Guang-Shan Tian

Coexistence of the ferromagnetic and antiferromagnetic long-range orders in the generalized antiferromagnetic Heisenberg model on a bipartite lattice

The concept of ferrimagnetism was first proposed by Neel to explain why some materials have a macroscopic magnetization but no ferromagnetic long-range order, when the temperature T is lower than a phase transition temperature Tc. In this article, based on a theorem of Lieb and Mattis, we show in a mathematically rigorous way that the global ground states of the generalized antiferromagnetic Heisenberg model on a bipartite lattice with unequal sublattice points have both ferromagnetic and antiferromagnetic long-range orders with the latter being predominant. Our rigorous results conform to Neels theory.

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

The author shows that the correlation function of a local operator Bi decays if there is another local operator Ai satisfying (H,Ai)= alpha Bi, 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 Nagaoka state in the one-hand Hubbard model with two and more holes

The infinite-U Hubbard model with two holes on a two-dimensional square lattice is explicitly studied. The author shows that the energy of the Nagaoka state and the exact ground state become degenerate in the thermodynamic limit, i.e. there exists no energy gap between the ground state and the first excited state. Finally, they discuss briefly how to generalize this result to cases in which there is a finite number of holes and the structure of the lattice is more complicated.

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

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.

A rigorous study on the charged gap in strongly correlated electron clusters

In this paper, we study a revised definition c of the charged gap for the strongly correlated electron models on small clusters, proposed by Nishino, in a mathematically rigorous way. By applying a simplified version of Liebs spin-reflection-positivity method, we show that this quantity is always positive for the half-filled Hubbard model, the periodic Anderson model and the Kondo lattice model. We also establish a model-dependent lower bound to the charged gap. Our results show explicitly the role played by electron repulsion in opening up a nonvanishing charged gap of a cluster.

Two rigorous theorems on the momentum distribution functions of the Hubbard model at half-filling

In this article, based on a recent theorem by Lieb et al. (1968), we shall prove two theorems on the momentum distribution functions of the half-filled Hubbard model on a d-dimensional simple cubic lattice in a mathematically rigorous way. More precisely, we shall first show that the half-filled positive-U and negative-U Hubbard models have the same momentum distribution functions n up arrow (q) and n down arrow (q). Then, we will show that nsigma (q) are symmetric functions about the value n= 1/2 . Finally, we shall briefly discuss some possible applications of these theorems to the further numerical investigations on the ground state of the Hubbard model at half-filling.

A rigorous theorem on off-diagonal long-range order of boson systems

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.

THE EFFECT OF AN EXTERNAL MAGNETIC-FIELD ON THE MLRO IN THE GLOBAL GROUND-STATES OF SOME ANTIFERROMAGNETIC MODELS

We study the effect of an external magnetic field on magnetic long-range order (MLRO) in the global ground states of the quantum XY model and the isotropic Heisenberg model on the simple cubic lattices. We shall rigorously prove that, while an external magnetic field (staggered in the antiferromagnetic case) which favours MLRO in a specific spin direction, say the x-direction, is turned on, it completely suppresses the MLRO in the perpendicular spin directions in the global ground states of these models.

The inseparability of resonating valence bond and on-site pairing long-range orders in doped Hubbard models

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 sum rule on Hofstadter spectrum and its application

Some time ago, in a remarkable paper Hofstadter (1976) showed that the energy spectrum of a spinless tight-binding electrons, moving in a constant magnetic field B, has a very complicated recursive structure. In this article, it is shown that, these energy levels satisfy a simple sum rule. Then, as an application of this sum rule, the author shows why a proposition on the absolute minimum of the energy of this system, which was made by Hasegawa et al (1989), holds in general cases.