arxiv: 2604.10775 · v1 · submitted 2026-04-12 · ❄️ cond-mat.str-el

Recognition: unknown

Thermalization Fronts in the Hubbard-Holstein Model

Antonio Picano, Marco Schiro

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:13 UTC · model grok-4.3

classification ❄️ cond-mat.str-el

keywords Hubbard-Holstein modelnonequilibrium DMFTthermalization frontselectron-phonon couplingquench dynamicsMigdal approximationpropagating frontsrelaxation crossover

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The pith

Thermalization spreads coherently in the Hubbard-Holstein model with fronts propagating at identical velocities in electrons and phonons.

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

The paper examines the nonequilibrium relaxation of the Hubbard-Holstein model after suddenly switching on the electron-phonon interaction. Using nonequilibrium DMFT with the Migdal approximation for phonons and second-order perturbation theory for electrons, it identifies a crossover from electron-dominated to phonon-dominated relaxation as coupling strength grows. In the Step-by-Step DMFT representation, thermalization appears as a sharp front in the plane of real time versus iteration number. Electronic fronts form quickly even for weak quenches, while phononic fronts are delayed at weak coupling but appear on similar timescales near and beyond the crossover. When both fronts are visible, they travel at the same speed.

Core claim

Thermalization is marked by a sharp propagating front in the plane of real time and DMFT iteration number. This front appears in electronic observables already for weak quenches within the simulated time window, whereas the phononic sector exhibits a visible front only at sufficiently strong coupling. Thus, at weak coupling the local dispersionless phonons show a delayed onset of front formation, while near and beyond the crossover the front develops on comparable timescales in both the electronic and phononic sectors. Whenever both fronts are resolved, they propagate with the same velocity, showing that thermalization spreads coherently through the coupled electron-phonon system.

What carries the argument

The Step-by-Step DMFT framework, which tracks dynamics in the real-time versus iteration-number plane to expose sharp propagating fronts that mark the microscopic onset of the thermal state.

Load-bearing premise

The self-consistent Migdal approximation together with second-order perturbation theory for the electron-electron interaction remains accurate enough to capture the microscopic buildup of the thermal state across the simulated quench strengths.

What would settle it

A calculation or experiment in which the resolved electronic and phononic thermalization fronts propagate at different velocities would falsify the claim of coherent spreading.

Figures

Figures reproduced from arXiv: 2604.10775 by Antonio Picano, Marco Schiro.

**Figure 1.** Figure 1: summarizes the relaxation dynamics. Panel (a) shows ∆n(t) for U/v∗ = 2 and several values of the final coupling gf . Here ∆n(t) quantifies the jump in the momentum distribution n(ϵk, t) = −iG< k (t, t) at the Fermi level, ϵ = 0. Because the initial state is at low but finite temperature and may already include finite Hubbard interaction, one has ∆n(0) ≲ 1, as also visible from the equilibrium curve at βi … view at source ↗

**Figure 2.** Figure 2: FIG. 2. Thermalization in infinite dimensions for the Hubbard–Holstein model at [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Thermalization in infinite dimensions for the Hubbard–Holstein model at [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Front velocity [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Time evolution of the kinetic energy [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. The electron spectral functions [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Thermalization in infinite dimensions for the Hubbard–Holstein model at [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Thermalization in infinite dimensions for the Hubbard–Holstein model at [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗

read the original abstract

We investigate the nonequilibrium dynamics of the weak-coupling Hubbard-Holstein model after a sudden switch-on of the electron-phonon interaction within nonequilibrium dynamical mean-field theory (DMFT). Using the self-consistent Migdal approximation for the electron-phonon coupling together with second-order perturbation theory for the electron-electron interaction, we show that the relaxation dynamics exhibits a crossover between electron-dominated and phonon-dominated regimes, extending to finite Hubbard interaction the scenario previously identified in the Holstein model. To investigate the microscopic buildup of the thermal state, we analyze the dynamics within the Step-by-Step DMFT framework. In the plane of real time and DMFT iteration number, thermalization is marked by a sharp propagating front. This front appears in electronic observables already for weak quenches within the simulated time window, whereas the phononic sector exhibits a visible front only at sufficiently strong coupling. Thus, at weak coupling the local dispersionless phonons show a delayed onset of front formation, while near and beyond the crossover the front develops on comparable timescales in both the electronic and phononic sectors. Whenever both fronts are resolved, they propagate with the same velocity, showing that thermalization spreads coherently through the coupled electron-phonon 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.

Desk Editor's Note private letter to a colleague

The paper shows thermalization fronts survive adding finite U to the Holstein model, with a crossover in regime dominance and equal front velocities when both sectors are visible.

read the letter

The main takeaway is that the propagating thermalization fronts previously seen in the pure Holstein model still appear once a finite Hubbard repulsion is turned on. The work tracks this in nonequilibrium DMFT by quenching the electron-phonon coupling and plotting observables in the real-time versus DMFT-iteration plane. At weak coupling the electronic front forms quickly while the phonon front is delayed; near and past the crossover both fronts develop on similar timescales and, when both are clear, they move at the same speed. This gives a concrete picture of how the two sectors reach equilibrium together under the stated approximations. What the paper does well is to carry the Step-by-Step DMFT visualization over to the Hubbard-Holstein case without changing the core numerical setup. The crossover behavior and the delayed phonon onset are direct, observable consequences of adding the second-order electron-electron term, and they extend the earlier Holstein-only results in a straightforward way. The claim that thermalization spreads coherently is tied to the equal-velocity observation rather than to any extra fitting. The soft spots are the usual ones for this level of approximation. Migdal plus second-order perturbation theory is reasonable at weak coupling, but the simulations reach the crossover region, so it is not obvious how much the front velocity or sharpness would shift if vertex corrections or higher-order diagrams were included. The abstract gives no quantitative error analysis or cutoff dependence, which leaves the robustness of the velocity equality a bit open. Everything stays inside DMFT, so spatial effects or longer-range correlations are outside the scope. This is a paper for people already working on nonequilibrium DMFT for electron-phonon systems. Readers who know the Holstein fronts will see the incremental extension clearly and can judge whether the added Hubbard term changes anything they care about. The central observation is internally consistent with the methods they chose. I would send it to peer review; the numerics deliver a clean, falsifiable picture that builds on prior work without obvious contradictions.

