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

This paper introduces and benchmarks two hybrid quantum-classical methods combining time-dependent Lanczos and matrix-product state approaches with the multi-trajectory Ehrenfest approximation to simulate electron-phonon systems, demonstrating that coupling strongly disordered interacting fermions to classical Einstein phonons induces delocalization and destabilizes many-body localization.

The Gist

Imagine a crowded dance floor where two types of dancers are trying to move together: Electrons (the fast, jittery dancers) and Phonons (the slow, heavy dancers who represent the vibrations of the floor itself).

In the real world, these two groups are constantly interacting. The electrons jump around, and their movement makes the floor vibrate. The floor's vibrations, in turn, push the electrons around. Physicists call this electron-phonon coupling.

The problem is that when you add disorder (like random obstacles on the dance floor) and strong interactions (where the electrons really don't like to be near each other), the math becomes impossibly complex. It's like trying to predict the path of every single dancer in a chaotic mosh pit while the floor is shaking.

This paper introduces a clever new way to simulate this chaos using a "Hybrid" approach. Here is the breakdown in simple terms:

1. The Problem: Too Many Variables

To simulate this perfectly, you would need a supercomputer the size of a planet. The electrons are quantum (fuzzy and probabilistic), and the phonons are also quantum. Tracking both simultaneously for a large system is currently impossible.

2. The Solution: The "Hybrid" Dance Floor

The authors created two new computer methods that split the work:

• The Electrons (The Quantum Part): They are treated with extreme precision using advanced math (Lanczos and Matrix-Product States). We track exactly how they interact and repel each other.
• The Phonons (The Classical Part): The floor vibrations are treated as "classical" objects, like heavy balls on springs. We don't track their quantum fuzziness; we just track their position and speed.

The Analogy: Imagine a video game where the main characters (electrons) are rendered in high-definition 4K with realistic physics, but the background scenery (the phonons) is rendered in simple, smooth 2D animation. It's a compromise that saves massive computing power while keeping the most important details accurate.

They use a technique called Multi-Trajectory Ehrenfest (MTE). Think of this as running the simulation thousands of times at once.

• In one run, the floor vibrates slightly left.
• In another, it vibrates slightly right.
• By averaging the results of all these "what-if" scenarios, they get a reliable picture of what actually happens.

3. The Experiment: The "Freeze" vs. The "Meltdown"

The researchers wanted to see what happens when you have a disordered system (a messy dance floor with random obstacles).

• The Baseline (No Phonons): If the floor is perfectly still, the electrons get stuck. They hit the obstacles and can't move. In physics, this is called Anderson Localization (or Many-Body Localization if they interact). It's like the dancers are frozen in place, unable to leave their spot.
• The Twist (Adding Phonons): When they turned on the floor vibrations (the phonons), something surprising happened. The vibrations acted like a "shaking table." Even though the obstacles were still there, the shaking helped the electrons hop over them.

The Result: The "frozen" dancers started moving again! The vibrations caused delocalization. The electrons began to spread out across the dance floor, though they moved slowly (a behavior called sub-diffusion).

4. Why This Matters

This is a big deal for understanding materials.

• The "Bad Metal" Mystery: In some materials, electricity doesn't flow well, but it's not a perfect insulator either. This paper suggests that the interaction between electrons and the vibrating lattice (phonons) might be the reason why these materials behave strangely.
• Stability: It shows that "Many-Body Localization" (a state where quantum systems refuse to thermalize or reach equilibrium) is actually unstable if you introduce these vibrations. The vibrations act as a "leak" that eventually lets the system relax and mix.

Summary Analogy

Imagine a room full of people (electrons) trying to walk through a hallway filled with random, heavy furniture (disorder).

• Without vibrations: Everyone gets stuck behind the furniture. No one moves.
• With vibrations: The floor starts shaking gently. The furniture rattles, creating tiny gaps. The people can now squeeze through, but they have to shuffle slowly and carefully. They aren't running freely, but they are no longer stuck.

The paper proves that even in a messy, crowded, and frozen system, the simple act of the environment "shaking" can wake everything up and allow movement to begin. The authors built two new "cameras" (Lanczos and TEBD methods) to film this process and confirmed that the shaking is the key to unlocking the system.

Technical Summary

Here is a detailed technical summary of the paper "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."

1. Problem Statement

The paper addresses the challenge of simulating the real-time dynamics of electron–phonon systems where strong electronic correlations are present.

• The Challenge: Standard quantum many-body methods (like Matrix Product States or Lanczos) struggle with electron–phonon coupling when phonon frequencies are small (the adiabatic regime) because the phonon Hilbert space becomes prohibitively large. Conversely, semi-classical methods often fail to capture strong electronic correlations accurately.
• Specific Context: The authors focus on one-dimensional systems with quenched disorder (random on-site potentials) and strong electron-electron interactions. The goal is to understand the stability of Many-Body Localization (MBL) and Anderson localization when coupled to a classical bath of phonons, specifically investigating the decay of Charge Density Wave (CDW) order.

2. Methodology

The authors propose and benchmark two hybrid quantum–classical methods that combine exact quantum treatment of electrons with classical treatment of phonons using the Multi-Trajectory Ehrenfest (MTE) approach.

