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Topological field theory approach to intermediate statistics
by W.L. Vleeshouwers, V. Gritsev
This is not the current version.
|As Contributors:||Ward Vleeshouwers|
|Date submitted:||2021-01-28 16:44|
|Submitted by:||Vleeshouwers, Ward|
|Submitted to:||SciPost Physics|
Random matrix models provide a phenomenological description of a vast variety of physical phenomena. Prominent examples include the eigenvalue statistics of quantum (chaotic) systems, which are characterized by the spectral form factor (SFF). Here, we calculate the SFF of unitary matrix ensembles of infinite order with the weight function satisfying the assumptions of Szegö’s limit theorem. We then consider a parameter-dependent critical ensemble which has intermediate statistics characteristic of ergodic-to-nonergodic transitions such as the Anderson localization transition. This same ensemble is the matrix model of $U(N)$ Chern-Simons theory on $S^3$ , and the SFF of this ensemble is proportional to the HOMFLY invariant of (2n,2)-torus links with one component in the fundamental and one in the antifundamental representation. This is one example of a large class of ensembles with intermediate statistics arising from topological field and string theories. Indeed, the absence of a local order parameter suggests that it is natural to characterize ergodic-to-nonergodic transitions using topological tools, such as we have done here.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 2021-2-23 (Contributed Report)
- Cite as: Anonymous, Report on arXiv:scipost_202101_00013v1, delivered 2021-02-23, doi: 10.21468/SciPost.Report.2599
NOTE: The following is from an invited referee who submitted the report by email rather than through the web interface.
At the point of the Anderson transition a system exhibits an "intermediate statistics", i.e. intermediate between Wigner-Dyson (ergodic phase) and Poisson (nonergodic phase). This statistics is universal, although there are several universality classes. The authors propose a correspondence between these universality classes and some "interacting topological states of matter" which occur in the Chern-Simons or string theories. As far as I can see, their logic goes like this: (i) Anderson transition is described by certain matrix ensembles. (ii) These ensembles resemble matrices which appear in some topological phases in string theory. Ergo, (iii) These very complicated phases are in correspondence with the relatively simple "intermediate statistics" of the Anderson transition (the latter is a single particle problem and in this sense it is "simple").
I have a problem with statement (i) in this logical chain. It is true that a number of authors have proposed various matrix ensembles which, supposedly, describe the Anderson transition. Those proposals, however, have little to do with the genuine Anderson transition and its intermediate statistics and, at best, can be considered as some "toy models". Indeed, the spatial dimensionality, d, plays a decisive role in the Anderson transition (the transition exists only for d>2 and the statistics is sensitive to d). But there is no "d" either in the banded matrices or in the log-Gaussian ensemble mentioned by the authors. Moreover, those ensembles don't even exhibit an Anderson transition in its usual sense, i.e. as a transition in the energy band when the energy crosses some critical value. What happens in those ensembles is that, when some parameter is changed, the eigenfunctions of the entire spectrum are changing from being localized to being extended. A simple way to see that The Anderson transition has little to do with those ensembles is simply to write down the random matrix for the original Anderson model, call it the Anderson ensemble (AE). The AE is NOT invariant under an orthogonal transformations, neither it has any similarity with the banded matrix ensemble. The main property of the AE is sparsity (the matrices in 3d are more sparse than those in 2d). Thus, I would argue that sparsity is the main property which determines the existence of the transition and the corresponding intermediate statistics.
In short, the whole discussion of the Anderson transition and the intermediate statistics in the paper is superficial and sometimes simply wrong. For instance, the statement "An important example of such a system is given by disordered conductors, where increasing the disorder strength leads to greater deviation from Wigner-Dyson universality" is incorrect. There is no gradual change of the Wigner-Dyson statistics with the increase of disorder. It holds all the way down to the mobility edge E_c, while below E_c the Poisson statistics sets in. The intermediate statistics takes place only at E_c (of course, one should carefully define the limiting procedure, starting with the large finite sample, where E_c is smeared into a narrow interval which however contains a macroscopic number of levels).
All of the above doesn't imply that the paper is useless. Let's forget the Anderson transition. Then, the main content of the paper is the above statement (ii), namely, that there exists a correspondence between matrix ensembles coming from different fields, and that perhaps topological phases in string theory can be understood with the help of some simple, previously studied, matrix ensembles. This might well be true, although it's not for me to judge since I don't know anything about those topological phases in string theory.
At the very least, in their discussion of the Anderson localization and the matrix ensembles the authors should clearly state that those ensembles do not really describe the genuine Anderson problem but can merely serve as toy models for that phenomenon.