Thermodynamic behavior of a one-dimensional Bose gas at low temperature

Abstract

We show that the chemical potential of a one-dimensional (1D) interacting Bose gas exhibits a non-monotonic temperature dependence which is peculiar of superfluids. The effect is a direct consequence of the phononic nature of the excitation spectrum at large wavelengths exhibited by 1D Bose gases. For low temperatures T , we demonstrate that the coefficient in T 2 expansion of the chemical potential is entirely defined by the zero-temperature density dependence of the sound velocity. We calculate that coefficient along the crossover between the Bogoliubov weaklyinteracting gas and the Tonks-Girardeau gas of impenetrable bosons. Analytic expansions are provided in the asymptotic regimes. The theoretical predictions along the crossover are confirmed by comparison with the exactly solvable Yang-Yang model in which the finite-temperature equation of state is obtained numerically by solving Bethe-ansatz equations. A 1D ring geometry is equivalent to imposing periodic boundary conditions and arising finite-size effects are studied in details. At T = 0 we calculated various thermodynamic functions, including the inelastic structure factor, as a function of the number of atoms, pointing out the occurrence of important deviations from the thermodynamic limit.

Ground-state energy and excitation spectrum of the Lieb-Liniger model : accurate analytical results and conjectures about the exact solution

“…Its exact solution has helped understand various aspects of the many-body problem in one dimension, the most appealing features being the effective fermionization of bosons at large interaction strength [3,9] and the existence of two branches of excitations [4], one of them reminiscent of the Bogoliubov dispersion [2,10], the other linked with a quantum analog of classical solitons [11]. The equilibrium grand canonical description of the LL model has been developed by Yang and Yang who introduced the Thermodynamic Bethe Ansatz [12], thereby opening new investigation lines, such as finite-temperature thermodynamics [13][14][15][16] or quantum statistics of the model [17]. Later on, Haldane used the Lieb-Liniger model as a testbed for the universal description of low-energy properties of gapless 1D systems within the bosonization technique [18], known as the Tomonaga-Luttinger liquid (TL) framework [19][20][21][22][23][24][25].…”

Mesoscopic density grains in a 1D interacting Bose gas from the exact Yang–Yang solution

“…This is shown in figure 4. Figure 4 shows the calculated behavior of the skewness of the density grain distribution s  using (57). There is also a simple correspondence between the values shown in figure 4 and the skewness of the total atom number N:…”

“…Top: contours of the skewness of the density grain distribution s  given by(57). Values are indicated on the plot.…”

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

“…Thermodynamic quantities can be measured more easily, but these generally exhibit a monotonic behavior to lowest order in temperature. For example, the phononic excitations are responsible for the linear increase with temperature of the specific heat [5], and for the quadratic growth of the chemical potential [21,22], for every interaction strength. As the temperature is increased, however, higher momenta get explored and the deviation of the spectrum from the simple linear behavior becomes important [23,24].…”

“…Within the LL theory, one retains only the phononic part of the BG dispersion, (p) ≈ v|p|, and obtains the universal result ∆µ LL =μT 2 , withT = k B T /(mv 2 ) andμ = πm 2 v 2 (∂v/∂n) L /(6 ) [22]. Expanding the BG spectrum to Figure 1.…”