pith. machine review for the scientific record. sign in

arxiv: 2605.02606 · v1 · submitted 2026-05-04 · ❄️ cond-mat.str-el

Recognition: 2 theorem links

· Lean Theorem

NESSi 2.0: The Non-Equilibrium Systems Simulation package version 2.0

Fabian K\"unzel , Michael Sch\"uler , Denis Gole\v{z} , Yuta Murakami , Sujay Ray , Christopher Stahl , Jiajun Li , Hugo U. R. Strand
show 2 more authors
Philipp Werner Martin Eckstein
Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:36 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords nonequilibrium Green's functionsKadanoff-Baym equationsmemory truncationnonequilibrium steady statesquantum many-body dynamicscomputational scalingNESSi package
0
0 comments X

The pith

Truncating memory integrals in the Kadanoff-Baym equations reduces nonequilibrium Green's function simulation cost from cubic to linear scaling with total time steps.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents NESSi version 2.0, an updated software package for simulating quantum many-body systems with nonequilibrium Green's functions. Standard approaches suffer from rapid growth in cost and memory because the equations of motion contain integrals over the entire history of the system. The main technical step limits those memory integrals to a finite cutoff of recent timesteps, which lowers the scaling provided the quantities of interest no longer change when the cutoff is enlarged. The release also adds support for computing time-translationally invariant nonequilibrium states that appear in steady transport or slowly relaxing regimes. This combination matters for processes that span many orders of magnitude in time, such as electron motion after a laser pulse in a solid.

Core claim

By truncating the memory integrals in the Kadanoff-Baym equations to a maximum of N_c timesteps, the computational complexity is reduced to O(N_t N_c^2), and the memory requirement to O(N_c^2). Provided that the results converge with respect to the cutoff N_c, memory truncation allows to extend the simulations to significantly longer times. Functionalities are introduced to describe nonequilibrium steady states, i.e. time-translationally invariant nonequilibrium states relevant for transport settings and approximate descriptions of slowly evolving prethermal states.

What carries the argument

Memory truncation of the integrals appearing in the Kadanoff-Baym equations to a finite number of timesteps N_c

If this is right

  • Simulations of laser-driven electron dynamics in solids can now reach from sub-femtosecond to picosecond scales within feasible compute budgets.
  • Nonequilibrium steady states become directly accessible for transport calculations without evolving the full transient.
  • Prethermal regimes can be approximated by steady-state solvers rather than requiring the complete long-time trajectory.
  • Perturbative methods already implemented in the package, such as nonequilibrium GW and dynamical mean-field theory, inherit the improved scaling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same truncation approach could be ported to other nonequilibrium Green's function codes that face similar memory bottlenecks.
  • The smallest N_c at which observables stabilize may serve as a practical measure of the intrinsic memory time of a given interaction or material.
  • Steady-state capabilities combined with truncation could enable hybrid simulations that switch from full transient evolution to steady-state description once transients have died out.

Load-bearing premise

Physical results converge with respect to the memory cutoff N_c for the target systems and timescales, so truncation does not introduce uncontrolled errors.

What would settle it

For a fixed physical system and total simulation length, increase the cutoff N_c and check whether key observables stabilize to a chosen numerical tolerance.

Figures

Figures reproduced from arXiv: 2605.02606 by Christopher Stahl, Denis Gole\v{z}, Fabian K\"unzel, Hugo U. R. Strand, Jiajun Li, Martin Eckstein, Michael Sch\"uler, Philipp Werner, Sujay Ray, Yuta Murakami.

