In spatially periodic Hermitian systems, such as electronic systems in crystals, the band structure is described by the band theory in terms of the Bloch wave functions, which reproduce energy levels for large systems with open boundaries. In this paper, we establish a generalized Bloch band theory in one-dimensional spatially periodic tight-binding models. We show how to define the Brillouin zone in non-Hermitian systems. From this Brillouin zone, one can calculate continuum bands, which reproduce the band structure in an open chain. As an example, we apply our theory to the non-Hermitian Su-Schrieffer-Heeger model. We also show the bulk-edge correspondence between the winding number and existence of the topological edge states.

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We investigate multilayers of a normal insulator and a Weyl semimetal using two models: an effective model and a lattice model. As a result, we find that the behavior of the multilayers is qualitatively different depending on the stacking direction relative to the displacement vector between the Weyl nodes. When the stacking direction is perpendicular to the displacement vector between the Weyl nodes, the system shows either the normal insulator or the Weyl semimetal phases depending on the thicknesses of the two layers. In contrast, when the stacking direction is parallel to that, the phase diagram is rich, containing the normal insulator phase, the Weyl semimetal phase and the quantum anomalous Hall phases with various values of the Chern number. As a superlattice period increases, the Chern number in the quantum anomalous Hall phases increases. Thus, one can design a quantum anomalous Hall system with various Chern numbers in a multilayer of a Weyl semimetal and a normal insulator. Applications to Weyl semimetal materials are discussed.I.

Energy bands of non-Hermitian crystalline systems are described in terms of the generalized Brillouin zone (GBZ) having unique features which are absent in Hermitian systems. In this paper, we show that in one-dimensional non-Hermitian systems with both sublattice symmetry and time-reversal symmetry such as the non-Hermitian Su-Schrieffer-Heeger model, a topological semimetal phase with exceptional points is stabilized by the unique features of the GBZ. Namely, under a change of a system parameter, the GBZ is deformed so that the system remains gapless. It is also shown that each energy band is divided into three regions, depending on the symmetry of the eigenstates, and the regions are separated by the cusps and the exceptional points in the GBZ.

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