Symmetry protected Z2-quantization and quaternionic Berry connection with Kramers degeneracy

As for a generic parameter dependent hamiltonian with the time reversal (TR) invariance, a non Abelian Berry connection with the Kramers (KR) degeneracy are introduced by using a quaternionic Berry connection. This quaternionic structure naturally extends to the many body system with the KR degeneracy. Its topological structure is explicitly discussed in comparison with the one without the KR degeneracy. Natural dimensions to have non trivial topological structures are discussed by presenting explicit gauge fixing. Minimum models to have accidental degeneracies are given with/without the KR degeneracy, which describe the monopoles of Dirac and Yang. We have shown that the Yang monopole is literally a quaternionic Dirac monopole. The generic Berry phases with/without the KR degeneracy are introduced by the complex/quaternionic Berry connections. As for the symmetry protected $\mathbb{Z}_2$ quantization of these general Berry phases, a sufficient condition of the $\mathbb{Z}_2$-quantization is given as the inversion/reflection equivalence. Topological charges of the SO(3) and SO(5) nonlinear $\sigma $-models are discussed in their relation to the Chern numbers of the $CP^1 $ and $HP^1$ models as well.


Introduction
Topological numbers have been important in physics, especially in quantum phenomena. They give a conceptual foundation of quantizations for various elementary degrees of freedom such as charges, fluxes, vortices and monopoles [1,2]. One of the milestones of the emerging topological numbers is a quantization of the Hall conductance where a response function is directly related to the topological quantum number as the first Chern number [3]- [7]. Its fundamental physical meaning has become clear by introducing an idea of the geometrical concept, which is known as the Berry connection today [8]. For quantum Hall (QH) states, the bulk is gapped and does not have any characteristic symmetry breaking. It results in the absence of a local order parameter and no low-energy excitations as Goldstone bosons. A class of such featureless systems is the (gapped) quantum liquid and the spin liquid. A possible effective field theory for gapped quantum liquids is topological field theory, where topological quantities play a central role. Then a corresponding new idea to describe the system is topological order [9,10]. It should be compared with the standard Ginzburg-Landau-Wilson scenario, where a local field theory to describe the fluctuation of a local order parameter is essential. As for a bulk topological ordered state, the degeneracy of the ground state depends on the topology of the physical space [9]. However, there were not so many quantities to describe the topological order. As an extension of the success for the QH state, we have successfully used the Berry connections and related topological quantities for characterization of the topological ordered states [11]- [17]. Also note that although the bulk QH state is featureless, the system with boundaries has characteristic localized states as the edge states [18,19]. Extending this observation, we propose an idea of the bulk-edge correspondence 3 which says that although the bulk is gapped and only characterized by topological quantities, there exist characteristic boundary states that reflect the topologically nontrivial bulk for the system with boundaries [7,20]. This 'bulk-edge correspondence' seems to be a universal feature of topological ordered states such as QH states, quantum spins [13,17,21], graphene [22], photonic crystals [23], cold atoms [24], characterization of localizations [25] and quantum spin Hall (QSH) systems [26]- [28].
The QSH state is an analogous state to the QH state, but it respects time reversal (TR) symmetry with the help of spins [26,27]. Then, it is natural that the Berry connection and TR invariance play fundamental roles. There have been substantial amounts of works on the topic [25]- [39]. In this paper, we present a self-contained description of Berry connections and related topological quantities with/without Kramers (KR) degeneracy. Especially we focus on its quaternioninc structure. Quaternions are fundamental in the description of the TR invariant system with KR degeneracy, which was first pointed out by Dyson a long time ago [29,30,40,41]. There is more than an analogy between the system with/without KR degeneracy. One can make a mapping between them by replacing the complex number by the quaternions [41]. We explicitly demonstrate this for topological quantities by introducing canonical minimum models, which are related to monopoles and accidental degeneracies.
As for topological quantities, there can be two classes. One includes topological invariants by their definition. Quantization for them is automatically guaranteed only by stability and a regularity of the Berry connections. Examples are Chern numbers, winding numbers and the Pontrjagin index. Additionally, we introduce a new class of quantized quantities as a generalization of the Z 2 Berry phase [13], which is geometrical. However, as for the quantization, one needs an additional symmetry requirement. We give a sufficient condition for this symmetry-protected Z 2 -quantization.
As for the application of a gauge-invariant description of Chern numbers, a relation between Chern numbers and topological charges of the SO(3) and SO(5) nonlinear σ -models is also presented.

