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Journal of the Physical Society of Japan

Bethe-Ansatz Analysis of the Extended AB Period in the One-Dimensional Hubbard Model

Koichi Kusakabe

Published Jul 15, 1997• 4 citations
Publisher

Abstract

The extended spectral flow at the normal-superconducting transition is investigated for the one-dimensional Hubbard model. Recombination of spectral flow lines at U =±0, which causes halving of the extended Aharonov-Bohm period for U <0 found in a recent paper [K. Kusakabe and H. Aoki: J. Phys. Soc. Jpn. 65 (1996) 2772], is exactly analyzed using the Bethe ansatz solution. Adiabatic motion of charge rapidities which are real for U >0 (normal Luttinger liquid) or complex for U <0 (superconducting Luther-Emery liquid) dominates sudden response of the spectral flow at the transition point. The extended period is obtained not only against the charge flux but also against the spin flux for both U >0 and U <0.

DOI
10.1143/jpsj.66.2075

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The extended spectral flow at the normal-superconducting transition is investigated for the one-dimensional Hubbard model. Recombination of spectral flow lines at U =±0, which causes halving of the extended Aharonov-Bohm period for U <0 found in a recent paper [K. Kusakabe and H. Aoki: J. Phys. Soc. Jpn. 65 (1996) 2772], is exactly analyzed using the Bethe ansatz solution. Adiabatic motion of charge rapidities which are real for U >0 (normal Luttinger liquid) or complex for U <0 (superconducting Luther-Emery liquid) dominates sudden response of the spectral flow at the transition point. The extended period is obtained not only against the charge flux but also against the spin flux for both U >0 and U <0.
The `extended Aharonov-Bohm (AB) period' recently proposed by Kusakabe and Aoki [J. Phys. Soc. Jpn (65), 2772 (1996)] is extensively studied numerically for finite size systems of strongly correlated electrons. While the extended AB period is the system length times the flux quantum for noninteracting systems, we have found the existence of the boundary across which the period is halved or another boundary into an even shorter period on the phase diagram for these models. If we compare this result with the phase diagram predicted from the Tomonaga-Luttinger theory, devised for low-energy physics, the halved period (or shorter periods) has a one-to-one correspondence to the existence of the pairing (phase separation or metal-insulator transition) in these models. We have also found for the t-J model that the extended AB period does not change across the integrable-nonintegrable boundary despite the totally different level statistics.
We investigated the one-dimensional extended Hubbard model with the on-site interaction U and the nearest-neighbor interaction V by using the numerical diagonalization method. Assuming the Luttinger liquid relations, we obtained the phase diagram of the ground state on the U - V plane. In the weak coupling region, the phase diagram agrees with the result of g -ology. On the other hand, in the strong coupling region for V ≫∣ t ∣, a novel superconducting phase appears both for positive and negative U . We also investigated the charge gap and the spin gap. The results suggest that the superconducting phase extends over the Tomonaga-Luttinger region (without spin gap) and the Luther-Emery region (with spin gap).
Motivated by Heilmann and Lieb's work [Ann. N. V. Acad. Sci. 172, 583 (1971)]. We discuss energy level crossings for the one-dimensional Hubbard model through the Bethe ansatz, constructing explicitly the degenerate eigenstates at the crossing points. After showing the existence of solutions for the Lieb--Wu equations with one-down spin, we solve them numerically and construct Bethe-ansatz eigenstates. We thus verify all the level crossings in the spectral flows observed by the numerical diagonalization method with one down-spin. For each of the solutions we obtain its energy spectral flow along the interaction parameter U. Then, we observe that some of the energy level crossings can not be explained in terms of U-independent symmetries. Dynamical symmetries of the Hubbard model are fundamental for identifying each of the spectral lines at the level crossing points. We show that the Bethe-ansatz eigenstates which degenerate at the points have distinct sets of eigenvalues of the higher conserved operators. We also show a twofold permanent degeneracy in terms of the Bethe-ansatz wave function.
The single-particle spectral properties near the Mott transition in the one-dimensional Hubbard model are investigated by using the dynamical density-matrix renormalization group method and the Bethe ansatz. The pseudogap, hole-pocket behavior, spectral-weight transfer, and upper Hubbard band are explained in terms of spinons, holons, antiholons, and doublons. The Mott transition is characterized by the emergence of a gapless mode whose dispersion relation extends up to the order of hopping t (spin exchange J) in the weak (strong) interaction regime caused by infinitesimal doping.
Inspired from Sutherland's work [Phys. Rev. Lett. {\bf 74}, 816 (1995)] on detecting bound spin waves, we propose that bound electron states can be detected from the dependence of interacting electron systems to the Aharonov-Bohm flux in the `extended zone' scheme, where the electron pairing halves the original period NaN_aNa​ flux quanta in a system of linear size NaN_aNa​. Along with the Bethe-ansatz analysis, a numerical implementation for keeping track of the adiabatic flow of energy levels is applied to the attractive/repulsive Hubbard models and the t−Jt-Jt−J ladder.
The authors introduce a new representation of the Bethe ansatz solutions for 1D Luttinger liquids which describes the spectral properties and asymptotic behaviour of the correlation functions of the 1D Hubbard model in terms of the renormalized interaction of charge and spin pseudoparticles. The study of the low-lying eigenstates is reduced to the familiar languages of band theory and of the Fermi liquid. The authors results provide a better understanding of the decoupling of charge and spin degrees of freedom in 1D interacting systems, generalize the concept of a Landau liquid and may be relevant to the physics of higher dimensional systems.