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Journal of the Physical Society of Japan

Spin Density Wave States in Two Dimensional Hubbard Model

Masahiko Ichimura, Mitsutaka Fujita, Kenji Nakao

Published Jun 1, 1992• 13 citations
Publisher

Abstract

For a wide range of electron filling we study the spin density wave states in the Hubbard model on a square lattice within a Hartree-Fock approximation. It is shown that the incommensurate helical spin density wave states appear for whole region of the electron filling and, especially, that the commensurate helical spin density wave state with wave vector (π, 0) is stable near the quarter-filling. We examine also the relative stability of the soliton and the vortex lattice states to the helical spin density wave states and obtain the results that near the half-filling case the soliton lattice states are stable states for intermediate interaction strength and that the vortex lattice states do not appear as the stable state, except one particular state with the wind-mill spin structure at the quarter-filling.

DOI
10.1143/jpsj.61.2027

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For a wide range of electron filling we study the spin density wave states in the Hubbard model on a square lattice within a Hartree-Fock approximation. It is shown that the incommensurate helical spin density wave states appear for whole region of the electron filling and, especially, that the commensurate helical spin density wave state with wave vector (π, 0) is stable near the quarter-filling. We examine also the relative stability of the soliton and the vortex lattice states to the helical spin density wave states and obtain the results that near the half-filling case the soliton lattice states are stable states for intermediate interaction strength and that the vortex lattice states do not appear as the stable state, except one particular state with the wind-mill spin structure at the quarter-filling.
We study the spin and charge textures of the two-dimensional Hubbard model for a wide range of electron filling at the ground state both on a square lattice and on a triangular lattice within a mean field approximation. We complete the overall phase diagram of the single-Q helical spin density wave state, which spreads widely in the electron filling and the Coulomb interaction plane, and discuss the relative stability of the soliton and the vortex lattice states on it. It is shown that the imperfect nesting condition, which is the intrinsic feature of two-dimensional systems, induces a rich variety of self-consistent solutions including collinear and noncollinear spin textures.
Spatially modulated magnetic phases are investigated within the mean field theory for an itinerant electron model, i.e. the Hubbard model on a two-dimensional square lattice. By numerically diagonalizing the Hamiltonian for finite-size systems under a periodic boundary condition, we examine relative stability and physical properties of several possible magnetic states. When the electron fillings are nearly half-full, the diagonally or vertically modulated spin density wave (SDW) state is stabilized over the uniform antiferromagnetic state and a crossover from the vertical to the diagonal states appears. The diagonal or vertical stripe state is characterized by the presence of the midgap band due to the soliton lattice formation inside the main SDW gap, being an insulator. The wave length λ SDW is linearly proportional to the excess carrier concentration. Excess carriers are accommodated in the form of the soliton lattice, forming a charge density wave whose wave length is λ SDW /2.
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.
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 Heisenberg model.
Using exact diagonalization, we calculate the density of states of the two-dimensional Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 square lattice at U/t=0.5, 4, and 10, and even number of electrons with filling factors n ranging from a quarter filling up to half filling. We compare the ground-state energy and density of states at U/t=0.5 and 4 with second-order perturbation theory in U/t in the paramagnetic phase, and find that while the agreement is reasonable at U/t=0.5, it becomes worse as the perturbatively determined (i.e., using Stoner's criterion) boundary of the paramagnetic to spin-density-wave instability is approached. In the strong coupling regime (U/t=10), we find reasonable agreement between the density of states of the Hubbard and the t-J model especially for low doping fractions. In general, we find that at half filling the filled states are separated from the empty states by a gap. At U/t=10, the density of states shows two bands clearly separated by a Mott-Hubbard gap of order U.
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.