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 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.
Highlights
Recent advances in magnetic materials and techniques allow manipulation of spin moments at nanoscale resolution
What has received much less attention is the contribution of quantum fluctuations, that become significant at nanoscale resolution and are crucial for controlling qubit devices [6]
The transition to chaos, at r = rcrit, appears at the same value as for the classical Lorenz system, only much more abrupt in the matrix case. Another way to characterize the quantum fluctuations is by studying the time evolution of the three commutators, Tr([X, Y ]), Tr([Y, Z]) and Tr([X, Z])
Summary
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