Symmetry reduction induced by anyon condensation: A tensor network approach

Topological ordered phases are related to changes in the properties of their quasiparticle excitations (anyons). We study these relations in the framework of projected entanglement pair states and show how condensing and confining anyons reduces a local gauge symmetry to a global on-site symmetry. We also study the action of this global symmetry over the quasiparticle excitations. As a byproduct, we observe that this symmetry reduction effect can be applied to one-dimensional systems as well and brings about appealing physical interpretations on the classification of phases with symmetries using matrix product states. The case of ${\mathbb{Z}}_2$ on-site symmetry is studied in detail.

Figure 1

(a) The parent tensor (blue) is invariant under the virtual action of a representation of the group $G$. This gauge symmetry endows the PEPS with topological order. (b) Some operators acting on the virtual level of the restricted tensor (pink) are equivalent to an action on the physical index of the tensor; they correspond to a global on-site symmetry. The elements of $G$ which still are gauge symmetry of the restricted tensor form a subgroup, that we name $G_{\mathrm{topo}}$, and characterize the topological order of the restricted PEPS.

Figure 2

Construction of an MPS with periodic boundary condition by placing maximally entangled pairs and applying the map $\mathcal{P}$. Let us note that the map acts on virtual particles belonging to different maximally entangled pairs.

Figure 3

(a) Diagram picture of the tensor $A_{\alpha \beta \gamma \delta }^i$ of a PEPS for a square lattice. (b) The PEPS constructed via the concatenation of the virtual d.o.f. of the tensors (without boundary conditions). (c) A single on-site unitary (which is a representation of the global symmetry) applied to the physical system is equivalent in an injective PEPS, as far as the tensor level is concerned, to the action of four unitaries, representing a group element, in the virtual d.o.f.

Figure 4

Relation between the lattice model of Ref. [17] and the tensor construction Eq. (4). The white dots represent the generalized spins living on the edges. Each tensor is placed on alternating plaquettes of the lattice, gathering the four surrounding spins. The physical index of the tensor corresponds to the tensor product of four local Hilbert spaces and this fact is reflected in Eq. (5). Diagonal gray edges represent PEPS virtual bonds.

Figure 5

Gauge symmetry of the tensor in the virtual d.o.f. Within the family of $G$-isometric PEPS, the invariance is achieved with tensor product operators: $\{L_g\otimes L_g\otimes L_g^{†}\otimes L_g^{†}:g\in G\}$. More generally the gauge symmetry is realised by matrix product operators [32].

Figure 6

(a) The action of the operator $S_g$ leaves invariant the tensor $A_G$. (b) The action of the operator $S_g$ over $A_G^{\mathrm{res}}$ change the tensor as depicted.

Figure 7

A subset of the lattice, in a background of $A_G$ tensors, having a flux excitation string. The red dots represent the operator $L_g$ acting on the virtual d.o.f. of the tensors. Each colored square drawn represents the place where a local Hamiltonian acts. Only the pink (dot-dashed line) plaquette is an excitation of the local Hamiltonian $h$: adds $+1$ to the total energy of the state.

Figure 8

A subset of the lattice, in a background of $A_G^{\mathrm{res}}$ tensors, having a string of $L_g$ operators acting on the virtual d.o.f. Each colored square drawn represents the place where a local Hamiltonian acts. In this situation, both plaquette are excitations of the local Hamiltonian $h^{\mathrm{res}}$: adds $+1$ to the total energy of the state.

Figure 9

(a) We apply the operator $S_g^{\otimes 3}$ on a background of $A_G^{\mathrm{res}}$ tensors; the result is depicted on part (b). (b) The confined loop of $L_{g^{\prime }}$ operators affects 7 plaquettes, corresponding each one to an excitation of the local Hamiltonian $h^{\mathrm{res}}$.

Figure 10

A subset of the lattice with an open string of virtual operators $L_k^G$, representing a flux excitation.

Figure 11

The previous flux excitation modified by the action of the symmetry operator $S_g^N$. The virtual operators defining the excitation are now $L_{k^{\prime }}$ with $k^{\prime }=gk{g}^{-1}$. Then, the flux-type particle belongs now to the conjugacy class $C\left[k^{\prime }\right]$ which represent a change in type of particle after the action of the symmetry.

Figure 12

The ground state corresponding to the pair conjugacy class of $(k,n)$ is depicted.

Figure 13

(a) The local symmetry of the tensor $A_G$ is represented. (b) The tensor $A_G^{\mathrm{res}}$ of Eq. (20) is depicted. The physical Hilbert space is decomposed into a tensor product form. (c) The action of the operators $S_g=L_g\otimes L_{g^{-1}}$ is translated into the virtual d.o.f. with a freedom on the choice of the element $g^{\prime }$ within coset $\left[g\right]$.

Figure 14

(a) Diagrammatic representation of a term appearing in the block $A_G^{\mathrm{res}}(\nu ,\mu )$; see Eq. (26). The gray dashed line is meant to indicate that the four boxes represent the operator ${\mathbb{I}}_{{\ell }_{\nu }}\otimes {\mathbb{I}}_{d_{\rho }}\otimes {\mathbb{I}}_{d_{\sigma }}\otimes {\mathbb{I}}_{{\ell }_{\mu }}$ corresponding to fixed values of $\rho ,\sigma$ in the sum. (b) Diagrammatic representation of the whole sum Eq. (26): each set of four tensors in a same plane parallel to the sheet relate to a same term of the sum, i.e., a same value of the pair $(\rho ,\sigma )$. The $\oplus$ symbol is meant to mark the summation over all possible values of $(\rho ,\sigma )$. (c) Decomposition of an irrep of $G$, described by the Eq. (30). We omit to represent the dependence of $V$ and $C$ on $g,\nu ,i,j$; see Eq. (30).

Figure 15

(a) Action of the irrep operator on the tensor according to Eq. (31). (b) Final result of the action on the restricted tensor given by Eq. (32).

Figure 16

The virtual representation of the dyonic quasiparticle is depicted. The red dots represent the operator $R_h$ acting on the virtual d.o.f. of the tensor network. The yellow rhombus corresponds to the operator $D_{\alpha }^m$ of Eq. (C1).

Figure 17

The operator of Eq. (C3) acts on four adjacent edges, denoted as $x_1,y_1,x_2,y_2$, and involves three vertices and three plaquettes. This operator depends on the orientation of each edge and we take the one represented by the arrows. The green and blue points identified the vertices and plaquettes excited, respectively.

Figure 18

Tensor network representation of the physical system involved in the creation of a dyon. The blue dots are depicted for comparison with Fig. 17 and the virtual index $r$ is depicted for clarification.