Figure 1
Figure 1. Figure 1: Green’s function domain and data structures on a discrete time grid. The Hermitian domain ( view at source ↗
Figure 2
Figure 2. Figure 2: Domains for data structures on a discrete time grid. The memory truncated domain, defined by Eqs. ( view at source ↗
Figure 3
Figure 3. Figure 3: The functions |Σ<(t, t′ )| (a) and |G<(t, t′ )| (b) in the initial time window 0 ≤ t, t′ ≤ 32. hid_t sub_group = create_group ( file_id ,"t"+ std :: to_string ( tstp ) ); G_t . write_timestep_to_hdf5 (0 , sub_group ,"G"); // save timestep 0 of moving window close_group ( sub_group ); In the Jupyter notebook, we can use the helper functions of readCNTRhdf5 to extract the data (timestep, size, retarded and l… view at source ↗
Figure 4
Figure 4. Figure 4: (a) The functions |ΣR,<(t, t′ )| and |GR,<(t, t′ )| as a function of the time difference t−t ′ at a given time slice (t = 32) on a logarithmic scale. (b) The function |GR k (t, t′ )| as a function of the time difference t − t ′ at a given time slice (t = 32) on a logarithmic scale, for different values of ϵk, where ϵk = 0 corresponds to the Fermi edge. Note the different vertical scale in the two plots. of… view at source ↗
Figure 5
Figure 5. Figure 5: Momentum occupation ρk(t) for selected values of ϵk, obtained with the memory-truncated time evolution with different tc. Different line colors represent different values of ϵk, close to the Fermi energy (ϵk = 0), in the middle of the band (ϵk = 1), and close the band edge (ϵk = 2). Different line-styles distinguish different values of tc. The on-site impurity Hamiltonian is Hloc = Un↑n↓ +  µ − U 2 + ϵd … view at source ↗
Figure 6
Figure 6. Figure 6: Anderson model for U = 6, and β = 20 (ϵd = 0). (a) Bath spectral function (88) and occupation functions (dashed) for voltage V = 2. (b) Current as a function of voltage V , for β = 20. (c) Interacting spectral function A(ω) (full blue line) and noninteracting spectral function A0(ω) (full red line) for V = 2. The dashed lines show the imaginary part of the corresponding lesser Green’s functions. (d) Occupa… view at source ↗
Figure 7
Figure 7. Figure 7: a) Convergence of the steady-state solution for fixed timestep view at source ↗
Figure 8
Figure 8. Figure 8: Steady-state DMFT loop (U = 4, β = 10, IPT), with Nft = 213 and a timestep hness = 0.02. (a) Convergence of the spectral function with iteration iter=0, ..., 30 (see color bar). (b) Same as (a), but for ImG<(ω). The inset in (b) shows the convergence error ϵiter = |Giter+1 − Giter|2 as a function of iteration; the difference is evaluated in the time-domain, as explained in Sec. 5.4.2 view at source ↗
Figure 9
Figure 9. Figure 9: Converged Green’s function G (a) and self-energy Σ (b) for the same parameters as in view at source ↗
Figure 10
Figure 10. Figure 10: Convergence of the NESS simulation with increasing view at source ↗
Figure 11
Figure 11. Figure 11: Memory-truncated time evolution of an equilibrium state, initialized with a NESS simulation. (Self-consistent view at source ↗
read the original abstract

Nonequilibrium Green's functions provide a powerful framework for studying quantum many-body dynamics including the laser-induced dynamics in solids. The Non-Equilibrium Systems Simulation package (NESSi) offers an efficient platform for such simulations, ranging from perturbative approaches like nonequilibrium $GW$ to nonequilibrium dynamical mean-field theory. However, simulations based on nonequilibrium Green's functions become computationally demanding when the dynamics span a large temporal range, such as from sub-femtosecond electron dynamics to the picosecond dynamics of collective modes. Due to the memory integral in the Kadanoff-Baym equations, which serve as equations of motion for nonequilibrium Green's functions, the computational cost scales as $\mathcal{O}(N_t^3)$ with the number of timesteps $N_t$, and the memory requirement scales as $\mathcal{O}(N_t^2)$. In this work, we extend NESSi by incorporating techniques that aim to overcome this bottleneck: (i) By truncating the memory integrals in the KBE to a maximum of $N_c$ timesteps, the computational complexity is reduced to $\mathcal{O}(N_tN_c^2)$, and the memory requirement to $\mathcal{O}(N_c^2)$. Provided that the results converge with respect to the cutoff $N_c$, memory truncation allows to extend the simulations to significantly longer times. (ii) We introduce functionalities to describe nonequilibrium steady states, i.e. time-translationally invariant nonequilibrium states. Such states are relevant for transport settings, and they provide an approximate description of slowly evolving (prethermal) nonequilibrium states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents NESSi 2.0, an update to the Non-Equilibrium Systems Simulation package for nonequilibrium Green's function (NEGF) calculations. It introduces (i) truncation of the memory integrals in the Kadanoff-Baym equations (KBE) to a maximum of N_c timesteps, reducing computational complexity from O(N_t^3) to O(N_t N_c^2) and memory requirements from O(N_t^2) to O(N_c^2), conditional on convergence with respect to N_c, and (ii) new functionalities to simulate nonequilibrium steady states (NESS) that are time-translationally invariant, relevant for transport and prethermal regimes.

