact, the HMI data show a strong increase of the Doppler velocity, indicating downflows, a sharp impulsive decrease of the magnetic field strength, which relaxed to a value lower the the pre-flare strength, and an increase in the continuum intensity brightness. All the variations in ‘Impact 2’ are stronger than in ‘Impact 1’, but this impact did not generate strong sunquake ripples. It seems that the main difference between these two places of the flare impact onto the photosphere of the Sun is that ‘Impact 1’ was located in a region of relatively weak ($\sim 400~$G) magnetic field contrary to ‘Impact 2’, which was in strong field ($\sim 2000~$G). In addition, ‘Impact 1’ was more variable and moving. It started near the inner boundary of the penumbra (bright point at the ‘Impact 1’ arrow in Fig. \[fig3\]a) and then moved into the penumbra, generating a localized motion in this part of the penumbra. The dynamic nature of the flare impact seems to be important for understanding the mechanism of sunquakes. The strong magnetic field in ‘Impact 2’ probably restricted wave motions, and, perhaps, this may explain the absence of helioseismic response from this impact.
The origin of photospheric impacts during the flare impulsive phase is yet to be understood. In this case, it is particularly puzzling that the initial ‘Impact 1’ occurred in the early impulsive phase, prior to the hard X-ray impulse in the energy range of 50-100 keV, and just at the beginning of the X-ray 25-50 keV impulse (Fig. \[fig5\]a). The traditional ‘thick-target’ mechanism of the energy transport in solar flares [e.g. @Hudson2011] assumes that most of the energy is released in the form of high-energy electrons, which heat the solar chromosphere generating a localized high-pressure zone. This zone explodes, and causes ‘chromospheric evaporation’ into the corona and the hydrodynamic impact in the photosphere, which leads to ‘sunquake’. However, in this case the photospheric impact apparently happened before the main particle acceleration phase. This requires a new mechanism of the energy and momentum transport into the low atmosphere during the early ‘pre-heating’ flare phase. Certainly, further investigations of the sunquake events, their energetics and dynamics, will provide new insight in the mechanisms of the flare energy release and transport.
Discussion
==========
The first observations of the sunquake event from SDO/HMI revealed very interesting properties of the flare impact onto the solar photosphere. The HMI data with the significantly higher resolution than the previous SOHO/MDI observations of sunquakes provide a new insight into the dynamics of the flare impact and the sunquake source. The preliminary analysis indicates that seismic flare waves are generated by the impact in the region of a relatively weak magnetic field of the sunspot penumbra. A significantly stronger impact in a region of high magnetic field strength did not generate helioseismic waves of a comparable magnitude.
A characteristic feature of this sunquake is anisotropy of the wave front: the observed wave amplitude is much stronger in one direction than in the others. This was observed also in previous events. In particular, the seismic waves excited during the October 28, 2003, flare had the greatest amplitude in the direction of the expanding flare ribbons. The wave anisotropy was attributed to the moving source of the hydrodynamic impact, which is located in the flare ribbons [@Kosovichev2006b; @Kosovichev2006c]. The motion of flare ribbons is often interpreted as a result of the magnetic reconnection processes in the corona. When the reconnection region moves up it involves higher magnetic loops, the footpoints of which are further apart. This may explain the expanding flare ribbons (as places of the photospheric flare impacts) and the association of sunquakes with the ribbon sources. In this event, the sunquake had a similar dynamical property: it started at an inner boundary of the sunspot penumbra and then quickly moved in the penumbra region. This was accompanied by a motion of this part of the penumbra. This is certainly an interesting phenomenon, which requires further investigation. Of course, there might be other reasons for the anisotropy of the wave front, such as inhomogeneities in temperature, magnetic field and plasma flows. However, the source motion seems to be quite important for generating sunquakes. In addition, the wave front traveling through the sunspot umbra is significantly delayed relative to the wave front traveling outside the sunspot. This delay may be related to the source motion and also to a lower wave speed in the sunspot umbra. Theoretical MHD modeling of the dynamic impact source and the wave propagation in sunspot models is necessary for the understanding of this phenomenon.
The comparison of the SDO/HMI observations with the X-ray observations from RHESSI shows that the photospheric impact, which led to the excitation of the helioseismic waves, occurred at the beginning of the flare impulsive phase, before the hard X-ray impulse in the energy range of 50-100 keV and before main particle acceleration phase. [ Perhaps,]{} the energy transport into the lower atmosphere may be provided by the saturated heat flux as recently suggested [ for chromospheric evaporation]{} by @Battaglia2009. Theoretical models of the heat flux-saturated (or flux-limited) energy transport in the solar atmosphere were previously studied by several authors [e.g. @Smith1986; @Karpen1987]. @Kosovichev1988 showed that this transport has wave properties with a sharp shock-like heat front. Alternatively, the difference in timing between the photospheric impact and the hard X-ray flux may be related to changes in anisotropy of high-energy electrons during the flare impulsive phase [V. Petrosian, private communication; e.g. @Leach1985]. It will be important to further investigate the role of high-energy particles, thermal and MHD effects in the initial phase of solar flares.
[9]{} natexlab\#1[\#1]{}
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![a) Images of AR 1158 during the impulsive phase of the X2.2 flare, at 2011.02.15 01:54:39 TAI: a) line-of-sight magnetic field $B$, c) continuum intensity $I_c$, and e) Doppler velocity $V$. The right panels show the differences between these images and the corresponding images taken 45 sec earlier. The range of the magnetogram color map is $\pm 1$ kG; the range of the Dopplergram is $\pm 1$ km/s. Arrows show positions of two analyzed sources of transient flare variations located along the same flare ribbon. Traces of the second ribbon can be seen in panel b) to the right and below of ‘Impact 2’. A powerful sunquake originated from ‘Impact 1’. ‘Impact 2’ is a place of a strong impulsive impact, but it did not generate a significant sunquake.[]{data-label="fig1"}](f1.eps){width="70.00000%"}
![Frequency-filtered Doppler velocity differences , $\Delta V$, at the moments of: a) the flare impact at ‘Impact 1’, calculated for 01:50:00 UT, and b) about 19 min later at 02:09:20 UT, showing the sunquake wavefront. These are two frames of the supplementary on-line movie. The movie is produced by interpolating the 45-sec cadence data into a new series with 20-sec cadence starting at 01:40 UT. This makes the movie slower and easier to watch. The original 45-sec cadence movie is available on-line in a RHESSI Science Nuggets article [@Kosovichev2011a].[]{data-label="fig2"}](f2.eps){width="70.00000%"}
![Illustration of positions of the flare ‘Impact 1’ (panel a) and the helioseismic fronts (b) observed in the Doppler-shift data, $\Delta V$, in the corresponding magnetogram, $B$, (c) and continuum intensity, $I_c$, (d) images. []{data-label="fig3"}](f3.eps){width="\textwidth"}
![Time-distance diagrams for: a) ‘Wave 1’ and b) ‘Wave 2’, both originating from ‘Impact 1’. In the duplicated diagrams c) and d), the dashed curves are the theoretical time-distance calculated for a standard solar model in the ray approximation. The locations of these curves are chosen to approximately match the leading wave fronts. The short solid line in panel c) indicates the source motion.[]{data-label="fig4"}](f4.eps){width="80.00000%"}
![Variations of the total X-ray fluxes from RHESSI (relative to the 6-12 keV flux), Doppler velocity, magnetic field and continuum intensity at: a) ‘Impact 1’; b) ‘Impact 2’. The X-ray fluxes are integrated for the whole flare region, and thus are identical in panels a) and b).[]{data-label="fig5"}](f5.eps){width="90.00000%"}
|
---
abstract: |
The rough Bergomi model, introduced by [@Bayer:2016], is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in equity markets. In the absence of analytical European option pricing methods for the model, we focus on reducing the runtime-adjusted variance of Monte Carlo implied volatilities, thereby contributing to the model’s calibration by simulation. We employ a novel composition of variance reduction methods, immediately applicable to any conditionally log-normal stochastic volatility model. Assuming one targets implied volatility estimates with a given degree of confidence, thus calibration [RMSE]{}, the results we demonstrate equate to significant runtime reductions—roughly 20 times on average, across different correlation regimes.
[*Keywords:*]{} Rough volatility, implied volatility, option pricing, Monte Carlo, variance reduction
[*2010 Mathematics Subject Classification:*]{} 91G60, 91G20
author:
- 'Ryan McCrickerd[^1]'
- 'Mikko S. Pakkanen[^2] [^3]'
date: |
First version: 8 August 2017\
This version: 16 March 2018
title: |
Turbocharging Monte Carlo pricing\
for the rough Bergomi model
---
Background
==========
Rough volatility is a new paradigm in quantitative finance, motivated by the statistical analysis of realised volatility by [@Gatheral:2014] and the theoretical results on implied volatility by [@Alos:2007] and [@Fukasawa:2011]. Rough volatility is generally characterised by the presence of a stochastic process *rougher* that Brownian motion driving the volatility dynamics—fractional Brownian motion with Hurst exponent $H \in \left(0, \frac{1}{2}\right)$, popularised by [@Mandlebrot:1968], is a convenient example of such a process. The rough Bergomi model (hereafter rBergomi) is the stochastic volatility pricing model developed by [@Bayer:2016], which is consistent with the realised volatility model of [@Gatheral:2014] by means of an elegant change of measure. This *rough stochastic volatility* pricing model outperforms classical counterparts by replicating implied volatility surface dynamics more accurately, being consistent with the stylised fact that the properties of volatility surfaces are essentially time-invariant, and by having fewer parameters—just three! The model is so named because of its relationship with the Bergomi variance curve model [@Bergomi:2005], and may be seen as a non-Markovian generalisation of the latter. Due to the lack of Markovianity or affine structure, conventional analytical pricing methods, such as [PDEs]{} or Fourier transform, do not apply, motivating our quest for fast Monte Carlo pricing of vanilla instruments through a composition of variance reduction methods. While our focus is on the rBergomi model, our approach is applicable to a wide class of stochastic volatility models.
We work throughout on a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{Q})$ that supports a two-dimensional Brownian motion $(W^1,W^2)$ with independent components, under the risk neutral measure $\mathbb{Q}$. The index $t$ will represent time in years from the present and we shall henceforth use the notation $\mathbb{E}[\cdot] = \mathbb{E}^{\mathbb{Q}}[\cdot|\mathcal{F}_0]$ unless we state otherwise. We let $S_t$ be an asset price process satisfying $\mathbb{E}[S_t] = 1$ for all $t\geq 0$, so define an *out-of-the-money* ([OTM]{}) European call/put option with maturity $t$ and log-strike $k$ by its payoff, $$\label{eqPayoff}
(S_t - e^k)^+ := \max\big(w(S_t - e^k), 0\big),\quad w := -\mathbbm{1}_{(-\infty,0]}(k) + \mathbbm{1}_{(0,\infty)}(k),$$ denoting its price observed today by $P(k,t)$.[^4] We define a Black–Scholes function $BS(\cdot)$ by $$BS(v;s,k) := w\big( s \mathcal{N}(wd_+) - e^k \mathcal{N}(wd_-) \big), \quad d_\pm := (\log s - k) /\sqrt{v} \pm \sqrt{v}/2,$$ where $\mathcal{N}(\cdot)$ represents the Gaussian cumulative distribution function.[^5] The implied volatility $\sigma_{BS}(k,t)$ of an observed price $P(k,t)$ is thus defined using the relationship $$\sigma^2_{BS}(k,t)t = BS^{-1}(P(k,t); 1, k).$$
The rBergomi model
------------------
We adopt the rBergomi model [@Bayer:2016] for the price process $S_t$, and define it here by $$\label{eqrBergomi}
\begin{aligned}
S_t & = \mathcal{E}\left( \int_0^\cdot \sqrt{V_u} \mathrm{d} \left( \rho W^1_u + \sqrt{1 - \rho^2} W^2_u \right) \right)_t, \\ V_t & = \xi_0(t) \exp\left( \eta W^\alpha_t -\frac{\eta^2}{2}t^{2\alpha + 1}\right),
\end{aligned}$$ where $\mathcal{E}(\cdot)$ denotes the stochastic exponential[^6] and $\eta>0$ and $\rho\in [-1,1]$ are parameters. We refer to $V_t$ as the variance process, and to $\xi_0(t) = \mathbb{E}[V_t] \in \mathcal{F}_0$ as the forward variance curve. In , $W^\alpha$ is a certain Volterra process, also known as the *Riemann-Liouville process*, defined by $$W_t^\alpha := \sqrt{2\alpha + 1} \int_0^t (t - u)^\alpha \mathrm{d} W^1_u$$ for $\alpha \in \left(-\frac{1}{2},0\right)$. This is a centred, locally $(\alpha + \frac{1}{2} - \epsilon)$-Hölder continuous, Gaussian process with $\mathrm{Var}[W_t^\alpha] = t^{2\alpha + 1}$, and *is not* a martingale, having negatively correlated increments, not even a semimartingale.
In order to simulate the process $W_t^\alpha$ efficiently and accurately, we utilise the first-order variant ($\kappa = 1$) of the *hybrid scheme* [@Bennedsen:2015b], which is based on the approximation $$\label{eqHybrid}
W_{\frac{i}{n}}^\alpha \approx \widetilde{W}_{\frac{i}{n}}^\alpha := \sqrt{2\alpha + 1} \left(\int^{\frac{i}{n}}_{\frac{i-1}{n}} \left(\frac{i}{n}-s\right)^\alpha \mathrm{d} W^1_u + \sum_{k=2}^i\left( \frac{b_k}{n} \right)^\alpha \Big(W^1_{\frac{i-(k-1)}{n}}-W^1_{\frac{i-k}{n}}\Big) \right),$$ where $$b_k := \left(\frac{k^{\alpha+1}-(k-1)^{\alpha+1}}{\alpha+1}\right)^{\frac{1}{\alpha}}.$$ Employing the fast Fourier transform to evaluate the sum in , which is a discrete convolution, a skeleton $\widetilde{W}_{0}^\alpha,\widetilde{W}_{\frac{1}{n}}^\alpha,\ldots,\widetilde{W}_{\frac{\lfloor nt \rfloor}{n}}^\alpha$ can be generated in $\mathcal{O}(n\log n)$ floating point operations.
We demonstrate Volterra sample paths in Figure \[figVolterra\], which lead directly to the rBergomi price sample paths of Figure \[figPrice\].[^7] The parameters of $\eta = 1.9$ and $\rho = -0.9$ there used are demonstrated by [@Bayer:2016] to be remarkably consistent with the SPX market on 4 February 2010, and form the basis for our experiment, along with the case $\rho = 0$, which is more applicable, generally speaking, to other asset classes that deserve our interest, such as [FX]{}. We refrain from formally naming these model parameters, but those seeking an intuitive understanding of their influence over implied volatilities might like *smile* for $\eta$, *skew* for $\rho$, and *explosion* (of smile and skew) for $\alpha$.
![Sample paths of the Volterra process $W^\alpha$ for $\alpha = 0$, for which the process coincides with Brownian motion, and $\alpha = -0.43$. Each are $\mathcal{N}(0,t^{2\alpha + 1})$-distributed, so coincide at $t = 1$. A much greater short-time, *i.e.* $t \ll 1$, variance is exhibited, however, when $\alpha = -0.43$. This *explosive* short-time variance, generated when $\alpha$ is close to $-\frac{1}{2}$, leads to short-time implied volatilities observed in practice. We present here antithetic paths on a 312-point time grid.[]{data-label="figVolterra"}](volterra0 "fig:"){width="0.425\linewidth"} ![Sample paths of the Volterra process $W^\alpha$ for $\alpha = 0$, for which the process coincides with Brownian motion, and $\alpha = -0.43$. Each are $\mathcal{N}(0,t^{2\alpha + 1})$-distributed, so coincide at $t = 1$. A much greater short-time, *i.e.* $t \ll 1$, variance is exhibited, however, when $\alpha = -0.43$. This *explosive* short-time variance, generated when $\alpha$ is close to $-\frac{1}{2}$, leads to short-time implied volatilities observed in practice. We present here antithetic paths on a 312-point time grid.[]{data-label="figVolterra"}](volterra1 "fig:"){width="0.425\linewidth"}
![Sample rBergomi price paths using $\xi = 0.235^2$, $\eta = 1.9$, $\rho$ and $\alpha$ as stated. The price process, despite being a continuous martingale, exhibits jump-like behaviour when the Volterra, thus variance, process peaks. These price paths are based on antithetic paths of $(W^1,W^2)$, again on a 312-point time grid.[]{data-label="figPrice"}](price0 "fig:"){width="0.425\linewidth"} ![Sample rBergomi price paths using $\xi = 0.235^2$, $\eta = 1.9$, $\rho$ and $\alpha$ as stated. The price process, despite being a continuous martingale, exhibits jump-like behaviour when the Volterra, thus variance, process peaks. These price paths are based on antithetic paths of $(W^1,W^2)$, again on a 312-point time grid.[]{data-label="figPrice"}](price1 "fig:"){width="0.425\linewidth"}
Implied volatility estimators
=============================
Accepting the representation $P(k,t) = \mathbb{E}[(S_t - e^k)^+]$ of [OTM]{} option prices, we proceed to consider price estimators $\hat{P}_n(k,t)$ of the following form under the rBergomi model $$\label{eqEstimators}
\hat{P}_n(k,t) := \frac{1}{n}\sum_{i=1}^n \left( X_i + \hat{\alpha}_n Y_i \right) - \hat{\alpha}_n \mathbb{E}[Y], \quad \hat{\sigma}^n_{BS}(k,t)^2 t = BS^{-1}\big(\hat{P}_n(k,t);1,k\big),$$ from which we derive implied volatility estimators $\hat{\sigma}^n_{BS}(k,t)$. Notice that these are always biased by the non-linearity of $BS(\cdot)$ and the requirement to take a square root.[^8] In , $X_i$ and $Y_i$ are samples of random variables to be specified. For example, our *Base* estimator shall be defined naturally by setting $$\label{eqBaseEstimator}
X = (S_t - e^k)^+,\quad Y = 0.$$ A rich variety of implied volatility smiles generated using this estimator are presented in Figure \[figSmiles\], which will further aid intuition for this model. The case $\alpha = 0$ is comparable to classical stochastic volatility models in the absence of time-dependent or randomised parameters, or jump processes. Some admirable recent efforts in the randomised case are [@Mechkov:2016] and [@Jacquier:2017], and for jumps [@Mechkov:2015]. On the contrary, when $\alpha = -0.43$, the explosions of skew and smile as $t \to 0$ are precisely as observed in practice.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Implied volatilities from a simulation of for maturities ranging from one day to one year, using the Base estimator defined by . Parameter values are $\xi_0(t) = \xi = 0.235^2$ and $\eta = 1.9$, with $\rho$ and $\alpha$ as stated. Log-strikes represent a range from 5 delta puts ($\mathcal{N}(-d_+) = 0.05$) to 5 delta calls ($\mathcal{N}(d_+) = 0.05$) in 5 delta increments (19 in total for each maturity). In the simulation, $400{,}000$ antithetic paths are used on $(W^1,W^2)$ and each maturity is separately discretised on a 312-point grid.[]{data-label="figSmiles"}](surface0 "fig:"){width="0.425\linewidth"} ![Implied volatilities from a simulation of for maturities ranging from one day to one year, using the Base estimator defined by . Parameter values are $\xi_0(t) = \xi = 0.235^2$ and $\eta = 1.9$, with $\rho$ and $\alpha$ as stated. Log-strikes represent a range from 5 delta puts ($\mathcal{N}(-d_+) = 0.05$) to 5 delta calls ($\mathcal{N}(d_+) = 0.05$) in 5 delta increments (19 in total for each maturity). In the simulation, $400{,}000$ antithetic paths are used on $(W^1,W^2)$ and each maturity is separately discretised on a 312-point grid.[]{data-label="figSmiles"}](surface1 "fig:"){width="0.425\linewidth"}
![Implied volatilities from a simulation of for maturities ranging from one day to one year, using the Base estimator defined by . Parameter values are $\xi_0(t) = \xi = 0.235^2$ and $\eta = 1.9$, with $\rho$ and $\alpha$ as stated. Log-strikes represent a range from 5 delta puts ($\mathcal{N}(-d_+) = 0.05$) to 5 delta calls ($\mathcal{N}(d_+) = 0.05$) in 5 delta increments (19 in total for each maturity). In the simulation, $400{,}000$ antithetic paths are used on $(W^1,W^2)$ and each maturity is separately discretised on a 312-point grid.[]{data-label="figSmiles"}](surface2 "fig:"){width="0.425\linewidth"} ![Implied volatilities from a simulation of for maturities ranging from one day to one year, using the Base estimator defined by . Parameter values are $\xi_0(t) = \xi = 0.235^2$ and $\eta = 1.9$, with $\rho$ and $\alpha$ as stated. Log-strikes represent a range from 5 delta puts ($\mathcal{N}(-d_+) = 0.05$) to 5 delta calls ($\mathcal{N}(d_+) = 0.05$) in 5 delta increments (19 in total for each maturity). In the simulation, $400{,}000$ antithetic paths are used on $(W^1,W^2)$ and each maturity is separately discretised on a 312-point grid.[]{data-label="figSmiles"}](surface3 "fig:"){width="0.425\linewidth"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In pursuit of a variance reducing estimator of the form , following [@Romano:1997], we consider the orthogonal separation of the rBergomi price process $S_t$ into $S^1_t$ and $S^2_t$, where $$S^1_t := \mathcal{E}\left( \rho \int_0^\cdot \sqrt{V_u} \mathrm{d} W^1_u \right)_t, \quad S^2_t := \mathcal{E}\left( \sqrt{1 - \rho^2} \int_0^\cdot \sqrt{V_u} \mathrm{d} W^2_u \right)_t,$$ which allows us to capitalise on its conditional log-normality.[^9] By conditional log-normality, we explicitly mean $$\label{eqCondLN}
\log S_t \ | \ \mathcal{F}^1_t \vee \mathcal{F}^2_0 \sim \mathcal{N} \left( \log S^1_t - \frac{1}{2}\left( 1 - \rho^2 \right) \int_0^t V_u \mathrm{d}u,\ \left( 1 - \rho^2 \right) \int_0^t V_u \mathrm{d}u \right),$$ where we use natural filtrations $\mathcal{F}^i_t = \sigma\{W^i_u:u \leq t\}$, $i = 1,2$. Since both $\int_0^t V_u \mathrm{d}u$ and $S_t^1$ are measurable with respect to $\mathcal{F}_t^1$, this representation becomes intuitively clear when we imagine $S^1_t$ as a spot price, and $\left( 1 - \rho^2 \right) \int_0^t V_u \mathrm{d}u$ as the integrated variance originating from $S_t^2$, as is described by [@Romano:1997] and [@Bergomi:2016] in wider stochastic volatility frameworks. This separation facilitates our *Mixed* estimator, which we define using with $$\label{eqMixtureEstimator}
X = BS\left( \left( 1-\rho^2 \right) \int_0^t V_u \mathrm{d}u; S_t^1, k \right),\quad Y = BS\left( \rho^2\left(\hat{Q}_n - \int_0^t V_u \mathrm{d}u\right); S^1_t, k \right),$$ where the estimated parameters $\hat{\alpha}_n$ and $\hat{Q}_n$ will be soon made explicit.
The Mixed estimator represents the composition of the conditional Monte Carlo method with a control variate, which we have found to be individually most effective in the regimes $\rho = 0$ and $\rho = -0.9$ respectively. This use of $X$ represents the simulation of a conditional expectation because, following , we have the representation $$X = \mathbb{E}\big[( S_t - e^k)^+ \,\big|\, \mathcal{F}^1_t \vee \mathcal{F}^2_0 \big].$$ The Tower property then ensures $\mathbb{E}[X]$ agrees with the expectation of the Base estimator. Amazingly, this eliminates all dependence on $W^2$, and in theory guarantees a variance reduction. The component $Y$ in the Mixed estimator admits a representation as the time $t$ price of a *Timer* option with variance budget $\rho^2\hat{Q}_n$, written on the parallel component $S^1_t$ of the price process.[^10] The process $Y = Y_t$ is clearly a martingale, because it has the representation $$Y_t = \mathbb{E}\big[(S^1_{\tau_{\hat{Q}_n}} - e^k)^+\,\big|\,\mathcal{F}^1_t\big],\quad \tau_{\hat{Q}_n} := \inf\left\{u>0:\int_0^u V_s \mathrm{d}s = \hat{Q}_n\right\},$$ as is the case for any tradeable asset. For all maturities $t$ we are therefore able to make use of the following expectation in , $$\mathbb{E}[Y] = \mathbb{E}[Y_0] = BS\big( \rho^2\hat{Q}_n ; 1, k \big).$$
We compute $\hat{\alpha}_n$ and $\hat{Q}_n$ post-simulation from sampled $X_i$, $Y_i$ and $\left(\int_0^t V_u \mathrm{d}u\right)_i$, using $$\label{eqParamDefs}
\hat{\alpha}_n := -\frac{\sum_{i=1}^n \left(X_i - \bar{X}_n \right)\left(Y_i - \bar{Y}_n \right)}{\sum_{i=1}^n \left(Y_i - \bar{Y}_n \right)^2},\quad \hat{Q}_n := \sup\left\{ \left(\int_0^t V_u \mathrm{d}u\right)_i : i=1,\dots,n \right\},$$ meaning that our variance reducing methods lose their relationship with hedging strategies in practice. The former is known to asymptotically minimise the variance of $\hat{P}_n(k,t)$ for any control variate, see for example @Asmussen:2007 [pp. 138–139]. The choice of $\hat{Q}_n$ might seem unnerving, but is the minimum that avoids the computation of stopping times when evaluating $Y$, which we find to be relatively computationally expensive.[^11] The choice otherwise ensures that $Y$ outperforms the more obvious martingale control variate $wS^1_t$, effectively because the following limit holds[^12] $$\lim_{Q \to \infty} Y = \lim_{Q \to \infty} BS\left( \rho^2\left(Q - \int_0^t V_u \mathrm{d}u\right); S^1_t, k \right) = wS^1_t.$$
Finally, we briefly explain our use of antithetic sampling for the Mixed estimator. We draw a path of $W^1$ over the interval $[0,t]$, and appeal to the symmetry in distribution of $S^{1,\pm}_t$, defined by $$\begin{aligned}
S^{1,\pm}_t & = \mathcal{E}\left\{\pm \rho \int_0^t \sqrt{V^\pm_u} \mathrm{d} W^1_u \right\}, & V^\pm_t & = \xi_0(t)\exp\left(-\frac{\eta^2}{2}t^{2\alpha + 1}\right)\left(V^\circ_t\right)^{\pm 1}, \\
V^\circ_t &= \exp\left( \eta W^\alpha_t \right).\end{aligned}$$ Notice that, besides providing an outright variance reduction, this immediately halves the number of required Volterra paths, reducing total runtime significantly. Now that the Mixed estimator is fully defined, we summarise the estimators from which it was developed in Table \[tabEstimators\]. The Conditional estimator and some methods related to our Controlled estimator, for example, the *Timer option-like algorithm*, may be found in @Bergomi:2016 [pp. 336–342] in a general stochastic volatility setting.
[|l|l|c|l|c|]{} Estimator & & $X$ & & $Y$\
Base & & $\left(S_t - e^k\right)^+$ & & $0$\
Conditional & & $BS\big((1 - \rho^2) \int_0^t V_u \mathrm{d}u; S^1_t, k\big)$ & & $0$\
Controlled & & $(S_t - e^k)^+$ & & $BS\big(\hat{Q}_n - \int_0^t V_u \mathrm{d}u;S_t, k\big)$\
Mixed & & $BS\big(( 1-\rho^2 ) \int_0^t V_u \mathrm{d}u; S_t^1, k \big)$ & & $BS\big( \rho^2 \big(\hat{Q}_n - \int_0^t V_u \mathrm{d}u \big); S^1_t, k \big)$\
In the next section, we conduct an experiment to compare implied volatilities derived from our Base and Mixed estimators. We use a relatively low number of paths, comparing resulting bias and variances with the higher quality data in Figure \[figSmiles\].[^13] Following this comparison, we proceed to briefly demonstrate the impact of our work on the rBergomi parameters driving smile and skew, $\eta$ and $\rho$, in an experiment assessing the calibration accuracy of those parameters by simulation. All of this is implemented in Python, although we use the NumPy library heavily to ensure C[++]{}-like runtimes. We use the default NumPy pseudo-random number generator (Mersenne Twister). The performance of all implied volatility estimators can be improved slightly by instead using quasi-random numbers (low-discrepancy sequences, e.g., Sobol), but our experiments with Sobol sequences, obtained using the Sobol Julia module, suggest that the improvement is not dramatic. With a focus on results, practical application and building intuition for the rBergomi model, we simply summarise results for the intermediate estimators, and point to [@Asmussen:2007] for some general theory underlying this work.
Variance reduction
==================
As is widely understood by practitioners of Monte Carlo methods, *the greatest gains from variance reduction techniques result from exploiting specific features of the problem at hand*—adapted from [@Glasserman:2004]. Although the theory of antithetic sampling, conditional Monte Carlo, and control variates are well understood, these methods are somewhat meaningless without refinement to our estimation of implied volatilities under the rBergomi model.
Experiment design
-----------------
We now fix the maturity $t = 0.25$, so may drop its reference, and rBergomi parameters $\xi = 0.235^2$, $\eta = 1.9$ and $\alpha = -0.43$. We consider the two correlation regimes of $\rho = -0.9$ and $\rho = 0$, and three log-strikes representing 10 delta put, [ATM]{}, and 10 delta call options in each regime.[^14] We consider sampling $\hat{\sigma}^n_{BS}(k)$ from $N$ times, in order to obtain a sequence $\{\hat{\sigma}^{n}_{BS}(k)_i\}_{i=1}^{N}$ of estimates. Given the following central limit theorem for estimated prices $$\sqrt{n}\big(\hat{P}^n(k,t) - P(k,t)\big) \xrightarrow[n\to\infty]{D} \mathcal{N}\left(0, v_\infty\right),$$ with $v_\infty: = \lim_{n\to\infty}\mathrm{Var}[X + \hat{\alpha}_n Y]$ and $X$, $Y$ as in , the Delta method provides the additional convergence $$\sqrt{n}\big(\hat{\sigma}^n_{BS}(k) - \sigma_{BS}(k)\big) \xrightarrow[n\to\infty]{D} \mathcal{N}\big(0, v_\infty \left(2t \sigma_{BS}(k,t)BS'(\sigma_{BS}^2(k,t)t;1,k) \right)^{-2}\big).$$ Fixing $n = 1{,}000$ and $N = 1{,}000$, we therefore plot histograms of the sampled sequences $\{\hat{\sigma}^n_{BS}(k)_i - \sigma_{BS}(k)\}_{i=1}^N$ alongside fitted normal distributions. Of course, we don’t truly know $\sigma_{BS}(k)$, hence the use of the results in Figure \[figSmiles\] as proxies. They are provided for the relevant [3M]{} maturity in the following table for clarity.
[|l|rrr|l|l|rrr|]{} $\rho = -0.9$ & & & & & $\rho = 0$ & & &\
$k$ & $-0.1787$ & 0.0000 & 0.1041 & & $k$ & $-0.1475$ & 0.0000 & 0.1656\
$\sigma_{BS}(k)$ & 29.61 & 20.61 & 15.76 & & $\sigma_{BS}(k)$ & 24.17 & 21.73 & 24.66\
In order to compare estimators in a manner which is both runtime-adjusted and weakly dependent on the choice of $n$, we take guidance from [@Glasserman:2004] when defining our measure of $\hat{\sigma}^n_{BS}(k)$ variance. To this end, we let $\tau$ denote the runtime in milliseconds to produce a single $\hat{\sigma}^n_{BS}(k)_i$ estimation.[^15] Considering log-strikes $\{k_i\}_{i = 1}^m$, we thus define the *mean squared error* and *mean runtime-adjusted squared error* measures of our estimators respectively by $$\label{eqMeasureDef}
\phi^2 := \frac{1}{m} \sum_{i=1}^{m} \hat{\sigma}^2_{n,N,k_i},\quad \psi^2 := \frac{\tau}{m} \sum_{i=1}^{m} \hat{\sigma}^2_{n,N,k_i},$$ where we simply estimate $$\hat{\sigma}^2_{n,N,k} := \frac{1}{N - 1}\sum_{i=1}^N \left(\hat{\sigma}^n_{BS}(k)_i - \sigma_{BS}(k) \right)^2.$$ Notice that $\psi^2$ is in theory asymptotically independent of $n$, since $\tau$ scales like $n$ and for each $k_i$, $\hat{\sigma}^2_{n,N,k_i}$ scales asymptotically like $1/n$. Having fixed $n$ and $N$, for ease of computations, we may therefore use ratios of estimator $\psi^2$ values in order to approximate the relative runtime to achieve a fixed $\phi$ value (corresponding to a calibration [RMSE]{}), since $\tau = \psi^2/\phi^2$. These observations are reflected in practice, certifying $\psi^2$ as a sensible means for comparison. We stress that our use of $n = 1{,}000$ is only for convenience, and to demonstrate the performance of the Mixed estimator with such few paths. Indeed, because we find that all estimators’ standard deviations adhere to the scaling suggested by the central limit theorem, one may predictably shrink observed confidence intervals by increasing $n$.
Results
-------
Histograms with fitted normal distributions, and implied volatility confidence intervals are shown for the Base estimator in Figures \[figBaseHist\] and \[figRMSEsBase\], and for the Mixed estimator in Figures \[figMixedHist\] and \[figRMSEsMixed\], respectively. Histograms are labelled with each applicable log-strike $k$, target *implied volatility* $\sigma_{BS}$ and *bias* (B) and *standard deviation* (S) of the sample $\{\hat{\sigma}^{n}_{BS}(k)_i\}_{i=1}^{N}$. We stress again that $\psi^2$ is the measure that should be used to determine relative estimator runtimes to achieve a given implied volatility confidence interval.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Implied volatility estimator $\hat{\sigma}^n_{BS}(k)$ expectations (red), with 95% confidence intervals (black), using the Base estimator of . Each $\hat{\sigma}^n_{BS}(k)$ is sampled $N = 1{,}000$ times, using $n = 1{,}000$ paths.[]{data-label="figRMSEsBase"}](histogram0-edit-crop "fig:"){width="0.9\linewidth"}
![Implied volatility estimator $\hat{\sigma}^n_{BS}(k)$ expectations (red), with 95% confidence intervals (black), using the Base estimator of . Each $\hat{\sigma}^n_{BS}(k)$ is sampled $N = 1{,}000$ times, using $n = 1{,}000$ paths.[]{data-label="figRMSEsBase"}](histogram1-edit-crop "fig:"){width="0.9\linewidth"}
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Implied volatility estimator $\hat{\sigma}^n_{BS}(k)$ expectations (red), with 95% confidence intervals (black), using the Base estimator of . Each $\hat{\sigma}^n_{BS}(k)$ is sampled $N = 1{,}000$ times, using $n = 1{,}000$ paths.[]{data-label="figRMSEsBase"}](bounds0-edit-crop.pdf "fig:"){width="0.425\linewidth"} ![Implied volatility estimator $\hat{\sigma}^n_{BS}(k)$ expectations (red), with 95% confidence intervals (black), using the Base estimator of . Each $\hat{\sigma}^n_{BS}(k)$ is sampled $N = 1{,}000$ times, using $n = 1{,}000$ paths.[]{data-label="figRMSEsBase"}](bounds1-edit-crop.pdf "fig:"){width="0.425\linewidth"}
The Base estimator results in Figure \[figBaseHist\] demonstrate standard deviations, around 1 percentage point (i.e., one Vega), which render it unfit for practical purposes. This, of course, is not surprising when using just $n = 1{,}000$ paths, and these results are nevertheless important for aiding comparison. Figure \[figRMSEsBase\] places these results and 95% confidence intervals over the equivalent implied volatilities from Figure \[figSmiles\], also showing root mean squared errors, $\phi$. In general, one finds greatest variances at the 10 delta call strike, but in the case of $\rho = -0.9$, this effect is dominated by the price process inheriting greater variances for low strikes.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Implied volatility estimator expectations, with 95% confidence intervals, using the Mixed estimator of , alongside data from Figure \[figSmiles\]. Each $\hat{\sigma}^n_{BS}(k)$ is sampled $N = 1{,}000$ times, using $n = 1{,}000$ paths.[]{data-label="figRMSEsMixed"}](histogram2-edit-crop "fig:"){width="0.9\linewidth"}
![Implied volatility estimator expectations, with 95% confidence intervals, using the Mixed estimator of , alongside data from Figure \[figSmiles\]. Each $\hat{\sigma}^n_{BS}(k)$ is sampled $N = 1{,}000$ times, using $n = 1{,}000$ paths.[]{data-label="figRMSEsMixed"}](histogram3-edit-crop "fig:"){width="0.9\linewidth"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Implied volatility estimator expectations, with 95% confidence intervals, using the Mixed estimator of , alongside data from Figure \[figSmiles\]. Each $\hat{\sigma}^n_{BS}(k)$ is sampled $N = 1{,}000$ times, using $n = 1{,}000$ paths.[]{data-label="figRMSEsMixed"}](bounds2-edit-crop.pdf "fig:"){width="0.425\linewidth"} ![Implied volatility estimator expectations, with 95% confidence intervals, using the Mixed estimator of , alongside data from Figure \[figSmiles\]. Each $\hat{\sigma}^n_{BS}(k)$ is sampled $N = 1{,}000$ times, using $n = 1{,}000$ paths.[]{data-label="figRMSEsMixed"}](bounds3-edit-crop.pdf "fig:"){width="0.425\linewidth"}
The Mixed estimator results in Figure \[figMixedHist\] demonstrate standard deviations much lower than 1 percentage point (i.e., one Vega). Even in the most uncertain case, the sampled implied volatility is within 1.1 percentage points of the known value 29.6%, 95% of the time. We consider this remarkable, evidently, considering the number of paths, $n = 1{,}000$, used. Figure \[figRMSEsBase\] places these results and 95% confidence intervals over the equivalent implied volatilities from Figure \[figSmiles\], also showing root mean squared errors, $\phi$.
The relative $\psi^2$ values for the Base and Mixed estimators in Figures \[figBaseHist\] and \[figMixedHist\] suggest a 13-fold runtime reduction in the $\rho = -0.9$ regime, and a 34-fold runtime reduction in the $\rho = 0$ regime, in order to match $\phi$ values, thereby a given implied volatility confidence interval. That is, roughly a 20-fold runtime reduction on average. Indeed, in Figure \[figRMSEsBase1\] we show another set of Base estimator results, which match the Mixed $\phi$ values, requiring $n = 8{,}000$ and $20{,}250$ paths respectively.
![A reproduction of the Base estimator results from Figure \[figRMSEsBase\], using instead $n = 8{,}000$ for the case $\rho = -0.9$ and $n = 20{,}500$ for the case $\rho = 0$, in order to match resulting $\phi$ values with those of the Mixed estimator when using $n = 1{,}000$. Ratios of previously observed $\phi^2$ values are used to predict the number of paths required to achieve this.[]{data-label="figRMSEsBase1"}](bounds4-edit-crop.pdf "fig:"){width="0.425\linewidth"} ![A reproduction of the Base estimator results from Figure \[figRMSEsBase\], using instead $n = 8{,}000$ for the case $\rho = -0.9$ and $n = 20{,}500$ for the case $\rho = 0$, in order to match resulting $\phi$ values with those of the Mixed estimator when using $n = 1{,}000$. Ratios of previously observed $\phi^2$ values are used to predict the number of paths required to achieve this.[]{data-label="figRMSEsBase1"}](bounds5-edit-crop.pdf "fig:"){width="0.425\linewidth"}
Before proceeding, we summarise standard deviations and runtimes for all estimators in Table \[tabSummary\], using $n = 1{,}000$. We have no practically meaningful bias to report. To aid a clearer comparison, the Conditional, Controlled and Mixed estimators all utilise antithetic sampling, hence their lower runtimes. The Mixed estimator adopts the variance reducing effects of the Conditional and Controlled estimators in the regimes $\rho = 0$ and $\rho = -1$, respectively. For $-1 < \rho < 0$, the Mixed estimator blends the effects of each, which is already observed in the case of $\rho = -0.9$. Experiment suggests that the Mixed estimator outperforms the Conditional and Controlled estimators best, in a joint sense, around the region $1- \rho^2 = \rho^2$.
[|l|l|cccc|c|cccc|]{} & & & &\
Estimator & & 10P & ATM & 10C & $\tau$ & & 10P & ATM & 10C & $\tau$\
Base & & 1.28 & 1.24 & 0.52 & 114 & & 0.94 & 1.03 & 1.25 & 115\
Antithetic & & 1.70 & 1.45 & 0.59 & 49 & & 0.92 & 0.74 & 1.25 & 49\
Conditional & & 1.19 & 1.02 & 0.34 & 68 & & 0.26 & 0.15 & 0.28 & 69\
Controlled & & 0.82 & 0.41 & 0.49 & 55 & & 0.70 & 0.56 & 0.82 & 55\
Mixed & & 0.55 & 0.27 & 0.26 & 71 & & 0.26 & 0.15 & 0.28 & 70\
Experiment assessing the accuracy of calibration
------------------------------------------------
We now briefly demonstrate the impact of these results on an example calibration by simulation of the rBergomi model. We stress that this is only really for illustrative purposes, since knowledge of (untraded) model parameter bounds seems somewhat meaningless without understanding the associated impact on (traded) implied volatility bounds, which we have covered directly. The specification of which rBergomi parameters should be calibrated by simulation is an open question and not a topic we intend to tackle here.
We assume, to aid this demonstration, that $\alpha$ and $\xi_0(t)$ are fixed by other means at $-0.43$ and $0.235^2$ respectively. This is consistent with the approach adopted by [@Jacquier:2017a] for a joint [SPX]{} and [VIX]{} calibration. Therein, $H = \alpha + \frac{1}{2}$ is calibrated pre-simulation to [VIX]{} futures, and $\xi_0(t)$ extracted from an e[SSVI]{} parameterisation [@Hendriks:2017] of an observed [SPX]{} implied volatility surface. A more asset class-indifferent approach might be to obtain $\alpha$ from historic time-series using, for example, the methods of [@Gatheral:2014] and [@Bennedsen:2015]. This is made possible, in theory, since $\alpha$ is preserved in the neat measure change from which the rBergomi model is derived. We suggest a natural approach across asset classes for obtaining $\xi_0(t)$ would be to utilise the elegant integrated variance representation summarised by [@Austing:2014], $$\int_0^t \xi_0(u) \mathrm{d}u = \mathbb{E}\left[\int_0^t V_u \mathrm{d}u\right] = \int_0^1 \sigma_{BS}(\Delta,t)^2 \mathrm{d}\Delta, \quad \Delta := \mathcal{N}(-d_-),$$ which follows from Fubini’s theorem and a change of variables. Clearly this requires an interpolation of observed $\sigma_{BS}(\cdot,t)$ in $\Delta$-space, and some parametric (or piece-wise parametric) assumption for $\xi_0(t)$. We find, however, that even a naïve cubic spline across $\sigma_{BS}(\Delta,t)$ and piece-wise constant $\xi_0(t)$ can produce impressive results. In Figure \[figDelta\], we reproduce Figure \[figSmiles\] in $\Delta$-space for the case of $\alpha = -0.43$, given that data sources like Bloomberg do similarly.
![A reproduction of Figure \[figSmiles\], in $\Delta$-space, integration of which may lead to $\xi_0(t)$. The near-inhomogeneity across maturities is intriguing. Notice that $\Delta = \mathcal{N}(-d_-)$, referred to as forward delta by [@Austing:2014], is not quite the same as the delta used to define log-strikes, $\mathcal{N}(-d_+)$.[]{data-label="figDelta"}](surface4 "fig:"){width="0.425\linewidth"} ![A reproduction of Figure \[figSmiles\], in $\Delta$-space, integration of which may lead to $\xi_0(t)$. The near-inhomogeneity across maturities is intriguing. Notice that $\Delta = \mathcal{N}(-d_-)$, referred to as forward delta by [@Austing:2014], is not quite the same as the delta used to define log-strikes, $\mathcal{N}(-d_+)$.[]{data-label="figDelta"}](surface5 "fig:"){width="0.425\linewidth"}
We proceed to calibrate the rBergomi skew and smile parameters $\rho$ and $\eta$, seeking a minimisation of absolute [RMSE]{}s for the 19 implied volatilites at the [3M]{} maturity in Figure \[figSmiles\]. Joint calibrated $\rho$ and $\eta$ distributions are presented in Figure \[figCalib\].
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![100 calibrations of $\rho$ and $\eta$ using the Base and Mixed estimators, with just $n = 1{,}000$ paths. The minimum of implied volatility absolute [RMSE]{}s is sought using the [L-BFGS-B]{} method of `scipy.optimize.minimize`, with bounds $\rho \in [-0.99,0.99]$, $\eta \in [1.00,3.00]$, allowing this to run for approximately 700 milliseconds. Despite actually making little difference, we initialised the solver for $\rho$ and $\eta$ at the known values in each case, so that the resulting calibrations observed here truly *represent the convergence of $\rho$ and $\eta$ to values away from these known values*—thereby measuring each estimator’s *failure* to produce the known distributive properties of the price process, and equivalently the known implied volatilities. The Mixed estimator substantially reduces calibrated $\rho$ and $\eta$ variance, with the Base estimator being somewhat aided in the $\rho = -0.9$ case by the lower bound of $-0.99$.[]{data-label="figCalib"}](calibration0 "fig:"){width="0.425\linewidth"} ![100 calibrations of $\rho$ and $\eta$ using the Base and Mixed estimators, with just $n = 1{,}000$ paths. The minimum of implied volatility absolute [RMSE]{}s is sought using the [L-BFGS-B]{} method of `scipy.optimize.minimize`, with bounds $\rho \in [-0.99,0.99]$, $\eta \in [1.00,3.00]$, allowing this to run for approximately 700 milliseconds. Despite actually making little difference, we initialised the solver for $\rho$ and $\eta$ at the known values in each case, so that the resulting calibrations observed here truly *represent the convergence of $\rho$ and $\eta$ to values away from these known values*—thereby measuring each estimator’s *failure* to produce the known distributive properties of the price process, and equivalently the known implied volatilities. The Mixed estimator substantially reduces calibrated $\rho$ and $\eta$ variance, with the Base estimator being somewhat aided in the $\rho = -0.9$ case by the lower bound of $-0.99$.[]{data-label="figCalib"}](calibration2 "fig:"){width="0.425\linewidth"}
![100 calibrations of $\rho$ and $\eta$ using the Base and Mixed estimators, with just $n = 1{,}000$ paths. The minimum of implied volatility absolute [RMSE]{}s is sought using the [L-BFGS-B]{} method of `scipy.optimize.minimize`, with bounds $\rho \in [-0.99,0.99]$, $\eta \in [1.00,3.00]$, allowing this to run for approximately 700 milliseconds. Despite actually making little difference, we initialised the solver for $\rho$ and $\eta$ at the known values in each case, so that the resulting calibrations observed here truly *represent the convergence of $\rho$ and $\eta$ to values away from these known values*—thereby measuring each estimator’s *failure* to produce the known distributive properties of the price process, and equivalently the known implied volatilities. The Mixed estimator substantially reduces calibrated $\rho$ and $\eta$ variance, with the Base estimator being somewhat aided in the $\rho = -0.9$ case by the lower bound of $-0.99$.[]{data-label="figCalib"}](calibration1 "fig:"){width="0.425\linewidth"} ![100 calibrations of $\rho$ and $\eta$ using the Base and Mixed estimators, with just $n = 1{,}000$ paths. The minimum of implied volatility absolute [RMSE]{}s is sought using the [L-BFGS-B]{} method of `scipy.optimize.minimize`, with bounds $\rho \in [-0.99,0.99]$, $\eta \in [1.00,3.00]$, allowing this to run for approximately 700 milliseconds. Despite actually making little difference, we initialised the solver for $\rho$ and $\eta$ at the known values in each case, so that the resulting calibrations observed here truly *represent the convergence of $\rho$ and $\eta$ to values away from these known values*—thereby measuring each estimator’s *failure* to produce the known distributive properties of the price process, and equivalently the known implied volatilities. The Mixed estimator substantially reduces calibrated $\rho$ and $\eta$ variance, with the Base estimator being somewhat aided in the $\rho = -0.9$ case by the lower bound of $-0.99$.[]{data-label="figCalib"}](calibration3 "fig:"){width="0.425\linewidth"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Concluding remarks
==================
We have demonstrated sample paths and the rich implied volatility surfaces generated from the rBergomi model in order to build intuition for its parameters. We have made Python code available on GitHub, from which one is able to replicate these surfaces and generate others. We believe that the potential of rough volatility models is evident and hope that the seeds for practical adoption are now sewn.
Drawing inspiration from [@Bergomi:2016], we have jumped towards the present requirement of rBergomi calibration by simulation, by carefully applying the conditional Monte Carlo method with a control variate and antithetic sampling. Specifically, we have provided a 20-fold runtime reduction on average for achieving a chosen European option implied volatility confidence interval, thus calibration [RMSE]{}.
Although there remain open questions (perhaps most significantly: which of the model’s parameters, if not all, can be reliably calibrated pre-simulation, and how best?), this is now a thriving area of research in academia, and we are full of resolute optimism. Having practical experience with a variety of stochastic volatility models, we cannot stress enough how central we believe rough processes, like the Volterra process, could be in the future of volatility modelling.
Acknowledgements {#acknowledgements .unnumbered}
================
M.S.P. acknowledges helpful discussions with Chithira Mamallan, who independently obtained results on the effectiveness of antithetic sampling and the Conditional estimator in the context of the rough Bergomi model in her MSci dissertation [@Mamallan:2017] at Imperial College London. He also thanks Christian Bayer for discussions on quasi-random numbers.
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[^1]: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK. E-mail: [](mailto:ryan.mccrickerd@jcrauk.com)
[^2]: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK. E-mail: [](mailto:m.pakkanen@imperial.ac.uk)
[^3]: Corresponding author.
[^4]: We must stress the importance of this first step towards variance reduction. The implied volatilities generated when exclusively considering call or put option estimators are significantly noisier when they are respectively *in-the-money*. This may be rationalised using the put-call parity, $\max\{S_t - e^k,0\} - \max\{e^k - S_t,0\} = S_t - e^k$. The methods we later employ remove this *in-the-money variance*, but it is avoidable from the outset by always evaluating [OTM]{} options. Instead setting $w := \pm 1$ in , perceived variance reductions increase dramatically.
[^5]: This later enables use of the famed result $\log S \sim \mathcal{N}\left(\log s - \frac{1}{2}v,v\right) \implies \mathbb{E}[(S - e^k)^+] = BS(v;s,k)$. The somewhat unusual implied definition $k := \log K$, for strike $K$, compared with $k := \log(K/s)$, is used so $k$ remains fixed when we later vary $s$ through time.
[^6]: Recall that for continuous semimartingale $X$, the stochastic exponential is defined $\mathcal{E}(X)_t:=\exp\left(X_t - X_0 - \frac{1}{2}[X]_t\right)$.
[^7]: We provide Python code on GitHub (<https://github.com/ryanmccrickerd/rough_bergomi>) and Jupyter notebooks that are able to reproduce sample paths and turbocharged implied volatilities.
[^8]: We later report some bias, but we find that even when using $n = 1{,}000$, it is never practically meaningful.
[^9]: For a conditionally Gaussian process, our methods could be adapted using, for example, Hull–White price evaluation in place of Black–Scholes.
[^10]: That analytical Timer option prices should be available under stochastic volatility models is intuitively clear, but a probabilistic interpretation of why is wonderful: $\log S_t + \frac{1}{2}\int_0^t V_u\mathrm{d}u = \int_0^t \sqrt{V_u}\mathrm{d}W^1_u$ is a continuous local martingale starting at zero on $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq 0},\mathbb{Q})$, so defining the stopping time $\tau_Q := \inf\{u>0:\int_0^u V_s \mathrm{d}s = Q\}$, the Dubins–Schwarz theorem provides $B_Q := \log S_{\tau_Q} + \frac{1}{2}Q$ is a Brownian motion on $(\Omega,\mathcal{F},\{\mathcal{F}_{\tau_Q}\}_{Q\geq 0},\mathbb{Q})$.
[^11]: For example, one might set $Y = BS \left( Q - \int_0^{t\wedge\tau_Q} V_u \mathrm{d}u; S_{t\wedge\tau_Q}, k \right)$, with $\tau_Q$ as defined above.
[^12]: It is worth appreciating that in the seemingly awkward limits of $\rho \to 0$ and $\rho \to \pm 1$, the Mixed estimator performs like the conditional Monte Carlo method and a control variate *independently*, respectively, by design.
[^13]: Number of paths is almost arbitrary, because we find our estimators adhere neatly to the scaling properties implied by the central limit theorem: to halve observed standard deviations, simply quadruple number of paths.
[^14]: Specifically, this means $\mathcal{N}(-d_+) = 0.10$, $k = 0$ and $\mathcal{N}(d_+) = 0.10$ respectively.
[^15]: Given the target application of this work, we must approximate a runtime which is indicative of the time taken by a minimisation routine of implied volatility [RMSE]{}s. This in itself is ambiguous, given, amongst other things, this time will be affected by which of the rBergomi parameters are being calibrated. Specifically, we let $\tau$ be the time to produce the sample $\{\hat{\sigma}^n_{BS}(k)_i - \sigma_{BS}(k)\}_{i=1}^N$, divided by $N$.
|
---
abstract: 'Fragmentation is the dominant mechanism for hadron production with high transverse momentum. For spin-triplet S-wave heavy quarkonium production, contribution of gluon fragmenting to color-singlet channel has been numerically calculated since 1993. However, there is still no analytic expression available up to now because of its complexity. In this paper, we calculate both polarization-summed and polarized fragmentation functions of gluon fragmenting to a heavy quark-antiquark pair with quantum number ${{{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}}$. Our calculations are performed in two different frameworks. One is the widely used nonrelativistic QCD factorization, and the other is the newly proposed soft gluon factorization. In either case, we calculate at both leading order and next-to-leading order in velocity expansion. All of our final results are presented in terms of compact analytic expressions.'
author:
- 'Peng Zhang$^{1}$'
- 'Yan-Qing Ma$^{1,2,3}$'
- 'Qian Chen$^{1}$'
- 'Kuang-Ta Chao$^{1,2,3}$'
title: 'Analytical calculation for the gluon fragmentation into spin-triplet S-wave quarkonium'
---
Introduction
============
As heavy quark mass $m_Q$ is much larger than the QCD nonperturbative scale $\Lambda _{QCD}$, the production of heavy quark-antiquark ($Q\bar{Q}$) pair is perturbatively calculable. Due to the binding energy of $Q\bar{Q}$ for a heavy quarkonium being at the order of $\Lambda _{QCD}$, hadronization of $Q\bar{Q}$ to heavy quarkonium is nonperturbative. Therefore, study of quarkonium production can help to understand both perturbative and nonperturbative physics in QCD. Nevertheless, more than 40 years after the discovery of the $J/\psi$, the production mechanism of heavy quarkonium, the simplest system under strong interaction, is still not well understood.
Recently, two of the present authors proposed a soft gluon factorization (SGF) theory to describe quarkonium production and decay [@Ma:2017xno; @machao]. On the one hand, SGF is as rigorous as the currently widely used nonrelativistic QCD factorization (NRQCD) [@Bodwin:1994jh], which means either both of them are correct to all orders in perturbation theory, or both of them are broken down at a sufficient large order in $\alpha_s$ expansion. On the other hand, it was argued that the convergence of velocity expansion in SGF should be much better than that in NRQCD [@Ma:2017xno]. Thus, SGF may resolve some difficulties encountered in NRQCD for quarkonium production. In this paper, we use SGF and NRQCD to compute the gluon fragmentation function (FF) to $J^{PC}=1^{--}$ quarkonia, which is useful for understanding the production of these quarkonia at high transverse momentum $p_T$. According to QCD collinear factorization [@Collins:1989gx], the inclusive production cross section of a specific hadron $H$ at very high $p_T$ is dominated by the fragmentation mechanism at leading power (LP) [^1], $${\mathrm{d}}\sigma _{A+B\rightarrow H(p_T)+X} = \sum_i {\mathrm{d}}\hat{\sigma} _{A+B \rightarrow i(p_T /z)+X} \otimes D_{i\to H}(z,\mu) + \co (1/p_T^2) \, ,$$ where $i$ sums over all quarks and gluons, and $z$ is the light-cone momentum fraction carried by $H$ with respect to the parent parton. The hard part $d \hat{\sigma} _{A+B \rightarrow i(p_T /z)+X}$ can be calculated perturbatively, while the FF $D_{i\to H}(z,\mu)$, describing the probability distribution of the hadronization from $i$ to $H$, is nonperturbative and universal. The dependence of FF on factorization scale $\mu$ is controlled by the DGLAP evolution equation[@Gribov:1972ri; @Altarelli:1977zs; @Dokshitzer:1977sg], $$\mu \frac{{\mathrm{d}}}{{\mathrm{d}}\mu} D_{i\to H}(z,\mu) = \sum_j \int_z^1 \frac{{\mathrm{d}}\xi}{\xi} P_{ij} \left( \frac{z}{\xi},\alpha_s(\mu) \right) D_{j\to H}(\xi,\mu) \, ,$$ where $P_{ij}$ are splitting functions that can be calculated perturbatively. Based on this evolution, FF at an arbitrary perturbative scale $\mu$ can be determined by FF at an initial scale $\mu_0$.
In the case that $H$ is a heavy quarkonium, there is an intrinsic hard scale $m_Q$ in FFs. Usually, we can choose the initial scale $\mu_0 \gtrsim 2 m_Q $, so that $\ln (\mu_0^2/m_Q^2)-$type logarithms are not large. As $m_Q\gg\Lambda_{\text{QCD}}$, FFs evaluated at $\mu_0$ can be further factorized as perturbative calculable short-distance coefficients (SDCs) multiplied by nonpertubative part at the scale $m_Q v$ and below. If one uses either SGF or NRQCD to do this factorization, one needs to sum over states of intermediate $Q\bar{Q}$ pair, which are usually expressed as spectroscopic notation ${{^{{2S+1}}\hspace{-0.6mm}L_{J}^{[c]}}}$ with $c=1~\text{or}~8$ to denote color singlet or color octet. For the gluon fragmentation to $1^{--}$ quarkonium, like the ${{J/\psi}}$, $\psi(2S)$ and $\Upsilon(nS)$, the dominant contribution comes from ${{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}$ intermediate state according to the velocity scaling rule [@Bodwin:1994jh].
In the SGF framework, calculation of fragmentation function from gluon to ${{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}$ intermediate state is still absent. In the NRQCD framework, SDCs of gluon fragmenting into polarization summed ${{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}$ intermediate state has been calculated numerically in Refs. [@Braaten:1993rw; @Braaten:1995cj] for $v^0$ contribution and in Ref. [@Bodwin:2003wh] for $v^2$ correction. However, analytical results are still absent because of the complexity of the problem. One reason is that there are two gluons emitted in the final state, so the phase space integral is similar to two-loops integral. The other reason is that light-cone momentum is involved in the definition of fragmentation function, which makes the phase space integral more complicated than usual. In Refs. [@Braaten:1993rw; @Braaten:1995cj], there is a four-dimensional integral left for numerical computing. In Ref. [@Bodwin:2003wh], the authors make some transformation of the variables and analytically integrate out two more dimensions, but there is still a two-dimensional integral that has to be calculated numerically. Besides, there is no calculation of polarized SDCs based on the definition of fragmentation function, of which the result is useful for understanding the polarization puzzle [@Abulencia:2007us; @Butenschoen:2012px; @Chao:2012iv; @Gong:2012ug; @Bodwin:2014gia].
In this paper, we analytically calculate SDCs of gluon fragmenting into ${{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}$ state separately in SGF and NRQCD frameworks. We include both $v^0$ contribution and $v^2$ contribution in nonrelativistic expansion. In all cases, we provide transversely polarized SDCs in addition to polarization summed SDCs, while longitudinally polarized SDCs can be obtained by their difference.
The rest of the paper is organized as follows. In Sec. \[sec:def\], we first introduce the definition of gluon FF, and then describe how to apply SGF and NRQCD to calculate the FF in detail. The resulting expressions are complicated phase space integrals. In Sec. \[sec:cal\], we use integration-by-part (IBP) method [@Chetyrkin:1981qh; @Smirnov:2012gma; @Smirnov:2014hma; @Lee:2013mka] to express these phase space integrals in terms of some bases, which are called master integrals. We then calculate these master integrals. Almost all master integrals can be easily calculated except one, which we calculate by constructing and solving a differential equation with a trivial initial condition. Then we exhibit the analytical results and the large $z$ behaviour of the FFs. Finally, we present numerical results and a discussion in Sec. \[sec:summary\]. Some coefficients of analytical results calculated in this paper are given in the Appendix.
Factorization of Quarkonium Fragmention Functions {#sec:def}
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Definition of fragmentation functions
-------------------------------------
In this paper, we use light-cone coordinates where a four-vector $V$ can be expressed as $$\begin{aligned}
\begin{split}
V & = (V^+,V^-,\boldsymbol{V}_{\perp})=(V^+,V^-,V^1,V^2) \, , \\
V^+ & = (V^0+V^3)/\sqrt{2} \, , \\
V^- & = (V^0-V^3)/\sqrt{2} \, .
\end{split}\end{aligned}$$ The scalar product of two four-vector $V$ and $W$ then becomes $$V \cdot W = V^+ W^- + V^- W^+ - \boldsymbol{V}_{\perp} \cdot \boldsymbol{W}_{\perp} \, .$$ We introduce a light-like vector $n=(0,1,\boldsymbol{0}_{\perp})$, so that $n\cdot V=V^+$.
The Collins-Soper definition of FF for a gluon fragmenting into a hadron (quarkonium) is given by [@Collins:1981uw] $$\begin{aligned}
\label{eq:defFF}
\begin{split}
D_{g \rightarrow H}(z,\mu_0)=
& \frac{-g_{\mu\nu}z^{D-3}}{2 \pi P_c^{+}(N_{c}^{2}-1)(D-2)} \int_{-\infty}^{+\infty}\mathrm{d}x^{-} e^{-i z P_c^{+} x^{-}} \\
& \times \langle 0 | G_{c}^{+\mu}(0) \mathcal{E}^{\dag}(0,0,\boldsymbol{0}_{\perp})_{cb} \mathcal{P}_{H(P_H)} \mathcal{E}(0,x^{-},\boldsymbol{0}_{\perp})_{ba} G_{a}^{+\nu}(0,x^{-},\boldsymbol{0}_{\perp}) | 0 \rangle \, ,
\end{split}\end{aligned}$$ where $G^{\mu\nu}$ is the gluon field-strength operator, $P_H$ and $P_c$ are respectively the momenta of the fragmenting hadron and initial virtual gluon, and $z$ is the “$+$” momentum fraction of the initial virtual gluon carried by the hadron. It is convenient to choose the frame so that the hadron has zero transverse momentum, $P_H = (z P_c^+, M_H^2/(2 z P_c^+),\boldsymbol{0}_{\perp})$, where $M_H$ is the mass of the hadron. The projection operator $\mathcal{P}_{H(P_H)}$ is given by $$\label{eq:projectH}
\mathcal{P}_{H(P_H)} = \sum_X |H(P_H)+X \rangle \langle H(P_H)+X|\,,$$ where $X$ sums over all unobserved particles. The gauge link $\mathcal{E}(x^{-})$ is an eikonal operator that involves a path-ordered exponential of gluon field operators along a light-like path, $$\mathcal{E}(0,x^{-},\boldsymbol{0}_{\perp})_{ba}= \mathrm{P} \, \text{exp} \left[+i g_s \int_{x^{-}}^{\infty}\mathrm{d}z^{-} A^{+}(0,z^{-},\boldsymbol{0}_{\perp}) \right]_{ba} \, ,$$ where $g_s=\sqrt{4 \pi \alpha_s}$ is the QCD coupling constant and $A^{\mu}(x)$ is the matrix-valued gluon field in the adjoint representation: $[A^{\mu}(x)]_{ac} = i f^{abc} A^{\mu}_{b}(x)$. In the light-cone gauge $A^+=A\cdot n=0$, the gauge link $\mathcal{E}(0,x^{-},\boldsymbol{0}_{\perp})$ becomes 1 and thus it does not show up in the Feynman diagrams. In fact, for the problem studied in this paper, the gauge link has no contribution even if we work in Feynman gauge.
Applying SGF to fragmentation functions
---------------------------------------
The one-dimensional SGF for gluon fragmenting to quarkonium $H$ is given by [@Ma:2017xno] $$\label{eq:SGF}
D_{g\to H}(z,\mu_0) = \sum_n\int dr\, d_{n} (z/r,M_H/r,m_Q,\mu_0)\,r F_n^H(r) \, ,$$ where $n={{^{{2S+1}}\hspace{-0.6mm}L_{J}^{[c]}}}$ is in spectroscopic notation to denote quantum numbers of the intermediate $Q\bar{Q}$ pair, $d_{n} (z/r,M_H/r,m_Q,\mu_0)$ are SDCs to produce a $Q\bar{Q}$ pair with invariant mass $M_H/r$ and quantum number $n$, $F_n^H(r)$ are one-dimensional soft gluon distributions (SGDs) defined by four-dimensional SGDs $$\begin{aligned}
F_{n}^H(r)=\int \frac{d^4P}{(2\pi)^4} \delta(r-\sqrt{P_H^2/P^2})\, F^H_{n}(P,P_H),\end{aligned}$$ and four-dimensional SGDs are defined as expectation values of bilocal operators in QCD vacuum, $$\begin{aligned}
\label{eq:SGDs}
\begin{split}
&F^H_{n}(P,P_H)=\int d^4x\, e^{iP\cdot x}\, \langle 0| \bar\psi(0)\Gamma_{n}^\prime \mathcal{E}^\dagger(0)\psi(0) \mathcal{P}_{H(P_H)} \bar\psi(x) \Gamma_{n}\mathcal{E}(x)\psi(x) |0\rangle,
\end{split}\end{aligned}$$ where $\Gamma$ and $\Gamma^\prime$ are color and angular momentum projection operators that define $n$, $\mathcal{E}(x)$ are gauge links that enable gauge invariance [@machao; @Ma:2017xno].
![One typical diagram for the gluon fragmenting into ${{^{3}\hspace{-0.6mm}S_{1}^{[1]}}} \ Q\bar{Q}$ pair in the light-cone gauge at LO order in $\alpha_s$. The other diagrams are obtained by permutation. \[fig:FeynmanDiagram\]](FeynmanDiagram.eps){width="40.00000%"}
As mentioned in the introduction, for gluon fragmenting to $1^{--}$ quarkonium in this paper we keep only $n={{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}$ intermediate $Q\bar{Q}$ state, we thus suppress the subscript $n$ in the rest of this paper. Then the lowest order in $\alpha_s$ expansion of $d (z/r,M_H/r,m_Q,\mu_0)$ is described by Feynman diagrams of a virtual gluon decaying to a $Q\bar{Q}$ pair with quantum number ${{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}$ combined with two more gluons, as shown in Fig. \[fig:FeynmanDiagram\] , which can be formally defined as the lowest order in $\alpha_s$ expansion of the following matrix element, $$\begin{aligned}
\label{eq:defsdc}
\begin{split}
& d^{\infty}(z,M,m_Q,\mu_0)= \frac{-g_{\mu\nu}z^{D-3}}{2 \pi P_c^{+}(N_{c}^{2}-1)(D-2)} \int_{-\infty}^{+\infty}\mathrm{d}x^{-} e^{-i z P_c^{+} x^{-}} \\
& \times \langle 0 | G_{c}^{+\mu}(0) \mathcal{E}^{\dag}(0,0,\boldsymbol{0}_{\perp})_{cb} |Q\bar{Q}({{^{3}\hspace{-0.6mm}S_{1}^{[1]}}})+g+g \rangle \langle Q\bar{Q}({{^{3}\hspace{-0.6mm}S_{1}^{[1]}}})+g+g| \mathcal{E}(0,x^{-},\boldsymbol{0}_{\perp})_{ba} G_{a}^{+\nu}(0,x^{-},\boldsymbol{0}_{\perp}) | 0 \rangle \, ,
\end{split}\end{aligned}$$ where $M$ is the invariant mass of the $Q\bar{Q}$ pair.
In the momentum space, we have the lowest order in $\alpha_s$ expansion $$\label{eq:sdcdef0}
d(z,M,m_Q,\mu_0) = \frac{N_{\mathrm{CS}}}{D-1} \int \mathrm{d} \Phi \, \left| \mathcal{M}(P, k_i, m_Q) \right|^2 \, ,$$ where $N_{\mathrm{CS}}=\frac{z^{D-2}}{(N_{c}^{2}-1)(D-2)} $, $D=4$ is the space-time dimension, $P$ is the total momentum of the $Q\bar{Q}$ pair, $k_i$ ($i=1,2$) is the momentum of the $i$th final-state gluon, and final-state phase space is defined as $$\begin{aligned}
\label{eq:phase}
\begin{split}
\mathrm{d} \Phi &=
\frac {1}{2!}
\delta \left( z - \frac{P^{+}}{P_{c}^{+}} \right)
(2\pi)^{D} \delta^{D} \left(
P_{c} - P - k_1-k_2 \right)
\frac{\mathrm{d}^{D} P_{c}}{(2\pi)^{D}}
\prod_{i=1}^{2} \frac{\mathrm{d} k_{i}^{+}}{4\pi k_{i}^{+}}
\frac{\mathrm{d}^{D-2} k_{i\perp}}{(2\pi)^{D-2}}
\theta(k_{i}^{+}) \\
&=
\frac {P^{+}}{z^{2} 2!}
\delta \left( \frac{1-z}{z} P^{+} - k_{1}^{+} - k_{2}^{+}\right)
\prod_{i=1}^{2} \frac{\mathrm{d} k_{i}^{+}}{4\pi k_{i}^{+}}
\frac{\mathrm{d}^{D-2} k_{i\perp}}{(2\pi)^{D-2}}
\theta(k_{i}^{+}) \,
\end{split}\end{aligned}$$ where $2!$ is the symmetry factor for identical gluons in the final state. The matrix elements $\mathcal{M}({P},{k}_{i},m_Q)$ are defined as $$\begin{aligned}
\left| \mathcal{M}({P},{k}_{i},m_Q) \right|^2=\sum_{\lambda \lambda_1 \lambda_2 \lambda_3} \left| \mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}(P,k_{i},m_Q) \right|^2,\end{aligned}$$ where $\lambda$ and $\lambda_i$ ($i=1, 2, 3$) are polarizations of the $Q\bar{Q}$ pair and gluons, respectively, and $\mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}(P,k_{i},m_Q)$ is the amplitude to produce a $Q\bar{Q}$ pair with momentum $P$ and quantum numbers ${{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}$. Summation over all polarizations of the heavy quark pair with momentum $P$ gives $$\begin{aligned}
\label{eq:polarsum}
I^{\alpha\beta}(P)=\sum_{\lambda=0,\pm 1} \epsilon_\lambda^\alpha& \epsilon_\lambda^{*\beta} = -g^{\alpha \beta}
+ \frac{P^{\alpha} P^{\beta}} {P^{2}} \, , \end{aligned}$$ and summation over all polarizations of gluons or summation over transverse polarizations of the heavy quark pair with momentum $k$ gives $$\begin{aligned}
\label{eq:polarsumT}
\sum_{\lambda_i=\pm 1} \epsilon_{\lambda_i}^\mu& \epsilon_{\lambda_i}^{*\nu} = -g^{\mu \nu}
+ \frac{k^{\mu} n^{\nu} + k^{\nu} n^{\mu}} {k^{+}}
- \frac{k^2 n^{\mu} n^{\nu}} {(k^{+})^{2}} \, .\end{aligned}$$ In the above, polarizations are quantized along the vector $n$.
The amplitude to produce a ${{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}$ state can be obtained by $$\begin{aligned}
\label{eq:amplitude}
\mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}(P,k_{i},m_Q)=\int {\mathrm{d}}^2 \Omega\, \text{Tr}\left[ \Gamma_\lambda \mathcal{M}_{\lambda_1 \lambda_2 \lambda_3}(P,k_{i},q,m_Q) \right]\, ,\end{aligned}$$ where $\mathcal{M}_{\lambda_1 \lambda_2 \lambda_3}(P,k_{i},q,m_Q)$ is the amplitude to produce an open $Q\bar{Q}$ pair, with momenta $p=P/2+q$ for $Q$ and $\overline{p}=P/2-q$ for $\bar Q$, and $\Gamma_\lambda $ is used to project the $Q\bar{Q}$ pair to ${{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}$ state, with definition $$\Gamma_{\lambda} = \frac{1} {\sqrt{N_{c}}}
\frac{1} {\sqrt{2 E} (E + m_Q)}
(\slashed{\overline{p}} - m_Q)
\frac{2 E - \slashed{P}}{4 E}
\slashed{\epsilon}_\lambda
\frac{2 E + \slashed{P}}{4 E}
(\slashed{p} - m_Q) \, ,$$ where $\epsilon_\lambda^\mu$ are polarization vectors. As both $p$ and $\overline{p}$ are approximated to be on mass shell [@machao; @Ma:2017xno], we have $$\begin{aligned}
\label{eq:relation}
P\cdot q =0, \quad \quad M^2=P^2=4E^2=4(m_Q^2-q^2).\end{aligned}$$As a result, the four-momentum $q$ has only two degrees of freedom, which are chosen to be the two-dimensional spatial angles $\Omega$ at the rest frame of $P$. After integration over spatial angles, the obtained $\mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}(P,k_{i},m_Q)$ in Eq. has no dependence on $q$ any more.
Note that the dominant contribution of Eq. comes from the region where $1\gg1-r^2=1-M_H^2/M^2\approx1-4m_Q^2/M^2=-4q^2/M^2$ [@machao; @Ma:2017xno], thus we can simplify SDCs by expanding $q^2/M^2$ . In the SGF framework, this expansion is obtained by first expressing $m_Q^2={M^2/4+q^2}$, and then fixing $M$ but expanding $q^2$ at the origin [^2]. Because neither phase space integration nor polarization vectors depend on $m_Q$ and $q$, the above expansion can be achieved by a similar expansion of the amplitude in Eq. , $$\label{eq:amplitudeExp}
\mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}(P,k_{i},m_Q) =
\mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}^{(0)}(P,k_{i})
-q^2 \mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}^{(2)}(P,k_{i})+O(q^4)\, ,$$ with $$\begin{aligned}
\label{eq:amplitudeExp2}
\begin{split}
\mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}^{(0)}(P,k_{i})
& = \text{Tr}\left[ \Gamma_\lambda \mathcal{M}_{\lambda_1 \lambda_2 \lambda_3}(P,k_{i},0,M/2) \right] \, ,
\\
\mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}^{(2)}(P,k_{i})
& = \frac{I^{\mu\nu}(P)}{2(D-1)}
\left\{\frac{\partial^2 }{\partial q^\mu \partial q^\nu}
\text{Tr}\left[ \Gamma_\lambda \mathcal{M}_{\lambda_1 \lambda_2 \lambda_3}\left(P,k_{i},q,\sqrt{\frac{M^2}{4}+q^2}\right)
\right]\Bigg|_{q=0}\right\}\, ,
\end{split}\end{aligned}$$ where $I^{\mu\nu}(P)$ is defined in Eq. . With this expansion, SDCs have expansion $$\begin{aligned}
\label{eq:sdcExp}
d(z,M,m_Q,\mu_0) =& \frac{N_{\mathrm{CS}}}{D-1} \int \mathrm{d} \Phi \, \sum_{\lambda \lambda_1 \lambda_2 \lambda_3} \left| \mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}^{(0)}(P,k_{i}) \right|^2 - q^2 \left[ \mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}^{(0)}(P,k_{i}) \mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}^{*(2)}(P,k_{i})+ c.c.\right] + O(q^4) \notag\\
\equiv& d^{(0)}(z, M,\mu_0 ) - \frac{4 q^2}{M^2} d^{(2)}(z, M ,\mu_0) + O(q^4).\end{aligned}$$
Similarly, if we sum over only transerve polarizations of the $Q\bar{Q}$ pair by using the projection operator in Eq. instead of that in Eq. , we can obtain transversely polarized SDCs $$\begin{aligned}
\label{eq:sdcTExp}
d_T(z,M,m_Q,\mu_0) = d_T^{(0)}(z, M,\mu_0 ) - \frac{4 q^2}{M^2} d_T^{(2)}(z, M ,\mu_0) + O(q^4).\end{aligned}$$ Longitudinal polarized SDCs can be obtained by subtracting out transversely polarized SDCs from corresponding polarization-summed SDCs.
Applying NRQCD to fragmentation functions
-----------------------------------------
While if applying the NRQCD, we get $$D_{g\to H}(z,\mu_0) = \sum_n d_n^O (z,2m_Q,\mu_0) {{\langle{{\mathcal O}}^{H}_n\rangle}}+ d_n^{P} (z,2m_Q,\mu_0) {{\langle{{\mathcal P}}^{H}_n\rangle}}+\cdots\, ,$$ where $d_{n}^{O,P} (z,2m_Q,\mu_0)$ are SDCs to produce a $Q\bar{Q}$ pair with invariant mass $2m_Q$ and quantum numbers $n$, and ${{\langle{{\mathcal O}}^{H}_n\rangle}}$ and ${{\langle{{\mathcal P}}^{H}_n\rangle}}$ are respectively NRQCD long-distance matrix elements (LDMEs) at first and second order in $v^2$ expansion [@Bodwin:1994jh], which can be expressed as the vacuum expectation value of a four-fermion operator in NRQCD vacuum $$\begin{aligned}
{{\langle{{\mathcal O}}^{H}_n\rangle}}&= \langle 0 | \chi ^{\dag} \kappa _n \psi \mathcal{P}_{H(P)} \psi ^{\dag} \kappa'_n \chi |0 \rangle \, ,\\
{{\langle{{\mathcal P}}^{H}_n\rangle}}&= \langle 0 |\frac{1}{2}\left[ \chi ^{\dag} \kappa _n \psi \mathcal{P}_{H(P)} \psi ^{\dag} \kappa'_n (-\frac{i}{2} \overleftrightarrow{\mathbf{D}})^2 \chi + h.c. \right]|0 \rangle \, ,\end{aligned}$$ where $\psi ^{\dag}$ and $\chi$ are the two-component operators to creat a heavy quark and a heavy antiquark, respectively, and $\kappa _n$ and $\kappa'_n$ are combinations of Pauli and color matrices. These LDMEs are defined in the rest frame of $H$ and expected to be universal. If the hadron $H$ is the free $Q \bar{Q}$ pair, we have ${{\langle{{\mathcal P}}^{H}_n\rangle}}=(-q^2/m_Q^2) {{\langle{{\mathcal O}}^{H}_n\rangle}}$. As mentioned above, we only consider $n={{{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}}$ intermediate state, and thus will drop the subscript $n$ in the following.
The calculations of $d^O$ and $d^P$ in NRQCD are very similar to the calculation of $d^{(0)}$ and $d^{(2)}$ in SGF defined in Eq. . The only difference is that, in the NRQCD, one expands $q^2$ with fixed $m_Q$ but not $M$, which implies that phase space also needs to be expanded. For this purpose, we first extract the dependence on $q$ explicitly by rescaling momenta in the delta function in Eq. by $M$ as following, $$\label{eq:dless}
\hat{P} = \frac {P}{M} \, , \quad
\hat{k}_i = \frac {k_i}{M} \, .$$ Thus the phase space in Eq. changes to $${\mathrm{d}}\Phi = M^{4} {\mathrm{d}}\hat{\Phi} \, ,$$ where ${\mathrm{d}}\hat{\Phi}$ is the same as ${\mathrm{d}}\Phi$ except that momenta in it have been changed to the dimensionless ones, and therefore it has no dependence on $q$. If we further denote $$\hat{\mathcal{M}}_{\lambda \lambda_1 \lambda_2 \lambda_3}(\hat{P},\hat{k}_{i},m_Q) = M^2 \mathcal{M}_{\lambda \lambda_1 \lambda_2 \lambda_3}(M \hat{P}, M \hat{k}_{i},m_Q) \, ,$$ we get a similar relation as that in Eq. , $$d(z,M,m_Q,\mu_0) = \frac{N_{\mathrm{CS}}}{D-1} \int \mathrm{d} \hat{\Phi} \, \left| \hat{\mathcal{M}}(\hat{P},\hat{k}_i, m_Q) \right|^2 \,.$$ Then the expansion of amplitude $\hat{\mathcal{M}}_{\lambda \lambda_1 \lambda_2 \lambda_3}(\hat{P},\hat{k}_{i},m_Q)$ can be achieved similarly as that in Eq. and Eq. , except that we express $M^2=4(m_Q^2-q^2)$ and fix $m_Q$. Eventually, we get $$\begin{aligned}
\label{eq:NRsdcExp}
\begin{split}
d(z,M,m_Q,\mu_0) = & \frac{N_{\mathrm{CS}} }{D-1} \int \mathrm{d} \hat\Phi\, \sum_{\lambda \lambda_1 \lambda_2 \lambda_3}
\left| \mathcal{\hat{M}}_{\lambda \lambda_1 \lambda_2 \lambda_3}^{(0)}(\hat{P},\hat{k}_{i},m_Q) \right|^2 \\
& \phantom{\frac{N_{\mathrm{CS}} }{D-1} \int \mathrm{d} \hat\Phi\, \sum_{\lambda \lambda_1 \lambda_2 \lambda_3} }
- q^2 \left[ \mathcal{\hat{M}}_{\lambda \lambda_1 \lambda_2 \lambda_3}^{(0)}(\hat{P},\hat{k}_{i},m_Q)
\mathcal{\hat{M}}_{\lambda \lambda_1 \lambda_2 \lambda_3}^{*(2)}(\hat{P},\hat{k}_{i},m_Q)+ c.c.\right] + O(q^4) \\
\equiv& d^{O}(z, 2m_Q,\mu_0 ) -\frac{q^2}{m_Q^2} d^{P}(z, 2m_Q ,\mu_0) + O(q^4) \, .
\end{split}\end{aligned}$$ Clearly, we have the relation $$\begin{aligned}
\label{eq:relationU}
d^{O}(z, M,\mu_0 )=d^{(0)}(z,M,\mu_0),\end{aligned}$$ but $d^{P}(z, M,\mu_0 )$ is different from $d^{(2)}(z,M,\mu_0)$.
Again, we can obtain transversely polarized SDCs $$\begin{aligned}
\label{eq:NRsdcTExp}
d_T(z,M,m_Q,\mu_0) = d_T^{O}(z, 2m_Q,\mu_0 ) -\frac{q^2}{m_Q^2} d_T^{P}(z, 2m_Q ,\mu_0) + O(q^4),\end{aligned}$$ where we also have $$\begin{aligned}
\label{eq:relationT}
d_T^{O}(z, M,\mu_0 )=d_T^{(0)}(z,M,\mu_0).\end{aligned}$$
Calculation of the short-distance coefficient {#sec:cal}
=============================================
For the process of gluon fragmenting to spin-triplet color-singlet S-wave quarkonium at LO in $\alpha_s$, there are two soft gluons in the final state as shown in Fig. \[fig:FeynmanDiagram\]. We denote the “$+$” component of the first gluon as $k_1^+ = (1-z) z_1 P_c^+ $, then for the second gluon we have $k_2^+ = (1-z)(1-z_1) P_c^+ $. To simplify our notation, we will use only dimensionless momenta defined in Eq. but omit the superscript “ $\hat{}$ ” in the rest of this paper.
According to Sec. \[sec:def\], calculation of SDCs can be decomposed into the sum of a series of integrals with the form $$\label{eq:iniff}
\int \mathrm{d} \Phi
f(z,z_{1})
\frac {(k_1 \cdot k_2)^{n_{5}}}{E_1^{n_{1}} E_2^{n_{2}} E_3^{n_{3}} E_4^{n_{4}}} \, ,$$ where $n_i>0(i=1,2,3,4,5)$, $f(z,z_{1})$ is fractional polynomials with respect to $z$ and $z_1$, and $$\label{eq:defdeno1}
E_1 = k_1 \cdot P \, , \
E_2 = k_2 \cdot P \, , \
E_3 = 2 k_1 \cdot k_2 + k_1 \cdot P + k_2 \cdot P \, , \
E_4 = 1 + 2 k_1 \cdot k_2 + 2 k_1 \cdot P + 2 k_2 \cdot P \, .$$ We note that $z_1$ does not appear in the denominators, which is because, as we pointed out, the gauge link in the definition of FFs has no contribution in our case.
Reduction to Master Integrals
-----------------------------
Calculating general integrations in Eq. analytically is not an easy task, and only numerical results are available in literature [@Braaten:1993rw; @Braaten:1995cj; @Bodwin:2003wh]. To perform them analytically, we employ the IBP reduction method [@Chetyrkin:1981qh; @Smirnov:2012gma; @Smirnov:2014hma; @Lee:2013mka] that are widely used for high loops calculation. Especially, we will use the program FIRE [@Smirnov:2014hma]. Feynman integrals that can be reduced by FIRE can be generally expressed as $$\label{eq:defhloop}
F(a_{1}, \ldots ,a_{n}) =
\idotsint \,
\frac{{\mathrm{d}}^{D} l_{1} \ldots {\mathrm{d}}^{D} l_{h}}
{D_{1}^{a_{1}} \ldots D_{n}^{a_{n}}} \, ,$$ where $a_i\,(i=1, \ldots, n)$ are integers that can be either positive or negative, and denominators $D_i \,( i=1, \ldots, n)$ are linear functions with respect to scalar products of loop momenta $l_{i}\, (i=1,\ldots,h)$ and external mementa. The program FIRE, by employing IBP, can reduce these complex integrals into limited number of simpler integrals which are called master integrals. Nevertheless, integrations in Eq. are not directly handleable by FIRE because there are delta functions in the phase space, which becomes more clearly if we rewrite the phase space as $$\label{eq:phase3}
\mathrm{d} \Phi =
\frac{\mathrm{d}^{D} k_1}{(2\pi)^{D}}
\frac{\mathrm{d}^{D} k_2}{(2\pi)^{D}}
\frac{P \cdot n}{z^{2} 2!}
\delta _+ (k_1^{2})
\delta _+ (k_2^{2})
\delta \left(
k_1 \cdot n + k_2 \cdot n - \frac{1-z}{z} P \cdot n
\right) \, ,$$ where subscript “$+$” of a delta function means that energy of the momentum inside the delta function is positive. To make delta functions handleable by FIRE, we rewrite a delta function as $$\delta (x) =
\frac{i}{2 \pi}
\lim_{\varepsilon \rightarrow 0}
\left(
\frac{1}{x + i \varepsilon}
- \frac{1}{x - i \varepsilon}
\right) \, ,$$ which changes the delta function to a propagator denominator. We can further identify $z_1=z \, k_1 \cdot n / (1-z) P \cdot n$, and choose the following notations $$\label{eq:defdeno2}
E_5 = k_1^2 + i \varepsilon \, , \
E_6 = k_2^2 + i \varepsilon \, , \
E_7 = k_1 \cdot n + k_2 \cdot n - \frac{1-z}{z} P \cdot n + i \varepsilon \, , \
E_8 = k_1 \cdot n \, ,$$ then integrals in Eq. are casted to $$\label{eq:intfire}
\int
\frac{\mathrm{d}^{D} k_1}{(2\pi)^{D}}
\frac{\mathrm{d}^{D} k_2}{(2\pi)^{D}}
f(z)
\frac{P \cdot n}{z^{2} 2!}
\left ( \frac{z}{(1-z) P \cdot n} \right)^{n_{6}}
\left ( \frac{i}{2 \pi} \right )^{3}
\frac{(k_1 \cdot k_2)^{n_{5}} E_8^{n_{6}}}{E_1^{n_{1}} E_2^{n_{2}} E_3^{n_{3}} E_4^{n_{4}} E_5 E_6 E_7 } \, ,$$ together with 7 other kinds of integrals with similar form except that some of small imaginary parts of $E_5,\, E_6,\, E_7$ change from “ $+i\varepsilon$ ” to “ $-i\varepsilon$ ”. Since IBP reduction is independent of the small imaginary part, these 8 kinds of integrals have similar reduced results. Therefore, after reduction, we can change $E_5,\, E_6,\, E_7$ back to corresponding delta functions, and thus we can obtain master integrals of Eq. . One important point is that any master integral with non-positive power of $E_5,\, E_6,\, E_7$ must be canceled by other master integrals reduced by the other 7 kinds of integrals. Combining with the fact that powers of $E_5,\, E_6,\, E_7$ can be always chosen to no larger than 1, our obtained master integrals have the same phase space integration as that in Eq. .
The denominators $E_1, \cdots, E_7$ in Eq. are linearly dependent, which can be easily changed to be linearly independent with the same integrations structure. We further add a denominator $E_8$ to some integrals to make them complete. After reduction by applying FIRE [@Smirnov:2014hma], SDC, say $d^{(0)}$, becomes $$\label{eq:sdcmi}
d^{(0)} (z,M,\mu_0) = \sum_{a=1}^{13} f_{a} (z, \epsilon)I_{a} \, ,$$ where coefficients $f_{a}$ are fractional polynomials in terms of $z$, which can be expanded in powers of $\epsilon$, and master integrals $I_{a}$ can be defined as $$\label{eq:defmi}
I_{a} = \int {\mathrm{d}}\Phi \, F_{a}
= \frac{1}{(4 \pi)^2 z (1 -z ) 2!} \int_{0}^{1} \frac{\mathrm{d} z_{1}}{z_{1} (1 - z_{1})}
\iint \frac{\mathrm{d}^{D-2} k_{1\perp}}{(2\pi)^{D-2}} \frac{\mathrm{d}^{D-2} k_{2\perp}}{(2\pi)^{D-2}}
F_{a} \, ,$$ with $F_{a} (a=1,\ldots,13) $ choosing from $$\label{eq:allmi}
\frac{1}{E_3} \, , \
\frac{1}{E_4} \, , \
\frac{1}{E_1 E_2} \, , \
\frac{1}{E_1 E_3} \, , \
\frac{1}{E_1 E_4} \, , \
\frac{E_2}{E_1 E_3} \, , \
\frac{E_4}{E_1 E_3} \, , \
\frac{E_2}{E_1 E_4} \, , \
\frac{E_3}{E_1 E_4} \, , \
\frac{1}{E_1 E_3^{2}} \, , \
\frac{1}{E_1^{2} E_4} \, , \
\frac{1}{E_3 E_4} \, , \
\frac{1}{E_1 E_2 E_4} \, ,$$ where $E_i (i=1,\ldots,4)$ are defined in Eq. .
Calculation of Master Integrals
-------------------------------
Calculation of SDCs is now reduced to calculation of the thirteen master integrals defined in Eq. . Among them, each of the first 11 master integrals involves only one denominator that has cross term $k_1\cdot k_2$. In this case, the cross term can be removed by shifting $k_2$, and then we can integrate over $k_{2\perp}$, $k_{1\perp}$ and $z_1$ sequentially. For the 12th master integral, as both of its denominators depend on $k_1\cdot k_2$, we can first do a Feynman parametrization, and then integrate over $k_{2\perp}$, $k_{1\perp}$, Feynman parameter, and $z_1$ sequentially. Although they are easy to calculate, expressions of the first 12 master integrals are quite long, we will not list them in this paper. The most complicated master integral is the last one, which is hard to integrate directly. We will find other way to get the analytical result. In this section, at first we discuss some difficulties encountered in the calculation of the first 12 master integrals, and then concentrate on calculating the last master integral.
For the 4th to 10th master integrals, after integrating over $k_{2\perp}$, there is still a term proportional to $$\label{eq:kuvdiv}
\int \frac {{\mathrm{d}}^{D-2} k_{1\perp}}{(2 \pi)^{D-2}}
\frac{1}{(k_{1\perp}^{2}+a) (k_{1\perp}^{2}+b)^{\epsilon}} \, ,$$ where $a$ and $b$ are both nonnegative functions of $z$ and $z_1$. This integral on the one hand is cumbersome to expand $\epsilon$ after the integration, and on the other hand is ultraviolet divergent and thus cannot expand $\epsilon$ at the integrand level. We rewrite Eq. as $$\int \frac {{\mathrm{d}}^{D-2} k_{1\perp}}{(2 \pi)^{D-2}}
\frac{b-a}{(k_{1\perp}^{2}+a) (k_{1\perp}^{2}+b)^{1+\epsilon}}
+ \int \frac {{\mathrm{d}}^{D-2} k_{1\perp}}{(2 \pi)^{D-2}}
\frac{1}{(k_{1\perp}^{2}+b)^{1+\epsilon}} \, ,$$ where the second term can be integrated and then expand $\epsilon$ easily, while the first term is ultraviolet finite and thus can expand $\epsilon$ at the integrand level. For the first term, we need to expand to second order in $\epsilon$ and thus results in one-dimensional integrals $$\int_{0}^{\infty} {\mathrm{d}}x
\frac{b-a}{(x+a)(x+b)}
=
\ln b - \ln a \, ,$$ and $$\int_{0}^{\infty} {\mathrm{d}}x
\frac{(b-a) \left[ \ln x + \ln (x+b) \right]}
{(x+a)(x+b)}
=
\mathrm{Li}_2 \left( 1-\frac{a}{b} \right)
- \ln a \, \ln b
- \frac{1}{2} \ln^{2} a
+ \frac{3}{2} \ln^2 b \, .$$ Then we can integrate over $z_1$ easily. For the 11th master integral, after integrating over $k_{2\perp}$, the master integral is proportional to $$\begin{aligned}
& \int_0^1 {\mathrm{d}}z_1 \int \frac {{\mathrm{d}}^{D-2} k_{1\perp}}{(2 \pi)^{D-2}}
\frac{z_1^{1+\epsilon} (1-z_1)^{-\epsilon} (1+a z_1)^{-1+2\epsilon} }{(k_{1\perp}^{2}+a^2 z_1^2)^2 (k_{1\perp}^{2}+a^2 z_1^2+a z_1)^{\epsilon}} \, ,\end{aligned}$$ with $a=(1-z)/z$, which is infrared divergent when integrating over $z_1$ near the region $z_1=0$. Thus one cannot expand $\epsilon$ at the integrand level. Yet, we can re-scale $k_{1\perp}$ by a factor of $z_1$, and we get $$\label{eq:irint}
\int_0^1 {\mathrm{d}}z_1 \, z_1^{-1-2\epsilon} \int \frac {{\mathrm{d}}^{D-2} k_{1\perp}}{(2 \pi)^{D-2}}
\frac{(1-z_1)^{-\epsilon} (1+a z_1)^{-1+2\epsilon} }{(k_{1\perp}^{2}+a^2)^2 (z_1 k_{1\perp}^{2}+a^2 z_1+a)^{\epsilon}}\, ,$$ which although is still infrared divergent, but we can expand the integrand other than $z_1^{-1-2\epsilon}$ as a power series of $\epsilon$.
Now let’s concentrate on the last master integral $$\begin{aligned}
\int {\mathrm{d}}\Phi \frac{1}{E_1 E_2 E_4} ,\end{aligned}$$ which is hard to calculate using the traditional integration method with Feynman parametrization. Yet we can calculate it by constructing and soloving a differential equation [@Kotikov:1990kg; @Remiddi:1997ny; @Argeri:2007up; @Henn:2014qga]. We define $$g(z) = \int {\mathrm{d}}\Phi \frac{z^2}{E_1 E_2 E_4}
= \int
\frac{\mathrm{d}^{D} k}{(2\pi)^{D}}
\frac{\mathrm{d}^{D} l}{(2\pi)^{D}}
\left ( \frac{i}{2 \pi} \right )^{3}
\frac{P \cdot n}{S}
\frac{1}{E_1 E_2 E_4 E_5 E_6 E_7}
+ \ldots \, ,$$ where we omit 7 other similar terms. Denominators of it do not contain $z$ except $E_7$. If we take the derivative of $g(z)$, it becomes $$\frac{{\mathrm{d}}g(z)}{{\mathrm{d}}z} =
\int
\frac{\mathrm{d}^{D} k}{(2\pi)^{D}}
\frac{\mathrm{d}^{D} l}{(2\pi)^{D}}
\left ( \frac{i}{2 \pi} \right )^{3}
\frac{- (P \cdot n)^2}{z^2 S}
\frac{1}{E_1 E_2 E_4 E_5 E_6 E_7^2 }
+ \ldots \, .$$ Then we can reduce the integrals again by using IBP and arrive at a differential equation about $g(z) $ $$\label{eq:dfe}
\frac{{\mathrm{d}}g(z)}{{\mathrm{d}}z} = \frac{2 (z-1) z }{2 z - 1} \epsilon \, g(z) + h(z) \, ,$$ where $h(z)$ is a linear combination of the first 12 master integrals, which gives $$h(z) = \frac{ (\ln z - \ln (1-z) )^2}{128 \pi ^4 (1-2 z)} \, .$$ It is easy to see that $g(z)$ has no divergence, and thus the term proportional to $\epsilon$ in Eq. can be safely omitted. Thus the differential equation can be solved by integrating $h(z)$ over $z$ combined with an initial value. A good choice of the initial value can be at $z=1$, where one gets $g(1) = 0$ as the integral over plus direction is suppressed. With this initial value, we eventually get $$\begin{aligned}
\label{eq:mi13}
\begin{autobreak}
I_{13} = - \frac{1}{128 \pi ^4 z^2}
\Bigg( \mathrm{Li}_3 \left( \frac{2 z-1}{z} \right)
+ \mathrm{Li}_3 \left( \frac{z}{z-1} \right)
+ \mathrm{Li}_3 \left( \frac{2 z - 1}{z-1} \right)
- \mathrm{Li}_2 (z) \ln \left( \frac{1-z}{z} \right)
+ \mathrm{Li}_2 \left( \frac{2 z-1}{z-1} \right) \ln \left( \frac{1-z}{z} \right)
+ \frac{\ln^3 \left( \frac{1-z}{z} \right)}{6}
- \frac{\ln z \, \ln (1-z) \, \ln \left( \frac{1-z}{z} \right)}{2}
- \zeta (3) \Bigg)
\end{autobreak}.\end{aligned}$$
Analytical results
------------------
Substituting analytical results for the thirteen master integrals into Eq. , we find all kinds of divergences are canceled, and finite result gives $$\label{eq:d0}
d^{(0)} (z,M,\mu_0) =
\frac{128 (N_c^2 - 4) \pi^3 \alpha_s^3}{3 N_c^2 M^3}
\left( C I_{13}
+\sum_{i=0}^{11} C_i \, L_i
\right) \, ,$$ where $I_{13}$ is given in Eq. , coefficients $C$ and $C_i(i=0,\ldots,11)$ are given in Eq. in the Appendix, and $L_i(i=0,\ldots,11)$ are defined as $$\begin{aligned}
\label{eq:d0logs}
\begin{split}
& L_0 = 1 \, , \
L_1 = \ln z \, , \
L_2 = \ln (1-z) \, , \
L_3 = \ln (2-z) \, , \
L_4 = \ln^2 z \, , \
L_5 = \ln^2 (1-z) \, , \
L_6 = \ln^2 (2-z) \, , \\
& L_7 = \ln z \, \ln (1-z) \, , \
L_8 = \ln z \, \ln (2-z) \, , \
L_9 = \li_2 (1-z) \, , \
L_{10} = \li_2 \left(\frac{z-1}{z-2}\right) \, , \
L_{11} = \li_2 \left(\frac{2 (z-1)}{z-2}\right) \, .
\end{split}\end{aligned}$$
For the transversely polarized SDC $d_T^{(0)} (z,M,\mu_0)$, we can express it similar to $d^{(0)} (z,M,\mu_0)$ in Eq. , but with different coefficients $C^T$ and $C^T_i(i=0,\ldots,11)$ given in Eq. . The relativistic correction SDC $d^{(2)}(z,M,\mu_0)$ and corresponding transverse polarized SDC $d_T^{(2)}(z,M,\mu_0)$ can also be expressed the same as that in Eq. with corresponding coefficients given in Eq. and Eq. .
As we note in Sec. \[sec:def\], LO SDCs $d^O(z,2m_Q,\mu_0)$ (similar for $d_T^O(z,2m_Q,\mu_0)$) in NRQCD factorization can be obtained by replacing $M$ in Eq. by $2m_Q$ and keeping other coefficients unchanged. Coefficients of the relativistic correction SDCs $d^P(z,2m_Q,\mu_0)$ and $d_T^P(z,2m_Q,\mu_0)$ are given in Eq. and Eq. .
Large z behaviour
-----------------
At hadron colliders, high $p_T$ quarkonium production is most sensitive to fragmentation function at large $z$ region. Thus we investigate SDCs obtained above at this region by expanding them around $z\to1$, and we get $$\begin{aligned}
\label{eq:largez}
\begin{autobreak}
d^{(0)} (z,M,\mu_0) = \frac{4 (N_c^2 - 4) \alpha_s^3}{3 \pi N_c^2 M^3}
\bigg((1-z) \big(-\ln (1-z)-3 \big)
+\frac{(1-z)^2}{18} \big(36 \ln ^2(1-z)
+18 \ln (1-z)
+4 \pi ^2
+93\big)
+O\big((1-z)^3\big) \bigg) \, ,
\end{autobreak}\notag\\
\begin{autobreak}
d_T^{(0)} (z,M,\mu_0) = \frac{4 (N_c^2 - 4) \alpha_s^3}{3 \pi N_c^2 M^3}
\bigg((1-z) \big(-\ln (1-z)-3 \big)
+\frac{(1-z)^2}{9} \big(18 \ln ^2(1-z)
+18 \ln (1-z)
+2 \pi ^2
+51\big)
+O\big((1-z)^3\big) \bigg) \, ,
\end{autobreak}\notag\\
\begin{autobreak}
d^{(2)} (z,M,\mu_0) = \frac{4 (N_c^2 - 4) \alpha_s^3}{3 \pi N_c^2 M^3}
\bigg(\frac{2}{135} \big(-15+22 \pi^2\big)
+\frac{1-z}{27} \big(-63 \ln ^2(1-z)
-81 \ln (1-z)
-239\big)
+O\big((1-z)^2\big) \bigg) \, ,
\end{autobreak}\notag\\
\begin{autobreak}
d_T^{(2)} (z,M,\mu_0) = \frac{4 (N_c^2 - 4) \alpha_s^3}{3 \pi N_c^2 M^3}
\bigg(\frac{2}{135} \big(-15+22 \pi ^2\big)
+\frac{1-z}{27} \big(-63 \ln ^2(1-z)
-81 \ln (1-z)
-257\big)
+O\big((1-z)^2\big) \bigg) \, ,
\end{autobreak}\notag\\
\begin{autobreak}
d^P (z,2m_Q,\mu_0) = \frac{4 (N_c^2 - 4) \alpha_s^3}{3 \pi N_c^2 (2m_Q)^3}
\bigg(\frac{2}{135} \big(-15+22 \pi^2\big)
+\frac{1-z}{54} \big(-126 \ln ^2(1-z)
-81 \ln (1-z)
-235\big)
+O\big((1-z)^2\big) \bigg) \, ,
\end{autobreak}\notag\\
\begin{autobreak}
d_T^P (z,2m_Q,\mu_0) = \frac{4 (N_c^2 - 4) \alpha_s^3}{3 \pi N_c^2 (2m_Q)^3}
\bigg(\frac{2}{135} \big(-15+22 \pi ^2\big)
+\frac{1-z}{54} \big(-126 \ln ^2(1-z)
-81 \ln (1-z)
-271\big)
+O\big((1-z)^2\big) \bigg) \, .
\end{autobreak}\end{aligned}$$ We find that, for all cases, polarization-summed SDC equals to transversely polarized SDC at lowest order in $1-z$ expansion, while there are differences at higher orders. Thus longitudinal polarized SDCs are negligible at large $z$ region. The physical reason is very simple. As the two final state gluons are very soft when $z\to1$, heavy quark spin symmetry ensures that soft gluon emission will not change the spin of heavy quark. Therefore, the finial state heavy quark pair has almost the same polarization as that of the fragmenting gluon, which is transversely polarized. The consequence is that, for high $p_T$ quarkonium production, contributions from gluon fragmentating to ${{{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}}$ channel are transversely polarized, for both SGF and NRQCD factorization.
Another information from Eq. is that, at large $z$ region, relativistic correction terms are much larger than corresponding lowest order terms. This is because nonrelativistic expansion enhances the power of heavy quark propagator denominators, which vanish as $z\to1$. If fact, there are even infrared divergences if one expands to $O(v^4)$ terms [@Bodwin:2012xc], and the divergences need to be removed by color-octet mechanism. Based on this, it makes no sense to compare the convergence of velocity expansion between SGF and NRQCD factorization for the current problem.
![Polarization-summed SDCs as functions of $z$. Solid curve corresponds to lowest order in $v^2$ expansion in either SGF (with superscript “$(0)$") or NRQCD (with superscript “$O$"), based on the relation in Eq. . Dashed curve corresponds to order $v^2$ expansion in SGF. Dash-dotted curve corresponds to order $v^2$ expansion in NRQCD. []{data-label="fig:cpall"}](compair-LSN.eps){width="95.00000%"}
\[fig:cpallT\]
![Transversely polarized SDCs as functions of $z$. Meaning of each curve is similar to that in Fig. \[fig:cpall\].[]{data-label="fig:cpallT"}](compair-LSNT.eps){width="95.00000%"}
![SDCs at lowest order in $v^2$ expansion in either SGF (with superscript “$(0)$") or NRQCD (with superscript “$O$"), based on relations in Eqs. and . Solid curve represents polarization summed SDCs, dashed curve represents transversely polarized SDCs, and dash-dotted curve represents longitudinally polarized SDCs. \[fig:cpLOTL\]](compair-LTL.eps){width="95.00000%"}
![SDCs at order $v^2$ expansion in SGF framework. Meaning of each curve is similar to that in Fig. \[fig:cpLOTL\]. \[fig:cpSGFTL\]](compair-STL.eps){width="95.00000%"}
![SDCs at order $v^2$ expansion in NRQCD framework. Meaning of each curve is similar to that in Fig. \[fig:cpLOTL\]. \[fig:cpNRQCDTL\]](compair-NTL.eps){width="95.00000%"}
Numerical results and discussion {#sec:summary}
================================
We plot our polarization-summed and transversely polarized SDCs in Fig. \[fig:cpall\] and Fig. \[fig:cpallT\], respectively. We find that our $d^O(z,2m_Q,\mu_0)$ is compatible with the numerical result in Refs. [@Braaten:1993rw; @Braaten:1995cj; @Bodwin:2003wh], and $d^P(z,2m_Q,\mu_0)$ is compatible with the numerical result in Ref. [@Bodwin:2003wh]. Our polarized SDC $d_T^O(z,2m_Q,\mu_0)$ seems to be not compatible with the result extracted from physical cross section in Ref.[@Qi:2007sf]. Other results calculated in this paper are new.
In Fig. \[fig:cpLOTL\], Fig. \[fig:cpSGFTL\] and Fig. \[fig:cpNRQCDTL\], we compare polarization-summed, transversely polarized, and longitudinal polarized SDCs for each case. As expected, polarization-summed SDCs approach transversely polarized SDCs as $z\to1$.
-- ----------- ----- ------- ------- ----- ------- ------- ----- ------- -------
$F$ $c_1$ $c_2$ $F$ $c_1$ $c_2$ $F$ $c_1$ $c_2$
Sum 1 9.07 1 13.9 1 18.4
Transvers 0.679 8.42 0.724 13.2 0.759 17.7
Sum 1 7.58 1 12.4 1 16.9
Transvers 0.679 7.41 0.724 12.1 0.759 16.5
-- ----------- ----- ------- ------- ----- ------- ------- ----- ------- -------
: Coefficients to estimate relative importance of each part of FFs calculated in either SGF or NRQCD framework. \[table:FFpro\]
To estimate the relative contribution of each term for cross section, we integrate FFs calculated in this paper with a test function, $$\begin{aligned}
\label{eq:numff}
\begin{split}
\int_0^1 {\mathrm{d}}z \, z^{n} \, D^{\textrm{SGF}}_{g\to H} (z)
& = F \cdot \lambda^3
\frac{\alpha_s^3}{m_Q^3} {\langle{{\mathcal O}}^{H}({{{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}})\rangle}
(c_1+c_2 \lambda^2 v^2+O(v^4) ) \, , \\
\int_0^1 {\mathrm{d}}z \, z^{n} \, D^{\textrm{NRQCD}}_{g\to H} (z)
& = F \cdot
\frac{\alpha_s^3}{m_Q^3} {\langle{{\mathcal O}}^{H}({{{^{3}\hspace{-0.6mm}S_{1}^{[1]}}}})\rangle}
(c_1+c_2 v^2+O(v^4) ) \, ,
\end{split}\end{aligned}$$ where we denote $\lambda = m_Q/E$ and $v^2=E^2/m_Q^2-1$. The factors $F$, $c_1$ and $c_2$ depend on $n$, polarization and factorization method. For $n=2, 4, 6$, corresponding factors are shown in the Table. \[table:FFpro\]. With larger $n$, the integration in Eq. probes larger $z$, we then find $c_2/c_1$ also becomes larger which is consistent with our observation of large $z$ behaviour.
We thank Haoyu Liu, Ce Meng, Chenyu Wang, Yujie Zhang and Huaxing Zhu for many useful communications and discussions. The work is supported in part by the National Natural Science Foundation of China (Grants No. 11475005 and No. 11075002), and the National Key Basic Research Program of China (No. 2015CB856700).
[^1]: When $p_T$ is not large enough, next-to-leading power (NLP) contribution will be also important. For quarkonium production, the NLP contribution can also be factorized in terms of perturbative hard part convoluting with double parton fragmentation function [@Kang:2011mg; @Fleming:2012wy; @Kang:2014tta], which will not be discussed in this paper.
[^2]: It is needed to pointed out that $q$ in the amplitude is not the same as that, say $q^\prime$, in the complex conjugate of the amplitude, but $q^2=q^{\prime2}$.
|
---
author:
- 'Herbert K. Dreiner$^{\ast,}$, Jean-François Fortin$^{\dagger,\$,}$, Jordi Isern$^{\S,\ddag,}$ and Lorenzo Ubaldi$^{\ast,}$'
bibliography:
- 'Neutralinos\_Draft\_Revised.bib'
date: March 2013
title: White Dwarfs constrain Dark Forces
---
Introduction {#Intro}
============
White dwarfs (WDs) are simple astrophysical objects whose cooling law is well understood. This fact makes them a good laboratory for testing new models of particle physics. Many such models predict the existence of light bosons or light fermions that interact very weakly with regular matter. If these new particles are produced in a WD, they will typically escape and accelerate the cooling of the star. Thus, determining the cooling law from astrophysical observations can be translated into constraints on particle physics beyond the Standard Model (BSM) [@Raffelt:1996wa].
We first give a brief review of WD cooling. Formally, the cooling evolution of white dwarfs can be written as: \[lwd\] where $L_\gamma$ and $L_\nu$ represent the photon and neutrino luminosities (energy per unit time). The first term on the r.h.s. is the well known contribution of the heat capacity of the star to the total luminosity, the second one represents the contribution of the change of volume. It is in general small since only the thermal part of the electronic pressure, the ideal part of the ions and the Coulomb terms other than the Madelung term contribute [@isern97]. The third term represents the contribution of the latent heat and gravitational readjustement of the white dwarf to the total luminosity at freezing. Finally, $L_x$ and $\dot\epsilon_x$ represent any extra energy sink or source of energy respectively. For many applications, this equation can be easily evaluated assuming an isothermal, almost completely degenerate core containing the bulk of the mass, surrounded by a thin, nondegenerate envelope.[^1]
The evolution of white dwarfs can be tested through the luminosity function (LF), $n(l)$, which is defined as the number of white dwarfs of a given luminosity or bolometric magnitude[^2] per unit of magnitude interval and unit volume: \[ewdlf\] where \[bc\] $T_G$ is the age of the Galaxy, $l\equiv-\log(L/{L_{\odot}})$, $M$ is the mass of the parent star (for convenience all white dwarfs are labeled with the mass of the main sequence progenitor), $t_\text{cool}$ is the cooling time down to luminosity $l$, $\tau_\text{cool}=dt/d{M_\text{bol}}$ is the characteristic cooling time, $M_\text{s}$ is the maximum mass of a main sequence star able to produce a white dwarf, and $M_\text{i}$ is the minimum mass of the main sequence stars able to produce a white dwarf of luminosity $l$, and $t_\text{PS}$ is the lifetime of the progenitor of the white dwarf. $\Phi(M)$ is the initial mass function, *i.e.* the number of main sequence stars of mass $M$ that are born per unit mass, and $\Psi(t)$ is the star formation rate, *i.e.* the mass per unit time and volume converted into stars. So the product $\Phi(M)\Psi(\tau)$ is the number of main sequence stars that were born at the right moment to produce a white dwarf of luminosiy $l$ now. Since the total density of white dwarfs is not well known, the computed luminosity function is usually normalized to the bin with the smallest error bar, traditionally the one with $l=3$, in order to compare theory with observations.
The star formation rate is not known, but fortunately the bright part of Eq. satisfies [@Isern:2008fs]: If $\Psi$ is a well behaved function and $T_G$ is large enough, the lower limit of the integral is not sensitive to the luminosity, and its value is absorbed by the normalization procedure in such a way that the shape of the luminosity function only depends on the averaged characteristic cooling time of white dwarfs.
![*Luminosity function of white dwarfs.* Red (Harris *et al.* [@Harris:2005gd]) and blue (Krzesinski *et al.* [@Krzesinski:2009]) points represent the luminosity function of all white dwarfs (DA and non-DA families). Magenta points [@DeGennaro:2007yw] represent the luminosity function of the DA white dwarfs alone. Both distributions have been normalized around ${M_\text{bol}}= 13$, see text. The dotted line represents the luminosity function obtained assuming Mestel’s approximation. The continuous lines correspond to full simulations assuming a constant star formation rate and an age of the Galaxy of 13 Gyr for the DA family (black line) and all, DA and non-DA, white dwarfs (blue line).[]{data-label="FigLF"}](figures/wdlf.pdf){width="70.00000%"}
It is important to realize that white dwarfs are divided into two broad categories, DA and non-DA. The DA white dwarfs exhibit hydrogen lines in their spectra caused by the presence of an external layer made of almost pure H. This hydrogen layer is absent in the case of non-DAs and, consequently, their spectra is free of the H spectral features. The main result is that the DAs cool down more slowly than the non-DAs [@Althaus:2010pi].
The observed LF is shown in Fig. \[FigLF\] for three different datasets. Note that moving from left to right along the horizontal axis we go from high luminosity (hot, young WDs) to low luminosity (cold, old WDs). The Harris *et al.* [@Harris:2005gd] (red) and the Krzesinski *et al.* [@Krzesinski:2009] (blue) data are representative of all, DAs and non-DAS, white dwarfs. The Harris *et al.* LF has been constructed using the reduced proper motion method which is accurate for cold WDs with ${M_\text{bol}}\gtrsim6$ but not appropriate for hot WDs with ${M_\text{bol}}\lesssim6$, and which have been thus removed from the sample. The Krzesinski *et al.* LF on the other hand has been built employing the UV-excess technique which is accurate for hot WDs with ${M_\text{bol}}\lesssim7$ but inappropriate for the colder ones. Since the datasets overlap and, assuming continuity, it is possible to construct a LF that extends from ${M_\text{bol}}\sim1.5$ to ${M_\text{bol}}\sim16$, although the cool end is affected by severe selection effects. The DeGennaro *et al.* [@DeGennaro:2007yw] sample was also obtained with the proper motion technique, which is why the hot end is not reliable and has been removed. Since the identification of DAs and non-DAs is not clear at low temperatures, the corresponding points of [@DeGennaro:2007yw] have been removed from Figure \[FigLF\]. See Isern *et al.* [@Isern:2012xf] for a detailed discussion. Since ultimately only the slope of the LF is of interest and the total density of WDs is quite uncertain, it is usually more convenient to normalize the LF with respect to one of its values, which is commonly chosen around $\log(L/{L_{\odot}})=-3$.
If the cooling were due only to photons and one assumes that Mestel’s approximation [@1952MNRAS.112..583M] holds (*i.e.* ions behave like an ideal gas and the opacity of the radiative envelope follows Kramer’s law), then the LF would be a straight line on this logarithmic plot, which already provides a reasonable fit to the data. Note, however, that the data show a dip for values of ${M_\text{bol}}$ around $6-7$. That is where the neutrinos enter the game: for the hotter WDs (to the left in Fig. \[FigLF\]), neutrino emission becomes more important than photon cooling. When neutrinos are included and the cooling is simulated with a full stellar evolution code the agreement becomes impressive (see the continuous lines of Fig. \[FigLF\]). This agreement can be used to bound the inclusion of new sources or sinks of energy [@Isern:2008fs].
Cooling mechanisms {#Cooling}
==================
In this section we review the various cooling mechanisms for WDs. The aim is to provide the reader with a simple understanding of what mechanism dominates in what regime.
Photons
-------
In WDs the thermal energy is mostly stored in the nuclei which form, to a good approximation, a classical Boltzmann gas. Taking into account the thermal conductance of the surface layers, one can relate the rate of energy loss at the surface to the internal temperature. Using Mestel’s approximation [@1952MNRAS.112..583M] one finds \[EqPhotons\] where $\epsilon_\gamma$ is the energy-loss rate per unit mass and $T_7\equiv\frac{T}{10^7\,\text{K}}$. This constitutes the main cooling for cold (${M_\text{bol}}\gtrsim7$) WDs. Realistic models indicate that $\epsilon_\gamma\propto T^{\beta}$, where $\beta \approx 7/2$, but varies slightly with temperature, chemical composition and mass of the white dwarf. We show in Fig. \[FigLTc\] what this energy loss as a function of the core temperature looks like for a realistic model as opposed to Mestel’s model.
![Luminosity (bolometric magnitude) versus core temperature for a realistic model (continuous line) and for Mestel’s model (dashed line). The photon luminosity $L_\gamma \simeq \epsilon_\gamma M_{\rm WD}$, with $M_{\rm WD}$ the WD mass, is related to ${M_\text{bol}}$ via Eq. (\[EqMbol\]).[]{data-label="FigLTc"}](figures/ltc.pdf){width="60.00000%"}
Light bosons vs light fermions in white dwarfs
----------------------------------------------
Additional light bosons and light fermions that interact very weakly can also contribute to the cooling of WDs, but their dominant production mechanisms are usually different.
First, consider the DFSZ axion [@Dine:1981rt; @Zhitnitsky:1980tq] as an example of a light boson. It would be mainly produced by the bremsstrahlung process $e+(Z,A)\to e+(Z,A)+a$ as shown in Fig. \[FigDiagrams\] (a). Raffelt gave an intuitive argument [@Raffelt:1985nj] to understand how the corresponding energy emission rate depends on the temperature. It goes as follows: The relevant interaction term in the Lagrangian is $iga\bar{e}\gamma_5e$, where $g=m_e/f_
\text{PQ}$, with $m_e$ the electron mass, $f_\text{PQ}\ge 10^9$ GeV the Peccei-Quinn scale, $a$ the axion field and $e$ the electron field. The axion emission by an electron is analogous to the emission of a photon but, due to the presence of the $\gamma_5$, there is an extra electron spin-flip in the amplitude. Whereas the usual photon bremsstrahlung cross section is proportional to $E_\gamma^{-1}$, the axionic analogue is proportional to $E_a$ due to the extra power $E_a^2$ from the spin-flip nature of the process. For the energy emission rate, we have to multiply the cross section by another factor of $E_a$, which makes it proportional to $E_a^2$. We still have to do the phase space integrals for the initial and final state electrons. Because electrons are degenerate in WDs, these integrals contribute a factor of $T/E_F$ each, with $E_F$ the electron Fermi energy. Combining the factors, the emission rate is proportional to $E_a^2(T/E_F)^2\propto T^4$, given that axion energies will be of the order of the temperature $T$.
Next, consider the bremsstrahlung of two fermions $\psi$, as depicted in Fig. \[FigDiagrams\] (b). The intuitive reasoning is analogous to what we just described for axions, but the difference is that the fermions from the electron line are produced in pairs (angular momentum conservation) as opposed to the single axion. This adds an extra factor of the energy ($\sim T$) in the cross section and an extra phase space integral. As a result, we have two more powers of $T$ in the final emission rate, which is therefore proportional to $T^6$.
When the calculations are done carefully one gets the following results for the energy-loss rates per unit mass in the two cases [@Raffelt:1996wa] \[EqBrem\] Here, $\alpha_{26}\equiv10^{26}\frac{g^2}{4\pi}$, with $g$ the coupling of the axion to electrons defined above;[^3] $G_\psi$ is the dimensionful coupling for the four-fermion interaction denoted by a red dot in Fig. \[FigDiagrams\], to be compared to the familiar Fermi constant, $G_\text{F}=1.166\times10^{-5}$ GeV$^{-2}$; $C_\psi$ is the effective coupling constant analogous to the effective neutral-current vector coupling constant $C_V=0.964$; $X_j$ is the mass fraction of the element $j$, with nuclear charge $Z_j$ and atomic mass number $A_j$, and the sum runs over the species of nuclei present in the WD. $F_a$ and $F_\psi$ are factors that take into account the effect of screening for Coulomb scattering in a plasma. In WDs $F_a$ and $F_\psi$ are of order one to a good approximation.
It is clear from expressions (\[EqPhotons\]) and (\[EqBrem\]) why the bremsstrahlung process for the neutrinos, where $C_\psi G_\psi=C_V
G_
\text{F}$, is completely irrelevant in WDs, with internal temperature of the order of $10^7$ K. First, the numerical coefficient in $\epsilon_\psi^\text{brem}$ is suppressed by four orders of magnitude compared to photons (and to axions if we take $\alpha_{26}$ of order one). Second, it has a steeper dependence on the temperature, which makes it less and less relevant as we go to lower temperatures (see Fig. \[FigCompareloss\]).
; ; ; ;
Unless we have a model in which $G_\psi$ is significantly bigger than $G_\text{F}$, this contribution is negligible. In fact, the dominant production mechanism of a pair of light fermions in WDs is not bremsstrahlung but is given by the so-called plasmon process [@1963PhRv..129.1383A], which is depicted in Fig. \[FigDiagrams\] (c). That is what we describe next.
In vacuum the photon is massless and can not decay into a pair of massive particles, no matter how light they are. But in a medium, as in the interior of a star, the photon dispersion relations are modified and this allows such a decay. What happens is that the photon also acquires a longitudinal polarization and is promoted to the so-called plasmon. One would be tempted to say that the photon becomes massive, but such a statement is strictly speaking incorrect. A better way to think about the plasmon decay, without ever referring to the mass of the photon, is the following: the propagation of an electromagnetic excitation (the plasmon) in the plasma is accompanied by an organized oscillation of the electrons, which in turn serve as a source for emitting a pair of light particles. Figs. \[FigDiagrams\] (c,d) are then understood as follows: the grey blob represents the medium response to the electromagnetic excitation; we can think of the black line outlining the blob as a loop of electrons, with the red dot denoting an effective interaction with the pair of light particles, that can be either fermions or bosons. This is a schematic description. The reader interested in more details is referred to the pedagogic treatment in chapter 6 of Ref. [@Raffelt:1996wa].
The calculation of the plasmon decay [@1963PhRv..129.1383A; @1965NCimA..40..502Z] is quite involved, due to the effects of the medium, and cannot be performed analytically. However, a good approximation, in the case of neutrinos as the products of the decay, was given in Ref. [@Haft:; @1993jt]. The result applies to a wide range of stellar temperatures and densities. Restricting ourselves to WDs, we can write it as \[EqEpsplasmon\] where numerically, to a good approximation \[Eqlambdagamma\] and \[Eqfs\] The plasmon decay depends in a complicated way on the photon dispersion relation in the medium. However its main features can be understood in an approximation where the photons are treated as particles with an effective mass equal to the plasma frequency, $\omega_p$, which in the zero-temperature limit is given by [@Raffelt:1996wa] $\omega_p^2 = 4\pi \alpha n_e/E_F$, with $\alpha$ the fine-structure constant, $n_e$ the electron density and $E_F$ the Fermi energy of the electrons. $\omega_p$ is of the order of a few tens of keV in WDs, slightly higher than the typical WD internal temperature, which is a few keV [@Raffelt:1996wa]. For the plasmon decay to happen, the decay products have to be kinematically accessible. Thus, when we talk about [*new light particles*]{} in this context we mean particles [*lighter than a few tens of keV*]{}.
It is not immediately obvious how the energy loss of Eq. compares to the previous ones because of its complicated form, but the differences can be easily visualized in the simple plot in Fig. \[FigCompareloss\]. For WDs whose internal temperature is below $4-5\times
10^7$ K, the cooling is dominated by photons, and perhaps axions. Above that temperature, the plasmon decay into two light particles becomes the main source of energy loss. The contribution from the bremsstrahlung of a pair of fermions is always negligible on the plot.
It is useful to translate from temperature to ${M_\text{bol}}$. From Eq. , multiplying by a typical WD mass, $M_{\rm WD}$, that we take to be 0.6 solar masses, we obtain the photon luminosity, $L_\gamma = M_{\rm WD} \epsilon_\gamma$. Plugging it into Eq. (\[EqMbol\]) we obtain an expression that relates the temperature $T_7$ to ${M_\text{bol}}$. Thus, a temperature of $4-5\times 10^7$ K corresponds to values of ${M_\text{bol}}$ between 6 and 7, which is indeed where we see the neutrino dip in Fig. \[FigLF\]. The plasmon decay into a pair of light particles constitutes the dominant cooling mechanism for ${M_\text{bol}}<6-7$, the exact figures depending on the mass of the WD and the properties of the envelope.
Note that in models where a light boson couples to the electrons through a Yukawa coupling, the important cooling mechanism is the bremsstrahlung where the boson is produced singly, as in Fig. \[FigDiagrams\] (a). Such is the case for the DFSZ axion. In other models, instead, the light bosons couple indirectly to the electrons through a mediator and can only be produced in pairs, as for example when the bosons are charged under a new symmetry. This is the case for models with a dark sector, for instance, which we study in section \[Examples\]. The dominant production for these bosons is then no longer the bremsstrahlung, but the plasmon decay.
A generic constraint on models with new light particles {#Generic}
=======================================================
This section describes a generic constraint on models with new light particles obtained from WD cooling and trapping. We also discuss analogous constraints from red giants (RGs) and big bang nucleosynthesis (BBN).
White dwarf cooling constraint
------------------------------
In this section we discuss generic constraints from WD cooling due to plasmon decay into new light particles, that can be either fermions or bosons. The only requirement is that they should be lighter than a few tens of keV, for the decay to be kinematically possible. As mentioned at the end of the previous section, such a process affects the LF for values of ${M_\text{bol}}$ below $6-7$. Particle physics models in which new plasmon decay channels are open will potentially be in tension with the data, given the remarkable agreement between standard cooling mechanisms, that include neutrino emission, and the observed LF [@Isern:2012xf]. We want to quantify how much the plasmon decay rate can deviate from the standard one, considering the neutrinos as the only decay products.
To achieve this goal, it is useful to introduce a unified formalism reminiscent of the Fermi interactions for fermions. In order to compare with the standard plasmon decay into neutrinos, it is necessary to describe the relevant interaction between neutrinos $\nu$ and electrons $e$. The interaction is given by \[EqLneutrino\] where the contribution from the effective neutral-current axial coupling constant $C_A$ is negligible for our purpose and can be ignored [@Braaten:1993jw]. From this Lagrangian one can compute the plasmon decay rate into two neutrinos. The result is [@1963PhRv..129.1383A; @1965NCimA..; @40..502Z; @1972PhRvD...6..941D] \[EqPlasmneutrino\] where $\alpha$ is the fine-structure constant, $Z_s$ is the plasmon wavefunction renormalization, $\pi_s$ is the effective plasmon mass which enters in the dispersion relation $\omega^2-k^2=\pi_s(\omega,k)$ for a plasmon with frequency $\omega$ and wave vector $k$, and the subscript $s=\{T,L\}$ denotes the plasmon polarizations (transverse and longitudinal, respectively). The explicit forms for $\pi_T$ and $\pi_L$ are involved. They can be found, for example, in Ref. [@Raffelt:1996wa]. We just point out for this discussion that $\pi_s$ is proportional to $\alpha$, so that $\Gamma_{\nu,s}$ goes to zero if we turn off the electromagnetic interaction, as expected. With the standard plasmon decay rate into neutrinos, Eq. , the energy-loss rate per unit mass is given by Eq. with $C_\psi G_\psi=C_VG_\text{F}$, *i.e.* \[Eqplasmons\] and the contribution to the luminosity that appears in Eq. (\[lwd\]) is simply $L_\nu=M_\text{WD}\,\epsilon_\nu^\text{plasmon}$.
Let us now turn to new neutrino-like cooling mechanisms for WDs. For BSM models with new light fermions $\psi$, the relevant interactions are given by \[EqLfermions\] which are the appropriate analogs of the four-fermion interaction. The quantities $C_\psi$ and $G_\psi$ have been described above. For new light bosons $\phi$ which must be produced in pairs \[see Fig. \[FigDiagrams\] (d)\], the interaction is \[EqLbosons\] with $\phi^\dagger\overleftrightarrow{\partial}^\mu\phi \equiv \phi^\dagger (\partial^\mu \phi) - (\partial^\mu \phi^\dagger)\phi $. The corresponding quantities are the effective coupling constant $C_\phi$ and the dimensionful parameter $G_\phi$, which is the analog of the Fermi constant. For both interactions the plasmon decay into two new light particles is \[EqPlasmnew\] where $\{C_x,G_x\}$ are given by $\{C_\psi,G_\psi\}$ for new light fermions or $\{C_\phi,G_\phi\}$ for new light bosons. The extra plasmon decay channel will lead to an extra energy-loss rate per unit mass as in Eq. with $C_\psi G_\psi=C_xG_x$, *i.e.* \[Eqplasmonbosons\] and an extra contribution to the total luminosity given by $L_x=M_\text{WD}\,\epsilon_x^\text{plasmon}$, as in Eq. (\[lwd\]). In the following the relevant constants will be denoted simply by $\{C_x,G_x\}$ both for new light fermions and bosons.
As already mentioned, in order not to upset the excellent agreement between standard WD cooling mechanisms [@Isern:2012xf], *i.e.* from photon emission and neutrino emission (relevant only for hotter WDs), and observational data, we postulate that plasmon decay into new light particles must not account for more than the plasmon decay into neutrinos. In the massless limit, both for new particles as well as neutrinos, this constraint can be stated simply as \[see Eqs. and \] \[EqConstraint\] In other words, any new sufficiently light particles (*i.e.* which are effectively massless in WDs), that can be produced through plasmon decay in WDs and can escape from WDs, generate extra cooling. This extra cooling must be subdominant compared to standard plasmon decay into neutrinos. To validate this order-one constraint, it is now necessary to properly quantify the agreement between the standard cooling mechanisms and observational data.
The standard cooling mechanisms relevant for WDs are photon cooling and plasmon decay into neutrinos. Since we are interested in constraining models which lead to extra neutrino-like cooling, we focus here only on the dataset of DeGennaro *et al.* which covers bolometric magnitudes between $5.5\lesssim{M_\text{bol}}\lesssim12.5$. This range is well understood and clearly exhibits the neutrino dip for ${M_\text{bol}}$ around $6-7$ (see Fig. \[FigLFJordi\]). Moreover, the dataset of DeGennaro *et al.* has the smallest error bars in this range and only contains DA WDs.
We start by minimizing the $\chi^2$ for the LF assuming standard cooling mechanisms and Mestel’s approximation. The free parameter is the WD birthrate. The best fit implies a birthrate $\sim 1.6\times 10^{-3}$ pc$^{-3}$ Gyr$^{-1}$, which is a reasonable local WD formation rate [@Liebert:2004bv], for $\chi_\text{min}^2=24.9$. Since there are $N_\text{exp}=18$ data points and $N_\text{th}=1$ free parameters, this provides a decent fit with a reduced chi-square $\chi_\text{red,min}^2=\chi_\text{min}^2/N_\text{dof}=1.47$, where $N_\text{dof}=N_\text{exp}-N_\text{th}=17$ is the total number of degrees of freedom.
Next we determine the 90% confidence level exclusion contours for extra cooling from plasmon decay into new light particles, assuming the latter are massless. Since the new plasmon decay channels are reminiscent of the standard plasmon decay into neutrinos, we take here $L_x=S_xL_\nu$, where $S_x$ determines the ratio of the new extra luminosity $L_x$ to the neutrino luminosity $L_\nu
$. Now we take $S_x$ as our only free parameter, leaving the WD birthrate fixed to the value determined above. Thus we still have $N_\text{dof}=17$. We then compute the new chi-square, $\chi^2$, including the $L_x$ contribution. From Fig. 36.1 in Ref. [@Nakamura:2010zzi] we find that $\chi^2$ must be such that \[EqDeltachi\] otherwise the extra cooling is excluded at 90% confidence level. Imposing the condition $\Delta\chi^2 <24.8$ translates into the constraint \[EqChiS\] which is equivalent to the one in Eq. , obtained from a simpler and more intuitive physical argument. We provide in Fig. \[FigLFJordi\] three curves for the LF obtained from realistic models. The top one includes only standard cooling (with neutrinos), while the two lower ones include extra neutrino-like contributions with $S_x=0.5$ and $S_x=1$ respectively.
![*Theoretical luminosity function for WDs.* The curves shown include the different contributions to Eq. (\[lwd\]) and correspond to values of $S_x=0, 0.5, 1$ from top to bottom for the $L_x = S_x L_\nu$ contribution. They are superimposed on the data points by DeGennaro *et al.* [@DeGennaro:2007yw]. []{data-label="FigLFJordi"}](figures/wdlfx.pdf){width="70.00000%"}
White dwarf trapping constraint
-------------------------------
One should also include the effects of trapping. Indeed, as $G_x$ increases the interactions between the new light particles and ordinary matter become stronger. For very large $G_x$ the interactions are too strong and the mean free path of the new light particles is too small for them to escape the WD and thus contribute to its cooling. To make an estimate, we compare the cross section for the scattering of new light particles on ordinary matter, $\sigma_x \propto C_x^2 G_x^2$, with the corresponding one for neutrinos, $\sigma_\nu \propto C_V^2 G_\text{F}^2$. Neutrinos have a mean free path of $\lambda_\nu=(n\sigma_\nu)^{-1}\simeq3000{R_{\odot}}$ in WDs [@Althaus:2010pi]. Requiring that the mean free path of our light particles is bigger than a typical WD radius, $R_\text{WD}\simeq0.019{R_{\odot}}$ [@Raffelt:1996wa], and comparing $\sigma_x$ and $\sigma_\nu$ we find the condition $C_xG_x\lesssim400 \ C_VG_\text{F}$. Combining this with Eq. implies that any new light particles produced in WDs are excluded by cooling considerations if \[EqWDConstraint\] Eq. is the main result of this paper and will be used in section \[Examples\] to constrain BSM models with new light particles.
Comparison to constraints from red giants and big bang nucleosynthesis
----------------------------------------------------------------------
In the same line of thoughts, it is possible to obtain cooling constraints from red giants (RGs). Following [@Raffelt:1996wa] the bound from RGs cooling can be translated into $S_x\lesssim2$, which corresponds to $C_xG_x\lesssim1.41C_VG_\text{F}$ and is comparable to, but slightly weaker than what we found in Eq. for WDs. Moreover, since the cores of RGs can be seen as WDs, trapping constraints in RGs will necessary be worse than in WD. Therefore, in this context RGs do not constrain new light particles as well as WDs.
Such new light particles could however be very tightly constrained by BBN. Given that we are interested in masses below a few tens of keV, if they were in thermal equilibrium with ordinary matter in the early universe until BBN, that happens at $T\sim 1$ MeV, they would contribute to the number of relativistic degrees of freedom, which is well constrained. To estimate this constraint, we follow [@Steigman:2013yua].
The reactions $e^+ e^- \leftrightarrow \psi \psi$ and $e \psi \leftrightarrow e \psi$, responsible for keeping the light particle, $\psi$, in thermal equilibrium, have a typical cross section $\sigma_x \propto C_x^2G_x^2T^2$, which leads to an interaction rate per particle of $\Gamma_x=n\sigma_x|v|\propto C_x^2G_x^2T^5$, since their number density is $n\propto T^3$. Comparing to the expansion rate $H\propto T^2/M_\text{Pl}$, the decoupling temperature can be estimated as $T_{x,\text{dec}}\propto(C_x^2G_x^2M_\text{Pl})^{-1/3}$, where $M_\text{Pl}$ is the Planck mass. This is completely analogous to the calculation for the neutrinos decoupling temperature, $T_{\nu,\text{dec}}\propto(C_V^2G_\text{F}^2M_\text{Pl})^{-1/3}$. Thus we can write \[EqTdec\] Following [@Steigman:2013yua] the effective number of neutrinos $N_\text{eff}$ is given by \[EqNeff\] where $g_s(T)$ is the ratio of the total entropy density to the photon entropy density and $\Delta N_\nu$ is the number of equivalent neutrinos, *i.e.* $\Delta N_\nu=2\times1$ for a Dirac fermion or $\Delta N_\nu=2\times4/7$ for a complex scalar. Demanding that the number of equivalent neutrinos be smaller than $4$ [@Ade:2013zuv] and taking $T_{\nu,\text{dec}}=3\ \text{MeV}$ [@Steigman:2013yua] leads to the constraint \[EqBBNbound\] which is three (two) orders of magnitude stronger than the WD bound Eq. for new light Dirac fermions (complex scalar bosons). From this analysis it would thus seem that BBN bounds are more competitive than WD bounds in constraining models with new light particles. Note, however, that there are caveats that could invalidate the BBN bounds without modifying the WD constraints. For example, a light \[$\sim\mathscr{O}(\text{MeV})$\] weakly-interacting massive particle (WIMP) whose annihilations heat up the photons but not the neutrinos would result in a lower $N_\text{eff}$ and thus leave more room for extra relativistic degrees of freedom [@Kolb:1986nf; @Serpico:2004nm; @Ho:2012br]. In such a scenario, the bound of Eq. would be relaxed to the extent that the WD constraint would be more competitive. Hence, the WD bound is robust because it is oblivious to possible caveats that would alter BBN considerations.
Three examples {#Examples}
==============
In this section we consider three examples of BSM scenarios. The first two are supersymmetric extensions of the Standard Model (SM): in the first, the light particle is the neutralino, while in the second, it is the axino. We show that WDs do not put competitive bounds on these models. The situation is different in the third example, where we consider models with a dark sector, in which case the WDs bounds are very competitive.
A light neutralino
------------------
The neutralino $\chi_0$ is often the lightest supersymmetric particle in the Minimal Supersymmetric Standard Model. It can be very light, even massless, and still evade all current experimental constraints [@Dreiner:2009ic; @Profumo:2008yg]. For the production of light neutralinos in WDs, that would predominantly occur via plasmon decay, we consider the four-fermion interaction obtained from integrating out the selectron $
\tilde{e}$ (see Fig. \[FigSUSY\])[^4], \[EqLneutralino\] where $G_{\tilde{e}}=\frac{e^2}{4\cos^2\theta_Wm_{\tilde{e}}^2}$ and $C_{\chi_0}=\frac{3}{4}$ [@Dreiner:2003wh], with $e$ the electric charge, $\theta_W$ the weak mixing angle and $m_{\tilde{e}}$ the selectron mass.
Since $G_\text{F}=\frac{\sqrt{2}e^2}{8\sin^2\theta_Wm_W^2}$ where $m_W$ is the $W$ gauge boson mass the constraint Eq. can be translated into a lower bound on the selectron mass of \[EqSelectron\] where $m_W=80.4$ GeV and $\sin^2\theta_W=0.23$. Thus, in order to have a significant impact on the LF one needs a selectron lighter than the $W$ gauge boson. The bound in Eq. applies to the case of a massless neutralino. Turning on a small neutralino mass has the effect of pushing the WD bound down to even lower selectron masses. Such light selectrons are already excluded by LEP searches [@Heister:2002jca]. Note that supernovae, contrary to WDs, provide a better arena to constrain the mass of a light neutralino [@Dreiner:2003wh]. Nevertheless, WD cooling bounds do not seem competitive for this process.
A light axino
-------------
A light axino is in principle very interesting in this context. It has already been argued that the inclusion of an axion gives a better fit to the LF [@Isern:2012ef]. If supersymmetry (SUSY) is realized in nature, the axion would be necessarily accompanied by its fermionic partner, the axino, which could also be very light (see *e.g.* [@Rajagopal:1990yx; @Chun; @Chun199543]). The axino could be pair-produced in the plasmon decay and contribute to the high luminosity part of the LF. When combined with the contribution of the axion one might hope to get an even better fit. Unfortunately, as we explain in the rest of this section, the axino interacts way too weakly so that its contribution to the LF turns out to be completely negligible.
Recall that the coupling of axions to electrons is given by $iga\bar{e}\gamma_5e$, with $g=m_e/f_\text{PQ}$. In SUSY there is a corresponding axino-electron-selectron interaction that can be written as $ig\tilde{e}\bar{e}\psi_a$, where $\psi_a$ denotes the axino. If we integrate out the selectron (see Fig. \[FigSUSY\]), the resulting four-fermion interaction between two electrons and two axinos is scalar-like \[*e.g.* $(\bar\psi_a
\psi_a) (\bar e e)$\] instead of vector-like \[*e.g.* $(\bar\psi_a \gamma^\mu \psi_a) (\bar e \gamma_\mu e)$\] and thus does not even allow plasmons to decay to pairs of axinos. Being more precise and starting from the derivative interaction between the axion and electrons instead, one obtains higher-dimensional operators after supersymmetrizing and integrating out the selectron, *i.e.* four-fermion interactions between two electrons and two axinos with extra derivatives, which are thus temperature-suppressed compared to the usual plasmon decay. Most importantly however, these interactions are always at least suppressed by $g^2$, which is incredibly tiny for reasonable $f_\text{PQ}\sim10^9-10^{12}$ GeV. Therefore, although the constraint Eq. cannot be applied directly here, the universal suppression just mentioned makes a possible production of axinos absolutely unobservable in WDs.
A dark sector
-------------
### The model
As seen in the two previous examples, WD cooling might not seem to lead to any strong bounds on new light fermions. The situation is however much more interesting when one considers models of BSM with massive dark photons [@Fayet:1980ad; @Fayet:1990wx]. In these models, which could be of relevance as models of dark matter, a dark sector $\mathscr{L}_\text{D}$ communicates with the SM, $\mathscr{L}_\text{SM}$, solely through kinetic mixing $\mathscr{L}_{\text{SM}\otimes\text{D}}$ [@Holdom:1985ag], *i.e.* \[EqLdark\] Above the electroweak scale the kinetic mixing occurs with strength $\varepsilon_Y$ between the hypercharge gauge group $U(1)_Y$, with the corresponding $F_{\mu\nu}^\text{SM} = \partial_\mu B_\nu - \partial_\nu B_\mu$, and a new Abelian gauge group $U(1)_\text{D}$, with $F_\text{D}
^{\mu
\nu} = \partial^\mu A_\text{D}^\nu - \partial^\nu A_\text{D}^\mu$, where $A_\text{D}^\mu$ is the $U(1)_\text{D}$ gauge boson, *i.e.* the dark photon. Below the electroweak scale the mixing involves instead the electromagnetic gauge group, and $\varepsilon=\varepsilon_Y\cos\theta_W$. The dimensionless parameter $\varepsilon$, which should be generated by integrating out massive states charged under both SM and dark gauge groups, is naturally small, $\varepsilon\sim10^{-4}-10^{-3}$. Thus, after rotating the fields appropriately such that gauge bosons have canonically-normalized kinetic terms, the SM fields become millicharged under the [*dark*]{} gauge group [@Cassel:2009pu; @Hook:2010tw], *i.e.* \[EqLSMD\] where $J_\mu^\text{SM}$ is the SM electromagnetic current.
Thus, in models with massive dark photons, WD plasmons could decay, through off-shell massive dark photons, to light dark sector particles if they are kinematically available. Note that both dark photon decays to bosons and fermions result in two-particle final states. Thus such plasmon decay through massive dark photons into light dark sector particles is reminiscent of plasmon decay into fermions (*e.g.* neutrinos) irrespective of the spin of the light dark sector particles \[see Fig. \[FigDiagrams\] (c,d)\]. Therefore the relevant constraints for plasmon decay in models with massive dark photons are equivalent to the constraint discussed in section \[Generic\].
We stress the fact that in the scenario we are contemplating, the dark $U(1)_{\rm D}$ gauge group is broken so that the dark photon is massive. Instead, when $U(1)_{\rm D}$ is unbroken, the corresponding gauge boson is commonly referred to as a paraphoton. In this latter case, dark sector particles acquire an electric millicharge, that is a tiny fractional charge under the visible $U(1)_{\rm EM}$, and the constraints are usually shown on the plane given by $\varepsilon$ versus the mass of the dark sector particle [@Davidson:2000hf]. In our case, with the broken dark $U(1)_{\rm D}$, there are no particles with an electric fractional charge. Rather, SM particles have a fractional charge under $U(1)_{\rm D}$, that is quite different.
### Excluded parameter region
In order to determine the resulting excluded parameter space it is necessary to integrate out the dark photon, as shown in Fig. \[FigDark\]. This leads to the interaction \[Eqelectronsdark\] where the dark constant is $G_\text{D}=\frac{4\pi\varepsilon\sqrt{\alpha\alpha_\text{D}}}{m_{A_\text{D}}^2}$ and $C_\psi=Q_\psi$, $C_\phi=\frac{Q_
\phi}
{2}$. Here $m_{A_\text{D}}$ is the dark photon mass, $\alpha_\text{D}$ is the dark fine-structure constant and $Q_{\psi,\phi}$ are the dark particle charges under the dark gauge group. Note that dark photon decay into a pair of dark gauge bosons is generically not kinematically accessible because the masses of the dark photon and of the other dark gauge bosons are usually of the same order, as, for example, the $Z$ and $W$ gauge bosons in the SM.
Comparing with plasmon decay to neutrinos as discussed in section \[Generic\], the constraint Eq. leads to \[EqDark\] where $C_\text{D}=C_{\psi,\phi}$. In Fig. \[FigExclusion\] we show the constraint Eq. and regions in parameter space which have already been explored or will be explored by future experiments [@Bjorken:2009mm], *i.e.* beam dump experiments at SLAC: E137, E141 and E774 [@Riordan:1987aw; @Bross:1989mp; @Andreas:2012mt]; $e^+e^-$ colliding experiments: BaBar [@Aubert:2009au; @Bjorken:; @2009mm] and KLOE [@Archilli:2011zc]; and fixed-target experiments: APEX [@Abrahamyan:2011gv], DarkLight [@Freytsis:2009bh], HPS [@Boyce:2012ym], MAMI [@Merkel:2011ze] and VEPP-3 [@Wojtsekhowski:2012zq]. Fig. \[FigExclusion\] also shows excluded regions from electron ($a_e$) and muon ($a_\mu$) anomalous magnetic moment measurements [@Pospelov:2008zw; @Davoudiasl:2012ig; @Endo:2012hp].
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; ;
; ;
; ;
;
(3,0.25) node; (2,2.6) node; (0.9,4) node; (0.3,4.3) node; (0.3,7.1) node; (5.3,7.3) node; (5.3,6.5) node\[red\]; (6.6,5.4) node\[purple\]; (7.4,6.7) node\[green\]; (7.4,4.2) node\[red\]; (4.3,5) node\[teal\]; (2.2,3.3) node\[pink\]; (4,4.3) node\[brown\]; (6.2,2.8) node\[brown\]; (7.65,1) node\[rotate=58\]; (6.8,1) node\[rotate=58\]; (5.95,1) node\[rotate=58\]; (0.85,2.8) node\[rotate=37\];
For reasonable dark sector parameters where $\alpha_\text{D}\sim\alpha$, one has $C_\text{D}\sim1$ and $C_\text{D}^2\alpha_\text{D}\sim10^{-3}-10^{-2}$, and thus all but a small fraction of the relevant dark sector parameter space is excluded by WD cooling and the some of the above-mentioned experiments become obsolete *if* dark photons couple to new dark sector fermions and/or bosons which are effectively massless in WDs, *i.e.* lighter than a few keV. In other words, to be viable models of dark photons, any model probed by the above-mentioned experiments cannot have dark sector fermions and/or bosons lighter than a few tens of keV due to WD cooling.
Note however that all of the experiments shown in Fig. \[FigExclusion\]—apart from DarkLight, VEPP-3 and the anomalous magnetic moment measurements—assume that dark photons decay predominantly back into the SM. Although this is not possible in WDs (dark photons could only decay back into electron-positron pairs which are not kinematically accessible, the decay to neutrino pairs is negligible), this assumption forbids either light dark sector particles, in which case the WD constraint presented here is irrelevant; or large dark fine-structure constant (relative to $\alpha\varepsilon^2$), for which dark photon decay rate into invisible channel dominates.
To investigate this last possibility, we include in Fig. \[FigExclusion\] (see dashed blue line) the WD cooling constraint for which the dark photon decay rate into visible channels dominates over the decay rate into invisible channels, *i.e.* $\Gamma_\text{invisible}\lesssim\Gamma_\text{visible}$ or $C_\text{D}^2\alpha_\text{D}\lesssim\alpha\varepsilon^2$. It is interesting to see that, for very weak dark fine structure constant, the experiments which are sensitive to invisible dark photon decays, *i.e.* DarkLight and VEPP-3, are still constrained by the WD cooling even when dark photons decay predominantly back into the SM.
Note that the constraint must be modified for a very light dark photon (again lighter than a few tens of keV), since it could be produced on-shell, which would result in an enhancement of the cooling rate. The resulting constraint would then be even tighter. For such a light dark photon bremsstrahlung might also become important.
Finally, it would be of interest to study astrophysical cooling constraints from more energetic objects, like supernovae, to relax the restriction on the masses of the dark particles produced.
Discussion and conclusion
=========================
We studied constraints from the WD LF on BSM models with new light particles. Whenever these light particles are produced in pairs, whether they are fermions or bosons, the dominant production mechanism in WDs is (usually) given by the plasmon decay. Such a decay is responsible also for the production of neutrino pairs, whose effect is well understood and clearly visible through the dip at ${M_\text{bol}}\sim6-7$ in the LF curve. Adding a significant decay into new light particles would deepen the dip, which would then be in disagreement with the data. This constrains part of the parameter space of these BSM models. More quantitatively, one needs to compare the strength of the interaction between the new light particles and the electrons with the interaction between neutrinos and electrons, *i.e.* the Fermi constant $G_\text{F}$, and require that the former do not exceed the latter.
We applied this constraint to three models. We first consider a supersymmetric model with a light neutralino and showed that the WD constraint is not competitive with existing collider bounds. The situation is analogous with an axino, whose interaction is even further suppressed with respect to the neutralino, and does not lead to any interesting constraint. We then explored models with a dark sector, for which the bounds are more relevant. That is due mainly to the fact that the dark photon, that mediates the interaction between the electrons and the light dark sector particles, can be light \[$\sim\mathscr{O}(\text{GeV})$\], which enhances the plasmon decay rate. It turns out that the limits on the dark sector parameter space from energy losses in WDs, as shown in Fig. \[FigExclusion\], are extremely competitive and render some experiments obsolete *if* the dark photon couples to light \[$\sim\mathscr{O}(10\ \text{keV})$\] dark sector particles. Said differently, the dark photon models which are probed by these experiments cannot have light dark sector fermions and/or bosons, due to WD cooling.
Such dark sector particles could contribute to the relativistic degrees of freedom, $N_\text{eff}$, in the early universe and alter BBN predictions. BBN bounds can indeed be stronger than those from WD cooling. However, they are subject to caveats and are not as robust.
*Note:* During completion of this work An *et al.* [@An:2013yfc] posted a paper on stellar constraints for dark photons. There is no overlap between our work and theirs since they consider dark photons with hard Stückelberg masses.
[^1]: The isothermal approximation is not valid when neutrinos are dominant, however the results are still reasonably good and provide a reasonable estimate of the luminosity. The ions do not follow the ideal gas law but the equation of state of a Coulomb plasma—for instance, in the region of interest the specific heat approaches the Dulong–Petit law—and crystallizes at low temperatures, around bolometric magnitude $12-13$, depending on the mass of the star.
[^2]: The bolometric magnitude and the luminosity are related through \[EqMbol\]
[^3]: For $\alpha_{26}$ of order one, axion cooling becomes comparable to photon cooling and one gets a better fit to the LF [@Isern:2012ef]. This fact can be taken as tentative evidence for the existence of axions, and it explains the choice of the power of 26 in the definition of $\alpha_{26}$.
[^4]: A very light neutralino, $m_{\chi_0}\ll 1$ GeV, is almost purely bino and does not couple to the $Z_0$ [@Choudhury].
|
---
abstract: 'The central regions of the three brightest members of the Leo I galaxy group – NGC 3368, NGC 3379, and NGC 3384 – are investigated by means of 2D spectroscopy. In all three galaxies we have found separate circumnuclear stellar and gaseous subsystems – more probably, disks – whose spatial orientations and spins are connected to the spatial orientation of the supergiant intergalactic H I ring reported previously by Schneider et al. (1983) and Schneider (1985, 1989). In NGC 3368 the global gaseous disk seems also to be inclined to the symmetry plane of the stellar body, being probably of external origin. Although the rather young mean stellar age and spatial orientations of the circumnuclear disks in NGC 3379, NGC 3384, and NGC 3368 could imply their recent formation from material of the intergalactic H I cloud, the time scale of these secondary formation events, of order 3 Gyr, does not support the collision scenario of Rood & Williams (1985), but is rather in line with the ideas of Schneider (1985, 1989) regarding tidal interactions of the galaxies with the H I cloud on timescales of the intergroup orbital motions.'
author:
- 'O. K. Sil’chenko'
- 'A. V. Moiseev, V. L. Afanasiev'
- 'V. H. Chavushyan, J. R. Valdes'
title: 'The Leo I Cloud: Secular nuclear evolution of NGC 3379, NGC 3384, and NGC 3368?'
---
Introduction
============
The origin of S0 galaxies is long-standing problem which was posed almost at the moment of the birth of the Hubble morphological classification scheme. However, this has been one of those rare cases when the first idea was correct, this being later confirmed more than once. From the beginning many investigators thought that lenticulars formed (more exactly, transformed) from spirals. Now with the advent of high-resolution imaging, including the HST as well as some ground-based work, using adaptive optics, there is direct evidence that rich clusters, presently populated mostly by S0’s, at $z=0.5 - 0.8$ contain a lot of spirals which are in the course being accreted by these clusters. Evidently, dense environments, or deep potential wells, or hot intracluster gas pressure, provides the conditions to transform the spirals into lenticulars. Two famous theoretical papers concerning particular mechanisms of this transformation must be mentioned here: @spitbaad propose collisions between spiral galaxies, which are frequent in dense clusters with high velocity dispersions, in which in a passage of one galaxy through another, all the gas is swept out of their galactic disks. The other paper is by @ltcs0 who analysed a less violent event during which tidal stripping removes a diffuse gaseous halo from a spiral galaxy. The secular disk building by gas accretion onto the equatorial plane having ceased, the gas remaining in the disk is typically consumed by star formation in a few billion years.
Therefore one can easily imagine – and even directly observe – how spirals transform into lenticulars in clusters. But clusters are not the only place where S0’s live; there are a lot of lenticulars in the field and in loose groups. What is their origin? There are suggestions that field S0s form in a different way from that of cluster S0s. There exists one loose galaxy group where the scenario of @spitbaad may apply: it is the Leo I group. Twenty years ago a unique supergiant intergalactic HI cloud was discovered in this galaxy group [@leoh1_1], and @roodwil suggested that this gas was swept out of galactic disks during a collision between NGC 3368 and NGC 3384 some $5 \times 10^8$ yrs ago. Now the HI cloud is located exactly half-way between two galaxies, and NGC 3384 looks like a bona-fide lenticular galaxy. But Schneider himself [@leoh1_2; @leoh1_3] gave another explanation to this phenomenon. The cloud has fainter filaments which encircle the galaxy pair NGC 3384/NGC 3379; the whole HI-complex may be treated as a clumpy gaseous ring with a radius of some 100 kpc. Velocities measured in several clumps imply a Keplerian rotation of the ring with a period of 4 Gyr, the SW-part of the ring being receding. If the intrinsic shape of the ring is circular, it is inclined to the line of sight with the line of nodes at $PA\approx 40^{\circ}$; interestingly, the close orientation and sense of rotation is repeated by the global stellar disk of NGC 3384. Even though symmetry (rotation) axes of the rest of the galaxies of the group are not aligned with those of the HI-ring and NGC 3384, @leoh1_2 [@leoh1_3] conclude that this supergiant, $1.7 \cdot 10^9$ $M_{\odot}$, HI-cloud may represent left-over primordial (pre-galactic) gas from which all the galaxies of the group have been formed.
As the HI-ring is rather massive, it contributes to the common potential; in particular, tidal interactions of the group galaxies with the ring are possible. @leoh1_3 have analysed the possible interaction of the ring with NGC 3368 – the nearest and the brightest neighbor of the NGC 3384/NGC 3379/HI-ring complex. He suggested that the intergroup orbit of NGC 3368 is inclined to the plane of the ring and that this galaxy has had two close passages near the ring during a Hubble time. In this case, the cycle time of interaction is about 5 Gyr, instead of $5 \cdot 10^8$ yrs as proposed in the model of @roodwil.
Well after the first simulations of tidal interactions of galaxies by @toomre, demonstrating spectacular outer tidal structures (tails, bridges, etc.), it was realized that an external tidal impulse affects also the innermost (circumnuclear) structure of the galaxy – see e.g. @noguchi. So a history of nuclear star formation may reflect in some way the history of galaxy interactions, especially if the galaxy possesses an early morphological type and lacks its own gas. In this paper we study the properties of nuclear stellar populations in NGC 3379, NGC 3384, and NGC 3368 – three of the four brightest galaxies of the Leo I group and three nearest neighbors of the unique intergalactic HI-ring. We are searching for signs of synchronous evolution of the nuclear stellar populations in these galaxies; obviously, the characteristic time of this evolution may help to discriminate between various scenarios of the origin of the HI-ring.
The global parameters of the galaxies are given in Table 1. The distance to the spiral galaxy NGC 3368 is determined from Cepheid observations [@distceph], and the distance to the high surface brightness elliptical NGC 3379 by the surface-brightness fluctuations method [@morshank]; the distance to NGC 3384 was assumed to be the same as to NGC 3379 as their planetary nebula systems seem to imply [@pndist].
The layout of the paper is as follows. We report our observations and other data which we use in this paper in Section 2. The radial variations of the properties of the stellar population are analysed in Section 3, and in Section 4 two-dimensional velocity fields obtained by means of 2D spectroscopy are presented. Section 5 presents a discussion and our conclusions.
[lccc]{} NGC & 3368 & 3379 & 3384\
Type (NED$^1$) & SAB(rs)ab & E1 & SB(s)0-:\
$R_{25}$, kpc (LEDA$^2$) & 12.6 & 9.8 & 9.8\
$B_T^0$ (RC3$^3$) & 9.80 & 10.18 & 10.75\
$M_B$ (LEDA) & –20.9 & –20.5 & –19.6\
$(B-V)_T^0$ (RC3) & 0.79 & 0.94 & 0.91\
$(U-B)_T^0$ (RC3) & 0.25 & 0.52 & 0.43\
$V_r$, $\mbox{km}\cdot \mbox{s}^{-1}$ (NED) & 897 & 911 & 704\
Distance, Mpc & 11.2 & 12.6 & 12.6\
Inclination (LEDA) & $55^{\circ}$ & $32^{\circ}$ & $90^{\circ}$\
[*PA*]{}$_{phot}$ (LEDA) & $5^{\circ}$ & $71^{\circ}$ & $53^{\circ}$\
\
\
Observations and data reduction
===============================
The spectral data which we analyse in this work were obtained with two different integral-field spectrographs. Integral-field spectroscopy is a rather new approach that was first proposed by Prof. G. Courtes some 15 years ago – for a description of the instrumental idea see e.g. @betal95. It allows one to obtain simultaneously a set of spectra in a wide spectral range from an extended area on the sky, for example, from a central part of a galaxy. A 2D array of microlenses provides a set of micropupils which are projected onto to the entrance window of a spectrograph. After reducing the full set of spectra corresponding to the individual spatial elements, one obtains a list of fluxes for the continuum and for the lines, line-of-sight velocities, both for stars and ionized gas, and of emission- and absorption-line equivalent widths which are usually expressed as indices in the well-formulated Lick system [@woretal]. This list can be transformed into two-dimensional maps of the above mentioned characteristics for the central part of the galaxy under study. Besides the panoramic view benefits, such an approach gives a unique opportunity to overlay various 2D distributions over each other without any difficulties with positioning. In this work we use the data from two 2D spectrographs: the fiber-lens Multi-Pupil Fiber Spectrograph (MPFS) at the 6m telescope BTA of the Special Astrophysical Observatory of the Russian Academy of Sciences (SAO RAS) and the international Tigre-mode SAURON at the 4.2m William Herschel Telescope at La Palma.
A new variant of the MPFS became operational at the prime focus of the 6m telescope in 1998 (http://www.sao.ru/$\sim$gafan/devices/mpfs/). With respect to the previous one, in the new variant the field of view is increased and the common spectral range is larger due to using fibers: they transmit light from $16\times 15$ square elements of the galaxy image to the slit of the spectrograph together with additional 16 fibers which transmit the sky background light taken away from the galaxy, so separate sky exposures are not necessary anymore. The size of one spatial element is approximately $1\arcsec \times 1\arcsec$; a CCD TK $1024 \times 1024$ detector is used. The reciprocal dispersion is 1.35 Å per pixel, with a rather stable spectral resolution of 5 Å. To calibrate the wavelength scale, we expose separately a spectrum of a hollow cathode lamp filled with helium, neon, and argon; an internal accuracy of linearizing the spectrum was typically 0.25 Å in the green and 0.1 Å in the red. Additionally we checked the accuracy and absence of a systematic velocity shift by measuring strong emission lines of the night sky, \[OI\]$\lambda$5577 and \[OI\]$\lambda$6300. We obtained the MPFS data mostly in two spectral ranges, the green one, 4300–5600 Å, and the red one, 5900–7200 Å. The green spectra are used to calculate the Lick indices H$\beta$, Mgb, Fe5270, and Fe5335 which are suitable to determine the mean (luminosity-averaged) metallicity, age, and Mg/Fe ratio of old stellar populations [@worth94]. Also, they are used for cross-correlating with a spectrum of a template star, usually of K0III–K3III spectral type, to obtain in such a way a line-of-sight velocity field for the stellar component and a map of stellar velocity dispersion. The red spectral range contains strong emission lines of H$\alpha$ and \[NII\]$\lambda$6583 and is used to derive line-of-sight velocity fields of the ionized gas. To calibrate the new MPFS index system onto the standard Lick one, we observed 15 stars from the list of @woretal during four observing runs and calculated the linear regression formulas to transform our index measurements to the Lick system; the rms scatter of points near the linear regime is about 0.2 Å for all 4 indices under consideration, i.e., within the observational errors quoted by @woretal. To correct the index measurements for the stellar velocity dispersion which is usually substantially non-zero in the centers of early-type galaxies, we smoothed the spectrum of the standard star HD 97907 by a set of Gaussians of various widths; the derived dependence of index corrections on $\sigma$ were approximated by 4th order polynomials and applied to the measured index values before their calibration to the Lick system.
The second 2D spectrograph which data we use in this work is a new instrument which is being operated at the 4.2m William Herschel Telescope on La Palma since 1999, named SAURON – for its detailed description see @betal01. We have taken the data for NGC 3379 and NGC 3384 from the open ING Archive of the UK Astronomy Data Centre. Briefly, the field of view of this instrument is $41\arcsec \times 33\arcsec$ with spatial element sizes of $0\farcs 94 \times 0\farcs 94$. The sky background at 2 arcminutes from the center of the galaxy is exposed simultaneously with the target. The fixed spectral range is 4800-5400 Å, the reciprocal dispersion is 1.11 Å-1.21 Å per pixel varying from the left to the right edge of the frame. The comparison spectrum is neon, and the linearization is made using a 2nd order polynomial with an accuracy of 0.07 Å. The index system is checked by using stars from the list of @woretal that have been observed during the same observing runs as the galaxies. The regressions describing the index system of the February-1999 run when NGC 3379 was observed can be found in our paper [@afsil02b], and the regressions for the March/April-2000 run are presented in Fig. 1. The relations between instrumental and standard-system indices were approximated by linear fits which were applied to our measurements to calibrate them on to the standard Lick system. The stellar velocity dispersion effect was corrected in the same manner as for the MPFS data.
The full list of exposures made for NGC 3368, NGC 3379, and NGC 3384 with two 2D spectrographs is given in Table 2.
Date Galaxy Exposure Configuration Field Spectral range Seeing
----------- ---------- ---------- ---------------------------------- ------------------------------- ---------------- -------------
20 Feb 99 NGC 3379 60 min WHT/SAURON+CCD $2k\times 4k$ $33\arcsec\times 41\arcsec$ 4800-5400 Å $1\farcs 5$
8 Feb 00 NGC 3368 45 min BTA/MPFS+CCD $1024 \times 1024$ $16\arcsec \times 15\arcsec $ 4200-5600 Å $1\farcs 4$
13 Feb 00 NGC 3368 40 min BTA/MPFS+CCD $1024 \times 1024 $ $16\arcsec \times 15\arcsec $ 6000-7200 Å $2\farcs 3$
28 Mar 00 NGC 3368 140 min BTA/MPFS+CCD $1024 \times 1024 $ $16\arcsec \times 15\arcsec $ 4840-6210 Å $2\farcs 5$
11 Dec 99 NGC 3384 45 min BTA/MPFS+CCD $1024 \times 1024$ $16\arcsec \times 15\arcsec $ 4200-5600 Å $1\farcs 6$
4 Apr 00 NGC 3384 120 min WHT/SAURON+CCD $2k\times 4k$ $33\arcsec\times 41\arcsec$ 4800-5400 Å $3\farcs 4$
: 2D spectroscopy of the galaxies studied
Date Exposure Spectral range Seeing
----------- ------------------- ----------------------------- --------
28 Feb 00 $32\times150$ sec around $H_\alpha$ 27
28 Feb 00 $32\times200$ sec around \[NII\]$\lambda6583$ 35
: IFP observations of NGC 3368 at the 6m telescope
Date Galaxy Filter Exposure time Scale,$\arcsec$ per px Seeing
----------- ---------- -------- --------------- ------------------------ -------------------------------
18 Apr 00 NGC 3368 $J$ 15 min 0.3 $1\farcs 5$
18 Apr 00 NGC 3368 $H$ 12 min 0.3 $1\farcs 5$
18 Apr 00 NGC 3368 $K'$ 12 min 0.3 $1\farcs 5$
12 Mar 01 NGC 3368 $J$ 15 min 0.85 $2\farcs 1 \times 1\farcs 5$
12 Mar 01 NGC 3368 $H$ 14 min 0.85 $2\farcs 4 \times 1\farcs 8$
16 Mar 01 NGC 3384 $J$ 12 min 0.85 $2\farcs 5 \times 1\farcs 85$
16 Mar 01 NGC 3384 $H$ 12 min 0.85 $2\farcs 2 \times 1\farcs 8 $
16 Mar 01 NGC 3384 $K'$ 12 min 0.85 $2\farcs 1 \times 1\farcs 65$
: Photometric observations of the galaxies studied
We have also observed the global kinematics of the ionized gas in NGC 3368 with the scanning Fabry-Perot Interferometer (IFP) at the 6m telescope. In contrast to the integral-field 2D spectrographs MPFS and SAURON, the IFP allows us to obtain spectral information over a large field of view, but over a relatively small spectral range. We use the IFP in interference order 235 (for $\lambda6563$Å).The IFP is installed at the pupil plane of a focal reducer attached to the f/4 prime focus of the 6m telescope. A brief description of this device is available from http://www.sao.ru/$\sim$gafan/devices/ifp/ifp.htm. The detector was a CCD TK $1024 \times 1024$ working with a binning of $2\times 2$ pixels. The resulting pixel size was $0\farcs 68$ and the field of view was about of $5\farcm 8$. During every object exposure we obtain 32 frames with interference rings for varying IFP gaps. The full spectral range (interfringe) was $28$Å, the spectral resolution was about $2.5$Å. A narrow-band filter with $FWHM\approx15$Å was used to select a spectral domain around the redshifted galactic emission lines H$\alpha$ and \[NII\]$\lambda6583\AA$. The log of the observations with the IFP is given in Table 3.
Besides the 2D spectral data, we have obtained NIR photometry for two of the galaxies under consideration. The observations were made at the 2.1m telescope of the National Astronomical Observatory of Mexico “San Pedro Mártir” with the infrared camera CAMILA. The camera is equipped with a NICMOS3 detector with a format of $256 \times 256$ pixels; mostly the mode with a scale of $0\farcs 85$ (f/4.5) has been used, except for NGC 3368 which has also been observed with a higher sampling. The details of the photometric observations are given in Table 4. Additionally, for all three galaxies we have retrieved the NICMOS/HST data from the HST Archive. NGC 3368 was observed on May 4, 1998, with the NIC2 camera, through the filters F110W and F160W during 128 sec in the framework of a program of Massimo Stiavelli (ID 7331), and on May 8, 1998, with the NIC2 camera through the F160W filter during 320 sec in the framework of a program of John Mulchaey (ID 7330). NGC 3379 was observed on June 14, 1998, with the NIC3 camera through the F160W filter during 192 sec as part of a program of William Sparks (ID 7919). NGC 3384 was observed on April 3, 1998, with the NIC2 camera through the F160W filter during 128 sec for the program of John Tonry (ID 7453).
Almost all the data, spectral and photometric, except the data obtained with the MPFS, have been reduced with software produced by V.V. Vlasyuk at the Special Astrophysical Observatory [@vlas]. Primary reduction of the data obtained with the MPFS was done in IDL with software created by one of us (V.L.A.). The Lick indices were calculated with our own FORTRAN program as well as by using a FORTRAN program written by A. Vazdekis. For the reduction of the IFP data we used our IDL software [@mois_ifp]; also the ADHOC software developed at the Marseille Observatory by J. Boulesteix (see http://www-obs.cnrs-mrs.fr/ADHOC/adhoc.html) was used. The observational data were converted into a “data cube” of 32 images. The data reduction includes subtraction of the night sky emission, channel-by-channel correction [@mois_ifp], wavelength calibration, and spatial and spectral gaussian smoothing. The velocity fields and monochromatic images in both emisison lines (H$\alpha$ and \[NII\]$\lambda$6583) have been constructed by Gauss fitting spectral line profiles; also, images in the red continuum close to the emission lines were calculated from the same IFP “data cubes”.
Lick indices and stellar population properties
==============================================
Up to now there have been a few attempts to map 2D distributions of Lick indices by means of integral-field spectroscopy; we mention here papers by @ems96 on NGC 4594, by @pel2dfis on NGC 3379, 4594, and 4472, and by @n4365sau on NGC 4365. In addition we mention our own papers[@sil99b; @we2000; @we2002; @afsil02a; @afsil02b; @s0_3] on NGC 7331, NGC 7217, NGC 4429, NGC 7013, NGC 5055, NGC 4138, NGC 4550, NGC 5574, and NGC 7457. A difficulty entering reliable mapping of Lick indices comes from the fact that we calibrate extended (panoramic) data but are using point-like calibration sources – Lick standard stars that are usually placed in the center of the field-of-view or frame. If the spectral resolution, or the spectral response, or both, vary over the frame, this may result in a systematic distortion of the Lick index distributions over the field of view. We check this effect by measuring Lick-index distributions in twilight exposures: the surface index distributions of the H$\beta$, Mgb, Fe5270, and Fe5335 calculated from twilight frames must be flat, and the mean level of every index distribution must be close to the values measured by us earlier: H$\beta _{\odot}=1.86\,$Å, Mgb$_{\odot}=2.59\,$Å, Fe5270$_{\odot}=2.04\,$Å, and Fe5335$_{\odot}=1.59\,$Å [@sharina]. Additionally, we compare azimuthally-averaged index radial profiles with well-calibrated long-slit measurements found in the literature. Figures 2 and 3 present such comparisons for NGC 3379 and NGC 3384, respectively.
The long-slit data along the major axis of NGC 3379 are taken from @vazd2. Their errors exceed the errors of the azimuthally-averaged SAURON data by several times; however, the Mgb-profile (Fig. 2, middle) allows us to conclude that there is no systematic shift between the SAURON- and Vazdekis et al.’s long-slit data for this particular index. It means that we have properly taken into account, or have justly neglected, the effects which are not related to the wavelength (spectral localization) of the features, i.e., the effects of the spectral resolution and stellar velocity dispersion broadening or contamination by one of the neighboring spectra which does not exceed 2%-3% according to our estimates. Meantime, the SAURON measurements of H$\beta$ are marginally too high (by less than 0.2 Å), and the SAURON measurements of Fe5270 are certainly too low by 0.3–0.4 Å.
The radial profiles of the Lick indices in NGC 3384 (Fig. 3) have been measured even more thoroughly than those in NGC 3379: we have plotted the azimuthally-averaged MPFS and SAURON data as well as major- and minor-axis long-slit measurements from @fish96. Again, the radial profile of Mgb looks the most reliable among all the indices – probably, because the magnesium line falls in the middle of the spectral range observed. The H$\beta$ measurements outside of the nucleus according to SAURON are higher by 0.3–0.4 Å as compared to the MPFS, and in this particular case the Fisher et al.’s measurements support the SAURON results. But as for Fe5270, the SAURON data are again too low by $\sim$0.4 Å, as seen already in the case of NGC 3379. @sau2 present preliminary SAURON results on NGC 3384 and noted this disagreement between their and @fish96 measurements, but they insisted that their results were more correct. Since our MPFS data for Fe5270 in NGC 3384 agrees with that of @fish96 and since the same systematic shift of Fe5270 is observed in NGC 3379, we suppose that the SAURON values of the iron index are systematically underestimated.
[|l|cc|]{} Lick index & $a$, Å per arcsec & $b$, Å \
\
Mgb & $-0.0261 \pm 0.0023$ & $4.59 \pm 0.002$\
Fe5270 & 0 & $2.64 \pm 0.01$\
\
Mgb & $-0.0381 \pm 0.0036$ & $4.04 \pm 0.03$\
Fe5270 & $-0.0142 \pm 0.0036$ & $2.61 \pm 0.03$\
However, intrinsically the SAURON azimuthally-averaged data are very precise: a typical error for every point in the profiles of Figs. 2 and 3 is $0.02 - 0.04$ Å (the accuracy of the MPFS azimuthally-averaged indices is about 0.1 Å). Their high quality allows us to diagnose chemically distinct cores in both galaxies NGC 3379 and NGC 3384: although the magnesium- and iron-index breaks are of rather low amplitude, we note a certain change of the profile slopes at $R\approx 4\arcsec$. Let us also note that we predicted the existence of a chemically distinct nucleus in NGC 3379 from the multi-aperture photometric data [@sil94]. If we approximate the ‘bulge’ data at $R \ge 4\arcsec$ by linear fits (the parameters of these fits are given in Table 5), we can obtain extrapolated ‘bulge’ indices at $R=0\arcsec$, or in the very centers of NGC 3379 and NGC 3384, and compare them to the real nuclear indices; the differences would characterize a chemical distinctness of the nuclei. In NGC 3384 (Fig. 3) $\Delta$Mgb=0.64 Å and $\Delta$Fe5270=0.42 Å, $\pm 0.08$ Å. If we treat the differences of metal-line indices between the nucleus and the extrapolated bulge as a difference of metallicity and apply the evolutionary synthesis models for old stellar populations of @worth94, we obtain $\Delta$\[Fe/H\]=+0.3, the nucleus being on average more metal-rich, the $\Delta$Mgb and $\Delta$Fe5270 results being consistent. In NGC 3379 (Fig. 2) the chemical distinctness of the core is more modest: $\Delta$Mgb=$0.26\pm 0.06$ Åcorresponds to $\Delta$\[m/H\]=+0.1, and $\Delta$Fe5270=0.08 Åimplies an even smaller $\Delta$\[Fe/H\]$\approx +0.05$, which is evidence for a marginally higher magnesium-to-iron ratio in the‘nucleus’ as compared to the ‘bulge’.
High-precision radial profiles of absorption-line indices provide a zero-order diagnostic of the central stellar substructures; but with 2D index distributions we are able to discuss a morphology of the chemically distinct entities and their relation to photometric substructures, in particular to compact circumnuclear disks. Let us consider in detail every galaxy of our small sample.
NGC 3379
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Figure 4 presents Lick-index maps for the central part of NGC 3379. As expected, the chemically distinct nucleus is better seen in Mgb; the central maximum of Fe5270 is of rather low contrast, and the distribution of H$\beta$ looks almost flat. The core distinguished by the enhanced magnesium line is well-resolved and has an elongated shape. After heavy smoothing (Fig. 4, upper right) we are able to determine an orientation of this elongated magnesium-rich core: $PA_{core}\approx 84^{\circ}$. The magnesium-index isolines trace a substructure which is more flat than the main stellar body: the axis ratio of the magnesium-index isolines is $b/a_{Mg} \approx 0.5$, whereas the isophote axis ratio is $b/a_{phot}\approx 0.9$. Is it a compact, $R \le 4\arcsec$, circumnuclear disk that is seen as a chemically distinct core?
Figure 5 allows us to quantify characteristics of the stellar populations in the center of NGC 3379 at different distances from the center. The magnesium-iron diagram (Fig. 5, top) reveals a systematic shift of the galactic measurements with respect to a model locus of the populations with the solar magnesium-to-iron ratio plotted according to @worth94. Although we know already from the analysis of Fig. 2 that the SAURON data of Fe5270 are slightly too low with respect to the standard Lick system, the systematic shift of the NGC 3379 measurements relative to Worthey’s models is so large that it cannot be explained by any index system bias. For example, the data of @trager for the very center of NGC 3379 and the sequence obtained by substitution of the @vazd2 long-slit data on Fe5270 in the radial profile table (Fig. 5, top) are also away from the model locus. We have to conclude that the nucleus (core) of NGC 3379 is magnesium overabundant, \[Mg/Fe\]$=+0.2 - +0.3$, and this enhanced Mg/Fe ratio holds to an approximately constant level (within 0.05 dex) as a function of radius.
So to determine an age of the stellar population in the center of NGC 3379, we must use the models with \[Mg/Fe\]$>0$; such as the models of @tantalo that are calculated in particular for \[Mg/Fe\]$=+0.3$. But the models of @tantalo involve the combined iron index $\langle \mbox{Fe} \rangle \equiv$(Fe5270+Fe5335)/2, and we have only Fe5270 which is unfortunately biased. By plotting an age-diagnostic diagram (Fig. 5, bottom), we have confronted the $\langle \mbox{Fe} \rangle$ measurements of @vazd2, which are well-calibrated on to the Lick system, though are not very precise, to our measurements of H$\beta$ at the same distances from the center, which on the contrary have a small error. Since the accuracy of the age estimates is determined mainly by the H$\beta$ accuracy, we obtain a certain mean (luminosity-weighted) age estimate of 8–9 Gyr for the very center ($R< 8\arcsec$) of NGC 3379, with possible variations within one Gyr with radius. The mean age of the stellar population in the core of NGC 3379 below 10 Gyr is in agreement with the recent result of @terlfor who have found an integrated value of 9.3 Gyr within an aperture of $R_e/8$.
NGC 3384
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Figure 6 presents Lick index maps for the center of NGC 3384 which have been obtained with the MPFS, and Fig. 7 presents similar maps obtained with SAURON for a larger area; note that the iron-index map covers the full area whereas the preliminary results of @sau2 showed only half of the field of view because of some problems they had with the data reduction. The spatial resolution of the former map is better by a factor of two than that of the latter, so in Fig. 6 we see a lot of subtle details which are not seen in Fig. 7. However, common features can also be noted: the magnesium index demonstrates a central maximum, rather compact and at high-contrast, and well-resolved. Also, the iron enhancement in the center of NGC 3384 is more diffuse and extended, so even from the analysis of Figs. 6 and 7 “by eye” it is suggestive that there is a gradient of the magnesium-to-iron ratio with radius. The metal-line enhanced areas are elongated as if they were produced by a compact circumnuclear disk appearance; however when we drew isolines of the Mgb distribution (Fig. 7, upper right), we convinced ourselves that their orientation, $PA_{Mg} \approx
20^{\circ} - 25^{\circ}$, differs substantially from the orientation of the inner isophotes, $PA_{inner}\approx 40^{\circ} - 45^{\circ}$ (see the next Section), and from the line of nodes, $PA_0=53^{\circ}$ (Table 1). The H$\beta$ distribution demonstrates an unresolved peak in the nucleus in the MPFS data only (Fig. 6, left), whereas the SAURON data is evidence for quite a flat distribution (Fig. 7, lower right).
The diagnostic diagrams for the average data at different radii are shown in Fig. 8, which includes only the MPFS data as they offer unbiased iron-index estimates. These diagrams present a slightly different picture from that in the center of NGC 3379. Indeed, there exists a Mg/Fe ratio gradient (Fig. 8, top): the nucleus is obviously magnesium-overabundant, $\mbox{[Mg/Fe]}_{nuc} \approx +0.3$ if one takes into account its very young mean age (see Fig. 8, bottom); and at $R \ge 5\arcsec$ the Mg/Fe ratio is about solar. Consequently, to determine a mean age, we must use two sets of stellar population models, with \[Mg/Fe\]$=+0.3$ and with \[Mg/Fe\]=0. Figure 8, bottom, presents the comparison of our data to the models of @tantalo for both values of \[Mg/Fe\]. One can see that the unresolved nucleus of NGC 3384 is rather young for a lenticular host: its mean stellar age is less than 5 Gyr, and probably close to 3 Gyr; the general metallicity is higher than the solar one: $\mbox{[m/H]}_{nuc} \approx +0.3 - +0.4$. In the nearest vicinity of the nucleus the metallicity drops to the quasi-solar value, and the mean age increases to 7–8 Gyr. Therefore, although in the center of NGC 3384 an extended region looks chemically distinct (the net dimensions of this region are difficult to determine because they are comparable to our resolution limit), the unresolved stellar nucleus has probably followed its own, quite separate evolution.
NGC 3368
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The maps of the metal-line Lick indices for the central part of NGC 3368 (Fig. 9) demonstrate a qualitative difference with respect to the maps of the centers of NGC 3379 and NGC 3384: instead of peaks in the nuclei they show ‘holes’ – net minima of Mgb and $\langle \mbox{Fe} \rangle$ just where the surface brightness has a maximum. These holes may be evidences for a metal deficiency of the nuclear stellar population, or more likely, for its extreme youth. Magnesium- and iron-index enhancements can also be seen in the maps of Fig. 9, especially on the February MPFS maps (Fig.9, top) which have a better spatial resolution, but these peaks are off-nuclear. The Mgb index has two maxima at the major axis located at $R\approx 4\arcsec$ symmetrically with respect to the nucleus, and the iron index $\langle \mbox{Fe} \rangle$ demonstrates something like a half-ring with a radius of $2\arcsec - 3\arcsec$. Such features associated usually to star-forming rings and ‘bright circumnuclear spots’ are clear signatures of a bar presence; indeed, the isophotal twist in the center of NGC 3368 can be noticed even over the very limited area covered by the MPFS frame. In the recent study of NGC 3368 structure and kinematics by @mois02 the presence of the minibar with an extension of about $5\arcsec$ has also been noted.
The age diagnostics in the center of NGC 3368 is complicated because of rather intense ionized-gas emission: the H$\beta$ absorption-line index is contaminated by the Balmer emission line. We have tried to take into account the effect of H$\beta$ in emission when calculating azimuthally-averaged index profiles: we have co-added separately the blue and the red spectra in the same concentric rings; then, by using the red spectra at various radii, we estimated equivalent widths of the H$\alpha$ emission line which is stronger than H$\beta$ and can be surely measured even inside a deep H$\alpha$ absorption line; after that we calculated the H$\beta$-index correction by involving the mean relation between H$\alpha$ and H$\beta$ emission lines in normal galaxies, $EW(H\beta em)=0.25EW(H\alpha em)$ [@sts2001]. The corrected indices taken as a function of radius are plotted in the diagnostic diagrams presented in Fig. 10. The correction applied to the H$\beta$ index is close to the minimal possible one (corresponding to pure radiative ionization), so the derived ages may be slightly overestimated. By considering both diagrams in Fig. 10 together, one can conclude that the nuclear stellar population has a mean age of 3 Gyr and a mild magnesium overabundance, \[Mg/Fe\]$\approx +0.2$. When moving toward $R\approx 10\arcsec$ in radius, the mean age strongly increases, and the magnesium-to-iron ratio drops to the solar value.
The ages that we determine by confronting the measured integrated-spectra indices to simple (one-age, one-metallicity) stellar population models are indeed mean luminosity-weighted ages: if one has a mix of populations of different ages, each of them would contribute to the integrated spectrum proportionally to its luminosity, and from the diagnostic diagrams like that in Fig. 10 (bottom), one could obtain an age estimate, intermediate between extreme individual stellar generation ages. So although we have obtained a mean age of 3 Gyr for the nuclear stellar population in NGC 3368, it does not mean that all stars there have been formed 3 Gyr ago; it means that we see a superposition of an intermediate-age stellar generation onto the older stellar bulge. To illustrate this idea, we have constructed a model experiment for the age-diagnostic diagram of Fig. 10: to a moderately metal-poor (\[Fe/H\]=-0.22), old ($T=12$ Gyr) bulge we added a relatively young “poststarburst” population with \[Fe/H\]$=+0.25$ and two trial ages – 1 and 3 Gyr, with a contribution from 1% to 80% of the total mass. The calculations were made with the ‘Dial-a-model’ machine of Guy Worthey available at his WEB-page (http://astro.wsu.edu/worthey/). Two model sequences corresponding to $T_{burst}=1$ and 3 Gyr were plotted in Fig. 10 (bottom), along which the contribution of the young population varies. One can see that although the former sequence crosses the SSP-model line for $T_{SSP}=3$ Gyr at a value of $\sim 20$% the overall trend of the $T=1$ Gyr young starburst deviates from the observational points, whereas the $T_{burst}=3$ Gyr sequence coincides exactly with the radial trend in the index in the center of NGC 3368. So we conclude that the age of the secondary nuclear star formation burst of 1 Gyr or less can be excluded and that whereas $T_{nuc}=3$ Gyr is the mean luminosity-weighted estimate, the true age of the secondary starburst does not differ strongly from this value.
Stellar and gaseous kinematics in the centers of the Leo I Group galaxies
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Since integral-field spectroscopy provides us with two-dimensional line-of-sight velocity fields, we are able now to analyse both the rotation and central structure of the galaxies. If we have an axisymmetric mass distribution and rotation on circular orbits, the direction of maximum central line-of-sight velocity gradient (we shall call it the “kinematical major axis”) should coincide with the line of nodes as well as the photometric major axis, whereas in a case of a triaxial potential the isovelocities align with the principal axis of the ellipsoid and the kinematical and photometric major axes generally diverge, showing twists in an opposite sense with respect to the line of nodes [@mbe92; @mm2000]. In a simple case of cylindric (disk-like) rotation we have a convenient analytical expression for the azimuthal dependence of central line-of-sight velocity gradients within the area of solid-body rotation:\
$dv_r/dr = \omega$ sin $i$ cos $(PA - PA_0)$,\
where $\omega$ is the deprojected central angular rotation velocity, $i$ is the inclination of the plane, and $PA_0$ is the orientation of the line of nodes, coinciding in the case of an axisymmetric ellipsoid (or a thin disk) with the photometric major axis. So by fitting azimuthal variations of the central line-of-sight velocity gradients with a cosine curve, we can determine the orientation of the kinematical major axis by its phase and the central angular rotation velocity by its amplitude. This is our main tool for kinematical analysis.
Let us note that the method of the analysis of line-of-sight velocity gradients works correctly only within the area of solid-body rotation. Therefore it cannot be used for a large-scale velocity field (beyond the central kpc region). For analysis of the IFP’s velocity fields we applied a method usually referred to as the ‘tilted-ring’ method [@begeman]. In the framework of this method the velocities were fitted in elliptical rings (elongated along the PA of the galactic disk major axis) by a model of pure circular rotation. As a first step we found and fixed the rotation center position which is the center of symmetry of the velocity fields. As a second step we calculated the radial dependence of the model parameters: systemic velocity $V_{sys}$, rotational velocity $V_{rot}$, disk inclination $i$ and position angle of the kinematical major axis $PA_0$. And finally, $i$ and $V_{sys}$ were fixed at their mean values and the run of $V_{rot}(r)$ and $PA_0(r)$ with radius were obtained. Radial variations of $PA_0$, if present, can be used for detecting various types of non-circular motions – see @mm2000 for details and references.
NGC 3379
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NGC 3379 was treated earlier as an example of a classic elliptical galaxy – round and homogeneous. However, recent kinematic studies by using a long-slit technique [@n3379ls; @it_3379] have proved that at least the central part of the galaxy consists of several subsystems including a highly inclined dust (and gaseous?) ring and that NGC 3379 may be a misclassified S0. Since we analyse the results of 2D kinematic mapping, new details showed up to complicate the picture.
Figure 11 shows the stellar line-of-sight velocity field and the stellar velocity dispersion field measured from the SAURON data. The former field demonstrates a rather fast (for an elliptical galaxy), quasi-axisymmetric rotation; the latter field reveals a prominent maximum in the center with a quasi-axisymmetric radial decrease of the velocity dispersion –both facts were already noted, e.g. by @n3379ls. We have measured an orientation of the kinematical major axis up to $R\approx 5\arcsec$, which is the approximate edge of the rapidly rotating area, and have found that it changes significantly from a $PA_{kyn}=265^{\circ}$ ($85^{\circ}$) at $R\approx 2\arcsec$ to $PA_{kyn}=256^{\circ}$ ($76^{\circ}$) at $R=4\arcsec - 5\arcsec$. We compare the orientation of the kinematical major axis to that of the photometric major axis according to the data of @frei and to the NICMOS/HST image analysis results in Fig. 12; we would like also to refer to the results of the analysis of the WFPC2/HST images of NGC 3379 by @it_3379. Within $R\le 15\arcsec$ the photometric major axis deviates in a positive sense from the tabulated outer isophote orientation $PA_0=70^{\circ}$ (see the Table 1) and agrees well with the kinematical major axis. Let us also remind here that the isolines of the enhanced magnesium index deviate from the outer isophote orientation too, although they are aligned with the kinematical major axis; and they have a more flattened distribution than the isophotes. Such coincidences are evidence for a compact stellar disk in the center of NGC 3379 which may be inclined to the main symmetry plane of the galaxy. There exists another inclined circumnuclear disk in NGC 3379 – a dust-gaseous one, with a radius of $2\arcsec$ and $PA_0=125^{\circ}$ [@it_3379]. Are they related? In our Fig. 12, and also in Fig. 3 of @it_3379, based on the high-resolution results of the HST/WFPC2 F555W and F814W isophote analysis, one can see a strong twist of the major axis when approaching the center, inside the central arcsecond. @it_3379 thought it to be an effect of the dust ring projection. But the dust ring with an orientation of $PA=125^{\circ}$ would force an isophote twist in a polar direction, so in a negative sense with respect to the line of nodes $PA_0 =70^{\circ}$. Such a turn is really observed in the radius range of $0\farcs 8 - 1\farcs 5$ (Fig. 12). However within $R=0\farcs 4$ the effect of the dust ring having a radius of $2\arcsec$ is negligible. We would think the circumnuclear isophote twist of $PA\approx 102^{\circ}$ in our measurements, or even up to $PA\approx 120^{\circ}$, as the rectified analysis of @it_3379 suggests, to be real and to represent a signature of the inclined circumnuclear stellar disk which is probably related to the dust-gaseous one. Some discrepancies of the position angle estimates can be explained by different spatial resolutions of the data.
NGC 3384
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Figure 13 presents the stellar line-of-sight velocity field and that of the stellar velocity dispersion in the center of NGC 3384, according to the data from MPFS (top) and SAURON (bottom). The stellar LOS-velocity field reveals a fast regular rotation, with isovelocity features typical for a disk embedded into a massive bulge (tight isovelocity crowding near the line of nodes, etc). The presence of the compact circumnuclear stellar disk with a radius of $6\arcsec$ in the center of NGC 3384 was found earlier from a photometric analysis by @it2_3384, so we confirm their finding from a kinematical point of view. Yet another exotic subcomponent – a stellar polar ring suggested by @fr_3384 and @polar_r – can be discarded with certainly based on the same kinematical arguments. This latter feature is localized in the radius range of $10\arcsec - 25\arcsec$, and the large SAURON velocity map for the stars in the center of NGC 3384 does not show any switch of the spin orientation (toward the polar direction) at $R=10\arcsec -
15\arcsec$. Hence we diagnose this morphological feature (identified as “EC” in the terminology of Busarello et al.) as a bar. Bars belong always to disks (cold dynamical subsystems), so we conclude that the inclination of $90^{\circ}$ given for NGC 3384 by LEDA (see the Table 1) is erroneous: one would not see a bar along the minor axis as an elongated structure under edge-on orientation of the global disk; in the meantime, the refined analysis of @it2_3384 has demonstrated a very elongated, almost “peanut”-shape structure, with $PA=132^{\circ}$, in this radius range. We show radial variations of the characteristics of global isophotes in NGC 3384 obtained through the NIR broad-band filters in Fig. 14. The sharp maximum of isophote position angle at $R=15\arcsec$ and the corresponding minimum of the ellipticity refer to a superposition of a flattened spheroid with $PA_0\approx 50^{\circ}$ and of a bar almost along the minor axis. An asymptotic ellipticity at $R>60\arcsec$, $1-b/a=0.45$, implies a possible disk inclination as low as $57^{\circ}$, (@it1_3384 obtained $i=63.5^{\circ}$). If the symmetry plane of NGC 3384 is inclined by some $60^{\circ}$ to the line of sight, the proposed “polar ring” is indeed a bar in the disk plane with a radial extension of $\sim 2.5$ kpc ($40\arcsec$). The presence of the bar in NGC 3384 is confirmed by the stellar velocity dispersion distribution in the center of the galaxy (Fig. 13): it has a strongly elongated shape with a flat maximum aligned roughly with the minor axis. As @vd97 have shown from dynamical modelling, within the bar potential the stellar velocity dispersion distributions are elongated along the bars. Just this picture is observed in the center of NGC 3384.
Let us quantify the kinematical arguments in favor of the circumnuclear disk. In Fig. 15 we compare orientations of the photometric (according to the HST data) and the kinematical (according to the MPFS and the SAURON data) major axes within $R\approx 5\arcsec$ from the center. The agreement is good within $1^{\circ} - 2^{\circ}$, or certainly to within our accuracy limits. However both photometric and kinematic major axes ($PA_{phot}=46^{\circ}$ ($226^{\circ}$) and $PA_{kyn}=225.5^{\circ} \pm 1.2^{\circ}$) deviate significantly (and consistently!) from the line of nodes, $PA_0=53^{\circ}$. Such behavior is evidence for an [*inclined*]{} circumnuclear stellar disk. Interestingly, as the orientation of the bar is $PA_{bar}=132^{\circ}$ [@it2_3384], obviously we deal with a disk [*which is polar with respect to the bar*]{}. Such a configuration can be often encountered when studying circumnuclear gas rotation within a triaxial potential [@we99; @we2000]. Here for the first time we have faced a similar, but completely stellar substructure.
NGC 3368
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The kinematical picture in the center of NGC 3368 can be presented in more detail than that in NGC 3379 and NGC 3384 because its intense emission lines provide us with the velocity field of the ionized gas. However more information does not clarify the situation but rather makes it quite puzzling.
Figure 16 presents the stellar velocity field and the stellar velocity dispersion distribution in the center of NGC 3368 according to the MPFS data of February 2000; the analogous data of the March 2000 observations are described by @mois02. The isovelocities demonstrate a regular solid-body rotation of stars; but the kinematical major axis is strongly different from the orientation of the inner isophotes which have a noticeable twist of the major axis over the radius range of $0\arcsec -8\arcsec$, unlike the isovelocities. The stellar velocity dispersion map is rich with subtle details and looks quite unusual: the nucleus is not distinguished, neither by a peak nor by a minimum of $\sigma _{*}$, but two arc-like regions – one of low stellar velocity dispersion at $R\approx 4\arcsec - 6\arcsec$ and another of high stellar velocity dispersion at a larger radius – prevent any reasonable interpretation of this data.
Figure 17 shows the ionized-gas velocity field and the 2D distributions of the emission-line surface brightness for \[NII\]$\lambda$6583 and H$\alpha$ which are obtained with MPFS. Again, the gas rotates rather regularly and as a solid body up to the edges of the area investigated. As for the emission maps, here there is a surprise: whereas the \[NII\] emission distribution repeats roughly the shape of the continuum isophotes, the H$\alpha$ emission-line intensity distribution is elongated orthogonally, approximately along the minor axis of the inner isophotes. Is this structure real? Let us appeal to high-resolution imagery. Figure 18 shows, firstly, the WFPC2/HST map of NGC 3368 in the visual regime and secondly, the NICMOS/HST color map (close to $J-H$). One can see a quasi-polar compact dust ring encircling the north-west part of the central elongated structure (of a minibar?). The dust (and gas?) density in this polar ring is so high that it is seen very well even at wavelengths longer than 1 mkm; it manifests itself at the color map as a red lane aligned in $PA\approx 30^{\circ} -
35^{\circ}$ to the north-west from the blue nucleus. Since this dust polar ring is traced by the H$\alpha$ emission, but is unseen in the \[NII\] emission line, we may suppose that it is a site of intense current star formation. Why is it polar? Inner gas polar structures are often met in the galaxies with a triaxial central potential as we have already mentioned when considering NGC 3384.
Let us quantify, as is usually done, a characteristic of rotation of the stars and ionized gas in the center of NGC 3368. In Fig. 19 we have plotted some selected azimuthal dependencies of the central LOS velocity gradients for the stars in radius ranges of $2\farcs 0 - 2\farcs 8$ and of $3\farcs
0 - 4\farcs 2$ and for the ionized gas in a radius range of $2\farcs 3 - 3\farcs 3$; for the stars we have plotted both the February and the March MPFS data, their agreement being excellent. The stars rotate slightly slower than the gas: $19\pm 2$ km/s/arcsec versus $26.4 \pm 4.2$ km/s/arcsec. The long-slit observations by @vega also confirm this kinematical feature. It is natural because the stellar velocity dispersion in the center of NGC 3368 is larger than 100 km/s, and the emission lines are narrow. However the orientation of the kinematical major axes for the ionized gas and for the stars are rather similar: $176^{\circ} \pm 2^{\circ}$ for the former and $167.5^{\circ} \pm
1.5^{\circ}$ for the latter subsystem. Indeed, we have no serious reasons to treat the gas and stellar rotations as different – they may rotate together. The coincidence of the kinematical major axes near $PA_{kyn} \approx 170^{\circ}$, an orientation which is not marked in any way in the central part of NGC 3368, is very strange. One could understand a rotation of the central ionized gas in the plane towards $PA=170^{\circ}$, because the outermost neutral hydrogen is distributed in an extended disk with just this line of nodes [@leoh1_3]. Meanwhile the molecular gas in the central part of the galaxy demonstrates elliptical isodensities elongated in $PA\approx 40^{\circ}$ [@co99] – the direction that coincides with the line of nodes of the dust polar ring; but the velocity field of the molecular gas over the entire area covered by mapping resembles rather a prolate rotation, with $PA_{kyn} \approx 170^{\circ}$ [@co99]. So, once more, for the ionized gas the orientation of the rotational plane at $PA_{kyn}=170^{\circ} -175^{\circ}$ is kinematically understandable; but among the visible [*stellar*]{} structures in NGC 3368 there are none with a distinguishing orientation at this position angle – see Figs. 20 and 21. Figure 20 shows a comparison of the NICMOS/HST isophote characteristics with orientation of the kinematical major axes. The $PA_{kyn}\approx 170^{\circ}$ holds in the radius range of $2\arcsec -
5\arcsec$ whereas the orientation of the isophotes holds at $PA_{phot}\approx 125^{\circ}$ (closer to the nucleus the isophotes twist by more than $100^{\circ}$, and we think this twist to be caused by an effect of the dust polar ring projection). @mois02 have argued from the azimuthal Fourier analysis of the large-scale surface brightness distributions that the line of nodes of the inner ($R<140\arcsec$) disk of NGC 3368 is $PA_0=135^{\circ}$ and that the value of $5^{\circ}$ given by LEDA may be related only to the very outermost part of the galaxy. So we are convinced that in the center of NGC 3368 we see a minibar with a semimajor axis of $5\arcsec$; however non-circular streaming motions around the bar with $PA_{bar}=125^{\circ}$ located in the disk with $PA_0=135^{\circ}$ would cause a twist of the kinematical major axis from the line of nodes in a positive sense, as observed, but only by about $10^{\circ}$ [@mbe92]. Meanwhile it turns by at least $30^{\circ}$, and therefore this turn cannot be explained by a simple dynamical effect of minibar triaxiality. Do we observe a rotation of a rather young faint stellar subsystem which is formed from the circumnuclear gas and shares its spin? It would be possible if this subsystem is much colder than the bulge dominating the surface brightness.
Fig. 21 shows large-scale radial variations of the isophote characteristics of NGC 3368 to illustrate its very complex structure. While the surface brightness profile of the galaxy looks very regular, with a single-scale exponential disk dominating the bulge at $R > 22\arcsec$ [@kent85], the isophote orientation and ellipticity change all the way up to the optical edges of the galaxy, implying either disk oval distortions or a strong disk warp. As mentioned in Section 2 we used the scanning Fabry-Perot data to study the large-scale ($r=100\arcsec -200\arcsec$) kinematics of the ionized gas in two emission lines (H$\alpha$ and \[NII\]$\lambda$6583). The velocity fields in both lines are shown in Fig. 22. Unfortunately stellar absorption and interference orders which overlap distort the emission line profiles in the H$\alpha$ datacube over the central region of the galaxy. We cannot resolve this problem because the free spectral range of the IFP is small ($\sim28$Å). Therefore the H$\alpha$ line-of-sight velocities were measured only in the star-forming ring (r=$50\arcsec -90\arcsec$) and in fragments of spiral arms. But the \[NII\] velocity field was constructed over the full range in radius (Fig. 22). The shape and orientation of the isovelocity contours at $r<7\arcsec$ demonstrate an agreement with the MPFS velocity field in \[NII\]$\lambda$6583 of Fig 17. The ‘tilted-ring’ analysis of our Fabry-Perot velocity fields confirms the stable orientation of the ionized gas kinematical major axis at $PA_{kyn}=170^{\circ} -175^{\circ}$ over the global disk of the galaxy out to a distance of $200\arcsec$ from the center. So the global dust-gaseous disk of the galaxy is probably decoupled completely from the stellar one, having its own orientation in space. Since an HI bridge is seen between the intergalactic gaseous ring and NGC 3368 [@leoh1_2], we suggest that all the gas in this spiral galaxy is of recent external origin and is not yet relaxed with respect to the main symmetry plane of the galaxy. We must stress the good agreement between the measurements of the $PA_{kyn}$ in two independent IFP velocity fields; but also we note a discrepancy of the rotational velocity amplitudes. The differences between the rotational velocities in the H$\alpha$ and \[NII\] (Fig.22) may be explained in the frame of the hypothesis of the inclined gaseous disk. As it is shown in Fig.22, the \[NII\] rotation velocity at $r=50\arcsec -90\arcsec$ is systematically slower by $\sim 50\, \mbox{km} \cdot \mbox{s}^{-1}$ than the H$\alpha$ one (under the assumption of a disk inclination of $i=48^\circ$). If the gaseous disk is inclined to the main galaxy plane then shock-wave fronts can exist at the cross-section of the global stellar and gaseous disks, because gas strikes the gravitational well. Moreover, the strong stellar spiral arms in this region intensify the contrast of the gravitational potential. Therefore the low \[NII\] velocities may be explained if the part of the nitrogen emission line is emitted by shock-excited gas slowed down by collisions with the stellar disk and spiral arms.
Discussion
==========
We have considered the three brightest galaxies in the vicinity of the intergalactic gaseous ring in the Leo I group. We have found that all three galaxies have peculiarities, or extra components, in their centers; and the space orientations of these extra components are related to the line of nodes of the intergalactic gaseous ring, $PA_{ring}\approx 40^{\circ}$, in all three cases. NGC 3384 has a circumnuclear stellar disk with the line of nodes at $PA=45^{\circ}$, inclined to the main symmetry plane of the galaxy but aligned with the global intergalactic gaseous ring. NGC 3368 has a polar (relative to the central minibar) dust-gaseous ring with a projected major axis at $PA\approx 35^{\circ}$, and its CO distribution is also elongated in $PA\approx 40^{\circ}$. NGC 3379 has a circumnuclear dust-(and gaseous) ring (and probably a stellar disk) with the major axis at $PA=120^{\circ} - 125^{\circ}$, orthogonal to the intergalactic ring orientation; but it is the only ellitical galaxy among ours, with a clear signature of triaxiality [@n3379ls], so its triaxial main body may provoke gas transfer on to a polar orbit when spiraling to the center. These are too many coincidences, if one takes into account the diversity of the global major axis orientations (see Table 1). We can suggest that all the circumnuclear structures mentioned above have the origin related to accretion of the gas from the intergalactic ring; in this case the spin conservation would provide alignment of the circumnuclear disks at $PA_0=40^{\circ}$, except in NGC 3379. Geometry and sizes of the circumnuclear structures must depend strongly on the global structures of the galaxies under consideration, in particular on their triaxiality and spheroid/disk ratios. And finally, the closely related mean ages, 3 Gyr, of the nuclear stellar populations in NGC 3368 and NGC 3384 (in NGC 3379 it may seem older because of the larger contribution of the old spheroid to the integrated spectrum) imply that a characteristic time between accretion events, or a timescale for interaction between the galaxies and the intergalactic ring, is of the order of several Gyrs, or of order the revolution time of the intergalactic ring which is 4 Gyr, or of order of the crossing time in the Leo I group. This result rejects conclusively the scenario of @roodwil who proposed a collision between NGC 3368 and NGC 3384 only 0.5 Gyr ago.
We are very grateful to Dr. Elias Brinks for editing the manuscript. Also we thank Dr. S.N. Dodonov for supporting the observations of NGC 3384 at the 6m telescope. The 6m telescope is operated with financial support from the Science Ministry of Russia (registration number 01-43); we thank also the Programme Committee of the 6m telescope for allocating observing time. During the data analysis stage we have used the Lyon-Meudon Extragalactic Database (LEDA) supplied by the LEDA team at the CRAL-Observatoire de Lyon (France) and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This work is partially based on data taken from the ING Archive of the UK Astronomy Data Centre and on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. The study of the young nuclei in lenticular galaxies is supported by a grant of the Russian Foundation for Basic Research 01-02-16767 and the study of the evolution of galactic centers – by the Federal Scientific-Technical Program – contract of the Science Ministry of Russia no.40.022.1.1.1101.
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abstract: '[The $1/N_c$ rotational corrections to the axial vector constant and the isovector magnetic moment of the nucleon are studied in the Nambu – Jona-Lasinio model. We follow a semiclassical quantization procedure in terms of path integrals in which we can include perturbatively corrections in powers of angular velocity $\Omega \sim \frac 1{N_c}$. We find non-zero $1/N_c$ order corrections from both the valence and the Dirac sea quarks. These corrections are large enough to resolve the long-standing problem of a strong underestimation of both $g_A$ and $\mu^{IV}$ in the leading order. The axial constant $g_A$ is well reproduced, whereas the isovector magnetic moment $\mu^{IV}$ is still underestimated by 25 %.]{}'
address: |
$^{1)}$Institut für Theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum, Germany\
$^{2)}$St.Petersburg Nuclear Physics Institute, Gatchina, St.Petersburg 188350,Russia\
$^{3)}$Department of Physics, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan\
$^{4)}$Department of Physics, Tokyo Metropolitan University, Hachiohji, Tokyo 192, Japan
author:
- |
Chr.V.Christov$^{1)}$[^1], K.Goeke${^1)}$, P.Pobilitsa$^{1,2)}$, V.Petrov$^{2)}$,\
M.Wakamatsu$^{1,3)}$ and T.Watabe$^{1,4)}$
title: |
$1/N_c$ Rotational Corrections to $g_A$ and\
Isovector Magnetic Moment of the Nucleon
---
\#1\#2[[\#1 \#2]{}]{} \#1\#2
\#1[\#1]{} \#1[\#1]{} \#1 \#1[\#1 ]{} \#1[ \#1 ]{} \#1\#2[\#1 \#2 ]{} \#1 ‘=11
\#1
‘=12
Among the effective low-energy models of baryons the chiral soliton model, based on a semibosonized version [@Eguchi76] of the Nambu – Jona-Lasinio lagrangean [@Nambu61], seems to play an essential role. In fact, an equivalent effective chiral quark-meson lagrangean can be derived [@Diakonov84] from the instanton model of the QCD vacuum. This chiral soliton model, which is frequently referred simply as NJL model, incorporates the general accepted phenomenological picture of the baryon as a bound state of $N_c$ valence quarks coupled to the polarized Dirac sea ($\overline qq$ pairs). Operationally in the model the nucleon problem is solved in two steps [@Diakonov89]. In the first step, motivated by the large $N_c$ (number of colors) limit, a static localized solution (soliton) of a hedgehog structure is found. This hedgehog solution does not preserve the spin and isospin. Making use of the rotational zero modes in a semiclassical quantization scheme one can assign proper spin and isospin quantum numbers to the soliton. The model is rather successful in describing both the static nucleon properties [@Diakonov89; @Reinhardt89W; @TMeissner89; @Goeke91; @Wakamatsu91] and the nucleon form factors [@TMeissner91; @Gorski92]. The only exceptions are the axial coupling constant $g_A$ and the isovector magnetic moment $\mu^{IV}$, which are strongly underestimated ($g_A\approx 0.8$ and $\mu^{IV}\approx 2.4$(n.m.)) in the leading order, compared to the experimental values of 1.25 and 4.71 (n.m.). This is not a problem just of the NJL model. In fact, most of the chiral models, if they assume physical values of the pion decay constant $\fpi$ and the pion mass $\mpi$, yield rather strong deviations from the experimental values for these quantities. In the case of the NJL model the simplest solution to this problem is to take into account the rotational corrections coming from the next to leading order terms.
In a very recent paper Wakamatsu and Watabe [@Wakamatsu93] estimated in the NJL model $1/N_c$ rotational corrections to $g_A$ thus providing a step in the right direction. They found a considerable non-zero valence contribution ($\approx 0.4$). In this work they made the important observation that in the case of $g_A$, after the canonical quantization, the collective operators do not commute. However, in their scheme the non-zero result is due to a particular order of the collective operators being not fully justified by path integrals or many-body techniques. As a consequence the valence contribution includes transitions between occupied levels, which violate the Pauli principle (even though, numerically, this Pauli violating contribution turns out to be only a tiny fraction of the valence contribution), and the Dirac sea $1/N_c$ contribution vanishes exactly. The latter is particularly puzzling since apart from the regularization there is no any principle difference between the valence and the sea quarks in the NJL model. Based on the formulae of Wakamatsu and Watabe [@Wakamatsu93] Alkofer and Weigel [@Alkofer93] studied the axial coupling constant in the context of the PCAC. From their values for $g_A$ one can estimate how far the PCAC holds up in the linear order in $\Omega$ and in particular, a violation of less than 2 % for a reasonable value of the constituent mass $M=400$ MeV can be guessed.
Obviously a satisfying theoretical and numerical treatment of higher order rotational corrections in the semiclassical quantization scheme is still missing. It is therefore the objective of the present work to evaluate the $1/N_c$ rotational corrections to both the axial coupling constant and the isovector magnetic moment in the NJL model. To that end we will follow the theoretical scheme of Diakonov et al. [@Diakonov89] elaborated in terms of path integrals.
We start with the definitions of axial and isovector magnetic currents of a fermion field $\Psi(x)$, $A_k^a(x)=\Psi^+(x)\gamma_0
\gamma_k\gamma_5\frac {\tau^a}2 \Psi(x)$ and $J_k^a
(x)=\Psi^+(x)\gamma_0\gamma_k \frac {\tau^a}2 \Psi(x)$, respectively. Here $k$ means space components and $a$ stands for the isospin index. We express the nucleon matrix element of a current operator $\hat A_k^a$ as a path integral including quark $\Psi,\Psi^+$ and meson fields $U$ in Minkowski space: $$\begin{aligned}
&\bra{N({\bf p\prime})}\Psi^+(0)\hat O_k^a\Psi(0)\ket{N({\bf p})}
\mathop{=}\limits_{T\to-i\infty} \frac 1Z\int d^3xd^3ye^{-i{\bf
p^\prime\cdot x}}e^{i{\bf p\cdot y}}\nonumber\\
&\times\int{\cal D}U\int {\cal D}\Psi {\cal D}\Psi^+
J_N(T/2,{\bf x})J_N^+(-T/2,{\bf y}) \Psi^+(0)\hat O_k^a\Psi(0)
e^{\textstyle i\int d^4z\Psi^+D(U)\Psi} \label{Eq2}\end{aligned}$$ The equality (\[Eq2\]) should be understood as a limit at large euclidean time separation. Here $Z$ is the normalization factor which is related to the same path integral but without the current operator $\Psi^+\hat O_k^a\Psi$, and $\hat O_k^a$ stands for the matrix part $\gamma_0\gamma_k(\gamma_5)\frac {\tau^a}2$ of the current. The Dirac operator $D(U)=i\partial_t-h(U)$ includes the single-particle hamiltonian $h(U)=\frac
{\bm{\alpha}\cdot\bm{\nabla}}i+M\beta U^{\gamma_5}+m_0\beta$ with meson fields $U^{\gamma_5}=e^{\textstyle i\gamma_5\bm{\tau}\cdot\bm{\pi}}$. Here $\bm{\alpha}$ and $\beta$ are the Dirac matrices and $m_0$ being the current quark mass. The composite operator: $$J_N(x)=\frac
1{N_c!}\varepsilon^{\beta_1\cdots\beta_{N_c}}\Gamma^{\{f_1\cdots
f_{N_c}\}}_{JJ_3,TT_3}
\Psi_{\beta_1f_1}(x)\cdots\Psi_{\beta_{N_c}f_{N_c}}(x),
\label{Eq6}$$ carries the quantum numbers $JJ_3,TT_3$ (spin, isospin) of the nucleon, where $\beta_i$ is the color index, and $\Gamma^{f_1\cdots f_{N_c}}_{
JJ_3,TT_3}$ is a symmetric matrix in flavor and spin indices $f_i$.
In eq.(\[Eq2\]) we can integrate the quarks out: $$\begin{aligned}
&\bra{N({\bf p^\prime})}\Psi^+(0)\hat O_k^a\Psi(0)\ket{N({\bf p})}=\frac 1Z
\Gamma^{\{g\}}_{JJ_3,TT_3}\Gamma^{\{f\}}_{JJ_3,TT_3}N_c \int d^3x d^3y
e^{-i{\bf p^\prime\cdot x}}e^{i{\bf p\cdot y}} \int{\cal D}U\nonumber\\
&\times\bigl\{\bra{T/2,{\bf x}}\frac iD \ket{0,0}_{f_1f^\prime}(\hat
O_k^a)_{f^\prime g^\prime}\bra{0,0} \frac iD \ket{-T/2,{\bf
y}}_{g^\prime g_1}-\Sp(\bra{0,0}\frac iD\ket{0,0}\hat O_k^a)\nonumber\\
&\times\bra{T/2,{\bf x}}\frac iD\ket{-T/2,{\bf y}}_{f_1g_1}\bigr\}
\prod\limits_{i=2}^{N_c}\bra{T/2,{\bf x}}\frac
iD\ket{-T/2,{\bf y}}_{f_ig_i}e^{\textstyle\tr\log D(U)}.
\label{Eq6a} \end{aligned}$$ In a natural way the result is split in a valence – the first term, and a Dirac sea contribution – the second one (see the diagrams on the l.h.s. of Fig.\[Figr1\] a) and b)).
In order to integrate over the meson fields $U$ we start from a stationary meson configuration of hedgehog structure $\bar
U(x)=e^{\textstyle i\bm{\tau}\cdot\hat{\bf x}\pi(x)}$ which minimizes the effective action. Then the integration over the meson fields $U$ in the path integral can be done in a saddle point approximation, which is motivated by the large $N_c$ limit. In the next step we should allow the system to fluctuate around the static hedgehog solution $\bar U(x)$ making use of the rotational zero modes. Since the fluctuations which correspond to the zero modes are not small they have to be treated “exactly” in the meaning of path integral. Operationally it can be done introducing a rotating meson fields of the form $U({\bf x},t)=R(t)\bar U({\bf x}) R^+(t)$, where $R(t)$ is a time-dependent rotation SU(2) matrix in the isospin space. It is easy to see that for such an ansatz one can transform the effective action $\quad\tr\log D(U)=\tr\log(D(\bar U)-\Omega)\quad$ as well as the quark propagator in the background meson fields $U$ $$\bra{T/2,{\bf x}}\frac i{D(U)} \ket{-T/2,{\bf y}}=R(T/2)\bra{T/2,{\bf
x}}\frac i{D(\bar U)-\Omega} \ket{-T/2,{\bf y}}R^+(-T/2),
\label{Eq10b}$$ where $\Omega=-iR^+(t)\dot R(t)=\frac 12\Omega_a\tau_a$ is the angular velocity matrix. Since $\Omega\sim \frac 1{N_c}$ (as can be seen below) one can consider $\Omega$ as a perturbation and evaluate any observable as a perturbation series in $\Omega$ which is actually an expansion in $\frac 1{N_c}$.
In this scheme the matrix element (eq.\[Eq2\]) of the current can be written as $$\begin{aligned}
&\bra{N({\bf p\prime})}\Psi^+(0)\hat O_k^a\Psi(0)\ket{N({\bf p})}=\frac 1Z
N_c\int
d^3xd^3y d^3ze^{-i{\bf p^\prime\cdot x}}e^{i{\bf p\cdot y}} e^{i{\bf
(p^\prime-p)\cdot z}}
\int{\cal D}R \nonumber\\
&\times{D^J_{-T_3J_3}}^*(R(T/2))
\bigl\{\bra{T/2,{\bf x}}\frac i{D-\Omega}\ket{0,{\bf z}}R^+(0) \hat
O_k^a R(0) \bra{0,{\bf z}}\frac i{D-\Omega} \ket{-T/2,{\bf y}}\nonumber\\
&-\Sp(\bra{0,{\bf z}}\frac i{D-\Omega}\ket{0,{\bf z}}R^+(0)\hat
O_k^aR(0))\bra{T/2,{\bf x}}\frac i{D-\Omega} \ket{-T/2,{\bf
y}}\bigr\}\nonumber\\
&\times\bra{T/2,{\bf x}}\frac i{D-\Omega}\ket{-T/2,{\bf y}}^{\textstyle 2}
D^J_{-T_3J_3}(R(-T/2)) e^{\textstyle\tr\log(D-\Omega)}.
\label{Eq12} \end{aligned}$$ Here, the finite rotation matrix $D^J_{-T_3J_3}$, which carries the spin and isospin quantum numbers of the nucleon, appears due to the rotations $R(t)$ of the valence quark propagators in eq.(\[Eq10b\]) correlated by the $\Gamma^{\{g\}}_{JJ_3,TT_3}$ matrices and the integral over ${\bf z}$ is due to the translational zero modes treated in the leading order.
Now we are ready to make an expansion in $\Omega$. For the effective action, it is well-known procedure [@Diakonov89; @Goeke91; @Wakamatsu91; @Gorski92], which yields up to the second order in $\Omega$: $$\tr\log{(D-\Omega)}\approx \tr\log{D}+i\frac {\Theta}2\int dt\Omega_a^2.
\label{Eq14}$$ Here $\Theta$ is the moment of inertia. The first term will be absorbed in $Z$ whereas the second one gives the evolution operator acting in the space of matrix $R$. Expanding the quark propagator $$\frac 1{D-\Omega}\longrightarrow\frac 1D +\frac 1D\Omega\frac 1D+...$$ we can separate the zero order ($\sim N_c^0$) and the linear order ($\sim \frac 1{N_c}$) corrections in $\Omega$. The expansion in $\Omega$ is illustrated in Fig.\[Figr1\] a) and b) for the valence contribution and for the Dirac sea one, respectively.
Henceforward we will concentrate on the linear order terms. In this case we are left with the following path integral over $R$: $$\int\limits_{R(-T/2)}^{R(T/2)}{\cal
D}R{D^J_{-T_3J_3}}^*(R(T/2))\frac 12\Sp(R^+(0)\tau^a
R(0)\tau^b)\Omega_c(t) D^J_{-T_3J_3}(R(-T/2))
e^{\textstyle i\frac{\Theta}2\int dt\Omega_c^2}. \label{Eq16}$$ Here we use the identity $\quad (R^+(0)\hat O_k^a R(0))_{fg}=\frac
12\Sp(R^+(0)\tau^a R(0)\tau^b)(\hat O_k^b)_{fg}\quad$ in order to separate the $R(t)$ dependent part of the current which does not carry flavor indices $fg$. The path integral (\[Eq16\]) can be taken [@Diakonov89] rigorously within the approximation (\[Eq14\]). We obtain the well-known canonical quantization rule $\quad\Omega_c\rightarrow J_c/\Theta$, where $J_a$ is the spin operator. The final result for path integral (\[Eq16\]) is a time ordered product: $$\vartheta(-t)D_{ab}(R(0))J_c+\vartheta(t)J_c D_{ab}(R(0)).
\label{Eq19}$$ which should be sandwiched between the nucleon rotational wave function. In order to obtain the result (\[Eq19\]) we essentially made use of the basic feature of the path integral: $$\int\limits_{q_1=q(T_1)}^{q_2=q(T_2)}{\cal D}q
F_1(q(t_1))\cdots F_n(q(t_n))e^{iS}=\bra{q_2,T_2}T\{\hat
F_1(q(t_1))\cdots\hat F_n(q(t_n)\}\ket{q_1,T_1},$$ namely that the path integral can be equivalently written as the expectation value of the time ordered product of the corresponding operators. Using (\[Eq19\]) and the standard spectral representation of the quark propagator it is straightforward to evaluate the matrix element of the current eq.(\[Eq12\]). In particular for the linear correction we get $$\bra{N(p^\prime)}\Psi^+(0)\hat O_k^a\Psi(0)\ket{N(p)}^{\Omega^1}=
\bra{J,J_3T_3}[\frac {J_c}\Theta,D_{ab}]\ket{J,J_3T_3}$$ -0.5cm $$\times N_c\sum\limits_{{n > val}\atop{m\leq val}}\frac
1{\epsilon_n-\epsilon_m}\bra{m}\tau_a\ket{n}\int d^3ze^{i{\bf
(p^\prime-p)\cdot z}}\Phi^+_n({\bf z})\hat O_k^b\Phi_m({\bf z}).
\label{Eq20}$$ Here $\Phi_n$ and $\epsilon_n$ are the eigenfunctions and eigenvalues of the single-particle hamiltonian $h$. Since the $\tau$-matrix element is asymmetric with respect to exchange of the states $m$ and $n$ a commutator appears in the collective matrix element of eq.(\[Eq20\]). It can be easily calculated: $$\bra{J=1/2,J_3T_3}[\frac {J_c}\Theta,D_{ab}]\ket{J=1/2,J_3T_3}=-\frac
13\frac {i}{\Theta} \varepsilon_{cb3}\delta_{a3} .
\label{Eq21}$$ Using that both the axial coupling constant and the isovector magnetic moment are related to the corresponding form factors at $q^2=0 $ finally we get for the $1/N_c$ rotational corrections: $$g_A(\Omega^1)=\frac {N_c}{9}\frac {i}{2\Theta} \sum_{\scriptstyle n >
val\atop {\scriptstyle m\leq val}}\frac
1{\epsilon_n-\epsilon_m}\bra{m}\tau_a\ket{n}
\bra{n}[\bm{\sigma}\times\bm{\tau}]_a\ket{m}
\label{Eq22}$$ and $$\mu^{IV}(\Omega^1)=\frac {N_c}9\frac {i}{2\Theta} \sum_{\scriptstyle
n > val\atop {\scriptstyle m\leq val}}\frac
1{\epsilon_n-\epsilon_m}\bra{m}\tau_a\ket{n}
\bra{n}\gamma_5[[\bm{\sigma}\times\bm{x}]\times\bm{\tau}]_a\ket{n}
\label{Eq23}$$ In both eqns.(\[Eq22\]) and (\[Eq23\]), a summation over $a$ is assumed. As should be expected both expressions have similar structure with transitions from occupied to non-occupied levels and back. They include also an essential non-zero contribution from the Dirac sea. In contrast to the leading order the above expressions are finite and one does not need to regularize them.
The parameters of the model are fixed in the meson sector to reproduce $\fpi=93$ MeV and $\mpi=139.6$ MeV. Similar to other works [@Reinhardt89W; @TMeissner89; @Wakamatsu91] we use a numerical procedure based on the method of Kahana and Ripka [@Ripka84]. The results for $g_A$ and $\mu^{IV}$ up to the first order terms in $\Omega$, are presented in Fig.\[Figr2\] and Fig.\[Figr3\] as a function of the constituent quark mass $M$. As can be seen in leading order ($\Omega^0$) they are almost independent of the constituent quark mass and the valence contribution is dominant. In the next to leading order ($\Omega^1$) the valence and Dirac sea contributions show much stronger and quite different mass dependence: with increasing constituent mass $M$ the valence part gets reduced whereas the Dirac sea part increases and becomes dominant. However, their sum shows almost no dependence on $M$. The result of Wakamatsu and Watabe for $g_A$ (labeled as W&W in Fig.\[Figr2\]) deviates from the present numbers. This is due to the fact that in their scheme the contribution of the Dirac sea to $g_A$ in the linear order in $\Omega$ vanishes exactly. In the present calculations for both quantities the enhancement due to the $1/N_c$ rotational corrections improves considerably the agreement with experiment. In the case of $g_A$ the experimental value is almost exactly reproduced and the inclusion of next order corrections will perhaps even overestimate it. In contrast to $g_A$ for the isovector magnetic moment $\mu^{IV}$ we are still below the experimental value by 25 %. It is interesting to notice that the enhancement for both quantities due to the $1/N_c$ rotational corrections is very close to the estimate $\frac {N_c+2}{N_c}$ [@Blaizot88].
To conclude, we have evaluated the axial coupling constant $g_A$ and the isovector magnetic moment $\mu^{IV}$ in the Nambu – Jona-Lasinio model in the next to leading order in the semiclassical quantization scheme. The $1/N_c$ rotational corrections are large enough to resolve the problem of strong underestimation of these two quantities in the leading order. In particular, $g_A$ is almost exactly reproduced. However, such large linear order corrections imply that in order to control the perturbation series in $\Omega$ the next order ($1/N_c^2$) corrections should be investigated as well.
[99]{}
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A.Gorski, F.Grümmer and K.Goeke, Phys.Lett.[**278B**]{}(1992)24; A.Gorski, Chr.V.Christov, F.Grümmer and K.Goeke, Bochum Preprint RUB-TP2-56/93 (submitted to Nucl.Phys.[**A**]{})
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[^1]: Permanent address:Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria
|
---
abstract: 'We present the results of the first multi-scale N-Body+SPH simulations of merging galaxies containing central supermassive black holes (SMBHs) and having a spatial resolution of only a few parsecs. Strong gas inflows associated with equal-mass mergers produce non-axisymmetric nuclear disks with masses of order $10^9 M_{\odot}$, resolved by about $10^6$ SPH particles. Such disks have sizes of several hundred parsecs but most of their mass is concentrated within less than $50$ pc. We find that a close SMBH pair forms after the merger. The separation of the two SMBHs then shrinks further owing to dynamical friction against the predominantly gaseous background. The orbits of the SMBHs decay down to the minimum resolvable scale in a few million years for an ambient gas temperature and density typical of a region undergoing a starburst. These results suggest the initial conditions necessary for the eventual coalescence of the two holes arise naturally from the merging of two equal-mass galaxies whose structure and orbits are consistent with the predictions of the $\Lambda$CDM model. Our findings have important implications for planned gravitational wave detection experiments such as [*LISA*]{}.'
author:
- 'Lucio Mayer, Stelios Kazantzidis, Piero Madau , Monica Colpi, Thomas Quinn'
- James Wadsley
title: 'Multi-scale simulations of merging galaxies with supermassive black holes'
---
Introduction {#sec:1}
============
Recent observations of molecular gas in the nuclear region of candidate merger remnants such as starbursting ultraluminous infrared galaxies (ULRIGs) reveal the presence of rotating gaseous disks which contain in excess of $10^9 M_{\odot}$ of gas within a few hundred parsecs (\[3\],\[5\]). Some of these galaxies, such as Mrk 231, also host a powerful AGN. The high central concentration of gas is likely the result of gaseous inflows driven by the prodigious tidal torques and hydrodynamical shocks occurring during the merger (\[1\], \[7\]), and possibly provides the reservoir that fuels the SMBHs. If both the progenitor galaxies host a SMBH the two holes may sink and eventually coalesce as a result of dynamical friction against the gaseous background. The sinking of two holes with an initial separation of $400$ pc in a nuclear disk described by a Mestel model has been studied by \[6\] and \[4\]. Here we avoid any assumption on the structure of the nuclear gaseous disk and initial separation of the pair: rather, we follow the entire merging process starting from when the cores of the two galaxies are hundreds of kiloparsecs apart up to the point where they merge, produce a nuclear disk, and leave a pair of SMBHs separated by the adopted force resolution of $10$ pc. Previous hydrodynamical simulations of merging galaxies with SMBHs have not followed the evolution of the central region below a scale of a few hundred parsecs due to their limited mass and force resolution (\[11\], \[7\]). In this study we are able for the to extend the dynamic range of previous works by orders of magnitude using the technique of particle splitting.
The Numerical Simulations {#sec:2}
=========================
Our starting point are the high-resolution simulations presented in \[7\] which, as the new simulations presented here, were performed with the parallel tree+SPH code GASOLINE \[13\]. In particular we refer to those simulations that followed the merger between two equal-mass, Milky-Way sized early type spirals having 10% of the mass of their exponential disk in a gaseous component and the rest in stars. The structural parameters of the two disks and their NFW dark matter halos, as well as their initial orbits, are motivated by the results of cosmological simulations. We apply the static splitting of SPH particles \[8\] to increase the gas mass resolution in such calculations (the stars and dark matter resolution remain the same), and reduce the gravitational softening accordingly as we increase the mass resolution. We select an output about $50$ Myr before the merger is completed, when the cores of the two galaxies are still $\sim 6$ kpc away, and split each SPH particle into $8$ children, reaching a mass resolution of about 3000 $M_{\odot}$ in the gas component (Figure 1). The gravitational softening of the gas particles is decreased from $200$ pc to either $40$ pc, $10$ pc, or $2$ pc (one run is performed for each of the three different softenings). With the new mass resolution even for the smallest among the gravitational softenings considered the number of SPH particles within a sphere of radius equal to the local Jeans length is much larger than twice the number of SPH neighbors (=32), thus avoiding spurious fragmentation (\[2\]). The two SMBHs are point masses with a softening length set equal to $10$ pc and a mass of $3 \times 10^6 M_{\odot}$. The simulations presented here do not include star formation but gas masses are only a factor of $3$ higher than those measured in the star formation simulations of \[7\] at the same evolutionary stage.
![Color coded density map of the nuclear region $50$ Myr before the merger (top panels) and just after the merger (bottom panels). The top panels show a box $30$ kpc on a side (left) and a zoom-in within the inner $6$ kpc (right).The bottom panels show the inner 300 pc, with the disk seen face-on (left) and edge-on (right). An adiabatic equation of state with $\gamma=7/5$ was used in this run.[]{data-label="fig:1"}](fig1.eps){height="12cm"}
We have run a suite of simulations with different prescriptions for the gas thermodynamics, and show here the results of two runs in which radiative cooling and heating processes are not included directly, rather an adiabatic equation of state with either $\gamma=7/5$ or $\gamma=5/3$ is adopted (irreversible shock heating, which is important during the merging phase, is included via an artificial viscosity term in the energy equation). According to the radiative transfer calculations of \[10\] the case $\gamma=7/5$ approximates quite well the balance between radiative heating and cooling in a starburst galaxy (in \[7\] a central starburst indeed does occur in the final phase of the merger that we are considering here). A stiffer equation of state such as that with $\gamma=5/3$ might instead be relevant when an additional strong heating source, for example AGN feedback, comes into play (\[11\]). Although one should follow directly the various cooling and heating mechanisms, this simple scheme can provide us with a guide of how gas thermodynamics can affect the results.
Results {#sec:3}
=======
Our simulations allow to assess in a self-consistent way the evolution of the nuclear region of a gas-rich remnant of a major merger as well as the orbital evolution of SMBHs in the nuclei of the two merging galaxies.
Gas Inflows and the Structure of the Nuclear Disks
--------------------------------------------------
About 80% of the gas originally belonging to the two galaxies is funneled to the central kiloparsec during the last stage of the merger and settles into two rotationally supported disks. When the cores of the two galaxies finally merge, the two disks also merge into a single gaseous core which rapidly becomes rotationally supported as radial motions are largely dissipated in shocks. The disk however remains non-axisymmetric, with evident bar-like and spiral patterns (Figure 1). A coherent thick disk forms independent of the relative inclination of the initial galactic disks, albeit the orientation of its angular momentum vector relative to the global angular momentum will change depending on those initial parameters (\[7\]). In the run with $\gamma=7/5$, that was designed to reproduce the thermodynamics of a starburst region, the disk has a vertical extent of about $20$ pc and $v_{rot}/\sigma > 1$ out to about $600$ pc. The thickness is about $5$ times higher in the run with $\gamma=5/3$. The scale height in the $\gamma=7/5$ run is comparable to that of the disks in the multi-phase simulations of a $1$ kpc-sized nuclear region performed by \[12\]. The simulations of \[12\] include radiative cooling and resolve the turbulence generated from supernovae explosions as well as gravitational instability, suggesting that our equation of state yields a characteristic pressure scale that accounts for the combined thermal and turbulent pressures.
![Dependence of gas mass inflow on force resolution. The curves show the cumulative radial gas mass profile (normalized to the total gas mass contained within a sphere of $10$ kpc in radius) for three runs using an adiabatic equation of state with $\gamma=7/5$ and different values of the gravitational softening of the gas particles (solid line=$40$ pc, dashed line=$10$ pc, dot-dashed line=$2$ pc).[]{data-label="fig:2"}](fig2.eps){height="9cm"}
Gravitational torques and the balance between gravity and the thermodynamical pressure at small scales depend ultimately on the adopted gravitational softening. We verified that the mass inflow seen in the simulations for a given value of $\gamma$ converges as the softening approaches $10$ pc (see Figure 2). Convergence in the disk vertical extent is also observed at such a spatial resolution. Figure 2 shows that at high resolution more than 60% of the mass piles up within as little as $30$ pc.
Pairing of SMBHs
----------------
The merger between the two galactic cores delivers a close but unbound pair of SMBHs. The pair is separated by about $100$ pc and is embedded within the newly formed nuclear gaseous disk. Up to this point the orbital decay of the two SMBHs had been equivalent to that of the cores in which they were embedded, and was driven by dynamical friction of the cores within the surrounding [*collisionless*]{} background of stars and dark matter. Once inside the massive gaseous disk the orbital decay of the two holes is dominated by dynamical friction in a [*gaseous*]{} background (\[9\]). The intensity of the drag is then higher or lower depending on whether the black holes move supersonically or subsonically with respect to the background, and increases also as the the characteristic density of the background increases. The run with $\gamma=7/5$ falls in the supersonic regime (the orbital velocity of the black holes is of order $300$ km/s, which corresponds to a temperature of about $10^6$ K), while the black holes move slightly subsonically in the run with $\gamma=5/3$. In the latter run the average background density is also a factor $\sim 2$ lower compared to the run with $\gamma=7/5$. These differences in the thermal and density structure of the remnants explain why in the run with $\gamma=7/5$ the two black holes reach a separation comparable to the force resolution limit ($10$ pc) in about $10$ Myr whereas they remain separated by a distance larger than $100$ pc in the other run (Figure 3). In neither case the SMBHs form a binary by the end of the simulation. However, with an even higher force resolution the formation of a bound pair is likely in the $\gamma=7/5$ since the orbital energy of the binary is only marginally positive at $t=5.128$ Gyr. Instead in the simulation with $\gamma=5/3$ the binding of the two SMBHs will be aborted because their orbital decay time is longer than the Hubble time on the last few orbits.
![Orbital separation of the two black holes as a function of time. Two runs with different values of $\gamma$ and a gas gravitational softening of $10$ pc are shown (see labels). The curves start from the time at which the galaxy cores have already merged.[]{data-label="fig:3"}](fig3.eps){height="9cm"}
As shown in Figure 3 the two black holes end up on moderately eccentric orbits ($e=0.3-0.5$) in all the simulations. However the orbits might circularize as the evolution proceeds further (\[4\], and these proceedings).
Conclusions {#sec:4}
===========
We have performed multi-scale hydrodynamical simulations of merging galaxies with SMBHs and shown that dense, rotationally supported nuclear disks are the natural outcome of dissipative mergers starting from cosmologically motivated initial conditions. The nuclear disks reported here likely provide the reservoir of gas that fuels the central SMBHs. The orbital evolution of the close pair of SMBHs formed at the center of the merger remnant is dominated by dynamical friction against the surrounding gaseous medium. The details of this process are extremely sensitive to thermodynamics of the gas. Our results indicate that the formation of a bound SMBH pair requires an equation of state not stiffer than that expected during a major starburst. This suggests that either AGN feedback has a minor impact on the gas in the disk or that strong AGN feedback has to be delayed for several million years after the galaxy merger is completed. In the second case the coalescence of the two black holes will occur when the merger remnant is a powerful starburst, such as a ULRIG, rather than a powerful AGN.
#### Acknowledgemnts
SK is supported by the Swiss National Science Foundation and by The Kavli Institute for Cosmological Physics (KICP) at The University of Chicago. We thank David Merritt for helpful comments. The simulations were performed on Lemieux at the Pittsburgh Supercomputing Center and on the Zbox and Zbox2 supercomputers at the University of Zürich.
|
[**Optical Gravitational Lensing Experiment.\
Photometry of the MACHO-SMC-1 Microlensing Candidate. [^1]**]{} 1.3cm [A. U d a l s k i$^1$, M. S z y m a [ń]{} s k i$^1$, M. K u b i a k$^1$,\
G. P i e t r z y ń s k i$^1$, P. W o ź n i a k$^2$, and K. Ż e b r u ń$^1$]{}
[ABSTRACT]{}
We present photometric observations of the MACHO-SMC-1 microlensing candidate collected by the OGLE-2 project. We show light curves of both components of the 1.6 arcsec blend: microlensed star and its optical companion. We find the contribution of the optical companion to the total flux to be 24% and confirm presence of the small amplitude periodic oscillations in the light curve of the lensed star with the period of 5.096 days and amplitude 0.05 mag. The lensed star is probably an ellipsoidal binary system.
Discovery of the first microlensing event candidate toward the Small Magellanic Cloud by the MACHO Collaboration (Alcock [*et al.*]{} 1997) opens a new direction in which probing of the Galactic halo with microlensing phenomena can be possible. Although it may take years before a significant sample of events in this direction will be collected allowing to draw sound conclusions about distribution of mass in the Galactic halo, the first candidate itself seems to be very interesting.
The microlensed source is a relatively bright (${V\sim 17.7}$ mag) main sequence star blended with a fainter object. Contribution of the companion was estimated to be 23–28% of the total blend light. The event duration was 242 days (Einstein diameter crossing time) and the magnification reached maximum of 2.4 on January 11, 1997 (Alcock [*et al.*]{} 1997; parameters derived taking into account blending). Symmetric light curve and good achromaticity make this brightening an excellent microlensing event candidate. However, it should be noted that the position of the lensed star on the color-magnitude diagram is dangerously close to the non-periodic variable stars which can mimic microlensing.
The MACHO-SMC-1 candidate was also observed by the EROS group (Palanque-Delabrouille [*et al.*]{} 1997). They covered mostly the rising branch of the event. They found that the lensed star exhibits small, periodic variations of light with the amplitude of a few percent and the period of 5.123 days. However, due to poor resolution they were only able to perform photometry of the entire blend and could not assign those possible variations to any of the blended components.
Parameters of the event, in particular its long duration, suggest that the lensing object could be a massive body (${\sim2.5{\rm M}_{\odot}}$) in the Galactic halo or alternatively an object in the SMC (Alcock [*et al.*]{} 1997).
The SMC is one of the targets of the second phase of the Optical Gravitational Lensing Experiment – OGLE-2 (Udalski, Kubiak and Szymański 1997). Although observations of the majority of fields in the SMC started in June 1997, well after the event reached maximum of light, we decided to analyze OGLE-2 data of the MACHO-SMC-1 event to clear up the questions concerning possible variability of the lensed star and contribution of the blend components to the total flux. In this paper we present our results.
The OGLE-2 project observations are carried out at the Las Campanas Observatory, Chile, which is operated by the Carnegie Institution of Washington, with the 1.3-m Warsaw telescope equipped with the “first generation” CCD camera. Details of the equipment and data reduction techniques can be found in Udalski, Kubiak and Szymański (1997).
Observations of the SMC are made in the driftscan mode with drifts in declination. One scan covers approximately ${14\times56}$ arcmins on the sky. Majority of scans are obtained with the $I$-band filter, with some measurements in $V$ and $B$-bands. The effective exposure time is 120 sec, 180 sec and 240 sec for $I$, $V$, and $B$-bands, respectively. Observations of the field in which the MACHO-SMC-1 candidate was identified – SMC\_SC8 – started on Jun. 27, 1997. The data presented in this paper cover the period through Oct. 9, 1997 (JD 2450730.6).
Fig. 1 shows a ${30\times30}$ arcsec subframe centered on the MACHO-SMC-1 candidate taken at 1.0 arcsec seeing conditions. As can be seen the blend is easily resolved and both components could be easily measured independently with the OGLE-2 software. The $I$-band light curves of both stars are presented in Figs. 2 and 3. The zero point was chosen as the normal (off-event) brightness of the lensed star, calculated by fitting to our observations the theoretical microlensing light curve with Alcock [*et al.*]{} (1997) parameters. Error bars correspond to the errors returned by the photometry program (DoPhot) rescaled by a factor of 1.3 to approximate the observational errors. Errors of both components, in particular the fainter component of the blend, are larger than for typical stars of such brightness due to relatively small separation. The distance between components is 1.6 arcsec.
As can be seen from Figs. 2 and 3, the magnified star was the brighter component of the blend. Although the observations begun more than 5 months after the maximum of brightness, the star was still slightly above its normal brightness and the slow fading can still be noticed. The fainter component of the blend was constant over the entire period of observations. The normal state brightness of the lensed star and the mean magnitude of the fainter companion suggest that, in the $I$-band, the brighter component contributes 76% to the total light. Both components of the blend have almost exactly the same ${V-I}$ colors. Thus, in the $V$-band contribution of the fainter component is very similar. Both stars are located among the SMC main sequence stars on the color magnitude diagram.
To check for periodic variability reported by Palanque-Delabrouille [*et al.*]{} (1997) we performed a period analysis of the light curves of both components with the [clean]{} algorithm (Roberts, Lehár and Dreher 1987). For the brighter star we first rectified the light curve by subtracting microlensing brightening. The analysis yields the periodicity of 5.096 days for the brighter star. No significant period was found for the fainter star. We also checked photometry of two nearby stars of similar brightness and again no significant periodicity was found.
Fig. 4 shows rectified observations of the brighter star folded with the period of 5.096 day. Clear sinusoidal variations can be noticed. We fit a sinusoid to the data. The full amplitude (peak to peak) of the best fit is about 0.05 mag. Elements of minimum of brightness are given by the following equation: $$
------------ --- ------- ------------- ------- -----------------
[JDhel.]{} = 2450625.612 + 5.096$\times P$
$\pm$ 0.050 $\pm$ 0.025.
------------ --- ------- ------------- ------- -----------------
$$
Analysis of the photometric data of the MACHO-SMC-1 microlensing event candidate collected in the course of the OGLE-2 program indicates that the star which underwent magnification was the brighter component of the blend separated by 1.6 arcsec. Light curve of the fainter component shows no significant light variations. The brighter star contributes 76% to the blend light in the $V$ and $I$-bands.
The light curve of the microlensing candidate shows additional small amplitude periodic variations with the period of 5.096 days – close to the value reported by the EROS group. Also the amplitude of sinusoidal variations is similar. As our data cover mostly the off-microlensing light curve while the EROS data were taken during the event, this may suggest that the amplitude is constant and is not related to microlensing. Most likely the star is a binary system with one or both components ellipsoidally distorted. Changing aspects cause small amplitude, sinusoidal variations similar to those observed in the light curve of the lensed star. The real period of the binary system would be twice of that derived in Section 3, that is 10.19 days. As the star is relatively bright, this hypothesis can be relatively easy verified by spectroscopic observations.
Photometry of the MACHO-SMC-1 candidate can be retrieved from the OGLE network archive: [*ftp://sirius.astrouw.edu.pl/ogle/ogle2/macho-smc-1.*]{}
We thank Prof. B. Paczyński for his comments and and Dr. I. Semeniuk for double checking period analysis. This paper was partly supported from the KBN BST grant. Partial support for the OGLE project was provided with the NSF grant AST-9530478 to B. Paczyński.
[Alcock, C., [*et al.*]{}]{} 1997, [* *]{}, [** **]{}, [preprint astro-ph/9708190]{}.
[Palanque-Delabrouille, N., [*et al.*]{}]{} 1997, [* *]{}, [** **]{}, [preprint astro-ph/9710194]{}.
[Roberts, D.H., Lehár, J., and Dreher, J.W.,]{} 1978, [**]{}, [**93**]{}, [968]{}.
[Udalski, A., Kubiak, M., and Szymański, M.]{} 1997, [**]{}, [**47**]{}, [319]{}.
[**Figure Captions**]{}
Fig. 1. ${30\times30}$ arcsec subframe centered on the MACHO-SMC-1 microlensing event candidate. North is up and East to the left.
Fig. 2. The $I$-band light curve of the microlensed star. Solid line shows the theoretical microlensing light curve of MACHO-SMC-1. Broken line indicates the normal brightness of the star.
Fig. 3. The $I$-band light curve of the star separated by 1.6 arcsec from the microlensed star. Broken line indicates its mean magnitude.
Fig. 4. Observations of the microlensed star phased with the period of 5.096 days. Thick, solid line indicates the best fit sinusoid. Two cycles are repeated for clarity.
[^1]: Based on observations obtained with the 1.3 m Warsaw telescope at the Las Campanas Observatory of the Carnegie Institution of Washington.
|
---
abstract: 'In this paper, we provide a pathwise spine decomposition for multitype superdiffusions with non-local branching mechanisms under a martingale change of measure. As an application of this decomposition, we obtain a necessary and sufficient condition (called the $L\log L$ criterion) for the limit of the fundamental martingale to be non-degenerate. This result complements the related results obtained in [@KLMR; @KM; @LRS09] for superprocesses with purely local branching mechanisms and in [@KP] for super Markov chains.'
author:
- '[**Zhen-Qing Chen**]{} [^1] [^2]'
title: '**$L\log L$ criterion for a class of multitype superdiffusions with non-local branching mechanisms**'
---
[**AMS Subject Classifications (2000)**]{}: Primary 60J80, 60F15; Secondary 60J25
**Keywords and Phrases:** multitype superdiffusion; non-local branching mechanism; switched diffusion; spine decomposition; martingale.
Introduction
============
Previous results
----------------
Suppose that $\{Z_n, n\ge 1\}$ is a Galton-Watson process with offspring distribution $\{p_n\}$. That is, each particle lives for one unit of time; at the time of its death, it gives birth to $k$ particles with probability $p_k$ for $k=0, 1, \cdots$; and $Z_n$ is the total number of particles alive at time $n$. Let $L$ be a random variable with distribution $\{p_n\}$ and $m:=\sum^{\infty}_{n=1}np_n$ be the expected number of offspring per particle. Then $Z_n/m^n$ is a non-negative martingale. Let $W$ be the limit of $Z_n/m^n$ as $n\to\infty$. Kesten and Stigum proved in [@KS] that, when $1<m<\infty$ (that is, in the supercritical case), $W$ is non-degenerate (i.e., not almost surely zero) if and only if $$\label{LLogL-GW}
E(L\log ^+L)=\sum^{\infty}_{n=1}p_nn\log n<\infty.$$ This result is usually called the Kesten-Stigum $L\log L$ criterion. In [@AH76a], Asmussen and Herring generalized this result to the case of branching Markov processes under some conditions.
In 1995, Lyons, Pemantle and Peres developed a martingale change of measure method in [@LPP] to give a new proof for the Kesten-Stigum $L\log L$ criterion for (single type) branching processes. Later this approach was applied to prove the $L\log L$ criterion for multitype and general multitype branching processes in [@BK; @KLPP].
In [@LRS09], the martingale change of measure method was used to prove an $L\log L$ criterion for a class of superdiffusions. In this paper, we will establish a pathwise spine decomposition for multitype superdiffusions with purely non-local branching mechanisms. Our non-local branching mechanisms are special in the sense that the types of the offspring are different from their mother, but their spatial locations at birth are the same as their mother’s spatial location immediately before her death. We will see below that, a multitype superdiffusion with a purely non-local branching mechanism given by below can also be viewed as a superprocess having a switched diffusion as its spatial motion and $\widehat \psi(x,i; \cdot)$ defined in as its (non-local) branching mechanism. Using a non-local Feynman-Kac transform, we prove that, under a martingale change of measure, the spine runs as a copy of an $h$-transformed switched-diffusion, which is a new switched diffusion. The non-local nature of the branching mechanism induces a different kind of immigration–the *switching-caused* immigration. That is to say, whenever there is a switching of types, new immigration happens and the newly immigrated particles choose their types according to a distribution $\pi$. The switching-caused immigration is a consequence of the non-local branching, and it does not occur when the branching mechanism is purely local. Note that in this paper we do not consider branching mechanism with a local term. Note also that our non-local branching mechanism is special in the sense that only the types, not the spatial locations at birth, are different from the mother’s. It is interesting to consider superprocesses with more general non-local branching mechanism and with local branching mechanism. For this case, one can see the recent preprint [@RSY], where the spine is a concatenation process.
Concurrently to our work, Kyprianou and Palau [@KP] considered super Markov chains with local and non-local branching mechanisms. Note that if particles do not move in space, our model reduces to the model considered in [@KP] with purely non-local branching mechanism. Kyprianou and Palau [@KP] also found that immigration happens when particle jumps (they call this immigration [*jump immigration*]{}), which corresponds to our switching-caused immigration.
Model: multitype superdiffusions
--------------------------------
For integer $K\geq 2$, a $K$-type superdiffusion is defined as follows. Let $S:=\{1,2,\cdots,K\}$ be the set of types. For each $k\in S$, ${{{\cal L}}}_k$ is a second order elliptic differential operator of divergence form: $$\label{def-L}{{{\cal L}}}_k=\sum^d_{i,j=1}\frac{\partial}{\partial x_i}\left(a^{(k)}_{i,j}\frac{\partial}{\partial
x_j}\right)\quad\mbox{ on }{\mathbb{R}}^d,$$ with $A^k(x)=(a^k_{ij}(x))_{1\leq i, j\leq d}$ being a symmetric matrix-valued function on ${\mathbb{R}}^d$ that is uniformly elliptic and bounded: $$\Lambda_1|v|^2\le\sum^d_{i,j=1}a^k_{i,j}(x)v_iv_j\le \Lambda_2|v|^2
\qquad\mbox{ for all } v\in{\mathbb{R}}^d\mbox{ and }x\in {\mathbb{R}}^d$$ for some positive constants $0<\Lambda_1\leq \Lambda_2<\infty$, where $a^k_{ij}(x)\in C^{2,\gamma}({\mathbb{R}}^d), 1\leq i, j\leq d$ for some $\gamma \in (0, 1)$. Throughout this paper, for $i=1, 2, \cdots$, $ C^{i,\gamma}({\mathbb{R}}^d)$ stands for the space of $i$ times continuously differentiable functions with all their $i$th order derivatives belonging to $C^\gamma({\mathbb{R}}^d)$, the space of $\gamma$-Hölder continuous functions on ${\mathbb{R}}^d$.
Suppose that for each $i\in S$, $\xi^i :=\{\xi^i_t, t\ge 0; \Pi^i_x, x\in {\mathbb{R}}^d\}$ is a diffusion process on ${\mathbb{R}}^d$ with generator ${{{\cal L}}}_i$. In this paper we will always assume that $D$ is a domain of finite Lebesgue measure in $\mathbb{R}^d$. For $x\in D$, denote by $\xi^{i, D}:= \{ \xi^{i, D}_t, t\ge 0; \Pi^i_x, x\in D\}$ the subprocess of $\xi^i$ killed upon exiting $D$; that is, $$\xi^{i,D}_t=\left\{\begin{array}{ll} \xi^i_t & \mbox{ if } t<\tau^i_D,\cr
\partial, & \mbox{ if } t\ge \tau^i_D,\cr
\end{array}
\right.$$ where $\tau^i_D =\inf\{t\ge0; \xi^{i}_t\notin D\}$ is the first exit time of $D$ and $\partial$ is a cemetery point.
Let ${\cal M}_1(S)$ denote the set of all probability measures on $S$, and ${\cal M}_F(\mathbb{R}^d\times S)$ denote the space of finite measures on $D\times S$. For any measurable set $E$, we use $B_b(E)$ (resp. $B^+_b(E)$) the family of bounded (resp. bounded positive) ${\cal B}(E)$-measurable functions on $E$. Any function $f$ on $D$ is automatically extended to $D_\partial := D\cup \{\partial\}$ by setting $f(\partial)=0$. Similarly, any function $f$ on $D\times S$ is automatically extended to $D_\partial \times S$ by setting $f(\partial, i)=0, i\in S$. If $f(t, x, i)$ is a function on $[0,+\infty)\times D\times S$, we say $f$ is *locally bounded* if $\sup_{t\in [0,T]}\sup_{(x, i)\in D\times S}|f(t, x, i)|<+\infty$ for every finite $T>0$. For a function $f(s, x, i)$ defined on $[0,+\infty)\times D\times S$ and a number $t\ge 0$, we denote by $f_t(\cdot)$ the function $(x, i)\mapsto f(t, x, i)$. For convenience we use the following convention throughout this paper: For any probability measure ${\mathbb{P}}$, we also use ${\mathbb{P}}$ to denote the expectation with respect to ${\mathbb{P}}$. When there is only one probability measure involved, we sometimes also use ${\mathbb{E}}$ to denote the expectation with respect to that measure.
We consider a multitype superdiffusion $\{\chi_t, t\ge 0\}$ on $D$, which is a strong Markov process taking values in ${\cal M}_F(D\times S)$. We can represent ${\chi}_t$ by $(\chi^1_t,\cdots, \chi^K_t)$ with $\chi^i_t \in {\cal M}_F (D)$ for $1\leq i\leq K$. For $f\in{B}^+_b(D\times S)$, we often use the convention $$f(x)=(f(x,1),\cdots, f(x,K))=(f_1(x),\cdots, f_K(x)),
\quad x\in D,$$ and $\langle f, \chi_t\rangle=\sum^K_{j=1}\langle f_j, {\chi}^j_t\rangle$ . Suppose that $F(x, i; du)$ is a kernel from $D\times S$ to $(0,\infty)$ such that, for each $i\in S$, the function $$m(x,i):=\int^\infty_0u F(x, i; du)$$ is bounded on $D$. Let $n$ be a bounded Borel function on $D\times S$ such that $n(x,i)\ge m(x, i)$ for every $(x, i)\in D\times S$, and $p^{(i)}_j(x)$, $i, j\in S$, be non-negative Borel functions on $D$ with $\sum^K_{j=1}p^{(i)}_j(x)=1$. Define $$\pi(x,i;\cdot)=\sum^K_{j=1}p^{(i)}_j(x)\delta_{(x,j)}(\cdot),$$ where $\delta_{(x,j)}$ denotes the unit mass at $(x,j)$. Then $\pi(x,i;\cdot)$ is a Markov kernel on $D\times S$. For any $f\in{B}^+_b(D\times S)$, we write $\pi(x,i;f)=\sum^K_{j=1}p^{(i)}_j(x)f_j(x).$ Define $$\zeta(x,i;f)=n(x,i)\pi(x,i;f)+
\int^\infty_0\left(1-e^{-u\pi(x,i;f)}-u\pi(x,i;f)\right)F(x,i;du).$$ Note that we can rewrite $\zeta(x,i;f)$ as $$\zeta(x,i;f)=\widetilde n(x,i)\pi(x,i;f)+
\int^\infty_0\left(1-e^{-u\pi(x,i;f)}\right)F(x,i;du),$$ where $$\label{def-tilde-d}
\widetilde n(x,i):=n(x,i)-m(x, i)\ge 0.$$ $\zeta(x,k;f)$ serves as the non-local branching mechanism, which is a special form of [@DGL (3.17)] with $d$ (corresponding to $n$ in the present paper) and $n$ (corresponding to $F$ in the present paper) independent of $\pi$, and $G(x, i; d\pi)$ being the unit mass at some $\pi(x,i;\cdot)\in {\cal M}_1(S)$, that is, the non-locally displaced offspring born at $(x,i)\in D\times S$ choose their types independently according to the (non-random) distribution $\pi(x,i; \cdot )$. Suppose $b(x,i)\in B^+_b(D\times S)$. Put $$\label{psi}
\psi(x,i;f)=b(x,i)\left(f_i(x)-\zeta(x,i;f)\right), \quad (x,i)\in D\times S, \
f\in B^+_b(D\times S).$$ Without loss of generality, we suppose that $p^i_i(x)=0$ for all $(x,i)\in D\times S$, which means that $\psi$ is a purely non-local branching mechanism. The Laplace-functional of $\chi$ is given by $$\label{Laplace}
{\mathbb{P}}_{\mu}\exp\langle -f, {\chi}_t\rangle =\exp\langle-
u^f_t(\cdot),\mu\rangle,$$ where $u^f_t(x, i)$ is the unique locally bounded positive solution to the evolution equation $$\label{int}\begin{array}{rl}
u^f_t(x,i)+&\displaystyle\Pi^i_{x}\left[
\int^t_0\psi(
\xi^{i,D}_s, i; u^f_{t-s})ds\right]=\Pi^i_xf_i(\xi^{i, D}_t), \quad\mbox{for }
t\ge 0,\end{array}$$ where we used the convention that $u^f_{t}(x)=(u^f_{t}(x, 1), \cdots,
u^f_{t}(x, K))$. This process is called an $(({{{\cal L}}}_1,\cdots {{{\cal L}}}_K), {\bf \psi})$-multitype superdiffusion in $D$. It is well known (see, e.g., [@F]) that for any non-negative bounded function $f$ on $D\times S$, the $u^f_t(x, i)$ in is a locally bounded positive solution to the following system of partial differential equations: for each $i\in S$, $$\label{pde}
\left\{
\begin{array}{rlll}&\displaystyle\frac{\partial{u^f(t,x, i)}}{\partial{t}}
& =& {{{\cal L}}}_i (t,x, i) - \psi(x, i; u^f_t )
\quad (t, x)\in (0, \infty)\times D\\
&\displaystyle u^f(0, x, i)&=& f_i(x),
\quad x\in D\\
& \displaystyle u^f(t, x, i)&=& 0 \quad
(t, x)\in (0, \infty)\times \partial D.
\end{array}\right.$$
Multitype superdiffusions can be obtained as a scaling limit of a sequence of multitype branching diffusions. See [@DGL] for details. The multitype superdiffusion $\chi$ considered in this paper are the scaling limits of multitype branching diffusions whose types can change only at branching times.
Define $$\label{def-m}
r_{il}(x)=n(x,i)p^{(i)}_l(x)\quad x\in D,\, i,l\in S.$$ Let $v(t, x,i)={\mathbb{P}}_{\delta_{(x,i)}}\langle f, \chi_t\rangle$. Using and , we see that for all $(t, x, i)\in (0, \infty)\times D\times S$, $$\label{mean-int}
\begin{array}{rl}
v_t(x,i)=&\displaystyle\Pi^i_xf_i(\xi^{i, D}_t)+\Pi^i_{x}
\int^t_0b(\xi_s, i)
\left(\sum^K_{l=1}r_{il}(\xi^{i,D}_s)
v_{t-s}(\xi^{i,D}_s,l)-v_{t-s}(\xi^{i,D}_s, i)\right)ds.\end{array}$$ Then $v(t,x,i)$ is the unique locally bounded solution to the following linear system (see, e.g., [@F]): for each $i\in S$, $$\label{pde-mean}
\left\{
\begin{array}{rlll}&\displaystyle\frac{\partial{v(t,x, i)}}{\partial{t}}&=&
{{{\cal L}}}_iv(t,x, i)+b(x, i)\sum^K_{l=1}(r_{il}(x)-\delta_{il})v(t,x,l),
\quad (t, x)\in (0, \infty)\times D\\
&\displaystyle v(0, x, i)&=&f_i(x),
\quad x\in D\\
&\displaystyle v(t, x, i)&=& 0, \quad
(t, x)\in (0, \infty)\times \partial D.
\end{array}\right.$$ Letting ${\bf v}(t, x)=(v(t,x, 1),\cdots, v(t,x, K))^T,$ we can rewrite the partial differential equations in as $$\label{pde-mean2}
\frac{\partial}{\partial t}
{\bf v}(t,x)={\cal L}{\bf v}(t,x)+
B(x) \cdot ( R(x)-I){\bf v}(t,x),$$ where $${\cal L}=\left(\begin{array}{llll}{{{\cal L}}}_1\quad &0&\cdots&\, 0\\
0\quad &{{{\cal L}}}_2\,&\cdots&\, 0\\
\vdots\quad&\vdots\,&\ddots&\, \vdots\\
0\quad &0&\cdots&\, {{{\cal L}}}_K\end{array}\right),$$ $$B(x)=\left(\begin{array}{llll}b(x, 1)\quad &0&\cdots&\,0\\
0\quad &b(x, 2)\,&\cdots&\, 0\\
\vdots\quad&\vdots\,&\ddots&\, \vdots\\
0\quad &0&\cdots&\, b(x, K)\end{array}\right)$$ and $$R(x)=\left(\begin{array}{llll}r_{11}(x)\quad &r_{12}(x)&\cdots&\, r_{1d}(x)\\
r_{21}(x)\quad &r_{22}(x)\,&\cdots&\,r_{2d}(x)\\
\vdots\quad&\vdots\,&\ddots&\, \vdots\\
r_{K1}(x)\quad &r_{K2}(x)&\cdots&\, r_{KK}(x)\end{array}\right).$$ In this paper we assume that $B(x)\cdot R(x)$ is symmetric, that is to say, $$\label{symm}
b(x,i)n(x,i)p^{(i)}_j(x)=b(x,j)n(x,j)p^{(j)}_i(x),\quad\mbox{ for all } i,j\in S, x\in D.$$ We assumed the symmetry of $B(x)\cdot R(x)$ and the symmetry of the operators ${{{\cal L}}}_k$ for simplicity. If the ${{{\cal L}}}_k$’s are of non-divergence form and $B(x)\cdot R(x)$ is not symmetric, we can use the intrinsic ultracontractivity introduced in [@KS1].
Note that $$\label{decom-M}
R(x)-I=R(x)-N(x)+(N(x)-I),$$ where $$N(x)=\mbox{diag}\left(n(x,1),\cdots n(x,K)\right),\quad x\in D.$$ Then by and , $$\label{decom-M'}
B(x)\cdot (R(x)-I)=\widehat B(x)\cdot \left( P(x)-I\right)
+B(x)\left(N(x)-I\right),$$ where $$\widehat B(x)
=\mbox{diag}\left(b(x, 1)n(x,1),\cdots, b(x, K)n(x,K) \right),$$ and $$P(x)=\left( p_{ij}(x)\right)_{{i,j}\in S},\quad p_{ij}(x)=p^{(i)}_j(x).$$ Put $Q(x)=(q_{ij}(x))_{{i,j}\in S}=\widehat B(x)\cdot (P(x)-I)$. We will assume that the matrix $Q$ is irreducible on $D$ in the sense that for any two distinct $k, l\in S$, there exist $k_0, k_1, \cdots, k_r\in S$ with $k_i\neq k_{i+1}$, $k_0=k$, $k_r=l$ such that $\{x\in D: q_{k_ik_{i+1}}(x)>0\}$ has positive Lebesgue measure for each $0\leq i \leq r-1$. Let $\{(X_t, Y_t), t\ge 0\}$ be a switched diffusion with generator ${\cal A}:= {{\cal L}}+ Q(x)$ killed upon exiting from $D\times S$, and $\Pi_{(x,i)}$ be its law starting from $(x,i)$. $\{(X_t, Y_t), t\ge 0\}$ is a symmetric Markov process on $D\times S$ with respect to $dx\times di$, the product of the Lebesgue measure on $D$ and the counting measure on $S$.
Define $$\label{def-zeta1}
\zeta_1(x,i;f)=n(x,i)\pi(x,i;f)=n(x,i)\sum^K_{i=1}p^{(i)}_l(x)f_l(x)=
\sum^K_{l=1}r_{il}(x)f_l(x)$$ and $$\label{def-zeta2}\zeta_2(x,i;f)=\int^\infty_0\left(1-e^{-u\pi(x,i;f)}-u\,\pi(x,i; f)\right)F(x,i;du).$$ Then $$\label{zeta-1+2}\zeta(x,i;f)=\zeta_1(x,i;f)+\zeta_2(x,i;f).$$ Letting $${u}^f(t,
x)=(u^f(t,x, 1),\cdots, u^f(t,x, K))^T \mbox{ and }
{\zeta}_2(x,f)=(\zeta_2(x,1; f),\cdots, \zeta_2(x,K;f))^T,$$ in view of we can rewrite the partial differential equation in as $$\label{pde2}
\frac{\partial}{\partial t}
{ u}^f(t,x)={\cal L}{u}^f(t,x)+
B(x) \cdot \left( R(x)-I\right){u}^f(t,x)+B(x) \cdot{\zeta}_2(x, u^f_t),$$ which, by , is equivalent to $$\begin{aligned}
\label{pde3}
\frac{\partial}{\partial t}{ u}^f(t,x)&=
\ {\cal L}{u}^f(t,x)+\widehat B(x)\cdot
\left(P(x)-I\right){u}^f(t,x)\nonumber\\
&\quad +\ B(x)\cdot\left[(N(x)-I){u}^f(t,x)+{\zeta}_2(x, u^f_t)\right].\end{aligned}$$ For $ f\in B^+_b({\mathbb{R}}^d\times S)$, define $$\label{hat-psi}
\widehat \psi(x,i;f):=-b(x, i)n(x,i)f_i(x)+b(x, i)(f_i(x)-\zeta_2(x,i;f)).$$ Then applying the strong Markov property of the switched diffusion process $(X, Y)$ at its first switching time and using the approach from [@CZ] (see in particular p.296, Proposition 2.2 and Theorem 2.5 there) and [@F], one can verify using that $u^f_t(x, i)$ satisfies $$\label{int-equi}\begin{array}{rl}
u^f_t(x,i)+&\displaystyle\Pi_{(x,i)}\left[
\int^t_0\widehat\psi(
X_s, Y_{s}; u^f_{t-s})ds\right]=\Pi_{(x,i)}f(X_t, Y_t), \quad
t\ge 0.\end{array}$$ This means that $\{\chi_t, t\ge 0\}$ can be viewed as a superprocess with the switched diffusion $(X_t, Y_t)$ as its spatial motion on the space $D\times S$ and $\widehat \psi(x,i; \cdot)$ as its (non-local) branching mechanism. See [@DKS] for a definition of superprocesses with general non-local branching mechanisms.
Main result
===========
It follows from [@CZ Theorem 5.3] that the switched diffusion $\{(X_t, Y_t), t\ge 0\}$ in $D$ has a transition density $p(t, (x, k), (y, l))$, which is positive for all $x, y\in D$ and $k, l\in S$. Furthermore, for any $k, l\in S$ and $t>0$, $(x, y)\mapsto p(t, (x, k), (y, l))$ is continuous. Let $\{P_t: t\ge 0\}$ be the transition semigroup of $\{(X_t, Y_t), t\ge 0\}$. For any $t>0$, $P_t$ is a compact self-adjoint operator. Let $\{e^{\nu_kt}: k=1, 2,\cdots\}$ be all the eigenvalues of $P_t$ arranged in decreasing order, each repeated according to its multiplicity. Then $\lim_{k\to \infty} \nu_k= -\infty$ and the corresponding eigenfunctions $\{\varphi_k\}$ can be chosen so that they form an orthonormal basis of $L^2(D\times S, dx\times di)$. All the eigenfunctions $\varphi_k$ are continuous. The eigenspace corresponding to $e^{\nu_1t}$ is of dimension 1 and $\varphi_1$ can chosen to be strictly positive.
Let $\{P^{{\cal A}+B\cdot (N-I)}_t,t\ge 0\}$ be the Feynman-Kac semigroup defined by $$P^{{\cal A}+B\cdot (N-I)}_tf(x,i)
:=\Pi_{(x,i)}\left[f(X_t, Y_t)\
\exp\left(\int^t_0b(X_s, Y_s)
(n(X_s, Y_s)-1) ds\right)\right].$$ Then, by , $P^{{\cal A}+B\cdot (N-I)}_tf(x,i)$ is the unique solution to and thus $$\label{expX}
{\mathbb{P}}_{\delta_{(x,i)}}\langle f, \chi_t\rangle=
P^{{\cal A}+B\cdot (N-I)}_tf(x,i).$$
Under the assumptions above, $P^{{\cal A}+B\cdot (N-I)}_t$ admits a density $\widetilde{p}(t,(x,i),(y,j))$, which is jointly continuous in $(x,y)\in D\times D$, such that $$P^{{\cal A}+B\cdot (N-I)}_tf(x,i)
=\sum_{j\in S}\int_D\widetilde{p}(t, (x,i), (y,j))f(y,j)dy,$$ for every $f\in{\cal B}^+_b(D\times S).$ $\{P^{{\cal A}+B\cdot (D-I)}_t, t\ge 0\}$ can be extended to a strongly continuous semigroup on $L^2(D\times S, dx\times di)$. The semigroup $\{ P^{{\cal A}+B\cdot (N-I)}_t, t\ge 0\}$ is symmetric in $L^2(D\times S, dx\times di)$, that is $$\sum_{i\in S}\int_Df(x,i)P^{{\cal A}+B\cdot (N-I)}_tg(x,i)dx
=\sum_{i\in S}\int_Dg(x,i) P^{{\cal A}+B\cdot (N-I)}_tf(x,i)dx$$ for $f,g\in
L^2(D\times S, dx\times di).$ For any $t>0$, $P^{{\cal A}+B\cdot (N-I)}_t$ is a compact self-adjoint operator. The generator of the semigroup $\{P^{{\cal A}+B\cdot (N-I)}_t\}$ is ${\cal A}+B\cdot (N-I)={\cal L}+B\cdot (R-I)$.
Let $\{e^{\lambda_kt}: k=1, 2,\cdots\}$ be all the eigenvalues of $P^{{\cal A}+B\cdot (N-I)}_t$ arranged in decreasing order, each repeated according to its multiplicity. Then $\lim_{k\to \infty} \lambda_k = -\infty$ and the corresponding eigenfunctions $\{\phi_k\}$ can be chosen so that they form an orthonormal basis of $L^2(D\times S, dx\times di)$. All the eigenfunctions $\phi_k$ are continuous. The eigenspace corresponding to $e^{\lambda_1t}$ is of dimension 1 and $\phi_1$ can chosen to be strictly positive. For simplicity, in the remainder of this paper, we will $\phi_1$ as $\phi$.
Throughout this paper we assume that $\{\chi_t, t\ge 0\}$ is supercritical and $\phi$ is bounded on $D\times S$; that is, we assume the following.
\[assume1\] $\lambda_1>0$ and its corresponding positive eigenfunction $\phi$ is bounded.
Define $$\label{e:dofRr}
R^\phi(x):=\left(r^\phi_{ij}(x)\right),
\quad r^\phi_{ij}(x):=r_{ij}(x)\frac{\phi(x, j))}{\phi(x, i)}
=n(x,i)\frac{p^{(i)}_j(x)\phi(x, j)}{\phi(x, i)}$$ and $$\label{e:dofpi(phi)}
\pi(\phi)(x,i):=\pi(x,i;\phi)=
\sum^K_{j=1}p^{(i)}_j(x)\phi(x,j),\quad (x,i)\in D\times S.$$
Let $\{{\cal E}_t; t\geq 0\}$ be the minimum augmented filtration generated by the switched diffusion $(X, Y)$ in $D$. Define a measure $\Pi_{(x,i)}^\phi$ by $$\label{Martingale switched diffusion}
\begin{array}{rl}
\displaystyle\frac{d\Pi_{(x,i)}^\phi}{d\Pi_{(x,i)}}\Big|_{\mathcal{E}_t}= &\displaystyle e^{-\lambda_1t}\frac{\phi(X_t, Y_{t})}{\phi(x,i)}\exp\left(\int^t_0b(X_s, Y_{s})(n(X_s, Y_s)
-1){\rm d}s\right).
\end{array}$$ Then $\{(X, Y), \Pi_{(x,i)}^\phi\}$ is a conservative Markov process which is symmetric with respect to the measure $\phi^2(x, i)dx\times di$. The process $\{(X, Y), \Pi_{(x,i)}^\phi\}$ has a transition density $p^\phi(t, (x,i),(y,j))$ with respect to $dy\times dj$ given by $$p^\phi(t,(x,i),(y,j))=\frac{e^{-\lambda_1t}\phi(y,j)}{\phi(x,i)}\ \widetilde{p}(t, (x, i),
(y, j)),\quad (x,i)\in D\times S.$$ Let $\{P^{\phi}_t: t\ge 0\}$ be the transition semigroup of $(X, Y)$ under $\Pi_{(x,i)}^\phi$. Then $\phi^2$ is the unique invariant probability density of $\{P^{\phi}_t: t\ge 0\}$, that is, for any $f\in B^+_b(D\times S)$, $$\sum^K_{i=1}\int_D\phi^2(x,i)P^{\phi}_tf(x,i){\rm d}x=\sum^K_{i=1}\int_Df(x,i)\phi(x,i)^2{\rm d}x.$$ Since the infinitesimal generator of $\{(X, Y), \Pi_{(x,i)}\}$ is ${\cal L}+\widehat B(x)\cdot(P(x)-I)$ with zero Dirichlet boundary condition on $\partial D \times S$, it follows from [@PR Theorem 4.2] that the generator of $\{ (X, Y), \Pi^\phi_{(x,i)}\}$ is $$\begin{aligned}
&&\frac{1}{{\bf \phi}}\left[{{\cal L}}({\bf u\phi})+\widehat B(x)\cdot( P(x)-I)({\bf{u\phi}})-{\bf u}({{\cal L}}({\bf \phi})+\widehat B(x)\cdot( P(x)-I){\bf{\phi}})\right]\\
&=&
\frac{1}{{\bf \phi}}\left[{{\cal L}}({\bf u\phi})+\widehat B(x)\cdot( P(x)-I)({\bf{u\phi}})+B(x)\cdot(N(x)-I)({\bf{u\phi}})-\lambda_1{\bf u}\phi\right]\\
&=&\frac{1}{{\bf \phi}}\left[{{\cal L}}({\bf u\phi})+
B(x)\cdot(R(x)-I)({\bf{u\phi}})\right]-\lambda_1{\bf u}\\
&=&{\cal L}^\phi {\bf u}+B(x)\cdot (R^\phi(x)-I){\bf u}-\lambda_1{\bf u},
\end{aligned}$$ where in the first equality above we used the fact that $\phi$ is an eigenfunction of $P^{{\cal A}+B\cdot (N-I)}_t$ and .
Note that $$\begin{array}{rll}\displaystyle B(x)\cdot (R^\phi-I)-\lambda_1
&=&\displaystyle\mbox{diag}\left(\frac{bn\pi(\phi)}{\phi}(x,1),\cdots \frac{bn\pi(\phi)}{\phi}(x,K)\right)(\widetilde P(x)-I)\\
&&\displaystyle+B(x)\left[\mbox{diag}\left(\frac{n\pi(\phi)}{\phi}(x,1),\cdots \frac{n\pi(\phi)}{\phi}(x,K)\right)-I\right]-\lambda_1
\\&=&\displaystyle\mbox{diag}\left(\frac{bn\pi(\phi)}{\phi}(x,1),\cdots \frac{bn\pi(\phi)}{\phi}(x,K)\right)(\widetilde P(x)-I).
\end{array}$$ Thus the generator of $\{(X, Y), \Pi^\phi_{(x,i)}\}$ is $$\label{transformed swithed diffu}
{\cal L}^\phi+\mbox{diag}\left(\frac{bn\pi(\phi)}{\phi}(x,1),\cdots \frac{bn\pi(\phi)}{\phi}(x,K)\right)(\widetilde P(x)-I)$$ which is the generator of a new switched diffusion, where $${\cal L}^\phi=\left(\begin{array}{llll}
{{\cal L}}^{\phi(\cdot, 1)}_1\quad &0&\cdots&\, 0\\
0\quad & {{\cal L}}^{\phi(\cdot,2)}_2\,&\cdots&\, 0\\
\vdots\quad&\vdots\,&\ddots&\, \vdots\\
0\quad &0&\cdots&\, {{\cal L}}^{\phi(\cdot, K)}_K\end{array}\right),$$ $${{\cal L}}^{\phi(\cdot, k)}_ku_k(x)=\frac{1}{\phi(x, k)}{{{\cal L}}}_k\left(\phi(x, k)u_k(x)\right),$$ $$\widetilde P(x)=\left(\widetilde p_{ij}(x)\right)_{i,j\in S},$$ and $$\widetilde p_{ij}(x)=\frac{\phi(x,i)}{n(x,i)\pi(x,i;\phi)}r^\phi_{ij}(x)
=\frac{p^{(i)}_j(x)\phi(x, j)}{\pi(x, i;\phi)},\quad i,j\in S, x\in D.$$
For any measure $\mu$ on $D\times S$ such that $\langle\phi,\mu\rangle<\infty$, define $$\Pi^\phi_{\phi\mu}=\frac{1}{\langle\phi,\mu\rangle}\int\phi(x,i)\Pi^\phi_{(x,i)}d\mu.$$ By , the jumping intensity of $(X, Y)$ under $\Pi^\phi_{\phi\mu}$ is $\frac{bn\pi(\phi)}{\phi}(x, i)$ at $(x,i)\in D\times S.$
Throughout this paper we assume that
\[assume2\] The first eigenfunction $\phi$ is bounded on $D\times S$. The semigroup $\{P_t:t\ge 0\}$ is intrinsically ultracontractive, that is, for any $t>0$, there exists $c_t>0$ such that $$p(t, (x, k), (y, l))\le c_t\phi(x, k)\phi(y, l), \qquad x, y\in D, k, l\in S.$$
It follows from [@DS Theorem 3.4] that the semigroup $\{P^{{\cal A}+B\cdot (N-I)}_t: t\ge 0\}$ is also intrinsically ultracontractive, that is, for any $t>0$, there exists $c_t>0$ such that $$\widetilde{p}(t, (x, k), (y, l))\le c_t\phi(x, k)\phi(y, l), \qquad x, y\in D, k, l\in S.$$ As a consequence, one can easily show (see, for instance, [@Ban]) that for any $t_0>0$, there exists $c>0$ such that for all $t\ge t_0$, $$\left|\frac{e^{-\lambda_1t}\widetilde{p}(t, (x, k), (y, l))}{\phi(x, k)\phi(y, l)}-1\right|\le
ce^{(\lambda_2-\lambda_1)t},
\qquad x, y\in D, k, l\in S.$$ Hence for any $\delta\in (0, 1)$, there exists $t_0>0$ such that for all $t\ge t_0$, $$\left|\frac{e^{-\lambda_1t}\widetilde{p}(t, (x, k), (y, l))}{\phi(x, k)\phi(y, l)}-1\right|\le \delta, \qquad x, y\in D, k, l\in S.$$ Thus for any $f\in B_b(D\times S)$, $t>t_0$ and $(x,i)\in D\times S$, $$\label{IU'}
\left| P^\phi_tf(x,i)-\int_{D\times S}f(y,j)\phi(y,j)^2 dydj \right|\le \delta \int_{D\times S}f(y,j)\phi(y,j)^2dydj.$$ It follows from that for any $f\in B^+_b(D\times S)\cap L^1(\phi^2(x,i) \, dx\times di)$, $t>t_0$ and $(x,i)\in D\times S$, $$\label{IU''}
(1-\delta)\int_{D\times S}f(y,j)\phi(y,j)^2dydj\le P^\phi_tf(x,i)\le (1+\delta)\int_{D\times S}f(y,j)\phi(y,j)^2dydj.$$
\[martingale\] Define $$\label{martingale-1}
W_t(\phi):=e^{-\lambda_1 t}\langle \phi, \chi_t\rangle.$$ Then $\{W_t(\phi), t\ge 0\}$ is a non-negative ${\mathbb{P}}_\mu$-martingale for each nonzero $\mu\in M_F(D\times S)$ and therefore there exists a limit $ W_{\infty}(\phi)\in[0,\infty)$ ${\mathbb{P}}_{\mu}$-a.s.
By the Markov property of $\chi$ and , and using the fact that $\phi$ is an eigenfunction corresponding $\lambda_1$, we get that for any nonzero $\mu\in M_F(D\times S)$, $$\begin{aligned}
{\mathbb{P}}_{\mu} \left[W_{t+s}(\phi) \big| {{\cal F}}_t\right] &=& \frac{1}{\langle\phi,
\mu\rangle}e^{-\lambda_1 t} {\mathbb{P}}_{\chi_t} \left[e^{-\lambda_1 s} \langle \phi, \chi_s\rangle\right]\\
& =&\frac{1}{\langle\phi, \mu\rangle}e^{-\lambda_1 t}
\left\langle e^{-\lambda_1 s}
P^{{\cal A}+B\cdot (N-I)}_s\phi, \, \chi_t\right\rangle \\
&=&\frac{1}{\langle\phi, \mu\rangle}e^{-\lambda_1 t} \langle\phi, \, \chi_t\rangle =
W_t(\phi).\end{aligned}$$ This proves that $\{W_t(\phi), t\ge 0\}$ is a non-negative ${\mathbb{P}}_\mu$-martingale and so it has an almost sure limit $W_\infty(\phi)\in[0,\infty)$ as $t\to \infty$.
We define a new kernel $F^{\pi(\phi)}(x,i; dr)$ from $D\times S$ to $(0,\infty)$ such that for any nonnegative measurable function $f$ on $(0,\infty),$ $$\int^\infty_0f(r)F^{\pi(\phi)}(x,i;dr)=\int^\infty_0f(\pi(x,i;\phi)r)F(x,i;dr),\quad (x,i)\in D\times S.$$
Define $$\label{def-l}
l(x,i):=\int_0^\infty r\log^+(r)F^{\pi(\phi)}(x,i; dr).$$
The main result of this paper is the following.
\[maintheorem\] Suppose that $\{\chi_t; t\ge 0\}$ is a multitype superdiffusion and that Assumptions \[assume1\] and \[assume2\] hold. Assume that $\mu\in M_F(D\times S)$ is non-trivial. Then $W_\infty(\phi)$ is non-degenerate under ${\mathbb{P}}_\mu$ if and only if $$\label{LlogL-BH}
\int_{D}{\phi}(x,i)b(x,i)l(x,i)dx<\infty \quad \hbox{for every } i\in S,$$ where $l$ is defined in . Moreover, when is satisfied, $W_t(\phi)$ converges to $W_\infty(\phi)$ in $L^1$ under ${\mathbb{P}}_\mu$.
Since does not depend on $\mu$, it is also equivalent to that $W_\infty(\phi)$ is non-degenerate under ${\mathbb{P}}_\mu$ for every non-trivial measure $\mu\in M_F(D\times S)$.
The proof of this theorem is accomplished by combining the ideas from [@LPP] with the “spine decomposition" of [@EK] and [@LRS09]. The new feature here is that we consider a different type of branching mechanisms. The new type of branching mechanisms considered here is non-local as opposed to the local branching mechanisms in [@EK] and [@LRS09]. The non-local branching mechanisms we consider here result in a kind of [*non-local*]{} immigration, as opposed to the local immigration in [@LRS09].
In the next section, we show that when $D$ is a bounded $C^{1,1}$ domain in ${\mathbb{R}}^d$, Assumption \[assume2\] holds. In Section \[s:spine\], we give our spine decomposition of the superdiffusion $\chi$ under a martingale change of measure with the help of Poisson point processes. In Section \[s:proof\], we use this spine decomposition to prove Theorem \[maintheorem\].
Intrinsic Ultracontractivity {#s:iu}
============================
In this section, we show that when $D$ is a bounded $C^{1, 1}$ domain in ${\mathbb{R}}^d$, Assumption \[assume2\] holds, that is, the semigroup $\{P_t:t\ge 0\}$ is intrinsically ultracontractive and the first eigenfunction is bounded.
Throughout this section, we assume that $D$ is a bounded $C^{1, 1}$ domain in ${\mathbb{R}}^d$. Let $p_0(t, x, y)$ be the transition density of the killed Brownian motion in $D$. For each $i\in S$, let $p_i(t, x, y)$ be the transition density of $\xi^{i, D}_t$, the process obtained by killing the diffusion with generator ${{{\cal L}}}_i$ upon exiting from $D$.
It is known (see [@CKP]) that there exist positive constants $C_i$, $i=1, 2, 3, 4$, such that for all $t\in (0, 1]$, $j=0, 1, \cdots K$ and $x, y\in D$, $$\begin{aligned}
&p_j(t, x, y)\ge C_1\left(\frac{\delta_D(x)}{\sqrt{t}}\wedge 1\right)
\left(\frac{\delta_D(y)}{\sqrt{t}}\wedge 1\right)t^{-d/2}e^{-\frac{C_2|x-y|^2}{t}},\label{e:lowerbnd}\\
&p_j(t, x, y)\le C_3\left(\frac{\delta_D(x)}{\sqrt{t}}\wedge 1\right)
\left(\frac{\delta_D(y)}{\sqrt{t}}\wedge 1\right)t^{-d/2}e^{-\frac{C_4|x-y|^2}{t}}.\label{e:upperbnd}\end{aligned}$$ Using these we can see that there exists $C_5>0$ such that for any $t\in (0, C_4/C_2]$ and $x, y\in D$, $$\label{e:compofhks}
p_j(t, x, y)\le C_5p_0\left({C_2t}/{C_4}, x, y\right).$$
It follows from [@CZ Theorem 5.3] that for any $x, y\in D$ and $k, l\in S$, $$\begin{aligned}
&p(t, (x, k), (y, l))\nonumber\\
&\quad = \delta_{kl}p_k(t, x, y)\nonumber\\
&\qquad+\sum^\infty_{n=0}\sum_{\stackrel{1\le l_1, l_2, \dots, l_n\le K}{l_1
\neq k, l_n\neq l, l_i\neq l_{i+1}}}\int\cdots\int_{0<t_1<t_2<\cdots<t_n<t}\int_D\cdots\int_Dp_k(t_1, x, y_1)q_{kl_1}(y_1)\nonumber\\
&\qquad\quad\times p_{l_1}(t_2-t_1, y_1, y_2)q_{l_1l_2}(y_2)\cdots q_{l_nl}(y_n)\nonumber\\
&\qquad\quad\times p_l(t-t_n, y_n, y)dy_n\cdots dy_1dt_n\cdots dt_1.
\label{e:CZdensity}\end{aligned}$$ Let $M>0$ be such that $$|q_{kl}(x)|\le M, \qquad x\in D, k, l\in S.$$ Then it follows from and that for $t\in (0, C_4/C_2]$, $x, y\in D$ and $k, l\in S$, $$\begin{aligned}
&p(t, (x, k), (y, l))\\
&\le C_5p_0\left( {C_2t}/{C_4}, x, y\right)\\
&\quad +\sum^\infty_{n=0}(MKC_5)^n\int\cdots\int_{0<t_1<t_2<\cdots<t_n<t}\int_D\cdots\int_Dp_0
\left({C_2t_1}/{C_4}, x, y_1\right)\\
&\quad\quad\times
p_0\left( {C_2(t_2-t_1)}/{C_4}, y_1, y_2\right)\cdots
p_0\left({C_2(t_-t_n)}/{C_4}, y_n, y\right)dy_n\cdots dy_1dt_n\cdots dt_1\\
&\le C_5p_0\left({C_2t}/{C_4}, x, y\right)+\sum^\infty_{n=0}\frac{(MKC_5t)^n}{n\!}p_0\left({C_2t}/{C_4}, x, y\right).\end{aligned}$$ Thus there exists $t_0\in (0, C_4/C_2)$ such that for $t\in (0, t_0]$, $x, y\in D$ and $k, l\in S$, $$\label{e:upbd4density}
p(t, (x, k), (y, l))\le C_6p_0( {C_2t}/{C_4}, x, y)$$ for some $C_6>0$.
