The spectral form factor in the ‘t Hooft limit – Intermediacy versus universality

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 $y\approx 1$.Comment: Changes: 1. Added a treatment of unfolding and revised our conclusions, changed title, abstract, introduction, and conclusion. 2. Removed comparison with linear fit to connected SFF 3. Changed commas to decimal points 4. Added figures on level density and unfolded SFF 5. Added references 6. Corrected typos 30 pages, 6 figure

Unitary matrix integrals, long-range random walks, and spectral statistics

Unitary matrix integrals play an important role in a wide variety of fields, ranging from gauge theory, enumerative combinatorics, number theory, and quantum chaos, to areas of telecommunication and quantitative finance. This thesis considers unitary matrix integrals over symmetric polynomials and presents novel, recursive expansions for these objects. These expressions generalize various long-standing results in ways that allow for a greater range and ease of application. These results are then applied to the study of non-intersecting long-range random walkers (LRRW’s), that is, hard-core bosons on a 1D lattice which can move over large distances, whose correlation functions are given by weighted unitary matrix integrals over Schur polynomials. These are remarkably rich systems, which have gained increasing attention in the last 15 years due in part to their experimental realizability in trapped ion systems. Various basic aspects of symmetric function theory lead directly to certain surprising results on LRRW’s, including physical dualities between certain LRRW systems, as well as between LRRW’s and long-range fermionic systems. The recursive expressions derived in this thesis can be directly applied to LRRW’s as well, leading to an expansion in powers of the time parameter which can be truncated at the desired order, with coefficients given explicitly in terms of the hopping parameters of the hamiltonian. These results are directly amenable to experimental checks, and further include procedures for benchmarking experimental setups such as trapped ion systems. We then consider the computation of the spectral form factor of a matrix model description of Chern-Simons theory, which was found by previous authors to describe the ‘intermediate’ statistics of quantum systems in between order and chaos. Despite a large collection of previous publications on this application, our calculations display an absence of intermediate statistics in this model, reproducing the statistics of fully chaotic quantum systems instead. These findings have precipitated a numerical follow-up study to clarify this apparent contradiction

Unitary matrix integrals, symmetric polynomials, and long-range random walks

Unitary matrix integrals over symmetric polynomials play an important role in a wide variety of applications, including random matrix theory, gauge theory, number theory, and enumerative combinatorics. We derive novel results on such integrals and apply these and other identities to correlation functions of long-range random walks (LRRW) consisting of hard-core bosons. We generalize an identity due to Diaconis and Shahshahani which computes unitary matrix integrals over products of power sum polynomials. This allows us to derive two expressions for unitary matrix integrals over Schur polynomials, which can be directly applied to LRRW correlation functions. We then demonstrate a duality between distinct LRRW models, which we refer to as quasi-local particle-hole duality. We note a relation between the multiplication properties of power sum polynomials of degree n and fermionic particles hopping by n sites. This allows us to compute LRRW correlation functions in terms of auxiliary fermionic rather than hard-core bosonic systems. Inverting this reasoning leads to various results on long-range fermionic models as well. In principle, all results derived in this work can be implemented in experimental setups such as trapped ion systems, where LRRW models appear as an effective description. We further suggest specific correlation functions which may be applied to the benchmarking of such experimental setups