Comparing the generalized Kadanoff-Baym ansatz with the full Kadanoff-Baym equations for an excitonic insulator out of equilibrium

We investigate out-of-equilibrium dynamics in an excitonic insulator (EI) with a finite-momentum pairing perturbed by a laser-pulse excitation and a sudden coupling to fermionic baths. The transient dynamics of the excitonic order parameter is resolved using the full nonequilibrium Green's function approach and the generalized Kadanoff-Baym ansatz (GKBA) within the second-order Born approximation. The comparison between the two approaches after a laser-pulse excitation shows a good agreement in the weak and the intermediate photodoping regime. In contrast, the laser-pulse dynamics resolved by the GKBA does not show a complete melting of the excitonic order after a strong excitation. Instead we observe persistent oscillations of the excitonic order parameter with a predominant frequency given by the renormalized equilibrium band gap. This anomalous behavior can be overcome within the GKBA formalism by coupling to an external bath, which leads to a transition of the EI system toward the normal state. We analyze the long-time evolution of the system and distinguish decay timescales related to dephasing and thermalization.

Figure 1

Model system schematic. (a) One-dimensional chains with nearest-neighbor hopping $J$, on-site energy $\pm \mathrm{\Delta }/2$, and interband Hubbard interaction $U$. (b) Noninteracting band structure of the two separate chains with $J=-1$ and $\mathrm{\Delta }=2$. Filling of the bands is set by the chemical potential at zero (dashed line) describing a hole pocket at the band edge for the lower band and an electron pocket at the band center for the upper band (shaded areas).

Figure 2

Energy- and momentum-resolved equilibrium spectral function for the (a) noninteracting system, and for the interacting systems described at the (b) Hartree-Fock, (c) second Born level with the GKBA, and (d) second Born level with the full KBE. The EI system parameters are $\mathrm{\Delta }=1.4, U=\{0.0,3.5\}$. The reduced Brillouin zone for the momentum axis is shown using backfolding. The noninteracting energy-band structure is superimposed onto the noninteracting spectral function in panel (a) with solid lines.

Figure 3

Time evolution of the absolute value of the order parameter $\left|\varphi \right(t\left)\right|$ with and without the excitation for the HF and 2B@GKBA propagation scheme. For the HF case the laser pulse is applied at $t=0$ while for the 2B@GKBA case the adiabatic preparation of the correlated equilibrium state is performed first, and the laser pulse is then applied at $t=t_0\equiv 150{\left|J\right|}^{-1}$.

Figure 4

Time evolution of the absolute value of the excitonic order parameter $\left|\varphi \right(t\left)\right|$ after applying a laser pulse with (a) fixed frequency and varying amplitude, and (b) fixed amplitude and varying frequency. (c) Fourier spectra of selected time-dependent data from panels (a) and (b) for the 2B@GKBA propagation (full lines). The Fourier spectrum for the HF solution is marked with the dashed line, and HF2 represents the spectrum for the lattice model with next-nearest-neighbor hopping; see text for details. The curves are shifted vertically for clarity.

Figure 5

Comparison of the absolute value of the excitonic order parameter $\left|\varphi \right(t\left)\right|$ evolution within the 2B@GKBA (solid lines) and the 2B@KBE (dashed lines) propagation schemes.

Figure 6

Energy-absorption spectrum (color map) in terms of the laser-pulse amplitude (horizontal axis) and frequency (vertical axis) for (a) the 2B@GKBA and (b) the full 2B@KBE solutions. See text for the discussion of the dashed region.

Figure 7

Energy- and momentum-resolved equilibrium spectral function at the 2B@GKBA level with a constant dipolar transition term of strength (a) $A=0.2$, (b) $A=0.6$, and (c) $A=2.0$.

Figure 8

Time evolution of the absolute value of the order parameter $\left|\varphi \right(t\left)\right|$ with and without the application of the fermionic bath for the HF and 2B@GKBA propagation scheme. For the HF case the coupling to the bath is applied at $t=0$ while for the 2B@GKBA case the adiabatic preparation of the correlated equilibrium state is performed first, and the bath coupling is applied at $t=t_0\equiv 150{\left|J\right|}^{-1}$.

Figure 9

Time evolution of the excitonic order parameter after the bath coupling with (a) fixed tunneling rate $\mathrm{\Gamma }=0.1$, fixed coupling duration $t_{\mathrm{bath}}=10$, and varying bias $V$; (b) fixed tunneling rate $\mathrm{\Gamma }=0.1$, fixed bias $V=\pm 0.2U$, and varying coupling duration $t_{\mathrm{bath}}$; (c) fixed bias $V=\pm 0$, fixed coupling duration $t_{\mathrm{bath}}=50$, and varying tunneling rate $\mathrm{\Gamma }$.

Figure 10

Exponential fits for the decay timescales of the excitonic order parameter shown in Fig. 9. The short-dashed lines are specified completely by $\sim e^{-\mathrm{\Gamma }t}$, while the long-dashed lines are obtained by fitting to the flat part [except in panel (a)].

Figure 11

Decay timescale exponents ($\sim e^{-t/\tau }$) as a function of the bias $V$ and varying tunneling rate (a) $\mathrm{\Gamma }=0.1$, (b) $\mathrm{\Gamma }=0.2$, and (c) $\mathrm{\Gamma }=0.3$. The statistical error bars are given by the numerical fitting procedure; for all ${\tau }_{\mathrm{de}}$ data points the error is smaller than the marker size.

Figure 12

Energy- and momentum-resolved nonequilibrium spectral function after coupling the baths with a fixed tunneling rate $\mathrm{\Gamma }=0.1$ and varying bias (a) $V=\pm 0$, (b) $V=\pm 0.1U$, and (c) $V=\pm 0.2U$.