Topological phases for free fermions in systems with crystal symmetry are classified by the topology of the valence band viewed as a vector bundle over the Brillouin zone. Additional symmetries, such as crystal symmetries which act non-trivially on the Brillouin zone, or time-reversal symmetry, endow the vector bundle with extra structure. These vector bundles are classified by a suitable version of K-theory. While relatively easy to define, these K-theory groups are notoriously hard to compute in explicit examples. In this paper we describe in detail how one can compute these K-theory groups starting with a decomposition of the Brillouin zone in terms of simple submanifolds on which the symmetries act nicely. The main mathematical tool is the Atiyah-Hirzebruch spectral sequence associated to such a decomposition, which will not only yield the explicit result for several crystal symmetries, but also sheds light on the origin of the topological invariants. This extends results that have appeared in the literature so far. We also describe examples in which this approach fails to directly yield a conclusive answer, and discuss various open problems and directions for future research.
Classification of crystalline topological insulators through -theory
Luuk Stehouwer, Jan de Boer, Jorrit Kruthoff, Hessel Posthuma,
Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 105-107, 1098 XG Amsterdam, The Netherlands
firstname.lastname@example.org, email@example.com, firstname.lastname@example.org, email@example.com
- 1 Introduction
- 2 Time-reversal only
- 3 The spectral sequence and applications
- 4 Generalizations
- 5 Discussion
- A Appendix
Topological phases of matter form an interesting playground for both experimental and theoretical physics. These phases have the remarkable property to be resilient against external perturbations such as weak disorder or weak interactions. This emerges from a gap in the spectrum, either between the ground state and first excited state or around the Fermi level and most of the physics is contained in the states below the gap, i.e. in the degeneracy of the ground state or the occupied states below the Fermi energy. For systems consisting of free fermions moving in a crystal the occupied states form the valence bands and it is the topology of this part of the spectrum that makes a topological phase (of free fermions) topological. Using topological invariants one can capture this topology and, in fact, characterise the topological phase. The most well-known example of this is the Chern number that characterises the integer quantum Hall (IQHE) plateaus.
We mentioned that a topological phase is resilient against external perturbations, but once these perturbations become too strong so that they cause the (band) gap to close, the topological phase is destroyed, either by becoming an ordinary insulator or by going to another topological phase. Deforming one topological phase into another allows us to understand how many topological phases there are and how to classify them. For free fermion systems with or without time-reversal symmetry and/or particle-hole symmetry, this program was initiated in [1, 2, 3]. In particular, Horava and Kitaev noticed an intricate relation between this classification and the classification of vector bundles using -theory. It was not until the work of Freed and Moore  that a complete proposal was formulated to classify topological phases of free fermions by including not only time-reversal or particle-hole symmetry, but also the crystal symmetries that these fermions experience.
The proposal of Freed and Moore involves the computation of a suitable twisted and equivariant -theory. It captures all topological invariants present for a given symmetry class and crystal. These invariants describe both global and local information of valence bands. Although we can treat these invariants in a unifying way, from a physics point of view, global and local invariants describe different aspects of the valence bands, which is why we will now discuss them separately.
The local invariants [5, 6] can be defined by carefully analyzing how crystal symmetries act on momentum space and the Hilbert space. They count the number of bands with a particular eigenvalue under the unbroken symmetry of the high-symmetry point in the Brillouin zone. For instance, when there is a fourfold rotation symmetry, the bands at the origin of the Brillouin zone are labelled by the four fourth roots of unity. The number of bands with a particular eigenvalue at the origin is a topological invariant, because changing it would either require closing the gap or breaking the symmetry. In particular this means that at the origin we already have four topological invariants, one for each eigenvalue. Repeating this procedure for other high-symmetry points of the Brillouin zone as well, yields other topological invariants, but these are not all independent. There are gluing conditions between representations associated to points and ones associated to other subspaces such as lines and planes. In other words, when going from a point with less symmetry to a point with more symmetry, the representations of the bigger stabilizer group have to restrict to the representations of the little group. This becomes especially visible when considering two-dimensional crystal groups with reflection symmetries that cause full circles to be fixed (or even 2-tori in three dimensions). On these circles there can then be special points at which the stabilizer group enhances.
Implementing these constraints consistently on the full set of topological invariants on each fixed point determines the local invariants. To complete the classification we also have to determine the global invariants. These invariants naturally live on circles or surfaces and at the same time use the global topology of the Brillouin zone in a non-trivial way. They generalize the more well-known invariants to the case where the phase is protected by additional crystal symmetries.
