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Journal of Physics A Mathematical and General

Massive particles in the relativistic limit of the non-half-filled 1D attractive Hubbard model

Ferenc Woynarovich

Published Jan 21, 1996• 3 citations
Publisher

Abstract

The continuum limit of the Bethe ansatz solution for an attractive Hubbard chain is considered in which the particle number per site n is kept finite (0<n<1). It is shown that by adjusting the hopping t, the Hubbard U and the lattice constant a in a proper way the excitations connected to unbound electrons will be relativistic with a finite mass. The higher level Bethe ansatz equations for these particles are given. The spectrum and the phase shifts of these particles show close analogy to those of the massive particles of the SU(2) chiral invariant Gross - Neveu model.

DOI
10.1088/0305-4470/29/2/003

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The continuum limit of the Bethe ansatz solution for an attractive Hubbard chain is considered in which the particle number per site n is kept finite (0<n<1). It is shown that by adjusting the hopping t, the Hubbard U and the lattice constant a in a proper way the excitations connected to unbound electrons will be relativistic with a finite mass. The higher level Bethe ansatz equations for these particles are given. The spectrum and the phase shifts of these particles show close analogy to those of the massive particles of the SU(2) chiral invariant Gross - Neveu model.
In this work, we present a proof of the existence of real and ordered solutions to the generalized Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any positive U to the limit of large interaction is also shown. This continuity property, when combined with the proof that the wavefunction obtained with the generalized Bethe Ansatz is normalizable, is relevant to the question of whether or not the solution gives us the ground state of the finite system, as suggested by Lieb and Wu. Lastly, for the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the well known solution of the 1D Hubbard model.
We study the phase diagram at finite temperature of a system of Fermi particles on the sites of the Bethe lattice with coordination number z and interacting through onsite U and nearest-neighbor V interactions. This is a physical realization of the extended Hubbard model in the atomic limit. By using the equations of motion method, we exactly solve the model. For an attractive intersite potential, we find, at half filling, a phase transition towards a broken particle-hole symmetry state. The critical temperature, as a function of the relevant parameters, has a re-entrant behavior as already observed in the equivalent spin-1 Ising model on the Bethe lattice.
Abstract. In this work, we present a proof of the existence of real and ordered solutions to the generalized Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any U> 0 to U = ∞ is also shown. This continuity property, when combined with the proof that the wavefunction obtained with the generalized Bethe Ansatz is normalizable, is relevant to the question of whether or not the solution gives us the ground state of the finite system, as suggested by Lieb and Wu. Lastly, for the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the well known solution of the 1D Hubbard model. PACS numbers: 71.10.Fd, 71.10.Hf, 71.30.+h. 1.
The thermodynamic Bethe-ansatz equations for the one-dimensional Hubbard model are derived for an attractive on-site interaction of the electrons, extending in this way Takahashi's result for the repulsive Hubbard chain. The results are discussed in several special limits, e.g., interaction much larger than the hopping, U\ensuremath{\rightarrow}0 case, and the high- and low-temperature limits. The dominating states at low temperatures consist of spin-paired electrons (resembling Cooper pairs) and bound states of such pairs. The excitation spectrum and its properties are discussed in the context of superconductivity and resonant valence bonds.
We consider the charge excitations of the half-filled Hubbard chain, which are related to electron pairs occupying the same lattice site. In the exact Bethe Ansatzeigen function, we separate the centre-off-mass motion by introducing the centre off-mass and relative coordinates and wave vectors. These excitations are then interpreted as interacting quasi-particles propagating as plane waves along the chain. The two-body problem is solved for these quasi-particles. The solution suggests the existence of bound pairs, namely particle-hole excitonic states. A connection to the magnon states of the ferromagnetic Heisenberg chain is also discussed.
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.