Topology of crystalline insulators and superconductors
Abstract
We complete a classification of topological phases and their topological defects in crystalline insulators and superconductors. We consider topological phases and defects described by noninteracting Bloch and Bogoliubov de Gennes Hamiltonians that support additional ordertwo spatial symmetry, besides any of ten classes of symmetries defined by timereversal symmetry and particlehole symmetry. The additional ordertwo spatial symmetry we consider is general and it includes global symmetry, mirror reflection, twofold 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 ordertwo spatial symmetry, in addition to the Bottperiodicity 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 bulkboundary correspondence in terms of the Ktheory.
pacs:
Contents
 I Introduction
 II Formalism
 III Kgroup in the presence of additional symmetry
 IV Properties of topological table and Kgroup

V Topological periodic table in the presence of additional
ordertwo symmetry

V.1 family
 V.1.1 Spin Chern insulator ( in class AII)
 V.1.2 Mirrorodd twodimensional topological superconductor ( in class D)
 V.1.3 Mirroreven twodimensional topological superconductor ( in class D)
 V.1.4 Superconducting nanowire with Rashba SO interaction and Zeeman fields (, in class D)
 V.1.5 Vortex in twodimensional superconductors with magnetic inplane reflection symmetry (, in class D)
 V.1.6 Zero mode in a magnetic inplane mirror reflection symmetric heterostructure ( in class D)
 V.1.7 Majorana point defect zero mode protected by magnetic rotation symmetry (, in class D)

V.2 family
 V.2.1 Topological number
 V.2.2 Second descendant index in real AZ classes with , , ,
 V.2.3 Mirror reflection symmetric vortex in threedimensional superconductors ( in class D)
 V.2.4 chiral doublet edge modes protected by the antiunitary reflection symmetry ( in class A)
 V.2.5 superconductor protected by emergent spinless reflection TRS ( in class D)
 V.2.6 Vortex in threedimensional superconductors with magnetic rotation symmetry ( in class D)

V.3 family
 V.3.1 rotation Chern number and rotation winding number
 V.3.2 topological insulator protected by the magnetic rotation symmetry ( in class A)
 V.3.3 HeB slab with parallel magnetic fields ( in class D)
 V.3.4 Inversion symmetric quantum (spin) Hall states ( in class A, in class AII)
 V.3.5 Odd parity superconductors in two dimensions ( in class D)
 V.4 family

V.1 family
 VI Weak crystalline topological indices
 VII Dimensional hierarchy with ordertwo additional symmetry
 VIII Topological classification of Fermi points with additional symmetry
 IX Majorana Ising spin character as a result of topological crystalline superconductivity
 X Anomalous topological pump
 XI Conclusion
 A Dimensional shift of Hamiltonians
 B Dimensional hierarchy of AZ classes
 C Classifying space of AZ classes with additional symmetry

D Topological invariants
 D.1 Topological invariants in zerodimension
 D.2 Chern number and winding number

