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Philosophical Magazine B

Exact solution of the non-crossing approximation for the strong coupling limit of the half-filled Hubbard model

Peter S. Riseborough

Published May 1, 1999• 1 citation
Publisher

Abstract

We present an analytic solution of the self consistent random phase approximation equations for the half-filled Hubbard model, in the strong coupling limit. In the half-filled limit the Hubbard model shows antiferromagnetic ordering. The self energy is calculated by considering processes whereby the electrons emit and absorb low frequency spin waves, which are the Goldstone modes of the broken symmetry phase. We examine the electronic spectrum, which is dominated by the broad incoherent portions of the spectrum associated with the upper and lower Hubbard bands. The quasi-particle spectrum has a divergent wave function renormalization, indicating a divergent renormalization of the quasi-particle mass and zero quasi-particle weight, as in a non-Fermi liquid. The self consistency equation determined from the Goldstone modes in the susceptibility shows that the antiferromagnetism is prohibited in one and two dimensions but not in three. The magnetic ordering temperature is determined by dimensionality and the magnitude of the Anderson kinetic exchange of the effective Heisenberg Hamiltonian.

DOI
10.1080/014186399257113

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Journal of Physics Condensed Matter•Aug 2, 2000•24 citations
We present an analytic solution of the self consistent random phase approximation equations for the half-filled Hubbard model, in the strong coupling limit. In the half-filled limit the Hubbard model shows antiferromagnetic ordering. The self energy is calculated by considering processes whereby the electrons emit and absorb low frequency spin waves, which are the Goldstone modes of the broken symmetry phase. We examine the electronic spectrum, which is dominated by the broad incoherent portions of the spectrum associated with the upper and lower Hubbard bands. The quasi-particle spectrum has a divergent wave function renormalization, indicating a divergent renormalization of the quasi-particle mass and zero quasi-particle weight, as in a non-Fermi liquid. The self consistency equation determined from the Goldstone modes in the susceptibility shows that the antiferromagnetism is prohibited in one and two dimensions but not in three. The magnetic ordering temperature is determined by dimensionality and the magnitude of the Anderson kinetic exchange of the effective Heisenberg Hamiltonian.
We study the low-temperature critical behavior of the one-dimensional Hubbard model near half filling caused by enhanced antiferromagnetic fluctuations. We use a mean-field-type approximation with a two-particle self-consistency renormalizing the bare interaction. It allows us to control a transition from high to low temperatures as well as from weak to strong-coupling. We show that there is a crossover temperature T0 = t exp{−1/Uρ(0)} for arbitrary interaction U > 0 and the bare density of states at the Fermi energy ρ(0) > 0. The solution at lower temperatures goes over to strong coupling and approaches a quantum critical point with the diverging staggered susceptibility and a gap in the excitation spectrum at zero temperature.
An analytic approach is presented of electronic one-particle spectra of the one-band Hubbard model at half filling in the antiferromagnetic phase. Starting from the strong-coupling regime U\ensuremath{\gg}:t, a projection technique is used to set up self-consistent coupled equations for the electron Green function, which are valid down to values U\ensuremath{\approx}t. The self-consistent equation for the hole propagator is a direct generalization of the one found from the t-J model. This gives further support to the ``string'' picture, where propagation of holes creates strings of overturned spins with which the holes interact. In the present work hopping of holes (or electrons) with up spin on the down sublattice is also taken into acount, as well as transitions between the lower and upper Hubbard bands. These are shown to change significantly the incoherent part of the t-J model spectra, by smearing out the shake-off peaks, reminiscent of higher bound string states due to multispin scattering. Coherent (quasiparticle) peaks exist at the band edges, on both sides of the insulating gap. We show that with decreasing U the quasiparticle concept loses its meaning for wave vectors at the center of the magnetic Brillouin zone (MBZ). For large values of U the dispersion of the quasiparticle is found to scale with its band width, which is of order J. Extrema are always found at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2). The weight of the quasiparticle at this k value decreases logarithmically with increasing U. In the strong-coupling limit the spectrum tends to be symmetric, i.e., to become an even function of the frequency around the chemical potential, for any wave vector. For small values of U the dispersion at the edge of the MBZ flattens away, as expected when approaching the noninteracting limit. The spectral function in this regime, for wave vectors away from the edge of the MBZ, is concentrated mainly on one side of the chemical potential.
