Spectral statistics in constrained many-body quantum chaotic systems

TL;DR

This paper develops a theoretical framework to analyze spectral statistics and Thouless time scaling in constrained many-body quantum systems with conserved multipole moments, using mappings to classical Markov circuits and field theories.

Contribution

It introduces a novel mapping of spectral form factors to classical Markov circuits and derives a field theory for RK-Hamiltonians in systems with conserved multipole moments.

Findings

Spectral form factor maps to a classical Markov circuit.

Thouless time scales subdiffusively as L^{2(m+1)} for systems with conserved m-th multipole moment.

Formalism generalizes to higher dimensions and other conserved components.

Abstract

We study the spectral statistics of spatially-extended many-body quantum systems with on-site Abelian symmetries or local constraints, focusing primarily on those with conserved dipole and higher moments. In the limit of large local Hilbert space dimension, we find that the spectral form factor $K(t)$ of Floquet random circuits can be mapped exactly to a classical Markov circuit, and, at late times, is related to the partition function of a frustration-free Rokhsar-Kivelson (RK) type Hamiltonian. Through this mapping, we show that the inverse of the spectral gap of the RK-Hamiltonian lower bounds the Thouless time $t_{\mathrm{Th}}$ of the underlying circuit. For systems with conserved higher moments, we derive a field theory for the corresponding RK-Hamiltonian by proposing a generalized height field representation for the Hilbert space of the effective spin chain. Using the field theory formulation, we obtain the dispersion of the low-lying excitations of the RK-Hamiltonian in the continuum limit, which allows us to extract $t_{\mathrm{Th}}$. In particular, we analytically argue that in a system of length $L$ that conserves the $m^{th}$ multipole moment, $t_{\mathrm{Th}}$ scales subdiffusively as $L^{2(m+1)}$. We also show that our formalism directly generalizes to higher dimensional circuits, and that in systems that conserve any component of the $m^{th}$ multipole moment, $t_{\mathrm{Th}}$ has the same scaling with the linear size of the system. Our work therefore provides a general approach for studying spectral statistics in constrained many-body chaotic systems.