The global invariants can be visualized in an intuitive way as follows. Let us start with a basic example, the Chern number, which is known to model the plateaus of the IQHE. While not really necessary for this case, we will for simplicity assume that the Brillouin zone is a two-dimensional sphere rather than a two-torus. With this assumption we are ignoring most of the topology of the Brillouin zone, with would correspond to a system which is neither protected by time-reversal symmetry nor by any other symmetry. The sphere can then for example be thought of as a compactified version of the entire two-dimensional space of momenta. We will frequently encounter spheres in what follows, which is why we choose this approximation here as well, but we emphasize once more that for this particular case the result does not depend on whether we take the Brillouin zone to be a two-sphere or a two-torus.
For the two-sphere (or two-torus), it is well known that there exists an infinite family of non-trivial band structures labelled by the integral of the Berry curvature . This integral is an integer and is known as the Chern number. Explicitly, it can be written as
with the Berry curvature two-form constructed out of a Berry connection , where , are the energy eigenstates of a Hamiltonian and derivatives along the directions of the sphere. The curvature is , which is again -valued. Let us focus on . In that case we have a -connection on the sphere. To see what type of connections correspond to non-trivial Chern numbers and hence to non-trivial topological invariants, we concentrate all the curvature on the north pole. This is possible as long as we do not change the integral. In fact, in the limit we simply have a delta function at the north pole with a coefficient such that the integral in 1.1 is an integer. On the level of the connection we can view this configuration as a vortex around the north pole. In local coordinates around that point, the connection will simply be
We can thus conclude that a non-trivial Chern number corresponds to a vortex in the Berry connection. The location of these vortices can be moved (the north pole is not special) but their vorticity is a topological invariant.
The situation becomes more interesting when we consider topological insulators with time-reversal symmetry, which come with their associated invariants. Using the vortex picture sketched above, we can understand this invariant as follows. When there is a time-reversal symmetry present which squares to minus one, the band structure will always consist of an even number of bands. Thus the minimal Berry connection is at least -valued. However, the curvature will always be zero, since time-reversal symmetry acts as an orientation reversing operation on the base space, i.e the sphere. Nevertheless, this does not mean that there is no other topological invariant. As shown by [8, 7], there still exists a invariant. If we focus on the case, the upshot of  is that there is still a way to define a Chern number associated to only one of the two energy eigenstates. Its parity then gives the invariant. The other energy eigenstate will then carry the opposite Chern number. In local coordinates around e.g. the north pole, the non-trivial connection will then look like
or any other odd vorticity in each block. The trivial connection instead has an even vorticity in both blocks. The non-trivial connection is thus one of a vortex-antivortex pair in the Brillouin zone. Notice that their position can be changed, but due to time-reversal symmetry they are always at antipodal points. Of course this is a simplified picture, but it serves as an intuitive and physical interpretation of the invariant. In particular, when other crystal symmetries are added, the vortices need to respect that symmetry too. This greatly constraints their position and in combination with representation theory, their vorticity .
1.1 Outline, summary of results and comparison
The objective of this paper is to formalise some of these ideas. In particular, in Section 2 we will introduce some basic terminology in algebraic topology by discussing an example of a crystal with only time-reversal symmetry. The next section, Section 3 contains the meat of the paper. We discuss in more detail what -theory we want to compute to classify topological insulators with time-reversal and crystal symmetry. To compute these K-theories we describe the construction of an Atiyah-Hirzebruch spectral sequence and compute two examples in detail. Section 4 is devoted to various other examples we computed, for example, we compute, for the first time, the full classification of a two dimensional crystal with time-reversal in class AII and a four fold rotation symmetry. Furthermore, we also determine the twisted representation rings, which are needed in the spectral sequence, in an algorithmic way. In Section 5 we mention various subtleties and future directions. Finally, in appendix A we have gathered various mathematical details on the spectral sequence construction, twists and twisted group algebras.
Computing twisted equivariant -theory groups using an Atiyah-Hirzebruch spectral sequence is not new. In previous works, [10, 11], an Atiyah-Hirzebruch spectral sequence was also proposed and used to compute the classification for certain symmetry groups and classes. In this work, we fill in certain gaps left open in these works and put the computation of the K-theory groups with an Atiyah-Hirzebruch spectral sequence on a firm mathematical footing. We have gathered most of these details in the appendix.
The K-theory groups we have computed match with known results in the literature, but also agree with a set of heuristic arguments given in [5, 9] in the cases where we have explicit results. In particular, for the Altland and Zirnbauer classes AI and AII with an order two symmetry in two dimensions, our results match with those in [12, 13, 14]. These works extended the analysis by Kitaev in  to include additional order two symmetries such as a reflection or two fold rotation symmetry. The basis of this analysis is Clifford algebras, which allow for a straightforward implementation of order two symmetries, but for more complicated symmetries, such a procedure is more difficult. In those cases one has to resort to more sophisticated computational methodes of which we outline one in this paper.
2 Time-reversal only
In this section we shall focus on topological insulators with only unbroken time-reversal symmetry on a two-dimensional lattice without any additional rotation or reflection symmetries. Such topological phases belong to either symmetry class AI or AII . In the former case the time-reversal operator squares to the identity, whereas in the latter case it squares to minus the identity. To classify such topological insulators, we need to know how many topologically distinct insulators there are with this symmetry. As was explained in the introduction, with distinct we mean that upon going from one to the other phase, either the gap closes or the symmetry is broken. For a more formal definition, see .
