Matrix product operators for symmetry-protected topological phases: Gauging and edge theories

Projected entangled pair states (PEPS) provide a natural ansatz for the ground states of gapped, local Hamiltonians in which global characteristics of a quantum state are encoded in properties of local tensors. We develop a framework to describe onsite symmetries, as occurring in systems exhibiting symmetry-protected topological (SPT) quantum order, in terms of virtual symmetries of the local tensors expressed as a set of matrix product operators (MPOs) labeled by distinct group elements. These MPOs describe the possibly anomalous symmetry of the edge theory, whose local degrees of freedom are concretely identified in a PEPS. A classification of SPT phases is obtained by studying the obstructions to continuously deforming one set of MPOs into another, recovering the results derived for fixed-point models [Chen et al., Phys. Rev. B 87, 155114 (2013)]. Our formalism accommodates perturbations away from fixed-point models, opening the possibility of studying phase transitions between different SPT phases. We also demonstrate that applying the recently developed quantum state gauging procedure to a SPT PEPS yields a PEPS with topological order determined by the initial symmetry MPOs. The MPO framework thus unifies the different approaches to classifying SPT phases, via fixed-point models, boundary anomalies, or gauging the symmetry, into the single problem of classifying inequivalent sets of matrix product operator symmetries that are defined purely in terms of a PEPS.

Figure 1

(a) A PEPS tensor on a trivalent vertex. (b) A right-handed MPO tensor.

Figure 2

The PEPS map $A_{\mathcal{R}}$ from virtual indices on edges in $\partial \mathcal{R}$ to physical indices on vertices in $\mathcal{R}$.

Figure 3

The symmetry of the PEPS map $A_{\mathcal{R}}$ on a region $\mathcal{R}$ containing five sites.

Figure 4

The bulk to boundary isometry, for a region $\mathcal{R}$ containing four sites, projected onto the injectivity subspace $P_{\partial \mathcal{R}}{\mathcal{W}}_{\mathcal{R}}^A\left[O\right]P_{\partial \mathcal{R}}=H^{+}{A}_{\mathcal{R}}^{†}O{A}_{\mathcal{R}}{H}^{+}$.

Figure 5

(a) A symmetry twist $(x,y)$ on an infinite plane. (b) Physical action of the aforementioned symmetry twist.

Figure 6

$(x,y)$ symmetry-twisted PEPS, for infinite or periodic boundary conditions.

Figure 7

(a) A symmetry twist along a cylinder PEPS. (b) A pair of monodromy defects in a PEPS.

Figure 8

(a) A simple example ${\varphi }_p={\varphi }_3{\varphi }_2^{-1}{\varphi }_1$. (b) Noncontractible cycles of the 2-torus.

Figure 9

A representative flat $\mathsf{G}$ connection labeled by $(x,y)$.

Figure 10

An $(x,y)$ symmetry-twisted PEPS on a torus.

Figure 11

A symmetry twist $g$ along an open path.

Figure 12

(a) A symmetry-twisted SPT PEPS on a cylinder. (b) A pair of monodromy defects on a sphere.

Figure 13

The process used to find the effect of braiding on the internal degrees of freedom of a single anyon.