Nonequilibrium dynamical cluster theory

We study the effect of spatially nonlocal correlations on the nonequilibrium dynamics of interacting fermions by constructing the nonequilibrium dynamical cluster theory, a cluster generalization of the nonequilibrium dynamical mean-field theory (DMFT). The formalism is applied to interaction quenches in the Hubbard model in one and two dimensions, and the results are compared with data from single-site DMFT, the time-dependent density matrix renormalization group, and lattice perturbation theory. Both in one and two dimensions the double occupancy quickly thermalizes, while the momentum distribution relaxes only on much longer time scales. For the two-dimensional square lattice we find a strongly momentum-dependent evolution of the momentum distribution around the Fermi energy, with a much faster relaxation near the momenta $(0,\pi )$ and $(\pi ,0)$ than near $(\pi /2,\pi /2)$. This result is interpreted as reflecting the momentum-anisotropic quasiparticle lifetime of the marginal Fermi liquid. The method is further applied to the two-dimensional Hubbard model driven by a dc electric field, where the damping of the Bloch oscillation of the current is found to be less effective than predicted by DMFT and lattice perturbation theory.

Figure 1

(a) The double occupancy and (b) the jump in the momentum distribution for a quench $U/J=0\to 1$ in the 1D Hubbard model calculated by DCA with cluster type A, and compared with other methods. The arrow in (a) indicates the thermal value of the double occupancy evaluated from the finite-temperature DMRG.

Figure 2

(a) The double occupancy and (b) the jump in the momentum distribution for a quench $U/J=0\to 1$ in the 1D Hubbard model calculated by DCA with cluster type B, and compared with other methods. The arrow in (a) indicates the thermal value of the double occupancy evaluated from the finite-temperature DMRG.

Figure 3

Averaging of the double occupancy obtained by DCA with cluster types A (Fig. 1) and B (Fig. 2) for $N_c=4$ (left) and $N_c=8$ (right).

Figure 4

(a) The double occupancy and (b) the jump in the momentum distribution for a quench $U/J=0\to 2$ in the 2D Hubbard model, obtained by DCA with the average of cluster types A and B, as compared with other methods. The arrow in (a) indicates the thermal value evaluated from DCA with $N_c=16\times 16$.

Figure 5

Top: Color-coded plot of $|\mathrm{Im}{\Sigma }^R(\mathbf{k},\omega )/\omega |$ at $\omega =0.3J$ for $U/J=2$ and $T=0$. Solid lines indicate the noninteracting Fermi surface. Bottom: Inverse of the quasiparticle lifetime $\tau \left(\mathbf{k}\right)$ estimated with the nonequilibrium DCA, along with that estimated from $\mathrm{Im}{\Sigma }^R(\mathbf{k},\omega )$ for three positions (A, B, and C on the left panel) in the Brillouin zone.

Figure 6

The current in the 2D Hubbard model driven by a dc electric field $E=4$ obtained with DCA for $N_c=8\times 8,16\times 16$ (red and blue curves, respectively) and with ${\Sigma }^{\left(2\right)}$ (dashed curve). The cluster type dependence (A or B) is negligible.