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REVIEW 2 major objections 6 minor 114 references

Generalized Keldysh formalism for nonequilibrium correlation functions and its application to fluctuation dynamics

T0 review · 2 major / 6 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Vertex corrections reverse short-time order-parameter fluctuation dynamics after a quench and grow both amplitude and lifetime near a nonthermal critical point.

desk verdict Solid computational advance: a matrix-free linear equation for vertex-corrected nonequilibrium two-particle functions, with a clean demonstration that vertices reverse short-time fluctuation dynamics near a nonthermal critical point. read the letter →

arxiv 2607.11055 v1 pith:MTETRFNK submitted 2026-07-13 cond-mat.str-el

Ken Inayoshi , Hiroshi Shinaoka , Yuta Murakami This is my paper
classification cond-mat.str-el
keywords nonequilibriumGreen’sfunctionsgeneralizedKeldyshformalismvertexcorrectionsquanticstensortrainsDMFTorder-parameterfluctuationsnonthermalcriticalpointinteractionquench
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Time-resolved spectroscopies now measure collective fluctuations in driven quantum materials, but theory has struggled to compute the needed two-particle correlation functions once vertex corrections (collective effects) are kept. This paper introduces a contour-dependent virtual probe field in a generalized Keldysh setup and converts the response into a single linear integral equation for the variation of the one-particle Green’s function, solved without building the four-time vertex kernel by a matrix-free Krylov method on quantics tensor trains. Combined with nonequilibrium dynamical mean-field theory for an interaction quench of the antiferromagnetic Hubbard model, the method shows that including vertex corrections qualitatively reverses the early-time evolution of the order-parameter fluctuation and that both the peak fluctuation and its decay time increase as the system approaches the nonthermal critical point. The result supplies a practical route to the correlation functions that experiments and entanglement diagnostics are beginning to demand far from equilibrium.

What carries the argument

The linear integral equation for the Green’s-function variation under a contour-dependent virtual probe (Eq. 3), solved by a matrix-free GMRES Krylov method that only needs the two-time action of the vertex kernel and represents two-time objects as quantics tensor trains.

What would settle it

Repeat the same interaction quench with a higher-order or non-perturbative impurity solver and check whether the short-time sign reversal of the fluctuation and the growth of peak amplitude and decay time near the nonthermal critical point survive.

Watch

Extended reading notes

Core claim

Once the unperturbed one-particle Green’s function is known, the full vertex-corrected nonequilibrium two-particle correlation function is obtained from the solution of a linear integral equation for its response to a contour-dependent virtual probe; applied to a quench into a nonequilibrium symmetry-broken state, that response shows that vertex corrections reverse the short-time fluctuation dynamics and that both the fluctuation amplitude and its decay time grow near the nonthermal critical point.

Load-bearing premise

The impurity self-energy is kept only at second-order Born level (Hartree–Fock plus second-order diagrams); a different truncation or a non-perturbative impurity solver could move the nonthermal critical point and change the size of the vertex corrections.

Editorial extensions

If this is right

  • Nonequilibrium spectra measured by time-resolved Raman or RIXS can be computed with the vertex corrections that control collective modes, rather than with the bubble approximation alone.
  • Order-parameter fluctuation amplitude and lifetime become practical diagnostics of nonthermal criticality, complementary to the order parameter itself.
  • The same linear-response equation can be fed any self-energy whose Jacobian-vector product is available, including diagrammatic or quantum-Monte-Carlo impurity solvers.
  • Quantics-tensor-train compression makes the nine-component generalized Keldysh storage feasible for multi-orbital and larger-lattice systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the nonthermal critical enhancement of fluctuations is generic, pump-probe experiments that track equal-time spin or charge variance should see a clear peak in both amplitude and recovery time as the drive strength is tuned through the nonthermal critical region.
  • Encoding the probe time as an extra quantics leg, as the authors sketch, would turn a sequence of independent linear solves into one compressed solve and could make dense frequency-domain spectra routine.
  • The same contour-probe construction applies immediately to other one-particle operators (current, pairing, orbital occupancy), so the method can target the full set of two-particle channels that enter quantum Fisher information and related entanglement witnesses.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The manuscript develops a generalized Keldysh framework for nonequilibrium two-particle correlation functions by introducing a contour-dependent virtual probe field. From the nonequilibrium Dyson equation it derives a linear integral equation (Eq. 3) for the Green’s-function variation G′ that incorporates vertex corrections without assembling the four-time kernel K, and solves it with a matrix-free GMRES/QTT scheme. Combined with nonequilibrium DMFT (second-order Born impurity self-energy) for an interaction quench of the half-filled Hubbard model on the Bethe lattice, the method shows that vertex corrections reverse the short-time sign of the order-parameter fluctuation σ_m and that both the peak fluctuation and its decay time grow as the quench approaches the nonthermal critical point.

