Toric code-like models from the parameter space of $3D$ lattice gauge theories
A state sum construction on closed manifolds á la Kuperberg can be used to construct the partition functions of lattice gauge theories based on involutory Hopf algebras, , of which the group algebras, , are a particular case. Transfer matrices can be obtained by carrying out this construction on a manifold with boundary. Various Hamiltonians of physical interest can be obtained from these transfer matrices by playing around with the parameters the transfer matrix is a function of. The quantum double Hamiltonians of Kitaev can be obtained from such transfer matrices for specific values of these parameters. A initial study of such models has been carried out in . In this paper we study other regions of this parameter space to obtain some new and known models. The new model comprise of Hamiltonians which “partially” confine the excitations of the quantum double Hamiltonians which are usually deconfined. The state sum construction allows for parameters depending on the position in obtaining the transfer matrices and thus it is natural to expect disordered Hamiltonians from them. Thus one set of known models consist of the disordered quantum double Hamiltonians. Finally we obtain quantum double Hamiltonians perturbed by magnetic fields which have been considered earlier in the literature to study the stability of topological order to perturbations.
Departmento de Física Matemática Universidade de São Paulo- USP
Topological Order - Topologically ordered systems have gained wide attention in recent years due to some of its consequences in topological quantum computation and emergence of new phases of matter among many others . Among the different types of systems exhibiting topological order the ones with long-ranged entangled (LRE) ground states are the ones which are thought to be most useful for quantum computation. The earliest proposals of such systems are the quantum double Hamiltonians of Kitaev [3, 4]. The toric code is the simplest example of a lattice systems which contains anyons as low energy excitations and have degenerate LRE states as ground states, this model consists of spin variables living on the links of a square lattice. These were further generalized by the Levin-Wen models  or the string-net models which described more general anyonic excitations by directly taking a unitary fusion category as inputs. These models are also quantum double models based on weak Hopf algebras as noted in  and can thus be constructed via the algorithm of Kitaev. Topological codes have also been considered on manifolds with boundary [27, 8]. Several other models inspired by the usefulness of the toric code as a stabilizer code have been constructed of which the topological color codes [9, 10, 11] are an example which have also been experimentally implemented .
State Sum Constructions and Statistical Mechanical Models - Both the Levin-Wen model and Toric code models can be thought of as Hamiltonian realizations of topological field theories (TQFTs) and then they can be formulated in terms of topological invariants. State sum constructions of TQFT’s  have been employed in realizing statistical mechanical models in the past [14, 15, 16]. Such methods have also been used to construct the Levin-Wen models  using the Turaev-Viro invariants  and chain-mail link invariants [18, 19]. Kitaev’s toric code has also been related to Turaev-Viro codes . The Levin-Wen model corresponds to a topological invariant called Barrett-Westbury invariant  and the toric code corresponds to a special case of the Kuperberg invariant . This has been especially noted in . We showed this explicitly in  where we embedded the quantum double models based on an involutory Hopf algebra, , in an enlarged parameter space (defined later), that of the generalized lattice gauge theories based on these algebras . In the special case where the algebra is taken to be the group algebra of a group , the generalized gauge theory can reproduce the lattice gauge theories familiar to physicists . The toric code occurs when we choose .
The Quantum Double Model - The quantum double model of a discrete group
where is the plaquette operator, the vertex operator and positive numbers. These operators are both projectors and also commute with each other for all vertex and plaquettes making energy levels discrete.
Consider a complete set of eigenvectors of with . If the system is put into a bath with temperature one can obtain thermodynamics properties of such a system by its partition function
with , being the Boltzmann constant. The matrix is called the transfer matrix. Since is given by equation 1, the partition function can also be written as
In the special case where this partition function can be rewritten as
where the sum runs over all the configurations and if the holonomy of the plaquette is flat and otherwise. The function is invariant under gauge transformation, and then is the partition function of a lattice gauge model. For it can also be shown that is a partition function of some lattice gauge model .
Partition Functions of Lattice Gauge Theories - We mentioned earlier that the quantum double Hamiltonian is related to the partition function of a lattice gauge model, as once we know the Hamiltonian the partition function is well defined. Now we can ask the question, if a partition function is given, is it possible to obtain a Hamiltonian such that the equation 2 is satisfied? The answer for this question is no, however if is restricted to be gauge invariant it can be done. The reason is that there is a way of building gauge invariant partition functions out of a topological invariant called the Kuperberg invariant , which is based on involutory Hopf algebras. Moreover this construction allows us to obtain partition functions that are more general than the lattice gauge ones, but they are still gauge invariant.
