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Physical review. B, Condensed matter

Applications of reflection positivity in strongly correlated electron systems

Shun-Qing Shen

Published Aug 15, 1996• 8 citations
Publisher

Abstract

In this paper, theorems are proved concerning the total spin and the sign of the spin-spin correlation function for positive (semi)definite states of correlated electrons at half filling. These theorems are applied to the ground states of the Anderson model, the Kondo model, and a generalized Hubbard model for ferromagnetism. It is found that there exists ferromagnetic or antiferromagnetic long-range order in several cases. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.4397

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In this paper, theorems are proved concerning the total spin and the sign of the spin-spin correlation function for positive (semi)definite states of correlated electrons at half filling. These theorems are applied to the ground states of the Anderson model, the Kondo model, and a generalized Hubbard model for ferromagnetism. It is found that there exists ferromagnetic or antiferromagnetic long-range order in several cases. \textcopyright{} 1996 The American Physical Society.
Theory of spin-reflection positivity developed in recent years is reviewed. This theory makes use of symmetries in an electron system and theory of matrix to investigate the ground state properties. Existences of anti- and ferromagnetic long-range orders in itinerant electron systems, and of off-diagonal long-range order are two successful applications of the theory. In this article, the author attempt to summarize exact results proved by utilizing this theory and related topics. First a general theory and basic theorems are introduced. Second, based on the band structures of conduction electrons, existences of a singlet state with strongly antiferromagnetic correlation, a state with both anti- and ferromagnetic long-range orders, and a fully saturated ferromagnetic state are proved. The theory is applied to several of the main theoretical models for strongly correlated electron systems, such as the Heisenberg model, the Hubbard model, the Anderson model, the single- and multichannel Kondo model, and the generalized Hubbard model, and a series of rigorous results are found in these models. Third, it is proved that off-diagonal long-range order and charge-density wave exist in the ground states of the attractive Hubbard model and the generalized Hubbard model. A relation between pseudospin symmetry and the uniform density theorem is introduced. Fourth, the theory is applied successfully to explain experimental observations of oscillatory interlayer magnetic coupling in ultrathin magnetic films. Finally several unsolved problems are discussed. All results introduced in this article are mathematically exact.
In the present chapter we review some rigorous results derived with Lieb’s spin-reflectionpositivity method in a pedagogical manner. To begin with, we emphasize the important role played by the sign rule satisfied by the ground-state wave function of a quantum many-body system. For example, we study some localized spin systems, such as the antiferromagnetic Heisenberg model, in detail. Then we argue why such a rule is difficult to establish for the ground-state wave functions of itinerant electron models. By using the spin-reflection-positivity method, we show that this problem can be partially remedied. A weaker form of the sign rule is uncovered. It enables us to prove several inequalities satisfied by either an electron pairing correlation function or a magnetic correlation function of the Hubbard model on a bipartite lattice. These inequalities imply that, if a bipartite lattice has macroscopically unequal numbers of sublattice sites, then the Hubbard model at half-filling is either a supersolid or a ferrimagnet. Similar inequalities can also be proven for the periodic Anderson model and the Kondo lattice model. Finally, by applying a generalized version of the spin-reflection-positivity method, we further establish an inequality satisfied by the charged and spin gaps of these models. Based on them, we make some audacious conjectures, and then we summarize this chapter.
We investigate the ground state of the Kondo-Hubbard model employing the recently introduced idea of spin-reflection positivity. It is shown exactly that the Kondo-Hubbard model for J<0J<0J<0 and U>∣J∣/4U>|J|/4U>∣J∣/4 at half filling has a unique ground state which can be ferromagnetic or singlet according to the lattice structure. In particular, for the Kondo lattice with U>∣J∣/4U>|J|/4U>∣J∣/4, the ground state is a singlet. We also show a theorem for the Kondo-Hubbard model with the anisotropic antiferromagnetic coupling ${J}_{\ensuremath{\perp}}>0$ and ${J}_{z}\ensuremath{\le}0$ for U>∣Jz∣/4U>|{J}_{z}|/4U>∣Jz​∣/4.
As an extension of our previous rigorous investigation on the spin correlations in the ground states of the half-filled Hubbard model and the periodic Anderson model [G. S. Tian, Phys. Rev. B 50, 6246 (1994)], in the present paper we study the behavior of these correlation functions at finite temperature. We show rigorously that, at any $T\ensuremath{\ne}0,$ the predominant spin correlations in these systems are antiferromagnetic. Furthermore, based on this result, we also show that a quasi-one (or two)-dimensional itinerant electron ferrimagnet must have a gapless branch of ferromagnetic excitations. This conclusion is consistent with the previous results derived by the spin-wave theory.
We examine the spin-reflection positivity of the ground state of the Kondo lattice model at half-filling with the antiferromagnetic and ferromagnetic exchange couplings J≠0J\ne0J=0. For every positive U>0U>0U>0, where U is the Coulomb interaction between the conduction electrons, we can show that the ground state is unique.