Now we prove a similar lower bound. It follows from that for any $t\in (0, 1]$, $x, y\in D$ and $k\in S$, $$\label{e:lowerbd1}
p(t, (x, k), (y, k))\ge p_k(t, x, y).$$ Now suppose $k\neq l$. Let $l_0, l_1, \cdots, l_n\in S$ with $l_i\neq l_{i+1}$, $l_0=k$, $l_n=l$ such that $\{x\in D: q_{l_il_{i+1}}(x)>0\}$ has positive Lebesgue measure for $i=0, 1, \cdots, n-1$. Then it follows from that $$\begin{aligned}
p(t, (x, k), (y, l))&\ge \int\cdots\int_{0<t_1<t_2<\cdots<t_n<t}\int_D\cdots\int_Dp_k(t_1, x, y_1)q_{kl_1}(y_1)\\
&\qquad \times p_{l_1}(t_2-t_1, y_1, y_2)q_{l_1l_2}(y_2)\cdots q_{l_nl}(y_n)\nonumber\\
&\qquad\times p_l(t-t_n, y_n, y)dy_n\cdots dy_1dt_n\cdots dt_1.\end{aligned}$$ Thus it follows from that there exists $C_7>0$ such that for any $t\in (0, 1]$, $x, y\in D$, $$\label{e:lowerbd2}
p(t, (x, k), (y, l))\ge C_7\left(\frac{\delta_D(x)}{\sqrt{t}}\wedge 1\right).$$ Combining and we get that for any $t\in (0, 1]$, $x, y\in D$ and $k, l\in S$, $$\label{e:lowerbd}
p(t, (x, k), (y, l))\ge C_8\left(\frac{\delta_D(x)}{\sqrt{t}}\wedge 1\right)$$ for some $C_8>0$.
It follows from and that there exists positive constants $C_{9}<C_{10}$ such that for all $(x, k)\in D\times S$, $$C_{9}\delta_D(x)\le \phi(x, k)\le C_{10}\delta_D(x).$$ Combining this with , and using the semigroup property, we immediately get the intrinsic ultarcontractivity of $\{P_t:t\ge 0\}$. The boundedness of $\phi$ is an immediate consequence of the display above.
Spine decomposition {#s:spine}
===================
Let ${\cal F}_t=\sigma(\chi_s;\ s\leq t)$. We define a probability measure $\widetilde{{\mathbb{P}}}_\mu$ by: $$\label{measure-change}
\frac{d\widetilde{{\mathbb{P}}}_\mu}{d{\mathbb{P}}_\mu}\Big|_{{\cal F}_t} =
\frac{1}{\langle\phi,\mu\rangle}W_t(\phi).$$ The purpose of this section is to give a spine decomposition of $\{\chi_t, t\ge 0\}$ under $\widetilde{{\mathbb{P}}}_\mu$. This decomposition will play an important role in proving Theorem \[maintheorem\]
The spine decomposition is roughly as follows: Under $\widetilde{{\mathbb{P}}}_\mu$, $\{\chi_t, t\ge 0\}$ has the same law as the sum of the following two independent measured-valued processes: the first process is a copy of $\chi$ under ${\mathbb{P}}_{\mu}$, and the second process is, roughly speaking, obtained by taking an “immortal particle" that moves according to the law of $\{(X, Y),\Pi^\phi_{\phi\mu}\}$ and spins off pieces of mass that continue to evolve according to the dynamics of $\chi$.
Define $$\label{e:dofeta}
\eta(x,i;\lambda)=\int_0^\infty e^{-u\lambda}uF(x,i;du),\quad \lambda\ge 0,\, (x,i)\in D\times S.$$ We first give a formula for the one-dimensional distribution of $\chi$ under $\widetilde{{\mathbb{P}}}_\mu$.
\[theorem1\] Suppose $\mu\in M_F(D\times S)$ and $g\in B^+_b(D\times S).$ Let $D_J$ be the set of jump times of $(X, Y)$. Then $$\begin{aligned}
\label{1inthm1}
&\widetilde{{\mathbb{P}}}_\mu\left(\exp\langle-g,\chi_t \rangle\right)\nonumber\\
&=\
{\mathbb{P}}_\mu\big(\exp\langle - g,\chi_t\rangle\big)
\Pi_{\phi\mu}^{\phi}\left[\exp\left(\sum_{s\in D_J, 0<s\le t}\ln\left(\frac{\eta(X_{s}, Y_{s};\pi(X_{s}, Y_{s};u^g_{t-s}))}{n(X_{s}, Y_{s})}+\frac{\widetilde n(X_s, Y_s)}{n(X_s, Y_s)}\right)\right) \right],\end{aligned}$$ where $u^g_{t-s}$ is the unique locally bounded positive solution of with $f$ replaced by $g$.
By , $$\label{exp-tilde-P}\begin{array}{rl}
\displaystyle \widetilde{{\mathbb{P}}}_\mu\big(\exp\langle-g,\chi_t \rangle\big) =&
\displaystyle\frac{e^{-\lambda_1 t}}{\langle\phi,\mu\rangle}{{\mathbb{P}}}_\mu\left(\langle \phi,\chi_t\rangle\exp\langle-g,\chi_t \rangle\right)\\
=&\displaystyle\left.\frac{e^{-\lambda_1 t}}{\langle\phi,\mu\rangle}\frac{\partial}{\partial\theta}
{{\mathbb{P}}}_\mu\left(\exp\langle-g-\theta\phi,\chi_t \rangle\right)\right |_{\theta=0}\\
=&\left.\displaystyle\frac{e^{-\lambda_1 t}}{\langle\phi,\mu\rangle}\frac{\partial}{\partial\theta}
\exp\langle-u^{g+\theta\phi}_t,\mu\rangle\right |_{\theta=0}\\
=&\displaystyle\frac{e^{-\lambda_1 t}}{\langle\phi,\mu\rangle}\exp\langle-u^g_t,\mu\rangle\left\langle\left.\frac{\partial}{\partial\theta}
u^{g+\theta \phi}_t\right|_{\theta=0},\mu\right\rangle.
\end{array}$$ Note that $\exp\langle-u^g_t,\mu\rangle ={\mathbb{P}}_\mu\exp\langle -g,\chi_t\rangle,$ and $u^{g+\theta \phi}_t$ is the unique locally bounded positive solution of the integral equation $$u^{g+\theta\phi}_t(x,i)+\Pi_{(x,i)}\left[
\int^t_0\widehat\psi(X_s, Y_{s}; u^{g+\theta\phi}_{t-s})ds\right]=\Pi_{(x,i)}\left[(g+\theta\phi)(X_t, Y_t)\right], \quad
t\ge 0.$$ Taking derivative with respect to $\theta$ on both sides of the above equation, and then letting $\theta=0$, we have that $v_t(x,i):=\left.\frac{\partial}{\partial\theta}
u^{g+\theta \phi}_t\right|_{\theta=0}$ satisfies $$\begin{aligned}
\label{int-v}
&v_t(x,i)-
\Pi_{(x,i)}\int^t_0b(X_s, Y_{s})\left(n(X_s, Y_{s})-1\right)v_{t-s}(X_s, Y_{s})ds\nonumber\\
&+\ \Pi_{(x,i)}\int^t_0b(X_s, Y_{s})\left(m(X_s, Y_{s})-\eta\left(X_s, Y_{s}; \pi(X_s, Y_{s}; u^{g}_{t-s})\right)\right)
\sum^K_{j=1}p^{(Y_{s})}_{j}(X_s)v_{t-s}(X_s,j)ds\nonumber\\
&=\ \Pi_{(x,i)}\left[\phi(X_t, Y_t)\right].
\end{aligned}$$ Let $$\label{def-J}
J((x, k), d(y, l))=\delta(x-y)q_{kl}(x)1_{\{k\neq l\}}dydl,
\qquad (x, k)\in D\times S,$$ where $dl$ stands for the counting measure on $S$. Then $(J((x, k), d(y, l)), t)$ is a Lévy system of $(X, Y)$. Define $$\label{def-F}
F(t-s, (x, i), (y, j)):=\ln \left(\frac{\eta(x, i; \pi(x, i; u^g_{t-s})}{n(x, i)}-
\frac{m(x, i)}{n(x, i)}+1\right)1_{i\neq j}.$$ Clearly, $F\le 0$. We would like to apply Lemma \[G-nonlocal-FK\] with $\xi=(X, Y)$, $q(t-s, (x,i))=b(x,i)(n(x,i)-1)$, $J$ given by and $F$ given by . Since $q_{ij}(x)$, $i,j\in S$, are bounded in $D$ and $D$ has finite Lebesgue measure, we have $\sup_{(x,i)\in D\times S}J((x,k), D\times S)<\infty$. By Remark \[rem6.2\](iii), conditions and are satisfied. Thus we can apply Lemma \[G-nonlocal-FK\] to get $$\begin{aligned}
\label{non-localFK'}
&v_t(x,i)\nonumber\\
&=\ \Pi_{(x,i)}\left[\exp\left\{\sum_{s\in D_J, 0<s\le t}\ln\left(\frac{\eta(X_{s}, Y_{s};\pi(X_{s}, Y_{s};u^g_{t-s}))}{n(X_{s}, Y_{s})}-\frac{m(X_s, Y_s)}{n(X_s, Y_s)}+1\right)\right.\right.\nonumber\\
&\qquad\qquad\qquad\left.\left.+\int^t_0b(X_s, Y_{s})
(n(X_s, Y_{s})-1){\rm d}s\right\}\phi(X_t, Y_t)\right]\nonumber\\
&=\ e^{\lambda_1t}\phi(x,i)\Pi_{(x,i)}^{\phi}\left[\exp\left\{\sum_{s\in D_J, 0<s\le t}\ln\left(\frac{\eta(X_{s}, Y_{s};\pi(X_{s}, Y_{s};u^g_{t-s}))}{n(X_{s}, Y_{s})}+\frac{\widetilde n(X_s, Y_s)}{n(X_s, Y_s)}\right)\right\}\right].\end{aligned}$$ Combining and , we obtain $$\begin{aligned}
&\widetilde{{\mathbb{P}}}_\mu\big(\exp\langle-g,\chi_t \rangle\big)\\
&={\mathbb{P}}_\mu\big(\exp\langle - g,\chi_t\rangle\big)\\
&\quad\cdot\Pi_{\phi\mu}^{\phi}\left[\exp\left\{\sum_{s\in D_J, 0<s\le t}\ln\left(\frac{\eta(X_{s}, Y_{s};\pi(X_{s}, Y_{s};u^g_{t-s}))}{n(X_{s}, Y_{s})}+\frac{\widetilde n(X_s, Y_s)}{n(X_s, Y_s)}\right)\right\}\right].\end{aligned}$$
Define $$\label{def-n}\widetilde F(x,i;du)=\frac{1}{n(x,i)}\left(\widetilde n(x,i)\delta_0+I_{(0,\infty)}u\, F(x,i;du)\right).$$ Then, by and , $\widetilde F(x,i;\cdot)$ is a probability measure on $[0,\infty)$ for any $(x,i)\in D\times S$ and $$\frac{\eta(x,i;\lambda)}{n(x, i)}+\frac{\widetilde n(x,i)}{n(x,i)}=
\int_{[0,\infty)}e^{-u\lambda}\widetilde F(x,i;du) \quad \hbox{for every } \lambda\ge 0.$$ Thus we may rewrite as $$\label{1inthm1'}\begin{array}{rl}&\displaystyle\widetilde{{\mathbb{P}}}_\mu\big(\exp\langle-g,\chi_t \rangle\big)\\
=&\displaystyle {\mathbb{P}}_\mu\big(\exp\langle - g,\chi_t\rangle\big)\cdot\Pi_{\phi\mu}^{\phi}\left[\prod_{s\in D_J, 0<s\le t}\int^\infty_0\exp(-u\pi(X_{s}, Y_{s};u^g_{t-s}))\widetilde F(X_{s}, Y_{s};du)\right].
\end{array}$$
From we see that the superdiffusion $\{\chi_t, t\ge 0; \widetilde{{\mathbb{P}}}_\mu\}$ can be decomposed into two independent parts. The first part is a copy of the original superdiffusion and the second part is an immigration process. To describe the second part precisely, we need to introduce another measure-valued process $\{\widehat{\chi}_t, t\ge 0\}$. Now we construct the measure-valued process $\{\widehat{\chi}_t, t\ge 0\}$ as follows:
[(i)]{} Suppose that $(\widehat X, \widehat Y)=\{(\widehat X_t, \widehat Y_t), t\ge 0\}$ is defined on some probability space $(\Omega, {\mathbb{P}}_{\mu, \phi})$, and $(\widehat X, \widehat Y)$ has the same law as $((X, Y);
\Pi^\phi_{\phi\mu})$. $(\widehat X, \widehat Y)$ serves as the spine or the immortal particle, which visits every part of $D\times S$ for large times since it is an ergodic process. Let $D_J$ be the set of jump points of $(\widehat X, \widehat Y)$. $D_J$ is countable.
[(ii)]{} Conditioned on $s\in D_J$, a measure-valued process $\chi^s$ started at $m_s\delta_{({\widehat X}_s,l)} (l\in S)$ is immigrated at the space position ${\widehat X}_s$ and the new immigrated particles choose their types independently according to the (nonrandom) distribution $\pi(x,i; \cdot)$. We suppose $\{m_s;s\in D_J\}$ is also defined on $(\Omega, {\mathbb{P}}_{\mu, \phi})$ such that, given $s\in D_J$ and $(\widehat X_s, \widehat Y_s)$, the distribution of $m_s$ is $\widetilde F({\widehat X}_s, {\widehat Y}_s;dr)$.
[(iii)]{} Once the particles are in the system, they begin to move and branch according to the $((X, Y),\, \widehat \psi(x,i,\cdot))$-superprocess independently.
We use $(\chi^{s}_t,\ t\ge s )$ to denote the measure-valued process generated by the mass immigrated at time $s$ and spatial position ${\widehat X}_s$. Conditional on $\{({\widehat X}_t, {\widehat Y}_t), t\ge 0; m_s, s\in D_J\}$, $\{\chi^{s}_t, t\ge s\}$ for different $s\in D_J$ are independent $((X, Y),\, \widehat \psi(x,i,\cdot))$-superprocesses. Set $$\label{X-hat}
\widehat{\chi}_t=\sum_{s\in(0,t]\cap D_J}\chi^s_t.$$ The Laplace functional of $\widehat{\chi}_t$ is described in the following proposition.
\[prop\] The Laplace functional of $\widehat{\chi}_t$ under ${\mathbb{P}}_{\mu,\phi}$ is equal to $$\Pi_{\phi\mu}^{\phi}\left\{\prod_{s\in(0,t]\cap D_J}\int_{[0,\infty)}\exp\left(-r\pi(X_s, Y_s; u^g_{t-s})\right)\widetilde F({X}_s, {Y}_s;dr)
\right\}.$$
For any $g\in B_b^+(D\times S)$, using , we have $$\begin{aligned}
{\mathbb{P}}_{\mu, \phi}\left[\exp(-\langle g,\widehat{\chi}_t\rangle)\right]
&=&{\mathbb{P}}_{\mu, \phi}\left\{ {\mathbb{P}}_{\mu, \phi}
\left[\exp(-\sum_{\sigma\in(0,t]\cap D_J}\langle g,
\chi_t^\sigma\rangle)\bigg|\sigma(({\widehat X}, {\widehat Y}), m)\right]\right\}\\
&=&{\mathbb{P}}_{\mu, \phi}\left[\prod_{s\in(0,t]\cap D_J}
\exp\left(-m_s\pi(\widehat X_s, \widehat Y_s,u^g_{t-s})\right)\right]\\
&=&{\mathbb{P}}_{\mu, \phi}\left\{{\mathbb{P}}_{\mu,
\phi}\left[\prod_{s\in(0,t]\cap D_J}\exp\left(-m_s\pi(\widehat X_s, \widehat Y_s, u^g_{t-s})\right)
\bigg|\sigma(({\widehat X},{\widehat Y}))\right]\right\}\\
&=&\Pi_{\phi\mu}^{\phi}\left\{\prod_{s\in(0,t]\cap D_J}\int_{[0,\infty)}\exp\left(-r\pi(X_s, Y_s, u^g_{t-s})\right)\widetilde F({X}_s, {Y}_s;dr)
\right\}.\end{aligned}$$
Without loss of generality, we suppose $\{\chi_t, t\ge 0; {\mathbb{P}}_{\mu,\phi}\}$ is a multitype superdiffusion defined on $(\Omega, {\mathbb{P}}_{\mu, \phi})$, having the same law as $\{\chi_t, t\ge 0;{\mathbb{P}}_{\mu}\}$ and independent of $\widehat\chi=\{\widehat \chi_t, t\ge 0\}$. Proposition \[prop\] says that we have the following decomposition of $\{\chi_t,t\ge 0\}$ under $\widetilde {\mathbb{P}}_{\mu}$: for any $t>0$, $$\label{decomp}
({\chi}_t, \widetilde {\mathbb{P}}_{\mu}) = (\chi_t+\widehat{\chi}_t,\
{\mathbb{P}}_{\mu,\phi})\quad\mbox{ in distribution}.$$ Since $\{\chi_t, t\ge 0;\widetilde {\mathbb{P}}_{\mu}\}$ is generated from the time-homogeneous Markov process $\{\chi_t, t\ge 0; {\mathbb{P}}_{\mu}\}$ via a non-negative martingale multiplicative functional, $\{\chi_t,t\ge 0;
\widetilde {\mathbb{P}}_{\mu}\}$ is also a time-homogeneous Markov process (see [@Sharpe Section 62]). From the construction of $\{
\widehat\chi_t, t\ge 0;{\mathbb{P}}_{\mu,\phi}\}$ we see that $\{
\widehat{\chi}_t, t\ge 0;{\mathbb{P}}_{\mu,\phi}\}$ is a time-homogeneous Markov process. For a rigorous proof of $\{ \widehat{\chi}_t, t\ge
0;{\mathbb{P}}_{\mu,\phi}\}$ being a time-homogeneous Markov process, we refer our readers to [@E]. Although the paper [@E] dealt with the representation of the superprocess conditioned to stay alive forever, one can check that the arguments there work in our case. Therefore, implies the following.
\[t:spine\] $$\label{decomp2}
\{{\chi}_t, t\ge 0; \widetilde {\mathbb{P}}_{\mu}\} = \{\chi_t+\widehat{\chi}_t, t\ge 0; {\mathbb{P}}_{\mu,\phi}\}\quad\mbox{ in law}.$$
$L\log L$ criterion {#s:proof}
===================
In this section, we give a proof of the main result of this paper, Theorem \[maintheorem\]. First, we make some preparations.
\[equi-deg\] Let $h(x,i)=\frac{1}{\phi(x,i)}{\mathbb{P}}_{\delta_{(x,i)}}(W_{\infty}(\phi))$. Then
[(i)]{} $h$ is a non-negative invariant function for the process $((X, Y); \Pi^{\phi}_{(x,i)})$.
[(ii)]{} Either $W_{\infty}$ is non-degenerate under ${\mathbb{P}}_{\mu}$ for all nonzero $\mu\in M_F(D\times S)$ or $W_{\infty}$ is degenerate under ${\mathbb{P}}_{\mu}$ for all $\mu\in M_F(D\times S)$.
\(i) By the Markov property of $\chi$, $$\begin{aligned}
h(x,i)&=&\frac{1}{\phi(x,i)}{\mathbb{P}}_{\delta_{(x,i)}}\left[\lim_{s\to\infty} \langle
e^{-\lambda_1(t+s)}\phi,
\chi_{t+s}\rangle\right]\\
&=&\frac{e^{-\lambda_1 t}}{\phi(x,i)}{\mathbb{P}}_{\delta_{(x, i)}}\left[{\mathbb{P}}_{\chi_t}
(\lim_{s\to\infty}\langle e^{-\lambda_1s}\phi,\chi_s\rangle)\right]\\
&=&\frac{e^{-\lambda_1 t}}{\phi(x,i)}{\mathbb{P}}_{\delta_{(x,i)}}\left[{\mathbb{P}}_{\chi_t}(
W_{\infty})\right]\ =\ \frac{e^{-\lambda_1
t}}{\phi(x,i)}{\mathbb{P}}_{\delta_{(x,i)}}\left[\langle
(h\phi), \chi_t\rangle\right]\\
&=&\frac{e^{-\lambda_1 t}}{\phi(x,i)}P^{{\cal A}+B\cdot(N-I)}_t(h\phi),\quad x\in D.\end{aligned}$$ By the definition of $\Pi^{\phi}_{(x,i)}$, we get that $h(x,i)=\Pi^{\phi}_{(x,i)}[h(X_t, Y_t)]$. So $h$ is an invariant function of the process $((X, Y); \Pi^{\phi}_{(x, i)})$. The non-negativity of $h$ is obvious.
\(ii) Since $h$ is non-negative and invariant, if there exists $(x_0, i)\in
D\times S$ such that $h(x_0,i)=0$, then $h\equiv 0$ on $D\times S$. Since ${\mathbb{P}}_{\mu}(W_{\infty}(\phi))=\langle h\phi, \mu\rangle,$ we then have ${\mathbb{P}}_{\mu}(W_{\infty}(\phi))=0$ for any $\mu\in M_F(D\times S)$. If $h>0$ on $D\times S$, then ${\mathbb{P}}_{\mu}(W_{\infty}(\phi))>0$ for any nonzero $\mu\in M_F(D\times S)$.
Using Proposition \[equi-deg\] we see that, to prove Theorem \[maintheorem\], we only need to consider the case $d\mu={\phi}(x,i)dxdi$, where $di$ is the counting measure on $S$. So in the remaining part of this paper we will always suppose that $d\mu={\phi}(x,i)dxdi$.
Recall from and that $$\pi(x,i;\phi)=\sum^K_{j=1}p^{i}_j(x)\phi(x,j),\quad (x,i)\in D\times S$$ and $$l(x,i)=\int_0^\infty r\log^+(r)F^{\pi(\phi)}(x,i; dr)=\int_0^\infty r\pi(x, i, \phi)
\log^+(r\pi(x, i, \phi))F(x,i; dr).$$
\[lemma1\] Let $(m_t;\ t\in D_J)$ be the Poisson point process constructed in Section 4, given the path of $(\widehat X_s, \widehat Y_s), s\ge 0$. Define $$\sigma_0=0,\quad \sigma_i=\inf\{s\in D_J;\ s>\sigma_{i-1},\
m_s\pi(\widehat X_s, \widehat Y_s;\phi)>1\}, \quad\eta_i=m_{\sigma_i},\quad
i=1,2,\cdots$$
[(i)]{} If $\displaystyle\sum^K_{i=1}\int_D{\phi}(y,i)b(y,i) l(y,i)dy<\infty$, then $$\label{e:5.1'}
\sum_{s\in D_J}\mbox{e}^{-\lambda_1s}m_s\pi(\widehat X_s, \widehat Y_s;\phi) < \infty, \quad
{\mathbb{P}}_{\mu,\phi}\mbox{-a.s.}$$
[(ii)]{} If $\displaystyle \sum^K_{i=1}\int_D{\phi}(y,i)b(y,i)l(y,i)dy=\infty$, then $$\label{e:5.1}
\limsup_{i\rightarrow\infty}e^{-\lambda_1\sigma_i}\eta_i
\pi(\widehat X_{\sigma_i}, \widehat Y_{\sigma_i};\phi) =\infty,\quad {\mathbb{P}}_{\mu,\phi}\mbox{-a.s.}$$
Since $\phi$ is bounded from above, $\sigma_i$ is strictly increasing with respect to $i$.
\(i) Suppose that $\sum_{i=1}^K\int_D\phi(y,i)b(y,i) l(y,i)dy<\infty$. For any $\varepsilon>0$, we write the sum above as $$\begin{aligned}
\label{sum}
&\sum_{s\in \mathcal D_J}\mbox{e}^{-\lambda_1s}m_s\pi(\widehat X_s, \widehat Y_s;\phi)\nonumber\\
&=\sum_{s\in \mathcal D_J}
\mbox{e}^{-\lambda_1s}m_s\pi(\widehat X_s, \widehat Y_s;\phi)
1_{\{m_s\pi(\widehat X_s, \widehat Y_s;\phi)\le\mbox{e}^{\varepsilon
s}\}}\nonumber\\
&\quad +\sum_{s\in \mathcal
D_J}\mbox{e}^{-\lambda_1s}m_s\pi(\widehat X_s, \widehat Y_s;\phi)
1_{\{m_s\pi(\widehat X_s, \widehat Y_s;\phi)>\mbox{e}^{\varepsilon
s}\}}\nonumber\\
&=\sum_{s\in \mathcal
D_J}\mbox{e}^{-\lambda_1s}m_s\pi(\widehat X_s, \widehat Y_s;\phi) 1_{\{
\pi(\widehat X_s, \widehat Y_s;\phi)m_s\le \mbox{e}^{\varepsilon s}
\}}\nonumber\\
&\quad+\sum_{i=1}^\infty\mbox{e}^{-\lambda_1\sigma_i}\eta_i
\pi(\widehat X_{\sigma_i}, \widehat Y_{\sigma_i};\phi)
1_{\{\eta_i\phi(\pi(\widehat X_{\sigma_i}, \widehat Y_{\sigma_i};\phi) >\mbox{e}^{\varepsilon
\sigma_i}\}}\nonumber\\&=I+II.\end{aligned}$$ Note that the jumping intensity of $\{(\widehat X, \widehat Y), {\mathbb{P}}_{\mu,\phi}\}$ is $\frac{bn\pi(\phi)}{\phi}(x, i)$ at $(x,i)\in D\times S$. Thus $$\begin{aligned}
&&\sum_{i=1}^\infty {\mathbb{P}}_{\mu,\phi}
\left(\eta_i\pi(\widehat X_{\sigma_i}, \widehat Y_{\sigma_i};\phi) > \mbox{e}^{\varepsilon
\sigma_i}\right)\\
&=&\sum_{i=1}^\infty
{\mathbb{P}}_{\mu,\phi}\left[{\mathbb{P}}_{\mu,\phi}\left(\eta_i
\pi(\widehat X_{\sigma_i}, \widehat Y_{\sigma_i};\phi) >\mbox{e}^{\varepsilon
\sigma_i}\big|\sigma({\widehat X},{\widehat Y})\right)\right]\\
&=&{\mathbb{P}}_{\mu,\phi}\left[{\mathbb{P}}_{\mu,\phi}\left(\sum_{i=1}^\infty
1_{\{\eta_i>\mbox{e}^{\varepsilon \sigma_i}
\pi(\widehat X_{\sigma_i}, \widehat Y_{\sigma_i};\phi) ^{-1}\}}\Big|
\sigma({\widehat X}, {\widehat Y})\right)\right]\\
&=&\Pi^{\phi}_{\phi\mu}\left[\int_0^\infty(bn\pi(\phi)/\phi)(X_{s}, Y_{s})
\left(\int^{\infty}_{\pi(X_{s}, Y_{s};\phi) ^{-1}\mbox{e}^{\epsilon s}}
\widetilde F(X_s, Y_s; dr)\right)ds\right].\end{aligned}$$ Recall that under $\Pi^{\phi}_{\phi\mu}$, $(X, Y)$ starts at the invariant measure $\phi^2(x,i)dxdi$. By the definition of $\widetilde F$ given in , $$\begin{aligned}
&&\sum_{i=1}^\infty {\mathbb{P}}_{\mu,\phi} \left(\eta_i\pi(X_{\sigma_i}, Y_{\sigma_i};\phi) > \mbox{e}^{\varepsilon \sigma_i}\right)\\
&=&\int_0^\infty
ds\sum^K_{j=1}\int_Ddy(b\phi)(y,j)\int^{\infty}_{\pi(y,j;\phi)^{-1}
\mbox{e}^{\epsilon s}}\pi(y,j;\phi)r\, F(y,j; dr)\\
&=& \sum^K_{j=1}\int_D(b \phi)(y,j)dy\int_{\pi(y,j;\phi)^{-1}}^\infty
\pi(y,j;\phi)r\, F(y,j; dr)\int^{\frac{\ln (r\pi(y,j;\phi))}{\epsilon}}_{0}ds\\
&=&\varepsilon^{-1}\sum^K_{j=1}\int_D(b{\phi})(y,j)l(y,j)dy.\end{aligned}$$ By the assumption that $\sum^K_{j=1}\int_D(b{\phi})(y,j) l(y,j)dy<\infty$ and the Borel-Cantelli Lemma, we get $$\label{io}
{\mathbb{P}}_{\mu,\phi}\Big(\eta_i\pi(\widehat X_{\sigma_i}, \widehat Y_{\sigma_i};\phi)>\mbox{e}^{\varepsilon \sigma_i} \mbox{ i. o.}\Big)=0$$ for all $\varepsilon>0$, which implies that $$\label{big}
II<\infty.\quad {\mathbb{P}}_{\mu,\phi}\mbox{-a.s.}$$ Meanwhile for $\varepsilon<\lambda_1$, $$\begin{array}{rl}
{\mathbb{P}}_{\mu,\phi}I&=\displaystyle {\mathbb{P}}_{\mu,\phi} \left[\sum_{s\in\mathcal
D_J} \mbox{e}^{-\lambda_1s}m_s\pi(\widehat X_s, \widehat Y_s;\phi) 1_{\{m_s\le
\mbox{e}^{\varepsilon s} \pi(\widehat X_s, \widehat Y_s;\phi) ^{-1}\}}\right]\\
&=\displaystyle \Pi_{\phi\mu}^\phi\int_0^\infty dt
\mbox{e}^{-\lambda_1t}\int_0^{\pi(X_t, {Y}_t;\phi)^{-1}
\mbox{e}^{\varepsilon t}}\frac{bn\pi(\phi)}{\phi}(X_t, Y_t)\pi(X_t, {Y}_t;\phi)r\widetilde F(X_t, {Y}_t;dr)\\
&\le \displaystyle
\Pi_{\phi\mu}^\phi\int_0^\infty dt
\mbox{e}^{-(\lambda_1-\varepsilon)t}\int_0^{\infty}\frac{b\pi(\phi)}{\phi}(X_t, Y_t)rF(X_t, {Y}_t; dr),
\end{array}$$ where for the inequality above we used the fact that $r\le\pi(X_t, {Y}_t,\phi)^{-1} \mbox{e}^{\varepsilon t}$ implies that $r\pi(X_t, {Y}_t,\phi)\le \mbox{e}^{\varepsilon t}$. By the assumption that $\sup_{(x,i)\in D\times S}\int_0^\infty r F(x,i, dr)<\infty$, we have $$\begin{array}{rll}{\mathbb{P}}_{\mu,\phi}I&\le&\displaystyle\frac{1}{\lambda_1-\epsilon}\sum^K_{i=1}\int_D b(y, i)\pi(y,i;\phi)\phi(y,i)\int_0^\infty rF(y,i, dr)dy\\
&\le&\displaystyle
\frac{1}{\lambda_1-\epsilon}\|\int_0^\infty rF(y,i, dr)\|_\infty
\|b\|_\infty<\infty.\end{array}$$ Thus $$\label{small}
I<\infty,\quad {\mathbb{P}}_{\mu,\phi}\mbox{-a.s.}$$ Combining , and , we obtain .
\(ii) Next, we assume $ \sum^K_{i=1}\int_D(b{\phi})(y,i)l(y,i)dy=\infty$. To establish , it suffices to show that for any $L>0$, $$\label{>K}
\limsup_{i\rightarrow\infty}e^{-\lambda_1\sigma_i}\eta_i
\pi(\widehat X_{\sigma_i}, \widehat Y_{\sigma_i};\phi)
>L,\quad {\mathbb{P}}_{\mu,\phi}\mbox{-a.s.}$$ Put $L_0:=1\vee(\max_{(x,i)\in D\times S}\phi(x,i))$. Then for $L\ge L_0$, $$L\inf_{(x,i)\in D\times S}\phi(x,i)^{-1}\geq 1.$$ Note that for any $T\in (0,\infty)$, conditional on $\sigma({\widehat X}, {\widehat Y})$, $$\sharp\{i: \sigma_i\in(0, T];
\eta_i>L\pi(\widehat X_{\sigma_i},
\widehat Y_{\sigma_i};\phi)^{-1}\mbox{e}^{\lambda_1\sigma_i}\}$$ is a Poisson random variable with parameter $$\int_0^Tdt(b\pi(\phi)/\phi)(\widehat X_t,{\widehat Y}_t)
\int_{L\pi(\widehat X_t, {\widehat Y}_t;\phi)^{-1}
\mbox{e}^{\lambda_1t}}^\infty r
F(\widehat X_t, {\widehat Y}_t; dr).$$ Since $(\widehat X, \widehat Y; {\mathbb{P}}_{\mu,\phi})$ has the same law as $(X, Y; \Pi^{\phi}_{\mu\phi})$, we have $$\begin{array}{rl}
&\displaystyle {\mathbb{P}}_{\mu,\phi}\int_0^Tdt
\frac{b\pi(\phi)}{\phi}(\widehat X_t, {\widehat Y}_t)
\int_{L\pi(\widehat X_t, {\widehat Y}_t;\phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
r F(\widehat X_t, {\widehat Y}_t;
dr)\\
=&\displaystyle\int^T_0dt\sum^K_{j=1}\int_Ddy(b\pi(\phi)\phi)(y,j)
\int_{L\pi(y,j;\phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
r F(y,j; dr)<\infty,\end{array}$$ thus $$\int_0^Tdt\frac{b\pi(\phi)}{\phi}(\widehat X_t, {\widehat Y}_t)
\int_{L\pi(\widehat X_t, {\widehat Y}_t;\phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
rF(\widehat X_t, {\widehat Y}_t;
dr)<\infty,\quad {\mathbb{P}}_{\mu,\phi}\mbox{-a.s.}$$ Consequently we have $$\label{number-finite}
\sharp\Big\{i: \sigma_i\in(0,T];
\eta_i>L\pi(\widehat X_t, {\widehat Y}_t;\phi)^{-1}\mbox{e}^{\lambda_1\sigma_i}
\Big\}<\infty,\quad
{\mathbb{P}}_{\mu,\phi}\mbox{-a.s.}$$ So, to prove , we need to prove $$\int_0^\infty dt\frac{b\pi(\phi)}{\phi}(\widehat X_t, {\widehat Y}_t)
\int_{L\pi(\widehat X_t, {\widehat Y}_t;\phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
rF(\widehat X_t, {\widehat Y}_t;
dr)=\infty,\quad {\mathbb{P}}_{\mu,\phi}\mbox{-a.s.}$$ which is equivalent to $$\begin{aligned}
\label{=infinity}
\int_0^\infty dt\frac{b\pi(\phi)}{\phi}(X_t, {Y}_t)
\int_{L\pi(X_t, Y_t;\phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
r F(X_t, {Y}_t; dr)=\infty,\quad \Pi^{\phi}_{\phi\mu}\mbox{-a.s.}\end{aligned}$$
For this purpose we first prove that $$\label{mean=infty}
\Pi^{\phi}_{\phi\mu}\left[\int_0^\infty dt\frac{b\pi(\phi)}{\phi}(X_t, {Y}_t)
\int_{L\pi(X_t, Y_t;\phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
rF(X_t, {Y}_t; dr)\right]=\infty.$$ Applying Fubini’s theorem, we get $$\begin{aligned}
&&\Pi^{\phi}_{\phi\mu}\left[\int_0^\infty dt\frac{b\pi(\phi)}{\phi}(X_t, {Y}_t)
\int_{L\pi(X_t, Y_t;\phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
r F(X_t, {Y}_t; dr)\right]\nonumber\\
&=& \sum^K_{j=1}\int_Db(y,j)\pi(y,j;\phi){\phi}(y,j)dy\int_0^\infty dt
\int_{L\pi(y,j;\phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty r F(y,j; dr)\\
&=&\sum_{j=1}^K \int_Db(y,j)\pi(y,j;\phi){\phi}(y,j)dy
\int_{L \pi(y,j;\phi)^{-1}}^\infty
rF(y,j;dr)\int_0^{\frac{1}{\lambda_1}
\ln(\frac{r\pi(y,j;\phi)}{L})}dt\\
&=&\sum_{j=1}^K\frac{1}{\lambda_1}\int_Db(y,j)\pi(y,j;\phi)
{\phi}(y,j)dy
\int_{L\pi(y,j;\phi)^{-1}}^\infty
(\ln[r\pi(y,j;\phi)]-\ln L)r F(y,j; dr)\\
&\ge&\sum_{j=1}^K\frac{1}{\lambda_1}\int_Db(y,j)\pi(y,j;\phi)
{\phi}(y,j)dy\left[ \int_{L\pi(y,j;\phi)^{-1}}^\infty
r\ln[r\pi(y,j;\phi)]F(y, j; dr)-A\right]\\
&=&\sum_{j=1}^K\frac{1}{\lambda_1}\int_D(b{\phi})(y,j)dy
\int_{L}^\infty
r\ln r\,F^{\pi(\phi)}(y,j; dr) -\sum_{j=1}^K
\frac{A}{\lambda_1}\int_D(b{\phi})(y,j)\pi(y, j;\phi)dy,\end{aligned}$$ for some constant $A>0$, where in the inequality we used the facts that $L\pi(y,j;\phi)^{-1}>1$ for any $(y,j)\in D\times S$ and $\sup_{(y, j)\in
D\times S}\int^\infty_1rF(y,j; dr)<\infty$. It is easy to see that $$\sum_{j=1}^K
\frac{A}{\lambda_1}\int_D(b{\phi})(y,j)\pi(y, j;\phi)dy\le\frac{A}{\lambda_1}\|b\|_\infty<\infty.$$ Since $$\sum_{j=1}^K\int_D(b{\phi})(y,j)dy\int_1^\infty r\ln r\,F^{\pi(\phi)}(y,j; dr)=\infty,$$ and $$\begin{aligned}
&&
\sum_{j=1}^K\int_D(b{\phi})(y,j)dy
\int_1^Lr\ln r\,F^{\pi(\phi)}(y,j; dr)\\
&\leq&L\ln L\sum^K_{j=1}\int_D (b{\phi})(y,j)F(y,j;
[\|\phi\|_{\infty}^{-1},\infty))dy<\infty,\end{aligned}$$ we get that $$\sum_{j=1}^K\int_D(b{\phi})(y,j)dy
\int_L^\infty r\ln r\,F^{\pi(\phi)}(y,j; dr)=\infty,$$ and therefore, holds.
By , there exists constant $t_0>0$ such that for any $t>t_0$ and any $f\in B^{+}_b(D\times S)$, $$\begin{aligned}
\label{domi-p}
\frac{1}{2}\int_{D\times S}\phi^2(y,j)f(y,j)dydi&\leq&
\int_{D\times S}p^{\phi}(t, (x,i), (y,j))f(y,j)dydi\nonumber\\
&\leq&
2\int_{D\times S}\phi^2(y,j)f(y,j)dydi\end{aligned}$$ holds for any $ (x, i)\in D\times S$. For $T>t_0$, we define $$\xi_T=\int_0^T dt\frac{b\pi(\phi)}{\phi}(X_t, {Y}_t)
\int_{L\pi(X_t, Y_t; \phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
r F(X_t, {Y}_t;dr)$$ and $$A_T=\sum_{j=1}^K
\int_{t_0}^Tdt\int_D(b{\phi})(y,j)dy\int_{L\mbox{e}^{\lambda_1t}}^\infty
r F^{\pi(\phi)}(y,j; dr).$$ Our goal is to prove , which is equivalent to $$\begin{aligned}
\xi_\infty:=\int_0^\infty dt\frac{b\pi(\phi)}{\phi}(X_t, {Y}_t)
\int_{L\pi(X_t, Y_t;\phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
r F(X_t, {Y}_t; dr)=\infty, \qquad \Pi_{\phi\mu}^\phi\mbox{-a.s.}\end{aligned}$$ Since $\left\{\xi_\infty=\infty\right\}$ is an invariant event, by the ergodic property of $\{(X, Y),\Pi^{\phi}_{\phi\mu}\}$, it is enough to prove $$\label{positive-prob}
\Pi^\phi_{\phi\mu}\left(\xi_\infty=\infty\right)>0.$$ Note that $$\label{domi-mean}
\Pi_{\phi\mu}^\phi \xi_T=\sum^K_{j=1}\int_0^Tdt\int_D
(b{\phi})(y,j)dy
\int_{L\mbox{e}^{\lambda_1t}}^\infty
r F^{\pi(\phi)}(y,j; dr)\ge A_T$$ and $$\begin{aligned}
\label{mean-limit}
&\lim_{T\rightarrow\infty}\Pi_{\phi\mu}^\phi \xi_T\ge
A_\infty=\sum^K_{j=1}
\int_{t_0}^\infty dt\int_D(b{\phi})(y,j)dy
\int_{L\mbox{e}^{\lambda_1t}}^\infty
rF^{\pi(\phi)}(y,j; dr)\nonumber\\
&=\sum^K_{j=1}\int_D(b\phi)(y,j)dy
\int_{Le^{\lambda_1t_0}}^\infty
\left(\frac{1}{\lambda_1}(\ln r-\ln L)-t_0\right)
r F^{\pi(\phi)}(y,j; dr)\nonumber\\
&\ge c\sum^K_{j=1}\int_D(b\phi)(y,j)l(y,j)dy=\infty,\end{aligned}$$ where $c$ is a positive constant. By [@D2 Exercise 1.3.8], $$\label{Durrett-domi}
\Pi_{\phi\mu}^\phi\left(\xi_T\geq \frac{1}{2}\Pi_{\phi\mu}^\phi
\xi_T\right)\geq \frac{(\Pi_{\phi\mu}^\phi
\xi_T)^2}{4\Pi_{\phi\mu}^\phi(\xi_T^2)}.$$ If we can prove that there is a constant $\widehat c>0$ such that or all $T>t_0$, $$\begin{aligned}
\label{uniform lower bound}
\frac{(\Pi_{\phi\mu}^\phi
\xi_T)^2}{4\Pi_{\phi\mu}^\phi(\xi_T^2)}\geq
\widehat c.\end{aligned}$$ Then by we would get $$\Pi_{\phi\mu}^\phi\left(\xi_T\geq \frac{1}{2}\Pi_{\phi\mu}^\phi
\xi_T\right) \geq \widehat c,$$ and therefore $$\begin{aligned}
&&\Pi^\phi_{\phi\mu}\left(\xi_\infty\geq
\frac{1}{2}\Pi_{\phi\mu}^\phi \xi_T\right)
\geq\Pi^\phi_{\phi\mu}\left(\xi_T\geq \frac{1}{2} \Pi_{\phi\mu}^\phi
\xi_T\right) \ge\widehat c>0.\end{aligned}$$ Since $\lim_{T\rightarrow\infty}\Pi_{\phi\mu}^\phi \xi_T=\infty$ (see ), the above inequality implies . Now we only need to prove . For this purpose we first estimate $\Pi_{\phi\mu}^\phi(\xi_T^2)$: $$\begin{aligned}
\Pi_{\phi\mu}^\phi
\xi_T^2&=&\Pi_{\phi\mu}^\phi\int_0^Tdt \int_{L\pi(X_t, Y_t; \phi)^{-1}
\mbox{e}^{\lambda_1t}}^\infty
\frac{b\pi(\phi)}{\phi}(X_t, {Y}_t)rF(X_t, {Y}_t; dr)\\
&&\quad\times\int_0^Tds
\int_{L\pi(X_s, Y_s;\phi)^{-1}\mbox{e}^{\lambda_1s}}^\infty
\frac{b\pi(\phi)}{\phi}(X_s, {Y}_s)uF(X_s, {Y}_s; du)\\
&=&2\Pi_{\phi\mu}^\phi\int_0^Tdt
\int_{L\pi(X_t, Y_t; \phi)^{-1}
\mbox{e}^{\lambda_1t}}^\infty\frac{b\pi(\phi)}{\phi}(X_t, {Y}_t)
r F(X_t, {Y}_t; dr)\\
&&\quad\times \int_t^Tds
\int_{L\pi(X_s, Y_s; \phi)^{-1}\mbox{e}^{\lambda_1s}}^\infty
\frac{b\pi(\phi)}{\phi}(\widehat X_s,\widehat{Y}_s)u F(X_s, {Y}_s; du)\\
&=&2\Pi_{\phi\mu}^\phi\int_0^Tdt
\int_{L\pi(X_t, Y_t; \phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
\frac{b\pi(\phi)}{\phi}(X_t, {Y}_t)r F(X_t, {Y}_t; dr)\\
&&\quad\times
\int_t^{(t+t_0)\wedge T}ds\int_{L\pi(X_s, Y_s; \phi)^{-1}
\mbox{e}^{\lambda_1s}}^\infty\frac{b\pi(\phi)}{\phi}(X_s, {Y}_s) uF(X_s, {Y}_s; du)\\
&&+2\Pi_{\phi\mu}^\phi\int_0^{ T}dt
\int_{L\pi(X_t, Y_t; \phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
\frac{b\pi(\phi)}{\phi}(X_t, {Y}_t) r F(X_t, {Y}_t; dr)\\
&&\quad\times
\int_{(t+t_0)\wedge T}^Tds\int_{L\pi(X_s, Y_s; \phi)^{-1}
\mbox{e}^{\lambda_1s}}^\infty \frac{b\pi(\phi)}{\phi}(X_s, {Y}_s)u F(X_s, {Y}_s; du)\\
&=&III+IV,\end{aligned}$$ where $$\begin{aligned}
III&=&\displaystyle 2\Pi_{\phi\mu}^\phi\int_0^Tdt
\int_{L\pi(X_t, Y_t; \phi)^{-1} \mbox{e}^{\lambda_1t}}^\infty
\frac{b\pi(\phi)}{\phi}(X_t, {Y}_t)r F(X_t,{Y}_t; dr)\\
&&\quad\times
\int_t^{(t+t_0)\wedge T}ds\int_{L\pi(X_s, Y_s; \phi)^{-1}
\mbox{e}^{\lambda_1s}}^\infty \frac{b\pi(\phi)}{\phi}(X_s, {Y}_s)u F(X_s, {Y}_s;du)\end{aligned}$$ and $$\begin{aligned}
IV= & 2\Pi_{\phi\mu}^\phi\int_0^{T}dt
\int_{L\pi(X_t, Y_t; \phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
\frac{b\pi(\phi)}{\phi}(X_t, {Y}_t)rF(X_t, {Y}_t; dr)\\
&\quad\times
\int_{(t+t_0)\wedge T}^Tds\int_{L\pi(X_s, Y_s; \phi)^{-1}
\mbox{e}^{\lambda_1s}}^\infty\frac{b\pi(\phi)}{\phi}(X_s, {Y}_s) u F(X_s, {Y}_s; du)\\
=& 2\sum^K_{j=1}\int_0^{T}dt\int_D(b\phi) (y,j)dy
\int_{L\pi(y,j;\phi)^{-1}\mbox{e}^{\lambda_1t}}^\infty
r\pi(y,j;\phi) F(y,j; dr)\\
& \ \ \ \times
\int_{(t+t_0)\wedge T}^Tds
\int_Dp^{\phi}(s-t,(y, j),(z,k))\frac{b\pi(\phi)}{\phi}(z,k)dz
\int_{L\pi(z,k;\phi)^{-1}\mbox{e}^{\lambda_1s}}^\infty u F(z,k;du).\end{aligned}$$ By our assumption we have $ \|\int_1^\infty
r\,F(\cdot;dr)\|_{\infty}<\infty$. Since $L\inf_{(x,i)\in D\times S}\phi(x,j)^{-1}\ge 1$, we have $$III\leq c_1\Pi^{\phi}_{\phi\mu}\xi_T,$$ for some positive constant $ c_1$ which does not depend on $T$. Using and the definition of $n^{\pi(\phi)}$, we get that $$\begin{array}{rl}
&\displaystyle
\int_{(t+t_0)\wedge T}^Tds
\int_Dp^{\phi}(s-t,(y,j),(z,k))\frac{b\pi(\phi)}{\phi}(z,k)dz
\int_{L\pi(z,k;\phi)^{-1}\mbox{e}^{\lambda_1s}}^\infty
u F(z,k; du)\\
\le&\displaystyle
2\int_{(t+t_0)\wedge T}^Tds\int_D(b\phi)(z,k)dz\int_{L\phi(z,k;\phi)^{-1}
\mbox{e}^{\lambda_1s}}^\infty \pi(z,k;\phi)u F(z,k; du)\\
\le&\displaystyle
2\int_{t_0}^Tds\int_D(b\phi)(z,k)dz\int_{Le^{\lambda_1s}}^\infty
rF^{\pi(\phi)}(z,k; dr)\\
=&\displaystyle2\sum^K_{k=1}
\int_{t_0}^Tds\int_D(b\phi)(z,k)dz\int_{Le^{\lambda_1s}}^\infty
rF^{\pi(\phi)}(z,k; dr)=2A_T.\end{array}$$ Then using , we have $$IV\leq 4 A_T\Pi^{\phi}_{\phi\mu}\xi_T\leq 4 (\Pi^{\phi}_{\phi\mu}
\xi_T)^2.$$ Combining the estimates above on $III$ and $IV$, we get that there exists a $c_2>0$ independent of $T$ such that for $T>t_0$, $$\Pi_{\phi\mu}^\phi(\xi_T^2)\le 4
(\Pi_{\phi\mu}^\phi(\xi_T))^2+
c_1\Pi_{\phi\mu}^\phi(\xi_T)\le
c_2(\Pi_{\phi\mu}^\phi(\xi_T))^2.$$ Then we have with $\widehat c=1/c_2$, and the proof of the theorem is now complete.
\[con-mart\] Suppose that $(\Omega, {\cal F}, {\mathbb{P}})$ is a probability space, $\{{\cal F}_t,t\ge 0\}$ is a filtration on $(\Omega, {\cal F})$ and that ${\cal G}$ is a sub-$\sigma$-field of ${\cal F}$. A real valued process $U_t$ on $(\Omega, {\cal F}, {\mathbb{P}})$ is called a ${\mathbb{P}}(\cdot |\
{\cal G})$-martingale (submartingale, supermartingale resp.) with respect to $\{{\cal F}_t,t\ge 0\}$ if (i) it is adapted to $\{{\cal
F}_t\vee{\cal G},t\ge 0\}$; (ii) for any $t\geq 0,\
{\mathbb{E}}(|U_t|\big|{\cal G})<\infty$ and (iii) for any $t>s$, $${\mathbb{E}}(U_t\big|{\cal F}_s\vee{\cal G})=(\ge,\le\mbox{ resp. })\
U_s,\quad{\rm a.s.}$$
We need the following result. For its proof, see [@LRS09 Lemma 3.3].
\[con-mart-conver\] Suppose that $(\Omega, {\cal F}, {\mathbb{P}})$ is a probability space, $\{{\cal F}_t,t\ge 0\}$ is a filtration on $(\Omega, {\cal F})$ and that ${\cal G}$ is a $\sigma$-field of ${\cal F}$. If $U_t$ is a ${\mathbb{P}}(\cdot|\ {\cal G})$-submartingale with respect to $\{{\cal
F}_t,t\ge 0\}$ satisfying $$\label{f-con-exp}
\sup_{t\ge 0} {\mathbb{E}}(|U_t|\big|{\cal G})<\infty\quad {\rm a.s.}$$ then there exists a finite random variable $U_{\infty}$ such that $U_t$ converges a.s. to $U_{\infty}$.
We are now in the position to prove the main result of this paper.
[**Proof of Theorem $\ref{maintheorem}$.**]{}
Recall that, by Proposition \[equi-deg\], to prove Theorem \[maintheorem\], we only need to consider the case $d\mu={\phi}(x,i)dxdi$, where $di$ is the counting measure on $S$.
We first prove that if $\sum^K_{i=1}\int_{D}{\phi}(x,i)b(x,i)l(x,i)dx<\infty $, then $W_{\infty}$ is non-degenerate under ${\mathbb{P}}_{\mu}$. Since $W_{t}(\phi)$ is a nonnegative martingale, to show it is a closed martingale, it suffices to prove ${\mathbb{P}}_{\mu}(W_\infty(\phi))={\mathbb{P}}_{\mu}(W_0 (\phi))= \langle\phi, \mu\rangle$. Since $W_t^{-1}(\phi)$ is a positive supermartingale under $\widetilde{{\mathbb{P}}}_\mu$, $W_t(\phi)$ converges to some nonnegative random variable $W_\infty(\phi)\in (0,\ \infty]$ under $\widetilde{{\mathbb{P}}}_\mu$. By [@D2 Theorem 5.3.3], we only need to prove that $$\label{to-prove}
\widetilde{{\mathbb{P}}}_\mu\left(W_\infty(\phi)<\infty\right)
=1.$$ By , $(\chi_t, t\ge 0;\widetilde{{\mathbb{P}}}_\mu)$ has the same law as $(\chi_t+\hat \chi_t, t\ge 0; {\mathbb{P}}_{\mu,\phi})$, where $\{\chi_t, t\ge 0;{\mathbb{P}}_{\mu,\phi})$ is a copy of $(\chi_t, t\ge 0; {\mathbb{P}}_{\mu})$, and $\hat \chi_t=\sum_{s\in(0,t]\cap D_J}\chi^s_t.$ Put $$\begin{aligned}
\label{definition of W_t}
M_t(\phi):=\sum_{s\in(0,t]\cap{\mathcal D_J}}\langle \phi,
\chi^s_t\rangle\ \mbox{e}^{-\lambda_1t}.\end{aligned}$$ Then $$\label{decomp3}
(W_t(\phi), t\ge 0; \widetilde {\mathbb{P}}_{\mu})
= (W_t(\phi)+M_t(\phi), t\ge 0; {\mathbb{P}}_{\mu,\phi})\quad \mbox{ in law},$$ where $\{W_t(\phi), t\ge 0\}$ is copy of the martingale defined in and is independent of $M_t(\phi)$. Let ${\cal
G}$ be the $\sigma$-field generated by $\{Y_t, m_t,
t\ge 0\}$. Then, conditional on ${\cal G}$, $(\chi^{s}_t, t\ge
s, {\mathbb{P}}_{\mu,\phi})$ has the same law as $(\chi_{t-s}, t\ge s, {\mathbb{P}}_{m_{s}\delta_{\widehat{Y}_{s}}} )$ and $(\chi^{s}_t, t\ge s, {\mathbb{P}}_{\mu,\phi})$ are independent for $s\in {\mathcal D}_J$. Then we have $$\begin{aligned}
\label{form2-W_t}
M_t(\phi)
\stackrel{d}{=}\sum_{s\in(0,t]\cap {\mathcal D}_J}
\mbox{e}^{-\lambda_1 s}
W_{t-s}^{s}(\phi),
\end{aligned}$$ where for each $s\in{ \mathcal D_J}$, $W^{s}_t(\phi)$ is a copy of the martingale defined by with $\mu=m_{s}\delta_{\widehat{Y}_{s}}$, and conditional on ${\cal G}$, $\{W^{s}_t(\phi),t\ge 0\}$ are independent for $s\in {\mathcal D}_J$. To prove , by , it suffices to show that $${\mathbb{P}}_{\mu,\phi}\left(\lim_{t\to\infty}[W_t(\phi)+M_t(\phi)]<\infty\right)
=1.$$ Since $(W_t(\phi), t\ge 0)$ is a nonnegative martingale under the probability ${\mathbb{P}}_{\mu,\phi}$, it converges ${\mathbb{P}}_{\mu,\phi}$ almost surely to a finite random variable $W_\infty(\phi)$ as $t\to\infty$. So we only need to prove $$\label{M-finite}
{{\mathbb{P}}}_{\mu,\phi}\big(\lim_{t\to\infty}M_t(\phi)<\infty\big)
=1.$$ Define ${\cal H}_t:={\cal G}\bigvee\sigma( \chi^{\sigma}_{(s-\sigma)}; \sigma\in[0,t]\cap {\mathcal D_m}, s\in[\sigma, t]).$ Then $(M_t(\phi))$ is a ${\mathbb{P}}_{\mu,\phi}(\cdot|{\cal G})$-nonnegative submartingale with respect to $({\cal H}_t)$. By and Lemma \[lemma1\], $$\begin{aligned}
\sup_{t\geq 0}{\mathbb{P}}_{\mu,\phi}\big(M_t(\phi)|{\cal G}\big)&=
\sup_{t\geq 0} \sum_{{s\in [0,\ t]\cap\mathcal
D_J}}\mbox{e}^{-\lambda_1s}m_s\phi({\widehat X}_{s},{\widehat Y}_{s})\\
& \leq
\sum_{{s\in\mathcal
D_J}}\mbox{e}^{-\lambda_1s}m_s\phi({\widehat X}_{s},{\widehat Y}_{s})<\infty,\
{\mathbb{P}}_{\mu, \phi}\mbox{-a.s.}\end{aligned}$$ Then by Lemma \[con-mart-conver\], $M_{t}(\phi)$ converges ${\mathbb{P}}_{\mu,\phi}$-a.s. to $M_{\infty}(\phi)$ as $t\to\infty$ and ${\mathbb{P}}_{\mu,\phi}(M_{\infty}(\phi)<\infty)=1$, which establishes .
Now we prove the other direction. Assume that $\sum^K_{i=1}\int_D{\phi}(y,i)b(y,i){l(y,i)}dy=\infty$. We are going to prove that $W_\infty(\phi):=\lim_{t\to\infty}W_t(\phi)$ is degenerate with respect to $ {\mathbb{P}}_{\mu}$. By [@HR Proposition 2], $\frac{1}{W_t(\phi)}$ is a supermartingale under $\widetilde {\mathbb{P}}_{\mu}$, and thus $1/[M_{t}(\phi)+W_t(\phi)]$ is a nonnegative supermartingales under ${\mathbb{P}}_{\mu,\phi}$. Recall that $W_{t}(\phi)$ is a nonnegative martingale under ${\mathbb{P}}_{\mu,\phi}$. Then the limits $\lim_{t\to\infty}W_t(\phi)$ and $1/\lim_{t\to\infty}[M_{t}(\phi)+W_t(\phi)]$ exist and finite ${\mathbb{P}}_{\mu,\phi}$-a.s. Therefore $\lim_{t\to\infty}M_t(\phi)$ exists in $[0, \infty]$ ${\mathbb{P}}_{\mu,\phi}$-a.s. Recall the definition of $(\eta_i,
\sigma_i; i=1,2,\cdots)$ in Lemma \[lemma1\], and note that $\lim_{i\to\infty}\sigma_i=\infty$. By Lemma \[lemma1\], $$\limsup_{t\rightarrow\infty}M_t(\phi)\geq\limsup_{i\rightarrow\infty
}M_{\sigma_i}(\phi) \geq\limsup_{i\rightarrow\infty}
\mbox{e}^{-\lambda_1\sigma_i}\eta_i\phi(\widehat X_{\sigma_i},
\widehat Y_{\sigma_i})=\infty\quad {\mathbb{P}}_{\mu,\phi}\mbox{-a.s.}$$ So we have $$\lim_{t\rightarrow\infty}M_t(\phi)=\infty \quad {\mathbb{P}}_{\mu,\phi}\mbox{-a.s.}$$ By , $$\widetilde{{\mathbb{P}}}_{\mu}(W_{\infty}(\phi)=\infty)=1.$$ It follows from [@D2 Theorem 5.3.3] that ${\mathbb{P}}_{\mu}(W_{\infty}=0)=1$.
Appendix: Non-local Feynman-Kac transform
=========================================
In this Appendix, we establish a result on time-dependent non-local Feynman-Kac transform, which has been used in the proof of Theorem \[theorem1\].
Let $E$ be a Lusin space and ${\cal B}(E)$ be the Borel $\sigma$-field on $E$, and let $m$ be a $\sigma$-finite measure on ${\cal B}(E)$ with ${\rm supp} [m] = E$. Let $\{\xi_t, t\ge 0; \Pi_{x}\}$ be an $m$-symmetric Borel standard process on $E$ with Lévy system $(J, t)$, where $J(x, dy)$ is a kernel from $(E, {\cal B}(E))$ to $(E\cup\{\partial\}, {\cal B}(E\cup\{\partial\}))$.
\[G-nonlocal-FK\] Suppose that $\{\xi_t, t\ge 0; \Pi_{x}\}$ is an $m$-symmetric Borel standard process on $E$ with Lévy system $(J, t)$. Assume that $q$ is a locally bounded function on $[0,\infty)\times E$ and that $F$ is a non-positive, ${\cal B}([0,\infty)\times E\times E)$-measurable function vanishing on the diagonal of $E\times E$ so that for any $x\in E$, $$\label{e:6.1}
\sum_{0< s\le t}F(t-s,\, \xi_{s-},\, \xi_s) > -\infty \quad \hbox{for every } t>0
\quad \Pi_x\mbox{-a.s.}$$ and $$\label{e:6.2}
\sup_{x\in E} \Pi_x \left[ \int_0^t \int_{E_\partial}(1-e^{F(t-s,\,\xi_{s},\, y)})J(\xi_s, dy) ds \right]
<\infty \quad \hbox{for every } t>0.$$ For any $x\in E$, $t\ge 0$ and $f\in B^+_b(E)$, define $$\label{non-local-FK}
h(t,x):=\Pi_{x}\left[e^{\int^t_0q(t-s, \xi_s)ds+\sum_{0< s\le t}F(t-s,\, \xi_{s-},\, \xi_s)}f(\xi_t)\right].$$ Then $h$ is the unique locally bounded positive solution of the following integral equation $$\begin{aligned}
\label{inteq-nonlocal-FK}h(t,x)=&\Pi_xf(\xi_t)+\Pi_x\int^t_0q(t-s,\xi_s)h(t-s, \xi_s)ds\nonumber\\
&+\Pi_{x}\left[\int^t_0\int_E(e^{F(t-s,\,\xi_{s},\, y)}-1)h(t-s, y)J(\xi_s, dy)ds\right].\end{aligned}$$
Note that under the locally boundedness assumption of $q(t, x)$ and , the function $h$ of is well defined and positive, and there exists $c>0$ such that $$h(t, x)\le e^{ct}\Pi_x[f(\xi_t)].$$ Thus $h(t,x)$ is bounded on $[0,T]\times E$ for any $T>0$. The assumption implies that the last term of is absolutely convergent and defines a bounded function on $[0, T]\times E$ for every $T>0$. For $s\le t$, define $$A_{s,t}=\int^t_sq(t-r, \xi_r)dr+\sum_{s<r\le t}F(t-r, \xi_{r-},\xi_r),$$ which is right continuous and has left limits as a function of $s$. Note that $$\begin{aligned}
e^{A_{0,t}}-1
&=& - \left( e^{A_{t,t}}-e^{A_{0,t}} \right) \\
&=&\int^t_0e^{A_{s-}, t}q(t-s, \xi_s)ds-\sum_{0<s\le t} \left( e^{A_{s,t}}-e^{A_{s-, t}} \right)\\
&=&\int^t_0 e^{A_{s, t}} q(t-s,\xi_s)ds+\sum_{0<s\le t} e^{A_{s,t}} \left( e^{F(t-s,\,\xi_{s-},\, \xi_s))}-1 \right).\end{aligned}$$ Hence we have $$\begin{aligned}
&\Pi_{x}\left[\left(e^{A_{0,t}}-1\right)f(\xi_t)\right]\\
=&\Pi_x\left[\int^t_0e^{A_{s, t}}q(t-s,\xi_s)f(\xi_t)ds\right]+\Pi_{x}\left[\sum_{0<s\le t}
e^{A_{s,t}}
\left( e^{F(t-s,\,\xi_{s-},\, \xi_s))}-1 \right) f(\xi_t)\right].\end{aligned}$$ By the Markov property of $\xi$ and the fact that $$\begin{aligned}
A_{s,t} =
\left( \int^{t-s}_0 q(t-s-r,\xi_r)dr+
\sum_{0< r\le t-s}
F(t-s-r,\, \xi_{r-},\, \xi_r ) \right)
\circ\theta_s ,\end{aligned}$$ we have $$\begin{aligned}
& h(t,x)\\
&= \Pi_xf(\xi_t)+\Pi_x\left[\int^t_0q(t-s,\xi_s)\Pi_{\xi_s}\left(e^{\int^{t-s}_0q(t-s-r,\xi_r)dr+
\sum_{0< r\le t-s}
F(t-s-r,\, \xi_{r-},\, \xi_r)}f(\xi_{t-s})\right)\right]\\
&\quad+\Pi_{x}\left[\sum_{0<s\le t} \left(e^{F(t-s,\,\xi_{s-},\, \xi_s))}-1 \right)
\Pi_{\xi_s}\left[e^{\int^{t-s}_0q(t-s-r, \xi_r)dr+
\sum_{0< r\le t-s}
F(t-s-r,\, \xi_{r-},\, \xi_r)}f(\xi_{t-s})\right]\right].
\\
&=\Pi_xf(\xi_t)+\Pi_x\int^t_0q(t-s,\xi_s)h(t-s, \xi_s)ds+\Pi_{x}\left[\sum_{0<s\le t}
\left( e^{F(t-s,\,\xi_{s-},\, \xi_s))}-1 \right) h(t-s, \xi_s)\right]
\\
&=\Pi_xf(\xi_t)+\Pi_x\int^t_0
q(t-s, \xi_s)
h(t-s, \xi_s)ds\\
&\quad+\Pi_{x}\left[\int^t_0\int_E \left( e^{F(t-s,\,\xi_{s},\, y)}-1 \right) h(t-s, z)
J(\xi_s, dy) ds\right].\end{aligned}$$ Thus $h(t,x)$ defined by is a locally bounded positive solution of .
It follows from [@Li Proposition 2.15] that has a unique locally bounded positive solution.
\[rem6.2\]
[(i)]{} Lemma \[G-nonlocal-FK\] can be easily extended to signed $F$ (with the same argument) by replacing condition - by $$\sum_{0< s\le t}F^-(t-s,\, \xi_{s-},\, \xi_s) < \infty \quad \hbox{for every } t>0
\quad \Pi_x\mbox{-a.s.} \eqno (6.1')$$ and $$\sup_{x\in E} \Pi_x \left[ \int_0^t \int_{E_\partial} \left| 1-e^{F(t-s,\,\xi_{s},\, y)} \right| (\xi_s, dy) ds \right]
<\infty \quad \hbox{for every } t>0. \eqno (6.2')$$
[(ii)]{} If $F$ does not depend on $t$, the above result follows easily from the results of [@CS03a].
[(iii)]{} If $\sup_{x\in E}J(x, E\cup\{\partial\})<\infty$, or if $$\sup_{x\in E} \Pi_x \left[ \int_0^t \int_{E_\partial} |F(t-s, \xi_s, y)| J(\xi_s, dy) ds \right]
<\infty \quad \hbox{for every } t>0,$$ then conditions and are satisfied.
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0.3truein
[**Zhen-Qing Chen**]{}
Department of Mathematics, University of Washington, Seattle, WA 98195, USA
E-mail: zqchen@uw.edu
LMAM School of Mathematical Sciences & Center for Statistical Science, Peking University, Beijing, 100871, P.R. China.
E-mail: yxren@math.pku.edu.cn
[**Renming Song:**]{}
Department of Mathematics, The University of Illinois, Urbana, IL 61801 U.S.A.,
E-mail: [rsong@illinois.edu]{}
[^1]: The research of this author is supported by NNSFC (Grant No. 11731009 and 11671017).
[^2]: Research supported in part by a grant from the Simons Foundation (\#429343, Renming Song).
|
---
abstract: 'We numerically investigate quantum rings in graphene and find that their electronic properties may be strongly influenced by the geometry, the edge symmetries and the structure of the corners. Energy spectra are calculated for different geometries (triangular, hexagonal and rhombus-shaped graphene rings) and edge terminations (zigzag, armchair, as well as the disordered edge of a round geometry). The states localized at the inner edges of the graphene rings describe different evolution as a function of magnetic field when compared to those localized at the outer edges. We show that these different evolutions are the reason for the formation of sub-bands of edge states energy levels, separated by gaps (anticrossings). It is evident from mapping the charge densities that the anticrossings occur due to the coupling between inner and outer edge states.'
address: 'Instituto de Física, Universidade Estadual de Campinas - UNICAMP, C.P. 6165, 13083-970, Campinas, Brazil'
author:
- 'D. A. Bahamon'
- 'A. L. C. Pereira'
- 'P. A. Schulz'
title: 'Inner and outer edge states in graphene rings: A numerical investigation'
---
I. Introduction
===============
Graphene is a bona fide two-dimensional material showing a great versatility due to its unconventional electronic properties and promising applications to nanoelectronics [@geim_rev; @rev_mod_phys]. Among the promises is the possibility of structuring graphene at a mesoscopic length. Indeed, some groups have already demonstrated that graphene can be cut in many different shapes and sizes, opening the door to the fabrication of graphene nano-devices through the impressive experimental obtention of graphene quantum dots [@novoselov; @ponomarenko; @ensslin], quantum rings [@morpurgo] and even antidot arrays [@schen]. This perspective leads to interesting scenarios since the electronic properties of graphene are deeply influenced by its size and shape: as it is well known for over 10 years, graphene nanoribbons have different properties depending on their width or on their edge terminations [@Nakada; @Wakabayashi; @Brey1; @cresti]. Besides the nanoribbons, some theoretical works have also addressed the effects of confinement on the electronic structure of graphene quantum dots (flakes) with different geometries, sizes and types of edge [@Yamamoto; @peeters; @akola]. Recently, the energy levels of graphene quantum rings [@recher; @abergel] and of graphene antidot lattices [@pedersen1; @pedersen2] have also been theoretically investigated.
In this paper we numerically analyze the energy spectra as a function of magnetic field ($B$) of graphene quantum dots and graphene quantum rings, with focus on the complex evolution of edge states in the graphene rings. Here we explore the effects of the interplay among different degrees of freedom given by size, geometry and edge symmetries on the electronic properties of these graphene nanostructures. We consider quantum dots and rings with six (hexagons), three (triangles) and two-fold (rhombus-shaped) rotational symmetry, with zigzag or armchair edges. Afterwards we consider round dots and rings, whose edges are not so simply: they are cut in a way to approach the circular geometry. Our attention is concentrated on the continuum limit [@ana] of the energy spectra as a function of the magnetic flux of these structures. Edge states appear with energies between consecutive Landau levels (LLs) in such spectra, as could be initially expected [@sivan]. However, the interplay between two different edges showing distinct local structures (the quantum rings can be seen as graphene structures containing an antidot, which introduces an inner edge to the system) leads to some surprising subtleties. We observe that the presence of the antidot introduces additional edge states, with a different evolution under $B$: their energies are increased with increasing $B$. For a better understanding of this behavior, the electronic densities of these states are mapped, and we show that edge states that rise in energy with $B$ are located in the internal edges of the ring structure. In this way, we show that inner and outer edge states give origin to the formation of sub-bands separated by energy gaps in the region of the spectra between LLs. The anticrossing of levels, which defines the sub-bands, occurs due to the inter-edge coupling of states. The formation of sub-bands is highly influenced by symmetry properties, and also by size effects, i.e., the relation between the ring width and the magnetic length.
As will be seen throughout the paper, the choice of quantum rings is strategic since the states within the edge states sub-bands can be perfectly associated to either inner or outer edges, or to a coupling of both edges of the ring structure, therefore enabling a good framework for studying the influence of the edges and edge junctions on the electronic structure and charge distribution.
II. Model
=========
We use a tight-binding model for a finite two-dimensional honeycomb lattice, considering nearest neighbors hoppings [@ana; @ana_valley]. The following non-interacting Hamiltonian is considered:
$$H = \sum_{i} \varepsilon_{i} c_{i}^{\dagger} c_{i}
+ t \sum_{<i,j>} (e^{i\phi_{ij}} c_{i}^{\dagger} c_{j} + e^{-i\phi_{ij}}
c_{j}^{\dagger} c_{i})$$
where $c_{i}$ is the fermionic operator on site $i$. The perpendicular applied magnetic field is included by means of Peierls substitution, which means a complex phase in the hopping parameter ($t$=2.7 eV): $\phi_{ij}= 2\pi(e/h) \int_{j}^{i} \mathbf{A} \! \cdot \! d \mathbf{l} \;$. In the Landau gauge, where the electromagnetic vector potential is defined as $\mathbf{A}=(0,Bx,0)$, one obtains $\phi_{ij}\!=\!0$ along the $x$ direction and $\phi_{ij}\!=\pm \pi (x/a) \Phi / \Phi_{0}$ along the $\mp y$ direction. The magnetic flux ($\Phi$) per magnetic flux quantum ($\Phi_{0}=h/e$) is defined as: $\Phi / \Phi_{0}=Ba^{2}\sqrt{3}e/(2h)$, and we use $a$=2.46[Å]{} as the lattice constant for graphene. The on-site energies are taken as $\varepsilon_{i}=0$.
To consider the ring geometries, a central region of absent atoms (antidot) is defined in the structure by setting the hopping parameters to zero for the absent atoms and the on-site energies at the position of these atoms equal to a large value outside the energy range of the spectra. The magnetic field we consider is not limited to the central region of the ring, but is homogeneously applied to the entire structure. By exact numerical diagonalization of the Hamiltonian, the energy spectrum as a function of magnetic flux is calculated for different geometries of quantum rings.
III. Edge States of Graphene Rings
==================================
A. Antidot effects on the energy spectrum
-----------------------------------------
We start by calculating the energy spectrum of the structure shown in Fig. 1(a): a finite hexagonal lattice forming an equilateral triangle with zigzag edges. Calling $N_{out}$ the number of individual hexagonal plaquetes along each side of the triangle, the total number of carbon atoms in this structure is $N^{2}_{out} + 4N_{out}+1$ [@Yamamoto]. The energy spectrum for such a triangular graphene quantum dot with $N_{out}$=45 is plotted in Fig. 1(c) as a function of magnetic flux. One can clearly observe the formation of the low energy LLs: the $n$=0 LL at zero-energy, and the $n$=+1 and $n$=+2, with their square-root dependence on magnetic field, typical from graphene systems (the spectrum is symmetrical with respect to the zero-energy). Also, one can see the expected presence of edge states between consecutive LLs and observe the evolution of these edge states with magnetic flux until coalescing to the LLs [@sivan]. The side length of the triangular structure is simply given by $aN_{out}$, where $a$=2.46[Å]{} is the lattice constant. So, for the case considered here of $N_{out}$=45, the side length of the triangular dot is $\approx$11nm.
![[**(a)**]{} Triangular graphene quantum dot with zigzag edges, with $N_{out}$=45. [**(b)**]{} Triangular graphene quantum ring, with $N_{out}$=45 and $N_{in}$=12. [**(c)**]{} Energy spectrum as function of the magnetic flux for the structure in (a). [**(d)**]{} Energy spectrum as function of the magnetic flux for the structure in (b) ](figura_1.pdf){width="8.5cm"}
We then take this nanostructure as a starting point to develop a triangular quantum ring just piercing a triangular hole (antidot) in the middle of it. The inner edges of this ring are also zigzag. To define the size of the antidot, we call $N_{in}$ the number of hexagons at each side of the internal removed triangle. The total number of atoms in this quantum ring is now $N^{2}_{out} + 4(N_{out} - N_{in})-N^{2}_{in} +6N_{in}$. In Fig. 1(b) there is a representation of such a ring for $N_{out}$=45 and $N_{in}$=12, and the corresponding energy spectrum is shown in Fig. 1(d). The interesting observation is that the presence of the antidot gives origin to additional edge states with a different evolution with magnetic field: states that go up in energy as the magnetic flux is increased. Indeed, looking at vacancies in graphene [@ana], the localized states around such defects also rises in energy with increasing magnetic field, since vacancies are actually minimal antidots. It is also clear that with the introduction of the antidot, the formation of the $n>$0 LLs starts at higher magnetic fields when compared to Fig. 1(c), due to the involved interplay between the inner and outer edge states. On the other hand, this interplay seems to anticipate the formation of the central LL, in these peculiar zero magnetic field limit showing edge states at the Dirac point due to the zigzag structure of the edge [@Wakabayashi].
To analyze in more details how the energy levels of the edge states evolve with magnetic field, in Fig. 2(a) we zoom in the energy scale of Fig. 1(d). It now becomes evident the formation of edge states sub-bands, separated by energy gaps that get smaller with increasing field. One can also note that each of these sub-bands contains three crossing energy levels for this triangular graphene ring.
B. Different evolutions for inner and outer edge states
-------------------------------------------------------
In order to gain a deeper understanding of this quite complex evolution of edge states, in Figs. 2(b-d) we look to the wave functions amplitudes of specific edge states. The arrow (b) in the spectrum in Fig. 2(a) is pointing to an edge state whose energy is reduced with increasing $B$. This state is mapped in Fig. 2(b), and clearly is localized at the outer edge of the triangular ring. The radii of the circles plotted are directly proportional to the wave function amplitude on each site, and we can observe a symmetrical and quasi homogeneous distribution over the edges, with higher concentrations at the outer most lattice sites (and always on the same sublattice, in this case of zigzag edges). A high charge density accumulation is also observed close to each corner forming part of a second charge density belt.
![ (color online) [**(a)**]{} Zoom in the energy scale of the spectrum shown in Fig. 1(d), showing now only the first few low-energy states and their evolution with magnetic flux. [**(b-d)**]{} Electronic charge distribution of the selected edge states indicated by the arrows and corresponding letters in the spectrum. Three typical behaviors are clearly defined: [**(b)**]{} an state whose energy is reduced with $B$ is an outer edge state. [**(c)**]{} an state whose energy is increased with $B$ is an inner edge state. [**(d)**]{} at the anticrossing levels the wave functions are distributed between the inner and the outer edges, indicating a coupling between both edges. The radii of the circles are proportional to the amplitude of the charge density.](figura_2.pdf){width="7.8cm"}
The arrow (c) in Fig. 2(a) points to one of those states that go up in energy with $B$, whose wave function is mapped in Fig. 2(c). In agreement with the observation that those states going up in energy appear only when the antidot in considered in the lattice, this is an edge state clearly located at the inner edge of the triangular quantum ring. Its electronic charge density is homogeneously distributed over the innermost lattice sites with decreasing amplitudes when approaching the corners.
This system has a three-fold rotational symmetry. The outermost atoms at the outer edges are all from the same sublattice except for the three atoms located at each corner. The atoms at the innermost edges are all from the same sublattice, including that located at the corners, and this is an important difference between the inner and outer edges in the triangular zigzag quantum ring. The charge density in the innermost edge is also sublattice modulated but with the charge density dominantly on a different sublattice than at the outermost edge.
C. Coupling between inner and outer edge states
-----------------------------------------------
An interesting observation emerges from mapping the charge density of a state situated at an anticrossing, like the state indicated in the spectrum by the arrow (d) and with the charge density depicted in Fig. 2(d). It can be observed that the wave function has amplitudes concentrated on both the inner and the outer edges of the ring. This indicates that a coupling between edge states from the inner edge and from the outer edge is taking place. Consistently to this picture of inner and outer edge states seeing each other and getting coupled, we observe that the higher the magnetic flux, the smaller are the energy gaps between the sub-bands. This can be attributed to the fact that increasing the magnetic flux $\Phi/\Phi_{0}$, the magnetic length $l_B$ of the states gets smaller, reducing the chances of coupling.
For a more quantitative comparison, the width (distance between outer and inner edges) of the triangular ring we are considering (with $N_{out}$=45 and $N_{in}$=12) is 24.1[Å]{} and the magnetic length is determined by: $l_B=\sqrt{\hbar /eB}$ = 0.913[Å]{}/$\sqrt{\Phi/\Phi_0}$. In this way, for a flux $\Phi/\Phi_0$=0.02, for which there are no energy gaps in the scale observed in the spectrum of Fig. 2(a), we have $l_B$=6.46Å. Reducing the flux, for example for $\Phi/\Phi_0$=0.01, where energy gaps already start to appear, the magnetic length is $l_B$=9.13Å. For $\Phi/\Phi_0$=0.005, a region of flux where the energy gaps are more clearly defined, we have $l_B$=12.9Å, a value corresponding to approximately half the width of the triangular ring, and so compatible with the suggested coupling between outer and inner edge states.
We recall the fact that we are showing typical charge density plots: any other chosen edge state that goes down (or up) in energy has a very similar charge density distribution to that shown in Fig. 2(b) (or 2(c)), while any state at an anticrossing shows wave function concentration on both inner and outer edges, similarly to the distribution observed in Fig. 2(d). It is interesting to notice that the coupling of both edges does not break the sublattice modulation of the charge density at each edge, but the state as a whole is now sublattice mixed.
IV. Widths, edges and corner effects on the quantum rings
=========================================================
A. Widths and sub-band gaps: tuning the inner and outer edge coupling
---------------------------------------------------------------------
![ (color online) [**(a)**]{} Hexagonal zigzag quantum ring with $N_{out}=21$ and $N_{in}=7$. [**(b)**]{} Thinner hexagonal zigzag quantum ring, with the same $N_{out}=21$, but with $N_{in}=12$. [**(c)**]{} Energy spectrum as function of the magnetic flux for the structure in (a). [**(d)**]{} Energy spectrum as function of the magnetic flux for the structure in (b).[]{data-label="Fig3"}](figura_3.pdf){width="8.3cm"}
We now turn our attention to hexagonal graphene quantum rings, first to compare the energy spectrum of this other geometry with the one from the triangular ring shown previously, and second to show the interesting effects of varying the width of the quantum ring on the formation of the edge states energy sub-bands. In Figs. 3(a) and 3(b) there are representations of two hexagonal quantum rings, with different widths. We once again consider a geometry with all zigzag edge terminations. The total number of atoms in an hexagonal zigzag graphene ring like these is $6N^{2}_{out}-6N^{2}_{in}$, where $N_{out}$ and $N_{in}$ are the number of hexagonal plaquetes along each side of the hexagon and removed hexagon, respectively. Again, the total length of each side of the structure is just given by the number $N_{out}$ or $N_{in}$ times the lattice constant $a$. For the ring in Fig. 3(a) we consider $N_{out}$=21 and $N_{in}$=7, while the ring in Fig. 3(b) has $N_{out}$=21 and $N_{in}$=12.
The energy spectra as a function of magnetic flux of these two rings are shown in Figs. 3(c) and 3(d), respectively. Comparing these spectra with one of an hexagonal quantum dot (without the antidot in the middle) [@peeters], it is evident that the ring geometry introduces energy sub-bands separated by energy gaps, exactly as in the case of the triangular quantum ring. However, it can now be observed that each band of the hexagonal structures has six energy levels instead of the three levels observed in the triangular structures. We note that this follows the rotational symmetry fold number of the hexagonal ring structure. Here the sublattice of the outer and inner most edges alternates from one sublattice to the other going from one arm of the hexagon to the next [@recher; @peeters].
The width of the ring (distance between outer and inner edges) in Fig. 3(a) is 29.8Å, while the width of the thinner ring in Fig 3(b) is 19.9Å. When observing the effects of varying the width in the energy spectra, these ring widths can be compared to the magnetic lengths for corresponding magnetic fluxes, as described in the previous section. Corroborating the idea that the coupling between inner and outer edge states is directly related to the appearance of the energy gaps between sub-bands, we clearly see that the thinner quantum ring shows energy gaps in the spectrum until higher values of magnetic fluxes (smaller magnetic lengths).
B. Zigzag versus armchairm edges: differences in the quantum ring spectra around the Dirac point
------------------------------------------------------------------------------------------------
![[**(a)**]{} Hexagonal graphene quantum dot with armchair edges, for which $N_{out}=13$. [**(b)**]{} Hexagonal graphene quantum ring with armchair edges, with $N_{out}=13$ and $N_{in}=6$. [**(c)**]{} Energy spectrum for the structure in (a). [**(d)**]{} Energy spectrum for the structure in (b), where the arrows indicate states whose charge densities are plotted in Figure 5.[]{data-label="Fig4"}](figura_4.pdf){width="9.1cm"}
![ (color online) Electronic charge distribution of the selected edge states indicated by the arrows and corresponding letters in Fig. 4(d). [**(a)**]{} an outer edge state. [**(b)**]{} an inner edge state. [**(c)**]{} coupling between the inner and the outer edges, at an anticrossing.[]{data-label="Fig5"}](figura_5.pdf){width="8.1cm"}
Having in mind the possible importance of the edge structure on the electronic structure of graphene quantum rings, we now look to hexagonal quantum dot and quantum ring systems with inner and outer armchair edges (Figs. 4(a) and 4(b)). The corresponding electronic structures as a function of magnetic field are shown in Figs. 4(c) and 4(d). The number of hexagonal plaquetes in each side of the hexagonal dot considered is $N_{out}=13$ (the counting for armchair edge terminations takes in account only the outermost plaquetes), corresponding to a total of 2814 atoms in the nanostructure. For the hexagonal ring, we considered $N_{out}=13$ and $N_{in}=6$, where $N_{in}$ is again the number of hexagonal cells in one side of the hexagon removed.
Recalling that at $B$=0 there are no states associated to armchair edges near the Dirac point [@Wakabayashi], the central part (around $E$=0) of the quantum ring spectrum here is completely different than in the case of zigzag edges (compare with Fig. 3). There are still edge states sub-bands defined, each one containing six energy levels, however the difference is that there is now a wide sub-band around the Dirac point. An interesting observation is the interchange between electron-like and hole-like states in this region, as a function of magnetic field. The huge difference in the electronic dispersion should be reflected in the related transport properties.
Similarly to the zigzag case, a clear and strong localization of the charge density at the inner and outer edge occurs in the decoupled edges limit (high magnetic field), as observed in the examples of Fig. 5(a) and Fig. 5(b), corresponding to the states pointed by the arrows (a) and (b) in the ring spectrum (Fig. 4(d)). Around an anticrossing, as for the state pointed by arrow (c), the charge density, observed in Fig. 5(c) is spread out on the two edges, indicating the edge coupling. As a difference between zigzag and armchair cases edges, we see that an armchair termination leads to a sublattice admixture of the charge density, different from the case of the zigzag edges, were there is a sublattice modulation [@Brey1].
C. Asymmetries introduced by the corners in diamond rings
---------------------------------------------------------
Next we consider a rhombus-shaped (diamond) graphene quantum ring which has only zigzag edges. This ring is interesting because of its two-fold rotational symmetry and because the upside outer edges of the diamond (in the perspective of the pictures in Figs. 6(b-d)) are from one sublattice meanwhile the downside outer edges are from the other sublattice. However, similarly to the triangular quantum ring, the up and down corners belong to a different sublattice of the neighboring edges. The same sublattice effects occur at the inner edges. The number of atoms for this kind of ring is given by $2(N^{2}_{out}-N^{2}_{in})+4(N_{out}-N_{in})-2$, within our definition. The energy spectrum of a diamond-like ring defined by $N_{out}=32$ and $N_{in}=10$ (1934 atoms) is plotted in Fig. 4(a). One can clearly observe the evolution of the two level bands, as expected from the symmetry of the structure.
![ (color online) [**(a)**]{} Energy spectrum as function of the magnetic flux for a rhombus-shaped quantum ring with zigzag outer and inner edges, containing $N_{out}=32$ and $N_{in}=10$ hexagonal plaquetes in each side length. Electronic charge densities: [**(b)**]{} going down state marked with the (b) arrow in the spectrum, [**(c)**]{} the going up state marked with the (c) arrow and [**(d)**]{} for an anticrossing state marked with the (d) arrow. The radii of the circles denote the magnitude of the charge density.[]{data-label="Fig6"}](figura_6.pdf){width="8.5cm"}
All the rings investigated in the present work with zigzag edges show similar energy spectra: at the low field limit sub-bands of energy states are formed, with the number of levels in each sub-band given by the ring symmetry and well defined outer and inner edge states at higher magnetic fields. Nevertheless, a closer look at the electronic charge density associated to different up and down going states, as well as at anticrossings, for these diamond-like rings reveals a further ingredient in the effect of edges on the electronic properties of quantum dots and rings in graphene, namely the edge junctions at the corners. In a diamond-like ring the junctions between the zigzag edges define single armchair-like units at the left and right (inner and outer) edges (Figs. 6 (b)-(d)), while the upper and lower corners remain zigzag like. One can see, in the sequences of electronic charge distributions in Fig. 6, the high density around the armchair like corners, independently from being a state at the outer edge, Fig. 6(b), inner edge, Fig. 6(c), or even at an anticrossing, Fig. 6(d).
This situation calls the attention to the possible role of the edge junctions on the localization of the electronic charge in graphene nanostructures, i.e., the localization of the electronic charge at a rough interface may depend on the symmetries at the corners that define the edge landscape.
D. Round rings - effects of irregular edges
-------------------------------------------
We then analyze the cases of a round graphene dot and ring. Here the edges of the structures are irregular and were defined in a way to best approach circular geometries for outer and inner edges, taking care not to leave edge atoms with only one nearest-neighbor [@akola], as observed in Figs. 7(a) and 7(b). Figures 7(c) and 7(d) show the corresponding energy-magnetic flux spectra for these two structures. For the round dot, the number of atoms that has been taken is 2283, defining a radius of $\approx47.1$ Å. For the round ring, the external radius is $\approx 47.1$ Å, and the internal radius is $\approx 7.3$ Å, containing a total of 2226 atoms.
![ [**(a)**]{} Round graphene quantum dot with 2283 atoms. [**(b)**]{} Round graphene quantum ring with 2226 atoms. [**(c)**]{} Energy spectrum for the structure in (a). [**(d)**]{} Energy spectrum for the structure in (b).[]{data-label="Fig7"}](figura_7.pdf){width="8.5cm"}
One can perceive from these spectra that, despite the irregularities of the edges, the main effects observed from the previous geometries are robust and keep present here. Comparing the spectra for the circular ring (Fig. 7(d)) with the one for the circular dot (Fig. 7(c)), it is again clear that the circular antidot introduces inner edge states whose energies are increased with increasing magnetic flux. In the low flux limit of the ring spectra, anticrossing levels are again observed in the edge states region, indicating the coupling of inner and outer edge states, exactly as observed and described for the structures with well defined edge structures (zigzag or armchair). The main difference is that, as for this geometry there is no rotational symmetry, we do not observe the formation of edge-states sub-bands with well defined number of energy levels.
V. Conclusions
==============
The present paper focus on the single particle electronic properties of finite graphene structures. The behavior of edge states in graphene rings is investigated, through the numerical calculation of the electronic energy spectra of these rings as a function of a perpendicular magnetic field and the mapping of charge density distributions. Several similar patterns may be found among quantum rings with different symmetries (triangular, hexagonal, diamond shapes), including the formation of sub-bands of edge states energy levels, separated by energy gaps (anticrossings). The choice of quantum rings revealed a strategic one because of the clear relation between the symmetry of the structure and the number of levels in the edge states sub-bands. Furthermore, the edge states levels within the sub-bands can be perfectly associated to either inner or outer edges, as well as the “bulk" region of the structure (coupling between edges), therefore enabling a good framework for studying the influence of the edges on the electronic structure and charge distribution. If edge terminations (zigzag or armchair) show to play an important role on the electronic properties, specially for the states around the Dirac point, the junction of the edges (corners) can also be crucial for charge density localization patterns.
Acknowledgments
===============
DABA acknowledges support from CAPES, ALCP acknowledges support from FAPESP. PAS received partial support from CNPq.
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abstract: 'The effect of domain walls on electron transport has been investigated in microfabricated Fe wires (0.65 to 20 $\mu m$ linewidths) with controlled stripe domains. Magnetoresistance (MR) measurements as a function of domain wall density, temperature and the angle of the applied field are used to determine the low field MR contributions due to conventional sources in ferromagnetic materials and that due to the erasure of domain walls. A negative domain wall contribution to the resistivity is found. This result is discussed in light of a recent theoretical study of the effect of domain walls on quantum transport.'
address:
- |
Department of Physics, New York University\
4 Washington Place, New York, New York 10003
- |
IBM Research Division, Almaden Research Center,\
San Jose, California 95120-6099
author:
- 'U. Ruediger, J. Yu, S. Zhang, A. D. Kent'
- 'S. S. P. Parkin'
date: 'January 8, 1998'
title: |
Negative Domain Wall Contribution to the Resistivity of\
Microfabricated Fe Wires
---
The interplay between the electron transport and magnetic properties of ferromagnetic nanowires and thin films has recently been the subject of an intense research effort. In mesoscopic ferromagnets an experimental aim has been to use magnetoresistance (MR) to study domain wall (DW) dynamics; in particular macroscopic quantum tunneling. These experiments have focused on the low temperature MR of nanowires of Ni [@Hong], Co, and Fe [@Otani]. Discontinuous changes in the wire conductance are observed as a function of the applied field. These are interpreted as the nucleation and movement of DWs which traverse the wire during magnetization reversal. In these experiments, nucleation of a DW appears to lead to a decrease in the wire’s resistivity. Independently a novel theoretical explanation has been proposed in which DWs destroy the electron coherence necessary for weak localization at low temperature, leading to such a negative DW contribution to the resistivity [@Tatara]. Another recent experiment suggests large MR effects due to DWs can be observed even at room temperature in simple ferromagnetic films [@Gregg]. A new physical mechanism has been proposed to explain these observations which is analogous to that operative in giant magnetoresistance (GMR). Within this model the resistivity in the presence of DWs is enhanced due to a mixing of minority and majority spin channels in a wall in the presence of spin dependent electron scattering [@Viret; @Levy]. This research points to the need for experiments over a range of temperatures on ferromagnetic wires with well characterized and controllable domain patterns to isolate the important contributions to the MR in small samples.
Here we report on such experiments. Expitaxially oriented micron scale Fe wires with controlled domain configurations have been realized to study the effect of DWs on magnetotransport properties. In order to isolate the DW contribution to the MR the conventional sources of low temperature MR in ferromagnets are characterized in detail $-$ both the anisotropic magnetoresistance (AMR) and Lorentz MR. As preliminary experiments on ferromagnetic nanowires suggest [@Hong; @Otani], we find that DWs enhance the wire conductance at low temperature. This remarkable effect, present in micron scale wires, is difficult to reconcile with the existing theories of DW scattering.
The starting point for these experiments are high quality thin (100 nm thick) expitaxially grown (110) oriented bcc Fe films. These films display a large in-plane uniaxial magnetocrystalline anisotropy, with the easy axis parallel to the \[001\] direction. They are grown on sapphire substrates as described in Ref. [@Kent]. The films are patterned using projection optical lithography and ion milling to produce micron scale wires (0.65 to 20 $\mu$m linewidths of $\sim 200\;\mu$m length) and wire arrays (0.65 to 20 $\mu$m linewidths of 3 mm length and 10 to 20 $\mu$m spacing) with the wires oriented perpendicular to the magnetic easy axis and parallel to the \[1${\bar
1}$0\] direction. The residual resistivity ratio of 30 and the residual resistance $\rho_{o} = 0.2 \; \mu\Omega$cm attest to the high crystalline quality of these films.
The competition between magnetocrystalline, exchange and magnetostatic interactions results in a pattern of regularly spaced stripe domains perpendicular to the wire axis. Varying the wire linewidth changes the ratio of these energies and hence the domain size. Fig. \[fig1\] shows magnetic force microscopy (MFM) measurements of a 2 $\mu$m wire in zero field performed at room temperature with a vertically magnetized tip. These images highlight the DWs and magnetic poles at the wire edges. For instance, clearly visible in Fig. \[fig1\]b are light and dark contrast along the DWs indicative of Bloch-like walls with sections of different chirality. The magnetic domain configurations are strongly affected by the magnetic history of the samples. Before imaging the wires were magnetized to saturation with a magnetic field transverse (Fig. \[fig1\]a) or longitudinal (Fig. \[fig1\]b) to the wire axis. In the transverse case the mean stripe domain length is 1.6 $\mu$m and much larger than in the longitudinal case, where it is 0.4 $\mu$m. The observed domain structure at $H=0$ is stable over observation times of at least several hours showing that the DWs are strongly pinned at room temperature.
In Fig. \[fig2\] the average domain wall separation is plotted as a function of wire linewidth and magnetic history. The DW density varies by an order of magnitude for the linewidths investigated. Differences between domain configurations after transverse and longitudinal saturation are observed for wires with linewidths between 1 and 10 $\mu$m. Dotted lines in Fig. \[fig1\]a illustrate the approximate domain structure. Since current is directed along the wire, there are domains with magnetization [**M**]{} oriented both parallel and perpendicular to the current density [**J**]{}. In order to estimate the MR contributions due to resistivity anisotropy the volume fraction of closure domains (with [**M**]{} $\parallel$ [**J**]{}) has been estimated. Fig. \[fig2\] also shows this fraction (labeled $\gamma$) determined from MFM images after magnetic saturation in either the transverse or longitudinal direction.
MR measurements were performed in a variable temperature high field cryostat with in-situ (low temperature) sample rotation capabilities. The applied field was in the plane of the film and oriented either longitudinal ($\parallel$) or transverse ($\perp$) to the wire axis. A 4 probe ac ($\sim$10 Hz) resistance bridge with low bias currents (10 to 40 $\mu$A) was employed and the magnetic history of the sample was carefully controlled. Fig. \[fig3\] shows representative MR results on a 2 $\mu$m linewidth wire at both a) high (270 K) and b) low temperature (1.5 K). There is structure to the MR in applied fields less than the saturation field ($H_{s\parallel}
= 0.035$ T and $H_{s\perp}= 0.085$T), at which point the MR slope changes, and the resistivity then increases monotonically with field. At 270 K the resistivity above the saturation field is larger in the longitudinal than in the transverse field orientation, while at 1.5 K this resistivity anisotropy is reversed, $\rho_{\perp}(H_s) > \rho_{\parallel}(H_s)$.
Evidently there are competing sources of resistivity anisotropy in these films which leads to this reversal of the resistivity anisotropy with temperature. Two predominant and well understood sources of low field low temperature MR must be considered to interpret this transport data. The first has its origins in spin-orbit coupling and is known as AMR $-$ the resistivity extrapolated back to zero internal field (B=0) depends on the relative orientation of ${\bf M}$ and ${\bf J}$ [@Campbell]. The second effect is due to the ordinary (Lorentz) magnetoresistance and is also in general anisotropic (i.e. dependent on relative orientation of [**J**]{} and [**B**]{}) [@Schwerer]. As Fe has a large magnetization and hence a large internal magnetic field ($4\pi M=2.2$ T) both factors are of importance. The resistivity of domains parallel and perpendicular to the current direction can be written as: $$\rho_{\perp}(B,T) = \rho_{\perp}(0,T)[1+F_{\perp}(B/\rho_{\perp}(0,T))]
\label{Gperp}$$ $$\rho_{\parallel}(B,T) =
\rho_{\parallel}(0,T)[1+F_{\parallel}(B/\rho_{\parallel}(0,T))]
\label{Gpar}$$ Here $B$ is the internal field in the ferromagnet; $B=4\pi M+H-H_d$, with $H$ the applied field and $H_d$ the demagnetization field. The AMR is equal to $(\rho_{\parallel}(0,T)-\rho_{\perp}(0,T))/\rho(0,T)$, where $\rho
(0,T)$ is the average resistivity. The function $F$ is known as the Kohler function and parametrizes the ordinary magnetoresistance for longitudinal and transverse field geometries in terms of $B/\rho \sim \omega_c\tau $, the cyclotron frequency times the relaxation time[@Ziman]. These scaling functions have been determined experimentally by performing MR measurements to large fields (6 T) as a function of temperature, as described in Ref. [@Schwerer]. The scaling relationships (Eqs. \[Gperp\] and \[Gpar\]) are shown in Fig. \[fig4\]. The inset displays both $\rho_{\perp}(0,T)$ and $\rho_{\parallel}(0,T)$ which result from this scaling analysis and which overlap on the scale shown. We find $\rho(0,T) \sim aT^2$ with $a=3 \times 10^{-4}
\mu \Omega cm/K^2$, as typically observed in 3d elemental ferromagnets [@Campbell]. The AMR is $\sim 4 \times 10^{-3}$ above $80$ K and decreases below this temperature. The reversal of the resistivity anisotropy at low temperatures ($\rho_{\perp}(H_s) >
\rho_{\parallel}(H_s)$, Fig. \[fig3\]b) is thus mainly a consequence of the increasing importance of the Lorentz MR ([*i.e.*]{} $F_{\perp}^\prime > F_{\parallel}^\prime $). At high temperature $\rho(0,T)$ is large and $F^\prime(x)_{x \rightarrow 0} \rightarrow 0$, so that the resistivity anisotropy is associated with the AMR as seen in Fig. \[fig3\]a.
As in all ferromagnetic materials the resistivity anisotropy is a source of low field MR. An applied field changes the domain configurations and domains with magnetization parallel and perpendicular to the current direction have different resistivities. Hence, this low field MR simply reflects the domain geometries during magnetization.
There are thus two ways to estimate the DW contribution to the resistivity. The first is to perform MR measurements at the temperature at which this resistivity anisotropy at $H=0$ vanishes. Since the AMR and Lorentz MR contributions to the resistivity anisotropy are of opposite sign there will be a temperature at which $\rho_{\parallel}(H=0,T_{comp})=\rho_{\perp}(H=0,T_{comp})$, which we denote the compensation temperature, $T_{comp}$. This occurs at 65.5 K and MR results are shown in Fig. \[fig5\] for a 2 $\mu$m wire. At this temperature the low field MR due to the resistivity anisotropy approaches zero. However, the measured resistivity at $H=0$ is lower in longitudinal than in the transverse field orientation. This correlates with DW density, which is larger after longitudinal magnetic saturation (Fig. \[fig1\]b). The magnitude of the effect also decreases systematically with increasing wire linewidth, (Fig. \[fig5\], left-hand inset) and, hence, decreasing DW density (Fig. \[fig2\]). The observed resistivity at $H=0$ is apparently suppressed in the presence of DWs with a magnitude which depends on the density of DWs.
A more definitive correlation between domain configurations, measured at room temperature using an MFM, and low temperature MR measurements has been established. To do this we warm the sample to room temperature, cycle the magnetic field to establish a known $H=0$ magnetic state, and cool. The resistivity at $H=0$ and the MR at low temperatures are unchanged for these samples in both longitudinal and transverse measurement geometries. This is strong evidence that the domain structure is not affected by temperature in this range and consistent with temperature dependent magnetic hysteresis-loop measurements on wire arrays which show no change of the remanent magnetization with temperature.
The temperature dependence of the DW contribution to the resistivity is estimated as follows. The effective resistivity in the $H=0$ magnetic state due to resistivity anisotropy can be written as [@KNote]: $$\rho_{eff}(H=0,T) =
\gamma \rho_{\parallel}(B_i,T) + (1-\gamma)\rho_{\perp}(B_i,T)$$ where $\gamma$ is the volume fraction of domains oriented longitudinally (see Fig. \[fig2\]) and $B_i$ is the field internal to these domains ($=4\pi
M-H_d$). We determine $\rho_{\perp}(B_i,T)$ and $\rho_{\parallel}(B_i,T)$ by extrapolation of the MR data above saturation (again, as indicated by the dashed and solid lines in Fig. \[fig5\]). The effective resistivity at $H=0$ is estimated with the MFM measurements of $\gamma$. Deviations from this $\rho_d=\rho(H=0)-\rho_{eff}(H=0)$, i.e., the measured $H=0$ resistivity minus this effective resistivity, are negative and depend systematically on domain wall density, increasing in magnitude with increasing domain wall density. They approach $1.3 \%$ of the wire resistivity at $1.5$ K for a 2 $\mu$m linewidth wire. We also find that $|\rho_d|$ decreases with increasing temperature approaching zero at $\sim 80$ K (Fig. \[fig5\] right-hand inset). This enhancement of the conductivity vanishes at $\sim 80 K$ for all the wire linewidths investigated.
There are few models of DWs scattering which predict enhancements in the conductivity in the presence walls. One is that of Tatara and Fukuyama based in weak localization phenomena [@Tatara]. They find that DWs contribute to the decoherence of conduction electrons which destroys weak localization. They introduce a wall decoherence time to parametrize this effect $\tau_w=\tau/(n_w/(6\lambda k_f^2)(\epsilon_f/\Delta)^2)$. Here $\tau$ is the momentum relaxation time, $n_w$ the domain wall density, $k_f$ the Fermi vector, $\lambda$ the domain wall thickness, and $\epsilon_f/\Delta$ the ratio of the Fermi energy to the exchange splitting of the band. With commonly used parameters for s electrons in Fe, $\epsilon_f/\Delta \sim 500$, $k_f \sim 1.7 $ Å$^{-1}$, $\lambda \sim 300$ Å, and with $n_w = 2.5 \; \mu$m$^{-1}$ we estimate $\tau_w \sim
60\tau$. Essential to observing such an effect is the absence of other decoherence mechanisms, such as inelastic scattering. Equating $\tau_w=\tau_{in}$ gives an upper temperature limit for the presence of weak localization phenomena. From the residual resistance $\tau=2.8
\times 10^{-14} s$ and with $\rho_{in}=\alpha T^2$ $(\alpha =3 \times 10^{-4}
\mu\Omega cm/K^2)$ we find $T_{max}=7$ K. From this point of view the suppression of weak localization due to DWs cannot explain our observation of enhanced conductivity up to $\sim 80$ K.
In summary, a new lithographic approach has been used to realize ferromagnetic wires with controlled magnetic interactions and hence domain configurations. This has enabled a detailed investigation of the low field MR in micron scale ferromagnetic wires and, in particular, a study of the effect of DWs on the resistivity. After considering the effects of conventional sources of low field MR (AMR and the Lorentz MR), a negative DW contribution to the resistivity is identified. While a negative contribution is consistent with a recent theory based on weak localization, results above $\sim 10$ K are difficult to reconcile with this theory. Further research of this type, on well characterized samples, is clearly warranted to elucidate the interplay between the transport and magnetic properties of mesoscopic ferromagnets.
The authors thank Peter M. Levy for helpful discussions of the work and comments on this manuscript. This research was supported by DARPA-ONR, Grant \# N00014-96-1-1207. We thank C. Noyan for x-ray characterization and M. Ofitserov for technical assistance. Microstructures were prepared at the CNF, project \#588-96.
K. Hong, and N. Giordano, Phys. Rev. B [**51**]{}, 9855 (1995).
Y. Otani, et. al, MRS Spring Meeting, San Francisco 1997 and Y. Otani, et al., submitted (1997).
G. Tatara and H. Fukuyama, Phys. Rev. Lett. [**78**]{}, 3773 (1997).
J. F. Gregg, et. al., Phys. Rev. Lett. [**77**]{}, 1580 (1996).
M. Viret, et. al., Phys. Rev. B [**53**]{}, 8464 (1996).
P. M. Levy and S. Zhang, Phys. Rev. Lett. [**79**]{}, 5110 (1997).
A. D. Kent, U. Ruediger, J. Yu, S. Zhang, P. M. Levy and S. S. P. Parkin, submitted (1997).
See, e.g., I. A. Campbell and A. Fert,“Transport Properties of Ferromagnets”, in Ferromagnetic Materials, ed. by E. P. Wohlfarth (North- Holland Pub. Co. 1982), Vol.3.
F. C. Schwerer and J. Silcox, Phys. Rev. Lett. [**20**]{}, 101 (1968), and J. Appl. Phys. [**39**]{}, 2047 (1968).
see, e.g. J. M. Ziman, Electrons and Phonons, (Clarendon Press, Oxford, England 1960), p. 490.
Since the resistivity anisotropy is small the current density in each domain is to a good approximation independent of the precise domain configurations. We also assume that the domain size is much greater than the mean free path.
|
---
abstract: 'We present Far-Infrared polarimetry observations of M82 at 53 and $154~\micron$ and NGC 253 at $89~\micron$, which were taken with HAWC+ in polarimetry mode on the Stratospheric Observatory for Infrared Astronomy (SOFIA). The polarization of M82 at $53~\micron$ clearly shows a magnetic field geometry perpendicular to the disk in the hot dust emission. For M82 the polarization at $154~\micron$ shows a combination of field geometry perpendicular to the disk in the nuclear region, but closer to parallel to the disk away from the nucleus. The fractional polarization at $53~\micron$ $(154~\micron)$ ranges from 7% (3%) off nucleus to 0.5% (0.3%) near the nucleus. A simple interpretation of the observations of M82 invokes a massive polar outflow, dragging the field along, from a region $\sim 700$ pc in diameter that has entrained some of the gas and dust, creating a vertical field geometry seen mostly in the hotter $(53~\micron)$ dust emission. This outflow sits within a larger disk with a more typical planar geometry that more strongly contributes to the cooler $(154~\micron)$ dust emission. For NGC 253, the polarization at $89~\micron$ is dominated by a planar geometry in the tilted disk, with weak indication of a vertical geometry above and below the plane from the nucleus. The polarization observations of NGC 253 at $53~\micron$ were of insufficient S/N for detailed analysis.'
author:
- Terry Jay Jones
- 'C. Darren Dowell'
- Enrique Lopez Rodriguez
- 'Ellen G. Zweibel'
- Marc Berthoud
- 'David T. Chuss'
- 'Paul F. Goldsmith'
- 'Ryan T. Hamilton'
- Shaul Hanany
- 'Doyal A. Harper'
- Alex Lazarian
- 'Leslie W. Looney'
- 'Joseph M. Michail'
- 'Mark R. Morris'
- Giles Novak
- 'Fabio P. Santos'
- Kartik Sheth
- 'Gordon J. Stacey'
- Johannes Staguhn
- 'Ian W. Stephens'
- Konstantinos Tassis
- 'Christopher Q. Trinh'
- 'C. G. Volpert'
- Michael Werner
- 'Edward J. Wollack'
title: 'SOFIA Far Infrared Imaging Polarimetry of M82 and NGC 253: Exploring the Super–Galactic Wind'
---
Introduction {#sec:intro}
============
Starburst galaxies are an important phenomenon in the universe due to the presence of enhanced star formation and the accompanying strong outflows into the intergalactic medium. This type of galaxy might be an important contributor to the magnetization of the intergalactic medium in the early Universe [e.g, @kron99; @bert06], but the generation and morphology of magnetic fields in starburst galaxies is poorly understood. Galactic scale winds are expected to be important at high redshift where starburst galaxies should be much more common than at the present epoch . Nearby starburst galaxies with massive outflows provide an excellent laboratory for the study of starburst–driven winds where we can spatially resolve the wind and study the magnetic field geometry in detail. The relationship between spiral arms, outflows and galaxy-galaxy interactions and the magnetic field geometry has been extensively investigated in the radio [see @beck15; @beck13 for a review]. Radio synchrotron emission arises from relativistic electrons and may not sample the same volume of gas as interstellar polarization, which is created by extinction or emission from asymmetric dust grains aligned with respect to the ambient magnetic field [e.g. @jowh15].
An early suggestion that galaxies with strong infrared emission may be undergoing intense star formation was made by [@harp73], based on Far-Infrared (FIR) observations of M82 and NGC 253. Because it is so well studied, M82 is considered the archtypical starburst galaxy [e.g. @tele88; @tele89] with an infrared luminosity of $3 \times 10^{10}$L$_ \odot$ and a star formation rate estimated at $13\rm{M_\odot}\;{\rm{y}}{{\rm{r}}^{{\rm{ - 1}}}}$. Based on extensive NIR integral field spectroscopy, [@fors03] find M82 is forming very massive stars ($\gtrsim$50-100 M$_\odot$). Their analysis suggests the global starburst activity in M82 occurred in two successive episodes. The first episode took place $10^7$ yr ago and was particularly intense at the nucleus, while the second episode occurred $5\times 10^6$ yr ago, predominantly in a circumnuclear ring and along what is believed to be a central stellar bar [e.g. @lark94].
Similar to many galaxies with intense starbursts in their nuclear regions, M82 has a bipolar superwind emanating from the central region, stretching well into the outer halo area [e.g. @shop98; @ohym02; @enge06]. The geometry of the magnetic field in the central starburst and in the superwind can be investigated using a number of different techniques. Classic interstellar polarization (in extinction) at $1.65~\micron$ in the Near-Infrared (NIR) by [@jone00] showed evidence for a near vertical field geometry at the nucleus. However, optical and NIR polarimetry is strongly contaminated by scattering of light from the very luminous nucleus of M82.
Radio synchrotron observations also trace magnetic fields. [@reut94] found that the center of M82 is largely depolarized due to differential Faraday rotation. They find evidence for a vertical field in the northern halo and a more planar geometry in the southwestern disk region. [@adeb17] also find a planar geometry in this region and propose a magnetic bar that stretches across the entire central region. They detect some polarization at the nucleus and find the field is vertical there and can be traced out to 600pc (35, assuming a distance of 3.6 Mpc [@kara06]) into the halo. They conclude that some of the non-detection of polarized emission 200pc North from the nucleus is most likely caused by canceling of polarization by the superposition of two perpendicular components of the magnetic field along the line of sight.
At a distance of 3.5 Mpc [@reko05], NGC253 is also a well studied starburst galaxy with a strong galactic wind [e.g. @shar10] and strong Mid-Infrared (MIR) and FIR emission [@riek80]. Unlike M82, which is seen nearly edge–on, NGC 253 has a visible tilt of about $12\degr$ [@deva58], exposing its nuclear regions and revealing a spiral pattern in the disk. The outflow, projected against the tilted disk, is not as prominent as in M82. However, as in M82, it is seen in several tracers, including emission from dense molecular gas [@bolatto13; @walt17]. Radio observations of NGC 253 do not show evidence for a vertical magnetic field geometry in the nucleus [@hees09]. Rather, the polarization map is consistent with a largely planar (disk) geometry.
Polarimetry at FIR wavelengths does not suffer from scattering effects. The albedo is $\gamma<10^{-5}$, [@drai84]. Faraday rotation, which is proportional to $\lambda^2$ [eq. 3-71 @spit78], is also insignificant. Also, it traces the column density of dust, which is much more closely tied with the total gas column density than the relativistic electrons producing the radio synchrotron emission. While the FIR–Radio emission correlation [@helo85] might suggest that the energy in cosmic ray electrons and the thermal dust emission are strongly associated, it is not clear that they trace the same magnetic field geometry. With the advent of a FIR polarimetric imaging capability on SOFIA via HAWC+ [@harp18], we can now map the magnetic field geometry in both the disk and central regions of M82 and NGC 253 with the goal of understanding the role of magnetic fields in starburst galaxies.
FAR–IR Polarimetric Observations {#sec:obs}
================================
M82 and NGC 253 were observed as part of the Guaranteed Time Observation program with the High-resolution Airborne Wideband Camera-plus (HAWC+) [@vail07; @dowe10; @harp18] on the 2.5-m Stratospheric Observatory For Infrared Astronomy (SOFIA) telescope. We made observations of the inner regions of these galaxies using the standard chop-nod polarimetry mode with the instrumental configurations shown in Table \[tbl:obs\]. HAWC+ polarimetric observations simultaneously measure two orthogonal components of linear polarization using a pair of detector arrays with 32 columns $\times$ 40 rows each. Observations were acquired with a sequence of four dither positions in a square pattern with half side length of three pixels. Integrations with four half-wave plate angles were taken at each dither position. Based on the morphology of the sources evident in Herschel maps [@rous10; @pere18], the chop throw and angle are sufficient to make the intensity in the chop reference beams negligible for the results reported here.
Data were reduced using the HAWC+ Data Reduction Pipeline v1.3.0beta3 (April 2018), but with some customizations to address these particular data sets. The data for all dither positions were screened for quality in the telescope tracking and basic instrument function, resulting in exclusion of one dither position for NGC 253 at 89 . As is standard with v1.3.0beta3, a $\chi^2$ test was performed by intercomparing dither sets. We found that the statistical uncertainties were underestimated by a small, typical amount, and we inflated the uncertainties to account for this. The inflation factors ranged from 1.19 to 1.34 for Stokes Q and U.
For subtraction of instrumental polarization, we used a revised calibration data product (v2, Aug. 2018); this is based on “polarization skydips” as is the previous standard calibration product and differs by only $\Delta q$ or $\Delta u \approx 0.1\%$ from it, but it is believed to be more accurate.
Close inspection of the M82 154 data revealed a likely crosstalk effect in the detector system, in which a bright source produces a small, artificial response in another detector in the same readout column. The magnitude of this effect is approximately 0.01-0.1% of the signal in the detector with the bright source, and it seems to affect a fraction of the rows which are read out following the bright source. This crosstalk was confirmed in separate observations of planets, and it may be similar to a crosstalk effect observed in certain calibration data from BICEP2 [@brevik12], which has some cryogenic and room-temperature detector readout circuit designs in common with HAWC+. We have not yet been able to develop a detailed model for the crosstalk, including its apparent variability over time; instead, we identify measurements (with granularity of “dither position” and “detector pair”) which are likely to be affected significantly by the crosstalk and discard them.
Specifically, we looked at positions away from the bright cores of the galaxies for inconsistency in the “$R+T$” total intensity signal among the 8 nods comprising the fundamental polarization measurement sequence (a single dither position with 4 half-wave plate angles). The $R+T$ detector pair sum [@harp18] should be constant over the 8 nods (independent of polarization) except for noise, calibration drifts, and artifacts in the detector system. For the M82 154 data, most of the suspect measurements identified and removed were found in the same column as bright galaxy emission and in rows which are read out afterward —- indicative of the crosstalk effect. For the other data sets, we found fewer inconsistent $R+T$ measurements, and the majority of them correspond to otherwise noisy detectors. We removed 0.03-0.7% of the measurements with this first cut, depending on the observation. We followed this with a general-purpose deglitcher which operates in the map domain, as described by @chus18. Approximately 0.1-0.7% of measurements were removed by the deglitcher. For the two epochs of the M82 154 observations, the parallactic angle differed by $\sim80\degr$, which improved the deglitching performance and map uniformity. For M82 at both wavelengths and NGC 253 at 89 , spatially-extended polarization was detected with statistical significance in excess of 10$\sigma$ in parts of each map. Our 53 observation of NGC 253 has significantly lower signal-to-noise in polarization, however, and since the effective integration time across the galaxy nucleus varies significantly due to the source falling on inoperational detectors for several of the dither positions, the polarization map is difficult to interpret. For this observation, we report only an integrated signal in Table \[tbl:integrated\].
[cccccccc]{}
M82 & 53 & 5.5 & 2016/12/08 & 180 & cross el. & 16 & 4243\
& 154 & 15.3 & 2017/10/24 & 180 & cross el. & 8 & 2141\
& & & 2018/09/27 & 180 & cross el. & 8 & 2089\
NGC253 & 53 & 5.5 & 2017/10/19 & 300 & -40 & 6 & 1660\
& 89 & 8.8 & 2017/10/19 & 300 & -40 & 7 & 2283\
\[tbl:obs\]
M82: Dust Temperature and Column Density {#sec:temp}
========================================
{width="2.9in"} {width="2.9in"}
Previous lower spatial resolution M82 observations at FIR wavelengths by [@kane10] find a dust mass of $2.3 \times {10^6}~{{\rm{M}}_ \odot }$ within the central $2\arcmin$ using a multitiple–dust temperature fit to their AKARI data. Using a gas-to-dust ratio of 100, this corresponds to a total mass (gas and dust) $M_{tot} \sim 2 \times 10^8~\rm{M_\odot}$. By subtracting the FIR emission from the starburst and the disk of M82, [@rous10] estimate the total dust mass in the wind and halo from their $500~\micron$ and dust temperature maps to be $1.1 \pm 0.5 \times 10^6 ~\rm{M}_\odot$, or $M_{tot} \sim 1 \times 10^8~\rm{M_\odot}$. Using FORCAST on SOFIA, [@niko12] were able to image the inner $75\arcsec$ of M82 at wavelengths from $6.6 - 37.1~\micron$ at an angular resolution $\sim 4\arcsec$ (70pc). Their analysis uses measurements of extinction and surface brightness to constrain the total mass and dust mass of the two central peaks seen at NIR and MIR wavelengths. They find $\rm{A_V}=18$ and $\rm{A_V}=9$ toward the main and secondary emission peaks with an estimated color temperature of 68 K at both peaks. The dust masses at the peaks within a $6\farcs6 \times 6\farcs6$ region were estimated to be $\sim 10^4 ~\rm{M_\odot}$, or $M_{tot} \sim 10^6~\rm{M_\odot}$. Based on the M82 rotation curve [@grec12], the total mass within a 100pc $(5.7\arcsec)$ radius of the center is $M_{100} \sim 5\times 10^7 \rm{M_\odot}$ and within 500pc $(28.6\arcsec)$ radius, $M_{500} \sim 1.5\times 10^9 \rm{M_\odot}$.
To support the polarimetric analysis, we made temperature and column density maps of M82. Specifically, we combine our 53 and $154~\micron$ HAWC+ observations with the publicly available 70, 160, and $250~\micron$ observations from the Herschel Space Observatory [@pilb2010] with the PACS instrument (Poglitsch et al. 2010) and SPIRE instrument [@grif2010]. These observations were taken as part of the Very Nearby Galaxies Survey (PI: Christine Wilson). We bin each observation to the pixel scale, $6\arcsec$, of the $250~\micron$ [*Herschel*]{} image. We then extract the intensity values of each pixel associated to the same part of the sky at each wavelength. Finally, we fit a modified blackbody function assuming a dust emissivity of $\epsilon_\lambda \propto \lambda^{-1.6}$ [e.g. @bose12], with the temperature and optical depth at $250~\micron$, $\tau_{250}$, left as free parameters. We compute ${N(H+H_2)} = {\tau _{250}}/\left( {k\mu {{\rm{m}}_{\rm{H}}}} \right)$ where $k=0.1$cm$^2/g$ [@hild83] and $\mu$ = 2.8. We use the HAWC+ data to both augment the Herschel data and help constrain the Wien side of the SED at $53~\micron$. Figure 1 shows both the temperature and column density maps within the same FOV.
The color temperature ranges from a peak of 40 K on the nucleus to 25 K at about $20\arcsec$ along the disk to the NE and SW. The computed column density peaks at $N(H+H_2) = 3 \times 10^{22}~ \rm{cm^{-2}}$, about $\rm{A_V} \sim 20$, somewhat higher than found by [@niko12]. Summing the column density in a $40\arcsec \times 20\arcsec$ box yields $M_{tot} \sim 8 \times 10^7~\rm{M_\odot}$. This is at least a factor of two less than seen in the molecular gas in the same region. Given the filling factor of 1% for the dense gas derived by [@nayl10], most of the molecular gas, even if the temperature of the dust on the surface of the dense cores was as high as 50 K, is probably not contributing significant flux to our HAWC+ maps. This means our HAWC+ observations do not sample regions of very dense, molecular cores, but rather, they sample the dust associated with the rest of the ISM in M82, including less dense $(\rho \la 100~\rm{cm^{-3}})$ molecular gas.
M82: Polarization Maps {#sec:maps}
======================
The SOFIA observations of M82 are shown in Figure 2, where we have plotted polarization vectors on a grid with one half beam width for the spacing and with position angles rotated $90\degr$ to represent the inferred magnetic field direction. In the top row, the vector length is proportional to the fractional polarization. Cuts for the fractional polarization were made at a S/N of 3.3/1 (debiased, see [@ward74]) and at an intensity of 0.21 Jy/$\square\arcsec$ at $53~\micron$ and 0.044 Jy/ at $154~ \micron$. Since HAWC+ is a relatively new instrument, we chose to be conservative in our S/N cuts. Also, since there is a large number of pixels outside these intensity contours $(\sim 1000)$, we used the cut in intensity to remove a few statistically insignificant pixels with no corroborating nearby position angles. We can likely trust the remaining vectors as being indicative of the field direction. In the second row, all vectors within these criteria are plotted with the same length to better clarify the position angle morphology. At a wavelength of $53~\micron$, the polarization fraction ranges from a high of 7% well off the nucleus to values as low as 0.5% at some locations along the plane (disk) of M82. The fractional polarization at the intensity peak is 2.2% , and it declines toward the east and west along the plane. Although the nominal systematic error in polarization for HAWC+ is 0.3% (1 $\sigma$) [@harp18], in this specific map, detections with $p$ as low as 0.5% appear to have position angles which fit the large scale pattern. Line Integral Contour maps using lower S/N data are shown in Figure 3 to better illustrate the mean position angle at greater distances from the disk.
Clearly evident in the $53~\micron$ image is the presence of a magnetic field geometry largely perpendicular to the plane. This geometry extends over a region at least 700pc ($40\arcsec$) along the disk and up to 350 pc above and below the plane. The stellar scale height for the thin disk in M82 is $h_z = 143$pc [@lim13], however the more extended distribution of AGB stars is much greater [@davi08]. We do not measure the magnetic field geometry in the outflow at kpc scales.
{width="2.9in"} {width="2.9in"} {width="2.9in"} {width="2.9in"}
![Line Integral Contour [@cabr93] maps of the polarization data. Notice the transition of the position angle from vertical to planar to the SW in the $53~\micron$ map and to both the NE and SE in the $154~\micron$ map. A cut in polarization S/N (debiased) of $2\sigma$ was used to form these images, which allows the general trend in position angle to be traced further into the halo and along the disk.](fig3L.eps "fig:"){width="2.9in"} ![Line Integral Contour [@cabr93] maps of the polarization data. Notice the transition of the position angle from vertical to planar to the SW in the $53~\micron$ map and to both the NE and SE in the $154~\micron$ map. A cut in polarization S/N (debiased) of $2\sigma$ was used to form these images, which allows the general trend in position angle to be traced further into the halo and along the disk.](fig3R.eps "fig:"){width="2.9in"}
For the polarization map at $154~\micron$, the polarization fraction ranges from a high of 3% well off the nucleus to 0.3% near the nucleus, and the vectors show more variation than in the $53~\micron$ map. The polarization vectors in the central region and to the North and Northwest are consistent with a vertical field. The vertical field is displayed by vectors with fractional polarization ranging from 4% in the Northwest to only 0.3% in the disk, so some caution is needed in interpreting the detailed field structure. The vectors to the Southwest and Northeast along the disk have larger magnitude (most $>$1%) and indicate a geometry closer to parallel to the plane of the disk. Using NIR polarimetry in extinction, [@jone00] found evidence for a vertical geometry at the nucleus, but a planar geometry to the SW of the central region. Our FIR observations agree with the geometry found by [@jone00], but the NIR observations were heavily contaminated by scattering polarization and were not conclusive. Our $154~\micron$ polarimetry is consistent with a planar disk geometry for the magnetic field visible in the polarization vectors to either side of the nuclear starburst. There is evidence for this in the $53~\micron$ map as well in a few vectors to the SW and perhaps the NE.
Using the SCUBA camera on the JCMT, @grea00 reported an 850 polarization map for M82 consisting of 22 polarization vectors with $>3\sigma$ significance and covering a $40\arcsec \times 50\arcsec$ region similar to ours. The main features of the map are a vertical magnetic field at the west nucleus, (inferred) low polarization at the east nucleus, and a loop or bubble shape to the field at the outskirts of the map. @matt09 reprocessed the $850~\micron$ data as part of an archive paper and produced a map with 16 vectors at $>2\sigma$ signficance. The vertical field at the west nucleus is still apparent, as is the (inferred) low polarization at the east nucleus, but the loop is no longer clear. The dust emitting at 850 should also contribute signficantly in the HAWC+ $154~\micron$ band. Comparing the maps at those two wavelengths, we find agreement regarding the vertical magnetic field toward the west nucleus and the decrease in fractional polarization toward the east. However, we do not observe the loop field, nor do we see other clear similarities. We performed a statistical test of agreement between each version of the 850 map and our $154~\micron$ map. For each reported 850 vector, we interpolated the $154~\micron$ Stokes parameters $(Q_{154},U_{154})$ and formed the dot-product-like quantity $S = Q_{154}Q_{850} + U_{154}U_{850}$. Positive values of $S$ indicate agreement of polarization angles within 45, and negative values indicate disagreement greater than 45. We observe a positive correlation for the 2-3 850 measurements toward the west nucleus, but for the remaining 13-20 measurements we find just as many positive as negative values of $S$. Of all the far-infrared/submillimeter polarization maps discussed in this section, the HAWC+ 53 and $154~\micron$ maps are the only ones that clearly show a correlation over an extended area.
[@jone00], observing polarization in extinction at $1.65~\micron$, interpreted the combination of a vertical position angle in the nucleus and a planar geometry in the disk as a mixture of two magnetic field geometries along the line of sight. The field in the central regions with the starburst is vertical, while the field in the surrounding disk is planar. In extinction, this causes a partial cancellation of the fractional polarization as the light from the nucleus first traverses the region with a vertical field and then traverses the disk with a planar magnetic field. In essence, the two regions act as crossed–polaroids (more accurately crossed-fields). Based on the expected level of fractional polarization for the measured extinction to the nucleus of M82 in the NIR [@jone89; @jone93], [@jone00] estimated that 2/3 of the dust along the line of sight to the nucleus has a vertical field, with the remainder of the dust lying in the disk passing in front of the nucleus.
The polarization in extinction is a function of the column depth of dust along the line of sight, but not the dust temperature. In emission at FIR wavelengths, hotter dust will radiate more effectively at shorter wavelengths than cooler dust. Referring to our dust temperature map, the $53~\micron$ emission is more sensitive to temperature than the $154~\micron$ emission because it is on the Wien side of the SED. Hence, the $53~\micron$ emission can dominate over the $154~\micron$ emission for regions along a path with warmer dust. Since the dust in the central region is hotter than in the disk (see Figure 1), the transition from vertical to planar position angle will take place more quickly at $154~\micron$ than at $53~\micron$ as the line of sight moves away from the nucleus along the disk. This is what we observe. If the dust temperature were constant everwhere in the disk, then the transition of the magnetic field geometry from planar to vertical would presumably take place at the same location for both wavelengths.
The vertical magnetic field geometry we see in the HAWC+ FIR polarimetry lies along the same direction seen in other measurements of the super–galactic wind in M82. Optical and H$\alpha$ images suggest a conical outflow with a fairly narrow launch point in the nucleus. Line splitting seen in CO observations of the molecular gas is interpreted as due to a conical outflow perpendicular to the disk with an opening angle of $20\degr$ that stretches up to 1.5kpc from the nucleus [@walt02]. [@mart18] show that the HI kinematics are inconsistent with a simple conical outflow centered on the nucleus, but instead require the more widespread launch of the HI over the $\sim 1$kpc extent of the starburst region. This result is consistent with our finding that the region in the disk with a vertical magnetic field is at least 700pc wide. There is some evidence that the polarization vectors in the $154~\micron$ map to the East line up with streamers S2, S4 and possibly S3 seen in the CO observations of [@walt02].
M82: The Threaded Field {#sec:thread}
=======================
The magnetic field in the ISM of spiral galaxies has both constant (threaded) and turbulent components [see a recent treatment by @plan18]. The effects of a turbulent component can be seen in both variations of the polarization position angle with position on the sky using a type of structure function [@kobu94; @hild99; @plan18] and the trend of fractional polarization with column depth [e.g. @joba15; @hild99]. We will examine the structure function in a later paper, but we can easily examine the trend of fractional polarization with optical depth. If the magnetic field geometry is perfectly constant with no bends or wiggles, the fractional polarization in emission will be constant [@joba15; @jowh15 for a review] with optical depth in the optically thin regime. If there is a region along the line of sight that has completely unaligned grains, it will add total intensity to the beam, but no polarized intensity, resulting in a slope of $P \propto \tau ^{ - 1}$ with increasing contribution from that region [@joba15].
If the dust grain alignment angle varies in a purely stochastic way along a line of sight, the fractional polarization in emission will decrease as $P\propto \tau^{-1/2}$ [@joba15], and there will be no correlation in position angle across the sky. A combination of a constant and a purely random component will cause the polarization to decrease with optical depth at a rate inbetween these two extremes, with the constant component dominating the position angle geometry after several decorrelation lengths [@zwei96]. If there is a coherent departure from purely constant component such as a spiral twist, paths with perpendicular magnetic fields, or other smooth variations of the projected field along the line of sight that depend on total column depth, the fractional polarization can drop **faster** than $P\propto \tau^{-1/2}$ due to strong cancellation of the polarization. Note, in this context, we are considering the field in the outflow of M82 to have a threaded component aligned with the outflow.
We are assuming the efficiency of the grain alignment mechanism is not a factor in our FIR polarimetry of M82 [see @ande15 for a review of grain alignment]. We argued in §\[sec:temp\] that our observations are sensitive to the warm dust in the diffuse ISM of M82, but are largely insensitive to contributions from very dense, molecular cloud cores. There is good evidence that grain alignment in the Milky Way is at its maximum in the diffuse ISM, and only in dense cloud cores with no internal radiation field is there a possible loss of grain alignment [@joba15]. Since our HAWC+ observations are not sensitive to very dense molecular cloud cores, we do not expect regions with unaligned grains to contribute to our FIR polarimetry. We can not rule out that regions of very high turbulence (scrambled field) on small scales may be present along some lines of sight, mimicking regions with unaligned grains (adding net total intensity, but no net polarized intensity).
Figure 4 plots the trend in fractional polarization with column density at $53~\micron$ and $154~\micron$. The $3.3 \sigma$ upper limits are plotted as green triangles. Simple power–law fits to the upper bound of the data points in Figure 4 are steeper than $P\propto \tau^{-1/2}$ at $53~\micron$ and about the same at $154~\micron$. We are concentrating on the upper bound in these plots because that delineates lines of sight where the minimum depolarization effects are present. Lines of sight with lower $P$ could suffer significant depolarization effects, but it is hard to point to a specific line of sight and conclude which effects are dominant. If a ‘crossed–field’ effect is at work along lines of sight through the plane of the disk, and it spans most of the area we have mapped, then it will lower the overall fractional polarization. Although the slope of $P \rm{vs.} N(H+H_2)$ at $53~\micron$ is about $P\propto \tau^{-1}$, the clear coherence of the position angles across the face of M82 indicates that a systematic cancellation of polarization is most likely taking place.
The fraction of the gas with a vertical field is difficult to determine. If the very low fractional polarization in the nucleus is due to simple cancellation of polarization by the superposition of a planar disk field and a polar nuclear field, then (using the value of 2/3 for the column depth corresponding to the vertical component in [@jone00]) approximately $5-6 \times 10^7~\rm{M_\odot}$ has entrained a vertical field. If, however, turbulence on a small scale relative to our beam dominates the field geometry in the warm dust, leaving only a modest fraction of the dust with a coherent polar field, then this must be considered an upper limit. If the estimates of the molecular gas mass in M82 are correct (see §\[sec:temp\]), then a sigificant fraction of the mass in the nuclear region of M82 is not detected by our FIR polarimetry.
{width="2.9in"} {width="2.9in"}
NGC253
======
In this section, we present the $89~\micron$ results of NGC 253, and these observations have an interesting contrast to those of M82.
![Polarization vector map of NGC 253 at $89~\micron$, with vectors rotated $90\degr$ to represent the inferred magnetic field direction. Left: Vectors with length proportional to the fractional polarization. Right: Position angle only, overlaying intensity contours. The first contour starts at $2.8 \times 10^4$ MJy/sr with increments of $2.8 \times 10^4$ MJy/sr. The dashed line indicates the long axis of the tilted disk [PA $= 51 \degr$, @penc80].](fig5L.eps "fig:"){width="2.9in"} ![Polarization vector map of NGC 253 at $89~\micron$, with vectors rotated $90\degr$ to represent the inferred magnetic field direction. Left: Vectors with length proportional to the fractional polarization. Right: Position angle only, overlaying intensity contours. The first contour starts at $2.8 \times 10^4$ MJy/sr with increments of $2.8 \times 10^4$ MJy/sr. The dashed line indicates the long axis of the tilted disk [PA $= 51 \degr$, @penc80].](fig5R.eps "fig:"){width="2.9in"}
The SOFIA/HAWC+ 89 observations of NGC 253 are shown in Figure 5, where we have plotted polarization vectors on a grid with half beam–width for the spacing and with position angles rotated $90\degr$ to represent the inferred magnetic field direction. M82 (d = 3.6Mpc) and NGC 253 (d = 3.5Mpc) are at very similar distances, so our maps in RA and DEC are on nearly the same physical scale. On the left, the vector length is proportional to the fractional polarization. On the right, all vectors are plotted with the same length to better clarify the position angle morphology. Cuts in fractional polarization are at the same S/N (3.3/1 debiased) as for M82, but with a cut at intensity contour of 0.38 Jy/at a wavelength of $89~\micron$. The polarization fraction ranges from a high of 2% well off the nucleus to 0.1 to 0.2% on the nucleus. The polarization at the nucleus is below our nominal systematic error of 0.3%, but the position angle is consistent with the vectors to the NE and SW along the major axis, not a vertical geometry. Unlike M82, the rotated polarization vectors lie largely along the long axis of the tilted disk in NGC 253. However, there is some evidence for a vertical field geometry above and below the plane to the NW and SE along the minor axis.
Radio synchrotron observations of NGC 253 [@hees09] show a magnetic field geometry that consists of disk and halo components. The disk component dominates the visible disk with the magnetic field orientation parallel to the disk at small distances from the midplane. Well out in the halo, the field (as measrued at radio wavelengths) shows the familiar X–shape seen in several nearly edge–on galaxies [@beck13]. The radio observations show no indication of a vertical component along the minor axis of the tilted disk. We do find several vectors away from the plane (dotted line in Figure 5) that might be indicative of a verctical component, but such a vertical field is much more obvious in M82. Compared to M82, NGC 253 can not have as large a fraction of the dust column depth containing a vertical field.
Figure 6 plots the trend of fractional polarization with surface brightness, similar to Figure 4 for M82, except we are using surface brightness as a proxy for optical depth. Upper limits are plotted as green triangles. The data show a decline in polarization with intensity, similar to M82. A rough fit to the slope of this trend is much steeper than $P\propto I^{-1/2}$. The steeper decline in polarization with intensity seen in NGC 253 must be due to greater large-scale cancellation effects with column depth in this galaxy. As with M82, if a ‘crossed–field’ effect is at work, and it spans most of the area we have mapped, then it will lower the overall fractional polarization. Since none of the vectors near the nucleus along the disk show a vertical geometry, any ‘crossed–field’ effect taking place must be dominated by the polarization in the disk, not the wind. For M82 we were able to use the NIR polarimetry to roughly estimate the fraction of the column depth that had vertical and planar fields. We do not have NIR polarimetry of NGC 253, but by analogy, the low net polarization in the disk of NGC 253 would indicate roughly 2/3 of the column depth contains a planar field, with no more than 1/3 associated with a vertical field and the super–galactic wind in this galaxy.
{width="5.0in"}
Discussion, Integrated Properties {#sec:disc}
=================================
The underlying mechanism for producing massive winds from the central regions of starburst galaxies is not understood. Detection of the wind of M82 inspired the pioneering work of [@chevalier85] on thermally driven winds. Later works explored the role of radiation pressure [e.g. @murr11; @krumholz18], cosmic ray driving [e.g. @everett08], and combinations of these effects [@hopkins12; @ruszkowski17]. Thermal models show a tight correlation between central temperature and asymptotic velocity, at least when the wind is sufficiently hot and/or tenuous that radiative cooling is insignificant [@bustard17]. Thus, $\sim$10$^8$K gas is required to reach the speeds of up to 2200 km s$^{-1}$ detected in the x-ray emitting gas in the wind of M82 [@strickland07]. The origin of the cooler and slower gas observed in the outflow is unclear, although it may form *in situ* through shock compression in the flow or through the effects of repeated supernova explosions driving supershells in the central regions of molecular disks [@fuji09].
It is expected that a wind, whatever its origin, as massive as the one in M82 will drag the magnetic fieldlines out along with it. In this sense, the transition to a near vertically-oriented field in the starburst core of M82 is not surprising. However, given that both the spectroscopic and imaging evidence for the wind is in warm to hot ionized gas, it is notable that the field is vertical in the warm dust, which presumably is situated in mostly molecular gas associated with star forming regions (but not dense cores). This suggests that the clouds and the intercloud medium are magnetically connected and that the field in the clouds is not overwhelmingly tangled by turbulence. [@walt02] find kinematic signatures of an outflow in observations of the molecular gas in M82, a characteristic in common with NGC 253 [@bolatto13]. It is interesting that the observations reported in this paper show weaker evidence for a vertical field in NGC 253, however.
Polarized emission from aligned dust grains provides information on the magnetic field geometry in the interstellar medium but does not directly measure magnetic field strength, and primarily traces the field in both the diffuse and molecular gas. Indirect methods of measuring field strength such as the Chandraskhar-Fermi method [@CF] or measures of the dispersion in position angle [see @houd16 and references therin], cannot be applied here, as there is simply too much averaging taking place in our 90pc beam. M82 is a strong emitter of both nonthermal radio radiation and $\gamma$-rays, and modeling their spectra can provide an estimate of the mean magnetic field strength [@yoas13]. Their best fit model assumes a total molecular gas mass of $\sim 4\times 10^8M_{\odot}$, a factor of 10 larger than the mass estimated here, to be threaded by the vertical field, and yields $B \sim 250~\mu {\rm{G}}$ and a wind speed of 500 km s$^{-1}$. Their derived magnetic field strength is somewhat above the field strength of $150~\mu {\rm{G}}$ used by [@dece09] under the assumption that the magnetic field energy density is in equipartition with the cosmic ray energy density. A range of models with larger fields and faster winds or smaller fields and slower winds fits the data nearly as well. The field strength is weighted by ISM properties in a complex way, with most of the synchrotron radiation being emitted in the low–density, large–filling–factor medium, but most of $\gamma$-rays and secondary leptons are produced in the high density clouds. A similar modeling attempt for NGC 253 [@yoas14] failed to produce a good joint fit for both the radio and $\gamma$-ray spectra. Whether the different outcomes for M82 and NGC 253 are related in any way to the different polarization properties reported here is beyond the scope of this paper.
The Galactic Center of the Milky Way also has a vertical magnetic field geometry immediately above and below the inner disk (which itself has a planar field) as evidended by the presence of numerous magnetized vertical filaments [@yuse04; @morr06]. If the magnetic field strength in these filaments is at the upper end of the range allowed by measurements, $\sim 1~$mG, stronger than model estimates for the wind in M82, then the vertical field dominates gas dynamics. Perhaps the vertical field in the Galactic Center could have an origin unrelated to winds, unlike our interpretation of the field geometry in M82. Note that if placed at the distance of M82, FIR polarimetry of the Galactic Center would likely be dominated by emission from dense molecular clouds with a planar field geometry [e.g. Fig. 1 in @chus05].
M82 and NGC 253 are nearby galaxies that allow us to map polarized dust emission on $80-100$pc scales. For comparison with future observations of more distant galaxies, we have computed the integrated properties of these two galaxies in much larger beams. The results are shown in Table \[tbl:integrated\]. Note that the net position angles from the integrated I, Q and U maps for M82 at $53~\micron$ and NGC 253 at $89~\micron$ preserve information on the magnetic field geometry relative to the disk and wind position angles. This implies that future observations of at least some more distant, unresolved (at FIR wavelengths) galaxies with known jet, wind, or disk geometries can still provide relevant information on the global magnetic field geometry. However, the large–scale planar and vertical components in the $154~\micron$ map of M82 cancel and produce very low fractional polarization in the integrated signal.
[ccccc]{}
M82 $53~\micron$ & 45 & 2110 & 1.5 & 81\
M82 $154~\micron$ & 60 & 1430 & 0.1 & –\
NGC253 $53~\micron$ & 15 & 960 & 0.1 & –\
NGC253 $89~\micron$ & 30 & 1450 & 0.4 & 160\
Conclusions
===========
We have presented FIR polarimetric imaging observations of M82 and NGC 253 using HAWC+ on SOFIA. Effects such as scattering seen at NIR wavelengths and Faraday rotation at radio frequencies are absent in the FIR polarimetry. Unlike radio synchrotron emission, the FIR emission is sensitive to the dust column density (weighted by temperature) along the line of sight, not the population of relativistic electrons. These observations of M82 are consistent with a vertical magnetic field in the central $40\arcsec \times 20\arcsec$ region where the starburst is located and a planar magnetic field in the surrounding disk. The fractional polarization is very low at the nucleus, but shows the same vertical field geometry. The low polarization could be due to the mutually perpendicular nuclear and disk fields partially canceling the polarization or this effect in combination with strong turbulence on scales much smaller than our beam. If the ‘crossed–field’ effect dominates, then $\sim 5-6 \times 10^7~ \rm{M_\odot}$ in the central region of M82 is threaded with a magnetic field perpendicular to the disk. For NGC 253 the observations at $89~\micron$ are consistent with a planar magnetic field geometry both in nucleus and to the NE and SW along the disk. There is some indication of a more vertical geometry along the minor axis, but off the nucleus. Compared to M82, NGC 253 can not have as much of the dust column depth containing a vertical field.
Acknowledgments
---------------
Based \[in part\] on observations made with the NASA/DLR Stratospheric Observatory for Infrared Astronomy (SOFIA). SOFIA is jointly operated by the Universities Space Research Association, Inc. (USRA), under NASA contract NAS2-97001, and the Deutsches SOFIA Institut (DSI) under DLR contract 50 OK 0901 to the University of Stuttgart. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. The LIC code was ported from publicly-available IDL source by Diego Falceta-Gonçalves.
Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.
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|
---
abstract: 'We prove an analogue of the Approximation Theorem of $L^2$-Betti numbers by Betti numbers for arbitrary coefficient fields and virtually torsionfree amenable groups. The limit of Betti numbers is identified as the dimension of some module over the Ore localization of the group ring.'
address:
- |
Department of Mathematics\
Virginia Tech\
Blacksburg, VA 24061-0123,\
USA
- |
Westfälische Wilhelms-Universität Münster\
Mathematisches Institut\
Einsteinstr. 62, D-48149 Münster, Germany
- |
Westfälische Wilhelms-Universität Münster\
Mathematisches Institut\
Einsteinstr. 62, D-48149 Münster, Germany
author:
- Peter Linnell
- Wolfgang Lück
- Roman Sauer
date: 'March 1, 2010'
title: 'The limit of $\mathbf{\IF_p}$-Betti numbers of a tower of finite covers with amenable fundamental groups'
---
Introduction
============
A *residual chain* of a group $G$ is a sequence $G=G_0\supset G_1\supset G_2\supset \cdots$ of normal subgroups of finite index such that $\bigcap_{i\ge 0}G_i=\{e\}$. The $n$-th $L^2$-Betti number of any finite free $G$-CW complex $X$ is the limit of the $n$-th Betti numbers of $G_i\backslash X$ normalized by the index $[G:G_i]$ for $i\to\infty$ [@Lueck(1994c)]. If we instead consider Betti numbers $b_n(G_i\backslash X;k)$ with respect to a field of characteristic $p>0$, the questions whether the limit exists, what it is, and whether it is independent of the residual chain are completely open for arbitrary residually finite $G$.
For $G={{\mathbb Z}}^k$ and every field $k$ Elek showed that $\lim _{i\to\infty} b_n(G_i\backslash X;k)$ exists and expresses it in terms of the entropy of $G$-actions on the Pontrjagin duals of finitely generated $kG$-modules [@Elek(2002)] – his techniques play an important role in this paper (see Section \[sec:Eleks\_dimension\_function\]). It was observed in [@Abert-Jaikin-Zapirain-Nikolov(2007)]\*[Theorem 17]{} that the mere convergence of the right hand side of \[the:dim\_approximation\_over\_fields:finitely\_presented\_modules\] in Theorem \[the:dim\_approximation\_over\_fields\] for every amenable $G$ and every field $k$ follows from a general convergence principle for subadditive functions on amenable groups [@Lindenstrauss-Weiss(2000)] and a theorem by Weiss [@Weiss(2001)].
The main purpose of this paper is to determine the limit $\lim _{i\to\infty} b_n(G_i\backslash X;k)$ in algebraic terms for a large class of amenable groups including virtual torsionfree elementary amenable groups. This makes the limit computable by homological techniques; see e.g., the spectral sequence argument of Example \[exa:s1\_actions\].
More precisely, the limit will be expressed in terms of the *Ore dimension*. The group ring $kG$ of a torsionfree amenable group satisfying the zero-divisor conjecture fulfills the Ore condition with respect to the subset $S=kG-\{0\}$ [@Lueck(2002) Example 8.16 on page 324]; we will review the Ore localization in Subsection \[sub:ore\_localization\]. The Ore localization $S^{-1}kG$ is a skew field containing $kG$. Therefore the following definition makes sense:
\[def:Ore\_dimension\] Let $G$ be a torsionfree amenable group such that $kG$ contains no zero-divisors. The *Ore dimension* of a $kG$-module $M$ is defined by $$\dim^{{\operatorname{Ore}}}_{kG}(M) = \dim_{S^{-1}kG}\bigl(S^{-1}kG \otimes_{kG} M\bigr).$$
The following theorem is our main result; we will prove a more general version, including virtually torsionfree groups, in Section \[sec:Extension\_to\_the\_virtually\_torsionfree\_case\].
\[the:dim\_approximation\_over\_fields\] Let $k$ be a field. Let $G$ be a torsionfree amenable group for which $kG$ has no zero-divisors[^1]. Let $(G_n)_{n\ge 0}$ be a residual chain of $G$. Then:
1. \[the:dim\_approximation\_over\_fields:finitely\_presented\_modules\] Consider a finitely presented $kG$-module $M$. Then $$\dim_{kG}^{{\operatorname{Ore}}}(M)
=
\lim_{n \to \infty} \frac{\dim_k\bigl(k \otimes_{kG_n} M\bigr)}{[G:G_n]};$$
2. \[the:dim\_approximation\_over\_fields:chain\_complexes\] Consider a finite free $kG$-chain complex $C_*$. Then we get for all $i \ge 0$ $$\dim_{kG}^{{\operatorname{Ore}}}\bigl(H_i(C_*)\bigr)
=
\lim_{n \to \infty} \frac{\dim_k\bigl(H_i(k \otimes_{kG_n} C_*)\bigr)}{[G:G_n]};$$
3. \[the:dim\_approximation\_over\_fields:CW-complexes\] Let $X$ be a finite free $G$-$CW$-complex. Then we get for all $i \ge 0$ $$\dim_{kG}^{{\operatorname{Ore}}} \bigl(H_i(X)\bigr)
=
\lim_{n \to \infty} \frac{\dim_k\bigl(H_i(G_n \backslash X;k)\bigr)}{[G:G_n]}.$$
\[rem:fields\_of\_characteristic\_zero\] Let $G$ be a group with a residual chain $(G_n)_{n \ge 0}$, and let $M$ be a finitely presented $kG$-module. Then the Approximation Theorem for $L^2$-Betti numbers says that $$\label{eq:limit_for_modules}
\dim_{{{\mathcal N}}(G)}\bigl({{\mathcal N}}(G) \otimes_{kG} M\bigr) = \lim_{n \to \infty} \frac{\dim_{k}\bigl(k \otimes_{kG_n} M\bigr)}{[G:G_n]}$$ provided $k$ is an algebraic number field. Here ${{\mathcal N}}(G)$ is the group von Neumann algebra, and $\dim_{{{\mathcal N}}(G)}$ is the von Neumann dimension. See [@Lueck(1994c)] for $k = \mathbb{Q}$ and [@Dodziuk-Linnell-Mathai-Schick_Yates(2003)] for the general case.
Let $k$ be a field of characteristic zero and let $u = \sum_{g \in G} x_g
\cdot g\in kG$ be an element. Let $F$ be the finitely generated field extension of ${{\mathbb Q}}$ given by $F = {{\mathbb Q}}(x_g \mid g \in G) \subset k$. Then $u$ is already an element in $FG$. The field $F$ embeds into ${{\mathbb C}}$: since $F$ is finitely generated, it is a finite algebraic extension of a transcendental extension $F'$ of ${{\mathbb Q}}$ [@Lang(2002_Algebra)]\*[Theorem 1.1 on p. 356]{}, and $F'$ has finite transcendence degree over ${{\mathbb Q}}$. Since the transcendence degree of ${{\mathbb C}}$ over ${{\mathbb Q}}$ is infinite, there exists an embedding $F'\hookrightarrow{{\mathbb C}}$ induced by an injection of a transcendence basis of $F'/{{\mathbb Q}}$ into a transcendence basis ${{\mathbb C}}/{{\mathbb Q}}$, which extends to $F\hookrightarrow{{\mathbb C}}$ because ${{\mathbb C}}$ is algebraically closed. This reduces the case of fields of characteristic zero to the case $k={{\mathbb C}}$. In [@Elek(2006strong)] Elek proved for amenable $G$ and $k={{\mathbb C}}$ (see also [@Pape(2008)]).
Moreover, if $G$ is a torsionfree amenable group such that ${{\mathbb C}}G$ contains no zero-divisors and $k$ is a field of characteristic zero, then $$\dim_{{{\mathcal N}}(G)}\bigl({{\mathcal N}}(G) \otimes_{kG} M\bigr) = \dim^{{\operatorname{Ore}}}_{kG}(M).$$ This follows from [@Lueck(2002) Theorem 6.37 on page 259, Theorem 8.29 on page 330, Lemma 10.16 on page 376, and Lemma 10.39 on page 388]. In particular, Theorem \[the:dim\_approximation\_over\_fields\] follows for $k$ of characteristic zero. So the interesting new case is the one of a field of prime characteristic.
Review of Ore localization and Elek’s dimension function {#sec:Review_of_Ore_localization_and_Eleks_dimension_function}
========================================================
Ore localization {#sub:ore_localization}
----------------
We review the Ore localization of rings. For proofs and more information the reader is referred to [@Stenstroem(1975)]. Consider a torsionfree group $G$ and a field $k$. Let $S$ be the set of non-zero-divisors of $k G$. This is a multiplicatively closed subset of $k G$ and contains the unit element of $k G$. Suppose that $k G$ satisfies the *Kaplansky Conjecture* or *zero-divisor conjecture*, i.e., $S=k G-\{0\}$. Further assume that $S$ satisfies the *left Ore condition*, i.e., for $r \in k G$ and $s \in S$ there exists $r' \in k G$ and $s' \in S$ with $s'r = r's$. Then we can consider the *Ore localization $S^{-1}k G$*. Recall that every element in $S^{-1}kG$ is of the form $s^{-1}\cdot r$ for $r \in k G$ and $s \in S$ and $s_0^{-1}\cdot r_0 =
s_1^{-1}\cdot r_1$ holds if and only there exists $u_0,u_1 \in R$ satisfying $u_0 r_0 = u_1r_1$ and $u_0s_0 = u_1s_1 $. Addition is given on representatives by $s_0^{-1}r_0 + s_1^{-1}r_1 = t^{-1}(c_0r_0 + c_1r_1)$ for $t = c_0s_0 =
c_1s_1$. Multiplication is given on representatives by $s_0^{-1}r_0 \cdot
s_1^{-1}r_1 = (ts_0)^{-1}cr_1$, where $cs_1 = t r_0$. The zero element is $e^{-1}\cdot 0$ and the unit element is $e^{-1}\cdot e$. The Ore localization $S^{-1}kG$ is a skew field and the canonical map $k G \to S^{-1}kG$ sending $r$ to $e^{-1} \cdot r$ is injective. The functor $S^{-1}kG \otimes_{kG} -$ is exact.
\[rem:The\_Ore\_condition\_for\_group\_rings\] If a torsionfree amenable group $G$ satisfies the Kaplansky Conjecture, i.e., $kG$ contains no zero-divisor, then for $S = kG-\{0\}$ the Ore localization $S^{-1}kG$ exists and is a skew field [@Lueck(2002)]\*[Example 8.16 on page 324]{}. Every torsionfree elementary amenable group satisfies the assumptions above for all fields $k$ . If the group $G$ contains the free group of rank two as subgroup, then the Ore condition is never satisfied for $kG$ [@Linnell(2006)]\*[Proposition 2.2]{}.
From the previous remark and the discussion above we obtain:
\[the:Ore-dimension\] Let $G$ be a torsionfree amenable group such that $kG$ contains no zero-divisors. Then the Ore dimension $\dim^{{\operatorname{Ore}}}_{kG}$ has the following properties:
1. $\dim^{{\operatorname{Ore}}}_{kG}(kG) = 1$;
2. For any short exact sequence of $kG$-modules $0 \to M_0 \to M_1 \to
M_2\to 0$ we get $$\dim^{{\operatorname{Ore}}}_{kG}(M_1) = \dim^{{\operatorname{Ore}}}_{kG}(M_0) + \dim^{{\operatorname{Ore}}}_{kG}(M_2).$$
Crossed products, Goldie rings, and the generalized Ore localization
---------------------------------------------------------------------
Throughout, let $G$ be a group, let $k$ be a skew field.
Let $R$ be a ring. The notion of crossed product generalizes the one of group ring. A *crossed product* $R\ast G=R\ast_{c,\tau} G$ is determined by maps $c:G\to{\operatorname{aut}}(R)$ and $\tau:G\times G\to R^\times$ such that, roughly speaking, $c$ is a homomorphism up to the $2$-cocycle $\tau$. We refer to the survey [@Lueck(2002)]\*[10.3.2 on p. 398]{} for details. If $G$ is an extension of $H$ by $Q$, then the group ring $kG$ is isomorphic to a crossed product $kH\ast Q$. Some results in this paper are formulated for crossed products, although we only need the case of group rings for Theorem \[the:dim\_approximation\_over\_fields\]. So the reader may think of group rings most of the time. However, crossed products show up naturally, e.g., in proving that the virtual Ore dimension is well defined.
We recall the following definition.
\[def:goldie\_ring\] A ring $R$ is *left Goldie* if there exists $d\in \mathbb{N}$ such that every direct sum of nonzero left ideals of $R$ has at most $d$ summands and the left annihilators $a(x)=\{r\in R;~rx=0\}$, $x\in R$, satisfy the maximum condition for ascending chains. A ring $R$ is *prime* if for any two ideals $A,B$ in $R$, $AB=0$ implies $A=0$ or $B=0$.
The subgroup of $G$ generated by its finite normal subgroups will be denoted by $\Delta^+(G)$. Then $\Delta^+(G)$ is also the set of elements of finite order which have only finitely many conjugates. We need the following three results:
\[lem:delta\_implies\_prime\] If $\Delta^+(G) = 1$, then $k*G$ is prime.
\[thm:prime\_Goldie\_matrix\] The set of non-zero-divisors in a prime left Goldie ring satisfies the Ore condition. The Ore localization $S^{-1}R$ is isomorphic to $\operatorname{M}_d(D)$ for some $d \in \mathbb{N}$ and some skew field $D$.
\[thm:elementary\_amenable\_goldie\] If $G$ is amenable and $k*G$ is a domain, then $k*G$ is a prime left Goldie ring. If $G$ is an elementary amenable group such that the orders of the finite subgroups are bounded, then $k*G$ is left Goldie.
If $G$ is amenable and $k\ast G$ is a domain, then $k\ast G$ satisfies the Ore condition [@Dodziuk-Linnell-Mathai-Schick_Yates(2003)]\*[Theorem 6.3]{}, thus its Ore localization with respect to $S=k\ast G-\{0\}$ is a skew field. By [@Passman(1977)]\*[Theorem 4.10 on p. 456]{} $k\ast G$ is a prime left Goldie ring. The second assertion is taken from [@Kropholler-Linnell-Moody(1988)]\*[Proposition 4.2]{}.
Next we extend the definition of Ore dimension to prime left Goldie rings. Let $R$ be such a ring. The functor $S^{-1}R \otimes_R -$ will still be exact [@Stenstroem(1975)]\*[Proposition II.1.4 on page 51]{}. If $M$ is a left $R$-module, then $S^{-1}R \otimes_R M$ will be a direct sum of $n$ irreducible $S^{-1}R$-modules for some non-negative integer $n$, and then the *(generalized) Ore dimension* of $M$ is defined as $$\dim_R^{{\operatorname{Ore}}}(M) = \frac{n}{d}.$$ Since $S^{-1}R\cong\operatorname{M}_d(D)$ (Theorem \[thm:prime\_Goldie\_matrix\]) and $\operatorname{M}_d(D)$ decomposes into $d$ copies of the irreducible module $D^d$, we have $\dim_R^{{\operatorname{Ore}}}(R)=1$.
Elek’s dimension function {#sec:Eleks_dimension_function}
-------------------------
Throughout this subsection let $G$ be a finitely generated amenable group. We review Elek’s definition [@Elek(2003c)] of a dimension function $\dim^{{\operatorname{Elek}}}_{kG}$ for finitely generated $kG$-modules.
Fix a finite set of generators and equip $G$ with the associated word metric $d_G$. A *Følner sequence* $(F_n)_{n \ge 0}$ is a sequence of finite subsets of $G$ such that for any fixed $R > 0$ we have $$\lim_{n \to \infty} \frac{|\partial_RF_n|}{|F_n|} = 0,$$ where $\partial_RF_n=\{g \in G \mid d(g,F_k) \le R \; \text{and}\; d(g,G\setminus F_k) \le R\}$.
Let $k$ be an arbitrary skew field endowed with the discrete topology and let $\mathbb{N}$ denote the positive integers $\{1,2,\dots\}$. Let $n\in{{\mathbb N}}$. We equip the space of functions ${\operatorname{map}}(G, k^n)=\prod_{g\in G}k^n$ with the product topology, which is the same as the topology of pointwise convergence. The natural right $G$-action on ${\operatorname{map}}(G, k^n)$ is defined by $$(\phi g)(x)=\phi(xg^{-1})\text{ for $g,x\in G,\phi\in{\operatorname{map}}(G,k^n)$.}$$ Also ${\operatorname{map}}(G,k^n)$ is a right $k$-vector space by defining $(\phi
k)(x) = \phi(x)k$. For any subset $S\subset G$ and any subset $W\subset{\operatorname{map}}(G,k^n)$ let $$W\vert_S=\{f:S\to k^n \mid \exists g\in W \;\text{with} \; g\vert_S=f\}.$$ A right $k$-linear subspace $V\subset{\operatorname{map}}(G,k^n)$ is called *invariant* if $V$ is closed and invariant under the right $G$-action.
Elek defines the *average dimension* $\dim_G^{{\mathcal A}}(V)$ of an invariant subspace $V$ by choosing a Følner sequence $(F_n)_{n\in{{\mathbb N}}}$ of $G$ and setting $$\label{eq:average_dimension}
\dim_G^{{\mathcal A}}(V)=\limsup_{n\to\infty}\frac{\dim_k \bigl(V\vert_{F_n}\bigr)}{|F_n|}.$$
The sequence in converges and its limit $\dim_G^{{\mathcal A}}(V)$ is independent of the choice of the Følner sequence.
Elek actually defines $\dim_G^{{\mathcal A}}(V)$ using Følner exhaustions, i.e. increasing Følner sequences $(F_{n \in \mathbb{N}})$ with $\bigcup_{n\in{{\mathbb N}}} F_n =G$. This makes no difference since the existence of the limit of $(\dim_k \bigl(V\vert_{F_n}\bigr)/|F_n|)_{n\in{{\mathbb N}}}$ for arbitrary Følner sequences (and thus its independence of the choice) follows from [@Lindenstrauss-Weiss(2000)]\*[Theorem 6.1]{}.
Let $M$ be a finitely generated left $kG$-module. The $k$-dual $M^\ast=\hom_k(M,k)$ (where $M$ and $k$ are viewed as left $k$-modules, and $(\phi a)m = \phi(am)$ for $\phi \in M^*$, $a \in k$ and $m\in M$) carries the natural right $G$-action $(\phi g)(m)=\phi(g m)$. The dual of the free left $kG$-module $kG^n$ is canonically isomorphic to ${\operatorname{map}}(G, k^n)$. Any left $kG$-surjection $f\colon kG^n\twoheadrightarrow M$ induces a right $kG$-injection $f^\ast\colon M^\ast\to{\operatorname{map}}(G, k^n)$ such that ${\operatorname{im}}(f^\ast)$ is a $G$-invariant $k$-subspace.
\[def:elek\_dimension\_function\] Let $M$ be a finitely generated left $kG$-module. Its *dimension in the sense of Elek* is defined by choosing a left $kG$-surjection $f\colon kG^n\twoheadrightarrow M$ and setting $$\label{eq:elek_dimension_and_average_dimension}
\dim^{{\operatorname{Elek}}}_{kG}(M) = \dim_G^{{\mathcal A}}\bigl({\operatorname{im}}(f^\ast)\bigr).$$
\[the:Eleks-dimension\] Let $G$ be a finitely generated amenable group. The definition of $\dim^{{\operatorname{Elek}}}_{kG}(M)$ is independent of the choice of the surjection $f$, and $\dim^{{\operatorname{Elek}}}_{kG}$ has the following properties:
1. $\dim^{{\operatorname{Elek}}}_{kG}(kG) = 1$;
2. For any short exact sequence of finitely generated $kG$-modules $0 \to M_0 \to M_1 \to M_2 \to 0$ we get $$\dim^{{\operatorname{Elek}}}_{kG}(M_1) = \dim^{{\operatorname{Elek}}}_{kG}(M_0) + \dim^{{\operatorname{Elek}}}_{kG}(M_2);$$
3. If the finitely generated $kG$-module $M$ satisfies $\dim^{{\operatorname{Elek}}}_{kG}(M) = 0$, then every quotient module $Q$ of $M$ satisfies $\dim^{{\operatorname{Elek}}}_{kG}(Q) = 0$.
The first two assertions are proved in [@Elek(2003c) Theorem 1]. Notice that the third condition does not necessarily follow from additivity since the kernel of the epimorphism $M \to Q$ may not be finitely generated. But the third statement is a direct consequence of the definition of Elek’s dimension.
\[rem:Pontrjagin\_duality\] Identify the left $kG$-module $kG^n$ with the finitely supported functions in ${\operatorname{map}}(G,k^n)$. Here we view ${\operatorname{map}}(G,k^n)$ as a left $k$-vector space by $(af)(g) = af(g)$, and the left $G$-action is given by $(hf)(g) = f(h^{-1}g)$ for $h,g \in G$ and $a \in k$. Let $$ \langle\_,\_\rangle\colon kG^n \times {\operatorname{map}}(G, k^n)$$ be the canonical pairing (evaluation) of $kG^n$ and its dual ${\operatorname{map}}(G, k^n)$. If we view an element $f\in kG^n$ as a finitely supported function $G\to k^n$ (in ${\operatorname{map}}(G,k^n)$), then the pairing of $f\in kG^n$ with $l\in {\operatorname{map}}(G, k^n)$ is given by $$ \langle f,l\rangle=\sum_{g\in G}(f(g),l(g)),$$ where $(\_,\_)$ denotes the standard inner product in $k^n$. For a subset $W\subset kG^n$ let $$W^\perp = \bigl\{f\in{\operatorname{map}}(G,k^n) \mid \langle x, f\rangle=0~\forall x\in W\bigr\}.$$ If $M$ is a finitely generated $kG$-module and $f\colon kG^n\twoheadrightarrow M$ is a left $kG$-surjection, then $f^* \colon M^* \hookrightarrow
{\operatorname{map}}(G,k^n)$ is a right $kG$-injection and $${\operatorname{im}}(f^\ast)=\ker(f)^\perp\subseteq{\operatorname{map}}(G,k^n).$$
Approximation for finitely presented $kG$-modules for Elek’s dimension function {#sec:Approximation_for_finitely_presented_kG-modules_for_Eleks_dimension_function}
===============================================================================
The main result of this section is:
\[the:Approximation\_for\_finitely\_presented\_kG-modules\_for\_Eleks\_dimension\_function\] Let $G$ be a finitely generated amenable group. Consider a sequence of normal subgroups of finite index $$G= G_0 \supseteq G_1 \supseteq G_2 \supseteq \cdots$$ such that $\bigcap_{n \ge 0} G_n = \{1\}$. Then every finitely presented $kG$-module $M$ satisfies $$\dim_{kG}^{{\operatorname{Elek}}}(M)= \lim_{n \to \infty} \frac{\dim_k\bigl(k
\otimes_{kG_n} M\bigr)}{[G:G_n]}.$$
Its proof needs some preparation.
Throughout, let $G$ be a finitely generated amenable group. For any subset $S\subset G$ let $k[S]$ be the $k$-subspace of $kG$ generated by $S\subset kG$. Let $j[S]\colon k[S]\to k[G]$ be the inclusion and ${\operatorname{pr}}[S]\colon kG\to k[S]$ be the projection given by $${\operatorname{pr}}[S](g)=\begin{cases}
g & \text{if $g\in S$;}\\
0 & \text{if $g\in G\backslash S$.}
\end{cases}$$
\[the:equality\_elek\_and\_approximative\_dimension\_of\_finitely\_presented\_modules\] Let $G$ be a finitely generated amenable group. Let $M$ be a finitely presented left $kG$-module $M$ with a presentation $kG^r\xrightarrow{f}kG^s\xrightarrow{p} M\to 0$. For every subset $S\subset G$ we define $$M[S]={\operatorname{coker}}\Bigl({\operatorname{pr}}[S]\circ f\circ j[S]\colon k[S]^r\to
k[S]^s\Bigr).$$ Let $(F_n)_{n \ge 0}$ be a Følner sequence of $G$. Then $$\dim^{{\operatorname{Elek}}}_{kG}(M)=\lim_{n\to\infty}\frac{\dim_k\left(M[F_n]\right)}{|F_n|}.$$
The map $f$ is given by right multiplication with a matrix $A\in
M_{r,s}(kG)$. Viewing $A$ as a map $G\to k^{r\times s}$ it is clear what we mean by the support ${\operatorname{supp}}(A)$ of $A$. Let $R >0$ be the diameter of ${\operatorname{supp}}(A)\cup{\operatorname{supp}}(A)^{-1}$. Since $$\lim_{n \to \infty} \frac{\bigl|\partial_R
F_n\bigr|}{\bigl|F_n\bigr|} = 0,$$ it is enough to show that for every $n\ge 1$ $$\label{eq:estimate_1}
\bigl|\dim_k(M[F_n])-\dim_k\bigl({\operatorname{im}}(p^\ast)\vert_{F_n}\bigr)\bigr| \le s\cdot \bigl|\partial_R F_n\bigr|.$$
For the definition of inner products $(\_,\_)$ and $\langle\_,\_\rangle$ we refer to Remark \[rem:Pontrjagin\_duality\]. Define the following $k$-linear subspaces of ${\operatorname{map}}(F_n,k^s)$: $$\begin{aligned}
W_n &= \bigl\{\phi\colon F_n\to k^s\mid
\langle{\operatorname{pr}}_n\circ f \circ j_n(x),\phi\rangle=0~\forall x\in k[F_n]^r\bigr\};\\
V_n &= \bigl\{\phi\colon F_n\to k^s \mid \exists \bar\phi:G\to k^s
\; \text{satisfying}\;
\bar\phi\vert_{F_n}=\phi, \langle f(y), \bar\phi\rangle=0~\forall y\in kG^r\bigr\};\\
Z_n &=\bigl\{\phi\colon F_n\to k^s \mid \phi\vert_{\partial_R
F_n}=0\bigr\}.
\end{aligned}$$
Since $\dim_k(M[F_n]) = \dim_k(W_n)$ and $\dim_k\bigl({\operatorname{im}}(p^\ast)\vert_{F_n}\bigr) = \dim_k(V_n)$, the desired estimate is equivalent to $$\label{eq:estimate_2}
\bigl|\dim_k(W_n)-\dim_k(V_n)\bigr| \le s\cdot \bigl|\partial_R F_n\bigr|.$$
By additivity of $\dim_k$ we obtain that $$\begin{aligned}
\dim_k (W_n\cap Z_n)&\ge
\dim_k(W_n)-\dim_k({\operatorname{map}}(F_n,k^s))+\dim_k(Z_n)\\
&\ge\dim_k(W_n)-s \cdot |F_n| + s \cdot (|F_n|-|\partial_R F_n|)\\
&=\dim_k(W_n)-s \cdot |\partial_R F_n|.
\end{aligned}$$ Similarly, we get $$\dim_k (V_n\cap Z_n)\ge \dim_k(V_n)-s \cdot |\partial_R F_n|.$$ To prove it hence suffices to show that $$\begin{aligned}
W_n\cap Z_n&\subset V_n; \label{eq:first_inclusion}\\
V_n\cap Z_n&\subset W_n. \label{eq:second_inclusion}
\end{aligned}$$ Let $\phi\in W_n\cap Z_n$. Extend $\phi$ by zero to a function $\bar\phi:G\to k^s$. Let $y\in kG^r$. Then we can decompose $y$ as $y=y_0+y_1$ with ${\operatorname{supp}}(y_0)\subset F_n$ and ${\operatorname{supp}}(y_1)\subset
G\backslash F_n$. By definition of the radius $R$ it is clear that ${\operatorname{supp}}(f(y_1))\subset G\backslash F_n\cup\partial_RF_n$. Because of $\phi\in Z_n$ we have $\langle f(y_1),\bar\phi\rangle=0$. The fact that $\phi\in W_n$ implies that $$\langle f(y_0), \bar\phi\rangle=\langle {\operatorname{pr}}_n\circ f\circ
j_n(y_0), \phi\rangle=0.$$ So we obtain that $\langle
f(y),\bar\phi\rangle=0$, meaning that $\phi\in V_n$. The proof of is similar.
The following theorem is due to Weiss. Its proof can be found in [@Deninger-Schmidt(2007) Proposition 5.5].
\[the:Weiss\_Theorem\] Let $G$ be a countable amenable group. Let $G_n\subset G$, $n\ge 1$, be a sequence of normal subgroups of finite index with $\bigcap_{n\ge 1}G_n = \{1\}$. Then there exists, for every $R\ge
1$ and every $\epsilon > 0$, an integer $M = M(R,\epsilon)\ge 1$ such that for $n \ge M$ there is a fundamental domain $Q_n\subset G$ of the coset space $G/G_n$ such that $$\frac{\bigl|\partial_R Q_n\bigr|}{\bigl|Q_n\bigr|} < \epsilon.$$
Now we are ready to prove Theorem \[the:Approximation\_for\_finitely\_presented\_kG-modules\_for\_Eleks\_dimension\_function\]
According to Theorem \[the:Weiss\_Theorem\] let $(Q_n)_{n \ge 0}$ be a Følner sequence of $G$ such that $Q_n\subset G$ is a fundamental domain for $G/G_n$. Choose a finite presentation of $M$: $$kG^r\xrightarrow{f}kG^s\to M\to 0.$$ Let $f_n=k[G/G_n]\otimes_{kG}f$. By right-exactness of tensor products we have the exact sequence $$k[G/G_n]^r\xrightarrow{f_n}k[G/G_n]^s\to k[G/G_n]\otimes_{kG} M\to
0.$$ The natural map $Q_n\subset G\rightarrow G/G_n$ induces an isomorphism $j_n\colon k[Q_n]\to k[G/G_n]$ of $k$-vector spaces. The map $f$ is given by right multiplication $f=R_A$ with a matrix $A\in
M_{r,s}(kG)$. Viewing $A$ as a map $G\to k^{r\times s}$ let ${\operatorname{supp}}(A)$ be the support of $A$. Let $R > 0$ be the diameter of ${\operatorname{supp}}(A)\cup{\operatorname{supp}}(A)^{-1}$ (with respect to the fixed word metric on $G$). Then $f$ restricts to a map $$f\vert_{Q_n\setminus \partial_RQ_n}\colon
k[Q_n\setminus \partial_RQ_n]^r\to k[Q_n]^s.$$ Hence there is precisely one $k$-linear map $g$ for which the following diagram of $k$-vector spaces commutes: $$\label{eq:approximation_diagram}
\xymatrix@C=16mm{k[G/G_n]^r\ar[r]^{f_n}& k[G/G_n]^s\ar[r]&k[G/G_n]\otimes_{kG}M\ar[r]&0\\
k[Q_n\setminus \partial_RQ_n]^r\ar[u]^{j_n\vert_{Q_n\setminus \partial_RQ_n}}
\ar[r]^-{f\vert_{Q_n\setminus \partial_RQ_n}} &
k[Q_n]^s\ar[u]^{j_n}_{\cong} \ar[r]^{{\operatorname{pr}}}&{\operatorname{coker}}(
f\vert_{Q_n\setminus \partial_RQ_n})\ar@{->>}[u]^g\ar[r]&0}$$ One easily verifies that $g$ is surjective and that $$\ker(g)\subset{\operatorname{im}}\bigl({\operatorname{pr}}\circ j_n^{-1}\circ f_n\colon
k[G/G_n]^r\to {\operatorname{coker}}( f\vert_{Q_n\setminus \partial_RQ_n})\bigr).$$ The map ${\operatorname{pr}}\circ j_n^{-1}\circ f_n$ descends to a map $${\operatorname{pr}}\circ j_n^{-1}\circ f_n\colon
{\operatorname{coker}}(j_n\vert_{Q_n\setminus \partial_RQ_n})\to {\operatorname{coker}}(
f\vert_{Q_n\setminus \partial_RQ_n}).$$ Note that $$\dim_k\bigl({\operatorname{coker}}(j_n\vert_{Q_n\setminus \partial_RQ_n})\bigr)=
r \cdot |\partial_R Q_n|.$$ Thus, $$\dim_k({\operatorname{coker}}( f\vert_{Q_n\setminus \partial_RQ_n}))-\dim_k\left(k[G/G_n]\otimes_{kG}M\right)
=\dim_k \ker(g)\le r\cdot \bigl|\partial_R Q_n\bigr|.$$ By replacing the upper row in diagram by $$k[Q_n]^r\xrightarrow{{\operatorname{pr}}[Q_n]\circ f\circ j[Q_n]}k[Q_n]^s\to M[Q_n]\to 0$$ and essentially running the same argument as before we obtain that $$\dim_k({\operatorname{coker}}( f\vert_{Q_n\setminus
\partial_RQ_n}))-\dim_k({\operatorname{coker}}(M[Q_n]))\le r\cdot \bigl|\partial_R Q_n\bigr|.$$ Since $$\frac{|\partial_R Q_n|}{[G/G_n]}=\frac{ |\partial_R Q_n|}{
|Q_n|}\xrightarrow{n\to\infty} 0$$ we get that $$\lim_{n\to\infty}\frac{\dim_k( k[G/G_n]\otimes_{kG}M)}{[G:G_n]}$$ exists if and only if $$\lim_{n\to\infty}\frac{\dim_k( M[Q_n])}{|Q_n|}$$ exists, and in this case they are equal. Now the assertion follows from Theorem \[the:equality\_elek\_and\_approximative\_dimension\_of\_finitely\_presented\_modules\].
Comparing dimensions {#sec:Comparing_dimensions}
====================
The main result of this section is:
\[the:Comparing\_dimensions\] Let $G$ be a group, let $k$ be a skew field, and let $k*G$ be a crossed product which is prime left Goldie. Let $\dim$ be any dimension function which assigns to a finitely generated left $k*G$-module a nonnegative real number and satisfies
1. $\dim(k*G) = 1$.
2. For every short exact sequence $0 \to M_0 \to M_1 \to M_2 \to 0$ of finitely generated left $k*G$-modules, we get $$\dim(M_1) = \dim(M_0) + \dim(M_2).$$
3. If the finitely generated left $k*G$-module $M$ satisfies $\dim(M) =
0$, then every quotient module $Q$ of $M$ satisfies $\dim(Q) = 0$.
Then for every finitely presented left $k*G$-module $M$, we get $\dim(M) = \dim_{k*G}^{{\operatorname{Ore}}}(M)$.
Let $S$ denote the non-zero-divisors of $k*G$. We have to show that for all $r,s \in \mathbb{N}$ and every $r\times
s$ matrix $A$ with entries in $k*G$ $$\dim_{k*G}^{{\operatorname{Ore}}}\bigl({\operatorname{coker}}\big(r_A \colon S^{-1}k*G^r \to
S^{-1}k*G^s\bigr) \bigr)=
\dim\bigl({\operatorname{coker}}\big(r_A \colon k*G^r \to k*G^s\bigr) \bigr),
\label{dim_Skg_is_dim}$$ where $r_A$ denotes the module homomorphism given by right multiplication with $A$. First note that we may assume that $r=s$. Indeed if $r<s$, replace $A$ with the $s \times s$ matrix which is $A$ for the first $r$ rows, and has 0’s on the bottom $s-r$ rows. On the other hand if $r>s$, replace $A$ with the $r \times r$ matrix $B$ with entries $(b_{ij})$ which is $A$ for the first $s$ columns, and has $b_{ij} =
\delta_{ij}$ if $i>s$, where $\delta_{ij}$ is the Kronecker delta.
We will often use the obvious long exact sequence associated to homomorphisms $f \colon M_0 \to M_1$ and $g \colon
M_1 \to M_2$ $$0 \to \ker(f) \to \ker(g\circ f) \to \ker(g)
\to {\operatorname{coker}}(f) \to {\operatorname{coker}}(g\circ f) \to {\operatorname{coker}}(g) \to 0.
\label{ker-coker-sequence}$$ We now assume that $A$ is an $r \times r$ matrix. Note that equation is true if $A$ is invertible over $S^{-1}k*G$; this is because then $\ker r_A = 0$ (whether $A$ is considered as a matrix over $k*G$ or $S^{-1}k*G$).
Next observe that if $U \in \operatorname{M}_r(k*G)$ which is invertible over $\operatorname{M}_r(S^{-1}k*G)$, then equation holds for $A$ if and only if it holds for $AU$, and also if and only if it hold for $UA$. This follows from , $\ker U = 0$, $\dim ({\operatorname{coker}}U) =
\dim_{k*G}^{{\operatorname{Ore}}} ({\operatorname{coker}}U) = 0$, and in the second case we use the third property of $\dim$.
We may write $S^{-1}k*G = \operatorname{M}_d(D)$ for some $d\in \mathbb{N}$ and some skew field $D$. By applying the Morita equivalence from $\operatorname{M}_d(D)$ to $D$ and back and doing Gaussian elimination over $D$ we see that there are invertible matrices $U,V \in \operatorname{M}_{rd}(S^{-1}k*G)$ such that $U\operatorname{diag}(A,\dots,A)V = J$, where there are $d$ $A$’s and $J$ is a matrix of the form $\operatorname{diag}(1,\dots,1,0,\dots,0)$. Now choose $u,v \in S$ such that $uU,vV \in
\operatorname{M}_{rd}(k*G)$. Then $(uU) \operatorname{diag}(A,\dots,A) (Vv) = uJv$, and the result follows.
\[the:Comparing\_Eleks\_dimension\_and\_the\_Ore\_dimension\] Let $G$ be a finitely generated group and let $k$ be a skew field. Suppose that $kG$ is a prime left Goldie ring. Then for any finitely presented left $kG$-module $M$ $$\dim^{{\operatorname{Elek}}}_{kG} (M) = \dim_{kG}^{{\operatorname{Ore}}}(M).$$
This follows from Theorem \[the:Comparing\_dimensions\] and Theorem \[the:Eleks-dimension\].
Proof of the main theorem {#sec:Proof_of_the_main_theorem}
=========================
\[the:dim\_approximation\_over\_fields:finitely\_presented\_modules\] In the first step we reduce the claim to the case, where $G$ is finitely generated. Consider a finitely presented left $kG$-module $M$. Choose a matrix $A \in M_{r,s}(kG)$ such that $M$ is $kG$-isomorphic to the cokernel of $r_A \colon kG^r \to kG^s$. Since $A$ is a finite matrix and each element in $kG$ has finite support, we can find a finitely generated subgroup $H \subseteq G$ such that $A \in
M_{r,s}(kH)$. Both $kG$ and $kH$ are prime left Goldie by Lemma \[lem:delta\_implies\_prime\] and Theorem \[thm:elementary\_amenable\_goldie\]. Consider the finitely presented $kH$-module $N := {\operatorname{coker}}\bigl(r_A \colon kH^r \to kH^s\bigr)$. Then $M = kG \otimes_{kH} N$. We can also consider the Ore localization $T^{-1}kH$ for $T$ the set of non-zero-divisors of $kH$. Put $H_n = H \cap G_n$. We obtain a residual chain $(H_n)_{n\ge 0}$ of $H$ and have: $$\begin{aligned}
\dim_{kG}^{{\operatorname{Ore}}}(M)
&=
\dim_{S^{-1}kG} \bigl(S^{-1}kG \otimes_{kG} M\bigr)\\
&=
\dim_{S^{-1}kG} \bigl(S^{-1}kG \otimes_{kG} kG \otimes_{kH} N\bigr)
\\
&=
\dim_{S^{-1}kG}\bigl(S^{-1}kG \otimes_{T^{-1}kH} T^{-1}kH \otimes_{kH} N\bigr)
\\
&=
\dim_{T^{-1}kH}\bigl(T^{-1}kH \otimes_{kH} N)
\\
&=
\dim_{kH}^{{\operatorname{Ore}}}(N).\end{aligned}$$ We compute $$\begin{aligned}
\frac{\dim_k\bigl(k\otimes_{kG_n} M\bigr)}{[G:G_n]}
&=
\frac{\dim_k\bigl(k[G/G_n]\otimes_{kG} M\bigr)}{[G:G_n]}\\
&=
\frac{\dim_k\bigl(k[G/G_n]\otimes_{kG} kG \otimes_{kH} N\bigr)}{[G:G_n]}
\\
&=
\frac{\dim_k\bigl(k[G/G_n]\otimes_{k[H/H_n]} k[H/H_n] \otimes_{kH} N\bigr)}{[G:G_n]}
\\
&=
\frac{[G/G_n:H/H_n] \cdot \dim_k\bigl(k[H/H_n] \otimes_{kH} N\bigr)}{[G:G_n]}
\\
&=
\frac{[G/G_n:H/H_n] \cdot \dim_k\bigl(k \otimes_{kH_n} N\bigr)}{[G/G_n:H/H_n] \cdot [H:H_n]}
\\
&=
\frac{\dim_k\bigl(k \otimes_{kH_n} N\bigr)}{[H:H_n]}.\end{aligned}$$ Therefore the claim holds for $M$ over $kG$ if it holds for $N$ over $kH$. Hence we can assume without loss of generality that $G$ is finitely generated.
Now apply Theorem \[the:Approximation\_for\_finitely\_presented\_kG-modules\_for\_Eleks\_dimension\_function\] and Theorem \[the:Comparing\_Eleks\_dimension\_and\_the\_Ore\_dimension\].\
\[the:dim\_approximation\_over\_fields:chain\_complexes\] We obtain from additivity, the exactness of the functor $S^{-1}kG \otimes_{kG}\_$ and the right exactness of the functor $k \otimes_{kG}\_$ that $$\begin{aligned}
\dim^{{\operatorname{Ore}}}_{kG}\bigl(H_i(C_*)\bigr)
&=
\dim^{{\operatorname{Ore}}}_{kG} \bigl({\operatorname{coker}}(c_{i+1})\bigr)
+\dim^{{\operatorname{Ore}}}_{kG}\bigl({\operatorname{coker}}(c_i)\bigr)
- \dim^{{\operatorname{Ore}}}_{kG}\bigl(C_{i-1}\bigr),\\
\dim_{k}\bigl(H_i(k \otimes_{kG_n} C_*)\bigr)
&=
\dim_{k}\bigl(k \otimes_{kG_n} {\operatorname{coker}}(c_{i+1})\bigr)
+\dim_{k}\bigl(k \otimes_{kG_n} {\operatorname{coker}}(c_i)\bigr)
\\ & \hspace{10mm} - \dim_{k}\bigl(k \otimes_{kG_n} C_{i-1}\bigr).\end{aligned}$$ Hence the claim follows from assertion \[the:dim\_approximation\_over\_fields:finitely\_presented\_modules\] applied to the finitely presented $kG$-modules ${\operatorname{coker}}(c_{i+1})$, ${\operatorname{coker}}(c_i)$ and $C_{i-1}$.\
\[the:dim\_approximation\_over\_fields:CW-complexes\] This follows from assertion \[the:dim\_approximation\_over\_fields:chain\_complexes\] applied to the cellular chain complex of $X$.
Extension to the virtually torsionfree case {#sec:Extension_to_the_virtually_torsionfree_case}
===========================================
Next we explain how Theorem \[the:dim\_approximation\_over\_fields\] can be extended to the virtually torsionfree case.
For the remainder of this section let $k$ be a skew field, let $G$ be an amenable group which possesses a subgroup $H$ of finite index with $\Delta^+(H) = 1$, and let $k*G$ be a crossed product such that $k*H$ is a left Goldie ring. We define the *virtual Ore dimension* of a $k*G$-module $M$ by $$\label{eq:virtual_Ore_dimension}
{\operatorname{vdim}}^{{\operatorname{Ore}}}_{k*G}(M) =
\frac{\dim^{{\operatorname{Ore}}}_{k*H}\bigl({\operatorname{res}}_{k*G}^{k*H} M\bigr)}{[G:H]},
$$ where ${\operatorname{res}}_{k*G}^{k*H} M$ is the $k*H$-module obtained from the $k*G$-module $M$ by restricting the $G$-action to $H$.
We have to show that this is independent of the choice of $H$. Since every subgroup of finite index contains a normal subgroup of finite index, it is enough to show that if $K$ is a normal subgroup of finite index in $H$ and $K \le H \le G$ with $H$ torsion free, then for every $k*H$-module $N$, $$\frac{\dim^{{\operatorname{Ore}}}_{k*K}\bigl({\operatorname{res}}_{k*H}^{k*K} N\bigr)}{[H:K]}
=
\dim^{{\operatorname{Ore}}}_{k*H}(N).
\label{Ore-dimension_and_restriction}$$ Let $T$ denote the set of non-zero-divisors of $k*H$ and write $S = (k*K) \cap T$. Note that $\Delta^+(K) = 1$, so $k*K$ is still a prime left Goldie ring and hence the ring $S^{-1}k*K$ exists. Then $S^{-1}k*H \cong (S^{-1}k*K)*[H/K]$ and there is a natural ring monomorphism $\theta\colon S^{-1}k*H \hookrightarrow T^{-1}k*H$. Since $S^{-1}k*K$ is a matrix ring over a skew field by Theorem \[thm:prime\_Goldie\_matrix\], we see that $(S^{-1}k*K)[H/K]$ is an Artinian ring, because $H/K$ is finite. But every element of $T$ is a non-zero-divisor in $(S^{-1}k*K)[H/K]$, and since every non-zero-divisor in an Artinian ring is invertible (compare [@Rowen(2008)]\*[Exercise 22 of Chapter 15 on p. 16]{}), we see that every element of $T$ is invertible in $S^{-1}k*H$ and we conclude that $\theta$ is onto and hence is an isomorphism. We deduce that $\dim_{S^{-1}k*K}(T^{-1}k*H) = [H:K]$ and that the natural map $S^{-1}N \to T^{-1}N$ induced by $s^{-1}n \mapsto
s^{-1}n$ is an isomorphism. This proves .
\[the:Extension\_to\_the\_virtually\_torsionfree\_case\] Let $G$ be an amenable group which possesses a subgroup $E \subseteq G$ of finite index such that $kE$ is left Goldie and $\Delta^+(E)=1$, and let $k$ be a skew field. Then assertions \[the:dim\_approximation\_over\_fields:finitely\_presented\_modules\], \[the:dim\_approximation\_over\_fields:chain\_complexes\] and \[the:dim\_approximation\_over\_fields:CW-complexes\] of Theorem \[the:dim\_approximation\_over\_fields\] remain true, provided we replace $\dim_{kG}^{{\operatorname{Ore}}}$ by ${\operatorname{vdim}}_{kG}^{{\operatorname{Ore}}}$ everywhere.
It suffices to prove the claim for assertion \[the:dim\_approximation\_over\_fields:finitely\_presented\_modules\] since the proof in Theorem \[the:dim\_approximation\_over\_fields\] that it implies the other two assertions applies also in this more general situation. Let $(G_n)_{n\ge 0}$ be a residual chain of $G$. To prove the result in general, we may assume that $G$ is finitely generated. Since $kE$ is left Goldie and $[G:E]<\infty$, the group ring $kG$ is also left Goldie. Further, every $kG_n$ is left Goldie. Since $\Delta^+(G)$ is finite (its order is bounded by $[G:E]$), there exists $N \in\mathbb{N}$ such that $G_N \cap \Delta^+(G) = 1$, and then $\Delta^+(G_N) = 1$ and $G_i \subseteq G_N$ for all $i \ge N$. Set $H = G_N$, so that $kH$ is prime by Lemma \[lem:delta\_implies\_prime\]. Then for a finitely presented $kH$-module $L$, $$\dim_{kH}^{{\operatorname{Ore}}}(L) =
\lim_{n \to \infty} \frac{\dim_k\bigl(k \otimes_{k[G_n\cap H]}
L\bigr)}{[H:H \cap G_n]}$$ by Theorems \[the:Approximation\_for\_finitely\_presented\_kG-modules\_for\_Eleks\_dimension\_function\] and \[the:Comparing\_dimensions\]. We have $[G:G_n] = [G:H] \cdot [H : H \cap G_n]$ for $n \ge N$. This implies for every finitely presented $kG$-module $M$ $$\begin{aligned}
{\operatorname{vdim}}_{kG}^{{\operatorname{Ore}}}(M)
=
\frac{\dim_{kH}^{{\operatorname{Ore}}}\bigl({\operatorname{res}}_{kG}^{kH} M\bigr)}{[G:H]}
& =
\lim_{n \to \infty} \frac{\dim_k\bigl(k \otimes_{k[H\cap G_n]}
{\operatorname{res}}_{kG}^{kH} M\bigr)}{[G:H] \cdot [H:H \cap G_n]}
\\
& =
\lim_{n \to \infty} \frac{\dim_k\bigl(k \otimes_{kG_n}
M\bigr)}{[G:G_n]}.\qedhere\end{aligned}$$
Because of Theorem \[thm:elementary\_amenable\_goldie\], Theorem \[the:dim\_approximation\_over\_fields\] is true in the case $k$ is a skew field and $G$ is an elementary amenable group in which the orders of the finite subgroups are bounded (clearly $\Delta^+(G_n) = 1$ for sufficiently large $n$). In particular Theorem \[the:dim\_approximation\_over\_fields\] is true for any virtually torsionfree elementary amenable group.
Examples {#sec:Examples}
========
\[rem:connection\_to\_L2\_for\_group\_C\] Let $(G_n)_{n\ge 0}$ be a residual chain of a group $G$. Let $X$ be a finite free $G$-$CW$-complex. Let $k$ be a field of characteristic ${\operatorname{char}}(k)$. For a prime $p$ denote by ${{\mathbb F}}_p$ the field of $p$ elements. Then we conclude from the universal coefficient theorem $$\begin{aligned}
\dim_{k}\bigl(H_i(G_n\backslash X;k)\bigr) & = \dim_{{{\mathbb Q}}}\bigl(H_i(G_n\backslash X;{{\mathbb Q}})\bigr) & {\operatorname{char}}(k) = 0;
\\
\dim_{k}\bigl(H_i(G_n\backslash X;k)\bigr) & = \dim_{{{\mathbb F}}_p}\bigl(H_i(G_n\backslash X;{{\mathbb F}}_p)\bigr) & p = {\operatorname{char}}(k) \not= 0;
\\
\dim_{{{\mathbb F}}_p}\bigl(H_i(G_n\backslash X;{{\mathbb F}}_p)\bigr) & \ge \dim_{{{\mathbb Q}}}\bigl(H_i(G_n\backslash X;{{\mathbb Q}})\bigr).& \end{aligned}$$ In particular we conclude from Remark \[rem:fields\_of\_characteristic\_zero\] that $$\liminf_{n \to \infty} \frac{\dim_{k}\bigl(H_i(G_n\backslash X;k)\bigr)}{[G:G_n]}
\ge
\lim_{n \to \infty} \frac{\dim_{k}\bigl(H_i(G_n\backslash X;{{\mathbb Q}})\bigr)}{[G:G_n]}
= b_i^{(2)}(X;{{\mathcal N}}(G)),$$ where the latter term denotes the $i$-th $L^2$-Betti number of $X$. In particular we get from Theorem \[the:dim\_approximation\_over\_fields\] for a torsionfree amenable group $G$ with no zero-divisors in $kG$ that $$\begin{aligned}
\dim^{{\operatorname{Ore}}}_{kG}\bigl(H_i(X;k)\bigr)
=
\lim_{n \to \infty} \frac{\dim_{k}(H_i(G_n\backslash X;k))}{[G:G_n]}
& \ge
\lim_{n \to \infty} \frac{\dim_{k}(H_i(G_n\backslash X;{{\mathbb Q}}))}{[G:G_n]}
\\
& =
b_i^{(2)}(X;{{\mathcal N}}(G))
\\
& =
\dim^{{\operatorname{Ore}}}_{{{\mathbb C}}G}\bigl(H_i(X;{{\mathbb C}})\bigr).\end{aligned}$$ This inequality is in general not an equality as the next example shows.
\[exa:S1\_wedge\_Sd\_cupDd\_plus\_1\] Fix an integer $d \ge 2$ and a prime number $p$. Let $f_p \colon S^d \to S^d$ be a map of degree $p$ and denote by $i \colon S^d \to S^1 \vee S^d$ the obvious inclusion. Let $X$ be the finite $CW$-complex obtained from $S^1 \vee
S^d$ by attaching a $(d+1)$-cell with attaching map $i \circ f^d \colon S^d
\to S^1 \vee S^d$. Then $\pi_1(X) = {{\mathbb Z}}$. Let $\widetilde{X}$ be the universal covering of $X$ which is a finite free ${{\mathbb Z}}$-$CW$-complex. Denote by $X_n$ the covering of $X$ associated to $n \cdot {{\mathbb Z}}\subseteq {{\mathbb Z}}$. The cellular ${{\mathbb Z}}C$-chain complex of $\widetilde{X}$ is concentrated in dimension $(d+1)$, $d$ and $1$ and $0$, the $(d+1)$-th differential is multiplication with $p$ and the first differential is multiplication with $(z-1)$ for a generator $z \in {{\mathbb Z}}$ $$0\to \cdots \to {{\mathbb Z}}[{{\mathbb Z}}] \xrightarrow{p} {{\mathbb Z}}[{{\mathbb Z}}] \to \cdots \to {{\mathbb Z}}[{{\mathbb Z}}] \xrightarrow{z-1} {{\mathbb Z}}[{{\mathbb Z}}].$$ If the characteristic of $k$ is different from $p$, one easily checks that $H_i(C_*) = 0$ and $$\dim^{{\operatorname{Ore}}}_{k {{\mathbb Z}}}\bigl(H_i(\widetilde{X};k)\bigr) = 0~\text{ for
$i \in \{d,d+1\}$}.$$ If $p$ is the characteristic of $k$, then $H_i(C_*) = k{{\mathbb Z}}$ and $$\dim^{{\operatorname{Ore}}}_{k{{\mathbb Z}}}\bigl(H_i(\widetilde{X};k)\bigr) = 1~\text{ for $i \in \{d,d+1\}$.}$$ Hence $\dim^{{\operatorname{Ore}}}_{kG}\bigl(H_i(\widetilde{X};k)\bigr)$ does depend on $k$ in general.
\[exa:s1\_actions\] Let $G$ be a torsionfree amenable group such that $kG$ has no zero-divisors. Let $S^1\to X\to B$ be a fibration of connected $CW$-complexes such that $X$ has fundamental group $\pi_1(X)\cong G$ and $\pi_1(S^1) \to \pi_1(X)$ is injective. Then $$\label{eq:fiber_bundles}
\dim^{{\operatorname{Ore}}}_{kG}\bigl(H_i(\widetilde X;k)\bigr)=0$$ for every $i\ge 0$.
Let $S=kG-\{0\}$ and $S_0=k{{\mathbb Z}}-\{0\}$. By looking at the cellular chain complex one directly sees that $$H_i\bigl(\widetilde S^1, S_0^{-1}k{{\mathbb Z}}\bigr)=0~\forall
i\ge 0,$$ thus $H_i\bigl(\widetilde S^1,S^{-1}kG\bigr)=S^{-1}kG\otimes_{S_0^{-1}k{{\mathbb Z}}}H_i\bigl(\widetilde S^1,S_0^{-1}k{{\mathbb Z}}\bigr) =0$ for every $i\ge 0$. The assertion is implied by the Hochschild-Serre spectral sequence that converges to $H_{p+q}(\widetilde X,S^{-1}kG)$ and has the $E^2$-term: $$E^2_{pq}=H_p\bigl(\widetilde B, H_q(\widetilde S^1, S^{-1}kG)\bigr).$$
Let $G$ be an infinite amenable group which possesses a subgroup $H$ of finite index such that $kH$ is left Goldie and $\Delta^+(H) = 1$, e.g., $G$ is a virtually torsionfree elementary amenable group. Let $k$ be a field. Let $(G_n)_{n\ge 0}$ be a residual chain of $G$. Denote by $b_i(G/G_n;K)$ the $i$-th Betti number of the group $G/G_n$ with coefficients in $k$. Then we get for every $i \ge 0$ $$\lim_{n \to \infty} \frac{b_i(G/G_n;k)}{[G:G_n]} = 0.$$ For $i=0$ this is obvious. For $i\ge 1$ this follows from Theorem \[the:Extension\_to\_the\_virtually\_torsionfree\_case\] and $H_i(EH;k)=H_i(H;k)= 0$.
[^1]: This assumption is satisfied if $G$ is torsionfree elementary amenable. See Remark \[rem:The\_Ore\_condition\_for\_group\_rings\]
|
---
abstract: |
A direct reconstruction algorithm based on Calderón’s linearization method for the reconstruction of isotropic conductivities is proposed for anisotropic conductivities in two-dimensions. To overcome the non-uniqueness of the anisotropic inverse conductivity problem, the entries of the unperturbed anisotropic tensors are assumed known *a priori*, and it remains to reconstruct the multiplicative scalar field. The quasi-conformal map in the plane facilitates the Calderón-based approach for anisotropic conductivities. The method is demonstrated on discontinuous radially symmetric conductivities of high and low contrast.
[**Keywords**]{}. Calderón’s problem, anisotropic, electrical impedance tomography, quasi-conformal map, exponential solutions, inverse conductivity problem, Dirichlet-to-Neumann map
address:
- 'Department of Mathematics, University of Helsinki, Finland'
- 'Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan'
- 'Department of Mathematics, Colorado State University, USA'
- 'Department of Mathematics and School of Biomedical Engineering, Colorado State University, USA'
author:
- Rashmi Murthy
- 'Yi-Hsuan Lin'
- Kwancheol Shin
- 'Jennifer L. Mueller'
bibliography:
- 'icebi\_refs.bib'
- 'ref.bib'
- 'ref1.bib'
- 'AnisotropyBib.bib'
title: 'A direct reconstruction algorithm for the anisotropic inverse conductivity problem based on Calderón’s method in the plane'
---
Introduction {#sec:intro}
============
The inverse conductivity problem was first proposed in Calderón’s pioneer work [@calderon2006inverse], in which the existence of a unique solution and a direct method of reconstruction of isotropic conductivities from the associated boundary measurements was given for the linearized problem in a bounded domain. This work inspired a large body of research on the global uniqueness question for the inverse conductivity problem and methods of reconstruction. Calderón made use of special exponentially growing functions known as *complex geometrical optics* solutions (CGOs), and these have proved useful for global uniqueness results for the inverse conductivity problem, leading also to a family of direct reconstruction algorithms known as D-bar methods. The reader is referred to [@Mueller2020; @MuellerBook] and the references therein for further reading on D-bar methods. Their relationship to Calderón’s method was investigated in [@knudsen2008reconstructions].
The inverse conductivity problem is the mathematical model behind a medical imaging technology known as *electrical impedance tomography* (EIT). EIT has the attributes of being inexpensive, non-invasive, non-ionizing and portable. Medical applications of EIT include pulmonary imaging [@panoutsos2007electrical; @frerichs2009assessment; @victorino2004imbalances; @Martins2019], breast cancer detection [@1344192; @cherepenin20013d; @kao20063d; @kerner2002electrical; @zou2003review], human head imaging [@yerworth2003electrical; @agnelli2020classification; @Boverman; @Malone2014], and others. See also the review articles [@lionheart2004eit; @cheney1999electrical; @hanke2003recent; @brown2003electrical] on EIT. The literature of reconstruction algorithms for the isotropic conductivity is extensive. Implementations of Calderón’s method for isotropic conductivities include [@bikowski20082d; @Muller1; @Peter_2017]. However, in reality, many tissues in the human body are anisotropic, meaning that electrical current will be conducted in spatially preferred directions. Anisotropic conductivity distributions are prevalent in the body, with significant differences in the transverse and lateral directions in the bone, skeletal and cardiac muscle, and in the brain [@abascal2008use; @barber1990quantification]. However, medical EIT imaging typically neglects anisotropic properties of the conductivity and reconstructs an isotropic approximation. This can lead to artifacts in the resulting images and incorrect estimates of conductivity values.
In this work, we want to study the anisotropic conductivity equation $$\begin{aligned}
\label{anisotropic conductivity equation in Sec 1}
L_A u :=\nabla \cdot (A(x)\nabla u)=\sum_{j,k=1}^2 {\partial}_{x_k}\left( A^{jk}(x){\partial}_{x_j}u\right)=0 \quad \text{ in }\Omega,\end{aligned}$$ where $\Omega\subset {{\mathbb R}}^2$ is a bounded Lipschitz domain and $A(x)=(A^{jk}(x))_{1\leq j,k\leq 2}$ is a positive definite symmetric matrix. The explicit regularity assumptions of $A(x)$ will be characterized later (see Hypothesis \[hypothesis of A\]). In order to study inverse problems in anisotropic media, the key step is to transform the anisotropic conductivity equation into an isotropic one, by using the *isothermal coordinates*, which are related to *quasi-conformal maps* [@ahlfors2006lectures]. This method is widely used in the study of the Calderón type inverse problems in the plane.
Contrary to the isotropic case, knowledge of the *Dirichlet-to-Neumann* (DN) map is not sufficient to recover an anisotropic conductivity [@greenleaf2003anisotropic; @kohn1983identification]. The non-uniqueness of the anisotropic problem stems from the fact that any diffeomorphism of $\Omega$ which keeps the boundary points fixed has the property of leaving the DN map unchanged (i.e., one can find a diffeomorphism with the boundary map being identity), despite the change in conductivity [@astala2006calderon]. However, a uniqueness result was proved in [@lionheart1997conformal] under the assumption that the conductivity is known up to a multiplicative scalar field. The uniqueness and stability results for an anisotropic conductivity of the form $ A = A(x,a(x)) $, where $a(x)$ is an unknown scalar function was proved in [@alessandrini2001determining]. In the same paper, the uniqueness result for the interior conductivities of $A$ were also proved by piecewise analytic perturbations of scalar term $a$. [*A priori*]{} knowledge of the preferred directions of conductivity, or the entries of the tensor of anisotropy, may be obtained, for example, from a diffusion tensor MRI, as discussed in [@abascal2011electrical].
In this work, we provide a direct reconstruction algorithm under the assumption that the anisotropic conductivity is a “small perturbation" (under the matrix notion) of a given $2\times 2$ positive definite matrix with constant entries in the plane. Other approaches to the reconstruction problem for anisotropic conductivities include [@Breckon; @Glidewell2D; @Glidewell3D; @abascal2008use; @HamiltonReyes2016; @Lionheart_2010]. The quasi-conformal map in the plane and *invariance* of the equations under a change of coordinates is the key to the algorithm. But, we note that in three spatial dimensions, there is no quasi-conformal map, and so our approach is not applicable.
CGO solutions have also been used to solve the inverse obstacle problem under various mathematical models. For the isotropic case, see, for example, [@KS2014; @KSU2011; @nakamura2007identification; @SW2006; @SY2012reconstruction; @UW2008]. For the anisotropic case in the three-dimensions, one can use the *oscillating-decaying solutions* to reconstruct unknown obstacles in a given medium, see [@KLS2015enclosure; @lin2014reconstruction; @NUW2005(ODS); @NUW2006]. It is worth mentioning that for the anisotropic case in the plane, one can also construct CGOs via the quasi-conformal map, and we refer readers to [@takuwa2008complex] for more detailed discussions.
The paper is organized as follows. Section \[sec:model\] contains the mathematical formulation of the anisotropic inverse conductivity problem. In Section \[sec:2\], we provide a rigorous mathematical analysis for the anisotropic elliptic problem. We prove that the linearization of the quadratic form is injective when evaluated at any positive definite $2\times 2$ constant matrix. The tool is to use the quasi-conformal map in the plane. In Section \[sec:3\], we provide a reconstruction methods based on Calderón’s approach for anisotropic conductivities. In addition, we also provide a numerical implementation for the inverse anisotropic conductivity problem in Section \[sec:4\].
Mathematical formulation {#sec:model}
========================
The mathematical model for the EIT problem with an anisotropic conductivity can be formulated as follows: Let $\Omega \subset {{\mathbb R}}^2$ be a simply connected domain with Lipschitz boundary $\partial \Omega$. Assume that the following conditions hold.
\[hypothesis of A\] Let $ A(x) = \left(A^{jk}(x)\right)_{1\leq j,k\leq 2}$ be the anisotropic conductivity, which satisfies:
- Symmetry: $A^{jk}(x)=A^{kj}(x)$ for $x\in \Omega$.
- Ellipticity: There is a universal constant $\lambda>0$ such that
$$\sum_{j,k=1}^2 A^{jk}(x)\xi_j \xi _k \geq \lambda |\xi|^2, \text{ for any }x\in \Omega \text{ and }\xi=(\xi_1,\xi_2) \in {{\mathbb R}}^2.$$
- Smoothness: The anisotropic conductivity $A\in C^1(\overline\Omega;{{\mathbb R}}^{2\times 2})$.
Let $ \phi \in H^{1/2}({\partial}\Omega)$ be the voltage given on the boundary. The electric field $u$ arising from the applied voltage $\phi$ on $\partial\Omega$ is governed by the following second order elliptic equation $$\begin{aligned}
\label{anisogenLap}
\begin{cases}
\nabla \cdot (A \nabla u) = \sum_{j,k = 1}^{2} \frac{{\partial}}{{\partial}x^j}\left( A^{jk} (x) \frac{{\partial}}{{\partial}x^k}u \right)= 0 &\text{ in } \Omega, \\
u= \phi &\text{ on } {\partial}\Omega.
\end{cases}\end{aligned}$$ Under Hypothesis \[hypothesis of A\] for $A $, given any Dirichlet data $\phi$ on ${\partial}\Omega$, it is known that is well-posed (for example, see [@gilbarg2015elliptic]). Therefore, the DN map is well-defined and is given by $$\begin{aligned}
\Lambda_A:H^{1/2}({\partial}\Omega)&\to H^{-1/2}({\partial}\Omega)\\
\Lambda_{A}: \phi &\mapsto \nu \cdot A\nabla u|_{{\partial}\Omega}= \left.\sum_{j,k =1}^{2} A^{jk}(x) \frac{{\partial}u}{{\partial}x^j}\nu_k\right|_{{\partial}\Omega} ,\end{aligned}$$ where $u \in H^{1} (\Omega)$ is the solution of , and $\nu=(\nu_1,\nu_2)$ is the unit outer normal vector on ${\partial}\Omega$. It is natural to consider the quadratic form $Q_{A, \Omega}(\phi)$ with respect to , which is defined by $$\label{poweraniso}
Q_{A,\Omega}(\phi) := \int_{\Omega} \sum_{j,k =1}^{2} A^{jk}(x) \frac{{\partial}u}{{\partial}x^j} \frac{{\partial}u}{{\partial}x^k}\, dx = \int_{{\partial}\Omega} \Lambda_{A} (\phi) \phi \, dS,$$ where $dS$ denotes the arc length on ${\partial}\Omega $ and we have utilized the integration by parts in the last equality. The quantity $Q_{A, \Omega}$ represents the power needed to maintain the potential $\phi$ on ${\partial}\Omega$. By the symmetry of the matrix $A $, knowledge of $Q_{A, \Omega}$ is equivalent to knowing $\Lambda_{A}$.
The inverse problem of anisotropic EIT is to ask whether $A$ is uniquely determined by the quadratic form $Q_{A,\Omega}$ and if so, how to calculate the matrix-valued function $A$ in terms of $Q_{A,\Omega}$.
Injectivity of the inverse problem {#sec:2}
==================================
Assume the conductivity $A(x)$ satisfies Hypothesis \[hypothesis of A\] and $A(x)$ is of the form $$A = a(x)A_0,$$ where $a(x)$ is a scalar function to be determined and $A_0$ is a known *constant* $2\times 2$ symmetric positive definite anisotropic tensor. Then by [@lionheart1997conformal], the inverse anisotropic conductivity problem of determining $a(x)$ has a unique solution.
Introduce the norms in the space of $A$ and in the space of quadratic forms $Q_{A}\equiv Q_{A,\Omega}(\phi)$ as follows. Let $$\label{norm of phi}
\| \phi \|^{2} := \int_{\Omega} |\nabla u|^2 dx,$$ where $u\in H^1(\Omega)$ is the solution of the following second order elliptic equation with constant matrix-valued coefficient, $$\begin{aligned}
\label{equ u}
\begin{cases}
\nabla \cdot (A_0 \nabla u) = 0 &\text{ in }\Omega,\\
u= \phi & \text{ on }{\partial}\Omega,
\end{cases}\end{aligned}$$ and $$\|Q_A \| = \sup_{ {\| \phi \| \leq 1}} \left|Q_A(\phi)\right|.$$
In the spirit of Calderón [@calderon2006inverse], the next step is to show that the mapping from conductivity to power, namely $$\label{phi}
\Phi : A \rightarrow Q_{A},$$ is analytic, where the conductivity to power map $Q_{A} $ is given by . The argument outlined below establishes that $\Phi$ is analytic.
Consider $A(x)$ as a perturbation from the constant matrix $A_0$ of the form $$\begin{aligned}
\label{definition of conductivity}
A(x): =(1+\delta(x))A_0,\end{aligned}$$ where $\delta(x)$ is regarded as a *scalar* perturbation function. We further assume that $\|\delta\|_{L^\infty(\Omega)} <1$ is sufficiently small so that the matrix $A$ is also positive definite in order to derive the well-posedness of the following second order elliptic equation $$\begin{aligned}
\label{equ w}
\begin{cases}
L_{A}(w):= \nabla \cdot (A(x) \nabla w) = 0 & \text{ in }\Omega, \\
w= \phi & \text{ on } {\partial}\Omega.
\end{cases}\end{aligned}$$ As in Calderón [@calderon2006inverse] for the isotropic case, we will use perturbation arguments. Let $w :=u+v $, where $w$ is the solution of , and $u$ is the solution of with the same boundary data $w=u=\phi$ on ${\partial}\Omega$. Then $$\label{eqn13}
L_A (w) = L_{(1+\delta(x))A_0} (u+v) = L_{A_0} v+L_{{\delta A_0}}v + L_{{\delta A_0}}u = 0 \quad \text{ in }\Omega,$$ and $v\vert_{\partial \Omega} = 0$. Then we have the following estimate for the function $v$.
The operator $L_{A_0} $ has a bounded inverse operator $G$ and $v$ has the following $H^1$ bound $$\label{v_bound}
\|v\|_{H^1(\Omega)} \leq \frac{\|G\|_{\mathcal L(H^{-1};H^1)} \|\delta\|_{\infty}\|A_0\|_F \|\phi\|}{1-\|G\| \|\delta\|_{\infty}\|A_0\| },$$ where $\|G\|_{\mathcal L(H^{-1};H^1)}$ denotes the operator norm from $H^{-1}$ to $H^1$, $\|A_0\|_F$ stands for the Frobenius norm of the matrix $A_0$ and $\|\phi\|$ is given by .
Since $u\in H^1(\Omega)$ is the unique solution of the boundary value problem $L_{A_0} u = 0$ in $\Omega$, $ u = \phi$ on ${\partial}\Omega$, the operator $L_{A_0}$ has a bounded inverse $G$.
Then from one sees, $$\label{relation 1}
G\left(L_{A_0}v +L_{\delta A_0} v +L_{\delta A_0} u \right)= 0.$$ That is, $$\label{representatiln of v}
\left(I+G L_{\delta A_0}\right)v = - G L_{\delta A_0} u.$$ Note that $$\label{nL}
\| L_{\delta A_0} w \|_{H^{-1}(\Omega)}= \sup_{\psi \in H^{1}_{0} (\Omega)} \frac{\left|\int_{\Omega} \nabla \cdot [(\delta(x) A_0) \nabla w] \psi \, dx\right|}{\| \psi\|_{H^1_0(\Omega)}},$$ and $$\begin{aligned}
\label{inequality 1}
\begin{split}
\left|\int_{\Omega} \nabla \cdot [ (\delta(x) A_0) \nabla w]\psi \, dx \right| = &\left| \int_{\Omega} \nabla \psi \cdot [(\delta(x) A_0) \nabla w] \, dx \right| \\
\leq& \| \delta(x) A_0 \|_{L^\infty(\Omega)} \|\nabla w \|_{L^2(\Omega)}\|\psi\|_{H^1_0(\Omega)}.
\end{split}\end{aligned}$$ Thus, from and , $$\label{relation 2}
\| L_{\delta A_0} w \|_{H^{-1}(\Omega)} \leq\| \delta(x) A_0 \|_{L^\infty(\Omega)} \|\nabla w \|_{L^2(\Omega)}.$$ Next, consider the operator norm $$\begin{aligned}
\|G L_{\delta A_0} \|_{\mathcal{L}(H^1;H^1)} = \sup_{w \neq 0} \frac{\| GL_{\delta A_0} w\|_{H^1(\Omega)}}{\|w\|_{H^1(\Omega)}}\leq \frac {\|G\|_{\mathcal{H^{-1};H^1}} \|L_{\delta(x) A_0} w \|_{H^{-1}(\Omega)}}{\| w \|_{H^1(\Omega)}}.\end{aligned}$$ That is, $$\|G L_{\delta A_0} \|_{\mathcal L(H^1;H^1)} \leq \|G\|_{\mathcal L(H^{-1};H^1)} \| \delta A_0 \|_{L^\infty(\Omega)},$$ where $\|\cdot \|_{\mathcal L(X;Y)}$ stands for the operator norm from the Banach space $X$ to $Y$. Moreover, when $$\| \delta(x) A_0 \|_{L^\infty(\Omega)} < \dfrac{1}{\|G\|_{\mathcal L(H^{-1};H^1)}} ,$$ the Neumann series $$\left[\sum_{j = 0}^{\infty} (-1)^j(GL_{\delta A_0})^j\right] \left(GL_{\delta A_0}u\right),$$ converges, and from , one has $$v = - \left[ \sum_{j = 0}^{\infty} (-1)^j (GL_{\delta A_0})^j \right](GL_{\delta A_0} u).$$ It is easy to see that $$\begin{aligned}
\label{relation 3}
\notag\|v+GL_{\delta A_0} v \|_{H^1(\Omega)} \geq & \|v\|_{H^1(\Omega)} - \|GL_{\delta A_0} v \|_{H^1(\Omega)} \\
\geq &\|v\|_{H^1(\Omega)}\left(1- \|G\|_{\mathcal L(H^{-1};H^1)} \|\delta(x) A_0\|_{L^\infty(\Omega)}\right).\end{aligned}$$ Finally, by using relations and , $$\begin{aligned}
&\quad \left(1-\|G\|_{\mathcal L(H^{-1};H^1)} \| \delta(x) A_0\|_{L^\infty(\Omega)}\right) \|v\|_{H^1(\Omega)} \\
&\leq \|v + G (-L_{\delta A_0}u - L_{A_0} v)\|_{H^1(\Omega)} \\
&= \|v - G L_{\delta A_0} u -v\|_{H^1(\Omega)}\\
&\leq \|G\|_{\mathcal L(H^{-1};H^1)} \|L_{\delta A_0} u \|_{H^{-1}(\Omega)}\\
&= \|G\|_{\mathcal L(H^{-1};H^1)} \| \delta(x) A_0 \|_{L^\infty(\Omega)} \| \phi\|,\end{aligned}$$ and substituting into the above inequality results in $$\label{estimate for v}
\|v \|_{H^1(\Omega)} \leq \frac{\|G\|_{\mathcal L(H^{-1};H^1)} \| \delta(x) A_0 \|_{L^\infty(\Omega)} \| \phi \|}{1- \|G\|_{\mathcal L(H^{-1};H^1)} \| \delta(x) A_0\|_{L^\infty(\Omega)}}.$$ Therefore, the above calculations allow us to conclude that the mapping $\Phi $ defined by is analytic at $A_{0}$, which completes the proof.
Next, let us linearize the map $Q_{A}(\phi) $ around a positive definite constant matrix $A(x) = A_0 $ as follows: $$\begin{aligned}
\notag Q_{(1+ \delta(x)) A_0}(\phi) = & \int_{\Omega} \left[(1 + \delta(x))A_0 \nabla w\right] \cdot \nabla w \, dx \\
\notag=& \int_{\Omega} \left[(A_0 +\delta(x) A_0)\nabla u\right] \cdot \nabla u \, dx + 2\int_{\Omega} (\delta(x) A_0\nabla u) \cdot \nabla v \, dx\\
&+\int_\Omega \left[(A_0+\delta(x) A_0)\nabla v\right]\cdot \nabla v \, dx,\end{aligned}$$ where we have used that $\nabla \cdot (A_0\nabla u)=0$ in $\Omega$. We now show that $$\int_{\Omega} (\delta(x) A_0 \nabla u) \cdot \nabla v \, dx \quad \text{ and } \quad \int_\Omega \left[(A_0+\delta(x) A_0)\nabla v\right]\cdot \nabla v \, dx$$ are of $\mathcal{O}(\| \delta \|) $. It is easy to see that $$\begin{aligned}
\label{inequality v 1}
\left|\int_\Omega (\delta(x) A_0)\nabla u\cdot \nabla v\, dx\right|\leq C_{A_0} \|\delta\|_{L^\infty(\Omega)}\|\phi \|\|v\|_{H^1(\Omega)},\end{aligned}$$ and $$\begin{aligned}
\label{ineqaulity v 2}
\left|\int_\Omega \left[(A_0+\delta(x) A_0)\nabla v\right]\cdot \nabla v \, dx\right|\leq C_{A_0}\left(1+\|\delta\|_{L^\infty(\Omega)}\right)\|v\|_{H^1(\Omega)}^2,\end{aligned}$$ for some constant $C_{A_0}>0$ independent of $\delta$. By inserting into , and taking $\|\delta\|_{L^\infty (\Omega)}$ sufficiently small, one obtains the desired result. Thus, the Fréchet derivative of the quadratic form $Q_A(\phi)$ at $A(x)=A_0$ is given by $$\label{dQ1}
dQ_{A}(\phi)\Big\vert_{A = A_{0}} = \int_\Omega \left((\delta(x) A_0 \right) \nabla u) \cdot \nabla u \, dx,$$ where $u\in H^1(\Omega)$ is a solution of $\nabla \cdot (A_0\nabla u)=0$ in $\Omega$ with $u=\phi$ on $\partial \Omega$.
The Fréchet derivative $ \left. dQ_A(\phi)\right\vert_{A = A_{0}}$ is injective.
In order to prove that $\left. dQ_A(\phi)\right|_{A = A_{0}}$ is injective, we only need to show that $$\begin{aligned}
\int_\Omega \left(\delta(x) A_0\nabla u \right) \cdot \nabla u\,dx=0 \quad \text{ implies that }\quad \delta \equiv 0,\end{aligned}$$ where $u \in H^1(\Omega)$ is a solution of $L_{A_0}u=0$ in $\Omega$ with $u=\phi$ on ${\partial}{\Omega}$. On the other hand, since the last integral in vanishes for all such $u$, then it is equivalent to prove $$\label{inj}
\int_\Omega ((\delta(x) A_0) \nabla u_1)\cdot \nabla u_2\ dx = 0,$$ where $u_1$, $u_2\in H^1(\Omega)$ are solutions of $L_{A_0}u_1 = L_{A_0}u_2 = 0$ in $\Omega$. Inspired by [@calderon2006inverse], we want to find special exponential solutions to prove it.
In order to achieve our aim, we utilize the celebrated quasi-conformal map in the plane. We first identify ${{\mathbb R}}^2$ with the complex plane ${{\mathbb C}}$. Recall that $\delta(x)\in C^1(\overline{\Omega})$ since the conductivity $A(x)$ is a $C^1(\overline{\Omega})$ matrix function. We now extend $\delta(x)$ to ${{\mathbb C}}$ (still denoted by $\delta(x)$) with $\delta(x)\in C^1_0 ({{\mathbb C}})$ by considering $\delta(x)\equiv 0$ for $|x|>r$, for some large constant $r>0$ such that $\Omega \Subset B_r(0)$. Meanwhile, we also extend the constant matrix $A_0$ to ${{\mathbb C}}$, denoted by $A_0(x)$, such that $A_0(x)\in C^1({{\mathbb C}})$ with $A_0(x)=I_2$ (a $2\times2$ identity matrix) for $|x|>r$.
From [@astala2005calderons Lemma 3.1] and [@takuwa2008complex Theorem 2.1], given a $C^1$-smooth anisotropic conductivity $A(x)$, it is known that there exists a $C^1$ bijective map $\Phi^{(A)}:\mathbb R^2 \to \mathbb R^2$ with $y=\Phi^{(A)}(x)$ such that $$\begin{aligned}
\label{scalar conductivity}
\Phi^{(A)}_*A = \left(\det A\circ (\Phi^{(A)})^{-1}\right)^{1/2}I_2,\end{aligned}$$ is a scalar conductivity, where $$\begin{aligned}
\label{transformation_formula}
\Phi^{(A)}_*A(y)=\left. \dfrac{\nabla \Phi^{(A)}(x)A(x)\nabla (\Phi^{(A)})^T(x)}{\det (\nabla \Phi^{(A)}(x))}\right|_{x=(\Phi^{(A)})^{-1}(y)}, \end{aligned}$$ with $$\begin{aligned}
&(\Phi^{(A)}_* A)^{i\ell}(y) \\
:=&\dfrac{1}{\det (\nabla \Phi^{(A)})}\sum _{j,k=1}^2{\partial}_{x^j}(\Phi^{(A)})^i(x) {\partial}_{x^k}(\Phi^{(A)})^\ell(x) A^{jk}(x)\Big\vert_{x=(\Phi^{(A)})^{-1}(y)}.\end{aligned}$$ Moreover, $\Phi^{(A)}$ solves the following Beltrami equation in the complex plane $\mathbb C$, $$\begin{aligned}
\label{Belt}
\overline{{\partial}}\Phi^{(A)}=\mu_A {\partial}\Phi^{(A)},\end{aligned}$$ where $$\begin{aligned}
\label{muA coeff}
\mu _A=\dfrac{A^{22}-A^{11}-2iA^{12}}{A^{11}+A^{22}+2\sqrt{\det A}},\end{aligned}$$ and $$\begin{aligned}
\overline{{\partial}}=\dfrac{1}{2}({\partial}_{x_1}+i{\partial}_{x_2}),\quad {\partial}=\dfrac{1}{2}({\partial}_{x_1}-i{\partial}_{x_2}).\end{aligned}$$ Here we point out that the coefficient $\mu_A$ defined in is supported in $B_r\subset {{\mathbb C}}$, since $\delta(x)=0$ and $A_0$ was extended $C^1$-smoothly to the identity matrix $I_2$ for the same domain $|x|>r$.
Next, for the given anisotropic tensor $A_0$, we can find a corresponding Beltrami equation for $A_0$. We do the same extension of $A_0$ as before, such that $A_0(x)\in {{\mathbb C}}^1({{\mathbb C}})$ with $A_0(x)=I_2$ for $|x|>r$ (for the same large number $r>0$). Now, from the representation formulas and , we know that there exists a quasi-conformal map $\Phi^{(A_0)}$ such that $$\begin{aligned}
\label{transformation formula 2}
\widetilde A_0:=&\Phi^{(A_0)}_* A_0=\left. \dfrac{\nabla \Phi^{(A_0)}(x)A_0\nabla (\Phi^{(A_0)})^T(x)}{\det (\nabla \Phi^{(A_0)}(x))}\right|_{x=(\Phi^{(A_0)})^{-1}(y)}=\sqrt{\det A_0}I_2 \end{aligned}$$ with $\det A_0>0$. Furthermore, $\Phi^{(A_0)}$ solves the Beltrami equation $$\overline{{\partial}}\Phi^{(A_0)}=\mu_{A_0} {\partial}\Phi^{(A_0)},$$ where $$\begin{aligned}
\label{muA_0 coeff}
\mu _{A_0}=\dfrac{A_0^{22}-A^{11}-2iA_0^{12}}{A^{11}+A_0^{22}+2\sqrt{\det A_0}}.\end{aligned}$$ Similarly, we also have that $\mu_{A_0}$ is supported in the same disc $B_r$ since $A_0=I_2$ for $|x|>r$. Note that $\delta=\delta(x)$ is a scalar function, then by using the formula and , one can see that $$\begin{aligned}
\Phi^{(A_0)}_* (\delta(x) A_0)=&\left. \dfrac{\nabla \Phi^{(A_0)}(x)\left(\delta(x)A_0\right)\nabla (\Phi^{(A_0)})^T(x)}{\det (\nabla \Phi^{(A_0)}(x))}\right|_{x=(\Phi^{(A_0)})^{-1}(y)} \\
=&\delta(x)|_{x=(\Phi^{(A_0)})^{-1}(y)}\sqrt{\det A_0}I_2.\end{aligned}$$ Let $\Phi\equiv\Phi^{(A_0)}$, $\widetilde {\Omega}:=\Phi(\Omega)$ and $\widetilde u_j(y):=u_j\circ (\Phi^{-1}(y)$ for $j=1,2$, by using and change of variables $y=\Phi(x)$ via the quasi-conformal map, then we obtain that $$\begin{aligned}
\label{transf-inj}
\int_{\widetilde \Omega}\Phi_*(\delta(x) A_0) \nabla_y \widetilde u_1 \cdot \nabla_y \widetilde u_2 \, dy=\int_\Omega ((\delta(x) A_0) \nabla u_1)\cdot \nabla u_2 \, dx =0,\end{aligned}$$ where $\widetilde u_j$ are solutions of $$\begin{aligned}
\label{A0_equation}
L_{\widetilde A_0}\widetilde u_1=L_{\widetilde A_0}\widetilde u_2=0 \text{ in }\widetilde \Omega.\end{aligned}$$
In fact, is equivalent to the Laplace equation $\Delta _y\widetilde u_j=0$ in $\widetilde \Omega$ for $j=1,2$ because $\widetilde A_0=\sqrt{\det A_0}I_2$ with $\det A_0$ being a positive constant. Based on Calderón’s constructions [@calderon2006inverse], we can consider two special exponential solutions in the transformed space as follows. Let $\xi \in \mathbb R^2$ be an arbitrary vector and $b\in \mathbb R^2$ such that $\xi \cdot b=0$ and $|\xi |=|b|$, then one can define $$\begin{aligned}
\label{exponential solution}
\widetilde u_1(y):=e^{\pi i(\xi \cdot y)+\pi (b\cdot y)}\quad\text{ and }\quad \widetilde u_2(y):=e^{\pi i(\xi \cdot y)-\pi (b\cdot y)},\end{aligned}$$ and it is easy to check that $\widetilde u_1$ and $\widetilde u_2$ are solutions of Laplace’s equation. By substituting these exponential solutions into , one has $$\begin{aligned}
2\pi |\xi |^2\int _{\widetilde \Omega} \left(\delta\circ \Phi^{-1}(y)\right) \sqrt{\det A_0}\,e^{2\pi i \xi \cdot y}\, dy=0, \text{ for any }\xi \in \mathbb R^2,\end{aligned}$$ which implies that $\delta=0$, due to the positivity of $\det A_0$. This proves the assertion.
It is worth mentioning that
- Due to the remarkable quasi-conformal mapping in the plane, one can reduce the anisotropic conductivity equation into an isotropic one. This method helps us to develop the reconstruction algorithm for the anisotropic conductivity equation proposed by Calderón [@calderon2006inverse].
- The method fails when the space dimension $n\geq 3$, because there are no suitable exponential-type solutions for the anisotropic case. For the three-dimensional case, we do not have complex geometrical optics solutions but we have another exponential solution, which is called the oscillating-decaying solution (see [@lin2014reconstruction]).
The linearized reconstruction method {#sec:3}
====================================
Since the tensor of anisotropy $A_0$ is known [*a priori*]{}, we can now transform the problem to the isotropic case, reconstruct the transformed conductivity on the transformed domain using Calderón’s method on an arbitrary domain as in [@Peter_2017], and then use the quasi-conformal map to transform the conductivity back to the original one. Recall that from the definitions of our choices of extensions for $A_0$ and $\delta(x)$, the representation formulations of and yield that $$\mu_A = \mu_{A_0} \text{ in }{{\mathbb C}}, \quad \text{ and }\quad \mu_A=\mu_{A_0}=0 \text{ for }|x|>r>0.$$ Thus, without loss of generality, we may assume that $\Phi^{(A_0)} = \Phi^{(A)}$ by using $\mu_{A_0}=\mu_A$. In the rest of this article, we simply denote the quasi-conformal map by $y=\Phi(x)\equiv\Phi^{(A_0)}(x) = \Phi^{(A)}(x)$. By utilizing the change of variables via the quasi-conformal map, we also have that $$\int_{\partial \Omega} u_1 \left( \Lambda_A u_2 \right) \, dS = \int_{\partial \widetilde{\Omega}} \widetilde{u}_1 \left( \Lambda _{\widetilde{A}} \widetilde{u}_2\right)\, dS,$$ where $\widetilde{\Omega}=\Phi(\Omega) $, $\widetilde{A}(y)=\Phi^{(A)}_*A(y) $, defined in , and $\widetilde u_j = u_j\circ \Phi^{-1}(y)$ for $j=1,2$.
Since the DN data is preserved under the quasi-conformal map (i.e., change of variables), we can reconstruct the scalar conductivity $\tilde{a}(x)$ from Calderón’s method as in [@Peter_2017], and then map it back. Defining Calderón’s linearized bilinear form for the isotropic problem by $$\begin{aligned}
B(\phi_1,\phi_2) = \int_{\partial \Omega} w_1 \left(\Lambda _A w_2 \right)\, dS,
\label{equB}\end{aligned}$$ $w_1|_{\partial \Omega} = \phi_1 = u_1|_{\partial \Omega}$ and $w_2|_{\partial \Omega} = \phi_2 = u_2|_{\partial \Omega}$, this can be computed directly from our measured data. Calderón proved [@calderon2006inverse] that the Fourier transform of an isotropic conductivity can be decomposed into two terms, one of which is negligible for small perturbations. Thus, denoting the Fourier transform of a function $f$ by $\widehat{f}$, by [@calderon2006inverse] we can write $$\begin{aligned}
\widehat{\tilde{a}}(z)=\widehat{F}(z)+R(z)\label{gammahat},\end{aligned}$$ where $$\begin{aligned}
\label{gammaFandR}
\begin{split}
\widehat{\tilde{a}}(z) =&-\frac{1}{ 2 \pi^2 |z|^2} \int_{\tilde{\Omega}} (1+\tilde{\delta}(x)) \nabla u_1\cdot \nabla u_2 \, dx \\
= &-\frac{1}{ 2 \pi^2 |z|^2}\int_{\tilde{\Omega}} \tilde{a}(x) e^{2\pi i(z\cdot x)}\,dx \\
\widehat{F}(z) =& -\frac{1}{2\pi^2|z|^2}B \left( e^{i\pi(z\cdot x)+\pi(b\cdot x)}, e^{i\pi(z\cdot x)-\pi(b\cdot x)}\right)
\\
R(z) =&\frac{1}{2\pi^2|z|^2} \int_{\tilde{\Omega}} \tilde{\delta} (\nabla u_1 \cdot \nabla v_2+ \nabla v_1 \cdot \nabla u_2) +(1+\tilde{\delta}) \nabla v_1 \cdot \nabla v_2 \, dx.
\end{split}\end{aligned}$$ For small $|z|$, the term $R(z)$ is small when the perturbation $\tilde{\delta}$ is small in magnitude and is to be neglected in numerical implementation. Thus, the isotropic conductivity can be approximated by the inverse Fourier transform of $\widehat{F}(z)$. Since the deformed domain $\tilde{\Omega}$ is not circular, we adopt the algorithm introduced in [@Peter_2017], in order to compute the function $\widehat{F}(z)$.
We will first invert $\widehat F(z)$ numerically to obtain a reconstruction of the scalar isotropic conductivity $\widetilde
A= \Phi_*{A(x)}$ on $\widetilde{\Omega}=\Phi^{(A_0)}\Omega $. Next, we compute the mapping $\Phi^{(A_0)}$ by solving the Beltrami equation . We then pull back the scalar conductivity from the deformed coordinates to the original coordinates by applying $(\Phi^{(A_0)})^{-1}$ to $\tilde{a}$ to obtain $a(x)$. Finally, we obtain the anisotropic conductivity $A(x)$ by multiplying $a(x)$ by the known matrix $A_0$.
Numerical implementation {#sec:4}
========================
Numerical solution of the forward problem for data simulation {#sec_forward}
-------------------------------------------------------------
A finite element method (FEM) implementation of the complete electrode model (CEM) [@Somersalo] for EIT was developed for data simulation. We first provide the equations of the CEM. Assume the anisotropic conductivity $A$ satisfies Hypothesis \[hypothesis of A\]. Then it satisfies the anisotropic generalized Laplace equation $$\nabla \cdot (A \nabla u) = \sum_{j,k = 1}^{2} \frac{{\partial}}{{\partial}x^j}\left( A^{jk} (x) \frac{{\partial}u}{{\partial}x^k} \right)= 0 \text{ in } \Omega.$$ The boundary conditions for the CEM with $L$ electrodes are defined as follows. The current $I_{l}$ on the $l$ the electrode is given by $$\int_{e_l} \sum_{j,k = 1}^{2} \left( A^{jk} (x) \frac{{\partial}u}{{\partial}x^k} \right) \, dS= I_{l}, \qquad l = 1,2,...,L,$$ $$\sum_{j,k = 1}^{2} \left( A^{jk} (x) \frac{{\partial}u}{{\partial}x^k} \right) = 0 \qquad \text{off} \qquad \bigcup_{l = 1}^{L} e_l,$$ where $e_l$ is the region covered by the $l$-th electrode, and $\nu$ is the outward normal to the surface of the body. The voltage on the boundary is given by $$u + z_{l} \sum_{j,k = 1}^{2} \left( A^{jk} (x) \frac{{\partial}u}{{\partial}x^k} \right) = U_l \qquad {\text{on}} \quad e_l \qquad \text{for} \quad l = 1,2,...,L,$$ where $z_l$ is the [*contact impedance*]{} corresponding to the $l$-th electrode. For a unique solution to the forward problem, one must specify the choice of ground, $$\label{ground}
\sum_{l = 1}^{L} U_l = 0,$$ and the current must satisfy Kirchhoff’s Law: $$\sum_{l = 1}^{L} I_l = 0.$$
Denote the potential inside the domain $\Omega$ by $u$ or $v$ and the voltages on the boundary by $U$ or $V$. The variational formulation of the complete electrode model is given by $$\label{var}
B_s((u, U), (v,V)) = \sum_{l =1}^L I_l \bar{V}_l,$$ where $v \in H^1(\Omega)$ and $ V \in \mathbb{C}^L$, and the sesquilinear form $B_s : H \times H \rightarrow \mathbb{C} $ is given by $$B_s((u, U), (v,V)) = \int_{\Omega} A \nabla u \cdot \nabla \bar{v}\, dx\, dy\, +\, \sum_{l =1}^L \frac{1}{z_l} \int_{e_l} (u -U_l) (v -\bar{V}_l) \, dS.$$
Discretizing the variational problem leads to the finite element formulation. The domain $\Omega$ is discretized into small triangular elements with $N$ nodes in the mesh. Suppose $(u,U)$ is a solution to the complete electrode model with an orthonormal basis of current patterns $\varphi_k$. Then a finite dimensional approximation to the voltage distribution inside $\Omega$ is given by: $$\label{voltinsideapp}
u^{h} (z) = \sum_{k =1}^{N} \alpha_k \varphi_k(z),$$ and on the electrodes by $$\label{voltonelecapp}
U^{h} (z) = \sum_{k = N+1}^{N+(L-1)} \beta_{(k-N)} \vec{n}_{(k-n)},$$ where discrete approximation is indicated by $h$ and the basis functions for the finite dimensional space $\mathcal{H} \subset H^{1} (\Omega)$ is given by $\varphi_{k}$, and $\alpha_{k}$ and $\beta_{(k-N)}$ are the coefficients to be determined. Let $$\vec{n}_{j} = (1,0,...,0,-1,0,...0)^T \in {{\mathbb R}}^{L \times 1},$$ where $-1$ is in the $(j-1)$st position. The choice of $ \vec{n}_{(k-N)} $ satisfies the condition for a choice of ground in , since $\vec{n}_{(k-N)}$ in results in $$U^{h}(z) = \Bigg ( \sum_{k =1}^{L-1} \beta_k, -\beta_1,... , -\beta_{L-1} \Bigg) ^T.$$ In order to implement the FEM computationally we need to expand using approximating functions and with $v =\varphi_{j}$ for $ j = 1,2,...N$ and $V = \vec{n}_{j} $ for $ j = N+1, N+2, ..... N+(L-1)$ to get a linear system $$\label{linsys}
M \vec{b} = \vec{f},$$ where $\overrightarrow{b} =(\overrightarrow{\alpha},\overrightarrow{\beta})^T \in \mathbb{C}^{N+L-1} $ with the vector $\overrightarrow{\alpha} = (\alpha_1,\alpha_2,.....,\alpha_N) $ and the vector $\overrightarrow{\beta} = (\beta_1,\beta_2,.....\beta_{L-1}) $, and the matrix $G \in \mathbb{C}^{(N+L-1)} $ is of the form
$$M=\left ( \begin{array}{c c}
B & C\\
\tilde{C} & D\end{array} \right )$$
The right-hand-side vector is given by $$\overrightarrow{f} = ({\bf 0}, \tilde{I})^T,$$ where ${\bf 0} \in \mathbb{C}^{1\times N} $ and $\tilde{I} = (I_1 -I_2, I_1 - I_3,....I_1-I_L) \in \mathbb{C}^{1\times (L-1)} $. The entries of $\overrightarrow{\alpha} $ represent the voltages throughout the domain, while those of $\overrightarrow{\beta}$ are used to find the voltages on the elctrodes by $$U^h = \mathcal{C} \overrightarrow{\beta}$$ where $\mathcal{C} $ is the $L\times (L-1) $ matrix $$\mathcal{C} = \left( \begin{matrix}
1 & 1 & 1 & \ldots & 1\\
-1 & 0 & 0 & \ldots & 0\\
0 & -1 & 0 & \ldots & 0\\
& & \ddots \\
0 & 0 & 0 & \ldots & -1 \end{matrix} \right).$$
The entries of the block matrix $B$ are determined in each of the following cases:
$\bullet$ Case (i) [$\bf 1 \leq k,j \leq N$]{}.
In this case $u^h \neq 0, U^h = 0, v \neq 0 $, but $V = 0 $. The sesquilinear form can be simplified to $$B_s((u^h, U^h), (v,V)) := \int_{\Omega} A \nabla u^h \cdot \nabla \bar{v} \, dx + \sum_{l =1}^L \frac{1}{z_l} \int_{e^l} u^h \bar{v} \, dS = 0.$$
Thus, the $(k,j) $ entry of the block matrix $B $ becomes, $$\label{Bmat}
B_{kj} = \int_\Omega A \nabla \phi_k \cdot \nabla \overline{\phi}_j \,dx +\sum_{l=1}^{L} \frac{1}{z_l} \int_{e_l} \phi_k \overline{\phi}_j \, dS.$$
The entries of the block matrix $C $ are determined as follows:
$\bullet$ Case (ii) [$ \bf 1 \leq k \leq N, N+1 \leq j \leq N+(L-1) $]{}.
In this case $u^h \neq 0, U^h = 0, v =0 $, and $V \neq 0 $. The sesquilinear form simplifies to $$B_s ((u^h, 0),(0,V)):= - \sum_{l = 1}^{L} \frac{1}{z_l} \int_{e^l} u^h \bar{V}_l \, dS = I_1 - I_{j+1}$$ Therefore, entries of C matrix becomes, $$\label{Cmat}
C_{kj} = - \bigg[ \frac{1}{z_l} \int_{e_l} \varphi_{k}(\overrightarrow{n}_{j})_{l} \, dS \bigg]$$
$\bullet$ Case (iii) The entries of the block matrix $\tilde{C} $ are determined as follows:
For [$\bf N \leq k \leq N+(L-1), 1 \leq j \leq N$]{}. Here $u^h = 0, U^h \neq 0, v \neq 0, V =0 $. The expression for sesquilinear is $$B_s ((0,U^h),(v,0)):= - \sum_{l = 1}^{L} \frac{1}{z_l} \int_{e^l} U^h \bar{v}_l \, dS= 0$$
Thus the [*kj*]{} entry of the $\tilde{C} $ is $$\tilde{C} = - \bigg[ \sum_{l =1}^{L} \int_{e_l} \overline{\varphi}_j \, dS - \frac{1}{z_l
+1} \int_{e_j +1} \overline{\varphi}_{j+1}\, dS \bigg]$$
$\bullet$ Case (iv) The entries of the block matrix $D $ are determined as follows:
For ${\bf N \leq k, j\leq N+(L-1)} $. Here $u^h =0, U^h \neq 0, v = 0, V \neq 0 $ The sequilinear form is given by $$B_s ((0, U^h),(0,V)):= \sum_{l = 1}^{L} \frac{1}{z_l} \int_{e^l} U^h \bar{V}_l \, dS = I_1 - I_{j+1}$$
Thus the entries of matrix D is given by $$\label{dmat}
D_{kj} = \begin{cases}
\frac{ \vert e_1 \vert}{z_1} + \frac{ \vert e_{j+1} \vert}{z_{j+1}}, & j = k-N\\
\frac{\vert e_1 \vert}{z_1}, & j \neq k-N.
\end{cases}$$
Solving gives us the coefficients $\beta_{(k-N)} $ required for the voltages $ U^h $ on the electrodes.
Computing the quasi-conformal map {#subsec:quasi_conf}
---------------------------------
A discrete approximation to the quasi-conformal map $\Phi^{(A_0)}$ was computed by solving the Beltrami equation by the method referred to as [*Scheme 1*]{} in [@Gaidashev2008]. Let $T[h]$ denote the Hilbert transform of a function $h$ $$T[h] = \frac{i}{2\pi}\lim_{\epsilon\rightarrow 0}\int\int_{{{\mathbb C}}\setminus B(z,\epsilon)}\frac{h(\xi)}{(\xi - z)^2}d\bar{\xi}\wedge \, d\xi,$$ and $P[h]$ denote the Cauchy transform of $h$ $$P[h] = \frac{i}{2\pi}\int\int_{{{\mathbb C}}}\frac{h(\xi)}{\xi - z} \frac{h(\xi)}{\xi}- d\bar{\xi}\wedge d\xi.$$ It is shown in [@Gaidashev2008] that a solution to equation can be computed as follows.
- Begin with an initial guess $h^0$ to the solution of the equation $$h^* = T[\mu h^*]+T[\mu]$$
- Compute the iterates $$h^{n+1} = T[\mu h^n]+T[\mu]$$ until the method converges to within a specified tolerance, and denote the solution by $h^*$.
- Compute the solution $f(z)$ from $$f(z) = P[\mu(h^*+1)](z) + z$$
The iterates converge since the map $h \mapsto T[\mu h]$ is a contraction in $L_p({{\mathbb C}})$, $p>2$ by the extended version of the Ahlfors-Bers-Boyarskii theorem [@Gaidashev2008; @Ahlfors1960].
Reconstruction
--------------
Once we compute $w = f(z)$ for all discretized values of $z \in \Omega$, by using the reconstruction algorithm in [@Peter_2017] and the data from the forward modeling in subsection \[sec\_forward\], we compute the isotropic conductivity $\tilde{\sigma}(w) = \tilde{\sigma}(f(z))$. Note that Calderón’s method is a pointwise reconstruction algorithm. Then we get $\sigma(z) = \tilde{\sigma}(f(z))$.
Examples
========
In this section we illustrate the method on a radially symmetric discontinuous conductivity on the unit disk with high and low contrast. This simple example was chosen to illuminate the features of the reconstruction and facilitate comparison with the reconstructions by the D-bar method and Calderón’s method in [@PropertiesPaper] of an isotropic conductivity of the same nature.
We consider four different conductivity tensors $A_0$ for the background on the unit disk $\Omega$: $$\begin{aligned}
A_0^1 =
\left ( \begin{array}{c c}
1 & 0\\
0 & 1.3\end{array} \right), \quad
A_0^2 =
\left ( \begin{array}{c c}
1.3 & 0\\
0 & 1\end{array} \right), \quad
A_0^3 =
\left ( \begin{array}{c c}
1 & 0\\
0 & 4\end{array} \right), \quad
A_0^4 =
\left ( \begin{array}{c c}
4 & 0\\
0 & 1\end{array} \right).\end{aligned}$$ The tensor $A_0^1$ corresponds to $\mu = 0.0655$, $A_0^2$ corresponds to $\mu = -0.0655$, $A_0^3$ corresponds to $\mu = 0.3333$, and $A_0^4$ corresponds to $\mu = -0.3333$, rounded to four digits after the decimal. The images of $\Omega$ under the quasi-conformal mapping computed by the method in Section \[subsec:quasi\_conf\] with initial guess $h^0= \mu{A_0}$ are found in Figure \[OmegaTilde\].
![The unit disk $\Omega$ and its image $\tilde{\Omega}$ (in blue) under the computed quasi-conformal mapping $\Phi^{(A_0)}$ for $A_0^1$ (upper left), $A_0^2$ (upper right), $A_0^3$ (lower left), $A_0^4$ (lower right).[]{data-label="OmegaTilde"}](Ellipse_Fig_1_1p3.jpg "fig:"){width=".35\textwidth"} ![The unit disk $\Omega$ and its image $\tilde{\Omega}$ (in blue) under the computed quasi-conformal mapping $\Phi^{(A_0)}$ for $A_0^1$ (upper left), $A_0^2$ (upper right), $A_0^3$ (lower left), $A_0^4$ (lower right).[]{data-label="OmegaTilde"}](Ellipse_Fig_1p3_1.jpg "fig:"){width=".35\textwidth"}\
![The unit disk $\Omega$ and its image $\tilde{\Omega}$ (in blue) under the computed quasi-conformal mapping $\Phi^{(A_0)}$ for $A_0^1$ (upper left), $A_0^2$ (upper right), $A_0^3$ (lower left), $A_0^4$ (lower right).[]{data-label="OmegaTilde"}](Ellipse_Fig_1_4.jpg "fig:"){width=".35\textwidth"} ![The unit disk $\Omega$ and its image $\tilde{\Omega}$ (in blue) under the computed quasi-conformal mapping $\Phi^{(A_0)}$ for $A_0^1$ (upper left), $A_0^2$ (upper right), $A_0^3$ (lower left), $A_0^4$ (lower right).[]{data-label="OmegaTilde"}](Ellipse_Fig_4_1.jpg "fig:"){width=".35\textwidth"}
The discontinuous radially symmetric isotropic conductivity $\sigma$ is defined by $$\sigma_M(x) = \left\{
\begin{array}{ll}
1, & 0.5<|x|\leq 1\\
M, & 0\leq |x| <0.5
\end{array} \right.$$
Four anisotropic conductivity distributions were then constructed by defining $A_1(x) \equiv \sigma_{1.3}(x)A_0^1$, $A_2(x) \equiv \sigma_{1.3}(x)A_0^2$, $A_3(x) \equiv \sigma_{4}(x)A_0^3$, and $A_4(x) \equiv \sigma_{4}(x)A_0^4$. Voltage data was simulated by the method described in Section \[sec:4\] with trigonometric current patterns defined by $$T_\ell^k=\left\{
\begin{array}{ll}
\cos(k\theta_\ell), & k=1,...,\frac{L}{2} \\
\sin\big((k-\frac{L}{2})\theta_\ell\big), & k=\frac{L}{2}+1,...,L-1,
\end{array}\right.
\label{eq:currpatt}$$ where $T_\ell^k$ specifies the current amplitude injected on electrode $\ell$ located at angular position $\theta_\ell$ for the $k$th pattern.
Reconstructions of $\sigma_{1.3}(x)$ with background tensor $ A_0^1$ and $ A_0^2$ are found in Figure \[fig:recons\_low\_contrast\], and reconstructions of $\sigma_{4}(x)$ with background tensor $ A_0^3$ and $ A_0^4$ are found in Figure \[fig:recons\_high\_contrast\].
![Top row: Cross sections along the x-axis of reconstructions of $\sigma_{1.3}(x)$ where $A_0 = A_0^1$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=1.8$ (left) and $R=2.0$ (right). Bottom row: Cross sections along the x-axis of reconstructions of $\sigma_{1.3}(x)$ where $A_0 = A_0^2$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=1.8$ (left) and $R=2.0$ (right). []{data-label="fig:recons_low_contrast"}](Recon_1_1p3_trunc1p8_v2.jpg "fig:"){width=".35\textwidth"} ![Top row: Cross sections along the x-axis of reconstructions of $\sigma_{1.3}(x)$ where $A_0 = A_0^1$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=1.8$ (left) and $R=2.0$ (right). Bottom row: Cross sections along the x-axis of reconstructions of $\sigma_{1.3}(x)$ where $A_0 = A_0^2$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=1.8$ (left) and $R=2.0$ (right). []{data-label="fig:recons_low_contrast"}](Recon_1_1p3_trunc2_v2.jpg "fig:"){width=".35\textwidth"}\
![Top row: Cross sections along the x-axis of reconstructions of $\sigma_{1.3}(x)$ where $A_0 = A_0^1$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=1.8$ (left) and $R=2.0$ (right). Bottom row: Cross sections along the x-axis of reconstructions of $\sigma_{1.3}(x)$ where $A_0 = A_0^2$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=1.8$ (left) and $R=2.0$ (right). []{data-label="fig:recons_low_contrast"}](Recon_1p3_1_trunc1p8.jpg "fig:"){width=".35\textwidth"} ![Top row: Cross sections along the x-axis of reconstructions of $\sigma_{1.3}(x)$ where $A_0 = A_0^1$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=1.8$ (left) and $R=2.0$ (right). Bottom row: Cross sections along the x-axis of reconstructions of $\sigma_{1.3}(x)$ where $A_0 = A_0^2$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=1.8$ (left) and $R=2.0$ (right). []{data-label="fig:recons_low_contrast"}](Recon_1p3_1_trunc2.jpg "fig:"){width=".35\textwidth"}
![Top row: Cross sections along the x-axis of reconstructions of $\sigma_{4}(x)$ where $A_0 = A_0^3$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=2.0$ (left) and $R=2.3$ (right). Bottom row: Cross sections along the x-axis of reconstructions of $\sigma_{4}(x)$ where $A_0 = A_0^4$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=2.0$ (left) and $R=2.3$ (right). []{data-label="fig:recons_high_contrast"}](Recon_1_4_trunc2.jpg "fig:"){width=".35\textwidth"} ![Top row: Cross sections along the x-axis of reconstructions of $\sigma_{4}(x)$ where $A_0 = A_0^3$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=2.0$ (left) and $R=2.3$ (right). Bottom row: Cross sections along the x-axis of reconstructions of $\sigma_{4}(x)$ where $A_0 = A_0^4$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=2.0$ (left) and $R=2.3$ (right). []{data-label="fig:recons_high_contrast"}](Recon_1_4_trunc2p3.jpg "fig:"){width=".35\textwidth"}\
![Top row: Cross sections along the x-axis of reconstructions of $\sigma_{4}(x)$ where $A_0 = A_0^3$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=2.0$ (left) and $R=2.3$ (right). Bottom row: Cross sections along the x-axis of reconstructions of $\sigma_{4}(x)$ where $A_0 = A_0^4$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=2.0$ (left) and $R=2.3$ (right). []{data-label="fig:recons_high_contrast"}](Recon_4_1_trunc2.jpg "fig:"){width=".35\textwidth"} ![Top row: Cross sections along the x-axis of reconstructions of $\sigma_{4}(x)$ where $A_0 = A_0^3$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=2.0$ (left) and $R=2.3$ (right). Bottom row: Cross sections along the x-axis of reconstructions of $\sigma_{4}(x)$ where $A_0 = A_0^4$ computed by Calderón’s method for anisotropic conductivities with truncation radius $R=2.0$ (left) and $R=2.3$ (right). []{data-label="fig:recons_high_contrast"}](Recon_4_1_trunc2p3.jpg "fig:"){width=".35\textwidth"}
Conclusions
===========
A direct reconstruction algorithm for reconstructing the multiplicative scalar field for 2-D anisotropic conductivities with known entries for the background anisotropic tensors was presented based on Calderón’s linearized method for isotropic conductivities. The quasi-conformal map was used to prove injectivity of the linearized problem in the plane. The map facilitates the reduction of the anisotropic problem to an isotropic problem that can then be solved by Calderón’s method on the image of the original domain under the mapping, and pulled back to obtain the multiplicative scalar field. The method is demonstrated on simple radially symmetric conductivities with jump discontinuity of high and low contrast. Further work is needed to determine the method’s practicality for more complicated conductivity distributions and experimental data. Also, the method presented here is not applicable to the three-dimensional case.
Acknowledgment {#acknowledgment .unnumbered}
==============
The project was supported by Award Number R21EB024683 from the National Institute Of Biomedical Imaging And Bioengineering. The content is solely the responsibility of the authors and does not necessarily represent the official view of the National Institute Of Biomedical Imaging And Bioengineering or the National Institutes of Health. Y.-H. L. is partially supported by the Ministry of Science and Technology Taiwan, under the Columbus Program: MOST-109-2636-M-009-006, 2020-2025.
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---
abstract: 'Quantum computation with quantum data that can traverse closed timelike curves represents a new physical model of computation. We argue that a model of quantum computation in the presence of closed timelike curves can be formulated which represents a valid quantification of resources given the ability to construct compact regions of closed timelike curves. The notion of self-consistent evolution for quantum computers whose components follow closed timelike curves, as pointed out by Deutsch \[Phys. Rev. D [**44**]{}, 3197 (1991)\], implies that the evolution of the chronology respecting components which interact with the closed timelike curve components is nonlinear. We demonstrate that this nonlinearity can be used to efficiently solve computational problems which are generally thought to be intractable. In particular we demonstrate that a quantum computer which has access to closed timelike curve qubits can solve NP-complete problems with only a polynomial number of quantum gates.'
author:
- Dave Bacon
bibliography:
- 'bigref.bib'
date: 'September 29, 2003'
title: Quantum Computational Complexity in the Presence of Closed Timelike Curves
---
The idea that [*information is physical*]{} has given rise to a series of discoveries which indicate that physics has much to say about the foundations of computer science. Computers which exploit coherent quantum evolution remarkably offer computational speedups over computers which evolve classically[@Shor:94a; @Grover:96a]. This discovery has lead to the development of a robust theory of computation with quantum elements: the theory of quantum computation[@Nielsen:00a]. Current theoretical work[@Aharonov:97a; @Gottesman:98a; @Kitaev:97b; @Knill:98a; @Preskill:98a; @Shor:96a] indicates that there is no fundamental physical obstacle toward the construction of a working quantum computer. The laws of physics appear to allow quantum computation.
The realization that the physicality of information has a profound effect on fundamental computer science challenges physics to understand the computational power of different physical theories. In this article we present an analysis of the consequences of one such theory. Morris, Thorne, and Yurtsever[@Morris:88a], asked the question of whether the laws of physics allow for the construction and maintenance of stable wormholes. The construction of such wormholes would necessarily lead to spacetimes with closed timelike curves(CTCs)[@Hawking:92a]. Without a theory of quantum gravity, however, there has been no conclusive resolution of the question of whether nature allows for CTCs[@Kim:91a; @Hawking:92a; @Deser:92a; @Visser:96a]. Despite this uncertainty, various authors have attempted to ascertain the status of the initial value problem on spacetimes with CTCs[@Friedman:90a; @Deutsch:91a; @Friedman:91a; @Hartle:94a; @Goldwirth:94a; @Politzer:94a; @Politzer:94b; @Cassidy:95a; @Hawking:95a]. Of particular importance in this initial value problem is the notion of self-consistent evolution[@Friedman:90a; @Deutsch:91a]. Previous arguments against the existence of CTCs which dictated that CTCs will always lead to paradoxical evolution[@Hawking:73a] now appear to be unfounded, especially in the context of quantum theory[@Deutsch:91a; @Hartle:94a]. For a given specification of initial data, there is always a self-consistent evolution of this data which does not give rise to any of the typical “patricidal paradoxes” usually associated with time travel.
In this article we examine the consequences of quantum computation in the presence of closed timelike curves. This work is complementary to work done by Brun[@Brun:02a] who demonstrated that a model of classical computation in the presence of CTCs could be used to solve hard computational problems in constant time. However, the world is not classical, and the status of the classical initial value problem in the presence of CTCs has no known generic solution[@Friedman:90a]. Thus in Brun’s model of computation, it is explicitly possible to write down programs which have no self-consistent evolution. Diverging from Brun’s approach we follow the formalism of Deutsch[@Deutsch:91a] who, soon after helping develop the theory of quantum computation, applied this formalism to the question of computation in the presence of CTCs. Deutsch was able to show that quantum computation in the presence of CTCs always allows self-consistent evolution. The evolution of chronology respecting systems interacting with systems which traverse CTCs, while locally unitary, Deutsch showed, is globally nonlinear. Nonlinearity in quantum computation has been shown by Abrams and Lloyd[@Abrams:97a] to be a powerful tool for solving hard computational problems. Both Deutsch and Brun conjectured that quantum computation in the presence of CTCs could solve hard problems. Here we show that this is indeed correct by demonstrating that the nonlinearity allowed by quantum computation in the presence of CTCs can be used to efficiently solve classically hard computational problems. We present specific cases of quantum evolution near CTCs which can be used to efficiently solve NP-complete problems. The efficient solution of such problems (the P=NP question) has long been doubted in classical computational complexity and it is also believed that quantum computers alone do not efficiently solve these important computer science problems. If nature allows for CTCs, then the theory of quantum computation in the presence of such CTCs provides for the efficient solution of computational problems previously thought to be intractable and therefore represents one of the most powerful physical models of computation known.
Given the extraordinary power of quantum computation in the presence of CTCs, however, one may wonder whether, as is the case with the similarly powerful models of analog computation[@Vergis:86a], this result is robust in the presence of noise or whether noise destroys the effect we are exploiting to solve the hard problem[^1]. This is particularly worrisome because we use nonlinear evolution to achieve an exponential increase in the distinguishablity of two nearly identical quantum systems: what is to keep the noise from growing exponentially along with this distinguishablity? We show, however, that the traditional methods of fault-tolerant quantum computation can be used to overcome at least some of the problems raised in this context. Thus, to the extent that the error mechanisms we consider encompass realistic errors for the model of quantum computation in the presence of CTCs, we find that a robust model of computation in the presence of CTCs can be formulated.
Quantum complexity theory with closed timelike curves
=====================================================
In physics, determination of the [*allowable*]{} manipulations of a physical system is of central importance. Computer science, on the other hand, has arisen in order to [*quantify*]{} what resources are needed in order to perform a certain algorithmic task. When one examines the computational consequences of a fundamental physical theory it is important that computer science’s quantification represent a reasonable application of physical resources. One such quantification of physical resources for a quantum computer is given by the quantum circuit model[@Deutsch:89a]. In the quantum circuit model, a series of gates are applied to a collection of qubits which have been prepared in an input state and are then measured to obtain the computation’s output. In order to be a realistic model of computation, it is usually assumed that there is some notion of locality among the quantum gates and further that these gates are generated by few-qubit interactions. The quantification of a quantum circuit model is then classified by the manner in which the quantum gates are used. There are various measures of complexity within this model which can be used: one can use the total number of gates, the depth of the circuit, or the breadth of the circuit. That this is a good qualification of resources has been argued elsewhere[@Deutsch:89a].
In order to deal with CTCs within the quantum circuit model, we make the simplifying assumptions enumerated by Deutsch[@Deutsch:91a] and Politzer[@Politzer:94b]: (a) The region of CTCs is a compact region of spacetime whose existence is generated by evolution from initial conditions prior to this compact region. (b) Two types of qubits in the quantum circuit model can be identified: those which traverse CTCs and those which do not. (c) Unitary evolution between the CTC qubits and the chronology respecting qubits is allowed. (d) Measurement and preparation of the CTC qubits is not allowed (see below however). (e) The evolution of the CTC qubits is determined by self-consistency. Deutsch[@Deutsch:91a] enumerates reasons for conjecturing that this model is universal in that any quantum evolution in the presence of CTCs can be mapped onto this model. Quantification of resources in this modified quantum circuit model then follows the same lines of reasoning as in the unmodified version. Now, however, gates between all qubits (CTC qubits and chronology respecting qubits) should be used in the quantification. It should be noted that it this model, the CTC qubits are a resource which cannot be reused: they are an expendable resource. Further, it is assumed that the resource cost of creating $n$ CTC qubits is not exponential in $n$. Finally, note that this quantification of resources implies that the naive method of using a CTC to perform a computation over and over again until the answer is arrived at is not quantified as a tractable use of resources. One could solve a hard problem by trying out a solution to the problem, sending one’s computer back in time, attempting a different solution to the problem, sending one’s computer back and time, etc. until a solution to the problem has been found. While only a single wormhole could be used for such an experiment and the total time required to obtain a solution will be constant, the number of computers (i.e. spatial resources) which need to exist to carry out this naive method is exponential in the problem size (when one sends a computer back in time, the previous versions of the computer still exist.) The goal of this paper is to show that a better alternative to the naive approach can actually be used to solve hard computational problems.
Self-consistent Quantum Evolution
=================================
Consider a system of $n$ qubits, $n-l$ of which are qubits which evolve along chronology respecting world lines and $l$ of which evolve along CTCs (see Fig. \[fig:st\]). The Hilbert space of these qubits is given by the tensor product ${\cal H} \equiv {\cal H}_A \otimes {\cal H}_B$ where ${\cal H}_A$ represents the chronology respecting qubits and ${\cal H}_B$ represents the qubits which traverse CTCs. Input into the quantum circuit comes from the initial conditions of the chronology respecting qubits. We now make two assumptions whose validity we discuss below: (a) There is a temporal origin of the CTCs which can be identified via the first interaction with the chronology respecting qubits. (b) The initial state of the chronology respecting and CTC qubits is initially uncorrelated. Let ${\bf U}$ be the unitary evolution operator of the entire system (made up of a series of gates), $\bmath{\rho}_{in}$ be the density matrix of the chronology respecting qubits, and $\bmath{\rho}$ be the density matrix of the CTC qubits at the temporal origin as defined above. In order to avoid logical inconsistency of quantum theory, one must invoke the principle of self-consistency: the state of the CTC qubits at the temporal origin should be the same as these same qubits after the evolution ${\bf U}$. Mathematically, we have $$\bmath{\rho}={\rm Tr}_A \left[ {\bf U} \left(\bmath{\rho}_{in} \otimes
\bmath{\rho} \right) {\bf U}^\dagger\right] \label{eq:consistent}$$ where ${\rm Tr}_A$ represents the trace over ${\mathcal H}_A$. Deutsch[@Deutsch:91a] demonstrated that there is always at least one solution $\bmath{\rho}$ to this self-consistency equation. What to do with multiple self-consistent solutions is discussed below. Given a self-consistent evolution of the CTC qubits, the output of the quantum circuit will be given by $$\bmath{\rho}_{out}={\rm Tr}_B \left[ {\bf U} \left( \bmath{\rho}_{in}
\otimes \bmath{\rho} \right) {\bf U}^\dagger \right] \label{eq:output}$$ where $\bmath{\rho}$ is a solution to Eq. (\[eq:consistent\]). Notice that the evolution from $\bmath{\rho}_{in}$ to $\bmath{\rho}_{out}$ is possibly nonlinear due to the consistency condition: the self-consistent solution to Eq. (\[eq:consistent\]) determines $\bmath{\rho}$ which in turn determines the final mapping Eq. (\[eq:output\]).
(0,0)(8,6) [(1cm,2.5cm)]{} (1.25,0.5)(1.25,1.52) (1.25,1.5)(1.25,2.5) (1.5,0.5)(1.5,1.52) (1.5,1.5)(1.5,2.5) (1.75,0.5)(1.75,1.52) (1.75,1.5)(1.75,2.5) (2.75,0.5)(2.75,1.52) (2.75,1.5)(2.75,2.5) (1.25,3.5)(1.25,4.52) (1.25,4.5)(1.25,5.5) (1.5,3.5)(1.5,4.52) (1.5,4.5)(1.5,5.5) (1.75,3.5)(1.75,4.52) (1.75,4.5)(1.75,5.5) (2.75,3.5)(2.75,4.52) (2.75,4.5)(2.75,5.5) (3.25,0.5)(3.25,1.77) (3.25,1.75)(3.25,2.5) (3.25,0.5)(5.25,0.5) (5.23,0.5)(7.25,0.5) (7.25,0.5)(7.25,3.02) (7.25,3)(7.25,5.5) (5.23,5.5)(7.25,5.5) (3.25,5.5)(5.25,5.5) (3.25,5.5)(3.25,4.48) (3.25,3.5)(3.25,4.5)
(3.5,0.75)(3.5,1.77) (3.5,1.75)(3.5,2.5) (3.5,0.75)(5.25,0.75) (5.23,0.75)(7,0.75) (7,0.75)(7,3.02) (7,3)(7,5.25) (5.23,5.25)(7,5.25) (3.5,5.25)(5.25,5.25) (3.5,5.25)(3.5,4.48) (3.5,3.5)(3.5,4.5)
(3.75,1)(3.75,1.77) (3.75,1.75)(3.75,2.5) (3.75,1)(5.25,1) (5.23,1)(6.75,1) (6.75,1)(6.75,3.02) (6.75,3)(6.75,5) (5.23,5)(6.75,5) (3.75,5)(5.25,5) (3.75,5)(3.75,4.48) (3.75,3.5)(3.75,4.5)
(4.75,1.75)(4.75,2.27) (4.75,2.25)(4.75,2.5) (4.75,1.75)(5.25,1.75) (5.23,1.75)(5.75,1.75) (5.75,1.75)(5.75,3.02) (5.75,3)(5.75,4.25) (5.75,4.25)(5.25,4.25) (5.25,4.25)(4.75,4.25) (4.75,4.25)(4.75,3.98) (4.75,3.5)(4.75,4)
(2,1.5)(2.25,1.5)(2.5,1.5) (2,4.5)(2.25,4.5)(2.5,4.5) (4,1.5)(4.25,1.6)(4.5,1.7) (4,4.5)(4.25,4.4)(4.5,4.3) (6,3)(6.25,3)(6.5,3)
(1.5,0.1)[$n$ qubits]{} (3.5,0.1)[$l$ CTC qubits]{}
Returning now to our assumptions we first discuss a problem previously unaddressed in the literature[@Deutsch:91a; @Politzer:94a; @Politzer:94b]. Above we have assumed that there is a temporal origin of the CTC evolution defined by the first interaction between the CTC qubits and the chronology respecting qubits (who have a unique time ordering). Now suppose that a gate ${\bf U}= \left( {\bf I} \otimes {\bf V}
\right) {\bf U}_0$ applied to this system. The consistency condition is then $$\bmath{\rho}={\rm Tr}_A \left[ \left( {\bf I} \otimes {\bf
V} \right) {\bf U}_0 \left(\bmath{\rho}_{in} \otimes
\bmath{\rho} \right) {\bf U}_0^\dagger\left( {\bf I} \otimes {\bf V}^\dagger \right) \right]$$ The temporal origin we have chosen for this consistency condition is now seen to be arbitrary in the following sense. Express ${\bf V}$ as a product of two evolutions ${\bf V}={\bf V}_2 {\bf V}_1$. The consistency condition is $$\bmath{\rho}_1={\rm Tr}_A \left[\left( {\bf I} \otimes {\bf
V}_2 {\bf V}_1 \right) {\bf U}_0 \left(\bmath{\rho}_{in} \otimes
\bmath{\rho}_1 \right) {\bf U}_0^\dagger \left( {\bf I} \otimes {\bf V}_1^\dagger {\bf
V}_2^\dagger \right) \right]$$ However there is no reason that the temporal origin should not be after ${\bf V}_1$ is applied such that the consistency condition is really $$\begin{aligned}
\bmath{\rho}_2&=&{\rm Tr}_A \left[ \left( {\bf I} \otimes {\bf V}_1
\right) {\bf U}_0 \left( {\bf I} \otimes {\bf V}_2 \right) \left(\bmath{\rho}_{in} \otimes
\bmath{\rho}_2 \right) \right. \nonumber \\
&&\left . \left( {\bf I} \otimes {\bf V}_2^\dagger \right)
{\bf U}_0^\dagger \left( {\bf I} \otimes {\bf V}_1^\dagger \right) \right] \\end{aligned}$$ Via Deutsch’s result, there are always self-consistent solutions to each of these different consistency conditions. However, in general these self-consistent solutions may be different and even more disturbing is that these different representations of the same physical process may lead to a different map between $\bmath{\rho}_{in}$ and $\bmath{\rho}_{out}$. This, however, is not the case. The two self-consistent solutions are related via a change of basis $\bmath{\rho}_2 = {\bf V}_2^\dagger \bmath{\rho}_1 {\bf V}_2$: the superoperator on the non-CTC qubits is therefore the same superoperator. Furthermore, the input-output relationship is unaffected by a change of basis of the CTC qubits due to the cyclic nature of the trace in Eq.( \[eq:output\]). Therefore, while the choice of a temporal origin is arbitrary up to the temporal ordering dictated by the chronology respecting system, every choice of a temporal origin results in the same input output relationship for the chronology respecting qubits.
(0,0)(8,6) [(4.5cm,2.5cm)]{} [(5.75cm,1.5cm)]{} [(5.75cm,4cm)]{} (5,0.5)(5,2) (5,1.98)(5,2.5) (5,3.5)(5,4.25) (5,4.23)(5,5.5) (6,0.5)(6,1) (6,0.98)(6,1.5) (6,2)(6,2.5) (6,3.52)(6,4) (6,4.52)(6,5) (6,4.98)(6,5.5) (6,5.5)(7,5.5) (7,5.5)(7,3) (7,3)(7,0.5) (7,0.5)(6,0.5) (4.5,1.25)(7.5,1.25)
[(0.5cm,1.5cm)]{} [(1.75cm,3cm)]{} [(1.75cm,4cm)]{} (1,0.5)(1,1) (1,0.98)(1,1.5) (1,2.5)(1,4) (1,3.98)(1,5.5) (2,0.5)(2,1) (2,0.98)(2,1.5) (2,2.5)(2,3) (2,3.52)(2,4) (2,4.52)(2,5) (2,4.98)(2,5.5) (2,5.5)(3,5.5) (3,5.5)(3,3) (3,3)(3,0.5) (3,0.5)(2,0.5) (0.5,1.25)(3.5,1.25)
(3.3,1.2)
----------
temporal
origin
----------
Next we turn to the assumption of an initially tensor product state $\bmath{\rho}_{in} \otimes \bmath{\rho}$. Politzer [@Politzer:94b] has argued that this assumption is not the most general assumption. The most general initial state which produces the correct input state is one which satisfies $\bmath{\rho}_{in}={\rm
Tr}_B \left[ \bmath{\rho}_0 \right]$ where $\bmath{\rho}_0$ is the initial state of both the chronology respecting and CTC systems. Politzer argues that it is wrong to assume that the state is initially a tensor product because the CTC system has interacted with the chronology respecting system in the CTC system’s past at any given event along the CTC qubits history. This withstanding, we note that there is always a factorizable self-consistent solution[@Deutsch:91a]. Thus if non-factorizable solutions are also allowed, the difference between these two must be in the initial value problem of the full chronology respecting plus CTC qubits. Thus the model we consider, with factorizable inputs, is at least as powerful as the non-factorizable model of Politzer.
Example of the consistency requirement
--------------------------------------
Consider the evolution of two qubits, the first chronology respecting and the second traversing a CTC under a controlled-phase gate followed by an exchange of the two qubits: ${\bf U}=|00\rangle \langle 00| +|01\rangle \langle 10| + |10\rangle \langle 01| -|11\rangle \langle 11
|$ (we use a basis where $|ab\rangle$ is the chronology respecting qubit in state $|a\rangle$ and the CTC qubit in state $|b\rangle$.) The initial state of the system is given by the general input density operator $\bmath{\rho}_{in}={1 \over 2} \left({\bf I}+ \vec{n} \cdot \vec{\bmath{\sigma}} \right)$ where $\vec{n}$ is the Bloch vector $|\vec{n}| \leq 1$ and $\vec{\bmath{\sigma}}$ is the three vector of the Pauli operators $\bmath{\sigma}_i$. Similarly, a self-consistent CTC qubit state is $\bmath{\rho}={1 \over 2} \left( {\bf I}+
\vec{m} \cdot \vec{\bmath{\sigma}} \right)$. The evolution of these qubits under ${\bf U}$ results in the consistency conditions, Eq. (\[eq:consistent\]), $$m_x=n_x n_z, \quad m_y=n_y n_z, \quad m_z=n_z.$$ In this case we see that the density operator of the CTC qubit is unique. The output of the chronology respecting qubit can similarly be calculated and found to be $$\bmath{\rho}_{out}={1 \over 2} \left( {\bf I} + n_z^2 n_x
\bmath{\sigma}_x + n_z^2 n_y \bmath{\sigma}_y + n_z \bmath{\sigma}_z \right)$$ Here we see that the evolution $\bmath{\rho}_{in} \rightarrow
\bmath{\rho}_{out}$ depends nonlinearly on the initial density matrix $\bmath{\rho}_{in}$.
Multiple self-consistent evolutions
-----------------------------------
In the previous example we have seen that there is a unique self-consistent solution for the CTC qubit. This, however, is not generally the case. Consider the evolution of the same two qubits, one chronology respecting and the other traversing a CTC, under a controlled-rotation gate ${\bf U}=|00\rangle \langle 00| +|01\rangle \langle 01 | +|10\rangle \langle 10 | + i|11 \rangle \langle
11 |$. Again take the initial state of the chronology respecting qubit and the CTC qubit to be ${1 \over 2}
\left( {\bf I}+\vec{n} \cdot \vec{\bmath{\sigma}} \right) \otimes {1 \over 2} \left( {\bf I}+ \vec{m} \cdot
\vec{\bmath{\sigma}} \right)$. In this case the consistency condition, Eq. (\[eq:consistent\]), yields the condition $$\begin{aligned}
m_x=0, \quad m_y=0, \quad m_z&=&{\rm unconstrained} \quad {\rm if}~n_z
\neq 1 \nonumber \\
m_x,m_y,m_z&=&{\rm unconstrained} \quad {\rm if }~n_z=1 \nonumber \\\end{aligned}$$ Clearly the CTC qubit is unconstrained. One may hope that while the CTC qubits state is unconstrained, this does not affect the observable evolution of the chronology respecting qubit. However, the output density matrix is given by $$\begin{aligned}
\bmath{\rho}_{out}={1 \over 2} &&\left( {\bf I}+ \left(n_x {1 + m_z
\over 2} + n_y {-1+m_z \over 2} \right) \bmath{\sigma}_x \nonumber
\right. \\
&&\left.+ \left( n_x {1 - m_z \over 2} + n_y {1 + m_z \over 2}\right) \bmath{\sigma}_y + n_z
\bmath{\sigma}_z \right) \nonumber \\\end{aligned}$$ We therefore see that the output of this interaction is dependent on the CTC qubit state.
Deutsch[@Deutsch:91a] has suggested that the self-consistent CTC density operator should be the chosen such that this density operator maximizes the entropy $-{\rm Tr} \left[\bmath{\rho} \ln \bmath{\rho} \right]$. We should point out, however, that this solution to choosing which self-consistent solution may itself be inconsistent: there may be multiple states maximizing the entropy which lead to different input output evolutions of the chronology respecting qubits. Further we note that there is another manner in which this consistency paradox can be alleviated: one can assume that the freedom in the density matrix of the CTC systems is an initial condition freedom. One recalls that there are initial conditions which evolve into the CTC qubit, i.e. the specification of conditions such that the compact region with CTCs is generated. It is not inconsistent to assume that some of the freedom in the initial conditions which produce this CTC qubit are exactly the freedoms in the consistency condition. Such a resolution to the multiple consistency problem puts the impetus of explaining the ambiguity on an as yet codified theory of quantum gravity. It is interesting to turn this around and to ask if understanding the conditions for a resolution of the multiple consistency problem can tell us something about the form of any possible theory of quantum gravity which admits CTCs?
Finally we note that not having a solution to the multiple self-consistent evolutions problem will not change our result concerning quantum computational complexity in the presence of CTCs. We will not encounter an ambiguity of this form in our results: any solution to the multiple self-consistent evolutions problem is compatible with our results.
Efficient solutions to NP-complete problems using closed timelike curves
========================================================================
Consider the following important example of a computation involving a single chronology respecting qubit and a single CTC qubit. In this example the unitary evolution of the two qubits is given by ${\bf U}=|00\rangle
\langle 00| + |10 \rangle \langle 01|+|11 \rangle \langle 10 | + |01 \rangle \langle 11|$ which corresponds to the process of a controlled-NOT (controlled by the chronology respecting qubit) followed by swapping the two qubits (this operation is also equivalent to two sequential controlled-NOT gates with alternating control qubits.) Again assuming the initial state to be ${1 \over 2} \left( {\bf I}+\vec{n} \cdot \vec{\bmath{\sigma}}
\right) \otimes {1 \over 2} \left( {\bf I}+ \vec{m} \cdot \vec{\bmath{\sigma}} \right)$, the evolution of the chronology respecting qubit is unambiguous if $n_x \neq 1$: $$\bmath{\rho}_{out}={1 \over 2} \left( {\bf I} + n_z^2 \bmath{\sigma}_z \right)$$ Examining this nonlinear evolution we can begin to see the power afforded by the nonlinearity. This map $n_z \rightarrow n_z^2$, when repeated, can lead to an exponential separation of states on the Bloch sphere which could normally not be distinguished. Let ${\tt S}$ denote this map $${\tt S} \left[ {1 \over 2} \left( {\bf I}+ \vec{n} \cdot
\vec{\bmath{\sigma}} \right) \right] = \left \{ \begin{array}{ll} {1
\over 2} \left( {\bf I}+ n_z^2 \bmath{\sigma}_z \right) & {\rm
if}~n_x \neq 1 \\ {\rm ambiguous} &{\rm if}~n_x=1
\end{array} \right.$$ which is not defined for $n_x=1$.
[*Efficient solutions to NP-complete problems.–*]{} The following problem is NP-complete:
> [**Satisfaction (SAT):** ]{} Given a boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$, specified in conjunctive normal form, does there exist a satisfying assignment ($\exists x | f(x)=1$)?
In order to efficiently solve this problem we will make use of the following oracle quantum gate acting on $n+1$ qubits $${\bf U}_f=\sum_{i=0}^{2^n-1} |i\rangle \langle i| \otimes \bmath{\sigma}_x^{f(i)}$$ This gate can be constructed using only polynomial resources in the size of the satisfaction problem. Without quantifying exactly how many resources are needed to enact this gate, we will simply show that only a polynomial number of queries to this quantum gate can be used in conjunction with ${\tt S}$ to solve the satisfaction problem.
The algorithm proceeds as follows. First the state $|\psi_0\rangle=\left({1
\over \sqrt{2^n}} \sum_{i=0}^{2^n-1} |i\rangle \right) \otimes
|0\rangle$ is prepared and acted upon by ${\bf U}_f$. This prepares the state ${1 \over \sqrt{2^n} }\sum_{i=0}^{2^n-1} |i \rangle \otimes
|f(i)\rangle $. The reduced density operator of the final qubit is now given by $$\bmath{\rho}_0={1 \over 2} \left( {\bf I} + \left(1 - {s
\over 2^{n-1}} \right)
\bmath{\sigma}_z \right) \label{eq:out}$$ where $s$ is the number of satisfying solutions to $f(x)=1$. We can assume that $s \neq 2^n$ for, if $s=2^n$, then we could easily solve this case by randomly querying a value of $f(x)$. We therefore wish to distinguish between $s=0$ and $0<s<2^n$.
Let $\gamma$ denote the $\bmath{\sigma}_z$ component of $\bmath{\rho}_0$. Initially this component is $\gamma=1
- {s \over 2^{n-1}}$. After applying the gate ${\tt S}$ $p>1$ times, the component of the $\bmath{\sigma}_z$ evolves to $\gamma_p= \left( 1- {s \over 2^{n-1}} \right)^{2^p}$. Notice if $s=0$, $\gamma_p=1$, and if $0<s<2^n$ then $\gamma_p$ tends to $0$ exponentially fast in $p$. After performing ${\tt S}$ $p$ times, one measures the qubit in the $\bmath{\sigma}_z$ basis. This whole procedure is then repeated $q$ times (for a total of $pq$ queries to ${\bf U}_f$). If any of the measurements during these $q$ runs yields ${\sigma}_z=-1$, then the algorithm outputs that there is a satisfying input. If none of the measurements yields $\sigma_z=-1$, then the algorithm outputs that there is no satisfying input. When there is no satisfying clause, this algorithm will always get the answer correct. When there is a satisfying clause, the algorithm will incorrectly identify this has having no satisfying clause with a probability $$P_{fail}={1 \over 2^q} \left( 1+ \left( 1- {s \over 2^{n-1}}
\right)^{2^p} \right)^q$$ With $p$ and $q$ polynomial in $n$ the probability of this algorithm failing is therefore exponentially small.
We therefore see that using the nonlinearity provided by the gate $\tt S$ we can amplify the probability in the quantum computer such that NP-complete problems can be efficiently computed. Via the definition of NP-completeness we therefore have shown that any problem in the class NP can be efficiently solved by our algorithm.
Error Correction
================
While we have demonstrated that an error-free quantum computer can, in the presence of CTCs, solve a hard problem, we do not have a fully convincing argument unless we can argue that the presence of noise or faulty components does not destroy this result. Here we argue that that the presence of noise in the system will not destroy our result.
Recall from the theory of fault tolerant quantum computation[@Aharonov:97a; @Gottesman:98a; @Kitaev:97b; @Knill:98a; @Preskill:98a; @Shor:96a] that a quantum circuit containing $p(n)$ gates can be simulated with a probability of error $\epsilon$ using $n^\prime(p(n),\epsilon)=O\left({\rm poly} \left(\log \left({p(n)\over \epsilon} \right)\right) p(n) \right)$ gates which fail with probability $p<p_{threshold}$ for some fixed $p_{threshold}$. One way to interpret this result is to say that if we want to define our density matrix up to probabilities of outcomes given by $\epsilon$, then we require an error correcting overhead $n^\prime(p(n),\epsilon)$ gates. We would like to use fault-tolerant methods on our construction of $\tt S$. The simulation of a quantum circuit in fault tolerant constructions occurs via encoding the quantum information into appropriate error correct code states and by acting with particular operations which fault-tolerantly act on this encoded quantum information. For $\tt S$ constructed above, this means that we need to have both the chronology respecting and the CTC systems evolve with encoded quantum information and for fault tolerant gates to act on both of these systems. For the chronology respecting qubits we can clearly arrange for the appropriate encoding. For the CTC qubits, however, it is not clear how to arrange for the appropriate encoding to occur. How can we use fault tolerant encoded methods when we cannot reach in to the CTC qubits and perform the appropriate encoding?
He we sketch a method to overcome this encoding difficulty. To simplify our discussion we will focus on fault tolerant methods which use the class of stabilizer codes known as Calderbank, Shor, and Steane (CSS) codes[@Calderbank:96a; @Steane:96a]. These codes are particularly nice for our construction of $\tt S$ because the encoded controlled-not operation can be implemented transversely (and hence fault-tolerantly) by a series of controlled-not’s from the encoded control qubits to the encoded target bits. The first observation which will allow us to perform fault tolerant methods on the chronology respecting and CTC qubits is to note that the consistency and evolution equations, Eq. \[eq:consistent\] and Eq. \[eq:output\], when considered over the error correcting codespace will yield identical evolution and consistency for the encoded quantum information as for the identical unencoded evolution when the fault tolerant operations preserve the codespaces. The transversal controlled-not for the CSS codes preserve the error correcting codespaces. Thus if in the unencoded evolution the CTC qubits are forced by consistency to be in the state $\bmath{\rho}$, then for the same encoded evolution, there is a self-consistent solution over the error correcting subspace which corresponds to the encoded version of $\bmath{\rho}$. The main problem then is that there may be other self-consistent evolutions which involve CTC qubits in states outside of the error correcting codespace.
The second observation we need is that there is a degeneracy in stabilizer coding which allows us to consider the full Hilbert space of the CTC qubits as divided into different [*equivalent*]{} error correcting code spaces. In particular, the error correcting subspace normally used corresponds to considering the subspace spanned by $+1$ eigenvalue eigenstates of a set of operators known as the generators of the stabilizer group[@Gottesman:97a]. However, one could equally well work with the subspace spanned by any fixed $\pm1$ eigenvalue eigenstates of the generators of the stabilizer group. Knowing the $\pm 1$ eigenvalues defines a codespace which is of equivalent error correcting capacity as the all $+1$ eigenvalue codespace. The operations which we perform for the $\pm 1$ eigenvalue codespace will be different than those if we used the all $+1$ eigenvalue codespace, but there is always an equivalent set of operations for this other $\pm 1$ eigenvalue codespace. Thus if we could encode into any one of these $\pm1$ eigenvalue codespaces, then, via our first observation, we could again guarantee correct encoded evolution. Finally we need the fact that all of the $\pm 1$ codespaces together span the entire space of unencoded qubits.
The three observations above allow us to perform the following procedure on our CTC qubits which effectively allows us to avoid the encoding problem on the CTC qubits. First, we perform fault-tolerant measurements of the stabilizer generators on the CTC qubits which put the result of these measurements in these chronology respecting qubits (as is done fault-tolerantly in [@Gottesman:97a].) Then, for all further operations, we classically control on this encoded measurement result the appropriate evolution (this needs to be done with a fault tolerant construction.) Thus, the self-consistent evolution will always be the appropriate encoded self-consistent evolution and the evolution will be the proper encoded evolution, but for the CTC qubits over a particular $\pm 1$ eigenvalue codespace.
Return now to the issue of using fault tolerance for quantum computation in the presence of CTC qubits. Notice that in our algorithm for efficiently solving NP-complete problems, we need to distinguish the $s=0$ state $\bmath{\rho}(s=0)={1 \over 2} \left( {\bf I}+\bmath{\sigma}_z\right)$ and the $s=1$ state $\bmath{\rho}(s=1)={
1 \over 2} \left( {\bf I} + \left( 1-{1 \over 2^{n-1}} \right) \bmath{\sigma}_z \right)$. Since the trace distance[@Fuchs:96a] between these two states is given by $D(\bmath{\rho}(s=0),\bmath{\rho}(s=1))={1 \over
2} {\rm Tr} \left( | \bmath{\rho}(s=0) - \bmath{\rho}(s=1)| \right)= {1 \over 2^{n-1}}$ then we clearly need to be able to use error correction to maintain at least the probability difference $\epsilon={1 \over 2^n}$. Using standard error correction this can be done using $O(\log\left(p(n) 2^n \right) p(n))=O(\log\left(p(n) n) p(n)
\right)$ faulty gates (operating below the threshold.) This is simply a polynomial increase in the size of the quantum circuit and therefore does not significantly slow down our CTC algorithm for NP problems.
