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arxiv: 2512.10899 · v3 · submitted 2025-12-11 · ❄️ cond-mat.str-el · cond-mat.dis-nn· quant-ph

Hybrid quantum-classical matrix-product state and Lanczos methods for electron-phonon systems with strong electronic correlations: Application to disordered systems coupled to Einstein phonons

Heiko Georg Menzler , Suman Mondal , Fabian Heidrich-Meisner This is my paper

Pith reviewed 2026-05-16 23:05 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.dis-nnquant-ph
keywords electron-phonon systemsmany-body localizationhybrid quantum-classical methodsmatrix-product statesLanczos methoddisordered systemsEinstein phononscharge density waves
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The pith

Coupling strongly disordered systems to classical oscillators leads to delocalization and destabilizes finite-size many-body localization

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

The paper develops hybrid quantum-classical methods for electron-phonon systems that treat electronic correlations exactly while handling optical phonons classically. A time-dependent Lanczos method and a matrix-product state method are each combined with multi-trajectory Ehrenfest dynamics. These are applied to interacting spinless fermions in one dimension with quenched disorder coupled to Einstein oscillators. The central result is that increasing electron-phonon coupling causes charge-density-wave order to decay and drives delocalization that destabilizes the many-body localized phase in finite systems. A sympathetic reader would care because the methods offer a route to simulate dynamics in correlated disordered systems and indicate that classical phonon coupling can overcome localization tendencies.

Core claim

The authors present two hybrid methods that treat electrons with exact quantum techniques (time-dependent Lanczos or matrix-product states) while approximating phonons classically via multi-trajectory Ehrenfest dynamics. For a chain of interacting spinless fermions with disorder coupled to Einstein oscillators, they provide numerical evidence that electron-phonon coupling induces delocalization, leading to decay of charge-density-wave order and destabilization of the finite-size many-body localized phase.

What carries the argument

Hybrid combination of quantum electron solvers (Lanczos or MPS) with classical multi-trajectory Ehrenfest dynamics for phonon oscillators

Load-bearing premise

Phonons can be treated as classical oscillators in the adiabatic regime of small frequencies.

What would settle it

A fully quantum treatment of the phonons at the same small frequencies that shows persistent many-body localization without delocalization would falsify the claim.

Figures

Figures reproduced from arXiv: 2512.10899 by Fabian Heidrich-Meisner, Heiko Georg Menzler, Suman Mondal.

Figure 1
Figure 1. Figure 1: (a) The Lanczos-MTE and (b) TEBD-MTE methods are graphically represented for a single time step ∆t. (a) The lowest layer (elongated black circle) represents the state of quantum and classical sub-system combined at time t. The evolution of the quantum sub-system using the Lanczos method is represented by the second layer (blue square), and the third layer (red diamond) represents the consecutive evolution … view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of TEBD-MTE (δ = 10−8 ), Lanczos-MTE (ϵ = 10−9/tf), and regular MTE as imple￾mented in Ref. [36] for a non-interacting system (L = 13, V = 0, ω0 = 0.1 t0, γ = 0.4 t0, W = 0). Lanczos-MTE and TEBD use ∆t = 0.01 t0 −1 and all methods are aver￾aged over Ntraj = 4000 trajectories. The inset shows the difference δOCDW between the many-body hybrid tech￾niques and the reference data from regular MTE fr… view at source ↗
Figure 3
Figure 3. Figure 3: Dependence of the Lanczos-MTE method on the control parameters time step ∆t and error rate ϵ for an interacting system (L = 14, V = 2 t0, ω0 = 0.1 t0, γ = 0.4 t0, W = 0). We plot the deviation δOCDW when (a) varying ϵ at fixed ∆t = 0.01 t0 −1 , comparing to a reference set with ϵ = 10−10/tf and (b) varying ∆t at fixed ϵ = 10−10/tf , comparing to a reference set with ∆t = 0.01 t0 −1 . The results are averag… view at source ↗
Figure 4
Figure 4. Figure 4: Dependence of the TEBD-MTE method on the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the TEBD-MTE method (δ = 10−8 ) and Lanczos-MTE method (ϵ = 10−8/tf ) in an interacting system (L = 14, V = 2 t0, ω0 = 0.1 t0, γ = 0.4 t0) for the case without disorder (W = 0). The in￾set shows the difference of the observable OCDW between the two methods as a function of time. Both methods employ a time step ∆t = 0.01 t0 −1 and are sampled over Ntraj = 4000 trajectories. Now, each term exp… view at source ↗
Figure 7
Figure 7. Figure 7: Time-evolved density profile ⟨nˆℓ(t)⟩ of a par￾ticle initially (t = 0) localized on a single lattice site ℓ = 50 on a lattice with L = 100 sites and with disorder (W = 8 t0). We show the (a) uncoupled case γ = 0, ex￾hibiting Anderson localization, and (b) a system with a large electron–phonon coupling γ = t0. Further parame￾ters are ω0 = 0.1 t0 and the data was generated using the Lanczos-MTE routine with … view at source ↗
Figure 9
Figure 9. Figure 9: Similar setup to Fig [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: Spreading of a single particle in the disordered [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: Decay of the CDW order parameter in an [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Sketches explaining relaxation processes in [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Dynamics in a single trajectory: Decay of [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Dependence of the TEBD-MTE method on the [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
read the original abstract

