A Hopf Algebras

2D Quantum Double Models From a 3D Perspective

Abstract

In this paper we look at 3D lattice models that are generalizations of the state sum model used to define the Kuperberg invariant of 3-manifolds. The partition function is a scalar constructed as a tensor network where the building blocks are tensors given by the structure constants of an involutary Hopf algebra . These models are very general and are hard to solve in its entire parameter space. One can obtain familiar models, such as ordinary gauge theories, by letting be the group algebra of a discrete group and staying on a certain region of the parameter space. We consider the transfer matrix of the model and show that Quantum double Hamiltonians are derived from a particular choice of the parameters. Such a construction naturally leads to the star and plaquette operators of the quantum double Hamiltonians, of which the toric code is a special case when . This formulation is convenient to study ground states of these generalized quantum double models where they can naturally be interpreted as tensor network states. For a surface , the ground state degeneracy is determined by the Kuperberg 3-manifold invariant of . It is also possible to obtain extra models by simple enlarging the allowed parameter space but keeping the solubility of the model. While some of these extra models have appeared before in the literature, our 3D perspective allows for an uniform description of them.

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Departmento de Física Matemática Universidade de São Paulo - USP

CEP 05508-090 Cidade Universitária, São Paulo - Brasil

1 Introduction

In recent years numerous efforts have gone towards finding systems exhibiting topological order [1]. Topological order is believed to classify new phases of matter which cannot be classified under Landau’s symmetry breaking scheme of classifying the states of matter. Their existence came to be known after the discovery of the fractional quantum Hall effect (see reviews by [2, 3]) and high temperature superconductivity  [4]. These have been spurred by the need to find viable realizations of protected qubits to be used in topological quantum computation [5, 6, 7]. In such attempts to find systems with topological order, the exactly soluble 2D lattice models of Levin and Wen [8] provide a general method for obtaining anyon models [9, 10, 11] using modular tensor categories. They exhibit the most general range of possible quasiparticle statistics of any known Hamiltonian lattice theory. Indeed, Hamiltonians of this kind are thought to exist for a large class of achiral anyon theories known as quantum double models. This class encompasses many of the previously studied anyon lattice models such as the Toric code [7], many of its generalizations [50, 13, 14] and doubled Chern-Simons theories [15].

Chain-Mail link invariants [16] have been used to realize such models from the spacetime perspective [17, 18], realizing the partition function of the Levin-Wen model as a knot invariant of a complicated link in three dimensional space. This provides a more physical picture of the Levin-Wen lattice model in terms of Wilson loops living on the edges of the three dimensional spacetime lattice. As the Hamiltonian of this model is made of commuting projectors the spectrum of the model can be obtained easily. The quasi particles are given a interpretation in the spacetime picture by adding additional strings to the chain-mail link. The chain-mail link invariant is known to be equivalent to the Turaev-Viro state sum invariant of 3-manifolds. Alternatively these models can also be related to the Turaev-Viro invariant of a closed 3-manifold. The invariant in this case is built out of a tensor network with values in a spherical category for a triangulation of the 3-manifold [19].

While the above models construct the Levin-Wen models starting from the Turaev-Viro 3-manifold invariants, in this paper we do the analogue for the toric code and its generalizations called quantum double models. Our starting point, however, will be the Kuperberg invariant [21] of 3-manifolds. This invariant uses a involutory Hopf algebra and a discrete presentation of the manifold known as the Heegaard splitting. In order to make contact with discretizations actually used in physical models, we use Heegaard splittings that come from triangular or cubic lattices. Our presentation is therefore similar to the one used in [20] to construct the same invariant.

The data needed to define the Kuperberg invariant is a Heegaard decomposition of the 3-manifold and a involutory Hopf algebra . It can be casted as a state sum model by using a Heegaard splitting coming from a triangulation of the manifold. From this point of view the Kuperberg invariant is essentially a partition function constructed out of weights associated to links and faces of a triangulated 3-manifold. These weights can be decomposed into the structure constants of an involutory Hopf algebra , where the antipode map squares to the identity. It is useful to regard the multiplication , the co-multiplication and the antipode of the Hopf algebra as tensors in . The invariant is a scalar constructed as a tensor network built out of and .

Turaev-Viro invariant is also a partition function of a state sum model. The relation between the Turaev-Viro and Kuperberg state sum models can be understood as follows. In the particular case when is the group algebra of a group , the states to be summed over are group elements associated to the links of the triangulation and the corresponding lattice field theory takes the familiar form of a gauge theory. The Turaev-Viro state sum model on the other hand starts from a fusion category . The degrees of freedom associated to links are elements of and therefore they are different lattice models. However, if one takes to be the category of representations of , it has been proved in [22] that . There is a closer relation between the two state sum models for the particular case when is a group algebra of a finite group . For this case one can show that the Turaev-Viro state sum model is a dual description of the Kuperberg state sum model when written in terms of spin network states where links are labeled by irreducible representations. This can be derived by using results discussed in [23] and [24].