A slightly more worrisome type of error is as follows. Suppose that in our algorithm the $\bmath{\rho}(s=0)$ state has a component of $\bmath{\sigma}_z$ which is different than $+1$, $\tilde{\bmath{\rho}}(s=0)={1 \over
2} \left( {\bf I}+(1-\mu) \bmath{\sigma}_z \right)$, due to some physical noise process. Then if we apply ${\tt
S}$ $p$ times to this state, it is possible, for large enough $\mu$ that our algorithm will incorrectly identify that the function has a satisfying assignment when, in fact the function does not. Suppose we run the first phase of our algorithm $p=n$ times. The state $\tilde{\bmath{\rho}}$ will then have a $\bmath{\sigma}_z$ component of $\left(1-\mu\right)^{2^n}$. Suppose $\mu > {b \over 2^{n^c}}$ for some constants $b$ and $c$ for large $n$. Then for large $n$, $$\left(1-\mu\right)^{2^n} \geq e^{-2^n \over 2^{n^c} +1 } \geq 1-{2^n \over {2^n}^c+1}.$$ If we choose some fixed $c > 1$, then for large $n$ the $\bmath{\sigma}_z$ component is exponentially close to $1$. Therefore repeated applications of ${\mathcal S}$ will improperly identify the $s=0$ case with only an exponentially small probability. This implies that we need only protect our system to accuracy $\epsilon=O\left({1 \over 2^{n^c}} \right)$ for some fixed $c$. This can be done using the threshold theorem using $O(\log(p(n))n^cp(n))$ gates, representing a polynomial slowdown, and thus error correction can be used to correct his form of error. While it is true that the nonlinearity we use to solve hard problems exponentially separates quantum states, it appears that we can design quantum error correcting circuits whose noise does not suffer a similar blowup. We have considered only limited errors in this paper, and then only as a sketch as to how more general errors can be dealt with in the presence of nonlinear quantum gates. It remains a challenge bring a fully rigorous treatment of errors in our model to be certain that our model is robust in the presence of noise. We have shown, however, that the problems for which using nonlinearly at first sight appear to be problematic are not problematic and we are thus hopeful that such a full rigorous theory could be developed.
Conclusion
==========
Since computation is physical, we need to examine physics in order to form the foundations of a theory of computational complexity. There are two direct ways in which one can go beyond the standard model of quantum computation for which physics might have something new to say about computational complexity. One possible manner to go beyond the standard model would occur if quantum theory needs to be replaced by a more fundamental theory of the evolution of physical systems. For example, proposals for deterministic nonlocal theories[@tHooft:01a; @Smolin:02a], might offer different computational complexities if the fundamental distinctions which make these models different from quantum theory are accessible. Another example is provided by Hawking’s conjecture[@Hawking:76a; @Preskill:92a] that quantum theory must be modified to solve the information paradox in black hole thermodynamics. The other path beyond the standard model of quantum computation is if the physical theories which are laid on top of quantum theory possess a computational power differing from the current understanding of these theories. For example, it is not entirely clear whether or not current versions of quantum gravity[@Smolin:03a] provide physics which is computationally equivalent to the standard model. If the physical theory of gravity is itself to provide a picture of spacetime, how does this modify the theory of computational complexity which is grossly constrained by the geometry of the computer? In this paper we have consider a possible hybrid method which will produce computational complexity which appears to be stronger than the standard model of quantum computation. We have considered that the possible existence of closed timelike curves might follow from a quantum theory of gravity and using the structure uncovered by Deutsch for how quantum theory itself must be modified in the presence of such curves to solve hard computational problems. There are of course many open issues left to be addressed, not the least whether a theory of quantum gravity exists which is compatible with CTCs. However, there are interesting possibilities which might also warrant consideration solely for their theoretical usefulness. For example, in computer science, the difference between space and time complexity is poorly understood[@Papadimitriou:95a]. Complexity classes which quantify reasonable amounts of spatial resources appear to be more powerful than complexity classes which quantify amounts of temporal resources. An obvious reason for this difference lies in the fact that spatial resources may be reused while temporal resources only be used once. Clearly if nature allows CTC’s this obstruction is partially remove. It is interesting to speculate that computational complexity with CTCs can lead to a simplified theory of computational complexity.
Finally, we would not be honest if we did not end this paper with the caveat that this work is at best a creature of eager speculation. Without a theory of quantum gravity, we cannot know whether CTCs can exist let alone whether they can be generated within the confines of the such a theory. Practical considerations are humorous at best. The surprising answer that quantum computation in the presence of CTCs is a powerful new model of quantum computation gives us reason, however, to pause and ponder the implications.
We thank Patrick Hayden for bringing to our attention concerns about error correction when using nonlinear quantum evolution. We also acknowledge useful conversations and communications with Todd Brun, Mike Cai, Aaron Denney, David Deutsch, Julia Kempe, John Preskill and Benjamin Toner. This work was supported in part by the National Science Foundation under grant EIA-0086038 through the Institute for Quantum Information.
[^1]: This point was first brought to our attention by Patrick Hayden.
|
---
abstract: 'The flow ansatz states that the single-particle distribution of a given event can be described in terms of the complex flow coefficients $V_n$. Multi-particle distributions can therefore be expressed as products of these single-particle coefficients; a property commonly referred to as factorization. The amplitudes and phases of the coefficients fluctuate from event to event, possibly breaking the factorization assumption for event-sample averaged multi-particle distributions. Furthermore, non-flow effects such as di-jets may also break the factorization assumption. The factorization breaking with respect to pseudorapidity $\eta$ provides insights into the fluctuations of the initial conditions of heavy ion collisions and can simultaneously be used to identify regions of the phase space which exhibit non-flow effects. These proceedings present a method to perform a factorization of the two-particle Fourier coefficients [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}which is largely independent of detector effects. AMPT model calculations of Pb–Pb collisions at ${\ensuremath{\sqrt{s_{\text{NN}}}}\xspace}= \SI{5.02}{TeV}$ are used to identify the smallest [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}necessary for the factorization assumption to hold. Furthermore, a possible [$\Delta\eta$]{}-dependent decorrelation effect in the simulated data is quantified using the empirical parameter $F_2^\eta$. The decorrelation effect observed in the AMPT calculations is compared to results by the CMS collaboration for Pb–Pb collisions at ${\ensuremath{\sqrt{s_{\text{NN}}}}\xspace}= \SI{2.76}{TeV}$.'
address: |
Niels Bohr Institute, University of Copenhagen\
Blegdamsvej 17, 2100 Copenhagen, Denmark
author:
- 'Christian Bourjau (for the ALICE Collaboration)'
bibliography:
- 'sources.bib'
title: 'Factorization of two-particle distributions in AMPT simulations of Pb–Pb collisions at $\mathbf{{\ensuremath{\sqrt{s_{\text{NN}}}}\xspace}} $ = 5.02 TeV'
---
Introduction
============
The Fourier coefficients of an event-sample averaged two-particle distribution are commonly described as $$\begin{aligned}
{\ensuremath{{\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)}\xspace}&= {\ensuremath{\left \langleV_n(\eta_a) V_n^*(\eta_b) \right \rangle}\xspace}, \\
\label{eq:average-written-out}
&= {\ensuremath{\left \langlev_n(\eta_a) v_n(\eta_b) e^{in(\psi_n(\eta_a) - \psi_n(\eta_b))} \right \rangle}\xspace},\end{aligned}$$ where $v_n$ are the flow coefficients and $\psi_n$ are the symmetry planes. Either of these two quantities may fluctuate from event to even due to varying initial conditions, thereby breaking the factorization of the sample average even for simulations of ideal hydrodynamics [@Gardim_2013]. By studying the factorization behavior of [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}one can therefore infer the properties of such fluctuations. Flow related analyses commonly assume that non-flow contributions decrease with an increasing $\eta$-separation of the particles. In order to minimize the impact of non-flow effects on the measurement a minimal longitudinal separation between particles, referred to as [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}, is therefore often applied. Under the assumption that non-flow effects do not factorize identically to anisotropic flow it is possible to identify regions of the phase space where non-flow effects become negligible [@1110.4809; @1002.0534]. Whether Eq. may be written in a factorized form depends on the correlations between the four quantities $v_n(\eta_a)$, $v_n(\eta_b)$, $\psi_n(\eta_a)$, and $\psi_n(\eta_b)$. These proceedings focus on the effect of symmetry plane decorrelation effects. The phases at $\eta_a$ and $\eta_b$ are commonly assumed to be correlated with each other through a common symmetry plane angle $\Psi_n$, but may fluctuate from event to event. The fluctuations are equally likely to occur in either direction which ensures that [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}is a real quantity. The observed average is attenuated due to these fluctuations which are therefore also referred to as *decorrelation* effects. If the fluctuations at $\psi_n(\eta_a)$ and $\psi_n(\eta_b)$ in Eq. exhibit a dependence on ${\ensuremath{\Delta\eta}\xspace}= \eta_a - \eta_b$ it may cause a factorization breaking of [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}.
Observable definition
=====================
The results presented here are exclusively based on the Monte-Carlo (MC) simulations and would therefore not require considering detector effects on the observables. However, in order to present a generally applicable method, the analysis presented here is constructed around an observable which is largely independent of uncorrelated detector deficiencies.
At its core, this analysis is based on the single- and two-particle distributions averaged over many events. The single-particle distribution $\hat{\rho}_1$ is given by $$\label{eq:rho1}
\hat{\rho}_1(\eta, \varphi) = \left \langle\frac{d^2N}{d\eta d\varphi} \right\rangle$$ where $N$ is the number of observed charged particles and $\varphi$ is the azimuthal coordinate. In experimental measurements any azimuthal anisotropies in $\hat{\rho}_1$ are exclusively caused by detector effects.
The distribution of particle-pairs $\hat{\rho}_2$ is given by $$\label{eq:pair-density}
\hat{\rho}_2(\eta_a, \eta_b, \varphi_a, \varphi_b) = {\ensuremath{\left \langle\frac{d^4N_{\text{pairs}}}{d\eta_a d\eta_ d\varphi_a d\varphi_b} \right \rangle}\xspace}$$ where $N_{\rm pairs}$ denotes the number of particle pairs observed at $\eta_a, \eta_b, \varphi_a, \varphi_b$.
The two distributions $\hat{\rho}_1$ and $\hat{\rho}_2$ can be used to construct the *normalized two-particle density* $r_2$ which is largely independent of uncorrelated detector effects [@Vechernin201521; @1311.3915]. $$\label{eq:r2}
r_2(\eta_a, \eta_b, \varphi_a, \varphi_b) = \frac{\hat{\rho}_2(\eta_a, \eta_b, \varphi_a, \varphi_b)}{\hat{\rho}_1(\eta_a, \varphi_a) \hat{\rho}_1(\eta_b, \varphi_b)}$$ The non-zero two-particle Fourier coefficients can then be computed by $$\label{eq:24}
{\ensuremath{{\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)}\xspace}= {\left ( \frac{1}{2\pi} \right )}^2\int_0^{2\pi} \int_0^{2\pi} r_2(\eta_a, \eta_b, \varphi_a, \varphi_b) e^{-in\varphi_a} e^{in\varphi_b} d\varphi_a d\varphi_b$$
Factorization {#sec:factorization-def}
=============
The functional form of [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}in the [($\eta_a, \eta_b$)-plane]{}is assessed with two different models. Both models focused on the flow ansatz [i.e.,]{} individual events can be described in terms of single particle distributions. Neither model attempts to describe any *non-flow* processes such as di-jets or weak decays.
Purely factorizing model (Model A) {#sec:purely-fact-model}
----------------------------------
This model assumes that the averaged two-particle coefficients may be described by a product of two identical functions $\hat{v}_n^A$ $$\begin{aligned}
{\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b) &= {\ensuremath{\left \langleV_n(\eta_a) {V_n^*(\eta_b)} \right \rangle}\xspace} \\
&= {\ensuremath{\left \langlev_n(\eta_a) v_n(\eta_b) e^{in(\psi_n(\eta_a) - \psi_n(\eta_b))} \right \rangle}\xspace} \\
\label{eq:factorization-def}
&= \hat{v}_n^A(\eta_a) \hat{v}_n^A(\eta_b)\end{aligned}$$ If $\psi_n(\eta_a)$ is always equal to $\psi_n(\eta_b)$ within each event and if the fluctuations of $v_n$ are uncorrelated along $\eta$, Eq. holds and $\hat{v}_n^A(\eta)$ is the mean value of the event-by-event flow coefficients $v_n(\eta)$. The degree to which the measured ${\ensuremath{{\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)}\xspace}$ is compatible with Model A provides a limit to the size of factorization-breaking fluctuations of the flow coefficients, event-plane decorrelations and non-flow effects.
The flow coefficients $\hat{v}_n^A(\eta)$ are fit to the observed [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}. The latter is computed as a histogram of finite bin size in $\eta_a$ and $\eta_b$. Eq. can thus be seen as a non-linear equation system $$\label{eq:equation-sys-mod-a}
V_{n\Delta}(\eta_a^i, \eta_b^j) = \hat{v}_n^A(\eta_a^i) \hat{v}_n^A(\eta_b^j)$$ where $i$ and $j$ are the bin-indices along $\eta_a$ and $\eta_b$ respectively. Graphically, Eq. can be represented as shown in Fig. \[fig:schema-model-a\] where [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}is a two-dimensional matrix and $v_n^A(\eta)$ a one-dimensional vector.
$$\begin{tikzpicture}[baseline=6.9ex, scale=0.5]
\draw(0,0) grid (5,5);
\foreach \x in {0,1,...,4} {
\draw[fill=mPurple, xshift=\x cm, yshift=1cm] (0, 0) -- (1,1) -- (0, 1) -- cycle;
}
\foreach \y in {0,1,...,4} {
\draw[fill=mOrange, xshift=3cm, yshift=\y cm] (0, 0) -- (1, 1) -- (1, 0) -- cycle;
}
\draw[->, thick, xshift=-0.5cm] (0,0) -- node[left] {$\eta_a$} (0, 5);
\draw[->, thick, yshift=-0.5cm] (0,0) -- node[below] {$\eta_b$} (5, 0);
\node[above] at (2.5, 5) {${\ensuremath{{\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)}\xspace}$};
\end{tikzpicture}
\hspace{0.8em}
=
\begin{tikzpicture}[baseline=6.9ex, scale=0.5]
\draw (0,0) grid (1,5);
\draw[fill=mPurple, yshift=1cm] (0, 0) rectangle (1,1);
\node[above] at (0.5, 5) {$\hat{v}_{n}^A(\eta_a)$};
\end{tikzpicture}
\cdot
\begin{tikzpicture}[baseline=6.9ex, scale=0.5]
\draw[yshift=2cm] (0,0) grid (5,1);
\draw[fill=mOrange, xshift=3cm, yshift=2cm] (0, 0) rectangle (1,1);
\node[above] at (2.5, 5) {$\hat{v}_{n}^A(\eta_b)$};
\node at (-0.5,0) {};
\node at (5.5,0) {};
\end{tikzpicture}$$
Solving Eq. for all points in [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}yields the “vector” $v_n^A(\eta)$ which best describes the observed data. Each value in $\hat{v}_n^A$ affects several points in the [($\eta_a, \eta_b$)-plane]{}.
Long-range decorrelating model (Model B) {#sec:long-range-decorr}
----------------------------------------
The second model presented here was suggested by the CMS collaboration, albeit based on a vastly different analysis method [@1503.01692]. The model is given by $$\label{eq:cms-model}
{\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b) = \hat{v}_n^B(\eta_a) \hat{v}_n^B(\eta_b) e^{-F_n^\eta |\eta_a - \eta_b|}$$ where the parameter $F_n^\eta$ is a measure for a $\Delta\eta = \eta_a - \eta_b$ dependent factorization breaking. Despite being empirical $F_n^\eta$ provides insights into longitudinal fluctuations during the early stages of the collision [@1011.3354; @1511.04131]. It should be noted that $\hat{v}_n^A(\eta) \neq \hat{v}_n^B(\eta)$ unless $F_n^\eta = 0$. Analogous to the previous model, the flow coefficients $\hat{v}_n^B(\eta)$ and the constant $F_n^\eta$ are found by solving $$\label{eq:equation-sys-mod-b}
V_{n\Delta}(\eta_a^i, \eta_b^j) = \hat{v}_n^B(\eta_a^i) \hat{v}_n^B(\eta_b^j) e^{-F_n^\eta |\eta_a^i - \eta_b^i|}$$ The graphical representation of Eq. is depicted in Fig. \[fig:schema-model-b\]. The exponential factor causes an attenuation of [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}along $|{\ensuremath{\Delta\eta}\xspace}|$ and is constant along $\eta_a + \eta_b$.
$$\begin{tikzpicture}[baseline=6.9ex, scale=0.5]
\draw(0,0) grid (5,5);
\foreach \x in {0,1,...,4} {
\draw[fill=mBlue, xshift=\x cm, yshift=1cm] (0, 0) -- (0.5, 0.5) -- (0.5,1) -- (0, 1) -- cycle;
}
\foreach \y in {0,1,...,4} {
\draw[fill=mGreen, xshift=3cm, yshift=\y cm] (1, 0) -- (0.5,0.5) -- (0.5, 1) -- (1, 1) -- cycle;
}
\draw[fill=mRed, xshift=3 cm, yshift=1 cm] (0, 0) -- (0.5,0.5) -- (1, 0) -- cycle;
\draw[fill=mRed, xshift=2 cm, yshift=0 cm] (0, 0) -- (0.5,0.5) -- (1, 0) -- cycle;
\draw[fill=mRed, xshift=4 cm, yshift=2 cm] (0, 0) -- (0.5,0.5) -- (1, 0) -- cycle;
\draw[->, thick, xshift=-0.5cm] (0,0) -- node[left] {$\eta_a$} (0, 5);
\draw[->, thick, yshift=-0.5cm] (0,0) -- node[below] {$\eta_b$} (5, 0);
\node[above] at (2.5, 5) {${\ensuremath{{\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)}\xspace}$};
\end{tikzpicture}
\hspace{0.8em}
=
\begin{tikzpicture}[baseline=6.9ex, scale=0.5]
\draw (0,0) grid (1,5);
\draw[fill=mBlue, yshift=1cm] (0, 0) rectangle (1,1);
\node[above] at (0.5, 5) {$\hat{v}_{n}^B(\eta_a)$};
\end{tikzpicture}
\cdot
\begin{tikzpicture}[baseline=6.9ex, scale=0.5]
\draw[yshift=2cm] (0,0) grid (5,1);
\draw[fill=mGreen, xshift=3cm, yshift=2cm] (0, 0) rectangle (1,1);
\node[above] at (2.5, 5) {$\hat{v}_{n}^B(\eta_b)$};
\node at (-0.5,0) {};
\node at (5.5,0) {};
\end{tikzpicture}
\cdot
{\color{mRed}{e^{-F_n^\eta|\eta_a - \eta_b|}}}$$
Particle pairs with a small separation in $\Delta\eta$ are commonly excluded from flow analyses as this region of the phases space is known to exhibit large non-flow contributions. Furthermore, an experimentally measured $V_{n\Delta}(\eta_a, \eta_b)$ may exhibit acceptance gaps if various detector systems are combined in order to maximize the $\eta$ coverage. Therefore, the procedure to numerically solve the equation systems in Eq. and Eq. needs to be able to be performed on arbitrary subsets of the [($\eta_a, \eta_b$)-plane]{}. A minimization of a weighted sum of squares fulfills this requirement. The weighted sum $S$ is given by $$\label{eq:fact-least-square}
S = \sum_{i,j=1}^{N_a^{\rm bin}, N_b^{{\rm bin}}} {
\left ( \frac{V_{n\Delta}(\eta_a^i, \eta_b^j) - M(\eta_a^i, \eta_b^j)}{\sigma_{n\Delta}(\eta_a^i, \eta_b^j)} \right )
}^2$$ where $N_a^{\rm bin}$ ($N_b^{\rm bin}$) represents the number of bins in the $\eta_a$ ($\eta_b$) and $M$ represent either Model A or Model B as defined in Eq. or Eq. respectively. The uncertainty associated with each point of ${\ensuremath{V_{n\Delta}}\xspace}(\eta_a^i, \eta_b^j)$ is given by $\sigma_{n\Delta}(\eta_a^i, \eta_b^j)$.
Factorization ratio {#sec:factorization-ratio-def}
-------------------
The agreement of [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}with either of the two models is assessed by means of a factorization ratio $f_n(\eta_a, \eta_b)$ defined by $$\label{eq:fact-ratio-def}
f_n(\eta_a, \eta_b) = \frac{{\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)}{M(\eta_a, \eta_b)}$$ where $M(\eta_a, \eta_b)$ represents either model with the parameters fitted to the observed [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}. Note that $f_n(\eta_a, \eta_b)$ can be computed for the entire [($\eta_a, \eta_b$)-plane]{}even if $S$ was only minimized for a subset of it.
Results
=======
Figure \[fig:V22\] presents [$V_{2\Delta}(\eta_a, \eta_b)$]{}obtained for collisions of 20–40% centrality for AMPT calculations of Pb–Pb collisions at ${\ensuremath{\sqrt{s_{\text{NN}}}}\xspace}= \SI{5.02}{TeV}$ with string melting enabled. Every two-particle Fourier coefficient in the [($\eta_a, \eta_b$)-plane]{}is computed independently with no a priori assumption about the event-by-event fluctuations. Figure \[fig:v2\_no\_gap\] (left) presents $\hat{v}_n^A(\eta)$ obtained from a fit to [$V_{2\Delta}(\eta_a, \eta_b)$]{}without the requirement of a [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}. Therefore, the factorization procedure included the short-range [$\Delta\eta$]{}region of the [($\eta_a, \eta_b$)-plane]{}which exhibits non-negligible non-flow contributions. Figure \[fig:v2\_no\_gap\] (right) shows the factorization ratio for the flow coefficients from the left panel. The short-range non-flow does not exhibit the same factorization behavior as the long-range regions. This caused the fitted solution to neither accurately describe the short-range nor the long range regions of the [($\eta_a, \eta_b$)-plane]{}highlighting the need of a [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}.
![ The two-particle Fourier coefficients [$V_{2\Delta}(\eta_a, \eta_b)$]{} from AMPT calculations of Pb–Pb collisions at with a centrality of 20–40%. The measurement of the two-particle Fourier coefficients is conducted for every point in the [($\eta_a, \eta_b$)-plane]{}independently. []{data-label="fig:V22"}](v22_eta_eta_cent3.pdf)
![ Factorization of [$V_{2\Delta}(\eta_a, \eta_b)$]{}according to Model A without the use of a [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}based on AMPT simulations. Left: The flow coefficients yielded by the minimization procedure. Right: Factorization ratio for the solution shown on the left. The found solution fails to describe the short-range and long-range pairs. []{data-label="fig:v2_no_gap"}](v2_no_gap_cent3.pdf "fig:"){width="0.5\linewidth"} ![ Factorization of [$V_{2\Delta}(\eta_a, \eta_b)$]{}according to Model A without the use of a [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}based on AMPT simulations. Left: The flow coefficients yielded by the minimization procedure. Right: Factorization ratio for the solution shown on the left. The found solution fails to describe the short-range and long-range pairs. []{data-label="fig:v2_no_gap"}](fact_ratio_ampt_no_twist_icent3_k0.pdf "fig:"){width="0.5\linewidth"}
Figure \[fig:fact\_ratio\_deta3\] (left) presents the factorization ratio for Model A if only particle pairs with $|{\ensuremath{\Delta\eta}\xspace}|>3$ are taken into account. By excluding the short-range pairs, good agreement to Model A is observed in the long-range region. However, the solution found for long-range pairs is not able to describe the short-range region of [$V_{2\Delta}(\eta_a, \eta_b)$]{}further corroborating that non-flow effects in this region of the phase space do not factorize identically to the long-range anisotropic flow.
![ Factorization ratio for Model A (left) and Model B (right) with $|{\ensuremath{\Delta\eta}\xspace}| > 3$ in both cases. Either model provides good agreement for $|{\ensuremath{\Delta\eta}\xspace}|>3$, while the short-range region is better described by Model B. []{data-label="fig:fact_ratio_deta3"}](fact_ratio_ampt_no_twist_icent3_k15.pdf "fig:"){width="0.5\linewidth"} ![ Factorization ratio for Model A (left) and Model B (right) with $|{\ensuremath{\Delta\eta}\xspace}| > 3$ in both cases. Either model provides good agreement for $|{\ensuremath{\Delta\eta}\xspace}|>3$, while the short-range region is better described by Model B. []{data-label="fig:fact_ratio_deta3"}](fact_ratio_ampt_twist_icent3_k15.pdf "fig:"){width="0.5\linewidth"}
In order to determine the minimal [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}necessary to exclude pairs originating from non-flow processes from the fitting procedure, a projection of the factorization ratios for Model A onto the [$\Delta\eta$]{}-axis is performed. The results for all analyzed centralities and for a [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}of $3$ are depicted in Fig. \[fig:deta\_projection\]. A centrality dependence for the factorization breaking in the short-range region is observed. The deviation from unity is most pronounced for the most central events, decreases to a minimum for the 20–40% centrality class and increases for more peripheral events thereafter. This centrality dependence may originate from the centrality dependence of $\hat{v}_2^A(\eta)$.
![ Projection of the factorization ratios of various centralities onto [$\Delta\eta$]{}. A [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}of $3$ was imposed during the factorization procedure with Model A. The shown uncertainties are statistical. []{data-label="fig:deta_projection"}](fact_ratio_deta_ngap15_no_twist.pdf){width="0.8\linewidth"}
Events of all centralities exhibit good agreement to Model A in the long-range region for ${\ensuremath{\Delta\eta}\xspace}> 3$. A further increase of the [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}does not significantly alter the extracted flow coefficients. A decrease of the [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}, includes short-range regions of the phase-space which are incompatible to the solution found in the long range region. Factorization procedures with a smaller [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}therefore decrease the fit quality in the long-range region. Analyses which implicitly rely on the factorization assumption to hold should thus apply a minimal longitudinal separation of ${\ensuremath{\Delta\eta_{\text{min}}}\xspace}\approx 3$ for the kinematic region here studied. The method presented here can be used to improve the precision of similar previously published results which found that the factorization assumption holds to within $10\%$ for Pb–Pb collisions at ${\ensuremath{\sqrt{s_{\text{NN}}}}\xspace}= \SI{2.76}{TeV}$ for ${\ensuremath{\Delta\eta}\xspace}> 2$ [@1203.3087].
Performing the factorization using Model B allows for the measurement of the decorrelation parameter $F_2^\eta$. The decorrelation parameters computed with the method presented here is shown in Fig. \[fig:F2\] for various [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}s and centralities. The data point for the 0–5% centrality bin is removed from the figure due to a poor statistical uncertainties. Results published by the CMS collaboration for Pb–Pb collisions at ${\ensuremath{\sqrt{s_{\text{NN}}}}\xspace}= \SI{2.76}{TeV}$ are included for comparison. The analysis used by the CMS collaboration corresponds to a [$|{\ensuremath{\Delta\eta}\xspace}|$-gap]{}of approximately $2.9$ in this analysis. The centrality dependence observed for Pb–Pb collisions is reproduced by the AMPT simulations at supporting previous model comparisons [@1511.04131] and studies of the energy dependence of $F_2^\eta$ [@1709.02301]. The centrality dependence of the decorrelation parameter is also found to reflect the centrality dependence of the short-range factorization breaking in Fig. \[fig:fact\_ratio\_deta3\]. However, a quantification of possible non-flow contributions to the centrality dependence of $F_2^\eta$ requires further research.
![Empirical decorrelation parameter $F_2^\eta$ for AMPT model calculations of Pb–Pb collisions at ${\ensuremath{\sqrt{s_{\text{NN}}}}\xspace}= \SI{5.02}{TeV}$. The simulation is compared to measurements by the CMS collaboration for ${\ensuremath{\sqrt{s_{\text{NN}}}}\xspace}= \SI{2.76}{TeV}$.[]{data-label="fig:F2"}](F_2_AMPT.pdf)
Summary
=======
Depending on the nature of the event-by-event fluctuations, the two-particle Fourier coefficients [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}may assume different functional shapes. In these proceedings two distinct models were studied: The first model is a purely factorizing one as it is implicitly assumed in most flow analyses. The second model is an extension to the former and allows for a [$\Delta\eta$]{}dependent attenuation of [${\ensuremath{V_{n\Delta}}\xspace}(\eta_a, \eta_b)$]{}. AMPT calculations of Pb–Pb collisions at ${\ensuremath{\sqrt{s_{\text{NN}}}}\xspace}= \SI{5.02}{TeV}$ were used as the basis for the presented results. The first model was use to estimate the longitudinal extent of short-range non-flow correlations under the assumption that such effects are not well described by the factorized solution found from long-range particle pairs. For the studied kinematic region a minimal [$\Delta\eta$]{}-separation of ${\ensuremath{\Delta\eta_{\text{min}}}\xspace}\approx 3$ is required for the factorization assumption to hold in the long-range region.
The second model was used to determine [$\Delta\eta$]{}-dependent decorrelation effects as they are to be expected from event-plane decorrelation effects. The empirical decorrelation parameter $F_2^\eta$ is qualitatively compatible to measurements by the CMS collaboration at ${\ensuremath{\sqrt{s_{\text{NN}}}}\xspace}= \SI{2.76}{TeV}$. This confirms previous studies suggesting that AMPT is able to reproduce the observed decorrelation effects as well as that these effects exhibit only a weak dependence on the center of mass energy [@1709.02301; @1511.04131].
The method presented here offers a new way to investigate possible non-flow contributions to the observed decorrelation effects and will help to better understand the three dimensional initial conditions of heavy ion collisions in the future.
|
---
abstract: 'The linear-$\sigma$ model has been widely used to describe the chiral phase transition. Numerically, the critical temperature $T_{c}$ of the chiral phase transition is in agreement with other effective theories of QCD. However, in the large-$N_{c}$ limit $T_{c}$ scales as $\sqrt{N_{c}}$ which is not in line with the NJL model and with basic expectations of QCD, according to which $T_{c}$ is –just as the deconfinement phase transition- $N_{c}$-independent. This mismatch can be corrected by a phenomenologically motivated temperature dependent parameter.'
address: 'Institute for Theoretical Physics, Johann Wolfgang Goethe University, Max-von-Laue-Str. 1, D–60438 Frankfurt am Main, Germany'
author:
- Achim Heinz
title: 'Phenomenological improvement of the linear-$\sigma$ model in the large-$N_{c}$ limit[^1]'
---
Introduction {#1}
============
The Quantum Chromodynamics (QCD) at finite temperature and finite density is a central topic in high energy physics. For small temperatures and densities the quark and gluon degrees of freedom are confined in hadrons. It is expected that there exists a region in the QCD phase diagram where quarks and gluons behave like a plasma (deconfinement) [@Cabibbo:1975ig; @Rischke:2003mt]. The QCD Lagrangian does not allow to directly calculate the confinement/deconfinement phase transition or the related chiral phase transition. The latter is mathematically well defined in the limit of zero quark masses, in which the QCD-Lagrangian is invariant under chiral symmetry transformation and the chiral condensate is an exact order parameter [@Casher:1979vw; @Cheng:2006qk; @Aoki:2006br]. Two effective models, the Nambu Jona-Lasino model (NJL) [@Nambu:1961tp; @Klevansky:1992qe; @Vogl:1991qt; @Hatsuda:1994pi] and the linear-$\sigma$ model [@Gell-Mann:1960ls; @Giacosa:2006tf; @Parganlija:2010fz], have been often used to study the properties of the chiral phase transition.
Beside these phenomenological approaches to QCD there is also the large-$N_{c}$ approximation [@'tHooft:1973jz; @Witten:1979kh]. The number of color degrees of freedom in the QCD Lagrangian is three, but a theory with an infinite large number of color degrees of freedom shows a behavior similar to the one of a theory with three colors. In the large-$N_{c}$ limit the gauge symmetry of the QCD is changed from $SU(3)$ to $SU(N_{c}\gg3)$. Enlarging the number of colors also leads to a modified QCD coupling $g_{QCD}$ in a way that for $N_{c}\rightarrow\infty$ the product $g_{QCD}^{2}N_{c}$ remains constant. Quarks and gluons are still present but gluons dominate the behavior of the theory. For low temperatures there still exists an confined phase, where the degrees of freedom are mesons and baryons [@Bonanno:2011yr]. For high enough $T$ it is believed that the theory is deconfined. Although also in the large-$N_{c}$ limit the theory is not solvable, it is significant simpler because only planar diagram survive.
Linear-$\sigma$ model in the large-$N_{c}$ limit {#3}
================================================
The linear-$\sigma$ model [@Gell-Mann:1960ls; @Giacosa:2006tf; @Parganlija:2010fz], is an effective theory which is able to describe the mass splitting of the pions and the sigma via spontaneous symmetry breaking. The model is built with terms which are invariant under chiral symmetry transformation. In the vacuum the chiral symmetry is spontaneously broken and the pions emerge as Goldstone bosons. In the original form there is no explicit $N_{c}$ dependency. From former studies one knows that the quark-antiquark meson masses are independent of the number of colors, but the coupling of three mesons is suppressed by a factor of $1/\sqrt{N_{c}}$ and the four mesons coupling by a factor of $1/N_{c}$ [@Witten:1979kh]. These scaling properties can be implemented by redefining the meson four point interaction $\lambda\rightarrow3\lambda
/4N_{c}$, while the parameter $\mu$ is not affected in the large-$N_{c}$ limit: $\mu\rightarrow\mu$. The Lagrangian of the $\sigma$-model as function of $N_{c}$ reads: $$\mathcal{L}_{\sigma}(N_{c})=\frac{1}{2}(\partial_{\mu}\Phi)^{2}+\frac{1}{2}%
\mu^{2}\Phi^{2}-\frac{\lambda}{4} \frac{3}{N_{c}} \Phi^{4} \text{ ,}
\label{ls1}%$$ where the scaler field $\sigma$ and the pseudoscalar pion triplet $\vec{\pi}$ are described by $\Phi^{t} = (\sigma, \vec{\pi})$. For $\mu^{2} > 0$ the chiral condensate is $\varphi_{0} = \varphi(T=0) = \mu\sqrt{N_{c}/ 3 \lambda} =
\sqrt{N_{c}/3} ~ f_{\pi}$. The only scale in the Lagrangian is $f_{\pi}$, to be identified with the pion decay constant. Note that the chiral condensate scales with $\varphi_{0} \propto\sqrt{N_{c}}$. The tree-level masses are not effected by the $N_{c}$-scaling and read $m_{\sigma}^{2} = 3
\lambda f_{\pi}^{2} - \mu^{2}$, $m_{\pi}^{2} = 0$.
The behavior at finite temperature can be analyzed using the Cornwall-Jackiw-Tomboulis (CJT) formalism [@Cornwall:1974vz]. The gap equation for the chiral condensate is $$0=\varphi(T)\left( \frac{3}{N_{c}}\lambda\varphi(T)^{2}-\mu^{2}+\frac
{9}{N_{c}}\lambda\int G_{\sigma}+\frac{9}{N_{c}}\lambda\int G_{\pi}\right)
\text{ .} \label{sigma_gap}%$$ The full propagators $G_{\sigma}$ and $G_{\pi}$ have the form: $$G_{i}=\int_{0}^{\infty}\frac{dk~k^{2}}{2\pi^{2}}\frac{1}{\sqrt{k^{2}
+m_{i}^{2}}} \left[ \exp\left( \sqrt{k^{2}+m_{i}^{2}} / T\right) -1 \right]
^{-1} \text{ .}%$$ The critical temperature $T_{c}$ is defined as the temperature where the condensate exactly vanishes $\varphi(T_{c}) = 0$. This leads to the following $N_{c}$ dependent scaling of the critical temperature: $$T_{c}(N_{c})=\sqrt{2}f_{\pi}\sqrt{\frac{N_{c}}{3}}\propto N_{c}^{1/2}\text{
.}\label{tcs}%$$ For the case $N_{c}=3$ obtains the known result $T_{c}=\sqrt{2}f_{\pi}$, but it also clearly emerges that $T_{c}$ increases with $N_{c}$. This means that for $N_{c}\gg3$ the chiral phase transition will not take place. The result is general for all hadronic models which do not include color degrees of freedom or temperature dependent parameters. This result, first noticed in Ref. [@Megias:2004hj], contradicts the results found in the NJL-model [@Klevansky:1992qe] where the critical temperature $T_c$ remains constant in the large-$N_{c}$ limit, and with the fact that the related confinement/deconfinement phase transition is expected to be proportional to $\Lambda_{QCD},$ which is a large-$N_{c}$ independent quantity.
Phenomenological modification of the linear-$\sigma$ model {#4}
==========================================================
In order to solve this discrepancy a phenomenological approach is proposed. In Refs. [@Gasser:1986vb; @Leupold:2010zz] it is argued that the $T^{2}$ scaling of order parameters is general. A phenomenological way to take this property into account is to make the parameter $\mu^{2}$ temperature dependent: $$\mu^{2}\rightarrow\mu(T)^{2}=\mu^{2}\left( 1-\frac{T^{2}}{T_{0}^{2}}\right)
\text{ .} \label{mod_mu}%$$ The parameter $T_{0}$ is a new temperature scale and should be of the order of $\Lambda_{QCD}$. The Eq. (\[mod\_mu\]) modifies the gap equation (\[sigma\_gap\]) and leads to a different critical temperature: $$T_{c}(N_{c})=T_{d}\left( 1+\frac{T_{d}^{2}}{2f_{\pi}^{2}}\frac{3}{N_{c}%
}\right) ^{-1/2}\text{ }\propto N_{c}^{0}\text{ .}%$$ The critical temperature is constant in the limit $N_{c}\rightarrow\infty$.
Now we turn back to the $N_{c} =3$ and study the case where the explicit symmetry breaking term $\epsilon\sigma$ is present. The complete Lagrangian including a second temperature scale reads: $$\mathcal{L}_{\sigma}(T_{0})=\frac{1}{2}(\partial_{\mu}\Phi)^{2}+\frac{1}{2}%
\mu^{2}\left( 1-\frac{T^{2}}{T_{0}^{2}}\right) \Phi^{2}-\frac{\lambda}%
{4}\Phi^{4}+\epsilon\sigma\text{ .}\label{ls2}%$$ All parameters are fixed via the masses and the pion decay constant ($\epsilon=f_{\pi}M_{\pi}^{2}$, $\lambda=(M_{\sigma}^{2}-M_{\pi}^{2}%
)/(2f_{\pi}^{2})$, $\mu^{2}=(M_{\sigma}^{2}-3M_{\pi}^{2})/2$). The quantity $T_{0}$ is set to a value of $T_{0}=0.27\text{~ GeV}$. The vacuum masses are chosen to be the following: the mass of the $\sigma$-field is $M_{\sigma
}=1.2\text{~ GeV}$ (for the discussion of the value of the $\sigma$ mass in the vacuum, see Refs [@d] and refs therein), of the $\pi$-field is $M_{\pi}=0.135\text{~GeV}$ and the value for the pion decay constant is $f_{\pi}=92.4\text{~MeV}$.
The finite temperature behavior for the masses, see Fig. 1, is similar to the one with no temperature dependent parameters. Until the critical temperature $T_{c}$ is reached the temperature dependency of the pion mass an the $\sigma$ mass varies slowly. Close to $T_{c}$ the mass of the $\sigma$ drops and slightly above $T_{c}$ the mass becomes degenerated with the pion mass. At high temperature both masses rise linearly.
Beside these similarities there are two remarkable properties that differ. First, the order of the phase transition is changed from a first order to a crossover phase transition. Second, the critical temperature is lowered to $T_{c}\approx200\text{~ MeV}$. Both phenomena can be seen in Fig. 2., where the case $N_{c}\rightarrow\infty$ is shown: in this limit the chiral symmetry is restored through the new temperature scale $T_{0}$ and not via mesonic loops.
Conclusions {#5}
===========
In this work the mismatch between the NJL-model and purely hadronic-models in the large-$N_{c}$ limit has been studied. We have found that the linear-$\sigma$ model implies a scaling of $T_{c}$ which is at odd with the NJL model and basic expectations [@Casher:1979vw].
In order to solve this issue we have introduced a phenomenologically motivated temperature dependent parameter. As a result the critical temperature remains constant in the large-$N_{c}$ limit. Moreover, for $N_{c}=3$ the critical temperature is lowered to $T_{c}\approx200$ MeV, a value which is in line with recent model and lattice results on the chiral phase transition.
Future studies should go beyond the simple phenomenological Ansatz presented in this work and include, for instance, the coupling of hadrons to the Polyakov loop [@Rischke:2003mt; @Dumitru:2000in]. Preliminary results [@me] show that this approach also leads to the correct large-$N_{c}$ scaling of the critical temperature $T_{c}\sim N_{c}^{0}$.
**Acknowledgment:** The author thanks F. Giacosa and D. H. Rischke for cooperation and discussions.
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[^1]: Presented by A. Heinz at the Excited QCD Workshop, 20.2.-25.2.2011, in Les Houches (France)
|
---
abstract: 'We report on an experiment demonstrating entanglement swapping of time-frequency entangled photons. We perform a frequency-resolved Bell-state measurement on the idler photons from two independent entangled photon pairs, which projects the signal photons onto a two-color Bell state. We verify entanglement in this heralded state using two-photon interference and observing quantum beating without the use of filters, indicating the presence of two-color entanglement. Our method could lend itself to use as a highly-tunable source of frequency-bin entangled single photons.'
author:
- 'Sofiane Merkouche$^{*1}$, Valérian Thiel$^2$, Alex O.C. Davis$^{2,3}$, and Brian J. Smith$^{1,2}$'
bibliography:
- 'biblio.bib'
title: 'Two-color Bell states heralded via entanglement swapping'
---
Introduction
============
Entanglement is one of the most distinguishing features of quantum mechanics, as well as being a crucial resource for quantum information science. An important technique to aid in harnessing this resource is entanglement swapping, which enables entanglement of distant quantum systems and thereby the long-range distribution of quantum correlations, in addition to shedding light on the fundamental nature and extent of quantum non-locality. Entanglement swapping has been experimentally demonstrated using photons entangled in their polarization [@Pan1998], spatial [@Zhang2017], and temporal [@Halder2007EntanglingTimemeasurement] degrees of freedom. Frequency entanglement swapping is rather more difficult due to the technical challenges involved in verifying frequency entanglement [@Barbieri2017].
Our frequency entanglement swapping experiment [@merkouche2018] is a close analogue to the time-bin experiment in reference [@Halder2007EntanglingTimemeasurement]. There, a time-resolved Bell-state measurement is performed on photons from continuously-entangled pairs, heralding a time-bin entangled Bell state. In our experiment, a frequency-resolved Bell-state measurement is performed with the use of narrowband filters, heralding a frequency-bin Bell state. The time-bin experiment, however, uses non-local Franson interference [@Franson1989] to verify entanglement in the heralded state, which for our analogue would require a time-varying phase modulation to implement a frequency shear [@Olislager2010]. Instead we use two-photon Hong-Ou-Mandel (HOM) interference, whereby observation of photon antibunching corresponds to an anti-symmetric two-photon wavefunction and confirms the presence of entanglement [@Fedrizzi2009].
Theory
======
The output state of a single spontaneous parametric down conversion (SPDC) source can be expressed as
$$\ket{\psi}=\sum_{n=0}^\infty \frac{\sqrt{\eta}^n}{n!}\Big(\int {\textrm{d}}\omega_s {\textrm{d}}\omega_i f(\omega_s, \omega_i) {\hat{a}^{\dagger}}(\omega_s) {\hat{b}^{\dagger}}(\omega_i)\Big)^n\ket{\mathrm{vac}}$$
where ${\hat{a}^{\dagger}}(\omega)\ ({\hat{b}^{\dagger}}(\omega))$ creates a photon with frequency ${\omega}$ in the signal (idler) mode, and $\eta$ is a parameter associated with the nonlinearity of the interaction and quantifies the probability of generating a photon pair from a pump photon. The function $f(\omega_s,\omega_i)$ is the normalized complex joint spectral amplitude (JSA), and its modulus squared, $|f(\omega_s,\omega_i)|^2$, is the two-photon probability density for detecting the signal photon at frequency $\omega_s$ and the idler photon at $\omega_i$. The state contains spectral entanglement when the JSA is not factorable; that is, when $f(\omega_s,\omega_i) \neq f_s(\omega_s)f_i(\omega_i)$. We assume that the process is single-mode in the polarization and transverse spatial degrees of freedom, so that only the time-frequency degrees of freedom are relevant.
The experiment we describe is depicted schematically in Fig. \[scheme\]. We consider two independent but identical SPDC sources, described by a tensor product state $\ket{\psi_1} \otimes \ket{\psi_2}$. Because this is a four-photon experiment, we restrict our attention to the second-order ($\mathcal{O}(\eta)$) contribution in the expansion. If we assume the sources to be mutually incoherent (see Supplemental), this contribution may be written as
$$\hat{\rho}_\eta = \frac{2}{3}\left(\ket{\psi_{12}}\bra{\psi_{12}} +\frac{1}{4}\ket{\psi_{11}}\bra{\psi_{11}}+\frac{1}{4}\ket{\psi_{22}}\bra{\psi_{22}}\right)$$
where $$\begin{gathered}
\ket{\psi_{mn}}=\int {\textrm{d}}\omega_s{\textrm{d}}\omega_i{\textrm{d}}\omega_s'{\textrm{d}}\omega_i' f(\omega_s,\omega_i)\hat{a}_m^{\dagger}(\omega_s)\hat{b}_m^{\dagger}(\omega_i)\\ \times f(\omega'_s,\omega'_i)\hat{a}_n^{\dagger}(\omega'_s)\hat{b}_n^{\dagger}(\omega'_i)\ket{\mathrm{vac}}
\end{gathered}$$ with $m,\ n \in \{1,2\}$, and where the subscripts 1 and 2 denote the first and second SPDC source, respectively. The first term corresponds to a single pair emission from each source, while the other two terms correspond to a double-pair emission from either one of the sources.
Entanglement swapping requires performing a partial Bell-state measurement (BSM) on the idler fields $b_1$ and $b_2$, which is achieved by interfering the fields at a 50:50 beamsplitter and performing a frequency-resolved coincident detection at the output, using narrowband filters centered at frequencies $\omega_r$ and $\omega_b$. This measurement projects the input idler fields onto the two-color singlet Bell state $\ket{\psi^-}=\frac{1}{\sqrt{2}} (\ket{\omega_r}_{b1}\ket{\omega_b}_{b2}-\ket{\omega_b}_{b1}\ket{\omega_r}_{b2})$, and the state heralded in the signal fields becomes (see Supplemental)
![Entanglement swapping scheme. Idler photons from two independent sources of frequency-entangled photon pairs are interfered at a 50-50 beamsplitter, and detected at frequencies $\omega_r$ and $\omega_b$, projecting the signal photons onto the singlet Bell state. Entanglement is verified by two-photon interference.[]{data-label="scheme"}](ES-scheme.pdf){width=".8\linewidth"}
$$\begin{gathered}
\label{state}
\hat{\rho}_H=\frac{1}{2}\ket{\Psi^-}\bra{\Psi^-}+\frac{1}{4}\ket{\Psi_1}\bra{\Psi_1}+\frac{1}{4}\ket{\Psi_2}\bra{\Psi_2}
\end{gathered}$$
where $$\begin{gathered}
\ket{\Psi^-}=\frac{1}{\sqrt{2}}\left(\ket{R}_1\ket{B}_2 - \ket{B}_1\ket{R}_2\right),\\
\ket{\Psi_j}=\ket{R}_j\ket{B}_j,
\end{gathered}$$ for $j \in \{1, 2\}$. Here we have defined $$\begin{gathered}
\ket{R}_j = \int {\textrm{d}}\omega \phi_R(\omega) {\hat{a}^{\dagger}}_j(\omega){\ket{\text{vac}}}, \\
\ket{B}_j = \int {\textrm{d}}\omega \phi_B(\omega) {\hat{a}^{\dagger}}_j(\omega){\ket{\text{vac}}},
\end{gathered}$$ where $\phi_{R(B)} (\omega) = f( \omega,\omega_{b(r)})$ can be approximated by Gaussian functions as $\phi_{R(B)}(\omega)=\exp{[-(\omega-\omega_{R(B)})^2/2\sigma^2]}$. The heralded two-color Bell-state is contained in the $\ket{\Psi^-}$ term of (\[state\]), which arises from the generation of a pair of photons in each source. The other terms, which are due to double-pair emissions, occur in all entanglement swapping experiments to date which rely on SPDC sources and linear optics [@Kok2000]. In previous experiments, the double-pair contributions are neglected because the entangled state is post-selected by four-photon coincidences. Because our entanglement verification method relies on two-photon interference of the signal fields, we are required to take these contributions into account.
![Experimental setup - see main text for description. PBS- polarizing beam splitter; PDC- parametric down conversion; SHG- second harmonic generation; APD: avalanche photodiode; FBS- fiber beamsplitter. Top: joint spectral intensities for sources 1 and 2.[]{data-label="fig:setup"}](experimental-scheme-VT.pdf){width="\linewidth"}
In order to show that the $\ket{\Psi^-}$ component of the state is indeed entangled and not merely anticorrelated, two-photon interference is used, whereby the heralded photons are detected in coincidence at the output of a 50:50 beamsplitter, as a function of relative arrival time delay $\tau$. For the input state $\rho_H$, the coincidence probability is given by (see Supplemental)
$$\label{coinc}
P_{\text{CC}}(\tau)=\big(\frac{1}{2} + \frac{1}{4}e^{-\tau^2 \sigma^2/2}\cos(\omega_B - \omega_R)\tau\big).$$
The oscillations at the frequency difference are due to coherence between the two terms of $\ket{\Psi^-}$. Note that these are observed without filtering in the signal modes. This is similar to other recent experiments producing frequency-bin entangled photons [@Ramelow2009] [@Jin2018], with the main difference being that the oscillations there are observed with unit visibility, since only the state $\ket{\Psi^-}$ contributes to coincidences. In our case, the terms due to double pair emission contribute incoherently to a background, reducing the maximum predicted visibility to 1/2. It should be noted that photon antibunching itself, where the coincidence probability exceeds that of the random baseline of 1/2, is indicative of antisymmetry, and thus entanglement, in the two-photon wavefunction [@Fedrizzi2009].
Experiment
==========
The light source for the experiment consists of ultrashort (100 fs) pulses from a titanium-doped sapphire (Ti:Sapph) laser oscillator at a central wavelength of 830 nm and a repetition rate of 80 MHz. These pulses are frequency-doubled in a 1-mm long birefringent $\mathrm{BiB_3O_6}$ crystal (BiBO) to generate the blue (415 nm) pump for the parametric downconversion (PDC) sources. PDC occurs at a second, 2.5-mm long BiBO, which is double-passed to probabilistically create a pair of frequency-entangled photons on the first pass (source 1), and on the second pass (source 2). This double-pass configuration ensures that the two sources are identical. Type II phase matching permits the deterministic separation of the signal and idler fields using polarizing beamsplitters (PBS), after the blue pump has been filtered out using dichroic mirrors. Signal and idler photons from both sources are collected into polarization-maintaining single-mode fibers (PM fibers) and directed to the remainder of the set-up for analysis and entanglement swapping. We measure a pair detection rate of 100 kHz from each source, uncorrected for losses.
![a) Simulated filtered joint spectral intensity using experimental data. b) Measured transfer functions of the filters after the idler FBS. c) Experimentally measured heralded signal state, which resembles a).[]{data-label="fig:herald"}](heralded-JSI.pdf){width="\linewidth"}
The joint spectral intensities of each source are measured efficiently using a time-of-flight spectrometer [@Davis2016], consisting of two highly dispersive chirped fiber Bragg gratings (CFBG) followed by two low-timing-jitter single photon detectors. The CFBG imparts a frequency-dependent delay on the incoming photons, thus mapping frequency onto time. Time-resolved coincidence detections at the output then provide a direct measure of the joint spectral intensity (see Fig. \[fig:setup\], top), which show frequency anticorrelation. The JSI shown in Fig. \[fig:setup\] are cropped because of the reflectivity bandwidth of the CFBG, but they extend further, since the bandwidth of the marginal spectra were measured at $9.0$ nm and $16.4$ nm full width at half maximum (FWHM) for the signal and idler photon, respectively, using a single-photon-sensitive spectrometer. Assuming negligible phase correlations [@davis2018], we estimate the amount of entanglement in the state by taking the square root of the JSI and calculating the Schmidt number [@Law2000], for which we obtain a value of $K \sim 5$.
The entangling partial BSM is performed on the idler photons by routing them to a polarization maintaining fiber beamsplitter (FBS) to ensure mode-matching and indistinguishability at the output, while temporal matching is achieved using a free-space delay line. The delays between the two sources were matched by monitoring the arrival time of the single photon packets with the fast APD, and matching them with delay lines placed in the both signal and idler paths of source 1. Sub-picosecond delay matching was subsequently achieved by monitoring the coincidences between the uncorrelated signals and idlers from both sources, at a low coincidence rate ($\sim 4$ kHz), until HOM interference is observed.
The visibility of the HOM dip was low ($< 25\%$) because the interfering photons are in mixed states due to the spectral entanglement with their respective twin photons. Since filtering the heralds increases the purity of the heralded photons, and since filters are used in the entanglement swapping configuration, we were able to obtain a lower bound on the purity of the heralded photons in the following manner. We angle-tuned our narrowband heralding filters (3 nm FWHM) to the same wavelength (830 nm), and measured the HOM dip visibility, and thus lower bound of the heralded purity, of the signal (idler) photons, when the idler (signal) photons are filtered. This measurement obtains a HOM interference visibility of 72% for the signal photons, and 80% for the idler photons, both of which are plotted in Fig. \[fig:homdips\]. For the entanglement swapping, the idler photons are detected in coincidence using SPCMs at the output of the FBS and with the narrowband filters angle-tuned to $\lambda_r=834.5$ nm and $\lambda_b=825$ nm. This measurement heralds the state $\hat{\rho}_H$ in the signal modes, with $\lambda_R=826.8\ \text{nm}$, $\sigma_\textrm{R}=2.2$ nm FWHM and $\lambda_B = 832.5\ \text{nm}$, $\sigma_\textrm{B}=2.3$ nm FWHM.
![Interferences between uncorrelated signal and idler photons. Square (blue) unheralded coincidences; Triangle (red) heralded coincidences where the other arm is spectrally filtered.[]{data-label="fig:homdips"}](heralded-hom-dips.pdf){width="\linewidth"}
To characterize the heralded state, we measure the populations in the $\ket{\Psi^-}$ subspace of the state by spectrally filtered coincidences on the separated signal fields, conditioned on the partial BSM. This measurement method was preferable to the time-of-flight spectrometer due to the prohibitive losses of the latter, combined with the low rate ($\sim$ 1 Hz) of fourfold coincidences. The results are plotted in Fig. \[fig:herald\], where frequency anticorrelations can be observed.
Finally, to verify that the heralded state is not only anticorrelated but entangled, we use two-photon interference by overlapping the signal modes at a 50:50 fiber beamsplitter and observe coincidences at the output while varying the relative arrival time $\tau$ at the input, conditioned on the partial BSM. Observation of modulation of the coincidence rate with a period of $2\pi/(\omega_B-\omega_R)$ as a function of $\tau$, as in expression (\[coinc\]), is indicative of coherence between the two terms in the $\ket{\Psi^-}$ subspace, and thus of frequency entanglement.
The two-photon interference measurements, collected over a period of 15 hours, are shown in Fig. \[fig:fringes\], where the predicted oscillations can be observed. A fit of the interferogram to the function $P(\tau)=A\big(1 + V e^{-\beta\tau^2}\cos\Delta\omega\tau\big)$ gives a period of $540\pm 30$ fs and with an envelope of $1.0 \pm 0.2$ ps, consistent with the measured values of $\omega_{R/B}$ and $\sigma_{R/B}$. The observed visibility of the oscillations V=$0.27 \pm 0.04$ is significantly less than the prediction of 0.5. This can be accounted for mostly by source mismatch, time delay drift, and beamsplitter ratio offset. However, it is sufficient to note that the central interference peak reveals anti-symmetry in the two-photon state, and hence entanglement [@Fedrizzi2009]. A second fit is also shown in Fig. \[fig:fringes\], which accounts for a relative phase in the heralded state, i.e. $\ket{\Psi^-} = \frac{1}{\sqrt{2}}(\ket{R}_1\ket{B}_2 + e^{i\phi}\ket{B}_1\ket{R}_2)$, due to a non-zero delay for the herald photons in the BSM. This manifests itself as a phase offset in the interferogram which corresponds to a temporal delay of merely $\tau_i=70$ fs and reducing fringes visibility by an additional 5%
![Two-photon interference fringes as a function of relative arrival time delay $\tau$ of the heralded signal photons at the beamsplitter, as described in Fig. \[scheme\].[]{data-label="fig:fringes"}](exptfringes.pdf){width=".85\linewidth"}
Discussion
==========
The observed two-photon interference is an unambiguous signature of the presence of frequency-bin entanglement [@Ramelow2009]. More precisely, the antibunching itself, where the coincidence probability exceeds 1/2, is a clear indication of anti-symmetric entanglement [@Fedrizzi2009]. As is the case in the time-bin experiment in reference [@Halder2007EntanglingTimemeasurement] and similar experiments [@Pan1998; @Zhang2017], our method is strictly *post-selected* entanglement swapping, where one can only claim an entangled state in the signal modes when separated detectors click in these modes. As has been noted in reference [@Kok2000], it is not correct to interpret the heralded state in equation (\[state\]) as a preparation of $\ket{\Psi^-}$ with probability $1/2$, as this interpretation commits the partition fallacy: the choice of basis in which to write the density matrix $\hat{\rho}_H$ is arbitrary and does not depend on the underlying physical processes. This is not a limitation for using this technique when the heralded photons are detected non-locally, because only the entangled component of the state contributes to such detections. On the other hand, generation of a pure heralded entangled photon pair from SPDC sources and linear optics is possible with the use of four herald photons [@Wagenknecht2010], or alternately by using a nonlinear interaction such as sum-frequency generation to jointly measure the two herald photons [@Sangouard2011FaithfulGeneration] [@Vitullo2018]. At the time of writing, a similar experiment has been reported [@Graffitti2019] where time-frequency Bell states are generated at the SPDC sources and swapped and verified using the same method we use.
acknowledgments
===============
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 665148, the United Kingdom Defense Science and Technology Laboratory (DSTL) under contract No. DSTLX-100092545, and the National Science Foundation under Grant No. 1620822.
Experimental frequency entanglement swapping: Supplemental materials
====================================================================
Deriving the heralded state
---------------------------
To derive the state $\hat{\rho}_H$ heralded by the BSM, we begin by writing the SPDC state due to two independent and identical sources as a tensor product
$$\begin{gathered}
\left\{\hat{1} + \sqrt{\eta} \int {\textrm{d}}\omega_s {\textrm{d}}\omega_i f(\omega_s,\omega_i){\hat{a}^{\dagger}}_1(\omega_s){\hat{b}^{\dagger}}_1(\omega_i) + \frac{\eta}{2}\left(\int {\textrm{d}}\omega_s {\textrm{d}}\omega_i f(\omega_s,\omega_i){\hat{a}^{\dagger}}_1(\omega_s){\hat{b}^{\dagger}}_1(\omega_i)\right)^2 + \dots \right\}\\
\otimes\left\{\hat{1} + \sqrt{\eta} \int {\textrm{d}}\omega_s {\textrm{d}}\omega_i f(\omega_s,\omega_i){\hat{a}^{\dagger}}_2(\omega_s){\hat{b}^{\dagger}}_2(\omega_i) + \frac{\eta}{2}\left(\int {\textrm{d}}\omega_s {\textrm{d}}\omega_i f(\omega_s,\omega_i){\hat{a}^{\dagger}}_2(\omega_s){\hat{b}^{\dagger}}_2(\omega_i)\right)^2 + \dots \right\}{\ket{\text{vac}}}.
\end{gathered}$$
We expand this and keep only terms of order $\eta$, which are responsible for the four-photon contribution:
$$\begin{gathered}
\ket{\Psi_{\mathrm{\eta}}}=\int {\textrm{d}}\omega_s {\textrm{d}}\omega_i {\textrm{d}}\omega_s' {\textrm{d}}\omega_i' f(\omega_s,\omega_i)f(\omega_s',\omega_i') {\hat{a}^{\dagger}}_1(\omega_s) {\hat{b}^{\dagger}}_1(\omega_i) {\hat{a}^{\dagger}}_2(\omega_s') {\hat{b}^{\dagger}}_2(\omega_i'){\ket{\text{vac}}}\\
+\frac{1}{2}\int {\textrm{d}}\omega_s {\textrm{d}}\omega_i {\textrm{d}}\omega_s' {\textrm{d}}\omega_i' f(\omega_s,\omega_i)f(\omega_s',\omega_i') {\hat{a}^{\dagger}}_1(\omega_s) {\hat{b}^{\dagger}}_1(\omega_i) {\hat{a}^{\dagger}}_1(\omega_s') {\hat{b}^{\dagger}}_1(\omega_i'){\ket{\text{vac}}}\\
+\frac{1}{2}\int {\textrm{d}}\omega_s {\textrm{d}}\omega_i {\textrm{d}}\omega_s' {\textrm{d}}\omega_i' f(\omega_s,\omega_i)f(\omega_s',\omega_i') {\hat{a}^{\dagger}}_2(\omega_s) {\hat{b}^{\dagger}}_2(\omega_i) {\hat{a}^{\dagger}}_2(\omega_s') {\hat{b}^{\dagger}}_2(\omega_i'){\ket{\text{vac}}}.
\end{gathered}$$
For convenience, we will denote these three terms $\ket{\Psi_{12}}$, $\ket{\Psi_{11}}$, and $\ket{\Psi_{22}}$, so that we have, with the proper renormalization
$$\ket{\Psi_\mathrm{\eta}} = \sqrt{\frac{2}{3}}\left(\ket{\Psi_{12}}+\frac{1}{2}\ket{\Psi_{11}} + \frac{1}{2}\ket{\Psi_{22}}\right), \label{eq:psip}$$
and the density matrix for this state is
$$\hat{\rho}_\mathrm{\eta} = \frac{2}{3}\left(\ket{\Psi_{12}}\bra{\Psi_{12}} + \frac{1}{4}\ket{\Psi_{11}}\bra{\Psi_{11}} + \frac{1}{4}\ket{\Psi_{22}}\bra{\Psi_{22}}\right) +\cancel{ \mathrm{cross\ terms}}.
\label{rhop}$$
The cross terms correspond to coherence between the terms in $\ket{\Psi_\eta}$, which is ultimately due to the optical phase of the pump. Because our sources are pumped by the same laser, we do indeed expect them to be mutually coherent. However, over the course of a measurement run (several hours), the phase drifts significantly, so it is reasonable to average over it, and thus these cross terms vanish.
A Bell state measurement is performed on the idler photons by interfering them at a beamsplitter and detecting coincidences at the output. The POVM element for this detection is given by
$$\hat{\Pi}_\mathrm{BSM} = \int {\textrm{d}}\omega {\textrm{d}}\omega' |t_r(\omega)|^2 |t_b(\omega')|^2 {\hat{c}^{\dagger}}(\omega){\hat{d}^{\dagger}}(\omega'){\ket{\text{vac}}}\bra{\mathrm{vac}}\hat{c}(\omega)\hat{d}(\omega')$$
where $t_{r(b)}(\omega)$ is a filter transmission amplitude centered at $\omega_{r(b)}$, and
$${\hat{c}^{\dagger}}(\omega) = \frac{{\hat{b}^{\dagger}}_1(\omega) + {\hat{b}^{\dagger}}_2(\omega)}{\sqrt{2}},\quad {\hat{d}^{\dagger}}(\omega') = \frac{{\hat{b}^{\dagger}}_1(\omega') - {\hat{b}^{\dagger}}_2(\omega')}{\sqrt{2}}$$
are the operators at the output of the beamsplitter. From this, we obtain the heralded state by tracing over the idler modes
$$\hat{\rho}_H = \mathrm{Tr}_b(\hat{\rho}_\eta\hat{\Pi}_\mathrm{BSM}) = \int {\textrm{d}}\omega {\textrm{d}}\omega' \bra{\mathrm{vac}}\hat{b}_1(\omega)\hat{b}_2(\omega') \hat{\rho}_\mathrm{p}\hat{\Pi}_\mathrm{BSM}{\hat{b}^{\dagger}}_1(\omega){\hat{b}^{\dagger}}_2(\omega'){\ket{\text{vac}}}\label{eq:trace}$$
We take the limit $t_{r(b)}(\omega) \rightarrow \delta(\omega-\omega_{r(b)})$, corresponding to a frequency-resolved measurement of the idler photons. Upon taking the trace, we define the functions
$$\phi_{R}(\omega) = \int {\textrm{d}}\omega' \delta(\omega-\omega_{b}) f(\omega,\omega'),\quad
\phi_{B}(\omega) = \int {\textrm{d}}\omega' \delta(\omega-\omega_r) f(\omega,\omega').$$
The frequency-resolved limit in the measurement of the idler photons implies that the signal photons are heralded in pure states described by the spectral amplitudes $\phi_{R(B)}(\omega)$. We take these to be Gaussian functions of the same width $\sigma$:
$$\phi_{R(B)}(\omega) = \frac{1}{\sqrt{\sigma\sqrt{\pi}}}\exp[-(\omega-\omega_{R(B)})^2/2\sigma^2]$$
Taking all this into account, we evaluate the trace term by term in equation \[rhop\]:
$$\begin{gathered}
\mathrm{Tr}_b(\ket{\Psi_{12}}\bra{\Psi_{12}}\hat{\Pi}_\mathrm{BSM}) = \frac{1}{2}\ket{\Psi^-}\bra{\Psi^-};\\
\ket{\Psi^-} = \frac{1}{\sqrt{2}}\int {\textrm{d}}\omega_1 {\textrm{d}}\omega_2 \big(\phi_R(\omega_1)\phi_B(\omega_2) - \phi_R(\omega_2)\phi_B(\omega_1)\big){\hat{a}^{\dagger}}_1(\omega_1){\hat{a}^{\dagger}}_2(\omega_2){\ket{\text{vac}}},
\end{gathered}$$
which gives the swapped entangled state, and
$$\begin{gathered}
\mathrm{Tr}_b(\ket{\Psi_{11(22)}}\bra{\Psi_{11(22)}}\hat{\Pi}_\mathrm{BSM}) = \ket{\Psi_{1(2)}}\bra{\Psi_{1(2)}};\\
\ket{\Psi_{1(2)}} = \int {\textrm{d}}\omega {\textrm{d}}\omega' \phi_R(\omega)\phi_B(\omega'){\hat{a}^{\dagger}}_{1(2)}(\omega){\hat{a}^{\dagger}}_{1(2)}(\omega'){\ket{\text{vac}}}\end{gathered},$$
which correspond to a pair of photons in each arm, one red and one blue. Combining these and renormalizing, we arrive at the heralded state
$$\hat{\rho}_{H} = \frac{1}{2}\ket{\Psi^-}\bra{\Psi^-} + \frac{1}{4}\ket{\Psi_1}\bra{\Psi_1} + \frac{1}{4}\ket{\Psi_2}\bra{\Psi_2}.$$
Calculating the coincidence probability
---------------------------------------
Given the heralded state $\hat{\rho}_H$ of the signal photons, conditioned on the Bell-state measurement on the idler photons, we can calculate the coincidence probability after the second beamsplitter. The measurement operator for an unresolving coincidence detection at the output of the beamsplitter is given by
$$\hat{\Pi} = \int {\textrm{d}}\omega {\textrm{d}}\omega' {\hat{x}^{\dagger}}(\omega){\hat{y}^{\dagger}}(\omega'){\ket{\text{vac}}}\bra{\mathrm{vac}}\hat{x}(\omega)\hat{y}(\omega'),$$
with
$${\hat{x}^{\dagger}}(\omega) = \frac{{\hat{a}^{\dagger}}_1(\omega) + e^{i\omega\tau}{\hat{a}^{\dagger}}_2(\omega)}{\sqrt{2}},\quad {\hat{y}^{\dagger}}(\omega') = \frac{{\hat{a}^{\dagger}}_1(\omega') - e^{i\omega'\tau}{\hat{a}^{\dagger}}_2(\omega')}{\sqrt{2}},$$
where a relative phase is obtained from a relative time delay $\tau$.
We recall that the single-photon wave packets are Gaussian functions normalized such that $\int {\textrm{d}}\omega |\phi_{R(B)}(\omega)|^2 = 1$. The coincidence probability is then given by
$$\begin{aligned}
P_{cc}(\tau) &= \mathrm{Tr}(\hat{\rho}_H\hat{\Pi}) = \int {\textrm{d}}\omega {\textrm{d}}\omega' \bra{\mathrm{vac}} \hat{a}_1(\omega)\hat{a}_2(\omega')\ \hat{\rho}_H\hat{\Pi}\ {\hat{a}^{\dagger}}_1(\omega){\hat{a}^{\dagger}}_2(\omega'){\ket{\text{vac}}}\nonumber \\
&= \frac{1}{4} + \frac{1}{4}e^{-\tau^2\sigma^2/2}\cos(\omega_B - \omega_R)\tau + \frac{1}{8}+\frac{1}{8} \nonumber \\
&=\ \frac{1}{2} + \frac{1}{4}e^{-\tau^2\sigma^2/2}\cos(\omega_B - \omega_R)\tau.\end{aligned}$$
Full model
----------
The previous derivation assumes that the inteference filters at the output of the idler FBS are delta functions that herald a pure state in the signal arm. In reality, these filters have a finite bandwidths, and therefore a more complete model should take into account the overlap between the modes at both beamsplitters. In this section we show the derivation that leads to a full description of the process. We also consider that both sources might have a different joint spectrum, leading to decrease in visibility.
In that model, equation still holds, but the individual terms are defined as:
$$\begin{aligned}
\ket{\Psi_{nm}}=\int {\textrm{d}}\omega_s {\textrm{d}}\omega_i {\textrm{d}}\omega_s' {\textrm{d}}\omega_i' f_n(\omega_s,\omega_i)f_m(\omega_s',\omega_i') {\hat{a}^{\dagger}}_1(\omega_s) {\hat{b}^{\dagger}}_1(\omega_i) {\hat{a}^{\dagger}}_2(\omega_s') {\hat{b}^{\dagger}}_2(\omega_i'){\ket{\text{vac}}},\end{aligned}$$
where we keep track of the source joint spectral amplitude. Computing the heralded state using Eq. now takes into account a certain filter transfer function as opposed to projecting in a single frequency bin. The filtered joint spectrum for source $n$ (where $n=1,2$) consists of two individual distributions $\phi_{A,n}(\omega_s,\omega_i)$ and $\phi_{B,n}(\omega_s,\omega_i)$ which contain a spectral phase (mostly linear phase, which is temporal delay). Following the development to the end under these considerations, we then obtain the coincidence probability:
$$\begin{aligned}
P_{cc} = \frac{1}{2}+\frac{1}{4} \textrm{Re}\Big\{ \int {\textrm{d}}^2\omega\ \phi_{A,1}^\ast(\omega_s,\omega_i) \phi_{A,2}(\omega_s,\omega_i) \int {\textrm{d}}^2\omega\ \phi_{B,1}(\omega_s,\omega_i) \phi_{B,2}^\ast(\omega_s,\omega_i) \Big\}
$$
which clearly shows that the interference pattern is defined by overlap integrals between the two individual spectral distributions from each source. By identifying these distributions as centered around their respective center frequencies on both signal and idler and considering some delay for each path of the interferometer, it is possible to show that the overlap integrals correspond to cross-correlations in time. This allows to model how mismatch between the two sources are impacting the measurement. Ultimately, the most important parameter that affects visibility of the fringes is found to be the delay between the two idler photons that are used to herald the measurement.
![a) Simulated coincidence probability as a function of delay in the signal arm, using experimental parameters and for different delays on the heralding (idler) arm. The delays corresponding to phase offset $\pi$ and $2\pi$ are respectively 130 and 260 fs. b) Heatmap of the interferogram of a) with respect to both signal and idler delays. The contours correspond to 90%, 50% and 10% of the envelope intensity.[]{data-label="fig:fringes"}](sim-fringes.pdf){width=".9\linewidth"}
By considering that the distributions have similar center wavelength $(\omega_{R,B},\omega_{b,r})$ and relative delays $\tau_s,\tau_i$, respectively for the signal and idler, the coincidence probability is simply given by:
$$\begin{aligned}
P_{cc}(\tau_s,\tau_i) = \frac{1}{2}+\frac{1}{4} \mathcal{E}_s(\tau_s)\mathcal{E}_i(\tau_i) \cos[\Delta\Omega\ \tau_i - \Delta\omega\ \tau_s] \label{eq:fringesfull}
$$
where $\mathcal{E}_{s,i}$ denote respectively the envelope of the cross correlation function between the two sources spectral distributions projected on the signal or idler axis and $\Delta\omega = \omega_B-\omega_R$, $\Delta\Omega = \omega_b-\omega_r$ are the frequency difference between the distributions. Note that this model works for non filtered sources where the frequency difference is zero on both dimensions.
The coincidence probability is plotted in the $(\tau_s,\tau_i)$ plane in Fig.\[fig:fringes\]b) for Gaussian distributions with moments that correspond to the experimental values. It shows that, for a temporal offset from zero in the idler arm, the fringes visibility and phase are affected. This is depicted clearly in Fig.\[fig:fringes\]a) where we plotted the probability for values of the idler delay of $0, 130, 260$ fs which correspond to a phase offset in of $0,\pi$ and $2\pi$. The fringes visibility then drops from 50% to 25% because of the envelope of the cross correlation function of the idler distributions. This shows how proper delay matching between the two independent idler photons is critical. The phase offset was observed during the experimental acquisition of the coincidence probability due to the difficulty to identify the location of the HOM dip in the idler arm of the interferometer. Moreover, we can see that optical path fluctuations that occur easily during a long acquisition at room temperature will result in a loss of contrast.
Note that the simulation from Fig.\[fig:fringes\] takes into account the background level due to two photon events from each sources but does not plot the high frequency fringes. These are incorporated in the model by adding an oscillating term $\cos[\Omega_+\ \tau_i - \omega\_+ \tau_s]$ where $\omega_+ = \omega_B+\omega_R$, $\Omega_+ = \omega_b+\omega_r$. Since that interferometric term is oscillating at a much higher frequency, any fluctuation in optical path would average it to zero, and hence it is reasonable to neglect that term.
|
---
abstract: |
Distributed learning platforms for processing large scale data-sets are becoming increasingly prevalent. In typical distributed implementations, a centralized master node breaks the data-set into smaller batches for parallel processing across distributed workers to achieve speed-up and efficiency. Several computational tasks are of sequential nature, and involve multiple passes over the data. At each iteration over the data, it is common practice to randomly re-shuffle the data at the master node, assigning different batches for each worker to process. This random re-shuffling operation comes at the cost of extra communication overhead, since at each shuffle, new data points need to be delivered to the distributed workers.
In this paper, we focus on characterizing the information theoretically optimal communication overhead for the distributed data shuffling problem. We propose a novel coded data delivery scheme for the case of no excess storage, where every worker can only store the assigned data batches under processing. Our scheme exploits a new type of coding opportunity and is applicable to any arbitrary shuffle, and for any number of workers. We also present information theoretic lower bounds on the minimum communication overhead for data shuffling, and show that the proposed scheme matches this lower bound for the worst-case communication overhead.
author:
- |
Mohamed Adel Attia Ravi Tandon\
Department of Electrical and Computer Engineering\
University of Arizona, Tucson, AZ, 85721\
Email: *{madel, tandonr}*@email.arizona.edu
bibliography:
- 'allerton2016-arxiv.bib'
nocite: '[@*]'
title: 'On the Worst-case Communication Overhead for Distributed Data Shuffling'
---
Introduction {#sec:Introduction}
============
Processing of large scale data-sets over a large number of distributed servers is becoming increasingly prevalent. The parallel nature of distributed computational platforms such as Apache Spark[@ZaChFrShSt2010], Apache Hadoop [@ShKuRaCh2010], and MapReduce [@DeGh2004] enables the processing of data-intensive tasks common in machine learning and empirical risk analysis. In typical distributed systems, a centralized node which has the entire data-set assigns different parts of the data to distributed workers for iterative processing.
Several practical computational tasks are inherently sequential in nature, in which the next iteration (or pass over the data) is dependent on the previous iteration. Of particular relevance are sequential optimization algorithms such as incremental gradient descent, stochastic gradient descent, and random reshuffling. The convergence of such iterative algorithms depends on the order in which the data-points are processed, which in turn depends on the skewness of the data. However, the *preferred ordering* of data points is unknown apriori and application dependent. One commonly employed practice is to perform *random reshuffling*, which involves multiple passes over the whole data set with different orderings at each iteration. Random reshuffling has recently been shown to have better convergence rates than stochastic gradient descent [@RR-arxiv-2015; @IoSz:2015].
Implementing random reshuffling in a distributed setting comes at the cost of an extra communication overhead, since at each iteration random data assignment is done for the distributed workers, and these data points need to be communicated to the distributed workers. This leads to a fundamental trade-off between the communication overhead, and storage at each worker. On one extreme case when each worker can store the whole data-set, no communication is necessary for any shuffle. On the other extreme, when the workers are just able to store the batches under processing, which is refereed to as the *no-excess storage* case, the communication overhead is expected to be maximum.
***Main Contributions:*** The main focus of this work is characterizing the information theoretic optimal communication overhead for the *no-excess storage* case. The main contributions of this paper are summarized as follows:
$\bullet$ We present an information theoretic formulation of the problem, and develop a novel approach of describing the communication problem through a shuffling matrix which describes the data-flow across the workers.
$\bullet$ We next present a novel coded-shuffling scheme which exploits a new type of coding opportunity in order to reduce the communication overhead, in contrast to existing approaches. Our scheme is applicable to any arbitrary shuffle, and for any number of distributed workers.
$\bullet$ We present information theoretic lower bounds on the communication overhead as a function of the shuffle matrix. Moreover, we show that the proposed scheme matches this lower bound on the worst-case communication overhead, thus characterizing the information theoretically optimal worst-case communication necessary for data shuffling.
***Related work:*** The benefits of coding to reduce communication overhead of shuffling were recently investigated in [@Kannan-2015], which proposes a probabilistic coding scheme. However, [@Kannan-2015] focuses on using the excess storage at the workers to increase the coding opportunities and reduce the average communication overhead. In our recent work [@AtRa:GC2016], we presented the optimal worst-case communication overhead for any value of storage for two and three distributed workers. In another interesting line of work, Coded MapReduce has been proposed in [@LiMaAv:2015], to reduce the communication between the mappers and reducers. However, the focus of this paper is significantly different, where we study the communication between the centralized master node and the distributed workers, motivated by the random reshuffling problem as initiated in [@Kannan-2015].
System Model {#sec:Model}
============
We consider a master-worker distributed system, where a master node possesses the entire data-set. The master node sends batches of the data-set to the distributed workers over a shared link in order to locally calculate some function or train a model in a parallel manner. The local results are then fed-back to the master node, for iterative processing. In order to enhance the statistical performance of the learning algorithm, the data-set is randomly permuted at the master node before each epoch of the distributed algorithm, and then the shuffled data-points are transmitted to the workers.
We assume a master node which has access to the entire data-set $A=[x_1^T,x_2^T,\ldots,x_N^T]^T$ of size $Nd$ bits, i.e., $A$ is a matrix containing $N$ data points, denoted by $x_1,x_2,\ldots,x_N$, where $d$ is the dimensionality of each data point. Treating the data points $\{x_n\}$ as independent and identically distributed (i.i.d.) random variables, we have
$$\begin{aligned}
H(x_n)=d, \;\;\forall n\in\{1,\ldots,N\}, \quad H(A)=Nd.\label{eq:data-set}\end{aligned}$$
At each iteration, indexed by $t$, the master node divides the data-set $A$ among $K$ distributed workers, given as $A^{t}_{1}, A^{t}_{2}, \ldots, A^t_K$, where the batch $A^t_k$ is designated to be processed by worker $w_k$, and these batches correspond to the random permutation of the data-set, $\pi^t:A\rightarrow\{A^{t}_{1}, \ldots, A^t_K\}$. Note that these data chunks are disjoint, and span the whole data-set, i.e.,
\[eq:data-batches\] $$\begin{aligned}
&A^t_i \cap A^t_j = \phi, \quad \forall i\neq j,\\
&A^t_1 \cup A^t_2 \cup \ldots \cup A^t_K =A, \quad \forall t.\label{eq:data-partitions}\end{aligned}$$
Hence, the entropy of any batch $A^t_k$ is given as $$\begin{aligned}
\label{eq:data-batches2}
H(A^t_k)= \frac{1}{K} H(A)= \frac{N}{K}d\quad ,\forall k\in\{1,\ldots,K\}.\end{aligned}$$
After getting the data batch, each worker locally computes a function (as an example, this function could correspond to the gradient or sub-gradients of the data points assigned to the $k$th worker) $f_k(A^t_k)$, in iteration $t$,. The local functions from the $K$ workers are processed later at the master node, to get an estimate of the function $f_t(A)$. For processing purposes, the data block $A^t_k$ is needed to be stored by the worker while processing, therefore, we assume that worker $w_k$ has a cache $Z^t_k$ with storage capability of size $sd$ bits, for some real number $s$, that must at least store the data block $A^t_k$ at time $t$, i.e., if we consider $Z^t_k$ and $A^t_k$ as random variables then the storage constraint is given by $$\begin{aligned}
\label{eq:cache-storage}
H(Z^t_k)=sd \geq H(A^t_k),\qquad \forall k\in\{1,\ldots,K\}.\end{aligned}$$
For the scope of this paper, we focus on the setting of *no-excess storage*, corresponding to $s= N/K$, in which each worker can exactly store $1/K$ fraction of the entire data, i.e., it only stores $s= N/K$ data points which are assigned to it in that iteration, therefore, the cache content at time $t$ for worker $w_k$ is given by $Z_k^t=A_k^t$, and the relationship in (\[eq:cache-storage\]) is satisfied with equality. Henceforth, we drop the notation $Z^t_k$ as the cache content and use the notation for the data batch $A^t_k$ instead since they are the same for the no-excess storage setting. In the next epoch $t+1$, the data-set is randomly reshuffled at the master node according to the random permutation $\pi^{t+1}: A\rightarrow\{A^{t+1}_1, A^{t+1}_2, \ldots, A^{t+1}_K\}$.
The main communication bottleneck occurs during since the master node needs to communicate some function of the data to all the workers $X_{(\pi_t,\pi_{t+1})}$ of size $R_{(\pi_t,\pi_{t+1})}d$ bits, where $R_{(\pi_t,\pi_{t+1})}$ is the rate of the shared link based on the shuffle $(\pi_t,\pi_{t+1})$. Each worker $w_k$ should be able to extract the data points designated for it out of the incoming data, $X_{(\pi_t,\pi_{t+1})}$ from the master node as well as its locally stored data, i.e., $A^{t}_k$.
We next proceed to describe the data delivery mechanism, and the associated encoding and decoding functions. The main process then can be divided into 2 phases, namely the data delivery phase and the storage update phase as described next: in the *data delivery phase*, the master node sends some function of the data to all the workers. Each worker should be able to extract the data points designated for it out of the incoming data from the master node as well as the data stored in its local cache storage. In the *cache update phase*, each worker stores the required data points for processing purposes, that can also be useful in reducing the communication overhead in subsequent epochs. At time $t+1$, the master node sends a function of the data batches for the subsequent shuffles $(\pi_t,\pi_{t+1})$, $X_{(\pi_t,\pi_{t+1})} = \phi(A^{t}_1, \ldots, A^{t}_K,A^{t+1}_1,\ldots, A^{t+1}_K)=\phi_{(\pi_t,\pi_{t+1})}(A)$ over the shared link, where $\phi$ is the data delivery encoding function $$\phi: \left[2^{\frac{N}{K}d}\right]^{2K} \rightarrow [2^{R_{(\pi_t,\pi_{t+1})}d}].$$ Since $X_{(\pi_t,\pi_{t+1})}$ is a function of the data set $A$, we have
$$\begin{aligned}
&H\left(X_{(\pi_t,\pi_{t+1})}|A\right)=0,\label{eq:transmit-const}\\
&H\left(X_{(\pi_t,\pi_{t+1})}\right) =R_{(\pi_t,\pi_{t+1})}d.\label{eq:transmit-load}
\end{aligned}$$
Each worker $w_k$ should decode the desired batch $A^{t+1}_k$ out of the transmitted function $X_{(\pi_t,\pi_{t+1})}$, and the data stored in the previous time slot denoted as $A^{t}_k$. Therefore, the desired data is given by $A^{t+1}_k =\psi(X_{(\pi_t,\pi_{t+1})}, A^{t}_k)$, where $\psi$ is the decoding function at the workers $$\psi: [2^{R_{(\pi_t,\pi_{t+1})}d}]\times [2^{sd}]\rightarrow [2^{\frac{N}{K}d}],$$ which also gives us the *decodability constraint* as follows $$\label{eq:decoding-const}
H\left(A^{t+1}_k|A^{t}_k, X_{(\pi_t,\pi_{t+1})}\right)=0 \quad ,\forall k\in\{1,\ldots,K\}.$$ The update procedure for the no-excess storage setting is rather straightforward: worker $w_k$ keeps the part that does not change in the new shuffle, i.e., $A_k^{t+1}\cap A_k^{t}$. Then it removes the remaining part of its previously stored content, i.e., $A_k^{t}\setminus A_k^{t+1}$, and stores instead the new part, i.e, $A_k^{t+1}\setminus A_k^{t}$.
Our goal in this work is to characterize the information theoretic bounds for optimal communication overhead $R_{(\pi_t,\pi_{t+1})}^*(K)$ for any arbitrary number of workers $K$, and any arbitrary shuffle $(\pi_t,\pi_{t+1})$, defined as $$R_{(\pi_t,\pi_{t+1})}^*(K)=\underset{(\phi,\psi)}{\min}\quad \;R_{(\pi_t,\pi_{t+1})}^{(\phi,\psi)}(K),$$ where $R^{(\phi,\psi)}_{(\pi_t,\pi_{t+1})}(K)$ is the rate of an achievable scheme defined by the encoding, and decoding functions $(\phi,\psi)$. Subsequently, the optimal worst-case overhead is defined as $$\begin{aligned}
&R_{\textsf{worst-case}} ^*(K)=\underset{(\pi_{t},\pi_{t+1})}{\max}\quad \;R_{(\pi_t,\pi_{t+1})}^*(K).
$$
Properties of Distributed Data Shuffling
========================================
Before presenting our main results on the communication overhead of shuffling, we present some fundamental properties that are satisfied for any two consecutive data shuffles give by $\pi_t: A \rightarrow \{A^t_1,\ldots ,A^t_K\}$, and $\pi_{t+1}: A\rightarrow \{A^{t+1}_1,\ldots ,A^{t+1}_K\}$. We start with the following definitions.
We define $$\begin{aligned}
\label{eq:S-defn}
S^{(\pi_t,\pi_{t+1})}_{i,j} \triangleq |A^{t}_{i}\cap A^{t+1}_j|,\end{aligned}$$ as the shuffle index representing the number of data points that are needed by worker $w_j$ at time $t+1$, and are available at worker $w_i$ from the previous shuffle $t$.
We also define the $K\times K$ shuffle matrix for the permutation pair $(\pi_t,\pi_{t+1})$ as $$\begin{aligned}
\label{eq:shuffle-matrix}
S^{(\pi_t,\pi_{t+1})}\triangleq \big[S^{(\pi_t,\pi_{t+1})}_{i,j}\big], \quad i,j\in\{1,\ldots,K\}.\end{aligned}$$
\[remark1\] The significance of $S^{\pi_t,\pi_{t+1}}_{i,i}$ is that it is the number of common data points between $A^t_i$, and $A^{t+1}_{i}$. Thus, these number of data points do not need to be transmitted to worker $w_i$, and are not involved in the data delivery process. Using the definition in (\[eq:S-defn\]), together with (\[eq:data-batches\]), it follows readily that $$\begin{aligned}
\label{eq:Sij-property}
&\overset{K}{\underset{i=1}{\sum}}S^{(\pi_t,\pi_{t+1})}_{i,j} = \sum_{i=1}^{K}|A^{t}_{i}\cap A^{t+1}_j| = |A^{t+1}_{j}| = \frac{N}{K},\nonumber\\
&\overset{K}{\underset{j=1}{\sum}}S^{(\pi_t,\pi_{t+1})}_{i,j} = \sum_{j=1}^{K}|A^{t}_{i}\cap A^{t+1}_j| = |A^{t}_{i}| = \frac{N}{K}.\end{aligned}$$ The properties in (\[eq:Sij-property\]) imply that the sum of elements across any row (or column) for the shuffling matrix $S^{\pi_t,\pi_{t+1}}$ is constant for any shuffle $(\pi_t,\pi_{t+1})$ and is equal to $\frac{N}{K}$.
We next state an important property satisfied by any shuffle, namely the data-flow conservation property: $$\label{eq:dataflow-conservation}
\underset{j\in \{1,\ldots,K\}\setminus i}{\sum}S^{(\pi_t,\pi_{t+1})}_{j,i}= \underset{j\in \{1,\ldots,K\}\setminus i}{\sum}S^{(\pi_t,\pi_{t+1})}_{i,j}.$$ The proof of this property follows directly from (\[eq:Sij-property\]), and has the following interesting interpretation: the total number of new data points that need to be delivered to worker $w_i$ (and are present elsewhere), i.e., $\sum_{j\neq i}S^{(\pi_t,\pi_{t+1})}_{j, i}$ is exactly equal to the total number of data points that worker $w_i$ has that are desired by the other workers, which is $\sum_{j\neq i}S^{(\pi_t,\pi_{t+1})}_{i, j}$.
We define the leftover index as the number of leftover data-points needed by worker $w_j$ at time $t+1$ and available at $w_i$ at time $t$ as $$\begin{aligned}
\label{eq:leftover-defn}
\Omega^{\pi_t,\pi_{t+1}}_{i,j}\triangleq S^{\pi_t,\pi_{t+1}}_{i,j}-\min(S^{\pi_t,\pi_{t+1}}_{i,j},S^{\pi_t,\pi_{t+1}}_{j,i}).\end{aligned}$$ The leftover matrix for the permutation pair $(\pi_t,\pi_{t+1})$ is defined as $$\begin{aligned}
\label{eq:leftover-matrix}
\Omega^{\pi_t,\pi_{t+1}}\triangleq [\Omega^{\pi_t,\pi_{t+1}}_{i,j}], \quad i,j\in\{1,\ldots,K\}.\end{aligned}$$
This definition and the significance of the leftover matrix will become clear in the subsequent sections, when we describe our proposed coded data delivery scheme. From the definition in (\[eq:leftover-defn\]), we note that the diagonal entries of the leftover matrix are all zero.
Analogous to the data-flow conservation property, we next show that the leftover indices also satisfy a similar leftover conservation property, as follows $$\label{eq:leftover-conservation}
\underset{j\in \{1,\ldots,K\}\setminus i}{\sum}\Omega^{(\pi_t,\pi_{t+1})}_{i,j}=\underset{j\in \{1,\ldots,K\}\setminus i}{\sum}\Omega^{(\pi_t,\pi_{t+1})}_{j,i}.$$
To prove the above property, we use the definition of leftovers in (\[eq:leftover-defn\]), to first compute the total leftovers at a worker $w_i$ as follows $$\begin{aligned}
\label{eq:leftovers-total}
&\underset{j\in \{1,\ldots,K\}\setminus i}{\sum} \Omega^{(\pi_t,\pi_{t+1})}_{i,j}= \underset{j\in \{1,\ldots,K\}\setminus i}{\sum} S^{(\pi_t,\pi_{t+1})}_{i,j} \\
&\qquad\qquad\qquad-\underset{j\in \{1,\ldots,K\}\setminus i}{\sum} \min(S^{(\pi_t,\pi_{t+1})}_{i,j},S^{(\pi_t,\pi_{t+1})}_{j,i}).\nonumber\end{aligned}$$ Similarly, we can also write the total number of leftover data points coming from all other workers to worker $w_i$ $$\begin{aligned}
\label{eq:needed-total}
&\underset{j\in \{1,\ldots,K\}\setminus i}{\sum} \Omega^{(\pi_t,\pi_{t+1})}_{j,i}= \underset{j\in \{1,\ldots,K\}\setminus i}{\sum} S^{(\pi_t,\pi_{t+1})}_{j,i}\\
&\qquad\qquad\qquad-\underset{j\in \{1,\ldots,K\}\setminus i}{\sum} \min(S^{(\pi_t,\pi_{t+1})}_{i,j},S^{(\pi_t,\pi_{t+1})}_{j,i}).\nonumber \end{aligned}$$ From the property in (\[eq:dataflow-conservation\]), we notice that the quantities in (\[eq:leftovers-total\]), and (\[eq:needed-total\]) are equal and hence we arrive at the proof of (\[eq:leftover-conservation\]). Using the leftover conservation property in (\[eq:leftover-conservation\]), we can show that the sum across rows or columns for the leftover matrix $\Omega$ is constant for any shuffle $(\pi_t,\pi_{t+1})$.
Subsequently, we refer to $R_{(\pi_t,\pi_{t+1})}$ as the rate for any achievable scheme $(\phi,\psi)$. We also drop the index $(\pi_t,\pi_{t+1})$ from $S^{(\pi_t,\pi_{t+1})}$, $\Omega^{(\pi_t,\pi_{t+1})}$, $R_{(\pi_t,\pi_{t+1})}$, and $X_{(\pi_t,\pi_{t+1})}$.
Main Results {#sec:Results}
============
The main contributions of this paper are presented next in the following three Theorems.
\[th:1\] *The optimal communication overhead $R^{*}(K)$ for a shuffle characterized by a shuffle matrix $S= [S_{i,j}]$ is upper bounded as* $$\begin{aligned}
\label{eq:thm1}
&R^*(K) \leq \overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}} \max(S_{i,j},S_{j,i})\nonumber \\
&\hspace{80pt} -\underset{k}{\max} \sum_{j\in \{1,\ldots, K\}\setminus \{k\}} \Omega_{k,j}.\end{aligned}$$
\[th:2\] *The optimal communication overhead $R^{*}(K)$, for any arbitrary shuffle matrix $S= [S_{i,j}]$ is lower bounded as* $$\label{eq:thm2}
R^*(K)\geq \overset{K-1}{\underset{i=1}{\sum}}\overset{K}{\underset{j=i+1}{\sum}} S_{\sigma_i,\sigma_j},$$ *for any permutation $\sigma$: $\{1,\ldots,K\}\rightarrow \{\sigma_1,\ldots,\sigma_K\}$ of the $K$ workers.*
\[th:3\] *The information theoretically optimal worst-case communication overhead for data shuffling is given by* $$R^*_{\textsf{worst-case}}(K) = \left(\frac{K-1}{K}\right)N.$$
Proof of Theorem \[th:1\]
==========================
In this section, we present an achievable scheme for the shuffling process, which gives an upper bound on the communication overhead as stated in Theorem \[th:1\]. We consider the random reshuffling process $(\pi_t,\pi_{t+1})$, characterized by a shuffle matrix $S=[S_{i,j}]$, from time $t$ given by the data batches $A^{t}_1,A^{t}_2,\ldots, A^{t}_K$, to time $t+1$ given by the data batches $A^{t+1}_1,A^{t+1}_2,\ldots, A^{t+1}_K$. We first describe the main idea of our scheme through a representative example.
------------------------------------------------------------------------
***Example 1:*** Consider $K=3$ workers (denoted as $\{w_1, w_2, w_3\}$) and $N=15$ be the total number of data points. Consider the following shuffle matrix $S=[S_{i,j}]$: $$\label{example:shuffle}
S=\left[\begin{array}{ccc}
S_{1,1} & S_{1,2} & S_{1,3}\\
S_{2,1} & S_{2,2} & S_{2,3}\\
S_{3,1} & S_{3,2} & S_{3,3}
\end{array} \right]= \left[\begin{array}{ccc}
2 & 1 & 2\\
2 & 1 & 2 \\
1 & 3 & 1
\end{array} \right]$$ The numbers in the diagonal represents the data points that remains unchanged across the workers, therefore, they do not participate in the communication process (see Remark \[remark1\]). For uncoded communication, the number of transmitted data points would be the sum of all non-diagonal entries, i.e., $R_{\textsf{uncoded}}=11.$
We first show how coding can be utilized to further reduce the communication overhead. For this example, worker $w_1$ needs $S_{2,1}=2$ data points from $w_2$. Let us denote these points as $\{x^{(1)}_{2,1}, x^{(2)}_{2,1}\}$. At the same time, $w_2$ needs $S_{1,2}=1$ data point from $w_1$ (denoted as $x_{1,2}$). Instead of uncoded transmission, the master node can send a coded symbol $x^{(1)}_{2,1}+ x_{1,2}$ which is simultaneously useful for both $w_1$, and $w_2$ as follows: $w_1$ has $x_{1,2}$, then it subtracts from the coded symbol to get the needed data-point $x^{(1)}_{2,1}$. Similarly, $w_2$ gets $x_{1,2}$ using $x^{(1)}_{2,1}$ and $x^{(1)}_{2,1}+ x_{1,2}$. This coded symbol is refereed to as an order-2 symbol, since it is useful for two workers at the same time.
By exploiting all such pairwise coding opportunities, we can send a total of $4$ order 2 symbols as follows: one coded symbol for $\{w_1,w_2\}$, one for $\{w_1,w_3\}$, and two for $\{w_2,w_3\}$. After having exhausted all pairwise coding opportunities, there are still some remaining data points, which we call as leftovers. The leftover matrix (defined in (\[eq:leftover-defn\]) and (\[eq:leftover-matrix\])), contains the number of leftover symbols after combining the order 2 symbols, is given as $$\label{example:leftover}
\Omega=\left[\begin{array}{ccc}
\Omega_{1,1} & \Omega_{1,2} & \Omega_{1,3}\\
\Omega_{2,1} & \Omega_{2,2} & \Omega_{2,3}\\
\Omega_{3,1} & \Omega_{3,2} & \Omega_{3,3}
\end{array} \right]=
\left[\begin{array}{ccc}
0 & 0 & 1\\
1 & 0 & 0 \\
0 & 1 & 0
\end{array} \right]$$ If the remaining $3$ leftover symbols (sum of all non-zero elements of $\Omega$) are sent uncoded, then, the total rate would be $R_{\textsf{paired-coding}}=R_{\textsf{coded-order2}}+R_{\textsf{uncoded-leftovers}}=4+3=7$, therefore, $R_{\textsf{paired-coding}}<R_{\textsf{uncoded}}$.
We now describe the main idea behind our proposed coding scheme which exploits a new type of coding opportunity as follows. Till this end, for each worker, we combine its incoming leftover symbols with its outgoing leftover symbols. By the leftover conservation property, these two are equal. Then, we have the three coded symbols as follows $$\begin{aligned}
\{x_{3,1}+x_{1,2}, \quad x_{1,2}+x_{2,3}, \quad x_{2,3}+ x_{3,1} \}.\end{aligned}$$ The key observation is that *any two out of these three* coded symbols are enough for all the workers to get the remaining leftovers. Two workers decode the needed points in one step, while the ignored worker decodes in two steps.
For example, if the master node transmits the first two coded symbols, i.e., $x_{3,1}+x_{1,2}$ and $x_{1,2}+x_{2,3}$, then the decoding works as follows: $w_1$, and $w_2$ have $x_{1,2}$, and $x_{2,3}$, respectively, then they can get the needed ones, $x_{3,1}$, and $x_{1,2}$, respectively. Worker $w_3$, however, decodes its desired symbol through a two step procedure as follows: since it has $x_{3,1}$, then it can get $x_{1,2}$ from the first symbol $x_{3,1}+x_{1,2}$ in the first step. In the second step, from the second symbol $x_{1,2}+x_{2,3}$, it then uses $x_{1,2}$ to finally obtain the needed data point $x_{2,3}$. As a summary, we are able to send 3 leftovers in 2 coded symbols only. Therefore, communication overhead of the proposed scheme reduces to $R_{\textsf{proposed-coded}}=R_{\textsf{coded-order2}}+R_{\textsf{coded-leftovers}}=4+2=6$, i.e., $R_{\textsf{proposed-coded}}<R_{\textsf{paired-coding}}$.
------------------------------------------------------------------------
We next present our proposed scheme for a general shuffle matrix and arbitrary number of workers $K$, which can be described in the following two phases, namely the first phase of transmitting order-2 symbols, and the second phase, which is what we call the leftover combining phase.
Phase 1: Order-2 symbols
------------------------
First we start by transmitting order-2 symbols, that are useful for two workers at the same time. If we consider two workers $w_i$, and $w_j$, then worker $w_i$ has some data points for worker $w_j$, given by $A_i^{t}\cap A_j^{t+1}$, which are $S_{i,j}=|A_i^{t}\cap A_j^{t+1}|$ data points in total. Similarly, $w_j$ has $S_{j,i}$ data points for $w_i$. Now, if we take all the data points $x_{i,j}\in A_i^{t}\cap A_j^{t+1}$, and combine them with the points $x_{j,i}\in A_j^{t}\cap A_i^{t+1}$ to transmit order-2 symbols jointly useful for $w_i$, and $w_j$, then we are limited by $\min(S_{i,j},S_{j,i})$ number of order-2 symbols for the pair $(i,j)$. Therefore, we can transmit total number of order-2 symbols for all possible $(i,j)$ pairs of workers as follows $$\label{eq:Phase1}
R_{\text{Phase 1}}=\overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}} \min(S_{i,j},S_{j,i}).$$
Phase 2: Coded Leftover Communication
-------------------------------------
Now, we consider a coded approach for sending the leftovers after combining the order-2 symbols at phase 1. For a pair of workers $(i,j)$, after combining $\min(S_{i,j},S_{j,i})$ symbols in phase 1, then we still have $\Omega_{i,j}=S_{i,j}-\min(S_{i,j},S_{j,i})$ leftover symbols that are still needed to be transmitted from $w_i$ to $w_j$. Similarly, the leftovers form $w_j$ to $w_i$ is given by $\Omega_{j,i}=S_{j,i}-\min(S_{i,j},S_{j,i})$. We notice that if $S_{i,j}>S_{j,i}$, then $\Omega_{i,j}= S_{i,j}-S_{j,i}>0$, and $\Omega_{j,i}=0$, and vice versa. This gives us the following properties $$\begin{aligned}
&\Omega_{i,j}+\Omega_{j,i}=\max(\Omega_{i,j},\Omega_{j,i})= |S_{i,j}-S_{j,i}|,\nonumber\\
&\min(\Omega_{i,j},\Omega_{j,i})=0.\label{eq:proberty-leftovers}\end{aligned}$$ Clearly, if $S_{i,j}=S_{j,i}$, then $\Omega_{i,j}=\Omega_{j,i}=0$, and there are no leftover symbols for the pair $(i,j)$. The property in (\[eq:proberty-leftovers\]) states that if a worker $w_i$ has some data points for $w_j$ in its leftovers ($\Omega_{i,j}\neq 0$), then $w_j$ has nothing in its leftovers needed by $w_i$ ($\Omega_{j,i}= 0$). Using the leftover data conservation property in (\[eq:leftover-conservation\]), we first state the following claim:
\[claim2\] *After combining the order-2 symbols in phase 1, the total number of symbols at a worker $w_i$ needed by other workers (outgoing leftovers) is equal to the total number of data points needed by the worker $w_i$ from other workers (incoming needed points).*
As a simple scheme, we can use Claim \[claim2\] to combine all the leftovers with the needed data points for every worker $w_i$. Therefore, each worker can use its own outgoing leftover data points to get the desired incoming points. However, it is obvious that this coded scheme achieves the same rate as if we are sending the leftovers uncoded.
We next present the following claim which is one of the novel contributions of this paper:
\[claim3\] *If we combine the leftovers with the needed data points for any $K-1$ workers, then under a certain combining condition (stated below) for the remaining *ignored worker*, say $w_k$, it can get its own needed data points without the need of being combined with its own leftovers.*
Before presenting the proof of Claim \[claim3\], we first state the combining condition. In order to ignore a worker $w_k$ from combining its leftovers with the needed points, the following condition must be satisfied while combining the leftovers with the needed points for other non-ignored workers:
*(Leftover Combining Condition for Ignoring $w_k$) \[def:combining-condition\] The needed data-points at the ignored worker $w_k$ from leftovers of other workers $x_{i,k}$, and independently the leftovers at $w_k$ needed by other workers $x_{k,j}$ should only be combined with the data-points $x_{j,i}$ as follows* $$\begin{aligned}
\left\{x_{k,j}+x_{j,i}, \:\: x_{j,i}+x_{i,k}\right\}.\end{aligned}$$
{width="95.00000%"}
In order to understand the combining condition, we use the following example. Let us consider the following three types of leftover data points: (i) a data point $x_{i,k}$ that is needed by an ignored worker $w_k$, and is available at worker $w_i$; (ii) a data point $x_{k,j}$ that is a leftover at $w_k$, and is needed by worker $w_j$; and (iii) a data point $x_{i,j}$ that is a leftover at $w_i$, and is needed by worker $w_j$.
In order for $w_k$ to decode $x_{i,k}$ using the leftover $x_{k,j}$, the leftover coded combining condition should be satisfied as follows
$\bullet$ While combining the leftovers with the needed points of $w_j$ at the master node, the needed data point $x_{k,j}$ (from $w_j$’s perspective) should only be combined with the leftover data point $x_{j,i}$ as follows: $$\begin{aligned}
\label{eq:coded-sym1}
x_{k,j}+x_{j,i}.\end{aligned}$$
$\bullet$ While combining the leftovers with the needed points of $w_i$ at the master node, the leftover data point $x_{i,k}$ (from $w_i$’s perspective) should only be combined with the needed data point $x_{j,i}$ as follows: $$\begin{aligned}
\label{eq:coded-sym2}
x_{j,i}+x_{i,k}.\end{aligned}$$
From the above coded combining, we notice the following: 1) Workers $w_i$, and $w_j$ still can decode the needed points $x_{j,i}$, and $x_{k,j}$, respectively. 2) Worker $w_k$ decodes in two steps: First, it uses $x_{k,j}$ to get $x_{j,i}$ from the coded symbol in (\[eq:coded-sym1\]). In the next step, from the second coded symbol in (\[eq:coded-sym2\]) it uses $x_{j,i}$ to decode the needed data point $x_{i,k}$.
Proof of Claim \[claim3\]
-------------------------
Now we need to prove formally the decodability at the ignored worker $w_k$. In order to complete the proof, we need to show that the number of intermediate points the ignored worker $w_k$ can get in the first step of decoding; are enough to decode the needed points in the next step of the decoding process.
We start by partitioning the leftover data points $\Omega_{i,j}$ into non-overlapping $(K-2)$ parts $\Omega_{i,j}^{(\ell)},\:\:\ell \in \{1,2,\ldots,K\}\setminus \{i, j\}$, where $\Omega^{(\ell)}_{i,j}\leq\Omega_{i,j}$ is defined as the number of intermediate (unintended since $\ell\neq\{i,j\}$) data points originally needed by $w_j$ that $w_\ell$ can get using its own leftovers needed for $w_i$ (through $w_i$).
Therefore, $\Omega_{i,j}$ can be written as $$\begin{aligned}
\label{eq:Omega-ij-struct}
\Omega_{i,j}=\underset{\ell\in\{1,\ldots,K\}\setminus\{i,j\}}{\sum}\Omega_{i,j}^{(\ell)}.\end{aligned}$$ As shown in Figure \[fig:leftover-combining\], $w_K$ for example uses its own leftovers needed by $w_1$ (through $w_1$), i.e., $\Omega_{K,1}$ points, to get unintended points (labelled with blue) that are needed by the other workers $\{2,3,\ldots,K-1\}$, i.e., $\Omega^{(K)}_{1,2},\ldots, \Omega^{(K)}_{1,K-1}$. Therefore, the total number of unintended (intermediate) data points recovered by $w_K$ using $\Omega_{K,1}$ data points is $$\begin{aligned}
\label{eq:Omega-ij-unintended}
\Omega_{K,1} = \overset{K-1}{\underset{j=2}{\sum}}\Omega^{(K)}_{1,j}.\end{aligned}$$ Generally, through the combined symbols for $w_i$, the number of unintended data points which worker $w_\ell$ can obtain is $$\begin{aligned}
\label{eq:dec-const1}
\Omega_{\ell,i} = \underset{j=\{1,\ldots,K\}\setminus\{i,\ell\}}{\sum}\Omega^{(\ell)}_{i,j}.\end{aligned}$$
Let us assume now without loss of generality, that the ignored worker is the last worker $w_K$. As shown in Figure \[fig:leftover-combining\], the ignored worker $w_K$ cannot get the needed data-points (colored chunks above the dotted lines) directly. Instead, $w_K$ uses its leftovers $\overset{K-1}{\underset{i=1}{\sum}}\Omega_{K,i}$ to get first unintended intermediate points (blue labelled points $\Omega^{(K)}_{1,j}$ through $w_1$, red labelled points $\Omega^{(K)}_{2,j}$ through $w_2$, etc.), which are shown below the solid line in the Figure.
In order for $w_K$ to make use of the intermediate symbols $\Omega^{(K)}_{i,j}$, $\{(i,j)\in\{1,\ldots ,K-1\},i\neq j\}$, every symbol $x_{i,j}$ of them should be paired up with data points useful for $w_K$ in the coded combining for $w_j$, i.e, $x_{i,j}+x_{j,K}$, which is satisfying the combining constraint in Definition \[def:combining-condition\]. Following the relation in (\[eq:dec-const1\]), the actual total number of unintended symbols $w_K$ can get in the first step of decoding is given by $$\begin{aligned}
\label{eq:decK-intermediate}
\overset{K-1}{\underset{i=1}{\sum}}\Omega_{K,i}& = \overset{K-1}{\underset{i=1}{\sum}} \underset{j=\{1,\ldots,K-1\}\setminus\{i\}}{\sum}\Omega^{(K)}_{i,j}\nonumber\\
&=\underset{\substack{(i,j)\in\{1,\ldots,K-1\} \\i\neq j}}{\sum}\Omega^{(K)}_{i,j}.\end{aligned}$$
Using the unintended symbols that $w_K$ gets through $w_i$ and are originally needed by $w_j$, i.e., $\Omega_{i,j}^{(K)}$, it should be able to decode the needed symbols $\Omega^{(i)}_{j,K}$. As an example, $w_K$ gets the blue unintended data points $\Omega^{(K)}_{1,2},\ldots, \Omega^{(K)}_{1,K-1}$ through $w_1$, then these data points are used to get the blue labelled needed points $\Omega^{(1)}_{2,K},\ldots, \Omega^{(1)}_{K-1,K}$ as shown above the solid line in Figure \[fig:leftover-combining\].
The minimum number of unintended symbols $w_K$ needs to decode out of $\Omega_{i,j}$ points in the first step, should be enough to decode (equal to) the needed part $\Omega^{(i)}_{j,K}$ in the next step of decoding. From the unintended data recovery condition in (\[eq:dec-const1\]), $\Omega^{(i)}_{j,K}$ is given by $$\begin{aligned}
\label{eq:dec-const1-2}
\Omega^{(i)}_{j,K}=\Omega_{i,j} - \underset{\ell=\{1,\ldots,K-1\}\setminus\{i,j \}}{\sum}\Omega^{(i)}_{j,\ell}.\end{aligned}$$
Therefore, the total number of unintended symbols that the worker $w_K$ should at least have in order to decode all the needed points in the next step is given by $$\begin{aligned}
&\overset{K-1}{\underset{j=1}{\sum}}\Omega_{j,K}\overset{(a)}{=}
\underset{\substack{(i,j)\in\{1,\ldots,K-1\} \\i\neq j}}{\sum}\Omega^{(i)}_{j,K}\nonumber\\
&\overset{(b)}{=}\underset{\substack{(i,j)\in\{1,\ldots,K-1\} \\i\neq j}}{\sum}\Omega_{i,j} - \underset{\substack{(i,j,\ell)\in\{1,\ldots,K-1\} \\i\neq j \neq\ell}}{\sum}\Omega^{(i)}_{j,\ell}\nonumber\\
&\overset{(c)}{=}\underset{\substack{(i,j)\in\{1,\ldots,K-1\} \\i\neq j}}{\sum}\Omega_{i,j} - \underset{\substack{(i,j,\ell)\in\{1,\ldots,K-1\} \\i\neq j \neq\ell}}{\sum}\Omega^{(\ell)}_{i,j}\nonumber\\
&=\underset{\substack{(i,j)\in\{1,\ldots,K-1\} \\i\neq j}}{\sum}\left[\Omega_{i,j} - \underset{ \ell\in\{1,
\ldots,K-1\}\setminus\{i,j\}}{\sum}\Omega^{(\ell)}_{i,j}\right]\nonumber\\
&\overset{(d)}{=}\underset{\substack{(i,j)\in\{1,\ldots,K-1\} \\i\neq j}}{\sum}\Omega^{(K)}_{i,j}\label{eq:decK-intermediate-min},\end{aligned}$$ where $(a)$ follows from (\[eq:Omega-ij-struct\]), $(b)$ follows from the constraint in (\[eq:dec-const1-2\]), $(c)$ by switching the sum indices, and $(d)$ from the definition in (\[eq:Omega-ij-struct\]). From (\[eq:decK-intermediate\]) and (\[eq:decK-intermediate-min\]), it now follows that the total number of intermediate points the ignored worker $w_K$ can decode in the first step is exactly equal to the minimum number it must decode in order to get the needed points in the second step, which completes the proof of Claim \[claim3\].
Hence, the total communication overhead of phase $2$ is the total of all leftover symbols (except the ignored worker $k$), and is given as: $$\begin{aligned}
&R_{\text{Phase 2}}=\overbrace{\underset{i\in\{1,\ldots,K\}\setminus \{k\}}{\sum}}^{\textsf{ignoring }w_k}\: \overbrace{\underset{j\in \{1,\ldots,K\}\setminus\{i\}}{\sum} \Omega_{i,j}}^{\textsf{leftovers at }w_i}\label{eq:variation}\\
&\:\:=\overset{K}{\underset{i=1}{\sum}}\: \underset{j\in \{1,\ldots,K\}\setminus\{i\}}{\sum} \Omega_{i,j}-\underset{j\in \{1,\ldots,K\}\setminus\{k\}}{\sum} \Omega_{k,j}\nonumber\\
&\:\:= \overset{K}{\underset{i=2}{\sum}}\: \overset{i-1}{\underset{j=1}{\sum}} \Omega_{i,j}+\overset{K-1}{\underset{i=1}{\sum}}\: \overset{K}{\underset{j=i+1}{\sum}} \Omega_{i,j}- \underset{j\in \{1,\ldots,K\}\setminus\{k\}}{\sum} \Omega_{k,j}\\
&\:\:\overset{(a)}{=} \overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}} \Omega_{j,i}+\overset{K-1}{\underset{i=1}{\sum}}\: \overset{K}{\underset{j=i+1}{\sum}} \Omega_{i,j}- \underset{j\in \{1,\ldots,K\}\setminus\{k\}}{\sum} \Omega_{k,j}\\
&\:\:= \overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}} \left(\Omega_{i,j}+\Omega_{j,i}\right)
- \underset{j\in \{1,\ldots,K\}\setminus\{k\}}{\sum} \Omega_{k,j}\\
&\:\:\overset{(b)}{=}\overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}}\max(\Omega_{i,j},\Omega_{j,i})-\underset{j\in \{1,\ldots,K\}\setminus\{k\}}{\sum}\Omega_{k,j},\label{eq:Phase2}\end{aligned}$$ where $(a)$ follows by swapping the indices $j$ and $i$ in the first summand, and $(b)$ follows from the property of leftovers in (\[eq:proberty-leftovers\]), which states that that $\min(\Omega_{i,j},\Omega_{j,i})=0$.
Hence, the total communication overhead of the proposed scheme is the total number of transmitted symbols over Phases $1$ and $2$, which is the sum of (\[eq:Phase1\]), and (\[eq:Phase2\]), and is given by $$\begin{aligned}
&R(K)= R_{\text{Phase 2}}+R_{\text{Phase 2}} \nonumber\\
&=\overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}} \min(S_{i,j},S_{j,i})+\overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}}\max(\Omega_{i,j},\Omega_{j,i})\nonumber\\
&\hspace{125pt} -\underset{j\in \{1,\ldots,K\}\setminus\{k\}}{\sum}\Omega_{k,j}\nonumber\\
&\overset{(a)}{=}\overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}} \max(S_{i,j},S_{j,i})-\underset{j\in \{1,\ldots,K\}\setminus\{k\}}{\sum}\Omega_{k,j},\label{eq:Rate}\end{aligned}$$ where $(a)$ follows from the property in (\[eq:proberty-leftovers\]). In order to get the lowest possible rate for this scheme, which is also an upper bound for the optimal communication overhead, the choice of the ignored worker $w_k$ can be optimized to have the maximum number of leftovers, which is given by $$\begin{aligned}
&R^*(K)\nonumber\\
&\leq \underset{k}{\min} \left( \:\:\overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}} \max(S_{i,j},S_{j,i})-\underset{j\in \{1,\ldots,K\}\setminus\{k\}}{\sum}\Omega_{k,j}\right)\nonumber\\
&=\overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}}\max(S_{i,j},S_{j,i})-\underset{k}{\max} \left(\underset{j\in \{1,\ldots,K\}\setminus\{k\}}{\sum}\Omega_{k,j}\right).
\label{eq:upper-bound}\end{aligned}$$ This completes the proof of Theorem \[th:1\].
Proof of Theorem \[th:2\]
==========================
In this section, we present the lower bound on the optimal communication overhead for any arbitrary random shuffle between two subsequent epochs $t$, and $t+1$ given by a shuffle matrix $S=[S_{i,j}]$, as stated in Theorem \[th:2\].
$$\begin{aligned}
\label{eq:lower-bound}
Nd &\overset{(a)}{=} H(A) \nonumber\\
&\overset{(b)}{=} I(A;A_1^{t},\ldots,A^{t}_K,X)+ H(A|A_1^{t},\ldots,A^{t}_K,X)\nonumber\\
&\overset{(c)}{=} H(A_1^{t},\ldots,A^{t}_K,X)-H(A_1^{t},\ldots,A^{t}_K,X|A)\nonumber\\
&\overset{(d)}{=} H(A_{\sigma_1}^{t},A^{t}_{\sigma_2},\ldots,A^{t}_{\sigma_K},X)\nonumber\\
&\overset{(e)}{=} H(A_{\sigma_K}^{t},X)+\overset{K-1}{\underset{i=1}{\sum}} H(A_{\sigma_i}^{t}|A_{\sigma_{i+1}}^{t},\ldots,A_{\sigma_K}^{t},X)\nonumber\\
&\overset{(f)}{\leq} H(A_{\sigma_K}^{t}) + H(X)+\overset{K-1}{\underset{i=1}{\sum}} H(A_{\sigma_i}^{t}|A_{\sigma_{i+1}}^{t+1},\ldots,A_{\sigma_K}^{t+1})\nonumber\\
&\overset{(g)}{\leq} \frac{Nd}{K}+Rd+\overset{K-1}{\underset{i=1}{\sum}} \left[\frac{Nd}{K}-I(A_{\sigma_i}^{t};A_{\sigma_{i+1}}^{t+1},\ldots,A_{\sigma_K}^{t+1})\right]\nonumber\\
&=Nd+Rd-\overset{K-1}{\underset{i=1}{\sum}}I(A_{\sigma_i}^{t};A_{\sigma_{i+1}}^{t+1},\ldots,A_{\sigma_K}^{t+1}),\end{aligned}$$
where $(a)$ follows from (\[eq:data-batches2\]), $(b)$ and $(c)$ are due to the fact that $I(A;B)=H(A)-H(A|B)=H(B)-H(B|A)$, and from (\[eq:data-partitions\]) where the data-batches at any time span $A$, $(d)$ from (\[eq:data-partitions\]) and (\[eq:transmit-const\]), where the data-batches and $X$ are all functions of the data-set $A$, and $\sigma$ is any permutation of the the set $\{1,\ldots,K\}$, $(e)$ from the chain rule of entropy, $(f)$ from the decoding constraint in (\[eq:decoding-const\]), the fact that conditioning reduces entropy, and the fact $H(A,B)\leq H(A)+H(B)$, and $(g)$ from (\[eq:data-batches2\]), (\[eq:transmit-load\]), and the fact $H(A|B)=H(A)-I(A;B)$. By rearranging the inequality in (\[eq:lower-bound\]), we arrive at $$\begin{aligned}
\label{eq:claim3}
Rd&\geq \overset{K-1}{\underset{i=1}{\sum}}I(A_{\sigma_i}^{t};A_{\sigma_{i+1}}^{t+1},\ldots,A_{\sigma_K}^{t+1})\nonumber\\
&= \overset{K-1}{\underset{i=1}{\sum}}\overset{K}{\underset{j=i+1}{\sum}}I(A_{\sigma_i}^{t};A_{\sigma_{j}}^{t+1})= \overset{K-1}{\underset{i=1}{\sum}}\overset{K}{\underset{j=i+1}{\sum}}S_{\sigma_i,\sigma_j} d.\end{aligned}$$ Therefore, the lower bound on the communication overhead is given by $R^*(K)\geq \overset{K-1}{\underset{i=1}{\sum}}\overset{K}{\underset{j=i+1}{\sum}}S_{\sigma_i,\sigma_j}$, completing the proof of Theorem \[th:2\].
Proof of Theorem \[th:3\]
=========================
In this section, we prove the optimality of our proposed scheme for the worst-case shuffle, which describes the maximum communication overhead across all possible shuffles.
Achievability
--------------
We start by using the upper bound described in Theorem \[th:1\], where we use a variation of the expression in (\[eq:thm1\]) by adding (\[eq:Phase1\]), and (\[eq:variation\]) as follows $$\begin{aligned}
\label{eq:upper-bound2}
R(K&)\overset{(a)}{=}\overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}} \min(S_{i,j},S_{j,i})+\underset{i\in\{1,\ldots,K\}\setminus \{k\}}{\sum}\:\overset{K}{\underset{j=1}{\sum}} \Omega_{i,j}\nonumber\\
&\overset{(b)}{=}\overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}} \min(S_{\sigma_i,\sigma_j},S_{\sigma_j,\sigma_i})+\overset{K-1}{\underset{i=1}{\sum}}\overset{K}{\underset{j=1}{\sum}} \Omega_{\sigma_i,\sigma_j}\nonumber\\
&\overset{(c)}{=}\overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}} \min(S_{\sigma_i,\sigma_j},S_{\sigma_j,\sigma_i})+\overset{K-1}{\underset{i=1}{\sum}}\overset{K}{\underset{j=1}{\sum}} S_{\sigma_i,\sigma_j}\nonumber\\
&\quad-\overset{K-1}{\underset{i=1}{\sum}}\overset{K}{\underset{j=1}{\sum}}\min(S_{\sigma_i,\sigma_j},S_{\sigma_j,\sigma_i})\nonumber\\
&\overset{(d)}{\leq} \overset{K-1}{\underset{i=1}{\sum}}\overset{K}{\underset{j=1}{\sum}} S_{\sigma_i,\sigma_j}\overset{(e)}{=}\overset{K-1}{\underset{i=1}{\sum}} \frac{N}{K}=\left(\frac{K-1}{K}\right)N,\end{aligned}$$ where $(a)$ holds because $\Omega_{i,i}=0$, $(b)$ follows by considering a permutation $\sigma=\{\sigma_1,\ldots\sigma_K\}$ of the workers, where $\sigma_K=k$ is the ignored worker, $(c)$ follows from the definition of $\Omega_{i,j}$ in (\[eq:leftover-defn\]), $(d)$ is due to the fact that $\min(S_{i,j},S_{j,i})\geq 0$, and $(e)$ from the property in (\[eq:Sij-property\]). Since this derived upper bound is found for any arbitrary shuffle, it is also an upper bound for the optimal worst-case communication overhead. Hence, we have $$\begin{aligned}
\label{eq:wc-upperbound}
R^*_{\textsf{worst-case}}(K)\leq \left(\frac{K-1}{K}\right)N.\end{aligned}$$
Converse
---------
We start by assuming a particular data shuffle, and then specialize our lower bound (obtained in Theorem \[th:2\]) for this particular shuffle. We use the fact that the worst-case overhead $R^{*}_{\textsf{worst-case}}(K)$ is lower bounded by the overhead of any shuffle $R(K)$, therefore the lower bound found for this given shuffle works as a lower bound for the worst-case as well, i.e., $$\label{eq:wc-to-any}
R^{*}_{\textsf{worst-case}}(K)\geq R^*(K).$$ We assume a data shuffle matrix $S$ described as follows: For some permutation of the $K$ workers given by $\sigma=\{\sigma_1,\sigma_2,\ldots,\sigma_K\}$, any worker $w_{\sigma_{i+1}}$ at time $t+1$ needs only all the data points that $w_{\sigma_i}$ has from the previous shuffle at time $t$, which can be described as $$\label{eq:wc-shuffle}
S_{\sigma_i,\sigma_j}=\left\{\begin{array}{cc}
\frac{N}{K}, & j=i+1,\\
0, & \text{otherwise}.
\end{array}\right.$$ Therefore, using the lower bound in Theorem \[th:2\] given by (\[eq:thm2\]), and using (\[eq:wc-to-any\]), the lower bound for this particular shuffle, and hence the optimal worst-case shuffle, can be found as $$\begin{aligned}
\label{eq:wc-lowerbound}
R^*_{\textsf{worst-case}}(K)&\geq R^*(K)\geq \overset{K-1}{\underset{i=1}{\sum}} \overset{K}{\underset{j=i+1}{\sum}} S_{\sigma_i,\sigma_j}\nonumber\\
&=\overset{K-1}{\underset{i=1}{\sum}} S_{\sigma_i,\sigma_{i+1}}=\overset{K-1}{\underset{i=1}{\sum}} \frac{N}{K}=\left(\frac{K-1}{K}\right)N.\end{aligned}$$ From (\[eq:wc-upperbound\]), and (\[eq:wc-lowerbound\]), it follows that the information theoretically optimal worst case communication overhead is $$\label{eq:wc-rate}
R^*_{\textsf{worst-case}}(K)=\left(\frac{K-1}{K}\right)N.$$
Conclusion {#sec:conclusion}
==========
In this paper, we presented new results on the minimum necessary communication overhead for the data shuffling problem. We proposed a novel coded-shuffling scheme which exploits a new type of coding opportunity, namely coded leftover combining in order to reduce the communication overhead. Our scheme is applicable to any arbitrary shuffle, and for any number of distributed workers. We also presented an information theoretic lower bound on the optimal communication overhead that is also applicable for any arbitrary shuffle. Finally, we showed that the proposed scheme matches this lower bound for the worst-case communication overhead across all shuffles, and thus characterizes the information theoretically optimal worst-case overhead.
|
---
abstract: 'The external-field QCD Sum Rules method is used to evaluate the coupling constants of the light-isoscalar scalar meson (“$\sigma$” or $\epsilon$) to the $\Lambda$, $\Sigma$, and $\Xi$ baryons. It is shown that these coupling constants as calculated from QCD Sum Rules are consistent with $SU(3)$-flavor relations, which leads to a determination of the $F/(F+D)$ ratio of the scalar octet assuming ideal mixing: we find $\alpha_s \equiv F/(F+D)=0.55$. The coupling constants with $SU(3)$ breaking effects are also discussed.'
author:
- 'G. Erkol'
- 'R. G. E. Timmermans'
- 'M. Oka'
- 'Th. A. Rijken'
title: 'Scalar-Meson-Baryon Coupling Constants in QCD Sum Rules'
---
Introduction
============
Hadronic interactions are in principle explained by quantum chromodynamics (QCD). Such a first-principles description of the hadron-hadron interaction, however, is highly complicated, particularly at low energy, where QCD is a nonperturbative theory. In practice, therefore, effective hadronic Lagrangians are often used. The coupling constants at the hadronic vertices are then among the most fundamental quantities that should be computed from QCD.
There is a long history of successful approaches of describing the two-baryon interaction using meson-exchange potentials. The values of the meson-baryon coupling constants have been empirically determined so as to reproduce the nucleon-nucleon (NN) [@Nag78; @Sto94], hyperon-nucleon (YN) [@Nag78x; @Mae89; @Rij98] and hyperon-hyperon (YY) interactions in terms of [*e.g.*]{} one-boson exchange (OBE) models. The scalar mesons play significant roles in such phenomenological potential models. In early OBE models for the NN interaction the exchange of an isoscalar-scalar “$\sigma$” meson with a mass of about 500 MeV was needed to obtain enough medium-range attraction and a sufficiently strong spin-orbit force. It was only later understood that the exchange of a broad isoscalar-scalar meson, the $\varepsilon$(760) [@Pro73; @Sve92], simulates the exchange of such a low-mass “$\sigma$” [@Bin72].
Existence of the “$\sigma$” meson is expected also from chiral symmetry of QCD. The $\sigma$ meson appears as a chiral partner of the Nambu-Goldstone boson, $\pi$, and thus plays the role of the “Higgs boson” in chiral symmetry breaking. Recent analyses of $\pi-\pi$ scattering have revealed a broad resonance at the mass around 600 MeV, called $f_0(600)$ by the Particle Data Group [@Eid04], which is considered to play the role of $\varepsilon$(760) in the boson exchange potential of the baryonic interactions.
In terms of OBE models for two-baryon interactions, the dominant contribution to $\Lambda\Lambda$ interaction comes from the scalar $\varepsilon(760)$ meson exchange [@Afn03; @Fer05]. The recent identification of $^6_{\Lambda\Lambda}He$ and the measurement of the $\Lambda\Lambda$ pair suggest that the binding energy of $\Lambda\Lambda$ ($\Delta B_{\Lambda\Lambda}\simeq 1.0\,$ MeV) is considerably smaller than the binding energy of $N\!N$ [@Tak01]. This is in contrast to the outcome of the earlier measurement which is $\Delta B_{\Lambda\Lambda}\simeq 4.7\,$ MeV [@Pro66]. This issue has also been examined within the framework of Nijmegen OBE potential model D (NHC-D) [@Nag76]. In this model, the pseudoscalar octet [$\pi$, $\eta$, $\eta^\prime$, $K$]{}, the vector octet [$\rho$, $\phi$, $\omega$, $K^\ast$]{} and the scalar singlet [$\epsilon$]{} are the exchanged mesons and the coupling constants are fitted to data while other physical properties of the particles are taken from experiment. The estimated value of $\Delta B_{\Lambda\Lambda}$ in this model implies a rather strong attractive $\Lambda\Lambda$. However, in the Nijmegen soft-core (NSC) potential models [@Mae89; @Rij98], where there is a scalar nonet instead of a scalar singlet, we have much weaker attractive potentials than in the case of NHC-D in the $\Lambda\Lambda$ systems. In this framework, since the scalar $\varepsilon(760)$ exchange plays the most crucial role in $\Lambda\Lambda$ interactions, it is necessary to determine the $\Lambda\Lambda\varepsilon$ coupling constant in a model independent way in order to understand the role of $\varepsilon$ exchange in the strangeness $S=-2$ sector.
The structure and even the status of the scalar mesons, however, have always been controversial [@Tim94; @Swa94]. In the quark model, the simplest assumption for the structure of the scalar mesons is the $^3P_0$ $q\bar{q}$ states. In this case, the scalar mesons might form a complete nonet of dressed $q\bar{q}$ states, resulting from [*e.g.*]{} the coupling of the $P$-wave $q\bar{q}$ states to meson-meson channels [@Bev86]. Explicitly, the unitary singlet and octet states, denoted respectively by $\varepsilon_1$ and $\varepsilon_8$, read \[octsing\] \_1 & = & (u|[u]{}+d|[d]{}+s|[s]{})/ ,\
\_8 & = & (u|[u]{}+d|[d]{}-2s|[s]{})/ . The physical states are mixtures of the pure $SU(3)$-flavor states, and are written as \[mix\] & = & \_s\_1+\_s\_8 ,\
f\_0 & = &-\_s\_1+\_s\_8 . For ideal mixing holds that $\tan\theta_s=1/\sqrt{2}$ or $\theta_s\simeq 35.3^\circ$, and thus one would identify \[psqqb\] (760) & = & (u|[u]{}+d|[d]{})/ ,\
f\_0(980) & = & -s|[s]{} . The isotriplet member of the octet is $a_0^{\pm,0}$(980), where \[azero\] a\_0\^0(980) = (u|[u]{}-d|[d]{})/ .
An alternative and arguably more natural explanation for the masses and decay properties of the lightest scalar mesons is to regard these as cryptoexotic $q^2\bar{q}^2$ states [@Jaf77]. In the MIT bag model, the scalar $q\bar{q}$ states are predicted around $1250$ MeV, while the attractive color-magnetic force results in a low-lying nonet of scalar $q^2\bar{q}^2$ mesons [@Jaf77; @Aer80]. This nonet contains a nearly degenerate set of $I=0$ and $I=1$ states, which are identified as the $f_0(980)$ and $a^{\pm,0}_0(980)$ at the $\bar{K}K$ threshold, where \[ps4q\] a\^0\_0(980) & = & (sd|[s]{}|[d]{}-su|[s]{}|[u]{})/ ,\
f\_0(980) & = & (sd|[s]{}|[d]{}+su|[s]{}|[u]{})/ , with the ideal-mixing angle $\tan\theta_s=-\sqrt{2}$ or $\theta_s\simeq
-54.8^\circ$ in this case. The light isoscalar member of the nonet is \[sigmaq4\] (760) & = & ud|[u]{}|[d]{} . The nonet is completed by the strange member $\kappa$(880), which like the $\varepsilon$(760) is difficult to detect because it is hidden under the strong signal from the $K^*$(892) [@Tim94; @Swa94]. We shall use in this paper the nomenclature $(a_0^{\pm,0},f_0,\sigma,\kappa)$ for the scalar-meson nonet, where one should identify $\sigma=\varepsilon(760)$.
One way to make progress with the scalar mesons is to study their role in the various two-baryon reactions (NN, YN, YY). Our aim in this paper is to calculate the $\Lambda\Lambda\sigma$, $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ coupling constants, using the QCD Sum Rules (QCDSR) method. The QCDSR method [@Shi79] is a powerful tool to extract qualitative and quantitative information about hadron properties [@Rei84; @Col00]. In this framework, one starts with a correlation function, which is constructed in terms of hadron interpolating fields. On the theoretical side, the correlation function is calculated using the Operator Product Expansion (OPE) in the Euclidian region. This correlation function is matched with an *Ansatz* which is introduced from the hadronic degrees of freedom on the phenomenological side, and this matching provides a determination of the hadronic parameters like baryon masses, magnetic moments, coupling constants of hadrons and so on.
There are different approaches in constructing the QCDSR (see [*e.g.*]{} Ref. [@Col00] for a review). One usually starts with the vacuum-to-vacuum matrix element of the correlation function that is constructed with the interpolating fields of two baryons and one meson. However, this three-point function method has as a major drawback that at low momentum transfer the OPE fails. Moreover, when the momentum of the meson is large, it is plagued by problems with higher resonance contamination [@Mal97]. The other method that is free from the above problems is the external-field method. There are two formulations that can be used to construct the external-field sum rules: In the vacuum-to-meson method, one starts with a vacuum-to-meson transition matrix element of the baryon interpolating fields, where some other transition matrix elements should be evaluated [@Rei84]. (This is also the starting point of the light-cone QCDSR method.) In Ref. [@Shi95], pion-nucleon coupling constant was calculated in the soft meson limit using this approach. Later it was pointed out that the sum rule for pion-nucleon coupling in the soft-meson limit can be reduced to the sum rule for the nucleon mass by a chiral rotation so the coupling was calculated again with a finite meson momentum [@Bir96]. These calculations were improved considering the coupling schemes at different Dirac structures and beyond the chiral limit contributions [@Lee98; @Kim98; @Kim99]. In this paper, we calculate the baryon-sigma meson coupling constants, using the external field QCDSR method [@Iof84]. We evaluate the vacuum to vacuum transition matrix element of two baryon interpolating fields in an external sigma field and construct the sum rules. This method has been used to determine the magnetic moments of baryons [@Iof84; @Bal83; @Chi86; @Chi85], the nucleon axial coupling constant [@Chi85; @Bel84], the nucleon sigma term [@Jin93], and baryon isospin mass splittings [@Jin95]. It has also been shown that at low momentum transfer, this method is very successful in evaluating the hadronic coupling constants. Recently, the $N\!N\sigma$ coupling constant, $g_{N\!N\sigma}$, was calculated using this method [@Erk05]. It has also been applied, previously, to the calculations of the strong and weak parity violating pion-nucleon coupling constants [@Hwa96; @Hwa97; @Hen96] and the coupling constants of the vector mesons $\rho$ and $\omega$ to the nucleon [@Wen97].
In the $SU(3)$ flavor symmetric one can classify the meson-baryon coupling constants in terms of two parameters, the $N\!Na_0$ coupling constant, $g_{N\!Na_0}$ and the $F/(F+D)$ ratio of the scalar octet, $\alpha_s$ [@Swa63]: \[relSU3\] g\_[NNa\_0]{}&=&g, g\_[NN\_8]{}=g(4\_s-1), g\_[\_8]{}=-g(1-\_s),\
g\_[\_8]{}&=&-g(1+2\_s), g\_[\_8]{}=g(1-\_s), g\_[a\_0]{}=g(2\_s-1),\
g\_[a\_0]{}&=&2g\_s, g\_[a\_0]{}=g(1-\_s), g\_[N]{}=-g(1+2\_s),\
g\_[N ]{}&=&g(1-2\_s).Considering the mixing between the singlet and the octet members of the scalar nonet, one obtains for $g_{B\!B\sigma}$ and $g_{B\!Bf_0}$, \[Noctsing\] g\_[BB]{} &=& \_s g\_1 + \_s g\_[BB\_8]{},\
g\_[BB f\_0]{} &=& - \_s g\_1 + \_s g\_[BB\_8]{}, where $g_1=g_{B\!B\varepsilon_1}$ is the flavor singlet coupling, and $\theta_s$ is the scalar mixing angle.
We shall first consider the sum rules in the $SU(3)$ flavor symmetric limit to see if the predicted values for the meson baryon coupling constants from the sum rules are consistent with the $SU(3)$ relations. We show that this is indeed the case which leads to a determination of the $F/(F+D)$ ratio of the scalar octet. Furthermore, keeping track of these coupling constants with the $SU(3)$ relations, we obtain the values of the other scalar meson-baryon coupling constants. For this purpose, we assume ideal mixing and make the analysis in both $q\bar{q}$ and $q^2\bar{q}^2$ pictures for the scalar mesons. As we move from the $S=0$ to the $S=-1$ and $S=-2$ sectors, the flavor $SU(3)$ breaking occurs as a result of the $s$-quark mass and the physical masses of the baryons and mesons. We also consider the $SU(3)$ breaking effects for the sum rules to estimate the amount of breaking, individually for each coupling.
We have organized our paper as follows: in Section \[secNSR\], we present the formulation of QCDSR with an external scalar field and construct the sum rules for the $\Lambda\Lambda\sigma$, $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ coupling constants. We give the numerical analysis and discuss the results in Section \[secAN\]. Finally, we arrive at our conclusions in Section \[secCONC\].
Baryon Sum Rules in an external sigma field {#secNSR}
===========================================
Construction of the Sum Rules
-----------------------------
In the external-field QCDSR method one starts with the correlation function of the baryon interpolating fields in the presence of an external constant isoscalar-scalar field $\sigma$, defined by the following: \[cor1\]\_[B]{}(q)=id\^4 x e\^[i qx]{} 0|[T]{}\[\_B(x)|\_B(0)\]|0\_ , where $\eta_B$ are the baryon interpolating fields which are chosen as [@Rei84] \[intfi\] \_&=&\_[abc]{}\[(s\_a\^T C \_s\_b)\_5\^u\_c\],\
\_&=&\_[abc]{}\[(u\_a\^T C \_u\_b)\_5\^s\_c\],\
\_&=&(2/3)\^[1/2]{}\_[abc]{}\[(u\_a\^T C \_s\_b)\_5\^d\_c-(d\_a\^T C \_s\_b)\_5\^u\_c\]. for $\Xi$, $\Sigma$ and $\Lambda$, respectively. Here $a, b, c$ denote the color indices, and $T$ and $C$ denote transposition and charge conjugation, respectively. For the interpolating field of each octet baryon, there are two independent local operators, but the ones in Eq. (\[intfi\]) are the optimum choices for the lowest-lying positive parity baryons (see [*e.g.*]{} Ref [@Jid96] for a discussion on negative-parity baryons in QCDSR).
The external sigma field contributes the correlation function in Eq. (\[cor1\]) in two ways: first, it directly couples the quark field in the baryon current and second, it modifies the condensates by polarizing the QCD vacuum. In the presence of the external scalar field there are no correlators that break the Lorentz invariance; however, the correlators already existing in the vacuum are modified by the external field: \[vaccon\]\_&& + g\^\_q ,\
g\_c |[q]{} q\_&& g\_c |[q]{} q+ g\^\_q \_G g\_c |[q]{} q ,where only the responses linear in the external-field are taken into account. Here, $g^\sigma_q$ is the quark-$\sigma$ coupling constant and $\chi$ and $\chi_G$ are the susceptibilities corresponding to quark and quark-gluon mixed condensates, respectively. In Eq. (\[vaccon\]), $\langle\bar{q}q\rangle$ represents either $\langle\bar{u}u\rangle$ or $\langle\bar{d}d\rangle$, as we have assumed that $\langle\bar{u}u\rangle \simeq \langle\bar{d}d\rangle$ and the responses of the up and the down quarks to the external isoscalar field are the same. Note that, here we assume ideal mixing in the scalar sector, that is, we take the sigma meson without a strange-quark content. Therefore, the sigma meson couples only to the $u$- or the $d$-quark in the baryon, where we take $g_u^\sigma=g_d^\sigma$ and $g_s^\sigma=0$.
In the Euclidian region, the OPE of the product of two interpolating fields can be written as follows: \[opex\]\_[B]{}(q)=\_[n]{} C\^\_n(q) O\_n ,where $C^\sigma_n(q)$ are the Wilson coefficients and $O_n$ are the local operators in terms of quarks and gluons. In order to calculate the Wilson coefficients, we need the quark propagator in the presence of the external sigma field. In coordinate space the full quark propagator takes the form: \[proptot\] S\_q(x)=S\_q\^[(0)]{}(x)+S\_q\^[()]{}(x) ,where,
\[prop0\] i S\_q\^[(0)ab]{}&&0|T\[q\^a(x) |[q]{}\^b(0)|0\_0\
\
&=& - G\_\^n (\^+ \^)-|[q]{}q-g\_c |[q]{} q\
\
&&--\_[ab]{}\^n g\_c G\_\^n\^ (-x\^2)- m\_q x\^2 (-x\^2)\
\
&&++g\_c |[q]{} q+O(\_s\^2,m\_q\^2) ,and
\[propE\] i S\_q\^[()ab]{} &&0|T\[q\^a(x) |[q]{}\^b(0)|0\_\
\
&=&g\^\_q +O(\^2) .Here, $G^{\mu\nu}$ is the gluon tensor and $g_c^2=4\pi\alpha_s$ is the quark-gluon coupling constant squared. Note that, in the quark propagator above, we have included the terms that are proportional to the quark masses, $m_q$, since these terms give non-negligible contributions to the final result as far as the strange quark mass is considered.
Using the quark propagator in Eq. (\[proptot\]), one can compute the correlation function $\Pi_{B\sigma}(q)$. The Lorentz covariance and parity implies the following form for $\Pi_{B\sigma}(q)$: \_[B]{}(q)=(\_[B0]{}\^1+\_[B0]{}\^q )+(\_[B]{}\^1+\_[B]{}\^q )+O(\^2),where $\hat{q}=q^\mu \gamma_\mu$ is the four momentum of the baryon. Here $\Pi_{B0}^1$ and $\Pi_{B0}^q$ represent the invariant functions in the vicinity of the external field, which can be used to construct the mass sum rules for the relevant baryons, and $\Pi_{B\sigma}^1$ and $\Pi_{B\sigma}^q$ denote the invariant functions in the presence of the external field. Using these invariant functions, one can derive the sum rules at the structures $1$ and $\hat{q}$. In Ref. [@Erk05] it was found that the sum rule at the structure $\hat{q}$ for the $N\!N\sigma$ coupling constant is more stable than the other sum rule at the structure $1$, with respect to variations in the Borel mass. Motivated with this, we here present only the sum rules at the structure $\hat{q}$ and use these for the determination of the coupling constants.
$\Lambda$ Sum Rules and $\Lambda\Lambda\sigma$ Coupling Constant
----------------------------------------------------------------
We shall first present the sum rules calculations for $\Lambda\Lambda\sigma$ coupling constant in detail and in the next subsection, we shall give the sum rules for $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ couplings. At the quark level, we have for $\Lambda$: \[cor2\]0|[T]{}\[\_(x)|\_(0)\]|0\_= \^[abc]{}\^[a\^b\^c\^]{} &&(Tr {S\_u\^[a a\^]{}(x) \_C \[S\_s\^[b b\^]{}(x)\]\^T C \_}\_5 \^S\_d\^[c c\^]{}(x)\^\_5\
&&+ Tr {S\_d\^[c c\^]{}(x) \_C \[S\_s\^[b b\^]{}(x)\]\^T C \_}\_5 \^S\_u\^[a a\^]{}(x)\^\_5\
&&-\_5 \_S\_d\^[c c\^]{}(x) \_C \[S\_s\^[b b\^]{}(x)\]\^T C \^S\_u\^[a a\^]{}(x) \^\_5\
&&-\_5 \_S\_u\^[a a\^]{}(x) \_C \[S\_s\^[b b\^]{}(x)\]\^T C \^S\_d\^[c c\^]{}(x) \^\_5) .
Using the quark propagator in Eq. (\[proptot\]), the invariant function at the structure $\hat{q}$ in the presence of the external field, $\Pi_{\Lambda\sigma}^q$, is calculated as:
\[qmom\] \^q\_(q)= g\_q\^ &&,where we have defined $f=\frac{\langle\bar{s}s\rangle}{\langle\bar{q}q\rangle}-1$, $a_q=-(2\pi)^2 \langle\bar{q}q\rangle$ and $\langle g_c\bar{q} {\bm\sigma} \cdot {\bm G} q\rangle = m_0^2 \langle\bar{q}q\rangle$.
In order to construct the hadronic side, we saturate the correlator in Eq.(\[cor1\]) with $\Lambda$ states and write, \[sat\] \_(q)= |, where $M_\Lambda$ is the mass of the $\Lambda$. The matrix element of the current $\eta_\Lambda$ between the vacuum and the $\Lambda$ state is defined as, \[overlap\] 0 | \_| = \_, where $\lambda_\Lambda$ is the overlap amplitude and $\upsilon$ is the Dirac spinor for the $\Lambda$, which is normalized as $\bar{\upsilon}\upsilon=2M_\Lambda$. Inserting Eq. (\[overlap\]) into Eq. (\[sat\]) and making use of the isoscalar scalar meson-baryon interaction Lagrangian density =-g\_ | , we obtain the hadronic part as: \[phpart\]-|\_|\^2 g\_.
We have also contributions coming from the excitations to higher $\Lambda$ states which are written as, \[phpartex\]-\_\_[\^]{} g\_[\^]{},and the ones coming from the intermediate states due to $\sigma$-$\Lambda$ scattering i.e. the [*continuum*]{} contributions. Note that the term that corresponds to the excitations to higher $\Lambda$ states also has a pole at the $\Lambda$ mass, but a single pole instead of a double one like in Eq. (\[phpart\]). This single pole term is not damped after the Borel transformation and should be included in the calculations. There is another contribution that comes from the response of the continuum to the external field, which is given as: \[cont\]\^\_0 (s-s\_0) ds ,where $s_0$ is the continuum threshold, $\Delta s_0$ is the response of the continuum threshold to the external field and $b(s)$ is a function that is calculated from OPE. When $\Delta s_0$ is large, this term should also be included in the hadronic part [@Iof95].
Matching the OPE side with the hadronic side and applying the Borel transformation, the sum rule for $\Lambda\Lambda\sigma$ coupling at the structure $\hat{q}$ is obtained as: \[sumq\] & {&-M\^4a\_q(1-f)E\^\_0+a\_q\^2(1+2f)L\^[4/9]{}- M\^2 m\_0\^2 a\_qL\^[-14/27]{}\
&-&(+\_G)a\_q\^2(1+2f)L\^[-2/27]{} +(2m\_s-m\_q)M\^6E\^\_1L\^[-8/9]{}\
&+& a\_q(2 m\_s-m\_q)M\^4 E\^\_0L\^[-4/9]{} +a\_q\^2 } e\^[M\_\^2/M\^2]{}\
& & =-\^2 g\_ + \_ + M\^2L\^[-4/9]{} e\^[(M\_\^2-s\_0\^)/M\^2]{} ,where we have defined $\lamt^2=32 \pi^4 \lambda_\Lambda^2$ and $M$ is the Borel mass. The continuum contributions are included by the factors E\^\_0&& 1- e\^[-s\_0\^/M\^2]{} ,\
E\^\_1&& 1- e\^[-s\_0\^/M\^2]{}(1+) , where $s_0^\Lambda$ is the continuum threshold. In the sum rule above, we have included the single pole contribution with the factors $\tilde{B}_\Lambda$. The third term on the right hand side (RHS) of Eq. (\[sumq\]) denotes the contribution that is explained in Eq. (\[cont\]). Note that this term is suppressed by the factor $e^{-(s_0^i-M_\Lambda^2)/M^2}$ as compared to the single pole term. We have incorporated the effects of the anomalous dimensions of various operators through the factor $L=\ln(M^2/\Lambda_{QCD}^2)/\ln(\mu^2/\Lambda_{QCD}^2)$, where $\mu$ is the renormalization scale and $\Lambda_{QCD}$ is the QCD scale parameter.
$\Xi$ and $\Sigma$ Sum Rules and $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ Coupling Constants
-------------------------------------------------------------------------------------------
One can apply the method explained in the previous subsection for the $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ couplings and derive the corresponding sum rules. Using the interpolating fields in Eq. (\[intfi\]), we obtain \[cor2sigcas\]0|[T]{}\[\_(x)|\_(0)\]|0\_&=& 2i \^[abc]{}\^[a\^b\^c\^]{} Tr {S\_s\^[a a\^]{}(x) \_C \[S\_s\^[b b\^]{}(x)\]\^T C \_}\_5 \^S\_u\^[c c\^]{}(x)\^\_5,\
0|[T]{}\[\_(x)|\_(0)\]|0\_&=& 2i \^[abc]{}\^[a\^b\^c\^]{} Tr {S\_u\^[a a\^]{}(x) \_C \[S\_u\^[b b\^]{}(x)\]\^T C \_}\_5 \^S\_s\^[c c\^]{}(x)\^\_5,at the quark level for $\Xi$ and $\Sigma$, respectively. Using the quark propagator in Eq. (\[proptot\]), the invariant functions at the structure $\hat{q}$ are calculated as: \[Cqmom\] \^q\_(q)= g\_q\^ ,
\[Sqmom\] \^q\_(q)= g\_q\^ &&.The sum rules are obtained by matching the OPE side with the hadronic side and applying the Borel transformation. As a result of this operation, we obtain: \[Csumq\] & e\^[M\_\^2/M\^2]{}\
& & =-\_\^2 g\_ + \_ + s\^\_0 M\^2L\^[-4/9]{} e\^[(M\_\^2-s\_0\^)/M\^2]{} ,\
and \[Ssumq\] & e\^[M\_\^2/M\^2]{}\
& & =-\_\^2 g\_ + \_ +\[(s\^\_0)\^2-2m\_s(f+1)a\_q\] M\^2L\^[-4/9]{} e\^[(M\_\^2-s\_0\^)/M\^2]{} ,for $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ couplings, respectively.
We would like to note that the sum rule for $\Xi\Xi\sigma$ coupling constant at the structure $\hat{q}$ is independent of the susceptibilities $\chi$ and $\chi_g$. Another feature of the sum rules above is that up to the dimension we consider, the terms involving the $s$-quark mass do not contribute to the OPE side. The contributions that come from the excited baryon states and the response of the continuum threshold are taken into account by the second and the third terms on the right-hand side (RHS) of the sum rules, respectively.
For the sake of completeness, here we also give the sum rule for $N\!N\sigma$ coupling constant at the structure $\hat{q}$ [@Erk05]: \[Nsumq\] & e\^[M\_N\^2/M\^2]{}\
&& =-\_N\^2 g\_[NN]{} + \_N + s\^N\_0 M\^2L\^[-4/9]{} e\^[(M\_N\^2-s\_0\^N)/M\^2]{} , which follows from a choice of the interpolating field [@Iof81] \[Nintfi\] \_N = \_[abc]{}\[u\_a\^T C\_u\_b\]\_5\^d\_c . Note that, in Eq. (\[Nsumq\]), the $N\!N\sigma$ sum rule in Ref. [@Erk05] has been improved including the quark mass terms.
Comparing the left-hand sides (LHS) of the sum rules in Eq. (\[Csumq\]), Eq. (\[Ssumq\]) and Eq. (\[Nsumq\]) one can derive a basic relation between the $\Xi\Xi\sigma$, $\Sigma\Sigma\sigma$ and $N\!N\sigma$ coupling constants in the $SU(3)$ limit, which is \[couprel\] g\_[NN]{}=g\_+g\_.This relation is quite natural because for the $\Xi\Xi\sigma$ coupling only the $u$-quark propagator outside of the trace and for the $\Sigma\Sigma\sigma$ coupling the $u$-quark propagators inside the trace involve the terms that are proportional to the external field. For the $N\!N\sigma$ coupling all the three quark propagators involve such terms and this implies that in the $SU(3)$ and isospin symmetric limit, the relation in Eq. (\[couprel\]) holds. It is interesting to note that this relation can also be derived from Eq. (\[relSU3\]) and Eq. (\[Noctsing\]) assuming the ideal mixing for the $q\bar{q}$ picture where the $N\!Nf_0$ coupling vanishes.
Analysis of the Sum Rules and Discussion {#secAN}
========================================
In this section we analyze the sum rules derived in the previous section in order to determine the values of the $\Lambda\Lambda\sigma$, $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ coupling constants. To proceed to the numerical analysis, we arrange the RHS of the sum rules in the form \[form\] f(M\^2) = A\_B + B\_B M\^2+C\_B M\^2 L\^[-4/9]{} e\^[(M\_B\^2-s\_0\^B)/M\^2]{} , and fit the LHS to $f(M^2)$. Here we have defined \[form2\] A\_B && -\_B\^2 g\_[BB]{} ,\
B\_B && , together with \[form2LC\] C\_&& \[(s\^\_0)\^2-4m\_sfa\_q\] ,\
C\_&& s\^B\_0 ,\
C\_&& \[(s\^\_0)\^2-4m\_s(f+1)a\_q\] ,for $\Lambda\Lambda\sigma$, $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ sum rules, respectively.
For the vacuum parameters, we adopt $a_{q}=0.51\pm 0.03 ~\text{GeV}^3$, and $m_0^2=0.8 ~\text{GeV}^2$ [@Ovc88]. We take the renormalization scale $\mu=0.5~\text{GeV}$ and the QCD scale parameter $\Lambda_{QCD}=0.1~\text{GeV}$. The value of the susceptibility $\chi$ has been calculated in Ref. [@Erk05] as $\chi= -10 \pm 1 ~\text{GeV}^{-1}$. The value of the susceptibility $\chi_G$ is less certain. Therefore, we consider $\chi_G$ to change in a wider range.
Scalar Meson-Baryon Coupling Constants in the $SU(3)$ Symmetric Limit
---------------------------------------------------------------------
We shall first consider the sum rules in the $SU(3)$ flavor symmetric limit, where we take $m_q=m_s=0$ and $f=0$. In this limit we also set the physical parameters of all the baryons equal to the ones of the nucleon; $M_B = M_N=0.94~\text{GeV}$, $\tilde{\lambda}_B = \tilde{\lambda}_N=2.1 ~\text{GeV}^6$ [@Iof84], $s^B_0 = s^N_0$.
In Figs. \[LambdaM\]-\[SigmaM\] we present the Borel mass dependence of the LHS and the RHS of the sum rules for $\Lambda\Lambda\sigma$, $\Sigma\Sigma\sigma$ and $\Xi\Xi\sigma$, respectively, for $s^B_0=2.3$ and $\chi_G \equiv\chi=-10$ GeV$^{-1}$. As stressed above, in the $SU(3)$ limit we choose the Borel window 0.8 GeV$^2$ $\leq M^2\leq 1.4$ GeV$^2$ which is commonly identified as the fiducial region for the nucleon mass sum rules [@Iof84]. It is seen from these figures that the LHS curves (solid) overlie the RHS curves (dashed). In order to estimate the contributions that come from the excited baryon states and the responses of the continuum threshold, we plot each term on the RHS individually. We observe that the single-pole terms (dotted) give very small contributions to the sum rules except to the one for $\Xi\Xi\sigma$ coupling. The responses of the continuum thresholds (dot-dashed) for all the couplings are quite sizable.
In order to see the sensitivity of the coupling constants on the continuum threshold and the susceptibility $\chi$, we plot in Figs. \[Lambdaks\] and \[Sigmaks\] the dependence of $g_{\Lambda\Lambda\sigma}/g_q^\sigma$ and $g_{\Sigma\Sigma\sigma}/g_q^\sigma$ on $\chi$ for three different values of the continuum thresholds, $s^B_0=2.0$, 2.3, and 2.5 GeV$^2$, and taking $\chi\equiv\chi_G$. One sees that these coupling constants change by approximately $10\%$ in the considered region of the susceptibility $\chi$. The values of the coupling constants are not very sensitive to a change in the continuum threshold, which gives an uncertainty of approximately $7\%$ to the final values.
Taking into account the uncertainties in $\chi$, $s_0^B$, and $a_q$, the predicted values for $N\!N\sigma$, $\Lambda\Lambda\sigma$, $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ coupling constants in terms of quark-$\sigma$ coupling constant read [^1]: \[mbSU3\] g\_[NN]{}/g\_q\^= 3.9 1.0, g\_/g\_q\^= 1.9 0.5, g\_/g\_q\^= 0.4 0.1, g\_/g\_q\^= 3.8 1.0.To determine the coupling constants, one next has to assume some value for the quark-$\sigma$ coupling constant $g_q^\sigma$. Adopting the value $g_q^\sigma=3.7$ as estimated from the sigma model [@Ris99], we obtain \[ccSU3\] g\_[NN]{}= 14.4 3.7, g\_=7.0 1.9, g\_= 1.5 0.4, g\_= 14.1 3.7.
Note that, the coupling constants in Eq. (\[ccSU3\]) are defined at $t=0$, [*i.e.*]{} $g_{B\!B\sigma} \equiv g_{B\!B\sigma}(t=0)$. As stressed above, the value of the susceptibility $\chi_G$ is less certain than the value of $\chi$. If we let $\chi_G$ change in a wider range, say $6 ~\text{GeV}^{-1} \leq -\chi_G \leq 14 ~\text{GeV}^{-1}$, this brings an additional $15 \%$ uncertainty to the values of $\Lambda\Lambda\sigma$ and $\Sigma\Sigma\sigma$ coupling constants but the $\Xi\Xi\sigma$ coupling constant remains intact because the sum rule is independent of the susceptibilities as stressed above. In order to keep consistent with the analysis in Ref. [@Erk05], here we also have taken $\Lambda_{QCD}=0.1~\text{GeV}$. A change in the value of this parameter, say an increase to $\Lambda_{QCD}=0.2~\text{GeV}$, does not have any considerable effect on $\Xi\Xi\sigma$ coupling constant, but the $N\!N\sigma$ and $\Sigma\Sigma\sigma$ coupling constants are increased by approximately $8\%$, while the increase in the value of $\Lambda\Lambda\sigma$ coupling constant is by $5\%$.
Our next concern is to investigate the $SU(3)$ relations for the scalar meson-baryon interactions and see if the coupling constants above as obtained from QCDSR are consistent with these relations. The values of three coupling constants as determined from QCDSR together with the first equation in Eq. (\[Noctsing\]) are sufficient to determine the three parameters of flavor $SU(3)$ structure of scalar meson-baryon couplings; namely $g_1$, $g$, and $\alpha_s$. For this purpose, we calculate the coupling constants in Eq. (\[ccSU3\]) with the average values of the parameters; $\chi\equiv\chi_g=-10~\text{GeV}^{-1}$, $a_q=0.51~\text{GeV}^2$ and $s^B_0=2.3~\text{GeV}^2$ where we obtain: \[mbavSU3\] g\_[NN]{}/g\_q\^= 4.0, g\_/g\_q\^= 1.7, g\_/g\_q\^= 0.3, g\_/g\_q\^= 3.6.We first assume $q\bar{q}$ structure with the ideal mixing angle $\theta_s\simeq 35.3^\circ$, and use $g_{N\!N\sigma}$, $g_{\Xi\!\Xi\sigma}$ and $g_{\Sigma\!\Sigma\sigma}$ in Eq. (\[mbavSU3\]). The $F/(F+D)$ ratio, $\alpha_s$, can directly be calculated via the relation, \[relalpha\] (g\_-g\_[NN]{})/(g\_-g\_[NN]{})=.With straightforward algebra, the values of the $F/(F+D)$ ratio, and the octet and the singlet couplings for the $q\bar{q}$ picture are determined as, \[cossu3\]\_s=F/(F+D)&=&0.55, g/g\_q\^=g\_[NNa\_0]{}/g\_q\^=3.3, g\_1/g\_q\^=3.2.
Inserting $\alpha_s$ and $g$ into the $SU(3)$ relations in Eq. (\[relSU3\]) and using the mixing scheme for the singlet and the octet couplings as in Eq. (\[Noctsing\]) with the value of $g_1$ in Eq. (\[cossu3\]), we observe that the coupling constants as determined from QCDSR in Eq. (\[mbavSU3\]) are consistent with the $SU(3)$ relations. This also gives $g_{N\!Nf_0}=0$ with the second equation in Eq. (\[Noctsing\]), which is justified by the non-strange content of the nucleon and by the ideal mixing scheme. In Table \[mbcq2\] we give all the scalar meson-baryon coupling constants, obtained from these relations, assuming $g_q^\sigma=3.7$.
M $N\!NM$ $\Lambda\Lambda M$ $\Xi\Xi M$ $\Sigma \Sigma M$ $\Lambda\Sigma M$ $\Sigma N M$ $\Lambda N M$
---------- --------- -------------------- ------------ ------------------- ------------------- -------------- ---------------
$\sigma$ 14.6 6.2 1.3 13.3
$f_0$ 0 $-12.0$ $-18.9$ $-1.8$
$a_0$ 12.0 1.3 13.3 6.2
$\kappa$ $-1.3$ $-14.7$
: The scalar meson-baryon coupling constants in the $SU(3)$ limit where the $q\bar{q}$ picture for the scalar mesons with the ideal mixing is assumed.
\[mbcq2\]
In the case of the $q^2\bar{q}^2$ picture, we have a quite distinct ideal mixing scheme for the scalar mesons from that for the $q\bar{q}$ picture. The ideal mixing angle corresponds to $\theta_s\simeq-54.8^\circ$ in this case. In this picture, we assume that the $u$- or the $d$-quark in the baryon couples to the $q\bar{q}$ component in the scalar meson but not to the $s\bar{s}$ component and the $\qqbar$ condensates are modified in the same way. Accordingly, the $s$-quark only couples to the $s\bar{s}$ component. Applying the same procedure as in the $q\bar{q}$ case with $\theta_s=-54.8^\circ$, we obtain the values of $F/(F+D)$ ratio, and the octet and the singlet couplings for the $q^2\bar{q}^2$ picture as: \[alpnnaq4\]\_s=F/(F+D)&=&0.55, g/g\_q\^=g\_[NNa\_0]{}/g\_q\^=-2.3, g\_1/g\_q\^=4.6.Inserting these values into the $SU(3)$ relations in Eq. (\[relSU3\]), we observe that the scalar meson-baryon coupling constants as found from QCDSR are consistent with the $SU(3)$ relations, as in the $q\bar{q}$ picture.
In Table \[mbcq4\], we present the scalar meson-baryon coupling constants $g_1$, $g_8$ and $\alpha_s$ in the $q^2\bar{q}^2$ picture, assuming $g_q^\sigma=3.7$. Comparing what we have found for the scalar meson-baryon couplings in the two pictures, the value of the $F/(F+D)$ ratio remains intact, as apparent from Eq. (\[relalpha\]), however, the values of $g$ and $g_1$ in the two pictures are quite different from each other. On the other hand, the $B\!Bf_0$ couplings differ very much with regard to the structure of the scalar mesons. In the $q^2\bar{q}^2$ picture for the scalar mesons, in contrary to the $q\bar{q}$ picture, $f_0$ strongly couples to the nucleon due to $\bar{u}u$ and $\bar{d}d$ components it has. The strengths of the $I=1$ couplings in the two pictures differ by a factor of $\sqrt{2}$.
M $N\!NM$ $\Lambda\Lambda M$ $\Xi\Xi M$ $\Sigma \Sigma M$ $\Lambda\Sigma M$ $\Sigma N M$ $\Lambda N M$
---------- --------- -------------------- ------------ ------------------- ------------------- -------------- ---------------
$\sigma$ 14.6 6.2 1.3 13.3
$f_0$ 10.3 16.2 19.7 11.3
$a_0$ $-8.5$ $-0.9$ $-9.4$ $-4.4$
$\kappa$ 0.9 10.3
: Same as Table \[mbcq2\] but for the $q^2\bar{q}^2$ picture for the scalar mesons.
\[mbcq4\]
Sigma Meson-Baryon Coupling Constants with $SU(3)$ Breaking Effects
-------------------------------------------------------------------
Now we turn to the effect of $SU(3)$ breaking, where we allow $m_s=0.15~\text{GeV}$ and $f=-0.2$, keeping $m_q=0$. We also restore the physical values for the parameters of baryons [@Hwa94; @Chi85]: \[parSU3b\] M\_&=&1.1 , M\_=1.3 , M\_=1.2 ,\
\_\^2&=&3.3 \^6, \^2\_=4.6 \^6, \^2\_=3.3 \^6,\
s\_0\^&=& 3.1 0.3 \^2, s\_0\^= 3.6 0.4 \^2, s\_0\^= 3.2 0.3 \^2.The corresponding Borel windows are chosen as: \[bwSU3b\], 1.0 \^2 M\^2 1.4 \^2,\
, 1.5 \^2 M\^2 1.9 \^2,\
, 1.2 \^2 M\^2 1.6 \^2,
In Figs. \[LambdaSU3b\]-\[SigmaSU3b\] we present the Borel mass dependence of the LHS and the RHS of the sum rules for $\Lambda\Lambda\sigma$, $\Sigma\Sigma\sigma$ and $\Xi\Xi\sigma$, respectively, with the $SU(3)$ breaking effects. We plot each term on the RHS individually as we did in the $SU(3)$ limit. We observe that the responses of the continuum thresholds for all the couplings are quite sizable. The contributions of the single pole terms to the $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ sum rules are large and opposite in sign to the third terms on the RHS. Therefore the contributions of these two terms tend to cancel each other which leads to a very stable sum rule for these couplings. The contribution of the single pole term for the $\Lambda\Lambda\sigma$ coupling is very small.
Taking into account the uncertainties in $\chi$, $s_0^B$, and $a_q$, the predicted values for $\Lambda\Lambda\sigma$, $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ coupling constants in terms of quark-$\sigma$ coupling constant with the $SU(3)$ breaking effects read: \[mbSU3b\] g\_/g\_q\^= 2.0 0.5, g\_/g\_q\^= 0.5 0.1, g\_/g\_q\^= 5.7 1.4.Adopting again the value $g_q^\sigma=3.7$ as estimated from the sigma model [@Ris99], we obtain \[ccSU3b\] g\_= 7.4 1.9, g\_= 1.9 0.4, g\_= 21.1 5.2 .
A few remarks are in order now. Comparing the values obtained from the sum rules in the $SU(3)$ symmetric limit and the ones beyond the $SU(3)$-limit, we observe that the introduction of the $SU(3)$ breaking effects does not change the $\Lambda\Lambda\sigma$ coupling constant, while the $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ couplings are modified by approximately $\%30$ and $\%50$, respectively. We also note that, the obtained value of $\Lambda\Lambda\sigma$ coupling constant is small as compared to the $N\!N\pi$ and $N\!N\sigma$ coupling constants. Since the $\sigma$ exchange gives the dominant contribution in the $\Lambda\Lambda$ system, this suggests that the $\Lambda\Lambda$ interaction is weak, in accordance with the recent experimental result.
Discussion and Conclusions {#secCONC}
==========================
We have calculated the $\Lambda\Lambda\sigma$, $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ coupling constants which play significant roles in OBE models of YN and YY interactions, employing the external field QCDSR method. The coupling constants can be determined in terms of quark-$\sigma$ coupling constant in this method. In order to compare our results with the others in the literature and keep as model-independent as possible, we find it useful to give the ratios of the coupling constants in the $SU(3)$ limit for the average values of the vacuum parameters, \[avmbdis\]=0.43, =0.08, =0.91.We observe that the $\Xi\Xi\sigma$ coupling constant is more than one order small as compared to $N\!N\sigma$ coupling constant. The $\Sigma\Sigma\sigma$ coupling constant is at the same order with $N\!N\sigma$ coupling constant and twice as large as the $\Lambda\Lambda\sigma$ coupling constant. We have shown that these coupling constants as determined from QCDSR satisfy the $SU(3)$ relations which lead to a determination of the $F/(F+D)$ ratio for the scalar octet. Although the scalar meson-baryon coupling constants depend on the picture assumed for the structure of the scalar mesons ($q\bar{q}$ or $q^2\bar{q}^2$), the $F/(F+D)$ ratio remains intact in the two pictures. We would like to also note that the third terms on the RHS’s of the sum rules which represent the responses of the continuum thresholds, affect only the values of the individual coupling constants, which receive contribution by the same factor. Therefore the ratios of the coupling constants and the value of $\alpha_{s}$ remain unchanged if this term is omitted.
All the Nijmegen soft-core OBE potential models have $\theta_s > 30^\circ$, which points to almost ideal mixing angles for the scalar $q\bar{q}$ states. Since the ideal mixing angle for the scalar mesons has been assumed in our QCDSR calculations as well, it is convenient to compare our results with the ones from NSC potential models. Our result for the ratio $g_{\Lambda\Lambda\sigma}/g_{NN\sigma}$ is half of the value found in Ref. [@Fer05], however, it qualitatively agrees with the one from NSC89 [@Mae89], which is $g_{\Lambda\Lambda\sigma}/g_{NN\sigma}=0.58$. The value we have obtained for the $\Lambda\Lambda\sigma$ coupling constant is small as compared to $N\!N\sigma$ coupling constant and this implies that $\Lambda\Lambda$ interaction is weak, since the sigma exchange gives the dominant contribution to this interaction in terms of OBE models. The value of the $F/(F+D)$ ratio, which is $0.55$ as obtained from QCDSR is about half of the values given in NSCa-f [@Rij98], which is $F/(F+D)\simeq1.1$.
In order to estimate the $SU(3)$ breaking in the couplings, we have restored the physical values of the parameters like the strange quark mass and the physical baryon masses. We observe that the $SU(3)$ breaking effects do not change the $\Lambda\Lambda\sigma$ coupling, while the $\Xi\Xi\sigma$ and $\Sigma\Sigma\sigma$ couplings are modified largely. It is also desirable to derive the sum rules for the $BBf_0$ and $BBa_0$ couplings in order to estimate the $SU(3)$ breaking in these coupling constants.
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![The Borel mass dependence of LHS and the fitted RHS of the sum rule for $\Lambda\Lambda \sigma$ coupling in Eq. (\[sumq\]) for $s^\Lambda_0=2.3$ GeV$^2$ and $\chi_G\equiv\chi=-10 $ GeV$^{-1}$. We also present the terms on the RHS individually. Note that the LHS curve (solid) overlies the RHS curve (dashed).[]{data-label="LambdaM"}](LambdaM.eps)
![Same as Fig. \[LambdaM\] but for the sum rule for $\Xi\Xi\sigma$ coupling in Eq. (\[Csumq\]).[]{data-label="CascadeM"}](CascadeM.eps)
![Same as Fig. \[LambdaM\] but for the sum rule for $\Sigma\Sigma\sigma$ coupling in Eq. (\[Ssumq\]).[]{data-label="SigmaM"}](SigmaM.eps)
![The dependence of $g_{\Lambda\Lambda\sigma}/g_q^\sigma$ on the susceptibility $\chi$ for three different values of $s_0^q=2.0$, 2.3, and 2.5 GeV$^2$; here we take $\chi\equiv\chi_G$.[]{data-label="Lambdaks"}](Lambdaks.eps)
![Same as Fig. \[Lambdaks\] but for the sum rule for the $g_{\Sigma\Sigma\sigma}/g_q^ \sigma$ coupling constant.[]{data-label="Sigmaks"}](Sigmaks.eps)
![The Borel mass dependence of LHS and the fitted RHS of the sum rule for $\Lambda\Lambda \sigma$ coupling in Eq. (\[sumq\]) with the $SU(3)$ breaking effects and for $\chi_G\equiv \chi=-10$ GeV$^{-1}$. We also present the terms on the RHS individually. Note that the LHS curve (solid) overlies the RHS curve (dashed).[]{data-label="LambdaSU3b"}](LambdaSU3b.eps)
![Same as Fig. \[LambdaSU3b\] but for the sum rule for $\Xi\Xi\sigma$ coupling in Eq. (\[Csumq\]).[]{data-label="CascadeSU3b"}](CascadeSU3b.eps)
![Same as Fig. \[LambdaSU3b\] but for the sum rule for $\Sigma\Sigma\sigma$ coupling in Eq. (\[Ssumq\]).[]{data-label="SigmaSU3b"}](SigmaSU3b.eps)
[^1]: We refer the reader to Ref. [@Erk05] for a detailed analysis of $N\!N\sigma$ coupling constant in QCDSR.
|
---
abstract: 'We introduce a gradient flow formulation of linear Boltzmann equations. Under a diffusive scaling we derive a diffusion equation by using the machinery of gradient flows.'
address:
- 'Giada Basile Dipartimento di Matematica, Università di Roma ‘La Sapienza’ P.le Aldo Moro 2, 00185 Roma, Italy'
- 'Dario Benedetto Dipartimento di Matematica, Università di Roma ‘La Sapienza’ P.le Aldo Moro 2, 00185 Roma, Italy'
- 'Lorenzo Bertini Dipartimento di Matematica, Università di Roma ‘La Sapienza’ P.le Aldo Moro 2, 00185 Roma, Italy'
author:
- Giada Basile
- Dario Benedetto
- Lorenzo Bertini
title: A gradient flow approach to linear Boltzmann equations
---
Introduction
============
The Boltzmann equation describes the evolution of the one-particle distribution on position and velocity of a rarefied gas. It has become a paradigmatic equation since it encodes most of the conceptual and technical issues in the description of the statistical properties for out of equilibrium systems. In particular, from a mathematical point of view, a global existence and uniqueness result is still lacking. In the kinetic regime, some transport phenomena can be described by linear Boltzmann equations. Typical examples are the charge (or mass) transport in the Lorentz gas [@Lo], the evolution of a tagged particle in a Newtonian system in thermal equilibrium [@Sp], and the propagation of lattice vibrations in insulating crystals [@BOS]. Since the evolution equations are linear, their analysis is simpler. From one side, these equations have been derived from an underlying microscopic dynamics globally in time [@Ga; @vBLLS; @BGS-R; @BOS]. From the other side, several results on the asymptotic behavior of the one-particle distribution have been obtained. In particular, by considering non degenerate scattering rates, under a diffusive rescaling the linear Boltzmann equation converges to the heat equation [@LK; @BLP; @BSS; @EP].
In the present paper, inspired by the general theory in [@AGS], we propose a formulation of linear Boltzmann equations in terms of gradient flows. Recently there have been some attempts to formulate the Fokker-Planck equation associated to continuous time reversible Markov chains, equivalently homogeneous linear kinetic equations, as gradient flows [@Ma; @Mi; @Er], essentially in terms of energy variational inequalities, and the potential applications of this approach have yet to be fully investigated. The case of homogeneous non linear Boltzmann equations is considered in [@Er2]. The present approach is based on an entropy dissipation inequality and can be applied naturally to the inhomogeneous case, that appears novel. In perspective, this approach could be adapted to the non linear, non homogeneous Boltzmann equation.
We consider a linear Boltzmann equation of the form $$\label{BE}
(\partial_t +b(v)\cdot \nabla_x )
f(t,x,v)=
\int\pi({\mathop{}\!\mathrm{d}}v')\sigma(v,v')\big[f(t,x,v')- f(t,x,v) \big]$$ where $\pi({\mathop{}\!\mathrm{d}}v)$ is a reference probability measure on the velocity space, $b$ is the drift, $\sigma(v,v')\geq 0$ is the scattering kernel and $f$ is the density of the one-particle distribution with respect to ${\mathop{}\!\mathrm{d}}x\,\pi({\mathop{}\!\mathrm{d}}v)$. We assume the detailed balance condition, i.e. $\sigma(v,v')=\sigma(v',v)$. The entropy $\mathcal{H}(f)=\int {\mathop{}\!\mathrm{d}}x\int\pi({\mathop{}\!\mathrm{d}}v)f\log f$ is a Lyapunov functional for the evolution , and the transport term do not affect its rate of decrease. This observation will allow to formulate as the following entropy dissipation inequality $$\label{EDI}
\mathcal{H}\big(f(T)\big) +\int_0^T {\mathop{}\!\mathrm{d}}t\, \mathcal{E}\big( f(t)\big) + \mathcal R_0 (f)\leq \mathcal H \big(f(0)\big),$$ where $\mathcal E\big( f\big)=\displaystyle\int {\mathop{}\!\mathrm{d}}x\,\iint \pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\sigma(v,v')\big[\sqrt{f(x,v')}- \sqrt{f(x,v)} \big]^2$ is the Dirichlet form of the square root of $f$ and $\mathcal R_0\geq
0$ is a kinematic term that will be defined later.
As an application of we discuss the diffusive limit of the linear Boltzmann equation. This is a classical result, but the gradient flow formulation provides a transparent proof and allows to consider more general initial conditions, which are only required to satisfy an entropy bound. More precisely, we will show that in the diffusive scaling limit the particle density converges to the solution of the heat equation. The proof will be achieved by taking the limit in the rescaled entropy dissipation inequality and deducing the corresponding inequality for the heat equation.
A gradient flow formulation
===========================
In this section we introduce a gradient flow formulation of non-homogeneous linear kinetic equations. Both for ease of presentation and for future use, we however first review the gradient flow formulation of the heat equation, that is here considered in somewhat different setting that includes the current as a dynamical variable.
Throughout the whole paper, the space domain is the $d$-dimensional torus ${{\mathbb T}}^d:= {{\mathbb R}}^d/{{\mathbb Z}}^d$ and we denote by ${\mathop{}\!\mathrm{d}}x$ the Haar measure on ${{\mathbb T}}^d$. The set of Borel probability measures on ${{\mathbb T}}^d$ is denoted by ${{\mathcal P}}({{\mathbb T}}^d)$ that we consider endowed with the (metrizable) topology induced by the weak convergence. Recall finally that the *entropy* is the convex lower semicontinuous functional $H\colon {{\mathcal P}}({{\mathbb T}}^d) \to [0,+\infty]$ defined by $H(\mu) =
\int\!{\mathop{}\!\mathrm{d}}x\, \rho \log \rho$ if ${\mathop{}\!\mathrm{d}}\mu=\rho\,{\mathop{}\!\mathrm{d}}x$ and $H(\mu)=+\infty$ otherwise.
Heat equation {#heat-equation .unnumbered}
-------------
We start by an informal discussion. Consider the heat equation on ${{\mathbb T}}^d$ $$\partial_t \rho=\nabla\cdot D\nabla \rho$$ where $\rho$ is a probability density and the diffusion coefficient $D$ is a positive symmetric $d\times d$ matrix. We introduce the currents as vector fields on ${{\mathbb T}}^d$, denoted by $j$. Given $\rho$, we define the associated current $j^\rho:=
-D\nabla\rho$. We can then rewrite the heat equation as $$\label{he0}
\begin{cases}
\partial_t\rho+\nabla \cdot j=0\\
j=j^\rho.
\end{cases}$$ We shall rewrite this system as a variational inequality that expresses the decrease of the entropy.
Fix $T>0$. On the set of paths $(\rho(t),j(t))$, $t\in [0,T]$, satisfying the continuity equation $\partial_t \rho+\nabla \cdot j=0$ consider the action functional $$I(\rho,j) = \frac 12 \int_0^T\!{\mathop{}\!\mathrm{d}}t \int\! {\mathop{}\!\mathrm{d}}x \frac 1{\rho(t)}
\big[ j(t) +D \nabla\rho(t) \big] \cdot D^{-1}
\big[ j(t) +D \nabla \rho(t) \big],$$ where $\cdot$ denotes the inner product in ${{\mathbb R}}^d$. This functional arises naturally by analyzing the large deviation asymptotics of $N$ independent Brownians [@DG; @KO] and its connection with the gradient flow formulation of the heat equation is discussed in [@ADPZ]. To be precise, the rate function in [@DG; @KO; @ADPZ] does not include the current as a dynamical variable but it can be extended to this case, see [@BDGJL] for a similar functional in the context of stochastic lattice gases.
Observe that $I\geq 0$ and $I(\rho, j)=0$ if and only if $j=j^\rho$. Hence the second equation in is equivalent to $I(\rho, j)\leq 0$. By expanding the square we deduce $$\label{ineq-calore}
\int_0^T\!{\mathop{}\!\mathrm{d}}t \int\!{\mathop{}\!\mathrm{d}}x\,
\Big[ \frac 12 \frac 1{\rho(t)} j(t) \cdot D^{-1} j(t)
+ \frac 12 \frac 1{\rho(t)} \nabla\rho(t) \cdot D \nabla \rho(t)
+ \frac 1{\rho(t)} \nabla \rho(t) \cdot j(t) \Big]\leq 0.$$ Since $(\nabla \rho)/\rho =\nabla \log\rho$, integrating by parts and using the continuity equation, the last term is the total derivative of $H(\rho(t))$.
We now introduce the *Fisher information* $E$ as the Dirichlet form of square root, namely $$E(\rho)=\frac 12\int\! {\mathop{}\!\mathrm{d}}x \frac 1{\rho} \nabla\rho \cdot D \nabla
\rho = 2\int \,{\mathop{}\!\mathrm{d}}x\, \nabla\sqrt\rho\cdot D\nabla\sqrt\rho.$$ Let also the *kinematic term* $R$ be the functional on the set the path $(\rho(t),j(t))$ defined by $$R (\rho, j)=\frac 12\int_0^T\! {\mathop{}\!\mathrm{d}}t\,\int\!{\mathop{}\!\mathrm{d}}x\,
\frac 1{\rho(t)} j(t) \cdot D^{-1} j(t),$$ then reads $$\label{gfhe}
H(\rho(T))+\int_0^T \!{\mathop{}\!\mathrm{d}}t\, E(\rho(t))+ R (\rho, j) \leq H(\rho(0))$$ which is the gradient flow formulation of the heat equation that we will use here.
We now specify the precise formulation in which we consider a family of probabilities $\mu_t({\mathop{}\!\mathrm{d}}x)=\rho(t,x){\mathop{}\!\mathrm{d}}x$, $t\in [0, T]$, while the currents are the vector valued measures $J({\mathop{}\!\mathrm{d}}t, {\mathop{}\!\mathrm{d}}x)=j(t,x){\mathop{}\!\mathrm{d}}t {\mathop{}\!\mathrm{d}}x$. Given $T>0$ let $C\big([0,T]; {{\mathcal P}} ({{\mathbb T}}^d)\big)$ be the set of continuous paths on ${{\mathcal P}}({{\mathbb T}}^d)$ endowed with the topology of uniform convergence. Let also ${{\mathcal M}}\big([0,T]\times {{\mathbb T}}^d; {{\mathbb R}}^d \big)$ be the set of vector valued Radon measures on $[0,T]\times {{\mathbb T}}^d$ endowed with the weak\* topology. Set $S:=C\big([0,T]; {{\mathcal P}} ({{\mathbb T}}^d)\big)\times {{\mathcal M}}\big([0,T]\times
{{\mathbb T}}^d; {{\mathbb R}}^d \big)$ endowed with the product topology.
Given a positive $d\times d$ matrix $D$, the Fisher information $E\colon {{\mathcal P}}({{\mathbb T}}^d)\to[0,\infty]$ can be defined by the variational formula $$\label{varD-calore}
E (\mu)=2\sup_{\phi\in C^2({{\mathbb T}}^d)}
\Big\{ -\int \!{\mathop{}\!\mathrm{d}}\mu \, e^{-\phi}\nabla \cdot D \nabla e^\phi\Big\},$$ which implies its lower semicontinuity and convexity. The kinematic term $ R \colon S\to [0,\infty]$ admits the variational representation $$\label{varR-calore}
R (\mu,J)=\sup_{w\in C([0,T]\times {{\mathbb T}}^d; {{\mathbb R}}^d)}
\Big\{J(w)-\frac 12 \int_0 ^T \!{\mathop{}\!\mathrm{d}}t\int \!{\mathop{}\!\mathrm{d}}\mu_t\,w\cdot Dw \Big\},$$ which implies its lower semicontinuity and convexity.
\[he\] Let $\nu\in{{\mathcal P}}({{\mathbb T}}^d)$ with $H(\nu)<+\infty$. A path $(\mu, J)\in
S$ is a solution of the heat equation with initial condition $\nu$ iff $\mu_0=\nu$ and $$\begin{aligned}
\label{eqcont}
&&\int_0^T \!{\mathop{}\!\mathrm{d}}t \,\mu_t(\partial_t\phi) +J(\nabla \phi)=0,\qquad
\phi\in C_c^1\big((0,T)\times {{\mathbb T}}^d \big)\\
\label{ineq1-calore}
&& H(\mu_T)+\int_0^T \!{\mathop{}\!\mathrm{d}}t\, E(\mu_t)+ R (\mu, J) \leq H(\nu).\end{aligned}$$
The standard formulation of the heat equation as gradient flow of the entropy is recovered from by projecting on the density. Indeed, by the Benamou-Brenier lemma [@BeBr], we deduce that if $(\mu,J)$ is a solution to the heat equation according to Definition \[he\], then $\mu=(\mu_t)_{t\in[0,T]}$ satisfies $$\label{hemu}
H(\mu_T)+\int_0^T \!{\mathop{}\!\mathrm{d}}t\, \Big\{ E(\mu_t)+
\frac 12 \big| \dot \mu_t \big|^2 \Big\} \leq H(\nu)$$ where $ \big| \dot \mu_t \big|$ is the metric derivative of $t\mapsto \mu_t$ with respect to the Wasserstein-$2$ distance, namely $\big| \dot \mu_t \big| =\lim_{h\to 0}
\mathrm{d}_{W_2}\big(\mu_{t+h},\mu_t\big)/h$, where $\mathrm{d}_{W_2}$ denotes the Wasserstein-$2$ distance on ${{\mathcal P}}({{\mathbb T}}^d)$.
Conversely, let $\mu$ be a solution to satisfying $\mu_0=\nu$. Introduce the functional $J^\mu$ on $C^1\big([0,T]\times
{{\mathbb T}}^d;{{\mathbb R}}^d\big)$ defined by $J^\mu(w) = \int_0^T\!{\mathop{}\!\mathrm{d}}t\,\mu_t \big(\nabla\cdot D w_t\big)$. Since $\int_0^T\!{\mathop{}\!\mathrm{d}}t \, E(\mu_t) \le H(\nu)$, the functional $J^\mu$ extends to an element of ${{\mathcal M}}\big( [0,T]\times {{\mathbb T}}^d;{{\mathbb R}}^d\big)$, still denoted by $J^\mu$. Using again the Benamou-Brenier lemma it is then straightforward to check that the pair $(\mu,J^\mu)$ is a solution to the heat equation in the sense of Definition \[he\].
The previous remarks, together with the existence and uniqueness result for the formulation in [@Gi], imply the following statement.
\[t:uniq\] For each $\nu\in{{\mathcal P}}({{\mathbb T}}^d)$, with $H(\nu)< \infty$, there exists a unique solution of the heat equation with initial condition $\nu$.
Linear Boltzmann equations {#linear-boltzmann-equations .unnumbered}
--------------------------
We do not need any particular hypotheses on the velocity space ${{\mathcal V}}$ that is assumed to be a Polish space, i.e. a metrizable complete and separable topological space. We denote by ${{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ the set of probabilities on ${{\mathbb T}}^d\times {{\mathcal V}}$, that we consider endowed with the topology of weak convergence. We suppose given a Borel probability measure $\pi$ on ${{\mathcal V}}$, a symmetric scattering kernel $\sigma$, i.e. a Borel function $\sigma\colon {{\mathcal V}}\times {{\mathcal V}} \to [0,+\infty)$ satisfying $\sigma(v,v')=\sigma(v',v)$, $v,v'\in
{{\mathcal V}}$, and a *drift* $b\colon {{\mathcal V}}\to {{\mathbb R}}^d$. Given $P\in {{\mathcal P}}({{\mathbb T}}^d\times{{\mathcal V}})$, we denote by ${{\mathcal H}}(P)$ the relative entropy of $P$ with respect to the probability ${\mathop{}\!\mathrm{d}}x\,\pi({\mathop{}\!\mathrm{d}}v)$ namely, ${{\mathcal H}}(P)
= \iint \!{\mathop{}\!\mathrm{d}}x \pi({\mathop{}\!\mathrm{d}}v)\, f \log f $ if ${\mathop{}\!\mathrm{d}}P=f \,{\mathop{}\!\mathrm{d}}x\,\pi({\mathop{}\!\mathrm{d}}v)$ and ${{\mathcal H}}(P)=+\infty$ otherwise.
Also in this case we start by an informal discussion. Fix $T>0$. Given a path $(P_t)_{t\in[0,T]}$ on ${{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ with ${\mathop{}\!\mathrm{d}}P_t = f(t,x,v) \, {\mathop{}\!\mathrm{d}}x
\,\pi({\mathop{}\!\mathrm{d}}v)$, we use the shorthand notation $f=f(t,x,v)$, $f'=f(t,x,v')$ and set $$\label{etaf}
\eta^f =\eta^f(t,x,v,v')
:= \sigma (f-f') = \sigma(v,v') \big[ f(t,x,v)-f(t,x,v')\big].$$ We then rewrite the linear Boltzmann equation in the form $$\label{continuita}
\begin{cases}
\big(\partial_t +b(v)\cdot \nabla_x \big) f(t,x,v)
+\int\! \pi(dv')\, \eta(t,x,v,v')=0\\
\eta = \eta^f
\end{cases}$$ We understand that the first equation has to be satisfied weakly and we shall refer to it as the *balance* equation. We are going to rewrite the condition $\eta=\eta^f$ as an inequality that expresses the decrease of the relative entropy ${{\mathcal H}}$. To this end, given $\varkappa\geq 0$ let $\Phi_\varkappa\colon {{\mathbb R}}_+\times{{\mathbb R}}_+\times {{\mathbb R}}\to [0,+\infty)$ be the convex function defined by $$\Phi_\varkappa (p,q;\xi) :=\sup_{\lambda\in{{\mathbb R}}}\Big\{\lambda \xi -
\varkappa p \big(e^\lambda -1) - \varkappa q \big(e^{-\lambda}
-1\big)\Big\}$$ observing that given $p,q\in{{\mathbb R}}_+$ the map $\xi\mapsto
\Phi_\varkappa(p,q;\xi)$ is positive (take $\lambda=0$), and equal to zero iff $\xi=\varkappa (p-q)$. Explicitly, as few computations shows, $\Phi_\varkappa$ reads $$\label{Phi}
\begin{split}
\Phi_\varkappa (p,q;\xi) &=
\xi \Big[ {\mathop{\rm ash}\nolimits}\frac \xi{2 \varkappa\sqrt{pq}}
-{\mathop{\rm ash}\nolimits}\frac {\varkappa (p-q)}{2 \varkappa\sqrt{pq}} \Big]
\\
& - \Big[ \sqrt{ \xi^2 + 4 \varkappa^2 pq} -
\sqrt{ \big[\varkappa(p-q)\big]^2 + 4 \varkappa^2 pq}\Big]
\end{split}$$ where we recall that ${\mathop{\rm ash}\nolimits}(z) = \log(z+\sqrt{1+z^2})$. We note that if $\varkappa=0$ then $\Phi_0(p,q;0)=0$ while $\Phi_0(p,q;\xi)=+\infty$ if $\xi\neq 0$.
Fix a path $(f(t),\eta(t))$, $t\in[0,T]$ satisfying $\eta(t,x,v,v')=-\eta(t,x,v',v)$ and the balance equation in . The condition $\eta(t)= \eta^{f(t)}$, $t\in[0,T]$ is equivalent to $$\label{jj1}
{{\mathcal I}} (f,\eta) := \int_0^T\!{\mathop{}\!\mathrm{d}}t \int\! {\mathop{}\!\mathrm{d}}x
\iint\! \pi({\mathop{}\!\mathrm{d}}v)\,\pi({\mathop{}\!\mathrm{d}}v')\,
\Phi_\sigma (f,f';\eta) \le 0.$$ This functional is connected with the large deviations asymptotic of a Markov chain on ${{\mathcal V}}$ with transition rates $\sigma(v',v)\pi({\mathop{}\!\mathrm{d}}v')$, see [@BFG; @MPR].
We next write $$\label{psi}
\Phi_\varkappa (p,q;\xi) = \Phi_\varkappa(p,q;0)
+\xi \frac{\partial}{\partial \xi}
\Phi_\varkappa(p,q;0) +\Psi_\varkappa(p,q;\xi).$$ By few explicit computations, $$\begin{split}
&\Phi_\varkappa(p,q;0) = \varkappa \big(\sqrt{p} -\sqrt{q}\big)^2\\
&\frac{\partial}{\partial \xi} \Phi_\varkappa(p,q;0)
= \frac 12 \log \frac qp\\
&\Psi_\varkappa(p,q;\xi) =
\xi {\mathop{\rm ash}\nolimits}\frac \xi{2 \varkappa\sqrt{pq}}
- \Big[ \sqrt{ \xi^2 + 4 \varkappa^2 pq} - 2\varkappa \sqrt{pq} \Big].
\end{split}$$ Observe that $\Psi_\varkappa$ has the variational representation $$\label{legpsi}
\Psi_\varkappa(p,q;\xi) =\sup_{\lambda\in {{\mathbb R}}} \Big\{ \lambda \xi -
2\varkappa \sqrt{pq} \big[ {\mathop{\rm ch}\nolimits}\lambda -1\big] \Big\}.$$ In particular, $\Psi_\varkappa\ge 0$. Moreover, while the map $(p,q;\xi)\mapsto \Psi_\varkappa(p,q;\xi)$ is convex, the map $\xi\mapsto
\Psi_\varkappa(p,q;\xi)$ is strictly convex. Finally, $\Psi_\varkappa(p,q;\xi)\sim \xi^2$ for $\xi$ small and $\Psi_\varkappa(p,q;\xi)\sim |\xi|\log|\xi|$ for $\xi$ large.
Observe now that for the path $(f(t), \eta(t))$, $t\in[0,T]$ satisfying the balance equation in we have $$\begin{split}
\frac{d}{dt} {{\mathcal H}}(f(t)) &= \int\! {\mathop{}\!\mathrm{d}}x \int\! \pi({\mathop{}\!\mathrm{d}}v) \, \log f
\Big[ - b(v)\cdot\nabla_x f
- \int \pi({\mathop{}\!\mathrm{d}}v') \, \eta(t,x,v,v')\Big]
\\
&= - \int\!{\mathop{}\!\mathrm{d}}x \iint \!
\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\, \eta\log f
\end{split}$$ since the first term is a total derivative in $x$. Hence, by the antisymmetry of $\eta$, $$\int\! {\mathop{}\!\mathrm{d}}x
\iint\! \pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v') \,
\eta \frac{\partial}{\partial \xi} \Phi_\varkappa(f,f';0)
= \frac{d}{dt} {{\mathcal H}}(f(t)).$$ for any $\varkappa >0$. Setting $\varkappa = \sigma$, inserting and integrating in time we obtain that, for any $(f(t), \eta(t))$, $t\in[0,T]$ satisfying the balance equation, it holds $$\label{uguh}
\begin{split}
{{\mathcal H}}(f(T)) + \int_0^T\!{\mathop{}\!\mathrm{d}}t \int\! {\mathop{}\!\mathrm{d}}x \iint\!
\pi({\mathop{}\!\mathrm{d}}v) \pi({\mathop{}\!\mathrm{d}}v') \,
\big[ \Phi_{\sigma}(f,f';0)+\Psi_{\sigma}(f,f';\eta)\big]
\\
= {{\mathcal H}}(f(0)) + \int_0^T\!{\mathop{}\!\mathrm{d}}t \int \!{\mathop{}\!\mathrm{d}}x \iint\!
\pi({\mathop{}\!\mathrm{d}}v) \pi({\mathop{}\!\mathrm{d}}v') \,
\Phi_{\sigma}(f,f';\eta).
\end{split}$$ Gathering the above computations we conclude that can be rewritten as $$\label{jj2}
{{\mathcal H}}(f(T)) + \int_0^T\!{\mathop{}\!\mathrm{d}}t \, {{\mathcal E}}({f(t)}) + {{\mathcal R}} (f,\eta)
\le {{\mathcal H}}(f(0))$$ where $$\label{dfsr}
{{\mathcal E}}({f}) =
\int \!{\mathop{}\!\mathrm{d}}x
\iint \!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\, \sigma(v,v')
\big[ \sqrt{f'} -\sqrt{f} \big]^2$$ and $$\label{metric}
{{\mathcal R}} (f,\eta) = \int_0^T\!{\mathop{}\!\mathrm{d}}t \int \!{\mathop{}\!\mathrm{d}}x
\iint\!\pi(dv)\pi(dv')\,
\Psi_\sigma (f,f'; \eta).$$ The inequality , formally analogous to , is the proposed gradient flow formulation of the linear Boltzmann equation .
We now discuss the precise formulation in which we introduce the measures ${\mathop{}\!\mathrm{d}}P = f(x,v) \, {\mathop{}\!\mathrm{d}}x\, \pi({\mathop{}\!\mathrm{d}}v)$ and $\Theta({\mathop{}\!\mathrm{d}}t,{\mathop{}\!\mathrm{d}}x,
{\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v')= \eta(t,x,v,v') \, {\mathop{}\!\mathrm{d}}t\, {\mathop{}\!\mathrm{d}}x\, {\mathop{}\!\mathrm{d}}v\, {\mathop{}\!\mathrm{d}}v'$. We first specify the hypotheses on the scattering rate $\sigma$ and the drift $b$ that are assumed to hold throughout the whole paper.
\[t:asb\]$\phantom{i}$
- The *scattering kernel* is a Borel function $\sigma\colon {{\mathcal V}} \times {{\mathcal V}}\to [0,+\infty)$ satisfying $\sigma(v,v')=\sigma(v',v)$, $(v,v') \in {{\mathcal V}}\times {{\mathcal V}}$.
- The *scattering rate* $\lambda\colon {{\mathcal V}} \to
[0,+\infty)$ is defined by $\lambda(v):=\int\!\pi({\mathop{}\!\mathrm{d}}v')\,
\sigma(v,v')$. We require that it has all exponential moments with respect to $\pi$ namely, $\pi\big[ e^{\gamma \lambda}\big]
<+\infty$ for any $\gamma\in {{\mathbb R}}_+$.
- The *drift* is a Borel function $b\colon {{\mathcal V}} \to
{{\mathbb R}}^d$. We require that it has all exponential moments with respect to $\pi$ namely, $\pi\big[ e^{\gamma |b| }\big]
<+\infty$ for any $\gamma\in {{\mathbb R}}_+$, where $|b|$ is the Euclidean norm of $b$.
Given $T>0$ let $C\big([0,T]; {{\mathcal P}} ({{\mathbb T}}^d\times {{\mathcal V}})\big)$ be the set of continuous paths on ${{\mathcal P}}({{\mathbb T}}^d \times {{\mathcal V}})$ endowed with the topology of uniform convergence. Denote by ${{\mathcal M}}_\mathrm{a}\big([0,T]\times {{\mathbb T}}^d\times {{\mathcal V}} \times {{\mathcal V}}\big)$ the set of finite Radon measures on $[0,T]\times {{\mathbb T}}^d\times {{\mathcal V}}\times
{{\mathcal V}}$ antisymmetric with respect to the exchange of the last two variables endowed with the weak\* topology. Set ${{\mathcal S}}:=C\big([0,T];
{{\mathcal P}} ({{\mathbb T}}^d \times {{\mathcal V}})\big) \times {{\mathcal M}}_\mathrm{a}
\big([0,T]\times {{\mathbb T}}^d \times {{\mathcal V}}\times {{\mathcal V}} \big)$ endowed with the product topology. Let also $C_\mathrm{be}\big([0,T]; {{\mathcal P}} ({{\mathbb T}}^d\times {{\mathcal V}})\big)$ the set of paths $(P_t)_{t\in [0.T]}$ in $C\big([0,T]; {{\mathcal P}} ({{\mathbb T}}^d\times {{\mathcal V}})\big)$ such that $\sup_{t\in[0,T]}\mathcal H(P_t)<+\infty$ and let finally ${{\mathcal S}}_\mathrm{be} :=C_\mathrm{be} \big([0,T];
{{\mathcal P}} ({{\mathbb T}}^d \times {{\mathcal V}})\big) \times {{\mathcal M}}_\mathrm{a}
\big([0,T]\times {{\mathbb T}}^d \times {{\mathcal V}}\times {{\mathcal V}} \big)$
If $P\in {{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ has finite entropy, the Dirichlet form of the square root ${{\mathcal E}}$ can be defined by the variational formula $$\label{varD}
{{\mathcal E}}(P) := \sup_{\phi \in C_\mathrm{b}({{\mathbb T}}^d\times {{\mathcal V}})}
\iint \! P({\mathop{}\!\mathrm{d}}x,{\mathop{}\!\mathrm{d}}v) \pi({\mathop{}\!\mathrm{d}}v') \, \sigma(v,v')
\Big[ 1 - e^{\phi(x,v')-\phi(x,v)} \Big].$$ Note indeed the right hand side is well defined for any $\phi\in C_\mathrm{b}({{\mathbb T}}^d\times {{\mathcal V}})$ in view of Assumption \[t:asb\] and the basic entropy inequality $P(\psi ) \le {{\mathcal H}} (P) + \log \int\!{\mathop{}\!\mathrm{d}}x
\pi({\mathop{}\!\mathrm{d}}v)\, e^\psi$, $\psi\colon {{\mathbb T}}^d\times {{\mathcal V}}\to \mathbb R$. The representation corresponds to the Donsker-Varadhan large deviation for the empirical measure of the continuous time Markov chain on ${{\mathcal V}}$ with transition rates $\sigma(v',v)\pi({\mathop{}\!\mathrm{d}}v')$ [@DV]. Indeed, ${{\mathcal E}}(P)=\sup_{\phi}\{-P(e^{-\phi}{{\mathcal L}} e^\phi) \}$, where $$\label{def:L}
{{\mathcal L}} g (v)=\int \pi({\mathop{}\!\mathrm{d}}v')\sigma(v',v)[g(v')-g(v)].$$
A variational representation for the kinematic term ${{\mathcal R}}$ is obtained by combining with the simple observation that for $p,q\in {{\mathbb R}}_+$ we have $-2\sqrt{pq} =\sup_{a>0}\big\{ -
a p - a^{-1} q \big\}$. We thus let ${{\mathcal R}} \colon {{\mathcal S}}_\mathrm{be} \to [0,+\infty]$ be the functional defined by $$\label{varR}
\begin{split}
{{\mathcal R}} (P,\Theta)
&:= \sup_{\zeta,\alpha} \bigg\{
\Theta (\zeta) -
\int_0^T\!{\mathop{}\!\mathrm{d}}t \iiint \! P_t({\mathop{}\!\mathrm{d}}x,{\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v') \,
\sigma(v,v')
\\ & \qquad \qquad
\times \big[ {\mathop{\rm ch}\nolimits}\zeta(t,x,v,v') -1 \big] \big[ \alpha(t,x,v,v') +
\alpha(t,x,v',v)^{-1} \big]
\bigg\},
\end{split}$$ where the supremum is carried out over the continuous functions $\zeta
\colon [0,T]\times {{\mathbb T}}^d\times {{\mathcal V}}\times {{\mathcal V}} \to {{\mathbb R}}$ with compact support and antisymmetric with respect to the exchange of the last two variables and the bounded continuous functions $\alpha \colon
[0,T]\times {{\mathbb T}}^d\times {{\mathcal V}}\times {{\mathcal V}} \to (0,+\infty)$ uniformly bounded away from zero. As before, the basic entropy inequality implies that ${{\mathcal R}}$ is well defined. At this point the gradient flow formulation of the linear Boltzmann equations is simply specified by the following entropy dissipation inequality.
\[t:dlbe\] Let $Q\in{{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ with ${{\mathcal H}}(Q)<+\infty$. An element $(P, \Theta)\in {{\mathcal S}}_\mathrm{be}$ is a solution to the linear Boltzmann equation with initial condition $Q$ iff $P_0=Q$ and $$\begin{aligned}
\label{beq}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\displaystyle{
\int_0^T \!{\mathop{}\!\mathrm{d}}t \,P_t(\partial_t\phi + b\cdot \nabla_x \phi)
= \frac 12 \int\!\Theta({\mathop{}\!\mathrm{d}}t,{\mathop{}\!\mathrm{d}}x,{\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v') \,
\big[\phi(t,x,v) -\phi(t,x,v')\big], }
\\
\label{ineq1-calore2}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\displaystyle{
{{\mathcal H}}(P_T)+\int_0^T \!{\mathop{}\!\mathrm{d}}t\, {{\mathcal E}}(P_t)+ {{\mathcal R}} (P, \Theta)
\leq {{\mathcal H}}(Q).}\end{aligned}$$ for all continuous functions $\phi\colon (0,T)\times {{\mathbb T}}^d\times {{\mathcal V}}$ with compact support and continuously differentiable in the first two variables.
\[t:rem\] If $(P,\Theta)$ is a solution to the linear Boltzmann equation in the time interval $[0,T]$ then it solves the same problem in the time interval $[0,t]$, $t\le T$ as well. This follows from the fact that any element $(P,\Theta)\in {{\mathcal S}}_\mathrm{be}$ satisfies, for $0\le s < t
\le T$, the inequality $$\label{jj1t}
{{\mathcal H}}(P_t)+\int_s^t \!{\mathop{}\!\mathrm{d}}u\, {{\mathcal E}}(P_u)+ {{\mathcal R}}^{s,t} (P, \Theta_{[s,t]})
\geq {{\mathcal H}}(P_s)$$ where $\Theta_{[s,t]}$ is the restriction of $\Theta$ to the interval $[s,t]$ and the kinematic term ${{\mathcal R}}^{s,t}$ is defined as in with the interval $[0,T]$ replaced by $[s,t]$. This inequality corresponds in fact to the trivial inequality ${{\mathcal I}}_{[s,t]}(P,\Theta)\ge 0$ where the action functional ${{\mathcal I}}_{[s,t]}$ is defined as in with the interval $[0,T]$ replaced by $[s,t]$. The actual proof of is detailed in Appendix \[app2\].
It is of course possible to obtain a formulation only in terms of the one particle distribution. More precisely, the formulation is obtained from simply by letting ${{\mathcal R}}_0
\colon C_\mathrm{be}([0,T]; {{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}}))\to [0,+\infty]$ be the functional defined by ${{\mathcal R}}_0(P) = \inf_{\Theta} {{\mathcal R}}(P,\Theta)$ where the infimum is carried out over all $\Theta\in {{\mathcal M}}_\mathrm{a}([0,T]\times {{\mathbb T}}^d\times {{\mathcal V}}\times {{\mathcal V}})$ such that the pair $(P,\Theta)$ satisfies the balance equation . It is however unclear to us whether ${{\mathcal R}}_0(P)$ could be represented by the metric derivative of $t\mapsto P_t$ with respect to a suitable distance on ${{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$.
We now state an existence and uniqueness result for the above formulation, together with a continuous dependence on the initial condition and the coefficients. In particular uniqueness implies that solutions to with bounded entropy are characterized by the gradient flow formulation in Definition \[t:dlbe\].
\[t:eu\] For each $Q\in{{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$, with ${{\mathcal H}}(Q)< +\infty$, there exists a unique solution $(P,\Theta)$ to the linear Boltzmann equation with initial condition $Q$. Furthermore
- Set $\Theta^P({\mathop{}\!\mathrm{d}}t,{\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v'):={\mathop{}\!\mathrm{d}}t\,\sigma
(v, v')\big[P_t({\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v')-\pi ({\mathop{}\!\mathrm{d}}v) P_t({\mathop{}\!\mathrm{d}}x,
{\mathop{}\!\mathrm{d}}v') \big]$. Then $\Theta=\Theta^P$.
- Let $\{Q^n\}\subset {{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ be such that $Q^n\to Q$ and ${{\mathcal H}}(Q^n)\to {{\mathcal H}}(Q)$ and denote by $(P^n,\Theta^n)\in {{\mathcal S}}_\mathrm{be}$ the solution to the linear Boltzmann equation with initial condition $Q^n$. Then the sequence $\{(P^n,\Theta^n)\}$ converges to $(P,\Theta)$.
- Fix $Q\in {{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ with ${{\mathcal H}}(Q) <+\infty$ and consider coefficients $b$, $\sigma$ together with sequences $b^n$, $\sigma^n$ all satisfying Assumption \[t:asb\]. Denote by $(P^n,\Theta^n)\in {{\mathcal S}}_\mathrm{be}$ the solution to the linear Boltzmann equation with initial condition $Q$ and coefficients $b^n$, $\sigma^n$. If $b_n\to b$ in $\pi$ probability, $\sigma^n\to \sigma$ in $\pi\times \pi$ probability, and $\lim_n \big\{ \log \pi \big[ e^{\gamma |b^n -b |}
\big] +\log \pi\big[ e^{\gamma |\lambda^n -\lambda|}\big] \big\}=
0$ for any $\gamma>0$ then the sequence $\{(P^n,\Theta^n)\}$ converges to $(P,\Theta)$.
While uniqueness will be proven by using the argument in [@Gi], the key ingredient for the continuity result is the following lemma. Its proof, whose details are omitted, is achieved by truncating with continuous and bounded functions and using the basic entropy inequality.
\[t:lem\] Let $\{P^n\}\subset {{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ be a sequence converging to $P$ and satisfying the entropy bound $\sup_n
{{\mathcal H}}(P^n) < +\infty$. Then $P^{n}(\phi) \to P(\phi)$ for any function $\phi$ having all exponential moments with respect to $\pi$. Moreover, if $\phi$ has all exponential moments and $\lim_n \log \pi \big[ e^{\gamma |\phi^n -\phi|} \big] =0$ for any $\gamma>0$ then $P^{n}(\phi^n) \to P(\phi)$.
In view of the variational definition , this lemma readily implies that the Dirichlet form ${{\mathcal E}}$ is lower semicontinuous on sublevel sets of the entropy. Analogously, recalling , the kinematic term ${{\mathcal R}}$ is lower semicontinuous on the sets $\big\{
(P,\Theta) \in {{\mathcal S}}_\mathrm{be} \colon \sup_{ t\in [0,T]} {{\mathcal H}}(P_t)
\le \ell \big\}$, $\ell \in {{\mathbb R}}_+$.
We start by proving uniqueness, and in particular by showing that if $(P^1, \Theta^1)$ and $(P^2, \Theta^2)$ are solutions then $P^1=P^2$. Assume by contradiction that there exists $t\in (0,T]$ such that $P^1_t\neq P^2_t$ and let $(\bar P, \bar \Theta)=\frac 1 2 \big (P^1, \Theta^1 \big)+\frac 1 2 (P^2, \Theta^2)$. In view of Remark \[t:rem\], by the convexity of $\mathcal E$, $\mathcal R$ and the strict convexity of ${{\mathcal H}}$ $${{\mathcal H}}(\bar P_t)+\int_0^t \!{\mathop{}\!\mathrm{d}}s\, {{\mathcal E}} (\bar P_s) + {{\mathcal R}} ^{0,t}(\bar P, \bar \Theta)< {{\mathcal H}} (Q),$$ which by provides the desired contradiction. Uniqueness is now concluded by observing that, for a given $P\in C_{\mathrm{be}}\big([0,T]; {{\mathcal P}}({{\mathbb T}}^d\times{{\mathcal V}})\big)$, the map $\Theta\mapsto {{\mathcal R}}(P, \Theta)$ is strictly convex, so that we can repeat the argument above with $P^1=P^2=P$ and deduce $\Theta^1=\Theta^2$.
Postponing the proof of the existence, we show item [(i)]{}. We write $\Theta=\Theta^P+\tilde{\Theta}$ and we observe that, in view of the balance equation , $\tilde\Theta(\zeta)=0$ if $\zeta(t,x,v,v')=z(t,x,v')-z(t,x,v)$ for some function $z$. By choosing in the variational formula $\zeta=z'-z$, where $z=z(t,x,v)$ and $z'=z(t,x,v')$, we get $$\begin{split}
{{\mathcal R}} (P,\Theta^P +\tilde\Theta)
\geq & \sup_{z,\alpha} \bigg\{
\Theta^P (z'-z) -
\int_0^T\!{\mathop{}\!\mathrm{d}}t \iiint \! P_t({\mathop{}\!\mathrm{d}}x,{\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v') \,
\sigma(v,v')
\\ & \qquad \qquad
\times \big[ {\mathop{\rm ch}\nolimits}(z'-z) -1 \big] \big[ \alpha(t,x,v,v') +
\alpha(t,x,v',v)^{-1} \big]
\bigg\}\\
= & {{\mathcal R}} (P,\Theta^P),
\end{split}$$ where the last equality follows by direct computation. We conclude by uniqueness.
We next prove item [(ii)]{}. Remark \[t:rem\] implies that $$\label{apb}
\varlimsup_{n\to \infty} \, \sup_{t\in [0,T]} \,
{{\mathcal H}}(P_t^n) \le {{\mathcal H}}(Q).$$ In view of the lower semicontinuity of ${{\mathcal H}}$ and the observation after Lemma \[t:lem\] regarding ${{\mathcal E}}$ and ${{\mathcal R}}$, using the uniqueness it is enough to show precompactness of the sequence $\{(P^n,\Theta^n)\}\subset {{\mathcal S}}_\mathrm{be}$. Observe indeed that, by Assumption \[t:asb\] and Lemma \[t:lem\], we can take the limit $n\to \infty$ in the balance equation .
To prove precompactness of $\{\Theta^n\}$, observe that from and the variational representation it follows $$\sup_n\, \sup_{\zeta\colon \|\zeta\|_\infty\le 1}
\Theta^n(\zeta) < +\infty$$ and we conclude by the Banach-Alaoglu theorem.
The bound implies, by the coercive properties of the relative entropy and Prohorov theorem, that there exist a compact ${{\mathcal K}}
\subset\subset {{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ such that $P^n_t\in {{\mathcal K}}$ for any $n$ and $t\in[0,T]$. Hence, by Ascoli-Arzelà theorem, to prove the precompactness of $\{P^n\}$ it is enough to show that for each continuous $g\colon {{\mathbb T}}^d\times {{\mathcal V}}\to {{\mathbb R}}$ with compact support and continuously differentiable with respect to $x$ we have $$\label{equic}
\lim_{\delta\downarrow 0} \, \sup_n\, \sup_{|t-s|<\delta}
\big| P^n_t(g) - P^n_s(g) \big| =0.$$ From the balance equation we deduce $$\begin{split}
& P^n_t(g) - P^n_s(g) =
- \int_s^t\!{\mathop{}\!\mathrm{d}}\tau\, P^n_\tau\big( b \cdot \nabla_x g \big)
\\
&\qquad \qquad
-\frac 12 \int_{[s,t]\times {{\mathbb T}}^d\times {{\mathcal V}}\times {{\mathcal V}}}
\Theta^n({\mathop{}\!\mathrm{d}}\tau, {\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v')
\big[ g(x,v) - g(x,v')\big].
\end{split}$$ By Assumption \[t:asb\], , and the basic entropy inequality, the first term on the right hand side vanishes as $|t-s|\to 0$ uniformly in $n$. On the other hand, by choosing $\alpha=1$ in the variational representation , $$\begin{split}
& \Big| \int_{[s,t]\times {{\mathbb T}}^d\times {{\mathcal V}}\times {{\mathcal V}}}
\Theta^n({\mathop{}\!\mathrm{d}}\tau, {\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v')
\big[ g(x,v) - g(x,v')\big] \Big|
\le {{\mathcal R}} (P^n,\Theta^n)
\\
& + \quad
2 \int_s^t\!{\mathop{}\!\mathrm{d}}\tau \iiint \! P_\tau({\mathop{}\!\mathrm{d}}x,{\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v') \,
\sigma(v,v')
\big\{ {\mathop{\rm ch}\nolimits}\big[ g(x,v) - g(x,v') \big] -1 \big\}.
\end{split}$$ Replacing $g$ by $\gamma g$ with $\gamma>0$, using , Assumption \[t:asb\], the basic entropy inequality, and ${{\mathcal R}} (P^n,\Theta^n) \le {{\mathcal H}}(Q^n)$, we obtain that there exists a constant $C$ independent on $n$, $t,s$ such that $$\begin{split}
& \Big| \int_{[s,t]\times {{\mathbb T}}^d\times {{\mathcal V}}\times {{\mathcal V}}}
\Theta^n({\mathop{}\!\mathrm{d}}\tau, {\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v')
\big[ g(x,v) - g(x,v')\big] \Big|
\\ & \qquad
\le \frac 1\gamma \, \sup_n {{\mathcal H}}(Q^n)
+ \frac C\gamma \, |t-s| \, \exp\{ 2 \gamma \| g\|_\infty\}.
\end{split}$$ By choosing $\gamma = (2 \|g\|_\infty)^{-1} \log
(1/|t-s|)$ when $|t-s|\le 1$ the bound follows.
In view of the second statement in Lemma \[t:lem\], the proof of item (iii) is achieved by the same arguments.
We finally prove the existence result. Consider first the case in which $b$ and $\sigma$ are continuous and bounded. If $Q({\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v) =f_0(x,v) {\mathop{}\!\mathrm{d}}x \pi({\mathop{}\!\mathrm{d}}v)$ for some continuous density $f_0$ uniformly bounded away from zero, by classical results, the linear Boltzmann equation has a continuous solution $f(t,x,v)$ uniformly bounded away from zero. Set $P_t({\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v) := f(t,x,v) {\mathop{}\!\mathrm{d}}x \pi({\mathop{}\!\mathrm{d}}v)$ and $\Theta({\mathop{}\!\mathrm{d}}t,
{\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v') := {\mathop{}\!\mathrm{d}}t {\mathop{}\!\mathrm{d}}x \pi({\mathop{}\!\mathrm{d}}v) \pi ({\mathop{}\!\mathrm{d}}v') \sigma
(v,v') \big[ f(t,x,v)- f(t,x,v')\big]$. It is then straightforward to justify the informal computations presented before and deduce that $(P, \Theta)$ solves the linear Boltzmann equation according to Definition \[t:dlbe\]. Existence in the general case of $Q$ satisfying the relative entropy bound ${{\mathcal H}} (Q) <+\infty$ and for coefficients $b$ and $\sigma$ satisfying Assumption \[t:asb\] is then achieved by items [(ii)]{} and [(iii)]{}.
Diffusive limit
===============
In this Section we discuss the asymptotic behavior of linear Boltzmann equation, showing that in the diffusive scaling limit the marginal distribution of the position evolves according to the heat flow. This is a classical topic and has been much investigated in the literature, see [@LK; @BLP; @BSS; @EP] and [@DGP] for a more general setting. We observe that this issue has natural counterpart in probabilistic terms namely, the central limit for additive functional of Markov chains. Indeed, the linear Boltzmann equation is the Fokker-Planck equation for the Markov process $(V_t,X_t)$ where $V_t$ is the continuous time Markov chain on ${{\mathcal V}}$ with transition rates $\sigma(v',v)\pi({\mathop{}\!\mathrm{d}}v')$ while $X_t$ is the ${{\mathbb R}}^d$-valued additive functional $X_t=\int_0^t\!{\mathop{}\!\mathrm{d}}s\,
b(V_s)$. We refer to [@KLO] for a recent monograph on this topic.
The gradient flow formulation of linear Boltzmann equations discussed before allows a novel approach to the analysis of the diffusive limit. According to a general scheme formalized in [@SaSe; @Se], a gradient flow formulation is particularly handy for analyzing asymptotic evolutions and it does not require a direct analysis of the dynamics. Indeed, by comparing Definition \[t:dlbe\] and Definition \[he\] we realize that the balance equation immediately leads to the continuity equation . Moreover, taking into account the convexity and lower semicontinuity of the entropy, in order to establish the diffusive limit we only need to prove two limiting variational inequalities comparing the Dirichlet form and the kinematic term for the linear Boltzmann equation with the corresponding ones for the heat flow. We shall prove these variational inequalities but, maybe surprisingly, the two terms exchange their role in the diffusive limit: the Dirichlet form ${{\mathcal E}}$ leads to $R$ while the kinematic term ${{\mathcal R}}$ leads to the Fisher information $E$.
To carry out the analysis of the diffusive limit of linear Boltzmann equations a few extra conditions, implying in particular homogenization of the velocity, are needed. As we show in the next section, this assumptions are satisfied for few natural models. To this end, recalling that the scattering rate $\lambda$ is defined by $\lambda(v)=\int\pi({\mathop{}\!\mathrm{d}}v')\sigma(v',v)$, let $\tilde\pi$ be the probability on ${{\mathcal V}}$ defined by $$\label{tpi}
\tilde \pi({\mathop{}\!\mathrm{d}}v) :=\frac{\lambda(v)}{\pi(\lambda)}\pi({\mathop{}\!\mathrm{d}}v).$$
$\phantom{i}$\[assumpt1\]
- The drift $b:{{\mathcal V}}\to\mathbb R^d$ is centered with respect to the measure $\pi$, namely $\pi(b)=0$.
- The scattering rate $\lambda$ satisfies $\pi[\lambda=0]=0$.
- $|b|^2/\lambda$ has all exponential moments, i.e. $\pi[\exp \{\gamma |b|^2/\lambda\}]< +\infty$ for any $\gamma>0$.
- There exists a constant $C_0>0$ such that for any $g\in L^2(\tilde \pi)$ $$\label{pin}
\int {\mathop{}\!\mathrm{d}}\tilde\pi \big[ g-\tilde\pi(g)\big]^2\leq C_0\iint\tilde\pi({\mathop{}\!\mathrm{d}}v)\tilde\pi({\mathop{}\!\mathrm{d}}v')\frac {\sigma(v,v')}
{\lambda(v)\lambda(v')}\big[g(v)-g(v') \big]^2.$$
We remark that, as in the case of phonon Boltzmann equation, in item (ii) we allow the case in which $\lambda(v)=0$ for some $v\in {{\mathcal V}}$. Item (iv) corresponds to the assumption that the continuous time Markov chain with transition rates $\frac
{\sigma(v,v')} {\lambda(v)\lambda(v')}\tilde\pi({\mathop{}\!\mathrm{d}}v')$ has spectral gap. The generator of this Markov chain is $(K-{{1 \mskip -5mu {\rm I}}})$, where $K$ is given by $$\label{def:K}
\big(K g\big)(v)=\pi(\lambda)\int \tilde\pi({\mathop{}\!\mathrm{d}}v')\frac{\sigma(v,v')}{ \lambda(v)\lambda(v')}g(v').$$ Observe that $$- \big({{\mathcal L}} f\big)(v)=\lambda(v) \big[\big({{1 \mskip -5mu {\rm I}}}-K\big) f\big](v),$$ where ${{\mathcal L}}$ is the generator of the original Markov chain as defined in . We emphasize that we do not assume the spectral gap of the generator ${{\mathcal L}}$, in fact the linear phonon Boltzmann equation, that will be discussed in the next section, meets the requirements in Assumption \[assumpt1\] but its generator has not spectral gap.
Assumption \[assumpt1\] implies that there exists $\xi\in L^2(\tilde \pi; \mathbb R^d)$ such that $-\mathcal L \xi =b$. Indeed, item (i) implies that $b/\lambda$ is centered with respect to $\tilde\pi$, item (ii) implies that $b/\lambda\in L^2(\tilde \pi; \mathbb R^d)$, and finally item (iii) implies that $\xi:=({{1 \mskip -5mu {\rm I}}}-K)^{-1}(b/\lambda)\in L^2(\tilde \pi; \mathbb R^d)$.
We need another technical condition on which we will rely to carry out a truncation on $\xi$.
\[assumpt3\] One of the following alternatives holds
- $(-{{\mathcal L}})^{-1}b$ is bounded, or
- there exists $C<\infty$ such that $\big\|\big({{1 \mskip -5mu {\rm I}}}-K\big)^{-1}f\big\|_{\infty}\leq C\|f\|_{\infty}$, for any $f$ such that $\tilde\pi(f)=0$.
Observe that if the map $(v,v')\mapsto\frac{\sigma(v,v')}{\lambda(v)\lambda(v')}$ is continuous and bounded then alternative (ii) holds.
For notation convenience, in this section we formulate the linear Boltzmann equation in terms of the density $(f,\eta)$, where ${\mathop{}\!\mathrm{d}}P_t=f(t){\mathop{}\!\mathrm{d}}x\pi({\mathop{}\!\mathrm{d}}v)$ and ${\mathop{}\!\mathrm{d}}\Theta= \eta {\mathop{}\!\mathrm{d}}t{\mathop{}\!\mathrm{d}}x\pi({\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v')$. Indeed, the boundedness of the entropy implies the existence of $f$, while the boundedness of the metric term ${{\mathcal R}}$ implies the existence of $\eta$.
Let ${\varepsilon}>0$ be a scaling parameter and consider the linear Boltzmann equation on a torus of linear size ${\varepsilon}^{-1}$, equipped with the uniform probability distribution, on the time interval $[0, {\varepsilon}^{-2}T]$. Under a diffusive rescaling of space and time, the rescaled solution $(f^{\varepsilon}, \eta^{\varepsilon})$ is defined on the torus of linear size one on the time interval $[0,T]$. It solves $$\begin{aligned}
\label{constr}
&&\partial_t f^{\varepsilon}(t,x,v)+\frac 1 {\varepsilon}b(v)\cdot \nabla_x f^{\varepsilon}(t,x,v)+\frac 1 {{\varepsilon}^2}\int\pi({\mathop{}\!\mathrm{d}}v')
\eta^{\varepsilon}(t,x,v,v')=0\qquad\\
\label{ineq}
&&\mathcal H(f^{\varepsilon}(T))+\frac 1 {{\varepsilon}^2}\int_0^T {\mathop{}\!\mathrm{d}}t\, \mathcal E(f^{\varepsilon}(t))+\frac 1{{\varepsilon}^2} \mathcal R (f^{\varepsilon},\eta^{\varepsilon})\leq \mathcal H(f^{\varepsilon}_0).\qquad\end{aligned}$$
We set $$\label{drJ}
\begin{split}
& \rho^{\varepsilon}(t,x):=\int\pi ({\mathop{}\!\mathrm{d}}v) f^{\varepsilon}(t,x,v) \\
& j^{\varepsilon}(t,x):=\frac 1 {\varepsilon}\int\pi({\mathop{}\!\mathrm{d}}v) f^{\varepsilon}(t,x,v)b(v).
\end{split}$$ Since $\eta^{\varepsilon}(t,x,v,v')$ is antisymmetric with respect to the exchange of $v$ and $v'$, by integrating with respect to $\pi(dv)$ we deduce the continuity equation $$\label{cont}
\partial_t \rho^{\varepsilon}+ \nabla\cdot j^{\varepsilon}= 0.$$
\[teo:3\] Assume that $\rho^{\varepsilon}_0\to\rho_0$ in $\mathcal P(\mathbb T^d)$ and $\lim_{{\varepsilon}\to 0} \mathcal H(f^{\varepsilon}_0)= H(\rho_0)$. Then the sequence $(\rho^{\varepsilon}, j^{\varepsilon})$ converges in $C([0,T]; \mathcal P(\mathbb
T^d))\times \mathcal M ([0,T]\times {{\mathbb T}}^d; \mathbb R^d)$ to the solution to the heat equation as in Definition \[he\], with initial datum $\rho_0$ and diffusion coefficient $$\label{D}
D=\pi \big(b \otimes(-{{\mathcal L}})^{-1}b\big).$$
Note that, by Assumption \[assumpt1\], $b/\lambda$ and $ \xi=(-{{\mathcal L}})^{-1} b$ are in $ L^2(\tilde\pi; \mathbb R^d)$, hence the diffusion coefficient $D=\pi(\lambda)\,\tilde\pi\big((b/\lambda)\otimes\xi\big)$ is finite.
The proof of this theorem will be achieved according to the following strategy. We first show precompactness of the sequence $\{(\rho^{\varepsilon}, j^{\varepsilon})\}$, we then consider a converging subsequence $(\rho^{\varepsilon}, j^{\varepsilon})\to (\rho, j)$ and take the inferior limit in the inequality . By the hypothesis of the theorem, ${{\mathcal H}}(f_0^{\varepsilon})\to H(\rho_0)$ and we prove that the inferior limit of the left hand side of majorizes the left hand side of . The statement follows by the uniqueness in Proposition \[t:uniq\]. We introduce the following notations. If $(f^{\varepsilon},\eta^{\varepsilon})$ satisfy , , recalling , we set $$\label{def:u}\begin{split}
& f^{\varepsilon}(t,x,v)=u_{\varepsilon}^2(t,x,v),\quad u_{\varepsilon}(t,x,v)=\bar u_{\varepsilon}(t,x) +
\tilde u_{\varepsilon}(t,x,v),\\
& \bar u_{\varepsilon}(t,x)=\tilde\pi (u_{\varepsilon}(t,x,\cdot)).
\end{split}$$ We will use the following bounds that hold uniformly for $t\in [0,T]$. By Cauchy-Schwarz inequality $$\label{ineq4}
\int\!{\mathop{}\!\mathrm{d}}x\, \bar u_{\varepsilon}^2(t) = \frac 1{\pi(\lambda)^2}
\int\!{\mathop{}\!\mathrm{d}}x
\bigg(\int\!d\pi\, \lambda \sqrt{f^{\varepsilon}(t)}\bigg)^2
\le \frac {\pi(\lambda^2 )}{\pi(\lambda)^2}.$$ Moreover, by the basic entropy inequality, for each $\gamma>0$ $$\label{ineq1}
\begin{split}
\int \!{\mathop{}\!\mathrm{d}}x \!\int\! {\mathop{}\!\mathrm{d}}\pi f^{\varepsilon}(t)
\frac {|b|^2} {\lambda}
&\le
\frac 1\gamma {{\mathcal H}} (f^{\varepsilon}(t)) + \frac 1\gamma \log \pi\Big(
\exp\Big\{ \gamma \frac{|b|^2}\lambda \Big\}\Big)
\\
&\le \frac 1\gamma {{\mathcal H}} (f^{\varepsilon}_0) + \frac 1\gamma \log \pi\Big(
\exp\Big\{ \gamma \frac{|b|^2}\lambda \Big\}\Big).
\end{split}$$
\[t:l4\] There exists a constant $C$ such that for any ${\varepsilon}\in (0,1)$ $$\begin{aligned}
\label{ineq2}
&& \displaystyle\int {\mathop{}\!\mathrm{d}}x\int {\mathop{}\!\mathrm{d}}\pi
\,\tilde u_{\varepsilon}(t)^2 \frac{|b|^2}{\lambda} < C,\qquad t\in[0,T],
\\
\label{ineq3}
&&\frac 1 {{\varepsilon}^2}\int_0^T {\mathop{}\!\mathrm{d}}t\int dx \int {\mathop{}\!\mathrm{d}}\tilde\pi \tilde u_{\varepsilon}(t)^2 <C .
\end{aligned}$$
In order to prove , since $\tilde u_{\varepsilon}=\sqrt{f^{\varepsilon}} -\bar
u_{\varepsilon}$, for each $t\in [0,T]$ $$ \begin{split}
\int\!{\mathop{}\!\mathrm{d}}x\!\int\!{\mathop{}\!\mathrm{d}}\tilde\pi\, \tilde
u_{\varepsilon}^2(t) \frac {|b|^2}{\lambda^2}
\le \frac 2{\pi(\lambda)}
\int{\mathop{}\!\mathrm{d}}x \!\int{\mathop{}\!\mathrm{d}}\pi\,
f^{\varepsilon}(t) \frac{|b|^2}{\lambda}
+ \frac 2{\pi(\lambda)}
\int{\mathop{}\!\mathrm{d}}x\, \bar u_{\varepsilon}^2(t)
\int{\mathop{}\!\mathrm{d}}\pi\, \frac {|b|^2} {\lambda}.
\end{split}$$ The first term on the right hand side is bounded by , while the second term is bounded, since $b^2/\lambda$ has finite exponential moments, by .
Regarding , by the Poincaré inequality , $$\begin{split}
& \frac 1 {{\varepsilon}^2} \int_0^T\!{\mathop{}\!\mathrm{d}}t\!\int\!{\mathop{}\!\mathrm{d}}x\!\int\!{\mathop{}\!\mathrm{d}}\tilde\pi\,
\tilde u_{\varepsilon}^2(t)
\\
&\qquad \le \frac {C_0}{{\varepsilon}^2 }\int_0^T
\!{\mathop{}\!\mathrm{d}}t\!\int{\mathop{}\!\mathrm{d}}x \!\iint\tilde\pi({\mathop{}\!\mathrm{d}}v)\tilde\pi({\mathop{}\!\mathrm{d}}v') \frac {\sigma(v,v')}
{\lambda(v)\lambda(v')} \big[u_{\varepsilon}(t,x,v)- u_{\varepsilon}(t,x,v')\big]^2
\\
&\qquad \le \frac{C_0}{{\varepsilon}^2} \pi (\lambda)^2\int_0^T
{\mathop{}\!\mathrm{d}}t\,\mathcal E(f^{\varepsilon}(t))\leq C \mathcal H(f^{\varepsilon}_0),
\end{split}$$ which concludes the proof.
\[teo:3.0\] The set $\{(\rho^{\varepsilon}, j^{\varepsilon})\}_{{\varepsilon}\in (0,1]} \subset C([0,T]; \mathcal P(\mathbb T^d))\times \mathcal M
([0,T]\times {{\mathbb T}}^d; \mathbb R^d) $ is precompact.
Given $0\le s <t \le T$, the restriction of the measure ${\mathop{}\!\mathrm{d}}J^{\varepsilon}= j^{\varepsilon}{\mathop{}\!\mathrm{d}}t {\mathop{}\!\mathrm{d}}x$ to $[s,t]\times {{\mathbb T}}^d$ is denoted by $J^{\varepsilon}_{s,t}$. We will prove the following bound. There exists a constant $C$ independent on $s,\,t$, such that for any ${\varepsilon}\in (0,1]$ and any $w\in C\big([s,t]\times {{\mathbb T}}^d;{{\mathbb R}}^d\big)$ $$\label{bonJ}
\big|J^{\varepsilon}_{s,t}(w)\big| =
\bigg|\int_s^t\!{\mathop{}\!\mathrm{d}}\tau\!\int{\mathop{}\!\mathrm{d}}x
\, j^{\varepsilon}(\tau,x)\cdot w(\tau,x)\bigg|
\le C \, \sqrt{t-s} \, \|w\|_\infty.$$ Let us first show that it implies the statement. Choosing $s=0$, $t=T$, and applying the Banach-Alaoglu theorem, directly yields the precompactness of $\{j^{\varepsilon}\}$. Since ${{\mathcal P}}({{\mathbb T}}^d)$ is compact, by the Ascoli-Arzelà theorem, to prove precompactness of $\{\rho^{\varepsilon}\}$ it is enough to show that for each $\phi\in C^1( {{\mathbb T}}^d)$ $$\label{1.5}
\lim_{\delta\downarrow 0} \, \sup_{{\varepsilon}\in (0,1]} \,
\sup_{\substack{t,s\in [0,T]\\ |t-s|<\delta}} \bigg| \int\!{\mathop{}\!\mathrm{d}}x\,
\big[\rho^{\varepsilon}(t,x) -\rho^{\varepsilon}(s,x)\big] \phi(x) \bigg| =0.$$ From the continuity equation we deduce $$ \int\!{\mathop{}\!\mathrm{d}}x\,
\big[\rho^{\varepsilon}(t,x) -\rho^{\varepsilon}(s,x)\big] \phi(x)
= \int_s^t\!{\mathop{}\!\mathrm{d}}\tau\!\int\!{\mathop{}\!\mathrm{d}}x\, j^{\varepsilon}(\tau,x)\cdot \nabla\phi(x)=
J^{\varepsilon}_{s,t}(\nabla\phi)$$ so that follows readily from .
To prove , using the decomposition , since $b$ has mean zero with respect to $\pi$, by definition of $j^{\varepsilon}$ we get $$\begin{split}
&J^{\varepsilon}_{s,t}(w) =
\frac 1{\varepsilon}\int_s^t\!{\mathop{}\!\mathrm{d}}\tau\!\int\!{\mathop{}\!\mathrm{d}}x\!\int\!\pi({\mathop{}\!\mathrm{d}}v)\,
f^{\varepsilon}(\tau,x,v) \, b(v)\cdot w(\tau,x)\\
&
= \frac {\pi(\lambda)}{\varepsilon}\int_s^t\!{\mathop{}\!\mathrm{d}}\tau\!\int\!{\mathop{}\!\mathrm{d}}x\!
\int\!\tilde\pi({\mathop{}\!\mathrm{d}}v)\,
\big[\tilde u_{\varepsilon}^2(\tau,x,v)
+ 2\bar u_{\varepsilon}(\tau,x) \tilde u_{\varepsilon}(\tau,x,v)\big]
\, \frac{b(v)}{\lambda(v)}\cdot w(\tau,x).
\end{split}$$ By Young’s inequality, for each $\gamma>0$ $$\begin{split}
& \frac 1{{\varepsilon}} \tilde u_{\varepsilon}^2 \frac {|b|}{\lambda}
\le\frac \gamma{2{\varepsilon}^2} \tilde u_{\varepsilon}^2 +
\frac 1{2\gamma} \tilde u_{\varepsilon}^2\frac {|b|^2}{\lambda^2} \\
&\frac 2{{\varepsilon}} \bar u_{\varepsilon}\,|\tilde u_{\varepsilon}| \frac {|b|}{\lambda}
\le \frac \gamma{{\varepsilon}^2} \tilde u_{\varepsilon}^2 +
\frac 1{\gamma} \bar u_{\varepsilon}^2\frac {|b|^2}{\lambda^2}
\end{split}$$ Then we obtain $$\label{split3}
\begin{split}
|J^{\varepsilon}_{s,t}(w)| \le
\|w\|_\infty & \left\{ \frac {3\gamma}{2{\varepsilon}^2}
\int_0^T\!{\mathop{}\!\mathrm{d}}\tau\!\int\!{\mathop{}\!\mathrm{d}}x\!\int\!{\mathop{}\!\mathrm{d}}\tilde\pi\, \tilde
u_{\varepsilon}^2 + \frac 1{2\gamma}
\int_s^t\!{\mathop{}\!\mathrm{d}}\tau\!\int\!{\mathop{}\!\mathrm{d}}x\!\int\!{\mathop{}\!\mathrm{d}}\tilde\pi\, \tilde
u_{\varepsilon}^2 \frac {|b|^2}{\lambda^2} + \right. \\
&+ \left. \frac 1{\gamma}
\int_s^t\!{\mathop{}\!\mathrm{d}}\tau\!\int\!{\mathop{}\!\mathrm{d}}x\, \bar
u_{\varepsilon}^2 \int\!{\mathop{}\!\mathrm{d}}\tilde\pi\, \frac {|b|^2}{\lambda^2}
\right\}.
\end{split}$$ Using , , the fact that $|b|^2/\lambda$ has finite exponential moments, and , we obtain that there exists $C$ such that $$|J^{\varepsilon}_{s,t}(w)| \le \frac C 2
\|w\|_\infty \left( \gamma + \frac{t-s}{\gamma} \right),$$ then is obtained by choosing $\gamma = \sqrt{t-s}$.
\[teo:3.1\] Assume that $\rho^{\varepsilon}\to \rho$. Then for each $t\in [0,T]$ $$\varliminf_{{\varepsilon}\to 0} \mathcal H(f^{\varepsilon}(t))\geq
H(\rho(t)).$$
The statement is a direct consequence of the convexity and lower semicontinuity of the relative entropy.
\[teo:3.2\] Assume that $(\rho^{\varepsilon}, j^{\varepsilon})\to (\rho, j)$. Then $$\varliminf_{{\varepsilon}\to 0}\frac 1 {{\varepsilon}^2}\int_0^T {\mathop{}\!\mathrm{d}}t \,\mathcal
E(f^{\varepsilon}(t))\geq R(\rho, j).$$
Assume first that condition (ii) in Assumption \[assumpt3\] holds. Recall and observe that, in view of item (iii) in Assumption \[assumpt1\], $b/\lambda\in L^2(\tilde\pi)$. Choose a sequence $\{a_n\}$, $a_n:{{\mathcal V}}\to\mathbb R^d$, converging to $b/\lambda$ in $L^2(\tilde\pi)$, such that: $a_n$ is bounded, $\tilde\pi(a_n)=0$ and $|a_n(v)|\leq |b(v)|/\lambda(v)$ for any $n\geq 1$. Upon extracting a subsequence, $a_n\to b/\lambda$ $\tilde\pi$-a.e. Set $\omega_{n}:=({{1 \mskip -5mu {\rm I}}}-K)^{-1}a_n$. By Poincaré inequality $({{1 \mskip -5mu {\rm I}}}-K)^{-1}$ is a bounded operator on the subspace of $L^2 (\tilde\pi)$ orthogonal to the constants; hence $\omega_n$ converges to $\xi=({{1 \mskip -5mu {\rm I}}}-K)^{-1}(b/\lambda)$ in $L^2(\tilde\pi)$. Moreover, by condition (ii) in Assumption \[assumpt3\], for each $n\geq 1$ $\omega_n$ is bounded.
Fix $w\in C([0,T]\times {{\mathbb T}}^d; \mathbb R^d)$. In the variational representation for $\mathcal E$ we chose the test function $\log\big(1+{\varepsilon}w(t,x)\cdot \omega_{n}(v)\big)$, with ${\varepsilon}$ small enough, and we deduce $$\begin{split}
\frac 1 {{\varepsilon}^2}\int_0^T \!{\mathop{}\!\mathrm{d}}t\, \mathcal E(f^{\varepsilon}(t))\geq \frac 1
{{\varepsilon}}\int_0^T \! {\mathop{}\!\mathrm{d}}t \int\!{\mathop{}\!\mathrm{d}}x \int\!{\mathop{}\!\mathrm{d}}\pi
f^{\varepsilon}\,\frac{w\cdot (-{{\mathcal L}})\omega_n}{1+{\varepsilon}w\cdot \omega_n}.
\end{split}$$ Since $\omega_{n}$ is bounded, by Taylor expansion we obtain $$\begin{split}
\varliminf_{{\varepsilon}\to 0}\frac 1 {{\varepsilon}^2}\int_0^T {\mathop{}\!\mathrm{d}}t\, \mathcal E(f^{\varepsilon}(t))\geq
\varliminf_{{\varepsilon}\to 0}\frac 1 {\varepsilon}\int_0^T {\mathop{}\!\mathrm{d}}t \int{\mathop{}\!\mathrm{d}}x
\int{\mathop{}\!\mathrm{d}}\pi f^{\varepsilon}\, w\cdot (-{{\mathcal L}})\omega_{n}\\
-
\varlimsup_{{\varepsilon}\to 0}\int_0^T {\mathop{}\!\mathrm{d}}t \int{\mathop{}\!\mathrm{d}}x \int{\mathop{}\!\mathrm{d}}\pi
f^{\varepsilon}\, w\cdot \omega_{n} w\cdot (-{{\mathcal L}})\omega_{n}.
\end{split}$$ Regarding the first term on the right hand side, we write $$\begin{split}
\frac 1 {\varepsilon}\int_0^T {\mathop{}\!\mathrm{d}}t \int{\mathop{}\!\mathrm{d}}x \int{\mathop{}\!\mathrm{d}}\pi f^{\varepsilon}\,
w\cdot (-{{\mathcal L}})\omega_{n}=
\int_0^T {\mathop{}\!\mathrm{d}}t \int{\mathop{}\!\mathrm{d}}x j^{\varepsilon}\cdot w + A_{{\varepsilon},n},
\end{split}$$ with $$A_{{\varepsilon},n}=\frac 1 {\varepsilon}\int_0^T {\mathop{}\!\mathrm{d}}t \int{\mathop{}\!\mathrm{d}}x \int{\mathop{}\!\mathrm{d}}\pi f^{\varepsilon}\, w\cdot \big((-{{\mathcal L}})\omega_{n} -b\big).$$ We will show that $$\label{lim_A}
\lim_{n\to\infty} \sup_{{\varepsilon}>0} |A_{{\varepsilon},n}|=0$$ and $$\label{D2}
\begin{split}
\varlimsup_{n\to\infty}\varlimsup_{{\varepsilon}\to 0}\int_0^T {\mathop{}\!\mathrm{d}}t
\int{\mathop{}\!\mathrm{d}}x \int{\mathop{}\!\mathrm{d}}\pi f^{\varepsilon}\, w\cdot \omega_{n}
w\cdot (-{{\mathcal L}})\omega_{n} \leq \frac 1 2 \int_0^T {\mathop{}\!\mathrm{d}}t \int{\mathop{}\!\mathrm{d}}x \rho w\cdot D\,w,
\end{split}$$ with $D$ given by . Then, by optimizing over $w$ and using the variational representation , the statement follows.
Postponing the proof of these two bounds, we consider the case that condition (i) in Assumption \[assumpt3\] holds. Fix $w\in C\big([0,T]\times\mathbb T^d;\,\mathbb R^d\big)$, then in the variational representation for $\mathcal E$ we choose the test function $\log\big(1+{\varepsilon}w(t,x)\cdot (-{{\mathcal L}})^{-1}b(v)\big)$, with ${\varepsilon}$ small enough. By Taylor expansion we deduce $$\begin{split}
\varliminf_{{\varepsilon}\to 0}\frac 1 {{\varepsilon}^2}\int_0^T {\mathop{}\!\mathrm{d}}t \,\mathcal E(f^{\varepsilon}(t))\geq
\varliminf_{{\varepsilon}\to 0}\frac 1 {\varepsilon}\int_0^T {\mathop{}\!\mathrm{d}}t \int{\mathop{}\!\mathrm{d}}x
\int{\mathop{}\!\mathrm{d}}\pi f^{\varepsilon}\, w\cdot b\\
-
\varlimsup_{{\varepsilon}\to 0}\int_0^T {\mathop{}\!\mathrm{d}}t \int{\mathop{}\!\mathrm{d}}x \int{\mathop{}\!\mathrm{d}}\pi
f^{\varepsilon}\, w\cdot (-{{\mathcal L}})^{-1}b\, w\cdot b.
\end{split}$$ Recalling and the variational representation , it suffices to show $$\label{Db2}
\begin{split}
\varlimsup_{{\varepsilon}\to 0}\int_0^T {\mathop{}\!\mathrm{d}}t
\int{\mathop{}\!\mathrm{d}}x \int{\mathop{}\!\mathrm{d}}\pi f^{\varepsilon}\,w\cdot b \,
w\cdot (-{{\mathcal L}})^{-1}b \leq \frac 1 2 \int_0^T {\mathop{}\!\mathrm{d}}t \int{\mathop{}\!\mathrm{d}}x \rho\, w\cdot D w.
\end{split}$$
*Proof of* . According to the decomposition of $f^{\varepsilon}$ in , we write $$\begin{split}
A_{{\varepsilon},n}=\frac 1{\varepsilon}\int_0^T {\mathop{}\!\mathrm{d}}t \int{\mathop{}\!\mathrm{d}}x \int
{\mathop{}\!\mathrm{d}}\pi \,\tilde u_{\varepsilon}^2\, w\cdot \big((-{{\mathcal L}})\omega_{n} -b\big)\\+ \frac 2
{\varepsilon}\int_0^T {\mathop{}\!\mathrm{d}}t \int{\mathop{}\!\mathrm{d}}x \int{\mathop{}\!\mathrm{d}}\pi\, \tilde u_{\varepsilon}\,
\bar u_{\varepsilon}\,w\cdot \big((-{{\mathcal L}})\omega_{n} -b\big),
\end{split}$$ where we used that $\pi[(-{{\mathcal L}})\omega_n-b]=0$. By Young’s inequality, for any $\gamma>0$ $$\begin{split}
&\frac 1{{\varepsilon}} \tilde u_{\varepsilon}^2 \,|w|\,\frac 1{\lambda}
\big|(-{{\mathcal L}})\omega_{n} -b\big|
\le \frac \gamma{2{\varepsilon}^2} \tilde u_{\varepsilon}^2 +
\frac 1{2\gamma} |w|^2\tilde u_{\varepsilon}^2 \frac 1{\lambda^2}
\big|(-{{\mathcal L}})\omega_{n} -b\big|^2 ,\\
&\frac 2 {\varepsilon}\frac 1{\lambda}\,\bar u_{\varepsilon}|\tilde u_{\varepsilon}|\,
\,|w|\, \big|(-{{\mathcal L}})\omega_{n} -b\big|
\le \frac \gamma{{\varepsilon}^2} \tilde u_{\varepsilon}^2+ \frac 1{\gamma}
|w|^2\bar u_{\varepsilon}^2 \frac 1{\lambda^2} \big|(-{{\mathcal L}})\omega_{n} -b\big|^2.
\end{split}$$ Then $$\label{stima:A}
\begin{split}
|A_{{\varepsilon},n}| \le & \frac {3\gamma}{2{\varepsilon}^2}
\int_0^T \!{\mathop{}\!\mathrm{d}}t \int \!{\mathop{}\!\mathrm{d}}x \int \!{\mathop{}\!\mathrm{d}}\tilde \pi\, \tilde u_{\varepsilon}^2 \\
&+
\frac {\pi[\lambda]^2} {2\gamma} \|w\|_{\infty}^2
\int_0^T \!{\mathop{}\!\mathrm{d}}t \int\! {\mathop{}\!\mathrm{d}}x
\int \!{\mathop{}\!\mathrm{d}}\tilde \pi \, \tilde u_{\varepsilon}^2 \frac 1 {\lambda^2}
\big| \big((-{{\mathcal L}})\omega_{n} -b\big)\big|^2
\\
& + \frac {\pi[\lambda]^2} {\gamma}
\|w\|_{\infty}^2\int_0^T \!{\mathop{}\!\mathrm{d}}t \int \!{\mathop{}\!\mathrm{d}}x \, \bar u_{\varepsilon}^2
\int \!{\mathop{}\!\mathrm{d}}\tilde \pi \, \frac 1 {\lambda^2} \big| \big((-{{\mathcal L}})\omega_{n} -b\big)\big|^2.
\end{split}$$ We claim that for each $\gamma >0$ the second and the third term on the right hand side vanishes as $n\to \infty$ uniformly in ${\varepsilon}$. Since the first term on the right hand side can be bounded by using , we then conclude taking the limit $\gamma \to 0$.
To prove the claim, observe that, by construction of the sequence $\{\omega_n\}$, $$\label{3.2}
\begin{split}
&[(-{{\mathcal L}})\omega_n](v) = \lambda (v) a_n(v) \to b(v)\quad \pi\textrm{-a.e.} \\
& \big|(-{{\mathcal L}})\omega_n(v) -b(v)\big| \le 2 |b(v)|.
\end{split}$$ As $ \int\!{\mathop{}\!\mathrm{d}}x\, \bar u_{\varepsilon}^2$ is bounded uniformly in ${\varepsilon}$ by , we conclude by dominated convergence and .
*Proof of* . It is enough to show that for each $n$ $$\label{3.4}
\lim_{{\varepsilon}\to 0}
\int_0^T \!{\mathop{}\!\mathrm{d}}t \int\!{\mathop{}\!\mathrm{d}}x
\int \!{\mathop{}\!\mathrm{d}}\pi \, \big(f^{\varepsilon}-\rho^{\varepsilon}\big)\, w\cdot \omega_{n}
w\cdot (-{{\mathcal L}})\omega_{n} =0.$$ Indeed, by construction of the sequence $a_n$ $$\begin{split}
& \lim_{n\to\infty}\pi\big(\omega_n \otimes (-{{\mathcal L}})\omega_n\big) =\lim_{n\to\infty}
\pi(\lambda)\tilde \pi\big( \omega_n \otimes a_n\big)
=\pi(\lambda)\tilde\pi\big(\xi\otimes \tfrac b \lambda\big)=D.
\end{split}$$
In order to prove , by using the decomposition of $f^{\varepsilon}$ in , $$f^{\varepsilon}-\rho^{\varepsilon}=
2 \bar u_{\varepsilon}\big[ \tilde u_{\varepsilon}- \pi\big(\tilde u_{\varepsilon}\big)
\big] + \tilde u_{\varepsilon}^2 - \pi \big(\tilde u_{\varepsilon}^2\big).$$ Since $\omega_n$ is bounded and $|(-{{\mathcal L}}) \omega_n (v)| \le |b(v)|$, it suffices $$\label{3.5}
\begin{split}
\lim_{{\varepsilon}\to 0}
\int_0^T\!{\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x\,
\Big\{
\bar u_{\varepsilon}\pi\big( |\tilde u_{\varepsilon}| {|b|}\big)
+ \bar u_{\varepsilon}\pi\big(|\tilde u_{\varepsilon}|\big)
\pi\big(|b| \big) + \pi\big( \tilde u_{\varepsilon}^2 |b|\big)
+ \pi\big( \tilde u_{\varepsilon}^2 \big)\pi\big(|b|\big)
\Big\} =0.
\end{split}$$ By Cauchy-Schwarz, Lemma \[t:l4\], and , we directly conclude that the first and third term vanishes as ${\varepsilon}\to 0$. To analyze the fourth term, given $\delta>0$ we write $$\label{g4}
\begin{split}
& \int_0^T {\mathop{}\!\mathrm{d}}t\,\int \!{\mathop{}\!\mathrm{d}}x\, \pi\big(\tilde u_{\varepsilon}^2\big)
=\int_0^T {\mathop{}\!\mathrm{d}}t\,\int\!{\mathop{}\!\mathrm{d}}x\, \pi\big(\tilde u_{\varepsilon}^2 \,\chi_{\{\lambda\geq \delta\}}\big)
+\int_0^T dt\,\int\!{\mathop{}\!\mathrm{d}}x\, \pi\big(\tilde u_{\varepsilon}^2 \,\chi_{\{\lambda<\delta\}}\big).
\end{split}$$ By , the first term on the right hand side vanishes as ${\varepsilon}\to 0$. It is therefore enough to show that the second term vanishes as $\delta\to 0$ uniformly in ${\varepsilon}$. To this end, recalling that $\tilde u_{\varepsilon}=u_{\varepsilon}-\bar u_{\varepsilon}$ with $u_{\varepsilon}^2=f^{\varepsilon}$, $$ \begin{split}
& \int\!{\mathop{}\!\mathrm{d}}x\, \int\!{\mathop{}\!\mathrm{d}}\pi \, \tilde u_{\varepsilon}^2 \,\chi_{\{\lambda< \delta\}}
=\int\!{\mathop{}\!\mathrm{d}}x\, \int\! {\mathop{}\!\mathrm{d}}\pi \, \big(u_{\varepsilon}-\bar u_{\varepsilon}\big)^2 \,\chi_{\{\lambda< \delta\}}\\
& \leq 2 \int\!{\mathop{}\!\mathrm{d}}x\, \int\!{\mathop{}\!\mathrm{d}}\pi \, f^{\varepsilon}\,\chi_{\{\lambda< \delta\}}
+ 2\,\pi\big(\lambda< \delta \big)\int\!{\mathop{}\!\mathrm{d}}x\, \bar u _{\varepsilon}^2 ,
\end{split}$$ and we conclude by using , the basic entropy inequality and the assumption $\pi (\lambda=0)=0$. To complete the proof of , we observe that by Schwartz inequality and the previous argument also implies that the second term vanishes as ${\varepsilon}\to 0$.
*Proof of* . As before, it suffices to show that $$\lim_{{\varepsilon}\to 0}
\int_0^T {\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x
\int\!{\mathop{}\!\mathrm{d}}\pi [f^{\varepsilon}-\rho^{\varepsilon}]\, w\cdot b\,
w\cdot (-{{\mathcal L}})^{-1}b =0.$$ Since $(-{{\mathcal L}})^{-1}b$ is bounded, this follows from .
\[teo:3.3\] Assume that $(\rho^{\varepsilon}, j^{\varepsilon})\to (\rho, j)$. Then $$\varliminf_{{\varepsilon}\to 0}\frac 1{{\varepsilon}^2}\mathcal R
(f^{\varepsilon},\eta^{\varepsilon})\geq \int_0^T dt\, E(\rho(t)).$$
Assume first that condition (ii) in Assumption \[assumpt3\] holds. Let $a_n$ and $\omega_n$ as in the previous lemma, and fix $\phi\colon (0,T)\times {{\mathbb T}}^d\to {{\mathbb R}}$ with compact support. In the variational formula we choose $\alpha=1$ and $\zeta(t,x,v,v') = {\varepsilon}\, \nabla \phi(t,x)\cdot \big(
\omega_n(v')-\omega_n(v)\big)$, then, by the antisymmetry of $\eta^{\varepsilon}$ with respect to the exchange of $v,v'$, $$\begin{split}
\frac 1 {{\varepsilon}^2} \mathcal R (f^{\varepsilon},\eta^{\varepsilon})\geq &
-\frac 2{{\varepsilon}}\int_0^T\! {\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x \, \nabla \phi(t,x) \cdot
\iint\!\pi(dv)\pi(dv') \eta^{\varepsilon}(t,x,v,v')
\omega_n(v)\\
&-\frac 2 {{\varepsilon}^2} \int_0^T\! {\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x \,
\iint\!\pi(dv)\pi(dv')
f^{\varepsilon}(t,x,v)\sigma(v,v')\\
& \qquad \times \big\{{\mathop{\rm ch}\nolimits}\big({\varepsilon}\, \nabla \phi(t,x)\cdot \big(
\omega_n(v')-\omega_n(v)\big)\big)-1 \big\}
\end{split}$$ By the balance equation , $$\begin{split}
&-\frac 2{{\varepsilon}}\int_0^T\! {\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x \, \nabla \phi(t,x) \cdot
\iint\!\pi(dv)\pi(dv') \eta^{\varepsilon}(t,x,v,v')
\omega_n(v)\\
&\quad =
-2 {\varepsilon}\int_0^T\!{\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x \, \partial_t\nabla \phi\cdot
\int\!{\mathop{}\!\mathrm{d}}\pi f^{\varepsilon}\, \omega_n
- 2 \int_0^T\!{\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x\int\!{\mathop{}\!\mathrm{d}}\pi f^{\varepsilon}\, \nabla\cdot \big[ \omega_n \cdot \nabla \phi \, b \big].
\end{split}$$ Since $\omega_n$ is bounded, the first term on the right hand side above vanishes as ${\varepsilon}\to 0$. Therefore, by Taylor expansion of ${\mathop{\rm ch}\nolimits}$, $$\begin{split}
&\varliminf_{{\varepsilon}\to 0}\frac 1{{\varepsilon}^2}
\mathcal R(f^{\varepsilon},\eta^{\varepsilon})
\\
&\geq - \varlimsup_{{\varepsilon}\to 0} 2 \int_0^T\!{\mathop{}\!\mathrm{d}}t\!\int\!{\mathop{}\!\mathrm{d}}x\!\int\!\pi({\mathop{}\!\mathrm{d}}v)
\,f^{\varepsilon}(t,x,v) \nabla\cdot \big( \omega_n(v) \cdot \nabla
\phi(t,x) \, b(v) \big)
\\
& - \varlimsup_{{\varepsilon}\to 0} \int_0^T\!{\mathop{}\!\mathrm{d}}t\!\int\!{\mathop{}\!\mathrm{d}}x\!\iint\!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')
\,f^{\varepsilon}(t,x,v) \sigma(v,v') \Big[ \nabla\phi(t,x) \cdot\big(\omega_n(v')-\omega_n(v)\big)\Big]^2.
\end{split}$$
We will show that $$\label{lim1}
\begin{split}
\varlimsup_{n\to\infty} \varlimsup_{{\varepsilon}\to 0} 2 \int_0^T\!{\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x\int\!{\mathop{}\!\mathrm{d}}\pi
f^{\varepsilon}\nabla\cdot \big[ \omega_n \cdot \nabla
\phi \, b \big]
\le
2\int_0^T\!dt\int\!dx\, \rho \nabla \cdot \big[D \nabla\phi\big]
\end{split}$$ and $$\label{lim2}
\begin{split}
&\varlimsup_{n\to\infty} \varlimsup_{{\varepsilon}\to 0}
\int_0^T\!{\mathop{}\!\mathrm{d}}t\!\int\!{\mathop{}\!\mathrm{d}}x\!\int\!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')
\,f^{\varepsilon}(t,x,v) \sigma(v,v')
\Big[ \nabla\phi(t,x)
\cdot\big(\omega_n(v')-\omega_n(v)\big)\Big]^2
\\
&\qquad\qquad \le 2\int_0^T\!{\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x\, \rho(t,x) \nabla\phi(t,x)\cdot D
\nabla\phi(t,x).
\end{split}$$ Then, by optimizing over $\phi$ and using the variational representation , the statement follows.
Assume now that condition (i) in Assumption \[assumpt3\] holds. Then we choose as test functions $\alpha=1$ and $\zeta(t,x,v,v')={\varepsilon}\nabla\phi(t,x)\cdot (-{{\mathcal L}})^{-1}\big(b(v')- b(v)\big)$, with a smooth $\phi\colon (0,T)\times\mathbb T^d\to\mathbb R^d$ with compact support. Using the fact that $({{\mathcal L}})^{-1}b$ is bounded we repeat the same arguments as above and therefore we have to show that $$\label{lim1b}
\begin{split}
\varlimsup_{{\varepsilon}\to 0} 2 \int_0^T\!{\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x\int\!{\mathop{}\!\mathrm{d}}\pi
f^{\varepsilon}\nabla\cdot \big[ (-{{\mathcal L}})^{-1}b \cdot \nabla
\phi \,b \big]
\le
2\int_0^T\!{\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x\, \rho \nabla \cdot \big[D \nabla\phi\big]
\end{split}$$ and $$\label{lim2b}
\begin{split}
&\varlimsup_{{\varepsilon}\to 0}
\int_0^T\!{\mathop{}\!\mathrm{d}}t\!\int\!{\mathop{}\!\mathrm{d}}x\!\iint\!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')
\,f^{\varepsilon}(t,x,v) \sigma(v,v')
\Big[ \nabla\phi(t,x)
\cdot(-{{\mathcal L}})^{-1}\big(b(v')-b(v)\big)\Big]^2
\\
&\qquad\qquad \le 2\int_0^T\!{\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x\, \rho(t,x) \nabla\phi(t,x)\cdot D
\nabla\phi(t,x).
\end{split}$$ *Proof of* . We claim that for each $n$ $$\lim_{{\varepsilon}\to 0}
\int_0^T\!{\mathop{}\!\mathrm{d}}t\int\!{\mathop{}\!\mathrm{d}}x\int\!{\mathop{}\!\mathrm{d}}\pi
\big[ f^{\varepsilon}-\rho^{\varepsilon}\big]
\nabla_\cdot \big[ \omega_n \cdot \nabla
\phi b \big] =0,$$ which is proven exactly as observing that there we used the bound $|(-{{\mathcal L}}) \omega_n|\le |b|$. Since $\rho^{\varepsilon}\to \rho$ and, by construction of the sequence $\{\omega_n\}$, $\lim_{n} \pi( \omega_n \otimes b)=
D$, we then conclude.
*Proof of* . We first show that for each $n$ $$\begin{split}
&\lim_{{\varepsilon}\to 0}
\int_0^T\!{\mathop{}\!\mathrm{d}}t\!\int\!{\mathop{}\!\mathrm{d}}x\!\int\!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')
\,\big[ f^{\varepsilon}(t,x,v) -\rho^{\varepsilon}(t,x)\big] \sigma(v,v')
\\
&\qquad\qquad\qquad\qquad \times
\Big[\nabla\phi(t,x)
\cdot\big(\omega_n(v')-\omega_n(v)\big)\Big]^2 = 0.
\end{split}$$ Since $\omega_n$ is bounded and $\lambda(v)=\int\!\pi(dv')
\sigma(v,v')$, it is enough to prove that $$\lim_{{\varepsilon}\to 0}
\int_0^T\!{\mathop{}\!\mathrm{d}}t\!\int\!{\mathop{}\!\mathrm{d}}x\!\int\!{\mathop{}\!\mathrm{d}}\pi \lambda
\,\big| f^{\varepsilon}-\rho^{\varepsilon}\big| = 0,$$ whose proof is achieved by the same arguments used in the proof of . We then conclude by observing that $$\begin{split}
&\lim_{n\to\infty} \iint\!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\sigma(v,v')\big(\omega_n(v')-\omega_n(v) \big)\otimes \big(\omega_n(v')-\omega_n(v) \big)\\
&\qquad =\lim_{n\to\infty}2\pi\big(\omega_n\times (-\mathcal L)\omega_n \big)=2D.
\end{split}$$
*Proofs of and* . These are achieved as the proofs of and with $(-{{\mathcal L}})^{-1}b$ instead of $\omega_n$.
Specific examples
=================
We consider three examples of linear Boltzmann equations and we show that they meet the requirements in the Assumptions \[t:asb\], \[assumpt1\] and \[assumpt3\]. For all of them the convergence to a diffusion is a classical result, therefore they are suitable testers for the machinery. We emphasizes however that we do not require the initial distribution to be in $L^2$, as it is usual in the classical approaches, but only with finite entropy with respect to the reference measure.
Boltzmann-Grad limit for the Lorentz gas with hard scatterers
-------------------------------------------------------------
The first example is the linear Boltzmann equation derived for the one particle distribution in [@Ga], starting from the Lorentz gas moving in a random array of fixed scatterers (hard spheres), in the Boltzmann-Grad limit. Since collisions are elastic, the kinetic energy is preserved, therefore the phase space is $\mathbb T^d\times S_{|v|}^{d-1}$, $d\geq 2$, where $S_{|v|}^{d-1}$ is the $d$-dimensional sphere with radius $|v|$. Without loss of generality, we assume $|v|=1$. The equation then reads $$\label{ex1}
\partial_t f(t,x,v)+v\cdot \nabla_x f(t,x,v)= \int_{S^{d-1}} {\mathop{}\!\mathrm{d}}\hat n\,[\hat n \cdot v]_+ \big[ f(t,x,v')-f(t,x,v)\big],$$ where $v'=v-2(v\cdot \hat n)\hat n$. The invariant measure is the uniform measure ${\mathop{}\!\mathrm{d}}\hat n$ on $S^{d-1}$, the scattering rate $\lambda$ is equal to $c$, for some constant $c$ depending on the dimension $d$. In order to identify the scattering kernel, we consider here the case $d=2$, referring to Appendix \[s:A\] for analogous computations if $d\geq 3$. By identifying the velocity $v\in S^1$ with the angle $\theta$, we rewrite the previous equation as $$\partial_t f(t,x,\theta)+b(\theta)\cdot \nabla_x f(t,x,\theta)=\frac 1 2 \int_{S^1} {\mathop{}\!\mathrm{d}}\theta'\,\Big|\sin\frac {\theta-\theta'}2 \Big|\big[ f(t, x,\theta')-f(t,x,\theta)\big],$$ with $b(\theta)=(\cos \theta,\, \sin\theta)$. In particular, the scattering kernel is $\sigma(\theta,\theta')=\Big|\sin\frac {\theta-\theta'}2 \Big|$. Recalling the definition of $\tilde\pi$ , the operator $K$ with kernel $\sigma/\lambda$ is compact in $L^2(\tilde\pi )$. Since $1$ is a simple eigenvalue of $K$, then the modified chain has spectral gap. Moreover, as shown in [@BNPP], Lemma 4.1, $L^{-1}v$ is bounded. Hence Assumptions \[t:asb\], \[assumpt1\] and the alternative (i) of Assumption \[assumpt3\] hold.
Rayleigh-Boltzmann equation
---------------------------
The Rayleigh-Boltzmann equation, also known as linear Boltzmann equation or Lorentz-Boltzmann equation, has been derived in the Boltzmann-Grad limit by looking at the distribution of a tracer particle in a gas of particles (hard spheres) in thermal equilibrium [@vBLLS]. The velocity space is then $\mathbb R^d$, $d\geq 2$, and the reference measure is the Maxwellian distribution with temperature $\beta^{-1}$, whose density with respect to the Lebesgue measure is denoted by $h_\beta$. The equation reads $$\label{ex3}\begin{split}
&\partial_t f(t,x,v)+v\cdot \nabla_x f(t,x,v)\\
&\qquad= \int_{\mathbb R^d}\!{\mathop{}\!\mathrm{d}}v_1\,
h_\beta(v_1)\int_{S^{d-1}}\!{\mathop{}\!\mathrm{d}}\hat n \,\big[\hat n\cdot(v-v_1)\big]_+\big[f(t,x,v')-f(t,x,v) \big],
\end{split}$$ where $v'= v-\hat n\cdot(v-v_1)\,\hat n$. As shown in the Appendix \[s:A\], he scattering rate is $$\lambda(v)=\chi\int_{\mathbb R^d}{\mathop{}\!\mathrm{d}}v_1 h_\beta(v_1)|v-v_1|,$$ where $\chi$ is the constant given by $\chi=\int_{S^{d-1}}\!{\mathop{}\!\mathrm{d}}\hat n [\hat n\cdot\hat v]_+$, in which $\hat v\in S^{d-1}$. In particular $\lambda$ is bounded away from $0$ and it has linear growth for large $|v|$. Therefore $|v|$ and $|v|^2/\lambda(v)$ have all the exponential moments with respect to $h_{\beta}(v){\mathop{}\!\mathrm{d}}v$. In Appendix \[s:A\] we identify the scattering kernel $\sigma$, see . From this expression and the properties of $\lambda$, recalling the definition , it follows that $K$ in has a kernel in $L^2(\tilde\pi\times\tilde \pi)$. Hence $K$ is compact in $L^2(\tilde\pi)$. Since $1$ is simple eigenvalue of $K$, then $({{1 \mskip -5mu {\rm I}}}-K)$ has spectral gap. The previous statements imply that Assumptions \[t:asb\] and \[assumpt1\] hold. The proof of alternative (i) in Assumption \[assumpt3\] is the content of Appendix \[s:A\].
Linear phonon Boltzmann equation
--------------------------------
The equation has been derived in the kinetic limit starting from an harmonic chain of oscillator perturbed by a stochastic conservative noise [@BOS]. It describes the evolution of the energy density of the normal modes, or phonons, identified by a wave-number $k\in\mathbb T^d$. The velocity space is then $\mathbb T^d$. Let $\omega$ be the dispersion relation of the harmonic lattice, i.e. $\omega(k)=\big(\nu +4\sum_{i=1}^d\sin^2(\pi k_i) \big)^{1/2}$, where $\nu\geq 0$ is the intensity of the pinning. A phonon with wave-number $k$ travels with velocity $\frac 1 {2\pi}\nabla\omega$, then it is scattered. The corresponding Fokker-Planck equation reads $$\label{ex2}
\partial_t f(t,x,k)+\frac 1 {2\pi}\nabla\omega(k)\cdot \nabla_x f(t,x,k)=\int_{\mathbb T^d}\!{\mathop{}\!\mathrm{d}}k'\,\sigma(k,k')\big[f(t,x,k') -f(t,x,k) \big],$$ where the scattering kernel has the form $\sigma(k, k')=
\sum_{i=1}^d \sin^2(\pi k_i)\sin^2(\pi k'_i)$ for $d\geq 2$. In dimension one it has a slightly different form, but despite the details the main features are that $\sigma$ is positive, bounded and symmetric in the exchange $k, k'$. Then the invariant measure $\pi$ is the Haar measure on the torus. Moreover $\sigma$ vanishes in zero, since $\sigma(k,k')\sim |k|^2$ for small $k$, and the scattering rate $\lambda$ has the same behavior. More precisely, $\lambda=c \sum_{i=1}^d \sin^2(\pi k_i)$, for some constant $c>0$. Since $\partial_i \omega(k)= 2\pi\sin(2\pi k_i)/\omega(k)$, $i=1,\dots,d$, in order to guarantee that $|\nabla\omega|^2/\lambda$ has exponential moments we have to restrict to the case $\nu>0$. This corresponds to assume that in the underlying harmonic chain the translational symmetry is broken, due to the presence of a on-site potential (pinning). Recalling the definition of $\tilde\pi$, by the properties of $\sigma$ and $\lambda$ we deduce that the operator $K$ defined in has a kernel of the form $p(k,k')\tilde\pi(dk')$, with $p$ strictly positive and bounded. Then $K$ is a compact operator in $L^2(\tilde\pi)$, and since $1$ is a simple eigenvalue, than the modified chain has spectral gap. The previous statements imply that Assumptions \[t:asb\] and \[assumpt1\] hold. Finally, since the modified chain satisfies the Doeblin condition, then for each $f\in L^\infty(\mathbb T^d)$ such that $\tilde\pi(f)=0$ we have $\|\sum_{n\geq 0} K^n f\|_{\infty}\leq c\|f\|_{\infty}$, which implies alternative (ii) in Assumption \[assumpt3\].
In the unpinned case $\nu=0$ the diffusion coefficient $D$ diverges in dimension $d=1,2$. In these cases the asymptotics of the linear phonon Boltzmann equation is in fact a superdiffusion when $d=1$ [@JKO; @BaBo] and a diffusion under an anomalous scaling, i.e. with logarithmic corrections, when $d=2$ [@Ba]; see also [@MMM] for other models with super-diffusive behavior. On the other hand, for $d\geq 3$ the diffusion coefficient $D$ is finite even if $|b|^2/\lambda$ does not have exponential moments. Moreover, as discussed in [@Ba], if the initial distribution satisfies strong integrability conditions that imply a Nash inequality, the diffusive scaling holds. As the gradient flow approach here introduced requires only an entropy bound on the initial condition, it does not cover this case. It is not clear if this is just a limitation of the approach or the diffusive limit fails if the integrability conditions are not satisfied. The main point is that phonons with small wave number are responsible of the ballistic transport which, low dimensions, induces the superdiffusion. If the initial conditions gives enough weight to those phonons, similar effects might occur also for $d\geq 3$.
Entropy balance {#app2}
===============
We here prove . We can assume that its left hand side is finite. Using also that $P_r$ has bounded entropy for each $r\in[s,t]$ we deduce that $P_r({\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v) = f_r(x,v) {\mathop{}\!\mathrm{d}}x \, \pi({\mathop{}\!\mathrm{d}}v)$ and $\Theta_{[s,t]} ({\mathop{}\!\mathrm{d}}r, {\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v') = \eta_r(x,v,v') {\mathop{}\!\mathrm{d}}r
\, {\mathop{}\!\mathrm{d}}x\, \pi({\mathop{}\!\mathrm{d}}v) \pi({\mathop{}\!\mathrm{d}}v')$. Moreover, recalling the function $\Psi_\kappa$ in , $$\begin{split}
& \int_s^t \!{\mathop{}\!\mathrm{d}}r\, {{\mathcal E}}(P_r) =
\int_s^t \!{\mathop{}\!\mathrm{d}}r \int\!{\mathop{}\!\mathrm{d}}x \iint \!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\,
\sigma(v,v') \Big[ \sqrt{f_r(x,v')} -\sqrt{f_r(x,v)}\Big]^2 \\
& {{\mathcal R}}^{s,t} (P, \Theta_{[s,t]})
=
\int_s^t \!{\mathop{}\!\mathrm{d}}r \int\!{\mathop{}\!\mathrm{d}}x \iint \!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\,
\Psi_{\sigma(v,v')} \big( f_r(x,v),f_r(x,v'); \eta_r(x,v,v')
\big)
\end{split}$$ and $$\label{intrep}
\int_s^t \!{\mathop{}\!\mathrm{d}}r\, {{\mathcal E}}(P_r) + {{\mathcal R}}^{s,t} (P, \Theta_{[s,t]}) < +\infty.$$
We claim that, for $f$ and $\eta$ as above, the following entropy balance holds $$\begin{split}
\label{entbal}
& {{\mathcal H}}(P_t) -{{\mathcal H}}(P_{s'})\\ & = \frac 12
\int_{s'}^t \!{\mathop{}\!\mathrm{d}}r\! \int\!{\mathop{}\!\mathrm{d}}x\! \iint \!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\,
\eta_r(x,v,v') \big[ \log f_r(x,v')- \log f_r(x,v) \big]
\end{split}$$ for any $0\leq s< s'<t\leq T$. Note that the last term is well defined by Legendre duality and . Informally, it is deduced by choosing the test function ${{1 \mskip -5mu {\rm I}}}_{ [s',t]}(r)\log f_r(x,v)$ in the balance equation . The actual proof is carried out by a truncation argument that is next detailed.
*Step 1. Approximation by space time convolutions.* For $n\in{{\mathbb N}}$ let now $\chi_n \colon {{\mathbb R}} \to {{\mathbb R}}_+$ be a smooth approximation of the identity with compact support contained in the positive axis, and $g_n\colon {{\mathbb T}}^d\to {{\mathbb R}}_+$ be a smooth approximation of the identity. For $0\leq s< s'\leq r \leq t \leq T$, by choosing $n$ such that the $\mathrm{supp}\chi_n \subset [0,s'-s]$, we define $$\begin{split}
f^n_r(x,v) := & \int\!{\mathop{}\!\mathrm{d}}r' \int {\mathop{}\!\mathrm{d}}y \, \chi_n(r-r') g_n(x-y)
f_{r'}(y,v) \\
\eta^n_r(x,v,v') := &\int\!{\mathop{}\!\mathrm{d}}r' \int {\mathop{}\!\mathrm{d}}y \, \chi_n(r-r') g_n(x-y)
\eta_{r'}(y,v,v').
\end{split}$$ As simple to check, the pair $(f^n,\eta^n)$ satisfies the balance equation and, by and convexity, there exists a constant $C$ such that $$\begin{split}
&\int_{s'}^t \!{\mathop{}\!\mathrm{d}}r \int\!{\mathop{}\!\mathrm{d}}x \iint \!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\,
\sigma(v,v') \Big[ \sqrt{f^n_r(x,v')} -\sqrt{f^n_r(x,v)}\Big]^2
\\
&\qquad +
\int_{s'}^t \!{\mathop{}\!\mathrm{d}}r \int\!{\mathop{}\!\mathrm{d}}x \iint \!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\,
\Psi_{\sigma(v,v')} \big( f^n_r(x,v),f^n_r(x,v'); \eta^n_r(x,v,v')
\big) \le C,
\end{split}$$ and $$\sup_{r\in[s',t]} {{\mathcal H}}(f^n_r)\leq \sup_{r\in[0,T]} {{\mathcal H}}(f_r)\leq C.$$
*Step 2. Truncation of $\log$.* The balance equation implies $$\begin{split}\label{beqt0}
& \int\! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v) f_t^n(x,v) \phi(t,x,v)-\int\! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v) f_{s'}^n(x,v) \phi(s',x,v)
\\ &-\int_{s'}^t \! {\mathop{}\!\mathrm{d}}r \int \! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v)
f_r^n(x,v)
\big\{
\partial_r \phi(r,x,v)
+b(v)\cdot \nabla \phi(r,x,v)
\big\}
\\
& =\frac 1 2 \int_{s'}^t\! {\mathop{}\!\mathrm{d}}r \int\! {\mathop{}\!\mathrm{d}}x\iint\! \pi({\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v')\eta^n(r,x,v,v')\big[\phi(r,x,v')- \phi(r,x,v) \big]
\end{split}$$ for all continuous functions $\phi\colon [s',t]\times {{\mathbb T}}^d\times {{\mathcal V}}$ with compact support in ${{\mathcal V}}$ and continuously differentiable in the first two variables. Recalling that $b$ has exponential moments, since $f^n$ has bounded entropy and $\eta^n\in L^1$ we can use $\phi$ bounded instead of compactly supported.
Given $0<\delta <L$ set $$\label{def:tlog}
\log_{\delta,\, L}(u)= \begin{cases}
\log \delta \quad \mbox{if } 0<u<\delta\\
\log u \quad \mbox{if } \delta \leq u \leq L\\
\log L \quad \mbox{if } u>L.
\end{cases}$$ By a straightforward approximation we can choose as test function in $\phi=\log_{\delta,\, L}(f^n)$, obtaining $$\begin{split}\label{beqt}
& \int\! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v) f_t^n(x,v) \log_{\delta,\, L}(f^n_t(x,v))
-\int\! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v) f_{s'}^n(x,v) \log_{\delta,\, L}(f^n_{s'}(x,v))
\\ &-\int_{s'}^t \! {\mathop{}\!\mathrm{d}}r \int \! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v) {{1 \mskip -5mu {\rm I}}}_{[\delta,L]}(f^n_r(x,v))
\big\{
\partial_r f^n_r(x,v)
+b(v)\cdot \nabla f^n_r(x,v)
\big\}
\\
& =\frac 1 2 \int_{s'}^t\! {\mathop{}\!\mathrm{d}}r \int\! {\mathop{}\!\mathrm{d}}x\iint\! \pi({\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v')\eta^n(r,x,v,v')
\big[\log_{\delta,\, L}(f^n_r(x,v))- \log_{\delta,\, L}(f^n_r(x,v')) \big].
\end{split}$$ We observe that $$\label{tra}\begin{split}
& \int_{s'}^t \! {\mathop{}\!\mathrm{d}}r \int \! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v) {{1 \mskip -5mu {\rm I}}}_{[\delta,L]}(f^n_r(x,v))
\big\{
\partial_r f^n_r(x,v)
+b(v)\cdot \nabla f^n_r(x,v)
\big\}\\
& =\int \! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v) \big(f^n_t(x,v)\wedge \delta \big)\vee L
-\int \! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v) \big(f^n_{s'}(x,v)\wedge \delta \big)\vee L.
\end{split}$$
*Step 3. Removing the convolution.* Since $\log_{\delta, L}$ is bounded, by dominated convergence we can remove regularization in space and time and we obtain $$\begin{split}\label{beqt1}
& \int\! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v) f_t(x,v) \log_{\delta,\, L}(f_t(x,v))
-\int\! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v) f_{s'}(x,v) \log_{\delta,\, L}(f_{s'}(x,v))
\\
&- \int \! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v) \big(f_t(x,v)\wedge \delta \big)\vee L
+\int \! {\mathop{}\!\mathrm{d}}x \int \pi({\mathop{}\!\mathrm{d}}v) \big(f_{s'}(x,v)\wedge \delta \big)\vee L.
\\
& =\frac 1 2 \int_{s'}^t\! {\mathop{}\!\mathrm{d}}r \int\! {\mathop{}\!\mathrm{d}}x\iint\! \pi({\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v')\eta(r,x,v,v')
\big[\log_{\delta,\, L}(f_r(x,v))- \log_{\delta,\, L}(f_r(x,v')) \big].
\end{split}$$
*Step 3. Removing the truncation of $\log $.*
Here we take the limit $\delta\downarrow 0$ and $L\uparrow +\infty$ in . For the left hand side this is accomplished by monotone convergence, in particular it converges to ${{\mathcal H}}(P_t) - {{\mathcal H}}(P_{s'})$. For the right hand side, it is enough to show that $$\label{van}\begin{split}
\frac 1 2 \int_{s'}^t\! {\mathop{}\!\mathrm{d}}r \int\! {\mathop{}\!\mathrm{d}}x\iint\! \pi({\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v')\eta(r,x,v,v')
\Big\{ \big[\log_{\delta,\, L}(f_r(x,v))- \log(f_r(x,v)) \big]\\
- \big[\log_{\delta,\, L}(f_r(x,v'))- \log(f_r(x,v')) \big] \Big\}
\end{split}$$ vanishes as $\delta \downarrow 0$ and $L\uparrow +\infty$. We apply Young inequality in the form $$p q \leq \psi_{\alpha}(p) +\psi_{\alpha}^*(q),$$ where, for $\alpha\geq 0$, $$\psi_\alpha(p)= p{\mathop{\rm ash}\nolimits}\frac p \alpha - \sqrt{p^2 +\alpha^2} +\alpha,\quad
\psi^*_\alpha(q)= \alpha \big(\cosh q -1\big).$$ Observe that $\psi_\alpha$ and $\psi_\alpha^*$ are even. By choosing $\alpha=2\sigma(v,v')\sqrt{f_r(x,v)f_r(x,v')}$, $p=\eta_r(x,v,v')$ and $q=\frac 12 \big[\log_{\delta,\, L}(f_r(x,v))- \log(f_r(x,v))
- \log_{\delta,\, L}(f_r(x,v'))+ \log(f_r(x,v')) \big]$, the first term is $$\begin{split}
& \int_{s'}^t\! {\mathop{}\!\mathrm{d}}r \int\! {\mathop{}\!\mathrm{d}}x\iint\! \pi({\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v')
\psi_{\alpha}(\eta)\big[1-{{1 \mskip -5mu {\rm I}}}_{[\delta, L]}(f_r(x,v)) \big]\\
\leq
& \int_{s'}^t \!{\mathop{}\!\mathrm{d}}r \int\!{\mathop{}\!\mathrm{d}}x \iint \!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\,
\Psi_{\sigma(v,v')} \big( f_r(x,v),f_r(x,v'); \eta_r(x,v,v')
\big),
\end{split}$$ which vanishes by dominated convergence since $\mathcal R (f,\eta)$ is finite. The second term has the following expression $$\begin{split}
& \int_{s'}^t\! {\mathop{}\!\mathrm{d}}r \int\! {\mathop{}\!\mathrm{d}}x\iint\! \pi({\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v') \psi_\alpha^*(q)\\
&= \int_{s'}^t\! {\mathop{}\!\mathrm{d}}r \int\! {\mathop{}\!\mathrm{d}}x\iint\! \pi({\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v')
\big[{{1 \mskip -5mu {\rm I}}}_{[0, \delta)}(f_r(x,v)){{1 \mskip -5mu {\rm I}}}_{[0, \delta)}(f_r(x,v'))\\
& \quad + {{1 \mskip -5mu {\rm I}}}_{(L, +\infty)}(f_r(x,v)){{1 \mskip -5mu {\rm I}}}_{(L, +\infty)}(f_r(x,v')) \big]
\sigma(v, v')\big(\sqrt {f_r(x,v)}- \sqrt {f_r(x,v')} \big)^2\\
& +2\int_{s'}^t\! {\mathop{}\!\mathrm{d}}r \int\! {\mathop{}\!\mathrm{d}}x\iint\! \pi({\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v')
{{1 \mskip -5mu {\rm I}}}_{[0, \delta)}(f_r(x,v)){{1 \mskip -5mu {\rm I}}}_{[\delta, L]}(f_r(x,v'))\\
& \quad\times
\sigma(v,v')\sqrt{f_r(x,v)f_r(x,v')}\Big( \frac {\sqrt \delta} {\sqrt{f_r(x,v)}}+\frac{\sqrt{f_r(x,v)}}{\sqrt \delta}-2 \Big)\\
& +2\int_{s'}^t\! {\mathop{}\!\mathrm{d}}r \int\! {\mathop{}\!\mathrm{d}}x\iint\! \pi({\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v')
{{1 \mskip -5mu {\rm I}}}_{[0, \delta)}(f_r(x,v)){{1 \mskip -5mu {\rm I}}}_{(L, +\infty)}(f_r(x,v'))\\
& \quad\times
\sigma(v,v')\sqrt{f_r(x,v)f_r(x,v')}
\Big( \frac {\sqrt{L f_r(x,v)}}{\sqrt{\delta f_r(x,v')}}
+\frac {\sqrt{\delta f_r(x,v')}}{\sqrt{L f_r(x,v)}}-2 \Big)\\
& + 2 \int_{s'}^t\! {\mathop{}\!\mathrm{d}}r \int\! {\mathop{}\!\mathrm{d}}x\iint\! \pi({\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v')
{{1 \mskip -5mu {\rm I}}}_{[\delta, L]}(f_r(x,v)){{1 \mskip -5mu {\rm I}}}_{(L, +\infty)}(f_r(x,v'))\\
& \quad\times \sigma(v,v')\sqrt{f_r(x,v)f_r(x,v')}
\Big( \frac {\sqrt L} {\sqrt{f_r(x,v')}}+\frac{\sqrt{f_r(x,v')}}{\sqrt L}-2 \Big)
\end{split}$$ We observe that the first integral on the right hand side vanishes as $\delta\downarrow 0$, $L\uparrow +\infty$ since $\int_{s'}^t {\mathop{}\!\mathrm{d}}r\, \mathcal E(f_r) < +\infty$. The term in the second integral is bounded by $\sigma(v,v')\sqrt{\delta}\sqrt{f_r(x,v')}$ and the term in the third is bounded by $\sigma(v,v')\sqrt{\delta/L}f_r(x,v') $, then the two integrals vanish as $\delta\downarrow 0$, $L\uparrow +\infty$ since the scattering rate $\lambda$ has all exponential moments and $f_r$ has finite entropy. Finally, the term in the last integral in bounded by $\sigma(v,v')f_r(x,v'){{1 \mskip -5mu {\rm I}}}_{(L, +\infty)}(f_r(x,v'))$, which vanishes by dominated convergence.