A. The Hybrid Framework (MTE)

• Concept: The phonon degrees of freedom (Einstein oscillators) are treated classically (coordinates $x_\ell$ and momenta $p_\ell$), while the electronic subsystem is treated quantum mechanically.
• Dynamics: The system evolves via coupled equations of motion:
  • Classical: Oscillators evolve under a mean-field force derived from the quantum expectation value of electron density $\langle \hat{n}_\ell \rangle$.
  • Quantum: The electronic wavefunction $|\psi(t)\rangle$ evolves under a Hamiltonian where the phonon coordinates act as time-dependent external potentials.
• Sampling: To account for quantum uncertainty, the method runs an ensemble of independent trajectories ($N_{traj}$), sampling initial phonon coordinates and momenta from the Wigner function of the initial state. Observables are averaged over this ensemble.

B. Two Implementation Schemes

The authors integrate MTE with two distinct quantum propagation techniques:

1. Lanczos-MTE: Uses the time-dependent Lanczos method to propagate the quantum state.
   • Mechanism: Constructs a Krylov subspace iteratively to approximate the time-evolution operator.
   • Strength: Highly accurate for small-to-medium system sizes ($L \approx 30$) and capable of reaching very long time scales.
2. TEBD-MTE: Uses Time-Evolving Block Decimation (TEBD) based on Matrix Product States (MPS).
   • Mechanism: Represents the wavefunction as an MPS and applies two-site gates via Trotter-Suzuki decomposition.
   • Strength: Can handle larger system sizes than Lanczos but is limited to shorter times due to entanglement growth (which increases the required bond dimension).

C. Benchmarking

The authors rigorously benchmarked both methods against each other and against standard MTE (using single-particle propagation) for non-interacting systems. They demonstrated that:

• Both methods converge to the same results within statistical error limits ($\sim 1/\sqrt{N_{traj}}$).
• The dominant error source is the statistical sampling of trajectories, not the quantum propagation algorithm, provided the time steps and truncation errors are sufficiently small.

3. Key Results

A. Stability of Anderson Localization (Non-Interacting Case, $V=0$)

• Setup: A single electron or finite density of non-interacting electrons in a disordered potential ($W$) coupled to classical phonons.
• Finding: In the absence of phonons ($\gamma=0$), the system exhibits Anderson localization (no transport).
• Effect of Coupling: Introducing electron–phonon coupling ($\gamma > 0$) leads to delocalization.
  • The system exhibits sub-diffusive dynamics (mean-square displacement $\sigma^2 \propto t^z$ with $z < 1$).
  • The dynamical exponent $z$ increases with coupling strength $\gamma$ but remains sub-diffusive ($z \approx 0.5$) for strong coupling.
  • This indicates that the classical phonon bath acts as a relaxation channel, destroying the Anderson insulator state.

B. Stability of Many-Body Localization (Interacting Case, $V > 0$)

• Setup: Interacting spinless fermions (t-V model) with disorder, coupled to phonons.
• Finding: The coupling to classical phonons destabilizes the MBL phase.
  • The Charge Density Wave (CDW) order parameter ($O_{CDW}$), which remains finite in a stable MBL phase, decays over time.
  • The decay follows a power law ($O_{CDW} \sim t^{-\alpha}$).
  • Non-monotonic behavior:
    • Weak Coupling ($\gamma \ll t_0$): Interactions accelerate the decay (relaxation is faster than in the non-interacting case).
    • Strong Coupling ($\gamma \gtrsim t_0$): Interactions inhibit relaxation. The formation of heavy polarons reduces particle mobility, effectively increasing the disorder strength seen by the particles, thus slowing down the delocalization process.

C. Physical Mechanism

The authors propose a mechanism for delocalization based on resonant local relaxation:

• The fluctuating classical phonon coordinates create a dynamical disorder potential that adds to the static disorder.
• Occasionally, these fluctuations cancel out the static energy difference between neighboring sites ($\Delta E \approx 0$), allowing electrons to resonantly tunnel.
• In the strong coupling regime, the formation of polarons (self-trapped states) makes particles less mobile, counteracting this relaxation mechanism.

4. Significance and Contributions

• Methodological Advancement: The paper establishes a robust framework for simulating strongly correlated electron–phonon systems in the adiabatic regime. It validates that hybrid quantum-classical methods can capture essential many-body physics (like MBL instability) at a fraction of the computational cost of full quantum treatments.
• Insight into MBL: It provides numerical evidence that MBL is unstable in the presence of a classical phonon bath (a generic relaxation channel in solids), leading to sub-diffusive transport rather than true localization.
• Polaron Physics: The study clarifies the competition between phonon-induced relaxation and polaron-induced localization, showing that strong coupling can paradoxically stabilize localized behavior by reducing mobility.
• Applicability: The methods are applicable to a wide range of problems, including pump-probe experiments, thermalization in disordered materials, and the study of variable-range hopping.

5. Conclusion

The authors successfully demonstrate that coupling disordered, interacting electronic systems to classical Einstein phonons leads to the breakdown of localization (both Anderson and Many-Body). The decay of CDW order is driven by resonant tunneling events facilitated by phonon fluctuations, resulting in sub-diffusive dynamics. The developed Lanczos-MTE and TEBD-MTE methods serve as powerful tools for exploring these non-equilibrium phenomena in regimes previously inaccessible to purely quantum simulations.