Fate of entanglement in magnetism under Lindbladian or non-Markovian dynamics and conditions for their transition to Landau-Lifshitz-Gilbert classical dynamics

TL;DR

This study explores how entanglement in quantum spin chains evolves under Lindbladian and non-Markovian dynamics, identifying conditions under which classical Landau-Lifshitz-Gilbert behavior emerges from quantum entangled states.

Contribution

It demonstrates that Markovian dynamics leads to entanglement decay and classical behavior, while non-Markovian dynamics can preserve entanglement, with specific conditions depending on spin magnitude and magnetic order.

Findings

Lindbladian dynamics causes entanglement to vanish, enabling classical LLG dynamics.

Non-Markovian dynamics can sustain some entanglement, preventing classical transition.

Entanglement decay depends on spin size and magnetic order, with exceptions for certain antiferromagnets.

Abstract

It is commonly assumed in spintronics and magnonics that localized spins within antiferromagnets are in the N\'{e}el ground state (GS), as well as that such state evolves, when pushed out of equilibrium by current or external fields, according to the Landau-Lifshitz-Gilbert (LLG) equation viewing localized spins as classical vectors of fixed length. On the other hand, the true GS of antiferromagnets is highly entangled, as confirmed by very recent neutron scattering experiments witnessing their entanglement. Although GS of ferromagnets is always unentangled, their magnonic low-energy excitation are superpositions of many-body spin states and, therefore, entangled. In this study, we initialize quantum Heisenberg ferro- or antiferromagnetic chains hosing localized spins $S=1/2$, $S=1$ or $S=5/2$ into unentangled pure state and then evolve them by quantum master equations (QMEs) of Lindblad or non-Markovian type, derived by coupling localized spins to a bosonic bath (such as due to phonons) or by using additional ``reaction coordinate'' in the latter case. The time evolution is initiated by applying an external magnetic field, and entanglement of time-evolving {\em mixed} quantum states is monitored by computing its logarithmic negativity. We find that non-Markovian dynamics maintains some degree of entanglement, which shrinks the length of the vector of spin expectation values, thereby making the LLG equation inapplicable. Conversely, Lindbladian (i.e., Markovian) dynamics ensures that entanglement goes to zero, thereby enabling quantum-to-classical (i.e., to LLG) transition in all cases -- $S=1/2$, $S=1$ and $S=5/2$ ferromagnet or $S=5/2$ antiferromagnet -- {\em except} for $S=1/2$ and $S=1$ antiferromagnet. We also investigate the stability of entangled antiferromagnetic GS upon suddenly coupling it to the bosonic bath.