Significance. If the truncation converges for the systems and timescales of interest, the work enables NEGF simulations (including nonequilibrium GW and DMFT) over significantly longer times, bridging sub-femtosecond electron dynamics to picosecond collective modes in solids. The NESS tools add practical value for steady-state transport problems. The conditional framing of the truncation and the focus on standard Volterra-integral reductions are appropriately stated.

major comments (2)
  1. [Memory truncation implementation] The central utility of the memory truncation rests on the assumption that observables converge with N_c (abstract). The manuscript should include quantitative benchmarks for at least one representative system (e.g., a Hubbard model under laser driving) that demonstrate both convergence of key quantities and the achieved error level as a function of N_c.
  2. [Nonequilibrium steady states section] For the NESS functionality, it is unclear how the time-translation invariance is numerically enforced when solving the KBE and whether the truncation is applied consistently to the steady-state self-energies and Green's functions to preserve the claimed scaling.
minor comments (2)
  1. [Abstract and scaling discussion] Ensure that all complexity statements in the main text match the abstract exactly, including the precise definition of N_c relative to the full timestep count N_t.
  2. [Software interface] Add a short table or figure caption that lists the new classes or functions introduced in version 2.0 for users migrating from NESSi 1.x.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of NESSi 2.0 and the constructive major comments. We address each point below and will incorporate clarifications and additional material into the revised manuscript.

read point-by-point responses

  1. Referee: The central utility of the memory truncation rests on the assumption that observables converge with N_c (abstract). The manuscript should include quantitative benchmarks for at least one representative system (e.g., a Hubbard model under laser driving) that demonstrate both convergence of key quantities and the achieved error level as a function of N_c.

    Authors: We agree that explicit quantitative benchmarks strengthen the presentation of the memory truncation. The revised manuscript will include a new subsection (or expanded results section) with benchmarks for the Hubbard model under laser driving. These will show convergence of observables such as the double occupancy and current as a function of N_c, together with the error relative to the full-memory reference calculation, thereby substantiating the conditional statement in the abstract. revision: yes

  2. Referee: For the NESS functionality, it is unclear how the time-translation invariance is numerically enforced when solving the KBE and whether the truncation is applied consistently to the steady-state self-energies and Green's functions to preserve the claimed scaling.

    Authors: We appreciate the request for clarification. In the NESS module, time-translation invariance is enforced by reformulating the Kadanoff-Baym equations in terms of the relative time τ = t − t′ only; the two-time Green's functions and self-energies are replaced by their steady-state counterparts G(τ) and Σ(τ). The equations are solved either by fixed-point iteration in real time (with a sufficiently long transient discarded) or via Fourier transformation to frequency space. The memory cutoff is applied uniformly by restricting the integral over τ to |τ| ≤ N_c Δt, which reduces the steady-state problem to an effective one-dimensional convolution whose cost scales as O(N_c²) per iteration, consistent with the claimed complexity. The revised manuscript will add a dedicated paragraph (with the relevant equations) in the NESS section to make this implementation explicit. revision: yes