This partition function is parametrized by four non-physical parameters, namely and , in other words will be a function of the form . The choice of such parameters leads to specific models. It is important to note that these parameters are not free such as for example in ordinary gauge theories. These parameters are fixed once the model is fixed. In particular, as it can be seen in , if the parameters are taken to be
and , where denotes the counit of the algebra . The partition function obtained by such a choice coincides with that of a pure lattice gauge theory with . Different choices of these parameters lead to different partition functions which may not be related with gauge theories at all, however if we restrict the parameters and to be elements of the centre of , whatever the partition function is, it will be gauge invariant . The action in this partition function may not be of physical interest but they will all be gauge invariant, and for that reason we will say that is the partition function of a generalized gauge theory. From now on we consider to be a discrete group, and the dimensional lattice a finite cubic lattice as a triangulation of a manifold of the form , where is a compact manifold of genus and is the one dimensional sphere.
Kuperberg’s Construction of Transfer Matrices - The way to build these generalized partition functions is by associating tensors, made up of the structure constants of the algebra, to the faces and links of the lattice leading to a very complicated tensor network that, fortunately, can be realized as the trace of a matrix by
where, by analogy, we can think of it as being the exponential of a Hamiltonian , i. e.
where the constant can be suppressed without loss of generality. The matrix can be thought of as an operator acting on a lattice (over the manifold ), but for that we have to make distinction between the timelike and spacelike directions on the original lattice
Models from the transfer matrix - We then obtain the Hamiltonian by taking the logarithm of . However the Hilbert space is very huge, which makes a very huge matrix and difficult to take its logarithm. But as we have done before  the matrix can be decomposed into a product of local operators acting on given by
where , and denotes plaquettes, vertices and links respectively with and operators acting on the degrees of freedom located on the links and and the plaquette and vertex operator which we previously encountered in the quantum double Hamiltonian given in Eq. 1. These vertex and plaquette operators satisfy the quantum double algebra [25, 26]
Other models using more parameters in the transfer matrix - Some examples include quasi-topological phases which result from a condensation of the excitations  of the quantum double phase of Kitaev. This leads to increased ground state degeneracy for the condensed phases. Examples of these phases were studied in . These phases including the quantum double phases of Kitaev were obtained when we considered the parameters and in the transfer matrix. In  we showed that we could obtain the quantum double phases of Kitaev by writing down models which included or .
Identifying topologically ordered phases - The models that exhibit topological order in D can be understood by their quasi-particle content called anyons. The data that determines the phase are the ground state degeneracy, their statistics and fusion parameters . If two models have different ground state degeneracies, or different fusion rules or statistics, they are not in the same topological phase.
In this paper we take this program further by considering more parameters in the transfer matrix which were not included in [1, 28]. We consider three types of models here. Two of them do not include and while the third includes them. The first two sets of models comprise of the disordered quantum double Hamiltonians of Kitaev and a new Hamiltonian which leads to “partial” confinement of the excitations of the quantum double phase of Kitaev. We use the term partial to emphasize the fact that the models are such that the excitations can be moved a few steps with an energy cost after which they become deconfined like in the usual quantum double models. According to the terms added we can confine the excitations for any number of steps that we wish to. We will also call these models -step confined models in the text to follow. The models which include the other two parameters, and are the quantum double Hamiltonians perturbed by magnetic fields. These can be thought of as local perturbations to the exactly solvable Hamiltonians of Kitaev. Due to the usefulness of these models to realize fault tolerant quantum computation, it is necessary to study the stability of the topological order to local perturbations [30, 31, 32]. These models have already been considered in the literature and we write them down here just for the sake of completion and to drive home the point that they are well within the parameter space of the three dimensional lattice gauge theories. Our focus is on exactly solvable Hamiltonians like the original toric code Hamiltonian and so phase transitions are out of the scope of this paper as we will then necessarily have to move through perturbed toric code Hamiltonians which are outside the exactly solvable regime.
The contents of the paper are organized as follows. Section 2 gives a brief review of the construction of the transfer matrix of the generalized lattice gauge theories. The section also includes an introduction to the mathematical preliminaries that go into the construction of the partition function and the transfer matrices. The algebra of operators, which include the quantum double relations between the vertex and plaquette operators, are written down. The models obtained from this transfer matrix are described in section 3. An outlook is presented in section 4.
