Abstract
Magnetic systems can be described by the classical Landau–Lifshitz (LL) equation or the fully quantum open generalized Heisenberg model. Using the Lindblad master equation and the mean-field approximation, we demonstrate that the open generalized Heisenberg model is reduced to a generalized LL equation. The open dynamic is modeled using spin-boson interactions with a common bosonic reservoir at thermal equilibrium. By tracing out the bosonic degrees of freedom, we obtain two different decoherence mechanisms: on-site dissipation and an effective spin–spin interaction mediated by bosons. Using our approach, we perform hysteresis calculations, closely connected with the Stoner–Wohlfarth theory. We compare the exact numerical master equation and the mean-field model, revealing the role of correlations originated by non-local interactions. Our work opens new horizons for the study of the LL dynamics from an open quantum formalism.
Highlights
Since its discovery in 1928 [1], the original Heisenberg exchange interaction between two spins JS1 · S2 has been extended to complex magnetic arrangements and successfully implemented in a variety of quantum systems
To shed more light on the magnetization dynamics of the system we introduce the magnetic moment of each particle through the relation μ(j) = −gsμBS(j)/, where μB = 9.27 × 10−24 JT−1 is the Bohr magneton, and gs ≈ 2 is the g-factor
In order to explain the mismatch between the mean-field approach and master equation, we remark that the meanfield approximation considers that the density matrix of the system can be written as a separable tensor product, as given in Eq (10)
Summary
Since its discovery in 1928 [1], the original Heisenberg exchange interaction between two spins JS1 · S2 has been extended to complex magnetic arrangements and successfully implemented in a variety of quantum systems. The latter is advantageous to future simulations of magnetic-like phenomena using quantum systems like trapped ions [14, 15], superconducting devices [8] and cavity QED [7] It allows the study of more general environments exhibiting memory effects usually described as non-Markovian master equations [50]. We called the superoperator Lnn(ρ) as the nearest-neighbours Lindbladian since it accounts for the effective energy-exchange between adjacent spins This dissipation’s source naturally appears in the master equation of multi-atomic systems coupled to light [70], and it is a pivotal result towards connecting the OHM with the LL theory, since Lnn(ρ) will reproduce the damping term M × (M × Beff ) in equation (2). We use the mean-field approximation to solve the OHM given in Eq (9), and after introducing the non-linear single-particle dynamics, we show its connection with the LL Eq (2)
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