Recoverable information and emergent conservation laws in fracton stabilizer codes

Schmitz, Albert T.; Ma, Han; Nandkishore, Rahul; Parameswaran, S. A.

doi:10.1103/physrevb.97.134426

Cited by 67 publications

(66 citation statements)

References 57 publications

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“…That is, the EH for these models can be mapped onto an effective subsystem symmetric Ising-like model only for those entanglement cuts consistent with the planar (fractal) subsystem symmetries of the X-cube (cubic code). Thus, we show that the ES serves as a clear entanglement measure distinguishing fracton order from topological order, adding to the existing diagnostics for fracton order [71,73,74]. We also provide strong evidence for a correspondence between the low-lying ES and the lowlying spectrum of physical edge states, extending the validity of the edge-ES correspondence to gapped fracton phases.…”

supporting

confidence: 62%

“…In certain cases, we rely upon results previously established in Refs. [70,71,75], and provide details only when required for the remainder. After reviewing the requisite background, we present a general method for deriving the entanglement spectrum of the ground state of a stabilizer Hamiltonian in the presence of arbitrary perturbations, allowing us to compare and contrast a myriad of phases within the same framework.…”

Section: Entanglement Spectra Of Stabilizer Code Hamiltoniansmentioning

confidence: 99%

“…After reviewing the requisite background, we present a general method for deriving the entanglement spectrum of the ground state of a stabilizer Hamiltonian in the presence of arbitrary perturbations, allowing us to compare and contrast a myriad of phases within the same framework. From the ES, one can also extract the entanglement entropy and recoverable information [71]; however, as the EE has been studied for both fracton order and SSPTs before, we only comment on this briefly.…”

Section: Entanglement Spectra Of Stabilizer Code Hamiltoniansmentioning

confidence: 99%

“…Since all members of S commute, stabilizers in S multiplicatively generate an Abelian group G = { s∈F O s : F ∈ P[S]}, where the power set of S, denoted P[S], is the set of all subsets of S. Not all stabilizers are independent, so S may over-determine G. We refer to any C ⊆ S such that s∈C O s = I as a constraint. Following [70,71,75], let d G = log 2 |G| and {O i } i≤d G ⊆ S be a complete, independent generating set for G. Elements g ∈ G can hence be labelled by a binary vector n = (n 1 , n 2 , . .…”

Section: A the Stabilizer Formalismmentioning

confidence: 99%

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Entanglement spectra of stabilizer codes: A window into gapped quantum phases of matter

2019

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The entanglement spectrum (ES) provides a barometer of quantum entanglement and encodes physical information beyond that contained in the entanglement entropy. In this paper, we explore the ES of stabilizer codes, which furnish exactly solvable models for a plethora of gapped quantum phases of matter. Studying the ES for stabilizer Hamiltonians in the presence of arbitrary weak local perturbations thus allows us to develop a general framework within which the entanglement features of gapped topological phases can be computed and contrasted. In particular, we study models harboring fracton order, both type-I and type-II, and compare the resulting ES with that of both conventional topological order and of (strong) subsystem symmetry protected topological (SSPT) states. We find that non-local surface stabilizers (NLSS), a set of symmetries of the Hamiltonian which form on the boundary of the entanglement cut, act as purveyors of universal non-local features appearing in the entanglement spectrum. While in conventional topological orders and fracton orders, the NLSS retain a form of topological invariance with respect to the entanglement cut, subsystem symmetric systems-fracton and SSPT phases-additionally show a non-trivial geometric dependence on the entanglement cut, corresponding to the subsystem symmetry. This sheds further light on the interplay between geometric and topological effects in fracton phases of matter and demonstrates that strong SSPT phases harbour a measure of quasi-local entanglement beyond that encountered in conventional SPT phases. We further show that a version of the edge-entanglement correspondence, established earlier for gapped two-dimensional topological phases, also holds for gapped three-dimensional fracton models.

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supporting

confidence: 62%

Section: Entanglement Spectra Of Stabilizer Code Hamiltoniansmentioning

confidence: 99%

Section: Entanglement Spectra Of Stabilizer Code Hamiltoniansmentioning

confidence: 99%

Section: A the Stabilizer Formalismmentioning

confidence: 99%

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Entanglement spectra of stabilizer codes: A window into gapped quantum phases of matter

2019

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“…That is, if we take out a sub-region, say of size R × R × R, and calculate its entanglement entropy, we would find an area law term which scales as R 2 and a sub-leading linear term which scales as R [69][70][71][72]. (One must take care to avoid any potential spurious contributions to the entanglement entropy, however [73].)…”

Section: A Product Of X Operators Over Links Along a Straight Line Anmentioning

confidence: 99%

Fracton phases of matter

Pretko

Xie

You

2020

Int. J. Mod. Phys. A

339

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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Topological entanglement entropy in d-dimensions for Abelian higher gauge theories

Ibieta-Jimenez

Petrucci

Xavier

et al. 2020

J. High Energ. Phys.

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We compute the topological entanglement entropy for a large set of lattice models in d-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy S A in a sub-region A is proportional to log(GSDÃ), where GSDÃ is the ground state degeneracy of a particular restriction of the full model to A. The quantity GSDÃ can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂A. There is also a topological contribution coming from A itself, that may be non-zero when A has a non-trivial homology. We present some examples and discuss how the topology of A affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension d. The result is in agreement with entanglement calculations for known topological models.

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Recoverable information and emergent conservation laws in fracton stabilizer codes

Cited by 67 publications

References 57 publications

Entanglement spectra of stabilizer codes: A window into gapped quantum phases of matter

Entanglement spectra of stabilizer codes: A window into gapped quantum phases of matter

Fracton phases of matter

Topological entanglement entropy in d-dimensions for Abelian higher gauge theories

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