The classification is most easily understood by translating the problem to momentum space, where discrete translations cause the momenta to only take values in a two-dimensional Brillouin torus . We visualize this torus as the square with opposite sides identified. Due to time-reversal symmetry, a non-trivial group acts on this two-torus which sends to , which is intuitively clear as reversing time should reverse the sign of momenta. Time-reversal symmetry can also easily be shown to be an anti-unitary symmetry. Another example of a possible anti-unitary symmetry (which we will not consider in this paper) is particle-hole symmetry, which acts trivially on the torus. Since we will ignore interactions the momenta are conserved quantities that can be used to label the states in our Hilbert space. The states with label are exactly the momentum Bloch waves. The collection of all these Hilbert spaces form a vector bundle. This vector bundle is the collection of all valence and conduction bands and since we are dealing with insulators here, there is a gap between them. In a (topological) insulator, only the valence bands are physically relevant. For the classification, we hence focus on this finite-dimensional sub-bundle.
The classification of topological insulators has now been translated into a mathematical question about the classification of vector bundles over the torus. In the absence of time-reversal symmetry, such a classification can be performed using standard (complex) -theory. With time-reversal symmetry, things get a little more exotic, since time-reversal is an antiunitary operator which in particular anticommutes with the imaginary unit . Nevertheless, Atiyah  generalized -theory to incorporate this symmetry and dubbed it Real -theory. Specifically, for class AI and AII, we need to compute . Here the is the two-dimensional Brillouin zone and the index labels the various Altland-Zirnbauer classes . In this situation we need as they indicate class AI and AII respectively. It is actually not too hard to compute these -theory groups [2, 4]. The result is
The conventional computation of these groups uses various basic properties of -theory, which cannot be generalized to include point group symmetries. Moreover, this computation is rather unsatisfactory as it gives no insight into what these invariants mean and where they come from. Part of the motivation of this work and of [5, 9] is to understand what the physical origin is of these invariants and what computational tool makes this physical origin manifest. In particular, we would like to see how the gluing of representations reveals itself in the computation. Looking ahead, we can interpret the result (2.1) as follows. The invariants are local in nature and just give the rank of the bundle, i.e. they represent the number of valence bands present. The more interesting invariant is a global two-dimensional strong topological invariant called the Fu-Kane-Mele invariant and is related to topologically protected edge states [16, 17].
In order to better understand the physical origin, we decompose the Brillouin zone into various parts that are easy for -theory to handle. Within -theory we have the freedom to consider a so-called stable equivalent space instead of the torus. Fortunately, there exists a nice space that is stably equivalent to the torus. This space is a certain wedge sum of one and two-dimensional spheres111The wedge sum of two spaces is the union of the two spaces but where one point of the first space is identified with one point of the second.. Moreover, -theory is additive under taking such wedge sums and hence we only have to compute the -theories of spheres, see the end of Section 3.1 for a more precise statement. Physically, this means that we are looking at properties of the band structure insensitive to (part of) the discrete translation symmetry. The two-dimensional sphere just represents the (compactified) momentum space of a topological insulator without translation symmetry and the -theory then gives the topological invariants associated with this Brillouin zone. For instance, the Chern number in the IQHE is just the complex -theory of the -sphere and is known not to rely on translational symmetry. After computing the -theories of all such pieces, one simply assembles all these pieces together by taking direct sums.
In two dimensions, going from the torus to the sphere can be accomplished by identifying the boundary of the square with a single point. Let us focus on this sphere for the moment. After this operation, the time-reversal action of on the Brillouin zone torus reduces to an action on the sphere that is still given by the formula if we view the sphere as . Now suppose we have a Hilbert bundle over the sphere with time-reversing operator , i.e. a bundle map , where denote the fibers of the bundle . There are two special points at and under the action of time-reversal at which the Bloch states with momentum are mapped to themselves. This gives vector space automorphisms on the corresponding fibers . In class AI (so ), the operator acts as an effective complex conjugation on the Bloch states of momenta and . In more mathematical jargon, there are canonical real structures on the vector spaces and . In class AII, when squares to , we instead have canonical quaternionic structures at and . In particular, we deduce that the space of Bloch waves at these special points is even dimensional, which is a manifestation of Kramer’s theorem. However, at a generic point on the sphere, the momenta are not preserved by , so that the state spaces at these points do not admit any extra structure.
