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Europhysics Letters (EPL)

Asymptotic Bethe-Ansatz Results for a Hubbard Chain with 1/sinh-Hopping

P.-A. Bares, Florian Gebhard

Published Mar 1, 1995• 10 citations
Publisher

Abstract

We investigate spin-(1/2) electrons with local Hubbard interaction and variable-range hopping amplitudes which decay like sinh (κ)/sinh (κ r ). Assuming integrability, the asymptotic-Bethe-ansatz approach allows us to derive the generalized Lieb-Wu integral equations from the two-particle phase shift. Due to the nesting property there is a metal-to-insulator transition at U c (κ > 0) = 0 + . The charge gap in the singular limit κ = 0 opens when the interaction strength equals the bandwidth, U c (κ = 0) = W > 0.

DOI
10.1209/0295-5075/29/7/011

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Asymptotic Bethe-Ansatz results for a Hubbard Chain with 1/sinh-Hopping

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We investigate spin-(1/2) electrons with local Hubbard interaction and variable-range hopping amplitudes which decay like sinh (κ)/sinh (κ r ). Assuming integrability, the asymptotic-Bethe-ansatz approach allows us to derive the generalized Lieb-Wu integral equations from the two-particle phase shift. Due to the nesting property there is a metal-to-insulator transition at U c (κ > 0) = 0 + . The charge gap in the singular limit κ = 0 opens when the interaction strength equals the bandwidth, U c (κ = 0) = W > 0.
We investigate spin-1/2 electrons with local Hubbard interaction and variable range hopping amplitudes which decay like~sinh⁡(κ)/sinh⁡(κr)\sinh(κ)/\sinh(κr)sinh(κ)/sinh(κr). Assuming integrability the Asymptotic Bethe Ansatz approach allows us to derive the generalized Lieb-Wu integral equations from the two-particle phase shift. Due to the nesting property there is a metal-to-insulator transition at Uc(κ>0)=0+U_c(κ>0)=0^+Uc​(κ>0)=0+. The charge gap in the singular limit κ=0κ=0κ=0 opens when the interaction strength equals the bandwidth, Uc(κ=0)=W>0U_c(κ=0)=W >0Uc​(κ=0)=W>0.
In this paper we study the exact solution of a one-dimensional model of spin-12 electrons composed by a nearest-neighbor triplet pairing term and the on-site Hubbard interaction. We argue that this model admits a Bethe ansatz solution through a mapping to a Hubbard chain with imaginary kinetic hopping terms. The Bethe equations are similar to that found by Lieb and Wu [3] but with additional twist phases which are dependent on the ring size. We have studied the spectrum of the model with repulsive interaction by exact diagonalization and through the Bethe equations for large lattice sizes. One feature of the model is that it is possible to define the charge gap for even and odd lattice sites and both converge to the same value in the infinite size limit. We analyze the finite-size corrections to the low-lying spin excitations and argue that they are equivalent to that of the spin-12 isotropic Heisenberg model with a boundary twist depending on the lattice parity. We present the classical statistical mechanics model whose transfer matrix commutes with the model Hamiltonian. To this end we have used the construction employed by Shastry [6], [7] for the Hubbard model. In our case, however, the building block is a free-fermion eight-vertex model with a particular null weight.
The gap equation at zero temperature for the half-filled Hubbard chain obtained by Lieb and Wu is rederived with use of the formalism of Yang and Yang and then the results are extended to the case of finite temperatures. For the values of the coupling constant c in the range 0<c\ensuremath{\le}1.5 the gap \ensuremath{\Delta}(T) exhibits BCS-like behavior \ensuremath{\Delta}(T)=${\ensuremath{\Delta}}_{0}$(1-T/Tc{T}_{c}Tc​)1/2{)}^{1/2})1/2 while for the larger values of c, the data are well approximated by the equation \ensuremath{\Delta}(T)=${\ensuremath{\Delta}}_{0}$(1-T/Tc{T}_{c}Tc​) with ${\ensuremath{\Delta}}_{0}$ being the zero-temperature result of Lieb and Wu for the gap and Tc{T}_{c}Tc​ some critical transition temperature.
We discuss the transition from a metal to charge- or spin-insulating phases characterized by the opening of a gap in the charge- or spin-excitation spectra, respectively. These transitions are addressed within the context of two exactly solvable Hubbard and t-J chains with long-range, 1/r hopping. We discuss the specific heat, compressibility, and magnetic susceptibility of these models as a function of temperature, the band filling, and the interaction strength. We then use conformal-field-theory techniques to extract ground-state correlation functions. Finally, by employing the g-ology analysis we show that the charge-insulator transition is accompanied by an infinite discontinuity in the Drude weight of the electrical conductivity. While the magnetic properties of these models reflect the genuine features of strongly correlated electron systems, the charge-transport properties, especially near the Mott-Hubbard transition, display a nongeneric behavior.
We consider the SU(N) generalization of the one-dimensional Hubbard model with arbitrary degeneracy N (spin and orbital degrees of freedom). This model is integrable and has several unusual properties at low temperatures. The Bethe-Ansatz equations at T=0 are analyzed in the thermodynamic limit in the absence of external fields. In the continuum limit, the effective interaction between the charge degrees of freedom corresponds to a potential of the form [sinh(ax)${]}^{\mathrm{\ensuremath{-}}2}$, where x is the distance between the particles involved and a is an inverse length scale. In the limit N\ensuremath{\rightarrow}\ensuremath{\infty} and in the continuum limit, the charges reduce to a Bose gas interacting via a \ensuremath{\delta}-function potential. We further address here the properties of the charge degrees of freedom for a band filling close to one electron per site. The charge excitations obey Fermi statistics. We find a Mott metal-insulator transition at a critical value Uc{\mathit{U}}_{\mathit{c}}Uc​ of the Coulomb repulsion. Uc{\mathit{U}}_{\mathit{c}}Uc​ depends on N (Uc{\mathit{U}}_{\mathit{c}}Uc​=0 for N=2). A qualitative change in the charge-rapidity distribution is found at Uc{\mathit{U}}_{\mathit{c}}Uc​. The Fermi velocity is finite for UUc{\mathit{U}}_{\mathit{c}}Uc​, diverges as U\ensuremath{\rightarrow}Uc{\mathit{U}}_{\mathit{c}}Uc​, and vanishes for U>Uc{\mathit{U}}_{\mathit{c}}Uc​.