pacs:71.10.-w, 02.20.Uw, 03.65.Fd
String-net models allow us to systematically construct and classify 2+1D topologically ordered states which can have gapped boundaries. We can use a simple ideal string-net wavefunction, which is described by a set of F-matrices [or more precisely, a unitary fusion category (UFC)], to study all the universal properties of such a topological order. In this paper, we describe a finite computational method – Q-algebra approach, that allows us to compute the non-Abelian statistics of the topological excitations [or more precisely, the unitary modular tensor category (UMTC)], from the string-net wavefunction (or the UFC). We discuss several examples, including the topological phases described by twisted gauge theory (i.e., twisted quantum double ). Our result can also be viewed from an angle of holographic bulk-boundary relation. The 1+1D anomalous topological orders, that can appear as edges of 2+1D topological states, are classified by UFCs which describe the fusion of quasiparticles in 1+1D. The 1+1D anomalous edge topological order uniquely determines the 2+1D bulk topological order (which are classified by UMTC). Our method allows us to compute this bulk topological order (i.e., the UMTC) from the anomalous edge topological order (i.e., the UFC).
- I Introduction
- II Quasiparticle excitations
III String-net models with tetrahedron-rotational symmetry
- III.1 String-net
- III.2 Evaluation and F-move
- III.3 Fixed-point Hamiltonian
- III.4 Cylinder ground states, quasiparticle excitations and Q-algebra
- III.5 String operators and quasiparticle statistics
- III.6 Examples
- IV Generalized string-net models
- V Boundary theory of string-net models
- VI Morita equivalence and fusion of excitations
- VII The mathematical structure of our construction
- VIII Conclusion
- A A brief introduction to algebras and modules
- B Q-algebras in string-net models with tetrahedron-rotational symmetry
- C The gauge transformation that fixes for or trivial and
- D Isomorphism between Q-algebra in Section IV.5 and twisted quantum double
- E Twisted () string-net model
A major problem of physics is to classify phases and phase transitions of matter. The problem was once thought to be completely solved by Landau’s theory of symmetry breakingLandau (1937), where the phases can be classified by their symmetries. However, the discovery of fractional quantum Hall (FQH) effectTsui et al. (1982) indicated that Landau’s theory is incomplete. There are different FQH phases with the same symmetry, and the symmetry breaking theory failed to distinguish those phases. FQH states are considered to possess new topological ordersWen (1989); Wen and Niu (1990); Wen (1990) beyond the symmetry breaking theory.
We know that all the symmetry breaking phases are labeled by two groups , where is the symmetry group of the Hamiltonian and is the symmetry group of the ground state. This fact motivates us to search for the complete “label” of topological order.
Here, the “label” that labels a topological order corresponds to a set of universal properties that can fully determine the phase and distinguish it from other phases. Such universal properties should always remain the same as long as there is no phase transition. In particular, they are invariant under any small local perturbations. Such universal properties are called topological invariants in mathematics.
In 2+1D, it seems that anyonic quasiparticle statistics, or the modular data matrices, are the universal properties. The set of universal properties that describes quasiparticle statistics is also referred to as unitary modular tensor category (UMTC). matrices (i.e., UMTC) can fully determine the topological phases, up to a bosonic FQH state.Wen (1990); Keski-Vakkuri and Wen (1993); Wen (2012); Zhang and Vishwanath (2013); Cincio and Vidal (2013) In Section II we will introduce topological quasiparticle excitations and their statistics, i.e. fusion and braiding data, in 2+1D topological phases and on 1+1D gapped edges.
Since the universal properties do not depend on the local details of the system, it is possible to calculate them from a simple renormalization fixed-point model. In this paper we will concentrate on a class of 2+1D fixed-point lattice model, the Levin-Wen string-net modelLevin and Wen (2005). As a fixed-point model, the building blocks of Levin-Wen models are effective degrees of freedom with the form of string-nets. The fixed-point string-net wavefunction is completely determined by important data – the F-matrices. The F-matrices are also referred to as unitary fusion category (UFC).
Therefore, a central question for string-net models is how to calculate the matrices from F-matrices (or how to calculate the UMTC from the UFC). In Ref. Levin and Wen, 2005 the matrices can be calculated by searching for string operators. String operators are determined by a set of non-linear algebraic equations involving the F-matrices. However, this algorithm is not an efficient one. The equations determining string operators have infinite many solutions and there is no general method to pick up the irreducible solutions. In this sense it is even not guarantied that one can find all the (irreducible) string operators.