2 Partition Function and Transfer Matrix of Generalized Lattice Gauge Theories
The partition function of lattice gauge theory is a well known example of a classical partition function built out of local weights associated to plaquettes of an oriented 3D lattice, where the gauge degrees of freedom are elements of a gauge group living on the edges of the lattice. A configuration is a choice of an element for each link of the lattice. For the gauge degrees of freedom are spin variables living on the links. The action of this model is defined by , where the sum runs over the plaquettes of the lattice and is the holonomy of a plaquette . The partition function which describes the model is given by
In above equation is the local weight for the model. Due to the invariance of the local action under cyclic permutation, the local weight is also invariant under this cyclic permutation, which makes it a class function, (equivalently, ). This construction can be generalized by choosing as being any class function . Moreover, we can associate local weights, , to the edges of the lattice, such that the partition function is now given by
We can reproduce the usual lattice gauge theories by making appropriate choices for and and also generate partition functions that are still gauge invariant but do not represent a physical model. The partition function in equation (10) is called the partition function of a generalized lattice gauge theory. Starting from this partition function we can obtain a transfer matrix whose logarithm gives us Hamiltonian operators, and thus dynamical quantum models defined over the Hilbert space defined before. These quantum models are parametrized by functions of the parameters of the generalized lattice gauge theories. In  it was shown that the quantum double Hamiltonians of Kitaev, of which the toric code is a special case, can be obtained from this approach. In other words it was shown how to embed such models in the parameter space of these generalized lattice gauge theories.
In  it was shown that this partition function can be build out of the structure constants of an involutory Hopf algebra and a 3-manifold of the form , where is some compact 2-manifold and is the 1-dimensional sphere. We did not consider all possible deformations of the generalized lattice gauge theory partition function in , working only with a specific kind of deformation (one parameter deformation) of the gauge theory partition function. Now we will allow other deformations by letting the parameters be any element of the center of the algebra and it’s dual algebra, . We only work with group algebras of a discrete group here. Nevertheless the methods presented here hold for any involutory Hopf algebra. We will go through the mathematical preliminaries beginning with the definition of the group algebra .
2.1 The Group Algebra
The group algebra of a discrete finite group is generated by the basis elements indexed by the group elements and let be the dual basis such that . In this basis the multiplication and co-multiplication are defined by
and we can easily see that the unit and the co-unit of the algebra are
where is the identity element of the group.
Finally the antipode map is defined by
It is not difficult to see that the group algebra structure constants satisfy all the axioms of Hopf algebras [26, 1]. An important thing about the group algebra is the fact that it is an involutory Hopf algebra, which means that .
As the simplest example lets consider the group algebra of . The group is defined by , and the product is the usual multiplication. The group algebra has as basis, and the product is given by
The dual basis is , and the coproduct defined by
The antipode in this case is trivial .
2.2 Constructing a Partition Function with an Involutory Hopf Algebra
Consider an oriented cubic lattice as a triangulation of a closed manifold of the form and let be a discrete finite group. The degrees of freedom are elements of the gauge group located on the links of this lattice. A configuration is a choice of one group element for each link of the lattice. The generalized partition function is then built by associating tensors for each face and link of the lattice which will play the role of local weights. These tensors contract with each other resulting in a scalar called generalized partition function .
For a plaquette, whose boundary links carry the group elements , , and , we associate a tensor defined by the algebra structure as shown in figure a. For a link with four plaquettes glued to it’s edges, labelled by , , and , we associate a tensor defined by the coalgebra structure as shown in figure b. It is very important to have an algebra and coalgebra structure so that the associated tensors can contract with each other on the closed manifold.
These tensors are parametrized by the elements of the center of the group algebra and it’s dual, and
where is the trace in the regular representation and . The partition function is obtained by contracting the indices of the tensors associated to the plaquettes and links. However we need to take care of the orientation of the lattice while performing this contraction. If the plaquette and link orientation matches the contraction is made directly, otherwise it is done through the antipode tensor . At the end the partition function will be of the form
where the sum runs over all the contracted indices of all tensors and runs over the links with mismatching orientations.
As an example take and choose and . Then the tensors will be of the form
Note that is the holonomy of the plaquete . In this case, due to Eq. 15, the sum over indices in the partition function in Eq. 13 will reduce to a sum of indices of the tensor (for all plaquettes), in other words, it will become a sum over configurations and the partition function will take the form
that is exactly the partition function of the lattice gauge theory defined in Eq. 9.
