Characterizing Topological Order with Matrix Product Operators

One of the most striking features of quantum phases that exhibit topological order is the presence of long range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of the corresponding ground state wavefunctions, in which the full wavefunction is encoded in terms of local tensors. Topological order is reflected in the symmetries of these tensors, and we give a characterization of those symmetries in terms of matrix product operators acting on the virtual level. This leads to a set of algebraic rules characterizing states with topological quantum order. The corresponding matrix product operators fully encode all topological features of the theory, and provide a systematic way of constructing topological states. We generalize the conditions of $\mathsf{G}$ and twisted injectivity to the matrix product operator case, and provide a complete picture of the ground state manifold on the torus. As an example, we show how all string-net models of Levin and Wen fit within this formalism, and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of certain matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string-nets, and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.

One of the most striking features of quantum phases that exhibit topological order is the presence of long range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of the corresponding ground state wavefunctions, in which the full wavefunction is encoded in terms of local tensors. Topological order is reflected in the symmetries of these tensors, and we give a characterization of those symmetries in terms of matrix product operators acting on the virtual level. This leads to a set of algebraic rules characterizing states with topological quantum order. The corresponding matrix product operators fully encode all topological features of the theory, and provide a systematic way of constructing topological states. We generalize the conditions of G and twisted injectivity to the matrix product operator case, and provide a complete picture of the ground state manifold on the torus. As an example, we show how all string-net models of Levin and Wen fit within this formalism, and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of certain matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string-nets, and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.
Classifying phases of matter is one of the most important problems in condensed matter physics. Landau's theory of symmetry breaking [1] has been extremely successful in characterizing phases in terms of local order parameters, but it has been known since the work of Wegner [2] that topological theories do not necessarily exhibit such a local order parameter, and hence that different topological phases cannot be distinguished locally. One of the main reasons to call such phases topological is the fact that the ground state degeneracy depends on the topology of the surface on which the system is defined [3]. Since the realization that quantum Hall systems exhibit topological quantum order [4], significant effort has been put into classifying all topological phases [5][6][7][8][9][10]. A very large class of models exhibiting topological order was constructed by Levin and Wen [11], which is conjectured to provide a complete characterization of non-chiral topological theories in two dimensions.
A recent development at the interface of quantum information and condensed matter theory is the growing use of projected entangled-pair states (PEPS), and more general tensor network states [12][13][14]. To construct a PEPS one associates a tensor, representing a map from some virtual vector space to the local physical Hilbert space, to each site of a lattice and performs tensor contractions on the virtual space according to the graph of the lattice. The resulting quantum state can then be used as an ansatz for the ground state of a local Hamiltonian on that lattice [15][16][17]. There are two immediate and very important properties of PEPS. Firstly, for every PEPS there exists a local, positive-semidefinite, frustration free operator called the parent Hamiltonian whose kernel contains the PEPS. Secondly, the entanglement entropy of a region R is upper bounded by |∂R| log D rather than the volume, where D is the virtual dimension and |∂R| is the number of virtual bonds crossing the boundary of the region. Hence PEPS are the ground states of local Hamiltonians and obey an area law (provided the bond dimension is upper bounded by a fixed constant D as the system size increases).
In this Letter, we propose a general framework for the exact description of topologically ordered ground states using tensor networks. To achieve this we generalize the concept of G injectivity [18], where the relevant subspace of the virtual space is invariant under a tensor product action of some symmetry group G, and its extension to twisted group actions [19]. The generalized notion of injectivity presented in this Letter provides a natural extension of these concepts that applies even when no group symmetry is involved. Furthermore, it allows for the consistent characterization of the invariant subspace on the virtual level across arbitrary lattice bipartitions in terms of local tensors that form a projection matrix product operator (MPO). Extending beyond the group case is significant as it is necessary for the description of more general topological orders including the string-net models.
