Topological classification of driven-dissipative nonlinear systems

TL;DR

This paper develops a topological classification framework for driven-dissipative nonlinear systems, revealing how topology influences phase transitions and dynamic responses in complex physical systems.

Contribution

It introduces a graph index based on topological vector analysis to classify the phases of nonlinear, non-Hermitian, driven-dissipative systems, extending topological methods beyond linear models.

Findings

Classifies the phase diagram of a nonlinear resonator system.

Identifies topological origins of phase transitions and damping responses.

Predicts topological phase transitions in symmetry-broken states.

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.