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Journal of Physics A Mathematical and General

A new method to show the absence of some long-range orderings

Guang-Shan Tian

Published May 21, 1993• 9 citations
Publisher

Abstract

The author shows that the correlation function of a local operator B i decays if there is another local operator A i satisfying (H,A i )= alpha B i , where H is the Hamiltonian of the many-body system under consideration and alpha is a constant. Finally, as an application of this theorem, he shall rigorously show that the RVB states, which were proposed by P. W. Anderson (1987) and his collaborators to explain high-temperature superconductivity, are absent in the Hubbard model at half-filling. He also gives an argument, which indicates that the existence of the RVB ground states in the doped cases is highly improbable.

DOI
10.1088/0305-4470/26/10/007

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The author shows that the correlation function of a local operator B i decays if there is another local operator A i satisfying (H,A i )= alpha B i , where H is the Hamiltonian of the many-body system under consideration and alpha is a constant. Finally, as an application of this theorem, he shall rigorously show that the RVB states, which were proposed by P. W. Anderson (1987) and his collaborators to explain high-temperature superconductivity, are absent in the Hubbard model at half-filling. He also gives an argument, which indicates that the existence of the RVB ground states in the doped cases is highly improbable.
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 show that in the translation invariant case and in the antiferromagnetic phase, the reduced density matrix ρ2 has no off-diagonal long-range order of on-site electron pairs for the single-band Hubbard model on a cubic lattice away from half filling at finite temperatures both for the positive coupling and for the negative coupling. In these cases the model can not give a mechanism for the superconductivity caused by the condensation of on-site electron pairs and the nearestneighbor electron pairs. PACS numbers: 71.10.+x, 74.20.-z, 05.30.-d. 1 It was proposed the possibility that the Hubbard model can give a mechanism to characterize high temperature superconductivity[1]. It has been shown that the η pairing states for the single band Hubbard model with constant hopping matrix element for nearest neighbor sites have ODLRO[2]. It has also been shown that in the ground state the model with negative coupling has off-diagonal long-range order (ODLRO) on some bipartite lattice[3]. However, as the temperature T> 0 whether the reduced density matrix ρ2 has ODLRO is inconclusive. The purpose of this work is to study the ODLRO of ρ2 in the single-band Hubbard model with constant hopping matrix element for nearest neighbor pairs on a cubic lattice at finite temperatures. In the translation invariant case we obtain an equation ( (10) in the following) which is satisfied by the long-range correlation functions of on-site electron pairs. From which we find that there is no ODLRO of on-site electron pairs. Then we consider the case with an antiferromagnetic background. In this case we will show that there is still no ODLRO of on-site electron pairs. The Hamiltonian of the model is H = −t ∑ (a † rar ′ + b † rb ∑
We describe a new method to numerically evaluate the properties of correlated superconducting wave functions. We have applied it to the resonating-valence-bond (RVB) wave function for the Hubbard model on the square lattice. For the half-filled case we find that the ddd-wave RVB state and the antiferromagnetic ordered state have the same energy within numerical accuracy. At 10% doping we find ddd-wave superconductivity, consistent with previous studies. We show that the superconducting order parameter is proportional to the number of holes, for small hole concentrations.
The extended Hubbard model with nearest neighbour inter-site correlations has been studied by mapping it onto a spin-Hamiltonian. In the case of a half-filled band the problem is equivalent to that of an anisotropic Heisenberg model in the absence of a field. The phases in the U-Q plane obtained here are in good agreement with those obtained by Monte Carlo simulation. The antiferromagnetic XY-ordered phase of the model is shown to be very similar to an RVB phase. Extending the analogy away from half-filling, a possible dependence of the critical temperature T c with the doping concentration delta in a high-T c system is suggested.