A Dimensional reduction with reflection symmetry

Bott periodicity for the topological classification of gapped states of matter with reflection symmetry


Using a dimensional reduction scheme based on scattering theory, we show that the classification tables for topological insulators and superconductors with reflection symmetry can be organized in two period-two and four period-eight cycles, similar to the Bott periodicity found for topological insulators and superconductors without spatial symmetries. With the help of the dimensional reduction scheme the classification in arbitrary dimensions can be obtained from the classification in one dimension, for which we present a derivation based on relative homotopy groups and exact sequences to classify one-dimensional insulators and superconductors with reflection symmetry. The resulting classification is fully consistent with a comprehensive classification obtained recently by Shiozaki and Sato [Phys. Rev. B 90, 165114 (2014)]. The use of a scattering-matrix inspired method allows us to address the second descendant phase, for which the topological nontrivial phase was previously reported to be vulnerable to perturbations that break translation symmetry.

I Introduction

The non-spatial symmetries time-reversal symmetry, particle-hole symmetry, chiral symmetry, and their combinations can be organized in two “complex” and eight “real” Altland-Zirnbauer symmetry classes. Altland and Zirnbauer (1997) For each of these classes the classification of the topological insulating (non-interacting) phases of matter was recently obtained. Schnyder et al. (2008); Kitaev (2009); Stone et al. (2011); Wen (2012); Abramovici and Kalugin (2012); Kennedy and Zirnbauer (2016) The classification has a cyclic dependence on the dimensionality, with a period-two sequence for the complex classes and a period-eight sequence for the real classes. This cyclic structure is known as “Bott periodicity” or “Bott clock”, and is firmly embedded in the mathematics of algebraic topology Bott (1959) and, in particular the eight “real” classes, in “K theory”.Atiyah (1966); Karoubi (2005)

An appealing physical construction that reproduces the period-two and period-eight cyclic structure was suggested recently by Fulga and coworkers.Fulga et al. (2012) Fulga et al. show that the reflection matrix of a gapped half-infinite system in dimensions with Hamiltonian can be naturally interpreted as the Hamiltonian of a gapped system in dimensions, but with a different symmetry, which precisely follows the period-two and period-eight Bott clocks of the complex and real Altland-Zirnbauer classes, respectively. Since topological classification reflection matrix essentially amounts to a classification of the boundary of the -dimensional Hamiltonian , the assumption of bulk-boundary correspondence — a topologically nontrivial bulk is accompanied with gapless boundary states in a unique way — then immediately gives the Bott periodicity for non-interacting gapped phases of matter.

The concept of topological phases has been extended to include (non-interacting) topological phases that are protected by a crystalline symmetry, in addition to the non-spatial symmetries of the Altland-Zirnbauer classification. The additional spatial symmetries give rise to gapless states at the boundary,Ryu and Hatsugai (2002); Sato (2006); Teo et al. (2008); Béri (2010); Yada et al. (2011); Fulga et al. (2012); Sato et al. (2011); Schnyder and Ryu (2011); Hatsugai (2010); Hughes et al. (2011); Fu and Berg (2010) provided the boundary is invariant under the symmetry operation. Although the topological crystalline phases rely on spatial symmetries, many topological phases associated with crystalline symmetries are robust to disorder that preserves these symmetries on the average. Fulga et al. (2014); Fu and Kane (2012); Ringel et al. (2012); Mong et al. (2012); Morimoto and Furusaki (2014); Obuse et al. (2014)

Because of the large number of possible spatial symmetries, especially in higher dimensions, the task of classifying crystalline topological phases is a formidable one. Although a comprehensive classification similar to that of the non-spatial symmetries is still lacking, considerable progress has been made. On the one hand, this includes complete classifications for all crystalline symmetries, but restricted to a single Altland-Zirnbauer class in two or three dimensions. Slager et al. (2013); Kruthoff et al. (2016); Fang et al. (2012, 2013); Liu et al. (2014); Alexandradinata et al. (2014); Zhang et al. (2013); Dong and Liu (2016); Jadaun et al. (2013); Teo and Hughes (2013); Benalcazar et al. (2014) On the other hand, there are classifications for all Altland-Zirnbauer classes and all spatial dimensions, but for a restricted set of crystalline symmetry operations, such as inversion symmetry Lu and Lee (2014, 2014) or reflection symmetry. Chiu et al. (2013); Morimoto and Furusaki (2013); Shiozaki and Sato (2014) The latter approach was found to yield period-two and period-eight dependencies on the dimension known from the classification without crystalline symmetries, Lu and Lee (2014); Chiu et al. (2013) including, in some cases, cyclic structures reminiscent of the “Bott clock”. Morimoto and Furusaki (2013); Shiozaki and Sato (2014)