Circularity Check

0 steps flagged

No circularity in the memory truncation or NESSi 2.0 implementation

full rationale

The paper presents a software package implementing standard numerical techniques for solving the Kadanoff-Baym equations (KBEs) of nonequilibrium Green's functions. The claimed complexity reduction from O(N_t^3) to O(N_t N_c^2) follows directly from the explicit truncation of the memory integrals at N_c timesteps, with the paper stating the conditional requirement that observables must converge with N_c. This is a conventional approximation for Volterra integral equations and does not involve self-definition, renaming of known results as new derivations, fitted parameters presented as predictions, or load-bearing self-citations. The derivation chain is self-contained against external numerical benchmarks for integral truncation and requires no internal reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard nonequilibrium Green's function formalism and the mathematical properties of truncated memory integrals; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • domain assumption Kadanoff-Baym equations serve as the equations of motion for nonequilibrium Green's functions
    Invoked in the abstract as the starting point for the memory-integral discussion.

pith-pipeline@v0.9.0 · 5631 in / 1267 out tokens · 32678 ms · 2026-05-08T18:36:43.616074+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

47 extracted references · 37 canonical work pages

  1. [1]

    Kadanoff, G

    L. Kadanoff, G. Baym, Quantum Statistical Mechanics: Green’s Function Methods in Equilibrium and Nonequilibrium Problems, Frontiers in physics, W.A. Benjamin, 1962

    1962

  2. [2]

    L. V. Keldysh, Diagram technique for nonequilibrium processes, Sov. Phys. JETP 20 (1965) 1018

    1965

  3. [3]

    Kamenev, Field Theory of Non-Equilibrium Systems, Cambridge University Press, Cambridge, 2011

    A. Kamenev, Field Theory of Non-Equilibrium Systems, Cambridge University Press, Cambridge, 2011

    2011

  4. [4]

    Stefanucci, R

    G. Stefanucci, R. van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction, Cambridge University Press, Cambridge, 2013

    2013

  5. [5]

    de la Torre, D

    A. de la Torre, D. M. Kennes, M. Claassen, S. Gerber, J. W. McIver, M. A. Sentef, Colloquium: Nonthermal pathways to ultrafast control in quantum materials, Rev. Mod. Phys. 93 (2021) 041002. doi:10.1103/RevModPhys.93.041002

    work page doi:10.1103/revmodphys.93.041002 2021

  6. [6]

    Giannetti, M

    C. Giannetti, M. Capone, D. Fausti, M. Fabrizio, F. Parmigiani, D. Mihailovic, Ultrafast optical spec- troscopy of strongly correlated materials and high-temperature superconductors: a non-equilibrium approach, Advances in Physics 65 (2) (2016) 58–238.doi:10.1080/00018732.2016.1194044

    work page doi:10.1080/00018732.2016.1194044 2016

  7. [7]

    Murakami, D

    Y. Murakami, D. Golež, M. Eckstein, P. Werner, Photoinduced nonequilibrium states in mott insulators, Rev. Mod. Phys. 97 (2025) 035001.doi:10.1103/tkjh-lr83

    work page doi:10.1103/tkjh-lr83 2025

  8. [8]

    Schüler, D

    M. Schüler, D. Golez, Y. Murakami, N. Bittner, A. Herrmann, H. U. Strand, P. Werner, M. Eckstein, Nessi: The non-equilibrium systems simulation package, Comput. Phys. Commun. 257 (2020) 107484. doi:https://doi.org/10.1016/j.cpc.2020.107484

    work page doi:10.1016/j.cpc.2020.107484 2020

  9. [9]

    Aryasetiawan, O

    F. Aryasetiawan, O. Gunnarsson, The gw method, Reports on Progress in Physics 61 (3) (1998) 237. doi:10.1088/0034-4885/61/3/002

    work page doi:10.1088/0034-4885/61/3/002 1998

  10. [10]

    Golež, P

    D. Golež, P. Werner, M. Eckstein, Photoinduced gap closure in an excitonic insulator, Phys. Rev. B 94 (2016) 035121.doi:10.1103/PhysRevB.94.035121

    work page doi:10.1103/physrevb.94.035121 2016

  11. [11]

