Dynamical instabilities and transient short-range order in the fermionic Hubbard model

We study the dynamics of magnetic correlations in the half-filled fermionic Hubbard model following a fast ramp of the repulsive interaction. We use Schwinger-Keldysh self-consistent second-order perturbation theory to investigate the evolution of single-particle Green's functions and solve the nonequilibrium Bethe-Salpeter equation to study the dynamics of magnetic correlations. This approach gives us new insights into the interplay between single-particle relaxation dynamics and the growth of antiferromagnetic correlations. Depending on the ramping time and the final value of the interaction, we find different dynamical behavior which we illustrate using a dynamical phase diagram. Of particular interest is the emergence of a transient short-range ordered regime characterized by the strong initial growth of antiferromagnetic correlations followed by a decay of correlations upon thermalization. The discussed phenomena can be probed in experiments with ultracold atoms in optical lattices.

Figure 1

(a) Linear interaction ramp $U\left(t\right)$ with ramping time $t_{\mathrm{r}}$ and final interaction $U_{\mathrm{f}}$. (b) Schematic equilibrium phase diagram showing the paramagnetic phase (PM), the critical temperature $T_c^{\mathrm{eq}}\left(U\right)$ for the antiferromagnetic (AFM) phase, and the relaxation trajectory to a final temperature $T_{\mathrm{f}}$. (c) Time scales and different regimes (see text). (d) Qualitatively different regimes for the evolution of the equal-time AFM correlations; NP: slow growth to normal phase result. TSO: transient short-range order; AFM correlations initially develop and later decay upon thermalization, $T_{\mathrm{f}}>T_c^{\mathrm{eq}}\left(U_{\mathrm{f}}\right)$. OP: AFM correlations grow and final thermalized state expected to be the ordered phase, $T_{\mathrm{f}}<T_c^{\mathrm{eq}}\left(U_{\mathrm{f}}\right)$. (e) The dynamical phase diagram showing different regimes after a ramp from an initial PM state at temperature $T_i=0$ as a function of $U_{\mathrm{f}}$ and $t_{\mathrm{r}}$. The dashed lines are meant as a guide to the eye.

Figure 2

Momentum distribution function $n_{\mathit{k}}\left(t\right)$ for $T_{\mathrm{i}}=0, U_{\mathrm{f}}=3$ and different ramping times. (a) $t_{\mathrm{r}}=0.5, T_{\mathrm{f}}\approx 0.29>T_c^{\mathrm{eq}}$, (b) $t_{\mathrm{r}}=1.5, T_{\mathrm{f}}\approx 0.15>T_c^{\mathrm{eq}}$, and (c) $t_{\mathrm{r}}=2, T_{\mathrm{f}}\approx 0.08<T_c^{\mathrm{eq}}$. The dotted lines show the final equilibrium $n_{\mathit{k}}$ at $T=T_{\mathrm{f}}$. (d) The time dependence of ${\chi }_{{\mathit{q}}_0}^{zz,\mathrm{K}}(t,t)$ for different $t_{\mathrm{r}}$ in TSO and OP regime (semilog plot). The long-time limit of the correlators in the TSO regime is shown on the plot from an equilibrium calculation at $T=T_{\mathrm{f}}$ [63].

Figure 3

Equal time spatial spin correlation function $i{\chi }^{\mathrm{K}}(t,x)$ for $t_{\mathrm{r}}=1$ (left, TSO), and $t_{\mathrm{r}}=2$ (right, OP) for $T_{\mathrm{i}}=0$ and $U_{\mathrm{f}}=3$ for different times $t$ and lattice spacing $a=1$. Notice the different scale on the vertical axis.