Topological and entanglement properties of resonating valence bond wave functions

We examine in details the connections between topological and entanglement properties of short-range resonating valence bond (RVB) wave functions using projected entangled pair states (PEPS) on kagome and square lattices on (quasi)infinite cylinders with generalized boundary conditions (and perimeters with up to 20 lattice spacings). By making use of disconnected topological sectors in the space of dimer lattice coverings, we explicitly derive (orthogonal) “minimally entangled” PEPS RVB states. For the kagome lattice, using the quantum Heisenberg antiferromagnet as a reference model, we obtain the finite-size scaling with increasing cylinder perimeter of the vanishing energy separations between these states. In particular, we extract two separate (vanishing) energy scales corresponding (i) to insert a vison line between the two ends of the cylinder and (ii) to pull out and freeze a spin at either end. We also investigate the relations between bulk and boundary properties and show that, for a bipartition of the cylinder, the boundary Hamiltonian defined on the edge can be written as a product of a highly nonlocal projector, which fundamentally depends upon boundary conditions, with an emergent (local) SU$\left(2\right)$-invariant one-dimensional (superfluid) $t$-$J$ Hamiltonian, which arises due to the symmetry properties of the auxiliary spins at the edge. This multiplicative structure, a consequence of the disconnected topological sectors in the space of dimer lattice coverings, is characteristic of the topological nature of the states. For minimally entangled RVB states, it is shown that the entanglement spectrum, which reflects the properties of the (gapless or gapped) edge modes, is a subset of the spectrum of the local Hamiltonian, e.g., half of it for the kagome RVB state, providing a simple argument on the origin of the topological entanglement entropy $S_0=-\ln 2$ of the ${\mathbb{Z}}_2$ spin liquid. We propose to use these features to probe topological phases in microscopic Hamiltonians, and some results are compared to existing density matrix renormalization group data.

Figure 1

Typical valence bond configurations on a $N_v\times N_h$ cylinder with periodic boundary conditions along the vertical ($v$) direction. Ellipses represent singlets of two spins 1/2. Open (a) or generalized (b) boundary conditions on the $B_L$ and $B_R$ ends of the cylinder are considered (GBC can be obtained physically by freezing some spins at the boundaries, e.g., with local magnetic fields). The RVB wave function is defined as the equal-weight superposition of all such configurations (for a fixed realization of $B_L$ and $B_R$).

Figure 2

Two valence bond configurations on a $4\times 2$ cylinder ($N_v=4$). The two configurations are obtained from each other by translating all dimers (in purple) along a (single) closed loop encircling the cylinder. Such configurations can be distinguished from the parities $G_h=\pm 1$ of the number of dimers cut by open lines along the $h_1$ and $h_2$ directions joining the two $B_L$ and $B_R$ ends of the cylinder and, hence, define two different topological sectors. Two RVB (variational) ground states with $\langle G_h\rangle =0$ can be constructed as equal-weight superpositions of all dimer coverings with $+$ or $-$ relative signs between the two topological sectors.

Figure 3

Two valence bond configurations on a $4\times 2$ cylinder ($N_v=4$ even). The two configurations are obtained from each other by translating all dimers (in purple) along a (single) open loop joining the two $B_L$ and $B_R$ ends of the cylinder (and adding extra spins). Such configurations can be distinguished from the parity $G_v=\pm 1$ of the number of dimers cut by any closed loop winding around the cylinder along the vertical direction and, hence, define two different “even” and “odd” topological sectors (and the corresponding RVB states).

Figure 4

Same as Fig. 3 for a $3\times 2$ cylinder ($N_v=3$ odd).

Figure 5

RVB wave functions on a cylindrical geometry: equal-weight superposition of hard-core dimer coverings [see, e.g., (a) and (c)] have simple representations in terms of PEPS [(b) and (d)]. The $B_L$ and $B_R$ boundary conditions of Figs. 1a and 1b can be realized by fixing the virtual variables going out of the cylinder ends; OBC (a) are defined by setting all boundary indices to “2” (b). Generalized boundary conditions (c) translate in the PEPS language by setting the boundary indices to 0 (spin $\downarrow$) or 1 (spin $\uparrow$) (d). A bipartition of the cylinder generates two $L$ and $R$ edges along the cut.

Figure 6

On the kagome lattice, an effective rank-5 tensor is constructed on each three-site unit cell. Three site tensors (red dots) carrying the physical indices and two 120${}^{\circ }$ tensors (in the center of the shaded triangles) are grouped together (a), (b) to construct the basic tensor (c). The kagome lattice is then mapped onto an effective square lattice. A partition of the cylinder in the vertical direction generates $L$ and $R$ edges (thick dotted line).