Significance. Computing vertex-corrected nonequilibrium two-particle correlators is a recognized bottleneck for interpreting time-resolved Raman and RIXS experiments. The linear equation for G′ together with the matrix-free QTT Krylov solver is a concrete, reusable algorithmic advance: it reduces memory from O(N_t^4) to two-time operations (or O(D_max^3) in QTT form) and extends naturally to denser kernels via Jacobian-vector products (End Matter). Short-time benchmarks against direct GCH pump-probe simulations (End Matter Fig. 4), public code (tensor4all-rs), and a controlled DMFT demonstration that vertex corrections are qualitatively essential strengthen the contribution. The physical finding that nonthermal criticality is visible in two-particle fluctuations is of independent interest within the nonequilibrium DMFT community.

major comments (2)
  1. The central methodological claim (Eq. 3 and the matrix-free solver) is sound and well supported by the short-time GCH pump-probe benchmark (End Matter Fig. 4). No load-bearing technical error was found in the derivation or the QTT/GMRES implementation.
  2. Figs. 2(e,f) and 3(b): the claim that fluctuation decay time τ_χ grows toward the nonthermal critical point U_nc_c rests on a narrow window of U_f (1.1–1.3) and a fit interval t∈[24,32) that coincides with the stated GMRES limit t_max=32. The text itself notes that the vertex-corrected τ_χ may appear to approach a different critical point and that longer times are needed. The abstract and conclusion should qualify this statement more carefully (e.g., “within the accessible window, both σ_m^max and τ_χ increase as U_f approaches U_nc_c”) so that the strongest physical claim is not overstated relative to the numerical reach.
minor comments (6)
  1. Fig. 1(c) caption and main text: the equal-time fluctuation is written σ_m(t_p)=−Im χ^<(t_p,t_p); a one-sentence reminder that this equals N(⟨m̂ m̂⟩−⟨m̂⟩²) (Eq. 6) would help readers who jump to the figure.
  2. End Matter / Algorithm 1: state the restarted GMRES subspace size m and the typical number of outer restarts used for the production runs in Figs. 2–3, so that the reported residual tolerances are reproducible.
  3. Supplemental Material Eqs. (S10)–(S11): the sparse structure of K for the second-order self-energy is useful; a brief cross-reference from the End Matter paragraph on computational cost would make the contrast with dense kernels (e.g., GW) clearer.
  4. Fig. 3(a): the linear fits used to extract U_nc_c from 1/τ and ω_H should report the fit ranges and uncertainties (or at least the numerical value of U_nc_c) in the caption or Supplemental Material.
  5. Notation: the same symbol χ is used for the full contour correlator and for its lesser component; a consistent superscript convention (already used in places) would reduce ambiguity.
  6. References: the recent functional-derivative Raman work on the Falicov–Kimball model (Ref. 36) is appropriately cited; a short sentence distinguishing the present linear-equation approach from that finite-difference GCH pump-probe scheme would help non-specialists.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the linear integral equation for G' and the fluctuation results are independent computational outputs, not inputs re-labeled as predictions.

full rationale

The paper's central methodological claim is the derivation of the linear integral equation (3) for the variation G' = δG/δh from the difference of the perturbed and unperturbed Dyson equations of a generalized contour-dependent Hamiltonian. That equation is solved matrix-free by GMRES acting only on two-time objects (via analytic Σ'_int or JVP), without constructing the four-time kernel K; the QTT representation is a numerical compression, not a physical ansatz. The physical demonstration (vertex corrections reverse the short-time sign of σ_m; peak amplitude and decay time grow near the nonthermal critical point) is an output of that solver applied inside a previously studied second-order Born DMFT quench setup. Short-time benchmarks against independent GCH pump-probe simulations (End Matter Fig. 4) and the RPA argument for the Hartree–Fock-driven drop further corroborate the results. Self-citations (to prior QTT-NEGF libraries and to the nonthermal-criticality literature) supply independent numerical infrastructure or known one-particle phenomenology; none of them is used as a uniqueness theorem that forces the two-particle conclusions. No parameter is fitted to the target fluctuation data and then re-presented as a prediction. The derivation chain is therefore self-contained and non-circular.

Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

The central claims rest on standard nonequilibrium Green’s-function identities, the DMFT locality assumption, a second-order diagrammatic truncation, and numerical controls of the QTT/GMRES solver. No new physical entities are postulated; free parameters are numerical cutoffs and the model parameters of the quench.

free parameters (5)
  • QTT max bond dimension D_max = 50–100
    Set by hand to 50–100; controls memory and accuracy of all two-time objects.
  • QTT truncation tolerance = ~1e-6
    Frobenius-norm cutoff ~10^{-6} applied after every contraction.
  • GMRES residual tolerance = 1e-5 to 1e-4
    Convergence criterion 10^{-5}–10^{-4}; affects long-time reliability.
  • Initial inverse temperature beta = 20
    Fixed at 20 for the equilibrium AFM state before the quench.
  • Quench values U_i -> U_f = U_i=2.0, U_f=1.0–1.9
    Chosen by hand to scan across the known nonthermal critical region.
assumptions (4)
  • domain assumption Generalized Keldysh contour and functional derivative of the contour-ordered expectation value yield the two-particle correlation function (Eq. 1).
    Standard extension of the Kubo formula to the Keldysh contour; cited from Canovi et al. and textbooks.
  • domain assumption Nonequilibrium DMFT locality: lattice self-energy is local and equal to the impurity self-energy.
    Exact only in infinite dimensions; used throughout the demonstration.
  • ad hoc to paper Impurity self-energy truncated at Hartree–Fock plus second-order Born diagrams.
    Explicit approximation stated in the Demonstration section and Supplemental Material; higher orders omitted.
  • domain assumption Quantics tensor-train representation of two-time Green’s functions preserves the necessary diagrammatic operations to controlled accuracy.
    Taken from prior QTT-NEGF literature of the same authors and collaborators.
invented entities (1)
  • perturbative GCH method (linear integral equation for G')
    purpose: Compute the variation of the Green’s function without self-consistent pump-probe simulations or storage of the four-time kernel.
    New algorithmic object introduced in this work; not a physical particle or force.

