Validity or not of the bulk-edge correspondence for a hydrodynamic model of equatorial waves.*

A characteristic feature of topological insulators is the surprising robustness to perturbation of the asymmetric transport observed along interfaces separating topologically distinct insulating bulks. This presentation first reviews a general classification of (elliptic) partial differential operators and establishes a fundamental principle, the bulk-edge correspondence (BEC), which relates a hard-to-compute, quantized, edge current observable modeling the robust interface transport to the easy-to-compute index of a Fredholm operator naturally associated to the insulating bulks.

While the BEC applies to many discrete and continuous systems, it it knows to fail for specific (non-elliptic) hydrodynamic models of equatorial waves. One major reason is the presence of flat bands in the spectral decomposition of the bulk Hamiltonians. Such flat bands are in fact also prevalent in many (mathematically more complex) topological plasma physics models. 

In this hydrodynamic model, the insulating phases are characterized by a non-vanishing Coriolis force parameter, which takes a positive value in the northern hemisphere and a negative value in the southern hemisphere. Only in the vicinity of the equator does this parameter vanish, which allows for asymmetric transport along the equator. A BEC would essentially state that the quantized asymmetric transport should be independent from the profile of the Coriolis parameter near the equator.  It is known to be incorrect. 

Our main result is to demonstrate that the BEC is verified when the Coriolis force parameter is sufficiently smooth. Note that this is simply a verification of the BEC, in the sense that the quantized edge current observable takes a value equal to two (2) as predicted from considerations of bulk invariants, but not a derivation. In fact, we show that the BEC may be arbitrarily violated when the Coriolis force parameter admits discontinuities. The theoretical results are confirmed by several numerical simulations.