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Topological phases in the theory of solid states A. G. Sergeev, E. Teplyakov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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     To the centenary of Vasilii Sergeevich Vladimirov IntroductionVasilii Sergeevich Vladimirov was always keenly interested in the last achievements in mathematics and theoretical physics. He always strongly recommended to his students to deeply understand such innovations and report on them at the seminar. It was Vladimirov who had suggested that I1 study the twistor theory and its applications in differential geometry and mathematical physics. In recent years, the advances in the theory of insulators and superconductors with involvement of topological methods have attracted my attention and the attention of my coauthor. This topic is elaborated in the present survey, which is devoted to the theory of topological phases, which is also based on the topological ideas. I think that this line of research would be also interesting for Vasilii Sergeevich, to whom we dedicate this survey written on the occasion of his centenary. This paper may be considered as a continuation of the papers [1], [2] devoted to mathematical aspects of the theory of topological insulators. Topological insulators represent a non-trivial example of the theory of topological phases underlain by the homotopy approach to the study of solid bodies. The role of topology in the solid state physics became clear, probably for the first time, in the investigation of the quantum Hall effect discovered in 1980. Topological methods play a leading role in the study of insulators characterized by the existence of the energy gap stable under small deformations. The availability of such a gap is an important condition also in the theory of topological phases considered in this survey. Let $G$ be the symmetry group. Consider the set $\mathrm{Ham}_G$ of classes of homotopy equivalence of $G$-symmetric Hamiltonians satisfying the above gap condition. On this set, it is possible to introduce a natural stacking operation with respect to which the set $\mathrm{Ham}_G$ is an Abelian monoid (that is, an Abelian semigroup with neutral element). The group of invertible elements of this monoid is known as the topological phase. It turns out that the family $(F_d)$ of $d$-dimensional topological phases forms an $\Omega$-spectrum, which, by definition, is a collection of topological spaces $F_d$ such that the loop space $\Omega F_{d+1}$ is homotopy equivalent to the space $F_d$. This paves the way for a wide use of algebraic topology in the study of topological phases. More precisely, to each $\Omega$-spectrum there corresponds a generalized cohomology theory determined by the functor $h^d$, which associates with the topological space $X$ the set $[X,F_d]$ of classes of homotopy equivalent maps $X\to F_d$. The spaces $[X,F_d]$ have been widely studied in algebraic topology, which gives us some hope that a complete classification of topological phases can be obtained on the basis of these results. Apart from the above approach to the study of topological phases, there are also other approaches based on the algebraic topology methods. The methods not considered in the present paper include interpretations of topological phases in terms of cobordisms [3], [4], the group cohomology theory [5]–[7], and the topological field theory [8]. The paper is organized as follows. In § 1, we give the definitions of fundamental concepts of topological phases and stacking operation. The important notions of $\mathrm{SRE}$-states and $\mathrm{SPT}$-phases are introduced. The bosonic Fock space is constructed for a description of many-particle states. An interpretation of topological phases in terms of lattices is given, and possible symmetry groups and their covariant and graded representations are described. In the presentation of § 1, we follow the paper [9] and the thesis [10]. In § 2, we give a formulation of the theory of topological phases in terms of $\Omega$-spectra and related generalized cohomology theories. We also present physically motivated examples of two- and three-dimensional systems of different symmetry types. The last § 3 is devoted to the relation