Nonequilibrium dynamical cluster approximation study of the Falicov-Kimball model

We use a nonequilibrium implementation of the dynamical cluster approximation (DCA) to study the effect of short-range correlations on the dynamics of the two-dimensional Falicov-Kimball model after an interaction quench. As in the case of single-site dynamical mean-field theory, thermalization is absent in DCA simulations, and for quenches across the metal-insulator boundary, nearest-neighbor charge correlations in the nonthermal steady state are found to be larger than in the thermal state with identical energy. We investigate to what extent it is possible to define an effective temperature of the trapped state after a quench. Based on the ratio between the lesser and retarded Green's function, we conclude that a roughly thermal distribution is reached within the energy intervals corresponding to the momentum-patch dependent subbands of the spectral function. The effectively different chemical potentials of these distributions, however, lead to a very hot, or even negative, effective temperature in the energy intervals between these subbands.

Figure 1

The sites in a cluster $\tilde{\mathbf{r}}$ form the unit cell of a superlattice $\mathbf{R}$ (left). The first Brillouin zone of the original lattice is split into patches of the size of the first Brillouin zone of the superlattice. The corresponding reciprocal vectors $\tilde{\mathbf{k}}$ are centered around the reciprocal vectors $\mathbf{K}$ of the periodized cluster (right).

Figure 2

Cluster geometries and patch layouts considered in this work. The leftmost column depicts different cluster geometries in real space. The remaining columns depict some possible choices of patch layouts in reciprocal space. Patch layouts that are equivalent due to symmetries in the dispersion ${\epsilon }_{\mathbf{k}}$ are grouped by color.

Figure 3

Comparison of the spectral function for different cluster geometries and interaction parameters in equilibrium. In all cases, the canonical patch layout was used.

Figure 4

Local and nonlocal cluster correlations in equilibrium at different temperatures and interaction strengths. The first column depicts the double occupation, and the next three columns depict nearest-neighbor density-density correlations between $c$- and $f$-, $c$- and $c$-, and $f$- and $f$-particles, respectively. The upper row corresponds to the $2\times 2$ cluster, and the lower row corresponds to the eight-site cluster. In both cases, the canonical patch layout was used.

Figure 5

Local and nonlocal correlations between $c$ and $f$ particles after an interaction ramp from $U_0=3$ to $U_q=4$, starting from an equilibrium state at $\beta =10$. Results are shown for different cluster geometries, averaged over different patch layouts. The uppermost plot depicts the local double occupation. The middle plot depicts nearest-neighbor density-density correlations between $c$ and $f$ particles. The lowermost plot depicts nearest-neighbor density-density correlations between $c$ particles. The triangles to the right indicate the expectation values in an equilibrium system with the same total energy.

Figure 6

Difference between the thermal expectation values and the steady-state observables [Eq. (32)] after an interaction ramp ($2\times 2$ cluster, canonical patch layout, initial $\beta =10$). The system is ramped from $U_0$ to $U_q$ according to Eq. (30), and the double occupancy and $c-f$ nearest-neighbor density-density correlation are measured. The color scale in the double-occupancy plot is cropped to match the scale of the nearest-neighbor correlation plot.

Figure 7

Spectral and occupation functions for different $K$ patches after ramping the interaction parameter from $U_0$ to $U_q$ within the time interval $[0,3]$. The thick red line is the spectral function, and the thin blue line is the occupation function. The green line in the right panels depicts the function $h\left(\omega \right)$ whose slope would correspond to the inverse temperature $\beta$ in a thermalized system. The thin green line that is laid over the spectral function shows the reciprocal of that slope.

Figure 8

First column: Spectral and occupation functions for different $f$-particle configurations after ramping the interaction parameter from $U_0$ to $U_q$ within the time interval $[0,3]$. The thick red line is the spectral function, and the thin blue line is the occupation function. Second column: Function $h\left(\omega \right)$, whose slope would correspond to the inverse temperature $\beta$ in a thermalized system. The thin green line that is laid over the spectral function in the left panels shows the reciprocal of that slope, i.e., $T_{\text{eff}}\left(\omega \right)$ (right axis). The following $f$-particle configurations are depicted from top to bottom: Unoccupied, one $f$ particle, two particles along an edge, two particles along a diagonal, three $f$ particles, fully occupied. The numbers in the upper right corner of each panel in the first column denote the multiplicity of that configuration due to symmetry and the corresponding configuration weight $w_{\alpha }$. Third column: Spectral and occupation functions for different $f$-particle configurations and momentum patches. The spectra in the first column are the normalized sum of these. Fourth column: Energy spectrum for different $f$-particle configurations and momentum patches in an isolated cluster at $U=2$. The $(0,\pi )$, and $(\pi ,0)$ spectra (green and blue curves) overlap in all but the third row.