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Cite this review

Pith. "Pith review of Generalized Keldysh formalism for nonequilibrium correlation functions and its application to fluctuation dynamics." pith.science (2026). https://pith.science/paper/MTETRFNK

@misc{pith2026260711055,
  author       = {Pith},
  title        = {Pith review of: Generalized Keldysh formalism for nonequilibrium correlation functions and its application to fluctuation dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MTETRFNK}},
  note         = {Machine review of arXiv:2607.11055}
}
read the original abstract

Recent advances in time-resolved spectroscopies provide increasing access to collective dynamics in correlated quantum materials. However, computing the corresponding nonequilibrium two-particle correlation functions remains a major challenge. Here, by introducing a contour-dependent virtual probe field within the generalized Keldysh formalism, we propose an approach that computes such correlation functions with the vertex corrections essential for describing collective dynamics. In particular, we introduce a linear integral equation that computes the correlation functions without explicitly constructing the four-time vertex kernel, and develop its matrix-free Krylov solver based on quantics tensor trains. Combining our method with nonequilibrium dynamical mean-field theory, we show that the fluctuation dynamics of the order parameter in a nonequilibrium symmetry-broken state depends significantly on whether vertex corrections are included, and that the fluctuation and its decay time grow near the nonthermal critical point. Our approach thus provides a practical route for evaluating nonequilibrium correlation functions, which are emerging as key observables for characterizing states far from equilibrium.

Figures

Figures reproduced from arXiv: 2607.11055 by the authors.

Figure 1
Figure 1. (a) Keldysh contour C and its forward real-time (C1), backward real-time (C2), and Matsubara (C3) branches. The blue Gaussian represents the contour-dependent external field 𝐹(𝑧) centered at the probe time 𝑧 𝑝. (b) Diagrammatic expression of the linear integral equation (3). The tensor network diagram is for 𝐺𝑖 𝑗, defined as 𝐺𝑖 𝑗(𝑡, 𝑡′ ) ≡ 𝐺(𝑡 ∈ C𝑖 , 𝑡′ ∈ C𝑗) for 𝑖, 𝑗 = 1, 2, 3. (c) Time evolution of the imaginary p… view at source ↗
Figure 2
Figure 2. Nonequilibrium dynamics of the staggered magnetization [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Various quantities used to characterize the qualitative [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Comparison of the spin-spin correlation function [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

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Works this paper leans on

114 extracted references · 6 linked inside Pith

  1. [1]

    H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and 6 P. Werner, Nonequilibrium dynamical mean-field theory and its applications, Rev. Mod. Phys.86, 779 (2014)

    2014

  2. [2]

    Giannetti, M

    C. Giannetti, M. Capone, D. Fausti, M. Fabrizio, F. Parmigiani, and D. Mihailovic, Ultrafast optical spectroscopy of strongly correlated materials and high-temperature superconductors: a non-equilibrium approach, Advances in Physics65, 58 (2016), https://doi.org/10.1080/00018732.2016.1194044

    doi:10.1080/00018732.2016.1194044 2016

  3. [3]

    de la Torre, D

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

    2021

  4. [4]

    Murakami, D

    Y. Murakami, D. Goleˇ z, M. Eckstein, and P. Werner, Pho- toinduced nonequilibrium states in mott insulators, Rev. Mod. Phys.97, 035001 (2025)

    2025

  5. [5]

    Caruso, M

    F. Caruso, M. A. Sentef, C. Attaccalite, M. Bonitz, C. Draxl, U. De Giovannini, M. Eckstein, R. Ernstorfer, M. Fechner, M. Gr¨ uning, H. H¨ ubener, J.-P. Joost, D. M. Juraschek, C. Kar- rasch, D. M. Kennes, S. Latini, I.-T. Lu, O. Neufeld, E. Perfetto, L. Rettig, R. R. Pela, A. Rubio, J. F. Rudzinski, M. Ruggen- thaler, D. Sangalli, M. Sch¨ uler, S. Shall...

    2025

  6. [6]

    J. A. Kash, J. C. Tsang, and J. M. Hvam, Subpicosecond time- resolved raman spectroscopy of lo phonons in gaas, Phys. Rev. Lett.54, 2151 (1985)

    1985

  7. [7]

    J.-A. Yang, S. Parham, D. Dessau, and D. Reznik, Novel electron-phonon relaxation pathway in graphite revealed by time-resolved raman scattering and angle-resolved photoemis- sion spectroscopy, Scientific Reports7, 40876 (2017)

    2017

  8. [8]

    R. B. Versteeg, J. Zhu, P. Padmanabhan, C. Boguschewski, R. German, M. Goedecke, P. Becker, and P. H. M. van Loosdrecht, A tunable time-resolved spontaneous raman spec- troscopy setup for probing ultrafast collective excitation and quasiparticle dynamics in quantum materials, Structural Dy- namics5, 044301 (2018)

    2018

Show all 114 references
  1. [9]

    J. Zhu, R. B. Versteeg, P. Padmanabhan, and P. H. M. van Loosdrecht, Dynamical resonance quench and fano interfer- ence in spontaneous raman scattering from quasiparticle and collective excitations, Phys. Rev. B99, 094305 (2019)

    2019

  2. [10]

    S. Han, C. Boguschewski, Y. Gao, L. Xiao, J. Zhu, and P. H. M. van Loosdrecht, Incoherent phonon population and exciton- exciton annihilation dynamics in monolayer ws2 revealed by time-resolved resonance raman scattering, Opt. Express27, 29949 (2019)

    2019

  3. [11]

    J.-A. Yang, N. Pellatz, T. Wolf, R. Nandkishore, and D. Reznik, Ultrafast magnetic dynamics in insulating yba2cu3o6.1 re- vealed by time resolved two-magnon raman scattering, Nature Communications11, 2548 (2020)