of topological phases with the $K$-theory, which was proposed in [11]. The definition of a $K$-functor, as given in this section, is based on the notion of spectral flattening of Hamiltonians. We consider several examples of systems with $\mathrm{CT}$-symmetries, and describe their graded representations, which are closely related to those of Clifford algebras. § 1. Topological phases1.1. The class of admissible HamiltoniansWe consider quantum mechanical systems described by Hamiltonians $H$ invariant under the action of the symmetry group $G$. A Hamiltonian $H$ is defined by a self-adjoint operator in a Hilbert space $\mathcal H$, and the group $G$ acts on $\mathcal H$ by unitary or anti-unitary operators. Apart from the $G$-invariance condition, the Hamiltonians $H$ may be subject to other constraints, of which the most important is the gap condition which requires that $\operatorname{Ker}H=\{0\}$. A $G$-symmetric gapped Hamiltonian will be called admissible. We next consider continuous deformations of admissible Hamiltonians, that is, continuous paths $H_t$, $0\leqslant t\leqslant 1$, in the class of admissible Hamiltonians. The admissible Hamiltonians are characterized by their ground states, that is, by the eigenstates with minimal energy. By this reason, it is natural to consider, along with admissible Hamiltonians, their corresponding admissible ground states. We introduce the stacking operation on the set of admissible ground states. Suppose we are given two admissible ground states $\Phi_0$ and $\Phi_1$ with associated admissible Hamiltonians $H_0$ and $H_1$ acting in Hilbert spaces $\mathcal H_0$ and $\mathcal H_1$, respectively. The stacking of these two states is the ground state of the form corresponding to the Hamiltonian $H$ acting on the tensor product The symmetry group $G$ acts on $\mathcal H$ as the tensor product of representations of $G$ in the Hilbert spaces $\mathcal H_0$ and $\mathcal H_1$, and the operator $H$ is given by the equality The ground state $\Phi$ and the Hamiltonian $H$ thus constructed are $G$-symmetric and gapped if so are the initial ground states $\Phi_0$, $\Phi_1$ and Hamiltonians $H_0$, $H_1$.1.2. The definition of topological phasesWe let $\mathrm{Ham}_G$ denote the set of classes of homotopy equivalent admissible Hamiltonians and the corresponding ground states. The above stacking operation is pushed down to a binary operation on $\mathrm{Ham}_G$. We denote by $[\Phi]$ the class in $\mathrm{Ham}_G$ containing the ground state $\Phi$, and by $[\Phi_1]+[\Phi_2]$ we denote the stacking of the ground states $[\Phi_1]$ and $[\Phi_2]$. This operation has the following properties. 1) associativity: for all admissible ground states $[\Phi_1]$, $[\Phi_2]$, $[\Phi_3]$,
$$
\begin{equation*}
([\Phi_1]+[\Phi_2])+[\Phi_3]=[\Phi_1]+([\Phi_2]+[\Phi_3]);
\end{equation*}
\notag
$$
2) commutativity: for all admissible ground states $[\Phi_1]$, $[\Phi_2]$, 3) for any admissible ground state $[\Phi]$, where $[0]$ is the neutral element.By an SRE (short range entangled)-state we will mean an admissible ground state which is homotopic to the trivial one in the class of admissible states. Keeping in mind the $G$-symmetry, we define the SPT (symmetry protected topological) phase as the class in $\mathrm{Ham}_G$, such that any of its representatives is an SRE-state (in other words, if each representative of this phase can be connected with the trivial state if one ignores the $G$-symmetricity condition). The topological phases can also be defined in the following more formal way. As pointed out above, the space $\mathrm{Ham}_G$, as equipped with the stacking operation, is an Abelian monoid. The group of invertible elements of the monoid $\mathrm{Ham}_G$ is known as an SPT-phase, and, therefore, an Abelian group relative to the stacking operation. Below, we will consider a concrete example of topological phases related to lattice systems. Here, we point out only an analogy of the given construction with the $K$-theory. We let $\operatorname{Vect}_s(X)$ denote the semigroup of vector bundles of finite rank over a topological space $X$ defined up to the stable equivalence (that is, up to the addition of trivial bundles). The functor $K(X)$ is identified with the Abelian Grothendieck group constructed from the semigroup $\operatorname{Vect}_s(X)$. In view of the analogy between the topological phases and the $K$-functor, the stacking operation should correspond to the direct sum of bundles and the trivial state, to the trivial bundle. 1.3. The Bosonic Fock spaceThe bosonic Fock space will be required for a description of many-particle states. The bosonic Fock space over a Hilbert space $\mathcal H$ is defined as the completion
