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

We show that the chemical potential of a one-dimensional (1D) interacting Bose gas exhibits a nonmonotonic 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 weakly interacting 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 detail. 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.

Figure 1

Sound velocity $v_s$ in units of Fermi velocity $v_F$ (solid line) as a function of the interaction parameter $\gamma$, calculated by solving the Lieb-Liniger equations. The Bogoliubov [dotted line, $v_s^{\mathrm{BG}}\left(\gamma \right)/v_F=\sqrt{\gamma }/\pi$] and Tonks-Girardeau (dashed line, $v_s^{\mathrm{TG}}=v_F$) limits, including their first-order corrections [thin solid line, $v_s(\gamma \ll 1)/v_F=v_s^{\mathrm{BG}}\left(\gamma \right)/v_F\sqrt{1-\sqrt{\gamma }/\left(2\pi \right)}$, and thin dot-dashed line, $v_s(\gamma \gg 1)/v_F=\sqrt{1-8/\gamma }$, respectively], are present too; see Secs. 3 and 4.

Figure 2

Lieb-Liniger excitation spectrum in the BG regime with $\gamma =4.52$ (left) and in the deep TG regime with $\gamma =\infty$ (right). The units are the Fermi energy $E_F$ and the Fermi momentum $p_F$. The shaded region represents the continuum of the excitations and is delimited by the upper (Lieb I) and the lower (Lieb II) branch of the spectrum. On the left, the Lieb I and II branches are not reported. On the left, the dashed line gives the Bogoliubov dispersion and the dotted line gives the mean-field soliton spectrum. In the limit $\gamma \to 0$, the Lieb I branch tends to be equal to the Bogoliubov dispersion, while the Lieb II one coincides with the soliton spectrum. The solid line is the Lieb-Liniger phononic spectrum calculated with $\gamma =4.52$. On the right, Lieb I and Lieb II branches are reported and they coincide with the particle and hole ideal Fermi gas excitations, respectively. The solid line is the phononic spectrum calculated with the Fermi velocity.

Figure 3

$\alpha \left(\gamma \right)$ (solid line) with the leading dependence [dashed line, ${\alpha }_{\mathrm{TG}}=1$, and dotted line, ${\alpha }_{\mathrm{BG}}\left(\gamma \right)=2\gamma /{\pi }^2$] and first-order corrections [thin dot-dashed line, $\alpha (\gamma \gg 1)=1-16/\left(3\gamma \right)$, and thin solid line, $\alpha (\gamma \ll 1)={\alpha }_{\mathrm{BG}}\left(\gamma \right)(1-\sqrt{\gamma }/\pi )$] for Tonks-Girardeau and Bogoliubov limits, respectively.

Figure 4

$\beta \left(\gamma \right)$ (solid line) with the Tonks-Girardeau (dashed line, ${\beta }_{\mathrm{TG}}={\pi }^2/12$) and Bogoliubov [dotted line, ${\beta }_{\mathrm{BG}}\left(\gamma \right)={\pi }^3/\left(24\sqrt{\gamma }\right)$] limits.

Figure 5

Chemical potential as a function of temperature $T$ in Fermi units for several values of $\gamma$ and at a fixed density $n|a_{1\mathrm{D}}|=2/\gamma$. The solid lines represent the Bethe-ansatz (BA) solutions for different values of $\gamma$. The dot-dashed lines are the low-temperature expansions of the chemical potential taking into account only the phononic contribution, Eq. (7). The phononic expansions for $\gamma \ge 1000$ are equal to the analytical Sommerfeld expansion of Eq. (21). Both the chemical potentials as a function of $T$ for the ideal Fermi (upper dashed line) and ideal Bose (lower dashed line) gas are also reported in Eqs. (10) and (11), respectively.

Figure 6

Chemical potential as a function of temperature in BG units for several values of $\gamma$. The solid lines represent the Bethe-ansatz (BA) solutions for different values of $\gamma$. The dashed lines are the low-temperature expansions of the chemical potential taking into account only the phononic contribution, Eq. (13). We notice also that the phononic expansion (13) does not hold for very small $\gamma$, like $\gamma =0.001$, for which value the low-temperature expansion is not reported.