Reflection symmetry is one of most often considered crystalline symmetries, and reflection-symmetric materials were the first experimental realizations of crystalline topological insulators. Hsieh et al. (2012); Tanaka et al. (2012); Xu et al. (2012) A complete classification of reflection-symmetric topological insulators and superconductors was reported by Chiu et al., employing the method of minimal Dirac Hamiltonians. Chiu et al. (2013) Morimoto and Furusaki, Morimoto and Furusaki (2013) using an approach based on Clifford algebras, Kitaev (2009) showed that the topological classes with reflection symmetry can be organized in period-two and period-eight cyclic structure, although for some sequences the cycles involve increasing the dimension in steps of two, rather than in unit steps, as in the case of the standard Bott periodicity. Shiozaki and Sato generalized these results to all order-two unitary and antiunitary crystalline symmetries Shiozaki and Sato (2014) and corrected some entries in the classification table obtained previously. Chiu et al. (2013); Morimoto and Furusaki (2013)

In this paper we show that the reflection-matrix-based dimensional reduction scheme of Fulga et al. can also be applied to reflection-symmetric topological insulators. Using the scheme of Fulga et al. naturally leads us to consider a “chiral reflection” operation, such that, once the chiral reflection-symmetric gapped Hamiltonians are included, all symmetry combinations can be grouped in period-two and period-eight cyclic sequences, for which the dimension is increased in unit steps. Our results are in complete agreement with the classification obtained by Shiozaki and Sato, Shiozaki and Sato (2014) who used the Hamiltonian dimensional reduction scheme of Teo and KaneTeo and Kane (2010) to obtain relations between the corresponding K groups.

Whereas the reflection-matrix-based dimensional reduction scheme allows one to obtain the classification for arbitrary dimension from the classification at in the absence of spatial symmetries,Fulga et al. (2012) in the presence of reflection symmetry this procedure ends already for dimension , since a one-dimensional system has no “boundary” that is mapped onto itself by reflection. To make this article self-contained and to provide an alternative to the existing classification schemes,Chiu et al. (2013); Morimoto and Furusaki (2013); Shiozaki and Sato (2014) we here present a classification of one-dimensional reflection-symmetric topological insulators based on relative homotopy groups and exact sequences, following the approach taken by Turner et al. in their classification of inversion-symmetric topological insulators.Turner et al. (2012) In combination with the reflection-matrix-based reduction scheme, the classification gives a complete classification for reflection-symmetric topological insulators in all dimensions . An additional advantage of the approach of Turner et al. is that it gives explicit expressions for topological invariants and for the generators of the topological classes (many examples are given in App. C).

Our approach allows us to address an issue related to stability of the topological phase of the second descendant of in the classes where reflection symmetry anticommutes with non-spatial symmetries. Chiu et al. and Morimoto and Furusaki argued that the topological index cannot be defined in these cases and that an eventual topologically non-trivial phase is always “weak”, i.e., it is instable to perturbations that break the lattice translation symmetry.Chiu et al. (2013); Morimoto and Furusaki (2013) While Shiozaki and Sato left open the possibility of a “subtle instability” to translation-symmetry-breaking perturbations, they insisted that the topological invariant is a “strong” one.Shiozaki and Sato (2014) Having the explicit form of the topological invariant at our disposal, we can confirm that it is invariant under a redefinition of the unit cell. Moreover, since our reflection-matrix based approach effectively classifies the boundary of the insulator, we can show explicitly that a nonzero topological invariant implies the existence of a topologically protected boundary state. We find no signs of the instability reported in Refs. Chiu et al., 2013; Morimoto and Furusaki, 2013.

