Kadanoff-Baym dynamics of Hubbard clusters: Performance of many-body schemes, correlation-induced damping and multiple steady and quasi-steady states

We present in detail a method we recently introduced [Phys. Rev. Lett. 103, 176404 (2009)] to describe finite systems in and out of equilibrium, where the evolution in time is performed via the Kadanoff-Baym equations within many-body perturbation theory. Our systems consist of small, strongly correlated clusters, described by a Hubbard Hamiltonian within the Hartree-Fock, second Born, $GW$, and $T$-matrix approximations. We compare the results from the Kadanoff-Baym dynamics to those from exact numerical solutions. The outcome of our comparisons is that, among the many-body schemes considered, the $T$-matrix approximation is superior at low electron densities while none of the tested approximations stands out at half filling. Such comparisons permit a general assessment of the whole idea of applying many-body perturbation theory, in the Kadanoff-Baym sense, to finite systems. A striking outcome of our analysis is that when the system evolves under a strong external field, the Kadanoff-Baym equations develop a steady-state solution as a consequence of a correlation-induced damping. This damping is present both in isolated (finite) systems, where it is purely artificial, as well as in clusters contacted to (infinite) macroscopic leads. The extensive numerical characterization we performed indicates that this behavior is present whenever approximate self-energies, which include correlation effects, are used. Another important result is that, for isolated clusters, the steady state reached is not unique but depends on how one switches on the external field. When the clusters are coupled to macroscopic leads, one may reach multiple quasisteady states with arbitrarily long lifetimes.

Figure 1

Keldysh contours.

Figure 2

Diagrammatic expansions of MBAs.

Figure 3

(Color online) The time-propagation square.

Figure 4

(Color online) Ground-state spectral functions for $M=6$ for weak interaction strength, $U=1$, for different fillings. The curves correspond to exact (black), TMA (red), BA (green), GWA (blue), and HFA (orange). The curves are shifted for clearer comparison and we have broadened the discrete spectra with a Lorentzian with a $\text{FWHM}=0.4$. The frequency is given in units of the hopping parameter of the central region, $V$.

Figure 5

(Color online) Ground-state spectral functions for $M=6$ for strong interaction strength, $U=4$, at different fillings. The curves correspond to exact (black), TMA (red), BA (green), GWA (blue), and HFA (orange). The curves are shifted for clearer comparison and we have broadened the discrete spectra with a Lorentzian with a $\text{FWHM}=0.4$. The frequency is given in units of the hopping parameter of the central region, $V$.

Figure 6

(Color online) Ground-state spectral functions for $M=6$ ordered as a ring at $n=1/6$ for $U=2$. The on-site energies vary periodically between 0 and $-1$, starting with 0 on site 1. The curves correspond to exact (black), TMA (red), BA (green), GWA (blue), and HFA (orange). The curves are shifted for clearer comparison and we have broadened the discrete spectra with a Lorentzian with a $\text{FWHM}=0.4$. The frequency is given in units of the hopping parameter of the central region, $V$.

Figure 7

(Color online) First-iteration TMA, and self-consistent TMA spectral function versus the exact one. $M=6$ and in (a) $n=1/6$ and in (b) $n=1/2$. The curves correspond to exact (black), first iteration TMA (green), and self-consistent TMA (thick blue). The curves are shifted for clearer comparison and we have broadened the discrete spectra with a Lorentzian with a $\text{FWHM}=0.4$. The frequency is given in units of the hopping parameter of the central region, $V$.

Figure 8

(Color online) Broadened (black) and unbroadened (red) spectral function in the TMA for $M=4$, $n=1/2$, and $U=2$. The broadening of the discrete spectra is done with a Lorentzian with a $\text{FWHM}=0.4$. The unbroadened spectral function has been scaled with a factor of 1.65 to adjust the height of the central peaks to those of the broadened one. The frequency is given in units of the hopping parameter of the central region, $V$.

Figure 9

(Color online) Time-dependent densities on site 1 for $M=6$. The curves correspond to exact (thick solid black), TMA (dashed red), BA (dotted green), GWA (thin solid blue), and HFA (brown dashed dot). The time is given in units of the inverse hopping parameter of the central region, $V^{-1}$.

