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Physical review. B, Condensed matter

Spiral states in the square-lattice Hubbard model

Sanjoy Kumar Sarker, Ciriyam Jayaprakash, H. R. Krishnamurthy, Wolfgang Wenzel

Published Apr 1, 1991• 51 citations
Publisher

Abstract

We present a variety of physical implications of a mean-field theory for spiral spin-density-wave states in the square-lattice Hubbard model for small deviations from half filling. The phase diagram with the paramagnetic metal, two spiral (semimetallic) states, and ferromagnet is calculated. The momentum distribution function and the (quasiparticle) density of states are discussed. There is a significant broadening of the quasiparticle bands when the antiferromagnetic insulator is doped. The evolution of the Fermi surface and the variation of the plasma frequency and a charge-stiffness constant with U/t and \ensuremath{\delta} are calculated. The connection to results based on the Schwinger-boson--slave-fermion formalism is made.

DOI
10.1103/physrevb.43.8775

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We present a variety of physical implications of a mean-field theory for spiral spin-density-wave states in the square-lattice Hubbard model for small deviations from half filling. The phase diagram with the paramagnetic metal, two spiral (semimetallic) states, and ferromagnet is calculated. The momentum distribution function and the (quasiparticle) density of states are discussed. There is a significant broadening of the quasiparticle bands when the antiferromagnetic insulator is doped. The evolution of the Fermi surface and the variation of the plasma frequency and a charge-stiffness constant with U/t and \ensuremath{\delta} are calculated. The connection to results based on the Schwinger-boson--slave-fermion formalism is made.
Using a path-integral formalism and decomposing the Hubbard interaction into charge and spin order parameters, we develop a 1/S loop expansion of the two-dimensional Hubbard model. At the saddle-point level, we present a detailed study of the energetics of twisted antiferromagnetic order. Doping the antiferromagnetic state leads to a rich mean-field phase diagram at zero temperature, which includes spiral phases, a column phase, and ferromagnetism. The on-site magnetization, the density of states, the plasma frequency, and the quasiparticle weights are evaluated for the twisted spin-density-wave (SDW) states as a function of doping and U/t. Doping-induced frustration remarkably leads to a closure of the Mott-Hubbard band gap for any U/t. At one-loop order, we delineate the collective spin and charge fluctuations of the mean-field SDW state. The action for the two transverse spin modes at half-filling maps to a nonlinear \ensuremath{\sigma} model, which tends to disorder upon doping. Twisted magnetic ground states exhibit a third Goldstone mode due to the complete breaking of rotational invariance. For the commensurate SDW, the charge fluctuations couple only to longitudinal spin fluctuations, whereas, for the incommensurate metallic SDW states, the charge modes are coupled nontrivially to the massless spin waves. This suggests a metallic state quite unlike traditional Fermi liquids.
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.
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.
We study the Hubbard model on the triangular lattice for general fillings by means of the slave-boson technique. In particular, we consider the stability regions of the paramagnetic, antiferromagnetic and linearly polarized spin-density-wave phases that are present at half-filling. The model presents a large anisotropy under particle or hole doping of the half-filled band, a consequence of the non-bipartite nature of the lattice. We also compute the helicity modulus in the antiferromagnetic phase, which characterizes the stiffness of the magnetic order.
The authors determine the ground state phase diagram of the Hubbard model on the square lattice allowing for homogeneous spiral, antiferromagnetic, ferromagnetic and paramagnetic phases. This is obtained from a saddle-point approximation of a spin-rotation-invariant form of the slave boson representation introduced by Kotliar and Ruckenstein. They obtain very good agreement in energy with exact diagonalization calculations and, concerning the magnetic structure, qualitative agreement with experimental data.