Classical fracton spin liquid and Hilbert space fragmentation in a 2D spin-$1/2$ model

TL;DR

This paper introduces a simple 2D spin model realizing classical fracton spin liquids with extensive ground state degeneracy, and explores how Hilbert space fragmentation affects quantum tunneling and order.

Contribution

It provides a straightforward discretization of higher-rank gauge theory to study fractonic properties and demonstrates the impact of Hilbert space fragmentation on quantum behavior.

Findings

Classical ground states form a manifold described by a tensor Gauss' law.

Quantum tunneling between ground states is suppressed, causing Hilbert space fragmentation.

The system exhibits either magnetic long-range order or a classical spin liquid depending on parameters.

Abstract

Classical U(1) fracton spin liquids feature an extensive ground state degeneracy and follow an effective description in terms of a tensor Gauss' law where charges, so-called fractons, have restricted mobility. Here we introduce a simple spin model that realizes such a state by straightforward discretization of the higher-rank gauge theory on a square lattice. The simplicity of this construction offers direct insights into the system's fundamental fractonic properties, such as real-space fracton configurations, height-field representation of the classical ground state manifold as well as properties of local and non-local fluctuations within the fracton-free subspace. By sampling classical Ising states from the extensive ground state manifold, we show that the effective tensor Gauss' law remains intact when explicitly enforcing the spin-1/2 length constraint, demonstrating the existence of a classical Ising fracton spin liquid. However, we observe that perturbative quantum effects are insufficient to efficiently tunnel between classical ground states, leading to severe Hilbert space fragmentation which obscures fractonic quantum behavior. Specifically, by simulating the spin-1/2 quantum model with Green function Monte Carlo as a function of the Rokhsar-Kivelson potential, we find that the system supports either magnetic long-range order or a classical spin liquid. Our findings highlight the crucial role of Hilbert-space fragmentation in fractonic spin systems but also indicate ways to mitigate such effects via increasing the spin magnitude to $S=1$, investigated in a companion paper.