We have now discussed how time-reversal acts on the Brillouin zone once the torus is reduced to a sphere. To include more complicated symmetries later on, it is convenient to view the sphere as being build up out of points, intervals and disks. We have chosen these particular building blocks because they are topologically trivial, i.e. contractible. Such a collection of building blocks is called a CW-complex. The building blocks themselves are called -cells where is the dimension of the block. When additional symmetries are present, such a CW-complex has to respect the symmetry. By this is meant that for each cell the symmetry must either fix it completely or map it to a different cell in the decomposition. In the case of time-reversal symmetry for instance, such a CW-complex is given in Figure 1. In this figure, we also gave the cells an orientation that is preserved by the symmetry, which is visualized by the direction of the arrows on the -cells. Note that the north and south pole are fixed by the action and hence constitute the -cells. The -cells are a line from to and its symmetry-related partner. The same is true for the -cells, which are the two hemispheres. This yields a practical setting to do the classification using -theory, because we can simply classify the bundles over these -cells and then glue them together consistently. Let us see how this works in more detail.
To start, we consider the complex and Real K-theory of spheres. It is a known fact that the K-theory of a point in degree is equal to the (reduced) K-theory of a -dimensional sphere. The results are given in Table 1. Bundles on the -cells, i.e the north and south pole in Figure 1, are classified by . In class AI, we have for each of the two fixed points. The now assigned to the north and south pole are simply given by the dimension of the fiber at those points. In class AII we get for each fixed point , which is given by the quaternionic dimension of the fiber. On the two intervals there is no real or quaternionic structure. Hence we should assign the (reduced) complex K-theory of the interval, where the boundary points of the interval are identified with each other. This K-theory is equal to the (reduced) complex K-theory of the circle, which is zero. The precise reason for this assignment is addressed in detail in the appendix. Finally to the two hemispheres, we assign the (reduced) complex K-theory of a sphere, which is . As before, the sphere appears here because we are identifying the boundary of the disc to a point. If our Hilbert space of states is to be preserved by the time-reversal symmetry, the bundle over the two 2-cells should come in pairs that are mapped into each other by the action of time-reversal symmetry. It is thus enough to know the bundle on one such 2-cell and hence under , the two copies of are identified. We thus have in zero dimensions (-cells), a in one dimension (-cells) and in two dimensions (-cells).
To get to a complete classification of topological insulators, we have to make sure that our assignment of bundles to cells is consistent. This can be done by imposing constraints in successive dimensions. For dimension zero, this means that when the fibers above the -cells are all extended to the -cells, the result should be consistent. In our case this means that the state spaces at the points and should have the same dimension, thereby reducing the we found before to .
This approach is intuitively clear and can easily be generalized to include point group symmetries. However, as advocated in the beginning, the approach of assigning representations to points is only part of the full classification. To get the other part, the global part, we should check consistency of assignments of bundles (not just representations) to higher-dimensional cells. This becomes a lot more difficult and it is hard to understand for generic crystal symmetries. In the case without time-reversal symmetry, these invariants are most of the time first Chern numbers, but there are exceptions . The invariants that can take any integer value can be understood by using the equivariant Chern character  or Segal’s formula , which also has an extension to the twisted case . However, for crystals invariant under time-reversal symmetry, the invariants are often torsion invariants and take only particular integer values. There is no systematic way of understanding them in the sense that there is no explicit formula for this piece. Instead, when assigning bundles to higher-dimensional cells, we have to check which bundles can be realized as a certain cohomological boundary and quotient out by these. The result will indeed give the -invariant of equation (2.1), but it requires some abstract mathematical theory to see this. Physically, however, there is a heuristic way of understanding these invariants as vortex-anti-vortex pairs in the connection on the bundle, which was presented in the beginning of this section.
Below we will formalize the heuristic arguments given above and put them on a firm mathematical footing. The example we have seen in this section will be computed again using machinery that allows for a generalization to more complicated crystal symmetries. To illustrate this, we compute the full classification for topological insulators in class AII on a two dimensional lattice with a twofold rotation symmetry.
3 The spectral sequence and applications
Now we come to the core of the paper. In the above we gave a heuristic classification of topological insulators with time-reversal symmetry. We will now make this classification precise and generalize to cases with non-trivial crystal symmetry. The strategy of this section will be to introduce all necessary tools. We will then reconsider the example without any crystal symmetry but with time-reversal symmetry. Whenever appropriate, we will mention the physical motivation and interpretation for these tools along the way.
Let us consider topological insulators in dimensions in class A, AI or AII, possibly with a point group symmetry. We denote the full classical symmetry group of the Brillouin zone by . If present, therefore contains time-reversal symmetries and point group symmetries but no translational symmetries. These are taken into account by the topology of our Brillouin zone torus. Let us denote by the space group and its (magnetic) point group that does not contain time-reversal symmetry, then the space group is a group extension
where represents the discrete lattice translations in spatial dimensions. When this extension is split, the space group is called symmorphic and non-symmorphic otherwise. We will focus on the former from now on and comment on the non-symmorphic case in the discussion. We will assume that there are no other symmetries, such as gauge symmetries with which the time-reversal operator could mix.