In this paper we try to fix this weak point. Motivated by the work of Kitaev and Kong Kitaev and Kong (2012); Kong (2012), we introduce the Q-algebra approach to compute quasiparticle statistics. The idea using Q-algebra modules to classify quasiparticles is analog to using group representations to classify particles. It is well known that in a system with certain symmetry the energy eigenspaces, including excited states of particles, form representations of the symmetry group. String-net models are fixed-point models thus renormalization can be viewed as generalized “symmetry”. Moreover we show that renormalization in string-net models can be exactly described by evaluation linear maps. This allows us to introduce the Q-algebra, which describes the renormalization of quasiparticle states. Quasiparticles are identified as the invariant subspaces under the action of the Q-algebra, i.e., Q-algebra modules.
Roughly speaking, the Q-algebra is the “renormalization group” of quasiparticles in string-net models, a linearized, weakened version of a group. The notions of algebra modules and group representations are almost equivalent. Modules over the group algebra are in one to one correspondence with group representations up to similarity transformations. The only difference is that “module” emphasizes on the subspace of states that is invariant under the action of the group or algebra, while “representation” emphasizes on how the group or algebra acts on the “module”.
The specific algorithm to compute the Q-algebra modules is also analog to that to compute the group representations. For a group, firstly, we write the multiplication rules. Secondly, we take the multiplication rules as the “canonical representation”. Thirdly, we try to simultaneously block-diagonalize the canonical representation. Finally, the irreducible blocks correspond to irreducible representations, or simple modules over the group algebra. The canonical representation of a group contains all types of irreducible representations of that group. This is also true for the Q-algebra. The multiplication rules of the Q-algebra are fully determined by the F-matrices (i.e., the UFC, see (49) and (86)). Therefore, following this block-diagonalization process we have a finite algorithm to calculate the quasiparticle statistics from F-matrices. We are guarantied to find all types of quasiparticles by block-diagonalizing the canonical representation of the Q-algebra. Simultaneous block-diagonalization is a straightforward algorithm, however, it is not a quite efficient way to decompose the Q-algebra. The algorithm used in this paper is an alternative one, idempotent decomposition.
The notions of algebra, module and idempotent play an important role in our discussion and algorithm. On the other hand, we think it a necessary step to proceed from “groups and group representations” to “algebras and modules”, since we are trying to extend our understanding from “symmetry breaking phases” to “topologically ordered phases”. We provide a brief introduction in Appendix A to these mathematical notions in case the reader is not familiar with them.
Another weak point of the original version of Levin-Wen model in Ref Levin and Wen, 2005 is that the F-matrices are assumed to be symmetric under certain index permutation. More precisely, the F-matrices have 10 indices which can be associated to a tetrahedron, 6 indices to the edges and 4 indices to the vertices. If we reflect or rotate the tetrahedron the indices get permuted and the F-matrices are assumed to remain the same. In this paper we find that such tetrahedral symmetry can be dropped thus the string-net model is generalized.
In Section III we will first drop the tetrahedron-reflectional symmetry of the F-matrices but keep the tetrahedron-rotational symmetry and reformulate the string-net model. We keep the tetrahedron-rotational symmetry because in this case the relation between string operators and Q-algebra modules is clear. We give the formula to compute quasiparticle statistics, the matrices from Q-algebra modules by comparing them to string operators.
Next, in Section IV we will drop the tetrahedron-rotational symmetry assumption, and generalize string-net models to arbitrary gauge. In arbitrary gauge the string operators are not naturally defined, but we can still obtain the formula of quasiparticle statistics by requiring the formula to be gauge invariant and reduce to the special case if we choose the tetrahedron-rotation-symmetric gauge.