TR and quaternions
Let us first introduce a quaternion notation for a TR-invariant bi-linear system [40]. Introducing where H mn is a 2 × 2 complex matrix and c n (n = 1, . . . , N ) is a spinor, a pair of fermion annihilation operators (summation over doubled indexes is assumed and n = 1, . . . , N ). Further let us impose an invariance under the TR operation for the Hamiltonian H. Since operates as c nσ → (−) (σ −1)/2 c n−σ (↑ = +1 and ↓ = −1, c n↑ → c n↓ and c n↓ → −c n↑ ) and taking a complex conjugate K, we have JH * , that is, H mn is identified as a quaternion H h mn by a standard equivalence which is the KR degeneracy. Its orthogonality, ψ † ψ = 0, is trivial here (generically, there are N KR pairs, = 1, . . . , N ). Let us write this KR pair as A linear canonical transformation of the fermions {c n } → {d }, c n = U n d , which is consistent with the TR invariance (written in {d }), requires that the 2 × 2 matrix U n does commute with the TR, that is,J U * n J = U n ∼ = u H n ∈ H. Supplementing the unitarity of this matrix U † U = UU † = E 2N , (U ) n = U n , U ∈ U (2N , C), which guarantees the fermion anticommutation relations of the {d σ }'s,

Quaternionic structure of the many-body system with KR degeneracy
The quaternionic structure introduced in section 2 is directly extended to the Fock space of the fermion many-body states as far as the total number of particles is conserved, since the TR operation commutes with the Sp(N ) unitary transformation among the fermion spinors {c † n } → {d † n } and the TR operation , c i↑ → c i↓ , c i↓ → −c i↑ and taking the complex conjugate, has a basis independent meaning. Then, it is also applicable for the S = 1 2 quantum spins by the standard representation S i = 1 2 c † i σ c i (extension to the general spins is trivial by introducing Hund coupling). Now let us consider a TR-invariant many-body Hamiltonian H, [H, ] = 0. When the state | is an eigen state of the Hamiltonian, its TR pair | = | is also an eigenstate. As mentioned before, we assume that the Hamiltonian preserves the total fermion number. Then one may discuss an M-particle sector separately. The TR operation for this M-particle sector then satisfies 2 |ψ = (−) M |ψ .

5
Let us further assume that the number of total fermions ( 1 2 spins) M is odd to have KR degeneracy. Then we have the following fundamental relation: A generic M-particle state is spanned by the Fock basis as where |O(i) and |E(i) represent a basis with an odd (even) number of spin-up fermions: They are orthonormalized as where i = 1, . . . , D F is a label of the Fock states. Since the total number of particles is odd, the basis with even up spins |E(i) is given by that of the odd as Therefore, one has (also it is confirmed directly) As for the generic state |ψ , the TR operation is given as Using this setup, one can directly extend the discussion in section 2. As for the eigenstate the KR multiplet of the many-body state is given as The orthogonality of the KR pair is also trivial. Similar to the discussion in section 2, we identity the KR multiplet as a single state of the quaternion as Then all of the discussion is trivially transformed into a discussion of the many-body states. For example, the quaternionic Berry connection for the many-body state is defined as a H = (ψ H ) † dψ H . All of the discussion in the paper can be applicable to the many-body system. Applications for electronic systems with electron-electron interaction will be given elsewhere.

