Boundary Hamiltonian theory for gapped topological phases on an open surface

In this paper we propose a Hamiltonian approach to gapped topological phases on open surfaces. Our setting is an extension of the Levin-Wen model to a 2d graph on an open surface, whose boundary is part of the graph. We systematically construct a series of boundary Hamiltonians such that each of them, when combined with the usual Levin-Wen bulk Hamiltonian, gives rise to a gapped energy spectrum which is topologically protected. It is shown that the corresponding wave functions are robust under changes of the underlying graph that maintain the spatial topology of the system. We derive explicit ground-state wavefunctions of the system on a disk as well as on a cylinder. For boundary quasiparticle excitations, we are able to construct their creation, annihilation, measuring and hopping operators etc. Given a bulk string-net theory, our approach provides a classification scheme of possible types of gapped boundary conditions by Frobenius algebras (modulo Morita equivalence) of the bulk fusion category; the boundary quasiparticles are characterized by bimodules of the pertinent Frobenius algebras. Our approach also offers a set of concrete tools for computations. We illustrate our approach by a few examples.

Two important characteristic properties of a 2d matter phase with an intrinsic topological order [1], crucial for topological quantum computation, are a finite set of topologically protected, degenerate ground states [2][3][4] and the corresponding anyon excitations obeying braid statistics [5]. While the former furnishes robust quantum memories [6], the latter delineates a logical Hilbert space that supports topological quantum computation via unitary braiding of anyonic quasiparticles [7][8][9][10]. It is also hopeful to realize or simulate Abelian [11,12] and non-Abelian anyons [13,14]. On a closed spatial 2-surface, the genus number and the fusion rules between anyon excitations determine the ground state degeneracy (GSD) [1,4,10,15,16]. In particular, on a torus, the GSD equals the number of anyon species. Nevertheless, realizing closed-surface material with topological order is difficult in experiments; it is much more natural to make finite open systems. Yet, any boundary massless modes that often appear must be gapped to have a well defined GSD. The gapping conditions of Abelian topological orders have recently been understood in terms of the concept of Lagrangian subsets [17][18][19][20][21][22][23][24], and subsequently the GSD of these Abelian phases on open surfaces with multiple boundaries were computed [23,25], based on the idea of anyon transport across boundaries. Experiments detecting and utilizing the topological degeneracy with gapped boundaries were proposed in [26,27]. The gapping conditions of non-Abelian topological orders have recently been tackled by the mechanism of anyon condensation [28] and equivalently by solving certain algebraic equations [29]. Gapped boundaries of topological orders can also be classified by Frobenius algebras [30], using the mechanism of anyon condensation.
Nevertheless, the rich studies and classifications of gapped boundaries of topological orders are, unfortunately, not practical enough, because they are based on abstract mathematical theories rather than explicit Hamiltonian models consisting of both bulk and boundary terms. This causes consequent issues. For example, ref. [28] offers a closed-form formula of computing the GSD of a topological order on a n-hole surface, in terms of the condensed anyons at the holes and their fusion rules. This formula, though mathematically complete and beautiful, cannot tell us how to realize the boundary conditions on a given Hamiltonian model of the topological order. Although it is known that a gapped boundary of a topological order corresponds to certain anyon condensation, it is not clear how bulk anyons interact with the boundary excitations. Hence, the proposal of using the generalized Laughlin-Tao-Wu charge-pumping argument [2,31] to braid anyons and realize topological quantum computation in ref. [28] would be impossible unless there is a concrete Hamiltonian model in which pumping a boundary excitation from one boundary to another can be studied in terms of physical operators.
More broadly, a dynamical theory in which one is not able to specify boundary conditions is not a complete dynamical theory. Therefore, Hamiltonian models of topological orders such as the Levin-Wen model [32], the Kitaev Model [7], and the twisted quantum double model [33] as a generalization of the Kitaev model are not complete dynamical models on open surfaces because they do not include boundary terms.

JHEP01(2018)134
In this paper, we develop a new systematic approach -in the framework of discrete models on a 2d graph with boundary -to gapped topological phases on open surfaces. Our approach leads to the following main results. 1. A large family of exactly solvable boundary Hamiltonian H bdry , using the data of a Frobenius algebra in the unitary fusion category (UFC) (or the fusion algebra associated with it), that defines the bulk Levin-Wen model.

A gapped energy spectrum for each boundary Hamiltonian together with its bulk
Hamiltonian, that is protected by the spatial topology, as in the original Levin-Wen model.
3. The topological ground-state wavefunctions on a disk and a cylinder in closed form, in terms of solely the input data (fundamental degrees of freedom of the model, whose generalization to surfaces with more boundaries is straightforward. 4. Explicit creation, measuring, and hopping operators for boundary excitations.

