Nonequilibrium sum rules for the retarded self-energy of strongly correlated electrons

We derive the first two moment sum rules of the conduction electron retarded self-energy for both the Falicov–Kimball model and the Hubbard model coupled to an external spatially uniform and time-dependent electric field (this derivation also extends the known nonequilibrium moment sum rules for the Green functions to the third moment). These sum rules are used to further test the accuracy of nonequilibrium solutions to the many-body problem; for example, we illustrate how well the self-energy sum rules are satisfied for the Falicov–Kimball model in infinite dimensions and placed in a uniform electric field turned on at time $t=0$. In general, the self-energy sum rules are satisfied to a significantly higher accuracy than the Green function sum rules.

Figure 1

The complex Kadanoff–Baym–Keldysh contour for the two-time Green functions in the nonequilibrium case.

Figure 2

(Color online) Constant piece of the retarded self-energy ${\Sigma }^R(\omega \to \infty )$ as a function of average time. The curves with symbols correspond to nonequilibrium calculations with different discretization sizes $\Delta t$ on the real part of the Kadanoff–Baym–Keldysh contour. The dashed line is the exact result (equal to 1.5 here) and the solid line with no symbols is the extrapolated result using a quadratic extrapolation with the three smallest discretization sizes. The parameters are $E=1$, $U=3$, and $1∕\beta =0.1$; the calculation is done at half-filling for both the delocalized and the localized particles. Note how the extrapolated result is accurate to better than 0.1%.

Figure 3

(Color online) Zeroth moment of the retarded self-energy $C_0^R\left(T\right)$ as a function of average time. The curves with symbols correspond to nonequilibrium calculations with different discretization sizes $\Delta t$ on the real part of the Kadanoff–Baym–Keldysh contour. The dashed line is the exact result (equal to 2.25 here) and the solid line with no symbols is the extrapolated result using a quadratic extrapolation with the three smallest discretization sizes. The parameters are $E=1$, $U=3$, and $1∕\beta =0.1$; the calculation is done at half-filling for both the delocalized and the localized particles. Note how the extrapolated result is accurate to better than 0.1%.

Figure 4

(Color online) First moment of the retarded self-energy $C_1^R\left(T\right)$ as a function of average time. The curves with symbols correspond to nonequilibrium calculations with different discretization sizes $\Delta t$ on the real part of the Kadanoff–Baym–Keldysh contour. The dashed line is the exact result (equal to 0 here) and the solid line with no symbols is the extrapolated result using a quadratic extrapolation with the three smallest discretization sizes. The parameters are $E=1$, $U=3$, and $1∕\beta =0.1$; the calculation is done at half-filling for both the delocalized and the localized particles. Note how the extrapolated result has high accuracy.