We can now extract a transfer matrix out of this partition function, but first we have to make a distinction between timelike and spacelike directions of the 3D lattice, as shown in figure 2 for a small piece of the lattice.
Since there is now a distinction between the timelike and spacelike parts there is nothing forcing the parameters of the timelike and spacelike plaquette weights to be the same. So for a more general description we have all spacelike plaquette weights parametrized by the element
2.3 The Transfer Matrix
From the partition function we have just defined we can get a transfer matrix such that its trace is equal to the partition function. This transfer matrix may depend on the same parameters as the partition function, namely . The operator acts on the links of the 2-dimensional lattice where the quantum states lives. A local Hilbert space associated to each link (with basis ). The Hilbert space of the full system is given by and a vector of this space is a linear combination of vectors of the form
The procedure to get such a transfer matrix was shown in  using a diagrammatic notation to manage the tensors that builds the partition function and also the transfer matrix. We are now considering a more general parametrization than the one considered in . The way to get the transfer matrix is similar to the procedure shown in , resulting in the same operators as in  but now in a bigger parameter space. We just write down the results in what follows. The most general transfer matrix including all the parameters is given by
where is an operator which acts on the links at edge of the plaquette , an operator which acts on the links sharing the vertex and is an operator which acts on a single link. The operators and are called the plaquette and vertex operators, as before. All the parameters in are central elements of and . It can be seen in  that the central elements of are written in terms of the conjugacy classes of as
while the central elements of we write in terms of the irreducible representations of as
where is the trace of in the representation . Hence the parameters of the transfer matrix can be written as
The plaquette and vertex operators are linear functions in its parameters. That is
where the operators and act on the links as shown in the picture in figure 3 and are given by
where if and if and the operators , and are operators which act on a single link defined as
These operators are linear on its parameters, in other words, (this property also holds for and . Sometimes we use the short notation , and . The link operator can also be written in terms of the and operator as we shall see next.
Thus the final expression for the transfer matrix is
2.4 Algebra of the operators
In order to find the algebra that the operators , and satisfy one should first look at the algebra of the operators , and . It is not difficult to see that the following relations holds
The set of operators is a complete basis of orthogonal projectors which generate the plaquette operators, which means
Same way the set of is a complete basis of orthogonal projectors which generate the vertex operator, in other words
The plaquette and vertex operator still commute for any choice of the parameters and , so we can write
The set of operators and are also complete sets of orthogonal projectors ( and ), however the operator does not commute with the link operator but it does commute with the plaquette operator, in the same way as the operator does not commute with the plaquette operator but it does commute with the vertex operator. Thus we can write
Therefore, the link operator commutes with the plaquette and vertex operators only for some choices of the parameters and , which means the quantum model obtained from the transfer matrix containing this link operator is in general not solvable.
3 Examples of Models from the Transfer Matrix
The fully parametrized transfer matrix obtained in the previous section helps us construct a number of other interesting models. We have seen that the quantum double Hamiltonians are only one special class of models in this parameter space. Here we will look at what other possibilities exist in the extended parameter space. The first set of examples consist of disordered quantum double Hamiltonians. These are quantum double Hamiltonians which do not have translational invariance. This is due to the appearance of coefficients, for the vertex and plaquette terms, which are not constant but depend on the vertex and plaquette . These models continue to be in the quantum double phase. The trace of the transfer matrix for these Hamiltonians also help us obtain their partition functions.
We can produce solvable models even in the presence of the parameters and . We discussed a class of such models in  where we exhibited exactly solvable models which continued to remain in the quantum double phase described by modified vertex operators or plaquette operators. Here we show another class of models that can be obtained from the transfer matrices of lattice gauge theories that continue to remain exactly solvable but the excitations are “partially” confined with respect to those of the original quantum double model. By this we mean that the deconfined quantum double excitations are now confined up to a few steps due to the addition of an extra term in the Hamiltonian. The number of steps for which they are confined can be controlled by adding the appropriate term in the Hamiltonian. We will call these models -step confined models. These models comprise our second set of examples. They are examples where the self-duality of the quantum double Hamiltonians is broken. We also discuss the ground states, it’s degeneracy apart from the excited states of the model.
