Spectral moment sum rules for strongly correlated electrons in time-dependent electric fields

We derive exact operator average expressions for the first two spectral moments of nonequilibrium Green’s functions for the Falicov-Kimball model and the Hubbard model in the presence of a spatially uniform, time-dependent electric field. The moments are similar to the well-known moments in equilibrium, but we extend those results to systems in arbitrary time-dependent electric fields. Moment sum rules can be employed to estimate the accuracy of numerical calculations; we compare our theoretical results to numerical calculations for the nonequilibrium dynamical mean-field theory solution of the Falicov-Kimball model at half-filling.

Figure 1

Kadanoff-Baym contour for the two-time Green’s functions in the nonequilibrium case. We take the contour to run from $-t_{\mathit{max}}$ to $t_{\mathit{max}}$ and back, and then extend it downward parallel to the imaginary axis for a distance of $\beta$. The field is usually turned on at $t=0$; i.e., the vector potential is nonzero only for positive times.

Figure 2

(Color online) DOS for the equilibrium retarded Green’s function for different values of $U$. The DOS is independent of temperature.

Figure 3

(Color online) Local lesser Green’s function in equilibrium at $\beta =10$ and for different values of $U$.

Figure 4

(Color online) (a) Imaginary part of the lesser Green’s function as a function of the relative time coordinate for different discretizations of the time contour. The model parameters are $U=0.5$, $\beta =10$, and $E=0$. The average time $T$ is set equal to zero. The parameters for the Kadanoff-Baym time-contour discretization are $t_{\mathit{max}}=15$ and $\Delta \tau =0.1$ (i.e., 100 points taken along the imaginary axis); the discretization along the real time axis is given by $\Delta t$ as shown in the figure. Note how the results systematically approach the exact result as the discretization goes to zero. (b) Imaginary part of the lesser Green’s function as a function of frequency for different discretizations of the time contour.

Figure 5

(Color online) (a) Imaginary part of the lesser Green’s function as a function of the relative time coordinate for different discretizations of the time contour. The model parameters are $U=2$, $\beta =10$, and $E=0$. The average time $T$ is set equal to zero. The parameters for the Kadanoff-Baym time-contour discretization are $t_{\mathit{max}}=15$ and $\Delta \tau =0.1$ (i.e., 100 points taken along the imaginary axis); the discretization along the real-time axis is given by $\Delta t$ as shown in the figure. Note how the results systematically approach the exact result as the discretization goes to zero. (b) Imaginary part of the lesser Green’s function as a function of frequency for different discretizations of the time contour; in the inset, the region around $\omega =0$ is blown up to show the gap development as a function of the discretization. Note how the gap region converges very slowly—instead we see the DOS go negative in the gap region. Properly determining that structure in the frequency domain requires the Green’s function over an extended time domain, which is not numerically feasible.

Figure 6

(Color online) Imaginary part of the retarded Green’s function as a function of the relative time coordinate at different values of the average time (starting at $T=0$ and running to $T=40$ in steps of 5). The field is switched on at the time $T=0$ and we take $\Delta t=0.1$ along the Kadanoff-Baym contour. The model parameters are $\beta =10$ and $E=1$. Panel (a) shows the noninteracting result $U=0$, while panel (b) shows the interacting result $U=0.5$ (note that the curves extend as far out in $t_{\mathit{rel}}$ as we have data; the cutoff in $t_{\mathit{rel}}$ comes from the finite time domains of our calculations). Note how the results appear to retrace themselves for different average times until the relative time becomes larger than approximately $2T$, where the Green’s function decays. For the noninteracting case, the pattern obviously becomes periodic in the Bloch period as $T\to \infty$, which leads to delta functions in the Fourier transform (with respect to $t_{\mathit{rel}}$), whereas the interacting case may or may not be approaching a periodic form for large relative time; the data does not extend far enough out to be able to make a conclusion about the asymptotic form.

Figure 7

(Color online) Density of states for different average times (running from $T=0$ to $T=40$ in steps of 5) for the interacting system, with $U=0.5$ and $E=1$. These results correspond to the Fourier transform of the data in Fig. 6. Note how the DOS seems to be approaching a limiting form even for this small a value of the average time. We cannot tell, however, whether there might be some low-weight delta functions appearing somewhere in the spectrum. In any case, it does appear that there are no sharp structures near the Bloch frequencies, which are at integer frequencies for $E=1$.

Figure 8

First moment of the local lesser function ${\tilde{\mu }}_1^{<}$ plotted as a function of average time T for $U=0.5$ and $E=1$. The field is turned on at $T=0$. The step size is $\Delta t=0.1$, and the moment is calculated from the numerical derivative of the data.