5.
A classification scheme of possible types of gapped boundary conditions by Moritaequivalent Frobenius algebras of the bulk fusion algebra, as well as that of boundary quasiparticles by bimodules of the pertinent Frobenius algebra.
These results enable one to study explicitly various topological properties, such as the GSDs, the topological entanglement entropy, excitations, bulk boundary correspondence, and so on, of topological orders on open surfaces.
Our approach is based on three physical ansatzes as follows.
1. Locality: the boundary Hamiltonian is a local one.
2. Asympototics: the boundary (interacting) theory is the asymptotics of the bulk (interacting) theory; hence, the boundary degrees of freedom should come from the bulk degrees of freedom.
3. Topological invariance: the ground-state Hilbert space is invariant under topologypreserving mutations of the graph. This leads to Frobenius algebras characterizing boundary interactions, as emergent structures of the bulk degrees of freedom.
The Locality ansatz is natural. The Asymptotics ansatz and the emergence of Frobenius algebras deserve more justification, as we now elaborate. It is well-known that for a continuum Chern-Simons gauge theory on a bounded spatial region in a plane, besides the usual Chern-Simons bulk term, the action contains an additional term on the boundary [34], to ensure the gauge invariance of the total action. The physical interpretation for adding a boundary action term is holography, i.e. a holographic correspondence between the bulk and the boundary, respected in a topological phase in two dimensions. Motivated by the desired holography in the discrete framework, the central idea of ours is to construct explicitly and systematically an appropriate boundary Hamiltonian, to be added to the Levin-Wen bulk Hamiltonian that was originally designed for a JHEP01(2018)134 closed surface. Thus, our new approach features the new Hamiltonian below, defined on a trivalent graph as in figure 1: where H LW is the usual Levin-Wen model Hamiltonian in the bulk of the graph, while H bdry is our boundary Hamiltonian defined along the boundary of the graph. As in the original Levin-Wen model, the bulk degrees of freedom, i.e., the string types labeled on the bulk edges, are simple objects of a unitary fusion category (UFC), such as irreducible representations of a finite group or a quantum group. The boundary Hamiltonian in our new approach however are associated with a Frobenius algebra that is a composite object in the bulk UFC, so that in the ground states the boundary degrees of freedom are restricted to be the string types appearing only in the pertinent Frobenius algebra. This consideration of Frobenius algebra is again motivated by the demand of exact solvability and by holography. First, when we restrict the boundary degrees of freedom to live in a subset of those in bulk described by the input UFC, and require the commutativity between the boundary operators and the bulk operators, we find that the boundary degrees of freedom would have to form a Frobenius algebra -an associative algebra with non-degenerate inner product. Second, restricted to bulk ground states only, the boundary theory of a topological order may be thought of as a (1 + 1)-d topological quantum field theory (TQFT) associated with the bulk (2 + 1)-d TQFT. It has been shown that (1 + 1)-d TQFTs are in one-to-one correspondence with commutative Frobenius algebras [35]. Third, in the case with finite groups, the Levin-Wen model is dual to the Kitaev quantum double model. Shor et al. have shown that the boundary degrees of freedom in the Kitaev model defined by a finite group G live in a subgroup of G [37], which in the dual Levin-Wen model corresponds to a Frobenius algebra in the UFC of the representations of G.
Compared to the existing approaches to gapped topological phases on open surfaces, our boundary Hamiltonian approach has the following advantages. First, the usual classification of topological boundaries by Lagrangian subsets and/or by anyon condensation, as mentioned above, is based on the data of the anyon species of excitations in topological orders, which is the output of the dynamics of the system. In contrast, our classification is characterized by the input data of the dynamical model. This is more in line with the spirit JHEP01(2018)134 of the usual Hamiltonian dynamics. Secondly, for a given Levin-Wen bulk Hamiltonian, the boundary terms of our model may not be unique, and the gapped energy spectrum of the whole bounded system depends on the choice of the boundary term accordingly. Hence, it is obvious that the boundary Hamiltonians can be used to characterize and classify the boundary conditions that give rise to gapped topological boundaries. Moreover, by solving the total Hamiltonian (bulk plus boundary terms) we can obtain the explicit wave functions of the ground and excited states, all in the form of tensor network states. This will provide us a very detailed, dynamic understanding of the stationary topological states of the whole bounded system, especially of what happens on and near the boundary. For example, our model enables us to study the boundary excitations explicitly. Also anyon condensation may be understood at more microscopic scales. These studies will be reported later separately.
We organize our paper as follows. In section 2 we review the core ingredients of the Levin-Wen model and set up notations. We extract some basic topological features of the Levin-Wen model on a closed surface, that are often overlooked except the original paper, and promote them into basic principles crucial to our development for open surfaces. Section 3 is devoted to the construction of the boundary Hamiltonian in our approach, with a graphical presentation developed along the way. Then the ground state on a disk of the total (bulk plus boundary) Hamiltonian is studied in section 4, while those on a cylinder are studied in section 6. The topological nature of the ground states is examined in section 5. We devote the next section 7 to the construction of relevant operators for studying boundary excitations, such as their creation, annihilation, measuring, and hopping operators etc. Section 8 presents a few explicit examples. In section 9 we discuss in what sense two apparently different boundary conditions are equivalent, resulting in a classification scheme of gapped boundary conditions by Morita equivalence of Frobenius algebras.
It is worth noting that there has been a few studies of the boundary Hamiltonians in the Kitaev model [36][37][38][39], as well as a study of the Levin-Wen model [17] with boundary in the language of module categories. While our approach is systematic and easier to access by the condensed matter community, we shall discuss in section 10 the relation between our approach and the one taken by Kitaev and Kong [17]. The appendices collect certain details and proofs.
Two of us also reported in a companion paper [40] a similar approach of constructing the boundary Hamiltonian of the twisted quantum double model. The present paper is a much expanded version, with many details and more results, of a short paper of three of us [41] that briefly reports the main ideas and some results.

