Adiabatic approximation in time-dependent reduced-density-matrix functional theory

With the aim of describing real-time electron dynamics, we introduce an adiabatic approximation for the equation of motion of the one-body reduced density matrix (one-matrix). The eigenvalues of the one-matrix, which represent the occupation numbers of single-particle orbitals, are obtained from the constrained minimization of the instantaneous ground-state energy functional rather than from their dynamical equations. The performance of the approximation vis-à-vis nonadiabatic effects is assessed in real-time simulations of a two-site Hubbard model. Due to Landau-Zener-type transitions, the system evolves into a nonstationary state with persistent oscillations in the observables. The amplitude of the oscillations displays a strongly nonmonotonic dependence on the strength of the electron-electron interaction and the rate of variation of the external potential. We interpret an associated resonance behavior in the phase of the oscillations in terms of “scattering” with spectator energy levels. To clarify the motivation for the minimization condition, we derive a sequence of energy functionals $E_v^{(n)}$, for which the corresponding sequence of minimizing one-matrices is asymptotic to the exact one-matrix in the adiabatic limit.

Figure 1

Bloch sphere and trajectory of the pseudospin $\mathrm{\gamma ⃗}$ with linear-time potential $\mathrm{V⃗}=(-2,0,3t)$, $U=\frac{7}{2}$, $t\in (-\infty ,4.7)$. The point of intersection shows $\mathrm{\gamma ⃗}$ at the time $t\approx -0.261$, where $\left|\mathrm{\gamma ⃗}\right|$ is minimum; the inner sphere shows $\left|\mathrm{\gamma ⃗}\right|{}_{\min }\approx 0.633$.

Figure 2

Time dependence of the one-matrix for the linear-time potential, $\tau =1$, $U=1$. Shown are ${\gamma }_1$ (top left), ${\gamma }_2$ (top right), ${\gamma }_3$ (bottom left), and the adiabatic many-body energies (solid lines) and adiabatic KS energies (dotted lines) (bottom right).

Figure 3

Time dependence of the one-matrix relative to the instantaneous ground-state one-matrix for the linear-time potential, $\tau =1$, $U=1$. Shown are ${\gamma }_1-{\gamma }_1^{(0)}$ (top left), ${\gamma }_3-{\gamma }_3^{(0)}$ (bottom left), $A$ (top right), and correlation energy $E_c$ (bottom right).

Figure 4

Time dependence of the one-matrix for the linear-time potential, $\tau =2$, $U=4$. Shown are ${\gamma }_1$ (top left), ${\gamma }_2$ (top right), $A-A^{(0)}$ (bottom left), and adiabatic energies (bottom right). Energies are the same as in Fig. 2 with the addition of HF eigenvalues (dashed lines).

Figure 5

$U=1$. Amplitude $\left|{\Delta }_1\right|$ and phase ${\Theta }_1$ of the asymptotic oscillations of ${\gamma }_1$. Arrows and vertical lines represent the resonances. Dashed vertical lines represent the resonances predicted by the two-level approximation, Eq. (49).

Figure 6

Linearization and semilinearization of the IONR approximation for the linear-time potential, $\tau =1$, $U=1.25$.

Figure 7

Linearization and semilinearization of ADFT for the linear-time potential, $\tau =1$, $U=1.25$.

Figure 8

Time dependence of the one-matrix for a pulse potential $V_3=1/\cosh t$, $U=3$.