Minimum dimensions for nontrivial Berry connections
To have a nontrivial topological structure with the Berry connection generically, there are some requirements for the dimension of the parameter space D, which we describe here. Let us first start from a generic consideration of the normalized m-dimensional multiplet = ( 1 , . . . , m ), † = E m and the corresponding m-dimensional non-Abelian Berry connection A = † d = † ∂ µ dx µ , which transforms under a gauge transformation g = g, g ∈ U (m), as A g = g −1 Ag + g −1 dg [8,42]. The nth Chern number C n of this connection is defined as where M 2n is a 2n-dimensional manifold without boundaries ∂M 2n = 0 [43,44]. Although the field strength F gets modified by the above gauge transformation as F g = g −1 Fg, the Chern number is invariant. As for the explicit discussion of the Berry connection, Zumino's generic construction of the topological quantities is quite useful. We summarize a part of them which we require in this paper [43,44]. They read as The transformation properties are given as where α 2 = Tr(Adgg −1 ). Although Zumino's construction is general for TrF n = dω 2n−1 (A), we just need n = 1 and 2, which one can explicitly confirm by a direct calculation.
The Chern number is gauge invariant and is explicitly given by the gauge-invariant projection P = † . It is given for the first Chern number [45], but is also given for the higher ones. By taking a differential of P, we have dP = d † + d † . Then the following direct calculation gives a useful formula for gauge-invariant quantities as It obeys the following observation: Then the Chern number is written as an explicit gauge-invariant form as As for the TR-invariant system with KR degeneracy, we identify the multiplet of dimension 2M to the quaternionic one with dimension M as = ( 1 , . . . , M ) ∼ = ψ H . Then a gauge transformation ψ H g = ψ H g, g ∈ Sp(M) preserves the TR-invariant linear space spanned by ψ H . Now the quaternionic Berry connection a H = (ψ H ) † dψ H and corresponding field strength f H = da H + (a H ) 2 are defined as usual. Their transformation properties are also standard as The nth Chern number with even n, C n , is defined as (since C n is intrinsically integer, it suggests vanishing C n for odd n) In the following, we omit the symbol H and simply use the lower character for quaternionic notation if the situation is clear.
Since the multiplet and the Berry connection have a gauge freedom, one needs to fix it for the connection to be well defined. As for the generic multiplet without KR degeneracy, the gauge is specified by an arbitrary but given multiplet as = P N −1/2 , where P is a gauge-independent projection, and the normalization matrix N = † P , which is also gauge invariant and semi-positive definite [11]. When one can use this single gauge over the whole parameter space, the Berry connection is trivial. Generically, however, the normalization matrix may have zero eigen values as det N ( ∃ x ) = 0. Then near this zero, x , this gauge is singular since one can not normalize. One needs to use the other gauge, say ψ , by taking . Since det N (x ) = 0, generically, one can express the projection by the multiplet explicitly as P = 8 One may write it as η = † when one does not need to specify the gauge. Now it is clear that the singularity is specified by Generically one does not have zeros when the dimension of the parameter space is too low and the Berry connection is trivial. To have a nontrivial topological structure, the dimension of the parameter space has to satisfy D D min = 2, since the condition to have the singularities is given by two real equations. A two-dimensional magnetic Brillouin zone to discuss the Hall conductance as the first Chern number is this minimum space where the singularities occur in points [6]. Note that the gauge transformation between the two gauges by taking and , = g , is explicitly given by The unitarity is confirmed as and g † g = E M similarly. As for a system with KR pairs, let us consider the simplest M = 1 case. Now starting from the gauge-invariant projection p into the degenerate KR space, the gauge is fixed by an arbitrary quaternion vector φ ∈ H N as (1)). Therefore, we do not need to specify the gauge for N (η φ ).
Near the singular point of this gauge, one needs to use the other gauge by taking φ . Then the condition of the vanishing norm N φ = N (η φ φ ) is expressed as It clearly shows that the singularity may occur in the parameter space of the dimension D D KR min = 4. The gauge transformation is also given as When the dimension of the parameter space is less than this minimum dimension, one can generically take a single patch over the whole parameter space. Since the base space to define the Chern numbers is assumed to be without boundaries, this implies that the Chern number is vanishing for dim M 2n = 2n < D KR min = 4. Then the natural quantities to have nontrivial topological structure by the Chern numbers are C 1 for the generic case and C 2 for the system with KR degeneracy.
Also note that the normalization of the KR pair in quaternion notation . This implies that the first Chern number vanishes, which is consistent with the generic argument [29,46]. Now let us focus on the second Chern number with KR degeneracy.