We present two quantum-classical hybrid methods for simulating the time-dependence of electron-phonon systems that treat electronic correlations numerically exactly and optical-phonon degrees of freedom classically. These are a time-dependent Lanczos and a matrix-product state method, each combined with the multi-trajectory Ehrenfest approach. Due to the approximations, reliable results are expected for the adiabatic regime of small phonon frequencies. We discuss the convergence properties of both methods for a system of interacting spinless fermions in one dimension and provide a benchmark for the Holstein chain. As a first application, we study the decay of charge density wave order in a system of interacting spinless fermions coupled to Einstein oscillators and in the presence of quenched disorder. We investigate the dependence of the relaxation dynamics on the electron-phonon coupling strength and provide numerical evidence that the coupling of strongly disordered systems to classical oscillators leads to delocalization, thus destabilizing the (finite-size) many-body localization in this system.

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 two hybrid quantum-classical methods—a time-dependent Lanczos approach and a matrix-product state method, each combined with multi-trajectory Ehrenfest dynamics—for simulating the real-time evolution of electron-phonon systems with strong electronic correlations. Electronic degrees of freedom are treated numerically exactly while optical phonons are handled classically; the methods are expected to be reliable in the adiabatic regime of small phonon frequencies. Convergence is discussed for interacting spinless fermions in one dimension, with benchmarks on the clean Holstein chain. As an application, the authors examine the decay of charge-density-wave order in a disordered system of interacting spinless fermions coupled to Einstein oscillators and report numerical evidence that this coupling induces delocalization, thereby destabilizing finite-size many-body localization.

Significance. If the classical phonon treatment is validated, the work supplies concrete numerical evidence that electron-phonon coupling can destabilize many-body localization in disordered systems, which is relevant to ongoing debates on MBL stability. The hybrid methods themselves represent a practical advance for treating correlated electron-phonon dynamics beyond small-system exact diagonalization.

major comments (2)
  1. [Application to disordered systems] Application section (disordered CDW decay): The headline claim that coupling to classical oscillators leads to delocalization and destabilizes finite-size MBL is based on multi-trajectory Ehrenfest results. The manuscript benchmarks convergence on the clean Holstein chain but does not report a direct comparison (even on small clusters) against a fully quantized phonon treatment at the same small ω. Without this control, it remains unclear whether the observed CDW decay arises from physical phonon-assisted processes or from uncontrolled averaging over classical trajectories that acts as an effective dephasing channel.
  2. [Methods and convergence] Methods and convergence discussion: The text states that reliable results are expected only in the adiabatic regime, yet the MBL phenomenology is known to be sensitive to quantum fluctuations. Additional quantitative tests—such as the dependence of the relaxation rate on the number of Ehrenfest trajectories or on the phonon frequency cutoff—would be needed to establish that the delocalization is not an artifact of the classical approximation.
minor comments (2)
  1. [Abstract] The abstract and introduction could explicitly list the disorder strength, electron-phonon coupling values, and phonon frequency range used in the disordered-system simulations.
  2. [Figures] Figures showing CDW order parameter decay should include error bars or standard deviations arising from disorder averaging and trajectory sampling.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major concerns point by point below, indicating the revisions we plan to implement.

read point-by-point responses

  1. Referee: [Application to disordered systems] Application section (disordered CDW decay): The headline claim that coupling to classical oscillators leads to delocalization and destabilizes finite-size MBL is based on multi-trajectory Ehrenfest results. The manuscript benchmarks convergence on the clean Holstein chain but does not report a direct comparison (even on small clusters) against a fully quantized phonon treatment at the same small ω. Without this control, it remains unclear whether the observed CDW decay arises from physical phonon-assisted processes or from uncontrolled averaging over classical trajectories that acts as an effective dephasing channel.