Now we show that holds. For $s'>s\geq 0$ it by applying again Young inequality with $p=-\eta_r(x,v,v')$ and $q=\frac 12 \big[\log (f_r(x,v))- \log(f_r(x,v'))\big]$ with the entropy balance . Finally the case $s'=s$ is achieved by the lower semi-continuity of $\mathcal H$.
Bounds on $(-{{\mathcal L}})^{-1}b$ for the Rayleigh gas {#s:A}
========================================================
We identify the scattering kernel $\sigma$ on the right hand side of . Setting $z = v-v_1$, the collision operator in becomes $$\label{Lmio}
{{\mathcal L}}f(v) =
\int_{\mathbb R^d}{\mathop{}\!\mathrm{d}}z\, h_\beta(v-z)\int_{S^{d-1}}{\mathop{}\!\mathrm{d}}\hat n \,
[\hat n\cdot z]_+\,\{f(v')-f(v)\}$$ where $$v'= v - (\hat n \cdot z) \, \hat n$$ Fixed $\hat n$, we can write $z = \alpha \hat n + z^\perp$, where $z^\perp$ lies in the hyperplane of dimension $d-1$ orthogonal to $\hat n$. We indicate with $v^\perp = v-(\hat n \cdot v)\, \hat n$, the projection of $v$ on this hyperplane. We have $[\hat n \cdot z]_+ = [\alpha]_+$, ${\mathop{}\!\mathrm{d}}z = {\mathop{}\!\mathrm{d}}\alpha \, {\mathop{}\!\mathrm{d}}z^\perp$, and $${{\mathcal L}}f(v) = \int_{S^{d-1}} {\mathop{}\!\mathrm{d}}\hat n
\int {\mathop{}\!\mathrm{d}}z^\perp h_\beta^{d-1}(v^\perp-z^\perp)
\int_0^{+\infty} {\mathop{}\!\mathrm{d}}\alpha \,\alpha \,h_\beta^1(v\cdot \hat n - \alpha)
\{f(v')-f(v)\}$$ where $h_\beta^k$ is the Maxwellian distribution in dimension $k$, and now $v' = v -\alpha \hat n$. The integral in ${\mathop{}\!\mathrm{d}}z^\perp$ gives 1. Choosing the new variable $w = v - \alpha \hat n$, we have $\alpha = |v-w|$, $v\cdot \hat n= v\cdot (v-w)/|(v-w)|$, $$v\cdot \hat n - \alpha = v\cdot (v-w)/|(v-w)| - |v-w| =
w \cdot (v-w)/|v-w|$$ and ${\mathop{}\!\mathrm{d}}w = \alpha^{d-1} {\mathop{}\!\mathrm{d}}\alpha\,{\mathop{}\!\mathrm{d}}\hat n = |v-w|^{d-1} {\mathop{}\!\mathrm{d}}\alpha \,
{\mathop{}\!\mathrm{d}}\hat n$. Then $${{\mathcal L}}f(v) =
\int_{{\mathbb{R}}^d}{\mathop{}\!\mathrm{d}}w
h_\beta^1( w\cdot (v-w) /|v-w|) \frac 1{|v-w|^{d-2}}\{f(w)-f(v)\}$$ which is of the form with $$\label{sfuturo}\begin{aligned}\sigma(v,w) &= \frac 1{|v-w|^{d-2}}
\frac {h_\beta^1( w\cdot (v-w) /|v-w|)}{h_\beta(w)} \\
&=
\left( \frac {\beta}{2\pi}\right)^{\frac {1-d}2}
\frac 1{|v-w|^{d-2}}
\exp \left\{
\frac {\beta}{2} \frac {|w|^2|v-w|^2 - (w\cdot (v-w))^2}{|v-w|^2}\right\}\\
&=
\left( \frac {\beta}{2\pi}\right)^{\frac {1-d}2}
\frac 1{|v-w|^{d-2}}
\exp \left\{
\frac {\beta}{2} \frac {|w|^2|v|^2 - (w\cdot v)^2}{|v-w|^2}
\right\}.
\end{aligned}$$ which is symmetric. We remark that for $d=3$ this expression has been obtained in [@LS].
In order to prove that $\xi = -{{\mathcal L}}^{-1} v$ is bounded, we decompose ${{\mathcal L}}$ in the gain and loss terms $$-({{\mathcal L}}f)(v) = \lambda(v) f(v) - (Gf)(v)$$ where $$\lambda(v)=(G1)(v) =
\int_{\mathbb R^d}{\mathop{}\!\mathrm{d}}w h_\beta(w)\sigma(v,w) =
\chi\int_{\mathbb R^d}{\mathop{}\!\mathrm{d}}v_1 h_\beta(v_1)|v-v_1|
= \lambda(|v|),$$ and $\chi = \int_{S^{d-1}}\! {\mathop{}\!\mathrm{d}}\hat n\, [\hat n\cdot \hat v]_+$ for any unit vector $\hat v$. Observe that $(Gf)(v) = \lambda(v) (Kf)(v)$, with $K$ defined in . Note that, for convexity, $$\label{dislambda}
\lambda(v) \ge \chi |v|.$$ We search for a bounded function $\gamma(|v|)$ such that $\xi (v) = \hat v \gamma(|v|)$. Then we have $$\begin{split} (G\xi)(v) &=
\int_{{\mathbb{R}}^d} {\mathop{}\!\mathrm{d}}w \, h_\beta(w) \sigma(v,w)
{\gamma(|w|)} \frac w{|w|} \\
&=\int_{{\mathbb{R}}^d} {\mathop{}\!\mathrm{d}}w \, h_\beta(w) \sigma(v,w)
\frac {\gamma(|w|)}{|w|}
\left[
(w - (\hat v \cdot w)\,
\hat v)+ (\hat v \cdot w )\, \hat v )\right]
\end{split}$$ where in the last step we decomposed $w$ into the component along $\hat v$ and the orthogonal part $w^\perp=w-(\hat v \cdot w) \hat v$. Since $|w|$ and $\sigma(v,w)$ are invariant in the exchange $w^\perp \to - w^\perp$, then $$(G\xi)(v) = \hat v \int_{{\mathbb{R}}^d}
{\mathop{}\!\mathrm{d}}w \, h_\beta(w) \sigma(v,w) (\hat v \cdot \hat w)
\gamma(|w|). $$ Since the integral is invariant under rotations of $v$, we can define the operator $\tilde G$ acting on functions on the positive half line by $$(\tilde Gf)(\rho) :=
\int_{{\mathbb{R}}^d} {\mathop{}\!\mathrm{d}}w \,
h_\beta(w) \sigma(\rho \hat v,w) (\hat v \cdot \hat w) f(|w|),$$ so that, for $\xi (v) = \hat v \gamma(|v|)$, $(G\xi)(v) = \hat v (\tilde G\gamma) (|v|)$. As $$(\tilde Gf)(\rho) =
$$ \_[w v>0]{} w h\_(w) ((v,w) - (v,-w)) (v w) (|w|)$$and, if $w \cdot v>0$, then $\sigma(v,w) \ge \sigma(v,-w)$, the operator $\tilde G$ has positive kernel.
Setting $\eta(\rho)=\lambda(\rho)\gamma(\rho)$, we look for the solution of the equation $$\label{Arho=}
\rho = \eta(\rho) - (A\eta) (\rho), \ \ \ \rho \in {\mathbb{R}}^+$$ where $$\label{AA}
\begin{aligned}
(A\eta)(\rho) &= \left({\tilde G}\frac {\eta}{\lambda}\right)(\rho) =
\int_{{\mathbb{R}}^d} {\mathop{}\!\mathrm{d}}w \, h_\beta(w) \sigma(\rho \hat v,w)
(\hat v \cdot \hat w)
\frac {\eta(|w|)}{\lambda(|w|)} \\
&=
\int_{{\mathbb{R}}^d}
{\mathop{}\!\mathrm{d}}v_1 \, h_\beta(v_1) \int_{S^{d-1}} {\mathop{}\!\mathrm{d}}\hat n \,
[\hat n \cdot (\rho \hat v-v_1)]_+ (\hat v \cdot \hat v')
\frac {\eta(|v'|)}{\lambda(|v'|)}
\end{aligned}$$ in which $v'= \rho \hat v - (\hat n \cdot (\rho \hat v -v_1) ) \hat n$. The operator $A$ is self-adjoint with respect to the scalar product $$(f,g) = \int_{{\mathbb{R}}^d}
{\mathop{}\!\mathrm{d}}v \, h_{\beta}(v) \frac 1{\lambda(|v|)} f(|v|)g(|v|),$$ defined for $f,g:[0,\infty)\to {\mathbb{R}}$. From the positivity of the kernel of the operator, it follows that if $f(\rho)\ge g(\rho)$ for any $\rho\ge 0$, then $(Af)(\rho) \ge (Ag) (\rho)$ for any $\rho\ge 0$. Moreover, if $\eta$ is continuous in $[0,\infty)$, $(A\eta)$ is continuous in $[0,\infty)$ as follow from .
By definition of $\lambda(\rho)$ $$(A\lambda) (\rho) =
\int_{{\mathbb{R}}^d} {\mathop{}\!\mathrm{d}}w \, h_\beta(w) \sigma(\rho \hat v,w)
(\hat v \cdot \hat w)
< \lambda(\rho)$$ and the inequality is strict for any $\rho$ because $\hat v \cdot \hat w < 1$ in a set of full measure. Observe that, using the definition of $v'$, $$\hat v \cdot \hat v'=
\frac {\rho(1-(\hat n \cdot \hat v)^2) + (\hat n \cdot v_1) (\hat n \cdot
\hat v)}{\sqrt{\rho^2 (1-(\hat n \cdot \hat v)^2) + (\hat n \cdot v_1)^2}}$$ which for fixed $v_1$ converges to $\sqrt{1- (\hat n \cdot \hat v)^2}$ when $\rho\to +\infty$, while $[\hat n\cdot (\rho\hat v-v_1)]_+/\rho
\to |\hat n
\cdot \hat v]_+$. By dominated convergence $$\lim_{\rho\to +\infty} \frac 1{\rho} (A\lambda)(\rho) =
\int_{{\mathbb{R}}^d}{\mathop{}\!\mathrm{d}}v_1 \, h_\beta(v_1)
\int_{S^{d-1}} {\mathop{}\!\mathrm{d}}\hat n \,
[\hat n \cdot \hat v]_+ \sqrt{1- (\hat n \cdot \hat v)^2} < \chi.$$ Since $\lim_{\rho \to +\infty} {\lambda(\rho)}/{\rho} =
\int_{S^{d-1}}{\mathop{}\!\mathrm{d}}\hat n \,[\hat n \cdot \hat v]_+ = \chi$, we then conclude that there exists a constant $0<z<1$ such that $$(A\lambda) (\rho) < z \lambda(\rho)$$ for any $\rho \ge 0$. Since $\lambda(\rho) \ge \chi \rho$ (see ), if $\zeta = \chi (1-z)$, then $$\lambda(\rho) \ge \zeta \rho + (A \lambda) (\rho).$$ Denoting by $\mathop{\rm id}:{\mathbb{R}}^+ \to {\mathbb{R}}^+$ the identity function ${\rm id}(\rho) = \rho$, and iterating the above expression, we get $$\lambda \ge
\zeta \mathop{\rm id} + \zeta A \mathop{\rm id}
+ A^2 \lambda \ge \dots \ge
\zeta\sum_{k=0}^n A^k \mathop{\rm id}
+ A^{n+1} \lambda,$$ which implies that $\eta = \sum_{k=0}^{+\infty} A^k \mathop{\rm id}$ is a well defined, positive, function, bounded by ${\lambda}/\zeta$. Since $\eta$ solves , then $\xi(v) = \hat v{\eta(|v|)}/{\lambda(|v|)}$, which is bounded by $1/\zeta$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Mauro Mariani for useful discussions about gradient flows and for his comments on an earlier version of the current manuscript. L. Bertini acknowledges the support by the PRIN 20155PAWZB “Large Scale Random Structures”.
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abstract: 'A unifying theory of the denaturation transition of DNA, driven by temperature $T$ or induced by an external mechanical torque $\Gamma$ is presented. Our model couples the hydrogen-bond opening and the untwisting of the helicoidal molecular structure. We show that denaturation corresponds to a first-order phase transition from B-DNA to d-DNA phases and that the coexistence region is naturally parametrized by the degree of supercoiling $\sigma$. The denaturation free energy, the temperature dependence of the twist angle, the phase diagram in the $T,\Gamma $ plane and isotherms in the $\sigma , \Gamma $ plane are calculated and show a good agreement with experimental data.'
address:
- '$^1$ Dipartimento di Scienze Biochimiche, Universit[à]{} di Roma “La Sapienza”, P.le A. Moro, 5 - 00185 Roma, Italy'
- '$^2$ CNRS-Laboratoire de Physique de l’ENS-Lyon, 46 All[é]{}e d’Italie, 69364 Lyon Cedex 07, France.'
- '$^3$ CNRS-Laboratoire de Physique Théorique de l’ENS, 24 rue Lhomond, 75005 Paris, France.'
author:
- 'Simona Cocco $^{1,2,3}$ and Rémi Monasson $^3$'
title: Statistical Mechanics of Torque Induced Denaturation of DNA
---
psfig.sty
.5cm PACS Numbers : 87.14.Gg, 05.20.-y, 64.10.+h .5cm Denaturation of the DNA, due to its essential relevance to transcription processes has been the object of intensive works in the last decades. Experiments on dilute DNA solutions have provided evidence for the existence of a [*thermally driven*]{} melting transition corresponding to the sudden opening of base pairs at a critical temperature $T_m$ [@Saen84]. Later, following the work of Smith et al.[@Smit92], micromanipulation techniques have been developed to study single-molecule behaviour under stress conditions and how structural transitions of DNA can be [*mechanically*]{} induced. While most single-molecule experiments have focused on stretching properties so far, the response of a DNA molecule to an external torsional stress has been studied very recently [@Stri99; @Chat99], sheding some new light on denaturation [@Stri99]. From a biological point of view, torsional stress is indeed not unusual in the living cell and may strongly influence DNA functioning[@Stri98; @Mark95].
For a straight DNA molecule with fixed ends, the degree of supercoiling $\sigma =(Tw-Tw_0)/Tw_0$ measures the twist $Tw$ ([*i.e.*]{} the number of times the two strands of the DNA double-helix are intertwined) with respect to its counterpart $Tw_0$ for an unconstrained linear molecule. In Strick et al. experiment [@Stri99], a $\lambda$ DNA molecule, in 10 mM PB, is attached at one end to a surface and pulled and rotated by a magnetic bead at the other end. At stretching forces of $\simeq 0.5$ pN, sufficient to eliminate plectonems by keeping the molecule straight, a torque induced transition to a partially denaturated DNA is observed. Beyond a critical supercoiling $\sigma_c \simeq -0.015$ and an associated critical torque $\Gamma_c \simeq -0.05 eV/\hbox{\rm rad}$ the twisted molecule separates into a pure B-DNA phase with $\sigma=\sigma_c$ and denaturated regions with $\sigma= -1$. Extra turns applied to the molecule increase the relative fraction of d-DNA with respect to B-DNA.
In this letter, we provide a unifying understanding of both thermally and mechanically induced denaturation transitions. We show that denaturation can be described in the framework of first-order phase transitions with control parameters being the temperature and the external torque. This is in close analogy to the liquid-gas transition, where control parameters are the temperature and the pressure. Our theory gives a natural explanation to the BDNA-dDNA phases coexistence observed in single molecule experiments [@Stri99]. We give quantitative estimates for the denaturation free-energy $\Delta G$, the temperature dependence of the average twist angle $\Delta \langle \theta \rangle /\Delta T$, the critical supercoiling $\sigma _ c $ and torque $\Gamma _c$ at room temperature in good agreement with the experimental data. Furthermore the dependence of the critical torque as a function of the temperature is predicted.
Our model reproduces the Watson-Crick double helix (B-DNA) as schematized fig 1. For each base pair ($n=1,\ldots ,N$), we consider a polar coordinate system in the plane perpendicular to the helical axis and introduce the radius $r_n$ and the angle $\varphi_n$ of the base pair [@Bar99]. The sugar phosphate backbone is made of rigid rods, the distance between adjacent bases on the same strand being fixed to $L =6.9 \AA$. The distance $h_n$ between base planes $n-1$ and $n$ is expressed in terms of the radii $r_{n-1}, r_n$ and the twist angle $\theta _n = \varphi _n
- \varphi _{n-1}$ as $$\label{h}
h_n (r _n , r_{n-1} , \theta _n ) =\sqrt{ L^2 - r_n ^2 -
r_{n-1}^2 +2 r_n r_{n-1} \cos \theta _n } \ .$$ The potential energy associated to a configuration of the degrees of freedom $(r_n ,
\varphi _n )$ is the sum of the following nearest neighbor interactions.
First, hydrogen bonds inside a given pair $n$ are taken into account through the short-range Morse potential [@Proh95; @PB89] $V_m (r_n)=D\, \left(e^{-a(r_n-R)}-1\right)^2$ with $R=10\AA$ . Fixing $a=6.3 \AA ^{-1}$ [@Daux95; @Zhan97], the width of the well amounts to $3a^{-1} \simeq 0.5 {\AA}$ in agreement with the order of magnitude of the relative motion of the hydrogen bonded bases [@Mac87]. A base pair with diameter $r > r_d = R + 6/a$ may be considered as open. The potential depth $D$, typically of the order of $0.1 eV$ [@Proh95; @Camp98] depends on the base pair type (Adenine-Thymine (AT) or Guanine-Citosine (GC)) as well as on the ionic strength.
Secondly, the shear force that opposes sliding motion of one base over another in the B-DNA conformation is accounted for by the stacking potential [@Saen84] $V_s(r_n,r_{n-1})= E\, e^{-b (r_n +
r_{n-1}-2R)}\; (r_n-r_{n-1})^2$. Due to the decrease of molecular packing with base pair opening, the shear prefactor is exponentially attenuated and becomes negligible beyond a distance $\simeq 5 b^{-1} =
10 {\AA}$, which coincides with the diameter of a base pair [@Daux95; @Zhan97; @Camp98; @Cule97].
Thirdly, an elastic energy $V_b (r _n , r_{n-1} , \theta _n ) = K [
h_n - H ]^2 $ is introduced to describe the vibrations of the molecule in the B phase. The helicoidal structure arises from $H <
L$: in the rest configuration $r_n=R$ at $T=0$K, $V_b$ is minimum and zero for the twist angle $\theta_n=2\pi/10$. Choosing $H=3\AA$, we recover at room temperature $T=298$K the thermal averages $\langle h_n
\rangle \simeq 3.4 \AA$ and $\langle \theta_n \rangle \simeq 2 \pi /
10.4$ [@Saen84]. The above definition of $V_b$ holds as long as the argument of the square root in (\[h\]) is positive, that is if $r_n , r_{n-1} , \theta _n$ are compatible with rigid rods having length $L$. By imposing $V_b =
\infty$ for negative arguments, unphysical values of $r _n , r_{n-1} ,
\theta _n$ are excluded. As the behaviour of a single strand ($r > r_d$) is uniquely governed by this rigid rod condition, the model does not only describe vibrations of helicoidal B-DNA but is also appropriate for the description of the denaturated phase.
As will be discussed later, the elastic constant $K=0.014 eV/\AA^2$ is determined to give back the torsional modulus $C$ of B-DNA estimated to $C=860 \pm 100 {\AA}$ [@Croq99; @Bouc97] at $T=298$K. The parameters of the Morse potential $D$ and of the stacking interaction $E$ we have set to fit the melting temperature $T_m=350$K of the homogeneous Poly(dGdT)-Poly(dAdC)-DNA at $20 mM\ Na^+$ [@Saen84], see inset of fig 3. This melting temperature coincides with the expected denaturation temperature of a heterogeneous DNA with a sequence GC/AT ratio equal to unity at $10
mM\ Na^+$ [@Saen84], as the $\lambda$-DNA in the experimental conditions of [@Stri99]. Among all possible pairs of parameters $(D,E)$ that correctly fit $T_m$, we have selected the pair $(D=0.16 eV , E=4 eV/\AA^2)$ giving the largest prediction for $\Delta G$, see inset of fig 2, that is in closest agreement with thermodynamical estimates of the denaturation free-energy.
When the molecule is fixed at one end and subject to a torque $\Gamma$ on the other extremity, an external potential $V_{\Gamma}(\theta
_n)=-\Gamma \,\theta _n$ has to be included. A torque $\Gamma
>0$ overtwists the molecule, while $\Gamma <0$ undertwists it.
The configurational partition function at inverse temperature $\beta$ can be calculated using the transfer integral method : $$\label{ZG}
Z_{\Gamma}= \int _{-\infty} ^\infty d\varphi _N \; \langle R, \varphi
_N | T^N| R , 0 \rangle$$ As in the experimental conditions, the radii of the first and last base pairs are fixed to $r_1=r_N=R$. The angle of the fixed extremity of the molecule is set to $\varphi_1=0$ with no restriction whereas the last one $\varphi _N$ is not constrained. The transfer operator entries read $<r,\varphi|T|r',\varphi'>\equiv X(r,r')\, \exp \{ -\beta
( V_b \,(r,r', \theta ) + V_\Gamma (\theta )) \} \, \chi (\theta )$ with $X(r,r')= \sqrt{rr'} \exp \{-\beta (V_m\,(r)/2+V_m\,(r')/2 +
V_s\,(r,r') ) \}$. The $\sqrt{rr'}$ factor in $X$ comes from the integration of the kinetic term; $\chi(\theta) =1$ if $0\leq \theta =
\varphi-\varphi' \leq \pi$ and 0 otherwise to prevent any clockwise twist of the chain. At fixed $r,r'$, the angular part of the transfer matrix $T$ is translationally invariant in the angle variables $\varphi$, $\varphi '$ and can be diagonalized through a Fourier transform. Thus, for each Fourier mode $k$ we are left with an effective transfer matrix on the radius variables $T_k(r,r')=X(r,r')\,Y_k(r,r')$ with $$\label{mu}
Y_k(r,r') =
\int_{0}^{\pi}\,d\theta \,e^{-\beta ( V_b\,(r,r',\theta) +
V_\Gamma (\theta ) )}e^{-ik\theta} \quad .$$ The only mode contributing to $Z_{\Gamma }$ is $k=0$ once $\varphi _N$ has been integrated out in (\[ZG\]). The eigenvalues and eigenvectors of $T_0$ will be denoted by $\lambda ^{(\Gamma )} _{q}$ and $\psi^{(\Gamma )}_{q}(r)$ respectively with $\lambda^{(\Gamma
)}_{0} \ge \lambda^{(\Gamma )}_{1} \ge \ldots$. In the $N\to\infty$ limit, the free-energy density $f^{(\Gamma )}$ does not depend on the boundary conditions on $r_1$ and $r_N$ and is simply given by $f
^{(\Gamma )}= -k_B T \ln \lambda ^{(\Gamma )} _{0}$.
Note that the above result can be straightforwardly extended to the case of a molecule with a fixed twist number $Tw = N \ell$, e.g. for circular DNA. Indeed, the twist density $\ell$ and the torque $\Gamma$ are thermodynamical conjugated variables and the free-energy at fixed twist number $\ell$ is the Legendre transform of $f_\Gamma$.
We have resorted to a Gauss-Legendre quadrature for numerical integrations over the range $r_{min} = 9.7 {\AA}< r < r_{max}$. The Morse potential $V_m$ increases exponentially with decreasing $r< R$ and may be considered as infinite for $r<9.7 \AA$ [@Zhan97]. The extrapolation procedure to $r_{max}\to \infty$ depends on the torque value $\Gamma$ and will be discussed below. Using Kellog’s iterative method [@Daux95], the eigenvalues $\lambda ^{(\Gamma )} _{q}$ and associated eigenvectors $\psi ^{(\Gamma )} _{q}(r)$ have been obtained for $q=0,1,2$. Like a quantum mechanical wave function, $\psi ^{(\Gamma )} _{0} (r)$ gives the probability amplitude of a base pair to be of radius $r$. Two quantities of interest are: the percentage of opened base pairs $P=\int_{r_d}^{\infty} dr\; |\psi
^{(\Gamma )} _{0} (r)|^2 $, the averaged twist angle $\langle \theta
\rangle = -\partial f_{\Gamma} /\partial \Gamma $.
Results for a freely swiveling molecule at room temperature are as follows. $\psi^{(\Gamma=0 )}_{0}$ is entirely confined in the Morse potential well and describes a closed molecule. Conversely the following eigenfunctions $\psi ^{(\Gamma=0 )}_{1}$, $\psi ^{(\Gamma=0
)}_{2}, \ldots$ correspond to an open molecule: they extend up to $r_{max}$ and vanish for $r<r_d$. They are indeed orthogonal to another family of excited states that are confined in the Morse potential with much lower eigenvalues. The shape of the open states are strongly reminiscent of purely diffusive eigenfunctions, $\psi _q ^{(\Gamma
=0)} (r) \simeq \sin ( q \pi (r - r_d) / ( r_{max} -r_d ))$ leading to a continuous spectrum in the limit $r_{max} \rightarrow \infty$.
This observation can be understood as follows. For $r,r' > r_d$, the transfer operator $T_0(r,r')$ is compared fig. 2 to the exact conditional probability $\rho (r,r')$ that the endpoint of a backbone rod of length $L$ is located at distance $r'$ from the vertical reference axis knowing that its other extremity lies at distance $r$ [@notar]. For fixed $r$, $T_0$ and $\rho$ both diverge in $r'=r\pm L$ and are essentially flat in between. The flatness of $T_0$ derive from the expression of $V_b$: a rigid rod with extremities lying in $r,r'$ may always be oriented with some angle $\theta ^*$ ($\to 0$ at large distances) at zero energetic cost $V_b (r,r', \theta ^*)=0$. As a conclusion, our model can reproduce the purely entropic denaturated phase.
As shown fig 2, at a critical temperature $T_m= 350$K, $\lambda ^{(\Gamma = 0)} _{0}$ crosses the second largest eigenvalue and penetrates, as in a first-order-like transition the continuous spectrum. For $T> T_m$, the bound state disappears and $\psi^{(\Gamma=0 )}_{1}$ in fig 2 becomes the eigenmode with largest eigenvalue [@nota4]. The percentage of opened base pairs $P$ exhibits an abrupt jump from 0 to 1 at $T_m$, reproducing the UV absorbance vs. temperature experimental curve for Poly(dGdT)-Poly(dAdC)-DNA [@Saen84]. The difference $\Delta G$ between the free energy $f_d ^{(\Gamma =0 )}$ of the open state ($q=1$) and the free energy $f_B ^{(\Gamma =0 )}$ of the close state ($q=0$) gives the denaturation free energy at temperature $T$, see fig. 2. At $T=298$K, we obtain $\Delta G =0.022 eV$ in good agreement with the free energy of the denaturation bubble formation $\Delta G \simeq 0.025 eV$ estimated in AT rich regions [@Saen84; @Stri98]. The thermal fluctuations in the B-DNA phase lead to an undertwisting $\Delta \langle\theta\rangle /\Delta T\simeq \, -1.4 \, 10^{-4} \hbox{\rm
rad/K}$ which closely agrees with experimental measures $\Delta \langle \theta \rangle /
\Delta T\simeq - 1.7 \, 10^{-4} \hbox{\rm rad/K}$ [@Dep75].
The presence of an overtwisting (respectively undertwisting) torque $\Gamma >0$ (resp. $\Gamma <0$) strongly affects $f_B ^{(\Gamma )}$, leaving almost unchanged the single strand free-energy $f_d ^{(\Gamma )}$. The denaturation transition takes place at $T_m
(\Gamma )$ [@nota3], see the phase diagram shown in the inset of fig 3. We expect a critical point at a high temperature and large positive torque such that $\psi^{\Gamma}_{1}$ is centered on $R$ [@nota3].
The supercoiling, induced by a torque at a given temperature smaller than $T_m (\Gamma =0) = 350$K, is the relative change of twist with respect to the value at zero torque in the B-DNA state, $\sigma(\Gamma)= ( \langle \theta \rangle _{\Gamma} -
\langle \theta \rangle _{\Gamma=0} )/\langle \theta \rangle
_{\Gamma=0}$. In fig 3, we have plotted the isotherms in the $\sigma,
\Gamma$ plane. Horizontals lines are critical coexistence regions between the B-DNA phase, on the left of the diagram and the denaturated phase on the right (with $\sigma=-1$). The left steep line is found to define a linear relation between $\Gamma$ and $\sigma$ : $\Gamma
= {K}_{\theta}\; (\langle \theta \rangle _{\Gamma} - \langle \theta
\rangle _{\Gamma =0} )$. The slope ${K}_{\theta}$ does not vary with temperature over the range 298 K$<T<$350 K and is related to the torsional modulus $C$ of B-DNA through $C={K}_{\theta} \langle h_n \rangle /
(k_B T)$[@Croq99]. The value of $K$ appearing in the elastic potential $V_b$ and given above was tuned to ensure that $C=860\AA$[@Croq99; @Bouc97]. At room temperature, critical coexistence between B-DNA and d-DNA arises at torque $\Gamma_c=-0.035 eV/\hbox{\rm rad}$ and supercoiling $\sigma_c=-0.01$. These theoretical results are in good agreement with the values $\Gamma_c=-0.05 eV/\hbox{\rm rad}$, $\sigma_c=-0.015$ obtained experimentally[@Stri99].
We plan to combine the present model with existing elasticity theories of DNA [@Smit92; @Bouc97] to understand the influence of an external stretching force on the structural transition studied in this paper. It would also be interesting to see how the above results are modified in presence of a heterogeneous sequence.
[**Acknowledgements**]{} : The present model is the fruit of a previous collaboration [@Bar99] of one of us (S.C.) with M. Barbi and M. Peyrard which we are particularly grateful to. We also thank B. Berge, D. Bensimon, C. Bouchiat, E. Bucci, A. Campa, A. Colosimo, V. Croquette, A. Giansanti for useful discussions.
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$ \rho (r,r')=\frac 1 {4\pi}\int_{0}^{2 \pi}d \theta\int_0^{\pi}
d \varphi \sin \varphi\, \delta (r'-x(\theta , \varphi ) )$ where $x(\theta , \varphi ) = \sqrt{r^2+ L^2 \sin^2 \varphi +
2 L r \sin \theta \sin \varphi }$.
The existence of this first-order transition, though numerically evident is not mathematically proven. However, due to the infinite barrier in $r=9.7 \AA$, the ground state is not necessarily bound (C. Cohen-Tannoudji, B. Diu, F. Laloë, [*Quantum Mechanics*]{}, John Wiley and Sons, New-York (1977)).
R.E. Depew, J.C. Wang, [*Proc. Natl. Acad. Sci.*]{} [**72**]{}, 4275-4279 (1975)
$\psi _1 ^{(\Gamma )}$ is centered around $r< r_{max} /2 $ (resp. $r>
r_{max} /2 $) for $\Gamma >0$ (respectively $\Gamma <0$). For positive or zero torque, the melting temperature $T_m (r_{max})$ remains unchanged as soon as $r_{max} > 150-200
{\AA}$. When $\Gamma <0$, $T_m (r_{max}) \simeq T_m + c/ r_{max
}$. $c$ increases with the torque strength and is related to the pressure exerted by the system walls.
{width="150pt"}
\[f1\]
\[f2\]
\[f3\]
|
16.cm
[INP-98-7/508, MPI/PhT/98-19, hep-ph/9802429, February 1998]{}
**Three-loop vacuum integrals in FORM and REDUCE**
P.A.BAIKOV[^1]
*Institute of Nuclear Physics, Moscow State University,*
*119 899, Moscow, Russia*
M.STEINHAUSER
*Max-Planck-Institut für Physik,Werner-Heisenberg-Institut,*
*D-80805 Munich, Germany*
**Abstract**
The implementation of an algorithm for three-loop massive vacuum integrals, based on the explicit solution of the recurrence relations, in REDUCE and FORM is described.
Introduction
============
The increasing experimental precision makes it mandatory to compute higher order corrections to physical quantities. This essentially requires the evaluation of multi-loop integrals. However, even at two loops it is very often not possible to solve the integrals exactly and one has to rely on approximations. A very powerful approach to get, nevertheless, reasonable results is based on expansions in small quantities. Recently a method was developed for three-loop polarization functions combining expansions from different kinematical regions with the help of conformal mapping and Padé approximation [@BaiBro95; @chetn]. An essential ingredient for this procedure are massive integrals with vanishing external momentum. For their computation usually recurrence relations are used which are based on the integration-by-parts technique [@ch-tk; @REC]. These relations connect Feynman integrals with different powers of their denominators. In many cases they provide a possibility to express an integral with given degrees of the denominators as a linear combination of a few so-called master integrals with prefactors which are rational functions of the space-time dimension $D$. The construction of such a procedure is a nontrivial problem even at two-loop level [@Tar97]. At three loops up to now only the case of vacuum integrals with one non-zero mass and various numbers of massless lines has been considered [@REC; @Avd]. For the problems of practical interest the direct application of these equations usually leads to intermediate expressions which need several hundred megabyte up to a few gigabyte of disk space.
For the two-loop massless master integral some years ago the recurrence procedure has been solved explicitly [@Tka83]. The solution, expressed in terms of multiple sums, is also used in practice [@mincer] — at least for those cases where the exponents of the propagators are not too small. Concerning the general multi-loop case in [@ES] a new approach to implement recurrence relations [@ch-tk] was suggested. There the factors in front of the master integrals, which in the following are called coefficient functions, are considered as independent solutions of the recurrence relations. In [@ES] integral representations for these solutions were obtained. It appeared, that in some cases these representations can be expressed in terms of Pochhammer symbols. As an example the vacuum integrals with four equal masses and two massless lines has been considered, and the efficiency of this approach was demonstrated by the calculation of the three-loop QED vacuum polarization.
In this paper we describe the algorithm, suggested in [@ES], in more details and discuss some peculiarities of its implementation in REDUCE [@REDUCE] and FORM [@FORM]. In Section \[secfor\] the general $L$-loop case is considered. In Section \[secthr\] the derived formulas are specified to three loops and explicit solutions are given. The implementation is discussed in Section \[secimp\] and finally Section \[seccon\] contains our conclusions.
\[secfor\]Basic formulas
========================
Let us in a first step derive the recurrence relations. The combinatoric structure of these relations becomes more transparent if we start with the general multi-loop case, keeping in mind, however, the application at three-loop level. Therefore let us consider $L$-loop vacuum integrals with $N=L(L+1)/2$ denominators (This number of denominators provides the possibility to express any scalar product of loop momenta as linear combination of the denominators; the diagrams of practical interest which usually have less number of denominators, can be considered as special cases with some exponents equal to zero.):
$$\begin{aligned}
B(\underline{n},D)\equiv
B(n_1,\ldots,n_N,D)=
{\mbox{$\frac{m^{2\Sigma n_i-LD}}{\big[\imath\pi^{D/2}\Gamma(3-D/2)\big]^L}$}}
\int \cdots \int {\mbox{$\frac{d^Dp_1\ldots d^Dp_L}{D_1^{n_1}\ldots D_N^{n_N}}$}},
\label{eqbn}
$$
where $p_i$ (${i=1,\ldots,L}$) are loop momenta and $D_a=
A^{ij}_a p_i\cdot p_j -\mu_a m^2$ (a=1,…,N; here and below the sum over repeated indices is understood). For convenience we set $m=1$ in the following. The recurrence relations are obtained by acting with $(\partial/\partial p_i)\cdot p_k$ on the integrand [@ch-tk]:
$$\begin{aligned}
D\delta_k^i B(\underline{n},D)&=&
2
\tilde{A}_{kl}^a({\bf I}^-_a+\mu_a)
A_d^{il}{\bf I}^{d+}
B(\underline{n},D),
\label{rr}\end{aligned}$$
where ${\bf I}^-_c B(\ldots, n_c,\ldots )\equiv B(\ldots, n_c-1,\ldots )$ and ${\bf I}^+_c B(\ldots, n_c,\ldots )\equiv n_c B(\ldots, n_c+1,\ldots)$ (here, no sum over $n_c$ may be performed). The factors $\tilde{A}_{kl}^a$ arise from the fact that the scalar products $p_i\cdot p_j$ in the numerator which appear as a result of the differentiation are expressed in terms of the denominators $D_a$ via: $$\begin{aligned}
p_k\cdot p_l&=&
\tilde{A}_{kl}^a(D_a+\mu_a)
\,.
\nonumber\end{aligned}$$
Using the identities $$\begin{aligned}
[{\bf I}^-_a,{\bf I}^{d+}]=\delta_a^d,
&&\qquad
A_a^{il} \tilde{A}_{kj}^a=
{\mbox{$\frac{1}{2}$}}(\delta^i_k\delta^l_j+\delta^i_j\delta^l_k)
\,,
\nonumber\end{aligned}$$ the recurrence relations (\[rr\]) can be represented as $$\begin{aligned}
(D-L-1)\delta_k^i B(\underline{n},D)&=&2
A_d^{il}{\bf I}^{d+}
\tilde{A}_{kl}^a({\bf I}^-_a+\mu_a)
B(\underline{n},D),
\label{rr1}\end{aligned}$$ where we have exploited that the matrices $A^a$ and $\tilde{A}^a$ can be chosen to be symmetrical. If we denote $$\begin{aligned}
\tilde{A}_{kl}^a({\bf I}^-_a+\mu_a)\equiv {\bf A}_{kl},
\qquad A_d^{il}{\bf I}^{d+}\equiv \mbox{\boldmath $\partial$}^{il}
\,,
\nonumber\end{aligned}$$ Eq. (\[rr1\]) will read: $$\begin{aligned}
\left[
\mbox{\boldmath $\partial$}^{il}\cdot{\bf A}_{kl}
-{\mbox{$\frac{D-L-1}{2}$}}\delta_k^i
\right]
B(\underline{n},D)&=&0.
\label{eq1}\end{aligned}$$
Let us now diagonalize these relations with respect to the operators $\mbox{\boldmath $\partial$}^{il}$. Therefore we multiply Eq. (\[eq1\]) with $\mbox{$\partial$}^{jk} \det({\bf A})$, which is the cofactor of the matrix element ${\bf A}_{kl}$, and sum afterwards over $k$. Finally we arrive at: $$\begin{aligned}
\left[
\mbox{\boldmath $\partial$}^{il}\cdot
\det({\bf A})
-{\mbox{$\frac{D-L-1}{2}$}}(\mbox{$\partial$}^{il}\det({\bf A}))
\right]
B(\underline{n},D)&=&0
\,.
\nonumber\end{aligned}$$
Our goal is to find the coefficient functions $f^k(\underline{n},D)$ which relate the given integral $B(\underline{n},D)$ with the basic master integrals $B(\underline{n}_{k},D)$: $$\begin{aligned}
B(\underline{n},D)&=&
f^k(\underline{n},D)B(\underline{n}_{k},D)
\,,
\nonumber\end{aligned}$$ where the coefficient functions have to fulfill the initial conditions $f^i(\underline{n}_k,D)=\delta^i_k$. The index $k$ is used to label different sets of indices $\underline{n}$ which are fixed by the choice of the master integrals. Assuming that the master integrals are algebraically independent one can conclude that the functions $f^k(\underline{n},D)$ should be independent solutions of the recurrence relations. This means that as soon as we find a set of independent solutions it is possible to construct the desired coefficient functions as linear combinations respecting the given initial conditions. To this end it turns out that it is useful to construct an auxiliary integral representation for these functions where the operators can be written in the form[^2]: ${\bf I}^{d+} \rightarrow \partial/\partial x_d$, ${\bf I}^-_d \rightarrow x_d$. Then the differential equation corresponding to Eq. (\[rr1\]) has the solution $g(x_a)=\det({\bf A})^{(D-L-1)/2}=P(x_a+\mu_a)^{(D-L-1)/2}$, where $P(x_a)$ is a polynomial in $x_a$ of degree $L$: $$\begin{aligned}
P(x_a)&=&\det(
\tilde{A}_{kl}^a x_a)
\,.
\nonumber\end{aligned}$$ Therefore it is tempting to consider the “Laurent” coefficients of the function $g(x_a)$ in order to determine the coefficient functions: $$\begin{aligned}
f^k(\underline{n},D)&=&
{\mbox{$\frac{1}{(2\pi\imath)^N}$}}
\oint \cdots \oint
{\mbox{$\frac{dx_1 \cdots dx_N}{x_1^{n_1} \cdots x_N^{n_N}}$}}
\det(\tilde{A}_{il}^a(x_a+\mu_a))^{(D-L-1)/2}
\,.
\label{solution}\end{aligned}$$ Here the integral symbols denote $N$ subsequent complex integrations. The contours depend on the index $k$ and will be specified below. Acting with Eq. (\[rr1\]) on (\[solution\]) leads (up to surface terms) to the corresponding differential operator acting on $g(x_a)$, which gives zero. The surface terms can be removed by a proper choice of the complex contours: either closed contours or integration paths which end in infinity. For the last case one has to consider analytical continuations of $D$ from large negative values. Note that Eq. (\[solution\]) is a solution of relation (\[rr\]) and thus the different choices of the contours correspond to different solutions which are enumerated with the index $k$.
The solutions (\[solution\]) satisfy by construction the following condition: $$\begin{aligned}
f^k(\underline{n},D)&=&P({\bf I}^-_a+\mu_a)f^k(\underline{n},D-2)
\,.
\label{rrD0}\end{aligned}$$ In general, however, we are interested in coefficient functions, $\tilde{f}^k(\underline{n},D)$, which correspond to a specific set of master integrals and hence a linear combination of $f^k(\underline{n},D)$ with coefficients depending on $D$ has to be considered. Then Eq. (\[rrD0\]) gets more complicated as a mixing among the different functions is possible: $$\begin{aligned}
\tilde{f}^k(\underline{n},D)&=&
S^k_i(D)P({\bf I}^-_a+\mu_a)\tilde{f}^i(\underline{n},D-2),
\nonumber
$$ where $S_i^k(D)$ are rational functions of the space-time dimension $D$ only.
\[secthr\]Three-loop case
=========================
In this section we specify the general $L$-loop formulas derived in the previous one to the three-loop case with four equal masses and two massless lines, i.e., $\mu_1=\mu_2=0,\mu_3=\mu_4=\mu_5=\mu_6=1$. Then Eq. (\[eqbn\]) becomes: $$\begin{aligned}
B(\underline{n},D)=
{\mbox{$\frac{m^{2\Sigma_1^6 n_i-3D}}{\big[\imath\pi^{D/2}\Gamma(3-D/2)\big]^3}$}}
\int\int\int {\mbox{$\frac{d^Dp\,d^Dk\,d^Dl}{D_1^{n_1}D_2^{n_2}D_3^{n_3}D_4^{n_4}D_5^{n_5}D_6^{n_6}}$}}
\,,
\nonumber
$$ with $$\begin{aligned}
&&
D_1=k^2,
D_2=l^2,
D_3=(p+k)^2-m^2,
D_4=(p+l)^2-m^2,
\nonumber\\&&
D_5=(p+k+l)^2-m^2,
D_6=p^2-m^2
\,.
\nonumber\end{aligned}$$ The coefficient functions from Eq. (\[solution\]) read: $$\begin{aligned}
f(\underline{n},D)&=&
{\mbox{$\frac{1}{(2\pi\imath)^6}$}}
\oint{\mbox{$\frac{dx_1}{x_1^{n_1}}$}}
\cdots
\oint{\mbox{$\frac{dx_6}{x_6^{n_6}}$}}
{P(x_1,x_2,x_3+1,\dots,x_6+1)^{D/2-2}}
\,,
\label{solution3}\end{aligned}$$ where the polynomial $P(x_1,\ldots,x_6)$ is given by $$\begin{aligned}
P(x_1,\dots,x_6)&=&
(x_1+x_2)(x_1x_2-x_3x_4-x_5x_6)+(x_3+x_4)(-x_1x_2+x_3x_4-x_5x_6)
\nonumber\\&&\mbox{}
+(x_5+x_6)(-x_1x_2-x_3x_4+x_5x_6)
+x_1x_3x_6+x_1x_4x_5+x_2x_3x_5
\nonumber\\&&\mbox{}
+x_2x_4x_6
\,.
\nonumber
\label{eqpx}\end{aligned}$$ In this equation we have omitted an overall factor $(-1/4)$ as it would lead to a trivial rescaling which cancels after considering initial conditions.
Let us now compute the coefficient functions which correspond to the choice of the master integrals given in [@REC; @Avd] where $B(\underline{n},D)$ has been written in the form: $$\begin{aligned}
B(\underline{n},D)&=&
N(\underline{n},D)B(0,0,1,1,1,1,D)+
M(\underline{n},D)B(1,1,0,0,1,1,D)
\nonumber\\&&\mbox{}
+
T(\underline{n},D)B(0,0,0,1,1,1,D)
\,.
\nonumber
$$ This leads us to the following normalization conditions: $$\begin{aligned}
N(0,0,1,1,1,1,D)=1,&\quad N(1,1,0,0,1,1,D)=0,&\quad
N(0,0,0,1,1,1,D)=0,\label{condN}\\
M(0,0,1,1,1,1,D)=0,&\quad M(1,1,0,0,1,1,D)=1,&\quad
M(0,0,0,1,1,1,D)=0,\nonumber\label{condM}
\\
T(0,0,1,1,1,1,D)=0,&\quad T(1,1,0,0,1,1,D)=0,&\quad
T(0,0,0,1,1,1,D)=1
\,.
\label{condT} \end{aligned}$$ In a first step the function $N(\underline{n},D)$ is considered. The last two conditions of Eq. (\[condN\]) are satisfied if the contours for the massive indices are chosen to be small circles around zero, i.e., $x_i=0$. In practice it is useful to perform a Taylor expansion in $x_3, x_4, x_5$ and $x_6$ where, according to the residuum theorem, only one term leads to a result different from zero (Here and in the following the proportional sign (“$\propto$”) is used as the $D$ dependent factor is fixed at the end in accordance with the normalization given in Eq. (\[condN\]).): $$\begin{aligned}
N(\underline{n},D)&\propto&\oint\oint
{\mbox{$\frac{dx_1dx_2}{x_1^{n_1}x_2^{n_2}}$}}
\left[{\mbox{$\frac{\partial_3^{n_3-1}\dots\partial_6^{n_6-1}}{(n_3-1)!\dots(n_6-1)!}$}}
P(x_1,x_2,x_3+1,\dots,x_6+1)^{D/2-2}
\right]
\bigg|_{x_3,\dots,x_6=0}.\nonumber\end{aligned}$$ The remaining integrals over $x_1$ and $x_2$, for which in general (because of the derivatives) the exponent of the polynomial $P(x_1,\ldots,x_6)$ is reduced, can easily be expressed in terms of Pochhammer symbols $(a)_n = \Gamma(a+n)/\Gamma(a)$: $$\begin{aligned}
\oint\oint
{\mbox{$\frac{dx_1dx_2}{x_1^{n_1}x_2^{n_2}}$}}
P(x_1,x_2,1,1,1,1)^{D/2-2-c}
\propto
{\mbox{$\frac{(D/2-1)_{-c}(4-3D/2)_{n_1+n_2+3c}}{4^{(n_1+n_2+3c)}(2-D/2)_{n_1+c}(2-D/2)_{n_2+c}}$}},
\nonumber\end{aligned}$$ where $\Gamma(x)$ is Eulers $\Gamma$ function. Let us next turn to the function $M(\underline{n},D)$ which can be treated in analogy. The only difference is that due to the symmetry $B(1,1,0,0,1,1,D)=B(1,1,1,1,0,0,D)$ it is profitable to consider the sum of the solutions with a Taylor expansion performed in the variables $(x_1,x_2,x_3,x_4)$ and $(x_1,x_2,x_5,x_6)$: $$\begin{aligned}
M(\underline{n},D)&=&
m(n_1,n_2,n_3,n_4,n_5,n_6,D)+m(n_1,n_2,n_5,n_6,n_3,n_4,D)
\,,
\nonumber\end{aligned}$$ where $m(\underline{n},D)$ is given by $$\begin{aligned}
m(\underline{n},D)&\propto&
\oint\oint
{\mbox{$\frac{dx_5dx_6}{x_5^{n_5}x_6^{n_6}}$}}
\left[{\mbox{$\frac{\partial_1^{n_1-1}\dots\partial_4^{n_4-1}}{(n_1-1)!\dots(n_4-1)!}$}}
P(x_1,x_2,x_3+1,\dots,x_6+1)^{D/2-2}
\right]
\bigg|_{x_1,\dots,x_4=0}
\,.
\nonumber\end{aligned}$$ Again the remaining integrals are easily performed with the result $$\begin{aligned}
\lefteqn{
\oint\oint
{\mbox{$\frac{dx_5dx_6}{x_5^{n_5}x_6^{n_6}}$}}
P(0,0,1,1,x_5+1,x_6+1)^{D/2-2-c}
}
\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad
\propto&
{\mbox{$\frac{(-)^{n_5+n_6}(3-D)_{n_5+2c}(3-D)_{n_6+2c}
(4-3D/2)_{n_5+n_6+3c}}{(2-D/2)_c(6-2D)_{n_5+n_6+4c}(3-D)_{n_5+n_6+2c}}$}}
\,.
\nonumber\end{aligned}$$
The case “T” is more complicated. The conditions (\[condT\]) show that we should find a solution which is non-zero if one of the massive indices is non-positive. It turns out that the following combination of “Taylor” solutions leads to the desired result: $$\begin{aligned}
T(n_1,n_2,n_3,n_4,n_5,n_6,D)&=&
t(n_1,n_2,n_3,n_4,n_5,n_6,D)+t(n_1,n_2,n_4,n_3,n_6,n_5,D)
\nonumber\\
&&\mbox{}
+t(n_1,n_2,n_5,n_6,n_3,n_4,D)+t(n_1,n_2,n_6,n_5,n_4,n_3,D),
\nonumber\end{aligned}$$ where $$\begin{aligned}
t(\underline{n},D)=0\ \quad\mbox{if}\quad\
n_4\ \mbox{or}\ n_5\ \mbox{or}\ n_6\ <\ 1.
\label{t3}\end{aligned}$$ Four terms are necessary in order to meet the symmetry properties of the initial integral. From Eq. (\[solution3\]) the following general representation for the solution with the property given in Eq. (\[t3\]) is obtained: $$\begin{aligned}
\overline{t}(\underline{n},D)&\propto&\oint\oint
{\mbox{$\frac{dx_1dx_2dx_3}{x_1^{n_1}x_2^{n_2}x_3^{n_3}}$}}
\nonumber\\&&\mbox{}
\quad\times
\left[{\mbox{$\frac{\partial_4^{n_4-1}\partial_5^{n_5-1}\partial_6^{n_6-1}}{(n_4-1)!(n_5-1)!(n_6-1)!}$}}
P(x_1,x_2,x_3+1,\dots,x_6+1)^{D/2-2}
\right]
\bigg|_{x_4,x_5,x_6=0}
\,,
\label{t31}\end{aligned}$$ where we have introduced the function $\overline{t}(\underline{n},D)$ as in general the expression given on the r.h.s. of Eq. (\[t31\]) leads to a mixture of the functions $N(\underline{n},D)$ and $t(\underline{n},D)$. We will moreover see below that $\overline{t}(\underline{n},D)$ obeys simple recurrence relations and in addition the extraction of $t(\underline{n},D)$ and $N(\underline{n},D)$ is quite simple. The resulting integral where $x_4=x_5=x_6=0$ is given by: $$\begin{aligned}
\overline{t}(n_1,n_2,n_3,D)&\equiv&
\overline{t}(n_1,n_2,n_3,1,1,1,D)
\nonumber\\&=&
{\mbox{$\frac{1}{(2\pi\imath)^3}$}}
\oint\oint\oint
{\mbox{$\frac{dx_1dx_2dx_3}{x_1^{n_1}x_2^{n_2}x_3^{n_3}}$}}
P(x_1,x_2,x_3+1,1,1,1)^{D/2-2}
\nonumber
\\&=&
{\mbox{$\frac{1}{(2\pi\imath)^3}$}}
\oint\oint\oint
{\mbox{$\frac{dx_1dx_2dx_3}{x_1^{n_1}x_2^{n_2}x_3^{n_3}}$}}
{(x_3^2-x_1x_2x_3+x_1x_2(x_1+x_2-4))^{D/2-2}}
\,.
\label{tbar}\end{aligned}$$ The function $\overline{t}(n_1,n_2,n_3,D)$ obeys the following recurrence relations which can be derived by using integration-by-parts in Eq. (\[tbar\]): $$\begin{aligned}
{\bf I}^+_1 \overline{t}(n_1,n_2,n_3,D+2)&=&
(D/2-1)\left[-{\bf I}^-_2{\bf I}^-_3 + 2{\bf I}^-_1{\bf I}^-_2
+ ({\bf I}^-_2)^2 -4 {\bf I}^-_2\right]\overline{t}(n_1,n_2,n_3,D),
\nonumber\\&&
\label{rn1}\\
{\bf I}^+_2 \overline{t}(n_1,n_2,n_3,D+2)&=&
(D/2-1)\left[-{\bf I}^-_1{\bf I}^-_3 + 2{\bf I}^-_1{\bf I}^-_2
+ ({\bf I}^-_1)^2 -4 {\bf I}^-_1\right]\overline{t}(n_1,n_2,n_3,D),
\nonumber\\&&
\\
{\bf I}^+_3 \overline{t}(n_1,n_2,n_3,D+2)&=&
(D/2-1)\left[2{\bf I}^-_3 - {\bf I}^-_1{\bf
I}^-_2\right]\overline{t}(n_1,n_2,n_3,D),
\label{eqrdrd}
\\
\overline{t}(n_1,n_2,n_3,D+2)&=&
\left[({\bf I}^-_3)^2
+{\bf I}^-_1{\bf I}^-_2(-{\bf I}^-_3 + {\bf I}^-_1 + {\bf I}^-_2 -4)\right]
\overline{t}(n_1,n_2,n_3,D)\label{rd}.\end{aligned}$$ The last equation immediately follows from (\[rrD0\]).
For $n_3<1$ the contribution from $N(\underline{n},D)$ vanishes as the integration around $x_3=0$ gives zero and only $t(\underline{n},D)$ survives. In this case with the simple change of variables, $x_3=\sqrt{y_3}+x_1x_2/2$, the integral (\[tbar\]) can be reduced to a sum over integrals which furthermore can be expressed in terms of Pochhammer symbols. Keeping in mind the normalization condition (\[condT\]) we get the desired solution for $t(\underline{n},D)$: $$\begin{aligned}
t(n_1,n_2,n_3<1,D-2c)
&=&{\mbox{$\frac{\overline{t}(n_1,n_2,n_3,D-2c)}{\overline{t}(0,0,0,D)}$}}
\nonumber\\
&=&{\mbox{$\frac{(D/2-1)_{-c}(D/2-1/2)_{-c}}{(-)^{n_3+c}\,2^{(2n_1+2n_2+6c+3n_3)}}$}}
{\mbox{$\frac{(2-D)_{(n_1+n_3+2c)}(2-D)_{(n_2+n_3+2c)}}{(3/2-D/2)_{(n_1+n_3+c)}
(3/2-D/2)_{(n_2+n_3+c)}}$}}
\nonumber\\&&\mbox{}
\times\sum_{k=0}^{[-n_3/2]}
{\mbox{$\frac{
(1/2)_k
(n_3)_{(-n_3-2k)}
(D/2-1/2-c)_k
}{
(-n_3-2k)!
(3/2-D/2+n_1+n_3+c)_k(3/2-D/2+n_2+n_3+c)_k
}$}}.
\label{n30}\end{aligned}$$
For $n_3>1$ one can reduce $n_3$ to 1 with the help of: $$\begin{aligned}
\overline{t}(n_1,n_2,n_3>1,D) & \propto &
\oint\oint\oint
{\mbox{$\frac{dx_1dx_2dx_3}{x_1^{n_1}x_2^{n_2}x_3}$}}
\nonumber\\&&\mbox{}\qquad\times
\left[
{\mbox{$\frac{\partial_3^{n_3-1}}{(n_3-1)!}$}}
{(x_3^2-x_1x_2x_3+x_1x_2(x_1+x_2-4))^{D/2-2}}
\right]
\,.
\label{tbar1}\end{aligned}$$ In connection with the reduction of $n_3$ two remarks are in order: First, we should mention that the reduction of $n_3>1$ to $n_3=1$ can also be expressed through a finite sum which is obvious form Eq. (\[tbar1\]). Second, also for $n_3<1$ the explicit sum in Eq. (\[n30\]) can be replaced by a recurrence procedure using Eqs. (\[eqrdrd\]) and (\[rd\]) in order to arrive at $n_3=0$. Afterwards $n_1$ and $n_2$ are reduced to zero using Eq. (\[n30\]) with $n_3=0$.
For the case $n_3>1$ one gets after the reduction to $n_3=1$ a set of functions $\overline{t}(n_1,n_2,1,D-2c)$ with various $n_1, n_2$ and $c$. The further reduction is done with the help of the relations: $$\begin{aligned}
\overline{t}(n_1,n_2,1,D)&=&{\mbox{$\frac{(D-4)}{(2n_1-D+2)}$}}
\left[\overline{t}(n_1-2,n_2-1,1,D-2)
-{\mbox{$\frac{1}{2}$}}\overline{t}(n_1-1,n_2-1,0,D-2)\right],
\nonumber\\&&
\label{rt1}\\
\overline{t}(n_1,n_2,1,D)&=&
{\mbox{$\frac{(2n_2-D+4)}{(2n_1-D+2)}$}}
\overline{t}(n_1-1,n_2+1,1,D)
+{\mbox{$\frac{(n_1-n_2-1)}{(2n_1-D+2)}$}}
\overline{t}(n_1,n_2+1,0,D)
\,,
\label{rt2}\end{aligned}$$ which can be obtained from Eqs. (\[rn1\])-(\[rd\]). Note that $\overline{t}(n_1,n_2,1,D)=\overline{t}(n_2,n_1,1,D)$. Therefore these equations can be used to reduce $(n_1,n_2)$ to $(0,-1)$, $(1,0)$, $(0,0)$ and further combinations where $n_3=0$. Then, using Eqs. (\[rn1\]) and (\[rd\]) for $n_1,n_2=-1,0,1,2$, respectively, we get: $$\begin{aligned}
\overline{t}(0,-1,1,D)&=&{\mbox{$\frac{4}{3}$}}\overline{t}(0,0,1,D)
+{\mbox{$\frac{1}{3}$}}\overline{t}(0,0,0,D),\label{s1}\\
\overline{t}(1,0,1,D)&=&{\mbox{$\frac{(3D-8)}{4(D-4)}$}}\overline{t}(0,0,1,D)
-{\mbox{$\frac{(D-2)^2}{8(D-3)(D-4)}$}}\overline{t}(0,0,0,D)
\,.\label{s2}\end{aligned}$$ Finally the shift in the last argument is fixed with the help of $$\begin{aligned}
\overline{t}(0,0,1,D+2)&=&
-{\mbox{$\frac{2(D-2)}{3(3D-2)(3D-4)}$}}\left[32(D-2)\overline{t}(0,0,1,D)
+(11D-16)\overline{t}(0,0,0,D)\right]
\,,
\nonumber\\&&
\label{s3}
\\
\overline{t}(0,0,0,D-2c)&=&
{\mbox{$\frac{(-)^c(D/2-1)}{4^c(D/2-1-c)}$}}
{\mbox{$\frac{(D/2-1/2)_{-c}}{(D/2)_{-c}}$}}
\,
\overline{t}(0,0,0,D)
\label{s4}
\,,\end{aligned}$$ which completes the reduction of $\overline{t}(n_1,n_2,n_3,D-2c)$ to $\overline{t}(0,0,1,D)$ and $\overline{t}(0,0,0,D)$. The result for $t(n_1,n_2,n_3,D-2c)$ is obtained from: $$\begin{aligned}
t(n_1,n_2,1,D) &=& \frac{1}{\overline{t}(0,0,0,D)}
\left[
\overline{t}(n_1,n_2,1,D)
-\overline{t}(0,0,1,D) N(n_1,n_2,1,1,1,1)
\right]
\,,\end{aligned}$$ which follows from Eq. (\[condT\]). This means that in practice from the final expression of the reduction procedure simply the coefficient of $\overline{t}(0,0,0,D)$ has to be extracted in order to get the result for $t(\underline{n},D)$. Note that due to Eq. (\[n30\]) the functions $\overline{t}(n_1,n_2,0,D-2c)$ in Eqs. (\[rt1\])-(\[s3\]) are rational in $D$, i.e., the recursion for $n_1, n_2$ and $c$ are simple one-parameter equations of the type $F(n)=a(n)F(n-1)+b(n)$ where $a(n)$ and $b(n)$ are known rational functions. Thus they can easily be solved and expressed through sums of Pochhammer symbols. At this point we refrain from listing the resulting formulas explicitly.
\[secimp\]Implementation
========================
In this section a possible implementation of the above formulas is discussed. We will mainly speak about the most difficult coefficient function (“T”), the others (“M” and “N”) constitute simple sub-cases and can be considered in complete analogy. The calculation of the “T” coefficient function implies subsequent reduction of:\
1) $n_4, n_5, n_6$ to 1,\
2) $n_3$ to 0 or 1,\
3) $n_1, n_2$ to (0,0), (1,0), (0,-1),\
4) $D-2c$ to $D$.\
The steps 1) and 2) demand a Taylor expansion of a polynomial raised to non-integer power. This can either be done by direct differentiation (see Eqs. (\[t31\]) and (\[tbar1\])), or by computing multiple combinatoric sums which arise after rewriting the action of the operators $\partial^n/\partial x_i^n$. Moreover, the reduction of each parameter can be done independently by one of these methods. The steps 3) and 4) can be performed either by using the recursion relations (\[rt1\])-(\[s4\]) or by explicitly evaluating the sums.
In practice, the quantities of physical interest can be expressed as linear combination of vacuum integrals, the number of which can be very large (up to $10^7$; see, e.g. [@chetn]). It is very natural to perform the reductions 1)-4) not for individual integrals, but for the whole expression as simplifications may take place in intermediate steps of the calculation. Of course, the method which gives the best performance strongly depends on the computer algebra system used. In the following we compare REDUCE 3.6 and FORM 2.3 both running on a DEC-Alpha-station with $266$ MHz.
REDUCE. Our experience shows that the method using differentiation in steps 1) and 2) works about two times faster, although the calculation of the multiple sums allows to save computer memory.
At step 3) the recurrence procedure is also about two times faster than the use of explicit sums. The higher demand concerning the memory is not essential at this point as for step 1) more memory is needed and thus the limitations are given from there. Step 4) needs negligible time and memory. Here in practice the recursion approach is used.
FORM. The explicit differentiation using FORM requires for the variables to be defined as “non-commuting” [@FORM]. This has the consequence that the sorting of the intermediate expressions becomes quite slow and needs in addition more main memory. Therefore in practice only the method relying on multiple sums are effective for step 1). At steps 2) and 3) we can choose between the combinatoric-sum and the recursion approach. It turned out that their performance is comparable. At step 4) we prefer to use a one-parameter table for the functions $t(0,0,1,D-2c)$.
The implementation in FORM is done in such a way that an expansion in $D-4$ is performed for the intermediate expressions. Thereby some tricks were used like the application of the “ACCU” function which collects polynomials in $D-4$ or the use of tables for the Pochhammer symbols [@FORM]. Note that explicit formulas are used for the realization. Therefore one has full control on the powers of intermediate poles in $D-4$. This actually is a problem in the approach based on recurrence relations only. There the expansion in $D-4$ has to be performed to higher order which has to be fixed after careful examination of the recurrence relations.
It is interesting to mention that the implementation within REDUCE, which even computes the full $D$-dependence, is about three times faster than the one using FORM. An explanation for this unexpected fact is the possibility to use direct differentiation within REDUCE. It might also be possible that during the calculation essential cancellations among long sums containing Pochhammer symbols take place. REDUCE with its ability to work with rationals can exploit this fact, however, FORM cannot. We should also mention that the attempt to expand in $D-4$ in the “REDUCE” version (using “wtlevel” declaration [@REDUCE]) takes more time but is still faster than FORM.
Within FORM we are also in the position to compare the recursion approach [@ch-tk; @REC] where, however, in addition to the recurrence relations also tables were used [@MATAD], with the one described in this paper. As an example we considered the contribution of the non-planar diagram to the photon propagator and performed an expansion up to ${\cal O}(q^{12})$ where $q$ is the external momentum. It turned out, that the “improved” recursion works better for planar vacuum integrals contributing to this diagram, however, worse for the non-planar ones. The joined variant which uses “improved” recursion for the planar diagrams and the new method for non-planar ones gives a gain of roughly $8$ % as compared to the use of only the “improved” recursion.
\[seccon\]Conclusions
=====================
The new approach described in this paper allows to solve the recurrence relations for three-loop vacuum integrals explicitly and expresses the result in terms of multiple sums containing Pochhammer symbols. Nevertheless, the final choice of the algorithm which leads to the best performance strongly depends on the peculiarities of the problem and also on the symbolic language used.
For the particular problem considered here — three-loop integrals with four massive and two massless lines — we get better results using REDUCE 3.6 than using FORM 2.3 from point of view of CPU time and in the presence of sufficient memory resources. Certainly, FORM stands beyond the competition as soon as the intermediate expressions become very large and enormous swapping to the hard disk is necessary.
Acknowledgment {#acknowledgment .unnumbered}
==============
We would like to thank J.H. Kühn and K.G. Chetyrkin for stimulating discussions. P.A.B. thanks the University of Karlsruhe for the hospitality during the visit when an important part of this work has been done.
[99]{}
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K.G.Chetyrkin, J.H.Kühn and M.Steinhauser, [Phys. Lett.]{} [**B371**]{} (1996) 93 [Nucl. Phys.]{} [**B482**]{} (1996) 213; [Nucl. Phys.]{} [**B505**]{} (1997) 40.
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F.V.Tkachov, Phys.Lett.[**B100**]{} (1981) 65.
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S.A.Larin, F.V.Tkachev and J.A.M.Vermaseren, NIKHEF Report No. NIKHEF–H/91–18 (September 1991).
P.A.Baikov, Phys.Lett.[**B385**]{}, (1996) 404; Nucl. Instr. & Methods [**A389**]{} (1997) 347.
A.C.Hearn, REDUCE User’s Manual, version 3.6.
J.A.M.Vermaseren, Symbolic Manipulation with FORM, version 2.
M. Steinhauser, Ph.D. thesis, Karlsruhe University (Shaker Verlag, Aachen, 1996).
[^1]: Supported in part by INTAS (grant 93–0744–ext), Volkswagen Foundation (contract No. I/73611)\
Email: baikov@theory.npi.msu.su
[^2]: Note that before the transition to this representation is performed the order of the operators has to be reversed.
|
---
abstract: 'Recent $\gamma$-ray observations suggest that the $\gamma$-ray millisecond pulsar (MSP) population is separated into two sub-classes with respect to the pair multiplicity. Here, we calculate the cosmic ray electron/positron spectra from MSPs. Based on the assumption of the equipartition in the pulsar wind region the typical energy of electrons/positrons ejected by a MSP with the pair multiplicity of order unity is $\sim50$ TeV. In this case, we find that a large peak at 10 - 50 TeV energy range would be observed in the cosmic ray electron/positron spectrum. Even if the fraction of pair starved MSPs is 10%, the large peak would be detectable in the future observations. We also calculate the contribution from MSPs with high pair multiplicity to the electron/positron spectrum. We suggest that if the multiplicity of dominant MSP population is $\sim 10^3$, electrons/positrons from them may contribute to the observed excess from the background electron/positron flux and positron fraction.'
author:
- |
Shota Kisaka$^{1 \ast}$, Norita Kawanaka$^{2 \star}$\
$^1$ Department of Physics, Hiroshima University, Higashi-Hiroshima 739-8526, Japan\
$^2$ Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem, 91904, Israel\
Email: $^\ast$ kisaka@theo.phys.sci.hiroshima-u.ac.jp, $^\star$ norita@phys.huji.ac.il
title: TeV Cosmic Ray Electrons from Millisecond Pulsars
---
stars: neutron — cosmic rays.
INTRODUCTION
============
The [*Fermi*]{} Gamma-Ray Space Telescope has detected $\gamma$-ray pulsed emissions from more than twenty millisecond pulsars (MSPs) [@Ab11], which have a rotation angular frequency $\Omega\sim 10^3$s$^{-1}$ and a stellar surface magnetic field $B_s\sim 10^{8.5}$G. The detection of the GeV emissions from a pulsar magnetosphere means that electrons and positrons are accelerated to more than $\sim$ TeV by the electric field parallel to the magnetic field, which arises in a depleted region of the Goldreich-Julian (GJ) charge density [@GJ69]. The $\gamma$-ray light curve is an important tool for probing the particle acceleration process in the pulsar magnetosphere. Therefore, the $\gamma$-ray emission region has been explored by comparing theoretical models such as polar cap [@DH96], outer gap [@CHR86] and slot gap models [@MH04] with the observed light curve (e.g., Venter, Harding & Guillemot 2009; Romani & Watters 2010; Kisaka & Kojima 2011).
@VHG09 fitted the pulse profiles of the [*Fermi*]{} detected MSPs with the geometries of $\gamma$-ray emission region predicted by different theoretical models. They found that the pulse profiles of six of eight MSPs could be fitted by the geometries of either the outer gap or the slot gap model, as was the case of canonical pulsars. They interpreted that copious pairs are produced in the magnetosphere of these MSPs. However, @VHG09 also found that the pulse profiles of remaining two MSPs show the unusual behavior in the $\gamma$-ray light curves and could not be fitted by the geometry of either the outer gap or the slot gap models. They proposed that these unusual light curves could be fitted by the pair starved polar cap model [@MH04b], in which the multiplicity of the pairs is not high enough to completely screen the electric field above the polar cap, and the particles are continuously accelerated up to high altitude over the entire open field line region. Thus, from the model fitting of the $\gamma$-ray light curves, @VHG09 suggested that the $\gamma$-ray MSP population is separated into two sub-classes.
The important fact is that radio pulsed emission is also detected from all currently detected $\gamma$-ray MSPs and remarkably similar to that from canonical pulsars. The pulsar radio emission is a highly coherent process because the brightness temperature is extremely high. In the theoretical models of the radio emission mechanisms, some authors have believed the conditions that there are a highly relativistic primary beam with the large Lorentz factor ($\sim 10^7$) and the number density nearly equal to the GJ density, and the secondary electron/positron plasma with relatively small bulk streaming Lorentz factor ($\sim 10$ - $10^3$) and the large pair multiplicity ($\sim 10^3$ - $10^5$) in the radio emitting region (e.g., Melrose 1995; Lyutikov, Blandford & Machabeli 1999; Gedalin, Gruman & Melrose 2002). However, the existence of pair starved MSPs suggests that the radio emission mechanisms should be insensitive to the particle number density down to sub-GJ number density. The pulsar radio emission mechanism is still poorly understood, so that the observationally-based constraints are valuable [@M95]. Therefore, another verification for the extent of the MSP multiplicity, especially the existence of the pair starved MSPs is important for the pulsar radio emission mechanisms.
Recently, HESS has discovered a new TeV source, which is located in the close vicinity of the globular cluster Terzan 5 [@HE11]. Several globular clusters, including Terzan 5 also emit GeV $\gamma$-ray [@Ab09b; @KHC10; @Ab10b; @Ta11], which may plausibly be due to a number of MSPs residing in these clusters [@HUM05; @VDC09]. Thus, inverse Compton scattering by the high-energy particles ejected from MSPs are proposed for the origin of the observed TeV emission [@BS07; @VDC09]. The high-energy electron/positron spectrum ejected from MSPs would be a useful probe for the multiplicity of the MSPs. However, only from the TeV spectra, we cannot distinguish two models [@BS07; @VDC09], which assume different pair multiplicities [@HE11]. Another way to investigate the electron/positron spectrum ejected from MSPs is its direct measurement. Since high-energy electrons/positrons can propagate only about a few kpc due to the energy losses by the synchrotron and the inverse Compton emission, the direct detection of the electrons/positrons ejected from MSPs in the globular clusters is unlikely. However, for the following reasons, we may detect those from nearby MSPs.
MSPs have much lower spin-down luminosity than canonical pulsars. @BVD08 investigated the possible contribution of the nearby MSP, PSR J0437-4715 to the cosmic ray electron/positron spectrum. They concluded that unlike canonical pulsars such as Geminga pulsar, the contribution from a MSP to the observed electron/positron flux is negligible. However, since the lifetime of MSPs is much longer than canonical pulsars ($>10^{10}$ yr), there should be much more nearby active MSPs. Furthermore, Kashiyama, Ioka & Kawanaka (2011; hereafter KIK11) pointed out that since white dwarf pulsars have long lifetime and continue to inject the electrons/positrons after the nebulae stop expanding, the adiabatic energy losses of electrons/positrons in the pulsar wind nebula region are negligible. Also the synchrotron cooling of electrons/positrons is so small and the high-energy electrons/positrons can escape the nebulae without losing much energy. Although they consider the case of the white dwarfs, their results are also applicable to MSPs. Therefore MSPs could potentially contribute to the observed high energy cosmic ray electrons/positrons and will be detectable by the next generation experiments, such as CALET [@To08] and CTA [@CTA].
In this paper, we investigate the contribution of electrons/positrons ejected from the MSPs to the observed cosmic ray spectrum. In section 2, we apply KIK11 model to the case of MSPs. We estimate the typical energy of electrons/positrons from the MSPs and show that during the propagation in pulsar wind nebulae, the adiabatic losses and radiative cooling of electrons/positrons are not so large. We also describe the propagation in the interstellar medium (ISM). In section 3, we calculate the energy spectrum of cosmic ray electrons/positrons from the MSPs and show the possibility that the electrons/positrons from these MSPs are detectable for the future observations.
THE MODEL
=========
Acceleration and cooling
------------------------
In order to estimate the energy of electrons/positrons available in the wind region and their adiabatic and radiative cooling in the shocked region, we adopt the model of KIK11. For a pulsar wind nebula formed by a MSP, we consider the conditions that the relativistic wind blasts off from the light cylinder $\sim R_{\rm lc}=c/\Omega$ where $c$ is the speed of light, and two shock fronts are formed between the supersonic pulsar wind and ISM. Since the energy from a MSP is continuously transported to the wind, the shock fronts are expanding until the pressure of the shocked region $P_{\rm sh}$ becomes equal to that of ISM $p$. Although KIK11 considered the case of white dwarf pulsars, the situations are similar to the case of MSPs because they have a long lifetime and the supernova shock front have already decayed. We assume that the effects of binary companion are negligible, because the fraction of solid angle occupied by companion is small ($<$1%). We also neglect radiative loss due to curvature radiation within light cylinder ($\sim 10$%). From now on we set fiducial parameters of the MSP’s surface magnetic field strength, angular frequency and radius as $B_0 = 10^{8.5}$G, $\Omega = 10^3$s$^{-1}$ and $R = 10^6$cm, respectively.
We assume the energy equipartition between particles and magnetic field, $\varepsilon_eN=B^2/8\pi$, and the conservation of the particle number flux, $4\pi r^2cN\sim$ constant, in the MSP wind region. Here, $N$ is the number density of electrons/positrons. The number density can be described as $$N=N_{\rm lc}\left(\frac{R_{\rm lc}}{r}\right)^2=\frac{B_{\rm lc}\Omega\kappa}{2\pi ce}\left(\frac{R_{\rm lc}}{r}\right)^2,$$ where $\kappa$ is the multiplicity of electrons/positrons, $N_{\rm lc}$ and $B_{\rm lc}$ are the number density and the magnetic field at the light cylinder, respectively. We assume the magnetic field configuration as pure dipole ($B\propto r^{-3}$) within light cylinder. Outside the light cylinder, we assume the conservation of the energy flux of the magnetic field, $Br\sim$ constant. Using these assumptions, the typical energy of electrons/positrons $\varepsilon_e$ can be described as $$\varepsilon_e = \frac{e\psi_{\max}}{\kappa} \sim 50\kappa^{-1}\left(\frac{B_0}{10^{8.5}{\rm G}}\right)\left(\frac{\Omega}{10^3{\rm s^{-1}}}\right)^2\left(\frac{R}{10^{6}{\rm cm}}\right)^3 {\rm TeV},
\label{eq:2.1}$$ where $\psi_{\rm max}$ is the electric potential difference across the open magnetic field lines described as $$\psi_{\max}=\frac{B_0\Omega^2R^3}{2c^2}\sim 5\times10^{13}\left(\frac{B_0}{10^{8.5}{\rm G}}\right)\left(\frac{\Omega}{10^3{\rm s^{-1}}}\right)^2\left(\frac{R}{10^{6}{\rm cm}}\right)^3 {\rm Volt}.
\label{eq:2.2}$$ These values are similar to those in the case of white dwarf pulsars (KIK11). Note that the typical energy depends on the pair multiplicity.
Next, we estimate the adiabatic and the radiative cooling of electrons/positrons in the shocked region. The outer shock of the pulsar wind nebula finally decays when the pressure of the shocked region $P_{\rm sh}$ becomes equal to that of the ISM $p$. If the outer shock decaying time is shorter than the lifetime of MSP, the adiabatic cooling is negligible. In order to estimate the outer shock decaying timescale, we solve the equation of motion and the energy conservation law at the outer shock front. The equation of motion can be described as $$\label{eos}
\frac{d}{dt} \left\{ \frac{4 \pi}{3} R_{\rm out}^3 \rho \frac{dR_{\rm out}}{dt} \right\} = 4\pi R_{\rm out}^2 P_{\rm sh},$$ where $R_{\rm out}$ is the radius of the outer shock front, $P_{\rm sh}$ is the pressure of the shocked region and $\rho$ is the density of ISM. The energy equation is $$\label{ece}
\frac{d}{dt} \left\{ \frac{4 \pi}{3} R_{\rm out}^3 \frac{3}{2} P_{\rm sh} \right\} = L_{\rm sd} - P_{\rm sh} \frac{d}{dt} \left\{ \frac{4 \pi}{3} R_{\rm out}^3 \right\} ,$$ where $L_{\rm sd}$ is the spin-down luminosity of MSP, $$L_{\rm sd} =\frac{B^2_0\Omega^4R^6}{2c^3} = 2\times 10^{33}\left(\frac{B_0}{10^{8.5}{\rm G}}\right)^2\left(\frac{\Omega}{10^3{\rm s^{-1}}}\right)^4\left(\frac{R}{10^{6}{\rm cm}}\right)^6 {\rm erg}\ {\rm s}^{-1}.$$ In the derivation of eq.(\[ece\]), we assume that in the shocked region the internal energy of particles is $3P_{\rm sh}/2$ because the energy of particles is the relativistic regime. Using the typical value for the density of ISM $\rho\sim 10^{-24}{\rm g}\ {\rm cm}^{-3}$, the pressure in ISM is $p\sim 10^{-13} {\rm dyn}\ {\rm cm}^{-2}$. Solving above equations, the outer shock decays at about $$t_{\rm dec}\sim 10^6 \left(\frac{B_0}{10^{8.5}{\rm G}}\right)\left(\frac{\Omega}{10^3{\rm s^{-1}}}\right)^2\left(\frac{R}{10^{6}{\rm cm}}\right)^3 \left(\frac{T}{10^{3}{\rm K}}\right)^{-5/4} {\rm yr},
\label{eq:2.3}$$ where $T$ is the temperature of ISM. The lifetime of a MSP $\tau$ can be estimated as $$\tau=\frac{E_{\rm rot}}{L_{\rm sd}},
\label{eq:2.4}$$ where $E_{\rm rot}$ is the rotation energy described as $$E_{\rm rot}\sim 10^{52}\left(\frac{M}{1.4M_{\odot}}\right)\left(\frac{R}{10^6{\rm cm}}\right)^2\left(\frac{\Omega}{10^3{\rm s^{-1}}}\right)^2 {\rm erg},
\label{eq:2.5}$$ where $M$ is the mass of a MSP. For the fiducial parameters of MSP $$\tau\sim 5\times 10^{10} \left(\frac{M}{1.4M_{\odot}}\right)\left(\frac{B_0}{10^{8.5}{\rm G}}\right)^{-2}\left(\frac{\Omega}{10^3{\rm s^{-1}}}\right)^{-2}\left(\frac{R}{10^{6}{\rm cm}}\right)^{-4} {\rm yr}.
\label{eq:2.7}$$ We find that the outer shock decays at a very early stage of the lifetime of MSP and does not expand after $t > t_{\rm dec}$. Therefore, in the similar to the case of white dwarf pulsars (KIK11), the adiabatic cooling shall give minor contributions to the cooling process of the high-energy electrons/positrons.
Also for the radiative cooling by the synchrotron radiation in the shocked region $R_{\rm in} < r < R_{\rm out}$, we follow the discussion in KIK11 based on the diffusion in the shocked region. We take the Bohm limit, where the fluctuation of the magnetic field is comparable to the coherent magnetic field strength. In this limit, the timescale $t_{\rm diff}$ for the electron/positron trapping in the shocked region is given by $$t_{\rm diff}=\frac{d^2}{2D_{\rm sh}}=\frac{3}{2}\frac{eBd^2}{\varepsilon_ec},$$ where $D_{\rm sh}=cr_{\rm g}/3$ is the diffusion coefficient under the Bohm limit, $d$ is the size of the shocked region and $r_g=\varepsilon_e/eB$ is the Larmor radius of electron/positron with energy $\varepsilon_e$. To estimate the diffusion timescale, we need to know the size and the magnetic field strength of the shocked region. Here we consider the time $t>t_{\rm dec}$, so that the size of the shocked region is an order of the radius of forward shock front at $t=t_{\rm dec}$, $d\sim R_{\rm out}(t=t_{\rm dec})\sim 3\times 10^{19}{\rm cm}$. For the radius of the inner shock front at $t>t_{\rm dec}$, we use the balance condition between the momentum transferred by wind and the pressure of ISM $$\frac{L_{\rm sd}}{4\pi R^2_{\rm in}c}=p.$$ For the fiducial parameters, $R_{\rm in}(t>t_{\rm dec})\sim 2\times 10^{17}$ cm. The strength of the magnetic field at the inner radius $B_{\rm in}$ can be estimated as $B_{\rm in}\sim 2 \times 10^{-6}$ G, which is almost the same as that of ISM. Then, the diffusion timescale is $$t_{\rm diff}\sim 2 \times 10^4 \left(\frac{\varepsilon_e}{50 {\rm TeV}}\right)^{-1} {\rm yr}.$$ The synchrotron energy loss of a particle with energy $\varepsilon_e$ is described as $$\frac{d\varepsilon_e}{dt}=-\frac{4}{3}\sigma_{\rm T}c\beta^2\frac{B^2}{8\pi}\left(\frac{\varepsilon_e}{m_ec^2}\right)^2,$$ where $\sigma_{\rm T}$ is the Thomson scattering cross section, $\beta$ is the particle velocity normalized by the speed of light and $m_e$ is the mass of electron/positron. Then, The typical energy loss of the electrons/positrons $\Delta\varepsilon_e$ with energy $\varepsilon_e$ can be estimated as $$\frac{\Delta\varepsilon_e}{\varepsilon_e}\sim 0.3\left(\frac{B_0}{10^{8.5}{\rm G}}\right)^2 \left(\frac{\Omega}{10^3{\rm s}^{-1}}\right)^4\left(\frac{R}{10^6{\rm cm}}\right)^6.
\label{eq:2.8}$$ This means that the high-energy electrons/positrons injected into the shocked region lose roughly 30% of the energy by the synchrotron radiation before diffusing out into ISM. Therefore, as in the case of white dwarf pulsars (KIK11), we can conclude that the radiative energy loss of electrons/positrons in the pulsar wind nebula is not so large.
The above expressions for the estimate of the energy losses are only applicable to the case that the velocity of a MSP is subsonic in ISM. The observed velocity of MSPs is less than that of canonical pulsars in average sense [@Ho05]. However, some MSPs have the large velocity and a few MSPs actually forms bow shock nebulae [@St03; @HB06]. In this case, the size of the bow shock is described as (e.g., Wilkin 1996) $$R_{\rm bow}=\left(\frac{L_{\rm sd}}{4\pi c\rho V^2}\right)^{1/2}\sim 10^{16}\left(\frac{B_0}{10^{8.5}{\rm G}}\right)\left(\frac{\Omega}{10^3{\rm s^{-1}}}\right)^{2}\left(\frac{R}{10^{6}{\rm cm}}\right)^{3}\left(\frac{V}{10^7{\rm cm\ s^{-1}}}\right)^{-1} {\rm cm},$$ where $V$ is the velocity of a MSP. Due to the assumption of the energy equipartition, the strength of the magnetic field is $$B_{\rm bow}=\left(\frac{2L_{\rm sd}}{cR^2_{\rm bow}}\right)^{1/2}\sim 50\left(\frac{V}{10^7{\rm cm\ s^{-1}}}\right) \mu {\rm G}.$$ The ratio of the Larmor radius of electrons/positrons to the bow shock radius is $$\frac{r_{\rm g}}{R_{\rm bow}}\sim 0.5 \kappa^{-1}.$$ The fact that $r_{\rm g}/R_{\rm bow}$ is close to unity supports that electrons/positrons may escape from the bow shock region. Therefore, in the case of $\kappa\sim 1$, high-energy electrons/positrons can escape with an efficiency of order unity [@B08; @BA10]. Even if we consider the case of $\kappa \gg 1$, the synchrotron loss can be estimated by using eqs. (11), (14) and (16) as $$\frac{\Delta\varepsilon_e}{\varepsilon_e}\sim 9\times 10^{-4}\left(\frac{B_0}{10^{8.5}{\rm G}}\right)^2 \left(\frac{\Omega}{10^3{\rm s}^{-1}}\right)^4\left(\frac{R}{10^6{\rm cm}}\right)^6\left(\frac{V}{10^7{\rm cm\ s^{-1}}}\right).$$ Therefore, we can conclude again that the radiative energy loss of electrons/positrons in the pulsar wind nebula is not so large.
Diffusion in Interstellar medium
--------------------------------
The observed electron/positron spectrum after the propagation in ISM is obtained by solving the diffusion equation $$\frac{\partial}{\partial t}f(t,r,\varepsilon_e)=D(\varepsilon_e)\nabla^2 f+\frac{\partial}{\partial \varepsilon_e}\left( P(\varepsilon_e)f \right) +Q(t,r,\varepsilon_e),$$ where $f(t,r,\varepsilon_e)$ is the energy distribution function of electrons/positrons, $D(\varepsilon _e)=D_0(1+\varepsilon_e/3{\rm GeV})^{\delta}$ is the diffusion coefficient, $P(\varepsilon_e)$ is the cooling function of the electrons/positrons which takes into account synchrotron emissions and inverse Compton scatterings during the propagation, and $Q(t,\varepsilon_e,r)$ is the injection term. Here we adopt $D_0=5.8\times 10^{28}{\rm cm}^2{\rm s}^{-1}$, $\delta=1/3$, which is consistent with the boron-to-carbon ratio according to the latest GALPROP code. Atoyan et al. (1995) showed a solution in the case of an instantaneous injection from a single point-like source, i.e. $Q(t, \varepsilon_e,r) \approx Q_0(\varepsilon_e) \delta(t-t_i)\delta(r)$. Then the observed spectrum $G(t,r,\varepsilon_e; \tilde{t})$ would be $$G(t,r,\varepsilon_e; \tilde{t})=\frac{Q_0(\varepsilon_{e,0})P(\varepsilon_{e,0})}{\pi ^{3/2} P(\varepsilon_e)d_{\rm diff}(\varepsilon_e, \varepsilon_{e,0})^3}\exp \left(-\frac{r^2}{d_{\rm diff}(\varepsilon_e, \varepsilon_{e,0})^2} \right),$$ where $\varepsilon_{e,0}$ is the energy of electrons/positrons at the time $\tilde{t} (<t)$ and which are cooled down to $\varepsilon_e$ at the time $t$, and $d_{\rm diff}$ is the diffusion length given by $$d_{\rm diff}=2\left[ \int_{\varepsilon_e}^{\varepsilon_{e,0}} \frac{D(x)dx}{P(x)} \right]^{-1/2}. \label{d_diff}$$
The cooling function $P(\varepsilon_e)$ can be described as $$P(\varepsilon_e)=\frac{4\sigma_T \varepsilon_e^2}{3m_e^2 c^3} \left[ \frac{B^2}{8\pi} + \int d\varepsilon_{\gamma} u_{\rm tot} (\varepsilon_{\gamma}) f_{\rm KN} \left( \frac{4\varepsilon_e \varepsilon_{\gamma}}{m_e^2 c^4} \right) \right]$$ where $u_{\rm tot}(\varepsilon_{\gamma})d\varepsilon_{\gamma}$ is the energy density of interstellar radiation fields with the photon energy between $\varepsilon_{\gamma}$ and $\varepsilon_{\gamma}+d\varepsilon_{\gamma}$ (including cosmic microwave background, starlight, and dust emission; Porter et al. 2008), and $B$ is the interstellar magnetic field with we here set as $1\mu {\rm G}$. Here the function $f_{\rm KN}(x)$ is the correction factor to include the Klein-Nishina (KN) effect, which approaches to unity when $x$ is much smaller than unity. The mathematical expression of $f_{\rm KN}$ can be found in Moderski et al. (2005).
As shown in the last section, in general MSPs have a very long lifetime ($\tau \sim 5\times 10^{10}{\rm yr}$) which is comparable with the cosmic age. In such a case that a point-like source with a finite duration, taking into account the time-dependence of an injection rate ($Q_0(\varepsilon_e)\rightarrow Q_0(\varepsilon_e, \tilde{t})$), the spectrum can be calculated by integrating $G(t,r,\varepsilon_e;\tau)$ for $\tau$:
$$f_1(t,r,\varepsilon_e; t_i)=\int_{t_i}^{t} d\tilde{t}~G\left( t,r,\varepsilon_e; \tilde{t} \right) ,$$
where $t_i$ is the time when the particle injection from a source has started. Here we assume that the electron/positron injection spectrum can be described as mono-energetic distribution $$\label{mono}
Q_0(\varepsilon_e,\tilde{t})\propto \left( 1+\frac{\tilde{t}-t_i}{\tau} \right)^{-2},$$ or power-law distribution $$\label{pl}
Q_0(\varepsilon_e,\tilde{t})\propto \varepsilon_e^{-\alpha}\exp \left( -\frac{\varepsilon_e}{\varepsilon_{\rm cut}}\right) \left( 1+\frac{\tilde{t}-t_i}{\tau} \right)^{-2},$$ where $\alpha$ is the intrinsic power-law index of an electron/positron spectrum and $\varepsilon_{\rm cut}$ is the maximum electron/positron energy from a source.
Now we can calculate the average electron/positron spectrum by considering the birth rate of MSPs as follows:
$$f_{\rm ave}(\varepsilon_e)=\int_0^{t_0} dt_i \int_0^{d_{\rm diff}(\varepsilon_e,\varepsilon_{e,i})} 2\pi r dr f_1(t_0,r,\varepsilon_e; t_i)R,$$
where $t_0$ is the cosmic age (i.e. the present time), and $R$ is the local pulsar birth rate (${\rm yr}^{-1}~{\rm kpc}^{-2}$). Here $\varepsilon_{e,i}$ is the energy at the time $t_i$ of CR electrons/positrons which are cooled down to $\varepsilon_e$ at $t$.
RESULTS AND DISCUSSIONS
=======================
First, we calculate the cosmic ray electron/positron spectra from the pair starved MSPs. We set the pair multiplicity $\kappa= 1$, the lifetime $\tau= 5\times 10^{10}$yr, the total energy $E_{\rm rot}= 10^{52}$ erg, the local birth rate $R = 3\times 10^{-9}$ yr$^{-1}$ kpc$^{-2}$ and the fraction of the lost energy due to synchrotron emission 30% for each MSP. We assume that each MSP has the same value of the parameters ($B_0=10^{8.5}{\rm G}, \Omega=10^3{\rm s}^{-1}, R=10^6{\rm cm}$), because most MSPs have the almost same spin-down luminosity. For the injection distribution function, we assume mono-energetic distribution eq.(\[mono\]) with the energy $\varepsilon_e=50$ TeV. Even if we consider the power-law distribution eq.(\[pl\]), the energy range of the distribution is small because the allowed range of the cutoff energy should be only 50 - 80 TeV due to the observed constraint by KASKADE/GRAPES/CASA-MIA [@KY09]. Thus the distribution should be nearly mono-energetic distribution. Note that our local MSP birth rate is based on the MSP local surface density of 38$\pm$ 16 pulsars kpc$^{-2}$ for 430 MHz luminosity above 1 mJy kpc$^2$ [@L08]. Actually, now 23 MSPs are detected within 1kpc [@ATNF]. We can only detect MSPs that have the radio beam directed toward us and the radio flux larger than the threshold of detectors. However, the cosmic ray electrons/positrons ejected from MSPs are distributed isotropically because of the effect of Galactic magnetic field, so that a large number of MSPs will contribute to the observed electron/positron spectrum. Therefore, our local MSP birth rate corresponds to the lower limit for the current radio observations.
In figure \[fig:3.1\], the electron/positron flux from multiple pair starved MSPs is shown. Thick solid, dashed and dash-dotted lines show the total electron/positron spectra if the fraction of the pair starved MSPs is 100%, 25% and 10%, respectively. Thin solid line shows the contribution of electrons/positrons from the pair starved MSPs when the fraction is 100%. The background flux is shown as a thin dashed line. We adopt the background model of an exponentially cutoff power law with an index of -3.0 and a cutoff at 1.5 TeV, which is similar to that shown in @Ah08 and reproduces the data in $\sim$10 GeV-1 TeV well. It is very interesting that there is a large peak at 10-50 TeV energy range. The existence of this peak cannot be ruled out from the current observations. The high-energy component is more enhanced for the long-duration sources [@AAV95; @KIN10]. This is because the longer the duration of injection is, the larger fraction of fresh electrons/positrons are expected to reach the Earth without losing their energy so much during the propagation. MSPs are the continuously injecting sources with the duration as long as the cosmic age, so that the spectrum from them has nearly the same shape with the injection spectrum with the soft energy tail component. This is the difference from other sources such as young pulsars, whose typical duration is only $\sim 10^4$-$10^5$ yrs. Fitting result of @VHG09 showed that the fraction of the pair starved MSPs is 25%, although they have only eight samples. Even if the fraction is 10%, the flux is $\sim 20$m$^{-2}$ s$^{-1}$ sr$^{-1}$ GeV$^2$ at 10 TeV. In this case, we can detect the electron/positron flux with near future missions such as CALET (we assume the geometrical factor times the observation time $\sim 5$ yrs as $\sim 220$ m$^2$ sr days) because the predicted electron/positron flux is sufficiently large. It was considered that the number of astrophysical sources contributing to the above several TeV energy range is quite small according to the birth rate of the supernovae and the canonical pulsars in the vicinity of the Earth [@Ko04; @KIN10; @KIOK11]. However, we find that it is possible for multiple pair starved MSPs to contribute to the 10 TeV energy range in the electron/positron spectrum. Therefore, if the anisotropy of the observed electrons/positrons are weak in the 10 TeV energy range, we suggest that the pair starved MSPs may contribute to the spectrum significantly.
Next, we also investigate the contribution of MSPs with high pair multiplicity to the observed cosmic ray electrons/ positrons. In this model, we assume that the injection function of these MSPs is power-law distribution with index $\alpha=2$, the cutoff energy $\varepsilon_{e, {\rm cut}}=1$TeV and the minimum energy $\varepsilon_{e, \min}=1$GeV. In this case, the pair multiplicity is $\sim 2000$. Other parameters are the same values as in the case of the pair starved MSPs. We assume that the fraction of MSPs with high multiplicity is 100%. The results are shown in figure \[fig:3.2\] for the electron/positron spectrum and in figure \[fig:3.3\] for the positron fraction. Both figures show that electrons/positrons ejected from MSPs with high multiplicity partially contribute to the excess from background flux observed by PAMELA, HESS and [*Fermi*]{}. In this energy range, the other sources such as canonical pulsars [@S70; @AAV95; @CCY96; @ZC01; @Ko04; @G07; @Bu08; @P08; @HBS09; @YKS09; @MCG09; @Gr09; @KIN10; @HGH10] would also contribute to the observed electron/positron spectrum. Note that even if other sources are dominant for the observed excess, the total spectrum added to the contribution of MSPs with high multiplicity does not significantly exceed the observed electron/positron spectrum.
SUMMARY
=======
In this paper, we show the possibility that the cosmic ray electrons/positrons from MSPs would significantly contribute to the observed spectrum. Although MSPs have relatively low spin-down luminosity and low birth rate, the lifetime is so long that there are many active MSPs in the vicinity of the Earth. Furthermore, such a long lifetime source continuously injects electrons/positrons after the nebula ceases expanding, so the adiabatic energy losses in a pulsar wind nebula region are negligible. The synchrotron cooling in the nebula is also small, so the high-energy electrons/positrons can escape the nebula without losing much energy. We calculate the diffusive propagation of high-energy electrons/positrons in the ISM taking into account the cooling via synchrotron emissions and inverse Compton scatterings, and predict their spectrum observed at the Earth.
In the case of the MSPs with multiplicity that is unity, the typical energy of the electrons/positrons produced should be $\sim 50$ TeV based on the assumption of the equipartition in the MSP wind region. Since the long duration of injection make a hard spectrum, the peak is enhanced at 10 - 50 TeV energy range. Even if the fraction of pair starved MSPs is as small as 10%, this peak would be detectable in the future missions such as CALET and CTA. Although a single young source can make the similar spectral feature in this energy range, in the case of pair starved MSPs the anisotropy of the electron/positron flux would be weaker because a number of sources contribute to it. If this peak is detected, that will be a great impact for on the studies of pulsar radio emission mechanisms because the existence of pair starved MSPs suggests that the radio emission mechanisms should be insensitive to the particle number density down to sub-GJ number density. The detection also suggests that the current outer gap model should be modified because @WH11 suggested that most MSPs locate above the pair death line of the outer gap and the multiplicity is larger than unity.
We also calculate the electron/positron spectrum from MSPs with high pair multiplicity. We suggest that if multiplicity of these MSPs is the order of $\sim 10^3$, electrons/positrons from them partially contribute to the observed excess of the total spectrum and the positron fraction.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank K. Ioka, K. Kashiyama, T. N. Kato, Y. Kojima, J. Takata and S. J. Tanaka for useful discussions and comments. This work was supported in part by the Grant-in-Aid for Scientific Research from the Japan Society for Promotion of Science (S.K.).
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![Electron/positron spectrum predicted from MSPs with the fraction of pair starved MSPs 100% (thin solid line), and its sum (thick solid line) with the background (thin dashed line). The injection distribution function is the mono-energetic distribution eq.(\[mono\]) with the energy $\varepsilon_e=50$ TeV and the multiplicity $\kappa = 1$. Data points correspond to measurements of ATIC (purple boxes, Chang et al. 2008), HESS (light-green triangles and black triangles, Aharonian et al. 2008; 2009), PPB-BETS (yellow triangles, Torii et al. 2008b), [*Fermi*]{} (blue circles, Ackermann et al. 2010) and GRAPES (black circle, Kistler & Yüksel 2009). We also show the total spectra from pair starved MSPs with the different fractions: 25% (thick dashed line) and 10% (dot-dashed line). We assume that the lifetime $\tau=5\times 10^{10}$ yr, the total energy $E_{\rm rot}=10^{52}$ erg, the local birth rate $R=3\times 10^{-9}$ yr$^{-1}$ kpc$^{-2}$ and the fraction of the energy loss is 30%.[]{data-label="fig:3.1"}](fig1.eps){width="160mm"}
![Electron/positron spectrum predicted from MSPs which have the power-law injection function with the cutoff energy $\varepsilon_{e,{\rm cut}}=1$ TeV, the minimum energy $\varepsilon_{e,{\rm min}} = 1$ GeV and index $\alpha=2$ (thin solid line). The thick solid line shows the total spectrum. For other parameters, we use the same values as in the case of Figure 1.[]{data-label="fig:3.2"}](fig2.eps){width="160mm"}
![Total positron fraction resulting from the spectrum (solid line) that have the same parameters as in Figure 2, and the background (dotted line), compared with the PAMELA data as red circles [@pamera]. Note that the solar modulation is important below $\sim 10$ GeV.[]{data-label="fig:3.3"}](fig3.eps){width="160mm"}
|
---
author:
- 'G. Galletta'
- 'V. Casasola'
- 'L. Piovan'
- 'E. Merlin'
- 'D. Bettoni'
date: 'Received ; Accepted 20 September 2006'
title: Relations between ISM tracers in galaxies
---
[We study the relations existing between fluxes emitted at CO(1-0) line, 60 and 100 $\mu$m wavelengths, B and soft X-ray wavebands for galaxies of all morphological types. The large set of data that we created allows to revisit some of known relations existing between the different tracers of the Interstellar Medium (ISM): the link between the FIR flux and the CO line emission, the relation between X-ray emission in non active galaxies and the blue or FIR luminosity.]{} [Using catalogues of galaxies and works presented in the literature, we collected fluxes in FIR, 21 cm, CO(0-1) line and soft X-ray for two samples, consisting of normal and interacting galaxies respectively. Joining together these samples, we have data for a total of 2953 galaxies, not all observed in the four above wavebands.]{} [ All the relations found are discussed in the frame of the star formation activity that is the link for most of them. We note that when an active star formation is present, it may link the galaxy fluxes at almost all wavelengths, from X to microwaves. On the contrary, in early-type galaxies where the current star formation rate has faded out the X-FIR fluxes link disappears. This result obtained for early-type galaxies is discussed and explained in detail in the frame of a suitable theoretical model, obtained coupling chemo-dynamical N-body simulations with a dusty spectrophotometric code of population synthesis.]{}
Introduction
============
The observations of galaxies at various wavelengths, going from radio to X-ray, allow to study the relationships existing between the various phases of the interstellar gas, and between gas, dust and stars. Some of these relations are already known since many years, such as that between CO and far infrared (FIR) luminosities [@mirabel; @sanders; @solomon; @devereux]. Others, connected with X-ray emission, have been studied more recently [@padovani; @david; @ranalli].
At present, different tracers of the gas are known, such as millimetric lines for the cold molecular gas, the 21 cm line for atomic hydrogen at $\sim$100 K, IR bands for molecules at thousands of degrees, UV lines and X-ray emission for hotter gas. The dust distribution is traced also by FIR emission at 60 and 100 $\mu$m if the grains are warm [@bregman] or at 170 $\mu$m, if they are colder [@popescu]. The diffusion of large archives of observations at the above wavelengths (except for molecular lines) allowed in the last years the compilation of catalogues containing a huge number of galaxies. Using these catalogues and the works presented in the literature, we collected fluxes in FIR, 21 cm, CO(0-1) line and soft X-ray for two wide samples of normal [@normal] and interacting [@interacting] galaxies. Joining together these samples, we have data for a total of 2953 galaxies, not all observed in the four above wavebands. The fluxes measured with the different tracers allow now a study of the link existing between dust, gas and stars based on hundreds of galaxies.
It is known that the fluxes emitted by a galaxy at very different wavelengths may be linked together by means of the star formation mechanism (see @david [@ranalli]). For instance, the formation of massive stars generates the heating of the dust clouds in which they are embedded, by absorption of their UV radiation, and produces a re-emission of this energy in the far infrared. This process links the current star formation rate to the IR emission at 60 and 100 $\mu$m [@thronson]. The ionizing radiation of stars may produce also the evaporation of the molecular clouds. Inside these clouds, where the particle density is great enough to produce a significant number of collisions between H$_2$ and CO molecules, these latter are excited and produce photons, but in optically thick regions. The warming by the UV stellar light makes these regions less dense, making visible the CO lines at their edge. Because of this mechanism, these lines are considered tracers of the cold molecular hydrogen that does not emit observable lines. The newly formed stars are also responsible of the X-ray emission, produced by very massive stars, by core-collapse SN, and by high mass X-ray binaries. According to the above described mechanisms, we expect that galaxies with active star formation will have a far infrared emission, but also CO and X-ray emissions induced by the more massive stars, linked together by means of different relations.
When the star formation decreases or vanishes, the far infrared emission decreases as well, but it may be fed by the stellar light absorbed and re-emitted in the infrared by dust (cirrus), while low-mass X-ray binaries and Type I SN contribute to the high energy galaxy spectrum. In addition, AGB stars, surrounded by dust, and the cooling flows of the interstellar medium ejected by supernovae may produce additional IR and X emission, between each other.
To study the activity of the galaxies at different wavebands, we collected data on galaxies starting from the original data of fluxes at 60, 100 $\mu$m, CO(1-0) 2.6 mm and soft X-ray used to compile our catalogues [@norm_cat; @inter_cat]. The merging of the two above catalogues produces 1764 known values of far infrared fluxes (1837 have 100 $\mu$m flux), 391 soft X-ray fluxes and 434 values of the CO(1-0) line luminosity. We extracted from LEDA catalogue [@leda] the values of the distance moduli, blue absolute magnitudes and morphological classification for all of them.
Galaxies with evident sign of interactions or disturbed morphologies according to the catalogues of @arp [@am; @vv] are 1038. We shall refer to them as “perturbed galaxies”. The remaining 1915 galaxies that appear neither morphologically nor dynamically perturbed are called “normal galaxies”. In our sample, we have 253 galaxies that have spectral classification of the nucleus and 231 of these appear to host an AGN (Seyfert 1, 2 or transition type, Seyfert 3 or Liners) according to the classifications of @ho and @veron. Most part of the remaining 2722 galaxies lacks of information about nuclear spectrum or have spectra of HII regions (22 starburst spectra). They are not included in any AGN catalogue and for this reason in the following discussion we refer to them as “non active galaxies" and to the others as “active galaxies". With all these data, we crossed the various tracers to understand and revisit the main relations existing between X, FIR, CO and B luminosities.
Cold gas and warm dust
======================
The relations existing between different cold components of the ISM such as the molecular gas and the dust have been studied since many years [@mirabel; @solomon; @bregman]. They find that the global galaxy luminosity derived from CO(1-0) line is directly related with the flux at 100 $\mu$m. With our large sample we can now test these relations using galaxies of different morphological types and activity or interaction.
In Figure \[CO\_100\] we plotted the logarithm of the flux measured from CO(1-0) line vs. the logarithm of the IRAS flux at 100 $\mu$m. In our plots, we have 193 galaxies with classification from E to Sb and 178 from Sbc to Sm. The relation found by @bregman for a sample of early-type galaxies, log S$_{CO}$=log S$_{100}$ - 1.76, is also plotted as comparison, as a dotted line.
The relations are evident, with this wider sample of galaxies. In these diagrams, active and non-active galaxies appear mixed together without clear differences and have been plotted together. The same behaviour appears for interacting and non interacting galaxies, that are not distinguished in our plots.
For all the galaxy types, we find: $$Log S_{CO}= 1.06\ Log S_{100} + 2.02
\label{eqCO_100}$$ with a correlation coefficient of 0.74 and a r.m.s. of 0.37. In the above formula, S$_{100}$ is in mJy and S$_{CO}$ is in Jy km/s.
Similar relations exist between the CO fluxes and the FIR magnitudes, defined as: $$m_{FIR} = -2.5\ Log(2.58\ S_{60}+S_{100})+ 22.25
\label{def_mfir}$$ where S$_{60}$ and S$_{100}$, the fluxes at 60 and 100 $\mu$m respectively, are in mJy. We find for all the galaxy types: $$Log S_{CO} = 0.41\ m_{FIR}+ 6.86
\label{CO_mfir}$$ with a correlation coefficient of 0.69 and a r.m.s of 0.40. The results are based on 179 early types and 170 late-type galaxies. For their similarity with Figures \[CO\_100\] these relations are not plotted in this paper.
We note that irregular galaxies are not fitted by these relations but have a wide spread. In our sample there are just 10 galaxies and their representative points have been not plotted in Figure \[CO\_100\].
X-ray component.
================
We are interested to understand what relations exist between L$_X$, the X-ray luminosity, and the other global galaxy properties. From the literature, it is known the existence of a proportionality between L$_X$ produced by discrete sources and L$_B$, the blue luminosity of the whole galaxy. This relation has been studied by @ciotti and compared by @beuing with soft X-ray fluxes measured by ROSAT satellite. It appears that late-type galaxies have a global X-ray luminosity directly proportional to L$_B$, while early-type systems are dominated by emission produced by hot diffuse gas and their $L_X$ is proportional to the square power of the blue luminosity, as discussed by @beuing. For this reason, the early and late-type galaxies are discussed separately.
Late-type galaxies
------------------
With our data, the X-ray luminosity of galaxies with morphological type later than Sb can be fitted by a linear relation as a function of L$_B$(dotted line in Fig. \[X1\], left panel). The direct proportionality is expressed by the equation: $$Log L_X = Log L_B - 3.85
\label{XBlate}$$ with a r.m.s. from observed data of $\sigma$=0.61 based on 63 galaxies. In this formula and in the following, all the luminosities are expressed in solar units.
If, instead of the blue luminosity, we use the galaxy area $D^2_{kpc}$, calculated from the apparent diameter measured at the 25 mag arcsec$^2$ isophote and converted in kpc$^2$, we discover that the relation is still present, but with a larger spread. It becomes: $$Log L_X = Log D^2_{kpc} + 3.83$$ ($\sigma$=0.80) for a sample of 64 galaxies.
A relation similar to that of @ciotti has been found by some authors [@padovani; @david; @ranalli], but using 60 $\mu$m fluxes or FIR luminosities. The values of $L_{FIR}$ are calculated using the formula: $$Log L_{FIR}= 2.59+Log(2.58\ S_{60}+S_{100})+2 Log\ d$$ where L$_{FIR}$ is in solar luminosities, fluxes are in mJy and the galaxy distance d is in Mpc.
From our data it is possible to find a relation between L$_X$ and L$_{FIR}$ that fits the values of late-type galaxies. We found L$_X \propto$ L$_{FIR}^{0.90}$, similar to the L$_X \propto$ L$_{FIR}^{0.88}$ found by @ranalli for fluxes between 0.5 and 2 keV and to L$_X \propto$ L$_{FIR}^{0.95}$ found by @david using fluxes between 0.5 and 4.5 keV. Forcing the relation to a linear proportionality between L$_X$ and L$_{FIR}$ we find: $$Log L_X = Log L_{FIR} - 3.18
\label{XFIR}$$ with a $\sigma$ of 0.47, based on 147 galaxies. This relation is plotted as a dashed line in the right panels of Figures from \[X1\] to \[X3\].
We note that the B and FIR luminosities are also connected in late-type galaxies by means of a linear relation fitted by: $$Log L_{FIR} = Log L_B - 0.38$$ with a r.m.s.=0.5. This equation, inserted into the relation (\[XBlate\]) gives: $$Log L_X = Log L_{FIR} -3.47$$ similar to the result of equation (\[XFIR\]) and to that found by @ranalli. This is an independent way to confirm our results and to verify the existence of a global link between L$_{FIR}$, B light and X-ray emission. The connection between B luminosity or galaxy area and X or FIR luminosities will be discussed in Section \[Discussion\].
Early-type galaxies
-------------------
When the early-type galaxies are considered in the above described relations involving X-ray emission, the correlations become less evident. Considering soft X-ray and B luminosities, we find a relation: $$Log L_X = 2\ Log L_B - 13.57
\label{XBearly}$$ ($\sigma$=0.73) based on 224 galaxies and plotted as full line in Fig. \[X2\], left panel. The above formula agrees with the expected relation for X-ray emission coming from hot diffuse gas, as discussed by @beuing.
The relation still hold if $D^2_{kpc}$ (kpc$^2$) is used. It becomes: $$Log L_X = 2 Log D^2_{kpc} + 1.51$$ ($\sigma$=0.85) for 226 early-type galaxies from E to Sb.
Many galaxies with high blue luminosity, indication of high masses and of a recent star formation, lie quite far from the mean line, with a behaviour different than that of late-type galaxies.
If the X-ray fluxes are compared with FIR luminosity, the disagreement with the behaviour found in late-type galaxies is more evident. The plot L$_X$ vs. L$_{FIR}$ for early-type galaxies shows the representative points of the galaxies above the relation (\[XFIR\]) for late-type galaxies (Fig.\[X2\], right panel). To understand this apparent disagreement, we should use a theoretical analysis of the far infrared emission, as explained in the next Section 4.
Active galaxies
---------------
Active galaxies (Seyfert 1, Seyfert 2 and Liners) have X-ray, B and FIR fluxes that are not linked together. This happens because, to the emission mechanisms stimulating the light emission at the different wavebands described for non active galaxies, adds an X-ray emission coming from nucleus. In fact, the points representative of these active galaxies are spread in the plot over the discrete sources line and around the diffuse gas line (see Fig.\[X3\], left side). In the L$_X$–L$_{FIR}$ diagram (Fig.\[X3\], right side) the spread is similar to that of early-type galaxies plotted in Fig.\[X2\], but we separately plotted the active galaxies because of the particular nature of their X-ray emission, due to the nuclear contribution.
Modelling $L_{X}$, $L_{B}$ and $L_{FIR}$ of early-type galaxies {#Modelling}
===============================================================
To cast light on the nature of the relations observed between $L_{X}$, **$L_{B}$** and $L_{FIR}$ for early-type galaxies, one has to consider the various components of a galaxy (stars, gas and dust) and to understand their mutual interactions as far as the spectral energy distribution (SED) is concerned. There are two basic schemes to model the formation and evolution of early type galaxies: (1) the semi-analytical models on which a great deal of our understanding of the chemo-spectro-photometric properties is derived, and (2) the N-Body Tree-SPH simulations which, in contrast, have been only occasionally used to study spectro-photometric properties of early type galaxies. In the following part of this section we will proceed as follows. First we will analyse the drawbacks of semi-analytical models, in particular dealing with the calculation of the infrared emission of early-type galaxies. Second, we will discuss how dynamical simulations and a dusty spectrophotometric code, when mixed together allow to move a step forward in the calculations of the SEDs properties. Third, we will show in detail how our model has been built and the coupling between dynamics and dusty population synthesis has been done.
The semi-analytical models and their drawbacks
----------------------------------------------
The semi-analytical models approximate a galaxy to a point mass system in which gas is turned into stars by means of suitable recipes for star formation and heavy elements are produced by stellar nucleosynthesis and stellar winds/explosions. The standard evolutionary population synthesis technique (EPS) is usually applied to derive the SED of the galaxy, with models able to explain many global features of early type galaxies, as amply described by many authors [@Arimoto87; @Arimoto89; @Bressan94; @Gibson97b; @Tantalo96; @Tantalo98]. There are three important and problematic issues of these models to be discussed for our purposes.
First, to determine the age at which the galactic wind sets [@Larson74; @Larson75], we need some hypothesis about Dark and Baryonic Matter with their relative distributions, and about the heating and cooling efficiency of the various mechanisms, to properly evaluate the total gravitational potential well and to describe the thermal history of the gas. In this scheme it comes out that the galactic wind occurs typically for ages $t_{GW} < 1$ Gyr, later in a massive early-type galaxy and much earlier in galaxies of lower mass [@Arimoto87; @Arimoto89; @Bressan94; @Gibson97b; @Tantalo96; @Tantalo98; @Chiosi98]. The maximum duration of star forming activity follows therefore in these models the trend $\Delta t_{SF} \propto M_{G}$. This trend of the SFH is, however, contrary to what required by the observes trend of the $\alpha$-enhancement for early type galaxies, which implies that the maximum duration of the star forming activity should decrease when the galaxy mass increases $\left( \Delta t_{SF}
\propto M_{G}^{-1}\right)$ [see @Bressan96; @Kuntschner00; @Trager00a; @Trager00b; @Tantalo04; @Thomas05 for more details on the enhancement in $\alpha$-elements and the SFH of early-type galaxies].
Second, after the galactic wind phase, star formation does no longer occur and the evolution is merely passive. However, AGB and RGB stars continue to loose gas in amounts that are comparable to those before the galactic wind [@Chiosi00]. What is the fate of this gas? One may imagine that the large amount of gas lost by stars will expand into the Dark Matter halo and heat up to an energy overwhelming the gravitational potential, it will escape the galaxy. Most likely a sort of dynamical equilibrium is reached in which gas is continuously ejected by stars and lost by the galaxy. It may happen therefore that some amount of gas is always present in the galaxy. The question is not trivial because if an early type galaxy is free of gas and contains only stars, the SED is expected to drop off long-ward of about $2 \mu m$ and no IR emission should be detected. However, as already pointed out long ago by @Guhathakurta86 [@Knapp89] (see also Fig. \[X2\]), many early-type galaxies of the local universe emit in the IR. The origin of this flux in the MIR/FIR is likely due to dust present in a diffuse ISM which, heated up by the galactic radiation field, emits at those wavelengths. Therefore to match the IR emission one has to allow for some amount of diffuse ISM. An interesting question to rise is therefore: how much gas can be present today in an elliptical galaxy and how is it distributed across the galaxy? Even if we can correctly estimate the amount of gas ejected by stars, the fate of this gas goes beyond the possibilities of classical semi-analytical models.
As a third point, note that when we fold many SSPs to calculate a galaxy SED using the classical EPS technique we simply convolve their fluxes with the SFH of the galaxy. Many classical spectrophotometric semi-analytical models of galaxies are built in this way: there is no dust at the level of SSPs and again no dust at the level of the galaxy model [see e.g. @Arimoto87; @Arimoto90; @Bruzual93; @Tantalo96; @Kodama97; @Tantalo98; @Buzzoni02; @Buzzoni05]. To calculate the emission by dust, a higher level of sophistication of the model is required. Indeed one has to develop a model in which the sources of radiation and the emitting/absorbing medium are distributed, to face and solve the problem of the radiative transfer simulating in a realistic way the interactions among the various physical components of a galaxy. Among recent models of this kind are those by @Silva98, @Devriendt99 and @Takagi03.
Improving upon semi-analytical models
-------------------------------------
Two drawbacks of the semi-analytical models concern therefore: (1) the description of galactic wind, which is supposed to occur within a finite time interval and (2) the star formation history that is reversed allowing longer SFH for more massive galaxies. These two problems, combined with a lack of geometrical information about the distribution of gas and dust, make semi-analytical models not suitable to calculate properly the IR emission of early type galaxies. To improve upon them we need to use the results obtained from dynamical simulations. They have shown to be able to properly model the ejection of gas by the galaxy as a sort of continuous process, taking place whenever a gas particle heated up by various mechanism has acquired a velocity greater than the escape velocity [see e.g. @Carraro98; @Kawata01; @Springel01; @Chiosi02]. They are able to reproduce the SF history of early-type galaxies both in the context of the monolithic collapse scenario [@Kawata01; @Chiosi02] and recently in the context of hierarchical scenario [@DeLucia06].
Finally, the galaxy is no more a mass point, but a fully three-dimensional structure of the galaxy is available with spatial distribution of stars and gas.
@Merlin06, with the aid of *N-Body Tree-SPH* simulations based on quasi-cosmological initial conditions in the standard-Cold Dark Matter scenario (S-CDM), modelled the formation and evolution of two early-type galaxies of different total mass (Dark + Baryonic Matter in the cosmological proportions 9:1). The total masses under considerations are $1.62\times 10^{12} M_\odot$ (Model A) and $0.03\times 10^{12} M_\odot$ (Model B). The galaxies have been followed from their separation from the global expansion of the universe to their collapse to virialized structures, the formation of stars and subsequent nearly passive evolution. They are followed for a long period of time, i.e. 13 Gyr (Model A) and 5 Gyr (Model B). In any case, well beyond the stages of active star formation which occurs within the first 3 to 4 Gyr (see below). The models take into account radiative cooling by several processes, heating by energy feed back from supernova explosions (both Type I and II) and chemical enrichment. All the models conform to the so-called *revised monolithic scheme*, because mergers of substructures have occurred very early in the galaxy life. Some parameters and results of the two models are summarized in Table \[tabcosmo\]. Note that the shape of the resulting galaxies is nearly spherical both in Dark Matter and stars.
----------------------------------------- ---------- ---------- --
Model A B
Cosmological background S-CDM S-CDM
Initial redshift 50 53
$\Omega_m$ 1 1
$H_0 = 50 \mbox{ } km Mpc^{-1} s^{-1} $ 50 50
Gas particles 13719 13904
CDM particles 13685 13776
Total Mass $ 1.62 $ $ 0.03 $
Initial baryonic mass fraction 0.10 0.10
Present gas mass 0.062 0.0004
Present star mass 0.091 0.0029
$M_{star}/M_{baryons}$ 0.556 0.82
Initial radius 33 9
Half-Mass radius of stars 7 1
Half-Mass radius of DM 52 15
Effective radius of stars 5.2 0.8
Present virial radius 300 41
Axial ratio b/a (stars) 1.08 1.04
Axial ratio c/a (stars) 1.07 1.00
Axial ratio b/a (Dark Matter) 1.14 1.14
Axial ratio c/a (Dark Matter) 1.17 0.96
Age of the last computed model 13 5
----------------------------------------- ---------- ---------- --
: Initial parameters for the dynamical simulations of @Merlin06 in the Standard CMD scenario. Masses are in units of $10^{12}M_\odot$, radii are in kpc and ages are in Gyr.
\[tabcosmo\]
The third drawback of classical semi-analytical model was the lack of the description of the dusty component, that for our purposes needs to be included. The semi-analytical chemo-spectro-photometric model developed by @Piovan06b allows us to overcome this issue. It takes into account not only the geometrical structure of galaxies of different morphological type, but also the effect of dust in converting the UV and Optical light in far IR radiation. In brief the @Piovan06b model follows the infall scheme, allows for the onset of galactic winds, and contains three main components: (i) the diffuse interstellar medium composed of gas and dust whose emission and extinction properties have been studied in detail by @Piovan06a, (ii) the large complexes of molecular clouds in which new stars are formed and (iii) the stars of any age and chemical composition. The total gas and star mass provided by the chemical model are distributed over the whole volume by means of suitable density profiles, one for each component and depending on the galaxy type (spheroidal, disk and disk plus bulge). The galaxy is then splitted in suitable volume elements to each of which the appropriate amounts of stars, molecular clouds and interstellar medium are assigned. Each elemental volume absorbs radiation from all other volumes and from the interstellar medium in between. The elemental volume also re-emits the absorbed light and produces radiation by the stars that it contains. On the other hand, the star formation, the initial mass function, the chemical enrichment of the @Piovan06b model are much similar to those by @Bressan94 [@Tantalo96; @Tantalo98; @Portinari98].
Coupling dynamical simulations and dusty population synthesis models
--------------------------------------------------------------------
The description of an early-type galaxy as far as predicting its spectro-photometric infrared properties can be therefore realized with a suitable combination of dynamical and spectro-photometric approaches. Coupling the dynamical models with spectro-photometric synthesis requires a number of steps that deserve some remarks.
### Radial density profiles.
Fig. \[RaggiViriali\] shows the cumulative distribution of gas and stars as a function of the radial galactocentric distance normalized to the virial radius for model A (top panel) and model B (bottom panel). The gas is generally distributed in the external regions of the galaxy and steeply decreases inward. In contrast the stars are more concentrated toward the centre. The gradients in the spherically averaged star- and gas- content provided by the dynamical models are the primary information to load into the spectro-photometric code of @Piovan06b. They allow us to infer the amount of gas contained within a given radius or within a given aperture. We fix the total dimension of portion of the average model producing the IR flux at a diameter $D_{gal} = 25$ kpc, consistent with the mean galaxy size of the observed sample.
As the spectro-photometric code of @Piovan06b suited to describe early-type galaxies is written in spherical symmetry, we have to derive suitable spherical distributions for the density of stars and gas to be used into the model. The task is facilitated by the nearly spherical shape of the dynamical models. To this aim, we consider the sphere of radius $R_{gal}$ centred at the centre of mass of the stellar component. The sphere is then divided in a number of thin spherical shells whose derived average density of stars and gas is shown in Fig. \[Fit\_RHO\]. Even if the centre of mass of the star and gas distributions may not be exactly coincident, this not relevant here, so that the same coordinate centre can be used for both components.
In order to secure a smooth behaviour at the galaxy radius $R_{gal}$ the star and gas density profiles are represented by the law:
$$\rho _{i}=\rho _{0i}\left[ 1+\left( \frac{r}{r_{c}^{i}}\right)
^{2}\right] ^{-\gamma _{i}} \label{rhostar_ell}$$
where $``i"$ stands for $``stars"$ or $``gas "$, $r_{c}^{i}$ are the corresponding core radii. The above representation is more suited to our aims than the classical King law. The fits are shown in Fig. \[Fit\_RHO\] (solid lines). They are normalized in such a way that the integral over the galaxy volume corresponds to the amount of gas contained inside $R_{gal}$.
### Star formation rate.
In the dynamical models, the period of intense star formation, during which most of the star mass is built up, is confined within the first 3 to 4 Gyr. In Model A this is followed by a long tail of minimal stellar activity which continues forever. If this activity would be real, we would expect a background of young stars giving rise to a significant emission in the UV-optical region up to the present, which is not compatible with the observed spectra of typical early-type galaxies. As already pointed out by @Merlin06 this minimal stellar activity is an artefact of the poor mass-resolution for the baryonic component, in other words the low number of particles considered in the numerical simulations. To cope with this, we simply set to zero the star formation rate when only one or two star particles are involved. This is equivalent to cut the star formation rate for ages older than about 5 Gyr. The problem does not occur with model B simply because the last computed models is at 6 Gyr.
### Checking dynamical models against chemo-spectro-photometric models.
To this aim we plug the star formation history (SFH) of dynamical models into the chemical code of @Portinari98. The closed-box approximation is adopted. The total baryonic mass of the chemical models is the same as in the dynamical ones. Equally for the initial mass function of the stars composing each star particle: @Kroupa98 in our case. In Fig. \[BIG\_SF\] we show the results obtained by inserting the SFH of Model A into a classical chemical model with total baryonic mass $M_B$ equal to $1.6 \cdot 10^{11} M_{\odot}$. The top panels display the adopted SFH (left) and the gas metallicity of the chemical model, respectively. The bottom left panel shows the temporal variation of the star mass $M_{star}$ and gas mass $M_{gas}$, whereas the bottom right panel shows the ratios $M_{star}/M_B$ and $M_{gas}/M_B$ for both the dynamical (thin lines) and chemical model (thick lines). The agreement is very good thus confirming the internal consistency between the descriptions of the same object. We also show the amount of gas at $13$ Gyr contained in the whole galaxy for both the dynamical (heavy dots) and the classical chemical models (open circles) and the amount of gas contained inside $R_{gal}$ (open squares). Indeed there is little gas left over inside the 25 kpc radius region. Similarly in Fig. \[SMALL\_SF\], we show the results obtained inserting the SFH of Model B into a classical chemical model with total baryonic mass of $3.5 \cdot 10^{9} M_{\odot}$. The only difference is that the maximum age of the dynamical model is 5 Gyr. This cross checking of the models is particularly significant because: first, it secures that the results of the analytical models fairly reproduce those of the dynamical simulations as far as some important features are concerned; second, it secures that we can safely use the result of chemical models to prolong the evolutionary history of Model B up to the present; third, that we can safely apply the population synthesis technique of @Piovan06b.
Knowing the amount of gas, we need to specify the fraction of it in form of dust to finally be able to derive the whole SED from X to FIR and look for relationships between the luminosity in the X, B and FIR pass-bands we want to interpret. Our models, both semi-analytical and chemo-dynamical, are not suitable to describe the evolution of the compositions and abundances of *both* gas and dust phases. The relative proportions of the various components of the dust would require the detailed study of the evolution of the dusty environment and the complete information on the dust yields, as in the models of @Dwek98 [@Dwek05]. This would lead to a better and more physically sounded correlation between the composition of dust and the star formation and chemical enrichment history of the galaxy itself, however at the price of increasing the complexity and the uncertainty of the problem.
The key parameter to calculate the amount of dust is the dust-to-gas ratio, defined as $\delta =M_{d}/ M_{H}$, where $M_d$ and $M_{H}$ are the total dust and hydrogen mass, respectively. For the Milky Way and the galaxies of the Local Group, $\delta$ is estimated to vary from about $1/100$ to $1/500$ and typical values $\delta =
0.01$, $\delta =0.00288$ and $\delta =0.00184$ are used for the Milky Way (MW) and the Large and Small Magellanic Clouds (LMC and SMC). These dust-to-gas mass ratios describe a decreasing sequence, going from the MW to the LMC and SMC. Since these galaxies also describe a sequence of decreasing metallicity, a simple assumption is to hypothesize $\delta \varpropto Z$ in such a way to match the approximate results for MW, LMC and SMC: $\delta
=\delta_{\odot}\left(Z/Z_{\odot}\right)$. This relation simply implies that the higher is the metal content of a galaxy, the higher is the abundance of grains per $H$ atom. However, the metallicity difference does not only imply a difference in the absolute abundance of heavy elements in the dust, but also a difference in the composition pattern as a function of the star formation history @Dwek98 [@Dwek05]. Despite these uncertainties [@Devriendt99], the relation $\delta \varpropto
Z$ is often adopted to evaluate the amount of dust in galaxy models [e.g. @Silva98] by simply scaling the dust content adopted for the ISM of the MW to the metallicity under consideration.
The $1.6 \cdot 10^{11}M_{\odot}$ and $3.5 \cdot 10^{9}
M_{\odot}$ galaxy models reach an average metallicity of solar and slightly more than twice solar, respectively. To describe them we have adopted the description of @Piovan06a [@Piovan06b] where a model of dusty ISM taking into account different metallicities is built. The problem however remained unsettled for metallicities higher than the solar one, where relative proportions holding good for the MW average diffuse ISM model have been adopted and the amount of dust scaled with $\delta \varpropto Z$. Therefore, for the $1.6 \cdot 10^{11}M_{\odot}$ galaxy with solar metallicity the MW diffuse ISM model has been adopted $\left(\delta =\delta_{\odot}
\right)$, while for the $3.5 \cdot 10^{9} M_{\odot}$ model we followed the $\delta \varpropto Z$ relation, using the MW average pattern of dust composition.
The connection between the results of this model and the observed diagrams are discussed in the following section.
Discussion {#Discussion}
==========
Our data confirm and extend the previous relations existing between various tracers of the ISM in galaxies of different morphological types.
In the literature the relation found by @bregman between S$_{CO}$ and S$_{100}$ indicates a direct proportionality (slope=1) between the two fluxes and differs from that of @solomon, that exhibits a steeper gradient. Our relation (\[eqCO\_100\]) agrees quite well with the proportionality found by @bregman, the slope we found being equal to 1.06. The similarity between the two curves in Figure \[CO\_100\] is evident. This link derives, as described in the introduction, from the excitation of gas clouds by the currently forming stars and by the warming of the dust present in the galaxy.
Late-type galaxies
------------------
In late-type galaxies (t$>$Sb) our data show the existence of a linear relation between soft X-ray fluxes and other indicators of recent and current star formation, such as the B and FIR luminosity respectively (equations \[XBlate\] and \[XFIR\]). This is known since the first X-ray observations of large samples @fabbiano0 and this connection between B and X-ray luminosity in late type galaxies has been interpreted as due to the contribution of discrete X-ray sources, whose number is proportional to the quantity of already formed stars [@ciotti; @beuing]. The recent work of @fabbiano, that is able to resolve the single X-ray binaries in 14 galaxies, indicates that the X-ray luminosity produced by discrete sources is related to B luminosity by a similar relation, with an intercept value of -3.63, similar to our -3.85 of equation \[XBlate\].
In addition to the interstellar radiation, that is proportional to the number of already formed stars, the X-ray emission is produced also by HII regions, where there is an ongoing vigorous star formation [@david]. This latter contribution appears more evident in FIR light and may explain the existence of a similar linear relation between L$_X$ and L$_{FIR}$.
Early-type galaxies
-------------------
In early type galaxies the behaviour of these relations is quite different. For most of these galaxies, the star formation is exhausted and it may be present in a few of them, eventually fed by gas accretion phenomena. Different mechanisms have been suggested to explain the X-ray emission in this kind of galaxies. In particular the main ones are the thermal emission due to hot ISM and the emission generated by a relatively old population of end objects of stellar evolution, composed by Type I supernovae remnants and low-mass X-ray binaries not yet evolved. In particular, for the fainter galaxies the X-ray emission is compatible with discrete sources and seems to be dominated by compact accreting systems, while for the brighter objects the emission from hot diffuse gas still present in the galactic potential well is present as additional component [@beuing]. The number size of this population of relatively old objects is well represented by the total blue luminosity of the galaxy. For this reason the X-ray fluxes are still linked in early type galaxies with the total blue luminosity, representing the more recent part of the history of star formation in the galaxy.
In the FIR however, since the star formation in most of these systems is almost exhausted, mechanisms different from the emission from warm dust heated by the newly born stars predominate. The FIR emission comes from circumstellar dusty shells around AGB stars and from an interstellar medium due to the outflow of dusty gas from AGB and RGB stars, as it has been described in Sect. \[Modelling\].
The key point to interpret the observed trends is that we deal with an emission coming from a more or less small amount of dust distributed over all the galaxy and heated by an average interstellar radiation field due to all the stars of any age. The situation is quite different from what happens for instance in starburst galaxies where high optical depth dusty regions reprocess the light coming from newly born stars embedded in the parental environment. We can therefore conclude that in most of our early-type galaxies the mechanism of IR emission is not strictly related to the star formation and the link between the younger generations of stars and dust emission is lost. For these reasons one may expect that the soft X-ray luminosity in early type galaxies is traced by the total blue luminosity [*but not*]{} by the FIR luminosity. With the end of the star formation, the far infrared emission of these galaxies has faded out and an early type galaxy with the same $L_X$ of a late type will have a lower $L_{FIR}$. This could explain the location of the points in Fig. \[X2\] (right panel), on the left side of the linear relation.
To check if this interpretation is correct we try to apply the detailed chemo-dynamical spectrophotometric model described in the previous section, in such a way to estimate the luminosities produced by the stars in connection with the various phenomena present inside the galaxy, taking into account the contribution by dust as well. Since the theoretical model can not derive the $L_{X}$ luminosity, we proceed in the following way.
The luminosities $L_{B}$ and $L_{FIR}$ are directly derived from the model. Then, we assume that the X-ray production of these galaxies is proportional to $L_{B}$ according to our relation (\[XBearly\]). In this way we may estimate the expected X-ray flux and define a representative point in the $L_{FIR}$ vs $L_{X}$ plot.
We start considering two template models, in which all the parameters are fixed using the clues coming from the dynamical simulations of @Merlin06, as described in Sect. \[Modelling\]. The King profiles represented in Fig. \[Fit\_RHO\] are similar for all the components, with $\gamma_{stars} \simeq
\gamma_{gas} \simeq 1.5$ and $r_{c}^{stars} \simeq r_{c}^{gas}
\simeq 0.5$ Kpc, while the dimension of the galaxy is an average one corresponding to most of the galaxies available in the catalogue. The SFH is exactly the one obtained by the dynamical simulations. The two values of $L_{FIR}$ and $L_{X}$ obtained for the $3.5 \cdot
10^{9} M_{\odot}$ and $1.6 \cdot 10^{11} M_{\odot}$ baryonic mass models are plotted in Fig. \[LxLfirES\]. The more massive galaxy fits well into the region defined by the really observed galaxies, while we can notice as the model of smaller mass, even if falling up to the linear relationship as we could expect, belongs to a region not covered by the observed data. The calculated levels of emission $L_{X}$ and $L_{FIR}$ of this galaxy are very low and for this reason they belong to a region where we do not have enough observations. The weak $L_{FIR}$ emission of this galaxy can be explained by the dynamical evolution in which almost all the gas is consumed to form stars and the galactic winds are very efficient [see @Chiosi02 for more details about galactic wind in low mass galaxies]. Therefore, even if the trend of this galaxy is the expected one for early-type galaxies (the model stays above the linear relation), nothing safer can be said, because we lack observed data in that region of the diagram.
Much more interesting is the model of higher mass. The calculated luminosities of the model, with its exhaustion of the star formation, seem to agree well with the observations of early-type galaxies. However, the model needs to be checked against other possibilities, at the purpose to understand the way in which the various parameters of the model influence the spreading of early-type galaxies into the observational data. First of all we have to check the effect of the geometrical parameters and of the masses of stars/gas.
### The galactic radius
In fig. \[Raggi\] we show the model of $1.6 \cdot 10^{11}
M_{\odot}$ baryonic mass at varying the galactic radius, keeping the galactic center in the center of mass of the stellar component. The radii taken into account range from $6$ Kpc to $50$ kpc. All the other parameters are fixed. Four models are represented (filled circles) and connected by a continuous line and the smaller and bigger models are marked using an arrow. For larger radii we observe an increase of both $L_{FIR}$ and $L_{X}$, with a more emphasized increase in $L_{X}$. Since the density profile is unchanged, both the increases in luminosity are simply due to the bigger amount of material considered taking into account larger radii in the dynamical simulation. The stronger increase in $L_{X}$ than $L_{FIR}$ can be simply explained. $L_{X}$ is linearly related to $L_{B}$, that is directly connected to the stellar luminosity. The stellar component is more massive and more concentrated toward the centre than the gaseous one (Fig. \[RaggiViriali\], upper panel). It follows that at increasing radius we introduce into the models more stars and more gas, but the added amount of stars is bigger than the gaseous one, shifting $L_{B}$ (and then the linearly related $L_{X}$) more than $L_{FIR}$. Finally, we observe how, even taking into account the smallest radius of $6$ Kpc, it is not possible to move the theoretical point near the linear relation holding for spirals.
### The masses of stars and gas
We also investigated in Fig. \[Raggi\] what happens if we forget about the clues coming from dynamical simulations on the masses of stars and gas and we arbitrarily start varying the amounts of stars or gas, keeping all fixed. Filled diamonds represent the shift of the model of lowest radius if we are changing the mass of stars inside $R_{gal}$, going in fraction from $f_{*}=0.2$ to $f_{*}=1.0$, with respect to the total amount of stars in the dynamical model. The effect is simply to move the point along a line about parallel to the linear relation. A smaller amount of stars imply directly a lower luminosity $L_{B}$ (and therefore a lower $L_{X}$), but also a lower $L_{FIR}$, because the weaker radiation field makes dust cooler and shifts the peak of dust emission to wavelengths longer than $100 \mu m$, with the result of a smaller $L_{FIR}$. Finally, with open circles we show in Fig. \[Raggi\] five models obtained at fixed amount of stars and at varying the mass of gas (and therefore of dust) from $f_{d}=0.2$ to $f_{d}=1.0$, in fraction respect to the total amount of gas in the dynamical model. The effect of this huge increase of the mass of diffuse gas and dust (in the original model at $R_{gal} = 6 $Kpc only $0.03 \%$ of the gas is inside $R_{gal}$) is to shift the models straight toward the linear relation. It can be explained in the following way. Increasing the amount of diffuse gas/dust (with all the parameters fixed and the star formation exhausted) implies more absorption of the stellar radiation and therefore a smaller $L_{B}$ (and $L_{X}$). On the other side, $L_{FIR}$ remains almost unchanged or becomes smaller. The reason is that the strongly increased mass of dust makes the average stellar radiation field weaker and therefore the increased emission of dust (due to the bigger mass) peaks at wavelengths longer than $100 \mu m$, leaving $L_{FIR}$ almost unchanged. Even if in this way we can shift the model toward the linear relation, the situation is physically unrealistic, requiring a huge amount of gas/dust concentrated in the centre of an early-type galaxy with exhausted star formation, which is not commonly observed and also not predicted by dynamical models.
### The scale radii
Further geometrical parameters that must be examined are the scale radii $r_{c}^{i}$ of the King’s laws - eqn. (\[rhostar\_ell\]) - that describe the distribution of the stellar and gaseous components. The averaged profiles showed in Fig. \[Fit\_RHO\] and used for the models of Figs. \[LxLfirES\] and \[Raggi\] are both characterized by $r_{c}^{i} \simeq 0.5$, allowing for a concentrated amount of stars and gas in the inner regions. Keeping all the other parameters fixed, we investigated what happens if we allow for a uniform distribution of one or both the physical components. Three cases have been considered: a uniform distribution of gas keeping fixed the stellar one $\left(r_{c}^{gas}
\rightarrow \infty, r_{c}^{stars} \simeq 0.5 \right)$, a uniform distribution of stars keeping fixed the gaseous one $\left(r_{c}^{stars} \rightarrow \infty, r_{c}^{gas} \simeq 0.5
\right)$ and, finally, a uniform distribution of both the components $\left(r_{c}^{stars} \rightarrow \infty, r_{c}^{gas} \simeq \infty
\right)$. The results are shown in Fig. \[Profili\], for two radii of the galaxy model, $R_{gal}=6$ Kpc and $R_{gal}=20$ Kpc, respectively. The three different distributions give a similar result: a weaker $L_{FIR}$, shifting the point to the left, and a slightly higher $L_{X}$.
This can be explained in the following way: for $r_{c}^{stars}$ and $r_{c}^{gas}$ both $\simeq 0.5$, the diffuse ISM and the stars are both concentrated in the inner region of the galaxy with a density of stars/gas of many order of magnitude bigger than the outer regions. This is the best condition to produce high $L_{FIR}$, because we have that the regions of higher density of dust are the same in which there is also the higher average radiation field heating dust. The spatial distribution of the ISM favors the interaction with the stellar radiation. When we destroy this coupling between stellar emission and density of gas, as we do allowing for a uniform distribution of gas or stars or both, the emission in the $L_{FIR}$ becomes weaker. The weakening of the dusty emission is stronger for the bigger radius of $20$ Kpc because in all the three cases one or both the components are distributed over a huge galactic volume and we have low density of gas eventually coupled with weak radiation field. For the $6$ Kpc model, even if the coupling in the central regions is destroyed, the galaxy is small enough to keep a good level of $L_{FIR}$, even when the matter is equally distributed across all the galaxy volume.
### The star formation history
Last and main point to be examined is how varying the star formation history affects the position of the galaxies into the $L_{FIR}$ vs $L_{X}$ plot. In Figs. \[Raggi\] and \[Profili\] the galaxies of different morphological type form a sequence that, going from systems in which the star formation got exhausted long ago to systems in which star formation is still active, moves toward the linear relation and suggests the key role played by the star formation. First of all we calculate the $L_{FIR}$ and $L_{X}$ obtained by the SEDs and the models by @Piovan06b of real galaxies of the local universe: three spiral galaxies $(M100, M51$ and $NGC6946)$ and two starburst galaxies $(Arp220$ and $M82)$. The key point is that the SFHs of these galaxies allow us to cover a good number of different star formation histories. All these SFHs, unlike the ones of the ellipticals obtained by dynamical simulations, never end and in the case of the two starbursters a strong burst of star formation is added in the last millions of years. A huge amount of $L_{FIR}$ comes therefore from the young and deeply obscured region of star formation and not only from the diffuse component.
The results, presented in Fig. \[RealGalaxies\], show that the three models of spirals stay near the linear relation, while the two starbursters stay below the line, with the model of $Arp220$, powered by a huge burst of star formation falling well below the linear relation. The stronger is the emission coming from the regions of star formation and the bigger is the shift toward higher $L_{FIR}$ and lower $L_{X}$ (due to the lower $L_{B}$). The results obtained from the models are quite similar to the observational data: for $M100$ we get $(L_{FIR},L_{X})=(10.28,7.29)$ with the observations giving $(10.37,7.01)$, for $Arp220$ we have $(L_{FIR},L_{X})=(11.92,7.17)$ compared with $(11.99,7.60)$ and for $M82$ we get $(L_{FIR},L_{X})=(10.15,6.45)$ against $(9.79,6.31)$. However, these galaxy models, even if they well represent real galaxies, differ in many parameters from the early-type galaxy model of $1.6 \cdot 10^{11} M_{\odot}$, like geometry and mass. These parameters, together with the SFH, obviously concur to determine the position of the models into the $L_{FIR}$ vs $L_{X}$ plot. To isolate the effect of the SFH, we first re-calculated the SFHs of the above five theoretical models, rescaled to the mass of $1.6 \cdot 10^{11} M_{\odot}$ of the early-type galaxy model. In Fig. \[SFH\] we can see four of the five SFH obtained. Second, we fixed all the geometrical parameters to the same values used for the average model of $1.6 \cdot 10^{11} M_{\odot}$ early type galaxy. The additional parameters, that is the escaping time of young stars from parental molecular clouds, the library of SEDs of young dusty regions and the mass of gas in the diffuse and molecular component, are fixed to the values used in @Piovan06b for spirals and starbursters as appropriate.
In Fig. \[varyingSFHs\] we finally show the results obtained as a function of the SFH of the galaxy of $1.6 \cdot 10^{11}
M_{\odot}$, keeping all the other parameters fixed. It is interesting to observe that since now the star formation never ends and the galactic wind is not included, the classical semi-analytical chemical evolution can be much more safely coupled to the spectro-photometric code. The effect of varying the SFH at fixed mass is to enhance the $L_{FIR}$, keeping almost fixed the $L_{X}$ and shifting the points toward the linear relation at higher infrared luminosities. This is ultimately due to the strong and efficient reprocessing of the light coming from very young stars, occurring into the dusty star-forming regions. As a consequence of this, models with starburst-like SFHs shift, as expected, toward higher $L_{FIR}$ luminosity than models with spiral-like SFH, because of the stronger star formation and therefore emission coming from young dusty regions. This can be also understood if we look in detail at the relative contribution to $L_{FIR}$ coming from the regions of star formation (let us define it $f_{MCs})$ and represent it as usual in $\log(L_{FIR}/L_{\odot}$) and from the diffuse interstellar medium $(f_{ISM})$. We get the following values: ($f_{SFR}=10.15,
f_{ISM}=10.48$), ($f_{SFR}=10.02, f_{ISM}=10.45$), ($f_{SFR}=9.98,
f_{ISM}=10.37$) for the three models with spiral-like SFHs, while we have ($f_{SFR}=10.97, f_{ISM}=10.47$) and ($f_{SFR}=11.93,
f_{ISM}=10.56$) for the models with starburst-like SFHs. The stronger is the contribution from star forming regions, the higher is $L_{FIR}$ keeping $L_{B}$ (and $L_{X}$) almost unchanged. Models slightly dominated by the ISM contribution, but with a significant contribution coming from obscured newly born stars are more suitable to agree with the linear relation of spirals.
It is worth noticing that in Fig. \[varyingSFHs\] we show both the results obtained applying the early-type linear relation between $L_{X}$ and $L_{B}$ - eqn. (\[XBearly\]) - and the late-type one - eqn. (\[XBlate\]). Since, however, the SFHs used (see Fig. \[SFH\]) are typical of late type galaxies (or starbursters), it’s more physically sounded to apply eqn. (\[XBlate\]) to obtain the $L_{X}$ luminosity. As last point we calculated also a sequence of models in which one of the SFHs of the spirals has been chosen (namely the one of NGC$6946$) with all the parameters fixed and only the mass is varied. As we see from Fig \[varyingSFHs\] the effect of varying the mass is to shift the object in diagonal almost along the relation. This is simply explained by the smaller amounts of stars/gas emitting radiation.
Conclusions
===========
We have been able to describe the relations existing in a galaxy between the various tracers of the ISM and to fix the coefficients of the relations existing between FIR, B and X-ray luminosity, both for early-type and late-type galaxies.
The large set of data we used allowed us to redefine more clearly the relation existing between the CO and the 100 $\mu$m fluxes. We found that the relation, first obtained by @bregman for early type galaxies, is valid also for late type galaxies. In these galaxies, the X-ray flux appears linked also to B and FIR emissions.
The only relation lacking from observations, i.e. the one between L$_X$ and L$_{FIR}$ has been studied by the use of the most recent chemo-dynamical models coupled with dusty evolutionary population synthesis.
The calculated luminosities of the models seem to confirm our hypothesis about a connection between the exhaustion of the star formation and the “migration” of the early type galaxies above the linear relation in the L$_X$ vs L$_{FIR}$ plot. In the frame of our assumptions, we may therefore conclude that the prediction of our dusty chemo-dynamical models of galaxy evolution is consistent with the observed lack of a direct relation between $L_{X}$ and $L_{FIR}$ for early type galaxies and is due to the different mechanisms of production of FIR light in galaxies where the active star formation is no longer active. In most of our early-type galaxies the mechanism of IR emission is no more strictly related to the ongoing star formation and to the reprocessing of the radiation in the dense regions where new stars are born. The FIR emission comes therefore most likely from circumstellar dusty shells around AGB stars and from an interstellar diffuse medium due to the outflow of dusty gas from AGB and RGB stars.
Finally, we can summarize that: (i) the SFH of the galaxies seems therefore to have the stronger effect on the position of early-type galaxies in the L$_X$ vs L$_{FIR}$ plot; (ii) other parameters, like the radius of the galaxy and the scale radii of stars and gas, play a secondary role, even if they can significantly contribute to the scatter of the models in the region above the linear relation; (iii) the mass is the main parameter explaining the scatter of the points along the linear relation.
This research has been partially funded by the University of Padua with Funds ex 60% 2005. We acknowledge Prof. C. Chiosi for useful discussions on theoretical subjects of this paper. L. Piovan is pleased to acknowledge the hospitality and stimulating environment provided by Max-Planck-Institut für Astrophysik in Garching where part of the work described in this paper has been made during his visit as EARA fellow on leave from the Department of Astronomy of the Padua University. We also thank the Referee for the detailed and useful comments about this topic.
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abstract: 'We present the observational results from the 43-GHz Very Long Baseline Array (VLBA) observations of 124 compact radio-loud active galactic nuclei (AGNs) that were conducted between 2014 November and 2016 May. The typical dimensions of the restoring beam in each image are about 0.5 mas $\times$ 0.2 mas. The highest resolution of 0.2 mas corresponds to a physical size of 0.02 pc for the lowest redshift source in the sample. The 43-GHz very long baseline interferometry (VLBI) images of 97 AGNs are presented for the first time. We study the source compactness on mas and sub-mas scales, and suggest that 95 sources in our sample are suitable for future space VLBI observations. By analyzing our data supplemented with other VLBA AGN surveys from literature, we find that the core brightness temperature increases with increasing frequency below a break frequency $\sim$7 GHz, and decreases between $\sim$7–240 GHz but increases again above 240 GHz in the rest frame of the sources. This indicates that the synchrotron opacity changes from optically thick to thin. We also find a strong statistical correlation between radio and $\gamma$-ray flux densities. Our correlation is tighter than those in literature derived from lower-frequency VLBI data, suggesting that the $\gamma$-ray emission is produced more co-spatially with the 43-GHz VLBA core emission. This correlation can also be extrapolated to the un-beamed AGN population, implying that a universal $\gamma$-ray production mechanism might be at work for all types of AGNs.'
author:
- 'X.-P. Cheng, T. An, S. Frey, X.-Y. Hong, X. He, K. I. Kellermann, M. L. Lister, B.-Q. Lao, X.-F. Li, P. Mohan, J. Yang, X.-C. Wu, Z.-L. Zhang, Y.-K. Zhang, W. Zhao'
title: 'Compact Bright Radio-loud AGNs – III. A Large VLBA Survey at 43 GHz'
---
Introduction {#intro}
============
Active galactic nuclei (AGNs) host the most powerful natural particle accelerators, producing also high-energy cosmic rays and neutrino emission. Compact jets emanating from around the central supermassive black holes (SMBHs) in radio-active AGNs are prominent sources in the widest range of electromagnetic wavebands, from radio wavelengths to $\gamma$-rays. AGN jets are in the forefront of modern multi-messenger astrophysical research . Surveys of AGNs supply a wealth of information for advancing our understanding of jet physics. Of particular importance are high-resolution radio interferometric surveys that zoom directly into the parsec (pc) and sub-pc scale regions of AGN jets which are closely related to the extreme astrophysical phenomena.
Recently, the source of high-energy cosmic neutrinos ($\sim$300 TeV) detected by IceCube on 2017 September 22 was identified as a distant $\gamma$-ray blazar, TXS 0506+056, which is an intermediate synchrotron-peaked BL Lac object at a redshift of $z=0.34$ [@2018arXiv181107439H; @2018Sci...361.1378Ia; @2018Sci...361..147Ib]. This first confirmed that blazars are sources of high-energy astrophysical-origin neutrinos, opening a new window of studying the Universe using the unobstructed messenger neutrino [@2018Sci...361..147Ib]. It is widely speculated that high-energy cosmic rays are accelerated in the magnetic fields in the innermost jets of blazars. The generated cosmic rays interact with nearby gas, photons or other cosmic rays producing charged mesons that decay into high-energy neutrinos, $\gamma$-rays and other particles [@1960ARNPS..10....1R].
Very long baseline interferometry (VLBI) is an elegant observing technique which provides the highest angular resolution. Blazars are among the most powerful objects in the Universe. Images from high-frequency high-resolution VLBI surveys of blazars are essential to test jet models [e.g., @1995PNAS...9211439M], and to investigate the innermost region of compact jets where the acceleration and collimation of the relativistic plasma flow takes place [@1969ApJ...155L..71K; @2004ApJ...613..794G; @2016AJ....152...12L] and where the high-energy neutrinos and $\gamma$-rays are produced [@2019arXiv191201743R].
The Large Area Telescope on-board the [*Fermi*]{} $\gamma$-ray space telescope ([*Fermi*]{}-LAT) has already detected 2683 AGNs, listed in the fourth [*Fermi*]{}-LAT catalog [@2019F]. Most of the sources are classified as blazars (comprising BL Lacs and optically violently variable quasars), which show prominent emission over a broad range of electromagnetic radiation, from radio to $\gamma$-ray energies. The spatial localization of the region where the $\gamma$-rays are emitted, $\gamma$-ray radiation mechanism in blazars, and correlation between radio and $\gamma$-rays are key questions to understand the blazar activity at multiple bands. Synchrotron radiation is responsible for the bump at radio to X-ray frequencies in the $\log \nu F_{\nu}$ versus $\log \nu$ spectral energy distribution (SED). Another dominant mechanism responsible for the bump from X-ray to TeV $\gamma$-ray regions is inverse-Compton (IC) radiation. Two possible scenarios are attributed to this: synchrotron self-Compton (SSC) radiation which results from IC scattering of synchrotron radiation by the same relativistic electrons that produced the synchrotron radiation, and external inverse-Compton (EC) radiation where the photons available for IC scattering in the inner jet are seeded from external sources such as the broad-line region [@1994ApJ...421..153S] and the accretion disk [@1993ApJ...416..458D]. One method to verify the process of SSC radiation is using the correlation between the mm-wavelength radio luminosity of the core and its $\gamma$-ray luminosity.
Some VLBI observations of blazars at high frequencies [e.g., @2008AJ....136..159L; @2010ApJS..189....1A; @2010ApJ...723.1150P] have revealed complex inner jet morphology and kinematics at sub-milliarcsecond (sub-mas) scales. New statistical studies show that the jets are accelerated in the sub-pc regions from the central engine [@2016ApJ...826..135L]. However, non-ballistic motion of the jet is found within a few parsec, e.g., NRAO 150 from observations with the Global mm-VLBI Array (GMVA) and the Very Long Baseline Array (VLBA) at 86 and 43 GHz, respectively. Thus, high-resolution VLBI observations and large complete surveys at high frequencies are important to study the jet structures and kinematics at sub-pc scales.
Ground-based VLBI observations show typical apparent core brightness temperature ($T_{\rm b}$) in the range of $10^{11}- 10^{12}$ K [e.g., @1996AJ....111.2174M]. However, the Japanese VLBI Space Observatory Programme [VSOP, @1998Sci...281.1825H] detected some AGNs with core brightness temperature higher than $10^{12}$ K [@2001ApJ...549L..55T; @2008ApJS..175..314D]. The highest brightness temperature that has been measured with VSOP is $5.8 \times 10^{13}$ K for AO0235+164 at 5 GHz [@2000PASJ...52..975F]. The maximum brightness temperature is determined by the longest baseline length, regardless of the observing frequency [@2005AJ....130.2473K]. The Russia-led [*RadioAstron*]{} mission [@2013ARep...57..153K] detected core brightness temperatures even higher than $10^{13}$ K [e.g., @2016ApJ...820L...9K; @2018MNRAS.474.3523P; @2018MNRAS.475.4994K], about two orders of magnitude above the equipartition [@1994ApJ...426...51R] and inverse Compton limits [@1969ApJ...155L..71K]. The interpretation of extremely high brightness temperatures is a challenge to AGN physics. Moreover, @2008AJ....136..159L showed that the core brightness temperature will be small at lower frequencies due to opacity effect between 2 GHz and 15 GHz. However, their 86-GHz core brightness temperatures are significantly lower than those measured at 15 GHz. Therefore 43-GHz VLBI observations, straddling 15 and 86 GHz, are crucial to explore the maximum core brightness temperature and its corresponding frequency and to study core brightness temperature distribution along the jet which can be used to test models of the inner jet [@1995PNAS...9211439M].
Despite more than 30 years of research of radio-loud AGNs at high frequencies with VLBI, only 163 sources were observed and imaged at 43 GHz [@2001ApJ...562..208L; @2010AJ....139.1713C; @2017ApJ...846...98J; @2018ApJS..234...17C] and 263 AGNs have been successfully imaged at 86 GHz . Most of these observations were carried out in snapshot mode, and high-quality images are scarce. Even fewer sources have multi-epoch imaging observations. In addition, the resolution is still not enough to explore the innermost jet emission region where the jet is formed and accelerated. Future space VLBI missions would leap forward in the direction of improving both the resolution and the imaging capability. A systematic survey of a large AGN sample is necessary to make progress of the future space VLBI proposals.
In this paper, we present the results of 43-GHz VLBA imaging of 124 AGNs and a statistical study of their compactness, core brightness temperatures, and the correlation between the radio and $\gamma$-ray emissions. The observations and the data reduction process are described in Sect. \[observ\]. We present the main results and comment on selected individual sources in Sect. \[results\]. Section \[discussion\] contains the discussion of the properties of our sample, and a summary appears in Sect. \[summary\]. Throughout this paper, the standard $\Lambda$CDM cosmological model with H$_{0}$ = 73 kms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm M} = 0.27$, and $\Omega_{\Lambda} = 0.73$ is adopted.
VLBA Observations and Data Reduction {#observ}
====================================
Sample Selection
----------------
As mentioned in Section \[intro\], a large sample of compact and bright AGNs is important for the successful detection and imaging to support the scientific operation of ground-based mm-wavelength and space VLBI. To enlarge and complete the existing database, we selected and observed 134 AGNs. The selection criteria are described in details in Paper I and Paper II [@2018ApJS..234...17C]. Ten bright AGNs were selected from our sample and observed at 43 and 86 GHz with long integration time and good $(u,v)$ coverage, in order to further study the inner jet structures [@2018ApJS..234...17C]. The remaining 124 sources consist of 97 quasars, 15 BL Lac objects, 6 radio galaxies, and 6 objects with no optical identification. We carried out an imaging survey at 43 GHz using the VLBA. Combined with the previous 43-GHz VLBI observations [@1992vob..conf..205K; @2001ApJ...562..208L; @2002ApJ...577...85M; @2010AJ....139.1695L], we obtain a large sample containing 260 AGNs to study brightness temperatures of AGN cores and to explore the jet acceleration mechanism. Table \[tab:sample\] lists general information of the sample: IAU source name (B1950.0), commonly used other source name, observation session, right ascension (RA), declination (Dec), redshift, optical classification, an indication of comments on the source in Sect. \[comments\], and $\gamma$-ray photon flux (where available). The photon flux of the $\gamma$-ray bright AGNs are provided by the [*Fermi*]{}-LAT in the 100 MeV – 300 GeV energy range [@2015ApJS..218...23A].
Observations
------------
To ensure the success of the observations, we split the sources into two sub-samples, based on the source flux densities measured at lower radio frequencies. The first sub-sample consists of 40 bright sources for which we acquired good images, while the second sub-sample includes 84 relatively weaker sources which were observed less thoroughly. These sub-samples were observed in the corresponding sessions B and C, as listed in Column 3 of Table \[tab:sample\]. Millimeter-wavelength VLBI observations are easily affected by tropospheric fluctuations, so the observing periods were carefully chosen to have exceptionally favorable weather conditions. The observations were spread over a period of almost one year from 2015 June to 2016 April as shown in Column 3 of Table \[tab:log\] and in Column 2 of Table \[tab:image\]. Each of the 40 sources in session B was observed for four scans of 7 min duration. Each source in session C was observed for two 7-min scans separated by long time spans in order to obtain relatively uniform $(u, v)$ coverage. Table \[tab:log\] summarizes the observation setups. All ten VLBA antennas were used for the 43-GHz observations. At the beginning and the end of each observation, calibrator scans were scheduled on some of these bright quasars: 3C84, 3C273, 3C279, 3C345, 3C454.3, 4C39.25, 1749+096, or BL Lac.
The data were recorded with 2-bit sampling at an aggregate data rate of 2 Gbps, using 8 intermediate frequency (IF) channels of 32 MHz bandwidth each. The observations were made in both left and right circular polarizations. One scan on fringe finder sources every 1.5 h was used to check the recording, pointing, and calibration. Figure \[uv\] shows a typical $(u,v)$ coverage for sources observed in the first (session B) and second (session C) samples in the left and right panels, respectively. The raw VLBA data were correlated using the DiFX software correlator [@2007PASP..119..318D; @2011PASP..123..275D] at the Array Operations Center in Socorro (New Mexico, U.S.), with 2 s averaging time, 128 frequency channels per IF, and uniform weighting. The correlated data were transferred to computer clusters in the China SKA Regional Centre prototype [@2019NatAs...3.1030A] where the calibration and imaging analysis were carried out.
Data Reduction
--------------
We processed the data in the standard way with the U.S. National Radio Astronomy Observatory (NRAO) Astronomical Imaging Processing Software ([AIPS]{}) package [@2003ASSL..285..109G]. First we loaded the data to AIPS by the task [fitld]{}, and flagged the bad data points (typically due to inclement weather) before proceeding further. We selected Pie Town (PT) as the reference antenna for most of the data. Fort Davis (FD) was used as the alternative reference telescope. In the amplitude calibration, we first removed the sampler bias with the task [accor]{}, then calibrated the correlator output with [apcal]{} and fitted for the ionospheric opacity correction using weather information, antenna system temperature and gain curve tables. Then we fringe-fitted a short (generally 1 min) scan of calibrator source data to determine the phase and single-band delay offsets and applied the solutions to the calibration table. After removing instrumental phase offsets from each IF, we performed a global fringe-fit using the task [fring]{} by combining all IFs to determine the frequency- and time-dependent phase corrections for each antenna and removed them from the data. To avoid false detections, fringe-fitting was done with a minimum signal-to-noise ratio (SNR) of 5. After fringe-fitting, the solutions for the sources were applied to their own data by linear phase connection using rates to resolve phase ambiguities. In the final step, we used the task [bpass]{} to calibrate the bandpass shapes by fitting a short data scan on a calibrator, and applied the solutions to all data. Then we made single-source calibrated data sets with [splat]{}/[split]{} and used the task [fittp]{} to export the calibrated visibility data files to the external work space in the clusters. These single source data files were then imported in the [Difmap]{} program [@1997ASPC..125...77S] to carry out self-calibration and imaging. A few runs of phase-only self-calibration were made to eliminate the residual phase errors. Amplitude self-calibration was performed with the amplitude normalization when the [clean]{} models reached an integrated flux density close to the correlated amplitude on the shortest baselines. The determined overall telescope gain correction factors were found to be small, typically within 10%, in agreement with other similar 43- and 86-GHz VLBA surveys .
Model Fitting
-------------
We used a number of circular Gaussian components to model the brightness distribution structure of each source in the visibility domain in [Difmap]{}. Typically 1 to 6 components were used to represent the detected features of the source structure. The radio core is identified as the bright and compact component at the apparent jet base. The fitted parameters are cataloged in Table \[model\]. The sizes of these compact jet components are usually smaller than the synthesized beam. The minimum resolvable size of a component in a VLBI observation is given by @2005astro.ph..3225L: $$d_{\rm lim}=\frac{\rm 2^{\rm 1+\rm \beta/2}}{\rm \pi}[\rm \pi \rm B_{\rm maj}B_{\rm min} \ln2 \ln\frac{(\rm S/N)}{(\rm S/N)-1}]^{\rm 1/2},$$ where $B_{\rm maj}$ and $B_{\rm min}$ are the major and minor axes of the restoring beam, respectively (full width at half-maximum, FWHM), S/N is the signal-to-noise ratio, and $\beta$ is the weighting function, which is 0 for natural weighting or 2 for uniform weighting. We took $d_{\rm lim}$ as the upper limit for the components with the fitted size $d < d_{\rm lim}$.
Results
=======
VLBA Images
-----------
Figure \[image-1\] shows contour plots of the final naturally weighted total intensity images for all the 124 sources. For 97 sources, the survey provides the first-ever VLBI image made at 43 GHz, extending the existing database by about 60%. Most of the sources shown here have secondary features or have complex structures. For these objects, we give brief comments on the characteristics of the radio structure, comparing our observations with other lower-frequency VLBI observations from the literature as discussed in the next subsection. The typical image size is 3 mas $\times$ 3 mas and the dimensions of the restoring beam in each image are 0.5 mas $\times$ 0.2 mas. The elliptical Gaussian restoring beam size is indicated in the bottom left corner of the maps in Fig. \[image-1\]. The peak intensity and the rms noise level are given in the bottom right corner of the maps. The lowest contour represents 3 times the off-source image noise level, and the contours are drawn at $-1$, 1, 2, 4, ..., $2^{\rm n}$ times the lowest level. Table \[tab:image\] provides the parameters relevant to the images: source name, observation date, beam size, integrated flux density, peak intensity, off-source rms noise in the [clean]{} image, correlated flux density on the shortest baselines, the length of the shortest baseline, correlated flux density on the longest baselines, and the length of the longest baseline.
Among the most important parameters derived from high-frequency VLBI surveys are the source compactness and core brightness temperature. Blazars often shows a one-sided core–jet structure, and the core is usually the most compact and bright component at one end of the jet. Eight other sources in our sample are identified as compact symmetric objects [CSOs, @2012ApJ...760...77A] or gigahertz-peaked spectrum sources: 0738+313, 0742+103, 0743$-$006, 1435+638, 2021+614, 2126$-$158, 2134+004, and 2209+236. All these sources, except 2021+614, are GPS sources, which show a sharp low-frequency spectral cutoff near 1 GHz. The exceptional source 2021+614 is a well-known CSO [@2003PASA...20...69P], but we are unable to identify the core component from our image. Comments on selected sources, including the designation of the core we used for calculating the core brightness temperature and the reliability of faint features in the images are given in Sec. \[comments\]. Table \[model\] lists the model-fitting parameters: source name, observation epoch, component identifier, model flux density, peak intensity, angular separation from the core, size of the components, position angle of the components.
Parsec-scale Morphology
-----------------------
The sources in our sample can be divided into four basic classes based on their morphology: simple core, one-sided core–jet, compact symmetric object, and complex structure. Eight sources (0847$-$120, 1049+215, 1257+519, 1329$-$049, 1417+385, 1657$-$261, 2325+093, 2342$-$161) have a single core component. One hundred and twelve sources show a core and a one-sided jet structure. However, 8 sources (0113$-$118, 0221+067, 0708+506, 0736+017, 0805$-$077, 1124$-$186, 1219+044, 2227$-$088) have faint extended radio emission and for 5 sources the cores (0605$-$085, 1036+054, 1045$-$188, 2021+317, 2223+210) are not the brightest but the most compact among the fitted components. Three sources (0241+622, 0743$-$006, 2021+614) show compact symmetric jet structure. We detect a component exceeding 7$\sigma$ noise level in the counter-jet direction in 0241+622. 0743$-$006 is GPS source that shows a triple structure. 2021+614 is a well-known CSO, showing two-sided structure. One source (0354+559) shows a complex structure, as shown in Sect. \[comments\].
Given our selection criteria, all sources have VLBI images at lower frequencies, at least at 2.3 and 8.4 GHz. Almost all sources also have radio images in literature showing their kpc-scale structure based on Very Large Array (VLA) observations. By comparing the radio structures from pc to kpc scales, we find that there are 6 sources showing completely oppositely-directed jet structures on mas and arcsec scales, and 23 sources display a relatively large change in the apparent jet direction.
Comments on Selected Individual Sources {#comments}
---------------------------------------
0048$-$071 (OB $-$082): Our 43-GHz image shows a single-sided jet to the northwest, in an opposite direction of the kpc-scale lobe (Kharb et al., in prep.)[^1].
0106$+$013 (4C $+$01.02): Our image shows that the innermost jet structure ($<1$ mas) is along the east–west direction (in a position angle $\sim -110\degr$), then it bends towards the southwest ($\sim -140\degr$) at about 2 mas from the core. This is in agreement with the low-frequency VLBI images [@2009AJ....138.1874L].
0113$-$118: This source is very compact at 43 GHz, we only detect a core and a faint extension within 1 mas. However, it was not detected on space–ground baselines in the 5-GHz VSOP AGN Survey [@2008ApJS..175..314D].
0119$+$041 (OC $+$033): The brightness temperature of the core is $1.6 \times 10^{10}$ K at 43 GHz, the derived Doppler factor is below unity which is consistent with slow jet motion found in the literature [@2009AJ....138.1874L; @2012ApJ...758...84P]. Neither the [*Compton*]{} Gamma Ray Observatory Energetic Gamma Ray Experiment Telescope (EGRET) nor [*Fermi*]{}/LAT has detected this source. The 15-GHz light curve from the Owens Valley Radio Observatory (OVRO) 40-m radio telescope shows that the flux density does not have significant variation from 2008 to mid-2017 [@2011ApJS..194...29R]. The VLBI structure (the core and the eastern jet component) does not show noticeable change [@2009AJ....138.1874L; @2012ApJ...758...84P]. The overall radio spectrum peaks at about 7 GHz. All these pieces of evidence imply that this is a GPS source, rather than a blazar.
0122$-$003 (UM 321): Our image shows a compact core and three jet components along a straight line to the west. It is in good agreement with the 5- and 15-GHz VLBA images [@2000ApJS..131...95F; @2013AJ....146..120L].
0130$-$171 (OC $-$150): The compact core is 0.08 mas in size. We detected a series of jet components extending to the southwest.
0149$+$218: This source shows a compact structure at 43 GHz. A relatively weak component extending to the north is detected.
0202$-$172: The jet shows a centrally symmetric S-shaped morphology within 5 mas. An extended feature is located at $\sim$2 mas from the core, sitting in the gap between the inner jet and the outer 3 mas jet.
0208$+$106 : The jet points to southeast up to 1 mas from the core, then it bends towards northeast seen in the 15-GHz VLBA image obtained in the Monitoring Of Jets in Active galactic nuclei with VLBA Experiments (MOJAVE) survey [@2009AJ....138.1874L], with a 70$\degr$ change in position angle. The outer ($>1$ mas) bent jet is diffuse at 43 GHz.
0221$+$067 (4C $+$06.11): The source shows a compact core–jet structure within 1 mas. Our high dynamic range image detected two weaker jet components to the west.
0224$+$671 (4C $+$67.05): The core is 0.13 mas in size. A jet component to the north is detected and has the similar flux density with the core at 43 GHz.
0229$+$131 (4C $+$13.14): The source displays considerable emission on both sides of the core at arcsec scales [@1995AJ....109.1555P]. Our image shows a bright core and an inner jet pointing to the northeast which is consistent with the high-resolution 5-GHz VSOP image [@2008ApJS..175..314D].
0239$+$108 (OD $+$166): It shows a compact core–jet structure. According to the total flux density light curve from the 15-GHz OVRO monitoring program [@2011ApJS..194...29R], our 43-GHz observation was made in a fading phase of the source.
0241$+$622 : This is a low-redshift ($z = 0.045$) Seyfert 1.2 galaxy . The eastern jet component is consistent with the 15-GHz VLBA image [@2009AJ....138.1874L]. However, there is another component appearing on the opposite (western) side of the core with an intensity in excess of 7$\sigma$ image noise, indicating a reliable detection. A detailed study of this component would require future high-resolution and high-sensitivity VLBI observations.
0306$+$102 (OE $+$110): The source is very compact in lower-frequency VLBI images. A faint radio emission is detected in the northeast.
0309$+$411 (NRAO 128): This is a strongly core-dominated broad-line radio galaxy showing core and double lobes morphology . Our 43 GHz VLBA image reveals a prominent core and a straight jet extending to 1 mas, aligning with the brighter and advancing kpc-scale jet.
0354$+$559: The source shows a complex jet in low-frequency VLBI images [@2003AJ....126.2562F]. In our 43-GHz image, the jet points to the northwest and then bends to the southwest within 1 mas. The source has rich structure even at mas and probably sub-mas scale, calling for a detailed study.
0400$+$258 (CTD 026): The core size is 0.21 mas. The jet extends to the southeast up to about 2 mas; this is also seen by @2008ApJS..175..314D. Further out, the jet becomes diffuse and bends to northeast, as seen only at larger scales with lower-frequency VLBI imaging [e.g., @2000ApJS..131...95F; @2009AJ....138.1874L].
0403$-$132 (OF $-$105): Our 43-GHz VLBA image shows a very compact core and faint jet emission extending to the southeast. @2007ApJS..171..376C detected an unresolved core and radio emission to the southwest. The VLBI data at 2.3 and 15 GHz exhibit a bright core and jet emission extending to the southern direction [@2000ApJS..131...95F; @2009AJ....138.1874L].
0405$-$123 (OF $-$109): This source is the second Seyfert 1.2 galaxy in our sample . The VLA image shows two hot spots in the north–south direction, and only the one in the northern lobe was detected in X-rays and optical [@2004ApJ...608..698S]. The lower-frequency VLBI images show a core–jet structure extending $\sim$30 mas to the north. Our image shows a resolved structure within 2 mas, with the south component corresponding to the core.
0507$+$179: identified it as a BL Lac object. The core brightness temperature is the maximum in our sample.
0529$+$075 (OG 050): Our image shows the inner jet extending to the northwest within 3 mas. However, the kpc-scale structure is pointing to the opposite direction [@2007ApJS..171..376C].
0605$-$085 (OC $-$010): The most important concern for this source is the core identification, considering that our observation took place during a flare [@2011ApJS..194...29R]. Two components along the east–west direction have equal brightness. Referring to the 15-GHz VLBA image [@2009AJ....138.1874L], we assume that the most compact and upstream component, i.e., the western component, is the core, even if it is not the brightest one. Further high-frequency VLBI observations are needed to confirm this.
0648$-$165: The source shows a large jet bending from the northwest as indicated by our image to the west-southwest which can be better seen in lower-frequency VLBI images
0657$+$172: The jet direction of the source at pc scale in our image (west-northwest) is different from that seen in low-frequency (2.3- and 8.6-GHz) VLBI images . Based also on radio spectrum data[^2], we suggest this source is a GPS or high-frequency peaker (HFP) candidate.
0723$-$008: Our image shows a jet pointing to the northeast at $\sim 35\degr$ position angle, in a good agreement with the 15-GHz VLBA image [@2009AJ....138.1874L].
0736$+$017 (OI $+$061): We only detect a compact core and a new weak jet component to the northeast at a distance of 0.41 mas.
0738$+$313 (OI $+$363): This is a GPS quasar . Our 43-GHz VLBA image shows a core–jet structure, consistently with the 15 GHz-image . Although the position of the radio core in this AGN is uncertain, we assumed the most compact feature at the base of the jet as the core. A more extended component appears at 3.5 mas south of the core.
0742$+$103 (OI $+$171): This is a high-redshift ($z=2.624$) GPS quasar which shows a large jet bending. We detect a bright component and two diffuse inner jet components in the northwest within 3 mas at 43 GHz. Further out, the jet bends to the northeast in the 15-GHz VLBA image [@2009AJ....138.1874L]. The 1.4-GHz[^3] and 2.3-GHz VLBA images show the jet bending from northeast to southeast. Although the position of the radio core in this GPS source is uncertain, we assumed the most compact feature at the base of the jet as the core. The coherent jet bending from northwest (inner $\sim$2 mas) to northeast (inner 4 mas), to east (12 mas), then to southeast ($\sim$200 mas) suggests a 180$\degr$ curved trajectory.
0743$-$006 (OI $-$072): This is a GPS quasar . Although the core position is uncertain, we assumed the most compact feature at the base of the jet as the core.
0838$+$133 (3C207): This is a powerful Fanaroff–Riley II (FR II) radio galaxy [@1983MNRAS.204..151L]. The 1.4- and 8.4-GHz VLA images show a fairly symmetric triple structure . Our 43-GHz VLBA image shows a bright core and a one-sided extended jet towards the east. This is in good agreement with what was found previously at 1.4 and 15 GHz [@2009AJ....138.1874L].
0859$-$140 (OJ $-$199): The source shows a compact core and two lobes aligned in the north–south direction at 408 MHz . Our image shows a faint and smoothly curved jet to the south-southeast within 2 mas, which is in good agreement with the previous 15-GHz VLBA image [@1998AJ....115.1295K].
0906$+$015 (4C $+$01.24): The VLA image at 1.6 GHz shows a compact core and a bright component $12\arcsec$ east of the core on kpc scale [@1993MNRAS.264..298M]. The 2.3-, 8.6-, and 15-GHz VLBI images show a jet toward the northeast from 5 to 30 mas [@2000ApJS..128...17F; @2009AJ....138.1874L]. There are 3 jet components detected towards the northeast within 2 mas in our 43-GHz image.
0945$+$408 (4C $+$40.24): The large-scale structure of this source is resolved into a very compact core with a one-sided jet extending over $4\arcsec$ ($\sim 18$ kpc) in the northeast direction using the VLA at 5 GHz and the Multi-Element Radio-Linked Interferometer Network (MERLIN) at 408 and 1666 MHz . The inner jet structure extends to southeast in our image, which appears to have a 90$\degr$ misalignment with respect to the large-scale structure.
1036$+$054: The 1.4-GHz VLA image shows a bright core and extended jet emission structure in the northeast–southwest direction until $\sim 18\arcsec$ [@2010ApJ...710..764K]. The 1.4-GHz VLBA image detected a one-sided jet pointing to the northeast over $\sim$150 mas. The position of the 15-GHz core [@2009AJ....138.1874L] coincides with the southernmost component in our image. The brightening jet component might be associated with the major outburst in late 2014 seen in the OVRO 40m light curve [@2011ApJS..194...29R]. Taking into account of the jet proper motion of 0.22 mas yr$^{-1}$ [@2019ApJ...874...43L] and the time interval between the outburst and our VLBA observation (2015 October 10), the flare-generated shock should have moved about 0.4 mas downstream, roughly consistently with the bright jet component seen in our 43-GHz image.
1045$-$188 (OL $-$176): The 1.4-GHz VLA image shows a bright core and extended jet emission structure in the northwest–southeast direction until $\sim 14\arcsec$ [@2010ApJ...710..764K]. The 1.4-GHz VLBA image shows a one-sided jet pointing to the northeast over $\sim$55 mas. The typical beam size at 15 GHz in the MOJAVE survey is 1.5 mas $\times$ 0.5 mas [@2009AJ....138.1874L]. In our new 43-GHz image, we detected two components within the area of the 15-GHz beam. We assumed the most compact and northernmost component as the core. Our J1 component corresponds to the bright 15-GHz core. More high-frequency VLBI observations are needed to clarify this.
1124$-$186 (OM $-$148): @2015AJ....150...58F only detect a compact core at 2.3 and 8.6 GHz. We detect a core and an extended faint emission feature to the south, in agreement with the 15-GHz image [@2009AJ....138.1874L].
1150$+$497 (4C $+$49.22): The 1.5-GHz VLA image shows a complex triple source with a halo in the north–south direction [@1981AJ.....86.1010U]. The source is not detected on space–Earth baselines at 5 GHz with the VSOP [@2008ApJS..175..314D]. In our image, we detect a bright core and a series of jet components to the southwest.
1219$+$044 (4C $+$04.42): The source shows a compact core and extended jet emission aligned in the north–south direction until $\sim 5\arcsec$ [@2010ApJ...710..764K]. The 15-GHz VLBA image only detected a one-sided jet pointing to the south until 7 mas [@2009AJ....138.1874L]. Our new image shows a bright core and a faint emission to the south.
1228$+$126 (M87): This is a well-known low-luminosity FR I radio galaxy [@1983MNRAS.204..151L]. The large-scale image of M87 observed with the VLA at 90 cm wavelength suggests that the outward flow from the nucleus extends well beyond the 2 kpc radio jet [@2000ApJ...543..611O]. The 15-GHz VLBA image displays an unresolved core and complex jet structure with an extent of 22 mas. Our 43-GHz VLBA image suggests a limb-brightening morphology with two ridge lines extending to the northwest and west directions. The jet opening angle, estimated from the northern and southern bright jet knots, is about 45 in projection, as is approximately consistent with the value reported in @2016ApJ...817..131H and @2019Galax...7...86Z.
1324$+$224: The deep VLA image at 1.4 GHz only detected a compact core [@2007ApJS..171..376C]. The 1.4-GHz VLBA image shows the jet pointing to the northwest up to 10 mas, then bending towards northeast until 90 mas from the core. In the 15-GHz VLBA image, the source shows a very compact core and a weak emission feature approximately 3 mas to the southwest [@2009AJ....138.1874L]. However, our new 43-GHz image indicates an inner jet towards the northwest on an angular scale of 2 mas, at a position angle consistent with the 1.4-GHz VLBA image.
1435$+$638 (VIPS 0792): The 1.4-GHz VLA image presents a faint lobe separated from the core by $15.4\arcsec$ in the southwest. It was not detected at 5 GHz . Previous studies presumed the radio core to lie at the northernmost end of the jet, based on the 5-GHz and 15-GHz VLBA maps [@2007ApJ...658..203H; @2016AJ....152...12L]. We also associated the northeastern component with the radio core in our image.
1504$-$166 (OR $-$107): The 1.4-GHz VLA map exhibits only radio core emission [@2010ApJ...710..764K]. The 1.4-GHz VLBA image shows the jet pointing to the west up to $\sim 150$ mas. The 8- and 15-GHz VLBI images show a compact core and extended structure to the south and southeast . Our new 43-GHz image shows the inner jet pointing to the south which suggests a jet bending.
1514$+$004 (4C $+$00.56): The source is a nearby radio galaxy $(z = 0.052)$. The 1.4-GHz NRAO VLA Sky Survey (NVSS) image shows a symmetric triple source extending in the northwest–southeast direction [@1998AJ....115.1693C]. In our image, we detect a core and bright jet component pointing to the northwest, in good agreement with the 15-GHz VLBA image [@2009AJ....138.1874L].
1548$+$056 (4C $+$05.64): @2010ApJ...710..764K detected a bright core and a relatively faint extended radio emission to the north with the VLA at 1.5 GHz. On mas scales, the source is dominated by a compact core with a jet extending to the north, as was seen previously at 1.4 GHz. In our image, we see a complex and curved jet extending to north and then bending to the northeast at 2 mas from the core.
1637$+$826 (NGC 6251): This is a well-studied FR I radio galaxy which shows both a bright core and large extended asymmetric jet emission [@1984ApJS...54..291P]. Lower-frequency VLBI observations only detected the jet extending to the southwest direction [@2004AJ....127.3587F; @2009AJ....138.1874L]. Our 43-GHz image shows the jet aligned well with the kpc-scale jet.
1716$+$686: The 4.5-GHz VLA observation shows a diffuse halo of 10 extension surrounding the core [@1996ApJS..107...37T]. @2009AJ....138.1874L presented a jet extending to the northwest up to $\sim 10$ mas, in a position angle in agreement with our image within 2 mas.
1926$+$611: The 1.5-GHz VLA image only detected a core [@1998AJ....115.1693C], but the 1.7-GHz VLBI image shows a bright core and a jet structure extended to the south [@1995ApJS...98....1P]. However, our image exhibits two jet components to the southeast, in agreement with the 15-GHz VLBA image [@2009AJ....138.1874L]. This indicates a jet bending from the south to the southeast, with nearly 70$\degr$ change in the position angle.
2007$+$777: The image made with the VLA at 1.5 GHz shows two-sided radio emission in the east–west direction [@1986AJ.....92....1A]. The eastern component is a prominent hot spot [@1993MNRAS.264..298M]. Our image shows the jet extending to the west, corresponding to the western side of the jet in the VLA image.
2021$+$317 (4C $+$31.56): The 1.4- and 15-GHz VLBA images display the jet extending towards the south [@2009AJ....138.1874L], consistently with our new 43-GHz VLBA image. However, we also detect a new component off-axis from the persistent jet, on the northwestern side. We believe this northwestern component is the core, and therefore the jet undergoes a sudden bending within 1 mas. Future observations can confirm the properties of this intriguing object.
2021$+$614 (OW $+$637): The source was classified as a CSO . No radio emission was detected on scales larger than 0.2 with the VLA at 1.4 GHz [@2010ApJ...710..764K]. @2009AJ....138.1874L identified the core feature located in the end of the southwestern component. We also use the same component as the core for our image, even though the J3 component at $\sim$2.5 mas is more compact than the core.
2029$+$121 (OW $+$149): The image made with the VLA in A-array at 1.4 GHz shows a one-sided, edge-brightened morphology pointing to the northwest [@2001AJ....122..565R]. The jet in our image extends to the southwest in $-130\degr$ position angle, in agreement with the 2.3- and 8.6-GHz VLBA images [@2015AJ....150...58F].
2126$-$158 (OX $-$146): This is the highest-redshift AGN in our sample ($z = 3.268$), a GPS quasar . The peak emission component in our image has a flat spectrum between 5 and 15 GHz and is also identified as the core by . Our image shows a bright core and a faint jet extending to the southwest.
2134$+$004 (OX $+$057): identified this source as a GPS quasar. found that the core of the source is located in the easternmost component. The core is also the brightest component in the 43-GHz radio structure.
2141$+$175 (OX $+$169): Large-scale VLA observations only detected a core at 1.4 GHz @1998AJ....115.1693C. In the 2.3- and 8.6-GHz VLBI images, the jet extends to the north, out to a distance of 25 mas ($\sim 85$ pc) [@2005AJ....129.1163P]. We detect the jet initially pointing to the west and then changing its position angle to the northwest, indicating that it has a large ($\sim 90\degr$) bending that starts at $\sim 0.2$ mas.
2155$-$152 (OX $-$192): The image made with the VLA at 1.4 GHz shows a triple structure with a size of $6\arcsec$ surrounding a central compact component in the north–south direction [@2007ApJS..171..376C]. Our image shows a jet towards the southwest up to 2 mas, in good agreement with the 5- and 15-GHz VLBA images [@2000ApJS..131...95F; @2009AJ....138.1874L].
2209$+$236: identified the source as a HFP. The 5-GHz VLBA image [@2000ApJS..131...95F] shows no indication of extended emission. @2016AJ....152...12L found a jet component to the northeast and determined a maximum apparent proper motion 1.35$c$. Our image shows the inner jet bending to the northeast.
2223$+$210 (DA 580): The jet structure extends to southwest in the 2.3-, 8.6-, and 15-GHz VLBI images [@2002ApJS..141...13B; @2009AJ....138.1874L]. We detect two compact components inside the low-frequency VLBI core region. Although the eastern one is not the brightest component, it appears in the upstream direction. Therefore we identify this as the radio core.
2227$-$088: The source shows an integrated flux density of 2.7 Jy in the core at 43 GHz, and only 21 mJy in the extended emission. @2003PASJ...55..351T found this source highly variable at 4.8 GHz with Australia Telescope Compact Array (ATCA). The many VLBI images made at lower frequencies[^4] indicate a faint, wiggling jet towards the north.
2234$+$282 (CTD 135): @2016AJ....152...12L identified the core in this BL Lac object with the northernmost jet feature in their 15-GHz VLBA images. Earlier @2016AN....337...65A claimed this source to be a rare $\gamma$-ray emitting CSO candidate with a two-sided jet, based on a comparison of VLBA maps at 8.4 and 15 GHz. In our new 43-GHz image, the southwestern component is the brightest and most compact, suggesting that this is the true core, instead of the northeastern feature [@2016AJ....152...12L]. Also, our new observations are at odds with the CSO interpretation [@2016AN....337...65A].
2243$-$123: The jet points to the northeast in the previous 5- and 15-GHz VLBA images [@2000ApJS..131...95F; @2009AJ....138.1874L]. In our image, the innermost section of the jet points closer to the north, within 4 mas from the core.
2318$+$049 (OZ $+$031): VLA images show a barely resolved structure along $-40\degr$ position angle [@1998PASP..110..111H]. In our VLBA image, the source shows a compact core–jet structure extending to the northwest in about the same direction. This is consistent with lower-frequency VLBI images [e.g., @2000ApJS..131...95F; @2009AJ....138.1874L]. Note that the 5-GHz VSOP Survey image also shows emission within 2 mas extended in the same direction, albeit with no clear indication of the core [@2004ApJS..155...33S].
Discussion
==========
Source Compactness
------------------
Only sources with cores that are compact enough can be successfully detected with the high angular resolution provided by high radio frequency VLBI observations. The source compactness is often expressed in two ways: the ratio of the VLBI core flux density to the total flux density, or alternatively, the ratio of the correlated flux density on the longest baselines to that on the shortest baselines [@2008AJ....136..159L; @2005AJ....130.2473K; @2018ApJS..234...17C]. Figure \[compactness\] shows the distributions of the total flux density $S_{\rm tot}$, the ratio $S_{\rm core}/S_{\rm tot}$, the correlated flux density on longest baselines derived from our data $S_{\rm L}$, and the ratio $S_{\rm L}/S_{\rm S}$, using the values listed in Table \[tab:image\]. $S_{\rm core}$ is obtained from fitting the brightest core component with a circular Gaussian model in [Difmap]{} (see Table \[model\]). $S_{\rm tot}$ is estimated by integrating the flux density contained in the emission region in the VLBI image. $S_{\rm L}$ and $S_{\rm S}$ are obtained from the correlated flux density on the longest and shortest baselines. The total flux density of these sources $S_{\rm tot}$ (Fig. \[compactness\]a) ranges from 0.10 to 2.91 Jy. The correlated flux density on the longest baselines $S_{\rm L}$ (Fig. \[compactness\]c) ranges from 0.02 to 2.38 Jy. The median values of the flux densities $S_{\rm tot}$ and $S_{\rm L}$ in our sample are 0.63 Jy and 0.26 Jy, respectively.
Figure \[compactness\]b gives the distribution of $S_{\rm core}/S_{\rm tot}$. Since most cores are even unresolved at 43 GHz with 0.5 mas $\times$ 0.2 mas resolution, the ratio $S_{\rm core}/S_{\rm tot}$ denotes the source compactness on mas scales, which varies in the range of 0.21–1.0 with a mean value of 0.85. This is also a measure of the core dominance in the VLBI image. For 112 sources (accounting for 90% of the sample), the compactness parameter $S_{\rm core}/S_{\rm tot}$ is larger than 0.5, indicating that a substantial fraction of the 43-GHz VLBI emission is from the core. There are 96 sources with a core flux density exceeding 0.30 Jy (Fig. \[compactness\]a). Figure \[compactness\]d gives the distribution of the source compactness $S_{\rm L}/S_{\rm S}$, ranging from 0.08 to 0.99 with a mean value of 0.65. The visibility amplitude on the longest baseline is associated with the most compact and unresolved part of the core, therefore it is used as an indicator of the compactness on sub-mas scales. A total of 86 sources have $S_{\rm L}/S_{\rm S} > 0.5$. In general, the ratios $S_{\rm core}/S_{\rm tot}$ are higher than $S_{\rm L}/S_{\rm S}$, consistent with the observational results that some cores are further resolved at higher sub-mas resolutions.
The source compactness parameters are useful for planning mm-wavelength ground-based and/or space VLBI observations. For the proposed Chinese SMVA project [@2014AcAau.102..217H], the expected highest angular resolution will be 20 $\mu$as at 43 GHz, and the baseline sensitivity is $\sim$17 mJy (1$\sigma$, assuming 512 MHz bandwidth, 60 s integration) when a space-borne 10-m radio telescope works in tandem with a 25-m VLBA telescope. To explore the innermost jets using space VLBI at even higher resolution, the extremely compact AGNs would be the preferred targets. However, some of the objects in this sample do not contain a bright, compact component and thus are not suitable for imaging with the future space VLBI network. Based on the flux densities and model-fitting results in Table \[tab:image\], 95 sources have $S_{\rm L} > 0.17$ Jy (i.e. about 10 times the baseline sensitivity of the SMVA), $S_{\rm core}/S_{\rm tot} > 0.5$ and $S_{\rm tot} > 0.30$ Jy. These sources remain compact at the longest available ground-based baselines and have sufficiently high flux densities. Therefore, combined with the previous 10 sources in paper II [@2018ApJS..234...17C], there are 105 sources suitable for future space VLBI missions [@2017ARep...61..310K; @2020An-ASR].
Core Brightness Temperature {#brightnesstemp}
---------------------------
We estimated the brightness temperature of the core components, and listed the values in Column 9 of Table \[model\]. We used the results of our brightness distribution model fitting, and the following formula: $$T_{\rm b} = 1.22\times10^{\rm 12}\frac{S}{ \nu^{2}d^{2}}(1+z),$$ where $S$ is the flux density of the core in Jy, $z$ is the redshift, $\nu$ is the observing frequency in GHz, and $d$ is the fitted Gaussian size (FWHM) of the component in mas.
The median value of the core brightness temperatures (Table \[model\]) is $7.92 \times 10^{10}$ K, with a maximum of $2.54 \times 10^{12}$ K. The maximum value is higher than the equipartition $T_{\rm b}$ limit [@1994ApJ...426...51R; @1994ApJ...429L..57B] but is approximately equal to the maximum brightness temperature set by the inverse Compton limit [@1969ApJ...155L..71K]. Apparent brightness temperatures exceeding the equipartition value are probably due to Doppler boosting of the relativistically beamed jet but could also be due to an intrinsic excess of particle energy over magnetic energy [e.g., @2000ARep...44..719K; @2002PASA...19...77K].
Figure \[Tb\]a shows the observed brightness temperatures ($T_{\rm b}$) as a function of frequency in the rest frame of source, $\nu$ = $\nu_{\rm obs} (1 + z)$. We adopted VLBI core brightness temperatures measured at observing frequencies $\nu_{\rm obs}$ at 2, 5, 8, 15, and 22 GHz from the literature , at 43 GHz (present data and Paper II), and at 86 GHz , to study the relation between brightness temperature and frequency. From Fig. \[Tb\]a, it is obvious that the core brightness temperatures at 43 and 86 GHz are much lower than those at 2, 5, 8, 15, and 22 GHz. The $T_{\rm b}$ values at lower frequencies, although with large scatter, seem to change only slightly with increasing frequency, until about 20 GHz. After that, a remarkable drop of $T_{\rm b}$ is seen above $\sim 30$ GHz. In order to quantify the variation of the core brightness temperature, we divided all 889 data points into 30 frequency bins, each containing 28–30 data points. Figure \[Tb\]b shows the data averaged within the bins with the error bars representing the scatter. The horizontal error bars indicate the frequency range within the bin, and the vertical error bars show the standard deviation of the mean values of core brightness temperature. The $T_{\rm b}$ distribution can be fitted with a smoothly broken power-law function between 2–240 GHz (eq. \[eqn:powerlaw\]), while the $T_{\rm b}$ values between 240–420 GHz (red-colored data points) seem to deviate from this relation and thus are excluded from the fitting: $$\label{eqn:powerlaw}
T_{\rm b}(\nu) = T_{\rm b,j}\left[\left(\frac{\nu}{\nu_{\rm j}}\right)^{\alpha_1 n}
+ \left(\frac{\nu}{\nu_{\rm j}}\right)^{\alpha_2 n}\right] ^{-1/n},$$ where $T_{\rm b,j}$ is the observed brightness temperature at the break frequency $\nu_{\rm j}$, $-\alpha_1$ and $-\alpha_2$ are the slopes at higher and lower frequencies than the break frequency, respectively, and $n$ is a numerical factor controlling the sharpness of the break. The frequencies (horizontal axis) have been corrected to the source rest frame by multiplying the observing frequencies by ($1+z$). We choose the numerical factor $n = 1.97 $ and use the nonlinear least-squares algorithm to fit the function. Figure \[Tb\]b shows the best fit with the following parameters: $T_{\rm b,j}=(218.14\pm5.27)\times10^{10}$ K, $\nu_{\rm j}=6.78\pm0.43$ GHz, $\alpha_1=1.91\pm0.03$, and $\alpha_2=-0.94\pm0.06$.
From the best fitting parameters, we find that the core brightness temperature is increasing below the break frequency $\sim$7 GHz, and decreasing between 7–240 GHz. $T_{\rm b,j}$ is $\sim 2\times10^{12}$ K at $\sim$7 GHz, which is slightly higher than the inverse Compton catastrophe limit [@1969ApJ...155L..71K] and significantly higher than the equipartition value [@1994ApJ...426...51R; @1994ApJ...429L..57B], consistent with Doppler boosting in AGN jets. The inferred broken power-law distribution can be explained in terms of opacity affecting the emission mechanism at different resolutions. Below 7 GHz, due to relatively lower resolutions, the core emission is mixed with jet emission, resulting in lower $T_{\rm b}$ at lower frequencies. From 2 to 7 GHz, with the resolution increasing, this dilution effect gradually becomes weaker, thus the $T_{\rm b}$ shows an increasing trend. The decreasing trend of $T_{\rm b}$ starting from 7 GHz to higher frequencies reflects the synchrotron opacity changing from optically thick to thin. A resulting lower brightness temperature has been found by opacity core-shift observations in many sources . Previous compactness studies suggest that most, if not all, AGN cores are resolved into sub-components at higher frequency and with higher resolution. However, the highest-frequency data (red-colored data point) in our Fig. \[Tb\]b show a clear deviation from the fitted trend (black line) and display a flattening. The inferred brightness temperatures are substantially higher than the values expected from the fit. We caution that this may be because of the sample selection bias caused by the flux density limited surveys. These high-frequency ($>240$ GHz in rest frame) data correspond to high-redshift ($z \gtrsim 3$) AGNs. Among those, the survey samples include the few brightest ones only, and the majority of the (much weaker) high-redshift population is missed. Future higher-sensitivity VLBI surveys at and above 86 GHz containing weaker high-redshift AGNs would be important for identifying the distribution trend at the very high frequency end of Fig. \[Tb\]b.
Correlation Between Radio and $\gamma$-ray Emission {#corelation}
---------------------------------------------------
We cross-matched our sample with the third [*Fermi*]{}-LAT AGN catalog [@2015ApJ...810...14A] and found 73 sources (59% of our sample) detected in $\gamma$-rays. Combining with our previously observed ten sources in Paper II, we obtain a total of 79 sources, including 61 QSOs, 14 BL Lac objects, 3 radio galaxies (M87, NGC6251, 3C371), and 1 object (0648$-$165) with no optical identification. In our sample, BL Lac objects have a higher $\gamma$-ray detection rate (88%) than quasars (58%).
Figure \[correlation\] presents the flux density correlation between $\gamma$-rays and radio for all the 79 sources. We used the Pearson correlation test to reveal the flux density correlation between the radio and $\gamma$-rays. That gives a correlation with $\tau = 0.550$ and $P<0.001$, where $\tau = 0$ means no correlation and $\tau = 1$ means strong correlation, and $P$ gives the probability of no correlation. We should note that the three non-beamed AGNs (radio galaxies: M87, NGC6251, and 3C371) in our sample also follow the same $\gamma$-ray/radio flux density correlation, indicating that probably a universal $\gamma$-ray production mechanism, regardless of their difference in the central engine and jet power, is at work for diverse types of AGNs.
According to @2010MNRAS.407..791G, there is a strong correlation between the radio flux density at 20 GHz and the $\gamma$-ray flux above 100 MeV. even found a more significant correlation between both the flux densities and luminosities in $\gamma$-rays and 37-GHz single-dish radio data. Recently @2016ApJS..226...20F found a correlation between $\gamma$-ray and 1.4-GHz monochromatic radio luminosities. @2011ApJ...741...30A found that there is a correlation between $\gamma$-ray and 8.4-GHz (VLA and ATCA) / 15-GHz (OVRO single dish) radio luminosities. In the above studies, the authors used single-dish or low-resolution interferometer measurements of the radio flux densities. In contrast, our analysis is based on high-resolution high-frequency VLBI data; the correlation derived from our study is consistent with the previous works cited above but shows higher correlation coefficient. All these studies support the notion that the $\gamma$-ray emission zone is close to or same with the compact VLBI core.
In conclusion, our results give support to the strong correlation between the $\gamma$-ray and radio emission in AGNs, suggesting that the $\gamma$-ray emission zone is in the parsec-scale jet revealed by the 43 and 86 GHz VLBI images. The correlation is valid for diverse types of AGN over six orders of magnitude in radio luminosity and eight orders of magnitude in $\gamma$-rays, indicating that the general AGN population have a common $\gamma$-ray production mechanism, regardless of their different central engine and jet power.
Summary
=======
We observed a sample of 124 bright and compact radio sources with the VLBA at 43 GHz between 2014 November and 2016 May. We achieved a highest angular resolution of $\sim$0.2 mas and a typical image noise level of 0.5 mJybeam$^{-1}$. In our sample, 8 sources remain unresolved and are only detected with a compact core, 112 sources show one-sided jet structure, 3 sources have compact symmetric structures, and one source (0354+559) shows complex structure. We present the 43-GHz contour images of all the 124 sources and give comments on selected individual sources to highlight their properties in the context with other information from literature. The majority of these radio-loud AGNs have not been previously imaged with VLBI at this frequency.
One of the main motivations of our project was to identify suitable target sources for the future mm-wavelength space VLBI program. From the distribution of source compactness on mas scales, $S_{\rm core}/S_{\rm tot}$, and sub-mas scales, $S_{\rm L}/S_{\rm S}$, 95 of the 124 sources have $S_{\rm L} > 0.17$ Jy (about 10 times the baseline sensitivity of the SMVA), $S_{\rm core}/S_{\rm tot} > 0.5$ and $S_{\rm tot} > 0.30$ Jy, and therefore there are 105 sources to be considered as targets for the SMVA.
For the 124 sources, we calculated the core brightness temperatures. Their median value is $7.92 \times 10^{10}$ K, with a maximum of $2.54 \times 10^{12}$ K at 43 GHz. This is somewhat higher than the inverse Compton catastrophe $T_{\rm b}$ limit and the equipartition limit, which can be explained by Doppler boosting of the relativistically beamed jet. We investigated the core brightness temperatures obtained from our project and other high-resolution VLBI observations from the literature at 2, 5, 8, 15, 22, 43, and 86 GHz. We find that the core brightness temperature is increasing below the break frequency $\sim 7$ GHz, and decreasing above 7 GHz. The break core brightness temperature value is $2 \times 10^{12}$ K at $\sim 7$ GHz. The change of brightness temperature with (source rest frame) observing frequency is related to the resolution and synchrotron opacity changes with frequency.
We used 79 sources to test the correlation between radio and $\gamma$-ray flux densities and found tighter correlation compared to previous works. Our result supports the scenario that the location of the $\gamma$-ray emission is close to the 43 GHz VLBI core. Moreover our result also indicate that the radio–$\gamma$-ray correlation is universal and can be applicable to different types of AGNs.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the SKA pre-research funding granted by the National Key R&D Programme of China (2018YFA0404602, 2018YFA0404603), the Chinese Academy of Sciences (CAS, 114231KYSB20170003), and the Hungarian National Research, Development and Innovation Office (grant 2018-2.1.14-TÉT-CN-2018-00001). X.-P. Cheng was supported by Korea Research Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (2019H1D3A1A01102564). The VLBA observations were sponsored by Shanghai Astronomical Observatory through an MoU with the NRAO (Project code: BA111). This research has made use of data from the MOJAVE database that is maintained by the MOJAVE team [@2009AJ....138.1874L]. The MOJAVE program is supported under NASA-[*Fermi*]{} grants NNX15AU76G and NNX12A087G. The Very Long Baseline Array is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This research has made use of data from the OVRO 40-m monitoring program [@2011ApJS..194...29R], which is supported in part by NASA grants NNX08AW31G, NNX11A043G and NNX14AQ89G and NSF grants AST-0808050 and AST-1109911. This work has made use of NASA Astrophysics Data System Abstract Service, and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
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[ccccccccc]{} 0048$-$071 & OB $-$082 & C & 00 51 08.20982 & $-$06 50 02.2291 & 1.975 & QSO & Y & 12\
0106$+$013 & 4C +01.02 & B & 01 08 38.77110 & $+$01 35 00.3713 & 2.099 & QSO & Y & 64\
0113$-$118 & & C & 01 16 12.52203 & $-$11 36 15.4348 & 0.671 & QSO & Y & 15\
0119$+$041 & OC +033 & C & 01 21 56.86170 & $+$04 22 24.7342 & 0.637 & QSO & Y & ...\
0122$-$003 & UM 321 & C & 01 25 28.84383 & $-$00 05 55.9322 & 1.077 & QSO & Y & ...\
0130$-$171 & OC $-$150 & B & 01 32 43.48746 & $-$16 54 48.5218 & 1.020 & QSO & Y & 21\
0149$+$218 & & C & 01 52 18.05904 & $+$22 07 07.6997 & 1.320 & QSO & Y & ...\
0202$-$172 & & C & 02 04 57.67434 & $-$17 01 19.8405 & 1.739 & QSO & Y & 7\
0208$+$106 & & C & 02 11 13.77363 & $+$10 51 34.7986 & 0.200 & BL Lac & Y & 26\
0221$+$067 & 4C +06.11 & B & 02 24 28.42819 & $+$06 59 23.3415 & 0.511 & QSO & Y & ...\
0224$+$671 & 4C +67.05 & C & 02 28 50.05149 & $+$67 21 03.0293 & 0.523 & QSO & Y & ...\
0229$+$131 & 4C +13.14 & C & 02 31 45.89406 & $+$13 22 54.7162 & 2.059 & QSO & Y & ...\
0239$+$108 & OD $+$166 & C & 02 42 29.17085 & $+$11 01 00.7280 & 2.680 & QSO & Y & ...\
0241$+$622 & & C & 02 44 57.69671 & $+$62 28 06.5157 & 0.045 & QSO & Y & ...\
0306$+$102 & OE +110 & C & 03 09 03.62350 & $+$10 29 16.3410 & 0.862 & QSO & Y & 15\
0309$+$411 & NRAO 128 & B & 03 13 01.96212 & $+$41 20 01.1835 & 0.136 & G & Y & ...\
0322$+$222 & & C & 03 25 36.81436 & $+$22 24 00.3655 & 2.066 & QSO & N & 11\
0354$+$559 & & B & 03 58 30.18819 & $+$56 06 44.4602 & ... & U & Y & ...\
0400$+$258 & CTD 026 & C & 04 03 05.58608 & $+$26 00 01.5027 & 2.109 & QSO & Y & ...\
0403$-$132 & & C & 04 05 34.00339 & $-$13 08 13.6908 & 0.571 & QSO & Y & ...\
0405$-$123 & & B & 04 07 48.43097 & $-$12 11 36.6593 & 0.573 & QSO & N & ...\
0420$+$417 & 4C +41.11 & C & 04 23 56.00979 & $+$41 50 02.7129 & ... & BL Lac & N & 29\
0451$-$282 & OF $-$285 & B & 04 53 14.64679 & $-$28 07 37.3265 & 2.559 & QSO & N & 19\
0507$+$179 & & C & 05 10 02.36913 & $+$18 00 41.5816 & 0.416 & QSO & Y & 12\
0529$+$075 & OG 050 & C & 05 32 38.99846 & $+$07 32 43.3449 & 1.254 & QSO & Y & 41\
0605$-$085 & OC $-$010 & B & 06 07 59.69923 & $-$08 34 49.9781 & 0.870 & QSO & Y & 18\
0627$-$199 & & B & 06 29 23.76185 & $-$19 59 19.7235 & 1.724 & BL Lac & N & 25\
0633$+$734 & & C & 06 39 21.96122 & $+$73 24 58.0403 & 1.850 & QSO & N & 6\
0648$-$165 & & C & 06 50 24.58186 & $-$16 37 39.7255 & ... & U & Y & 14\
0657$+$172 & & B & 07 00 01.52553 & $+$17 09 21.7014 & ... & U & Y & ...\
0708$+$506 & & B & 07 12 43.68355 & $+$50 33 22.7069 & 0.502 & BL Lac & N & 17\
0723$-$008 & & B & 07 25 50.63996 & $-$00 54 56.5441 & 0.128 & BL Lac & Y & 11\
0730$+$504 & & C & 07 33 52.52059 & $+$50 22 09.0621 & 0.720 & QSO & N & 7\
0736$+$017 & OI 061 & C & 07 39 18.03390 & $+$01 37 04.6177 & 0.189 & QSO & Y & 25\
0738$+$313 & OI 363 & C & 07 41 10.70331 & $+$31 12 00.2292 & 0.631 & QSO & Y & ...\
0742$+$103 & & C & 07 45 33.05952 & $+$10 11 12.6922 & 2.624 & QSO & Y & ...\
0743$-$006 & OI $-$072 & C & 07 45 54.08232 & $-$00 44 17.5399 & 0.996 & QSO & Y & ...\
0754$+$100 & OI +090.4 & C & 07 57 06.64295 & $+$09 56 34.8522 & 0.266 & BL Lac & N & 20\
0805$+$410 & & C & 08 08 56.65204 & $+$40 52 44.8888 & 1.418 & QSO & N & ...\
0805$-$077 & & B & 08 08 15.53603 & $-$07 51 09.8862 & 1.837 & QSO & N & 150\
0808$+$019 & OJ 014 & C & 08 11 26.70732 & $+$01 46 52.2201 & 1.148 & BL Lac & N & 13\
0821$+$394 & 4C +39.23 & C & 08 24 55.48386 & $+$39 16 41.9040 & 1.216 & QSO & N & 4\
0838$+$133 & 3C 207 & C & 08 40 47.58843 & $+$13 12 23.5637 & 0.681 & QSO & Y & 8\
0847$-$120 & & C & 08 50 09.63563 & $-$12 13 35.3762 & 0.566 & QSO & N & 31\
0859$-$140 & OJ $-$199 & C & 09 02 16.83092 & $-$14 15 30.8753 & 1.333 & QSO & Y & ...\
0906$+$015 & 4C +01.24 & C & 09 09 10.09160 & $+$01 21 35.6177 & 1.024 & QSO & Y & 32\
0917$+$624 & OK 630 & C & 09 21 36.23107 & $+$62 15 52.1803 & 1.447 & QSO & N & 8\
0925$-$203 & & B & 09 27 51.82431 & $-$20 34 51.2324 & 0.348 & QSO & N & 6\
0945$+$408 & 4C +40.24 & B & 09 48 55.33815 & $+$40 39 44.5869 & 1.249 & QSO & Y & 4\
1032$-$199 & & B & 10 35 02.15530 & $-$20 11 34.3595 & 2.198 & QSO & N & ...\
1036$+$054 & & C & 10 38 46.77988 & $+$05 12 29.0865 & 0.473 & QSO & Y & ...\
1038$+$064 & 4C +06.41 & C & 10 41 17.16250 & $+$06 10 16.9235 & 1.265 & QSO & N & 11\
1045$-$188 & & C & 10 48 06.62060 & $-$19 09 35.7268 & 0.595 & QSO & Y & ...\
1049$+$215 & 4C +21.28 & C & 10 51 48.78907 & $+$21 19 52.3137 & 1.300 & QSO & N & ...\
1124$-$186 & OM $-$148 & C & 11 27 04.39245 & $-$18 57 17.4418 & 1.048 & QSO & Y & 34\
1149$-$084 & & B & 11 52 17.20951 & $-$08 41 03.3138 & 2.367 & QSO & N & 12\
1150$+$497 & 4C +49.22 & B & 11 53 24.46663 & $+$49 31 08.8301 & 0.334 & QSO & Y & 8\
1213$-$172 & & C & 12 15 46.75176 & $-$17 31 45.4032 & ... & U & N & ...\
1219$+$044 & 4C +04.42 & B & 12 22 22.54962 & $+$04 13 15.7761 & 0.966 & QSO & Y & 13\
1228$+$126 & 3C 274 & B & 12 30 49.42338 & $+$12 23 28.0438 & 0.004 & G & Y & 14\
1243$-$072 & & C & 12 46 04.23210 & $-$07 30 46.5748 & 1.286 & QSO & N & ...\
1257$+$519 & & B & 12 59 31.17401 & $+$51 40 56.2607 & 0.405 & G & N & ...\
1306$+$360 & & B & 13 08 23.70914 & $+$35 46 37.1639 & 1.055 & QSO & N & 11\
1324$+$224 & & C & 13 27 00.86131 & $+$22 10 50.1628 & 1.398 & QSO & Y & 19\
1329$-$049 & OP $-$050 & B & 13 32 04.46467 & $-$05 09 43.3056 & 2.150 & QSO & N & 50\
1354$+$195 & 4C +19.44 & C & 13 57 04.43666 & $+$19 19 07.3723 & 0.720 & QSO & N & ...\
1417$+$385 & & C & 14 19 46.61376 & $+$38 21 48.4750 & 1.831 & QSO & N & 5\
1435$+$638 & VIPS 0792 & C & 14 36 45.80216 & $+$63 36 37.8663 & 2.068 & QSO & Y & ...\
1502$+$106 & OR 103 & C & 15 04 24.97978 & $+$10 29 39.1984 & 1.838 & QSO & N & 401\
1504$-$166 & & C & 15 07 04.78696 & $-$16 52 30.2671 & 0.876 & QSO & Y & ...\
1514$+$004 & & C & 15 16 40.21905 & $+$00 15 01.9087 & 0.052 & G & Y & ...\
1514$+$197 & & C & 15 16 56.79616 & $+$19 32 12.9919 & 1.070 & BL Lac & N & 5\
1532$+$016 & & C & 15 34 52.45368 & $+$01 31 04.2064 & 1.425 & QSO & N & ...\
1546$+$027 & & C & 15 49 29.43685 & $+$02 37 01.1631 & 0.414 & QSO & N & 18\
1548$+$056 & 4C +05.64 & B & 15 50 35.26924 & $+$05 27 10.4484 & 1.417 & QSO & Y & 10\
1606$+$106 & 4C +10.45 & C & 16 08 46.20319 & $+$10 29 07.7756 & 1.232 & QSO & N & 16\
1636$+$473 & 4C +47.44 & C & 16 37 45.13056 & $+$47 17 33.8310 & 0.735 & QSO & N & 18\
1637$+$826 & NGC 6251 & C & 16 32 31.96989 & $+$82 32 16.3999 & 0.024 & G & Y & 11\
1639$-$062 & & C & 16 42 02.17772 & $-$06 21 23.6952 & 1.514 & BL Lac & N & 11\
1642$+$690 & 4C +69.21 & B & 16 42 07.84850 & $+$68 56 39.7564 & 0.751 & QSO & N & ...\
1655$+$077 & & C & 16 58 09.01147 & $+$07 41 27.5403 & 0.621 & QSO & N & ...\
1656$+$477 & & C & 16 58 02.77959 & $+$47 37 49.2307 & 1.615 & QSO & N & ...\
1657$-$261 & & B & 17 00 53.15406 & $-$26 10 51.7253 & ... & U & N & ...\
1659$+$399 & & B & 17 01 24.63481 & $+$39 54 37.0915 & 0.507 & BL Lac & N & ...\
1716$+$686 & & B & 17 16 13.93800 & $+$68 36 38.7446 & 0.777 & QSO & Y & 12\
1725$+$044 & & C & 17 28 24.95272 & $+$04 27 04.9137 & 0.293 & QSO & N & 13\
1726$+$455 & & C & 17 27 27.65080 & $+$45 30 39.7312 & 0.717 & QSO & N & 15\
1800$+$440 & & C & 18 01 32.31482 & $+$44 04 21.9002 & 0.663 & QSO & N & 4\
1806$+$456 & & B & 18 08 21.88588 & $+$45 42 20.8663 & 0.830 & QSO & N & ...\
1807$+$698 & 3C 371 & B & 18 06 50.68064 & $+$69 49 28.1085 & 0.051 & BL Lac & N & 38\
1842$+$681 & & C & 18 42 33.64169 & $+$68 09 25.2277 & 0.472 & QSO & N & ...\
1849$+$670 & & C & 18 49 16.07228 & $+$67 05 41.6802 & 0.657 & QSO & N & 74\
1920$-$211 & OV $-$235 & B & 19 23 32.18981 & $-$21 04 33.3330 & 0.874 & QSO & N & 83\
1926$+$611 & & B & 19 27 30.44262 & $+$61 17 32.8792 & ... & BL Lac & Y & 9\
1957$-$135 & & C & 20 00 42.14510 & $-$13 25 33.5338 & 0.222 & QSO & N & ...\
1958$-$179 & & C & 20 00 57.09045 & $-$17 48 57.6727 & 0.652 & QSO & N & 25\
2007$+$777 & & B & 20 05 30.99853 & $+$77 52 43.2475 & 0.342 & BL Lac & Y & 11\
2021$+$317 & 4C +31.56 & C & 20 23 19.01734 & $+$31 53 02.3060 & 0.356 & BL Lac & Y & ...\
2021$+$614 & OW +637 & B & 20 22 06.68174 & $+$61 36 58.8047 & 0.227 & G & Y & ...\
2022$-$077 & NRAO 0629 & C & 20 25 40.66041 & $-$07 35 52.6892 & 1.388 & QSO & N & 101\
2023$+$335 & & C & 20 25 10.84211 & $+$33 43 00.2144 & 0.219 & QSO & N & 41\
2029$+$121 & OW +149 & B & 20 31 54.99427 & $+$12 19 41.3403 & 1.215 & QSO & Y & 12\
2126$-$158 & OX $-$146 & C & 21 29 12.17590 & $-$15 38 41.0416 & 3.268 & QSO & Y & ...\
2134$+$004 & & B & 21 36 38.58630 & $+$00 41 54.2129 & 1.941 & QSO & Y & ...\
2135$+$508 & & C & 21 37 00.98621 & $+$51 01 36.1290 & ... & U & N & ...\
2141$+$175 & OX 169 & B & 21 43 35.54457 & $+$17 43 48.7874 & 0.213 & QSO & Y & 45\
2142$+$110 & & B & 21 45 18.77507 & $+$11 15 27.3123 & 0.548 & QSO & N & ...\
2144$+$092 & & C & 21 47 10.16297 & $+$09 29 46.6723 & 1.113 & QSO & N & 35\
2155$-$152 & & B & 21 58 06.28190 & $-$15 01 09.3278 & 0.672 & QSO & Y & 11\
2201$+$171 & & C & 22 03 26.89368 & $+$17 25 48.2476 & 1.076 & QSO & N & 65\
2209$+$236 & & C & 22 12 05.96631 & $+$23 55 40.5437 & 1.125 & QSO & Y & 10\
2216$-$038 & 4C $-$03.79 & C & 22 18 52.03773 & $-$03 35 36.8796 & 0.901 & QSO & N & ...\
2223$+$210 & DA 580 & B & 22 25 38.04713 & $+$21 18 06.4150 & 1.959 & QSO & Y & ...\
2227$-$088 & PHL 5225 & C & 22 29 40.08434 & $-$08 32 54.4356 & 1.560 & QSO & Y & 43\
2234$+$282 & CTD 135 & C & 22 36 22.47085 & $+$28 28 57.4132 & 0.790 & BL Lac & Y & 51\
2243$-$123 & & C & 22 46 18.23198 & $-$12 06 51.2775 & 0.632 & QSO & Y & ...\
2308$+$341 & & C & 23 11 05.32880 & $+$34 25 10.9056 & 1.817 & QSO & N & 43\
2318$+$049 & & C & 23 20 44.85660 & $+$05 13 49.9524 & 0.623 & QSO & Y & ...\
2320$-$035 & & C & 23 23 31.95375 & $-$03 17 05.0239 & 1.411 & QSO & N & 21\
2325$+$093 & OZ 042 & C & 23 27 33.58058 & $+$09 40 09.4626 & 1.841 & QSO & N & 20\
2342$-$161 & & C & 23 45 12.46232 & $-$15 55 07.8345 & 0.621 & QSO & N & 42\
2344$+$092 & & C & 23 46 36.83855 & $+$09 30 45.5147 & 0.677 & QSO & N & ...\
2345$-$167 & OZ $-$176 & C & 23 48 02.60853 & $-$16 31 12.0222 & 0.576 & QSO & N & 16\
2351$-$154 & OZ $-$187 & C & 23 54 30.19518 & $-$15 13 11.2129 & 2.675 & QSO & N & ...\
[ccccccccc]{}
A & BA111A – BA111F & 2014/11/21 – 2016/05/06 & 43/86 & 80 & 10 & @2018ApJS..234...17C\
B & BA111G – BA111I & 2015/06/19 – 2016/04/22 & 43 & 28 & 40 & This paper\
C & BA111J – BA111M & 2015/10/10 – 2016/02/09 & 43 & 14 & 84 & This paper\
[cccccccccccc]{} 0048$-$071 & 2015-11-30 & 0.67 & 0.21 & 16.7 & 0.480 & 0.412 & 0.22 & 0.48 & 27 & 0.41 & 270\
0106$+$013 & 2015-06-19 & 0.44 & 0.18 &$-$4.4 & 1.202 & 0.801 & 0.70 & 1.09 & 24 & 0.75 & 406\
0113$-$118 & 2015-11-30 & 0.67 & 0.21 & 12.7 & 0.698 & 0.568 & 0.85 & 0.70 & 27 & 0.55 & 260\
0119$+$041 & 2015-11-30 & 0.52 & 0.21 & 14.2 & 0.546 & 0.345 & 0.40 & 0.54 & 30 & 0.33 & 296\
0122$-$003 & 2015-11-30 & 0.63 & 0.19 & 18.4 & 0.425 & 0.256 & 0.17 & 0.40 & 27 & 0.30 & 286\
0130$-$171 & 2015-06-19 & 0.44 & 0.16 &$-$5.1 & 0.597 & 0.390 & 0.58 & 0.59 & 20 & 0.23 & 438\
0149$+$218 & 2015-11-30 & 0.49 & 0.20 & 16.8 & 1.160 & 0.970 & 1.46 & 1.14 & 33 & 0.81 & 336\
0202$-$172 & 2015-11-30 & 0.72 & 0.20 & 12.6 & 1.015 & 0.676 & 1.20 & 0.82 & 31 & 0.65 & 244\
0208$+$106 & 2015-11-30 & 0.51 & 0.22 & 13.4 & 0.530 & 0.314 & 0.30 & 0.49 & 33 & 0.25 & 306\
0221$+$067 & 2015-06-19 & 0.42 & 0.17 & $-$2.5 & 1.020 & 0.693 & 0.40 & 1.01 & 29 & 0.34 & 438\
0224$+$671 & 2015-11-30 & 0.44 & 0.24 & $-$22.1 & 0.519 & 0.294 & 0.89 & 0.49 & 33 & 0.21 & 360\
0229$+$131 & 2015-11-30 & 0.54 & 0.25 & 15 & 0.692 & 0.442 & 0.48 & 0.66 & 27 & 0.48 & 306\
0239$+$108 & 2015-11-30 & 0.71 & 0.29 & 24.3 & 0.240 & 0.189 & 0.17 & 0.24 & 26 & 0.14 & 293\
0241$+$622 & 2015-11-30 & 0.45 & 0.26 & $-$30 & 0.825 & 0.693 & 0.70 & 0.83 & 32 & 0.44 & 385\
0306$+$102 & 2015-11-30 & 0.59 & 0.26 & 17.3 & 0.899 & 0.776 & 0.35 & 0.90 & 26 & 0.67 & 291\
0309$+$411 & 2015-06-19 & 0.44 & 0.16 & 10.3 & 0.735 & 0.388 & 0.33 & 0.74 & 31 & 0.17 & 510\
0322$+$222 & 2015-11-30 & 0.49 & 0.26 & 7.7 & 0.553 & 0.487 & 0.34 & 0.55 & 30 & 0.46 & 325\
0354$+$559 & 2015-06-19 & 0.44 & 0.17 & 19.7 & 0.521 & 0.211 & 0.40 & 0.42 & 28 & 0.21 & 495\
0400$+$258 & 2015-11-30 & 0.47 & 0.21 & 10.7 & 0.237 & 0.112 & 0.23 & 0.23 & 31 & 0.11 & 331\
0403$-$132 & 2015-11-30 & 0.84 & 0.26 & 15.9 & 0.337 & 0.280 & 0.70 & 0.34 & 20 & 0.24 & 227\
0405$-$123 & 2015-06-19 & 0.43 & 0.14 & $-$7.3 & 0.801 & 0.389 & 0.95 & 0.80 & 23 & 0.14 & 356\
0420$+$417 & 2015-11-30 & 0.45 & 0.25 & $-$5.3 & 0.300 & 0.222 & 0.64 & 0.28 & 33 & 0.23 & 376\
0451$-$282 & 2015-06-19 & 0.46 & 0.17 & $-$4.4 & 0.958 & 0.740 & 0.89 & 0.96 & 23 & 0.64 & 400\
0507$+$179 & 2015-11-30 & 0.52 & 0.26 & 10.9 & 2.487 & 2.440 & 1.20 & 2.48 & 29 & 2.46 & 308\
0529$+$075 & 2015-11-30 & 0.58 & 0.27 & 13.7 & 1.316 & 1.040 & 0.80 & 1.31 & 27 & 0.85 & 277\
0605$-$085 & 2015-06-19 & 0.47 & 0.17 & $-$9.9 & 2.047 & 0.784 & 1.70 & 2.04 & 24 & 0.49 & 407\
0627$-$199 & 2015-06-19 & 0.44 & 0.17 & $-$3.6 & 0.423 & 0.340 & 0.80 & 0.43 & 22 & 0.29 & 440\
0633$+$734 & 2015-11-30 & 0.53 & 0.23 & $-$16.7 & 0.411 & 0.380 & 0.45 & 0.41 & 30 & 0.34 & 310\
0648$-$165 & 2015-11-30 & 0.94 & 0.23 & 20.9 & 0.501 & 0.350 & 0.53 & 0.50 & 23 & 0.33 & 217\
0657$+$172 & 2015-06-19 & 0.43 & 0.17 & $-$4.5 & 0.335 & 0.184 & 0.48 & 0.33 & 32 & 0.13 & 460\
0708$+$506 & 2015-06-19 & 0.42 & 0.17 & 7.3 & 0.370 & 0.330 & 0.27 & 0.37 & 30 & 0.28 & 540\
0723$-$008 & 2015-06-19 & 0.48 & 0.16 & $-$11.8 & 1.771 & 0.573 & 1.60 & 1.78 & 29 & 0.45 & 400\
0730$+$504 & 2015-10-10 & 0.39 & 0.17 & 15.8 & 0.215 & 0.127 & 0.50 & 0.21 & 29 & 0.10 & 460\
0736$+$017 & 2015-10-10 & 0.46 & 0.17 & $-$11.5 & 0.706 & 0.424 & 1.06 & 0.71 & 30 & 0.18 & 415\
0738$+$313 & 2015-10-10 & 0.45 & 0.19 & 2.7 & 1.142 & 0.467 & 1.00 & 0.99 & 32 & 0.23 & 473\
0742$+$103 & 2015-10-10 & 0.62 & 0.21 & $-$14.1 & 0.412 & 0.257 & 0.50 & 0.35 & 32 & 0.23 & 368\
0743$-$006 & 2015-10-10 & 0.40 & 0.17 & $-$1.4 & 0.491 & 0.177 & 0.48 & 0.42 & 30 & 0.14 & 403\
0754$+$100 & 2015-10-10 & 0.43 & 0.17 & $-$5.8 & 0.669 & 0.480 & 0.38 & 0.63 & 32 & 0.31 & 420\
0805$+$410 & 2015-10-10 & 0.45 & 0.18 & 2.3 & 0.844 & 0.643 & 0.98 & 0.85 & 31 & 0.37 & 490\
0805$-$077 & 2015-06-19 & 0.43 & 0.18 & 1.1 & 0.601 & 0.534 & 0.70 & 0.61 & 29 & 0.46 & 404\
0808$+$019 & 2015-10-10 & 0.46 & 0.18 & $-$6.7 & 1.175 & 1.100 & 0.66 & 1.17 & 30 & 1.05 & 414\
0821$+$394 & 2015-10-10 & 0.42 & 0.18 & 0.3 & 0.320 & 0.106 & 0.37 & 0.32 & 31 & 0.12 & 500\
0838$+$133 & 2015-10-10 & 0.40 & 0.17 & $-$0.4 & 0.711 & 0.436 & 0.65 & 0.69 & 33 & 0.27 & 428\
0847$-$120 & 2015-10-10 & 0.45 & 0.18 & $-$2.4 & 0.837 & 0.743 & 0.75 & 0.84 & 28 & 0.68 & 422\
0859$-$140 & 2015-10-10 & 0.49 & 0.18 & $-$9.2 & 0.198 & 0.086 & 0.55 & 0.20 & 27 & 0.10 & 430\
0906$+$015 & 2015-10-10 & 0.46 & 0.18 & $-$7.8 & 0.789 & 0.402 & 0.70 & 0.75 & 23 & 0.32 & 402\
0917$+$624 & 2015-10-10 & 0.43 & 0.17 & $-$18.4 & 0.572 & 0.420 & 0.50 & 0.57 & 32 & 0.39 & 406\
0925$-$203 & 2015-06-19 & 0.43 & 0.16 & $-$4.7 & 0.370 & 0.249 & 0.70 & 0.38 & 23 & 0.27 & 424\
0945$+$408 & 2015-06-19 & 0.43 & 0.16 & 1.8 & 0.308 & 0.152 & 0.58 & 0.30 & 33 & 0.05 & 455\
1032$-$199 & 2015-06-19 & 0.43 & 0.17 & $-$1.7 & 0.335 & 0.234 & 0.42 & 0.31 & 27 & 0.24 & 432\
1036$+$054 & 2015-10-10 & 0.52 & 0.17 & $-$10.8 & 0.332 & 0.198 & 0.90 & 0.32 & 28 & 0.22 & 415\
1038$+$064 & 2015-10-10 & 0.63 & 0.18 & $-$15.7 & 0.381 & 0.326 & 0.39 & 0.37 & 27 & 0.23 & 414\
1045$-$188 & 2015-10-10 & 0.49 & 0.16 & $-$9.6 & 0.738 & 0.423 & 0.90 & 0.74 & 20 & 0.19 & 420\
1049$+$215 & 2015-10-10 & 0.43 & 0.16 & $-$1.3 & 0.246 & 0.128 & 0.30 & 0.25 & 30 & 0.05 & 460\
1124$-$186 & 2015-10-10 & 0.49 & 0.17 & $-$7.7 & 1.269 & 1.100 & 1.20 & 1.28 & 22 & 0.93 & 440\
1149$-$084 & 2015-06-19 & 0.42 & 0.17 & $-$5.6 & 0.326 & 0.126 & 0.48 & 0.32 & 28 & 0.06 & 410\
1150$+$497 & 2015-06-19 & 0.45 & 0.17 & 7.8 & 0.837 & 0.617 & 0.44 & 0.81 & 31 & 0.52 & 480\
1213$-$172 & 2015-10-10 & 0.44 & 0.14 & $-$3.7 & 0.560 & 0.158 & 0.58 & 0.54 & 25 & 0.19 & 370\
1219$+$044 & 2015-06-20 & 0.44 & 0.17 & $-$5.1 & 0.628 & 0.563 & 0.30 & 0.63 & 25 & 0.37 & 418\
1228$+$126 & 2015-06-20 & 0.42 & 0.18 & $-$3.5 & 1.331 & 0.604 & 0.68 & 1.26 & 28 & 0.29 & 442\
1243$-$072 & 2016-02-09 & 0.73 & 0.20 & 21.6 & 0.928 & 0.754 & 0.40 & 0.91 & 62 & 0.65 & 207\
1257$+$519 & 2015-06-20 & 0.42 & 0.18 & $-$2.1 & 0.339 & 0.289 & 0.18 & 0.34 & 31 & 0.26 & 530\
1306$+$360 & 2015-06-20 & 0.41 & 0.18 & $-$3.5 & 0.669 & 0.625 & 0.28 & 0.67 & 33 & 0.54 & 520\
1324$+$224 & 2016-02-09 & 0.53 & 0.20 & 25.4 & 0.314 & 0.281 & 0.24 & 0.31 & 84 & 0.29 & 248\
1329$-$049 & 2015-06-20 & 0.47 & 0.17 & $-$6.5 & 0.811 & 0.557 & 0.52 & 0.68 & 25 & 0.40 & 422\
1354$+$195 & 2016-02-09 & 0.53 & 0.20 & 26.1 & 1.254 & 0.583 & 0.28 & 1.06 & 82 & 0.43 & 244\
1417$+$385 & 2016-02-09 & 0.49 & 0.31 & 32.1 & 0.172 & 0.145 & 0.14 & 0.17 & 82 & 0.15 & 268\
1435$+$638 & 2016-02-09 & 0.56 & 0.28 & 40.7 & 0.267 & 0.099 & 0.45 & 0.25 & 28 & 0.16 & 368\
1502$+$106 & 2016-02-09 & 0.54 & 0.25 & 26.7 & 2.909 & 2.770 & 0.48 & 2.90 & 32 & 2.85 & 234\
1504$-$166 & 2016-02-09 & 0.75 & 0.22 & 15 & 0.553 & 0.511 & 0.24 & 0.55 & 44 & 0.52 & 204\
1514$+$004 & 2016-02-09 & 0.60 & 0.22 & 21.7 & 0.529 & 0.299 & 0.35 & 0.39 & 68 & 0.38 & 210\
1514$+$197 & 2016-02-09 & 0.53 & 0.21 & 28.1 & 0.628 & 0.501 & 0.12 & 0.63 & 32 & 0.59 & 253\
1532$+$016 & 2016-02-09 & 0.56 & 0.19 & 19.5 & 0.419 & 0.223 & 0.35 & 0.44 & 30 & 0.24 & 291\
1546$+$027 & 2016-02-09 & 0.55 & 0.22 & 12.6 & 2.516 & 2.060 & 0.98 & 2.51 & 31 & 1.71 & 266\
1548$+$056 & 2015-06-20 & 0.43 & 0.18 & $-$1.8 & 1.056 & 0.448 & 0.59 & 0.84 & 31 & 0.23 & 414\
1606$+$106 & 2016-02-09 & 0.50 & 0.21 & 16.6 & 0.549 & 0.496 & 0.23 & 0.57 & 58 & 0.48 & 287\
1636$+$473 & 2016-02-09 & 0.46 & 0.21 & 16.4 & 0.462 & 0.374 & 0.48 & 0.45 & 85 & 0.30 & 333\
1637$+$826 & 2016-02-09 & 0.56 & 0.22 & 4.4 & 0.432 & 0.258 & 0.21 & 0.43 & 25 & 0.11 & 267\
1639$-$062 & 2016-02-09 & 0.66 & 0.21 & 21.1 & 0.709 & 0.628 & 0.24 & 0.70 & 69 & 0.59 & 268\
1642$+$690 & 2015-06-20 & 0.57 & 0.21 & 7.4 & 0.780 & 0.489 & 1.40 & 0.76 & 27 & 0.42 & 312\
1655$+$077 & 2016-02-09 & 0.55 & 0.23 & 15.5 & 0.480 & 0.348 & 0.73 & 0.44 & 78 & 0.30 & 295\
1656$+$477 & 2016-02-09 & 0.44 & 0.22 & 1.0 & 0.594 & 0.512 & 0.30 & 0.54 & 81 & 0.44 & 370\
1657$-$261 & 2015-06-20 & 0.44 & 0.16 & $-$7.0 & 1.122 & 1.080 & 1.30 & 1.13 & 27 & 1.04 & 406\
1659$+$399 & 2015-06-20 & 0.46 & 0.17 & 2.6 & 0.103 & 0.075 & 0.17 & 0.11 & 33 & 0.05 & 420\
1716$+$686 & 2016-04-22 & 0.49 & 0.18 & 2.9 & 0.232 & 0.100 & 0.22 & 0.24 & 29 & 0.05 & 320\
1725$+$044 & 2016-02-09 & 0.54 & 0.20 & 18.4 & 0.421 & 0.290 & 0.46 & 0.41 & 76 & 0.27 & 293\
1726$+$455 & 2016-02-09 & 0.44 & 0.22 & 3.0 & 0.887 & 0.526 & 0.21 & 0.88 & 83 & 0.33 & 360\
1800$+$440 & 2016-02-09 & 0.43 & 0.21 & 8.2 & 1.025 & 0.807 & 0.55 & 1.02 & 33 & 0.54 & 356\
1806$+$456 & 2016-04-22 & 0.51 & 0.15 & 12.8 & 0.389 & 0.279 & 0.45 & 0.39 & 31 & 0.16 & 415\
1807$+$698 & 2016-04-22 & 0.52 & 0.16 & 16.8 & 0.887 & 0.541 & 1.30 & 0.83 & 29 & 0.54 & 320\
1842$+$681 & 2016-02-09 & 0.46 & 0.22 & 6.5 & 0.179 & 0.126 & 0.25 & 0.18 & 30 & 0.12 & 350\
1849$+$670 & 2016-02-09 & 0.48 & 0.23 & 3.2 & 1.449 & 1.250 & 0.89 & 1.45 & 30 & 1.03 & 350\
1920$-$211 & 2016-04-22 & 0.44 & 0.16 & $-$8.9 & 1.099 & 0.958 & 0.24 & 1.09 & 27 & 0.78 & 430\
1926$+$611 & 2016-04-22 & 0.50 & 0.16 & 4.8 & 0.488 & 0.224 & 0.26 & 0.48 & 30 & 0.29 & 460\
1957$-$135 & 2016-01-18 & 0.42 & 0.17 & 2.3 & 1.605 & 1.360 & 0.80 & 1.56 & 26 & 1.15 & 408\
1958$-$179 & 2016-01-18 & 0.40 & 0.16 & $-$0.8 & 2.830 & 2.230 & 0.59 & 2.45 & 25 & 1.82 & 412\
2007$+$777 & 2016-04-22 & 0.49 & 0.17 & 15.6 & 0.703 & 0.440 & 0.37 & 0.66 & 28 & 0.26 & 356\
2021$+$317 & 2016-01-18 & 0.50 & 0.17 & 8.5 & 0.892 & 0.443 & 0.51 & 0.83 & 33 & 0.35 & 425\
2021$+$614 & 2016-04-22 & 0.51 & 0.19 & 11.7 & 0.578 & 0.193 & 0.60 & 0.49 & 29 & 0.04 & 375\
2022$-$077 & 2016-01-18 & 0.43 & 0.17 & 2.5 & 1.548 & 1.430 & 0.45 & 1.54 & 27 & 1.22 & 415\
2023$+$335 & 2016-01-18 & 0.48 & 0.17 & 8.1 & 1.113 & 0.762 & 0.50 & 1.12 & 33 & 0.75 & 426\
2029$+$121 & 2016-04-22 & 0.43 & 0.16 & $-$5.0 & 0.546 & 0.347 & 0.48 & 0.54 & 30 & 0.31 & 414\
2126$-$158 & 2016-01-18 & 0.47 & 0.18 & 4.6 & 0.359 & 0.219 & 0.35 & 0.36 & 29 & 0.26 & 375\
2134$+$004 & 2016-04-22 & 0.75 & 0.15 & $-$19.0 & 2.294 & 0.779 & 1.50 & 1.55 & 32 & 0.79 & 260\
2135$+$508 & 2016-01-18 & 0.52 & 0.17 & 17.5 & 0.189 & 0.163 & 0.11 & 0.17 & 30 & 0.14 & 445\
2141$+$175 & 2016-04-22 & 0.41 & 0.16 & 2.1 & 0.546 & 0.344 & 0.48 & 0.44 & 33 & 0.26 & 404\
2142$+$110 & 2016-04-22 & 0.43 & 0.16 & $-$9.1 & 0.219 & 0.183 & 0.35 & 0.23 & 28 & 0.18 & 400\
2144$+$092 & 2016-01-18 & 0.46 & 0.18 & 8.4 & 0.762 & 0.604 & 0.17 & 0.76 & 32 & 0.56 & 406\
2155$-$152 & 2016-04-22 & 0.62 & 0.17 & $-$11.4 & 1.856 & 1.100 & 2.70 & 1.80 & 22 & 0.59 & 402\
2201$+$171 & 2016-01-18 & 0.45 & 0.18 & 12.2 & 0.849 & 0.805 & 0.43 & 0.85 & 33 & 0.74 & 426\
2209$+$236 & 2016-01-18 & 0.47 & 0.18 & 14.8 & 0.482 & 0.244 & 0.29 & 0.48 & 20 & 0.09 & 425\
2216$-$038 & 2016-01-18 & 0.43 & 0.17 & 2.7 & 0.900 & 0.574 & 0.43 & 0.80 & 28 & 0.50 & 408\
2223$+$210 & 2016-04-22 & 0.47 & 0.16 & $-$6.0 & 0.475 & 0.300 & 0.33 & 0.48 & 30 & 0.29 & 412\
2227$-$088 & 2016-01-18 & 0.44 & 0.17 & 2.5 & 2.732 & 2.590 & 1.25 & 2.73 & 27 & 2.26 & 410\
2234$+$282 & 2016-01-18 & 0.51 & 0.17 & 0.1 & 1.139 & 0.871 & 0.70 & 1.09 & 33 & 0.82 & 385\
2243$-$123 & 2016-01-18 & 0.42 & 0.17 & 3.0 & 1.673 & 1.190 & 0.86 & 1.62 & 26 & 0.29 & 412\
2308$+$341 & 2016-01-18 & 0.51 & 0.18 & $-$6.8 & 0.440 & 0.346 & 0.50 & 0.41 & 33 & 0.29 & 395\
2318$+$049 & 2016-01-18 & 0.47 & 0.19 & $-$4.3 & 0.548 & 0.333 & 0.20 & 0.54 & 26 & 0.24 & 397\
2320$-$035 & 2016-01-18 & 0.44 & 0.18 & $-$0.1 & 1.049 & 0.739 & 0.57 & 1.04 & 22 & 0.74 & 412\
2325$+$093 & 2016-01-18 & 0.46 & 0.18 & $-$3.8 & 2.019 & 1.850 & 0.45 & 2.02 & 26 & 1.70 & 409\
2342$-$161 & 2016-01-18 & 0.44 & 0.18 & 3.2 & 1.393 & 1.190 & 0.70 & 1.40 & 18 & 0.84 & 420\
2344$+$092 & 2016-01-18 & 0.47 & 0.21 & 0.9 & 0.124 & 0.084 & 0.42 & 0.12 & 26 & 0.06 & 410\
2345$-$167 & 2016-01-18 & 0.47 & 0.18 & $-$0.6 & 2.026 & 1.470 & 0.37 & 2.03 & 17 & 1.18 & 376\
2351$-$154 & 2016-01-19 & 0.44 & 0.18 & 0.7 & 0.355 & 0.182 & 0.38 & 0.35 & 17 & 0.12 & 420\
[cccccccccc]{} 0048$-$071... & C & C & 0.424 & 0.412 & ... & ... & 0.05 & 33.1\
& & J3 & 0.028 & 0.024 & 0.43 & $-$39.97 & 0.13 &\
& & J2 & 0.021 & 0.014 & 0.90 & $-$52.83 & 0.21 &\
& & J1 & 0.007 & 0.002 & 2.36 & $-$57.45 & 0.53 &\
0106$+$013... & B & C & 0.869 & 0.801 & ... & ... & 0.05 & 70.7\
& & J2 & 0.118 & 0.103 & 0.35 & $-$116.37 & 0.10 &\
& & J1 & 0.215 & 0.036 & 1.36 & $-$136.52 & 1.04 &\
0113$-$118... & C & C & 0.657 & 0.569 & ... & ... & 0.12 & 5.0\
& & J1 & 0.041 & 0.011 & 0.35 & $-$48.46 & 0.44 &\
0119$+$041... & C & C & 0.480 & 0.345 & ... & ... & 0.18 & 1.6\
& & J2 & 0.048 & 0.022 & 1.11 & 85.24 & 0.34 &\
& & J1 & 0.018 & 0.005 & 1.57 & 93.71 & 0.76 &\
0122$-$003... & C & C & 0.225 & 0.256 & ... & ... & 0.05 & 12.3\
& & J3 & 0.162 & 0.149 & 0.17 & $-$99.15 & 0.18 &\
& & J2 & 0.016 & 0.004 & 0.99 & $-$98.53 & 0.50 &\
& & J1 & 0.022 & 0.005 & 2.37 & $-$93.12 & 0.61 &\
0130$-$171... & B & C & 0.438 & 0.391 & ... & ... & 0.08 & 9.1\
& & J2 & 0.084 & 0.015 & 0.34 & $-$110.92 & 0.57 &\
& & J1 & 0.075 & 0.018 & 1.01 & $-$108.83 & 0.47 &\
0149$+$218... & C & C & 1.071 & 0.969 & ... & ... & 0.09 & 20.1\
& & J1 & 0.089 & 0.198 & 0.42 & $-$6.17 & 0.13 &\
0202$-$172... & C & C & 0.757 & 0.676 & ... & ... & 0.10 & 13.6\
& & J4 & 0.101 & 0.055 & 0.88 & $-$6.04 & 0.27 &\
& & J3 & 0.061 & 0.011 & 2.75 & 4.99 & 0.73 &\
& & J2 & 0.035 & 0.038 & 4.37 & $-$0.73 &$<$0.12 &\
& & J1 & 0.061 & 0.045 & 4.76 & 2.87 & 0.17 &\
0208$+$106... & C & C & 0.325 & 0.314 & ... & ... & 0.06 & 7.1\
& & J3 & 0.138 & 0.111 & 0.35 & 125.01 & 0.15 &\
& & J2 & 0.019 & 0.006 & 0.76 & 104.84 & 0.56 &\
& & J1 & 0.048 & 0.008 & 1.43 & 49.64 & 1.54 &\
0221$+$067... & B & C & 0.911 & 0.696 & ... & ... & 0.06 & 25.1\
& & J2 & 0.077 & 0.279 & 0.23 & $-$76.07 & 0.20 &\
& & J1 & 0.032 & 0.020 & 1.21 & $-$78.87 & 1.36 &\
0224$+$671... & C & C & 0.352 & 0.294 & ... & ... & 0.13 & 2.1\
& & J1 & 0.166 & 0.106 & 0.67 & 10.41 & 0.22 &\
0229$+$131... & C & C & 0.460 & 0.442 & ... & ... & 0.07 & 18.8\
& & J1 & 0.232 & 0.101 & 0.84 & 63.58 & 0.40 &\
0239$+$108... & C & C & 0.215 & 0.189 & ... & ... & 0.14 & 2.6\
& & J2 & 0.018 & 0.019 & 0.51 & 117.76 &$<$0.12 &\
& & J1 & 0.007 & 0.004 & 2.83 & 137.52 & 0.36 &\
0241$+$622... & C & C & 0.802 & 0.693 & ... & ... & 0.13 & 3.3\
& & J1 & 0.023 & 0.016 & 0.54 & 99.16 & 0.22 &\
& & JW & 0.007 & 0.006 & 0.55 & $-$100.79 &$<$0.12 &\
0306$+$102... & C & C & 0.882 & 0.776 & ... & ... & 0.12 & 7.5\
& & J1 & 0.017 & 0.016 & 0.73 & 60.58 & 0.11 &\
0309$+$411... & B & C & 0.331 & 0.388 & ... & ... & 0.06 & 6.9\
& & J4 & 0.218 & 0.353 & 0.14 & $-$57.36 & 0.10 &\
& & J3 & 0.111 & 0.261 & 0.23 & $-$55.18 & 0.11 &\
& & J2 & 0.064 & 0.082 & 0.42 & $-$54.16 & 0.29 &\
& & J1 & 0.011 & 0.008 & 0.84 & $-$56.88 & 0.22 &\
0322$+$222... & C & C & 0.509 & 0.487 & ... & ... & 0.08 & 16.1\
& & J2 & 0.032 & 0.158 & 0.24 & 128.47 & 0.26 &\
& & J1 & 0.012 & 0.006 & 1.06 & 90.61 & 0.50 &\
0354$+$559... & B & C & 0.202 & 0.211 & ... & ... & 0.05 & 10.6\
& & J2 & 0.155 & 0.107 & 0.22 & $-$67.95 & 0.18 &\
& & J1 & 0.164 & 0.042 & 0.42 & $-$140.71 & 0.35 &\
0400$+$258... & C & C & 0.175 & 0.112 & ... & ... & 0.21 & 0.8\
& & J2 & 0.051 & 0.019 & 0.95 & 87.28 & 0.39 &\
& & J1 & 0.011 & 0.009 & 1.43 & 103.90 & 0.16 &\
0403$-$132... & C & C & 0.300 & 0.281 & ... & ... & 0.10 & 3.1\
& & J2 & 0.026 & 0.025 & 2.53 & 149.29 &$<$0.13 &\
& & J1 & 0.011 & 0.011 & 3.17 & 149.86 &$<$0.14 &\
0405$-$123... & B & C & 0.434 & 0.389 & ... & ... & 0.06 & 12.4\
& & J1 & 0.367 & 0.285 & 0.50 & 11.98 & 0.10 &\
0420$+$417... & C & C & 0.266 & 0.222 & ... & ... & 0.14 & 1.8\
& & J1 & 0.034 & 0.034 & 0.56 & $-$97.76 &$<$0.15 &\
0451$-$282... & B & C & 0.764 & 0.741 & ... & ... & 0.05 & 71.4\
& & J2 & 0.182 & 0.097 & 0.57 & $-$1.95 & 0.33 &\
& & J1 & 0.012 & 0.011 & 1.96 & $-$7.53 & 0.08 &\
0507$+$179... & C & C & 2.464 & 2.440 & ... & ... & 0.03 & 254.4\
& & J1 & 0.024 & 0.024 & 0.61 & $-$105.16 & 0.31 &\
0529$+$075... & C & C & 1.232 & 1.042 & ... & ... & 0.08 & 28.5\
& & J2 & 0.084 & 0.014 & 1.24 & $-$8.14 & 0.89 &\
& & J1 & 0.036 & 0.028 & 1.92 & $-$10.21 & 0.26 &\
0605$-$085... & B & C & 0.622 & 0.464 & ... & ... & 0.08 & 4.7\
& & J2 & 1.356 & 0.639 & 0.44 & 81.08 & 0.16 &\
& & J1 & 0.069 & 0.075 & 0.77 & 119.99 & 0.11 &\
0627$-$199... & B & C & 0.359 & 0.341 & ... & ... & 0.05 & 25.7\
& & J1 & 0.064 & 0.049 & 0.41 & 25.26 & 0.13 &\
0633$+$734... & C & C & 0.386 & 0.381 & ... & ... & 0.05 & 28.9\
& & J2 & 0.018 & 0.081 & 0.45 & $-$11.51 & 0.17 &\
& & J1 & 0.007 & 0.006 & 1.62 & $-$5.61 & 0.14 &\
0648$-$165... & C & C & 0.343 & 0.352 & ... & ... & 0.06 & 12.5\
& & J2 & 0.106 & 0.087 & 0.48 & $-$3.07 & 0.54 &\
& & J1 & 0.052 & 0.011 & 0.83 & $-$67.81 & 0.94 &\
0657$+$172... & B & C & 0.202 & 0.184 & ... & ... & 0.09 & 3.3\
& & J1 & 0.133 & 0.031 & 0.38 & $-$61.04 & 0.47 &\
0708$+$506... & B & C & 0.344 & 0.330 & ... & ... & 0.07 & 6.9\
& & J1 & 0.026 & 0.054 & 0.20 & 99.33 & 0.22 &\
0723$-$008... & B & C & 0.651 & 0.558 & ... & ... & 0.09 & 5.7\
& & J3 & 0.104 & 0.084 & 0.48 & $-$40.07 & 0.12 &\
& & J2 & 0.344 & 0.105 & 1.75 & $-$39.71 & 0.41 &\
& & J1 & 0.672 & 0.152 & 2.65 & $-$34.62 & 0.51 &\
0730$+$504... & C & C & 0.189 & 0.127 & ... & ... & 0.08 & 3.3\
& & J1 & 0.026 & 0.007 & 1.03 & $-$139.55 & 0.42 &\
0736$+$017... & C & C & 0.680 & 0.424 & ... & ... & 0.18 & 1.6\
& & J1 & 0.026 & 0.023 & 0.41 & $-$42.26 &$<$0.12 &\
0738$+$313... & C & C & 0.497 & 0.467 & ... & ... & 0.07 & 10.9\
& & J3 & 0.201 & 0.226 & 0.45 & 177.20 & 0.09 &\
& & J2 & 0.101 & 0.111 & 0.83 & $-$178.13 & 0.14 &\
& & J1 & 0.343 & 0.059 & 3.47 & 174.32 & 0.43 &\
0742$+$103... & C & C & 0.286 & 0.257 & ... & ... & 0.09 & 8.4\
& & J2 & 0.101 & 0.005 & 2.18 & $-$9.50 & 1.75 &\
& & J1 & 0.025 & 0.008 & 3.04 & $-$34.22 & 0.49 &\
0743$-$006... & C & C & 0.190 & 0.177 & ... & ... & 0.09 & 3.1\
& & J4 & 0.071 & 0.066 & 0.21 & 46.52 & 0.35 &\
& & J3 & 0.095 & 0.071 & 0.91 & 57.87 & 0.15 &\
& & J2 & 0.038 & 0.044 & 1.14 & 52.67 & 0.21 &\
& & J1 & 0.097 & 0.048 & 1.46 & 48.98 & 0.25 &\
0754$+$100... & C & C & 0.387 & 0.480 & ... & ... &$<$0.04 & $>$20.1\
& & J4 & 0.151 & 0.401 & 0.18 & 4.23 & 0.07 &\
& & J3 & 0.080 & 0.151 & 0.40 & 9.19 & 0.16 &\
& & J2 & 0.039 & 0.023 & 1.38 & 16.62 & 0.20 &\
& & J1 & 0.012 & 0.009 & 1.79 & 16.86 & 0.26 &\
0805$+$410... & C & C & 0.844 & 0.643 & ... & ... & 0.18 & 4.1\
0805$-$077... & B & C & 0.554 & 0.534 & ... & ... & 0.05 & 41.3\
& & J1 & 0.051 & 0.141 & 0.39 & $-$11.74 & 0.11 &\
0808$+$019... & C & C & 1.152 & 1.106 & ... & ... & 0.05 & 101.5\
& & J2 & 0.013 & 0.041 & 0.53 & $-$174.60 &$<$0.11 &\
& & J1 & 0.010 & 0.011 & 0.74 & $-$165.27 &$<$0.12 &\
0821$+$394... & C & C & 0.283 & 0.206 & ... & ... & 0.12 & 2.9\
& & J2 & 0.028 & 0.050 & 0.34 & $-$54.69 & 0.28 &\
& & J1 & 0.009 & 0.002 & 1.09 & $-$52.76 & 0.64 &\
0838$+$133... & C & C & 0.552 & 0.436 & ... & ... & 0.12 & 4.2\
& & J2 & 0.113 & 0.024 & 0.32 & 93.04 & 0.51 &\
& & J1 & 0.046 & 0.018 & 0.97 & 98.55 & 0.30 &\
0847$-$120... & C & C & 0.837 & 0.743 & ... & ... & 0.08 & 13.4\
0859$-$140... & C & C & 0.092 & 0.086 & ... & ... & 0.10 & 1.4\
& & J3 & 0.036 & 0.055 & 0.23 & 130.41 & 0.09 &\
& & J2 & 0.020 & 0.011 & 0.86 & 139.42 & 0.23 &\
& & J1 & 0.050 & 0.010 & 1.73 & 152.31 & 0.57 &\
0906$+$015... & C & C & 0.437 & 0.402 & ... & ... & 0.08 & 9.1\
& & J3 & 0.207 & 0.141 & 0.38 & 44.18 & 0.17 &\
& & J2 & 0.113 & 0.059 & 0.85 & 45.91 & 0.24 &\
& & J1 & 0.032 & 0.007 & 1.63 & 41.81 & 0.47 &\
0917$+$624... & C & C & 0.451 & 0.420 & ... & ... & 0.05 & 29.0\
& & J4 & 0.028 & 0.026 & 0.55 & $-$31.91 & 0.17 &\
& & J3 & 0.026 & 0.015 & 1.05 & $-$36.22 & 0.19 &\
& & J2 & 0.013 & 0.007 & 1.39 & $-$49.29 & 0.23 &\
& & J1 & 0.054 & 0.011 & 2.03 & $-$40.78 & 0.54 &\
0925$-$203... & B & C & 0.257 & 0.249 & ... & ... & 0.05 & 9.1\
& & J1 & 0.113 & 0.058 & 0.87 & $-$30.23 & 0.21 &\
0945$+$408... & B & C & 0.227 & 0.152 & ... & ... & 0.15 & 1.5\
& & J3 & 0.014 & 0.011 & 0.52 & 96.89 & 0.11 &\
& & J2 & 0.013 & 0.007 & 1.10 & 104.19 & 0.27 &\
& & J1 & 0.054 & 0.011 & 1.61 & 110.17 & 0.55 &\
1032$-$199... & B & C & 0.236 & 0.233 & ... & ... & 0.05 & 19.8\
& & J2 & 0.051 & 0.032 & 0.49 & $-$141.11 & 0.18 &\
& & J1 & 0.048 & 0.008 & 1.12 & $-$142.63 & 0.58 &\
1036$+$054... & C & C & 0.072 & 0.070 & ... & ... &$<$0.04 & $>$4.3\
& & J3 & 0.197 & 0.199 & 0.60 & $-$19.61 & 0.05 &\
& & J2 & 0.029 & 0.114 & 0.85 & $-$14.21 & 0.15 &\
& & J1 & 0.034 & 0.019 & 1.78 & $-$15.38 & 0.22 &\
1038$+$064... & C & C & 0.363 & 0.326 & ... & ... & 0.08 & 8.4\
& & J1 & 0.018 & 0.007 & 1.42 & 178.29 & 0.36 &\
1045$-$188... & C & C & 0.152 & 0.126 & ... & ... & 0.12 & 1.1\
& & J1 & 0.586 & 0.422 & 0.75 & 157.72 & 0.21 &\
1049$+$215... & C & C & 0.246 & 0.128 & ... & ... & 0.22 & 0.8\
1124$-$186... & & C & 1.238 & 1.10 & ... & ... & 0.21 & 3.8\
& & J1 & 0.031 & 0.103 & 0.64 & 160.97 & 0.41 &\
1149$-$084... & B & C & 0.185 & 0.126 & ... & ... & 0.16 & 1.6\
& & J1 & 0.141 & 0.107 & 0.29 & 109.53 & 0.14 &\
1150$+$497... & B & C & 0.661 & 0.617 & ... & ... & 0.09 & 7.1\
& & J4 & 0.059 & 0.155 & 0.46 & $-$163.18 & 0.15 &\
& & J3 & 0.078 & 0.054 & 0.91 & $-$151.19 & 0.16 &\
& & J2 & 0.028 & 0.013 & 1.55 & $-$152.13 & 0.28 &\
& & J1 & 0.011 & 0.010 & 1.91 & $-$157.46 & 0.18 &\
1213$-$172... & C & C & 0.357 & 0.155 & ... & ... & 0.35 & 0.4\
& & J1 & 0.203 & 0.077 & 0.48 & 80.74 & 0.27 &\
1219$+$044... & B & C & 0.622 & 0.563 & ... & ... & 0.10 & 8.1\
& & J1 & 0.006 & 0.006 & 0.64 & $-$179.81 & 0.05 &\
1228$+$126... & B & C & 0.724 & 0.604 & ... & ... & 0.13 & 2.8\
& & J4 & 0.148 & 0.291 & 0.19 & $-$52.42 & 0.24 &\
& & J3 & 0.202 & 0.033 & 0.54 & $-$89.42 & 0.62 &\
& & J2 & 0.069 & 0.016 & 1.30 & $-$51.87 & 0.54 &\
& & J1 & 0.188 & 0.006 & 1.74 & $-$91.03 & 1.82 &\
1243$-$072... & C & C & 0.731 & 0.754 & ... & ... & 0.05 & 43.9\
& & J1 & 0.197 & 0.162 & 0.28 & $-$121.06 & 0.47 &\
1257$+$519... & B & C & 0.308 & 0.289 & ... & ... & 0.09 & 3.5\
& & J1 & 0.031 & 0.110 & 0.16 & $-$108.9 & 0.11 &\
1306$+$360... & B & C & 0.653 & 0.625 & ... & ... & 0.08 & 13.8\
& & J1 & 0.016 & 0.027 & 0.46 & $-$11.64 & 0.12 &\
1324$+$224... & C & C & 0.295 & 0.281 & ... & ... & 0.06 & 12.9\
& & J1 & 0.019 & 0.004 & 0.61 & $-$60.96 & 0.69 &\
1329$-$049... & B & C & 0.811 & 0.557 & ... & ... & 0.14 & 8.6\
1354$+$195... & C & C & 0.671 & 0.583 & ... & ... & 0.11 & 6.3\
& & J2 & 0.451 & 0.307 & 0.43 & 115.69 & 0.19 &\
& & J1 & 0.132 & 0.033 & 1.41 & 128.88 & 0.44 &\
1417$+$385... & C & C & 0.172 & 0.145 & ... & ... & 0.16 & 1.2\
1435$+$638... & C & C & 0.108 & 0.095 & ... & ... & 0.08 & 0.8\
& & J1 & 0.159 & 0.101 & 1.72 & $-$128.04 & 0.20 &\
1502$+$106... & C & C & 2.827 & 2.766 & ... & ... & 0.06 & 146.2\
& & J1 & 0.082 & 0.011 & 0.51 & 118.79 & 0.08 &\
1504$-$166... & C & C & 0.537 & 0.511 & ... & ... & 0.07 & 13.5\
& & J2 & 0.012 & 0.189 & 0.52 & $-$174.56 & 0.28 &\
& & J1 & 0.004 & 0.003 & 1.93 & 177.99 & 0.33 &\
1514$+$004... & C & C & 0.348 & 0.299 & ... & ... & 0.12 & 1.7\
& & J1 & 0.081 & 0.059 & 0.66 & $-$30.51 & 0.18 &\
1514$+$197... & C & C & 0.617 & 0.501 & ... & ... & 0.11 & 6.9\
& & J1 & 0.011 & 0.002 & 1.09 & $-$17.81 & 0.67 &\
1532$+$016... & C & C & 0.304 & 0.223 & ... & ... & 0.16 & 1.9\
& & J1 & 0.115 & 0.019 & 0.95 & $-$129.83 & 0.69 &\
1546$+$027... & C & C & 2.485 & 2.061 & ... & ... & 0.15 & 10.2\
& & J1 & 0.031 & 0.011 & 1.02 & $-$176.68 & 0.45 &\
1548$+$056... & B & C & 0.558 & 0.448 & ... & ... & 0.08 & 13.8\
& & J3 & 0.188 & 0.103 & 1.21 & $-$7.89 & 0.23 &\
& & J2 & 0.205 & 0.031 & 2.06 & $-$6.35 & 0.66 &\
& & J1 & 0.105 & 0.007 & 3.06 & 7.41 & 1.06 &\
1606$+$106... & C & C & 0.523 & 0.496 & ... & ... & 0.07 & 15.6\
& & J1 & 0.026 & 0.002 & 1.06 & $-$52.14 & 1.01 &\
1636$+$473... & C & C & 0.445 & 0.374 & ... & ... & 0.08 & 7.9\
& & J1 & 0.017 & 0.005 & 1.11 & $-$11.66 & 0.47 &\
1637$+$826... & C & C & 0.365 & 0.258 & ... & ... & 0.17 & 0.8\
& & J2 & 0.036 & 0.036 & 0.41 & $-$72.25 & 0.29 &\
& & J1 & 0.031 & 0.014 & 0.92 & $-$64.21 & 0.41 &\
1639$-$062... & C & C & 0.677 & 0.628 & ... & ... & 0.09 & 13.6\
& & J1 & 0.032 & 0.005 & 0.41 & 101.62 & 0.80 &\
1642$+$690... & B & C & 0.592 & 0.489 & ... & ... & 0.13 & 4.1\
& & J1 & 0.188 & 0.108 & 1.21 & $-$158.27 & 0.25 &\
1655$+$077... & C & C & 0.393 & 0.348 & ... & ... & 0.11 & 3.5\
& & J1 & 0.087 & 0.026 & 0.55 & $-$46.81 & 0.35 &\
1656$+$477... & C & C & 0.558 & 0.512 & ... & ... & 0.11 & 7.9\
& & J2 & 0.013 & 0.021 & 0.63 & 0.49 & 0.24 &\
& & J1 & 0.023 & 0.011 & 1.44 & $-$1.21 & 0.30 &\
1657$-$261... & B & C & 1.122 & 1.079 & ... & ... & 0.06 & 40.9\
1659$+$399... & B & C & 0.083 & 0.075 & ... & ... & 0.08 & 1.3\
& & J1 & 0.020 & 0.013 & 0.49 & $-$161.78 & 0.21 &\
1716$+$686... & B & C & 0.152 & 0.100 & ... & ... & 0.09 & 2.2\
& & J2 & 0.056 & 0.054 & 0.49 & $-$21.09 & 0.19 &\
& & J1 & 0.024 & 0.017 & 0.89 & $-$26.30 & 0.29 &\
1725$+$044... & C & C & 0.387 & 0.290 & ... & ... & 0.10 & 3.3\
& & J1 & 0.034 & 0.009 & 0.45 & 119.59 & 0.52 &\
1726$+$455... & C & C & 0.500 & 0.526 & ... & ... & 0.06 & 15.6\
& & J1 & 0.387 & 0.321 & 0.31 & $-$137.26 & 0.21 &\
1800$+$440... & C & C & 0.742 & 0.805 & ... & ... & 0.05 & 32.4\
& & J1 & 0.283 & 0.404 & 0.35 & $-$148.77 & 0.14 &\
1806$+$456... & B & C & 0.355 & 0.279 & ... & ... & 0.12 & 3.0\
& & J2 & 0.028 & 0.027 & 0.53 & 176.89 & 0.22 &\
& & J1 & 0.006 & 0.006 & 0.96 & $-$173.55 & 0.29 &\
1807$+$698... & B & C & 0.623 & 0.541 & ... & ... & 0.09 & 5.3\
& & J2 & 0.169 & 0.066 & 0.36 & $-$106.82 & 0.31 &\
& & J1 & 0.095 & 0.039 & 1.14 & $-$105.22 & 0.30 &\
1842$+$681... & C & C & 0.130 & 0.126 & ... & ... & 0.07 & 2.6\
& & J2 & 0.040 & 0.046 & 0.27 & 133.08 & 0.07 &\
& & J1 & 0.009 & 0.016 & 0.51 & 135.84 & 0.26 &\
1849$+$670... & C & C & 1.306 & 1.252 & ... & ... & 0.06 & 39.4\
& & J1 & 0.143 & 0.091 & 0.51 & $-$21.91 & 0.34 &\
1920$-$211... & B & C & 1.012 & 0.958 & ... & ... & 0.05 & 49.8\
& & J2 & 0.039 & 0.056 & 0.49 & 1.89 & 0.28 &\
& & J1 & 0.048 & 0.002 & 2.44 & $-$15.52 & 1.33 &\
1926$+$611... & B & C & 0.220 & 0.224 & ... & ... & 0.05 & 11.5\
& & J3 & 0.189 & 0.179 & 0.17 & 84.83 &$<$0.09 &\
& & J2 & 0.055 & 0.017 & 0.58 & 118.94 & 0.18 &\
& & J1 & 0.024 & 0.003 & 1.41 & 121.81 & 0.68 &\
1957$-$135... & C & C & 1.452 & 1.360 & ... & ... & 0.13 & 6.9\
& & J2 & 0.083 & 0.064 & 1.11 & 17.79 & 0.12 &\
& & J1 & 0.070 & 0.046 & 1.67 & 17.82 & 0.16 &\
1958$-$179... & C & C & 2.805 & 2.233 & ... & ... & 0.13 & 18.1\
& & J1 & 0.025 & 0.025 & 0.44 & 100.22 &$<$0.12 &\
2007$+$777... & B & C & 0.569 & 0.440 & ... & ... & 0.06 & 13.9\
& & J3 & 0.023 & 0.013 & 0.41 & $-$97.44 & 0.29 &\
& & J2 & 0.052 & 0.033 & 0.81 & $-$103.29 & 0.19 &\
& & J1 & 0.059 & 0.017 & 1.53 & $-$89.68 & 0.42 &\
2021$+$317... & C & C & 0.101 & 0.086 & ... & ... & 0.09 & 0.9\
& & J5 & 0.491 & 0.441 & 0.41 & 126.62 & 0.11 &\
& & J4 & 0.213 & 0.196 & 0.80 & $-$155.18 & 0.18 &\
& & J3 & 0.056 & 0.011 & 1.65 & $-$160.32 & 0.56 &\
& & J2 & 0.020 & 0.003 & 2.74 & $-$165.69 & 0.61 &\
& & J1 & 0.011 & 0.011 & 4.02 & $-$170.19 & 0.01 &\
2021$+$614... & B & C & 0.347 & 0.193 & ... & ... & 0.25 & 0.5\
& & J3 & 0.129 & 0.103 & 0.59 & 8.00 & 0.16 &\
& & J2 & 0.083 & 0.035 & 2.01 & 31.39 & 0.32 &\
& & J1 & 0.019 & 0.011 & 7.09 & 31.83 & 0.18 &\
2022$-$077... & C & C & 1.479 & 1.435 & ... & ... &$<$0.04 &$>$144.8\
& & J2 & 0.062 & 0.020 & 1.01 & $-$13.84 & 0.35 &\
& & J1 & 0.007 & 0.005 & 2.33 & 7.67 & 0.14 &\
2023$+$335... & C & C & 0.875 & 0.762 & ... & ... & 0.11 & 5.8\
& & J2 & 0.178 & 0.101 & 0.39 & $-$51.83 & 0.22 &\
& & J1 & 0.060 & 0.021 & 0.71 & $-$13.03 & 0.56 &\
2029$+$121... & B & C & 0.326 & 0.347 & ... & ... &$<$0.04 &$>$29.6\
& & J2 & 0.174 & 0.174 & 0.20 & $-$132.37 & 0.11 &\
& & J1 & 0.046 & 0.017 & 0.81 & $-$159.29 & 0.32 &\
2126$-$158... & C & C & 0.317 & 0.219 & ... & ... & 0.35 & 0.7\
& & J1 & 0.042 & 0.028 & 1.03 & $-$165.82 & 0.18 &\
2134$+$004... & B & C & 0.940 & 0.779 & ... & ... & 0.11 & 14.9\
& & J2 & 0.412 & 0.152 & 0.56 & $-$41.01 & 0.41 &\
& & J1 & 0.942 & 0.149 & 2.22 & $-$92.78 & 0.70 &\
2135$+$508... & C & C & 0.178 & 0.163 & ... & ... & 0.76 & 0.04\
& & J2 & 0.004 & 0.033 & 0.48 & 28.04 & 0.12 &\
& & J1 & 0.007 & 0.001 & 3.49 & 50.43 & 0.72 &\
2141$+$175... & B & C & 0.348 & 0.344 & ... & ... & 0.06 & 7.7\
& & J2 & 0.165 & 0.087 & 0.19 & $-$91.04 & 0.24 &\
& & J1 & 0.033 & 0.038 & 0.44 & $-$63.15 & 0.20 &\
2142$+$110... & B & C & 0.165 & 0.183 & ... & ... & $<$0.04&$>$10.5\
& & J2 & 0.038 & 0.129 & 0.19 & 159.96 & 0.05 &\
& & J1 & 0.016 & 0.018 & 0.57 & 169.28 & 0.12 &\
2144$+$092... & C & C & 0.610 & 0.604 & ... & ... & 0.04 & 52.9\
& & J2 & 0.142 & 0.105 & 0.27 & 59.37 & 0.17 &\
& & J1 & 0.010 & 0.003 & 0.88 & 67.39 & 0.33 &\
2155$-$152... & B & C & 1.715 & 1.103 & ... & ... & 0.18 & 5.8\
& & J1 & 0.141 & 0.139 & 1.71 & $-$140.30 & $<$0.05&\
2201$+$171... & C & C & 0.840 & 0.805 & ... & ... & 0.08 & 17.9\
& & J1 & 0.009 & 0.010 & 0.56 & 34.74 & $<$0.10&\
2209$+$236... & C & C & 0.265 & 0.244 & ... & ... & 0.08 & 5.9\
& & J3 & 0.086 & 0.104 & 0.22 & 86.41 & 0.13 &\
& & J2 & 0.110 & 0.092 & 0.48 & 61.73 & 0.09 &\
& & J1 & 0.021 & 0.006 & 1.48 & 29.82 & 0.46 &\
2216$-$038... & C & C & 0.638 & 0.573 & ... & ... & 0.13 & 4.7\
& & J2 & 0.086 & 0.056 & 0.62 & $-$161.68 & 0.17 &\
& & J1 & 0.176 & 0.054 & 1.74 & $-$158.70 & 0.37 &\
2223$+$210... & B & C & 0.072 & 0.033 & ... & ... & $<$0.05&$>$5.6\
& & J3 & 0.036 & 0.030 & 0.19 & $-$95.18 & 0.09 &\
& & J2 & 0.334 & 0.072 & 0.45 & $-$97.74 & 0.07 &\
& & J1 & 0.033 & 0.004 & 1.37 & $-$115.18 & 0.72 &\
2227$-$088... & C & C & 2.711 & 2.588 & ... & ... & 0.06 & 126.5\
& & J1 & 0.021 & 0.018 & 1.70 & $-$3.83 & 0.08 &\
2234$+$282... & C & C & 0.850 & 0.870 & ... & ... & $<$0.04&$>$39.9\
& & J2 & 0.110 & 0.270 & 0.15 & 58.54 & 0.12 &\
& & J1 & 0.179 & 0.061 & 0.86 & 39.06 & 0.34 &\
2243$-$123... & C & C & 1.599 & 1.185 & ... & ... & 0.28 & 2.2\
& & J2 & 0.036 & 0.044 & 0.71 & 5.02 & 0.05 &\
& & J1 & 0.038 & 0.012 & 3.43 & 9.09 & 0.38 &\
2308$+$341... & C & C & 0.374 & 0.345 & ... & ... & 0.31 & 0.7\
& & J3 & 0.042 & 0.058 & 0.54 & $-$4.44 & 0.24 &\
& & J2 & 0.007 & 0.008 & 0.61 & 26.79 & 0.09 &\
& & J1 & 0.017 & 0.007 & 1.96 & $-$0.44 & 0.31 &\
2318$+$049... & C & C & 0.487 & 0.330 & ... & ... & 0.16 & 2.1\
& & J1 & 0.061 & 0.053 & 0.34 & $-$46.28 & 0.49 &\
2320$-$035... & C & C & 0.982 & 0.737 & ... & ... & 0.18 & 4.8\
& & J1 & 0.067 & 0.004 & 1.91 & $-$27.87 & 1.09 &\
2325$+$093... & C & C & 2.019 & 1.851 & ... & ... & 0.11 & 31.1\
2342$-$161... & C & C & 1.393 & 1.185 & ... & ... & 0.17 & 5.1\
2344$+$092... & C & C & 0.105 & 0.084 & ... & ... & 0.14 & 0.6\
& & J1 & 0.019 & 0.012 & 1.31 & 39.55 & 0.21 &\
2345$-$167... & C & C & 1.822 & 1.472 & ... & ... & 0.18 & 5.8\
& & J2 & 0.051 & 0.039 & 0.57 & 91.98 & 0.14 &\
& & J1 & 0.153 & 0.051 & 1.83 & 120.24 & 0.38 &\
2351$-$154... & C & C & 0.217 & 0.182 & ... & ... & 0.18 & 1.6\
& & J3 & 0.098 & 0.063 & 0.47 & $-$63.63 & 0.18 &\
& & J2 & 0.029 & 0.007 & 1.12 & $-$34.04 & 0.51 &\
& & J1 & 0.011 & 0.003 & 2.68 & $-$25.95 & 0.46 &\
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![Example $(u,v)$ coverages at 43 GHz for a source in the first sample (left panel) and in the second sample (right panel). The labels on the vertical and horizontal axes are in units of million times the observing wavelength ($\lambda$).[]{data-label="uv"}](uvplot_left.pdf "fig:"){width="50.00000%"} ![Example $(u,v)$ coverages at 43 GHz for a source in the first sample (left panel) and in the second sample (right panel). The labels on the vertical and horizontal axes are in units of million times the observing wavelength ($\lambda$).[]{data-label="uv"}](uvplot_right.pdf "fig:"){width="50.00000%"}
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![Naturally weighted total intensity VLBA images of 124 sources at 43 GHz. The legends in the image: P – peak intensity (in units of $\rm Jy \, beam^{-1}$); L – rms noise (in units of $\rm mJy \, beam^{-1}$). The lowest contours are at $\pm 3$ times rms noise and further positive contours are drawn at increasing steps of 2. The image parameters are given in Table \[tab:image\]. The gray ellipse in the lower left corner of each image is the restoring beam.[]{data-label="image-1"}](0048-071.pdf "fig:"){width="30.00000%"} ![Naturally weighted total intensity VLBA images of 124 sources at 43 GHz. The legends in the image: P – peak intensity (in units of $\rm Jy \, beam^{-1}$); L – rms noise (in units of $\rm mJy \, beam^{-1}$). The lowest contours are at $\pm 3$ times rms noise and further positive contours are drawn at increasing steps of 2. The image parameters are given in Table \[tab:image\]. The gray ellipse in the lower left corner of each image is the restoring beam.[]{data-label="image-1"}](0106+013.pdf "fig:"){width="30.00000%"} ![Naturally weighted total intensity VLBA images of 124 sources at 43 GHz. The legends in the image: P – peak intensity (in units of $\rm Jy \, beam^{-1}$); L – rms noise (in units of $\rm mJy \, beam^{-1}$). The lowest contours are at $\pm 3$ times rms noise and further positive contours are drawn at increasing steps of 2. The image parameters are given in Table \[tab:image\]. The gray ellipse in the lower left corner of each image is the restoring beam.[]{data-label="image-1"}](0113-118.pdf "fig:"){width="30.00000%"}
![Naturally weighted total intensity VLBA images of 124 sources at 43 GHz. The legends in the image: P – peak intensity (in units of $\rm Jy \, beam^{-1}$); L – rms noise (in units of $\rm mJy \, beam^{-1}$). The lowest contours are at $\pm 3$ times rms noise and further positive contours are drawn at increasing steps of 2. The image parameters are given in Table \[tab:image\]. The gray ellipse in the lower left corner of each image is the restoring beam.[]{data-label="image-1"}](0119+041.pdf "fig:"){width="30.00000%"} ![Naturally weighted total intensity VLBA images of 124 sources at 43 GHz. The legends in the image: P – peak intensity (in units of $\rm Jy \, beam^{-1}$); L – rms noise (in units of $\rm mJy \, beam^{-1}$). The lowest contours are at $\pm 3$ times rms noise and further positive contours are drawn at increasing steps of 2. The image parameters are given in Table \[tab:image\]. The gray ellipse in the lower left corner of each image is the restoring beam.[]{data-label="image-1"}](0122-003.pdf "fig:"){width="30.00000%"} ![Naturally weighted total intensity VLBA images of 124 sources at 43 GHz. The legends in the image: P – peak intensity (in units of $\rm Jy \, beam^{-1}$); L – rms noise (in units of $\rm mJy \, beam^{-1}$). The lowest contours are at $\pm 3$ times rms noise and further positive contours are drawn at increasing steps of 2. The image parameters are given in Table \[tab:image\]. The gray ellipse in the lower left corner of each image is the restoring beam.[]{data-label="image-1"}](0130-171.pdf "fig:"){width="30.00000%"}
![Naturally weighted total intensity VLBA images of 124 sources at 43 GHz. The legends in the image: P – peak intensity (in units of $\rm Jy \, beam^{-1}$); L – rms noise (in units of $\rm mJy \, beam^{-1}$). The lowest contours are at $\pm 3$ times rms noise and further positive contours are drawn at increasing steps of 2. The image parameters are given in Table \[tab:image\]. The gray ellipse in the lower left corner of each image is the restoring beam.[]{data-label="image-1"}](0149+218.pdf "fig:"){width="30.00000%"} ![Naturally weighted total intensity VLBA images of 124 sources at 43 GHz. The legends in the image: P – peak intensity (in units of $\rm Jy \, beam^{-1}$); L – rms noise (in units of $\rm mJy \, beam^{-1}$). The lowest contours are at $\pm 3$ times rms noise and further positive contours are drawn at increasing steps of 2. The image parameters are given in Table \[tab:image\]. The gray ellipse in the lower left corner of each image is the restoring beam.[]{data-label="image-1"}](0202-172.pdf "fig:"){width="30.00000%"} ![Naturally weighted total intensity VLBA images of 124 sources at 43 GHz. The legends in the image: P – peak intensity (in units of $\rm Jy \, beam^{-1}$); L – rms noise (in units of $\rm mJy \, beam^{-1}$). The lowest contours are at $\pm 3$ times rms noise and further positive contours are drawn at increasing steps of 2. The image parameters are given in Table \[tab:image\]. The gray ellipse in the lower left corner of each image is the restoring beam.[]{data-label="image-1"}](0208+106.pdf "fig:"){width="30.00000%"}
![Naturally weighted total intensity VLBA images of 124 sources at 43 GHz. The legends in the image: P – peak intensity (in units of $\rm Jy \, beam^{-1}$); L – rms noise (in units of $\rm mJy \, beam^{-1}$). The lowest contours are at $\pm 3$ times rms noise and further positive contours are drawn at increasing steps of 2. The image parameters are given in Table \[tab:image\]. The gray ellipse in the lower left corner of each image is the restoring beam.[]{data-label="image-1"}](0221+067.pdf "fig:"){width="30.00000%"} ![Naturally weighted total intensity VLBA images of 124 sources at 43 GHz. The legends in the image: P – peak intensity (in units of $\rm Jy \, beam^{-1}$); L – rms noise (in units of $\rm mJy \, beam^{-1}$). The lowest contours are at $\pm 3$ times rms noise and further positive contours are drawn at increasing steps of 2. The image parameters are given in Table \[tab:image\]. The gray ellipse in the lower left corner of each image is the restoring beam.[]{data-label="image-1"}](0224+671.pdf "fig:"){width="30.00000%"} ![Naturally weighted total intensity VLBA images of 124 sources at 43 GHz. The legends in the image: P – peak intensity (in units of $\rm Jy \, beam^{-1}$); L – rms noise (in units of $\rm mJy \, beam^{-1}$). The lowest contours are at $\pm 3$ times rms noise and further positive contours are drawn at increasing steps of 2. The image parameters are given in Table \[tab:image\]. The gray ellipse in the lower left corner of each image is the restoring beam.[]{data-label="image-1"}](0229+131.pdf "fig:"){width="30.00000%"}
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![Histograms of the total flux density, $S_{\rm tot}$ (panel a), the compactness parameter on mas scales $S_{\rm core}/S_{\rm tot}$ (panel b), the correlated flux density on the longest baselines $S_{\rm L}$ (panel c), and the compactness parameter on sub-mas scales $S_{\rm L}/S_{\rm S}$ (panel d).[]{data-label="compactness"}](compactness.pdf "fig:"){width="100.00000%"}
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![Distribution of core brightness temperature as a function of frequency (left panel) and the best fit of the relationship (right panel). Coloured symbols represent core brightness temperatures measured at various observed frequencies: yellow , purple [5 GHz, @2004ApJ...616..110H], green , blue [15 GHz, @2005AJ....130.2473K], cyan [22 GHz, @1996AJ....111.2174M], red [43 GHz, @2018ApJS..234...17C and this paper], black . The frequencies have been converted into the source rest frame. The fitting in the right panel is represented with a broken power-law function as described in Sect. \[brightnesstemp\].[]{data-label="Tb"}](Tb.pdf "fig:"){width="48.00000%"} ![Distribution of core brightness temperature as a function of frequency (left panel) and the best fit of the relationship (right panel). Coloured symbols represent core brightness temperatures measured at various observed frequencies: yellow , purple [5 GHz, @2004ApJ...616..110H], green , blue [15 GHz, @2005AJ....130.2473K], cyan [22 GHz, @1996AJ....111.2174M], red [43 GHz, @2018ApJS..234...17C and this paper], black . The frequencies have been converted into the source rest frame. The fitting in the right panel is represented with a broken power-law function as described in Sect. \[brightnesstemp\].[]{data-label="Tb"}](Tb-fit.pdf "fig:"){width="45.00000%"}
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![Flux density correlations between radio 43 GHz and $\gamma$-ray bands. The correlation coefficient is 0.550. The error bars are omitted for the clarity of the plot. []{data-label="correlation"}](logS.pdf "fig:"){width="45.00000%"}
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[^1]: http://www.physics.purdue.edu/astro/MOJAVE/sourcepages/0048-071.shtml
[^2]: NASA/IPAC Extragalactic Database, http://ned.ipac.caltech.edu/
[^3]: Here and elsewhere, when we refer to 1.4-GHz VLBA images, we used the unpublished data observed in the project BG196 (PI: D. Gabuzda); calibrated data were downloaded from the Astrogeo database (http://astrogeo.org/).
[^4]: see http://astrogeo.org/
|
---
abstract: 'This paper concerns contravariant functors from the category of rings to the category of sets whose restriction to the full subcategory of commutative rings is isomorphic to the prime spectrum functor $\operatorname{Spec}$. The main result reveals a common characteristic of these functors: every such functor assigns the empty set to ${\mathbb{M}}_n({\mathbb{C}})$ for $n \geq 3$. The proof relies, in part, on the Kochen-Specker Theorem of quantum mechanics. The analogous result for noncommutative extensions of the Gelfand spectrum functor for $C^*$-algebras is also proved.'
address: |
Department of Mathematics\
University of California, San Diego\
9500 Gilman Drive \#0112\
La Jolla, CA 92093-0112
author:
- 'Manuel L. Reyes'
bibliography:
- 'EmptySpec.bib'
date: 'June 2, 2011'
title: |
Obstructing extensions of the functor Spec\
to noncommutative rings
---
[^1]
Introduction {#introduction section}
============
The prime spectrum of commutative rings and the Gelfand spectrum of commutative $C^*$-algebras play a foundational role in the classical link between algebra and geometry, since these spectra form the underlying point-sets of the spaces attached to a commutative ring or $C^*$-algebra. It is tempting to hope that one could extend these spectra to the noncommutative setting in order to construct the “underlying set of a noncommutative space.” The main results of this paper (Theorems \[main theorem\] and \[main C\* theorem\] below) hinder naive attempts to do so by obstructing the existence of functors that extend these spectra.
In order to produce an obstruction, one must first fix the desired properties of the “noncommutative spectrum” in question. Consider the prime spectrum $\operatorname{Spec}$. From the viewpoint of $\operatorname{Spec}$ as an underlying point-set, two facts of key importance are (1) the spectrum of every nonzero commutative ring is nonempty, and (2) the prime spectrum construction can be regarded as a contravariant functor from the category of commutative rings to the category of sets, $$\operatorname{Spec}\colon \operatorname{\mathsf{CommRing}}\to \operatorname{\mathsf{Set}}.$$ (For commutative rings, $\operatorname{Spec}$ is easily made into a functor because the inverse image of a prime ideal under a ring homomorphism is again prime.)
Over the years, many different extensions of the prime spectrum to noncommutative rings have been studied. Let $F$ be a rule assigning to each ring $R$ a set $F(R)$, such that for every commutative ring $C$ one has $F(C) \cong \operatorname{Spec}(C)$. There are two desirable properties that such an invariant may possess.
Property A
: *For every nonzero ring $R$, the set $F(R)$ is nonempty.*
Property B
: *The invariant $F$ can be made into a set-valued functor extending $\operatorname{Spec}$,* in the sense that the assignment $R \mapsto F(R)$ is the object part of a functor $F$ whose restriction to the category of commutative rings is isomorphic to $\operatorname{Spec}$.
Examples of invariants that satisfy Property A include the set of prime ideals of a noncommutative ring, Goldman’s prime torsion theories [@Goldman], and the “left spectrum” of Rosenberg [@Rosenberg1]. (These invariants satisfy Property A because they all have elements corresponding to maximal one- or two-sided ideals.) Some invariants that satisfy Property B are the spectrum of the “abelianization” $R \mapsto \operatorname{Spec}(R/[R,R])$, the set of completely prime ideals, and the “field spectrum” of Cohn [@Cohn].
Each of the different “noncommutative spectra” listed above possess only one of the two properties. Our first main result states that this situation is inevitable.
\[main theorem\] Let $F$ be a contravariant functor from the category of rings to the category of sets whose restriction to the full subcategory of commutative rings is isomorphic to $\operatorname{Spec}$. Then $F({\mathbb{M}}_n({\mathbb{C}})) = \varnothing$ for any $n \geq 3$.
Next we state the analogous result in the context of $C^*$-algebras. For our purposes, we define the *Gelfand spectrum* of a commutative unital $C^*$-algebra $A$ to be the set $\operatorname{Max}(A)$ of maximal ideals of $A$; these are necessarily closed in $A$. The set $\operatorname{Max}(A)$ is in bijection with the set of *characters* of $A$, which are the nonzero multiplicative linear functionals (equivalently, unital algebra homomorphisms) $A \to {\mathbb{C}}$; the correspondence associates to each character its kernel (see [@Davidson Thm. I.2.5]). This is easily given the structure of a contravariant functor $$\operatorname{Max}\colon \operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}\to \operatorname{\mathsf{Set}}.$$ With appropriate topologies taken into account, the Gelfand spectrum functor provides a contravariant equivalence between the category of commutative unital $C^*$-algebras and the category of compact Hausdorff spaces.