Finally, in Section V we briefly discuss the boundary theoryKitaev and Kong (2012) of generalized string-net models which shows the holographic bulk-edge relation. In 2+1D there are many different kinds of topological orders, classified by the non-Abelian statistics of the quasiparticles plus the chiral central charge of the edge state. Mathematically, the non-Abelian statistics, or the fusion and braiding data of quasiparticles form a UMTC. On the other hand, in 1+1D, there is only trivial topological order.Verstraete et al. (2005); Chen et al. (2011) However, if we consider anomalous topological orders that only appear on the edge of 2+1D gapped states, we will have nontrivial anomalous 1+1D topological orders. In these anomalous 1+1D topological orders, the fusion of quasiparticles is also described by a set of F-matrices. Mathematically, the F-matrices give rise to a UFC, and anomalous 1+1D topological orders are classified by UFCs. The F-matrices we use to determine a string-net ground state wavefunction turn out to be the same F-matrices describing the fusion of quasiparticles on one of the edges of the string-net model.Kitaev and Kong (2012); Kong and Wen (2014) Thus, our algorithm calculating the bulk quasiparticle statistics (UMTC) from the F-matrices (UFC) can also be understood as calculating the bulk topological order (UMTC) from the anomalous boundary topological order (UFC). Since the same bulk topological order may have different gapped boundaries, it is a natural consistency question: Do these different gapped boundaries lead to the same bulk? The answer is “yes”.Kitaev and Kong (2012) Mathematically, we give an algorithm to compute the Drinfeld center functor that maps a UFC (that describes a 1+1D anomalous topological order) to a UMTC (that describes a 2+1D topological order with zero chiral central charge).Müger (2003) Different gapped boundaries of a 2+1D topological phase are described by different UFCs, but they share the same Drinfeld center UMTC. In Appendix E we discuss the twisted string-net model in detail to illustrate this holographic relation.
Ii Quasiparticle excitations
ii.1 Local quasiparticle excitations and topological quasiparticle excitations
Topologically ordered states in 2+1D are characterized by their unusual particle-like excitations which may carry fractional/non-Abelian statistics. To understand and to classify particle-like excitations in topologically ordered states, it is important to understand the notions of local quasiparticle excitations and topological quasiparticle excitations.
First we define the notion of “particle-like” excitations. Consider a gapped system with translation symmetry. The ground state has a uniform energy density. If we have a state with an excitation, we can measure the energy distribution of the state over the space. If for some local area, the energy density is higher than ground state, while for the rest area the energy density is the same as ground state, one may say there is a “particle-like” excitation, or a quasiparticle, in this area (see Figure 1).
Among all the quasiparticle excitations, some can be created or annihilated by local operators, such as a spin flip. This kind of particle-like excitation is called local quasiparticle. However, in topologically ordered systems, there are also quasiparticles that cannot be created or annihilated by any finite number of local operators (in the infinite system size limit). In other words, the higher local energy density cannot be created or removed by any local operators in that area. Such quasiparticles are called topological quasiparticles.
From the notions of local quasiparticles and topological quasiparticles, we can further introduce the notion topological quasiparticle type, or simply, quasiparticle type. We say that local quasiparticles are of the trivial type, while topological quasiparticles are of nontrivial types. Two topological quasiparticles are of the same type if and only if they differ by local quasiparticles. In other words, we can turn one topological quasiparticle into the other one of the same type by applying some local operators.
ii.2 Simple type and composite type
To understand the notion of simple type and composite type, let us discuss
another way to define quasiparticles:
Consider a gapped local Hamiltonian qubit system defined by a local Hamiltonian in dimensional space without boundary. A collection of quasiparticle excitations labeled by and located at can be produced as gapped ground states of where is non-zero only near ’s. By choosing different we can create all kinds of quasiparticles. We will use to label the type of the quasiparticle at .
The gapped ground states of may have a degeneracy which depends on the quasiparticle types and the topology of the space . The degeneracy is not exact, but becomes exact in the large space and large particle separation limit. We will use to denote the space of the degenerate ground states.
If the Hamiltonian is not gapped, we will say (i.e., has zero dimension). If is gapped, but if also creates quasiparticles away from ’s (indicated by the bump in the energy density away from ’s), we will also say . (In this case quasiparticles at ’s do not fuse to trivial quasiparticles.) So, if , only creates quasiparticles at ’s.
If the degeneracy cannot not be lifted by any small local perturbation near , then the particle type at is said to be simple. Otherwise, the particle type at is said to be composite. The degeneracy for simple particle types is a universal property (i.e., a topological invariant) of the topologically ordered state.
ii.3 Fusion of Quasiparticles
When is composite, the space of the degenerate ground states has a direct sum decomposition:
where , , , etc. are simple types. To see the above result, we note that when is composite the ground state degeneracy can be split by adding some small perturbations near . After splitting, the original degenerate ground states become groups of degenerate states, each group of degenerate states span the space or etc. which correspond to simple quasiparticle types at . We denote the composite type as
When we fuse two simple types of topological particles and together, it may become a topological particle of a composite type:
where are simple types and is a composite type. In this paper, we will use an integer tensor to describe the quasiparticle fusion, where label simple types. When , the fusion of and does not contain . When , the fusion of and contain one : . When , the fusion of and contain two ’s: . This way, we can denote that fusion of simple types as
In physics, the quasiparticle types always refer to simple types. The fusion rules is a universal property of the topologically ordered state. The degeneracy is determined completely by the fusion rules .
