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arXiv:1202.0947 · Physical Review B

Topological and entanglement properties of resonating valence bond wave functions

Didier Poilblanc, Norbert Schuch, David Pérez-Garcı́a, J. I. Cirac

Published Jul 6, 2012• 140 citations
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Abstract

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.

DOI
10.1103/physrevb.86.014404

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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)(2)(2)-invariant one-dimensional (superfluid) ttt-JJJ 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 S0=\ensuremath−ln2{S}_{0}=\ensuremath{-}\mathrm{ln}2S0​=\ensuremath−ln2 of the Z2{\mathbb{Z}}_{2}Z2​ 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.
We construct a short-range resonating-valence-bond (RVB) state on the ruby lattice, using projected entangled-pair states with bond dimension D=3D=3D=3. By introducing nonlocal moves to the dimer patterns on the torus, we distinguish four distinct sectors in the space of dimer coverings, which is a signature of the topological nature of the RVB wave function. Furthermore, by calculating the reduced density matrix of a bipartition of the RVB state on an infinite cylinder and exploring its entanglement entropy, we confirm the topological nature of the RVB wave function by obtaining a nonzero topological contribution, $\ensuremath{\gamma}=\ensuremath{-}\mathrm{ln}\phantom{\rule{4pt}{0ex}}2$, consistent with that of a Z2{\mathbb{Z}}_{2}Z2​ topological quantum spin liquid. We also calculate the ground-state energy of the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg model on the ruby lattice and compare it with the RVB energy. Finally, we construct a quantum dimer model for the ruby lattice and discuss it as a possible parent Hamiltonian for the RVB wave function.
We investigate the relations between the bulk and boundaries properties of short-range resonating valence bond wave functions using Projected Entangled Pair States. We find a gapless (gapped) entanglement spectrum for kagome (square) lattices, which reflects the properties of the edge modes rather than those of the bulk correlations. We also derive the topological entropy and show that the topological nature of the wave function manifests itself in a non-local many-body boundary Hamiltonian which we identify completely. We propose to use this feature as a probe to identify topological phases in microscopic Hamiltonians.
Gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids have been proposed as candidates for the ground state of the S=1/2S=1/2S=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 22%--3%2 in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological Z2{\mathbb{Z}}_{2}Z2​ 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.
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+idd+i\phantom{\rule{0.16em}{0ex}}dd+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 (Z2{\mathbb{Z}}_{2}Z2​) 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.
We construct a class of projected entangled pair states which is exactly the resonating valence bond wave functions endowed with both short range and long range valence bonds. With an energetically preferred resonating valence bond pattern, the wave function is simplified to live in a one-parameter variational space. We tune this variational parameter to minimize the energy for the frustrated spin-1/2 J[subscript 1]-J[subscript 2] antiferromagnetic Heisenberg model on the square lattice. Taking a cylindrical geometry, we are able to construct four topological sectors with an even or odd number of fluxes penetrating the cylinder and an even or odd number of spinons on the boundary. The energy splitting in different topological sectors is exponentially small with the cylinder perimeter. We find a power law decay of the dimer correlation function on a torus, and a ln⁡L correction to the entanglement entropy, indicating a gapless spin-liquid phase at the optimum parameter.
We study resonating valence bond (RVB) states in the projected entangled pair states (PEPS) formalism. Based on symmetries in the PEPS description, we establish relations between the toric code state, the orthogonal dimer state, and the SU(2)\mathrm{SU}(2)SU(2) singlet RVB state on the kagome lattice: We prove the equivalence of toric code and dimer state, and devise an interpolation between the dimer state and the RVB state. This interpolation corresponds to a continuous path in Hamiltonian space, proving that the RVB state is the fourfold degenerate ground state of a local Hamiltonian on the (finite) kagome lattice. We investigate this interpolation using numerical PEPS methods, studying the decay of correlation functions, the change of overlap, and the entanglement spectrum, none of which exhibits signs of a phase transition.