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The spectral form factor in the `t Hooft limit  Intermediacy versus universality
by W. L. Vleeshouwers, V. Gritsev
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Submission summary
Authors (as registered SciPost users):  Ward Vleeshouwers 
Submission information  

Preprint Link:  https://arxiv.org/abs/2201.07841v3 (pdf) 
Date accepted:  20220922 
Date submitted:  20220909 15:52 
Submitted by:  Vleeshouwers, Ward 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
The Spectral Form Factor (SFF) is a convenient tool for the characterization of eigenvalue statistics of systems with discrete spectra, and thus serves as a proxy for quantum chaoticity. This work presents an analytical calculation of the SFF of the ChernSimons Matrix Model (CSMM), which was first introduced to describe the intermediate level statistics of disordered electrons at the mobility edge. The CSMM is characterized by a parameter $ 0 \leq q\leq 1$, where the Circular Unitary Ensemble (CUE) is recovered for $q\to 0$. The CSMM was later found as a matrix model description of $U(N)$ ChernSimons theory on $S^3$, which is dual to a topological string theory characterized by string coupling $g_s=\log q$. The spectral form factor is proportional to a colored HOMFLY invariant of a $(2n,2)$torus link with its two components carrying the fundamental and antifundamental representations, respectively. We check that taking $N \to \infty$ whilst keeping $q<1$ reduces the connected SFF to an exact linear ramp of unit slope, confirming the main result from arXiv:2012.11703 for the specific case of the CSMM. We then consider the `t Hooft limit, where $N \to \infty$ and $q \to 1^$ such that $y = q^N $ remains finite. As we take $q\to 1^$, this constitutes the opposite extreme of the CUE limit. In the `t Hooft limit, the connected SFF turns into a remarkable sequence of polynomials which, as far as the authors are aware, have not appeared in the literature thus far. A gap opens in the spectrum and, after unfolding by a constant rescaling, the connected SFF approximates a linear ramp of unit slope for all $y$ except $y \approx 1$, where the connected SFF goes to zero. We thus find that, although the CSMM was introduced to describe intermediate statistics and the `t Hooft limit is the opposite limit of the CUE, we still recover WignerDyson universality for all $y$ except $y\approx 1$.
Published as SciPost Phys. Core 5, 051 (2022)
Author comments upon resubmission
List of changes
1. Added a treatment (including plots) of numerically unfolded SFF's for N=10 and N=20.
2. Corrected typos