Chiral Projected Entangled-Pair State with Topological Order

We show that projected entangled-pair states (PEPS) can describe chiral topologically ordered phases. For that, we construct a simple PEPS for spin-$1/2$ particles in a two-dimensional lattice. We reveal a symmetry in the local projector of the PEPS that gives rise to the global topological character. We also extract characteristic quantities of the edge conformal field theory using the bulk-boundary correspondence.

Figure 1

Left: Construction of the PEPS defined via Eqs. (1) and (2). $|P⟩$ is defined on two physical fermionic modes (red squares) and eight virtual Majorana modes (blue disks) for each site. Sites are marked by dashed circles. Afterwards, the projectors ${\omega }_{\mathbf{r},\mathbf{r}+\mathbf{x}}$ and ${\omega }_{\mathbf{r},\mathbf{r}+\mathbf{y}}$ are applied on the four Majorana modes located between neighboring sites, as indicated by light blue boxes, yielding the PEPS $|\mathrm{\Psi }⟩$. Right: Noncontractible loops on the torus.

Figure 2

The entanglement spectra for a bipartition of a $N_v=14$ infinite cylinder in four topological sectors. The momentum in the horizontal direction is extended beyond its $2\pi$ periodicity to distinguish the spectra corresponding to different particle-number subspaces. The entanglement spectra with the same particle number obey the degeneracy pattern $\{1,1,2,3,5,\dots \}$, a fingerprint for $c=1$ chiral Luttinger liquid theory. Insets: Zoom-in of the low-lying part of the spectra.

Figure 3

Conformal dimensions of the primary fields (a) $\mathit{s}$ and (b) $\mathit{v}$ through exponential decay fitting.

Figure 4

Entanglement entropy versus perimeter $N_v$ for different topological sectors. The dashed lines are linear fits based on the average of the four data points for $N_v=8-14$, $N_v=10-14$, and $N_v=12-14$, respectively.

Figure 5

Finite-size scaling of the gap of the transfer operators with and without flux. The fitting function is $a/N_v^2+b/N_v+c$, with $c$ given in the plot (for the “no flux” sectors, fits for two different ranges of $N_v$ are given), and suggests that the gap vanishes as $1/N_v$ in the thermodynamic limit.