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Physical review. B, Condensed matter

Spiral states in the three-band Hubbard model: A Hartree-Fock and variational quantum Monte Carlo approach

Fakher F. Assaad

Published Apr 1, 1993• 14 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.47.7910

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•Jan 1, 1995•2 citations
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.
The Gutzwiller wave function for the three-band Hubbard model on a two-dimensional CuO2{\mathrm{CuO}}_{2}CuO2​ plane is studied by using the variational Monte Carlo (VMC) method in the limit Ud{\mathit{U}}_{\mathit{d}}Ud​\ensuremath{\rightarrow}\ensuremath{\infty}, where Ud{\mathit{U}}_{\mathit{d}}Ud​ is the Coulomb repulsion between holes on a Cu site. The VMC results for the energy and the fraction of d holes are compared with those of the Rice-Ueda type Gutzwiller approximation (GA). The difference between the VMC and the GA results are most pronounced at half-filling, and away from half-filling the two results agree well in both the hole-doped and the electron-doped cases. The doping dependence of the momentum distribution function is also studied.
Choosing the Gutzwiller-type variational wave function and using the variational Monte Carlo (VMC) method, we have calculated the ground-state energies of two-dimensional (2D) Hubbard squares at temperature T=0 and have obtained the phase diagram in the small doping region (\ensuremath{\delta}=0--0.15). We do not find the spiral phase until at least U\ensuremath{\approxeq}20 in the half-filled and dilute-doping cases. This is different from the results obtained by C. Jayaprakash et al., who calculated the phase diagram of the 2D t-J model at temperature T=0 with the slave-fermion mean-field theory and found the spiral phase at much lower values of U/t.
The Hartree-Fock ground-state phase diagram of the one-dimensional Hubbard model is calculated in the μ−Uμ-Uμ−U plane, restricted to phases with no charge density modulation, extending the results presented in cond-mat/9511116. This allows antiferromagnetism, saturated ferromagnetism, spiral spin density waves and a collinear structure with unit cell ↑↑↓↓\uparrow \uparrow \downarrow \downarrow↑↑↓↓. The spiral phase is unstable against phase separation near quarter-, half- and three-quarter-filling. For large UUU this occurs at hole (or electron) doping of (3t/π2U)1/3(3t/π^{2}U)^{1/3}(3t/π2U)1/3 from half filling.
We study, within the Hartree-Fock approximation, the relative stability of different classes of states for the single-band Hubbard model on a square lattice in the presence of a nearest-neighbor Coulomb repulsion V and away from half-filling. We consider (i) homogeneous metallic states (including spiral phases), (ii) states that exhibit phase separation into hole-rich and antiferromagnetic regions, and (iii) insulating (longitudinally polarized) spin-density-wave states with domain walls that trap holes. For V=0, the spiral states have the lowest energy for hole concentration \ensuremath{\delta}>${\mathrm{\ensuremath{\delta}}}_{\mathit{s}}$. At ${\mathrm{\ensuremath{\delta}}}_{\mathit{s}}$\ensuremath{\sim}0.25--0.3, there is a transition to domain walls for small to intermediate U/t. At larger U/t the transition is to a phase-separated state, with ${\mathrm{\ensuremath{\delta}}}_{\mathit{s}}$ decreasing slowly with increasing U/t. We find that even a small V destabilizes the domain walls relative to the spiral states; e.g., for V/t\ensuremath{\sim}0.2, the domain-wall states are unstable for all U/t and all \ensuremath{\delta}\ensuremath{\ge}0.05. The effect of V on the phase-separated state is much less drastic; its region of stability relative to the spiral states also shrinks to smaller \ensuremath{\delta}, but rather slowly. The last results are in agreement with the findings from a Schwinger-boson treatment of the t-J model.
The Hartree-Fock ground state phase diagram of the one-dimensional Hubbard model is calculated, constrained to uniform phases, which have no charge density modulation. The allowed solutions are saturated ferromagnetism (FM), a spiral spin density wave (SSDW) and a double spin density wave} (DSDW). The DSDW phase comprises two canted interpenetrating antiferromagnetic sublattices. FM occurs for small filling, SSDW in most of the remainder of the phase diagram, and DSDW in a narrow tongue near quarter (and three-quarter) filling. Itinerant electrons lift the degeneracy with respect to canting angle in the DSDW. The Hartree-Fock states are metallic except at multiples of a quarter filling. Near half filling the uniform SSDW phase is unstable against phase separation into a half-filled antiferromagnetic phase and a hole-rich SSDW phase. The dependence of the ground state wave number on chemical potential is conjectured to be a staircase. Comparison is made with higher dimensional Hubbard models and the J1−J2J_{1}-J_{2}J1​−J2​ Heisenberg model.