$$
\begin{equation*}
\mathcal B(\mathcal H)=\overline{\mathfrak S(\mathcal H)}
\end{equation*}
\notag
$$
of the algebra of symmetric polynomials $\mathfrak S(\mathcal H)$ with respect to the natural norm generated by the inner product on $\mathfrak S(\mathcal H)$. This inner product, which is generated by the inner product on $\mathcal H$, is constructed as follows. On the monomials of the same degree, it is given by the formula
$$
\begin{equation*}
(v_1\otimes\dots\otimes v_p, v'_1\otimes\dots \otimes v'_p) = \sum_{\sigma}(v_1,v'_{i_1})\cdot\ldots\cdot(v_p,v'_{i_p}),
\end{equation*}
\notag
$$
where the summation is over all permutations $\sigma=\{i_1,\dots,i_p\}$ of the set $\{1,\dots,p\}$ (the inner product of monomials of different degrees is zero). The inner product on monomials is extended by linearity to the whole algebra $\mathfrak S(\mathcal H)$ of symmetric polynomials. Now the bosonic Fock space $\mathcal B(\mathcal H)$ is defined as the completion of the algebra $\mathfrak S(\mathcal H)$ with respect to the above norm. If $\{w_n\}$, $n=1,2,\dots$, is an orthonormal basis for $\mathcal H$, then, as an orthonormal basis for the Fock space $\mathcal B(\mathcal H)$, we can take for the collection of monomials
$$
\begin{equation*}
P_K(v)=\frac1{\sqrt{K!}}\,(v,w_1)^{k_1}\cdot\ldots\cdot(v,w_n)^{k_n},
\end{equation*}
\notag
$$
where $v\in\mathcal H$, $K=(k_1,\dots,k_n)$, $k_i\in\mathbb N$, and $K!=k_1!,\dots,k_n!$. We let $a_i^\unicode{8224}$ denote the operator of creation of a particle in the state $w_i$, as given by the operator of multiplication by the inner product with $w_i$. The adjoint operator of annihilation of a particle in the state $w_i$ coincides with the operator $-\partial_{w_i}$, where $\partial_{w_i}$ is the operator of differentiation in the direction $w_i$. These operators satisfy the standard commutation relation (the other commutators being zero). An arbitrary linear operator $O\colon \mathcal H \to \mathcal H$ extends to the linear operator $\widehat O\colon \mathcal B(\mathcal H)\to\mathcal B(\mathcal H)$ given on monomials by the formula
$$
\begin{equation*}
\widehat O(v_1\otimes\dots\otimes v_p)=(Ov_1)\otimes\dots\otimes(Ov_p)
\end{equation*}
\notag
$$
with subsequent extension by linearity and completion to the whole space $\mathcal B(\mathcal H)$. In terms of creation and annihilation operators, this operator can be written as In parallel with the bosonic Fock space, we will need its fermionic analog. The fermionic Fock space is defined similarly (for a definition, see, for example, [2]). For our purposes it is important that the fermionic Hamiltonian $H$ is defined (like the bosonic one) by the sum of the form (1) over fermionic creation and annihilation operators acting in the fermionic Fock space $\mathcal F(\mathcal H)$. 1.4. Lattice interpretationSuppose that we are given a lattice $\mathcal L$ in $\mathbb R^d$, that is, a discrete Abelian group isomorphic to $\mathbb Z^d$ and which acts on $\mathbb R^d$ by translations by vectors $\gamma\in\mathcal L$. We let $G$ denote the symmetry group of the Hamiltonian. In this case, the class of admissible Hamiltonians $H$ consists of $d$-dimensional local $G$-symmetric gapped self-adjoint operators acting on the Hilbert space $\mathcal H$ and on the Fock space $\mathcal B(\mathcal H)$. To each site $\gamma$ of the lattice $\mathcal L$ there corresponds the Hilbert space $\mathcal H_\gamma\equiv\mathcal H$ so that $H$ is the sum of tensor products of operators on the copies $\mathcal H_\gamma$. The admissible operators are given by (1) in which the number of terms is majorized by a common constant $k$ (the locality condition). The trivial state, also called the trivial product, is the state of the form
$$
\begin{equation*}
\bigotimes_{\gamma\in\mathcal L}\Phi_\gamma\in \bigotimes_{\gamma\in\mathcal L}\mathcal H_\gamma.
