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Physical review. B, Condensed matter

One-dimensional backward-scattering fermion model with built-in cutoff

S. Rudin

Published Oct 15, 1983• 7 citations
Publisher

Abstract

A built-in cutoff for a fermion model in one dimension is proposed, consisting of (1) a cubic term in the kinetic energy and (2) an "optical potential" counterterm. This model is shown to be soluble by the Bethe ansatz and the mass gap is obtained as a function of coupling constant. This functional dependence reproduces a result obtained by methods other than the Bethe ansatz.

DOI
10.1103/physrevb.28.4825

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A built-in cutoff for a fermion model in one dimension is proposed, consisting of (1) a cubic term in the kinetic energy and (2) an "optical potential" counterterm. This model is shown to be soluble by the Bethe ansatz and the mass gap is obtained as a function of coupling constant. This functional dependence reproduces a result obtained by methods other than the Bethe ansatz.
We present a variational solution of the sine-Gordon theory, and at the same time the Luther-Emery backward-scattering model, for the range of coupling constants $0<\frac{{\ensuremath{\beta}}^{2}}{4\ensuremath{\pi}}\ensuremath{\le}1$ or $\ensuremath{-}1<\frac{{U}_{\ensuremath{\parallel}}}{2\ensuremath{\pi}{v}_{F}}<~\frac{\ensuremath{-}3}{5}$, respectively. This yields a renormalized mass for the fundamental fermion of the massive Thirring model ${m}_{f}=(\frac{\ensuremath{\pi}}{\sqrt{c}})(\frac{1}{{\ensuremath{\beta}}^{2}}){(\frac{{\ensuremath{\alpha}}_{0}}{{\ensuremath{\Lambda}}^{\frac{{\ensuremath{\beta}}^{2}}{4\ensuremath{\pi}}}})}^{\ensuremath{\nu}}$ and an energy gap $\ensuremath{\Delta}={v}_{F}{c}^{\ensuremath{\nu}\ensuremath{-}1}{[4+{(\frac{{\ensuremath{\beta}}^{2}}{4\ensuremath{\pi}})}^{2}]}^{\ensuremath{\nu}\ensuremath{-}1}(\frac{1}{{\ensuremath{\alpha}}_{L}}){(\frac{{U}_{\ensuremath{\perp}}}{2\ensuremath{\pi}{v}_{F}})}^{\ensuremath{\nu}}$ in the fermion spectrum for the backward-scattering model where $\ensuremath{\nu}=\frac{1}{(2\ensuremath{-}\frac{{\ensuremath{\beta}}^{2}}{4\ensuremath{\pi}})}$ and $\frac{{\ensuremath{\beta}}^{2}}{8\ensuremath{\pi}}={[\frac{(1+\frac{{U}_{\ensuremath{\parallel}}}{2\ensuremath{\pi}{v}_{F}})}{(1\ensuremath{-}\frac{{U}_{\ensuremath{\parallel}}}{2\ensuremath{\pi}{v}_{F}})}]}^{\frac{1}{2}}$.
A one-dimensional gas mixture consisting of bosons and fermions without spin interacting via a repulsive [Formula: see text]-function potential is considered. The model is integrable and soluble via two nested Bethe ansatz, if all particles are assumed to have equal masses and the interaction strength between the bosons and among the bosons and fermions is the same. The low energy excitation spectrum is a two-component Luttinger liquid and can be parametrized by a conformal field theory with conformal charges c = 1. In the low-energy limit, where the band curvature terms in the dispersion can be neglected, the linear dispersion of a Luttinger liquid is asymptotically exact. The spectral function, however, displays deviations from the Luttinger behavior for higher energy excitations. In the neighborhood of the single-particle (hole) energy, the spectral function is represented by an effective X-ray edge type model. Expressions of the critical exponents for the single-hole Green’s function are obtained using the Bethe ansatz solution in the limit of the bosonic gas. The results could be of relevance in the context of ultracold atoms confined to an elongated optical trap.
We show that the one-dimensional Luttinger model generalized to include spin and backward scattering is equivalent to a two-dimensional Coulomb gas. Scaling equations are derived and correlation functions are given simple physical interpretation in terms of the Coulomb gas; e.g., existence of an energy gap can be understood in terms of Debye screening. We conclude that an energy gap exists for ${U}_{\ensuremath{\parallel}}<|{U}_{\ensuremath{\perp}}|$ so that triplet excitations are nondivergent, and we provide physical arguments to support the exponents proposed by Luther and Emery for singlet excitations for general coupling constant.
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.