Hilbert subspace ergodicity

TL;DR

This paper investigates how mechanisms like quantum many-body scars and Hilbert space fragmentation influence complete Hilbert space ergodicity, revealing the emergence of Hilbert Subspace Ergodicity and its implications for quantum state exploration.

Contribution

It introduces the concept of Hilbert Subspace Ergodicity, showing how ETH-violation mechanisms lead to ergodic behavior within subspaces, expanding understanding of quantum thermalization.

Findings

ETH-violation mechanisms induce subspace ergodicity

Hilbert Subspace Ergodicity enables t-designs in subspaces

Fragmentation affects quantum state exploration

Abstract

Ergodicity has been one of the fundamental concepts underpinning our understanding of thermalization in isolated systems since the first developments in classical statistical mechanics. Recently, a similar notion has been introduced for quantum systems, termed complete Hilbert space ergodicity (CHSE), in which the evolving quantum state explores all of the available Hilbert space. This contrasts with the eigenstate thermalisation hypothesis (ETH), in which thermalisation is formulated via the properties of matrix elements of local operators in the energy eigenbasis. In this work we explore how ETH-violation mechanisms, including quantum many-body scars and Hilbert space fragmentation can affect complete Hilbert space ergodicity. We find that the presence of these mechanisms leads to CHSE in decoupled subspaces, a phenomenon we call Hilbert Subspace Ergodicity, and which represents a protocol for constructing t-designs in subspaces.