The familiar unrestricted Hartree-Fock variational principles is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices and these, in turn, provide a convenient formulation of a generalized Hartree-Fock variational principle, which includes the BCS theory as a special case. While this generalization is not new, it is not well known and we begin by elucidating it. The Hubbard model, with its particle-hole symmetry, is well suited to exploring this theory because BCS states for the attractive model turn into usual HF states for the repulsive model. We rigorously determine the true, unrestricted minimizers for zero and for nonzero temperature in several cases, notably the half-filled band. For the cases treated here, we can exactly determine all broken and unbroken spatial and gauge symmetries of the Hamiltonian.

Generalized Hartree-Fock theory and the Hubbard model

The familiar unrestricted Hartree-Fock variational principle is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices and these, in turn provide a convenient formulation of a generalized Hartree-Fock variational principle, which includes the BCS theory as a special case. While this generalization is not new, it is not well known and we begin by elucidating it. The Hubbard model, with its particle-hole symmetry, is well suited to exploring this theory because BCS states for the attractive model turn into usual HF states for the repulsive model. We rigorously determine the true, unrestricted minimizers for zero and for nonzero temperature in several cases, notably the half-filled band. For the cases treated here, we can exactly determine all broken and unbroken spatial and gauge symmetries of the Hamiltonian.

Generalized Hartree-Fock Theory and the Hubbard Model

Intermolecular Coulomb repulsion, V, of the highest occupied molecular orbitals (HOMO) of BEDT-TTF (bis(ethylenedithio)tetrathiafulvalene) is calculated for various molecular geometries. The bare V is a quantity that is easily estimated under the point charge approximation. As far as the screened V in actual crystals is proportional to the calculated bare V, the usual θ-phase prefers the horizontal or diagonal stripe, whereas the vertical stripe becomes comparatively stable in the limit of the small dihedral angle (in the metallic limit). The phase diagrams of the θ-phase are discussed under the combination of the static charge distribution (the atomic limit) and the Stoner model (the extended Stoner model). The model contains two order parameters: the spin polarization, Sz, and the charge order, n-1/2. This model explains why the insulating state of the Rb salt below 190 K is a paramagnetic charge-ordered state, while the Cs salt has a different insulating phase below 20 K. The lattice dimerization of th...

Estimation of Off-Site Coulomb Integrals and Phase Diagrams of Charge Ordered States in the θ-Phase Organic Conductors

Normal states of the attractive Hubbard model, mainly in two dimensions, are studied in the context of a transition from a Fermi liquid to an insulating or gapped state. A series of variational Monte Carlo calculations with better statistics was carried out to estimate accurately the expectation values of several many-body wave functions. Although a relatively clear crossover is observed, even for the plain Gutzwiller wave function, the states in both regimes are metallic. Meanwhile, a substantial metal-insulator transition takes place at ‖U‖ ∼ W (band width) for an improved wave function, into which an intersite correlation is introduced by taking account of virtual states in the second-order perturbation in the infinite ‖U‖ limit. The critical value compares favorably with recent results of the dynamical-mean-field approximation. In contrast, a conventional Jastrow-type wave function scarcely improves the normal state. In addition, the issue of the Brinkman-Rice metal-insulator transition is reconsidered with much larger systems.

Variational Monte Carlo studies of attractive Hubbard model. I - Normal-state properties

Saddle Point Mean Field Calculation in the Hubbard Model

The ground state energy of the neutral Hubbard model is calculated by BCS methods for all values of total spinSz. Numerical results are given for the simple cubic and for the body centred cubic lattice. Antiferromagnetic ordering and a finite paramagnetic susceptibility is found for all values of the coupling constantV0.

The ground state of the neutral Hubbard model

By use of Bogoljubov's variational principle the free energy of the neutral Hubbard model is calculated as a function of the total spinSz. A transition from an antiferro-magnetic to a paramagnetic phase is found for all values of the coupling constantV0. The calculations are performed for the simple cubic and the body-centred cubic lattice; some numerical results are given for the transition temperature, the free energy, the sublattice magnetization, the transversal magnetic susceptibility and the response to a finite magnetic field.

H. Koppe

Papers

The ground state of the neutral Hubbard model

Thermodynamics of the neutral Hubbard model