This article is organized as follows: In Sec. II we review the constraints that reflection symmetry poses on the Hamiltonian of a gapped system in dimensions. In Sec. III we review the reflection-matrix-based method of dimensional reduction originally proposed by Fulga et al.Fulga et al. (2012) and we show how the method can be generalized to reflection-symmetric topological insulators. The topological classification of one-dimensional topological insulators with reflection symmetry using the method of relative homotopy groups and exact sequences is given in Sec. IV. We discuss the controversial second-descendant phase in Sec. V. We conclude with a brief summary in Sec. VI. Four appendices contain details of the dimensionless reduction scheme, an extension of the classification to higher dimensions (i.e., without the assumption of bulk-boundary correspondence, which underlies the reflection-matrix-based dimensional reduction scheme), explicit examples for topological invariants of one-dimensional reflection-symmetric topological insulators, and supporting details for the second-descendant phase.

Ii Symmetries

We consider a Hamiltonian in dimensions, with . For definiteness we take the reflection plane to be perpendicular to the th unit vector, so that reflection maps the wavevector to . Reflection also affects the basis states in the unit cell, so that for the Hamiltonian reflection symmetry takes the form


with a -independent unitary matrix. We require to fix the phase freedom in the definition of .

The reflection symmetry exists possibly in combination with time-reversal (), particle-hole (), and/or chiral () symmetries. These symmetries take the form


where , , and are -independent unitary matrices. If time-reversal symmetry and particle-hole symmetry are both present, . Further, the unitary matrices , , and satisfy , , , and . [The condition is not fundamental, since multiplication of with a phase factor results in the same chiral symmetry relation (4). We will use this condition to fix signs in intermediate expressions for the general derivation of the Bott clock from scattering theory, but not in the discussion of examples for specific symmetry classes.]

The three non-spatial symmetry operations , , and define the ten Altland-Zirnbauer classes.Altland and Zirnbauer (1997) The two “complex” classes have no symmetries linking to ; the remaining eight “real” classes have time-reversal symmetry, particle-hole symmetry, or both. Following common practice in the field, we use Cartan labels to refer to the corresponding symmetry classes, see Table 1.

Class Cartan
A 0 0 0 0
AIII 0 0 1 0
AI 1 0 0
BDI 1 1 1
D 0 1 0 0
DIII -1 1 1 0
AII -1 0 0 0
CII -1 -1 1 0 0
C 0 -1 0 0 0
CI 1 -1 1 0
Table 1: The ten Altland-Zirnbauer classes are defined according to the presence or absence of time-reversal (), particle-hole symmetry (), and chiral symmetry (). A nonzero entry indicates the square of the antiunitary symmetry operations or .

How the presence of a reflection symmetry affects the topological classification depends on whether the reflection operation commutes or anticommutes with the non-spatial symmetry operations , , and . (The condition ensures that the reflection operation has well-defined algebraic relations with , , and .) Following Ref. Chiu et al., 2013, to distinguish the various cases, we use the symbol to denote the presence of reflection symmetry in the absence of any spatial symmetries, , , or to denote a reflection symmetry that commutes (“”) or anticommutes (“”) with the non-spatial symmetry operation , , or if only one non-spatial symmetry is present present, and for a reflection symmetry that commutes/anticommutes with time-reversal symmetry and particle-hole symmetry if all three non-spatial symmetries are present. (If neither commutes nor anticommutes with the fundamental non-spatial symmetries, the Hamiltonian can be brought into block-diagonal form, such that there are well-defined commutation or anticommutation relations between and , , and for each of the blocks.) The commutation or anticommutation relations between and , , or imply the algebraic relations , , and between the unitary matrices implementing these operations.

Iii Dimensional reduction

We now describe how one can construct a dimensional reduction scheme consistent with Bott periodicity using reflection matrices. We first review how this method works in the absence of reflection symmetry, as discussed by Fulga et al.,Fulga et al. (2012) and then show how to include the presence of reflection symmetry.

iii.1 Reflection matrix-based method

The key step in the method of Ref. Fulga et al., 2012 is the construction a -dimensional gapped Hamiltonian for each dimensional gapped Hamiltonian . The Hamiltonians and have different symmetries, but the same (strong) topological invariants. Fulga et al. show how the Hamiltonian can be constructed from the reflection matrix if a gapped system with Hamitonian is attached to an ideal lead with a -dimensional cross section.

