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Physical review. B, Condensed matter

Rigorous theorems on off-diagonal long-range order in the negative-UHubbard model

Guang-Shan Tian

Published Feb 1, 1992• 37 citations
Publisher

Abstract

In this article, we study off-diagonal long-range order in the negative-U Hubbard model. We rigorously prove that all the elements of the two-particle reduced density matrix are positive, which may indicate a Bose condensation at q=0. Furthermore, we shall show that the ground state at half filling has pseudospin number 0. It implies that a sufficient condition proposed by Yang and Zhang [Int. J. Mod. Phys. B 5, 977 (1991)], for a state \ensuremath{\Psi} having off-diagonal long-range order, may not be a necessary condition.

DOI
10.1103/physrevb.45.3145

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In this article, we study off-diagonal long-range order in the negative-U Hubbard model. We rigorously prove that all the elements of the two-particle reduced density matrix are positive, which may indicate a Bose condensation at q=0. Furthermore, we shall show that the ground state at half filling has pseudospin number 0. It implies that a sufficient condition proposed by Yang and Zhang [Int. J. Mod. Phys. B 5, 977 (1991)], for a state \ensuremath{\Psi} having off-diagonal long-range order, may not be a necessary condition.
In this paper, we show rigorously that there exists the ferrimagnetic long-range order in the ground state of the positive-U Hubbard model at half filling on some bipartite lattices. When NA{\mathit{N}}_{\mathit{A}}NA​>NB{\mathit{N}}_{\mathit{B}}NB​(NA{\mathit{N}}_{\mathit{A}}NA​ and NB{\mathit{N}}_{\mathit{B}}NB​ are the total site numbers of two sublattices A and B), except for the ferromagnetism which was found by Lieb [Phys. Rev. Lett. 62, 1201 (1989)], there also exists the antiferromagnetic long-range order in the ground state. This result only requires U>0 and is independent of the dimension of the lattices.
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 ∑
For a strongly interacting boson system, it has been proposed that the concept of off-diagonal long-range ordering (ODLRO) is equivalent to Bose-Einstein condensation for a free boson system. Therefore, the existence of ODLRO in the ground state implies superfluidity. The author checks the validity of this proposition from another point of view. He rigorously shows that, for a hard-core lattice-boson system with a short-ranged interaction ODLRO is suppressed when a charged excitation gap develops and, hence, the boson system becomes a Mott-insulator. Finally, by using his theorem, he shows some interesting properties of the antiferromagnetic Heisenberg model, which can be taken as a hard-core lattice-boson system.