Degeneracies and monopoles with/without KR degeneracy
As pointed out by Berry, the generic degeneracy of a complex Hamiltonian has a co-dimension d C = 3 [8]. That is, the minimum Hamiltonian (N = 2) to describe the degeneracy (at E = Similarly, the system with KR degeneracy has a co-dimension d H = 5, as pointed out by Avron et al [29,46]. Then the minimum model (N = 2, E = Tr H H = 0) is realized by the following quaternionic Hermite Hamiltonian: parameters of the minimum model with KR degeneracy. The above observation suggests a strong analogy between systems with and without KR degeneracy, which we pursue in this paper. There is also topological correspondence, as discussed below (see figure 1). Actually, it is more than analogy and there exists a mapping by R 3 → Q 5 and z(∈ C) → q(∈ H), as one can see. The origins of the parameter spaces R = 0 and Q = 0 give degeneracies that bring singularities for each of the Berry connections. They are the Dirac monopole [1] and the Yang monopole [27], [30]- [32], [47]- [49]. The Yang monopole is literally a quaternionic Dirac monopole up to its topological structure.

Dirac monopole and the first Chern number
Owing to a simple observation, H 2 C = R 2 E 2 , R = |R|, the energies of H C are ±R = |z| 2 + R 2 3 . Then degeneracy occurs at the origin in the three-dimensional R space R 3 . Away from this degeneracy, the eigenstate of the energy ±R subspace is well defined by the projection P ± = 1 2 (1 ± H C /R). As for the base manifold to define the first Chern number, for simplicity let us take the two-sphere S 2 = {R|R = 1} ⊂ R 3 as for M 2n , n = 1 (figure 1). Then the possible singularities of the Berry connection can be points on S 2 by the generic consideration before. When one considers a generic base space in R 3 , these singularities form lines, which correspond to Dirac strings [50]. The gauge-invariant projection into each eigen subspace is explicitly given as In the following, let us consider a positive energy subspace P = P + . Taking a gauge by the normalized state on S 2 (|z| 2 + R 2 3 = 1) is given as N = P N N Since this gauge is only singular at the south pole R 3 = −1, we can safely use for the north hemisphere S 2 N (R 3 0). For the south hemisphere, we need to use the other gauge, say, by taking Then the normalized state is given similarly as which is regular everywhere on the south hemisphere S 2 S (R 3 0). The gauge transformation, g C S N , between them, N = S g C S N , is given by the generic formula before as This is regular except for the north and south poles R 3 = ±1.
The first Chern number of the Berry connection is easily evaluated using these two gauges and the gauge transformation, is a winding number of the map from the one-sphere (circle) This winding number can be evaluated in several ways. Since it is invariant against a rotation in S 1 (g → e iθ g), we write it in a local coordinate near R 1 = 0 and R 2 = 1 as g = +i, dg = −dR 1 as Considering S 2 as a boundary of the solid sphere V 3 (∂ V 3 = S 2 ), naive application of the Stokes (Gauss) theorem, C 1 = V 3 dF, suggests that i 2π dF = −δ (3) (R) since dF = d 2 A = 0 as far as the Berry connection is well defined except at the origin. This is the Dirac monopole at the origin of the three-dimensional R space where the degeneracy of the generic complex Hamiltonian occurs [1].

