Topological invariants and corner states

Topological invariants and corner states for Hamiltonians on a three dimensional lattice

Abstract.

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.

1. Introduction

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 [TKNdN82]. This invariant is called the TKNN number, which is the first Chern number of the Bloch bundle [Koh85]. 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 [Bel86, BvESB94], -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 [SBKR00, KRSB02]. There are many mathematical works (see a comprehensive account [PSB16] and references therein).

Apart from the study of topological phases, the analysis of Toeplitz operators was developed (see [Dou98, 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,

(1)

where is the quarter-plane Toeplitz algebra (the meaning of each symbols are explained in Sect. 2.3). Park studied quarter-plane Toeplitz algebras by using -theory for -algebras [Par90, 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 3.10)

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, ASBVB13, GP13, BCE15, Hay16, Kub16, MT16].

There also is a bulk-edge correspondence for systems with symmetry (time-reversal symmetry for quantum spin Hall systems, for example) [ASBVB13, GP13, Kub16, 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. 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.

Acknowledgement

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.

2. Preliminaries

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 [Mur90, WO93, Bla98, 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.

The map is called the exponential map. If and are unital -algebras, and is a closed ideal in , then is expressed in the following way. For , we can take its self-adjoint lift , and then we have . Note that the following diagram is commutative.

2.2. Spectral Flow and Winding Number

In this subsection, we discuss some relationship between the spectral flow and the winding number [AS69, APS75, Phi96].

Let (resp. ) be the set of strictly positive (resp. negative) real numbers. For a separable Hilbert space , we consider the space of bounded linear operators on with norm topology. Let be the subspace of consists of all self-adjoint Fredholm operators whose essential spectrum1 does not contained neither nor . We consider the following subspace of ,

Let be the inclusion. Then is a homotopy equivalence. Let be the inductive limit of a sequence , where is the unitary group of degree , and the map is given by . We have a map given by . The map also is a homotopy equivalence.2 Thus we have . For a continuous loop in , its spectral flow is defined. This is, roughly speaking, the net number of crossing points of eigenvalues with zero counted with multiplicity. Then the following is commutative diagram.

(2)

where the map is given by the spectral flow, and the map is given by taking the winding number of the determinant. All arrows are group isomorphisms.

2.3. Quarter-Plane Toeplitz -algebras

In this subsection, we collects basic facts about quarter-plane Toeplitz operators which are used later.

Let be the Hilbert space . For a pair of integers , let be the element of that is at and elsewhere. For such , let be the translation operator defined by . We choose real numbers , and let and be closed subspaces of spanned by and , respectively. Let be their intersection (see Figure .).

Figure 1. Half planes and , and the quarter-plane in

We can take or , but not both. Let and be orthogonal projections of onto and , respectively. Then be the orthogonal projection of onto . We define the quarter-plane Toeplitz -algebra to be the -algebra generated by . We also define half-plane Toeplitz -algebras and to be -algebras generated by and , respectively. We have surjective -homomorphisms and , which map to and , respectively. We also have surjective -homomorphisms and which map to and to , respectively, where . Well-definedness of and is proved in [Par90], and that of and is proved in [CD71]. Let , and be bounded linear maps given by compression, that is, , and , respectively. We define a -algebra to be the pullback of and along , that is, There is a -homomorphism given by . Let be the -algebra of compact operators on .

Theorem 2.1 (Park[Par90]).

There is the following short exact sequence,

which has a linear splitting given by .

The following theorem follows immediately by using Atkinson’s theorem.3

Theorem 2.2 (Douglas-Howe[Dh71], Park[Par90]).

An operator in is a Fredholm operator if and only if and are both invertible elements in and , respectively.

By taking a tensor product of the above sequence and , we have the short exact sequence,4 Associated to this sequence, we have the following six-term exact sequence.

Note that is isomorphic to .

Lemma 2.3.

The map is surjective.

Proof.

Let us take a base point of , then we have isomorphisms5 and . Consider the following commutative diagram,

The group is isomorphic to [Par90], and the map is an isomorphism [Par90, Jia95]. Thus the left map is surjective. ∎

Remark 2.4.

Note that the group is calculated as follows [Par90].

In this sense, depends delicately on angles , of edges. By the proof of Lemma 2.3, this component maps to zero by .

3. Bulk-Edge and Corner Correspondence

In this section, we consider some “gapped” Hamiltonians, and define two topological invariants for them. A correspondence between these two is proved.

3.1. Bulk-Edge Invariant

Let be a finite dimensional Hermitian vector space and denote the complex dimension of by . Let , , , , and . We consider a continuous family of bounded linear operators, where, for each in , the operator is a self-adjoint multiplication operator generated by a continuous map.6 We call bulk Hamiltonians. By using the Fourier transform , we have a continuous family of self-adjoint bounded linear operators. We write these operators by using the same symbol.

Example 3.1.

