Symmetry and localisation in causally constrained quantum operator dynamics

TL;DR

This paper investigates how causality constraints in quantum many-body systems lead to algebraic structures that cause operator localization, resulting in non-ergodic behavior and specific entanglement properties.

Contribution

It introduces a formal framework linking causality, algebraic invariance, and operator localization, providing new insights into non-ergodic quantum dynamics and chaos suppression.

Findings

Causally independent subsystems arise from invariant sub-algebras.

Local constraints induce an entanglement area-law.

Spectral correlations differ from universal chaotic ensembles.

Abstract

This paper explores the connection between causality and many-body dynamics by studying the algebraic structure of tri-partite unitaries ('walls') which permanently arrest local operator spreading in their time-periodic evolution. We show that the resulting causally independent subsystems arise from the invariance of an embedded sub-algebra in the system (ie. a generalised symmetry) that leads to the splitting of operator space into commuting sub-algebras. The commutant structure of the invariant algebra is then used to construct local conserved quantities. Using representation theory of finite matrix algebras, the general form of wall gates is derived as unitary automorphisms. Taking causal independence as a minimal model for non-ergodic dynamics, we study its effect on probes of many-body quantum chaos. We prove an entanglement area-law due to local constraints and we study its stability against projective measurements. In a random ensemble exhibiting causal independence, we compare spectral correlations with the universal (chaotic) ensemble using the spectral form factor. Our results offer a rigorous understanding of locally constrained quantum dynamics from a quantum information perspective.