# Topological field theory approach to intermediate statistics

### Submission summary

 As Contributors: Ward Vleeshouwers Preprint link: scipost_202101_00013v2 Date accepted: 2021-06-08 Date submitted: 2021-06-04 13:08 Submitted by: Vleeshouwers, Ward Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Mathematical Physics Approach: Theoretical

### Abstract

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.

Published as SciPost Phys. 10, 146 (2021)

We resubmit our work after making the revisions which were suggested by the referees.

### List of changes

We made the revisions suggested by the referees. In particular, we corrected our previous formulation to say that the genuine Anderson localization transition is sharp, and stressed the phenomenological nature of the matrix model considered in this paper and its inability to capture the dimension-dependence that is present in the genuine Anderson model.
Further, we commented briefly on the GUE limit as q goes to 1 in the Hermitian version of the matrix model, as suggested by the second referee report. As noted in the revised version of our paper, we aim to consider this in more detail in a future work.
Lastly, we clarified a few sentences and corrected some spelling errors.