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Physical review. B, Condensed matter

Topological long-range order for resonating-valence-bond structures

Douglas J. Klein, Tomislav P. Živković, Roser Valentí

Published Jan 1, 1991• 22 citations
Publisher

Abstract

Resonating-valence-bond structures involving pairings of sites not too distant from one another are considered. It is shown that under fairly general conditions these structures fall into different classes manifesting different ``topological'' long-range orders. Implications for ``winding numbers'' resulting from the superposition of pairs of valence-bond structures are noted.

DOI
10.1103/physrevb.43.723

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Resonating-valence-bond structures involving pairings of sites not too distant from one another are considered. It is shown that under fairly general conditions these structures fall into different classes manifesting different topological'' long-range orders. Implications for winding numbers'' resulting from the superposition of pairs of valence-bond structures are noted.
The concept of the resonating valence bond (RVB) state was originally proposed by Anderson (1987) for the ground state of the positive-U Hubbard model. However, the on-site pairing long-range order is expected to exist in the ground state of the negative-U Hubbard model and to be incompatible with the RVB order. In this article, we shall rigorously prove that, in fact, these two long-range orderings either coexist or are suppressed simultaneously in the global ground states of the doped Hubbard models.
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.
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.