Chiral topological spin liquids with projected entangled pair states

Topological chiral phases are ubiquitous in the physics of the fractional quantum Hall effect. Nonchiral topological spin liquids are also well known. Here, using the framework of projected entangled pair states (PEPS), we construct a family of chiral spin liquids on the square lattice, which are generalized spin-1/2 resonating valence bond (RVB) states obtained from deformed local tensors with $d+id$ symmetry. On a cylinder, we construct four topological sectors with even or odd number of spinons on the boundary and even or odd number of (${\mathbb{Z}}_2$) fluxes penetrating the cylinder, which, we argue, remain orthogonal in the limit of infinite perimeter. The analysis of the transfer matrix provides evidence of short-range (long-range) triplet (singlet) correlations as for the critical (nonchiral) RVB state. The entanglement spectrum exhibits chiral edge modes, which we confront to predictions of the conformal field theory, and the corresponding entanglement Hamiltonian is shown to be long-ranged.

Figure 1

(a) Local tensor $A$. (b) Square lattice with (nearest-neighbor) singlet bonds oriented from sublattice A to sublattice B. (c) Under a ${180}^{\circ }$ spin rotation on, e.g., the B sublattice, oriented singlets are transformed into symmetric $|\uparrow \uparrow \rangle +|\downarrow \downarrow \rangle$ maximally entangled pair states. (d) Transfer matrix on a periodic ring connecting the (noncontracted) BK virtual indices on the left of the ring to the ones on its right.

Figure 2

(a) Leading eigenvalue ${\tilde{\gamma }}_{ee}$ (normalized by ${\gamma }_{ee}$) of the ${\tilde{\mathrm{\Gamma }}}_{ee}$ block of the transfer matrix as a function of ${\lambda }_{\mathrm{chiral}}$ (for ${\lambda }_1={\lambda }_2=1$) and for $N_v=4$, 6, and 8 (as shown on plot). (b) Finite size scaling of the leading eigenvalues of the diagonal blocks (${\mathrm{\Gamma }}_{ee}$ and ${\mathrm{\Gamma }}_{oo}$) and of the off-diagonal blocks (${\tilde{\mathrm{\Gamma }}}_{ee}$ and ${\mathrm{\Gamma }}_{eo}$) of the transfer matrix (after proper normalization). The diagonal (off-diagonal) blocks have $S_z^{\mathrm{BK}}=0$ ($S_z^{\mathrm{BK}}=1/2$) quantum numbers. (c) Finite size scaling of the second leading eigenvalues ${\gamma }_{\mu \nu }^{\left(2\right)}$ of the ${\mathrm{\Gamma }}_{ee}, {\mathrm{\Gamma }}_{oo}$, and ${\mathrm{\Gamma }}_{eo}$ blocks (after proper normalization).

Figure 3

Rescaled entanglement spectrum vs momentum along the edge (for ${\lambda }_1={\lambda }_2={\lambda }_{\mathrm{chiral}}=1$). (a)–(c) show different sectors of the $Z$ component of the virtual spin on the edge. The dashed lines show the chiral edge modes. In (a), the momentum of the states of the chiral mode is defined mod $\pi$. The multiplicities 1, 1, 2, and 3 of the quasidegenerate states in the boxes—expected for a CFT—are shown on the plot.

Figure 4

(a) Entanglement entropy $S$ vs cylinder perimeter in the two topological sectors with (open symbols) and without (full symbols) ${\mathbb{Z}}_2$ flux (for ${\lambda }_1={\lambda }_2={\lambda }_{\mathrm{chiral}}=1$). Linear fits (area law) $S=a{N}_v+b$ are shown. (b) Weights vs $N$ of the entanglement Hamiltonian decomposed in terms of $N$-body contributions, for $N_v=6$ and $N_v=8$.

Figure 5

Finite size scaling of the leading eigenvalues of the diagonal blocks (${\mathrm{\Gamma }}_{ee}$ and ${\mathrm{\Gamma }}_{oo}$) and of the off-diagonal blocks (${\tilde{\mathrm{\Gamma }}}_{ee}$ and ${\mathrm{\Gamma }}_{eo}$) of the transfer matrix (after proper normalization). Same notation as Fig. 2 in the main text. The three panels correspond to different choices of the parameters of the PEPS as indicated on the plot. (a) is a nonchiral RVB state (${\tilde{\gamma }}_{ee}=1$), (b) does not contain NN valence bonds, and (c) is a repetition of Fig. 2 for convenience.

Figure 6

Entanglement spectrum vs momentum along the edge for $N_v=4,6,8$ (for ${\lambda }_1={\lambda }_2={\lambda }_{\mathrm{chiral}}=1$). (a)–(c) show $S_z=0, |S_z|=1$, and $|S_z|=1/2$, respectively. The dashed lines show a fit of the chiral edge modes for $N_v=8$. (d) Finite size scaling of the lowest eigenvalue of the ES.