Quantum Magnets and Matrix Lorenz Systems

TL;DR

This paper explores quantum effects in magnetization dynamics by representing them with matrices in Lie algebras, revealing conditions for chaos and quantum fluctuations, and analyzing their complex attractor structures.

Contribution

It introduces a matrix-based approach to quantum magnetization dynamics, connecting Lie algebra properties with chaotic behavior and quantum fluctuations.

Findings

Identification of a criterion for non-linear terms based on invariant tensors

Discovery of knotted attractor structures in the dynamics

Observation of bimodal Lyapunov exponent distribution

Abstract

The Landau--Lifshitz--Gilbert equations for the evolution of the magnetization, in presence of an external torque, can be cast in the form of the Lorenz equations and, thus, can describe chaotic fluctuations. To study quantum effects, we describe the magnetization by matrices, that take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can describe chaotic fluctuations. We identify a criterion, for the appearance of such non-linear terms. This depends on whether an invariant, symmetric tensor of the algebra can vanish or not. This proposal is studied in detail for the fundamental representation of $\mathfrak{u}(2)=\mathfrak{u}(1)\times\mathfrak{su}(2)$. We find a knotted structure for the attractor, a bimodal distribution for the largest Lyapunov exponent and that the dynamics takes place within the Cartan subalgebra, that does not contain only the identity matrix, thereby can describe the quantum fluctuations.