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arXiv:2509.22177 · Physical review. B./Physical review. B

Predictor-corrector method based on dynamic mode decomposition for tensor-train nonequilibrium Green's function calculations

M. Środa, Ken Inayoshi, Michael Schüler, Hiroshi Shinaoka, Philipp Werner

Published Feb 6, 2026• 2 citations
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Abstract

The nonequilibrium Green's function (NEGF) formalism is a powerful tool to study the nonequilibrium dynamics of correlated lattice systems, but its applicability to realistic system sizes and long timescales is limited by unfavorable memory scaling. While compressed representations, such as the recently introduced quantics tensor train (QTT) format, alleviate the memory bottleneck, the efficiency of QTT-NEGF calculations is hindered by poor initializations and slow or unstable convergence of globally updated self-consistent iterations. Here, we introduce a predictor-corrector solver for QTT-NEGF simulations that combines dynamic mode decomposition (DMD) extrapolation with the recently proposed causality-preserving block-time-stepping updates. The DMD predictor supplies accurate initial guesses that reduce the iteration count of the calculation, while the block-time-stepping correction ensures stable convergence even for long propagation intervals. Applying this method to the Hubbard model on a $32\times 32$ lattice within the nonequilibrium $GW$ approximation, we demonstrate stable propagation up to times of $t_\mathrm{max}=512$ inverse hoppings, surpassing the capabilities of both matrix-based implementations and previous QTT solvers. Our contribution is twofold. (i) We integrate tensor dynamic mode decomposition with the QTT representation, which establishes a general framework that is not limited to NEGFs. (ii) We demonstrate its practical benefits in NEGF simulations, where it enables stable and efficient access to unprecedented timescales at high momentum resolution, thereby advancing controlled studies of long-time dynamics and nonequilibrium steady states in correlated lattice systems.

DOI
10.1103/xnf8-kntz

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The nonequilibrium Green's function formalism provides a versatile and powerful framework for numerical studies of nonequilibrium phenomena in correlated many-body systems. For calculations starting from an equilibrium initial state, a standard approach consists of discretizing the Kadanoff-Baym contour and implementing a causal time-stepping scheme in which the self-energy of the system plays the role of a memory kernel. This approach becomes computationally expensive at long times, because of the convolution integrals and the large amount of computer memory needed to store the Green's functions. A recent idea for the compression of nonequilibrium Green's functions is the quantics tensor train representation. Here, we explore this approach by implementing equilibrium and nonequilibrium simulations of the two-dimensional Hubbard model with a second-order weak-coupling approximation to the self-energy. We show that calculations with compressed two-time functions are possible without any loss of accuracy, and that the quantics tensor train implementation shows a much improved scaling of the computational effort and memory demand with the length of the time contour.
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 multicycle 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.
Correlated real-time dynamics in large, spatially inhomogeneous quantum systems remain difficult to access with nonequilibrium many-body methods. Two-time nonequilibrium Green functions (NEGF) retain dynamical correlations but their computational runtime grows cubically with the number of time steps NtN_\mathrm{t}Nt​. This scaling bottleneck could recently be overcome by introducing the G1--G2 scheme that is linear in NtN_\mathrm{t}Nt​, but requires propagation of a two-particle correlation function and may suffer from numerical instabilities. This has restricted simulations to small systems with Nb∼102N_\mathrm{b} \sim 10^2Nb​∼102 basis states. Here we introduce a quantum-fluctuation formulation of nonequilibrium Green functions, denoted δδδNEGF, that represents dynamical two-particle correlations through fluctuations of field-operator products, δG^δ\hat GδG^. This guarantees stable dynamics by preserving the positivity of the reduced density matrices, avoids the explicit storage of the two-particle Green function, and reduces the propagation to a finite ensemble of Hartree-Fock-like trajectories. Combined with a stochastic low-rank decomposition of the correlation functions, the method retains time-linear scaling while extending dynamical GWGWGW and particle-particle and particle-hole TTT-matrix simulations to basis sizes of order Nb∼104N_\mathrm{b}\sim 10^4Nb​∼104. We benchmark δδδNEGF against exact and HF-GKBA results for lattice systems, finding stable correlated dynamics also at strong coupling. We further demonstrate large-scale simulations of diffusion in two-dimensional Hubbard lattices and ultrafast relaxation in graphene nanoribbon heterostructures with long-range Coulomb interactions. These results establish δδδNEGF as a scalable route to dynamical self-energy simulations of large, spatially inhomogeneous correlated quantum systems beyond the reach of existing NEGF implementations.
The nonequilibrium interplay of electrons and phonons plays an important role in the thermalization of solids, yet the microscopic picture of transient states and relaxation pathways remains incomplete. Previous nonequilibrium Green's function (NEGF) studies with full two-time dependence were restricted to local phonons and local self-energy approximations, leaving momentum-dependent phonon dynamics largely unexplored. Here, we demonstrate the recently developed quantics-tensor-train (QTT) NEGF framework via large-scale lattice simulations of models with arbitrary phonon dispersions. QTTs provide a memory-efficient representation of full two-time Green's functions, enabling momentum-resolved simulations with full electron-phonon feedback on lattices up to 256x256 sites. Comparing optical and acoustic phonon models, we reveal a hierarchy of relaxation bottlenecks that extends the well-known phonon-window bottleneck effect. For optical phonons, we confirm the main phonon-energy window and uncover an additional reduced window separating momentum-space regions with excess and deficit electronic population. We also identify a separate phonon-thermalization bottleneck rooted in momentum-dependent coupling to the particle-hole continuum. High momentum and frequency resolution makes the phonon--charge correspondence directly visible. For acoustic phonons, the phonon-energy window acquires pronounced momentum dependence dictated by energy and momentum conservation. The reduced window becomes asymmetric, and directional scattering between Brillouin-zone regions produces a persistent bottleneck for low-momentum phonon modes. Our results establish diagrammatic QTT-NEGF simulations as a scalable framework for quantitative nonequilibrium electron-phonon dynamics, overcoming previous lattice-size and propagation-time limitations and providing accurate reference data for time-resolved spectroscopies.
The Hartree-Fock generalized Kadanoff-Baym ansatz (HF-GKBA) offers an approximate numerical procedure for propagating the two-time nonequilibrium Green's function (NEGF). Here, using the GWGWGW self-energy, we compare the HF-GKBA to exact results for a variety of systems with long- and short-range interactions, different two-body interaction strengths, and various nonequilibrium preparations. We find excellent agreement between the HF-GKBA and exact time evolution in models when more realistic long-range exponentially decaying interactions are considered. This agreement persists for long times and for intermediate to strong interaction strengths. In large systems, HF-GKBA becomes prohibitively expensive for long-time evolutions. For this reason, we look at the use of dynamical mode decomposition (DMD) to reconstruct long-time NEGF trajectories from a sample of the initial trajectory. Using no more than 16% of the total time evolution, we reconstruct the total trajectory with high fidelity. Our results show the potential for DMD to be used in conjunction with HF-GKBA to calculate long-time trajectories in large-scale systems.
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.