    2020

  4. [12]

    Pellatz, S

    N. Pellatz, S. Roy, J.-W. Lee, J. L. Schad, H. Kandel, N. Arndt, C. B. Eom, A. F. Kemper, and D. Reznik, Relaxation timescales and electron-phonon coupling in optically pumped yba2cu3o6+𝑥 revealed by time-resolved raman scattering, Phys. Rev. B104, L180505 (2021)

    2021

  5. [13]

    T.-H. Chou, M. F ¨orst, M. Fechner, M. Henstridge, S. Roy, M. Buzzi, D. Nicoletti, Y. Liu, S. Nakata, B. Keimer, and A. Cavalleri, Ultrafast raman thermometry in driven yba2cu3o6.48, Phys. Rev. B109, 195141 (2024)

    2024

  6. [14]

    Reuveni, M

    G. Reuveni, M. L. Greenberg, M. Menahem, O. Hellman, and O. Yaffe, Resolving transient electron-phonon coupling with time-resolved spontaneous raman spectroscopy (2026), arXiv:2603.09755 [cond-mat.mtrl-sci]

    2026

  7. [15]

    M. P. M. Dean, Y. Cao, X. Liu, S. Wall, D. Zhu, R. Mankowsky, V. Thampy, X. M. Chen, J. G. Vale, D. Casa, J. Kim, A. H. Said, P. Juhas, R. Alonso-Mori, J. M. Glownia, A. Robert, J. Robin- son, M. Sikorski, S. Song, M. Kozina, H. Lemke, L. Patthey, S. Owada, T. Katayama, M. Yab...

    2016

  8. [16]

    Y. Cao, D. G. Mazzone, D. Meyers, J. P. Hill, X. Liu, S. Wall, and M. P. M. Dean, Ultrafast dynamics of spin and orbital correlations in quantum materials: an energy- and momentum-resolved perspective, Philosoph- ical Transactions of the Royal Society A: Mathemati- cal, Physic...

    2019 doi

  9. [17]

    Mitrano, S

    M. Mitrano, S. Lee, A. A. Husain, L. Delacretaz, M. Zhu, G. de la Pe˜na Munoz, S. X.-L. Sun, Y. I. Joe, A. H. Reid, S. F. Wandel, G. Coslovich, W. Schlotter, T. van Driel, J. Schnee- loch, G. D. Gu, S. Hartnoll, N. Goldenfeld, and P. Abba- monte, Ultrafast time-resolved x-ray ...

    2019

  10. [18]

    Parchenko, E

    S. Parchenko, E. Paris, D. McNally, E. Abreu, M. Dantz, E. M. Bothschafter, A. H. Reid, W. F. Schlotter, M.-F. Lin, S. F. Wandel, G. Coslovich, S. Zohar, G. L. Dakovski, J. J. Turner, S. Moeller, Y. Tseng, M. Radovic, C. Saathe, M. Agaaker, J. E. Nordgren, S. L. Johnson, T. Sc...

    2020

  11. [19]

    Mitrano and Y

    M. Mitrano and Y. Wang, Probing light-driven quantum ma- terials with ultrafast resonant inelastic x-ray scattering, Com- munications Physics3, 184 (2020)

    2020

  12. [20]

    D. G. Mazzone, D. Meyers, Y. Cao, J. G. Vale, C. D. Dashwood, Y. Shi, A. J. A. James, N. J. Robinson, J. Lin, V. Thampy, Y. Tanaka, A. S. Johnson, H. Miao, R. Wang, T. A. Assefa, J. Kim, D. Casa, R. Mankowsky, D. Zhu, R. Alonso-Mori, S. Song, H. Yavas, T. Katayama, M. Yabashi,...

    2021

  13. [21]

    Mitrano, S

    M. Mitrano, S. Johnston, Y.-J. Kim, and M. P. M. Dean, Explor- ing quantum materials with resonant inelastic x-ray scattering, Phys. Rev. X14, 040501 (2024)

    2024

  14. [22]

    Merzoni, L

    G. Merzoni, L. Martinelli, S. Parchenko, S. F. R. TenHuisen, V. Lebedev, L. Adriano, R. Carley, N. Gerasimova, L. Mer- cadier, M. Teichmann, B. E. van Kuiken, Z. Yin, A. Alic, D. R. Baykusheva, S. G. Chiuzbaian, S. D. Conte, O. Dogadov, A. F¨ohlisch, M. W. Haverkort, M. Kusch,...

    2025 arXiv

  15. [23]

    D. Jost, J. Li, J. Hales, J. Sobota, G. Merzoni, L. Mar- tinelli, S. Ding, K. Xu, J. Schlappa, A. Scherz, R. Carley, B. E. V. Kuiken, T. C. Asmara, L. P. Hoang, L. Mercadier, S. Parchenko, M. Teichmann, P. S. Kirchmann, G. Ghir- inghelli, B. Moritz, Z.-X. Shen, T. P. Devereaux...

    2025

  16. [24]

    Padma, P

    H. Padma, P. Sharma, S. F. R. TenHuisen, F. Glerean, A. Roll, P. Zhou, S. Kundu, A. Romaguera, E. Skoropata, H. Ueda, B. Liu, E. Paris, Y. Wang, S. H. Lee, Z. Mao, M. P. M. Dean, E. W. Huang, E. Razzoli, Y. Wang, and M. Mitrano, Light- induced charge order mode in a metastable...

