Deep thermalization under charge-conserving quantum dynamics

TL;DR

This paper investigates how charge conservation symmetries influence deep thermalization in quantum many-body systems, revealing universal wavefunction distributions dependent on initial states and measurement choices, supported by rigorous proofs, analytical calculations, and simulations.

Contribution

It introduces a universal ansatz for the limiting wavefunction distribution under charge-conserving dynamics, extending understanding of deep thermalization beyond traditional thermalization.

Findings

Universal wavefunction distributions depend on initial charge fluctuations and measurement basis.

A universal ansatz is proposed, based on maximum-entropy principles and coarse-graining.

Rigorous proof and analytical calculations support the ansatz, complemented by numerical simulations.

Abstract

"Deep thermalization" describes the emergence of universal wavefunction distributions in quantum many-body dynamics, appearing on a local subsystem upon measurement of its environment. In this work, we study in detail the effect of continuous internal symmetries and associated conservation laws on deep thermalization. Concretely, we consider quantum spin systems with a $U(1)$ symmetry associated with the conservation of magnetization (or `charge'), and analyze how the choice of initial states (specifically, their degree of charge fluctuations) and the choice of measurement basis (specifically, whether or not it can reveal information about the local charge density) determine the ensuing universal wavefunction distributions. We put forth a universal ansatz for the limiting form of the projected ensemble, motivated by maximum-entropy principles rooted in statistical physics and quantum information theory. This limiting form depends on a polynomial amount of data on the initial state and measurement basis, a `coarse-graining' that is an essential feature of bona fide thermodynamic ensembles. We support our ansatz with three complementary approaches: (i) a rigorous proof in the simplest case of no charge fluctuations in either the initial state or the measurement basis; (ii) analytical calculations using a `replica limit' approach, applicable when charge fluctuations are allowed in either the input state or the measurement basis but not both; (iii) extensive numerical simulations of finite-sized systems in the most general case. Our findings demonstrate a rich interplay between symmetries and the information extracted by measurements, which allows deep thermalization to exhibit a range of universal behaviors far beyond regular thermalization.