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Journal of Physics A Mathematical and General

Boundary susceptibilities of the Hubbard model in open chains

Hitoshi Asakawa, Masuo Suzuki

Published Dec 21, 1996• 22 citations
Publisher

Abstract

Boundary contributions to the magnetization and the electron density are evaluated analytically, near the half-filling and non-magnetic region. Boundary susceptibilities in the spin and the charge sectors are also calculated. The boundary magnetization approaches zero logarithmically as the magnetic field goes to zero. The boundary charge susceptibility diverges like the bulk compressibility when the filling goes to unity.

DOI
10.1088/0305-4470/29/24/008

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Boundary contributions to the magnetization and the electron density are evaluated analytically, near the half-filling and non-magnetic region. Boundary susceptibilities in the spin and the charge sectors are also calculated. The boundary magnetization approaches zero logarithmically as the magnetic field goes to zero. The boundary charge susceptibility diverges like the bulk compressibility when the filling goes to unity.
The magnetic susceptibility of the one-dimensional Hubbard model with open boundary conditions at arbitrary filling is obtained from field theory at low temperatures and small magnetic fields, including leading and next-leading orders. Logarithmic contributions to the bulk part are identified as well as algebraic-logarithmic divergences in the boundary contribution. As a manifestation of spin-charge separation, the result for the boundary part at low energies turns out to be independent of filling and interaction strength and identical to the result for the Heisenberg model. For the bulk part at zero temperature, the scale in the logarithms is determined exactly from the Bethe ansatz. At finite temperature, the susceptibility profile as well as the Friedel oscillations in the magnetisation are obtained numerically from the density-matrix renormalisation group applied to transfer matrices. Agreement is found with an exact asymptotic expansion of the relevant correlation function.
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.
The magnetic properties of a Hubbard chain with on-site attraction are studied as a function of the band filling. Magnetization curves are calculated numerically for several values of the interaction and band filling. An unexpected feature of these curves is that the susceptibility is always finite except for the empty and half-filled band. The gap in the magnetic excitation spectrum is also calculated as a function of the band filling. The numerical results are compared with analytical results, which are obtained for the nearly empty and nearly-half-filled band.
The phase diagram, the Bethe-ansatz ground-state properties, including the chemical potential μ, the spin (magnetic) and charge susceptibilities, are calculated within the one-dimensional Hubbard model in entire range of interaction strength (-∞< U/t<+∞), magnetic field (h≥ 0) and all electron concentrations (0≤n≤1). The continuous and smooth variation of μ with n and h in the vicinity of n=1 points on the gapless character of charge excitations at U<0 and provides rigorous upper and lower bounds for μ. The spin (magnetic) susceptability χ at half-filling changes discontinuously as U→0 and is strongly enhanced by electron repulsion, comparing with that of the non-interactig case. The compressibility κ ch increases with n at U<0 and shows non-monotonous behavior with a dramatic increase at U>0. Variations of κ ch -1 in both repulsive and attractive cases qualitatively well reproduces corresponding behavior of charge stiffness.