Let us then consider the fusion of 3 simple quasiparticles . We may first fuse , and then with , . We may also first fuse and then with , . This requires that . If we further consider the degenerate states , it is not hard to see fusion in different orders means splitting the space as different direct sums of subspaces. Thus, fusion in different orders differ by basis changes of . The F-matrices are nothing but the data to describe such basis changes.
For 1+1D anomalous topological orders (gapped edges of 2+1D topological orders), the quasiparticles can only fuse but not braiding. So, the fusion rules and the F-matrices are enough to describe 1+1D anomalous topological orders. Later, we will see fusion rules and F-matrices are also used to determine a string-net wavefunction, which may seem confusing. However, as we have mentioned, this is a natural result of the holographic bulk-edge relation. Intuitively, one may even view the string-net graphs in 2D space as the 1+1D space-time trajectory of the edge quasiparticles.
For 2+1D topological orders, the quasiparticles can also braid. We also need data to describe the braiding of the quasiparticles in addition to the fusion rules and the F-matrices, as introduced in the next two subsections.
ii.4 Quasiparticle intrinsic spin
If we twist the quasiparticle at by rotating at by 360 (note that at has no rotational symmetry), all the degenerate ground states in will acquire the same geometric phase provided that the quasiparticle type is a simple type. We will call the intrinsic spin (or simply spin) of the simple type , which is a universal property of the topologically ordered state.
ii.5 Quasiparticle mutual statistics
If we move the quasiparticle at around the quasiparticle at , we will generate a non-Abelian geometric phase – a unitary transformation acting on the degenerate ground states in . Such a unitary transformation not only depends on the types and , but also depends on the quasiparticles at other places. So, here we will consider three quasiparticles of simple types , , on a 2D sphere . The ground state degenerate space is . For some choices of , , , , which is the dimension of . Now, we move the quasiparticle around the quasiparticle . All the degenerate ground states in will acquire the same geometric phase . This is because, in , the quasiparticles and fuse into , the anti-quasiparticle of . Moving quasiparticle around the quasiparticle plus rotating and respectively by 360 is like rotating by 360. So, moving quasiparticle around the quasiparticle generates a phase . We see that the quasiparticle mutual statistics is determined by the quasiparticle spin and the quasiparticle fusion rules . For this reason, we call the set of data quasiparticle statistics.
It is an equivalent way to describe quasiparticle statistics by matrices. The matrix is a diagonal matrix. The diagonal elements are the quasiparticle spins
The matrix can be determined from the quasiparticle spin and quasiparticle fusion rules [see Eq. (223) in Ref. Kitaev, 2006] :
where is the largest eigenvalue of the matrix , whose elements are .
On the other hand matrix determines the fusion rules via the Verlinde formula[see (60) in Section III.5]. So, and fully determine the quasiparticle statistics , and the quasiparticle statistics fully determines and .
We want to emphasize that the fusion rules and F-matrices of bulk quasiparticles and edge quasiparticles are different. In this paper we use only the F-matrices of edge quasiparticles, which is also the F-matrices describing the bulk string-net wavefunctions. Although our Q-algebra module algorithm can be used to compute the F-matrices of bulk quasiparticles, we did not explain in detail how to do this, because calculating the matrices is enough to distinguish and classify 2+1D topological orders with gapped boundaries.
Iii String-net models with tetrahedron-rotational symmetry
The string-net condensation was suggested by Levin and Wen as a mechanism for topological phases.Levin and Wen (2005) We give a brief review here.
The basic idea of Levin and Wen’s construction was to find an ideal fixed-point ground state wave function for topological phases. Such an ideal wave function can be fully determined by a finite amount of data. The idea is not to directly describe the wave function, but to describe some local constraints that the wave function must satisfy. These local constraints can be viewed as a scheme of ground state renormalization.
Let us focus on lattice models. We put the lattice on a sphere so that there is no nontrivial boundary conditions. Since renormalization will change the lattice, we will consider a class of ground states on arbitrary lattices on the sphere. One way to obtain “arbitrary lattices” is to triangulate the sphere in arbitrary ways. There may be physical degrees of freedom on the faces, edges, as well as vertices of the triangles. Any two triangulations can be related by adding, removing vertices and flipping edges. The ideal ground state must renormalize coherently when re-triangulating.
