SciPost Phys. Core 5, 051 (2022) ·
published 1 December 2022

· pdf
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 $y\approx
1$.
SciPost Phys. 10, 146 (2021) ·
published 16 June 2021

· pdf
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 parameterdependent critical ensemble which has intermediate statistics characteristic of ergodictononergodic transitions such as the Anderson localization transition. This same ensemble is the matrix model of $U(N)$ ChernSimons 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 ergodictononergodic transitions using topological tools, such as we have done here.
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