\end{equation*}
\notag
$$
For every pair of such states, there exists a continuous path connecting them in the space of trivial states. 1.5. Symmetry groupsRecall that the symmetry group $G$ acts on the Hilbert space $\mathcal H$ by the unitary or anti-unitary transformations. It is convenient to introduce the homomorphism $\phi\colon G\to\{\pm1\}$ which indicates that, for $\phi(g)=+1$, the element $g\in G$ acts on $\mathcal H$ as a unitary operator, while for $\phi(g)=-1$ it acts as an anti-unitary operator. In addition, the group $G$ may contain the symmetry with respect to time inversion, as given by the homomorphism $T\colon G\to\{\pm1\}$, and the symmetry of the charge conjugation, as given by the homomorphism $C\colon G\to\{\pm1\}$. The $G$-symmetry condition is closely related to the functoriality property of $G$-protected SPT-phases. The latter condition means that, given a homomorphism $\varphi\colon G'\to G$ of the symmetry groups, the composition of $\varphi$ with a representation of the group $G$ in the Hilbert space $\mathcal H$ generates the representation of the group $G'$ in the same Hilbert space. This, in turn, defines the $G'$-protected SPT-phase. We generalize now the original problem statement by augmenting it with a $C^*$-algebra $\mathcal A$. Consider the pairs $(G,\mathcal A)$ in which the action of the group $G$ on the algebra $\mathcal A$ is given by the homomorphism $\alpha\colon G\to\operatorname{Aut}(\mathcal A)$ into the group of linear $*$-automorphisms of the algebra $A$. A covariant representation of a pair $(G,\mathcal A)$ is a non-degenerate $*$-representation $\theta$ of the algebra $\mathcal A$ by bounded linear operators in the Hilbert space $\mathcal H$. Suppose now that the algebra $\mathcal A$ is graded, that is, $\mathcal A=\mathcal A_0\oplus\mathcal A_1$, where $\mathcal A_0$, $\mathcal A_1$ are self-adjoint closed subspaces satisfying
$$
\begin{equation*}
\mathcal A_i\mathcal A_j\subset\mathcal A_{(i+j)(\operatorname{mod}2)}.
\end{equation*}
\notag
$$
Let $\operatorname{Aut}(\mathcal A)$ be the group of even $*$-automorphisms of the algebra $\mathcal A$, that is, $*$-automorphisms of the algebra $\mathcal A$ which preserve the decomposition $\mathcal A=\mathcal A_0\oplus\mathcal A_1$. The graded covariant representation of the system $(G,\mathcal A,c)$, where $c$ is the homomorphism $G\to\{\pm1\}$, is the graded $*$-representation of the algebra $\mathcal A$ in the graded Hilbert space $\mathcal H=\mathcal H_0\oplus\mathcal H_1$ such that $\theta(g)$ is an even operator for $c(g)=+1$ and an odd operator for $c(g)=-1$.