Both Kuperberg and Turaev-Viro state sum models are 3D lattice topological QFTs. They describe the low energy limit of fully dynamical lattice models such as the quantum double models. One could ask what 3D lattice models reproduces (in the Hamiltonian formalism) not only the low energy states but the entire spectrum of these models. Such a 3D field theory can not be a lattice TQFT. We show that a 3D model that resembles a traditional lattice gauge theory is the answer to this question. Instead of constructing Hamiltonians with a given topological phase as it is done in the string net approach, the present work, however, goes in an opposite direction. We start from a generalization of familiar 3D lattice gauge theory with a certain parameter space and look for topological order. The 3D model defined on the paper is inspired by the state sum model used to define Kuperberg invariant but it is not topological. Our model and the Kuperberg state sum model share the same state space but it has a non-trivial dynamics. It generalizes ordinary gauge theories in a way that a Hopf algebra plays an analogous role to the gauge group. If one chooses to be the group algebra of a group , ordinary Wilson lattice gauge theory corresponds to a curve in the parameter space. The quantum double model appears when the model is restricted to a 2D surface of the parameter space. The Kuperberg TQFT appears as a point in the parameter space. Going away from this 2D surface leads to deformations of the quantum double model. Some of these deformations are still exactly soluble lattice models and some are not.

A precise relation between the Turaev-Viro model, the quantum double model and the Levin-Wen models have also been studied from a mathematical point of view in [25, 26, 27]. For the specific case of the quantum double models that concern us, they are able to set a correspondence that goes beyond the ground state by considering Turaev-Viro model on surfaces with boundaries. The particle excitations of the quantum double model correspond to punctures on the 2D surface. This correspondence can only be made at the level of Hilbert space of states since the Turaev-Viro model is topological whereas the quantum double model has a nontrivial Hamiltonian. The 3D model considered in this paper, on the other hand, is not topological. In our case, the correspondence with quantum double models is also dynamic in the sense that the the quantum double Hamiltonian is the logarithm of transfer matrix of the model.

The quantum double model phase is not the only quantum phase present in the parameter space. We give an example of this fact in the simplest case of . We show that for a certain choice of parameters, the corresponding Hamiltonian is a modification of the usual toric code with the following features. The model is exactly soluble and have essentially the same particles as the toric code. However, some of the original dyons from the toric code are now bound states with zero energy. As a consequence, the Ground State Degeneracy (GSD) is not the same as the toric code. Parameters can be fixed such that GSD is increased by a factor proportional to the exponential of the area (number of plaquettes) of the surface. This is a simple example of a quantum phase that is described in the low energy limit not by a TQFT but by a quasi-TQFT. This example opens the question of classifying all quantum phases, not all of them topological, for the 3D model for the case of a generic involutory Hopf algebra. Although we do not have a solution to this problem, we can nevertheless expect some limitations in the list of topological phases of the model. That comes from the fact that more general algebras, such as non-involutory Hopf algebras and weak Hopf algebras, are excluded from the model in its present formulation. It means that not all topological phases classified by fusion category theory will be present. In particular, phases described by twisted quantum doubles as in [28, 29]. However, the model not only embeds quantum double models, but gives rise to interesting deformations as one explores the parameter space.

Perturbations of the quantum double model have been considered before, for example in [51, 52]. One of their motivations was to investigate phase transitions between different topological phases via a mechanism of charge condensations. The perturbations in section 6 are not meant to achieve such an effect but are intended to give examples of how to depart from the quantum double models while remaining in the parameter space of the 3D model. This parameter space, however, is large enough to accommodate other types of deviations from the quantum double model. A simple example is given in Section 7. As mentioned before, it leads to a quantum phase that is not exactly topological. A complete analysis of this kind of model, especially in the non-Abelian case, is beyond the scope of the present paper.

We now outline our formalism using Kuperberg invariants. The models are parametrized by an element in the center of and an element in the center of the dual algebra . The partition function is proportional to the Kuperberg invariant only in the limit ,  [30, 31]. Here and denote the unit and co-unit of the Hopf algebra . Such techniques are especially useful for quasi-topological field theories, which arise by relaxing some of the conditions required for the theory to be topological [32, 33]. In addition, the observation that is a scalar constructed as a tensor network have been used too generalize the Kramers and Wannier dualities of lattice models [34]. We have also observed that partition functions of classical statistical mechanical models like the 3D Ising model and the lattice gauge theories can be constructed in a similar fashion.

These results are for three dimensional lattice models. Here we extend this formalism by considering the three dimensional manifold as spacetime. In other words, is of the form where is a 2D surface and is a time interval. In particular, we consider a discretization of such that is one single lattice step. The tensor network analogue to is no longer a scalar since it will have one free leg for each link in the lattice . That is precisely the transfer matrix for the model. In order to relate with the quantum double models, however, we need to take into account the splitting of into space and time directions. The weights associated to both directions are not necessarily the same. Just as depends on two parameters, will be parametrized by a pair of elements in the center of and another pair in the center of the dual . The labels and refer to space and time directions respectively.

The transfer matrix is very general. It encompasses models such as ordinary lattice gauge theories with matter in the regular representation. It is necessary to look at particular subsets of the parameter space if one is interested in soluble models. In this paper we set , . The algebra can be any involutory Hopf algebra, not necessarily the group algebra . In this paper we derive some of the models that result from setting or without going, however, into much detail. These new models will be carefully analyzed in another paper [35].