The string-net picture is dual to the triangulation picture. As an intuitive example, one can consider the strings as electric flux lines through the edges of the triangles. Like the triangulation picture, there are some basic local transformations of the string-nets, which we call evaluations. Physically, evaluations are related to the so-called local unitary transformationsChen et al. (2010), and states related by local unitary transformations belong to the same phase. If we evaluate the whole string-net on the sphere, or in other words, we renormalize the whole string-net so that no degrees of freedom are left, we should obtain just a number. We require that this number remains the same no matter how we evaluate the whole string-net. This gives rise to the desired local constraints of the ideal ground state wave function. We now demonstrate in detail the formulation of the string-net model with the tetrahedron-rotational symmetry.
A string-net is a 2-dimensional directed trivalent graph. The vertices and edges (strings) are labeled by some physical degrees of freedom. By convention, we use for string labels and for vertex labels. We assume that the string and vertex label sets are finite.
A fully labeled string-net corresponds to a basis vector of the Hilbert space. If a string-net is not labeled, it stands for the ground state subspace in the total Hilbert space spanned by the basis string-nets with all possible labellings. A partially labeled string-net corresponds to the projection of the ground state subspace to the subspace of the total Hilbert space where states on the labeled edges/vertices are given by the fixed labels. This way, we have a graph representation of the ground state subspace, which will help us to actually compute the ground state subspace.
There is an involution of the string label set, satisfying , corresponding to reversing the string direction
When an edge is vacant, or not occupied by any string, we say it is a trivial string. The trivial string is labeled by 0 and . Trivial strings are usually omitted or drawn as dashed lines
In addition we assume that trivial strings are totally invisible, i.e., can be arbitrarily added, removed and deformed without affecting the ideal ground state wave function. To understand this point, suppose we have a unlabeled string-net on a graph. It corresponds to a subspace of the total Hilbert space on the graph. Now, we add a trivial string to the string-net which give us a partially labeled string-net on a new graph (with an extra string carrying the label ). Such a partially labeled string-net on a new graph corresponds to subspace of the total Hilbert space on the new graph. The two subspaces and are very different belonging to different total Hilbert spaces. The statement that trivial strings are totally invisible implies that the two subspaces are isomorphic to each other . In other words, there exists a local linear map from to , such that the map is unitary when restricted on . Such a map is called an evaluation, which will be discussed in more detail below.
iii.2 Evaluation and F-move
A string-net graph represents a subspace, which corresponds to the ground state subspace on that graph. When we do wavefunction renormalization, we change the graph on which the string-net is defined. However, the ground state subspace represented by the string-net, in some sense, is not changed since the string-net represents a fixed-point wavefunction under renormalization. To understand such a fixed-point property of the string-net wavefunction, we need to compare ground state subspaces on different graphs. This leads to the notion of evaluation.
We do not directly specify the ground state subspace represented by a string-net. Rather, we specify several evaluations (i.e. several local linear maps). Those evaluations will totally fix the ground state subspace of the string-net for every graph.
Consider two graphs with total Hilbert space and . Assume that the two graphs differ only in a local area and dim dim. An evaluation is a local linear map from to . Here “local” means that the map is identity on the overlapping part of the two graphs. Note that the evaluation maps a Hilbert space of higher dimension to a Hilbert space of lower dimension. It reduces the degrees of freedom and represents a wave function renormalization.
Although evaluation depends on the two graphs with and , since the graphs before and after evaluation are normally shown in the equations, we will simply use to denote evaluations. We will point out the two graphs only if it is necessary.
Let us list the evaluations that totally fix the ground state subspace. For a single vertex, we have the following evaluation
We note that the above evaluation does not change the graph and thus . The evaluation is a projection operator in whose action on the basis of is given by (9).
The vertex with is called a stable vertex. is the dimension of the stable vertex subspace, called fusion rules. To determine the order of the labels, one should first use (7) to make the three strings going inwards, then read the string labels anticlockwise. If one thinks of strings as electric flux lines, enforces the total flux to be zero for the ground state.
The next few evaluations are for 2-edge plaquettes, -graphs, and closed loops:
where is called the quantum dimension of the type string. When is self-dual , the phase factor corresponds to the Frobenius-Schur indicator. Otherwise can be adjusted to 1 by gauge transformations. is because for any closed string-net on the sphere, the half loop on the right can be moved to the left across the other side of the sphere. Those evaluations change the graph. They are described by how every basis vector of is mapped to a vector in .
The last evaluation is called F-move. It changes the graph. In fact, the F-move is the most basic graph changing operation acting on local areas with two stable vertices. It is given by
It is equivalent to flipping edges in the triangulation picture. The rank 10 tensor are called F-matrices. are considered as column indices and as row indices. is zero if any of the four vertices is unstable. Otherwise, is a unitary matrix.
