Topological classification of driven-dissipative nonlinear systems

Abstract: In topology, one averages over local geometrical details to reveal robust global features. This approach proves crucial for understanding quantized bulk transport and exotic boundary effects of linear wave propagation in (meta-)materials. Moving beyond linear Hamiltonian systems, the study of topology in physics strives to characterize open (non-Hermitian) and interacting systems. Here, we establish a framework for the topological classification of driven-dissipative nonlinear systems. Specifically, we define a graph index for the Floquet semiclassical equations of motion describing such systems. The graph index builds upon topological vector analysis theory and combines knowledge of the particle-hole nature of fluctuations around each out-of-equilibrium stationary state. To test this approach, we divulge the topological invariants arising in a micro-electromechanical nonlinear resonator subject to forcing and a time-modulated potential. Our framework classifies the complete phase diagram of the system and reveals the topological origin of driven-dissipative phase transitions, as well as that of under- to over-damped responses. Furthermore, we predict topological phase transitions between symmetry-broken phases that pertain to population inversion transitions. This rich manifesting phenomenology reveals the pervasive link between topology and nonlinear dynamics, with broad implications for all fields of science.