By conveniently splitting this transfer matrix at each link, we write down the transfer matrix as a product of operators acting on the vertices and operators acting on the spacelike plaquettes of . In other words

where and denotes a vertex and a plaquette respectively. By writing the transfer matrix as , we obtain the Hamiltonian by taking the logarithm of both sides, namely

where is the identity matrix. The vertex operator and the plaquette operator are projectors and are precisely the ones occurring in the Hamiltonian of the quantum double model.

Our approach relies on and extends the diagrammatic notation of [21]. Such notation is an efficient representation of the type of 3D tensor networks describing the transfer matrix and other operators relevant to this paper. It is also very useful for finding the ground states of the quantum double models. We exhibit two ground states, one of which coincides with the one found in [36] and the other is different from the first when on a surface with non-trivial topology but is the same as the first one on the 2-sphere. This ground state was written in [54]. Their representation in terms of a tensor network comes out naturally from the formalism. The ground state degeneracy is shown to be equal to the Kuperberg 3-manifold invariant of divided by , for any involutory Hopf algebra.

We organize the paper as follows: we start section 2 with a brief review on gauge theories by building its partition function using the same notation as in the constraining of Kuperberg’s invariant. In order to get the transfer matrix from the partition function a diagrammatic presentation of a tensor network, called Kuperberg diagrams is defined in section 2. The Kuperberg invariant is made of two systems of curves such that their weights are built out of the structure constants of an involutory Hopf algebra. In section 3 we get the transfer matrix as well as the plaquette and vertex operators of the quantum double models, by using a set of properties of these systems of curves, also described in section 3. The method to obtain the Hamitlonian from the transfer matrix is discussed in section 4. The section 5 is dedicated to the study of the ground states of such models. In this section we explicitly exhibit two ground states and also get the ground state degeneracy for any involutory Hopf algebra. In section 6 we write down other models we can obtain using the formalism developed in this paper which we analyze in detail in [35]. We close this work with some remarks in section 7.

2 The 3D model and its relation to Lattice Gauge Theories

The partition function of the 3D model we are about to construct is a generalization of the invariant defined in [20, 21]. Lattice gauge theories are particular cases of this more general 3D field theories. We build these theories using the same set of data which are used in [20, 21], namely a lattice discretization of a 3D manifold and the structure constants of an involutory Hopf algebra .

The first step is to define and describe the discretization. We begin by introducing a cubic lattice of some 3-manifold without boundary. The first step will be to encode the 3D lattice structure into what is called the Heegaard diagram. One advantage of using a Heegaard diagram is that they are two-dimensional. Using 2D diagrams instead of 3D lattices helps in making some of the computations more transparent. More importantly, Heegaard diagrams are more flexible and allow for manipulations that are hard to describe or juts do not make sense when the model is written in a conventional lattice. This is specially true when one investigate topological invariance.

The second step is to introduce a set of weights to build the partition function . They are given by a set of tensors with covariant and contravariant indices and is a scalar defined by a certain tensor network. It has become standard in physics to use a graphical notation to describe tensor networks [37]. In this paper we will use the diagrams introduced by Kuperberg [21, 37].

2.1 Diagrams and Lattices

We recall here how one can canonicaly associate a 2D diagram to a 3D lattice discretization . All information contained in will be encoded in . For convenience we will use cubic lattices but the procedure can be applied for any lattice such as triangulations by tetrahedra.

Consider to be a manifold without a boundary . Let be a lattice discretization of . This lattice is made of cubes glued together by their faces, as illustrated in figure 1.

Figure 1: An example of two cubes glued together by their faces.

The diagram which represents the lattice comprise of two sets of curves drawn on a 2D surface . One type of curve is represented by red curves and the other by blue curves.

(a) The skeleton associated with one cube.
(b) The tubular neighborhood of the skeleton of figure a.
Figure 2: Defining the space where the systems of curves lie.

The surface is given as follows. Consider the 1-skeleton of , in other words, just the skeleton composed of links and vertexes of and then consider its tubular neighborhood , as shown in figure 2.

The space is compose of balls (one for each vertex) and cylinders (one for each link) glued accordingly. Now we draw one red curve on each cylinder of (figure a) to represent the links. We also draw a blue curve for each plaquette (face) of as indicated in figure b).

(a) The red curves associated with the links.
(b) The diagram associated with a cube of .
Figure 3: Systems of curves on the surface .

As one can see, the read curves are placed around the links and the blue curves follows the boundary of each plaquette.

Let be the entire diagram made of a lot of cells like the one show in figure 3 glued together. Notice that a face with four sides corresponds to a blue curve crossed by four read curves as indicated in figure a. In a similar way, a link where four faces are joined is depicted by a red curve that crosses four blue curves as shown in b.

(a) A blue curve associated with a plaquette of .
(b) A red curve associated with a link of .
Figure 4: Definition of the two system of curves.

A final remark about orientations is in order. The models to be defined here are generalizations of usual lattice gauge theories. In order to write down the partition function we need to choose an orientation for each face and each link of . However the partition function does not depend on the choice of orientation, in the same as it also happens for usual lattice gauge theories. The orientation of a face is encoded by an orientation of the corresponding blue curve. In the same way, we can read the orientation of a link by looking at the corresponding red curve. The rule should be clear from figure 4.