In order to classify topological phases in the sense of Freed and Moore , we have to compute a joint generalization of Real and equivariant -theory. In particular, we want to take two additional things into account. First of all, we want to keep track of which elements in act antiunitarily or not. For this we will use a map which sends an element of to if it is unitary and to if it is antiunitary. Moreover, we want to know how elements of acting on the Brillouin zone lift to elements acting on the fiber. This is most easily accounted for by a twist , a suitable group two-cocycle. This twist encodes the action of the symmetries on the quantum Hilbert space. For example, it prescribes whether . But it also provides the signs coming from taking the spin of particle into account. For example, an -fold rotation operator for spin particles satisfies . This minus sign is also encoded in .
Let us for a moment describe the situation in more precise abstract mathematical terms. Assume we have the following data:
a finite group acting on a space (in our case , the Brillouin zone);
a homomorphism ;
a group -cocycle with values in the circle group with -action .
Such a cocycle is a special case of the more general twists defined by Freed and Moore , called -twisted central extensions. Using such data, Freed and Moore  defined a version of twisted equivariant -theory denoted by
which was further studied in . It was also argued that this -theory group classifies free fermion topological insulators protected by the quantum symmetry defined by and .
To connect with more common language used in the physics literature, we describe the and that occur in the classification of crystalline topological insulators. Firstly, a class A topological insulator with point group simply has and and are both trivial. For class AI, will instead be the magnetic point group, i.e. it will contain both point symmetries and time-reversal symmetry. We will only consider magnetic point groups of the form , with the action of time-reversal symmetry on the Brillouin zone, even though the mathematical machinery developed here can handle more general point groups as well. For instance, one could also consider cases in which the time-reversal operator is a combination of the usual time-reversal operator with a lattice translation or point group symmetry in . For in class AI, will simply be projection onto the second factor and will be trivial. Finally, for class AII, we again take to be the magnetic point group and the same projection, but now we pick in such a way that the twisted group action represents the desired action on the quantum Hilbert space. In particular, we pick so that time-reversal squares to , reflections square to and rotation by equals . To assure a consistent choice, a precise construction of for a given point group is given at the end of appendix A.1.
It is shown in [21, Thm 3.11] that the groups satisfy certain equivariant versions of the homotopy, excision, additivity and exactness axioms of Eilenberg and Steenrod. The fact that our twisting class is defined by a group cocycle implies that these axioms are exactly the axioms for an equivariant cohomology theory on the category of -spaces as defined in Bredon [22, §I.2]. This is what makes the following computations mathematically sound; as explained in [22, §IV.4] the axioms guarantee the existence of the Atiyah–Hirzebruch spectral sequence. Moreover, the orbifold point of view advocated in  allows us to change the group and the space as long as the quotient space remains the same and we keep the same stabilizer. This is useful in some computations, see the end of Section 3.3. For more details on how the -theory we use is defined, see appendix A.1.
We are therefore left with the task to compute the -theory of the Brillouin zone dressed with and . The technique to compute these groups goes along the lines that we have discussed in the previous section. We first decompose the Brillouin zone into cells and view them as a CW-complex. Non-trivial symmetries have to leave these complexes invariant. Such complexes are equivariant -CW complexes, which is nothing more than an upgraded version of the unit cell in momentum space. After having found this -CW complex, we use an Atiyah-Hirzebruch spectral sequence to compute , which are assembled to give . Let us now formalize this computational method.
3.1 A general method: the Atiyah-Hirzebruch spectral sequence
The spectral sequence for the computation of the twisted equivariant -theory of a space is constructed by using a decomposition as a -CW complex , where the superscript on the cells indicates the dimension of the subspace. For the applications considered in this paper, is either a torus or a sphere (we will remark on how to reduce the computation of the -theory of the torus to the -theory of a sphere at the end of this section). A spectral sequence is a successive approximation method converging to the desired answer in a number of steps. For us these steps will always be finite and in fact, most of the time only two steps are necessary. These steps are usually referred to as pages. The first page of the spectral sequence, just as in the last example, is given by equivariant assignments of the -theory of spheres to the cells of . For the -cells , this means that we assign to each a twisted representation of the stabilizer group of that point. These representations, which are twisted using and the map , are conveniently packaged in the twisted representation ring . These objects are actually not rings, but since they are equal to the usual representation ring of in case and are trivial, we will keep on referring to them as twisted representation rings. Details about twists and twisted representations can be found in appendix A.1. So maps to an element of the twisted representation ring of the corresponding stabilizer group . By equivariance is meant that preserves the symmetry in the following sense: is required to map to the resulting conjugate representation in . More generally, we equivariantly assign higher representation rings (i.e. the higher degree twisted equivariant -theory of a point, see appendix A.2 for details) to -cells . These classify twisted -equivariant bundles over -spheres, instead of over just a point. The grid of such assignments of representation rings for each and form the first page of the spectral sequence and is denoted by . Those assignments can be shown to be equivalent to Bredon -cochains with values in the coefficient functor . In appendix A.3 we define these coefficient functors and present a derivation of this result. Intuitively, the functor keeps track of both the (higher) representations at fixed loci and how they restrict to each other. For Bredon -cochains, this functor will pick out the stabilizer group of the -cells and assign degree twisted representation rings to the -cells. It should be noted here that the action of group elements on the higher representation rings can be tedious to determine explicitly in certain examples, so that the equivariance of can result in nontrivial results. One example of this is given in Section 3.2.