Yang monopole as a quaternionic Dirac monopole
The discussion with KR degeneracy can be done analogously. Let us again start from a simple observation H 2 H = Q 2 E 5 , Q = | Q|, which implies that eigen energies of the KR multiplets are ±Q = ± |q| 2 + Q 2 5 , |q| = √ N (q) ∈ R, and the additional degeneracy to KR degeneracy occurs at the origin in the five-dimensional Q space R 5 (figure 1). A projection into the positive energy KR multiplet is defined as p = 1 2 (1 + H H /Q). Similar to the discussion above, let us take a foursphere S 4 = { Q|Q = 1} ⊂ R 5 as the base space M 2n (n = 2) to define the second Chern number C 2 . Then the generic singularities of the KR multiplet are again points on S 4 , which make lines in R 5 when one considers a generic four-dimensional surface as a base space ('Yang' strings). To be more specific, let us take a gauge by taking a quaternion vector with two components Then the normalized KR multiplet is given, in the north pole gauge (regular in the northern hemisphere S 4 N (Q 5 0)), as . This gauge is only singular at the south pole Q 5 = −1 on S 4 . The other gauge by also defines the multiplet (in the south pole gauge) . This is regular in the southern hemisphere S 4 S (Q 5 0). The gauge transformation between them is also calculated as Now let us calculate the second Chern number in quaternionic notation as is an equator, ω 3 (a) = T (ada + 2 3 a 3 ) and W S 3 (g H S N ) is the Pontrjagin number of the map S 3 → Sp(1) ∼ = S 3 that is a covering degree, which is intrinsically integer. Here we used S 3 dα 2 = ∂ S 3 α 2 = 0 since the gauge is regular on S 3 , which does not have boundaries. This Pontrjagin number is explicitly evaluated [51]. Since it is invariant for the change q → qξ , |ξ | = 1 that induces a rotation of S 3 , it is enough to evaluate it near q = 1 (q 0 = 1, q 1 = q 2 = q 3 = 0), where T (q −1 dq) 3 = 3!T (i H j H k H )dq 1 dq 2 dq 3 = −12dq 1 dq 2 dq 3 . Then we have C 2 ( Q) = W S 3 (g H S N ) = 1 24π 2 (−12 × 2π 2 ) = −1, where 2π 2 is a volume of S 3 .
Again writing S 4 as a surface of a five-dimensional solid sphere V 5 = { Q|| Q| 1}, ∂ V 5 = S 4 , one may write symbolically dT ( f 2 ) = −δ (5) ( Q) by a simple application of the Stokes away from the origin where the singularity exists. The origin of the five-dimensional Q space, Q = 0, is a singular point for the Berry connection due to the additional degeneracy (four-fold) and it induces the Yang monopole in five dimensions [47], which locates at Q = 0 (the charge is −1). This explicitly demonstrates that the Yang monopole is a quaternioninc Dirac monopole.

