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arXiv:0711.2697 · Physical Review Letters

Exact Exponents of Edge Singularities in Dynamic Correlation Functions of 1D Bose Gas

Adilet Imambekov, L. I. Glazman

Published May 21, 2008• 115 citations
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Abstract

The spectral function and dynamic structure factor of bosons interacting by contact repulsion and confined to one dimension exhibit power-law singularities along the dispersion curves of the collective modes. We find the corresponding exponents exactly, by relating them to the known Bethe ansatz solution of the Lieb-Liniger model. Remarkably, the Luttinger liquid theory predictions for the exponents fail even at low energies, once the immediate vicinities of the edges are considered.

DOI
10.1103/physrevlett.100.206805

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The spectral function and dynamic structure factor of bosons interacting by contact repulsion and confined to one dimension exhibit power-law singularities along the dispersion curves of the collective modes. We find the corresponding exponents exactly, by relating them to the known Bethe ansatz solution of the Lieb-Liniger model. Remarkably, the Luttinger liquid theory predictions for the exponents fail even at low energies, once the immediate vicinities of the edges are considered.
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.
Recently, Imambekov and Glazman [Phys. Rev. Lett. 100, 206805 (2008)] showed that the dynamic structure factor (DSF) of the 1D Bose gas demonstrates power-law behaviour along the limiting dispersion curve of the collective modes and calculated the corresponding exponents exactly. Combining these recent results with a previously obtained strong-coupling expansion we present an interpolation formula for the DSF of the 1D Bose gas. The obtained expression is further consistent with exact low energy exponents from Luttinger liquid theory and shows nice agreement with recent numerical results.
We discuss approximate formulas for the dynamic structure factor of the one-dimensional Bose gas in the Lieb-Liniger model that appear to be applicable over a wide range of the relevant parameters such as the interaction strength, frequency, and wave number. The suggested approximations are consistent with the exact results known in limiting cases. In particular, we encompass exact edge exponents as well as the Luttinger-liquid and perturbation theoretic results. We further discuss derived approximations for the static structure factor and the pair distribution function g(x)g(x)g(x). The approximate expressions show excellent agreement with numerical results based on the algebraic Bethe ansatz.
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.