Loop Charges and Fragmentation in Pairwise Difference Conserving Circuits

TL;DR

This paper introduces pairwise-difference-conserving circuits (PDC), revealing how loop charges induce Hilbert-space fragmentation and nonergodic dynamics in quantum cellular automata across arbitrary graphs.

Contribution

The work defines PDC circuits on arbitrary graphs, uncovers their loop charge symmetries, and demonstrates their role in Hilbert-space fragmentation and nonergodic behavior.

Findings

Loop charges form an extensive set of conserved quantities.

Hilbert space fragments into dynamically disconnected sectors.

Nonergodic dynamics observed in ladder geometries.

Abstract

In this work, we introduce a broad class of circuits, or quantum cellular automata, which we call 'pairwise-difference-conserving circuits' (PDC). These models are characterized by local gates that preserve the pairwise difference of local operators (e.g. particle number). Such circuits can be de- fined on arbitrary graphs in arbitrary dimensions for both quantum and classical degrees of freedom. A key consequence of the PDC construction is the emergence of an extensive set of loop charges associated with closed walks of even length on the graph. These charges exhibit a one-dimensional character reminiscent of 1-form symmetries and lead to strong Hilbert-space fragmentation. As a case study, we analyze a quasi one-dimensional ladder geometry, where we characterize all dynam- ically disconnected sectors by the loop-charge symmetries, providing a complete decomposition of the Hilbert space. For the ladder geometry, we observe clear signatures of nonergodic dynamics even within the largest symmetry sector.