Foliated fracton order in the checkerboard model

Shirley, Wilbur; Slagle, Kevin; Xie, Changsheng

doi:10.1103/physrevb.99.115123

Cited by 50 publications

(39 citation statements)

References 48 publications

(59 reference statements)

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“…By calculating the universal properties of foliated fracton phases as discussed in Ref. [25][26][27][28], we see that the two models could actually be in the same foliated fracton phase, and the explicit mapping discussed in section VI and section VII further confirms this result.…”

Section: Discussionsupporting

confidence: 66%

Foliated fracton order in the Majorana checkerboard model

2019

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We establish the presence of foliated fracton order in the Majorana checkerboard model. In particular, we describe an entanglement renormalization group transformation which utilizes toric code layers as resources of entanglement, and furthermore discuss entanglement signatures and fractional excitations of the model. In fact, we give an exact local unitary equivalence between the Majorana checkerboard model and the semionic X-cube model augmented with decoupled fermionic modes. This mapping demonstrates that the model lies within the X-cube foliated fracton phase. arXiv:1904.01111v1 [cond-mat.str-el] 1 Apr 2019

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Section: Discussionsupporting

confidence: 66%

Foliated fracton order in the Majorana checkerboard model

2019

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“…As discussed in Ref. [94], excitations of this model can also be characterized through their QSS, as was the case with the X-Cube model. A convenient choice for the generating set for the QSS sectors is given by fracton excitations labelled by the ordered pair c X c Z , indicating the violated X c (Z c ) stabilizer in the c X (c Z ) = r, g, b sublattice.…”

Section: Checkerboard Model With On-site Hadamard Symmetrymentioning

confidence: 93%

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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“…However, it is by now a common observation that discrete lattice geometry seems to play a key role in the physics of fracton phases, and it appears that some geometrical structure may be a necessary ingredient of a tractable theory. An important example is the theory of foliated fracton phases [24][25][26][27][28][29], described in more detail below, where geometrical information is provided by a certain kind of foliation structure. Lattice translation imposes a different kind of geometrical structure complementary to that provided by foliation.…”

Section: Introductionmentioning

confidence: 99%

Fracton fusion and statistics

Pai

Hermele

2019

Phys. Rev. B

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We introduce and develop a theory of fusion and statistical processes of gapped excitations in Abelian fracton phases. The key idea is to incorporate lattice translation symmetry via its action on superselection sectors, which results in a fusion theory endowed with information about the non-trivial mobility of fractons and sub-dimensional excitations. This results in a description of statistical processes in terms of local moves determined by the fusion theory. Our results can be understood as providing a characterization of translation-invariant fracton phases. We obtain simple descriptions of the fusion theory in the X-cube and checkerboard fracton models, as well as for gapped electric and magnetic excitations of some gapless U(1) tensor gauge theories. An alternate route to the X-cube model fusion theory is provided by starting with a system of decoupled twodimensional toric code layers, and giving a description of the p-string condensation mechanism within our approach. We discuss examples of statistical processes of fractons and sub-dimensional excitations in the X-cube and checkerboard models. As an application of the ideas developed, we prove that the X-cube and semionic X-cube models realize distinct translation-invariant fracton phases, even when the translation symmetry is broken corresponding to an arbitrary but finite enlargement of the crystalline unit cell. CONTENTS

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Foliated fracton order in the checkerboard model

Cited by 50 publications

References 48 publications

Foliated fracton order in the Majorana checkerboard model

Foliated fracton order in the Majorana checkerboard model

Gauging permutation symmetries as a route to non-Abelian fractons

Fracton fusion and statistics

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