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arXiv:cond-mat/9605178 · Physical review. B, Condensed matter

Perturbation study on the spin and charge susceptibilities of the two-dimensional Hubbard model

Takashi Hotta, Satoshi Fujimoto

Published Aug 15, 1996• 17 citations
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Abstract

We investigate the spin and charge susceptibilities of the two-dimensional Hubbard model based upon the perturbative calculation in the strength of correlation U. For U comparable to a bare bandwidth, the charge susceptibility decreases near half filling as hole doping approaches zero. This behavior suggesting the precursor of the Mott-Hubbard gap formation cannot be obtained without the vertex corrections beyond the random-phase approximation. In the low-temperature region, the spin susceptibility deviates from the Curie-Weiss-like law and finally turns to decrease with the decrease of temperature. This spin-gap-like behavior originates from the van Hove singularity in the density of states. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.5381

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We investigate the spin and charge susceptibilities of the two-dimensional Hubbard model based upon the perturbative calculation in the strength of correlation U. For U comparable to a bare bandwidth, the charge susceptibility decreases near half filling as hole doping approaches zero. This behavior suggesting the precursor of the Mott-Hubbard gap formation cannot be obtained without the vertex corrections beyond the random-phase approximation. In the low-temperature region, the spin susceptibility deviates from the Curie-Weiss-like law and finally turns to decrease with the decrease of temperature. This spin-gap-like behavior originates from the van Hove singularity in the density of states. \textcopyright{} 1996 The American Physical Society.
We investigate the charge susceptibility in strongly correlated electron systems by using a Bethe-Salpeter equation in a generalized cumulant expansion for the p{d model, at varying values of the doping and temperature.We nd that for small values of the correlation Ud, the charge susceptibility decreases near the half{ lling as hole{doping approaches zero, this behavior suggesting the precursor of the Mott{Hubbard gap formation.Particular attention is paid to the role played by the Van Hove singularity near the Fermi level in the density of states, in connection with the unusual normal state properties of the hole{doped cuprates.
Using quantum Monte Carlo techniques we study the magnetic susceptibility of the Hubbard model on a two-dimensional square lattice for values of the coupling U/t ranging from 4 to 20, and as a function of temperature and doping. For U/t=4 the magnetic susceptibility has a maximum at half filling and decreases with doping, in qualitative agreement with recent random-phase-approximation calculations. However, for U/t=10 the susceptibility increases with hole doping near half filling resembling the experimental results obtained for ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$Srx{\mathrm{Sr}}_{\mathit{x}}Srx​CuO4{\mathrm{CuO}}_{4}CuO4​. At low electronic density the temperature dependence of the susceptibility is compatible with that of a Fermi liquid.
We study the weak-coupling limit of the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ Hubbard model on a two-dimensional square lattice using a direct perturbative approach. Aided by symbolic computational tools, we compute the longitudinal density-density correlation functions in the ${\ensuremath{\chi}}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}$ and ${\ensuremath{\chi}}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}$ basis from which we can obtain the dynamical spin and charge susceptibilities at arbitrary doping and temperature. We find that for nonzero ${t}^{\ensuremath{'}}$, the zero-frequency commensurate $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ spin and charge excitations are each strongest at different densities and we observe a clear behavioral change that appears tied to the van Hove singularity of the noninteracting dispersion upon which the perturbative expansion is built. We find a strongly reduced compressibility in the vicinity of the van Hove singularity as well as a behavioral change in the double occupancy. For finite ${t}^{\ensuremath{'}}$, the observed van Hove singularity occurs away from half filling, leading us to conclude that this reduction in compressibility is distinct from the Mott insulating physics that one expects in the strong-coupling regime. We compute the full dynamical spin and charge excitations and observe distinct structure for electron- and hole-doped scenarios, in agreement with experiments on cuprate materials. Finally, we observe a peculiar splitting in spin and charge excitations in the vicinity of the van Hove singularity, the origin of which is traced to a splitting near the bottom of the band.
We calculate the charge compressibility and uniform spin susceptibility for the two-dimensional (2D) Hubbard model slightly away from half filling within a two-loop renormalization group scheme. We find numerically that both those quantities flow to zero as we increase the initial interaction strength from weak to intermediate couplings. This result implies gap openings in both charge and spin excitation spectra for the latter interaction regime. When this occurs, the ground state of the lightly doped 2D Hubbard model may be interpreted as an insulating spin liquid as opposed to a Mott insulating state.
The Landau parameters F 0 s , F 0 a , F 1 s and F 1 a in the two-dimensional Hubbard model are calculated by the second order perturbation method of Abrikosov-Khalatnikov. The following points are revealed as the half-filling is approached: (1) F 0 a changes its sign from negative to positive, if the Coulomb interaction U is strong enough, i.e., U/t≳2.0U/t\gtrsim 2.0 U/t≳2.0. This means that the spin density oscillations, zero spin-sound, can propagate there; (2) F 1 s changes its sign from positive to negative, so that we can understand, on the basis of the anisotropic Fermi liquid theory, the behavior of the Drude weight whose magnitude decreases as the half-filling is approached even if there were no mass enhancement; (3) From the results of F 0 s and F 0 a , it is shown that both the charge and spin susceptibility decrease rapidly close to the half-filling. The latter is regarded as the so-called spin-gap behavior.
We analyze the density dependence of the spin stiffnesses and the stability of magnetic order with respect to quantum fluctuations in the two-dimensional Hubbard model close to half-filling. The stiffnesses are computed from the spin susceptibility obtained from a random-phase approximation in a magnetically ordered state. For a sizable next-to-nearest-neighbor hopping amplitude and a moderate Hubbard interaction, the mean-field ground state is a Néel antiferromagnet in the electron-doped regime at and above half-filling and a planar circular spiral state in the hole-doped regime below half-filling. Upon electron doping, the Néel stiffness decreases smoothly and not very steeply. By contrast, the in-plane and out-of-plane stiffnesses in the spiral state drop abruptly at half-filling. The out-of-plane stiffness even drops to zero and then increases again very slowly upon increasing hole doping. At finite temperatures, the Néel-to-spiral transition is shifted into the hole-doped regime, the stiffnesses are continuous functions of the density, and they vanish at the transition. For small hole doping, the spin stiffnesses describe the quantum fluctuations only in a small momentum range, which shrinks to zero upon approaching half-filling. Using the above results, we show that the quantum ground state of the lightly electron-doped Hubbard model remains Néel ordered, while quantum fluctuations probably destroy the spiral long-range order in the hole-doped regime, giving rise to a quantum disordered state with a spin gap.