§ 2. Spectra and generalized cohomology theories2.1. The notion of $\Omega$-spectrumRecall that by definition an $\Omega$-spectrum is the family of pointed topological spaces $(T_n)$, $n\in\mathbb Z$, with the following property: for any $n\in\mathbb Z$, the pointed topological spaces are homotopy equivalent, where $\Omega T_{n+1}$ is the loop space of the topological space $T_{n+1}$ considered as a pointed space.The notion of an $\Omega$-spectrum is well known in topology (see, for example, [12]), where it plays an important role. Namely, with each $\Omega$-spectrum one associates the generalized cohomology theory determined by a contravariant functor $h^n$. This functor associates with each pair $(X,Y)$ of pointed topological spaces $(X,Y)$, $Y\subset X$, the Abelian group where on the right we have the set of homotopy classes of continuous maps $(X,Y)\to (T_n,*)$ sending $Y$ to the marked point $*$.In order to take into account the action of the symmetry group $G$, we suppose that it acts on the pair $(X,Y)$ by a continuous homeomorphism $\varphi$. In this case, we can introduce the $G$-invariant generalized cohomology theory given by the functor where $EG\to BG$ is the classifying bundle which is a principal $G$-bundle over the classifying space $BG$, and the space $EG$ is contractible. The space $EG\times_GX$ itself is identified with the quotient $(EG\times X)/G$. In particular, for $X=*$, we get2.2. SPT-phases and $\Omega$-spectraThe notion of an $\Omega$-spectrum is well suited for a description of the theory of topological phases. Namely, let $F_d$ be the space of $d$-dimensional SRE-states. It is claimed that, for such spaces,
$$
\begin{equation}
F_d\sim\Omega F_{d+1}\quad\text{for}\quad d\geqslant0.
\end{equation}
\tag{2}
$$
The idea that this property should hold for topological phases $(F_d)$ was formulated by Kitaev [13]. The physical arguments in favour of this claim were expressed in a series of physical papers (see, for example, [10]), however we did not found a rigorous mathematical proof. Nevertheless, (2) looks quite plausible and, in what follows, we will describe results which can be obtained with the help of this relation. If this property holds for $d\geqslant0$, then it allows one to define by induction the spaces $F_d$ for all $d\in\mathbb Z$. So, the family of the spaces $(F_d)_{d\in\mathbb Z}$ forms an $\Omega$-spectrum. Let us recall some known facts about the homotopy groups of the spaces $F_d$. The group $\pi_0(F_d)$ classifies the $d$-dimensional SPT-phases without symmetry. In lower dimensions, it is equal to
$$
\begin{equation*}
\pi_0(F_0)=0,\quad \pi_0(F_1)=0,\quad \pi_0(F_2)=\mathbb Z,\quad \pi_0(F_3)=0
\end{equation*}
\notag
$$
(the group $\mathbb Z$ in dimension $2$ is generated by the so called $E_8$-phase). The space $F_0$ is identified with the infinite-dimensional projective space and the other spaces $F_d$ of lower dimensions can be described in terms of the Eilenberg–Mac Lane spaces $K(\mathbb Z,n)$ as follows:
$$
\begin{equation*}
F_1=K(\mathbb Z,3),\quad F_2=K(\mathbb Z,4)\times\mathbb Z,\quad F_3=K(\mathbb Z,5)\times\mathrm{U}(1).
