Memory-Efficient Nonequilibrium Green's Function Framework Built On Quantics Tensor Trains

Abstract

One of the challenges in diagrammatic simulations of nonequilibrium phenomena in lattice models is the large memory demand for storing momentum-dependent two-time correlation functions. This problem can be overcome with the recently introduced quantics tensor train (QTT) representation of multivariable functions. Here, we demonstrate nonequilibrium Green's function simulations within the $GW$ and Migdal approximations with high momentum resolution, up to times which exceed the capabilities of standard implementations and are long enough to study, e.g., transient Floquet physics during multi-cycle electric field pulses and thermalization dynamics. The self-consistent calculation on the three-leg Kadanoff-Baym contour is fully self-contained, employing only QTT-compressed functions and input functions which are either generated directly in QTT form or obtained via quantics tensor cross interpolation.

Figures (6)

• Figure 1: (a) Kadanoff-Baym contour. (b) QTT representing a Green's function component. Same length scales are ordered as neighbors Shinaoka2023. (c) Occupations $n_\mathbf{k}(t)$ for a $U=0\to 1$ quench and $N_k=64$, $\beta=100$, where $\mathbf{k}=(\pi,\pi)\to(\pi,0)$ [with $(\pi,0)$, $(\pi,2\pi/N_k)$ not shown in the main panel]. The inset shows the same data as a color map, with the dashed lines guiding the eye. The gray line in the main panel plots the change in the double occupation $d(t)$ with respect to the initial value $0.25$. (d) Maximum bond dimension $D$ across all components and $\mathbf{k}$ points vs $t_\mathrm{max}$. Here, $N_k=32$. Usually, the lesser component contributes the largest $D$. (e) The same as in (d) but as a function of linear lattice size $N_k$ with $t_\mathrm{max}=250$. All calculations use $\epsilon_\mathrm{cutoff}=10^{-11}$ during the self-consistency loop.
• Figure 2: (a) Electric field $E(t)$ ($\Omega=3\pi$, $E_0=3\Omega$) and double occupation $d(t)$. (b) Time dependence of the fermionic spectrum $A(\omega,t)$ (lines) and the occupation $N(\omega,t)$ (shaded regions). (c),(d) Time dependence of $-\mathrm{Im}W^R(\omega,t)$ and $\mathrm{Re}W^R(\omega,t)$. The curves in panel (b) are offset by 1.33, in (c) by 0.06 and in (d) by 0.04. The parameters are: $U=1$, $N_k=32$, $\beta=10$. $G_{0\mathbf{k}}$ is prepared by tensor cross interpolation with the maximum-norm tolerance $10^{-5}$ and then truncated with $\epsilon_\mathrm{cutoff}=10^{-8}$. In the self-consistent loop, $\epsilon_\mathrm{cutoff}=10^{-11}$, $D_\mathrm{max}=140$.
• Figure 3: Electron-phonon coupling $g$ quench in the electron-phonon model \ref{['eq:holstein']} with $N_k=32$ and $\beta=20$. The main panel shows the change in the electron, phonon, and interaction energies after a sudden $g=0 \to 0.5$ quench. The sum of the energy differences is shown in gray. The dotted dark gray line shows $a\cos(2\Omega_r t)+b$ with $a$, $b$ constants and $\Omega_r\approx 0.69$ the renormalized phonon frequency estimated from the position of the maximum of the local phonon spectral function $B(\omega,t)$. The inset shows the momentum-resolved phonon spectral function $B_\mathbf{q}(\omega,t=25)$ with the dotted white line marking the bare phonon frequency $\Omega=0.8$.
• Figure S1: Convergence analysis for the quench calculations. (a) Conservation of energy $\mathcal{E}(t)$ and particle density per spin $n_\sigma(t)$. (b) Fulfillment of the fermionic sum rules for the zeroth and first moments. For the analysis of sum rules, we evaluate the fermionic spectral function $A(\omega,t)$ with Eq. \ref{['eq:trelft']} instead of Eq. \ref{['eq:pes']}. The first moment is given by $\int\!d\omega\,\omega A(\omega,t)=U(\sum_\sigma n_\sigma -1)/2$White1991, which evaluates to 0 at half-filling.
• Figure S2: Convergence analysis for the electric-field calculations. (a) Conservation of the particle density per spin: $n_\sigma(t)$ is the density for an interacting system, whereas $n_{0\sigma}(t)$ is for the noninteracting one. (b) Conservation of the energy $\mathcal{E}(t)$. The electric field $\mathbf{E}(t)$ pumps energy into the system, hence the energy conservation is expressed as $\dot{\mathcal{E}}(t) = \mathbf{E}(t)\mathbf{j}(t)$, where $\mathbf{j}(t)=j(t)(1,1)$ is the current. After integration, one gets $\mathcal{E}(t)-\mathcal{E}(0) = \int_0^t d\bar{t} \,\mathbf{E}(\bar{t})\mathbf{j}(\bar{t})$, which is more convenient to check numerically. We perform the integration on the very fine grid within the quantics-tensor-train format by an appropriate tensor-network contraction. After the pulse, we also show how well the energy is conserved with respect to the final value $\mathcal{E}(t_\mathrm{max})$. (c) Fulfillment of the fermionic sum rules for the zeroth and first moments. For the analysis of sum rules, we evaluate the fermionic spectral function $A(\omega,t)$ with Eq. \ref{['eq:trelft']} instead of Eq. \ref{['eq:pes']}. The first moment is given by $\int\!d\omega\,\omega A(\omega,t)=U(\sum_\sigma n_\sigma -1)/2$White1991, which evaluates to 0 at half-filling.