We first define MPO injectivity, proceed by formulating a set of algebraic conditions that have to be satisfied by valid MPOs, and then show how the ground state degeneracy and topological order is determined by those MPOs. We go on to illustrate that all ground states of the string-net models satisfy the proposed algebraic conditions, and that the key pulling through condition for these models is implied by the pentagon equation for the F -symbols. We conclude by providing an outlook to-wards possible extensions of the framework to fermionic models and higher dimensional theories, and a discussion of the potential relevance of our formalism to the development of more efficient PEPS contraction schemes.
MPO injectivity -Consider a tensor A for building a PEPS on a lattice with coordination number three, which can be understood as a linear map from the virtual space to the physical space, A : By blocking several tensors, we can define a mapping from the virtual boundary space (C D ) ⊗|∂V | to the physical bulk (C d ) ⊗|V | for arbitrary regions V of the lattice. There always exists some maximal subspace S V of the virtual space (C D ) ⊗|∂V | on which this mapping is injective. Denoting the projection onto S V by 1 1 S V , the key condition of MPO injectivity is that 1 1 S V can be represented by a MPO on the boundary ∂V , built from copies of a single local tensor M . Assuming now that we already have this property for a single site e, there must exist a pseudo-inverse A + such that A + A = 1 1 Se for some MPO projector 1 Throughout the remainder of this Letter we make use of standard tensor diagram notation, depicting each tensor as a point (or shape) with a leg emerging for each vector space it acts upon, and where a leg joining two tensors implies contraction of the associated indices.
By applying the pseudoinverse A + to the physical leg of PEPS tensor A we obtain a one-to-one correspondence between virtual and physical degrees of freedom within the subspace S. Pictorially, where we make the convention that a PEPS tensor is associated to each intersection of black lines, and a MPO tensor, denoted by M on the right hand side of Eq. (1) where M : C D ⊗ C m → C D ⊗ C m , to each intersection of a black and red line (denoting C m ). Algebraic Rules for Topological Order -The main aim of MPO injectivity is to characterize gapped quantum phases at zero temperature using the PEPS framework, including all models that give rise to a topological degeneracy of the ground state. In light of the now well established understanding of two dimensional topological models [20] and topology dependent ground state subspaces [21] we seek to describe general topological phases using string-like operators that can be arranged along noncontractable loops of the surface. The definition of MPO injectivity places the physical degrees of freedom in one-to-one correspondence with the virtual degrees of freedom in a certain subspace, allowing us to import the properties of string operators and ground states to MPOs on the virtual level. This approach is advantageous for the description of models away from a renormalization fixed point as we can essentially treat string operators on the virtual level as if we were at a fixed point, avoiding the need to broaden the string operators over some correlation length.
Except for at open endpoints, the MPOs must be free to move through the lattice to ensure that they are locally unobservable. This requirement can be satisfied locally by imposing the following condition.
Pulling-through condition: MPOs are free to pass through the PEPS tensor A on the virtual level . ( By applying the pseudo inverse A + and Eq. (1), we can express this condition purely in terms of MPO tensors. We also require the following condition for a trivial MPO loop , which guarantees that any closed MPO is a projector, as required by the definition of MPO injectivity. This can easily be seen by taking two elementary loops of MPOs, pulling one through the other onto a single leg (using (1) and (2)) and then applying (3). Generalized inverse: The final condition required for a consistent definition of the invariant subspace S on arbitrary regions of the lattice is the existence of a tensor X, for which (5) holds. Note that there is no tensor associated to the intersection of blue and red lines. Together with the pulling through condition [Eq.
(2)], Eq. (5) implies that the MPO injectivity condition is stable when multiple PEPS tensors are concatenated. It further implies that the ground space of the parent Hamiltonian on a contiguous region (which is given by a sum of local terms) is frustration free and is spanned by the concatenated PEPS tensors on the region with arbitrary states on the virtual boundary, which is known as the intersection property. Proofs of the stability under concatenation and the intersection property are provided in the supplementary material.
