Dynamics of strongly correlated fermions: Ab initio results for two and three dimensions

Quantum transport of strongly correlated fermions is of central interest in condensed matter physics. While the stationary expansion dynamics have recently been measured with cold atoms in 2D optical lattices, ab initio simulations have been limited to 1D setups so far. Here, we present the first precise fermionic quantum dynamics simulations for 2D and 3D. The simulations are based on nonequilibrium Green functions and incorporate strong correlations via $T$-matrix self-energies. The simulations predict the short-time dynamics, and we discover a universal scaling of the expansion velocity with the particle number. Our predictions can be verified experimentally using the recently developed fermionic atom microscopes.

Figure 1

(Top) Expansion dynamics of 74 circularly confined fermions in a 2D $19\times 19$ Hubbard lattice (1) following the removal of the confinement at time $t=0$, for $U=1\left(5\right)$, first (second) row. Snapshots of the density $n_{\mathit{s}}$ for four time points during the early stage of the expansion ($t$ in units of the inverse hopping rate, color code: $\sqrt{n_{\mathit{s}}}$). (Bottom) Asymptotic core [defined as density HWHM] expansion velocity $C_{\exp }$. Plus signs: experimental results for different lattice depths in units of the recoil energy $E_{\mathrm{r}}$ [10]; gray dashed line: RTA model [4]; red circles: present NEGF results. The black line is a fit through the experimental points to guide the eye.

Figure 2

(Left) Time evolution of the expansion velocity for $N=58$ in 2D and various $U$. (Right) Evolution of the independent-particle and correlation parts of the entropy and energy, for $U=4$. Symbols mark the respective inflection points. Shaded areas correspond to the three phases of the evolution (see text).

Figure 3

Fermion expansion dynamics in a 2D $19\times 19$ Hubbard lattice (1) at $U=4$. Top three rows: square root of density $n_s$ for $N=2,26$, and 74, respectively. Rows $4–6$: square root of double occupation, entropy density $S_{\mathit{s}}$, and the pair correlation function $\delta n_{\mathit{s}}^{\uparrow \downarrow }=n_{\mathit{s}}^{\uparrow \downarrow }-n_{\mathit{s}}^{\uparrow }{n}_{\mathit{s}}^{\downarrow }$.

Figure 4

Momentum distribution $p\left(k\right)$, Eq. (11) at $t=9.5$, for a 1D system of $N_s=65$ for $N=2,...,42$. (Top) $U=3$. (Bottom) $U=-3$. The dashed lines denotes the uniform initial distribution $p_0\left(k\right)$. (Insets) $N$-dependence of amplitude $a$. Symbols: data points, lines: linear fits.

Figure 5

(Left) Asymptotic expansion velocity vs particle number for $D=1,...,3$ and $U=1,...,3$. Symbols correspond to data points, errors are smaller than symbol size. Dashed lines: linear extrapolation $N\to \infty$ according to Eq. (13). (Right) Corresponding slope $\chi$ vs bandwidth-normalized interaction.

Figure 6

Macroscopic expansion velocity $V_{\exp }$ for varying $U$. Comparison between Hartree-Fock (HF) and the correlated $T$-matrix (TMA) results.

Figure 7

Dependence of the asymptotic expansion velocity on $N$ for confinement potentials of different curvature ${\gamma }_k\left(N\right)$. Insets show the shape of the initial density profile.