Topology of crystalline insulators and superconductors

Topology of crystalline insulators and superconductors

Ken Shiozaki Department of Physics, Kyoto University, Kyoto 606-8502, Japan    Masatoshi Sato Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
July 5, 2019

We complete a classification of topological phases and their topological defects in crystalline insulators and superconductors. We consider topological phases and defects described by non-interacting Bloch and Bogoliubov de Gennes Hamiltonians that support additional order-two spatial symmetry, besides any of ten classes of symmetries defined by time-reversal symmetry and particle-hole symmetry. The additional order-two spatial symmetry we consider is general and it includes global symmetry, mirror reflection, two-fold rotation, inversion, and their magnetic point group symmetries. We find that the topological periodic table shows a novel periodicity in the number of flipped coordinates under the order-two spatial symmetry, in addition to the Bott-periodicity in the space dimensions. Various symmetry protected topological phases and gapless modes will be identified and discussed in a unified framework. We also present topological classification of symmetry protected Fermi points. The bulk classification and the surface Fermi point classification provide a novel realization of the bulk-boundary correspondence in terms of the K-theory.


I Introduction

Symmetry and topology have been two important principles in physics, both of which result in quantum numbers and the conservation laws. In many-body systems, symmetry can be broken spontaneously as a collective phenomenon. The spontaneous symmetry breaking, which is characterized by local order parameters, describes many quantum phases such as ferromagnetism and superconductivity.

Topology also describes quantum phases that are not captured by spontaneously symmetry breaking. Instead of local order parameters, those quantum phases are characterized by topological numbers of wave functions. Such quantum phases are called as topological phase. Volovik (2003) Integer and fractional quantum Hall systems are two representative examples of topological phases. 111 More specifically, these two states are classified into two different categories of topological phase: Integer quantum Hall states belong to a short range entangled topological phase, but fractional quantum Hall states belong to a long range entangled one. Whereas short range entangled topological phases does not have topological degeneracy, i.e. they have a unique ground state on a closed real space manifold, long range entangled ones show topological degeneracy. The presence of symmetry crucially enriches possible short range entangled topological phases, which referred to as symmetry protected topological phase. Chen et al. (2013) In particular, those in non-interacting fermionic system are called as topological insulator and superconductor, which we will discuss in this paper. The ground state wave functions of these quantum Hall states host non-zero Chern numbers, which directly explain the quantization of the Hall conductivity. Thouless et al. (1982); Kohmoto (1985); Niu et al. (1985) In general, a topologically nontrivial phase can not adiabatically deform into a topologically trivial one, and it is robust under perturbations and/or disorders unless the bulk gap closes.

It has been recently discovered that topological phases are enriched by general symmetries of time-reversal and charge conjugation Hasan and Kane (2010); Qi and Zhang (2011); Volovik (2011); Tanaka et al. (2012a); Ando (2013); Budich and Trauzettel (2013); Fruchart and Carpentier (2013). Those non-spatial symmetries can persist even in the presence of disorders and/or perturbations. For instance, non-magnetic disorders retain time-reversal symmetry (TRS), and thus a non-trivial topological phase accompanied by TRS is robust against non-magnetic disorders. Quantum spin Hall states Kane and Mele (2005a, b); Bernevig and Zhang (2006) and topological insulators Moore and Balents (2007); Fu et al. (2007); Fu and Kane (2007); Roy (2009a); Qi et al. (2008) support such topological phases protected by TRS. In a similar manner, charge conjugation symmetry specific to superconductivity makes it possible to realize a novel topological state of matters, topological superconductor. Read and Green (2000); Ivanov (2001); Kitaev (2001); Sato (2003); Fu and Kane (2008); Linder et al. (2010); Qi et al. (2009); Schnyder et al. (2008); Sato (2009, 2010); Tanaka et al. (2009a); Sato and Fujimoto (2009); Sato et al. (2009); Tanaka et al. (2010); Sau et al. (2010); Alicea (2010); Sato et al. (2010); Sato and Fujimoto (2010); Lutchyn et al. (2010); Oreg et al. (2010); Tanaka et al. (2009b); Nakosai et al. (2012); Qi et al. (2013); Hosur et al. (2014); Goswami and Roy (2014); Foster et al. (2014); Wan and Savrasov (2014) Topological phases enriched by those general non-spatial symmetries are classified for non-interacting fermionic systems, Schnyder et al. (2008); Kitaev (2009); Teo and Kane (2010); Stone et al. (2011); Abramovici and Kalugin (2012); Wen (2012) in terms of the Altland-Zirnbauer (AZ) tenfold symmetry classes. Altland and Zirnbauer (1997)