Review of the Levin-Wen model
Let us start with a brief review of the Levin-Wen model [32] on a closed surface without boundary, setting up our notations and conventions. This model is a lattice Hamiltonian model defined by a set of input data -a unitary fusion category (UFC) C -that specifies the Hilbert space of the model. In this work, we will use the tensor description of C in terms of 6j-symbols. (For a physicist-readable introduction see, e.g., ref. [42].) JHEP01(2018)134 j 10 j 11 j 12 j 13 j 14 j 15  Figure 2. A configuration of string types on a directed trivalent graph. The configuration (b) and hence the associated Hilbert space is regarded the same as (a), with some of the directions of some edges reversed and the corresponding labels j conjugated j * .
The model is defined on a trivalent graph embedded in a closed, oriented surface. The Hilbert space is spanned by the degrees of freedom on edges (see figure 2), which are the simple objects (called string types) in C and are labeled by j that runs over a finite set of integers L = {j = 0, 1, . . . , N }. (For example, string types could be irreducible representations of a finite or quantum group.) The Hilbert space is spanned by all configurations of the labels on edges. Each label j has a "conjugate" j * , which is also an integer and satisfies j * * = j. If we reverse the direction of one edge labeled by j and replace the label by j * , we require the state remains the same. See figure 2. There is a unique "trivial" label j = 0 with 0 * = 0.
As simple objects in a UFC C, the string types are subject to fusion rules. A fusion rule on L is a function N : L × L × L → N such that for a, b, c, d ∈ L, A fusion rule is multiplicity-free if N c ab ∈ {0, 1} for all a, b, c ∈ L. We restrict to the multiplicity-free case throughout this paper unless stated otherwise. We define δ abc := N c * ab , with the symmetric properties: δ abc = δ bca and δ abc = δ c * b * a * . A triple (a, b, c) is admissible if δ abc = 1.
Given a fusion rule on L, a quantum dimension is a map d : (2.4) In particular, d 0 = 1. Let α j = sgn(d j ), which take values of ±1 for each label j and satisfy Given fusion rules and quantum dimensions on L, we may define 6j-symbols, often denoted as G. A tetrahedral symmetric unitary 6j-symbol is a map G : L 6 → C satisfying JHEP01(2018)134 these conditions: where the second equation above is the pentagon identity.
The input data of the Levin-Wen model is such a set: {d j , δ ijk , G ijm klm } that can be derived from the representation theory of a finite group or a quantum group, and more generally, such a set of data is from a UFC. For instance, we may take the labels j to be the irreducible representations of a finite group H. The trivial label 0 is the trivial representation. The fusion rules indicate whether the tensor product j 1 ⊗ j 2 ⊗ j 3 contains the trivial representation or not. Each number α j is the Frobenius-Schur indicator telling whether the representation j is real, complex, or pseudoreal. The relation d j = α j dim(j) holds, where dim(j) is the dimension of the corresponding representation space. The 6jsymbols G ijm kln are identified with the (symmetrized) Racah 6j-symbols of the group H. In this example, the Levin-Wen model can be mapped to the Kitaev quantum double model.
One important property of the 6j-symbols is that To prove this, one can rewrite the orthogonality condition as wherec stands for the complex conjugate of a complex number c. When q = i, the above equality implies that G mli kp * n must vanish unless δ mlq δ k * ip = 1. By the tetrahedral symmetry, one arrives at eq. (2.7), where v j = d j is a choice of a square root of the quantum dimension. The number v j is either real or pure imaginary, depending on the α j = sgn(d j ), and is determined up to a sign that can be fixed as follows. From the conditions (2.6), we have (G ijk 0kj v j v k ) 2 = δ ijk , and it is possible to fix the sign of v j such that In particular, v 0 = 1 because d 0 = 1 (from eq. (2.4)), and thus G 000 000 = 1 from eq. (2.6). Indeed, we can verify v 2 j = d j directly from the orthogonality condition in eq. (2.6) together with d 0 = 1. The definition in eq. (2.9) also implies that and the orthogo-