2.2 Partition Function

The way we will construct the partition function is based on the procedure used in [20, 21]. We associate weights given by tensors to each curve of and define as a scalar costructed out of these tensors. The data we need to define the weights is given by the structure constants of an involutory Hopf algebra. Thus let be an involutory Hopf algebra, in other words, , where and are the multiplication and co-multiplication maps, and are the unit and co-unit of and is the antipode such that . In appendix A one can find a definition of a Hopf algebra as well as the proof of the identities on Hopf algebras relevant for this work. In this paper, we consider only square lattices and so all the blue and red curves of have exactly four crossings.

Let be a basis of the algebra . For each blue curve (plaquette of ) we associate a covariant tensor , like the one shown in figure a, here , , and represent the links which cross this curve. The tensor is defined by

(1)

where tr means the trace in the regular representation and is some element belonging to the center of the algebra (see figure a).

(a) The weight associated with a blue curve.
(b) The weight associated with a red curve.
Figure 5: The weights associated with the system of curves.

Analogously we associate a contra-variant tensor for each red curve of (figure b), where , , and represent the faces which share this link. The tensor is defined by

(2)

where is a basis of such that and is an element which belongs to the co-center of . Note that we have placed a small rhombus on the curves. They carry labels or as in figures a and b and indicate the particular elements entering in the definition of weights (1) and (2). We say that that curves are colored by an element or . In the special case where is the unit of the algebra (or is the co-unit of the algebra), we represent this as a single curve without these little rhombus, see figure 6. When a curve is colored with unit (or co-unit) we refer to them as being trivially colored. The weighs defined in [20, 21] are trivially colored.

Figure 6: In 6 a trivial red curve, colored by co-unit of , and in 6 a trivial blue curve, colored by unit of .

The partition function we are building is made of contractions between these two kinds of tensors, but before we go ahead we have to take into account the orientation on each curve. As stated before, curve orientation encode lattice orientation. We also need to fix a orientation for the surface where the curves are lying. By convention let us take the normal vector to the surface pointing out. For each crossing between a blue and a red curve we have to contract the corresponding index of each curve. This contraction can be direct or indirect according to the convention shown in figure 7. The vector is the normal vector to the surface, so when is parallel to we contract the curves as shown in figure 7, otherwise we use the antipode map to make the contraction, as shown in figure 7.

Figure 7: The contraction rule for the tensors and .

The contraction of the tensors associated with all the curves gives us the partition function

(3)

Notice that since all indices are contracted is a scalar. Such a scalar can be viewed as given by a tensor network constructed out of , and . The pattern of contractions defining the network is determined by the lattice or, equivalently, by the crossings of blue and red curves of . We recal once more that the models defined in [20, 21] correspond to and .

2.3 Gauge Theories

Let us briefly discuss how the partition function of lattice gauge theories are obtained form (3). In this case, fields living on the links are elements of the gauge group . Therefore has to be the group algebra . For simplicity, let us assume that is finite dimensional and let be the basis of . The weight associated to the faces is the Boltzmann factor

(4)

where , , and are the variables which live on the boundary of some face. Note that the tensor is invariant under cyclic permutation of their indices. For pure gauge theories there is no weight associated to the links, but faces shared by the same link have to agree on the same variable, which means that the tensor has to be

(5)

where , , and are the variables at faces which share the same link. If we want to define a gauge theory on the diagrams, instead of associating a weight to the faces, we have to associate a weight to the blue curves. But the way to do that is straightforward we just associate the same weight to the blue curves. The next step is to choose and so as to reproduce the tensors and .

Consider as an example . In this case the group algebra has only two elements in its basis . If we want to reproduce the tensors (4) and (5) we just need to make the following choices for and [31].

and then the partition function becomes

where .

In the special case where and , the function (3) is proportional to the topological invariant defined in [20, 21]. In other words, the gauge theories, in the limit , become topological [31].

2.4 Kuperberg’s Diagrams

So far we have considered only a manifold without boundaries. In the following we start to look at field theories in . That means that our manifold is of a particular kind which is a 3-manifold with boundary. The way we will do that is by defining the transfer matrix such that its trace is the partition function of the system.

It will be convenient to regard as a scalar given by a tensor network. As such, a graphical notation turns out to be useful. In this section we introduce a notation which will be well adapted to our purposes. This notation is essentially the same as the one adopted for example in [37]. The main difference being the distinction between covariant and contravariant indices. Since it has been introduced in Kuperberg’s work [21] we call this Kuperberg’s notation. In particular, the tensors given by the structure constants of a Hopf algebra can also be represented in this fashion. It turns out that this diagramatic notation simplifies some of the algebraic manipulations we need to perform.

2.5 Kuperberg’s Notation

A Kuperberg diagram is defined in the following way: consider a generic tensor which belongs to the space . We associate a diagram to this tensor. We represent each covariant index by an arrow coming into the diagram , and each contravariant index by an arrow going out of the diagram . By convention, the arrows coming in are enumerated counter clockwise and the arrows going out are enumerated clockwise. See figure 8.

Figure 8: (a) The Kuperberg diagram associated with the tensor . (b) The contraction rule between two tensors.

The contraction rule in terms of Kuperberg diagrams involves connecting the arrows which are contracted. Consider the following contraction: , the corresponding Kuperberg diagram is the one shown in figure 8. Once summed, we do not need to write them down on the diagram (see figure 8).

