Topological Order in the Projected Entangled-Pair States Formalism: Transfer Operator and Boundary Hamiltonians

We study the structure of topological phases and their boundaries in the projected entangled-pair states (PEPS) formalism. We show how topological order in a system can be identified from the structure of the PEPS transfer operator and subsequently use these findings to analyze the structure of the boundary Hamiltonian, acting on the bond variables, which reflects the entanglement properties of the system. We find that in a topological phase, the boundary Hamiltonian consists of two parts: A universal nonlocal part which encodes the nature of the topological phase and a nonuniversal part which is local and inherits the symmetries of the topological model, which helps to infer the structure of the boundary Hamiltonian and thus possibly of the physical edge modes.

Figure 1

Tensor networks for entanglement spectra and for topological models; see the text for details.

Figure 2

(a) Maximal eigenvalues ${\gamma }_{p\varphi }^{p^{\prime }{\varphi }^{\prime }}$ for the TC with string tension (relative to ${\gamma }_{e0}^{e0}$) for $N_v=12$. (b),(c) Splitting (b) $1-|{\gamma }_{p\pi }^{p\pi }/{\gamma }_{p0}^{p0}|$ and (c) $1-|{\gamma }_{e0}^{o0}/{\gamma }_{p0}^{p0}|$ for $\lambda =0.65,0.66,\dots$ (blue dashed lines, topological phase) and $\lambda =0.64,0.63,\dots$ (green solid lines, trivial phase). The red dotted line is $\lambda =0.644\approx {\lambda }_{\mathrm{crit}}$. The calculations (as well as the ones in Figs. 3, 4) have been carried out using exact column-wise contraction (cf. Refs. 9, 20, 21) and are thus exact.

Figure 3

(a) Interaction range of $H_0$ for the TC with string tension ($N_v=12$), obtained using Eq. (2) with equal weights, which is the correct interpretation for the topological sector. Interactions in the topological phase (black lines, $\lambda =0.7,0.75,\dots$) decay exponentially. In the trivial phase (gray lines, $\lambda =0.60,0.55,\dots$), the interpretation is no longer valid, seemingly leading to a nonlocal Hamiltonian. (The red dashed line is ${\lambda }_{\mathrm{crit}}$.) (b) By changing to the interpretation of the transfer operator which is valid in the trivial phase (see the text), we obtain a $H_0$ which is local in the trivial phase. The blue solid line shows $\Vert {\sigma }_{\mathrm{triv}}-\mu (B){\Vert }_1/N_v$ for $N_v=12$ (right scale), cf. the text, and the red dashed line shows the corresponding $B$ (left scale). The comparison with $\Vert {\sigma }_{\mathrm{triv}}-\mu (B=0){\Vert }_1/N_v$ (green dotted line) and $\Vert {\sigma }_{\mathrm{topo}}-\mu (B){\Vert }_1/N_v$ [yellow dash-dotted line, cf. Eq. (2)] shows that the boundary Hamiltonian is well approximated by $B\sum _{}^{}{X}_i$ in the trivial phase, while the decay in the topological phase is due to the high temperature. (c) Comparison of $H_0$ and $H_{\pi }$. The plot shows $\Vert H_0-\mathcal{F}(H_{\pi }){\Vert }_{\mathrm{op}}/\Vert H_0-H_{\pi }{\Vert }_{\mathrm{op}}$, where $\mathcal{F}$ flips the sign of all terms which change the parity across the boundary (see the text); in the topological phase, the difference converges to 0 exponentially in $N_v$.

Figure 4

RVB to TC interpolation. (a) $|{\gamma }_{p\varphi }^{p^{\prime }{\varphi }^{\prime }}|$ for the off-diagonal blocks as a function of the interpolation parameter $\theta$ (with $\theta =0$ the RVB and $\theta =1$ the TC), with normalization ${\gamma }_{e0}^{e0}=1$, for $N_v=10$. Inset: Maximal splitting of the diagonal blocks $|{\gamma }_{p\varphi }^{p\varphi }|$ (from top: $\theta =0,0.1,0.2,\dots$). (b) Interaction range along the interpolation for $N_v=6$, 8 (small or large symbols).