To go to the next page of the spectral sequence, we have to take the cohomology of the first page with respect with the first differential, which in our case is known as the Bredon differential. In fact, the first differential maps to and is given by the differential of Bredon cohomology, which is
with denoting the set of -cells of and a Bredon -cochain. Here means that we take the higher twisted representation of that assigns to and restrict it to a representation of . The notation stands for an integer factor that tells us in which way intersects the boundary of the -cell . In general the behavior and computation of this number can be quite complicated, but if our CW-complex is sufficiently nice, this number is usually just a sign depending on a fixed orientation. For example, if we have a line (1-cell) oriented from the endpoint to the other endpoint , i.e. , then we simply have
and of course if is not an endpoint of . If instead is a -cell that lies in a disk surrounded by a couple of intervals , then depending on whether the orientations of the line coincide with the orientation of . In more general situations, where there is nontrivial gluing present, it can be computed as the degree of a certain map between spheres. This map is exactly the same as for the cellular boundary map in ordinary cellular homology, which can be found for example in Hatcher’s book .
The second page is the cohomology of the first page with respect to the differential given in (3.3). Mathematically speaking the second page entry therefore equals the degree Bredon equivariant cohomology of with coefficient functor . For the third and higher order pages, we need to know the higher differentials, which are much more abstractly defined and no explicit form is known. Therefore, until more is known about this it is not possible to fully classify topological phases for general point groups using this method. It is however often the case in practice that we can arrive at a definite answer without knowing explicit expressions for the higher differentials. At least it is known that the th differential is of bidegree , so . Therefore, for -dimensional spaces, the th differential is zero for all . For more details on the construction of the spectral sequence and explicit definitions, see appendix A.3.
But how do we construct the twisted equivariant -theory of from the data of the spectral sequence? After taking the cohomology with respect to the th differential, we arrive at the final page, . We can construct the -theory by extensions out of , so by equating and . In two dimensions, this means that there exist exact sequences
see the final paragraph of appendix A.3 for the details. Unfortunately, these sequences do not split in general. Therefore the -theory is not always fully determined by the spectral sequence (unless of course we would explicitly determine the maps in these sequences, which is a tedious exercise). We will call this the problem of non-unique extensions, which unfortunately is intrinsic to our approach. An example of this phenomenon will be addressed in Section 3.3.
Now that the spectral sequence is contained in our toolbox, we will explain how to reduce the computation of equivariant -theory of the Brillouin torus to the computation of the -theory of spheres. For this we use an equivariant stable homotopy equivalence that generalizes [4, Thm 11.8]. This equivalence adresses the decomposition of the Brillouin torus in terms of spheres. Indeed, if the action of on can be realized as the restriction of an action of , where acts on and permutes the copies of , then is equivariantly stably homotopy equivalent to a wedge of spheres. More explicitly, this means in two dimensions that there is an isomorphism
Here the tilde indicates the reduced -theory and is a space that looks like the figure , which is nothing but the boundary of the Brillouin zone torus seen as a square with opposite sides identified. Note that the symmetry could potentially interchange the two ’s of the figure eight, for example in case there is a fourfold rotation symmetry. If there is no group element permuting the two copies of the circle, the -theory decomposes further as
where we used that
A similar isomorphism as in (3.8) exists in three dimensions under the given assumptions. The relation between reduced and unreduced -theory is
When using the equivariant splittings (3.7) and (3.8), we can thus compute the unreduced -theory and then strip of the -part to obtain the reduced -theory. Note that the assumption that the action of comes from some action of does not always hold, so that we cannot always use (3.7). If for example three-fold rotations are present, we seem to be bound to applying the Atiyah-Hirzebruch spectral sequence to the Brillouin zone torus directly.
Since the -theory of a one-dimensional space is easy to compute, the isomorphism (3.8) effectively reduces computations of the -theory of a two-dimensional torus to a two-dimensional sphere. Indeed, for one-dimensional spaces all higher differentials vanish and , so that the exact sequences (3.5) and (3.6) reduce to a single exact sequence. Because the twisted representation ring is torsion free (for ), so is . Hence the resulting sequence splits, giving us
Despite the absence of torsion in the first term, the second term can give rise to torsion of which we will see examples below. The torsion in was anticipated before in  for systems in class AII with a reflection symmetry in one dimension.
