Two-spinon effects on the thermal Tonks-Girardeau gas

TL;DR

The paper addresses the finite-temperature one-body correlator $\mathcal{G}(x,t)$ of the Tonks-Girardeau gas (the infinitely repulsive Lieb-Liniger model). It develops a two-spinon excitation framework on top of a representative thermal state to efficiently expand the spectral sum, showing two-spinon excitations dominate at low $T$. Analytically, adding particle-hole excitations on top of two-spinon excitations further reduces the form-factor weights, aiding convergence; numerically, both the Fredholm determinant and the two-spinon sum converge at low $T$. The results indicate that a small set of excitations captures the essential finite-temperature correlation physics in 1D integrable Bose gases, offering a practical route to accurate predictions.

Abstract

We study the effects of the two-spinon excitations on the field-field correlator of the Tonks-Girardeau gas. While these excitations have been previously examined in the ground state of the system, their role at finite temperatures remains unexplored. Here, we extend the analysis to the one-dimensional interacting Bose gas at thermal equilibrium, focusing on the one-body correlation function of the infinitely repulsive Lieb-Liniger model. We demonstrate that two-spinon excitations, characterized by two holes within the rapidity distribution, constitute the dominant contribution to the field-field correlator at low temperatures. Furthermore, we analytically show that incorporating additional particle-hole excitations diminishes their contribution, highlighting the efficacy of the two-spinon framework in capturing the essential physics of the system. Numerical evaluations of both the Fredholm determinant and the spectral sum stemming from the two-spinon program, with the addition of particle-hole excitations, reveal convergence at low temperatures.

Figures (4)

• Figure 1: On the left-hand side we have the thermal ($\beta=1$) rapidities distribution in red, following the discretisation \ref{['eq.discretisation']} for $N=200$ particles measured by the distribution $\rho(\lambda)$ in blue. On the right-hand size, the correlator \ref{['eq.correlator']} evaluated at $t=0$ both in terms of the Fredholm determinant representation from \ref{['eq.Fredholm']} and by means of the two-spinon program from \ref{['eq.2sp']} with two particle-hole excitations on top, for two inverse temperatures $\beta=1,10$.
• Figure 2: Prefactor scaling as $\Omega \sim e^{-\xi(\beta) N}$. On the left-hand side, we see the relation between the prefactor and the number of particles in log scale displaying its exponential decay, while on the right-hand side the scaling parameter $\xi$ as function of the inverse temperature.
• Figure 3: On the left-hand side we plot the sigma term \ref{['eq.sum_ff']} as a function of the number of particles $N$ for the inverse temperature $\beta=100$. On the right-hand side, we plot the sigma term as function of the inverse temperature $\beta$ for $N=100,150,200$ particles.
• Figure 4: Rapidities distribution in terms of their counting numbers following the interpolation method described.