Advances in Mathematical Physics

Volume 2010 (2010), Article ID 671039, 18 pages

http://dx.doi.org/10.1155/2010/671039

## Microscopic Description of 2D Topological Phases, Duality, and 3D State Sums

^{1}Institute for Scientific Interchange Foundation, Villa Gualino, Viale Settimio Severo 75, 10131 Torino, Italy^{2}Dipartimento di Fisica Nucleare e Teorica, Istituto Nazionale di Fisica Nucleare, Universita degli Studi di Pavia, Sezione di Pavia, via A. Bassi 6, 27100 Pavia, Italy^{3}Dipartimento di Fisica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received 7 September 2009; Accepted 23 January 2010

Academic Editor: Debashish Goswami

Copyright © 2010 Zoltán Kádár et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1, 2]. Perhaps the most recent territory, where they appeared to describe real physical systems, is that of topological phases of matter, being, for example, responsible for the fractional quantum Hall effect [3]. Since the idea of fault-tolerant quantum computation appeared in the literature [4], TQFTs are also important in quantum information theory. These new applications also enhanced the mathematical research, and led to classification of the simplest models [5].

Due to their topological nature, TQFTs 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 form and the curvature -form of a gauge field [6]. We will deal here with a special class of three-dimensional theories, which describe doubled topological phases and restrict our attention to discrete gauge groups . The context they appeared in, in recent physics literature [4, 7], is Hilbert spaces of states in two dimensions and dynamics therein, which are boundary Hilbert spaces of the relevant TQFTs. Operators acting on correspond to three-dimensional amplitudes on the thickened surface. In this paper we will explain this correspondence, which was proved for the ground-state projection recently [8], and provide the geometric interpretation of the ribbon operators, which create quasiparticle excitations from the ground state. This is a step towards extending the correspondence to identify the ribbon operators as invariants of manifolds with coloured links embedded in them in the TQFT.

The emergence of topological phases from a description of microscopic degrees of freedom is modeled by the lattice models of Kitaev [4] and Levin and Wen [7]. Since they generically have degenerate ground states and quasi-particle excitations insensitive to local disturbances, they are also investigated in the theory of quantum computation [9], their continuum limit being closely related to the spin network simulator [10, 11]. The ground states were extensively studied in the literature; their MERA (multiscale entanglement renormalization ansatz) [12, 13] and tensor network representations [14] have been constructed to study for example, their entanglement properties [15, 16]. Finding the explicit root of these structures in lattice gauge theory and TQFT can help to understand their physical properties.

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 [17]. To name an application, this description turned out to provide a convenient basis for most approaches to modern quantum gravity theories [18, 19]. In this paper we will show in detail how these dual descriptions give rise to Kitaev’s quantum double models in one hand and the spin net models of Levin and Wen on the other. Then the ribbon operators in both models and their identification will be discussed.

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 3, we briefly introduce the string net models of Levin and Wen on the honeycomb lattice in the surface and recall the proof [8] that the ground-state projection is given by the Turaev-Viro amplitude on . The boundary triangulations of are given by the dual graphs of decorated by the labels inherited from the “initial” and “final” spin nets. In Section 4 the duality between the states of the Kitaev model and the string nets will be shown by changing the basis from the group algebra to the Fourier one. By using this duality and an additional projection, we will obtain the electric constraint operators of the string net models. The matrix elements of the magnetic constraints are also recovered provided that the local rules of Levin and Wen hold. We explain that they do in all BF theories, which is a strong motivation in their favor for the case when the gauge group is finite. In Section 5, we discuss ribbon operators and give their three-dimensional geometric interpretation in terms of framed links in the Turaev-Viro picture. Finally, a summary is given with a list of questions for future research.

#### 2. Turaev-Viro Models

In three dimensions both the (the field strength) and fields of BF theory can be considered to be forms valued in the Lie algebra of the gauge group . The action can then be written as with being an invariant nondegenerate bilinear form on the Lie algebra and is a smooth, oriented, closed three-manifold. We may start from the case when is a semisimple Lie group relevant in particle physics theories and gravity, being the connection in the principal -bundle over . In three dimensions the “space-time” separated form of the Lagrangian has the structure where are spatial indices. (There is not necessarily physical time in the theory; one can do this decomposition for Euclidean signature as well.) The first term is the standard kinetic term, the second implies the (Gauss) constraint of gauge invariance, the third stands for the vanishing of the (two-dimensional) field strength ( and are Lagrange multipliers), while and stand for the covariant and the exterior derivatives, respectively.

Since locally the solution of the constraints is given by a pure gauge ( smooth function, with being the spatial hypersurface), one may discretize the theory by introducing a lattice on the spatial surface and quantize the remaining degrees of freedom: the holonomies (elements of ) describing the coordinate change between faces of the lattice. They correspond to the edges of the dual lattice, which is constructed by placing a vertex inside each face and connecting new vertices, which were put inside neighbouring faces. This dual lattice is the starting point of the models in [4], the electric constraints are the remainders of the Gauss constraint and the magnetic ones are the remainders of the flatness constraint. For a detailed exposition see, for example, [20].