    Authors: We agree that a direct comparison to a fully quantized phonon treatment on small disordered clusters would provide stronger validation. Such benchmarks are computationally intensive even for small systems because of the enlarged Hilbert space from quantized phonons. Our existing benchmarks on the clean Holstein chain already show good agreement with exact results in the adiabatic limit. In that regime the multi-trajectory Ehrenfest approach captures the leading phonon-assisted delocalization physics, consistent with known mechanisms in the literature. We will add an explicit discussion of this limitation together with a small-cluster comparison (disorder without interactions) where feasible, and we will tone down the headline claim to emphasize that the evidence is within the classical-phonon approximation. revision: partial

  2. Referee: [Methods and convergence] Methods and convergence discussion: The text states that reliable results are expected only in the adiabatic regime, yet the MBL phenomenology is known to be sensitive to quantum fluctuations. Additional quantitative tests—such as the dependence of the relaxation rate on the number of Ehrenfest trajectories or on the phonon frequency cutoff—would be needed to establish that the delocalization is not an artifact of the classical approximation.

    Authors: We accept the need for additional quantitative convergence data. We have already verified that the relaxation dynamics stabilize with increasing trajectory number; we will include explicit plots of the CDW decay rate versus number of trajectories. We will also add results for several smaller phonon frequencies to demonstrate that the delocalization effect remains robust inside the adiabatic regime. These figures and accompanying text will be inserted in the methods and application sections. revision: yes

Circularity Check

0 steps flagged

No circularity: results from direct numerical simulation

full rationale

The paper introduces hybrid quantum-classical methods (time-dependent Lanczos and MPS combined with multi-trajectory Ehrenfest) and applies them to compute time-dependent observables in electron-phonon models. All reported results, including the decay of CDW order and the destabilization of finite-size MBL, are obtained from explicit numerical propagation of the equations of motion under the stated approximations. No derivation chain exists that reduces a claimed prediction to a fitted parameter, self-definition, or self-citation load-bearing step. Benchmarks on the clean Holstein chain and convergence checks are independent of the disordered-system conclusions. The central claim is therefore an output of the simulation protocol rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that classical phonon treatment is valid in the adiabatic limit; no free parameters or invented entities are identifiable from the abstract alone.

axioms (1)
  • domain assumption Phonons can be treated classically for small frequencies in the adiabatic regime
    Explicitly stated as the regime where reliable results are expected

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Reference graph

Works this paper leans on

115 extracted references · 115 canonical work pages

  1. [1]

    embarrass- ingly parallel

    Basic idea and algorithm The multi-trajectory Ehrenfest method (MTE) can be derived as a special case of the quantum–classical Liouville (QCL) equation [ 31, 33], with two essential approximations: The coupling between the electronic subsystem and the phonon environment is treated in a mean-field approximation and the phonon dynamics is treated classicall...

    work page 2000

  2. [2]

    Time propagation The standard way to solve the quantum and classical equations of motion numerically is an alternating time propagation of each subsystem from time t→t+ ∆t (∆t is the time step), with the other subsystem, i.e., |ψ(t)⟩ or{xℓ,pℓ}kept constant, i.e., a version of a splitting method [ 72]. This scheme can be viewed as a Trotter- Suzuki breakup...

    work page

  3. [3]

    Some limitations can be understood in the Born- Oppenheimer picture

    Discussion of strength and weaknesses of MTE MTE is one of the simplest trajectory based meth- ods [ 32, 73, 74], stable, and straightforward to imple- ment. Some limitations can be understood in the Born- Oppenheimer picture. For instance, in MTE, electrons ex- perience the average force from several Born-Oppenheimer surfaces and do not see the effects o...

    work page

  4. [4]

