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

被引：40

作者：

Hayashi, Shin ^{[1
,2
]}

机构：

[1] Tohoku Univ, Math Adv Mat Open Innovat Lab, AIST, Adv Inst Mat Res,Aoba Ku, 2-1-1 Katahira, Sendai, Miyagi 9808577, Japan

[2] Univ Tokyo, Grad Sch Math Sci, 3-8-1 Komaba, Tokyo 1538914, Japan

来源：

COMMUNICATIONS IN MATHEMATICAL PHYSICS | 2018年 / 364卷 / 01期

关键词：

D O I：

10.1007/s00220-018-3229-2

中图分类号：

O4 [物理学];

学科分类号：

0702 ;

摘要：

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.

引用

页码：343 / 356

页数：14

共 39 条

[1]

Atiyah M.F., Singer I.M., Index theory for skew-adjoint Fredholm operators, Inst. Hautes Études Sci. Publ. Math., 37, pp. 5-26, (1969)

[2]

Avila J.C., Schulz-Baldes H., Villegas-Blas C., Topological invariants of edge states for periodic two-dimensional models, Math. Phys. Anal. Geom., 16, 2, pp. 137-170, (2013)

[3]

Bellissard J., -Algebras in Solid State Physics. Statistical Mechanics and Field Theory: Mathematical Aspects (Groningen, 1985), pp. 99-156, (1986)

[4]

Bellissard J., van Elst A., Schulz-Baldes H., The noncommutative geometry of the quantum Hall effect, J. Math. Phys., 35, 10, pp. 5373-5451, (1994)

[5]

Benalcazar W.A., Bernevig B.A., Hughes T.L., Quantized electric multipole insulators, Science, 357, pp. 61-66, (2017)

[6]

Blackadar B., K-theory for operator algebras: 2nd edn, Mathematical Sciences Research Institute Publications, 5, (1998)

[7]

Brown A., Pearcy C., Spectra of tensor products of operators, Proc. Am. Math. Soc., 17, pp. 162-166, (1966)

[8]

Bottcher A., Silbermann B., Analysis of Toeplitz Operators, 2nd edn, (2006)

[9]

Bourne C., Carey A.L., Rennie A., The bulk-edge correspondence for the quantum Hall effect in Kasparov theory, Lett. Math. Phys., 105, 9, pp. 1253-1273, (2015)

[10]

Bourne C., Kellendonk J., Rennie A., The K-theoretic bulk-edge correspondence for topological insulators, Ann. Henri Poincaré, 18, 5, pp. 1833-1866, (2017)

← 1 2 3 4 →