Whereas the classification based on the non-spatial symmetries successfully captures topological nature of general systems, real materials often have other symmetries specific to their structures such as translational and point group symmetries. Those additional symmetries also give rise to a non-trivial topology of bulk wave functions and gapless states on boundaries. Volovik (1987); Ryu and Hatsugai (2002); Sato (2006); Teo et al. (2008); Béri (2010); Yada et al. (2011); Sato et al. (2011); Schnyder and Ryu (2011); Wan et al. (2011); Yang et al. (2011); Burkov and Balents (2011); Fu and Kane (2007); Hatsugai (2010); Hughes et al. (2011); Turner et al. (2010); Sato (2009, 2010); Fu and Berg (2010) It had been naively anticipated that the gapless boundary modes are fragile against disorders because these specific symmetries are microscopically sensitive to small perturbations, but recent studies of topological crystalline insulators have shown that if the symmetries are preserved on average, then the existence of some gapless boundary states is rather robust. Mong et al. (2012); Ringel et al. (2012); Fu and Kane (2012); Fulga et al. (2012) Moreover, surface gapless states protected by the mirror reflection crystal symmetry have been observed experimentally. Fu (2011); Hsieh et al. (2012); Tanaka et al. (2012b); Dziawa et al. (2012); Xu et al. (2012) Motivated by those progresses, various symmetries and corresponding topological phases have been elucidated in insulators Fang et al. (2012); Slager et al. (2012); Jadaun et al. (2013); Fang et al. (2013a); Chiu et al. (2013); Liu and Zhang (2013); Alexandradinata et al. (2014) and superconductors. Teo and Hughes (2013); Ueno et al. (2013); Zhang et al. (2013); Liu and Law (2013); Benalcazar et al. (2013) In particular, various symmetry protected Majorana fermions have been predicted in spinful unconventional superconductors or superfluids Ueno et al. (2013); Zhang et al. (2013); Tsutsumi et al. (2013); Sato et al. (2014).

In this paper, we complete a topological classification of crystalline insulators and superconductors that support additional order-two spatial symmetry besides ten classes of discrete AZ symmetries. Our classification reproduces previous results for additional reflection symmetry Chiu et al. (2013); Morimoto and Furusaki (2013), but the symmetry we consider is general, and it also includes global symmetry, two-fold rotation, and inversion. Furthermore, the additional symmetry can be anti-unitary. Although ordinary point group symmetries are given by unitary operators, systems in a magnetic field or with a magnetic order often support an anti-unitary symmetry as a magnetic point group symmetry. The magnetic symmetry also has been known to provide non-trivial topological phases in various systems Sato and Fujimoto (2009); Mong et al. (2010); Mizushima et al. (2012); Mizushima and Sato (2013); Fang et al. (2013b); Liu (2013); Kotetes (2013); Fang et al. (2014); Zhang and Liu (2014).

Our approach here provides a unified classification of topological phases and defects in crystalline insulators and superconductors with additional order-two symmetry. The topological classification we obtain indicates that topological defects can be considered as boundary states in lower dimensional systems. The resultant topological periodic table shows a novel periodicity in the number of the flipped coordinates under the order-two additional spatial symmetry, in addition to the Bott-periodicity in the space dimensions. Using the new topological periodic table, various symmetry protected topological gapless modes at topological defects are identified in a unified manner.

In addition, we also present a topological classification of Fermi points in the crystalline insulators and superconductors. The bulk topological classification and the Fermi point classification show the bulk-boundary correspondence in terms of the K-theory.

The organization of this paper is as follows. In Sec. II, we explain the formalism we adapt in this paper. In this paper, we use the approach based on the K-theory. Atiyah et al. (1964); Atiyah (1966); Dupont (1969); Karoubi (2008) Our main results are summarized in Sec. III. We show relations between K-groups with different order-two additional spatial symmetries and dimensions. The derivation and proof are given in Sec. VII. In Sec. IV, we discuss properties of the obtained K-groups in the presence of additional symmetry. A novel periodicity in the number of flipped coordinates under the additional symmetry is pointed out. We also find that the K-groups naturally implement topological defects as boundaries of lower dimensional crystalline insulators/superconductors. Crystalline weak topological indices are argued in Sec. VI. In Sec. V, we present topological classification tables of crystalline insulators/superconductors and their defect zero modes with order-two additional spatial symmetry. The topological periodic tables are classified into four families. Various symmetry protected topological phases and their gapless defect modes are identified in a unified framework. We also apply our formalism to a classification of Fermi point protected by additional order-two symmetry in Sec. VIII. By combing the results in Secs. III and VIII, the bulk-boundary correspondence of K-groups are presented. In Sec. IX, we demonstrate that the Ising character of Majorana fermions is a result of symmetry protected topological phases. In Sec. X, we apply our theory to anomalous topological pumps in Josephson junctions, in which crystalline symmetry is not essential to lead to a new topological classification. We conclude the present paper with some discussions in Sec. XI.

