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arXiv:2412.14032 · Physical Review Letters

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

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

Published Oct 14, 2025• 8 citations
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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 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.

DOI
10.1103/dxfb-b3l5

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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.
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.
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 32×3232\times 3232×32 lattice within the nonequilibrium GWGWGW approximation, we demonstrate stable propagation up to times of tmax=512t_\mathrm{max}=512tmax​=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.
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.
We present an application of quantics tensor trains (QTTs) and tensor cross interpolation (TCI) to the solution of a full set of self-consistent equations for multivariate functions, the so-called parquet equations. We show that the steps needed to evaluate the equations (Bethe-Salpeter equations, parquet equation, and Schwinger-Dyson equation) can be decomposed into basic operations on the QTT-TCI compressed objects. The repeated application of these operations does not lead to a loss of accuracy beyond a specified tolerance and the iterative scheme converges even for numerically demanding parameters. As examples, we take the Hubbard model in the atomic limit and the single impurity Anderson model, where the basic objects in parquet equations, the two-particle vertices, depend on three frequencies, but not on momenta. The results show that this approach is able to overcome major computational bottlenecks of standard numerical methods. The applied methods allow for an exponential increase of the number of grid points included in the calculations, and a corresponding exponential reduction of the computational error, for a linear increase in computational cost.
The correlation functions of quantum systems-central objects in quantum field theories-are defined in high-dimensional space-time domains.Their numerical treatment thus suffers from the curse of dimensionality, which hinders the application of sophisticated many-body theories to interesting problems.Here, we propose a multiscale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains (QTTs), "qubits" describing exponentially different length scales.The ansatz then assumes a separation of length scales by decomposing the resulting high-dimensional tensors into tensor trains (also known as matrix product states).We numerically verify the ansatz for various equilibrium and nonequilibrium systems and demonstrate compression ratios of several orders of magnitude for challenging cases.Essential building blocks of diagrammatic equations, such as convolutions or Fourier transforms, are formulated in the compressed form.We numerically demonstrate the stability and efficiency of the proposed methods for the Dyson and Bethe-Salpeter equations.The QTT representation provides a unified framework for implementing efficient computations of quantum field theories.
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.