We conjecture that in two spatial dimensions all gapped, topologically ordered ground states admitting a PEPS description can be constructed from MPO injective [Eq. (1)] tensors, with the MPO arising as a solution of Eq. (2) and Eq. (3) for which there exists a tensor X satisfying Eq. (5).
Ground state subspace -As a result of the algebraic rules, the ground state subspace of a MPO injective PEPS parent Hamiltonian is spanned by a finite number of states. If the tensor network is closed on a torus, we find (see supplementary material) that the ground state subspace is spanned by states obtained from tensors Q that satisfy where the dots indicate periodic boundary conditions, and the equivalent equation for the vertical direction. This ensures that the resultant physical ground states are translation invariant and the location of the closure is of no significance. The ground state degeneracy on the torus is then given by the number of linearly independent physical states arising from the solution of Eq. (6) that remain distinct and normalizable in the thermodynamic limit (see appendix).
As this formalism was set up in order to characterize all gapped quantum phases, it naturally also includes states with discrete symmetry breaking (i.e. cat states). In that case, the different ground states can have local order parameters and the degeneracy will be independent of the topology. Throughout the remainder of this letter, we focus on the more interesting case of topological phases.
To determine whether any of the resulting degeneracy is truly topological in nature, one must compare the ground state degeneracy of the tensor network on the topological manifold of interest to that arising on a topologically trivial manifold (such as the sphere). This provides a deterministic recipe for checking, for any given model, the topological dependency of the ground state subspace.
Identifying the topological order -Let us now discuss how to identify the topological order in MPO injective PEPS. Since MPO injectivity is stable under concatenation, for any contiguous region the virtual indices at the boundary are supported on the invariant subspace of the MPO. This is on the one hand reflected in the low-energy excitations at the edge, which are in one-to-one correspondence with admissible boundary conditions. The edge dynamics are thus restricted to the invariant subspace of the MPO [22], which provides topological protection to the edge and allows to infer the structure of the MPO from the edge physics. At the same time, it is reflected in the entanglement spectrum and the corresponding entanglement Hamiltonian [23]. The entanglement spectrum is also restricted to the invariant subspace, and therefore, the entanglement Hamiltonian contains a universal term with infinite strength which restricts the system to be in the invariant subspace [24]. A consequence of this restriction (together with the MPO injectivity) is that the number of non-zero eigenvalues is equal to the dimension of the invariant subspace of the MPO, which gives rise to a topological correction to the zero Rényi entropy. In the case of RG fixed points, the correction should not depend on the Rényi index (as has been shown for string-net models [25]) which implies a corresponding topological correction to the entanglement entropy [26,27].
The topological correction does not fully characterize the topological phase. To this end, one must obtain the modular S and T matrices which contain all the relevant information about the topological excitations such as the mutual and self braiding statistics [28]. The fusion rules of the topological excitations can be obtained from the S matrix via the Verlinde formula. An advantage of the MPO formalism is that it allows for an unambiguous definition of modular transformations on the ground states of a lattice system on a torus, obtained by solving Eq. (6). The 90 • rotation can be performed directly on the ground state tensors Q i defined in Eq. (6). The Dehn twist, on the other hand, corresponds to increasing the winding number of the MPO along the twisting direction by one. If one uses A + A for the PEPS tensors then the overlap matrix of the original ground states with the rotated (twisted) ground states will only contain universal information and therefore correspond to the S (T ) matrix [29][30][31].
The solutions of Eq. (6) will in general not correspond to the minimally entangled states (MES), i.e. the states that have the physical interpretation of being threaded with a definite anyon flux through one of the holes of the torus [29]. Thus, one first has to find a unitary basis transformation of the ground state subspace (corresponding to a basis transformation of the tensors Q) that makes T diagonal and S symmetric [32] to be able to read off the topological properties of the excitations. Note that by wrapping the MPO around the torus in one direction, we can always construct a state with a topological flux corresponding to some Abelian anyon threaded through the hole in the orthogonal direction, since these are states with maximal topological entropy 2γ (while for general topological fluxes, the correction is 2γ − log(d 2 i ) with d i the quantum dimension). In the case that all anyons are non-Abelian this MES clearly corresponds to the one with a trivial flux. But since this construction works for any anyon theory it is very likely that it will always lead to the MES with a trivial flux.
