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Physical review. B, Condensed matter

One-flipped-spin sector of the Hubbard model: A search for non-Fermi-liquid behavior

David M. Frenkel

Published Dec 15, 1992• 11 citations
Publisher

Abstract

One flipped spin in the spin-polarized sector of the Hubbard model away from half-filling is considered. Aspects other than stability of the Nagaoka solution or viability of the Gutzwiller approximation (considered previously by other authors) are focused on. It is argued that understanding the phase-coherence properties of the flipped spin may be a good starting point in an attempt to prove or disprove the violation of the Fermi-liquid-like behavior in the two-dimensional (2D) Hubbard model, an outcome that has been strongly advocated by P. W. Anderson. The system is analyzed in the weak-coupling limit via a variational wave function in 1D and 2D. The variational problem is reduced to solving a nonlinear Schr\"odinger equation in both of these dimensionalities. It is found that a solitonic type of solution emerges in weak coupling in 1D, but not in 2D. This implies a conventional Fermi-liquid-like behavior in our system in 2D in the weak-coupling limit. A discussion of possible loopholes in our approach is presented, and the remaining possibility of non-Fermi-liquid-like behavior in the strong-coupling limit at finite densities in 2D is noted. An interesting connection with the existing work on heavy mobile impurities in metals and in $^{3}\mathrm{He}$ is pointed out.

DOI
10.1103/physrevb.46.15008

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One flipped spin in the spin-polarized sector of the Hubbard model away from half-filling is considered. Aspects other than stability of the Nagaoka solution or viability of the Gutzwiller approximation (considered previously by other authors) are focused on. It is argued that understanding the phase-coherence properties of the flipped spin may be a good starting point in an attempt to prove or disprove the violation of the Fermi-liquid-like behavior in the two-dimensional (2D) Hubbard model, an outcome that has been strongly advocated by P. W. Anderson. The system is analyzed in the weak-coupling limit via a variational wave function in 1D and 2D. The variational problem is reduced to solving a nonlinear Schr"odinger equation in both of these dimensionalities. It is found that a solitonic type of solution emerges in weak coupling in 1D, but not in 2D. This implies a conventional Fermi-liquid-like behavior in our system in 2D in the weak-coupling limit. A discussion of possible loopholes in our approach is presented, and the remaining possibility of non-Fermi-liquid-like behavior in the strong-coupling limit at finite densities in 2D is noted. An interesting connection with the existing work on heavy mobile impurities in metals and in 3He^{3}\mathrm{He}3He is pointed out.
The applicability of the Hartree-Fock and random phase approximations to models of strongly correlated electrons is discussed. The 2D Hubbard model is analyzed. An antiferromagnetic phase (at half filling) and Fermi liquid behavior (at low electron densities) are found. In addition, the system shows an unconventional regime near half filling. It is characterized by the formation of spin and charge textures. Local magnetic moments and spin waves are also present, and the quasiparticle pole at the Fermi energy vanishes. The charge excitation spectrum is gapless. The overall features resemble those of the 1D Luttinger liquid.
The two-dimensional Hubbard model with a single spin-up electron interacting with a finite density of spin-down electrons is studied by use of the quantum Monte Carlo technique, the standard second-order perturbation theory and a very good variational wave function introduced recently by Edwards. We also present a new conjugate gradient scheme for the evaluation of this variational state. We performed simulations up to 242 sites at U/t=4 reaching the zero-temperature properties with no ``fermion-sign problem'' and found a surprisingly good accuracy of the Edwards state at low density or low doping. The conjugate gradient method was then applied to the system up to 3698 sites and infinite U. Fermi-liquid theory seems to remain stable in two dimensions for all cases studied with the exception of the half-filling case where a Luttinger-like behavior survives in the Hubbard model, yielding a vanishing quasiparticle weight in the thermodynamic limit.
The ground state of the 1D and of the 2D (square lattice) finite-size Hubbard model is investigated for variable filling using a novel quantum simulation method. We have studied up to 256 sites for both 1D and 2D. Away from half filling the 2D antiferromagnetic order is initially destroyed, albeit without any clear sign of a Fermi liquid behaviour. A metallic jump in n(k) appears only very far from half filling. In the 1D case, by contrast, a Fermi liquid-like jump in n(k) is obtained even very close to half filling.
We study an extended Hubbard model in the limit of infinite dimensions and its zero-dimensional counterpart, a generalized asymmetric Anderson model. In the impurity model we find three kinds of mixed valence states: (a) the usual strong-coupling state in which a resonance forms at the Fermi level; (b) a weak-coupling state in which neither the impurity spin nor the impurity charge degrees of freedom are quenched; and (c) an intermediate-coupling state where the spin but not the charge degree of freedom is quenched. The corresponding phases of the extended Hubbard model in infinite dimensions are (a) a Fermi liquid; (b) a non-Fermi-liquid state with incoherent charge and spin excitations; and (c) a non-Fermi-liquid state with incoherent charge but coherent spin excitations. The non-Fermi-liquid phases are incoherent metallic states with vanishing quasiparticle residue, self-similar local correlation functions, and asymptotically decoupled charge and spin excitations. The non-Fermi-liquid phases occur for a wide range of parameters.
We study the phase diagram of the one-dimensional Hubbard model with next-nearest-neighbor hopping using exact diagonalization, the density-matrix renormalization group, the Edwards variational ansatz, and an adaptation of weak-coupling calculations on the two-chain Hubbard model. We find that a substantial region of the strong-coupling phase diagram is ferromagnetic, and that three physically different limiting cases are connected in one ferromagnetic phase. At a point in the phase diagram at which there are two Fermi points at weak coupling, we study carefully the phase transition from the paramagnetic state to the fully polarized one as a function of the on-site Coulomb repulsion. We present evidence that the transition is second order and determine the critical exponents numerically. In this parameter regime, the system can be described as a Luttinger liquid at weak coupling. We extract the Luttinger-liquid parameters and show how their behavior differs from that of the nearest-neighbor Hubbard model. The general weak-coupling phase diagram can be mapped onto that of the two-chain Hubbard model. We exhibit explicitly the adapted phase diagram and determine its validity by numerically calculating spin and charge gaps using the density-matrix renormalization group.