Note that the evaluations can be done recursively. When two graphs within and are connected by different sequences of evaluations, the induced maps from to by different sequences must be the same. Firstly the F-matrices must satisfy the well known pentagon equations
We also assume the tetrahedron-rotational symmetry. The tetrahedron-rotational symmetry is actually the symmetry of the evaluation, not of the graphs. For example, if one rotates the graphs in (15)by , the result of the evaluation should be and the tetrahedron-rotational symmetry requires that . In general, with tetrahedron-rotational symmetry, doing evaluation is “rotation-invariant”. When the evaluation of tetrahedron graphs, and simpler graphs such as -graphs or closed loops, is rotation-invariant, the evaluation of all graphs is rotation-invariant. Therefore, we call it tetrahedron-rotational symmetry.
The tetrahedron-rotational symmetry puts the following constraints on the F-matrices. Firstly, it is necessary that the trivial string is totally invisible. So, if in (20) we set the label to 0, the corresponding F-matrix elements should be 1 when the labels match and 0 otherwise, i.e.,
Secondly consider the tetrahedron graphs. After one step of F-move, the tetrahedron graphs have only 2-edge plaquettes. Thus, the amplitude can be expressed by , and , i.e.,
where the F-move is performed in the boxed area. These four results must be the same. Thus, we got another constraint on the F-matrices.
Note that (28) is different from that in Ref. Levin and Wen, 2005 because we do not allow reflection of the tetrahedron. This is necessary to include cases of fusion rules like , for example the finite group model with a non-Abelian group [see Section III.6.4]. It turns out that the conditions above are sufficient for evaluation of any string-net graph to be rotation-invariant.
With these consistency conditions, given any two string-net graphs with total Hilbert spaces and , , there is a unique evaluation map from to , given by the compositions of simple evaluations listed above. Thus, evaluation depends on only the graphs before and after, or and , not on the way we change the graphs. As we mentioned before, usually it is not even necessary to explicitly point out and , since they are automatically shown in the equations and graphs.
We want to emphasize that the fusion rules (9-13), the F-move (20), and the pentagon equation (21) are the most fundamental ones. The rest of the equations (14-19)(23)(28) are either normalization conventions, gauge choices, or conditions of the tetrahedron-rotational symmetry. With the tetrahedron-rotational symmetry, are encoded in F-matrices. In (28) set some indices to 0, and we have
Moreover, in (21) set to 0 and one can get
This implies that is an eigenvalue of the matrix , whose entries are , and the corresponding eigenvector is .
iii.3 Fixed-point Hamiltonian
Does the evaluation defined above really describe the renormalization of some physical ground states? What is the corresponding Hamiltonian? A sufficient condition for the string-nets to be physical ground states is that the F-move is unitary, or that the F-matrices are unitary
which implies that , are real numbers, or , and , i.e., if
Hence has to be the largest eigenvalue (Perron-Frobenius eigenvalue) of the matrix and the corresponding eigenvector is .
To find the corresponding Hamiltonian, note that
where is the total quantum dimension.
For a local area with plaquettes, consider the evaluation that removes all the plaquettes and results in a tree graph, as sketched in Figure 2. Since F-move does not change the number of plaquettes, we can first use F-move to deform the local area and make all the plaquettes 2-edge plaquettes. Thus, we have
which means that first use to remove all the plaquettes in the local area, and then use to recreate the plaquettes and go back to the original graph. It is easy to see that . Thus, is a Hermitian projection. Like evaluation, can also act on any local area of the string-net. We can take the Hamiltonian as the sum of local projections acting on every vertex and plaquette
which is the fixed-point Hamiltonian.
We see that is exactly the projection onto the ground state subspace. acting on a single vertex projects onto the stable vertex; acting on a plaquette is equivalent to the operator.Levin and Wen (2005); Kitaev and Kong (2012); Kong (2012); Lan (2012) The operator is more general because there may be “nonlocal” plaquettes, for example when the string-net is put on a torus, in which case evaluation cannot be performed. But in this paper we will not consider such “nonlocal” plaquettes. Evaluation is enough for our purpose.
If we evaluate the whole string-net, the evaluated tree-graph string-net represents the ground state. For a fixed lattice on the sphere with plaquettes, the evaluated tree graph is just the void graph, or the vacuum. Therefore, the normalized ground state is
Generically the ground state subspace is
iii.4 Cylinder ground states, quasiparticle excitations and Q-algebra
Now we have defined the string-net models with tetrahedron-rotational symmetry. We continue to study the quasiparticles excitations.
