Linear dynamics of classical spin as Möbius transformation

Area of Science:

• Physics
• Non-equilibrium thermodynamics
• Quantum mechanics

Background:

• Most natural processes are non-equilibrium, yet theoretical frameworks for non-equilibrium phase transitions are limited.
• Recent advances utilize non-Hermitian quantum mechanics for open dissipative systems, identifying phase transitions linked to parity-time symmetry loss.

Purpose of the Study:

• To describe the time evolution of classical spin systems using Möbius transformations.
• To identify the parity-time symmetry-breaking phase transition in spin-transfer torque-driven systems.
• To connect non-equilibrium phase transitions to topological transitions in configuration space.

Main Methods:

• Modeling the time evolution of a classical spin using the Landau-Lifshitz-Gilbert-Slonczewski equation.
• Employing complex stereographic coordinates to represent spin dynamics as Möbius transformations.
• Analyzing the transition between hyperbolic and loxodromic Möbius transformation classes.

Main Results:

• The time evolution of a classical spin is shown to be a Möbius transformation.
• The parity-time symmetry-breaking phase transition is identified as a transition between hyperbolic and loxodromic Möbius transformations.
• The critical point of this transition corresponds to a parabolic Möbius transformation.

Conclusions:

• Non-equilibrium phase transitions in linear spin systems can be understood as topological transitions.
• This work provides a new perspective on non-equilibrium phase transitions using the framework of Möbius transformations.
• The findings bridge classical spin dynamics with concepts from non-Hermitian quantum mechanics and topology.