The following analogue of Theorem \[main theorem\] provides a similar obstruction to any noncommutative extension of the Gelfand spectrum functor.
\[main C\* theorem\] Let $F$ be a contravariant functor from the category of unital $C^*$-algebras to the category of sets whose restriction to the full subcategory of commutative unital $C^*$-algebras is isomorphic to $\operatorname{Max}$. Then $F({\mathbb{M}}_n({\mathbb{C}})) = \varnothing$ for any $n \geq 3$.
Of course, the statements of Theorems \[main theorem\] and \[main C\* theorem\] with the category of sets replaced by the category $\operatorname{\mathsf{Top}}$ of topological spaces follow as immediate corollaries.
There are plenty of results stating that a *particular* spectrum of a ring or algebra is empty. For instance, it is easy to find examples of noncommutative $C^*$-algebras that have no characters. In the realm of algebra, one can think of rings that have no homomorphisms to any division ring as having empty spectra. For one more example, S.P. Smith suggested a notion of “closed point” such that every infinite dimensional simple ${\mathbb{C}}$-algebra has no closed points [@Smith p. 2170]. Notice that each of these examples assumes a fixed notion of spectrum. The main feature setting Theorem \[main theorem\] apart from the arguments mentioned above is that it applies to *any* notion of spectrum satisfying Properties A and B mentioned above, and similarly for Theorem \[main C\* theorem\]. Indeed, these spectra need not be defined in terms of ideals (either one-sided or two-sided) or modules at all.
**Outline of the proof.** The proofs of Theorems \[main theorem\] and \[main C\* theorem\] proceed roughly as follows: (1) construct a functor that is “universal” among all functors whose restriction to the commutative subcategory is the spectrum functor; (2) show that this functor assigns the empty set to ${\mathbb{M}}_n({\mathbb{C}})$; (3) by universality, conclude that *every* such functor does the same.
It is perhaps surprising that a key tool used for step (2) above is the *Kochen-Specker Theorem* [@KochenSpecker] of quantum mechanics, which forbids the existence of certain hidden variable theories. Recently this result has surfaced in the context of noncommutative geometry in the *Bohrification* construction introduced by C. Heunen, N. Landsman, and B. Spitters in [@HeunenLandsmanSpitters Thm. 6]. Those authors use the Kochen-Specker Theorem to show that a certain “space” associated to the $C^*$-algebra of bounded operators on a Hilbert space of dimension $\geq 3$ has no points. This is obviously close in spirit to Theorems \[main theorem\] and \[main C\* theorem\]. A common theme between that paper and the present one is the *focus on commutative subalgebras of a given algebra*, and we acknowledge the inspiration and influence of that work on ours.
In the ring-theoretic case, step (1) is achieved in Section \[partial algebras section\]. The universal functor $\operatorname{\mathit{p}-Spec}$ is defined in terms of *prime partial ideals*, which requires an exposition of partial algebras along with their ideals and morphisms. Step (2) is carried out in Section \[Kochen-Specker section\], where we establish a connection between prime partial ideals and the Kochen-Specker Theorem. The proof of Theorem \[main theorem\] (basically Step (3) above) is given in Section \[proof section\], and it is accompanied by some corollaries. In Section \[C\* section\] we prove Theorem \[main C\* theorem\] by quickly following Steps (1)–(3) in the context of $C^*$-algebras, and we state a few of its corollaries.
**Generalizations and positive implications.** Since the present results were announced, stronger obstructions to spectrum functors have been proved by B. van den Berg and C. Heunen in [@BergHeunen2]. Those results hinder the extension of $\operatorname{Spec}$ and $\operatorname{Max}$ even when they are considered as functors whose codomains are over-categories of $\operatorname{\mathsf{Top}}$, such as the categories of locales and toposes. However, one can view these obstructions in a positive light: it seems that the actual construction of contravariant functors extending the classical spectra necessitates a creative choice of target category ${\mathcal{C}}$ that contains $\operatorname{\mathsf{Top}}$ (or $\operatorname{\mathsf{Set}}$, if one forgets the topology). From this perspective, the construction of “useful” noncommutative spectrum functors extending the classical ones seems to remain an interesting issue.
**Conventions.** All rings are assumed to have identity and ring homomorphisms are assumed to preserve the identity, except where explicitly stated otherwise. The categories of unital rings and unital commutative rings are respectively denoted by $\operatorname{\mathsf{Ring}}$ and $\operatorname{\mathsf{CommRing}}$. We will consider $\operatorname{Spec}$ as a contravariant functor from the category of commutative rings to the category of sets, instead of topological spaces, unless indicated otherwise. A contravariant functor $F \colon {\mathcal{C}}_1 \to {\mathcal{C}}_2$ can also be viewed as a covariant functor out of the opposite category $F \colon {\mathcal{C}}_1^{{\textnormal{op}}} \to {\mathcal{C}}_2$. For the most part, we will view contravariant functors as functors that reverse the direction of arrows, in order to avoid dealing with “opposite arrows.” But when it is convenient we will occasionally change viewpoint and consider contravariant functors as covariant functors out of the opposite category. Given a category ${\mathcal{C}}$, we will often write $C \in {\mathcal{C}}$ to mean that $C$ is an object of ${\mathcal{C}}$. When there is danger of confusion, we will write the more precise expression $C \in \operatorname{Obj}({\mathcal{C}})$.
A universal Spec functor from prime partial ideals {#partial algebras section}
==================================================
In this section we will define a functor $\operatorname{\mathit{p}-Spec}$ that is universal among all candidates for a “noncommutative $\operatorname{Spec}$.” We set the stage for its construction by describing the universal property that we seek.
Given categories ${\mathcal{C}}$ and ${\mathcal{C}}'$, we let $\operatorname{Fun}({\mathcal{C}}, {\mathcal{C}}')$ denote the category of (covariant) functors from ${\mathcal{C}}$ to ${\mathcal{C}}'$ whose morphisms are natural transformations. (This category need not have small Hom-sets.) The inclusion of categories $\operatorname{\mathsf{CommRing}}\hookrightarrow \operatorname{\mathsf{Ring}}$ induces a *restriction* functor $$\begin{aligned}
{\mathfrak{r}}\colon \operatorname{Fun}({\operatorname{\mathsf{Ring}}^{{\textnormal{op}}}}, \operatorname{\mathsf{Set}}) &\to \operatorname{Fun}({\operatorname{\mathsf{CommRing}}^{{\textnormal{op}}}}, \operatorname{\mathsf{Set}}) \\
F &\mapsto F|_{{\operatorname{\mathsf{CommRing}}^{{\textnormal{op}}}}},\end{aligned}$$ which is defined in the obvious way on morphisms (i.e., natural transformations). Now we define the “fiber category” over $\operatorname{Spec}\in \operatorname{Fun}({\operatorname{\mathsf{CommRing}}^{{\textnormal{op}}}}, \operatorname{\mathsf{Set}})$ to be the category ${\mathfrak{r}}^{-1}(\operatorname{Spec})$ whose objects are pairs $(F, \phi)$ with $F \in \operatorname{Fun}({\operatorname{\mathsf{Ring}}^{{\textnormal{op}}}}, \operatorname{\mathsf{Set}})$ and $\phi \colon {\mathfrak{r}}(F) \overset{\sim}{\longrightarrow} \operatorname{Spec}$ an isomorphism of functors, in which a morphism $\psi \colon (F, \phi) \to (F', \phi')$ is a morphism $\psi \colon F \to F'$ of functors such that $\phi' \circ {\mathfrak{r}}(\psi) = \phi$, i.e. the following commutes: $$\xymatrix{
{\mathfrak{r}}(F) \ar[rr]^{{\mathfrak{r}}(\psi)} \ar[dr]_{\phi} & & {\mathfrak{r}}(F') \ar[dl]^{\phi'} \\
& \operatorname{Spec}&.
}$$ (Our use of the terminology “fiber category” and notation ${\mathfrak{r}}^{-1}$ is slightly different from other instances in the literature. The main difference is that we are considering objects that map to $\operatorname{Spec}$ under ${\mathfrak{r}}$ *up to isomorphism*, rather than “on the nose.”)
The category ${\mathfrak{r}}^{-1}(\operatorname{Spec})$ is of fundamental importance to us; we are precisely interested in those contravariant functors from $\operatorname{\mathsf{Ring}}$ to $\operatorname{\mathsf{Set}}$ whose restriction to $\operatorname{\mathsf{CommRing}}$ is isomorphic to $\operatorname{Spec}$. The “universal $\operatorname{Spec}$ functor” $\operatorname{\mathit{p}-Spec}$ that we seek is a terminal object in this category. The rest of this section is devoted to defining this functor and proving its universal property.
The functor $\operatorname{\mathit{p}-Spec}$ to be constructed is best understood in the context of partial algebras, whose definition we recall here. The notion of a partial algebra was defined in [@KochenSpecker §2]. (A more precise term for this object would probably be *partial commutative algebra,* but we retain the historical and more concise terminology in this paper.)
A *partial algebra* over a commutative ring $k$ is a set $R$ with a reflexive symmetric binary relation ${\perp}\, \subseteq R \times R$ (called *commeasurability*), partial addition and multiplication operations $+$ and $\cdot$ that are functions ${\perp}\, \to R$, a scalar multiplication operation $k \times R \to R$, and elements $0,1 \in A$ such that the following axioms are satisfied:
1. For all $a \in R$, $a {\perp}0$ and $a {\perp}1$;
2. The relation ${\perp}$ is preserved by the partial binary operations: for all $a_1, a_2, a_3 \in R$ with $a_i {\perp}a_j$ ($1 \leq i,j \leq 3$) and for all $\lambda \in k$, one has $(a_1+a_2) {\perp}a_3$, $(a_1 a_2) {\perp}a_3$, and $(\lambda a_1) {\perp}a_2$;
3. If $a_i {\perp}a_j$ for $1 \leq i,j \leq 3$, then the values of all (commutative) polynomials in $a_1$, $a_2$, and $a_3$ form a commutative $k$-algebra.
A *partial ring* is a partial algebra over $k = \mathbb{Z}$.
The third axiom of a partial algebra appears as stated in [@KochenSpecker p. 64]. While the axiom is succinct, it can be instructive to unravel its meaning. The third axiom is equivalent to the following collection of axioms:
- The element $0 \in R$ is an additive identity and $1 \in R$ is a multiplicative identity;
- Addition and multiplication are commutative when defined: if $a {\perp}b$ in $R$, then $a+b = b+a$ and $ab = ba$;
- Addition and multiplication are associative on commeasurable triples: if $a {\perp}b$, $a {\perp}c$, and $b {\perp}c$ in $R$, then $(a + b) + c = a + (b + c)$ and $(a \cdot b) \cdot c = a \cdot (b \cdot c)$;
- Multiplication distributes over addition on commeasurable triples: if $a {\perp}b$, $a {\perp}c$, and $b {\perp}c$ in $R$, then $a \cdot (b + c) = a \cdot b + a \cdot c$;
- Each element $a \in R$ is commeasurable to an element $-a \in R$ that is an additive inverse to $a$ and such that $a {\perp}r \implies -a {\perp}r$ for all $r \in R$ (see the paragraph before Lemma \[eigenvalue lemma\] for a discussion of uniqueness of inverses);
- Multiplication is $k$-bilinear.
A *commeasurable subalgebra* of a partial $k$-algebra $R$ is a subset $C \subseteq R$ consisting of pairwise commeasurable elements that is closed under $k$-scalar multiplication and the partial binary operations of $R$. (Thus the operations of $R$ restricted to $C$ endow $C$ with the structure of a commutative $k$-algebra.)
In particular, given any $a \in R$ one can evaluate every polynomial in $k[x]$ at $x = a$ to obtain commeasurable $k$-subalgebra $k[a] \subseteq R$. More generally, any set of pairwise commeasurable elements of $R$ is contained in a commeasurable $k$-subalgebra of $R$. Notice also that $R$ is the union of its commeasurable $k$-subalgebras.
When we need to distinguish between a $k$-algebra and a partial $k$-algebra, we shall refer to the former as a “full” algebra. As the following example shows, every full algebra can be considered as a partial algebra in a standard way.
Let $R$ be a (full) algebra over a commutative ring $k$. We may define a relation ${\perp}\, \subseteq R \times R$ by $a {\perp}b$ if and only if $ab = ba$ (i.e., $[a,b]=0$). This relation along with the addition, multiplication, and scalar multiplication inherited from $R$ make $R$ into a partial algebra over $k$. For us, this is the prototypical example of a partial algebra. We will refer to this as the “standard partial algebra structure” on $R$.
Considering a full algebra $R$ as a partial algebra is, in effect, a way to restrict our attention to *only* the commutative subalgebras of $R$. This is further amplified when one applies the notions (defined below) of morphisms of partial algebras and partial ideals to the algebra $R$.
Another important example of a partial algebra is considered in [@KochenSpecker]. Let $A$ be a unital $C^*$-algebra, and let $A_{sa}$ denote the set of self-adjoint elements of $A$. Notice that the sum and product of two commuting self-adjoint elements is again self-adjoint, and that real scalar multiplication preserves $A_{sa}$. So if ${\perp}\, \subseteq A_{sa} \times A_{sa}$ is the relation of commutativity (as in the previous example), then $A_{sa}$ forms a partial algebra over $\mathbb{R}$.
Just as one may study ideals of a $k$-algebra, we will consider “partial ideals” of a partial $k$-algebra.
Let $R$ be a partial algebra over a commutative ring $k$. A subset $I \subseteq R$ is a *partial ideal* of $R$ if, for all $a, b \in R$ such that $a {\perp}b$, one has:
- $a, b \in I \implies a+b \in I$;
- $b \in I \implies ab \in I$.
Equivalently, a partial ideal of $R$ is a subset $I \subseteq R$ such that, for every commeasurable subalgebra $C \subseteq R$, the intersection $I \cap C$ is an ideal of $C$. If $R$ is a (full) $k$-algebra, then a partial ideal of $R$ is a partial ideal of the standard partial algebra structure on $R$.
To better understand the set of partial ideals of an arbitrary (full or partial) algebra, it helps to consider some general examples. Let $R$ be an algebra over a commutative ring $k$. If $I$ is a left, right, or two-sided ideal of $R$, then $I$ is a clearly a partial ideal of $R$. Furthermore, when $R$ is commutative the partial ideals of $R$ are precisely the ideals of $R$.
\[improper partial ideal\] Let $I$ be a partial ideal of a partial $k$-algebra $R$. Then $I = R$ if and only if $1 \in I$.
(“If” direction.) If $1 \in I$, then $1 {\perp}R$ gives $R = (R \cdot 1) \subseteq I$. Hence $I = R$.
\[division ring partial ideals\] Let $D$ be a division ring. Then the only partial ideals of $D$ are $0$ and $D$.
Suppose that $I \subseteq D$ is a nonzero partial ideal, and let $0 \neq a \in I$. Then $a {\perp}a^{-1}$, so $1 = a^{-1} \cdot a \in I$. It follows from Lemma \[improper partial ideal\] that $I = D$.
Yet another example of a partial ideal in an arbitrary ring $R$ is the set $N \subseteq R$ of nilpotent elements of $R$. Indeed, for any commutative subring $C$ of $R$, $N \cap C$ is the nilradical of $C$ and hence is an ideal of $C$. It is well-known that the set of nilpotent elements of a ring $R$ is not even closed under addition for many noncommutative rings $R$. In fact, it is hard to find *any* structural properties that this set possesses for a general ring $R$, making this observation noteworthy. (This example also illustrates that ring theorists must take particular care not to impose their usual mental images of ideals upon the notion of a partial ideal.)
We now introduce a notion of prime partial ideal, which will provide a type of “spectrum.”
\[prime partial ideal definition\] A partial ideal $P$ of a partial $k$-algebra $R$ is *prime* if $P \neq R$ and whenever $x {\perp}y$ in $A$, $xy \in P$ implies that either $x \in P$ or $y \in P$. Equivalently, a partial ideal $P$ of $R$ is prime if $P \subsetneq R$ and for every commeasurable subalgebra $C \subseteq R$, $P \cap C$ is a prime ideal of $C$. The set of prime partial ideals of a (full) $k$-algebra $R$ is denoted $\operatorname{\mathit{p}-Spec}(R)$.
If $R$ is a commutative $k$-algebra, then the prime partial ideals of $R$ are precisely the prime ideals of $R$. Now the fact that $\operatorname{Spec}\colon \operatorname{\mathsf{CommRing}}\to \operatorname{\mathsf{Set}}$ defines a (contravariant) functor depends on the fact prime ideals behave well under homomorphisms of commutative rings. It turns out that prime partial ideals behave just as well, provided that one uses the “correct” notion of a morphism of partial algebras. This is proved in Lemma \[preimage lemma\] below. The following definition was given in [@KochenSpecker §2].
Let $R$ and $S$ be partial algebras over a commutative ring $k$. A *morphism of partial algebras* is a function $f \colon R \to S$ such that, for every $\lambda \in k$ and all $a, b \in R$ with $a {\perp}b$,
- $f(a) {\perp}f(b)$,
- $f(\lambda a) = \lambda f(a)$,
- $f(a + b) = f(a) + f(b)$,
- $f(ab) = f(a) f(b)$,
- $f(0) = 0$ and $f(1) = 1$.
(In other words, $f$ preserves the commeasurability relation and its restriction to every commeasurable subalgebra $C \subseteq R$ is a homomorphism of commutative $k$-algebras $f|_C \colon C \to f(C)$.)
Of course, any algebra homomorphism $R \to S$ of $k$-algebras is also a morphism of partial algebras when $R$ and $S$ are considered as partial algebras.
\[preimage lemma\] Let $f \colon R \to S$ be a morphism of partial $k$-algebras, and let $I$ be a partial ideal of $S$.
1. The set $f^{-1}(I) \subseteq R$ is a partial ideal of $R$.
2. If $I$ is prime, then $f^{-1}(I)$ is also prime.
This holds, in particular, when $R$ and $S$ are (full) algebras, $f$ is a $k$-algebra homomorphism, and $I$ is a (prime) partial ideal of $S$.
Let $a, b \in R$ be such that $a {\perp}b$. Then $f(a) {\perp}f(b)$. If $a, b \in f^{-1}(I)$ then $f(a),f(b) \in I$. Thus $f(a+b) = f(a) + f(b) \in I$, so that $a + b \in f^{-1}(I)$. On the other hand if $a \in R$ and $b \in f^{-1}(I)$, then $f(a) \in S$ and $f(b) \in I$. This means that $f(ab) = f(a) f(b) \in I$, whence $ab \in f^{-1}(I)$. Thus $f^{-1}(I)$ is a partial ideal of $R$.
Now suppose that $I$ is prime. The fact that $I \neq S$ implies that $f^{-1}(I) \neq R$, thanks to Lemma \[improper partial ideal\]. If $a {\perp}b$ in $R$ are such that $ab \in f^{-1}(I)$, then $f(a) {\perp}f(b)$ and $f(a) f(b) = f(ab) \in I$. Because $I$ is prime, either $f(a) \in I$ or $f(b) \in I$. In other words, either $a \in f^{-1}(I)$ or $b \in f^{-1}(I)$. This proves that $f^{-1}(I)$ is prime.
The rule assigning to each ring $R$ the set $\operatorname{\mathit{p}-Spec}(R)$ of prime partial ideals of $R$, and to each ring homomorphism $f \colon R \to S$ the map of sets $$\begin{aligned}
\operatorname{\mathit{p}-Spec}(S) &\to \operatorname{\mathit{p}-Spec}(R) \\
P &\mapsto f^{-1}(P),\end{aligned}$$ is a contravariant functor from the category of rings to the category of sets. We denote this functor by $\operatorname{\mathit{p}-Spec}\colon \operatorname{\mathsf{Ring}}\to \operatorname{\mathsf{Set}}$, extending the notation introduced in Definition \[prime partial ideal definition\].
Notice immediately that the restriction of $\operatorname{\mathit{p}-Spec}$ to $\operatorname{\mathsf{CommRing}}$ is equal to $\operatorname{Spec}$, and therefore the functor $\operatorname{\mathit{p}-Spec}$ gives an object of the category ${\mathfrak{r}}^{-1}(\operatorname{Spec})$ defined earlier in this section. Of course, this functor could be defined on the category of all partial algebras and partial algebra homomorphisms. But because our primary interest is in the category of rings, we have chosen to restrict our definition to that category.
\[domain example\] Recall that an ideal $P \lhd R$ is *completely prime* if $R/P$ is a domain; that is, $P \neq R$ and for $a,b \in R$, $ab \in P$ implies that either $a \in P$ or $b \in P$. Certainly every completely prime ideal of a ring is a prime partial ideal. Thus every domain has a prime partial ideal: its zero ideal. Recalling Proposition \[division ring partial ideals\] we conclude that the zero ideal of a division ring $D$ is its unique prime partial ideal, so that $\operatorname{\mathit{p}-Spec}(D)$ is a singleton.
The universal property of $\operatorname{\mathit{p}-Spec}$ will be established in Theorem \[universal Spec theorem\] below. In preparation, we observe that a partial ideal of a ring is equivalent to a choice of ideal in every commutative subring. For a partial algebra $R$ over a commutative ring $k$, we let ${\mathscr{C}}_k(R)$ denote the partially ordered set of all commeasurable subalgebras of $R$. (In case $R$ is a ring, ${\mathscr{C}}(R) := {\mathscr{C}}_{{\mathbb{Z}}}(R)$ is the poset of commutative subrings of $R$.) Recall that a subset ${\mathcal{S}}$ of a partially ordered set $X$ is *cofinal* if for every $x \in X$ there exists $s \in {\mathcal{S}}$ such that $x \leq s$.
\[data determining partial ideal\] Each of the following data uniquely determines a partial ideal of a partial algebra $R$:
1. A rule $I$ that associates to each commeasurable subalgebra $C \subseteq R$ an ideal $I(C) \lhd C$ such that, if $C \subseteq C'$ are commeasurable subalgebras of $R$, then $I(C) = I(C') \cap C$;
2. A rule $I$ that associates to each commeasurable subalgebra $C \subseteq R$ an ideal $I(C) \lhd C$ such that, if $C_1$ and $C_2$ are commeasurable subalgebras of $R$, then $I(C_1) \cap C_2 = C_1 \cap I(C_2)$;
3. For a cofinal set ${\mathcal{S}}$ of commeasurable subalgebras of $R$, a rule $I$ that associates to each $C \in {\mathcal{S}}$ an ideal $I(C) \lhd C$ such that, if $C_1$ and $C_2$ are in ${\mathcal{S}}$, then $I(C_1) \cap C_2 = C_1 \cap I(C_2)$;
4. A rule $I$ that associates to each maximal commeasurable subalgebra $C \subseteq R$ an ideal $I(C) \lhd C$ such that, if $C_1$ and $C_2$ are maximal commeasurable subalgebra of $R$, then $I(C_1) \cap C_2 = C_1 \cap I(C_2)$.
First notice that the rules described in (1) and (2) are equivalent. For if $I$ satisfies (1), then for any $C_1,C_2 \in {\mathscr{C}}_k(R)$ we have $$\begin{aligned}
I(C_1) \cap C_2 &= I(C_1) \cap (C_1 \cap C_2) \\
&= I(C_1 \cap C_2) \\
&= I(C_2) \cap (C_1 \cap C_2) \\
&= I(C_2) \cap C_1.\end{aligned}$$ Thus $I$ satisfies (2). Conversely, if $I$ satisfies (2) and if $C,C' \in {\mathscr{C}}(R)$ are such that $C \subseteq C'$, then $$I(C) = I(C) \cap C' = C \cap I(C'),$$ proving that $I$ satisfies (1).
The equivalence of the rules described in (2)–(4) is straightforward to verify. To complete the proof, we show that the data described in (1) uniquely determines a partial ideal of $R$. Given a rule $I$ as in (1), the set $J = \bigcup_{C \in {\mathscr{C}}_k(R)} I(C) \subseteq R$ is certainly a partial ideal of $R$. Conversely, given a partial ideal $J$ of $R$, the assignment $I$ sending $C \mapsto I(C) := J \cap C$ satisfies (1). Clearly these maps $I \mapsto J$ and $J \mapsto I$ are mutually inverse.
A choice of a prime ideal in each commutative subring of a ring $R$ can be viewed as an element of the product $\prod_{C \in {\mathscr{C}}(R)} \operatorname{Spec}(C)$. The above characterization (1) of partial ideals says that the prime partial ideals can be identified with those elements $(P_C)_{C \in {\mathscr{C}}(R)}$ of this product such that for every $C,C' \in {\mathscr{C}}(R)$ with $C \subseteq C'$, one has $P_{C'} \cap C = P_C$. This fact is used below.
The last step before the main result of this section is to give an alternative description of $\operatorname{\mathit{p}-Spec}$ as a certain limit. We recall the “product-equalizer” construction of limits in the category of sets (see [@MacLane V.2]). Let $D \colon J \to \operatorname{\mathsf{Set}}$ be a diagram (i.e., $D$ is a functor and $J$ is a small category). Then the limit of $D$ can be formed explicitly as $${\varprojlim}_J D = \left\{ \left. (x_j) \in \prod_{j \in \operatorname{Obj}(J)}D(j) \, \right| \,
\text{$D(f)(x_i) = x_j$ for all $i,j \in \operatorname{Obj}(J)$ and all $f \colon i \to j$ in $J$} \right\},$$ with the morphisms ${\varprojlim}D \to D(j)$ defined for each $j \in \operatorname{Obj}(J)$ via projection.
For a ring $R$, we view the partially ordered set ${\mathscr{C}}(R)$ defined above as a category by considering each element of ${\mathscr{C}}(R)$ as an object and each inclusion as a morphism. (The appropriate analogue of this category in the context of $C^*$-algebras makes a key appearance in the definition of the Bohrification functor [@HeunenLandsmanSpitters Def. 4] of Heunen, Landsman, and Spitters.) The functor that is shown to be isomorphic to $\operatorname{\mathit{p}-Spec}$ in the following proposition is very close to one defined by van den Berg and Heunen in [@BergHeunen1 Prop. 5].
\[isomorphic functors\] The contravariant functor $\operatorname{\mathit{p}-Spec}\colon \operatorname{\mathsf{Ring}}\to \operatorname{\mathsf{Set}}$ is isomorphic to the functor defined, for a given ring $R$, by $$R \mapsto {\varprojlim}_{C \in {\mathscr{C}}(R)^{{\textnormal{op}}}} \operatorname{Spec}(C).$$ This isomorphism preserves the isomorphism of functors $\operatorname{\mathit{p}-Spec}|_{\operatorname{\mathsf{CommRing}}} \cong \operatorname{Spec}$.
For any ring $R$, we have the following isomorphisms of sets: $$\begin{aligned}
{\varprojlim}\nolimits_{C \in {\mathscr{C}}(R)^{{\textnormal{op}}}} \operatorname{Spec}(C)
&= \left\{ \left. (P_C) \in \prod_{C \in {\mathscr{C}}(R)} \operatorname{Spec}(C) \, \right| \,
\begin{gathered}
\text{for all inclusions } i \colon C \hookrightarrow C',\\ \operatorname{Spec}(i)(P_{C'}) = P_{C}
\end{gathered}
\right\} \\
&= \left\{ \left. (P_C) \in \prod_{C \in {\mathscr{C}}(R)} \operatorname{Spec}(C) \, \right| \,
\begin{gathered}
\text{for all inclusions } C \subseteq C', \\ \ P_{C'} \cap C = P_{C}
\end{gathered}
\right\} \\
&\cong \operatorname{\mathit{p}-Spec}(R),\end{aligned}$$ where the last isomorphism comes from Lemma \[data determining partial ideal\] (and the discussion that followed). These isomorphisms are natural in $R$ and thus provide an isomorphism of functors.
We will now show that $\operatorname{\mathit{p}-Spec}$ is our desired “universal $\operatorname{Spec}$ functor.” In fact, we prove a stronger result stating that it is universal among all contravariant functors $\operatorname{\mathsf{Ring}}\to \operatorname{\mathsf{Set}}$ whose restriction to $\operatorname{\mathsf{CommRing}}$ has a natural transformation to $\operatorname{Spec}$ that is not necessarily an isomorphism. This is made precise below.
Given functors $K \colon {\mathcal}{A} \to {\mathcal}{B}$ and $S \colon {\mathcal}{A} \to {\mathcal}{C}$, we recall that the *(right) Kan extension of $S$ along $K$* is a functor $R \colon {\mathcal}{B} \to {\mathcal}{C}$ along with a natural transformation $\varepsilon \colon RK \to S$ such that for any other functor $F \colon {\mathcal}{B} \to {\mathcal}{C}$ with a natural transformation $\eta \colon FK \to S$ there is a unique natural transformation $\delta \colon F \to R$ such that $\eta = \epsilon \circ (\delta K)$. (The “composite” $\delta K \colon FK \to RK$ of a functor with a natural transformation is a common shorthand for the *horizontal composite* $\delta \circ \mathbf{1}_{K}$ of the identity natural transformation $\mathbf{1}_{K} \colon K \to K$ with $\delta$, so that $\delta K(X) = \delta(K(X)) \colon FK \to RK$ for any $X \in {\mathcal}{A}$; see [@MacLane II.5] for information on horizontal composition.) When $K \colon {\mathcal}{A} \to {\mathcal}{B}$ is an inclusion of a subcategory ${\mathcal}{A} \subseteq {\mathcal}{B}$, notice that $FK = F|_{{\mathcal}{A}}$ is the restriction. In this case, the natural transformation $\delta K \colon FK \to RK$ is the induced natural transformation of the restricted functors $\delta|_{{\mathcal}{A}} \colon F|_{{\mathcal}{A}} \to R|_{{\mathcal}{A}}$.
\[universal Spec theorem\] The functor $\operatorname{\mathit{p}-Spec}\colon {\operatorname{\mathsf{Ring}}^{{\textnormal{op}}}}\to \operatorname{\mathsf{Set}}$, along with the identity natural transformation $\operatorname{\mathit{p}-Spec}|_{{\operatorname{\mathsf{CommRing}}^{{\textnormal{op}}}}} \to \operatorname{Spec}$, is the Kan extension of the functor $\operatorname{Spec}\colon {\operatorname{\mathsf{CommRing}}^{{\textnormal{op}}}}\to \operatorname{\mathsf{Set}}$ along the embedding ${\operatorname{\mathsf{CommRing}}^{{\textnormal{op}}}}\subseteq {\operatorname{\mathsf{Ring}}^{{\textnormal{op}}}}$. In particular, $\operatorname{\mathit{p}-Spec}$ is a terminal object in the category ${\mathfrak{r}}^{-1}(\operatorname{Spec})$.
Let $F \colon \operatorname{\mathsf{Ring}}\to \operatorname{\mathsf{Set}}$ be a contravariant functor with a fixed natural transformation $\eta \colon F|_{\operatorname{\mathsf{CommRing}}} \to \operatorname{Spec}$. We need to show that there is a unique natural transformation $\delta \colon F \to \operatorname{\mathit{p}-Spec}$ that induces $\eta$ upon restriction to $\operatorname{\mathsf{CommRing}}\subseteq \operatorname{\mathsf{Ring}}$. To construct $\delta$, fix a ring $R$. For every commutative subring $C$ of $R$, the inclusion $C \subseteq R$ gives a morphism of sets $F(R) \to F(C)$ and $\eta$ provides a morphism $\eta_C \colon F(C) \to \operatorname{Spec}(C)$; these compose to give morphisms $F(R) \to \operatorname{Spec}(C)$. By naturality of the morphisms involved, these maps out of $F(R)$ collectively form a cone over the diagram obtained by applying $\operatorname{Spec}$ to the diagram ${\mathscr{C}}(R)$ of commutative subrings of $R$. By the universal property of the limit, there exists a unique dotted arrow making the square below commute for all $C \in {\mathscr{C}}(R)$: $$\begin{tikzpicture}
\path
(0,0) node (FR) {$F(R)$} +(3.5,0) node (limSpec) {${\varprojlim}\limits_{C \in {\mathscr{C}}(R)} \operatorname{Spec}(C)$} + (7,0) node(pSpecR) {$\operatorname{\mathit{p}-Spec}(R)$}
+(0,-2) node (FC) {$F(C)$} +(3.5,-2) node (SpecC) {$\operatorname{Spec}(C)$.}
;
\draw[dashed,->] (FR) -- (limSpec);
\draw[->] (FR) -- (FC);
\draw[->] (limSpec) -- (SpecC);
\draw[->] (FC) edge node[above]{$\eta_C$} (SpecC);
\draw[->] (limSpec) edge node[above=-3pt]{$\sim$} (pSpecR);
\draw[->] (FR) edge[bend left=20] node[above]{$\delta_R$} (pSpecR);
\end{tikzpicture}$$ These morphisms $\delta_R$ form the components of a natural transformation $\delta \colon F \to \operatorname{\mathit{p}-Spec}$. By construction, $\delta$ induces $\eta$ when restricted to $\operatorname{\mathsf{CommRing}}$. Uniqueness of $\delta$ is guaranteed by the uniqueness of dotted arrow used to define $\delta_R$ above.
The second sentence of the theorem follows from the first by applying the universal property of the Kan extension in the special case where $F \colon {\operatorname{\mathsf{Ring}}^{{\textnormal{op}}}}\to \operatorname{\mathsf{Set}}$ is a functor with a natural transformation $\eta \colon F|_{{\operatorname{\mathsf{CommRing}}^{{\textnormal{op}}}}} \to \operatorname{Spec}$ that is an isomorphism.
Morphisms of partial algebras and the Kochen-Specker Theorem {#Kochen-Specker section}
============================================================
Having defined our universal functor $\operatorname{\mathit{p}-Spec}$ extending $\operatorname{Spec}$, we must now determine its value on the algebra ${\mathbb{M}}_n({\mathbb{C}})$. The first result of this section establishes a connection between the partial prime ideals of this algebra and certain morphisms of partial algebras.
We recall a relevant fact from commutative algebra. Let $C$ be a finite dimensional commutative algebra over an algebraically closed field $k$. Such an algebra is artinian, so all of its prime ideals are maximal. Given a maximal ideal $\mathfrak{m} \subseteq C$, the factor $k$-algebra $C/\mathfrak{m}$ is a finite dimensional field extension of the algebraically closed field $k$ and thus is is isomorphic to $k$. Hence $\operatorname{Spec}(C)$ is in bijection with the set of $k$-algebra homomorphisms $C \to k$. This situation is generalized below.
\[finite dimensional partial Spec\] Let $R$ be partial algebra over an algebraically closed field $k$ such that every element of $R$ is algebraic over $k$ (e.g., $R$ is a finite dimensional $k$-algebra). Then there is a bijection between the set $\operatorname{\mathit{p}-Spec}(R)$ and the set of all morphisms of partial $k$-algebras $f \colon R \to k$, which associates to each such morphism $f$ the inverse image $f^{-1}(0)$.
Because $R$ consists of algebraic elements, every element of $R$ generates a finite dimensional commeasurable subalgebra. In other words, $R$ is the union of its finite dimensional commeasurable subalgebras.
Given a morphism $f \colon R \to k$ of partial $k$-algebras, the set $P_f := f^{-1}(0) \subseteq R$ is a prime partial ideal of $R$ according to Lemma \[preimage lemma\]. Furthermore, for each finite dimensional commeasurable subalgebra $C \subseteq R$, the prime ideal $C \cap P_f \lhd C$ is maximal. Thus the restriction $f|_C$ must be equal to the canonical homomorphism $C \twoheadrightarrow C/(P_f \cap C) \overset{\sim}{\longrightarrow} k$.
Conversely, suppose that $P \subseteq R$ is a prime partial ideal. We define a function $f \colon R \to k$ as follows. As before, for each finite dimensional commeasurable subalgebra $C \subseteq R$ the prime ideal $P \cap C$ of $C$ is a maximal ideal. Thus we may define $g_C \colon C \to k$ via the quotient map $C \twoheadrightarrow C/(P \cap C) \overset{\sim}{\longrightarrow} k$. Notice that for finite dimensional commeasurable subalgebras $C \subseteq C'$, the following diagram commutes: $$\xymatrix{
C \ar@{^{(}->}[d] \ar@{->>}[r] & C/(P \cap C) \ar[d] \ar[r]^-{\sim} & k \ar@{=}[d] \\
C' \ar@{->>}[r] & C'/(P \cap C') \ar[r]^-{\sim} & k.
}$$ Thus there is a well-defined function $f_P \colon R \to k$ given, for any $r \in R$, by $f_P(r) = g_C(r)$ for any finite dimensional commeasurable subalgebra $C$ of $R$ containing $r$ (such as $C = k[r] \subseteq R$). It is clear from the construction of $f_P$ that $f_P^{-1}(0) = P$.
We have defined maps $P \mapsto f_P$ and $f \mapsto P_f$. The last sentences of the previous two paragraphs show that these assignments are mutually inverse, completing the proof.
Thus the proof of Theorem \[main theorem\] is reduced to understanding the morphisms of partial ${\mathbb{C}}$-algebras ${\mathbb{M}}_n({\mathbb{C}}) \to {\mathbb{C}}$. The *Kochen-Specker Theorem* provides just the information that we need. This theorem, due to S. Kochen and E. Specker [@KochenSpecker], is a “no-go theorem” from quantum mechanics that rules out the existence of certain types of hidden variable theories. Probability is an inherent feature in the mathematical formulation of quantum physics; only the evolution of the probability amplitude of a system is computed. A hidden variable theory is, roughly speaking, a theory devised to explain quantum mechanics by predicting outcomes of all measurements *with certainty*.
The observable quantities of a quantum system are mathematically represented by self-adjoint operators in a $C^*$-algebra. The Heisenberg Uncertainty Principle implies that if two such operators do not commute, then the exact values of the corresponding observables cannot be simultaneously determined. On the other hand, commuting observables have no uncertainty restriction imposed upon them by Heisenberg’s principle. In [@KochenSpecker] Kochen and Specker argued that a hidden variable theory should assign a real value to each observable of a quantum system in such a way that values of the sum or product of commuting observables is equal to the sum or product of their corresponding values. That is to say, Kochen and Specker’s notion of a hidden variable theory is a morphism of partial $\mathbb{R}$-algebras from the partial algebra of observables to $\mathbb{R}$. With this motivation, Kochen and Specker showed that no such morphism exists.
\[Kochen-Specker theorem\] Let $n \geq 3$, and for $A := {\mathbb{M}}_n(\mathbb{C})$ let $A_{sa} \subseteq A$ denote the subset of self-adjoint elements of $A$. There does not exist a morphism of partial $\mathbb{R}$-algebras $f \colon A_{sa} \to \mathbb{R}$.
Actually, [@KochenSpecker] establishes this result for $n=3$, but it is often cited in the literature for $n \geq 3$. Because the reduction to the case $n=3$ is straightforward, we include it below.
We assume that the result holds for $n=3$, as proved in [@KochenSpecker]. Let $n>3$, and assume for contradiction that there is a morphism of partial algebras $f \colon A_{sa} \to \mathbb{R}$. Let $P_i = E_{ii} \in A_{sa}$ be the orthogonal projection onto the $i$th basis vector. Then $\sum P_i = I$ and $P_i P_j = \delta_{ij} P_i$. In particular, because $f$ is a morphism of partial algebras we have $\sum f(P_i) = f(\sum P_i) = 1$. Furthermore, each $f(P_i) = f(P_i^2) = f(P_i)^2$ must equal either $0$ or $1$. So the values $f(P_i)$ are all equal to $0$, except for one $P_j$ with $f(P_j) = 1$.
Choose two of the other projections $P_i$ to get a set of three distinct projections $P_j$, $P_k$, and $P_{\ell}$. Then $E := P_j + P_k + P_{\ell}$ is an orthogonal projection, so there is an isomorphism of the corner algebra $EAE \cong {\mathbb{M}}_3(\mathbb{C})$ that preserves self-adjoint elements. Now the restriction of $f$ to $(EAE)_{sa} = EAE \cap A_{sa}$ satisfies all properties of a morphism of partial $\mathbb{R}$-algebras, except possibly the preservation of the multiplicative identity. But the multiplicative identity of $(EAE)_{sa}$ is $E$ and $f(E) = f(P_j) + f(P_k) + f(P_{\ell}) = 1$, proving that $f$ is a morphism of partial algebras. This contradicts the Kochen-Specker Theorem in dimension 3.
In Corollary \[Kochen-Specker corollary\] below, we will establish an analogue of the Kochen-Specker Theorem that is more suitable for our purposes. First we require one preparatory result. Given an element $x$ of a partial ring $R$, we will say that another element $y \in R$ is *an inverse of $x$* if $x \perp y$ and $xy = 1$. (Such an element need not be unique! An example of an element with two inverses is easily constructed by taking two copies of a Laurent polynomial ring $k[x_1, x_1^{-1}]$, $k[x_2, x_2^{-1}]$, “gluing” them by identifying $k[x_1]$ with $k[x_2]$, and declaring $x_i^{-1} {\perp}x_1 = x_2$ but with the $x_i^{-1}$ not commeasurable to one another. An inverse $y$ of $x$ is unique if $y$ is commeasurable to all elements of $R$ that are commeasurable to $x$. We thank George Bergman for these observations.) The following argument is a standard one. It basically appeared in [@KochenSpecker pp.81–82], and it even has roots in the theory of the Gelfand spectrum of $C^*$-algebras.
\[eigenvalue lemma\] Let $R$ be a partial algebra over a commutative ring $k \neq 0$, and let $f \colon R \to k$ be a morphism of partial $k$-algebras. Then for any $r \in R$, the element $r-f(r) \in R$ does not have an inverse. In particular, if $k$ is a field and $R = {\mathbb{M}}_n(k)$, then $f(r) \in k$ is an eigenvalue of $r$.
If $r-f(r)$ has an inverse $u \in R$, then $k=0$ by the following equation: $$\begin{aligned}
1 &= f(1) \\
&= f((r - f(r))u) \\
&= f(r-f(r)) f(u) \\
&= (f(r) - f(f(r) \cdot 1)) f(u) \\
&= (f(r) - f(r)) f(u) \\
&= 0. \qedhere\end{aligned}$$
We now have the following reformulation of the Kochen-Specker Theorem that is more appropriate to our needs. (One could think of it as a “complex-valued,” rather than “real-valued,” Kochen-Specker Theorem.) Together with Proposition \[finite dimensional partial Spec\], this constitutes the final “key result” used in the proof of Theorem \[main theorem\].
\[Kochen-Specker corollary\] For any $n \geq 3$, there is no morphism of partial $\mathbb{C}$-algebras ${\mathbb{M}}_n(\mathbb{C}) \to \mathbb{C}$.
Let $A = {\mathbb{M}}_n(\mathbb{C})$. Every self-adjoint matrix in $A$ has real eigenvalues, so Lemma \[eigenvalue lemma\] implies that a morphism of partial ${\mathbb{C}}$-algebras $A \to {\mathbb{C}}$ restricts to a morphism of partial $\mathbb{R}$-algebras $A_{sa} \to \mathbb{R}$. But such morphisms are forbidden by the Kochen-Specker Theorem \[Kochen-Specker theorem\].
It is natural to ask what is the status of Corollary \[Kochen-Specker corollary\] in the case $n=2$. Regarding their original theorem, Kochen and Specker demonstrated the existence of a morphism of partial $\mathbb{R}$-algebras ${\mathbb{M}}_2(\mathbb{C})_{sa} \to \mathbb{R}$ in [@KochenSpecker §6], showing that the Kochen-Specker Theorem does not extend to $n = 2$. Similarly, Corollary \[Kochen-Specker corollary\] does not extend to $n = 2$. There exist morphisms of partial algebras ${\mathbb{M}}_2({\mathbb{C}}) \to {\mathbb{C}}$, and we can describe all of them as follows. Incidentally, this result also shows that the statement of Theorem \[main theorem\] is not valid in the case $n = 2$; the functor $F = \operatorname{\mathit{p}-Spec}$ assigns a nonempty set (of cardinality $2^{2^{\aleph_0}}$, in fact!) to ${\mathbb{M}}_2({\mathbb{C}})$.
\[the case n = 2\] Let $k$ be an algebraically closed field, and let $\mathcal{I} \subseteq A := {\mathbb{M}}_2(k)$ be any set of idempotents such that the set of all idempotents of $A$ is partitioned as $$\{ 0, 1\} \sqcup \mathcal{I} \sqcup \{ 1-e : e \in \mathcal{I} \}.$$ Then for every function $\alpha \colon \mathcal{I} \to \{0,1\}$ there is a morphism of partial $k$-algebras $f_{\alpha} \colon A \to k$ such that the restriction of $f$ to $\mathcal{I}$ is $\alpha \colon \mathcal{I} \to \{0,1\} \subseteq k$. Moreover, there are bijective correspondences between:
- the set of functions $\alpha \colon \mathcal{I} \to \{0,1\}$;
- the set of morphisms of partial $k$-algebras $A \to k$; and
- the set of prime partial ideals of $A$;
given by $\alpha \leftrightarrow f_{\alpha} \leftrightarrow f_{\alpha}^{-1}(0)$.
First we construct a commutative $k$-algebra $B$ with a morphism of partial $k$-algebras $h \colon A \to B$. Let $\mathcal{N}$ be a set of nonzero nilpotent elements of $A$ such that every nonzero nilpotent matrix in $A$ has exactly one scalar multiple in $\mathcal{N}$. Let $B$ be the commutative $k$-algebra $B := k[ x_e, x_n : e \in \mathcal{I}, n \in \mathcal{N}]$ with relations $x_e^2 = x_e$ for $e \in \mathcal{I}$ and $x_n^2 = 0$ for $n \in \mathcal{N}$.
A result of Schur [@Schur] (also proved more generally by Jacobson [@Jacobson Thm. 1]) implies that every maximal commutative subalgebra of $A$ is has $k$-dimension 2. Thus the intersection of two distinct commutative subalgebras of $A$ is the scalar subalgebra $k \subseteq A$. This makes it easy to see that a function $h \colon A \to B$ is a morphism of partial $k$-algebras if and only if its restriction to every 2-dimensional commutative subalgebra of $A$ is a $k$-algebra homomorphism.
Now define a function $h \colon A \to B$ as follows. For each scalar $\lambda \in k \subseteq A$, we set $h(\lambda) = \lambda \in k \subseteq B$. Now assume $a \in A \setminus k$. Then $k[a]$ is a 2-dimensional commutative subalgebra of $A$. Because the only 2-dimensional algebras over the algebraically closed field $k$ are $k \times k$ and $k[\varepsilon]/(\varepsilon^2)$, there exists $b \in \mathcal{I} \sqcup \mathcal{N}$ such that $k[a] = k[b]$. The careful choice of the sets $\mathcal{I}$ and $\mathcal{N}$ ensures that this $b$ is unique. Thus it suffices to define $h$ on each $k[b]$. But for $b \in \mathcal{I} \sqcup \mathcal{N}$, the map $k[b] \to B$ defined by sending $b \mapsto x_b$ is clearly a homomorphism of $k$-algebras. We define the restriction of $h$ to $k[a] = k[b]$ to be this homomorphism, which in particular defines the value $h(a)$.
Certainly $h$ is well-defined, and it is a morphism of partial algebras because its restriction to every 2-dimensional subalgebra is an algebra homomorphism. Thus we have constructed a morphism of partial algebras $h \colon A \to B$.
Given a function $\alpha \colon \mathcal{I} \to \{0,1\}$, there exists a $k$-algebra homomorphism $g_{\alpha} \colon B \to k$ given by sending $x_e \mapsto \alpha(e) \in k$ for $e \in \mathcal{I}$ and $x_n \mapsto 0$ for $n \in \mathcal{N}$. So the composite $f_{\alpha} := g_{\alpha} \circ h$ is a morphism of partial $k$-algebras whose restriction to $\mathcal{I}$ is equal to $\alpha$. The bijection between the three sets in the statement of the proposition follows directly from Proposition \[finite dimensional partial Spec\] above and Lemma \[values on idempotents\] below.
\[values on idempotents\] Let $R$ be a partial algebra over an algebraically closed field $k$ in which every element is algebraic (e.g., $R$ is a finite dimensional $k$-algebra). A morphism of partial algebras $R \to k$ is uniquely determined by its restriction to the set of idempotents of $R$.
Let $f \colon R \to k$ be a morphism of partial $k$-algebras, and let $C \subseteq R$ be a finite dimensional commeasurable subalgebra of $R$. Because $R$ is the union of its finite dimensional commeasurable subalgebras, it is enough to show that the restriction of $f$ to $C$, which is a $k$-algebra homomorphism $C \to k$, is uniquely determined by its values on the idempotents of $C$.
Because $C$ is finite dimensional it is artinian and thus is a finite direct sum of local $k$-algebras. Write $C = A_1 \oplus \cdots \oplus A_n$ where each $(A_i, M_i)$ is local and the identity element of $A_i$ is $e_i$, an idempotent of $C$. Since $k$ is algebraically closed, each of the residue fields $A_i/M_i$ is isomorphic to $k$ as a $k$-algebra. Thus each $A_i = ke_i \oplus M_i$. Because $A_i$ is finite dimensional, its Jacobson radical $M_i$ is nilpotent and hence is in the kernel of $f|_C$. It now follows easily that the restriction of $f$ to $C$ is determined by the values $f(e_i)$.
Proof and consequences of the main result {#proof section}
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We are now prepared to prove Theorem \[main theorem\], the main ring-theoretic result of the paper.
Fix $n \geq 3$ and let $A = {\mathbb{M}}_n(\mathbb{C}))$. According to Theorem \[universal Spec theorem\] there exists a natural transformation $F \to \operatorname{\mathit{p}-Spec}$. By Proposition \[finite dimensional partial Spec\], $\operatorname{\mathit{p}-Spec}(A)$ is in bijection with the set of morphisms of partial ${\mathbb{C}}$-algebras $A \to \mathbb{C}$. No such morphisms exist according to Corollary \[Kochen-Specker corollary\] of the Kochen-Specker Theorem, so $\operatorname{\mathit{p}-Spec}(A) = \varnothing$. The existence of a function $F(A) \to \operatorname{\mathit{p}-Spec}(A) = \varnothing$ now implies that $F(A) = \varnothing$.
It seems appropriate to mention some partial positive results that contrast with Theorem \[main theorem\]. One might hope that restricting to certain well-behaved ring homomorphisms could allow the functor $\operatorname{Spec}$ to be “partially extended.” In this vein, Procesi has shown [@Procesi Lem. 2.2] that if $f \colon R \to S$ is a ring homomorphism such that $S$ is generated over $f(R)$ by elements centralizing $f(R)$, then for every prime ideal $Q \lhd S$ the inverse image $f^{-1}(Q)$ is again prime. Furthermore, he showed in [@Procesi Thm. 3.3] that if $R$ is a Jacobson PI ring and $f \colon R \to S$ is a ring homomorphism such that $S$ is generated by the image $f(R)$ and finitely many elements that centralize $f(R)$, then for every maximal ideal $M \lhd S$ the inverse image $f^{-1}(M)$ is a maximal ideal of $R$. (Although he only stated these results for injective $f$, they are easily seen to hold more generally.)
On the other hand, one may try to replace functions between prime spectra by “multi-valued functions,” which may send a single element of one set to many elements of another set. For instance, one might consider a functor that maps each homomorphism $R \to S$ of noncommutative rings to a *correspondence* $\operatorname{Spec}(S) \to \operatorname{Spec}(R)$, which sends a single prime ideal of $S$ to some nonempty finite set of prime ideals of $R$. This notion was introduced by Artin and Schelter in [@ArtinSchelter §4] and studied in further detail by Letzter in [@Letzter]. There is an appropriate notion of “continuity” of a correspondence, and it is shown in [@Letzter Cor. 2.3] (see also [@ArtinSchelter Prop. 4.6]) that if $f \colon R \to S$ is a ring homomorphism and $S$ is a PI ring, then the associated correspondence is continuous. However, there exist homomorphisms between noetherian rings whose correspondence is not continuous [@Letzter §2.5].
We now present a few corollaries of Theorem \[main theorem\]. The first is a straightforward generalization of that theorem replacing ${\mathbb{M}}_n({\mathbb{C}})$ with ${\mathbb{M}}_n(R)$ where $R$ is any ring containing a field isomorphic to ${\mathbb{C}}$.
\[matrix algebra corollary\] Let $F \colon \operatorname{\mathsf{Ring}}\to \operatorname{\mathsf{Set}}$ be a contravariant functor whose restriction to the full subcategory of commutative rings is isomorphic to $\operatorname{Spec}$. If $R$ is any ring with a homomorphism ${\mathbb{C}}\to R$, then $F({\mathbb{M}}_n(R)) = \varnothing$ for $n \geq 3$.
The homomorphism $\mathbb{C} \to R$ induces a homomorphism ${\mathbb{M}}_n(\mathbb{C}) \to {\mathbb{M}}_n(R)$. Thus we have a set map $F({\mathbb{M}}_n(R)) \to F({\mathbb{M}}_n(\mathbb{C}))$. If $n \geq 3$ then by Theorem \[main theorem\] the latter set is empty; hence the former set must also be empty.
In the corollary above, $R$ can be any complex algebra. But rings that contain ${\mathbb{C}}$ as a non-central subring, such as the real quaternions, are also allowed. On the other hand, suppose that $R$ is a complex algebra such that $R \cong {\mathbb{M}}_n(R)$ for some $n \geq 2$. It follows that $R \cong {\mathbb{M}}_n(R) \cong {\mathbb{M}}_n({\mathbb{M}}_n(R)) \cong {\mathbb{M}}_{n^2}(R)$, so we may assume that $n \geq 4$. Then the corollary implies that for functors $F$ as above, $F(R) \cong F({\mathbb{M}}_n(R)) = \varnothing$. For instance, if $V$ is an infinite dimensional ${\mathbb{C}}$-vector space and $R$ is the algebra of ${\mathbb{C}}$-linear endomorphisms of $V$, then the existence of a vector space isomorphism $V \cong V^{\oplus n}$ (any $n \geq 2$) implies the existence of an algebra isomorphism $R \cong {\mathbb{M}}_n(R)$.
An attempt to extend the ideas above suggests one possible algebraic generalization of the Kochen-Specker Theorem. Suppose that $\operatorname{\mathit{p}-Spec}({\mathbb{M}}_n({\mathbb{Z}})) = \varnothing$ for some integer $n \geq 3$. For any ring $R$ the canonical ring homomorphism ${\mathbb{Z}}\to R$ induces a morphism ${\mathbb{M}}_n({\mathbb{Z}}) \to {\mathbb{M}}_n(R)$. Then one would have $\operatorname{\mathit{p}-Spec}({\mathbb{M}}_n(R)) = \varnothing$. It would follow that any contravariant functor $F \colon \operatorname{\mathsf{Ring}}\to \operatorname{\mathsf{Set}}$ whose restriction to $\operatorname{\mathsf{CommRing}}$ is isomorphic to $\operatorname{Spec}$ must assign the empty set to ${\mathbb{M}}_n(R)$ for any ring $R$. This highlights the importance of the following question.
Do there exist integers $n \geq 3$ such that $\operatorname{\mathit{p}-Spec}({\mathbb{M}}_n({\mathbb{Z}})) = \varnothing$?
If $\operatorname{\mathit{p}-Spec}({\mathbb{M}}_n({\mathbb{Z}}))$ were in fact empty for all sufficiently large values of $n$, then this would be a sort of “integer-valued” Kochen-Specker Theorem.
The next corollary of Theorem \[main theorem\] concerns certain functors sending rings to commutative rings. Consider the functor $\operatorname{\mathsf{Ring}}\to \operatorname{\mathsf{CommRing}}$ that sends each ring $R$ to its “abelianization” $R/[R,R]$. Rings whose abelianization is zero are easy to find, and this functor necessarily destroys all information about these rings. One could try to abstract this functor by considering any functor $\operatorname{\mathsf{Ring}}\to \operatorname{\mathsf{CommRing}}$ whose restriction to $\operatorname{\mathsf{CommRing}}$ is isomorphic to the identity functor. The following result says that every such “abstract abelianization functor” necessarily destroys matrix algebras.
\[abelianization corollary\] Let $\alpha \colon \operatorname{\mathsf{Ring}}\to \operatorname{\mathsf{CommRing}}$ be a functor such that the restriction of $\alpha$ to $\operatorname{\mathsf{CommRing}}$ is isomorphic to the identity functor. Then for any ring $R$ with a homomorphism ${\mathbb{C}}\to R$ and any $n \geq 3$, one has $\alpha({\mathbb{M}}_n(R)) = 0$. In particular, $\alpha$ is not faithful.
Because $\alpha$ restricts to the identity functor on $\operatorname{\mathsf{CommRing}}$, the contravariant functor $F := \operatorname{Spec}\circ \alpha \colon \operatorname{\mathsf{Ring}}\to \operatorname{\mathsf{Set}}$ satisfies $F|_{\operatorname{\mathsf{CommRing}}} \cong \operatorname{Spec}$. For $n \geq 3$, Corollary \[matrix algebra corollary\] implies that $\operatorname{Spec}(\alpha({\mathbb{M}}_n(R)) = F({\mathbb{M}}_n(R)) = \varnothing$. Hence the commutative ring $\alpha({\mathbb{M}}_n(R))$ is zero.
To see that $\alpha$ is not faithful, fix $n \geq 3$ and consider that $\alpha$ induces a function $$\operatorname{Hom}_{\operatorname{\mathsf{Ring}}}({\mathbb{M}}_n({\mathbb{C}}), {\mathbb{M}}_n({\mathbb{C}})) \to
\operatorname{Hom}_{\operatorname{\mathsf{CommRing}}}(\alpha({\mathbb{M}}_n({\mathbb{C}})), \alpha({\mathbb{M}}_n({\mathbb{C}})) = \operatorname{Hom}_{\operatorname{\mathsf{CommRing}}}(0, 0).$$ The latter set is a singleton, while the former set is not a singleton (because ${\mathbb{M}}_n({\mathbb{C}})$ has nontrivial inner automorphisms). So the function above is not injective, proving that $\alpha$ is not faithful.
Interestingly, this result does not hold in the case $n = 2$; we thank George Bergman for this observation. Let $\alpha \colon \operatorname{\mathsf{Ring}}\to \operatorname{\mathsf{CommRing}}$ be the functor sending each ring to the colimit of the diagram of its commutative subrings. Certainly $\alpha|_{\operatorname{\mathsf{CommRing}}}$ is isomorphic to the identity functor on $\operatorname{\mathsf{CommRing}}$. One can check that for an algebraically closed field $k$, the commutative ring $\alpha({\mathbb{M}}_2(k))$ is isomorphic to the algebra $B$ constructed in the proof of Proposition \[the case n = 2\]; in particular, $\alpha({\mathbb{M}}_2(k)) \neq 0$. (At the very least, it is not hard to verify from the universal property of the colimit that there exists a homomorphism $\alpha({\mathbb{M}}_2(k)) \to B$, confirming that $\alpha({\mathbb{M}}_2(k)) \neq 0$.) Furthermore, one can show that this functor is initial among all “abstract abelianization functors,” but the details will not be presented here.
The final corollary of Theorem \[main theorem\] to be presented in this section is a rigorous proof that the rule that assigns to each (not necessarily commutative) ring $R$ the set $\operatorname{Spec}(R)$ of prime ideals of $R$ is “not functorial.” (Recall that an ideal $P \lhd R$ is *prime* if, for all ideals $I, J \lhd R$, $IJ \subseteq P$ implies that either $I \subseteq P$ or $J \subseteq P$.) The fact that this assignment “is not a functor” seems to be common wisdom. (Specific mention of this idea in the literature is not widespread, but see [@vanOystaeyenVerschoren pp. 1 and 36] or [@Letzter §1] for examples.) It is easy to verify that this assignment is not a functor in the natural way; that is, if $f \colon R \to S$ a ring homomorphism and $P \lhd S$ is prime, one can readily see that the ideal $f^{-1}(P) \lhd R$ need not be prime. However, we are unaware of any rigorous statement or proof in the literature of the precise result below.
\[prime ideals not functorial\] There is no contravariant functor $F \colon \operatorname{\mathsf{Ring}}\to \operatorname{\mathsf{Set}}$ whose restriction to the full subcategory $\operatorname{\mathsf{CommRing}}$ is isomorphic to $\operatorname{Spec}$ and such that, for every ring $R$, the set $F(R)$ is in bijection with the set of prime ideals of $R$.
Assume for contradiction that such $F$ exists. Fix $n \geq 3$. Because the zero ideal of ${\mathbb{M}}_n({\mathbb{C}})$ is (its unique) prime, the assumption on $F$ implies $F({\mathbb{M}}_n({\mathbb{C}})) \neq \varnothing$, violating Theorem \[main theorem\].
This corollary can also be derived from an elementary argument that avoids using Theorem 1.1. In fact, the statement can even be strengthened as follows.
\[Morita invariant Spec\] There is no contravariant functor $F\colon\operatorname{\mathsf{Ring}}\to\operatorname{\mathsf{Set}}$ whose restriction to $\operatorname{\mathsf{CommRing}}$ is isomorphic to $\operatorname{Spec}$ and such that $F$ satisfies either of the following conditions:
1. For some field $k$ and some integer $n \geq 2$, the set $F({\mathbb{M}}_n(k))$ is a singleton;
2. $F$ is Morita invariant in the following sense: for any Morita equivalent rings $R$ and $S$, one has $F(R)\cong F(S)$.
First notice that if $F$ satisfies condition (2) above, then it satisfies condition (1) because ${\mathbb{M}}_n(k)$ is Morita equivalent to $k$, which would mean that $F({\mathbb{M}}_n(k)) \cong F(k) \cong \operatorname{Spec}(k)$ is a singleton. So assume for contradiction that there exists a functor $F$ as above satisfying (1).
Fix $k$ and $n$ as in condition (1). Define $\pi := (1\ 2\ \cdots \ n) \in S_n$, a permutation of the set $\{1,2, \dots, n\}$. Let $\rho$ be the automorphism of $k^n$ given by $(a_i) \mapsto (a_{\pi(i)})$, let $P \in {\mathbb{M}}_n(k)$ be the permutation matrix whose $i$th row is the $\pi(i)$th standard basis row vector, and let $\sigma$ be the inner automorphism of ${\mathbb{M}}_n(k)$ given by $\sigma(A) = PAP^{-1}$. For the final piece of notation, let $\iota \colon k^n \hookrightarrow {\mathbb{M}}_n(k)$ be the diagonal embedding.
The following equality of algebra homomorphisms $k^n \to {\mathbb{M}}_n(k)$ is elementary: $$\iota \circ \rho = \sigma \circ \iota.$$ Applying the contravariant functor $F$ to this equation gives $F(\rho) \circ F(\iota) = F(\iota) \circ F(\sigma)$. By hypothesis the set $F({\mathbb{M}}_n(k))$ is a singleton. Hence the automorphism $F(\sigma)$ of $F({\mathbb{M}}_n(k))$ is the identity. It follows that $$\label{trouble equation}
F(\rho) \circ F(\iota) = F(\iota).$$ On the other hand $F(k^n) \cong \operatorname{Spec}(k^n) = \{1, \dots, n\}$, and under this isomorphism $F(\rho)$ acts as $\operatorname{Spec}(\rho) = \pi^{-1}$ which has no fixed points. Thus the image of the unique element of $F({\mathbb{M}}_n(k))$ under $F(\iota)$ is distinct from its image under $F(\rho) \circ F(\iota)$, contradicting above.
Because the set of prime ideals of a noncommutative ring is Morita invariant (for instance, see [@Lectures (18.45)]) the proposition above implies Corollary \[prime ideals not functorial\]. Notice that Proposition \[Morita invariant Spec\] with $k = {\mathbb{C}}$ and $n = 2$ cannot be derived from Theorem \[main theorem\] because that theorem does not apply to the algebra ${\mathbb{M}}_2({\mathbb{C}})$, as explicitly shown in Proposition \[the case n = 2\].
Of course, there are many important examples of invariants of rings extending $\operatorname{Spec}$ of a commutative ring that respect Morita equivalence, aside from the set of prime two-sided ideals of a ring. Two examples are the prime torsion theories introduced by O. Goldman in [@Goldman] and the spectrum of an abelian category defined by A. Rosenberg in [@Rosenberg2]. (Incidentally, both of these spectra arise from the theory of noncommutative localization.) Each of these invariants is certainly useful in the study of noncommutative algebra, and they have appeared in different approaches to noncommutative algebraic geometry. Thus we emphasize that Proposition \[Morita invariant Spec\] does not in any way suggest that such invariants should be avoided. It simply reveals that we cannot hope for such invariants to be functors to $\operatorname{\mathsf{Set}}$.
The analogous result for $C^*$-algebras {#C* section}
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In this section we will prove Theorem \[main C\* theorem\], which obstructs extensions of the Gelfand spectrum functor to noncommutative $C^*$-algebras. We begin by reviewing some facts and setting some conventions about the category of $C^*$-algebras. (Many of these basics can be found in [@Davidson §I.5] and [@KadisonRingrose §4.1].) We emphasize that *all $C^*$-algebras considered in this section are assumed to be unital*. Let $\operatorname{\mathsf{C^*}\mathsf{Alg}}$ denote the category whose objects are unital $C^*$-algebras and whose morphisms are identity-preserving $*$-homomorphisms. Such morphisms do not increase the norm and are norm-continuous. The only topology on a $C^*$-algebra to which we will refer is the norm topology. A closed ideal of a $C^*$-algebra is always $*$-invariant, and the resulting factor algebra is a $C^*$-algebra. A *$C^*$-subalgebra* of a $C^*$-algebra $A$ is a closed subalgebra $C \subseteq A$ that is invariant under the involution of $A$; such a subalgebra inherits the structure of a Banach algebra with involution from $A$ and is itself a $C^*$-algebra with respect to this inherited structure. If $f \colon A \to B$ is a $*$-homomorphism, then the image $f(A) \subseteq B$ is always a $C^*$-subalgebra. The full subcategory of $\operatorname{\mathsf{C^*}\mathsf{Alg}}$ consisting of commutative unital $C^*$-algebras is denoted by $\operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}$. Finally, the reader may wish to see Section \[introduction section\] for the definition of the (contravariant) Gelfand spectrum functor $\operatorname{Max}\colon \operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}\to \operatorname{\mathsf{Set}}$.
As in the ring-theoretic case, we define an appropriate category of functors in which we seek a universal functor. The inclusion of categories $\operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}\hookrightarrow \operatorname{\mathsf{C^*}\mathsf{Alg}}$ induces a *restriction* functor between functor categories $$\begin{aligned}
{\mathfrak{r}}\colon \operatorname{Fun}(\operatorname{\mathsf{C^*}\mathsf{Alg}}^{{\textnormal{op}}}, \operatorname{\mathsf{Set}}) &\to \operatorname{Fun}(\operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}^{{\textnormal{op}}}, \operatorname{\mathsf{Set}}) \\
F &\mapsto F|_{\operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}^{{\textnormal{op}}}}.\end{aligned}$$ Again we define the “fiber category” over $\operatorname{Max}\in \operatorname{Fun}(\operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}^{{\textnormal{op}}}, \operatorname{\mathsf{Set}})$ to be the category ${\mathfrak{r}}^{-1}(\operatorname{Max})$ of pairs $(F, \phi)$ where $F \in \operatorname{Fun}(\operatorname{\mathsf{C^*}\mathsf{Alg}}^{{\textnormal{op}}}, \operatorname{\mathsf{Set}})$ and $\phi \colon {\mathfrak{r}}(F) \overset{\sim}{\longrightarrow} \operatorname{Max}$ is an isomorphism of functors; a morphism $\psi \colon (F, \phi) \to (F', \phi')$ in ${\mathfrak{r}}^{-1}(\operatorname{Max})$ is a natural transformation $\psi \colon F \to F'$ such that $\phi' \circ {\mathfrak{r}}(\psi) = \phi$. Our first goal is to locate a final object of the category ${\mathfrak{r}}^{-1}(\operatorname{Max})$, which we view as a “universal noncommutative Gelfand spectrum functor.”
First we define the analogue of the spectrum $\operatorname{\mathit{p}-Spec}$ of prime partial ideals. To do so, it will be useful to think in terms of partial $C^*$-algebras, as defined by van den Berg and Heunen in [@BergHeunen1 §4].
A partial $C^*$-algebra $P$ is a partial ${\mathbb{C}}$-algebra with an involution $* \colon P \to P$ and a function $\| \cdot \| \colon P \to {\mathbb{R}}$ such that any set $S \subseteq P$ of pairwise commeasurable elements is contained in a set $T \subseteq P$ such that the restricted operations of $P$ endows $T$ with the structure of a commutative $C^*$-algebra. Such a subset $T \subseteq P$ is called a *commeasurable $C^*$-subalgebra* of $P$. A *$*$-morphism* of partial $C^*$-algebras $f \colon P \to Q$ is a morphism of partial ${\mathbb{C}}$-algebras satisfying $f(a^*) = f(a)^*$ for all $a \in P$.
It is very important to note that, unlike the ring-theoretic case, a $C^*$-algebra with the commeasurability relation of commutativity is generally *not* a partial $C^*$-algebra. What is true is that for any $C^*$-algebra $A$, the set $N(A) = \{a \in A : aa^* = a^* a\}$ of normal elements (with commutativity as commeasurability) is always a partial $C^*$-algebra. This makes use of the fact that any normal element of a $C^*$-algebra has the property that its centralizer is a $*$-subalgebra; see [@Fuglede]. The assignment $A \mapsto N(A)$ defines a functor from the category of $C^*$-algebras to the category of partial $C^*$-algebras, defined on a morphism $f \colon A \to B$ by restricting and corestricting $f$ to $N(f) \colon N(A) \to N(B)$; see [@BergHeunen1 Prop. 3]. (Because of this subtlety regarding normal elements, we will typically use $P,Q$ to denote partial $C^*$-algebras and $A,B$ to denote full $C^*$-algebras.)
A *partial closed ideal* of a partial $C^*$-algebra $P$ is a subset $I \subseteq P$ such that, for every commeasurable $C^*$-subalgebra $C \subseteq P$, the intersection $I \cap C$ is a closed ideal of $C$. If, for every commeasurable $C^*$-subalgebra $C$ one has that $I \cap C$ is a maximal ideal of $C$, then $I$ is a *partial maximal ideal* of $N$.
Let $A$ be a $C^*$-algebra. We say that a subset $I \subseteq A$ is a *partial closed (resp. maximal) ideal* of $A$ if $I \subseteq N(A)$ and $I$ is a partial closed (resp. maximal) ideal of the partial $C^*$-algebra $N(A)$.
Because we require a partial closed ideal $I$ of a $C^*$-algebra $A$ to consist of normal elements, $I$ is completely determined by its intersection with all commutative subalgebras in the sense that $I = \bigcup_C (I \cap C)$, where $C$ ranges over all commutative $C^*$-subalgebras of $A$. This is true because an element of $A$ is normal if and only if it is contained in a commutative $C^*$-subalgebra of $A$.
As in the ring-theoretic case, partial closed ideals behave well under homomorphisms.
Let $f \colon P \to Q$ be a $*$-homomorphism of $C^*$-algebras, and let $I$ be a partial closed (resp. maximal) ideal of $Q$. The set $f^{-1}(I) \subseteq P$ is a partial closed (resp. maximal) partial ideal of $P$.
In particular, if $f \colon A \to B$ is a $*$-homomorphism between $C^*$-algebras and $I \subseteq N(B) \subseteq B$ is a partial closed (resp. maximal) ideal, then $f^{-1}(I) \cap N(A) \subseteq A$ is a partial closed (resp. maximal) ideal of $A$.
Let $C \subseteq P$ be a commeasurable $C^*$-subalgebra. We wish to show that $f^{-1}(I) \cap C$ is a closed (resp. maximal) ideal of $C$. First notice that since $C$ consists of pairwise commeasurable elements, so does $f(C) \subseteq Q$. Thus there is a commeasurable $C^*$-subalgebra $D \subseteq Q$ such that $f(C) \subseteq D$. But then since $f$ (co)restricts to a $*$-homomorphism of (full) $C^*$-algebras $C \to D$, its image $f(C) \subseteq D$ is a $C^*$-subalgebra. It follows that $f$ (co)restricts to a $*$-homomorphism of (full, commutative) $C^*$-algebras $f|_C \colon C \to f(C)$.
Now $I \cap f(C)$ is a closed (resp. maximal) ideal in $f(C)$ by hypothesis, so its preimage under $f|_C$ is a closed (resp. maximal) ideal of $C$. On the other hand, $(f|_C)^{-1}(I \cap f(C))$ is easily seen to be equal to $f^{-1}(I) \cap C$. Hence the latter is a closed (resp. maximal) ideal, as desired.
This allows us to define a “partial Gelfand spectrum” functor.
We define a contravariant functor $\operatorname{\mathit{p}-Max}\colon \operatorname{\mathsf{C^*}\mathsf{Alg}}\to \operatorname{\mathsf{Set}}$ by assigning to every $C^*$-algebra $A$ the set $\operatorname{\mathit{p}-Max}(A)$ of partial maximal ideals of $A$, and by assigning to each morphism $f \colon A \to B$ in $\operatorname{\mathsf{C^*}\mathsf{Alg}}$ the function $$\begin{aligned}
\operatorname{\mathit{p}-Max}(B) &\to \operatorname{\mathit{p}-Max}(A) \\
M &\mapsto f^{-1}(M) \cap N(A).\end{aligned}$$ (The only potential hindrance to functoriality is the preservation of composition of morphisms, but this is easily verified. Alternatively, this is seen to be a functor because it is the composite of the functor from $C^*$-algebras to partial $C^*$-algebras $A \mapsto N(A)$ with the contravariant functor from partial $C^*$-algebras to sets that sends a partial algebra to the set of its partial maximal ideals.)
Notice that the restriction of $\operatorname{\mathit{p}-Max}$ to $\operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}$ is equal to the Gelfand spectrum functor $\operatorname{Max}$, so that $\operatorname{\mathit{p}-Max}$ is an object of the category ${\mathfrak{r}}^{-1}(\operatorname{Max})$.
As in Proposition \[isomorphic functors\], the functor $\operatorname{\mathit{p}-Max}$ can be recovered through a limit construction. For a $C^*$-algebra $A$, we let ${\mathscr{C}}^*(A)$ denote the partially ordered set of its commutative $C^*$-subalgebras, viewed as a category in the usual way.
\[isomorphic C\* functors\] The contravariant functor $\operatorname{\mathit{p}-Max}\colon \operatorname{\mathsf{C^*}\mathsf{Alg}}\to \operatorname{\mathsf{Set}}$ is isomorphic to the functor defined, for a given $C^*$-algebra $A$, by $$A \mapsto {\varprojlim}_{C \in {\mathscr{C}}^*(A)^{{\textnormal{op}}}} \operatorname{Max}(C).$$ This isomorphism preserves the isomorphism of functors $\operatorname{\mathit{p}-Max}|_{\operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}} = \operatorname{Spec}$.
We will not include a proof of this fact, but we will mention the main subtlety. The appropriate analogue of Lemma \[data determining partial ideal\] (replacing each occurrence of “commeasurable subalgebra” with “commeasurable $C^*$-subalgebra”) still holds, and is used as before to prove the present result. Here it is crucial to recall that a partial closed ideal $I \subseteq A$ consists of normal elements, for this ensures that $I$ is determined by its intersection with all commutative $C^*$-subalgebras of $A$.
Just as before, this allows one to show that $\operatorname{\mathit{p}-Max}$ is a “universal Gelfand spectrum functor.”
\[universal Gelf theorem\] The functor $\operatorname{\mathit{p}-Max}\colon {\operatorname{\mathsf{C^*}\mathsf{Alg}}^{{\textnormal{op}}}}\to \operatorname{\mathsf{Set}}$, with the identity natural transformation $\operatorname{\mathit{p}-Max}|_{\operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}} \to \operatorname{Max}$, is the Kan extension of the functor $\operatorname{Max}\colon {\operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}^{{\textnormal{op}}}}\to \operatorname{\mathsf{Set}}$ along the embedding ${\operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}^{{\textnormal{op}}}}\subseteq {\operatorname{\mathsf{C^*}\mathsf{Alg}}^{{\textnormal{op}}}}$. In particular, $\operatorname{\mathit{p}-Max}$ is a terminal object in the category ${\mathfrak{r}}^{-1}(\operatorname{Max})$.
Our next goal is to connect the functor $\operatorname{\mathit{p}-Max}$ to the Kochen-Specker Theorem, in a manner similar to that of Section \[Kochen-Specker section\]. We have the following analogue of Proposition \[finite dimensional partial Spec\]. Its proof is completely analogous, and thus is omitted.
\[partial Gelf and morphisms\] Let $P$ be a partial $C^*$-algebra. There is a bijection between the set of partial maximal ideals of $P$ and the set of $*$-morphisms of partial $C^*$-algebras $f \colon P \to {\mathbb{C}}$, which associates to each such morphism $f$ the inverse image $f^{-1}(0)$.
In particular, if $A$ is a $C^*$-algebra then there is a bijection between $\operatorname{\mathit{p}-Max}(A)$ and the set of $*$-morphisms of partial $C^*$-algebras $f \colon N(A) \to {\mathbb{C}}$.
We have effectively reduced the proof of Theorem \[main C\* theorem\] to a question of the existence of morphisms of partial $C^*$-algebras. Thus we are in a position to apply the Kochen-Specker Theorem. The following corollary to Kochen-Specker is proved just like Corollary \[Kochen-Specker corollary\], relying upon Lemma \[eigenvalue lemma\].
\[Kochen-Specker C\* corollary\] For any $n \geq 3$, there is no $*$-morphism of partial $C^*$-algebras $N({\mathbb{M}}_n({\mathbb{C}})) \to {\mathbb{C}}$.
We are now ready to give a proof of Theorem \[main C\* theorem\], obstructing extensions of the Gelfand spectrum functor.
Fix $n \geq 3$ and let $A = {\mathbb{M}}_n({\mathbb{C}}) \in \operatorname{\mathsf{C^*}\mathsf{Alg}}$. By Theorem \[universal Gelf theorem\] there is a natural transformation $F \to \operatorname{\mathit{p}-Max}$. The set $\operatorname{\mathit{p}-Max}(A)$ is in bijection with the set of $*$-morphisms of partial $C^*$-algebras $N(A) \to {\mathbb{C}}$ according to Proposition \[partial Gelf and morphisms\]. By Corollary \[Kochen-Specker C\* corollary\] of the Kochen-Specker Theorem there are no such $*$-morphisms. Thus $\operatorname{\mathit{p}-Max}(A) = \varnothing$, and the existence of a function $F(A) \to \operatorname{\mathit{p}-Max}(A)$ gives $F(A) = \varnothing$.
The corollaries to Theorem \[main theorem\] given in Section \[proof section\] all have analogues in the setting of $C^*$-algebras. For the most part we will omit the proofs of these results because they are such straightforward adaptations of those given in Section \[proof section\]. First we provide an analogue of Corollary \[matrix algebra corollary\], and we include its proof only to illustrate how our restriction to unital $C^*$-algebras comes into play.
Let $F \colon \operatorname{\mathsf{C^*}\mathsf{Alg}}\to \operatorname{\mathsf{Set}}$ be a contravariant functor whose restriction to the full subcategory of commutative $C^*$-algebras is isomorphic to $\operatorname{Max}$. Then for any $C^*$-algebra $A$ and integer $n \geq3$, one has $F({\mathbb{M}}_n(A)) = \varnothing$.
Because $A$ is unital, there is a canonical morphism of $C^*$-algebras ${\mathbb{C}}\to {\mathbb{C}}\cdot 1_A \subseteq A$. This induces a $*$-morphism ${\mathbb{M}}_n({\mathbb{C}}) \to {\mathbb{M}}_n(A)$. Thus there is a function of sets $F({\mathbb{M}}_n(A)) \to F({\mathbb{M}}_n({\mathbb{C}}))$, and the latter set is empty by Theorem \[main C\* theorem\]. Hence $F({\mathbb{M}}_n(A)) = \varnothing$.
As in the discussion following Corollary \[matrix algebra corollary\], this result shows that if $A$ is a unital $C^*$-algebra for which there is an isomorphism $A \cong {\mathbb{M}}_n(A)$ for some $n \geq 2$, then for any functor $F$ as above, $F(A) = \varnothing$. As an example, we may take $A$ to be the algebra of bounded operators on an infinite-dimensional Hilbert space.
Next is the appropriate analogue of Corollary \[abelianization corollary\].
Let $\alpha \colon \operatorname{\mathsf{C^*}\mathsf{Alg}}\to \operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}$ be a functor whose restriction to $\operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}$ is isomorphic to the identity functor. Then for every $C^*$-algebra $A$ and every $n \geq 3$, one has $\alpha({\mathbb{M}}_n(A)) = 0$
Finally, there is the following analogue of Corollary \[prime ideals not functorial\].
There is no contravariant functor $F \colon \operatorname{\mathsf{C^*}\mathsf{Alg}}\to \operatorname{\mathsf{Set}}$ whose restriction to the full subcategory $\operatorname{\mathsf{Comm}\mathsf{C^*}\mathsf{Alg}}$ is isomorphic to $\operatorname{Max}$ and such that, for every $C^*$-algebra $A$, the set $F(R)$ is in bijection with the set of primitive ideals of $A$.
This corollary can be obtained as a consequence either of Theorem \[main C\* theorem\] or of the obvious analogue of Proposition \[Morita invariant Spec\]. In fact, the proof of the latter proposition (with $k = {\mathbb{C}}$) extends directly to the setting of $C^*$-algebras because all of the homomorphisms used in its proof are actually $*$-homomorphisms.
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to Andre Kornell and Matthew Satriano for several stimulating conversations in the early stages of this work, George Bergman for many insightful comments, Lance Small for helpful references to the literature, and Theo Johnson-Freyd for advice on creating Ti*k*z diagrams. Finally, I thank the referee for a number of suggestions and corrections that improved the readability of the paper.
[^1]: The author was supported by a Ford Foundation Predoctoral Diversity Fellowship at the University of California, Berkeley, and a University of California President’s Postdoctoral Fellowship at the University of California, San Diego.
|
---
abstract: 'We prove the boundedness of complements modulo two conjectures: Borisov-Alexeev conjecture and effective adjunction for fibre spaces. We discuss the last conjecture and prove it in two particular cases.'
address:
- 'Yu. G. Prokhorov: Department of Algebra, Faculty of Mathematics, Moscow State University, Moscow 117234, Russia'
- 'V. V. Shokurov: The Johns Hopkins University, Department of Mathematics, Baltimore, Maryland 21218, USA Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina str. 8, 119991, Moscow, Russia'
author:
- 'Yu. G. Prokhorov'
- 'V. V. Shokurov'
title: Towards the second main theorem on complements
---
Introduction
============
This paper completes our previous work [@Prokhorov-Shokurov-2001] modulo two conjectures [[\[BAB\]]{}]{} and [[\[conj-main-adj\]]{}]{}. The first one relates to Alexeev’s, A. and L. Borisov’s conjecture:
\[BAB\] Fix a real number ${\varepsilon}>0$. Let $(X,B=\sum b_iB_i)$ be a $d$-dimensional log canonical pair with nef $-(K_X+B)$, that is, $(X,B)$ is a log semi-Fano variety (cf. Definition [[\[definition\]]{}]{}). Assume also that
1. $K+B$ is ${\varepsilon}$-lt; and
2. $X$ is FT that is $(X,\Theta)$ is a klt log Fano variety with respect to some boundary $\Theta$.
Then $X$ is bounded in the moduli sense, i.e., it belongs to an algebraic family $\mathcal{X}({\varepsilon},d)$.
This conjecture was proved in dimension $2$ by V. Alexeev [@Alexeev-1994] and in toric case by A. Borisov and L. Borisov [@Borisov-Borisov] (see also [@Nikulin-1990-e], [@Borisov-1996], [@Borisov-2001], [@McKernan-2002-AG]).
We hope that Conjecture \[BAB\] can be generalized by weakening condition (ii). For example, we hope that instead of (ii) one can assume that
1. $X$ is rationally connected, cf. [@Zhang-Qi-2006].
Recall that a log pair $(X,B)$ is said to be *${\varepsilon}$-log terminal* (or simply *${\varepsilon}$-lt*) if ${\operatorname{totaldiscr}}(X,B)>-1+{\varepsilon}$, see Definition \[def-ep-lt\] below.
\[BAB\_th\] Conjecture [[\[BAB\]]{}]{} holds in dimension two.
The second conjecture concerns with Adjunction Formula and will be discussed in Section [[\[sect-Adj\]]{}]{}.
Our main result is the following.
\[main-result\] Fix a finite subset ${{\mathfrak{R}}}\subset [0,\, 1]\cap {{\mathbb Q}}$. Let $(X,B)$ be a klt log semi-Fano variety of dimension $d$ such that $X$ is FT and the multiplicities of $B$ are contained in $\Phi({{\mathfrak{R}}})$ (see [[\[not-hyperstand\]]{}]{}). Assume the LMMP in dimension $\le d$. Further, assume that Conjectures [[\[BAB\]]{}]{} and [[\[conj-main-adj\]]{}]{} hold in dimension $\le d$. Then $K+B$ has bounded complements. More precisely, there is a positive integer $n=n(d,{{\mathfrak{R}}})$ divisible by denominators of all $r\in {{\mathfrak{R}}}$ and such that $K+B$ is $n$-complemented. Moreover, $K+B$ is $nI$-complemented for any positive integer $I$.
In particular, $$\left|-nK-nS -{\left\lfloor (n+1)D\right\rfloor}\right|\neq \emptyset,$$ where $S:={\left\lfloor B\right\rfloor}$ and $D:=B-S$. For $B=0$, $|-nK|\neq \emptyset$, where $n$ depends only on $d$.
Note that the last paragraph is an immediate consequence of the first statemet and the definition of complements.
In the case when $K+B$ is numerically trivial our result is stronger. For the definition of $0$-pairs we refer to [[\[definition\]]{}]{}.
\[main-result0\] Fix a finite subset ${{\mathfrak{R}}}\subset [0,\, 1]\cap {{\mathbb Q}}$. Let $(X,B)$ be a $0$-pair of dimension $d$ such that $X$ is FT and the multiplicities of $B$ are contained in $\Phi({{\mathfrak{R}}})$. Assume the LMMP in dimension $\le d$. Further, assume that Conjectures [[\[BAB\]]{}]{} and [[\[conj-main-adj\]]{}]{} hold in dimension $\le d$. Then there is a positive integer $n=n(d,{{\mathfrak{R}}})$ such that $n(K_X+B)\sim 0$.
\[addendum-1\] Our proofs show that we do not need Conjecture [[\[BAB\]]{}]{} in dimension $d$ in all generality. We need it only for some special value ${\varepsilon}^o:={\varepsilon}^o(d,{{\mathfrak{R}}})>0$ (cf. Corollary [[\[main-result-3-Cor-1-b\]]{}]{}).
We will prove Conjecture [[\[conj-main-adj\]]{}]{} in §\[section-Adj-conj-part-cases\] in dimension $\le 3$ under additional assumption that the total space is projective and FT.
\[main-result-3-Cor-1-b\] Fix a finite rational subset ${{\mathfrak{R}}}\subset [0,\, 1]\cap {{\mathbb Q}}$ and let $I$ be the least common multiple of denominators of all $r\in {{\mathfrak{R}}}$. Let $(X,B)$ be a klt log semi-Fano threefold such that $X$ is FT and the multiplicities of $B$ are contained in $\Phi({{\mathfrak{R}}})$. Assume that Conjecture [[\[BAB\]]{}]{} holds in dimension $3$ for ${\varepsilon}^o$ as in Addendum [[\[addendum-1\]]{}]{}. Then $K+B$ has a bounded $n$-complement such that $I\mid n$. In particular, there exists a positive integer number $n$ such that $I\mid n$ and $$\left|-nK-nS -{\left\lfloor (n+1)D\right\rfloor}\right|\neq \emptyset,$$ where $S:={\left\lfloor B\right\rfloor}$ and $D:=B-S$; $n$ depends only on ${{\mathfrak{R}}}$ and ${\varepsilon}^o$ for Addendum [[\[addendum-1\]]{}]{}. For $B=0$, $|-nK|\neq \emptyset$, where $n$ depends only on ${\varepsilon}^o$.
Immediate by Addendum [[\[addendum-1\]]{}]{}, Theorem [[\[main-result\]]{}]{}, and Corollary [[\[corollary-815\]]{}]{}.
\[main-result-2\] Fix a finite subset ${{\mathfrak{R}}}\subset [0,\, 1]\cap {{\mathbb Q}}$ and let $I$ be the least common multiple of denominators of all $r\in {{\mathfrak{R}}}$. Let $(X,B)$ be a klt log semi-del Pezzo surface such that $X$ is FT and the multiplicities of $B$ are contained in $\Phi({{\mathfrak{R}}})$. Then $K+B$ has a bounded $n$-complement such that $I\mid n$. In particular, there exists a positive integer $n$ such that $I\mid n$ and $$\left|-nK-nS -{\left\lfloor (n+1)D\right\rfloor}\right|\neq \emptyset,$$ where $S:={\left\lfloor B\right\rfloor}$ and $D:=B-S$; $n$ depends only on ${{\mathfrak{R}}}$. For $B=0$, $|-nK|\neq \emptyset$, where $n$ is an absolute constant.
Immediate by Theorems [[\[main-result\]]{}]{}, [[\[BAB\_th\]]{}]{}, and [[\[th-n-n-1\]]{}]{}.
The following corollaries are consequences of our techniques. The proofs will be given in \[sketch-proof-main-result-3-Cor-1-aC\], \[sketch-proof-main-result-3-Cor-1-a\], and \[sketch-proof-main-result-3-Cor-2div\].
\[main-result-3-Cor-1-a\] Fix a finite rational subset ${{\mathfrak{R}}}\subset [0,\, 1]\cap {{\mathbb Q}}$. Let $(X,D)$ be a three-dimensional $0$-pair such that $X$ is FT and the multiplicities of $B$ are contained in $\Phi({{\mathfrak{R}}})$. Assume that $(X,D)$ is not klt. Then there exists a positive integer $n$ such that $n(K+D)\sim 0$; this $n$ depends only on ${{\mathfrak{R}}}$.
\[main-result-3-Cor-1-2div\] Fix a finite rational subset ${{\mathfrak{R}}}\subset [0,\, 1]\cap {{\mathbb Q}}$ and let $I$ be the least common multiple of denominators of all $r\in {{\mathfrak{R}}}$. Let $(X,B)$ be a klt log semi-Fano threefold such that $X$ is FT and the multiplicities of $B$ are contained in $\Phi({{\mathfrak{R}}})$. Then there exists a real number $\bar {\varepsilon}$ such that $K+B$ has a bounded $n$-complement with $I\mid n$ if there are two divisors $E$ (exceptional or not) with discrepancy $a(E,X,B)\le -1+\bar {\varepsilon}$; this $\bar {\varepsilon}$ depends only on ${{\mathfrak{R}}}$.
\[main-result-3-Cor-1-aC\] Fix a finite rational subset ${{\mathfrak{R}}}\subset [0,\, 1]\cap {{\mathbb Q}}$. Let $(X,D)$ be a two-dimensional $0$-pair such that the multiplicities of $B$ are contained in $\Phi({{\mathfrak{R}}})$. Then there exists a positive integer $n$ such that $n(K+D)\sim 0$; this $n$ depends only on ${{\mathfrak{R}}}$.
We give a sketch of the proof of our main results in Section \[sect-reduction\]. One can see that our proof essentially uses reduction to lower-dimensional global pairs. However it is expected that an improvement of our method can use reduction to local questions in the same dimension. In fact we hope that the hypothesis in our main theorem \[main-result\] should be the existence of local complements and Conjecture [[\[BAB\]]{}]{} for ${\varepsilon}$-lt Fano varieties (without a boundary), where ${\varepsilon}\ge {\varepsilon}^o>0$, ${\varepsilon}^o$ is a constant depending only on the dimension (cf. Addendum \[addendum-1\] and Corollary \[main-result-3-Cor-1-b\]). If $\dim X=2$, we can take ${\varepsilon}^o=1/7$. Note also that our main theorem \[main-result\] is weaker than one can expect. We think that the pair $(X,B)$ can be taken arbitrary log-semi-Fano (possibly not klt and not FT) and possible boundary multiplicities can be taken arbitrary real numbers in $[0,1]$ (not only in $\Phi({{\mathfrak{R}}})$). The only hypothesis we have to assume is the existence of an ${{\mathbb R}}$-complement $B^+\ge B$ (cf. [@Shokurov-2000]). However the general case needs actually a [*finite*]{} set of natural numbers for complements, and there are no such universal number for all complements (cf. [@Shokurov-1992-e Example 5.2.1]).
Acknowledgements {#acknowledgements .unnumbered}
----------------
The work was conceived in 2000 when the first author visited the Johns Hopkins University and finished during his stay in Max-Planck-Institut für Mathematik, Bonn in 2006. He would like to thank these institutes for hospitality. Finally both authors are grateful to the referee whose constructive criticism helped us to revise the paper very much.
Preliminaries
=============
Notation
--------
All varieties are assumed to be algebraic and defined over an algebraically closed field $\Bbbk$ of characteristic zero. Actually, main results holds for any $\Bbbk$ of characteristic zero not necessarily algebraically closed since they are related to singularities of general members of linear systems (see [@Shokurov-1992-e 5.1]). We use standard terminology and notation of the Log Minimal Model Program (LMMP) [@KMM], [@Utah], [@Shokurov-1992-e]. For the definition of complements and their properties we refer to [@Shokurov-1992-e], [@Shokurov-2000], [@Prokhorov-2001] and [@Prokhorov-Shokurov-2001]. Recall that a *log pair* (or a *log variety*) is a pair $(X,D)$ consisting of a normal variety $X$ and a *boundary* $D$, i.e., an ${{\mathbb R}}$-divisor $D=\sum d_i D_i$ with multiplicities $0\le
d_i\le 1$. As usual $K_X$ denotes the canonical (Weil) divisor of a variety $X$. Sometimes we will write $K$ instead of $K_X$ if no confusion is likely. Everywhere below $a(E,X,D)$ denotes the discrepancy of $E$ with respect to $K_X+D$. Recall the standard notation: $$\begin{array}{lll}
{\operatorname{discr}}(X,D)&=&\inf_E\{ a(E,X,D)\mid {\operatorname{codim}}{\operatorname{Center}}_X(E)\ge 2\},
\vspace{6pt}\\
{\operatorname{totaldiscr}}(X,D)&=&\inf_E\{ a(E,X,D)\mid
{\operatorname{codim}}{\operatorname{Center}}_X(E)\ge 1\}.
\end{array}$$ In the paper we use the following strong version of ${\varepsilon}$-log terminal and ${\varepsilon}$-log canonical property.
\[def-ep-lt\] A log pair $(X,B)$ is said to be *${\varepsilon}$-log terminal* (*${\varepsilon}$-log canonical*) if ${\operatorname{totaldiscr}}(X,B)>-1+{\varepsilon}$ (resp., ${\operatorname{totaldiscr}}(X,B)\ge-1+{\varepsilon}$).
Usually we work with ${{\mathbb R}}$-divisors. An ${{\mathbb R}}$-divisor is an ${{\mathbb R}}$-linear combination of prime Weil divisors. An ${{\mathbb R}}$-linear combination $D=\sum \alpha_i L_i$, where the $L_i$ are integral Cartier divisors is called an ${{\mathbb R}}$-Cartier divisor. The pull-back $f^*$ of an ${{\mathbb R}}$-Cartier divisor $D=\sum \alpha_i L_i$ under a morphism $f\colon Y\to X$ is defined as $f^*D:=\sum \alpha_i f^*L_i$. Two ${{\mathbb R}}$-divisors $D$ and $D'$ are said to be *${{\mathbb Q}}$- (resp., ${{\mathbb R}}$-)linearly equivalent* if $D-D'$ is a ${{\mathbb Q}}$- (resp., ${{\mathbb R}}$-)linear combination of principal divisors. For a positive integer $I$, two ${{\mathbb R}}$-divisors $D$ and $D'$ are said to be *$I$-linearly equivalent* if $I(D-D')$ is an (integral) principal divisor. The ${{\mathbb Q}}$-linear (resp., ${{\mathbb R}}$-linear, $I$-linear) equivalence is denoted by ${\mathbin{\sim_{\scriptscriptstyle{{{\mathbb Q}}}}}}$ (resp., ${\mathbin{\sim_{\scriptscriptstyle{{{\mathbb R}}}}}}$, ${\mathbin{\sim_{\scriptscriptstyle{I}}}}$). Let $\Phi\subset {{\mathbb R}}$ and let $D=\sum d_iD_i$ be an ${{\mathbb R}}$-divisor. We say that $D\in \Phi$ if $d_i\in \Phi$ for all $i$.
Let $f\colon X\to Z$ be a morphism of normal varieties. For any ${{\mathbb R}}$-divisor $\Delta$ on $Z$, define its *divisorial pull-back* $f^\bullet\Delta$ as the closure of the usual pull-back $f^*\Delta$ over $Z\setminus V$, where $V$ is a closed subset of codimension $\ge 2$ such that $V\supset {\operatorname{Sing}}Z$ and $f$ is equidimensional over $Z\setminus V$. Thus each component of $f^\bullet\Delta$ dominates a component of $\Delta$. It is easy to see that the divisorial pull-back $f^\bullet\Delta$ does not depend on the choice of $V$. Note however that in general $f^\bullet$ does not coincide with the usual pull-back $f^*$ of ${{\mathbb R}}$-Cartier divisors.
\[definition\] Let $(X,B)$ be a log pair of global type (the latter means that $X$ is projective). Then it is said to be
- *log Fano* variety if $K+B$ is lc and $-(K+B)$ is ample;
- *weak log Fano* (WLF) variety if $K+B$ is lc and $-(K+B)$ is nef and big;
- *log semi-Fano* (ls-Fano) variety if $K+B$ is lc and $-(K+B)$ is nef;
- *$0$-log pair* if $K+B$ is lc and numerically trivial[^1].
In dimension two we usually use the word *del Pezzo* instead of *Fano*.
\[CY->F\] Let $X$ be a normal projective variety. We say that $X$ is *FT* (*Fano type*) if it satisfies the following equivalent conditions:
1. there is a ${{\mathbb Q}}$-boundary $\Xi$ such that $(X,\Xi)$ is a klt log Fano;
2. there is a ${{\mathbb Q}}$-boundary $\Xi$ such that $(X,\Xi)$ is a klt weak log Fano;
3. there is a ${{\mathbb Q}}$-boundary $\Theta$ such that $(X,\Theta)$ is a klt $0$-pair and the components of $\Theta$ generate $N^1(X)$;
4. for any divisor $\Upsilon$ there is a ${{\mathbb Q}}$-boundary $\Theta$ such that $(X,\Theta)$ is a klt $0$-pair and ${\operatorname{Supp}}\Upsilon\subset {\operatorname{Supp}}\Theta$.
Similarly one can define relative FT and $0$-varieties $X/Z$, and the results below hold for them too.
Implications (i) $\Longrightarrow$ (iv), (iv) $\Longrightarrow$ (iii), (i) $\Longrightarrow$ (ii) are obvious and (ii) $\Longrightarrow$ (i) follows by Kodaira’s lemma (see, e.g., [@KMM Lemma 0-3-3]). We prove (iii) $\Longrightarrow$ (i). Let $(X,\Theta)$ be such as in (iii). Take an ample divisor $H$ such that ${\operatorname{Supp}}{H}\subset{\operatorname{Supp}}{\Theta}$ and put $\Xi=\Theta-{\varepsilon}H$, for $0<{\varepsilon}\ll 1$. Clearly, $(X,\Xi)$ is a klt log Fano.
Recall that for any (not necessarily effective) ${{\mathbb R}}$-divisor $D$ on a variety $X$ a $D$-MMP is a sequence $X=X_1 \dashrightarrow X_N$ of extremal $D$-negative divisorial contractions and $D$-flips which terminates on a variety $X_N$ where either the proper transform of $D$ is nef or there exists a $D$-negative contraction to a lower-dimensional variety (see [@Utah 2.26]).
\[CY-MMP\] Let $X$ be an FT variety. Assume the LMMP in dimension $\dim X$. Then the $D$-MMP works on $X$ with respect to any ${{\mathbb R}}$-divisor $D$.
Immediate by Lemma \[CY->F\], (iv). Indeed, in the above notation we may assume that ${\operatorname{Supp}}D\subset{\operatorname{Supp}}\Theta$. It remains to note that the $D$-LMMP is is nothing but the LMMP with respect to $(X,\Theta+{\varepsilon}D)$ some $0<{\varepsilon}\ll 1$.
\[lemma-FT\]
1. Let $f\colon X\to Z$ be a (not necessarily birational) contraction of normal varieties. If $X$ is FT, then so is $Z$.
2. The FT property is preserved under birational divisorial contractions and flips.
3. Let $(X,D)$ be an ls-Fano variety such that $X$ is FT. Let $f\colon Y\to X$ be a birational extraction such that $a(E,X,D)< 0$ for every $f$-exceptional divisor $E$ over $X$. Then $Y$ is also FT.
We need the following result of Ambro [@Ambro-2005 Th. 0.2] which is a variant of Log Canonical Adjunction (cf. \[conj-main-adj\], [@Fujino-1999app]).
\[th-ambro\] Let $(X,D)$ be a projective klt log pair, let $f\colon X\to Z$ be a contraction, and let $L$ be a ${{\mathbb Q}}$-Cartier divisor on $Z$ such that $$K+D{\mathbin{\sim_{\scriptscriptstyle{{{\mathbb Q}}}}}} f^*L.$$ Then there exists a ${{\mathbb Q}}$-Weil divisor $D_Z$ such that $(Z,D_Z)$ is a log variety with Kawamata log terminal singularities and $L {\mathbin{\sim_{\scriptscriptstyle{{{\mathbb Q}}}}}} K_Z+D_Z$.
First note that (ii) and the birational case of (i) easily follows from from \[CY->F\] (iii). To prove (i) in the general case we apply Theorem \[th-ambro\]. Let $\Theta=\sum_i \theta_i\Theta_i$ be a ${{\mathbb Q}}$-boundary on $X$ whose components generate $N^1(X)$ and such that $(X, \Theta)$ is a klt $0$-pair. Let $A$ be an ample divisor on $Z$. By our assumption $f^*A\equiv \sum_i \delta_i \Theta_i$. Take $0<\delta\ll 1$ and put $\Theta':= \sum_i (\theta_i-\delta \delta_i) \Theta_i$. Clearly, $K+\Theta'\equiv -\delta f^*A$ and $(X,\Theta')$ is a klt log semi-Fano variety. By the base point free theorem $K+\Theta'{\mathbin{\sim_{\scriptscriptstyle{{{\mathbb Q}}}}}} -\delta f^*A$. Now by Theorem \[th-ambro\] there is a ${{\mathbb Q}}$-boundary $\Theta_Z$ such that $(Z,\Theta_Z)$ is klt and $K_Z+\Theta_Z {\mathbin{\sim_{\scriptscriptstyle{{{\mathbb Q}}}}}} -\delta A$. Hence $(Z, \Theta_Z)$ is a klt log Fano variety. This proves (i).
Now we prove (iii). Let $\Xi$ be a boundary such that $(X,\Xi)$ is a klt log Fano. Let $D_Y$ and $\Xi_Y$ be proper transforms of $D$ and $\Xi$, respectively. Then $(Y,D_Y)$ is an ls-Fano, $(Y,\Xi_Y)$ is klt and $-(K_Y+\Xi_Y)$ is nef and big. However $\Xi_Y$ is not necessarily a boundary. To improve the situation we put $\Xi':=(1-{\varepsilon})D_Y+{\varepsilon}\Xi_Y$ for small positive ${\varepsilon}$. Then $(Y,\Xi')$ is a klt weak log Fano.
\[def-compl\] Let $X$ be a normal variety and let $D$ be an ${{\mathbb R}}$-divisor on $X$. Then a *${{\mathbb Q}}$-complement* of $K_X+D$ is a log divisor $K_X+D'$ such that $D'\ge D$, $K_X+D'$ is lc and $n(K_X+D')\sim 0$ for some positive integer $n$.
Now let $D=S+B$, where $B$ and $S$ have no common components, $S$ is an effective integral divisor and ${\left\lfloor B\right\rfloor}\le 0$. Then we say that $K_X+D$ is *$n$-complemented*, if there is a ${{\mathbb Q}}$-divisor $D^+$ such that
1. $n(K_X+D^+)\sim 0$ (in particular, $nD^+$ is integral divisor);
2. $K_X+D^+$ is lc;
3. $nD^+\ge nS+{\left\lfloor (n+1)B\right\rfloor}$.
In this situation, $K_X+D^+$ is called an *$n$-complement* of $K_X+D$.
Note that an $n$-complement is not necessarily a ${{\mathbb Q}}$-complement (cf. Lemma \[lemma-PPP-n-1\]).
\[rem-complne\] Under (i) and (ii) of \[def-compl\], the condition (iii) follows from the inequality $D^+\ge D$. Indeed, write $D=\sum d_iD_i$ and $D^+=\sum d_i^+D_i$. We may assume that $d_i^+<1$. Then we have $$nd_i^+\ge {\left\lfloor nd_i^+\right\rfloor}={\left\lfloor (n+1)d_i^+\right\rfloor}\ge {\left\lfloor (n+1)d_i\right\rfloor}.$$
\[cor-complne\] Let $D^+$ be an $n$-complement of $D$ such that $D^+\ge D$. Then $D^+$ is also an $nI$-complement of $D$ for any positive integer $I$.
For basic properties of complements we refer to [@Shokurov-1992-e §5] and [@Prokhorov-2001], see also §\[sect-hyper\].
Fix a class of (relative) log pairs $(\mathcal{X}/\mathcal{Z}\ni o,\,
\mathcal{B})$, where $o$ is a point on each $Z\in \mathcal Z$. We say that this class has *bounded complements* if there is a constant $\operatorname{Const}$ such that for any log pair $(X/Z,B)\in (\mathcal{X}/\mathcal{Z},\,
\mathcal{B})$ the log divisor $K+B$ is $n$-complemented near the fibre over $o$ for some $n\le\operatorname{Const}$.
Notation
--------
Let $X$ be a normal $d$-dimensional variety and let ${{\mathscr{B}}}=\sum_{i=1}^r B_i$ be any reduced divisor on $X$. Recall that $Z_{d-1}(X)$ usually denotes the group of Weil divisors on $X$. Consider the vector space $\mathfrak{D}_{{{\mathscr{B}}}}$ of all ${{\mathbb R}}$-divisors supported in ${{\mathscr{B}}}$: $$\mathfrak{D}_{{{\mathscr{B}}}}:=\bigl\{D\in Z_{d-1}(X)\otimes {{\mathbb R}}\mid
{\operatorname{Supp}}{D}\subset{{\mathscr{B}}}\bigr\}=\sum_{i=1}^r {{\mathbb R}}\cdot B_i.$$ As usual, define a norm in $\mathfrak{D}_{{{\mathscr{B}}}}$ by $$\|B \|=\max(|b_1|,\dots, |b_r|),$$ where $B=\sum_{i=1}^rb_iB_i\in \mathfrak{D}_{{{\mathscr{B}}}}$. For any ${{\mathbb R}}$-divisor $B=\sum_{i=1}^r b_i B_i$, put $\mathfrak{D}_{B}:=\mathfrak{D}_{{\operatorname{Supp}}{B}}$.
Hyperstandard multiplicities {#sect-hyper}
============================
Recall that *standard multiplicities* $1-1/m$ naturally appear as multiplicities in the divisorial adjunction formula $(K_X+S)|_S=K_S+{\operatorname{Diff}}_S$ (see [@Shokurov-1992-e §3], [@Utah Ch. 16]). Considering the adjunction formula for fibre spaces and adjunction for higher codimensional subvarieties one needs to introduce a bigger class of multiplicities.
\[ex-Kodaira\] Let $f\colon X\to Z\ni P$ be a minimal two-dimensional elliptic fibration over a one-dimensional germ ($X$ is smooth). We can write a natural formula $K_X=f^*(K_Z+D{_{\operatorname{div}}{}})$, where $D{_{\operatorname{div}}{}}=d_P P$ is an effective divisor (cf. \[def-cW\] below). From Kodaira’s classification of singular fibres (see [@Kodaira-1963]) we obtain the following values of $d_P$:
[p[35pt]{}|p[35pt]{}p[25pt]{}p[25pt]{} p[25pt]{}p[25pt]{}p[25pt]{}p[25pt]{}p[25pt]{}]{} [Type]{}& $m\mathrm{I}_n$&$\mathrm{II}$&$\mathrm{III}$&$\mathrm{IV}$& $\mathrm{I}^*_b$&$\mathrm{II}^*$&$\mathrm{III}^*$&$\mathrm{IV}^*$\
\
$d_P$& $1-\frac1m$&$\frac16$&$\frac14$&$\frac13$&$\frac12$ &$\frac56$&$\frac34$&$\frac23$\
Thus the multiplicities of $D{_{\operatorname{div}}{}}$ are not necessarily standard.
{#not-hyperstand}
Fix a subset ${{\mathfrak{R}}}\subset {{\mathbb R}}_{\ge 0}$. Define $$\Phi({{\mathfrak{R}}}):=\left\{\left. 1-\frac rm \quad \right|\quad m\in {\mathbb Z},
\quad m>0 \quad
r\in {{\mathfrak{R}}}\right\} \bigcap \Bigl[0,\, 1\Bigr].$$ We say that an ${{\mathbb R}}$-boundary $B$ has *hyperstandard multiplicities* with respect to ${{\mathfrak{R}}}$ if $B\in \Phi({{\mathfrak{R}}})$. For example, if ${{\mathfrak{R}}}=\{0,1\}$, then $\Phi({{\mathfrak{R}}})$ is the set of *standard multiplicities*. The set ${{\mathfrak{R}}}$ is said to be *rational* if ${{\mathfrak{R}}}\subset {{\mathbb Q}}$. Usually we will assume that ${{\mathfrak{R}}}$ is rational and finite. In this case we denote $$I({{\mathfrak{R}}}):=
{\operatorname{lcm}}\bigl(
\text{denominators of $r\in{{\mathfrak{R}}}\setminus \{0\}$}
\bigr).$$
[**([.]{})**]{}
\[para-def-N\] Denote by ${{\mathscr{N}}}_d({{\mathfrak{R}}})$ the set of all $m\in {\mathbb Z}$, $m>0$ such that there exists a log semi-Fano variety $(X,D)$ of dimension $\le d$ satisfying the following properties:
1. $X$ is FT and $D\in\Phi({{\mathfrak{R}}})$;
2. either $(X,D)$ is klt or $K_X+D\equiv 0$;
3. $K_X+D$ is $m$-complemented, $I({{\mathfrak{R}}}) \mid m$, and $m$ is minimal under these conditions.
Since any nef divisor is semiample on FT variety, for any log semi-Fano variety $(X,D)$ satisfying (i) and (ii), there exists some $m$ in (iii). Put $$N_{d}=N_{d}({{\mathfrak{R}}}):=\sup {{\mathscr{N}}}_d({{\mathfrak{R}}}), \quad
{\varepsilon}_{d}={\varepsilon}_{d}({{\mathfrak{R}}}):=1/(N_{d}+2).$$ We expect that ${{\mathscr{N}}}_d({{\mathfrak{R}}})$ is bounded whenever ${{\mathfrak{R}}}$ is finite and rational, see Theorems \[main-result\] and \[main-result0\]. In particular, $N_{d}<\infty$ and ${\varepsilon}_{d}>0$. For ${\varepsilon}\ge 0$, define also the set of *semi-hyperstandard* multiplicities $$\Phi({{\mathfrak{R}}},{\varepsilon}):=
\Phi({{\mathfrak{R}}}) \cup [1-{\varepsilon},\, 1].$$
Fix a positive integer $n$ and define the set ${{\mathscr{P}}}_n\subset {{\mathbb R}}$ by $$\label{def_PPP_n}
\alpha\in{{\mathscr{P}}}_n\quad\Longleftrightarrow\quad 0\le\alpha\le
1\quad\text{and}\quad {\left\lfloor (n+1)\alpha\right\rfloor}\ge n\alpha.
\index{${{\mathscr{P}}}_n$}$$ This set obviously satisfies the following property:
\[lemma-PPP-n-1\] If $D\in {{\mathscr{P}}}_n$ and $D^+$ is an $n$-complement, then $D^+\ge D$.
Taking \[cor-complne\] into account we immediately obtain the following important.
\[cor-imp\] Let $D\in {{\mathscr{P}}}_n$ and let $D^+$ be an $n$-complement of $D$. Then $D^+$ is an $nI$-complement of $D$ for any positive integer $I$.
\[lemma-PPP-n\] If ${{\mathfrak{R}}}\subset [0,\, 1]\cap {{\mathbb Q}}$, $I({{\mathfrak{R}}}) \mid n$, and $0\le {\varepsilon}\le 1/(n+1)$, then $${{\mathscr{P}}}_n\supset \Phi ({{\mathfrak{R}}},{\varepsilon}).$$
Let $1\ge \alpha\in \Phi ({{\mathfrak{R}}},{\varepsilon})$. If $\alpha\ge 1-{\varepsilon}$, then $$(n+1)\alpha > n+1-{\varepsilon}(n+1)\ge n.$$ Hence, ${\left\lfloor (n+1)\alpha\right\rfloor}\ge n\ge n\alpha$ and $\alpha\in {{\mathscr{P}}}_n$. Thus we may assume that $\alpha\in \Phi({{\mathfrak{R}}})$. It is sufficient to show that $$\label{(n+1)}
{\left\lfloor (n+1)\left(1-\frac rm\right)\right\rfloor}\ge n\left(1-\frac rm\right)$$ for all $r\in{{\mathfrak{R}}}$ and $m\in{\mathbb Z}$, $m>0$. We may assume that $r>0$. It is clear that is equivalent to the following inequality $$\label{(n+1)1}
(n+1)\left(1-\frac rm\right)\ge k\ge n\left(1-\frac rm\right),$$ for some $k\in{\mathbb Z}$ (in fact, $k={\left\lfloor (n+1)\left(1-\frac
rm\right)\right\rfloor}$). By our conditions, $N:=nr\in{\mathbb Z}$, $N>0$. Thus can be rewritten as follows $$\label{(n+1)2}
mn-N+m-r\ge mk\ge mn-N.$$ Since $m-r\ge m-1$, inequality has a solution in $k\in {\mathbb Z}$. This proves the statement.
\[cor-pull-back\_compl-I\] Let $f\colon Y\to X$ be a birational contraction and let $D$ be an ${{\mathbb R}}$-divisor on $Y$ such that
1. $K_Y+D$ is nef over $X$,
2. $f_*D\in {{\mathscr{P}}}_n$ (in particular, $f_*D$ is a boundary).
Assume that $K_X+f_*D$ is $n$-complemented. Then so is $K_Y+D$.
\[prodolj\] Let $(X/Z\ni o,D=S+B)$ be a log variety. Set $S:={\left\lfloor D\right\rfloor}$ and $B:={\left\{ D \right\}}$. Assume that
1. $K_X+D$ is plt;
2. $-(K_X+D)$ is nef and big over $Z$;
3. $S\ne 0$ near $f^{-1}(o)$;
4. \[prop-2-iv\] $D\in {{\mathscr{P}}}_n$.
Further, assume that near $f^{-1}(o)\cap S$ there exists an $n$-complement $K_S+{\operatorname{Diff}}_ S(B)^+$ of $K_S+{\operatorname{Diff}}_ S(B)$. Then near $f^{-1}(o)$ there exists an $n$-complement $K_X+S+B^+$ of $K_X+S+B$ such that ${\operatorname{Diff}}_S(B)^+={\operatorname{Diff}}_S({B^+})$.
Adjunction on divisors (cf. [@Shokurov-1992-e Cor. 3.10, Lemma 4.2]) {#adjunction-on-divisors-cf.-cor.-3.10-lemma-4.2 .unnumbered}
--------------------------------------------------------------------
Fix a subset ${{\mathfrak{R}}}\subset {{\mathbb R}}_{\ge 0}$. Define also the new set $${\overline{{{\mathfrak{R}}}}}:= \left\{\left. r_0-m \sum_{i=1}^s (1-r_i)\ \right|\
r_0,\dots, r_s\in{{\mathfrak{R}}}, \ m\in {\mathbb Z}, \ m>0 \right\}\cap {{\mathbb R}}_{\ge 0}.$$ It is easy to see that ${\overline{{{\mathfrak{R}}}}}\supset {{\mathfrak{R}}}$. For example, if ${{\mathfrak{R}}}=\{0,1\}$, then ${\overline{{{\mathfrak{R}}}}}={{\mathfrak{R}}}$.
1. If ${{\mathfrak{R}}}\subset [0,\, 1]$, then ${\overline{{{\mathfrak{R}}}}}\subset [0,\, 1]$.
2. If ${{\mathfrak{R}}}$ is finite and rational, then so is ${\overline{{{\mathfrak{R}}}}}$.
3. $I({{\mathfrak{R}}})=I({\overline{{{\mathfrak{R}}}}})$.
4. Let $\mathfrak{G}\subset {{\mathbb Q}}$ be an additive subgroup containing $1$ and let ${{\mathfrak{R}}}={{\mathfrak{R}}}_{\mathfrak{G}} :=\mathfrak{G}\cap [0,1]$. Then ${\overline{{{\mathfrak{R}}}}}={{\mathfrak{R}}}$.
5. If the ascending chain condition (a.c.c.) holds for the set ${{\mathfrak{R}}}$, then it holds for ${\overline{{{\mathfrak{R}}}}}$.
(i)-(iv) are obvious. We prove (v). Indeed, let $$q^{(n)}=r_0^{(n)}-m^{(n)} \sum_{i=1}^{s^{(n)}} (1-r_i^{(n)})\in
{\overline{{{\mathfrak{R}}}}}$$ be an infinite increasing sequence, where $r_i^{(n)}\in{{\mathfrak{R}}}$ and $m^{(n)}\in{\mathbb Z}_{>0}$. By passing to a subsequence, we may assume that $m^{(n)} \sum_{i=1}^{s^{(n)}} (1-r_i^{(n)})>0$, in particular, $s^{(n)}>0$ for all $n$. There is a constant ${\varepsilon}={\varepsilon}({{\mathfrak{R}}})>0$ such that $1-r_i^{(n)}>{\varepsilon}$ whenever $r_i^{(n)}\neq 1$. Thus, $0\le
q^{(n)}\le r_0^{(n)}- m^{(n)}s^{(n)}{\varepsilon}$ and $m^{(n)}s^{(n)}\le
(r_0^{(n)}- q^{(n)})/{\varepsilon}$. Again by passing to a subsequence, we may assume that $m^{(n)}$ and $s^{(n)}$ are constants: $m^{(n)}=m$, $s^{(n)}=s$. Since the numbers $r_i^{(n)}$ satisfy a.c.c., the sequence $$q^{(n)}=r_0^{(n)}+m \sum_{i=1}^{s} r_i^{(n)}-ms$$ is not increasing, a contradiction.
\[prop-prop\] Let ${{\mathfrak{R}}}\subset [0,\, 1]$, $1\in {{\mathfrak{R}}}$, ${\varepsilon}\in [0,\, 1]$, and let $(X,S+B)$ be a plt log pair, where $S$ is a prime divisor, $B\ge 0$, and ${\left\lfloor B\right\rfloor}=0$. If $B\in \Phi({{\mathfrak{R}}}, {\varepsilon})$, then ${\operatorname{Diff}}_S(B)\in \Phi({\overline{{{\mathfrak{R}}}}}, {\varepsilon})$.
Write $B=\sum b_i B_i$, where the $B_i$ are prime divisors and $b_i \in \Phi({{\mathfrak{R}}}, {\varepsilon})$. Let $V\subset S$ be a prime divisor. By [@Shokurov-1992-e Cor. 3.10] the multiplicity $d$ of ${\operatorname{Diff}}_S(B)$ along $V$ is computed using the following relation: $$\label{eq-adji-div-coeff}
d=1-\frac1{n}+\frac1n\sum_{i=0}^s k_i b_i=1-\frac{\beta}{n},$$ where $n$, $k_i\in {\mathbb Z}_{\ge 0}$, and $\beta:=1-\sum k_i b_i$. It is easy to see that $d\ge b_i$ whenever $k_i>0$. If $b_i\ge 1-{\varepsilon}$, this implies $d\ge 1-{\varepsilon}$. Thus we may assume that $b_i\in \Phi({{\mathfrak{R}}})$ whenever $k_i>0$. Therefore, $$\beta=1-\sum k_i\left(1-\frac{r_i}{m_i}\right),$$ where $m_i\in{\mathbb Z}_{>0}$, $r_i\in {{\mathfrak{R}}}$. Since $(X,S+B)$ is plt, $d<1$. Hence, $\beta>0$. If $m_i=1$ for all $i$, then $$\beta=1- \sum k_i(1-r_i)\in {\overline{{{\mathfrak{R}}}}}.$$ So, $d\in \Phi({\overline{{{\mathfrak{R}}}}})$ in this case. Thus we may assume that $m_0>1$. Since $1-\frac{r_i}{m_i}\ge 1-\frac{1}{m_i}$, we have $m_1=\cdots=m_s=1$ and $k_0=1$. Thus, $$\beta= \frac{r_0}{m_0}-
\sum _{i=1}^s k_i(1-r_i)= \frac{r_0-m_0\sum_{i=1}^sk_i(1-r_i)}{m_0}$$ and $m_0\beta=r_0-m_0\sum_{i=1}^sk_i(1-r_i)\in {\overline{{{\mathfrak{R}}}}}$. Hence, $d=1-\frac{m_0\beta}{m_0n}\in \Phi({\overline{{{\mathfrak{R}}}}})$.
\[lemma-compl-P1-bound-s\] Let $1\in {{\mathfrak{R}}}\subset [0,\, 1]$ and let $(X,B)$ be a klt log semi-Fano of dimension $\le d$ such that $X$ is FT. Assume the LMMP in dimension $d$. If $B\in \Phi({{\mathfrak{R}}},{\varepsilon}_{d})$, then there is an $n$-complement $K+B^+$ of $K+B$ for some $n\in {{\mathscr{N}}}_d({{\mathfrak{R}}})$. Moreover, $B\in {{\mathscr{P}}}_n$, and so $B^+\ge B$.
In the proposition we do not assert that ${{\mathscr{N}}}_d({{\mathfrak{R}}})$ is finite. However later on we use the proposition in the induction process when the set of indices is finite (cf. the proof of Lemma \[lemma-2-compl-\] and see \[subsectio-reduction\] – \[sufficient-condition\]).
If ${\varepsilon}_{d}=0$, then $\Phi({{\mathfrak{R}}},{\varepsilon}_{d})=\Phi({{\mathfrak{R}}})$ and there is nothing to prove. So we assume that ${\varepsilon}_{d}>0$. If $X$ is not ${{\mathbb Q}}$-factorial, we replace $X$ with its small ${{\mathbb Q}}$-factorial modification. Write $B=\sum b_iB_i$. Consider the new boundary $D=\sum d_i B_i$, where $$d_i=
\begin{cases}
b_i&\text{if $b_i< 1-{\varepsilon}_{d}$,}
\\
1-{\varepsilon}_{d}&\text{otherwise.}
\end{cases}$$ Clearly, $D\in \Phi({{\mathfrak{R}}})$. Since $D\le B$, there is a klt ${{\mathbb Q}}$-complement $K+D+\Lambda$ of $K+D$ (by definition, $\Lambda\ge 0$). Run $-(K+D)$-MMP. Since all the birational transformations are $K+D+\Lambda$-crepant, they preserve the klt property of $(X,D+\Lambda)$ and $(X,D)$. Each extremal ray is $\Lambda$-negative, and therefore is birational. At the end we get a model $(\bar X,\bar D)$ which is a log semi-Fano variety. By definition, since $\bar D\in \Phi({{\mathfrak{R}}})$ and $X$ is FT, there is an $n$-complement $\bar D^+$ of $K_{\bar X}+\bar D$ for some $n\in {{\mathscr{N}}}_d({{\mathfrak{R}}})$. Note that $I({{\mathfrak{R}}}) \mid n$, so $\bar D\in \Phi({{\mathfrak{R}}})\subset {{\mathscr{P}}}_n$. By Proposition \[cor-pull-back\_compl-I\] we can pull-back this complement to $X$ and this gives us an $n$-complement of $K_X+B$. The last assertion follows by Lemmas \[lemma-PPP-n\] and \[lemma-PPP-n-1\].
General reduction {#sect-reduction}
=================
In this section we outline the main reduction step in the proof of our main results \[main-result\] and \[main-result0\]. First we concentrate on the klt case. The non-klt case of \[main-result0\] will be treated in \[subsect-1\] and \[subsetion-0-pairs-proof\]. Note that in Theorem \[main-result\] it is sufficient to find only one integer $n=n(d,{{\mathfrak{R}}})$ divisible by $I({{\mathfrak{R}}})$ and such that $K+B$ is $n$-complemented. Other statements immediately follows by Corollary \[cor-imp\], Lemma \[lemma-PPP-n\] and by the definition of complements.
Setup {#eq-reduction-setup}
-----
Let $(X,B=\sum b_iB_i)$ be a klt log semi-Fano variety of dimension $d$ such that $B\in \Phi({{\mathfrak{R}}})$. In particular, $B$ is a ${{\mathbb Q}}$-divisor. Assume that $X$ is FT. By induction we may assume that Theorems \[main-result\] and \[main-result0\] hold in dimension $d-1$. So, by this inductive hypothesis, ${\varepsilon}_{d-1}({\overline{{{\mathfrak{R}}}}})>0$ whenever ${{\mathfrak{R}}}\subset [0,\, 1]$ is finite and rational. Take any $0<{\varepsilon}'\le {\varepsilon}_{d-1}({\overline{{{\mathfrak{R}}}}})$. Put also $I:=I({{\mathfrak{R}}})$.
[**([.]{})**]{}
\[para-rem-Noetherian\] First assume that the pair $(X,B)$ is ${\varepsilon}'$-lt. Then the multiplicities of $B$ are contained in the finite set $\Phi({{\mathfrak{R}}})\cap [0,\, 1-{\varepsilon}']$. By Conjecture [[\[BAB\]]{}]{} the pair $(X,B)$ is bounded. Hence $(X,{\operatorname{Supp}}B)$ belongs to an algebraic family and we may assume that the multiplicities of $B$ are fixed. Let $m:=nI$. The condition that $K+B$ is $m$-complemented is equivalent to the following $$\exists\ {\overline{B}}\in \left|-K-{\left\lfloor (m+1)B\right\rfloor}\right|\hspace{4pt} \text{such that}
\hspace{4pt}
\left(X, \frac 1m \left({\left\lfloor (m+1)B\right\rfloor}+\bar B\right)\right)\hspace{4pt} \text{is lc}$$ (see \[def-compl\], [@Shokurov-1992-e 5.1]). Obviously, the last condition is open in the deformation space of $(X,{\operatorname{Supp}}B)$. By Proposition \[prop-1\] below and Noetherian induction the log divisor $K+B$ has a bounded $nI$-complement for some $n\le C(d,{{\mathfrak{R}}})$. From now on we assume that $(X,B)$ is not ${\varepsilon}'$-lt.
{#sub-reduction-1}
We replace $(X,B)$ with log crepant ${{\mathbb Q}}$-factorial blowup of all divisors $E$ of discrepancy $a(E,X,B)\le -1+{\varepsilon}'$, see [@Utah 21.6.1]. Condition $B\in \Phi({{\mathfrak{R}}})$ will be replaced with $B\in \Phi({{\mathfrak{R}}},{\varepsilon}')\cap {{\mathbb Q}}$. Note that our new $X$ is again FT by Lemma \[lemma-FT\]. From now on we assume that $X$ is ${{\mathbb Q}}$-factorial and $$\label{eq-discr-2-B-ep}
{\operatorname{discr}}(X,B)>-1+{\varepsilon}'.$$
[**([.]{})**]{}
\[reduct-FT-Lambda\] For some $n_0\gg 0$, the divisor $n_0B$ is integral and the linear system $|-n_0(K+B)|$ is base point free. Let ${\overline{B}}\in |-n_0(K+B)|$ be a general member. Put $\Theta:=B+\frac1 {n_0} {\overline{B}}$. By Bertini’s theorem ${\operatorname{discr}}(X,\Theta)={\operatorname{discr}}(X,B)$. Thus we have the following
- $K_X+\Theta$ is a klt ${{\mathbb Q}}$-complement of $K_X+B$,
- ${\operatorname{discr}}(X,\Theta)\ge 1+{\varepsilon}'$, and
- $\Theta-B$ is supported in a movable (possibly trivial) divisor.
Define a new boundary $D$ with ${\operatorname{Supp}}D={\operatorname{Supp}}B$: $$\label{eq-D}
D:=\sum d_iB_i,\quad\text{ where}\quad d_i=
\begin{cases}
1 & \text{if}\quad b_i\ge 1-{\varepsilon}',
\\
b_i &
\text{otherwise}.
\end{cases}$$ Here the $B_i$ are components of $B$. Clearly, $D\in \Phi({{\mathfrak{R}}})$, $D>B$, and by we have ${\left\lfloor D\right\rfloor}\neq 0$.
\[lemma-2-compl-\] Fix a finite set ${{\mathfrak{R}}}\subset [0,\, 1]\cap {{\mathbb Q}}$. Let $(X\ni o,D)$ be the germ of a ${{\mathbb Q}}$-factorial klt $d$-dimensional singularity, where $D\in \Phi ({{\mathfrak{R}}},{\varepsilon}_{d-1}({\overline{{{\mathfrak{R}}}}}))$. Then there is an $n$-complement of $K_X+D$ with $n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$.
Recall that according to our inductive hypothesis, ${{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$ is finite.
Consider a plt blowup $f\colon \tilde X\to X$ of $(X,D)$ (see [@Prokhorov-Shokurov-2001 Prop. 3.6]). By definition the exceptional locus of $f$ is an irreducible divisor $E$, the pair $(\tilde X, \tilde D+E)$ is plt, and $-(K_{\tilde X}+\tilde D+E)$ is $f$-ample, where $\tilde D$ is the proper transform of $D$. We can take $f$ so that $f(E)=o$, i.e., $E$ is projective. By Adjunction $-(K_E+ {\operatorname{Diff}}_E(\tilde D))$ is ample and $(E, {\operatorname{Diff}}_E(\tilde D))$ is klt. By Proposition \[prop-prop\] we have ${\operatorname{Diff}}_E(\tilde D)\in \Phi({\overline{{{\mathfrak{R}}}}}, {\varepsilon}_{d-1}({\overline{{{\mathfrak{R}}}}}))$. Hence there is an $n$-complement of $K_E+{\operatorname{Diff}}_E(\tilde D)$ with $n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$, see Proposition \[lemma-compl-P1-bound-s\]. This complement can be extended to $\tilde X$ by Proposition \[prodolj\].
\[claim-0\] The pair $(X,D)$ is lc.
By Lemma \[lemma-2-compl-\] near each point $P\in X$ there is an $n$-complement $K+B^+$ of $K+B$ with $n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$. By Lemma \[lemma-PPP-n\], we have ${{\mathscr{P}}}_n\supset \Phi ({{\mathfrak{R}}},{\varepsilon})$. Hence, by Lemma \[lemma-PPP-n-1\], $B^+\ge B$. On the other hand, $nB^+$ is integral and for any component of $D-B$, its multiplicity in $B$ is $\ge {\varepsilon}'>1/(n+1)$. Hence, $B^+\ge D$ and so $(X,D)$ is lc near $P$.
{#subs-2-inductive-steps}
Run $-(K+D)$-MMP (anti-MMP). If $X$ is FT, this is possible by Corollary \[CY-MMP\]. Otherwise $K+B\equiv 0$ and $-(K+D)$-MMP coincides with $K+B-\delta (D-B)$-MMP for some small positive $\delta$.
It is clear that property $B\in \Phi({{\mathfrak{R}}},{\varepsilon}')$ is preserved on each step. All birational transformations are $(K+\Theta)$-crepant. Therefore $K+\Theta$ is klt on each step. Since $B\le \Theta$, $K+B$ is also klt. By Claim \[claim-0\] the log canonical property of $(X,D)$ is also preserved and $X$ is FT on each step by \[lemma-FT\].
\[red-not-contr\] None of components of ${\left\lfloor D\right\rfloor}$ is contracted.
Let ${\varphi}\colon X\to \bar X$ be a $K+D$-positive extremal contraction and let $E$ be the corresponding exceptional divisor. Assume that $E\subset {\left\lfloor D\right\rfloor}$. Put $\bar D:={\varphi}_* D$. Since $K_X+D$ is ${\varphi}$-ample, we can write $$K_X+D={\varphi}^*(K_{\bar X}+\bar D)-\alpha E,\qquad \alpha> 0.$$ On the other hand, since $(\bar X,\bar D)$ is lc, we have $$-1 \le a(E,\bar X,\bar D)=a(E,X,D)-\alpha=-1-\alpha<-1,$$ a contradiction.
\[cor-reduct-discrs\] Condition holds on each step of our MMP.
Note that all our birational transformations are $(K+\Theta)$-crepant. Hence by , it is sufficient to show that none of the components of $\Theta$ of multiplicity $\ge 1-{\varepsilon}'$ is contracted. Assume that on some step we contract a component $B_i$ of multiplicity $b_i\ge 1-{\varepsilon}'$. Then by $B_i$ is a component of ${\left\lfloor D\right\rfloor}$. This contradicts Claim \[red-not-contr\].
Reduction {#subsectio-reduction}
---------
After a number of divisorial contractions and flips $$\label{MMP}
X\dashrightarrow X_1\dashrightarrow \cdots\dashrightarrow X_N=Y,$$ we get a ${{\mathbb Q}}$-factorial model $Y$ such that either
[**([.]{})**]{}
\[cases-1\] there is a non-birational $K_{Y}+D_Y$-positive extremal contraction ${\varphi}\colon Y\to Z$ to a lower-dimensional variety $Z$, or
[**([.]{})**]{}
\[cases-2\] $-(K_{Y}+D_Y)$ is nef.
Here $\square_Y$ denotes the proper transform of $\square$ on $Y$.
\[claim-1\] In case , $Z$ is a point, i.e., $\rho(Y)=1$ and $-(K_Y+B_Y)$ is nef.
Let $F={\varphi}^{-1}(o)$ be a general fibre. Since $\rho(Y/Z)=1$ and $-(K+B)\equiv \Theta-B\ge 0$, the restriction $-(K+B)|_F$ is nef. It is clear that $B|_F\in \Phi({{\mathfrak{R}}},{\varepsilon}')$. Assume that $Z$ is of positive dimension. Then $\dim F<\dim X$. By our inductive hypothesis and Proposition \[lemma-compl-P1-bound-s\] there is a bounded $n$-complement $K_F+B|_F^+$ of $K_F+B|_F$ for some $n\in {{\mathscr{N}}}_{d-1}({{\mathfrak{R}}})\subset {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$. By Lemmas \[lemma-PPP-n-1\] and \[lemma-PPP-n\], we have $B|_F^+\ge D|_F \ge B|_F$. On the other hand, $(K_X+D)|_F$ is ${\varphi}$-ample, a contradiction.
{#case}
Therefore we have a ${{\mathbb Q}}$-factorial FT variety $Y$ and two boundaries $B_Y=\sum b_iB_i$ and $D_Y=\sum d_i B_i$ such that ${\operatorname{discr}}(Y,B_Y)>-1+{\varepsilon}'$, $B_Y\in \Phi({{\mathfrak{R}}},{\varepsilon}')$, $D_Y\in \Phi({{\mathfrak{R}}})$, $D_Y\ge B_Y$, and $d_i>b_i$ if and only if $d_i=1$ and $b_i\ge -1+{\varepsilon}'$. Moreover, one of the following two cases holds:
[**([.]{})**]{}
\[case-rho=1\] $\rho(Y)=1$, $K_Y+D_Y$ is ample, and $(Y,B_Y)$ is a klt log semi-Fano variety, or
[**([.]{})**]{}
\[case-nef\] $({Y},D_Y)$ is a log semi-Fano variety with ${\left\lfloor D_Y\right\rfloor}\neq 0$. Since $D>B$, this case does not occur if $K+B\equiv 0$.
These two cases will be treated in sections [[\[sec-rho=1\]]{}]{} and [[\[sec-nef\]]{}]{}, respectively.
Outline of the proof of Theorem \[main-result\] {#sufficient-condition}
-----------------------------------------------
Now we sketch the basic idea in the proof of boundedness in case . By we may assume that $(X,B)$ is not ${\varepsilon}'$-lt. Apply constructions of \[sub-reduction-1\], \[subs-2-inductive-steps\] and \[subsectio-reduction\]. Recall that on each step of we contract an extremal ray which is $(K+D)$-positive. By Proposition [[\[cor-pull-back\_compl-I\]]{}]{} we can pull-back $n$-complements with $n\in {{\mathscr{N}}}_d({{\mathfrak{R}}})$ of $K_{Y}+D_Y$ to our original $X$. However it can happen in case $\rho(Y)=1$ that $K_{Y}+D_Y$ has no any complements. In this case we will show in Section \[sec-rho=1\] below that the multiplicities of $B_Y$ are bounded from the above: $b_i<1-c$, where $c>0$. By Claim [[\[red-not-contr\]]{}]{} divisorial contractions in do not contracts components of $B$ with multiplicities $b_i\ge
1-{\varepsilon}'$. Therefore the multiplicities of $B$ also are bounded from the above. Combining this with ${\operatorname{discr}}(X,B)>-1+{\varepsilon}'$ and Conjecture [[\[BAB\]]{}]{} we get that $(X,{\operatorname{Supp}}B)$ belong to an algebraic family. By Noetherian induction (cf. ) we may assume that $(X,{\operatorname{Supp}}B)$ is fixed. Finally, by Proposition [[\[prop-1\]]{}]{} we have that $(X,B)$ has bounded complements.
Case will be treated in Sect. [[\[sec-nef\]]{}]{}. In fact in this case we study the contraction $f\colon Y\to Z$ given by $-(K+D)$. When $Z$ is a lower-dimensional variety, $f$ is a fibration onto varieties with trivial log canonical divisor. The existence of desired complements can be established inductively, by using an analog of Kodaira’s canonical bundle formula (see Conjecture [[\[conj-main-adj\]]{}]{}).
The proof of Theorem \[main-result0\] in case when $(X,B)$ is not klt is based on the following
\[lemma-Adj-hor-mult-1\] Let $(X,B=\sum b_iB_i)$ be a $0$-pair of dimension $d$ such that $B\in \Phi({{\mathfrak{R}}},{\varepsilon}')$, where ${\varepsilon}':={\varepsilon}_{d-1}({\overline{{{\mathfrak{R}}}}})$. Assume the LMMP in dimension $d$. Further, assume either
1. $(X,B)$ is not klt and Theorems [[\[main-result\]]{}]{}-[[\[main-result0\]]{}]{} hold in dimension $d-1$, or
2. Theorem [[\[main-result0\]]{}]{} holds in dimension $d$.
Then there exists $\lambda:=\lambda(d,{{\mathfrak{R}}})>0$ such that either $b_i=1$ or $b_i\le 1-\lambda$ for all $b_i$.
The proof is by induction on $d$. Case $d=1$ is well-known; see, e.g., Corollary \[lemma-compl-P1-bound\]. If $B=0$, there is nothing to prove. So we assume that $B>0$. Assume that, for some component $B_i=B_0$, we have $1-\lambda<b_0<1$.
First consider the case when $(X,B)$ is not klt. Replace $(X,B)$ with its ${{\mathbb Q}}$-factorial dlt modification so that ${\left\lfloor B\right\rfloor}\neq 0$. If $B_0\cap {\left\lfloor B\right\rfloor}=\emptyset$, we run $K+B-b_0B_0$-MMP. After several steps we get a $0$-pair $(\hat X,\hat B)$ such that $\hat {{\left\lfloor B\right\rfloor}}\neq 0$, $\hat B_0\neq 0$ and one of the following holds:
1. $\hat B_0\cap \hat {{\left\lfloor B\right\rfloor}}\neq \emptyset$, or
2. there is an extremal contraction $\hat f\colon \hat X\to \hat Z$ to a lower-dimensional variety such that both $\hat{{\left\lfloor B\right\rfloor}}$ and $\hat B_0$ are $\hat f$-ample, and $\hat B_0\cap \hat {{\left\lfloor B\right\rfloor}}= \emptyset$. (In particular, $\hat Z$ is not a point.)
In the second case we can apply induction hypothesis restricting $\hat B$ to a general fibre of $\hat f$. In the first case, replacing our original $(X,B)$ with a dlt modification of $(\hat X,\hat B)$, we may assume that $B_0\cap {\left\lfloor B\right\rfloor}\neq \emptyset$. Let $B_1\subset {\left\lfloor B\right\rfloor}$ be a component meeting $B_0$. Then $(B_1,{\operatorname{Diff}}_{B_1}(B-B_1))$ is a $0$-pair with ${\operatorname{Diff}}_{B_1}(B-B_1)\in \Phi({\overline{{{\mathfrak{R}}}}},{\varepsilon}')$ (see Lemma \[lemma-PPP-n\]). Write ${\operatorname{Diff}}_{B_1}(B-B_1)=\sum \delta_i\Delta_i$ and let $\Delta_0$ be a component of $B_0\cap B_1$. Then the multiplicity $\delta_0$ of $\Delta_0$ in ${\operatorname{Diff}}_{B_1}(B-B_1)$ is computed as follows: $\delta_0=1-1/r+\sum_l k_lb_l/r$, where $r$ and $k_l$ are non-negative integers and $r,\, k_0>0$ (see [@Shokurov-1992-e Corollary 3.10]). Since $b_0>1-{\varepsilon}'>1/2$, we have $k_0=1$ and $k_l=0$ for $l\neq 0$. Hence, $\delta_0=1-1/r+b_0/r$. By induction we may assume either $\delta_0=1$ or $\delta_0\le 1-\lambda(d-1,{\overline{R}})$. Thus we have either $b_0=1$ or $$b_0\le 1-r\lambda(d-1,{\overline{R}})\le 1-\lambda(d-1,{\overline{R}})$$ and we can put $\lambda(d, R)=\lambda(d-1,{\overline{R}})$ in this case.
Now consider the case when $(X,B)$ is klt. Replace $(X,B)$ with its ${{\mathbb Q}}$-factorialization and again run $K+B-b_0B_0$-MMP: $(X,B) \dashrightarrow (X',B')$. Clearly at the end we get a $B_0'$-positive extremal contraction $\varphi \colon (X',B')\to W$ to a lower-dimensional variety $W$. If $W$ is not a point, we can apply induction restricting $B'$ to a general fibre. Thus replacing $(X,B)$ with $(X',B')$ we may assume that $X$ is ${{\mathbb Q}}$-factorial, $\rho(X)=1$ and $(X,B-b_0 B_0)$ is a klt log Fano variety. In particular, $X$ is FT. By Theorem \[main-result0\] and Proposition \[lemma-compl-P1-bound-s\] the log divisor $K+B$ is $n$-complemented for some $n \in {{\mathscr{N}}}_{d}({{\mathfrak{R}}})$. For this complement $B^+$, we have $B^+\ge B$. Since $K_{X}+B\equiv 0$, $B^+=B$. In particular, $nB$ is integral. Thus we can put $\lambda:=1/{\operatorname{lcm}}({{\mathscr{N}}}_{d-1}({{\mathfrak{R}}}))$. This proves the statement in case (ii).
\[cor-Adj-hor-mult-1\] Notation and assumptions as in Lemma [[\[lemma-Adj-hor-mult-1\]]{}]{}. Let $E$ be any divisor (exceptional or not) over $X$. Then either $a(E,X,B)=1$ or $a(E,X,B)\le 1-\lambda$.
We can take $\lambda<{\varepsilon}'$. If $1-\lambda<a(E,X,B)<1$, replace $(X,B)$ with a crepant blowup of $E$, see [@Utah 21.6.1] and apply Lemma [[\[lemma-Adj-hor-mult-1\]]{}]{}.
Proof of Theorem \[main-result0\] in the case when $(X,B)$ is not klt {#subsect-1}
---------------------------------------------------------------------
Let $(X,B)$ be a $0$-pair such that $X$ is FT and $B\in \Phi({{\mathfrak{R}}})$. We assume that $(X,B)$ is not klt. Replace $(X,B)$ with its ${{\mathbb Q}}$-factorial dlt modification. Then in particular, $X$ is klt. Moreover, ${\left\lfloor B\right\rfloor}\neq 0$ (and $B\in \Phi({{\mathfrak{R}}})$). Let $\lambda(d,{{\mathfrak{R}}})$ be as in Lemma \[lemma-Adj-hor-mult-1\] and let $0<\lambda<\lambda(d,{{\mathfrak{R}}})$. If $X$ is not $\lambda$-lt, then for each exceptional divisor $E$ of discrepancy $a(E,X,0)< -1+\lambda$ by Corollary \[cor-Adj-hor-mult-1\] we have $a(E,X,B)=1$. Hence as in \[sub-reduction-1\] replacing $(X,B)$ with blowup of all such divisors $E$, see [@Utah 21.6.1], we get that $X$ is $\lambda$-lt and $(X,B)$ is a $0$-pair with $B\in \Phi({{\mathfrak{R}}})$ and ${\left\lfloor B\right\rfloor}\neq 0$. Run $K$-MMP:$X\dashrightarrow X'$ and let $B'$ be the birational transform of $B$. Since $B\neq 0$, $X'$ admits a $K$-negative Fano fibration $X'\to Z'$ over a lower-dimensional variety $Z'$. By our construction $X'$ is $\lambda$-lt and $(X',B')$ is a non-klt $0$-pair with $B'\in \Phi({{\mathfrak{R}}})$. By Proposition \[cor-pull-back\_compl-I\] we can pull-back $n$-complements from $X'$ to $X$ if $I({{\mathfrak{R}}}) \mid n$. If $Z'$ is a point, then $\rho(X')=1$ and $X'$ is a klt Fano variety. In this case, arguing as in \[para-rem-Noetherian\] we get that $(X',{\operatorname{Supp}}B')$ belong to an algebraic family. By Proposition \[prop-1\] and Noetherian induction the log divisor $K+B$ has a bounded $nI$-complement for some $n\le C(d,{{\mathfrak{R}}})$. Finally, if $\dim Z'>0$, then we apply Proposition \[prop-D-not-big\] below. This will be explained in \[subsetion-0-pairs-proof\]. Theorem \[main-result0\] in the non-klt case is proved.
Approximation and complements {#approximation}
=============================
The following Lemma [[\[open\]]{}]{} shows that the existence of $n$-complements is an open condition in the space of all boundaries $B$ with fixed ${\operatorname{Supp}}{B}$.
Notation
--------
Let ${{\mathscr{B}}}$ be a finite set of prime divisors $B_i$. Recall that $\mathfrak{D}_{{{\mathscr{B}}}}$ denotes the ${{\mathbb R}}$-vector space all ${{\mathbb R}}$-Weil divisors $B$ with ${\operatorname{Supp}}B=\sum_{B_i\in {{\mathscr{B}}}} B_i$, where the $B_i$ are prime divisors. Let $$\mathfrak{I}_{{{\mathscr{B}}}}:=\left\{\sum
\beta_iB_i\in\mathfrak{D}_{{{\mathscr{B}}}}\mid 0\le\beta_i\le 1,\ \forall
i\right\}$$ be the unit cube in $\mathfrak{D}_{{{\mathscr{B}}}}$.
\[open\] Let $(X,B)$ be a log pair where $B$ is an ${{\mathbb R}}$-boundary. Assume that $K+B$ is $n$-complemented. Then there is a constant ${\varepsilon}={\varepsilon}(X,B,n)>0$ such that $K+B'$ is also $n$-complemented for any ${{\mathbb R}}$-boundary $B'\in \mathfrak{D}_B$ with $\|B-B'\|<{\varepsilon}$.
Let $B^+=B^{\sharp}+\Lambda$ be an $n$-complement, where $\Lambda$ and $B$ have no common components and $B^{\sharp}\in\mathfrak{D}_B$. Write $B=\sum b_iB_i$, $B'=\sum
b_i'B_i$, $B^{\sharp}=\sum b_i^+B_i$. Take ${\varepsilon}$ so that $$0<(n+1){\varepsilon}<\min(1-{\left\{ (n+1)b_i \right\}} \mid 1\le i\le r,\quad b_i<1).$$ We claim that $B^+$ is also an $n$-complement of $B'$ whenever $\|B-B'\|<{\varepsilon}$. If $b_i=1$, then obviously $b_i^+=1$. So, it is sufficient to verify the inequalities $nb_i^+\ge {\left\lfloor (n+1)b_i'\right\rfloor}$ whenever $b_i^+<1$ and $b_i<1$. Indeed, in this case, $${\left\lfloor (n+1)b_i'\right\rfloor}\le
{\left\lfloor (n+1)b_i+(n+1)(b_i'-b_i)\right\rfloor}={\left\lfloor (n+1)b_i\right\rfloor}\le nb_i^+.$$ (because $(n+1)(b_i'-b_i)<(n+1){\varepsilon}<\min(1-{\left\{ (n+1)b_i \right\}})$). This proves the assertion.
\[corollary-open\] For any $D\in Z_{d-1}(X)$, the subset $$\mathfrak{U}^n_D:=\{B\in \mathfrak{I}_D\mid \text{$K+B$ is
$n$-complemented}\}$$ is open in $\mathfrak{I}_D$.
\[prop-1\] Fix a positive integer $I$. Let $X$ be an FT variety such that $K_X$ is ${{\mathbb Q}}$-Cartier and let $B_1,\dots,B_r$ are ${{\mathbb Q}}$-Cartier divisors on $X$. Let ${{\mathscr{B}}}:=\sum_{i=1}^r B_i$. Then for any boundary $B\in\mathfrak{I}_{{{\mathscr{B}}}}$ such that $K+B$ is lc and $-(K+B)$ is nef, there is an $n$-complement of $K+B$ for some $n\le{\operatorname{Const}}\left(X,{{\mathscr{B}}}\right)$ and $I\mid n$.
\[rem-prop-1\] In notation of Proposition [[\[prop-1\]]{}]{} the following holds.
[**([.]{})**]{}
\[condition\] [(Effective base point freeness)]{} There is a positive integer $N$ such that for any integral nef Weil ${{\mathbb Q}}$-Cartier divisor of the form $mK+\sum m_iB_i$ the linear system $\left|N(mK+\sum m_iB_i)\right|$ is base point free.
Indeed, we have ${\operatorname{Pic}}(X)\simeq {\mathbb Z}^{\rho}$ (see, e.g., [@Iskovskikh-Prokhorov-1999 Prop. 2.1.2]). In the space ${\operatorname{Pic}}(X)\otimes {{\mathbb R}}\simeq {{\mathbb R}}^{\rho}$ we have a closed convex cone $\operatorname{NEF}(X)$, the cone of nef divisors. This cone is dual to the Mori cone ${{\overline{\operatorname{NE}}}}(X)$, so it is rational polyhedral and generated by a finite number of semiample Cartier divisors $M_1,\dots, M_s$. Take a positive integer $N'$ so that all the linear systems $|N'M_i|$ are base point free, and $N'K$, $N'B_1,\dots, N'B_r$ are Cartier. Write $$N'K{\mathbin{\sim_{\scriptscriptstyle{{{\mathbb Q}}}}}} \sum_{i=1}^s \alpha_{i,0}M_i,\quad N'B_j{\mathbin{\sim_{\scriptscriptstyle{{{\mathbb Q}}}}}} \sum_{i=1}^s
\alpha_{i,j}M_i,\qquad \alpha_{i,j}\in{{\mathbb Q}},\qquad \alpha_{i,j}\ge
0.$$ Let $N''$ be the common multiple of denominators of the $\alpha_{i,j}$. Then $$\begin{gathered}
{N'}^2N''\left(mK+\sum_{j=1}^r m_jB_j\right)\sim
N'N''m\left(\sum_{i=1}^s \alpha_{i,0}M_i\right)+\\ \sum_{j=1}^r
N'N''m_j \left(\sum_{i=1}^s \alpha_{i,j}M_i\right)= \sum_{i=1}^s
\left(mN''\alpha_{i,0}+\sum_{j=1}^r N''m_j
\alpha_{i,j}\right)N'M_i\end{gathered}$$ The last (integral) divisor generates a base point free linear system, so we can take $N={N'}^2N''$.
In the proof of Proposition [[\[prop-1\]]{}]{} we follow arguments of [@Shokurov-2000 Example 1.11], see also [@Shokurov-1992-e 5.2].
Define the set $$\label{def-MMM}
{{\mathscr{M}}}={{\mathscr{M}}}_{{{\mathscr{B}}}}:=\left\{B\in \mathfrak{I}_{\bar
B}\quad\left|\quad \text{$K+B$ is lc and $-\left(K+B\right)$ is
nef}\right.\right\}$$ Then ${{\mathscr{M}}}$ is a closed compact convex polyhedron in $\mathfrak{I}_{{{\mathscr{B}}}}$. It is sufficient to show the existence of some $n$-complement for any $B\in {{\mathscr{M}}}$. Indeed, then ${{\mathscr{M}}}\subset
\bigcup\limits_{n\in{\mathbb Z}_{>0}}\mathfrak{U}^n_{{{\mathscr{B}}}}$. By taking a finite subcovering ${{\mathscr{M}}}\subset
\bigcup\limits_{n\in\mathcal{S}}\mathfrak{U}^n_{{{\mathscr{B}}}}$ we get a finite number of such $n$.
Assume that there is a boundary $B^o=\sum_{i=1}^r b_i^oB_i\in
{{\mathscr{M}}}$ which has no any complements. By [@Cassels-1957 Ch. 1, Th. VII] there is infinite many rational points $(m_1/q,\dots,m_r/q)$ such that $$\max\left(\left|\frac{m_1}{q}-b_1^o\right|, \dots,
\left|\frac{m_r}{q}-b_r^o\right|\right)
<\frac{r}{(r+1)q^{1+1/r}}<\frac1{q^{1+1/r}}.$$ Denote $b_i:=m_i/q$ and $B:=\sum b_iB_i$. Thus, $\|B-B^o\|<1/q^{1+1/r}$. Then our proposition is an easy consequence of the following
For $q\gg 0$ one has
[**([.]{})**]{}
\[claim-11\] ${\left\lfloor (qN+1)b_i^o\right\rfloor}\le qNb_i$ whenever $b_i<1$;
[**([.]{})**]{}
\[claim-2\] $B\equiv B^o$ and $-(K+B)$ is nef; and
[**([.]{})**]{}
\[claim-3\] $K+B$ is lc.
Indeed, by the linear system $|-qN(K+B)|$ is base point free. Let $F\in |-qN(K+B)|$ be a general member. Then $K+B+\frac1{qN}F$ is an $qN$-complement of $K+B^o$, a contradiction.
By the construction $${\left\lfloor (qN+1)b_i^o\right\rfloor}=m_iN+{\left\lfloor b_i^o+qN(b_i^o-b_i)\right\rfloor}$$ Put $c:=\max\limits_{b_i^o<1}\{ b_1^o,\dots b_r^o\}$. Then for $b_i<1$ we have $b_i^o<c<1$ and for $q\gg 0$, $$b_i^o+qN(b_i^o-b_i)<c+\frac{qN}{q^{1+1/r}}<1.$$ This proves .
Further, let $L_1,\dots, L_r$ be a finite set of curves generating $N_1(X)$. We have $$\begin{gathered}
\left|L_j\cdot(B-B^o)\right|= \left|\sum_i\frac{m_i}q(L_j\cdot
B_i)- \sum_i b_i^o(L_j\cdot B_i)\right| \\ < \frac1{q^{1+1/r}}
\sum_i(L_j\cdot B_i).\end{gathered}$$ If $q\gg 0$, then the right hand side is $\ll 1/q$ while the left hand side is from the discrete set $\pm L_j\cdot
(K+B^o)+\frac{1}{qN}{\mathbb Z}$ (because $qB$ is an integral divisor and by our assumption ). Hence the left hand side is zero and $B\equiv B^o$. This proves .
Finally, we have to show that $K+B$ is lc. Assume the converse. By the divisor $qN\left(K+B\right)$ is Cartier. So there is a divisor $E$ of the field $\Bbbk(X)$ such that $a(E,X,B)\le
-1-1/qN$ and $a(E,X,B^o)\ge -1$. On the other hand, $a(E,X,\sum
\beta_iB_i)$ is an affine linear function in $\beta_i$: $$\frac{1}{qN}\le a(E,X,B^o)-a(E,X,B)=\sum
c_i(b_i^o-b_i)<\frac{{\operatorname{Const}}}{q^{1+1/r}}$$ which is a contradiction.
The following is the first induction step to prove Theorem \[main-result\].
\[lemma-compl-P1-bound\] Fix a finite set ${{\mathfrak{R}}}\subset [0,\, 1]\cap {{\mathbb Q}}$ and a positive integer $I$. Then the set ${{\mathscr{N}}}_1({{\mathfrak{R}}})$ is finite.
Let $(X,B)$ be a one-dimensional log pair satisfying conditions of . Since $X$ is FT, $X\simeq{{\mathbb P}}^1$. Since $B\in \Phi({{\mathfrak{R}}})$ and ${{\mathfrak{R}}}$ is finite, we can write $B=\sum_{i=1}^r b_iB_i$, where $b_i\ge \delta$ for some fixed $\delta>0$. Thus we may assume that $r$ is fixed and $B_1,\dots,B_r$ are fixed distinct points. Then by Proposition \[prop-1\] we have a desired complements.
Let $X\simeq {{\mathbb P}}^1$. If ${{\mathfrak{R}}}=\{0,\, 1\}$, then $I({{\mathfrak{R}}})=1$ and $\Phi({{\mathfrak{R}}})$ is the set of standard multiplicities. In this case, it is easy to compute that ${{\mathscr{N}}}_1({{\mathfrak{R}}})=\{1,\,2,\,3,\,4,\,6\}$ [@Shokurov-1992-e 5.2]. Consider more complicated case when ${{\mathfrak{R}}}=\{0,\, \frac12,\, \frac 23,\, \frac 34,\, \frac 56,\,
1\}$. Then $I=12$ and one can compute that $${{\mathscr{N}}}_1({{\mathfrak{R}}})=12\cdot \{1,\, 2,\, 3,\, 4,\, 5,\, 7,\, 8,\, 9,\, 11\}.$$ Indeed, assume that $(X,D)$ has no any $12n$-complements for $n\in $ $\{1, $ $2,$ $3, $ $4, $ $ 5, $ $7,$ $ 8,$ $ 9,$ $ 11\}$. Write $D=\sum_{i=1}^r d_iD_i$, where $D_i\neq D_j$ for $i\neq j$. It is clear that the statement about the existence of an $n$-complement $D^+$ such that $D^+\ge D$ is equivalent to the following inequality $$\label{eq-compl-P1}
\sum_i {\left\lceil nd_i\right\rceil}\le 2n.$$ Since $d_i=1-r_i/m_i$, where $r_i\in {{\mathfrak{R}}}$, $m_i\in {\mathbb Z}_{>0}$, we have $d_i\ge 1/6$ for all $i$. We claim that at least one denominator of $d_i$ does not divide $24$. Indeed, otherwise $24D$ is an integral divisor and $D^+:=D+\frac 1{24}\sum_{j=1}^k D_j$ is an $24$-complement, where $D_j\in X$ are general points and $k=24(2-\deg D)$. Thus we may assume that the denominator of $d_1$ does not divide $24$. Since $d_1=1-r_1/m_1$, where $r_1\in {{\mathfrak{R}}}$, we have $m_1\ge 3$ and the equality holds only if $r_1=2/3$ or $5/6$. In either case, $d_1\ge 13/18$.
Recall that a log pair $(X,D)$ of global type is said to be *exceptional* if at has at least one ${{\mathbb Q}}$-complement and any ${{\mathbb Q}}$-complement is klt. If $(X,D)$ is not exceptional, we can increase $d_1$ by putting $d_1=1$. Then as above $13/18\le d_2\le 5/6$, so $r=3$ and $d_3\le 5/18$. Now there are only a few possibilities for $d_2$ and $d_3$:
--------- ----------------- -------------- -------------- ----------------- -------------- ----------------- --------------
$d_2$ $\frac{13}{18}$ $\frac34$ $\frac79$ $\frac{19}{24}$ $\frac45$ $\frac{13}{16}$ $\frac56$
\[8pt\]
$d_3$ $\le\frac5{18}$ $\le\frac14$ $\le\frac29$ $\le\frac5{24}$ $\le\frac15$ $\le\frac3{16}$ $\le\frac16$
--------- ----------------- -------------- -------------- ----------------- -------------- ----------------- --------------
In all cases $K+D$ has a $12n$ complement for some $n\in \{1,\, 2,\, 3,\, 4,\, 5\}$. In the exceptional case, there is a finite number of possibilities for $(d_1,\dots,d_r)$. However the computations are much longer. We omit them.
The main theorem: Case $\rho=1$ {#sec-rho=1}
===============================
{#rho=1-begin}
Now we begin to consider case . Thus we assume that $(X,B)$ is klt but not ${\varepsilon}'$-lt, where we take ${\varepsilon}'$ so that ${\varepsilon}'N=1$ for some integer $N\ge N_{d-1}({\overline{{{\mathfrak{R}}}}})+2$. Then obviously, ${\varepsilon}_{d-1}({\overline{{{\mathfrak{R}}}}})\ge {\varepsilon}'>0$. Further, assume that applying the general reduction from Section \[sect-reduction\] we get a pair $(Y,B_Y)$, where $Y$ is FT and ${{\mathbb Q}}$-factorial, $\rho(Y)=1$, the ${{\mathbb Q}}$-divisor $-(K_Y+B_Y)$ is nef, and $${\operatorname{discr}}(Y,B_Y)>-1+{\varepsilon}'.$$ Moreover, $K_Y+D_Y$ is ample and ${\left\lfloor D_Y\right\rfloor}\neq 0$, where $D_Y$ is a boundary satisfying . In particular, $B_Y\neq 0$.
{#section-3}
Assume that the statement of Theorems \[main-result\] and \[main-result0\] is false in this case. Then there is a sequence of klt log pairs $(X^{(m)},B^{(m)})$ as in \[rho=1-begin\] and such that complements of $K_{X^{(m)}}+B^{(m)}$ are unbounded. More precisely, for each $K_{X^{(m)}}+B^{(m)}$, let $n_m$ be the minimal positive integer such that $I({{\mathfrak{R}}})\mid n_m$ and $K_{X^{(m)}}+B^{(m)}$ is $n_m$-complemented. We assume that the sequence $n_m$ is unbounded. We will derive a contradiction.
{#section-4}
By Corollary \[lemma-compl-P1-bound\] we have $\dim X^{(m)}\ge 2$. By our hypothesis we have a sequence of birational maps $X^{(m)}\dashrightarrow Y^{(m)}$, where $Y^{(m)}$, $X^{(m)}$, $B^{(m)}$ and $D^{(m)}$ are as above for all $m$. Recall that by Corollary \[cor-reduct-discrs\] $$\label{eq-rho=1-discr}
{\operatorname{discr}}(Y^{(m)})\ge {\operatorname{discr}}(Y^{(m)},B^{(m)}_Y)
\ge{\operatorname{discr}}(X^{(m)},B^{(m)})> -1+{\varepsilon}',$$ where $0<{\varepsilon}' \le {\varepsilon}_{d-1}({\overline{{{\mathfrak{R}}}}})$. Note that $-(K_{Y^{(m)}}+B^{(m)})$ is nef, so by Conjecture \[BAB\] the sequence of varieties $Y^{(m)}$ is bounded. By Noetherian induction (cf. ), we may assume that $Y^{(m)}$ is fixed, that is, $Y^{(m)}=Y$.
Let $Y\hookrightarrow{{\mathbb P}}^N$ be an embedding and let $H$ be a hyperplane section of $Y$. Note that the multiplicities of $B_Y^{(m)}=\sum b_i^{(m)}B_i^{(m)}$ are bounded from below: $b_i^{(m)}\ge \delta_0>0$, where $\delta_0:=\min \Phi({{\mathfrak{R}}})\setminus \{0\}$. Then, for each $B_i^{(m)}$, $$\delta_0 H^{d-1}\cdot B_i^{(m)}\le H^{d-1}\cdot B^{(m)}_Y\le -H^{d-1}\cdot K_Y.$$ This shows that the degree of $B_i^{(m)}$ is bounded and $B_i^{(m)}$ belongs to an algebraic family. Therefore we may assume that ${\operatorname{Supp}}{B_Y^{(m)}}$ is also fixed: $B_i^{(m)}=B_i$.
{#subsect-rho=1-case1}
Assume that the multiplicities of $B^{(m)}_Y$ are bounded from $1$, i.e., $b_i^{(m)}\le 1-c$, where $c>0$. Then we argue as in [[\[sufficient-condition\]]{}]{}. By Claim [[\[red-not-contr\]]{}]{} divisorial contractions in do not contract components of $B^{(m)}$ with multiplicities $b_i^{(m)}\ge
1-{\varepsilon}'$. Therefore the multiplicities of $B^{(m)}$ on $X^{(m)}$ are also bounded from the $1$. Combining this with ${\operatorname{discr}}(X,B)>-1+{\varepsilon}'$ and Conjecture [[\[BAB\]]{}]{} we get that $(X,{\operatorname{Supp}}B)$ belong to an algebraic family. By Noetherian induction (cf. ) we may assume that $(X,{\operatorname{Supp}}B)$ is fixed. Finally, by Proposition [[\[prop-1\]]{}]{} we have that $(X,B)$ has a bounded $n$-complement such that $I({{\mathfrak{R}}})\mid n$.
Thus by our construction the only possibility is the case below.
{#section-5}
From now on we consider the remaining case when some multiplicity of $B^{(m)}_Y$ is accumulated to $1$ and we will derive a contradiction. Since ${\operatorname{Supp}}B^{(m)}_Y$ does not depend on $m$, by passing to a subsequence we may assume that the limit $B^{\infty}_Y:=\lim\limits_{m\to\infty} B^{(m)}_Y$ exists and ${\left\lfloor B^{\infty}_Y\right\rfloor}\neq 0$. As above, write $B^{\infty}_Y=\sum b^{\infty}_i B_i$. Up to permutations of components we may assume that $b^{\infty}_1=1$. It is clear that $-(K_Y+B^{\infty}_Y)$ is nef and ${\operatorname{discr}}(Y,B^{\infty}_Y)\ge -1+{\varepsilon}'$. In particular, $(Y,B^{\infty}_Y)$ is plt.
\[claim-bi\] Under the above hypothesis, we have $b_j^{\infty}\le 1-{\varepsilon}'$ for all $1< j\le r$. Moreover, by passing to a subsequence we may assume the following:
1. If $b_j^{\infty}=1$, then $j=1$ and $b_1^{(m)}$ is strictly increasing.
2. If $b_j^{\infty}<1-{\varepsilon}'$, then $b_j^{(m)}=b_j^{\infty}$ is a constant.
3. If $b_j^{\infty}=1-{\varepsilon}'$, then $b_j^{(m)}$ is either a constant or strictly decreasing.
In particular, $B^{\infty}_Y\in\Phi({{\mathfrak{R}}})$, $B^{\infty}_Y$ is a ${{\mathbb Q}}$-boundary, and $D_Y^{(m)}\ge B^{\infty}_Y$ for $m\gg 0$.
Since $\rho(Y) =1$ and $Y$ is ${{\mathbb Q}}$-factorial, the intersection $B_1\cap B_j$ on $Y$ is of codimension two and non-empty. For a general hyperplane section $Y\cap H$, by we have the inequality $${\operatorname{discr}}(Y\cap H,B^{\infty}\cap H)\ge -1+{\varepsilon}'.$$ Thus by Lemma [[\[pair-discr\]]{}]{} below, we have $b_1^{\infty}+b_j^{\infty}\le 2-{\varepsilon}'$, i.e., $b_j^{\infty}\le 1-{\varepsilon}'$ for all $1< j\le r$. The rest follows from the fact that the set $\Phi({{\mathfrak{R}}})\cap [0,1-{\varepsilon}']$ is finite.
\[pair-discr\] Let $(S\ni
o,\Lambda=\sum \lambda_i\Lambda_i)$ be a log surface germ. Assume that ${\operatorname{discr}}(S,\Lambda)\ge -1+{\varepsilon}$ at $o$ for some positive ${\varepsilon}$. Then $\sum \lambda_i\le 2-{\varepsilon}$.
Locally near $o$ there is an étale outside of $o$ Galois cover $\pi\colon S'\to S$ such that $S'$ is smooth. Let $\Lambda':=\pi^*\Lambda$ and $o':=\pi^{-1}(o)$. Then ${\operatorname{discr}}(S',\Lambda')\ge{\operatorname{discr}}(S,\Lambda)\ge -1+{\varepsilon}$ at $o$ (see, e.g., [@Utah Proposition 20.3]). Consider the blow up of $o'\in S'$. We get an exceptional divisor $E$ of discrepancy $$-1+{\varepsilon}\le
a(E,S',\Lambda')=1-\sum \lambda_i.$$ This gives us the desired inequality.
$b_j^{\infty}=1-{\varepsilon}'$ for some $j$.
Indeed, otherwise $D^{(m)}_Y=B^{\infty}_Y$ for $m\gg 0$ and $-(K_{Y}+D^{(m)}_Y)$ is nef, a contradiction.
We claim that the log divisor $K_{Y}+B^{\infty}_Y$ is $n$-complemented, where $n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$. Recall that $b_1^{\infty}=1$. Put $B':=B^{\infty}-B_1$. By the last corollary $B'\neq 0$. By Proposition \[prop-prop\] we have ${\operatorname{Diff}}_{B_1}(B')\in \Phi({\overline{{{\mathfrak{R}}}}})$. Recall that $-(K_Y+B^\infty)$ is nef. Since $(Y,B^{\infty}_Y)$ is plt, the pair $(B_1,{\operatorname{Diff}}_{B_1}(B'))$ is klt. Further, $-(K_{B_1}+{\operatorname{Diff}}_{B_1}(0))$ is ample (because $\rho(Y)=1$), so $B_1$ is FT. Thus by the inductive hypothesis there is an $n$-complement $K_{B_1}+{\operatorname{Diff}}_{B_1}(B')^+$ of $K_{B_1}+{\operatorname{Diff}}_{B_1}(B')$ for some $n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$. By Lemma \[lemma-PPP-n\] $B'\in {{\mathscr{P}}}_n$. Take a sufficiently small positive $\delta$ and let $j$ be such that $b_j^{\infty}=1-{\varepsilon}'$. We claim that $B'-\delta B_j\in {{\mathscr{P}}}_n$. Indeed, otherwise $(n+1)b_j^\infty$ is an integer. On the other hand, $$n+1> (n+1)b_j^\infty=(n+1)(1-{\varepsilon}')\ge n+1-(n+1)/N.$$ where $N\ge N_{d-1}({\overline{{{\mathfrak{R}}}}})+2\ge n+2$. This is impossible. Thus, $B'-\delta B_j\in {{\mathscr{P}}}_n$. Since $-(K_Y+B^\infty-\delta B_j)$ is ample, by Proposition \[prodolj\] the $n$-complement $K_{B_1}+{\operatorname{Diff}}_{B_1}(B')^+$ of $K_{B_1}+{\operatorname{Diff}}_{B_1}(B'-\delta B_j)$ can be extended to $Y$. So there is an $n$-complement $K_Y+B^{+}$ of $K_Y+B^{\infty}-\delta B_j$. Write $B^+=\sum b^+_i B_i$. Since $B-\delta B_j\in {{\mathscr{P}}}_n$, we have $B^{+}\ge B^{\infty}-\delta B_j$. Moreover, since $nb_j^+$ is an integer and $1\gg \delta>0$, we have $b_j^+\ge b_j^\infty$. Hence $B^+\ge B^{\infty}_Y$ and $B^+$ is also an $n$-complement of $K_Y+B^{\infty}$ (see Remark \[rem-complne\]). By Lemma \[open\] $K_Y+B^{+}$ is also an $n$-complement of $K_Y+B^{(m)}$ for $m\gg 0$.
By Lemmas \[lemma-PPP-n-1\] and \[lemma-PPP-n\] we have $B^+\ge B^{(m)}$ for $m\gg 0$. More precisely, $$b_i^+\quad
\begin{cases}
\quad=1& \text{if $b_i^{\infty}\ge 1-{\varepsilon}'$,}
\\
\quad\ge b_i^{(m)}=b_i^{\infty}& \text{if $b_i^{\infty}< 1-{\varepsilon}'$.}
\end{cases}$$ By the construction of $D$ we have $D^{(m)}\le B^+$. Hence $-(K_Y+D^{(m)})$ is nef, a contradiction. This completes the proof of Theorems \[main-result\] and \[main-result0\] in case .
Effective adjunction {#sect-Adj}
====================
In this section we discuss the adjunction conjecture for fibre spaces. This conjecture can be considered as a generalization of the classical Kodaira canonical bundle formula for canonical bundle, see [@Kodaira-1963], [@Fujita-1986ZE], [@Kawamata-1997-Adj], [@Kawamata-1998], [@Ambro-PhD], [@Fujino-1999app], [@Fujino-Mori-2000], [@Fujino-2003-AG], [@Ambro-2004S], [@Ambro-2005].
The set-up {#not-adj}
----------
Let $f\colon X\to Z$ be a surjective morphism of normal varieties and let $D=\sum d_iD_i$ be an ${{\mathbb R}}$-divisor on $X$ such that $(X,D)$ is lc near the generic fibre of $f$ and $K+D$ is ${{\mathbb R}}$-Cartier over the generic point of any prime divisor $W\subset Z$. In particular, $d_i\le 1$ whenever $f(D_i)=Z$. Let $d:=\dim X$ and $d':=\dim Z$.
For any divisor $F=\sum \alpha_i F_i$ on $X$, we decompose $F$ as $F=F^{\mathrm{h}}+F^{\mathrm{v}}$, where $$F^{\mathrm{h}}:=\sum\limits_{f(F_i)=Z} \alpha_i F_i, \qquad
F^{\mathrm{v}}:=\sum\limits_{f(F_i)\neq Z} \alpha_i F_i.$$ These divisors $F^{\mathrm{h}}$ and $F^{\mathrm{v}}$ are called the *horizontal* and *vertical* parts of $F$, respectively.
Construction {#def-cW}
------------
For a prime divisor $W\subset Z$, define a real number $c_W$ as the log canonical threshold over the generic point of $W$: $$\label{cW}
c_W:=\sup\left\{c \mid (X,D+cf^\bullet W)\quad \text{is lc over the
generic point of $W$}\right\}.$$ It is clear that $c_W\in{{\mathbb Q}}$ whenever $D$ is a ${{\mathbb Q}}$-divisor. Put $d_W:=1-c_W$. Then the ${{\mathbb R}}$-divisor $$D{_{\operatorname{div}}{}}:=\sum_W d_WW$$ is called the *divisorial part* of adjunction (or *discriminant* of $f$) for $K_X+D$. It is easy to see that $D{_{\operatorname{div}}{}}$ is a divisor, i.e., $d_W$ is zero except for a finite number of prime divisors.
\[rem-adj-2\]
1. Note that the definition of the discriminant $D{_{\operatorname{div}}{}}$ is a codimension one construction, so computing $D{_{\operatorname{div}}{}}$ we can systematically remove codimension two subvarieties in $Z$ and pass to general hyperplane sections $f_H\colon X\cap f^{-1}(H)\to Z\cap H$.
2. Let $h\colon X'\to X$ be a birational contraction and let $D'$ be the crepant pull-back of $D$: $$K_{X'}+D'=h^*(K_X+D),\quad h_*D'=D.$$ Then ${D'}{_{\operatorname{div}}{}}=D{_{\operatorname{div}}{}}$, i.e., the discriminant $D{_{\operatorname{div}}{}}$ does not depend on the choice of crepant birational model of $(X,D)$ over $Z$.
The following lemma is an immediate consequence of the definition.
\[lemma-adj-prelim\] Notation as in [[\[not-adj\]]{}]{}.
1. [(effectivity, cf. [@Shokurov-1992-e 3.2])]{} If $D$ is boundary over the generic point of any prime divisor $W\subset Z$, then $D{_{\operatorname{div}}{}}$ effective.
2. [(semiadditivity, cf. [@Shokurov-1992-e 3.2])]{} Let $\Delta$ be an ${{\mathbb R}}$-divisor on $Z$ and let $D':=D+f^\bullet\Delta$. Then ${D'}{_{\operatorname{div}}{}}=D{_{\operatorname{div}}{}}+\Delta$.
3. $(X,D)$ is klt (resp., lc) over the generic point of $W$ if and only if $d_W< 1$ (resp., $d_W\le 1$).
4. If $(X,D)$ is lc and $D$ is an ${{\mathbb R}}$- (resp., ${{\mathbb Q}}$-)boundary, then $D{_{\operatorname{div}}{}}$ is an ${{\mathbb R}}$- (resp., ${{\mathbb Q}}$-)boundary.
Construction {#construction-adj-def-mod}
------------
From now on assume that $f$ is a contraction, $K_X+D$ is ${{\mathbb R}}$-Cartier, and $K+D{\mathbin{\sim_{\scriptscriptstyle{{{\mathbb R}}}}}} f^* L$ for some ${{\mathbb R}}$-Cartier divisor $L$ on $Z$. Recall that the latter means that there are real numbers $\alpha_j$ and rational functions ${\varphi}_j\in \Bbbk(X)$ such that $$\label{eq-def-dm}
K+D-f^*L = \sum \alpha_j\, ({\varphi}_j).$$ Define the *moduli part* $D{_{\operatorname{mod}}}$ of $K_X+D$ by $$\label{eq-def-dm-3}
D{_{\operatorname{mod}}}:=L-K_Z-D{_{\operatorname{div}}{}}.$$ Then we have $$\label{eq-adj-first-p-R}
K_X+D=f^*(K_Z+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})+\sum \alpha_j\, ({\varphi}_j).$$ In particular, $$K_X+D{\mathbin{\sim_{\scriptscriptstyle{{{\mathbb R}}}}}} f^*(K_Z+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}}).$$ Clearly, $D{_{\operatorname{mod}}}$ depends on the choice of representatives of $K_X$ and $K_Z$, and also on the choice of $\alpha_j$ and ${\varphi}_j$ in . Any change of $K_X$ and $K_Z$ and change of $\alpha_j$ and ${\varphi}_j$ gives a new $D{_{\operatorname{mod}}}$ which differs from the original one modulo ${{\mathbb R}}$-linear equivalence.
If $K+D$ is ${{\mathbb Q}}$-Cartier, the definition of the moduli part is more explicit. By our assumption there is a positive integer $I_0$ such that $I_0(K+D)$ is linearly trivial on the generic fibre. Then for some rational function $\psi\in \Bbbk(X)$, the divisor $M:=I_0(K+D)+(\psi)$ is vertical (and ${{\mathbb Q}}$-linearly trivial over $Z$). Thus, $$M-I_0f^*L=(\psi)+\sum I_0\alpha_j\, ({\varphi}_j),\qquad \alpha_j\in {{\mathbb Q}}.$$ Rewrite it in a more compact form: $M-I_0f^*L=\alpha\, ({\varphi})$, $\alpha\in {{\mathbb Q}}$, ${\varphi}\in \Bbbk(X)$. The function ${\varphi}$ vanishes on the generic fibre, hence it is a pull-back of some function $\upsilon\in \Bbbk(Z)$. Replacing $L$ with $L+\frac{\alpha}{I_0}\, (\upsilon)$ we get $M=I_0f^*L$ and $$\label{eq-def-dm-Q}
K_X+D-f^*L =\frac1{I_0}(\psi),
\qquad \psi\in \Bbbk(X).$$ In other words, $K+D{\mathbin{\sim_{\scriptscriptstyle{I_0}}}} f^*L$. Here $L$ is ${{\mathbb Q}}$-Cartier. Then again we define the moduli part $D{_{\operatorname{mod}}}$ of $K_X+D$ by , where $L$ is taken to satisfy . In this case, $D{_{\operatorname{mod}}}$ is ${{\mathbb Q}}$-Cartier and we have $$\label{eq-adj-first-p-Q}
K_X+D=f^*(K_Z+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})+\frac 1{I_0}\, (\psi).$$ In particular, $$K_X+D{\mathbin{\sim_{\scriptscriptstyle{I_0}}}}f^*(K_Z+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}}).$$ As above, $D{_{\operatorname{mod}}}$ depends on the choice of representatives of $K_X$ and $K_Z$, and also on the choice of $I_0$ and $\psi$ in . Note that $I_0$ depends only on $f$ and the horizontal part of $D$. Once these are fixed, we usually will assume that $I_0$ is a constant. Then any change of $K_X$, $K_Z$, and $\psi$ gives a new $D{_{\operatorname{mod}}}$ which differs from the original one modulo $I_0$-linear equivalence.
By Lemma \[lemma-adj-prelim\] $(D+f^\bullet \Delta){_{\operatorname{mod}}}=D{_{\operatorname{mod}}}$. Roughly speaking this means that “the moduli part depends only on the horizontal part of $D$”.
For convenience of the reader we recall definition of b-divisors and related notions, see [@Iskovskikh-2003-b-div] for details.
Let $X$ be a normal variety. Consider an infinite linear combination ${\mathbf D}:=\sum_P d_P P$, where $d_P\in {{\mathbb R}}$ and $P$ runs through all discrete valuations $P$ of the function field. For any birational model $Y$ of $X$ define the trace of ${\mathbf D}$ on $Y$ as follows ${\mathbf D}_Y:=\sum\limits_{{\operatorname{codim}}_Y P=1} d_P P$. A *b-divisor* is a linear combination ${\mathbf D}=\sum_P d_P P$ such that the trace ${\mathbf D}_Y$ on each birational model $Y$ of $X$ is an ${{\mathbb R}}$-divisor, i.e., only a finite number of multiplicities of ${\mathbf D}_Y$ are non-zero. In other words, a b-divisor is an element of $\underleftarrow{\lim} {\operatorname{Div}}_{{{\mathbb R}}} (Y)$, where $Y$ in the inverse limit runs through all normal birational models $f\colon Y\to X$, ${\operatorname{Div}}_{{{\mathbb R}}} (Y)$ is the group of ${{\mathbb R}}$-divisors of $Y$, and the map ${\operatorname{Div}}_{{{\mathbb R}}} (Y)\to {\operatorname{Div}}_{{{\mathbb R}}} (X)$ is the push-forward. Let $D$ be a ${{\mathbb R}}$-Cartier divisor on $X$. The *Cartier closure* of $D$ is a b-divisor ${\overline{D}}$ whose trace on every birational model $f\colon Y\to X$ is $f^*D$. A b-divisor ${\mathbf D}$ is said to be b-Cartier if there is a model $X'$ and a ${{\mathbb R}}$-Cartier divisor $D'$ on $X'$ such that ${\mathbf D}={\overline{D'}}$. A b-divisor ${\mathbf D}$ is said to be b-nef (resp. b-semiample, b-free) if it is b-Cartier and there is a model $X'$ and a ${{\mathbb R}}$-Cartier divisor $D'$ on $X'$ such that ${\mathbf D}={\overline{D'}}$ and $D'$ is nef (resp. semiample, integral and free).
${{\mathbb Q}}$- and ${\mathbb Z}$-versions of b-divisors are defined similarly.
\[rem-adj-3\] Let $g\colon Z'\to Z$ be a birational contraction. Consider the following diagram $$\label{eq-Diff-sq-diag}
\begin{CD}
X'@>h>> X
\\
@V{f'}VV @V{f}VV
\\
Z'@>{g}>> Z
\end{CD}$$ where $X'$ is a resolution of the dominant component of $X\times _{Z} Z'$. Let $D'$ be the crepant pull-back of $D$ that is $K_{X'}+D'=h^*(K_X+D)$ and $h_*D'=D$. By Remark \[rem-adj-2\] we have $g_*D'{_{\operatorname{div}}{}}=D{_{\operatorname{div}}{}}$. Therefore, the discriminant defines a b-divisor ${\mathbf D}{_{\operatorname{div}}{}}$.
For a suitable choice of $K_X'$, we can write $$h^* (K_X+D)=K_{X'}+D',$$
Now we fix the choice of $K$, $\alpha_j$ and ${\varphi}_j$ in (resp. $K$ and $\psi$ in ) and induce them naturally to $X'$. Then $D{_{\operatorname{mod}}}$ and $D'{_{\operatorname{mod}}}$ are uniquely determined and $g_*D'{_{\operatorname{mod}}}=D{_{\operatorname{mod}}}$. This defines a b-divisor ${\mathbf D}{_{\operatorname{mod}}}$.
We can write $$K_{Z'}+D'{_{\operatorname{div}}{}}+D'= g^*(K_{Z}+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})+E,$$ where $E$ is $g$-exceptional. Since $$h^* f^*(K_Z+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}}) \equiv K_{X'}+D' \equiv
f'^*(K_{Z'}+D'{_{\operatorname{div}}{}}+D'{_{\operatorname{mod}}}),$$ we have $E=0$ (see [@Shokurov-1992-e 1.1]), i.e., $g$ is $(K_Z+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})$-crepant: $$\label{eq-Diff-sq}
K_{Z'}+D'{_{\operatorname{div}}{}}+D'{_{\operatorname{mod}}}= g^* (K_Z+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}}).$$
Let us consider some examples.
Assume that the contraction $f$ is birational. Then by the ramification formula [@Shokurov-1992-e §2], [@Utah Prop. 20.3] and negativity lemma [@Shokurov-1992-e 1.1] we have $D{_{\operatorname{div}}{}}=f_*D$, $K+D=f^*(K_Z+D{_{\operatorname{div}}{}})$, and $D{_{\operatorname{mod}}}=0$.
Let $X=Z\times {{\mathbb P}}^1$ and let $f$ be the natural projection to the first factor. Take very ample divisors $H_1,\dots, H_4$ on $Z$. Let $C$ be a section and let $D_i$ be a general member of the linear system $|f^*H_i+C|$. Put $D:=\frac12\sum D_i$. Then $K_X+D$ is ${{\mathbb Q}}$-linearly trivial over $Z$. By Bertini’s theorem $D+f^*P$ is lc for any point $P\in Z$. Hence $D{_{\operatorname{div}}{}}=0$. On the other hand, $$K_X+D=f^*K_Z-2C+\frac12f^*\sum H_i +2C= f^*\left(K_Z+\frac12\sum
H_i\right).$$ This gives us that $D{_{\operatorname{mod}}}{\mathbin{\sim_{\scriptscriptstyle{{{\mathbb Q}}}}}}\frac12\sum H_i$.
Let $X$ be a hyperelliptic surface. Recall that it is constructed as the quotient $X=(E\times C)/G$ of the product of two elliptic curves by a finite group $G$ acting on $E$ and $C$ so that the action of $G$ on $E$ is fixed point free and the action on $C$ has fixed points. Let $$f\colon X=(E\times C)/G\to {{\mathbb P}}^1=C/G$$ be the projection. It is clear that degenerate fibres of $f$ can be only of type $m\mathrm{I}_0$. Using the classification of such possible actions (see, e.g., [@BPV-1984 Ch. V, Sect. 5]) we obtain the following cases:
[p[85pt]{}p[100pt]{}p[100pt]{}]{} [Type]{}&[singular fibres]{}&\
\
$\mathrm{a)}$($2K_X\sim 0$)& $2\mathrm{I}_0$, $2\mathrm{I}_0$, $2\mathrm{I}_0$, $2\mathrm{I}_0$& $\frac12P_1+\frac12P_2+\frac12P_3+\frac12P_4$\
$\mathrm{b)}$($3K_X\sim 0$)& $3\mathrm{I}_0$, $3\mathrm{I}_0$, $3\mathrm{I}_0$& $\frac23P_1+\frac23P_2+\frac23P_3$\
$\mathrm{c)}$($4K_X\sim
0$)& $2\mathrm{I}_0$, $4\mathrm{I}_0$, $4\mathrm{I}_0$& $\frac12P_1+\frac34P_2+\frac34P_3$\
$\mathrm{d)}$($6K_X\sim
0$)& $2\mathrm{I}_0$, $3\mathrm{I}_0$, $6\mathrm{I}_0$& $\frac12P_1+\frac23P_2+\frac56P_3$\
In all cases the moduli part $D{_{\operatorname{mod}}}$ is trivial.
\[assumpt-adj-\*\] Under the notation of \[not-adj\] and \[construction-adj-def-mod\] assume additionally that $D$ is a ${{\mathbb Q}}$-divisor and there is a ${{\mathbb Q}}$-divisor $\Theta$ on $X$ such that $K_X+\Theta$ is ${{\mathbb Q}}$-linearly trivial over $Z$ and $(F,(1-t)D|_F +t \Theta|_F)$ is a klt log pair for any $0<t\le 1$, where $F$ is the generic fibre of $f$. In particular, $\Theta$ and $D$ are ${{\mathbb Q}}$-boundaries near the generic fibre. In this case, both $D{_{\operatorname{div}}{}}$ and $D{_{\operatorname{mod}}}$ are ${{\mathbb Q}}$-divisors.
The following result is very important.
\[th-Ambro-m\] Notation and assumptions as in [[\[not-adj\]]{}]{} and [[\[construction-adj-def-mod\]]{}]{}. Assume additionally that $D$ is a ${{\mathbb Q}}$-divisor, $D$ is effective near the generic fibre, and $(X,D)$ is klt near the generic fibre. Then we have.
- The b-divisor ${\mathbf K}+{\mathbf D}{_{\operatorname{div}}{}}$ is b-Cartier.
- The b-divisor ${\mathbf D}{_{\operatorname{mod}}}$ is b-nef.
According (ii) of Theorem \[th-Ambro-m\] the b-divisor ${\mathbf D}{_{\operatorname{mod}}}$ is b-nef for $D\ge 0$ and $(X,D)$ is klt near the generic fibre (see also [@Kawamata-1998 Th. 2], [@Fujino-1999app]). We expect more.
\[conj-main-adj\] Let notation and assumptions be as in [[\[not-adj\]]{}]{} and [[\[assumpt-adj-\*\]]{}]{}. We have
[**([.]{})**]{}
\[conj-main-adj-1\] (Log Canonical Adjunction) ${\mathbf D}{_{\operatorname{mod}}}$ is b-semiample.
[**([.]{})**]{}
\[conj-main-adj-21\] (Particular Case of Effective Log Abundance Conjecture) Let $X_\eta$ be the generic fibre of $f$. Then $I_0(K_{X_\eta}+ D_\eta)\sim 0$, where $I_0$ depends only on $\dim X_\eta$ and the multiplicities of $D^{\mathrm{h}}$.
[**([.]{})**]{}
\[conj-main-adj-2\] (Effective Adjunction) ${\mathbf D}{_{\operatorname{mod}}}$ is effectively b-semiample, that is, there exists a positive integer $I$ depending only on the dimension of $X$ and the horizontal multiplicities of $D$ (a finite set of rational numbers) such that $I{\mathbf D}{_{\operatorname{mod}}}$ is very b-semiample, that is, $I{\mathbf D}{_{\operatorname{mod}}}={\overline{M}}$, where $M$ is a base point free divisor on some model $Z'/Z$.
Note that by we may assume that $$\label{eq-adj-(*)}
K+D{\mathbin{\sim_{\scriptscriptstyle{I}}}} f^*(K_Z+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}}).$$
\[rem-Adj\_conj-compl\] We expect that hypothesis in [[\[conj-main-adj\]]{}]{} can be weakened as follows.
1. It is sufficient to assume that $K+D$ is lc near the generic fibre, the horizontal part $D^{{\mathrm{h}}}$ of $D$ is an ${{\mathbb R}}$-boundary, $K+D$ is ${{\mathbb R}}$-Cartier, and $K+D\equiv f^*L$.
2. $D^{{\mathrm{h}}}$ is a ${{\mathbb Q}}$-boundary and $K+D\equiv 0$ near the generic fibre.
3. $D^{{\mathrm{h}}}$ is a ${{\mathbb Q}}$-boundary, $K+D$ is ${{\mathbb R}}$-Cartier, and $K+D\equiv f^*L$.
This however is not needed for the proof of the main theorem.
In the notation of we have $K_{X_\eta}+D_\eta{\mathbin{\sim_{\scriptscriptstyle{{{\mathbb Q}}}}}} 0$, where $X_\eta$ is the generic fibre of $f$. Assume that
1. $X$ is FT, and
2. LMMP and conjectures [[\[BAB\]]{}]{} and [[\[conj-main-adj\]]{}]{} hold in dimensions $\le \dim X-\dim Z$.
Then the pair $(X_\eta, D_\eta)$ satisfies the assumptions of Theorem [[\[main-result0\]]{}]{} with ${{\mathfrak{R}}}$ depending only on horizontal multiplicities of $D$. Hence $I_0(K_{X_\eta}+ D_\eta)\sim 0$, where $I_0$ depends only on $\dim X_\eta$ and horizontal multiplicities of $D$. Thus holds automatically under additional assumptions (i)-(ii).
\[ex-elliptic\] Let $f\colon X\to Z$ be a fibration satisfying [[\[construction-adj-def-mod\]]{}]{} whose generic fibre is an elliptic curve. Then $D^{\mathrm{h}}=0$, $D=D^{\mathrm{v}}$, and $I_0=1$. Thus we can write $K_X+D=f^*L$. The $j$-invariant defines a rational map $J\colon Z \dashrightarrow {{\mathbb C}}$. By blowing up $Z$ and $X$ we may assume that both $X$ and $Z$ are smooth and $J$ is a morphism: $J\colon Z\to {{\mathbb P}}^1$. Let $P$ be a divisor of degree $1$ on ${{\mathbb P}}^1$. Take a positive integer $n$ such that $12n$ is divisible by the multiplicities of all the degenerate fibres of $f$. In this situation, there is a generalization of the classical Kodaira formula [@Fujita-1986ZE]: $$12n(K_X+D)=f^*\left(12nK_Z+12nD{_{\operatorname{div}}{}}+nJ^*P\right),$$ We can rewrite it as follows $$\label{eq-2}
K_X+D= f^*\left(K_Z+D{_{\operatorname{div}}{}}+\frac1{12}J^*P\right).$$ Here $D{_{\operatorname{mod}}}=\frac1{12}J^*P$ is semiample and the multiplicities of $D{_{\operatorname{div}}{}}$ are taken from the table in Example [[\[ex-Kodaira\]]{}]{} if $D=0$ over such divisors in $Z$ or $D$ is minimal as in Lemma \[lemma-adj-minimal\]. (Otherwise to compute $D{_{\operatorname{div}}{}}$ we can use semiadditivity Lemma \[lemma-adj-prelim\], (ii).)
Fix a positive integer $m$. Let $(E,0)$ be an elliptic curve with fixed group low and let $e_m\in E$ be an $m$-torsion. Define the action of ${{\boldsymbol\mu}}_m:=\left\{ \sqrt[m]{1}\right\}$ on $E\times {{\mathbb P}}^1$ by $${\varepsilon}(e,z)=(e+e_m,{\varepsilon}z),\qquad e\in E,\ z\in {{\mathbb P}}^1,$$ where ${\varepsilon}\in {{\boldsymbol\mu}}_m$ is a primitive $m$-root. The quotient map $$X:=(E\times {{\mathbb P}}^1)/{{\boldsymbol\mu}}_m\longrightarrow {{\mathbb P}}^1/{{\boldsymbol\mu}}_m\simeq{{\mathbb P}}^1$$ is an elliptic fibration having exactly two fibres of types $m
\mathrm{I}_0$ over points $0$ and $\infty\in {{\mathbb P}}^1$. Using the Kodaira formula one can show that $$K_X=f^*K_Z+(m-1)F_0+(m-1)F_{\infty},$$ where $F_0:=f^{-1}(0)_{{\operatorname{red}}}$ and $F_{\infty}:=f^{-1}(\infty)_{{\operatorname{red}}}$. Hence, in we have $I_0=1$ and $$L=K_Z+\left(1-\frac1m\right)\cdot 0+ \left(1-\frac1m\right)\cdot
\infty.$$ Clearly, $$D{_{\operatorname{div}}{}}=\left(1-\frac1m\right)\cdot 0+ \left(1-\frac1m\right)\cdot
\infty.$$ Hence $D{_{\operatorname{mod}}}=0$, $I=I_0=1$, and $K_X=f^*(K_Z+D{_{\operatorname{div}}{}})$.
\[cor\_Inv\_aDj\] Let notation and assumptions be as in [[\[not-adj\]]{}]{}, [[\[construction-adj-def-mod\]]{}]{}, and [[\[assumpt-adj-\*\]]{}]{} (cf. [[\[rem-Adj\_conj-compl\]]{}]{}).
1. If $(Z,D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})$ is lc and $\mathbf D{_{\operatorname{mod}}}$ is effective, then $(X,D)$ is lc.
2. Assume that holds. If $(X,D)$ is lc, then so is $(Z,D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})$ for a suitable choice of $D{_{\operatorname{mod}}}$ in the class of ${{\mathbb Q}}$-linear equivalence (respectively $I$-linear equivalence under ). Moreover, if $(X,D)$ is lc and any lc centre of $(X,D)$ dominates $Z$, then $(Z,D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})$ is klt.
For a log resolution $g\colon Z'\to Z$ of the pair $(Z, D{_{\operatorname{div}}{}})$, consider base change . Thus ${\operatorname{Supp}}D'{_{\operatorname{div}}{}}$ is a simple normal crossing divisor on $Z'$.
\(i) Put $D_t:=(1-t)D +t \Theta$ (see [[\[assumpt-adj-\*\]]{}]{}). Assume that $(X,D)$ is not lc. Then $(X,D_t)$ is also not lc for some $0<t\ll 1$. Let $F$ be a divisor of discrepancy $a(F,X,D_t)<-1$. Since $(X,D_t)$ is klt near the generic fibre, the centre of $F$ on $Z$ is a proper subvariety. By Theorem \[th-Ambro-m\] we can take $g$ so that ${\mathbf K}+(\mathbf D_t){_{\operatorname{div}}{}}={\overline{K_{Z'}+(D'_t){_{\operatorname{div}}{}}}}$ and $(\mathbf D_t){_{\operatorname{mod}}}={\overline{(D'_t){_{\operatorname{mod}}}}}$. Moreover, by [@Kollar-1996-RC Ch. VI, Th. 1.3] we can also take $g$ so that the centre of $F$ on $Z'$ is a prime divisor, say $W$. Put $(D_t)_Z:=(D_t){_{\operatorname{div}}{}}+(D_t){_{\operatorname{mod}}}$. By we have $$-1\le a(W,Z,(D_t)_Z)=a(W,Z', (D'_t){_{\operatorname{div}}{}}+(D'_t){_{\operatorname{mod}}})\le a(W,Z',(D'_t){_{\operatorname{div}}{}}).$$ Therefore $(X',D'_t)$ is lc over the generic point of $W$ (see ). In particular, $a(F,X',D'_t)=a(F,X,D_t)\ge -1$, a contradiction.
\(ii) By our assumption $\mathbf D{_{\operatorname{mod}}}$ is b-Cartier, so we can take $g$ so that $\mathbf D{_{\operatorname{mod}}}={\overline{D'{_{\operatorname{mod}}}}}$ and ${\mathbf K}+\mathbf D{_{\operatorname{div}}{}}={\overline{K_{Z'}+D'{_{\operatorname{div}}{}}}}$. Moreover by (respectively by ) we can take $g$ so that ${D'{_{\operatorname{mod}}}}$ (respectively ${ID'{_{\operatorname{mod}}}}$) is semiample (respectively linearly free). By $g$ is $(K+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})$-crepant. If $(Z',D'{_{\operatorname{div}}{}})$ is lc (resp. klt), then replacing ${D'{_{\operatorname{mod}}}}$ with an effective general representative of the corresponding class of ${{\mathbb Q}}$-linear equivalence we obtain $${\operatorname{discr}}(Z,D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})={\operatorname{discr}}(Z',D'{_{\operatorname{div}}{}}+D'{_{\operatorname{mod}}})={\operatorname{discr}}(Z',D'{_{\operatorname{div}}{}})
\ge -1.$$ (resp. $>-1$). We can suppose also that ${\left\lfloor D{_{\operatorname{mod}}}\right\rfloor}=0$. Hence $(Z,D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})$ is lc (resp. klt) in this case. Thus we assume that $(Z',D'{_{\operatorname{div}}{}})$ is not lc (resp. not klt). Let $E$ be a divisor over $Z$ of discrepancy $a(E,Z',D'{_{\operatorname{div}}{}})\le -1$. Clearly, we may assume that ${\operatorname{Center}}_{Z'} E \not\subset {\operatorname{Supp}}D'{_{\operatorname{mod}}}$. Then $a(E,Z',D'{_{\operatorname{div}}{}}+D'{_{\operatorname{mod}}})=a(E,Z',D'{_{\operatorname{div}}{}})\le -1$. Replacing $Z'$ with its blowup we may assume that $E$ is a prime divisor on $Z'$ (and again ${\operatorname{Center}}_{Z'} E \not\subset {\operatorname{Supp}}D'{_{\operatorname{mod}}}$). Since $(X',D')$ is lc and by , $c_E=0$, $d_E=1$, and $a(E,Z',D'{_{\operatorname{div}}{}})=-1$. Then $(Z',D'{_{\operatorname{div}}{}})$ is lc. Furthermore, by the pair $(X', D'+cf^{\prime \bullet}E)$ is not lc for any $c>0$. This means that $f^{-1}({\operatorname{Center}}_Z (E))$ contains an lc centre.
The following example shows that the condition $\mathbf D{_{\operatorname{mod}}}\ge 0$ in (i) of Corollary \[cor\_Inv\_aDj\] cannot be omitted.
\[ex-Inv-Adj-contr\] Let $f\colon X\to Z={{\mathbb C}}^2$ be a standard conic bundle given by $x^2+uy^2+vz^2$ in ${{\mathbb P}}^2_{x,y,z}\times {{\mathbb C}}^2_{u,v}$. The linear system $ |-n K_X|$ is base point free for $n\ge 1$. Let $H\in |-2 K_X|$ be a general member. Now let $\Gamma(t):= \Gamma_1+t\Gamma_2$, where $\Gamma_1:=\{u=0\}$ and $\Gamma_2:=\{v=0\}$. Put $D(t):=\frac12 H+f^*\Gamma(t)$. Then $2(K+D)= 2f^*\Gamma(t)$ and $D(t){_{\operatorname{div}}{}}=\Gamma(t)$. Since $K_Z=0$, we have $D{_{\operatorname{mod}}}=0$.
For $t=1$, the log divisor $K_Z+\Gamma(t)$ is lc but $K+D(t)= f^*(K_Z+\Gamma(t))$ is not. Indeed, in the chart $z\neq
0$ there is an isomorphism $$\label{ex-m-cp}
(X,f^*\Gamma)\simeq ({{\mathbb C}}^3_{x,y,u}, \{u(x^2+uy^2)=0\}).$$
The explanation of this fact is that the b-divisor ${\mathbf D}{_{\operatorname{mod}}}$ is non-trivial. To show this we consider the following diagram [@Sarkisov-1980-re §2]: $$\xymatrix{
X'\ar@{-->}[r]^{\chi}\ar[dd]^{f'}&\tilde X\ar[dr]^{h}&
\\
&&X\ar[d]^{f}
\\
Z'\ar[rr]^{g}&&Z
}$$ where $h$ is the blowup the central fibre $f^{-1}(0)_{{\operatorname{red}}}$, $\chi$ is the simplest flop, $g$ is the blowup of $0$, and $f'$ is again a standard conic bundle. Put $t=1/2$ and let $\tilde D$ and $D'$ be the crepant pull-backs of $D:=D(t)$ on $\tilde X$ and $X'$, respectively. The $h$-exceptional divisor $F$ appears in $\tilde D$ with multiplicity $1/2$. Let $F'$ be the proper transform of $F$ on $X'$. Then $F'=f'^* E$, where $E$ is the $g$-exceptional divisor. It is easy to see from that the pair $(X,D)$ is lc but not klt at the generic point of $f^{-1}(0)_{{\operatorname{red}}}$. So is $(\tilde X, \tilde D)$ at the generic point of the flopping curve. This implies that $(X', D')$ is lc but not klt over the generic point of $E$. Therefore, $D'{_{\operatorname{div}}{}}=E+\Gamma'$, where $\Gamma'$ is the proper transform of $\Gamma$. On $Z'$, we have $K_{Z'}=E$ and $K+D'=f'^*g^*\Gamma$, so $D'{_{\operatorname{mod}}}=g^*\Gamma-E-D'{_{\operatorname{div}}{}}=-\frac12 E$. Thus $D'{_{\operatorname{mod}}}\le 0$ and $2D'{_{\operatorname{mod}}}$ is free.
Two important particular cases of Effective Adjunction {#section-Adj-conj-part-cases}
======================================================
Using a construction and a result of [@Kawamata-1997-Adj] we prove the following.
\[th-n-n-1\] Conjectures [[\[conj-main-adj\]]{}]{} hold if $\dim X=\dim Z+1$.
We expect that in this case one can take $I=12q$, where $q$ is a positive integer such that $qD^{{\mathrm{h}}}$ is an integral divisor.
We may assume that a general fibre of $f$ is a rational curve (see Example \[ex-elliptic\]). Thus the horizontal part $D^{\mathrm{h}}$ of $D$ is non-trivial. First we reduce the problem to the case when all components of $D^{\mathrm{h}}$ are generically sections. Write $D=\sum d_iD_i$ and take $$\delta:= \min \{ d_i \mid \text{$D_i$ is horizontal and $d_i>0$}\}.$$ (we allow components with $d_i=0$). Let $D_i$ be a horizontal component and let $D_i\to \hat Z\stackrel{g}{\longrightarrow} Z$ be the Stein factorization of the restriction $f|_{D_i}$. Let $n_i:=\deg g$. Let $l$ be a general fibre of $f$. Since $d_iD_i\cdot l\le D\cdot l=-K\cdot l=2$, we have $$\label{eq-Adj-3-2-n-i}
n_i= D_i\cdot l\le 2/d_i\le 2/\delta.$$ Assume that $n_i>1$. Consider the base change $$\begin{CD}
\hat X@>{h}>>X
\\
@V{\hat f}VV@V{f}VV
\\
\hat Z@>{g}>>Z
\end{CD}$$ where $\hat X$ is the normalization of the dominant component of $X\times_{Z} \hat Z$. Define $\hat D$ on $\hat X$ by $$\label{eq-3-1-crepant-pb}
K_{\hat X}+\hat D=h^*(K_X+D).$$ More precisely, $\hat D=\sum_{i,j} \hat d_{i,j}\hat D_{i,j}$, where $h(\hat D_{i,j})=D_i$, $1-\hat d_{i,j}=r_{i,j}(1-d_i)$, and $r_{i,j}$ is the ramification index along $\hat D_{i,j}$. By construction, the ramification locus $\Lambda$ of $h$ is $\hat f$-exceptional, that is $\hat f (\Lambda)\neq \hat Z$. Therefore, $\hat D$ is a boundary near the generic fibre. Similarly, we define $\hat \Theta$ as the crepant pull-back of $\Theta$ from \[assumpt-adj-\*\]. Thus the pair $(\hat X,\hat D)$ satisfies assumptions of [[\[not-adj\]]{}]{} and [[\[assumpt-adj-\*\]]{}]{}. It follows from that $$K_{\hat X}+\hat D=\hat f^* g^*(K_Z+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}}).$$ According to [@Ambro-PhD Th. 3.2] for the discriminant $\hat D{_{\operatorname{div}}{}}$ of $\hat f$ we have $$K_{\hat Z}+\hat D{_{\operatorname{div}}{}}=g^*(K_Z+D{_{\operatorname{div}}{}}).$$ For a suitable choice of $\hat D{_{\operatorname{mod}}}$ in the class of $n_i$-linear equivalence, we can write $$K_{\hat Z}+\hat D{_{\operatorname{div}}{}}+\hat D{_{\operatorname{mod}}}=g^*(K_Z+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}}).$$ Therefore, $\hat D{_{\operatorname{mod}}}=g^* D{_{\operatorname{mod}}}$. If $\hat I \hat D{_{\operatorname{mod}}}$ is free for some positive integer $\hat I$, then so is $n_i \hat I D{_{\operatorname{mod}}}$. Thus we have proved the following.
\[claim-Adj-3-2-a\] Assume that Conjecture [[\[conj-main-adj\]]{}]{} holds for $\hat f\colon \hat X\to \hat Z$ with constant $\hat I$. Then this conjecture holds for $f\colon X\to Z$ with $I:=n_i \hat I$.
Note that the restriction $\hat f|_{h^{-1} (D_i)}\colon h^{-1} (D_i)\to \hat Z$ is generically finite of degree $n_i$. Moreover, $h^{-1} (D_i)$ has a component which is a section over the generic point. Applying Claim \[claim-Adj-3-2-a\] several times and taking into account we obtain the desired reduction to the case when all the horizontal $D_i$’s with $d_i>0$ are generically sections.
{#subs-Adj-conic-sect}
Further by making a birational base change and by blowing up $X$ we can get the situation when
1. $Z$ and $X$ are smooth,
2. the $D_i$’s are regular disjointed sections,
3. the morphism $f$ is smooth outside of a simple normal crossing divisor $\Xi\subset Z$,
4. $f^{-1}(\Xi)\cup {\operatorname{Supp}}D$ is also a simple normal crossing divisor.
Let $n$ be the number of horizontal components of $D$. Note that we allow sections with multiplicities $d_i=0$ on this step.
Let $\mathcal M_n$ be the moduli space of $n$-pointed stable rational curves, let $f_n\colon \mathcal U_n\to \mathcal M_n$ be the corresponding universal family, and let $\mathcal P_1,\dots,\mathcal P_n$ be sections of $f_n$ which correspond to the marked points (see [@Knudsen-1983-2]). It is known that both $\mathcal M_n$ and $\mathcal U_n$ are smooth and projective. Take $d_i\in [0,\, 1]$ so that $\sum d_i=2$ and put $\mathcal D:=\sum d_i\mathcal P_i$. Then $K_{\mathcal U_n}+\mathcal D$ is trivial on the general fibre. However, $K_{\mathcal U_n}+\mathcal D$ is not numerically trivial everywhere over $\mathcal M_n$, moreover, it is not nef everywhere over $\mathcal M_n$:
\[th-Knud\]
1. There exist a smooth projective variety $\bar {\mathcal{U}}_n$, a ${{\mathbb P}}^1$-bundle $\bar f_n\colon \bar {\mathcal{U}}_n\to \mathcal M_n$, and a sequence of blowups (blowdowns) with smooth centres $$\sigma\colon {\mathcal{U}}_n= \mathcal{U}^{1}\to \mathcal{U}^{2}\to \cdots\to
\mathcal{U}^{n-2} = \bar {\mathcal{U}}_n.$$
2. For $\bar{\mathcal D}:=\sigma_*\mathcal D$, the (discrepancy) divisor $$\mathcal F:=K_{{\mathcal{U}}_n}+\mathcal D-\sigma^*(K_{\bar {\mathcal{U}}_n}+\bar {\mathcal D})$$ is effective and *essentially exceptional* on $\mathcal M_n$.
3. There exists a semiample ${{\mathbb Q}}$-divisor $\mathcal L$ on $\mathcal M_n$ such that $$K_{\bar {\mathcal{U}}_n}+\bar {\mathcal D}=
\bar f_n^*(K_{\mathcal M_n}+\mathcal L).$$ Therefore, $$K_{{\mathcal{U}}_n}+\mathcal D -\mathcal F=
f_n^*(K_{\mathcal M_n}+\mathcal L).$$
Recall that for any contraction ${\varphi}\colon Y\to Y'$, a divisor $G$ on $Y$ is said to be *essentially exceptional* over $Y'$ if for any prime divisor $P$ on $Y'$, the support of the divisorial pull-back ${\varphi}^\bullet P$ is not contained in ${\operatorname{Supp}}G$.
In the above notation we have $$(\mathcal D-\mathcal F){_{\operatorname{div}}{}}=0,\qquad
(\mathcal D-\mathcal F){_{\operatorname{mod}}}=\mathcal L.$$
Moreover, the proof of Theorem \[th-n-n-1\] implies that the b-divisor $\mathbf G$, the b-divisor of the moduli part of $\mathcal D-\mathcal F$, stabilizes on $\mathcal M_n$, that is, $\mathbf G={\overline{(\mathcal D-\mathcal F){_{\operatorname{mod}}}}}$.
See Example \[exam-mod-mod\] below.
Since the horizontal components of $D$ are sections, $(X/Z,D^{{\mathrm{h}}})$ is generically an $n$-pointed stable curve [@Knudsen-1983-2]. Hence we have the induced rational maps $$\xymatrix{
X\ar@{-->}[r]^{\beta}\ar[d]^{f}&\mathcal U_n\ar[d]^{f_n}
\\
Z\ar@{-->}[r]^{\phi}&\mathcal M_n
}$$ so that $f_n{\mathrel{\scriptstyle{\circ}}}\beta=\phi {\mathrel{\scriptstyle{\circ}}}f$ and $\beta (D_i)\subset \mathcal P_i$. Let $\Xi\subset Z$ as above (see \[subs-Adj-conic-sect\], (iii)). Thus $f$ is a smooth morphism over $Z\setminus \Xi$. Replacing $X$ and $Z$ with its birational models and $D$ with its crepant pull-back we may assume additionally to \[subs-Adj-conic-sect\] that $\beta$ and $\phi$ are regular morphisms. Now take the $\mathcal D=\sum d_i\mathcal P_i$ so that it corresponds to the horizontal part $D^{{\mathrm{h}}}=\sum_{f(D_i)=Z} d_i D_i$. Consider the following commutative diagram $$\xymatrix{
X\ar[r]^{\mu}\ar@/^2pc/[rrr]^{\beta}
\ar[ddr]^{f}
&\hat X\ar[rr]^{\psi}\ar[dd]^{\hat f}&&{\mathcal U}_n
\ar[dd]^{f_n}\ar@/^0.3pc/[dr]^{\sigma}
\\
&&&&\bar {{\mathcal U}_n}\ar@/^0.3pc/[dl]^{\bar f_n}
\\
&Z\ar[rr]^{\phi}&&{\mathcal M}_n
}$$ where $\hat X:= Z\times_{\mathcal M_n} {\mathcal U}_n$.
\[Adj-conic-hat-plt\] Since the fibres of $f_n$ are stable curves, near every point $u\in \mathcal U_n$ the morphism $f_n$ is either smooth or in a suitable local analytic coordinates is given by $$(u_1,\, u_2,\dots,u_{n-2}) \longmapsto
(u_1u_2,\, u_2,\dots,u_{n-2}).$$ Then easy local computations show that $\hat X$ is normal and has only canonical singularities [@Kawamata-1997-Adj]. Moreover, the pair $(\hat X,\hat D^{{\mathrm{h}}}=\psi^*\mathcal D)$ is canonical because $f_n$ is a smooth morphism near ${\operatorname{Supp}}\mathcal D$.
We have $$K_{X}+D= f^*(K_{Z}+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}}).$$ Put $\hat D:=\mu_*D$. Then $K_{X}+D=\mu^*(K_{\hat X}+\hat D)$, so $$\hat D{_{\operatorname{div}}{}}=D{_{\operatorname{div}}{}}, \qquad \hat D{_{\operatorname{mod}}}=D{_{\operatorname{mod}}},$$ $$K_{\hat X}+\hat D= \hat f^*(K_{Z}+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}}).$$
Let ${\varphi}\colon Y\to Y'$ be any contraction, where $\dim Y'\ge 1$. We introduce $G^{\bot}= G- {\varphi}^\bullet G_{\neg}$, where $G_{\neg}$ is taken so that the vertical part $(G^{\bot})^{\mathrm{v}}$ of $G^{\bot}$ is essentially exceptional and $G_{\neg}$ is maximal with this property. In particular, $(G^{\bot})^{\mathrm{v}}\le 0$ over an open subset $U'\subset Y'$ such that ${\operatorname{codim}}(Y'\setminus U')\ge 2$. Note that our construction of $G_{\neg}$ and $G^{\bot}$ is in codimension one over $Y'$, i.e., to find $G_{\neg}$ and $G^{\bot}$ we may replace $Y'$ with $Y'\setminus W$, where $W$ is a closed subset of codimension $\ge 2$.
\[lemma-adj-minimal\] Let ${\varphi}\colon Y\to Y'$ be a contraction and let $G$ be an ${{\mathbb R}}$-divisor on $Y$. Assume that $\dim Y'\ge 1$. Assume that $(Y/Y',G)$ satisfies conditions [[\[not-adj\]]{}]{}. The following are equivalent:
1. $G^{{\mathrm{v}}}-{\varphi}^\bullet G{_{\operatorname{div}}{}}$ is essentially exceptional,
2. $G{_{\operatorname{div}}{}}=G_\neg$,
3. $(G^\bot){_{\operatorname{div}}{}}=0$.
Implications (ii) $\Longleftrightarrow$ (iii) $\Longrightarrow$ (i) follows by definition of $G{_{\operatorname{div}}{}}$ and semiadditivity (Lemma \[lemma-adj-prelim\]). Let us prove (i) $\Longrightarrow$ (ii). Assume that $G^{{\mathrm{v}}}-{\varphi}^\bullet G{_{\operatorname{div}}{}}$ is essentially exceptional. Then by definition $G{_{\operatorname{div}}{}}\le G_\neg$. On the other hand, for any prime divisor $P\subset Y'$, the multiplicity of $G^{\bot}$ along some component of ${\varphi}^\bullet P$ is equal to $0$. Hence the log canonical threshold of $(K+G^\bot, {\varphi}^\bullet P)$ over the generic point of $P$ is $\le 1$. So by definition of the divisorial part and Lemma \[lemma-adj-prelim\] we have $0\le (G^{\bot}){_{\operatorname{div}}{}}=G{_{\operatorname{div}}{}}-G_\neg$.
\[exam-mod-mod\] Clearly, for $f_n\colon \mathcal U_n \to \mathcal M_n$, the discrepancy divisor $\mathcal F$ is essentially exceptional. Hence, $(\mathcal D-\mathcal F){_{\operatorname{div}}{}}\ge 0$. On the other hand, by construction every fibre of $f_n$ is reduced. Hence, for every prime divisor $W\subset Z$, the divisorial pull-back $f_n^\bullet W$ is reduced and ${\operatorname{Supp}}(f_n^\bullet W + \mathcal D)$ is a simple normal crossing divisor over the generic point of $W$. This implies that $c_W\ge 1$ and so $(\mathcal D-\mathcal F){_{\operatorname{div}}{}}=0$.
It is sufficient to show that $D{_{\operatorname{mod}}}=\phi^*\mathcal L=\phi^*(\mathcal D-\mathcal F){_{\operatorname{mod}}}$ (we replace $Z$ with its blowup if necessary). Then the b-divisor $\mathbf D{_{\operatorname{mod}}}$ automatically stabilizes on $Z$, i.e., $\mathbf D{_{\operatorname{mod}}}={\overline{D{_{\operatorname{mod}}}}}$. In this situation $D{_{\operatorname{mod}}}$ is effectively semiample because $N\mathcal L$ is an integral base point free divisor for some $N$ which depends only on $n$. Since and $\phi$ is a regular morphism, to show $D{_{\operatorname{mod}}}=\phi^*\mathcal L$ we will freely replace $Z$ with an open subset $U\subset Z$ such that ${\operatorname{codim}}(Z\setminus U)\ge 2$. Thus all the statements below are valid over codimension one over $Z$. In particular, we may assume that $D{_{\operatorname{mod}}}=(D^\bot){_{\operatorname{mod}}}$. Replacing $D$ with $D^\bot$ we may assume that $D_\neg=0$ (we replace $Z$ with $U$ as above). Thus $D^{\mathrm{v}}\le 0$ and $D^{\mathrm{v}}$ is essentially exceptional. In particular, $D{_{\operatorname{div}}{}}\ge 0$.
On the other hand, by construction the fibres $(\hat f^*(z),\hat D^{{\mathrm{h}}}=\psi^*\mathcal D)$, $z\in Z$ are stable (reduced) curves. In particular, they are slc (semi log canonical [@Kollar-ShB-1988 §4], [@Utah Ch. 12]). By the inversion of adjunction [@Shokurov-1992-e §3], [@Utah Ch. 16-17] for every prime divisor $W\subset Z$ and generic hyperplane sections $H_1,\dots,H_{\dim Z-1}$ the pair $(\hat X, \hat D^{{\mathrm{h}}}+
\hat f^\bullet W+\hat f^\bullet H_1+\cdots+\hat f^\bullet H_{\dim Z-1})$ is lc. Since $\hat D^{{\mathrm{v}}}\le 0$, so is the pair $(\hat X, \hat D+\hat f^\bullet W)$. This implies that $c_W\ge 1$ and so $D{_{\operatorname{div}}{}}=\hat D{_{\operatorname{div}}{}}=0$.
We claim that $\hat D^{{\mathrm{v}}}=\mu_*D^{{\mathrm{v}}}$ is essentially exceptional. Indeed, otherwise $\hat D^{{\mathrm{v}}}$ is strictly negative over the generic point of some prime divisor $W\subset Z$, i.e., $\mu$ contracts all the components $E_i$ of $f^\bullet W$ of multiplicity $0$. By \[Adj-conic-hat-plt\] the pair $(\hat X,\hat D+{\varepsilon}\hat f^\bullet W)$ is canonical over the generic point of $W$ for some small positive ${\varepsilon}$. On the other hand, for the discrepancy of $E_i$ we have $a(E_i,\hat X,\hat D+{\varepsilon}\hat f^\bullet W)=
a(E_i,X,D+{\varepsilon}f^\bullet W)=-{\varepsilon}$. The contradiction proves our claim.
For relative canonical divisors we have $$K_{\hat X/Z}=\psi^*K_{{\mathcal U}_n/{\mathcal M}_n}$$ (see, e.g., [@Hartshorn-1977-ag Ch. II, Prop. 8.10]). Taking $\hat D^{{\mathrm{h}}}=\psi^*\mathcal D$ into account we obtain $$K_{\hat X/Z}+\hat D^{{\mathrm{h}}}- \psi^*\mathcal F=\psi^*
(K_{{\mathcal U}_n/{\mathcal M}_n}+\mathcal D-\mathcal F)
=\psi^* f_n^* \mathcal L=\hat f^* \phi^* \mathcal L.$$ Hence, $$-\hat D^{{\mathrm{v}}}-\psi^*\mathcal F{\mathbin{\sim_{\scriptscriptstyle{{{\mathbb R}}}}}}
K_{\hat X}+\hat D^{{\mathrm{h}}}-\psi^*\mathcal F
{\mathbin{\sim_{\scriptscriptstyle{{{\mathbb R}}}}}} \hat f^* \phi^* \mathcal L+\hat f^*K_Z$$ over $Z$, i.e., $\hat D^{{\mathrm{v}}}+\psi^*\mathcal F$ is ${{\mathbb R}}$-linearly trivial over $Z$.
Since $\psi^*\mathcal F$ is also essentially exceptional over $Z$, by Lemma \[11a\] below we have $\hat D^{{\mathrm{v}}}= - \psi^*\mathcal F$ and $$\hat f^*D{_{\operatorname{mod}}}=\hat f^*(D{_{\operatorname{mod}}}+D{_{\operatorname{div}}{}})=K_{\hat X/Z}+\hat D=\hat f^* \phi^* \mathcal L.$$ This gives us $D{_{\operatorname{mod}}}=\phi^*\mathcal L=\phi^*(\mathcal D-\mathcal F){_{\operatorname{mod}}}$. Therefore $D{_{\operatorname{mod}}}$ is effectively semiample. This proves Theorem \[th-n-n-1\].
\[11a\] Let ${\varphi}\colon Y\to Y'$ be a contraction with $\dim Y'\ge 1$ and let $A$, $B$ be essentially exceptional over $Y'$ divisors on $Y$ such that $A\equiv B$ over $Y'$ and $A,\, B\le 0$ (both conditions are over codimension one over $Y'$). Then $A=B$ over codimension one over $Y'$.
The statement is well-known in the birational case (see [@Shokurov-1992-e §1.1]), so we assume that $\dim Y'<\dim Y$. As in [@Prokhorov-2003d-e Lemma 1.6], replacing $Y'$ with its general hyperplane section $H'\subset Y'$ and $Y$ with ${\varphi}^{-1} (H')$ we may assume that $\dim {\varphi}({\operatorname{Supp}}A) =0$ and $\dim {\varphi}({\operatorname{Supp}}B) \ge 0$. The essential exceptionality of $A$ and $B$ is preserved.
We may also assume that $Y'$ is a sufficiently small affine neighbourhood of some fixed point $o\in Y'$ (and ${\varphi}({\operatorname{Supp}}A) =o$). Further, all the conditions of lemma are preserved if we replace $Y$ with its general hyperplane section $H$. If $\dim Y' > 1$, then we can reduce our situation to the case $\dim Y=\dim Y'$. Then the statement of the lemma follows by [@Shokurov-1992-e §1.1] and from the existence of the Stein factorization. Finally, consider the case $\dim Y'=1$ (here we may assume that $\dim Y=2$ and ${\varphi}$ has connected fibres). By the Zariski lemma $A=B+a{\varphi}^*o$ for some $a\in {{\mathbb Q}}$. Since $A$ and $B$ are essentially exceptional and $\le 0$, $a=0$.
Assume that all the components $D_1,\dots,D_r$ of $D^{{\mathrm{h}}}$ are sections. If $r=3$, then since $\mathcal M_3$ is a point, we have $D{_{\operatorname{mod}}}=0$. For $r=4$ the situation is more complicated: $\mathcal M_4\simeq{{\mathbb P}}^1$,$\mathcal U_4$ is a del Pezzo surface of degree $5$, and $f_4\colon \mathcal U_4\to \mathcal M_4={{\mathbb P}}^1$ is a conic bundle with three degenerate fibres. Each component of degenerate fibre meets exactly two components of $\mathcal D$. Hence $\bar {\mathcal D}$ is a normal crossing divisor. It is easy to see that $\sigma$ contracts a component of a degenerate fibre which meets $\mathcal D_i$ and $\mathcal D_j$ with $d_i+d_j\le 1$. Clearly, $\bar {\mathcal U}_4\simeq \mathbb F_e$ is a rational ruled surface, $e= 0$ or $1$. We can write $\bar {\mathcal D}_i\sim \Sigma+a_i F$, where $\Sigma$ is the minimal section and $F$ is a fibre of $\bar {\mathcal U}_4=\mathbb F_e\to {{\mathbb P}}^1$. Up to permutation we may assume that $\bar {\mathcal D}_i\neq \Sigma$ for $i=2,3,4$. Taking $\sum d_i=2$ into account we get $$K_{\bar {\mathcal U}_4}+\bar {\mathcal D}
\sim -2\Sigma-(2+e)F+\sum d_i(\Sigma+a_i F)=
\left( \sum d_i a_i-e\right) F+\bar f_n^*K_{\mathcal M_4}.$$ Therefore, $$\deg \mathcal L= \sum d_i a_i-e\ge e\sum d_i- ed_1-e\ge 0.$$
{#section-6}
Now we consider the case when the base variety $Z$ is a curve.
\[Prop-dimz=1\] Assume Conjectures [[\[BAB\]]{}]{} and [[\[conj-main-adj\]]{}]{} in dimensions $\le d-1$ and LMMP in dimension $\le d$. If $X$ is FT (and projective) variety of dimension $d$, then Conjecture [[\[conj-main-adj\]]{}]{} holds in dimension $d$.
\[corollary-815\] Conjecture [[\[conj-main-adj\]]{}]{} holds true in the following cases:
1. $\dim X=\dim Z+1$,
2. $\dim X=3$ and $X$ is FT.
Immediate by Theorem \[th-n-n-1\] and Proposition \[Prop-dimz=1\].
The rest of this section is devoted to the proof of Proposition \[Prop-dimz=1\]. Thus from now on and through the end of this section we assume that the base variety $Z$ is a curve. First we note that $Z\simeq {{\mathbb P}}^1$ because $X$ is FT.
\[lemma-local-comlpl-31\] Fix a positive integer $N$. Let $f\colon X\to Z\ni o$ be a contraction to a curve germ and let $D$ be an ${{\mathbb R}}$-divisor on $X$. Let $D^{{\mathrm{h}}}$ be the horizontal part of $D$. Assume that
1. $\dim X\le d$ and $X$ is FT over $Z$,
2. $D^{\mathrm{h}}$ is a ${{\mathbb Q}}$-boundary and $ND^{{\mathrm{h}}}$ is integral,
3. $K_X+D$ is lc and numerically trivial over $Z$.
Assume LMMP in dimension $\le d$. Further assume that the statement of Theorem [[\[main-result\]]{}]{} holds in dimensions $\le d-1$. Then there is an $n$-complement $K+D^+$ of $K+D$ near $f^{-1}(o)$ such that $N \mid n$, $n\le {\operatorname{Const}}(N,\dim X)$, and $a(E,X,D^+)=-1$ for some divisor $E$ with ${\operatorname{Center}}_Z E=o$.
Take a finite set ${{\mathfrak{R}}}\subset [0,\, 1]\cap {{\mathbb Q}}$ and a positive integer $I$ so that $D^{{\mathrm{h}}}\in {{\mathfrak{R}}}$, $I({{\mathfrak{R}}})\mid I$, and $N\mid I$. Replacing $D$ with $D+\alpha f^*o$ we may assume that $(X,D)$ is maximally lc. Next replacing $(X,D)$ with its suitable blowup we may assume that $X$ is ${{\mathbb Q}}$-factorial and the fibre $f^{-1}(o)$ has a component, say $F$, of multiplicity $1$ in $D$. Run $-F$-MMP over $Z$. This preserves the ${{\mathbb Q}}$-factoriality and lc property of $K+D$. Clearly, $F$ is not contracted. On each step, the contraction is birational. So at the end we get a model with irreducible central fibre: $f^{-1}(o)_{{\operatorname{red}}}=F$. Then $D\in \Phi({{\mathfrak{R}}})$. Applying $D^{{\mathrm{h}}}$-MMP over $Z$, we may assume that $D^{{\mathrm{h}}}$ is nef over $Z$. We will show that $K+D$ is $n$-complemented for some $n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$. Then by Proposition \[cor-pull-back\_compl-I\] we can pull-back complements to our original $X$. Note that the $f$-vertical part of $D$ coincides with $F$, so it is numerically trivial over $Z$. Since $X$ is FT over $Z$, $-K_X$ is big over $Z$. Therefore $D\equiv D^{{\mathrm{h}}}$ is nef and big over $Z$. Now apply construction of [@Prokhorov-Shokurov-2001 §3] to $(X, D)$ over $Z$. There are two cases:
1. $(X,F)$ is plt,
2. $(X,F)$ is lc but not plt (recall that $F\le D$).
Consider, for example, the second case (the first case is much easier and can be treated in a similar way). First we define an auxiliary boundary to localise a suitable divisor of discrepancy $-1$. By Kodaira’s lemma, for some effective $D^\mho$, the divisor $D-D^\mho$ is ample. Put $D_{{\varepsilon},\alpha}:=(1-{\varepsilon})D+\alpha D^\mho$. Then $K_X+D_{{\varepsilon},\alpha}\equiv -{\varepsilon}D+\alpha D^\mho$. So $(X,D_{{\varepsilon},\alpha})$ is a klt log Fano over $Z$ for $0<\alpha \ll {\varepsilon}\ll 1$. Take $\beta=\beta({\varepsilon},\alpha)$ so that $(X,D_{{\varepsilon},\alpha}+\beta F)$ is maximally lc and put $G_{{\varepsilon},\alpha}:=D_{{\varepsilon},\alpha}+\beta F$. Thus $(X,G_{{\varepsilon},\alpha})$ is a lc (but not klt) log Fano over $Z$.
Let $g\colon \widehat X\to X$ be an *inductive blowup* of $(X,G_{{\varepsilon},\alpha})$ [@Prokhorov-Shokurov-2001 Proposition 3.6]. By definition $\widehat X$ is ${{\mathbb Q}}$-factorial, $\rho(\widehat X/X)=1$, the $g$-exceptional locus is a prime divisor $E$ of discrepancy $a(E,X,G_{{\varepsilon},\alpha})=-1$, the pair $(\widehat X,E)$ is plt, and $-(K_{\widehat X}+E)$ is ample over $X$. Since $(X,G_{{\varepsilon},\alpha}-\gamma F)$ is klt for $\gamma>0$, ${\operatorname{Center}}_Z(E)=o$. Note that, by construction, $E$ is not exceptional on some fixed log resolution of $(X,{\operatorname{Supp}}G_{{\varepsilon},\alpha})$. Hence we may assume that $E$ and $g$ do not depend on ${\varepsilon}$ and $\alpha$ if $0<{\varepsilon}\ll 1$. In particular, $a(E,X,D)=-1$.
By (iii) of Lemma \[lemma-FT\] $\widehat X$ is FT over $Z$. Let $\widehat D$ and $\widehat G_{{\varepsilon},\alpha}$ be proper transforms on $\widehat X$ of $D$ and $G_{{\varepsilon},\alpha}$, respectively. Then $$\begin{array}{rll}
0\equiv &g^*(K_X+D)&=K_{\widehat X}+\widehat D+E,
\\[8pt]
&g^*(K_X+G_{{\varepsilon},\alpha})&=K_{\widehat X}+\widehat G_{{\varepsilon},\alpha}+E,
\end{array}$$ where $-(K_X+G_{{\varepsilon},\alpha})$ is ample over $Z$. Run $-(K_{\widehat X}+E)$-MMP starting from $\widehat X$ over $Z$: $$\xymatrix{
&\widehat X\ar[dl]_{g}\ar@{-->}[dr]^{}&
\\
X\ar[dr]^{f}&&{\overline{X}}\ar[dl]_{\bar f}
\\
&Z&
}$$ Since $-(K_{\widehat X}+E)\equiv \widehat D$, we can contract only components of $\widehat D$. At the end we get a model $({\overline{X}},{\overline{D}}+{\overline{E}})$ such that $-(K_{{\overline{X}}}+{\overline{E}})$ is nef and big over $Z$, $K_{{\overline{X}}}+{\overline{E}}+{\overline{D}}\equiv 0$, and $({\overline{X}},{\overline{E}}+{\overline{D}})$ is lc.
We claim that the plt property of $K_{\widehat X}+E$ is preserved under this LMMP. Indeed, for $0<t\ll 1$, the log divisor $K_{\widehat X}+(1-t)\widehat G_{{\varepsilon},\alpha}+E$ is a convex linear combination of log divisors $K_{\widehat X}+\widehat G_{{\varepsilon},\alpha}+E$ and $K_{\widehat X}+E$. The first divisor is anti-nef and is trivial only on one extremal ray $R$, the ray generated by fibres of $g$. The second one is strictly negative on $R$. Since $\widehat X$ is FT over $Z$, the Mori cone ${{\overline{\operatorname{NE}}}}(\widehat X/Z)$ is polyhedral. Therefore $K_{\widehat X}+(1-t)\widehat G_{{\varepsilon},\alpha}+E$ is anti-ample (and plt) for $0<t\ll 1$. By the base point free theorem there is a boundary $M\ge (1-t)\widehat G_{{\varepsilon},\alpha}+E$ such that $(\widehat X,M)$ is a plt $0$-pair. Since $E$ is not contracted, this property is preserved under our LMMP. Hence $({\overline{X}},{\overline{M}})$ is plt and so is $({\overline{X}},{\overline{E}})$. This proves our claim. In particular, ${\overline{E}}$ is normal and FT.
Take $\delta:=1/m$, $m\in {\mathbb Z}$, $m\gg 0$. For any such $\delta$, the pair $({\overline{X}},(1-\delta){\overline{D}}+{\overline{E}})$ is plt and $-(K_{{\overline{X}}}+(1-\delta){\overline{D}}+{\overline{E}})$ is nef and big over $Z$. By our inductive hypothesis there is an $n$-complement $K_{{\overline{E}}}+{\operatorname{Diff}}_{{\overline{E}}}({\overline{D}})^+$ of $K_{{\overline{E}}}+{\operatorname{Diff}}_{{\overline{E}}}({\overline{D}})$ with $n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$. Clearly, this is also an $n$-complement of $K_{{\overline{E}}}+{\operatorname{Diff}}_{{\overline{E}}}((1-\delta){\overline{D}})$. Note that $nD$ is integral. We claim that $(1-\delta){\overline{D}}\in {{\mathscr{P}}}_n$. Indeed, the vertical multiplicities of $(1-\delta){\overline{D}}$ are contained in $\Phi({{\mathfrak{R}}})$. Let $d_i$ be the multiplicity of a horizontal component of ${\overline{D}}$. Then $nd_i\in {\mathbb Z}$. If $d_i=1$, then obviously $(1-\delta)d_i\in {{\mathscr{P}}}_n$. So we assume that $d_i<1$. Then ${\left\lfloor (n+1)d_i\right\rfloor}=nd_i$ and ${\left\lfloor (n+1)(d_i-\delta)\right\rfloor}=nd_i\ge n(d_i-\delta)$ for $\delta\ll 1$. This proves our claim. Now the same arguments as in [@Prokhorov-Shokurov-2001 §3] shows that $K_X+(1-\delta)D$ is $n$-complemented near $f^{-1}(o)$. Since $D\in {{\mathscr{P}}}_n$, there is an $n$-complement $K_X+D^+$ of $K_X+D$ near $f^{-1}(o)$ and moreover, $a(E,X,D^+)=-1$.
\[cor-mult-fs\] Notation as in Proposition [[\[Prop-dimz=1\]]{}]{}. The multiplicities of $D{_{\operatorname{mod}}}$ are contained in a finite set.
Consider a local $n$-complement $D^+$ of $K+D$ near $f^{-1}(o)$. Then $n(K_Z+D^+{_{\operatorname{div}}{}}+D^+{_{\operatorname{mod}}})$ is integral at $o$. By construction, $(X,D^+)$ has a centre of log canonical singularities contained in $f^{-1}(o)$. Hence $D^+{_{\operatorname{div}}{}}=0$. By semiadditivity (see Lemma \[lemma-adj-prelim\]) we have $D^+{_{\operatorname{mod}}}=D{_{\operatorname{mod}}}$. Thus $nD{_{\operatorname{mod}}}$ is integral at $o$.
The statement of follows by Theorem \[th-Ambro-m\] (cf. [@Kawamata-1998]). Indeed, for any $0<t<1$ we put $D_t:=(1-t)D+t\Theta$, where $\Theta$ is such as in [[\[assumpt-adj-\*\]]{}]{}. Then by Theorem \[th-Ambro-m\] $(D_t){_{\operatorname{mod}}}$ is semiample. Hence so is $D{_{\operatorname{mod}}}$.
Assertion follows by Theorem \[main-result0\] (in lower dimension).
Finally for we note that by Corollary \[cor-mult-fs\] $ID{_{\operatorname{mod}}}$ is integral and base point free for a bounded $I$ because $Z\simeq {{\mathbb P}}^1$.
It is possible that Proposition [[\[Prop-dimz=1\]]{}]{} can be proved by using results of [@Fujino-Mori-2000], [@Fujino-2003-AG]. In fact, in these papers the authors write down the canonical bundle formula (for arbitrary $\dim Z$) in the following form (we change notation a little): $$b(K+D)= f^*(bK_Z+L_{X/Z}^{log, ss})+ \sum_P s_P^D f^*P+B^D.$$ Here $D{_{\operatorname{div}}{}}=\frac1b \sum_P s_P^D P$, $D{_{\operatorname{mod}}}=\frac 1b L_{X/Z}^{log, ss}$, and ${\operatorname{codim}}f(B^D)\ge 2$, so the term $B^D$ is zero in our situation. Under the additional assumption that $D$ is a boundary it is proved that the denominators of $D{_{\operatorname{mod}}}$ are bounded (and $D{_{\operatorname{mod}}}$ is semiample because it is nef on $Z={{\mathbb P}}^1$), see [@Fujino-Mori-2000 Theorem 4.5], [@Fujino-2003-AG Theorem 5.11]. This should imply our Proposition \[Prop-dimz=1\]. We however do not know how to avoid the effectivity condition of $D$.
The main theorem: Case $-(K+D)$ is nef {#sec-nef}
======================================
In this section we prove Theorem \[main-result\] in case and Theorem \[main-result0\] in the case when $(X,B)$ is not klt. Thus we apply reduction from §\[sect-reduction\] and replace $(X,B)$ with $(Y,B_Y)$ and put $D:=D_Y$. The idea of the proof is to consider the contraction $f\colon X\to Z$ given by $-(K+D)$ and use Effective Adjunction to pull-back complements from $Z$. In practice, there are several technical issues which do not allow us to weaken the last assumption in Theorem \[main-result\], that is, we cannot omit the klt condition when $K+B\not \equiv 0$. Roughly speaking the inductive step work if the following two conditions hold:
1. $0<\dim Z<\dim X$, and
2. the pair $(Z,D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})$ satisfies assumptions of Theorem \[main-result\].
The main technical step of the proof is Proposition \[prop-D-not-big\]. The proof is given in \[subsetion-0-pairs-proof\] and \[subsect-1-1\].
Setup {#not-last-first}
-----
Let $(X,D)$ be an lc log pair and let $f\colon X\to Z$ be a contraction such that $K+D {\mathbin{\sim_{\scriptscriptstyle{{{\mathbb Q}}}}}}f^* L$ for some $L$ and $X$ is FT. Further, assume the LMMP in dimension $d:=\dim X$. Our proof uses induction by $d$. So we also assume that Theorems \[main-result\] and \[main-result0\] hold true for all $X$ of dimension $<d$.
By Lemma \[lemma-Adj-hor-mult-1\] we have the following.
\[cor-Adj-hor-mult\] In notation of [[\[not-last-first\]]{}]{} assume that $\dim Z>0$. Fix a finite rational set ${{\mathfrak{R}}}\subset [0,1]$ and let $D\in \Phi({{\mathfrak{R}}})$. Then the multiplicities of horizontal components of $D$ are contained into a finite subset $M\subset \Phi({{\mathfrak{R}}})$, where $M$ depends only on $\dim X$ and ${{\mathfrak{R}}}$.
Restrict $D$ to a general fibre and apply Lemma \[lemma-Adj-hor-mult-1\].
Now we verify that under certain assumptions and conjectures the hyperstand multiplicities transforms to hyperstandard ones after adjunction.
For a subset ${{\mathfrak{R}}}\subset [0,1]$, denote $${{\mathfrak{R}}}(n):=\left({\overline{{{\mathfrak{R}}}}}+\frac 1n{\mathbb Z}\right)\cap [0,1],
\qquad {{\mathfrak{R}}}':= \bigcup_{n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})} {{\mathfrak{R}}}(n)\subset [0,\, 1].$$ These sets are rational and finite whenever so is ${{\mathfrak{R}}}$.
\[prop-R-prime\] In notation of [[\[not-last-first\]]{}]{}, fix a finite rational set ${{\mathfrak{R}}}\subset [0,1]$.
1. If $D\in \Phi({{\mathfrak{R}}})$, then $D{_{\operatorname{div}}{}}\in \Phi({{\mathfrak{R}}}')$.
2. If $D\in \Phi({{\mathfrak{R}}}, {\varepsilon}_{d-1})$, then $D{_{\operatorname{div}}{}}\in \Phi({{\mathfrak{R}}}',{\varepsilon}_{d-1})\subset \Phi({{\mathfrak{R}}}',{\varepsilon}_{d-2})$.
By taking general hyperplane sections we may assume that $Z$ is a curve. Furthermore, we may assume that $X$ is ${{\mathbb Q}}$-factorial. Fix a point $o\in Z$. Let $d_o$ be the multiplicity of $o$ in $D{_{\operatorname{div}}{}}$. Then $d_o=1-c_o$, where $c_o$ is computed by . It is sufficient to show that $d_o\in \Phi({{\mathfrak{R}}}(n)) \cup [1-{\varepsilon}_{d-1},\, 1]$ for any point $o\in Z$ and some $n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$. Clearly, we can consider $X$ and $Z$ small neighbourhoods of $f^{-1}(o)$ and $o$, respectively. We also may assume that $c_o>0$, so $f^{-1}(o)$ does not contain any centres of log canonical singularities of $(X,D)$. By our assumptions in \[not-last-first\] and Lemma \[lemma-local-comlpl-31\] there is an $n$-complement $K_X+D^+$ of $K_X+D$ near $f^{-1}(o)$ with $n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$ and moreover, $a(E,X,D^+)=-1$.
Now we show that $d_o\in \Phi({{\mathfrak{R}}}(n)) \cup [1-{\varepsilon}_{d-1},\, 1]$. By Lemma [[\[lemma-PPP-n\]]{}]{} $D\in {{\mathscr{P}}}_n$. Hence, $D^+\ge D$, i.e., $D^+=D+D'$, where $D'\ge 0$. Let $F\subset f^{-1}(o)$ be a reduced irreducible component. Since $K_X+D$ is ${{\mathbb R}}$-linearly trivial over $Z$, $D'$ is vertical and $D'= c_of^*P$. Let $d_F$ and $\mu$ be multiplicities of $F$ in $D$ and $f^*o$, respectively ($\mu$ is a positive integer). Since $(X,D+D')$ is lc and $n(D+D')$ is an integral divisor, the multiplicity of $F$ in $D+D'$ has the form $k/n$, where $k\in{\mathbb Z}$, $1\le k\le n$. Then $k/n=d_F+c_o\mu$ and $$c_o=\frac1\mu\left(\frac kn-d_F\right),\qquad
d_o=1-\frac1\mu\left(\frac kn-d_F\right).$$ Consider two cases.
a\) $d_F\in \Phi({{\mathfrak{R}}})$, so $d_F=1-r/m$ ($r\in{{\mathfrak{R}}}$, $m\in{\mathbb Z}$, $m>0$). Then we can write $$d_o=1-\frac{km+rn-nm}{nm\mu}=1-\frac{r'}{m\mu}<1,$$ where $$0\le r'=r+\frac{km}n-m= \frac{km+rn-nm}{n}\le \frac{nm+rn-nm}n\le 1.$$ Therefore, $d_o\in \Phi({{\mathfrak{R}}}(n))$, where $0\le r'=r+\frac{km}n-m\le 1$. This proves, in particular, (i).
b\) $d_F>1-{\varepsilon}_{d-1}$. In this case, $$1>d_o=1-\frac1\mu\left(\frac kn-d_F\right)>
1-\frac1\mu\left(\frac kn-1+{\varepsilon}_{d-1}\right)>
1-{\varepsilon}_{d-1}.$$ This finishes the proof of (ii).
\[prop-D-not-big\] Fix a finite rational subset ${{\mathfrak{R}}}\subset [0,\, 1]$ and a positive integer $I$ divisible by $I({{\mathfrak{R}}})$. Let $(X,D)$ be a log semi-Fano variety of dimension $d$ such that $X$ is ${{\mathbb Q}}$-factorial FT and $D\in \Phi({{\mathfrak{R}}})$. Assume that there is a $(K+D)$-trivial contraction $f\colon X\to Z$ with $0<\dim Z<d$. Fix the choice of $I_0$ and $\psi$ in [[\[construction-adj-def-mod\]]{}]{} so that $\mathbf D{_{\operatorname{mod}}}$ is effective. We take $I$ so that $I_0$ divides $I$. Assume the LMMP in dimension $d$. Further, assume that Conjectures [[\[BAB\]]{}]{} and [[\[conj-main-adj\]]{}]{} hold in dimension $d-1$ and $d$, respectively. If $K_Z+D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}}$ is $Im$-complemented, then so is $K_X+D$.
Put $D_Z:=D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}}$. Apply (i) of Conjecture [[\[conj-main-adj\]]{}]{} to $(X,D)$. We obtain $$K+D=f^*(K_Z+D_Z),$$ and $(Z,D_Z)$ is lc, where $D{_{\operatorname{div}}{}}\in\Phi({{\mathfrak{R}}}')$. By $I''D{_{\operatorname{mod}}}$ is integral for some bounded $I''$. Thus replacing ${{\mathfrak{R}}}'$ with ${{\mathfrak{R}}}'\cup \{1/I'', 2/I'',\dots, (I''-1)/I''\}$ we may also assume that $D{_{\operatorname{mod}}}\in \Phi({{\mathfrak{R}}}')$. Then $D_Z \in \Phi({{\mathfrak{R}}}')$. Furthermore, by Lemma \[lemma-FT\] $Z$ is FT and by the construction, $-(K_Z+D_Z)$ is nef. By our inductive hypothesis $K_Z+D_Z$ has bounded complements.
Let $K_Z+D_Z^+$ be an $n$-complement of $K_Z+D_Z$ such that $I\mid n$. Then $D_Z^+\ge D_Z$ (see Lemmas \[lemma-PPP-n-1\] and \[lemma-PPP-n\]). Put $H_Z:=D_Z^+-D_Z$ and $D^+:=D+f^*H_Z$. Write $D^+=\sum d_i^+D_i$. By the above, $d_i^+\ge d_i$. We claim that $K+D^+$ is an $n$-complement of $K+D$. Indeed, since $K+D{\mathbin{\sim_{\scriptscriptstyle{I}}}} f^*(K_Z+D_Z)$, we have $$\begin{aligned}
n(K+D^+)=&n(K+D+f^*H_Z)=
\\
&(n/I)I(K+D)+(n/I)If^*H_Z\sim
\\
&(n/I)f^*I(K_Z+D_Z)+(n/I)f^*IH_Z=
\\
&(n/I)f^*I(K_Z+D_Z^+)=
f^*n(K_Z+D_Z^+)\sim f^*0=0.\end{aligned}$$ Thus, $n(K+D^+)\sim 0$. Further, since $nd_i^+$ is a nonnegative integer and $d_i^+\ge d_i$, the inequality $$nd_i^+= {\left\lfloor (n+1)d_i^+\right\rfloor}\ge {\left\lfloor (n+1)d_i\right\rfloor}$$ holds for every $i$ such that $0\le d_i<1$. Finally, by Corollary \[cor\_Inv\_aDj\] the log divisor $K+D^+=f^*(K_Z+D_Z)$ is lc. This proves our proposition.
Proof of Theorem \[main-result0\] in the case when $(X,B)$ is not klt (continued) {#subsetion-0-pairs-proof}
---------------------------------------------------------------------------------
To finish the proof Theorem \[main-result0\] in the non-klt case we have to consider the following situation (see \[subsect-1\]). $(X',B')$ is a non-klt $0$-pair such that $B'\in \Phi({{\mathfrak{R}}})$, $X'$ is $\lambda$-lt and $X'$ is FT, where $\lambda$ depends only on ${{\mathfrak{R}}}$ and the dimension of $X'$. Moreover, there is a Fano fibration $X'\to Z'$ with $0<\dim Z'<\dim X'$. The disired bounded $nI({{\mathfrak{R}}})$-complements exist by Proposition \[prop-D-not-big\] and inductive hypothesis.
Proof of Theorem \[main-result\] in Case {#subsect-1-1}
-----------------------------------------
To finish our proof of the main theorem we have to consider the case when $(X,B)$ is klt and general reduction from Section \[sect-reduction\] leads to case , i.e., $-(K_Y+D_Y)$ is nef. Replace $(X,B)$ with $(Y,B_Y)$ and put $D:=D_Y$. Recall that in this situation $X$ is FT and $B\in \Phi({{\mathfrak{R}}}, {\varepsilon}')$, where $0<{\varepsilon}'\le {\varepsilon}_{d-1}({\overline{{{\mathfrak{R}}}}})$. By there is a boundary $\Theta\ge B$ such that $(X,\Theta)$ is a klt $0$-pair. For the boundary $D$ defined by we also have $D\in \Phi({{\mathfrak{R}}})$ and ${\left\lfloor D\right\rfloor}\neq 0$ by . All these properties are preserved under birational transformations in \[subs-2-inductive-steps\]. By our assumption at the end we have case , i.e., $-(K+D)$ is nef (and semiample). Therefore it is sufficient to prove the following.
\[prop-inductive\] Fix a finite rational subset ${{\mathfrak{R}}}\subset [0,\, 1]$. Let $(X,D=\sum d_i D_i)$ is a $d$-dimensional log semi-Fano variety such that
1. $D\in \Phi({{\mathfrak{R}}})$, $(X,D)$ is not klt and $X$ is FT,
2. there is boundary $B=\sum b_i D_i\le D$ such that either $b_i=d_i< 1-{\varepsilon}'$ or $b_i\ge 1-{\varepsilon}'$ and $d_i=1$, where $0<{\varepsilon}'\le {\varepsilon}_{d-1}({\overline{{{\mathfrak{R}}}}})$,
3. $(X,\Theta)$ is a klt $0$-pair for some $\Theta\ge B$.
Assume the LMMP in dimension $d$. Further, assume that Conjectures [[\[BAB\]]{}]{} and [[\[conj-main-adj\]]{}]{} hold in dimension $d$. Then $K+D$ has a bounded $n$-complement such that $I({{\mathfrak{R}}}) \mid n$.
The idea of the proof is to reduce the problem to Proposition \[prop-D-not-big\] by considering the contraction $f\colon X\to Z$ given by $-(K+D)$. But here two technical difficulties arise. First it may happen that the divisor $-(K+D)$ is big and then $f$ is birational. In this case one can try to extend complements from ${\left\lfloor D\right\rfloor}$ but the pair $(X,D)$ is not necessarily plt and the inductive step (Proposition \[prodolj\]) does not work. We have to make some perturbations and birational transformations. Second to apply inductive hypothesis to $(Z,D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})$ we have to check if this pair satisfies conditions of Theorem \[main-result\]. In particular, we have to check the klt property of $(Z,D{_{\operatorname{div}}{}}+D{_{\operatorname{mod}}})$. By Corollary \[cor\_Inv\_aDj\] this holds if any lc centre of $(X,D)$ dominates $Z$. Otherwise we again need some additional work.
Note that we may replace $B$ with $B_t:=tB+(1-t)D$ for $0<t<1$. This preserves all our conditions (i)-(iii). Indeed, (i) and (ii) are obvious. For (iii), we note that $(X,D^{\lozenge})$ is a $0$-pair for some $D^{\lozenge}\ge D$ (because $-(K+D)$ is semiample). Hence one can replace $\Theta$ with $\Theta_t:=t\Theta+(1-t)D^{\lozenge}$.
Let $\mu\colon (\tilde X,\tilde D)\to (X,D)$ be a dlt modification of $(X,D)$. By definition, $\mu$ is a $K+D$-crepant birational extraction such that $\tilde X$ is ${{\mathbb Q}}$-factorial, the pair $(\tilde X,\tilde D)$ is dlt, and each $\mu$-exceptional divisor $E$ has discrepancy $a(E,X,D)=-1$ (see, e.g., [@Utah 21.6.1], [@Prokhorov-2001 3.1.3]). In particular, $\tilde D\in \Phi({{\mathfrak{R}}})$ and $\tilde X$ is FT by Lemma \[lemma-FT\]. Let $\tilde B$ be the crepant pull-back of $B$. One can take $t$ so that the multiplicities in $\tilde B$ of $\mu$-exceptional divisors are $\ge 1-{\varepsilon}_{d-1}$. Thus for the pair $(\tilde X,\tilde D)$ conditions (i)-(iii) hold. Therefore, we may replace $(X,D)$ with $(\tilde X,\tilde D)$ (and $B$, $\Theta$ with their crepant pull-backs).
Let $f\colon X\to Z$ be the contraction given by $-(K+D)$. By Theorem \[main-result0\] (see \[subsect-1\] and \[subsetion-0-pairs-proof\]) we may assume that $\dim Z>0$. We apply induction by $N:=\dim X-\dim Z$.
First, consider the case $N=0$. Then $-(K+D)$ is big. We will show that $K+D$ is $n$-complemented for some $n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$.
Fix $n_0\gg 0$, and let $\delta:=1/n_0$. Then $D_\delta:=D-\delta {\left\lfloor D\right\rfloor}\in \Phi({{\mathfrak{R}}})$. It is sufficient to show that $K_X+D_\delta$ is $n$-complemented for some $n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$. We will apply a variant of [@Prokhorov-Shokurov-2001 Th. 5.1] with hyperstandard multiplicities. To do this, we run $-(K+D_\delta)$-MMP over $Z$. Clearly, this is equivalent ${\left\lfloor D\right\rfloor}$-MMP over $Z$. This process preserve the ${{\mathbb Q}}$-factoriality and lc (but not dlt) property of $K+D$. At the end we get a model $(X',D')$ such that $-(K_{X'}+D'_\delta)$ is nef over $Z$. Since $X'$ is FT, the Mori cone ${{\overline{\operatorname{NE}}}}(X')$ is rational polyhedral. Taking our condition $0<\delta\ll 1$ into account we get that $-(K_{X'}+D'_\delta)$ is nef. Since $$-(K_{X'}+D'_\delta)= -(K_{X'}+D')+\delta {\left\lfloor D'\right\rfloor},$$ where ${\left\lfloor D'\right\rfloor}$ is effective, $-(K_{X'}+D'_\delta)$ is also big. Note that $(X',D')$ is lc but not klt. By our assumptions, $$D'_\delta=(1-\delta)D'+\delta B' \le (1-\delta)D'+\delta\Theta'$$ and $(X',(1-\delta)D'+\delta\Theta')$ is klt. Therefore so is $(X',D'_\delta)$. Now we apply [@Prokhorov-Shokurov-2001 Th. 5.1] with $\Phi=\Phi({{\mathfrak{R}}})$. This says that we can extend complements from some (possibly exceptional) divisor. By Proposition \[prop-prop\] the multiplicities of the corresponding different are contained in ${\overline{{{\mathfrak{R}}}}}$. We obtain an $n$-complement of $K_{X'}+D'_\delta$ for some $n\in {{\mathscr{N}}}_{d-1}({\overline{{{\mathfrak{R}}}}})$. By Proposition \[cor-pull-back\_compl-I\] we can pull-back this complement to $X$ (we use the inclusion $D'_\delta\in \Phi({{\mathfrak{R}}})\subset {{\mathscr{P}}}_n$).
Now assume that Proposition \[prop-inductive\] holds for all $N'<N$. Run ${\left\lfloor D\right\rfloor}$-MMP over $Z$. After some flips and divisorial contractions we get a model on which ${\left\lfloor D\right\rfloor}$ is nef over $Z$. Since $X$ is FT, the Mori cone ${{\overline{\operatorname{NE}}}}(X)$ is rational polyhedral. Hence $-(K+D-\delta {\left\lfloor D\right\rfloor})$ is nef for $0<\delta \ll 1$. As above, put $D_\delta :=D-\delta {\left\lfloor D\right\rfloor}$. We can take $\delta =1/n_0$, $n_0\gg 0$ and then $D_\delta \in \Phi({{\mathfrak{R}}})$. On the other hand, $D_\delta\le (1-\delta)D +\delta B$ for some $\delta>0$. Therefore, $(X,D_\delta)$ is klt. Now let $f^\flat\colon X\to Z^\flat$ be the contraction given by $-(K+D_\delta)$. Since $-(K+D_\delta)=-(K+D)+\delta {\left\lfloor D\right\rfloor}$, there is decomposition $f\colon X \stackrel {f^\flat} \longrightarrow Z^\flat \longrightarrow Z$.
If $\dim Z^\flat=0$, then $Z^\flat=Z$ is a point, a contradiction. If $\dim Z^\flat<\dim X$, then by Corollary \[cor\_Inv\_aDj\] $(Z^\flat, (D_\delta){_{\operatorname{div}}{}}+(D_\delta){_{\operatorname{mod}}})$ is a klt log semi-Fano variety. We can apply Proposition \[prop-D-not-big\] to the contraction $X\to Z^\flat$ and obtain a bounded complement of $K+D_\delta$. Clearly, this will be a complement of $K+D$.
Therefore, we may assume that $-(K+D_\delta)$ is big, $f^\flat$ is birational, and so ${\left\lfloor D\right\rfloor}$ is big over $Z$. In particular, the horizontal part ${\left\lfloor D\right\rfloor}^{{\mathrm{h}}}$ of ${\left\lfloor D\right\rfloor}$ in non-trivial.
Replace $(X,D)$ with its dlt modification. Assume that ${\left\lfloor D\right\rfloor}^{{\mathrm{h}}}\neq{\left\lfloor D\right\rfloor}$. As above, run ${\left\lfloor D\right\rfloor}^{{\mathrm{h}}}$-MMP over $Z$. For $0<\delta \ll 1$, the divisor $-(K+D-\delta {\left\lfloor D\right\rfloor}^{{\mathrm{h}}})$ will be nef. Moreover, it is big over $Z$. Therefore, $-(K+D-\delta {\left\lfloor D\right\rfloor}^{{\mathrm{h}}})$ defines a contraction $f'\colon X\to Z'$ with $\dim Z'>\dim Z$. By our inductive hypothesis there is a bounded complement.
It remains to consider the case when ${\left\lfloor D\right\rfloor}^{{\mathrm{h}}}={\left\lfloor D\right\rfloor}$. Then any lc centre of $(X,D)$ dominates $Z$. By Corollary \[cor\_Inv\_aDj\] and Proposition \[prop-D-not-big\] there is a bounded complement of $K+D$.
This finishes the proof of Theorem \[main-result\]. Corollaries \[main-result-3-Cor-1-b\] and \[main-result-2\] immediately follows by this theorem, Corollary \[corollary-815\], and [@Alexeev-1994].
Proof of Corollary \[main-result-3-Cor-1-aC\] {#sketch-proof-main-result-3-Cor-1-aC}
---------------------------------------------
Replacing $(X,D)$ with its log terminal modification we may assume that $(X,D)$ is dlt. If $D=0$, we have $nK\sim 0$ for some $n\le 21$ by [@Blache-1995]. Thus we assume that $D\neq 0$. Run $K$-MMP. We can pull-back complements by Proposition \[cor-pull-back\_compl-I\]. The end result is a $K$-negative extremal contraction $(X',D')\to Z$ with $\dim Z\le 1$. If $Z$ is a curve, then either $Z\simeq{{\mathbb P}}^1$ or $Z$ is an elliptic curve. In both cases we apply Proposition \[prop-D-not-big\] (the FT property of $X$ is not needed). Otherwise $X'$ is a klt log del Pezzo surface with $\rho(X')=1$. In particular, $X'$ is FT. In this case the assertion follows by Theorem \[main-result0\].
Proof of Corollary \[main-result-3-Cor-1-a\] {#sketch-proof-main-result-3-Cor-1-a}
--------------------------------------------
First we construct a crepant dlt model $(\bar X,\bar D)$ of $(X,D)$ such that each component of $\bar D$ meets ${\left\lfloor \bar D\right\rfloor}$. Replacing $(X,D)$ with its log terminal modification we may assume that $(X,D)$ is dlt, $X$ is ${{\mathbb Q}}$-factorial, and ${\left\lfloor D\right\rfloor}\neq 0$. If ${\left\lfloor D\right\rfloor}=D$, we put $(\bar X,\bar D)=(X,D)$. Otherwise, run $K+D-{\left\lfloor D\right\rfloor}$-MMP. Note that none of connected components of ${\left\lfloor D\right\rfloor}$ is contracted. Moreover, the number of connected components of ${\left\lfloor D\right\rfloor}$ remains the same (cf. [@Utah Prop. 12.3.2], [@Shokurov-1992-e Th. 6.9]). At the end we get an extremal contraction $(X',D')\to Z$ with $\dim Z\le 2$. If $Z$ is not a point, we can apply Proposition \[prop-D-not-big\]. Otherwise, $\rho(X')=1$, ${\left\lfloor D'\right\rfloor}$ is connected, and each component of $D'$ meets ${\left\lfloor D'\right\rfloor}$. The same holds on a log terminal modification $(\bar X,\bar D)$ of $(X',D')$ because $X$ is ${{\mathbb Q}}$-factorial.
By Corollary \[main-result-3-Cor-1-aC\], for each component $\bar D_i\subset {\left\lfloor \bar D\right\rfloor}$, the log divisor $K_{\bar D_i}+{\operatorname{Diff}}_{\bar D_i}(\bar D-\bar D_i)$ has bounded complements, i.e., there is $n_0=n_0({{\mathfrak{R}}})$ such that $n_0(K_{\bar D_i}+{\operatorname{Diff}}_{\bar D_i}(\bar D-\bar D_i))\sim 0$. Thus we may assume that $n_0(K_{\bar X}+\bar D)|_{{\left\lfloor \bar D\right\rfloor}}\sim 0$. Recall that the multiplicities of ${\operatorname{Diff}}_{\bar D_i}(\bar D-\bar D_i)
=\sum \delta_j\Delta_j$ are computed by the formula $\delta_j=1-1/m_j+(\sum_l k_l d_l)/m_j$, where $m_j,\, k_l\in {\mathbb Z}$, $m_j>0$, $k_l\ge 0$, $d_l$ are multiplicities of $\bar D$, and $\sum_l k_l d_l\le 1$. Since $d_l\in \Phi({{\mathfrak{R}}})$, there is only a finite number of possibilities for $d_l$ with $k_l\neq 0$. Further, since $n_0\delta_j\in {\mathbb Z}$, there is only a finite number of possibilities for $m_j$. Thus we can take $n_1=n_1({{\mathfrak{R}}})$ such that $n_1\bar D$ is an integral divisor and $n_1(K_{\bar X}+\bar D)|_{{\left\lfloor \bar D\right\rfloor}}\sim 0$. Since $\bar X$ is FT, there is an integer $n_2$ such that $n_1n_2(K_{\bar X}+\bar D)\sim 0$ on $\bar X$. This defines a cyclic étale over ${\left\lfloor \bar D\right\rfloor}$ cover $\pi\colon \hat X\to \bar X$. Let $\hat D:=\pi^*\bar D$. Then $(\hat X,\hat D)$ is a $0$-pair such that ${\left\lfloor \hat D\right\rfloor}$ has at least $n_2$ connected components. On the other hand, the number of connected componets of a $0$-pair is at most two (see [@Fujino-2000-ab 2.1], cf. [@Shokurov-1992-e 6.9]). Thus, $n(K_{\bar X}+\bar D)\sim 0$, where $n=2n_1$. This proves our corollary.
Proof of Corollary \[main-result-3-Cor-1-2div\] {#sketch-proof-main-result-3-Cor-2div}
-----------------------------------------------
In notation of \[eq-reduction-setup\], take $0<\bar {\varepsilon}<{\varepsilon}_{2}({\overline{{{\mathfrak{R}}}}})/2$. We may assume that $(X,B)$ is such as in \[sub-reduction-1\], so there are (at least) two components $B_1$ and $B_2$ of multiplicities $b_i\ge 1-\bar {\varepsilon}$ in $B$. Then by Lemma \[pair-discr\] and components $B_1$, $B_2$ do not meet each other and by Corollary \[cor-reduct-discrs\] this holds on each step of the LMMP as in \[subs-2-inductive-steps\]. Therefore, we cannot get a model with $\rho=1$. In particular, case is impossible.
Consider case . If the divisor $-(K_Y+D_Y)$ is big, we can argue as in the proof of Proposition \[prop-inductive\]. Then we do not need Conjecture \[BAB\]. If $K_Y+D_Y\equiv 0$, we can use Corollary \[main-result-3-Cor-1-a\]; it is sufficient to have only one divisor $E$ (exceptional or not) with $a(E,X,D)\le -1+\bar {\varepsilon}$. In other cases we use induction to actual fibrations (Proposition \[prop-D-not-big\]), that is, with the fibres and the base of dimension $\ge 1$ and by our assumptions with dimensions $\le 2$.
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[^1]: Such a log pair can be called also a *log Calabi-Yau variety*. However the last notion usually assumes some additional conditions such as $\pi_1(X)=0$ or $q(X)=0$.
|
---
abstract: 'Utilizing a non-equilibrium Green function like the generalized Kadanoff-Baym ansatz, a systematic perturbative method is presented to calculate the expectation value of an arbitrary physical quantity under the restriction that the Wigner distribution function is fixed. It is shown that, in the diagrammatic expression of the quantity, a certain part of contributions can be eliminated due to the restriction. Together with the quantum kinetic equation, this method provides a basis for the kinetic-theoretical description.'
address: 'Department of Physics, Faculty of Science and Technology, Keio University, Yokohama 223-8522, japan '
author:
- Jun Koide
title: ' Kinetic-Theoretic Description based on Closed-Time-Path Formalism '
---
Introduction {#intro}
============
The non-equilibrium state of a dilute gas system is considered to be described by the one-particle distribution function (1PDF), and such an approach to the non-equilibrium system is called the ‘kinetic theory’ [@Zubarev]. In the kinetic theory, the 1PDF is treated as the independent dynamical variable of the system, and all the physical quantities are determined by the 1PDF. The kinetic equation is an equation of motion of the 1PDF, and the dynamics in the kinetic theory is described by this equation.
A lot of works has been done on the derivation of the quantum kinetic equation (QKE) in the framework of the Green function technique. Perhaps, the most popular approach is the generalized Kadanoff-Baym (GKB) formalism [@KB; @Lipavsky] which utilizes an ansatz for the non-equilibrium Green function called the GKB ansatz. The GKB ansatz can be expressed by the usual equilibrium Keldysh Green function in which the equilibrium 1PDF is replaced by a non-equilibrium one.
An alternative approach is the counter-term method based on the CTP formalism [@Lawrie; @Niegawa], or on the thermo-field dynamics [@Umezawa]. In this approach, a counter-term, in which the non-equilibrium properties are included, is first introduced into the CTP or thermo-field Lagrangian, and then the unperturbed propagator gets the similar structure as the GKB ansatz.
Although the QKE can be derived in these approaches, the kinetic theory is not completely constructed because they do not give a proper method to express the expectation values of physical quantities in terms of the 1PDF. Of course, the expectation value of a physical quantity of interest can be calculated perturbatively by the usage of the non-equilibrium propagators mentioned above, and it becomes a functional of the 1PDF. But if we want to obtain a functional of the 1PDF, the value of the 1PDF must be fixed from the exterior, and the integrations over the microscopic fields should be carried out under the restriction due to the fixing of the 1PDF. This restriction has not been considered in the above formalisms, and hence they do not give a complete basis for the kinetic theory.
In this paper, we present a systematic perturbative method to calculate the expectation value of any physical quantity as a functional of the Wigner distribution function (WDF), which plays the role of the 1PDF in quantum theory. Our approach, is somewhat different from the GKB or the counter-term method. It is based on the inversion method [@Fukuda; @suppl]. An inversion-method approach to derive the QKE was presented in [Refs. ]{}, and the problem of calculating the physical quantities in terms of the WDF is partly solved there; We first introduce an external source $J$ to probe the WDF $z$, and calculate the physical quantity as a functional of the source $Q[J]$. Then we express the source as a functional of the WDF ($J=J[z]$ which is an inversion of $z=z[J]$), and by substituting it into the above calculated $Q[J]$, we can write the quantity by the WDF. In this approach, the propagator is a functional of the external source $J$, and the perturbative calculation can be done without the restriction of fixing the WDF. The calculations in our formulation indeed generates different results from those obtained by the perturbative calculation using, e.g., the GKB ansatz. By the substitution of the inverted relation, some contributions from the diagrams, which will be present in the calculations with other formalisms, are canceled.
We show that the contributions which are canceled can be expressed by a corresponding time-ordered diagrams: The contributions from a diagram in the non-equilibrium theory can be classified by the temporal order of the vertices in the diagram, and to each temporal ordering of the vertices, a time-ordered diagram (called a configuration in the article) corresponds. Then if an obtained configuration can be separated into two parts by cutting two propagators at the same instant, the contribution from that configuration is canceled. Because the propagator used in our method has the form of the GKB ansatz, this will also provide a basis for the kinetic theoretic description in the GKB formalism.
In the course of proving this property, we reformulate the inversion method approach in the framework of the Legendre transformation [@suppl; @CTP; @Chou]. The definition of non-equilibrium generating functional is slightly modified in a way characteristic to the non-equilibrium theory, and the effective action is defined as the Legendre transformation of it. Then the diagrammatic rule for the effective action discussed in [Ref. ]{} can be utilized with an extension to the non-equilibrium case. By virtue of this, the QKE can also be expressed in a compact form which is finally given as (\[QKE\]).
In the next section, we summarize the inversion method approach to the QKE, and reformulate it in the terminology of Legendre transformation. Then in [\[diagrammer\]]{}, the diagrammatic rule for the kinetic theory is discussed: The rule to calculate the expectation value as a functional of the WDF is presented in [\[Q-rule\]]{}, and the rule to derive the QKE is in [\[QKE-rule\]]{}.
Inversion method approach to the kinetic equation {#inversion}
=================================================
In this section, we describe the inversion method approach to the QKE [@Koide; @Koide2; @Koide3].
Probing source and the Green function {#source-inhom}
-------------------------------------
The system to be considered is the same as in [Ref. ]{}; a non-relativistic bosonic field described by the Hamiltonian $
\hat{H} = \hat{H}_{0}+\hat{H}_{\rm int}
$ with $$\hat{H}_{0}
= \sum_{{{ {\bf k} }}} {\epsilon_{{{ {\bf k} }}}}{\hat{\psi}}^{{ \mbox{\footnotesize \dag} }}_{{{ {\bf k} }}} {\hat{\psi}}_{{{ {\bf k} }}}
,
\label{H0}
\hspace{2em}
\hat{H}_{\rm int}
= \frac{\lambda}{4}\sum_{{{ {\bf k} }},{{ {\bf k} }'},{{ {\bf q} }}}
{\hat{\psi}}^{{ \mbox{\footnotesize \dag} }}_{{{ {\bf k} }}+{{ {\bf q} }}} {\hat{\psi}}^{{ \mbox{\footnotesize \dag} }}_{{{ {\bf k} }'}-{{ {\bf q} }}}
{\hat{\psi}}_{{{ {\bf k} }}} {\hat{\psi}}_{{{ {\bf k} }'}}
,
\label{Hint}$$ and a spatially inhomogeneous initial density matrix $\hat{\rho}$. We consider the case that the interaction $\hat{H}_{\rm int}$ can be treated perturbatively, and for simplicity, the initial correlation is not taken into account. See [Ref. ]{} for the treatment of the initial correlation.
In quantum statistical physics, the natural alternative of the 1PDF will be the Wigner distribution function (WDF) defined as $$f_{{{ {\bf K} }}}({{ {\bf X} }},t)
= \int\frac{{{\rm d}}{{\mit \Delta}}{{ {\bf x} }}}{V} \,{{\rm e}}^{-{{\rm i}}{{ {\bf K} }}\cdot{{\mit \Delta}}{{ {\bf x} }}}
{\left<\, {\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}({{ {\bf X} }}-\frac{{{\mit \Delta}}{{ {\bf x} }}}{2},t)
{\hat{\psi}}({{ {\bf X} }}+\frac{{{\mit \Delta}}{{ {\bf x} }}}{2},t)
\,\right>}
,
\label{Wigner}$$ where $
{\hat{\psi}}({{ {\bf x} }})
= \frac{1}{\sqrt{V}} \sum_{{{ {\bf k} }}} {{\rm e}}^{{{\rm i}}{{ {\bf k} }}\cdot{{ {\bf x} }}}{\hat{\psi}}_{{{ {\bf k} }}}
$ and the angular bracket implies the average over initial density matrix $\hat{\rho}$; ${\left<\, \cdots \,\right>}={{\rm Tr}}\hat{\rho}\cdots$. As in [Ref. ]{}, for the sake of perturbative calculation, it is more convenient to work with the Fourier transform of the WDF defined as $${z_{{{ {\bf k} }}, {{ {\bf q} }}}}(t)
\equiv
{\left<\, {\hat{\psi}^{{ \mbox{\footnotesize \dag} }}_{{{ {\bf q} }}}}(t) {\hat{\psi}_{{{ {\bf k} }}}}(t) \,\right>}
= \int {{\rm d}}{{ {\bf x} }}\:
{{\rm e}}^{-{{\rm i}}({{ {\bf k} }}-{{ {\bf q} }}) \cdot {{ {\bf x} }}} f_{\frac{{{ {\bf k} }}+{{ {\bf q} }}}{2}}({{ {\bf x} }},t)
,
\label{defz}
\label{n2z}$$ to which we refer simply as the WDF in the following. Note that ${z^{\ast}_{{{ {\bf q} }}, {{ {\bf k} }}}}={z_{{{ {\bf k} }}, {{ {\bf q} }}}}$ holds due to the hermitian property of $\hat{\rho}$.
Within the CTP formalism, Eq. (\[defz\]) can be represented as $${z_{{{ {\bf k} }}, {{ {\bf q} }}}}(t)
\propto
\int{\left[ {{\rm d}}\psi_{1} {{\rm d}}\psi_{2} \right]}
\psi^{\ast}_{{{ {\bf q} }}}(t) \psi_{{{ {\bf k} }}}(t)
{{\rm e}}^{ \frac{{{\rm i}}}{\hbar}{{\int_{{t_{{{\rm I}}}}}^{t}\hspace{-0.2em}}}{{\rm d}}s
{\left( L(\psi_{1})-L(\psi_{2}) \right)}
}
{\left<\, \psi_{1,{{\rm I}}} {\left|\, \hat{\rho} \,\right|} \psi_{2,{{\rm I}}} \,\right>}
.
\label{z}$$ In the inversion method approach to the kinetic theory, we introduce a probing source $J$ for $z$, and calculate $z[J]$ as a functional of the source. By inverting the relation as $J=J[z]$, the QKE is obtained as an equation of motion for $z$ by setting $J=0$. According to [Ref. ]{}, the proper way to introduce the source ${J_{{{ {\bf k} }},{{ {\bf q} }}}}$ is that the source is built into the quadratic form of the free part of the Lagrangian in (\[z\]) $
L_{0}(\psi_{1})-L_{0}(\psi_{2})
= \sum_{{{ {\bf k} }}{{ {\bf q} }},ij}\psi^{\ast}_{{{ {\bf k} }},i} {{\cal D}}_{{{ {\bf k} }}{{ {\bf q} }},ij}\psi_{{{ {\bf q} }},j}
$ by $$\begin{aligned}
{{\cal D}_{{{ {\bf k} }},{{ {\bf q} }}}}&=& {\left( \begin{array}{cc}
({{\rm i}}\hbar {\partial_{t}}-{\epsilon_{{{ {\bf k} }}}}) {\delta_{{{ {\bf k} }},{{ {\bf q} }}}}+{{\rm i}}{J_{{{ {\bf k} }},{{ {\bf q} }}}}(t)
& -{{\rm i}}{J_{{{ {\bf k} }},{{ {\bf q} }}}}(t)
\\
-{{\rm i}}{J_{{{ {\bf k} }},{{ {\bf q} }}}}(t)
& -({{\rm i}}\hbar {\partial_{t}}-{\epsilon_{{{ {\bf k} }}}}) {\delta_{{{ {\bf k} }},{{ {\bf q} }}}}+{{\rm i}}{J_{{{ {\bf k} }},{{ {\bf q} }}}}(t)
\\
\end{array}
\right)}
.
\nonumber\\
\label{D1-inhom}\end{aligned}$$ An inverse of this matrix leads to the $2\times2$-Green function, which is a functional of the source $J$. From the relation ${{\cal D}}{G^{(0)}}=-{{\rm i}}\hbar$, we get $${G^{(0)}}_{{{ {\bf k} }},{{ {\bf q} }}}[t,s;J]
= -\theta(t-s){{\rm e}}^{-{{\rm i}}{\omega_{{{ {\bf k} }}}}(t-s)}
{\left( \begin{array}{cc}
{\bar{z}^{(0)}_{{{ {\bf k} }}, {{ {\bf q} }}}}(s) & {z^{(0)}_{{{ {\bf k} }},{{ {\bf q} }}}}(s)\\
{\bar{z}^{(0)}_{{{ {\bf k} }}, {{ {\bf q} }}}}(s) & {z^{(0)}_{{{ {\bf k} }},{{ {\bf q} }}}}(s)\\
\end{array}
\right)}
-\theta(s-t){{\rm e}}^{ {{\rm i}}{\omega_{{{ {\bf q} }}}}(s-t)}
{\left( \begin{array}{cc}
{z^{(0)}_{{{ {\bf q} }},{{ {\bf k} }}}}(t) & {z^{(0)}_{{{ {\bf q} }},{{ {\bf k} }}}}(t) \\
{\bar{z}^{(0)}_{{{ {\bf q} }}, {{ {\bf k} }}}}(t) & {\bar{z}^{(0)}_{{{ {\bf q} }}, {{ {\bf k} }}}}(t) \\
\end{array}
\right)}
,
\label{Go0}$$ where ${z^{(0)}_{{{ {\bf k} }},{{ {\bf q} }}}}$ is an unperturbed WDF as a functional of $J$ given by $${z^{(0)}_{{{ {\bf k} }},{{ {\bf q} }}}}{\left[ t;J \right]}
= {{\rm e}}^{-{{\rm i}}({\omega_{{{ {\bf k} }}}}-{\omega_{{{ {\bf q} }}}})(t-s)} {z^{(0)}_{{{ {\bf k} }},{{ {\bf q} }}}}({t_{{{\rm I}}}})
+\frac{1}{\hbar}{{\int_{{t_{{{\rm I}}}}}^{t}\hspace{-0.2em}}}{{\rm d}}s\, {{\rm e}}^{-{{\rm i}}({\omega_{{{ {\bf k} }}}}-{\omega_{{{ {\bf q} }}}})(t-s)} {J_{{{ {\bf k} }},{{ {\bf q} }}}}(s)
,
\label{z0}$$ and we have used ${\bar{z}^{(0)}_{{{ {\bf k} }}, {{ {\bf q} }}}}= {z^{(0)}_{{{ {\bf k} }},{{ {\bf q} }}}}+\delta_{{{ {\bf k} }},{{ {\bf q} }}}$ and ${\omega_{{{ {\bf k} }}}}={\epsilon_{{{ {\bf k} }}}}/\hbar$. The unperturbed WDF satisfies an equation of motion $${J_{{{ {\bf k} }},{{ {\bf q} }}}}(t) = {\left\{ \hbar {\partial}_{t}+{{\rm i}}({\epsilon_{{{ {\bf k} }}}}-{\epsilon_{{{ {\bf q} }}}}) \right\}} {z^{(0)}_{{{ {\bf k} }},{{ {\bf q} }}}}(t)
,
\label{J0}$$ and if we replace ${z^{(0)}}$ by $z$ in (\[J0\]), this gives a functional expression of the source $J$ in terms of $z$ in the lowest order of the perturbative inversion. We denote it as ${J^{(0)}}[z]$. Note that, whenever the source has an index of the perturbative order as $J^{(i)}$, it should be understand as a functional of $z$. By the inversion method, we can calculate a perturbative correction ${{{\mit \Delta}}J}[z]$ to (\[J0\]), and up to the second order of the perturbation, the source is expressed by $z$ as $$\begin{aligned}
{J_{{{ {\bf k} }},{{ {\bf q} }}}}(t)
&=& {J^{(0)}_{{{ {\bf k} }},{{ {\bf q} }}}}[z] +{{{\mit \Delta}}J_{{{ {\bf k} }},{{ {\bf q} }}}}[z]
\nonumber\\
&=& {\left( \hbar {\partial}_{t} +{{\rm i}}{\left( {\epsilon_{{{ {\bf k} }}}}-{\epsilon_{{{ {\bf q} }}}}\right)} \right)}{z_{{{ {\bf k} }}, {{ {\bf q} }}}}(t)
+{{\rm i}}\lambda \sum_{{{ {\bf q} }'},{{ {\bf m} }}}
{\left\{ {z^{\ast}_{{{ {\bf q} }}, {{ {\bf q} }'}}}{z^{\ast}_{{{ {\bf m} }}-{{ {\bf k} }}, {{ {\bf m} }}-{{ {\bf q} }'}}}-{z_{{{ {\bf k} }}, {{ {\bf q} }'}}}{z_{{{ {\bf m} }}-{{ {\bf q} }}, {{ {\bf m} }}-{{ {\bf q} }'}}}\right\}}(t)
\nonumber\\
& &-\frac{\lambda^{2}}{2\hbar}\sum_{{{ {\bf l} }},{{ {\bf m} }}}
{{\int_{{t_{{{\rm I}}}}}^{s}\hspace{-0.2em}}}{{\rm d}}s
{\left\{ {{\rm e}}^{ {{\rm i}}{\omega^{(2)}_{{{ {\bf q} }},{{ {\bf k} }},{{ {\bf l} }},{{ {\bf m} }}}}(t-s)} {Z^{(2)}_{{{ {\bf q} }},{{ {\bf k} }},{{ {\bf l} }},{{ {\bf m} }}}}(s)
+{{\rm e}}^{-{{\rm i}}{\omega^{(2)}_{{{ {\bf k} }},{{ {\bf q} }},{{ {\bf l} }},{{ {\bf m} }}}}(t-s)} {Z^{(2)\ast}_{{{ {\bf k} }},{{ {\bf q} }},{{ {\bf l} }},{{ {\bf m} }}}}(s) \right\}}
,
\label{J2}\end{aligned}$$ where $
{\omega^{(2)}_{{{ {\bf q} }},{{ {\bf k} }},{{ {\bf l} }},{{ {\bf m} }}}}= {\omega_{{{ {\bf q} }}}}+{\omega_{{{ {\bf m} }}-{{ {\bf k} }}}}-{\omega_{{{ {\bf l} }}}}-{\omega_{{{ {\bf m} }}-{{ {\bf l} }}}}$ and $${Z^{(2)}_{{{ {\bf q} }},{{ {\bf k} }},{{ {\bf l} }},{{ {\bf m} }}}}= \sum_{{{ {\bf q} }'},{{ {\bf l} }'},{{ {\bf m} }'}}
{\left\{ {\bar{z}^{\ast}_{{{ {\bf q} }}, {{ {\bf q} }'}}}{\bar{z}^{\ast}_{{{ {\bf m} }}-{{ {\bf k} }}, {{ {\bf m} }'}-{{ {\bf q} }'}}}{z_{{{ {\bf l} }}, {{ {\bf l} }'}}}{z_{{{ {\bf m} }}-{{ {\bf l} }}, {{ {\bf m} }'}-{{ {\bf l} }'}}}-{z^{\ast}_{{{ {\bf q} }}, {{ {\bf q} }'}}}{z^{\ast}_{{{ {\bf m} }}-{{ {\bf k} }}, {{ {\bf m} }'}-{{ {\bf q} }'}}}{\bar{z}_{{{ {\bf l} }}, {{ {\bf l} }'}}}{\bar{z}_{{{ {\bf m} }}-{{ {\bf l} }}, {{ {\bf m} }'}-{{ {\bf l} }'}}}\right\}}
.
\label{ZZdef}$$ A QKE follows from (\[J2\]) after the removal of the source. This QKE is reduced to the usual Boltzmann equation after the Markovian and local approximations [@Koide3].
kinetic theoretic description in the inversion method {#problem}
-----------------------------------------------------
In the kinetic theory, the 1PDF is considered to be an independent dynamical variable and the kinetic equation describes its dynamics. This means a coarse graining from the microscopic field variables to the 1PDF. All other quantities should be expressed in terms of the 1PDF, and their dynamics should follow from the kinetic equation. Since such physical quantities may be defined microscopically by a temporary-local functions of the field variables as $\hat{Q}=Q(\hat{\psi})$, in order to obtain a complete framework of kinetic theory, we must express their expectation values as functionals of the 1PDF.
In the inversion method approach, this can be realized by first calculating perturbatively the expectation value $Q(t)=\langle \hat{Q}(t) \rangle$ with the use of the propagator $-{G^{(0)}}[J]$ in previous subsection, and then by substituting the source written by the WDF as in (\[J2\]) into the obtained functional. The former procedure will provide us with the expectation value as a functional of the source $J$, and the latter reduces it into a functional of the WDF $z$.
From some explicit calculations [@thesis], we can see the following fact: After the substitution of $J={J^{(0)}}[z]+{{{\mit \Delta}}J}[z]$ in the second step, if we expand the obtained expression around ${J^{(0)}}[z]$, the contributions due to the perturbative correction ${{{\mit \Delta}}J}[z]$ cancels some part of the unperturbed contributions. Such an expansion can be expressed diagrammatically by the usage of a propagator $-{G^{(0)}}[J={J^{(0)}}[z]]$, and the above mentioned cancellation implies that some part of the diagram should be omitted if we want to evaluate the expectation value as a functional of the WDF.
From this observation, it is expected that a simplified procedure to calculate the expectation value $Q[z]$ exists. Our problem in this paper is to find what kind of the diagram should be retained in the evaluation of the $Q[z]$ if the diagrams are written with the propagator $-{G^{(0)}}[J={J^{(0)}}]$. In fact, the propagator $-{G^{(0)}}[J={J^{(0)}}[z]]$ has the form of the GKB ansatz with the free particle approximation of the spectral function; it can be obtained by replacing ${z^{(0)}}[J]$ in (\[Go0\]) by $z$. So our consideration here also provides the way to calculate $Q[z]$ in the GKB formalism.