3.2 Time-reversal only: revisited
Let us now illustrate how the Atiyah-Hirzebruch spectral sequence formalizes the intuitive approach of the last section. So again we will take with the -action with the -CW-structure as given in Figure 1. We will consider the classes AI () and AII () simultaneously and note the distinctions along the way. Mathematically, we distinguish between the two classes by picking the twist to be trivial in class AI and nontrivial in class AII. The higher twisted representation rings of and the trivial subgroup are given in Table 1. Note that the stabilizers of the 0-cells are both , while for the other cells the stabilizer is trivial.
We will start by computing all Bredon cohomology groups that are necessary for obtaining the -theory group from the spectral sequence. These cohomology groups are the ones that correspond to second page entries of the spectral sequence which could possibly influence the three desired entries and of the final page occurring in the exact sequences of equation (3.5) and (3.6). Because the second differential (which is of bidegree ) is the only possibly nonzero higher differential, we have . Therefore we have to compute for equal to and .
First we have to find all necessary Bredon equivariant cochains, as they constitute the first page . We start with . So we consider the equivariant -cochains with values in , which here are the equivariant maps from the set to for both twists. Because the -cells are completely fixed by the group, all -cochains are equivariant. Therefore the equivariant -cochains are spanned by two basis elements and over :
Here maps to and to . For the roles of and are interchanged. In more basic terms: assigns a state space of dimension one to and a zero space to , while assigns a zero space to and a one-dimensional space to .
Going up to , there is only one equivariant -cochain, so that
Indeed, from Table 1 it is clear the this cochain is an equivariant map from to . By equivariance, it is uniquely specified by specifying its value on , which we take to be for . In case the reader is interested in the actual value of , simply note that the action of on the representation ring is just complex conjugation. In case acts on the representation ring of a nontrivial group this will result in the complex conjugation of nontrivial representations, which are in general not isomorphic to the original representation. However, a complex vector space is noncanonically isomorphic to its complex conjugation since it has the same dimension. Hence the automorphism that induces on is simply the identity, thus . Along the way we will see that this automorphism is not always trivial and acts with minus the identity on the higher representation ring of degree . This is the heart of the matter, since it is the aspect that creates torsion in this example.
For , the situation simplifies, since the degree representation ring of the trivial group equals zero. Hence there are no equivariant -cochains or -cochains with values in . The degree representation ring of depends on whether the twist is taken trivial (class AI) or nontrivial (class AII). For trivial it equals and for nontrivial it equals . In class AII, we therefore also have no equivariant -cochains with values in . In class AI instead, the equivariant -cochains are spanned by and , just as for . However, this time they are a basis over :
Here maps to the nontrivial element of and to the trivial element, while for it is the other way around.
Finally, for there are some subtleties. The degree representation ring of the trivial group is equal to . Analogously to , we get that the -cochains are spanned over by a single element with . Similarly, the -cochains are spanned by a single element with . However, unlike for , we have that
This is because the action of on the degree representation ring is , as can be shown by an explicit analysis using Clifford algebras, using the explicit definitions in appendix A.2. We can conclude that the relevant part of the first page of the spectral sequence for class AI and class AII respectively is given in the following table:
With the information we have gathered now, we can construct the second page, consisting of Bredon cohomologies. Let us start off by computing . For this we need to compute the kernel of the Bredon differential
On the -cochain it acts as
where the symbol denotes the restriction of the representation to the trivial group. In class AI, this restriction maps complex vector spaces with a real structure to their underlying complex vector space. Since all complex vector spaces admit a real structure, this implies that the restriction map is the identity. In class AII, where , the restriction is multiplication by two, because only complex vector spaces of even dimension admit a quaternionic structure. Hence we get
Using the orientation we analogously get that . In both class AI and AII, we see that the degree zero cohomology equals
In general, this cohomology group contains all local topological invariants. More precisely, the zeroth degree cohomology group is actually a mathematical formalization of the heuristic method of consistently assigning representations to point sketched in Section 2 and used extensively in  and .
The next row of the second page is easily deduced from the first page. In class AII, all cochains vanish and therefore so do the cohomology groups. In class AI nontrivial -cochains exist, but not in higher degrees. Therefore the differential is necessarily zero and the second page equals the first page.
The final relevant cohomology group is . In this case, induced a non-trivial automorphism on given by , so that
hence for both and , so that . The kernel of acting on -cochains is , since we are in top degree. Thus .
Summarizing the results by filling in the second page of the spectral sequence, we get the following tables for and respectively:
Since is a free group, the sequence splits. This gives us . Moreover, using the spectral sequence it can easily be shown that (for an example of a computation of the -theory of a one-dimensional space using the spectral sequence, see the next section). By the equivariant splitting (3.8), the -theory of the torus is thus
which confirms the result using a different approach, see equation (2.1). It is worth noting that the torsion invariant managed to appear because of the nontrivial action of induced by complex conjugation and not by the torsion in the -theory of a point as it does when computing using the methods of for example Freed and Moore .
