Hilbert Space Fragmentation and Gauge Symmetry

TL;DR

This paper explores Hilbert space fragmentation in lattice gauge theories and spin chains, revealing emergent gauge symmetries and their role in quantum simulations of gauge theories.

Contribution

It introduces an emergent gauge symmetry in a spin chain with Hilbert space fragmentation, enabling exact quantum simulation of a gauge theory without explicit gauge invariance.

Findings

Hilbert space fragmentation leads to many disconnected sectors.

Emergent gauge symmetry labels exponentially many sectors.

Simulating a non-gauge-invariant Hamiltonian reproduces gauge theory dynamics.

Abstract

The Hamiltonian formulation of lattice gauge theories plays a central role in quantum simulations of gauge theories, and understanding their spectrum and other properties is expected to become crucial in the upcoming years. The relevant Hamiltonians in this framework possess local symmetry at each lattice site and may exhibit higher-form symmetries. There are then an exponentially large number of dynamically disconnected symmetry sectors, most of which are not translation-invariant. An exponential number of dynamically disconnected sectors, i.e., Hilbert space fragmentation, can also occur in systems in which no such symmetries have been identified. In this contribution, we describe an emergent gauge symmetry that is valid only in a subset of sectors of the fragmented $S=1$ dipole-conserving spin chain. These non-invertible symmetries can label exponentially many of the model's sectors. Simulating this Hamiltonian, which is not gauge-invariant, yields an exact quantum simulation of a gauge theory.