TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhysCore.5.4.051
TI  - The spectral form factor in the 't Hooft limit – Intermediacy versus universality
PY  - 2022/12/01
UR  - https://www.scipost.org/SciPostPhysCore.5.4.051
JF  - SciPost Physics Core
JA  - SciPost Phys. Core
VL  - 5
IS  - 4
SP  - 051
A1  - Vleeshouwers, Ward
AU  - Gritsev, Vladimir
AB  - 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 Chern-Simons 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 $U(N)$ Chern-Simons 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 Wigner-Dyson universality for all $y$ except $y\approx 1$.
ER  -

@Article{10.21468/SciPostPhysCore.5.4.051,
	title={{The spectral form factor in the 't Hooft limit – Intermediacy versus universality}},
	author={Ward L. Vleeshouwers and Vladimir Gritsev},
	journal={SciPost Phys. Core},
	volume={5},
	pages={051},
	year={2022},
	publisher={SciPost},
	doi={10.21468/SciPostPhysCore.5.4.051},
	url={https://scipost.org/10.21468/SciPostPhysCore.5.4.051},
}