# Linear dynamics of classical spin as Mobius transformation

## 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 Mobius 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 Mobius transformations, with the critical point of the transition corresponding to the parabolic transformation. However, this establishes the understanding of non-equilibrium phase transitions as topological transitions in configuration space.

- Authors:

- Univ. of Chicago, Chicago, IL (United States); Argonne National Lab. (ANL), Argonne, IL (United States)

- Publication Date:

- Research Org.:
- Argonne National Lab. (ANL), Argonne, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22), Materials Sciences and Engineering Division

- OSTI Identifier:
- 1372900

- Grant/Contract Number:
- AC02-06CH11357

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Scientific Reports

- Additional Journal Information:
- Journal Volume: 7; Journal Issue: 1; Journal ID: ISSN 2045-2322

- Publisher:
- Nature Publishing Group

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Galda, Alexey, and Vinokur, Valerii Ð.
```*Linear dynamics of classical spin as Mobius transformation*. United States: N. p., 2017.
Web. doi:10.1038/s41598-017-01326-x.

```
Galda, Alexey, & Vinokur, Valerii Ð.
```*Linear dynamics of classical spin as Mobius transformation*. United States. doi:10.1038/s41598-017-01326-x.

```
Galda, Alexey, and Vinokur, Valerii Ð. Wed .
"Linear dynamics of classical spin as Mobius transformation". United States.
doi:10.1038/s41598-017-01326-x. https://www.osti.gov/servlets/purl/1372900.
```

```
@article{osti_1372900,
```

title = {Linear dynamics of classical spin as Mobius transformation},

author = {Galda, Alexey and Vinokur, Valerii Ð.},

abstractNote = {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 Mobius 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 Mobius transformations, with the critical point of the transition corresponding to the parabolic transformation. However, this establishes the understanding of non-equilibrium phase transitions as topological transitions in configuration space.},

doi = {10.1038/s41598-017-01326-x},

journal = {Scientific Reports},

number = 1,

volume = 7,

place = {United States},

year = {Wed Apr 26 00:00:00 EDT 2017},

month = {Wed Apr 26 00:00:00 EDT 2017}

}