For another lesson is to be learned from this example. Namely, note that as long as we do not know any expression for the second differential , we cannot uniquely determine the -theory group by the spectral sequence method. However, we know from other methods that so that this differential must be surjective. If in future research an explicit expression for the second differential is found, it would be interesting to compute it in this example.
3.3 Time-reversal and a twofold rotation symmetry
For a more exciting example, we now also include a rotation by . So we take the symmetry group . We twist the group so that the twisted group algebra satisfies the desirable physical situation on the quantum level, namely and . This thus represents spinful fermions on a two-dimensional square lattice with twofold rotation symmetry and hence the wallpaper group is . On the Brillouin torus these symmetries act as and .
Before the topological computations, we first have to compute the twisted Bredon coefficients, i.e. the representation rings and the relevant maps between them. Note that the only stabilizers that occur are and , so we only have to compute twisted representations for these groups. Because of this exceptional role played by it is useful to set and forget about for the moment. Note that in the twisted group algebra, , and . The twisted group algebras are abstractly isomorphic to matrix algebras:
where the last isomorphism follows because the twisted group algebra is . Therefore the twisted group algebra of is Morita equivalent to the algebra , while the twisted group algebra of is Morita equivalent to the algebra . The representation rings are therefore
see appendix A.2 for details on higher degree representation rings. The restriction map in degree zero
is just given by mapping a complex vector space to its underlying real space and hence it is given by multiplication by two. For the restriction map can only be zero, since . For restriction is a map
so it can either be zero or reduction mod . It is possible to explicitly check which it is by using explicit Clifford modules, but it turns out that we do not need to know which one it is in order to compute the -theory. The only remaining map between representation rings is the action of on the representation ring. This action is given by conjugating modules over with . Since is in the center of , the automorphism on resulting from this is trivial. On the two relevant higher degree representation rings
the action of is trivial as well because has no nontrivial automorphisms.
In order to compute the full twisted equivariant -theory of the Brillioun zone torus, we first use the equivariant splitting method, giving the isomorphism 3.8. Secondly we apply the spectral sequence on the components. Note that the circles occuring in the isomorphism, i.e. and in the Brillouin zone, have identical group actions. Hence they give isomorphic -theory groups and we only have to compute one. Next we have to decide on -CW decompositions of our new spaces and . Since the action of is the same as the action of , we can reuse the -CW structure of the last example for as given in Figure 1. For the circle we use the one-dimensional sub--CW complex of the -CW structure on .
Let us start by computing the twisted equivariant K-theory of the circle. We compute the -theory by using 3.11 for one-dimensional spaces. The zeroth-cohomology is analogous to the example in the previous subsection. We can define -bases of equivariant -cochains and -cochains . In contrast with the last example, we now have complex vector spaces on the fixed points and real vector spaces on the -cells for . Recall that the restriction map sends a complex vector space to its underlying real space and therefore this map is given by multiplication by two. The Bredon differential is thus given by
Similarly Hence . Notice that for the first cohomology group the twisted representation ring of vanishes in te corresponding degree, so that the differential equals zero. Hence this cohomology group is equal to the group of equivariant -cochains , which equals the twisted representation ring of in degree . We conclude that . Via equation (3.11), we arrive at
Since , we see that for both circles in the splitting of the torus. These are precisely the invariants proposed by Lau et al. in  and when non-trivial represent a Möbius twist in the Hilbert space of states along the invariant circles at and . Our K-theory computation thus provides a mathematical proof of the existence of this invariant.
Now we turn to the computation of the twisted equivariant -theory of the 2-sphere. We use the same bases of equivariant cochains and as in the last example. For the zeroth cohomology, , the computation is equivalent to the one in the previous subsection, hence . Going to , we see that there are no -cochains, since . The differential on -cochains gives
since necessarily acts trivially on . Hence the cohomology groups are equal to the cochain groups:
Since for the -cochains and -cochains are exactly the same, the above computation also applies to the computation of the cohomology in degree . Therefore it follows that equals as well. The relevant part of the second page is thus conveniently summarized in the following table.
The second differential is either zero or reduction modulo . Independent of this distinction, the kernel of is abstractly isomorphic to . Hence the relevant part of the final page of the spectral sequence agrees with the diagonal in the table above. The exact sequences (3.5) and (3.6) that follow from the spectral sequence now reduce to
Note that the second sequence splits. Unfortunately, the first exact sequence implies only that or . Hence the Atiyah-Hirzebruch spectral sequence gives that is either or , depending on whether the first exact sequence splits or not. We can conclude from equation 3.8 that
To determine which of these two is the correct one, we employ the equivariant Mayer-Vietoris exact sequence. We can focus on the sphere since two possibilities for the -theory originated there. Take open