Chiral symmetry and topological stability of the doubled Dirac cones
For simplicity, we have assumed the two-sphere and the four-sphere as parameter spaces M 2 and M 4 . In a generic situation, let us consider the Chern numbers of models H C (R(x)) (x ∈ M 2 ) and H H ( Q(x)) (x ∈ M 4 ). Assuming that the energy gaps never collapse, images R(M 2 ) ⊂ R 3 and Q(M 4 ) ⊂ R 5 are deformed into spheres S 2 and S 4 without changing the Chern numbers. This is topological stability and these topological numbers are given by the covering degrees of the maps as [52] To have well-defined Chern numbers, the gap has to be always open. However, in some situations, the gap may collapse. Generically speaking, this is accidental (accidental degeneracy). In other words, one may need to fine-tune physical parameters that occur at a quantum critical point. By imposing some restriction by symmetry, the situation may change and the gap closing has topological stability. Let us here impose 'chiral symmetry' and restrict the parameter space. The chiral operator in the minimum model is given by = σ 3 , 2 = 1. The Hamiltonians of the minimum models satisfy {H C , } = 2R 3 , {H H , } = 2Q 5 . That is, the equators (S 1 and S 3 , respectively) are characterized as chiral symmetrical spaces  figure 2). These Dirac cones are generically topologically stable, that is, stable against small but finite perturbation, since the images R(M 2 ) ⊂ R 2 and Q(M 4 ) ⊂ R 4 . The topological stability of Dirac cones in two/four dimensions is discussed in relation to graphene and the Nielsen-Ninomiya theorem [53]- [55].
Here we comment on our quaternionic description of the Yang monopole to SU(2) gauge theory. The gauge structure of the TR-invariant system is symplectic and the gauge group of the simplest situation is Sp(1), which is mathematically equivalent to SU (2). Therefore, the results of the Sp(1) gauge structure we have described are directly related to SU(2) gauge theory. Our description of the Yang monopole is closely related to the instanton of the SU(2) gauge theory [30], [47]- [49], [51]. By the mathematical equivalence Sp(1) = SU(2), the quaternionic description of the Yang monopole here can be understood as a re-description of a known SU(2) gauge structure. However, our parallel discussion of the U(1) and Sp(1) gauge structures without/with KR degeneracy simplifies the physical understanding of the SU(2) gauge structure, especially with our explicit gauge fixing and treatment of the gauge singularities of the Berry connections. Finally, note that the SU(2) gauge structure originating from the Sp(1) gauge structure of the TR-invariant system is not related to the conservation of spins. The spin is not a conserved quantity in most of the TR-invariant system such as the QSH systems with spin-orbit interaction. The multi-component structure by the spin is necessary to have the nontrivial Sp(1) gauge structure, but the spin itself is not a conserved quantity.

Symmetry-protected Z 2 -quantization
As discussed, Chern numbers are gauge invariant and intrinsically integer and apparently have topological stability. This implies that the quantization is stable for small but finite perturbation for the Hamiltonian. This topological stability does play a crucial role, for example, in the theory of quantized Hall effects. Note that the dimension of parameter space to define Chern numbers is necessarily even. The winding number W S 1 and the Pontrjagin index W S 3 are also topological and defined for spaces with odd dimensions. In odd dimensions, one may also define quantized quantities if one imposes additional symmetry requirements. They are generalizations of the Berry phase and generically gauge dependent as a phase of the wavefunction [8,13]. This implies that these quantities are essentially quantum mechanical and do not have any classical correspondents. They also have a fundamental advantage in the identification of topological ordered states [12,13]. An example is a Z 2 -quantization of the Berry phase for the TR-invariant system without KR degeneracy 2 = 1 [13]- [16]. The focus of this section is to extend the idea and supply a generic condition for the Z 2 -quantization. Now let us start by defining generic Berry phases γ 1 (A) and γ 3 (a) as where γ 1 (A) is for a generic system (without degeneracy M = 1) and γ 3 (a) is for a system with KR degeneracy using quaternionic notation. Note here that the same topological quantity by the integral of the Chern-Simons form is discussed in several papers [28,37,56]. They are not invariant for the gauge transformation A g = g −1 Ag + g −1 dg (g ∈ U (1)) and a g = g −1 ag + g −1 dg (g ∈ Sp(1)). Therefore, they are not well defined (as they are) but are gauge independent and well defined in modulo 1 as [13] since the gauge dependence is due to a nontrivial large gauge transformation. These contributions are topological and integers as W S 1 (g) ∈ Z and W S 1 (g) ∈ Z [13] as far as the gauge transformations are regular over S 1 and S 3 . A phase factor of the Berry phase e i2γ (γ = 2πγ 1 ) is gauge independent and is a well-defined quantity (observed as a geometrical phase), but the phase γ itself is gauge dependent [8,13]. Generically speaking, these generic Berry phases γ 1 and γ 3 may take any real values even in modulo 1. However they can be quantized when the system obeys some symmetry requirement, which we discuss below.