Referee Report

0 major / 2 minor

Summary. The manuscript investigates the nonequilibrium dynamics of the weak-coupling Hubbard-Holstein model after a sudden switch-on of the electron-phonon interaction within nonequilibrium dynamical mean-field theory (DMFT). Using the self-consistent Migdal approximation for the electron-phonon coupling together with second-order perturbation theory for the electron-electron interaction, the authors show that the relaxation dynamics exhibits a crossover between electron-dominated and phonon-dominated regimes, extending prior Holstein-model results. They analyze the microscopic buildup of the thermal state using the Step-by-Step DMFT framework, identifying sharp propagating fronts in the real-time versus DMFT-iteration plane. Electronic fronts appear for weak quenches, while phononic fronts require stronger coupling; when both are resolved, the fronts propagate at identical velocities, indicating coherent thermalization spread through the coupled system.

Significance. If the numerical results hold under the stated approximations, this work meaningfully extends the Holstein-model scenario to finite Hubbard U, offering a microscopic view of how thermalization propagates coherently in coupled electron-phonon systems. The Step-by-Step DMFT diagnostic, which treats DMFT iteration number as a spatial-like coordinate, provides a clear, parameter-free visualization of front formation and velocity equality. Direct observation of matched velocities in electronic and phononic sectors strengthens the central claim of coherent spread.

minor comments (2)

The abstract refers to 'the simulated time window' and 'sufficiently strong coupling' without quoting specific numerical ranges for U, electron-phonon coupling strength, or quench amplitudes; adding these values would allow readers to assess the extent of the reported crossover and the regimes where both fronts are resolved.
The Step-by-Step DMFT framework is introduced without a brief definition or citation on first use; a short explanatory sentence or reference would improve accessibility for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The provided summary accurately captures our investigation of nonequilibrium dynamics in the Hubbard-Holstein model, the crossover between electron- and phonon-dominated regimes, and the identification of propagating thermalization fronts via the Step-by-Step DMFT framework.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results follow from direct numerical solution of the nonequilibrium DMFT equations under the self-consistent Migdal approximation for electron-phonon coupling and second-order perturbation theory for electron-electron interactions, within the Step-by-Step DMFT framework. The propagating thermalization fronts and their equal velocities (when both are resolved) are reported as observations from the simulated dynamics in the real-time vs. DMFT-iteration plane, extending prior Holstein-model behavior to finite Hubbard U without any self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the claim to its inputs. The derivation chain remains self-contained against the stated approximations and numerical outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the Migdal and second-order perturbative approximations inside nonequilibrium DMFT for the Hubbard-Holstein model; no new entities are introduced.

axioms (2)

domain assumption The self-consistent Migdal approximation accurately describes the electron-phonon interaction in the weak-coupling regime.
Explicitly adopted in the abstract for the phonon sector.
domain assumption Second-order perturbation theory is sufficient for the electron-electron interaction.
Stated as the treatment used alongside Migdal for the Hubbard term.

pith-pipeline@v0.9.0 · 5504 in / 1318 out tokens · 62375 ms · 2026-05-10T15:13:46.561806+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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conserving approximation

Electronic problem: Second-order perturbation theory In this work, we focus on the weakly correlated electronic regime. We employ the self-consistent 4 second-order perturbation theory (SPT) as an impurity solver [55] which, being a conserving approximation, is particularly suited to studying thermalization in isolated quantum systems and the energy trans...
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Phonon problem: Self-consistent Migdal approximation In the Migdal approximation, one treats within a Gaussian approximation the quantum fluctuations of the phonons around the average order parameter, repre- sented by the displacement⟨X⟩of the atoms [35]. The self-consistent Migdal approximation is the lowest or- der approximation for the electronic self-...
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(9) to fix the hybridiza- tion function of the impurity model: ∆(t, t′) =v 2 ∗G(t, t′) (18)

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We implement the self-consistent Migdal approxima- tion for the electron-phonon interaction: (a) Calculate the phonon self-energy Π (polariza- tion operator) with Eq

Solve the impurity model to get Σ el-ph[G, D]. We implement the self-consistent Migdal approxima- tion for the electron-phonon interaction: (a) Calculate the phonon self-energy Π (polariza- tion operator) with Eq. (13) in order to in- clude the back-action of the electrons on the phonons: Π(t, t′) =−i2g(t)g(t ′)G(t, t′)G(t′, t) (19) (b) Solve the Dyson eq...
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With SPT as an impu- rity solver, we apply Eq

Solve the impurity model for the electron-electron interaction to get Σ e-e[G]. With SPT as an impu- rity solver, we apply Eq. (15): Σe-e(t, t′) =U(t)U(t ′)G(t, t ′)G(t ′, t)G(t, t ′) (22)
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Step-by-Step DMFT atU= 0 For completeness, we briefly discuss the corresponding Step-by-Step DMFT results in the Holstein limitU/v ∗ = 0, shown in Figs. 7 and 8. We keep the same parameters as in the main text, namely half filling,ω 0 = 0.7, and initial inverse temperatureβ i = 100. Figure 7 shows the representative caseg f = 0.5. Pan- els (a) and (b) con...