JHEP01(2018)134
There are two types of local operators, Q v defined at a vertex v and B s p (indexed by the string types s = 0, 1, . . . , N ) on a plaquette p. On a trivalent graph, a Q v acts on the labels j 1 , j 2 , and j 3 on three edges incident at the vertex v, such that where the tensor δ j 1 j 2 j 3 determines whether the triple {j 1 , j 2 , j 3 } is admissible or not at v. Since δ j 1 j 2 j 3 is invariant under permutations of its three indices, the ordering in the triple An operator B s p acts on the boundary edges of the plaquette p, and has the following matrix elements on a triangular plaquette. (2.12) The action of B s p on a quadrangle, a pentagon, or a hexagon, etc, is similar. Note that the matrix of B s p is non-diagonal only on the labels of the boundary edges (such as j 1 , j 2 , and j 3 in the above graph). The operators B s p have the properties which can be verified by using conditions (2.6). The operators defined above comprise the Hamiltonian of the model: where the sum run over vertices v and plaquettes p of the trivalent graph, and D = j d 2 j is the total quantum dimension. It turns out that all Q v and B p involved are mutually-commuting projectors. Namely, Thus the Hamiltonian is exactly soluble. The elementary energy eigenstates are given by common eigenvectors of all these projectors. The ground states satisfies the constraints Q v = B p = 1 for all v and p, while the excited states violate these constraints for certain plaquettes and/or vertices.
In cases where the input data {d, δ, G} arises from the representations of groups or quantum groups, we have δ rst * = δ srt * . Then the operators B s p also meet the following commutation relation, which can be verified by the conditions (2.6), with p 1 and p 2 two neighboring plaquettes, and by eq. (2.14), together with δ rst * = δ srt * when p 1 = p 2 .

Topological feature
We briefly review the topological nature of the bulk ground state(s). Any two given trivalent graphs Γ (1) and Γ (2) can be mutated into each other by a composition of three elementary moves, call Pachner moves. There are unitary linear maps [42] associated to these moves: This provides a linear transformation H (1) → H (2) between the two Hilbert spaces. Instead of T 3→1 , one may use another move: squeezing a "bubble", which is a composition of T 3→1 and T 2→2 . The topological nature of the bulk ground states is described as follows. The groundstate Hilbert space is invariant under arbitrary composition of T 2→2 , T 1→3 , and T 3→1 . Moreover, given initial and final trivalent graphs there are multiple ways to compose T 2→2 , T 1→3 , T 3→1 , but different ways result in one and the same transformation on the ground-state Hilbert space.
The topological feature is similar in the Hilbert space H Q=1 of simultaneous eigenvectors of Q = 1 at all vertices. All states in H Q=1 are invariant under any transformation composed of T 2→2 . An example that yields the Pentagon identity (A.1) is presented in appendix A.1.
Transformations involving T 2→2 , T 1→3 and T 3→1 that moves from one given graph to another are generally not unique; however, there are subsets of all possible sequences of T 1→3 and T 3→1 moves, each of which leads to a unique transformation. Such a subset is obtained by specifying (using a '×') in the initial graph the plaquettes to be killed by T 3→1 moves, and in the final graph the plaquettes (using a '·') to be created by T 1→3 moves. In the rest of the paper, we will follow the above convention. That is, we suppress the choice of sequence of Pachner moves behind any transformation, and denote the transformation by T hereafter. In this convention, the B p operator is compactly expressed by

The boundary Hamiltonian
In this section, we explicitly construct the boundary Hamiltonian for the Levin-Wen model on open surfaces. To understand our systematic construction, we will first introduce certain necessary mathematical structures.

Frobenius algebra as input data
The topological feature in the bulk is a consequence of the conditions (2.6) on the 6jsymbols. We expect the gapped boundaries to have similar topological feature; hence, it is natural to look for ingredients in the input data that may play a role similar to that of the 6j-symbols in the bulk. We shall need a mathematical structure -Frobenius algebra of simple objects in a UFC -to construct the boundary terms to be added to the Levin-Wen bulk Hamiltonian. Let G be a symmetric 6j-symbol over the label set L. A Frobenius algebra is a subset We can normalize (as described in appendix A.2) the non-degeneracy condition to Due to the symmetry conditions (2.6) of symmetric 6j-symbols [42], the multiplication meets the following defining properties. (unit) Then the above equation is expressed by which, when compared with eq. (2.4), implies that symbols ∆ abc play a role similar to that of the fusion coefficients δ ijk . The definition above can be illustrated graphically by the Pachner moves. The associativity condition (3.1) graphically reads where in the initial and final states, each vertex is associated with a multiplication f . The strong condition (3.3) may also be understood graphically as 8) or, equivalently, as To save writing, we can suppress the coefficients f and related summations. For example, let us express the associativity and strong conditions compactly as: The rule is to put a thick dot at any vertex associated with an f and draw an unlabeled thick line indicating a summation. We shall call this rule the thick-line convention and JHEP01(2018)134 Figure 3. Boundary is a wall carrying tails. j's are bulk labels and a are tail labels.
follow it hereafter. It is sometimes natural and handy to refer to the multiplication f of a Frobenius algebra A as the Frobenius algebra without causing any confusion.
To summarize, a Frobenius algebra f determines a state (on a trivalent graph) with an f at each vertex, and a factor √ D/d A at an internal plaquette. Such a state is invariant under Pachner moves T 2→2 , and T 3→1 but not invariant under T 1→3 that creates plaquettes. In other words, it is invariant under moves from bigger/dense graphs to smaller/sparse graphs.
For computational convenience, we set f ijk = 0 for any i, j, k in L\L A , so that f is defined for all labels.