Formulation with Legendre transformation
----------------------------------------
In order to discuss the problem settled in the previous subsection, it is convenient to rewrite the inversion method in the framework of the Legendre transformation using the ‘physical representation’ of the CTP formalism. The physical representation of the CTP formalism is introduced by a simple transformation of the variables from $\psi_{1}$ and $\psi_{2}$ to ${\psi_{{{\rm C}}}}$ and ${\psi_{{{\rm \Delta}}}}$, which is defined as $${\psi_{{{\rm C}}}}=\frac{\psi_{1}+\psi_{2}}{2},
\hspace{2em}
{\psi_{{{\rm \Delta}}}}=\psi_{1}-\psi_{2}.$$ Then the free part of the Lagrangian is rewritten as $$L_{0}({\psi_{{{\rm C}}}},{\psi_{{{\rm \Delta}}}})
= \sum_{{{ {\bf k} }},{{ {\bf q} }}}
\begin{array}{c}
{\left( {\psi^{\ast}_{{{ {\bf k} }},{{\rm C}}}}{\psi^{\ast}_{{{ {\bf k} }},{{\rm \Delta}}}}\right)} \\
{} \\
\end{array}
{\left( \begin{array}{cc}
0 & ({{\rm i}}\hbar {\partial_{t}}-{\epsilon_{{{ {\bf k} }}}}) {\delta_{{{ {\bf k} }},{{ {\bf q} }}}}\\
({{\rm i}}\hbar {\partial_{t}}-{\epsilon_{{{ {\bf k} }}}}) {\delta_{{{ {\bf k} }},{{ {\bf q} }}}}& {{\rm i}}{J_{{{ {\bf k} }}{{ {\bf q} }},{{\rm C}}}}(t) \\
\end{array}
\right)}
{\left( \begin{array}{c}
{\psi_{{{ {\bf q} }},{{\rm C}}}}\\
{\psi_{{{ {\bf q} }},{{\rm \Delta}}}}\\
\end{array}
\right)}
,
\label{L0CD}$$ where we have denoted the source $J$ in (\[D1-inhom\]) as ${J_{{{\rm C}}}}$ for convenience. We can see the source ${J_{{{\rm C}}}}$ is simply coupled to ${\psi^{\ast}_{{{\rm \Delta}}}}{\psi_{{{\rm \Delta}}}}$. Correspondingly, the Green function ${G^{(0)}}[{J_{{{\rm C}}}}]$ becomes $${G^{(0)}_{{{ {\bf k} }}, {{ {\bf q} }}}}[t,s;{J_{{{\rm C}}}}]
= {\left( \begin{array}{cc}
{g^{{{\rm C}}}_{{{ {\bf k} }},{{ {\bf q} }}}}[t,s;{J_{{{\rm C}}}}] & {g^{\rm R}_{{{ {\bf k} }},{{ {\bf q} }}}}(t,s) \\
{g^{\rm A}_{{{ {\bf k} }},{{ {\bf q} }}}}(t,s) & 0 \\
\end{array}
\right)}
,
\label{Go}$$ where the respective components are defined as $$\begin{aligned}
{g^{{{\rm C}}}_{{{ {\bf k} }},{{ {\bf q} }}}}[t,s;{J_{{{\rm C}}}}]
&=&-\theta(t-s){{\rm e}}^{-{{\rm i}}{\omega_{{{ {\bf k} }}}}(t-s)} {\left( {z^{(0)}_{{{ {\bf k} }},{{ {\bf q} }}}}[s;{J_{{{\rm C}}}}]+\frac{1}{2} \right)}
-\theta(s-t){{\rm e}}^{ {{\rm i}}{\omega_{{{ {\bf q} }}}}(s-t)} {\left( {z^{(0)}_{{{ {\bf q} }},{{ {\bf k} }}}}[t;{J_{{{\rm C}}}}]+\frac{1}{2} \right)}
,
\label{gC}
\\
{g^{\rm R}_{{{ {\bf k} }},{{ {\bf q} }}}}(t,s)
&=&-\theta(t-s){{\rm e}}^{-{{\rm i}}{\omega_{{{ {\bf k} }}}}(t-s)} \delta_{{{ {\bf k} }},{{ {\bf q} }}}
,
\label{gR}
\\
{g^{\rm A}_{{{ {\bf k} }},{{ {\bf q} }}}}(t,s)
&=& \theta(s-t){{\rm e}}^{ {{\rm i}}{\omega_{{{ {\bf q} }}}}(s-t)} \delta_{{{ {\bf q} }},{{ {\bf k} }}}
.
\label{gA}\end{aligned}$$ To use the Legendre transformation formalism, we must introduce another source ${J_{{{\rm \Delta}}}}$ coupled to ${\psi^{\ast}_{{{\rm C}}}}{\psi_{{{\rm C}}}}$, and define the generating functional $W$ as $$\begin{aligned}
{{\rm e}}^{\frac{{{\rm i}}}{\hbar}W[{J_{{{\rm C}}}},{J_{{{\rm \Delta}}}},{I_{{{\rm \Delta}}}}]}
&\equiv&
\int{\left[ {{\rm d}}{\psi_{{{\rm C}}}}{{\rm d}}{\psi_{{{\rm \Delta}}}}\right]}
{\left<\, \psi_{{{\rm C}},{{\rm I}}}+{{\textstyle \frac{1}{2} }}\psi_{{{\rm \Delta}},{{\rm I}}}
{\left|\, \hat{\rho} \,\right|}
\psi_{{{\rm C}},{{\rm I}}}-{{\textstyle \frac{1}{2} }}\psi_{{{\rm \Delta}},{{\rm I}}}
\,\right>}
\nonumber\\
& &\times
{{\rm e}}^{\frac{{{\rm i}}}{\hbar}{{\int_{{t_{{{\rm I}}}}}^{{t_{{{\rm F}}}}}\hspace{-0.2em}}}{{\rm d}}t
{\left\{ L_{0}({\psi_{{{\rm C}}}},{\psi_{{{\rm \Delta}}}})-V({\psi_{{{\rm \Delta}}}},{\psi_{{{\rm C}}}})
+\sum_{{{ {\bf k} }},{{ {\bf q} }}} {J_{{{ {\bf k} }}{{ {\bf q} }},{{\rm \Delta}}}}{\psi^{\ast}_{{{ {\bf k} }},{{\rm C}}}}{\psi_{{{ {\bf q} }},{{\rm C}}}}\right\}}
}
\nonumber\\
& &\times
{{\rm e}}^{\frac{{{\rm i}}}{\hbar}{{\int_{{t_{{{\rm I}}}}}^{{t_{{{\rm F}}}}}\hspace{-0.2em}}}{{\rm d}}t \,{I_{{{\rm \Delta}}}}Q({\psi_{{{\rm C}}}})}
,
\label{W0}\end{aligned}$$ where $V$ is the interaction part of the CTP Lagrangian $H_{\rm int}(\psi_{1})-H_{\rm int}(\psi_{2})$ written in terms of ${\psi_{{{\rm C}}}}$ and ${\psi_{{{\rm \Delta}}}}$. The source ${J_{{{\rm \Delta}}}}$ is unphysical in the sense that the expectation value of an hermitian operator is not guaranteed to be real under the existence of this source. It is just introduced so that we can write the WDF $z$ by a derivative of the generating functional, and should be removed after all the calculation. For the same reason, we have introduced the source ${I_{{{\rm \Delta}}}}$ which is coupled to $Q({\psi_{{{\rm C}}}})$; replacement of ${\hat{\psi}}$ by ${\psi_{{{\rm C}}}}$ in the time-local composite operator $Q({\hat{\psi}})$ in which we are interested. Note that all the integrands in the exponent of (\[W0\]) are local in time.
Here we define two variables $${z_{{{ {\bf k} }}{{ {\bf q} }},{{\rm C}}}}(t)
\equiv \frac{\delta W[{J_{{{\rm \Delta}}}},{J_{{{\rm C}}}},{I_{{{\rm \Delta}}}}]}{\delta {J_{{{ {\bf k} }}{{ {\bf q} }},{{\rm \Delta}}}}(t)}
,
\hspace{1em}
{z_{{{ {\bf k} }}{{ {\bf q} }},{{\rm \Delta}}}}(t)
\equiv \frac{\delta W[{J_{{{\rm \Delta}}}},{J_{{{\rm C}}}},{I_{{{\rm \Delta}}}}]}{\delta {J_{{{ {\bf k} }}{{ {\bf q} }},{{\rm C}}}}(t)}
.
\label{zdef}$$ When the sources are removed, ${z_{{{\rm C}}}}$ is reduced to $
\langle T{\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}{\hat{\psi}}+{\hat{\psi}}{\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}+{\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}{\hat{\psi}}+\tilde{T}{\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}{\hat{\psi}}\rangle
= {z_{{{ {\bf k} }}, {{ {\bf q} }}}}+\delta_{{{ {\bf k} }},{{ {\bf q} }}}/2
$, and ${z_{{{\rm \Delta}}}}$ is to $
{{\rm i}}\langle T{\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}{\hat{\psi}}-{\hat{\psi}}{\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}-{\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}{\hat{\psi}}+\tilde{T}{\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}{\hat{\psi}}\rangle
= 0
$. Note that we regard $\psi^{\ast}\psi$ as $\psi^{\ast}(t+0)\psi(t)$ in the course of the path integration. Particularly, ${z_{{{\rm \Delta}}}}=0$ is realized by removing only the unphysical source ${J_{{{\rm \Delta}}}}$, and in this case, ${z_{{{\rm C}}}}$ becomes a functional of ${J_{{{\rm C}}}}$ as ${z_{{{ {\bf k} }}, {{ {\bf q} }}}}[{J_{{{\rm C}}}}]+\delta_{{{ {\bf k} }},{{ {\bf q} }}}/2$. Of course, the non-equilibrium expectation value of symmetrized $\hat{Q}$ is obtained as a functional of the sources ${J_{{{\rm C}}}}$ by $$Q[t;{J_{{{\rm C}}}}]
= {\left. {\frac{\delta W[{J_{{{\rm \Delta}}}},{J_{{{\rm C}}}},{I_{{{\rm \Delta}}}}]}{\delta {I_{{{\rm \Delta}}}}(t)}} \,\right|_{{J_{{{\rm \Delta}}}}={I_{{{\rm \Delta}}}}=0}}
.
\label{Q1}$$ To use the variables ${z_{{{\rm C}}}}$ and ${z_{{{\rm \Delta}}}}$ as the independent variables, we define the Legendre transformation of $W$ by $${{\mit \Gamma}}[{z_{{{\rm C}}}},{z_{{{\rm \Delta}}}};{I_{{{\rm \Delta}}}}]
\equiv W[{J_{{{\rm \Delta}}}},{J_{{{\rm C}}}},{I_{{{\rm \Delta}}}}]
-\sum_{{{ {\bf k} }},{{ {\bf q} }}}{{\int_{{t_{{{\rm I}}}}}^{{t_{{{\rm F}}}}}\hspace{-0.2em}}}{{\rm d}}t {\left( {J_{{{ {\bf k} }}{{ {\bf q} }},{{\rm \Delta}}}}{z_{{{ {\bf k} }}{{ {\bf q} }},{{\rm C}}}}+{J_{{{ {\bf k} }}{{ {\bf q} }},{{\rm C}}}}{z_{{{ {\bf k} }}{{ {\bf q} }},{{\rm \Delta}}}}\right)}
,
\label{Gamma0}$$ where ${J_{{{\rm \Delta}}}}$ and ${J_{{{\rm C}}}}$ are functional of ${z_{{{\rm C}}}}$ and ${z_{{{\rm \Delta}}}}$, which are obtained by solving (\[zdef\]). Now we can obtain the expectation value of $\hat{Q}$ as a functional of the WDF ${z_{{{\rm C}}}}$ by $$Q[t;{z_{{{\rm C}}}}]
= {\left. {\frac{\delta {{\mit \Gamma}}[{z_{{{\rm C}}}},{z_{{{\rm \Delta}}}};{I_{{{\rm \Delta}}}}]}{\delta {I_{{{\rm \Delta}}}}(t)}} \,\right|_{{z_{{{\rm \Delta}}}}={I_{{{\rm \Delta}}}}=0}}
.
\label{Q2}$$ Moreover, from an identity of the Legendre transformation, we have $${J_{{{ {\bf k} }}{{ {\bf q} }},{{\rm C}}}}(t)
= -\frac{\delta {{\mit \Gamma}}[{z_{{{\rm C}}}},{z_{{{\rm \Delta}}}};{I_{{{\rm \Delta}}}}]}{\delta {z_{{{ {\bf k} }}{{ {\bf q} }},{{\rm \Delta}}}}(t)}
,
\hspace{1em}
{J_{{{ {\bf k} }}{{ {\bf q} }},{{\rm \Delta}}}}(t)
= -\frac{\delta {{\mit \Gamma}}[{z_{{{\rm C}}}},{z_{{{\rm \Delta}}}};{I_{{{\rm \Delta}}}}]}{\delta {z_{{{ {\bf k} }}{{ {\bf q} }},{{\rm C}}}}(t)}
.
\label{Jderiv}$$ If we remove the unphysical source ${J_{{{\rm \Delta}}}}$, the first equation become an equation of motion of ${z_{{{ {\bf k} }}{{ {\bf q} }},{{\rm C}}}}={z_{{{ {\bf k} }}, {{ {\bf q} }}}}[{J_{{{\rm C}}}}]+\delta_{{{ {\bf k} }},{{ {\bf q} }}}/2$ which corresponds to (\[J2\]), and finally the removal of ${J_{{{\rm C}}}}$ reduces the equation of motion to the QKE. In this sense, ${{\mit \Gamma}}$ is referred to as the effective action.
Diagrammatic rule for kinetic theory {#diagrammer}
====================================
Diagrammatic expression of the effective action ${{\mit \Gamma}}$ is well investigated. For an expectation value of non-local product $\langle {\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}(t){\hat{\psi}}(s) \rangle$, the effective action is expressed simply by the two particle irreducible (2PI) diagrams [@DM]. Here, 2PI diagram is a diagram which cannot be separated by cutting any pair of propagators. For the expectation value of a local product, such as the 1PDF, the situation is more complicated. In this subsection, we utilize the rules presented in [Ref. ]{} with a non-equilibrium extension, and clarify the meaning of the rule. By use of the rule, the QKE can also be rewritten in a compact form. For notational simplicity, the time arguments and wave-number indices will not explicitly be written if it is not misleading.
Diagrammatic expression of the effective action
-----------------------------------------------
First we consider a diagrammatic expansion of the generating functional $W$. The building blocks of the diagram are a $2\times2$-propagator $-{\tilde{G}^{(0)}}$ given below, an interaction vertex $V({\psi_{{{\rm \Delta}}}},{\psi_{{{\rm C}}}})$ given in (\[W0\]) and an external leg $Q({\psi_{{{\rm C}}}})$ coupled to ${I_{{{\rm \Delta}}}}$. In the diagram, an arrow expresses the contraction operator $$-\sum_{{{ {\bf k} }},{{ {\bf q} }}}\int{{\rm d}}t {{\rm d}}s
\begin{array}{c}
{\left( \frac{\delta}{\delta{\psi_{{{ {\bf k} }},{{\rm C}}}}(t)} \:
\frac{\delta}{\delta{\psi_{{{ {\bf k} }},{{\rm \Delta}}}}(t)}
\right)} \\
{} \\
\end{array}
{\tilde{G}^{(0)}}_{{{ {\bf k} }},{{ {\bf q} }}}(t,s)
{\left( \begin{array}{c}
\frac{\delta}{\delta\psi_{{{ {\bf q} }},{{\rm C}}}^{\ast}(s)} \\
\frac{\delta}{\delta\psi_{{{ {\bf q} }},{{\rm \Delta}}}^{\ast}(s)}\\
\end{array}
\right)}
.$$ The $2\times2$-Green function ${\tilde{G}^{(0)}}$ is defined as an inverse of the matrix in bilinear form of the exponent in (\[W0\]); $${\tilde{G}^{(0)}}[{J_{{{\rm C}}}},{J_{{{\rm \Delta}}}}]
\equiv
\frac{{{\rm i}}}{\hbar}
{\left( \begin{array}{cc}
{J_{{{\rm \Delta}}}}& {{\rm i}}\hbar {\partial_{t}}-\epsilon \\
{{\rm i}}\hbar {\partial_{t}}-\epsilon & {{\rm i}}{J_{{{\rm C}}}}\\
\end{array}
\right)}^{-1}
= {\left( \begin{array}{cc}
{\tilde{g}^{{{\rm C}}}}& {\tilde{g}^{\rm R}}\\
{\tilde{g}^{\rm A}}& {\tilde{g}^{{{\rm \Delta}}}}\\
\end{array}
\right)}
,
\label{Gotil}$$ where the tilde implies the unphysical case ${J_{{{\rm \Delta}}}}\neq 0$. Using the physical case ${J_{{{\rm \Delta}}}}=0$ given in (\[gC\])-(\[gA\]), the components of (\[Gotil\]) can be written as $$\begin{aligned}
{\tilde{g}^{{{\rm C}}}}[{J_{{{\rm \Delta}}}},{J_{{{\rm C}}}}]
&\equiv& {\left( 1-{g^{{{\rm C}}}}[{J_{{{\rm C}}}}]\frac{{J_{{{\rm \Delta}}}}}{{{\rm i}}\hbar} \right)}^{-1} {g^{{{\rm C}}}}[{J_{{{\rm C}}}}]
,
\label{gCtil}
\\
{\tilde{g}^{\rm R}}[{J_{{{\rm \Delta}}}},{J_{{{\rm C}}}}]
&\equiv& {\left( 1+{\tilde{g}^{{{\rm C}}}}[{J_{{{\rm C}}}},{J_{{{\rm \Delta}}}}]\frac{{J_{{{\rm \Delta}}}}}{{{\rm i}}\hbar} \right)} {g^{\rm R}},
\\
{\tilde{g}^{\rm A}}[{J_{{{\rm \Delta}}}},{J_{{{\rm C}}}}]
&\equiv& {g^{\rm A}}{\left( 1+\frac{{J_{{{\rm \Delta}}}}}{{{\rm i}}\hbar}{\tilde{g}^{{{\rm C}}}}[{J_{{{\rm C}}}},{J_{{{\rm \Delta}}}}] \right)}
,
\\
{\tilde{g}^{{{\rm \Delta}}}}[{J_{{{\rm \Delta}}}},{J_{{{\rm C}}}}]
&\equiv& {g^{\rm A}}{\left( \frac{{J_{{{\rm \Delta}}}}}{{{\rm i}}\hbar}
-\frac{{J_{{{\rm \Delta}}}}}{{{\rm i}}\hbar}{\tilde{g}^{{{\rm C}}}}[{J_{{{\rm C}}}},{J_{{{\rm \Delta}}}}]
\frac{{J_{{{\rm \Delta}}}}}{{{\rm i}}\hbar}
\right)}
{g^{\rm R}},
\label{gDtil}\end{aligned}$$ with a short-hand notation. Of course ${\tilde{G}^{(0)}}$ is reduced to ${G^{(0)}}$ in (\[Go\]) by setting ${J_{{{\rm \Delta}}}}=0$. Note that retarded or advanced nature of ${g^{\rm R}}$ or ${g^{\rm A}}$, respectively, is recovered only in the physical case ${J_{{{\rm \Delta}}}}=0$.
Then the generating functional $W$ can be expressed as $$\frac{{{\rm i}}}{\hbar}W[J,{I_{{{\rm \Delta}}}}] = {{\rm Tr}}\ln {\tilde{G}^{(0)}}[J] +\kappa[J,{I_{{{\rm \Delta}}}}]
,
\label{W2}$$ where $J$ expresses the set of ${J_{{{\rm \Delta}}}}$ and ${J_{{{\rm C}}}}$, and $\kappa$ is the sum of all the connected diagrams constructed by the propagator $-{\tilde{G}^{(0)}}[J]$, the vertex $V$ and the external leg ${I_{{{\rm \Delta}}}}$. For simplicity, we suppress the argument ${I_{{{\rm \Delta}}}}$ in this subsection.
Next we evaluate (\[W2\]) at $J={J^{(0)}}[z]+{{{\mit \Delta}}J}[z]$ (cf. (\[J2\])), and substitute it into the definition (\[Gamma0\]) of the effective action ${{\mit \Gamma}}$. Expanding ${{\mit \Gamma}}$ around $J={J^{(0)}}[z]$ in terms of ${{{\mit \Delta}}J}[z]$, the terms linear in ${{{\mit \Delta}}J}$ are canceled, and we obtain $$\begin{aligned}
\frac{{{\rm i}}}{\hbar}{{\mit \Gamma}}[z]
&=& {{\rm Tr}}\ln {\tilde{G}^{(0)}}[{J^{(0)}}+{{{\mit \Delta}}J}] +\kappa[J[z]]
-\frac{{{\rm i}}}{\hbar}
{\left\{ {\left( {J^{(0)}_{{{\rm \Delta}}}}+{{{\mit \Delta}}J_{{{\rm \Delta}}}}\right)}{z_{{{\rm C}}}}+{\left( {J^{(0)}_{{{\rm C}}}}+{{{\mit \Delta}}J_{{{\rm C}}}}\right)}{z_{{{\rm \Delta}}}}\right\}}
\\
&=& \frac{{{\rm i}}}{\hbar}{{\mit \Gamma}^{(0)}}[z]
-\frac{1}{2\hbar^{2}}
\begin{array}{c}
{\left( {{{\mit \Delta}}J_{{{\rm \Delta}}}}{{{\mit \Delta}}J_{{{\rm C}}}}\right)} \\
{} \\
\end{array}
\Delta_{2}
{\left( \begin{array}{c}
{{{\mit \Delta}}J_{{{\rm \Delta}}}}\\
{{{\mit \Delta}}J_{{{\rm C}}}}\\
\end{array}
\right)}
+\kappa'[z]
,
\label{Gamma2}\end{aligned}$$ where ${{\mit \Gamma}^{(0)}}$, $\Delta_{2}$ and $\kappa'$ are defined respectively by $$\begin{aligned}
\frac{{{\rm i}}}{\hbar}{{\mit \Gamma}^{(0)}}[z]
&\equiv& {{\rm Tr}}\ln {\tilde{G}^{(0)}}[{J^{(0)}}]
-\frac{{{\rm i}}}{\hbar}{\left\{ {J^{(0)}_{{{\rm \Delta}}}}{z_{{{\rm C}}}}+{J^{(0)}_{{{\rm C}}}}{z_{{{\rm \Delta}}}}\right\}}
,
\\
\Delta_{2}(t,s)
&\equiv&
-{\left( \begin{array}{cc}
{\tilde{g}^{{{\rm C}}}}(t,s){\tilde{g}^{{{\rm C}}}}(s,t) & {{\rm i}}{\tilde{g}^{\rm R}}(t,s){\tilde{g}^{\rm A}}(s,t) \\
{{\rm i}}{\tilde{g}^{\rm A}}(t,s){\tilde{g}^{\rm R}}(s,t) & -{\tilde{g}^{{{\rm \Delta}}}}(t,s){\tilde{g}^{{{\rm \Delta}}}}(s,t) \\
\end{array}
\right)}
,
\label{Del2def}
\\
\kappa'[z]
&\equiv& \kappa[J[z]]
+{{\rm Tr}}\sum_{k\geq 3}\frac{1}{k}
{\left\{ \frac{1}{{{\rm i}}\hbar}{\tilde{G}^{(0)}}[{J^{(0)}}]
{\left( \begin{array}{cc}
{{{\mit \Delta}}J_{{{\rm \Delta}}}}& 0 \\
0 & {{\rm i}}{{{\mit \Delta}}J_{{{\rm C}}}}\\
\end{array}
\right)}
\right\}}^{k}
.\end{aligned}$$ Eq. (\[Gamma2\]) corresponds to Eq. (3.21) in [Ref. ]{}, and then, as it is proved in [Ref. ]{}, the effective action can be expressed as $$\frac{{{\rm i}}}{\hbar}{{\mit \Gamma}}[z]
= {{\cal R}}_{2}{\left( \frac{{{\rm i}}}{\hbar}W[{J^{(0)}}[z]] \right)}
-\frac{{{\rm i}}}{\hbar}\sum_{{{ {\bf k} }},{{ {\bf q} }}}{{\int_{{t_{{{\rm I}}}}}^{{t_{{{\rm F}}}}}\hspace{-0.2em}}}{{\rm d}}t
{\left( {J^{(0)}_{{{ {\bf k} }}{{ {\bf q} }},{{\rm \Delta}}}}[z]{z_{{{ {\bf k} }}{{ {\bf q} }},{{\rm C}}}}+{J^{(0)}_{{{ {\bf k} }}{{ {\bf q} }},{{\rm C}}}}[z]{z_{{{ {\bf k} }}{{ {\bf q} }},{{\rm \Delta}}}}\right)}
.
\label{Gamma3}$$ Here, ${{\cal R}}_{2}$ is a diagrammatic operation defined by the following process.
1. \[2PRcutJoint\] The first process of ${{\cal R}}_{2}$ can be expressed schematically as $$\raisebox{-3.5mm}{\epsfile{width=22mm,file=R2orig.eps}}
\Longrightarrow
\raisebox{4mm}{$
{\left( \raisebox{-4mm}{\epsfile{width=29.7mm,file=R2raw.eps}} \right)}
$}
\Delta_{2}^{-1}
{\left( \raisebox{-10mm}{\epsfile{width=13.1mm,file=R2col.eps}} \right)}
,
\label{cutpatch}$$ to which we refer as the ‘cut-and-patch’ operation: If there is a 2PR part in the diagram, separate the graph into two pieces by cutting the corresponding pair of the propagators. In each of the separated diagrams, make the resultant two external lines to contract ${\psi_{{{\rm C}}}}^{\ast}(t){\psi_{{{\rm C}}}}(t)$ or ${{\rm i}}{\psi_{{{\rm \Delta}}}}^{\ast}(t){\psi_{{{\rm \Delta}}}}(t)$, which we call the ${z_{{{\rm C}}}}$- or ${z_{{{\rm \Delta}}}}$-leg, respectively. Then reconnect the two diagrams by contracting their $z$-legs with $\Delta_{2}^{-1}$.
2. The second step is to carry out the procedure \[2PRcutJoint\] in all possible ways, and sum up all the resulting diagrams including the original one.
For example, $${{\cal R}}_{2}{\left( \raisebox{-5mm}
{\epsfile{width=10.65mm,file=R2-0.eps}}
\right)}
= \raisebox{-5mm}{\epsfile{width=10.65mm,file=R2-0.eps}}
+\raisebox{-6mm}{\epsfile{width=14.75mm,file=R2-1.eps}}
+\raisebox{-7mm}{\epsfile{width=12.75mm,file=R2-2.eps}}
+\raisebox{-7mm}{\epsfile{width=18.75mm,file=R2-3.eps}}
,$$ where ${{\rm \Delta}}_{2}^{-1}$ is expressed by doubled lines. It should be emphasized that $W$ is evaluated at $J={J^{(0)}}[z]$ in (\[Gamma3\]): In the diagrammatic expression of $W$, the propagators are $-{G^{(0)}}[{J^{(0)}}[z]]$ which acquire the form of the GKB ansatz when we set ${z_{{{\rm \Delta}}}}=0$.
As it was discussed in [Ref. ]{}, the operation ${{\cal R}}_{2}$ cancels some part of the 2PR diagrams, and in this sense, ${{\mit \Gamma}}$ has a modified 2PI property. It is reduced to the usual 2PI when we discuss an effective action of non-local operator ${\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}(t)\psi(s)$. In the next subsection, we will clarify what is the contents of this modified 2PI property.
Diagrammatic Rule for $Q[{z_{{{\rm C}}}}]$ {#Q-rule}
------------------------------------------
Recovering the argument ${I_{{{\rm \Delta}}}}$ in (\[Gamma3\]), the expectation value $Q$ as a functional of the WDF is obtained from (\[Q2\]) as $$Q[t;{z_{{{\rm C}}}}]
= {\left. { {{\cal R}}_{2}{\left( \frac{\delta W[{J^{(0)}}[z],{I_{{{\rm \Delta}}}}]}{\delta{I_{{{\rm \Delta}}}}(t)} \right)} } \,\right|_{ {z_{{{\rm \Delta}}}}={I_{{{\rm \Delta}}}}=0 }}
.
\label{Q3}$$ Note that ${J^{(0)}}[z]$ in (\[Gamma3\]) does not depend on ${I_{{{\rm \Delta}}}}$. Diagrammatically, inside the operation ${{\cal R}}_{2}$ is a sum of all the connected diagrams with one external point expressing $Q({\psi_{{{\rm C}}}})$. For definiteness, we consider the case $\hat{Q} = {\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}_{{{ {\bf q} }}}{\hat{\psi}^{{ \mbox{\footnotesize \dag} }}}_{{{ {\bf q} }'}}{\hat{\psi}}_{{{ {\bf k} }'}}{\hat{\psi}}_{{{ {\bf k} }}}$ as an example. In the following, since we have set ${z_{{{\rm \Delta}}}}={I_{{{\rm \Delta}}}}=0$ in (\[Q3\]), the form of the Green function ${\tilde{G}^{(0)}}[{J^{(0)}}[z]]$ is reduced to that of the GKB ansatz; Eq.(\[Go\]) in which ${z^{(0)}}[{J_{{{\rm C}}}}]$ is replaced by $z$.
### Time-ordered configuration and causality {#Causality}
Before considering the operation ${{\cal R}}_{2}$, we define terminologies ‘time-ordered configuration’ and ‘causality’.
In the non-equilibrium Green function technique, because the propagator depends explicitly on time, the evaluation of a diagram may be carried out as a function of time as follows. For all possible ways of time ordering of the vertices, the diagram is arranged in such a way that the vertices are put on the time axis from right to left. Then assigning the factors of propagators and vertices, each time ordering gives different contribution. In the following, we refer to the diagram with a fixed time ordering of the vertices as the ‘time-ordered configuration’ or simply the configuration. For example, the diagram in [Fig. \[L4ex\]]{} (a) can be arranged as the eight configurations shown in [Fig. \[L4ex\]]{} (b) and (c). Other possible configurations can be eliminated by the following mechanism.
\
The vertices in the diagram expresses $H_{\rm int}(\psi_{1})-H_{\rm int}(\psi_{2})$ rewritten by ${\psi_{{{\rm \Delta}}}}$ and ${\psi_{{{\rm C}}}}$, which is odd in ${\psi_{{{\rm \Delta}}}}$ for generic $H_{\rm int}$, and contains at least one ${\psi_{{{\rm \Delta}}}}$ or $\psi_{{{\rm \Delta}}}^{\ast}$. Then, we can conclude that the time-ordered configuration like [Fig. \[causal\]]{}, where a vertex is on the latest time, vanishes: Assuming the vertex of [Fig. \[causal\]]{} is on time $t$, ${\psi_{{{\rm \Delta}}}}(t)$ or $\psi_{{{\rm \Delta}}}^{\ast}(t)$ therein must be contracted by ${g^{\rm A}}(t,s)$ or ${g^{\rm R}}(s,t)$ ($t>s$), respectively, (Recall that we are working in the physical case ${J_{{{\rm \Delta}}}}=0$.) and their advanced or retarded character leads to the vanishing. This implies that, when we calculate an expectation value of a physical quantity at time $t$, the interaction at time later than $t$ does not contribute since the configuration like [Fig. \[causal\]]{} can not be avoided. In other word, the time-ordered configuration of the diagram for $Q$ must have the external point $Q$ on the latest time within the diagram. In this article, we call such a fact the ‘causality’.
### Meaning of the operation ${{\cal R}}_{2}$
Now we consider the meaning of the operation ${{\cal R}}_{2}$ in (\[Q3\]). For this sake, we first examine the cut-and-patch operation. Since we are considering the physical case ${z_{{{\rm \Delta}}}}=0$, the ${{\rm \Delta}}{{\rm \Delta}}$-component of ${\tilde{G}^{(0)}}$ as well as the ${{\rm \Delta}}{{\rm \Delta}}$-component of ${{\rm \Delta}}_{2}$ vanish. Then, because ${{\rm \Delta}}_{2}^{-1}$ can be written as $$\begin{aligned}
{{\rm \Delta}}_{2}^{-1}
&=&-{\left( \begin{array}{cc}
{g^{{{\rm C}}}}{g^{{{\rm C}}}}& {{\rm i}}{g^{\rm R}}{g^{\rm A}}\\
{{\rm i}}{g^{\rm A}}{g^{\rm R}}& 0 \\
\end{array}
\right)}^{-1}
\nonumber\\
&=& {\left( \begin{array}{cc}
0 &
{{\rm i}}\{{g^{\rm A}}{g^{\rm R}}\}^{-1}
\\
{{\rm i}}\{{g^{\rm R}}{g^{\rm A}}\}^{-1} &
-\{ {g^{\rm R}}{g^{\rm A}}\}^{-1} {g^{{{\rm C}}}}{g^{{{\rm C}}}}\{ {g^{\rm A}}{g^{\rm R}}\}^{-1}
\\
\end{array}
\right)}
,
\label{Del2inv}\end{aligned}$$ the ${{\rm C}}{{\rm C}}$-component of ${{\rm \Delta}}_{2}^{-1}$ disappears. Thus, in (\[cutpatch\]), the connection of two ${z_{{{\rm C}}}}$-legs is absent.
Moreover, in (\[Q3\]), the connection of two ${z_{{{\rm \Delta}}}}$-legs is forbidden by the following reason. Since there is only one external point $Q$ in each diagram for (\[Q3\]), the external point belongs to only one of the two sub-diagrams connected by ${{\rm \Delta}}_{2}^{-1}$. Then, if both of the two sub-diagrams are connected with their ${z_{{{\rm \Delta}}}}$-legs, the one which does not contain the external point $Q$ must vanish due to the causality: The ${z_{{{\rm \Delta}}}}$-leg at time $t$ is produced by a pair of Green functions ${g^{\rm R}}(s,t)$ and ${g^{\rm A}}(t,s')$, which implies they must be connected to vertices at time $s$ and $s'$ later than $t$. So, if the sub-diagram does not have an external point, any time-ordered configuration of the sub-diagram must have a vertex which possesses at the latest time as shown in [Fig. \[zDleg2\]]{}, and this gives a vanishing contribution due to the causality discussed in the previous subsection.
Thus it is enough to consider the connection of ${z_{{{\rm \Delta}}}}$-leg and ${z_{{{\rm C}}}}$-leg, where the external point belongs to the sub-diagram with ${z_{{{\rm \Delta}}}}$-leg. In contrast to ${z_{{{\rm \Delta}}}}$-leg, the ${z_{{{\rm C}}}}$-leg must be possessed on the latest time within the sub-diagram: Otherwise, some vertex must be possessed on the latest time because the sub-diagram with ${z_{{{\rm C}}}}$-leg does not have the external point, and the configuration in [Fig. \[causal\]]{} cannot be avoided. As the result, the time-ordered configurations for $Q$ has a generic form shown in [Fig. \[Q4\]]{}: In the sub-diagram with ${z_{{{\rm \Delta}}}}$-leg, the external point $Q$ is on the latest time, and the ${z_{{{\rm \Delta}}}}$-leg is connected to the vertices on the later time (need not be on the earliest time of the sub-diagram). On the other hand, the sub-diagram with ${z_{{{\rm C}}}}$-leg has the leg on the latest time.
Let us see the joint of $z_{{{ {\bf k} }'}{{ {\bf k} }},{{\rm \Delta}}}$-leg at $t$ and $z_{{{ {\bf q} }}{{ {\bf q} }'},{{\rm C}}}$-leg at $s$ by ${{\rm i}}\{{g^{\rm R}}{g^{\rm A}}\}^{-1}_{{{ {\bf k} }}{{ {\bf k} }'},{{ {\bf q} }}{{ {\bf q} }'}}(t,s)$. The $z_{{{ {\bf k} }'}{{ {\bf k} }},{{\rm \Delta}}}$-leg at time $t$ is produced by a pair of propagators which can be written as $$\begin{aligned}
{{\rm i}}{g^{\rm R}}_{{\tilde{{ {\bf k} }}},{{ {\bf k} }}}(t',t) {g^{\rm A}}_{{{ {\bf k} }'},{\tilde{{ {\bf k} }}'}}(t,t'')
&=& {{\rm i}}\theta(t'-t''){{\rm e}}^{-{{\rm i}}\omega_{{\tilde{{ {\bf k} }}}}(t'-t'')}
{g^{\rm R}}_{{\tilde{{ {\bf k} }}},{{ {\bf k} }}}(t'',t) {g^{\rm A}}_{{{ {\bf k} }'},{\tilde{{ {\bf k} }}'}}(t,t'')
\nonumber\\
& & +{{\rm i}}\theta(t''-t'){{\rm e}}^{{{\rm i}}\omega_{{\tilde{{ {\bf k} }}'}}(t''-t')}
{g^{\rm R}}_{{\tilde{{ {\bf k} }}},{{ {\bf k} }}}(t'',t) {g^{\rm A}}_{{{ {\bf k} }'},{\tilde{{ {\bf k} }}'}}(t,t'')
.
\label{gRgA}\end{aligned}$$ On the other hand, the $z_{{{ {\bf q} }}{{ {\bf q} }'},{{\rm C}}}$-leg at time $s$ is produced by a pair of propagators, one of which is $-{g^{\rm R}}_{{{ {\bf q} }},{\tilde{{ {\bf q} }}}}(s,s')$ or $-{g^{{{\rm C}}}}_{{{ {\bf q} }},{\tilde{{ {\bf q} }}}}(s,s')$ and the other is $-{g^{\rm A}}_{{\tilde{{ {\bf q} }}'},{{ {\bf q} }'}}(s'',s)$ or $-{g^{{{\rm C}}}}_{{\tilde{{ {\bf q} }}'},{{ {\bf q} }'}}(s'',s)$. As discussed above, ${z_{{{\rm C}}}}$-leg is non-zero only when $s > s',s''$, and the following relations hold in this case; $$\begin{aligned}
\theta(t'-s){{\rm e}}^{-{{\rm i}}\omega_{{{ {\bf q} }}}(t'-s)}
g^{{\rm R}/{{\rm C}}}_{{{ {\bf q} }},{\tilde{{ {\bf q} }}}}(s,s')
&=& g^{{\rm R}/{{\rm C}}}_{{{ {\bf q} }},{\tilde{{ {\bf q} }}}}(t',s')
,
\label{gRC}
\nonumber\\
\theta(t''-s){{\rm e}}^{{{\rm i}}\omega_{{{ {\bf q} }'}}(t''-s)}
g^{{\rm A}/{{\rm C}}}_{{\tilde{{ {\bf q} }}'},{{ {\bf q} }'}}(s'',s)
&=& g^{{\rm A}/{{\rm C}}}_{{\tilde{{ {\bf q} }}'},{{ {\bf q} }'}}(s'',t'')
.
\label{gAC}\end{aligned}$$ With the aid of Eqs. (\[gRgA\])-(\[gAC\]), we have $$\sum_{{{ {\bf k} }},{{ {\bf k} }'},{{ {\bf q} }},{{ {\bf q} }'}} \int {{\rm d}}t {{\rm d}}s\,
{{\rm i}}{g^{\rm R}}_{{\tilde{{ {\bf k} }}},{{ {\bf k} }}}(t',t) {g^{\rm A}}_{{{ {\bf k} }'},{\tilde{{ {\bf k} }}'}}(t,t'')
{{\rm i}}\{-{g^{\rm R}}{g^{\rm A}}\}^{-1}_{{{ {\bf k} }}{{ {\bf k} }'},{{ {\bf q} }}{{ {\bf q} }'}}(t,s)\,
g^{{\rm R}/{{\rm C}}}_{{{ {\bf q} }},{\tilde{{ {\bf q} }}}}(s,s')\,
g^{{\rm A}/{{\rm C}}}_{{\tilde{{ {\bf q} }}'},{{ {\bf q} }'}}(s'',s)
= -g^{{\rm R}/{{\rm C}}}_{{\tilde{{ {\bf k} }}},{\tilde{{ {\bf q} }}}}(t',s')
g^{{\rm A}/{{\rm C}}}_{{\tilde{{ {\bf q} }}'},{\tilde{{ {\bf k} }}'}}(s'',t'')
,$$ where $t',t''>s',s''$ holds. This implies that the joint of ${z_{{{\rm \Delta}}}}$- and ${z_{{{\rm C}}}}$-legs can simply be expressed as $$\raisebox{-4mm}{\epsfile{width=13mm,file=zDleg.eps}}
\:{{\rm i}}\{{g^{\rm R}}{g^{\rm A}}\}^{-1}\:
\raisebox{-4mm}{\epsfile{width=13mm,file=zCleg.eps}}
=
-\,
\raisebox{-4mm}{\epsfile{width=24mm,file=zD-zC2.eps}}
,
\label{zD-zC}$$ Note that, on the rhs of (\[zD-zC\]), the time order of the vertices is restricted unlike usual diagrams: The vertices originally connected to the ${z_{{{\rm \Delta}}}}$-leg are on later time than those originally connected to ${z_{{{\rm C}}}}$-leg. This implies that the diagram on the rhs of (\[zD-zC\]) contains only the configurations which can be separated into two parts by cutting the pair of propagators at the same instant. In this sense, we call such a time-ordered configuration ‘instantaneous-2PR configuration’. For instance, the configurations shown in [Fig. \[L4ex\]]{} (b) are instantaneous-2PR, and those in (c) are instantaneous-2PI. Summarizing, the cut-and-patch operation (\[cutpatch\]) extracts an instantaneous-2PR configuration from the original diagram with the opposite signature.
The second process of ${{\cal R}}_{2}$ is to operate cut-and-patch in all possible ways and to sum up the resultant diagrams. This process ensures that the instantaneous-2PR configuration is precisely canceled out after ${{\cal R}}_{2}$ is carried out. As it was shown above, the cut-and-patch process just restricts the time ordering of the vertices, and except the signature, the contribution produced by cut-and-patch is included in the original diagram. Then we should count how many times the same contribution appears throughout ${{\cal R}}_{2}$. Considering a configuration which is instantaneous-2PR with respect to $N$ pairs of propagators, such a configuration appears in a diagram where $k$ of the corresponding $N$ pairs of the propagators are cut-and-patched. There are ${}_{N}{\rm C}_{k}$ ways of choosing $k$ pairs, and the signature $(-1)^{k}$ is assigned. Thus through the total process of ${{\cal R}}_{2}$, the instantaneous-2PR configuration appears $\sum_{k}(-1)^{k}{}_{N}{\rm C}_{k}=0$ times.
As the result,
> the operation ${{\cal R}}_{2}$ on a diagram implies that we can eliminate the instantaneous-2PR configurations which will be produced from the original diagram.
For example, considering (\[Q3\]) in the fourth order of the perturbation, when we evaluate the diagram shown in [Fig. \[L4ex\]]{} (a), we only need to calculate the contributions of the instantaneous-2PI configurations shown in [Fig. \[L4ex\]]{} (b), and can eliminate the instantaneous-2PR ones in (b). More simple example can be seen in [Ref. ]{} (explicit calculations are shown in [Ref. ]{}), where the four-point function is calculated up to the first order of the perturbation. There appear tadpole diagrams, but their contributions are canceled when the four-point function is expressed in terms of the WDF. From the view point of our rule, the contribution from the tadpole diagrams in [Ref. ]{} can be eliminated by the ${{\cal R}}_{2}$ operation in (\[Q3\]) because all of the time-ordered configurations produced from those diagrams are instantaneous-2PR.
Quantum kinetic equation {#QKE-rule}
------------------------
Finally, we summarize the rule for deriving the QKE. The physical source ${J_{{{\rm C}}}}$ as a functional of ${z_{{{\rm C}}}}$ is obtained by setting ${z_{{{\rm \Delta}}}}=0$ in the first equation of (\[Jderiv\]) since this condition is equivalent to ${J_{{{\rm \Delta}}}}=0$. With the use of (\[Gamma3\]), it can be expressed as $$\begin{aligned}
{J_{{{\rm C}}}}[t;{z_{{{\rm C}}}}]
&=& {J^{(0)}_{{{\rm C}}}}[t;{z_{{{\rm C}}}}]
- \int {{\rm d}}s
{\left. { \frac{\delta {J^{(0)}_{{{\rm \Delta}}}}(s)}{\delta {z_{{{\rm \Delta}}}}(t)} } \,\right|_{ {z_{{{\rm \Delta}}}}=0 }}
{\left\{ {{\cal R}}_{2}{\left( \frac{\delta W}{\delta {J_{{{\rm \Delta}}}}(s)}
\right)}_{ J={J^{(0)}}[{z_{{{\rm \Delta}}}}=0] }
-{z_{{{\rm C}}}}(s)
\right\}}
,
\label{J3}\end{aligned}$$ and the QKE for the WDF ${z_{{{\rm C}}}}$ is obtained by setting ${J_{{{\rm C}}}}[t;{z_{{{\rm C}}}}]=0$.
To obtain the explicit expression of $\delta{J_{{{\rm \Delta}}}}/\delta{z_{{{\rm \Delta}}}}$ in (\[J3\]), we differentiate the identity $z={z^{(0)}}[{J^{(0)}}[z]]$ with respect to $z$, and obtain $${\left( \begin{array}{cc}
\frac{\delta {J^{(0)}_{{{\rm \Delta}}}}}{\delta {z_{{{\rm C}}}}} &
\frac{\delta {J^{(0)}_{{{\rm \Delta}}}}}{\delta {z_{{{\rm \Delta}}}}}
\\
\frac{\delta {J^{(0)}_{{{\rm C}}}}}{\delta {z_{{{\rm C}}}}} &
\frac{\delta {J^{(0)}_{{{\rm C}}}}}{\delta {z_{{{\rm \Delta}}}}}
\end{array}
\right)}_{{z_{{{\rm \Delta}}}}=0}
= {\left( \begin{array}{cc}
\frac{\delta {z_{{{\rm C}}}}^{(0)}}{\delta {J_{{{\rm \Delta}}}}} &
\frac{\delta {z_{{{\rm C}}}}^{(0)}}{\delta {J_{{{\rm C}}}}}
\\
\frac{\delta {z_{{{\rm \Delta}}}}^{(0)}}{\delta {J_{{{\rm \Delta}}}}} &
\frac{\delta {z_{{{\rm \Delta}}}}^{(0)}}{\delta {J_{{{\rm C}}}}}
\end{array}
\right)}^{-1}_{ J={J^{(0)}}[{z_{{{\rm \Delta}}}}=0] }
.
\label{dJ/dz}$$ Because ${z_{{{\rm C}}}}^{(0)}[t;{J_{{{\rm \Delta}}}},{J_{{{\rm C}}}}]$ and ${z_{{{\rm \Delta}}}}^{(0)}[t;{J_{{{\rm \Delta}}}},{J_{{{\rm C}}}}]$ are given by $-{\tilde{g}^{{{\rm C}}}}(t,t)$ and $-{{\rm i}}{\tilde{g}^{{{\rm \Delta}}}}(t,t)$, respectively, their derivatives can be calculated using the definitions (\[gCtil\]) and (\[gDtil\]). Then we can see that the rhs of (\[dJ/dz\]) is nothing but ${{\rm \Delta}}_{2}^{-1}$ (multiplied by ${{\rm i}}\hbar$) given in (\[Del2inv\]), and $\delta{J^{(0)}_{{{\rm \Delta}}}}/\delta{z_{{{\rm \Delta}}}}$ is reduced to $$\begin{aligned}
{\left. { \frac{\delta {J^{(0)}}_{{{ {\bf q} }'}{{ {\bf k} }'},{{\rm \Delta}}}(s)}{\delta {z_{{{ {\bf k} }}{{ {\bf q} }},{{\rm \Delta}}}}(t)} } \,\right|_{ {z_{{{\rm \Delta}}}}=0 }}
&=&-\hbar{\left\{ {g^{\rm A}}{g^{\rm R}}\right\}}^{-1}_{{{ {\bf k} }}{{ {\bf q} }},{{ {\bf k} }'}{{ {\bf q} }'}}(t,s)
\nonumber\\
&=&-{\left\{ \hbar{\partial}_{t}+{{\rm i}}{\left( {\epsilon_{{{ {\bf k} }}}}-{\epsilon_{{{ {\bf q} }}}}\right)} \right\}}
\delta_{{{ {\bf k} }},{{ {\bf k} }'}}\delta_{{{ {\bf q} }},{{ {\bf q} }'}}\delta(t-s)
.\end{aligned}$$ Thus, in the rhs of (\[J3\]), the last term in the braces compensates for the first term (cf. (\[J0\])), and the QKE can simply be written as $${\left\{ \hbar{\partial}_{t}+{{\rm i}}{\left( {\epsilon_{{{ {\bf k} }}}}-{\epsilon_{{{ {\bf q} }}}}\right)} \right\}}
{\left\{ {{\cal R}}_{2}{\left( {z_{{{ {\bf k} }}{{ {\bf q} }},{{\rm C}}}}[t;{J_{{{\rm C}}}}={J^{(0)}_{{{\rm C}}}}[{z_{{{\rm C}}}}]] \right)} \right\}}
= 0
.
\label{QKE}$$ The QKE can be derived by calculating the instantaneous-2PI configurations of the diagrams for ${z_{{{ {\bf k} }}{{ {\bf q} }},{{\rm C}}}}$, and by operating ${\left\{ \hbar{\partial}_{t}+{{\rm i}}{\left( {\epsilon_{{{ {\bf k} }}}}-{\epsilon_{{{ {\bf q} }}}}\right)} \right\}}$. This rule can be confirmed by the example in [Ref. ]{} (details are in [Ref. ]{}). There, it is explicitly shown that the tadpole diagrams for ${z_{{{\rm C}}}}$ are canceled through the process of inversion. According to our rule, the tadpole diagrams in [Ref. ]{} must vanish because they necessarily lead to instantaneous-2PR configurations.
Discussions
===========
We have presented a systematic method to to calculate an expectation value $Q(t)$ of some physical quantity $\hat{Q}$ as a functional of the WDF $z$. Using the propagator $-{G^{(0)}}[{J^{(0)}}[z]]$, which has a form of the GKB ansatz, the precise expression of $Q[z]$ is obtained by eliminating the instantaneous-2PR configurations from the calculations. This is due to a restriction which must be taken into account in the course of the perturbative calculation: the integration over the microscopic field variable must be carried out in a way so that the value of the WDF is fixed. Together with the QKE, which can also be expressed in a compact form by the use of instantaneous-2PI property (cf. (\[QKE\]), this method provides us with a complete framework for the quantum kinetic theory.
As pointed out in [\[problem\]]{}, the method presented here can straight forwardly be used in the GKB formalism. What we have used for the propagator is a GKB ansatz with the free-particle approximation of the spectral function $a(t,s)={g^{\rm R}}(t,s)-{g^{\rm A}}(t,s)$. The GKB ansatz is defined for a more general form of the spectral function, which implies a corresponding renormalization of the free part of the Lagrangian. Even using more generic form of the spectral function, our method is applicable if the conditions (\[gRgA\])-(\[gAC\]) are held with the replacement of the free-particle spectral function ${{\rm e}}^{-{{\rm i}}\omega(t-s)}$ by renormalized one $a(t,s)$. (These conditions are nothing but the semi-group property discussed in [Ref. ]{}.) Other parts of the proof are based on the retarded or advanced character of the propagators which is not affected by the use of generic spectral function. Thus, even in the generic GKB formalism, where the diagrammatic rule may be different due to the renormalization, the instantaneous-2PR configuration can be eliminated if the semi-group property is held for the GKB ansatz.
Note that our method is not valid for the time correlation function of $\hat{Q}$ such as $\langle \hat{Q}(t)\hat{Q}(s) \rangle$ because we have used the condition that the external point expressing $Q({\psi_{{{\rm C}}}})$ appears only once in the diagram. For the calculation of the time correlation function of the composite operator, some of the instantaneous-2PR configuration may not be canceled.
Acknowledgements {#acknowledgements .unnumbered}
================
The author is very grateful to Prof. R. Fukuda for helpful comments.
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|
---
abstract: 'We discuss ROSAT HRI X-ray observations of 33 very nearby galaxies, sensitive to X-ray sources down to a luminosity of approximately $10^{38}$ erg s$^{-1}$. The galaxies are selected from a complete, volume limited sample of 46 galaxies with $d < 7$ Mpc for which we have extensive multi-wavelength data. For an almost complete sub-sample with $M_{B} < -14$ (29/31 objects) we have HRI images. Contour maps and source lists are presented within the central region of each galaxy, together with nuclear upper limits where no nuclear source was detected. Nuclear X-ray sources are found to be very common, occurring in $\sim 35\%$ of the sample. Nuclear X-ray luminosity is statistically connected to host galaxy luminosity - there is not a tight correlation, but the probability of a nuclear source being detected increases strongly with galaxy luminosity and the distribution of nuclear luminosities seems to show an upper envelope that is roughly proportional to galaxy luminosity. While these sources do seem to be a genuinely nuclear phenomenon rather than nuclear examples of the general X-ray source population, it is far from obvious that they are miniature Seyfert nuclei. The more luminous nuclei are very often spatially extended, and HII region nuclei are detected just as often as LINERs. Finally, we also note the presence of fairly common super-luminous X-ray sources in the off-nuclear population – out of 29 galaxies we find 9 sources with a luminosity larger than 10$^{39}$ erg s$^{-1}$. These show no particular preference for more luminous galaxies. One is already known to be a multiple SNR system, but most have no obvious optical counterpart and their nature remains a mystery.'
author:
- |
P. Lira,$^1$ A. Lawrence,$^2$ R.A. Johnson,$^3$\
$^1$ Department of Physics & Astronomy, University of Leicester, Leicester LE1 7RH, UK\
$^2$ Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, Scotland\
$^2$ Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK\
bibliography:
- 'paper\_figs.bib'
title: 'Multiwavelength study of the nuclei of a volume-limited sample of galaxies I: X-ray observations'
---
galaxies: general – galaxies: active – galaxies: nuclei – X-rays: galaxies.
Introduction
============
Evidence has been growing that both (i) weak nuclear activity, and (ii) the presence of quiescent black holes, are common features of normal galaxies. The best evidence to date for (i) comes from the very thorough high S/N spectroscopic survey of 486 nearby galaxies by . They found that 86% of galaxies have emission lines; roughly a third have Seyfert 2 or LINER spectra, indicating but not proving some kind of weak AGN; and 10% have weak broad emission lines, so are almost certainly AGN [@hfs3]. These latter objects are, however, three orders of magnitude less luminous than classic Seyfert galaxies and six orders of magnitude less luminous than the most luminous quasars. They have become known as ‘dwarf AGN’. The evidence for (ii) comes from dynamical studies. The review by found evidence for ‘Massive Dark Objects’ (MDOs) in 20% of galaxies, but this can be considered a lower limit because of the difficulty of rigorously demonstrating the requirement for a dark mass. With a more relaxed analysis, have claimed that there is strong evidence for an MDO in essentially all galaxies.
Most interestingly, and also claim a correlation between the MDO mass and the stellar bulge mass, with $M_{MDO}/M_{bulge} \sim 0.003 -
0.006$. Likewise, various authors have shown evidence that AGN activity correlates with galaxy size [@gehren+etal; @hutchings+etal; @dibai+zasov; @sadler; @huchra+burg; @mcleod3]. A weakness with all these studies is that only fairly luminous galaxies are considered. As well as investigating whether feeble AGN are present in normal galaxies, it is very important to know whether AGN occur at all in feeble galaxies. Is there a minimum galaxy size below which quasar-like nuclei are simply not present? There are two known examples of AGN in relatively small galaxies: a Seyfert 1 nucleus in NGC4395 [@filippenko+sargent; @lira] and a Seyfert 2 nucleus in G1200-2038 [@kunth+etal]. Both these galaxies have $M_{B} \sim
-18$, an order of magnitude less luminous than typical $L^{\ast}$ galaxies and two orders of magnitude less luminous than the giant ellipticals which dominate the sample. Are they rare freaks, or the tip of the iceberg? Furthermore, in the local field there are are plenty of spiral galaxies an order of magnitude less luminous still than NGC4395 and below that we have true dwarf galaxies, dwarf irregulars, dwarf ellipticals, and dwarf spheroidals, which continue all the way down to $M_{B} \sim -7$. Where does AGN activity stop?
To address the issues of (a) how common weak AGN are, and (b) how activity is connected to host galaxy size and type, we have undertaken a multi-wavelength study of a complete volume-limited sample of 46 very nearby AGN, with (broad-band and emission line) optical, IR, X-ray and radio imaging, and long slit spectroscopy. In terms of survey size and spectroscopic quality it would be hard to improve on the survey. Instead, our study has several key features. (i) The multiwavelength approach gives the greatest chance of finding a weak AGN, being sensitive to a variety of signatures, and gives us diagnostic information on the nature of the objects seen. (ii) All our galaxies are closer than 7 Mpc, maximizing spatial resolution, and minimizing the detectable luminosity. (iii) The volume-limited nature (as opposed to magnitude-limited) of the sample means that we have a fairly even distribution of galaxy sizes and types, going down to extremely small galaxies. (iv) In many cases we found that the location of the true nucleus is not entirely obvious. Multi-wavelength imaging gave us a ‘shooting list’ of multiple targets for spectroscopy to ensure we did not miss the true nucleus.
This paper (Paper I) focuses on the analysis of the ROSAT HRI X-ray data for the galaxies in our sample and presents some preliminary scientific results. Section 2 describes the main characteristics of the sample. In section 3 the acquisition and analysis of the X-ray data are described. The atlas of X-ray images is found in section 4. In Section 5 we describe detailed results for individual galaxies. In section 6 we summarize and discuss some of the most interesting results and examine the correlation of X-ray luminosity and host galaxy luminosity.
In subsequent papers we will present optical-IR imaging and radio maps (Paper II), long-slit spectroscopy (Paper III), and a final analysis of results (Paper IV).
The volume-limited sample
=========================
----------------- ----------------- ---------- ------------ --------- ------- --------- --------- ------- --------
Nuclear
Galaxy Position (2000) $-M_{B}$ Morph D (Mpc) Radio Near-IR Optical X-ray Spect.
\[10pt\] NGC147 00 30 +48 14 13.71 dE5 0.65
ANDIII 00 33 +36 14 10.17 dE 0.65
NGC185 00 36 +48 04 13.94 dE3 0.65
NGC205–M110 00 38 +41 25 15.24 S0/E5 0.65
NGC221–M32 00 40 +40 36 15.06 E2 0.65
NGC224–M31 00 40 +40 60 19.69 Sb 0.65
ANDI 00 43 +37 44 10.17 dE3 0.65
NGC247 00 45 -21 02 18.32 Sc(s) 3.69
NGC253 00 45 -25 34 20.26 Sc(s) 4.77
SCULPTOR 00 58 -33 58 10.38 dE3 –
NGC404 01 07 +35 27 17.07 S0 4.34
ANDII 01 14 +33 09 10.17 dE 0.65
NGC598–M33 01 31 +30 24 18.43 Sc(s) 0.87
MAFFEII 02 33 +59 26 13.87 E3 5.43
FORNAX 02 38 -34 44 12.64 dE0 0.22
MAFFEIII 02 38 +59 23 8.08 Sbc 5.21
IC342 03 42 +67 56 19.50 S(B)cd(rs) 6.08
NGC1560 04 27 +71 46 15.95 Sd(s) 4.34
NGC2366 07 24 +69 19 17.53 SBm 6.29
NGC2403 07 32 +65 43 20.25 Sc(s) 6.73
NGC2976 09 43 +68 09 16.98 Sd 3.69
A0951+68 09 51 +68 50 11.21 dE 1.52
NGC3031–M81 09 52 +69 18 19.22 Sb(r) 2.60
LEOI 10 06 +12 33 10.87 dE3 0.22
LEOB$^{a}$ 11 11 +22 26 8.98 dE0 0.22
UGC6456 11 25 +79 16 12.19 peculiar 2.39
NGC3738 11 33 +54 48 16.78 Sd 6.08
NGC4136 12 07 +30 12 17.63 Sc(r) 6.94
NGC4144 12 07 +46 44 16.83 Scd 5.86
NGC4150 12 08 +30 41 15.30 S0/a 3.47
NGC4236 12 14 +69 45 17.50 SBd 3.26
NGC4244 12 15 +38 05 17.59 Scd 4.34
UGC7321 12 15 +22 49 14.82 Scd 5.43
NGC4395 12 23 +33 50 17.89 Sd 5.21
NGC4605 12 38 +61 53 17.80 Sc(s) 5.64
NGC4736–M94 12 49 +41 24 20.00 RSab(s) 6.08
NGC4826 12 54 +21 57 19.39 Sab(s) 5.64
NGC5204 13 28 +58 41 17.39 Sd 6.73
NGC5238 13 33 +51 52 15.79 S(B)dm 6.73
NGC5236–M83 13 34 -29 37 20.16 SBc(s) 5.43
NGC5457–M101 14 01 +54 36 21.22 Sc(s) 7.60
URSAMINOR 15 08 +67 23 7.40 dE4 –
DRACO 17 19 +57 58 7.00 dE0 –
NGC6503 17 50 +70 10 18.28 Sc(s) 6.94
NGC6946 20 34 +59 59 19.66 Sc(s) 7.38
NGC7793 23 55 -32 52 18.43 Sd(s) 4.12
----------------- ----------------- ---------- ------------ --------- ------- --------- --------- ------- --------
To study the occurrence of nuclear activity in galaxies to the lowest possible level, a volume-limited sample of galaxies was defined. The sample contains the nearest examples of various morphological types of galaxies, and a representative range of intrinsic luminosities (Johnson 1997; see also Paper II).
The starting point was the Kraan-Korteweg & Tamman Catalogue [@kraan-korteweg+tamman; @kraan-korteweg] with all galaxies with V $<$ 500 km/s giving a distance-complete sample within $d=0.35 \times
d_{\rm Virgo}$ (after applying a virgocentric flow model with an infall velocity of the Local Group equal to 220 km s$^{-1}$). With $d_{\rm Virgo} = 21.5$ Mpc, the sample is complete for nearby galaxies within 7.6 Mpc. The catalogue is not complete for intrinsically faint galaxies without velocity information, but it includes all galaxies down to $M_{B} \sim -13$. Comparing the distribution of absolute magnitudes of the galaxies in the KKT with the galaxies in the Revised Shapley Ames (magnitude-limited) catalogue [@sandage+tamman] it is found that the RSA sample of galaxies presents a strong peak around $M_{B}=-21$ and has less than 10% of the objects below $M_{B}=-18$, while the KKT sample shows a broad distribution with a maximum around $M_{B}=-16$ and a significant number of galaxies down to magnitude $M_{B}=-13$.
The observations were restricted to objects with $\delta >
-35^{\circ}$. Because the aim was to observe objects with well-formed nuclei, galaxies classified as Sdm, Sm and Irr were removed from the sample. This gave a basic sample of 46 nearby galaxies (26 spirals, 5 E/S0, 12 dwarf ellipticals and 3 blue compact dwarfs, of which two are classified as Sm spirals and one is a peculiar object). The assumed distances for the galaxies are based on the observed recession velocities and the assumed Virgo flow model.
Table \[tab:allobs\] shows the resulting sample on which our study is based, the properties of the galaxies concerned, and a summary of the observations available to date. Note that only 34/46 galaxies have available X-ray data. However, we can define a very useful sub-sample of all those galaxies with $M_{B} < - 14$ (these are all disk systems, except for the elliptical galaxy NGC221). Of these, 29/31 have X-ray data – only UGC7321 and NGC7793 are missing. In addition this paper describes observations of five further dwarf galaxies: NGC147, NGC185, UGC6456, LeoB, and A0951+68. Only one of these has a detected X-ray source. The analysis we present in section 6 is based on the almost-complete subsample with $M_{B} < -14$.
High resolution X-ray imaging
=============================
Observations
------------
The data reported here come from pointed observations carried out with the ROSAT HRI. Table \[tab:hriobs\] shows those galaxies in the volume-limited sample with available data. Observations for 17 galaxies were awarded to this project through Announcement of Opportunity (AO) calls. For 13 objects data were retrieved from the public archives using [arnie]{} (a World Wide Web interface to the databases and catalogues supported by the Leicester Database and Archive Service, [ledas]{}). Galaxies are identified in table \[tab:hriobs\] with an ‘N’ (new data allocated to this project) or a ‘P’ (public data retrieved from the archive). Since NGC4826 has both new and archive observations a total of 29 objects are covered. The ROSAT HRI data have been reduced and analyzed as described below.
----------------- ------------ ---------------- ---------- -------- --------- ------------ ---------------- ---------- --------
Galaxy ROR Number Observing Date Livetime Origin Galaxy ROR Number Observing Date Livetime Origin
\[10pt\] NGC147 400744 19/01/95 14616.6 P NGC4236 600763 01/12/94 7543.2 N
NGC185 400743 19/01/95 20881.3 P 600763 10/04/95 3390.6 N
NGC205 600816 04/08/96 27842.5 P NGC4244 702724 20/06/96 8630.9 N
NGC221 600600 19/07/94 12533.5 P NGC4395 702725 23/06/96 11252.5 N
NGC247 600622 29/06/94 33701.1 P NGC4605 702729 16/06/96 2158.4 N
600622 12/06/95 17681.8 P NGC4736 600678 07/12/94 111870.5 P
NGC404 703894 04/01/97 23562.6 N 600769 25/12/94 27033.8 P
IC342 600022 13/02/91 18990.4 P NGC4826 600715 09/07/95 10042.8 P
NGC1560 702727 08/03/96 17287.5 N 703900 12/06/97 9248.9 N
NGC2366 702732 31/03/96 31465.5 N 703900 08/01/98 5342.2 N
NGC2403 600767 18/09/95 26244.7 P NGC5204 702723 31/12/95 14683.0 N
NGC2976 600759 04/10/94 26238.1 P 702723 02/05/96 13554.1 N
600759 12/04/95 23244.3 P NGC5236 600024 20/01/93 23316.5 P
A0951+68 703895 29/03/97 14240.2 N 600024 30/07/94 23920.5 P
UGC6456 703896 01/04/97 13027.7 N NGC5238 702733 13/06/96 23163.1 N
NGC3738 703897 01/11/97 13711.2 N NGC5457 600092 09/01/92 18452.8 P
NGC4136 702734 25/06/96 1951.0 N 600383 10/12/92 32361.1 P
NGC4144 703898 11/11/97 11083.4 N NGC6503 600618 08/03/94 14640.6 N
NGC4150 600762 25/12/94 3730.1 N NGC6946 600501 14/05/94 59885.3 P
600762 23/05/95 10353.8 N 600718 13/08/95 21514.3 P
----------------- ------------ ---------------- ---------- -------- --------- ------------ ---------------- ---------- --------
For several galaxies more than one exposure was available. This could be because several observations were requested, or because a single request was scheduled as several observations months apart. In table \[tab:hriobs\] observations of the same galaxy associated with different requests are distinguished by their ROR (ROSAT Observation Request) numbers. If a single request was scheduled in more than one observation only one ROR is given but the different observing dates distinguish between the individual observations.
Data analysis
-------------
The aim of the analysis is to extract fluxes and upper limits for all the sources within a $\sim 6\arcmin \times 6 \arcmin$ region centered on the position of the galaxy. In this way the X-ray field matches the size of the optical frames obtained using the Jacobus Kapteyn Telescope (JKT) (Johnson 1997; Paper II). For each galaxy an isointensity contour map was overlaid onto the optical image. No attempt has been made to systematically study the temporal behavior of the sources, although some interesting individual cases will be mentioned later (section 5).
The reduction and analysis of the ROSAT HRI data was done using [pros]{}, an X-ray analysis software system designed to run under the Image Reduction and Analysis Facility ([iraf]{}).
For each galaxy with more than one exposure, the different images were coadded, merging their respective livetimes. Using the sources in the field, the images were inspected and corrected for systematic shifts in the positions before co-adding.
### Source identification
For the identification of all point sources in the images we used the ROSAT Standard Analysis Software System (SASS) source list as a starting point. The following steps were followed to identify all source candidates in the X-ray images:
1. the images were binned into 2 pixels and smoothed using gaussians with $\sigma = 2\arcsec, 4\arcsec$ and $8\arcsec$; the smoothed images were inspected visually to evaluate the existence of sources not reported by SASS and to exclude obvious spurious sources;
2. whenever more than one exposure was available for a target the different files were compared so that variable sources that might appear weak in the coadded image could be recognized in the individual frames where they might have been more luminous;
3. optical images with X-ray isocontours were generated in order to look for weak sources with optical counterparts.
For each source candidate a background subtracted count number was obtained from the coadded imaged. An aperture of 10 was adopted for all point sources which should encircle $\sim$ 99% of the photons at 0.2 keV and $\sim$ 86% of the photons at 1.7 keV for nearly on-axis sources [@david+etal]. The pixel coordinates were obtained from the SASS report or by using a separate centroid algorithm. To estimate the background one or two large circular regions free of evident X-ray sources and away from the galaxy were used. These regions normally lay outside the $\sim 6\arcmin \times 6
\arcmin$ central image.
A final list of sources for each galaxy was created with all sources that comply with at least one of the following criteria:
1. have a signal-to-noise ratio above 2.5 in the coadded image,
2. have a signal-to-noise ratio above 2.5 in at least one of the individual images,
3. or have a low signal-to-noise ratio (between 1.5 and 2.5) and an optical counterpart.
For sources with an extended component an additional, larger aperture was used to estimate its contribution. The size of the aperture was determined from the radial profile of the source. For galaxies without nuclear X-ray source detections, upper limits were established using a 10 aperture located at the best estimate of the nuclear position from our optical images. They were computed as $2\sigma$ limits (ie., 0.9772 CL) assuming Bayesian statistics [@kraft]. To find the count rates the total livetime of the coadded images was used.
----------------- ------------- -------------- ---------------- ---------------- ---------------- ----------------
Galaxy Size $\log N_{H}$
(deg$^{2}$) (cm$^{-2}$) $kT = 0.1$ keV $kT = 1.0$ keV $\alpha = 1.5$ $\alpha = 1.0$
\[10pt\] NGC147 0.091 21.02 24.71 1.21 2.26 1.42
NGC185 0.098 21.04 25.02 1.22 2.27 1.42
NGC205 0.148 20.82 10.87 1.17 2.07 1.37
NGC221 0.038 20.81 10.58 1.17 2.07 1.37
NGC247 0.129 20.17 1.96 1.01 1.36 1.14
NGC404 0.016 20.70 8.28 1.15 1.98 1.34
IC342 0.270 21.48 43.83 1.30 2.48 1.48
NGC1560 0.017 21.06 25.34 1.22 2.27 1.43
NGC2366 0.023 20.59 4.62 1.11 1.78 1.29
NGC2403 0.170 20.62 5.23 1.12 1.83 1.30
NGC2976 0.011 20.65 6.05 1.13 1.89 1.32
A0951+68 – 20.63 5.48 1.12 1.85 1.31
UGC6456 0.001 20.56 4.15 1.10 1.74 1.27
NGC3738 0.004 20.01 1.67 0.99 1.27 1.10
NGC4136 0.014 20.20 2.03 1.02 1.38 1.15
NGC4144 0.008 20.16 1.94 1.01 1.36 1.14
NGC4150 0.004 20.19 2.01 1.02 1.38 1.15
NGC4236 0.112 20.26 2.20 1.03 1.43 1.16
NGC4244 0.036 20.27 2.23 1.03 1.43 1.17
NGC4395 0.123 20.12 1.85 1.00 1.33 1.13
NGC4605 0.011 20.26 2.20 1.03 1.43 1.16
NGC4736 0.087 20.15 1.92 1.01 1.35 1.13
NGC4826 0.044 20.42 2.88 1.06 1.57 1.22
NGC5204 0.013 20.18 1.98 1.01 1.37 1.14
NGC5236 0.100 20.63 5.48 1.12 1.85 1.31
NGC5238 0.100 20.04 1.71 0.99 1.29 1.11
NGC5457 0.618 20.07 1.76 1.00 1.30 1.11
NGC6503 0.013 20.61 5.01 1.12 1.82 1.30
NGC6946 0.094 21.31 31.90 1.26 2.37 1.45
----------------- ------------- -------------- ---------------- ---------------- ---------------- ----------------
### Astrometry checks
In order to check the absolute astrometry of the HRI observations we used bright X-ray sources in the field of view of each image and looked for optical counterparts that would signal the presence of large shifts in the astrometric solutions. However, in many cases the only bright sources available were found at large off-axis angles. Given the dependancy of the PSF with distance from the centre of the HRI optical axis [@david+etal], it is important to establish if the position of these sources could be reliably recovered.
To establish the accuracy with which the position of X-ray sources could be recovered we examined two HRI observations for which a large number of optical couterparts for the X-ray sources has been found: one of the Lockman Hole observations (ROR 701867) and one of the ROSAT Deep Surveys centered in the direction $\alpha = 13^{h} 34^{m} 36^{s}$ and $\delta = 37\degr 54\arcmin 36\arcsec$ (ROR 900717). The optical identifications of the X-ray sources were obtained from and .
We used a total of 12 sources from the Lockman Hole and the ROSAT Deep Survey image, with off-axis angles ranging from 4 to 14.5 arcmin, and count numbers ranging from $\la$ hundred to a few thounsands. It was found that the position of the X-ray sources could be recovered with an accuracy better than 5 arcsec (average of 2.8 arcsec) regardless of the count numbers and off-axis angle of the source. Therefore, by using optical identifications (IDs) of the X-ray sources found in the field of view of our HRI observations, it should be possible to detect shifts in the astrometric solutions larger than 5 arcsec.
We checked the astrometry of our HRI observations for all those images that showed X-ray sources that could be associated with the galaxies. To look for optical counterparts large DSS images were used and inspected carefully to identify as many candidate IDs as possible. The celestial positions for the IDs were obtained usign a centroid algorithm and compared with the positions of the X-ray sources. The results can be grouped into two categories as follows: fields for which at least two secure optical IDs were found (NGC247, NGC2976, NGC5457, NGC6503, NGC6946, UGC6456); fields for which only one secure ID was found (IC342, NGC221, NGC404, NGC2403, NGC4136, NGC4150, NGC4236, NGC4395, NGC4736, NGC4826, NGC5204, NGC5236).
The analysis showed that most astrometric solutions were accurate to within 5 arcsec. Exceptions were NGC2976, for which the observation obtained on the 12/04/1995 presented a shift of $\sim 6$ arcsec, NGC4236 for which both observations presented a significant shift (notice however, that this result is based on only one optical ID), and NGC6946 – ROR 600718, which was corrected by a shift $\sim 7$ arcsec.
### X-ray fluxes and luminosities
The conversion of the HRI count rates to fluxes was done assuming a Bremsstrahlung spectrum with $kT = 5$ keV, an energy range 0.1–2.4 keV, and a Galactic line of sight absorption derived from the 21 cm line of atomic hydrogen [@stark+etal], listed in table \[tab:nh\_conv\]. This choice of parameters is justified by typical galactic soft X-ray spectral properties [@kim+fabbiano+trinchieri]. Fluxes were converted to luminosities assuming the distances in table \[tab:allobs\]. Fluxes and luminosities inferred from a Bremsstrahlung spectrum with $kT = 1.0$ or 0.1 keV, or a power law model with index $\alpha = 1.0$ or 1.5, can be calculated using the conversion factors in table \[tab:nh\_conv\] for each galaxy.
Contour maps
------------
Isointensity contour maps were produced for all the coadded images. All the files were binned into $2\arcsec \times 2\arcsec$ pixels and then smoothed using a gaussian with $\sigma = 4\arcsec$. The binning and gaussian sizes were chosen so that the representation of the sources was consistent with the statistical asssesment of their strength. Contours were drawn at $2.5^{n}$ times the standard deviation in the smoothed background, where $n = 1, 2, 3$, and so on. With this selection of contour intensities bright sources do not present contour crowding. For each map, the background fluctuation was calculated for the same region used earlier (when measuring source fluxes in the raw data).
The atlas
=========
--------------------- -------- ------- ----------------------------------- ---------------------------------- -----------------------------------------------------
Galaxy Source S/N Corrected Flux Luminosity Comments
(ergs s$^{-1}$ cm$^{-2}$) (ergs s$^{-1}$)
\[10pt\] NGC205 X1 3.82 $7.93 {\mathrel{\times}}10^{-14}$ $4.01 {\mathrel{\times}}10^{36}$ Probably not associated with NGC205
\[6pt\] NGC221 X1 10.60 $6.37 {\mathrel{\times}}10^{-13}$ $3.22 {\mathrel{\times}}10^{37}$
\[6pt\] NGC247 X1 22.39 $7.37 {\mathrel{\times}}10^{-13}$ $1.20 {\mathrel{\times}}10^{39}$
X2 5.99 $9.15 {\mathrel{\times}}10^{-14}$ $1.49 {\mathrel{\times}}10^{38}$
\[6pt\] NGC404 X1 3.34 $7.48 {\mathrel{\times}}10^{-14}$ $1.69 {\mathrel{\times}}10^{38}$ Nuclear
\[6pt\] IC342$^{a}$ X1 2.31 $9.58 {\mathrel{\times}}10^{-14}$ $4.24 {\mathrel{\times}}10^{38}$ Low S/N (see text); source 6 in BCT
X2 9.27 $6.07 {\mathrel{\times}}10^{-13}$ $2.68 {\mathrel{\times}}10^{39}$ Nuclear; marginally extended; source 8 in BCT
X3 6.48 $3.44 {\mathrel{\times}}10^{-13}$ $1.52 {\mathrel{\times}}10^{39}$ Possible optical counterpart; source 9 in BCT
\[6pt\] NGC2403 X1 15.19 $5.56 {\mathrel{\times}}10^{-13}$ $3.01 {\mathrel{\times}}10^{39}$
X2 2.92 $5.67 {\mathrel{\times}}10^{-14}$ $3.07 {\mathrel{\times}}10^{38}$
X3 7.88 $1.89 {\mathrel{\times}}10^{-13}$ $1.02 {\mathrel{\times}}10^{39}$ Possible very faint optical counterpart
X4 2.08 $4.29 {\mathrel{\times}}10^{-14}$ $2.33 {\mathrel{\times}}10^{38}$ Low S/N; giant HII region [@drissen+roy]
\[6pt\] NGC2976 X1 8.75 $1.32 {\mathrel{\times}}10^{-13}$ $2.15 {\mathrel{\times}}10^{38}$
\[6pt\] A0951+68 X1 4.79 $1.06 {\mathrel{\times}}10^{-13}$ $4.00 {\mathrel{\times}}10^{37}$ Probably not associated with A0951+68
\[6pt\] UGC6456 X1 13.85 $8.82 {\mathrel{\times}}10^{-13}$ $6.03 {\mathrel{\times}}10^{38}$
\[6pt\] NGC3738 X1 4.74 $1.18 {\mathrel{\times}}10^{-13}$ $5.22 {\mathrel{\times}}10^{38}$ Probably not associated with NGC3738
X2 2.46 $5.25 {\mathrel{\times}}10^{-14}$ $2.32 {\mathrel{\times}}10^{38}$ Probably not associated with NGC3738
\[6pt\] NGC4136 X1 2.15 $2.19 {\mathrel{\times}}10^{-13}$ $1.26 {\mathrel{\times}}10^{39}$ Diffuse blue optical counterpart
\[6pt\] NGC4144 X1 3.09 $8.77 {\mathrel{\times}}10^{-14}$ $3.60 {\mathrel{\times}}10^{38}$
\[6pt\] NGC4150 X1 15.84 $1.02 {\mathrel{\times}}10^{-12}$ $1.88 {\mathrel{\times}}10^{45}$ Background quasar ($z=0.52$); see section 5;
flux obtained assuming a power law spectrum
with $\alpha=1.0$
\[6pt\] NGC4236 X1 2.61 $7.24 {\mathrel{\times}}10^{-14}$ $9.21 {\mathrel{\times}}10^{37}$
\[6pt\] NGC4395 X1 1.67 $3.73 {\mathrel{\times}}10^{-14}$ $1.21 {\mathrel{\times}}10^{38}$ Nuclear; low S/N; flux obtained assuming a
power law spectrum with $\alpha=1.0$; see chapter 4
\[6pt\] X2 11.85 $6.46 {\mathrel{\times}}10^{-13}$ $2.10 {\mathrel{\times}}10^{39}$
NGC4736 X1 5.78 $3.40 {\mathrel{\times}}10^{-14}$ $1.51 {\mathrel{\times}}10^{38}$
X2 18.78 $1.46 {\mathrel{\times}}10^{-13}$ $6.46 {\mathrel{\times}}10^{38}$
X3 65.32 $1.38 {\mathrel{\times}}10^{-12}$ $6.11 {\mathrel{\times}}10^{39}$ Nuclear
X3 84.49 $4.70 {\mathrel{\times}}10^{-12}$ $2.08 {\mathrel{\times}}10^{40}$ Total nuclear emission (r = 100)
NGC4826 X1 9.58 $2.41 {\mathrel{\times}}10^{-13}$ $9.19 {\mathrel{\times}}10^{38}$ Nuclear
X1 13.03 $7.73 {\mathrel{\times}}10^{-13}$ $2.94 {\mathrel{\times}}10^{39}$ Total nuclear emission (r = 40)
X2 2.27 $3.98 {\mathrel{\times}}10^{-14}$ $1.52 {\mathrel{\times}}10^{38}$ Probably not associated with NGC4826
\[6pt\] NGC5204 X1 25.23 $1.08 {\mathrel{\times}}10^{-12}$ $5.86 {\mathrel{\times}}10^{39}$ Faint optical counterpart
\[6pt\] NGC5236 X1 4.46 $5.83 {\mathrel{\times}}10^{-14}$ $2.06 {\mathrel{\times}}10^{38}$ Variable
X2 6.07 $8.23 {\mathrel{\times}}10^{-14}$ $2.90 {\mathrel{\times}}10^{38}$
X3 26.69 $8.75 {\mathrel{\times}}10^{-13}$ $3.09 {\mathrel{\times}}10^{39}$ Nuclear
X3 38.48 $3.01 {\mathrel{\times}}10^{-12}$ $1.06 {\mathrel{\times}}10^{40}$ Total nuclear emission (r = 100)
X4 5.09 $6.69 {\mathrel{\times}}10^{-14}$ $2.36 {\mathrel{\times}}10^{38}$
X5 3.78 $4.95 {\mathrel{\times}}10^{-14}$ $1.75 {\mathrel{\times}}10^{38}$
X6 8.60 $1.31 {\mathrel{\times}}10^{-13}$ $4.61 {\mathrel{\times}}10^{38}$
X7 5.01 $8.45 {\mathrel{\times}}10^{-14}$ $2.98 {\mathrel{\times}}10^{38}$ Variable
X8 4.13 $5.38 {\mathrel{\times}}10^{-14}$ $1.90 {\mathrel{\times}}10^{38}$ Variable
\[6pt\] NGC5457 X1 4.13 $3.88 {\mathrel{\times}}10^{-14}$ $2.68 {\mathrel{\times}}10^{38}$ Nuclear
X2 3.52 $3.32 {\mathrel{\times}}10^{-14}$ $2.30 {\mathrel{\times}}10^{38}$
X3 4.80 $4.56 {\mathrel{\times}}10^{-14}$ $3.15 {\mathrel{\times}}10^{38}$
X4 3.31 $3.16 {\mathrel{\times}}10^{-14}$ $2.19 {\mathrel{\times}}10^{38}$
--------------------- -------- ------- ----------------------------------- ---------------------------------- -----------------------------------------------------
\[tab:fluxes\]
------------------ -------- ------- ----------------------------------- ---------------------------------- -----------------------------------------
Galaxy Source S/N Corrected Flux Luminosity Comments
(ergs s$^{-1}$ cm$^{-2}$) (ergs s$^{-1}$)
\[10pt\] NGC6503 X1 3.01 $7.86 {\mathrel{\times}}10^{-14}$ $4.53 {\mathrel{\times}}10^{38}$
X2 2.51 $6.47 {\mathrel{\times}}10^{-14}$ $3.73 {\mathrel{\times}}10^{38}$ Probably off-nuclear
X2 3.47 $2.69 {\mathrel{\times}}10^{-13}$ $1.55 {\mathrel{\times}}10^{39}$ Total emission (r = 30)
\[6pt\] NGC6946 X1 11.85 $2.05 {\mathrel{\times}}10^{-13}$ $1.34 {\mathrel{\times}}10^{39}$
X2 2.67 $4.19 {\mathrel{\times}}10^{-14}$ $2.73 {\mathrel{\times}}10^{38}$
X3 10.73 $1.77 {\mathrel{\times}}10^{-13}$ $1.16 {\mathrel{\times}}10^{39}$ Nuclear
X3 12.80 $4.49 {\mathrel{\times}}10^{-13}$ $2.92 {\mathrel{\times}}10^{39}$ Total nuclear emission (r = 50)
X4 4.37 $6.10 {\mathrel{\times}}10^{-14}$ $3.98 {\mathrel{\times}}10^{38}$
X5 3.06 $4.59 {\mathrel{\times}}10^{-14}$ $2.99 {\mathrel{\times}}10^{38}$
X6 5.48 $7.62 {\mathrel{\times}}10^{-14}$ $4.97 {\mathrel{\times}}10^{38}$
X7 13.25 $2.45 {\mathrel{\times}}10^{-13}$ $1.59 {\mathrel{\times}}10^{39}$
X8 32.22 $1.15 {\mathrel{\times}}10^{-12}$ $7.53 {\mathrel{\times}}10^{39}$ Known SNR (see text); faint red optical
counterpart
X9 5.13 $7.11 {\mathrel{\times}}10^{-14}$ $4.64 {\mathrel{\times}}10^{38}$
------------------ -------- ------- ----------------------------------- ---------------------------------- -----------------------------------------
In this section we present an atlas of X-ray images, and tabulate all X-ray sources found satisfying the conditions described in the previous section, for the 29 galaxies with ROSAT HRI data found in table \[tab:hriobs\].
The atlas consists of maps of isointensity contour levels overlaid on optical images. The size of the images is $\sim 6\arcmin \times 6
\arcmin$ centered on the position of the galaxy. Whenever an optical JKT image was not available, a $6\arcmin \times 6 \arcmin$ Digital Sky Survey plate was used. The atlas is ordered by increasing values of right ascension.
All detected X-ray point sources fluxes are given in table \[tab:fluxes\]. The sources are ordered by increasing values of right ascension. For each source in table \[tab:fluxes\] successive columns list the measured signal to noise ratio, the flux corrected for Galactic absorption in the 0.1–2.4 keV band, and the luminosity found assuming the distances shown in table \[tab:allobs\]. Comments on particular sources are given in the last column (nuclear sources are also noted in this way). When there is evidence of an extended component, the source is listed twice: the flux measured in a 10 radius aperture is followed by the total flux observed in a larger aperture (the radius of the second aperture is given in the comments).
Upper limits ($2\sigma$) were obtained for galaxies without detected nuclear sources using a 10 aperture located at the nuclear positions and can be found in table \[tab:upperlim\].
In addition to the galaxies listed in table \[tab:hriobs\] the results for four galaxies with comprehensive studies of their X-ray emission from HRI observations have been obtained directly from the literature. Luminosities for the detected point sources in these galaxies (and located within a $\sim 6\arcmin \times 6 \arcmin$ region centered on the nuclei) are shown in table \[tab:literature\]. These values will be used during the analysis in chapter 6.
----------------- --------------------------- -----------------
Galaxy Corrected Upper Limit Log luminosity
(ergs s$^{-1}$ cm$^{-2}$) (ergs s$^{-1}$)
\[10pt\] NGC147 $6.05\times10^{-14}$ 36.49
NGC185 $5.28\times10^{-14}$ 36.43
NGC205 $6.80\times10^{-14}$ 36.54
NGC221 $5.59\times10^{-14}$ 36.45
NGC247 $2.38\times10^{-14}$ 37.59
NGC1560 $6.00\times10^{-14}$ 38.13
NGC2366 $2.92\times10^{-14}$ 38.14
NGC2403 $4.73\times10^{-14}$ 38.41
NGC2976 $4.56\times10^{-14}$ 37.87
A0951+68 $3.30\times10^{-14}$ 36.96
UGC6456 $5.79\times10^{-14}$ 37.60
NGC3738 $5.13\times10^{-14}$ 38.36
NGC4136 $2.12\times10^{-13}$ 39.09
NGC4144 $5.96\times10^{-14}$ 38.39
NGC4150 $1.51\times10^{-13}$ 38.34
NGC4236 $7.51\times10^{-14}$ 37.98
NGC4244 $9.35\times10^{-14}$ 38.33
NGC4605 $3.66\times10^{-13}$ 39.15
NGC5204 $4.53\times10^{-14}$ 38.39
NGC5238 $3.45\times10^{-14}$ 38.27
NGC6503 $5.19\times10^{-14}$ 38.48
----------------- --------------------------- -----------------
------------- ------- ---------- ------- ---------- ------- ---------- -------
source log L source log L source log L source log L
\[10pt\] 27 36.46 23 37.36 14 36.91 4 37.78
31 37.27 25 37.36 15 36.89 5 37.38
32 36.41 26 36.85 16$^{a}$ 39.04 7 37.26
35 38.25 28 37.26 16$^{c}$ 39.11 9 38.22
36 36.63 29 36.90 10 37.24
37 37.49 32 37.04 12 37.24
38 36.23 33 38.47 13$^{a}$ 39.64
39 36.84 34$^{a}$ 38.82 14 37.41
40 36.44 34$^{b}$ 39.51 16 37.56
41 36.92 35 37.32
42 36.88 36 37.90
43 37.11 40 37.97
44$^{a}$ 37.32 42 37.43
46 36.25 44 36.95
47 37.35
48 37.20
49 36.83
50 37.30
52 37.29
54 37.58
55 36.50
57 37.54
60 37.35
65 36.62
67 37.00
------------- ------- ---------- ------- ---------- ------- ---------- -------
Notes on individual objects
===========================
In this section we briefly review the results on all 34 objects with high resolution X-ray data. This includes objects for which ROSAT HRI data is analyzed in this paper (tables \[tab:fluxes\] and \[tab:upperlim\]), objects for which we have used ROSAT HRI results from the literature (table \[tab:literature\]), and LeoB, which was observed with the Einstein HRI but has not been observed by the ROSAT HRI.
[**NGC147**]{}: NGC147 was observed for the first time in X-rays using the ROSAT HRI and no emission was detected. These observations have also been reported by . They give a $2\sigma$ upper limit for the flux from a point source located in the galaxy of $6
\times 10^{-14}$ ergs s$^{-1}$ cm$^{-2}$ in the 0.1–2.5 keV band-pass, which is in good agreement with the upper limit reported in table \[tab:upperlim\].
[**NGC185**]{}: As with NGC147, this galaxy was observed for the first time in X-rays and no emission was detected. These observations have also been reported by . A $2\sigma$ upper limit for the flux of a point source was found to be $4 \times
10^{-14}$ ergs s$^{-1}$ cm$^{-2}$ in the 0.1–2.5 keV band-pass, again in good agreement with the value quoted in table \[tab:upperlim\].
[**NGC205 - M110**]{}: This small elliptical galaxy was observed with the Einstein HRI and no X-ray sources were detected with a flux $> 1.8 \times 10^{-12}$ ergs s$^{-1}$ cm$^{-2}$ in the 0.5–4.0 keV band-pass [@fabbiano+etal; @markert]. The ROSAT HRI observations of NGC205 reported here give an upper limit of $6.78 \times 10^{-14}$ ergs s$^{-1}$ cm$^{-2}$ for the flux of any point source in the nuclear region. The only source seen in figure \[fig:n205xcon\] lies 2.7 away from the nucleus and is probably not associated with the galaxy.
[**NGC221 - M32**]{}: A strong off-nuclear source in NGC221 has been found with the HRI on both Einstein and ROSAT. The flux reported here is consistent with the Einstein measurements ($F_{X} = 9.1 {\mathrel{\times}}10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ in the 0.5–4.0 keV band-pass [@fabbiano+etal]). The X-ray source lies $\sim 7\arcsec$ away from the NGC221 nucleus and has no optical counterpart. ASCA observations of this source reported by reveal a hard spectrum and a flux decrease of 25 percent in two weeks, favoring the identification of the source as a single XRB. Variability by a factor 3 has also been detected during ROSAT PSPC observations [@supper]. An upper limit for the nuclear X-ray emission is given in table \[tab:upperlim\]. The position of the aperture used to compute the upper limit was shifted slightly from the exact position of the galactic nucleus to avoid contamination from the off-nuclear source.
Dynamical studies of the stellar rotation velocities in this galaxy have revealed the presence of a central dark massive object, probably a black hole, with mass $3 \times 10^{6} M_{\odot}$ [@bender; @vandermarel+etal], which corresponds to an Eddington luminosity of $\sim 10^{44}$ ergs s$^{-1}$. The X-ray upper limit in table \[tab:upperlim\] shows that the central object is emitting at most at $\la 10^{-8} L_{\rm Edd}$.
[**NGC224 - M31**]{}: No analysis of X-ray data for this galaxy has been done in this paper. Einstein and ROSAT HRI observations of M31 have detected over 100 individual sources with luminosities $10^{36}
\la L_{x} \la 10^{38}$ ergs s$^{-1}$ [@primini; @trinchieri+fabbiano]. The ROSAT PSPC, which is a more sensitive than the HRI (but with worse spatial resolution) gives an upper limit for a diffuse component associated with the galactic bulge of $2.6 \times 10^{38}$ ergs s$^{-1}$ [@supper].
The point sources are associated with two components, the disk and the bulge, and are strongly concentrated towards the centre. PSPC data show that the bulge and disk components account for about one and two thirds of the total emission, respectively [@supper]. Ginga (2–20 keV) and BeppoSAX (0.1–10 keV) observations have shown that this emission from the galaxy is consistent with a population of LMXRB [@makishima; @trinchieri+etal3]. A list of the sources found within $\sim 6\arcmin$ from the nucleus is shown in table \[tab:literature\]. The source coincident with the galactic nucleus has been reported to vary [@primini] and is probably an XRB.
[**NGC247**]{}: Two strong X-ray sources are seen in the southern region of the ROSAT HRI image (see figure \[fig:n247xcon\]). A $2\sigma$ upper limit for a point source located in the nuclear region of the galaxy can be found in table \[tab:upperlim\]. report the detection of 5 ROSAT PSPC sources associated with the galaxy, 3 of which lie outside our optical image. The remaining two correspond to our sources X1 and X2. No obvious optical couterparts are seen at these positions, although X1 is located at the edge of a bright star-forming region. Notice that the astrometry of the observations is thought to be accurate (section 3.2.2).
From a fit in the 0.1–2.0 keV range, found that the PSPC spectrum of X1 indicates a very soft and obscured source ($kT = 0.12$ keV, $N_{H} = 6.4 \times 10^{21}$ cm$^{-2}$). Using their spectral parameters we find that the HRI count rate corresponds to a flux in the 0.1–2.0 keV energy range, (corrected by Galactic absorption only) of $5.0 \times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$, compared with the PSPC flux of $2.1 \times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ given by . This corresponds to an increase in the source luminosity by more than a factor 2 between the PSPC and HRI observations. From the HRI count rate, the intrinsic luminosity of the source is found to be $\sim 10^{42}$ ergs s$^{-1}$ for an assumed distance to NGC247 of 3.69 Mpc, although variations in the adopted parameters can change this significantly. For example, changing the hydrogen column and temperature by 1$\sigma$ each (using the error bars in , ie., $kT = 0.15$ keV, $N_{H} = 4.7 \times 10^{21}$ cm$^{-2}$) decreases the luminosity by an order of magnitude. The very soft spectral distribution, extremely high intrinsic luminosity, and detected variability make this source a good candidate for a stellar accreting black hole.
also report PSPC observations and find a faint nuclear source with $L_{X} = 1 \times 10^{36}$ ergs s$^{-1}$, which is well below the detection limit of our HRI observations. However, due to the poor spatial resolution of the PSPC, it is not clear whether this source is coincident with the galaxy nucleus.
[**NGC253**]{}: No analysis of X-ray data for this galaxy has been done in this paper. This starburst galaxy has very complex X-ray emission. Thirty one point sources have been detected with the Einstein and ROSAT HRI [@vogler+pietsch] with luminosities of up to a few times $10^{38}$ ergs s$^{-1}$ (see table \[tab:literature\]). Analysis of HRI and PSPC observations show that the source located in the nuclear region is extended, soft and possibly variable [@vogler+pietsch]. Spectral fits to some disk sources using PSPC observations show that they are consistent with the spectra of absorbed XRBs [@read].
Extended emission is detected well above and below the galactic plane of this galaxy (see figure 11 and 12 in ) and NGC253 is a prototype object for the study of X-ray emission from starburst galaxies. Combined ASCA (FWHM $\sim 3\arcmin$) and PSPC observations show that three components are required to fit the integral spectrum of the galaxy: a power law ($\Gamma \sim 1.9$) and a two-temperatur plasma ($kT \sim 0.3$ keV and $kT \sim 0.7$) [@dahlem+etal]. The determination of more detailed physical parameters is hampered by the difficulty of measuring multiple absorbing columns, as discussed by . Their analysis of the halo diffuse emission detected with the PSPC shows that the spectrum is well fitted by a two-temperatur plasma ($kT \sim 0.1$ keV and $kT \sim 0.7$) if solar abundances are assumed. BeppoSAX observations of NGC253 show the presence of a $\sim 300$ eV FeK line at 6.7 KeV, probably of thermal origin [@persic]. Other prominent lines are also resolved with ASCA [@ptak+etal].
[**NGC404**]{}: A weak X-ray nuclear source ($F_{X} = 7.5 \times
10^{-12}$ ergs s$^{-1}$) has been detected by the ROSAT HRI in the LINER nucleus of this galaxy. An ASCA 2–10 keV upper limit of $3
\times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ has been reported by , which implies that the ROSAT flux is either dominated by very soft emission, or that the source is variable. The former idea is supported by recent UV observations of the NGC404 nucleus which show that the spectrum is dominated by stellar absorption features from massive young stars [@maoz+etal] and not by the blue, featureless continuum expected from an active nucleus.
[**NGC598 - M33**]{}: No analysis of X-ray data for this galaxy has been done in this paper. Several point sources have been detected with the Einstein and ROSAT HRI in this spiral galaxy [@markert+rallis; @schulman+bregman]. Those sources located within the central $\sim 6\arcmin \times 6\arcmin$ region are listed in table \[tab:literature\]. The most striking source is the nucleus, with a luminosity of $\ga 10^{39}$ ergs s$^{-1}$ which makes it a good AGN candidate. However, the nucleus is not detected at radio wavelengths and shows very little line emission in the optical (Schulman & Bregman 1995, Paper III). Nuclear dynamical studies also give a strict limit for a central black hole mass of $\la 5 \times 10^{4} M_{\odot}$ [@kormendy+mcclure]. ASCA observations show that the X-ray emission from this source is much softer than the typical AGN spectrum and it does not show signs of variability to within 10% on time scales of 100 minutes [@takano]. This almost certainly excludes the possibility of an active nucleus in NGC598. In fact, a disk blackbody fit to the ASCA data shows that its emission is similar to Galactic black hole candidates [@takano; @colbert]. Diffuse emission around the nucleus has been detected in ROSAT HRI as well as PSPC observations [@schulman+bregman; @long+etal; @read].
[**IC342**]{}: We have re-analyzed the ROSAT HRI data already discussed by . They found evidence for a diffuse nuclear component that could be explained as a hot interstellar medium generated by a very young nuclear starburst.
![Observed profile and model PSF for IC342 X-2 and an off-nuclear source. The solid line corresponds to a model PSF [@david+etal] scaled to match the peak of the observed profile. The dashed line corresponds to the same model PSF but with a scale factor found by minimizing a chi-squared fit to the data.[]{data-label="fig:psf.ic342"}](ic342.psf.ps "fig:") ![Observed profile and model PSF for IC342 X-2 and an off-nuclear source. The solid line corresponds to a model PSF [@david+etal] scaled to match the peak of the observed profile. The dashed line corresponds to the same model PSF but with a scale factor found by minimizing a chi-squared fit to the data.[]{data-label="fig:psf.ic342"}](ic342.psf_J.ps "fig:")
Figure \[fig:ic342xcon\] shows that three sources lie within the limits of the optical JKT image of the galaxy. The brightest source is coincident with the nucleus and visual inspection of the image suggests that it is marginally resolved. A comparison of the azimuthally averaged profile of the source and the HRI model PSF can be seen in figure \[fig:psf.ic342\] and a clear deviation from the model PSF is visible at a radius of between 7 and 13 arcsec from the centre. For comparison, an off-nuclear source is also shown, which is in good agreement with the model PSF, confirming the diffuse nuclear component.
X3 is located close to a faint knot of optical emission while X1 has no obvious optical counterpart. Notice however, that only one source was used to check the astrometry of this observation (section 3.2.2). X1 has a S/N of 2.3 (see table \[tab:fluxes\]), but is included here because it was reported by (they found a S/N of 2.8, probably due to a different background estimate).
analyzed Einstein IPC observations of the nuclear region in IC342 and argued that the emission is consistent with starburst activity. Unfortunately, the IPC was not able to resolve the three sources detected with the HRI.
[**NGC1560**]{}: This galaxy was observed for the first time in X-rays and no emission was detected in the ROSAT HRI data. A $2\sigma$ upper limit for a point source located in the nuclear region of the galaxy can be found in table \[tab:upperlim\].
[**NGC2366**]{}: No emission was detected in the ROSAT HRI observations of this galaxy. A $2\sigma$ upper limit for a point source located in the nuclear region of the galaxy can be found in table \[tab:upperlim\].
[**NGC2403**]{}: This galaxy was observed with the Einstein HRI and IPC instruments [@fabbiano+trinchieri]. Three prominent sources were identified in the IPC observations with no emission from the nuclear region of the galaxy.
The ROSAT HRI observations reported here show a total of 4 point sources associated with the galaxy. Sources X1 and X3 in figure \[fig:n2403xcon\] correspond to two of the sources reported by (their third source lies outside the JKT image). Sources X2 and X4 are about one order of magnitude fainter than X1 and X3, and were probably below the sensitivity threshold of the Einstein observations. It is confirmed that no nuclear source is present in the galaxy and a $2\sigma$ upper limit for a nuclear point source is given in table \[tab:upperlim\].
A search for SNRs in NGC2403 has yielded 35 detections [@matonick]. The position of remnant number 15 in table 2 of is coincident with the X-ray source X3 reported here (see table \[tab:fluxes\]). A very faint optical counterpart is observed at this position. If the identification is correct the SNR would belong to a class of super-luminous (probably young) remnants [@schlegel3].
Source X4 is coincident with a giant HII region. A photometric study of this region (N2403-A) reveals more than 1400 detected stars, among them 800 O-type stars and a lower limit of 23 WR stars [@drissen+roy].
The coincidence of X3 and X4 with previously known sources in NGC2403 gives support to the astrometric checks carried out for this galaxy (section 3.2.2).
[**NGC2976**]{}: This galaxy was observed for the first time in X-rays and no emission was detected in the ROSAT HRI data. A $2\sigma$ upper limit for a point source located in the nuclear region of the galaxy can be found in table \[tab:upperlim\].
The only detected X-ray source has no obvious counterpart. Notice, however, that the astrometry of the observations is highly uncertain given that only one optical identification was found in the field of view of the HRI and a significant shift was applied to one of the images (section 3.2.2).
[**A0951+68**]{}: This galaxy has been observed in X-rays for the first time. As can be seen in figure \[fig:a0951xcon\], the only detected source is probably a foreground or background object.
[**NGC3031**]{}: No analysis of X-ray data for this galaxy has been done in this paper. Nine and 30 point sources have been detected in this galaxy from observations with the Einstein and ROSAT HRI, respectively [@fabbiano; @roberts+warwick]. All Einstein sources within the galaxy D$_{25}$ isophote were detected by ROSAT with the exception of the Einstein source X1. ROSAT sources located within the central $\sim
6\arcmin \times 6\arcmin$ region are listed in table \[tab:literature\].
The central emission is dominated by the nuclear source, coincident with a low-luminosity active nucleus, with a luminosity $\la 10^{40}$ ergs s$^{-1}$ in the 0.2–4.0 keV band-pass. Einstein IPC data show that the spectrum of the nuclear source is soft, with a good fit given by thermal emission with $kT \sim 1$ keV or by a power law with index $\alpha \sim 2$ [@fabbiano]. Broad band observations obtained with BBXRT (0.5–10 keV) and ASCA (0.2–10 keV) show, however, that the nuclear emission is consistent with a power law distribution with index $\alpha \sim 1$ [@petre+etal; @ishisaki; @serlemitsos+ptak+yaqoob]. The ASCA data also revealed the presence of a broad iron K emission line similar to those seen in more luminous Seyfert galaxies [@ishisaki; @serlemitsos+ptak+yaqoob]. It must be kept in mind that due to the coarse spatial resolution of ASCA some contamination is expected from nearby sources, although estimated this to be less than 10 percent. X-ray long term and fast variability by significant factors have been reported for the nuclear source [@petre+etal; @ishisaki; @serlemitsos+ptak+yaqoob].
[**LeoB**]{}: No analysis of X-ray data for this galaxy has been done in this paper. LeoB was observed with the Einstein HRI. report that no sources were detected.
![Background quasar in NGC4150.[]{data-label="n4150_qso"}](n4150_qso.ps)
[**UGC6456**]{}: This is a blue compact galaxy. These galaxies are characterized by low metallicities, high gas content and vigorous star formation. In UGC6456, evidence of both a recent episode of strong star formation (600–700 Myr) and an older stellar population has been found [@lynds].
The JKT optical images show that the galaxy has numerous bright knots of emission surrounded by a low surface brightness outer envelope (Johnson 1997; Paper II). The knots of emission are displaced south from the geometrical center of the envelope and it is not clear whether they represent the true nuclear region of the galaxy (figure \[fig:u6456.i\]).
ROSAT PSPC observations of this galaxy show a central X-ray core and three extended structures connected to the central source [@papaderos]. This morphology was interpreted as outflows from the central region of the galaxy, powered by starburst activity. The total PSPC flux within a circular aperture of radius 3 is $1.6 {\mathrel{\times}}10^{-13}$ ergs s$^{-1}$ cm$^{-2}$.
The high resolution data reported here show a strong X-ray source located at the north-most limit of the optical emission knots, but displaced to the west with respect to the geometrical centre of the outer envelope (figure \[fig:u6456.i\]). Notice also that the astrometry of the HRI observation is fairly well established (section 3.2.2). There is no evidence of extended X-ray emission in the observations, probably due to the lower sensitivity of the HRI. The HRI flux of the point source is $\sim 9 \times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$, about 5 times more luminous than the PSPC observations. A very luminous XRB ($L_{X} \ga 10^{38}$ ergs s$^{-1}$) could be responsible for this flux variation.
[**NGC3738**]{}: This galaxy has an irregular optical appearance with several bright knots of emission, although the outer parts are quite regular. None of the bright knots seems to coincide with the geometrical center of the galaxy (Johnson 1997; Paper II).
NGC3738 has been observed in X-rays for the first time and no point sources associated with the galaxy have been found. An upper limit for the X-ray emission can be found in table \[tab:upperlim\]. Source X1 in figure \[fig:n3738xcon\] is coincident with a faint knot of optical emission and is probably a foreground or background object. Source X2 is coincident with a bright point-like object and probably corresponds to a foreground star.
[**NGC4136**]{}: This spiral galaxy has been observed for the first time in X-rays. No sources associated with the nuclear region have been found. An upper limit to the flux from a nuclear point source can be found in table \[tab:upperlim\]. A strong X-ray source is coincident with one of the spiral arms of the galaxy where several knots of emission can be seen in the optical image (see figure \[fig:n4136xcon\]). The remnant of the historic Type II supernova SN1941C seen in NGC4136 is not located close to this X-ray source [@vandyk].
[**NGC4144**]{}: This galaxy was observed for the first time in X-rays and no emission was detected in the ROSAT HRI data. A $2\sigma$ upper limit for a point source located in the nuclear region of the galaxy can be found in table \[tab:upperlim\].
[**NGC4150**]{}: A strong point-like source coincident with this galaxy was detected in the ROSAT All-Sky Survey with $F_{X} = 6
\times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ and a photon index $\Gamma =
1.41$. The emission was assumed to be from the nucleus of NGC4150 [@moran; @boller]. The high resolution image seen in figure \[fig:n4150xcon\] shows, however, that the X-ray source is more than 15 away from the galactic nucleus and has a position consistent with a knot of optical emission. Spectroscopy of the optical counterpart shows that the source is a background quasar at redshift 0.52 (figure \[n4150\_qso\]).
Figure \[fig:n4150xcon\] shows that the X-ray contours of the source are elongated in the north west direction, suggesting that [*some*]{} emission might be coming from the nuclear region of the galaxy. An estimate of the nuclear emission was obtained using a 10radius aperture located as shown in figure \[fig:n4051.ap\]. Although the observed counts have a S/R $\sim 2.6$ (and so would be considered a significant detection by the criteria defined in section 3.2.2) the measurement will be treated as an upper limit because of contamination from the nearby quasar.
[**NGC4236**]{}: This galaxy has a very low surface brightness and no obvious nucleus (Johnson 1997; Paper II). From the ROSAT HRI observations reported here no X-ray sources have been found in the central region of the galaxy. The only detected source (X1) is located in the galactic plane and might have a faint optical counterpart. Notice, however, that the astrometry of the observations is highly uncertain given that only one optical identification was found in the field of view of the HRI and a significant shift was applied to both coadded images (section 3.2.2). A $2\sigma$ upper limit for a point source located in the nuclear region of the galaxy can be found in table \[tab:upperlim\].
[**NGC4244**]{}: No emission was detected in the ROSAT HRI observations of this galaxy. A $2\sigma$ upper limit for a point source located in the nuclear region of the galaxy can be found in table \[tab:upperlim\].
[**NGC4395**]{}: This galaxy contains the faintest and nearest Seyfert 1 nucleus known today [@filippenko+sargent; @lira]. Its nuclear X-ray source is highly variable and the continuum is well fitted by a power-law distribution with photon index $\Gamma = 1.7$ [@iwasawa]. The bright source seen in figure \[fig:n4395xcon\] (X2) has no obvious optical counterpart.
![Observed profile and model PSF for the nuclear X-ray source in NGC4736. Models as in figure \[fig:psf.ic342\].[]{data-label="fig:psf.n4736"}](n4736.psf.ps)
![Observed profile and model PSF for the nuclear X-ray source in NGC4826. Models as in figure \[fig:psf.ic342\].[]{data-label="fig:psf.n4826"}](n4826.psf.ps)
[**NGC4605**]{}: This galaxy was observed for the first time in X-rays and no emission was detected in the ROSAT HRI data. A $2\sigma$ upper limit for a point source located in the nuclear region of the galaxy can be found in table \[tab:upperlim\].
[**NGC4736 - M94**]{}: Strong X-ray emission is associated with the LINER nucleus of this galaxy. An extended nuclear source can be seen in figure \[fig:n4736xcon\]. The azimuthally averaged profile of this source and the HRI model PSF are shown in figure \[fig:psf.n4736\].
The galaxy was previously imaged with the Einstein HRI. A nuclear flux of $2.0\times 10^{-12}$ ergs s$^{-1}$ cm$^{-2}$ was measured within an aperture of 60 radius in the 0.5–4.0 keV band-pass [@fabbiano+etal]. The fluxes measured from the ROSAT HRI observation using a small (r = 10) and a large aperture (r = 100) are $1.4 \times 10^{-12}$ and $4.7 \times 10^{-12}$ ergs s$^{-1}$ cm$^{-2}$ respectively (see table \[tab:fluxes\]). Since the emission is highly concentrated towards the nucleus, the difference between the 60 Einstein and 100 ROSAT fluxes cannot be explained by the different aperture sizes alone. The change in the total observed flux could be explained, however, if the central source is variable or if a substantial fraction of the emission is radiated in the very soft X-rays ($kT \la 0.5$ keV).
have recently reported on ASCA and ROSAT PSPC and HRI observations for NGC4736. They find that the nuclear emission is consistent with an unresolved source plus an extended component. PSF modeling shows that the unresolved source accounts for more than 50% of the detected emission [@roberts+etal]. The 0.1–10 keV ASCA spectrum of the emission is consistent with a power-law with index $\alpha \sim 1$ plus a softer thermal component ($kT \sim 0.1-0.6$ keV) which dominates below 2 keV.
[**NGC4826**]{}: The nucleus of NGC4826 has been classified as a transition object (a combination of a LINER and an HIIR nucleus) by . The galaxy was observed with the Einstein IPC and a nuclear flux of $7.89 \times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ was measured within a 4.5 radius aperture [@fabbiano+etal], in good agreement with the flux found in table \[tab:fluxes\]. Figure \[fig:psf.n4826\] shows the azimuthally averaged profile as observed by the ROSAT HRI. Significant extended emission is observed within $\ga 20\arcsec$ of the central peak.
[**NGC5204**]{}: Einstein IPC observations of this galaxy show strong X-ray emission with a total flux of $9.6 \times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ [@fabbiano+etal] in good agreement with the measurement given here (see table \[tab:fluxes\]). Population I XRBs were thought to be responsible for this emission since the number of OB stars inferred from IUE observations were in agreement with the X-ray luminosity [@fabbiano+panagia]. However, the HRI image (see figure \[fig:n5204xcon\]) shows that the X-ray emission is consistent with a single off-nuclear point source $\sim 17\arcsec$ away from the nucleus. Since only one optical ID was found in the field of view of the HRI, the astrometric solution of these observations is quite uncertain. However, the very good agreement in the pointing of the two coadded exposures argues in favor of a fairly accurate astrometry.
Three SNRs have been identified in NGC5204, but none of them is coincident with the position of the X-ray source [@matonick2]. Although there is no obvious optical counterpart for the source, several nearby optical knots can be seen in figure \[fig:n5204xcon\]. The spectrum of one of the candidates, which corresponds to a star forming region, will be presented in Paper III.
The extremely high luminosity of the X-ray source, together with the lack of variability observed between the Einstein and the ROSAT observations, favor an identification as a SNR from a class of super-luminous remnants (the SASS report does not find conclusive evidence of variability during the HRI observations, either). However, the null detection by argues against this hypothesis, unless the remnant has an unusually low SII/H$\alpha$ ratio (a value $\geq 0.45$ was adopted by as selection criteria to identify SNRs) or unsually high $L_{X}$/H$\alpha$ ratio (the super-luminous remnant in NGC6946 - see below - has $L_{X}$/H$\alpha \sim 15$, which is the highest value seen in these type of objects; the SNR in NGC5204 would require $L_{X}$/H$\alpha > 550$).
------------- -------------------- --------------------
Source CR (20/01/93) CR (30/07/94)
(counts ks$^{-1}$) (counts ks$^{-1}$)
\[10pt\] X1 $1.27 \pm 0.31$ $0.38 \pm 0.22$
X2 $1.18 \pm 0.31$ $1.39 \pm 0.32$
X3$^{a}$ $15.20 \pm 0.86$ $17.57 \pm 0.91$
X4 $0.75 \pm 0.27$ $0.88 \pm 0.28$
X5 $0.67 \pm 0.26$ $0.64 \pm 0.25$
X6 $2.08 \pm 0.37$ $2.43 \pm 0.39$
X7 $1.91 \pm 0.36$ $0.05 \pm 0.19$
X8 $1.31 \pm 0.32$ $0.17 \pm 0.20$
------------- -------------------- --------------------
: Count rates for the point sources observed in NGC5236 from two ROSAT HRI observations obtained in January 1993 and July 1994. $a$: nuclear source.[]{data-label="tab:n5236"}
![Observed profile and model PSF for the nuclear X-ray source in NGC5236. Models as in figure \[fig:psf.ic342\].[]{data-label="fig:psf.n5236"}](n5236.psf.ps)
[**NGC5236 - M83**]{}: The very complex X-ray emission in this galaxy can be appreciated in figure \[fig:n5236xcon\]. Several X-ray knots are distributed on top of bright, uneven extended emission. Comparison between the two ROSAT HRI observations reveals that at least three of the eight detected point sources are variable. Table \[tab:n5236\] shows the count rates observed in January 1993 and July 1994. Sources X1, X7 and X8 are not detected during the observations obtained in July 1994, but are among the brightest objects seen in January 1993.
![Observed profile and model PSF for the nuclear X-ray source in NGC5457. Models as in figure \[fig:psf.ic342\].[]{data-label="fig:psf.n5457"}](n5457.psf.ps)
reported on the Einstein HRI observations of this galaxy. From their observations (obtained in January 1980 and February 1981) only 3 sources were detected in the nuclear region of the galaxy. Two correspond to the ROSAT sources X3 (the nucleus) and X6, which are the brightest sources observed in the ROSAT HRI data (see table \[tab:n5236\]). Their fluxes in the Einstein (0.5–3.0 keV) band-pass are in good agreement with the fluxes given in table \[tab:fluxes\]. The third source had a luminosity of $2.3 \times 10^{38}$ ergs s$^{-1}$ in the Einstein band-pass (for a distance to NGC5236 of 3.75 Mpc as assumed by ), and it is not detected in the ROSAT data. It probably corresponds to a transient XRB.
report on ROSAT PSPC observations of NGC5236 obtained between January 1992 and January 1993. They find a luminosity for the nuclear source of $7 \times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$, in good agreement with the measurement in table \[tab:fluxes\]. They also detect all the point sources seen in the HRI observations, with the exception of X5, which corresponds to the faintest source in the nuclear region. This suggests that the variable sources X1, X7 and X8 were detectable for at least a year (from the beginning of 1992 until the beginning of 1993), before fading away, becoming undetectable by July 1994.
The diffuse emission from the central region of NGC5236 can be appreciated in figure \[fig:psf.n5236\]. From the PSPC observations find that the soft 0.1-0.4 keV diffuse component accounts for almost half of the total X-ray emission and argue that most of it is due to hot gas in a super bubble with radius $\sim 10-15$ kpc. Evidence of vigorous starburst activity comes from observations of the nuclear and circumnuclear regions of NGC5236 which have intricate morphologies in the UV, optical and infrared [@bohlin; @johnson; @gallais].
Finally, several historic supernovae have been observed in this galaxy, but none of them is consistent with the positions of the point X-ray sources.
[**NGC5238**]{}: This galaxy was observed for the first time in X-rays and no emission was detected in the ROSAT HRI data. A $2\sigma$ upper limit for a point source located in the nuclear region of the galaxy can be found in table \[tab:upperlim\].
[**NGC5457 - M101**]{}: Observations of this galaxy with the Einstein IPC and ROSAT PSPC have been widely reported [@trinchieri+etal2; @murphy; @snowden]. Results from an ultra-deep (229 ks) ROSAT HRI observation have become available recently [@wang+etal]. The data reported here were the result of combining 2 of the 4 images used to produce the ultra-deep observation reported by .
find a total of 51 point sources down to fluxes $\sim 6\times 10^{-15}$ ergs s$^{-1}$ cm$^{-2}$, of which about half are thought to be associated with the galaxy. The X-ray emission beautifully traces the spiral arms of the galaxy and 5 of the individual sources are associated with giant HIIRs [@murphy], but they are located outside the $\sim 6\arcmin \times 6\arcmin$ optical JKT image shown in figure \[fig:n5457xcon\].
From IPC data find a nuclear X-ray flux of $3
\times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ within a circle of 90radius for the Einstein (0.2–4.0 keV) band-pass. From the ROSAT HRI observations a nuclear flux of $4 \times 10^{-14}$ ergs s$^{-1}$ cm$^{-2}$ is obtained, an order of magnitude fainter than the IPC flux. The difference can be explained if sources X2 and X3, which are not resolved by the IPC, were contained within the large aperture used by , or by the effect of luminous and highly variable XRBs. The ROSAT HRI count rate found by for the nuclear source is identical to our value (0.67 counts ks$^{-1}$), although the inferred fluxes disagree by a factor $\sim 1.5$, probably because different spectral models were assumed.
The presence of a soft diffuse component is discussed by and . They find conclusive evidence in ROSAT PSPC observations for extended emission within the inner 7 of the galaxy. The radial profile obtained from the ROSAT HRI observations shown in figure \[fig:psf.n5457\] suggests some patchiness in the X-ray. The better signal to noise data analyzed by confirm this result.
An astonishing total of 93 SNRs have been identified in the galaxy [@matonick2]. Remnants 57 and 54 are consistent with the positions of sources X2 and X3 seen in figure \[fig:n5457xcon\]. detected X-ray emission from two further remnants, but they fall outside our JKT image.
![Observed profile and model PSF for the nuclear X-3 source (left) and the off-nuclear X-7 source (right) in NGC6946. Models as in figure \[fig:psf.ic342\].[]{data-label="fig:psf.n6946"}](n6946.psf_L.ps "fig:") ![Observed profile and model PSF for the nuclear X-3 source (left) and the off-nuclear X-7 source (right) in NGC6946. Models as in figure \[fig:psf.ic342\].[]{data-label="fig:psf.n6946"}](n6946.psf_P.ps "fig:")
[**NGC6503**]{}: The nucleus of this galaxy has been classified as a combination of a transition object and a Seyfert 2 nucleus by . The ROSAT HRI observations in figure \[fig:n6503xcon\] show extended and elongated emission close to the central region of the galaxy. This diffuse emission can be better appreciated in figure \[fig:n6503xcon.4\] where the X-ray image has been smoothed using a Gaussians with $\sigma = 8\arcsec$.
The extended source lies $\sim 10\arcsec$ away from the galactic nucleus. The astrometric solution of the HRI observation has been confirmed using two bright X-ray sources in the field with optical counterparts and therefore the location of the diffuse emission is probably off-nuclear. Notice, however, that the astrometric checks carried out in section 3.2.2 assumed a point source distribution of counts, while the source seen in NGC6503 is clearly extended, implying possible errors in the determination of the centroid of the emission. The shape and energetics of the diffuse emission could be explained within the super-wind model for starburst galaxies. More detailed observations are necessary to confirm this hypothesis.
[**NGC6946**]{}: Einstein IPC observations showed X-ray emission associated with the whole body of this galaxy. Two peaks of emission were detected, one coincident with its starburst nucleus and the other associated with a prominent northern spiral arm [@fabbiano+trinchieri]. The spectral fit to these data was consistent with a soft and a hard component, probably associated with diffuse emission and individual accreting sources respectively. The presence of diffuse emission across the disk of the galaxy was confirmed by ROSAT PSPC observations [@schlegel2] and can also be appreciated in our HRI image (figure \[fig:n6946xcon\]). The PSPC also resolved the nucleus into three sources which correspond to X3, X4 and X7 in the HRI data. Figure \[fig:psf.n6946\] shows the profile of the nuclear source X3 and of an off-nuclear source (X7). Comparing both plots it is clear that the nuclear source is extended up to $\sim 20\arcsec$ away from the central peak. From the analysis of ASCA and PSPC data, found that a composite spectrum (a Raymond-Smith plasma with $kT \sim 1$ keV plus a power-law with photon index $\Gamma \sim 2.5$) was a good fit to the observations. It should be remembered, however, that due to the poor spatial resolution of ASCA most of the sources seen in figure \[fig:n6946xcon\] were probably contained in the extraction apertures of the X-ray spectra.
Six historic SNs have been seen in this spiral galaxy (SNs 1917A, 1939C, 1948B, 1968D, 1969P, and 1980K) [@barbon+etal]. A total of 27 remnants (not including the historic SNs) have been detected by . SN1980K has been observed in X-rays [@schlegel1], but the source lies outside the $\sim 6\arcmin
\times 6 \arcmin$ JKT optical image of the galaxy. The extreme northern source seen in this galaxy (X8 in figure \[fig:n6946xcon\]) seems to belong to an extreme group of SNRs with $\ga 10^{39}$ ergs s$^{-1}$ [@schlegel3]. A faint, red ($R = 18.81; B-V = 1.43$) counterpart is seen at the position of this source in our optical images, which also corresponds to the object No 16 in the ’s list of SNRs. Recent PC2 HST images show that the optical source has an intricate morphology, with a small bright shell and what seem to be two outer loops or arcs, as can be seen in the narrow filter image (F673N) shown in figure \[fig:n6946\_snr\]. Based on this morphology, suggested that the source could be explained as two interacting remnants of different age, with the smaller shell being a young remnant colliding with the outer, older shells. However, have found no high velocity components in their optical high-dispersion spectra of the source, which is not consistent with the presence of a very young and compact SNR. From the H$\alpha$ flux they derived a kinetic energy of $\sim 7 \times
10^{50}$ ergs for the shocked gas. Although this value is somewhat larger than what is normally assumed for supernova explosions, it is still within the range of normal events.
Discussion
==========
First we summarize the most striking results, before discussing some of the key issues more carefully.
\(i) Of the 34 galaxies with X-ray data, 12 have X-ray sources associated with their nuclear regions: a 35% success rate. The first result, then, is that nuclear X-ray sources are very common. (It should be noticed that due to the astrometric uncertainties discussed in section 3.2.2, the aligment of the sources with the galactic nuclei is a tentative result for NGC404, IC342, and NGC4395. However, the presence of an active nucleus in NGC4395 – see section 5 – gives further support to this identification).
\(ii) However, it is clear that the detection rate changes markedly with host galaxy luminosity. All 12 detections are in the sub-sample of 29 galaxies with $M_{B} < -14$. This effect is discussed in more detail below.
\(iii) Of the galaxies spectroscopically classified as Seyfert, LINER, or transition object, 5/7 have detected nuclear X-ray sources, whereas only 7/22 objects with HIIR or absorption-line spectra are detected (spectroscopic classifications will be discussed in Paper III, but the same result holds using the classifications of ). However, this apparent preference for AGN only reflects the fact that nearly all the smaller galaxies are classified as HIIR. Amongst detected nuclear X-ray sources, 5/12 are classified as AGN of some kind, and 7/12 as HIIR or absorption line.
\(iv) Some of the nuclear sources are only just at the limit of detectability, but many are considerably more luminous, up to $10^{40}$ erg s$^{-1}$. Of these more luminous sources, a large fraction are clearly extended on a scale of 10 arcsec or more, corresponding to $> 150$ pc or so at the typical distance of our sample galaxies.
\(v) Many sources outside the nucleus are detected – most interestingly nine off-nuclear sources are found with luminosities exceeding $10^{39}$ erg s$^{-1}$, not easily explained as individual X-ray binaries or SNRs.
Dwarf galaxies
--------------
Of the twelve galaxies classified as dwarfs in the sample (see table \[tab:allobs\]), only five have Einstein or ROSAT HRI observations (NGC147, NGC185, A0951+68, LeoB and UGC6456). Of these only one has a positive detection of an X-ray source (UGC6456) and even this one is not nuclear. For the rest of section 6, we do not include these dwarfs in the analysis, concentrating on the sub-sample of 29 galaxies with $M_{B} < -14$.
Super-luminous X-ray off-nuclear sources
----------------------------------------
------------ --------- ---- ----------------- ---------------------------------------------
Number Host ID $\log L_{X}$ Remarks
(ergs s$^{-1}$)
\[10pt\] 1 NGC247 X1 39.08 Possible counterparts at the edge
of a bright star-forming region
2 IC342 X3 39.18 Possible ($R \sim 16$) counterpart
3 NGC2403 X1 39.47 Possible faint counterpart
4 NGC2403 X3 39.01 Possible very faint counterpart - known SNR
5 NGC4136 X1 39.10 Diffuse blue counterpart
6 NGC5204 X1 39.77 Several faint counterparts
7 NGC6946 X1 39.13 No obvious optical counterpart
8 NGC6946 X7 39.21 Possible very faint counterparts
9 NGC6946 X8 39.87 Known SNR; faint red counterpart
------------ --------- ---- ----------------- ---------------------------------------------
Outside the nucleus, remarkably luminous sources ($L > 10^{38}$ erg s$^{-1}$) are seen very frequently in galaxies spanning 2 orders of magnitude in luminosity. These are plotted against host galaxy luminosity in figure \[fig:lx\_off\_host\]. The most luminous sources ($L > 10^{39}$ erg s$^{-1}$) have been labeled with numbers and brief comments about them can be found in table \[tab:offnuc\]. Figure \[fig:lx\_off\_host\] shows that there is no obvious correlation with galaxy size.
![X-ray luminosity of off-nuclear sources as a function of the host absolute blue magnitude. The dotted line across the figure shows an estimation of the average sensitivity limit for the HRI observations. Sources with luminosities above $10^{39}$ ergs s$^{-1}$ (dash-dotted line) have been labeled. Comments on their identifications can be found in table \[tab:offnuc\].[]{data-label="fig:lx_off_host"}](lx_off_host.ps)
![Nuclear X-ray luminosity as a function of the host blue absolute magnitude. Solid symbols show unresolved detections: starlight dominated nuclei (), HIIR nuclei (), transition objects (), LINERs (), and Seyfert nuclei ($\blacktriangle$). Empty circles show the addition of X-ray flux from an extended component. Arrows correspond to 2$\sigma$ upper limits.[]{data-label="fig:lx_host"}](lx_host.ps)
The population of X-ray sources in the Milky Way and M31 does not include luminosities above $10^{38}$ ergs s$^{-1}$. The presence of luminous ($L_{X} \ga 10^{38}$ ergs s$^{-1}$) sources in the Magellanic Clouds was assumed to be a metallicity effect [@helfand; @vanparadijs+mcclintock]. However, as more galaxies were surveyed using the capabilities of the Einstein and ROSAT satellites it became clear that extremely luminous objects ($L_{X} \ga 10^{39}$ ergs s$^{-1}$) were not rare [@fabbiano2].
Next we assess the possibility that the ultra-luminous sources are actually background objects. Using the results from the ROSAT Deep Survey in the Lockman Field [@hasinger] we estimate that no more than three background sources with fluxes of $10^{-12} - 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ would be found in a 1 deg$^{2}$ field. If these targets are distant, and so obscured by the intervening galaxy, their intrinsic luminosities must be higher and the corresponding number densities even lower. Given the average projected size of the galaxies on the sky (see table \[tab:nh\_conv\] for the area contained within the D$_{25}$ ellipses) the probability that all the sources correspond to background objects is extremely small.
At least one super-luminous source has been identified as a multiple object formed by interacting SNRs in the disk of NGC6946 [@blair] (see figure \[fig:n6946\_snr\]). has recently suggested that some of the X-ray bright remnants observed in M101 could correspond to [*‘Hypernova Remnants’*]{}, a term introduced by Paczyński (1998) for super-energetic $\gamma$-ray bursts and the associated afterglow event - but see also . Other multiple object systems could be formed from XRBs, SNRs, and diffuse emission from hot bubbles of interstellar gas. Alternatively, the sources could be super-Eddington XRBs with luminosities several times greater than the Eddington limit for a $\sim 1.4 M_{\odot}$ neutron star. The high luminosities in this case are explained as anisotropic emission from neutron stars in binary systems with very strong magnetic fields [@vanparadijs+mcclintock], or as black hole XRBs.
![Nuclear H$\alpha$ luminosities versus host galaxy absolute magnitude for galaxies with HII region nuclei. Data from .[]{data-label="fig:hiir_la_host"}](hiir_la_host.ps)
![Nuclear broad H$\alpha$ luminosities versus host galaxy absolute magnitude for galaxies with Seyfert nuclei. Data from .[]{data-label="fig:seyfert.b_la_host"}](sey.b_la_host.ps)
![Nuclear narrow H$\alpha$ luminosities versus host galaxy absolute magnitude for galaxies with Seyfert nuclei. Data from .[]{data-label="fig:seyfert_la_host"}](sey.n_la_host.ps)
![Nuclear radio power as a function of host galaxy absolute magnitude (adapted from ).[]{data-label="fig:lr_host"}](lr_host.ps)
have recently analyzed ASCA data from three nearby spiral galaxies. One of the galaxies is M33 and these observations have already been discussed in section 5. The other two galaxies (NGC1313 and NGC5408) each harbor a luminous source ($L_{X} \ga
10^{39}$ ergs s$^{-1}$) displaced $\sim 50 \arcsec$ from the nucleus. The fit to the X-ray spectra of these sources shows that at least two components, a steep power law and a ‘Disk Black Body’, are required to explain the observations, suggesting that these are accretion driven systems. The parameters from the model imply BH masses of $100 - 10,000 M_{\sun}$, intermediate between those seen in XRB and AGN. Given the observed X-ray luminosities and the estimated masses, these objects are not super Eddington sources.
It is clear, then, that super-luminous point sources are common. However, unlike nuclear sources discussed in the next section, there seems to be no correlation between the probability of having a super-luminous off-nuclear source and the brightness of the parent galaxy.
Correlation between nuclear X-ray luminosity and host galaxy luminosity
-----------------------------------------------------------------------
Figure \[fig:lx\_host\] shows the nuclear X-ray luminosities and upper limits for the galaxies in the sample as a function of the blue absolute magnitude of the host. Different symbols have been used to show the optical classification of the nuclear emission as starlight, HII regions, LINERs, transition objects (i.e., a HIIR and LINER composite), and Seyferts (Paper III). As was suggested before, it is found that the probability of detection of a nuclear X-ray source correlates strongly with host galaxy luminosity: the rate of detection is 0% (0/7) below $M_{B} \sim -17$, 25% (3/12) for $-17 < M_{B} <
-19$, and 90% (9/10) for galaxies with $M_{B} < -19$.
have collected data from ROSAT HRI observations for 39 nearby spiral and elliptical galaxies, of which 9 objects are common to our sample. Their results confirm what is seen in figure \[fig:lx\_host\]: for those sources detected within $\sim 6$ arcsecs from the optical center of the galaxy, and therefore likely to be nuclear sources, the detection rate is nearly 100 percent for hosts with $M_{B} \la -20$.
The distribution of points does not suggest a straightforward correlation, but rather a large spread with an upper envelope. In other words, large galaxies can have luminous or feeble nuclei, but small galaxies can only have feeble nuclei. A similar upper-envelope effect has been claimed for the host galaxy luminosities of quasars and Seyferts [@mcleod3]. Given this correlation, the lack of detection of nuclear sources in small galaxies does not necessarily mean they cannot form AGN at all – it may be that they are exceedingly weak nuclear sources.
It is not clear how to model the envelope-like relationship between nuclear and host galaxy luminosities. However, a simple visual comparison with other related samples reveals a rather striking effect. The upper envelope for our sample seems to go roughly as $L_{x} \propto L_{\rm host}^{1.5}$. Figures \[fig:hiir\_la\_host\], \[fig:seyfert.b\_la\_host\] and \[fig:seyfert\_la\_host\] show nuclear emission line data taken from the study of . The nuclear H$\alpha$ luminosity from galaxies classified as HII regions shows a very similar correlation with host galaxy luminosity (figure \[fig:hiir\_la\_host\]) and once again an upper envelope with slope 1.5 is a good fit. Likewise for dwarf Seyferts, the broad H$\alpha$ component shows a correlation with a slope of 1.5 (figure \[fig:seyfert.b\_la\_host\]). However, for the [*narrow-line*]{} component of H$\alpha$ for Seyfert nuclei, the correlation is much steeper, with a slope of 3.0 (figure \[fig:seyfert\_la\_host\]). Figure \[fig:lr\_host\] shows data from the radio survey of elliptical galaxies by . Here we find that nuclear radio luminosity shows a very strong correlation with an upper envelope slope of 3.0. ( do a more careful analysis showing that the 30th percentile goes as $L_{\rm
host}^{2.2}$). So it seems that narrow emission lines in AGN are intimately connected with radio emission. On the other hand, the relation seen for our X-ray sources is equally consistent with either AGN or star formation.
Nuclear X-ray sources as disk sources
-------------------------------------
Some of the nuclear X-ray sources we have seen are consistent with the most luminous known X-ray binaries. Furthermore, we have seen that super-luminous off-nuclear sources do occur quite often. Is it possible, then, that the nuclear sources are simply examples of the general X-ray source population that happen to be located in the nucleus? Where we see a clear extended source, this cannot be the case. This argument assumes that the observed extended emission is truly diffuse in nature. Of course, we cannot exclude a contribution of unresolved sources to the extended component, but this cannot account for the bulk of the emission for the more conspicuous cases (see , , , and ). It could also be that more luminous galaxies are simply more likely to have at least one example of a particularly luminous source. The strongest argument against this is that amongst the off-nuclear sources we do not see the upper-envelope correlation with host galaxy luminosity (figure \[fig:lx\_off\_host\]). We can also crudely estimate the likely size of such an effect if we know the luminosity distribution of X-ray sources.
The [*combined*]{} luminosity distribution of disk (ie, off-nuclear) X-ray sources from a sample of 83 spiral galaxies has been established recently by . The sample was defined as those objects surveyed by that had archive ROSAT HRI observations. The distribution determined by reaches luminosities for off-nuclear sources just below $10^{41}$ ergs s$^{-1}$, ie, it extends well into the realm of the super-luminous sources discussed in the previous section. The low luminosity end of the distribution was found using the ROSAT PSPC deep observations of NGC224 (M31) published by . find that the (differential) luminosity distribution is well fitted by a power law slope of $-1.8$, with a flattening for luminosities $\la
10^{36}$ ergs s$^{-1}$. It is not clear whether this change in the slope is due to incompleteness or to an intrinsic variation in the faint source population.
Since the luminosity distribution determined by corresponds to an average distribution a direct comparison between the properties of the populations in individual galaxies and their hosts is not possible. Moreover, as their sample is drawn from the flux-limited sample surveyed by , with an under-representation of low luminosity galaxies, it is not possible to confirm the lack of correlation between the probability of finding a super-luminous off-nuclear source and the luminosity parent galaxy seen in figure \[fig:lx\_off\_host\].
The luminosity distribution of off-nuclear sources predicts that about 8 disk sources with $L_{X} \geq 10^{37}$ ergs s$^{-1}$ will be found in a $10^{10} \times L_{\sun}$ ($M \sim -20$) galaxy. Of these sources only one will have a luminosity $\geq 10^{38}$ ergs s$^{-1}$. The probability of that source being located within the nucleus of the parent galaxy can be expressed as the ratio between the total luminosity and the luminosity of a ‘nuclear region’ of 100 pc (equal to the HRI spatial resolution at a distance of 4 Mpc). Using the well established exponential disk profile in spirals, and assuming a scale length for the disk equal to 4 kpc [@simien] this ratio is only $\sim 10^{-4}$.
For more luminous sources the situation is even worse, since the luminosity distribution predicts that about $8
\times 10^{10} \times L_{\sun}$ galaxies have to be surveyed in order to find a $L_{X} \geq 10^{39}$ ergs s$^{-1}$ X-ray source. This prediction does not agree with our results, however. In our sample of 29 galaxies we have detected 9 sources with luminosities above $10^{39}$ ergs s$^{-1}$ and all but one of the host galaxies (NGC2403) have $M_{B} > -20$ (see figure \[fig:lx\_off\_host\]). A closer look at the results reveals that two effects could be responsible for this difference: (1) statistic fluctuations introduced in the results drawn from our smaller sample due to the distances adopted for the parent galaxies; (2) the fairly large threshold radius (25 arcsecs) adopted by to discriminate between nuclear and off-nuclear sources (as an example, the ultra-luminous off-nuclear source in NGC5204 was labeled as nuclear by ). As pointed by , individual galaxies can also show important deviations from the determined luminosity distribution. The X-ray population in NGC5457, for example, shows the same power-law index determined by for the combined sample, but the normalization is 4 times larger.
In summary, even with a large error (or fluctuation) in the normalization of the luminosity distribution of off-nuclear sources, it is clear that nuclear sources cannot be explained as disk sources located [*by chance*]{} in the nuclei of galaxies. There is another possibility, however. Nuclear sources could be explained if massive X-ray emitting systems, such as black hole binaries, suffer considerable dynamical friction against field stars and so migrate to the centre of their parent galaxies. In this scenario the upper envelope seen in figure \[fig:lx\_host\] could be the result of smaller galaxies having shallower potential wells and therefore being less efficient in dragging the heavy objects to their centres. However, it is well known that the scale time for a significant change in the orbit of any stellar system is comparable to the local relaxation time which, in the case of disks, is much longer than the Hubble time [@binney+tremaine]. Therefore, galactic disks are too young for a source to have migrated significantly from its original location.
Nuclear X-ray sources as bulge sources
--------------------------------------
An alternative to the scenario described above is that the massive XRBs could have originally been located in the bulge of the galaxy, where larger background stellar densities make mass segregation a much more efficient mechanism. However, the anticorrelation between the time it takes for a massive object to migrate to the galactic nucleus and the mass of the object implies that an XRB would need a mass of over $100 M_{\sun}$ to fall to the galactic centre from a distance of only 10 pc within the lifetime of the parent galaxy (1 Gyr) [@morris]. Therefore, dynamical friction will only be efficient for very heavy or very nearby stars. Indeed, show that only extremely massive objects such as $10^{6} M_{\sun}$ globular clusters would have spiralled into the centre of their parent galaxies in less than a Hubble time from a significantly extra-nuclear (2 kpc) distance.
It could also be argued that the X-ray emission from the central regions corresponds to bulge sources which appear coincident with the galactic nuclei. This is unlikely, however, given the spatial resolution of our observations (ranging from $\sim 16$ pc for NGC224 to $\sim 190$ pc for NGC5457, with most galaxies with detected nuclear sources being observed at a resolution $\ga 100$ pc), implying that a very large number of highly centrally concentrated bulge sources would be necessary to explain the observations. NGC224 indeed shows a high concentration of bulge sources when compared with our own Galaxy, with 4 times more sources seen in NGC224 within the central 5 arcmins [@primini]. However, only two sources are found within the innermost 100 pc, giving a total luminosity of $4 \times
10^{37}$ ergs s$^{-1}$. The probability of those sources having $L_{X}
\ga 10^{38}$ ergs s$^{-1}$ is extremely small, as is shown in the bulge luminosity distribution determined of . Also, as we have already shown, the hypothesis that particularly heavy (and luminous) binary systems are located within this region because of mass segregation is not viable. Therefore, most of the galaxies with $M_{B} < -18$ seen in figure \[fig:lx\_host\] would need to harbor a much larger population of bulge sources that the one seen in NGC224 in order to explain their observed nuclear luminosities, which is a very unlikely scenario.
The nature of the nuclear X-ray sources
---------------------------------------
The very different correlations between X-ray luminosity and host galaxy absolute magnitude seen in figures \[fig:lx\_off\_host\] and \[fig:lx\_host\] for nuclear and off-nuclear sources imply two different populations. Nuclear sources are not disk or bulge sources located in the nuclear region by chance. Instead, nuclear sources have a particular [*nuclear*]{} nature.
The next step is to try to unveil this nature. What are these sources? Are they connected with stellar processes in the nuclear region of the galaxies? Could they be an expression of nuclear activity that somehow escapes detection at optical wavelengths?
An interesting pattern can be seen in figure \[fig:lx\_host\] between the optical (spectral) classification of the galaxies and their X-ray luminosities. Both LINER objects (NGC404 and NGC4258) lie close to the upper envelope of the distribution of detected sources, as does the Seyfert galaxy NGC3031. The second Seyfert nucleus, NGC4395, is heavily absorbed below 3 keV and its intrinsic soft X-ray luminosity is estimated to be an order of magnitude larger than the values found from ROSAT observations [@iwasawa]. Introducing this last correction all objects classifies as AGN would be found near the top of the upper envelope in figure \[fig:lx\_host\].
![Nuclear X-ray luminosity as a function of the host visual absolute magnitude of the bulge component. Symbols as in figure \[fig:lx\_host\]. The dash-dotted line represents $10^{-4}$ times the predicted X-ray emission from central black hole masses at the Eddington limit assuming the bulge to BH relationship determined by and an X-ray to bolometric flux ratio of 0.1 (see text).[]{data-label="fig:lx_blg_host"}](lx_host_blg.ps)
The distribution of X-ray sources with galaxy absolute magnitude, shown in figure \[fig:lx\_host\], can be bounded by a nearly linear upper envelope. A similar linear upper envelope has been found in the correlation between nuclear (AGN) luminosity and host galaxy luminosity for samples of Seyfert galaxies and low redshift quasars at IR and optical wavelengths [@yee.92; @mcleod]. This suggests that there is a maximum allowed AGN luminosity which is an increasing function of the luminosity of the parent galaxy. The limit could be the result of a correlation between the total mass of the host galaxy and the mass of the central engine. The observed range of nuclear luminosities would be given by different accretion rates, with sources located at the bound of the region, showing the maximum possible luminosity. Indeed, from their study of nearby quasars and show that if the correlation between black hole mass and galaxy size claimed by is correct, then the upper envelope is consistent with the most luminous AGN radiating at roughly 20% of the Eddington limit. If the smaller ratio of black hole mass to galaxy mass suggested by is used then the closeness of the upper envelope to the Eddington prediction is even more remarkable.
However, in the case of the comparatively weak nuclear X-ray sources we have found here, it is not clear what process determines the maximum possible luminosity. If the observed X-ray emission is driven by accretion of matter onto a BH, then the Eddington limit for the central mass seems the most straight-forward mechanism to explain the existence of the linear upper envelope. However, this cannot be the case. The only two Seyfert galaxies in the sample, NGC3031 (M81) and NGC4395, are thought to be extremely sub-Eddington [@hfs.96; @lira], with their bolometric luminosities being only $10^{-4} - 10^{-3} \times L_{\rm Edd}$.
Figure \[fig:lx\_blg\_host\] makes this point graphically. We use the relation given by , $\log M_{BH} = -1.96 + \log
L$, where $L$ is the $V$-band luminosity of the bulge and all quantities are expressed in solar units. The bulge magnitudes for the galaxies in our sample can be found from the total magnitudes shown in table \[tab:allobs\] using the relantionship obtained by as a function of Hubble Type, while a colour $B-V =1.0$ was assumed for all objects to convert the blue magnitudes to visual magnitudes. Using the relationship above we can infer the X-ray output for a central BH emitting at the Eddington limit as a function of its host galaxy bulge luminosity, assuming that the soft X-ray emission corresponds to $\sim 10\%$ of the bolometric luminosity. It is found that the predicted X-ray luminosities (seen as a dotted line in figure \[fig:lx\_blg\_host\]) are about 4 orders of magnitude brighter than what is observed.
The nuclear-host correlation we have seen cannot, therefore, be simply interpreted as the upper envelope relation seen amongst luminous AGN, because the X-ray sources are so feeble. However, it should be borne in mind that the objects discussed in have been pre-selected as AGN, whereas we are looking at galaxies as a whole. For example, it is possible that the probability distribution $P(L_{x}/L_{\rm Edd})$ is a universal function for all galaxies, declining continuously from large values at small $L_{x}/L_{\rm Edd}$ to small values at large $L_{x}/L_{\rm Edd}$, but with a cutoff at $L_{x}/L_{\rm Edd} = 1$. The envelope seen in AGN samples then traces the cutoff, but the envelope seen in complete galaxy samples is a statistical effect, tracing in each host-galaxy-bin the nuclear luminosity beyond which the expected number of objects is just less than one. With a larger sample, the density of points would be greater, so the envelope would move up, but the slope might stay the same.
Most of the preceding discussion has centred round the possibility that the weak nuclear X-ray sources are AGN. However, this is not obvious. They seem to show no particular preference for AGN-like optical spectra, and the correlation with host galaxy luminosity is exactly like that shown by the emission line luminosity for HII nuclei (see figure \[fig:hiir\_la\_host\]). Furthermore many of the most luminous sources are extended on scales of hundreds of parsecs or more. It seems equally likely that the X-ray emission is connected with star formation, but with the current data we cannot go much further. In later papers in this series we examine the correlation between X-ray, optical, and emission-line luminosities, and test specific AGN and star formation models.
Although it is unclear which physical process causes the observed distributions, the upper envelopes give strong evidence for a correlation between the host galaxy luminosity and the level of nuclear activity. The fact that an upper envelope is also observed in the distribution of the nuclear HII regions might suggest that the correlations are governed by the amount of gas available in the nuclear regions. If bigger galaxies are more efficient in dragging gas to their nuclear regions they could show more vigorous star formation and, potentially, feed their massive BHs more efficiently.
Summary
=======
X-ray sources in the nuclei of galaxies are very common and the luminosity of these sources is strongly connected to the luminosity of the host galaxy. The highest luminosities reach $\sim 10^{40}$ ergs s$^{-1}$ which is $10^{4}$ times less than the Eddington limit for the massive black holes that may be present in such galaxies. The most luminous nuclear X-ray sources are frequently extended on scales of hundreds of parsecs. It is, so far, unclear whether these nuclear X-ray sources are miniature AGN of some kind or a phenomenon connected with normal star formation. Outside the nuclei of galaxies, extremely luminous sources ($L > 10^{39}$ erg s$^{-1}$, compared with ‘normal’ $\sim 10^{37}$ erg s$^{-1}$ SNRs or $\sim 10^{38}$ erg s$^{-1}$ XRBs) are also quite common, occurring in a third of all galaxies. The nature of these sources is currently also unclear.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Gordon Stewart and John Arabadjis for insightfull discussion on this paper. This research has made use of data obtained from the Leicester Database and Archive Service at the Department of Physics and Astronomy, Leicester University, UK. The Jacobus Kapteyn Telescope is operated on the island of La Palma on behalf of the UK Particle Physics and Astronomy Research Council (PPARC) and the Nederlanse Organisatie voor Wetenschappelijk Onderzoek (NWO) and is the located at the Spanish Observatorio del Roque de los Muchachos. The observatory is hosted by the Instituto de Astrofísica de Canarias. The DSS plates used in this paper were based on photographic data of the National Geographic Society – Palomar Observatory Sky Survey (NGS-POSS) obtained using the Oschin Telescope on Palomar Mountain. The NGS-POSS was funded by a grant from the National Geographic Society to the California Institute of Technology. The plates were processed into the present compressed digital form with their permission. The Digitized Sky Survey was produced at the Space Telescope Science Institute under US Government grant NAG W-2166. The data presented in this paper were reduced using Starlink facilities.
|
---
abstract: 'A direct WIMP (Weakly Interacting Massive Particle) detector with a neutron veto system is designed to better reject neutrons. An experimental configuration is studied in the present paper: 984 Ge modules are placed inside a reactor neutrino detector. The neutrino detector is used as a neutron veto device. The neutron background for the experimental design has been estimated using the Geant4 simulation. The results show that the neutron background can decrease to O(0.01) events per year per tonne of high purity Germanium and it can be ignored in comparison with electron recoils.'
author:
- |
XiangPan Ji, Ye Xu[^1], JunSong Lin, Yulong Feng,\
JunCheng Wang, QiKun Li, DanNing Di, HaoLin Li
title: Evaluation of the neutron background in a direct WIMP detector with HPGe target using a reactor neutrino detector as a neutron veto system
---
School of Physics, Nankai University, Tianjin 300071, China
Dark matter, Neutron background, Neutrino detector, High purity germanium
PACS numbers: 95.35.+d, 95.55.Vj, 29.40.Mc
Introduction
============
It is indicated by seven year Wilkinson Microwave Anisotropy Probe data combined with measurements of baryon acoustic oscillations and Hubble constant that $\sim$$83\%$ of the matter content in the Universe is non-baryonic dark matter [@WMAP2011; @BAQ2010; @H2009]. Weakly Interacting Massive Particles (WIMPs) [@Steigman1985], predicted by extensions of the Standard Model of particle physics, are a well-motivated class of candidates for dark matter. They are distributed in the halo surrounding the Milky Way. WIMPs may be directly detected through measuring nuclear recoils in terrestrial detectors produced by their scattering off target nuclei [@Goodman1985; @Jungman1996; @Gaitskell2004]. The nuclear reoils is expected to have a roughly exponential energy distribution with a mean energy in a few tens of keV [@Jungman1996; @Bertone2005; @Lewin1996].
In direct searches for WIMPs, there are three different methods used to detect the nuclear recoils, including collecting ionization, scintillation and heat signatures induced by them. The background of this detection is made up of electron recoils produced by $\gamma$ and $\beta$ scattering off electrons, and nuclear recoils produced by neutrons scattering elastically off target nucleus. It is very efficient discriminating nuclear recoils from electron recoils with pulse shape discrimination, hybrid measurements and so on. The rejection powers of these techniques can even reach $\>$$10^6$ [@CDMSII2010; @LAr2008]. For example, the CDMS-II [@CDMSII2010] and EDELWEISS-II [@EDELWEISS-II2011] experiments measure both ionization and heat signatures using cryogenic germanium detectors in order to discriminate between nuclear and electron recoils, and the XENON100 [@XENON2012] and ZEPLIN-III [@ZEPLIN-III2011] experiments measure both ionization and scintillation signatures using two-phase xenon detectors. However, it is very difficult to discriminate between nuclear recoils induced by WIMPs and by neutrons. This discrimination is one of the most important tasks in direct dark matter searches.
The cross-sections of neutron-nuclei interactions are much larger than those of WIMP-nuclei, so the multi-interactions between neutrons and detector components are applied to tag neutrons and thus separate WIMPs from neutrons. In the ZEPLIN-III experiment, the $0.5\%$ Gadolinium (Gd) doped polypropylene is used as the neutron veto device, and its maximum tagging efficiency for neutrons reaches about $80\%$ [@ZEPLIN-III2010]. In Ref. [@GdWater2008], the $2\%$ Gd-doped water is used as the neutron veto, and its neutron background can be reduced to 2.2 (1) events per year per tonne of liquid xenon (liquid argon). In our past work [@GdLS2010], the reactor neutrino detector with $1\%$ Gd-doped liquid scintillator (Gd-LS) is used as the neutron veto system, and its neutron background can be reduced to about 0.3 per year per tonne of liquid xenon. These neutron background events are mainly from the spontaneous fission and ($\alpha$, n) reactions due to $^{238}U$ and $^{232}Th$ in the photomultiplier tubes (PMTs) in the liquid xenon.
Because of its advantages of the low background rate, energy resolutions and low energy threshold, high purity Germanium (HPGe) is widely applied in dark matter experiments. It makes their neutron background much less than in the case of Xenon target that there are no PMTs in HPGe detectors. The feasibility of direct WIMPs detection with the neutron veto based on the neutrino detector had been validated in our past work[@GdLS2010]. So, in the present paper, a neutrino detector with Gd-LS ($1\%$ Gd-doped) is still used as a neutron-tagged device and WIMP detectors with HPGe targets(called Ge modules) are placed inside the Gd-LS. Here we designed an experimental configuration: 984 Ge modules are individually placed inside four reactor neutrino detector modules which are used as a neutron veto system. The experimental hall of the configuration is assumed to be located in an underground laboratory with a depth of 910 meter water equivalent (m.w.e.), which is similar to the far hall in the Daya Bay reactor neutrino experiment[@DayabayProp2007]. Collecting ionization signals is considered as the only method of the WIMPs detection in our work. The neutron background for this design are estimated using the Geant4 [@Geant4] simulation.
The basic detector layout will be described in the section 2. Some features of the simulation in our work will be described in the section 3. The neutron background of the experimental configuration will be estimated in the section 4. The contamination due to reactor neutrinos will be discussed in the section 5. We give a conclusion in the section 6.
Detector description
====================
Four identical WIMP detectors with HPGe targets are individually placed inside four identical neutrino detector modules. The experimental hall of this experimental configuration is assumed to be located in an underground laboratory with a depth of 910 m.w.e., which is similar to the far hall in the Daya Bay reactor neutrino experiment. The detector is located in a cavern of 20$\times$20$\times$20 $m^3$. The four identical cylindrical neutrino modules (each 413.6 cm high and 393.6 cm in diameter) are immersed into a 13$\times$13$\times$8 $m^3$ water pool at a depth of 2.5 meters from the top of the pool and at a distance of 2.5 meters from each vertical surface of the pool. The detector configuration is shown in Fig.\[fig:detector\].
Each neutrino module is partitioned into three enclosed zones. The innermost zone is filled with Gd-LS, which is surrounded by a zone filled with unload liquid scintillator (LS). The outermost zone is filled with transparent mineral oil[@DayabayProp2007]. 366 8-inch PMTs are mounted in the mineral oil. These PMTs are arranged in 8 rings of 30 PMTs on the lateral surface of the oil region, and 5 rings of 24, 18, 12, 6, 3 on the top and bottom caps.
Each WIMP detector (136.6 cm height, 75.8 cm in diameter) consists of two components: the upper component is a cooling system with liquid Nitrogen (30 cm height, 52.8 cm in diameter) and the lower one is an active target of 246 Ge modules arranged in 6 columns (each column includes 4 rings of 20, 14, 6, 1) inside a 0.5 cm thick copper vessel (83.6 cm height, 53.8 cm in diameter). Each Ge module is made up of a copper vessel and a HPGe target: there is a HPGe target(6.2 cm height, 6.2 cm in diameter, $\sim$1 kg) in a 0.1 cm thick copper vessel (12.6 cm height, 6.4 cm in diameter).
Some features of simulation
===========================
The Geant4 (version 8.2) package has been used in our simulations. The physics list in the simulations includes transportation processes, decay processes, low energy processes, electromagnetic interactions (multiple scattering processes, ionization processes, scintillation processes, optical processes, cherenkov processes, Bremsstrahlung processes, etc.) and hadronic interactions (lepton nuclear processes, fission processes, elastic scattering processes, inelastic scattering processes, capture processes, etc.). The cuts for the production of gammas, electrons and positrons are 1 mm, 100 $\mu$m and 100$\mu$m, respectively. The quenching factor is defined as the ratio of the detector response to nuclear and electron recoils. The Birks factor for protons in the Gd-LS is set to 0.01 g/cm$^{2}$/MeV, corresponding to the quenching factor 0.17 at 1 MeV, in our simulations.
Neutron background estimation
=============================
The recoil energies for WIMP interactions with Ge nuclei was set to a range from 10 keV to 100 keV[@CDMSII2010] in this work. Proton recoils induced by neutrons and neutron-captured signals are used to tag neutrons which reach the Gd-LS. The energy deposition produced by proton recoils is close to a uniform distribution. Neutrons captured on Gd and H lead to a release of about 8 MeV and 2.2 MeV of $\gamma$ particles, respectively. Due to the instrumental limitations of the Gd-LS, we assume neutrons will be tagged if their energy deposition in the Gd-LS is more than 1 MeV, corresponding to 0.17 MeVee (electron equivalent energy). In the Gd-LS, it is difficult to distinguish signals induced by neutrons from electron recoils, which are caused by the radioactivities in the detector components and the surrounding rocks. But these radioactivities can be controlled to less than $\sim$ 50 Hz according to the Daya Bay experiment[@DayabayProp2007]. If we assume a 100 $\mu$s for neutron tagging time window, the indistinguishable signals due to the radioactivities will result in a total dead time of less than 44 hours per year.
Neutrons are produced from the detector components and their surrounding rock. For the neutrons from the surrounding rock there are two origins: first by spontaneous fission and ($\alpha$, n) reactions due to U and Th in the rock (these neutrons can be omitted because they are efficiently shielded, see Sec.4.2), and secondly by cosmic muon interactions with the surrounding rock.
We estimated the number of neutron background in the Ge target of one tonne. This number has been normalized to one year of data taking and are summarized in Tab.\[tab:bg\].
Neutron background from detector components
-------------------------------------------
Neutrons from the detector components are induced by ($\alpha$, n) reactions due to U and Th. According to Mei et al.[@Mei2009], the differential spectra of neutron yield can be expressed as
$\displaystyle Y_{i}(E_{n})=N_i{
\sum_{j}\frac{R_{\alpha}(E_{j})}{S_{i}^{m}(E_{j})}}$$\displaystyle
\intop_{0}^{E_{j}}\frac{d\sigma(E_{\alpha},E_{n})}{dE_{\alpha}}dE_{\alpha}$
where $N_i$ is the total number of atoms for the $i^{th}$ element in the host material, $R_\alpha$$(E_j)$ refers to the $\alpha$-particle production rate for the decay with the energy $E_j$ from $^{232}Th$ or $^{238}U$ decay chain, $E_{\alpha}$ refers to the $\alpha$ energy, $E_n$ refers to the neutron energy, and $S_{i}^{m}$ is the mass stopping power of the $i^{th}$ element.
### Neutrons from copper vessels
In the copper vessels, neutrons are produced by the U and Th contaminations and emitted with their average energy of 0.81 MeV[@Mei2009]. Their total volume is about $5.4\times10^4$ $cm^3$. The radioactive impurities Th can be reduced to $2.5\times10^{-4}$ ppb in some copper samples[@Martin2009] If we conservatively assume a 0.001 ppb U/Th concentrations in the copper material[@Exo200_2007], a rate of one neutron emitted per $4\times10^4$ $cm^3$ per year is estimated[@GdWater2008]. Consequently, there are 1.3 neutrons produced by the all copper vessels per year.
The simulation result is summarized in Tab.\[tab:bg\]. 0.38 neutron events/(ton$\cdot$yr) reach the HPGe targets and their energy deposition falls in the same range as that of the WIMP interactions (see Tab.\[tab:bg\]). As 0.01 of them are not tagged in the Gd-LS, these background events cannot be eliminated. The uncertainty of the neutron background from the copper vessels are from the binned neutron spectra in the Ref.[@Mei2009]. But the neutron background errors from the statistical fluctuation (their relative errors are less than 1$\%$) are too small to be taken into account.
### Neutrons from other components
The U and Th contaminations in other detector components also contribute to the neutron background in our experiment setup. Neutrons from the Aluminum reflectors are emitted with the average energy of 1.96 MeV[@Mei2009]. The U and Th contaminations in the Carbon material are considered as the only neutron source in the Gd-LS/LS. Neutrons from the Gd-LS/LS are emitted with the average energy of 5.23 MeV[@Mei2009]. The U and Th contaminations in the $SiO_2$ material are considered as the only neutron source in the PMTs in the oil. Neutrons from PMTs are emitted with the average energy of 2.68 MeV[@Mei2009]. The U and Th contaminations in the Fe material are considered as the only neutron source in the stainless steel tanks. Neutrons from the stainless steel tanks are emitted with the average energy of 1.55 MeV[@Mei2009]. We evaluated the neutron background from the above components using the Geant4 simulation. All the nuclear recoils in the HPGe targets, which is in the same range as that of the WIMP interactions, are tagged. The neutron background from these components can be ignored.
Neutron background from natural radioactivity in the surrounding rock
---------------------------------------------------------------------
In the surrounding rock, almost all the neutrons due to natural radioactivity are below 10 MeV [@GdWater2008; @Carson2004]. Water can be used for shielding neutrons effectively, especially in the low energy range of less than 10 MeV [@Carmona2004]. The Ge detectors are surrounded by about 2.5 meters of water and more than 1 meter of Gd-LS/LS, so these shields can reduce the neutron contamination from the radioactivities to a negligible level.
Neutron background due to cosmic muons
--------------------------------------
Neutrons produced by cosmic muon interactions constitute an important background component for dark matter searches. These neutrons with a hard energy spectrum extending to several GeV energies, are able to travel far from produced vertices.
The total cosmogenic neutron flux at a depth of 910 m.w.e. is evaluated by a function of the depth for a site with a flat rock overburden [@Mei2006], and it is 1.31$\times$$10^{-7}$ $cm^{-2}s^{-1}$ . The energy spectrum (see Fig.\[fig:energy\]) and angular distribution of these neutrons are evaluated at the depth of 910 m.w.e. by the method in [@Mei2006; @Wang2001]. The neutrons with the specified energy and angular distributions are sampled on the surface of the cavern, and the neutron interactions with the detector are simulated with the Geant4 package. Tab.\[tab:bg\] shows that 30 neutron events/(ton$\cdot$yr) reach the HPGe targets and their energy deposition is in the same range as that of the WIMP interactions. 0.3 of them are not tagged by the Gd-LS/LS. Muon veto systems can tag muons very effectively, thereby most cosmogenic neutrons can be rejected. For example, using the muon veto system, the neutron contamination level could be reduced by a factor of about 10[@Carson2004]. In the Daya Bay experiment, the contamination level can even be reduced by a factor of more than 30[@DayabayProp2007]. We conservatively assume the neutron contamination level from cosmic muons decreases by a factor of 10 using a muon veto system. This could lead to the decrease of cosmogenic neutron contamination to 0.03 events/(ton$\cdot$yr). The uncertainties of the cosmogenic neutron background in Tab.\[tab:bg\] are from the statistical fluctuation.
Contamination due to reactor neutrino events
============================================
Since neutrino detectors are fairly close to nuclear reactors (about 2 kilometers away) in reactor neutrino experiments, a large number of reactor neutrinos will pass through the detectors, and nuclear recoils will be produced by neutrino elastic scattering off target nucleus in the Ge detectors. Although neutrinos may be a source of background for dark matter searches, they can be reduced to a negligible level by setting the recoil energy threshold of 10 keV[@Jocely2007]. Besides, nuclear recoils may also be produced by low energy neutrons produced by the inverse $\beta$-decay reaction $\bar{\nu_{e}}+p\rightarrow e^{+}+n$. But their kinetic energies are almost below 100 keV[@chooz2003], and their maximum energy deposition in the WIMP detectors is as large as a few keV. Thus the neutron contamination can be reduced to a negligible level by the energy threshold of 10 keV.
Conclusion and discussion
=========================
The neutron background can be effectively suppressed by the neutrino detector used as the neutron veto system in direct dark matter searches. Tab.\[tab:bg\] shows the total neutron contamination are 0.04 events/(ton$\cdot$yr). And compared to Ref.[@GdLS2010], it is reduced by a factor of about 8. This decrease is caused by the reason that the neutron contamination is mainly from the PMTs in the Xe detector, but there are no photomultiplier tubes (PMTs) in the HPGe detector. Compared to electron recoils[@CoGeNT], the estimated neutron contamination in the paper can be ignored. After finishing a precision measurement of the neutrino mixing angle $\theta_{13}$, we can utilize the existing experiment hall and neutrino detectors. This will not only save substantial cost and time for direct dark matter searches, but the neutron background could also decrease to O(0.01) events per year per tonne of HPGe.
Acknowledgements
================
This work was supported by the National Natural Science Foundation of China (NSFC) under the contract No. 11235006 and the Fundamental Research Funds for the Central Universities No. 65030021.
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10keV<$E_{recoil}$<100keV Not Tagged
------------------ --------------------------------- ----------------
copper vessels 0.38$\pm$0.07 0.01$\pm$0.002
cosmic muons 30.0$\pm$1.73 0.3$\pm$0.17
muon veto 3.0$\pm$0.55 0.03$\pm$0.055
total(muon veto) 3.4$\pm$0.55 0.04$\pm$0.055
: Estimation of neutron background from different sources for an underground laboratory at a depth of 910 m.w.e. The column labeled “10keV<$E_{recoil}$<100keV” identifies the number of neutrons whose energy deposition in the Ge is in the same range as WIMP interactions. The column labeled “Not Tagged” identifies the number of neutrons which are misidentified as WIMP signatures (their energy deposition in the Ge is in the same range as WIMP interactions while their recoil energies in the Gd-LS/LS are less than the energy threshold of 1 MeV). The row labeled “copper vessel” identifies the number of neutrons from the copper vessels. The row labeled “cosmic muons” identifies the number of cosmogenic neutrons in the case of not using the muon veto system. The row labeled “muon veto” identifies the number of cosmogenic neutrons in the case of using the muon veto system. We assume that neutron contamination level from cosmic muons decreases by a factor of 10 using a muon veto system. Only the total background in the case of using the muon veto system is listed in this table. The terms after $\pm$ are errors.[]{data-label="tab:bg"}
![Top left: Ge Module with HPGe material. Top right: WIMP detector with 246 Ge Modules. Bottom: four WIMP detectors individually placed inside four neutrino detectors in a water shield.[]{data-label="fig:detector"}](detector){width="70.00000%"}
![The energy spectrum of cosmogenic neutrons at depth of 910 m.w.e.[]{data-label="fig:energy"}](energy){width="80.00000%"}
[^1]: Corresponding author, e-mail address: xuye76@nankai.edu.cn
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---
abstract: 'The superposition of a Gaussian mode and a Laguerre-Gauss mode with $\ell=0,p\neq0$ generates the so-called bottle beam: a dark focus surrounded by a bright region. In this paper, we theoretically explore the use of bottle beams as an optical trap for dielectric spheres with a refractive index smaller than that of their surrounding medium. The forces acting on a small particle are derived within the dipole approximation and used to simulate the Brownian motion of the particle in the trap. The intermediate regime of particle size is studied numerically and it is found that stable trapping of larger dielectric particles is also possible. Based on the results of the intermediate regime analysis, an experiment aimed at trapping living organisms in the dark focus of a bottle beam is proposed.'
author:
- 'B. Melo'
- 'I. Brandão'
- 'B. Pinheiro da Silva'
- 'R. B. Rodrigues'
- 'A. Z. Khoury'
- 'T. Guerreiro'
bibliography:
- 'main.bib'
title: Optical Trapping in a Dark Focus
---
\[sec:introduction\] Introduction
=================================
Tightly focused laser beams can be used to exert forces upon dielectric particles. If the particle’s refractive index is *larger* than that of its surroundings, the laser pulls it to regions of higher intensity of light. This technique, introduced by Arthur Ashkin in 1986 [@Ashkin1986] and known today as optical tweezing, allows one to hold and manipulate very tiny objects and finds applications in a large number of fields ranging from biology [@Fazal2011; @Nussenzveig2017; @S.Araujo2019; @Pontes2013] to fundamental physics [@Monteiro2020; @Ether2015; @Arvanitaki2013; @Geraci2010; @Moore2014]. In standard optical tweezers, Gaussian beams are used to create the trapping focus. To a good approximation, the trap can be described as a three dimensional quadratic potential.
Notably it was also pointed out by Ashkin that air droplets immersed in water were pushed away from the Gaussian focus [@Ashkin1970]. This is a consequence of the fact that when the refractive index of the particle is *smaller* than that of its surroundings, the particle is repelled from the region of high intensity. One can then envision an *inverted* optical trap, in which an engineered beam of light has a high-intensity boundary and a dark focus. A particle with the appropriate refractive index will be trapped within the dark focus by the absence of light [@Ahluwalia2006]. We refer to this type of beam in general as *bottle beams*. A bottle beam is one example in a myriad of engineered optical traps aimed at different purposes such as circular Airy beams [@Lu2019; @Cheng2010; @Jiang2013], Bessel beams [@Arlt2001], radially polarized beams [@Yan2007; @Shu2013], frozen waves [@Suarez2020] and many others [@Zhao2007; @Zhao2009; @Zhao2009a; @Zhang2015; @Zhao2011; @Zhan2003].
Several techniques can be employed to create bottle beams, such as the generation of Bessel beams using axicons [@Wei2005; @Lin2007; @Du2014], the interference of Gaussian beams of different waists [@Isenhower2009] and the superposition of different modes [@Silva2020; @Ahluwalia2004; @Xu2010; @Zhang2011] created using Spatial Light Modulators [@Matsumoto2008; @Ohtake2007; @Rhodes2006; @Ando2008]. Here, we focus on the bottle beam created by the superposition of a Gaussian beam and a Laguerre-Gauss beam with $\ell=0, p\neq0$ and a relative phase of $\pi$ presented in [@Arlt2000] and study the optical forces it exerts upon low refractive index particles.
Because optical trapping can be applied to particles in a wide size range [@Hansen2005; @Alinezhad2019], we analyse both the cases of small Rayleigh particles and of larger micron-sized particles. In the former, the optical forces and potential are derived from the dipole approximation and thoroughly analysed under different assumptions, which are verified by simulating the motion of the trapped particle in a viscous medium. In the latter, generalized Lorenz–Mie theory is employed to calculate the forces caused by the beam, with the aid of the tools introduced in [@Nieminen2007]. Constraints on the numerical aperture, particle size and relative refractive index are found.
Understanding particle dynamics under the influence of a bottle beam can lead to striking applications. Notably, the bottle is an interesting tool for trapping experiments requiring little or no light scattering upon the trapped object. This is of particular interest in biology, where trapping a living cell or organelles within the cell without the constant influence of laser light might be crucial to reveal mechanical properties of the organism without excessive heating and laser interference [@Liu1995; @Peterman2003; @BlazquezCastro2019]. We thus propose a set of experimental parameters that could be used to trap living organisms in the dark focus of a bottle beam.
The dipole approximation {#sec:dipole}
========================
We begin by investigating the optical forces acting on a Rayleigh particle with a refractive index lower than its surrounding medium under the dipole approximation. This is valid when the radius of the trapped particle is much smaller than the wavelength of the trapping laser $(R\lesssim\lambda/10)$ [@Li2013].
The optical bottle beam
-----------------------
To generate a dark focus surrounded by a bright region we superpose a Laguerre-Gauss beam with $\ell=p=0$ - a Gaussian beam - and a Laguerre-Gauss beam with $\ell=0, p\neq0$ and a relative phase of $\pi$. The electric field magnitude of a Laguerre-Gauss beam is $$\begin{aligned}
\label{eq:LGbeam2}
\hspace*{-2em}&\,&E^{LG}_{\ell,p}(\rho,\phi,z) =\sqrt{\frac{4P_0}{c\epsilon\pi\omega(z)^2}}\sqrt{\frac{p!}{(\vert \ell \vert+p)!}}\times\nonumber\\&&\left( \frac{\sqrt{2}\rho}{\omega(z)} \right)^{\vert\ell\vert}L^{\vert\ell\vert}_p\left( \frac{2\rho^2}{\omega(z)^2} \right)\exp\left[-\frac{\rho^2}{\omega(z)^2}\right]\times\nonumber\\&&\exp[ik_mz+ik_m\frac{\rho^2}{2R(z)}-i\zeta(z)+i\ell\phi],\end{aligned}$$ where $c$ is the speed of light, $\epsilon$ is the medium’s permittivity, $P_0$ is the laser power, $k_m$ is the wavenumber in the medium and $\omega(z)$, $R(z)$, $\zeta(z)$ and $L^{\vert\ell\vert}_p$ are the beam width, the wavefront radius, the Gouy phase and the Associated Laguerre polynomial. These quantities are respectively given by $$\begin{aligned}
\omega(z)&=&\omega_0\sqrt{1+\frac{z^2}{z^2_R}};\\
R(z) &=& z\left(1+\frac{z_R^2}{z^2} \right);\\
\zeta(z)&=&(2p+\vert\ell\vert+1)\arctan \frac{z}{z_R};\\
L_p^{\vert\ell\vert}(x) &=& \sum^p_{i=0}\frac{1}{i!}\binom{p+\vert\ell\vert}{p-i}(-x)^i
$$ where the Rayleigh range ($z_R$) and the beam waist ($\omega_0$) are defined as $$\begin{aligned}
\omega_0=\frac{\lambda_0}{\pi \textrm{NA}}\,,\quad z_R = \frac{n_{m}\lambda_0}{\pi \textrm{NA}^2} \label{eq:waist_and_Rayleigh_range}\end{aligned}$$
with $\lambda_0$ the wavelength in vacuum, $n_{m}$ the medium refractive index and $\textrm{NA}$ the numerical aperture. Throughout this work we will consider linearly polarized electric fields only.
The intensity of the bottle beam reads $$\begin{aligned}
\label{eq:exact_intensity}
&\,&I_p(\rho,z)=I_0\frac{\omega_0^2}{\omega(z)^2}\exp\left[-\frac{2 \rho^2}{\omega(z)^2}\right]\times\nonumber\\
&&\bigg[1-\hspace{-0.5mm}2\cos\left(2p\arctan\frac{z}{z_R}\right) L^0_p\left(\frac{2\rho^2}{\omega(z)^2}\right)+\nonumber\\&&L^0_p\left(\frac{2\rho^2}{\omega(z)^2}\right)^2 \bigg]\end{aligned}$$ where $I_0=2P_0/\pi\omega_0^2$ is the intensity at the origin of the Gaussian beam. Figures \[fig:intensity\_examples\](a) and \[fig:intensity\_examples\](b) shows the intensity as a function of the transverse coordinate $x$ and the longitudinal coordinate $z$. The potential landscape in the $xz$ plane is shown in Figures \[fig:intensity\_examples\](c) for the cases $p=1$, and \[fig:intensity\_examples\](d) $p=2$. A dielectric particle with the appropriate refractive index placed at the origin would be trapped in the dark focus, since it would be repelled in all directions by the surrounding regions of higher electromagnetic intensity.
![Intensity in the (a) radial and (b) axial directions for bottle beams with $p=1$ and $p=2$. Intensity landscape in the $xz$ plane for bottle beams with (c) $p=1$ and (d) $p=2$. Due to the normalization of $x$, $z$ and $I$, these plots depend only on $ p $, and are independent from the remaining beam parameters.[]{data-label="fig:intensity_examples"}](Figure1.pdf){width="\linewidth"}
Dimensions of the bottle
------------------------
We can define the width $W$ (height $H$) of the bottle as the distance between the two intensity maxima surrounding the dark region along the $x$ axis ($z$ axis). These values can be found by solving $$\begin{aligned}
\label{eq:width}
dI_p(x,0,0)/dx\vert_{x=W/2}=0\, ,\\
\label{eq:height}
dI_p(z,0,0)/dz\vert_{z=H/2}=0\, .\end{aligned}$$
The above equations admit analytical solutions for small $p$, yielding $W=2\omega_0,H=2z_R$ for $p=1$ and $W=2\sqrt{2-\sqrt{2}}\omega_0,H=\sqrt{2}z_R$ for $p=2$.
To gain insight into $H$ and $W$ it is useful to make the change of variables $\rho/\omega_0\rightarrow\rho', z/z_R\rightarrow z'$ in the intensity given in Eq.(\[eq:exact\_intensity\]). The function $I_p(\rho',z')$ has no explicit dependence on any of the beam’s parameters other than $p$ and $I_0$, with its associated re-scaled width $W'$ and height $H'$. The pre-factor $I_0$ does not alter the distance between maxima along the $x'$ and $z'$ axis, meaning that $W' = W'(p)$ and $H' = H'(p)$ depend only on $ p $. Going back to the original variables we find that $W=\omega_0W'(p)$ and $H=z_RH'(p)$. From Eq. (\[eq:waist\_and\_Rayleigh\_range\]), we observe that the width of the bottle scales with $\textrm{NA}^{-1}$ and the height scales with $\textrm{NA}^{-2}$. Hence an increase in $\textrm{NA}$ causes the bottle to become overall smaller and compressed along the $z$ direction.
Radiation forces
----------------
When a Rayleigh particle is placed in an electromagnetic field, there are three forces that act on it [@Jones2015]. The first, called spin-curl force, is a result of polarisation gradients [@Albaladejo2009] and can be disregarded in the case of uniform linear polarization we are interested in. The second is called scattering force, and is proportional to the Poynting vector. Near the origin the scattering force points in the direction of propagation of the beam. Finally, the gradient force is proportional to the gradient of the potential energy of the particle under the influence of the electromagnetic field. From the intensity given by Eq. (\[eq:exact\_intensity\]), the scattering force $\vec{F}_{p}^{(scat)}(\vec{r})$, the gradient force $\vec{F}_{p}^{(grad)}(\vec{r})$ and the optical potential $V_p(\vec{r})$ acting on a trapped particle with radius $R$ and refractive index $n_p$ can be readily calculated using [@Li2013] $$\begin{aligned}
\vec{F}_{p}^{(scat)}(\vec{r}) &=& \hat{z}\frac{128\pi^5R^6}{3c\lambda_0^4}\left(\frac{m^2-1}{m^2+2}\right)^2n_{m}^5I_p(\vec{r})\\
\vec{F}_{p}^{(grad)}(\vec{r})&=&\frac{2\pi n_{m}R^3}{c}\left( \frac{m^2-1}{m^2+2}\right)\nabla I_p(\vec{r})\\
\label{eq:potential_dipole}
V_p(\vec{r})& =& -\frac{2\pi n_{m}R^3}{c}\left(\frac{m^2-1}{m^2+2} \right)I_p(\vec{r})\end{aligned}$$ where $m=n_p/n_m$ is the particle-medium refractive index ratio. We are interested in situations in which the $ m $ parameter is smaller than $1$, in such a way that the particle is repelled by light.
The forces acting on a spherical water droplet ($n_p=$1.33) with radius trapped in oil ($n_m=$1.46) by a bottle beam ($\lambda=$, $P_0=$ for each beam in the superposition) focused by an objective lens (NA=0.5) are displayed in Figure \[fig:force\_examples\]. As expected, the gradient forces point to the origin. Note that the scattering force, which points along the propagation direction, is null at the equilibrium position. This is in strong contrast to standard Gaussian traps and presents an advantage since the imbalance between scattering and gradient forces often poses challenges to optical trapping [@Nieminen2008].
Another interesting feature of the bottle beam trap is the flat bottom of the intensity well in the $z=0$ plane, seen in Figure \[fig:intensity\_examples\](a), and the approximate null derivative of the force along the radial direction at the origin. This can be understood by looking at the potential near the origin ($\rho\ll\omega_0$, $z\ll z_R$). It can be approximated to $4^{th}$ order as $$\frac{V_p(\rho,z)}{V_0}\approx \underbrace{\frac{4p^2}{\omega_0^4}\rho^4}_{T_{\rho^4}}-\underbrace{\frac{8p^2(p+1)}{\omega_0^2z_R^2}\rho^2z^2}_{T_{\rho^2 z^2}}+\underbrace{\frac{4p^2}{z_R^2}z^2}_{T_{z^2}}, \label{eq:intensity_approx}$$ where $V_0=[2\pi n_mR^3(m^2-1)/c(m^2+2)]I_0$ and the term of order $\mathcal{O}((z /z_R)^4)$ has been neglected since $(z /z_R)^4\ll(z /z_R)^2$ for $z\ll z_R$. At the plane $z=0$ the potential scales with $\rho^4$. Therefore, the force scales with $\rho^3$ and has vanishing first and second derivatives. For $z\neq0$, Eq. has a crossed term $\rho^2z^2$ that couples motion along the axial and radial directions. Because the scattering force is proportional to the intensity, it also has null derivatives at the equilibrium position and hence vanishes for a particle placed at and near the origin.
Finally, the potential in the $xz$ plane is displayed in Figures \[fig:force\_examples\](c) and \[fig:force\_examples\](d) for the cases of $p=1$ and $p=2$. As it can be seen, a trapped particle does not need to go through the high intensity peaks along the $x$ or $z$ axis in order to escape the trap. Smaller potential barriers have to be climbed if the particle undergoes paths like the yellow dashed ones. We will call the lowest potential energy needed for the particle to leave the trap $V_{min}$. Because the potential scales with $V_0$, we have $V_{min}\propto V_0$.
Decoupling approximation
------------------------
The axial and radial movements can be decoupled if the coupling term in Eq.(\[eq:intensity\_approx\]) is much smaller than the remaining terms. The conditions under which this assumption holds true can be found by estimating the magnitude of the particle’s displacements under the influence of the trap. Neglecting the cross term and considering thermal equilibrium we may write $$\begin{aligned}
\label{eq:rho4integral}
\langle \rho^4\rangle &=& \frac{1}{Z_0}\int d^3\vec{r} \rho^4 \exp{\left[-\frac{4V_0p^2}{k_BT}\left(\frac{\rho^4}{\omega_0^4}+\frac{z^2}{z_R^2}\right)\right]}\\
\label{eq:z2integral}
\langle z^2\rangle &=&\frac{1}{Z_0}\int d^3\vec{r} z^2 \exp{\left[-\frac{4V_0p^2}{k_BT}\left(\frac{\rho^4}{\omega_0^4}+\frac{z^2}{z_R^2}\right)\right]},\end{aligned}$$ where $k_B$ is the Boltzmann constant, $T$ is the temperature and $Z_0$ is given by $$\label{eq:Z0integral}
Z_0 = \int d^3\vec{r} \exp{\left[-\frac{4V_0p^2}{k_BT}\left(\frac{\rho^4}{\omega_0^4}+\frac{z^2}{z_R^2}\right)\right]}.$$
From Eqs. - we find that $$\begin{aligned}
\label{eq:rho4}
\sqrt[4]{\langle\rho^4}\rangle&=&\sqrt[4]{\frac{\omega_0^4k_BT}{8p^2V_0}}\\
\label{eq:z2}\sqrt{\langle z^2\rangle}&=& \sqrt{\frac{z_R^2k_BT}{8p^2V_0}}.\end{aligned}$$ Although Eqs. (\[eq:rho4\]) and (\[eq:z2\]) were derived by neglecting the cross term, they can be used to estimate the magnitude of the three different terms in Eq. (\[eq:intensity\_approx\]). Through simple scaling we are led to $$\frac{T_{\rho^2z^2}}{T_{\rho^4}}\sim \frac{T_{\rho^2z^2}}{T_{z^2}}\sim \frac{1+p}{\sqrt{2}p} \left(\frac{V_0}{k_BT}\right)^{-1/2}.$$ Because $V_{min}/k_BT\gg1$ is required for the particle to be confined in the presence of a thermal bath [@Li2013] and $V_{min}\propto V_0$, fulfillment of the decoupling condition is associated with increased trap stability.
In the decoupling regime the optical potential becomes $$V_p(\rho,z)\approx \frac{k_\rho^{(3)}}{4}\rho^4+\frac{k_z}{2}z^2,
\label{eq:intensity_approx2}$$ with the constants $k_\rho^{(3)}$ and $k_z$ given by, $$\begin{aligned}
\label{eq:k_rho}
k_\rho^{(3)}&=&\frac{64n_mP_0R^3}{c}\left(\frac{\pi{\rm{NA}}}{\lambda_0}\right)^6\left(\frac{1-m^2}{2+m^2}\right)p^2 \\
k_z&=&\frac{32P_0R^3\lambda_0^2}{\pi^2n_mc}\left(\frac{\pi{\rm{NA}}}{\lambda_0}\right)^6\left(\frac{1-m^2}{2+m^2}\right)p^2.\end{aligned}$$
Trapped particle dynamics
-------------------------
To further evaluate the validity of the above estimates and approximations, it is useful to simulate the dynamics of a particle trapped by the potential of a bottle beam in its exact form, calculated from Eqs. (\[eq:exact\_intensity\]) and (\[eq:potential\_dipole\]). The equation of motion for a spherical particle under this condition is $$M\ddot{\vec{r}}(t)=-\gamma \dot{\vec{r}}(t)-\nabla V(\vec{r}(t))+\sqrt{2\gamma k_BT}\vec{W}(t),
\label{eq:eom}$$ where $\eta$ is the medium’s viscosity, $\gamma=6\pi\eta R$ is the drag coefficient and $M$ is the particle’s mass. The environmental fluctuations are modelled using a Gaussian, white and isotropic stochastic process $\vec{W}(t)=(W_x(t), W_y(t), W_z(t))$, with zero mean and no correlations among different directions. We have that $$\langle W_i(t) W_i(t') \rangle= \delta(t-t') \ ,$$ where $\delta(t-t')$ is the Dirac delta in the time-domain.
For a sufficiently small particle the inertial term $M \ddot{\vec{r}}$ is negligible in comparison to the viscous term $\gamma \dot{\vec{r}}$. In this so-called *over-damped* regime we can numerically integrate equation using $$\vec{r}(t+\Delta t)=\vec{r}(t)-\frac{\nabla V(\vec{r}(t))}{\gamma}\Delta t+\sqrt{\frac{2k_BT\Delta t}{\gamma}}\vec{W}(t)$$ where $\Delta t=\tau/n$ is the time interval between iterations, $\tau$ the total time of simulation and $n$ the total number of iterations.
Numerical integration of the motion of a water droplet ($n_p=1.33$, $R=70$nm) trapped in oil ($n_m=1.46$) by a bottle beam ($p=1$, $\lambda=780$nm) focused using an objective lens (NA=0.5) was performed for different trapping powers. Note that the total trapping power is two times larger than the power $P_0$ of each beam. See Appendix A for details.
The motion was simulated for a period of 10s using time steps of 0.5$\mu$s. This resulted in $20\times10^6$ position values for each trapping power. The values of $\sqrt[4]{\langle x^4\rangle}$ and $\sqrt{\langle z^2\rangle}$ obtained from this simulation of the exact potential and the curves predicted using the approximated potential in Eq. together with Eqs. (\[eq:rho4integral\]) and (\[eq:z2integral\]) are displayed in Figure \[fig:curves\]. The largest values of $\sqrt[4]{\langle x^4\rangle}/\omega_0$ and $\sqrt{\langle z^2\rangle}/z_R$ obtained are approximately 0.27 and 0.067, respectively. This justifies the fourth order approximation leading to Eq. for the entire simulated range of trapping powers.
Moreover, we can see from Figure \[fig:curves\] that agreement between the simulated dynamics of the exact potential and the approximate potential of Eq. increases with $P_0$. For $P_0>1$W, exact and approximate values differ by less than 3%, and hence Eq. can be considered a good approximation of the potential. This behavior is consistent with the previous estimate that the ratios $T_{\rho^2z^2}/T_{\rho^4}$ and $T_{\rho^2z^2}/T_{z^2}$ scale with $V_0^{-1/2}$, and hence, the larger the trapping power the smaller the cross term in comparison to the remaining relevant terms.
This can be further verified in Figure \[fig:ratios\], where we plot the ratios $$r_1 = \frac{\left\langle T_{\rho^2 z^2}\right\rangle}{\left\langle T_{\rho^4}\right\rangle}\,, \quad
r_2 = \frac{\left\langle T_{\rho^2 z^2}\right\rangle}{\left\langle T_{z^2}\right\rangle},$$ obtained from the simulations. The decreasing behavior of $r_1$ and $r_2$ with respect to $P_0$ confirms that increasing the trapping power is an effective way of decoupling the radial and axial directions.
Another consequence of the interplay between a radial quartic and a longitudinal quadratic potential is that elongation of the trap can be adjusted by tuning the laser power. This is illustrated in Figure \[fig:motion\], in which the positions of the trapped particle obtained from the numerical simulation are displayed in a scatter plot, for $P_0=100$mW and $P_0=5$W. As it can be seen, the trap is appreciably compressed along the z axis in the latter case, but not in the former. This feature is not present in regular Gaussian tweezers: since the potential is quadratic along the three axis, the expected value of the displacement along all axes scale equally with $\sqrt{P}$. We note that this compression is different from the one caused by an increase in numerical aperture, mentioned previously. In that case we have a compression of the overall shape of the intensity landscape along the $z$ axis, which happens in the case of a Gaussian beam due to the scaling of $\omega_0$ with $\textrm{NA}^{-1}$ and of $z_R$ with $\textrm{NA}^{-2}$. In contrast, an increase in the trapping power of a bottle beam compresses the region visited by the particle over time.
In summary, we conclude that in the dipole regime Eq. is a good approximation for the optical potential generated by a bottle beam for a wide range of trapping powers, as it only relies on $\rho^4/\omega_0^4\ll1$ and $z^4/z_R^4\ll1$. On the other hand, decoupling of the radial and axial motions only occurs for high trapping powers that make the cross term in Eq. negligible. This allows approximating the potential by Eq. . Furthermore, increasing the trapping power causes squashing along the axial direction of the accessible region for a trapped particle.
Decoupling by addition of an extra mode
---------------------------------------
An alternative way to decouple the radial and longitudinal dependencies of the potential is to add an extra Laguerre-Gauss mode with $\ell_2=0$ and $p_2\neq0$ to the superposition. Consider the intensity $I(\rho, z)$ of the following superposition: $$\begin{aligned}
\label{}
\hspace{-4mm}E(\rho, z)\!=\!E^{LG}_{0, 0}(\rho,z)\!+\! \alpha_1E^{LG}_{0,1}(\rho,\!z)\!+\!\alpha_2 E^{LG}_{0,p_2}(\rho,z),\end{aligned}$$ where $\alpha_{j=1,2}$ are complex amplitudes. The condition to have a bottle beam is that $I(\rho, z)$ vanishes at the focus. The light intensity at the beam focus is proportional to $$\begin{aligned}
\label{eq:bottle_condition}
I(\mathbf{0}) = \mathcal{N}^2
\big|1 + \alpha_1 + \alpha_2\big|^2,\end{aligned}$$ where $\mathcal{N}=\sqrt{4P_0/c\epsilon\pi\omega_0^2}$ .
With the appropriate approximations (see Appendix B for details) and the bottle beam condition in Eq., we obtain the approximate intensity $I(\rho, z)$, $$\begin{aligned}
\!\!\!\!\mathcal{N}^2\!\!\left[
4|B|^2\!\left(\frac{z^2}{z_R^2} \!+\!\frac{\rho^4}{w_0^4}\right)\!\!-8\!\left[|B|^2\!+\!\!\mathrm{Re}\left(AB^*\right)\right]\!\frac{z^2\!\rho^2}{z_R^2w_0^2}\right]\!\!\!\,,\end{aligned}$$ where $A = \alpha_1 + \alpha_2 p_2^2$ and $B = \alpha_1 + \alpha_2p_2$. Decoupling of the radial and longitudinal motions can then be achieved by choosing $\alpha_1$ and $\alpha_2$ such that $$\begin{aligned}
|B|^2 + \mathrm{Re}\left(AB^*\right)=0\qquad (B\neq 0).\end{aligned}$$
As an example, let us choose $p_2=2$. A decoupled bottle beam can be obtained by the superposition coefficients $$\begin{aligned}
\alpha_1 &=& -3/2, \\ \alpha_2 &=& 1/2.\end{aligned}$$ Using Eq. we can find the decoupled potential near the origin ($\rho\ll\omega_0$, $z\ll z_R$). This is given to $4^{th}$ order by $$\begin{aligned}
\label{eq:potencial_dipole_2}
V(\rho, z) \approx V_0
\left(\frac{z^2}{z_R^2} + \frac{\rho^4}{w_0^4}\right),\end{aligned}$$ which has the same form as Eq.(\[eq:intensity\_approx2\]). Moreover, this solution yields the maximum trap stiffness of the three mode configuration, as demonstrated in Appendix B.
Calibration of the optical trap
-------------------------------
In laboratory conditions, quantitative measurements using optical tweezers rely on knowledge of the trap’s parameters. In the case of a bottle trap defined by the potential in Eq. , the relevant parameters are $k_z$ and $k^{(3)}_\rho$. To properly operate the tweezer these must be found by measuring the particle’s position during a finite interval of time. This yields a time series $\vec{r}_m(t)=\vec{\beta}\cdot\vec{r}(t)$, where $\vec{\beta}=(\beta_x, \beta_y, \beta_z)$ are conversion factors between position displacements and the measured quantity, such as the voltage in a position sensitive detector. For simplicity, we will assume $\beta_x=\beta_y=\beta_\rho$. For a bottle beam trap the particle’s position can be measured using a high speed camera [@Gibson2008], or alternatively by applying a purely Gaussian beam at a different wavelength with respect to the bottle beam. The second beam can be focused onto the trapped particle by the same objective lens used for the bottle, and collected by a second objective lens after separation from the trapping beam by a dichroic mirror. The collected Gaussian light can then be directed onto a Quadrant Photo Detector, where the usual forward scattering measurement is performed [@Pralle1999]. The Gaussian power should be kept significantly weaker then the Bottle power to avoid disturbances due to the presence of this auxiliary Gaussian trap.
In the decoupled regime, movement along the $z$ axis is independent from movement along the $x$ and $y$ axes and the equations of motion can be separated from Eq. , yielding $$-\gamma \dot{z}(t)-k_zz(t)+\sqrt{2\gamma k_BT}W_z(t)=0,$$ where once again we assume the inertial term is negligible. The constants $k_z$ and $\beta_z$ can be found using the standard procedure of analysing the autocorrelation function [@Alves2012] or the power spectral density [@BergSorensen2004; @Melo2020] of the measured axial displacements $z_m(t)=\beta_zz(t)$.
To find the remaining relevant constants we need two independent equations. Using Eqs.(\[eq:rho4integral\]) and (\[eq:k\_rho\]) we may write $$\frac{k_\rho^{(3)}\langle\rho^4\rangle}{4} = \frac{k_BT}{2}\rightarrow\langle\rho^4\rangle = \frac{2k_BT}{k_\rho^{(3)}},$$ leading to the relation, $$\label{eq:calibrateX1}
\langle\rho_m^4\rangle = \beta_\rho^4 \frac{2k_BT}{k_\rho^{(3)}}.$$
A second equation can be obtained from an active method of calibration consisting of moving the sample in which the particle is immersed with a known velocity $\vec{v}_{drag}$ [@Simmons1996; @Brouhard2003]. This will cause a constant drag force $\gamma \vec{v}_{drag}$ on the particle, and taking $\vec{v}_{drag}=v_{drag}\hat{x}$ the equation of motion along the $x$ axis becomes $$\label{eq:eomX}
\gamma v_{drag}-\gamma \dot{x}(t)-k_\rho^{(3)}x(t)\rho(t)^2+W_x(t)=0 \ .$$ After a transient time the particle reaches an equilibrium position displaced with respect to the trap’s center, with $\langle \dot{x}(t)\rangle = 0 $. Taking the time average of Eq.(\[eq:eomX\]) leads to $$\gamma v_{drag}-k^{(3)}_\rho\langle x(t)\rho(t)^2\rangle=0,$$ which can then be used to obtain the relation $$\label{eq:calibrateX2}
\langle x_m(t)\rho_m(t)^2\rangle = \beta_\rho^3 \frac{\gamma v_{drag}}{k_\rho^{(3)}}.$$
Eqs.(\[eq:calibrateX1\]) and (\[eq:calibrateX2\]) together with the standard autocorrelation procedure for the axial motion enables the measurement of the four parameters $\beta_\rho,\beta_z, k_\rho^{(3)}$ and $k_z$ in the decoupled approximation.
Intermediate regime
===================
In many applications it is desirable to trap ‘large’ micron-sized particles such as living cells [@Zhong2013; @Liang2019]. This presents an intermediate regime, in which the size of the particle is comparable to the wavelength of the trapping beam ($R\approx\lambda$) and neither the dipole ($R\ll\lambda$) nor geometric optics ($R\gg\lambda$) approximations can be used to calculate the optical forces. Instead, the forces must be calculated using the so-called generalized Lorenz–Mie theory (GLMT), for which we provide a brief introduction following the treatment presented in [@Nieminen2007].
{width="\linewidth"}
Generalized Lorenz-Mie Theory
-----------------------------
Regardless of the size of the trapped particle, optical forces arise from the exchange of momentum with the photons from the trapping beam. Therefore, the total momentum transferred to the particle is equal to the change in momentum of the scattered electromagnetic field. It is then useful to separate the field in incoming $\vec{E}_{in}$ and outgoing $\vec{E}_{out}$ parts, which in turn can be expanded in terms of vector spherical wave-functions (VSWFs) defined in a coordinate system centered at the particle’s center, $$\begin{aligned}
\label{eq:expasion_ein}
\vec{E}_{in} = \sum^\infty_{i=1}\sum^i_{j=-i}a_{ij}\vec{M}_{ij}^{(2)}(k\vec{r})+b_{ij}\vec{N}^{(2)}_{ij}(k\vec{r}),\\
\label{eq:expasion_eout}
\vec{E}_{out} = \sum^\infty_{i=1}\sum^i_{j=-i}p_{ij}\vec{M}_{ij}^{(1)}(k\vec{r})+q_{ij}\vec{N}^{(1)}_{ij}(k\vec{r}),\end{aligned}$$ where $\vec{M}_{ij}^{(1)},\vec{N}_{ij}^{(1)},\vec{M}_{ij}^{(2)}$ and $\vec{N}_{ij}^{(2)}$ are the VSWFs, with the upper index (1) standing for outward-propagating transverse electric and transverse magnetic multipole fields and (2) for the corresponding inward-propagating multipole fields.
The coefficients $a_{ij}$ and $b_{ij}$ can be calculated for the incident beam and used to obtain the $p_{ij}$ and $q_{ij}$ coefficients for the scattered field by a simple matrix-vector multiplication between the so-called $T$-matrix and a vector containing the coefficients of the incoming field. The $T$-matrix depends only on the characteristics of the trapped particle, which we assume spherical. Once the coefficients are calculated, the force along the axial direction $z$ is given by $$\begin{aligned}
\label{eq:force_z_mie}
F_z&=&\frac{2n_{md}P}{cS}\sum_{i=1}^\infty\sum_{j=-1}^i\frac{j}{i(i+1)}\textrm{Re}(a_{ij}^*b_{ij}-p_{ij}^*q_{ij})-\nonumber\\&&\frac{1}{i+1}\sqrt{\frac{i(i+2)(i-j+1)(i+j+1)}{(2i+1)(2i+3)}}\times\nonumber\\
&&\hspace{-1mm}\textrm{Re}(a_{ij}a_{i+1,j}^*\hspace{-1mm}+\hspace{-1mm}b_{ij}b_{i+1,j}^*\hspace{-1mm}-\hspace{-1mm}p_{ij}p_{i+1,j}^*\hspace{-1mm}-\hspace{-1mm}q_{ij}q_{i+1,j}^*)\end{aligned}$$ with $$S=\sum_{i=1}^\infty\sum_{j=-i}^i(\vert a_{ij}\vert^2+\vert b_{ij}\vert^2).$$
Forces acting along the $x$ and $y$ axis have more complicated formulae and can be more easily calculated by rotating the coordinate system. The effect of displacing the particle can be taken into account by appropriate translations of the trapping beam.
Due to the linearity of Eqs. and , the expansion coefficients for a superposition of different beams can be found by adding the expansion coefficients for each beam, and subsequently substituted in Eq. to calculate the resultant force. We shall use the latest version of the toolbox developed in [@Nieminen2007] to perform these computations for the case of a particle trapped by a bottle beam.
Optical forces from a bottle beam
---------------------------------
Optical forces generated by the superposition of a Gaussian beam and a Laguerre-Gauss beam with $\ell=0, p\neq0$ are obtained with the aid of [@Nieminen2007; @Lenton2020]. For simplicity, we focus on the $p=1$ case and a particle of refractive index $n_{p}=1.33$ trapped by a 500 mW beam at $\lambda_0=780$ nm immersed in oil of refractive index $n_{m}=1.46$.
Figure \[fig:many\_intermediate\] shows the plots of $F_z(z)$ and $F_x(x)$ divided by the particle’s mass for four different NA’s and four different particle radii. The force in the $z$ direction is evaluated for $x=y=0$, while $F_x(x)$ is evaluated at $y=0, z=z_{eq}$, where $z_{eq}$ is the equilibrium coordinate along the $z$ direction, i.e., $$\begin{cases}
F_z(z_{eq})=0\\
dF_z(z)/dz\vert_{z=z_{eq}}<0
\end{cases}$$ When no equilibrium position exists, $F_x(x)$ is evaluated at $z=0$.
Some general trends can be extracted from Figure \[fig:many\_intermediate\]. First, we note that if the sphere is small ($R=\lambda_0/4$) and the numerical aperture is low (NA $=0.3,0.5$), the force in the $x$ direction resembles the one calculated using the dipole approximation, i.e., it appears to scale with $x^3$ around the origin. As R or NA increases, this cubic dependence starts to vanish, giving place to a linear dependence.
We can also notice that the size of the particle and the numerical aperture play an important role on the existence of an equilibrium position in the axial direction, with large radius $R$ and large NA being detrimental to the trap stability along the $z$ axis. For NA = 0.7, for instance, there is an equilibrium position if $R=\lambda_0/4$ or $R=\lambda_0/2$, but not if $R$ is larger. For a fixed $R=\lambda_0$, $z_{eq}$ doesn’t exist for NA $>0.5$. This is rather different from what happens in the regular Gaussian trap, in which increasing the NA is associated with an increase in trap stability [@Li2013].
Limitations of trapping in a dark focus
---------------------------------------
The trends observed in Figure \[fig:many\_intermediate\] can be understood qualitatively by recalling that a bottle beam is a dark region surrounded by a finite bright light boundary. If the particle is small enough it will fit inside the dark region and will be repelled by the boundary. In contrast, if the particle is too big it does not fit inside the bottle and the dark focus becomes irrelevant, with the beam effectively pushing the particle away.
This can be seen for in Figures \[fig:many\_intermediate\](e)-(h): the dimensions of the bottle when NA $=0.5$ are $W=0.99\mu$m and $H=2.9\mu$m. Therefore, a particle of diameter $0.5\lambda$ fits entirely inside the bottle and is free within the dark region, causing the force in the $x$ direction to have vanishing derivative near the origin. When $R=\lambda_0$, the particle no longer fits in the dark focus, and the influence of light gives a linear scaling to $F_x(x)$ around the equilibrium position. When $R=2\lambda_0$ the particle has an increased overlap with the light intensity and no longer encounter an equilibrium position.
Similarly, an increase in numerical aperture causes the dark focus to shrink. When the bottle becomes too small to comprise the particle the situation in the third column of Figure \[fig:many\_intermediate\] is reached and the forces eventually turns into non-restorative ones.
This qualitative reasoning is confirmed in Figure \[fig:limiting\], in which the equilibrium position $z_{eq}$ and the derivative along the $x$ direction near the equilibrium position are displayed as a function of the particle’s radius and the numerical aperture. Two main regions can be identified in each of the plots. The first of them, is the region for which $z_{eq}$ was not found in the range of inspected axial coordinates $-6\lambda_0<z<6\lambda_0$. In this region, the derivative along the $x$ axis was not evaluated. The remaining areas are the ones in which an axial equilibrium position exists.
Because we wish to trap the particle in the dark focus, we need to avoid equilibrium situations as the ones described in [@Gahagan1996] in the context of vortex beams, in which the scattering force is balanced by the repelling gradient force before the focus. To exclude trapping positions outside the bottle, the regions in which $z_{eq}>H/2$ were displayed in dark grey and the regions in which $z_{eq}<-H/2$ were displayed in light grey. In the latter case, the derivative of the radial force was found to be positive and hence non-restorative, while in the former this derivative was found to be negative. The coloured region, then, is the region for which stable trapping inside the bottle is possible.
We can then conclude that for a given $R$, there is a maximum numerical aperture that can be used to form a stable trap. Conversely, for a given NA, there is a limit on the size of the particles that can be trapped. Figure \[fig:refractive\](a) shows how this limit varies for different refractive indices of the medium and a fixed NA $=0.5$. The curves were chopped when $z_{eq}$ became larger than $H/2$, and we can clearly see that the closer the refractive index gets to that of the particle, the larger the radius of the particle that can be trapped. Figure \[fig:refractive\] confirms that the radial force is restorative for the entire range of $R$ and $n_m$ we considered. It also shows that while decreasing $n_m$ can help trapping larger particles, it also diminishes the force experienced by the particle, and hence plays an important role in the trap’s stability.
Trapping living organisms in the dark
-------------------------------------
![Intermediate regime simulations for different values $n_m$: (a) Axial equilibrium coordinate and (b) first derivative of the force in the radial direction as a function of particle’s radius. Points for which $z_{eq}>H/2$ are not displayed. The parameters used in the simulation were $\lambda_0=780$nm, $P=500$mW, $n_m=1.46$, $n_p=1.33$, $p=1$.[]{data-label="fig:refractive"}](Figure8.pdf){width="\linewidth"}
The bottle beam trap finds promising applications in biology. For instance it has been reported that organelles with a refractive index lower than its surroundings are repelled from standard Gaussian optical tweezers [@Zhang2008]. The bottle beam could then be used to manipulate such organelles within a cell.
Similarly, a dark optical trap could also be employed to trap living organisms without excessive laser damage onto the cell by appropriate choice of a surrounding medium. Iodixanol has been reported as a non-toxic medium for different organisms, with high water solubility, in which the refractive index can be linearly tuned in the visible to near-IR range from $\sim1.33$ to $\sim 1.40$ by changing concentration [@Boothe2017]. Assuming a mean refractive index for a living cell to be within the range $\sim1.36$ to $\sim 1.39 $ [@Liu2016; @Rappaz2005] it is expected that stable trapping in a dark focus can be attained.
Mycoplasma are known to be among the smallest living organisms, and perhaps the simplest cells [@Morowitz1984]. With radii around $\sim 0.3 \mu$m, these organisms lack a cell wall [@Krause2001], being protected from the surrounding environment solely by their cellular membrane. This may present interesting mechanical and elastic properties which could be probed with the bottle beam. We propose investigating the trapping of Mycoplasma cells immersed in a non-toxic mixture of refractive index $~1.40$. Iodixanol presents a possible such medium, but further empirical tests must be carried over to fully determine how it affects living Mycoplasma cells. Figure \[fig:myco\] shows the simulated forces acting on a trapped Mycoplasma when the parameters shown in Table \[tab:exp\] are used. As it can be seen, forces along radial and axial directions are restorative and should provide stable trapping inside the bottle.
Parameter Units Value
--------------------------------- -------- -----------
Particle refractive index $n_p$ - 1.36-1.39
Medium refractive index $n_m$ - 1.40
Particle radius $R$ $\mu$m 0.3
Laser wavelength $\lambda_0$ nm 1064
Numerical aperture NA - 0.7
Laser power mW 500
Index $p$ - 1
: Proposed values for trapping a Mycoplasma cell using a bottle beam. \[tab:exp\]
It is known that direct incidence of focused laser light onto living cells can affect their division and growth [@Zhang2008]. As an interesting application of the bottle beam one could observe the process of cell division without directly sending a focused beam onto the trapped particle. The following experiment could be performed: at each round of measurement, a cell undergoing division is trapped in the dark focus by a given laser power and the complete cycle of the division process is observed. The power is then incrementally increased in every round of the experiment with a new cell, until a threshold value is reached at which cell division is significantly affected or perhaps even precluded. With the proper tweezer calibration presented in the previous section, the threshold power provides information on the forces acting during the process of cell division.
![Forces acting on a trapped Mycoplasma cell, shown here centered at the origin for size comparison, (a) along the $x$ direction as a function of radial displacement and (b) along the $z$ direction as a function of axial displacement. The solid and the dashed curves correspond to a medium’s refractive index of 1.36 and 1.39, respectively, while the yellow area correspond to $1.36\leq n_m\leq 1.39$. The other parameters used in the simulation are displayed in Table \[tab:exp\].[]{data-label="fig:myco"}](Figure9.pdf){width="\linewidth"}
Conclusions
===========
We have theoretically analysed the optical forces acting on a particle of lower refractive index than its surrounding medium trapped in the dark focus of a bottle beam generated by the superposition of a Gaussian beam and a Laguerre-Gauss beam with $\ell=0$ and $p\neq0$. Because the size of trapped particles commonly range from tens of nanometers [@Hansen2005] to several microns [@Alinezhad2019] we analysed such forces both for particles much smaller than and with dimensions comparable to the wavelength of the trapping beam.
In the case of small particles, the dipole approximation was applied, resulting in a number of distinguishing features of the investigated trap. Scattering was found to be null at the focus of the beam, eliminating imbalance between gradient and scattering forces [@Nieminen2008]. The optical potential, on the other hand, coupled the motion along the radial and axial directions. It was shown that these could be decoupled by using a sufficiently high trapping power. The approximated decoupled potential turns out to be quartic and quadratic in the radial and axial directions, respectively. To test the validity of the approximated potential, motion of a particle trapped by the exact potential in a viscous medium was simulated and the results were confronted with those expected from the approximation as a function of laser power. We have also shown that by superposing a third mode, motion along the axial and radial directions can be decoupled independently of the trap power. To guide future experimental realizations, a calibration method was proposed.
In the case of larger objects, for which the dipole approximation is not valid, the tools developed in [@Nieminen2007; @Lenton2020] were used to calculate the optical forces. Equilibrium positions after the focus were found, indicating a trapping regime different form the one described in [@Gahagan1996]. The interplay between the numerical aperture and the sphere’s radius were explored and led to the conclusion that there is an upper bound for both of these quantities when using bottle beams for optical trapping. These limitations were interpreted in terms of the size of the optical bottle in comparison to the size of the particle, and were found to be eased by choosing a medium with refractive index close to that of the particle.
Finally, the findings obtained through exploration of the intermediate regime led to an experimental proposal to trap a living organism using the bottle beam. Considering values of refractive index reported in the literature, it is expected that trapping of small cells such as the Mycoplasma immersed in a non toxic high-refractive index medium in a dark focus is within reach. This could be applied to situations in which focusing a high laser power onto the scrutinized cell is detrimental [@BlazquezCastro2019], as in the case of cellular division [@Zhang2008].
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to acknowledge Lucianno Defaveri for fruitful discussions regarding the statistical mechanics aspects of this work.
In 2019, T.G. attended the Prospects in Theoretical Physics program at the Institute for Advanced Studies in Princeton. The meeting was centered on “Great Problems in Biology for Physicists” and had an important impact in the development of this work.
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (Faperj, Scholarships No. E-26/200.270/2020 and E-26/202.830/2019), Instituto Serrapilheira (Serra-1709-21072) and Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-CNPq).
Appendix A: total power of a bottle beam {#appendix-a-total-power-of-a-bottle-beam .unnumbered}
========================================
Throughout Section \[sec:dipole\], we used the power $P_0$ of each beam in the superposition as a measure of the trapping power. To find the exact relation between $P_0$ and the total power of the beam, we need to integrate equation \[eq:exact\_intensity\] along some plane orthogonal to the propagation of the beam. Choosing the plane $z=0$, $$\begin{aligned}
P&=&\int_{0}^\infty d\rho \int_{0}^{2\pi}(d\theta\rho)I_0e^{-2\rho^2/\omega_0^2}\nonumber\\&\times&\left[1-2L^0_p\left(\frac{2\rho^2}{\omega_0^2}\right)+L^0_p\left(\frac{2\rho^2}{\omega_0^2}\right)^2 \right]\nonumber\\&=&2\pi I_0\int_{0}^\infty \frac{\omega_0^2du}{4} e^{-u}[1-2L_p^0(u)+L_p^0(u)^2]\nonumber\\&=&P_0\int_{0}^\infty du\,\, e^{-u}[1-2L_p^0(u)+L_p^0(u)^2]\label{eq:Ptotal}.\end{aligned}$$
The Laguerre-Polynomials satisfy $$\int_0^\infty x^\ell e^{-x} L_p^{\vert\ell\vert}(x)L_q^{\vert\ell\vert}(x)=\frac{(p+\ell)!}{p!}\delta_{p,q},$$ which implies that $$\begin{aligned}
\label{eq:int1}
\int_0^\infty du\,e^{-u}&=&\int_0^\infty du\,e^{-u}L^0_0(u)L^0_0(u)=1\\
\label{eq:int2}
\int_0^\infty du\,e^{-u}L_p^0(u)&=&\int_0^\infty du\,e^{-u}L^0_0(u)L^0_p(u)=0\\
\label{eq:int3}
\int_0^\infty du\,e^{-u}L_p^0(u)^2&=&\int_0^\infty du\,e^{-u}L^0_p(u)L^0_p(u)=\hspace{-1mm}1.\end{aligned}$$ Finally, substituting Eqs. (\[eq:int1\])-(\[eq:int3\]) into Eq.(\[eq:Ptotal\]), we find $$P=2P_0.$$
Appendix B: The decoupling mode {#appendix-b-the-decoupling-mode .unnumbered}
===============================
In this section we derive the conditions that must be fulfilled by a three-mode superposition in order to provide a decoupled potential for the trapped particles, while keeping the bottle beam structure. Let us consider the superposition $$\label{superp}
E (\mathbf{r}) = E^{LG}_{0,0}(\mathbf{r}) + \alpha_1 E^{LG}_{0,p_1}(\mathbf{r}) + \alpha_2 E^{LG}_{0,p_2}(\mathbf{r})\,.
$$ We want to obtain a relationship between the complex coefficients $\alpha_1$ and $\alpha_2\,$, and the radial orders $p_1$ and $p_2$ that achieve the desired decoupling and bottle-beam profile. The LG modes with zero orbital angular momentum (OAM) can be written as $$\label{LG}
E^{LG}_{0, p}(\rho, \phi, z) = \mathcal{N} e^{- \frac{\bar{\rho}^2}{2}} L^{0}_p \left( \bar{\rho}^2 \right)
\left(\frac{1-i\bar{z}}{\sqrt{1+\bar{z}^2}}\right)^{2p+1}\,,
$$ where we defined $\bar{\rho}=\sqrt{2}\rho/w(z)\,$, $\bar{z}=z/z_R$ and the last term is the Gouy phase.
The trapping potential is proportional to the light intensity distribution and, therefore, to the square modulus of the electric field $$\begin{aligned}
\label{intensity}
\vert E (\mathbf{r})\vert^2 &=& \mathcal{N}^2 e^{-\bar{\rho}^2}
\bigg\vert1 + \alpha_1 L^{0}_{p_1} \left( \bar{\rho}^2 \right) \left(\frac{1-i\bar{z}}{\sqrt{1+\bar{z}^2}}\right)^{2p_1}
\nonumber\\&+& \alpha_2 L^{0}_{p_2} \left( \bar{\rho}^2 \right) \left(\frac{1-i\bar{z}}{\sqrt{1+\bar{z}^2}}\right)^{2p_2}\bigg\vert^2
\,.
$$ We seek an approximate expression for the trapping potential around the beam focus, which can be obtained from a power series expansion around the beam focus. The bottle-beam condition requires that the light intensity vanishes at this point. Note that $L^{0}_{p} (0) = 1\,$, so the light intensity at the beam focus is proportional to $$\label{intensity0}
\vert E (\mathbf{0})\vert^2 = \mathcal{N}^2
\vert 1 + \alpha_1 + \alpha_2\vert^2
\,,$$ which implies $$\label{bb-condition}
\vert 1 + \alpha_1 + \alpha_2\vert^2 = 0\,.$$ This condition cancels out the zero order contribution to the power series expansion. We will keep terms up to $\bar{\rho}^4$ and $\bar{z}^2\,$, which are the first non vanishing contributions to the power series. Since the zero order term vanishes, it will be easier to expand first the expression inside the square modulus in Eq. and keep terms up to $\bar{\rho}^2$ and $\bar{z}^2\,$. The following approximations are assumed $$\begin{aligned}
\label{approx}
e^{-\bar{\rho}^2} &\approx& 1 - \bar{\rho}^2\,,
\\
L^{0}_{p} \left( \bar{\rho}^2 \right) &\approx& 1 - p\bar{\rho}^2\,,
\\
\left(\frac{1-i\bar{z}}{\sqrt{1+\bar{z}^2}}\right)^{2p} &\approx& 1 - 2ip\bar{z} -2p^2 \bar{z}^2
\,.\end{aligned}$$ Applying the approximations above together with the bottle-beam condition , we find $$\begin{aligned}
\label{approx-intensity}
\vert E (\mathbf{r})\vert ^2 &\approx&
\mathcal{N}^2 \hspace{-0.75mm}
\left(1 \hspace{-0.75mm}- \hspace{-0.75mm}\bar{\rho}^2\right)
\hspace{-0.75mm}
\vert 2 A \bar{z} (\bar{z} \hspace{-0.75mm}- \hspace{-0.75mm}i\bar{\rho}^2) + B (\bar{\rho}^2 \hspace{-0.75mm} + \hspace{-0.75mm} 2i\bar{z})\vert ^2
\\
&\approx& \mathcal{N}^2
\left[
\vert B\vert ^2 (4\bar{z}^2 + \bar{\rho}^4)
-4
C
\bar{\rho}^2 \bar{z}^2
\right]
\\
&\approx& \mathcal{N}^2
\left[
4 \vert B\vert ^2 \left(\frac{z^2}{z_R^2} + \frac{\rho^4}{w_0^4}\right)
-8
C
\frac{z^2\,\rho^2}{z_R^2\,w_0^2}
\right]
\,.\end{aligned}$$ where we defined $$\begin{aligned}
\label{defs}
A &=& \alpha_1 p_1^2 + \alpha_2 p_2^2\,,
\\
B &=& \alpha_1 p_1 + \alpha_2 p_2,
\,
$$ and $C=\vert B\vert ^2 + \mathrm{Re}\left(A B^*\right)$. The two-mode bottle-beam potential is recovered by making $\alpha_1 = -1$ and $\alpha_2 = 0\,$.
Decoupling between the radial ($\rho$) and longitudinal ($z$) dependencies is achieved by choosing $\alpha_1$ and $\alpha_2$ such that $$\begin{aligned}
\label{decouple-condition}
\vert B\vert^2 + \mathrm{Re}\left(A B^*\right)=0
\,.\end{aligned}$$ We can write this condition in terms of the real and imaginary parts of the coefficients $\alpha_j = a_j + ib_j\,$. Including the bottle-beam condition, the following equations must hold $$\begin{aligned}
\label{cond1}
&& 1 + a_1 + a_2 = 0\,,
\\
\label{cond2}
&& b_1 + b_2 = 0\,,
\\
\label{cond3}
&& (a_1^2 + b_1^2) (p_1^2 + p_1^3) + (a_2^2 + b_2^2) (p_2^2 + p_2^3) \nonumber\\&&\,\,\,\,+ p_1 p_2 (p_1 + p_2 + 2) (a_1 a_2 + b_1 b_2) = 0
\,.\end{aligned}$$ By using and in , and defining $c=a_1^2 + b_1^2\,$, we derive the following condition $$\begin{aligned}
\label{condfinal}
&& c \left[p_1^2 (p_1+1) + p_2^2 (p_2+1) - p_1 p_2 (p_1 + p_2 + 1)\right] +
\\
\nonumber
&& a_1 \left[2 p_2^2 (p_2+1) - p_1 p_2 (p_1 + p_2 + 1)\right] + p_2^2 (p_2 + 1) = 0
\,.\end{aligned}$$ This condition has infinite solutions. A simple example is obtained by setting $p_1 = 2\,$, $p_2 = 1$ and $c=1\,$, which means that the coefficient $\alpha_1$ is a phase factor. The following coefficients are then obtained $$\begin{aligned}
\label{coefficients}
\alpha_1 &=& \frac{2 + i\sqrt{5}}{3}\,,
\\
\nonumber
\alpha_2 &=& -\left(\frac{5 + i\sqrt{5}}{3}\right)
\,.\end{aligned}$$ These coefficients define a beam with a dark focus that also decouples the radial and longitudinal variations of the trapping potential. The resulting expression for the square modulus of the electric field is $$\begin{aligned}
\label{decouple-intensity}
\vert E (\mathbf{r})\vert^2 &\approx&
\frac{8}{3}\,\mathcal{N}^2
\left(\frac{z^2}{z_R^2} + \frac{\rho^4}{w_0^4}\right)
\,.\end{aligned}$$
Appendix B: The decoupling mode {#appendix-b-the-decoupling-mode-1 .unnumbered}
===============================
In this section we derive the conditions that must be fulfilled by a three-mode superposition in order to provide a decoupled potential for the trapped particles, while keeping the bottle beam structure. Let us consider the superposition $$\label{superp}
E (\rho, z) = E^{LG}_{0,0}(\rho, z) + \alpha_1 E^{LG}_{0,p_1}(\rho, z) + \alpha_2 E^{LG}_{0,p_2}(\rho, z)\,.
$$ We want to obtain a relationship between the complex coefficients $\alpha_1$ and $\alpha_2\,$, and the radial orders $p_1$ and $p_2$ that achieve the desired decoupling and bottle-beam profile. The LG modes with zero OAM are given by $$\begin{aligned}
\label{LG}
\!\!E^{LG}_{0, p}(\bar{\rho}, \bar{z}) &=&\!\frac{\mathcal{N}}{\sqrt{1 + \bar{z}^2}} e^{- \frac{\bar{\rho}^2}{2}} L^{0}_p \left( \bar{\rho}^2 \right)
\left(\frac{1-i\bar{z}}{\sqrt{1+\bar{z}^2}}\right)^{2p+1}\nonumber\\
&\times&\exp\left[ikz+ik\frac{\rho^2}{2R(z)}\right],
$$ where we defined $\mathcal{N}=\sqrt{4P_0/c\epsilon\pi\omega_0^2}$, $\bar{\rho}=\sqrt{2}\rho/w(z)\,$, $\bar{z}=z/z_R$ and the last term is the Gouy phase.
The trapping potential is proportional to the light intensity distribution and, therefore, to the square modulus of the electric field $$\begin{aligned}
\label{intensity}
I(\bar{\rho}, \bar{z}) &=& \frac{I_0}{1 + \bar{z}^2} \,e^{-\bar{\rho}^2}
\Bigg|1 + \alpha_1 L^{0}_{p_1} \left( \bar{\rho}^2 \right) \left(\frac{1-i\bar{z}}{\sqrt{1+\bar{z}^2}}\right)^{2p_1}\nonumber\\&+&\alpha_2 L^{0}_{p_2} \left( \bar{\rho}^2 \right) \left(\frac{1-i\bar{z}}{\sqrt{1+\bar{z}^2}}\right)^{2p_2}\Bigg|^2.\end{aligned}$$ We seek an approximate expression for the trapping potential around the beam focus, which can be obtained from a power series expansion around this point. The bottle-beam condition requires that the light intensity vanishes at the focus. Note that $L^{0}_{p} (0) = 1\,$, so the light intensity at the beam focus is proportional to $$\label{intensity0}
I(\mathbf{0}) = I_0
\big|1 + \alpha_1 + \alpha_2\big|^2,$$ which implies $$\label{bb-condition}
\big|1 + \alpha_1 + \alpha_2\big|^2 = 0\,.$$ This condition cancels out the zero order contribution to the power series expansion. We will keep terms up to $\bar{\rho}^4$ and $\bar{z}^2\,$, which are the first non vanishing contributions to the power series. Since the zero order term vanishes, it will be easier to expand first the expression inside the square modulus in Eq. and keep terms up to $\bar{\rho}^2$ and $\bar{z}^2\,$. The following approximations are assumed $$\begin{aligned}
\label{approx}
e^{-\bar{\rho}^2} &\approx& 1 - \bar{\rho}^2\,,
\\
L^{0}_{p} \left( \bar{\rho}^2 \right) &\approx& 1 - p\bar{\rho}^2\,,
\\
\left(\frac{1-i\bar{z}}{\sqrt{1+\bar{z}^2}}\right)^{2p} &\approx& 1 - 2ip\bar{z} -2p^2 \bar{z}^2
\,.\\
\frac{1}{1+\bar{z}^2}&\approx& 1 - \bar{z}^2\end{aligned}$$ Applying the approximations above together with the bottle-beam condition , we find the following approximate expression for the trapping intensity $$\begin{aligned}
\label{approx-intensity}
I(\Bar{\rho}, \Bar{z}) &\approx&
I_0 \left(1-\bar{\rho}^2\right)
\big|2 A \bar{z} (\bar{z}\!-\!i\bar{\rho}^2)\!+\!B (\bar{\rho}^2 + 2i\bar{z})\big|^2
\nonumber\\
&\approx& I_0
\left[
|B|^2 (4\bar{z}^2\!+\!\bar{\rho}^4)-4
\{|B|^2\!+\!\mathrm{Re}\left(A B^*\right)\}
\bar{\rho}^2 \bar{z}^2
\right]
\nonumber\\
&\approx& I_0\!\!\left[
4|B|^2\!\left(\frac{z^2}{z_R^2} \!+\!\frac{\rho^4}{w_0^4}\right)\!\!-8\!\left[|B|^2\!+\!\!\mathrm{Re}\left(AB^*\right)\right]\!\frac{z^2\!\rho^2}{z_R^2\!w_0^2}\right] ,
\nonumber\end{aligned}$$ where we defined $$\begin{aligned}
\label{defs}
A &=& \alpha_1 p_1^2 + \alpha_2 p_2^2\,,
\\
B &=& \alpha_1 p_1 + \alpha_2 p_2
\,.
$$ The two-mode bottle-beam potential is recovered by making $\alpha_1 = -1$ and $\alpha_2 = 0\,$.
Decoupling between the radial ($\rho$) and longitudinal ($z$) dependencies is achieved by choosing $\alpha_1$ and $\alpha_2$ such that $$\begin{aligned}
\label{decouple-condition}
|B|^2 + \mathrm{Re}\left(AB^*\right)=0\qquad (B\neq 0)\,.\end{aligned}$$ We can write this condition in terms of the real and imaginary parts of the coefficients $\alpha_j = a_j + ib_j\,$. Including the bottle-beam condition, the following equations must hold $$\begin{aligned}
\label{cond1}
&&1 + a_1 + a_2 = 0\,,\\
\label{cond2}
&&b_1 + b_2 = 0\,,\\
\label{cond3}
&&(a_1^2 + b_1^2)(p_1^2 + p_1^3) + (a_2^2 + b_2^2)(p_2^2 + p_2^3)
\nonumber\\
&& + p_1 p_2 (p_1 + p_2 +2)(a_1 a_2 + b_1 b_2) = 0\,.\end{aligned}$$ By using and in , we derive the following condition: $$\begin{aligned}
\label{condfinal}
&&(a_1^2 + b_1^2) \left[p_1^2 (p_1+1) + p_2^2 (p_2+1) - p_1 p_2 (p_1 + p_2 + 2)\right]
\nonumber \\
&&+a_1 \left[2 p_2^2 (p_2+1) - p_1 p_2 (p_1 + p_2 + 2)\right] + p_2^2 (p_2 + 1) = 0\,.
\nonumber\\\end{aligned}$$ For example, let us set $p_1 = 1$ and $p_2 = 2\,$, giving $$\begin{aligned}
\label{cond-example}
\!\!\!\!\!\!\!\!\!\!
a_1^2\!+\!b_1^2\!+\!\frac{7}{2}a_1\!+\!3 = 0\;\Rightarrow\;
b_1^2 = -\!\left(\!a_1^2\!+\!\frac{7}{2} a_1\!+\!3\right)\!\geq\!0
\,.\end{aligned}$$ This condition has infinite solutions in the interval $-2\leq a_1\leq -3/2\,$. Its limits provide real solutions for the superposition coefficients:
- $\alpha_1 = -2\;, \alpha_2 = 1\,$.
- $\alpha_1 = -3/2\;, \alpha_2 = 1/2\,$.
Note that the first real solution is useless, since it gives $B=0$ and cancels out all terms up to $\rho^4$ and $z^2$ in the trapping potential. The other real solution gives $A=1/2$ and $B=-1/2\,$, resulting in the following expression for intensity of the electric field $$\begin{aligned}
\label{decouple-intensity}
I(\rho, z) &\approx&
I_0
\left(\frac{z^2}{z_R^2} + \frac{\rho^4}{w_0^4}\right)
\,,\end{aligned}$$ which provides the desired bottle-beam configuration with decoupled dynamics along the transverse and longitudinal directions. Moreover, we can easily show that this solution is optimal. Under the bottle-beam and decoupling condition, the trapping strength is $$\begin{aligned}
\label{Bfinal}
4\vert B\vert ^2 = 2a_1 + 4\,,\end{aligned}$$ which is a linear function of $a_1$ with positive slope. Therefore, its maximum value is obtained at the upper limit $a_1=-3/2\,$.
|
---
abstract: 'We present the first measurement of the mass of the top quark in a sample of $t\bar{t}\rightarrow\ell\bar{\nu} b\bar{b} q\bar{q}$ events (where $\ell = e$, $\mu$) selected by identifying jets containing a muon candidate from the semileptonic decay of heavy-flavor hadrons (soft muon $b$-tagging). The $p\bar{p}$ collision data used corresponds to an integrated luminosity of 2 fb$^{-1}$ and was collected by the CDF II detector at the Fermilab Tevatron. The measurement is based on a novel technique exploiting the invariant mass of a subset of the decay particles, specifically the lepton from the $W$ boson of the $t\rightarrow Wb$ decay, and the muon from a semileptonic $b$ decay. We fit template histograms, derived from simulation of $t\bar{t}$ events and a modeling of the background, to the mass distribution observed in the data and measure a top quark mass of $180.5\pm12.0({\rm stat.})\pm3.6({\rm syst.})~{\rm GeV}/c^2$, consistent with the current world average.'
title: 'Measurement of the Top Quark Mass Using the Invariant Mass of Lepton Pairs in Soft Muon $b$-tagged Events'
---
A massive top quark plays an important role in the standard model (SM). The mass of the top quark ($m_t$) enters electroweak (EW) precision observables as an input parameter via quantum effects, i.e. loop corrections, and its large numerical value gives rise to sizable corrections that behave as powers of $m_t$ [@veltman]. For example, in the theoretical prediction of the $W$ boson mass ($m_W$) within the SM, when these corrections are combined with the logarithmic dependence on the mass of the postulated Higgs boson ($m_H$), a relationship emerges that provides a constraint on $m_H$ from experimental determinations of $m_W$ and $m_t$ [@pdg]. Indeed, the strong dependence of the SM radiative corrections on $m_t$ made it possible to predict the value of $m_t$ [@predict] prior to its experimental determination [@predictions1; @predictions2]. Thus, a precision value of $m_t$ is crucial for constraining SM parameters, for high-sensitivity searches for effects of new physics and for stringent consistency tests of models beyond the SM (e.g. supersymmetry). Furthermore, independent measurements of $m_t$ in all final states of $t\bar{t}$ decay provide an important consistency check of the top quark sector of the SM, and might reveal new physics with top-like signatures.
Significant progress has been made recently in reducing the uncertainty in measurements of $m_t$ and in devising alternative and independent techniques. The current best single measurement is determined by reconstructing the full decay chain and computing the invariant mass of the decay products in $t\bar{t}\rightarrow \ell \bar{\nu} b \bar{b}q\bar{q}$ events, and yields $m_t=172.1\pm1.6$ GeV/$c^2$ [@bestmt; @bestmt2]. However, this and all the most precise of the current techniques are limited by the common systematic uncertainty in the calorimeter jet energy calibration (jet energy scale, JES). To provide independent measurements, several techniques with minimal dependence on the JES have been proposed. For example, the flight distance of the $b$-hadron from the top decay can be used to infer the mass of the top quark [@lxy], but this method also requires precision track reconstruction to determine the decay length. A proposal has been made [@cms] for exploiting the correlation between $m_t$ and the invariant mass of the system composed of a $J/\psi$ (from the decay of a $b$ hadron) and the lepton from the $W$ decay. The advantage is a stronger correlation of this system-mass with $m_t$ than that of individual decay products of the top quark, and thus a better sensitivity to the top quark mass, but the overall branching ratio for this final state is only $\mathcal{O}(10^{-5})$. We present the first measurement of the mass of the top quark in a sample of $t\bar{t}\rightarrow\ell\bar{\nu} b\bar{b} q\bar{q}$ events (where $\ell = e$, $\mu$) selected by identifying $b$-jets with a candidate muon from semileptonic decay of heavy-flavor hadrons. We have developed a novel technique that exploits the invariant mass of the lepton from the $W$ boson of the $t\rightarrow Wb$ decay, and the muon from a semileptonic $b$ decay. The selection method is complementary to that taking advantage of the long lifetime of $b$-hadrons through the presence of a decay vertex displaced from the primary interaction. Since only $\sim 50\%$ of the sample of $t\bar{t}$ candidates with a semileptonic $b$ decay overlaps the top samples selected by the identification of a displaced vertex, and a still smaller fraction is in common with traditional samples that require all four jets for the mass reconstruction, our technique provides an essentially independent measurement of $m_t$ from these data. Moreover, our observable is largely independent of the JES, because the calorimeter information is used solely for the selection of event candidates, and therefore the result can add a significant amount of information when averaged with those from other measurements. Including sequential decays of charm, the branching fraction for $b\rightarrow\mu\nu X\simeq 20$% [@pdg] is sizable and since this technique does not require precision secondary vertex reconstruction to suppress backgrounds, it could be an attractive option for the early phase of experiments at the Large Hadron Collider (LHC). Finally, the observable has a higher correlation to the top quark mass than the momentum of the lepton from the $W$ decay alone. A partial reduction in sensitivity will arise from $b$-$W$ mis-pairing, when the lepton from the $W$ decay and the muon from the $b$ semileptonic decay do not originate from the same top quark. Top quarks are produced at the Tevatron proton-antiproton collider predominantly in pairs of $t$ and $\bar{t}$, and are identified by the SM decay $t\rightarrow Wb$, providing a final state that includes two $W$ bosons and two bottom quarks. $W$’s are identified through their decay to leptons or quarks. Quarks hadronize and are observed as jets of charged and neutral particles. The CDF II detector is described in detail elsewhere [@CDF]. The components relevant to this analysis include the central outer tracker (COT), the central electromagnetic and hadronic calorimeters, the central muon detectors and the luminosity counters. The data sample, produced in $p\bar{p}$ collisions at $\sqrt{s}=1.96~{\rm TeV}$ during Run$~$II of the Fermilab Tevatron, was collected between March 2002 and May 2007 and corresponds to an integrated luminosity of $2.0\pm0.1~{\rm fb}^{-1}$. We select events where one of the $W$ bosons decays to an isolated electron (muon) carrying large transverse energy ($E_T$) (momentum ($p_T$)) [@cdfsys] with respect to the beam line, plus a neutrino. We refer to these high-$p_T$ electrons or muons as primary leptons (PL). The neutrino escapes the detector causing an imbalance of total transverse energy vector, referred to as missing $E_T$ ($\slashed{E}_T$). The other $W$ boson in the event decays hadronically to a pair of quarks. We take advantage of the semileptonic decay of $B$ hadrons by searching for muons within final-state jets (soft-lepton tagging, or SLT), in order to identify those jets that result from hadronization of the bottom quarks.
The event selection starts with an inclusive lepton trigger requiring an electron (muon) with $E_T>18~{\rm GeV}$ ($p_T>18~{\rm GeV}/c$). Further selection requires that candidate electron (muon) PLs are isolated and have $E_T>20$ GeV ($p_T>20$ GeV/$c$) and $|\eta|<1.1$. We define an isolation parameter, $I$, as the calorimeter transverse energy in a cone of opening $\Delta R\equiv \sqrt{(\Delta\eta)^2+(\Delta\phi)^2}=0.4$ around the lepton (not including the lepton energy itself) divided by the electron $E_T$ or muon $p_T$. We select isolated electrons (muons) by requiring $I<0.1$. The event must have $\slashed{E}_T>30$ GeV, consistent with the presence of a neutrino from the $W$ boson decay. Jets are identified using a fixed-cone algorithm with a cone opening of $\Delta R=0.4$ and are constrained to originate from the $p\bar{p}$ collision vertex. Muons inside jets are identified by matching the tracks of the jet, as measured in the COT, with track segments in the muon detectors. Such a muon with $p_T>3$ GeV/$c$ and within $\Delta R<0.6$ of a jet axis is called an SLT$\mu$ [@sltprd]. The probability of misidentifying a hadron as an SLT$\mu$, denoted as the SLT$\mu$ mistag probability, is measured using a data sample of pions, kaons and protons from $D^*$ and $\Lambda^0$ decays. A Monte Carlo (MC) simulation of $W$+light flavor events is used to model the $\pi$, $K$ and $p$ admixture in light-quark jets. The SLT$\mu$ mistag probability is parametrized as a function of the track $p_T$ and $\eta$, and is seen to describe within $\pm$5% the number of false SLT$\mu$ tags in light flavor jets of QCD multijet and $\gamma+{\rm jet}$ events. To reduce background from dimuon resonances and double-semileptonic $B$ hadron decays, we remove events in which the PL muon and SLT$\mu$ are oppositely charged and have an invariant mass consistent with a $Z$, $\Upsilon$ or, irrespectively of the PL flavor, less than 5 ${\rm GeV}/c^2$. We further reject events as candidate radiative Drell-Yan and $Z$ bosons if the tagged jet has an electromagnetic energy fraction above 0.8 and only one track with $p_T>1.0$ GeV/$c$ within a cone of $\Delta R=0.4$ about the jet axis The jet energies are corrected to account for variations of the detector response in $\eta$ and time, calorimeter gain drifts, non linearity of calorimeter energy response, multiple $p\bar{p}$ interactions in an event and for energy loss in un-instrumented regions [@jetcorr]. Finally, the sample is partitioned according to the number of jets with $E_T>20$ GeV and $|\eta|<2.0$ in the event, and at least one jet is required to contain an SLT$\mu$ (defining the $SLT\mu$-tagged $W$+$n$ jets sample). The subset of $W$ plus at least 3 jets is the $t\bar{t}$ candidate sample, and to reduce background from QCD production of $W$ with multiple jets, we additionally require the total transverse scalar energy in the event ($H_T$ [@ht]) to be greater than 200 GeV. Standard model processes that result in the same signature as the $t\bar{t}$ signal are backgrounds to this measurement. There are three dominant backgrounds: the largest one is mistags of $W$+light flavor events, and a smaller contribution is due to $W$ boson in association with heavy flavor jets ($Wb\bar{b}$, $Wc\bar{c}$, $Wc$). Events without $W$ bosons that pass the event selection are typically QCD multijet events where one jet has been reconstructed as a high-$p_T$ lepton, mismeasured jet energies produce apparent $\slashed{E}_T$ and an additional jet contains an SLT$\mu$. A fraction of these events is from $b\overline{b}$ and $c\overline{c}$, where the candidate PL may result from a semileptonic decay of one of the fragmenting heavy quark and the SLT$\mu$ from a semileptonic decay of the other. Other minor backgrounds that can mimic a $W$ boson and an SLT$\mu$ signature include diboson ($WW$, $ZZ$, $WZ$), Drell-Yan$\rightarrow\tau\tau$, single top quark, and residual Drell-Yan$\rightarrow\mu\mu$ events not removed by the dimuon resonance removal. The composition of the data sample used in this analysis has been studied extensively in [@sltprd], where we have measured the production cross section for $p\bar{p}\rightarrow t\bar{t}X$, and is summarized in Table \[tab:sample\]. The $W$+jets, QCD multijet and Drell-Yan background are determined using the data, while the remaining backgrounds are estimated from MC simulations. The $W$+1,2 jets samples contain little $t\bar{t}$ events and have a composition similar to the background of the $t\bar{t}$ candidate sample. The simulation of $t\bar{t}$ events is performed using [pythia]{} [@pythia] and [herwig]{} [@herwig]. The generators are used with the [CTEQ5L]{} [@cteq5] parton distribution functions (PDF). Modeling of $b$ and $c$ hadron decay is provided by [evtgen]{} [@evtgen]. Modeling of $W+{\rm jets}$ production is performed using [alpgen]{} [@alpgen], coupled with [pythia]{} for the shower evolution and [evtgen]{} for the heavy flavor hadron decays. Diboson production ($WW$, $ZZ$, $WZ$) and Drell-Yan$\rightarrow\tau\tau$ are determined using [pythia]{}. Drell-Yan$\rightarrow\mu\mu+{\rm jets}$ events are modeled using [alpgen]{} while single top production is modeled with [madevent]{} [@madevent], both with [pythia]{} showering. The CDF II detector simulation models the response of the detector to particles produced in $p\bar{p}$ collisions. The detector geometry used in the simulation is the same as that used for reconstruction of the collision data. Details of the CDF II simulation, based on the [geant3]{} package, can be found in [@cdfIIsim].
Source $W$+1 jet $W$+2 jet $W+\geq3$ jets
----------------------------------------------- --------------- --------------- ----------------
$W$+light flavor 622$\pm$31 226$\pm$12 52.3$\pm$2.6
$W$+heavy flavor 145$\pm$55 66.6$\pm$25.2 14.3$\pm$5.4
QCD multijet 91.9$\pm$16.5 44.9$\pm$10.4 6.9$\pm$1.5
$WW+WZ+ZZ$ 3.8$\pm$0.4 7.0$\pm$0.7 1.9$\pm$0.3
Drell-Yan$\rightarrow\tau\tau$ 2.6$\pm$0.6 1.5$\pm$0.4 0.6$\pm$0.3
Drell-Yan$\rightarrow\mu\mu$ 6.0$\pm$1.2 4.1$\pm$0.9 0.8$\pm$0.5
Single top 4.4$\pm$0.4 9.0$\pm$0.7 2.7$\pm$0.2
Total background 876$\pm$54 359$\pm$24 79.5$\pm$5.3
$t\bar{t}$ ($\sigma_{t\bar{t}}=9.1~{\rm pb}$) 3.5$\pm$0.2 31.8$\pm$1.0 168.5$\pm$5.3
Data 892 384 248
\[tab:sample\]
We compute the invariant mass ($M_{\ell \mu}$) between the PL and the SLT$\mu$ in the $t\bar{t}$ candidates sample. In rare cases where there is more than one SLT$\mu$ tag in the same jet, or more than one SLT$\mu$ tagged jet in the same event, we use the SLT$\mu$ candidate that has the best match between the COT track and the track segment in the muon detectors. No attempt is made to choose the correct pairing from the decay chain of the two top-quarks. The electric charge of the SLT$\mu$ for instance is not an effective flavor selector due to abundant sequential $b\rightarrow c\rightarrow\mu$ decays. When the wrong pairing is chosen, there is still sensitivity to the top quark mass due to the boost of the SLT$\mu$ and the PL. The distribution of $M_{\ell \mu}$ is given by the contribution of $t\bar{t}$ and background events. For the background, the $M_{\ell \mu}$ distribution of QCD multijet events is derived from the data themselves in the kinematic-region of $I>0.15$, $\slashed{E}_T>30$ GeV, topologically close to the signal region, while for other background sources we use MC simulation. We check the background model in $W$+1,2 jet SLT$\mu$-tagged data events, a sample with a similar composition as the background to $t\bar{t}$ candidates. We find the predicted and observed distributions of $M_{\ell \mu}$ (Figure \[fig:Control\_IM\_12Ja\]) to be in agreement with a $p$-value of 55%, as given by the Kolmogorov-Smirnov test.
![[The predicted and observed $M_{\ell \mu}$ distributions in the sample of $W$+1,2 jet SLT$\mu$-tagged events. The predicted distributions are stacked.]{}[]{data-label="fig:Control_IM_12Ja"}](figure1.eps){width="6.5cm"}
![[The correlation between the mean value of the $M_{\ell \mu}$ histograms from simulated $t\bar{t}$ and background samples, and the input $m_t$. The continuous line shows a linear fit to the points.]{}[]{data-label="fig:Control_IM_12Jb"}](figure2.eps){width="6.5cm"}
We construct a set of template histograms of the $M_{\ell \mu}$ distribution using the background model and a simulation of $t\bar{t}$ events. The $t\bar{t}$ samples are generated with different top quark mass values in the range 150–195 GeV/$c^2$, incrementing by steps of up to 0.5 GeV/$c^2$, and the full $M_{\ell \mu}$ spectra are determined by adding the signal and expected background histograms in the ratio shown in Table \[tab:sample\]. Figure \[fig:Control\_IM\_12Jb\] shows the mean value of the $M_{\ell \mu}$ distributions versus the input top quark mass, indicating a linear relationship between the two quantities. Also shown is $<M_{\ell \mu}>=35.6\pm1.1({\rm stat.})$ GeV/$c^{2}$, measured in the data. We perform a binned-likelihood fit to the $M_{\ell \mu}$ histogram of the data, in 20 bins between 4–100 GeV/$c^2$, with the binning and range chosen a priori appropriately to the size of the data sample. The likelihood is defined as: $$\begin{aligned}
-{\rm ln}L(m_t) & = & -\sum_{i=1}^{\rm N_{bins}} n_{i}^{\rm data}~{\rm ln}\left[\frac{n_i^{\rm TP}(m_t)}{n_{\rm tot}^{\rm TP}}\right],
% & \equiv & -\sum_{i=1}^{\rm Nbins} n_{i}^{\rm data}~{\rm ln}[ {\cal{P}}_i (m_t)],\end{aligned}$$
where $n_i^{\rm data}$ and $n_i^{\rm TP}(m_t)$ are the number of entries in each $i$-bin of the data and template histograms respectively, the total number of entries is ${n_{\rm tot}^{\rm TP}}={n_{\rm tot}^{\rm data}}$, and ${n_i^{\rm TP}(m_t)}/{n_{\rm tot}^{\rm TP}}\equiv{{\cal{P}}_i(m_t)}$ is the probability of the $i$-th bin, normalized such that $\sum_i { {\cal{P}}_i}=1$. The background normalization is fixed and its value is varied in the evaluation of the systematic uncertainty. A parabolic function is fit to the values of ${\rm ln}L(m_t)$ derived from each mass template, and the measured top quark mass is determined from the minimum of the likelihood function, while the statistical uncertainty is given by the range corresponding to an increase in the $-{\rm ln}L$ of 0.5 units above the minimum. For each mass point within the full mass range, we generate 5000 pseudoexperiments with the same sample size as that of the data and verify that the fitting procedure is unbiased and that the statistical uncertainty returned by the fits represents the 68% confidence level. From 248 $t\bar{t}$ candidate events, we measure: $$m_t=180.5\pm12.0({\rm stat.})\pm3.6({\rm syst.})~{\rm GeV}/c^2.$$
Figure \[fig:LKmeasured\] shows the $M_{\ell \mu}$ distribution of the data, the background, and the templates corresponding to the best fit and the statistical uncertainty.
![[The distribution of invariant mass $M_{\ell\mu}$ of the lepton from the $W$ decay and the SLT$\mu$, from a sample of 248 candidate $t\bar{t}$ events with 79.5 background.]{}[]{data-label="fig:LKmeasured"}](figure3.eps "fig:"){width="6.5cm"}\
The sources of systematic uncertainty that affect the measured value of the top quark mass are summarized in Table \[tab:syst\]. The limited size of the $t\bar{t}$ samples simulated with different values of $m_t$, input to the fitting procedure, yields an uncertainty of $\pm$0.3 GeV/$c^2$. Several components enter the uncertainty on the modeling of the background. The uncertainty on the $W+$ heavy and light flavor normalizations yields an uncertainty of $\pm$0.5 GeV/$c^2$. The uncertainty on the shape of the $W$+jets histogram is evaluated by varying the distribution, to within the statistical accuracy associated with the comparison in the $W$+1,2 jets sample between the data and the background model, and yields an uncertainty of $\pm$1.4 GeV/$c^2$. The normalization of the QCD multijet background contributes $\pm$0.8 GeV/$c^2$. The shape of the QCD multijet distribution accounts for $\pm$0.6 GeV/$c^2$, as determined by replacing the nominal sample with dijet enriched data selected by $I<0.1$ and $\slashed{E}_T<15$ GeV, and by varying the distribution according to its statistical uncertainty. The shift on the measured top quark mass due to the uncertainties on the remaining backgrounds is negligible. The total uncertainty from background modeling is $\pm$1.9 GeV/$c^2$.
Monte Carlo modeling of the signal $M_{\ell \mu}$ distributions includes effects of PDFs, initial-state radiation (ISR), final-state radiation (FSR), and JES. The uncertainty due to the MC modeling of $t\bar{t}$ production and decay, including $b$ fragmentation, is determined by comparing the simulation using [pythia]{} with that using [herwig]{} and gives $\Delta m_t=\pm$2.1 GeV/$c^2$. The PDF uncertainty is evaluated by adding in quadrature the contribution of four effects: variations of the PDFs according to the 20 CTEQ eigenvectors [@pdf], the difference between the standard $t\bar{t}$ simulation using the CTEQ5L PDF and one derived using MRST98 [@mrst] in the default configuration or with two alternative choices for $\alpha_s$, and the variation of the contribution of gluon fusion in $t\bar{t}$ production between 5 and 20%. The overall estimated uncertainty from PDF is $\pm$1.0 GeV/$c^2$. We vary both ISR and FSR simultaneously in the $t\bar{t}$ Monte Carlo simulation, within constraints set by studies of radiation in Drell-Yan events in the data, and assign a systematic uncertainty on $m_t$ of $\pm1.3$ GeV/$c^2$.
The jet reconstruction is used in this analysis only for the selection of event candidates and therefore the uncertainty on the calibration of the jet energies enters the measurement solely through the event selection, via the jet counting and the $\slashed{E}_T$ requirement. The uncertainty due to the JES is measured by shifting the energies of the jets in $t\bar{t}$ MC simulation by $\pm1\sigma$ of the JES [@jetcorr] and results in $\Delta m_t=\pm$0.3 GeV/$c^2$. The uncertainty of $\pm$1% on the difference between data and simulation of the PL energy and momentum scales gives an uncertainty of $\pm$0.9 GeV/$c^2$. The differences in the data versus simulation for the SLT$\mu$ $p_T$ spectrum depends on the $b$-quark fragmentation modeling and the momentum calibration. In addition to the different fragmentation models in [herwig]{} versus [pythia]{}, we consider comparisons of the data with MC simulation of the muon $p_T$ spectra in $B\rightarrow\mu^-D^0X$ [@kraus] and $b\bar{b}\rightarrow\mu\mu X$ [@paolo] which indicate an uncertainty on the muon $p_T$ of $\sim\pm0.8$%, corresponding to $\Delta m_t=\pm0.9$ GeV/$c^2$. The uncertainty on the $p_T$ dependence of the SLT$\mu$ tagging efficiency yields a shift on the top quark mass of $\pm$0.2 GeV/$c^2$. Finally, a source of systematic uncertainty is due to the modeling of pile-up events from multiple $p\bar{p}$ interactions and it is estimated to affect the measured mass by $\leq \pm0.5$ GeV/$c^2$.
[r]{}
Source $\Delta m_{\rm t}$ \[GeV/$c^2$\]
--------------------------------------- ----------------------------------
MC $t\bar{t}$ samples statistics $\pm$0.3
Background $\pm$1.9
$t\bar{t}$ production and decay model $\pm$2.1
Parton distribution functions $\pm$1.0
Initial- and final-state radiation $\pm$1.3
Jet energy scale $\pm$0.3
PL energy/momentum scale $\pm$0.9
SLT$\mu$ momentum $\pm$0.9
Pileup $\pm$0.5
Total $\pm$3.6
: Summary of systematic uncertainties.
\[tab:syst\]
In summary, we have performed the first measurement of the top quark mass in a sample of $t\bar{t}\rightarrow \ell\bar{\nu} b \bar{b}q\bar{q}$ events selected by identifying $b$-jets with a muon candidate from the semileptonic decay of heavy-flavor hadrons. The result, $m_t=180.5\pm12.0({\rm stat.})\pm3.6({\rm syst.})~{\rm GeV}/c^2$, is in agreement with the current world average value of 173.1$\pm$1.3 GeV/$c^2$ [@bestmt], providing a consistency check of the top quark sector with soft muon $b$-tagged events. Our measurement technique exploits the correlation between the parent top quark mass and the invariant mass of the system composed of the lepton from the $W$ decay and the muon from the semileptonic $B$ decay. The uncertainty at present is dominated by the statistical component. The method has a minimal dependence on the jet energy calibration, making it suitable for averaging the result with those from other techniques, and its dominant systematic uncertainties are likely reducible, e.g. by improving the calibration of the leptons’ $p_T$ to better than 1% with $J/\psi$, $\Upsilon$ and $Z$ resonances, by using improved tuning for the MC modeling of $t\bar{t}$ production and decay, and with high statistics data samples for the background model. We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture, Sports, Science and Technology of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the Bundesministerium für Bildung und Forschung, Germany; the Korean Science and Engineering Foundation and the Korean Research Foundation; the Science and Technology Facilities Council and the Royal Society, UK; the Institut National de Physique Nucleaire et Physique des Particules/CNRS; the Russian Foundation for Basic Research; the Ministerio de Ciencia e Innovación, and Programa Consolider-Ingenio 2010, Spain; the Slovak R&D Agency; and the Academy of Finland.
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---
abstract: 'We consider the simultaneous production of two heavy-flavoured hadrons – particularly D mesons – at the LHC. We base our calculations on collinearly factorized QCD at next-to-leading order, using the contemporary parton distribution functions and D-meson fragmentation functions. The contributions of double-parton scatterings are included in the approximation of independent partonic interactions. Our framework benchmarks well with the available proton-proton data from the LHCb collaboration giving us confidence to make predictions for proton-lead collisions. Our results indicate that the double D-meson production in proton-lead collisions should be measurable at the LHCb kinematics with the already collected Run-II data, and should provide evidence for double-parton scattering at perturbative scales with a nuclear target.'
address:
- 'University of Jyvaskyla, Department of Physics, P.O. Box 35, FI-40014 University of Jyvaskyla, Finland'
- 'Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland'
author:
- Ilkka Helenius
- Hannu Paukkunen
bibliography:
- 'doubleD.bib'
title: 'Double D-meson production in proton-proton and proton-lead collisions at the LHC'
---
Open heavy-flavour production ,double-parton scattering
Introduction {#Introduction}
============
The recent measurements of inclusive open heavy-flavour – particularly D and B mesons – in proton-proton (p-p) collisions at the CERN Large Hadron Collider (LHC) [@Aaij:2013mga; @Aaij:2015bpa; @Aaij:2016jht; @Aaij:2017qml; @Acharya:2017jgo; @Acharya:2019mgn; @Aad:2015zix; @Sirunyan:2017xss] provide opportunities to expose different facets of Quantum Chromodynamics (QCD) [@Andronic:2015wma]. On one hand, due to the heavy-quark mass which serves as a hard interaction scale, the perturbative QCD calculations [@Nason:1989zy; @Mangano:1991jk; @Cacciari:1998it; @Frixione:2003ei; @Kniehl:2004fy; @Helenius:2018uul] can be extended e.g. to very small transverse momenta ($p_{\rm T}$) where the calculations with massless quarks become inherently invalid. As the measurements at low $p_{\rm T}$ are statistically very precise, they offer an ideal testbed to benchmark perturbative calculations at low interaction scales. On the other hand, open heavy-flavour production can be used as a tool to probe non-perturbative aspects of heavy-quark fragmentation [@Anderle:2017cgl] and the quark-gluon structure of protons and nuclei [@Eskola:2019bgf; @Zenaiev:2015rfa; @Gauld:2015yia; @Gauld:2016kpd; @Gauld:2015lxa; @Lansberg:2016deg; @Kusina:2017gkz; @Cacciari:2015fta]. The low-$p_{\rm T}$ open heavy-flavour production in proton-lead (p-Pb) collisions [@Adam:2016ich; @Aaij:2017gcy; @Aaij:2019lkm; @Aaij:2018ogq] may also open prospects to disentangle non-linear saturation [@Fujii:2013yja; @Ducloue:2016ywt] and collinearly factorized QCD pictures in a regime where both should be valid descriptions.
The inclusive production of two D mesons provides exciting further opportunities. While the heavy-quarks are predominantly produced in pairs, the experimental overall reconstruction efficiencies for two D-meson final states are low and, roughly, only one out of million primarily produced double-D events can be reconstructed. Nevertheless, simultanous production of two D mesons has been observed in p-p [@Aaij:2012dz] and p-$\overline{\rm p}$ [@Reisert:2007zza] collisions. This offers possibilities to test e.g. the heavy-quark vs. heavy-antiquark asymmetries [@Gauld:2015qha] and, in particular, to study the double-parton scattering (DPS) [@Belyaev:2017sws; @Blok:2016lmd; @Cazaroto:2016nmu; @Maciula:2016wci; @vanHameren:2014ava]. While the formal theory of factorization in DPS has recently advanced significantly [@Diehl:2017wew; @Diehl:2017kgu; @Bartalini:2017jkk], we still know relatively little of the non-perturbative structure of e.g. the double-parton distributions (dPDFs) [@Gaunt:2009re] which would be required in precise phenomenological applications. Thus, simplifying assumptions concerning DPS have to be made which can lead to apparent shortcomings. For instance, measurements are often interpreted in terms of an *effective cross section* $\sigma_{\rm eff}$ whose inverse is proportional to the DPS probability. Its value has been observed to differ significantly depending from which observable it is extracted [@Aaboud:2018tiq]. It is conceivable that this is due to overly simplifying the problem of DPS or, alternatively, overlooking the contributions of single-parton scattering (SPS) [@Karpishkov:2019vyt]. A complementary approach to hard DPS is provided by Monte-Carlo event generators in which the soft or semi-hard multiparton interactions (MPIs) give rise to the underlying event found necessary to describe the multiplicity distributions in hadronic high-energy collisions [@Bartalini:2017jkk; @Bartalini:2011jp].
The generic prediction is that in proton-nucleus (p-$A$) collisions the DPS signal gets enhanced in comparison to p-p case, due to the possibility of proton to interact with two or more nucleons simultaneously [@Strikman:2001gz; @dEnterria:2012jam; @dEnterria:2013mrp; @dEnterria:2017yhd; @Huayra:2019iun]. As in p-p collisions, multiple interactions are necessary to explain the multiplicity distributions in collisions involving heavy nuclei [@Deng:2010mv; @Bierlich:2018xfw], but a clean experimental confirmation for hard DPS processes is still lacking. As we will conclude later in this letter, it seems realistic that the double D-meson production could provide the first direct evidence of DPS in p-Pb collisions at clearly perturbative scales, and that the signal should be visible already in the collected Run-II data. In reaching this conclusion we have first confronted our QCD framework with the LHCb p-p data and a reasonble agreement there encourages us to apply it to p-Pb collisions. Before getting into the actual results we will first describe our theoretical framework in the next two sections.
Double-inclusive production in nuclear collisions {#DoubleInclusive}
=================================================
We will estimate the double-inclusive cross sections in collision of two nuclei, $A$ and $B$, in terms of inclusive per-nucleon SPS cross sections $\sigma^{\rm sps}_{\rm nn \rightarrow \mathcal{O} + X}$ as $$\begin{aligned}
& \frac{\mathrm{d}\sigma_{AB \rightarrow a + b + X}}{\mathrm{d}^3\vec{p}^{\, a} \mathrm{d}^3\vec{p}^{\, b}} =
AB \left[ \frac{\mathrm{d}\sigma^{\rm sps}_{nn \rightarrow a + b + X}}{\mathrm{d}^3\vec{p}^{\, a} \mathrm{d}^3\vec{p}^{\, b}} +
\frac{m}{\sigma_{\rm eff}^{AB}} \frac{\mathrm{d}\sigma^{\rm sps}_{nn \rightarrow a + X}}{\mathrm{d}^3 \vec{p}^{\,a} } \frac{\mathrm{d}\sigma^{\rm sps}_{nn \rightarrow b + X}}{\mathrm{d}^3\vec{p}^{\,b} } \right]
\label{eq:finaldouble} \end{aligned}$$ where $\vec{p}^{\, a}$ and $\vec{p}^{\, b}$ refer to the momenta of the produced particles $a$ and $b$. If $a$ and $b$ are identical particles $m=1/2$, and $m=1$ otherwise. In the case of independent partonic interactions, the effective cross section $\sigma_{\rm eff}^{AB}$ in $A$-$B$ collision is process independent and can be interpreted as a purely geometric object, $$\begin{aligned}
\frac{1}{\sigma_{\rm eff}^{AB}} \equiv \Bigg\{ \frac{ 1}{\sigma_{\rm eff}}
& + \frac{(B-1)}{B^2} \int \mathrm{d}^2\vec{B} \left[ T_{nB}\left( \vec{B}\right) \right]^2 \label{eq:sigmaeffAB} \\
& + \frac{(A-1)}{A^2} \int \mathrm{d}^2\vec{B} \left[ T_{nA}\left( \vec{B}\right) \right]^2 \nonumber \\
& + \frac{(A-1)(B-1)}{(AB)^2} \int \mathrm{d}^2\vec{B} \left[ T_{AB}\left( \vec{B}\right) \right]^2 \Bigg\} \nonumber \,.\end{aligned}$$ Here, $$\frac{1}{\sigma_{\rm eff}} = \int \mathrm{d}^2\vec{b} \left[ t_{\rm nn}\left( \vec{b}\right) \right]^2 \,,$$ where $t_{\rm nn}(\vec{b})$ is the overlap function between two nucleons at fixed impact parameter $\vec{b}$. In geometric sense, we would write $$t_{\rm nn}(\vec {b}) \equiv \int_{-\infty}^{\infty} \mathrm{d}^2{\vec s} \, t_{\rm n}({\vec s} + {\vec b}/2)t_{\rm n}({\vec s} - {\vec b}/2) \,,$$ where $t_{\rm n}({\vec s})$ is the transverse profile of nucleons obtained by integrating the density of nucleons $\rho^{\rm n}$ over the longitudinal spatial component, $$t_{\rm n}({\vec s}) \equiv \int_{-\infty}^{\infty} \mathrm{d}z\, \rho^{\rm n}(\vec{s},z) \,.$$ The overlap functions $T_{\mathrm{n}A}(\vec{B})$ and $T_{AB}(\vec{B})$ at fixed impact parameter $\vec{B}$ are here defined as [@Florkowski:2010zz] $$\begin{aligned}
T_{\mathrm{n}A}(\vec{B}) & \equiv \int_{-\infty}^{\infty} \mathrm{d}^2{\vec s} \, t_{\rm nn}({\vec s} + {\vec B}/2)T_{A}({\vec s} - {\vec B}/2) \\
& \approx T_{A}({\vec B}) \,, \nonumber\end{aligned}$$ where the approximation holds for point-like nucleons, and $$\begin{aligned}
T_{AB}(\vec{B}) & \equiv \int_{-\infty}^{\infty} \mathrm{d}^2{\vec s^{\, A}} \mathrm{d}^2{\vec s^{\, B}} \,
T_{A}(\vec s^{\, A})
T_{B}(\vec s^{\, B})\,
t_{\rm nn}({\vec B} + {\vec s^B} - {\vec s^A}) \nonumber \\
& \approx
\int_{-\infty}^{\infty} \mathrm{d}^2{\vec s} \,
T_{A}(\vec s + {\vec B}/2)
T_{B}(\vec s - {\vec B}/2) \,,\end{aligned}$$ where $T_{A}(\vec{S})$ is the standard nuclear thickness function $$T_{A}(\vec{s}) \equiv \int_{-\infty}^{\infty} \mathrm{d}z\, \rho^{A}(\vec{s},z) \,,$$ and $\rho^{A}$ denotes the density of nuclei. In our notation, the normalization is $$\int \mathrm{d}^2\vec{s} \, T_{A}(\vec{s}\,) = A. \label{eq:norm}$$
Typically, the DPS contribution in Eq. (\[eq:finaldouble\]) is derived [@dEnterria:2012jam; @dEnterria:2013mrp; @dEnterria:2017yhd; @Diehl:2011yj] by writing the DPS cross section in terms of dPDFs, and assuming that the dPDFs factorize into a product of single-parton PDFs and that the partonic cross sections for the two subprocesses are unrelated and spatially independent. Alternatively, Eq. (\[eq:finaldouble\]) can be derived from an eikonal model for multiparton interactions. Indeed, in a Glauber-type approach, the total cross section for a collision of nuclei A and B can be written in the impact-parameter space as $$\sigma_{AB}^{\rm total} = \int \mathrm{d}^2\vec{B} \sum_{k=1}^{AB} \mathcal{P}_k ( \vec B ) \label{eq:totAB} \,,$$ where $\mathcal{P}_k ( \vec B )$ is the probability of exactly $k$ nucleon-nucleon interactions at fixed impact parameter $ \vec B $, $$\begin{aligned}
\mathcal{P}_k ( \vec B ) & = \int \,
\left[ \prod_{i=1}^{A} \mathrm{d}\vec{S}^{A}_i \frac{T_{A}\left(\vec{S}^{A}_i \right)}{A} \right]
\left[ \prod_{i=1}^{B} \mathrm{d}\vec{S}^{B}_i \frac{T_{B} \left(\vec{S}^{B}_i \right)}{B} \right] \label{eq:tot} \\
&\hspace{-2.8em} \times \hspace{-0.3em} \sum_{\alpha_{11}=0}^1 \hspace{-0.4em} \ldots \hspace{-0.4em} \sum_{\alpha_{AB}=0}^1
\left[ p_{11}({\alpha_{11}}) \, p_{12}({\alpha_{12}}) \, \cdots \, p_{AB}({\alpha_{AB}}) \right] \delta_{k, \alpha_{11}+\ldots+\alpha_{AB}}
\nonumber \,.\end{aligned}$$ In this expression, we have defined $$\begin{aligned}
p_{ij}({\alpha_{ij}}) \equiv
\left( t_{ij} \, \sigma_{\rm nn}^{\rm total} \right)^{\alpha_{ij}}
\left(1 - t_{ij} \, \sigma_{\rm nn}^{\rm total} \right)^{1-\alpha_{ij}} \end{aligned}$$ where $t_{ij}$ is an abbreviation for the overlap function between two nucleons $$t_{ij} \equiv t_{\rm nn}\left( \vec B + \vec{S}^{A}_i - \vec{S}^{B}_j \right) \,.$$ The second line in Eq. (\[eq:tot\]) thus corresponds to the probability of getting exactly $k$ nucleon-nucleon interactions (and $AB-k$ missing ones) at fixed geometric configuration. The total cross section in a single nucleon-nucleon collisions $\sigma_{\rm nn}^{\rm total}$ is given by $$\sigma_{\rm nn}^{\rm total} = \int {\mathrm{d}^2}\vec{b} \sum_{k=1}^\infty p_k (\vec{b}) \,,$$ where the probability $p_k (\vec{b})$ for $k$ partonic interactions is considered to be Poissonian, $$p_k (\vec{b}) = \exp \left[ -t_{\rm nn}(\vec{b}) \sigma_{\rm nn} \right] \frac{\left[t_{\rm nn}(\vec{b}) \sigma_{\rm nn}\right]^k}{k!} \,. \label{eq:poisson}$$ The quantity $\sigma_{\rm nn}$ appearing in Eq. (\[eq:poisson\]) is the integrated inclusive cross sections, $$\begin{aligned}
\sigma_{\rm nn} & = \sum_f \int \mathrm{d}{\rm PS}_f \frac{d\sigma^{\rm sps}_{\rm nn \rightarrow f}}{\mathrm{d}{\rm PS}_f} \,, \ \ \mathrm{d}{\rm PS}_f = \prod_{i \in f} \mathrm{d}^3\vec{p}_i \end{aligned}$$ where the summation is over all *exclusive* final states $f$. We will always make a distinction between the (intensive) total cross section like $\sigma_{\rm nn}^{\rm total}$, and (extensive) integrated cross section like $\sigma_{\rm nn}$. The double-inclusive cross section can now be written as $$\begin{aligned}
& \frac{\mathrm{d}\sigma_{AB\rightarrow a + b + X}}{\mathrm{d}\vec p^{\, a} \mathrm{d}\vec p^{\, b}} = \int \mathrm{d}\vec{B} \sum_{k=1}^{AB} \mathcal{P}_k ( \vec B ) \label{eq:masterdoublydiff} \\
&
\prod_{r=1}^k
\int {\mathrm{d}^2}\vec{b}_r \sum_{k_r=1}^\infty \frac{p_{k_r} (\vec{b}_r) }{\sigma_{\rm nn}^{\rm total}}
\prod_{\ell=1}^{k_r}
\left[\frac{1}{\sigma_{\rm nn}} \sum_{f_{r\ell}} \int \mathrm{d}{\rm PS}_{f_{r\ell}} \frac{\mathrm{d}\sigma^{\rm sps}_{nn \rightarrow f_{r\ell}}}{\mathrm{d}{\rm PS}_{f_{r\ell}}} \right]
\nonumber \\
& \left[ \sum_{i=1}^{k} \sum_{j=1}^{k_r} \sum_n \delta^{\,(3)} \left(\vec p^{\, a} - \vec p_{ij}^{\, a_n} \right)
\times \left\{
\begin{array}{c}
1, \ {\rm if} \ a_n \in f_{ij}\\
0, \ {\rm if} \ a_n \notin f_{ij}
\end{array}
\right\}
\right] \nonumber \\
& \left[ \sum_{i=1}^{k} \sum_{j=1}^{k_r} \sum_n \delta^{\,(3)} \left(\vec p^{\, b} - \vec p_{ij}^{\, b_n} \right)
\times \left\{
\begin{array}{c}
1, \ {\rm if} \ b_n \in f_{ij}\\
0, \ {\rm if} \ b_n \notin f_{ij}
\end{array}
\right\}
\right] \,. \nonumber \end{aligned}$$ In the equation above, each term is a product of the form $${\mathcal{P}_k ( \vec B )} \ \times \ \left[ \prod_{r=1}^k {p_{k_r} (\vec{b}_r)} \right] \ \times \ \left[ \prod_{\ell=1}^{k_r} \frac{1}{\sigma_{\rm nn}} \frac{\mathrm{d}\sigma^{\rm sps}_{nn \rightarrow f_{r\ell}}}{\mathrm{d}{\rm PS}_{f_{r\ell}}} \right] \,,$$ corresponding to the total probability density of having $k$ nucleon-nucleon interactions, each with exactly $k_{r=1,\ldots,k}$ partonic interactions resulting with a specific (exclusive) final state $f_{r\ell}$. The last two lines in Eq. (\[eq:masterdoublydiff\]) simply select those final states which contain the desired particles carrying the momenta $\vec{p}^{\, a}$ and $\vec{p}^{\, b}$, and the summation over $n$ accounts for the fact that the final state can contain several a or b particles. With some combinatorics, Eq. (\[eq:masterdoublydiff\]) simplifies to Eq. (\[eq:finaldouble\]) when we identify $$\begin{aligned}
\frac{{\mathrm{d}}\sigma^{\rm sps}_{nn \rightarrow a + X}}{\mathrm{d}^3\vec{p}^{\, a}} & \equiv \sum_f \int \mathrm{d}{\rm PS}_f \frac{\mathrm{d}\sigma^{\rm sps}_{nn \rightarrow f}}{\mathrm{d}{\rm PS}_f} \label{eq:defsingle} \\
& \times \sum_i \delta^{\,(3)} \left(\vec p^{\, a} - \vec p^{\, a_i} \right) \times \left\{
\begin{array}{c}
1, \ {\rm if} \ a_i \in f\\
0, \ {\rm if} \ a_i \notin f
\end{array}
\right\} \,, \nonumber\end{aligned}$$ and $$\begin{aligned}
\frac{{\mathrm{d}}\sigma^{\rm sps}_{nn \rightarrow a + b + X}}{\mathrm{d}^3\vec{p}^{\, a} \mathrm{d}^3\vec{p}^{\, b}} & \equiv \sum_f \int \mathrm{d}{\rm PS}_f \frac{\mathrm{d}\sigma^{\rm sps}_{nn \rightarrow f}}{\mathrm{d}{\rm PS}_f} \\
& \times \sum_i \delta^{\,(3)} \left(\vec p^{\, a} - \vec p^{\, a_i} \right) \times \left\{
\begin{array}{c}
1, \ {\rm if} \ a_i \in f\\
0, \ {\rm if} \ a_i \notin f
\end{array}
\right\} \nonumber \\
& \times \sum_i \delta^{\,(3)} \left(\vec p^{\, b} - \vec p^{\, b_i} \right) \times \left\{
\begin{array}{c}
1, \ {\rm if} \ b_i \in f\\
0, \ {\rm if} \ b_i \notin f
\end{array}
\right\} \,. \nonumber\end{aligned}$$ From the same formalism, also three-particle (in general $n$-particle) inclusive cross sections [@dEnterria:2016ids; @dEnterria:2016yhy; @dEnterria:2017yhd] can be derived.
Perturbative-QCD framework for open heavy flavour {#QCDFramework}
=================================================
In this paper we will be mostly concerned in the D-meson production at $p_{\rm T} > 3\,{\rm GeV}$, which is the kinematic region considered in the LHCb double-D measurement [@Aaij:2012dz]. In this region the inclusive production of D mesons can be reliably described within general-mass variable-flavour-number scheme (GM-VNFS). Schematically, the cross sections are convolutions of PDFs $f_i(x,\mu_{\rm fact}^2)$, partonic cross sections $d\hat\sigma$, and fragmentation functions [(FFs)]{} $D_{k \rightarrow h}(z, \mu_{\rm frag}^2)$, $$\begin{aligned}
\mathrm{d}\sigma^{\rm sps}_{nn \rightarrow a + X} = \sum_{ijk} f_i(\mu_{\rm fact}^2) & \otimes \mathrm{d}\hat\sigma_{ij \rightarrow k + X}(\mu_{\rm fact}^2, \mu_{\rm ren}^2, \mu_{\rm frag}^2) \\
& \otimes f_j(\mu_{\rm fact}^2) \otimes D_{k \rightarrow a}(\mu_{\rm frag}^2) \nonumber \,.\end{aligned}$$ For single-inclusive D-meson production this has been considered at next-to-leading order (NLO) QCD first in Ref. [@Kniehl:2004fy] within the so-called SACOT scheme [@Kramer:2000hn; @Guzzi:2011ew]. In the SACOT scheme, the partonic cross sections for contributions in which the partonic subprocess is initiated by a charm quark or the fragmenting parton is a light one, are independent of the charm-quark mass $m_{\rm charm}$. This leads, in general, to diverging cross sections towards $p_{\rm T} \rightarrow 0$. In an alternative SACOT-$m_{\rm T}$ scheme [@Helenius:2018uul] this unphysical behaviour is resolved by accounting for the underlying kinematic constraint of heavy-quark production. In this work we use the SACOT-$m_{\rm T}$ variant, albeit in the considered $p_{\rm T} > 3 \, {\rm GeV}$ region, both schemes should be equivalent within the scale uncertainties. Our default choice for factorization ($\mu_{\rm fact}$), fragmentation ($\mu_{\rm frag}$) and renormalization ($\mu_{\rm ren}$) scales is $\mu^2_{\rm fact} = \mu^2_{\rm frag} = \mu^2_{\rm ren} = {p_{\rm T}^2 + m_{\rm charm}^2}$, where $p_{\rm T}$ refers to the D-meson transverse momentum.
The SPS contribution in which the two D mesons, $h_1$ and $h_2$, are simultaneously produced is of the form, $$\begin{aligned}
\mathrm{d}\sigma^{\rm sps}_{\mathrm{pp} \rightarrow a + b + X} & = \sum_{ijkl} f_i(\mu_{\rm fact}^2) \otimes \mathrm{d} \hat\sigma_{ij \rightarrow k + l + X}(\mu_{\rm fact}^2, \mu_{\rm ren}^2, \mu_{\rm frag}^2) \nonumber \\
& \otimes f_j(\mu_{\rm fact}^2) \otimes D_{k \rightarrow a}(\mu_{\rm frag}^2) \otimes D_{l\rightarrow b}(\mu_{\rm frag}^2) \,.\end{aligned}$$ For this process, no GM-VFNS calculation is available. Thus, we will resort to the zero-mass approximation available in the NLO <span style="font-variant:small-caps;">diphox</span> [@Binoth:2001vm] (v.1.2) code. Taking into account the large scale uncertainties, this approximation should be sufficiently precise in the considered $p_{\rm T} > 3 \, {\rm GeV}$ region. However, the kinematical cuts applied in the considered LHCb measurement [@Aaij:2012dz] ($p_{\rm T} > 3 \, {\rm GeV}$ and $2 < y < 4$) include also a problematic configuration in which the two D mesons are collinear. In a full SACOT-$m_{\rm T}$ description this contribution would be finite, scaling as $\log(m_{\rm charm}^2)$ where the remaining $\log(m_{\rm charm}^2)$ terms would still need to be resummed via *di-hadron FFs* [@Majumder:2004wh]. In a zero-mass calculation, however, the cross sections diverge in the collinear configuration. Here, as a proxy for the full SACOT-$m_{\rm T}$ treatment, we have regulated our calculations by imposing a physical cut $(\hat p_1 + \hat p_2)^2 > 4m_{\rm charm}^2$ for the fragmenting partons’ four momenta $\hat p_{1,2}$. Our central choice for the QCD scales here is the average $p_{\rm T}$ of the produced two D mesons.
{width="0.495\linewidth"} {width="0.495\linewidth"} {width="0.495\linewidth"} {width="0.495\linewidth"}
The dominant uncertainty in our calculations comes from the unknown higher-order (NNLO and beyond) contributions. As usual, we estimate the potential size of these corrections by varying the QCD scales as $$0.5 \leq \frac{\mu_{\rm fact}}{\mu_{\rm ren}} \leq 2, \ \ \ 0.5 \leq \frac{\mu_{\rm frag}}{\mu_{\rm ren}} \leq 2 \,,$$ around the central scale choices and finding the combinations that give the highest and lowest prediction for each considered observable. As default, we do the scale variations in sync for the two contributions in Eq. (\[eq:finaldouble\]), the single-inclusive and double-inclusive SPS cross sections (17 scale configurations in total). We use NNPDF3.1pch PDFs [@Ball:2017nwa] in which the intrinsic charm component is zero at the mass threshold $\mu_{\rm fact} = m_{\mathrm{c}} = 1.51 \, {\rm GeV}$. The fragmentation functions for D$^0$ and D$^+$ are taken from the KKKS08 analysis [@Kneesch:2007ey] (see Ref. [@Salajegheh:2019nea] for a very recent alternative). The KKKS08 FFs have been fitted to $e^+e^-$ data from different experiments. We have checked that while the fits to BELLE [@Seuster:2005tr] and OPAL [@Alexander:1996wy] data give essentially equally good descriptions of the inclusive LHCb D$^0$ and D$^+$ cross sections at $\sqrt{s}=7\,{\rm TeV}$ [@Aaij:2013mga], the FFs fitted to CLEO data [@Artuso:2004pj] clearly overshoot the LHCb data at high $p_{\rm T}$. This is demonstrated in the upper panels of Figure \[fig:peeteespektrit\]. However, we have found that the D$^0$-to-D$^+$ ratios which are almost exclusively sensitive to the FFs are clearly best described by the OPAL variant, which also gives a better description than the BELLE FFs of the CMS midrapidity data [@Sirunyan:2017xss] at very-high $p_{\rm T}$ [@HPtalk]. Thus, in this paper, we adopt the OPAL FFs from the KKKS08 package. For D$_{\mathrm{s}}^\pm$ and $\Lambda_{\mathrm{c}}^\pm$ FFs we use BKK05 [@Kniehl:2006mw] analysis. While the LHCb and ALICE single-inclusive D$_{\rm s}^\pm$ data [@Aaij:2013mga; @Acharya:2019mgn] are well consistent with these FFs, the $\Lambda_{\mathrm{c}}^\pm$ data [@Aaij:2013mga] are underestimated by the BKK05 $\Lambda_{\mathrm{c}}^\pm$ FFs. Our comparisons with the LHCb data on $D_{\mathrm{s}}^\pm$ and $\Lambda_{\mathrm{c}}^\pm$ are shown in the bottom panels of Figure \[fig:peeteespektrit\].
The KKKS08 and BKK05 FFs do not discriminate between charge-conjugate states, but are given as a sum. (e.g. $D^{{\rm D}^0 + \overline{\rm D^0}}_{i}$). In what follows, however, we will need the D-meson FFs one by one. Taking the D$^0$ states here as an example, we will use the following prescription for the charm-quark containing state, $$\begin{aligned}
D^{\rm D^0}_{\mathrm{c}} & = D^{{\rm D}^0 + \overline{\rm D^0}}_{\mathrm{c}/\overline{\mathrm{c}}} \,, \\
D^{\rm D^0}_{\overline{\mathrm{c}}} & = 0 \,, \\
D^{\rm D^0}_i & = \frac{1}{2} D^{{\rm D}^0 + \overline{\rm D^0}}_{i}, \ {i \neq \mathrm{c}/\overline{\mathrm{c}}} \,,\end{aligned}$$ and an analogous one for the antiquark-containing state, $$\begin{aligned}
D^{\overline{\rm D}^0}_{\mathrm{c}} & = 0 \,, \\
D^{\overline{\rm D}^0}_{\overline{\mathrm{c}}} & = D^{{\rm D}^0 + \overline{\rm D^0}}_{\mathrm{c}/\overline{\mathrm{c}}} \,, \\
D^{\overline{\rm D}^0}_i & = \frac{1}{2} D^{{\rm D}^0 + \overline{\rm D^0}}_{i}, \ {i \neq \mathrm{c}/\overline{\mathrm{c}}} \,.\end{aligned}$$
In addition to the NLO QCD framework described above, we present the predictions from <span style="font-variant:small-caps;">Pythia</span> 8 Monte-Carlo event generator using the standard “Monash 2013 tune” [@Skands:2014pea]. A sample of minimum-bias events were generated, including also MPIs, from which the different D-meson combinations within the LHCb acceptance were picked up to obtain the cross sections for each pair. In line with the LHCb measurements, each pair of D mesons is counted separately. In Figure \[fig:peeteespektrit\] we also show the <span style="font-variant:small-caps;">Pythia</span> predictions for the inclusive D mesons and $\Lambda_{\mathrm{c}}^\pm$, generated with the provided Rivet analysis [@Buckley:2010ar]. In general, the <span style="font-variant:small-caps;">Pythia</span> setup overpredicts the LHCb D-meson measurements, and the disagreement is stronger for D$^\pm$ and D$^\pm_{\mathrm{s}}$ than for D$^0$. A similar behaviour has been recently observed in the case of jets containing a ${\rm D}^0$ meson [@Acharya:2019zup]. The measured $\Lambda_{\mathrm{c}}^\pm$ cross sections are, in turn, underestimated by <span style="font-variant:small-caps;">Pythia</span>. In the Monash tune the parameters related to charm fragmentation were constrained using a limited set of LEP data. Partly the interpretation of these data is hindered by the large feed-down from B-mesons. Furthermore, the data is not sensitive to $\mathrm{g} \rightarrow \mathrm{c\overline{c}}$ branchings that are abundant at the LHC. Thus the observed disagreement could potentially be cured by re-tuning the relevant parameters using a larger sample of charm-production data from LEP, HERA and LHC.
Results {#Results}
=======
We will now compare our results for double D-meson production with the LHCb p-p data [@Aaij:2012dz], and make predictions for p-Pb collisions. As for $\sigma_{\rm eff}$, we will consider the variation $10~\text{mb} < \sigma_{\rm eff} < 25~\text{mb}$ which is roughly the range deduced from jet, W$^\pm$ and photon measurements [@Aaboud:2018tiq]. The uncertainty estimates shown in the plots combine the scale uncertainty and the variation in $\sigma_{\rm eff}$.
![The integrated double-D cross sections for opposite-sign (upper panel) and like-sign (lower panel) cases. The coloured bands denote the combined scale and $\sigma_{\rm eff}$ uncertainty in NLO calculations. The inner darker bands include only the variation in $\sigma_{\rm eff}$. The <span style="font-variant:small-caps;">Pythia</span> predictions are shown as blue dashed lines. The data are from Ref. [@Aaij:2012dz].[]{data-label="fig:ppyields"}](LHCb_pp_pythia.pdf){width="0.95\linewidth"}
![The integrated double-D cross sections for opposite-sign (upper panel) and like-sign (lower panel) cases. The coloured bands denote the combined scale and $\sigma_{\rm eff}$ uncertainty in NLO calculations. The inner darker bands include only the variation in $\sigma_{\rm eff}$. The <span style="font-variant:small-caps;">Pythia</span> predictions are shown as blue dashed lines. The data are from Ref. [@Aaij:2012dz].[]{data-label="fig:ppyields"}](LHCb_pp2_pythia.pdf){width="0.95\linewidth"}
p-p collisions {#ppResults}
--------------
In the case of p-p collision, Eqs. (\[eq:finaldouble\]) and (\[eq:sigmaeffAB\]) reduce to $$\frac{{\mathrm{d}}\sigma_{\mathrm{pp} \rightarrow a + b + X}}{\mathrm{d}\vec{p}^{\, a} \mathrm{d}\vec{p}^{\, b}} = \label{eq:doublepp}
\frac{{\mathrm{d}}\sigma^{\rm sps}_{\mathrm{pp} \rightarrow a + b + X}}{\mathrm{d}\vec{p}^{\, a} \mathrm{d}\vec{p}^{\, b}} \ +
\frac{m}{\sigma_{\rm eff}} \frac{{\mathrm{d}}\sigma^{\rm sps}_{\mathrm{pp} \rightarrow a + X}}{\mathrm{d} \vec{p}^{\,a} } \frac{{\mathrm{d}}\sigma^{\rm sps}_{\mathrm{pp} \rightarrow b + X}}{\mathrm{d} \vec{p}^{\,b} } \,.$$ Our results for the integrated cross sections within the LHCb acceptance are shown in Figure \[fig:ppyields\]. For the opposite-sign D mesons (upper panel), the SPS contribution is clearly larger than the DPS one, and the agreement with the data is very good. The measured systematics among different combinations of D (and $\Lambda_{\rm c}$) species is well reproduced by the used set of FFs. As the cross section accumulates from the lower end of the considered $p_{\mathrm{T}}$ range, the scale uncertainty is sizable and dominates over the variation in $\sigma_{\rm eff}$. For like-sign final states (lower panel) the DPS becomes the dominant production mechanism. Again, the calculation agrees with the data within the scale uncertainties, though our central scale choice seem to somewhat overestimate the cross sections. The disagreement between the data and <span style="font-variant:small-caps;">Pythia</span> results is considerably larger than in the single-inclusive case (Figure \[fig:peeteespektrit\]). Apart from pairs including $\Lambda_{\rm c}$ the predicted cross sections are 4–8 times higher than the data. For pairs including $\Lambda_{\rm c}$ the systematic is again the opposite.
More insight can be obtained from Figure \[fig:ppratios\] where we show cross-section ratios. The upper panel shows ratios between the double like-sign vs. opposite-sign cross sections, $${\sigma^{ab}}/{\sigma^{a\overline{b}}} \equiv \frac{\sigma_{\mathrm{pp} \rightarrow a + b + X}}{\sigma_{\mathrm{pp} \rightarrow a + \overline{b} + X}} \,. \label{eq:kivasuhde2}$$ These measure essentially the ratio between the DPS and SPS contributions. There is clearly a fair data-to-theory agreement within the scale and $\sigma_{\rm eff}$ uncertainties. Our central predictions somewhat overestimate the measured values which is consistent with Figure \[fig:ppyields\]. The scale uncertainties do not cancel out since the partonic channels for like-sign and opposite-sign production are different (e.g. $\rm c \overline{\rm c}$ pair production is significant for $\rm D^0\overline{D^0}$ final state but not for $\rm D^0D^0$). Interestingly, the <span style="font-variant:small-caps;">Pythia</span> results are in excellent agreement with the LHCb data even though the absolute cross sections are way off. Since the numerator in Eq. (\[eq:kivasuhde2\]) is sensitive to DPS (or MPIs in general), we conclude that the good agreement here suggest that the inconsistencies observed in Figure \[fig:ppyields\] are indeed due to poorly-constrained charm fragmentation, rather than the MPI modelling in <span style="font-variant:small-caps;">Pythia</span> [@Sjostrand:1987su; @Sjostrand:2004pf; @Sjostrand:2017cdm].
The bottom panel of Figure \[fig:ppratios\] shows ratios $$\begin{aligned}
{\sigma^a\sigma^b}/{\sigma^{ab}} & \equiv
m \, \frac{\sigma_{\mathrm{pp} \rightarrow a + X} \times \sigma_{\mathrm{pp} \rightarrow b + X}}{\sigma_{\mathrm{pp} \rightarrow a + b + X}}
\,. \label{eq:kivasuhde}\end{aligned}$$ From Eq. (\[eq:doublepp\]) we see that in the absence of SPS, this ratio would be equal to $\sigma_{\rm eff}$, but if there is a contribution from SPS, the ratio will be below $\sigma_{\rm eff}$. In general, our predictions for the opposite-sign case match very well with the data, but tend to underestimate the measured like-sign ratios. This is well in line with our earlier observations and also here a better overall agreement would be obtained if the DPS cross section would be somewhat smaller. Thus, the double-charm production data would prefer a somewhat larger phenomenological $\sigma_{\rm eff}$ than what other measurements indicate [@Aaboud:2018tiq]. The <span style="font-variant:small-caps;">Pythia</span> predictions are here well compatible with our NLO calculations, though they somewhat undershoot the measured ratios both for like- and opposite-sign ratios. This further supports our conclusion that the disagreement observed in Figures \[fig:peeteespektrit\] and \[fig:ppyields\] arise from the fragmentation scheme in <span style="font-variant:small-caps;">Pythia</span>.
