Effects of first- and second-order topological phases on equilibrium crystal shapes

We study equilibrium crystal shapes of a topological insulator (TI), crystalline (TCI) protected by mirror symmetry, and second-order (SOTI) inversion symmetry. By adding magnetic fields to the three-dimensional TI, we can realize mirror-symmetry-protected TCI inversion-symmetry-protected SOTI. They each have boundary states in different positions: has gapless on surfaces that are invariant under symmetry operation, SOTI at intersections between certain surfaces. In this paper, discuss how these affect surface energies terms calculations simple tight-binding model using Wulff construction. comparing changes TI trivial through process applying fields, show presence/absence affects emergence specific facets way from insulator.