    Lanczos-MTE TEBD-MTE 0 10 20 30 4010−6 10−4 10−2 δOCDW Figure 2: Comparison of TEBD-MTE ( δ= 10−8), Lanczos-MTE (ϵ= 10−9/tf), and regular MTE as imple- mented in Ref

    0 t [1/t 0] t [1/t 0] OCDW MTE, ten Brink et al. Lanczos-MTE TEBD-MTE 0 10 20 30 4010−6 10−4 10−2 δOCDW Figure 2: Comparison of TEBD-MTE ( δ= 10−8), Lanczos-MTE (ϵ= 10−9/tf), and regular MTE as imple- mented in Ref. [ 36] for a non-interacting system (L = 13, V = 0, ω0 = 0.1 t0, γ= 0.4 t0, W = 0). Lanczos-MTE and TEBD use ∆ t = 0.01 t0−1and all methods ar...

    work page

  5. [5]

    For this, we can draw from well-known many- body techniques such as the time-dependent Lanczos 6 method [79]

    Algorithm Our goal is to deal with quantum systems having di- rect electron–electron interactions which warrants using a many-body quantum state to represent the electronic subsystem. For this, we can draw from well-known many- body techniques such as the time-dependent Lanczos 6 method [79]. In the time-dependent Lanczos method, a Krylov subspace is cons...

    work page

  6. [6]

    The combined state of the whole system (|ψ{xℓ,pℓ}(t)⟩,{xℓ, pℓ}) is graphically represented in the bottom layer of Fig

    Algorithm In TEBD-MTE, the quantum many-body state |ψ(t)⟩ of the electronic sub-system is described by a matrix product state (MPS). The combined state of the whole system (|ψ{xℓ,pℓ}(t)⟩,{xℓ, pℓ}) is graphically represented in the bottom layer of Fig. 1(b). Figure 1(c) describes one graph element associated with a site ℓ, which contains information about ...

    work page

  7. [7]

    The in- set shows the difference of the observable OCDW between the two methods as a function of time

    0 t [1/t 0] t [1/t 0] OCDW Lanczos-MTE TEBD-MTE 0 100 200 300 400 500 10−6 10−4 10−2 δOCDW Figure 5: Comparison of the TEBD-MTE method (δ= 10−8) and Lanczos-MTE method ( ϵ= 10−8/tf) in an interacting system ( L = 14, V = 2 t0, ω0 = 0.1 t0, γ= 0.4 t0) for the case without disorder (W = 0). The in- set shows the difference of the observable OCDW between the...

    work page 2000

  8. [8]

    Bardeen, L

    J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev.108, 1175 (1957)

    work page 1957

  9. [9]

    Fr¨ ohlich, Electrons in lattice fields, Advances in Physics3, 325 (1954)

    H. Fr¨ ohlich, Electrons in lattice fields, Advances in Physics3, 325 (1954)

    work page 1954

  10. [10]

    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 ap- proach, Adv. Phys.65, 58 (2016)

    work page 2016

  11. [11]

    Pastor, M

    E. Pastor, M. Sachs, S. Selim, J. R. Durrant, A. A. Bakulin, and A. Walsh, Electronic defects in metal oxide photocatalysts, Nature Reviews Materials7, 503 (2022). 15 Figure 13: Dynamics in a single trajectory: Decay of the CDW in the non-interacting system (L= 100 , V= 0 , γ= 0.5t0, W= 8t 0, ω0 = 0.1t0). We compare trajecto- ries with slightly different ...

    work page 2022

  12. [12]

    Bloch, A

    J. Bloch, A. Cavalleri, V. Galitski, M. Hafezi, and A. Ru- bio, Strongly-correlated electron–photon systems, Nature 606, 41 (2022)

    work page 2022

  13. [13]

    M. Rini, R. Tobey, N. Dean, J. Itatani, Y. Tomioka, Y. Tokura, R. W. Schoenlein, and A. Cavalleri, Control of the electronic phase of a manganite by mode-selective vibrational excitation, Nature449, 72 (2007)

    work page 2007

  14. [14]

    R. I. Tobey, D. Prabhakaran, A. T. Boothroyd, and A. Cavalleri, Ultrafast electronic phase transition in La1/2Sr3/2MnO4 by coherent vibrational excitation: Ev- idence for nonthermal melting of orbital order, Phys. Rev. Lett.101, 197404 (2008)

    work page 2008

  15. [15]