In the case when is a linear transformation, from a vector space on itself, we write as a tensor with one covariant and one contravariant index , therefore we represent this as a diagram with one arrow coming in and one arrow going out, as shown in figure 9. The identity map is the one shown in figure 9.

Figure 9: (a) The Kuperberg diagram of a linear transformation. (b) The Kuperberg diagram of the identity map of .

There is a very important quantity which is the trace of an operator, the Kuperberg diagram associated to this quantity is a diagram which has an arrow that goes out and comes in to the same diagram. For a linear operator the trace is given by

it means that the indices of are being contracted. Therefore the Kuperberg diagram associated with the trace of a tensor is the one in figure 10. We represent the trace of the identity map as a single closed curve, as shown in figure 10.

Figure 10: (a) The Kuperberg diagram of the trace of an arbitrary linear map. (b) The trace of the identity map of .

We now write all the weights associated with the curves in terms of Kuperberg’s notation.

2.6 Weights in Kuperberg’s Notation

Consider the blue curve in the figure 11. The weight associated to this curve is the tensor defined in equation (1) and since it is the trace in the regular representation it can be written in terms of the structure constants of the algebra (see proposition (1) in appendix A). Therefore its weight is the one written in figure 11, where we can still use the associativity (appendix A, figure 45) of the algebra in order to write this tensor in a more symmetric way, as in figure 11.

Figure 11: In (a), the little rhombus with the element hanging on it means that this curve is colored by the element . In (b) and (c) the tensor associated with a blue curve with four crossings.

Also the weight associated to the links can be written in terms of Kuperberg’s diagrams. The curve drawn in figure 12 represents one red curve colored by an element and its weight is the one in figure 12, where again we have used proposition (1) to write it in terms of the structure constants of the co-algebra. Using co-associativity we can also write this in a more symmetric way, as shown in figure 12.

Figure 12: In (a), the little rhombus with the element hanging on it means that this curve is colored by the element . In (b) and (c) the tensor associated with a red curve with four crossings.

3 Lattices with boundaries and the Transfer Matrix

Consider a manifold of the form where the compact “time” direction has being discretized into steps. From the partition function we can derive a Hamiltonian operator such that

where is the transfer matrix. This can be thought of as being the time evolution operator in . For the purpose of obtaining the Hamiltonian, it is enough to consider . In Kitaev’s model there is one quantum state ( or in the case of .) living on each link of the lattice, such states belongs to a Hilbert space . For the quantum double model, a basis for is . In any case, there is one vector associated to each link. The Hilbert space for the entire lattice is , where is the number of links for the lattice discretization of . The transfer matrix is a map and therefore can be viewed as a tensor in . In terms of Kuperberg’s notation it will be represented by a diagram with arrows coming in and arrows going out.

The processes of obtaining can be visualized as follows. The partition function for is given by some tensor network represented by on figure 13. Each link of will contribute with a contraction as indicated in the figure. The operator is the splitting of such network along the links of . As explained ahead in this section, the network can be encoded in a diagram of blue and read curves. It will be convenient to also encode with a similar diagram of curves. That can be done provided we improve the diagram in order to include more tensors other then just and .

Figure 13: The partition function as the transfer matrix trace.

We already know that each red curve cross exactly four blue curves. Let us represent these crossings by blue dots on the red curve, as shown in figure 14. The associated weight is shown in figure 14. Thus the weight associated to the red curve has one free arrow going out for each blue dot on it. In the same way we can build a red curve with blue and red dots, where blue dots mean arrows going out and red dots mean arrows coming in, as illustrated in figure 14. The tensor in figure 14 is the co-multiplication tensor. Here the orientation of the curve is very important, the tensor has indices ordered clockwise starting from the arrow that comes in.

Figure 14: The blue dots mean arrows going out of the tensor while the red one means one arrow coming in.

In this notation we can combine curves of the same color, just contracting them by dots of different colors. For example we can combine the two red curves of the figure 14 and 14, as shown in 15 and 15.

Figure 15: Joining two red curves by two dots with different colors. If the picture is read from left to write it shows the splitting of a red curve.

We can also read figure 15 from left to right. That corresponds to a splitting of a loop into a pair of curves. This spliting will be used in order to factorize the transfer matrix. Note that in the case of figure 15 both orientations agree, hence the resultant curve is the one shown in figure 15. But, if they do not agree in orientation the contraction has to be done as shown in figure 16.

Figure 16: Combination of red curves with different orientations.

In the same way we can put red and blue dots on the blue curves, the meaning of these dots is the same of the ones in the red curves. In figure 17 we can see the tensor associated with a blue curve with red and blue dots.

Figure 17: The red dots mean arrows coming into the tensor while the blue one means one going out.

Note that the indices of the tensor is ordered counter-clockwise. The rules for composing blue curves is similar to the one for red curves. On figure 18 we show the gluing and splitting of blue curves with the same orinetation.

Figure 18: Joining and splitting of blue curves with the same orientations.

