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Advanced series in mathematical physics

Spin excitations in a Hubbard chain

Ferenc Woynarovich

Published May 1, 1994• 0 citations
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Abstract

The spin excitations of a Hubbard chain are calculated on the basis of the Lieb-Wu equations (1968). A system of equations, which contains the parameters of the excitation only, is derived. It is found that both the energy and the momentum of the spin excitations depend only on the positions of the holes in the real gamma distribution. The energy-momentum dispersion for a half-filled band is given. The structure and degeneracy of the lowest energy excitations are also discussed.

DOI
10.1142/9789812798268_0006

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The spin excitations of a Hubbard chain are calculated on the basis of the Lieb-Wu equations (1968). A system of equations, which contains the parameters of the excitation only, is derived. It is found that both the energy and the momentum of the spin excitations depend only on the positions of the holes in the real gamma distribution. The energy-momentum dispersion for a half-filled band is given. The structure and degeneracy of the lowest energy excitations are also discussed.
Those excited states of a half-filled 1D Hubbard chain that are connected with electron pairs occupying the same site are studied. It is argued that these states should be described by solutions of the Lieb-Wu equations (1968) in which some of the wavenumbers are complex. Solutions of this type have been found that correspond to states in which the spin part is not excited. The energy-momentum dispersion is also calculated. The gap in the spectrum of the single excitations equals the discontinuity of the chemical potential calculated by Lieb and Wu.
The ground state of the one-dimensional Hubbard Hamiltonian is discussed on the basis of the Bethe ansatz solution of Lieb and Wu. A simple analytical representation of the distribution functions for charge and spin degrees of freedom is derived. The elementary excitations are associated with electron-hole pairs of two pseudo-fermion systems, one for the charge, the other for the magnetic excitations. Spinons and holons correspond to ground state configurations of odd-numbered rings. It is argued that holons will form bound pairs in weakly coupled chains.
The structure and energy of the eigenstates of a 1D Hubbard Hamiltonian with negative U is studied. Solutions for the Lieb-Wu equations (1968) corresponding to U 0. It is found that the ground state is continuous, while the ground-state energy, although continuous, is not analytic at the U=0 point. In the excited states, in contrast to the U>0 case, both kinds of excitation are connected with the charge distribution and the state of the spins does not affect the energy explicitly. It is also found that the distribution of parameters can change discontinuously in the excited states as U crosses zero.
We have extended calculations of the zero-temperature excitation spectrum of the one-dimensional Hubbard model to the case where the number of electrons is less than the number of sites in the chain. The results are computed as a function of the ratio Ut\frac{U}{t}tU​, where UUU represents the on-site Coulomb repulsion and ttt is the transfer integral, assumed to be nonzero only for nearest neighbors. Exact calculations are made for the energy and momentum of excitations having single-particle character. Unlike the situation for the half-filled band, we find no gap in the excitation spectrum. We have also considered excitations of the spin-wave type. These are shown to vary linearly with momentum for small momentum. The group velocity for small momentum is found to be inversely proportional to the magnetic susceptibility.