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arXiv:1112.4917 · Physical Review Letters

Resonating Valence Bond Wave Functions and Classical Interacting Dimer Models

Kedar Damle, Deepak Dhar, Kabir Ramola

Published Jun 15, 2012• 30 citations
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Abstract

We relate properties of nearest-neighbor resonating valence-bond (NNRVB) wave functions for $SU(g)$ 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 $n$-body potentials ${V}_{n}$, which are recursively determined from the NNRVB wave function on finite subgraphs of the original lattice. The magnitude of the $n$-body interaction ${V}_{n}$ ($n>1$) 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 ${V}_{2}(g)$ favoring two parallel dimers on elementary plaquettes. For $SU(2)$ spins, using our calculated value of ${V}_{2}(g=2)$, 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 $\ensuremath{\alpha}\ensuremath{\approx}1.20$.

DOI
10.1103/physrevlett.108.247216

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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 O(g\ensuremath−(n\ensuremath−1))\mathcal{O}({g}^{\ensuremath{-}(n\ensuremath{-}1)})O(g\ensuremath−(n\ensuremath−1)) for small g\ensuremath−1{g}^{\ensuremath{-}1}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/∣r\ensuremath→∣\ensuremathα1/|\stackrel{\ensuremath{\rightarrow}}{r}{|}^{\ensuremath{\alpha}}1/∣r\ensuremath→∣\ensuremathα with \ensuremathα\ensuremath≈1.22\ensuremath{\alpha}\ensuremath{\approx}1.22\ensuremathα\ensuremath≈1.22. This result is in remarkable quantitative agreement with earlier direct numerical studies of the corresponding wave function, which give \ensuremathα\ensuremath≈1.20\ensuremath{\alpha}\ensuremath{\approx}1.20\ensuremathα\ensuremath≈1.20.
We study bond and spin correlations of the nearest-neighbor resonating valence bond (RVB) wave function for a SU(2) symmetric S=1/2S=1/2S=1/2 antiferromagnet on the honeycomb lattice. We find that spin correlations in this wave function are short ranged, while the bond energy correlation function takes on an oscillatory power-law form $D(\stackrel{P\vec}{r})\ensuremath{\sim}cos(\mathbf{Q}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{r})/{|\stackrel{P\vec}{r}|}^{{\ensuremath{\eta}}_{w}(2)}$, where $\mathbf{Q}=(2\ensuremath{\pi}/3,\ensuremath{-}2\ensuremath{\pi}/3)$ is the wave vector corresponding to ``columnar'' valence-bond solid order on the honeycomb lattice, and ${\ensuremath{\eta}}_{w}(2)\ensuremath{\approx}1.49(3)$. We use a recently introduced large-ggg expansion approach to relate bond-energy correlators of the SU(g)\mathrm{SU}(g)SU(g) wave function to dimer correlations of an interacting fully packed dimer model with a three-dimer interaction of strength $V(g)=\ensuremath{-}ln(1+1/{g}^{2})$. Putting g=2g=2g=2, we find numerically that the dimer correlation function ${D}^{d}(\stackrel{P\vec}{r})$ of this dimer model has power-law behavior ${D}^{d}(\stackrel{P\vec}{r})\ensuremath{\sim}cos(\mathbf{Q}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{r})/{|\stackrel{P\vec}{r}|}^{{\ensuremath{\eta}}_{d}(2)}$ with ${\ensuremath{\eta}}_{d}(2)\ensuremath{\approx}1.520(15)$, in rather good agreement with the wave function results. We also study the same quantities for g=3,4,10g=3,4,10g=3,4,10 and find that the bond-energy correlations in the SU(g)\mathrm{SU}(g)SU(g) wave function are consistently well reproduced by the corresponding dimer correlations in the interacting dimer model.
We use Monte Carlo simulations to study properties of Anderson's resonating-valence-bond (RVB) spin-liquid state on the square lattice (i.e., the equal superposition of all pairing of spins into nearest-neighbor singlet pairs) and compare with the classical dimer model (CDM). The latter system also corresponds to the ground state of the Rokhsar-Kivelson quantum dimer model at its critical point. We find that, although spin-spin correlations decay exponentially in the RVB, four-spin valence-bond-solid correlations are critical, qualitatively like the well-known dimer-dimer correlations of the CDM, but decaying more slowly (as $1/{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\approx}1.20$, compared with $\ensuremath{\alpha}=2$ for the CDM). We also compute the distribution of monomer (defect) pair separations, which decay by a larger exponent in the RVB than in the CDM. We further study both models in their different winding-number sectors and evaluate the relative weights of different sectors. Like the CDM, all the observed RVB behaviors can be understood in the framework of a mapping to a ``height'' model characterized by a gradient-squared stiffness constant KKK. Four independent measurements consistently show a value ${K}_{\mathrm{RVB}}\ensuremath{\approx}1.6{K}_{\mathrm{CDM}}$, with the same kinds of numerical evaluations of KCDM{K}_{\mathrm{CDM}}KCDM​ giving results in agreement with the rigorously known value ${K}_{\mathrm{CDM}}=\ensuremath{\pi}/16$. The background of a nonzero winding-number gradient W/LW/LW/L introduces spatial anisotropies and an increase in the effective KKK, both of which can be understood as a consequence of anharmonic terms in the height-model free energy, which are of relevance to the recently proposed scenario of ``Cantor deconfinement'' in extended quantum dimer models. In addition to the standard case of short bonds only, we also studied ensembles in which fourth-neighbor (bipartite) bonds are allowed at a density controlled by a tunable fugacity, resulting (as expected) in a smooth reduction of KKK.
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 consider a class of singlet resonanting-valence-bond wave functions on a square lattice, with the bond-length distribution as a variational parameter. This class contains the two limiting cases of the dimer wave function and of the N\'eel state. We present numerical calculations of the energy and the spin-spin correlation functions up to very large lattice sizes (180\ifmmode\times\else\texttimes\fi{}180) both for disordered states with exponentially decaying correlation functions and for ordered states. The energy of a disordered state can be within 0.1% of our best ordered state ($\frac{\ensuremath{-}0.3344J}{\mathrm{bond}}$).
Numerical results are presented for a generalized resonance-valence-bond state which includes both ionic and covalent contributions in each bond; non-nearest-neighbor bonds are also considered. Variational calculations have been performed and the space-group symmetry has been taken into account. The results for the Hubbard Hamiltonian are compared with the exact ones for the 2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2 and the 4\ifmmode\times\else\texttimes\fi{}4 lattices. The agreement is quite satisfactory for large values of the interaction UUU (for U=40U=40U=40 the overlap with the exact ground-state wave function is 0.999). Several correlation functions are also compared.
We consider a resonating-valence-bond state as a trial wave function for the ground state of the antiferromagnetic Heisenberg model on the square lattice. The energy and the staggered spin-spin correlation functions are calculated on a small lattice and are compared with the results of the exact diagonalization of Oitmaa and Betts. The trial energy is only about 4.7% different from the exact ground-state energy on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The importance of the singlet pairs between the next-nearest antiferromagnetic sites is inferred. An argument against the previous interpretation of the numerical data as being evidence of long-range order is presented.