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On the Hubbard Model in the Limit of Vanishing Interaction

Holger Frahm, Markus P. Pfannmüller

Published Jan 1, 1995• 2 citations
OpenAlex

Abstract

We address the question how a correspondence between the particle like excitations in the one dimensional Hubbard model (i.e. “holons ” and “spinons”) and the free fermionic picture can be estabilished in the limit of vanishing interaction by studying the finite size spectrum in the framework of the Bethe Ansatz. Special attention has to be paid to the case of a vanishing magnetic field where the two bands of excitations in either description are degenerate. The interaction lifts this degeneracy.

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We address the question how a correspondence between the particle like excitations in the one dimensional Hubbard model (i.e. “holons ” and “spinons”) and the free fermionic picture can be estabilished in the limit of vanishing interaction by studying the finite size spectrum in the framework of the Bethe Ansatz. Special attention has to be paid to the case of a vanishing magnetic field where the two bands of excitations in either description are degenerate. The interaction lifts this degeneracy.
The authors study the critical behaviour of the SU(N) generalization of the one-dimensional Hubbard model with arbitrary degeneracy N. Using the integrability of this model by Bethe ansatz they are able to compute the spectrum of the low-lying excitations in a large but finite box for arbitrary values of the electron density and of the Coulomb interaction. This information is used to determine the asymptotic behaviour of correlation functions at zero temperature in the presence of external fields lifting the degeneracy. The critical exponents depend on the system parameters through an N*N dressed charge matrix implying the relevance of the interaction of charge- and spin-density waves.
We study the critical behaviour of the SU(N) generalization of the one-dimensional Hubbard model with arbitrary degeneracy N. Using the integrability of this model by Bethe Ansatz we are able to compute the spectrum of the low-lying excitations in a large but finite box for arbitrary values of the electron density and of the Coulomb interaction. This information is used to determine the asymptotic behaviour of correlation functions at zero temperature in the presence of external fields lifting the degeneracy. The critical exponents depend on the system parameters through a N × N dressed charge matrix implying the relevance of the interaction of charge- and spin-density waves.
We use the Bethe ansatz solution for the one-dimensional Hubbard model with open boundary conditions and applied boundary fields to study the spectrum of bound states at the boundary. Depending on the strength of the boundary potentials, one finds that the true ground state contains a single charge or, for boundary potentials comparable with the Hubbard interaction, a pair of electrons in a bound state. If these are left unoccupied one finds holon and spinon bound states. We compute the finite size corrections to the low-lying energies in this system and use the predictions of boundary conformal field theory to study the exponents related to the orthogonality catastrophe.
We use the Bethe Ansatz solution for the one dimensional Hubbard model with open boundary conditions and applied boundary fields to study the spectrum of bound states at the boundary. Depending on the strength of the boundary potentials one finds that the true ground state contains a single charge or, for boundary potentials comparable to the Hubbard interaction, a pair of electrons in a bound state. If these are left unoccupied one finds holon and spinon bound states. We compute the finite size corrections to the low lying energies in this system and use the predictions of boundary conformal field theory to study the exponents related to the orthogonality catastrophe.
We use thermodynamic Bethe ansatz to study nonrelativistic scattering theory of low energy excitations of 1D Hubbard model, using the SSS-matrices proposed by Eßler and Korepin. This model can be described by two types of excitation states, holons and spinons, as asymptotic states. In the attractive regime, the spinon is massive while the holon is massless. The situation is reversed with a repulsive coupling. We derive that the central charge of the Hubbard model in the IR limit is c=1c=1c=1 while it vanishes in the UV limit. The contribution is due to the massless degree of freedom, i.e. the holon for the attractive regime, and the massive mode decouples completely. This result is consistent with various known results based on lattice Bethe ansatz computations. Our results make it possible to use the SSS-matrices of the excitations to compute more interesting quantities like correlation functions.
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.