D.3 topological invariant
 D.3.1 Dimensional reduction
 D.3.2 MooreBalents argument for second descendant index
 D.3.3 ChernSimons invariant for first descendant index in odddimensional nonchiral real class
 D.3.4 FuKane invariant for first and second descendant indices in evendimensional TRS class
 D.3.5 Constrained ChernSimons invariant for second descendant index in odddimensional chiral TRS class
I Introduction
Symmetry and topology have been two important principles in physics, both of which result in quantum numbers and the conservation laws. In manybody 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. ^{1}^{1}1 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 noninteracting 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 nonzero 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 timereversal 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 nonspatial symmetries can persist even in the presence of disorders and/or perturbations. For instance, nonmagnetic disorders retain timereversal symmetry (TRS), and thus a nontrivial topological phase accompanied by TRS is robust against nonmagnetic 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 nonspatial symmetries are classified for noninteracting 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 AltlandZirnbauer (AZ) tenfold symmetry classes. Altland and Zirnbauer (1997)
Whereas the classification based on the nonspatial 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 nontrivial 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 ordertwo 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, twofold rotation, and inversion. Furthermore, the additional symmetry can be antiunitary. Although ordinary point group symmetries are given by unitary operators, systems in a magnetic field or with a magnetic order often support an antiunitary symmetry as a magnetic point group symmetry. The magnetic symmetry also has been known to provide nontrivial 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 ordertwo 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 ordertwo additional spatial symmetry, in addition to the Bottperiodicity 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 bulkboundary correspondence in terms of the Ktheory.
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 Ktheory. Atiyah et al. (1964); Atiyah (1966); Dupont (1969); Karoubi (2008) Our main results are summarized in Sec. III. We show relations between Kgroups with different ordertwo additional spatial symmetries and dimensions. The derivation and proof are given in Sec. VII. In Sec. IV, we discuss properties of the obtained Kgroups 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 Kgroups 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 ordertwo 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 ordertwo symmetry in Sec. VIII. By combing the results in Secs. III and VIII, the bulkboundary correspondence of Kgroups 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 ordertwo 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 bandinsulators 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)
(2.1) 
Here the base space of the Hamiltonian is composed of momentum , defined in the dimensional Brillouin zone , and realspace coordinates of a dimensional sphere surrounding a defect. For instance, the Hamiltonian of a point defect in threedimensions is given by , where are the coordinates of a twodimensional sphere surrounding the point defect. Another example is a line defect in threedimensions, 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 AltlandZirnbauer 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, particlehole symmetry (PHS) and/or chiral symmetry (CS). The AZ symmetries, TRS, PHS, and CS, imply
(2.2) 
respectively, where and are antiunitary operators and is a unitary operator. For spin1/2 fermions, timereversal operator is given by with the Pauli matrix in the spin space and the complex conjugation operator , which obeys . In the absence of the spinorbit interaction, spin rotation symmetry allow a different timereversal 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 spinrotation 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 timereversal invariance and particlehole 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 redefinition 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  
1  AIII  
0  AI  
1  BDI  
2  D  
3  DIII  
4  AII  
5  CII  
6  C  
7  CI 
ii.2.2 OrderTwo Spatial Symmetry
In addition to the AZ symmetries, we assume an additional symmetry of Hamiltonians. As an additional symmetry, we consider general ordertwo spatial symmetry. Ordertwo symmetry implies that the symmetry operation in twice trivially acts on the Hamiltonian,
(2.3) 
where can be either unitary or antiunitary . The ordertwo unitary symmetry includes reflection, twofold spatial rotation and inversion. It also permits global symmetry such as a twofold spin rotation. The antiunitary case admits ordertwo magnetic point group symmetries.
Under an ordertwo spatial symmetry, the momentum in the base space of the Hamiltonian transforms as
(2.4) 
with an orthogonal matrix satisfying . Note that like timereversal operator, the antilinearity of results in the minus sign of the transformation law of . In a diagonal basis of , this transformation reduces to
(2.5) 
with and .
In contrast to nonspatial AZ symmetries, the spatial coordinate of the dimensional sphere surrounding a topological defect also transforms nontrivially under ordertwo 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
(2.6) 
into itself, inducing the transformation
(2.7) 
where is an orthogonal matrix with . The transformation of can be rewritten as
(2.8) 
with and in a diagonal basis . When , we can introduce the coordinate of the dimensional sphere by the stereographic projection of
(2.9) 
which gives a simple transformation law of as
(2.10) 
with and . Below, we assume , since the bulkboundary correspondence for topological defects works only in this case.
Now the ordertwo unitary symmetry is expressed as
(2.11) 
and the ordertwo antiunitary symmetry is
(2.12) 
We suppose that
(2.13) 
and commutes or anticommutes with coexisting AZ symmetries,
(2.14) 
where , , and . For a faithful representation of ordertwo symmetry, the sign of must be , but a spinor representation of rotation makes it possible to obtain . For instance, twofold 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 ordertwo antisymmetry defined by
(2.15)  
(2.16) 
where can be either unitary or antiunitary . Such an antisymmetry can be realized by combining any of ordertwo symmetries with CS or PHS. In a similar manner as , we define , , and by
(2.17) 
ii.3 Stable equivalence and Kgroup
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 Kgroup. 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)
(2.18)  
(2.19) 
where () denotes the Kgroup 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 ordertwo spatial symmetry gives additional constraints on the classifying space. In the subsequent sections, we provide the resulting Kgroup of the homotopy classification.
Iii Kgroup in the presence of additional symmetry
In this section, we present the Kgroups for topological crystalline insulators/superconductors and their topological defects protected by an additional ordertwo symmetry. The derivation and proof are given in Sec. VII.
iii.1 Complex AZ classes (A and AIII) with additional ordertwo 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,
(3.1) 
Now we impose an additional ordertwo symmetry ,
(3.2) 
or ordertwo antisymmetry
(3.3) 
on complex AZ classes. Since there is no antiunitary 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/anticommutation 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 Kgroup by
(3.4) 
Here () is the total space dimension (defect codimension), 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  (, )  (, ) 
In Sec. VII, we prove the following relation:
(3.5) 
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 0dimensional Kgroup reduce to complex Clifford algebra, and we can obtain
(3.6) 
where represents the classifying space of complex AZ classes. (See Table 1.)
iii.2 Complex AZ classes (A and AIII) with additional ordertwo antiunitary symmetry
Next, we consider ordertwo antiunitary symmetry or as an additional symmetry:
(3.7)  
(3.8) 
As listed in Table 3, two different ordertwo antiunitary symmetries and their corresponding antisymmetries 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 antiunitary symmetries, , and their corresponding antisymmetries, , where represents the sign of or and indicates the commutation () or the anticommutation () 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 antiunitary 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 antiunitary symmetry can be mapped into a real AZ class, as summarized in Table 3. As a result, the Kgroup of Hamiltonians with the symmetry ( (mod 8)) of Table 3
(3.9) 
reduces to the Kgroup of real AZ classes in Eq. (2.19),
(3.10) 
where () is the total space dimension (defect codimension), and () is the number of the flipping momentum (defect surrounding parameter) under the additional symmetry transformation. From Eq. (2.19), we have
(3.11) 
with
(3.12) 
s  AZ class  Coexisting symmetry  Mapped AZ class 

0  A  AI  
1  AIII  (, )  BDI 
2  A  D  
3  AIII  (,)  DIII 
4  A  AII  
5  AIII  (, )  CII 
6  A  C  
7  AIII  (, )  CI 
iii.3 Real AZ classes with additional ordertwo symmetry
Hamiltonians in eight real AZ classes are invariant under TRS,
(3.13) 
and/or PHS,
(3.14) 
In addition to TRS and/or PHS, we enforce one of ordertwo unitary/antiunitary spatial symmetries, , , , and on the Hamiltonians,
(3.15)  
(3.16)  
(3.17)  
(3.18) 
In class AI and AII, which support TRS, we have the following equivalence relations between the additional symmetries,
(3.19)  
(3.20) 
where the superscript of denotes the sign of , and the subscript of specifies the commutation () or anticommutation relation between and . In a similar manner, in class D and C, PHS leads to the following equivalence relations
(3.21)  
(3.22) 
where the superscript denotes the sign of and the subscript denotes the commutation or anticommutation relation between and . Finally, in class BDI, DIII, CII and CI, we obtain
(3.23)  