# SKM 2023 – wissenschaftliches Programm

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# SYQC: Symposium Topology in Quantum and Classical Physics – From Topological Insulators to Active Matter

## SYQC 1: Topology in Quantum and Classical Physics – From Topological Insulators to Active Matter

### SYQC 1.3: Hauptvortrag

### Mittwoch, 29. März 2023, 16:00–16:30, HSZ 01

**Topological Phase Transitions in Population Dynamics** — •Erwin Frey — LMU Muenchen, Theresienstrasse 37, 80333 Muenchen, Germany

In this talk, I discuss how topological phases determine the behavior of nonlinear dynamical systems that arise, for example, in population dynamics. We have shown that topological phases can be realized with the antisymmetric Lotka-Volterra equation (ALVE). It governs, for example, the evolutionary dynamics of zero-sum games, such as the rock-paper-scissors game [1]. It also describes the condensation of non-interacting bosons in driven-dissipative setups [2]. We have shown that robust polarization emerges at the chain's edge for the ALVE, defined on a one-dimensional chain of rock-paper-scissors cycles [3]. The system undergoes a transition from left to right polarization as the control parameter passes through a critical value. We found that the polarization states are topological phases and that this transition is indeed a topological phase transition. Remarkably, this phase transition falls into symmetry class D within the "ten-fold way" classification scheme of gapped free-fermion systems. Beyond the observation of topological phases in the ALVE, it might be possible to generalize the approach of our work to other dynamical systems in biological physics whose attractors are nonlinear oscillators or limit cycles.

[1] J. Knebel, T. Krüger, M. F. Weber, and E. Frey, Phys. Rev. Lett. 110, 168106 (2013). [2] J. Knebel, M. F. Weber, T. Krüger, and E. Frey, Nature Communications 6, 6977 (2015). [3] J. Knebel, P. M. Geiger, and E. Frey, Phys. Rev. Lett. 125, 258301 (2020).