Figure 7

Value of the dimensionless coefficient $\delta \left(\gamma \right)$ of the low-temperature expansion of the adiabatic inverse compressibility (solid line). The BG and TG analytical limits are also shown: ${\delta }_{\mathrm{BG}}\left(\gamma \right)=5{\pi }^3/\left(16\sqrt{\gamma }\right)$ (dotted line) and ${\delta }_{\mathrm{TG}}={\pi }^2/2$ (dashed line).

Figure 8

Absolute value of the dimensionless coefficient $\eta \left(\gamma \right)$ of the low-temperature expansion of the isothermal inverse compressibility (solid line). The BG and TG analytical limits are also shown: ${\eta }_{\mathrm{BG}}\left(\gamma \right)=-{\pi }^3/\left(16\sqrt{\gamma }\right)$ (dotted line) and ${\eta }_{\mathrm{TG}}=-{\pi }^2/6$ (dashed line).

Figure 9

Comparison of the numerical series $G\left(y\right)$ (43) (solid line) and its analytical expansion (44) (dashed line) holding for $y\gg 1$. The dot-dashed line represents the thermodynamic value.

Figure 10

Comparison of the ground-state energy per particle, in BG units, as a function of $y=\gamma N^2$ in the thermodynamic limit of Bogoliubov theory (16) (dot-dashed line), the Bethe-ansatz (BA) calculation (circle), the Bogoliubov expression (41) (solid line), and the $y\gg 1$ expansion (45) (dashed line), for several values of the interaction parameter $\gamma$.

Figure 11

Comparison of the numerical series $F\left(y\right)$ (47) (solid line) and its analytical expansion (48) (dashed line) holding for $y\gg 1$. The dot-dashed line represents the thermodynamic value.

Figure 12

Comparison of the chemical potential in BG units as a function of $y=\gamma N^2$ in the thermodynamic limit (17) (dot-dashed line), the Bethe-ansatz (BA) calculation (symbols), and the Bogoliubov expression (46) (solid line), for several values of the interaction parameter $\gamma$. For the BA, ${\mu }_{+}=E_0(N+1)-E_0\left(N\right)$ (square), ${\mu }_{-}=E_0\left(N\right)-E_0(N-1)$ (star), and $\overline{\mu }=({\mu }_{+}+{\mu }_{-})/2$ (circle).

Figure 13

Energy per particle in units of the TG gas energy $E_{\mathrm{TG}}={\pi }^2{\hslash }^2{n}^2/\left(6m\right)$ as a function of $N$. Bethe-ansatz (BA) results (symbols) with different values of the interaction parameter $\gamma$ are compared with the TG gas (54) (solid line) and the hard-sphere (HS) model (55) (dashed and dotted lines).

Figure 14

Chemical potential at $T=0$ in Fermi units as a function of the number of particles $N$ in the TG regime [Eq. (A5)] for fixed values of $\gamma$ (solid line). The dashed lines correspond to the thermodynamic limit $[1+2/\left(3\gamma \right)]/{(1+2/\gamma )}^3$ of the TG model [Eq. (A6)]. The symbols correspond to the Bethe-ansatz (BA) calculation: ${\mu }_{+}=E_0(N+1)-E_0\left(N\right)$ (square), ${\mu }_{-}=E_0\left(N\right)-E_0(N-1)$ (star), and $\overline{\mu }=({\mu }_{+}+{\mu }_{-})/2$ (circle).

Figure 15

Static structure factor at $T=0$ in the Tonks-Girardeau limit for different number of particles (solid lines). Values of momenta (38) $k_i=2\pi n_i/L$ corresponding to standing waves on a ring are shown with circles. The dashed line represents the phononic law $S\left(k\right)=\hslash \left|k\right|/\left(2m{v}_F\right)$, which coincides with the static structure factor in the thermodynamic limit for $\left|k\right|<2\pi n$ in the TG regime.

Figure 16

Static structure factor at $T=0$ and $\gamma =1$ for different number of particles (solid lines). Values of momenta (38) $k_i=2\pi n_i/L$ corresponding to standing waves on a ring are shown with circles.