Full dyon excitation spectrum in extended Levin-Wen models

Hu, Yuting; Geer, Nathan; Wu, Yong‐Shi

doi:10.1103/physrevb.97.195154

Cited by 60 publications

(113 citation statements)

References 40 publications

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“…Additionally the techniques discussed here may allow a generalization to recently introduced topological invariants for 4-dimensional manifolds, introduced by Bärenz and Barrett [3]. For the model discussed here we construct different bases, which generalize the fusion bases for the (2 + 1)-dimensional extended topological quantum field theories [4][5][6], and reveal a fascinating duality. These bases will allow a wide range of further applications, e.g.…”

Section: Introductionmentioning

confidence: 99%

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

Dittrich

2017

J. High Energ. Phys.

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We apply the recently suggested strategy to lift state spaces and operators for (2 + 1)-dimensional topological quantum field theories to state spaces and operators for a (3 + 1)-dimensional TQFT with defects. We start from the (2 + 1)-dimensional TuraevViro theory and obtain a state space, consistent with the state space expected from the Crane-Yetter model with line defects.This work has important applications for quantum gravity as well as the theory of topological phases in (3 + 1) dimensions. It provides a self-dual quantum geometry realization based on a vacuum state peaked on a homogeneously curved geometry. The state spaces and operators we construct here provide also an improved version of the Walker-Wang model, and simplify its analysis considerably.We in particular show that the fusion bases of the (2 + 1)-dimensional theory lead to a rich set of bases for the (3 + 1)-dimensional theory. This includes a quantum deformed spin network basis, which in a loop quantum gravity context diagonalizes spatial geometry operators. We also obtain a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian.Furthermore, the construction presented here can be generalized to provide state spaces for the recently introduced dichromatic four-dimensional manifold invariants.

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Section: Introductionmentioning

confidence: 99%

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

Dittrich

2017

J. High Energ. Phys.

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“…Its advantages are multiple: first, its language is closer to that of the Turaev Viro based representation [28]. Second, it translates a range of techniques used in the context of string-net models [18,32] to an holonomy-based formalism. Finally, it provides a lattice-independent description of the Kitaev model [41], which can in turn be mapped onto an 'extended' string-net model [19,20].…”

Section: An Alternative Description Of the Bf Representationmentioning

confidence: 99%

“…An explicit definition for SU (2) q , at q root of unity, was given in [31], which is easily generalizable to modular fusion categories (see also [28]). The fusion basis for more general fusion categories appeared -albeit only implicitly -in [32].…”

Section: Jhep02(2017)061mentioning

confidence: 99%

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Fusion basis for lattice gauge theory and loop quantum gravity

Delcamp

Dittrich

Riello

2017

J. High Energ. Phys.

133

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We introduce a new basis for the gauge-invariant Hilbert space of lattice gauge theory and loop quantum gravity in (2 + 1) dimensions, the fusion basis. In doing so, we shift the focus from the original lattice (or spin-network) structure directly to that of the magnetic (curvature) and electric (torsion) excitations themselves. These excitations are classified by the irreducible representations of the Drinfel'd double of the gauge group, and can be readily "fused" together by studying the tensor product of such representations. We will also describe in detail the ribbon operators that create and measure these excitations and make the quasi-local structure of the observable algebra explicit. Since the fusion basis allows for both magnetic and electric excitations from the onset, it turns out to be a precious tool for studying the large scale structure and coarse-graining flow of lattice gauge theories and loop quantum gravity. This is in neat contrast with the widely used spin-network basis, in which it is much more complicated to account for electric excitations, i.e. for Gauß constraint violations, emerging at larger scales. Moreover, since the fusion basis comes equipped with a hierarchical structure, it readily provides the language to design states with sophisticated multi-scale structures. Another way to employ this hierarchical structure is to encode a notion of subsystems for lattice gauge theories and (2 + 1) gravity coupled to point particles. In a follow-up work, we have exploited this notion to provide a new definition of entanglement entropy for these theories.

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“…This will avoid certain technical inconveniences of the (undeformed) BF representation, in particular the need to resort to a Bohr compactification of the dual of the group ( ) SU 2 . Furthermore, the TV model in its extended form has been quite recently developed mathematically [54][55][56][57] and has also found widespread applications in condensed matter and quantum computation [58][59][60][61]. In particular, the structure of the excitations for this model is very well understood.…”

Section: Introductionmentioning

confidence: 99%

Quantum gravity kinematics from extended TQFTs

2017

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In this paper, we show how extended topological quantum field theories (TQFTs) can be used to obtain a kinematical setup for quantum gravity, i.e. a kinematical Hilbert space together with a representation of the observable algebra including operators of quantum geometry. In particular, we consider the holonomy-flux algebra of (2+1)-dimensional Euclidean loop quantum gravity, and construct a new representation of this algebra that incorporates a positive cosmological constant. The vacuum state underlying our representation is defined by the Turaev-Viro TQFT. This vacuum state can be thought of as being peaked on connections with homogeneous curvature. We therefore construct here a generalization, or more precisely a quantum deformation at root of unity, of the previously introduced SU(2) BF representation. The extended Turaev-Viro TQFT provides a description of the excitations on top of the vacuum. These curvature and torsion excitations are classified by the Drinfeld center category of the quantum deformation of SU(2), and are essential in order to allow for a representation of the holonomies and fluxes. The holonomies and fluxes are generalized to ribbon operators which create and interact with the excitations. These excitations agree with the ones induced by massive and spinning particles, and therefore the framework presented here allows automatically for a description of the coupling of such matter to + ( ) 2 1 -dimensional gravity with a cosmological constant. The new representation constructed here presents a number of advantages over the representations which exist so far. In particular, it possesses a very useful finiteness property which guarantees the discreteness of spectra for a wide class of quantum (intrinsic and extrinsic) geometrical operators. Also, the notion of basic excitations leads to a so-called fusion basis which offers exciting possibilities for the construction of states with interesting global properties, as well as states with certain stability properties under coarse graining. In addition, the work presented here showcases how the framework of extended TQFTs, as well as techniques from condensed matter, can help design new representations, and construct and understand the associated notion of basic excitations. This is essential in order to find the best starting point for the construction of the dynamics of quantum gravity, and will enable the study of possible phases of spin foam models and group field theories from a new perspective.Several such realizations of quantum geometry are now known. Each realization is based on and can be characterized by a different kinematical vacuum state. So far, all representations have in common that this vacuum is totally squeezed, i.e. sharply peaked either on the flux operators or on the holonomy operators. The operators of quantum geometry act on this vacuum state, and generate thereby all the excitations of the Hilbert space underlying the representation of the holonomy-flux algebra. The vacuum is therefore cyclic with respect to the set of ...

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Full dyon excitation spectrum in extended Levin-Wen models

Cited by 60 publications

References 40 publications

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

Fusion basis for lattice gauge theory and loop quantum gravity

Quantum gravity kinematics from extended TQFTs

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