Polylogs, thermodynamics and scaling functions of one-dimensional quantum many-body systems

TL;DR

This paper introduces an analytic method using polylog functions to calculate the thermodynamics of one-dimensional Lieb-Liniger bosons, simplifying the process and enabling exploration of quantum criticality and Tomonaga-Luttinger liquid physics.

Contribution

It presents a novel analytic approach with polylog functions for thermodynamics of 1D quantum systems, avoiding numerical solutions of the Bethe ansatz equations.

Findings

Analytic expressions for the equation of state and scaling functions.

Mapping of quantum criticality in Lieb-Liniger bosons.

Application of the method to various Bethe ansatz integrable systems.

Abstract

We demonstrate that the thermodynamics of one-dimensional Lieb-Liniger bosons can be accurately calculated in analytic fashion using the polylog function in the framework of the thermodynamic Bethe ansatz. The approach does away with the need to numerically solve the thermodynamic Bethe ansatz (Yang-Yang) equation. The expression for the equation of state allows the exploration of Tomonaga-Luttinger liquid physics and quantum criticality in an archetypical quantum system. In particular, the low-temperature phase diagram is obtained, along with the scaling functions for the density and compressibility. It has been shown recently by Guan and Ho (arXiv:1010.1301) that such scaling can be used to map out the criticality of ultracold fermionic atoms in experiments. We show here how to map out quantum criticality for Lieb-Liniger bosons. More generally the polylog function formalism can be applied to a wide range of Bethe ansatz integrable quantum many-body systems which are currently of theoretical and experimental interest, such as strongly interacting multi-component fermions, spinor bosons and mixtures of bosons and fermions.