Toeplitz operators on concave corners and topologically protected corner states

  • Shin HayashiEmail author


We consider Toeplitz operators defined on a concave corner-shaped subset of the square lattice. We obtain a necessary and sufficient condition for these operators to be Fredholm. We further construct a Fredholm concave corner Toeplitz operator of index one. By using this, a relation between Fredholm indices of quarter-plane and concave corner Toeplitz operators is clarified. As an application, topological invariants and corner states for some bulk-edges gapped Hamiltonians on two-dimensional (2-D) class AIII and 3-D class A systems with concave corners are studied. Explicit examples clarify that these topological invariants depend on the shape of the system. We discuss the Benalcazar–Bernevig–Hughes’ 2-D Hamiltonian and see that there still exist topologically protected corner states even if we break some symmetries as long as the chiral symmetry is preserved.


Toeplitz operators on concave corners Topologically protected corner states Bulk-edge and corner correspondence K-theory and index theory 

Mathematics Subject Classification

19K56 47B35 81V99 



The author would like to thank Takeshi Nakanishi and Yukinori Yoshimura for showing him the result of a numerical calculation, which convinced him about the content of this paper. He also would like to thank Ken-Ichiro Imura and Ryo Okugawa for many discussions concerning [1] and Max Lein for sharing the information regarding [25]. The author acknowledges the support of the Erwin Schrödinger Institute where part of this work was conducted. He would like to thank organizers of the workshop “Bivariant K-theory in Geometry and Physics” for their hospitability. This work was supported by JSPS KAKENHI Grant Nos. JP17H06461 and JP19K14545.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Mathematics for Advanced Materials-OIL c/o AIMR Tohoku UniversityNational Institute of Advanced Industrial Science and TechnologySendaiJapan

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