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arXiv:cond-mat/9912128 · Journal of the Physical Society of Japan

Luttinger-Liquid Parameter of Hubbard Chain and Hubbard Ladder

Kazuhiro Sano

Published Apr 15, 2000• 8 citations
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Abstract

We study the Luttinger-liquid parameter $K_ρ$ of the Hubbard chain and the Hubbard ladder models by the ordinary perturbation method combined with the Luttinger-liquid relation. According to the Luttinger-liquid relation, the critical exponent $K_ρ$ is related to the charge susceptibility $χ_c$ and the Drude weight D by $ K\sbρ={1/2}(πχ_c D)^{1/2}$. By calculating these quantities with the perturbation method, we obtain $K_ρ$ at the first-order analytically and up to the second-order numerically. We compare these results with results of the Bethe ansatz for the Hubbard chain and that of the numerical diagonalization for the Hubbard ladder. It shows that the perturbation calculation of $K_ρ$ is reliable in the weak coupling regime.

DOI
10.1143/jpsj.69.1000

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We study the 1D extended Hubbard model with a weak repulsive short-range interaction in the non-half-filled band case, using non-perturbative Renormalization Group methods and Ward Identities. At the critical temperature, T = 0, the response functions have anomalous power-law decay with multiplicative logarithmic corrections. The critical exponents, the susceptibility and the Drude weight verify the universal Luttinger liquid relations. Borel summability and (a weak form of) Spin-Charge separation is established.
We investigate two-leg Hubbard ladders in the Luttinger liquid regime by the density-matrix renormalization group method. Applicability of the bond concept in this region is discussed in a quantitative comparison of power-law decay exponents, energy components and anti-bond occupations between ladders and chains. The interaction between bond and anti-bond bands induced by finite Hubbard repulsion U is emphasized by our numerical results. Our quantitative results are useful for the justification of the validity of various analytical approaches.
We present simple derivation of the Luttinger liquid relation for the 1D Hubbard model both for finite UUU and in the U=∞U=\inftyU=∞ limit. We describe the simple solution of the Hubbard model in the infinite repulsion limit and use it to calculate the correlators of the model in this limit in a simple and a physical way using the Bosonization technique. We then calculate the asymptotics of the correlators of the model at arbitrary UUU through the single parameter, which can be calculated from the Bethe Ansatz solution. Our derivation of the critical exponents is simple and allows one to express different physical operators of the Hubbard model through the charge and spin Bose fields in a direct and a physically transparent way.
Abstract We present a simple derivation of the Luttinger liquid relation for the one-dimensional Hubbard model, considering a finite U and in the U = ∞ limit. We describe a simple solution of the Hubbard model in the infinite repulsion limit and use it to calculate the correlators of the model in this limit in a simple and physical way using the bosonization technique. We then calculate the asymptotics of the correlators of the model at arbitrary U through a single parameter, which can be calculated from the Bethe Ansatz solution. Our derivation of the critical exponents is simple and allows one to express different physical operators of the Hubbard model through the charge and spin Bose fields in a direct and physically transparent way.
Antiferromagnetic Heisenberg spin chains in a sufficiently strong magnetic field are Luttinger liquids, whose parameters depend on the actual magnetization of the chain. Here we present precise numerical estimates of the Luttinger liquid dressed charge Z, which determines the critical exponents, by calculating the magnetization and quadrupole operator profiles for S=1/2S=1/2S=1/2 and S=1S=1S=1 chains using the density matrix renormalization group method. Critical amplitudes and the scattering length at the chain ends are also determined. Although both systems are Luttinger liquids the characteristic parameters differ considerably.
We show that the zero-temperature susceptibility of the one-dimensional, dimerized Hubbard model at quarter-filling can be accurately determined on the basis of exact diagonalization of small clusters. The best procedure is to perform a finite-size scaling of the spin velocity ${\mathit{u}}_{\mathrm{\ensuremath{\sigma}}}$, and to calculate the susceptibility from the Luttinger-liquid relation \ensuremath{\chi}=2/\ensuremath{\pi}${\mathit{u}}_{\mathrm{\ensuremath{\sigma}}}$. We show that these results are reliable by comparing them with the analytical results that can be obtained in the weak- and strong-coupling limits. We have also used quantum Monte Carlo simulations to calculate the temperature dependence of the susceptibility for parameters that should be relevant to the Bechgaard salts. This shows that, used together, these numerical techniques are able to give precise estimates of the low-temperature susceptibility of realistic one-dimensional models of correlated electrons.