In topological insulators,massive surface states resulting from local symmetry breaking were thought to exhibit a half-quantized Hall conductance,obtained from the low-energy effective model in an infinite Brillouin z...In topological insulators,massive surface states resulting from local symmetry breaking were thought to exhibit a half-quantized Hall conductance,obtained from the low-energy effective model in an infinite Brillouin zone.In a lattice model,the surface band is composed of a combination of surface states and bulk states.The massive surface states alone may not be enough to support an exact one-half quantized surface Hall conductance in a finite Brillouin zone and the whole surface band always gives an integer quantized Hall conductance as enforced by the TKNN theorem.To explore this,we investigate the band structures of a lattice model describing the magnetic topological insulator film that supports the axion insulator,Chern insulator,and semi-magnetic topological insulator phases.We reveal that the gapped and gapless surface bands in the three phases are characterized by an integer-quantized Hall conductance and a half-quantized Hall conductance,respectively.We propose an effective model to describe the three phases and show that the low-energy dispersion of the surface bands inherits from the surface Dirac fermions.The gapped surface band manifests a nearly half-quantized Hall conductance at low energy near the center of Brillouin zone,but is compensated by another nearly half-quantized Hall conductance at high energy near the boundary of Brillouin zone because a single band can only have an integer-quantized Hall conductance.The gapless band hosts a zero Hall conductance at low energy but is compensated by another half-quantized Hall conductance at high energy,and thus the half-quantized Hall conductance can only originate from the gapless band.Moreover,we calculate the layer-resolved Hall conductance of the system.The conclusion suggests that the individual gapped surface band alone does not support the half-quantized surface Hall effect in a lattice model.展开更多
基金supported by the Research Grants CouncilUniversity Grants Committee+3 种基金Hong Kong(Grant Nos.C7012-21G,and 17301220)the National Key R&D Program of China(Grant No.2019YFA0308603)the National Natural Science Foundation of China(Grant No.12304195)the Chutian Scholars Program in Hubei Province。
文摘In topological insulators,massive surface states resulting from local symmetry breaking were thought to exhibit a half-quantized Hall conductance,obtained from the low-energy effective model in an infinite Brillouin zone.In a lattice model,the surface band is composed of a combination of surface states and bulk states.The massive surface states alone may not be enough to support an exact one-half quantized surface Hall conductance in a finite Brillouin zone and the whole surface band always gives an integer quantized Hall conductance as enforced by the TKNN theorem.To explore this,we investigate the band structures of a lattice model describing the magnetic topological insulator film that supports the axion insulator,Chern insulator,and semi-magnetic topological insulator phases.We reveal that the gapped and gapless surface bands in the three phases are characterized by an integer-quantized Hall conductance and a half-quantized Hall conductance,respectively.We propose an effective model to describe the three phases and show that the low-energy dispersion of the surface bands inherits from the surface Dirac fermions.The gapped surface band manifests a nearly half-quantized Hall conductance at low energy near the center of Brillouin zone,but is compensated by another nearly half-quantized Hall conductance at high energy near the boundary of Brillouin zone because a single band can only have an integer-quantized Hall conductance.The gapless band hosts a zero Hall conductance at low energy but is compensated by another half-quantized Hall conductance at high energy,and thus the half-quantized Hall conductance can only originate from the gapless band.Moreover,we calculate the layer-resolved Hall conductance of the system.The conclusion suggests that the individual gapped surface band alone does not support the half-quantized surface Hall effect in a lattice model.
