Simplex ${\mathbb{Z}}_2$ spin liquids on the kagome lattice with projected entangled pair states: Spinon and vison coherence lengths, topological entropy, and gapless edge modes

Gapped ${\mathbb{Z}}_2$ spin liquids have been proposed as candidates for the ground state of the $S=1/2$ quantum antiferromagnet on the kagome lattice. We extend the use of projected entangled pair states to construct (on the cylinder) resonating valence bond (RVB) states including both nearest-neighbor and next-nearest-neighbor singlet bonds. Our ansatz—dubbed “simplex spin liquid”—allows for an asymmetry between the two types of triangles (of order $2\%–3\%$ in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological ${\mathbb{Z}}_2$ structure is still preserved and, by considering the presence or the absence of spinon and vison lines along an infinite cylinder, we explicitly construct four orthogonal RVB minimally entangled states. The spinon and vison coherence lengths are extracted from a finite size scaling with regard to the cylinder perimeter of the energy splittings of the four sectors and are found to be of the order of the lattice spacing. The entanglement spectrum of a partitioned (infinite) cylinder is found to be gapless, suggesting the occurrence, on a cylinder with real open boundaries, of gapless edge modes formally similar to Luttinger liquid (nonchiral) spin and charge modes. When inversion symmetry is spontaneously broken, the RVB spin liquid exhibits an extra Ising degeneracy, which might have been observed in recent exact diagonalization studies.

Figure 1

(a) Kagome lattice with a vertical cut (dashed line). If the inversion center is lacking, right (light shading) and left (darker shading) triangles become nonequivalent but all sites (blue dots) keep the same environment. (b), (c) PEPS representation of the RVB state: Three $S=1/2$ sites are grouped together (b) to construct a rank-5 tensor (c) involving $2^3=8$ physical states and $D=3$ virtual states on the four outgoing bonds. (d), (e) Two dimer configurations around a “defect triangle” (in blue). Red ellipses represent singlets between two sites. The $\alpha$ parameter (see text) controls the relative weights between (d) and (e).

Figure 2

(a) Variational energy vs energy difference in left and right triangles (varying $\alpha$, at fixed $\beta ={\beta }^{*}$). The two parabolic “branches” are obtained by interchanging the role of left and right triangles in the PEPS construction giving two orthogonal wave functions $|{\Phi }_{\mathrm{L}}\rangle$ and $|{\Phi }_{\mathrm{R}}\rangle$. (b) Energy of the (fixed) optimized state vs NNN AF coupling $J_2$ ($N_v=6$, YC12) compared to DMRG data.

Figure 3

The four topological sectors ($G_v=\pm 1$, ${\mathcal{W}}_v=\pm 1$) of the ${\mathbb{Z}}_2$ spin liquids on a thin torus (a) or on a thin cylinder (b). Zero or one ${\mathbb{Z}}_2$ flux can be threaded through the torus and cylinder (dashed green line). In (b) spinons (visons) are shown by black (orange) dots. For $G_v=-1$ two spinons s and $\overline{\mathrm{s}}$ (with opposite spins) are localized at the cylinder ends. For each case, a schematic plot of the low-energy spectrum is shown. When inversion symmetry is spontaneously broken, each level becomes doubly degenerate.

Figure 4

(a) Finite size scaling of the RVB energy in the four topological sectors. The energy averaged between even ($G_v=+1$) and odd ($G_v=-1$) boundary conditions (dots) can be fitted as $e_{\infty }\pm C\exp (-N_v/\xi )$ (dashed lines). (b) The splittings in (a) are plotted vs $N_v$ on a logarithmic scale, revealing two different coherence lengths.

Figure 5

(a) Bipartitions of the long RVB cylinders into A and B semicylinders used to compute the EE. By optionally inserting spinon and/or vison lines, four disconnected topological sectors can be constructed. VN entanglement entropy of the symmetric (b) and simplex (c) ${\mathbb{Z}}_2$ RVB liquids in the four topological sectors as the function of the cylinder perimeter $N_v$. The dashed lines are linear fits based on the averages of the four data points for the two largest (YC12 and YC16) cylinders providing $\ln \mathcal{D}$ with an accuracy of ${10}^{-4}$.

Figure 6

Entanglement spectra for a bipartition of a $N_v=8$ (infinite) cylinder in the $G_v=-1,W_v=+1$ (a) and $G_v=+1,W_v=+1$ topological sectors, for $\alpha ={\alpha }^{*}$ and $\beta ={\beta }^{*}$. The levels are labeled according to their total spin $S$ (see legends) and momentum $K$ along the cut. (c) String representation of the edge states of an infinite semicylinder.