Let us first discuss the generic properties of quasiparticle excitations from a different point of view. By definition, a quasiparticle is a local area with higher energy density, labeled by , surrounded by the ground state area (see Figure 3). We want to point out that, a topological quasiparticle is scale invariant. If we zoom out, put the area and ground state area together, and view the larger area as a single quasiparticle area , then should be the same type as . Moreover, if we are considering a fixed-point model such as the string-net model, the excited states of the quasiparticle won’t even change no matter how much surrounding ground state area is included. Intuitively, we may view this renormalization process as “gluing” a cylinder ground state to the quasiparticle area. “Gluing a cylinder ground state” is then an element of the “renormalization group” that acts on (renormalizes) the quasiparticle states. Thus, quasiparticle states form “representations” of the “renormalization group”. Of course “renormalization group” is not a group at all, but the idea to identify quasiparticles as “representations” still works. We develop this idea rigorously in the following. We will define the “gluing” operation, introduce the algebra induced by gluing cylinder ground states and show that quasiparticles are representations of, or modules over this algebra. This algebra is nothing but the “renormalization group”.
Since any local operators acting inside the area will not change the quasiparticle type, we do not quite care about the degrees of freedom inside the area, Instead, the entanglement between the ground state area and the area is much more important, and should capture all the information about the quasiparticle types and statistics. Since we are considering systems with local Hamiltonians, the entanglement should be only in the neighborhood of the boundary between the ground state area and the area.
To make things clear, we would first forget about the entanglement and study the properties of ground states on a cylinder with the open boundary condition. Here open boundary condition means that setting all boundary Hamiltonian terms to zero thus strings on the boundary are free to be in any state. Later we will put the entanglement back by “gluing” boundaries and adding back the Hamiltonian terms near the “glued” boundaries.
On a cylinder with the open boundary condition, the ground states form a subspace of the total Hilbert space. should be scale invariant, i.e., not depend on the size of the cylinder. We want to show that, the fixed-point cylinder ground states in allows a cut-and-glue operation.
Given a cylinder, we can cut it into two cylinders with a loop, as in Figure 4. The states in the two cylinders are entangled with each other; but again, the entanglement is only near the cutting loop. If we ignore the entanglement for the moment, in other words, imposing open boundary conditions for both cylinders, by scale invariance, the ground state subspaces on the two cylinders should be both . Next, we add back the entanglement (this can be done, e.g., by applying proper local projections in the neighborhood of the cutting loop), which is like “gluing” the two cylinders along the cutting loop, and we should obtain the ground states on the bigger cylinder before cutting, but still states in . Therefore, gluing two cylinders by adding the entanglement back gives a map
It is a natural physical requirement that such gluing is associative, . Thus, it can be viewed as a multiplication. Now, the cylinder ground state subspace is equipped with a multiplication, the gluing map. Mathematically, forms an algebra [see Appendix A].
We can also enlarge a cylinder by gluing another cylinder onto it. Note that when two cylinders are cut from a larger one as in Figure 4, there is a natural way to put them back together, however, when we arbitrarily pick two cylinders, simply putting them together may not work. To glue, or enforce entanglements between two cylinders, we need to first put them in such a way that there is an overlapping area between their glued boundaries (see Figure 5). In this overlapping area, we identify degrees of freedom from one cylinder with those from the other cylinder; this way we “connect and match” the boundaries. Next, we apply proper local projections in the neighborhood of the overlapping area, such that the two cylinders are well glued. But, the ground state subspace remains the same, i.e., “multiplying” by is still ,
Now, we put back the quasiparticle . Since the entanglement between and the ground state area is restricted in the neighborhood of the boundary, it can be viewed as imposing some nontrivial boundary conditions on the cylinder. Equivalently, we may say that the quasiparticle picks a subspace of . should also be scale invariant. If we enlarge the area by gluing a cylinder onto it, in other words, multiply by , remains the same, Mathematically, is a module over the algebra . In this way, the quasiparticle is identified with the module over the algebra . A reducible module corresponds to a composite type of quasiparticle, and an irreducible module corresponds to a simple type of quasiparticle (see section II.2).