Boundary Hamiltonian
A section of a generic boundary of our model is depicted in figure 3. The boundary is a domain wall separating the bulk (in gray in the figure) and the vacuum. The bulk edges are labeled by j 1 , j 2 , . . . , which take value in L, the set of objects of the input UFC. The boundary degrees of freedom, also taking value in L, inhabit the tails, i.e. open (or dangling) edges, labeled by a 1 , a 2 . . . . In ground states the boundary degrees of freedom are required to be restricted to a Frobenius algebra L A ⊆ L; this will be implemented by projection operators comprising the boundary Hamiltonian, to be explained shortly. The Hilbert space of the model thus consists of all possible configurations of the bulk and boundary degrees of freedom.
The boundary Hamiltonian comprises two sets of operators as follows.
Here, Q n is a boundary edge operator acting on open edge n, which projects the boundary degrees of freedom to L A ⊆ L: (3.14) And B p is an operator comprised of operators B t p : If one of the neighboring open edges a n or a n+1 / ∈ L A , then we have f t * a 2 * a 2 f a 1 ta 1 * = 0, and hence B t p = 0. (3.17) Note that our definition of the boundary Hamiltonians needs a Frobenius algebra A with L A ⊂ L as input data.

Graphical presentation
We have reviewed unitary transformations associated with Pachner moves on 2D graphs on a closed surface, in terms of 6j-symbol. These transformations quantitatively describe the topological feature in the bulk. Likewise, we can use Frobenius algebras to associate unitary transformations to Pachner moves on 1D boundary part of the graphs on an open surface.
Similar to transformations (2.17), we can use the Frobenius algebra A to define unitary transformations associated with 1D Pachner moves on the boundaries of a graph: (with where u a = √ v a (sign of square root may be arbitrarily chosen but if fixed once then for all).

JHEP01(2018)134
Alternative to T 1→2 and T 2→1 , the boundary Pachner moves can be defined as (3.20) The T 1→2 and T 2→1 can be derived by composing the bulk Pachner moves and these alternative T moves. The action of B p can be expressed in terms of T moves: To see this, we expanded in terms of f and G: (3.21) Recall the thick-line convention that a thick dot stands for an f , an unlabeled thick line stands for a summation, and a × marks the plaquette to be killed by Pachner moves. The detailed derivation of the above equation can be found in appendix A.3. More compactly, we can write B p as (3.23) On a generic open plaquette, the formula above should be sandwiched between a sequence of moves T 2→2 . . . that turns the generic open plaquette to a simple plaquette and another sequence of moves T 2→2 . . . that turns the simple plaquette back to the shape of the original generic plaquette. That is, we have B p = T 2→2 . . . T 1→2 T 2→1 T 2→2 . . . on a generic open plaquette.
One can use the graphical representation of the boundary terms to prove that: (i) the boundary terms Q n and B p commute with bulk terms Q v and B p , and (ii) the boundary plaquette operators B p are mutual commuting projections. Appendix A.4 records the details of the proof. Therefore, the total Hamiltonian -the sum of the bulk and boundary Hamiltonians -is exactly solvable.

Ground state on a disk
Now we proceed to consider the sector of states on a disk without quasiparticles in the bulk, which contains the ground state of the whole system. Effectively, on such a disk, we can apply the T transformation to shrink the bulk graph to a single plaquette, bounded The boundary Hamiltonian takes the form H bdry = − n B (n,n+1) , where (n, n+1) labels the boundary plaquette p sandwiched by the links (a n , a n+1 ), and = u an u a n+1 u a n u a n+1 v ln v l n f

JHEP01(2018)134
A topologically ordered system on the disk has exactly one ground state, which is the simultaneous +1 eigenvector of B p and N n=1 B (n,n+1) . To find the unique ground state, we need to first understand the notion of local ground states on the boundary, which boils down to solving the eigen-problem of n B (n,n+1) = 1. It turns out that the local eigenvectors are characterized by A-modules over the Frobenius algebra A, which is defined as follows.
A (right) module over Frobenius algebra A (or, a A-module) is a subset L M ⊆ L of labels equipped with an action tensor ρ a j 1 j 2 , with a ∈ L A and j 1 , j 2 ∈ L M . Note that L M is not anything ad hoc but to be obtained by solving the tensor equations of ρ a j 1 j 2 . The tensor ρ a j 1 j 2 vanishes outside of these subsets and satisfies the following condition.
which can be understood in terms of Pachner moves: , where the f factor in the last row should be understood as the tensor of the trivial module to be defined below eq. (4.7). Let us again take the thick-line convention. We also suppress the indices of the action tensor ρ and put it in a box. In this box-notation, the condition (4.3) takes the following compact form: (4.4) Here a boxed ρ at a vertex means that the tensor ρ is associated with the vertex (e.g., ρ c j 1 j 2 on the r.h.s. , with a thick-line summation). We denote the collection of all modules over A as M od A . The unit condition on the Frobenius algebra A implies the unit condition on A-modules: If we set a = 0, a 1 = a * 2 = a, j 2 = j 1 = j, and j = k in eq. (4.3), we get which can be verified directly as in appendix A.5. The ground state on a disk has neither bulk nor boundary quasiparticle excitations. It is easily shown to be non-degenerate. Using the local basis found above, the unique ground state on the disk is expressed as The unique ground state on a disk can also be expressed in terms of f ijk and 6j-symbols: .