    E. T. Nibbering, H. Fidder, and E. Pines, Ultrafast chemistry: Using time-resolved vibrational spectroscopy for interrogation of structural dynamics, Annual Review of Physical Chemistry56, 337 (2005)

    work page 2005

  16. [16]

    Dagotto, T

    E. Dagotto, T. Hotta, and A. Moreo, Colossal magne- toresistant materials: The key role of phase separation, Physics Reports344, 1 (2001)

    work page 2001

  17. [17]

    Tokura, Critical features of colossal magnetoresistive manganites, Reports on Progress in Physics69, 797 (2006)

    Y. Tokura, Critical features of colossal magnetoresistive manganites, Reports on Progress in Physics69, 797 (2006)

    work page 2006

  18. [18]

    Weber and J

    M. Weber and J. K. Freericks, Real-time evolution of static electron–phonon models in time-dependent electric fields, Phys. Rev. E105, 025301 (2022)

    work page 2022

  19. [19]

    Cohen-Stead, S

    B. Cohen-Stead, S. M. Costa, J. Neuhaus, A. T. Ly, Y. Zhang, R. Scalettar, K. Barros, and S. Johnston, Smo- QyDQMC.jl: A flexible implementation of determinant quantum Monte Carlo for Hubbard and electron–phonon interactions, SciPost Phys. Codebases , 29 (2024)

    work page 2024

  20. [20]

    Picano, F

    A. Picano, F. Grandi, P. Werner, and M. Eck- stein, Stochastic semiclassical theory for nonequilibrium electron–phonon coupled systems, Phys. Rev. B108, 035115 (2023)

    work page 2023

  21. [21]

    Jankovi´ c and N

    V. Jankovi´ c and N. Vukmirovi´ c, Spectral and thermody- namic properties of the Holstein polaron: Hierarchical equations of motion approach, Phys. Rev. B105, 054311 (2022). 0 1 2 3 4 10−6 10−4 10−2 100 (a) δOCDW δ = 10 −6 δ = 10 −7 δ = 10 −8 δ = 10 −9 0 1 2 3 4 10−4 10−2 100 (b) t [1/t 0] δOCDW ∆ t = 0. 5/t 0 ∆ t = 0. 4/t 0 ∆ t = 0. 3/t 0 ∆ t = 0. 2/t 0...

    work page 2022

  22. [22]

    Bonˇ ca, S

    J. Bonˇ ca, S. A. Trugman, and I. Batisti´ c, Holstein po- laron, Phys. Rev. B60, 1633 (1999)

    work page 1999

  23. [23]

    Dorfner, L

    F. Dorfner, L. Vidmar, C. Brockt, E. Jeckelmann, and F. Heidrich-Meisner, Real-time decay of a highly excited charge carrier in the one-dimensional Holstein model, Phys. Rev. B91, 104302 (2015)

    work page 2015

  24. [24]

    G. L. Goodvin and M. Berciu, Momentum average ap- proximation for models with electron–phonon coupling dependent on the phonon momentum, Phys. Rev. B78, 235120 (2008)

    work page 2008

  25. [25]

    Jeckelmann and S

    E. Jeckelmann and S. R. White, Density-matrix renormalization-group study of the polaron problem in the Holstein model, Phys. Rev. B57, 6376 (1998)

    work page 1998

  26. [26]

    C. Guo, A. Weichselbaum, J. von Delft, and M. Vojta, Critical and strong-coupling phases in one- and two-bath spin-boson models, Phys. Rev. Lett.108, 160401 (2012)

    work page 2012

  27. [27]

    Brockt, F

    C. Brockt, F. Dorfner, L. Vidmar, F. Heidrich-Meisner, and E. Jeckelmann, Matrix-product-state method with a dynamical local basis optimization for bosonic systems out of equilibrium, Phys. Rev. B92, 241106 (2015)

    work page 2015

  28. [28]

    Paeckel, T

    S. Paeckel, T. K¨ ohler, and S. R. Manmana, Automated construction of U(1)-invariant matrix-product operators from graph representations, SciPost Phys.3, 035 (2017)

    work page 2017

  29. [29]

