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arXiv:cond-mat/9403042 · Physical Review Letters

Resonating Valence Bond Theory of Coupled Heisenberg Chains

Steven R. White, R. M. Noack, Douglas J. Scalapino

Published Aug 8, 1994• 405 citations
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Abstract

Using numerical results from a density matrix renormalization group study as a guide, we develop a resonating valence bond (RVB) theory for coupled Heisenberg chains. We argue that simple topological effects mandate a short-range RVB description of systems with an even number of chains ${n}_{c}$, with a spin gap, short-range correlations, and confinement of topological spin defects. Odd-${n}_{c}$ systems have long-range RVB ground states, no gap, and power-law correlations.

DOI
10.1103/physrevlett.73.886

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Using numerical results from a density matrix renormalization group study as a guide, we develop a resonating valence bond (RVB) theory for coupled Heisenberg chains. We argue that simple topological effects mandate a short-range RVB description of systems with an even number of chains nc{n}_{c}nc​, with a spin gap, short-range correlations, and confinement of topological spin defects. Odd-nc{n}_{c}nc​ systems have long-range RVB ground states, no gap, and power-law correlations.
Resonating valence bond (RVB) states are of crucial importance in our intuitive understanding of quantum spin liquids in 2D. We systematically classify short-range bosonic RVB states into symmetric or nematic spin liquids by examining their flux patterns. We further map short-range bosonic RVB states into projected BCS wave functions, on which we perform large-scale Monte Carlo simulations without the minus sign problem. Our results clearly show that both spin and dimer correlations decay exponentially in all the short-range frustrated (nonbipartite or Z2) bosonic RVB states we studied, indicating that they are gapped Z2 quantum spin liquids. Generically, we conjecture that all short-range frustrated bosonic RVB states in 2D have only short-range correlations.
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.
Resonating valence bond (RVB) states are of crucial importance in our intuitive understanding of quantum spin liquids in 2D. We systematically classify short-range bosonic RVB states into symmetric or nematic spin liquids by examining their flux patterns. We further map short-range bosonic RVB states into projected BCS wave functions, on which we perform large-scale Monte Carlo simulations without the minus sign problem. Our results clearly show that both spin and dimer correlations decay exponentially in all the short-range frustrated (non-bipartite or Z2Z_2Z2​) bosonic RVB states we studied, indicating that they are gapped Z2Z_2Z2​ quantum spin liquids. Generically, we conjecture that all short-range frustrated bosonic RVB states in 2D have only short-range correlations.
A fully variational resonance-valence-bond (RVB) study is performed for two-, three- and four-legged isotropic spin-1/2 Heisenberg ladders under periodic boundary conditions along the rungs and periodic/twisted boundary conditions along the chains. By using the full lattice groups of these systems very accurate results have been obtained for the ground-state bulk limit properties in the case of the dimer RVB basis. The results reported here represent the best one can do within the dimer RVB ansatz and should be useful when comparing with results obtained from an a priori fixed structure of the wavefunction. Further results are presented for some finite-sized systems, extending beyond the dimer RVB approach by including RVB functions containing spin pairings between non-nearest-neighbour sites. These results indicate that the three-legged ladder with periodic boundary conditions along the rungs falls into the same universality class of ladders possessing short-range RVB wavefunctions as the two- and four-legged ladders.