Communications in Mathematical Physics

, Volume 364, Issue 1, pp 343–356 | Cite as

Topological Invariants and Corner States for Hamiltonians on a Three-Dimensional Lattice

  • Shin HayashiEmail author


We consider periodic Hamiltonians on a three-dimensional (3-D) lattice with a spectral gap not only on the bulk but also on two edges at the common Fermi level. By using K-theory applied for the quarter-plane Toeplitz extension, two topological invariants are defined. One is defined for the gapped bulk and edge Hamiltonians, and the non-triviality of the other means that the corner Hamiltonian is gapless. We prove a correspondence between these two invariants. Such gapped Hamiltonians can be constructed from Hamiltonians of 2-D type A and 1-D type AIII topological insulators, and its corner topological invariant is the product of topological invariants of these two phases.


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This work is part of a Ph.D. thesis, defended at the University of Tokyo in 2017. The author would like to expresses his gratitude for the support and encouragement of his supervisor, Mikio Furuta. This work was inspired by the author’s collaborative research with Mikio Furuta, Motoko Kotani, Yosuke Kubota, Shinichiroh Matsuo, and Koji Sato. He would like to thank them for many stimulating conversations and much encouragement. He also would like to thank Christopher Bourne, Ken-Ichiro Imura, Takeshi Nakanishi, and Yukinori Yoshimura for their many discussions, and thank Emil Prodan for sharing the information regarding [5]. This work was supported by JSPS Grant-in-Aid for Scientific Research on Innovative Areas, “Discrete Geometric Analysis for Materials Design”: Grant Number JP17H06461.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics for Advanced Materials – Open Innovation Laboratory, AIST, c/o Advanced Institute for Materials ResearchTohoku UniversitySendaiJapan
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan

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