Entanglement Properties of Gauge Theories from Higher-Form Symmetries
arxiv(2023)
摘要
We explore the relationship between higher-form symmetries and entanglement
properties in discrete lattice gauge theories, which can exhibit both
topologically ordered phases and higher-form symmetry-protected topological
(SPT) phases. Our study centers on generalizing the Fradkin-Shenker model,
where the Gauss law constraint can be either emergent or exact. The phase
diagram includes a topologically ordered phase and a non-trivial SPT phase
protected by a 1-form and a 0-form symmetry. We obtain the following key
findings: First, the entanglement properties depend on whether the 1-form
symmetries and the Gauss law are exact or emergent. For the emergent Gauss law,
the entanglement spectrum (ES) of the non-trivial 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 is 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
non-trivial SPT phase, we find that spontaneous higher-form symmetry breaking
removes "half" of the ES levels, leading to a non-degenerate 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 fragile when
the 1-form symmetry is exact.
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