Non-reciprocity is necessary for robust dimensional reduction and strong responses in stochastic topological systems

Abstract

Topological theory predicts the necessary conditions for robust dimensional reduction in a host of quantum and classical systems. Models have recently been proposed for stochastic systems which describe many biological and chemical phenomena. However, general theoretical principles are lacking for this class of systems, exemplified by the breakdown of the celebrated bulk-edge correspondence. We prove that contrary to topological phases in quantum and many classical systems, stochastic systems require non-reciprocal (or non-Hermitian) transitions for robust edge responses, which holds across all dimensions and geometries. We propose a novel explanation of hybridization that destroys edge responses in reciprocal (Hermitian) stochastic systems. Further, we show that stochastic steady states grow dramatically with non-reciprocity, in contrast to their quantum counterparts which plateaus. We analyze the resulting theoretical and physical consequences and how non-reciprocity mitigates the effects of hybridization towards robust edge states in equilibrium and non-equilibrium steady states. These results establish the crucial role of non-reciprocal interactions for responses that are robust to random perturbations in active and living matter.

19 pages, 10 figures. In v3: introduced a novel explanation of hybridization earlier in the manuscript. In v2: Added new section IV about the consequence of non-Hermiticity for topological classification and physical observables. Added new Appendix C with analysis of edge currents in a kagome lattice

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• Topology of the simplest gene switch