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arXiv:1204.3195 · Physical Review Letters

Coexistence of Long-Range and Algebraic Correlations for Short-Range Valence-Bond Wave Functions in Three Dimensions

A. Fabricio Albuquerque, Fabien Alet, Roderich Moessner

Published Oct 4, 2012• 21 citations
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Abstract

We investigate nearest-neighbor valence-bond wave functions on bipartite three-dimensional lattices. By performing large-scale Monte Carlo simulations, we find that long-range magnetic order coexists with dipolar four-spin correlations on the cubic lattice, this latter feature being reminiscent of the Coulomb phase for classical dimers on the same geometry. Similar properties are found for the lower-coordination diamond lattice. While this suggests that the coexistence of magnetic order and dipolar four-spin correlations is generic for such states on bipartite three-dimensional lattices, we show that simple generalizations of these wave functions can encode different ordering behaviors.

DOI
10.1103/physrevlett.109.147204

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We investigate nearest-neighbor valence-bond wave functions on bipartite three-dimensional lattices. By performing large-scale Monte Carlo simulations, we find that long-range magnetic order coexists with dipolar four-spin correlations on the cubic lattice, this latter feature being reminiscent of the Coulomb phase for classical dimers on the same geometry. Similar properties are found for the lower-coordination diamond lattice. While this suggests that the coexistence of magnetic order and dipolar four-spin correlations is generic for such states on bipartite three-dimensional lattices, we show that simple generalizations of these wave functions can encode different ordering behaviors.
We investigate the arguably simplest SU(2)SU(2)SU(2)-invariant wave functions capable of accounting for spin-liquid behavior, expressed in terms of nearest-neighbor valence-bond states on the square lattice and characterized by different topological invariants. While such wave functions are known to exhibit short-range spin correlations, we perform Monte Carlo simulations and show that four-point correlations decay algebraically with an exponent 1.16(4). This is reminiscent of the classical dimer problem, albeit with a slower decay. Furthermore, these correlators are found to be spatially modulated according to a wave vector related to the topological invariants. We conclude that a recently proposed spin Hamiltonian that stabilizes the here considered wave function(s) as its (degenerate) ground state(s) should exhibit gapped spin and gapless nonmagnetic excitations.
We study classical hard-core dimer models on three-dimensional lattices using analytical approaches and Monte Carlo simulations. On the bipartite cubic lattice, a local gauge field generalization of the height representation used on the square lattice predicts that the dimers are in a critical Coulomb phase with algebraic, dipolar correlations, in excellent agreement with our large-scale Monte Carlo simulations. The nonbipartite fcc and Fisher lattices lack such a representation, and we find that these models have both confined and exponentially deconfined but no critical phases. We conjecture that extended critical phases are realized only on bipartite lattices, even in higher dimensions.
We relate properties of nearest-neighbor resonating valence-bond (NNRVB) wave functions for SU(g)SU(g)SU(g) spin systems on two-dimensional bipartite lattices to those of fully packed interacting classical dimer models on the same lattice. The interaction energy can be expressed as a sum of nnn-body potentials Vn{V}_{n}Vn​, which are recursively determined from the NNRVB wave function on finite subgraphs of the original lattice. The magnitude of the nnn-body interaction Vn{V}_{n}Vn​ (n>1n>1n>1) is of order $\mathcal{O}({g}^{\ensuremath{-}(n\ensuremath{-}1)})$ for small ${g}^{\ensuremath{-}1}$. The leading term is a two-body nearest-neighbor interaction V2(g){V}_{2}(g)V2​(g) favoring two parallel dimers on elementary plaquettes. For SU(2)SU(2)SU(2) spins, using our calculated value of V2(g=2){V}_{2}(g=2)V2​(g=2), we find that the long-distance behavior of the bond-energy correlation function is dominated by an oscillatory term that decays as $1/|\stackrel{\ensuremath{\rightarrow}}{r}{|}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\approx}1.22$. This result is in remarkable quantitative agreement with earlier direct numerical studies of the corresponding wave function, which give $\ensuremath{\alpha}\ensuremath{\approx}1.20$.
We report on a valence bond projector Monte Carlo simulation of the cubic lattice quantum Heisenberg model with additional higher-order exchange interactions in each unit cell. The model supports two different valence bond solid (VBS) ground states. In one of these states, the dimer pattern is a three-dimensional analogue of the columnar pattern familiar from two dimensions. In the other, the dimers are regularly arranged along the four main diagonals in 1/8 of the unit cells. The phases are separated from one another and from a Néel phase by strongly first-order boundaries. Our results strengthen the case for exotic transitions in two dimensions, where no discontinuities have been detected at the Heisenberg Néel-VBS transition driven by four-spin plaquette interactions.
We construct a family of short-range resonating-valence-bond wave functions on a layered cubic lattice, allowing for a tunable anisotropy in the amplitudes assigned to nearest-neighbour valence bonds along one axis. Monte Carlo simulations reveal that four phases are stabilized over the full range of the anisotropy parameter. They are separated from one another by a sequence of continuous quantum phase transitions. An antiferromagnetic phase, centred on the perfect isotropy point, intervenes between two distinct quantum spin liquid states. One of them is continuously deformable to the two-dimensional U(1) spin liquid, which is known to exhibit critical bond correlations. The other has both spin and bond correlations that decay exponentially. The existence of this second phase is proof that, contrary to expectations, neither a bipartite lattice structure nor a conventional Marshall sign rule is an impediment to realizing a fully gapped quantum spin liquid.
We consider (spin-s generalizations of) Anderson's short-range resonating-valence-bond (RVB) state and derive rigorous results that suggest that the properties of the RVB state depend crucially on the basic lattice structures. In an arbitrary bipartite lattice with coordination number z, we prove that the spin-s RVB state has exponentially decaying correlation function when z\ensuremath{\le}1+ \ensuremath{\surd}2s+1 . We also prove that the spin-s RVB state on the Cayley tree with coordination number z exhibits long-range N\'eel order when z>2s+2 but does not when z\ensuremath{\le}2s+2.