The variational approach initiated by Gutzwiller is extended to include antiferromagnetism. A simplified assumption corresponding to that adopted by him is used in order to evaluate the ground state energy of the Hubbard model and to draw the phase diagram similar to those constructed by Penn on the basis of the Hartree-Fock approximation. It is found that the antiferromagnetic state in the phase diagram is restricted to a narrow region bounded by threshold values for both the coupling constant and electron density. For sufficiently large intraatomic repulsion paramagnetic state becomes more stable than antiferromagnetic state except for the case of one electron per site. The phase transition between paramagnetic and antiferromagnetic states is of first order and the coexistence of these two states with different density occurs.
Two methods are used, Hartree-Fock and a variational wave function, to investigate spiral states in the three-band Hubbard model. Our variational wave function is a Gutzwiller-projected Hartree-Fock spiral state. We use quantum Monte Carlo methods to calculate the variational energy exactly. In order to avoid boundary defects in the spiral spin structure, we work in a frame where the spin-quantization axis is locally rotated. Both methods show that upon doping the antiferromagnetic structure present at half-band-filling is unstable against a (1,1) spiral state. For moderate (large) values of the Cu on-site Hubbard repulsion, there is a transition to a (1,0) spiral (ferromagnetic) state. The Hartree-Fock calculations show that most of the (1,1) spiral phase is unstable. On the other hand, the (1,0) spiral state is stable.
We studied ferromagnetism in the one-dimensional Hubbard model with doubly degenerate atomic orbitals by means of the density-matrix renormalization-group method and obtained the ground-state phase diagrams. It was found that ferromagnetism is stable from low to high
(0<n<1.75) electron density when the interactions are sufficiently strong. The quasi-long-range order of triplet superconductivity coexists with the ferromagnetic order for a strong Hund coupling region, where the interorbital interaction ${U}^{\ensuremath{'}}\ensuremath{-}J$ is attractive. At quarter-filling
(n=1), the insulating ferromagnetic state appears, accompanying orbital quasi-long-range order. For low densities
(n<1), ferromagnetism occurs owing to the ferromagnetic exchange interaction caused by virtual hoppings of electrons, the same as in the quarter-filled system. This comes from separation of the charge and spin-orbital degrees of freedom in the strong-coupling limit. This ferromagnetism is fragile against variation of band structure. For high densities
(n>1), the phase diagram of the ferromagnetic phase is similar to that obtained in infinite dimensions. In this case, the double exchange mechanism is operative for stabilizing the ferromagnetic order and this long-range order is robust against variation of the band dispersion. A partially polarized state appears in the density region $1.68\ensuremath{\lesssim}n\ensuremath{\lesssim}1.75$ and phase separation occurs for n just below the half-filling
(n=2).The ground-state phase diagrams of the three-orbital
t2g Hubbard model are studied using a Hartree-Fock approximation. First, a complete set of multipolar order parameters for
t2g models defined in terms of the effective total angular momentum
jeff is theoretically derived. These order parameters can classify off-diagonal orders between
jeff=1/2 and
jeff=3/2 manifolds. Second, through extensive Hartree-Fock calculations, the ground-state phase diagrams in the space of (1) the on-site Coulomb repulsion
U, (2) the spin-orbit coupling $\ensuremath{\lambda}$, and (3) the number of electrons
n are mapped out. A variety of nontrivial quantum phases with
jeff-diagonal and
jeff--off-diagonal multipole orders are found. Finally, future studies using more numerically expensive methods, such as dynamical mean-field theory, are discussed.
We explore the phase diagram of the Hubbard model by generalizing the Gutzwiller variational wave function to take into account nearest-neighbor spin correlations. The energies of various variational states are calculated and compared. At half filling the lowest energy state has long-range antiferromagnetic correlations, in agreement with previous results. Away from half filling, when the antiferromagnet no longer has the lowest energy, the Gutzwiller approximation yields a lower energy than that found for small spin polarons.
We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (
t∣IH) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For
t∣IH=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero
t∣IH, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of
t∣IH.