[Paper Review] Avalanches in the Relaxation Dynamics of Electron Glasses

Martin Goethe, Matteo Palassini|arXiv (Cornell University)|Aug 4, 2018

Quantum and electron transport phenomena1 references3 citations

TL;DR

This study investigates avalanche dynamics in electron glasses using zero-temperature relaxation simulations with single-electron hops. It finds that displacement-induced avalanches exhibit scale-free, power-law size distributions with exponent τ ≈ 1.5, while injection-induced avalanches show diverging mean avalanche size without a power-law limit, both well described by a branching process model, indicating fundamental differences from infinite-range spin glasses due to the absence of a pseudogap in electron-hole excitations.

ABSTRACT

We study the zero-temperature relaxation dynamics of an electron glass model with single-electron hops. We find numerically that in the charge rearrangements (avalanches) triggered by displacing an electron, the number of electron hops has a scale-free, power-law distribution up to a cutoff diverging with the system size $N$, independently of the disorder strength and provided hops of arbitrary length are allowed. In avalanches triggered by the injection of an extra electron, the distribution does not have a power-law limit, but its mean diverges non-trivially with $N$. In both cases, the avalanche statistics is well reproduced by a branching process model that assumes independent hops. Qualitative differences with avalanches in infinite-range spin glasses and related systems are discussed.

Motivation & Objective

• To understand the nature of relaxation dynamics in disordered electron systems with long-range Coulomb interactions.
• To investigate whether avalanche behavior—characterized by scale-free size distributions—emerges in electron glasses similar to infinite-range spin glasses.
• To determine how the statistics of electron-hopping avalanches depend on system size, disorder strength, and injection vs. displacement triggers.
• To test whether a branching process model accurately captures the observed avalanche statistics in electron glasses.

Proposed method

• Simulates the Efros model of electron glasses on 2D and 3D cubic lattices with periodic boundary conditions and Ewald summation for long-range Coulomb interactions.
• Implements zero-temperature dynamics with energy-lowering single-electron hops, where transition rates decay exponentially with hop distance (Γ ∝ exp(−2r/ξ)).
• Performs two distinct protocols: (1) displacement of a single electron to trigger avalanches, and (2) injection of an electron into the softest available site.
• Analyzes avalanche size distributions p(S) and uses a branching process model to describe the statistics, assuming independent hops across generations.
• Applies finite-size filtering by focusing on 'mid-generation' hops to reduce boundary-induced finite-size effects in the statistics.
• Fits avalanche size distributions to a generalized branching process model (Eq. S11) to extract parameters λ_L (offspring mean) and ρ_L (initial offspring mean), and tests scaling with system size L.

Experimental results

Research questions

• RQ1 Does the relaxation dynamics of electron glasses exhibit scale-free avalanche behavior similar to that in infinite-range spin glasses?
• RQ2 How does the avalanche size distribution p(S) differ between electron displacement and electron injection protocols?
• RQ3 To what extent can the avalanche statistics be described by a branching process model assuming independent, uncorrelated hops?
• RQ4 What is the scaling of the mean avalanche size ⟨S⟩ with system size N in the injection protocol, and how does it relate to the pseudogap exponent δ?
• RQ5 How does the cutoff in the avalanche size distribution depend on system size in two and three dimensions?

Key findings

• Displacement-induced avalanches exhibit a power-law size distribution p(S) ∼ S^−τ with τ ≈ 1.5, consistent with the mean-field prediction, and the cutoff S_c increases linearly with system size L in 3D.
• Injection-induced avalanches do not display a power-law limit in the thermodynamic limit, but the mean avalanche size ⟨S⟩ diverges with system size, scaling approximately as log L in 2D and linearly in 3D.
• The branching process model accurately reproduces both the size distribution and the scaling of ⟨S⟩, with the offspring mean λ_L ≈ 1 for mid-generation hops, indicating critical branching dynamics.
• The parameter ρ_L, representing the initial number of first-generation hops, increases linearly with L in 3D and logarithmically with L in 2D, consistent with mean-field estimates.
• Finite-size effects from boundary re-entry are filtered out by focusing on mid-generation hops, yielding a Poisson-distributed offspring count with mean ≈1, supporting the critical branching process assumption.
• The pseudogap exponent δ does not govern the scaling of ⟨S⟩ with system size, indicating that the avalanche process in electron glasses is qualitatively different from that in spin glasses due to the absence of a pseudogap in electron-hole excitations.