A causality-based divide-and-conquer algorithm for nonequilibrium Green's function calculations with quantics tensor trains

TL;DR

This work tackles the challenge of long-time nonequilibrium Green's function simulations in strongly correlated systems, where memory and compute costs scale unfavorably with the simulated duration. It introduces a causality-based divide-and-conquer scheme that augments the quantics tensor train (QTT) representation to extend the time domain blockwise while preserving stability and dramatically reducing data storage. Implemented within nonequilibrium DMFT for the Hubbard model on a Bethe lattice in an antiferromagnetic state, the method solves the Dyson equation via a linear-equation solver and updates self-energies self-consistently across extended time blocks; convergence is achieved despite slow long-time relaxation and significant memory savings. The approach enables long-time tracking of slow relaxation dynamics, with potential improvements via better initial guesses and refined masking strategies, and lays groundwork for applying QTT-NEGF to larger systems and more complex orbital/momentum structures.

Abstract

We propose a causality-based divide-and-conquer algorithm for nonequilibrium Green's function calculations with quantics tensor trains. This algorithm enables stable and efficient extensions of the simulated time domain by exploiting the causality of Green's functions. We apply this approach within the framework of nonequilibrium dynamical mean-field theory to the simulation of quench dynamics in symmetry-broken phases, where long-time simulations are often required to capture slow relaxation dynamics. We demonstrate that our algorithm allows to extend the simulated time domain without a significant increase in the cost of storing the Green's function.

Figures (14)

• Figure 1: (a) Kadanoff-Baym contour, which consists of the forward real-time branch $\mathcal{C}_1$, the backward real-time branch $\mathcal{C}_2$, and the Matsubara imaginary-time branch $\mathcal{C}_3$. (b) QTT representations of the retarded ($R$) and lesser ($<$) Green's functions.
• Figure 2: Flow of the divide-and-conquer algorithm. 1. Solve the Dyson equation with the linear equation solver for $t\le t_{\mathrm{max}}$ and get the converged Green's function. 2. Extend the time domain by an interval of width $\Delta t$ and prepare the initial guess in the extended domain. 3. Calculate the self-energy in each block. 4. Solve the Dyson equation in each block. 5. Fuse the Green's functions in the four blocks. 6. Perform an affine transformation.
• Figure 3: Schematic illustration of how the Green's functions are divided into blocks: (a) For retarded and lesser Green's functions, the time domain is partitioned into four blocks corresponding to $(t_1, t'_1) = (0,0), (0,1), (1,0), (1,1)$. (b) For the left-mixing Green's function, the single real-time index is divided into two blocks corresponding to $t_1 = 0$ and $t_1 = 1$.
• Figure 4: Particle number and energy conservation in the equilibrium AFM state. (a-d) The absolute error from the initial state $t=0$ for $n_{\uparrow}$, $n_{\downarrow}$, and $n_{\uparrow} + n_{\downarrow} - 1$. (e-h) The absolute error from the initial state $t=0$ of the electron kinetic, interaction, and total energies $E_{\mathrm{kin}}$, $E_{\mathrm{int}}$, and $E_{\mathrm{tot}}$, respectively. (a) and (e) are for $\epsilon_{\mathrm{cutoff}} = 10^{-12}$, $\epsilon_{\mathrm{conv}}\sim 10^{-4}$. (b) and (f) are for $\epsilon_{\mathrm{cutoff}} = 10^{-16}$, $\epsilon_{\mathrm{conv}}\sim 6 \times 10^{-7}$. (c) and (g) are for $\epsilon_{\mathrm{cutoff}} = 10^{-18}$, $\epsilon_{\mathrm{conv}}\sim 8 \times 10^{-8}$. (d) and (h) are for $\epsilon_{\mathrm{cutoff}} = 10^{-20}$, $\epsilon_{\mathrm{conv}}\sim 5 \times 10^{-9}$.
• Figure 5: Physical quantities in the equilibrium AFM state for the QTT-NEGF and conventional methods. (a) The number of electrons with up and down spins, $n_{\uparrow}$ and $n_{\downarrow}$, and the order parameter $m = n_{\uparrow} - n_{\downarrow}$. (b) The electron kinetic, interaction, and total energies, $E_{\mathrm{kin}}$, $E_{\mathrm{int}}$, and $E_{\mathrm{tot}}$, respectively. The gray dashed lines are the reference data calculated by the conventional method using the NESSi NESSi2020. (c) The absolute error of the number of electrons between the results of QTT-NEGF and NESSi. We also plot the absolute error of $n_{\uparrow} + n_{\downarrow} - 1$. (d) The absolute error of energy between the results of QTT-NEGF and NESSi.