The partition function is made of a lot of cells, like the one shown in figure b, glued one besides the other. Note that faces and links are either timelike or spacelike, so let us call and the elements of the center which color timelike and spacelike faces, respectively. Let us call and the elements of the co-center which color timelike and spacelike links, respectively. It means that the partition function has different weights associated to timelike faces and spacelike faces, and for the links as well, or in other words,

The elements ’s and ’s parametrize the theory. In order to obtain the unperturbed Kitaev’s model we have to make a specific choice for these parameters. As we will see later, the following choice is enough to reproduce such a model

(6)
(7)

where and are the unit and co-unit of , and are the integral and co-integral of and are real parameters which belong to the interval . In other words we are fixing timelike blue curves and spacelike red curves as being colorless. With this choice of parameters we can reproduce quantum double models.

At this point we are ready to use these tools to split the transfer matrix as a product of operators which acts on links, vertices and plaquettes. To do this we define the transfer matrix operator such that the partition function described above can be written as . For that consider one single cell of the diagram for the partition function, shown on figure 19. Consider the diagram obtained by repeating this single cell on the with the appropriate boundary condition for the surface . This is a lattice with a single link on the “time” direction. There are red dots on the bottom and blue dots on the top. The corresponding tensor has a pair of arrows in and out for each link of . That is precisely the transfer matrix . It is straightforward to see that the partition function for is obtained by stacking copies of and connecting the corresponding red dots at the bottom to the blue dots at the top.

The diagram is not planar but it will be useful to draw its projection on the plane. That can be achieved by selecting a region of the surface as illustrated on figure 19. The corresponding projection can be seen on figure 20. Some of the curves are not completely contained in this projection and are represented as line segments instead of closed paths. To complete the loops their ends have to be connected. In figure 20 we see a smaller portion of figure 20 where the loops have been completed. The the pattern we see on figure 20 correspond to a plaquette of . It is clear that we only need to analyze the portion contained in figure 20.

Figure 19: In (a) the transfer matrix . The corresponding tensor has arrows coming going in corresponding to blue and red dots. In (b) we select a portion of the surface enough to contain all information in the diagram.
Figure 20: A projection (bottom view) of the shaded region of figure 19 is in (a). Some of curves are cuted and their ends have to be identified. Figure (b) shows a detailed view of the same diagram.

In the following we will see how we can write the transfer matrix as a product of operators which acts on the plaquettes, vertices and links. This factorization is achieved by repeating the splitting of curves described by figures 15 and 18. The sequence of figures below gives us a prescription of how it can be done. Note that in figure 20 we are just looking at one single link of . But all the modifications we will perform are local and can be repeated for the entire graph. The first step is to split the blue loop of figure a as in figure b. The splitting points will be numbered in order to keep track of them.

(a) One spacelike link of the transfer matrix.
(b) The blue curve in the middle has been broken into two blue curves with red and blue extra dots.
Figure 21: Spliting the diagram to get the operators I.

The next step is to slice the right blue loop of figure b. The result is shown in figure a. After that we perform the same sequence of splittingsd to the red loop in the middle. The first steps is shown by figure b.

(a) One more split in the middle blue curve.
(b) We repeat the same steps in the red curve in the middle.
Figure 22: Spliting the diagram to get the operators II.

Finally we get the diagram shown in figure 23.

Figure 23: All the curves after being split.

After applying the same procedure for the entire graph one can see that the transfer matrix is written as a product of the operators shown in figure 24.

(a) This operator is called link operator . It acts on a link of the lattice.
(b) This operator calls star operator . It acts on an vertex of the lattice.
(c) This is the plaquette operator . It acts on a plaquette of the lattice.
Figure 24: The operators which generate the transfer matrix.

Each vertex, link and plaquette of contributes with a star, link and plaquette operator respectively. Any two link operators commute because they act on different links. The same argument goes for the star and plaquette operator. Thus we can write the following commutation relations

(8)

where , and are some link, site and plaquette of the lattice, respectively.

We now discuss each of these operators.

The link operator shown in figure a is the simplest one. This operator acts on a link of the lattice, therefore there is one link operator for each link. In terms of Kuperberg diagrams is shown in figure a. But due to proposition (6) appendix A it is easy to see that this operator is proportional to the identity map.

(a) Link operator written in terms of Kuperberg diagrams.
(b) The link operator is proportional to the identity map.
Figure 25: Link Operator.

Thus we can write it down as shown in figure b. In figure 23 we can see that this operator plays a role in connecting one plaquette and one star operator which act on the same link, but since this operator is trivial we just connect them directly.

The star operator given in figure b acts on a vertex of the lattice. The action changes the states living on the links which share the vertex . The corresponding tensor network is given as a Kuperberg diagram in figure 26.

Figure 26: Star operator written in terms a of Kuperberg diagram.

The plaquette operator represented by figure c acts on a plaquette of the lattice. Its Kuperberg diagram is given in figure 27. The action depends on the central element . For the case of it is a sum of projectors which project onto the different conjugacy classes of  [35].

Figure 27: Plaquette operator written in terms of a Kuperberg diagram.

4 Obtaining the Hamiltonian from the Transfer Matrix

In the previous section the partition function was written down as the trace of the transfer matrix . This operator is the product of all plaquette operators and all star operators. It turns out that the star and plaquette operators commute with each other. This fact is very easy to see for the case but becomes more elaborated in general or even for group algebras of non-abelian groups. A proof can be found in appendix E. The commutation of plaquette and star operators allows us to write the transfer matrix as a product of plaquette and star operators in the most convenient way. Thus consider the transfer matrix

(9)

where is the total number of links. The Hamiltonian H can now be found from

by taking the logarithm of .

