Non-Hermitian topology for stochastic systems and relevant symmetries

Establish the relationship between non-Hermitian topological classifications and master-equation-based stochastic systems, and identify the symmetry classes that are relevant for biological networks.

Background

Topological classifications originating from random matrix theory are well developed for Hermitian systems, and recent work has begun extending them to non-Hermitian settings.

Stochastic systems mapped from master equations are inherently non-Hermitian for exhibiting edge states, but a dedicated classification that directly captures their topological phases and the relevant symmetries for biological applications remains to be established.

References

While some topological classifications of non-Hermitian Hamiltonians have been recently proposed , the relation of non-Hermitian topology to stochastic systems and the relevant symmetries for biological systems are questions that remain open.

— Topological phases in discrete stochastic systems  (2406.03925 - Agudo-Canalejo et al., 2024) in Section 6.5 (Topological classification using symmetries)

Finally, the repeated reentrance has been demonstrated here for one minimal network. Determining which graph structures, rate asymmetries, and non-Abelian representations support finite-time Chern-number transitions remains an open problem.

— Non-Abelian Spin Counting of Ordered Stochastic Trajectories: Reentrant Finite-Time Chern Numbers  (2608.23533 - Du, 24 Aug 2026) in Discussion, paragraph beginning “Several limitations point to natural extensions.”

In addition, the topological classification of stochastic systems with separable eigenvalue bands (where separability is defined by eigenvalues satisfying \lambda_n(\bm{k}) \neq \lambda_m(\bm{k}) for all m \neq n and all \bm{k}) remains open.

— Classification of topological phases of matter in stochastic systems  (2609.38171 - Li et al., 29 Sep 2026) in Section Conclusion