Example: String-net models -In this section, we show that the set of models described by MPO injective PEPS contains all string-net models [11]-the largest set of many-body bosonic lattice models exhibiting topological order available in the literature. Details about the PEPS description of string-net models [33,34] are provided in the supplementary material. We now proceed to show that for the tensor network description of stringnet condensed states, both MPO injectivity [Eq. (1)] and the pulling through condition [Eq. (2)] are implied by the pentagon equation for the F -symbols. For simplicity, we demonstrate this in the multiplicity free case and note that the direct generalization applies to all stringnet models. We start from the definition of the string-net PEPS tensor where the i, j, k legs are copied to the physical level, and the MPO tensor note that we explicitly depict all tensors as 2D shapes for the string-net PEPS. The G-symbol is a symmetrized version of the F -symbol, defined in Eq. (52), that is invariant under simultaneous cyclic permutation of the upper and lower indices. These diagrams use the convention that a pair of tensor legs i, i , that are connected through the body of a tensor corresponds to a Kronecker delta on the associated indices, i.e. T {j},i,i =T {j},i δ i,i ; we therefore use a single label in the pictures. As a final convention, we always associate a multiplicative factor of the quantum dimension d λ to each term in a sum over any index λ (appearing as a closed loop in the diagrams below) [35].
We are now in a position to define the pseudo-inverse and demonstrate that it satisfies equation (1) for the string-net PEPS and MPO tensors defined in Eq. (7) and Eq. (8). We then move on to show that the pulling through condition in Eq. (2) also holds for these PEPS and MPO tensors. We point out that both of these identities are implied by the pentagon equation (53), which appears as a compatibility condition for the Fsymbols [11] and is thus guaranteed to be true for any string-net model.
The pseudo-inverse is given by whence MPO injectivity (1) follows from The loop condition [Eq. (3)] for string-net PEPS follows from the unitarity of the F -symbols as a basis transformation. The existence of a generalized inverse X ∈ C m ⊗ C m ⊗ C D ⊗ C D , for which Eq. (4) holds, follows from Eq. (10) (i.e. the pentagon equation) and the unitarity of the F -symbols. One can readily verify that a closed string-net MPO constructed from the tensors in Eq. (8), with a weighting of the normalized quantum dimension d i /D 2 associated to the internal index forming a closed loop, is a projector for any length. Since the MPO is a projector its rank is easily obtained by calculating the trace. By examining the behavior of this rank as the length increases it is clear that the topological entanglement entropy is log(D 2 ), originating from the overall normalization factor D −2 .
Conclusions -In this Letter, we have presented a general framework for the characterization of topological phases in two dimensions using the PEPS formalism. The key ingredient is a generalized notion of injectivity, in which central object is a MPO fulfilling a fundamental pulling through condition. Both G injectivity [18] and twisted injectivity [19] turn out to be special cases of MPO injectivity, which can be verified directly by constructing the relevant MPOs. As a very general example, we illustrated that all string-net models satisfy our axioms by explicitly constructing the appropriate MPOs and elucidating the correspondence between the pentagon equation and our pulling-through condition.
The characterization of topological order in terms of MPOs opens up the possibility of classifying topological phases via their MPOs, similar to results on 2D symmetry-protected topological (SPT) phases [36]. In fact, for string-net models with abelian group elements as local degrees of freedom on edges and with group multiplication as the trivalent vertex constraints, the pentagon equation reduces to a 3-cocycle condition. This close connection between the boundary symmetry MPOs of SPT states and the virtual gauge symmetry MPOs of intrinsic topological states is explicitly described by a gauging duality in the PEPS picture [37].
