Subsystem symmetry protected topological order

You, Yizhi; Devakul, Trithep; Burnell, F. J.; Sondhi, S. L.

doi:10.1103/physrevb.98.035112

Cited by 149 publications

(216 citation statements)

References 94 publications

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“…The toy model we introduce here is a subsystem symmetric topological state [131][132][133][134][135][136] with gapless edge modes, which we refer to as topological plaquette Ising model (TPIM).…”

Section: Subsystem Symmetry Protected Topological Phasesmentioning

confidence: 99%

Fracton phases of matter

Pretko

Xie

You

2020

Int. J. Mod. Phys. A

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343

269

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Fractons are a new type of quasiparticle which are immobile in isolation, but can often move by forming bound states. Fractons are found in a variety of physical settings, such as spin liquids and elasticity theory, and exhibit unusual phenomenology, such as gravitational physics and localization.The past several years have seen a surge of interest in these exotic particles, which have come to the forefront of modern condensed matter theory. In this review, we provide a broad treatment of fractons, ranging from pedagogical introductory material to discussions of recent advances in the field. We begin by demonstrating how the fracton phenomenon naturally arises as a consequence of higher moment conservation laws, often accompanied by the emergence of tensor gauge theories.We then provide a survey of fracton phases in spin models, along with the various tools used to characterize them, such as the foliation framework. We discuss in detail the manifestation of fracton physics in elasticity theory, as well as the connections of fractons with localization and gravitation.Finally, we provide an overview of some recently proposed platforms for fracton physics, such as Majorana islands and hole-doped antiferromagnets. We conclude with some open questions and an outlook on the field.

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Section: Subsystem Symmetry Protected Topological Phasesmentioning

confidence: 99%

Fracton phases of matter

Pretko

Xie

You

2020

Int. J. Mod. Phys. A

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343

269

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“…A new chapter in the book of three-dimensional (3D) quantum phases of matter was opened with the discovery of fracton models [1][2][3][4][5][6][7][8], characterized by the presence of topological excitations with restricted mobility. These peculiar particles have attracted significant recent interest, thereby revealing intriguing connections to quantum information processing [9][10][11][12], topological order [13][14][15][16][17][18][19][20][21][22], sub-system symmetries [23][24][25][26][27][28][29][30], and slow quantum dynamics [1,3,[31][32][33]. Much of the phenomenology of fractons can also be realized in tensor gauge theories [34][35][36][37][38][39][40][41][42][43][44] with higher moment conservation laws, unveiling further connections of fractons with elasticity …”

Section: Introductionmentioning

confidence: 99%

Gauging permutation symmetries as a route to non-Abelian fractons

Prem

Williamson

2019

SciPost Phys.

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We discuss the procedure for gauging on-site Z 2 global symmetries of threedimensional lattice Hamiltonians that permute quasi-particles and provide general arguments demonstrating the non-Abelian character of the resultant gauged theories. We then apply this general procedure to lattice models of several well known fracton phases: two copies of the X-Cube model, two copies of Haah's cubic code, and the checkerboard model. Where the former two models possess an on-site Z 2 layer exchange symmetry, that of the latter is generated by the Hadamard gate. For each of these models, upon gauging, we find non-Abelian subdimensional excitations, including non-Abelian fractons, as well as non-Abelian looplike excitations and Abelian fully mobile pointlike excitations. By showing that the looplike excitations braid non-trivially with the subdimensional excitations, we thus discover a novel gapped quantum order in 3D, which we term a "panoptic" fracton order. This points to the existence of parent states in 3D from which both topological quantum field theories and fracton states may descend via quasi-particle condensation. The gauged cubic code model represents the first example of a gapped 3D phase supporting (inextricably) non-Abelian fractons that are created at the corners of fractal operators. Contents B Gauging the Hadamard symmetry of the 2D color code 30References 33 1 Here, by subdimensional we refer to excitations that are fully immobile (fractons), mobile only along lines (lineons), or mobile only along planes (planons) of the 3D lattice.2 Non-Abelian excitations are those which participate in multiple fusion channels and can thus encode quantum information non-locally throughout the system. See e.g., Refs. [77,78]. 3 We interchangeably refer to this global symmetry as swap, layer-swap, or layer exchange symmetry. 4 This gauging technique, when applied to two copies of a quantum double D(G), leads to a model in the same phase as D(G), withG = (G × G) Z2 which is non-Abelian even when the underlying group G is Abelian.

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“…Using the cocycle conditions of ω and α in Eqs. (25) and (28), one finds that the above equation is indeed satisfied. To show constraint (B11), consider a general state |a i−1 ⊗ |b i ⊗ |c i+1 .…”

Section: (B7)mentioning

confidence: 82%

Searching for fracton orders via symmetry defect condensation

Tantivasadakarn

Vijay

2020

Phys. Rev. B

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We propose a set of constraints on the ground-state wavefunctions of fracton phases, which provide a possible generalization of the string-net equations 1 used to characterize topological orders in two spatial dimensions. Our constraint equations arise by exploiting a duality between certain fracton orders and quantum phases with "subsystem" symmetries, which are defined as global symmetries on lower-dimensional manifolds, and then studying the distinct ways in which the defects of a subsystem symmetry group can be consistently condensed to produce a gapped, symmetric state. We numerically solve these constraint equations in certain tractable cases to obtain the following results: in d = 3 spatial dimensions, the solutions to these equations yield gapped fracton phases that are distinct as conventional quantum phases, along with their dual subsystem symmetryprotected topological (SSPT) states. For an appropriate choice of subsystem symmetry group, we recover known fracton phases such as Haah's code 2 , along with new, symmetry-enriched versions of these phases, such as non-stabilizer fracton models which are distinct from both the X-cube model and the checkerboard model in the presence of global time-reversal symmetry, as well as a variety of fracton phases enriched by spatial symmetries. In d = 2 dimensions, we find solutions that describe new weak and strong SSPT states, such as ones with both line-like subsystem symmetries and global time-reversal symmetry. In d = 1 dimension, we show that any group cohomology solution for a symmetry-protected topological state protected by a global symmetry, along with lattice translational symmetry necessarily satisfies our consistency conditions. CONTENTS

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Subsystem symmetry protected topological order

Cited by 149 publications

References 94 publications

Fracton phases of matter

Fracton phases of matter

Gauging permutation symmetries as a route to non-Abelian fractons

Searching for fracton orders via symmetry defect condensation

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