(4.11)
Here p is the only plaquette of bulk in the simplest reduced diagram.

JHEP01(2018)134
We now prove that |Φ in eq. (4.9) is a ground state on the disk. It suffices to show that B p |Φ = |Φ . We apply unitary Pachner moves on |Φ and get which can be evaluated using eq. (A.17) which is a B p = 1 eigenvector. Hence |Φ is a ground state on disk.
Certain useful proofs can be found in appendix A.5. We often abuse the notation by referring to M as an A-module.

Topological feature of ground states
In this section, we study the topological nature of the ground state. Namely, we show that the ground states of our Hamiltonian are invariant under Pachner moves.
The bulk topological feature in the case with boundaries is the same as that in the case without boundaries. We then need only to show the boundary topological feature via the 1+1D Pachner moves on boundaries, which are defined in eqs. (3.18) through (3.20).
With boundary, the topological feature can be described as follows. The ground state is invariant under any transformation composed by T 2→2 , T 1→3 , T 3→1 in the bulk and T 1→2 , T 2→1 on the boundary. Moreover, such transformation is unique: different ways to composing T 's results in the same transformation.
To show the uniqueness of the transformation, we consider boundary Pachner moves T 1→2 , T 2→1 . Take example of a transformation from N 1 tails to N 2 tails. The composition of T 1→2 , T 2→1 amounts a graph structure with N 1 input edges and N 2 output edges, where each trivalent vertex is attached with a multiplication f . 6 Ground states on a cylinder A topologically ordered system on a cylinder has two boundaries. We can specify the two boundary Hilbert spaces and define the two boundary Hamiltonians by two Frobenius algebras over L A and L B , respectively. The corresponding multiplications are denoted by f k ij for i, j, k ∈ L A and g c ab for a, b, c ∈ L B . If we consider the states without any bulk quasiparticles, we can completely shrink the bulk graph by Pachner moves to make it disappear, so that the cylinder graph becomes a ring with open edges on both sides of the ring, as in figure 4. Consider the Hilbert subspace spanned by all the labels in the graph. The total Hamiltonian contains two boundary Hamiltonians defined by two Frobenius algebras.
The ground states are characterized by the A-B-bimodules, as will be defined shortly. Each bimodule P M gives rise to a ground-state wavefunction: An A-B-bimodule is a subset L M equipped with an action tensor P ab ijk , satisfying Here the tensor P M is expressed by a box, whose meaning is as follows:

JHEP01(2018)134
The r.h.s. is independent of j which is summed; hence, j does not appear on l.h.s.. Note that in this work modules and bimodules are multiplicity free (see appendix A.8 for details). The A-B-bimodules are subject to the orthonormality and completeness conditions, respectively as follows.
Both conditions can be proved in a fashion similar to that in the case of Amodules. Now the ground state Φ cyl M characterized by the bimodule P M can be expressed graphically as (6.6) Let us prove this. First, similar to the disk case, we study the local basis of the ground states on a cylinder. By local we mean a piece of the ring comprising two neighboring tensors as follows.
Detailed derivation can be found in appendix A.6. Hence, Φ cyl support topological quasiparticles. In this section we characterize the excitations and topological quasiparticles, by studying the algebra of local operators B t p . We show that topological quasiparticles are classified by the bimodules over A.
The main result is that topological quasiparticles are classified by A-B-bimodules, which are solutions to eq. (6.2). Particularly, if A = B, the topological quasiparticles and cylinder ground states are classified by the A-A-bimodules, which are also solutions to eq. (6.2) for A = B. The GSD on the cylinder is identical to the number of quasiparticle species on the boundaries.
There are three kinds of important operators to characterize quasiparticles. One is a set of orthonormal projection operators as measuring operators to identify quasiparticles. Another is the set of creation operators to create quasiparticle pairs (quasiparticles can not be singly created). The third is a set of hopping operators that can hop a quasiparticle along a boundary. We will construct these three kinds of operators in the following three subsections. Then we discuss the topological feature of quasiparticles in terms of hopping operators. We also discuss fluxons as a special subset of quasiparticles.