The framework set forth in this paper can be easily generalized to fermionic PEPS [38], as well as to higher dimensional systems by replacing MPOs with their higher dimensional generalization, Projected Entangled Pair Operators (PEPOs); it thus provides a systematic way to understand both topological phases of interacting fermions and exotic topological order in three dimensions such as the Haah code [39].
A natural question is whether our framework contains topological phases outside the string-net picture. Since the excitations of the doubled phases described by stringnets all have a Lagrangian subgroup we know that their edge modes can be gapped out [40]. Thus, to obtain phases outside string-nets we need to look at models with protected gapless edge modes (or models which do not correspond to a TQFT). Given the close connection between edge physics and the MPO, this amounts to understanding which MPOs give rise to protected gapless edge modes; indeed, in the recently discovered chiral fPEPS [41], fermionic MPOs satisfying a pulling-through condition have been identified [42].
Finally, an equation closely related to the pulling through condition [Eq. (2)] could yield an algorithm to bring 2D PEPS into a normal form that facilitates the calculation of physical observables. Intuitively, this is because once the algorithm has converged we find an MPO that approximates the transfer matrix of the model. Hence, contracting the whole PEPS with a physical observable reduces to contracting the PEPS in a local region around the observable and using a MPO to approximate the boundary. We leave these directions to future work.
within the ground subspace are given by a unique tensor network representation built from the original PEPS tensors A in the bulk with arbitrary tensors closing the network at the virtual boundary.
The parent Hamiltonian consists of a sum of 2 × 2 plaquette terms that each project locally onto the subspace spanned by the PEPS on that plaquette with arbitrary virtual boundary tensors. Here we consider the mutual ground state subspace of two neighbouring plaquette terms, any state within this subspace must be of the following form for some boundary tensors A and B, note that we are free to choose B to be invariant under a loop of MPO on the virtual boundary. By applying the pseudoinverse to all sites we find which, after pulling through and applying a generalized inverse, leads to The application of another generalized inverse yields and after applying a coarse-grained generalized inverse over two legs we find We define a new boundary tensor and find that B must take the following particular form where we have used the invariance of B under a loop of MPO on the virtual boundary of the right plaquette. Hence all states within the mutual ground space of neighboring plaquette terms are of the form C (25) for some boundary tensor C.
It is possible to iterate this argument to show that the ground space of the parent Hamiltonian on any simply connected region with a nontrivial boundary is spanned by the PEPS on that region with an arbitrary virtual boundary tensor.
Closure on a torus: In the previous section we have shown that the ground state subspace on any simply connected region of the lattice with nontrivial boundary is spanned by a tensor network built from the PEPS tensor A in the bulk and closed by an arbitrary tensor on the virtual boundary. If we proceed to close the region on a compact manifold, the additional plaquette terms of the parent Hamiltonian now crossing the boundary will further restrict the possible form of the boundary tensor in a way that depends on the topology of the manifold.