Figure 7

Illustration of the energy splitting between the four (variational) RVB wave functions for the kagome HAF. From left to right, the cylinder length (at fixed perimeter $N_v$ even) and then its perimeter are increased to infinity.

Figure 8

Schematic patterns of the exchange interaction on the triangles in the center of (quasi)infinite cylinders. For $N_v$ even ($N_v$ odd), the system is uniform (dimerized).

Figure 9

(a) Finite-size scaling of the energy (per site) of the four RVB wave functions on infinite kagome cylinders versus inverse perimeter $N_v$. ‘Even” and “odd” refer to the parity of the number of spins frozen on the cylinder boundaries. “$+$” and “$-$” states differ by the absence or presence of a vison horizontal line, respectively. The energies of the fixed parity ($G_h=\pm 1$) states are obtained by averaging the $+$ and $-$ energies (since the cross terms vanish) separately in the even and odd sectors. The energies of the four nonequivalent triangles of odd-perimeter infinite cylinders (with no vison) are also included. Averages over the even and odd energies separately in the no-vison ($+$) and vison ($-$) sectors are also shown. (b) Corresponding energy splittings (normalized per three-site unit cell) vs $N_v$ [see Fig. 7b]. We also include the dimerization energy of the ${\Psi }_{\mathrm{RVB}}^{+,1,2}$ states on odd-perimeter infinite cylinders, defined as the energy difference between even and odd columns. DMRG data (S. R. White, Ref. 9 and private communication) for the dimerization of the YC6 cluster ($N_v=3$) or the spittings of the YC4 ($N_v=2$) and YC8 ($N_v=4$) clusters are shown for comparison. Dashed straight lines correspond to exponential fits of the form $A_0\exp (-N_v/\xi )$.

Figure 10

Boundary operator ${\sigma }_L$ obtained by contracting all physical indices (wavy lines connecting the tensors in the front and in the back) of the left half-cylinder. Here, arbitrary boundary conditions have been chosen for $B_L$.

Figure 11

Weights of the projectors ${\mathcal{P}}_{\Delta }$, $\Delta =0,1,2,$ and 3 (square lattice, $N_v=6$) and ${\mathcal{P}}_{\mathrm{even}}$ (kagome lattice, $N_v=8$) expended in terms of $N$-body operators. Same for $H_{\mathrm{local}}$ for the RVB wave function on the square (a) and kagome (b) lattices, on $N_v=6$ (small symbols) and $N_v=8$ (large symbols) infinite cylinders.

Figure 12

Weights $|d_{\lambda \mu }{\left(r\right)|}^2$ of the two-body operators appearing in $H_{\mathrm{local}}$ for the square (a) and kagome (b) lattices. In (a), the diagonal interaction ${\widehat{x}}_2^i{\widehat{x}}_2^{i+r}$ shows a long-range behavior.

Figure 13

ES (w.r.t. the same $S_z=0$ GS energy ${\xi }_0$ at $K=0$) of an infinitely long kagome cylinder of perimeter $N_v=8$. Eigenstates with half-integer (a) and integer (b) spins correspond to odd and even sectors, respectively (see text).

Figure 14

Same as Fig. 13 for the critical RVB state (square lattice). Eigenstates with half-integer (integer) spins correspond to $\Delta$ odd (even) (see text). For OBC, one gets a subset of (b) ($\Delta =0$ sector).

Figure 15

Entanglement entropy versus perimeter $N_v$ for specific sectors (open symbols) or when summing over all sectors (shaded symbols). (a) Square lattice ($\Delta =0$ is obtained with OBC); (b) kagome lattice (no ${\mathbb{Z}}_2$ flux through the cylinder, $G_h=0$).

Figure 16

ES of a cylinder with fixed perimeter $N_v=8$ and increasing length $N_h$ ranging from 2 to 50. OBC are used for $B_L$ and $B_R$. Kagome (top) and square (bottom) lattices.

Figure 17

Finite-size scaling of the lowest-energy levels marked by arrows in the ES of Fig. 16. Excellent convergence is found when $N_h\to \infty$ (at constant $N_v=8$). Note the alternating behavior according to the parity of $N_h/2$ for the square lattice, in contrast to the kagome lattice showing a straight exponential convergence.

Figure 18

ES of infinite cylinders with increasing perimeter $N_v$. OBC are used for $B_L$ and $B_R$. Kagome (top) and square (bottom) lattices.

Figure 19

Finite-size scaling of low excitation energies of infinite cylinders vs inverse perimeter, suggesting a vanishing (finite) gap in the thermodynamic limit for the kagome (square) lattice. For the square lattice, the average (divided by 3) between the lowest $K=\pi$ singlet and triplet excitations is shown.