We assume that we are given, for each , an endomorphism on , which satisfies . We consider an operator, defined by,

We assume that is a self-adjoint operator. Then its partial Fourier transform gives an example of our family.

We consider following half-plane Toeplitz operators,

They are bounded self-adjoint operators. We call and edge Hamiltonians. We take an orthonormal frame of . Then, since and are compression of the same operator , the pair defines a self-adjoint element of the -algebra . We now take a real number such that does not contained neither nor for any in . Then the element is invertible in .

Remark 3.2.

Such does exists. Actually we can take sufficiently large or small. However, if we choose such , our topological invariants are zero (see also Remark 3.9). Non-trivial invariants appear if operators and have a common spectral gap at the Fermi level . Note that, in this case, our Hamiltonian also has a spectral gap at since is contained in .

By Remark 3.2, we further assume that does not contained neither nor . We refer to our assumption about the choice of as the spectral gap condition.

We next define some topological invariants for our operators. We consider following subspaces of the complex plane.

Let be a continuous function which is on and on .

Definition 3.3.

By the continuous functional calculous, we have a projection in . We denote the element in the -group by , and call the bulk-edge invariant.7

Remark 3.4.

Note that, in order to define the bulk-edge invariant, we use just the information of “bulk and edge”, and do not use the information of “corner”. This is a justification of the name of the “bulk-edge invariant”.

3.2. Corner Invariant

We consider following quarter-plane Toeplitz operators,

We call corner Hamiltonians.

Definition 3.5.

By Theorem 2.2, we have a continuous family of bounded self-adjoint Fredholm operators. By our spectral gap condition, this family defines an element of the -group . We call the corner invariant.

3.3. Correspondence

The following is the main theorem of this paper.

Theorem 3.6.

The map maps the bulk-edge invariant to the corner invariant, that is,

Proof.

Let . Since , we have , where is a self-adjoint lift of . By our spectral gap condition and Theorem 2.1, we have . By considering a spectral deformation which collapses eigenvalues in some small neighborhoods of and to points and , respectively, we can deform into an element of . Thus we obtain . Let us consider the isomorphisms . Then we have the following relation.

Since two loops and are homotopic in , we have . ∎

Remark 3.7.

By the diagram (2), the map maps our invariants to the spectral flow of . The spectral flow of is defined by counting wave functions localized near the corner. Thus, if the corner invariant is non-trivial, there exists topologically protected corner states. Note that the bulk-edge invariant cannot change unless the spectral gap of edges closes. By Theorem 3.6, this stability also holds for corner invariants.

Remark 3.8.

Since the map is not an isomorphism (Remark 2.4), bulk-edge invariants may have more information than corner invariants.

Remark 3.9.

If the spectrum of is contained in or , then we can define the bulk-edge invariant in the same way. The spectral flow of the family is also defined.8 In this case, it is easy to see that (see the proof of Theorem 3.6) and . Thus we still have such “bulk-edge and corner” correspondence.

Remark 3.10.

Since is just a parameter space in our formulation, we can generalize the parameter space to other spaces. Let be a compact Hausdorff space, and consider a continuous family of bounded self-adjoint multiplication operators generated by continuous functions . Then we have edge Hamiltonians and the corner Hamiltonian. If we assume our spectral gap condition, we can define the bulk-edge invariant and the corner invariant in the same way as elements of and , respectively. As in Theorem 3.6, it is easily checked that the bulk-edge invariant maps to the corner invariant by the exponential map . In this case we lack the understanding of the corner invariant as the spectral flow, but we still have a relation with our invariants and corner states. It is easily checked that if the corner invariant is non-trivial, then there are topologically protected corner states. In particular, if we take , then this argument gives a “bulk-edge and corner” correspondence for such four dimensional systems.

Footnotes

  1. In this paper, we denote the spectrum of an operator by , and the essential spectrum of by . For a self-adjoint operator , the essential spectrum of consists of accumulation points of and isolated points of with infinite multiplicity (see Proposition of [HR00], for example).
  2. The subspace and the commutativity of the diagram (2) was discussed explicitly in [Phi96], while the homotopy equivalence between and was essentially proved in [AS69]. We here follow the notations used in [Phi96].
  3. Such a necessary and sufficient condition was first obtained by Douglas-Howe [DH71] for the special case by expressing the algebra as a tensor product of two Toeplitz algebras. Park obtained such conditions for the general in a different way [Par90].
  4. Since is an abelian -algebra, is a nuclear -algebra by Takesaki’s theorem. Since is nuclear, this sequence is exact (see [Mur90], for example).
  5. See Example 7.5.1 of [Mur90], for example.
  6. Let be a continuous map. Then the operator on defined by is called the multiplication operator generated by .
  7. In order to define the element of the -group , we took the orthonormal frame of . However, this element does not depend on the choice.
  8. In this case, the family is not contained in , and does not define an element of the -group .

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