Topological invariants and corner states for Hamiltonians on a three dimensional lattice
Periodic Hamiltonians on a three dimensional lattice which have a spectral gap not only on the bulk but also on two edges at the common Fermi level are considered. By using -theory applied for quarter-plane Toeplitz algebras, two topological invariants are defined for such gapped Hamiltonians. One is defined for the bulk and edges, and the other corresponds to wave functions localized near the corner. A correspondence of these two invariants is proved.
Key words and phrases:bulk-edge invariants, bulk-edge and corner correspondence, quarter-plane Toeplitz operators, -theory for -algebras
2010 Mathematics Subject Classification:Primary 19K56; Secondary 47B35, 81V99.
In condensed matter physics, it is known a correspondence between topological invariants defined for a gapped Hamiltonian of an infinite system without edge and that for a Hamiltonian of a system with edge. This correspondence is called the bulk-edge correspondence. In the theoretical study of the quantum Hall effect, an integer valued topological invariant for a gapped Hamiltonian of an infinite system without edge was introduced by D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs [TKNN82]. This invariant is called the TKNN number, which is the first Chern number of the Bloch bundle [Ko85]. Y. Hatsugai considered such phenomena on a system with edge, and defined a topological invariant as a winding number counted on a Riemann surface, in other words, the spectral flow of a family of self-adjoint Fredholm Toeplitz operators [Hat93a]. The equality between these two topological invariants (bulk-edge correspondence) was proved by Hatsugai [Hat93b].
Since the work of J. Bellissard [Be86, BvES94], -theory for -algebras is known to be very useful in this context. J. Kellendonk, T. Richter and H. Schulz-Baldes proved the bulk-edge correspondence by using the six-term exact sequence of -Theory for -algebras associated to the following Toeplitz extension,
where is the -algebra of compact operators on , is the Toeplitz algebra and is the -algebra consists of continuous functions on the unit circle in the complex plane with uniform norm [HKR00, HKR02]. There are many mathematical works (see a comprehensive account [PS16] and references therein).
Apart from the study of topological phases, the analysis of Toeplitz operators was developed (see [Do98, BS06], for example). We here focus on the theory of quarter-plane Toeplitz operators. Such operators were first studied by R. G. Douglas and R. Howe, and many results were obtained [DH71, CDSS71, CDS72]. Among other things, E. Park showed that there is a short exact sequence for -algebras,
where is the quarter-plane Toeplitz algebra (the meaning of each symbols are explained in Sect. LABEL:sec:2.3). Park studied quarter-plane Toeplitz algebras by using -theory for -algebras [Pa90, PS91].
In this paper, we mainly consider a system with corner which appears as the intersection of two edges. We consider a periodic Hamiltonian on the three dimensional lattice , which has a spectral gap not only on the bulk (which means that our Hamiltonian is gapped) but also on two edges (which means that restrictions of our Hamiltonian onto two semigroups of our lattice which correspond to two edges, are gapped) at the common Fermi level. For such systems, we define two topological invariants. One is defined for the bulk and edges, and the other is defined for the corner. Both invariants are defined as elements of some -groups. In our settings, topological invariants considered in the case of the bulk-edge correspondence are zero since our edges are also gapped. In this sense, invariants considered in this paper can be seen as secondary invariants. We next show a correspondence of these two invariants, by using the six-term exact sequence associated to (1). We have maps from these -groups to , and our invariants corresponds to a spectral flow of a one-parameter family of self-adjoint Fredholm quarter-plane Toeplitz operators. Although we mainly consider some three dimensional systems, our method can also be applied to some systems of other dimensions (Remark LABEL:remark311)
Note that, except for the use of the sequence (1), our invariants are defined and the correspondence is proved as in the case of the bulk-edge correspondence. Although our study of such invariants are modeled on Kellendonk–Richter–Schulz-Baldes’ proof of the bulk-edge correspondence, we note that there are many other proofs of the bulk-edge correspondence [EG02, ASV13, GP13, BCR1, Hayashi, Ku16, MT16].
There also is a bulk-edge correspondence for systems with symmetry (time-reversal symmetry for quantum spin Hall systems, for example) [ASV13, GP13, Ku16, MT16, BKR17]. So it is natural to expect such a “bulk-edge and corner” correspondence for systems with symmetries, which is not treated in this paper.
This paper is organized as follows. In Sect. 2, some basic facts about -theory for -algebras, spectral flow and quarter-plane Toeplitz operators, which are used in this paper, is collected. In Sect. LABEL:sec:3, conditions for “gapped” Hamiltonians which are considered in this paper are fixed. Two topological invariants for such Hamiltonians are defined, and the correspondence of these two is proved.
The author expresses gratitude to his supervisor Mikio Furuta for his support and encouragement. This work is motivated by 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 encouragements.
In this section, we collects some basic facts and calculations needed in this paper. Throughout this paper, all algebras and Hilbert spaces are considered over the complex field , and all operators are complex linear.
2.1. -theory for -algebras
In this paper, we use -theory for -algebras in order to define some topological invariants and also to prove our main theorem. This subsection collects some basic facts from -theory for -algebras without proof. We refer the reader to [Mu90, We93, Bl98, HR00, RLL00] for the details.
Let be a unital -algebra, that is a Banach -algebra with multiplicative unit which satisfies for any in . An element in is called a projection if , and an element in is said to be unitary if . For a positive integer , let be the matrix algebra of all matrices with entries in . As in the case of , the matrix algebra has a natural -algebra structure. We know that has a (unique) norm making it a -algebra. We denote the set of all projections in and let by . For and , we write if and only if there exists some such that and . The relation is an equivalence relation on . We define a binary operation on by . Then induces an addition on equivalence classes , and is a commutative monoid. The -group for a unital -algebra is defined to be the group completion (Grothendieck group) of the commutative monoid . We denote the class of in by . For a non-unital -algebra , we define its -group to be the kernel of the map , where is the unitization of , and the map is induced by the projection onto the second component. Let be the group of unitary elements in , and let . We consider the set . Let be a binary operation on defined as above. For and , we write if and only if there exists some such that and are homotopic in . The relation is an equivalence relation on , and induces an addition on equivalence classes . Then is an abelian group. We denote this group by and the class of in by . For a non-unital -algebra , its -group is defined by using its unitization, . Note that, by using the polar decomposition, an invertible element in defines an element in . -homomorphisms between -algebras are said to be homotopic if there exists a path of -homomorphisms for such that the map defined by is continuous for each , and . In this sense, and are the additive covariant homotopy functor from the category of -algebras to the category of abelian groups. Let be a separable Hilbert space and be the set of compact operators on . For a -algebra , we have the stability property, that is, where and is the -algebraic tensor product.
Let be a -algebra. The suspension of is the -algebra , where is the -algebra of continuous functions from to , and is the -algebra of complex valued continuous functions which vanish at infinity. Then there is an isomorphism . We also have a map , called the Bott map. If is a unital -algebra, is given by where . By the Bott periodicity theorem, is an isomorphism. For a short exact sequence of -algebras , there associates the following six-term exact sequence.