Since the reflection matrix depends on the properties of the surface of the -dimensional insulator, this dimensional reduction method assumes that the boundary properties can be used to classify the bulk, i.e., it assumes a bulk-boundary correspondence. This is the case for the standard Altland-Zirnbauer classes (without additional symmetries). It is also the case

in the presence of a mirror symmetry, provided the surface contains the normal to the mirror plane.Chiu et al. (2013)

To be specific, following Ref. Fulga et al., 2012 we consider a -dimensional gapped insulator with Hamiltonian , occupying the half space , see Fig. 1. The half space consists of an ideal lead with transverse modes labeled by the dimensional wavevector . The amplitudes and of outgoing and incoming modes are related by the reflection matrix ,


Since is gapped, is unitary. Time-reversal symmetry, particle-hole symmetry, or chiral symmetry pose additional constraints on . These follow from the action of these symmetries on the amplitudes and ,


where , , , , , are -independent unitary matrices that satisfy , , and . Systems with both time-reversal and particle-hole symmetry also have a chiral symmetry, with and . For the reflection matrix one then finds that the presence of time-reversal symmetry, particle-hole symmetry, and/or chiral symmetry leads to the constraints

Figure 1: Schematic picture of a -dimensional gapped insulator occupying the half space (blue) with twisted boundary conditions applied along -dimension (black line), coupled to an ideal lead (red) with a -dimensional cross section. The reflection matrix relates the amplitudes of outgoing and incoming modes in the lead.

The effective Hamiltonian is constructed out of in different ways, depending on the presence or absence of chiral symmetry. With chiral symmetry one sets


using Eq. (11) to verify that is indeed hermitian. (Recall that since .) Without chiral symmetry one defines as


which is manifestly hermitian and satisfies a chiral symmetry with .

Bulk-boundary correspondence implies that the bulk, which is described by the Hamiltonian , and the boundary, which determines the reflection matrix , have the same topological classification. Since is one-to-one correspondence with the Hamiltonian , this implies that and have the same topological classification (provided bulk-boundary correspondence applies). Inspecting the symmetries of the sequence of Hamiltonians that results upon stepwise lowering the dimension , one recovers two periodic sequences of Hamiltonians with the same topological classification. The appearance of a period-two sequence for the complex classes

follows immediately from the alternating presence and absence of chiral symmetry in the sequence of Hamiltonians constructed above. To establish the period-eight sequence one needs to inspect how the dimensional reduction affects the symmetries of if time-reversal symmetry and/or particle-hole symmetry are present. If has both time-reversal and particle-hole symmetry, is given by Eq. (12). From Eqs. (9), (10), and (12) one derives that time-reversal symmetry and particle-hole symmetry of the reflection matrix yield identical symmetry constraints for the Hamiltonian ,


which has the form of a particle-hole symmetry if , and of a time-reversal symmetry otherwise. In both cases the symmetry operation squares to . If has time-reversal symmetry but no particle-hole symmetry, one verifies that satisfies




which has the form of a time-reversal symmetry squaring to and a particle-hole symmetry squaring to , whereas if has particle-hole symmetry but no time-reversal symmetry, satisfies the symmetry constraints




which has the form of a time-reversal symmetry squaring to and a particle-hole symmetry squaring to . Combining everything, one arrives at the sequence of symmetry classes


which is the well-known period-eight Bott periodicity known from the classification of topological insulators and superconductors.Schnyder et al. (2008); Kitaev (2009); Stone et al. (2011); Wen (2012); Abramovici and Kalugin (2012); Kennedy and Zirnbauer (2016)

iii.2 reflection symmetry

The dimensional reduction based on the calculation of reflection matrices can also be applied in the presence of a reflection symmetry. As in Sec. II, we take the reflection plane to be perpendicular to the axis, so that the reflection operator maps the lead-system interface onto itself. As with the non-spatial symmetries, the action of the reflection operation on the amplitudes and of incoming and outgoing states in the leads involves multiplication with -independent unitary matrices,


where denotes the reflected mode vector. The matrices and satisfy . The presence of reflection symmetry leads to a constraint on the reflection matrix,


The algebraic relations involving the matrices , depend on whether the reflection operation commutes or anticommutes with the non-spatial symmetry operations , , and , , , , , , and .

