Symmetry-resolved properties of the trace distance in thermalizing SU(2) systems

TL;DR

This paper investigates how symmetry-resolved trace distances can diagnose thermalization in quantum systems with SU(2) symmetry, revealing exponential suppression of certain fluctuations and dominance of configurational differences in large systems.

Contribution

It introduces a symmetry-resolved trace distance based on block structure, extending ETH diagnostics to non-Abelian symmetries and providing a detailed analysis of intra-sector fluctuations.

Findings

Spin-sector probability fluctuations are exponentially suppressed with system size.

Numerical results support that configurational trace distance dominates in the thermal regime.

The approach generalizes ETH diagnostics to systems with non-Abelian symmetry.

Abstract

We study diagnostics of thermalization in quantum many-body systems with global SU(2) symmetry, where the standard eigenstate thermalization hypothesis (ETH) is generalized to its non-Abelian form. As an eigenstate-level probe, we introduce a symmetry-resolved trace distance constructed from the block structure of the reduced density matrix. This block structure separates spin-sector probabilities from configurational fluctuations within each sector, naturally leading to a decomposition into a probability trace distance and a configurational trace distance. The microcanonical average of the former is bounded by fluctuations of the corresponding spin-sector probabilities within a microcanonical energy window, whereas the latter captures finer intra-sector fluctuations. In non-Abelian thermalizing systems, these spin-sector-probability fluctuations are constrained by the non-Abelian ETH and therefore become exponentially suppressed with system size. Numerical studies of the one-dimensional \(J_1\)--\(J_2\) Heisenberg chain are consistent with this picture and suggest that, in the thermal regime, the trace distance is asymptotically dominated by the configurational trace distance.