Detecting the Néel vector of altermagnets in heterostructures with a topological insulator and a crystalline valley-edge insulator

We investigate topological phases in a bilayer system composed of an altermagnet and a two-dimensional topological insulator described by the Bernevig-Hughes-Zhang model. A topological phase transition occurs from a first-order topological insulator to a trivial insulator at a certain critical altermagnetization if the N\'eel vector of altermagnet is along the $x$ or $y$ axis. It is intriguing that valley-protected edge states emerge along the N\'eel vector in this trivial insulator, which are as stable as the topological edge states. We name it a crystalline valley-edge insulator. On the other hand, the system turns out to be a second-order topological insulator when the N\'eel vector is along the $z$ axis. The tunneling conductance has a strong dependence on the N\'eel vector. In addition, the band gap depends on the N\'eel vector, which is measurable by optical absorption. Hence, it is possible experimentally to detect the N\'eel vector by measuring tunneling conductance and optical absorption.