    N. E. Bickers, D. J. Scalapino, S. R. White, Conserving approximations for strongly correlated electron systems: Bethe-salpeter equation and dynamics for the two-dimensional hubbard model, Phys. Rev. Lett. 62 (1989) 961–964.doi:10.1103/PhysRevLett.62.961

    work page doi:10.1103/physrevlett.62.961 1989

  12. [12]

    Sayyad, N

    S. Sayyad, N. Tsuji, A. Vaezi, M. Capone, M. Eckstein, H. Aoki, Momentum-dependent relaxation dynamics of the doped repulsive hubbard model, Phys. Rev. B 99 (2019) 165132.doi:10.1103/ PhysRevB.99.165132

    2019

  13. [13]

    Stahl, M

    C. Stahl, M. Eckstein, Electronic and fluctuation dynamics following a quench to the superconducting phase, Phys. Rev. B 103 (2021) 035116.doi:10.1103/PhysRevB.103.035116

    work page doi:10.1103/physrevb.103.035116 2021

  14. [14]

    H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, P. Werner, Nonequilibrium dynamical mean-field theory and its applications, Rev. Mod. Phys. 86 (2014) 779–837.doi:10.1103/RevModPhys.86.779

    work page doi:10.1103/revmodphys.86.779 2014

  15. [15]

    A. Stan, N. E. Dahlen, R. van Leeuwen, Time propagation of the kadanoff-baym equations for inhomo- geneous systems, J. Chem. Phys. 130 (22) (2009) 224101.doi:10.1063/1.3127247. 44

    work page doi:10.1063/1.3127247 2009

  16. [16]

    J. K. Freericks, V. M. Turkowski, V. Zlatić, Nonequilibrium dynamical mean-field theory, Phys. Rev. Lett. 97 (2006) 266408.doi:10.1103/PhysRevLett.97.266408

    work page doi:10.1103/physrevlett.97.266408 2006

  17. [17]

    Balzer, M

    K. Balzer, M. Bonitz, Nonequilibrium Green’s Functions Approach to Inhomogeneous Systems, Lecture Notes in Physics, Springer Berlin Heidelberg, 2012

    2012

  18. [18]

    J. Kaye, D. Golež, Low rank compression in the numerical solution of the nonequilibrium Dyson equa- tion, SciPost Phys. 10 (2021) 091.doi:10.21468/SciPostPhys.10.4.091

    work page doi:10.21468/scipostphys.10.4.091 2021

  19. [19]

    Shinaoka, M

    H. Shinaoka, M. Wallerberger, Y. Murakami, K. Nogaki, R. Sakurai, P. Werner, A. Kauch, Multiscale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains, Phys. Rev. X 13 (2023) 021015.doi:10.1103/PhysRevX.13.021015

    work page doi:10.1103/physrevx.13.021015 2023

  20. [20]

    Środa, K

    M. Środa, K. Inayoshi, H. Shinaoka, P. Werner, Memory-efficient nonequilibrium green’s function frame- work built on quantics tensor trains, Phys. Rev. Lett. 135 (2025) 226501.doi:10.1103/dxfb-b3l5. URLhttps://link.aps.org/doi/10.1103/dxfb-b3l5

    work page doi:10.1103/dxfb-b3l5 2025

  21. [21]

    Meirinhos, M

    F. Meirinhos, M. Kajan, J. Kroha, T. Bode, Adaptive numerical solution of Kadanoff-Baym equations, SciPost Phys. Core 5 (2022) 030.doi:10.21468/SciPostPhysCore.5.2.030

    work page doi:10.21468/scipostphyscore.5.2.030 2022

  22. [22]

    J. Lang, S. Sachdev, S. Diehl, Numerical renormalization of glassy dynamics, Phys. Rev. Lett. 135 (2025) 247101.doi:10.1103/z64g-nqs6

    work page doi:10.1103/z64g-nqs6 2025

  23. [23]