Consider the plaquette operator in figure 27, where is the one chosen in (6). Since the tensor is defined as the trace in the regular representation it is a linear function of , so we can write this operator in the following way

(10)

The operator it is colored by the unit element of . It is represented by a single blue curve without a rhombus and will be denoted by . The operator is the plaquette operator colored with the co-integral, and it is proportional to the identity map. These operators fulfil property equations

(11)
(12)

A proof can be found in appendix C. Using equation (12) in (10) the operator can be written in the following way

where and . The operator obeys the following property

(13)

Now we are ready to compute the hamiltonian by taking the logarithm of the transfer matrix. For we obtain

(14)

Using the Taylor expansion of and equation (4) we can compute the logarithm of the plaquette operator

(15)

All the arguments used for plaquette operator hold for the star operator, since they are constructed in a analogous way. This is a consequence of the duality for Hopf algebras (between the algebra and co-algebra structures). Therefore we can write

(16)

where and . Finally using equations (15) and (16) in (14) gives us the Hamiltonian

(17)

where the parameters , and are given by

Notice that the transfer matrix is a product of and but the Hamiltonian is a linear combination of operators and . Since , it is not difficult to see that and . This Hamiltonian is the one we got from a deformed manifold invariant, its partition function () is not topological for all values of the parameters and , but in the limit () we get the Hamiltonian of Kitaev’s model and its partition function becomes topological. Such a partition function is the one which comes from the non-deformed Kuperberg invariant. The constant is a finite constant, for , and it represents just a shift in the energy levels. From now on we will ignore this term in the Hamiltonian.

In the following sections we are going to present a ground state in terms of diagrams and we are also going to study the degeneracy of the ground state.

5 Ground States

The Hamiltonian defined in (17) is made of a sum of commuting operators, so it is enough to look at the spectrum of each operator separately.

The operators and are projectors, which implies they have the same spectrum which is 4. Consider a basis of eigenvectors of and , it is not difficult to see that the Hamiltonian has the lowest eigenvalue when

(18)
(19)

therefore the lowest eigenvalue of is . Thus the eigenstates of are the states such that the equations (18) and (19) holds or equivalently if the equation below holds,

(20)

The statement in equation (20) gives us the energy eigenvalue of the ground state but it says nothing about the structure of the ground state. In this section we are going to construct two tensor networks that are exact ground states of (17).

Before we proceed, we need to establish the behaviour of plaquette and vertex operators under change of curve orientation. One can show that and are mapped to and . A proof for this statement can be found in appendix F.

The plaquette operator we have previously defined is the one shown in figure 28 for . But it can also be written as the ones shown in figures 28 and 28. This change of orientation will not affect the coloured operators provided that .

Figure 28: Possibles curves orientation for the plaquette operator.

The same thing happens for the star operator. In figure 29 all the operators are equal despite the differences in orientation.

Figure 29: Possibles curves orientation for the star operator.

5.1 Constructing

The way we will construct a ground state is by taking the product of all plaquette operators (the one drawn in figure 28), and then feeding the input arrows with the co-integral. In other words the ground state will be of the form

of course it is a eigenstate for all the plaquette operators, since an operator commutes with all ’s, which means that

(21)

It remains to show that it is also an eigenstate of the star operator.

In the following we are going to shown a diagrammatic way to represent such a state. First we draw a diagram for the product of all plaquette operators, as shown in figure 30, then we feed all the red dots in figure 30 with the co-integral (which is the trace in the regular representation) and after that we get the state drawn in figure 31.

Figure 30: Product of all the plaquette operators.
Figure 31: The ground state built from the product of plaquette operators.

Note that the state drawn in figure 31 has one blue curve for each plaquette and one red curve for each link, as illustrated in figure 32, where we removed the orientation of the curves for simplicity.

Figure 32: The ground state build from the product of plaquette operators, it has one blue curve for each plaquette and one red curve for each link.

We are going to use the curves diagram to show that the state drawn in figure 32 is a ground state for the Hamiltonian (17). Let us start by deriving equation (21) one more. The result of applying a plaquette operator on such a state is shown on figure 33.

Figure 33: Applying the plaquette operator on a plaquette .

We end up in the diagram in figure 34 after removing the outside blue curve using the sliding and the two-point moves described in appendix B.

Figure 34: Here we can see that is an eigenstate of the plaquette operator with eigenvalue .

Exactly the same computation holds for all the plaquettes.

We still have to prove that is also an eigenstate of the star operator. For that let us apply the star operator on the vertex , as described by figure 35. In this case we will have to use a slightly different procedure to remove the red curve in the center of the diagram. Instead of one slide move in the plaquette operator case, we will have to use four slide moves, one over each red curve associated to the vertex , see figure 36. The same procedure holds for all vertex operators. Thus these computations show us that the state (figure 32) is in fact one ground state of the Hamiltonian (17).

Figure 35: The star operator acting on a vertex of the lattice.
Figure 36: The star operator acting on a vertex of the lattice, after three slides of red curves. In this figure we are not drawing the blue dots in the blue curves just for simplicity.
Figure 37: After four slides of red curves we can use two-point moves to get rid of some crossings.

