Abstract This paper describes a group theoretical classification of the unrestricted Hartree–Fock (UHF) solution of the extended Hubbard model including nearest‐neighbor repulsion and exchange interactions. We derived systematically all types of the broken symmetry solution with a single‐order parameter for the ordering vector Q = (π,π). It is shown that there are four nonmagnetic states: a charge density wave (CDW), a charge current wave (CCW), and two bond‐order waves (BOW), and five magnetic states: an ordinary antiferromagnetic state (AF), a spin current wave (SCW), two axial spin bond‐order waves (ASBOW), and an helical spin bond‐order wave (HSBOW). We showed that these phases can be derived naturally from the irreducible decomposition of the Fourier transformed interaction and the self‐consistent field (SCF) condition. Some numerical results are given for the half‐filled case.
Abstract This paper describes a group theoretical classification of the unrestricted Hartree–Fock (UHF) solution of the extended Hubbard model including nearest‐neighbor repulsion and exchange interactions. We derived systematically all types of the broken symmetry solution with a single‐order parameter for the ordering vector Q = (π,π). It is shown that there are four nonmagnetic states: a charge density wave (CDW), a charge current wave (CCW), and two bond‐order waves (BOW), and five magnetic states: an ordinary antiferromagnetic state (AF), a spin current wave (SCW), two axial spin bond‐order waves (ASBOW), and an helical spin bond‐order wave (HSBOW). We showed that these phases can be derived naturally from the irreducible decomposition of the Fourier transformed interaction and the self‐consistent field (SCF) condition. Some numerical results are given for the half‐filled case.
This paper describes a group-theoretical classification of the mean-field solutions of the t-J model and the extended t-J model on a planar square lattice. It is shown that the mean-field solutions can be classified into four categories, spin density wave (SDW), charge density wave (CDW), bond order wave (BOW) and charve current wave (CCW) with distinct behavior under spatial inversion, spin rotation and time reversal. We derived systematically all types of broken-symmetry solutions, each of which is characterized by a single order parameter, for the ordering vector q= Q≡(π, π) and those for q= q1≡(π, 0). It is shown that for the t-J model there are four states (flux, Peierls, kite and SDW) with q= Q and six states (two CDWs, two BOWs and two CCWs) with q= Q and six states (two CDWs, two BOWs and two CCWs) with q= q1. For the t-J-J′ model including next-nearest-neighbour Heisenberg coupling, other four states (one CDW, one BOW and two CCWs) with q= Q and seven states (three SDWs, two BOWs and two CCWs) with q= q1 are possible. These phases are derived naturally from the irreducible spectral decomposition of the Σ JSiSj term and the self-consistent condition of the renormalized mean-field theory. The role of local SU(2) symmetry present at half filling is also discussed.
This paper describes a group-theoretical classification of the mean-field solutions of the t-J model and the extended t-J model on a planar square lattice. It is shown that the mean-field solutions can be classified into four categories, spin density wave (SDW), charge density wave (CDW), bond order wave (BOW) and charve current wave (CCW) with distinct behavior under spatial inversion, spin rotation and time reversal. We derived systematically all types of broken-symmetry solutions, each of which is characterized by a single order parameter, for the ordering vector q= Q≡(π, π) and those for q= q1≡(π, 0). It is shown that for the t-J model there are four states (flux, Peierls, kite and SDW) with q= Q and six states (two CDWs, two BOWs and two CCWs) with q= Q and six states (two CDWs, two BOWs and two CCWs) with q= q1. For the t-J-J′ model including next-nearest-neighbour Heisenberg coupling, other four states (one CDW, one BOW and two CCWs) with q= Q and seven states (three SDWs, two BOWs and two CCWs) with q= q1 are possible. These phases are derived naturally from the irreducible spectral decomposition of the Σ JSiSj term and the self-consistent condition of the renormalized mean-field theory. The role of local SU(2) symmetry present at half filling is also discussed.
The ground states of the two-dimensional repulsive Hubbard model are studied within the unrestricted Hartree-Fock (UHF) theory. Magnetic and charge properties are determined by systematic, large-scale, exact numerical calculations, and quantified as a function of electron doping, h. In the solution of the self-consistent UHF equations, multiple initial configurations and simulated annealing are used to facilitate convergence to the global minimum. New approaches are employed to minimize finite-size effects in order to reach the thermodynamic limit. At low to moderate interacting strengths and low doping, the UHF ground state is a linear spin-density wave (l-SDW), with antiferromagnetic order and a modulating wave. The wavelength of the modulating wave is 2/h. Corresponding charge order exists but is substantially weaker than the spin order, hence holes are mobile. As the interaction is increased, the l-SDW states evolve into several different phases, with the holes eventually becoming localized. A simple pairing model is presented with analytic calculations for low interaction strength and small doping, to help understand the numerical results and provide a physical picture for the properties of the SDW ground state. By comparison with recent many-body calculations, it is shown that, for intermediate interactions, the UHF solution provides a good description of the magnetic correlations in the true ground state of the Hubbard model.
The two-dimensional extended Hubbard model that includes a nearest-neighbor Heisenberg interaction is studied using a mean-field theory where quasiparticles are defined by an U(8) group of canonical transformations permitting both broken gauge, spin, and sublattice symmetry. The theory is further extended to incorporate a possible twist in the spin-quantization axis, so that the competition between superconductivity, charge density waves, and N\'eel and spiral antiferromagnetic order can be monitored within one single theory. Our results for positive Hubbard U and Heisenberg exchange J suggest that antiferromagnetic ordering dominates close to half-filling, while spiral states and d-wave superconductivity compete when doping is introduced. For moderate values of J, we find a phase diagram where a phase transition occurs from an antiferromagnet to a d-wave superconductor as doping is increased. A narrow region of (s+id)-wave superconductor is found for some values of J and U.
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.