As for string-net models, recall that ground state subspaces can be represented by evaluated tree graphs. The actual ground state subspace can always be obtained by applying to the space of evaluated tree graphs. Thus, we can find out by examining the possible tree graphs on a cylinder. A typical tree graph on a cylinder is like Figure 6. Assuming that there are legs on the outer boundary and legs on the inner boundary, we denote the space of these graphs by . As evaluated graphs, all the vertices in the graphs in must be stable. In principle can take any integer numbers. But note that if , we can add trivial legs on the outer boundary, and can be viewed as a subspace of . Similarly for . Therefore, we know the largest space is .
We find that the gluing of cylinder ground states can be captured by the spaces . The gluing is nothing but adding back the entanglement. For string-net model the proper local projections are just . But before doing evaluation we have to “connect and match” the boundaries. i.e., make sure the strings are well connected. Note that and acting inside each cylinder do not affect the boundary legs. can be glued onto from the outer side only if . We need to first connect the legs on the inner boundary of with those on the outer boundary of and make their labels match each other’s; broken strings are not allowed inside a ground state area. This defines a map , where w.c. means restriction to the subspace in which the strings are well connected. Thus, is the desired gluing if there are plaquettes in . Recall that evaluation can be performed in any sequence. We know the following diagram
commutes. Thus, gluing with to obtain ground states in can be done by first considering the evaluation of the tree graphs, and then applying to get the actual ground states.
However, it is impossible to deal with an infinite-dimensional algebra . We want to reduce it to an algebra of finite dimension. Again our idea is to do renormalization. When we glue the cylinder ground states, we renormalize along the radial direction. Now, we renormalize along the tangential direction, or reduce the number of boundary legs, to reduce the dimension of the algebra.
More rigorously, our goal is to study the quasiparticles, which correspond to modules over , rather than the algebra itself. So, if we can find some algebra such that its modules are the “same” as those over (here “same” means that the categories of modules are equivalent), this algebra can also be used to study the quasiparticles. Mathematically, two algebras are called Morita equivalentMorita (1958); Kong (2012) if they have the “same” modules. Thus, we want to find finite dimensional algebras that are Morita equivalent to .
Note that with the multiplication forms an algebra. From (45) we also know that and are isomorphic algebras (the isomorphisms are just and ). It turns out that all the algebras are Morita equivalent for [see Section VI]. Thus, we know and have the “same” modules. We choose the algebra to study the quasiparticles of string-net models for has the lowest dimension among the algebras . Now, we reduced the infinite-dimensional algebra to the finite-dimensional . Since a graph in is like a letter Q, and describes the physics of quasiparticles, we name it the Q-algebra, denoted by
The subtlety of Morita equivalence will be discussed further in Section VI.
In detail, the natural basis of is
The notation looks like a tensor. But denotes a basis vector rather than a number. On one hand, represents a cylinder ground state ; on the other hand, when glued onto other cylinder ground states, can be viewed as a linear operator . Both of and are incomplete and misleading. That is why we choose the simple notation ; just keep in mind that it stands for a vector/operator. As an evaluated graph, the two vertices are stable, . Thus, the dimension of the Q-algebra is
In terms of the natural basis, the multiplication is
We know that the identity is
We can study the quasiparticles by decomposing the Q-algebra. The simple quasiparticle types correspond to simple -modules. The number of quasiparticle types is just the number of different simple -modules. As of the Morita equivalence of algebras, we also want to mention that the centers [see Appendix A] of Morita equivalent algebras are isomorphic. Thus, the center is an invariant. We argue that is exactly the ground state subspace on a torus and is the torus ground state degeneracy, also the number of quasiparticle types.
We give a more detailed discussion on the Q-algebra in Appendix B.
Assume that we have obtained the module over the Q-algebra, or the invariant subspace , that corresponds to the quasiparticle . Since , it is possible to choose the basis vectors of from respectively. Such a basis vector can be labeled by , namely,
Then we can calculate the representation matrix of with respect to this basis
where is still the map that connects legs and matches labels. We know that the representation matrix of is , which is a block matrix. Since is an idempotent, . Later we will see that the representation matrices are closely related to the string operators, and can be used to calculate the quasiparticle statistics.
iii.5 String operators and quasiparticle statistics
The string operatorLevin and Wen (2005) is yet another way to study the quasiparticles. A string operator creates a pair of quasiparticles at its ends (see Figure 7). It is also the hopping operator of the quasiparticles, i.e., a quasiparticle can be moved around with the corresponding string operator. First recall the matrix representations of string operators. For consistency we still label the string operator with ,
where is zero when either vertex is unstable.
For a longer string operator, one can apply (53) piece by piece, and contract the or labels at the connections. In particular, , since means simply extend the string operator. We define , which means the number of type strings the string operator