TY - JOUR

T1 - Ferromagnetism of the hubbard model at strong coupling in the hartree-fock approximation

AU - Bach, Volker

AU - Lieb, Elliott H.

AU - Travaglia, Marcos V.

N1 - Funding Information:
The authors are grateful to Alessandro Giuliani for very helpful discussions and comments about an earlier version of this paper. They also thank Manfred Salmhofer, Jürg Fröhlich and Daniel Ueltschi for useful discussions. M.T. thanks the German student exchange service DAAD for a generous stipend, which supported two-thirds of his graduate studies. V.B. and M.T. gratefully acknowledge financial support from grant no. HPRN-CT-2002-00277 of the European Union and grant no. Ba 1477/3-3 of the Deutsche Forschungsgemeinschaft. E.L. gratefully acknowledges
Funding Information:
support from the Alexander von Humboldt Foundation of a fellowship, the U.S. National Science Foundation, grant no. PHY-0133984, and the hospitality of the Mathematics Departments of the University of Mainz and the Technical University of Berlin. The authors appreciate the careful and helpful work of the referee.

PY - 2006/6

Y1 - 2006/6

N2 - As a contribution to the study of the Hartree-Fock theory, we prove rigorously that the Hartree-Fock approximation to the ground state of the d-dimensional Hubbard model leads to saturated ferromagnetism when the particle density (more precisely, the chemical potential μ) is small and the coupling constant U is large, but finite. This ferromagnetism contradicts the known fact that there is no magnetization at low density, for any U, and thus shows that HF theory is wrong in this case. As in the usual Hartree-Fock theory, we restrict attention to Slater determinants that are eigenvectors of the z-component of the total spin, S*z = ∑x n x,↑ -nx,↓, and we find that the choice 2S*Z = N = particle number gives the lowest energy at fixed 0 < μ, < 4d.

AB - As a contribution to the study of the Hartree-Fock theory, we prove rigorously that the Hartree-Fock approximation to the ground state of the d-dimensional Hubbard model leads to saturated ferromagnetism when the particle density (more precisely, the chemical potential μ) is small and the coupling constant U is large, but finite. This ferromagnetism contradicts the known fact that there is no magnetization at low density, for any U, and thus shows that HF theory is wrong in this case. As in the usual Hartree-Fock theory, we restrict attention to Slater determinants that are eigenvectors of the z-component of the total spin, S*z = ∑x n x,↑ -nx,↓, and we find that the choice 2S*Z = N = particle number gives the lowest energy at fixed 0 < μ, < 4d.

KW - Ferromagnetism

KW - Hartree-Fock theory

KW - Hubbard model

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U2 - 10.1142/S0129055X06002735

DO - 10.1142/S0129055X06002735

M3 - Article

AN - SCOPUS:33747039530

VL - 18

SP - 519

EP - 543

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

SN - 0129-055X

IS - 5

ER -