Topological Properties of Two-Dimensional Resonating-Valence-Bond States

Indrani Bose

Walter Metzner

Didier Poilblanc, Norbert Schuch, David Pérez-Garcı́a +1 more

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$(2)$-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 ${S}_{0}=\ensuremath{-}\mathrm{ln}2$ 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.

Dmitri A. Ivanov, Vladimir Lebedev, M. V. Feigel’man

The notion of Z2 topological order in two‐dimensional resonating‐valence‐bond systems is reviewed. Using quantum dimer models as an example, I discuss general properties of the topological order and the resulting vortex‐like excitations (visons). Some connections to theories of frustrated magnetism and of high‐temperature superconductivity are pointed out.

Bill Sutherland

We have suggested that the properties of the resonating-valence-bond (RVB) ground state can usefully be studied first by a correspondence with a lattice statistics model, and second by embedding this RVB point into a larger family of lattice statistical models---the loop gas. The nature of this loop gas---correlations, phase transitions, and order parameters---is studied in this paper by means of Monte Carlo techniques, on lattices of up to 4608 sites or 9216 bonds. An ordered phase is identified with divergent susceptibility and an order parameter which has the topology of a circle. Since the model is two dimensional, the phase transition is expected to be of Kosterlitz-Thouless (KT) type. The RVB point is within this ordered KT phase.

Emmanuel V. C. Lopes