\end{equation*}
\notag
$$
If $\mathrm{SPT}^d(G)$ denotes the Abelian group of $d$-dimensional $G$-protected SPT-phases and $H^n(G,\mathbb Z)$ is the $n$-dimensional cohomology group of the group $G$, then the lower groups $\mathrm{SPT}^d(G)$ are computed as follows:
$$
\begin{equation*}
\begin{gathered} \, \mathrm{SPT}^0(G)=H^2(G,\mathbb Z),\quad \mathrm{SPT}^1(G)=H^3(G,\mathbb Z), \\ \mathrm{SPT}^2(G)=H^4(G,\mathbb Z)\oplus H^0(G,\mathbb Z),\quad \mathrm{SPT}^3(G)=H^5(G,\mathbb Z)\oplus H^1(G,\mathbb Z). \end{gathered}
\end{equation*}
\notag
$$
2.3. ExamplesWe give a series of examples of topological phases borrowers from the physical papers (see, for example, [10]). We start from the fermionic systems featuring the hourglass symmetries. By such systems, one means the symmetry groups including the charge conjugation symmetry $\mathrm{U}(1)$, the time reversion symmetry $T$ with $T^2=-1$, and the glide symmetry given by the composition of the translation with reflection with respect to the half-period. As an example of systems with glide symmetry we can take a three-dimensional system in which the planes with fixed coordinate $x\in\mathbb Z$ are occupied by two-dimensional systems (quantum spin Hall insulators), and the planes with fixed coordinate $x\in\mathbb Z+1/2$ are occupied by their mirror reflections. The system thus obtained is invariant under the glide given by the map: $(x,y,z)\mapsto (x+1/2,-y,z)$. This procedure will be called the alternating fibres construction. This construction can be visualized as the diagram which connects the topological insulators in the dimensions two and three. In the dimension $2$, the generator of the first group $\mathbb Z_2$ is the quantum spin Hall insulator (the QSH-phase). We can associate with it the above three-dimensional system corresponding to the group $\mathbb Z_4$ and having the hourglass symmetry. The transition from the group $\mathbb Z_4$ to the second group $\mathbb Z_2$ is by “forgetting” the glide symmetry.This construction can be extended to arbitrary symmetry groups $G$. Let, as above, $\mathrm{SPT}^d(G)$ be the Abelian group of $d$-dimensional $G$-protected SPT-phases. We have the following exact sequence of homomorphisms:
$$
\begin{equation}
0 \to\mathrm{SPT}^{d-1}(G)/2\mathrm{SPT}^{d-1}(G) \xrightarrow{\alpha}\mathrm{SPT}^d(\mathbb Z\times G) \xrightarrow{\beta} \{[c]\in\mathrm{SPT}^d(G)\colon 2[c]=0\}\to 0.
\end{equation}
\tag{3}
$$
Here, the homomorphism $\beta$ is generated by the forgetting map, and the homomorphism $\alpha$ is generated by the alternating fibres construction. As another example of application of the developed methods, we give a description of the Wigner–Dyson class A. In this case $d=3$, and the group $G=\mathrm{U}(1)$ corresponds to charge preservation. In the two-dimensional case, the fermionic phases of this type are classified by the first Chern class of the Bloch bundle over the Brilluoin zone (cf. [1]). The phases with odd Chern classes represent non-trivial elements in the first term of the exact sequence (3). Such phases may be pulled up with the help of the alternating fibres construction to the three-dimensional phases including the gliding. The three-dimensional phases thus obtained are characterized by a topological invariant $\kappa\in\mathbb Z_2$. The phases with non-trivial invariant $\kappa$ are known as Möbius phases. Using the exact sequence (3), we can obtain a complete classification of two- and three-dimensional $\mathrm{U}(1)$-protected fermionic SPT-phases. Namely,
$$
\begin{equation*}
\mathrm{SPT}^2(\mathrm{U}(1))\cong\mathbb Z\oplus\mathbb Z,\quad \mathrm{SPT}^3(\mathrm{U}(1))\cong0,
\end{equation*}
\notag
$$
where the first group $\mathbb Z$ is generated by the phase with zero Chern class, the second group $\mathbb Z$ corresponds to the phase $E_8$. Consider another Wigner–Dyson class AII. Here again $d=3$ and the group $G$ is generated by the $\mathrm{U}(1)$-symmetry and $T$-symmetry. In this case, the fermionic $G$-protected phases admit the following classification:
$$
\begin{equation*}
\mathrm{SPT}^2(G)\cong\mathbb Z_2,\quad \mathrm{SPT}^3(G)\cong\mathbb Z_2\oplus\mathbb Z_2\oplus\mathbb Z_2,
\end{equation*}
\notag
$$
here the group $\mathbb Z_2$ in the two-dimensional case is generated by the QSH-phase and the three groups $\mathbb Z_2$ in the three-dimensional case correspond to the topological insulators.