Figure 10

(Color online) Time-dependent densities on site 1 for a ring-shaped cluster with $M=6$ sites and filling $n=1/6$, for $U=2$ and $w_0=5$. The on-site energies vary periodically between 0 and $-1$, starting with 0 on site 1. The curves correspond to exact (thick solid black), TMA (dashed red), BA (dotted green), GWA (thin solid blue), and HFA (brown dashed dot). The time is given in units of the inverse hopping parameter of the central region, $V^{-1}$.

Figure 11

(Color online) Time-dependent $v_{eff}$ on site 1 for $M=6$, $U=4$, and $w_0=5$. The curves correspond to exact (thick solid black), TMA (dashed red), BA (dotted green), GWA (thin solid blue), and HFA (brown dashed dot). The time is given in units of the inverse hopping parameter of the central region, $V^{-1}$.

Figure 12

(Color online) Time-dependent densities for $M=4,U=1.5,w_0=5$ and $n=1/4$. The curves correspond to exact (black), GWA (dotted red), SGWA (thick blue), and TMA (dashed green). The time is given in units of the inverse hopping parameter of the central region, $V^{-1}$.

Figure 13

(Color online) Densities for $M=2,n=1/2,U=1,w_0=5$, exact (black), GWA (dashed red), and GWA initialized with $G_0$ (blue). In (a): damping of GWA density versus exact solution. In (b): time-dependent densities for the GWA, initialized with the self-consistent GWA ground state and the noninteracting $G_0$. The time is given in units of the inverse hopping parameter of the central region, $V^{-1}$.

Figure 14

(Color online) Conservation laws and time reversal in the GWA for $M=2,n=1/2,U=1,w_0=5$. In (a): relative change in particle number. In (b): total energy for different time steps. In (c): density propagated forward in time from $-2$ to 3 and then backward from 3 to $-2$. The time is given in units of the inverse hopping parameter of the central region, $V^{-1}$.

Figure 15

(Color online) Time-dependent spectral functions in the GWA for $M=2,n=1/2,U=1,w_0=2$, (a): in energy space and (b): in time space. The different curves correspond to $T_p=10$ (black), $T_p=35$ (red), and $T_p=100$ (blue). The time is given in units of the inverse hopping parameter of the central region, $V^{-1}$.

Figure 16

(Color online) Time-dependent densities for Born approximations with different levels of self-consistency for $M=2,n=1/2,U=1,w_0=5$; ${\text{BA}}_0$ (black), ${\text{BA}}_{\text{HFA}}$ (red), and BA (blue). The time is given in units of the inverse hopping parameter of the central region, $V^{-1}$.

Figure 17

(Color online) Time-dependent densities for a cluster isolated in the TMA with sudden and slow switch on for $M=2,U=1,w_0=8$. Sudden switch on (thin black), slow switch on $\left(t_{max}=40\right)$ (thick red), and ground-state density of the final Hamiltonian (dashed red). The time is given in units of the inverse hopping parameter of the central region, $V^{-1}$.

Figure 18

(Color online) Time-dependent densities for a cluster isolated and coupled to two leads in the TMA with sudden and slow switch on. $M=2,U=1,w_0=8$. In the contacted case, the on-site $\text{energies}=-U/2$. Lead parameters: $V_{L_{\alpha }}=1$, $V_{L_{\alpha }C}=0.5$, $\text{filling}=0.5$, and $\text{bias}=0$. (a) Sudden switch on: isolated cluster (thin black), coupled to leads (thick red). (b) Slow switch on $\left(t_{max}=40\right)$: isolated cluster (thin black), coupled to leads (thick red); ground-state densities of the final Hamiltonian: isolated cluster (dashed black), coupled to leads (dashed thick red). Inset: expanded view of the long-time limit behavior. The time is given in units of the inverse hopping parameter of the central region, $V^{-1}$.

Figure 19

(Color online) Time-dependent densities for a cluster with different couplings to two unbiased leads in the BA. $M=2$, $U=1$, on-site $\text{energies}=-U/2$, $w_0=5$, $V_{L_{\alpha }}=5$, and $\text{filling}=0.5$ in the leads. The time is given in units of the inverse hopping parameter of the central region, $V^{-1}$.