ABSTRACT

Richard T. Scalettar Physics Department, University of California, Davis, California 95616, U.S.A.

George G. Batrouni INLN, Universite´ de Nice - Sophia Antipolis, CNRS; 1361 route des Lucioles, 06560 Valbonne, France

Originally introduced almost fifty years ago, the Hubbard Hamiltonian,

H = −t ∑ 〈ij〉

∑ i

(ni↑− 12)(ni↓− 1 2 )−μ

∑ i

(ni↑+ni↓), (11.1)

has been widely, and very successfully, used to model many-body effects and quantum phase transitions (QPTs) in condensed matter, despite the simplifications it embodies [1-4]; see also Chap. 1. The Hamiltonian, given by Eq. (11.1), contains a kinetic energy term t which describes the hopping of two sets of fermionic particles, labeled by σ, on near-neighbor sites 〈ij〉 of the lattice.1 These fermions feel an interaction energy U if they occupy the same site, a term written here in a symmetric form which makes chemical potential μ = 0 correspond to half-filling ρ = 〈∑σ niσ〉 = 1 at any value of t, U or temperature T , as long as the lattice is bipartite. Many different geometries have been studied, including 1D chains, 2D square, triangular, and honeycomb structures, and 3D cubic lattices.