Topological Phase Transition in Coupled Rock-Paper-Scissors Cycles

A hallmark of topological phases is the occurrence of topologically protected modes at the system’s boundary. Here, we find topological phases in the antisymmetric Lotka-Volterra equation (ALVE). The ALVE is a nonlinear dynamical system and describes, for example, the evolutionary dynamics of a rock-paper-scissors cycle. On a one-dimensional chain of rock-paper-scissor cycles, topological phases become manifest as robust polarization states. At the transition point between left and right polarization, solitary waves are observed. This topological phase transition lies in symmetry class $D$ within the “tenfold way” classification as also realized by 1D topological superconductors.

Figure 1

One-dimensional chain of rock-paper-scissors cycles. The interactions on the $S$ sites of the RPS chain (one single RPS cycle highlighted) are captured by the antisymmetric matrix $A$ in Eq. (2). An arrow from one site to another indicates that mass is transported in this direction at a rate of $r_1,r_2,r_3>0$ following the ALVE (1); the skewness $r=r_2/r_3$ defines the control parameter. The auxiliary site $S+1$ facilitates periodic boundary conditions (dashed lines) within the framework of topological band theory.

Figure 2

Polarization of mass to the boundary. (a)–(c) (i) Temporal mass averages $⟨x_{\alpha }{⟩}_T$ from single realizations of the ALVE (1) are depicted on the RPS chain (disk size encodes magnitude) and on lin-log scale for $T=5000$. Mass becomes polarized to the right for $r<1$ and to the left for $r>1$. For $r=1$, mass is uniformly distributed on the whole chain. Polarization is no state of rest (nonvanishing fluctuations ${\sigma }_{\alpha ,T}$ around the averages; insets). (a)–(c) (ii) Polarization is robust against perturbations (${\epsilon }_{\alpha \beta }$ uniformly sampled in $[-0.30,0.20]$) with the same qualitative characteristics as without perturbations.

Figure 3

Solitary waves at the transition point $r=1$. The suitably prepared wave package ($t=0$) remains localized and travels along the RPS chain ($S=69$) without changing its shape. In an interaction with another solitary wave ($t=120$), the shapes of both wave packages remain unchanged afterwards ($t=240$). The initial wave packages were numerically obtained from the dispersion of a mass peak at a single site.

Figure 4

Topological phase transition on the RPS chain. Point-symmetric band structure of the RPS Hamiltonian (white dots denote eigenvalues of $H$ for $S=14$) with a spectral gap for $r\ne 1$ (gray shade). The band structures for $r<1$ and $r>1$ are topologically distinct in how the eigenvectors change within the Brillouin zone, which is quantified by the topological invariant $\mathcal{M}$ and visualized by whether $\mathbf{h}(k)$ winds around 0 [34].