Categories of quantum liquids I

Kong, Liang; Zheng, Hao

doi:10.1007/jhep08(2022)070

Cited by 22 publications

(11 citation statements)

References 68 publications

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“…Such states are short-range entangled. The SPT states are also short-range entangled [29], in contrast to topologically ordered states [30][31][32][33][34][35][36][37][38][39][40] that are long-range entangled. Thus, short-range entangled phases are generally symmetry broken (described by the Landau paradigm), SPT (outside of the Landau paradigm), or support simultaneous coexistence of symmetry breaking order and SPT order.…”

Section: Introductionmentioning

confidence: 97%

$Z_3$ and $(\times Z_3)^3$ symmetry protected topological paramagnets

Topchyan¹,

Iugov²,

Mirumyan³

et al. 2022

Preprint

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We identify three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by (×Z 3 ) 3 ≡ Z 3 × Z 3 × Z 3 symmetry and smaller Z 3 symmetry. We study microscopic models for the gapless edge and discuss the corresponding conformal field theories and their central charges. These are the edge states of s = 1 paramagnets protected by Z 3 ×Z 3 ×Z 3 and Z 3 symmetries, in analogy with the free fermion XX model edge discussed by Levin and Gu for the spin-1/2 Z 2 Ising paramagnet. We show that in spin-1 magnets, there are numerous options to obtain self-dual Hamiltonians and gapless edge modes. These models form the basis for the realization of a variety of symmetry-protected topological phases, opening a window for studies of the unconventional quantum criticalities between them.

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Section: Introductionmentioning

confidence: 97%

$Z_3$ and $(\times Z_3)^3$ symmetry protected topological paramagnets

Topchyan¹,

Iugov²,

Mirumyan³

et al. 2022

Preprint

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“…To be more precise, the mathematical structure on a 1d gapped/gapless boundary can be split into two parts: a local quantum symmetry U which encodes the information of local observables; and a Mod U -enriched fusion category Mod U M which encodes all the topological defects. The enriched fusion category Mod U M is called the topological skeleton [22].…”

Section: Jhep06(2023)010 1 Introductionmentioning

confidence: 99%

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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“…To be more precisely, the mathematical structure on a 1d gapped/gapless boundary can be split into two parts: a local quantum symmetry V which encodes the information of local observables; and a Mod V -enriched fusion category Mod V M which encodes all the topological data. The enriched fusion category Mod V M is called the topological skeleton [KZ20a].…”

Section: Introductionmentioning

confidence: 99%

The boundary phase transitions of the 2+1D $\mathbb{Z}_N$ topological order via topological Wick rotation

Yan¹,

Holiverse²

2022

Preprint

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In this work, we construct an enriched fusion category to describes 1d self-dual boundary phase transition between two gapped boundaries of the Z N topological order. The boundary phase transition actually corresponds to a gappable non-chiral gapless boundary of Z N topological order. To prove the idea, we also construct a lattice model to recover the mathematical data of this enriched fusion category. The construction process further shows that the symmetry of the boundary is determined by the topological sector in the bulk. This work as a concrete example shows that the mathematical theory developed in arxiv.1905.04924 and arxiv.1912.01760 is powerful tool to study more general phase transitions.

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Categories of quantum liquids I

Cited by 22 publications

References 68 publications

$Z_3$ and $(\times Z_3)^3$ symmetry protected topological paramagnets

$Z_3$ and $(\times Z_3)^3$ symmetry protected topological paramagnets

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

The boundary phase transitions of the 2+1D $\mathbb{Z}_N$ topological order via topological Wick rotation

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