    2026

  17. [25]

    Hauke, M

    P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, Measur- ing multipartite entanglement through dynamic susceptibili- ties, Nature Physics12, 778 (2016)

    2016

  18. [26]

    Mathew, S

    G. Mathew, S. L. L. Silva, A. Jain, A. Mohan, D. T. Adroja, V. G. Sakai, C. V. Tomy, A. Banerjee, R. Goreti, A. V. N., R. Singh, and D. Jaiswal-Nagar, Experimental realization of multipartite entanglement via quantum fisher information in a uniform antiferromagnetic quantum sp...

    2020

  19. [27]

    Scheie, P

    A. Scheie, P. Laurell, A. M. Samarakoon, B. Lake, S. E. Nagler, G. E. Granroth, S. Okamoto, G. Alvarez, and D. A. Tennant, Witnessing entanglement in quantum magnets using neutron scattering, Phys. Rev. B103, 224434 (2021)

    2021

  20. [28]

    Laurell, A

    P. Laurell, A. Scheie, C. J. Mukherjee, M. M. Koza, M. Enderle, Z. Tylczynski, S. Okamoto, R. Coldea, D. A. Tennant, and G. Alvarez, Quantifying and controlling entanglement in the quantum magnet cs2cocl4, Phys. Rev. Lett.127, 037201 (2021)

    2021

  21. [29]

    Laurell, A

    P. Laurell, A. Scheie, E. Dagotto, and D. A. Ten- nant, Witnessing entanglement and quantum cor- relations in condensed matter: A review, Ad- vanced Quantum Technologies8, 2400196 (2025), https://advanced.onlinelibrary.wiley.com/doi/pdf/10.1002/qute.202400196

    2025 doi

  22. [30]

    Scheie, P

    A. Scheie, P. Laurell, W. Simeth, E. Dagotto, and D. A. Ten- nant, Tutorial: Extracting entanglement signatures from neu- tron spectroscopy, Materials Today Quantum5, 100020 (2025)

    2025

  23. [31]

    D. R. Baykusheva, M. H. Kalthoff, D. Hofmann, M. Claassen, D. M. Kennes, M. A. Sentef, and M. Mitrano, Witnessing nonequilibrium entanglement dynamics in a strongly corre- lated fermionic chain, Phys. Rev. Lett.130, 106902 (2023)

    2023

  24. [32]

    Hales, U

    J. Hales, U. Bajpai, T. Liu, D. R. Baykusheva, M. Li, M. Mi- trano, and Y. Wang, Witnessing light-driven entanglement us- ing time-resolved resonant inelastic x-ray scattering, Nature Communications14, 3512 (2023)

    2023

  25. [33]

    Y. Wang, T. P. Devereaux, and C.-C. Chen, Theory of time- resolved raman scattering in correlated systems: Ultrafast en- gineering of spin dynamics and detection of thermalization, Phys. Rev. B98, 245106 (2018)

    2018

  26. [34]

    O. P. Matveev, A. M. Shvaika, T. P. Devereaux, and J. K. Freericks, Stroboscopic tests for thermalization of electrons in pump-probe experiments, Phys. Rev. Lett.122, 247402 (2019)

    2019

  27. [35]

    Werner, M

    P. Werner, M. Eckstein, and N. Tsuji, Nonequilibrium dmft approach to time-resolved raman spectroscopy, Phys. Rev. B 108, 245157 (2023)

    2023

  28. [36]

    O. P. Matveev, A. M. Shvaika, and J. K. Freericks, Func- tional derivative approach to raman response in nonequilib- rium many-body systems described by dynamical mean-field theory, Phys. Rev. B113, 195148 (2026)

    2026

  29. [37]

    Y. Chen, Y. Wang, C. Jia, B. Moritz, A. M. Shvaika, J. K. Fre- ericks, and T. P. Devereaux, Theory for time-resolved resonant inelastic x-ray scattering, Phys. Rev. B99, 104306 (2019)

    2019

  30. [38]

    Y. Wang, Y. Chen, C. Jia, B. Moritz, and T. P. Devereaux, Time-resolved resonant inelastic x-ray scattering in a pumped mott insulator, Phys. Rev. B101, 165126 (2020)

    2020

  31. [39]

    Zawadzki, L

    K. Zawadzki, L. Yang, and A. E. Feiguin, Time-dependent approach to inelastic scattering spectroscopies in and away from equilibrium: Beyond perturbation theory, Phys. Rev. B 102, 235141 (2020)

    2020

  32. [40]

    Eckstein and P

    M. Eckstein and P. Werner, Simulation of time-dependent reso- nant inelastic x-ray scattering using nonequilibrium dynamical mean-field theory, Phys. Rev. B103, 115136 (2021)

    2021

  33. [41]

    Werner, S

    P. Werner, S. Johnston, and M. Eckstein, Nonequilibrium-dmft based rixs investigation of the two-orbital hubbard model, Eu- rophysics Letters133, 57005 (2021)

    2021

  34. [42]

    Eckstein and E

    M. Eckstein and E. Paprotzki, Floquet x-ray scattering as a probe of hidden electronic orders (2026), arXiv:2604.06496 [cond-mat.str-el]

    2026 arXiv

  35. [43]

    Rinc ´on and A

    J. Rinc ´on and A. E. Feiguin, Nonequilibrium optical response of a one-dimensional mott insulator, Phys. Rev. B104, 085122 (2021)

    2021

  36. [44]

    Kadanoff and G

    L. Kadanoff and G. Baym,Quantum Statistical Mechanics (W.A. Benjamin Inc., New York, 1962)

    1962

  37. [45]

    L. V. Keldysh, Diagram technique for nonequilibrium pro- cesses, J. Exptl. Theoret. Phys.47, 1515 (1964), [Sov. Phys. JETP,20, 1018 (1965)]

