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Journal of Physics Condensed Matter

Two-dimensional spin models with resonating valence bond ground states

Indrani Bose

Published Jun 18, 1990• 11 citations
Publisher

Abstract

Two spin models in two dimensions are constructed for which the exact ground states can be written down in a certain parameter regime of the Hamiltonian. The ground state of the first model is a local resonating valence bond (RVB) state in which not all the spins participate in resonance. The ground state is highly degenerate and there is a gap in the excitation spectrum. The second model is shown to have a ground state which is a quantum spin liquid with all the spins taking part in resonance. The ground state obtained is non-degenerate and there is no gap in the excitation spectrum.

DOI
10.1088/0953-8984/2/24/018

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Two spin models in two dimensions are constructed for which the exact ground states can be written down in a certain parameter regime of the Hamiltonian. The ground state of the first model is a local resonating valence bond (RVB) state in which not all the spins participate in resonance. The ground state is highly degenerate and there is a gap in the excitation spectrum. The second model is shown to have a ground state which is a quantum spin liquid with all the spins taking part in resonance. The ground state obtained is non-degenerate and there is no gap in the excitation spectrum.
A quasi-one-dimensional spin model that consists of a chain of octahedra of spins has been suggested, for which in a certain parameter regime of the Hamiltonian the ground state can be written down exactly. The ground state is highly degenerate and can be other than a singlet. Also, several excited states can be constructed exactly. The ground state is a local resonating valence bond state for which resonance is confined to rings of spins. Some exact numerical results for an octahedron of spins have also been reported.
A spin-12\frac{1}{2}21​ system with double orbital degeneracy may possess SU(4) symmetry. According to the group theory a global SU(4) singlet state can be expressed as a linear combination of all possible configurations consisting of four-site SU(4) singlets. Following Andersion's idea for spin-12\frac{1}{2}21​ system, we propose that the ground state for the antiferromagnetic SU(4) model is an SU(4)-resonating valence-bond (RVB) state. A short-range SU(4)-RVB state is a spin and orbital liquid, and its elementary excitations has an energy gap. We construct a series of solvable models that ground states are short-range SU(4)-RVB states. The results are generalized to the antiferromagnetic SU(N) models.
The resonating-valence-bond (RVB) theory for two-dimensional quantum antiferromagnets is shown to be the correct paradigm for large enough "quantum frustration." This scenario, proposed a long time ago but never confirmed by microscopic calculations, is strongly supported by a new type of variational wave function, which is extremely close to the exact ground state of the J(1)-J(2) Heisenberg model for 0.4 less than approximately J(2)/J(1) less than approximately 0.5. This wave function is proposed to represent the generic spin-half RVB ground state in spin liquids.
We construct a family of rotationally invariant, local, S=1∕2S=1∕2S=1∕2 Klein Hamiltonians on various lattices that exhibit ground-state manifolds spanned by nearest-neighbor valence bond states. We show that with selected perturbations such models can be driven into phases modeled by well-understood quantum dimer models on the corresponding lattices. Specifically, we show that the perturbation procedure is arbitrarily well controlled by a new parameter which is the extent of decoration of the reference lattice. This strategy leads to Hamiltonians that exhibit (i) Z2{Z}_{2}Z2​ resonating valence bond (RVB) phases in two dimensions, (ii) U(1)U(1)U(1) RVB phases with a gapless ``photon'' in three dimensions, and (iii) a Cantor deconfined region in two dimensions. We also construct two models on the pyrochlore lattice, one model exhibiting a Z2{Z}_{2}Z2​ RVB phase and the other a U(1)U(1)U(1) RVB phase.
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.