Some technical details are presented in Appendices. In Appendix A, following Ref. Teo and Kane, 2010, we introduce useful maps between Hamiltonians in different dimensions. The isomorphic maps introduced here are used in Sec. VII. We review the dimensional hierarchy of AZ classes in the absence of additional symmetry in Appendix B. The classifying spaces of AZ classes with additional order-two symmetry are summarized in Appendix C. The definition and the basic properties of Chern numbers, winding numbers, and topological numbers which are used in this paper, are given in Appendix D. Throughout this paper, we use the notation , and to represent the Pauli matrices in the spin, Nambu and orbital spaces, respectively.

Ii Formalism

In this section, we briefly give our set up of the classification problem. The reader who only concerns the classification table with an additional symmetry, please see Sec. V.

ii.1 Spatially Modulated Hamiltonian

In this paper, we consider band-insulators and superconductors which are described by Bloch and Bogoliubov de Gennes (BdG) Hamiltonians, respectively. In addition to uniform ground states, we also consider topological defects of these systems. Away from the topological defects, the systems are gapped, and they are described by spatially modulated Bloch and BdG Hamiltonians, Volovik (2003); Teo and Kane (2010)


Here the base space of the Hamiltonian is composed of momentum , defined in the -dimensional Brillouin zone , and real-space coordinates of a -dimensional sphere surrounding a defect. For instance, the Hamiltonian of a point defect in three-dimensions is given by , where are the coordinates of a two-dimensional sphere surrounding the point defect. Another example is a line defect in three-dimensions, in which the Hamiltonian is where is a parameter of a circle enclosing the line defect. The case of corresponds to a uniform system.

As mentioned above, the exact base space is , but instead we consider a simpler space in the following. This simplification does not affect on “strong” topological nature of the system. Although the difference of the base space may result in “weak” topological indices of the system, they can be obtained as“strong” topological indices in lower dimensions, as will be argued in Sec.VI. Therefore, generality is not lost by the simplification.

Below, we treat and in the Hamiltonian as classical variables, i.e. momentum operators and coordinate operators are commute with each other. This semiclassical approach is justified if the characteristic length of the spatial inhomogeneity is sufficiently longer than that of the quantum coherence. A realistic Hamiltonian would not satisfy this semiclassical condition, but if there is no bulk gapless mode, then the Hamiltonian can be adiabatically deformed so as to satisfy the condition. Because the adiabatic deformation does not close the bulk energy gap, it retains the topological nature of the system. Niemi and Semenoff (1986); Teo and Kane (2010); Sato et al. (2011); Shiozaki et al. (2012)

ii.2 Symmetries

ii.2.1 Altland-Zirnbauer Symmetry Classes

In the present paper, we classify the topological phases that have an additional symmetry, beside any of the ten AZ symmetry classes. Here we briefly review the AZ symmetry classes.

The AZ symmetry classes are defined by the presence or absence of TRS, particle-hole symmetry (PHS) and/or chiral symmetry (CS). The AZ symmetries, TRS, PHS, and CS, imply


respectively, where and are anti-unitary operators and is a unitary operator. For spin-1/2 fermions, time-reversal operator is given by with the Pauli matrix in the spin space and the complex conjugation operator , which obeys . In the absence of the spin-orbit interaction, spin rotation symmetry allow a different time-reversal symmetry with . PHS is naturally realized in superconductors as with the Pauli matrix acting on the Nambu space of the BdG Hamiltonian, where , but again spin-rotation symmetry can introduce another PHS with . Finally, CS can be obtained by combination of TRS and PHS, . With a suitable choice of the phase , one can always place the relation .

In terms of the sign of and , the Hamiltonians are classified into ten symmetry classes listed in Table 1. The AZ symmetry classes are further divided into two complex classes and eight real classes: In the absence of time-reversal invariance and particle-hole symmetry, the Hamiltonian belongs to one of two complex classes, A or AIII. The presence of the anti unitary symmetries and introduces a real structure of the Hamiltonian, and thus the remaining eight classes are called as real AZ classes.