We consider the specific case of closure on a 2 × 2 torus below, and note that a direct generalization of the argument to any size of torus leads to the same conclusion. By examining several different possible closures we refine the description of the boundary tensors that lead to a linearly independent set of ground states. We begin by looking at states in the intersection of two subspaces obtained from the following two different closures A B (26) which must be of this form, for some A and B. We utilize the pulling through condition twice and apply two generalized inverses to achieve A B . (27) Next we apply a coarse-grained generalized inverse over two legs and find A B . (28) We note the following equality, attained after using coarse-grained pulling through twice, B B (29) and define the tensor B ,

B B'
, (30) for which we have the following equality It is possible to repeat the preceding arguments to find states in the intersection of the 90 • rotated versions of the above boundary configurations. For states in the triple intersection we must have both Eq. (31) and the following A C , (32) and hence

B'
C (33) for some A, B and C. Viewing the tensors in Eq. (33) as linear maps from the vertical to horizontal indices we have where C V and B H are MPOs of length two. Writing B + H for the pseudo-inverse of B H we have that , for some possibly different boundary tensors Q i . Ground state tensors: The ground state tensors Q are only defined up to transformations that do not affect the physical state. We first note that the equality of physical states for two different tensors Q and Q on the same plaquette, is equivalent to equality of the physical states arising from the same tensors only involving the virtual bonds they directly act upon, by utilizing the pseudo-inverse on the topologically trivial region not acted upon by the MPO. We are assuming periodic boundary conditions in the above two equalities and for all lattices throughout the remainder of this section. By further utilizing the RG moves we find that this is logically equivalent to equality of the states formed by Q and Q on the smallest possible torus. Hence we use this condition [Eq. (40)] to define an equivalence relation on four index tensors, whose equivalence classes capture all tensors that lead to the same physical state, note there are periodic boundary conditions for the left equality and open boundary conditions for the right equivalence relation. By the arguments of the previous section we know that it is possible to close the PEPS tensor network, with a possibly site dependent Q tensor, on any plaquette of the lattice to achieve the same physical state. We now compare the closures at different points, first considering Q tensors at two different locations along the same row of the dual lattice that give rise to the same physical state where we have pulled through the MPO such that the boundary regions match. Now by employing the pseudoinverse on the bulk, followed by RG moves, we arrive at the equation Q Q' (42) which must be satisfied by Q and Q if they give rise to the same physical state. Now consider a third Q at a different plaquette along the same row, we proceed to compare each of the original tensors Q, Q to the new one Q" via two different maps (constructed from RG moves) to arrive at two similar conditions Q Q" (43) and Q' Q" (44) which, together, imply equality of the two physical states that arise from Q and Q on the same plaquette of the PEPS, i.e., Q ∼ Q . Note, a similar argument applies to boundary conditions shifted in the vertical direction, which then implies (in combination with the horizontal) that any two tensors closing the PEPS tensor network (possibly on different plaquettes) to give the same physical state must be equivalent. Hence on the level of equivalence classes we are searching for tensor solutions of the elementary pulling through equation [Eq. (6)] in both the horizontal and vertical direction and it suffices to consider a particular representative Q for each class. This ensures that the resulting tensor networks, with the same PEPS tensor on every site and a Q tensor on the virtual level, will be translation invariant. To determine the degeneracy on the torus we must then look at the dimension of the subspace spanned by physical ground states coming from all the different Q tensor solutions. Since the RG maps yield linear transformations between MPO injective PEPS on lattices of different sizes, which are invertible on the subspaces spanned by states of the form given in Eq. (40) and Eq.(38), we can be sure that the exact degeneracy does not change for any finite system size. However, it is possible that as the system grows in size any number of states within the ground state subspace may converge to a single ground state or to zero in the thermodynamic limit. Hence one must examine the stability of the subspace as the system grows.
Finally we note that these closure arguments imply that any transformation preserving the ground state subspace can implicitly be rewritten as a transformation directly upon the Q tensors, although there is no explicit formula in general.
Tensor Network description of String-net Condensed States: As first described in [33,34], the ground states of the string-net models have exact PEPS representations. Inside of every hexagon there is one virtual degree of freedom, these are connected to one another and to the degrees of freedom on the edges by tensors that sit on every vertex. The ground state is represented by the following tensor network , where the tensor sitting on the vertices is d s = v 2 s is the quantum dimension for sector s and D = s d 2 s is the total quantum dimension. In the tensor network description, we make the convention that every closed loop comes with the multiplicative factor a s = d s /D and the middle legs that connect each pair of tensors are copied to physical degrees of freedom on the adjacent vertices. In the above expressions G is a six index tensor, known as the symmetric F -symbol. For the sake of completeness, we define these symbols and describe their symmetry properties which have been used in proving that the string-nets satisfy our axioms. The F -symbol is defined to be a scalar map (when the branching is multiplicity free, i.e., N ijk is either 0 or 1) from one fusion path to another