Resonating valence bond wavefunctions and classical interacting dimer models

TL;DR

This paper establishes a connection between nearest-neighbour resonating valence bond wavefunctions for SU(g) spin systems and classical interacting dimer models, deriving effective interactions and analyzing their long-distance correlations.

Contribution

It introduces a cluster expansion of the potential energy in classical dimer models derived from RVB wavefunctions, revealing dominant two-body interactions and their impact on correlation decay.

Findings

Two-body interactions dominate at leading order

Correlation functions decay as a power law with exponent ~1.22

Results agree with previous numerical studies

Abstract

We relate properties of nearest-neighbour resonating valence bond (nnRVB) wavefunctions for $SU(g)$ spin systems on two dimensional bipartite lattices to those of fully-packed classical dimer models with potential energy $V$ on the same lattice. We define a cluster expansion of $V$ in terms of $n$-body potentials $V_n$, which are recursively determined from the nnRVB wavefunction on {\em finite subgraphs} of the original lattice. The magnitude of the $n$-body interaction $V_n$ ($n>1$) is of order ${\mathcal O}(g^{-(n-1)})$ for small $g^{-1}$, while $V_1$ reduces to a constant due to the fully-packed nature of the model. At leading non-trivial order on the square lattice, the interacting dimer model only has two-body interactions $V_2(g)$ that favour two parallel dimers on elementary plaquettes. Setting $g=2$ and using the results of earlier work on this interacting dimer model, we find that the long-distance behaviour of the bond-energy correlation function is dominated by an oscillatory term that decays as $ 1/|\vec{r}|^{\alpha}$ with $\alpha \approx 1.22$ for SU(2) spins. This result is in remarkable quantitative agreement with earlier direct numerical studies of the corresponding wavefunction, which give $\alpha \approx 1.20$.