Measure quasiparticles
In this subsection we construct a set of orthonormal projection operators as measuring operators to identify quasiparticles.
Given a bimodule M , define the corresponding measuring operator Π M by Using the orthonormal condition (6.4) and completeness condition (6.5), we verify that the set {Π M j } is orthonormal

Fluxons
We consider a subclass of quasiparticles called fluxons. Thus, we can restrict to the Hilbert subspace of v Q v = 1. We find that the local operators B t p form an algebra The quasiparticles occupied at plaquette p are then identified by the orthonormal projection operators n x p n y p = δ x,y n x p , (7.8)

JHEP01(2018)134
where Y x t satisfies the following conditions, as can be derived from eq. (7.6): Particularly, n x=0 p = B p with Y 0 t = 1 for all t ∈ L A . For an excitation ψ with n x p = 1 we say ψ supports an x-type fluxon at position p. Fluxons are a subclass of the full set of topological quasiparticles identified by the bimodules. Indeed, let Y x t = [P M ] tt * 0t0 , then eq. (7.9) is identified with (6.4). Hence, fluxon is a special type of quasiparticle identified by those modules (M, P M ) in which M contains 0.

Charge boundary
For any input fusion category, there is always a trivial Frobenius algebra A 0 = 0, such that B p is trivial and hence the boundary Hamiltonian reduces to The A 0 -modules (and A 0 -A 0 -bimodules) are the entire label set L, with [ρ j ] 0 jj = 1 ([ρ j ] 00 jjj = 1), j ∈ L. Boundary quasiparticles are then characterized by labels j ∈ L.
There are two Frobenius algebras, one is the trivial one A 0 = 0, which defines a charge boundary condition. Quasiparticles on the charge boundary are identified with 1 and e, with e a Z 2 charge.
The nontrivial Frobenius algebra is A 1 = 0 ⊕ 1, with L A = {0, 1} = L. This is a flux boundary. The boundary quasiparticles are identified with 1, m with m a Z 2 -flux. The charge-boundary is referred to z-boundary or smooth boundary (respectively, the flux-boundary to x-boundary or rough boundary) in some literature (e.g., see ref. [36]).
Cylindrical model has GSD = 2 with the charge-charge or flux-flux boundary conditions, and GSD = 1 with the charge-flux boundary condition. We shall explain the flux-flux boundary case in detail below.
Consider the model with the flux boundary condition (with the algebra A 1 ) on a disk illustrated in figure 5. The total Hamiltonian is  where is a positive constant, and Seen in figure 5, examples of these operators are and If we consider states without quasiparticles in the bulk, we can simplify the problem with the effective theory on the disk as defined on a chain, see figure 6. The Hilbert space is spanned by N + 1 spins: on N external edges and one internal edge denoted by 0 (the spins on all other internal edges are determined by the fusion rules). We require a global constraint The boundary Hamiltonian is The extra σ x 0 in last term is due to the nontrivial action of B p on the spin at edge 0.
One verifies that the GSD = 2, as expected. The two ground states are and Here σ x n = ±1 denotes the simultaneous eigenvalues of σ x n = ±1 for all n.
The only nontrivial multiplication reads A 1 has two modules: (1) M 0 = 0 ⊕ 2, i.e., A 1 itself, with action morphism being the multiplication ρ a jk = f ak * j ; (2). M 1 = 2, with action morphism given by The two algebras A 0 and A 1 are Morita equivalent, hence giving rise to the same boundary condition.

Equivalent boundary conditions
The boundary conditions are classified by A-A-bimodules, in the sense that boundary elementary excitations with good quantum numbers are identified with equivalent bimodules. In this section, however, we discuss a situation where two different Frobenius algebras in a unitary fusion category give rise to equivalent boundary conditions. Two Frobenius algebras A and B are Morita equivalent if category M od A of A-modules is equivalent to M od B . [43] By the previous analysis, the local ground state basis is characterized by modules. Hence, Morita equivalent Frobenius algebras define equivalent boundary conditions.
For any A-module M , k ⊗ M is also a right module. But k ⊗ M is reducible; hence, we can decompose k ⊗ M into a direct sum of irreducible modules. To do so, we need to study For example, in Fibonacci case, the two Frobenius algebras are Morita equivalent. One can easily verify that the fusion rules by mapping M 0 → N 1 and M 1 → N 0 . Hence, the two Frobenius algebras are Morita equivalent and give rise to the same boundary conditions. In the Ising case, the two Frobenius algebras are also Morita equivalent by mapping M 1 → N 2 , M 2 → N 1 and M 3 → N 3 . One verifies the fusion rules are preserved under the mapping. Hence, the two Frobenius algebras are Morita equivalent and give rise to the same boundary conditions.

Relation to the Kitaev-Kong formulation
We used a Frobenius algebra to define the Boundary theory in this paper. This formulation is closely related to Kitaev and Kong's work [17] that formulates boundary theories using module categories over C. In this section we will discuss the relation between our approach and the Kitaev-Kong (KK) formulation. In our approach, we take boundary degrees of freedom from the labels of the input UFC -the same degrees of freedom as in the bulk, and we start with local boundary Hamiltonians. To write down a "good" boundary Hamiltonian we examine the (unitary representation of) 1+1D boundary Pachner moves. The desired form of the Hamiltonian will be one such that the ground-state Hilbert space is invariant under all bulk and boundary Pachner moves. This invariance leads to a Frobenius algebra structure appearing in the boundary Hamiltonian operators.