To see how the presence of reflection symmetry affects the dimensional reduction we first consider the complex classes A and AIII. Starting from a Hamiltonian in symmetry class A with reflection symmetry we construct a Hamiltonian in class AIII as described above and find that reflection symmetry imposes the constraint


on , with


Since commutes with , we conclude that dimensional reduction maps the class A to AIII. A similar procedure can be applied to a Hamiltonian in class AIII with reflection symmetry , with . In this case, one finds that dimensional reduction leads to a Hamiltonian in class A with the additional constraint


This constraint has the form of a reflection symmetry if , i.e., if commutes with , but not if , i.e., if anticommutes with . Instead, if the constraint (24) represents the product of a reflection symmetry and a chiral symmetry. We denote such a combined symmetry operation with the symbol “”. To complete the analysis, we consider a Hamiltonian in class A with the symmetry constraint,


where . On the level of the reflection matrix the symmetry is implemented as


where and . Performing the dimensional reduction scheme to this Hamiltonian , one immediately finds that satisfies the constraint




Since anticommutes with , the constraint Eq. (27) has the form of a reflection symmetry that anticommutes with the chiral symmetry . Combining everything, we conclude that, once the symmetry operation is added, the dimensional reduction scheme for the complex classes with reflection symmetry leads to two period-two sequences,


The symmetry operation naturally appears in the dimensional reduction scheme for the real classes as well. As with the standard reflection symmetry we have to distinguish between the cases that the symmetry operation commutes (”) or anticommutes (“”) with the time-reversal or particle-hole symmetry operations, if one of these symmetries is present. (If both symmetries are present, there is no need to consider as a separate symmetry operation.) The relations (25) and (26) also apply to the general case. If chiral symmetry is present, one has and . One further has the algebraic relations , , , . Proceeding as above, one verifies that the dimensional reduction scheme then leads to four period-eight sequences,


Details of the derivation are given in Appendix A. The above sequences were first derifed by Morimoto and Furusaki but “skipping” the classes containing symmetry.Morimoto and Furusaki (2013) Shiozaki and Sato obtained the relations between K groups that give all the sequences derived here as a special case.Shiozaki and Sato (2014)

Iv Topological classification with reflection symmetry

Having established the dimensional reduction scheme, it is sufficient to consider the case in order to completely classify gapped Hamiltonians with reflection symmetry. (The dimensional reduction scheme can not be used down to because there can be no reflection-invariant lead-system interface in one dimension.) Various methods have been used in the literature to accomplish this task,Chiu et al. (2013); Morimoto and Furusaki (2013); Shiozaki and Sato (2014) as discussed in the introduction or in the review article Ref. Chiu et al., 2016.

To make this paper self-contained, we here include a systematic classification of reflection-symmetric gapped Hamiltonians for . We have chosen to use a different method than used in Refs. Chiu et al., 2013; Morimoto and Furusaki, 2013; Shiozaki and Sato, 2014, which makes use of concepts from algebraic topology, using relative homotopy groups and exact sequences. This method was used by Turner et al. for their classification of topological insulators with inversion symmetry.Turner et al. (2012) In App. B we discuss how this classification method can be directly applied to reflection-symmetric Hamiltonians in dimensions , without the use of the reflection matrix-based dimensional reduction scheme (and, hence, without the implicit assumption of bulk-boundary correspondence).

The construction of a topological classification for the Hamiltonians requires a mathematical formalism that endows the space of Hamiltonians with a group structure. The theory of vector bundles and the “Grothendieck group” provides such a formal structure, essentially using the diagonal addition of Hamiltonians as the group addition operation. Both concepts are reviewed in a language accessible to physicists, e.g., in Ref. Budich and Trauzettel, 2013 and in the appendix of Ref. Turner et al., 2012. We here employ a more informal language, noting that a formally correct formulation requires an interpretation of our statements in the framework of the vector bundles and the Grothendieck group. As in the previous Section we use the Cartan labels to denote the space of hermitian matrices with a gapped spectrum for the two complex and eight real Altland-Zirnbauer symmetry classes, see Table 1.

In one dimension, we are interested in in periodic, functions , with a gapped Hamiltonian, where the antiunitary symmetry operations and as well as the reflection operations and relate and . It is then sufficient to consider the Hamiltonian on the interval only. For generic only symmetries that relate to itself play a role. These symmetries confine for to one of the classifying spaces of table 1. We use the symbol to denote this space. The momenta and are mapped to themselves under , so that and satisfy additional symmetries. We use to denote the classifying space of Hamiltonians that also satisfy these additional symmetry constraints. Figure 2 schematically illustrates the spaces and .

In general a Hamiltonian can be block-decomposed as , where is topologically “trivial”. The -independent Hamiltonian has topological indices characteristic of the zero dimensional case. These indices become weak indices of one-dimensional Hamiltonian . The classification of the Hamiltonians gives the strong topological indices.

