Quantum Magnets and Matrix Lorenz Systems

3rd International Conference on Mathematical Modeling in Physical Sciences

物理学;数学

Tranchida, J.^1,2 ; Thibaudeau, P.^1 ; Nicolis, S.^2

CEA DAM/Le Ripault, BP 16, Monts

F-37260, France^1

CNRS-Laboratore de Mathématiques et Physique Thdorique (UMR7350), Fédération de Recherche denis Poisson (FR2964), Université François Rabelais de Tours Parc Grandmont, Tours

37200, France^2

关键词: Bimodal distribution;    Chaotic fluctuations;    Identity matrices;    Landau-Lifshitz-Gilbert equations;    Largest Lyapunov exponent;    Lorenz equation;    Quantum fluctuation;    Symmetric tensors;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/574/1/012146/pdf
DOI  :  10.1088/1742-6596/574/1/012146

来源: IOP

【 摘 要 】

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 u(2) u(1) × 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.