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arXiv:2301.05261 · Physical review. B./Physical review. B

Exact emergent higher-form symmetries in bosonic lattice models

Salvatore D. Pace, Xiao-Gang Wen

Published Nov 27, 2023• 54 citations
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Abstract

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 ${\mathbb{Z}}_{N}$ and $\mathrm{U}(1)\phantom{\rule{4pt}{0ex}}p$-gauge theory, finding regions of parameter space with exact emergent (anomalous) higher-form symmetries. Furthermore, we perform a generalized Landau analysis of a $2+1\mathrm{D}$ lattice model that gives rise to ${\mathbb{Z}}_{2}$ 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 ${\mathbb{Z}}_{2}$ 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 $p$-form symmetries with $p\ensuremath{\ge}2$ are.

DOI
10.1103/physrevb.108.195147

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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.
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.
Global symmetry (0-symmetry) acts on the whole space while higher kkk-symmetry acts on all the codimension-kkk closed subspaces. The usual condensed matter lattice theories do not include dynamical electromagnetic (EM) field and do not have higher symmetries (unless we engineer fine-tuned toy models). However, for gapped systems, (anomalous) higher symmetries can emerge from the usual condensed matter theories at low energies (usually in a spontaneously broken form). We pointed out that the emergent spontaneously broken higher symmetries are nothing but a kind of topological order. Thus the study of emergent spontaneously broken higher symmetries is a study of topological order. The emergent (anomalous) higher symmetries can be used to constrain possible phase transitions and possible phases induced by certain types of excitations in topological orders. (Anomalous) higher symmetry can also emerge in gapless systems if the gapless excitations contain gapless gauge fields. In particular, EM condensed matter systems that include the dynamical EM field have an emergent anomalous U(1)U(1)U(1)-1-symmetry below the energy gap of the magnetic monopoles. So EM condensed matter systems can realize some physical phenomena of anomalous higher symmetry. In particular, any gapped liquid phase of an EM condensed matter system (induced by arbitrary fluctuations and condensations of electric charges and photons) must have a nontrivial bosonic topological order.
Slow mixing and emergent one-form symmetries in three-dimensional Z2\mathbb{Z}_2Z2​ gauge theory
Symmetry-breaking order at low temperatures is often accompanied by slow relaxation dynamics, due to diverging free-energy barriers arising from interfaces between different ordered states. Here, we extend this correspondence to classical topological order, where the ordered states are locally indistinguishable, so there is no notion of interfaces between them. We study the relaxation dynamics of the three-dimensional (3D) classical Z2\mathbb{Z}_2Z2​ lattice gauge theory (LGT) as a canonical example. We prove a lower bound on the mixing time in the deconfined phase, tmix=exp⁡[Ω(L)]t_{\text{mix}} = \exp [Ω(L)]tmix​=exp[Ω(L)], where L is the linear system size. This bound applies even in the presence of perturbations that explicitly break the one-form symmetry between different long-lived states. This perturbation destroys the energy barriers between ordered states, but we show that entropic effects nevertheless lead to diverging free-energy barriers at nonzero temperature. Our proof establishes the LGT as a robust finite-temperature classical memory. We further prove that entropic effects lead to an emergent one-form symmetry, via a notion that we make precise. We argue that the exponential mixing time follows from universal properties of the deconfined phase, and numerically corroborate this expectation by exploring mixing time scales at the Higgs and confinement transitions out of the deconfined phase. These transitions are found to exhibit markedly different dynamic scaling, even though both have the static critical exponents of the 3D Ising model. We expect this novel entropic mechanism for memory and emergent symmetry to also bring insight into self-correcting quantum memories.
We explore exact generalized symmetries in the standard 2+1d lattice \mathbb{Z}_2 ℤ 2 gauge theory coupled to the Ising model, and compare them with their continuum field theory counterparts. One model has a (non-anomalous) non-invertible symmetry, and we identify two distinct non-invertible symmetry protected topological phases. The non-invertible algebra involves a lattice condensation operator, which creates a toric code ground state from a product state. Another model has a mixed anomaly between a 1-form symmetry and an ordinary symmetry. This anomaly enforces a nontrivial transition in the phase diagram, consistent with the “Higgs=SPT” proposal. Finally, we discuss how the symmetries and anomalies in these two models are related by gauging, which is a 2+1d version of the Kennedy-Tasaki transformation.
Generalized symmetries often appear in the form of emergent symmetries in low energy effective descriptions of quantum many-body systems. Non-invertible symmetries are a particularly exotic class of generalized symmetries, in that they are implemented by transformations that do not form a group. Such symmetries appear in large families of gapless states of quantum matter and constrain their low-energy dynamics. To provide a UV-complete description of such symmetries, it is useful to construct lattice models that respect these symmetries exactly. In this paper, we discuss two families of one-dimensional lattice Hamiltonians with finite on-site Hilbert spaces: one with (invertible) S_3 S 3 symmetry and the other with non-invertible \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) symmetry. Our models are largely analytically tractable and demonstrate all possible spontaneous symmetry breaking patterns of these symmetries. Moreover, we use numerical techniques to study the nature of continuous phase transitions between the different symmetry-breaking gapped phases associated with both symmetries. Both models have self-dual lines, where the models are enriched by so-called intrinsically non-invertible symmetries generated by Kramers-Wannier-like duality transformations. We provide explicit lattice operators that generate these non-invertible self-duality symmetries. We show that the enhanced symmetry at the self-dual lines is described by a 2+1d symmetry-topological-order (SymTO) of type JK_4\boxtimes \overline{JK}_4 J K 4 ⊠ J K ¯ 4 . The condensable algebras of the SymTO determine the allowed gapped and gapless states of the self-dual S_3 S 3 -symmetric and \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) -symmetric models.