Beyond-Luttinger-liquid thermodynamics of a one-dimensional Bose gas with repulsive contact interactions

We present a thorough study of the thermodynamics of a one-dimensional repulsive Bose gas, focusing in particular on corrections beyond the Luttinger-liquid description. We compute the chemical potential, pressure, and contact as a function of temperature and gas parameter with an exact thermal Bethe ansatz. In addition, we provide interpretations of the main features in the analytically tractable regimes, based on a variety of approaches (Bogoliubov, hard core, Sommerfeld, and virial). The beyond-Luttinger-liquid thermodynamic effects are found to be nonmonotonic as a function of gas parameter. Such behavior is explained in terms of nonlinear dispersion and “negative excluded volume” effects, for weak and strong repulsion, respectively, responsible for the opposite sign corrections in the thermal next-to-leading term of the thermodynamic quantities at low temperatures. Our predictions can be applied to other systems including super Tonks-Girardeau gases, dipolar and Rydberg atoms, helium, quantum liquid droplets in bosonic mixtures, and impurities in a quantum bath.

Figure 1

Thermal shift of the chemical potential, $\mathrm{\Delta }\mu =\mu -{\mu }_0$, vs temperature. The symbols denote numerical TBA results for several interaction strengths $\gamma$, and the lines correspond to various theories described in the text. The temperature is normalized to the typical energy associated with phonons, $\overline{T}=k_{B}T/\left(m{v}^2\right)$, and we measure shifts in terms of the LL result $\mathrm{\Delta }{\mu }_{\mathrm{LL}}$, so that deviations from unity directly quantify beyond-LL effects. In these units, the BG result, given by Eq. (7), is independent of $\gamma$.

Figure 2

Thermal shift of the pressure. Symbols denote numerical TBA results, and lines correspond to various theories, as in Fig. 1. Deviations from unity of $\mathrm{\Delta }P/\mathrm{\Delta }{P}_{\mathrm{LL}}$ indicate beyond-LL effects.

Figure 3

Thermal shift of the Tan's contact. Symbols denote numerical TBA results and lines correspond to various theories, as in Fig. 1. Deviations from unity of $\mathrm{\Delta }\mathcal{C}/\mathrm{\Delta }{\mathcal{C}}_{\mathrm{LL}}$ indicate beyond-Luttinger-liquid corrections.