Temperature-driven crossover in the Lieb-Liniger model

The large-distance behavior of the density-density correlation function in the Lieb-Liniger model at finite temperature is investigated by means of the recently derived nonlinear integral equations characterizing the correlation lengths. We present extensive numerical results covering all the physical regimes from weak to strong interaction and all temperatures. We find that the leading term of the asymptotic expansion becomes oscillatory at a critical temperature which decreases with the strength of the interaction. As we approach the Tonks-Girardeau limit the asymptotic behavior becomes more complex with a double crossover of the largest and next-largest correlation lengths. The crossovers exist only for intermediate couplings and vanish for $\gamma =0$ and $\infty$.

Figure 1

Upper panel: $\text{Re}[1/{\xi }_0]$ with ${\xi }_0$ the leading correlation length as a function of temperature for various values of $\gamma$. Lower panel: $2k_{nF}=\text{Im}[1/{\xi }_0]$ for the same parameters. The critical temperature $T_o\left(\gamma \right)$ is the value for which $2k_{nF}$ becomes nonzero. Note the kinks of $\text{Re}[1/{\xi }_0]$ at $T_o$. (All quantities are in units of $1/l_0$ and $T_0$ [28].)

Figure 2

$\text{Re}[1/{\xi }_1]$ with ${\xi }_1$ the next-leading correlation length as a function of temperature for various values of $\gamma$. Inset: $2k_F=\text{Im}[1/{\xi }_1]$ for the same parameters. (All quantities are in units of $1/l_0$ and $T_0$.)

Figure 3

$\text{Re}[1/{\xi }_0]$ (thick line) and $\text{Re}[1/{\xi }_1]$ (thin line) as functions of temperature for various values of $\gamma$. For $\gamma =20$ and 100 a complex crossover scenario can be seen. (All quantities are in units of $1/l_0$ and $T_0$.)

Figure 4

$\text{Re}\left[{\xi }_1\right]/\text{Re}\left[{\xi }_0\right]$ as a function of temperature for various values of $\gamma$. For $\gamma =20$ and 100 in the region of temperatures where this ratio is greater than 1, ${\xi }_1$ becomes the leading correlation length. (Temperature is in units of $T_0$.)