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Physical review. B, Condensed matter

Collective and single-particle excitations in a Hubbard antiferromagnet

Prasenjit Sen, Avinash Singh

Published Dec 1, 1993• 2 citations
Publisher

Abstract

The spectrum of spin-wave and single-particle excitations is studied in a half-filled-band Hubbard antiferromagnet for interaction strength values in the intermediate and weak-coupling regimes. The spin-wave-energy scale, which goes essentially as ${\mathit{t}}^{2}$/U in the strong- and intermediate-coupling regimes, exhibits a crossover and goes down with decreasing interaction strength in the weak-coupling regime. This crossover is found to occur at U/t=3.26, and it demarcates the Heisenberg behavior in the strong- and intermediate-coupling regimes from the spin-density-wave behavior in the weak-coupling regime. The spin-wave-energy scale gets compressed by the charge gap in the weak-coupling regime, and consequently the spin-wave branch, which approaches the single-particle excitation band quite closely, never actually merges into it.

DOI
10.1103/physrevb.48.15792

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The spectrum of spin-wave and single-particle excitations is studied in a half-filled-band Hubbard antiferromagnet for interaction strength values in the intermediate and weak-coupling regimes. The spin-wave-energy scale, which goes essentially as t2{\mathit{t}}^{2}t2/U in the strong- and intermediate-coupling regimes, exhibits a crossover and goes down with decreasing interaction strength in the weak-coupling regime. This crossover is found to occur at U/t=3.26, and it demarcates the Heisenberg behavior in the strong- and intermediate-coupling regimes from the spin-density-wave behavior in the weak-coupling regime. The spin-wave-energy scale gets compressed by the charge gap in the weak-coupling regime, and consequently the spin-wave branch, which approaches the single-particle excitation band quite closely, never actually merges into it.
The authors re-examine the triplet (S=1) 'spin-wave' excitations of the exactly solvable spin- 1 / 2 1D Hubbard chain. They indicate that a continuous spectrum exists for all couplings U. In the strong coupling U to infinity limit, the spin-wave continuum spectrum of the isotropic Heisenberg antiferromagnet is recovered, for the half-filled band. Results for the quarter-filled band are also obtained by numerical solution of a set of linear integral equations. In addition they compare the particle-hole spectrum obtained by Coll (1974) with the Stoner excitations for a spiral spin density wave Hartree-Fock theory and found good qualitative agreement.
We describe a low-temperature approach to the two-dimensional half-filled Hubbard model which allows us to study both antiferromagnetism and single-particle properties. This approach ignores amplitude fluctuations of the antiferromagnetic (AF) order parameter and is valid below a crossover temperature TX{T}_{X}TX​ which marks the onset of AF short-range order. Directional fluctuations (spin waves) are described by a nonlinear sigma model $(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ that we derive from the Hubbard model. The parameters of the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$---the spin stiffness and spin-wave velocity---are calculated as a function of the Coulomb repulsion U. The $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ is solved by a saddle-point approximation within the CP1{\mathrm{CP}}^{1}CP1 representation where the N\'eel field is parametrized by two Schwinger bosons. At zero temperature, there is always Bose condensation of the Schwinger bosons, which signals AF long-range order for any value of the Coulomb repulsion. At finite temperature, the AF long-range order is suppressed (in agreement with the Mermin-Wagner theorem), but the AF correlation length remains exponentially large. In the CP1{\mathrm{CP}}^{1}CP1 representation, the fermion field is naturally expressed as the product of a Schwinger boson and a pseudofermion whose spin is quantized along the (fluctuating) N\'eel field. This allows us to write the fermion Green's function as the product (in direct space) of the Schwinger boson propagator (which is derived from the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ and the pseudofermion propagator. At zero temperature and weak coupling, our results are typical of a Slater antiferromagnet. The AF gap is exponentially small; there are well-defined Bogoliubov quasiparticles (QP's) (carrying most of the spectral weight) coexisting with a high-energy incoherent excitation background. As U increases, the Slater antiferromagnet progressively becomes a Mott-Heisenberg antiferromagnet. The Bogoliubov bands evolve into Mott-Hubbard bands separated by a large AF gap. A significant fraction of spectral weight is transferred from the Bogoliubov QP's to incoherent excitations. At finite temperature, there is a metal-insulator transition between a pseudogap phase at weak coupling and a Mott-Hubbard insulator at strong coupling. Finally, we point out that our results straightforwardly translate to the half-filled attractive Hubbard model, where the $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge and q=0\mathbf{q}=0q=0 pairing fluctuations combine to form an order parameter with SO(3) symmetry.
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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.
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