Random Matrix Model Reveals Crossover in Impurity Charge Distribution and Power-Law Behaviour

The behaviour of electrons trapped within disordered materials presents a long-standing challenge in condensed matter physics, influencing the properties of many modern electronic devices. Maxime Debertolis from the Institute of Physics, University of Bonn, and Serge Florens from Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, and their colleagues, now shed new light on this problem by investigating how electrons distribute themselves around imperfections within a material. Their research introduces a simplified model, based on random matrix theory, that accurately captures key features of electron behaviour previously observed in more complex systems, revealing a transition from predictable to strikingly uneven charge distributions. This work not only provides a deeper theoretical understanding of electron behaviour in disordered materials, but also suggests potential avenues for experimental verification using nanoscale devices and chaotic electronic reservoirs, potentially paving the way for more efficient and controllable electronic components.

The model reproduces salient features of the impurity charge distribution observed in previous studies of interacting disordered impurity models. Researchers compute the impurity charge distribution using numerical sampling and find a crossover from a Gaussian distribution, centred on half a charge unit, at strong hybridization to a bimodal distribution, centred on both zero and full charge occupations, at weak hybridization. In the bimodal regime, a universal (−3/2) power-law is also observed within the data, aligning well with an analytic surmise computed with a single random electron level in the bath, providing strong agreement between theory and simulation. The team also derives an exact functional integral for the general case, extending the analytical framework.

Random Matrix Impurity Probability Distribution Derivation

This research details the derivation of the probability distribution for a quantum impurity embedded within a random matrix environment, aiming to determine the probability of finding a specific energy level and square amplitude associated with the impurity. The derivation establishes a joint probability distribution, P(E, z), describing the probability of observing a particular energy (E) and amplitude (z), subject to constraints including normalization of square amplitudes and a relationship between energy levels. The team expresses the probability distribution mathematically, incorporating these constraints through delta functions to ensure model accuracy, relying heavily on random matrix theory and the Kondo problem.

Bimodal Charge Distribution in Quantum Impurities

Researchers have gained new insight into how quantum impurities behave in disordered materials, revealing a surprising transition in their electronic charge distribution. Focusing on a simplified model, a single quantum dot coupled to a chaotic environment, the team discovered that the probability distribution of the charge on the quantum dot changes dramatically depending on the strength of the connection between the dot and its surroundings. When the connection is strong, the charge distribution follows a predictable Gaussian pattern, centred around half a charge unit, but as the connection weakens, the distribution becomes bimodal, forming two peaks indicating no charge or full charge. This bimodal behaviour is unexpected and suggests a fundamental shift in how the impurity interacts with its environment.

Importantly, the researchers found a universal power-law governing the distribution in this bimodal regime, meaning the specific details of the chaotic environment are less important than the overall strength of the connection. The team’s analytical approach allowed them to create a remarkably accurate surmise, a simplified mathematical description, that captures the entire transition between the Gaussian and bimodal distributions, effectively replacing the complex environment with a single random energy level. The accuracy of this surmise, even in predicting the power-law behaviour at weak connections, is a significant achievement with implications for the design of nanoscale devices and the understanding of materials where disorder plays a crucial role.

Impurity Charge Distribution Reveals Disorder Crossover

This research investigates the behaviour of a quantum impurity, a localized electronic level, coupled to a disordered system described by random matrix theory. By examining the distribution of charge on the impurity, the team uncovered a crossover from a bimodal distribution, favouring either zero or full occupation at weak coupling, to a Gaussian distribution as the coupling strengthens. This transition reflects how the impurity moves from being sensitive to charge fluctuations to becoming effectively diluted within the surrounding disordered environment, supported by both numerical simulations and analytical calculations. The study successfully derives a general mathematical framework to describe the probability distribution of energy levels within this model, offering insights into the statistical properties of the system. While the model simplifies real materials, the researchers confirmed its robustness by applying the same analysis to a more realistic disordered system, suggesting the core findings are broadly applicable. Future work could explore the transition between different statistical regimes, particularly concerning the highest energy level in the system, and extend the model to include spin effects and interactions between electrons, with potential applications in understanding charge distribution in nanoscale devices.