We present fully self-consistent calculations in the fluctuation exchange approximation for the half-filled Hubbard model in two dimensions. A non-Fermi-liquid state evolves with decreasing temperature in this self-consistent model of coupled spin fluctuations and quasiparticles. The mean-field phase transition to long-range antiferromagnetic order is suppressed and we find no evidence of a phase transition to long-range magnetic order. We show that the real part of the self-energy at zero energy shows a positive slope and the imaginary part of the self-energy shows a local minimum. The scale of this structure is set by the zero-temperature gap in mean-field theory. The growth of spin fluctuations is reflected in the evolution of sharply peaked structure in the spin fluctuation propagator around zero energy and Q = (π, π). We present calculations for the Hubbard model in one dimension in a two-particle self-consistent parquet approximation. A second-moment sum rule suggests that vertex corrections to the self-energy are responsible for the greater accuracy of the parquet approximation compared with other self-consistent perturbation theories.
We present an approach to the magnetic properties of the half-filled Hubbard model, based on an approximate mapping of its low-energy transverse spin excitations on to those of an effective underlying Heisenberg model, but with effective spin interactions which are self-consistently determined and not confined solely to nearest-neighbor couplings. The mapping is exact in strong-coupling and is found to be accurate over a very wide range of interaction strengths, down to weak coupling. At zero temperature, it permits ready evaluation at finite U of the one-loop effects of zero-point spin fluctuations on, e.g., the sublattice magnetization. At finite temperatures, thermodynamic properties of the system in the thermal paramagnet are studied via a physically transparent Onsager reaction field approach, which amounts to a self-consistent treatment of paramagnetic spin correlations. This is central not only in recovering the correct dimensionality dependence of antiferromagnetic long-ranged order, but also for the d=3 case of primary interest here yields a N\'eel temperature in close agreement with known strong- and weak-coupling limits. Spin correlation functions and magnetic susceptibilities also show very good agreement with quantum Monte Carlo calculations over an appreciable temperature range in which the low-lying transverse spin excitations are thermally dominant. \textcopyright{} 1996 The American Physical Society.
The single-particle Green's function for the strong-coupling limit of the single-band Hubbard model ( t-J model) at half-filling is calculated numerically using a Lanczos algorithm. Two-dimensional systems of up to 18 sites with periodic boundary conditions are examined, and evidence is presented for a coherent quasi-particle band of width J which emerges at the low-energy edge of the incoherent spectrum. We find the quasi-particle dispersion to be consistent with ε k = ( J /2)[cos ( k x ) + cos ( k y )] 2 , corresponding to effective next-nearest-neighbour hopping.
The exact phase diagrams of the one-dimensional Hubbard model, both attractive and repulsive, in the presence of an arbitrary magnetic field h for various electron concentrations n and on-site interaction strengths U <0 or U >0 are calculated and investigated. The exact ground-state properties, namely, the ground-state energy, the average spin (magnetization), the concentration of the doubly occupied sites, the kinetic energy, the chemical potential, the spin (magnetic) susceptibility and the charge compressibility, are calculated and examined over a wide range of interaction strengths U for various h and n . It is found that the spin susceptibility at half-filling is non-analytic and changes discontinuously as U →0. The exact theory shows the absence of a charge energy gap in the U ⩽0 region for all n and provides, for the chemical potential, the rigorous upper and lower bounds for half-filled and empty bands respectively. The analytical results derived for the weak-coupling limit and asymptotic expansions for strong coupling are in full agreement with the numerical calculations.