    Jansen, J

    D. Jansen, J. Bonˇ ca, and F. Heidrich-Meisner, Finite- 16 temperature density-matrix renormalization group method for electron–phonon systems: Thermodynamics and Holstein-polaron spectral functions, Phys. Rev. B 102, 165155 (2020)

    work page 2020

  30. [30]

    Stolpp, T

    J. Stolpp, T. K¨ ohler, S. R. Manmana, E. Jeckelmann, F. Heidrich-Meisner, and S. Paeckel, Comparative study of state-of-the-art matrix-product-state methods for lat- tice models with large local Hilbert spaces without U(1) symmetry, Computer Physics Communications269, 108106 (2021)

    work page 2021

  31. [31]

    Jansen, J

    D. Jansen, J. Bonˇ ca, and F. Heidrich-Meisner, Finite- temperature optical conductivity with density-matrix renormalization group methods for the Holstein polaron and bipolaron with dispersive phonons, Phys. Rev. B 106, 155129 (2022)

    work page 2022

  32. [32]

    Jansen and F

    D. Jansen and F. Heidrich-Meisner, Thermal and optical conductivity in the Holstein model at half filling and finite temperature in the Luttinger-liquid and charge- density-wave regime, Phys. Rev. B108, L081114 (2023)

    work page 2023

  33. [33]

    S. R. Manmana, R. Noack, and A. Muramatsu, Time evo- lution of one-dimensional quantum many body systems, AIP Conf. Proc.789, 269 (2005)

    work page 2005

  34. [34]

    Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Ann

    U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (N. Y.)326, 96 (2011)

    work page 2011

  35. [35]

    Zhang, E

    C. Zhang, E. Jeckelmann, and S. R. White, Density matrix approach to local Hilbert space reduction, Phys. Rev. Lett.80, 2661 (1998)

    work page 1998

  36. [36]

    Giustino, Electron–phonon interactions from first prin- ciples, Rev

    F. Giustino, Electron–phonon interactions from first prin- ciples, Rev. Mod. Phys.89, 015003 (2017)

    work page 2017

  37. [37]

    Lively, S

    K. Lively, S. A. Sato, G. Albareda, A. Rubio, and A. Kelly, Revealing ultrafast phonon mediated inter- valley scattering through transient absorption and high harmonic spectroscopies, Phys. Rev. Res.6, 013069 (2024)

    work page 2024

  38. [38]

    I. V. Aleksandrov, The statistical dynamics of a sys- tem consisting of a classical and a quantum subsystem, Zeitschrift f¨ ur Naturforschung A36, 902 (1981)

    work page 1981

  39. [39]

    J. C. Tully, Mixed quantum–classical dynamics, Faraday Discussions110, 407 (1998)

    work page 1998

  40. [40]

    Nielsen, R

    S. Nielsen, R. Kapral, and G. Ciccotti, Statistical me- chanics of quantum-classical systems, The Journal of Chemical Physics115, 5805 (2001)

    work page 2001

  41. [41]

    X. Li, J. C. Tully, H. B. Schlegel, and M. J. Frisch, Ab initio Ehrenfest dynamics, The Journal of Chemical Physics123, 084106 (2005)

    work page 2005

  42. [42]

    Polkovnikov, Phase space representation of quantum dynamics, Annals of Physics325, 1790 (2010)

    A. Polkovnikov, Phase space representation of quantum dynamics, Annals of Physics325, 1790 (2010)

    work page 2010

  43. [43]

    ten Brink, S

    M. ten Brink, S. Gr¨ aber, M. Hopjan, D. Jansen, J. Stolpp, F. Heidrich-Meisner, and P. E. Bl¨ ochl, Real-time non- adiabatic dynamics in the one-dimensional Holstein model: Trajectory-based vs exact methods, The Journal of Chemical Physics156, 234109 (2022)

    work page 2022

  44. [44]

    Manawadu, D

    D. Manawadu, D. J. Valentine, M. Marcus, and W. Bar- ford, Singlet triplet-pair production and possible singlet- fission in carotenoids, The Journal of Physical Chemistry Letters13, 1344 (2022)

    work page 2022

  45. [45]

    Manawadu, D

    D. Manawadu, D. J. Valentine, and W. Barford, Dy- namical simulations of carotenoid photoexcited states using density matrix renormalization group techniques, The Journal of Physical Chemistry A127, 3714 (2023), pMID: 37054397

    work page 2023

  46. [46]

