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arXiv:1712.05262 · SciPost Physics

The Inhomogeneous Gaussian Free Field, with application to ground state correlations of trapped 1d Bose gases

Yannis Brun, Jérôme Dubail

Published Jun 25, 2018• 69 citations
arXivHTMLPDF

Abstract

This formalism is then applied to the study of ground state correlations of the Lieb-Liniger gas trapped in an external potential V(x) V ( x ) . Relations with previous works on inhomogeneous Luttinger liquids are discussed. The main innovation here is in the identification of local observables \hat{O} (x) O ̂ ( x ) in the microscopic model with their field theory counterparts \partial_x h, e^{i h(x)}, e^{-i h(x)} ∂ x h , e i h ( x ) , e − i h ( x ) , etc., which involve non-universal coefficients that themselves depend on position — a fact that, to the best of our knowledge, was overlooked in previous works on correlation functions of inhomogeneous Luttinger liquids —, and that can be calculated thanks to Bethe Ansatz form factors formulae available for the homogeneous Lieb-Liniger model. Combining those position-dependent coefficients with the correlation functions of the IGFF, ground state correlation functions of the trapped gas are obtained. Numerical checks from DMRG are provided for density-density correlations and for the one-particle density matrix, showing excellent agreement.

DOI
10.21468/scipostphys.4.6.037

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This formalism is then applied to the study of ground state correlations of the Lieb-Liniger gas trapped in an external potential V(x) V ( x ) . Relations with previous works on inhomogeneous Luttinger liquids are discussed. The main innovation here is in the identification of local observables \hat{O} (x) O ̂ ( x ) in the microscopic model with their field theory counterparts \partial_x h, e^{i h(x)}, e^{-i h(x)} ∂ x h , e i h ( x ) , e − i h ( x ) , etc., which involve non-universal coefficients that themselves depend on position — a fact that, to the best of our knowledge, was overlooked in previous works on correlation functions of inhomogeneous Luttinger liquids —, and that can be calculated thanks to Bethe Ansatz form factors formulae available for the homogeneous Lieb-Liniger model. Combining those position-dependent coefficients with the correlation functions of the IGFF, ground state correlation functions of the trapped gas are obtained. Numerical checks from DMRG are provided for density-density correlations and for the one-particle density matrix, showing excellent agreement.
Abstract We provide a thorough characterisation of the zero-temperature one-particle density matrix of trapped interacting anyonic gases in one dimension, exploiting recent advances in the field theory description of spatially inhomogeneous quantum systems. We first revisit homogeneous anyonic gases with point-wise interactions. In the harmonic Luttinger liquid expansion of the one-particle density matrix for finite interaction strength, the non-universal field amplitudes were not yet known. We extract them from the Bethe Ansatz formula for the field form factors, providing an exact asymptotic expansion of this correlation function, thus extending the available results in the Tonks–Girardeau limit. Next, we analyse trapped gases with non-trivial density profiles. By applying recent analytic and numerical techniques for inhomogeneous Luttinger liquids, we provide exact expansions for the one-particle density matrix. We present our results for different confining potentials, highlighting the main differences with respect to bosonic gases.
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.
Ultracold gases are a versatile platform to simulate condensed matter physics, as virtually any parameter is experimentally tunable. In particular, highly anisotropic traps allow the realization of low-dimensional systems, where the role of quantum fluctuations is enhanced. I investigate the paradigmatic model of a one-dimensional Bose gas with contact interactions, a.k.a the Lieb-Liniger model, using powerful analytical tools such as Bethe Ansatz, Conformal Field Theory and the Tomonaga-Luttinger liquid formalism, as well as the Bose-Fermi mapping. The effect of an additional harmonic trap in the longitudinal direction is studied within the Local Density Approximation. These methods allow to investigate with high accuracy its ground-state thermodynamics of the gas, its excitation spectra and correlation functions, from the k-body local correlations to the momentum distribution and the dynamical structure factor. These quantities characterize the coherence and superfluidity of the gas. In particular, I study the drag force exerted by a weak gaussian barrier stirred into the gas within linear response theory. Then, I investigate the appearance of a multimode structure obtained by releasing a transverse trapping, and the dimensional crossover in the limit of an infinite number of transverse modes.
The physics of the attractive one-dimensional Bose gas (Lieb–Liniger model) is investigated with techniques based on the integrability of the system. Combining a knowledge of particle quasi-momenta to exponential precision in the system size with determinant representations of matrix elements of local operators coming from the algebraic Bethe ansatz, we obtain rather general analytical results for the zero-temperature dynamical correlation functions of the density and field operators. Our results thus provide quantitative predictions for possible future experiments in atomic gases or optical waveguides.
Abstract. The physics of the attractive one-dimensional Bose gas (Lieb-Liniger model) is investigated with techniques based on the integrability of the system. Combining a knowledge of particle quasi-momenta to exponential precision in the system size with determinant representations of matrix elements of local operators coming from the Algebraic Bethe Ansatz, we obtain rather general analytical results for the zero-temperature dynamical correlation functions of the density and field operators. Our results thus provide quantitative predictions for possible future experiments in atomic gases or optical waveguides. Dynamics of the attractive 1D Bose gas: analytical treatment from integrability 2
Ultracold bosonic and fermionic quantum gases confined to quasi-one-dimensional (1D) geometry are promising candidates for probing fundamental concepts of Luttinger liquid (LL) physics. They can also be exploited for devising applications in quantum information processing and precision measurements. Here, we focus on 1D dipolar Bose gases, where evidence of super-strong coupling behavior has been demonstrated by analyzing the low-energy static and dynamical structures of the fluid at zero temperature by a combined reptation quantum Monte Carlo (RQMC) and bosonization approach. Fingerprints of LL behavior emerge in the whole crossover from the already strongly interacting Tonks–Girardeau at low density to a dipolar density wave regime at high density. We have also shown that a LL framework can be effectively set up and utilized to describe this strongly correlated crossover physics in the case of confined 1D geometries after using the results for the homogeneous system in LL hydrodynamic equations within a local density approximation. This leads to the prediction of observable quantities such as the frequencies of the collective modes of the trapped dipolar gas under the more realistic conditions that could be found in ongoing experiments. The present paper provides a description of the theoretical framework in which the above results have been worked out, making available all the detailed derivations of the hydrodynamic Luttinger equations for the inhomogeneous trapped gas and of the correlation functions for the homogeneous system.