Establishing equivalence between singular quantum kicked rotors and power-law random banded matrices (PRBM) proved difficult due to deterministic correlations within the rotor system. A quantum kicked rotor, possessing either a power-law or logarithmic singularity, reproduces the long-range Anderson transition observed in PRBM after appropriate matching of symmetry class and coupling convention. An equivalence exists between two distinct physical systems: a chaotic system called a quantum kicked rotor and disordered systems described by random matrices.

This finding builds on earlier theoretical work suggesting these connections through detailed mathematical analysis of how energy spreads within each system. Establishing this link refines existing models used to study complex behaviour in areas such as condensed matter physics and quantum chaos where understanding disorder is key. A quantum kicked rotor mirrors behaviour seen in power-law random banded matrices (PRBM), according to researchers at the University of Warwick and KTH Royal Institute of Technology.

This connection stems from similarities in how energy spreads within each system but has been hampered by deterministic correlations inherent to rotors. Understanding this relationship is vital for refining models used to study complex phenomena like those found in condensed matter physics where disorder plays a key role. Floquet matrix elements, essentially the building blocks describing how a periodic ‘kick’ affects a particle’s momentum, much like repeatedly nudging a swing, decay algebraically in both systems.

Establishing equivalence via systematic elimination of deterministic influences using renormalisation group methods

Renormalization-group analysis proved central to establishing equivalence between singular quantum kicked rotors and power-law random banded matrices. The technique allows understanding how physical properties evolve across different scales, much like zooming in or out on an image reveals new details. It was used to systematically eliminate deterministic effects inherent within the rotor system which previously obscured direct comparison with disordered systems.

Analysing interactions as they were ‘coarse grained’, effectively averaging over small variations, revealed underlying universal behaviour independent of microscopic specifics. Detailed examination of interactions at varying scales compared singular quantum kicked rotors with power-law random banded matrices up to two-loop order. Determining that algebraically decaying momentum space Floquet matrix elements describing energy changes over time within the rotor system were negligible enabled a direct comparison with disordered systems such as those modelled by PRBMs and their associated critical exponents.

Two-loop renormalisation confirms equivalence of kicked rotors and power-law random banded matrices

An unprecedented level of precision in modelling quantum systems has been achieved, extending renormalization group analysis to two-loop order for singular kicked rotors, a sharp improvement on previous one-loop methods. This advancement proved key because it allowed direct comparison with power-law random banded matrices (PRBM), revealing equivalence between previously disparate physical models after careful matching of defining characteristics. Demonstrably irrelevant deterministic correlations within the rotor system masked underlying similarities but are negligible at large distances allowing conditions where such effects vanish to be pinpointed.

Equivalence extends to detailed predictions regarding spectral statistics; both localised and critical phases in the rotor system exhibit identical descriptions of localisation volume as their PRBM counterparts. However, it remained unclear whether deterministic correlations within the rotor become irrelevant at large distances or under what conditions its behaviour matches that of PRBM systems.

From a quantum kicked rotor possessing either a power-law or logarithmic singularity, researchers derived a nonlocal supersymmetric nonlinear sigma model. Renormalization-group analysis up to two loops demonstrated that after aligning symmetry class and coupling convention, the rotor reproduces the long-range Anderson transition observed in corresponding PRBM models across localised, critical, and extended infrared regimes.

Deterministic chaos and random matrix theory reveal shared foundations through symmetry analysis

This work offers compelling evidence for underlying connections between seemingly disparate areas of physics; chaotic systems governed by deterministic rules and disordered ones described through random matrices promise new insights into complex behaviour. Careful consideration of symmetry and interactions is required when establishing perfect equivalence but this doesn’t diminish its significance, it clarifies a fundamental link between order and disorder in physical systems.

Developing a theoretical framework using a nonlinear sigma model, a tool to simplify complex physical interactions, they showed correlations inherent within the rotor system become insignificant at larger scales.

The research established that a quantum kicked rotor reproduces the long-range Anderson transition seen in power-law random banded matrix (PRBM) systems. This finding suggests similarities exist between chaotic and disordered physics despite their differing origins, clarifying relationships between order and disorder.

By deriving a nonlocal supersymmetric nonlinear sigma model from the rotor and performing renormalization-group analysis to two loops, researchers showed correlations within the rotor become irrelevant at larger distances. The results indicate this system behaves similarly to its randomly ordered counterpart across localised, critical, and extended infrared regimes.

👉 More information
🗞 Long-range Nonlinear Sigma Model for a Singular Quantum Kicked Rotor
✍️ Weitao Chen and Yunxiang Liao
🧠 ArXiv: https://arxiv.org/abs/2608.17649

More like this