    J. Yin, Y. hao Chan, F. H. da Jornada, D. Y. Qiu, S. G. Louie, C. Yang, Using dynamic mode decomposition to predict the dynamics of a two-time non-equilibrium green’s function, Journal of Computational Science 64 (2022) 101843.doi:https://doi.org/10.1016/j.jocs.2022.101843

    work page doi:10.1016/j.jocs.2022.101843 2022

  24. [24]

    Y. Zhu, J. Yin, C. C. Reeves, C. Yang, V. Vlček, Predicting nonequilibrium green’s function dynamics and photoemission spectra via nonlinear integral operator learning, Machine Learning: Science and Technology 6 (1) (2025) 015027.doi:10.1088/2632-2153/ada99d

    work page doi:10.1088/2632-2153/ada99d 2025

  25. [25]

    andˇSpiˇ cka, V

    P. Lipavský, V. Špička, B. Velický, Generalized kadanoff-baym ansatz for deriving quantum transport equations, Phys. Rev. B 34 (1986) 6933–6942.doi:10.1103/PhysRevB.34.6933

    work page doi:10.1103/physrevb.34.6933 1986

  26. [26]

    & Bonitz, M

    N. Schlünzen, J.-P. Joost, M. Bonitz, Achieving the scaling limit for nonequilibrium green functions simulations, Phys. Rev. Lett. 124 (2020) 076601.doi:10.1103/PhysRevLett.124.076601

    work page doi:10.1103/physrevlett.124.076601 2020

  27. [27]

    Schüler, M

    M. Schüler, M. Eckstein, P. Werner, Truncating the memory time in nonequilibrium dynamical mean field theory calculations, Phys. Rev. B 97 (2018) 245129.doi:10.1103/PhysRevB.97.245129

    work page doi:10.1103/physrevb.97.245129 2018

  28. [28]

    Stahl, N

    C. Stahl, N. Dasari, J. Li, A. Picano, P. Werner, M. Eckstein, Memory truncated kadanoff-baym equations, Phys. Rev. B 105 (2022) 115146.doi:10.1103/PhysRevB.105.115146

    work page doi:10.1103/physrevb.105.115146 2022

  29. [29]

    Picano, M

    A. Picano, M. Eckstein, Accelerated gap collapse in a slater antiferromagnet, Phys. Rev. B 103 (2021) 165118.doi:10.1103/PhysRevB.103.165118

    work page doi:10.1103/physrevb.103.165118 2021

  30. [30]

    Dasari, J

    N. Dasari, J. Li, P. Werner, M. Eckstein, Photoinduced strange metal with electron and hole quasipar- ticles, Phys. Rev. B 103 (2021) L201116.doi:10.1103/PhysRevB.103.L201116

    work page doi:10.1103/physrevb.103.l201116 2021

  31. [31]

    Lange, Z

    F. Lange, Z. Lenarčič, A. Rosch, Pumping approximately integrable systems, Nature Communications 8 (1) (Jun. 2017).doi:10.1038/ncomms15767

    work page doi:10.1038/ncomms15767 2017

  32. [32]

    J. Li, M. Eckstein, Nonequilibrium steady-state theory of photodoped mott insulators, Phys. Rev. B 103 (2021) 045133.doi:10.1103/PhysRevB.103.045133

    work page doi:10.1103/physrevb.103.045133 2021

  33. [33]

    Künzel, A

    F. Künzel, A. Erpenbeck, D. Werner, E. Arrigoni, E. Gull, G. Cohen, M. Eckstein, Numerically ex- act simulation of photodoped mott insulators, Phys. Rev. Lett. 132 (2024) 176501.doi:10.1103/ PhysRevLett.132.176501. 45

    2024

  34. [34]

    R. E. V. Profumo, C. Groth, L. Messio, O. Parcollet, X. Waintal, Quantum monte carlo for correlated out-of-equilibrium nanoelectronic devices, Phys. Rev. B 91 (2015) 245154.doi:10.1103/PhysRevB. 91.245154

    work page doi:10.1103/physrevb 2015

  35. [35]

