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International Journal of Modern Physics B

EXACT RESULTS IN STRONGLY CORRELATED ELECTRONS — SPIN-REFLECTION POSITIVITY AND THE PERRON-FROBENIUS THEOREM

Takashi Yanagisawa, Yukihiro Shimoi

Published Nov 15, 1996• 35 citations
Publisher

Abstract

This article is devoted to discuss two exact arguments in strongly-correlated electron systems: one is the method of reflection positivity in the spin space and the other is the Perron-Frobenius theorem in matrix theory. For several specific systems, we can show that the ground state is unique and we can predict the total spin of the ground state.

DOI
10.1142/s0217979296001835

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This article is devoted to discuss two exact arguments in strongly-correlated electron systems: one is the method of reflection positivity in the spin space and the other is the Perron-Frobenius theorem in matrix theory. For several specific systems, we can show that the ground state is unique and we can predict the total spin of the ground state.
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 eigen-functional theoretical framework, exact/accurate treatment of high-dimensional strongly correlated electron systems may be realized. The two key steps of this method are that under the path integral formulation we first change a D-dimensional quantum many-particle system into an (D+1)-dimensional (time-dependent) effective "single-particle" problem, then by solving the eigen-equation of the propagator operator of the particles, we can obtain the ground state energy functional and action, respectively. Under the eigen-functional theory, the problems of quantum many-particle systems end in to solve the equation of phase fields that are completely determined by the electron interaction. In practice this equation of the phase fields can be numerically solved for large lattice sites. After replacing the real time by the imaginary time, this method can also be applied for finite temperature cases.
In this article we discuss the reflection positivity in spin space for the Heisenberg model and the Kondo Hamiltonian. We show that the ground state is unique for J ≠0 where J is the strength of the exchange interaction in the Heisenberg model or the exchange coupling between the localized spins and conduction electrons in the Kondo model. We introduce the Coulomb interaction U >0 between the conduction lectrons for the Kondo Hamiltonian to show the uniqueness of the ground state. We write the Hamiltonian in terms of fermions to express the exchange interactions and impose constraints. We employ the Lagrange-multiplier method to deal with the constraints in this work.