The $1/N$ Expansion and Spin Correlations in Constrained Wavefunctions

TL;DR

This paper develops a large-N expansion for Gutzwiller projected spin states, providing a simpler approach to analyze valence bond correlations and spin fluctuations in quantum antiferromagnets, with exact identities and improved understanding of large-N methods.

Contribution

It introduces a novel large-N expansion for Gutzwiller projected states, simplifying analysis and deriving identities and sum rules applicable to quantum antiferromagnets.

Findings

Mean field, order 1/N, and exact correlations are proportional for bosons.

1/N correction enhances zone edge singularity in fermions.

Comparison with N=2 results improves understanding of large-N approximations.

Abstract

We develop a large-N expansion for Gutzwiller projected spin states. We consider valence bonds singlets, constructed by Schwinger bosons or fermions, which are variational ground states for quantum antiferromagnets. This expansion is simpler than the familiar expansions of the quantum Heisenberg model, and thus more instructive. The diagrammatic rules of this expansion allow us to prove certain identities to all orders in 1/N. We derive the on-site spin fluctuations sum rule for arbitrary N. We calculate the correlations of the one dimensional Valence Bonds Solid states and the Gutzwiller Projected Fermi Gas upto order 1/N. For the bosons case, we are surprised to find that the mean field, the order 1/N and the exact correlations are simply proportional. For the fermions case, the 1/N correction enhances the zone edge singularity. The comparison of our leading order terms to known results for N=2, enhances our understanding of large-N approximations in general.