    1964

  38. [46]

    Bonitz,Quantum Kinetic Theory, 2nd ed

    M. Bonitz,Quantum Kinetic Theory, 2nd ed. (Springer, 2016)

    2016

  39. [47]

    Kamenev,Field Theory of Non-Equilibrium Systems, 2nd ed

    A. Kamenev,Field Theory of Non-Equilibrium Systems, 2nd ed. (Cambridge University Press, 2023)

    2023

  40. [48]

    Stefanucci and R

    G. Stefanucci and R. van Leeuwen,Nonequilibrium Many- body Theory of Quantum Systems: A Modern Introduction, 2nd ed. (Cambridge University Press, 2025)

    2025

  41. [49]

    Eckstein and M

    M. Eckstein and M. Kollar, Measuring correlated electron dy- namics with time-resolved photoemission spectroscopy, Phys. Rev. B78, 245113 (2008)

    2008

  42. [50]

    J. K. Freericks, H. R. Krishnamurthy, and T. Pruschke, Theoret- ical description of time-resolved photoemission spectroscopy: Application to pump-probe experiments, Phys. Rev. Lett.102, 136401 (2009)

    2009

  43. [51]

    Freericks and A

    J. Freericks and A. F. Kemper, What do the two times in two-time correlation functions mean for interpreting tr-arpes?, Journal of Electron Spectroscopy and Related Phenomena251, 147104 (2021)

    2021

  44. [52]

    Sentef, A

    M. Sentef, A. F. Kemper, B. Moritz, J. K. Freericks, Z.-X. Shen, and T. P. Devereaux, Examining electron-boson coupling using time-resolved spectroscopy, Phys. Rev. X3, 041033 (2013)

    2013

  45. [53]

    A. F. Kemper, M. A. Sentef, B. Moritz, J. K. Freericks, and T. P. Devereaux, Direct observation of higgs mode oscillations in the pump-probe photoemission spectra of electron-phonon mediated superconductors, Phys. Rev. B92, 224517 (2015)

    2015

  46. [54]

    Goleˇ z, P

    D. Goleˇ z, P. Werner, and M. Eckstein, Photoinduced gap clo- sure in an excitonic insulator, Phys. Rev. B94, 035121 (2016)

    2016

  47. [55]

    Randi, D

    F. Randi, D. Fausti, and M. Eckstein, Bypassing the energy- time uncertainty in time-resolved photoemission, Phys. Rev. B 95, 115132 (2017)

    2017

  48. [56]

    A. F. Kemper, M. A. Sentef, B. Moritz, T. P. Devereaux, and J. K. Freericks, Review of the theoretical description of time-resolved angle-resolved photoemission spectroscopy in electron-phonon mediated superconductors, Annalen der Physik529, 1600235 (2017)

    2017

  49. [57]

    A. F. Kemper, O. Abdurazakov, and J. K. Freericks, General principles for the nonequilibrium relaxation of populations in quantum materials, Phys. Rev. X8, 041009 (2018)

    2018

  50. [58]

    Perfetto, S

    E. Perfetto, S. Bianchi, and G. Stefanucci, Time-resolved arpes spectra of nonequilibrium excitonic insulators: Reveal- ing macroscopic coherence with ultrashort pulses, Phys. Rev. B101, 041201(R) (2020)

    2020

  51. [59]

    M. Eckstein, Time-resolved photoemission spectroscopy on correlated electrons: Insights from dynamical mean-field the- ory, Journal of Electron Spectroscopy and Related Phenomena 253, 147108 (2021)

    2021

  52. [60]

    Sch¨ uler and M

    M. Sch¨ uler and M. A. Sentef, Theory of subcycle time- 8 resolved photoemission: Application to terahertz photodress- ing in graphene, Journal of Electron Spectroscopy and Related Phenomena253, 147121 (2021)

    2021

  53. [61]

    Boschini, M

    F. Boschini, M. Zonno, and A. Damascelli, Time-resolved arpes studies of quantum materials, Rev. Mod. Phys.96, 015003 (2024)

    2024

  54. [62]

    Perfetto, D

    E. Perfetto, D. Sangalli, A. Marini, and G. Stefanucci, Nonequi- librium bethe-salpeter equation for transient photoabsorption spectroscopy, Phys. Rev. B92, 205304 (2015)

    2015

  55. [63]

    Kumar and A

    A. Kumar and A. F. Kemper, Higgs oscillations in time- resolved optical conductivity, Phys. Rev. B100, 174515 (2019)

    2019

  56. [64]

    Canovi, P

    E. Canovi, P. Werner, and M. Eckstein, First-order dynamical phase transitions, Phys. Rev. Lett.113, 265702 (2014)

    2014

  57. [65]

    I. V. Oseledets, Approximation of matrices with logarithmic number of parameters, Doklady Mathematics80, 653 (2009)

    2009

  58. [66]

    B. N. Khoromskij, O(dlog n)-quantics approximation of n-d tensors in high-dimensional numerical modeling, Constructive Approximation34, 257 (2011)

    2011

  59. [67]

    Shinaoka, M

    H. Shinaoka, M. Wallerberger, Y. Murakami, K. Nogaki, R. Sakurai, P. Werner, and A. Kauch, Multiscale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains, Phys. Rev. X13, 021015 (2023)

    2023

  60. [68]

    Murray, H

    M. Murray, H. Shinaoka, and P. Werner, Nonequilibrium dia- grammatic many-body simulations with quantics tensor trains, Phys. Rev. B109, 165135 (2024)

    2024

  61. [69]