§ 3. Relation to $K$-theory3.1. Spectrally flat HamiltoniansConsider the Hamiltonians $H$ acting in the Hilbert space $\mathcal H$ and satisfying the gap condition $\operatorname{Ker}H=\{0\}$. Denote by $\Gamma$ the spectral flattening $\operatorname{sgn}H$ of a Hamiltonian $H$. In other words, $\Gamma$ is the grading operator belonging to the same phase as $H$, and whose spectrum consists of two points $\{+1,-1\}$. (A continuous deformation of the Hamiltonian $H$ to its spectral flattening $\Gamma=\operatorname{sgn}H$ can be constructed explicitly via the spectral theorem.) The space of grading operators $\operatorname{sgn}H$ in the Hilbert space $\mathcal H$, is denoted by $\operatorname{Grad}(\mathcal H)$. Two grading operators $\Gamma_1$, $\Gamma_2$ are called homotopic if they can be connected by a continuous path inside $\operatorname{Grad}(\mathcal H)$. The triple $(\mathcal H,\Gamma_1,\Gamma_2)$, where $\Gamma_1,\Gamma_2\in\operatorname{Grad}(\mathcal H)$, will be called the ordered difference between the grading operators $\Gamma_1$, $\Gamma_2$ or the corresponding Hamiltonians $H_1$, $H_2$. If, in this triple, $\Gamma_1$ is homotopic to $\Gamma_2$, then we say that $(\mathcal H,\Gamma_1,\Gamma_2)$ is trivial. This definition can be generalized by including into consideration the symmetry group $G$. Namely, denote by $\mathcal A$ the $C^*$-algebra on which the group $G$ acts by the representation $\alpha\colon G\to\operatorname{Aut}\mathcal A$. Let $W$ be a finitely generated $\mathcal A$-module and let $\operatorname{Grad}_\mathcal A(W)$ denote the space of $\mathcal A$-compatible grading operators acting in $W$. The above definitions, related to $\operatorname{Grad}(\mathcal H)$, immediately extend to the case $\operatorname{Grad}_\mathcal A(W)$. The direct sum operation provides $\operatorname{Grad}_\mathcal A(W)$ with the Abelian monoid structure. 3.2. Definition of $K$-functorThiang [11] proposed the following definition of $K$-functor. Denote by $K_0(\mathcal A)$ the quotient of the monoid $\operatorname{Grad}_\mathcal A(W)$ with respect to the equivalence relation determined by trivial triples. To be more precise, a triple $(W,\Gamma_1,\Gamma_2)$ is equivalent to a triple $(W',\Gamma'_1,\Gamma'_2)$ if there exist trivial triples $(V,\Delta_1,\Delta_2)$ and $(V',\Delta'_1,\Delta'_2)$ such that
$$
\begin{equation*}
(W\oplus V,\Gamma_1\oplus\Delta_1,\Gamma_2\oplus\Delta_2)= (W'\oplus V',\Gamma'_1\oplus\Delta'_1,\Gamma'_2\oplus\Delta'_2)
\end{equation*}
\notag
$$
in $\operatorname{Grad}_\mathcal A(W)$. The group $K_0(\mathcal A)$ is called the group of differences of $\mathcal A$-compatible gapped Hamiltonians. This group is Abelian, and $-[W,\Gamma_1,\Gamma_2]=[W,\Gamma_2,\Gamma_1]$. This group also satisfies
$$
\begin{equation*}
[W,\Gamma_1,\Gamma_2]+[W,\Gamma_2,\Gamma_3]=[W,\Gamma_1,\Gamma_3] \quad \text{in}\ K_0(\mathcal A).
