The realization of quantum spin Hall (QSH) effect in HgTe quantum wells (QWs) is considered a milestone in the discovery of topological insulators. The QSH edge states are predicted to allow current to flow at the edges of an insulating bulk, as demonstrated in various experiments. A key prediction of QSH theory that remains to be experimentally verified is the breakdown of the edge conduction under broken time reversal symmetry (TRS). Here we first establish a rigorous framework for understanding the magnetic field dependence of electrostatically gated QSH devices. We then report unexpected edge conduction under broken TRS, using a unique cryogenic microwave impedance microscopy (MIM), on a 7.5 nm HgTe QW device with an inverted band structure. At zero magnetic field and low carrier densities, clear edge conduction is observed in the local conductivity profile of this device but not in the 5.5 nm control device whose band structure is trivial. Surprisingly, the edge conduction in the 7.5 nm device persists up to 9 T with little effect from the magnetic field. This indicates physics beyond simple QSH models, possibly associated with material- specific properties, other symmetry protection and/or electron-electron interactions.
For an ideal I with a positive dimensional real variety, based on moment relaxations, we study how to compute a Pommaret basis which is simultaneously a Groebner basis of an ideal J generated by the kernel of a truncated moment matrix and nesting between I and its real radical ideal. We provide a certificate consisting of a condition on coranks of moment matrices for terminating the algorithm. For a generic delta-regular coordinate system, we prove that the condition is satisfiable in a large enough order of moment relaxations.