    ´Sroda, K

    M. ´Sroda, K. Inayoshi, H. Shinaoka, and P. Werner, Memory- efficient nonequilibrium green’s function framework built on quantics tensor trains, Phys. Rev. Lett.135, 226501 (2025)

    2025

  62. [70]

    Inayoshi, M

    K. Inayoshi, M. ´Sroda, A. Kauch, P. Werner, and H. Shi- naoka, A causality-based divide-and-conquer algorithm for nonequilibrium Green’s function calculations with quantics tensor trains, SciPost Phys.20, 077 (2026)

    2026

  63. [71]

    ´Sroda, K

    M. ´Sroda, K. Inayoshi, M. Sch¨ uler, H. Shinaoka, and P. Werner, Predictor-corrector method based on dynamic mode decompo- sition for tensor-train nonequilibrium green’s function calcula- tions, Phys. Rev. B113, 165113 (2026)

    2026

  64. [72]

    ´Sroda and P

    M. ´Sroda and P. Werner, Nonequilibrium electron-phonon dynamics with high momentum resolution: Thermalization bottlenecks and the effects of phonon dispersion (2026), arXiv:2606.28855 [cond-mat.str-el]

    2026 arXiv

  65. [73]

    Metzner and D

    W. Metzner and D. Vollhardt, Correlated lattice fermions in 𝑑=∞dimensions, Phys. Rev. Lett.62, 324 (1989)

    1989

  66. [74]

    Georges and G

    A. Georges and G. Kotliar, Hubbard model in infinite dimen- sions, Phys. Rev. B45, 6479 (1992)

    1992

  67. [75]

    Georges, G

    A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dy- namical mean-field theory of strongly correlated fermion sys- tems and the limit of infinite dimensions, Rev. Mod. Phys.68, 13 (1996)

    1996

  68. [76]

    Schmidt and H

    P. Schmidt and H. Monien, Nonequilibrium dynamical mean- field theory of a strongly correlated system (2002), arXiv:cond- mat/0202046 [cond-mat.str-el]

    2002

  69. [77]

    J. K. Freericks, V. M. Turkowski, and V. Zlati ´c, Nonequilib- rium dynamical mean-field theory, Phys. Rev. Lett.97, 266408 (2006)

    2006

  70. [78]

    Eckstein,Nonequilibrium dynamical mean-field theory, doctoralthesis, Universit¨at Augsburg (2011)

    M. Eckstein,Nonequilibrium dynamical mean-field theory, doctoralthesis, Universit¨at Augsburg (2011)

    2011

  71. [79]

    Kubo, Statistical-mechanical theory of irreversible pro- cesses

    R. Kubo, Statistical-mechanical theory of irreversible pro- cesses. i. general theory and simple applications to magnetic and conduction problems, Journal of the Physical Society of Japan12, 570 (1957), https://doi.org/10.1143/JPSJ.12.570

    1957 doi

  72. [80]

    J. W. Negele and H. Orland,Quantum Many-particle Systems (Westview Press, 1998)

    1998

  73. [81]

    Eckstein and M

    M. Eckstein and M. Kollar, Theory of time-resolved optical spectroscopy on correlated electron systems, Phys. Rev. B78, 205119 (2008)

    2008

  74. [82]

    A. N. Rubtsov, V. V. Savkin, and A. I. Lichtenstein, Continuous-time quantum monte carlo method for fermions, Phys. Rev. B72, 035122 (2005)

    2005

  75. [83]

    Werner, A

    P. Werner, A. Comanac, L. de’ Medici, M. Troyer, and A. J. Millis, Continuous-time solver for quantum impurity models, Phys. Rev. Lett.97, 076405 (2006)

    2006

  76. [84]

    Werner and A

    P. Werner and A. J. Millis, Hybridization expansion impurity solver: General formulation and application to kondo lattice and two-orbital models, Phys. Rev. B74, 155107 (2006)

    2006

  77. [85]

    E. Gull, P. Werner, A. Millis, and M. Troyer, Performance anal- ysis of continuous-time solvers for quantum impurity models, Phys. Rev. B76, 235123 (2007)

    2007

  78. [86]

    E. Gull, P. Werner, O. Parcollet, and M. Troyer, Continuous- time auxiliary-field monte carlo for quantum impurity models, Europhysics Letters82, 57003 (2008)

    2008

  79. [87]

    M¨ uhlbacher and E

    L. M¨ uhlbacher and E. Rabani, Real-time path integral approach to nonequilibrium many-body quantum systems, Phys. Rev. Lett.100, 176403 (2008)

    2008

  80. [88]

    Werner, T

    P. Werner, T. Oka, and A. J. Millis, Diagrammatic monte carlo simulation of nonequilibrium systems, Phys. Rev. B79, 035320 (2009)

    2009

  81. [89]

    Schir ´o and M

    M. Schir ´o and M. Fabrizio, Real-time diagrammatic monte carlo for nonequilibrium quantum transport, Phys. Rev. B79, 153302 (2009)

    2009

  82. [90]

    Eckstein, M

    M. Eckstein, M. Kollar, and P. Werner, Thermalization after an interaction quench in the hubbard model, Phys. Rev. Lett.103, 056403 (2009)

    2009

  83. [91]

    Werner, T

    P. Werner, T. Oka, M. Eckstein, and A. J. Millis, Weak-coupling quantum monte carlo calculations on the keldysh contour: The- ory and application to the current-voltage characteristics of the anderson model, Phys. Rev. B81, 035108 (2010)

    2010

  84. [92]

    E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Continuous-time monte carlo meth- ods for quantum impurity models, Rev. Mod. Phys.83, 349 (2011)