\end{equation*}
\notag
$$
3.3. The symmetry groupsAs pointed out above, the main types of symmetries arising in the theory of solid bodies are the time reversal symmetry $T$, the charge conjugation symmetry $C$, and the chiral symmetry $S=CT=TC$. Denote by $G$ be symmetry group of the Hamiltonian. We equip $G$ with the following homomorphisms: 1) $\phi\colon G\to\{\pm1\}$ is responsible for the unitarity of the action of the element $g\in G$: this action is unitary if $\phi(g)=+1$, and is anti-unitary if $\phi(g)=-1$; 2) $c\colon G\to\{\pm1\}$ is responsible for the charge preservation: the action of $g\in G$ commutes with the Hamiltonian if $c(g)=+1$, and anti-commutes if $c(g)=-1$; 3) $\tau\colon G\to\{\pm1\}$ is responsible for the time direction preservation: the action of $g \in G$ preserves the time direction if $\tau(g)=+1$, and inverts it if $\tau(g)=-1$. A concrete example of a symmetry group $G$ is given by the so-called CT-group. This group is generated by the unit and three generators $T$, $C$, $S$, where 1) $\phi(T)=-1$, $c(T)=+1$; 2) $\phi(C)=-1$, $c(C)=-1$; 3) $\phi(S)=+1$, $c(S)=+1$. We are interested in graded representations of the CT-group and its subgroups. Let $\widehat T$, $\widehat C$ and $\widehat S$ be, respectively, the operators corresponding to the generators of the group $G$. There are the following possibilities: $\widehat T^2=\pm1$, $\widehat C^2=\pm1$ and $\widehat S=\widehat C\widehat T=\widehat T\widehat C$. The family of pairwise anti-commuting odd operators $\{\widehat C,i\widehat C,i\widehat C\widehat T\}$ generates the graded representation of the real Clifford algebra $\operatorname{Cl}_{r,s}$, where $r$ (respectively, $s$) is the number of the negative (respectively, positive) defined self-adjoint generators so that the representation of the full CT-group $G$ coincides with the graded $*$-representation of the Clifford algebra $\operatorname{Cl}_{r,s}$. Consider now the representations of the subgroups $A$ of the full CT-group. In the case of the subgroup $A=\{1,C\}$, we take for the odd generators of the representation the operators $\{\widehat C,i\widehat C\}$ with $\widehat C^2=\pm1$. They generate the graded representation of the Clifford algebras $\operatorname{Cl}_{0,2}$ and $\operatorname{Cl}_{2,0}$. In the case of the subgroup $A=\{1,S\}$, we necessarily have the relation $\widehat S^2=+1$, and so the obtained representation coincides with the graded representation of the complex Clifford algebra $\mathbb C\mathrm{l}_1$. In the case of the subgroup $A=\{1,T\}$, there are two possibilities for $\widehat T^2=\pm1$. The family of operators $\{i,\widehat T,i\widehat T\widehat\Gamma\}$, where $\widehat\Gamma$ is the grading operator, generates the (non-graded) representation of the Clifford algebra $\operatorname{Cl}_{1,2}$ for $\widehat T^2=+1$ and of the algebra $\operatorname{Cl}_{3,0}$ for $\widehat T^2=-1$. A complete classification of the groups $K_0(\mathcal A)$ for the CT-group and its subgroups was given in [11]. Another example of a symmetry group arising in the theory of solid bodies is the symmetry groups of zonal insulators of the form $G=\mathcal L\rtimes P$, where $\mathcal L$ is the group of translations of the lattice $\mathcal L$, and $P$ is a compact symmetry group acting at points of $\mathcal L$. In this case, the Pontryagin dual group $\widehat{\mathcal L}$ coincides with the Briilouin torus $\mathbb T^d$, and the finitely generated graded $\mathcal A$-module is equal to the graded $C(\mathbb T^d)$-module of sections of the Bloch bundle over $\mathbb T^d$ (see [1], [2]). If the group $G=\mathcal L\rtimes A$, where $A$ is one of the subgroups of CT-group so that the algebra $\mathcal A=\mathbb C\rtimes G$, then the group $K_0(\mathcal A)$ coincides, in the case of complex Clifford algebras, with $K^{-n}(\mathbb T^d)$, and, in the case of real Clifford algebras $\operatorname{Cl}_{r,s}$, this group coincides with $KR^{(r-s)(\operatorname{mod}8)}(\mathbb T^d)$.
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