    S. R. White, Density matrix formulation for quan- tum renormalization groups, Phys. Rev. Lett.69, 2863 (1992)

    work page 1992

  47. [47]

    S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B48, 10345 (1993)

    work page 1993

  48. [48]

    Paeckel, T

    S. Paeckel, T. K¨ ohler, A. Swoboda, S. R. Manmana, U. Schollw¨ ock, and C. Hubig, Time-evolution meth- ods for matrix-product states, Ann. Phys. (N. Y.)411, 167998 (2019)

    work page 2019

  49. [49]

    J. E. Subotnik, A. Jain, B. Landry, A. Petit, W. Ouyang, and N. Bellonzi, Understanding the surface hopping view of electronic transitions and decoherence, Annual Review of Physical Chemistry67, 387 (2016)

    work page 2016

  50. [50]

    D. V. Shalashilin, Quantum mechanics with the basis set guided by Ehrenfest trajectories: Theory and application to spin-boson model, The Journal of Chemical Physics 130, 244101 (2009)

    work page 2009

  51. [51]

    D. V. Shalashilin, Nonadiabatic dynamics with the help of multiconfigurational ehrenfest method: Improved the- ory and fully quantum 24d simulation of pyrazine, The Journal of Chemical Physics132, 244111 (2010)

    work page 2010

  52. [52]

    W. C.-W. Huang, S. Mu, G. von Witte, Y. S. Li, F. Kurtz, S.-H. Hung, H.-T. Jeng, K. Rossnagel, J. G. Horstmann, and C. Ropers, Ultrafast optical switching to a het- erochiral charge-density wave state, arXiv preprint , arXiv:2405.20872 (2024)

    work page arXiv 2024

  53. [53]

    Hashimoto and S

    H. Hashimoto and S. Ishihara, Photoinduced charge- order melting dynamics in a one-dimensional interacting Holstein model, Phys. Rev. B96, 035154 (2017)

    work page 2017

  54. [54]

    Stolpp, J

    J. Stolpp, J. Herbrych, F. Dorfner, E. Dagotto, and F. Heidrich-Meisner, Charge-density-wave melting in the one-dimensional Holstein model, Phys. Rev. B101, 035134 (2020)

    work page 2020

  55. [55]

    Nandkishore, Many-body localization proximity effect, Phys

    R. Nandkishore, Many-body localization proximity effect, Phys. Rev. B92, 245141 (2015)

    work page 2015

  56. [56]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys.91, 021001 (2019)

    work page 2019

  57. [57]

    F. X. Bronold and H. Fehske, Anderson localization of polaron states, Phys. Rev. B66, 073102 (2002)

    work page 2002

  58. [58]

    A. A. Franz X. Bronold and H. Fehske, Anderson local- ization in strongly coupled disordered electron–phonon systems, Philosophical Magazine84, 673 (2004)

    work page 2004

  59. [59]

    A. N. Das and S. Sil, Thermodynamic properties of Holstein polarons and the effects of disorder, Journal of Physics: Condensed Matter20, 345222 (2008)

    work page 2008

  60. [60]

    Berciu, A

    M. Berciu, A. S. Mishchenko, and N. Nagaosa, Holstein polaron in the presence of disorder, Europhysics Letters 89, 37007 (2010)

    work page 2010

  61. [61]

    Ebrahimnejad and M

    H. Ebrahimnejad and M. Berciu, Perturbational study of the lifetime of a Holstein polaron in the presence of weak disorder, Phys. Rev. B86, 205109 (2012)

    work page 2012

  62. [62]

    O. R. Tozer and W. Barford, Localization of large po- larons in the disordered Holstein model, Phys. Rev. B 89, 155434 (2014)

    work page 2014

  63. [63]

    Kloss, D

    B. Kloss, D. R. Reichman, and R. Tempelaar, Multiset matrix product state calculations reveal mobile Franck- Condon excitations under strong Holstein-type coupling, Phys. Rev. Lett.123, 126601 (2019)

    work page 2019

  64. [64]

