Tridiagonal Hamiltonians modeling the density of states of the Double-Scaled SYK model

TL;DR

This paper constructs a finite-dimensional tridiagonal Hamiltonian to model the density of states in the Double-Scaled SYK model, providing analytical and numerical insights into its spectral properties.

Contribution

It introduces a novel Hamiltonian construction that accurately reproduces the DOS of the DSSYK model and analytically characterizes the Lanczos coefficients.

Findings

Analytical expression for bulk Lanczos coefficients as a q-deformation of the logarithm

Numerical results confirm the analytical predictions

Semi-analytical and random matrix approaches support the findings

Abstract

By analyzing the global density of states (DOS) in the Double-Scaled Sachdev-Ye-Kitaev (DSSYK) model, we construct a finite-dimensional Hamiltonian that replicates this DOS. We then tridiagonalize the Hamiltonian to determine the mean Lanczos coefficients within the parameter range. The bulk Lanczos coefficients, especially the Lanczos descent can be analytically expressed as a particular $q$-deformation of the logarithm. Our numerical results are further corroborated by semi-analytical findings, a random matrix potential construction in the bulk, and the analytic results at the edge of the Lanczos spectra using the method of moments.