JHEP01(2018)134
With Hilbert space spanned by labels of the input UFC, all operators are explicitly expressed using these labels. Our approach is convenient for computational purposes yet rigorous in characterizing the topological properties. The reader can compute the ground states and excitations by solving the Hamiltonian eigen-problems without knowledge of categories.
Given the bulk Levin-Wen model with input fusion category C, in the KK formulation, the input data to specify the boundary degrees of freedom and boundary operators is a module category M over C. The topological feature of the boundary ground states comes from the compatibility conditions between the bulk degrees of freedom in C on the left side and boundary degrees of freedom in M. Here and after we assume the bulk is on the left of a boundary. This is always possible if one tracks along the boundary clockwise.
In the following we will build up the correspondence between the Hilbert space structures in KK formulation and our formulation, by studying the eigen-problem of B p = 1, where p label the boundary plaquettes.
Given a boundary Γ, the local basis of boundary ground states (i.e., the p B p = 1 eigenstates) has been discussed in previous sections and has the form for some M ∈ M od A . This basis is defined for simple boundary plaquettes but can be generalized to cover the cases with generic boundary plaquettes, where bulk edges must also be taken into account. We write the generic form of potential basis vectors as where we assume two potentially different modules M and N and a tensor η connected to bulk edge k, to be determined by the condition B p = 1. Acting B p on such states yields Figure 8. The above is a B p = 1 eigenvector if and only if This condition is equivalent to the defining property (9.1) of the morphisms in Hom(k ⊗ M, N ) of the category M od A . Therefore, on a generic boundary graph, the local basis of boundary ground states is characterized by modules M 's in M od A and morphisms η in M od A . It is known that M od A is equivalent to a module category. We can use such module category data M and η as input degrees of freedom to describe the ground states. See figure 8. Hence, we build up the mapping between the Hilbert space in our formulation and that in the KK formulation on the level of ground states.
This mapping is two-way, which follows from a mathematical theorem: the category of right modules over an algebra A in C is equivalent to the right module category over (unitary fusion) C [43]. The mapping is many to one. Namely, two Frobenius algebras A and B are Morita equivalent if M od A is equivalent to M od B , and they specify the same boundary condition.
In the KK formulation, the boundary excitations are constructed using module functors F un(M, M) of the input module category M. In our formulation, however, the elementary excitations are identified with the bimodules M od A|A . In this paper, we are not going to directly prove the equivalence of elementary excitations in the two formulations. Nevertheless, it is known that [43] if M is taken to be M od A then the category F un(M, M) is equivalent to M od A|A . Therefore, we expect our formulation also agrees with the KK formulation on boundary elementary excitations.
By above analysis, we show that our approach with an Frobenius algebra A is equivalent to the KK formulation with input module category M = M od A . : By the pentagon identity (2.6), eq. (A.1) is identified with eq. (A.2). We can simply write both transformations as T . In fact, any transformation composed by T 2→2 moves only depends on the initial and final graphs (with the same topology) and can be written as T , without specifying the choice of the sequence of T 2→2 moves.
Transformations involving T 2→2 , T 1→3 and T 3→1 moves from one graph to another are not unique. So we introduce the cross-dot notation as in section 2.1, see figure 10 for an example. In figure 10(a), the initial graph has two plaquettes marked by ×, while the final graph has one plaquette marked by ·. In this case, all possible sequences of T 2→2 , T 1→3 , and T 3→1 moves result in the same transformation between the Hilbert spaces associated with the initial and final graphs. one such sequence is shown in figure 10(b). where the second equality follows from the association condition. We also state that boundary plaquette operators B p are mutual commuting projections: which is verified by directly computation: where in the second equality uses the association condition, and third the strong condition.

JHEP01(2018)134
A.5 Ground state on a disk We first prove (4.8) and then state the proof for orthonormality and completeness for ρ M 's.
A.7 W M |Φ is an eigenvector of Π M = 1 We now verify that W M |Φ is an eigenvector of Π M = 1 in the following. It suffices to verify that where the last equality is due to the completeness condition (6.5). Hence W M |Φ is a Π M = 1 eigenvector.

A.8 Bimodules with multiplicity
In general, the action tensor of modules ρ and bimodules P carries extra indices, say, α and β. The action tensor of a bimodule P M is now expressed by In defining property (6.2), for a thick line on the l.h.s. we should also sum over appropriate α indices of the two action tensors. Similarly, in general we need to put extra indices to a module tensor action and follow the same convention. Nevertheless, the discussion and derivation throughout the paper remains true when we add the extra indices to tensor actions and add the corresponding summation rule to the thick line convention. Therefore, in the rest of paper, we suppress the α indices for simplicity.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.