Dynamical generalization of Yetter’s model based on a crossed module of discrete groups

Bochniak, Arkadiusz; Hadasz, Leszek; Ruba, Błażej

doi:10.1007/jhep03(2021)282

Cited by 4 publications

(9 citation statements)

References 53 publications

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“…Our model has four phases, each corresponding to one attractive renormalization group fixed point. In each of the fixed points local dynamics becomes trivial, but some nonlocal observables remain important: We remark that the four renormalization group fixed points described here may be identified with four integrable hamiltonians described in [25].…”

Section: Phase Diagram Proposal For D =mentioning

confidence: 84%

“…Its hamiltonian realizations resembling the Kitaev's toric code were constructed in [22][23][24]. In [25] a common generalization of the Yetter's model, 2-form Z n electrodynamics and lattice Yang-Mills theory has been proposed. It is a genuinely dynamical model, formulated in the hamiltonian formalism, which reduces to a TQFT only in certain limits.…”

Section: Jhep09(2021)068mentioning

confidence: 99%

“…In order to make the paper more accessible, we have decided to define everything explicitly using notations standard in lattice gauge theory. We refer to [25] for an exposition of the slightly more involved formalism of crossed modules and proofs of various algebraic facts used in the present text.…”

Section: Jhep09(2021)068mentioning

confidence: 99%

“…This observation follows from the fact that the presented proof relies only on general properties of gauge transformations and constraint-preserving moves. These were discussed in detail in [25].…”

Section: Jhep09(2021)068mentioning

confidence: 99%

“…In this appendix we use terminology and notations from [25]. Thus in contrast to the remainder of the paper, this part is not fully self-contained.…”

Section: A Non-spherical Wilson Surfacesmentioning

confidence: 99%

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Dynamics of a lattice 2-group gauge theory model

Bochniak

Hadasz

Korcyl

et al. 2021

J. High Energ. Phys.

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We study a simple lattice model with local symmetry, whose construction is based on a crossed module of finite groups. Its dynamical degrees of freedom are associated both to links and faces of a four-dimensional lattice. In special limits the discussed model reduces to certain known topological quantum field theories. In this work we focus on its dynamics, which we study both analytically and using Monte Carlo simulations. We prove a factorization theorem which reduces computation of correlation functions of local observables to known, simpler models. This, combined with standard Krammers-Wannier type dualities, allows us to propose a detailed phase diagram, which form is then confirmed in numerical simulations. We describe also topological charges present in the model, its symmetries and symmetry breaking patterns. The corresponding order parameters are the Polyakov loop and its generalization, which we call a Polyakov surface. The latter is particularly interesting, as it is beyond the scope of the factorization theorem. As shown by the numerical results, expectation value of Polyakov surface may serve to detects all phase transitions and is sensitive to a value of the topological charge.

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Section: Phase Diagram Proposal For D =mentioning

confidence: 84%

Section: Jhep09(2021)068mentioning

confidence: 99%

Section: Jhep09(2021)068mentioning

confidence: 99%

Section: Jhep09(2021)068mentioning

confidence: 99%

“…In this appendix we use terminology and notations from [25]. Thus in contrast to the remainder of the paper, this part is not fully self-contained.…”

Section: A Non-spherical Wilson Surfacesmentioning

confidence: 99%

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Dynamics of a lattice 2-group gauge theory model

Bochniak

Hadasz

Korcyl

et al. 2021

J. High Energ. Phys.

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Study of a lattice 2-group gauge model

Bochniak¹,

Hadasz²,

Korcyl³

et al. 2022

Proceedings of the 38th International Symposium on Lattice Field Theory — PoS(LATTICE2021)

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Gauge theories admit a generalisation in which the gauge group is replaced by a finer algebraic structure, known as a 2-group. The first model of this type is a Topological Quantum Field Theory introduced by Yetter. We discuss a common generalisation of both the Yetter's model and Yang-Mills theory and in particular we focus on the lattice formulation of such model for finite 2-groups. In the second part we present a particular realization based on a 2-group constructed from Z 4 groups. In the selected model, independent degrees of freedom are associated to both links and faces of a four-dimensional lattice and are subject to a certain constraint. We present the details of this construction, discuss the expected dynamics in different regions of phase space and show numerical results from Monte Carlo simulations corroborating these expectations.

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Exactly solvable models for 2+1D topological phases derived from crossed modules of semisimple Hopf algebras

Koppen¹,

Martins²,

Martin³

2021

Preprint

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We define an exactly solvable model for 2+1D topological phases of matter on a triangulated surface derived from a crossed module of semisimple finite-dimensional Hopf algebras, the Hopf-algebraic higher Kitaev model. This model generalizes both the Kitaev quantum double model for a semisimple Hopf algebra and the full higher Kitaev model derived from a 2-group, and can hence be interpreted as a Hopfalgebraic discrete higher gauge theory.We construct a family of crossed modules of semisimple Hopf algebras) that depends on four finite groups E, G, X and Y . We calculate the ground-state spaces of the resulting model when G = E = {1} and when Y = {1}, both properly Hopf-algebraic constructions; prove that they are canonically independent of the triangulations; and find a 2+1D TQFT whose state spaces on surfaces give the ground-state spaces. These TQFTs are particular cases of Quinn's finite total homotopy TQFT and hence the state spaces assigned to surfaces are free vector spaces on sets of homotopy classes of maps from a surface to homotopy finite spaces, in this case obtained as classifying spaces of finite groupoids and finite crossed modules of groupoids.We leave it as an open problem whether the ground-state space of the Hopfalgebraic higher Kitaev model on a triangulated surface is independent of the triangulation for general crossed modules of semisimple Hopf algebras, whether a TQFT always exists whose state space on a surface gives the ground-state space of the model, and whether the ground-state space of the model obtained from E, G, X, Y can always be given a homotopical explanation.

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Dynamical generalization of Yetter’s model based on a crossed module of discrete groups

Cited by 4 publications

References 53 publications

Dynamics of a lattice 2-group gauge theory model

Dynamics of a lattice 2-group gauge theory model

Study of a lattice 2-group gauge model

Exactly solvable models for 2+1D topological phases derived from crossed modules of semisimple Hopf algebras

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