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

TL;DR

The paper addresses how strong electronic correlations and disorder interact with phonons to influence localization and relaxation in low-dimensional systems. It develops two hybrid quantum–classical approaches, Lanczos–MTE and TEBD–MTE, that couple exact many-body electron dynamics to a classical phonon bath via multi-trajectory Ehrenfest dynamics. The methods are benchmarked against pure MTE and applied to Anderson insulators and interacting, disordered systems, revealing bath-induced delocalization and subdiffusive transport, with charge-density-wave order decaying according to a power law. The work provides scalable tools to study electron–phonon dynamics in disordered, correlated systems and suggests that a classical phonon bath can destabilize many-body localization in the adiabatic regime, with implications for thermalization and transport in realistic materials.

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.

Figures (14)

• Figure 1: \ref{['fig:method_lanczos_mte']} The Lanczos-MTE and \ref{['fig:method_tebd_mte']} TEBD-MTE methods are graphically represented for a single time step $\Delta t$. \ref{['fig:method_lanczos_mte']} 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 for the classical sub-system. Similarly, in \ref{['fig:method_tebd_mte']}, the bottom layer stands for the MPS representation of the quantum sub-system. The second layer represents the TEBD step for the quantum sub-system and the last layer evolves the classical-sub system. \ref{['fig:method_mte_tensor']} Element of the bottom layer of \ref{['fig:method_tebd_mte']} on site $\ell$. This contains the information about the MPS in $A^{\sigma_\ell}_{a_{\ell-1,\ell}}$ and the coordinates and momenta for the phonons ($\lbrace x_\ell,~p_\ell \rbrace$) as external parameters.
• Figure 2: Comparison of TEBD-MTE (${\delta = 10^{-8}}$), Lanczos-MTE ($\epsilon = 10^{-9} / t_\text{f}$), and regular MTE as implemented in Ref. TenBrink2022 for a non-interacting system ($L=13$, $V=0$, $\omega_0=\qty{0.1}{\hopping}$, $\gamma=\qty{0.4}{\hopping}$, $W=0$). Lanczos-MTE and TEBD use $\Delta t = \qty{0.01}{\per\hopping}$ and all methods are averaged over $N_\mathrm{traj}=4000$ trajectories. The inset shows the difference $\delta \mathcal{O}_\mathrm{CDW}$ between the many-body hybrid techniques and the reference data from regular MTE from Ref. TenBrink2022.
• Figure 3: Dependence of the Lanczos-MTE method on the control parameters time step $\Delta t$ and error rate $\epsilon$ for an interacting system ($L=14$, $V=\qty{2}{\hopping}$, $\omega_0=\qty{0.1}{\hopping}$, $\gamma=\qty{0.4}{\hopping}$, $W=0$). We plot the deviation $\delta \mathcal{O}_\mathrm{CDW}$ when \ref{['fig:convergence_lanczos_errorrate']} varying $\epsilon$ at fixed $\Delta t = \qty{0.01}{\per\hopping}$, comparing to a reference set with $\epsilon = 10^{-10}/t_f$ and \ref{['fig:convergence_lanczos_timestep']} varying $\Delta t$ at fixed $\epsilon = 10^{-10} / t_f$, comparing to a reference set with $\Delta t = \qty{0.01}{\per\hopping}$. The results are averaged over $N_\mathrm{traj} = 4000$ trajectories and the final time $t_f = \qty{500} {\per\hopping}$.
• Figure 4: Dependence of the TEBD-MTE method on the discarded weight $\delta$ and the time step $\Delta t$ in an interacting system ($L=14$, $V=\qty{2}{\hopping}$, $\omega_0=\qty{0.1}{\hopping}$, $\gamma=\qty{0.4}{\hopping}$, $W=0$). We plot the deviation $\delta \mathcal{O}_\mathrm{CDW}$ when \ref{['fig:convergence_tebd_cutoff']} varying $\delta$ at a fixed time step $\Delta t = \qty{0.01}{\per\hopping}$, comparing with a reference set using $\delta = 10^{-8}$ and \ref{['fig:convergence_tebd_timestep']} varying $\Delta t$ at fixed cutoff ${\delta = 10^{-8}}$, comparing with reference set using ${\Delta t = \qty{0.01}{\per\hopping}}$. The results are averaged over $N_\mathrm{traj} = 4000$ trajectories.
• Figure 5: Comparison of the TEBD-MTE method (${\delta = 10^{-8}}$) and Lanczos-MTE method ($\epsilon = 10^{-8} / t_f$) in an interacting system ($L=14$, $V=\qty{2}{\hopping}$, $\omega_0=\qty{0.1}{\hopping}$, $\gamma=\qty{0.4}{\hopping}$) for the case without disorder ($W=0$). The inset shows the difference of the observable $\mathcal{O}_\mathrm{CDW}$ between the two methods as a function of time. Both methods employ a time step $\Delta t = \qty{0.01}{\per\hopping}$ and are sampled over $N_\mathrm{traj} = 4000$ trajectories.