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Physical review. B, Condensed matter

One-dimensional fermions with δ-function repulsion in the Brueckner theory

Florin D. Buzatu

Published Jan 15, 1997• 1 citation
Publisher

Abstract

The single-particle energies for a one-dimensional system of interacting fermions via a repulsive \ensuremath{\delta}-function potential are self-consistently calculated using the Brueckner and Gammel method. The breakdown of the usual Fermi liquid picture is signaled in this approach by the occurrence of a negative gap at \ifmmode\pm\else\textpm\fi{}${\mathrm{k}}_{\mathrm{F}}$ between the lowest energy of a particle state (|k|>${\mathrm{k}}_{\mathrm{F}}$) and the highest energy of a hole state (|k|${\mathrm{k}}_{\mathrm{F}}$); a normal ground-state configuration with all the hole states double occupied is thus unstable. Two possible pictures getting account of the rearrangement of the particles in a stable configuration are discussed: (1) an effective potential simulating a normal Fermi liquid behavior; (2) a condensate of fermions and a reduced Fermi sea normally occupied. The ground-state energy computed within these schemes is in remarkable agreement with the Bethe-Ansatz results for all values of the model parameters.

DOI
10.1103/physrevb.55.2114

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The single-particle energies for a one-dimensional system of interacting fermions via a repulsive \ensuremath{\delta}-function potential are self-consistently calculated using the Brueckner and Gammel method. The breakdown of the usual Fermi liquid picture is signaled in this approach by the occurrence of a negative gap at \ifmmode\pm\else\textpm\fi{}kF{\mathrm{k}}_{\mathrm{F}}kF​ between the lowest energy of a particle state (|k|>kF{\mathrm{k}}_{\mathrm{F}}kF​) and the highest energy of a hole state (|k|kF{\mathrm{k}}_{\mathrm{F}}kF​); a normal ground-state configuration with all the hole states double occupied is thus unstable. Two possible pictures getting account of the rearrangement of the particles in a stable configuration are discussed: (1) an effective potential simulating a normal Fermi liquid behavior; (2) a condensate of fermions and a reduced Fermi sea normally occupied. The ground-state energy computed within these schemes is in remarkable agreement with the Bethe-Ansatz results for all values of the model parameters.
The ground-state energy of a one-dimensional fermion system interacting via a repulsive $\ensuremath{\delta}$-function potential is calculated rigorously within the "ladder" approximation as a function of the coupling constant and the results compared with those given by the exact solution of Yang. Comparison with other many-body approximate theories has also been made. The density-density response function has also been calculated in an approximation in which the particle-hole interaction in the polarization bubble is taken into account to all orders. Results of this dynamic theory are compared with those of other approximations.
We consider a gas of fermions with parabolic dispersion and N spin components (or spin S, N=2S+1) with SU(N) symmetry in one dimension interacting via a delta -function potential. The model is integrable and its solution has been obtained by Sutherland in terms of N nested Bethe ansatze. The ground-state Bethe ansatz integral equations are solved numerically for both repulsive and attractive interactions to obtain the energy, the chemical potential, and the magnetic susceptibility as a function of the band filling and the interaction strength. For the repulsive interaction the Fermi gas has the properties of a Luttinger liquid. In the attractive case, on the other hand, the fermions in the ground-state form bound states of up to N fermions of different spin components. The spectrum of elemental charge and spin excitations is derived for the repulsive and attractive situations. The spectrum is discussed in the limits of vanishing interaction strength and very strong coupling. For the repulsive interaction the low-lying charge excitations can be characterized by the Fermi momentum and the Fermi velocity. The range of the spin-wave excitations is correlated with the Fermi momentum of the charges. The spin wave velocity is inversely proportional to the magnetic susceptibility. The spin-wave excitations become soft in the infinite-repulsive-coupling limit. In the attractive case in zero field all excitation branches except that of bound states of N fermions have an energy gap. It requires a finite energy to break these bound states and hence there is no response to a field smaller than a critical field. The low-T specific heat is proportional to the temperature.
One parametric one-dimensional spinless fermion models is formulated and solved exactly by the Bethe ansatz method. The Hamiltonian contains the kinetic terms describing nearest-neighbor hopping of electrons with different hopping integrals for many particle configurations. The Luttinger liquid behavior is studied for the case repulsion effective interactions. The ground state energy, the critical exponent, describing the asymptotic behavior of the density-density correlation function at long distances, the Fermi velocity have been calculated numerically for an arbitrary density of electrons. The model displays the condensation of particles in the case of the attractive interaction.
The properties of a one-dimensional system of bosons interacting via a repulsive delta-function potential are studied here using a modified Bogolyubov method. The motivation for the modification is twofold: First, to cure an internal inconsistency that predicts a single-particle ground-state occupancy of minus infinity when the Bogolyubov scheme is applied to this model. Second, to abandon the initial assumption---at odds with the exact result---of a macroscopically occupied single-particle ground-state. Whereas the Bogolyubov scheme uses the lowest-energy-single-particle state as a reservoir for the other states, our method uses, for each single-particle state, all single-particle states of lower energy collectively as a reservoir. This not only removes the inconsistency but also allows us to estimate the ground-state energy in much better agreement with the exact result.
One parametric one-dimensional spinless fermion model is formulated and solved exactly by the Bethe ansatz method. The Hamiltonian contains the kinetic terms describing nearest-neighbor hopping of electrons with different hopping integrals for many-particle configurations. The Luttinger liquid behavior is studied for the case repulsion effective interaction. The ground-state energy, the critical exponent, describing the asymptotic behavior of the density-density correlation function at long distances, the Fermi velocity have been calculated numerically for an arbitrary density of electrons. The model displays the condensation of particles in the case of the attractive interaction.
The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.