Expansion of one-dimensional lattice hard-core bosons at finite temperature

We develop an exact approach to study the quench dynamics of hard-core bosons initially in thermal equilibrium in one-dimensional lattices. This approach is used to study the sudden expansion of thermal states after confining potentials are switched off. We find that a dynamical fermionization of the momentum distribution occurs at all temperatures. This phenomenon is studied for low initial site occupations, for which the expansion of the cloud is self-similar. In this regime, the occupation of the natural orbitals allows one to distinguish hard-core bosons from noninteracting fermions. We also study the free expansion of initial Mott insulating domains at finite temperature and show that the emergence of off-diagonal one-body correlations is suppressed gradually with increasing temperature. Surprisingly, the melting of the Mott domain is accompanied by an effective cooling of the system. We explain this phenomenon analytically using an equilibrium description based on an emergent local Hamiltonian.

Figure 1

(a) Momentum ($n_k$) and (b) site ($n_i$) occupations for $N=25$ hard-core bosons during the expansion in a lattice. The system is initially prepared in a harmonic trap with characteristic density $\tilde{\rho }=0.1$ and temperature $T=0.02$. The inset in panel (a) shows a comparison between $n_k$ at $\tau =0$ for $T=0.02$ and $T=0$, and highlights the effect of $T\ne 0$. The inset in panel (b) makes apparent the self-similar behavior of the site occupations $n_i$ during the expansion.

Figure 2

Decrease of the relative difference $\delta n_k$ during the expansion for different initial characteristic densities: (a) $\tilde{\rho }=0.1$ and (b) $\tilde{\rho }=0.32$. Overlapping thick (color) and thin (black) lines show results for $N=25$ and $N=50$, respectively, and different values of $T$. The topmost black line depicts the ground-state result for $N=50$. Horizontal dotted lines mark $\delta n_k=0.05$. The values of $\tau /N$ vs $T$ for $\delta n_k=0.05$ are reported in Fig. 3.

Figure 3

(a) Time ${\tau }_F$ at which $\delta n_k=0.05$ (see Fig. 2) as a function of the effective temperature $T/\tilde{\rho }$. The solid lines are exponential fits to the numerical results. (b) Ratio $R_F$ between the full width at half maximum (FWHM) at ${\tau }_F$ and the FWHM at $\tau =0$ as a function of $T/\tilde{\rho }$. The solid lines are a guide to the eye.

Figure 4

(a) Natural orbital occupations for the same initial parameters and times for which momentum and site occupations are reported in Fig. 1. The black dashed line depicts the natural orbital occupations of the noninteracting fermions, which do not change with time. (b) Natural orbital occupations at $\tau =250$ for different initial temperatures $T$. All black dotted lines are ${\eta }^{-4}$ fits to the tails.

Figure 5

(a) One-body density matrix at different times for $N=25, \tilde{\rho }=0.1$, and $T=0.1$. (b) Scaled results for the one-body density matrix (main panel), and the lowest natural orbital wave function (inset).

Figure 6

Contour plots of the momentum distribution as a function of the expansion time. Panels (a)–(c) correspond to initial temperatures $T=0.1, T=0.5$, and $T=1.0$, respectively. These systems contain $N=200$ particles and, initially, $V_1=0.125$. Open boundaries are considered to, after the linear potential is turned off, constrain the expansion of the Mott domain to the right.

Figure 7

Maximal values (a) ${\left[n_k^m\right]}_{\text{max}}$ and (b) ${\left[{\lambda }_0\right]}_{\text{max}}$ during the expansion (symbols) as a function of the number of bosons in the system. The calculations are done for initial states with constant $\tilde{\rho }=50$, and different $T=0.1$, 0.5, and 1.0. We also report the occupation of (a) the zero-momentum mode $n_{k=0}$ (labels on the right) and (b) the lowest natural orbital ${\lambda }_0$ in homogeneous lattices with open boundary conditions at half-filling (lines) at the same initial temperatures of the systems that undergo expansion.

Figure 8

Comparison between the emergent local Hamiltonian description [lines in the legend in panel (a)] and the exact results from the dynamics (thin continuous lines). The calculations were done in a system with $N=500$, an initial potential with $V_1=0.05$, and $T=1.0$, and results are reported for four times. (a) Momentum distributions (main panel) and site occupation profiles (inset). (b) Absolute values of one-body correlations measured from site 500 (main panel) and the corresponding phases (inset).

Figure 9

(a) Momentum distributions in the reference system [lines in the legend in panel (a)] compared to those during the expansion [thin lines, see also Fig. 8]. (b) Results after shifting $n_k$ during the expansion by the peak momentum $k_0$. (Inset) Peak momentum $k_0$ as a function of time. The calculations were done in a system with $N=500$, an initial potential with $V_1=0.05$, and $T=1.0$, and results are reported for four times (see also Fig. 8).