Boundary-obstructed topological phases of a Dirac fermion in a magnetic field.

It is known that in some higher-order topological (HOT) insulators, HOT phases are distinguished not by gap-closings of bulk states but by those of edge states, which are called boundary-obstructed topological phases (BOTP). In this paper, we investigate the BOTP of two-dimensional (2D) Su-Schrieffer-Heeger (SSH) model in a uniform magnetic field. At $\pi$-flux per plaquette, this model corresponds to the typical model of HOT insulators proposed by Benalcazar-Bernevig-Hughes (BBH). The BBH model can be approximated by Dirac fermions with two kinds of mass terms, which will be referred to as BBH Dirac fermion. To clarify the BOTP of the 2D SSH model around $\pi$ flux, we study the BBH Dirac model including a magnetic field. In continuum Dirac models, boundary conditions associated with the hermiticity of Hamiltonians are known to play a crucial role in determining the edge states. We first demonstrate that for the conventional Dirac fermion with a single mass term, such boundary conditions indeed determine the edge states even in the presence of a magnetic field. Next, imposing boundary conditions consistent to the lattice terminations, symmetries, and hermiticity of the Hamiltonian, we obtain the edge states of the BBH Dirac fermion in a magnetic field, and clarify its BOTP. In particular, we show that the unpaired Landau levels, which cause the spectral asymmetry, yield the edge states responsible for the BOTP.