Figure 2: Schematic illustration of the spaces and . The solid dot indicates the trivial element. The thick curve shows a path in that starts at the trivial element and ends in . Equivalence classes of such paths form the relative homotopy group .

In view these considerations, our goal is to classify functions on the interval , such that is gapped, is trivial, , and otherwise. The space of equivalence classes (defined with respect to continuous deformations) of such functions is known as the relative homotopy group Nash and Sen (1988); Turner et al. (2012) The group gives the topological classification of gapped Hamiltonians with the desired symmetries. A function with these constraints can be interpreted as a continuous “path” in , starting at the trivial point, and ending somewhere in , see Fig. 2.

The relative homotopy group can be calculated from the zeroth and first homotopy groups of and , where we recall that the zeroth homotopy groups labels the connected components of a topological space , whereas the first homotopy group contain equivalence classes of “closed loops” in that begin and end at the trivial reference point. This calculation makes use of an “exact sequence” of mappingsTurner et al. (2012)


where a sequence of mappings is called “exact” if the image of each mapping is the kernel of the subsequent one. In the sequence (35), the maps , and are inclusion maps where the same object is interpreted as an element of a larger space. The map is the “boundary map”, mapping an equivalence class of “pathes” in to the connected component of their endpoint in . Since the groups and , as well as the image of and the kernel of are known, the relative homotopy group and its structure follow immediately from the exactness of the sequence (35). Similarly, generators for can be constructed by application of the inclusion map and a suitable inverse of the boundary map . Table 1 lists the groups and for the classifying spaces and .

To classify one-dimensional gapped Hamiltonians with reflection symmetry, the spaces and are identified for each symmetry class, see Tables 2 and 3, for the two period-two “complex” sequences and for the four period-eight “real” sequences, respectively. The relative homotopy group , which classifies the gapped Hamiltonians with reflection or symmetry, is then calculated from the exact sequence (35). The results of this classification are shown in Tables 2 and 3. In addition to the classification for , the table also lists the results for , , and , following the Bott clock structure outlined in the previous Section. The assignment of the spaces and for the different symmetry classes and the details on the resolution of the exact sequence in the nontrivial cases is discussed in detail in appendix C.

Table 2: The complete classification for the complex Altland-Zirnbauer classes with reflection symmetry.
Table 3: The complete classification for the real Altland-Zirnbauer classes with reflection symmetry.

V The second descendant phase

Chiu et al.Chiu et al. (2013) and Morimoto and FurusakiMorimoto and Furusaki (2013) argued that the class CII of reflection-symmetric topological superconductors in two dimensions () has gapless boundary states that not protected against perturbations that lift the discrete translation symmetry of the underlying lattice. On the other hand, Shiozaki and Sato point out that this class has a well-defined strong index, although they nevertheless allow for a “subtle instability” of the topologically nontrivial state.Shiozaki and Sato (2014)

The dimensional reduction scheme links class CII with to class AII in one dimension, i.e., the reflection matrix of a two-dimensional Hamiltonian in class CII is a one-dimensional object with symmetries characteristic of class AII. In this Section we show that the definition of the topological invariant for class AII is robust to the addition of perturbations that break the discrete translation symmetry, consistent with the observation of Shiozaki and Sato that there is a well-defined topological index.Shiozaki and Sato (2014) We then use our scattering approach to show that a nontrivial value of the invariant implies the existence of gapless states at the boundary of the two-dimensional system. As explained in App. E, we believe the fact that Refs. Chiu et al., 2013 and Morimoto and Furusaki, 2013 observe a gap opening for edge states is because the perturbation considered there includes a long-range hopping term with a hopping amplitude decaying inversely proportional to distance, which does not result in a continuous Bloch Hamiltonian as a function of .

The class AII has time-reversal symmetry with . Combining the reflection and time-reversal symmetries we arrive at


with since and anticommute. Without loss of generality we may represent by complex conjugation and by , so that is a real symmetric matrix with the additional condition . We conclude that is the class AI, whereas at the reflection symmetric points , , is of the form


with () real symmetric (antisymmetric). Such a structure forms a two-dimensional representation of the complex numbers, so that we find that is the space of gapped Hamiltonians in class A. Following the general procedure of Sec. IV we write , where is topologically trivial. This gives the exact sequence


with . The topological structure is inherited from the left part of the exact sequence. The index can be calculated as the standard invariant classifying loops of real symmetric matrices,Fu and Kane (2007)