The partition function of the above BF theory is formally obtained by taking the functional integral over the fields an of the phase associated to the classical action It is not so easy to give this definition a precise sense, but for the moment it is not necessary to go into further details. What does matter is that there exists a consistent way to discretize the partition function by considering an oriented triangulation of the manifold by assigning two Lie algebra elements to each edge in . The generator can be thought as the integral of along the edge , whereas as the logarithm of the group element corresponding to the holonomy around the edge . (To be more precise, one needs to introduce the dual complex by putting vertices inside every tetrahedron, connecting those vertices which were put in neighbouring tetrahedra and a vertex should be singled out on the boundary of each dual face. Then the procedure to get is the following: take the dual face corresponding to . Multiply the holonomies along the boundary edges of this dual face starting from the vertex singled out in a circular direction determined by the orientation of (say, by the right-hand rule). The logarithm of this group element is .) Then the Feynman integral in (1) can be replaced by . The integrals will yield Dirac deltas and one can now proceed with decompositions in terms of irreps of the gauge group. This way one ends up with a discrete state sum instead of the original Feynman integral, where each state is the triangulation coloured with irreps and its weight is given by the precise final form of the amplitude (examples are given below). The structure of the partition function (amplitude) for a prototypical theory, the Ponzano-Regge model [21] corresponding to , reads where is the Wigner symbol of depending on the irreps decorating the edges of the tetrahedron , is the dimension of the irreps assigned to the edge , and the sum ranges over all states, that is, all possible colourings of the edges with irreps. It turns out that this type of state sum is well-defined and independent of the chosen triangulation for a large class of models. (This is one way to define a TQFT rigorously.) For a systematic derivation of this state sum from action functionals, see [22] or [23 Section ].

The Ponzano-Regge partition function is formally independent of , but is divergent. However, the Turaev-Viro (TV) model [24], a regularized version thereof, has a well defined partition function, given by where the underlying algebraic structure is the quasitriangular Hopf algebra with fixed, denotes the irreps of (with nonzero trace), is the so-called quantum dimension of , the constant is defined by , the quantity in the brackets is the quantum symbol, and is the number of vertices of the triangulation. One finds the precise definitions of all the quantities along with the algebraic properties assuring consistency and triangulation independence [21] of the amplitudes in [24]. We will briefly mention the origin of the latter property in the next section (Note, that in the case of the Turaev-Viro model based on these properties hold, as in the case of the Ponzano-Regge model, for arbitrary , but it is only for being a root of unity, when the partition function is finite.). The final fact for this introductory section is about the form of the amplitude for manifolds with nonempty boundary. The associated boundary triangulation, whose edges are decorated by labels , derives from a given triangulation in the 3D bulk and is kept fixed. The amplitude reads where is the number of internal vertices, the index ranges over internal edges, over boundary edges, over boundary vertices (each boundary vertex is the endpoint of an internal edge; is its colour), and over all tetrahedra, and the summation is done for internal edge labels only, while those on the boundary are kept fixed.

Note that there is a quasi-triangular Hopf algebra associated to finite groups as well, the so-called Drinfeld (quantum) double [25]. There, the dimensions as well as the symbols can be obtained from the representation theory of the group .

#### 3. String Nets

Levin and Wen [7] started off from the algebraic structure underlying the above models (consistent set of symbols and quantum dimensions), which serves as the algebraic data in defining TQFT’s. Taking these data for granted, they constructed a two-dimensional lattice model, which we will now introduce briefly. Consider a surface with a fixed oriented honeycomb lattice embedded in it. The Hilbert space is spanned by all possible decorations of the edges with labels ; we will refer to them as irreps (of or the finite group ) as we will not need to treat the most general TQFT’s. The Hamiltonian is a sum of two families of mutually commuting constraint operators: where the first sum is over all vertices and the second is over all plaquettes of the lattice. for being the irreps decorating the edges adjacent to the vertex (the numbers are referred to as fusion coefficients between the irreps: ). The magnetic constraints are written as a sum over irreps and the action of the individual terms is while its action on the rest of the state supported on the honeycomb lattice is trivial. The numbers are the symbols, part of the algebraic data of a TQFT; denotes the irreps dual to . Changing the orientation of an edge is equivalent to changing its label to its dual . Levin and Wen use a different normalisation that of , : Before proceeding, let us write down an important algebraic property of the symbols: This identity is called the Biedernharn-Elliot identity or pentagon equation, which holds in every TQFT. In the concrete examples mentioned above, they can be proved by the definition of the symbols as connecting the two different fusion channels of recoupling irreps (graphically encoded by in Section 4.2) (by means of using two different ways of coupling five irreps.)

##### 3.1. Reconstructing 3D Geometry

In our work [8] we recovered a three-dimensional Turaev-Viro invariant [26, 27] from the algebra of Levin and Wen. We associated geometric tetrahedra to the algebraic symbols, where the edges are decorated with irreps from the symbols. In that a convention needs to be adopted; for example, the upper row should correspond to a (triangular) face of the tetrahedron and labels in the same column should correspond to opposite edges. In the examples we are looking at there is always a normalization of the ’s such that they have the same symmetry as the tetrahedron. Orientation of edges can also be taken care of in a consistent manner; we will however omit them for most of what follows. Now we can translate the Biedernharn-Elliot identity to geometry. Then Figure 1 arises: where the two configurations (three tetrahedra joined at the edge and two tetrahedra glued along the triangle ) correspond to the left-hand and right-hand sides of (9), respectively. This is a cornerstone in proving triangulation independence of the amplitudes (3) and (4) and shows that whenever tetrahedra are glued labels corresponding to internal edges have to be summed over.