Finally we write down models that are obtained by using the remaining two parameters and along with and . The transfer matrix is now significantly modified as new single qudit operators, acting on individual links, appear. They do not commute with the vertex and plaquette operators in general. However for certain special values of parameters they commute with products of vertex and plaquette operators as we shall see when we consider these models later in this section. The models obtained at these values are quantum double Hamiltonians perturbed by generalized “magnetic” fields. They have the interpretation of magnetic fields in the case the input algebra is . For the remaining values of the parameters we obtain more complicated terms in the Hamiltonian which we will briefly touch upon. These models are not exactly solvable and are outside the phase described by the quantum double Hamiltonians, at least for sufficiently large values of the parameters.
3.1 Disordered Quantum Double Hamiltonians (QDH)
The transfer matrices used to obtain these models only use and . The other two parameters are set to and for all the timelike plaquettes and spacelike links. To obtain the disordered QDH models we associate a different central element of the algebra and it’s dual to every spacelike plaquette and timelike link respectively. This leads to the following transfer matrix
where and are the parameters for the plaquette and vertex respectively. The plaquette and vertex operators commute with each other in this transfer matrix and each of the operators is a sum of projectors. Thus it is easy to take the logarithm of these matrices to obtain the disordered QDHs.
Let us look at these Hamiltonians in the case when . Denote the basis elements of by with , and the basis elements of the dual by with the product . The central elements of and can be written as and respectively. By assigning each spacelike plaquette and timelike link with and respectively we obtain the transfer matrix as
where the plaquette operators are given by
and the vertex operators are given by
The action is shown in figure (3).
The disordered QDH can now be written as
As it can be seen this Hamiltonian breaks translational invariance and is made up of a sum of commuting projectors. When translational invariance is restored we recover the familiar toric code Hamiltonian.
The ground state for this Hamiltonian is easily obtained by projecting on to the smallest of and for every vertex and and for every plaquette . The ground state degeneracy is the same as the usual toric code and the winding operators,
where the non-contractible loops are defined on the direct and dual lattice respectively as shown in the figure 4, commute with the Hamiltonian.
The excitations correspond to the other value of the coefficients for each vertex and plaquette. The string operators (ribbon operators in the case of the non-Abelian groups ) creating the excitations are the same as in the QDH case. For the specific case of we have the string operators, creating charge or vertex excitations at the end points of the string along the direct lattice, as
and those of the fluxes or plaquette excitations at the end points of the string along the dual lattice as
These are shown in figure 5. As is well known these excitations are deconfined by which we mean that there is no cost in energy for moving them around by stretching the string creating them. Moreover the fusion rules and braiding statistics are the same as in the translationally invariant toric code. Thus we conclude that the disordered QDH given in Eq. (43) continues to remain in the toric code phase.
3.2 Quantum Double Hamiltonian with -Step Confined Excitations
We first consider the case with . To obtain this we write down an example of exactly solvable Hamiltonian made up of the QDH vertex and plaquette operators along with new terms made of these operators which have 2-step confined low energy excitations unlike the QDH case where all the excitations are completely deconfined. The transfer matrix used to obtain these models contain only the and parameters.
The transfer matrix can be written as
In the case of we can write down the Hamiltonian with the 2-step confined charges and fluxes as follows
where and are nearest neighbor vertices in the direct and dual lattices respectively. All the terms in this Hamiltonian commute with each and other and are sums of projectors. The ground states are given by the usual toric code Hamiltonian. The degeneracy does not change as the winding operators in the toric code case, given by Eq.(44) and Eq.(45), continue to commute with this Hamiltonian and thus help create the new states from a given ground state. In particular on a torus the degeneracy is four.
The interesting feature of this model occurs when we look at the excitations. As in the toric code case the string operators creating charge and flux excitations are given by Eq.(46) and Eq.(47) respectively. However in this model when we create a charge or flux excitation we also excite the direct or the dual link given by the or the
It is easy to see that the dyonic excitations are also confined in a similar manner. Thus we have a model based on lattice gauge theory which is exactly solvable, has ground state degeneracy and has excitations which are confined up to two steps or can be thought of as being partially confined.
There is a natural way to increase the number of steps for which these particles are confined. This is achieved by coupling more number of vertex and plaquette operators. For example to obtain three-step confinement we add terms of the form where are nearest neighbors on the direct lattice. The corresponding plaquette terms are where are nearest neighbor vertices on the dual lattice. The energy of these excitations are more when compared to the original QDH as we violate more terms in the Hamiltonian to obtain them. This argument can easily be extended to any number of steps. The corresponding figures of the -step confined excitations will have a larger shaded region where they are confined when compared to the