Below, we choose a convention that and commutes with each other, i.e. : Because Eq.(2.2) yields for any Hamiltonians with TRS and PHS, the unitary operator should be proportional to the identity, . The phase can be removed by a re-definition of the relative phase between and without changing the sign of and , which leads to .

s AZ class TRS PHS CS or classifying space or
0 A
0 AI
2 D
6 C
7 CI
Table 1: AZ symmetry classes and their classifying spaces. The top two rows ( (mod 2)) are complex AZ classes, and the bottom eight rows ( (mod 8)) are real AZ classes. The second column represents the names of the AZ classes. The third to fifth columns indicate the absence (0) or the presence of TRS, PHS and CS, respectively, where means the sign of and . The sixth column shows the symbols of the classifying space.

ii.2.2 Order-Two Spatial Symmetry

In addition to the AZ symmetries, we assume an additional symmetry of Hamiltonians. As an additional symmetry, we consider general order-two spatial symmetry. Order-two symmetry implies that the symmetry operation in twice trivially acts on the Hamiltonian,


where can be either unitary or anti-unitary . The order-two unitary symmetry includes reflection, two-fold spatial rotation and inversion. It also permits global symmetry such as a two-fold spin rotation. The anti-unitary case admits order-two magnetic point group symmetries.

Under an order-two spatial symmetry, the momentum in the base space of the Hamiltonian transforms as


with an orthogonal matrix satisfying . Note that like time-reversal operator, the anti-linearity of results in the minus sign of the transformation law of . In a diagonal basis of , this transformation reduces to


with and .

In contrast to non-spatial AZ symmetries, the spatial coordinate of the -dimensional sphere surrounding a topological defect also transforms non-trivially under order-two spatial symmetry. To determine the transformation law, we specify the coordinate of the -dimensional sphere. First, to keep the additional symmetry, the topological defect should be invariant under . Therefore, the additional symmetry maps the -dimensional sphere (with a radius ) given by


into itself, inducing the transformation


where is an orthogonal matrix with . The transformation of can be rewritten as


with and in a diagonal basis . When , we can introduce the coordinate of the -dimensional sphere by the stereographic projection of


which gives a simple transformation law of as


with and . Below, we assume , since the bulk-boundary correspondence for topological defects works only in this case.

Now the order-two unitary symmetry is expressed as


and the order-two anti-unitary symmetry is


We suppose that


and commutes or anticommutes with coexisting AZ symmetries,


where , , and . For a faithful representation of order-two symmetry, the sign of must be , but a spinor representation of rotation makes it possible to obtain . For instance, two-fold spin rotation obeys . Note that when , we can set by multiplying by the imaginary unit , but this changes the (anti-)commutation relations with and/or at the same time.

Our classification framework also works even for order-two anti-symmetry defined by


where can be either unitary or anti-unitary . Such an anti-symmetry can be realized by combining any of order-two symmetries with CS or PHS. In a similar manner as , we define , , and by


ii.3 Stable equivalence and K-group

In principle, the classification of topological insulators and superconductors are provided by a homotopy classification of maps from the base space to the classifying space of Hamiltonians , subject to a given set of symmetries: If the maps are smoothly connected to each other, they belong to the same topological phase, but if not, they are in topologically different phases.

Hamiltonians we consider here support an energy gap separating positive and negative energy bands, relative to the Fermi level. Such Hamiltonians are adiabatically deformed so that the all empty (occupied) bands have the same energy +1 (-1). If there are no symmetries, the flattened Hamiltonians are characterized by unitary matrices that diagonalize the Hamiltonians, divided by unitary rotations of the conduction bands and valence bands. The classifying space is therefore . Symmetries impose some constraints on the classifying space.

Following the idea of stable equivalence, we extend the classifying space by adding extra trivial bands Kitaev (2009): Two sets of Hamiltonians , are stable equivalent , if they can be continuously deformed into each other by adding extra trivial bands. One can then identify a family of Hamiltonians that are stable equivalent to each other. We use a notation to represent a set of Hamiltonians that are stable equivalent to . The stable equivalence classes make it possible to supply addition in the classifying space of Hamiltonians: , where implies the direct sum of matrices. The identity expresses the trivial insulating Hamiltonian, and is ensured to be . The last relation yields that the inverse of is . As a result, the stable equivalent classes form an Abelian group, which is called the K-group. From the definition, it is evident that the stable equivalence retains topological natures. The extended classifying spaces subject to AZ symmetries are listed in Table 1.

For topological insulators and superconductors in ten AZ symmetry classes, the following relations summarize their classification Teo and Kane (2010)


where () denotes the K-group of maps from to the extended classifying space of complex (real) AZ class in Table 1. The case of corresponds to the bulk topological classification, and the presence of topological defects shifts the dimension of the system.

