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arXiv:2311.16235 · Physical Review X

Entanglement Properties of Gauge Theories from Higher-Form Symmetries

Wen-Tao Xu, Tibor Rakovszky, Michael Knap, Frank Pollmann

Published Jan 2, 2025• 16 citations
arXivHTMLPDF

Abstract

We explore the relationship between higher-form symmetries and entanglement properties in lattice gauge theories with discrete gauge groups, which can exhibit both topologically ordered phases and higher-form symmetry-protected topological (SPT) phases. Our study centers on a generalization of the Fradkin-Shenker model describing Z 2 lattice gauge theory with matter, where the Gauss law constraint can be either emergent or exact. The phase diagram includes a topologically ordered deconfined phase and a nontrivial SPT phase protected by a 1-form and a 0-form symmetry, among others. We obtain the following key findings: First, the entanglement properties of the model depend on whether the 1-form symmetries and the Gauss law constraint are exact or emergent. For the emergent Gauss law, the entanglement spectrum (ES) of the nontrivial SPT phase exhibits degeneracies, which are robust at low energies against weak perturbations that explicitly break the exact 1-form symmetry. When the Gauss law and the 1-form symmetry are both exact, the ES degeneracy is extensive. This extensive degeneracy turns out to be fragile and can be removed completely by infinitesimal perturbations that explicitly break the exact 1-form symmetry while keeping the Gauss law exact. Second, we consider the ES in the topologically ordered phase where 1-form symmetries are spontaneously broken. In contrast to the ES of the nontrivial SPT phase, we find that spontaneous higher-form symmetry breaking removes “half” of the ES levels, leading to a nondegenerate ES in the topologically ordered phase, in general. Third, we derive a connection between spontaneous higher-form symmetry breaking and the topological entanglement entropy (TEE). Using this relation, we investigate the entanglement entropy that can be distilled in the deconfined phase of the original Fradkin-Shenker model using gauge-invariant measurements. We show that the TEE is robust against the measurement when the 1-form symmetry is emergent but it is fragile when the 1-form symmetry is exact. Our results demonstrate the advantage of higher-form symmetries for understanding entanglement properties of gauge theories.

DOI
10.1103/physrevx.15.011001

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We explore the relationship between higher-form symmetries and entanglement properties in lattice gauge theories with discrete gauge groups, which can exhibit both topologically ordered phases and higher-form symmetry-protected topological (SPT) phases. Our study centers on a generalization of the Fradkin-Shenker model describing Z 2 lattice gauge theory with matter, where the Gauss law constraint can be either emergent or exact. The phase diagram includes a topologically ordered deconfined phase and a nontrivial SPT phase protected by a 1-form and a 0-form symmetry, among others. We obtain the following key findings: First, the entanglement properties of the model depend on whether the 1-form symmetries and the Gauss law constraint are exact or emergent. For the emergent Gauss law, the entanglement spectrum (ES) of the nontrivial SPT phase exhibits degeneracies, which are robust at low energies against weak perturbations that explicitly break the exact 1-form symmetry. When the Gauss law and the 1-form symmetry are both exact, the ES degeneracy is extensive. This extensive degeneracy turns out to be fragile and can be removed completely by infinitesimal perturbations that explicitly break the exact 1-form symmetry while keeping the Gauss law exact. Second, we consider the ES in the topologically ordered phase where 1-form symmetries are spontaneously broken. In contrast to the ES of the nontrivial SPT phase, we find that spontaneous higher-form symmetry breaking removes “half” of the ES levels, leading to a nondegenerate ES in the topologically ordered phase, in general. Third, we derive a connection between spontaneous higher-form symmetry breaking and the topological entanglement entropy (TEE). Using this relation, we investigate the entanglement entropy that can be distilled in the deconfined phase of the original Fradkin-Shenker model using gauge-invariant measurements. We show that the TEE is robust against the measurement when the 1-form symmetry is emergent but it is fragile when the 1-form symmetry is exact. Our results demonstrate the advantage of higher-form symmetries for understanding entanglement properties of gauge theories.
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Although condensed matter systems usually do not have higher-form symmetries, we show that, unlike 0-form symmetry, higher-form symmetries can emerge as exact symmetries at low energies and long distances. In particular, emergent higher-form symmetries at zero temperature are robust to arbitrary local UV perturbations in the thermodynamic limit. This result is true for both invertible and noninvertible higher-form symmetries. Therefore emergent higher-form symmetries are exact emergent symmetries: they are not UV symmetries but constrain low-energy dynamics as if they were. Since phases of matter are defined in the thermodynamic limit, this implies that a UV theory without higher-form symmetries can have phases characterized by exact emergent higher-form symmetries. We demonstrate this in three lattice models, the quantum clock model and emergent ZN{\mathbb{Z}}_{N}ZN​ and U(1)p\mathrm{U}(1)\phantom{\rule{4pt}{0ex}}pU(1)p-gauge theory, finding regions of parameter space with exact emergent (anomalous) higher-form symmetries. Furthermore, we perform a generalized Landau analysis of a 2+1D2+1\mathrm{D}2+1D lattice model that gives rise to Z2{\mathbb{Z}}_{2}Z2​ gauge theory. Using exact emergent 1-form symmetries accompanied by their own energy/length scales, we show that the transition between the deconfined and Higgs/confined phases is continuous and equivalent to the spontaneous symmetry-breaking transition of a Z2{\mathbb{Z}}_{2}Z2​ symmetry, even though the lattice model has no symmetry. Also, we show that this transition line must always contain two parts separated by multi-critical points or other phase transitions. We discuss the physical consequences of exact emergent higher-form symmetries and contrast them to emergent 0-form symmetries. Lastly, we show that emergent 1-form symmetries are no longer exact at finite temperatures, but emergent ppp-form symmetries with $p\ensuremath{\ge}2$ are.
Higher-form symmetries act on subdimensional spatial manifolds of a quantum system. They can emerge as an exact symmetry at low energies even when they are explicitly broken at the microscopic level, making them difficult to characterize. In this work, we propose that the emergence of 1-form symmetries is information-theoretic in nature, characterized by the preservation of information about a specific bare (microscopic) 1-form symmetry. As a consequence, the loss of the emergent 1-form symmetry is an information-theoretic transition which we argue to be revealed from the long-range entanglement in the ensemble of post-measurement states. We analytically determine the regimes in which a 1-form symmetry emerges in product states on one- and two-dimensional lattices. In analytically intractable regimes, we demonstrate how to efficiently detect 1-form symmetries with a global quantum error correction (QEC) decoder and numerically examine the information-theoretic transition of the 1-form symmetry, including systems with Z 2 topological order. As a practical application of our framework, we show that once the 1-form symmetry is detected to exist, a topological quantum phase transition characterized by the spontaneous breaking of the 1-form symmetry can be accurately determined by a disorder parameter. We further argue that our proposed theory for emergent 1-form symmetries offers new perspectives on particle condensation and suggests sharp information-theoretic phase boundaries between Higgs and confining regimes in the Z 2 lattice gauge theory.
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