Pauli Stabilizer Models of Twisted Quantum Doubles

Ellison, Tyler D.; Chen, Yu-An; Dua, Arpit; Shirley, Wilbur; Tantivasadakarn, Nathanan; Williamson, Dominic J.

doi:10.1103/prxquantum.3.010353

Cited by 27 publications

(23 citation statements)

References 63 publications

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“…As a first step, our results for fractionalization in Z N topological order should straightforwardly extend to any finite Abelian group G. From here, one can use the results of Ref. [52] to obtain fractionalized models with arbitrary nonchiral Abelian topological order, equivalent to the Abelian twisted quantum double models. One topic that was not covered in this paper is the idea of anomalous symmetry fractionalization.…”

Section: Discussionmentioning

confidence: 90%

Fractionalization of subsystem symmetries in two dimensions

et al. 2022

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Section: Discussionmentioning

confidence: 90%

Fractionalization of subsystem symmetries in two dimensions

et al. 2022

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“…48 gives a dimer model for DS order; can we generate this effective resonance using local anyon fluctuations? Moreover, recent work [129] has shown that using a 4state system one can even write down a stabilizer model for DS order; is this also possible using our formalism? Quadrupolar models.…”

Section: Discussionmentioning

confidence: 97%

Unifying Kitaev magnets, kagome dimer models and ruby Rydberg spin liquids

Verresen¹,

Vishwanath²

2022

Preprint

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The exploration of quantum spin liquids (QSLs) has been guided by different approaches including the resonating valence bond (RVB) picture, deconfined lattice gauge theories and the Kitaev model. More recently, a spin liquid ground state was numerically established on the ruby lattice, inspired by the Rydberg blockade mechanism. Here we unify these varied approaches in a single parent Hamiltonian, in which local fluctuations of anyons stabilize deconfinement. The parent Hamiltonian is defined on kagomé triangles-each hosting four RVB-like states-and includes only Ising interactions and single-site transverse fields. In the weak-field limit, the ruby spin liquid and exactly soluble kagomé dimer models are recovered, while the strong-field limit reduces to the Kitaev honeycomb model, thereby unifying three seemingly different approaches to QSLs. We similarly obtain the chiral Yao-Kivelson model, honeycomb toric code and a new spin-1 quadrupolar Kitaev model. The last is shown to be in a QSL phase by a non-local mapping to the kagomé Ising antiferromagnet. We demonstrate various applications of our framework, including (a) an adiabatic deformation of the ruby lattice model to the exactly soluble kagomé dimer model, conclusively establishing the QSL phase in the former; and (b) demystifying the dynamical protocol for measuring off-diagonal strings in the Rydberg implementation of the ruby lattice spin liquid. More generally, we find an intimate connection between Kitaev couplings and the repulsive interactions used for emergent dimer models. For instance, we show how a spin-1/2 XXZ model on the ruby lattice encodes a Kitaev honeycomb model, providing a new route toward realizing the latter in cold-atom or solid-state systems. CONTENTSI. Introduction 1 II. Model 3 A. Emergent dimer model in a tensor product Hilbert space 3 B. Hamiltonian 6 C. Brief summary of the phase diagram(s) 8 III. Quantum liquids from e-anyon fluctuations 9 A. Weak-field limit: dimer liquid stabilizer model 9 B. Arbitrary field: duality to spin-1/2 transverse field Ising model on the kagomé lattice 9 C. Strong-field limit: spin-1 quadrupolar Kitaev model 10 IV. Quantum liquids from f -anyon fluctuations 11 A. Weak-field limit: distinct dimer liquid SETs 11 B. Exact solution for arbitrary field: Majorana Fermi surfaces and chiral non-Abelian Ising topological order 12 C. Strong-field limit: spin-1/2 Kitaev model 13 V. Applications 13 A. Adiabatically connecting ruby lattice spin liquid to solvable stabilizer Hamiltonian 13 B. The Hadamard transformation for a dimer model 14 C. A natural family of spin-1 Kitaev models 15 D. Kagomé dimer model as honeycomb toric code 16 VI. Local rewriting of spin-3/2 model as two-body spin-1/2 'grandparent' models 17 A. Ruby lattice 17 B. Star lattice 18 VII. Conclusions and future directions 19 Acknowledgments 21 References 21 A. Spin-3/2 Hilbert space 25 1. Algebra: Majoranas and bosonic operators 25 2. Z 3 isomorphisms 26 3. Z 2 Hadamard 26 4. Reduction to spin-1/2 26 5. Matrix representation 27 6. Connection t...

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“…The tube algebra represents D ω (G), the twisted quantum double of G, introduced by Dijkgraaf et al in Ref. [33] and plays an important role in the study of (finite) gauge theory models for topological phases and its applications to topological stabilizer codes [6,30,34,35].…”

Section: Topological Gauge Theory Modelsmentioning

confidence: 99%

“…It is known that any Abelian twisted quantum double of a finite group can be obtained from boson condensation in an untwisted quantum double of a group of larger cardinality [35,40]. The condensation process can be viewed as a non-invertible domain wall from the parent phase (an untwisted quantum double of an Abelian group) to the condensate (the twisted quantum double in question).…”

Section: Condensation Domain Walls For Abelian Phasesmentioning

confidence: 99%

“…( 48), with n 1 = n 2 = n 12 =: n. For N = 2, it describes the six-semion phase where all elementary fluxes have semionic self-exchange statistics, see Ref. [35]. This twisted quantum double can also be reppresented as a non-trivial condensate of the untwisted quantum double of Z N 2 × Z N 2 .…”

Section: Condensation Domain Walls Between Fully Twisted and Untwiste...mentioning

confidence: 99%

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Bulk-to-boundary anyon fusion from microscopic models

Fuente¹,

Eisert²,

Bauer³

2023

Preprint

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Topological quantum error correction based on the manipulation of the anyonic defects constitutes one of the most promising frameworks towards realizing fault-tolerant quantum devices. Hence, it is crucial to understand how these defects interact with external defects such as boundaries or domain walls. Motivated by this line of thought, in this work, we study the fusion events between anyons in the bulk and at the boundary in fixed-point models of 2+1-dimensional nonchiral topological order defined by arbitrary fusion categories. Our construction uses generalized tube algebra techniques to construct a bi-representation of bulk and boundary defects. We explicitly derive a formula to calculate the fusion multiplicities of a bulk-to-boundary fusion event for twisted quantum double models and calculate some exemplary fusion events for Abelian models and the (twisted) quantum double model of S 3 , the simplest non-Abelian group-theoretical model. Moreover, we use the folding trick to study the anyonic behavior at non-trivial domain walls between twisted S 3 and twisted Z 2 as well as Z 3 models. A recurring theme in our construction is an isomorphism relating twisted cohomology groups to untwisted ones. The results of this work can directly be applied to study logical operators in two-dimensional topological error correcting codes with boundaries described by a twisted gauge theory of a finite group.

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Pauli Stabilizer Models of Twisted Quantum Doubles

Cited by 27 publications

References 63 publications

Fractionalization of subsystem symmetries in two dimensions

Fractionalization of subsystem symmetries in two dimensions

Unifying Kitaev magnets, kagome dimer models and ruby Rydberg spin liquids

Bulk-to-boundary anyon fusion from microscopic models

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