    Prelovˇ sek, J

    P. Prelovˇ sek, J. Bonˇ ca, and M. Mierzejewski, Transient and persistent particle subdiffusion in a disordered chain coupled to bosons, Phys. Rev. B98, 125119 (2018)

    work page 2018

  65. [65]

    Murakami, P

    Y. Murakami, P. Werner, N. Tsuji, and H. Aoki, Inter- action quench in the Holstein model: Thermalization crossover from electron- to phonon-dominated relaxation, Phys. Rev. B91, 045128 (2015). 17

    work page 2015

  66. [66]

    Di Sante, S

    D. Di Sante, S. Fratini, V. Dobrosavljevi´ c, and S. Ciuchi, Disorder-driven metal-insulator transitions in deformable lattices, Phys. Rev. Lett.118, 036602 (2017)

    work page 2017

  67. [67]

    Fleishman, D

    L. Fleishman, D. C. Licciardello, and P. W. Anderson, Elementary excitations in the Fermi glass, Phys. Rev. Lett.40, 1340 (1978)

    work page 1978

  68. [68]

    D. A. Huse, R. Nandkishore, F. Pietracaprina, V. Ros, and A. Scardicchio, Localized systems coupled to small baths: From Anderson to Zeno, Phys. Rev. B92, 014203 (2015)

    work page 2015

  69. [69]

    Gopalakrishnan, K

    S. Gopalakrishnan, K. R. Islam, and M. Knap, Noise- induced subdiffusion in strongly localized quantum sys- tems, Phys. Rev. Lett.119, 046601 (2017)

    work page 2017

  70. [70]

    T. L. M. Lezama and Y. B. Lev, Logarithmic, noise- induced dynamics in the Anderson insulator, SciPost Phys.12, 174 (2022)

    work page 2022

  71. [71]

    Brighi, A

    P. Brighi, A. A. Michailidis, K. Kirova, D. A. Abanin, and M. Serbyn, Localization of a mobile impurity inter- acting with an Anderson insulator, Phys. Rev. B105, 224208 (2022)

    work page 2022

  72. [72]

    Brighi, A

    P. Brighi, A. A. Michailidis, D. A. Abanin, and M. Ser- byn, Propagation of many-body localization in an An- derson insulator, Phys. Rev. B105, L220203 (2022)

    work page 2022

  73. [73]

    Sierant, M

    P. Sierant, M. Lewenstein, A. Scardicchio, and J. Za- krzewski, Stability of many-body localization in Floquet systems, Phys. Rev. B107, 115132 (2023)

    work page 2023

  74. [74]

    Schreiber, S

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨ uschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of inter- acting fermions in a quasi-random optical lattice, Science 349, 842 (2015)

    work page 2015

  75. [75]

    Vidal, Efficient simulation of one-dimensional quan- tum many-body systems, Phys

    G. Vidal, Efficient simulation of one-dimensional quan- tum many-body systems, Phys. Rev. Lett.93, 040502 (2004)

    work page 2004

  76. [76]

    Grunwald, A

    R. Grunwald, A. Kelly, and R. Kapral, Quantum dy- namics in almost classical environments (Springer Berlin Heidelberg, 2009) pp. 383–413

    work page 2009

  77. [77]

    Wigner, On the quantum correction for thermody- namic equilibrium, Physical Review40, 749 (1932)

    E. Wigner, On the quantum correction for thermody- namic equilibrium, Physical Review40, 749 (1932)

    work page 1932

  78. [78]

    ˙Imre, E

    K. ˙Imre, E. ¨Ozizmir, M. Rosenbaum, and P. F. Zweifel, Wigner method in quantum statistical mechanics, Jour- nal of Mathematical Physics8, 1097 (1967)

    work page 1967

  79. [79]

    R. I. McLachlan and G. Reinout W. Quispel, Splitting methods, Acta Numerica11, 341 (2002)

    work page 2002

  80. [80]

    Ehrenfest, Bemerkung ¨ uber die angen¨ aherte G¨ ultigkeit der klassischen Mechanik innerhalb der Quanten- mechanik, Zeitschrift f¨ ur Physik45, 455 (1927)

    P. Ehrenfest, Bemerkung ¨ uber die angen¨ aherte G¨ ultigkeit der klassischen Mechanik innerhalb der Quanten- mechanik, Zeitschrift f¨ ur Physik45, 455 (1927)

    work page 1927

Showing first 80 references.