    Erpenbeck, E

    A. Erpenbeck, E. Gull, G. Cohen, Quantum monte carlo method in the steady state, Phys. Rev. Lett. 130 (2023) 186301.doi:10.1103/PhysRevLett.130.186301

    work page doi:10.1103/physrevlett.130.186301 2023

  36. [36]

    Eckstein, Solving quantum impurity models in the non-equilibrium steady state with tensor trains (2024).arXiv:2410.19707

    M. Eckstein, Solving quantum impurity models in the non-equilibrium steady state with tensor trains (2024).arXiv:2410.19707

    work page arXiv 2024

  37. [37]

    A. J. Kim, P. Werner, Strong coupling impurity solver based on quantics tensor cross interpolation, Phys. Rev. B 111 (2025) 125120.doi:10.1103/PhysRevB.111.125120

    work page doi:10.1103/physrevb.111.125120 2025

  38. [38]

    Bl´ azquez-Salcedo, C

    E. Arrigoni, M. Knap, W. von der Linden, Nonequilibrium dynamical mean-field theory: An auxiliary quantum master equation approach, Phys. Rev. Lett. 110 (2013) 086403.doi:10.1103/PhysRevLett. 110.086403

    work page doi:10.1103/physrevlett 2013

  39. [39]

    Frigo, S

    M. Frigo, S. Johnson, The design and implementation of fftw3, Proceedings of the IEEE 93 (2) (2005) 216–231.doi:10.1109/JPROC.2004.840301

    work page doi:10.1109/jproc.2004.840301 2005

  40. [40]

    Press, Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press, 2007

    W. Press, Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press, 2007

    2007

  41. [41]

    S. G. J. Matteo Frigo, FFTW online manual, Massachusetts Institute of Technology (2020) [cited 30.09.2025]. URLhttps://www.fftw.org/fftw3.pdf

    2020

  42. [42]

    Chowdhury, A

    D. Chowdhury, A. Georges, O. Parcollet, S. Sachdev, Sachdev-ye-kitaev models and beyond: Window into non-fermi liquids, Rev. Mod. Phys. 94 (2022) 035004.doi:10.1103/RevModPhys.94.035004

    work page doi:10.1103/revmodphys.94.035004 2022

  43. [43]

    Moeckel, S

    M. Moeckel, S. Kehrein, Interaction quench in the hubbard model, Phys. Rev. Lett. 100 (2008) 175702. doi:10.1103/PhysRevLett.100.175702

    work page doi:10.1103/physrevlett.100.175702 2008

  44. [44]

    Eckstein, M

    M. Eckstein, M. Kollar, P. Werner, Thermalization after an interaction quench in the hubbard model, Phys. Rev. Lett. 103 (2009) 056403.doi:10.1103/PhysRevLett.103.056403

    work page doi:10.1103/physrevlett.103.056403 2009

  45. [45]

    Werner, T

    P. Werner, T. Oka, M. Eckstein, A. J. Millis, Weak-coupling quantum monte carlo calculations on the keldysh contour: Theory and application to the current-voltage characteristics of the anderson model, Phys. Rev. B 81 (2010) 035108.doi:10.1103/PhysRevB.81.035108

    work page doi:10.1103/physrevb.81.035108 2010

  46. [46]

    A.Georges, G.Kotliar, W.Krauth, M.J.Rozenberg, Dynamicalmean-fieldtheoryofstronglycorrelated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68 (1996) 13–125.doi:10.1103/ RevModPhys.68.13

    1996

  47. [47]

    Physical Review B81, 195107 (2010) https://doi.org/10.1103/PhysRevB.81

    M. Eckstein, M. Kollar, P. Werner, Interaction quench in the hubbard model: Relaxation of the spectral function and the optical conductivity, Phys. Rev. B 81 (2010) 115131.doi:10.1103/PhysRevB.81. 115131. 46

    work page doi:10.1103/physrevb.81 2010