Categorical symmetries at criticality

Wu, Xiao-Chuan; Ji, Wenjie; Xu, Cenke

doi:10.1088/1742-5468/ac08fe

Cited by 25 publications

(22 citation statements)

References 25 publications

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“…The quantity order diagnosis operator (ODO) was introduced in Ref. [82] to characterize the behavior of the explicit and inexplicit symmetries, especially the notion of spontaneous symmetry breaking of both the explicit and the inexplicit symmetries defined above. The ODO reduces to previously introduced concepts in specific cases.…”

Section: Discussionmentioning

confidence: 99%

“…[15] for systems without subsystem symmetries. But for systems with a more exotic subsystem symmetries [82] the proper form of the ODO is not always defined on a simple patch of the lattice.…”

Section: Discussionmentioning

confidence: 99%

“…In order to describe the behavior of theZ N symmetry, Ref. [16] introduced the "order diagnosis operator"Õ C . Represented in terms of lattice operators, the ODO for the dual Z…”

Section: Example 1: Z N Order-disorder Transitionmentioning

confidence: 99%

“…The notion of categorical symmetry unifies the explicit symmetry of a model and the inexplicit symmetry of its dual model, and treat them on an equal footing [15]. To diagnose the behavior of the categorical symmetries, and most importantly to diagnose the explicit symmetry and the inexplicit dual symmetry on an equal footing, a concept of "order diagnosis operator" (ODO) was introduced, whose expectation value reduces to the correlation function between order parameters for an explicit 0-form symmetry, and reduces to a Wilson loop for an explicit 1form symmetry [16]. The ODO was also referred to as the "patch operator" in Ref.…”

Section: Introductionmentioning

confidence: 99%

“…ODOs for systems with special symmetries such as subsystem symmetries may have special forms and behaviors, and examples with these special symmetries were discussed in Ref. [16]. The expectation value of O i j andÕ C in the 2d quantum Ising system characterizes different phases of the system.…”

Section: Introductionmentioning

confidence: 99%

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Universal features of higher-form symmetries at phase transitions

Jian

2021

SciPost Phys.

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We investigate the behavior of higher-form symmetries at various quantum phase transitions. We consider discrete 1-form symmetries, which can be either part of the generalized concept “categorical symmetry” (labelled as \tilde{Z}_N^{(1)}Z̃N(1)) introduced recently, or an explicit Z_N^{(1)}ZN(1) 1-form symmetry. We demonstrate that for many quantum phase transitions involving a Z_N^{(1)}ZN(1) or \tilde{Z}_N^{(1)}Z̃N(1) symmetry, the following expectation value \langle \left( O_\mathcal{C}\right)^2 \rangle⟨(O𝒞)2⟩ takes the form \langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim - \frac{A}{\epsilon} P + b \log P⟨(logO𝒞)2⟩∼−AϵP+blogP, where O_\mathcal{C}O𝒞 is an operator defined associated with loop \mathcal{C}𝒞 (or its interior \mathcal{A}𝒜), which reduces to the Wilson loop operator for cases with an explicit Z_N^{(1)}ZN(1) 1-form symmetry. PP is the perimeter of \mathcal{C}𝒞, and the b \log PblogP term arises from the sharp corners of the loop \mathcal{C}𝒞, which is consistent with recent numerics on a particular example. bb is a universal microscopic-independent number, which in (2+1)d(2+1)d is related to the universal conductivity at the quantum phase transition. bb can be computed exactly for certain transitions using the dualities between (2+1)d(2+1)d conformal field theories developed in recent years. We also compute the "strange correlator" of O_\mathcal{C}O𝒞: S_{\mathcal{C}} = \langle 0 | O_\mathcal{C} | 1 \rangle / \langle 0 | 1 \rangleS𝒞=⟨0|O𝒞|1⟩/⟨0|1⟩ where |0\rangle|0⟩ and |1\rangle|1⟩ are many-body states with different topological nature.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…In order to describe the behavior of theZ N symmetry, Ref. [16] introduced the "order diagnosis operator"Õ C . Represented in terms of lattice operators, the ODO for the dual Z…”

Section: Example 1: Z N Order-disorder Transitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Universal features of higher-form symmetries at phase transitions

Jian

2021

SciPost Phys.

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Spontaneously broken subsystem symmetries

Distler

Karch

Raz

2022

J. High Energ. Phys.

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We investigate the spontaneous breaking of subsystem symmetries directly in the context of continuum field theories by calculating the correlation function of charged operators. Our methods confirm the lack of spontaneous symmetry breaking in some of the existing continuum field theories with subsystem symmetries, as had previously been established based on a careful analysis of the spectrum. We present some novel continuum field theory constructions that do exhibit spontaneous symmetry breaking whenever allowed by general principles. These interesting patterns of symmetry breaking occur despite the fact that all the theories we study are non-interacting.

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The boundary phase transitions of the 2+1D ℤN topological order via topological Wick rotation

Yan

Holiverse

2023

J. High Energ. Phys.

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In this work, we show that a critical point of a 1d self-dual boundary phase transition between two gapped boundaries of the ℤN topological order can be described by a mathematical structure called an enriched fusion category. The critical point of a boundary phase transition can be viewed as a gappable non-chiral gapless boundary of the ℤN topological order. A mathematical theory of the gapless boundaries of 2d topological orders developed by Kong and Zheng (arXiv:1905.04924 and arXiv:1912.01760) tells us that all macroscopic observables on the gapless boundary form an enriched unitary fusion category, which can be obtained by a holographic principle called the “topological Wick rotation.” Using this method, we obtain the enriched fusion category that describes a critical point of the phase transition between the e-condensed boundary and the m-condensed boundary of the ℤN topological order. To verify this idea, we also construct a lattice model to realize the critical point and recover the mathematical data of this enriched fusion category. The construction further shows that the categorical symmetry of the boundary is determined by the topological defects in the bulk, which indicates the holographic principle indirectly. This work shows, as a concrete example, that the mathematical theory of the gapless boundaries of 2+1D topological orders is a powerful tool to study general phase transitions.

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Categorical symmetries at criticality

Cited by 25 publications

References 25 publications

Universal features of higher-form symmetries at phase transitions

Universal features of higher-form symmetries at phase transitions

Spontaneously broken subsystem symmetries

The boundary phase transitions of the 2+1D ℤN topological order via topological Wick rotation

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