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Doubled topological phases introduced by Kitaev, Levin, and Wen supported on two-dimensional lattices are Hamiltonian versions of three-dimensional topological quantum field theories described by the Turaev-Viro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state is shown in case of the honeycomb lattice and the gauge group being a finite group by means of the well-known duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three-dimensional geometrical interpretation are given.

Topological quantum field theories (TQFTs) in three dimensions describe a variety of physical and toy models in many areas of modern physics. The absence of local degrees of freedom is a great simplification it often leads to complete solvability [

Due to their topological nature, TQFT's admit discretization yet remaining an exact description of the theory given by an action functional on a continuous manifold. One large class thereof is the so-called BF theories, whose Lagrangian density is given by the wedge product of a

The emergence of topological phases from a description of microscopic degrees of freedom is modeled by the lattice models of Kitaev [

Lattice gauge theories admit seemingly very different descriptions. A state can be represented by assigning elements of the gauge group to edges of the lattice. The dual description in terms of spin network states where edges are labelled by irreducible representations (irreps) of the gauge group and vertices by invariant intertwiners are also well known since the publication of [

The organization of the paper is as follows. In the next section, we introduce the Turaev-Viro models via the example of BF theories. In Section

In three dimensions both the

Since locally the solution of the constraints is given by a pure gauge

The partition function of the above BF theory is formally obtained by taking the functional integral over the fields

The Ponzano-Regge partition function (

Note that there is a quasi-triangular Hopf algebra associated to finite groups as well, the so-called Drinfeld (quantum) double

Levin and Wen [

In our work [

Now, one can try to find the geometric counterpart of the operator (

Note that

Now, we will restrict our attention to the case when the TQFT is given by the structure of the double

The Hilbert space of the Kitaev model (and that of a lattice gauge theory) is spanned by the group algebra basis

The basic idea is the well-known expansion of any function

Let us recall the electric constraints of the Kitaev model. They are written in terms of the following operators:

The transformation rule of

A shorter way to arrive at invariant spin network states is to consider a generic gauge invariant state supported on

To recall the construction of the magnetic operators of the Kitaev model, we define auxiliary operators associated to pairs

Let us summarize what we have achieved. If we have a Lie group

Nevertheless, for finite groups the local rules are, even if well motivated, postulates. The magnetic operator has been derived in a more direct way by introducing some auxiliary degrees of freedom in the very recent paper in [

In Section

A general ribbon in the spin net model is a string running along a certain path in the honeycomb lattice. The corresponding operator has the following structure:

In Figure

We may interpret the above in the following way. There is the path

The observables in the TV model are typically ribbon graphs, fat graphs or links embedded in a manifold, over the labels of which, there is no summation in the amplitude [

The ribbon operators are present in the Kitaev model as well [

A prototypical example shown in the Figure

In this paper we have been studying the lattice models of Levin, Wen, and Kitaev from two perspectives. On one hand we identified the ground states and the constraint operators of these models in case the underlying lattice is the honeycomb and the gauge group is a finite group. This has been achieved by changing the basis from that of the group algebra, that is, when edges are decorated by group elements, to the Fourier basis. This basis is spanned by the matrix elements of the irreps. A special linear combination by means of invariant intertwiners at the vertices has been shown to provide the range of all electric constraints and the projection at individual vertices has been identified with the projection to the invariant subspace. Then, the magnetic operators in the group algebra basis have been shown to correspond to those in the spin net model once the local rules postulated in the latter are satisfied. We gave an argument in favour of them from lattice gauge theory with continuous gauge group.

A second focus of the paper was on mapping the spin net to the Turaev-Viro state sum. We have used the idea of building up simplicial manifolds by tetrahedra with edges decorated with irreps corresponding to

One would also like to match these ribbon operators also in the model of Kitaev and the spin net of Levin and Wen. However, finding generalized spin network representations of the previous so that one could reduce them to the spin network basis is not straightforward.

In a series of papers [

The remarks above make it manifest that the efficient (approximate) quantum algorithms proposed in [

Z. Kádár would like to thank Dirk Schlingemann and Zoltán Zimborás for helpful discussions.