Figure 32 is a graphical representation of a tensor network. The corresponding Kuperberg diagram is given in figure 38.

Figure 38: State written as a tensor network in the Kuperberg notation.

5.2 Constructing

The ground state we have just discussed was made of product of all plaquette operators with all its input arrows fed with co-integrals. Using the same line of thought one can write a ground state made with a product of vertex operators fed with some element of the algebra. In fact, there is another ground state we can get by feeding the product of vertex operators with a tensorial product of the unit of the algebra (such a state was also written down in [54]). That is

This new state is for sure an eigenstate of the vertex operator, due to the fact that any vertex operator commutes with all the others and the fact that is a projector

(22)

The product of all the star operators, in terms of diagrams, is the one shown in figure 39. In order to get the state we just attach one unit element in each free red dot, as shown in figure 40.

Figure 39: Product of all the vertex operators.
Figure 40: The ground state built from the product of vertex operators.

A graphical proof that is a eigenstate with eigenvalue equal to for both the plaquette and vertex operators is completely analogous to the one presented for . All steps are the same if we exchange blue and red curves.

As for the ground state , the ground state has a geometrical meaning. It has one red curve for each link of the lattice and one blue curve for each link, as illustrated in figure 41. In the figure we have not drawn the orientations for simplicity. The corresponding tensor network is given in figure 42.

Figure 41: The ground state built from the product of vertex operators. It has one red curve for each vertex and one blue curve for each link.
Figure 42: State written as a tensor network in the Kuperberg notation.

5.3 Comparing and

We can understand the difference between the ground states and by considering them in the loop representation [40]. As is obtained by acting with the product of the plaquette operators on the tensor product of co-integrals on all the links, it is precisely the sum of all closed loops. This is because the plaquette operator projects just these states which contribute to the ground state. If we are working on a surface with non-zero genus then this contains the sum of terms belonging to the different topological sectors. Consider for example the toric code on the 2-torus . We have

(23)

where are the ground states belonging to the different sectors distinguished by the different non-contractible loops of .

On the other hand while obtaining we started with a configuration which chose a particular sector, namely , and then acted with the product of gauge transformations, that is the product of the star operators which do not mix the different topological sectors. Thus on we have

(24)

These two states coincide on the 2-sphere, as there are no non-contractible loops on .

The loop representation for the ground states is only true for the abelian quantum double models as the excitations in these models have abelian fusion rules. In the case of non-abelian models the fusion rules are more complicated leading to a representation for the ground states not just in terms of closed loops but also other structures depending on the non-abelian group.

5.4 Ground State Degeneracy

Consider a basis of eigenstates of both and . The ground state space is defined by

In this section we are going to compute the ground state degeneracy of (17) defined on a surface . We will show that is determined by the 3-manifold invariant defined in [21] computed on .

Let us denote by the transfer matrtix (9) computed at . Then

which means that, in the basis , the operator is of the form

where in the null matrix and . If we take the trace of we get

(25)

A similar answer for the ground state degeneracy in terms of the trace of a transfer matrix was obtained in [38].

But is the partition function computed for and

Here is the Kuperberg invariant, is the genus of the Heegaard spliting, and are the numbers of blue and red curves, respectively [21]. Therefore

(26)

The factor multiplying seems to depend on the size of the lattice. However that is not the case. Looking at figure 20 we can see that and are related to the parameters , and of the manifold , by

Besides, the parameters , and satisfy the Euler equation

where is the Euler caracteristic of . This way we get

Now the last step is to write down as a function of , and . Let us do this looking at the dual Heegaard diagram. In figure 43 we can see the dual Heegaard diagram of , shown in figure 43. Each grey rod highlighted (figure 43) represents a spacelike plaquette.

Figure 43: This figure illustrate how to relate with the parameters of .

Note that there is one handle on the Heegaard diagram for each vertex of , see figure 43. Here we are essentially counting the number of non contractible loops and since has genus , it has more non-contractible loops. Besides, since we took the trace of the transfer matrix, each plaquette is actually glued to itself, as shown in figure 43, and it shows us that each plaquette increases the number of handles by one unit as well. Therefore we can write

where the factor is due to the fact we can always stretch one face over around the entire diagram, decreasing the genus by one unit. Therefore the ground state degeneracy becomes

(27)

which is clearly a topological invariant.

5.5 The ground state degeneracy of quantum double models

Quatum double models with have its ground state degeneracy known [7] to be given by

(28)

Such a result can be obtained from the expression we got for the ground state degeneracy as a special case. In this case the plaquette and vertex operators can be written in terms of the Pauli matrices in the following way

The degeneracy, according to equation (25) is given by

which means we just have to evaluate the trace of the product of all the plaquette and vertex operators. Both plaquette and vertex operators are made of a tensorial product of identities all over the place except in four specific positions which correspond to the link where they are acting, in these specific positions instead of an identity map they have an operator ( for the vertex operator and for the plaquette operator) acting on it. It is not difficult to convince ourselves that the product of all plaquette (vertex) operator will have exactly two terms which is a tensorial product of identity maps all over the place. All the remaining terms will have an operator () in at least one position of such a tensorial product. Let us write this statement in the following way