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Condensed Matter

Exponents of Spectral Functions in the One-Dimensional Bose Gas

P. Schlottmann

Published Oct 23, 2018• 3 citations
Publisher

Abstract

The one-dimensional gas of bosons interacting via a repulsive contact potential was solved long ago via Bethe's ansatz by Lieb and Liniger [Phys. Rev. {\bf 130}, 1605 (1963)]. The low energy excitation spectrum is a Luttinger liquid parametrized by a conformal field theory with conformal charge $c=1$. For higher energy excitations the spectral function displays deviations from the Luttinger behavior arising from the curvature terms in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this problem. The ``impurity'' term is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-particle and one-hole hole Green's function correctly. We show that the exponents obtained via the finite size corrections to the ground state energy are identical to those obtained through the shift function.

DOI
10.3390/condmat3040035

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The one-dimensional gas of bosons interacting via a repulsive contact potential was solved long ago via Bethe's ansatz by Lieb and Liniger [Phys. Rev. {\bf 130}, 1605 (1963)]. The low energy excitation spectrum is a Luttinger liquid parametrized by a conformal field theory with conformal charge c=1c=1c=1. For higher energy excitations the spectral function displays deviations from the Luttinger behavior arising from the curvature terms in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this problem. The ``impurity'' term is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-particle and one-hole hole Green's function correctly. We show that the exponents obtained via the finite size corrections to the ground state energy are identical to those obtained through the shift function.
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.
Bariev's model for correlated hopping in one dimension is integrable and soluble within the framework of two nested Bethe Ansätze. For repulsive interaction and finite magnetization the low energy excitation spectrum is a two-component Luttinger liquid with conformal charges c=1. In zero magnetic field the spectrum of spin excitations is gapped. For higher energy excitations the single-hole (single-particle) spectral functions display edge singularities that deviate from those expected from a Luttinger liquid. The system can be described as a mobile impurity coupled to the Luttinger liquid yielding an effective x-ray edge type model. Using the Bethe Ansatz solution for repulsive interaction we obtain expressions for the critical exponents for the single-particle and single-hole Green's functions in zero magnetic field.
Conformal field theory and Bethe ansatz are used to investigate the low energy features of the spectral function in one dimensional models which exhibit a gap in the spin or in the charge excitation spectrum. Exotic behavior is found in the superconducting case, where the Green function displays momentum dependent Tomonaga-Luttinger liquid exponents. The predictions of the formalism are confirmed by Lanczos diagonalizations in the $t\ensuremath{-}J$ model up to 32 sites. These results may be relevant in connection to photoemission experiments in quasi-one-dimensional insulators or superconductors.
We consider the one-dimensional gas of fermions with spin S interacting via an attractive δ -function potential using the Bethe Ansatz solution. In zero magnetic field the atoms form bound states of N = 2 S + 1 fermions, i.e. generalized Cooper states with each atom having a different spin component. For low energy excitations the system is a Luttinger liquid and is properly described by a conformal field theory with conformal charge c = 1 . The linear dispersion of a Luttinger liquid is asymptotically exact in the low-energy limit where the band curvature terms in the dispersion are irrelevant. For higher energy excitations, however, the spectral function displays deviations in the neighborhood of the single-particle (hole) energy, which can be described by an effective X-ray edge type model. Using the Bethe Ansatz solution we obtain expressions for the critical exponents for the single-particle (hole) Green's function. This model can be relevant in the context of ultracold atoms with effective total spin S confined to an elongated optical trap.
A continuum of excitations in interacting one-dimensional systems is bounded from below by a spectral edge that marks the lowest possible excitation energy for a given momentum. We analyze short-range interactions between Fermi particles and between Bose particles (with and without spin) using Bethe-ansatz techniques and find that the dispersions of the corresponding spectral edge modes are close to a parabola in all cases. Based on this emergent phenomenon we propose an empirical model of a free, nonrelativistic particle with an effective mass identified at low energies as the bare electron mass renormalized by the dimensionless Luttinger parameter KKK (or ${K}_{\ensuremath{\sigma}}$ for particles with spin). The relevance of the Luttinger parameters beyond the low-energy limit provides a more robust method for extracting them experimentally using a much wider range of data from the bottom of the one-dimensional band to the Fermi energy. The empirical model of the spectral edge mode complements the mobile impurity model to give a description of the excitations in proximity of the edge at arbitrary momenta in terms of only the low-energy parameters and the bare electron mass. Within such a framework, for example, exponents of the spectral function are expressed explicitly in terms of only a few Luttinger parameters.