The spectral form factor in the 't Hooft limit – Intermediacy versus universality

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 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 0≤q≤1, where the Circular Unitary Ensemble (CUE) is recovered for q\to 0 q→0. The CSMM was later found as a matrix model description of U(N) U(N) Chern-Simons theory on S^3 S3 , which is dual to a topological string theory characterized by string coupling g_s=-\log q gs=−logq. The spectral form factor is proportional to a colored HOMFLY invariant of a (2n,2) (2n,2)-torus link with its two components carrying the fundamental and antifundamental representations, respectively. We check explicitly that taking N \to \infty N→∞ whilst keeping q<1 q<1 reduces the connected SFF to an exact linear ramp of unit slope, thereby confirming the main result from for the specific case of the CSMM. We then consider the ’t Hooft limit, where N \to \infty N→∞ and q \to 1^- q→1− such that y=q^N y=qN remains finite. As we take q\to 1^- q→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 y except y \approx 1 y≈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 y except y\approx 1 y≈1.