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Journal of Physics A Mathematical and General

A sufficient condition for two long-range orders coexisting in a lattice many-body system

Guang-Shan Tian

Published Feb 7, 1997• 7 citations
Publisher

Abstract

In this paper, we prove a sufficient condition for two long-range orders being either present or absent simultaneously in the absolute ground state of a lattice many-body boson or fermion model. As an application of this theorem, we give a simplified proof on the coexistence of the resonating valence bond (RVB) long-range order and the on-site-pairing long-range order in the ground state of the Hubbard model.

DOI
10.1088/0305-4470/30/3/010

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In this paper, we prove a sufficient condition for two long-range orders being either present or absent simultaneously in the absolute ground state of a lattice many-body boson or fermion model. As an application of this theorem, we give a simplified proof on the coexistence of the resonating valence bond (RVB) long-range order and the on-site-pairing long-range order in the ground state of the Hubbard model.
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.
A two-dimensional square-lattice Hubbard model is investigated. A slave-fermion formulation, in which the electron operator is expressed by a combination of bose operators for the spin freedom, and fermi operators for the charge freedom, is used. The phase diagram for the resonating valence bond (RVB) state is obtained. The RVB state shows long-range antiferromagnetic order at T =0, the period of which deviates from that of the commensurate Néel order as the system deviates from the half-filled case.
We give a criterion for the simultaneous existence or non-existence of two long-range orders for two observables, at finite temperatures, for quantum lattice many body systems. Our analysis extends previous results of Tian (1997 30 841) limited to the ground state of similar models. The proof involves an inequality of Dyson-Lieb-Simon (1978 J. Stat. Phys. 18 335) which connects the Duhamel two-point function to the usual correlation function. An application to the special case of the Holstein model is discussed.
We derive the singlet versions of the N\'eel states for certain n-sublattice systems, including the two-dimensional square lattice (previously considered by Carlson and Sethna) and the triangular lattice. These states are written explicitly in resonating-valence-bond form through the help of the Young-tableau formalism. Since these states are exact ground states for infinite-range models, they can be used as a test for the fermion and Schwinger boson mean-field theories (MFT). We show that the fermion MFT is incompatible with the ground state, while the boson MFT gives the exact ground state. Implications of the results with respect to the ability of the MFT's to describe antiferromagnetic long-range order are discussed.
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.