The existence of an order-two spatial symmetry gives additional constraints on the classifying space. In the subsequent sections, we provide the resulting K-group of the homotopy classification.

Iii K-group in the presence of additional symmetry

In this section, we present the K-groups for topological crystalline insulators/superconductors and their topological defects protected by an additional order-two symmetry. The derivation and proof are given in Sec. VII.

iii.1 Complex AZ classes (A and AIII) with additional order-two unitary symmetry

The complex AZ classes, A and AIII, are characterized by the absence of TRS and PHS. Whereas no AZ symmetry is imposed on Hamiltonians in class A, Hamiltonians in class AIII is invariant under CS,


Now we impose an additional order-two symmetry ,


or order-two antisymmetry


on complex AZ classes. Since there is no anti-unitary symmetry, a phase factor of and do not change the topological classification, and thus the sign of and can be fixed to be . For class AIII, we specify the commutation/anti-commutation relation between and ( and ) by (). Note that in class AIII is essentially the same as because they can be converted to each other by the relation .

We denote the obtained K-group by


Here () is the total space dimension (defect co-dimension), and () is the number of the flipping momenta (defect surrounding parameters) under the additional symmetry transformation, as was introduced in Sec.II.2.2. The label (mod 2) indicates the AZ class ( for class A and for class AIII) to which the Hamiltonian belongs, and (mod 2) specifies the coexisting additional unitary symmetry as in Table. 2.

AZ class
0 A
1 AIII (, ) (, )
Table 2: Possible types ( (mod 2)) of order-two additional unitary symmetries in complex AZ class ( (mod 2)). and represent symmetry and antisymmetry, respectively. The subscript of and specifies the relation . Symmetries in the same parenthesis are equivalent.

In Sec. VII, we prove the following relation:


This relation implies that topological natures of the system can be deduced from those in -dimension. As we show in Appendix C, the classifying spaces of the 0-dimensional K-group reduce to complex Clifford algebra, and we can obtain


where represents the classifying space of complex AZ classes. (See Table 1.)

iii.2 Complex AZ classes (A and AIII) with additional order-two antiunitary symmetry

Next, we consider order-two anti-unitary symmetry or as an additional symmetry:


As listed in Table 3, two different order-two anti-unitary symmetries and their corresponding anti-symmetries are possible in class A, depending on the sign of or , i.e. , . In a similar manner, class AIII have two different types of additional anti-unitary symmetries, , and their corresponding anti-symmetries, , where represents the sign of or and indicates the commutation () or the anti-commutation () relation between and or those between and . Note that and are equivalent in class AIII since they can be related to each other as .

The existence of the anti-unitary symmetry introduces real structures in complex AZ classes. Actually, by regarding as “momenta”, and as “spatial coordinates”, and can be considered as TRS and PHS, respectively. From this identification, a system in complex AZ class with an additional anti-unitary symmetry can be mapped into a real AZ class, as summarized in Table 3. As a result, the K-group of Hamiltonians with the symmetry ( (mod 8)) of Table 3


reduces to the K-group of real AZ classes in Eq. (2.19),


where () is the total space dimension (defect co-dimension), and () is the number of the flipping momentum (defect surrounding parameter) under the additional symmetry transformation. From Eq. (2.19), we have



s AZ class Coexisting symmetry Mapped AZ class
0 A AI
1 AIII (, ) BDI
2 A D
5 AIII (, ) CII
6 A C
7 AIII (, ) CI
Table 3: Possible types ( (mod 8)) of order-two additional anti-unitary symmetries in complex AZ class. and represent symmetry and anti-symmetry, respectively. The superscript of and represent the sign of the square , and the subscript of specifies the (anti-)commutation relation . Symmetries in the same parenthesis are equivalent.

iii.3 Real AZ classes with additional order-two symmetry

Hamiltonians in eight real AZ classes are invariant under TRS,


and/or PHS,


In addition to TRS and/or PHS, we enforce one of order-two unitary/antiunitary spatial symmetries, , , , and on the Hamiltonians,


In class AI and AII, which support TRS, we have the following equivalence relations between the additional symmetries,


where the superscript of denotes the sign of , and the subscript of specifies the commutation () or anti-commutation relation between and . In a similar manner, in class D and C, PHS leads to the following equivalence relations


where the superscript denotes the sign of and the subscript denotes the commutation or anti-commutation relation between and . Finally, in class BDI, DIII, CII and CI, we obtain