Abstract
Though the overwhelming majority of natural processes occur far from the equilibrium, general theoretical approaches to non-equilibrium phase transitions remain scarce. Recent breakthroughs introduced a description of open dissipative systems in terms of non-Hermitian quantum mechanics enabling the identification of a class of non-equilibrium phase transitions associated with the loss of combined parity (reflection) and time-reversal symmetries. Here we report that the time evolution of a single classical spin (e.g. monodomain ferromagnet) governed by the Landau-Lifshitz-Gilbert-Slonczewski equation in the absence of magnetic anisotropy terms is described by a Möbius transformation in complex stereographic coordinates. We identify the parity-time symmetry-breaking phase transition occurring in spin-transfer torque-driven linear spin systems as a transition between hyperbolic and loxodromic classes of Möbius transformations, with the critical point of the transition corresponding to the parabolic transformation. This establishes the understanding of non-equilibrium phase transitions as topological transitions in configuration space.
Highlights
The interest to dissipative spin-transfer torque (STT)-driven dynamics of a spin, described by Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation[1,2,3] is two-fold
We find that the PT symmetry-breaking phase transition occurring in STT-driven linear spin systems is a transition between hyperbolic and loxodromic classes of Möbius transformations governing the spin dynamics
We have shown that the time evolution of linear classical single-spin systems has a simple interpretation in terms of Möbius transformations of 2, provided magnetic anisotropies are negligibly small
Summary
Though the overwhelming majority of natural processes occur far from the equilibrium, general theoretical approaches to non-equilibrium phase transitions remain scarce. We identify the parity-time symmetry-breaking phase transition occurring in spin-transfer torque-driven linear spin systems as a transition between hyperbolic and loxodromic classes of Möbius transformations, with the critical point of the transition corresponding to the parabolic transformation. This establishes the understanding of non-equilibrium phase transitions as topological transitions in configuration space. It has recently been shown that nonequilibrium classical spin dynamics described by the LLGS equation naturally follows from the non-Hermitian extension of Hamiltonian formalism[10] Within this framework, the nonconservative effects of Gilbert damping and applied Slonczewski STT3 originate from the imaginary part of the system’s Hamiltonian. The linear form of the spin dynamics equation provides a solid foundation for the study of nonlinear effects in single and coupled spin systems, including chaotic dynamics[11, 12], spin-wave instabilities[13], and solitons[14]
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