![Upper panel: like-sign vs. opposite-sign ratios, see Eq. (\[eq:kivasuhde2\]). Lower panel: Product of two single-inclusive D-meson cross sections divided by the double-D cross sections, see Eq. (\[eq:kivasuhde\]). The coloured bars denote the combined scale and $\sigma_{\rm eff}$ uncertainty, and the inner darker bands include only the variation in $\sigma_{\rm eff}$. The dashed lines correspond to what <span style="font-variant:small-caps;">Pythia</span> predicts. The upper set of bands/lines/data points correspond to like-sign D mesons, and the lower set to opposite-sign combinations. []{data-label="fig:ppratios"}](LHCb_pp5.pdf){width="0.99\linewidth"}
![Upper panel: like-sign vs. opposite-sign ratios, see Eq. (\[eq:kivasuhde2\]). Lower panel: Product of two single-inclusive D-meson cross sections divided by the double-D cross sections, see Eq. (\[eq:kivasuhde\]). The coloured bars denote the combined scale and $\sigma_{\rm eff}$ uncertainty, and the inner darker bands include only the variation in $\sigma_{\rm eff}$. The dashed lines correspond to what <span style="font-variant:small-caps;">Pythia</span> predicts. The upper set of bands/lines/data points correspond to like-sign D mesons, and the lower set to opposite-sign combinations. []{data-label="fig:ppratios"}](LHCb_pp3.pdf){width="0.99\linewidth"}
p-Pb collisions {#pPbResults}
---------------
The reasonble description of the p-p data gives us confidence to apply the framework in p-$A$ collisions. In this case, Eqs. (\[eq:finaldouble\]) and (\[eq:sigmaeffAB\]) reduce to $$\begin{aligned}
& \frac{{\mathrm{d}}\sigma_{\mathrm{p}A \rightarrow a + b + X}}{\mathrm{d}\vec{p}^{\, a} \mathrm{d}\vec{p}^{\, b}} = \label{eq:doublepA} %\\
%&
A \left[ \frac{{\mathrm{d}}\sigma^{\rm sps}_{nn \rightarrow a + b + X}}{\mathrm{d}\vec{p}^{\, a} \mathrm{d}\vec{p}^{\, b}} \ + %\\
%&
\frac{m}{\sigma_{\rm eff}^{\mathrm{p}A}} \frac{{\mathrm{d}}\sigma^{\rm sps}_{nn \rightarrow a + X}}{\mathrm{d} \vec{p}^{\,a} } \frac{{\mathrm{d}}\sigma^{\rm sps}_{nn \rightarrow b + X}}{\mathrm{d} \vec{p}^{\,b} } \right] %\nonumber\end{aligned}$$ with $$\begin{aligned}
\frac{1}{\sigma_{\rm eff}^{\mathrm{p}A}} \equiv \frac{ 1}{\sigma_{\rm eff}} \times \Bigg\{1
& + \sigma_{\rm eff} \frac{A-1}{A^2} \int \mathrm{d}^2\vec{B} \left[ T_{\mathrm{n}A}( \vec{B}\,) \right]^2 \Bigg\} \label{eq:sigmaeffpA}
\,.\end{aligned}$$ The impact-parameter integral for $A=208$ (Pb) gives $$\int \mathrm{d}^2\vec{B} \left[ T_{\rm nPb}\left( \vec{B}\right) \right]^2 \approx 31.66~{\rm mb}^{-1}$$ taking $d=0.54\,{\rm fm}$ and $r=6.49\,{\rm fm}$ in the Woods-Saxon profile, $$\rho^A(\vec s, z) = {n_0}\left[{1 + \exp \left(\frac{\sqrt{\vec s^{\,2}+z^2} - r}{d}\right)}\right]^{-1} \,,$$ and fixing $n_0$ by the normalization condition of Eq. (\[eq:norm\]). With $\sigma_{\rm eff} = 10\ldots 25 ~{\rm mb}$, we find $$\begin{aligned}
\frac{1}{\sigma_{\rm eff}^{\mathrm{p{Pb}}}} \approx \frac{2.5 \ldots 4.8}{\sigma_{\rm eff}} \end{aligned}$$ in full consistency e.g. with Ref. [@dEnterria:2012jam]. That is, the DPS signal is enhanced approximately by a factor of three in comparison to p-p scattering. Our results for the integrated cross sections within the LHCb kinematics are shown in Figure \[fig:pPbyields\]. Here, we have only considered D$^0$ production which has the largest cross sections, see Figure \[fig:ppyields\], and the $y$ acceptance refers to that in the center-of-mass frame of the p-Pb collision. When computing the per-nucleon cross sections $\sigma^{\rm sps}_{nn \rightarrow a + b + X}$ and $\sigma^{\rm sps}_{nn \rightarrow a/b + X}$, we have used the EPPS16 nuclear modifications [@Eskola:2016oht] for Pb. At the LHCb kinematics this leads to a $\sim 20\%$ suppression for p-Pb (forward) SPS cross sections, but since this is squared in DPS contribution, the suppression can reach $\sim 40\%$ in DPS case. For Pb-p configuration (backward) the nuclear-PDF effects are smaller. In comparison to the p-p case in Figure \[fig:ppyields\] the impact of enhanced DPS contribution is clear: Whereas in p-p case the DPS contribution to the opposite-sign yield was rather small in comparsion to SPS, in p-Pb collisions the two are comparable. For the like-sign yields the SPS contribution in p-Pb collisions is entirely overpowered by the DPS part, whereas in the p-p case the SPS still had a 20% contribution or so. Due to the additional contribution from the $T^2_{{\rm n}A}(\vec B)$ integral in Eq. (\[eq:sigmaeffpA\]), the variation in $\sigma_{\rm eff}$ plays only a minor role as indicated in Figure \[fig:pPbyields\]. The $\sim$30% differences between forward and backward cross sections are due to the EPPS16 nuclear effects. Thus, by a suitable measurement where other theoretical uncertainties would cancel out, e.g. a forward-to-backward ratio for double D-meson production, further constraints for nuclear PDFs could, perhaps, be obtained.
An interesting question is whether these cross sections are large enough to be measured with the already collected Run-II data. In Run-II data taking the luminosities collected by the LHCb were $12.2\,{\rm nb}^{-1}$ for p-Pb (forward) and $18.6\,{\rm nb}^{-1}$ Pb-p (backward) collisions [@Aaij:2019lkm]. The overall detection efficiency $\epsilon$ for ${\rm D}^0\overline{{\rm D^0}}$ and ${\rm D}^0{{\rm D}}^0$ final states in the LHCb p-p measurement [@Aaij:2012dz] was approximately $\epsilon \approx 1.2 \times 10^{-6}$. Using these luminosities and efficiencies with our central theoretical predictions we calculate the expected number of events $N$ from which the statistical uncertainty is obtained as $\sqrt{N}/N$. These estimates are also shown in Figure \[fig:pPbyields\]. Within the scale uncertainties we expect approximately $10\dots40$ ${\rm D}^0\overline{{\rm D}}^0$ pairs in p-Pb collisions (forward), and $20\ldots80$ in Pb-p configuration (backward). For the like-sign case the corresponding numbers are $2\dots20$ ${\rm D}^0{\rm D}^0$ pairs in p-Pb collisions (forward), and $4\ldots40$ in Pb-p configuration (backward). Thus, we are led to conclude that the double D-meson production – at least the opposite-sign case – should be observable at the LHCb with the Run-II luminosity. Lowering the minimum-$p_{\rm T}$ cut below $3\,{\rm GeV}$ would easily increase the yields to a definitely measurable level, but towards lower $p_{\rm T}$ our predictions become increasingly uncertain.
![Upper panel: Integrated cross sections for ${\rm D}^0\overline{\rm D^0}$ and ${\rm D}^0{\rm D}^0$ cross sections in p-Pb collisions within the LHCb acceptance at $\sqrt{s}=8.16 \, {\rm TeV}$. The coloured bars denote the combined scale and $\sigma_{\rm eff}$ uncertainty, and the inner darker bands include only the variation in $\sigma_{\rm eff}$. The LHCb projections correspond to $12.2\,{\rm nb}^{-1}$ (forward) and $18.6\,{\rm nb}^{-1}$ (backward) luminosities assuming the predicted central value and overall efficiency of $1.2 \times 10^{-6}$. Lower panel: A sketch of the relative azimuthal-angle dependence in p-p and projected p-Pb collisions.[]{data-label="fig:pPbyields"}](LHCb_pPb_8TeV.pdf){width="0.99\linewidth"}
![Upper panel: Integrated cross sections for ${\rm D}^0\overline{\rm D^0}$ and ${\rm D}^0{\rm D}^0$ cross sections in p-Pb collisions within the LHCb acceptance at $\sqrt{s}=8.16 \, {\rm TeV}$. The coloured bars denote the combined scale and $\sigma_{\rm eff}$ uncertainty, and the inner darker bands include only the variation in $\sigma_{\rm eff}$. The LHCb projections correspond to $12.2\,{\rm nb}^{-1}$ (forward) and $18.6\,{\rm nb}^{-1}$ (backward) luminosities assuming the predicted central value and overall efficiency of $1.2 \times 10^{-6}$. Lower panel: A sketch of the relative azimuthal-angle dependence in p-p and projected p-Pb collisions.[]{data-label="fig:pPbyields"}](LHCb_azimuth.pdf){width="0.99\linewidth"}
As is well known, the increased importance of the DPS contribution in p-Pb collisions may significantly affect the kinematic distributions [@Strikman:2010bg; @Lappi:2012nh]. Particularly interesting observable is the relative azimuthal-angle $\Delta\phi$ distribution of the two D mesons [@Vogt:2018oje]. In p-p collisions [@Aaij:2012dz] the $\Delta\phi$ distribution for ${\rm D}^0\overline{{\rm D^0}}$ peaks at $\Delta\phi=0$ for the logarithmically enhanced $g \rightarrow c\overline{c}$ splitting, and at $\Delta\phi=\pi$ due to the leading-order contributions that are back-to-back in transverse plane. These are commonly referred to as the *near-side peak* and the *away-side peak*, respectively. The disappearance of the away-side peak has long been predicted to be the smoking gun of saturation physics [@Albacete:2010pg; @Albacete:2014fwa]. However, the enhanced DPS contribution in p-Pb collisions will generate a $\Delta\phi$-independent contribution which levels off these peaks. Unfortunately our NLO QCD framework cannot reliably predict the $\Delta\phi$ dependence near $\Delta\phi=\pi$ but a soft-gluon resummation encoded e.g. in parton showers, would be required. To estimate the effect, we have fitted the $\Delta\phi$ dependence of the LHCb ${\rm D}^0\overline{{\rm D}^0}$ data [@Aaij:2012dz] in p-p collisions assuming a negligible contribution from DPS. This assumption is consistent both with our results (see Figure \[fig:ppyields\]) and also with Ref. [@Vogt:2018oje] where it has been shown that for $0 \lesssim \Delta\phi \lesssim \pi/2$ the fixed-order QCD quite correctly predicts the $\Delta\phi$ dependence. Our sketchy estimate for p-Pb is then a linear combination $$\frac{\mathrm{d}\sigma^{\rm pPb}}{\mathrm{d}\Delta\phi} \propto \frac{\mathrm{d}\sigma^{\rm pp}}{\mathrm{d}\Delta\phi}{\big|}_{\rm fitted} + \beta$$ where the constant $\beta$ is determined by the relative importance of integrated SPS and DPS cross sections. Depending on the scale choices, we estimate the DPS contribution to be roughly between 20%…40%. The resulting projection for the $\Delta\phi$ dependence is shown in the lower panel of Figure \[fig:pPbyields\] where the coloured band comes from the scale and $\sigma_{\rm eff}$ uncertainties. We see that both the near- and away-side peaks become less pronounced in p-Pb than what they are in p-p. Thus, we can confirm that the DPS contributions should be considered when interpreting the possible (probable?) weakening of the away-side two-particle correlations in terms of e.g. saturation physics. Since the DPS contribution should be nearly the same for ${\rm D}^0\overline{{\rm D}^0}$ and ${\rm D}^0{\rm D^0}$ final states (in our calculations they are equal), the difference $$\frac{\mathrm{d}\sigma_{\mathrm{pPb} \rightarrow {\rm D}^0\overline{{\rm D}^0}+X}}{\mathrm{d}\Delta\phi} - \frac{\mathrm{d}\sigma_{\mathrm{pPb} \rightarrow {\rm D}^0{\rm D}^0 + X}}{\mathrm{d}\Delta\phi} \approx \frac{\mathrm{d}\sigma^{\mathrm{sps}}_{\mathrm{pPb} \rightarrow {\rm D}^0\overline{{\rm D}^0}+ X}}{\mathrm{d}\Delta\phi}$$ should serve to subtract the “pedestal” DPS yield in a rather model-independent way and, as indicated above, correspond very closely to the SPS contribution in ${\rm D}^0\overline{{\rm D}^0}$ production. Even though the presented calculations are for double D-meson production, we would expect a similar reduction of the near- and away-side peaks also for light-flavour hadrons (such as $\pi^+ \pi^-$) in due to enhanced DPS contribution in p-Pb collisions.
Summary
=======
We have explored the double-inclusive D-meson production at the LHC with focus on the forward LHCb kinematics. The contributions of double-parton scatterings were included in the approximation of independent parton-parton collisions, and the required single-parton cross sections were computed within the collinearly factorized QCD at an NLO level. We confronted our predictions with the LHCb p-p data finding a good, or least an acceptable agreement within the QCD scale uncertainties and reasonable variation in the effective cross $\sigma_{\rm eff}$. As a whole, the LHCb data would prefer a rather large $\sigma_{\rm eff}$ compared to values derived from other final states. We also compared the LHCb p-p data with <span style="font-variant:small-caps;">Pythia</span> predictions. We found that the absolute cross sections for single- and double-inclusive open-charm production are not well reproduced by the widely used Monash tune. However, the cross-section ratios, which are less sensitive to the details in charm fragmentation, are described equally well or even better than what our NLO calculations do. Since the ratios are more sensitive to the multi-parton dynamics than the heavy-quark fragmentation model, it seems that the latter will need some further tuning to establish an agreement with the absolute cross sections.
In addition, we applied our framework to the case of p-Pb collisions in which the contribution from double-parton scattering is predicted to get significantly enhanced due to multiple nucleon-nucleon interactions. Our calculations, accounting for realistic reconstruction efficiencies, indicate that the yields should be high enough to be measured with the already-collected LHC Run-II data at the LHCb. This should provide a clear evidence for the hard double-parton scattering in p-Pb. As the contributions from single- and double-parton scatterings to opposite-sign double-D pair become comparable, also the azimuthal correlations are significantly altered. Therefore it seems necessary to take the double-parton scattering component into account when interpreting e.g. the possible complete or probable partial disappearance of the away-side peak.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Michael Winn for clarifying us details of the LHCb p-p measurements and Peter Skands for discussions related to the Monash-tune applied in <span style="font-variant:small-caps;">Pythia</span> simulations. Our work was financed by the Academy of Finland, project n.o. 308301. The Finnish IT Center for Science (CSC) is acknowledged for computing resources within the project jyy2580 of T. Lappi.
|
---
abstract: 'Using continuation methods from the integrable Ablowitz-Ladik lattice, we have studied the structure of numerically exact mobile discrete breathers in the standard Discrete Nonlinear Schrödinger equation. We show that, away from that integrable limit, the mobile pulse is dressed by a background of resonant plane waves with wavevectors given by a certain selection rule. This background is seen to be essential for supporting mobile localization in the absence of integrability. We show how the variations of the localized pulse energy during its motion are balanced by the interaction with this background, allowing the localization mobility along the lattice.'
address:
- 'Dpt. de Física de la Materia Condensada and Instituto de Biocomputación y Física de los Sistemas Complejos (BIFI), Universidad de Zaragoza, 50009 Zaragoza, Spain.'
- 'Dept. de Teoría y Simulación de Sistemas Complejos, Instituto de Ciencia de Materiales de Aragón (ICMA), C.S.I.C.-Universidad de Zaragoza, 50009 Zaragoza, Spain.'
author:
- 'J. Gómez-Gardeñes'
- 'F. Falo'
- 'L.M. Floría'
title: Mobile Localization in Nonlinear Schrödinger Lattices
---
,
and
Nonlinear dynamics, Localized modes 05.45.-a; 63.20.Pw
Introduction {#s.intro}
============
The phenomenon of intrinsic localization (collapse to self-localized states) due to nonlinearity in discrete systems governed by Schrödinger equations is of fundamental interest in Nonlinear Physics [@Scott; @Cai], and is the subject of current active experimental research in several areas like nonlinear optics [@Christo], Bose-Einstein condensate arrays [@Strecker; @Cataliotti; @Smerzi], polaronic effects in biomolecular processes, and local (stretching) modes in molecules and molecular crystals (see [@Scott; @Eilbeck; @Chaos] and references therein). Discrete Non-Linear Schrödinger equations (NLS lattices for short) provide the theoretical description of these systems, where pulse-like (self-localized) states are observed.
The standard Discrete Nonlinear Schrödinger (DNLS) equation is the (simplest) discretization of the one-dimensional continuous Schrödinger equation with cubic nonlinearity in the interaction term, [*i.e.*]{}, $${\mbox i} \dot{\Phi}_n= -(\Phi_{n+1} + \Phi_{n-1}) - \gamma |\Phi_n|^2 \Phi_n\;,
\label{DNLS}$$ where $\Phi_n(t)$ is a complex function of time. The first term on the right takes account of the dispersion and the second of the nonlinearity, the parameter $\gamma$ is the ratio between them. For the Bose-Einstein condensate lattices dealt with in [@Strecker; @Cataliotti; @Smerzi] one can think of $\Phi_n$ as the boson condensate wavefunction in the $n$-th (optical) potential well, and $\gamma$ would thus be related to the so-called $s$-wave scattering length [@Leggett]. The self-focussing effect of local nonlinearity balanced by the opposite effect of the dispersive coupling makes possible the existence of localized boson states in the Schrödinger representation of the condensate lattice (Gross-Pitaevskii equation). In a localized state (discrete breather) of the boson lattice the profile of $|\Phi_n|^2$ decays exponentially away from the localization center. These solutions have an internal frequency, $\Phi_n=|\Phi_n|\exp({\mbox i}\omega_b t)$, so that the discreteness is essential to avoid resonances with the phonon band and keeping localized the energy. Pinned (immobile) localized solutions of eq. (\[DNLS\]) have been rigorously characterized [@MackayAubry] and extensively studied by highly accurate numerical [@Johansson] and analytical approximations. However, for exact [*mobile*]{} discrete breathers no rigorous formal proof of existence in standard DNLS is available nowadays although lot of works have studied these kind of solutions (see e.g. [@CretegnyAubry; @Duncan; @Kladko; @Musslimani]).
The translational motion of discrete breathers introduces a new time scale (the inverse velocity) into play, so generically a moving breather should excite resonances with the plane wave band expectra. In a hamiltonian system, these radiative losses would tend to delocalize energy and some compensating mechanism is needed in order to sustain exact stationary states of breather translational motion. To address the problem we use unbiased ([*i.e.*]{} not based on ansatze on the expected functional form of the exact solution) and precise numerical methods which allow observations of numerically exact non-integrable mobility, paving the way to further physical (and mathematical) insights.
In this letter, after explaining in section \[s.model\] the basis of the numerical method (fixed point continuation from the integrable Ablowitz-Ladik limit [@AL]) and its relevant technical details briefly, we will discuss the structure of the discrete NLS breathers in \[s.mdb\]. They are found to be the exact superposition of a travelling exponentially localized oscillation (the [*core*]{}), and an extended “background” built up of finite amplitude plane waves $A \exp[{\mbox i}(k n-\omega t)]$. These resonant plane waves fit well simple (thermodynamic limit) predictions based on discrete symmetry requirements. Finally in section \[s.bckg\] we show how the resonant background is seen to be an indispensable part of the solution. In this regard we present the mechanism through which the interaction core-background compensates the variations of the core energy (no longer an invariant of motion away from the integrable limit), during the translational motion.
Salerno Model and Continuation Method {#s.model}
=====================================
The method used here makes use of the following NLS lattice, originally introduced by Salerno [@Salerno], $${\mbox i} \dot{\Phi}_n= -(\Phi_{n+1} + \Phi_{n-1})\left[ 1 + \mu |\Phi_n|^2 \right]
- 2 \nu \Phi_n |\Phi_n|^2\;.
\label{Salerno}$$ This lattice, though non-integrable for $\nu \neq 0$, provides a Hamiltonian interpolation between the standard DNLS equation (\[DNLS\]), for $\mu = 0$ and $\nu = \gamma/2$, and the integrable Ablowitz-Ladik lattice [@AL], A-L for short, when $\mu = \gamma/2$ and $\nu = 0$. The A-L model is a remarkable integrable lattice possessing a family of exact moving breather solutions: $$\begin{aligned}
\Phi_n (t) &=&\sqrt{\frac{2}{\gamma}} \sinh \beta \; \mbox{sech}
[\beta (n-x_0(t))] \exp [{\mbox i}(\alpha (n-x_0(t)) +\Omega(t))],
\label{A-Lbreather}\end{aligned}$$ the two parameters $\omega_b$ and $v_b$ are the breather frequency and velocity $$\omega_b\equiv\dot{\Omega}(t)=2\cosh \beta \; \cos \alpha \;+\; \alpha v_b\;\; , \;\;\;\;\;
v_b\equiv\dot{x}_{0}(t)=\frac{2}{\beta}\sinh \beta \; \sin \alpha
\label{omegav}$$ where $-\pi \leq \alpha \leq \pi$ and $0 < \beta < \infty$. The equation (\[Salerno\]) has the following conserved quantities, namely the Hamiltonian ${H}$ and the norm ${N}$: $$\begin{aligned}
{H}&=& -\sum_{n}(\Phi_{n}\overline{\Phi}_{n+1} +
\overline{\Phi}_{n}\Phi_{n+1}) -2\frac{\nu}{\mu}\sum_{n}
|\Phi_n|^2 +2\frac{\nu}{\mu^2}\sum_{n}\ln(1+\mu|\Phi_n|^2)
\label{Ham}
\\
{N} &=& \frac{1}{\mu}\sum_{n}\ln(1+\mu|\Phi_n|^2)
\label{Norm}\end{aligned}$$ where $\overline{\Phi}_{n}$ denotes the complex conjugate of $\Phi_{n}$. In what follows we will fix the value $\gamma = 2$ in eq. (\[DNLS\]) and $\mu + \nu = 1$ in eq. (\[Salerno\]), as usual.
Perturbative inverse scattering transform [@Vakhnenko], as well as collective coordinate methods [@Claude; @Cai; @MackaySep], have been used to study moving breathers of the Salerno equation (\[Salerno\]) near the integrable A-L limit ($\nu\simeq0$). The numerical procedure that we explain below has the advantage of being unbiased and not restricted to small values of the non-integrability parameter $\nu$, at the expense of restricting attention to those solutions (\[A-Lbreather\]) which are [*resonant*]{}, meaning that the two breather time scales are commensurate $2\pi v_b/\omega_b =p/q$ (rational time scales ratio). A resonant ($p/q$) moving breather $\hat{\Phi}_n(t)$ is numerically represented as a fixed point of the map ${M} = {L}^{p} {T}^{q}$, where ${L}$ is the lattice translation operator ${L}(\{\Phi_n(t)\})=\{\Phi_{n+1}(t)\}$, and ${T}$ is the $T_b$-evolution map ($T_b = 2\pi /\omega_b$), ${T}(\{\Phi_n(t)\})=\{\Phi_{n}(t+T_b)\}$; explicitly $$\hat{\Phi}_n(t) = \hat{\Phi}_{n+p}(t+qT_b) \;\;\;\;\;\mbox{ for all }\;n
\label{fixed point}$$
Let us briefly present the numerical method. The implicit function theorem [@Ledermann] ensures a unique continuation of a fixed point solution of ${M}$ for parameter ($\nu$) variations, provided the Jacobian matrix $J=D({M} - I)$ is invertible: with this proviso the Newton method [@Marin] is an efficient numerical algorithm to find the uniquely continued fixed point. In other words, continuation from a resonant A-L breather along the Salerno model is possible if one restricts the Jacobian matrix $J$ to the subspace orthogonal to its center (null) subspace. The center subspace turns out to be spanned by two continuous symmetries of the Salerno model, namely, [*time translation*]{} and [*gauge*]{} (uniform phase rotation) invariances. Using Singular Value Decomposition (SVD) techniques [@NumRec], one then obtains numerically accurate continued resonant moving breathers along the Salerno model until conditions for continuation cease to hold. A (SVD)-regularized Newton algorithm was already used by Cretegny and Aubry in [@CretegnyAubry] to refine moving breathers of Klein-Gordon lattices with Morse potentials obtained by other means. From the methodological side what is novel here is the systematic use of it in order to obtain the family of moving Schrödinger breathers of the NLS lattice (\[Salerno\]), for different values of $2\pi v_{b}/\omega_{b}=0,\; 1/2,\; 3/4,\; 1,...$ and a fine grid of frequency values $\omega_{b}$ and the nonintegrability parameter $\nu$.
Mobile Discrete Breathers {#s.mdb}
=========================
Let first start with a few remarks on immobile breathers ($p=0$). Some nonintegrable issues, that affect mobile solutions, can be shown continuing the immobile ones along the Salerno Model ($\nu = 0$,[…]{}, $1$). First we remark that the uniquely continued solution of standard DNLS ($\nu = 1$) is equal to the pinned discrete breather uniquely continued from the anticontinuous limit ($\gamma \rightarrow \infty$) [@MackayAubry]. Second, only inmobile breathers which are centered either at a site ($n$) or at a bond ($n\pm1/2$) persist; this is due to the emergence of Peierls-Nabarro barriers away from integrability ($\nu\not=0$), a well-known result of collective variable theory [@Cai; @MackaySep]. The breather centered at a site is stable while the one centered at a bond is unstable [@Cai; @note]; its energy difference is the Peierls-Nabarro barrier. This energy difference acts as a barrier to mobile breathers for travelling along the lattice; the numerical computations of this barrier nicely fit with collective variable predictions.
Our main interest, however, focusses on mobile solutions, [*i.e.*]{} $p \neq 0$. How are Peierls-Nabarro barriers to mobility overcome by the fixed point solution? Our results show clearly that the uniquely continued $p/q$-resonant fixed point for $\nu \neq 0$ is spatially asymptotic to an [*extended background*]{}, whose amplitude increases from zero (at $\nu =0$) with increasing non-integrability $\nu$, superposed to the moving (A-L)-like core, see figure (\[fig:solution\]). In order to reveal the structure of this extended background, we have to pay attention to spatially extended solutions of the Salerno model.
----- -----
(a) (b)
(c) (d)
----- -----
The Salerno equation (\[Salerno\]) admits extended solutions of the plane wave form, $\Phi_n(t) = A \exp[{\mbox i}(k n-\omega t)]$, provided the following (nonlinear) dispersion relation holds: $$\omega = -2 [1+(1-\nu)|A|^{2}]\cos k - 2\nu |A|^{2}
\label{dispersion}$$ A $p/q$-resonant plane wave satisfies $\Phi_n(t) = \Phi_{n+p}(t+qT_b)$, and therefore, for a $p/q$-resonant plane wave the following condition also holds: $$\frac{\omega}{\omega_b} = \frac{1}{q}\left(\frac{p}{2\pi} k -m \right)
\label{resonant}$$ $m$ being any integer. Equations (\[dispersion\]) and (\[resonant\]) can be solved for $k$ and one obtains a finite number of branches $k_j(|A|)$ of $p/q$-resonant wavenumbers in the first Brillouin zone, $-\pi \leq k_j \leq \pi$. The simplest case of a unique branch for fixed $\nu$ (as well as $\omega_b$ and $p/q$) and $A$ small, is represented in a) and b) of figure (\[fig:FFT\]). For example, for $A$ small and $\omega_b > 4$, for any value of $0<\nu<1$ and $A$, there is a unique $1/1$-resonant wavenumber branch $k_0(\nu, A)$.
For the general situation where several branches $k_j$ ($j=0, {\ldots},\;s-1 $) of resonant plane waves solve (\[dispersion\]) and (\[resonant\]), the power spectrum of a background site $n$, $S(\omega)={\mid\int_{-\infty}^{\infty}{\Re(\Phi_{n}(t))\exp\lbrack{\mbox
i}\omega t\rbrack dt}\mid^2}$, reveals $s$ peaks at the values $\omega_j$ corresponding to the resonant wavevector branches. The background is, up to numerical accuracy, a linear superposition of $p/q$-resonant plane waves, namely $$\sum_{j=0}^{s-1} A_j \exp[{\mbox i}(k_j n-\omega_j t)]$$ The amplitudes $A_j$ differ typically orders of magnitude, [*i.e.*]{} $|A_0| \gg |A_1| \gg |A_2| {\ldots} $, so that only a few frequencies are dominant, for most practical purposes. One would speak of localization in $k$-space to describe the extended background of the $p/q$-resonant fixed point.
----- -----
(a) (b)
(c) (d)
----- -----
Once the values of $\omega_b$, $v_b$ and $\nu$ are given, the “selection rule” provided by equations (\[dispersion\]) and (\[resonant\]), does not determine directly the resonant wavenumbers $k_j$, but only branches $k_j(A)$. This reflects the inherent nonlinearity of the NLS lattice, wherefrom the frequency of the plane wave depends on both wavenumber and amplitude in equation (\[dispersion\]). Along the parametric continuation path the fixed point “adjusts” the planewave content ($k_j$) of the background, so that it remains $p/q$-resonant under the changes in the amplitudes of the background plane waves ($A_j$).
Background relevance to Mobility {#s.bckg}
================================
Along the Newton continuation path to the standard DNLS equation the background amplitudes have a monotone increasing behavior with $\nu$, see figure (\[fig:bckg\_Ham\].a). High frequency solutions cannot be continued up to that limit, and the continuation stops for values of $\nu\;<\;1$. This result correlates well with the collective variable (particle perspective) predictions [@Claude; @Kundu] where the non-persistence of travelling solutions is related to the growth of the Peierls-Nabarro barrier. In this respect, one observes a sudden increase in the background amplitude near the continuation border. This result reinforces the interpretation of the background as an energy support to the core for surpassing the (nonintegrable) Peierls-Nabarro barriers to mobility, and so its unavoidable presence for the existence of mobile breathers in the absence of integrability.
--------- --
[(a)]{}
[(b)]{}
--------- --
The role of the background in the localized core mobility can be analysed as follows. As the solution is unambiguosly found to be $\hat{\Phi}=\hat{\Phi}^{core}+\hat{\Phi}^{bckg}$, the energy ${H}$, equation (\[Ham\]), of a mobile breather can be written as $${H}={H}[\hat{\Phi}^{core}]+{H}[\hat{\Phi}^{bckg}]+{H}^{int}
\label{ham_dec}$$ where ${H}^{int}$ is the interaction energy, [*i.e.*]{} the crossed terms of $\hat{\Phi}^{core}$ and $\hat{\Phi}^{bckg}$ in the Hamiltonian. In the simplest case in which the background has a single resonant plane wave, its energy is a constant of motion (along with the total energy), so one obtains $$\frac{\partial {H}[\hat{\Phi}^{core}]}{\partial t}=
-\frac{\partial {H}^{int}}{\partial t}
\label{balance}$$ [*i.e.*]{} the variations of the core energy along the motion are balanced by the variations in the core-background interaction energy. Equation (\[balance\]) dictates the dynamics of any (eventual) effective (collective) variables intended to describe the mobile core in a particle-like description of the breather.
One can compute the core energy variations directly from the numerical integration of a solution by substracting (at each time step) the background from it. In figure (\[fig:bckg\_Ham\].b) we plot the evolution of the core energy as a function of the core localization center, $x_{0}$, defined by using the norm ${N}$ (equation (\[Norm\])) of the Salerno model (\[Salerno\]) as $$x_{0}=\frac{\sum_{n}{n\ln(1+\mu|\Phi_{n}^{core}|^{2})}}{\mu{N}}.
\label{loc_center}$$ One observes that the core has extracted the maximum energy from the interaction term when the core passes over $x_{0}=n\pm 1/2$ (maxima of the PN barrier) and has given it back to the interaction term at $x_{0}=n$ (minima of the PN barrier). The bigger the PN barrier, the larger the interaction term (directly proportional to background amplitude) is. This result illustrate the role of the resonant background on the core mobility and the interpetation of its amplitude increase with $\nu$. [*The increase of nonintegrability, and the subsequent growth of the PN barrier, demands aditional support of energy from the interaction term, which is achieved by an increase of the background amplitude*]{}.
Conclusions {#s.conc}
===========
We have used a (SVD)-regularized Newton algorithm to continue mobile discrete breathers in the Salerno model from the integrable Ablowitz-Ladik limit. Our results indicate that, away from integrability, a description of these solutions based exclusively on localized (collective) variables is incomplete. The solutions are composed by a localized core and a linear superposition of plane waves, the background, whose amplitudes differ orders of magnitude. The background plays an important role in the translational motion of the localized core. Exact mobile localization only exist over finely tuned extended states of the nonlinear lattice. Mobile “pure” ([*i.e.*]{} zero background) localization must be regarded as very exceptional (integrability).
Acknowledgments
===============
Financial support came from MCyT (BFM2002-00113, I3P-BPD2002-1) and LOCNET HPRN-CT-1999-00163. The authors acknowledge A.R. Bishop for arising our interest in DNLS, P. Kevrekidis, J.L. García-Palacios, S. Flach, R.S. Mackay, M. Peyrard and G.P. Tsironis for discussions on our numerics and sharing his intuitions.
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---
abstract: 'We compute the value distributions of the eigenfunctions and spectral determinant of the Schrödinger operator on families of star graphs. The values of the spectral determinant are shown to have a Cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related. This is in contrast to the spectral determinants of random matrices, for which the logarithm is known to satisfy a Gaussian limit distribution. The value distribution of the eigenfunctions also differs from the corresponding random matrix result. We argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of Šeba-type billiards.'
author:
- 'J. P. Keating, J. Marklof and B. Winn'
date: 'Received: 29 October 2002'
title: Value distribution of the eigenfunctions and spectral determinants of quantum star graphs
---
Introduction
============
The study of quantum graphs as model systems for quantum chaos was initiated by Kottos and Smilansky [@kot:1], [@kot:pot], who observed that the spectral statistics of fully-connected graphs are typical of those associated with generic classically chaotic systems. The relative simplicity of quantum graphs, together with the existence of an exact trace formula, has lead to the suggestion that their study might provide insights into some of the fundamental problems of quantum chaos [@kot:2]. This has motivated many works considering a variety of aspects of quantum graphs, [@bar:lsd], [@berk:3], [@bolte:1], [@desbois:1], [@kurasov:1], [@prot:1], [@pascaud:1], [@schanz:1], [@schanz:2], [@tanner:1].
Studies [@berk:2], [@berk:1] of a special class of graphs - the so-called “hydra” graphs or star graphs - have revealed spectral statistics that are not typically associated with quantum chaotic systems. These have been dubbed “intermediate statistics” in recent works, [@bogomolny:1], [@bogomolny:3], [@bogomolny:2] and have been observed in a number of systems. We are motivated to investigate this further by studying the value distributions of the eigenfunctions and the spectral determinant of the Schrödinger operator on quantum star graphs.
A [*star graph*]{} consists of a single central vertex together with $v$ outlying vertices each of which is connected only to the central vertex by a bond (figure \[fig:0\]). Hence there are $v$ bonds. We associate to each bond a length $L_j, j=1,\ldots,v$. We will often refer to the vector of bond lengths ${{\mathbf L}}:=(L_1,\ldots,L_v)$.
![A star graph with 5 bonds[]{data-label="fig:0"}](star.eps){width="3.0cm" height="2.5cm"}
The Schrödinger operator on a star graph takes the form of the Laplacian $-{{\mathrm d}}^2/{{\mathrm d}}x^2$ acting on the space of functions defined on the bonds of the graph that are twice-differentiable and satisfy the following matching conditions at the vertices: $$\begin{aligned}
\psi_j(0)=\psi_i(0)&=:&\Psi,\qquad j,i=1,\ldots,v\\
\sum_{j=1}^v \psi_j^{\prime}(0)&=&\frac{1}{\lambda}\Psi\\
\psi_j^{\prime}(L_j)&=&0,\qquad j=1,\ldots,v.\end{aligned}$$ Here $\psi_j$ is the component of the function defined on the $j^{\rm th}$ bond of the graph, and $\psi_j:[0,L_j]\to{\mathbb R}$ with the convention that $\psi_j(0)$ is the value of the function at the central vertex of the star graph. $\lambda$ is a parameter that allows us to vary the boundary conditions at the central vertex.
The Schrödinger operator so-defined is self-adjoint, so there exists a discrete unbounded set of values $0\leq k_0 < k_1 \leq k_2 \leq \cdots \to
\infty$ such that $k_n^2$ is an eigenvalue. It can be shown that $k=\pm k_n$ corresponds to an eigenvalue if and only if it is a solution of $Z(k,{{\mathbf L}})=0$, where $$Z(k,{{\mathbf L}}):=\sum_{j=1}^v \tan kL_j-\frac{1}{k\lambda}.
\label{spec:det}$$ We refer to $Z(k,{{\mathbf L}})$ as the [*spectral determinant*]{}. Note that $Z(k,{{\mathbf L}})$ has poles at $k=(2n+1)\pi/2L_j$ for each $n\in{\mathbb Z}$ and $j=1,\ldots,v$. The zeros and poles of $Z(k,{{\mathbf L}})$ interlace. The usual definition of the spectral determinant would require these poles to be factored out.
For simplicity we henceforth consider the case $1/\lambda=0$. We shall employ the notation $$Z'(k,{{\mathbf L}})=\frac{\partial Z}{\partial k}(k,{{\mathbf L}})=\sum_{j=1}^v
L_j\sec^2 kL_j.$$
The eigenfunction corresponding to the $n^{\rm th}$ eigenvalue is found to be $$\psi_i^{(n)}(x)=A^{(n)}\frac{\cos k_n(x-L_i)}{\cos k_nL_i}.$$ The constant $A^{(n)}$ is determined by the normalisation $$\sum_{j=1}^v \int_0^{L_j} |\psi_j^{(n)}(x)|^2 {{\mathrm d}}x=1,
\label{norm:const}$$ to be $$A^{(n)}=\left(\frac{2}{\sum_{j=1}^v L_j\sec^2 k_nL_j}\right)^{\frac{1}{2}}.$$ The value distribution of the eigenfunctions is determined by these normalisation constants. For definiteness, we shall focus here on the maximum amplitude squared of the eigenfunctions on a single bond, $$\begin{aligned}
A_i(n,{{\mathbf L}};v)&:=&\sup_{x\in[0,L_i]}\{|\psi_{i}^{(n)}(x)|^2\}\\
&=&(A^{(n)}\sec k_nL_i)^2\\
&=&\frac{2\sec^2 k_nL_i}{\sum_{j=1}^v L_j\sec^2 k_nL_j}.\end{aligned}$$
We now state our main results.
\[thm:1\] For any fixed ${\bar{L}}> 0$ $$\lim_{k\to\infty} \frac{1}{(\Delta L)^v}
{\mathop{\rm meas}}\left\{ {{\mathbf L}}\in[{\bar{L}},{\bar{L}}+\Delta L]^v:\frac{1}{v}Z(k,{{\mathbf L}})<y
\right\}=\frac{1}{\pi}
\int_{-\infty}^y \frac{1}{1+x^2} {{\mathrm d}}x,$$ provided that $k\Delta L\to\infty$ as $k\to\infty$.
We emphasise that in theorem \[thm:1\] we do not require that $\Delta L\to 0$. In some later results we shall make this stipulation.
\[thm:2\] Suppose that the components of ${{\mathbf L}}$ are fixed and linearly independent over ${\mathbb Q}$. Then $$\lim_{K\to\infty} \frac{1}{K}{\mathop{\rm meas}}\left\{k\in[0,K]:\frac{1}{v}Z(k,{{\mathbf L}})<y
\right\}=\frac{1}{\pi}\int_{-\infty}^y\frac{1}{1+x^2}{{\mathrm d}}x.$$
Theorems \[thm:1\] and \[thm:2\] demonstrate the equivalence of taking a $k$-average and a bond-length average at large $k$ for the distribution of values taken by the function $Z(k,{{\mathbf L}})$. Such a correspondence was noted in [@bar:lsd] for the spacing distribution of the eigenvalues of quantum graphs.
In [@snaith:1] it was shown that the value distributions of the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix drawn from the circular ensembles of random matrix theory tend independently to a Gaussian distribution in the limit as the matrix size tends to infinity, subject to appropriate normalisation. The distribution that appears in theorems \[thm:1\] and \[thm:2\] is known as the Cauchy distribution. It is related to the Gaussian distribution by the fact that both are examples of a larger class of distributions known as stable distributions. Such distributions share the property that the sum of two random variables from a stable distribution is distributed like a random variable from the same distribution. Theorem \[thm:1\] is a consequence of this fact. We also note that the density in theorems \[thm:1\] and \[thm:2\] is independent of $v$ when $Z(k,{{\mathbf L}})$ is normalised as indicated.
We next consider the distribution of values taken by $Z'(k)$ when $k=k_n$, $n=1,2,\ldots$.
\[thm:3\] Let the components of ${{\mathbf L}}$ be linearly independent over ${\mathbb Q}$. Then there exists a probability density $P_v(y)$, depending on ${{\mathbf L}}$, such that $$\lim_{N\to\infty}\frac{1}{N}{\#}\!\left\{n\in\{1,\ldots,N\} : \frac{1}{v^2}
Z'(k_n,{{\mathbf L}})<R\right\}=\int_{-\infty}^{R} P_v(y){{\mathrm d}}y,$$ with $P_v(y)=0$ for $y<0$.
\[thm:4\] For each $v$ let the bond lengths $L_j$, $j=1,\ldots,v$ lie in the range $[{\bar{L}},{\bar{L}}+\Delta L]$ and be linearly independent over ${\mathbb Q}$. If $v\Delta L\to0$ as $v\to\infty$ then for any $R\in{\mathbb R}$, $$\int_{-\infty}^R P_v(y){{\mathrm d}}y\to\int_{-\infty}^R P(y){{\mathrm d}}y$$ as $v\to\infty$. The limiting density is given by the continuous function $$P(y)=\left\{
\begin{array}{lr}
\displaystyle \frac{\sqrt{{\bar{L}}}}{4\pi y^{3/2}}\int_{-\infty}^{\infty}
\exp\left(-\frac{\xi^2}{4}-\frac{{\bar{L}}m(\xi)^2}{4y}\right)m(\xi)
{{\mathrm d}}\xi,
&\mbox{$y>0$}\\
0, & \mbox{$y\leq0$},
\end{array}
\right.$$ where $$m(\xi):=\frac{2}{\sqrt{\pi}}{{\mathrm e}}^{-\xi^2/4}+\xi{\mathop{\rm erf}}(\xi/2).$$
By comparison, the value distribution for the logarithm of the derivative of the characteristic polynomial of a matrix drawn from the CUE of random matrix theory, evaluated at an eigenvalue in the limit as matrix size tends to infinity, is Gaussian [@hug:rmt].
The following results refer to the value distribution of $A_i(n,{{\mathbf L}};v)$.
\[thm:5\] Assume the conditions of theorem \[thm:3\] are satisfied. Then there exists a probability density $Q_v(\eta)$ such that $$\lim_{N\to\infty}\frac{1}{N}{\#}\!\left\{ n\in\{1,\ldots,N\}: v^2A_i
(n,{{\mathbf L}};v)<R\right\}=\int_0^{R} Q_v(\eta){{\mathrm d}}\eta$$ where the density $Q_v(\eta)$ is independent of the choice of bond $i$ but depends on ${{\mathbf L}}$.
\[thm:6\] Assume the conditions of theorem \[thm:4\] are satisfied. Then for each $R>0$ $$\int_{0}^R Q_v(\eta){{\mathrm d}}\eta\to\int_0^R Q(\eta){{\mathrm d}}\eta.$$ as $v\to\infty$. The limiting density is given by the function $$\label{eq:Q:def}
Q(\eta)=\frac{1}{2\pi^{3/2}\eta}{{\mathfrak{Im}}}\int_{-\infty}^{\infty}\exp\left(-\frac{\xi^2}{4}-\frac{{\bar{L}}\eta m(\xi)^2}
{8}\right){\mathop{\rm erfc}}\left( \frac{\sqrt{{\bar{L}}\eta}m(\xi)}{2{{\mathrm i}}\sqrt{2}}\right){{\mathrm d}}\xi$$ which is continuous on $(0,\infty)$. Here $m(\xi)$ is as in theorem \[thm:4\]. $Q(\eta)$ has asymptotic expansion $$\label{eq:Q:asympt}
Q(\eta)=\frac{\sqrt{2}}{\sqrt{{\bar{L}}}\pi^2\eta^{3/2}}\int_{-\infty}^{\infty}
\frac{{{\mathrm e}}^{-\xi^2/4}}{m(\xi)}{{\mathrm d}}\xi
+{{\mathrm O}}(\eta^{-5/2})\qquad$$ as $\eta\to\infty$.
The proofs of theorems \[thm:3\]–\[thm:6\] rely on an equidistribution result of Barra and Gaspard [@bar:lsd]. We review this work in section \[sec:3\].
The limit $v\to\infty$ is analogous to the semiclassical limit $\hbar\to 0$ [@kot:pot]. We note that theorem \[thm:6\] describes the wave functions on a vanishingly small fraction of the graph. It thus goes beyond the information provided by the Schnirelman theorem. It instead corresponds to the Gaussian value distribution for the wave functions of classically chaotic systems implied by the random wave model [@ber:rwm].
The value distribution of the eigenvector components of asymptotically large random matrices are particular cases of the $\chi^2_{\beta}$ density $$P_{\chi_{\beta}^2}(\eta)=\left(\frac{\beta}{2}\right)^{\beta/2}
\eta^{\beta/2-1}\Gamma^{-1}\left(\frac{\beta}{2}\right){{\mathrm e}}^{-\beta\eta/2},$$ where the parameter $\beta$ takes the values $1$, $2$ and $4$ in, respectively, the orthogonal, unitary and symplectic ensembles (see for example [@haa:rmt]). When $\beta=1$ the density is called the Porter-Thomas density. It is characterised by ${{\mathrm O}}(\eta^{-1/2})$ behaviour as $\eta\to0$ and ${{\mathrm O}}(\eta^{-1/2}{{\mathrm e}}^{-\eta/2})$ as $\eta\to\infty$. The limiting distribution we find in theorem \[thm:6\] completely determines the value distribution of the star graph eigenfunctions (see the appendix) and has a significantly different shape (c.f. equation [(\[eq:Q:asympt\])]{} and figure \[fig:9:5\] below). Other quantum systems for which the value distribution of the eigenfunctions has a non-random-matrix limit are the Cat Maps [@par:cat].
In [@berk:2] a correspondence was noted between the two-point spectral correlation functions for star graphs and a class of systems known as Šeba billiards. The original Šeba billiard [@seba:1] was a rectangular quantum billiard perturbed by a point singularity. More generally, we describe any integrable system perturbed in such a way as belonging to the same class [@seba:2]. We conjecture that the results derived in the present work will also apply to systems in the Šeba class.
The remainder of this paper is structured as follows. In section \[sec:2\] we prove theorems \[thm:1\] and \[thm:2\]. In section \[sec:3\] we treat the finite $v$ cases, theorem \[thm:3\] and \[thm:5\]. In sections \[sec:4\] and \[sec:5\] we prove, respectively, theorems \[thm:4\] and \[thm:6\], developing the necessary machinery in section \[sec:4\]. Section \[sec:6\] is devoted to numerical computations that illustrate our results. We develop more fully the connections between the present work and Šeba billiards in section \[sec:7\].
The value distribution of $Z({{\mathbf L}},k)$ {#sec:2}
==============================================
\[lem:1\] Let ${\bar{L}}>0$ and $\zeta$ be real constants, then $$\lim_{k\to\infty}\frac{1}{\Delta L}{\int_{{\bar{L}}}^{{\bar{L}}+\Delta L}}\exp({{\mathrm i}}\zeta\tan kL){{\mathrm d}}L
={{\mathrm e}}^{-|\zeta|}$$ uniformly for $k\Delta L\to\infty$ as $k\to\infty$.
[*Proof.*]{}
By the periodicity of the integrand we may shift the range of integration by multiples of $\pi/k$ so that without loss of generality we may take ${\bar{L}}$ in the range $0\leq{\bar{L}}\leq \pi/k$. We write $\Delta L=\pi(n+p)/k$ where $n\in{\mathbb Z}$ and $0\leq p < 1$. Then by the periodicity of the integrand, $$\begin{aligned}
\int_{{\bar{L}}}^{{\bar{L}}+\pi n/k+\pi p/k}\exp&({{\mathrm i}}\zeta\tan kL){{\mathrm d}}L \\
&= \int_{{\bar{L}}}^{{\bar{L}}+p\pi/k}\exp({{\mathrm i}}\zeta\tan kL){{\mathrm d}}L
+n\int_0^{\pi/k}\exp({{\mathrm i}}\zeta\tan{kL}){{\mathrm d}}L
\\
&= \int_{{\bar{L}}}^{{\bar{L}}+p\pi/k}\exp({{\mathrm i}}\zeta\tan kL){{\mathrm d}}L
+\frac{n}{k}\int_{-\infty}^{\infty}
\frac{{{\mathrm e}}^{{{\mathrm i}}\zeta z}}{1+z^2}{{\mathrm d}}z\end{aligned}$$ where the substitution $z=\tan kL$ has been made. We note now that $n/k\Delta L\to\pi^{-1}$ as $k\to\infty$ and $$\left|\frac{1}{\Delta L}\int_{{\bar{L}}}^{{\bar{L}}+p\pi/k}\exp({{\mathrm i}}\zeta\tan{kL})
{{\mathrm d}}L\right| \leq \frac{\pi}{k\Delta L} \to 0\quad \mbox{as $k\to\infty$.}$$ A simple application of Cauchy’s residue theorem allows us to evaluate the integral $$\int_{-\infty}^{\infty}\frac{{{\mathrm e}}^{{{\mathrm i}}\zeta z}}{1+z^2}{{\mathrm d}}z=
\pi{{\mathrm e}}^{-|\zeta|}.
\label{eqn:10}$$
$\square$
[*Proof of theorem \[thm:1\].*]{}
We use here the characteristic function. With bond lengths chosen from a uniform distribution, $$\begin{aligned}
{\mathbb E}_{\rm L}(\exp({{\mathrm i}}\zeta Z(k,{{\mathbf L}})))&=&\frac{1}{(\Delta L)^v}
{\int_{{\bar{L}}}^{{\bar{L}}+\Delta L}}\mspace{-20mu}
\cdots\!{\int_{{\bar{L}}}^{{\bar{L}}+\Delta L}}\mspace{-15mu}\exp\left({{\mathrm i}}\zeta\textstyle\sum_{j=1}^v
\tan kL_j\right){{\mathrm d}}L_1\cdots{{\mathrm d}}L_v \nonumber \\
&=&\left(\frac{1}{\Delta L}{\int_{{\bar{L}}}^{{\bar{L}}+\Delta L}}\exp({{\mathrm i}}\zeta\tan kL){{\mathrm d}}L
\right)^v.\end{aligned}$$ The subscript ${\rm L}$ indicates that the expectation is with respect to an average over bond lengths. Since the map $t\mapsto t^v$ is continuous, lemma \[lem:1\] together with [(\[eqn:10\])]{} allows us to deduce that $$\lim_{k\to\infty}{\mathbb E}_{\rm L}(\exp({{\mathrm i}}\zeta Z(k,{{\mathbf L}})))=
{{\mathrm e}}^{-v|\zeta|}.
\label{eqn:11}$$ The limiting density corresponding to the characteristic function on the right hand side is given by $$P_Z(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\exp(-{{\mathrm i}}\zeta x-v|\zeta|)
{{\mathrm d}}\zeta =\frac{1}{\pi}\frac{v}{v^2+x^2}.$$ The theorem follows now from the classical continuity theorem for characteristic functions ([@fel:ipt] chapter XV).
$\square$
The proof of theorem \[thm:2\] uses Weyl’s Equidistribution theorem. This celebrated result [@wey:ube] has numerous applications in analysis and number theory. We state here the form most convenient for application to our current work. Let ${\mathbb T}^v$ be the $v$-dimensional torus, with sides of length $\pi$.
\[thm:weyl\] Let $f\in C({\mathbb T}^v)$, and let the components of ${{\mathbf L}}$ be linearly independent over ${\mathbb Q}$. Then $$\lim_{K\to\infty}\frac{1}{K}\int_0^K f(L_1k,\ldots,L_vk){{\mathrm d}}k=
\frac{1}{\pi^v}\int_{{\mathbb T}^v} f({{\mathbf x}}){{\mathrm d}}{{\mathbf x}}$$ where $\displaystyle {{\mathrm d}}{{\mathbf x}}={{\mathrm d}}x_1\cdots{{\mathrm d}}x_v$ denotes Lebesgue measure.
We shall use Weyl’s theorem as our main tool to relate $k$-averages to bond length averages. It is for this reason that it is crucial that the bond lengths are incommensurate.
We remark that theorem \[thm:weyl\] can also apply to more general functions such as piecewise continuous functions through an argument similar to the one in the following lemma.
\[lem:2\] Theorem \[thm:weyl\] can also be applied to the function $$f({{\mathbf x}}):=\exp\left(\textstyle {{\mathrm i}}\zeta\sum_{j=1}^v\tan x_j\right).$$
[*Proof.*]{}
We treat the real and imaginary parts of $f$ separately. The functions $$\begin{aligned}
f_1({{\mathbf x}})&:=\cos\left(\textstyle \zeta\sum_{j=1}^{v}\tan x_j\right)\\
f_2({{\mathbf x}})&:=\sin\left(\textstyle \zeta\sum_{j=1}^{v}\tan x_j\right)\end{aligned}$$ are smooth everywhere apart from at an essential singularity when $x_i=\pi/2$ for some $i$, which we tame in the following way. Let $\epsilon>0$ . We can construct functions $\phi$ and $\psi$ satisfying the conditions of theorem \[thm:weyl\] such that $$\begin{aligned}
\psi({{\mathbf x}})=-1 & &\mbox{if $|x_i-\pi/2|<\pi\epsilon^{1/v}/8$ for some
$i=1,\ldots,v$,} \nonumber\\
\phi({{\mathbf x}})=1 & &\mbox{if $|x_i-\pi/2|<\pi\epsilon^{1/v}/8$ for some
$i=1,\ldots,v$,} \nonumber\\
\psi({{\mathbf x}})=\phi({{\mathbf x}})=f_1({{\mathbf x}})& &\parbox{2.5in}{if
$\pi\epsilon^{1/v}/4<|x_i-\pi/2|<\pi/2$ for some
$i=1,\ldots,v$,} \nonumber\\
-1\leq\psi({{\mathbf x}})\leq f_1({{\mathbf x}})\leq\phi({{\mathbf x}})\leq 1
& &\mbox{for all ${{\mathbf x}}\in{\mathbb T}^v$.}
\label{eqn:13}\end{aligned}$$ This implies $$\frac{1}{\pi^v}\int_{{\mathbb T}^v} \left(\phi({{\mathbf x}})-\psi({{\mathbf x}})
\right){{\mathrm d}}{{\mathbf x}}
\leq \frac{1}{\pi^v}\left(2\frac{\pi}{2}\epsilon^{1/v}\right)^v =\epsilon.$$ From [(\[eqn:13\])]{} and theorem \[thm:weyl\], $$\begin{aligned}
\frac{1}{\pi^v}\int_{{\mathbb T}^v} \psi({{\mathbf x}}){{\mathrm d}}{{\mathbf x}} \leq
\liminf_{K\to\infty}&\frac{1}{K}\int_0^K f_1(k{{\mathbf L}}){{\mathrm d}}k \\
&\leq\limsup_{K\to\infty}\frac{1}{K}\int_0^K f_1(k{{\mathbf L}}){{\mathrm d}}k \leq
\frac{1}{\pi^v}\int_{{\mathbb T}^v} \phi({{\mathbf x}}){{\mathrm d}}{{\mathbf x}}.\end{aligned}$$ The ends of this inequality differ by $\epsilon$ which can be made arbitrarily small, so we see that $\lim_{K\to\infty} K^{-1}\int_0^K
f_1(k{{\mathbf L}}){{\mathrm d}}k$ exists and is equal to $$\frac{1}{\pi^v}\int_{{\mathbb T}^v}
f_1({{\mathbf x}}){{\mathrm d}}{{\mathbf x}}.$$ The extension to $f_2$ and hence $f$ is obvious.
$\square$
[*Proof of theorem \[thm:2\].*]{}
We begin in the same way as in the proof of theorem \[thm:1\]. In this case $k$ is chosen uniformly from the interval $[0,K]$ with $K>0$, so that the characteristic function with respect to this uniform distribution is $${\mathbb E}_K(\exp({{\mathrm i}}\zeta Z(k,{{\mathbf L}})))=\frac{1}{K}\int_0^K \exp\left({{\mathrm i}}\zeta \textstyle \sum_{j=1}^v \tan kL_j\right) {{\mathrm d}}k.$$ By lemma \[lem:2\] we can write this integral as an average over the torus as $K\to\infty$: $$\begin{aligned}
\lim_{K\to\infty}{\mathbb E}_K(\exp({{\mathrm i}}\zeta Z(k,{{\mathbf L}})))&=&
\frac{1}{\pi^v}
\int_{{\mathbb T}^v}\exp\left( \textstyle{{\mathrm i}}\zeta\sum_{j=1}^v \tan x_j\right)
{{\mathrm d}}{{\mathbf x}}\\
&=&\left( \frac{1}{\pi}\int_0^{\pi} \exp({{\mathrm i}}\zeta\tan x){{\mathrm d}}x \right)^v.\end{aligned}$$ Following the substitution $z=\tan x$ in this final integral, we have $$\lim_{K\to\infty}{\mathbb E}_K(\exp({{\mathrm i}}\zeta Z(k,{{\mathbf L}}))={{\mathrm e}}^{-v|\zeta|}$$ and the theorem follows from the same arguments used in the end of the proof of theorem \[thm:1\].
$\square$
An equidistribution theorem {#sec:3}
===========================
Barra and Gaspard [@bar:lsd] observed that the condition for $k$ to be an eigenvalue of a graph can be written in the form $$G(k{{\mathbf L}})=0,$$ where $G$ is a function that is periodic in each variable. For star graphs $G$ is defined on ${\mathbb T}^v$ by $$G({{\mathbf x}})=\tan x_1+\cdots+\tan x_v.$$ The equation $$G({{\mathbf x}})=0$$ defines a surface $\Sigma$ embedded in ${\mathbb T}^v$. A flow $\phi^k, k\in{\mathbb R}$ can be defined on ${\mathbb T}^v$ by $$\phi^{k}({{\mathbf x}}_0)={{\mathbf x}}_0+k{{\mathbf L}} \quad\mbox{(mod $\pi$)}.$$
Since $k=0$ is an eigenvalue for star graphs with Neumann boundary conditions considered here, we take ${{\mathbf x}}_0=0$ in this case.
At each value $k=k_n$ we have an intersection of this flow with the surface $\Sigma$. We note that the angle between the normal to the surface $\Sigma$ and the flow $\phi^k$ is given by $$\cos\theta=\frac{|{{\mathbf L}}\cdot\nabla G|}{\|{{\mathbf L}}\|\|\nabla G\|}.$$ For star graphs, $$\nabla G({{\mathbf x}})=(\sec^2 x_1,\ldots,\sec^2 x_v).$$ Hence there exists a constant $c_1>0$ such that $\cos\theta> c_1$. This means that the angle between the flow and the surface $\Sigma$ is uniformly bounded away from 0. We can therefore parameterise $\Sigma$ locally by $v-1$ real variables ${{\boldsymbol \xi}}=(\xi_1,\ldots,\xi_{v-1})$ so that for ${{\mathbf x}}\in\Sigma$, $$x_i=s_i({{\boldsymbol \xi}})$$ and $G(s_1({{\boldsymbol \xi}}),\ldots,s_v({{\boldsymbol \xi}}))=0$.
The central result of Barra and Gaspard is the existence of an invariant measure on the surface $\Sigma$.
\[thm:bar:gas\] Let $f$ be a piecewise continuous function $\Sigma\to{\mathbb R}$. Then $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(k_n{{\mathbf L}})=\int_{\Sigma}
f({{\boldsymbol \xi}}){{\mathrm d}}\nu({{\boldsymbol \xi}})
\label{eq:bar:gas}$$ where the measure $\nu$ is given by $${{\mathrm d}}\nu({{\boldsymbol \xi}})=\frac{J({{\boldsymbol \xi}}){{\mathrm d}}{{\boldsymbol \xi}}}
{\int_{\Sigma}J({{\boldsymbol \xi}}){{\mathrm d}}{{\boldsymbol \xi}}}$$ and ${{\mathrm d}}{{\boldsymbol \xi}}={{\mathrm d}}\xi_1\cdots{{\mathrm d}}\xi_{v-1}$ is Lebesgue measure. $J$ is the Jacobian determinant $$J({{\boldsymbol \xi}})=\left|
\begin{array}{ccc}
L_1 & \cdots & L_v \\
\frac{\partial s_1}{\partial \xi_1} & \cdots &
\frac{\partial s_v}{\partial \xi_1}\\
\vdots & \ddots & \vdots \\
\frac{\partial s_1}{\partial \xi_{v-1}} & \cdots &
\frac{\partial s_v}{\partial \xi_{v-1}}
\end{array} \right|$$
For completeness, we sketch a proof of theorem \[thm:bar:gas\] for star graphs with $v$ bonds.
[*Proof.*]{}
Let $\tilde{f}:{\mathbb T}^v\to{\mathbb R}$ be an extension of $f$ to ${\mathbb T}^v$, so that $\tilde{f}\big|_{\Sigma}=f$, i.e.$$\tilde{f}({{\mathbf x}})=f({{\mathbf x}})\qquad\mbox{for all ${{\mathbf x}}\in\Sigma$.}$$ We let $\tilde{f}$ be constructed in such a way that for all ${{\boldsymbol \xi}}\in\Sigma$, $\tilde{f}(\phi^{k}({{\boldsymbol \xi}}))$ is a differentiable function of $k$ with compact support in some neighbourhood of $k=0$.
Let $\epsilon>0$. We construct the set $\Sigma_{\epsilon,{{\mathbf L}}}$ which is a thickening of $\Sigma$ in the direction of the flow $\phi^k$, $$\Sigma_{\epsilon,{{\mathbf L}}}:=\{{{\mathbf x}}\in{\mathbb T}^v :\exists{{\boldsymbol \xi}}\in
\Sigma, k\in[-\epsilon,\epsilon]:{{\mathbf x}}=\phi^k({{\boldsymbol \xi}})\}\subseteq
{\mathbb T}^v.$$ We define ${{1\!\!1}}_A$, the indicator function of a set $A$, by $${{1\!\!1}}_A(x):=\left\{
\begin{array}{lr}
1, & x\in A\\
0, & x\not\in A
\end{array}\right.$$ The indicator function ${{1\!\!1}}_{\Sigma_{\epsilon,{{\mathbf L}}}}({{\mathbf x}})$ is piecewise constant.
By the differentiability properties of $\tilde{f}$ we can write for every ${{\mathbf x}}\in\Sigma$ $$f({{\mathbf x}})=\tilde{f}({{\mathbf x}})=
\frac{1}{2\epsilon}\int_{-\epsilon}^{\epsilon}
\tilde{f}(k{{\mathbf L}}+{{\mathbf x}}){{\mathrm d}}k+{{\mathrm O}}(\epsilon)$$ as $\epsilon\to 0$. The implied constant does not depend on ${{\mathbf x}}$. Setting ${{\mathbf x}}=k_n{{\mathbf L}}$ gives $$\frac{1}{N}\sum_{n=1}^N f(k_n{{\mathbf L}})=\frac{1}{2\epsilon N}\sum_{n=1}^N
\int_{-\epsilon}^{\epsilon}\tilde{f}((k+k_n){{\mathbf L}}){{\mathrm d}}k+{{\mathrm O}}(\epsilon).$$ Let the mean density of zeros of $Z(k,{{\mathbf L}})$ be $\bar{d}$: $$\bar{d}:=\lim_{K\to\infty}\frac{\#\{n:k_n\leq K\}}{K}.$$ Then $$\begin{aligned}
\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(k_n{{\mathbf L}})&=&\frac{1}
{2\epsilon\bar{d}}\lim_{K\to\infty}\frac{1}{K}\int_0^{K} \tilde{f}(k{{\mathbf L}})
{{1\!\!1}}_{\Sigma_{\epsilon,{{\mathbf L}}}}(k{{\mathbf L}}){{\mathrm d}}k+{{\mathrm O}}(\epsilon) \\
&=&\frac{1}{\pi^v \bar{d}}\int_{{\mathbb T}^v} \tilde{f}({{\mathbf x}})
{{1\!\!1}}_{\Sigma_{\epsilon,{{\mathbf L}}}}({{\mathbf x}}){{\mathrm d}}{{\mathbf x}} +{{\mathrm O}}(\epsilon),\end{aligned}$$ applying theorem \[thm:weyl\] to the piecewise continuous function $\tilde{f}({{\mathbf x}})
{{1\!\!1}}_{\Sigma_{\epsilon,{{\mathbf L}}}}({{\mathbf x}})$. Changing to the system of coordinates $(t,{{\boldsymbol \xi}})$ on ${\mathbb T}^v$ via the change of variables $$x_i=L_i t+s_i({{\boldsymbol \xi}}),$$ gives $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(k_n{{\mathbf L}})=\frac{1}
{\pi^v\bar{d}}\int_{\Sigma}\frac{1}{2\epsilon}\int_{-\epsilon}^{\epsilon}
\tilde{f}(t,{{\boldsymbol \xi}})J({{\boldsymbol \xi}}){{\mathrm d}}t{{\mathrm d}}{{\boldsymbol \xi}}+{{\mathrm O}}(\epsilon).$$ Since this is true for all $\epsilon>0$, we deduce that $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(k_n{{\mathbf L}})=\frac{1}
{\pi^v\bar{d}}\int_{\Sigma}{f}({{\boldsymbol \xi}})J({{\boldsymbol \xi}}){{\mathrm d}}{{\boldsymbol \xi}}.$$ By setting $f=1$, we see that $$\bar{d}=\frac{1}{\pi^v}\int_{\Sigma}J({{\boldsymbol \xi}}){{\mathrm d}}{{\boldsymbol \xi}},$$ to complete the proof.
$\square$
We note incidentally that $\bar{d}$ can be evaluated using spectral methods [@kot:pot] to give $$\bar{d}=\frac{1}{\pi}\sum_{j=1}^v L_j=\frac{v{\bar{L}}}{\pi}+{{\mathrm O}}(v\Delta L)
\qquad\mbox{as $v\Delta L\to 0$.}$$
We observe that the right hand side of equation [(\[eq:bar:gas\])]{} can formally be written in the form $$\int_{\Sigma}
f({{\boldsymbol \xi}}){{\mathrm d}}\nu({{\boldsymbol \xi}})
=\frac{1}{2\pi^{v+1}
\bar{d}}\int_{-\infty}^{\infty}
\int_0^{\pi}\!\cdots\!\int_0^{\pi} f({{\mathbf x}})[{{\mathbf L}}\cdot\nabla
G({{\mathbf x}})]{{\mathrm e}}^{{{\mathrm i}}\zeta G}{{\mathrm d}}x_1\cdots{{\mathrm d}}x_v{{\mathrm d}}\zeta,
\label{eq:states}$$ where now $f$ is a function ${\mathbb T}^v\to{\mathbb R}$ of an appropriate class. This follows from writing $\int_{\Sigma}
f({{\boldsymbol \xi}}){{\mathrm d}}\nu({{\boldsymbol \xi}})$ in the equivalent form, $$\frac{1}{\pi^v\bar{d}}
\int_{\Sigma}\int f(t,{{\boldsymbol \xi}})[{{\mathbf L}}\cdot\nabla G] \delta(G(t,{{\boldsymbol \xi}}))
J({{\boldsymbol \xi}}){{\mathrm d}}t{{\mathrm d}}{{\boldsymbol \xi}}.
\label{eq:del:seq}$$ We then write the $\delta$-function as the limit of the sequence $$\delta_{m}(x):=\frac{1}{2\pi}\int_{-m}^{m}{{\mathrm e}}^{{{\mathrm i}}\zeta x}{{\mathrm d}}\zeta=
\frac{\sin mx}{\pi x}
\label{eq:delta}$$ as $m\to\infty$, and changing back to the usual Cartesian coordinates on ${\mathbb T}^v$. We need to show that $\delta_m$ is an appropriate $\delta$-sequence for the function $f$ that we consider. We will check this point directly in the calculations where the identity [(\[eq:states\])]{} is used. We shall also justify taking the $\zeta$-integral in [(\[eq:delta\])]{} outside the integral over ${\mathbb T}^v$.
For a star graph with $v$ bonds and the parameterisation $$\begin{aligned}
s_i=\xi_i\qquad\mbox{$i=1,\ldots,v-1$}\\
s_v=-\tan^{-1}(\tan \xi_1+\cdots+\tan\xi_{v-1})\end{aligned}$$ $J({{\boldsymbol \xi}})$ takes the following form $$J({{\boldsymbol \xi}})=\frac{L_1\sec^2\xi_1+\cdots+L_{v-1}\sec^2{\xi_{v-1}}}
{1+(\tan \xi_1+\cdots+\tan \xi_{v-1})^2} +L_v
\label{eqn:j}$$
[*Proof.*]{}
Differentiating gives $$\frac{\partial s_i}{\partial \xi_j}=\delta_{ij}$$ for $i<v$ and $$\frac{\partial s_v}{\partial \xi_j}=\frac{-\sec^2{\xi_j}}{1+(\tan\xi_1+
\cdots+\tan\xi_{v-1})^2}.$$ For ease of notation we write $D:={1+(\tan\xi_1+
\cdots+\tan\xi_{v-1})^2}$. Thus we have $$J({{\boldsymbol \xi}})=\left|
\begin{array}{ccccc}
L_1 & L_2 & \cdots & L_{v-1} & L_v \\
1 & 0 & \cdots & 0 & \frac{-\sec^2\xi_1}{D} \\
0 & 1 & \cdots & 0 & \frac{-\sec^2\xi_2}{D} \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & 1 & \frac{-\sec^2\xi_{v-1}}{D}
\end{array}\right|.$$ To complete the proof we employ the identity $$\left| \begin{array}{ccccc}
\alpha_1 & \alpha_2 & \cdots & \alpha_{n-1} & \alpha_n \\
1 & 0 & \cdots & 0 & \beta_1 \\
0 & 1 & \cdots & 0 & \beta_2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & 1 & \beta_{n-1}
\end{array}\right|
=(-1)^n\left( -\alpha_n+\sum_{k=1}^{n-1}\alpha_k\beta_k\right),$$ which may be readily checked by induction.
$\square$
We note in passing that the explicit form of $J({{\boldsymbol \xi}})$ given above together with theorem \[thm:bar:gas\] provides a convenient representation for use numerical studies of eigenvalues and eigenfunctions because the zeros of $Z(k,{{\mathbf L}})$ do not need to be computed explicitly.
[*Proof of theorem \[thm:3\].*]{}
We take as the function $f$ in theorem \[thm:bar:gas\] $$f({{\mathbf x}})={{1\!\!1}}_{(-\infty,R]}\left(\frac{1}{v^2}\sum_{j=1}^vL_j\sec^2{x_j}\right)$$ Then we define $P_v(y)$ by $$\begin{aligned}
\int_{-\infty}^R P_v(y){{\mathrm d}}y=\frac{1}{\pi^v\bar{d}}\int_0^{\pi}
\!\cdots\!\int_0^{\pi}f(\xi_1,\ldots,\xi_{v-1},-\tan^{-1}(\tan \xi_1+
\cdots+\tan \xi_{v-1}))\\ \times J({{\boldsymbol \xi}}){{\mathrm d}}\xi_1\cdots{{\mathrm d}}\xi_{v-1},\end{aligned}$$ where $J({{\boldsymbol \xi}})$ is defined by [(\[eqn:j\])]{}. Since $\sec^2x>0$ for all $x\in{\mathbb R}$, it follows that $P_v(y)=0$ for $y<0$.
$\square$
[*Proof of theorem \[thm:5\].*]{}
In this case, we take as the function $f$ in theorem \[thm:bar:gas\], $$f({{\mathbf x}})={{1\!\!1}}_{[0,R]}\left(\frac{2v^2\sec^2x_i}{\sum_j L_j\sec^2{x_j}}\right).$$ We take as the parameterisation of $\Sigma$ $$\begin{aligned}
s_j&=&\xi_j,\quad\mbox{$1\leq j<i$}\\
s_i&=&-\tan^{-1}(\tan\xi_1+\cdots+\tan\xi_{v-1}) \\
s_j&=&\xi_{j-1},\quad\mbox{$i<j\leq v$}.\end{aligned}$$ This does not change the form of $J({{\boldsymbol \xi}})$ from that in [(\[eqn:j\])]{}, but introduces extra symmetry in $f$. Then define, as before, $$\int_0^R P_v(\eta){{\mathrm d}}\eta=\frac{1}{\pi^v\bar{d}}
\int_0^{\pi}\!\cdots\!\int_0^{\pi}f(s_1({{\boldsymbol \xi}}),\ldots,s_v({{\boldsymbol \xi}}))
J({{\boldsymbol \xi}}){{\mathrm d}}\xi_1\cdots{{\mathrm d}}\xi_{v-1}.$$ Since $f({{\mathbf s}}({{\boldsymbol \xi}}))$ is symmetric in $\xi_1,\ldots,\xi_{v-1}$ we see that $P_v(\eta)$ is independent of the choice of bond $i$.
$\square$
Value distribution of $Z'(k_n)$ in the limit $v\to\infty$ {#sec:4}
=========================================================
We take as the function $f$ in [(\[eq:states\])]{} the characteristic function for the value distribution of the derivative of $Z(k,{{\mathbf L}})$, $$Z'(k,{{\mathbf L}})=\sum_{j=1}^v L_j\sec^2 kL_j.
\label{eq:5:1}$$ Since we are stipulating that $v\Delta L\to0$ as $v\to\infty$, we replace $L_j$ by ${\bar{L}}$ where it does not multiply $k$ in [(\[eq:5:1\])]{} and take as our $f$ $$f({{\mathbf x}})=\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}\sum_{j} \sec^2x_j\right).$$ Let the quantity in which we are interested be denoted $E_v(\beta)$. Then $$\begin{aligned}
E_v(\beta)&:=&\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}
\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}\sum_j \sec^2 k_nL_j\right) \nonumber\\
&=&\frac{{\bar{L}}}{2\pi\bar{d}v}\int_{-\infty}^{\infty}\frac{1}{\pi^v}
\int_0^{\pi}\!\cdots\!\int_0^{\pi}
\left(\sum_j \sec^2{x_j}\right) \nonumber \\
& &\times\prod_{j=1}^v\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}\sec^2 x_j+
\frac{{{\mathrm i}}\zeta}{v}\sum_j\tan{x_j}\right)
{{\mathrm d}}{{\mathbf x}}{{\mathrm d}}\zeta. \label{jon:23}\end{aligned}$$ We have made the re-scaling $\zeta\mapsto\zeta/v$. This is a natural normalisation since $Z(k,{{\mathbf L}})$ is a sum of $v$ terms.
We exploit the symmetry in the integral in [(\[jon:23\])]{} to write $$E_v(\beta)=\frac{1}{2v}\int_{-\infty}^{\infty} I_1(\beta,\zeta)
(I_2(\beta,\zeta))^{v-1}{{\mathrm d}}\zeta,$$ where we have replaced $\bar{d}$ by ${\bar{L}}v/\pi$ and defined the integrals $$I_1(\beta,\zeta):=\frac{1}{\pi}\int_0^{\pi}\sec^2{x}\exp\left(-
\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}
\sec^2{x}+\frac{{{\mathrm i}}\zeta}{v}\tan x\right){{\mathrm d}}x$$ and $$I_2(\beta,\zeta):=\frac{1}{\pi}\int_0^{\pi}\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}}
{v^2}\sec^2{x}+\frac{{{\mathrm i}}\zeta}{v}\tan x\right){{\mathrm d}}x.$$ We note that $I_2$ is uniformly convergent in $\zeta$ but that $I_1$ is not.
The integrals $I_1$ and $I_2$
-----------------------------
The substitution $z=\tan x$ gives $$\begin{aligned}
I_1(\beta,\zeta)&=&\frac{1}{\pi}\int_{-\infty}^{\infty} \exp\left(
-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}(1+z^2)+\frac{{{\mathrm i}}\zeta z}{v}\right){{\mathrm d}}z \nonumber
\\
&=&\frac{v}{\sqrt{\pi}}\frac{1}{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}
\exp\left(-\frac{\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}} -\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}\right)\end{aligned}$$ quoting a standard integral.
That $\delta_m$ defined in [(\[eq:delta\])]{} is a $\delta$-sequence for the function $\exp({{\mathrm i}}\alpha z^2)$ follows from the equality $$\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} {{\mathrm e}}^{{{\mathrm i}}\alpha z^2+{{\mathrm i}}\zeta(z-w)}{{\mathrm d}}z{{\mathrm d}}\zeta=\exp({{\mathrm i}}\alpha w^2),$$ which can be checked by direct evaluation of the integrals. This and uniform convergence of $I_2$ in $\zeta$ justifies the operations that lead to identity [(\[eq:states\])]{}.
We can treat $I_2$ in a similar manner to that in which we treated $I_1$. $$I_2(\beta,\zeta)=\frac{1}{\pi}
\int_{-\infty}^{\infty}\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}(1+z^2)+\frac{{{\mathrm i}}\zeta z}{v}\right)\frac{{{\mathrm d}}z}{1+z^2}.$$ We first write $$\frac{1}{1+z^2}=\frac{1}{2{{\mathrm i}}}\left(\frac{1}{z-{{\mathrm i}}}-\frac{1}{z+{{\mathrm i}}}\right)$$ so that $I_2$ can be decomposed into a difference of two similar integrals $$I_2(\beta,\zeta)=:I^{-}_2(\beta,\zeta)-I^{+}_2(\beta,\zeta).$$ Observing that $$\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}(1+z^2)+\frac{{{\mathrm i}}\zeta z}{v}\right)=
\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}-\frac{\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}-
\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}\left(z+\frac{\zeta v}{2\beta{\bar{L}}}\right)^2\right),$$ we can write $$\begin{aligned}
I_2^{-}(\beta,\zeta)&:=&\frac{1}{2\pi{{\mathrm i}}}\int_{-\infty}^{\infty}\exp\left(
-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}(1+z^2)+\frac{{{\mathrm i}}\zeta z}{v}\right)\frac{{{\mathrm d}}z}
{z-i}\\
&=&\frac{1}{2\pi{{\mathrm i}}}\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}+\frac{{{\mathrm i}}\zeta^2}
{4\beta{\bar{L}}}\right)
\int_{-\infty}^{\infty}\frac{\exp(-{{\mathrm i}}\beta{\bar{L}}y^2/v^2)}
{y-{{\mathrm i}}-\zeta v/2\beta{\bar{L}}}{{\mathrm d}}y\end{aligned}$$ via $y=z+\zeta v/2\beta{\bar{L}}$. We make the change of variable $$r=\frac{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}y.$$ This is permitted, because it rotates the contour of integration into the second and fourth quadrants of the complex plane, where the analytic function ${{\mathrm e}}^{-{{\mathrm i}}z^2}$ decays rapidly.
In the case $\zeta v/2\beta{\bar{L}}> -1$ the new contour of integration avoids the pole at $y=\zeta v/2\beta{\bar{L}}+{{\mathrm i}}$ (figure \[fig:1\]) and Cauchy’s Theorem yields $$I_2^{-}(\beta,\zeta)=
\frac{1}{2\pi{{\mathrm i}}}\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}+\frac{{{\mathrm i}}\zeta^2}
{4\beta{\bar{L}}}\right)
\int_{-\infty}^{\infty}\frac{{{\mathrm e}}^{-r^2}}{r-\frac{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}(
{{\mathrm i}}+\frac{\zeta v}{2\beta{\bar{L}}})}{{\mathrm d}}r.$$ This integral is standard, and may be found in, for example, [@abr:ste] (Equation [**7.1.4**]{}): $$\frac{{{\mathrm i}}}{\pi}\int_{-\infty}^{\infty} \frac{{{\mathrm e}}^{-t^2}}{z-t} {{\mathrm d}}t
={{\mathrm e}}^{-z^2}{\mathop{\rm erfc}}(-{{\mathrm i}}z),\qquad\mbox{for ${{\mathfrak{Im}}}\:z>0$.}$$ The result we get is $$\begin{aligned}
I^{-}_2(\beta,\zeta)&=&\frac{1}{2}\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}+
\frac{{{\mathrm i}}\zeta^2}{4\beta{\bar{L}}}\right)\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}
\left({{\mathrm i}}+\frac{\zeta v}{2\beta{\bar{L}}}\right)^2\right)\\
& &\times{\mathop{\rm erfc}}\left(
\frac{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}\left(1-\frac{{{\mathrm i}}\zeta v}{2\beta{\bar{L}}}
\right)\right)\\
&=&\frac{1}{2}\exp\left(\frac{\zeta}{v}\right){\mathop{\rm erfc}}\left(\frac{\zeta}{2\sqrt{
{{\mathrm i}}\beta{\bar{L}}}}+\frac{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}\right).\end{aligned}$$
(1,1) (0,0)[![Deforming the contour of integration avoiding pole[]{data-label="fig:1"}](contour1.eps "fig:"){width="7.0cm" height="6cm"}]{} (0.74,0.55)[$y=\frac{\zeta v}{2\beta{\bar{L}}}+{{\mathrm i}}$]{}
(1,1) (0,0)[![Deforming the contour of integration enclosing pole[]{data-label="fig:2"}](contour2.eps "fig:"){width="7.0cm" height="6cm"}]{}
If $\zeta v/2\beta{\bar{L}}<-1$ then the contour encloses a pole (figure \[fig:2\]). In this case, $$\int_{-\infty}^{\infty}\frac{\exp(-{{\mathrm i}}\beta{\bar{L}}y^2/v^2)}
{y-{{\mathrm i}}-\zeta v/2\beta{\bar{L}}}{{\mathrm d}}y=2\pi{{\mathrm i}}R +\int_{-\infty}^{\infty}
\frac{{{\mathrm e}}^{-r^2}}{r-\frac{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}({{\mathrm i}}+\frac{\zeta v}
{2\beta{\bar{L}}})}{{\mathrm d}}r$$ where $R$ is the residue at the pole $$\begin{aligned}
R&=&\left.\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}y^2}{v^2}\right)\right|_{y={{\mathrm i}}+
\zeta v /2\beta{\bar{L}}}\\
&=&\exp\left(-\frac{{{\mathrm i}}\zeta^2}{4\beta{\bar{L}}}+\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}+
\frac{\zeta}{v}\right),\end{aligned}$$ so that we also get in this case $$I_2^{-}(\beta,\zeta)=\frac{1}{2}\exp\left(\frac{\zeta}{v}\right)
{\mathop{\rm erfc}}\left(\frac{\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}+\frac{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}
{v}\right).$$
Treating $I_2^{+}$ in a similar way, yields an expression for $I_2$, $$I_2(\beta,\zeta)=\frac{1}{2}{{\mathrm e}}^{\zeta/v}{\mathop{\rm erfc}}\left(
\frac{\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}+\frac{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}\right)
+ \frac{1}{2}{{\mathrm e}}^{-\zeta/v}{\mathop{\rm erfc}}\left(
\frac{-\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}+\frac{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}\right).
\label{eq:5:2}$$
Some estimates
--------------
Let $-\sqrt{v}\leq\zeta\leq\sqrt{v}$, then $$\begin{aligned}
I_1(\beta,\zeta)(I_2(\beta,\zeta))^{v-1}=\frac{v}{\sqrt{\pi}}\frac{1}
{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}\exp\left(\frac{-\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}\right)
M(\beta,\zeta)\nonumber \\
\times\left(1+{{\mathrm O}}(v^{-1/2})+{{\mathrm O}}(\zeta^2/v)\right)
\label{eq:5:30}\end{aligned}$$ as $v\to\infty$, where $$M(\beta,\zeta):=\exp\left(-\frac{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{\sqrt{\pi}}
\exp\left(\frac{-\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}\right)-
{\zeta}{\mathop{\rm erf}}\left(\frac{\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}\right) \right).$$
[*Proof.*]{}
A key step will be to make a uniform expansion of $I_2(\beta,\zeta)$ as $v\to\infty$. By Taylor’s theorem, $$\exp\left(\pm\frac{\zeta}{v}\right)=1\pm\frac{\zeta}{v}+{{\mathrm O}}(
\zeta^2{v^{-2}}) \qquad\mbox{as $v\to\infty$.}\label{eq:5:17}$$ A second application of Taylor’s theorem yields $${\mathop{\rm erfc}}\left(\frac{\pm\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}+
\frac{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}\right)={\mathop{\rm erfc}}\left(\frac{\pm\zeta}{2
\sqrt{{{\mathrm i}}\beta{\bar{L}}}}\right)-\frac{2}{\sqrt{\pi}}\frac{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}
{v}\exp\left(-\frac{\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}\right)+\frac{R_{1}(\zeta)}
{v^2},
\label{eq:erfc}$$ where the remainder term is $$R_1(\zeta)=-{{{\mathrm i}}\beta{\bar{L}}}\int_0^1 \left.\frac{{{\mathrm d}}^2}
{{{\mathrm d}}z^2}{\mathop{\rm erfc}}(z)\right|_{z=\frac{\pm\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}+\sigma
\frac{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}}(1-\sigma){{\mathrm d}}\sigma.$$ The second derivative of ${\mathop{\rm erfc}}$ is $$\frac{{{\mathrm d}}^2}{{{\mathrm d}}z^2}{\mathop{\rm erfc}}(z)=\frac{4}{\sqrt{\pi}}z{{\mathrm e}}^{-z^2}.$$ This allows us to estimate $$\begin{aligned}
|R_1|&\leq&\frac{4\beta{\bar{L}}}{\sqrt{\pi}}\int_0^1 \left(\frac{|\zeta|}
{2\sqrt{\beta{\bar{L}}}}+\sigma\frac{|\sqrt{\beta{\bar{L}}}|}{v}\right)
{{\mathrm e}}^{\mp\sigma\zeta/v}(1-\sigma){{\mathrm d}}\sigma \nonumber\\
&\leq&\frac{4\beta{\bar{L}}}{\sqrt{\pi}}{{\mathrm e}}^{|\zeta|/v}\left(\frac{|\zeta|}
{2\sqrt{\beta{\bar{L}}}}+{{\mathrm O}}(v^{-1})\right) \nonumber \\
&=&{{\mathrm O}}(\sqrt{v}) \qquad\mbox{as $v\to\infty$,}\label{eq:5:21}\end{aligned}$$ uniformly for $|\zeta|<\sqrt{v}$. Substituting [(\[eq:5:17\])]{} and [(\[eq:erfc\])]{} into [(\[eq:5:2\])]{} gives $$I_2(\beta,\zeta)=1-\frac{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v\sqrt{\pi}}
\exp\left(\frac{-\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}\right)-
\frac{\zeta}{v}{\mathop{\rm erf}}\left(\frac{\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}\right)
+\frac{R_2(\zeta)}{v^2},
\label{eq:5:22}$$ where $R_2$ is a combination of the errors in [(\[eq:5:17\])]{} and [(\[eq:5:21\])]{}. So $$|R_2(\zeta)|={{\mathrm O}}(1+\zeta^2)\qquad\mbox{as $v\to\infty$.}
\label{eq:5:222}$$
We thus have that $$\begin{aligned}
\frac{1}{\sqrt{v}}\Bigg|\frac{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{\sqrt{\pi}}
&\exp\left(\frac{-\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}\right)+
{\zeta}{\mathop{\rm erf}}\left(\frac{\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}\right)
-\frac{R_2(\zeta)}{v} \Bigg| \nonumber \\
&\leq\frac{2\sqrt{\beta{\bar{L}}}}{\sqrt{\pi v}}+\frac{|\zeta|}{\sqrt{v}}
+\frac{|R_2|}{v^{3/2}} \nonumber \\
&<2\qquad\mbox{for all $\zeta\in[-\sqrt{v},\sqrt{v}]$ and $v$ sufficiently
large.}
\label{eq:5:6}\end{aligned}$$
We are now in a position to understand the asymptotics of $(I_2(\beta,\zeta))^{v-1}$. We first consider $$\ln\left(1-\frac{a}{v}\right)^{v-1}=(v-1)\left(-\frac{a}{v}
+\frac{R_3(a)}{v^2}\right), \label{eq:5:7}$$ where $$|R_3(a)|\leq\frac{|a|^2}{1-|a|/v}$$ provided that $|a|<v$. We take $$a=\frac{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{\sqrt{\pi}}
\exp\left(\frac{-\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}\right)+
{\zeta}{\mathop{\rm erf}}\left(\frac{\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}\right)
+\frac{R_2(\zeta)}{v}$$ which satisfies $|a|/\sqrt{v}<2$ for $v$ sufficiently large by [(\[eq:5:6\])]{} and $a={{\mathrm O}}(|\zeta|)$ as $|\zeta|\to\infty$ for $|\zeta|<\sqrt{v}$ by [(\[eq:5:222\])]{}. We see that $$\frac{|R_3(a)|}{v}={{\mathrm O}}(|a|^2/v)\qquad\mbox{as $v\to\infty$,}$$ so exponentiation of [(\[eq:5:7\])]{} gives $$\left(1-\frac{a}{v}\right)^{v-1}={{\mathrm e}}^{-a}\left(1+{{\mathrm O}}(v^{-1/2})\right)
\left(1+{{\mathrm O}}((1+\zeta^2)/v)\right),
\label{eq:5:28}$$ where the error estimates are uniform for $|\zeta|<\sqrt{v}$ as $v\to\infty$ and $${{\mathrm e}}^{-a}=\exp\left(-\frac{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{\sqrt{\pi}}
\exp\left(\frac{-\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}\right)-
{\zeta}{\mathop{\rm erf}}\left(\frac{\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}\right) \right)
\left[1+{{\mathrm O}}((1+\zeta^2)/v)\right].$$ We can also write $$I_1(\beta,\zeta)=\frac{v}{\sqrt{\pi}}\frac{1}{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}
\exp\left(\frac{-\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}\right)\left(1+{{\mathrm O}}(v^{-2})\right).
\label{eq:5:29}$$ Thus, taking [(\[eq:5:29\])]{} together with [(\[eq:5:22\])]{} and [(\[eq:5:28\])]{} gives the required estimate.
$\square$
\[lem:5:7\] Let $|\zeta|>\sqrt{v}$, then $$|I_1(\beta,\zeta)(I_2(\beta,\zeta))^{v-1}|\leq\frac{1}{\pi}\left(
\frac{\beta{\bar{L}}}{v^2}+\frac{\zeta^2}{\beta{\bar{L}}}\right)^{-1}+{{\mathrm O}}(v\zeta^{-3})
\qquad\mbox{as $|\zeta|\to\infty$.}$$
[*Proof.*]{}
We make use of the asymptotic expression $${\mathop{\rm erfc}}(z)=\frac{1}{\sqrt{\pi}}{{\mathrm e}}^{-z^2}\left(\frac{1}{z}+{{\mathrm O}}(z^{-3})\right)$$ as $z\to\infty$, valid for $|\arg z|<3\pi/4$. We see that $$\begin{aligned}
{{\mathrm e}}^{\zeta/v}{\mathop{\rm erfc}}\left(\frac{\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}+\frac{
\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}\right)=\frac{1}{\sqrt{\pi}}\exp\left(\frac{-\zeta^2}
{4{{\mathrm i}}\beta{\bar{L}}}+\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}\right) \\
\times\left[\left(
\frac{\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}+\frac{
\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}\right)^{-1}+{{\mathrm O}}(\zeta^{-3})\right].\end{aligned}$$ Similarly, $$\begin{aligned}
{{\mathrm e}}^{-\zeta/v}{\mathop{\rm erfc}}\left(\frac{-\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}+\frac{
\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}\right)=\frac{1}{\sqrt{\pi}}\exp\left(\frac{-\zeta^2}
{4{{\mathrm i}}\beta{\bar{L}}}+\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}\right) \\
\times\left[\left(
\frac{-\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}+\frac{
\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{v}\right)^{-1}+{{\mathrm O}}(\zeta^{-3})\right].\end{aligned}$$ Adding these gives the estimate $$|I_2(\beta,\zeta)|\leq\frac{\sqrt{\beta{\bar{L}}}}{v\sqrt{\pi}}\left(
\frac{\beta{\bar{L}}}{v^2}+\frac{\zeta^2}{\beta{\bar{L}}}\right)^{-1}
+{{\mathrm O}}(\zeta^{-3})\qquad\mbox{as $|\zeta|\to\infty$.}
\label{eq:5:3}$$ Taking [(\[eq:5:3\])]{} together with the estimates $|I_2(\beta,\zeta)|\leq 1$ and $$|I_1(\beta,\zeta)|\leq\frac{v}{\sqrt{\pi\beta{\bar{L}}}}$$ gives us the estimate we require.
$\square$
\[prop:5:9\] With the notation above, $$\lim_{v\to\infty}\frac{1}{2v}
\int_{-\infty}^{\infty} I_1(\beta,\zeta) (I_2(\beta,\zeta))^{v-1}
{{\mathrm d}}\zeta=\frac{1}{2\sqrt{\pi}}\int_{-\infty}^{\infty}
\frac{1}
{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}\exp\left(\frac{-\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}\right)
M(\beta,\zeta){{\mathrm d}}\zeta.$$
[*Proof.*]{}
We split the region of integration as follows $$\begin{aligned}
\frac{1}{2v}\int_{-\infty}^{\infty} I_1(\beta,\zeta)(I_2(\beta,\zeta))^{v-1}
{{\mathrm d}}\zeta = \frac{1}{2v}\int_{-\sqrt{v}}^{\sqrt{v}} I_1(\beta,\zeta)
(I_2(\beta,\zeta))^{v-1} {{\mathrm d}}\zeta \nonumber \\
+\frac{1}{2v}\int_{|\zeta|>\sqrt{v}}I_1(\beta,\zeta)(I_2(\beta,\zeta))^{v-1}
{{\mathrm d}}\zeta. \label{eq:5:16}\end{aligned}$$ The function $M$ is bounded and satisfies $$|M(\beta,\zeta)|\leq\exp\left(\frac{2\sqrt{\beta{\bar{L}}}}{\sqrt{\pi}}-\frac
{|\zeta|}{2}\right)$$ for $\zeta$ sufficiently large. Integrating [(\[eq:5:30\])]{} gives $$\begin{aligned}
\frac{1}{2v}\int_{-\sqrt{v}}^{\sqrt{v}} I_1(\beta,\zeta)
(I_2(\beta,\zeta))^{v-1}& {{\mathrm d}}\zeta =
\frac{1}{2\sqrt{\pi}}\int_{-\sqrt{v}}^{\sqrt{v}}\frac{1}
{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}\exp\left(\frac{-\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}\right)
M(\beta,\zeta){{\mathrm d}}\zeta \nonumber \\
&+{{\mathrm O}}\left[\int_{-\sqrt{v}}^{\sqrt{v}}\exp\left(-\frac{|\zeta|}{2}\right)
\left(\frac{1}{\sqrt{v}}+\frac{\zeta^2}{v}\right){{\mathrm d}}\zeta\right]
\label{eq:5:35}\end{aligned}$$ and the integral in the remainder term converges as we let $v\to\infty$.
To deal with the integral $$\frac{1}{v}\int_{|\zeta|>\sqrt{v}}I_1(\beta,\zeta)(I_2(\beta,\zeta))^{v-1}
{{\mathrm d}}\zeta$$ we make use of the result of lemma \[lem:5:7\], giving $$\begin{aligned}
\Bigg|\frac{1}{v}\int_{|\zeta|>\sqrt{v}}I_1(\beta,\zeta)
(I_2(\beta,\zeta&))^{v-1}{{\mathrm d}}\zeta\Bigg| \nonumber \\
&\leq \frac{1}{\pi}\int_{|\zeta|>\sqrt{v}} \left(
\frac{\beta{\bar{L}}}{v^2}+\frac{\zeta^2}{\beta{\bar{L}}}\right)^{-1}{{\mathrm d}}\zeta
+{{\mathrm O}}(v^{-1}) \label{eq:5:41}\\
&\to0 \qquad\mbox{as $v\to\infty$.} \nonumber\end{aligned}$$
Hence substituting [(\[eq:5:35\])]{} and [(\[eq:5:41\])]{} into [(\[eq:5:16\])]{} and taking the limit $v\to\infty$ gives the required result.
$\square$
Properties of $M(\beta,\zeta)$
------------------------------
We wish to rotate the variable $\zeta$. However, we need to check that the function $M$ does not blow up for large $|\zeta|$.
If $\zeta=R{{\mathrm e}}^{{{\mathrm i}}\theta}$, then $${\mathop{\rm erf}}\left(\frac{\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}\right)={\mathop{\rm erf}}\left(
\frac{R}{2\sqrt{\beta{\bar{L}}}}{{\mathrm e}}^{{{\mathrm i}}(\theta-\pi/4)}\right)=1+{{\mathrm O}}(R^{-1})
\label{eq:7:7}$$ as $R\to\infty$, provided that $0<\theta<\pi/2$ [@abr:ste]. So, $$\begin{aligned}
\left|\exp\left(-\zeta{\mathop{\rm erf}}\left(\frac{\zeta}{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}\right)
\right)\right|&=&\exp\left(-R\;{{\mathfrak{Re}}}\left[{{\mathrm e}}^{{{\mathrm i}}\theta}{\mathop{\rm erf}}\left(\frac{\zeta}{2\sqrt{{{\mathrm i}}\beta
{\bar{L}}}}\right)\right]\right)\\
&=&{{\mathrm e}}^{-R\cos\theta}{{\mathrm O}}(1)\\
&\to&0\qquad\mbox{as $R\to\infty$ provided $0<\theta<\pi/2$}\end{aligned}$$ and convergence is exponentially fast. Similarly, $$\begin{aligned}
\left|\exp\left(-\frac{2\sqrt{{{\mathrm i}}\beta{\bar{L}}}}{\sqrt{\pi}}\exp
\left(-\frac{\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}\right)\right)\right|
&=&\exp\left(\frac{-2\sqrt{\beta{\bar{L}}}}{\sqrt{\pi}}\exp\left(-\frac{R^2}{4\beta
{\bar{L}}}\sin2\theta\right)\right.\\
& &\left.\times\cos\left(\frac{R^2}{4\beta{\bar{L}}}\cos2\theta+\frac{\pi}
{4}\right)\right)\\
&\to&1\qquad\mbox{as $R\to\infty$} \end{aligned}$$ provided $0<\theta<\pi/2$.
Hence, if $0<\theta<\pi/4$ then $$\lim_{R\to\pm\infty} |RM(\beta,R{{\mathrm e}}^{{{\mathrm i}}\theta})|=0$$ and we can make the change of variables $\zeta=\xi\sqrt{{{\mathrm i}}\beta{\bar{L}}}$: $$\int_{-\infty}^{\infty}
\frac{1}
{\sqrt{{{\mathrm i}}\beta{\bar{L}}}}\exp\left(\frac{-\zeta^2}{4{{\mathrm i}}\beta{\bar{L}}}\right)
M(\beta,\zeta){{\mathrm d}}\zeta
=
\int_{-\infty}^{\infty}{{\mathrm e}}^{-\xi^2/4}M(\beta,\xi\sqrt{{{\mathrm i}}\beta{\bar{L}}}){{\mathrm d}}\xi. \label{eq:5:10}$$ We observe that $$\begin{aligned}
M(\beta,\xi\sqrt{{{\mathrm i}}\beta{\bar{L}}})&=&\exp\left(-\frac{2}{\sqrt{\pi}}
\sqrt{{{\mathrm i}}\beta{\bar{L}}}{{\mathrm e}}^{-\xi^2/4}-\xi\sqrt{{{\mathrm i}}\beta{\bar{L}}}{\mathop{\rm erf}}\left(\frac{\xi}{2}\right)\right)\\
&=&\exp\left(-\sqrt{{{\mathrm i}}\beta{\bar{L}}}m(\xi)\right)\end{aligned}$$ where $$m(\xi):=\frac{2}{\sqrt{\pi}}{{\mathrm e}}^{-\xi^2/4}+\xi{\mathop{\rm erf}}(\xi/2).$$
[*Proof of theorem \[thm:4\]*]{}
--------------------------------
$m$ satisfies the bound $m(\xi)\geq2/\sqrt{\pi}$, so $M( \beta,\xi\sqrt{{{\mathrm i}}\beta{\bar{L}}})$ is bounded for all $\beta$. By the Weierstrass $M$-test the integral in [(\[eq:5:10\])]{} is uniformly convergent and hence $$\lim_{v\to\infty} E_v(\beta)$$ is a continuous function of $\beta$. We appeal, once again, to the continuity theorem for characteristic functions to deduce that the limiting density $P(y)$ exists and is given by $$\begin{aligned}
P(y)&=&\frac{1}{4\pi^{3/2}}\int_{-\infty}^{\infty}\!\int_{-\infty}^{\infty}
{{\mathrm e}}^{-\xi^2/4}M(\beta,\xi\sqrt{{{\mathrm i}}\beta{\bar{L}}}){{\mathrm e}}^{{{\mathrm i}}\beta y}{{\mathrm d}}\xi{{\mathrm d}}\beta \nonumber \\
&=&\frac{1}{2\pi^{3/2}}{{\mathfrak{Re}}}\int_{0}^{\infty}\!\int_{-\infty}^{\infty}
{{\mathrm e}}^{-\xi^2/4}M(\beta,\xi\sqrt{{{\mathrm i}}\beta{\bar{L}}}){{\mathrm e}}^{{{\mathrm i}}\beta y}{{\mathrm d}}\xi{{\mathrm d}}\beta.\label{eq:4:66}\end{aligned}$$ The integrand is dominated by $$\exp\left(-\xi^2/4-\sqrt{\pi{\bar{L}}\beta}\right),$$ so Fubini’s theorem allows us to switch the order of integration. We quote the standard integral $$\int_0^{\infty}{{\mathrm e}}^{ax+b\sqrt{x}}{{\mathrm d}}{x}=-\frac{1}{a}-\frac{b}{2a}\sqrt
{\frac{\pi}{-a}}\exp\left(\frac{-b^2}{4a}\right){\mathop{\rm erfc}}\left(\frac{-b}
{2\sqrt{-a}}\right)$$ valid for ${{\mathfrak{Re}}}\: a<0$ and use this to perform the $\beta$ integral in [(\[eq:4:66\])]{}. This leads to the result $$P(y)=\frac{\sqrt{{\bar{L}}}}{4\pi y^{3/2}}{{\mathfrak{Re}}}\int_{-\infty}^{\infty}
\exp\left(-\frac{\xi^2}{4}-\frac{{\bar{L}}m(\xi)^2}{4y}\right)m(\xi)
{\mathop{\rm erfc}}\left(\frac{\sqrt{{\bar{L}}}m(\xi)}{2{{\mathrm i}}y}\right){{\mathrm d}}\xi,$$ which reduces to the form given in the statement of the theorem upon noticing that ${{\mathfrak{Re}}}\{ {\mathop{\rm erfc}}({{\mathrm i}}\theta)\}=1$ for all $\theta\in{\mathbb R}$.
$\square$
Value distribution of the eigenfunctions in the limit $v\to\infty$ {#sec:5}
==================================================================
To prove theorem \[thm:6\] we use a standard approximation argument. We introduce the smoothed $\delta$-function $$\delta_{\epsilon}(x):=\frac{1}{2\pi}\int_{-\infty}^{\infty}
{{\mathrm e}}^{-\epsilon|\beta|+{{\mathrm i}}\beta x}{{\mathrm d}}\beta=
\frac{\epsilon}{\pi(\epsilon^2+x^2)}.$$
\[prop:5:100\] Let $\tilde{Q}_v(\eta)$ be related to $Q_v(\eta)$ by $$\tilde{Q}_v(\eta):=\frac{1}{\eta^2}Q_v\left(\frac{1}{\eta}\right).$$ For any fixed $\epsilon$, $$\lim_{v\to\infty}\int_0^{\infty}\delta_{\epsilon}(\eta-\eta')
\tilde{Q}_v(\eta'){{\mathrm d}}\eta'=\int_0^{\infty} \delta_{\epsilon}(\eta-\eta')\tilde{Q}(\eta'){{\mathrm d}}\eta',
\label{eq:5:100}$$ uniformly for $\eta$ in compact intervals, where $$\tilde{Q}(\eta)=\frac{1}{2\pi^{3/2}\eta}{{\mathfrak{Im}}}\int_{-\infty}^{\infty}
\exp\left(-\frac{\xi^2}{4}
-\frac{{\bar{L}}m(\xi)^2}{8\eta}\right){\mathop{\rm erfc}}\left(\frac{\sqrt{{\bar{L}}}m(\xi)}
{2\sqrt{2\eta}{{\mathrm i}}}\right){{\mathrm d}}\xi.$$
[*Proof.*]{}
Let $$\tilde{Q}_{\epsilon,v}(\eta):=\int_{0}^{\infty}\delta_{\epsilon}(\eta-\eta')
\tilde{Q}_v(\eta'){{\mathrm d}}\eta'
=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}
\delta_{\epsilon}\left(\eta-\frac{1}{v^2A_i(n,{{\mathbf L}};v)}\right).$$ We use the identity $$\delta_{\epsilon}\left(\eta-\frac{A}{B}\right)=B\delta_{\epsilon|B|}(B\eta
-A)$$ with $$A=\frac{{\bar{L}}}{v^2}\sum_{j=1}^v \sec^2{kL_j}$$ and $$B=2\sec^2kL_i.$$ Thus $$\begin{aligned}
\delta_\epsilon\left(\eta-\frac{\sum_j
{\bar{L}}\sec^2 kL_j}{2v^2\sec^2kL_i}\right)
&=&\frac{2\sec^2kL_i}{\pi}{{\mathfrak{Re}}}\int_{0}^{\infty}
\exp\left({{\mathrm i}}\beta\left(2\eta-\frac{{\bar{L}}}{v^2}+2{{\mathrm i}}\epsilon\right)
\sec^2{kL_i}\right)\\
& &\times
\prod_{j\neq i}\exp\left(-\frac{{{\mathrm i}}\beta{\bar{L}}}{v^2}\sec^2 kL_j\right)
{{\mathrm d}}\beta.\end{aligned}$$ Applying identity [(\[eq:states\])]{} to this function and following the method in section \[sec:5\] leads to $$\begin{aligned}
\frac{2{\bar{L}}}{\pi^2\bar{d}v}{{\mathfrak{Re}}}\int_{0}^{\infty}\int_{-\infty}^{\infty}
I_4(\beta,\zeta)&((I_2(\beta,\zeta))^{v-1}\\
&+(v-1)I_3(\beta,\zeta)I_1(\beta,\zeta)
(I_2(\beta,\zeta))^{v-2}{{\mathrm d}}\zeta{{\mathrm d}}\beta
=:\tilde{P}_{\epsilon}(\eta)\end{aligned}$$ where the new integrals are $$I_3(\beta,\zeta):=\frac{1}{\pi}\int_{0}^{\pi}\sec^2x\exp\left({{\mathrm i}}\beta
\left(2\eta-\frac{{\bar{L}}}{v^2}+2{{\mathrm i}}\epsilon\right)\sec^2x+
\frac{{{\mathrm i}}\zeta}{v}\tan x
\right){{\mathrm d}}x$$ and $$I_4(\beta,\zeta):=\frac{1}{\pi}\int_{0}^{\pi}\sec^4x\exp\left({{\mathrm i}}\beta
\left(2\eta-\frac{{\bar{L}}}{v^2}+2{{\mathrm i}}\epsilon\right)
\sec^2x+\frac{{{\mathrm i}}\zeta}{v}\tan x
\right){{\mathrm d}}x.$$ $I_3$ and $I_4$ converge uniformly in $\zeta$ and $\eta$. Integral $I_3$ closely resembles integral $I_1$. $$I_3(\beta,\zeta)=\frac{{{\mathrm e}}^{2{{\mathrm i}}\beta\eta-2\epsilon\beta}}{\sqrt{\beta}
\sqrt{2\pi\epsilon-2\pi{{\mathrm i}}\eta}}+{{\mathrm O}}_{\epsilon}(v^{-1})\qquad\mbox{as $v\to\infty$.}$$ By making the substitution $z=\tan x$, $I_4$ reduces to $$\begin{aligned}
I_4(\beta,\zeta)&=&\frac{1}{\pi}\int_{-\infty}^{\infty} (1+z^2)\exp\left(
{{\mathrm i}}\beta\left(2\eta-\frac{{\bar{L}}}{v^2}+2{{\mathrm i}}\epsilon\right)
(1+z^2)+\frac{{{\mathrm i}}\zeta z}{v}
\right){{\mathrm d}}z\\
&=&{{\mathrm O}}_{\epsilon}(1)\qquad\mbox{as $v\to\infty$.}\end{aligned}$$ This estimate ensures that $\tilde{Q}_{\epsilon,v}(\eta)$ is dominated by the second term. Since $I_3$ is also bounded as $v\to\infty$, the analysis of proposition \[prop:5:9\] holds and $$\begin{aligned}
&\lim_{v\to\infty}\tilde{Q}_{\epsilon,v}(\eta)\\
&=\frac{2}{\pi^2}{{\mathfrak{Re}}}\frac{1}{\sqrt{2\epsilon-2{{\mathrm i}}\eta}}
\int_0^{\infty}\!
\int_{-\infty}^{\infty}\frac{1}{\beta\sqrt{{{\mathrm i}}{\bar{L}}}}\exp\left
(2{{\mathrm i}}\beta\eta-\frac{\zeta^2}
{4{{\mathrm i}}\beta{\bar{L}}}-2\epsilon\beta\right)
M(\beta,\zeta){{\mathrm d}}\zeta{{\mathrm d}}\beta\\
&=\frac{2}{\pi^2}{{\mathfrak{Re}}}\left[\frac{1}{\sqrt{2\epsilon-2{{\mathrm i}}\eta}}
\int_{-\infty}^{\infty}\int_0^{\infty}
\exp\left(-\frac{\xi^2}{4}-\sqrt{{{\mathrm i}}\beta{\bar{L}}}m(\xi)
+2{{\mathrm i}}\beta\eta-2\epsilon\beta
\right)\frac{{{\mathrm d}}\beta}{\sqrt{\beta}}{{\mathrm d}}\xi\right],\end{aligned}$$ where we have made, once again, the substitution $\zeta=\xi\sqrt{{{\mathrm i}}\beta{\bar{L}}}$ in the final line. Putting $\theta^2=\beta$ reduces the $\beta$ integral to $$\begin{aligned}
\int_0^{\infty}{{\mathrm e}}^{-\sqrt{{{\mathrm i}}{\bar{L}}}\theta m(\xi)-(2\epsilon-2{{\mathrm i}}\eta)
\theta^2}
{{\mathrm d}}\theta\mspace{-100mu}&\\
&= \frac{\sqrt{\pi}}{2\sqrt{2\epsilon-2{{\mathrm i}}\eta}}
\exp\left(\frac{-{\bar{L}}m(\xi)^2}
{8(\eta+{{\mathrm i}}\epsilon)}\right)
{\mathop{\rm erfc}}\left(\frac{\sqrt{{\bar{L}}}m(\xi)}{2{{\mathrm i}}\sqrt{2\eta+2{{\mathrm i}}\epsilon}}\right),\end{aligned}$$ so $$\begin{aligned}
\lim_{v\to\infty}&\tilde{Q}_{\epsilon,v}(\eta)\\
&=\frac{1}{2\pi^{3/2}}{{\mathfrak{Im}}}\left[
\frac{1}{\eta+{{\mathrm i}}\epsilon}
\int_{-\infty}^{\infty}\exp\left(-\frac{\xi^2}{4}-\frac{{\bar{L}}m(\xi)^2}
{8(\eta+{{\mathrm i}}\epsilon)}\right){\mathop{\rm erfc}}\left(\frac{\sqrt{{\bar{L}}}m(\xi)}{2{{\mathrm i}}\sqrt{2\eta+2{{\mathrm i}}\epsilon}}\right){{\mathrm d}}\xi\right]\\
&=\int_{0}^{\infty}\delta_{\epsilon}(\eta-\eta')\tilde{Q}(\eta'){{\mathrm d}}\eta'\end{aligned}$$
$\square$
We note that $\tilde{Q}(\eta)$ is a continuous probability density on $(0,\infty)$.
[*Proof of theorem \[thm:6\].*]{}
Let $0<a<b$ be fixed. Then let $${{1\!\!1}}^{\epsilon}_{[a,b]}(\eta):=\int_{a}^{b}
\delta_{\epsilon}
(\eta-y){{\mathrm d}}y.$$ ${{1\!\!1}}^{\epsilon}_{[a,b]}(\eta)$ converges pointwise as $\epsilon\to 0$ to the function ${{1\!\!1}}_{[a,b]}(\eta)$ everywhere apart from at the end-points $a$ and $b$. Given $\sigma>0$, consider the function $$\chi_1(\eta):={{1\!\!1}}^{\epsilon}_{[a-\sigma,b+\sigma]}(\eta)+\sigma.$$ There exists $\epsilon>0$ such that:-
- $0\leq\chi_1(\eta)\leq 2\sigma$ for $\eta\leq a-2\sigma$ and $\eta\geq b+2\sigma$,
- $1\leq\chi_1(\eta)\leq 1+\sigma$ for $a\leq\eta\leq b$,
- $0\leq\chi_1(\eta)\leq 1+\sigma$ for $a-2\sigma\leq\eta\leq a$ and $b\leq\eta\leq b+2\sigma$.
This construction is illustrated in figure \[fig:6\]
(1,0.9) (-0.25,0.1)[![Approximating ${{1\!\!1}}_{[a,b]}$ from above[]{data-label="fig:6"}](estimate.eps "fig:"){width="9.0cm" height="5cm"}]{} (-0.32,0.15)[$2\sigma$]{} (0.1,0.03)[$a$]{} (0.68,0.03)[$b$]{} (-0.09,0.7)[$\sigma$]{} (0.04,0.85)[$2\sigma$]{} (0.4,0.8)[$\chi_1$]{}
Similarly, the function $$\chi_2(\eta)={{1\!\!1}}^{\epsilon}_{[a+\sigma,b-\sigma]}(\eta)-\sigma$$ satisfies for $\epsilon$ sufficiently small:-
- $-\sigma\leq\chi_2(\eta)\leq 0$ for $\eta\leq a$ and $\eta\geq b$,
- $1-2\sigma\leq\chi_2(\eta)\leq 1$ for $a+2\sigma\leq\eta\leq b-2\sigma$,
- $-\sigma\leq\chi_2(\eta)\leq 1$ for all $a\leq\eta\leq a+2\sigma$ and $b-2\sigma\leq\eta\leq b$.
So that for all $\eta\in[0,\infty)$ $$\chi_2(\eta)<{{1\!\!1}}_{[a,b]}(\eta)<\chi_1(\eta).
\label{eq:5:69}$$ Also, $$\begin{aligned}
\int_0^{\infty}\left[\chi_1(\eta)-
\chi_2(\eta)\right]\tilde{Q}(\eta){{\mathrm d}}\eta\leq
3\sigma\int_{0}^{\infty}\tilde{Q}(\eta){{\mathrm d}}\eta+(1+2\sigma)
\int_{a-2\sigma}^{a
+2\sigma}\tilde{Q}(\eta){{\mathrm d}}\eta\\
+(1+2\sigma)\int_{b-2\sigma}^{b+2\sigma}\tilde{Q}(\eta){{\mathrm d}}\eta,\end{aligned}$$ which can be made arbitrarily small because $\tilde{Q}$ is a continuous probability density. It follows from proposition \[prop:5:100\] that $$\lim_{v\to\infty}\int_0^{\infty}\chi_1(\eta)\tilde{Q}_v(\eta)
{{\mathrm d}}\eta=\int_0^{\infty}\chi_1(\eta)\tilde{Q}(\eta){{\mathrm d}}\eta$$ and similarly for $\chi_2$. Hence, we can use the argument of lemma \[lem:2\] [*mutatis mutandis*]{}, to deduce that $$\lim_{v\to\infty}\int_a^b\tilde{Q}_v(\eta){{\mathrm d}}\eta=\int_{a}^{b}\tilde{Q}(\eta)
{{\mathrm d}}\eta.$$ Making the substitution $\eta\mapsto 1/\eta$ then completes the proof of convergence.
Expanding the error function in [(\[eq:Q:def\])]{} as $${\mathop{\rm erfc}}\left(\frac{\sqrt{{\bar{L}}\eta}m(\xi)}{2{{\mathrm i}}\sqrt{2}}\right)=
\frac{1}{\sqrt{\pi}}\exp\left(\frac{{\bar{L}}\eta m(\xi)^2}{8}\right)
\left(\frac{2{{\mathrm i}}\sqrt{2}}{\sqrt{{\bar{L}}\eta}m(\xi)}+{{\mathrm O}}(\eta^{-3/2})\right),$$ where the implied constant does not depend on $\xi$, yields $$Q(\eta)=\frac{b}{\eta^{3/2}}+{{\mathrm O}}(\eta^{-5/2})\qquad
\mbox{as $\eta\to\infty$,}$$ where the constant $b$ is $$b=\frac{\sqrt{2}}{\sqrt{{\bar{L}}}\pi^2}\int_{-\infty}^{\infty}
\frac{{{\mathrm e}}^{-\xi^2/4}}{m(\xi)}{{\mathrm d}}\xi
\approx\frac{0.348}{\sqrt{{\bar{L}}}}.$$
$\square$
The algebraic decay of $Q(\eta)$ is in contrast to the exponential decay of the $\chi_1^2$ density.
Numerical Results {#sec:6}
=================
The results presented above show close agreement with numerical computations. We present these computations now by way of illustration.
In all the figures in this section, the choice of ${\bar{L}}=2$ has been made.
Figure \[fig:9:3\] shows a comparison between a numerical evaluation of values taken by the spectral determinant and the Cauchy distribution. The numerical evaluation was based on a star graph with 7 randomly chosen bond lengths, and 100,000 samples of $k$.
![The value distribution for the spectral determinant[]{data-label="fig:9:3"}](niceplot.eps){width="8.0cm" height="6cm"}
Figure \[fig:9:4\] shows a comparison between the distribution of values taken by the derivative of the spectral determinant at its zeros, and the corresponding numerical evaluation. Plotted is numerical data for a 70-bond star graph, together with the $v\to\infty$ limiting density given in theorem \[thm:4\]. Once again we see good agreement.
![The value distribution of $Z'(k)$[]{data-label="fig:9:4"}](niceplot1.eps){width="8.0cm" height="6cm"}
In figure \[fig:9:5\] we compare a numerical evaluation of the density of values taken by the maximum norm of eigenvectors of a 50-bond graph to the $v\to\infty$ limiting density given in theorem \[thm:6\]. Also plotted for comparison is the density of the $\chi^2_1$ distribution associated with the COE of random matrices.
![The value distribution of $A_i(n,{{\mathbf L}};v)$[]{data-label="fig:9:5"}](niceplot2.eps){width="8.0cm" height="6cm"}
Connections with the Šeba Billiard {#sec:7}
==================================
The correspondence between the spectral statistics of quantum star graphs and those of Šeba billiards with periodic boundary conditions has already been noted [@berk:2]. This is due to the fact that the spectral determinant for the star graphs [(\[spec:det\])]{} may be re-written in a form similar to the spectral determinant of a Šeba billiard: $$Z_{\rm Seba}(E)=\sum_{k=1}^{\infty}\frac{1}{E_k^{(0)}-E}$$ where the $E_k^{(0)}$ are the energy levels of the unperturbed system. Both spectral determinants have infinitely many poles of first order, which separate the energy levels of the perturbed system. We therefore expect the value distribution of the spectral determinant of a Šeba billiard to be Cauchy to be consistent with theorems \[thm:1\] and \[thm:2\].
This conjecture is supported by figure \[fig:10:1\] which is a plot of the density of values given taken by the function $$\pi\langle d\rangle\sum_{k=1}^{K}\frac{1}{E_k^{(0)}-E}
\label{eq:conj:1}$$ for $K=3000$ unperturbed levels of a rectangular quantum billiard with Neumann boundary conditions, with $E$ distributed uniformly between $E_{1000}^{(0)}$ and $E_{2000}^{(0)}$. The constant $\langle d\rangle$ is the mean density of levels of the system and it takes the place of the constant $v^{-1}$ in theorems \[thm:1\] and \[thm:2\]. The fit to a Cauchy density is convincing.
![The value distribution for the spectral determinant of a Šeba billiard[]{data-label="fig:10:1"}](sebaplot1.eps){width="8.0cm" height="6cm"}
If we treat the unperturbed levels in [(\[eq:conj:1\])]{} as independent identically distributed random variables with a uniform density then the random variables $$\frac{1}{E_k^{(0)}-E}$$ have a distribution that falls into the domain of attraction of the stable Cauchy density. That the limiting density is Cauchy is then a classical result of probability theory [@fel:ipt].
We now present an argument which suggests that the normalisation constant associated with the wave functions of the Šeba billiard also shares significant features with the normalisation constant of the star graphs [(\[norm:const\])]{}.
The wave functions of a general Šeba billiard [@seba:2] can be written in the form $$\psi_n({{\mathbf x}})=A_n\sum_{k=1}^{\infty} \frac{\psi_k^{(0)}({{\mathbf x}}_0)
\psi_k^{(0)}({{\mathbf x}})}{E_k^{(0)}-E_n}$$ where $E_n$ is the $n^{\rm th}$ energy level and $E_k^{(0)}$ and $\psi_k^{(0)}$ are, respectively, the energy levels and wave functions of the original integrable system of which the Šeba problem is a perturbation. The Berry-Tabor Conjecture [@berry:1] asserts that the unperturbed levels $E_k^{(0)}$ are distributed like Poisson variables; that is independent and random. We fix the usual normalisation $$\int|\psi_n({{\mathbf x}})|^2 {{\mathrm d}}{{\mathbf x}}=1,$$ which leads to a value for the constant $A_n$ $$A_n^2=\left(\sum_{k=1}^{\infty} \frac{|\psi_k^{(0)}({{\mathbf x}}_0)|^2}
{(E_k^{(0)}-E_n)^2}\right)^{-1}.
\label{seba:norm}$$ In the case that the unperturbed system is a rectangular quantum billiard with Neumann boundary conditions and sides of length $\alpha^{1/4}$ and $\alpha^{-1/4}$ the wavefunctions are $$\psi_{n,m}^{(0)}(x,y)=2\cos\left(\frac{n\pi x}{\alpha^{1/4}}\right)
\cos(m\pi y\alpha^{1/4}),\qquad\mbox{$n,m=0,1,2,\ldots$.}$$ If we position the scatterer at the origin, then $|\psi_{n,m}^{(0)}({{\mathbf x}}_0)|=2$. This billiard problem is equivalent to the billiard with periodic boundary conditions desymmetrised to remove degeneracies in the spectrum. Provided that the constant $\alpha$ satisfies certain diophantine conditions (see [@marklof:1; @eskin:1] for details) then $A_n^2$ in [(\[seba:norm\])]{} is the reciprocal of a sum of functions with poles of second order distributed independently. These poles play the rôle of the singularities of the functions $\sec kL_j$ which appear in the normalisation of the quantum graphs. Such poles determine the rate of decay of the tails of the relevant probability distributions, and this implies that the analysis performed in the present work also holds for this billiard problem. In particular, we conjecture that the distribution of the square of the $i^{\rm th}$ coefficient of the eigenfunctions in the basis $|\psi_k^{(0)}\rangle$ is the same as the limiting distribution of $A_i(n,{{\mathbf L}};v)$.
We present in figure \[fig:10:2\] the distribution of values taken by $$c\frac{(E_i^{(0)}-E_n)^{-2}}{\sum_{k=1}^{K} (E_k^{(0)}-E_n)^{-2}}$$ where $n$ is now a random variable uniformly distributed on $\{1000,\ldots,2000\}$ and we take $K=3000$, $i=1500$ and $\alpha=(\sqrt{5}-1)/2$. The constant $c$, which in general may be expected to depend on $K$ and the distribution of $n$, is required to ensure that the sum of terms in the denominator is normalised and to compensate for the fact that the functions are not periodic. In order to compare with the corresponding results for star graphs we require that the tail of the distribution of $c(E_i^{(0)}-E_n)^{-2}$ is asymptotic to the tail of the distribution of $(2/{\bar{L}})\sec^2k_nL_i$. Assuming $n$ to be distributed between $n_{\rm max}$ and $n_{\rm min}$, a heuristic examination of these densities leads to the association $$c=\frac{2}{{\bar{L}}}{(E_{\rm max}-E_{\rm min})^2}{\langle d\rangle^2}$$ where $E_{\rm max}$ and $E_{\rm min}$ are respectively the energy levels corresponding to $n=n_{\rm max}$ and $n=n_{\rm min}$. For the data in figure \[fig:10:2\], we get $c\approx9.75\times10^5$.
![The value distribution for the eigenfunctions of a Šeba billiard[]{data-label="fig:10:2"}](sebaplot2.eps){width="8.0cm" height="6cm"}
Acknowledgements {#acknowledgements .unnumbered}
================
JM is supported by an EPSRC Advanced Research Fellowship, the Nuffield Foundation (Grant NAL/00351/G), and the Royal Society (Grant 22355). BW is supported by an EPSRC studentship (Award Number 0080052X). Additionally, we are grateful for the financial support of the European Commission under the Research Training Network (Mathematical Aspects of Quantum Chaos) HPRN-CT-2000-00103 of the IHP Programme.
Appendix {#appendix .unnumbered}
========
We here show that the distribution of the maximum amplitude $A_i(n,L;v)$ completely determines the value distribution of the eigenfunctions on the $i^{\rm th}$ bond, which is described by $$\label{one}
\frac{1}{N L_i} \sum_{n=1}^N \int_0^{L_i} f\big( \psi_i^{(n)}(x)\big) \, {{\mathrm d}}x$$ where $f$ is an arbitrary bounded continuous function. Let us in fact consider the more general joint distribution, $$\frac{1}{N L_i} \sum_{n=1}^N \int_0^{L_i}
F\big( \cos k_n(x-L_i),v^2 A_i(n,L;v)\big) \, {{\mathrm d}}x$$ where $F$ is a bounded continuous function in two variables. We obtain the expression (\[one\]) for the choice $F(t,\eta)=f(t \sqrt{\eta})$ provided $f$ is even (which, as will become clear below, we may assume w.l.o.g.).
We begin with the special case when $F$ factorizes, i.e., $F(t,\eta)=f_1(t)\,f_2(\eta)$ where $f_1,f_2$ are arbitrary bounded continuous functions. Then $$\begin{aligned}
\nonumber
\frac{1}{L_i} \int_0^{L_i}
f_1\big( \cos k_n(x-L_i)\big) \, {{\mathrm d}}x
&= \int_0^1 f_1\big(\cos(2\pi x)\big)\, {{\mathrm d}}x + {{\mathrm O}}(k_n^{-1}) \\
&= \frac1\pi \int_{-1}^1 f_1(t) \frac{{{\mathrm d}}t}{\sqrt{1-t^2}} + {{\mathrm O}}(k_n^{-1}),
\label{uno}\end{aligned}$$ and, by theorem \[thm:5\], $$\label{due}
\frac{1}{N} \sum_{n=1}^N f_2\big(v^2 A_i(n,L;v)\big)
\to \int_0^\infty f_2(\eta) Q_v(\eta)\, {{\mathrm d}}\eta$$ as $N\to\infty$. Since the mean density of the eigenvalues $k_n$ is constant, we have $ \sum_{n\leq N} {{\mathrm O}}(k_n^{-1}) = {{\mathrm O}}(\log N)$ and thus from (\[uno\]) and (\[due\]) $$\begin{gathered}
\label{final}
\lim_{N\to\infty}\frac{1}{N L_i} \sum_{n=1}^N \int_0^{L_i}
F\big( \cos k_n(x-L_i),v^2 A_i(n,L;v)\big) \, {{\mathrm d}}x \\
=
\frac1\pi \int_{-1}^1 \int_0^\infty F(t,\eta) \,
Q_v(\eta)\, \frac{{{\mathrm d}}\eta\, {{\mathrm d}}t}{\sqrt{1-t^2}} .\end{gathered}$$ This holds for functions $F=f_1\, f_2$ and, by linearity, also for finite linear combinations of such functions. Given any $\epsilon>0$ we can approximate any bounded continuous $F$ from above and below by such finite linear combinations $F_+$ and $F_-$, respectively, such that $$\frac1\pi \int_{-1}^1 \int_0^\infty \big[F_+(t,\eta)-F_-(t,\eta)\big] \,
Q_v(\eta)\, \frac{{{\mathrm d}}\eta\, {{\mathrm d}}t }{\sqrt{1-t^2}} < \epsilon .$$ Since $\epsilon$ can be arbitrarily small, (\[final\]) holds in fact for any bounded continuous $F$.
We can therefore choose $F(t,\eta)=f(t \sqrt{\eta})$ as a test function, and we find $$\lim_{N\to\infty}
\frac{1}{N L_i} \sum_{n=1}^N \int_0^{L_i} f\big( \psi_i^{(n)}(x)\big) \, {{\mathrm d}}x
=\int_{-\infty}^\infty f(r) \, R_v(r)\, {{\mathrm d}}r$$ with the limiting distribution $$R_v(r)=\frac{1}{\pi} \int_{r^2}^{\infty}
Q_v(s) \frac{{{\mathrm d}}s}{\sqrt{s-r^2}}.$$
The limit $v\to\infty$ can be handled in an analogous way and leads to the same formulas for the limit $R(r)$ of $R_v(r)$ with $Q_v(s)$ replaced by $Q(s)$ in the above. It follows from the asymptotic expansion [(\[eq:Q:asympt\])]{} that $R(r)$ also decays with an algebraic tail.
[31]{}
|
---
abstract: 'The ignorability assumption and the overlap condition are key assumptions in causal inference. They are commonly made, but often violated in observational studies. In this paper, we investigate a local version of the ignorability assumption for continuous treatments, where potential outcomes are independent of the treatment assignment only in a neighborhood of the current treatment assignment. Similarly, we introduce a local version of the overlap condition, where the positivity assumption only holds in a neighborhood of observations. Under these local assumptions and a smoothness condition, we show that the effect of shifting a continuous treatment variable by a small amount across the whole population (termed average partial effect or incremental causal effect) is still identifiable, and that the incremental causal effect can be estimated via the average derivative. Moreover, we prove that in certain regression settings, estimating the incremental effect is easier than estimating the average treatment effect in terms of asymptotic variance. In addition, we show that estimation of incremental causal effects is often more robust than estimating average treatment effects if the ignorability assumption is slightly violated. For high-dimensional settings, we develop a simple feature transformation that allows for doubly-robust estimation and doubly-robust inference of incremental causal effects. Finally, we compare the behaviour of estimators of the incremental treatment effect and average treatment effect in experiments including data-inspired simulations.'
author:
- Dominik Rothenhäusler
- Bin Yu
bibliography:
- 'references.bib'
title: Incremental Causal Effects
---
Introduction
============
The estimation of treatment effects has a long history in many disciplines and is of central interest in many data science (both scientific and business) endeavours. Often, data from a randomized experiment is not available and one has to resort to observational data. In this situation, practitioners usually mitigate the problem using regression adjustment, matching, inverse probability weighting or instrumental variables regression.
The overall goal in causal inference is to estimate the effect of intervening on a treatment $T$ on an outcome $Y$. Average treatment effect and conditional average treatment effect are often the quantities of interest in causal inference, where a treatment variable $T$ is set to the same level across a certain population. In the Neyman-Rubin model, if $Y(t)$ denotes the potential outcome of a unit with treatment assignment $t$ [@rubin1974estimating; @splawa1990application], this can be expressed as
$$\mathbb{E}[Y(t')] - \mathbb{E}[ Y(t)], \text{ for some fixed $t,t' \in \mathbb{R}$},$$
where the expectation is taken over a superpopulation. However, there also exist other notions of interventions. For continuous treatment one may want to estimate the effect of shifting a pre-interventional (potentially random) assignment $T$ by an infinitesimal amount across a certain population. Mathematically, this can be expressed as
$$\label{eq:14}
\frac{\mathbb{E}[Y(T+\delta)] - \mathbb{E}[Y(T)]}{\delta} \text{ for deterministic $\delta >0 $ close to zero,}$$
where the expectation is taken over a superpopulation of units. This notion of intervention is much less used in parts of the causal inference community. The effect in equation (\[eq:14\]) corresponds to average partial effects in the econometrics literature, which refer to the effect of an infinitesimal shift in structural equation models [@powell1989semiparametric; @cameron2005microeconometrics; @wooldridge2005unobserved]. More precisely, in the literature sometimes “average partial effect” refers to the causal estimand, sometimes it refers to the functional $\mathbb{E}[\partial_{t} \mathbb{E}[Y|X,T]]$. To distinguish the causal estimand from the functional $\mathbb{E}[\partial_{t} \mathbb{E}[Y|X,T]]$, following @kennedy2018nonparametric, we refer to the estimand in equation as an *incremental* treatment effect.
Practitioners have to decide how to specifically formulate a domain question and which notion of intervention to employ to answer the domain question. Of course, these notions correspond to different domain questions, and the domain question is one of the most important factors for the choice of intervention notion. We would argue that issues of identification, robustness to confounding and difficulty of the estimation task (asymptotic error) should also play important roles in the choice of intervention notion.
As mentioned previously, treatment effects are often estimated under the assumptions of weak ignorability and the overlap condition. However, these assumptions can be unrealistic or hard to justify if the data is observational. We introduce a local ignorability assumption and a local overlap condition, which are weaker than their more general counterparts. Roughly speaking, the local ignorability assumption states that potential outcomes are independent of the current treatment assignment in a neighborhood of observations. It will turn out that incremental treatment effects are identifiable under these assumptions, while average treatment effects are not. To the best of our knowledge, tailor-made assumptions to identify incremental treatment effects for continuous treatments have not been defined previously.
In situations where the practitioner suspects some latent confounding, it would be sensible to employ a notion which is the least sensitive to confounding, if possible. We will see that incremental effects are often more robust under worst-case additive confounding than average treatment effects if the weak ignorability assumption is slightly violated. If the signal-to-noise ratio is low, then estimation of average treatment effects might be highly variable and thus uninformative. We will discuss situations in which incremental treatment effects can be estimated with lower asymptotic error than average treatment effects. This includes situations where the signal to noise ratio is low. Thus, estimating incremental treatment effects is potentially more informative than estimating average treatment effects if incremental treatment effects help solve the domain problem.
Causal inference from observational data is known to be challenging and prone to mistakes @rosenbaum2002, with sometimes devastating effects for human lives. Revisiting the conceptual foundations of the field may help us distinguish situations in which some notions of interventions can be estimated more reliably than others and are thus more informative. In this work, we investigate advantages and disadvantages of incremental treatment effects compared to average treatment effects under the local ignorability assumption and overlap condition. We hope that our work aids practical decisions on choice of intervention notions to answer a specific domain question.
Related work {#sec:relatedwork}
------------
For discrete treatments, @kennedy2018nonparametric defines incremental propensity score interventions by multiplying the odds of receiving treatment. The author shows that the overlap assumption is not necessary for identifying incremental propensity score interventions and develops an efficiency theory with corresponding nonparametric estimators. Asymptotic equivalence of several estimators of the average partial effect in parametric settings has been shown in @stoker1991equivalence. To the best of our knowledge, semiparametric estimation of average derivatives has first been discussed in @powell1989semiparametric. More recently, @chernozhukov2018double and @hirshberg2017augmented derived semiparametrically efficient estimators of linear functionals of the conditional expectation function. In @hirshberg2017augmented, this is achieved under Donsker assumptions, while @chernozhukov2018double employ a sparsity assumption on either the approximation of the Riesz representer or the approximation to the regression function. @hirano2004propensity generalize the unconfoundedness assumption to continuous treatments. This allows to identify the dose-response function using a generalized propensity score. Under the assumption of our framework, the dose-response function is generally not identifiable.
When there exists a binary instrumental variable, under non-compliance average treatment effects are usually not identifiable. In [@angrist1996identification], the authors show that under a monotonicity assumption, interventions on the subgroup of compliers, the so-called local average treatment effect is still identifiable. In fact, estimating the effect of this “weaker” notion of treatment effect in cases where the average treatment effect is not identifiable is increasingly popular in certain parts of the causal inference community [@imbens2010better].
In models based on structure equations, incremental interventions can be seen as a special case of “parametric” or “imperfect” interventions [@Eberhardt2007; @korb2004; @Tian2001; @pearl2014external]. @korb2004 argue that naively estimating (deterministic) causal effects using Bayesian networks can be misleading and discuss different types of indeterministic interventions. In the context of structure learning, @Eberhardt2007 have shown that causal systems of $N$ variables can be identifiable using one parametric intervention. @Tian2001 do structure learning based on local mechanism changes, which are a more general notion of intervention than incremental effects.
Our contribution
----------------
We show that the effect of incremental interventions is identifiable under a local ignorability assumption and a local overlap condition, which can be seen as relaxed versions of their more general counterparts. To our best knowledge, this is the first time that one deals with identifiability issues for incremental treatment effects in the potential outcome framework.
In regression settings, the distribution of the treatment variable given the covariates is Gaussian, using nonlinear basis expansions, we show that incremental subpopulation causal effects can be estimated efficiently with usually lower asymptotic error than average treatment effects. Furthermore, we show that the estimation of incremental causal effect in a regression setting in the population case is usually less sensitive to additive unobserved confounding compared to the estimation of average treatment effects. We propose a feature transformation “incremental effect orthogonalization” that facilitates estimation and inference of high-dimensional incremental causal effects. Our method is based on a feature transformation in the first step and running a de-sparsified lasso on the transformed data. The de-sparsifying technique is similar to the ones developed for high-dimensional linear regression [@zhang2014confidence; @javanmard2014confidence; @van2014asymptotically; @belloni2014inference] but due to the randomness in treatment assignment the asymptotic variance formula differs from existing approaches for inference in high-dimensional regression models. We modify a robust version of the desparsifying approach which was introduced in [@buhlmann2015high]. Our novel technique has a double-robustness property, in the sense that if one of two models is well-specified, we obtain asymptotically valid estimation and inference. The main advantage compared to existing approaches for the estimation of incremental treatment effects is that after a simple feature transformation, off-the-shelf software for estimation in high-dimensional linear models can be used. To substantiate the claims of our theoretical results, we compare the behaviour of estimators of the incremental treatment effect and average treatment effect on simulated data. We also cover simulations settings where the assumptions of some of our theoretical results are violated and discover that the conclusions are fairly robust. The rest of the paper is organized as follows. In Section \[sec:motivation-setup\] we introduce the model class. In Section \[sec:ident-incr-caus\], we discuss identification of the incremental treatment effect under a local ignorability and local overlap condition. In Section \[sec:prop-our-estim\], we discuss models in which estimating incremental treatment effects is more robust under confounding and can be done with lower asymptotic error than estimating average treatment effects. Furthermore, we introduce a feature transformation that facilitates doubly robust estimation and inference in high-dimensional settings. Finally, in Section \[sec:sim\] we validate our theoretical results on simulated data, including set-ups based on real-world data.
Motivation and setup {#sec:motivation-setup}
====================
We are interested in the causal effect of a continuous treatment variable $T$ on an outcome $Y$ in the presence of some covariates $X \in \mathbb{R}^{d}$. We use the Neyman-Rubin potential outcome framework [@rubin1974estimating; @splawa1990application] and assume a super population or distribution $\mathbb{P}$ of $(Y(t)_{t \in \mathbb{R}},T,X)$ from which $N$ independent draws $(Y_{i}(T_{i}),T_{i},X_{i})$ are given, where $Y_{i}(t)$ is the potential outcome of $Y$ under treatment or dose $T=t$. Without any assumptions or adjustment, an observed association between $T$ and $Y$ might simply be due to some confounding variable that affects both the treatment and the outcome. A commonly made assumption in such a setting is weak ignorability [@rosenbaum1983assessing], which states that the treatment assignment is independent of the outcome, conditional on some covariates $X$. Formally, this assumption is often written as $$\label{eq:1}
\{Y(t), t \in \mathcal{T} \} \perp T | X,$$ where $Y(t)$ is defined as the potential outcome of $Y$ under treatment $T=t$ and $\mathcal{T}$ is the set of treatment levels. To avoid issues of measurability, we assume that $Y(t)$ is continuous. In addition, it is common to assume that the overlap condition holds, which can be written as $$\label{eq:globaloverlap}
p(t|x) > 0 \text{ for all $t,x$, }$$ where $p(t|x)$ is the conditional density of $T$ given $X$ and $p(x)$ is the density of $X$. If both weak ignorability and the overlap condition holds, the average treatment effect $\mathbb{E}[Y(t)] - \mathbb{E}[Y(t')]$ for some choice of $t,t' \in \mathcal{T}$ can be estimated via regression, matching, inverse probability weighting or combinations of these methods, see for example @rosenbaum2002. In particular, $$\mathbb{E}[Y(t)] - \mathbb{E}[Y(t')] = \mathbb{E}[\mathbb{E}[Y|X,T=t]]-\mathbb{E}[\mathbb{E}[Y|X,T=t']],$$ where on the right-hand side the outer expectation is taken over the distribution of $X$. In the following we will show that incremental causal effects are identifiable in scenarios where the ignorability assumptions holds locally in subgroups of patients with similar treatments but not necessarily across all patients. In addition, we show that the overlap condition can be weakened as well. Average treatment effects and the dose-response function are generally not identifiable under these assumptions. Throughout the paper we assume that the Stable Unit Treatment Value Assumption [@rubin1980discussion; @hernan2010causal] holds, which says that the potential outcomes are well-defined and that there is no interference between units, i.e. that the potential outcome of one individual is only a function of the treatment assignment to this individual and not of the others. Formally, this assumption can be expressed as $Y_{i}(\mathbf{T})=Y_{i}(T_{i})$ for $\mathbf{T} = (T_{1},\ldots,T_{n})$.
Local ignorability
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In practice there are cases where the weak ignorability assumption in equation might be unrealistic. For example, if $T$ describes the dose of a medication, the patients receiving high doses of the medication may be a completely different population than the patients receiving low doses of the medication, even conditional on the covariates $X$. If $X = \text{``severity of symptoms''}$ and there is a patient that has severe symptoms and receives a a very low dose, this might be due to the doctor making an exceptional decision due to an exceptional circumstance that is not encoded in the data set. This exceptional circumstance might also affect the outcome. It could also be that the patient willingly decides to take only a lower dose than usual. Consequentially, this patient may make other decisions that affect $Y(t)$ that are not encoded in $X$. In the following, we introduce a localized version of the ignorability assumption which allows for some unobserved heterogeneity.
From the ignorability assumption, by conditioning on $T$, the following condition follows immediately. Ignoring measure-theoretic special cases where the condition below is not well-defined, equation is practically equivalent to the assumption below. The reformulation is similar to @hernan2010causal [p. 15].
Assume that $$Y(t) | \{X=x, T=t'\} \stackrel{d}{=} Y(t)| \{ X=x,T=t'' \} \text{ for all $t,t',t''$.}$$
Thus, one way to think about the ignorability assumption is that, conditionally on $X=x$, the distribution of the potential outcomes is the same across patients with different treatment assignments. In the following, to deal with certain violations of equation as discussed in the previous chapter, we will define a local version of this assumption.
\[ass:local-ignorability\] Assume that for almost all $(x,t_0)$ with $p(x,t_0)>0$ $$\label{eq:1-easy}
Y(t) | \{X=x, T=t'\} \stackrel{d}{=} Y(t) | \{ X=x,T=t''\},$$ for all $\max(|t - t_0|,|t'-t_0|,| t'' - t_0|) \le \delta_0$, where $\delta_{0}>0$ can depend on $x$ and $t_0$.
Roughly speaking, we assume that patients are “comparable” (i.e. have the same potential outcome in distribution), if they have a treatment assignment that is sufficiently similar. In other words, we assume that treatment assignments are randomized locally but not necessarily globally. This can happen, for example, if a doctor has discrete groups of patients that he treats differently. Then the treatment assignment might be (approximately) random between patients with similar treatments, but not between patients with very different treatments. An example is given in Figure \[fig:pic\].
![In this example, assume that the weak ignorability assumption holds for each of the patient groups A, B, C and D. In general, conditionally on $X$, the weak ignorability assumption is thus violated for the total patient population. On the other hand, as the boundaries of the groups are a null set, the local ignorability assumption holds.[]{data-label="fig:pic"}](Example.pdf)
Even if the group assignment is not observed (unobserved confounder), the incremental causal effect can be identifiable, as we will see below. Thus, the local ignorability assumption allows for some unobserved heterogeneity.
Local overlap
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Now let us turn to weakening the overlap condition. If the overlap condition is not satisfied, parts of the population are observed with probability zero for some treatment $T=t$. Estimating a causal effect for this part of the population amounts to extrapolating from other parts of the population, which is naturally prone to errors. In practice, the overlap condition is often not satisfied. In the example discussed above, ethical considerations, among others, might prevent doctors to give a very low dose to patients with severe symptoms and a very high dose to patients with minor symptoms. If $X = \text{``severity of symptoms''}$, then by conditioning on a large value $X=x$ we will have only very few patients with low doses, or no patients at all. This makes it exceedingly difficult to estimate the effect of giving a low dose to this group of patients. In finite samples, estimation of the average treatment effects is difficult if there exists regions where assignment variables $t$ has low density and the other treatment assignment $t'$ has high density. This issue can be exacerbated if the covariate vector $X$ is high-dimensional and the data is observational [@d2017overlap]. Roughly speaking, due to the curse of dimensionality, in high-dimensions there will often be regions where one of the treatments has low density. A similar problem appears in practice when trying to match subjects on many covariates. The more covariates we have, the harder it is to find pairs of subjects that are similar in all covariates.
High-dimensional covariates are potentially also challenging for the estimation of incremental effects. However, for the estimation of incremental effects, we can relax the overlap condition. In the following we assume that the density functions $p(t,x)$ and $p(t|x)$ exists.
\[ass:local-overlap\] Assume that $p(t,x)$ and $p(t|x)$ are continuous.
Roughly speaking, we assume that if there is a patient with severity $X=x$ that gets treatment $T=t$ then the probability of another patient with slightly different severity and treatment is nonzero. An example is given in Figure \[fig:local\_overlap\].
![Let the ellipsoid denote the region of positive density, $\{(x,t): p(x,t)>0\}$. In this example, the overlap condition is violated as there are no patients that have strong symptoms $X=x$ but get a low dose $T=t$. On the other hand, the local overlap condition holds if the densities are continuous.[]{data-label="fig:local_overlap"}](local_overlap.pdf)
### Regularity assumption
Here we make an assumption to guarantee that the conditional expectations used in this paper exist. In practice, these assumptions can be thought of as putting a smoothness condition on the potential outcomes.
\[ass:regul-assumpt\] Assume that the potential outcomes $Y(t)$ are bounded and that the derivative $Y'(t) := \partial_{t} Y(t)$ is continuous and bounded.
This assumption can be slightly relaxed. Instead of asking for smoothness of the potential outcomes, we could assume a smoothness condition for the regression surface $\mathbb{E}[Y|X=x,T=t]$. We will work with the regularity condition as defined above for reasons of simplicity.
Identification of incremental causal effects {#sec:ident-incr-caus}
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Note that if we only have the local ignorability assumption or the local overlap condition, it is generally not possible to consistently estimate (or identify) the average treatment effect $$\mathbb{E}[Y(t)] - \mathbb{E}[Y(t')] \text{ for some $t,t' \in \mathbb{R}$.}$$ However, we will now show that the effect of shifting $t$ by a small amount around the current treatment can still be identifiable. In the following, we assume that Assumption \[ass:regul-assumpt\] holds, i.e. we make the assumption that $Y(t)$ is bounded and continuously differentiable in $t$ with bounded derivative. Together with the local ignorability assumption and the local overlap condition, it implies that $\mathbb{E}[Y|X=x,T=t]$ exists and is differentiable in $t$ for all $(x,t)$ with $p(x,t)>0$. The short proof of the following result can be found in Section \[sec:proofmain\] in the Appendix.
\[prop:ident\] If Assumption \[ass:local-ignorability\] (local ignorability), Assumption \[ass:local-overlap\] (local overlap) and Assumption \[ass:regul-assumpt\] (regularity) are satisfied, then for almost all $(x,t)$ with $p(x,t)>0$, $$\begin{aligned}
\begin{split}
\mathbb{E}[ Y'(t) |X=x,T=t] = \partial_t \mathbb{E}[Y|X=x,T=t].
\end{split}\end{aligned}$$
Note that Assumption \[ass:regul-assumpt\], which assumes that $Y(t)$ is bounded and differentiable with bounded derivative, is made for expositional clarity and can be weakened slightly. In the following we have to differentiate between the incremental effect for the superpopulation and the incremental effect for the sample. In the latter case, all statements are conditional on the units that we observe. More information for the difference between population causal effects and finite sample causal effects can be found in [@imbens2015causal], Section 1. More specifically, we will investigate estimation and inference for the expected effect of an infinitesimal shift intervention for a finite sample $(y_{i},t_i,x_i)$, $i=1,\ldots,n$, $$\label{eq:globsa}
\theta_\text{fs} := \frac{1}{n}\sum_{i=1}^n \mathbb{E}[ Y'(t_{i})|X=x_i,T=t_i],
$$ where the expectation is taken with respect to the potential outcomes $Y$ given $X=x$ and $T=t$. We allow for randomness in $Y$ after conditioning on $X=x$ and $T=t$. This can account for within-subject variability of treatment effects. For example, for a given subject we allow the treatment effect to vary across time. In words, $\theta_{\text{fs}}$ corresponds to the expected effect of an incremental intervention conditionally on $(t_{i},x_{i})_{i=1,\ldots,n}$. We condition on the $(t_{i})_{i=1,\ldots,n}$ as we aim to estimate the effect of slightly shifting the current treatment assignments $(t_{i})_{i=1,\ldots,n}$. Note that we do not condition on $(y_{i})_{i=1,\ldots,n}$ as is sometimes done in the causal inference literature for binary treatments [@imbens2015causal]. Conditioning on the $y_{i}$, in addition to conditioning on the $t_{i}$ and $x_{i}$, would result in a deterministic sample and is thus not practical. For $n \rightarrow \infty$, under some regularity conditions and a super population model, $\theta_\text{fs}$ will converge to the super population effect, $$\label{eq:glob}
\theta_\text{sp} := \mathbb{E}[Y'(T)].
$$ Here, the expectation is taken over $T$, $X$ and $Y$. In words, $\theta_{\text{sp}}$ corresponds to the expected effect of an incremental intervention on the superpopulation. We want to emphasize that the interpretation of these causal effects is different from the most common notion of interventions, so-called average treatment effects, surgical interventions or do-interventions. The population incremental causal effect answers the question: “How will the average outcome change if we change the treatment of all patients by a small amount, i.e. use treatment assignment $T' = T + \delta$ across all patients” for some $\delta$ close to zero. Note that the quantities of interest are in general also different from $$\mathbb{E}[Y'(t)],$$ which corresponds to first setting the value of $T$ to $t$ across the whole population and than varying that intervention by a small amount. In the next section we will discuss general properties of estimation and inference of the effect in equation and equation .
Estimating incremental effects in regression settings {#sec:prop-our-estim}
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In this section we discuss various aspects of estimation and inference for incremental causal effects for the sample incremental effect $\theta_{\text{fs}}$ and the population incremental effect $\theta_{\text{sp}}$. In Section \[sec:smaller\] we discuss regression settings for which the error of efficiently estimating the effect of shift interventions is lower than the error of efficiently estimating average treatment effects. This is achieved if the conditional distribution $p(t|x)$ is Gaussian. In Section \[sec:sens-addit-conf\] we show that estimation of incremental interventions is usually less sensitive to worst-case confounding than estimation of average treatment effects. In Section \[sec:high-dimens-conf\] we discuss a feature transformation that facilitates obtaining confidence statements in high-dimensional scenarios. The main advantage compared to existing approaches [@powell1989semiparametric; @hirshberg2017augmented; @chernozhukov2018double] is that after a simple feature transformation, off-the-shelf software for estimation and inference in high-dimensional linear models can be used.
Variance comparison {#sec:smaller}
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In this section we will see that in some standard regression settings, the squared error of estimating the sample average treatment effect is often larger than the squared error of estimating the sample effect of incremental interventions for regression estimators. We assume that we observe $n$ i.i.d. observations $(x_i,t_i,y_i)$, $i=1,\ldots,n$, of distribution $\mathbb{P}$ that follow the additive noise model $$\label{eq:model}
Y = f^0(T,X) + \epsilon,$$ where $\epsilon$ is independent of $T$ and $X$, $\mathbb{E}[\epsilon]=0$ and $\text{Var}(\epsilon)= \sigma_\epsilon^2 > 0$. We assume that $f^{0}$ is differentiable in $t$, i.e. we make a smoothness assumption on how the treatment affects the outcome. As an example, $y$ could be the math score of a student in a test, $t$ the study time after class and $x$ pre-treatment covariates such as age and gender. $\epsilon$ could be the influence of some other unmeasured factors that are independent of $X$ and $T$ but have an influence on the math score. In the easiest case, the function $f^{0}$ is linear in its arguments, but often there will be interactions between the pre-treatment covariates and study time after class. If $x = \text{``doing at least two hours of sports per week''}$, then the effect of studying on the outcome $y$ might be stronger if the student does enough sports. Under the assumptions of Proposition \[prop:ident\], we have $\theta_\text{fs} = \frac{1}{n} \sum_{i=1}^n \partial_t f^0 (t_i,x_i)$. We can estimate $\theta_\text{fs}$ via a two-stage procedure in non-parametric settings. First, choose differentiable basis functions $b_{1},\ldots, b_{p}$. These functions can for example include linear terms, polynomials, radial basis functions or wavelets. Ideally, the choice of basis functions is guided by domain knowledge. In cases where domain knowledge is not available, we recommend using linear and quadratic terms. First, we solve the least-squares problem $$\hat \beta = \arg \min_{\beta} \sum_{i=1}^n \left(y_i - \sum_{k=1}^{p}b_{k}(t_{i},x_i) \beta_{k} \right)^2.$$ Secondly, we estimate the derivative via $\hat \theta = \frac{1}{n} \sum_i \sum_{k} \hat \beta_{k} \partial_t b_{k}(t_{i},x_i)$. For some $t' \neq t$, we compare the performance of this estimator with a naive estimator $$\label{eq:defhattheta}
\hat \tau(t,t') = \frac{1}{n}\sum_{i=1}^n \frac{\hat f(t,x_i) - \hat f(t',x_i)}{t-t'},$$ where $\hat{f} = \sum_{k} \hat \beta_{k} b_{k}(t_{i},x_{i})$. Under weak ignorability, this is an estimator of the (normalized) sample average treatment effect $$\begin{aligned}
\tau_\text{fs}(t,t') := \frac{1}{n} \sum_{i=1}^n \frac{\mathbb{E}[Y(t)|X=x_i,T=t] - \mathbb{E}[Y(t')|X=x_i,T=t']}{t-t'}.\end{aligned}$$ We normalize the average treatment effect such that if the treatment effect is linear, $\tau_{\text{fs}}$ and $\theta_{\text{fs}}$ agree. Define $\mathcal{B}$ as the linear span of $b_{1},\ldots,b_{p}$. Write $\textbf{X}_{j;i} = b_{j}(t_{i},x_{i})$. Here and in the following, to be able to use partial integration we tacitly assume that $b_{k}(t,x) p(t|x) \rightarrow 0$ and $\partial_{t} b_{k}(t,x) p(t|x) \rightarrow 0$ for fixed $x$ and $ |t| \rightarrow \infty$.
\[thm:variance\_comp\] Assume that the data $(y_i,t_i,x_i)$, $i=1,\ldots,n$ are i.i.d. and follow the model in equation . Furthermore, assume that $ \partial_t \log p(t,x) \in \mathcal{B}$ and $f^{0} \in \mathcal{B}$. Let $b_{1}(t,x),\ldots,b_{p}(t,x)$ be differentiable and let $(b_k(T,X))_{k=1,\ldots,p}$ and $(\partial_{t} b_{k}(T,X))_{k=1,\ldots,p}$ have finite second moments. If the conditional distribution $p(t|x)$ suffices $\partial_{t}^{2} \log p(t|x) \equiv 0$ for all $t,x$, then for all $t,t' \in \mathbb{R}$, $$\limsup_{n \rightarrow \infty} \frac{\mathbb{E}[(\hat \theta - \theta_{\text{fs}})^{2}|\mathcal{D}_{\text{feat}}]}{\mathbb{E}[(\hat \tau - \tau_{\text{fs}})^{2}|\mathcal{D}_{\text{feat}}]} \le 1,$$ where $\mathcal{D}_{\text{feat}} = \{(x_{i},t_{i}),i=1,\ldots,n\}$.
The theorem implies that the simple plug-in estimator $\hat \theta$ has asymptotically lower error or the same error as $\hat \tau$ if $\mathcal{B}$ is large enough and $p(t|x)$ is Gaussian. The difference in asymptotic variance can be drastic, as we will see in the simulation section. Hence, the concept of incremental interventions might be helpful in situations where the signal-to-noise ratio is too low for drawing any conclusions from $\hat \tau$.
The estimators above are both efficient for estimating their respective target quantities within the class of unbiased and linear estimators [@van2000asymptotic]. In this sense, the result above shows that in low-dimensional scenarios, estimating sample incremental causal effects $\theta_{\text{fs}}$ is easier in terms of optimal asymptotic variance. Thus, if the domain problem is adequately addressed in the incremental causal effect formulation, estimating incremental causal effects might be more informative than estimating average treatment effects. We will discuss scenarios in which estimation of average treatment effects results in lower asymptotic variance than estimating incremental treatment effects in Section \[sec:sim\]. While the theoretical result above makes relatively strong assumptions on the distribution of $T$, in practice the effect seems fairly robust.
Sensitivity to additive confounding {#sec:sens-addit-conf}
-----------------------------------
In this section we discuss how estimation of average treatment effects and incremental causal effects behaves in the super-population case if the ignorability assumption is slightly violated. Analysing the sensitivity of causal effects with respect to violations of assumptions plays a central role in observational studies, see for example @cornfield1959smoking [@rosenbaum1983assessing; @rosenbaum2002]. Roughly speaking, we will see that estimation of incremental treatment effects is relatively little affected by confounding if the variation of the treatment variable $T$ is high compared to the variation of confounding $H$. In addition, we will show that average treatment effects are usually more sensitive to worst-case additive confounding than incremental causal effects in this setting. Before we proceed, we need some additional notation. Let $\mathbb{P}_{\text{unconf}}$ denote the distribution of $(Y,T,X)$ and assume that the weak ignorability and overlap condition holds under $\mathbb{P}_{\text{unconf}}$. Then, the (relative) average treatment effect is identifiable via $\tau_{\text{sp}} = \tau(\mathbb{P}_{\text{unconf}})$, where $$\tau(\mathbb{P}) = \frac{\mathbb{E}[\mathbb{E}[Y|X=X,T=t]] -\mathbb{E}[\mathbb{E}[Y|X=X,T=t']]}{t-t'},$$ for some $t \neq t'$. Analogously, the incremental effect can be identified via $\theta_{\text{sp}} = \theta(\mathbb{P}_{\text{unconf}})$, where $$\theta(\mathbb{P}) = \mathbb{E}[ - \partial_{t} \log p \cdot Y] = \mathbb{E}[\partial_t \mathbb{E}[Y|X,T]].$$ In the following we investigate how the functionals $\tau(\cdot)$ and $\theta(\cdot)$ behave under additive confounding.
We assume that there is some additive confounding $H$ in $Y$, i.e. we assume that $(Y,T,X)$ has the same distribution under $\mathbb{P}_{\text{conf}}$ as $(Y+ H, T,X)$ under $\mathbb{P}_{\text{unconf}}$. Let $\mathcal{P}$ be the set of distributions $\mathbb{P}_{\text{conf}}$ for which the second moment of $H$ is bounded by $\epsilon$. For all $\epsilon >0$, define $$\begin{aligned}
\mathrm{sens}_{\text{ATE}}(\epsilon) &= \max_{ \mathbb{P}_{\text{conf}} \in \mathcal{P}} |\tau(\mathbb{P}_{\text{unconf}}) - \tau(\mathbb{P}_{\text{conf}}) |, \text{ and }\\
\mathrm{sens}_{\text{incr}}(\epsilon) &= \max_{ \mathbb{P}_{\text{conf}} \in \mathcal{P}} |\theta(\mathbb{P}_{\text{unconf}}) - \theta(\mathbb{P}_{\text{conf}}) |.
\end{aligned}$$ In words, $\mathrm{sens}_{\text{ATE}}(\epsilon)$ and $\mathrm{sens}_{\text{incr}}(\epsilon)$ quantify how robust the identification strategies $\theta(\mathbb{P})$ and $\tau(\mathbb{P})$ are under slight violations of the ignorability assumption. The proof of the following result can be found in the Appendix.
\[thm:robustn-addit-conf\] For all $\epsilon >0$ we have $$\begin{aligned}
\mathrm{sens}_{\mathrm{incr}}(\epsilon) &= \sqrt{\epsilon \mathbb{E}[-\partial_{t}^{2} \log p(X,T) ]},\\
\mathrm{sens}_{\mathrm{ATE}}(\epsilon) &= \infty.\end{aligned}$$
As an example, if $p(t|x)$ is Gaussian with variance $\sigma^{2}$, then $\mathrm{sens}_{\text{incr}}(\epsilon) = \frac{\sqrt{\epsilon}}{\sigma}$. Intuitively, if there is a lot of variation in $T$ compared to the variation of $H$, then estimation of the incremental causal effect is relatively little affected. This result shows that estimating the average treatment effect from observational data using the ignorability assumption is oftentimes less robust to confounding than estimating the effect of incremental interventions if the ignorability assumption is slightly violated. Of course, this result is conservative as we are dealing with worst-case additive confounding in an $\epsilon$-ball around the unconfounded distribution $\mathbb{P}_{\text{unconf}}$. Robustness of regression estimators of incremental treatment effects and the average treatment effect in a simulation setting where the ignorability assumption is violated can be found in Section \[sec:conf\].
Doubly-robust estimation and confidence intervals under sparsity {#sec:high-dimens-conf}
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In this section we describe a simple procedure to derive asymptotically valid confidence intervals using the lasso. “Doubly robust” is meant in the sense that the method yields asymptotically valid confidence intervals if the function class $\mathcal{B}$ either contains $\partial_{t} \log p(t,x)$ or $f^{0}$. The method we describe is based on the method on page 6 in [@buhlmann2015high]. However, we will transform the features in a pre-processing step. This transformation will depend on the observations $(t_{i},x_{i})$ and change for every $n$, which adds some complexity. The benefit of this pre-processing step is to reduce the problem of estimating incremental treatment effects to a problem of estimating a single component in a high-dimensional, potentially misspecified linear model. As a result, existing software such as the `R`-package `hdi` [@hdi] can be used to efficiently estimate incremental treatment effects. Of course, the results below also extend to the low-dimensional case, i.e. to the case where the number of features $p$ is fixed and the number of observations $n$ goes to infinity.
As above, consider basis functions $b_{1}(t,x),\ldots,b_{p}(t,x)$ of $\mathcal{B}$ which are potentially non-linear. To make notation simpler without loss of generality we assume that $b_{1}(t,x)=t$. Assume we have an i.i.d. sample $(y_{i},t_{i},x_{i})$, $i=1,\ldots,n$, and define the feature matrix $\mathbf{X}$ via $\mathbf{X}_{k;i} = b_{k}(t_{i},x_{i})$, the target vector $\mathbf{Y} = (y_{1},\ldots,y_{n})^{\intercal}$ and the transformed feature matrix $\tilde{\mathbf{X}}$ via $$\tilde{\mathbf{X}}_{k;i} = \begin{cases} t_{i} & \text{ for $k=1$,} \\
b_{k}(t_{i},x_{i}) - t_{i} \frac{1}{n} \sum_{i=1}^{n} \partial_{t} b_{k}(t_{i},x_{i}) & \text{ for $k>1$}.
\end{cases}$$ This can be thought of as an orthogonalization step. Thus, we call this technique “incremental effect orthogonalization”. By construction, for $k>1$ the features $(\tilde{\mathbf{X}}_{k;i})_{i=1,\ldots,n}$ are asymptotically uncorrelated of the Riesz representer of the average partial effect, $ (-\partial_{t} \log p(t_{i},x_{i}))_{i=1,\ldots,n}$. Using implicit or explicit orthogonality properties is common in semiparametric approaches. For general functionals the orthogonalization step can be more involved, see for example @chernozhukov2018double. The intuition behind the orthogonalization step is as follows: If we define the transformed functions $\tilde b_{k} = b_{k}(t,x) - t \alpha_{k}$ with $\alpha_{k} = \frac{1}{n} \sum_{i=1}^{n} \partial_{t} b_{k}(t_{i},x_{i})$ for $k > 1$ and $\tilde b_{k} = t$ for $k=1$, then also $\tilde b_{1},\ldots,\tilde b_{p}$ is a basis of $\mathcal{B}$. Hence, if $f^{0} \in \mathcal{B}$, we can write it as $f^{0} = \sum_{k} \beta_{k}^{0} \tilde b_{k}$. And in particular, we have the average derivative $$\theta_{\text{fs}} = \frac{1}{n} \sum_{i=1}^{n} \partial_{t} f^{0}(t_{i},x_{i}) = \sum_{k=1}^{p} \beta_{k}^{0} \frac{1}{n} \sum_{i=1}^{n} \partial_{t} \tilde b_{k} = \beta_{1}^{0}.$$ The following proposition is a direct result of the observation above.
Assume that $b_{1},\ldots,b_{p}$ is a basis of the function class $\mathcal{B}$. Assume that $Y = f^{0}(T,X) + \epsilon$ for $\epsilon$ independent of $(T,X)$ and that $f^{0} \in \mathcal{B}$. Let $(y_{i},t_{i},x_{i},\epsilon_{i})_{i=1,\ldots,n}$ be i.i.d. with the same distribution as $(Y,T,X,\epsilon)$ and denote $\mathbb{E}_{\epsilon}$ the expectation with respect to the $\epsilon_{i}$, $i=1,\ldots,n$. Then, $$\theta_{\text{fs}} = \beta_{1}^{0},$$ where $$\beta^{0} \in \arg \min_{\beta} \mathbb{E}_{\epsilon} \left[ \frac{1}{n} \sum_{i=1}^{n} (y_{i} - \sum_{k} \tilde b_{k}(t_{i},x_{i}) \beta_{k})^{2} \right].$$
Thus, we have transformed the problem of estimation and inference for subpopulation or sample incremental causal effects $\theta_{\text{fs}}$ to the problem of doing estimation and inference of one component in a (potentially high-dimensional) linear model. Several approaches have been developed to do inference in high-dimensional linear models [@memepb09; @liu2013asymptotic; @zhang2014confidence; @javanmard2014confidence; @van2014asymptotically; @belloni2014inference]. However, using one of these methods with the transformed data $(\tilde{\mathbf{X}},\mathbf{Y})$ does not guarantee that we still have asymptotically valid inference if the Riesz representer $\partial_{t} \log p(t,x) \in \mathcal{B}$, but $f^{0} \not \in \mathcal{B}$ for $f^{0} := \mathbb{E}[Y|X=x,T=t] $. In other words, this approach does not automatically guarantee a double robustness property if the outcome model is misspecified. In the following we discuss how to obtain double robustness properties for estimating the super population effect $\theta_{sp}$. Compared to existing approaches [@powell1989semiparametric; @hirshberg2017augmented; @chernozhukov2018double], the main difference is that the orthogonalization is done in a simple pre-processing step which allows us to rely on commonly used Lasso software for high-dimensional regression models.
Define the two Lasso estimators $$\begin{aligned}
\hat \gamma &:= \arg \min_{\gamma} \| \mathbf{X}_{1} - \tilde{\mathbf{X}}_{-1} \gamma \|_{2}^{2}/n + 2 \lambda_{X} \| \gamma \|_{1}, \\
\hat \beta &:= \arg \min_{\beta} \| \mathbf{Y} - \tilde{\mathbf{X}} \beta \|_{2}^{2}/n + 2 \lambda \| \beta \|_{1}.\end{aligned}$$ Furthermore, define the desparsified estimator for the first component $$\begin{aligned}
\hat \beta_{1}^{\text{despar}} &= \frac{\tilde{\mathbf{Z}}^{\intercal} \mathbf{Y}}{\tilde{\mathbf{Z}}^{\intercal} \mathbf{X}_{1}} - \sum_{k > 1}\frac{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{k}}{\tilde{\mathbf{Z}}^{\intercal} \mathbf{X}_{1}} \hat \beta_{k}, \text{ where }\\
\tilde{\mathbf{Z}} &= \mathbf{X}_{1} - \tilde{\mathbf{X}}_{-1} \hat \gamma.
\end{aligned}$$ Note that this desparsified estimator is defined analogously as in the regression literature [@zhang2014confidence; @javanmard2014confidence; @van2014asymptotically; @belloni2014inference]. To formulate the assumptions and the result, we need some additional notation. **Notation.** Define $$\begin{aligned}
\tilde{\mathbf{X}}_{k;i}^{0}&= \begin{cases} t_{i} & \text{ for $k=1$}, \\ b_{k}(t_{i},x_{i}) - t_{i} \mathbb{E}[ \partial_{t} b_{k}(t_{1},x_{1})], & \text{ for } k > 1, \end{cases} \\
\gamma^{0} &= \arg \min_{\gamma} \mathbb{E}[\| \mathbf{X}_{1} - \tilde{\mathbf{X}}_{-1}^{0} \gamma \|_{2}^{2}], \\
\beta^{0} &= \arg \min_{\beta} \mathbb{E}[\| \mathbf{Y} - \tilde{\mathbf{X}}^{0} \beta \|_{2}^{2}], \\
\tilde{\mathbf{Z}}^{0} &= \mathbf{X}_{1} -\tilde{\mathbf{X}}_{-1}^{0} \gamma^{0}, \\
\hat \epsilon &= \mathbf{Y} - \tilde{\mathbf{X}} \hat \beta, \\
\epsilon &= \mathbf{Y} - \tilde{\mathbf{X}}^{0} \beta^{0}. \\
$$ **Assumptions.** As mentioned before, our work builds on [@buhlmann2015high]. In comparison to their work, the main difference in terms of assumptions is that we added assumption (A8), see below. As in our setting $\log(p)/n \rightarrow 0$, assumption (A8) means that the $\ell_{1}$-norm of the population regression coefficient $\beta^{0}$ is bounded and that the $\ell_{1}$ norm of $\gamma^{0}$ grows slower than $\sqrt{n/\log(p)}$. Thus, we consider the added assumption (A8) as rather weak. Viewed in total, the assumptions are strong and in particular require that the nonlinear functions $b_{k}$ and $\partial_{t} b_{k}$ and the error terms are bounded.
1. $ \mathbb{E}[(\tilde{\mathbf{X}}^{0})^{\intercal} \tilde{\mathbf{X}}^{0}]/n$ has smallest eigenvalue lower bounded by $C_{1} >0$.
2. We assume that there exists a constant $C_{2}$ such that $\mathbb{P}[|b_{k}(t_{i},x_{i})| > C_{2}] = 0$ and the $\mathbb{P}[|\partial_{t}b_{k}(t_{i},x_{i})|>C_{2}] = 0$ for all $k$.
3. $ \| \tilde{\mathbf{Z}}^{0} \|_{\infty} \le C_{3} < \infty$.
4. $s_{1} = |\{k : \gamma_{k}^{0} \neq 0\}| = o(\sqrt{n}/\log(p))$
5. $s_{0} = |\{k : \beta_{k}^{0} \neq 0\}| = o(\sqrt{n}/\log(p))$
6. The normalized asymptotic error is bounded from below: for $$\begin{aligned}
u^{2} = \text{Var} \left( \frac{\epsilon_{1} \tilde{\mathbf{Z}}_{1}^{0}}{\mathbb{E}[\tilde{\mathbf{Z}}_{1}^{0} \tilde{\mathbf{X}}_{11}^{0}]
} + \sum_{k} \partial_{t} b_{k}(t_{1},x_{1}) \beta_{k}^{0} \right),
\end{aligned}$$ we have $u^{2} \ge C_{4} > 0$.
7. The error is bounded $\|\epsilon\|_{\infty} \le V$.
8. $\| \beta^{0} \|_{1}$ is bounded by a constant and $\|\gamma^{0}\|_{1} = o(\sqrt{n/\log(p)})$.
The assumptions (A5) and (A7) can be relaxed. For details, see [@buhlmann2015high]. Now let us turn to the result.
\[thm:doubly-robust-conf\] Let $(y_{i},x_{i},t_{i})$, $i=1,\ldots,n$ be i.i.d. and assume that (A1)–(A8) holds. Then, for $\lambda_{X} = D_{2} \sqrt{\log(p)/n}$ and $\lambda = D_{1} \sqrt{\log(p)/n}$ with constants $D_{1}, D_{2}$ sufficiently large and $\sqrt{\log(p)/n} \rightarrow 0$, $$\label{eq:16}
\frac{\sqrt{n} (\hat \beta_{1}^{\text{despar}} - \beta_{1}^{0})}{\hat u} \rightharpoonup \mathcal{N}(0,1),$$ where $\hat u^{2}$ is the empirical variance of $$\begin{aligned}
\frac{\hat \epsilon_{i} \tilde{\mathbf{Z}}_{i}}{(\tilde{\mathbf{Z}})^{\intercal} \tilde{\mathbf{X}}_{1}/n
} + \sum_{k} \partial_{t} b_{k}(t_{i},x_{i}) \hat \beta_{k} , \text{ for $i=1,\ldots,n$.}
\end{aligned}$$
Note that we neither assumed $f^{0} \in \mathcal{B}$ nor $\partial_{t} \log p(t,x) \in \mathcal{B}$. The variance has two components. Loosely speaking, one part of the variance is induced by the randomness in the treatment assignment $T$, the other component comes from the randomness in $\epsilon$. The following result shows under which conditions we have $\beta_{1}^{0}=\mathbb{E}[\partial_{t} \mathbb{E}[Y|X,T]]$. The result is a variation of well-known results for doubly robust estimation of causal parameters, see for example @bang2005doubly. For reasons of completeness, we include a proof in the Appendix.
\[lem:trick\] If $f^{0} = \mathbb{E}[Y|X=x,T=t] \in \mathcal{B}$ or $\partial_{t} \log p(t|x) \in \mathcal{B}$, then $\beta_{1}^{0} = \mathbb{E}[\partial_{t} \mathbb{E}[Y|X,T]]$.
This shows that estimation and inference is doubly robust: if one of the two functions $f^{0}$ or $\partial_{t} \log p$ lie in $\mathcal{B}$, we obtain consistency for the average partial effect, i.e. $\hat \beta_{1}^{\text{despar}} \rightarrow \mathbb{E}[\partial_{t} \mathbb{E}[Y|X,T]]$ with asymptotically valid confidence intervals. If both $f^{0} \in \mathcal{B}$ and $\partial_{t} \log p(t|x) \in \mathcal{B}$ then the proposed estimator reaches the semiparametric efficiency bound, $$\text{Var}(\partial_{t} f^{0}) + \text{Var}(\epsilon) \mathbb{E}[(\partial_{t} \log p(T|X))^{2}].$$ A proof of this result can be found in the Appendix, Lemma \[lem:semieffic\].
Simulations {#sec:sim}
===========
In this section, we validate our theoretical results on simulated data, including settings where the assumptions of our theoretical results are violated and set-ups based on real-world data. In Section \[sec:synthetic-data-set\] we discuss a very simple low-dimensional model, where the features are drawn from a Gaussian or $t$-distribution. In the first setting, the assumptions of Theorem \[thm:variance\_comp\] are satisfied, whereas in the second setting the assumptions of Theorem \[thm:variance\_comp\] are violated. In Section \[sec:enhancer-data-set\], to increase the realism of our simulation study, the covariates are taken from the enhancer data set. In Section \[sec:conf\] we investigate robustness of incremental treatment effects under the aforementioned low-dimensional model under varying additive confounding. Finally, in Section \[sec:opp\] we discuss a setting in which it is challenging to estimate incremental treatment effects. The results indicate that the conclusions of Theorem \[thm:variance\_comp\] are relatively robust under violations of the assumptions and that estimating incremental effects can be challenging if the error variance is large at the edge of the observation space.
Synthetic data set {#sec:synthetic-data-set}
------------------
In this section we compare estimation and statistical inference for the sample incremental causal effects $\theta_\text{fs}$ and the (relative) sample average treatment effect $\tau_\text{fs}$ as defined in Section \[sec:smaller\] in two simple scenarios. The first setting was chosen such that the assumptions of Theorem \[thm:variance\_comp\] are satisfied, whereas the second setting was chosen such that the assumptions of Theorem \[thm:variance\_comp\] are violated. In both cases, the regression surfaces were chosen such that they exhibit moderate curvature across the observation space. In the first scenario we generate $n$ i.i.d. observations according to the following equations: $$\begin{aligned}
\begin{split} \label{sim1}
t & \sim \mathcal{N}(0,1)\\
\epsilon &\sim \text{Unif}(0,1) \\
y & = 3t + t^2 + \epsilon
\end{split}\end{aligned}$$ We fit a cubic model $y \sim t + t^2 + t^{3}$ using ordinary least squares and then we use the simple plug-in estimators $\hat \theta$ to estimate $\theta_\text{fs}$ and $\hat \tau$ to estimate $\tau_\text{fs}$ with $t=\frac{1}{2}$, $t'=-\frac{1}{2}$ as defined in Section \[sec:smaller\].
As we are fitting a linear model, asymptotically valid confidence intervals for both the incremental treatment effect and the average treatment effect can be readily computed using standard formulae. Coverage, average length of the resulting confidence intervals and root mean-squared error for varying sample size $n$ are given in Table \[tab:1\] below. Note that the model is well-specified, thus we expect asymptotically correct coverage for both estimators for their specific target quantities.
n=10 n=20 n=50
------------------------- ----------------- ----------------- -----------------
CI coverage sample incr 0.97 $\pm$ 0.01 0.96 $\pm$ 0.01 0.95 $\pm$ 0.01
CI coverage sample ATE 0.94 $\pm$ 0.01 0.94 $\pm$ 0.02 0.95 $\pm$ 0.01
CI length sample incr 1.01 $\pm$ 0.04 0.4 $\pm$ 0.01 0.19 $\pm$ 0
CI length sample ATE 1.27 $\pm$ 0.07 0.57 $\pm$ 0.01 0.29 $\pm$ 0
RMSE sample incr 0.24 $\pm$ 0.02 0.1 $\pm$ 0 0.05 $\pm$ 0
RMSE sample ATE 0.28 $\pm$ 0.03 0.14 $\pm$ 0.01 0.07 $\pm$ 0
: \[tab:1\]Coverage of confidence intervals, length of confidence intervals and root mean-squared error for estimating sample incremental treatment effects and sample average treatment effects using a cubic model. The data is generated according to equation .
In both cases, coverage is approximately correct. However, note that the average length of confidence intervals for the effect of the incremental effect is smaller than the respective length for the average treatment effect. This is expected and in line with our theory as the assumptions for Theorem \[thm:variance\_comp\] hold. Now let us turn to a case where the model is misspecified. We generate $n$ samples according to the equations below. $$\begin{aligned}
\begin{split} \label{sim2}
t & \sim t_{4} \\
\epsilon &\sim \text{Unif}(-1,1)\\
y &= 3t + t^2 + t^3 + |t| + \epsilon,
\end{split} \end{aligned}$$ where $t_4$ denotes a $t$-distribution with $4$ degrees of freedom. Then, we fit a cubic model $y \sim t + t^2 + t^{3}$ using ordinary least squares and then use simple plug-in estimators for the sample incremental effect and the average treatment effect as above. However, in this case, neither of the confidence intervals are asymptotically valid due to model misspecification. Coverage, average length of the (asymptotically invalid) confidence intervals and root mean-square error are given in Table \[tab:sec\].
n=10 n=20 n=50
------------------------- ----------------- ----------------- -----------------
CI coverage sample incr 0.82 $\pm$ 0.02 0.72 $\pm$ 0.03 0.62 $\pm$ 0.03
CI coverage sample ATE 0.9 $\pm$ 0.02 0.74 $\pm$ 0.03 0.25 $\pm$ 0.03
CI length sample incr 0.81 $\pm$ 0.03 0.32 $\pm$ 0.01 0.17 $\pm$ 0
CI length sample ATE 1.08 $\pm$ 0.04 0.48 $\pm$ 0.01 0.25 $\pm$ 0
RMSE sample incr 0.24 $\pm$ 0.01 0.14 $\pm$ 0.01 0.09 $\pm$ 0
RMSE sample ATE 0.27 $\pm$ 0.02 0.2 $\pm$ 0.01 0.26 $\pm$ 0.01
: Coverage of confidence intervals, length of confidence intervals and root mean-squared error for estimating sample incremental treatment effects and sample average treatment effects using a cubic model. The data is generated according to equation .[]{data-label="tab:sec"}
Notably, while both confidence intervals have asymptotically incorrect coverage, incremental causal effects have lower root mean-squared error and lower asymptotic variance. Thus, while the assumptions for Theorem \[thm:variance\_comp\] do not hold in this case, estimating incremental effects is still easier in terms of asymptotic mean squared-error for our choices of estimators. Of course, there exist also scenarios in which it is harder to estimate incremental than average treatment effects. This is further discussed in Section \[sec:opp\].
Simulations based on an enhancer data set {#sec:enhancer-data-set}
-----------------------------------------
We aim to investigate estimation of the average derivative using the transformation proposed in Section \[sec:high-dimens-conf\]. To increase the realism of our simulation study, we consider features from a real-world data set. More specificaly, we consider the activity of $36$ transcription factors in Drosophila embryos on $n=7809$ segments of the genome [@li2008transcription; @macarthur2009developmental]. As the features are heavy-tailed, a square-root transform was performed. The activity of the transcription factors is obtained using the following approach: A transcription-specific antibody is used to filter segments of DNA from the embryo. The filtered segments are measured using microarrays and mapped back to the genome, resulting in a genome-wide map of DNA binding for each transcription factor. Then, $n=7809$ segments of the genome are selected based on background knowledge about enhancer activity. The main effects and interactions of transcription factors form a $7809 \times 666$-dimensional feature matrix $\mathbf{X}$. We consider two simulation settings which differ in the way the vector $\beta^{0}$ is formed. In one case, the non-zero entries of $\beta^{0}$ are constant $1$, whereas in the other case the non-zero entries of $\beta^{0}$ are sampled from an exponential distribution:
**Exponential $\beta^{0}$:** As a feature vector, we randomly select a subset of size eight of the main effects and interactions of the transcription factors, $S \subset \{1,\ldots,666\}$. For each of the effects $k \in S \cup \{1\}$, we draw $\beta_{k}^{0} \sim \exp(\lambda)$ with $\lambda = \sqrt{10}$ and $\beta_{k}^{0} = 0$ otherwise.\
**Constant $\beta^{0}$:** For the feature vector we randomly select a subset $S \subset \{1,\ldots,666\}$ of size eight. For each of the selected $k \in S \cup \{1\}$, we set $\beta_{k}^{0} = 1$, and set $\beta_{k}^{0} = 0$ for $k \not \in S$.
We draw $n$ samples $\epsilon_{i}$ from a standard Gaussian distribution with unit variance. Then, we form observations $$y_{i} = \mathbf{X}_{i,\bullet} \beta + \epsilon_{i}.$$ We report both the mean-squared error for $\theta_{\text{fs}}$ corresponding to the first component for varying sample size using the method described in Section \[sec:high-dimens-conf\]. The tuning parameters $\lambda$ and $\lambda_{X}$ are chosen via cross-validation. As treatment effect, we consider $$\tau(t,t'),$$ where $t$ and $t'$ are both randomly drawn from the empirical distribution of $T$. For fairness of comparison, we also use the desparsified lasso to estimate the sample average treatment effect. The results can be found in Table \[tab:third\] and Table \[tab:fourth\].
n=200 n=400 n=600 n=1000
------------------ ----------------- ----------------- ----------------- -----------------
RMSE sample incr 0.14 $\pm$ 0.01 0.12 $\pm$ 0.01 0.1 $\pm$ 0.01 0.08 $\pm$ 0
RMSE sample ATE 0.16 $\pm$ 0.01 0.14 $\pm$ 0.01 0.13 $\pm$ 0.01 0.11 $\pm$ 0.01
: Root mean-squared error of estimating the sample average treatment effect and sample incremental treatment effect. The estimator of incremental causal effects exhibits consistently lower error variance. The noise is drawn from a centered Gaussian distribution with unit variance. For the feature vector, a subset of size eight is randomly selected from $ S \subset \{1,\ldots,666\}$. For each of the $k \in S \cup \{1 \}$, we draw $\beta_k^0 \sim \exp(\lambda)$ with $\lambda = \sqrt{10}$ and set $\beta_k^0 = 0$ otherwise. \[tab:third\]
n=200 n=400 n=600 n=1000
------------------ ----------------- ----------------- ----------------- -----------------
RMSE sample incr 0.59 $\pm$ 0.02 0.45 $\pm$ 0.02 0.3 $\pm$ 0.01 0.13 $\pm$ 0.01
RMSE sample ATE 0.67 $\pm$ 0.02 0.63 $\pm$ 0.02 0.52 $\pm$ 0.02 0.31 $\pm$ 0.02
: Root mean-squared error of estimating the sample average treatment effect and sample incremental treatment effect. The estimator of incremental causal effects exhibits consistently lower error variance. The noise is drawn from a centered Gaussian distribution with unit variance. For the feature vector, a subset of size eight is randomly selected from $ S \subset \{1,\ldots,666\}$. For each of the $k \in S \cup \{1\}$, we set $\beta_k^0=1$ and for $k \not \in S \cup \{1\}$ we set $\beta_k^0 = 0$. \[tab:fourth\]
Evidently, in this simulation setting, estimating the average treatment effect results in higher asymptotic variance as estimating the incremental effect. This is in line with the theory presented in Section \[sec:smaller\]. This phenomenon seems relatively stable across varying choices of simulation parameters as can be seen in the Appendix.
Robustness to local confounding {#sec:conf}
-------------------------------
In Section \[sec:sens-addit-conf\], we discussed that identification of incremental causal effects is often more robust under worst-case confounding than identification of average treatment effects in regression settings. In this section, we investigate the behavior of the plug-in estimators $\hat \tau$ and $\hat \theta$ under confounding. Roughly speaking, we add hidden confounding $h$ so that for subjects with treatment assignment $T \in [a,b]$, the local ignorability assumption does not hold. We call $r = \mathbb{P}[a \le T \le b]$ the ratio of confounding as it corresponds to the ratio of subjects for which the local ignorability assumption does not hold. We chose the simulation such that the regression surface exhibits moderate curvature. Specifically, we generate $n=100$ i.i.d. observations according to the following equations: $$\begin{aligned}
\xi,\xi' &\sim \mathcal{N}(0,1/2) \\
\epsilon &\sim \text{Unif}(-.5,.5) \\
t &= \xi + \xi' \\
y &= 3t + t^2 + h_{\text{conf}}(t) \sqrt{2}\xi + \epsilon,\end{aligned}$$ where the confounder $\xi$ only acts locally, $$h_{\text{conf}}(t) = \begin{cases}
0 & \text{for $t<a$}, \\
t-a & \text{for $a \le t < b$}, \\
b-a & \text{for $b \le t$}.
\end{cases}$$ A cubic model $y \sim t + t^2 + t^{3}$ is fitted using ordinary least squares. We use the plug-in estimators $\hat \theta$ and $\hat \tau$ as defined in Section \[sec:smaller\] to estimate $\theta_\text{fs}$ and $\tau_\text{fs}$ with $t=\frac{1}{2}$, $t'=-\frac{1}{2}$. We investigate two cases. In Figure \[fig:av\], we choose the endpoints of the interval $[a,b]$ randomly and average the mean-squared error over all intervals with fixed ratio of confounded subjects $r = \mathbb{P}[a < X < b]$. In Figure \[fig:max\], we consider the worst-case mean-squared error, where the maximum is taken over all intervals $[a,b]$ with the same ratio of confounded subjects $r$. In both cases, the RMSE increases under confounding and the estimator for the incremental causal effect is more robust than the estimator for the average treatment effect. Under worst-case additive confounding, the gap in RMSE between $\hat \theta$ and $\hat \tau$ widens. Theoretical underpinnings of this discrepancy in robustness can be found Section \[sec:sens-addit-conf\].
![Root mean-squared error under confounding. For a ratio $r \in [0,1]$ of the subjects, the local ignorability assumption is violated. The plug-in estimator $\hat \theta$ for the incremental treatment effect $\theta_{\text{fs}}$ is more robust under confounding than the plug-in estimator $\hat \tau$ for the average treatment effect $\tau_{\text{fs}}$ in this setting.[]{data-label="fig:av"}](low_dim_sim/averageRMSE.pdf)
![Root mean-squared error under worst-case confounding. For a ratio $r \in [0,1]$ of the subjects, the local ignorability assumption is violated. The plug-in estimator $\hat \theta$ for the incremental treatment effect $\theta_{\text{fs}}$ is more robust under confounding than the plug-in estimator $\hat \tau$ for the average treatment effect $\tau_{\text{fs}}$ in this setting. Theoretical underpinnings for the discrepancy in robustness can be found Section \[sec:sens-addit-conf\].[]{data-label="fig:max"}](low_dim_sim/worstRMSE.pdf)
Challenges of estimating incremental treatment effects {#sec:opp}
------------------------------------------------------
Of course, there exist scenarios where estimating the effect of incremental interventions is considerably harder than estimating average treatment effects. For example, performance can suffer if the error variance is larger at the tails of $T$ than in the bulk of the observations. In this case, estimating an average treatment effect $\tau(t,t')$, where $t$ and $t'$ are in the bulk of the observations, is relatively easy compared to estimating incremental causal effects. As an example, we generate i.i.d. observations according to the following equations: $$\begin{aligned}
t &\sim \text{Unif}(-.5,1.5) \\
\epsilon &\sim \text{Unif}(-.5,.5) \\
y &= t^2 + |t| \cdot\epsilon\end{aligned}$$ As before, a cubic model $y \sim t + t^2 + t^{3}$ is fitted using ordinary least squares. Then, the plug-in estimators estimators $\hat \theta$ and $\hat \tau(.5,-.5)$ are calculated as in Section \[sec:smaller\]. The root mean-squared error under varying sample size is reported in Table \[tab:chall\]. As expected, for large $n$ the mean-squared error $\mathbb{E}[(\hat \tau - \tau_{\text{fs}})^{2}]$ is smaller than the mean-squared error $\mathbb{E}[(\hat{\theta} - \theta_{\text{fs}} )^{2}]$. If the error variance for subjects at the edge of the observation space is very large, we recommend estimating average and incremental treatment effects for a subgroup that exhibits lower error variance.
n=100 n=200 n=500
------------------ ------------------- ------------------- -------------------
RMSE sample incr 0.097 $\pm$ 0.004 0.067 $\pm$ 0.003 0.04 $\pm$ 0.002
RMSE sample ATE 0.059 $\pm$ 0.003 0.041 $\pm$ 0.002 0.026 $\pm$ 0.001
: Root mean-squared error under heteroscedasticity. Estimating incremental treatment effects is difficult as the error variance at the edge of the observation space is large. In this scenario, estimating the average treatment effect $\tau(.5, -.5)$ is relatively easy as both $t=.5$ and $t'=-.5$ are in regions where the error variance is low.[]{data-label="tab:chall"}
Conclusion
==========
The estimation of treatment effects is of central interest in many disciplines. Often, treatment effects are estimated under the assumption of weak ignorability and the overlap condition. Both of these assumptions are strong and easily violated if the data is observational. In this paper, we have shown that these assumptions can be substantially weakened for identification of *incremental* treatment effects. We introduced a local ignorability assumption and a local overlap condition and show that incremental treatment effects are identifiable under these two new local conditions. As an example, treatment assignment might be randomized locally within subgroups of patients but not across all patients. In simulation studies, we have seen some evidence indicating that the estimation of the average treatment effect using a plug-in estimator has often higher variance than a comparable estimator for the incremental treatment effect. If the distribution of the treatment given the covariates is Gaussian, we have shown that this difference in asymptotic error is indeed systematic. Moreover, we have shown that estimation of incremental treatment effects is usually more robust under worst-case additive confounding than estimation of average treatment effects.
We discussed how to obtain asymptotically valid confidence intervals that are doubly robust both in terms of estimation and inference using a two-step procedure. In the first step, a feature transformation is performed. We call this feature transformation “incremental effect orthogonalization”. In the second step, an ordinary lasso regression is performed.
Causal inference from observational data is known to be unreliable and has to be done with extreme care. In high-stake scenarios such as healthcare this can have devastating effects on human lives. We have identified situations where incremental effects can be more reliably estimated than average treatment effects. In those settings, estimating incremental effects might be more informative for practitioners than estimating the average treatment effect. We hope that our work aids decisions on choice of intervention notions to reliably answer domain questions.
Acknowledgements
================
Partial supports are gratefully acknowledged from ARO grant W911NF1710005, ONR grant N00014-17-1-2176, NSF grants DMS-1613002 and IIS 1741340, and the Center for Science of Information (CSoI), a US NSF Science and Technology Center, under grant agreement CCF-0939370. BY is a Chan Zuckerberg Biohub investigator.
Appendix
========
Additional simulation results
-----------------------------
n=200 n=400 n=600 n=1000
------------------ ----------------- ----------------- ----------------- -----------------
RMSE sample incr 0.23 $\pm$ 0.02 0.17 $\pm$ 0.01 0.15 $\pm$ 0.01 0.13 $\pm$ 0.01
RMSE sample ATE 0.24 $\pm$ 0.02 0.2 $\pm$ 0.02 0.18 $\pm$ 0.01 0.16 $\pm$ 0.01
: Root mean-squared error of estimating the sample average treatment effect and sample incremental treatment effect. The estimator of incremental causal effects exhibits lower error. The noise is drawn from a $t$-distribution with three degrees of freedom. For the feature vector, a subset of size eight is randomly selected from $ S \subset \{1,\ldots,666\}$. For each of the $k \in S \cup \{1\}$, we draw $\beta_k^0 \sim \exp(\lambda)$ with $\lambda = \sqrt{10}$ and set $\beta_k^0 = 0$ otherwise. \[tab:second\]
n=200 n=400 n=600 n=1000
------------------ ----------------- ----------------- ----------------- -----------------
RMSE sample incr 0.73 $\pm$ 0.28 0.48 $\pm$ 0.02 0.38 $\pm$ 0.02 0.21 $\pm$ 0.01
RMSE sample ATE 0.68 $\pm$ 0.02 0.64 $\pm$ 0.02 0.56 $\pm$ 0.02 0.38 $\pm$ 0.02
: Root mean-squared error of estimating the sample average treatment effect and sample incremental causal effect. The estimator of incremental causal effects exhibits lower error for $n>200$. The noise is drawn from a centered $t$-distribution with 3 degrees of freedom. For the feature vector, a subset of size eight is randomly selected from $ S \subset \{1,\ldots,666\}$. For each of the $k \in S \cup \{1\}$, we set $\beta_k^0=1$ and for $k \not \in S \cup \{1\}$ we set $\beta_k^0 = 0$. \[tab:fifth\]
Proof of Proposition \[prop:ident\] {#sec:proofmain}
-----------------------------------
Without loss of generality we will drop “conditional on $X$”. In the following, choose $t$ with $p(t) > 0$. As $Y(t)$ is continuously differentiable with derivative $Y'(t)$ there exists a random variable $\xi_\delta \in [t,t+\delta]$ such that $$\frac{Y(t+\delta) - Y(t)}{\delta} = Y'(\xi_\delta).$$ As the derivative $Y'(t)$ is continuous and bounded, by dominated convergence, $$\begin{aligned}
\begin{split}
\lim_{\delta \rightarrow 0} \frac{ \mathbb{E} \left[ Y(t+\delta)|T=t] - \mathbb{E}[Y(t) |T=t\right]}{\delta} = \mathbb{E}[Y'(t)|T=t].
\end{split}\end{aligned}$$ Now choose $\delta_{0} > 0$ small enough such that both the local overlap condition and local ignorability is satisfied, i.e. such that $p(t') >0$ for all $|t'-t|\le \delta_{0}$ and that $Y(t+\delta) | T=t \stackrel{d}{=} Y(t+\delta) | T=t + \delta$ for all $|\delta| \le \delta_{0}$. Hence, for $\delta$ close to zero, $$\begin{aligned}
\mathbb{E}[Y(t+\delta)-Y(t)|T=t] &= \mathbb{E}[Y(t+\delta)|T=t]-\mathbb{E}[Y(t)|T=t]\\ &=\mathbb{E}[Y(t+\delta)|T=t+\delta]-\mathbb{E}[Y(t)|T=t].\end{aligned}$$ Dividing by $\delta$, $$\begin{aligned}
\frac{\mathbb{E}[Y(t+\delta)-Y(t)|T=t]}{\delta} &= \frac{ \mathbb{E}[Y(t+\delta)|T=t+\delta]-\mathbb{E}[Y(t)|T=t]}{\delta}.\end{aligned}$$ As shown above, for $\delta \rightarrow 0$, the limit of the quantity on the left exists and is equal to $\mathbb{E}[Y'(t)|T=t]$. Thus, $\mathbb{E}[Y|T=t]$ is differentiable and the quantity on the left converges to $\partial_{t} \mathbb{E}[Y|T=t]$. Taking the limit on both sides concludes the proof. In particular, as $Y(t)$ is continously differentiable with bounded derivative, $t \mapsto \mathbb{E}[Y|T=t]$ is also continuously differentiable with bounded derivative in neighborhoods where $p(t) > 0$.
Proof of Theorem \[thm:variance\_comp\]
---------------------------------------
Take an orthogonal basis $b_1,\ldots,b_p$ of $\mathcal{B}$, such that $b_1 = - \partial_t \log p(t|x) $. For $n$ large enough, the estimator $\hat f$ can be written as $\hat f = \sum_k \hat{\alpha}_k b_k$ with unique $\hat \alpha_1,\ldots,\hat \alpha_p$. Define $\mathbf{X}_{j;i} = b_{j}(t_{i},x_{i})$. Note that conditionally on $\mathcal{D}_{\text{feat}}$, $\hat \theta_{\text{fs}}$ and $\hat \tau_{\text{fs}}$ are unbiased estimators of $\theta_{\text{fs}}$ and $\tau_{\text{fs}}$. Thus, in the following we will derive the conditional asymptotic variance of these estimators. The conditional variance of the vector $\hat \alpha$ can be written as $$(\mathbf{X}^\intercal \mathbf{X} )^{-1} \mathbb{E}[\epsilon^{2}]$$ The conditional variance of $\hat \alpha$ can thus be written as $$\frac{1}{n} \cdot \begin{pmatrix}
\frac{ \mathbb{E}[\epsilon^{2}]}{\mathbb{E}[b_1^2]} & 0 & \ldots & 0 \\
0 & * & \ldots & * \\
\vdots & \vdots & \ddots & \vdots \\
0 & * & \ldots & *
\end{pmatrix} + o_{P} \left( \frac{1}{n}\right).$$ Here we used that by choice of $b_1,\ldots,b_p$ we have that $\mathbb{E}[b_1 b_k] = 0$ for all $k>1$. In addition, we used the formula for block-wise inversion and multiplication of matrices. Hence $\hat \theta_{\text{fs}} = \frac{1}{n} \sum_{i=1}^n \sum_k \hat \alpha_k \partial_t b_k(t_i,x_i) = \sum_k \hat \alpha_k \frac{1}{n} \sum_{i=1}^n \partial_t b_k(t_i,x_i)$ has asymptotic conditional variance $$\frac{1}{n} \cdot \begin{pmatrix} \mathbb{E}[\partial_t b_1] & \ldots & \mathbb{E}[\partial_t b_p] \end{pmatrix} \begin{pmatrix}
\frac{ \mathbb{E}[\epsilon^2]}{\mathbb{E}[b_1^2]} & 0 & \ldots & 0 \\
0 & * & \ldots & * \\
\vdots & \vdots & \ddots & \vdots \\
0 & * & \ldots & *
\end{pmatrix}\begin{pmatrix} \mathbb{E}[\partial_t b_1] \\ \vdots \\ \mathbb{E}[\partial_t b_p] \end{pmatrix} + o_{P}\left( \frac{1}{n} \right).$$ Using that by partial integration, $\mathbb{E}[\partial_t b_k] = \mathbb{E}[b_k b_1] = 0$ for all $k > 1$, $\hat \theta_{\text{fs}}$ has asymptotic conditional variance $$\frac{1}{n} \mathbb{E}[\partial_t b_1]^2 \frac{ \mathbb{E}[\epsilon^{2}]}{\mathbb{E}[b_1^2]}.$$ We can now use that $\mathbb{E}[\partial_t b_1]^2 = \mathbb{E}[b_1^2]^2$. This gives us asymptotic conditional variance $$\frac{\mathbb{E}[ b_1^2 ]\mathbb{E}[\epsilon^{2}]}{n} + o_{P}\left( \frac{1}{n} \right).$$ Through analogous argumentation we obtain that the conditional variance of $\hat \tau$ is asymptotically $$\frac{1}{n} \cdot \begin{pmatrix} v_1 & \ldots & v_p \end{pmatrix} \begin{pmatrix}
\frac{ \mathbb{E}[\epsilon^{2}]}{\mathbb{E}[b_1^2]} & 0 & \ldots & 0 \\
0 & * & \ldots & * \\
\vdots & \vdots & \ddots & \vdots \\
0 & * & \ldots & *
\end{pmatrix}\begin{pmatrix} v_1 \\ \vdots \\ v_p\end{pmatrix} + o_{P}\left( \frac{1}{n} \right),$$ where $v_k := \frac{\mathbb{E}[b_k(t,X)] - \mathbb{E}[b_k(t',X)]}{t - t'}$. Using that the submatrix denoted by “\*” is positive semidefinite, the variance of $\hat \tau_{\text{fs}}$ is asymptotically lower bounded by $$\frac{1}{n} \left(\frac{\mathbb{E}[b_1(t,X)] - \mathbb{E}[b_1(t',X)]}{t - t'} \right)^2 \frac{\mathbb{E}[ \epsilon^2]}{\mathbb{E}[b_1^2]} + o_{P}\left( \frac{1}{n} \right).$$ Recall that we assume that the second derivative of the log-density given $X$ is constant. In this case, $$\left(\frac{\mathbb{E}[b_1(t,X)] - \mathbb{E}[b_1(t',X)]}{t - t'} \right)^2 = \mathbb{E}[\partial_{1} b_{1}]^2$$ Hence, in this case, the variance of $\hat \tau_{\text{fs}}$ is asymptotically lower bounded by $$\frac{\mathbb{E}[ b_1^2 ] \mathbb{E}[\epsilon^{2}]}{n} + o_{P}\left( \frac{1}{n} \right).$$ Thus, $\liminf_{n \rightarrow \infty} \text{Var}(\hat \tau_{\text{fs}}|\mathcal{D}_{\text{feat}}) / \text{Var}(\hat \theta_{\text{fs}} | \mathcal{D}_{\text{feat}}) \ge 1$.
Proof of Theorem \[thm:robustn-addit-conf\]
-------------------------------------------
We will first prove the result for incremental treatment effects. Set $f_{*} = - \partial_{t} \log p(T,X)$. Then, $$\begin{aligned}
\text{sens}_{\text{incr}}(\epsilon) &= \max_{\mathbb{E}[H^{2}] \le \epsilon} \mathbb{E}_{\text{conf}}[ f_{*} Y] - \mathbb{E}_{\text{unconf}}[ f_{*} Y]\\
&= \max_{\mathbb{E}[H^{2}] \le \epsilon} \mathbb{E}_{\text{unconf}}[ f_{*} (Y+H)] - \mathbb{E}_{\text{unconf}}[ f_{*} Y] \\
&= \max_{\mathbb{E}[H^{2}] \le \epsilon} \mathbb{E}_{\text{unconf}}[ f_{*} H]\\
&= \sqrt{\epsilon} \sqrt{\mathbb{E}[f_{*}^{2}]}\end{aligned}$$ Now, use that $\mathbb{E}[f_{*}^{2}] = \mathbb{E}[\partial_{t} f_{*}] = \mathbb{E}[-\partial_{t}^{2} \log p(t,x)]$. In the case of the average treatment effect use $$H = \sqrt{\epsilon} \frac{f((T-t)/\sigma)}{\sqrt{\mathbb{E}_{\text{unconf}}[f((T-t)/\sigma )^{2}]}},$$ where $f$ is the density function of a standard Gaussian random variable. For $\sigma \rightarrow 0$, $\mathbb{E}_{\text{conf}}[Y|X=x,T=t] \rightarrow \infty$, whereas $\mathbb{E}_{\text{conf}}[Y|X=x,T=t'] \rightarrow \mathbb{E}_{\text{unconf}}[Y|X=x,T=t']$. This concludes the proof.
Proof of Theorem \[thm:doubly-robust-conf\] and auxiliary results
-----------------------------------------------------------------
The proof follows closely @buhlmann2015high with some modifications. Before we proceed, we show that the following auxiliary results hold:
- $$\begin{aligned}
\max_{k } |\epsilon^{\intercal} \tilde{\mathbf{X}}_{k}/n| &= O_{P}(\sqrt{\log(p)/n}) \\
\max_{k } | \epsilon^{\intercal} (\tilde{\mathbf{X}}_{k} - \tilde{\mathbf{X}}_{k}^{0})/n| &= O_{P}(\sqrt{\log(p)}/n) \\
\max_{k \neq 1} | (\tilde{\mathbf{X}}_{k}^{\intercal} \tilde{\mathbf{Z}}^{0})/n| &= O_{P}(\sqrt{\log(p)/n})\\
\max_{k \neq 1} | (\tilde{\mathbf{X}}_{k} - \tilde{\mathbf{X}}_{k}^{0})^{\intercal} \tilde{\mathbf{Z}}^{0}/n| &= O_{P}(\sqrt{\log(p)}/n)
\end{aligned}$$
- $\| \hat \gamma(\lambda_X) - \gamma^{0} \|_{1} = o_{P}(1/\sqrt{\log(p)})$
- $ \|\hat \beta(\lambda) - \beta^{0} \|_{1} = o_{P}(1/\sqrt{\log(p)})$
\[le:error\] Assume (A2), (A3) and (A7). Then, (D1) holds.
We will prove the first part of the statement. The other parts of the statement can be proven analogously. First, note that $$\begin{aligned}
\label{eq:8}
\mathbb{E}[\max_{1 \le k \le p} |n^{-1} \epsilon^{\intercal} \tilde{\mathbf{X}}_{k} |^{2}] \le 2 \mathbb{E}[\max_{1 \le k \le p} |n^{-1} \epsilon^{\intercal} \mathbf{X}_{k} |^{2}] + 2 \mathbb{E}[\max_{1 \le k \le p} | \delta_{k} |^{2}],\end{aligned}$$ where $\delta_{1} = 0$ and $\delta_{k} = n^{-1} \sum_{i} \epsilon_{i} t_{i}
\cdot n^{-1}\sum_{i}\partial_{t} b_{k}(t_{i},x_{i}) $ for $k>1$. Using Nemirowski’s inequality [@pbvdg11 Lemma 14.24] we obtain: $$\label{eq:6}
\mathbb{E}[\max_{1 \le k \le p} |n^{-1} \epsilon^{\intercal} \mathbf{X}_{k} |^{2}] \le 8 \log(2p) C_{2}^{2} V^{2}/n = O(\log(p)/n),$$ and similarly $$\label{eq:7}
\mathbb{E}[\max_{1 \le k \le p} | \delta_{k} |^{2}] \le 8 \log(2p) C_{2}^{4} V^{2}/n = O(\log(p)/n).$$ Using equation and equation in equation we obtain $$\mathbb{E}[\max_{1 \le k \le p} |n^{-1} \epsilon^{\intercal} \tilde{\mathbf{X}}_{k} |^{2}] = O(\log(p)/n).$$ In the next step, we can use Markov’s inequality and $\mathbb{E}[\epsilon^{\intercal} \tilde{\mathbf{X}}_{k}] = 0$ to conclude that $$\begin{aligned}
\mathbb{P}[\max_{k=1,\ldots,p}|n^{-1} \epsilon^{\intercal} \tilde{\mathbf{X}}_{k} | > c] &\le \frac{\mathbb{E}[ \max_{k=1,\ldots,p} |n^{-1} \epsilon^{\intercal} \tilde{\mathbf{X}}_{k} |]}{c} \\
&\le \frac{\sqrt{\mathbb{E}[ \max_{k=1,\ldots,p} |n^{-1} \epsilon^{\intercal} \tilde{\mathbf{X}}_{k} |^{2}]}}{c} \\ &= O(\sqrt{\log(p)/n}).\end{aligned}$$
\[lem:D2D3\] Assume (A1) and (A2) and $\sqrt{\log(p)/n} \rightarrow 0$.
1. If (A3) and (A4) hold, then for $\lambda_{X} = D_{2} \sqrt{\log(p)/n}$ with $D_{2}$ sufficiently large we have (D2).
2. If (A7) and (A5) holds, then for $\lambda = D_{1} \sqrt{\log(p)/n}$ with $D_{1}$ sufficiently large, we have (D3).
The proof proceeds mostly as in [@buhlmann2015high]. However, there is one slight complication that the rows of $\tilde{\mathbf{X}}$ are not i.i.d. We will prove statement (1). The proof for statement (2) proceeds analogously.
First, we will prove the compatibility condition for the transformed data $\tilde{\mathbf{X}}$. To this end, note that $$\label{eq:9}
\frac{1}{n}\tilde{\mathbf{X}}_{j}^{\intercal} \tilde{\mathbf{X}}_{k} = \frac{1}{n} \mathbf{X}_{j}^{\intercal} \mathbf{X}_{k} - \frac{1}{n} \sum_{i=1}^{n} \mathbf{X}_{i,j} \mathbf{X}_{i,1} \delta_{k} - \frac{1}{n}
\sum_{i=1}^{n}\mathbf{X}_{i,1} \mathbf{X}_{i,k} \delta_{j} + \delta_{k} \delta_{j} \frac{1}{n} \sum_{i=1}^{n}\mathbf{X}_{i,1}^{2},$$ where $\delta_{j} = \frac{1}{n} \sum_{i'} \partial_{t} b_{j}(t_{i'},x_{i'}) $ for $j>1$ and $\delta_{j}=0$ for $j=1$. Using assumption (A2) and sub-Gaussian tail bounds [@boucheron2013concentration Chapter 2], the terms $$\begin{aligned}
&\frac{1}{n} \mathbf{X}_{j}^{\intercal} \mathbf{X}_{k} - \frac{1}{n}\mathbb{E}[\mathbf{X}_{j}^{\intercal} \mathbf{X}_{k}], \\
& \delta_{k} - \mathbb{E}[\delta_{k}], \end{aligned}$$ are uniformly of the order $O_{P}(\sqrt{\log(p)/n})$. Hence, using equation , the term $$\max_{j,k} \left| \frac{1}{n}\tilde{\mathbf{X}}_{j}^{\intercal} \tilde{\mathbf{X}}_{k} - \frac{1}{n} \mathbb{E}[(\tilde{\mathbf{X}}_{j}^{0})^{\intercal} \tilde{\mathbf{X}}_{k}^{0}] \right| $$ is of order $O_{P}(\sqrt{\log(p)/n})$.
The sparsity assumption (A4) combined with (A1) imply that the compatibility condition holds with probability converging to one, c.f. @pbvdg11 [Chapter 6.12]. Using Lemma \[le:error\], we obtain $\| \tilde{\mathbf{X}}_{-1}^{\intercal} \tilde{\mathbf{Z}}^{0} \|_{\infty} \le O_{P}(\sqrt{n \log(p)})$. Invoking an inequality for the lasso [@pbvdg11 Chapter 6.1] with assumption (A4), we obtain statement (1).
\[prop-defw\] Assume (A1), (A2), (A3), (A6), (A7) and (A8). Write $$\begin{aligned}
u^{2} = \text{Var} \left( \frac{\epsilon_{1} \tilde{\mathbf{Z}}_{1}^{0}}{\mathbb{E}[\tilde{\mathbf{Z}}_{1}^{0} \tilde{\mathbf{X}}_{11}^{0}]
} + \sum_{k} \partial_{t} b_{k}(t_{1},x_{1}) \beta_{k}^{0}) \right)\end{aligned}$$ Then, $$\label{eq:12}
\sqrt{n} \left( \frac{ \frac{\epsilon^{\intercal} \tilde{\mathbf{Z}}^{0}/n}{\mathbb{E}[\tilde{\mathbf{Z}}^{0}_{1} \tilde{\mathbf{X}}_{11}^{0}]
} - \sum_{k} (\mathbb{E}[\partial_{t} b_{k} \beta_{k}^{0} ] - \hat{\mathbb{E}}[\partial_{t} b_{k} \beta_{k}^{0} ] ) }{ u }\right) \rightharpoonup \mathcal{N}(0,1).$$
By assumption (A6), $u$ is bounded from below. In addition, note that due to (A1), $\mathbb{E}[(\tilde{\mathbf{Z}}^{0})^{\intercal} \tilde{\mathbf{X}}_{1}^{0}]/n$ is bounded away from zero and due to (A2) and (A3) it is bounded from above. Due to (A2) and (A8), $\sum_{k} \partial_{t} b_{k} \beta_{k}^{0}$ is bounded. The proof then proceeds analogously to the proof in [@buhlmann2015high] using the Lindeberg condition.
\[prop-defw2\] Assume $\sqrt{\log(p)/n} \rightarrow 0$, (A1), (A2), (A3), (A6), (A7), (A8), (D1), (D2) and (D3). Then: $$\label{eq:13}
\sqrt{n} \left( \frac{\frac{\tilde{\mathbf{Z}}^{\intercal} \epsilon}{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}} - \sum_{k} ( \mathbb{E}[\partial_{t} b_{k} \beta_{k}^{0} ] - \hat{\mathbb{E}}[\partial_{t} b_{k} \beta_{k}^{0}] )}{u} \right) \rightharpoonup \mathcal{N}(0,1)$$
We have to show that the difference between equation and equation is of order $o_{P}(1)$. Note that due to (A6), the quantity $u$ is bounded away from zero and can be ignored. The difference between equation and equation , up to bounded factors, is $$\sqrt{n} \left( \frac{\tilde{\mathbf{Z}}^{\intercal} \epsilon}{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}} - \frac{\epsilon^{\intercal} \tilde{\mathbf{Z}}^{0}/n}{\mathbb{E}[(\tilde{\mathbf{Z}}_{1}^{0})^{\intercal} \tilde{\mathbf{X}}_{11}^{0}] } \right).$$ We want to show that this terms goes to zero. Let us assume for a moment that
1. $| \epsilon^{\intercal}(\tilde{\mathbf{Z}}^{0} - \tilde{\mathbf{Z}}) / \sqrt{n}| = o_{P}(1)$,
2. $ \tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}/n - \mathbb{E}[(\tilde{\mathbf{Z}}_{1}^{0})^{\intercal} \tilde{\mathbf{X}}_{11}^{0}] = o_{P}(1) $,
3. $ \epsilon^{\intercal} \tilde{\mathbf{Z}}^{0}/\sqrt{n} = O_{P}(1)$,
4. $\mathbb{E}[(\tilde{\mathbf{Z}}_{1}^{0})^{\intercal} \tilde{\mathbf{X}}_{11}^{0}]$ is bounded away from zero.
Then, $$\begin{aligned}
& \, \, \sqrt{n} \left( \frac{\tilde{\mathbf{Z}}^{\intercal} \epsilon}{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}} - \frac{\epsilon^{\intercal} \tilde{\mathbf{Z}}^{0}/n}{\mathbb{E}[(\tilde{\mathbf{Z}}_{1}^{0})^{\intercal} \tilde{\mathbf{X}}_{11}^{0}] } \right)\\ &= \epsilon^{\intercal} \tilde{\mathbf{Z}}^{0}/\sqrt{n} \left(\frac{1}{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}/n} - \frac{1}{\mathbb{E}[(\tilde{\mathbf{Z}}_{1}^{0})^{\intercal} \tilde{\mathbf{X}}_{11}^{0}]}\right) \\
&\, \,+ ( \epsilon^{\intercal} \tilde{\mathbf{Z}}^{0}/\sqrt{n} - \epsilon^{\intercal} \tilde{\mathbf{Z}}/\sqrt{n} ) \frac{1}{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}/n} \\
&= o_{P}(1),\end{aligned}$$ which is the desired result. Thus, it remains to show that the claims (1)–(4) hold. Let us first show claim (1). $$\begin{aligned}
| \epsilon^{\intercal}(\tilde{\mathbf{Z}}^{0} - \tilde{\mathbf{Z}}) / \sqrt{n}| &\le | \epsilon^{\intercal} \tilde{\mathbf{X}}_{-1} (\hat \gamma - \gamma^{0})|/\sqrt{n} + |\epsilon^{\intercal} (\tilde{\mathbf{X}}_{-1} - \tilde{\mathbf{X}}_{-1}^{0}) \gamma^{0}|/\sqrt{n} \\
&\le \| \epsilon^{\intercal} \tilde{\mathbf{X}}_{-1} \|_{\infty} \| \hat \gamma - \gamma^{0} \|_{1} / \sqrt{n} + \|\epsilon^{\intercal} (\tilde{\mathbf{X}}_{-1} - \tilde{\mathbf{X}}_{-1}^{0}) \|_{\infty} / \sqrt{n} \| \gamma^{0} \|_{1}\end{aligned}$$ Now we can use (D1), (D2) and (A8) to conclude that this term goes to zero in probability for $n \rightarrow \infty$. This proves claim (1). Now let us turn to claim (2). Similarly as proving claim (1) we can show that $$\begin{aligned}
|\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}/n - (\tilde{\mathbf{Z}}^{0})^{\intercal} \tilde{\mathbf{X}}_{1}/n | &= | \frac{1}{n} \tilde{\mathbf{X}}_{1}^{\intercal} \tilde{\mathbf{X}}_{-1} (\hat \gamma - \gamma^{0})| + | \frac{1}{n} \tilde{\mathbf{X}}_{1}^{\intercal} (\tilde{\mathbf{X}}_{-1}^{0} - \tilde{\mathbf{X}}_{-1} ) \gamma^{0} |\\
&= o_{P}(1).\end{aligned}$$ As $\tilde{\mathbf{X}}_{1} = \tilde{\mathbf{X}}_{1}^{0}$, $$\label{eq:19}
|\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}/n - (\tilde{\mathbf{Z}}^{0})^{\intercal} \tilde{\mathbf{X}}_{1}^{0}/n | = o_{P}(1).$$ By (A2) and the law of large numbers, $$(\tilde{\mathbf{Z}}^{0})^{\intercal} \tilde{\mathbf{X}}_{1}^{0}/n - \mathbb{E}[(\tilde{\mathbf{Z}}_{1}^{0})^{\intercal} \tilde{\mathbf{X}}_{11}^{0}] = o_{P}(1).$$ Using equation proves claim (2). $$\left(\frac{1}{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}/n} - \frac{1}{\mathbb{E}[(\tilde{\mathbf{Z}}_{1}^{0})^{\intercal} \tilde{\mathbf{X}}_{11}^{0}]}\right) = o_{P}(1).$$ Due to (A3), (A7) and the definition of $\epsilon$, $ \epsilon^{\intercal} \tilde{\mathbf{Z}}^{0}/\sqrt{n} = O_{P}(1)$. This proves claim (3). Claim (4) follows from assumption (A1).
\[prop:almost\] Assume (A1), (A2), (A3), (A6), (A7), (A8), (D1), (D2) and (D3). Then, for $\lambda_{X} = D_{2} \sqrt{\log(p)/n}$ with $D_{2}$ sufficiently large and $\sqrt{\log(p)/n}\rightarrow 0$, $$\frac{\sqrt{n}(\hat \beta_{1}^{\text{despar}} - \beta_{1}^{0})}{u} \rightharpoonup \mathcal{N}(0,1),$$ where $u$ is defined as in Proposition \[prop-defw\].
Let us recall the definition $$\hat \beta_{1}^{\text{despar}} = \frac{\tilde{\mathbf{Z}}^{\intercal} \mathbf{Y}}{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}} - \sum_{k \neq 1}\frac{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{k}}{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}} \hat \beta_{k}.$$ Then, using that $\mathbf{Y} = \epsilon + \tilde{\mathbf{X}}^{0} \beta^{0} = \epsilon + \tilde{\mathbf{X}} \beta^{0} - \sum_{k \neq 1}\tilde{\mathbf{X}}_{1} (\mathbb{E}[\partial_{t} b_{k} \beta_{k}^{0}]-\hat{\mathbb{E}}[\partial_{t} b_{k} \beta_{k}^{0}])$, $$\begin{aligned}
& \, \, \sqrt{n} \frac{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}}{n} (\hat \beta_{1}^{\text{despar}} - \beta_{1}^{0}) \\
&= \frac{1}{\sqrt{n}} \left( \tilde{\mathbf{Z}}^{\intercal} \mathbf{Y} - \sum_{ k \neq 1} \tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{k} \hat \beta_{k} - \tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1} \beta_{1}^{0} \right) \\
&= \frac{1}{\sqrt{n}} \left( \tilde{\mathbf{Z}}^{\intercal} (\epsilon - \tilde{\mathbf{X}}_{1} \sum_{k \neq 1} ( \mathbb{E}[\partial_{t} b_{k}] - \hat{\mathbb{E}}[\partial_{t} b_{k}] ) \beta_{k}^{0} + \sum_{k \neq 1} \tilde{\mathbf{X}}_{k} (\beta_{k}^{0} - \hat \beta_{k})) \right) \\
&= \frac{1}{\sqrt{n}} \left( \tilde{\mathbf{Z}}^{\intercal} (\epsilon - \tilde{\mathbf{X}}_{1} \sum_{k \neq 1} ( \mathbb{E}[\partial_{t} b_{k}] - \hat{\mathbb{E}}[\partial_{t} b_{k}] ) \beta_{k}^{0}+ \tilde{\mathbf{X}}_{-1} (\beta_{-1}^{0} - \hat \beta_{-1})) \right).
\end{aligned}$$ The latter quantity in this term can be bounded, $$\begin{aligned}
& \, \, \left| \frac{1}{\sqrt{n}} \tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{-1} (\beta_{-1}^{0} - \hat \beta_{-1}) \right| \\
&\le \frac{1}{\sqrt{n}} \left\| \tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{-1} \right\|_{\infty} \left\| (\beta_{-1}^{0} - \hat \beta_{-1}) \right\|_{1} \\
\end{aligned}$$ The KKT conditions for the regression of $\tilde{\mathbf{X}}_{1}$ on $\tilde{\mathbf{X}}_{-1}$ read as $$\tilde{\mathbf{X}}_{-1}^{\intercal} \tilde{\mathbf{Z}}/n + \lambda_{X} \hat \kappa = 0,$$ for $\hat \kappa \in [-1,1]^{p-1}$. Thus, $\| \tilde{\mathbf{X}}_{-1}^{\intercal} \tilde{\mathbf{Z}} \|_{\infty} = O(\sqrt{n\log(p)}) $. Furthermore, by assumption $ \| \beta^{0} - \hat \beta \|_{1} = o_{P}(1/\sqrt{\log(p)})$. Thus, $$\, \, \left| \frac{1}{\sqrt{n}} \tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{-1} (\beta_{-1}^{0} - \hat \beta_{-1}) \right| = o_{P}(1)$$ Thus, $$\begin{aligned}
& \, \, \sqrt{n} \frac{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}}{n} (\hat \beta_{1}^{\text{despar}} - \beta_{1}^{0}) \\
&= \frac{1}{\sqrt{n}} \left( \tilde{\mathbf{Z}}^{\intercal} (\epsilon - \tilde{\mathbf{X}}_{1} \sum_{k \neq 1} ( \mathbb{E}[\partial_{t} b_{k}] - \hat{\mathbb{E}}[\partial_{t} b_{k}] ) \beta_{k}^{0}) \right) + o_{P}(1).
\end{aligned}$$ Rearranging and using property (2) and (4) from the proof of Proposition \[prop-defw2\] yields $$\begin{aligned}
& \, \, \sqrt{n} (\hat \beta_{1}^{\text{despar}} - \beta_{1}^{0}) \\
&= \sqrt{n} \left( \frac{\tilde{\mathbf{Z}}^{\intercal} \epsilon}{\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}} - \sum_{k \neq 1} ( \mathbb{E}[\partial_{t} b_{k}] - \hat{\mathbb{E}}[\partial_{t} b_{k}] ) \beta_{k}^{0} \right) + o_{P}(1). \end{aligned}$$ Using Proposition \[prop-defw2\] completes the proof.
\[prop:varianceestimation\] Assume $\sqrt{\log(p)/n} \rightarrow 0$, (A1), (A2), (A3), (A6), (A7), (A8), (D2) and (D3). Then, $$\begin{aligned}
\hat u^{2} = u^{2} +o_{P}(1),
\end{aligned}$$ where $\hat u^{2}$ is the empirical variance of $$\begin{aligned}
\frac{\hat \epsilon_{i} \tilde{\mathbf{Z}}_{i}}{(\tilde{\mathbf{Z}})^{\intercal} \tilde{\mathbf{X}}_{1}/n
} - \sum_{k} \hat{\mathbb{E}}[\partial_{t} b_{k} ]\hat \beta_{k} - \partial_{t} b_{k}(t_{i},x_{i}) \hat \beta_{k} ,
\end{aligned}$$ for $i=1,\ldots,n$ and $u^{2}$ is the variance of $$\begin{aligned}
\frac{\epsilon_{1} \tilde{\mathbf{Z}}_{1}^{0}}{\mathbb{E}[(\tilde{\mathbf{Z}}_{1}^{0})^{\intercal} \tilde{\mathbf{X}}_{1;1}^{0}]
} - \sum_{k} \mathbb{E}[\partial_{t} b_{k} \beta_{k}^{0} ] - \partial_{t} b_{k}(t_{1},x_{1}) \beta_{k}^{0}.
\end{aligned}$$
Define $$\begin{aligned}
\xi_{i}^{0} = \frac{\epsilon_{i} \tilde{\mathbf{Z}}_{i}^{0}}{\mathbb{E}[(\tilde{\mathbf{Z}}_{1}^{0})^{\intercal} \tilde{\mathbf{X}}_{1;1}^{0}]
} - \sum_{k} \mathbb{E}[\partial_{t} b_{k} \beta_{k}^{0} ] - \partial_{t} b_{k}(t_{i},x_{i}) \beta_{k}^{0},
\end{aligned}$$ and $$\begin{aligned}
\xi_{i} = \frac{\hat \epsilon_{i} \tilde{\mathbf{Z}}_{i}}{(\tilde{\mathbf{Z}})^{\intercal} \tilde{\mathbf{X}}_{1}/n
} - \sum_{k} \hat{\mathbb{E}}[\partial_{t} b_{k} ]\hat \beta_{k} - \partial_{t} b_{k}(t_{i},x_{i}) \hat \beta_{k}.\end{aligned}$$ By assumption (A1), $\mathbb{E}[(\tilde{\mathbf{Z}}_{1}^{0})^{\intercal} \tilde{\mathbf{X}}_{1;1}^{0}]$ is bounded away from zero. Thus, by (A2), (A3), (A7) and (A8), the $\xi_{i}^{0}$ are bounded. Using the law of large numbers, $$\begin{aligned}
\frac{1}{n} \sum_{i} \xi_{i}^{0} &= \mathbb{E}[\xi_{1}^{0}] + o_{P}(1), \\
\frac{1}{n} \sum_{i} (\xi_{i}^{0})^{2} &= \mathbb{E}[(\xi_{1}^{0})^{2}] + o_{P}(1).
\end{aligned}$$ Thus, it suffices to show that $$\begin{aligned}
\frac{1}{n} \sum_{i} \xi_{i}^{0} - \xi_{i} &= o_{P}(1), \\
\frac{1}{n} \sum_{i} (\xi_{i}^{0})^{2} - \xi_{i}^{2} &= o_{P}(1).
\end{aligned}$$ Note that we have $$\begin{aligned}
\label{eq:21}
\begin{split}
\left| \frac{1}{n} \sum_{i} \xi_{i}^{0} - \xi_{i} \right|| &\le \max_{i} \left| \xi_{i}^{0} - \xi_{i} \right| \\
\left| \frac{1}{n} \sum_{i} (\xi_{i}^{0})^{2} - \xi_{i}^{2} \right| &\le \max_{i} | \xi_{i}^{0} - \xi_{i} | (\max_{i} | \xi_{i}^{0} - \xi_{i} | + \max_{i} | \xi_{i}^{0} |)
\end{split}
\end{aligned}$$ As the $\xi_{i}^{0}$ are bounded, using equation it suffices to show that $$\max_{i}| \xi_{i} - \xi_{i}^{0}| = o_{P}(1).$$ We will do this in two steps. $$\begin{aligned}
\label{eq:24}
\begin{split}
& \, \, \max_{i}| \xi_{i} - \xi_{i}^{0}| \\
&\le \max_{i} \left| \frac{\epsilon_{i} \tilde{\mathbf{Z}}_{i}^{0}}{\mathbb{E}[(\tilde{\mathbf{Z}}_{1}^{0})^{\intercal} \tilde{\mathbf{X}}_{1;1}^{0}]
} - \frac{\hat \epsilon_{i} \tilde{\mathbf{Z}}_{i}}{(\tilde{\mathbf{Z}})^{\intercal} \tilde{\mathbf{X}}_{1}/n }\right|
\\
& + \max_{i} \left| \sum_{k} \mathbb{E}[\partial_{t} b_{k} \beta_{k}^{0} ] - \partial_{t} b_{k}(t_{i},x_{i}) \beta_{k}^{0}
- \sum_{k} \hat{\mathbb{E}}[\partial_{t} b_{k} ]\hat \beta_{k} - \partial_{t} b_{k}(t_{i},x_{i}) \hat \beta_{k} \right|.
\end{split}
\end{aligned}$$ By assumption, $\xi_{i}^{0}$ and $\xi_{i}$ are bounded. Note that $$\begin{aligned}
\max_{i} \left| \sum_{k} \partial_{t} b_{k}(t_{i},x_{i}) (\beta_{k}^{0}- \hat \beta_{k} ) \right| \le \max_{i} \max_{k} |b_{k}(t_{i},x_{i})|^{2} \|\beta^{0} - \hat \beta \|_{1}.
\end{aligned}$$ Due to assumption (A2), the $b_{k}$ are bounded. Recall that due to (D3), $\| \hat \beta - \beta^{0} \|_{1} = o_{P}(\sqrt{1/\log{p}})$. Thus, $$\begin{aligned}
\label{eq:22}
\max_{i} \left| \sum_{k} \partial_{t} b_{k}(t_{i},x_{i}) (\beta_{k}^{0}- \hat \beta_{k} ) \right| = o_{P}(1).
\end{aligned}$$ Using (A2) and that $\| \beta^{0} \|_{1}$ is bounded, using a sub-Gaussian tail inequality [@boucheron2013concentration Chapter 2], $$\begin{aligned}
\label{eq:23}
\left| \sum_{k} \mathbb{E}[\partial_{t} b_{k}] \beta_{k}^{0} - \sum_{k} \frac{1}{n} \sum_{i} \partial_{t} b_{k}(t_{i},x_{i}) \beta_{k}^{0} \right| = O_{P}(\sqrt{\log(p)/n}).
\end{aligned}$$ Combining equation and equation , as $\sqrt{\log(p)/n} \rightarrow 0$, $$\begin{aligned}
\left| \sum_{k} \mathbb{E}[\partial_{t} b_{k} \beta_{k}^{0} ] - \partial_{t} b_{k}(t_{i},x_{i}) \beta_{k}^{0}
- \sum_{k} \hat{\mathbb{E}}[\partial_{t} b_{k} ]\hat \beta_{k} - \partial_{t} b_{k}(t_{i},x_{i}) \hat \beta_{k} \right| = o_{P}(1).
\end{aligned}$$ Using equation , it remains to show that $$\begin{aligned}
\max_{i} \left| \frac{\epsilon_{i} \tilde{\mathbf{Z}}_{i}^{0}}{\mathbb{E}[\tilde{\mathbf{Z}}_{1}^{0} \tilde{\mathbf{X}}_{1;1}^{0}]
} - \frac{\hat \epsilon_{i} \tilde{\mathbf{Z}}_{i}}{(\tilde{\mathbf{Z}})^{\intercal} \tilde{\mathbf{X}}_{1}/n|} \right| = o_{P}(1)
\end{aligned}$$ Expanding the terms, $$\begin{aligned}
\label{eq:25}
\begin{split}
&\left| \frac{\epsilon_{i} \tilde{\mathbf{Z}}_{i}^{0}}{\mathbb{E}[\tilde{\mathbf{Z}}_{1}^{0} \tilde{\mathbf{X}}_{1;1}^{0}]
} - \frac{\hat \epsilon_{i} \tilde{\mathbf{Z}}_{i}}{(\tilde{\mathbf{Z}})^{\intercal} \tilde{\mathbf{X}}_{1}/n} \right| \\
&\le \left| \frac{\epsilon_{i} \tilde{\mathbf{Z}}_{i}^{0} - \hat \epsilon_{i} \tilde{\mathbf{Z}}_{i}}{\mathbb{E}[\tilde{\mathbf{Z}}_{1}^{0} \tilde{\mathbf{X}}_{1;1}^{0}]
} \right| + \left| \hat \epsilon_{i} \tilde{\mathbf{Z}}_{i} \left( \frac{1}{(\tilde{\mathbf{Z}})^{\intercal} \tilde{\mathbf{X}}_{1}/n} - \frac{1}{\mathbb{E}[\tilde{\mathbf{Z}}_{1}^{0} \tilde{\mathbf{X}}_{1;1}^{0}]}
\right) \right| \\
&\le \left| \frac{\epsilon_{i} \tilde{\mathbf{Z}}_{i}^{0} - \hat \epsilon_{i} \tilde{\mathbf{Z}}_{i}}{\mathbb{E}[\tilde{\mathbf{Z}}_{1}^{0} \tilde{\mathbf{X}}_{1;1}^{0}]
} \right| + \left| \left( \hat \epsilon_{i} \tilde{\mathbf{Z}}_{i} - \epsilon_{i} \tilde{\mathbf{Z}}_{i}^{0} \right)\left( \frac{1}{(\tilde{\mathbf{Z}})^{\intercal} \tilde{\mathbf{X}}_{1}/n} - \frac{1}{\mathbb{E}[\tilde{\mathbf{Z}}_{1}^{0} \tilde{\mathbf{X}}_{1;1}^{0}]}
\right) \right| \\
&+\left| \epsilon_{i} \tilde{\mathbf{Z}}_{i}^{0} \left( \frac{1}{(\tilde{\mathbf{Z}})^{\intercal} \tilde{\mathbf{X}}_{1}/n} - \frac{1}{\mathbb{E}[\tilde{\mathbf{Z}}_{1}^{0} \tilde{\mathbf{X}}_{1;1}^{0}]}
\right) \right|.
\end{split}
\end{aligned}$$ We have shown in Proposition \[prop-defw2\] that $$\tilde{\mathbf{Z}}^{\intercal} \tilde{\mathbf{X}}_{1}/n = \mathbb{E}[\tilde{\mathbf{Z}}_{1}^{0} \tilde{\mathbf{X}}_{11}^{0}] + o_{P}(1),$$ and that the latter quantity is bounded away from zero. Furthermore, by assumption $ \max_{i} | \epsilon_{i} \tilde{\mathbf{Z}}_{i}^{0}|$ is bounded. Using equation it suffices to show that $$\max_i |\epsilon_{i} \tilde{\mathbf{Z}}_{i}^{0} - \hat \epsilon_{i} \tilde{\mathbf{Z}}_{i}| = o_{P}(1).$$ To this end note that $$\begin{aligned}
\begin{split}
& \max_{i}|\epsilon_{i} \tilde{\mathbf{Z}}_{i}^{0} - \hat \epsilon_{i} \tilde{\mathbf{Z}}_{i}| \\
&\le \max_{i}|\epsilon_{i} (\tilde{\mathbf{Z}}_{i}^{0} - \tilde{\mathbf{Z}}_{i})| + \max_{i}| (\epsilon_{i} - \hat \epsilon_{i}) (\tilde{\mathbf{Z}}_{i}- \tilde{\mathbf{Z}}_{i}^{0})| + \max_{i}| (\epsilon_{i} - \hat \epsilon_{i}) \tilde{\mathbf{Z}}_{i}^{0})|
\end{split}\end{aligned}$$ Now use the following inequalities $$\begin{aligned}
\begin{split}
\|\tilde{\mathbf{Z}}^{0} \|_{\infty} &\le C_{3} < \infty \\
\| \tilde{\mathbf{Z}} - \tilde{\mathbf{Z}}^{0}\|_{\infty} & = \| \tilde{\mathbf{X}}_{-1} \hat \gamma - \tilde{\mathbf{X}}_{-1}^{0} \gamma^{0} \|_{\infty} \\
&\le \| \tilde{\mathbf{X}}_{-1} (\hat \gamma - \gamma^{0}) \|_{\infty} + \| (\tilde{\mathbf{X}}_{-1} - \tilde{\mathbf{X}}_{-1}^{0}) \gamma^{0} \|_{\infty} \\
&\le \|\tilde{\mathbf{X}}_{-1} \|_{\infty} \| \hat \gamma - \gamma^{0} \|_{1} + \| \tilde{\mathbf{X}}_{-1} - \tilde{\mathbf{X}}_{-1}^{0} \|_{\infty} \| \gamma^{0} \|_{1} \\
&= o_{P}(1) \\
\|\hat \epsilon - \epsilon \|_{\infty}
&\le \| \tilde{\mathbf{X}} (\hat \beta - \beta^{0}) \|_{\infty} + \| (\tilde{\mathbf{X}} - \tilde{\mathbf{X}}^{0}) \beta^{0} \|_{\infty}\\
& \le \| \tilde{\mathbf{X}} \|_{\infty} \| \hat \beta - \beta^{0} \|_{1} + \| \tilde{\mathbf{X}} - \tilde{\mathbf{X}}^{0} \|_{\infty} \| \beta^{0} \|_{1} \\
&= o_{P}(1)
\end{split}
\end{aligned}$$ Here we used that by a sub-Gaussian tail bound [@boucheron2013concentration Chapter 2], (A2) implies $\| \tilde{\mathbf{X}} - \tilde{\mathbf{X}}^{0} \|_{\infty} = O_{P}(\sqrt{\log(p)/n})$. Furthermore, we used that by (A2), $\| \tilde{\mathbf{X}} \|_{\infty} = O_{P}(1)$, that by (D3) we have $\| \hat \beta - \beta^{0} \|_{1} = O_{P}(1/\sqrt{\log(p)})$, by (D2) we have $\|\hat \gamma - \gamma^{0}\|_{1} = O_{P}(1/\sqrt{\log(p)})$ and by assumption $\| \beta^{0} \|_{1} = O(1)$ and $\| \gamma^{0} \|_{1} = o(\sqrt{n/\log(p)})$. Hence, we have shown that $$\begin{aligned}
\frac{1}{n} \sum_{i} \xi_{i}^{0} - \xi_{i} &= o_{P}(1), \\
\frac{1}{n} \sum_{i} (\xi_{i}^{0})^{2} - \xi_{i}^{2} &= o_{P}(1).
\end{aligned}$$ As argued above, this concludes the proof.
### Proof of Theorem \[thm:doubly-robust-conf\]
Combine Lemma \[le:error\] and Lemma \[lem:D2D3\] with Proposition \[prop:almost\] and Proposition \[prop:varianceestimation\]. Note that due to assumption (A6), $u$ is bounded away from zero. Thus, $$\frac{\hat u^{2}}{u^{2}} = 1 + o_{P}(1),$$ which completes the proof.
Proof of Lemma \[lem:trick\]
----------------------------
Define $\tilde b_{k} = b_{k} - t \mathbb{E}[\partial_{t} b_{k}]$ for $k>1$ and $\tilde b_{1} = t$. By definition of $\beta^{0}$, $$\beta^{0} = \arg \min_{\beta} \mathbb{E}[\| \mathbf{Y} - \tilde{\mathbf{X}}^{0} \beta \|_{2}^{2}] = \arg \min_{\beta} \mathbb{E}[ ( Y - \sum_{k} \tilde b_{k}(T,X) \beta )^{2}].$$ Now use Lemma \[lem:doublyrobust\]. This implies that $$\mathbb{E}[\partial_{t} \mathbb{E}[Y|X=x,T=t]] = \mathbb{E}[\sum_{k} \partial_{t} \tilde b_{k} \beta_{k}^{0}].$$ Expanding the definition, $$\begin{aligned}
\mathbb{E}[\partial_{t} \mathbb{E}[Y|X=x,T=t]] &= \sum_{k}(\mathbb{E}[\partial_{t} b_{k}] - 1_{k > 1} \partial_{t} t \mathbb{E}[\partial_{t} b_{k}]) \beta_{k}^{0} \\
&= \sum_{k}(\mathbb{E}[\partial_{t} b_{k}] - 1_{k > 1} \mathbb{E}[\partial_{t} b_{k}]) \beta_{k}^{0} \\
&= \beta_{1}^{0}.\end{aligned}$$ This concludes the proof.
Proof of Lemma \[lem:doublyrobust\] {#sec:doublyrobust}
-----------------------------------
\[lem:doublyrobust\] Define $$b^{0} = \arg \min_{b \in \mathcal{B}} \mathbb{E}[(f^{0}(T,X)- b(T,X))^2],$$ and $$b_{*} = \arg \min_{b \in \mathcal{B}} \mathbb{E}[(f_*(T,X)- b(T,X))^2],$$ where $f_{*} = - \partial_t \log p(t|x)$. If, $\mathbb{P}$-a.s. we have $$b^{0}(T,X) = f^{0}(T,X) \text{ or } b_{*}(T,X) = f_*(T,X),$$ then $\mathbb{E}[\partial_t f^{0} ] =\mathbb{E}[\partial_t b^{0}]$.
As $\mathbb{P}$-a.s. we have $b^{0} = f^{0}$ or $b_{*} = f_*$, we also have that $\mathbb{P}$-a.s. $(b^{0} - f^{0} ) ( b_{*} - f_*) = 0$. Hence, $$\begin{aligned}
0 = \, &\mathbb{E}[ (b^{0} - f^{0} ) ( b_{*} - f_*) ] \\
=\, & \mathbb{E}[f^{0} f_*] + \mathbb{E}[b^{0} b_{*}] - \mathbb{E}[b^{0}f_{*}] -\mathbb{E}[f^{0}b_{*}] \\
= \, & \mathbb{E}[f^{0} f_*] + \mathbb{E}[b^{0} b_{*}] -\mathbb{E}[b^{0} b_{*}] -\mathbb{E}[b^{0} b_{*}] \\
= \, & \mathbb{E}[f^{0} f_*] - \mathbb{E}[b^{0} b_{*}] \\
= \, & \mathbb{E}[f^{0} f_*] - \mathbb{E}[b^{0} f_{*}]\end{aligned}$$ Here, we used repeatedly that $\mathbb{E}[f^{0} b_{*}] = \mathbb{E}[b^{0} b_{*}]$ and that $\mathbb{E}[b^{0} f_{*}] = \mathbb{E}[b^{0} b_{*}]$. Now, using that $$0 = \mathbb{E}[f^{0} f_*] - \mathbb{E}[b^{0} f_{*}] = \mathbb{E}[\partial_t f^{0} ] -\mathbb{E}[\partial_t b^{0}],$$ completes the proof.
Proof of Lemma \[lem:semieffic\]
--------------------------------
\[lem:semieffic\] Let the assumptions of Theorem \[thm:doubly-robust-conf\] hold. If $f^{0} \in \mathcal{B}$ and $\partial_{t} \log p(t|x) \in \mathcal{B}$, then the asymptotic variance of $\sqrt{n}(\hat \beta_{1}^{\text{despar}} - \beta_{1}^{0})$ is equal to $\text{Var}(\partial_{t} f^{0}) + \text{Var}(\epsilon) \mathbb{E}[(\partial_{t} \log p(T|X))^{2}]$, which is the semiparametric efficiency bound [@stoker1991equivalence; @powell1989semiparametric].
First, by Proposition \[prop:varianceestimation\], the asymptotic variance of $\sqrt{n}(\hat \beta_{1}^{\text{despar}} - \beta_{1}^{0})$ is $$\frac{\text{Var}(\epsilon_{1}) \text{Var}(\tilde{\mathbf{Z}}_{1}^{0})}{\text{Var}(\tilde{\mathbf{Z}}_{1}^{0})^{2}} + \text{Var}(\partial_{t} f^{0})
$$ Thus, it suffices to show that $$\text{Var}(\mathbf{Z}_{i}^{0}) = 1/\mathbb{E}[(\partial_{t} \log p)^{2}] \text{ for $i=1,\ldots,n$}.$$ Write $f_{*} = -\partial_{t} \log p$. Let us first consider a univariate regression of $T$ on $ f_{*} - t \mathbb{E}[\partial_{t} f_{*}]$. Then, the residual variance is $$\min_{\alpha} \mathbb{E}[(T + \alpha(f_{*}-T \mathbb{E}[\partial_{t}f_{*}]))^{2}] = \min_{\alpha} \mathbb{E}[(T(1- \alpha \mathbb{E}[\partial_{t}f_{*}]) + \alpha f_{*})^{2}]$$ Expanding, and using that $\mathbb{E}[T f_{*}] = 1$, $$\begin{aligned}
& \, \, \mathbb{E}[(T(1- \alpha \mathbb{E}[\partial_{t}f_{*}]) + \alpha f_{*})^{2}] \\
& = \mathbb{E}[T^{2}] (1 - 2 \alpha \mathbb{E}[\partial_{t}f_{*}] + \alpha^{2} \mathbb{E}[\partial_{t}f_{*}]^{2} ) \\
& \, \, + 2 \alpha (1- \alpha \mathbb{E}[\partial_{t}f_{*}]) + \alpha^{2} \mathbb{E}[f_{*}^{2}].
\end{aligned}$$ Now we can use that $\mathbb{E}[f_{*}^{2}] = \mathbb{E}[\partial_{t}f_{*}]$. Taking the derivative with respect to $\alpha$, we obtain $$\begin{aligned}
-2\mathbb{E}[T^{2}] \mathbb{E}[f_{*}^{2}] + 2 \alpha \mathbb{E}[T^{2}] \mathbb{E}[f_{*}^{2}]^{2} + 2 - 4 \alpha \mathbb{E}[f_{*}^{2}] + 2 \alpha \mathbb{E}[f_{*}^{2}]
\end{aligned}$$ Setting this term to zero and rearranging $$\begin{aligned}
-2\mathbb{E}[T^{2}] \mathbb{E}[f_{*}^{2}] +2 &= 2 \alpha \mathbb{E}[f_{*}^{2}] - 2 \alpha \mathbb{E}[T^{2}] \mathbb{E}[f_{*}^{2}]^{2} \\
&= \alpha \mathbb{E}[f_{*}^{2}] ( 2 - 2\mathbb{E}[f_{*}^{2}] \mathbb{E}[T^{2}])
\end{aligned}$$ Thus, the solution is $\alpha = 1/\mathbb{E}[f_{*}^{2}]$ and the resulting residual variance is $$\min_{\alpha} \mathbb{E}[(T + \alpha(f_{*}-T \mathbb{E}[\partial_{t}f_{*}]))^{2}] = \mathbb{E}[(f_{*}/\mathbb{E}[f_{*}^{2}])^{2}] = 1/\mathbb{E}[f_{*}^{2}]$$ By definition, $f_{*}$ is uncorrelated with $\tilde b_{k}$ for all $k>1$. Thus, $$\min_{\alpha_{1},\ldots,\alpha_{p}} \mathbb{E}[(T - \sum_{k>1} \alpha_{k}\tilde b_{k})^{2}]= 1/\mathbb{E}[f_{*}^{2}].$$
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---
address: 'CERN, EP-division, 1211 Geneva 23, Switzerland'
author:
- THORSTEN WENGLER
title: 'HOW WELL DOES QCD WORK FOR PHOTON-PHOTON COLLISIONS?'
---
Jet Production
==============
Differential di-jet cross sections have been measured by DELPHI [[@bib-delphi-dijet]]{} and OPAL [[@bib-opal-dijet]]{}. DELPHI has analysed 220 of data taken at =192-202 GeV using a cone algorithm with R=1 to define the jets. The first preliminary results of the di-jet cross sections as a function of the jet transverse energy and the jet pseudorapidity are found to be consistent with a previous measurement by OPAL [[@bib-opal-old-dijet]]{}. OPAL has used 593 from =189 GeV to 209 GeV to study the performance of NLO perturbative QCD and the hadronic structure of the photon in di-jet production. The jets are reconstructed using an inclusive clustering algorithm [[@bib-ktclus]]{}. The two jets are used to estimate the fraction of the photon momentum participating in the hard interaction as $ \xgpm \equiv {{\sum_{\rm jets=1,2}
(E^\mathrm{jet}{\pm}p_z^\mathrm{jet})}}/
{{\sum_{\rm hfs}(E{\pm}p_z)}} $ where $p_z$ is the momentum component along the $z$ axis of the detector and $E$ is the energy of the jets or objects of the hadronic final state (hfs). In single (double) resolved processes, where one (both) of the photons interacts via a fluctuation into a hadronic state, the hard interaction is accompanied by one or two remnant jets, and one or both are smaller than one. Differential cross sections as a function of or in regions of are therefore a sensitive probe of the structure of the photon. The left plot in Figure [\[fig1\]]{} shows the differential di-jet cross section as a function of the mean transverse energy of the di-jet system for the full -range, for either or $<$1 (dominated by single resolved processes), or $<$1 (dominated by double resolved processes). The prediction of perturbative QCD in NLO [@bib-ggnlo] using the [@bib-grv] parton densities is compared to the data after hadronisation corrections have been applied to the calculation. The calculation is in good agreement with the data, except for being to low at small for $<$1. In this region the contribution of the so called underlying event as described by the concept of multiple parton interactions (MIA) is expected to be largest. This contribution is not included in the NLO calculation. The three plots on the right hand side of Figure [\[fig1\]]{} show the differential cross section as a function of for the three regions in --space described above. The shaded histogram on the bottom of each of the three plots indicates the contribution of MIA to the cross section as obtained from the PYTHIA [[@bib-pythia]]{} MC generator. It is evident especially for $<$1 that the MIA contribution is of about the same size as the discrepancy between the measurement and the NLO prediction. Furthermore it is interesting to observe that there is next to no MIA contribution to the cross section if either or is required to be less than one, while the sensitivity to the photon structure at small is retained. As one would expect also the agreement of the NLO calculation with the measurement is best in this case. With these measurements one is therefore able to disentangle the hard subprocess from soft contributions and make the firm statement that NLO perturbative QCD is adequate to describe di-jet production in photon-photon collisions. At the same time a different sub-set of observables can be used to study in more detail the nature of the soft processes leading to the underlying event.
Transverse Momentum Spectra of Light Hadrons
============================================
The L3 collaboration has recently presented new results on the inclusive production of K$^0_S$ and neutral and charged pions in 414 of data taken at from 189 GeV to 202 GeV [[@bib-l3-inclhad]]{}. Compared to the jet measurements described above, inclusive hadron production offers a complementary signature to study the dynamics in hadronic photon-photon collisions. Predictions are available in NLO perturbative QCD which make use of fragmentation functions to translate the partonic cross sections to the observables measured. The uncertainty associated with the hadronisation correction commonly applied to the calculation of partonic NLO jet cross sections can hence be avoided, but is of course replaced by any uncertainty attached to the determination of the fragmentation functions. The new measurements can be shown to be consistent with results obtained previously by the OPAL collaboration [[@bib-opal-had]]{} where both measurements overlap in phase space. The new L3 data significantly extends the older measurements towards high transverse momenta for the production of charged and neutral pions. As can be seen in Figure [\[fig2\]]{} it is in this region that a significant discrepancy between the measurement and the QCD prediction occurs. The left plot in Figure [\[fig2\]]{} shows the differential cross section for inclusive [$\pi^o$]{}-production as a function of the transverse momentum $p_T$ of the [$\pi^o$]{}. For $p_T>$ 8 GeV the data exhibits a significantly harder spectrum than predicted by NLO QCD. This behaviour is confirmed by the independent measurement of charged pions. This discrepancy is particularly remarkable as it occurs at high transverse momenta of the produced hadrons, which in turn suggests an underlying high momentum partonic process. Under these conditions one expects the perturbative calculation to be reliable due to the relatively small value of the strong coupling constant. The discrepancy also appears to be in contradiction with the good agreement of the NLO QCD calculation obtained for the jet measurements described above, which should be sensitive to a similar set of processes and mainly differs in that fragmentation functions are not needed here. A possible direction for a further investigation of this anomaly may be derived from the OPAL measurement [[@bib-opal-had]]{}. Here the data is presented in four regions of the invariant hadronic mass $W$. For $W<$ 50 GeV a similar tendency of a harder $p_T$ spectrum in the data then in the theory can be observed. For $W>$ 50 GeV, however, the agreement is good. It would therefore be interesting to repeat these measurements for the full LEP2 data set in several regions of $W$.
Heavy Flavour Production
========================
Yet another test of the performance of perturbative QCD in photon-photon collisions is the production of heavy quarks. Due to the large physical scale set by the charm or beauty mass one can expect the theoretical prediction form perturbative QCD to be reliable. A significant part of the cross section at LEP2 energies is predicted to be from resolved processes, in particular the photon-gluon fusion [$\gamma g \rightarrow c\bar{c} (b\bar{b})$]{}. The production rate of charm and beauty quarks therefore depends not only on their mass but also on the gluon density in the photon. The latest results from the LEP collaborations on charm production use identified charged D$^*$ meson for charm tagging. L3 [[@bib-l3-cc]]{} has analysed 683 at =183-209 GeV, DELPHI [[@bib-delphi-cc]]{} 458 at =183-209 GeV, OPAL [[@bib-opal-cc]]{} 428 at =183-202 GeV, and ALEPH [[@bib-aleph-cc]]{} 236 at =183-189 GeV. The smallest uncertainties are to be expected if one compares the measurements and the theoretical predictions for the restricted phase space corresponding to the experimental acceptances, as any extrapolation to the total charm production cross section introduces additional assumptions. The differential cross section of charged D$^*$ production as a function of the D$^*$ transverse momentum is in good agreement among the four experiments, with the possible exception of a slightly harder spectrum for ALEPH. NLO perturbative QCD using massive quarks [[@bib-cc-nlo]]{} and a charm mass of m$_c$=1.5 GeV describes the data well. This translates to similar agreement of the total charm production cross section with the QCD prediction [[@bib-cc-nlo]]{} as can be seen in the right plot of Figure [\[fig2\]]{}. In the same figure the results are shown for the total production cross section of b$\bar{\mathrm{b}}$ as obtained by the L3 and the OPAL collaborations. Both measurements are based on the higher transverse momentum of the lepton in semileptonic b-decays as compared to those of lighter quarks. L3 has analysed 410 at =189-202 GeV using both the electron and the muon signatures [[@bib-l3-bb]]{}, OPAL has analysed 371 at =189-202 GeV using the muon channel [[@bib-opal-bb]]{}. The plot demonstrates that there is good agreement between the experimental results. The prediction of perturbative QCD in NLO however significantly underestimates the measurements. Similar deficiencies have been observed in b-production in p$\bar{\mathrm{p}}$- and ep-collisions. A recent analysis of the b-fragmentation functions used in these calculations suggests that much of this discrepancy can be recovered at least for p$\bar{\mathrm{p}}$-collisions [[@bib-cacciari]]{}. This analysis considers also the higher moments of the differential distributions of the average b-quark energy fraction carried by the B-meson. Using recent LEP and SLD data in this way to fix the b-fragmentation the agreement of the perturbative calculation with the p$\bar{\mathrm{p}}$-data becomes acceptable. One might hope that a similar technique applied to ep- and photon-photon-collisions will also improve the agreement of the theory with the measurements.
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==========
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abstract: 'The difficulty is analysed in evaluating fluctuations in phase transition of finite-size system at temperature far below the critical point. Film system is discussed with one-component order parameter $\phi^4$ model for phase transition. Non-trivial vacuum state corresponding to minimum Hamiltonian is given approximately for various boundary conditions. It is shown that the spontaneous symmetry breaking plays an important role for such systems, and that perturbative calculations can be done safely when the effect of the vacuum state or the local spontaneous symmetry breaking is taken into consideration.'
address: ' Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430079, China[^1]'
author:
- 'C.B. Yang and X. Cai'
title: 'Local Spontaneous Symmetry Breaking for Film System Within Scalar $\phi^4$ Model for Phase Transition'
---
Finite-size effects near critical points have been remained over the past two decades to be an important topic of the active research both theoretically and experimentally \[1\]. Nowadays, the experimental sample are usually so pure and so well shielded from perturbing fields that the correlation length can grow up to several thousand angstroms as the critical point is approached. When one or more dimensions of a bulk system is reduced to near or below a certain characteristic length scale, the associated properties are modified reflecting the lower dimensionality. It is believed that finite-size effects are precursors of the critical behavior of the infinite system and can be exploited to extract the limiting behavior. A central role plays the finite-size scaling behavior predicted by both the phenomenological \[2\] and renormalization group \[3\] theories. Those theories allowed a systematic discussion of the finite-size effects and, consequently, form the cornerstone of our current understanding of the way in which the singularities of an infinite system are modified by the finiteness of the system in some or all of the dimensions. Of course, the exact form of scaling functions can’t be given in those scaling theories.
In 1985, Brézin and Zinn-Justin (BZ) \[4\] and Rudnick, Guo and Jasnow (RGJ) \[5\] developed two field-theoretical perturbation theories for the calculation of the finite-size scaling functions within the $\phi^4$ model which corresponds to the Ising model. Most applications of these theories to three-dimensional systems have been restricted to $T$ higher than the bulk critical temperature $T_C$ \[6\] with a few calculations in region below $T_C$ \[7\]. In recent years the $\phi^4$ and the extended $\phi^6$ models have been used to investigate the multiplicity fluctuations in the final states for first- and second-order phase transitions of quark gluon plasma \[8, 9\], under the approximation similar to the so-called zero-mode approximation. However, some limitations exist in the theories of Ref. \[4,5\]. As pointed out in the first paper in Ref. \[10\], the theory of BZ is not applicable for $T<T_C$ and the results from RGJ theory are not quantitatively reliable in the same temperature region since the coefficients of the Gaussian terms in the integrals are negative for those temperatures. In Ref. \[11\] the order parameter is expanded into sum of eigenfunctions of $\bigtriangledown^2$ for various boundary conditions. Again, the functional integral is turned out into normal integrals. But the fluctuations can be evaluated only for temperature not too far below the critical point. Authors of Ref. \[10\] tried to avoid the difficulty mathematically, but they failed to account for the origin of the difficulty physically. Although the modified perturbation method in Ref. \[10\] can be used for both $T>T_C$ and $T<T_C$, the calculation is lengthy and can be done only to the first order in practice. Since one does not know the exact order of values of higher order terms, theoretical results have large uncertainty.
It should be pointed out that all perturbation theories mentioned above are based on Fourier decomposition of the order parameter. This method is natural because the decomposition enables one to transform the functional integral into an infinite product of tractable normal integrals. Although such decomposition has simple physical explanation which is very fruitful for the understanding of properties of infinite systems and can deduce reliable physical results, as in the case of usual field theories in particle physics, it brings about a great deal of calculations for systems with finite-size. This is not surprising. As is well-known, quantities complicated in coordinate space may have simple momentum spectra thus look simpler in momentum space, but those obviously nonzero only in a finite range must have puzzling momentum spectra. Therefore, for the study of properties of finite-size systems, calculations in coordinate space might be simpler and more effective. The point here is that one must calculate the complicated functional integral which is very difficult to be evaluated directly.
It should be asked that which physical effect causes the failure of direct perturbative calculation of fluctuations for finite-size system with temperature below $T_C$. In our opinion, the real origin of the difficulty lies in the lack of knowledge about the spontaneously symmetry breaking for finite-size systems. It is well-known that an infinite system will have non-zero mean order parameter $\phi_0$, which is called vacuum state of the system in this Letter since it corresponds to minimum of the Hamiltonian $H$, if the temperature is below the critical one, and everyone knows that the difficulty of negative coefficient for the Gaussian term can be overcome by shifting the order parameter, $\phi\to\phi+\phi_0$. This phenomenon is known as the spontaneous symmetry breaking because of the fact that $\phi_0$ does not have the same symmetry as $H$ does. This kind of spontaneous symmetry breaking for infinite system can be called global since the shift $\phi_0$ is the same for every point in the space. For finite-size system, such a simple shift of the order parameter does not work because of the existence of specific boundary conditions for the systems. Anyway, fluctuations of the system, in their own sense, should be around certain vacuum state which corresponds to minimum Hamiltonian $H$, and they can be approximated by Gaussian terms in most cases if they are not very large. Thus one sees that the vacuum state plays an determinative role in the study of fluctuations in the phase transitions. For infinite system, the vacuum state $\phi_0$ is constant and can easily be calculated. But for finite-size systems, the vacuum is surely not constant nor it is easy to be obtained. So, the spontaneous symmetry breaking for finite size system can be called local one. Therefore, the solution for the vacuum state is non-trivial and necessary, and one has reason to hope that the difficulty mentioned above can be overcome once the vacuum state is known.
In this Letter, we first calculate the vacuum states for $\phi^4$ model of phase transition with one-component order parameter under various boundary conditions. Then, with the vacuum states, the Hamiltonian of the system is reexpressed as Gaussian term and higher order fluctuations of a locally shifted order parameter. And it is shown that the perturbative calculation can be done with the new Hamiltonian for temperatures far below the bulk critical point.
In a $\phi^4$ model for phase transition with a one-component order parameter, the partition function can be expressed as a functional integral of exponential of the Hamiltonian $H$ of the system $$Z=\int {\cal D}\phi \exp(-H)=\int {\cal D}\phi \exp\left\{-\int d^3\,x\left[{
\gamma\over 2}\phi^2+\frac{1}{2}(\bigtriangledown \phi)^2+\frac{u}{4!}\phi^4
\right]\right\}\ ,$$
in which $\gamma=a(T-T_C)$, $a$ and $u$ are temperature dependent positive constants, $\phi$ is the order parameter of the system. In the following, we are limited only to systems of a film with thickness $L$. Since we are interested only in the temperature region $T<T_C$ or $\gamma<0$, the Hamiltonian $H$ can be standardized by introducing correlation length $\xi=\sqrt{-1/\gamma}$, new order parameter $\Psi=\phi/\phi_0$ with $\phi_0=\sqrt{-6\gamma/u}$ the vacuum state for bulk system, scaled coordinates ${\bf r}^\prime={\bf r}/L$, and reduced thickness $l=L/\xi$, into $$H=\int d^3 x^\prime {L^3\phi_0^2\over \xi^2}\left[
{1\over 2l^2}(\bigtriangledown^\prime \Psi)^2-\frac{1}{2}\Psi^2+\frac{1}{4}
\Psi^4\right]\ .$$
From this expression one can get the equation for the vacuum state by ${\delta H\over \delta\Psi}=0$. The vacuum state satisfies $${1\over l^2}{d^2\Psi_0\over dx^2}=-\Psi_0+\Psi_0^3\ .$$
In the equation we have used $x$ instead of $x^\prime$ in the range (0, 1) to denote the coordinate along the thickness direction. Derivatives in other directions do not appear in the equation since any state with non-zero derivatives in other directions does not correspond to minimum $H$. But if the system in fully limited in all directions, last equation should have $\bigtriangledown^2$ in place of $d^2/dx^2$. In Ref. \[12\] last equation is solved analytically for Dirichlet boundary conditions $\Psi(0)=\Psi(1)=0$. The exact solution is $$\Psi_0(x)={\sqrt{2}k\over \sqrt{1+k^2}}{\rm sn}(2xF(k), k) \ ,$$
in which $k$ is determined by $l$ through $l=2\sqrt{1+k^2} F(k)$. Here, $F(k)$ is the first kind of complete elliptic integral, ${\rm sn}(x,k)$ is elliptic sine function. Unfortunately, no simple compact solution is found yet for other boundary conditions. One can easily see that the main obstacle comes from the nonlinear term $\Psi^3$ in the differential equation of $\Psi$. To find approximate solutions of $\Psi$ for other boundary conditions, the following method can be used. First of all, we replace $\Psi^3$ by $\lambda\Psi$ and get a solution satisfying the same boundary condition. For Dirichlet boundary conditions, the solution is $$\Psi_0=A\sin\pi x\ , \hbox{\hspace*{0.8cm}and } \lambda=1.0-\pi^2/l^2.$$
The constant $A$ can be determined by requiring the mean square of the deviation caused by the replacement, i.e., the integral $\int_0^1 dx (\Psi_0^3-\lambda\Psi_0)^2$, to be minimum. Thus one gets $$\Psi_0(x)=\sqrt{{4\over 3}\left(1-{\pi^2\over l^2}\right)}\sin\pi x\ .$$
Now one can see that the requirement of minimum deviation caused by the replacement is equivalent to retaining $\sin\pi x$ term but neglecting terms with higher frequency in $\Psi_0^3$. Thus, this approximation is equivalent to the standard functional variation method. The virtue of this method is that it can be used more simply and in a step-by-step way. As discussed in Ref. \[12\], the vacuum state $\Psi_0=0$ if the reduced thickness $l$ of the film is less than $\pi$. The existence of minimum reduced thickness of the film implies a shift of the critical temperature for the finite system from the bulk one. The exact solutions and the approximate ones are compared in Fig. 1 for $l/\pi$=1.05, 1.10, 1.15, and 1.20. A very good approximation can be seen. For larger $l$, the same approximative method can be used further after shift $\Psi_0=\Psi_0^\prime+\sqrt{4(1-\pi^2/l^2)/3}\sin \pi x$ in Eq. (3).
For Neumann boundary conditions, $\Psi_0^\prime(0)=\Psi_0^\prime(1)=0$, the vacuum state can also be approximately obtained. The result is $$\Psi_0=\left\{\begin{array}{ll}
0\ ,& \hbox{\ \ \ for\ \ \ }\ l\le \pi\cr
\sqrt{4(1-\pi^2/l^2)/3}\cos\pi x\ ,& \hbox{\ \ \ for\ \ \ } \pi<l\le 2\pi\cr
\sqrt{4(1-\pi^2/l^2)/5-3/5}\pm\sqrt{8/5-4(1-\pi^2/l^2)/5}\cos\pi x
\ .\hspace*{0.3cm} & \hbox{\ \ \ for\ \ \ } l>2\pi
\end{array}
\right.$$
The two solutions for $l>2\pi$ can be connected through $x\leftrightarrow 1-x$.
Then one can consider mixed boundary conditions $\Psi_0(0)=0,
\Psi_0^\prime(1)=0$. The first order approximation of the solution for vacuum state is $$\Psi_0(x)=\sqrt{{4\over 3}\left(1-{\pi^2\over 4l^2}\right)}\sin{\pi
x\over 2}\ .$$
As a final example, we give the vacuum state for periodic boundary condition $\Psi_0(x)=\Psi_0(1+x)$. The approximate vacuum state is $$\Psi_0(x)=\left\{
\begin{array}{ll}
0\ ,& \hbox{\ \ \ for\ \ \ }l \le 2\pi\cr
\sqrt{4(1-4\pi^2/l^2)/5}(\cos2\pi x\pm \sin 2\pi x)\ ,
& \hbox{\ \ \ for\ \ \ }4\sqrt{6}\pi/3\ge l>2\pi \cr
\sqrt{3(1-32\pi^2/3l^2)/11}\pm\sqrt{4(
1+4\pi^2/l^2)/11}(\cos2\pi x\pm \sin2\pi x) \ . \hspace*{0.3cm}&
\hbox{\ \ \ for\ \ \ }
l>4\sqrt{6}\pi/3
\end{array}
\right.$$
It should be pointed out that $-\Psi_0$ is also a vacuum state of the system. Then the fluctuations of the system can be around either $\Psi_0$ or $-\Psi_0$. This is the copy for finite systems of spontaneous symmetry breaking in $\phi^4$ model. With the vacuum state $\Psi_0$, one can shift the order parameter $\Psi=
\Psi^\prime+\Psi_0$, then the Hamiltonian $H$ turns out to be $$H=H[\Psi_0]+{L^3\phi_0^2\over \xi^2}\int d^3 x
{1\over 2}\left[{1\over l^2}(\bigtriangledown \Psi^\prime)^2-{\Psi^\prime}^2+
3\Psi_0^2{\Psi^\prime}^2
+2\Psi_0{\Psi^\prime}^3+{1\over 2}{\Psi^\prime}^4\right] \ .$$
In this expression, $H[\Psi_0]$ has the same form as $H$ in Eq. (2) with $\Psi_0$ in place of $\Psi$. Now the quadratic part of fluctuation $\Psi$ is positive definite for $l$ larger than characteristic length, or for temperature enough below the critical point. Then one sees that the new Hamiltonian can be safely used to calculate perturbatively fluctuations at low temperature region for finite systems. For the sake of easier perturbative calculation, one can use $\langle \Psi_0^2\rangle$ in place of $\Psi_0^2$, and treat all other terms, $H_I={L^3\phi^2\over \xi^2}\int d^3x \left[
{3\over 2}(\Psi_0^2-\langle \Psi_0^2\rangle){\Psi^\prime}^2
+\Psi_0{\Psi^\prime}^3
+{1\over 4}{\Psi^\prime}^4\right]$, as small perturbations. In the lowest order, ignoring contributions from $H_I$, one can get one-point and two-point and other correlation functions in terms of $\Psi_0$ and Green’s function $G(x,y)$, the inverse of operator $-\bigtriangledown^2/l^2-1
+3\langle\Psi_0^2\rangle$. For example, $$\begin{array}{l}
\langle \phi(x)\rangle=L^3\phi_0\Psi_0(x)\\
\langle\phi(x)\phi(y)\rangle=L^6\phi_0^2(\Psi_0(x)\Psi_0(y)+{\xi^2\over
L^3\phi_0^2}G(x,y))
\end{array}$$
For system with Dirichlet boundary conditions and under the condition that $l$ is a little larger than $\pi$, the Green’s function $G(x,y)$ is $$G(x;y)=\int {d^2p\over (2\pi)^2}
\sum_{i=1}^\infty
{\sin i\pi x_1 \sin i\pi y_1 e^{i{\bf p}\cdot {\bf (r-r^\prime)}}\over
2(1-\pi^2/l^2)-(1-i^2\pi^2/l^2-{\bf p}^2/l^2)}$$
Here, $x_1, y_1$ are components of coordinates in finite size direction of two points $x, y$, ${\bf r}$ and ${\bf r}^\prime$ are vectors in other directions, ${\bf p}$ is the corresponding momentum. An important feature of the Green’s function is that the translational invariance in the direction with finite length is violated. This result is natural for finite system. In this time the Green’s function cannot be written as function of the difference of the coordinates of two points. The physical reason is simple. The Green’s function is the response at $x^\prime$ of the system to a source at $x$. When both $x$ and $x^\prime$ are translated in the same way, the influence of the boundary response changes. Thus, the net response to the source also changes. So translational invariance is surely violated. From Eq. (10) one can easily get the vertices needed. Then a perturbative calculation can be done readily, which is beyond the scope of this Letter.
In summary, we showed the importance of local spontaneous symmetry breaking for finite system in the calculation of fluctuations in phase transition in low temperature region for such system. The vacuum states are approximately given for various boundary conditions for film system within scalar $\phi^4$ model for phase transition. With the vacuum state, perturbative calculations can be done safely.
This work was supported in part by the NNSF, the Hubei SF and the SECF in China.
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[Figure Caption]{}
[**Fig. 1**]{} Comparison between exact solutions and approximate ones for Eq. (3) under Dirichlet boundary conditions for $l/\pi$=1.05, 1.10, 1.15, and 1.20. The solid curves correspond to exact solutions, dotted curves are drawn according to Eq. (6).
[^1]: Mailing address:yangcb@iopp.ml.org; yangcb@sgi31.rmki.kfki.hu
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abstract: 'Simple quark models for the low lying vector mesons suggest a mixing between the $u$– and $d$–flavors and a violation of the isospin symmetry for the $\rho-\omega$ system much stronger than observed. It is shown that the chiral dynamics, especially the QCD anomaly, is responsible for a restoration of the isospin symmetry in the $\rho-\omega$ system.'
address: |
Sektion Physik, Ludwig–Maximilians–Universität München,\
Theresienstrasse 37, D–80333 München
author:
- 'H. Fritzsch and A.S. Müller'
title: 'Isospin Symmetry Breaking and the $\rho$–$\omega$–System'
---
We should like to report on an interesting phenomenon, thus far unrecognized, which we found in the vector meson sector and which is directly related to the violation of isospin symmetry and its description whithin the QCD framework. It points towards a dynamical restoration of isospin symmetry in the low energy sector of QCD.\
Although there are no doubts that all observed strong interaction phenomena can be described within the theory of QCD, a quantitative description of the strong interaction phenomena in the low energy sector (e.g in the energy range $0...2$ GeV) is still lacking, although some features of the low energy phenomena have been partially understood by the lattice gauge theory approach. Nevertheless a number of features of the strong interaction phenomena at low energies can be related to basic symmetries like isospin or $SU(3)$, to symmetry breaking effects and to basic properties derived in simple phenomenological models.\
The low energy sector of the physics of the strong interactions is dominated by the low–lying pseudoscalar mesons ($\pi, K,\eta,\eta'$) and the low–lying vector mesons ($\rho,\omega, K^{*},\phi$). It is well–known that the structures of the quark wave functions of the pseudoscalar mesons ($0^{-+}$) and of the vector mesons ($1^{--}$) differ substantially.\
In the vector meson channel there is a strong mixing between the eights component of the $SU(3)$ octet (wave function: $(\bar u u +\bar d d -2\bar s s)/(\sqrt 6)$) and of the $SU(3)$ singlet (wave function: $(\bar u u +\bar d d +\bar s s)/(\sqrt 3)$). The mixing strength is such that the mass eigenstates are nearly the state $(\bar u u +\bar d d)/(\sqrt 2)$), the $\omega$–meson, and the state $\bar s s$, the $\phi$–meson. While this feature looks peculiar, when viewed upon from the platform of the underlying $SU(3)$ symmetry, it finds a simple interpretation, if one takes into account the Zweig rule [@okub63], which states that the mixing must take place in such a way that quark lines are neither destroyed nor created.\
On the other hand the pseudoscalar mesons follow the pattern prescribed by the $SU(3)$ symmetry in the absence of singlet–octet mixing. The neutral mass eigenstates $\eta$ and $\eta'$ are nearly an $SU(3)$–octet or $SU(3)$–singlet: $$\begin{aligned}
\quad\eta &\approx& \frac{1}{\sqrt 6}\,(\bar u u +\bar d d -2\bar s s)
\quad\mbox{or}\\
\quad \eta' &\approx&\frac{1}{\sqrt
3}\,(\bar u u +\bar d d + \bar s s)\,. \nonumber\end{aligned}$$ This indicates a large violation of the Zweig rule in the $0^{-+}$ channel [@fritzsch75] [@venez89]. Large transitions between the various ($\bar q q$)–configurations must take place. In QCD the strong mixing effects are related to the spontaneous breaking of the chiral $U(1)$ symmetry normally attributed to instantons. Effectively the mass term for the pseudoscalar mesons can written as follows, neglecting the effects of symmetry breaking in the gluonic mixing term [@fritzsch73] [@hooft76] [@shifm86]: $$\quad M^2_{\bar q q} = \left( \begin{array}{ccc}
M^2_u & 0 & 0\\ 0 & M^2_d & 0\\ 0 & 0 & M^2_s \end{array} \right)\,$$ $$\quad\quad\quad\quad\quad+\,\,
\lambda \left( \begin{array}{ccc} 1&1 &1\\ 1&1 &1\\ 1&1 &1\\
\end{array} \right)\,,$$ where $M^{2}_{u},M^{2}_{d}$ and $M^{2}_{s}$ are the $M^{2}$–values of the masses of quark composition $\bar u u, \bar d d$ and $\bar s s$ respectively.\
It is well–known that the mass and mixing pattern of the $0^{-+}$–mesons is described by such an ansatz [@fritzsch75]. The parameter $\lambda$, which describes the mixing strength due to the gluonic forces, is essentially given by the $\eta'$–mass: $\lambda \cong 0.24$ GeV$^{2}$. Since $\lambda$ is large compared to the strength of $SU(3)$ violation given by the $s$–quark mass, large mixing phenomena are present in the $0^{-+}$ channel, as seen in the corresponding wave functions.\
The situation is different in the vector meson $1^{--}$ channel. Here the gluonic mixing term is substantially smaller than the strength of $SU(3)$ violation such that the Zweig rule is valid to a good approximation. If one describe the mass matrix for the vector mesons in a similar way as for the pseudoscalar, we have $$M_{\bar q q} = \left( \begin{array}{ccc} M(\bar u u) & 0 & 0\\ 0 &
M(\bar d d) & 0\\ 0 & 0 & M(\bar s s) \end{array}\right)\,$$ $$\quad\quad\quad\quad\quad\quad\quad\quad\,
+\,\,\tilde \lambda \left( \begin{array}{ccc} 1&1 &1\\
1&1 &1\\ 1&1 &1\\ \end{array} \right)\,,$$ here $M(\bar q q)$ denotes the mass of a vector meson with quark composition $\bar q q$ in the absence of the mixing term. The magnitude of the mixing term $\tilde \lambda$ can be obtained in a number of different ways, e.g by considering the $\rho_{0}$–$\omega$ mass difference. Neglecting the isospin violation caused by the $m_{d}$–$m_{u}$ mass splitting, the gluonic mixing term is responsible for the $\rho_{0}$–$\omega$ mass shift: $$\quad M_{\omega}-M_{\rho}= 2 \tilde \lambda\quad ,$$ $$\quad\tilde \lambda \cong 6.0 \pm 0.5
\,\,\mbox{MeV}\,.$$ The decay $\phi\rightarrow 3\pi$ proceeds via the $(\bar u u+\bar d d)$–admixture in the $\phi$ wave function. Using the observed branching ratio $$\quad\,\,\frac{\Gamma (\phi\rightarrow 3\pi)}{\Gamma (\omega\rightarrow
3\pi)}\,\,\simeq 0.09 \,,$$ on finds a gluonic mixing term of the same order of magnitude.\
In QCD the isospin symmetry is violated by the mass splitting between the $u$– and $d$–quark. Typical estimates give: $$\quad \frac{m_{d}-m_{u}}{\frac{1}{2}(m_{d}+m_{u})}\cong 0.58\,.$$ The observed smallness of isospin breaking effects is usually attributed to the fact that the mass difference $m_{d}$ – $m_{u}$ is small compared to the QCD scale $\Lambda_{QCD}$. However in the case of the vector mesons the QCD interaction enters in two different ways:\
\
a) In the chiral limit of vanishing quark masses the masses of the vector mesons are solely due to the QCD interaction, i.e. $M= \mbox{const}\cdot\Lambda_{QCD}$.\
\
b) The QCD mixing term will lead to a mixing among the various flavour
components such that the $SU(3)$ singlet (quark composition $(\bar u u+
\bar d d +\bar s s)/\sqrt 3$) is lifted upwards compared to the two other neutral components given by the wave functions $(\bar u u - \bar d
d)/\sqrt 2$ and $(\bar u u + \bar d d-2 \bar s s)/\sqrt 6$. The corresponding mass shift is given by $3\tilde \lambda$.\
\
We approach the real world by first introducing the mass of the strange quark. As soon as $m_{s}$ becomes larger than $3\tilde \lambda$, substantial singlet–octet mixing sets in, and the mass of one vector meson increases until it reaches the observed value of the $\phi$–mass. At the same time the Zweig rule, which is strongly violated in the chiral $SU(3)_{L}\times SU(3)_{R}$ limit becomes more and more valid.\
The validity of the Zweig rule is determined by the ratio $m_{s}/\tilde\lambda$. If this ratio vanishes, the Zweig rule is violated strongly. In reality, taking $m_{s}$ (1GeV) $\approx$ 150 MeV, the ratio $m_{s}/\tilde\lambda$ is about $25$ implying that the Zweig rule is nearly exact.\
In a second step we introduce the light quark masses $m_{u}$ and $m_{d}$. We concentrate on the non–strange vector mesons. If the gluonic mixing interaction were turned off, the mass eigenstates would be $v_{u}=|\bar u u\rangle$ and $v_{d}=|\bar d d\rangle$. The masses of these mesons are given by: $$\quad M(v_{u}) = \langle v_{u}|\, H^{0}+ m_{u}\,\bar u u\,|
\,v_{u}\rangle\,,$$ $$\quad M(v_{d}) = \langle v_{d}|\, H^{0}+ m_{d}\,\bar d d\,|
\,v_{d}\rangle\,.$$ Here $H^{0}$ is the QCD–Hamiltonian in the chiral limit $m_{u}=m_{d}=0$. We have assumed, as expected in simple valence quark models that the matrix elements $\langle v_{u}|\,\bar d d\,|\,
v_{u}\rangle$ and $\langle v_{d}|\, \bar u u\,|\,v_{d}\rangle$ are very small and can be neglected. Thus the masses can be written as $$\quad M(v_{u}) = M_{0}+ 2m_{u}\cdot c\,,$$ $$\quad M(v_{d})= M_{0}+ 2m_{d}\cdot c\,.$$ (c: constant, given by the expectation value of $\bar q q$). The introduction of the light quark masses induces positive mass shifts for both $v_{u}$ and $v_{d}$. These mass shifts can be estimated by considering the corresponding mass shifts of the charged $K^{*}$–mesons: $$\begin{aligned}
\Delta \tilde M & = & \tilde M(K^{*0})- \tilde M(K^{*+})\\
& = & (m_{d}-m_{u})\cdot c = 4.44\,\,\mbox{MeV}\, , \nonumber\end{aligned}$$ where $\tilde M$ is the mass of the vector meson in the absence of electromagnetism. Taking the electromagnetic mass shift into account [@scadr84]. $$\begin{aligned}
\Delta M^{2}(K^{*})_{elm}&\cong&
\frac{2}{3}\Delta M^{2}(\rho)\\
\Delta M^{2}(\rho)&=& \Delta M^{2}(K^{*})- 3\Delta M^{2}(K) \nonumber\\
& & +\,\frac{9}{2} \Delta M^{2}(\pi)\, , \nonumber\end{aligned}$$ we find $M(K^{*})_{elm}\cong -3,59 $ MeV and $\Delta \tilde M = (m_{d}-m_{u})\cdot c = 0.85$ MeV. Thus we obtain for the mass shift of the neutral vector mesons: $$\begin{aligned}
M(v_{d})-M(v_{u}) & \cong & 2\,(m_{d}-m_{u})\cdot c \\
& \cong & 1.7 \,\,\mbox{MeV}. \nonumber\end{aligned}$$ We like to emphasise that our way of relating the mass differences between the $(\bar u u)$ and $(\bar d d)$ vector mesons to the mass differences between the $(\bar u s)$ and $(\bar d s)$ vector mesons
is more than using isospin symmetry, since the first two mesons are members of an isotriplet, while the second two mesons form an isodoublet. Using simple $SU(6)$ type quark models or using $SU(3)$ symmetry with the additional input that the $\bar q q$–operator has a pure F–coupling, in accordance with observation in the case of the baryons, the two mass terms are indeed related, as we stated, i.e. $M(\bar d d)-M (\bar u u)= 2(M(\bar s d)- M(\bar s u))$.\
It is remarkable that this mass shift is of similar order of magnitude as the mass shift between the isosinglet and isotriplet state in the chiral limit, where isospin symmetry is valid. This implies that the strength of the gluonic
mixing term is comparable to the $\Delta I = 1$ mass term. If follows that the eigenstates of the mass operator taking both the violation of isospin and the gluonic mixing into account will not be close to being eigenstates of the isospin symmetry.\
\
For the $\rho_{0}$–$\omega$ system the mass operator takes the form: $$\quad M = \left( \begin{array}{cc} M(v_u) & 0 \\ 0 & M(v_d)
\end{array} \right)\,$$ $$\quad\quad\quad\quad\quad\quad \,+\,\,\tilde \lambda
\left( \begin{array}{cc} 1&1 \\1&1 \end{array} \right)\,.$$ Using $M(v_{u})= M(\bar u u), M(v_{d})= M(\bar d d)$ and $\tilde \lambda= 5.9$ MeV, we find $$\begin{aligned}
|\rho_{0}\rangle = |\bar u u\rangle - |\bar d d\rangle \nonumber\end{aligned}$$ $$\begin{aligned}
&=& \mbox{cos}\, \alpha \,|\frac{1}{\sqrt 2}(\bar u u -\bar d d)\rangle
- \mbox{sin}\, \alpha \, |\frac{1}{\sqrt 2} (\bar u u +\bar d
d)\rangle\nonumber\\
&=& 0.997 |\frac{1}{\sqrt 2}(\bar u u -\bar d
d)\rangle + 0.071 |\frac{1}{\sqrt 2}(\bar u u +\bar d
d)\rangle\nonumber\end{aligned}$$ $$\begin{aligned}
|\omega\rangle = |\bar u u\rangle + |\bar d
d\rangle \nonumber\end{aligned}$$ $$\begin{aligned}
&=&\,\,\,\mbox{sin}\, \alpha \,|\frac{1}{\sqrt 2}(\bar u u -\bar d d)\rangle +
\mbox{cos}\, \alpha\, |\frac{1}{\sqrt 2}(\bar u u +\bar d
d)\rangle\nonumber\\
&=& -0.071 |\frac{1}{\sqrt 2}(\bar u u -\bar d
d)\rangle + 0.997 \,|\frac{1}{\sqrt 2}(\bar u u +\bar d
d)\rangle\nonumber\end{aligned}$$ $$\begin{aligned}
M(\omega)-M(\rho_{0}) &=& \sqrt{(M(v_{u})-M(v_{d}))^{2}
+4\tilde \lambda^{2}} \nonumber \\ &=& 2.02\,\,\tilde \lambda\end{aligned}$$ The mixing angle $\alpha$ discribing the strength of the triplet–singlet mixing is about $-4.1^{o}$, i.e. a sizeable violation of isospin symmetry is obtained. Neither is the $\rho_{0}$–meson an isospin triplet, nor is the $\omega$–meson an isospin singlet.\
\
The conclusions we have derived follow directly from the observed smallness of the gluonic mixing in the vector meson channel and the $m_{u}-m_{d}$ mass splitting, as observed e.g. in the mass spectrum of the $K^{*}$–mesons. Nevertheless they are in direct conflict with observed facts. According to eq. (13), the probability of the $\rho_{0}$–meson to be an $I=|\frac{1}{\sqrt 2}(\bar u u +\bar d d)\rangle$–state is $\mbox{sin}^{2}\alpha \cong 0.51\%$. Taking into account the observed branching ratio for the decay $\omega \rightarrow
\pi^{+}\pi^{-}$, BR $\cong(2.21\pm 0.30)\%$, this probability is bound to be less than $0.12\%$, in disagreement with the value derived above. Obviously our theoretical estimate cannot be correct.\
We consider the discrepancy described above as a serious challenge for our understanding of the low energy sector of QCD. It arrives since the strength of gluonic mixing is comparable to the estimated mass difference between the $|\bar u u\rangle$– and $|\bar d d \rangle$–state. We can envisage two possible solutions.\
a) The strength of the gluonic mixing in the $1^{--}$–channel is much
larger than envisaged. This would lead to a substantial violation of the
Zweig rule and to a $\omega$–$\rho$ mass difference larger than observed. Thus an increase of $\tilde \lambda$ is excluded.\
b) The mass difference $\Delta M = M(v_{d})-M(v_{u})$ must be smaller than estimated above. In order to reproduce the observed branching ratio for the decay $\omega \rightarrow \pi^{+}\pi^{-}$, $\Delta M$ cannot exceed $0.82$ MeV.\
We believe that this is the correct solution of the problem, for the following reasons. We consider the following two–point functions $$\begin{aligned}
u_{\mu\nu}&=& \langle 0|\bar u (x)\gamma_{\mu}u(x)\,
\bar u (y)\gamma_{\nu} u(y)|0\rangle\,, \nonumber \\
d_{\mu\nu}&=& \langle 0|\bar d (x)\gamma_{\mu}d(x)\,
\bar d (y)\gamma_{\nu} d(y)|0\rangle \\
m_{\mu\nu}&=& \langle 0|\bar d (x)\gamma_{\mu}d(x)\,
\bar u (y)\gamma_{\nu} u(y)|0\rangle\,. \nonumber\end{aligned}$$ The mixed spectral function $m_{\mu\nu}$ is expected to be essentially zero in the low energy region, since the two different currents can communicate only via intermediate gluonic mesons. In perturbative QCD these states would be represented by three gluons. The vanishing of $m_{\mu\nu}$ implies the validity of the Zweig rule.\
The spectral functions $u_{\mu\nu}$ and $d_{\mu\nu}$ are strongly dominated at low energies by the $\rho_{0}$– and $\omega$–resonances. The actual
intermediate states contributing to the two-point functions are $2
\pi$– and $3 \pi$–states. However, a violation of the isospin symmetry due to
the $u-d$–quark mass splitting does not show up in the $\pi$-meson spectrum. The $\pi^+- \pi^{\circ}$ mass splitting is due to the electromagnetic interaction. It follows that resonant $\left( 2 \pi \right)$ of $\left( 3 \pi \right)$ states, i. e. the $\rho$–$\omega$–resonances, cannot display the effects of the isospin violation either, and the mass difference $\Delta M = M \left( v_d \right) - M \left( v_u \right)$ must be very small.\
\
Although the isospin symmetry is broken explicitly by the $u-d$ mass terms, this symmetry violation does not show up in the $\rho$–$\omega$ sector. The isospin symmetry breaking is shielded by the pion dynamics.\
\
Effectively the symmetry is restored by dynamical effects. Here the gluon anomaly plays an important role. It might be that similar symmetry restoration effects are present in other situations, for example in the electroweak sector, which is sensitive to the dynamics in the TeV region.\
\
[10]{} S. Okubo, Phys. Lett. 5 (1963) 163; G. Zweig, CERN Report No. 8419/TH 414 (1964); J. Iizuka, Prog. Theor. Phys. Suppl. 37–8 (1996) 21. H. Fritzsch, P. Minkowski, Nuovo Cim. 30 (1975) 393. G. Veneziano, Mod. Phys. Lett. A4 1605 (1989); G. M. Shore, G. Veneziano, Nucl. Phys. B381 23 (1992). H. Fritzsch, M. Gell–Mann, H. Leutwyler, Phys. Lett. B47 (1973)365. G. t’ Hooft, Phys. Rev. D14 3432 (1976). M.A. Shifman, Phys. Rep. 209 (1986) 341. M. D. Scadron, Phys. Rev. D29 (1984) 2076.
|
---
abstract: 'We formalize a notion of discrete Lorentz transforms for Quantum Walks (QW) and Quantum Cellular Automata (QCA), in $(1+1)$-dimensional discrete spacetime. The theory admits a diagrammatic representation in terms of a few local, circuit equivalence rules. Within this framework, we show the first-order-only covariance of the Dirac QW. We then introduce the Clock QW and the Clock QCA, and prove that they are exactly discrete Lorentz covariant. The theory also allows for non-homogeneous Lorentz transforms, between non-inertial frames.'
author:
- Pablo Arrighi
- Stefano Facchini
- Marcelo Forets
bibliography:
- '../Bibliography/biblio.bib'
title: Discrete Lorentz covariance for Quantum Walks and Quantum Cellular Automata
---
Introduction
============
[*Symmetries in Quantum Walks.*]{} For the purpose of quantum simulation (on a quantum device) as envisioned by Feynman [@FeynmanQC], or for the purpose of exploring the power and limits of discrete models of physics, a great deal of effort has gone into discretizing quantum physical phenomena. Most of these lead to Quantum Walk (QW) models of the phenomena. QWs are dynamics having the following features:
- The underlying spacetime is a discrete grid;
- The evolution is unitary;
- It is causal, i.e. information propagates strictly at a bounded speed.