    2011

  85. [93]

    A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, Automatic differentiation in machine learning: a survey, Jour- nal of machine learning research18, 1 (2018)

    2018

  86. [94]

    tensor4all, tenferro-rs: A rust-native tensor & autodiff stack for scientific computing,https://github.com/tensor4all/ tenferro-rs(2026)

    2026

  87. [95]

    Sch¨ uler, D

    M. Sch¨ uler, D. Goleˇ z, Y. Murakami, N. Bittner, A. Herrmann, H. U. Strand, P. Werner, and M. Eckstein, Nessi: The non- equilibrium systems simulation package, Computer Physics Communications257, 107484 (2020)

    2020

  88. [96]

    Nessi,https://nessi.readthedocs.io/en/latest/ index.html

  89. [97]

    Kaye and D

    J. Kaye and D. Goleˇ z, Low rank compression in the numerical solution of the nonequilibrium Dyson equation, SciPost Phys. 10, 091 (2021)

    2021

  90. [98]

    Blommel,Numerical Integration of the Kadanoff-Baym Equations, Phd thesis, University of Michigan (2024)

    T. Blommel,Numerical Integration of the Kadanoff-Baym Equations, Phd thesis, University of Michigan (2024)

    2024

  91. [99]

    Blommel, J

    T. Blommel, J. Kaye, Y. Murakami, E. Gull, and D. Goleˇ z, Chirped amplitude mode in photoexcited superconductors, Phys. Rev. B111, 094502 (2025)

    2025

  92. [100]

    Blommel, J

    T. Blommel, J. Kova ˇcevi´c, J. Kaye, E. Gull, J. Vuˇciˇcevi´c, and D. Goleˇ z, H-nessi: The hierarchical non-equilibrium systems simulation package (2026), arXiv:2604.05319 [cond-mat.str- el]

    2026 arXiv

  93. [101]

    Saad and M

    Y. Saad and M. H. Schultz, Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing7, 856 (1986), https://doi.org/10.1137/0907058. 9

    1986 doi

  94. [102]

    tensor4all, tensor4all-rs: A rust implementation of ten- sor networks,https://github.com/tensor4all/ tensor4all-rs(2026), rust port of the tensor4all tensor network ecosystem with C-API

    2026

  95. [103]

    Y. N. Fern ´andez, M. K. Ritter, M. Jeannin, J.-W. Li, T. Kloss, T. Louvet, S. Terasaki, O. Parcollet, J. von Delft, H. Shinaoka, and X. Waintal, Learning tensor networks with tensor cross interpolation: New algorithms and libraries, SciPost Phys.18, 104 (2025)

    2025

  96. [104]

    tensor4all,https://tensor4all.org/

  97. [105]

    Werner, N

    P. Werner, N. Tsuji, and M. Eckstein, Nonthermal symmetry- broken states in the strongly interacting hubbard model, Phys. Rev. B86, 205101 (2012)

    2012

  98. [106]

    Tsuji, M

    N. Tsuji, M. Eckstein, and P. Werner, Nonthermal antiferro- magnetic order and nonequilibrium criticality in the hubbard model, Phys. Rev. Lett.110, 136404 (2013)

    2013

  99. [107]

    Tsuji and P

    N. Tsuji and P. Werner, Nonequilibrium dynamical mean- field theory based on weak-coupling perturbation expansions: Application to dynamical symmetry breaking in the hubbard model, Phys. Rev. B88, 165115 (2013)

    2013

  100. [108]

    Picano and M

    A. Picano and M. Eckstein, Accelerated gap collapse in a slater antiferromagnet, Phys. Rev. B103, 165118 (2021)

    2021

  101. [109]

    Eckstein, M

    M. Eckstein, M. Kollar, K. Byczuk, and D. Vollhardt, Hopping on the bethe lattice: Exact results for densities of states and dynamical mean-field theory, Phys. Rev. B71, 235119 (2005)

    2005

  102. [110]

    Grosso, M

    G. Grosso, M. K. Ritter, S. Rohshap, S. Badr, A. Kauch, M. Wallerberger, J. von Delft, and H. Shi- naoka, Adaptive patching for tensor train computations (2026), arXiv:2602.22372 [physics.comp-ph]

    2026 arXiv

  103. [111]

    K¨ unzel, M

    F. K¨ unzel, M. Sch¨ uler, D. Goleˇ z, Y. Murakami, S. Ray, C. Stahl, J. Li, H. U. R. Strand, P. Werner, and M. Eckstein, Nessi 2.0: The non-equilibrium systems simulation package version 2.0 (2026), arXiv:2605.02606 [cond-mat.str-el]

    2026 arXiv

  104. [112]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calculations, SciPost Phys. Codebases , 4 (2022)

    2022

  105. [113]

    N´ u˜nez Fern´andez, M

    Y. N´ u˜nez Fern´andez, M. Jeannin, P. T. Dumitrescu, T. Kloss, J. Kaye, O. Parcollet, and X. Waintal, Learning feynman dia- grams with tensor trains, Phys. Rev. X12, 041018 (2022)

    2022

  106. [114]

    Generalized Keldysh formalism for nonequilibrium correlation functions and its application to fluctuation dynamics

    M. K. Ritter, Y. N´ u˜nez Fern ´andez, M. Wallerberger, J. von Delft, H. Shinaoka, and X. Waintal, Quantics tensor cross in- terpolation for high-resolution parsimonious representations of multivariate functions, Phys. Rev. Lett.132, 056501 (2024). 10 End Matter Evaluation ofΣ...

    2024

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