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arXiv:2306.06853 · New Journal of Physics

A hydrodynamical description of Bose liquid with fractional exclusion statistics

Zhaoyu Fei, Yu Chen

Published Feb 1, 2024• 2 citations
arXivHTMLPDF

Abstract

Abstract Hydrodynamical systems are usually taken as chaotic systems with fast relaxations. It is counter intuitive for ‘ideal’ gas to have a hydrodynamical description. We find that a hydrodynamical model of one-dimensional | Φ | 6 theory shares the same ground state density profile, density-wave excitation, as well as the similar dynamical and statistical properties with the Calogero–Sutherland model in thermodynamic limit when their interaction strengths matches each other. The interaction strength g 0 in | Φ | 6 theory is then the index of fractional statistics. Although the model is interacting in Bose liquid sense, but it shows integrability with periodical coherent evolution. We also discussed the fractional statistics emerges from the | Φ | 6 theory.

DOI
10.1088/1367-2630/ad2bae

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Abstract Hydrodynamical systems are usually taken as chaotic systems with fast relaxations. It is counter intuitive for ‘ideal’ gas to have a hydrodynamical description. We find that a hydrodynamical model of one-dimensional | Φ | 6 theory shares the same ground state density profile, density-wave excitation, as well as the similar dynamical and statistical properties with the Calogero–Sutherland model in thermodynamic limit when their interaction strengths matches each other. The interaction strength g 0 in | Φ | 6 theory is then the index of fractional statistics. Although the model is interacting in Bose liquid sense, but it shows integrability with periodical coherent evolution. We also discussed the fractional statistics emerges from the | Φ | 6 theory.
We discuss recent results on the relation between the strongly interacting one-dimensional Bose gas and a gas of ideal particles obeying nonmutual generalized exclusion statistics (GES). The thermodynamic properties considered include the statistical profiles, the specific heat and local pair correlations. In the strong coupling limit γ → ∞, the Tonks-Girardeau gas, the equivalence is with Fermi statistics. The deviation from Fermi statistics during boson fermionization for finite but large interaction strength γ is described by the relation α ≈ 1 - 2/γ, where α is a measure of the GES. This gives a quantitative description of the fermionization process. In this sense the recent experimental measurement of local pair correlations in a 1D Bose gas of87Rb atoms also provides a measure of the deviation of the GES parameter α away from the pure Fermi statistics value α=1. Other thermodynamic properties, such as the distribution profiles and the specific heat, are also sensitive to the statistics. They also thus provide a way of exploring fractional statistics in the strongly interacting 1D Bose gas.
I discuss ideal and interacting quantum gases obeying general fractional exclusion statistics. For systems with constant density of single-particle states, described in the mean field approximation, the entropy depends neither on the microscopic exclusion statistics, nor on the interaction. Such systems are called thermodynamically equivalent and I show that the microscopic reason for this equivalence is a one-to-one correspondence between the excited states of these systems. This provides a method, different from the bosonization technique, of transforming between systems of different exclusion statistics. In the last section the macroscopic aspects of this method are discussed. In appendix A, I calculate the fluctuation of the ground-state population of a condensed Bose gas in a grand-canonical ensemble and mean field approximation, while in appendix B I show a situation where although the system exhibits fractional exclusion properties on microscopic energy intervals, a rigorous calculation of the population of single-particle states reveals a condensation phenomenon. This also implies a malfunction of the usual and simplified calculation technique of the most probable statistical distributions.
The Calogero-Sutherland model represents a paradigmatic example of an integrable quantum system with applications ranging from cold atoms to random matrix theory. Combining sum rules with the Monte Carlo technique, we introduce a stochastic method that allows one to compute the dynamic structure factor and obtain an exact description of excitations beyond the conventional Luttinger liquid regime. We explore a broad range of interaction regimes, including weak interactions, where a Bogoliubov-type spectrum emerges, the Tonks-Girardeau regime, where excitations resemble those of an ideal Fermi gas, and strong interactions, where umklapp scattering leads to a Brillouin zone structure, typical of a crystal. Additionally, we discuss the connection between the hydrodynamic description of one-dimensional quantum gases, liquids, and solids with the Calogero-Sutherland wave function. The model's universality extends beyond atoms in waveguides, with implications for disordered systems and random matrix theory.
One-dimensional model of nonrelativistic particles with inverse-square interaction potential known as the Calogero-Sutherland model is shown to possess fractional statistics. Using the theory of Jack polynomials the exact dynamical density-density correlation function and the hole propagator part of one-particle Green's function at any rational interaction coupling constant are obtained and used to show clear evidences of the fractional exclusion statistics in the sense of Haldane's "generalized Pauli exclusion principle." This model is also endowed with the corresponding natural exchange statistics.