- It is homogeneous, i.e. translation-invariant and time-independent.
By definition, therefore, they have several of the fundamental symmetries of physics, built-in. But can they also have Lorentz covariance? The purpose of this article is to address this question.
[*Summary of results.*]{} Lorentz covariance states that the laws of physics remain the same in all inertial frames. Lorentz transforms relate spacetimes as seen by different inertial frames. This paper formalizes a notion of discrete Lorentz transforms, acting upon wavefunctions over discrete spacetime. It formalizes the notion of discrete Lorentz covariance of a QW, by demanding that a solution of the QW be Lorentz transformed into another solution, of the same QW.
Before the formalism is introduced, the paper investigates a concrete example: the Dirac QW [@BenziSucci; @Bialynicki-Birula; @MeyerQLGI; @ArrighiDirac]. The Dirac QW is a natural candidate for being Lorentz covariant, because its continuum limit is the covariant, free-particle Dirac equation [@StrauchPhenomena; @ArrighiDirac; @bisio2013dirac]. This example helps us build our definitions. However, the Dirac QW turns out to be first-order covariant only. In order to obtain exact Lorentz covariance, we introduce a new model, the Clock QW, which arises as the quantum version of a covariant classical Random Walk [@wall1988discrete]. However, the Clock QW requires an observer-dependent dimension for the internal state space. In order to overcome this problem, the formalism is extended to multiple-walkers QWs, i.e. Quantum Cellular Automata (QCA). Indeed, the Clock QCA provides a first finite-dimensional model of an exactly covariant QCA. We use numerous figures to help our intuition. In fact, the theory admits a simple diagrammatic representation, in terms of a few local, circuit equivalence rules. The theory also allows for non-homogeneous Lorentz transforms, a specific class of general coordinate transformations, and yet expressive enough to switch between non-inertial frames.
[*Related works.*]{} Researchers have tried to reconcile discreteness and Lorentz covariance in several ways.
In the causal set approach, only the causal relations between the spacetime events is given. Without a background spacetime Lorentz covariance is vacuous. If, however, the events are generated from a Poissonian distribution over a flat spacetime, then covariance is recovered in a statistical sense [@dowker2004quantum].
Researchers working on Lattice Boltzmann methods for relativistic hydrodynamics also take a statistical approach: the underlying model breaks Lorentz covariance, but the statistical distributions generated are covariant [@mendoza2010derivation].
Loop Quantum Gravity offers a deep justification for the statistical approach. By interpreting spacetime intervals as the outcome of measurements of quantum mechanical operators, one can obtain covariance for the mean values, while keeping to a discrete spectrum [@rovelli2003reconcile; @livine2004lorentz].
The idea of interpreting space and time as operators with a Lorentz invariant discrete spectrum goes back to Snyder [@snyder1947quantized]. This line of research goes under the name of Doubly Special Relativity (DSR). Relations between DSR and QWs are discussed in [@bibeau2013doubly]. In the DSR approach, a deformation of the translational sector of the Poincaré algebra is required.
Instead of deforming the translation operator algebra, one could look at dropping translational invariance of the QW evolution. Along these lines, models have been constructed for QWs in external fields, including specific cases of gravitational fields [@MolfettaDebbasch2014Curved; @MolfettaDebbasch].
Another non-statistical, early approach is to restrict the class of allowed Lorentz transforms, to a subgroup of the Lorentz group whose matrices are over the integers numbers [@schild1948discrete]. Unluckily, there are no non-trivial integral Lorentz transforms in (1+1)-dimensions. Moreover, interaction rules that are covariant under this subgroup are difficult to find [@tHoofttwodimensional; @das1960cellular].
[*Approach.*]{} The approach of the present paper is non-statistical: we look for exact Lorentz covariance. Spacetime remains undeformed, always assumed to be a regular lattice, and the QW remains homogeneous. While keeping to $1+1$ dimensions and integral transforms, we allow for a global rescaling, so that we can represent all Lorentz transforms with rational velocity. The basic idea is to map each point of the lattice to a lightlike rectangular spacetime patch, as illustrated in Fig. \[fig:boostedaxes\] and \[fig:transformingall\].
[*Plan of the paper.*]{} The remainder of this paper is organized as follows. In Section \[sec:Preliminaries\] we set notations by recalling the Dirac QW and the proof of covariance of the Dirac Eq. In Section \[sec:aLorentz transform\] we discuss the first-order-only covariance of the Dirac QW. In Section \[sec:Lorentztransforms\] we formalize discrete Lorentz transforms, covariance, and discuss non-homogeneous Lorentz transforms. In Sections \[sec:ClockQW\] and \[sec:ClockQCA\] apply this theory to the Clock QW and the Clock QCA respectively. We finish with a discussion in Section \[sec:Conclusions\].
Preliminaries {#sec:Preliminaries}
=============
Finite Difference Dirac Eq. and the Dirac QW {#subsec:DiracQW}
--------------------------------------------
[*The Dirac Equation.*]{} The (1+1)-dimensional free particle Dirac equation is (with Planck’s constant and the velocity of light set to one): $$\begin{aligned}
\partial_t \psi &= -\ii m\sigma_1{\psi} - \partial_x \sigma_3 \psi, \label{eq:Dirac1D}\end{aligned}$$ where $m$ is the mass of the particle, $\psi=\psi(t,x)$ is a spacetime wavefunction from $\mathbb{\R}^{1+1}$ to $\mathbb{C}^2$ and $\sigma_j$ ($j=0,\ldots,3$) are the Pauli spin matrices, with $\sigma_0$ the identity. Eq. (\[eq:Dirac1D\]) corresponds to the Weyl (or spinor) representation [@Thaller].
In order to study covariance, it is always a good idea to switch to lightlike coordinates $r=(t+x)/2$ and $l=(t-x)/2$, in which a Lorentz transform is just a rescaling of the coordinates. Redefine the wavefunction via $\psi(r+l,r-l)\rightarrow \psi(r,l)$, then Eq. becomes $$\begin{aligned}
\diag{\partial_r}{\partial_l} {\psi} &= -\ii m\sigma_1{\psi}. \label{eq:LightDirac1D}\end{aligned}$$
In this paper $e^{\varepsilon\partial_\mu}$ will be used as a notation for the translation by $\varepsilon$ along the $\mu$-axis (with $\mu=0,1$), i.e. $(e^{\varepsilon\partial_\mu}{\psi})(x_\mu) = {\psi(x_\mu+\varepsilon)}$.
Using the first order expansion of the the exponential, the spacetime wavefunction $\psi$ is a solution of the Dirac equation if and only if, as $\varepsilon \rightarrow 0$, $$\begin{aligned}
\diag{e^{\varepsilon\partial_r}}{e^{\varepsilon\partial_l}}{\psi}
&=\left(\Id+\diag{\varepsilon\partial_r}{\varepsilon\partial_l}\right){\psi}+O(\varepsilon^2) \nonumber \\
&=(\Id - \ii m \varepsilon \sigma_1){\psi}+O(\varepsilon^2).\label{eq:FDApprox}\end{aligned}$$
Equivalently, if we denote $\psi=(\psi_+,\psi_-)^\mathsf{T}$, then $\psi$ is a solution of the Dirac equation if and only if, to first order in $\varepsilon$ and as $\varepsilon \rightarrow 0$, $$\begin{aligned}
&\psi_+(r+\varepsilon,l)=\psi_+(r,l)-\ii m \varepsilon \psi_-(r,l) \nonumber \\
\textrm{and}\quad &\psi_-(r,l+\varepsilon)=\psi_-(r,l)-\ii m \varepsilon \psi_+(r,l). \label{eq:FDDirac1D}\end{aligned}$$ If we now suppose that $\varepsilon$ is fixed, and consider that ${\psi}$ is a spacetime wavefunction from $(\varepsilon\mathbb{Z})^{2}$ to $\mathbb{C}^2$, then Eq. defines a Finite-difference scheme for the Dirac equation (FD Dirac). As a dynamical system, this FD Dirac is illustrated in Fig. \[fig:conventions\] with: $$C=\left(\begin{array}{cc}
1 & -\ii\varepsilon m\\
-\ii\varepsilon m & 1
\end{array} \right). \label{eq:FDQWCoin}$$
We could have gone a little further with Eq. (\[eq:FDApprox\]). Indeed, by recognizing in the right-hand side of the equation the first order expansion of an exponential, we get: $$\begin{aligned}
\diag{e^{\varepsilon\partial_r}}{e^{\varepsilon\partial_l}}{\psi} &= e^{-\ii m \varepsilon \sigma_1}{\psi} +O(\varepsilon^2). \label{eq:DiracQW}\end{aligned}$$
In fact, $\psi$ is a solution of the Dirac equation if and only if, as $\varepsilon\rightarrow 0$, Eq. (\[eq:DiracQW\]) is satisfied. See [@StrauchPhenomena; @ArrighiDirac] for a rigorous, quantified proof of convergence.
If we again say that $\varepsilon$ is fixed, and so that ${\psi}$ is a discrete spacetime wavefunction, then Eq. defines a Quantum Walk for the Dirac equation (Dirac QW) [@BenziSucci; @Bialynicki-Birula; @MeyerQLGI; @bisio2013dirac; @ArrighiDirac]. Indeed, as a dynamical system, this Dirac QW is again illustrated in Fig. \[fig:conventions\] but this time taking: $$C=e^{-\ii m \varepsilon \sigma_1} = \left(\begin{array}{cc}
\cos(\varepsilon m) & -\ii\sin(\varepsilon m)\\
-\ii\sin(\varepsilon m) & \cos(\varepsilon m)
\end{array} \right), \label{eq:DiracQWCoin}$$ which is exactly unitary, i.e. to all orders in $\varepsilon$.
In the original $(t,x)$ coordinates, both the FD Dirac and the Dirac QW evolutions are given by $\psi(t+\varepsilon,x) = TC\psi(t,x)$, where $T=e^{-\varepsilon \partial_x \sigma_3}$ is the shift operator and $C$ is the matrix appearing in Eq. (\[eq:FDQWCoin\]) or Eq. (\[eq:DiracQWCoin\]) respectively (see [@ArrighiDirac] for details). In the case of the Dirac QW, $W=TC$ is referred to as the walk operator: it is shift-invariant and unitary. $C$ is referred to as the coin operator, acting over the ‘coin space’, which is ${\cal H} \cong \C^2$ for the Dirac QW. Eq. (\[eq:DiracQW\]) reads as follows: the top and bottom components of the coin space get mixed up by the coin operator, and then the top component moves at lightspeed towards the right, whereas the bottom component goes in the opposite direction.
\[rk:detregion\] Let $\alpha, \beta$ be arbitrary positive integers. Notice that knowing the value of the scalars $\psi_-(r,l),\ldots,\psi_-(r+(\alpha-1)\varepsilon,l)$ carried by the right-incoming wires, together with the scalars $\psi_+(r,l),\ldots,\psi_+(r,l+(\beta-1)\varepsilon)$ carried by the left-incoming wires, fully determines $\psi(r+i\varepsilon,l+j\varepsilon)$ for $0\leq i \leq (\alpha-1)$ and $0\leq j \leq (\beta-1)$, as made apparent in Fig. \[fig:determined\]. We denote by $\overline{C}(i,j)$ the operator which, given the vectors $$\begin{aligned}
\overline{\psi}_-(r,l)=
\left(\begin{array}{c} \psi_-(r,l)\\\vdots\\\psi_-(r+(\alpha-1)\varepsilon,l)\end{array}\right)\end{aligned}$$ and $$\begin{aligned}
\overline{\psi}_+(r,l)=
\left(\begin{array}{c} \psi_+(r,l)\\\vdots\\\psi_+(r,l+(\beta-1)\varepsilon)\end{array}\right)\end{aligned}$$ combined as $$\begin{aligned}
\overline{\psi}(r,l)=
\left(\begin{array}{c} \overline{\psi}_+(r,l)\\ \overline{\psi}_-(r,l)\end{array}\right),\end{aligned}$$ yields $\psi(r+i\varepsilon,l+j\varepsilon)$, i.e. $\psi(r+i\varepsilon,l+j\varepsilon)=\overline{C}(i,j)\overline{\psi}(r,l).$ Moreover, notice that those values also determine the right outcoming wires $\psi_+(r+\alpha\varepsilon,l+j\varepsilon)$ for $0\leq j \leq (\beta-1)$, which we denote by $\overline{C}_+\overline{\psi}(r,l)$, and the left outcoming wires $\psi_-(r+i\varepsilon,l+\beta\varepsilon)$ for $0\leq i \leq (\alpha-1)$, which we denote be $\overline{C}_-\overline{\psi}(r,l)$. More generally, we denote by $\overline{C}$ the circuit made of $(\alpha\beta)$ gates shown in Fig. \[fig:determined\], i.e. $$\overline{C}~\overline{\psi}(r,l)=\left(\overline{C}_+ \oplus \overline{C}_-\right)\overline{\psi}(r,l).$$ We write $\overline{C}_m$ for the operator, instead of $\overline{C}$, when we want to make explicit its dependency upon the parameter $m$.
Scaled Lorentz transforms and covariance {#subsec:conttransform}
----------------------------------------
Let us review the covariance of the Dirac equation in a simple manner, that will be useful for us later. Consider a change of coordinates $r'=\alpha r$, $l'=\beta l$. This transformation is proportional by a factor of $\sqrt{\alpha\beta}$ to the Lorentz transform $$\Lambda = \left(\begin{array}{cc}
\sqrt{\dfrac\alpha\beta} & 0 \\
0 & \sqrt{\dfrac\beta\alpha}
\end{array}\right)$$ whose velocity parameter is $u=(\alpha-\beta)/(\alpha+\beta)$. Let us define $\widetilde{\psi}(r',l')=\widetilde{\psi}(\alpha r,\beta l)=\psi(r,l)$. A translation by $\varepsilon$ along $r$ (resp. $l$) becomes a translation by $\alpha \varepsilon$ along $r'$ (resp. $\beta \varepsilon$ along $l'$). Hence the Dirac equation now demands that as $\varepsilon\rightarrow 0$, $$\begin{aligned}
{\diag{e^{\alpha \varepsilon\partial_{r'}}}{e^{\beta \varepsilon\partial_{l'}}}}{\widetilde{\psi}}
&=
\left(\begin{array}{cc}
1 & -\ii\varepsilon m\\
-\ii\varepsilon m & 1
\end{array} \right)
{\widetilde{\psi}}+O(\varepsilon^2).\end{aligned}$$ Equivalently, to first order in $\varepsilon$ and as $\varepsilon \rightarrow 0$, $$\begin{aligned}
\widetilde{\psi}_+(r'+\alpha \varepsilon,l')&=\widetilde{\psi}_+(r',l')-\ii m \varepsilon \widetilde{\psi}_-(r',l')\\
\widetilde{\psi}_-(r',l'+\beta \varepsilon)&=\widetilde{\psi}_-(r',l')-\ii m \varepsilon \widetilde{\psi}_+(r',l')\end{aligned}$$ Unfortunately, whenever $\alpha\neq\beta$, this is not in the form of a Dirac equation. In other words the coordinate change alone does not take the Dirac equation into the Dirac equation.
In Section \[sec:ClockQW\] we will study the Clock QW, inspired by: $$\begin{aligned}
\quad \left(\begin{array}{cc}
e^{\alpha\varepsilon\partial_r} & 0\\
0 & e^{\beta\varepsilon\partial_l}
\end{array} \right) \psi
=e^{-\ii m \varepsilon \sigma_1} \psi.\end{aligned}$$
![\[fig:determined\] [*Lightlike rectangular patches of spacetime*]{} (in this example $\alpha=4$, $\beta=3$) are fully-determined by the incoming wires.](determinedregion.pdf){width="\columnwidth"}
Meanwhile, notice that in the first order, the top and bottom $\varepsilon$ can be taken to be different, leading to
$$\begin{aligned}
&{\diag{e^{\varepsilon\partial_{r'}}}{e^{\varepsilon\partial_{l'}}}}{\widetilde{\psi}}
=
\left(\begin{array}{cc}
1 & -\ii \varepsilon m/\alpha\\
-\ii\varepsilon m/\beta & 1
\end{array} \right)
{\widetilde{\psi}}+O(\varepsilon^2).\end{aligned}$$
$$\begin{aligned}
&{\diag{e^{\varepsilon\partial_{r'}}}{e^{\varepsilon\partial_{l'}}}}
\left(\begin{array}{c}
\widetilde{\psi}_+ / \sqrt{\beta}\\
\widetilde{\psi}_- / \sqrt{\alpha}
\end{array} \right)= \\
&\left(\begin{array}{cc}
1 & -\ii \varepsilon m/\sqrt{\alpha\beta}\\
-\ii\varepsilon m/\sqrt{\alpha\beta} & 1
\end{array} \right)
\left(\begin{array}{c}
\widetilde{\psi}_+/\sqrt{\beta}\\
\widetilde{\psi}_-/\sqrt{\alpha}
\end{array} \right)+O(\varepsilon^2).\end{aligned}$$
Let us define $$\begin{aligned}
S=\left(\begin{array}{cc}
1/\sqrt{\beta} & 0\\
0 & 1/\sqrt{\alpha}
\end{array} \right)\qquad\textrm{and}\qquad \psi'=S\widetilde{\psi}.\end{aligned}$$ Call this $\psi'$ the Lorentz transformed of $\psi$, instead of $\widetilde{\psi}$. Now we have: $$\begin{aligned}
&{\diag{e^{\varepsilon\partial_{r'}}}{e^{\varepsilon\partial_{l'}}}}{\psi'}
=
\left(\begin{array}{cc}
1 & -\ii\varepsilon m/\sqrt{\alpha\beta}\\
-\ii\varepsilon m/\sqrt{\alpha\beta} & 1
\end{array} \right)
{\psi'}\\\end{aligned}$$ i.e. the Dirac equation just for a different mass $m' = m / \sqrt{\alpha \beta}$. This different mass is due to the fact that the transformation to primed coordinates that we considered was a scaled a Lorentz transform. In the special case where $\alpha \beta = 1$, the above is just the proof of Lorentz covariance of the Dirac equation.
A discrete Lorentz transform for the Dirac QW {#sec:aLorentz transform}
=============================================
Normalization problem and its solution {#subsec:normalization}
--------------------------------------
Take $\psi(r,l)$ a solution of the Dirac QW such that the initial condition is normalized and localized at single point e.g. $\psi(0,0) = (1,0)^\mathsf{T}$ and $\psi(r,l)=(0,0)^\mathsf{T}$ for $t=r+l=0$. Then, after applying the Lorentz transform described in Subsection \[subsec:conttransform\], the initial condition is $\psi'(0,0) = (1/\sqrt\beta,0)^\mathsf{T}$ and $\psi'(r,l)=(0,0)^\mathsf{T}$ for $t=r+l=0$ which is not normalized for any non-trivial Lorentz transform, see Fig. \[fig:normalizationproblem\]$(a)$. Hence, we see that the Lorentz transform described in Subsection \[subsec:conttransform\], i.e. that used for the covariance of the continuous Dirac equation, is problematic in the discrete case: the transformed observer sees a wavefunction which is not normalized. This seems a paradoxical situation since in the limit when $\varepsilon\to 0$, the discrete case tends towards the continuous case, which does not have such a normalization issue. In order to fix this problem, let us look more closely at how normalization is preserved in the continuous case.
Now take $\psi(r,l)$ a solution of the massless Dirac equation such that the initial condition is the normalized, right-moving rectangular function, i.e. $\psi(r,l)=(1/\sqrt{2},0)^\mathsf{T}$, for $0 \leq l < 1$ and $\psi(r,l)= (0,0)^\mathsf{T}$ elsewhere. The Lorentz transformed of $\psi$ is $$\psi'(r',l') = S\psi(r'/\alpha, l'/\beta) = \left\{ \begin{array}{ll} \left(\begin{array}{c} 1/\sqrt{2\beta}\\0\end{array}\right) & 0 \leq l' < \beta \\ \left(\begin{array}{c} 0\\0\end{array}\right) & \mbox{elsewhere},\end{array}\right.$$ which is normalized. We see that the $S$ matrix is no longer a problem for normalization, but rather it is needed to compensate for the larger spread of the wavefunction, see Fig. \[fig:normalizationproblem\]$(b)$. This suggests that the normalization problem for the localized initial condition in the discrete case could be fixed, by allowing the discrete Lorentz transform to spread out the initial condition.
Intuitively, we could think of defining the discrete Lorentz transform as the missing arrow “$\textrm{Discrete } \Lambda?$” that would make the following diagram commute: $$\begin{tikzcd}
Dirac \arrow{r}{\Lambda} & Dirac' \arrow{d}{\textrm{Sample}} \\
QW \arrow{r}{\textrm{Discrete }\Lambda?} \arrow{u}{\textrm{Interpolate}} & QW'
\end{tikzcd}.$$ In other words, $$\textrm{Discrete }\Lambda:= \textrm{Sample} \circ \Lambda \circ \textrm{Interpolate}$$ The discrete Lorentz transform that we propose next originates from this idea, even though it is phrased directly in the discrete setting. Later in Section \[sec:Lorentztransforms\] we provide a more general and diagrammatic definition of discrete Lorentz transform and discrete Lorentz covariance.
A discrete Lorentz transform {#subsec:DiracLorentzTransform}
----------------------------
In the continuous case we had $\psi'(r',l')=S\psi(r,l)$. Hence $\psi'(r',l')=S\psi(r'/\alpha,l'/\beta)$. In the discrete case, however ${\psi}$ is a spacetime wavefunction from $(\varepsilon\mathbb{Z})^{2}$ to $\mathbb{C}^2$, as in Fig. \[fig:conventions\]$(b)$. Hence, demanding, for instance, that $\psi'(\varepsilon,0)=S\psi(\varepsilon/\alpha,0)$ becomes meaningless, because $\psi(\varepsilon/\alpha,0)$ is undefined. The normalization issues and the related discussion of Subsection \[subsec:normalization\] suggests setting $\psi_-'(\varepsilon,0)$ to $S\psi_-(0,0)$, and not to $0$. More generally, we will take: $$\forall r'\in \varepsilon\alpha \mathbb{Z},\quad\psi_+'(r',l')=\frac{\psi_+( r'/\alpha,\lfloor l'/\beta\rfloor_\varepsilon)}{\sqrt{\beta}}$$ and $$\forall l'\in \varepsilon\beta \mathbb{Z},
\quad \psi_-'(r',l')=\frac{\psi_-(\lfloor r'/\alpha\rfloor_\varepsilon, l'/\beta)}{\sqrt{\alpha}}.$$ where $\lfloor .\rfloor_\varepsilon$ takes the closest multiple of $\varepsilon$ that is less or equal to the number. Notice that this implies that for all $r'\in \varepsilon\alpha \mathbb{Z}$ and $l'\in \varepsilon\beta \mathbb{Z}$, we have $\psi'(r',l')=S\psi(r'/\alpha,l'/\beta)$, as in the continuous case. However, what if we have neither $r'\in \alpha\varepsilon\mathbb{Z}$ nor $l'\in \beta\varepsilon\mathbb{Z}$? As was illustrated in Fig. \[fig:determined\], this spacetime region is now fully determined, i.e. we set $$\begin{aligned}
\forall r',l'\in \varepsilon\mathbb{Z},\quad \psi'(r',l')=\overline{C}_{m'}(i,j)\overline{\psi'}(\lfloor r'\rfloor_{\alpha\varepsilon},\lfloor l'\rfloor_{\beta\varepsilon}) \label{eq:Lorentz transform}\end{aligned}$$ with $m' = m / \sqrt{\alpha \beta}$, $i\varepsilon=r'-\lfloor r'\rfloor_{\alpha\varepsilon}$, $j\varepsilon=l'-\lfloor l'\rfloor_{\beta\varepsilon}$, $\overline{C}_{m'}(i,j)$ as defined in Remark \[rk:detregion\], and $\overline{\psi'}(\lfloor r'\rfloor_{\alpha\varepsilon},\lfloor l'\rfloor_{\beta\varepsilon})$ again as defined in Remark \[rk:detregion\], namely $$\begin{aligned}
\overline{\psi'}_+(\lfloor r'\rfloor_{\alpha\varepsilon},\lfloor l'\rfloor_{\beta\varepsilon})
&=\left(\begin{array}{c} \psi'_+(\lfloor r'\rfloor_{\alpha\varepsilon},\lfloor l'\rfloor_{\beta\varepsilon})\\\vdots\\ \psi'_+(\lfloor r'\rfloor_{\alpha\varepsilon},\lfloor l'\rfloor_{\beta\varepsilon}+(\beta-1)\varepsilon ) \end{array}\right) \\
&=\left(\begin{array}{c} \dfrac{\psi_+(\lfloor r'\rfloor_{\alpha\varepsilon},\lfloor l'\rfloor_{\beta\varepsilon})}{\sqrt{\beta}}\\\vdots\end{array}\right) \end{aligned}$$ and similarly $$\begin{aligned}
\overline{\psi'}_-(\lfloor r'\rfloor_{\alpha\varepsilon},\lfloor l'\rfloor_{\beta\varepsilon})&=
\left(\begin{array}{c} \dfrac{\psi_-(\lfloor r'\rfloor_{\alpha\varepsilon},\lfloor l'\rfloor_{\beta\varepsilon})}{\sqrt{\alpha}}\\\vdots\end{array}\right) \end{aligned}$$ and finally $$\begin{aligned}
\overline{\psi'}(\lfloor r'\rfloor_{\alpha\varepsilon},\lfloor l'\rfloor_{\beta\varepsilon})=
\left(\begin{array}{c} \overline{\psi'}_+(\lfloor r'\rfloor_{\alpha\varepsilon},\lfloor l'\rfloor_{\beta\varepsilon})\\ \overline{\psi'}_-(\lfloor r'\rfloor_{\alpha\varepsilon},\lfloor l'\rfloor_{\beta\varepsilon})\end{array}\right).\end{aligned}$$ This finishes to define a discrete Lorentz transform $L_{\alpha,\beta}$, which is illustrated in Fig. \[fig:Lorentz transform\].
\
An equivalent, more concise way of specifying this discrete Lorentz transform $L_{\alpha,\beta}$ is as follows. First, consider the isometry $E_\beta$ (resp. $E_\alpha$) which codes $\psi_+(r,l)$ (resp. $\psi_-(r,l)$) into the more spread out $\widecheck{\psi}_+(r,l)=E_\beta\psi_+(r,l)=\overline{\psi'}_+(\alpha r, \beta l)$ (resp. $\widecheck{\psi}_-(r,l)=E_\alpha\psi_-(r,l)= \overline{\psi'}_-(\alpha r, \beta l)$), and let $\widecheck{\psi}(r,l)=\widecheck{\psi}_+(r,l)\oplus \widecheck{\psi}_-(r,l)$, and $m' = m / \sqrt{\alpha \beta}$. Second, construct $\psi'=L_{\alpha,\beta}\psi$ by replacing every spacetime point $\psi(r,l)$ with the lightlike rectangular spacetime patch $\left(\overline{C}_{m'}(i,j)\widecheck{\psi}(r,l)\right)_{i=0\ldots(\alpha-1),j=0\ldots(\beta-1)}$.
Does this discrete Lorentz transform fix the normalization problem of Subsection \[subsec:normalization\]? Let us evaluate this question.
From continuous to discrete current and norm {#subsec:discretecurrent}
--------------------------------------------
### Continuous current and norm
In order to evaluate the norm of a spacetime wavefunction $\psi$ in the continuous setting, we need the following definition. We say that a surface $\sigma$ is a [*Cauchy surface*]{} if it intersects every causal curve exactly once (a causal curve being a curve whose tangent vector is always timelike or lightlike). The relativistic current $j^\mu=(j^0,j^1)$ is equal to $j^{\mu}=(|\psi_+|^2 + |\psi_-|^2, |\psi_+|^2 - |\psi_-|^2)$, and in lightlike coordinates becomes $j^{s} = (|\psi_+|^2, |\psi_-|^2)$, $s=\pm$. The norm of $\psi$ along a Cauchy surface $\sigma$ is defined by integrating the current $j^s$ along $\sigma$ $$||\psi||_\sigma^2 = \int_\sigma j^s n_s d\sigma$$ where $n_s$ is the unit normal vector to $\sigma$ in $r,l$ coordinates.
If $\psi$ is a solution of the Dirac equation, then this definition does not actually depend on the surface $\sigma$ (for a proof see for instance [@schweber1961introduction], Chap. 4), and so in this case we can write $||\psi||_\sigma^2=||\psi||^2$.
This definition of norm is Lorentz invariant, indeed: $$\begin{aligned}
||\psi||_{\sigma}^2 &= \int_\sigma j^s n_s d\sigma \nonumber \\
&= \int_\sigma \left(\frac{|\psi_+|^2}{\beta}\beta dl + \frac{|\psi_-|^2}{\alpha} \alpha dr\right) \nonumber \\
&= \int_{\sigma'} (|\psi'_+|^2dl' + |\psi'_-|^2 dr') \nonumber \\
&= \int_{\sigma'} j'^s n'_s d\sigma' \nonumber \\
&= ||\psi'||_{\sigma'}^2 \label{eq:normlorentzinv}\end{aligned}$$
### Discrete Cauchy surfaces
We now provide discrete counterparts to the above notions, beginning with [*discrete Cauchy surfaces*]{}. Let us consider a function $\sigma: \mathbb{Z} \to \{R,L\}$, and an origin $(r_0,l_0)$. Together, they describe a piecewise linear curve made up of segments of the following form (in gray):\
$$\begin{tikzpicture}
\path[draw] (1,1) -- (2,2);
\draw[fill=white] (1,1) circle(.08);
\draw[fill=white] (2,2) circle(.08);
\path[draw,color=gray] (1,2) -- (2,1);
\node[xshift=31*\gs,yshift=15*\gs,color=gray]{$R$};
\end{tikzpicture}
\quad
\begin{tikzpicture}
\path[draw] (2,1) -- (1,2);
\draw[fill=white] (2,1) circle(.08);
\draw[fill=white] (1,2) circle(.08);
\path[draw,color=gray] (1,1) -- (2,2);
\node[xshift=11*\gs,yshift=15*\gs,color=gray]{$L$};
\end{tikzpicture}$$ i.e. this curve intersects the spacetime lattice in two ways, labeled $R$ and $L$ (right, left). The centering on the origin is done as in Fig. \[fig:cauchysurface\] (a).
$(a)$
(1,1) – (2,2) – (3,1); (1,1) circle(.08); (2,2) circle(.08); (3,1) circle(.08); (1,2) – (2,1) – (3,2); ; ; ;
$(b)$
(0,2) – (2,4); (1,1) – (4,4); (0,4) – (3,1); (2,4) – (4,2); (4,4) – (5,3);
(0,2) circle(.08); (0,4) circle(.08); (1,1) circle(.08); (1,3) circle(.08); (2,2) circle(.08); (2,4) circle(.08); (3,1) circle(.08); (3,3) circle(.08); (4,2) circle(.08); (4,4) circle(.08); (5,3) circle(.08);
(0,3) – (2,1) – (5,4); (1,2) circle(.05); (2,1) circle(.05); (3,2) circle(.05); (4,3) circle(.05);
; ; ;
We say that such a curve is a [*discrete Cauchy surface*]{} if it does not contain infinite sequences of contiguous $R$ or $L$. One can easily see that such a surface must intersect every lightlike curve exactly once. For concreteness, notice that the discrete equivalent to the continuous constant-time $t=0$ Cauchy surface, is described by: $$\sigma(n) = \left\{
\begin{array}{ll}
L & \quad \mbox{for even } n \\
R & \quad \mbox{for odd } n
\end{array}
\right.$$ with origin $(0,0)$.
### Discrete current and discrete norm
Similarly, let us define the discrete current carried by a wavefunction $\psi$. At each wire connecting two points of the discrete lattice, the current is given by: $$\begin{tikzpicture}
\path[draw] (2*\gs,\gs) -- (\gs,2*\gs);
\node[xshift=72*\gs,yshift=45*\gs]{$j=|\psi_-(r,l+\varepsilon)|^2$};
\draw[fill=white] (2*\gs,\gs) circle(.08);
\draw[fill=white] (\gs,2*\gs) circle(.08);
\node[xshift=20*\gs,yshift=60*\gs]{$(r,l+\varepsilon)$};
\node[xshift=65*\gs,yshift=25*\gs]{$(r,l)$};
\end{tikzpicture}$$
$$\begin{tikzpicture}
\path[draw] (\gs,\gs) -- (2*\gs,2*\gs);
\node[xshift=75*\gs,yshift=45*\gs]{$j=|\psi_+(r+\varepsilon,l)|^2$};
\draw[fill=white] (\gs,\gs) circle(.08);
\draw[fill=white] (2*\gs,2*\gs) circle(.08);
\node[xshift=65*\gs,yshift=60*\gs]{$(r+\varepsilon,l)$};
\node[xshift=20*\gs,yshift=25*\gs]{$(r,l)$};
\end{tikzpicture}$$ In analogy with the continuous case, we can evaluate the norm of $\psi$ along a surface $\sigma$ as follows $$||\psi||_\sigma^2 = \sum_{i\in \mathbb{Z}} j(i)$$ where $j(i)$ is the current of the wire at intersection $i$. For instance, for the discrete constant-time surface the above expression evaluates to the usual $L_2$-norm of a spacelike wavefunction $$\begin{aligned}
||\psi||_{t=0}^2 &= \sum_{i\in \mathbb{Z}} j(i)\\
&= \sum_{i\in 2\mathbb{Z}+1}|\psi_+(i,-i)|^2 + \sum_{i\in 2\mathbb{Z}}|\psi_-(i,-i)|^2 \\
&= \sum_i ||\psi(i,-i)||^2=||\psi||^2.\end{aligned}$$
### Cauchy surface independence of the discrete norm
If $\psi$ is a solution of a QW, then just like the continuous case, the discrete norm does not depend on the discrete Cauchy surface chosen for evaluating it. The proof outline is as follows. First, two Cauchy surfaces $\sigma$ and $\sigma'$ can be made to coincide on an arbitrary large region using only a finite sequence of swap moves[^1]:
$$\begin{array}{c}
\begin{array}{c}
\begin{tikzpicture}
\path[draw] (1*\gs,1*\gs) -- (2*\gs,2*\gs);
\path[draw] (2*\gs,1*\gs) -- (1*\gs,2*\gs);
\draw[fill=white] (1*\gs,1*\gs) circle(.08);
\draw[fill=white] (2*\gs,2*\gs) circle(.08);
\draw[fill=white] (1*\gs,2*\gs) circle(.08);
\draw[fill=white] (2*\gs,1*\gs) circle(.08);
\draw[fill=white] (1.5*\gs,1.5*\gs) circle(.08);
\path[draw,color=gray] (1*\gs,1.5*\gs) -- (1.5*\gs,2*\gs) -- (2*\gs,1.5*\gs);
\node[xshift=20*\gs,yshift=60*\gs]{$(r,l+\varepsilon)$};
\node[xshift=65*\gs,yshift=60*\gs]{$(r+\varepsilon,l)$};
\node[xshift=50*\gs,yshift=42*\gs]{$(r,l)$};
\node[xshift=20*\gs,yshift=25*\gs]{$(r-\varepsilon,l)$};
\node[xshift=65*\gs,yshift=25*\gs]{$(r,l-\varepsilon)$};
\end{tikzpicture}
\end{array} \\
\quad \upharpoonleft \downharpoonright \quad \\
\begin{array}{c}
\begin{tikzpicture}
\path[draw] (\gs,\gs) -- (2*\gs,2*\gs);
\path[draw] (2*\gs,\gs) -- (\gs,2*\gs);
\draw[fill=white] (\gs,\gs) circle(.08);
\draw[fill=white] (2*\gs,2*\gs) circle(.08);
\draw[fill=white] (\gs,2*\gs) circle(.08);
\draw[fill=white] (2*\gs,\gs) circle(.08);
\draw[fill=white] (1.5*\gs,1.5*\gs) circle(.08);
\path[draw,color=gray] (\gs,1.5*\gs) -- (1.5*\gs,\gs) -- (2*\gs,1.5*\gs);
\node[xshift=20*\gs,yshift=60*\gs]{$(r,l+\varepsilon)$};
\node[xshift=65*\gs,yshift=60*\gs]{$(r+\varepsilon,l)$};
\node[xshift=50*\gs,yshift=42*\gs]{$(r,l)$};
\node[xshift=20*\gs,yshift=25*\gs]{$(r-\varepsilon,l)$};
\node[xshift=65*\gs,yshift=25*\gs]{$(r,l-\varepsilon)$};
\end{tikzpicture}
\end{array}.
\end{array}$$
Second, swap moves leave the norm invariant, because of unitarity of the $C$ gate (see Fig. \[fig:conventions\]): $$|\psi_+(r+\varepsilon,l)|^2 + |\psi_-(r,l+\varepsilon)|^2 = |\psi_+(r,l)|^2 + |\psi_-(r,l)|^2.$$ Third, take a positive $\delta$. By having $\sigma$ to coincide with $\sigma'$ on a large enough region, we obtain that $|(||\psi||_\sigma-||\psi||_{\sigma'})|\leq \delta$. Lastly, since $\delta$ is arbitrary, $||\psi||_\sigma=||\psi||_{\sigma'}$.
### Lorentz invariance of the discrete norm
Finally, we will prove the analogue of equation in the discrete setting. First of all we define how a discrete Cauchy surface $\sigma$ transforms under a discrete Lorentz transform with parameters $\alpha, \beta$. The sequence $\sigma'$ is constructed from $\sigma$ by replacing each $L$ by $L^\alpha$ and each $R$ with $R^\beta$, starting from the center. The origin $(r_0,l_0)$ is mapped to the point $(r'_0,l'_0) = (\alpha r_0, \beta l_0)$. For instance, the piece of surface in Fig. \[fig:cauchysurface\](a) is transformed as in Fig. \[fig:cauchysurface\](b). We obtain (where $\mathcal{R}_\sigma$ and $\mathcal{L}_\sigma$ are the sets of right and left intersections respectively): $$\begin{aligned}
||\psi||^2 &= \sum j(i) \\
&= \sum_{i\in \mathcal{R}_\sigma}|\psi_+(r_i,l_i)|^2 + \sum_{i\in \mathcal{L}_\sigma}|\psi_-(r_i,l_i)|^2 \\
&= \sum_{i\in \mathcal{R}_\sigma}\beta \left|\frac{\psi_+(r_{i},l_{i})}{\sqrt{\beta}}\right|^2 + \sum_{i\in \mathcal{L}_\sigma}\alpha\left|\frac{\psi_-(r_{i},l_{i})}{\sqrt{\alpha}}\right|^2 \\
&= \sum_{i'\in \mathcal{R}_{\sigma'}}|\psi'_+(r_{i'},l_{i'})|^2 + \sum_{i'\in \mathcal{L}_{\sigma'}}|\psi'_-(r_{i'},l_{i'})|^2 \\
&= \sum j'(i') = ||\psi'||^2.\end{aligned}$$
The First-order-only Lorentz covariance of the Dirac QW {#subsec:focovariance}
-------------------------------------------------------
In Subsection \[subsec:DiracQW\] we defined the Dirac QW, and explained when a spacetime wavefunction $\psi$ is a solution for it. In Subsection \[subsec:DiracLorentzTransform\] we defined a discrete Lorentz transform, taking a spacetime wavefunction $\psi$ into another spacetime wavefunction $\psi'$. In Subsection \[subsec:discretecurrent\] we showed that this transformation preserves the norm, i.e. $||\psi||_\sigma^2=||\psi'||_{\sigma'}^2$. The question that remains is whether the Dirac QW is Lorentz covariant with respect to this discrete Lorentz transform. In other words, is it the case that $\psi'$ is itself a solution of the Dirac QW, for some $m'$? This demand is concrete translation of the main principle of special relativity, stating that the laws of physics (here, the Dirac QW) remain the same in all inertial reference frames (here, those of $\psi$ and $\psi'$).
Recall that the discrete Lorentz transform works by replacing each point of the spacetime lattice by a lightlike rectangular patch of spacetime, which can be understood as a “biased, zoomed in version" of that point, see Fig. \[fig:Lorentz transform\]. Internally, each patch is a piece of spacetime solution of the Dirac QW by construction, see Eq. . But is it the case that the patches match up, to form the entire spacetime wavefunction of a solution? After all, there could be inconsistencies in between patches: values carried by the incoming wires to the next patches, e.g. $\widecheck{\psi}_+(r+\varepsilon, l)$ (resp. $\widecheck{\psi}_-(r,l+\varepsilon)$) could be different from those carried by the wires coming out of the preceding patch, i.e. $\widehat{\psi}_+(r,l)=\overline{C}_+\widecheck{\psi}(r,l)$ (resp. $\widehat{\psi}_-(r,l)=\overline{C}_-\widecheck{\psi}(r,l)$). More precisely, we need both $\widehat{\psi}_+(r,l)=\widecheck{\psi}_+(r+\varepsilon,l)$ and $\widehat{\psi}_-(r,l)=\widecheck{\psi}_-(r,l+\varepsilon)$ for every $r,l$. This potential mismatch is represented by the discontinuations of the wires of Fig. \[fig:Lorentz transform\]$(b)$. Clearly, the patches making up $\psi'$ match up to form the spacetime wavefunction of a solution if and only if there are no such inconsistencies. We now evaluate these inconsistencies.
In the first order, the Dirac QW and the Finite-Difference Dirac equation are equivalent, as shown in Subsection \[subsec:DiracQW\]. This makes it easier to compute the outcoming values of the patches, which should match the corresponding incoming wires (see Fig. \[fig:FirstOrderCovariance\] in Appendix \[App:FirstOrderOnlyProof\]). Let $m'=m/\sqrt{\alpha\beta}$. In general, we obtain (to first order in $\varepsilon$, for $i=0\ldots\beta-1$, $j=0\ldots\alpha-1$): $$\begin{aligned}
\widehat{\psi}_+(r,l)_i &= \left(\overline{C}_+\widecheck{\psi}(r,l)\right)_i \\
&=\frac{\psi_+(r,l)}{\sqrt{\beta}}-\alpha \ii m'\varepsilon \frac{\psi_-(r,l)}{\sqrt{\alpha}} \\
&=\frac{\psi_+(r,l)-\ii m\varepsilon\psi_-(r,l)}{\sqrt{\beta}} \\
&= \widecheck{\psi}_+(r+\varepsilon,l)_i\\
\textrm{and}\qquad\qquad\widehat{\psi}_-(r,l)_j &= \left(\overline{C}_-\widecheck{\psi}(r,l)\right)_j \\
&=\frac{\psi_-(r,l)}{\sqrt{\alpha}}-\beta \ii m'\varepsilon \frac{\psi_+(r,l)}{\sqrt{\beta}} \\
&=\frac{\psi_-(r,l)-\ii m\varepsilon\psi_+(r,l)}{\sqrt{\alpha}} \\
&= \widecheck{\psi}_-(r,l+\varepsilon)_j.\end{aligned}$$ Hence, the wires do match up in the first order. However, the second order cannot be fixed, even if we allow for arbitrary encodings. The proof of this statement is left for Appendix \[App:FirstOrderOnlyProof\].
The lack of second order covariance of the Dirac QW can be interpreted in several ways. First, as saying that the Dirac QW is not a realistic model. This interpretation motivates us to explore, in the next sections, the question whether other discrete models (QWs or QCA) could not suffer this downside, and be exactly covariant. Second, as an indication that Lorentz covariance breaks down at Planck scale. Third, as saying that we have no choice but to view $\varepsilon$ as an infinitesimal, so that we can ignore its second order. In this picture, the Dirac QW would be understood as describing an infinitesimal time evolution, but in the same formalism as that of discrete time evolutions, i.e. in an alternative language to the Hamiltonian formalism. Formulating an infinitesimal time quantum evolution in such a way has an advantage: it sticks to the language of unitary, causal operators [@ArrighiUCAUSAL] and readily provides a quantum simulation algorithm.
Transformation of velocities {#subsec:velocities}
----------------------------
In Subsection \[subsec:DiracLorentzTransform\] we defined a discrete Lorentz transform, which takes a spacetime wavefunction $\psi$ into a Lorentz transformed wavefunction $\psi'$. In Subsection \[subsec:focovariance\] we proved that the Dirac QW is first-order covariant. Is it the case that the velocity of $\psi$ is related to the velocity of $\psi'$ according to the transformation of velocity rule of special relativity? We will show that it is indeed the case, so long as we transform the “local velocity field” $v(r,l)$, defined as: $$v(r,l) = \frac{|\psi_+(r,l)|^2 - |\psi_-(r,l)|^2}{||\psi(r,l)||^2} \label{Eq:LocalVelocityDiracQW}.$$ In order to see how we arrive at this formula, let us first recall the definition of velocity in the continuous case.
For the Dirac equation, the velocity operator is obtained via the Heisenberg formula, $d \hat{x} /dt = \ii [ H, \hat{x}] = \sigma_3$ (see Eq. (\[eq:Dirac1D\])). Thus, in the discrete setting it is natural to define the velocity operator as $\Delta X = X - W X W^\dagger$, where $X$ is the position operator, $X=\sum_x x P_x = \sum_x x \ket{x}\bra{x}$ and $W=TC$ is the walk operator. We have $$\begin{aligned}
\Delta X & = X - W XW^\dagger = X - T C X C^\dagger T^\dagger = X - T X T^\dagger \\
& = \begin{pmatrix} \sum_x xP_x & 0 \\ 0 & \sum_x xP_x \end{pmatrix} - \begin{pmatrix} \sum_x xP_{x+1} & 0 \\ 0 & \sum_x xP_{x-1} \end{pmatrix} \\
& = \sigma_3\end{aligned}$$ Thus the expected value of $\Delta X$ at the time slice $t=0$ is, as in the continuous case, $$\begin{aligned}
\langle \sigma_3 \rangle_{\psi} & = \sum_{i \in \Z} |\psi_+(i,-i)|^2 - |\psi_-(i,-i)|^2 \nonumber\\
& = \sum_{i\in\Z}p(i,-i)v(i,-i) \label{Eq.GlobalVelocity}\end{aligned}$$ where $p(r,l) = ||\psi(r,l)||^2$. It should be noted that it is not a constant of motion. This is in fact a manifestation of the Zitterbewegung, whose connection with the continuous case was well studied by several authors [@StrauchPhenomena; @bisio2013dirac; @kurzynski2008relativistic]. Eq. justifies our definition of local velocity.
Let us now consider the case of a walker which at $t=0$, $x=0$, has internal degree of freedom $\psi = (\psi_+,\psi_-)^\mathsf{T}$. We will relate $v = v(0,0)$ and $v' = v'(0,0)$ as calculated from a Lorentz transformed observer with parameters $\alpha, \beta$. We have $v= (|\psi_+|^2 - |\psi_-|^2)/||\psi||^2$. We can deduce $|\psi_+|^2=||\psi||^2(1+v)/2$ and $|\psi_-|^2=||\psi||^2(1-v)/2$. Now, let us apply a discrete Lorentz transform. At point $(0,0)$, it takes $\psi$ into $\psi'=S\psi$, whose corresponding velocity is: $$\begin{aligned}
v'&=\frac{|\psi'_+|^2 - |\psi'_-|^2}{||\psi'||^2} = \frac{\alpha|\psi_+|^2 - \beta|\psi_-|^2}{\alpha|\psi_+|^2 + \beta|\psi_-|^2}\\
&= \frac{\alpha||\psi||^2(1+v) - \beta||\psi||^2(1-v)}{\alpha||\psi||^2(1+v) + \beta||\psi||^2(1-v)} \\
&= \frac{v+\frac{\alpha-\beta}{\alpha+\beta}}{1+v\frac{\alpha-\beta}{\alpha+\beta}} \\
&= \frac{v+u}{1+vu}\end{aligned}$$ where $u=(\alpha-\beta)/(\alpha+\beta)$ is the velocity that corresponds to the discrete Lorentz transform with parameters $\alpha, \beta$. Thus the local velocity associated to a spacetime wavefunction $\psi$ is related to the local velocity of the corresponding Lorentz transformed $\psi'$ by the rule of addition of velocities of special relativity.
Formalization of Discrete Lorentz covariance in general {#sec:Lorentztransforms}
=======================================================
We will now provide a formal, general notion of discrete Lorentz transform and Lorentz covariance for Quantum Walks and Quantum Cellular Automata.
Over Quantum Walks {#subsec:LorentztransformsQWs}
------------------
Beforehand, we need to explain which general form we assume for Quantum Walks.
### General form of Quantum Walks
Intuitively speaking, a Quantum Walk (QW) is a single particle or walker moving in discrete-time steps on a lattice. Axiomatically speaking, QWs are shift-invariant, causal, unitary evolutions over the space $\bigoplus_{\mathbb{Z}} {\cal H}_{c}$, where $c$ is the dimension of the internal degrees of freedom of the walker. Constructively speaking, in turns out [@GrossNesmeVogtsWerner] that, at the cost of some simple recodings, any QW can be put in a form which is similar to that of the circuit for the Dirac QW shown Fig. \[fig:conventions\]$(c)$. In general, however, $c$ may be larger than $2$ (the case $c$ equal $1$ is trivial [@MeyerQLGI]). But it can always be taken to be even, so that the general shape for the circuit of a QW can be expressed as in Fig. \[fig:QWGeneral\]. Notice how, in this diagram, each wire carries a $d$-dimensional vector $\psi_\pm(r,l)$. We will say that the QW has ‘wire dimension’ $d$. Incoming wires get composed together with a direct sum, to form a $2d$-dimensional vector $\psi(r,l)$. The state $\psi(r,l)$ undergoes a $2d\times2d$ unitary gate $C$ to become some $\psi'(r,l)=\psi_+'(r+\varepsilon,l)\oplus \psi_-'(r,l+\varepsilon)$, etc. The unitary gate $C$ is called the ‘coin’. Algebraically speaking, this means that a QW can always be assumed to be of the form: $$\diag{e^{\varepsilon\partial_r}\Id_d}{e^{\varepsilon\partial_l}\Id_d}{\psi}=C{\psi}.$$
![\[fig:QWGeneral\] [*The circuit for a general QW.*]{} The wire dimension is $d$, meaning that $\psi_+(r,l)=(\psi^1_+(r,l),\ldots,\psi^d_+(r,l))^\mathsf{T}$, etc.](conventions4.pdf){width="\columnwidth"}
### Lorentz transforms for QW
The formalization of a general notion of Lorentz transform for QWs generalizes that presented in Section \[sec:aLorentz transform\]. Consider a QW having wire dimension $d$, and whose $2d\times 2d$ unitary coin is $C_m$, where the $m$ are parameters. A Lorentz transform $L_{\alpha,\beta}$ is specified by:
- a function $m'=f_{\alpha,\beta}(m)$, such that $f_{\alpha'\alpha, \beta'\beta}=f_{\alpha',\beta'}\circ f_{\alpha,\beta} $.
- a family of isometries $E_\alpha$ from ${\cal H}_{d}$ to $\bigoplus_\alpha{\cal H}_{d}$, such that $(\bigoplus_\alpha E_{\alpha'})E_\alpha = E_{\alpha'\alpha}$.
Above we used the notation $\bigoplus_\alpha{\cal H}_d = \bigoplus_{i=1\ldots \alpha} {\cal H}_d$. Consider ${\psi}$ a spacetime wavefunction (at this stage it is not necessary to assume that it is a solution of the QW). Switching to lightlike coordinates, its Lorentz transform $\psi'=L_{\alpha,\beta}\psi$ is obtained by:
- for every $(r,l)$, computing: $\widecheck{\psi}_+=E_\beta\psi_+$, $\widecheck{\psi}_-=E_\alpha\psi_-$, and $\widecheck{\psi}=\widecheck{\psi}_+\oplus\widecheck{\psi}_-=\overline{E} {\psi}$.
- for every $(r,l)$, replacing: the point $(r,l)$ by the lightlike $\alpha\times\beta$ rectangular patch of spacetime $$\begin{aligned}
\left(\overline{C}_{m'}(i,j)\widecheck{\psi}(r,l)\right)_{i=0\ldots\alpha-1,j=0\ldots\beta-1}\label{eq:genLorentz transform}\end{aligned}$$ with $\overline{C}_{{m}'}(i,j)$ as in Remark \[rk:detregion\] and Fig. \[fig:determined\].
Again, Fig. \[fig:Lorentz transform\] illustrated an example of such a transformation.
### Lorentz covariance for QW
The formalization of a general notion of Lorentz covariance for QWs generalizes that presented in Subsection \[subsec:focovariance\]. Consider a QW having wire dimension $d$ whose $2d\times 2d$ and unitary coin unitary coin $C_m$, where the $m$ are parameters. Consider $\psi$ a spacetime wavefunction which is a solution of this QW. We just gave the formalization of a discrete general notion of Lorentz transform taking a spacetime wavefunction $\psi$ into another spacetime wavefunction $\psi'=L_{\alpha,\beta}\psi$, and parameters ${m}$ into ${m}'$. Is it the case, for any $\alpha$ and $\beta$, that the spacetime wavefunction $\psi'$ is a solution of the same QW, but with parameters ${m}'$? If so, the QW is said to be covariant with respect to the given discrete Lorentz transform. Now, the above-defined discrete Lorentz transform is obtained by replacing each point with a lightlike $\alpha\times\beta$ rectangular patch of spacetime, which, by definition, is internally a piece of spacetime solution of the Dirac QW see Eq. . But again, is it the case that the patches match up to form the entire spacetime wavefunction of a solution? Let us again define $$\widehat{\psi}_+(r,l)=(\overline{C}_{m'})_+\widecheck{\psi}(r,l)\quad\textrm{and}\quad \widehat{\psi}_-(r,l)=(\overline{C}_{m'})_-\widecheck{\psi}(r,l).$$ We need: $\widehat{\psi}_+(r,l)=\widecheck{\psi}_+(r+\varepsilon,l)$ and $\widehat{\psi}_-(r,l)=\widecheck{\psi}_-(r,l+\varepsilon)$. An equivalent, algebraic way of stating these two requirements is obtained as follows: $$\begin{aligned}
\widecheck{\psi}_+(r+\varepsilon,l) \oplus \widecheck{\psi}_-(r,l+\varepsilon) = \widehat{\psi}_+(r,l) \oplus \widehat{\psi}_-(r,l) \nonumber\end{aligned}$$ Equivalently, $$\begin{aligned}
& \left(E_\beta\oplus E_\alpha \right)\left(\psi_+(r+\varepsilon,l)\oplus \psi_-(r,l+\varepsilon)\right) = \nonumber \\
& \left(\overline{C}_{{m}'}(\alpha,\cdot)\oplus \overline{C}_{{m}'}(\cdot,\beta)\right)\left(E_\beta\oplus E_\alpha \right){\psi}(r,l)\nonumber\\
\Leftrightarrow & \left(E_\beta\oplus E_\alpha \right)C_m \psi(r,l) = \overline{C}_{{m}'}\left(E_\beta\oplus E_\alpha \right){\psi}(r,l) \nonumber\\
\Leftrightarrow & \left(E_\beta\oplus E_\alpha \right)C_m = \overline{C}_{{m}'}\left(E_\beta\oplus E_\alpha \right)\nonumber\\
\Leftrightarrow & \overline{E} C_m = \overline{C}_{{m}'}\overline{E}. \label{eq:consistency}\end{aligned}$$
This expresses discrete Lorentz covariance elegantly, as a form of commutation relation between the evolution and the encoding. Diagrammatically this is represented by Fig. \[fig:CovRules\]$(a)$. The isometry of the $E_\alpha$ can also be represented diagrammatically, cf. Fig. \[fig:CovRules\]$(b)$. Combining both properties straightforwardly leads to $$\begin{aligned}
C_m &= \overline{E}^\dagger \overline{C}_{{m}'} \overline{E}.\label{eq:pointconsistency}\end{aligned}$$ This is represented as Fig. \[fig:OtherRules\]$(a)$, which of course can be derived diagrammatically from Fig. \[fig:CovRules\]. Is this diagrammatic theory powerful enough to be considered an abstract, pictorial theory of Lorentz covariance, in the spirit of [@CoeckeBoxes]?
### Diagrammatic Lorentz covariance for QW
Combining the diagrammatic equalities of Fig. \[fig:CovRules\], we can almost rewrite the spacetime circuit of a QW with coin $C_m$, into its Lorentz transformed version, for any parameters $\alpha,\beta$…but not quite. A closer inspection shows that this can only be done over regions such as past cones, by successively: 1/ Introducing pairs of encodings via rule Fig. \[fig:CovRules\]$(b)$ along the border of the past cone; 2/ Pushing back towards the past the bottom $E$ via rule Fig. \[fig:CovRules\]$(a)$, thereby unveiling the Lorentz transformed past cone. Whilst this limitation to past-cone-like regions may seem surprising at first, there is a good intuitive reason for that. Indeed, the diagrammatic equalities of Fig. \[fig:CovRules\] tell you that you can locally zoom into a spacetime circuit; but you can only locally zoom out if you had zoomed in earlier, otherwise there may be a loss of information. This asymmetry is captured by the fact that Fig. \[fig:CovRules\]$(a)$ cannot be put upside-down, time-reversed. It follows that you should not be able to equalize an entire spacetime circuit with its complete Lorentz transform, at least not without using further hypotheses. And indeed, when we local Lorentz transform an entire past cone, its border is there to keep track of the fact that this region was locally zoomed into, and that we may later unzoom from it, if we want.\
 
![\[fig:OtherRules\] [*Completed covariance rules*]{}. $(a)$ is a theorem, derived from the diagrams of Fig. \[fig:CovRules\], see also Eqs and . It expresses the idea of a Lorentz transform being a zoom in. The dashed line is optional, it is an indication which results from using this rule: it tells us that the state of these wires belongs to the subspace $S_\alpha$. The gray and white dots stand for same unitary interaction, but with different parameters. $(b)$ is a conditional rule: the thicker dashed line is a precondition for the equality to hold. Again it follows from the isometry of the encodings used for the discrete Lorentz transform, see also Eq. .](covariancesplitting.pdf "fig:") ![\[fig:OtherRules\] [*Completed covariance rules*]{}. $(a)$ is a theorem, derived from the diagrams of Fig. \[fig:CovRules\], see also Eqs and . It expresses the idea of a Lorentz transform being a zoom in. The dashed line is optional, it is an indication which results from using this rule: it tells us that the state of these wires belongs to the subspace $S_\alpha$. The gray and white dots stand for same unitary interaction, but with different parameters. $(b)$ is a conditional rule: the thicker dashed line is a precondition for the equality to hold. Again it follows from the isometry of the encodings used for the discrete Lorentz transform, see also Eq. .](covariancesubspace.pdf "fig:")
{width="\textwidth"}
Now, could we add a further diagrammatic rule which would allow us to perform an complete Lorentz transformation, perhaps at the cost of annotating our spacetime circuit diagrams with information on whom has been zoomed into? Those annotations are the dashed lines of Fig. \[fig:CovRules\] and Fig. \[fig:OtherRules\]$(a)$. Clearly, as we use those rules, we know whether some bunch of wires lives in the subspace $S_{\alpha}$ of the projector $E_\alpha E_\alpha^\dagger$, and we can leave that information behind. Moreover, on this subspace, it is the case that $$\begin{aligned}
E_\alpha E_\alpha^\dagger=\Id_{S_{\alpha}}.\label{eq:subspace}\end{aligned}$$ Then, representing this last equation in rule Fig. \[fig:OtherRules\]$(b)$, which is conditional on the annotation being there (the other rules are non-conditional, they provide the annotations), we reach our purpose. Indeed, in order to perform a complete Lorentz-transform we can now apply the rule Fig. \[fig:OtherRules\]$(a)$ everywhere, leading to Fig. \[fig:transformingall\], and then remove the encoding gates everywhere via Fig. \[fig:OtherRules\]$(b)$. Thus, it could be said that the rewrite rules of Fig. \[fig:OtherRules\] provide an abstract, pictorial theory of Lorentz covariance. They allow to equalize, spacetime seen by a certain observer, with spacetime seen by another, inertial observer. Besides their simplicity, the local nature of the rewrite rules is evocative of the local Lorentz covariance of General Relativity. This is explored a little further in Subsection \[subsec:nonhomog\].
### Inverse transformations and equivalence upon rescaling
In analogy with the continuum case, we would like the inverse of a Lorentz transform $L_{\alpha,\beta}$ to be $L_{\beta,\alpha}$, i.e. $$\label{eq:inverselorentz}
L_{\alpha,\beta}L_{\beta,\alpha} = \Id.$$ However, according our definitions of $L_{.,.}$, we know that $L_{\alpha,\beta}L_{\beta,\alpha}$ is a transformation such that
- each point $(r,l)$ is replaced by the lightlike $\alpha\beta\times\alpha\beta$ square patch of spacetime, with left-incoming wires $F\psi_+(r,l)$, right-incoming wires $F\psi_-(r,l)$, right-outgoing wires $F\psi_+(r+\varepsilon,l)$ and left-outgoing wires $F\psi_-(r,l+\varepsilon)$, where $$F = \left(\bigoplus_\alpha E_\beta\right)E_\alpha = E_{\beta\alpha} = E_{\alpha\beta} = \left(\bigoplus_\beta E_\alpha\right)E_\beta$$
- the coin parameter $m$ is mapped to $m'=f_{\alpha\beta,\alpha\beta}(m)$.
Hence, if we are to claim we need to identify any two spacetime diagrams which satisfy these relations. This is achieved as a special case in the completed diagrammatic theory of Fig. \[fig:OtherRules\].
Over Quantum Cellular Automata {#subsec:LorentztransformsQCA}
------------------------------
### General form of Quantum Cellular Automata
Intuitively speaking, a Quantum Cellular Automata (QCA) is a multiple walkers QW. The walkers may or may not interact, their numbers may or may not be conserved. Axiomatically speaking, a QCA is a shift-invariant, causal, unitary evolution over the space $``\bigotimes_{\mathbb{Z}} {\cal H}_{c}"$, where $c$ is the dimension of the internal degrees of freedom of each site. Actually, care must be taken when defining such infinite tensor products, but two solutions exist [@SchumacherWerner; @ArrighiLATA; @ArrighiIJUC]. Constructively speaking, it turns out [@SchumacherWerner; @ArrighiLATA; @ArrighiIJUC] that, at the cost of some simple recodings, any QCA can be put in the form of a quantum circuit. This circuit can then be simplified [@ArrighiPQCA] to bear strong resemblance with the circuit of a general QW seen in Fig. \[fig:QWGeneral\]. In particular $c$ can always be taken to be $d^2$, so that the general shape for the quantum circuit of a QCA is that of Fig. \[fig:QCAGeneral\]. Notice how, in this diagram, each wire carries a $d$-dimensional vector $\psi_\pm(r,l)$. We will say that the QCA has ‘wire dimension’ $d$. Incoming wires get composed together with a tensor product, to form a $d^2$-dimensional vector $\psi(r,l)$. The state $\psi(r,l)$ undergoes a $d^2\times d^2$ unitary gate $U$ to become some $\psi_+(r+\varepsilon,l)\otimes \psi_-(r,l+\varepsilon)$, etc. The unitary gate $U$ is called the ‘scattering operator’. Notice how, to some extent, the QCA are alike QW up to replacing $\oplus$ by $\otimes$. Algebraically speaking, the above means that one time-step of a QCA can always be assumed to be of the form: $$\begin{aligned}
{\psi}\mapsto
&\left(\bigotimes_{2\mathbb{Z}+1}U\right)\left(\bigotimes_{2\mathbb{Z}}U\right){\psi}.\end{aligned}$$
![\[fig:QCAGeneral\] [*The circuit for a general QCA.*]{}](conventions5qca.pdf){width="\columnwidth"}
### Lorentz transforms for QCA
The formalization of a general notion of Lorentz transform for QCA is obtained from that over QW essentially by changing occurrences of $\oplus$ into $\otimes$. Indeed, consider a QCA having wire dimension $d$, and whose $d^2\times d^2$ unitary scattering operator $U$ has parameters $m$. A Lorentz transform $L_{\alpha,\beta}$ is specified by:
- a function ${m}'=f_{\alpha,\beta}({m})$ such that $f_{\alpha'\alpha, \beta'\beta}=f_{\alpha',\beta'}\circ f_{\alpha,\beta} $.
- a family of isometries $E_{\alpha}$ from ${\cal H}_{d}$ to $\bigotimes_\alpha {\cal H}_{d}$, such that $(\bigotimes_\alpha E_{\alpha'})E_\alpha = E_{\alpha'\alpha}$.
There is a crucial difference with QWs, however, which is that we cannot easily apply this discrete Lorentz transform to a spacetime wavefunction. Indeed, consider ${\psi}$ a spacetime wavefunction. For every time $t$, the state $\psi(t)$ may be a large entangled state across space. What meaning does it have, then, to select another spacelike surface? What meaning does it have to switch to lightlike coordinates? Unfortunately the techniques which were our point of departure for QWs, no longer apply. Fortunately, the algebraic and diagrammatic techniques which were out point of arrival for QWs, apply equally well to QCA, so that we may still speak of Lorentz-covariance.
### Lorentz covariance for QCA
Again, the formalization of the notion of Lorentz-covariance for QCA cannot be given in terms of $\psi'$ being a solution if $\psi$ was a solution, because we struggle to speak of $\psi'$. Instead, we define Lorentz-covariance straight from the algebraic view: $$\begin{aligned}
\left(E_\beta\otimes E_\alpha \right)U_m &= \overline{U}_{{m}'}\left(E_\beta\otimes E_\alpha \right)\nonumber\\
\textrm{i.e. }\quad\overline{E} U_m &= \overline{U}_{{m}'}\overline{E}.\label{eq:consistency2}\end{aligned}$$ Diagrammatically this is represented by the same figure as for QWs, namely Fig. \[fig:CovRules\]$(a)$. The isometry of the $E_\alpha$ is again represented by Fig. \[fig:CovRules\]$(b)$. Algebraically speaking, combining both properties again leads to $$\begin{aligned}
U_m &= \overline{E}^\dagger \overline{U}_{{m}'} \overline{E}.\label{eq:pointconsistency2}\end{aligned}$$ Which diagrammatically this is again represented as Fig. \[fig:OtherRules\]$(a)$. For the same reasons, the conditional rule Fig. \[fig:OtherRules\]$(b)$ again applies: the whole diagrammatic theory carries through unchanged from QWs to QCA.
Non-homogeneous discrete Lorentz transforms and non-inertial observers {#subsec:nonhomog}
----------------------------------------------------------------------
{width="\textwidth"}
Nothing in the above developed diagrammatic theory forbids us to apply different local discrete Lorentz transforms to different points of spacetime, so long as point $(r,l)$ and point $(r+\varepsilon,l)$ (resp. point $(r,l+\varepsilon)$) have the same parameter $\beta$ (resp. $\alpha$). This constraint propagates along lightlike lines, so that there can be, at most, one different $\alpha_r$ (resp. $\beta_l$) per right-moving (resp. left-moving) lightlike line $r$ (resp. $l$). We call this a non-homogeneous discrete Lorentz transform of parameters $(\alpha_r)$, $(\beta_l)$.
The circuit which results from applying such a non-homogeneous discrete Lorentz transform is, in general, a non-homogeneous QWs (resp. QCA), as it may lack shift-invariance in time and space. This is because the coin $C_m$ (resp. scattering unitary $U_m$) of the point $(r,l)$ gets mapped into lightlike $\alpha_r\beta_l$-rectangular patch of spacetime $\overline{C}_{m'}$ (resp. $\overline{U}_{m'}$), whose parameters $m'=f_{\alpha_r,\beta_l}(m)$ may depend, in general, upon the position $(r,l)$. This problem is avoided if $f_{\alpha\beta}=f$ does not depend upon $\alpha$ and $\beta$, as in the example which will be introduced in Section \[sec:ClockQCA\]. Provided that the condition $f_{\alpha\beta}=f$ is met, we can now transform between non-inertial observers by a non-homogeneous discrete Lorentz transform. Figure \[fig:nonhomogeneoussimple\] illustrates this with the simple example of an observer which moves one step right, one step left, until it reaches point $(0,0)$ where it gets accelerated, and continues moving two steps right, one step left etc. We choose $\beta_{l}=1$ for $l < 0$,$\beta_{l}=2$ for $l\geq 0$ and $\alpha_{r}=1$ for all $r$. This has the effect of slowing down the observer just beyond the point $(0,0)$. All along his trajectory, he now has to move two steps right for every two steps left that he takes, so that he is now at rest.
In general, suppose that an observer moves $a_k$ steps to the right, $b_k$ steps left, $a_{k+1}$ steps right, etc. He does this starting from position $r_k = r_{k-1}+a_k$ and $l_k=l_{k-1}+b_k$. For every $k$, let $M_k$ be the least common multiple of $a_k$ and $b_k$. We choose $\alpha_r=M_k/a_k$ for $r_{k-1} \leq r < r_{k}$ and $\beta_l=M_k/b_k$ for $l_{k-1} \leq l < l_{k}$. Let us perform the non-homogeneous discrete Lorentz transform of parameters $(\alpha_r)$, $(\beta_l)$. Then, the observer now moves $M_k$ steps right for every $M_k$ steps left he takes, and then $M_{k+1}$ steps right for every $M_{k+1}$ steps left, etc.
The Clock QW {#sec:ClockQW}
============
Equipped with a formal, general notion of Lorentz transform and Lorentz covariance for QW, we can now seek for an exactly covariant QW.
Definition
----------
In the classical setting, covariance of random walks has already been explored [@wall1988discrete]. The random walk of [@wall1988discrete] uses a fair coin, but is nonetheless biased in the following way: after a (fair) coin toss the walker moves during $\rs$ time steps to the right (resp. during $\ls$ time steps to the left). There is a reference frame in which the probability distribution is symmetric, namely that with velocity $u = (\rs - \ls)/(\rs + \ls)$. Changing the parameters $\rs$ and $\ls$ corresponds to performing a Lorentz transform of the spacetime diagram.
Now we will make an analogous construction in the quantum setting. The main point is to enlarge the coin space so that the coin operator is idle during $\rs$, or $\ls$, time steps. The coin space will be ${\mathcal{H}}_C = \mathcal{H}_C^+ \oplus \mathcal{H}_C^-$, where $\mathcal{H}_C^+ \cong \mathcal{H}_C^- = \ell^2(\mathbb Q^{\geq 0})$. The Hilbert space of the quantum walk is then ${\mathcal{H}} = \ell^2(\Z) \otimes {\mathcal{H}_C}$, whose basis states will be indicated by $\ket{x,h^s}$, with $h\in\mathbb Q^{\geq0}$, $s=\pm$.
This $\mathcal{H}_C^\pm$ will act as a “counter”. When $h>0$, the walker moves without interaction and the counter is decreased. When the counter reaches $0$, the effective coin operator is applied and the counter is reset.
The evolution of the Clock QW with parameters $\rs,\ls$ is defined on the subspace $\mathcal{H}_C^{\rs,\ls}$ of $\mathcal{H}_C$ spanned by the $\rs + \ls$ vectors $\{\ket{\frac{i}{\rs}^+},\ket{\frac{j}{\ls}^-}\}$ with $i=0,\dots,\rs - 1$ and $j=0,\dots,\ls - 1$, as follows:
$$W_{\rs,\ls}\ket{x,h^s} = \left\{
\begin{array}{ll}
\ket{x+1,(h-\frac1{\rs})^+} & \mbox{for } s=+, \quad 0 < h \leq 1-\frac1{\rs}\\
\ket{x-1,(h-\frac1{\ls})^-} & \mbox{for } s=-, \quad 0 < h \leq 1-\frac1{\ls}\\
a \ket{x+1,(1-\frac1{\rs})^+} + b \ket{x-1,(1-\frac1{\ls})^-} & \mbox{for } s=+, \quad h=0\\
c \ket{x+1,(1-\frac1{\rs})^+} + d \ket{x-1,(1-\frac1{\ls})^-} & \mbox{for } s=-, \quad h=0 \\
\end{array} \right. \label{eq:ClockQW_Map}$$
This map is unitary provided that the $2\times 2$ matrix $C$ of coefficients $C_{11}=a$, $C_{12}=b$, $C_{21}=c$, $C_{22}=d$ is unitary. For instance we could choose, as for the Dirac QW, $a=d=\cos(m \varepsilon)$, $b=c=-\ii \sin(m \varepsilon)$.
The Clock QW with parameters $\rs$ and $\ls$ will only be used over $\ell^2(\Z)\otimes \mathcal{H}_C^{\rs,\ls}$ where it admits a matrix form which we now provide (over the rest of $\mathcal{H}_C$ it can be assumed to be the identity). From Eq. (\[eq:ClockQW\_Map\]) we can write $W_{\rs,\ls} = T_{\rs,\ls}C_{\rs,\ls}$ where $T_{\rs,\ls}$ is the shift operator, $$T_{\rs,\ls} = \text{diag} \left(\overbrace{e^{-\varepsilon \partial_x},\dots,e^{-\varepsilon \partial_x}}^{\rs\mbox{ \footnotesize times}},\overbrace{e^{\varepsilon \partial_x},\dots,e^{\varepsilon \partial_x}}^{\ls\mbox{ \footnotesize times}} \right)$$ and $C_{\rs,\ls}$ is the coin operator: $$C_{\rs,\ls} = \left(\begin{array}{cc|cc}
0 & \Id_{\rs-1} & 0 & 0 \\
a & 0 & b & 0 \\
\hline
0 & 0 & 0 & \Id_{\ls - 1} \\
c & 0 & d & 0
\end{array}\right)$$
Hence, the Clock QW has an effective coin space of finite dimension $\rs+\ls$. However, we will see that this dimension changes under Lorentz transforms.
Covariance
----------
In order to prove covariance, we need to find isometries satisfying the equation expressed by Fig. \[fig:CovRules\]$(a)$. Let us consider isometries $E_{\alpha}: \mathcal{H}_C \to \bigoplus_{\alpha} \mathcal{H}_C$ defined by: $$\begin{aligned}
E_{\alpha} \ket{h^s} = (\ket{h^s} \oplus \overbrace{0 \oplus \dots \oplus 0}^{\mbox{$\alpha-1$ times}}) \\\end{aligned}$$ (the Hilbert spaces in the direct sum are ordered from the bottom wire to the top one, as in remark \[rk:detregion\]). In Fig. \[fig:ClockQWCov\] it is proved that this choice actually satisfies the covariance relation $\overline{E}C_{\rs,\ls} = \overline{C}_{\rs', \ls'}~\overline{E}$, where the coin operator parameters have been rescaled as $\rs'=\alpha\rs$ and $\ls'=\beta\ls$. Intuitively, the Lorentz transformation rescales the fractional steps of the Clock QW by $\alpha$ (resp. $\beta$), while adding $\alpha-1$ (resp. $\beta-1$) more points to the lattice. In this way, the counter will reach $0$ just at the end of the patch, as it did before the transformation.
Continuum limit of the Clock QW
-------------------------------
The Clock QW does not have a continuum limit because its coin operator is not the identity in the limit $\varepsilon \rightarrow 0$. However, by appropriately sampling the spacetime points, it is possible to take the continuum limit of a solution of the Clock QW and show that it converges to a solution of the Dirac equation, subject to a Lorentz transform with parameters $\rs$, $\ls$. Indeed, the limit can be obtained as follows. First, we divide the spacetime in lightlike rectangular patches of dimension $\rs\times\ls$. Second, we choose as representative value for each patch the point where the interaction is non-trivial, averaged according to the dimensions of the rectangle: $$\psi'(r,l) = \left( \begin{array}{c} \dfrac{\psi_+(\floor{r/\rs}_\varepsilon,\floor{l/\ls}_\varepsilon)}{\sqrt{\ls}} \\ \dfrac{\psi_-(\floor{r/\rs}_\varepsilon,\floor{l/\ls}_\varepsilon)}{\sqrt{\rs}} \end{array} \right).$$ Finally, by letting $\varepsilon\to 0$ we obtain $$\psi'(r,l) = S\psi(r/\rs,l/\ls)$$ where now the $r,l$ coordinates are to be intended as continuous.
Since $\psi'$ is of course a solution of the Dirac equation (with a rescaled mass), this proves that the continuum limit of the Clock QW evolution, interpreted as described above, is again the Dirac equation itself.
Decoupling of the QW and the Klein-Gordon equation
--------------------------------------------------
The Clock QW does not have a proper continuum limit unless we exclude the intermediate computational steps. Still, as we shall prove in this section, its decoupled form [*has*]{} a proper limit, which turns out to be the Klein-Gordon Equation with a rescaled mass. By a decoupled form, we mean the scalar evolution law satisfied by each component of a vector field, individually (see [@ArrighiKG]). In the following, we give the decoupled form of the Clock QW. The evolution matrix $W$ is sparse and allows for decoupling by simple algebraic manipulations, leading to: $$\left[T^{\ls+\rs} - a\tau^{-\ls} T^\rs - d\tau^{\rs} T^\ls + \det(C) \tau^{\rs-\ls}\right]\psi = 0$$ (where $T=e^{\varepsilon\partial_t}$ and $\tau=e^{\varepsilon\partial_x}$). This is a discrete evolution law which gives the value of the current step depending on three previous time steps, namely the ones at $t=-\rs$, $t=-\ls$ and $t=-\rs-\ls$.
By expanding in $\varepsilon$ the displacement operators and assuming that the coin operator verifies: $$\det(C) = 1, \quad a = d = 1 + \frac{\varepsilon^2m^2}{2} + O(\varepsilon^3)$$ (which is the case if $a=d=\cos(m \varepsilon)$) we obtain the continuum limit: $$\left(\partial_t^2 - \partial_x^2 + \frac{m^2}{\rs \ls}\right)\psi = 0.$$ Up to redefinition of the mass $m' = m/\sqrt{\rs \ls}$, this is the Klein-Gordon Equation. This reinforces the interpretation of the Clock QW as model for a relativistic particle of mass $m'$.
The Clock Quantum Cellular Automata {#sec:ClockQCA}
===================================
One downside of the Clock QW is the fact that the dimension of the coin space varies according to the observer. Equipped with a formal, general notion of Lorentz transform and Lorentz covariance for QCA, we can now seek for an exactly covariant QCA of fixed, small, internal degree of freedom.
From the Clock QW to the Clock QCA
----------------------------------
The idea of the Clock QW was to let the walker propagate during a number of steps to the right (resp. to the left), without spreading to the left (resp. to the right). In the absence of any other walker, this had to be performed with the help of an internal clock. In the context of QCA, however, the walker can be made to cross “keep going” signals instead.\
The Clock QCA has wire dimension $d=3$, with orthonormal basis $\ket{q}$, $\ket{0}$, $\ket{1}$. Both $\ket{q}$ and $\ket{0}$ should be understood as vacuum states, but of slightly different natures as we shall see next. $\ket{1}$ should be understood as the presence of a particle.\
Thus, the Clock QCA has scattering unitary a $9\times 9$ matrix $U$, which we can specify according to its action over the nine basis vectors. First we demand that the vacuum states be stable, i.e. $$\begin{aligned}
\ket{q}\otimes\ket{q}&\mapsto \ket{q}\otimes\ket{q},\\
\ket{q}\otimes\ket{0}&\mapsto \ket{0}\otimes\ket{q},\\
\ket{0}\otimes\ket{q}&\mapsto \ket{q}\otimes\ket{0},\\
\ket{0}\otimes\ket{0}&\mapsto \ket{0}\otimes\ket{0}.\end{aligned}$$ Second we demand that multiple walkers do not interact: $$\begin{aligned}
\ket{1}\otimes\ket{1}&\mapsto \ket{1}\otimes\ket{1}.\end{aligned}$$ Third we demand that the interaction between $\ket{1}$ and $\ket{q}$ be dictated by a massless Dirac QW, or “Weyl QW”, i.e. the single walker goes straight ahead: $$\begin{aligned}
\ket{1}\otimes\ket{q}&\mapsto \ket{q}\otimes\ket{1},\\
\ket{q}\otimes\ket{1}&\mapsto \ket{1}\otimes\ket{q}.\end{aligned}$$ Last we demand that the interaction between $\ket{1}$ and $\ket{0}$ be dictated by: $$\begin{aligned}
\ket{1}\otimes\ket{0}&\mapsto a(\ket{0}\otimes\ket{1})+b(\ket{1}\otimes\ket{0}),\\
\ket{0}\otimes\ket{1}&\mapsto c(\ket{0}\otimes\ket{1})+ d(\ket{1}\otimes\ket{0}).\end{aligned}$$ This map is unitary provided that the $2\times 2$ matrix $C$ of coefficients $C_{11}=a$, $C_{12}=b$, $C_{21}=c$, $C_{22}=d$ is unitary. For instance we could choose, as for the Dirac QW, $a=d=\cos(m \varepsilon)$, $b=c=-\ii \sin(m \varepsilon)$.\
The Clock QCA is covariant, even though its wire dimension is fixed and small, as we shall see.\
Covariance of the Clock QCA
---------------------------
In order to give a precise meaning to the statement according to which the Clock QCA is covariant, we must specify our Lorentz transform. According to Section \[subsec:LorentztransformsQCA\] we must provide a function $f$, which we take to be the identity, and an encoding $E_\alpha: {\cal H}_d \longrightarrow {\cal H}^{\otimes\alpha}_d$ which we take to be: $$\ket{a}\mapsto \ket{a}\otimes \bigotimes_{\alpha-1} \ket{q},$$ written from the bottom wire to the top wire as was the convention for QWs. The intuition is that the $(\alpha-1)$ ancillary wires are just there to stretch out this lightlike direction, but given that $\ket{q}$ interacts with no one, this stretching will remain innocuous to the physics of the QCA.
Let us prove that things work as planned:
$$\begin{aligned}
\overline {U} ~ \overline{E} (\ket{a}\otimes\ket{b}) &= \left(\prod_{i=0\ldots\alpha-1,j=0\ldots\beta-1} U_{m'}\right) (\ket{a}^{0,0}\otimes \bigotimes_{i=1\ldots\alpha} \ket{q}^{i,0}) \otimes (\ket{b}^{0,0}\otimes \bigotimes_{j=1\ldots\beta} \ket{q}^{0,j})\\
&= \left(\prod^{(i,j)\neq(0,0)}_{i=0\ldots\alpha-1,j=0\ldots\beta-1} U_{m'}\right) U(\ket{a}^{0,0}\otimes\ket{b}^{0,0})\otimes \bigotimes_{i=1\ldots\alpha} \ket{q}^{i,0} \otimes \bigotimes_{j=1\ldots\beta} \ket{q}^{0,j})\\
&= U(\ket{a}^{0,\beta-1}\otimes\ket{b}^{\alpha-1,0}) \otimes \bigotimes_{i=1\ldots\alpha} \ket{q}^{i,\beta-1} \otimes \bigotimes_{j=1\ldots\beta} \ket{q}^{\alpha-1,j}\\
&= \overline{E} U (\ket{a}\otimes\ket{b}).\end{aligned}$$
Hence, the Clock QCA is Lorentz covariant. Notice that things would have worked equally well if $E_{\alpha}$ had placed $\ket{a}$ differently amongst the $\ket{q\ldots}$. It could even have spread out $\ket{a}$ evenly across the different positions, in a way that is more akin to the Lorentz transform for the Dirac QW.
Conclusions {#sec:Conclusions}
===========
In the context of QW and QCA, we have formalized a notion of discrete Lorentz transform of parameters $\alpha, \beta$, which consists in replacing each spacetime point with a lightlike $\alpha\times\beta$ rectangular spacetime patch, $\overline{C}_{m'}\overline{E}$, where $\overline{E}$ is an isometric encoding, and $\overline{C}_{m'}$ is the repeated application of the unitary interaction $C_{m'}$ throughout the patch (see Fig. \[fig:Lorentz transform\]). We then formalized discrete Lorentz covariance as a form of commutativity: $\overline{E} C_m = \overline{C}_{m'} \overline{E}$. This commutation rule as well as the fact the $E$ is isometric can be expressed diagrammatically in terms of a few local, circuit equivalence rules (see Fig. \[fig:CovRules\] and \[fig:OtherRules\]), à la [@CoeckeBoxes]. This simple diagrammatic theory allows for non-homogeneous Lorentz transforms (Fig. \[fig:nonhomogeneoussimple\]), which let you switch between non-inertial observers. Actually, it would be interesting to compare the respective powers of covariance under non-homogeneous Lorentz transformations versus general covariance under diffeomorphisms plus local Lorentz covariance.
First we considered the Dirac QW, a natural candidate, given that it has the Dirac equation as continuum limit, which is of course covariant. Unfortunately, we proved the Dirac QW to be covariant only up to first-order in the lattice spacing $\varepsilon$. This is inconvenient if $\varepsilon$ is considered a physically relevant quantity, i.e. if spacetime is really thought of as discrete. But if $\varepsilon$ is thought of as an infinitesimal, then the second-order failure of Lorentz-covariance is irrelevant. Thus, this result encourages us to take the view that $\varepsilon$ is akin to infinitesimals in non-standard analysis. Then, the Dirac QW would be understood as describing an infinitesimal time evolution, but in the same formalism as that of discrete time evolutions. As an alternative language to the Hamiltonian formalism, it has the advantage of sticking to local unitary interactions [@ArrighiJCSS], and that of providing a quantum simulation algorithm.
Exact Lorentz covariance, however, is possible even for finite $\varepsilon$. This paper introduces the Clock QW, which achieves this property. However the effective dimension of its internal degree of freedom depends on the observer. Furthermore, the Clock QW does not admit a continuum limit, unless we appropriately sample the points of the lattice. Yet, its decoupled form does have a continuum limit, which is the Klein-Gordon Equation. It is interesting to see that there is a QW evolution which can be interpreted as a relativistic particle (since it satisfies the KG Equation), and yet not have a continuum limit for itself.
Finally, we introduced the Clock QCA, which is exactly covariant and has a three dimensional state space for its wires.
We leave the following question open: is there a systematic method which given a QW with coin operator $C$, decides whether it exists a Lorentz transform $E_{\alpha}$, $f_{\alpha,\beta}$ such that $\overline{E} C_m = \overline{C}_{m'} \overline{E}$, i.e. such that the QW is covariant? The same question applies to QCA; answering it would probably confirm the intuition that covariant QWs are scarce amongst QWs.
The simple theory presented here can be criticized on several grounds. First, one may wish for more explicit comparison with the continuum theory. This may be done along the lines of [@ArrighiDirac; @DAriano]: by letting the lattice spacing $\varepsilon$ go to zero, the convergence of the spacetime wavefunction solution of the Dirac QW can be shown to tend to the solution of the Dirac equation, in a manner which can be quantified. Second, one may argue that the very definition of the Lorentz transform should not depend on the QW under consideration. Similarly, one may argue that the transformed wave function should be a solution of the original QW, without modifications of its parameters. However, recall that $1+1$ dimensional, integral Lorentz transforms are trivial unless we introduce a global rescaling. Thus the discrete Lorentz transform of this paper may be thought of as a biased zooming in. In order to fill in the zoomed in region, one generally has to use the QW in a weakened, reparameterized manner.
On the one hand, this paper draws it inspiration from Quantum Information and a perspective for the future would be to discuss relativistic quantum information theory [@aharonov1984quantum; @peres2004quantum] within this framework. On the other hand, it forms part of a general trend seeking to model quantum field theoretical phenomena via discrete dynamics. For now, little is known on how to build QCA models from first principles, which admit physically relevant Hamiltonians [@DAriano3D; @elze2013action; @farrelly2013discrete; @t2013duality] as emergent. In this paper we have identified one such first principle, namely the Lorentz covariance symmetry. We plan on studying another fundamental symmetry, namely isotropy, thereby extending this work to higher dimensions.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are indebted to A. Joye, D. Meyer, V. Nesme and A. Werner. This work has been funded by the ANR-10-JCJC-0208 CausaQ grant.
First-order-only covariance of the Dirac QW {#App:FirstOrderOnlyProof}
===========================================
[*Uniqueness of encodings.*]{} Here we prove that the only encoding compatible with first-order covariance is the flat one, as described in section \[subsec:DiracLorentzTransform\]. In general, the encoding isometries $E_\alpha$, $E_\beta$ can be defined in terms of normalized vectors, $\mathbf v_\pm$ as follows (remember that for the Dirac QW, $\psi_+$ and $\psi_-$ are just scalars): $$E_\beta\psi_+ = \psi_+ \mathbf v_+, \qquad E_\alpha\psi_- = \psi_- \mathbf v_-.$$ In order to require covariance, we need to calculate the terms appearing in the commutation relation . The r.h.s of the relation is (see Fig. \[fig:FirstOrderCovariance\] and Subsection \[subsec:focovariance\]): $$\overline{C}_{m'}\overline{E} = \left(\begin{array}{c}
\psi_+\mathbf v_+ - im'\varepsilon (\sum \mathbf v_-) \psi_- \mathbf 1_\beta \\
\psi_-\mathbf v_- - im'\varepsilon (\sum \mathbf v_+) \psi_+ \mathbf 1_\alpha
\end{array}\right) + O(\varepsilon^2)$$ where $\mathbf 1_d = (1,\dots,1)^{\mathsf T}$ is the $d$-dimensional uniform vector, and $\sum \mathbf v = \sum_i v_i$. On the other hand the l.h.s. is: $$\overline{E}C_m = \left(\begin{array}{c}
\psi_+\mathbf v_+ - im\varepsilon\psi_-\mathbf v_+ \\
\psi_-\mathbf v_- - im\varepsilon\psi_+\mathbf v_-
\end{array}\right) + O(\varepsilon^2).$$ Requiring first-order covariance, one obtains $$\begin{aligned}
m\mathbf v_+ = m' \left(\sum\mathbf v_-\right) \mathbf 1_\beta, \qquad m\mathbf v_- = m' \left(\sum\mathbf v_+\right) \mathbf 1_\alpha\end{aligned}$$ which, together with the normalization of $\mathbf v_\pm$, gives $$\begin{aligned}
m' = \frac{m}{\sqrt{\alpha\beta}}, \quad \mathbf v_+ = \frac{e^{i\lambda_+}}{\sqrt{\beta}}\mathbf 1_\beta, \quad \mathbf v_- = \frac{e^{i\lambda_-}}{\sqrt{\alpha}}\mathbf 1_\alpha.\end{aligned}$$ thereby proving that the only possible encoding compatible with first-order covariance is the flat one (up to irrelevant phases).
{width="\textwidth"}
![\[fig:FailsSecondOrder\] [*Failure of covariance at the second order for the Dirac QW.*]{} The outcoming wires of a patch do not match the incoming wires of the next patch.](determinedregion4.pdf){width="\columnwidth"}
The Dirac QW can similarly be expanded to the second order. This time, however, the patches that make up $\psi'$ do not match up. A simple counter-example supporting this fact arises with $\alpha=2$ and $\beta=1$ already, as illustrated in Fig. \[fig:FailsSecondOrder\]. Notice that we ought to have $\overline{C}(0,1)\widecheck{\psi}_-(0,0)=\overline{C}(1,1)\widecheck{\psi}_-(0,0)$, if we want those outcoming wires to match up with the corresponding incoming wires of the next patch $\widecheck{\psi}_-(0,1)_0=\widecheck{\psi}_-(0,1)_1= \psi_-(0,1)/\sqrt{2}$. But it turns out that those outcoming wires verify $\overline{C}(0,1)\widecheck{\psi}_-(0,0)\neq \overline{C}(1,1)\widecheck{\psi}_-(0,0)$ due a term in $\varepsilon^2$.
[^1]: We take the convention that if the swap move is applied to a segment around the origin, the origin moves along.
|
---
abstract: 'The mechanical and electronic properties of transition metal dichalcogenide (TMD) monolayers corresponding to transition groups IV, VI, and X are explored under mechanical bending from first principles calculations using the strongly constrained and appropriately normed (SCAN) meta-GGA (MGGA). SCAN provides an accurate description of the phase stability of the TMD monolayers. Our calculated lattice parameters and other structure parameters agree well with experiment. We find that bending stiffness (or flexural rigidity) increases as the transition metal group goes from IV to X to VI, with the exception of PdTe$_2$. Variation in mechanical properties (local strain, physical thickness) and electronic properties (local charge density, band structure) with bending curvature is discussed. The local strain profile of these TMD monolayers under mechanical bending is highly non-uniform. The mechanical bending tunes not only the thickness of the TMD monolayers, but also the local charge distribution as well as the band structures, adding more functionalization options to these materials.'
author:
- 'Niraj K. Nepal$^1$'
- Liping Yu$^2$
- Qimin Yan$^1$
- Adrienn Ruzsinszky$^1$
title: 'First-principles study of mechanical and electronic properties of bent monolayer transition metal dichalcogenides'
---
Introduction
============
Layered transition metal dichalcogenides (TMD) offer a wide variety of physical and chemical properties from metal to insulator [@WY69; @XLSC13; @KT16] and are extensively studied [@TAN14; @JS12; @CSELLZ13; @YYR17]. An increasing interest and recent progress towards these materials led to a variety of improved applications such as sensors, energy storage, photonics, optoelectronics, and spintronics [@CHS12; @APH14; @WKKCS12; @WKKCS12]. In particular, atomically thin monolayer TMDs have attracted most of the attention due to the unique mechanical and electronic properties related to their high flexibility [@HPMS13; @CSBAPJ14; @LZZZT16]. A large scope of flexible electronics has been realized via applications such as flexible displays [@JLK09; @ZWWSPJ06; @R16; @WS09], wearable sensors [@SC14; @KGLR12; @WW13], and electronic skins [@STM13; @PKV14; @HCTTB13]. Each TMD (TX$_2$) monolayer consisting of 3 atomic layers (X-T-X stacking) can undergo bending deformation, possessing higher flexural rigidity than graphene (D$_{MoS_2}$ $\sim$ 7-8 D$_{Graphene}$ [@AB17]). The bending behavior (curvature effect) of 2D TMD monolayers, especially of MoS$_2$, has been studied both theoretically [@JQPR13; @XC16] and experimentally [@ZDLHSR15; @CSBAPJ14]. For 2D materials such as MoS$_2$, the bending can induce localization or delocalization in the electronic charge distribution. This change in the charge distribution results in changes in electronic properties such as the Fermi level, effective mass, and band gap [@YRP16]. However, the bending behavior of other TMD monolayers is largely unexplored at least from first-principles. Quantitatively, the resistance of a material against bending is characterized by the bending stiffness. The bending stiffness or flexural rigidity of the TMD monolayers can be estimated using first-principles as in Refs. . Most of the earlier studies used nanotubes of different radii created by rolling an infinitely extended sheet to estimate the bending stiffness of 2D monolayers [@RBM92; @AB04; @HWH06]. However, such a scheme has several limitations. (1) It does not mimic the edges present in the monolayer. (2) The nanoribbons unfolded from differently sized nanotubes have different widths which contribute to different quantum confinement effects along with the curvature effect. By utilizing the bending scheme similar to the bending of a thin plate, we restore the edges as well as fix the width of the nanoribbon, thereby eliminating the quantum confinement effect resulting from difference in width between various configurations of nanoribbons from flat to bent ones. However, the edge effects due to their finite width may not be completely eliminated.\
Here we report a comprehensive first-principles study of the structural, mechanical, and electronic properties of flat and bent monolayer TMD compounds, i.e., TX$_2$ (T = transition metal, X = chalcogen atom). As in Ref. , we represent each TMD (TX$_2$) with its transition metal group. For example, d$^0$ for group IV, d$^2$ for group VI, and d$^6$ for group X. Their layer structures have been observed in experiment: group IV (T = Ti, Zr or Hf; X = S, Se or Te) and group X (PdTe$_2$ and PtX$_2$) TMDs prefer the 1T phase, while group VI TMDs crystallize in the 1H (T = Mo or W; X = S, Se) as well as the distorted T (1T$'$) phase (WTe$_2$) [@WY69]. We first investigate the relative stability of a monolayer in three different phases (1H, 1T, 1T$'$). The mechanical and electronic properties have been studied only for those most stable phases. The organization of the rest of the paper is as follows. The computational details are presented in Sec. II. Section III presents our results, followed by some discussion and conclusions in Sec. IV.\
Computational Details
=====================
![Rectangular unit-cells of types 1H, 1T, and 1T$'$ (WTe$_2$) used in the calculations. The first row represents the top view (a-c) while second (d-f) corresponds to the side view; d(T-X) is metal-chalcogen distance, $\angle$XTX is an angle made by two d(T-X) sides, and d(X-X) (or d$_{X-X}$) is the distance between the outer and inner layer of flat monolayer bulk TMDs.[]{data-label="fig:struc"}](Figure1.pdf)
The ground state calculations were performed using the Vienna ab initio simulation package (VASP) [@VASP] with projected augmented wave (PAW) [@B94] pseudo-potentials (PS) [@KJ99] as implemented in the VASP code [@K01], modified to include the kinetic energy density required for meta-GGA (MGGA) calculations. We used pseudo-potentials recommended in VASP for all elements except for tungsten (W), where we used a pseudopotential such that the valence electron configuration includes 6s$^1$5d$^5$ electrons. The exchange-correlation energy was approximated using the strongly constrained and appropriately normed (SCAN) MGGA [@SRP15]. It can describe an intermediate range of dispersion via the kinetic energy density and is proven to deliver sufficiently accurate ground state properties for diversely bonded systems [@NBR18; @SRZ16; @SSP18; @BLBRSB17], as compared to local density approximation (LDA) and the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE). The unit-cell calculations for all pristine TMD monolayers were carried out using a rectangular supercell consisting of two MX$_2$-units with three different configurations 1H, 1T, and 1T$'$-WTe$_2$ to determine the most stable ground state. We used the energy cutoff of 550 eV and 24 $\times$ 16 $\times$ 1 and 16 $\times$ 24 $\times$ 1 Gamma-centered Monkhorst-Pack k-meshes [@MP76] to sample the Brillouin zone. Periodic boundary conditions were applied along the in-plane direction, while a vacuum of about 20 $\AA$ was inserted along the out-of-plane direction. The geometry optimization of the mono-layer unit-cell was achieved by converging all the forces and energies within 0.005 eV/$\AA$ and 10$^{-6}$ eV respectively. To estimate the bending stiffness, we relaxed our nano-ribbons having a width of 3-4 nm ([ Supplementary Table S1]{}) with forces less than 0.01 eV/$\AA$, using an energy cutoff of 450 eV. The Brillouin zone was sampled using Gamma-centered Monkhorst-Pack k-meshes of 8 $\times$ 1 $\times$ 1 and 1 $\times$ 8 $\times$ 1.\
To estimate the in-plane stiffness, we applied strain along one direction (say the x-direction) and relaxed the system along the lateral direction (i.e., the y-direction) or vice versa (See Figure \[fig:struc\]). An in-plane stiffness then can be estimated using $$Y_{2D} =\frac{1}{A_0} \frac{\partial^2 E_s}{\partial \epsilon^2},
\label{Y2D}$$ where E$_s$ $=$ E($\epsilon = s$) - E($\epsilon = 0$) is the strain energy, $\epsilon = \frac{\textnormal{Change in length ($\Delta l$)}} {\textnormal{equilibrium length ($l_0$)}} $ is the linear strain, and A$_0$ is an equilibrium area of an unstrained supercell. We also applied a 5% axial strain and relaxed the rectangular supercell in the transverse direction to estimate the lateral strain and hence found the Poisson’s ratio. We first relaxed the flat ribbon using various edge schemes. The choices of edges are mainly due to either relaxation of the flat nanoribbon or to satisfy the condition, areal bending energy density u($\kappa $)$=$ $\frac{E_{bent} - E_{flat}}{Area (A)}$ $\rightarrow$ 0 as the bending curvature $\kappa$ $=$ $\frac{1}{\textnormal{radius of curvature (R)}}$ $\rightarrow 0$ (Figure \[fig:ribbon\] (IV)). We have taken stoichiometric (n(X):n(T)$=$ 2:1) nano-ribbons ([ Supporting Figure S4]{}) for most of the calculations in which TiTe$_2$, MoTe$_2$-1T$'$, and WX$_2$ (X = S, Se, or Te) were stabilized using hydrogen passivated edges whereas others were relaxed without hydrogen passivation. We also relaxed TiSe$_2$, HfS$_2$, PdTe$_2$, and PtSe$_2$ nano-ribbons in symmetric configuration (Figure \[fig:ribbon\] II). Finally, the bent structures of different bending curvatures were created by relaxing the ribbons along the infinite length direction, while keeping the transition atoms fixed at the opposite end, and applying strain along the width direction. A 20 $\AA$ of vacuum was introduced along the y- and z- direction to eliminate an interaction between the system and its image ([ Supplementary Figure S4]{}). The areal bending energy density (u($\kappa$)) vs bending curvature ($\kappa$) curve were fitted with a cubic polynomial to capture the non-linear behavior (Figure \[fig:ribbon\] (IV)). The quadratic coefficient of the cubic fitting was utilized to estimate the bending stiffness, $$S_b = \frac{\partial^2 u(\kappa)}{\partial \kappa^2} |_{\kappa = 0}.$$
Results
=======
Relative Stability
------------------
Experimentally, it is largely known which phase is preferred in the bulk layer structure. However, the relative stability of their monolayer structures remained elusive. We have performed relative stability analysis of monolayer TX$_2$ among 3 different phases, namely 1H, 1T, and 1T$'$-WTe$_2$, to test the predictive power of SCAN. Energies of TMDs in different phases relative to the 1T phase are presented in Figure \[fig:stability\]. Among two different phases, 1H and 1T, group (IV) and (X) TMD monolayers prefer the 1T phase. We could not find a distorted phase (1T$'$) for these TMD monolayers. In addition to the 1H and 1T phase, group (VI) TMDs MoTe$_2$ and WTe$_2$ also crystallize in the distorted (1T$'$) form. Our relative stability analysis shows that TX$_2$ with X$=$S or Se prefers the 1H phase, while it depends on the transition metal for X$=$Te, consistent with the experimental predictions [@WY69]. WTe$_2$ prefers the 1T$'$ phase while the cohesive energies of 1H and 1T$'$ phases of MoTe$_2$ are almost identical (favoring the 1H phase by 5 meV), leading to an easy modulation between 2 phases [@WNGJDY17]. Satisfying 17 exact known constraints, SCAN accurately captures the necessary interactions present in these TMD monolayers and predicts the correct ground state structure.
![Stability (relative to the 1T phase) from SCAN calculations for TMDs between the 3 experimentally observed phases 1H, 1T, and 1T$'$-WTe$_2$. The *x*-axis represents the TMD with a phase corresponding to the minimum ground state (GS) energy, and the relative GS energies per atom of the TMDs of any phase with respect to corresponding GS of 1T phase are presented on the *y*-axis. The straight line parallel to the *x*-axis passing through the origin represents the GS energies of 1T phases. SCAN correctly predicts the ground state for these compounds. Also, MoTe$_2$ seems to be iso-energetic between 1H and 1T$'$-WTe$_2$ phases.[]{data-label="fig:stability"}](Figure3.pdf){height="3.5in" width="4.5in"}
Structural properties
---------------------
Comparison has been made for the estimated in-plane lattice constant of monolayers with the experimental bulk results in Figure \[fig:lattice\]. The lattice constants are in good agreement with the experimental results with a mean absolute error (MAE) and a mean absolute percentage error (MAPE) of 0.03 $\AA$ and 0.7% respectively. The results for the structural parameters related to the monolayer bulk are in good agreement with reference values [@CHS12]. The structural parameters related to the lattice constant such as d$_{T-X}$, d$_{X-X}$, and $\theta_{X-T-X}$ increase from S to Se to Te. The decreasing cohesive energies from S to Se to Te make them more loosely bound, thereby increasing the lattice parameters.\
![Comparision of the SCAN-calculated in-plane lattice constants of various TMD mono-layers in the ground state with respect to the bulk lattice constants available in the literature [@WY69; @AMTTRD16; @LWBR73].[]{data-label="fig:lattice"}](Figure4.pdf){height="3in" width="5in"}
In-plane stiffness and Poisson’s ratio
--------------------------------------
The strength of a material is crucial for a device’s performance and its durability. As a measure of the strength, we computed an in-plane stiffness or 2D Young’s modulus (Eq. \[Y2D\]) of the most stable ground state and tabulated it in Table \[tab:eff-t\]. Similar to the cohesive energy, the in-plane stiffness decreases from S to Se to Te, indicating a softening of TMD monolayers from S to Te under an application of linear strain. The estimated 2D in-plane stiffness of MoS$_2$ is 141.59 N/m, which is in close agreement with the experimental value of 180 $\pm$ 60 N/m [@BBK11].\
Under Poisson’s effect, materials tend to expand (or contract) in a direction perpendicular to the axis of compression (or expansion). It can be measured using Poisson’s ratio $\nu_{ij} = -\frac{d\epsilon_j}{d\epsilon_i}$, where $d\epsilon_j$ and $d\epsilon_i$ are transverse and axial strains respectively. The in-plane (-$\frac{d\epsilon_y}{d\epsilon_x}$ or $-\frac{d\epsilon_x}{d\epsilon_y}$) and an out of plane Poisson’s ratio (-$\frac{d\epsilon_z}{d\epsilon_x}$) are also calculated and tabulated. The in-plane Poisson’s ratio is different than that of the out of plane Poisson’s ratio for 1T compounds. For example, PtS$_2$ has $\nu_{xy} = 0.29$ and $\nu_{xz} = 0.58$. However, the Poisson’s ratio of 1H monolayers is almost isotropic ($\nu_{xy} \approx \nu_{xz}$ ).
Mechanical bending
------------------
The primary focus of this study is to understand the response of the TMD monolayers to mechanical bending. We have calculated the bending stiffness and studied the change in various physical and electronic properties due to bending. Since previous studies [@YRP16; @ZDLHSR15] showed that the bending stiffness is independent of the type of the armchair or zigzag edges (chiral), we only utilized armchair-edge nanoribbons for the 1H structures. The bending stiffness of 20 TMDs are compared and tabulated in Table \[tab:eff-t\]. Unlike the in-plane stiffness, the overall bending stiffness increases from S to Se to Te (Table \[tab:eff-t\]), indicating a hardening of the nanoribbons from S to Se to Te. The d$^0$ compounds, especially S and Se, along with the PdTe$_2$ have lower ($<$ 3 eV) bending stiffness. The lower flexural rigidity of these compounds can result in enormous changes in their local strain as well as the charge density profile under mechanical bending. The 1H compounds have higher bending stiffness, possessing higher flexural rigidity against mechanical bending. The estimated bending stiffness of 12.29 eV for MoS$_2$ agrees with the experimental values of 6.62-13.24 eV [@CSBAPJ14] as well as 10-16 eV [@ZDLHSR15]. To explore the trend of mechanical strengths with respect to transition metal, one can look into the d-band filling of valence electrons. The filling of the d band increases from transition metal group IV ($\sim$ sparsely-filled) to VI ($\sim$ half-filled) to X ($\sim$ completely filled) within the same row in periodic table. Both quantities Y$_{2D}$ and S$_b$ increase as the number of valence d electrons increases until the shell becomes nearly half-filled. To facilitate the claim further, we have estimated the in-plane stiffness and bending stiffness of 1H-NbS$_2$ and 1H-TaS$_2$ corresponding to group V (d$^1$) transition metals. The in-plane stiffness of NbS$_2$ and TaS$_2$ were found to be 95.74 N/m and 115.04 N/m respectively. In addition, the bending stiffness was obtained as 4.87 eV and 6.43 eV respectively for NbS$_2$ and TaS$_2$. Comparing TMDs (TX$_2$) having the same chalcogen atom, we can see the trend d$^0$ $<$ d$^1$ $<$ d$^2$ for both stiffness. However, there is a decrement in both Y$_{2D}$ and S$_b$ while going from half-filled (d$^2$) to nearly completely filled (d$^6$) d-band transition metal. Moreover, the large bending stiffness of group VI compounds decreases on changing phase from 1H to distorted 1T phase, for instance, 1H to 1T$^\prime$ transformation in MoTe$_2$.\
We utilized $$\begin{aligned}
t_{eff} = \sqrt{12S_b/Y_{2D}} \\
\shortintertext{and}
Y_{3D} = Y_{2D}/t_{eff}
\end{aligned}$$ to estimate the effective thickness as well as the 3D Young’s modulus. An effective thickness is the combination of d$_{X-X}$ distance and the total effective decay length of electron density into the vacuum. Experimentally, it is difficult to define the total effective decay length of the electronic charge distribution. Therefore, it is a common practice to take a range from the d$_{X-X}$ distance to the inter layer metal-metal distance within the bulk structure as the effective thickness, which gives the range for both in-plane stiffness and bending stiffness [@CSBAPJ14; @ZDLHSR15]. Using equation 3, one can estimate a reasonable value for the effective thickness for a wide range of TMDs. However, the computed effective thicknesses t$_{eff}$ of certain TX$_2$ (T=Ti, Zr, Hf; X=S, Se) are less than their d$_{X-X}$ distance (Figure 1), which means that bending is much easier than stretching. Similar underestimation was found for the effective thickness of a carbon monolayer estimated by various methods [@YS97; @W04; @KGB01; @Z00]. Utilizing eq. (3), Yakobson et al. [@YS97], Wang [@W04], and Yu et al.[@YRP16] estimated the effective thickness of the carbon monolayer to be around 0.7-0.9 $\AA$, which is much less than 3.4 $\AA$, the normal spacing between sheets in graphite. Such huge underestimation indicates the possible breakdown of the expression (3) to estimate the effective thickness in the case of atomically thin carbon layer [@W04]. The 3D Young’s modulus (eq.4) allows us to compare the strength between various 2D and 3D materials, for instance, MoS$_2$ against steel. Similar to 2D in-plane stiffness, the 3D Young’s modulus of TMD monolayers decreases from S to Se to Te. Due to the larger underestimation of the effective thickness, there is a huge overestimation in the 3D Young’s modulus of group IV compounds with sulfur as the chalcogen atom. With that in mind, one can conclude that MoS$_2$ as well as WS$_2$ have large 3D Young’s moduli of 347.03 GPa and 351.02 GPa respectively, agreeing with the experimental value of 270$\pm$100 GPa [@BBK11] for MoS$_2$.\
Effect of bending on physical properties
----------------------------------------
**I. Local Strain**\
Local strain ($\epsilon$ $=$ $\frac{\delta - \delta_{flat}}{\delta_{flat}}$) projected on the y-z plane (see b-c plane in Figure \[fig:ribbon\] (II)) of different TMD nano-ribbons corresponding to the bending curvature around 0.09 $\AA^{-1}$ are presented in [Supplementary Figure S1]{}. The inner layer gets contracted while the outer layer gets expanded, and this is consistent with the elastic theory of bending of a thin plate [@LL86]. The expansion of the outer layer is close to the contraction of the inner layer for 1T compounds, while the expansion dominates the contraction in the case of 1H compounds ([Supplementary Figure S1]{}). The middle metal layer is expanded up to 2% in the case of 1T while it is 5-10% for 1H, indicating that the middle layer is closer to the neutral axis for 1T than that of the 1H compounds. For 1T$'$ compounds (MoTe$_2$ and WTe$_2$), the outer layer is expanded more as compared to the contraction of the inner layer with a distortion represented by the zigzag structure in the strain profile ([ Supplementary Figure S1]{}).\
To study the effect of bending on the local strain profiles, we compare the local strain profiles of the PtS$_2$ nano-ribbon projected on y-z plane, as shown in Figure \[fig:strain-PtS\]. The inner layer is contracted while the outer layer gets expanded. This effect increases upon increasing the bending curvature. For PtS$_2$, the middle layer is expanded within 2-3%, while the expansion is 16-20% for the inner and the outer layer. Such large local strain can induce a highly non-uniform local potential and hence affect the charge distribution. Both lattice expansion in the outer layer and the lattice contraction in the inner layer could be applicable in tuning adsorption (binding distance and energy) of the 2D materials, similar to the linear strain modulated adsorption properties of various semiconducting or metallic surfaces [@KDCF14; @CW16; @MHN98]. The tensile strain strengthens the hydrogen adsorption in TMD surfaces, while a compressive strain weakens it [@CW16]. By utilizing both the concave (compressive strain) and convex (tensile strain) surfaces of a bent monolayers, one can tune the Gibb’s free energy of hydrogen adsorption to zero when it is respectively more negative and more positive.\
![Local strain ($\epsilon$ $=$ $\frac{\delta - \delta_{flat}}{\delta_{flat}}$) with respect to the inner chalcogen-chalcogen ($\epsilon^{inner}_{X-X}$), metal-metal ($\epsilon_{T-T}$), and outer chalcogen-chalcogen distance ($\epsilon^{outer}_{X-X}$) projected in the y-z plane for PtS$_2$. Strain at metal indices “i" (see 2$^{nd}$ subfigure) is calculated with respect to the distance between two metals at indices i-1 and i where i = 1, 2, ...10 (or 11)[]{data-label="fig:strain-PtS"}](Figure5.pdf)
**II. Physical thickness**\
![The strain with respect to the physical thickness of the bent nano-ribbon around 0.09 $\AA^{-1}$ for various TMD compounds; t$_{tot}$ (t$_{up}$ + t$_{dn}$, [ blue]{}) is outer-inner layer thickness; t$_{up}$ ([ red]{}) and t$_{dn}$ ([ green]{}) are measured between outer-middle and middle-inner layers respectively (see Figure \[fig:ribbon\] (III)).[]{data-label="fig:compare-t"}](Figure6.pdf)
The behavior of different layers within the TMD nano-ribbon under mechanical bending can be understood by looking at the variation of the physical thickness (t$_{tot}$, defined later in this section and shown in Figure \[fig:compare-t\]) with respect to bending curvature. Moreover, tuning of the physical thickness can be particularly useful in nano-electronic applications due to an enhancement of the electron confinement in 2D materials with an out-of-plane compression [@BB18; @GDKF17]. A percentage change in the thickness (t$_{tot}$, t$_{up}$, or t$_{dn}$) at the middle of various bent nano-ribbons with respect to the unbent ones is presented in [Supplementary Figure S2]{}. t$_{tot}$ represents an outer-inner chalcogen atom layer thickness at the vertex of a bent ribbon, while t$_{up}$ and t$_{dn}$ correspond to outer-middle and middle-inner layers respectively. We fitted a 6$^{th}$ order polynomial to each layer of the bent nanoribbon to estimate the thickness ([Supplementary Figure S3]{}). The thickness measured between outer and inner chalcogen layers is described by t$_{tot}$ (t$_{up}$ + t$_{dn}$, [ blue]{}) while t$_{up}$ ([ red]{}) and t$_{dn}$ ([ green]{}) are measured between the outer-to-middle and middle-to-inner layers respectively (see Figure \[fig:compare-t\]). When a thin plate is bent, it undergoes both compression (z’ to N, t$_{dn}$) and expansion (N to z’+h, t$_{up}$) with “N" being the neutral surface [@LL86] (see figure \[fig:ribbon\] (III)). As the middle layer does not mimic the neutral surface (N), t$_{up}$ and t$_{dn}$ do not respectively increase and decrease with the bending curvature. For most of the compounds, t$_{up}$ decreases on increasing the bending curvature. On the other hand, t$_{dn}$ slightly increases for d$^0$-1T compounds, but depends on the bending curvature for d$^2$-1H and d$^6$-1T compounds ([Supplementary Figure S2]{}). For a quantitative comparison among different materials, we plot the thicknesses for various TMDs around the bending curvature of 0.09 $\AA^{-1}$ as shown in Figure \[fig:compare-t\]. Group IV compounds have a lower flexural rigidity, therefore have more of a decrement in the physical thickness (t$_{tot}$) than group VI and X compounds.\
Effect of the bending on electronic properties
----------------------------------------------
**I. Local electronic charge density**\
Along with the change in physical properties, mechanical bending also affects the electronic properties. The local charge density (average over ab-plane, \[Figure \[fig:ribbon\] (I)\]) is computed and plotted against distance along an out-of-plane direction (*c*- axis) \[Figure \[fig:ribbon\] (II)\]. The different nature of the local charge distribution of flat WX$_2$ (X=S, Se, Te) ribbon with two equal local maxima may be related to the different pseudopotential used in the calculation. We choose a narrow window (within 2 black vertical lines) at the middle of a nano-ribbon (for both flat and bent) to study the local charge distribution near the surface-vacuum interface as shown in [Supplementary Figure S4]{}. We define 3 different quantities Width, Max, and an Area of the local charge density (left) and compared among the flat nano-ribbons of various TMDs (right), as shown in Figure \[fig:local-charge\]. The “Width" represents the distance over which the charge density decays to a smaller non-zero value ($\epsilon < 10^{-4}$) in vacuum ([ Supplementary Figure S4]{}) which also gives a tentative idea about the total effective decay length of electron density. In addition, the areal density ($\int_{0}^{Width}\rho(z)dz$, an area under the curve) represents the average number of electrons per unit area, as shown in Figure \[fig:local-charge\].\
For the flat nano-ribbons, the width increases whereas Max and the Area decrease as we go from S to Se to Te for a given transition metal. Increasing the width from S to Se to Te indicates an increase in the total effective decay length of electron density, hence the effective thickness. Also, the width corresponding to flat 1H nano-ribbons is shifted upward by atleast 0.5 $\AA$ compared to that of 1T flat nano-ribbons which then contributes to an effective thickness giving larger bending stiffness. Our results suggest that the overall bending stiffness follows the trend of the width of an electron density and hence the effective thickness. The variation of the local charge density along an out of plane direction for different TMD nano-ribbons with the bending curvature is presented in [Supplementary Figure S5]{}. When a nanoribbon is bent, the local charge density shrinks with the bending curvature within an outer layer-vacuum interface while expanding near the inner layer-vacuum interface leaving the total width unaffected. However, both the Max and the Area decrease with increasing bending curvature for most of the TMD compounds except for TiTe$_2$ and WX$_2$. For WX$_2$, the max. value of local maximum closer to the surface-vacuum interface decreases on increasing the bending curvature (Circular region in the [Supplementary Figure S5]{}) whereas the other local maxima have an opposite trend. To study the effect of bending on the aforementioned local maximum (Max) and areal density (Area) among different materials, we estimate their percentage change with respect to the flat ribbon, as in Figure \[fig:local-charge\]. The bending produces noticeable changes in the charge distribution within the surface-vacuum interfaces.\
\[tab:eff-t\]
**II. Band structure**\
The band structure plots of group IV, VI, and X TMDs with respect to vacuum with various bending curvatures are shown in [Supplementary Figures S6, S7, and S8]{} respectively. The dashed lines in the band structure plots indicate the SCAN estimated Fermi energy with respect to vacuum (“-ve" of the work function) while the red bands correspond to in-gap edge states. The edge states are identified by comparing the band structures of the ribbon with that of the monolayer bulk, and are highlighted by red color. The bulk band-gap (E$_g$ (eV)) (excluding edge states) and the work function ($\phi$ (eV)) of our flat nano-ribbons are extracted and tabulated in [ Supplementary Table S1]{}. Out of TMD nano-ribbons considered, ZrX$_2$, HfX$_2$, MoY$_2$, and WX$_2$ (X = S, Se; Y = S, Se, Te) are semiconductors. To study the changes in the band structure of these semiconductors with respect to bending, we utilized the hydrogen passivated edges. A few of the low band-gap semiconductors such as TiY$_2$, TTe$_2$ (T=Zr, Hf) and group (X) indirect band-gap semiconductors (PtX$_2$) become metallic due to the edge states. We did not observe any substantial effect of bending on metallic compounds. An effect of the mechanical bending on the band-gap is of particular interest for semiconductors, due to a wide range of applications in nano-electronics. One each from the 1T and the 1H group, respectively ZrS$_2$ and MoS$_2$, are chosen to study the effect of bending on the band structure as shown in Figure \[fig:band-semi\].\
![Variation of band edges with respect to the bending curvature for ZrS$_2$ (left) and MoS$_2$ (right); CBM and VB1 are the conduction band minimum and edge state VB (valence band) respectively; CB1 (CBM), CB2, VB1 (VBM), and VB2 respectively are edge state CB (conduction band), bulk CB, edge state VB (valence band), and bulk VB. For flat MoS$_2$ ribbon, VB1 represents the VBM while for higher bending curvature ($\kappa$ $=$ $0.09 \AA^{-1}$) VB2 switches to VBM.[]{data-label="fig:band-semi"}](Figure8.pdf "fig:") ![Variation of band edges with respect to the bending curvature for ZrS$_2$ (left) and MoS$_2$ (right); CBM and VB1 are the conduction band minimum and edge state VB (valence band) respectively; CB1 (CBM), CB2, VB1 (VBM), and VB2 respectively are edge state CB (conduction band), bulk CB, edge state VB (valence band), and bulk VB. For flat MoS$_2$ ribbon, VB1 represents the VBM while for higher bending curvature ($\kappa$ $=$ $0.09 \AA^{-1}$) VB2 switches to VBM.[]{data-label="fig:band-semi"}](Figure9.pdf "fig:")
The nature of edge states is different for 1T and 1H semiconductors. The 1T nanoribbon has edge states only below the Fermi level while both the edge states above and below the Fermi level are present in the 1H nanoribbon. The horizontal black dashed lines represent water redox potentials with respect to the vacuum level, -4.44 eV for the reduction (H$^+$/H$_2$), and -5.67 eV for the oxidation (O2/H$_2$O) at pH 0 [@CAAWSS07]. When the band edges straddle these potentials, materials possess good water splitting properties. The band edges CB2, VB1 (VBM), and VB2 of MoS$_2$ straddle the water redox potentials while only the edge state CB1 stays within the gap. As semilocal DFT functionals underestimate the band gap [@P85], a correction is always expected at the G$_0$W$_0$ level ([ Supplementary Table S1]{}), which shifts the bands above and below the Fermi level even further up and below respectively [@YRP16]. However, it is known that such correction for localized states (in the case of point defects) is less considerable than that for the delocalized bulk states [@ABP08].\
**(a) Tuning of band edges**\
The band edges (conduction band minimum (CBM) and valence band maximum (VBM)) of ZrS$_2$ and other 1T semiconductors increase on increasing the bending curvature, while this varies from one band edge to another for MoS$_2$ and other 1H semiconductors. For example, shifting of the band energies with respect to vacuum is negligible for edge states as compared to the bulk ones for MoS$_2$. The shifting of band edges also leads to changing of the Fermi level as well as the band gap ([ Supplementary Figure S10)]{}. For MoS$_2$, VB2 increases while VB1 decreases on increasing the bending curvature and eventually results in the removal of some of the edge states, though, complete elimination might not be possible. Since the mechanical bending shifts the band edges only by a little, the photocatalytic properties of MoS$_2$ and WS$_2$ is preserved even for a larger bending curvature. On the other hand, bending can shift the band edges of 1T semiconductors by a considerable amount for bending curvature up to 0.06 $\AA^{-1}$, but shift downward for higher bending curvature. For example, one can shift the band edges of ZrS$_2$ upward by 0.25 eV when applying the bending curvature of 0.06 $\AA^{-1}$. Moreover, the G$_0$W$_0$ calculated band structure shows that the CBM (-4.58 eV and -4.53 eV respectively) of ZrS$_2$ and HfS$_2$ is slightly lower than the reduction potential (-4.44 eV) while the VBM (-7.15 eV and -6.98 eV) is significantly lower than the water oxidation potential (-5.67 eV) [@ZH13]. Mechanical bending can shift the band edges in the upward direction to straddle the water redox potentials, enhancing the photocatalytic activity. The effect of bending on the band edges of 1H-TSe$_2$ semiconductors is different than that of 1H-TS$_2$ ([ Supplementary Figure S9]{}), especially in the bulk valence band maximum (VB2). The VB2 is almost constant for lower bending curvature for TSe$_2$, while there is an appreciable increase in the case of TS$_2$.\
**(b) Charge localization and Conductivity**\
In this section, we describe the effect of bending on band edges in terms of localization or delocalization of the charge carriers at those band edges. The variation of an isosurface of the partial charge (electrons or holes) density with respect to bending curvature are presented in Figures \[fig:band-charge\] and \[fig:cbm\]. Using the mechanical bending, one can tune the conductivity of TMD monolayers [@YRP16]. Before bending, the charge carriers (holes) of ZrS$_2$ at VB2 are delocalized over the whole ribbon width, decreasing in magnitude from S-edge to Zr-edge. The mechanical bending localizes the charges towards the S-edge while depleting along the Zr-edge, reducing the conductivity from one edge to the other. On the other hand, the charge density on top of VB1 does not change much with the bending for lower bending curvatures. However, at $\kappa$ $=$ 0.09 $\AA^{-1}$ some charges accumulate at the Zr-edge, thereby changing the trend of band energy with respect to vacuum (see Figure \[fig:band-semi\]). Unlike ZrS$_2$, the charge carriers (holes) of MoS$_2$ at VB2 are delocalized over the whole width, decrease in magnitude from the center of the ribbon to either side of edges symmetrically. With bending, the charge carriers localize at the middle of the ribbon and deplete at the edges, reducing the conductivity due to holes from one edge to the other [@YRP16]. At a higher bending curvature beyond $\kappa > 0.065 \AA^{-1}$, edge state VB1 crosses the bulk-VB and becomes VB2 and vice versa. Similar to VB1, CB1 also has the same behavior before and after bending, except it does not cross the CB2. Instead, it is also shifted down as VB1 does.\
Conversely, the charge carriers (electrons) of ZrS$_2$ at the CBM (CB2) decrease in magnitude from Zr-edge to the S-edge. Again, mechanical bending localizes the electrons towards the Zr-edge. On the other hand, the electronic conductivity does not change even for larger bending curvature for MoS$_2$. The electrons are delocalized uniformly over the whole ribbon width which remains unaffected for a wide range of bending curvature. The conductivity of a semiconductor is the sum of conductivity of both electrons and holes. The mechanical bending reduces both types of conductivity in 1T semiconductors, while it only reduces hole-type conductivity in 1H semiconductors.\
![[Variation in the isosurface of partial charge densities at VB1 and VB2 (holes) with respect to the bending curvature; (a) ZrS$_2$; (b) MoS$_2$; (c) Variation in the isosurface of the partial charge densities (donor-like) of MoS$_2$ at CB1 with bending curvatures.]{}[]{data-label="fig:band-charge"}](Figure11.pdf)
![Variation in the isosurface of partial charge density (electrons) with respect to the bending curvature at bulk conduction band minimum; (a) CBM for ZrS$_2$; (b) CB2 for MoS$_2$.[]{data-label="fig:cbm"}](Figure12.pdf)
**G. Stability of nano-ribbons and finite width effect**\
Based on our calculation, we have found that the stability of the flat nano-ribbons also depends on the type of edge. We have taken stoichiometric (n(X):n(T)$=$ 2:1) nano-ribbons ([ Supplementary Figure S4]{}) for most of the calculations. However, we could not relax TiSe$_2$, HfS$_2$, PdTe$_2$, and PtSe$_2$ nano-ribbons in this configuration. We confirm that the instability of these flat ribbons cannot be removed simply by increasing the width of the ribbon. We chose a symmetric edge nano-ribbon by removing 2 dangling X (S, Se or Te) atoms from one of the edges for these compounds (Figure \[fig:ribbon\] II). Our calculation shows that the TMD nano-ribbons are stable against mechanical bending for a wide range of bending curvature, except for WTe$_2$. The bond breaking at the curvature region is observed for $\kappa$ $>$ 0.086 $\AA^{-1}$, as shown in Figure \[fig:WTe\]. Upon bending, one of the chalcogen atoms in the curvature region moves towards the middle layer, causing a further separation of the 2 metal atoms, as shown inside the circle, creating a sudden jump, as shown in an areal bending energy density vs curvature plot (See Figure \[fig:WTe\] (III)).\
We utilized the thin plate bending model in our assessment in which we fix the width between flat and bent nanoribbons. It eliminates the quantum confinement effect present in the nanotube method due to dissimilarity of the width between flat and bent nanoribbons of the different radii of curvatures. However, the edge effects due to the finite width may remain uneliminated. Rafael I. Gonz$\acute{a}$lez et al. [@GVRVSKM18], using classical molecular dynamics simulation, reported that the bending stiffness of MoS$_2$ estimated with a 0.95 nm width nanoribbon is only 46% of those estimated using a 8 nm width nanoribbon. But, it recovers 88-93% of bending stiffness when the width increases up to 3-4 nm, leaving the overall trend unaffected. We believe that such an accuracy would be a reasonable tradeoff to the computational complexity that arises while using a larger width. Moreover, we expect that the finite size effect would be less present in our results than in those calculated from MD simulation, as the quantum effects are more properly treated.
![(I)-(II): Structures for 2 different bending curvatures, showing the breaking of the ribbon within the curvature region; The figure on left is for $\kappa$ $=$ 0.086 $\AA^{-1}$ while the one on the right is for $\kappa$ $=$ 0.093 $\AA^{-1}$. (III) An areal bending energy density with respect to bending curvature for WTe$_2$, showing the breaking of structure.[]{data-label="fig:WTe"}](Figure10.pdf)
conclusion and discussion
=========================
The 2D materials offer a wide range of electronic properties efficiently applicable in sensors, energy storage, photonics, and optoelectronic devices. The higher flexural rigidity and strain-tunable properties of these compounds make them potential functional materials for future flexible electronics. In this work, we have employed the SCAN functional to explore the physical and mechanical properties of the 2D transition metal dichalcogenide (TMD) monolayers under mechanical bending. SCAN performs reasonably well in predicting the correct ground state phase as well as the geometrical properties. Also, a wide variety of flexural rigidities can be observed while scanning the periodic table for TMDs. The in-plane stiffness decreases from S to Se to Te, while the bending stiffness has the opposite trend. Overall, the bending stiffness also depends on the d band filling in the transition metal. The bending stiffness increases on increasing the filling of the d band from sparsely-filled (d$^0$) to nearly half-filled (d$^2$). However, decrease in bending stiffness is observed on moving from nearly half-filled (d$^2$) to completely-filled (d$^6$) d band. The out-of-plane Poisson’s ratios are found to be different from the in-plane Poisson’s ratio for 1T and 1T$'$ monolayers, while the difference is negligible in the case of 1H compounds, showing an anisotropic behavior of 1T and 1T$'$ monolayers.\
Despite the extraordinary physical and electronic properties of TMDs, there are still challenges to make use of TMD semiconductors in nanoelectronics. The strong Fermi level pinning and high contact resistance are key bottlenecks in contact-engineering which are mainly due to in-plane, in-gap edge states and do not depend too much on the work function of a contact metal [@KMLCAN17]. Thanks to mechanical bending, tuning of various properties of monolayer TMDs is possible, including band edges, thickness, and local strain. Bending deformation produces highly non-uniform local strain up to 40% ([ Supplementary Figure S1]{}), which is almost impossible with a linear strain ($\epsilon$). The high out-of-plane compressive strain developed within the layers due to bending reduces the mechanical thickness and makes the materials thinner in the curvature region. Moreover, one can remove strong Fermi-level pinning while using it in contact-engineering. Besides that, the optimal band alignment with the HER redox potential can be achieved for 1T semiconductors ZrS$_2$ and HfS$_2$ under mechanical bending, which are not present in an unbent monolayer. Furthermore, both electron and hole conductivities are affected in 1T semiconductors, while only the hole conductivity is affected in 1H semiconductors [@YRP16]. Similar to graphene [@YS97; @W04; @KGB01; @Z00], the estimated effective thickness of group IV TMDs, especially sulfide and selenide, is underestimated as compared to chalcogen-chalcogen distance (d$_{X-X}$), which is quite puzzling and needs further investigation.
Acknowledgement
===============
We thank Prof. John P. Perdew for useful comments on the manuscript. This research was supported as part of the Center for Complex Materials from First Principles (CCM), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award No. DE-SC0012575. Computational support was provided by National Energy Research Scientific Computing Center (NERSC). Some of calculations were carried out on Temple University’s HPC resources and thus was supported in part by the National Science Foundation through major research instrumentation grant number 1625061 and by the US Army Research Laboratory under contract number W911NF-16-2-0189.
**Supplementary material\
**
TMDs Width ($\AA$) E$^{QP}_g$ (eV)[@ZH13]
---------------------- --------------- ------- ------- ------------------------
TiS$_2$ 32.04 – 5.236
TiSe$_2$ 32.62 – 5.508
TiTe$_2$ 34.11 – 5.09
ZrS$_2$ 34.64 1.549 5.886 2.56
ZrSe$_2$ 35.55 1.025 5.549 1.54
ZrTe$_2$ 37.10 – 5.084
HfS$_2$ 34.18 1.751 5.919 2.45
HfSe$_2$ 35.01 1.092 5.386 1.39
HfTe$_2$ 36.77 – 4.938
MoS$_2$ 29.85 1.836 5.376 2.36
MoSe$_2$ 30.95 1.709 4.952 2.04
MoTe$_2$ 32.62 1.349 4.631 1.54
MoTe$_2$-1T$^\prime$ 33.72 – 4.795
WS$_2$ 29.80 2.094 5.126 2.64
WSe$_2$ 30.73 1.893 4.736 2.26
WTe$_2$ 33.26 – 4.584
PdTe$_2$ 37.07 – 4.4
PtS$_2$ 35.36 – 5.482
PtSe$_2$ 34.32 – 4.958
PtTe$_2$ 37.20 – 4.466
: Width of the relaxed flat nanoribbons used in the calculations ($\AA$); The bulk band gap (excluding edge states) of the semiconducting unbent nano-ribbons (E$_g$); Workfunction ($\phi$ (eV)); The G$_0$W$_0$ quasi-particle gap of monolayer semiconductors (E$^{QP}_g$) is shown for comparison.
\[tab:band\]
Atom
-- ------ --------- ----- ---------
Mo1 0.01712 0 0.50454
Mo2 0.37804 0.5 0.49545
Te1 0.27688 0 0.39764
Te2 0.62769 0 0.57361
Te3 0.11724 0.5 0.60234
Te4 0.76664 0.5 0.42641
W1 0.02032 0 0.50509
W2 0.37395 0.5 0.49487
Te1 0.27811 0 0.39653
Te2 0.62559 0 0.57281
Te3 0.11669 0.5 0.60346
Te4 0.76895 0.5 0.42722
: The calculated structure parameters for the rectangular 1T$^\prime$ monolayer unit cell of WTe$_2$-type having 2 TX$_2$ units: For MoTe$_2$, a $=$ 3.43439, b $=$ 6.31457 $\AA$. Similarly for WTe$_2$, a $=$ 3.45822, b $=$ 6.24802 $\AA$ (Figure 1 in the main text). The combined-fractional coordinates X, Y, and Z represent the position of the corresponding atom in the same row. Lattice constants can be estimated as $b/\sqrt3$. Chalcogen-chalcogen distances d$_{X-X}$ for the distorted MoTe$_2$ and WTe$_2$ are (2.9486 $\AA$ and 4.0940 $\AA$) and (2.9118 $\AA$ and 4.1386 $\AA$) respectively.
\[tab:eff-2t\]
![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/TiS2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/TiSe2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/TiTe2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/ZrS2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/ZrSe2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/ZrTe2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/HfS2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/HfSe2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/HfTe2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/MoS2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/MoSe2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/MoTe2-1H-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/WS2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/WSe2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/PtS2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/PtSe2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/PtTe2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/PdTe2-strain-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/MoTe2-dT-eps-converted-to.pdf "fig:") ![The local strain projected onto the bending plane (bc) for the bending curvature around 0.09 $\AA^{-1}$. The upper and lower plots correspond to outer and inner chalcogen layers respectively, while the middle one corresponds to the metallic layer.[]{data-label="lab:local-strain"}](strain/WTe2-dT-eps-converted-to.pdf "fig:")
![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/TiS2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/TiSe2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/TiTe2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/ZrS2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/ZrSe2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/ZrTe2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/HfS2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/HfSe2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/HfTe2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/MoS2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/MoSe2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/MoTe2-1H.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/WS2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/WSe2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/PtS2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/PtSe2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/PtTe2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/PdTe2.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/MoTe2-dT.pdf "fig:") ![The change in the physical thickness with respect to the flat nano-ribbon for the bending curvature around 0.09 $\AA^{-1}$. We utilized a 6$^{th}$ order polynomial fit to estimate the physical thickness. The [ blue]{}, [ red]{}, and [ green]{} plots represent t$_{tot}$ (t$_{up}$ + t$_{dn}$), t$_{up}$, and t$_{dn}$ respectively (see figure 2 (III) in the main paper).[]{data-label="lab:thickness"}](thickness/WTe2.pdf "fig:")
![Estimating the physical thickness at the curvature (vertex) region for 1H structure. (a) The layer is fitted with a n$^{th}$ order polynomial and the shortest distance from a point at the middle (A and B) to the fitted curve; t$_{tot}$ is the shortest distance from point A to the inner layer; t$_{up}$ is the shortest distance from point A to the middle layer; t$_{dn}$ is the shortest distance from point B to the inner layer.(b) an absolute error with respect to polynomial order; t$_{X-X}$ is the distance between point A and C. A sixth order polynomial is sufficient to estimate the physical thickness for 1H, 1T, and 1T$^\prime$ structures.[]{data-label="fig:1T-BG"}](fitting.pdf)
![A narrow window is taken at the middle of the nano-ribbon. The local electronic charge density is calculated along the out-of-plane direction (c- axis) within the narrow window. The first, second, and the third figure correspond to 1T, 1H, and 1T$^\prime$ respectively.[]{data-label="fig:window"}](new2/area1.pdf)
![A narrow window is taken at the middle of the nano-ribbon. The local electronic charge density is calculated along the out-of-plane direction (c- axis) within the narrow window. The first, second, and the third figure correspond to 1T, 1H, and 1T$^\prime$ respectively.[]{data-label="fig:window"}](new2/1H.png "fig:") ![A narrow window is taken at the middle of the nano-ribbon. The local electronic charge density is calculated along the out-of-plane direction (c- axis) within the narrow window. The first, second, and the third figure correspond to 1T, 1H, and 1T$^\prime$ respectively.[]{data-label="fig:window"}](new2/axis.png "fig:") ![A narrow window is taken at the middle of the nano-ribbon. The local electronic charge density is calculated along the out-of-plane direction (c- axis) within the narrow window. The first, second, and the third figure correspond to 1T, 1H, and 1T$^\prime$ respectively.[]{data-label="fig:window"}](new2/dT.png "fig:")
               \[fig:log-charge\]
    
![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-flat-TiS.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-TiS-2.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-TiS-4.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-TiS-5.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-flat-TiSe.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-TiSe-2.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-TiSe-4.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-TiSe-7.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-flat-TiTe.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-TiTe-2.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-TiTe-4.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-TiTe-7.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-flat-ZrS.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-ZrS-2.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-ZrS-4.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-ZrS-7.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-flat-ZrSe.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-ZrSe-2.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-ZrSe-4.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-ZrSe-7.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-flat-ZrTe.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-ZrTe-2.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-ZrTe-4.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-ZrTe-7.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-flat-HfS.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-HfS-2.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-HfS-4.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-HfS-7.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-flat-HfSe.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-HfSe-2.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-HfSe-4.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-HfSe-7.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-flat-HfTe.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-HfTe-2.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-HfTe-4.pdf "fig:"){height="0.9in" width="1.5in"} ![Band structures with respect to vacuum corresponding to group IV TMDs; **TiS$_2$**, **TiSe$_2$**, **TiTe$_2$**, **ZrS$_2$**, **ZrSe$_2$**, **ZrTe$_2$**, **HfS$_2$**, **HfSe$_2$**, and **HfTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-IV"}](EIGENVAL/EIGENVAL-HfTe-7.pdf "fig:"){height="0.9in" width="1.5in"}
![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-flat-MoS.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-MoS-2.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-MoS-4.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-MoS-7.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-flat-MoSe.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-MoSe-2.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-MoSe-4.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-MoSe-7.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-flat-MoTe-S.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-MoTe-5S.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-MoTe-7S.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-MoTe-8.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-flat-WS.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-WS-2.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-WS-4.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-WS-7.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-flat-WSe.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-WSe-2.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-WSe-5.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-WSe-7.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-flat-MoTe.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-MoTe-2.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-MoTe-4.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-MoTe-7.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-flat-WTe.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-WTe-2.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-WTe-4.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group VI TMDs; **MoS$_2$**, **MoSe$_2$**, **MoTe$_2$**, **MoTe$_2$-1T$^\prime$**, **WS$_2$**, **WSe$_2$**, and **WTe$_2$**. $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-VI"}](EIGENVAL/EIGENVAL-WTe-7.pdf "fig:"){height="1in" width="1.5in"}
![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-flat-PdTe.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-PdTe-2.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-PdTe-4.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-PdTe-7.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-flat-PtS.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-PtS-2.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-PtS-4.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-PtS-7.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-flat-PtSe.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-PtSe-2.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-PtSe-4.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-PtSe-7.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-flat-PtTe.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-PtTe-2.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-PtTe-4.pdf "fig:"){height="1in" width="1.5in"} ![Band structures with respect to vacuum for group X TMDs; PdTe$_2$ and PtY$_2$ (Y = S, Se, Te). $\kappa$ is the bending curvature ($\AA^{-1}$).[]{data-label="fig:band-X"}](EIGENVAL/EIGENVAL-PtTe-7.pdf "fig:"){height="1in" width="1.5in"}
![Band edges with respect to vacuum; CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text). The horizontal dotted lines represent water redox potentials: reduction (H$^+$/H2; -4.44 eV), and oxidation (H2O/O2; -5.67 eV).[]{data-label="fig:1T-BE"}](ZrSe2-gap.pdf "fig:") ![Band edges with respect to vacuum; CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text). The horizontal dotted lines represent water redox potentials: reduction (H$^+$/H2; -4.44 eV), and oxidation (H2O/O2; -5.67 eV).[]{data-label="fig:1T-BE"}](HfS2-gap.pdf "fig:") ![Band edges with respect to vacuum; CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text). The horizontal dotted lines represent water redox potentials: reduction (H$^+$/H2; -4.44 eV), and oxidation (H2O/O2; -5.67 eV).[]{data-label="fig:1T-BE"}](HfSe2-gap.pdf "fig:") ![Band edges with respect to vacuum; CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text). The horizontal dotted lines represent water redox potentials: reduction (H$^+$/H2; -4.44 eV), and oxidation (H2O/O2; -5.67 eV).[]{data-label="fig:1T-BE"}](MoSe2-gap.pdf "fig:") ![Band edges with respect to vacuum; CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text). The horizontal dotted lines represent water redox potentials: reduction (H$^+$/H2; -4.44 eV), and oxidation (H2O/O2; -5.67 eV).[]{data-label="fig:1T-BE"}](WS2-gap.pdf "fig:") ![Band edges with respect to vacuum; CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text). The horizontal dotted lines represent water redox potentials: reduction (H$^+$/H2; -4.44 eV), and oxidation (H2O/O2; -5.67 eV).[]{data-label="fig:1T-BE"}](WSe2-gap.pdf "fig:")
![Band gaps as a function of the bending curvatures. CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text). Band gap remains almost constant for 1T semiconductors, while it decreases with the bending curvature for 1H compounds.[]{data-label="fig:1T-BG"}](band-Hf-Zr.pdf "fig:") ![Band gaps as a function of the bending curvatures. CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text). Band gap remains almost constant for 1T semiconductors, while it decreases with the bending curvature for 1H compounds.[]{data-label="fig:1T-BG"}](band-Hf-Zr-gap.pdf "fig:") ![Band gaps as a function of the bending curvatures. CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text). Band gap remains almost constant for 1T semiconductors, while it decreases with the bending curvature for 1H compounds.[]{data-label="fig:1T-BG"}](MoS2-band-gap.pdf "fig:") ![Band gaps as a function of the bending curvatures. CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text). Band gap remains almost constant for 1T semiconductors, while it decreases with the bending curvature for 1H compounds.[]{data-label="fig:1T-BG"}](MoSe2-band-gap.pdf "fig:") ![Band gaps as a function of the bending curvatures. CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text). Band gap remains almost constant for 1T semiconductors, while it decreases with the bending curvature for 1H compounds.[]{data-label="fig:1T-BG"}](WS2-band-gap.pdf "fig:") ![Band gaps as a function of the bending curvatures. CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text). Band gap remains almost constant for 1T semiconductors, while it decreases with the bending curvature for 1H compounds.[]{data-label="fig:1T-BG"}](WSe2-band-gap.pdf "fig:")
![An angular momentum decomposed wavefunction character of different bands corresponding to a different layer and their variation with the bending curvature; (a) - (c): ZrS$_2$; (d) - (g): MoS$_2$. The inner, middle, outer represent the different layers of the nanoribbon. CBM, CB1, VB1, CB2, VB2 are the band edges defined in Figure 8 (Main text).[]{data-label="fig:wavefunc"}](Figure13.pdf)
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|
---
author:
- 'Zs. Regály'
- 'Zs. Sándor'
- 'C. P. Dullemond'
- 'R. van Boekel'
bibliography:
- 'regaly.bib'
date: 'Received March 15, 2010; accepted July 6, 2010'
title: |
Detectability of giant planets in protoplanetary disks\
by CO emission lines
---
[Planets are thought to form in protoplanetary accretion disks around young stars. Detecting a giant planet still embedded in a protoplanetary disk would be very important and give observational constraints on the planet-formation process. However, detecting these planets with the radial velocity technique is problematic owing to the strong stellar activity of these young objects.]{} [We intend to provide an indirect method to detect Jovian planets by studying near infrared emission spectra originating in the protoplanetary disks around TTauri stars. Our idea is to investigate whether a massive planet could induce any observable effect on the spectral lines emerging in the disks atmosphere. As a tracer molecule we propose CO, which is excited in the ro-vibrational fundamental band in the disk atmosphere to a distance of $\sim 2-3\,\mathrm{AU}$ (depending on the stellar mass) where terrestrial planets are thought to form.]{} [We developed a semi-analytical model to calculate synthetic molecular spectral line profiles in a protoplanetary disk using a double layer disk model heated on the outside by irradiation by the central star and in the midplane by viscous dissipation due to accretion. 2D gas dynamics were incorporated in the calculation of synthetic spectral lines. The motions of gas parcels were calculated by the publicly available hydrodynamical code FARGO which was developed to study planet-disk interactions.]{} [We demonstrate that a massive planet embedded in a protoplanetary disk strongly influences the originally circular Keplerian gas dynamics. The perturbed motion of the gas can be detected by comparing the CO line profiles in emission, which emerge from planet-bearing to those of planet-free disk models. The planet signal has two major characteristics: a permanent line profile asymmetry, and short timescale variability correlated with the orbital phase of the giant planet. We have found that the strength of the asymmetry depends on the physical parameters of the star-planet-disk system, such as the disk inclination angle, the planetary and stellar masses, the orbital distance, and the size of the disk inner cavity. The permanent line profile asymmetry is caused by a disk in an eccentric state in the gap opened by the giant planet. However, the variable component is a consequence of the local dynamical perturbation by the orbiting giant planet. We show that a forming giant planet, still embedded in the protoplanetary disk, can be detected using contemporary or future high-resolution near-IR spectrographs like VLT/CRIRES and ELT/METIS.]{}
Introduction
============
According to the general consensus, planets and planetary systems form in circumstellar disks. These disks consist of gaseous and solid dust material. There are two major mechanisms proposed for the formation of giant planets. The first is the disk instability, which requires a gravitationally unstable disk in which giant planets form by direct collapse of the gas as a consequence of its self-gravity, originally suggested by @Kuiper1951 [@Cameron1978] and more recently by @Boss2001. Later on, the gas giant may collect dust, which after settling to its center, forms a solid core. The second mechanism is the core-accretion process [@BodenheimerPollack1986; @Pollacketal1996], which is the final stage of planet formation in the planetesimal hypothesis [@Safronov1972]. In this hypothesis dust coagulates first and forms planetesimals (meter to kilometer-sized objects). The subsequent growth of planetesimals consists of two stages, the runaway [@WetherillStewart1989] and the oligarchic growth [@KokuboIda1998], in which the formed bodies are massive enough to increase their masses by gravity-assisted collisional accretion processes. In this way planetary embryos and, through their consecutive collisions, protoplanets, terrestrial planets, and planetary cores of giant planets can be formed. During the gas accretion phase the precursor of the proto giant planet reaches a critical mass (where the planetary envelope contains more material than its core), and at about the order of $10\,M_{\oplus}$ starts to contract, causing an increased accretion rate, which in turn raises radiative energy losses resulting in a runaway gas accretion. In this model the planetary core rapidly builds up a massive envelope of gas from its surrounding disk, in less than $10\,\mathrm{Myr}$, and eventually a gas giant forms. Because the disk dispersal time is presumably about $5\,\mathrm{Myr}$ [@Haischetal2001; @Hillenbrand2005], this model should also incorporate planetary migration to explain giant planets observed well inside the snow line, see @Alibertetal2004.
Both planetary formation processes have their weak points. Disk instability assumes a very massive disk that is gravitationally unstable or only marginally stable. Another as yet unsolved problem is that during the gravitational collapse an effective cooling mechanism should work to ensure the formation of gravitationally bound clumps. While the radiative cooling is not effective enough and the recently invoked thermal convection is also ineffective, the disk instability as a common planet-formation process is questionable [@Klahr2008].
Regarding the giant planet formation in the planetesimal hypothesis, @Mordasinietal2009 convincingly presented the success of the core accretion model to produce giant planets within the disk lifetime in a planet population synthesis model, but the artificial slow-down of the type I migration was necessary. Type I migration has such a short time scale [@Ward1997] that the planet inevitably will be engulfed by the host star within the disk lifetime. Another unsolved issue of the planetesimal hypothesis is the so-called “meter-sized barrier problem”: i.e. the formation of meter-sized bodies is impeded by their quick inward drift to the star [@Weidenschilling1977], and by mutual disruptive collisions due to their high relative velocities above $1\,\mathrm{m/s}$ [@BlumandWurm2008]. If no other physical processes are taking place, these two mechanisms would impede the formation of the meter-sized and consequently the larger planetesimals.
A wealth of information about the planet formation process would clearly be provided if newly formed planets, or at least their signatures, could be observed in still gas-rich protoplanetary disks. There are already attempts to discover signatures of giant planet-formation assuming the disk instability mechanism in the works of @Narayananetal2006 and @Jang-CondellBoss2007. In the first study, for instance, the authors suggest observations of $\mathrm{HCO}^{+}$ emission lines as a tracer of the accumulation of large clumps of cold material, arguing that with high-resolution millimeter and sub-millimeter interferometers the cold clumps can be directly imaged. The observability of an already formed planet embedded in a circumstellar disk has also been studied recently by @WolfDAngelo2005 and @Wolfetal2007. In these works the authors investigated whether the influence of a forming planet on the protoplanetary disk can be detected by analyzing the spectral energy distribution (SED) coming from the star-disk system. It was found that by studying only the SED it is impossible to infer the presence of an embedded planet. On the other hand, it was also concluded that the dust re-emission in the hot regions near the gap could certainly be detected and mapped by ALMA for the nearby (less than $100\,\mathrm{pc}$ in distance) and approximately face-on protoplanetary disks.
Another interesting attempt was presented by @ClarkeArmitage2003, where the authors have investigated the possibility of detection of giant planets by excess CO overtone emission originating from the planetary accretion inflow in FUOri type objects. Nevertheless, the periodic line profile distortions are observable only in tight systems in which the giant planet is orbiting within $0.25\,\mathrm{AU}$. Another possibility to detect embedded giant planets is based on the radial velocity measurements, which favor high-mass, short-period companions. The detectability limit of an embedded giant planet by radial velocity measurements is $\sim10\,M_\mathrm{J}$ or $\sim 2\,M_\mathrm{J}$ ($M_\mathrm{J}$ is the mass of Jupiter) orbiting at 1AU or 0.5AU, respectively, due to heavy variability in the optical spectra of young ($<10\,\mathrm{Myr}$) stars (e.g., @PaulsonYelda2006 [@Pratoetal2008]). Note that although there were attempts to detect planets by radial velocity measurements in young systems, no firm detection has yet been repeated see, e.g., @Setiawanetal2008 for TWHyab, debated later by @Huelamoetal2008. The discovery of planetary systems by direct imaging nowadays is restricted to large separations $>10-100\,\mathrm{AU}$ [@Verasetal2009] and favors higher mass planets. As of today nine exoplanetary systems have been directly imaged in the optical or near-infrared band, but only five of them are still embedded: GQ Lup [@Neuhauseretal2005] and CT Cha [@Schmidtetal2008], but they are rather hosting a stellar companion as the companion masses are, because high as $21.5\,\mathrm{M_{J}}$ and $17\,\mathrm{M_{J}}$, respectively; 2M1207 [@Chauvinetal2004; @Chauvinetal2005a] hosting a $4\,\mathrm{M_{J}}$ mass planet; UScoCTIO 108 [@Kashyapetal2008] hosting a $14\,\mathrm{M_{J}}$ mass planet; $\beta\,\mathrm{Pic}$ [@Lagrangeetal2009a; @Lagrangeetal2009b] hosting an $8\,\mathrm{M_{J}}$ mass planet. ABPic [@Chauvinetal2005b], HR 8799 [@Maroisetal2008] (triple system), SR 1845 [@Billeretal2006] and Fomalhaut [@Kalasetal2008] are mature planetary systems with ages of about 30Myr, 60Myr, 100Myr, and 200Myr. Note that centimeter wavelength radio observations of a very young ($<0.1\,\mathrm{Myr}$) disk around HLTau, presented by @Greavesetal2008, revealed the possibility of a 12 Jupiter mass giant planet being in formation at $\sim75\,\mathrm{AU}$.
We investigate a new possibility to detect young giant planets embedded in protoplanetary disks of TTauri stars via line profile distortions in its near-IR spectra. Contrary to the studies mentioned above, instead of dealing with the disk’s density perturbations, we investigate the gas *dynamics* in the disk, particularly near the gap opened by the planet. We have found that there is a reasonable possibility to discover giant planets embedded in the protoplanetary disk of young stars with our approach.
Our idea has been inspired by recent findings of @KleyDirksen2006, who showed that a massive giant planet embedded in a protoplanetary disk produces a transition of the disk from the nearly circular state into an eccentric state. The mass limit for this transition is $3\,M_{\mathrm{J}}$ for a disk with $\nu=10^{-5}$ uniform kinematic viscosity (measured in dimensionless units, see details in Sect. \[sect:Hydro\]). The most visible manifestation of the disk’s eccentric state is that the outer rim of the gap opened by the giant planet has an elliptic shape. After performing a series of hydrodynamical simulations with the code FARGO [@Masset2000], we can also confirm the results of @KleyDirksen2006 even using $\alpha$-type viscosity. Note that beside the elliptic geometry that is clearly visible in the disk surface-density distribution, the giant planet also disturbs the motion of the gas parcels near the gap. Using CO as tracer molecule of gas, which is excited to $\sim3-5\,\mathrm{AU}$ (depending on the stellar mass) in the disk atmosphere [@Najitaetal2007], we investigate how the CO emission line profiles at $4.7\,\mathrm{ \mu m}$ are distorted by the gas dynamics. With our semi-analytical synthetic spectral model we explore the influence of the mass of the giant planet, the disk inclination angle, the mass of the hosting star, the orbital distance of planet, and finally, the size of inner cavity on the line profile distortions.
The paper is structured as follows: in Sect. 2 we review the basic physics of our synthetic spectral line models. In Sect. 3 we present disk models and details of numerical simulations. Our results of the CO ro-vibrational line profile distortions, the detectability of an embedded giant planet, and its observability constraints are presented in Sect. 4. The paper closes with the discussion of the results and concluding remarks.
Synthetic spectral line model
=============================
To calculate the CO ro-vibrational spectra emerging from protoplanetary disks we developed a semi-analytical line spectral model. The thermodynamical model of the disk is based on the two-layer flared disk model, originally proposed by @ChiangGoldreich1997, which describes the heating by stellar irradiation in a disk with a flared geometry. In our model we use the flared disk approximation, and assume the disk to consist of two layers: an optically thick interior, producing continuum radiation and an optically thin atmosphere, producing line emission or absorption in the spectra.[^1] The boundary between the two layers lies where the dust becomes optically thick in the visual wavelengths, i.e. at optical depth $\tau_\mathrm{V}=1$ along the line of the incident stellar irradiation. The irradiation from the boundary layer formed at the inner edge of the disk [@Pophametal1993] is neglected though because its existence had not yet been confirmed conclusively by observations yet. Nevertheless, the emission from the inner disk rim is substantial, resulting in strong near-IR bump in Herbig Ae/Be [@Nattaetal2001] and weaker one in TTauri SEDs [@Muzerolleetal2003]. In addition it was found by @Monnieretal2006 that the simple vertical wall assumption for the inner rim is inconsistent with the interferometric measurements. Indeed, the shape of the disk inner rim is presumably rounded-off, resulting in less inclination-angle-dependent near-IR excess emission [@IsellaNatta2005]. A schematic representation of the disk model is shown in Fig.\[fig:disk-model\]. If the disk atmosphere is superheated by the stellar irradiation with respect to its interior, we may expect emission lines. Conversely, if the disk interior is heated above the temperature of the disk atmosphere by viscous dissipation, for example due to an abrupt increase in accretion rate, spectral lines are expected in absorption.
![Double-layer flared disk model used in our simulations, after @ChiangGoldreich1997. In our model the following emission components are included: the optically thin atmosphere emission, with a temperature $T_\mathrm{atm}$ above the optically thick disk interior, with the temperature $T_\mathrm{int}$; the continuum emission of the rounded-off disk inner rim, with a temperature $T_\mathrm{rim}$ and the stellar continuum assumed to be a blackbody with a temperature $T_*$. The accretion heating of disk interior is also accounted for.[]{data-label="fig:disk-model"}](disk_model.pdf){width="\columnwidth"}
The vertical structure of a geometrically thin disk can be derived by considering vertical hydrostatic equilibrium [@ShakuraSunyaev1973]. Ignoring any contribution from the gravitational force of the disk itself, the vertical density profile would be set by the equilibrium of gas pressure, the stellar gravitation and centrifugal force owing to orbital motion, resulting in a Gaussian vertical density profile. Because the thickness of the disk atmosphere in the double layer model is principally determined by the grazing angle of the stellar irradiation, which is narrow ($\delta(R)\ll 1$) in our computational domain ($R\leq 5\,\mathrm{AU}$), we assume that the disk atmosphere has uniform vertical density distribution. Note that the surface density in the disk atmosphere according to Eq. (\[eq:dens-surf\]) does not depend on the surface density ($\Sigma(R)$) distribution by definition.
The expected flux emerging from the double-layer disk is the result of the radiation of gas emission from the optically thin disk atmosphere above the continuum emission of dust in the optically thick disk interior and inner rim. Because the stellar continuum contributes considerably to the line flux in the near-IR band compared to the disk continuum, especially in disks with an inner cavity, we have to also take into account the stellar continuum in the total flux. In our model the stellar radiation is taken to be blackbody radiation of the stellar effective surface temperature. Because the gas temperature is regulated by collisions with dust grains and stellar X-ray heating (which is significant for a young star), the gas and dust components may be thermally uncoupled in the tenuous disk atmosphere below a critical density of $n_\mathrm{cr}\sim 10^{13}-–10^{14}\,\mathrm{cm^{-3}}$ [@ChiangGoldreich1997]. As a result the gas temperature can be as high as $\sim 4000-5000\,\mathrm{K}$ in a region where the column density is below $\sim 10^{21}\,\mathrm{cm^{-2}}$ [@Glassgoldetal2004]. Because the gas column density is $\sim 10^{22}\,\mathrm{cm^{-2}}$ in our model disk atmosphere, we adopt thermal coupling of dust and gas, i.e. $T_\mathrm{g}(R)=T_\mathrm{d}(R)=T_\mathrm{atm,irr}(R)$, where $T_\mathrm{atm,irr}(R)$ is the atmospheric temperature defined by Eq. (\[eq:temp-surf\]). Although @Glassgoldetal2004 and @KampDullemond2004 suggest the existence of an overheated layer above the superheated disk atmosphere, we did not consider this for simplicity’s sake. In this model, assuming local thermodynamic equilibrium, the monochromatic intensity at frequency $\nu$ emitted by gas parcels at a distance $R$ to the star and an azimuthal angle $\phi$ observed at inclination angle $i$ ($i=0^{\circ}$ meaning face on disk) can be given by $$\begin{aligned}
\label{eq:intensity}
I(\nu,R,\phi,i)&=&B(\nu,T_{\mathrm{int}}(R))e^{-\tau(\nu,R,\phi,i)}+ \nonumber \\
&&B(\nu,T_{\mathrm{ atm}}(R))\left(1-e^{-\tau(\nu,R,\phi,i)}\right),\end{aligned}$$ where $\tau(\nu,R,\phi,i)$ is the monochromatic optical depth of the disk atmosphere at a frequency $\nu$ along the line of sight, $B(\nu,T_\mathrm{int}(R))$ and $B(\nu,T_\mathrm{atm}(R))$ are the Planck functions of the dust temperature $T_\mathrm{int}(R)$ in the disk interior, and the gas temperature $T_\mathrm{atm}(R)$ in disk atmosphere, respectively. Regarding the disk inner edge, which has a temperature $T_\mathrm{rim}(R_0)$, it is assumed that it radiates as a blackbody ($I_\mathrm{rim}(\nu)=B(\nu,T_\mathrm{rim}(R_0))$), see Appendix \[apx:sinner-rim\] for details. The total flux emerging from the protoplanetary disk seen by inclination angle $i$ at frequency $\nu$ can thus be given as $$\begin{aligned}
\label{eq:flux-disk}
F_{\mathrm{disk}}(\nu,i)&=&\int _{R_{0}}^{R_{1}} \int _{0}^{2\pi} I(\nu,R,\phi,i)\frac{R dR d\phi}{D^2}\cos(i)+\nonumber \\
&&\frac{I_\mathrm{rim}(\nu,R)}{D^2}A_\mathrm{rim}(i)\cos(90-i),\end{aligned}$$ where $R_0$ and $R_1$ are the disk inner and outer radii and $D$ is the distance of the source to the observer. In Eq. \[eq:flux-disk\] $A_\mathrm{rim}\cos(90-i)$ is the visible area of the disk inner rim. If the disk inner rim were a perfect vertical wall, its surface area would be $A_\mathrm{rim}(i)=4\pi h(R_0)R_0^2$, where $h(R_0)$ is the disk aspect ratio at the inner edge, which is taken to be 0.05. In order to be consistent with the observations regarding the inclination-angle dependence of the rim flux [@IsellaNatta2005], the rim is taken to be a vertical wall seen under a constant 60 degree inclination angle.
To calculate the synthetic spectra numerically, we first set up the two-dimensional computational domain with $N_\mathrm{R}$ logarithmically distributed radial and $N_\phi$ equidistantly distributed azimuthal grid cells. For a given set of stellar parameters, shown in Table \[table:1\], first the grazing angle of the incident stellar irradiation is determined according to Eq. (\[eq:grazing-angle-flaring\]). The temperature distributions in the disk atmosphere and interior are calculated according to Eq. (\[eq:temp-surf\]) and Eqs. (\[eq:temp-int-irr\], \[eq:acc-temp\], \[eq:temp-rim\], \[eq:temp-int-mod\]), respectively. After determining the dust surface-density of the disk atmosphere with Eq. (\[eq:dens-surf\]), the optical depth along the line of sight is calculated by Eq. (\[eq:tau\_nu\_2\]) using the gas opacity given by Eq. (\[eq:gas-kappa\]) and considering Eqs. (\[eq:ilp\])-(\[eq:Doppler-shift\]) to calculate appropriate intrinsic line profiles and Doppler shifts. In order to determine the mass absorption coefficient of the $^{12}\mathrm{C}^{16}\mathrm{O}$ molecule we used the transition data provided by @Goorvitch1994, such as transition probability $A$, lower and upper state energy $E_\mathrm{l},\,E_\mathrm{u}$ and statistical weight $g_\mathrm{u}$. Knowing the optical depth ($\tau(\nu,R,\phi,i)$), the temperature in disk atmosphere and interior in the 2D computational domain the expected flux along the line of sight can be calculated by applying Eq. (\[eq:intensity\]) and Eq. (\[eq:flux-disk\]) at each frequency in the vicinity of the investigated transition.
Disk models
===========
In this section we describe in detail our disk models that fed into the synthetic spectral line model. As a first step, we calculated the disk surface-density and velocity distribution perturbed by a massive embedded planet with the publicly available hydrodynamical code FARGO of @Masset2000. A very important difference to the planet-free case is that the disk surface-density distribution is very heavily modified. It shows a clear elliptic character of the gap opened by the giant planet. According to @KleyDirksen2006, @DAngeloetal2006, and also to our calculations (see below), the disk becomes eccentric after several hundred orbits of the embedded planet. With regard to our synthetic spectral line model, the most relevant are those effects that are originating from the gas dynamics caused by the giant planet. It is well known that a pure Keplerian motion of the gas parcels on circular orbits results in symmetric double-peaked emission line profiles [@HorneMarsh1986]. As the CO emission is strongly depressed $2-3\,\mathrm{AU}$ in our models, the origin of the double-peaked profiles is clear. Any deviation of the gas dynamics from the pure rotation will break this symmetry, resulting in the distortion of line profiles. If the distorting effect is strong enough, the line profile distortion could be significant to be detected by high-resolution spectroscopic observations.
Indeed several TTauri stars show symmetric but centrally peaked CO line profiles that can be explained by radially extent CO emission overriding the double peaks [@Najitaetal2003; @Brittainetal2009]. Stronger CO emission can be expected if UV fluorescence plays a role [@Krotkovetal1980], or the gas temperature is not well coupled to the dust [@KampDullemond2004; @Glassgoldetal2004]. In the tenuous region above the disk atmosphere the gas can be significantly hotter than is predicted by the double-layer model. As a consequence the slowly rotating distant disk parcels could produce a substantial contribution to the low-velocity part of the line profile, resulting in a centrally peaked profile. Nevertheless, for simplicity’s sake here we do not consider any of the above mentioned effects.
Hydrodynamical setup {#sect:Hydro}
--------------------
We adopt dimensionless units in hydrodynamical simulations, for which the unit of length and mass is taken to be the orbital distance of the planet, and the mass of the central star, respectively. The unit of time, $t_0$, is taken to be the reciprocal of the orbital frequency of the planet, resulting in $t_0=1/2\pi$, setting the gravitational constant to unity. In each disk model we assigned four planetary masses to the embedded planet, which are in increasing order $q=m_\mathrm{pl}/m_{*} = 0.001$, $0.003$, $0.005$, and $0.008$, where $m_\mathrm{pl}$ and $m_*$ are the planetary and stellar masses. Regarding the disks geometry, the flat thin disk approximation was assumed with an aspect ratio of $H(R)/R=0.05$. The disk extends between $0.2-5$ dimensionless units. This computational domain was covered by 256 radial and 500 azimuthal grid cells. The radial spacing was logarithmic, while the azimuthal spacing was equidistant. This results in practically quadratic grid cells, because the approximation $\Delta R\sim R\Delta\phi$ is valid at each radius. The disks are driven by $\alpha$-type viscosity [@ShakuraSunyaev1973], which is consistent with our thermal disk model with an intermediate value of $\alpha$, which was taken to be $1\times10^{-3}$. Note that assuming a cold disk (therefore virtually non-ionized) the viscosity generated by magnetohydrodynamical turbulence is hard to explain, although, according to @Mukhopadhyay2008, the transient growth of two or three-dimensional pure hydrodynamic elliptic-type perturbations could result in $\alpha\simeq 10^{-1}-10^{-5}$, depending on the disk scale height (decreasing $\alpha$ with increasing scale height). The disk’s initial surface density profile is given by a power law $\Sigma(R)=\Sigma_0 R^{-1/2}$, where the surface density at 1 distance unit $\Sigma_0$ is taken to be $2.15\times 10^{-5}$ in dimensionless units. For simplicity, a locally isothermal equation of state is applied for the gas, and the disk self-gravity is neglected. While the inner boundary of the disk is taken to be open, allowing the disk material to leave the disk on accretion timescale, the outer boundary is closed, i.e. no mass supply is allowed.
Below we assign physical units to the dimensionless quantities. If the stellar mass is assumed to be $m_*=1M_{\sun}$, the planetary masses in our models correspond to $1M_{\mathrm{J}}$, $3M_{\mathrm{J}}$, $5M_{\mathrm{J}}$, and $8 M_{\mathrm{J}}$. Thanks to the dimensionless calculations it is possible to scale the results to different stellar masses. For different stellar masses, the mass of the giant planet scales with the stellar mass, i.e $m_{\mathrm{pl}}= 1\times 10^{-3}\,m_{*}$, $3\times 10^{-3}\,m_{*}$, $5\times 10^{-3}\,m_{*}$, $8\times 10^{-3}\,m_{*}$, and the stellar masses are taken to be $m_*=0.5\,M_{\sun}$, $1\,M_{\sun}$, $1.5\,M_{\sun}$. Furthermore, the distance unit is set to the distance of the planet to the star, i.e., the planet orbits at 1AU. Below we extended our models to tight and wide systems, where the giant planets are orbiting at 0.5 and 2AU, respectively. In $1\,M_{\sun}$ stellar mass disk models the surface density at 1AU is $\Sigma_0=2.15\times 10^{-5} M_{\sun}\,\mathrm{AU^{-2}}\simeq 191\,\mathrm{g/cm^2}$ at the beginning of simulation, which is also scaled with the mass of the central star. But note that the solution to the hydrodynamical system of equations is independent of $\Sigma_0$, if there is no back-reaction to the planet, see the vertically integrated Navier-Stokes equations in @Kley1999. For $0.5\,M_{\sun}$, $1\,M_{\sun}$ and $1.5\,M_{\sun}$ stellar mass model the disk mass in the computational domain ($0.2\,\mathrm{AU}\leq R \leq 5\,\mathrm{AU}$) is $5\times10^{-4}\,M_{\sun}$, $1\times10^{-3}\,M_{\sun}$ and $1.5\times10^{-3}\,M_{\sun}$. These values refer to the mass of the inner disk only, as the whole disk may extend to several tens or hundreds of AU, and the total disk mass is in the conventional range of $0.01-0.1\,M_{\sun}$. More details about the models can be found in Table \[table:1\].
Model No. $m_*\,(M_{\sun})$ $T_{*}\,(\mathrm{K})$ $R_*\,(R_{\sun})$ $i\,(^\circ)$ $m_\mathrm{p}\,(M_\mathrm{J})$
----------- ------------------- ----------------------- ------------------- --------------- --------------------------------
\#1 0.5 3760 1.4 20,40,60 0.5
\#2 0.5 3760 1.4 20,40,60 1.5
\#3 0.5 3760 1.4 20,40,60 2.5
\#4 0.5 3760 1.4 20,40,60 4
\#5 1 4266 1.83 20,40,60 1
\#6 1 4266 1.83 20,40,60 3
\#7 1 4266 1.83 20,40,60 5
\#8 1 4266 1.83 20,40,60 8
\#9 1.5 4584 2.22 20,40,60 1.5
\#10 1.5 4584 2.22 20,40,60 4.5
\#11 1.5 4584 2.22 20,40,60 7.5
\#12 1.5 4584 2.22 20,40,60 12
: Stellar and disk parameters in our hydrodynamical models. The stellar parameters were taken from a publicly available tool [@Siessetal2000]. The disk extends between $0.2-5\,\mathrm{AU}$ and the orbital distance of the massive planets is $1\,\mathrm{AU}$.[]{data-label="table:1"}
Synthetic spectral calculation setup {#sec:spectra-calc-setup}
------------------------------------
We calculate the spectral line of a fundamental band CO transition ($\mathrm{V}=1\rightarrow0\mathrm{P}(10)$, $\lambda_0=4.7545\,\mathrm{\mu m}$), which is not blended by the higher excitation V=2$\rightarrow$1 and V=3$\rightarrow$2 lines, although in our thermal model the temperature is not high enough to excite the higher vibrational levels at all.[^2] The CO rotational and vibrational level populations were calculated in local thermodynamical equilibrium. The line profiles were calculated in 200 wavelengths and in the same numerical domain that was used in hydrodynamical simulations. According to our preliminary tests on “circularly Keplerian”[^3] disks, the CO fundamental band spectra did not show significant changes with increasing outer disk boundary. This can be explained by the fact that a substantial part of the CO fundamental band emission is arising in the inner disk, consequently the outer part of the disk ($R>3-5\,\mathrm{AU}$, depending on the stellar mass) does not contribute to the disk emission at 4.7 micron. Contrary to this, the distance of the inner boundary of the disk to the central star heavily influences the CO fundamental band spectra, i.e. the closer the disk inner boundary to the star the line-over-continuum is weaker and broader. This is because the disk innermost regions give stronger contribution to the continuum, which in turn weakens the CO emission normalized to continuum, although the CO emission itself is also getting stronger due to an increased amount of hot CO. The line broadening is a natural consequence of the larger orbital velocity of gas orbiting closer to the star.
Because we investigate disks with already formed planets, it is reasonable to assume that a cavity is formed at the very inner disk like the ones found in disks of several TTauri stars [@Akesonetal2005; @Eisneretal2005; @Eisneretal2007; @Salyketal2009]. If dust evaporation is takes place close to the star, the CO will be depleted too, due to the disappearance of the dust which protects the CO molecules against UV photo dissociation. In this way, the observed inner radius of CO ($R_\mathrm{CO}$) should intuitively be similar or slightly larger than the dust sublimation radius $R_\mathrm{sub}$. Indeed, $R_\mathrm{CO}$ was found to be larger than $R_\mathrm{sub}$ by a factor of a few in several disks, e.g., see Fig.5. of @Eisneretal2007 or Fig.10 in @Salyketal2009. Contrary to this, @Carr2007 found that the CO inner radius is $\sim 0.7$ that of the dust. If the gas disk extends as far as 0.05AU, there would be an additional broad component to the emission that is not modeled here. Taking these argumentations into account it is reasonable to assume that the CO inner radius is set by the dust sublimation. The dust sublimation radius $R_\mathrm{sub}$ is given by $$\label{eq:sublimation-radius}
R_{\mathrm{sub}}\simeq0.03\left(\frac{T_*}{T_{\mathrm{sub}}}\right)^{5/2}\frac{R_{*}}{R_{\sun}}\mathrm{ AU},$$ where Eq. (\[eq:temp-surf\]) with standard dust opacity law, $\beta=1$ [@Rodmannetal2006] was used. Assuming $T_\mathrm{sub}=1500\,\mathrm{K}$ for dust sublimation temperature, $T_*\sim4000\,\mathrm{K}$ and $R_{*}=1.8\,R_{\sun}$ for stellar surface effective temperature and radius, appropriate for a 2.5Myr solar mass star, we obtain $R_\mathrm{sub}=0.05\,\mathrm{AU}$. To keep this simple, we set the disk inner boundary (namely the CO inner radius) fixed to $R_\mathrm{CO}\equiv4R_\mathrm{sub}=0.2\,\mathrm{AU}$, which is an acceptable value in disks hosted by TTauri stars with 3500-5000K surface temperature and $1.4-2.2\,R_{\sun}$ radius. Note that $R_\mathrm{sub}$ was fixed throughout our models (except the ones where we investigated the effect of the size of the inner cavity) to make them comparable, neglecting the effect of the change in stellar luminosity on the $R_\mathrm{sub}$.
![Logarithmic density (*green shaded* in $g/cm^3$) and temperature contours (*red dashed contours* in K) of the disk. The density contours are stopped at a gas density $\rho=10^{-34}\,\mathrm{g/cm^3}$ to avoid color crowding, while the maximum is $\rho=10^{-11}\,\mathrm{g/cm^3}$. The height of the disk atmosphere, i.e the range where the radial optical depth is less than 1 is shown with *white dotted lines* in the disk inner region. Comparing the unperturbed case (*top*) to the disk in which a gap exists at $0.5\,\mathrm{AU}<R<2\,\mathrm{AU}$, after depleting the density by 1/1000 (*bottom*), it is appreciable that however the disk atmosphere is decreased in height, the temperature distribution is not considerably changed.[]{data-label="fig:RADMC-output"}](radmc-unp.pdf "fig:"){width="\columnwidth"}\
![Logarithmic density (*green shaded* in $g/cm^3$) and temperature contours (*red dashed contours* in K) of the disk. The density contours are stopped at a gas density $\rho=10^{-34}\,\mathrm{g/cm^3}$ to avoid color crowding, while the maximum is $\rho=10^{-11}\,\mathrm{g/cm^3}$. The height of the disk atmosphere, i.e the range where the radial optical depth is less than 1 is shown with *white dotted lines* in the disk inner region. Comparing the unperturbed case (*top*) to the disk in which a gap exists at $0.5\,\mathrm{AU}<R<2\,\mathrm{AU}$, after depleting the density by 1/1000 (*bottom*), it is appreciable that however the disk atmosphere is decreased in height, the temperature distribution is not considerably changed.[]{data-label="fig:RADMC-output"}](radmc-gap.pdf "fig:"){width="\columnwidth"}
It is reasonable to assume that a disk that is $2.5\,\mathrm{Myr}$ old contains a considerable amount of gas and tracer CO also in the inner disk, hence the stellar age is taken to be $2.5\,\mathrm{Myr}$ in all models. Note that according to @Haischetal2001, half of the observed young stars in nearby embedded clusters (NGC2024, Trapezium, IC348, NGC2264, NGC2362, NGC1960), with ages of about $3\,\mathrm{Myr}$, show near-IR excess, which is related to the excess emission of dust in disks. Furthermore, @Hillenbrand2005 also concluded that the median lifetime of the optically thick inner disk is between $2-3\,\mathrm{Myr}$.
In order to simplify the models it is assumed that the dust consists of pure silicates with $0.1\,\mathrm{\mu m}$ grain size [@DrainLee1984]. According to this the mass absorption coefficient of the dust is taken to be $2320\,\mathrm{cm^2/g}$ at visual wavelength. Note that neither the effect of the coagulation nor the settling out of tenuous atmosphere of grains with a large size are taken into account. The size distribution and therefore the overall mass absorption coefficient of the dust is taken to be the same in the interior and the atmosphere. In this way the dust opacity is slightly overestimated. According to Eq. (\[eq:tau\_nu\_2\]) the atmospheric gas density and thus gas line emission strength is slightly underestimated. The dust-to-gas and CO-to-gas mass ratios are assumed to be constant throughout the disk and are taken to be $10^{-2}$ and $4\times10^{-4}$ (measured per gram of gas+dust mass) in the disk atmosphere and interior, respectively.
For the planet-free disks circular Keplerian velocity distribution is applied to determine the observed line center shift, given by Eq. (\[eq:Doppler-shift\]). For a massive planet-bearing disk observed under inclination angle $i$, the Doppler shift of the emission from a single patch of gas in the disk compared to its fundamental frequency $\nu_0$ emerging from a given $R,\phi$ point is calculated by $$\begin{aligned}
\Delta\nu(R,\phi,i)&=&\frac{\nu_{0}}{c}\left\{u_\mathrm{R}(R,\phi)\left[\sin(\phi)+\cos(\phi)\right]\right. +\nonumber\\
&&\left. u_\mathrm{\phi}(R,\phi)\left[\cos(\phi)-\sin(\phi)\right]\right\}\sin(i),
\label{eq:Doppler-shift-pertdisk}\end{aligned}$$ where the radial $u_{\mathrm{R}}(R,\phi)$ and azimuthal $u_{\mathrm{\phi}}(R,\phi)$ velocity components of gas parcels are provided by the hydrodynamic simulations.
The perturbations in the surface density distribution of the disk interior and atmosphere were not taken into account, because the disk atmosphere density given by Eq. (\[eq:dens-surf\]) is independent of the density distribution of the disk even in the gap, as long as there is enough dust in the gap to keep it optically thick. To test the optical thick assumptions, let us for a moment neglect the effect of disk shelf-shadowing that occurres in the inner edge of the gap. Applying the dust density in the disk $\Sigma_\mathrm{d}(R=1\,\mathrm{AU})\simeq\Sigma_{0}X_d$, where $X_d=0.01$ is the dust-to-gas ratio and the dust opacity $\kappa_\mathrm{V}=2320\,\mathrm{cm^2/g}$, in the gap, depleted to 0.1% of the surrounding density, the optical depth at visual wavelengths along the incident stellar irradiation is $\tau_\mathrm{V}=\Sigma_\mathrm{d}(R=1\,\mathrm{AU})\kappa_\mathrm{V}/\delta(R=1\,\mathrm{AU})\simeq1000$. To argue that the shelf shadowing can be neglected we calculated the height of the disk atmosphere and temperature distribution in a planet-free and planet-bearing disk by a 2-D Monte-Carlo code RADMC for dust continuum radiative transfer [@DullemondDominik2004] with the same dust prescription as in our synthetic spectral model. In the planet-bearing disk model the gap is taken to be extended from 0.5AU to 2AU and artificially depleted in density to 0.1% that of the planet-free case. The amount of gap depletion is an average value measured in our hydrodynamic calculations. The results are shown in Fig. \[fig:RADMC-output\]. It is evident that although the height of the disk atmosphere ($\tau_\mathrm{V}=1$ along the line of sight of stellar irradiation) above the mid-plane is decreased in the gap, the temperature in the disk atmosphere is not substantially changed. Because the temperature is not significantly decreased we neglect the surface density perturbations. Note that the effect of the distance of the gap from the central star on the shelf shadowing is not considered. The dust and gas temperature decoupling was also not taken into account. Because the gas has a temperature well in excess of the dust in depleted regions, where the gas column density is $\ll 10^{22}$ [@Glassgoldetal2004; @KampDullemond2004], we can expect increased contribution to the CO line flux originating from the gap. Contrary to this, if the base of the gap is shadowed completely by the inner wall, the gap does not contribute to the CO emission. Thus a gap could cause a strong permanent line profile asymmetry due to its asymmetric geometry (see Fig. \[fig:FARGO-output\]), which requires further investigation.
Because the accretion heating has been taken into account according to Eq. (\[eq:acc-temp\]), we had to set an appropriate accretion rate. The accretion rate measured in hydrodynamical simulations gives a value of about $2\times10^{-8}\,M_{\sun}/\mathrm{yr}$. Note that according to @Fangetal2009 the accretion rate inferred from $\mathrm{H}\alpha$ emission luminosity in the young star population of Lynds1630N and 1641 clouds in the OrionGMC with an age about $2-3\,\mathrm{Myr}$, is between $10^{-10}-10^{-8}\,M_{\sun}/\mathrm{yr}$. In order to be consistent with the latter, the accretion rate is taken to be $5\times 10^{-9}\,M_{\sun}/\mathrm{yr}$ for all our models. However, note that at this accretion rate the contribution of the viscous dissipation to the disk continuum is weak in the near-IR band.
Results
=======
Effect of massive planets on the disk structure
-----------------------------------------------
{width="\columnwidth"} {width="\columnwidth"}
Three groups of models were computed through 2000 planetary orbits, with stellar and planetary parameters listed in Table \[table:1\]. In each run the planet was kept fixed on circular orbit during the first 1000 orbits. After the 1000th orbit the planet was released. It thus felt the (gravitational) backreaction of the disk, resulting in its inward migration. In a good accordance with the expectations, we found that a more massive planet opened a broader and deeper gap. The depletion is 0.25%-0.1% for planets with mass in the range of $1\,M_\mathrm{J}-8\,M_\mathrm{J}$. After the first 1000 orbits a quasi steady state disk structure has developed in all simulations, which slightly changed during the following 1000 orbits, when the planet orbit was not fixed anymore.
To shed light on how the giant planet distorts the originally circularly Keplerian gas flow, several snapshots of density and velocity distributions were taken during the 2000th planetary orbit. Figure\[fig:FARGO-output\] shows four snapshots of the density and the radial velocity component of the orbital velocity of gas for an $8\,M_\mathrm{J}$ mass planet orbiting an $1\,M_{\sun}$ star (model \#8) during one orbit. It is evident that in a disk with an embedded massive planet the overall orbits of gas parcels are non–circularly Keplerian because the radial components of their orbital velocity distribution are strongly departing from zero. Moreover, we found that the elliptic gap, while preserving its shape, precesses slowly retrograde with a period of about $150$ planetary orbits.
As we already mentioned, @KleyDirksen2006 found that an originally circular disk with an embedded giant planet can reach an *eccentric* equilibrium state if the mass of the embedded planet is larger than a certain limit ($3\,M_{\mathrm{J}}$ for a $1\,M_{\sun}$ star). Kley & Dirksen found that for sufficiently wide gaps the growth of the eccentricity is induced by the interaction of the planet’s gravitational potential with the disks material at the radial location of the 1:3 (outer) Lindblad resonance. For smaller planetary masses, this effect is damped mainly by the co-orbital and the 1:2 Lindblad resonances. If the gap is deep and wide enough, which is the case for a giant planet, the above resonances cannot damp the eccentricity-exciting effect appearing at the 1:3 Lindblad resonance, and the disk becomes eccentric. For a more detailed explanation of disk eccentricity growth see @Lubow1991 and @KleyDirksen2006. The eccentric state of the disk in our simulations can also be clearly seen in Fig. \[fig:FARGO-output\] where the shape of the outer rim of the gap becomes elliptic in the surface density distribution.
![Azimuthally averaged eccentricities as the functions of radii after 2000 orbits of the giant planet. The eccentricity curves in models \#5-\#8 (where the planetary masses are $1\,M_\mathrm{J}$, $3\,M_\mathrm{J}$, $5\,M_\mathrm{J}$ and $8\,M_\mathrm{J}$ for $1\,M_{\sun}$ stellar mass) are shown with dot-dashed, dashed, dotted, and solid lines, respectively.[]{data-label="fig:FARGO-ecc"}](ecc.pdf){width="\columnwidth"}
The departure of the velocities of the gas parcels from the pure circularly Keplerian circular revolution can be characterized by calculating their eccentricities. Considering the eccentric equilibrium state of the disk, we expect that each gas parcel would move on a non-circular orbit, which may be characterized most conveniently by an average eccentricity value. Therefore, for each disk radius between 0.2AU and 5AU, the azimuthally averaged eccentricities of the orbits of the gas parcels were calculated in the following way $$\label{eq:ecc}
e(R)=\int_0^{2\pi}\sqrt{1+2h(R,\phi) c(R,\phi)^2}d\phi.$$ In Eq. (\[eq:ecc\]) $c(R,\phi)$ and $h(R,\phi)$ stand for $$c(R,\phi)=x(R,\phi)v_\mathrm{y}(R,\phi)-y(R,\phi)v_\mathrm{x}(R,\phi)$$ and $$h(R,\phi)=\frac{v_\mathrm{x}(R,\phi)^2+v_\mathrm{y}(R,\phi)^2}{2}-\frac{1}{\sqrt{x(R,\phi)^2+y(R,\phi)^2}},$$ where $v_\mathrm{x}(R,\phi)$, $v_\mathrm{y}(R,\phi)$ and $x(R,\phi)$, $y(R,\phi)$ are the Cartesian velocity components and coordinates at point $R,\phi$. We found that the averaged eccentricities differ considerably from zero, and each eccentricity curve reaches its maximum near the outer boundary of the gap (Fig. \[fig:FARGO-ecc\]). A priori one would expect that the averaged eccentricity values are higher for more massive planets. Indeed we found that the maximum of the azimuthally averaged eccentricity is monotonically increasing with a planetary mass in the range of $1\,M_\mathrm{J}\le m_\mathrm{pl} \le 8\,M_\mathrm{J}$. The eccentricity curve for the $8\,M_\mathrm{J}$ mass planet peaks about $e_{\mathrm{max}}\sim 0.3$, while the peak stays well bellow 0.1 for a $1\,M_\mathrm{J}$ mass planet. Note that a very similar behavior of the disk eccentricity has already been found by @KleyDirksen2006. In their cases, however, the maximum of the eccentricity curves is somewhat lower than in our cases, and at least $3\,M_\mathrm{J}$ is required to set the disk into eccentric state. This can be the consequence because contrary to @KleyDirksen2006, we used an $\alpha$-type viscosity in our simulations. The higher eccentricities we found are plausible inasmuch as @KleyDirksen2006 found that the eccentricity of the disk is increasing with decreasing viscosity and in our approach the kinematic viscosity $\nu(R)=\alpha H^2 \Omega_\mathrm{K}(R)$ measured in dimensionless units is $2.5\times10^{-6}$ at $R=1$, which is smaller than the $\nu=1\times 10^{-5}$ used by @KleyDirksen2006.
![Offset of the line of sight component of the velocity distribution from circularly Keplerian values on the disk surface in the hydrodynamic simulations, in the same model presented in Fig. \[fig:FARGO-output\]. Snapshots are calculated for four different azimuthal positions of the planet of the 2000th orbit. The velocity difference is in the range of $-7.92-4.88\,\mathrm{km/s}$. Here the inclination angle is taken to be $i=40^{\circ}$ and the viewing angle is taken to be $-90^{\circ}$, i.e the disk is seen from the bottom of Fig. \[fig:FARGO-output\]. The planetary orbit is shown with a red dashed circle. Two permanent high-velocity regions can be seen at position angle $0^\circ$ and $180^\circ$, and a variable pattern in the vicinity of the planet orbits. The strength of the variable component can reach that of the permanent one, e.g, in the panel at the bottom left (planet is at position angle $180^\circ$).[]{data-label="fig:FARGO-output-lsvel"}](gaslsvelplot.pdf){width="\columnwidth"}
As one can see, a giant planet has substantial impact on the density and velocity distributions of its host disk. It is an essential question, whether the velocity perturbations appear in the line-of-sight velocity with substantial strength. Figure\[fig:FARGO-output-lsvel\] shows the subtracted distribution of the line-of-sight velocity in the perturbed and circularly Keplerian case. It is evident that the line-of-sight velocities show a significant departure from the circularly Keplerian fashion because the difference is non-vanishing. The radial velocity component of the orbital velocity and more importantly the line-of-sight velocity distributions have variable patterns following the planet on the top of a permanent excess seen at $0^\circ$ and $180^\circ$ position angle. Note that the the permanent pattern precesses slowly (with $\sim 150$ orbital periods), retrograde to the planet. The disk inclination angle $i$ is taken to be $40^{\circ}$ in the calculation of Fig. \[fig:FARGO-output-lsvel\] and the disk was rotated to the line-of-sight in a way that the line-of-sight velocity component of dynamically perturbed gas parcels is maximized. Consequently we had expected that not only significant distortions appear in the line profiles, but that they vary in time within the orbital time scale of the giant planet. Because the deviation from circularly Keplerian velocity is in the range of $-7.92-4.88\,\mathrm{km/s}$, the width of variable component in the line profile should be $\sim10\,\mathrm{km/s}$, depending on the inclination angle.
Distortion of CO lines
----------------------
Below we address the following questions: (i) Does the line-of-sight component of the non-circularly Keplerian velocity distribution (Fig. \[fig:FARGO-output-lsvel\]) result in significant distortions in the CO spectral line profiles? (ii) Are these distortions varying in the planets orbital timescale? (iii) How do the distortions depend on the inclination angle, the planetary and stellar mass, the orbital distance of the planet, the size of the inner cavity and the disk geometry (if the disk is flared or not)? (iv) Are these distortions strong enough to be detected by high-resolution near-IR spectral measurements? A complete set of answers to the above questions may give us a novel method to detect massive planets embedded in protoplanetary accretion disks.
![Asymmetric CO ro-vibrational line profile in a circularly Keplerian disk with a gap between $0.8-2.8\,\mathrm{AU}$ (similar to model \#8), where the gas is flowing in eccentric orbit with $e=0.2$. The gas outside the gap is in Keplerian circular orbit. It is evident that the line profiles are asymmetric. For comparison, we display the symmetric double-peaked line profiles (dashed lines) emerging from a planet-free disk as well.[]{data-label="fig:Ms1-e02(08-28)"}](egV1-0P10.pdf){width="\columnwidth"}
As was shown in Fig. \[fig:FARGO-ecc\] the disk eccentricity is considerable near the gap in all planet-bearing disk models. If the gas parcels move on pure elliptic orbits in the gap, the CO line profile becomes permanently asymmetric. This effect is illustrated in Fig. \[fig:Ms1-e02(08-28)\], where we have calculated the emerging line profile of a disk with an eccentricity $e=0.2$ in the gap between $0.8\,\mathrm{AU}-2.8\,\mathrm{AU}$ hosted by a $1\,M_{\sun}$ star viewed $20^{\circ}$, $40^{\circ}$ and $60^{\circ}$ degree of inclination angles. Thus we can expect that the non-circularly Keplerian property of the velocity field shown in Figs. \[fig:FARGO-output\] and \[fig:FARGO-output-lsvel\] will break the symmetry of the spectral lines. Although the orbits of the gas parcels can be characterized by average eccentricities, their orbits are more complicated than those of a pure elliptic motion, because of the perturbations induced by the planet.
First we examined the CO emission line profile distortions from a disk in which a 8$M_\mathrm{J}$ planet orbits a 1$M_{\sun}$ star (model \#8). As expected, the strongly non-circularly Keplerian velocity flow significantly modified the CO line profiles. In Fig. \[fig:V1-0P10-profiles\](a) we display the V=1-0P(10) line profiles for three different inclination angles. The line profiles show a strongly asymmetric double-peaked shape. Studying Fig. \[fig:V1-0P10-profiles\] (a) two distortion patterns of the spectral line profiles can be recognized. An excess can be seen near the red peak of the lines at about $11\,\mathrm{km/s}$, $21\,\mathrm{km/s}$ and $29\,\mathrm{km/s}$, for inclinations $i=20^{\circ}$, $40^{\circ}$ and $60^{\circ}$, respectively. Moreover, a deficiency appears around the blue peak of each spectral line at slightly lower velocities. The above velocities correspond to the orbital velocity of the gas revolving around a $1\,M_{\sun}$ star at 0.8-0.9AU, i.e. near the inner boundary of the gap, taking into account the appropriate inclination angles. Smaller-scale distortions also appear in the red and blue peaks as well, corresponding to the 0.8-2.9AU region, i.e. the whole gap.
Calculating the line profiles at different orbital phases during one orbit, we found that the shape of the already distorted line profile varies in time, yielding a clear dependence on the orbital position of the planet, see results in Fig. \[fig:variation-Ms1-Mb8-V1-0P10\] for model \#8. The permanent asymmetry in lines is caused by the permanent velocity pattern seen at PA $0^\circ$ and $180^\circ$ in Fig. \[fig:FARGO-output-lsvel\]. Note that the expression “permanent” is not entirely correct because the velocity pattern slowly ($\sim 150$ orbital periods of planet) precesses retrograde to the planet, but we still use it henceforward. The variable component indicated with arrows is moving in regions between $\sim\pm 25\,\mathrm{km/s}$ due to the variable pattern following the planet. Note that the maximum width of variable component is $\sim 10\,\mathrm{km/s}$, as is expected, which is resolvable by CRIRES, whose maximum resolution is $3\,\mathrm{km/s}$. In this particular model the time scale of variations is on the order of weeks, because the orbital period of the planet is one year, thus the time elapsed between snapshots is approximately 18 days.
Planetary and stellar masses
----------------------------
To gain further insights, it is useful to study the influence of the masses on the strength of the spectral line distortions. First we have investigated the cases in which a $1\,M_{\sun}$ star hosts a $1\,M_\mathrm{J}$, $3\,M_\mathrm{J}$, $5\,M_\mathrm{J}$, and $8\,M_\mathrm{J}$ mass planet at 1 AU, corresponding to models \#5-\#8. Parts of the results are presented in Fig. \[fig:V1-0P10-profiles\](b) for model \#5 ($m_\mathrm{pl}=1\,M_\mathrm{J}$, $m_*=1\,M_{\sun}$) and in Fig. \[fig:V1-0P10-profiles\](a) and Fig. \[fig:variation-Ms1-Mb8-V1-0P10\] for model \#8. One can conclude that for a less massive planetary companion the line profile distortions are weaker. The same conclusion can be obtained by the growing influence of the increasing planetary mass on the disk eccentricity, see Fig. \[fig:FARGO-ecc\].
A natural way to study the effect of the mass of the central star on the line profile distortions would be that the planetary mass is kept fixed, and the stellar mass is increased. On the other hand, we recall that dimensionless units were used in hydrodynamical simulations thus the mass of the planet is expressed in stellar mass units. Thus changing either the planet mass or the stellar mass is equivalent to changing the ratio between the planetary and the stellar mass. Therefore, one could conclude from the previous results that the larger the stellar mass, the stronger the line profile distortions. The situation is however a bit more complicated; if we change, for instance, the stellar mass, the luminosity of the star changes, which has an influence on the temperature distribution in the disk atmosphere and interior as well. The effect of the stellar mass can be therefore studied if the planetary-to-stellar mass ratio is kept fixed, while the mass along with the surface temperature and the stellar radius is changed in the synthetic spectral model. Due to the effect of the increased/decreased flux of stellar irradiation in case of the larger/smaller stellar mass, the outer boundary of the CO emitting region will be moved farther/closer to the star, and increased/decreased in size. To investigate the dependence of the observable line profile distortions on the stellar mass, we have recalculated the above presented line profiles for a $0.5\,M_{\sun}$ (models \#1-\#4) and a $1.5\,M_{\sun}$ (models \#9-\#12) central stars. Part of the results are shown in Fig. \[fig:V1-0P10-profiles\](c) (for model \#4) and Fig. \[fig:V1-0P10-profiles\](d) (for model \#12), respectively. Comparing the line profiles to those presented in Fig. \[fig:V1-0P10-profiles\](a) (for model \#8), it is evident that the profiles are considerable narrower for $0.5\,M_{\sun}$ and broader for $1.5\,M_{\sun}$ models, due to the change in the Keplerian angular velocity ($\Omega_\mathrm{K}(R)=(Gm_{*}/R^3)^{1/2}$). Moreover, the line-to-continuum ratio at the maximum is slightly decreased and increased about a same amount for disks with 0.5 and $1.5\,M_{\sun}$ star. The change in line-to-continuum ratio can be easily explained: for a smaller mass star, the disk temperature is also decreased due to the decrease in stellar luminosity in a way that the ratio of the CO line to the disk plus star continuum is decreased too. The opposite is true for larger stellar masses. Finally, we can conclude that for a given planet-to-star mass ratio the larger the central stellar mass, the larger the observable CO line profile distortion caused by the giant planet. Here we have to note that although the observational data on TTauri stars [@Akesonetal2005] show that the size of the disk inner cavity is increasing with increasing stellar luminosity (which can be the consequence of the increased dust sublimation radius, see Eq. (\[eq:sublimation-radius\])), resulting in smaller continuum and stronger relative line strength, in our cases the size of inner cavity stayed fixed in order to make models comparable.
Analyzing Fig. \[fig:V1-0P10-profiles\](b) we conclude that the line profile distortions in model \#5 ($m_\mathrm{pl}=1\,M_{\mathrm{J}}$, $m_*=1\,M_{\sun}$, 1AU orbital distance) are so small that the giant planet signal below $1\,M_{\mathrm{J}}$ is strongly suppressed. Note that a disk with a $1\,M_\mathrm{J}$ mass planet embedded into it will not get into the eccentricity state at all, see Fig. \[fig:FARGO-ecc\]. Planets orbiting the host star closer than 1AU, however, can induce stronger distortions, as shown in the following section.
Orbital distance of the planet
------------------------------
In this section we investigate the effect of the orbital distance of the planet to the strength of the spectral line distortion. One would initially expect that the spectral line distortions are stronger/weaker as the orbital distance of the planet decreases/increases. We recalculated the hydrodynamical simulations in “tight” and “wide” models, where the planets orbit at 0.5 and 2AU distances from the central star, respectively. In good agreement with the expectations, we found that the planet signature in the spectral line was strongly suppressed in the “wide” versions of models \#1-\#8, where $m_*=0.5\,M_{\sun}$ and $m_*=1\,M_{\sun}$. The reason for this is obvious: in wide systems the planet is orbiting in cold regions (the atmosphere temperature is varied between 200-350K at 2AU, depending on the stellar mass), where the atmospheric CO emission measured to the continuum is weak. As can be seen in Fig. \[fig:V1-0P10-profiles\](e), the “wide” version of model \#8 ($m_\mathrm{pl}=8\,M_\mathrm{J}$, $m_*=1\,M_{\sun}$, 2AU orbital distance) shows considerably weaker planet signatures in the CO line profile than the original 1AU model, see Fig. \[fig:V1-0P10-profiles\](a) for comparison. The distortion patterns appear at lower velocities owing to lower orbital velocities in $0.5\,M_{\odot}$ models. On the other hand, we found about the same strength of the planet signal in the “wide” version of model \#12 ($m_\mathrm{pl}=12\,M_\mathrm{J}$, $m_*=1.5\,M_{\sun}$, 2AU orbital distance) as in model \#8. This is expected, since in the latter case the luminosity of the host star is high enough to produce sufficient CO excitation even at 2AU.
The resulting line profile calculated in the “tight” version of the model \#8 ($m_\mathrm{pl}=8\,M_\mathrm{J}$, $m_*=1\,M_{\sun}$, 0.5AU orbital distance) is shown in Fig. \[fig:V1-0P10-profiles\](f). Because the giant planet orbits closer to the host star, where the disk atmosphere is heated to higher temperatures (the atmosphere temperature varies between 450-700K at 0.5AU, depending on the stellar mass), the emission emerging from the perturbed regions is stronger than in the original 1AU models. It is also evident that the distorted peaks are slightly shifted to higher velocities compared to the original 1AU models, because the orbital velocity of the gas parcels in dynamically perturbed orbits is higher in “tight” models. The disk is more eccentric on average in the latter case (Fig. \[fig:ecc-temp\]), but the strength of the line profile distortion is increasing as the planetary orbital distance decreases.
) than in the “wide” (see Fig. \[fig:V1-0P10-profiles\](e)) model, because the disk atmosphere is considerably hotter at 0.5AU than 2AU, where the disk is dynamically perturbed by the planet.[]{data-label="fig:ecc-temp"}](ecc-temp.pdf){width="\columnwidth"}
In order to demonstrate that smaller mass planets can also be detected in closer orbits we have calculated the line profile in a “tight” version of model \#2, $m_{\mathrm{pl}}=1.5\,M_\mathrm{J}$, $m_*=0.5\,M_{\sun}$, 0.5AU orbital distance. We find that one could observe roughly the same strength line profile distortions as in model \#8 ($m_\mathrm{pl}=8\,M_\mathrm{J}$, $m_*=1\,M_{\sun}$, 1AU orbital distance).
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Inner cavity size
-----------------
Now we turn our attention to the question under which circumstances planets can be detected orbiting farther away from the star (farther than 2AU for instance). In Sect. 3.2 we argued that there is a region around the star called inner cavity from which the CO is already depleted. In our simulations presented so far, we assumed that the radius of the inner cavity is fixed at 0.2AU irrespective of the stellar luminosity. On the other hand, the size of the inner cavity is presumably dependent on the stellar mass, and indeed this is confirmed by observations [@Akesonetal2005]. As we mentioned in Sect. 3.2, the size of the inner cavity has a significant influence on the overall line-to-continuum ratio. In disks dynamically perturbed by planets orbiting at 1AU and with a significant amount of CO lying inside 0.2AU, the majority of CO flux is emerging from the unperturbed innermost regions, which eventually could smear out the planet signatures. On the contrary, if the size of the inner cavity is larger, the contribution of the perturbed regions to the total CO flux becomes stronger. In Fig. \[fig:V1-0P10-profiles\](g) we present the line profiles obtained in model \#8, in which the inner cavity is increased to 0.4AU in size, while the $8\,M_\mathrm{J}$ mass planet was still orbiting at 1AU. As can be seen, the overall line-to-continuum ratio is decreased due to the absence of hot gas, but the level of the asymmetric pattern profile is more apparent than in models with an inner cavity extending only to 0.2AU, see Fig. \[fig:V1-0P10-profiles\](a) for comparison. For larger inner cavities the continuum level of the disk interior and inner rim are decreased too, resulting in a moderate weakening of the line-to-continuum ratio and slight strengthening of planet signal. In this case the detection possibility of a planet orbiting at larger distances ($>2\,\mathrm{AU}$) is growing.
Disk geometry
-------------
It is well known that many TTauri sources present flatter than $\lambda F_\lambda \sim \lambda^{-4/3}$ SEDs. One attempt to explain this SED flattening was to assume that the disk is flaring [@KenyonHartmann1987], resulting in a geometrically thicker disk atmosphere $H(R)\sim R^{\gamma+1}$, where $\gamma$ is the flaring index. In the approach of @ChiangGoldreich1997 the flaring index is $\gamma=2/7$. In order to investigate the effect of disk geometry on the strength of the line profile distortions we recalculated several models using flat thermal disk approximation (the hydrodynamical inputs were the same). An example of the flat version of model \#8 ($m_\mathrm{pl}=8\,M_\mathrm{J}$, $m_{*}=1\,M_{\sun}$, 1AU orbital distance) is shown in Fig. \[fig:V1-0P10-profiles\](h). Note that in order to have the same peak line-to-continuum ratio in flat as in flared models, the amount of emitting CO is arbitrary increased by decreasing the dust-to-gas ratio to $1.2\times 10^{-3}$ and keeping the disk mass unchanged. The reason for this is that although the atmospheric temperature does not differ in flared and flat disk models (see the atmospheric temperature given by Eq. (\[eq:temp-surf\])), the atmospheric surface density of CO is substantially smaller in flat than flared disk models (see the dependence of surface density given by Eq. (\[eq:dens-surf\]) on the grazing angle, given by the first term of Eq. (\[eq:grazing-angle-flaring\]) in flat disk models). It is evident that the total line-to-continuum ratio is higher in the flat than in the flared disk model, but the line width at the foot of the line profile does not change. According to our calculations the line profile distortions in flat models compared to the flared one are less significant in “wide” systems where planets orbit at larger distances. This can be explained by the effect of flaring getting stronger with increasing distance to the host star.
Observational considerations
----------------------------
, but superimposed with an artificial noise expected for one hour of integration on $M_\mathrm{mag}=6$ brightness source with VLT/CRIRES.[]{data-label="fig:noise-Ms1-Mb8-V1-0P10"}](N-Ms1-Mb8-0017V1-0P10.pdf "fig:"){width="\columnwidth"}\
, but superimposed with an artificial noise expected for one hour of integration on $M_\mathrm{mag}=6$ brightness source with VLT/CRIRES.[]{data-label="fig:noise-Ms1-Mb8-V1-0P10"}](V1-0P10_trm.pdf "fig:"){width="\columnwidth"}
![Goodness-of-fits of distorted line profiles contaminated with artificial noise obtained in planet-free (dashed lines) and giant planet-bearing disk (solid lines) models versus the integration time for $20^{\circ}$, $40^{\circ}$ and $60^{\circ}$ inclinations, represented by blue, green and red colors, respectively. It is visible that for a reasonable signal-to-noise ratio, the planet-free models provide always fits with less confidence than the giant planet models, i.e the $\chi^2$ is always larger. Moreover, the smaller the inclination angle, the larger the required minimal integration time ($t_\mathrm{int}>3187\,\mathrm{sec}$, $1909\,\mathrm{sec}$ and $2271\,\mathrm{sec}$ for the $20^{\circ}$, $40^{\circ}$ and $60^{\circ}$ inclinations, respectively) to distinguish the planet-free and planet-bearing disk models.[]{data-label="fig:noise-goodness"}](noise_fit.pdf){width="\columnwidth"}
To observe the planet-induced distortions of the CO line profiles that we simulated, a spectroscopic facility is required that meets two basic demands: it needs sufficient spectral resolution to properly resolve the sub-structure in the line profile and sufficient sensitivity to bring out the low-contrast line profile distortions. In this section, we investigate the observability of the modeled effect in the context of contemporary facilities, and give a brief outlook into the ELT era. We present a quantitative example for the *Cryogenic High Resolution Echelle Spectrograph* (CRIRES) at the VLT (Kaeufl et al. 2004), which is currently the most powerful high-resolution spectrograph covering the M-band around $4.8\,\mathrm{\mu m}$, capable of observing CO ro-vibrational spectra.
The main limiting factor for ground-based thermal infrared observations is the Earth atmosphere, which strongly absorbs the radiation from astronomical sources and causes a high thermal background. Particularly strong telluric absorption and emission is present at the wavelengths of low-excitation CO transitions. Within $\sim 10\,\mathrm{km/s}$ from the line center the telluric transmission is so low and the background emission so high that these spectral regions are not accessible. However, depending on the location of the source with respect to the ecliptic, the CO features are Doppler shifted by up to $\pm 30\,\mathrm{km/s}$ due to the Earth’s orbital motion, making the whole CO line accessible over the course of several months, in principle. Note though that we can never measure the entire CO line profile simultaneously, there will always be an approximately $20\,\mathrm{km/s}$ wide “gap” in our spectral coverage at any observing epoch.
In order to test whether we would be able to detect the planet-induced permanent profile asymmetries with currently available instrumentation, we have simulated a VLT/CRIRES observation of model \#8. We have assumed the source to have a brightness of $M=6\,\mathrm{mag}$, and calculated the achieved $SNR$ as a function of wavelength using the VLT/CRIRES exposure-time calculator[^4]. This calculation takes into account the telluric absorption and emission, the quality of the adaptive optics correction, the system throughput and detector characteristics. We note that our simulation only considers the signal-to-noise ratio, and assumes that systematic effects due to, e.g., time-variable telluric absorption, can be well calibrated. The achieved $SNR$ can be scaled according to $$SNR \propto 10^{−0.4M} \sqrt{T_{\rm{int}}},$$ where $M$ denotes the M-band magnitude and $T_{\rm{int}}$ the integration time. In Fig. \[fig:noise-Ms1-Mb8-V1-0P10\] we show the emerging V=1-0P(10) line profiles for model \#8 (upper figure) and the atmospheric transmission (lower figure) for an assumed integration time of 1 hour on a source of $M=6\,\mathrm{mag}$. The model spectrum has been convolved with the 3km/s CRIRES instrumental resolution. We have assumed zero relative velocity between the source and the Earth at the time of the observation. The effects of the telluric CO line are clearly seen around the line center, where no meaningful measurement can be obtained. The line wings, where the effects of the planet are most prominent, can be well measured.
In order to quantify the significance with which a planetary signal (the most prominent permanent asymmetric pattern) may be detected, we have tried to fit the modeled observation with both planet-free and planet-bearing models. In Fig. \[fig:noise-goodness\] we show the goodness-of-fits (reduced $\chi^2$) as a function of integration time for both cases (with dashed line for planet-free and solid line for planet-bearing disk models) for $20^\circ$, $40^\circ$, and $60^\circ$ inclination angles. Above a certain integration time (presented by vertical lines) the planet-bearing models yield a more than 50% better reduced $\chi^2$ fit to the simulated data than those of the planet-free models. The asymmetric pattern induced by an embedded planet increase and thus become easier to detect with disk inclination.
Because the fundamental band ro-vibration R(0), R(1), R(12), R(13), R(19)-(21), R(25), R(26) and R(30)-(33), P(6-12) and P(22-32) CO lines are not blended by the higher vibrational lines[^5], the line profiles can be averaged, using appropriate scaling, resulting in an increased signal-to-noise ratio. Note that the unblended lines with higher $J > 25$ or lower $J < 2$ rotational quantum numbers are too weak for averaging: using them would introduce only additional noise to the averaged profiles. Also note that, since the higher vibrational levels are not excited at all in disk atmospheres similar to that used in our models, the blended lines can also be used for averaging, resulting in an even more increased signal-to-noise ratio.
We conclude that in the confines of our model, the permanent asymmetric signals produced by an $8\,M_\mathrm{J}$ giant planet orbiting at $\sim 1\,\mathrm{AU}$ embedded in the disk of a TTauri star of brightness $M=6\,\mathrm{mag}$ would be detectable with VLT/CRIRES with an integration time of 1hour. Moreover, because the variable component at the top of the permanent asymmetry has a width of $\sim 5-10\,\mathrm{km/s}$, exceeding the CRIRES $\sim 3\,\mathrm{km/s}$ resolution, and $\sim 10$% magnitude, there is a certain possibility to strengthen its planetary origin.
With the next generation of giant, ground-base telescopes our sensitivity to low-contrast features in infrared spectra will increase greatly. We have used the ESO exposure-time calculator for the ELT (version 2.14) to simulate an observation with the proposed first generation infrared instrument *Mid-infrared E-ELT Imager and Spectrograph* (METIS) [@Brandletal2008]. The exposure-time calculator necessarily incorporates planned telescope and instrument specifications, because this facility is yet to be built. We find that ELT/METIS yields an increase in sensitivity of a factor of $\sim$30 compared to VLT/CRIRES. Thus also sources that are substantially fainter or have weaker spectral features than those modeled here in the context of VLT/CRIRES are within reach of the observational facilities of the next decade.
Conclusions
===========
We have shown that our semi-analytical 2D double-layer thermal disk model (in which the disk layers are heated mainly by stellar irradiation, and the emission of the disk inner rim is accounted for) is capable to reproduce the double-peaked Keplerian line profiles in CO fundamental ro-vibrational band in young (about $2.5\,\mathrm{Myr}$ old) TTauri-type protoplanetary disks assuming canonical dust and gas properties. However, note that several TTauri stars show centrally peaked symmetric profiles instead, which can be attributed to an enhanced CO flux formed in distant regions (up to 10AU), presumably above the disk atmosphere. This can be explained by non-LTE heating, such as UV fluorescence [@Krotkovetal1980], or dust gas temperature decoupling [@Glassgoldetal2004; @KampDullemond2004], resulting in highly excited CO emissions. These effects were not taken into account in our calculations. Thus the gas temperature at disk atmosphere provided by our thermal model is presumably somewhat lower than the realistic one. Our predictions, however, demonstrate a “the worst case” because the planet signal is getting stronger with increasing atmospheric temperature. It is revealed that significant line profile distortions appear in the CO fundamental ro-vibrational band caused by a giant planet. As we have demonstrated, a giant planet with a mass of at least $1\,M_\mathrm{J}$ can perturb the disk sufficiently to be detected by permanent asymmetry in line profiles and its short timescale variations. The permanent asymmetry in line profiles can be interpreted by the disk being in an eccentric state only in the gap, but the short timescale variability is certainly connected to the local dynamical perturbations of the orbiting planet. Depending on the signal-to-noise ratio and the resolution of the spectra, even lower-mass giant planets can be detected revolving on close orbits in disks observed with high-inclination angles and with large size inner cavities. The summary of our findings is:
1. [The dynamical perturbation induced by a giant planet significantly distorts the CO ro-vibrational fundamental emission line profiles.]{}
2. [The line profile distortions are more apparent for larger inclination angles.]{}
3. [The main component of line profile distortions is a permanent asymmetry. Its position in the line profile depends on the stellar mass and orbital distance of the planet. The permanent asymmetry shape depends on the orientation of the gas elliptic orbit with respect to the line of sight.]{}
4. [The line profiles are changing with time, correlated to the orbital phase of the giant planet. The timescale of the variation is on the order of weeks, depending on the orbital period of the planet, i.e, the orbital distance and the mass of the host star. The magnitude of variable component is $\sim10\%$, its width is $\sim 10\,\mathrm{km/s}$, depending on the inclination angle.]{}
5. [The planet signal becomes stronger with increasing planetary mass.]{}
6. [The planet signal strengthens/weakens with increasing/decreasing mass of the host star, neglecting the effect of the stellar luminosity on the size of the inner cavity]{}
7. [The planet signal strengthens/weakens with decreasing/increasing orbital distance of the planet.]{}
8. [The size of the inner cavity has a strong influence on the giant planet observability. If the size of the inner cavity is substantially smaller than 0.2AU, we can detect a giant planet only at close (about 0.5AU) orbit. On the contrary, if the cavity is larger (0.4AU), a distant planet (in 2AU orbital distance) can also be detected.]{}
9. [The influence of disk geometry on the planet signal is modest. With a decreasing flaring index the strength of planet signal does not change although the line profiles are strengthened, except in “wide” flat models where the planet signal substantially suppressed.]{}
10. [The lowest mass of giant planet orbiting at 1AU that still can be detected is $0.5\,M_\mathrm{J}$ for a $0.5\,M_{\sun}$ star in the confines of our models.]{}
11. [By high-resolution near-IR spectroscopic monitoring with VLT/CRIRES, giant planets with $\geq 1\,M_\mathrm{J}$ mass orbiting within 0.2-3AU in a young disks may be detectable, if other phenomena do not confuse, mimic, or obscure the planet signature. The level of confidence is growing with increasing inclination angle.]{}
In the light of our findings, we propose to observe TTauri stars that shows an appreciable amount of CO in emission with high-resolution near-IR spectrograph to search for giant planet companions. The asymmetric and time-varying double-peaked line profiles can be explained by strongly non-circularly Keplerian gas flow in the disk caused by the giant planets, as was presented. Although higher $V\geq 2$ vibrational states of CO are not excited in our models, overlaps can occur, resulting in asymmetric line profile. To avoid this, one should consider only non-blended lines for profile averaging such as P(6)-P(12), and R(0), R(1), R(12), R(13), R(19)-(21), R(25), R(26) and R(30)-(33). Another possible source of permanent line profile distortions could be a large companion deforming the circumprimary disk to fully eccentric in mid-separation binaries (e.g. @Kleyetal2008; Regaly et al. in prep.). Last but not least, the distorted PSF, caused by tracking errors of the telescope or unstable active optics during an exposure, can induce artificial signals. A real signal from a bipolar structure will change sign when observed using antiparallel slit position angles whereas the artificial signal will not. Thus artificial signatures can be successfully identified with the comparison of the spectra of disks obtained at two antiparallel slit position angles, e.g. $0^\circ$ and $180^\circ$ [@Branniganetal2006]. By detecting short time-scale (weeks to months) variations of the asymmetric profiles with similar patterns as presented in Fig. \[fig:variation-Ms1-Mb8-V1-0P10\], we could infer the presence of a giant planet in formation.
The proposed instrument METIS for the ELT [@Brandletal2008], providing high-resolution ($\lambda/\Delta\lambda=10^5$) infrared spectroscopy with an increase in sensitivity of a factor of $\sim 30$ to that of VLT/CRIRES will give the opportunity to even decrease more the mass and brightness limits of possible giant planets. If we revealed for instance the distance of the observed planets to the star as a function of the age of the star-disk system, we would be able to constrain planetary migration models. Moreover, determining the youngest star host planet-bearing disks could also provide constraints on planet-formation models.
Although our assumptions are reasonable from the point of view of finding a simple model of the signal of dynamical perturbations, it can be developed further of course. For example, if the gap is cleared by the planet to a fraction of the surrounding density, the disk is not directly irradiated, but is shadowed by the inner wall of the gap [@Varniereetal2006]. This mild heating causes a temperature drop at the inner gap wall, which results in a depressed CO flux. On the other hand, the outer edge of the gap is directly illuminated by the star, resulting in a slight temperature increase, which produces stronger CO emission from the regions where the eccentricity reaches its maximum (see e.g. Fig.\[fig:FARGO-ecc\]). According to our results the gap is strongly non-axisymmetric, thus 3D radiative transfer using 3D density structure needed to calculate the real temperature profile in perturbed disks. Moreover, as the gap is depleted in large grains because of the trapping of dust grains larger than $\sim 10\,\mathrm{\mu m}$ [@Riceetal2006], the CO flux normalized to the continuum is strengthened. The gas and dust can be thermally uncoupled (see e.g., @Glassgoldetal2004 [@KampDullemond2004; @Woitkeetal2009]) in the tenuous gap, which may result in CO being overheated to the dust. These effects could cause additional permanent line profile distortions due to the elliptic shape of the gap seen in density distribution (Fig.\[fig:FARGO-output\]). If we consider an excess emission originating from the planetary accretion flow suggested by @ClarkeArmitage2003, or the density waves in disk surface caused by the orbiting planet, it would cause a varying planet signal. To these caveats, one might also consider the disk turbulence (e.g., due to MRI effects), which may add random asymmetries and “blobbiness” to the line profiles. We will investigate these effects in a forthcoming paper using a more sophisticated thermal disk model.
CO exists inside the dust sublimation radius in some TTauri stars [@Carr2007] or HerbigAe/Be stars [@Brittainetal2009], where the disk is optically thin in the absence of dust. One possibility to excite CO in the optically thin inner disk close to the stellar surface is the UV fluorescence, which is far from LTE [@Krotkovetal1980] applied in our calculations. Depending on the efficiency of X-ray heating the gas temperature can be as high as $\sim 4000-5000\,\mathrm{K}$. Somewhat below this temperature, where CO still exists, the high-excitation vibrational levels of CO such as $V\geq 2$ are populated, resulting in significant $V=2\rightarrow 1$, $V=3\rightarrow 2$, etc. ro-vibrational lines. According to @Skrutskieetal1990, the inner cavity in transitional disk may be formed by a close giant planet. Indeed, @LubowDAngelo2006 found that an $1-5M_\mathrm{J}$ mass planet significantly lowers the accretion of dust and gas to 10%-25% of the accretion rate outside the gap. We expect that the planetary signal forming in the optically thin inner disk is also detectable, which is strongly supported by the revealed line profile distortion strengthening with decreasing orbital distance of the planet. The UV pumping requires considerable UV flux from the central star, which is a characteristic of Herbig Ae/Be stars rather than smaller TTauri stars. Considering that the CO may be excited up to to 10AU by UV fluorescence [@Brittainetal2007; @Brittainetal2009] in HebigAe/Be stars, planet orbiting at larger distances than 2AU may be also detectable by CO ro-vibrational line profile distortions.
Spectroscopy of TTauri stars shows emission of molecules such as $\mathrm{H_2O}$, $\mathrm{OH}$, $\mathrm{HCN}$, $\mathrm{C_2H_2}$, and $\mathrm{CO_2}$, as well. Nevertheless CO is more abundant than these molecules by $\sim 10$ times, only $\mathrm{H_2O}$ could reach the abundance of CO, predicted by recent models that calculate the vertical chemical structure of the gas in disk atmosphere (e.g. @Glassgoldetal2004, @KampDullemond2004, and @Woitkeetal2009). Indeed in some cases, like AATau [@CarrNajita2008], and AS205A and DRTau [@Salyketal2008], the rotational transitions of $\mathrm{H_2O}$ dominate the mid-infrared ($10-20\,\mathrm{\mu m}$) spectra, suggesting that $\mathrm{H_2O}$ is abundant in disk atmospheres. The strong water emission could be the consequence of turbulent mixing that carries molecules from the disk midplane, where they are abundant, to the disk atmosphere [@CarrNajita2008], or the effects of an enhanced mechanical heating of the atmosphere [@Glassgoldetal2009]. While the $\mathrm{H_2O}$ ro-vibrational lines probe the inner regions of disk out to radii $\geq 2\,\mathrm{AU}$, the rotational lines are produced between radii 10-100AU [@Meijerinketal2008]. The rotational and ro-vibrational line profiles of water are also subject to distortions caused by the dynamical perturbations of a giant planet. Thus it is reasonable to search for signature of giant planets in the high-resolution spectra of $\mathrm{H_2O}$ in the mid-infrared band (for planets orbiting int the inner disk), and rotational spectra of $\mathrm{H_2O}$ in the far-infrared band (for planets orbiting in the outer disk), as well. Because $\mathrm{H_2O}$ is heated by stellar X-rays and sub-thermally populated beyond 0.3AU, X-ray-heating and non-LTE level population treatment is needed to calculate water lines (e.g., @Meijerinketal2008 [@Kampetal2010]).
Finally, we point out some noticeable resemblance to our findings revealed in observed CO line profiles. The V836Tau transitional disk shows strongly distorted $4.7\,\mathrm{\mu m}$ CO ro-vibrational line profile presented by @Najitaetal2008. The averaged CO line profiles shows very similar features to our results. According to @Najitaetal2008 the possibility of a massive planet is also restricted by current limits on the stellar radial velocity of V836Tau, which constrain the mass of a companion within $0.4-1\,\mathrm{AU}$ to $5-10\,M_\mathrm{J}$. Because the disk extents to a limited range of radii ($\sim 0.05-0.4\,\mathrm{AU}$), the dynamical perturbation of a close orbiting planet with a mass smaller than $5\,M_\mathrm{J}$ presumably could cause the observed line profile distortions. Note that radial velocity measurements indicate that no planet is present larger than $1-2\,M_\mathrm{J}$ closer than $\sim 0.5\,\mathrm{AU}$ [@Pratoetal2008]. The transitional disks that encompass the young sources SR21 and HD135344B have been observed by VLT/CRIRES [@Pontoppidanetal2008] and they already show clear line profile asymmetries in CO fundamental ro-vibrational band. Our scenario of a planet is a reasonable explanation of the asymmetry. The presence of a planet was also proposed by @Gradyetal2009 for HD135344B and @Eisneretal2009 for SR21, based on SED and visibility data, respectively. EXLupi (prototype of EXor type young variable stars) showed remarkable line profile variations during its 2008 outburst (Goto et al. 2010, in prep.). Goto et al. revealed that the CO emission has a spatially multiple origin. The quiescent component forms in the outer optically thick disk, while the outburst component in the inner optically thin gas disk, where the higher vibrational levels of CO are presumably excited by UV pumping. These high-excitation $V=2\rightarrow 1$ and $V=3\rightarrow 2$ lines are double peaked, strongly asymmetric and variable on a short (weekly) time scale. Thus for example, by measuring short timescale (on the order of weeks or less) periodic variations in the CO ro-vibrational spectra of these sources, by mid-IR monitoring observations would eventually indirectly detect an embedded Jupiter-like planet in birth.
It is a pleasure to acknowledge helpful discussions with Ewine van Dischoeck, Klaus Pontoppidan and Attila Juhász. This research has bee supported in part by DAAD-PPP mobility grant P-MÖB/841/ and “Lendület” Young Researcher Program of the HAS. We are grateful to L. Kiss who helped us to substantially clarify the text. We also thank the anonymous referee for thoughtful comments that helped to significantly improve the quality of the paper.
Temperature distribution in the disk {#apx:temp-dist}
====================================
For the sake of completeness we present here the derivation of the dust temperature distributions in the disk interior and atmosphere according to @ChiangGoldreich1997. Below we use the local thermodynamical equilibrium (LTE) assumption everywhere. The stellar flux in the disk atmosphere at a distance $R$ from the stellar surface is $$F_{*}(R)\simeq\frac{\delta(R)}{2}\left(\frac{R_{*}}{R}\right)^{2}\sigma T_{*}^4,$$ where $\delta(R)$ is the grazing angle of the incident irradiation, $R_{*}$ and $T_{*}$ are the stellar radius and surface temperature, respectively, and $\sigma$ is the Stefan–Boltzmann constant. Here we assumed that only half of the stellar surface is visible from a given point at a distance $R$ from the star in the disk atmosphere. According to @ChiangGoldreich1997 the grazing angle can be given by $$\label{eq:grazing-angle-flaring}
\delta(R)=\frac{2}{5}\left(\frac{R_{*}}{R}\right)+\frac{8}{7}\left(\frac{T_*}{T_\mathrm{g}}\right)^{4/7}\left(\frac{R_*}{R}\right)^{-2/7},$$ in a flared disk assumed to be in hydrostatic equilibrium. In Eq. (\[eq:grazing-angle-flaring\]) $T_\mathrm{g}$ is the gravitational temperature at which the thermal energy of gas parcel balances the gravitational energy at the stellar surface, which can be given by $$T_\mathrm{g}=\frac{GM_*m_\mathrm{H}}{kR_*},$$ where $m_\mathrm{H}$ is the mean molecular weight of the gas, $G$ and $k$ is the gravitational constant and Boltzmann constant, respectively. Note that for a non-flared, i.e flat disk the grazing angle is defined by only the first term of Eq. (\[eq:grazing-angle-flaring\]).
By definition the optical depth of the disk atmosphere along the incident stellar irradiation is $\tau_\mathrm{V}=1$ in the optical wavelengths. The optical depth of the atmosphere at near-IR heated to $T_\mathrm{atm,irr}(R)$ perpendicular to the disk plane is $\tau_\mathrm{V}\delta(R)\epsilon_\mathrm{atm}(R)$, where $\epsilon_\mathrm{atm}(R)=(T_\mathrm{atm,irr}(R)/T_\mathrm{*})^\beta$ is the dust emissivity, and $\beta$ is the power law index of the absorption coefficient of the dust, see Appendix \[apx:emissivity\] for details. The total flux emitted inward and upward by the optical thin atmosphere can be given by $$F_\mathrm{atm}(R)=2\delta(R)\epsilon_\mathrm{atm}(R)\sigma T_\mathrm{atm,irr}(R)^4.$$ In LTE the absorbed flux of atmosphere equals to the emitted one ($F_{*}(R)=F_\mathrm{atm,irr}(R)$), thus the disk atmosphere is heated to the temperature $$\label{eq:temp-surf}
T_\mathrm{atm,irr}(R)=\left(\frac{1}{4}\right)^{1/(4+\beta)}\left(\frac{R_{*}}{R}\right)^{2/(4+\beta)}T_{*}$$ by stellar irradiation.
Assuming that the disk interior with a temperature $T_\mathrm{int,irr}(R)$ is optically thick to its radiation everywhere in our computational domain,[^6] the flux $F_\mathrm{int,irr}(R)$ emitted by the disk interior is $$F_\mathrm{int,irr}(R)=\sigma T_\mathrm{int,irr}(R)^{4}.$$ Considering that the absorbed stellar flux will be re-emitted by the disk atmosphere upward and downward only half of this flux heats the interior, i.e $0.5F_\mathrm{atm}(R)=0.5F_{*}(R)=F_\mathrm{int,irr}(R)$. Accordingly, the temperature of the disk interior set by the stellar irradiation can be given by $$\label{eq:temp-int-irr}
T_\mathrm{int,irr}(R)=\left(\frac{\delta(R)}{4}\right)^{1/4}\left(\frac{R_{*}}{R}\right)^{1/2}T_{*}.$$
Owing to the accretion process a significant amount of gravitational potential energy has to be dissipated by a viscous processes. According to @Lynden-BellPringle1974, in a steady state disk[^7] with a constant accretion rate $\dot M$, the flux $F_\mathrm{acc}(R)$ released by the disk mid-plane due to the change of potential energy can be given by $$\label{eq:acc-flux}
F_\mathrm{acc}(R)=\frac{3GM_{*}\dot{M}}{8\pi}\left(1-\left(\frac{R_\mathrm{*}}{R}\right)^{1/2}\right) R^{-3}.$$ In this way the disk mid-plane is heated to the temperature $$\label{eq:acc-temp}
T_\mathrm{acc}(R)=\left[\frac{3GM_{*}\dot{M}}{8\pi\sigma}\left(1-\left(\frac{R_\mathrm{*}}{R}\right)^{1/2}\right)\right]^{1/4}R^{-3/4}.$$ Here we have to note that only half of the total accretion power is involved in Eq. (\[eq:acc-flux\]), the remaining is stored in the kinetic energy of orbiting gas. This energy should be radiated away by the disk boundary layer [@Pophametal1993; @Pophametal1995] or by the accreting material in the funnel flow formed along the magnetic field lines in magnetospheric accretion model, see @Hartmannetal1994 [@Bouvieretal2007]. Because the disk boundary layer and the funnel flows are confined into such small volumes that the gas temperature reaches about 10000K emitting in the UV band, its radiation is not taken into account.
Now let us take into account the heating due to viscous dissipation in disk interior using the superposition principle, i.e the radiation at a given location is the sum of the radiations corresponding to different heating sources. To determine the disk interior temperature $T_\mathrm{irr,acc}(R)$ caused by the viscous dissipation first assume that the flux emitted by the optical thick disk interior is $$F_\mathrm{int,acc}(R)=\sigma T_\mathrm{int,acc}(R)^4.$$ Because the disk mid-plane radiates the accretion flux into two directions $0.5F_\mathrm{acc}(R)=F_\mathrm{int,acc}(R)$, the disk interior is heated to $$\label{eq:temp-acc}
T_\mathrm{int,acc}(R)=(1/2)^{1/4}T_\mathrm{acc}(R)$$ by accretion. Taking into account the heating of the disk interior by irradiation of atmosphere and viscous dissipation together (simply summing the radiation fluxes), the resulting temperature of disk interior is $$\label{eq:temp-int}
T_\mathrm{int}(R)=\left(T_\mathrm{int,irr}(R)^4+T_\mathrm{int,acc}(R)^4\right)^{1/4}.$$
Emission from the disk inner rim {#apx:sinner-rim}
================================
To determine the temperature profile of the disk interior close to the disk inner edge, where the simple double layer assumption cannot be applied, we first assume that the rim is a perfect vertical wall. Any given optically thick vertical slab with a thickness of $dR$ at $R+dR$ distance from the central star is irradiated by the neighboring hotter slab located at $R$. Assuming that half of its radiation is received by the one at $R+dR$, the emitted and irradiated fluxes are in equilibrium in LTE, i.e. $$\frac{1}{2}\sigma T_\mathrm{rim}(R)^4=\sigma\epsilon(R) T_\mathrm{rim}(R+dR)^4,$$ where $\epsilon(R)=(T_\mathrm{rim}(R+dR)/T_\mathrm{rim}(R))^\beta$ is the emissivity of the slab at $R+dR$, see in Appendix \[apx:emissivity\]. Note that here we neglect the irradiation of the neighboring slab at $R+2dR$ with a lower temperature than the slab at $R+dR$. To calculate $T_\mathrm{rim}(R)$, we first approximate $T_\mathrm{rim}(R+dR)$ with $T_\mathrm{rim}(R)+T_\mathrm{rim}^\prime (R)dR$. This results in an ODE for $T(R)$, which has the solution $$\label{eq:temp-rim}
T_\mathrm{rim}(R)=T_\mathrm{rim}(R_0)\exp\left[-\left(1-\left(\frac{1}{2}\right)^{1/(4+\beta)}\right)\left (R-R_0 \right)\right],$$ where $R_0$ is the radius of the disk inner edge. To take into consideration the additional irradiation of the rim at the opposite side, we assume that the temperature at the innermost slab is $T_\mathrm{rim}(R_0)=qT_\mathrm{int,irr}(R_0)$, where $q\simeq1.2$ according to results of our simulations done by 2D RADMC [@DullemondDominik2004]. Thus incorporating the additional heating by the disk rim, the disk interior temperature, previously given by Eq. (\[eq:temp-int\]), can be given by $$\label{eq:temp-int-mod}
T_\mathrm{int}(R)=\left(T_\mathrm{int,irr}(R)^4+T_\mathrm{int,acc}(R)^4+T_\mathrm{rim}(R)^4\right)^{1/4}.$$ where the superposition rule is applied.
Dust emissivity {#apx:emissivity}
===============
In this section the derivation of dust emissivity at a specific temperature is given, assuming optically thick environment. Let us define the dust emissivity at a temperature $T_\mathrm{dust}$ heated by a blackbody of the temperature $T_\mathrm{irr}$ as the ratio of the Rosseland mean opacities $$\label{eq:emissivity}
\epsilon_\mathrm{dust}=\frac{\left<\kappa(T_\mathrm{dust})\right>_\mathrm{R}}{\left<\kappa(T_\mathrm{irr})\right>_\mathrm{R}}.$$ The Rosseland mean opacity is defined by $$\label{eq:R-mean-opac}
\left<\kappa(T)\right>_\mathrm{R}=\frac{\int_0^\infty dB(\nu,T)/dTd\nu}{\int_0^\infty dB(\nu,T)/dT\kappa(\nu,T)^{-1}d\nu},$$ where $B(\nu,T)$ is the Planck function. The dust used in our model generates the $\kappa(\nu)=\kappa_0 (\nu/\nu_0)^{\beta}$ opacity law, where $\beta>0$. Note that wee use $\beta=1$ throughout all our models [@Rodmannetal2006]. Following the calculations of @StahlerPalla2005 in Appendix G, we assume that the dust absorption coefficient can be given by $$\kappa(\nu,T)=\kappa_0\left(\frac{kT}{h\nu_0}\right)^{\beta} x^{\beta},$$ where $k$ and $h$ are Boltzmann and Planck constants, respectively, and $x=h\nu/kT$. Supposing that $$\partial B(\nu,T)/\partial Td\nu=Af(x)dx,$$ where A is an appropriate dimensional constant, and substituting this into the definition of Rosseland mean opacity, Eq. (\[eq:R-mean-opac\]), leads to $$\label{eq:Rosseland-mo}
\left<\kappa(T)\right>_\mathrm{R}=\kappa_0\left(\frac{kT}{h\nu_0}\right)^{\beta}\frac{\int f(x)dx}{\int x^{-\beta}f(x)dx}.$$ Because the quotient of integrals is a pure number, the Rosseland mean opacity is proportional to $T^\beta$. Applying \[eq:emissivity\] and \[eq:Rosseland-mo\] we find that the dust emissivity can be given by $$\epsilon_\mathrm{dust}=\left(\frac{T_\mathrm{dust}}{T_\mathrm{irr}}\right)^\beta.$$
Optical depth of the disk atmosphere
====================================
Assuming that the disk is observed with an inclination angle $i$, the monochromatic optical depth of the disk atmosphere is the sum of optical depths of dust and gas along the line of sight, i.e.: $$\label{eq:tau_nu_1}
\tau({\nu,R,\phi},i)=\frac{1}{\cos(i)}\left( \kappa_\mathrm{d}(\nu)\Sigma_\mathrm{d}(R)+\kappa_\mathrm{g}(\nu,R,\phi,i)\Sigma_{g}(R)\right),$$ where $\kappa_\mathrm{d}(\nu)$ and $\kappa_\mathrm{g}(\nu,R,\phi,i)$ are the dust and gas opacity at frequency $\nu$, while $\Sigma_\mathrm{d}(R)$ and $\Sigma_\mathrm{g}(R)$ are the dust and gas surface densities in the disk atmosphere. Note that because the dust opacity ($\kappa_\mathrm{d}$) is taken to be uniform throughout the disk, the optical depth of the disk atmosphere depends on $R$ and $\phi$ via the gas opacity characterized by the temperature distribution and the line-of-sight component of the orbital velocity of gas parcels. By definition the optical depth at the bottom of the disk atmosphere is unity at optical wavelengths along the stellar irradiation. Assuming that the mere opacity source at visual wavelengths is the dust, the dust surface-density of disk atmosphere is $$\label{eq:dens-surf}
\Sigma_\mathrm{d}(R)=\frac{\delta(R)}{\kappa_\mathrm{V}},$$ where $\kappa_\mathrm{V}$ is the overall opacity of dust at visual wavelengths. Assuming that the gas- and dust-mass ratio to the total mass ($X_\mathrm{g}$ and $X_\mathrm{d}$, respectively) is constant throughout the disk (i.e. there are no vertical or radial variations in the mass ratios), the surface density of gas in the disk atmosphere is $$\label{eq:dens-gas}
\Sigma_\mathrm{g}(R)=X_\mathrm{g}\Sigma(R)=\frac{X_\mathrm{g}}{X_\mathrm{d}}\Sigma_\mathrm{d}(R),$$ using the dust surface-density Eq. (\[eq:dens-surf\]). Thus, the optical depth in the disk atmosphere along the line of sight can be given by $$\label{eq:tau_nu_2}
\tau(\nu,R,\phi,i)=\frac{1}{\cos(i)}\left(\kappa_\mathrm{d}(\nu)\frac{\delta(R)}{\kappa_\mathrm{V}}+\frac{X_\mathrm{g}}{X_\mathrm{d}}\frac{\delta(R)}{\kappa_\mathrm{V}}\kappa_\mathrm{g}(\nu,R,\phi,i)\right),$$ where we used Eqs. (\[eq:tau\_nu\_1\]-\[eq:dens-gas\]).
Monochromatic opacities
=======================
In LTE the monochromatic opacity of the emitting gas at frequency $\nu$ with a molecular mass of $m_\mathrm{CO}$ due to transitions between states u$\rightarrow$l can be given by $$\begin{aligned}
\label{eq:gas-kappa}
\kappa_\mathrm{g}({\nu},R,\phi,i)&=&\frac{1}{m_\mathrm{CO}8\pi}\frac{1}{Q(T_\mathrm{atm}(R))}\left(\frac{c}{\nu_{0}}\right)^{2}A_\mathrm{ul} g_\mathrm{u} \nonumber \\
&\times&\left( \exp\left[-\frac{E_\mathrm{l}}{k T_\mathrm{atm}(R)}\right] - \exp\left[-\frac{E_\mathrm{u}}{k T_\mathrm{atm}(R)}\right]\right)\\
&\times&\Phi(\nu,R,\phi,i)\end{aligned}$$ where $\nu_{0}$ is the fundamental frequency of the transition, $Q(T_\mathrm{atm}(R))$ is the partition sum at the gas temperature $T_\mathrm{atm}(R)$, $A_\mathrm{ul}$ and $g_\mathrm{u}$ are the probability of transition (i.e the Einstein $A$ coefficient of the given transition) and the statistical weight of the upper state, respectively, while $c$ is the the light speed. The partition sum can be given by $$Q(T_\mathrm{atm}(R))=\sum_\mathrm{i}g_\mathrm{i}\exp\left[-\frac{E_\mathrm{i}}{k T_\mathrm{atm}(R)}\right],$$ where $g_\mathrm{i}$ and $E_\mathrm{i}$ are the statistical weight and energy level of the $i$th excitation state. In Eq. (\[eq:gas-kappa\]) the $\Phi(\nu,R,\phi,i)$ is the local intrinsic line profile originating by the natural thermal and the local turbulent broadening acting together. If the pressure of gas is negligible, the intrinsic line profile can be represented by a normalized Gauss function $$\label{eq:ilp}
\Phi(\nu,R,\phi,i)=\frac{1}{\sigma(R)\sqrt{\pi}}\exp\left[-\left(\frac{\nu-\nu_{0}+\Delta\nu(R,\phi,i)}{\sigma(R)}\right)^2\right],$$ where $\sigma(R)$ is the line width, $\Delta\nu(R,\phi,i)$ is the line center shift due to Doppler shift caused by the apparent motion of the gas parcels along the line of sight. The line width is determined by the natural thermal broadening $$\label{eq:natural-width}
\sigma_\mathrm{therm}(R)=\frac{\nu_{0}}{c}\sqrt{\frac{2 k T_\mathrm{atm}(R)}{m_\mathrm{CO}}},$$ and the local turbulent broadening $$\label{eq:turbulent-width}
\sigma_\mathrm{turb}(R)=\frac{\nu_{0}}{c}\chi\sqrt{\frac{\gamma k T_\mathrm{atm}(R)}{m_\mathrm{H}}},$$ assuming that the speed of turbulent motions is $\chi$ times the local sound speed. In Eqs. (\[eq:natural-width\]) and (\[eq:turbulent-width\]) $m_\mathrm{CO}$ and $m_\mathrm{H}$ are the molecular masses of H and CO and $\gamma$ is the adiabatic index of the main constituent of the gas, i.e that of hydrogen. Considering the thermal and local turbulent broadening, the resulting profile will be the convolution of the two Gaussian line profiles, i.e $$\sigma(R)=\sqrt{\sigma_\mathrm{therm}(R)^2+\sigma_\mathrm{turb}(R)^2}.$$ In a planet-free disk, in which the gas parcels are moving on circularly Keplerian orbits, the line center at $\nu_0$ fundamental frequency shifts due to the Doppler shift is $$\label{eq:Doppler-shift}
\Delta\nu(R,\phi,i)=\frac{\nu_0}{c}\sqrt{\frac{GM_{*}}{R}}\cos(\phi)\sin(i).$$ Here we neglect that the massive planet and the host star are orbiting the common center of mass, instead the center of mass is set to the center of host star. Moreover, the influence of the gas pressure on the angular velocity of gas parcels, which causes slightly sub-Keplerian orbital velocities due to radial pressure support, is also not taken into account.
[^1]: We only consider the inner part of disk ($R\leq 5\,\mathrm{AU}$). A significant part of the fundamental band CO ro-vibrational emission arises in $R= 2-3\,\mathrm{AU}$, however the line-to-continuum flux at $4.7\,\mathrm{\mu m}$ is sensitive to material that lies up to 5AU in our models. In this way the outer part of disk, where the disk interior becomes optically thin to its own and even to the radiation from superheated atmosphere, is not considered.
[^2]: In our thermal model the temperature stayed below 1500K everywhere in the computational domain. At this temperature the $V\geq 2$ vibrational levels are not thermally excited.
[^3]: We note that the term “Keplerian”, widely used in the literature, here means “circularly Keplerian” motion. On the other hand, elliptic motion is also Keplerian, but below we will use the widely accepted nomenclature for circular motion, though we think it is not entirely correct.
[^4]: We used version 3.2.8 of the ETC, assumed a water vapor column of 2.3mm, an airmass of 1.4, a seeing of $1\farcs0$, a slit width of $0\farcs2$, and adaptive optics correction using a guide star of $R=10.0\,\mathrm{mag}$ and spectral type of M0V.
[^5]: Assuming that the abundance of $\mathrm{^{12}C^{16}O}$ isotopologues are low, thus their fundamental ${\mathrm{V=1\rightarrow 0}}$ contribution to lines is negligible.
[^6]: As mentioned before, in the computational domain, the disk interior remains optically thick to its own radiation.
[^7]: The disk being in steady state means that the density distribution corresponding to the surface density distribution does not considerably change in time.
|
---
abstract: 'By using the concept of negativity, we investigate entanglement in (1/2,1) mixed-spin Heisenberg systems. We obtain the analytical results of entanglement in small isotropic Heisenberg clusters with only nearest-neighbor (NN) interactions up to four spins and in the four-spin Heisenberg model with both NN and next-nearest-neighbor (NNN) interactions. For more spins, we numerically study effects of temperature, magnetic fields, and NNN interactions on entanglement. We study in detail the threshold value of the temperature, after which the negativity vanishes.'
author:
- 'Zhe Sun, XiaoGuang Wang, AnZi Hu, and You-Quan Li'
title: 'Entanglement properties in (1/2,1) mixed-spin Heisenberg systems'
---
Introduction
============
Entanglement, an essential feature of the quantum mechanics, has been introduced in many fields of physics. In the field of quantum information, the entanglement has played a key role. The study of entanglement properties in many-body systems have attracted much attention [@M_Nielsen]-[@QPT_GVidal]. The Heisenberg chains, widely studied in the condensed matter field, display rich entanglement features and have many useful applications such as in the quantum state transfer [@M_Sub].
Most of the systems considered in previous studies are spin-half systems as there exists a good measure of entanglement of two spin-halves, the concurrence [@Conc], which is applicable to an arbitrary state of two spin halves. On the other hand, the entanglements in mixed-spin or higher spin systems are not well-studied due to the lack of good operational entanglement measures. There are several initial studies along this direction [@Schliemann; @Yi; @Zhu], however these works are restricted to the case of two particles.
For the case of higher spins, a non-entangled state has necessarily a positive partial transpose according to the Peres-Horodecki criterion [@PH]. In the case of two spin halves, and the case of (1/2,1) mixed spins, fortunately, a positive partial transpose is also sufficient. Thus, the sufficient and necessary condition for entangled state in (1/2,1) mixed spin systems is that it has a negative partial transpose. This allows us to investigate entanglement features of the mixed spin system.
The Peres-Horodecki criterion give a qualitative way for judging whether the state is entangled or not. The quantitative version of the criterion was developed by Vidal and Werner [@Vidal]. They presented a measure of entanglement called negativity that can be computed efficiently, and the negativity does not increase under local manipulations of the system. The negativity of a state $\rho$ is defined as $${\cal N(\rho)}=\sum_i|\mu_i|,$$ where $\mu_i$ is the negative eigenvalue of $\rho^{T_1}$, and $T_1$ denotes\
the partial transpose with respect to the first system. The negativity ${\cal N}$ is related to the trace norm of $\rho^{T_1}$ via $${\cal N(\rho)}=\frac{\|\rho^{T_1}\|_1-1}{2},$$ where the trace norm of $\rho^{T_1}$ is equal to the sum of the absolute values of the eigenvalues of $\rho^{T_1}$. In this paper, we will use the concept of negativity to study entanglement in (1/2,1) mixed-spin systems.
As shown in most previous works, models with the NN exchange interactions are considered and it is not easy to have pairwise entanglement between the NNN spins [@M_Osborne; @M_Osterloh]. It is true that there exist some quasi-one-dimension compounds offering us systems with NNN interactions. Bose and Chattopadhyay [@Ibose] and Gu et al. [@Gu] have investigated entanglement in spin-half Heisenberg chain with NNN interactions. In our paper here, we study entanglement properties not only in the (1/2,1) mixed-spin systems only with NN interactions, but also in the system with NNN interactions.
Entanglement in a system with a few spins displays general features of entanglement with more spins. For instance, in the anisotropic Heisenberg model with a large number of qubits, the pairwise entanglement shows a maximum at the isotropic point [@Gu]. This feature was already shown in a small system with four or five qubits [@Wang04]. So, the study of small systems is meaningful in the study of entanglement as they may reflect general features of larger or macroscopic systems. Also, due to the limitation of our computation capability, we only concentrate on small systems such as 4, 5 and 6-spin models.
The paper is organized as follows. In Sec. II, we study the systems with only NN interactions. The analytical results of negativity for the cases of two and three spins are given. The relation between entanglement and the macroscopic thermodynamical function, the internal energy is revealed. Also we numerically compute the negativity in more general mixed-spin models up to eight spins, and consider the effects of magnetic fields in this section. In Sec. III, the system with NNN interaction is discussed. For the four-spin case, we analytically calculate the eigenenergy of the system from which we get the analytical results of the negativity of the NN spins. We numerically study negativities versus NNN exchanging coupling, and the case of finite temperature is also considered. For larger system up to eight-spin system, we get some numerical results. The conclusion is given in Sec. IV.
Entanglement in Heisenberg chain only with nearest-neighbor interaction
=======================================================================
Analytical results of Entanglement in Heisenberg models
-------------------------------------------------------
We study entanglement of states of the system at thermal equilibrium described by the density operator $\rho(T)=\exp(-\beta
H)/Z$, where $\beta=1/k_BT$, $k_B$ is the Boltzmann’s constant, which is assumed to be one throughout the paper , and $Z=\text{Tr}\{\exp(-\beta H)\}$ is the partition function. The entanglement in the thermal state is called thermal entanglement.
We consider two kinds of spins, spin $\frac{1}{2}$ and $1$, alternating on a ring with antiferromagnetic exchange coupling. The Hamiltonian is given by $$\begin{aligned}
H_0&=&\sum_{i=1}^{N/2} ({\bf s}_i\cdot{\bf S}_{i}+{\bf S}_{i}\cdot
{\bf
s}_{i+1}), (N\in \text{even})\label{H01}\\
H_0&=&\sum_{i=1}^{(N-1)/2} ({\bf s}_i\cdot{\bf S}_{i}+{\bf S}_{i}\cdot {\bf s}_{i+1})+
{\bf s}_{\frac{(N+1)}{2}}\cdot{\bf s}_{1},\nonumber\\
&&(N\in \text{odd}),\label{H02}\end{aligned}$$ where ${\bf s}_i$ and ${\bf S}_i$ are spin-1/2 and spin-1 operators, respectively. The exchange interactions exist only between nearest neighbors, and they are of the same strength which are set to one. We adopt the periodic boundary condition. In Fig. 1, we give the schematic representation of the above Hamiltonian. Next, we first consider the models with two and three spins, and aim at getting analytical results of entanglement.
### Two-spin case
For the two-spin case, the Hamiltonian (\[H01\]) reduces to $H_0={\bf s}_1\cdot{\bf S}_1$. To have a matrix representation of the Hamiltonian, we choose the following basis $$\Big\{ |-\frac{1}{2},-1\rangle, |\frac{1}{2}, 0\rangle,
|-\frac{1}{2}, 1\rangle, |\frac{1}{2} , -1\rangle, |-\frac{1}{2},
0\rangle, |\frac{1}{2} ,1\rangle\Big\},$$ where $|m,M\rangle$ is the eigenstate of ${s}_z$ and $S_z$ with the corresponding eigenvalues given by $m$ and $M$, respectively.
{width="35.00000%"}
In the above basis, the Hamiltonian can be written as a block-diagonal form with the dimension of each block being at most $2\times 2$. Thus, the density matrix $\rho_{12}$ for the thermal state is obtained as $$\rho_{12}=\left(
\begin{array}{llllll}
a_1&0 & 0 &0 & 0 & 0\\
0 &a_2&b_1&0 & 0 & 0 \\
0& b_1&a_3&0 & 0 & 0 \\
0& 0 &0 &a_4&b_2&0\\
0& 0 & 0 & b_2&a_5&0\\
0& 0 & 0 & 0 & 0 & a_6
\end{array}
\right) , \label{rho}$$ with the partition function and the matrix elements given by $$\begin{aligned}
Z&=&2 e^\beta+4e^{-\frac{1}{2}\beta},\label{para1}\\
a_1&=&a_6=e^{-\frac{1}{2}\beta}/Z,\label{para22}\\
a_2&=&a_5=\frac{1}{6},\label{para222}\\
a_3&=&a_4=\frac{1}3\left(2e^\beta+e^{-\frac{1}{2}\beta}\right)/Z,\\
b_1&=&b_2=\frac{\sqrt{2}}{3}\left(e^{-\frac{1}{2}\beta}-e^\beta\right)/Z,\nonumber\\
&=&\sqrt{2}(a_1-a_2). \label{para2}\end{aligned}$$ After the partial transpose with respect to the first spin-half subsystem, we can get $\rho_{12}^{T_1}$ $$\rho_{12}^{T_1}=\left(
\begin{array}{llllll}
a_1&b_2&0 &0 &0 &0\\
b_2&a_2&0 &0 &0 &0\\
0 &0 &a_3&0 &0 &0\\
0 &0 &0 &a_4&0 &0\\
0 &0 &0 &0 &a_5&b_1\\
0 &0 &0 &0 &b_1&a_6
\end{array}
\right), \label{rhoT1}$$ which is still of the block-diagonal form, and computation of its eigenvalues is straightforward. There are only two eigenvalues which are possibly negative. The negativity is thus given by $$\begin{aligned}
{\cal N}(\rho_{12})=&\frac{1}2\max \big[0,\sqrt{(a_1-a_2)^2+4b_2^2}-a_1-a_2)\big]\nonumber\\
+&\frac{1}2\max\big[0,\sqrt{(a_5-a_6)^2+4b_1^2}-a_5-a_6\big].\label{N}\end{aligned}$$
Substituting Eqs. (\[para1\])–(\[para2\]) leads to the analytical result of negativity $$\begin{aligned}
{\cal N}(\rho)=&\max
\big[0,\sqrt{(a_1-a_2)^2+4b_2^2}-a_1-a_2)\big]\nonumber\\
=&2\max[0,a_2-2a_1]\nonumber\\
=&\frac{1}{3}\max\Big[0,\frac{e^\beta-4e^{-\frac{1}{2}\beta}}
{e^\beta+2e^{-\frac{1}{2}\beta}}\Big],\label{N1}\end{aligned}$$ where the second equality follows from Eq. (\[para2\]).
We can see that the negativity is a function of the single parameter $\beta$. In the limit of $T\rightarrow 0$, the negativity becomes 1/3 and the ground state is entangled. From Eq. (\[N1\]), it is direct to check that the negativity is a monotonically decreasing function when temperature increases. After a certain threshold value of the temperature, the entanglement disappears. This threshold value $T_\text{th}$ can be obtained as $$T_\text{th}=3/(4\ln2)\approx 1.0820.$$ For a ring of spin-half particles interacting via the Hisenberg Hamiltonian, it was shown that the pairwise thermal entanglement is determined by the internal energy [@WangPaolo]. It is natural to ask if similar relations exist in the present mixed-spin system. The internal energy can be obtained from the partition function as $$U=-\frac{1}{Z}\frac{\partial Z}{\partial\beta}.$$ Substituting Eq. (\[para1\]) into the above equation leads to $$U=\frac{\displaystyle
-e^\beta+e^{-\frac{1}{2}\beta}}{\displaystyle
e^\beta+2e^{-\frac{1}{2}\beta}}. \label{U}$$ From Eqs. (\[N1\]) and (\[U\]), we obtain a quantitative relation between the negativity and the internal energy $${\cal N(\rho)}=\frac{1}{3}\max[0,-1-2U].\label{U1}$$ The above equation builds a connection between the microscopic entanglement and the macroscopic thermodynamical function, the internal energy. The internal energy completely determine the thermal entanglement. From the equation, we can also read that the thermal state becomes entangled if and only if the internal energy $U<-1/2$. Since $U=\langle H\rangle=\langle \bf s_1\cdot\bf
S_1\rangle$, we have $${\cal N(\rho)}=\frac{1}{3}\max[0,-1-2\langle\bf s_1\cdot\bf
S_1\rangle],\label{U2}$$ which is consistent with the result obtained in Ref. [@Schliemann] by the group-theoretical technique.
### Three-spin case
We now consider the three-spin case and the schematic representation of the corresponding Hamiltonian is given by Fig. 1. In this situation, there are two types of pairwise entanglement, the entanglement between spin 1/2 and spin 1 and the entanglement between two spin halves.
The eigenvalue problem can be solved analytically, and after tracing out the third spin-half system the reduced density matrix $\rho_{12}$ is still of the same form as in Eq. (\[rho\]) with matrix elements given by $$\begin{aligned}
a_1&=&\left(\frac{5}{4}e^{-\frac{5}{4}\beta}+\frac{3}{4}e^{\frac{3}{4}\beta}\right)/Z,\nonumber\\
a_2&=&\left(\frac{5}{6}e^{-\frac{5}{4}\beta}+e^{\frac{3}{4}\beta}+\frac{1}{6}e^{\frac{7}{4}\beta}\right)/Z,\nonumber\\
a_3&=&\left(\frac{5}{12}e^{-\frac{5}{4}\beta}+\frac{5}{4}e^{\frac{3}{4}\beta}+\frac{1}{3}e^{\frac{7}{4}\beta}\right)/Z,\nonumber\\
b_1&=&\left(\frac{5\sqrt{2}}{12}e^{-\frac{5}{4}\beta}-\frac{\sqrt{2}}{4}e^{\frac{3}{4}\beta}-\frac{\sqrt{2}}{6}e^{\frac{7}{4}\beta}\right)/Z,\nonumber\\
Z&=&5e^{-\frac{5}{4}\beta}+6e^{\frac{3}{4}\beta}+e^{\frac{7}{4}\beta}.\end{aligned}$$ Substituting the above equations to Eq. (\[N\]) leads to the negativity $${\cal N}(\rho_{12})=\max\Big[0,
{-\frac{10}{3}e^{-\frac{5}{4}\beta}-e^{\frac{3}{4}\beta}+\frac{1}{3}e^{\frac{7}{4}\beta}}\Big]/Z.$$ It is evident that the negativity becomes 1/3 in the limit of $T\rightarrow 0$. From the expression of the negativity, the threshold value of temperature after which the entanglement vanishes can be estimated as $$T_{\text{th}}\approx 1/\ln3.2719\approx0.7609.$$
To examine the entanglement between two spin halves, we trace out the spin 1 system and get the reduced density matrix $\rho_{13}$ as follows $$\rho_{13}=\left(
\begin{array}{llll}
a_1& 0& 0& 0\\
0& a_2&b& 0\\
0& b& a_2& 0\\
0& 0& 0& a_1
\end{array} \right),\label{rho13}$$ with the matrix elements given by $$\begin{aligned}
a_1&=&\left(\frac{5}{3}e^{-\frac{5}{4}\beta}+e^{\frac{3}{4}\beta}+\frac{1}{3}e^{\frac{7}{4}\beta}\right)/Z\nonumber\\
a_2&=&\left(\frac{5}{6}e^{-\frac{5}{4}\beta}+2e^{\frac{3}{4}\beta}+\frac{1}{6}e^{\frac{7}{4}\beta}\right)/Z\nonumber\\
b&=&\left(\frac{5}{6}e^{-\frac{5}{4}\beta}-e^{\frac{3}{4}\beta}+\frac{1}{6}e^{\frac{7}{4}\beta}\right)/Z.
\label{ele}\end{aligned}$$ After taking the partial transpose, we can get $\rho^T_{13}$ $$\rho^T_{13}=\left(
\begin{array}{llll}
a_1& 0& 0& b\\
0& a_2&0& 0\\
0& 0& a_2& 0\\
b& 0& 0& a_1
\end{array} \right).\label{rhoT13}$$ Then, the negativity is readily obtained as $${\cal N}(\rho_{13})=\max[0,|b|-a_1].$$ It is straightforward to check that the negativity is always zero. Or, from another way, all the eigenvalues of the matrix $\rho^T_{13}$ are obtained as $$\begin{aligned}
\lambda_1&=&\left(\frac{5}{2}e^{-\frac{5}{4}\beta}+\frac{1}{2}e^{\frac{7}{4}\beta}\right)/Z,\nonumber\\
\lambda_2&=&\left(\frac{5}{6}e^{-\frac{5}{4}\beta}+2e^{\frac{3}{4}\beta}+\frac{1}{6}e^{\frac{7}{4}\beta}\right)/Z,\nonumber\\
\lambda_3&=&\lambda_4=a_2.\end{aligned}$$ Obviously the negativity vanishes here, in other words there is no entanglement between the two spin halves.
The ground-state negativity ${\cal N}_{12}={\cal
N}(\rho_{12})$=1/3 and ${\cal N}_{13}={\cal N}(\rho_{13})$=0, here ${\cal N}_{12}$ denotes the negativity between the $1_\text{th}$ and $2_\text{th}$ spin on the chain and ${\cal N}_{13}$ denotes the one between the $1_\text{th}$ and $3_\text{th}$ spin. The equation above can be obtained from the non-degenerate ground state given by: $$\begin{aligned}
|\psi_0\rangle&=&\frac{\sqrt{6}}{6}\left(|\frac{1}{2},0,-\frac{1}{2}\rangle+|-\frac{1}{2},0,\frac{1}{2}\rangle\right)\nonumber\\
&-&\frac{\sqrt{3}}{3}\left(|\frac{1}{2},-1,\frac{1}{2}\rangle+|-\frac{1}{2},1,-\frac{1}{2}\rangle\right).\end{aligned}$$ It is interesting to see that the ground-state entanglement between the spin half and spin 1 in the three-spin case is the same as that in the two-spin case.
Due to the SU(2) symmetry in our system, there are following relations between correlation functions and negativities [@Schliemann] $$\begin{aligned}
{\cal N}_{12}=&-\frac{1}3-\frac{2}3\langle{\bf s}_1\cdot{\bf
S}_1\rangle,\label{N12}\nonumber\\
{\cal N}_{23}=&-\frac{1}3-\frac{2}3\langle{\bf S}_1\cdot{\bf
s}_2\rangle,\nonumber\\
{\cal N}_{31}=&-\frac{1}4-\langle{\bf s}_2\cdot{\bf s}_1\rangle,\end{aligned}$$ where we have removed the max function in the negativity, implying that the negative value of ${\cal N}$ indicates no entanglement. Then, we have the relation between the internal energy and the negativities $$\begin{aligned}
U=&-\frac{5}4-{\cal N}_{13}-\frac{3}2({\cal N}_{12}+{\cal N}_{23})
\nonumber\\
=&-\frac{5}4-{\cal N}_{13}-{3}{\cal N}_{12}. \label{relation}\end{aligned}$$ The second equality follows from the exchange symmetry, namely, the Hamiltonian is invariant when exchanging two spin halves. So, for the three-spin case, the internal energy is related to two negativities.
To apply the above result, we consider the the ground-state properties ($T=0$). The Hamiltonian can be rewritten as $$H=\frac{1}{2}[({\bf s}_1+{\bf S}_1+{\bf s}_2)^2-{\bf s}_1^2-{\bf
S}_1^2-{\bf s}_2^2].$$ Then, by the angular momentum coupling theory, the ground-state energy is obtained as $E_0=-7/4$. Substituting the ground-state energy and ${\cal N}_{12}=1/3$ to Eq. (\[relation\]), we obtain ${\cal N}_{13}=-1/2$, indicating that there exists no entanglement between two spin halves. Next, we consider more general situations, i.e., the case of even $N$ sites.
### The case of even $N$ spins
Except for the SU(2) symmetry in the system, there exists exchange symmetry for the case of even spins. For instance, in the four-spin model, the Hamiltonian is invariant when exchanging two spin halves or two spin ones. Thus, for even-spin model, the entanglements between the two nearest-neighbor spins and the correlation functions $\langle{\bf s}_i\cdot{\bf S}_{i}\rangle$ are independent on index $i$. Therefore, the internal energy per spin is equal to the correlation function $\langle{\bf s}_i\cdot{\bf S}_{i}\rangle$ $$u=U/N=\langle{\bf s}_1\cdot{\bf S}_{1}\rangle. \label{uuu}$$ From Eqs. (\[N12\]) and (\[uuu\]), we have $${\cal N}_{12}=-\frac{1}3-\frac{2}3 u. \label{N1212}$$ This equation indicates that for the case of even spins the entanglement between two nearest neighbors is solely determined by the internal energy per spin. And for the case of zero temperature, the entanglement is determined by the ground-state energy. The less the energy, the more the entanglement.
We now apply the above result to the study of ground-state entanglement. We rewrite the four-spin Hamiltonian as follows $$H=\frac{1}{2}[({\bf s}_1+{\bf S}_1+{\bf s}_2+{\bf S}_2)^2-({\bf
s}_1+{\bf s}_2)^2-({\bf S}_1+{\bf S}_2)^2].$$ Then, by the angular momentum coupling theory, we obtain the ground-state energy per site $e_0=E_0/4=-3/4$. Thus, from Eq. (\[N1212\]), we have ${\cal N}_{12}=1/6$. For $N\ge 5$, it is hard to get analytical results. We next numerically calculate the entanglement for the case of more spins, and also consider the effects of magnetic fields.
Numerical results
-----------------
Having obtained analytical results of entanglement in the Heisenberg model with a few spins, we now numerically examine the entanglement behaviors in more general Hamiltonian including more spins and magnetic fields.
### Entanglement versus temperature
We consider the entanglement versus temperature for different number of spins $N$, and the numerical results are plotted in Fig. 2 and Fig. 3. It is clear to see that the ground state exhibits maximal entanglement, and with the increase of temperature, the entanglement monotonically decreases until it reaches zero. The decrease of entanglement is due to the mixture of less entangled excited states when increasing the temperature. The existence of the threshold temperature is also evident. For the case of even number of sites (Fig. 2), the threshold temperature decreases with the increase of $N$. In contrast to this behavior, for the case of odd number of spins, as seen from Fig. 3, the threshold temperature increases with the increase of $N$.
{width="45.00000%"}
{width="45.00000%"}
### Effects of magnetic fields
We now examine the effect of magnetic fields on entanglement. The Heisenberg Hamiltonian with a magnetic field along $z$ direction is given by $$H_1=H_0+B\sum_{i=1}^{N/2} \big({s}_{iz}+{S}_{i,z}\big), (N\in
\text{even}).$$ For $N=2$, the analytical result of negativity can be obtained (see Appendix A).
{width="45.00000%"}
In Fig. 4, we plot the negativity versus the magnetic field at a low temperature. For the two-spin case, with the increase of the magnetic field, the negativity rapidly reach a platform, and then after a certain magnetic field $B_\text{th}$, it jumps down to zero, indicating no entanglement. When $B=0$, the ground state is two-fold degenerate with ${\cal N}=1/3$. For a small $B>0$, the system is no longer degenerate, and the ground state is given by $$|\psi_0\rangle=\frac{\sqrt{6}}{3}|\frac{1}{2},-1\rangle-\frac{\sqrt{3}}{3}|-\frac{1}{2},0\rangle,$$ which is of the Schmidt form. For a state written as its Schmidt form $$|\psi\rangle=\sum_n c_n|e_n\rangle|f_n\rangle,$$ the negativity is obtained as [@Vidal] $${\cal N}=\left[\big(\sum_n c_n\big)^2-1\right]/2.$$ Here, $C_n$ are the Schmidt coefficients, and $\{|e_n\rangle\}$ and $\{|f_n\rangle\}$ are bases for subsystems 1 and 2, respectively. Applying this formula to the ground state, we immediately have ${\cal N}=\sqrt{2}/3$. When $B=B_\text{th}=3/2$, the ground state is two-fold degenerate, and when $B>B_\text{th}$ the ground state becomes non-degenerate and the corresponding wave function is given by $$|\psi_0\rangle=|-\frac{1}2,-1\rangle,$$ which is obviously of no entanglement. Then, the ground-state negativity forms a platform when $0<B<B_\text{th}$. The jump of negativity at $B=B_\text{th}$ is due to the level crossing. For $N>2$, the effects of magnetic fields on entanglement can be also explained by level crossing. For instance, for $N=4$, there are two level crossing, and the entanglement displays two jumps.
Effects of next-nearest-neighbor interactions on entanglement
=============================================================
We have studied the effects of finite temperature and magnetic fields on entanglement, and now consider the model containing two kinds of spins, spin $\frac{1}{2}$ and $1$, alternating on a ring with antiferromagnetic exchange coupling between both the NN spins and the NNN spins. The Hamiltonian can be expressed as $$H=J_1\sum^{N/2}_{i=1}\big({\bf s}_i\cdot {\bf S}_{i}+ {\bf
S}_{i}\cdot {\bf s}_{i+1}\big)+J_2\sum^{N/2}_{i=1}\big({\bf
s}_i\cdot {\bf s}_{i+1}+ {\bf S}_{i}\cdot {\bf S}_{i+1}\big),$$ where the ${\bf s}_i$ and ${\bf S}_i$ are spin-1/2 and spin-1 operators in the $i$th cell. $J_1$ characterizes the NN exchange coupling and $J_2$ the NNN coupling. We consider the antiferromagnetic interaction by taking $J_1,J_2>0$. $N$ is the total number of spins and here we choose it be even. Also we adopt the periodic boundary condition.
Four-spin model
---------------
### Eigenenergy and ground-state entanglement
The model with four spins is the simplest model with NNN interactions. We first solve the eigenvalue problem of this model. The key step is to write the four-spin Hamiltonian in the following form, $$\begin{aligned}
\label{H4}
H&=&\frac{1}{2}\big\{J_1{\bf S}^2+(2J_2-J_1)[({\bf s}_1+{\bf
s}_2)^2+({\bf S}_1+{\bf S}_2)^2]\nonumber\\
&-&2J_2({\bf s}_1^2+{\bf s}_2^2+{\bf S}_1^2+{\bf S}_2^2 )\big\},\end{aligned}$$ where ${\bf S}={\bf S}_1+{\bf S}_2+{\bf s}_1+{\bf s}_2$ denotes the total spin. From the above form, by angular momentum coupling theory, one can readily obtain all eigenvalues of the system as $$\begin{aligned}
E_1&=&J_2n(n+1)-\frac{11}{2}J_2,\nonumber\\
E_2&=&nJ_1+n(n+1)J_2-\frac{7}{2}J_2,\nonumber\\
E_3&=&-J_1+n(n+1)J_2-\frac{7}{2}J_2,\nonumber\\
E_4&=&-(n+1)J_1+n(n+1)J_2-\frac{7}{2}J_2, \label{E}\end{aligned}$$ where parameter $n=0,1,2$ in expressions $E_1$ and $E_2$ and $n=1,2$ in expressions $E_3$ and $E_4$, respectively. Then, from Eq. (\[E\]), we may find the ground-state energy as $$\label{Egs}
E_\text{GS}=\left\{
\begin{array}{ll}
-3J_1+\frac{5}{2}J_2 &\; \text{when}\;\;\; 0\leq J_2<J_1/4, \\
-2J_1-\frac{3}{2}J_2 &\; \text{when}\;\;\; J_1/4<J_2< J_1/2, \\
-\frac{11}{2}J_2 &\;\text{when} \;\;\;J_2>J_1/2,
\end{array}
\right.$$ Clearly, there are two level-crossing points, which will greatly affect behaviors of ground-state entanglement.
From Eq. (\[H4\]), it is obvious that in addition to the the SU$(2)$ symmetry, there also exists an exchange symmetry, namely, exchanging two spin halves or two spin ones yields invariant Hamiltonian. Then, the correlator $\langle {\bf s}_i\cdot{\bf
S}_i \rangle$ between any NN spins are the same. Therefore, we can get the correlator $\langle {\bf s}_i\cdot{\bf S}_i \rangle$ from the ground-state energy via the Hellmann-Feynman theorem.
When $0\leq J_2<J_1/4$, after applying the Hellmann-Feynman theorem to the ground-state energy, we obtain $$\langle{\bf s}_1\cdot{\bf S}_1\rangle=\frac{1}{4}\frac{\partial
E_0} {\partial J_1}=-\frac{3}{4}.$$ Due to the SU$(2)$ symmetry in our system, we have the following relation between the negativity and the correlator $\langle{\bf
s}_1\cdot{\bf S}_1\rangle$[@Schliemann], $${\cal N}_{1/2,1}=\max\big\{0, -\frac{1}3-\frac{2}3\langle{\bf
s}_1\cdot{\bf S}_1\rangle \big \},\label{cor}$$ where we use ${\cal N}_{1/2,1}$ to denote the negativity between the NN spins. Thus we obtain $${\cal N}_{1/2,1}=1/6.$$ In other regions, we find that ${\cal N}_{1/2,1}=0$ for $J_2>J_1/4$.
From the above analytical results, we can find that the negativity between the NN spins is not a continuous function of parameter $J_2$. It jumps from the value $\frac{1}{6}$ down to zero at $J_2=0.25$. Hence, we can see that the NNN interaction may deteriorate the entanglement of the NN spins. The critical point of $J_2=0.25$ may be considered to be a threshold value, after which the NN entanglement vanishes.
Having studied ground-state entanglement, now we make a short discussion of entanglement of excited states. We consider the first excited state. In the region $J_1/4<J_2<3J_1/8$, the first excited energy is given by $$E_1=-3J_1+\frac{5}{2}J_2,$$ from that we can get $\langle{\bf s}_1\cdot{\bf S}_1\rangle=-3/4$ and $N_{1/2,1}=1/6$. While in the rest region the negativity of the first excited state is zero.
### Thermal entanglement
Next, we consider the thermal entanglement. From Eq. (\[E\]), the partition function can be obtained as $$\begin{aligned}
\label{PF}
Z&=&5e^{-\frac{1}{2}\beta J_2}+6e^{\frac{7}{2}\beta
J_2}+e^{\frac{11}{2}\beta J_2}\nonumber\\
&+&e^{-\frac{5}{2}\beta J_2}(7e^{-2\beta J_1}+5e^{\beta J_1}+3e^{3\beta J_1})\nonumber\\
&+&e^{\frac{3}{2}\beta J_2}(5e^{-\beta J_1}+3e^{\beta J_1}+e^{2\beta
J_1}),\end{aligned}$$ The correlator $\langle{\bf s_1}\cdot {\bf S_1} \rangle$ at finite temperature can be computed from the partition function via the following relation $$\label{r1}
\langle{\bf s_1}\cdot {\bf S_1} \rangle=-\frac{1}{4\beta
Z}\frac{\partial Z}{\partial J_1},$$
Substituting (\[PF\]) to Eq. (\[r1\]) yields $$\begin{aligned}
\langle{\bf s_1}\cdot {\bf S_1} \rangle&=&-\frac{1}{4Z}\big [
e^{-\frac{5}{2}\beta J_2}(-14e^{-2\beta J_1 }+5e^{\beta J_1}+9e^{3\beta J_1})\nonumber\\
&+&e^{\frac{3}{2}\beta J_2}(-5e^{-\beta J_1}+3e^{\beta
J_1}+2e^{2\beta J_1})\big ].\end{aligned}$$ After substituting the above equation into (\[cor\]), we may get analytical expression of the negativity ${\cal N}_{1/2,1}$ at finite temperatures. The negativity is a function of $J_1$, $J_2$ and $T$.
*Low-temperature case*:
{width="45.00000%"}
We now make numerical calculations of entanglement and first consider the low-temperature case. We take $J_1=1$ in all the following plots. In our system there exist three kinds of negativity, the negativity ${\cal N}_{1/2,1}$ between spin-1 and spin-1/2, ${\cal N}_{1/2,1/2}$ between two spin-1/2, and ${\cal
N}_{1,1}$ between two spin-1.
In Fig. 5, we plot the negativity versus $J_2$ in four-spin system at a low temperature of $T=0.008$. It is clear to see that ${\cal
N}_{1/2,1}$ keeps a value about $1/6$ when $J_2$ increases from zero until it reaches the critical point, at which the ${\cal
N}_{1/2,1}$ displays a jump to zero. This behavior of entanglement is consistent with that at zero temperature from the analytical results. It is natural to see that increase of NNN exchange interaction will suppress the entanglement of NN spins, and at last completely erase the entanglement.
In comparison with ${\cal N}_{1/2,1}$, the negativities ${\cal
N}_{1,1}$ and ${\cal N}_{1/2,1/2}$ behave distinctly. We see that near the point of $J_2=1/4$, ${\cal N}_{1,1}$ increases quickly to a steady value about $1/3$, and when $J_2$ reaches the value about $1/2$, ${\cal N}_{1,1}$ jumps another steady value 1. These two jumps result from the two level crossing as seen clearly from the figure inserted. The second level crossing also leads to a small dip in the curve of ${\cal N}_{1,1}$. The negativity ${\cal
N}_{1/2,1/2}$ displays a jump to a steady value of 1/2 near $J_2=1/2$. It is evident that the entanglement between NNN spins is enhanced by increasing NNN interactions. The competition between NN and NNN interactions leads to rich behaviors of quantum entanglement. Another observation is that there is a range of $J_2$, at which negativities ${\cal N}_{1/2,1}$ and ${\cal
N}_{1/2,1/2}$ are zero, and only ${\cal N}_{1,1}$ is not zero. This indicates that the NNN interaction must be strong enough to build up the entanglement of two spin halves.
*Entanglement versus $J_2$ and $T$*: As temperature increases the entanglement will decrease due to the mixing of less entangled excited states to the thermal state. It is obvious that there exists a threshold temperature after which the negativity is zero. In the frustrated system, there exists the parameter $J_2$, and with its increase, the negativity ${\cal N}_{1/2,1}$ will decrease to zero, while ${\cal N}_{1,1}$ and ${\cal N}_{1/2,1/2}$ increase from zero to their maxima. So it is clear that there also exists a threshold $J_\text{2th}$ corresponding to the boundary between zero and nonzero negativities.
{width="45.00000%"}
In Fig. 6, we show the negativity ${\cal N}_{1/2,1}$ versus the temperature and $J_2$. When the temperature approaches zero, the ${\cal N}_{1/2,1}$ reaches its maximum, and then with the temperature increasing, ${\cal N}_{1/2,1}$ decreases to zero. On the $J_2-T$ plane, there is a curve along which the negativity just turns to be zero. It is possible to consider that the curve describes the threshold $J_\text {2th}$ versus the temperature. Obviously, $J_\text {2th}$ does not behave as a monotonous function of the temperature, and it displays a peak at about $T=0.178$, This behavior is in big contrast with the case of non-mixed qubit systems [@Gu]. When the temperature rises, the weight of excited states will increase and it may strongly affect the negativity. This behavior of $J_\text {2th}$ results from both the mixture of excited states to the thermal state and the intrinsic properties of the mixed-spin system. In addition, after crossing the temperature about $T=1.082$, ${\cal N}_{1/2,1}$ will vanish, irrespective of the value of $J_2$.
From another point of view, we can read the threshold temperature $T_\text {th}$ versus different $J_2$ from the curve in the $J_2-T$ plane. When $J_2$ increases, $T_\text {th}$ decreases, and when $J_2$ crosses about $0.3758$, ${\cal N}_{1/2,1}$ will disappear at any temperature.
Next, we consider the entanglement between NNN spins. In Fig. 7, we plot the negativity ${\cal N}_{1/2,1/2}$ as a function of the temperature and $J_2$. We can see that, before $J_2$ reaches the value about $J_2=0.5$, ${\cal N}_{1/2,1/2}$ keeps being zero at any temperature. And in the region $J_2>0.5$, the ${\cal
N}_{1/2,1/2}$ can be enhanced by the increasing NNN interaction. This is a result from the competition of two kinds of exchange interactions. The thermal fluctuation all along suppresses the entanglement. So, from the curve lying on the $J_2-T$ plane which corresponds to the boundary of the nonzero and zero values of ${\cal N}_{1/2,1/2}$, we may find that the higher the temperature is, the larger the threshold $J_\text {2th}$ will be. From another point of view, the $T_\text {th}$ increases as $J_2$ increases.
{width="45.00000%"}
In Fig. 8, we plot the negativity ${\cal N}_{1,1}$ versus $T$ and $J_2$. In the region of $J_2<0.25$, ${\cal N}_{1,1}$ is zero at any temperature. When $J_2>0.25$, the increasing NNN exchange interaction $J_2$ enhances the negativity and exhibits two particular flat roofs. With the temperature rises, ${\cal
N}_{1,1}$ is suppressed to zero. Also we can consider the threshold $T_\text {th}$ and $J_\text {2th}$ from the critical curve on the $J_2-T$ plane, and the $J_\text {2th}$ also behaves as an increasing function of the temperature. We can see the nonzero region of ${\cal N}_{1,1}$ is much larger than ${\cal
N}_{1/2,1/2}$. But here it should be pointed out that, because ${\cal N}_{1,1}>0$ only gives a sufficient condition for entangled state, we can not definitely say that the state in the area of zero negativity is not entangled.
{width="45.00000%"}
Numerical results of negativity for more spins
----------------------------------------------
In this section, we present numerical results of negativity for more spins, and first consider the low-temperature case.
### Low-temperature case
{width="45.00000%"}
{width="45.00000%"}
In Fig. 9, we plot the negativity versus $J_2$ for the case of six spins at a lower temperature. The negativity ${\cal
N}_{1/2,1}$ behaves as a decreasing function of $J_2$. It decreases to zero at about $J_2=0.27$, which is the special point corresponding to the energy level crossing. On the contrary, around the point $J_2=0.27$, ${\cal N}_{1,1}$ jumps up to a nonzero value, and then increases gradually until approaching the limit about ${\cal N}_{1,1}=0.33$. This behavior is quite different from that in the four-spin model.
We also see that ${\cal N}_{1/2,1/2}$ is zero all the time. It can be understood as follows. In the six-site system there are three spin halves with the NNN interaction. Even for a pure homogeneous three-qubit system, there is no entanglement between two spins, irrespective of the strength of the exchange interactions [@M_Three]. Now, in addition to the interaction among three spin halves, there are also interaction between spin halves and spin ones. So, it is reasonable that the entanglement between two spin halves is zero.
The negativity versus $J_2$ in the eight-site case is shown in Fig. 10. After the first sharp jump to a value (not zero) at about $J_2=0.25$, ${\cal N}_{1/2,1}$ goes down to zero gradually. At approximately $J_2=0.55$, the negativity is zero. On the contrary, the negativity ${\cal N}_{1,1}$ jumps up at about $J_2=0.25$, and then goes up gradually and almost linearly until $J_2$ reaches about $0.55$. Then there happens a sharp decrease to nearly zero, and after that it begins to increase gradually. The negativity ${\cal N}_{1/2,1/2}$ keeps zero until $J_2$ reaches about $0.67$, and then it goes up until reaches a steady value.
In Fig. 10, we find that the three kinds of negativity exhibit different properties. From figure inserted, i.e., the energy levels of the eight-spin system, we can see that in the region from about $J_2=0.55$ to $J_2=0.67$ the first excited energy is quite close to the ground energy. It is known that the energy level crossing can greatly affect the entanglement. Here, the two close energy levels also play an important role in the behavior of entanglement. The approximate degenerate energy levels may remarkably change the probability distribution even at a very low temperature, thereby affect the negativity.
### Entanglement versus $J_2$ and $T$
{width="45.00000%"}
{width="45.00000%"}
Now, we present the entanglement versus $J_2$ and $T$ in the six-spin model. The NN negativity ${\cal N}_{1/2,1}$ is shown in Fig. 11. On the $J_2-T$ plane, similar to the four-spin case, the $J_\text {2th}$ does not behave as a monotonous function of $T$ and it reaches its maximum at about $T=0.27$. From the figure, we may find that the entanglement only exists in the region approximately $T<0.925$ and $J_2<0.418$. The strong NNN interaction and thermal fluctuation will suppress the NN entanglement to zero.
We do not plot the negativity ${\cal N}_{1/2,1/2}$ as it is zero all along for any $J_2$ and $T$. We give the NNN negativity ${\cal
N}_{1,1}$ in Fig. 12. In the region of $J_2<0.282$, there is no negativity ${\cal N}_{1,1}$ at any temperature. The threshold $J_\text {2th}$ is increased by the increasing temperature, similar to the four-spin case.
conclusion
==========
In conclusion, we have studied the entanglement properties of the (1/2,1) mixed-spin systems described by the Heisenberg model. In the systems only with NN exchange interactions, for two-spin and three-spin cases analytical results of the negativity have been obtained, which facilitate our discussions of entanglement. The analytical expression of threshold temperature after which the entanglement vanishes are obtained for the two-site case. For the case of even number of particles, it has been found that the pairwise thermal entanglement is solely determined by the internal energy, and thus builds an interesting relation between the microscopic quantity, entanglement, and the macroscopic thermal dynamical function, the internal energy in the mixed-spin systems. For the odd number of particles such as the three-site case, we also provide a relation between the internal energy and negativities. We have numerically studied the effects of different finite temperature and magnetic fields on entanglement. As a conclusion, the thermal fluctuation suppresses the entanglement, and entanglement may change evidently at some critical points of magnetic field.
In the systems also with NNN interactions, by applying the angular momentum coupling method, we obtained analytical results of all the eigenenergies of the four-spin system, based on which the negativity of the NN spins has been obtained. We also considered the excited-state entanglement. At finite temperature, from the partition function, the analytical results of negativity has been given. We have numerically studied the effects of the NNN interaction on the NN entanglement and NNN entanglement. It is natural to see that the NNN interaction suppresses the NN entanglement, and enhances the NNN entanglement. We found that the negativity between two spin ones is sensitive to $J_2$ and displays some interesting properties. At finite temperature, the thermal fluctuation suppresses both the NN entanglement and the NNN entanglement. The threshold values $J_\text {2th}$ and $T_\text {th}$ are studied in detail. The entanglement displays some peculiar properties, which are quite different from those of the spin-half model. These are due to inherent mixed-spin character of our system. It is more interesting to study entanglements in other mixed systems and explore some universal properties, which are under consideration.
Two-site Heisenberg model with a magnetic field
===============================================
The Hamiltonian of the two-site Heisenberg model with a magnetic field is written explicitly as $$H_1=s_{1x}\otimes S_{2x}+s_{1y}\otimes S_{2y}+s_{1z}\otimes
S_{2z}+B(s_{1z}+S_{2z}).\nonumber$$ Following the same way as the discussions of subsection II.A, the density matrix of the thermal state is given by Eq. (\[rho\]) with the matrix elements $$\begin{aligned}
a_1=&a_6e^{3\beta B}=e^{\frac{\beta}{2}(3B-1)},\nonumber\\
a_5=&a_2e^{\beta B}= \frac{1}3e^{\frac{\beta
B}{2}}\Big(e^\beta+2e^{-\frac{\beta}2}\Big),
\nonumber\\
a_4=&a_3e^{\beta B}=\frac{1}3e^{\frac{\beta
B}{2}}\Big(2e^\beta+e^{-\frac{\beta}2}\Big),
\nonumber\\
b_2=&b_1e^{\beta B}=-\frac{\sqrt{2}}3e^{\frac{\beta
B}{2}}\Big(e^\beta-e^{-\frac{\beta}2}\Big) ,\nonumber\\\end{aligned}$$ and the partition function $$\begin{aligned}
Z&=&e^{\frac{\beta}{2}(3B-1)}+e^{-\frac{\beta}{2}(3B+1)}\nonumber\\
&+&e^{\frac{1}{4}\beta}\cosh\big(\frac{3\beta}{4}\big)
\cosh\big(\frac{\beta B}2\big).\end{aligned}$$ Having obtained the analytical expressions of the matrix, we directly obtain the negativity after substituting the matrix elements to Eq. (\[N\]).
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