Z 2 -quantization of a TR-invariant system without KR degeneracy
Let us first consider a TR-invariant system without KR degeneracy [13]- [16]. This is realized for quantum systems with an even number of quantum spins. Since the Hamiltonian H does commute with the TR operator , which is anti-unitary, [H, ] = 0, Owing to the uniqueness of the state, ψ and ψ are only different in phase, that is, the corresponding Berry connections A = ψ † dψ and A = (ψ ) † dψ are transformed into each other by some gauge transformation g, A = g −1 A θ g + g −1 dg, as γ 1 (A) ≡ γ 1 (A ) mod 1, since the gauge transformation is, generically, well defined on the parameter space x ∈ S 1 . Also the TR operation for the many-spin state ψ is written as = U K with some parameter-independent unitary transformation U . Then the Berry connection is written as Therefore, allowed values of the Berry phase are restricted to two as γ 1 (A) = 0, 1 2 . This is the Z 2 -quantization of the Berry phase for the unique TR-invariant state.
In most of the application [13]- [16], we have used a U (1) twist e iθ , θ : 0 → 2π as a parameter. In this case, the condition of the Z 2 -quantization is reformulated from a more generic point of view (see below).

Z 2 -quantization by inversion/reflection equivalence
Similar quantizations protected by symmetry occur for the generic Berry phases γ 1 and γ 3 when the system (with parameter) satisfies the following inversion/reflection equivalence. Inversion/reflection equivalence implies the existence of the unitary operator U I or U R , is a complex or a quaternionic Hamiltonian for x ∈ S 1 or x ∈ S 3 , respectively. Inversion in parameter space is defined as x I = −x and reflection is one of the following three: As for the x ∈ S 1 case, reflection is the same as the inversion. This is a sufficient condition for Z 2 -quantization.
Although we use the quaternion notation with the reflection below (with KR degeneracy), it is also true for the inversion and the complex cases. The isolated KR multiplet, denoted as ψ( where ψ R (x) = U R ψ(x R ). Since the unitary equivalence between H (x) and H (x R ) implies that all of the eigenvalues are equal to each other, we may generically assume E(x R ) = E(x) supplementing a unitary transformation of reshuffling the KR degenerated eigenspaces. Now, as for the isolated eigenspace of the KR multiplet, ψ(x) and ψ R (x) are different just in Sp(1) phase, which implies that the corresponding Berry connections are gauge equivalent, Then the generic Berry phases satisfy γ 1 (A R ) ≡ γ 1 (A) and γ 3 (a R ) ≡ γ 3 (a) in modulo 1. Here note that γ 1 and γ 3 are defined by the integral over the odd dimensional spaces S 1 and S 3 . Therefore, the generic Berry phases γ 1 and γ 3 are odd by the inversion/reflection of parameter spaces S 2 and S 3 , . Therefore, we have a Z 2 -quantization of the Berry phases as

Chiral symmetry for minimum models
The chiral symmetry of the minimum models discussed before is a typical example of systems with inversion equivalence because the anti-commutators between the unitary operator = where the models are defined on the equators as R ∈ S 1 and Q ∈ S 3 . This is what we need for the Z 2 -quantization of γ 1 and γ 3 . We explicitly confirm it by direct calculations below. Let us first consider a generic case without KR degeneracy. In the north pole gauge, the multiplet at the equator R 3 = 0, |z| = 1 is given as

Then we have
If we take the south pole gauge, we have This implies γ 1 (A S ) = + 1 2 ≡ γ 1 (A N ) (mod 1), which is consistent with the general consideration and Z 2 -quantization.

Reflection and TR invariance without KR degeneracy
The quantization of the Z 2 Berry phase discussed in section 6.1 [13] can be considered as the quantization due to the reflection equivalence discussed in section 6.2 when the parameter introduced is the U (1) twist e ix and the other parameters are all real. It is simply due to the following observation of TR invariance: where U is a unitary operator to change c iσ → (−) (1−σ )/2 c i−σ for the fermions and the spins . This is just the inversion or reflection equivalence as discussed in section 6.2.

Topological charge and nonlinear σ -models
Finally in this section, let us calculate topological charges of the nonlinear σ -model [30,49], [57]- [60] as applications of the gauge-invariant forms of Chern numbers C 1 and C 2 in section 4. [57,58,61] Let us start by considering an x-dependent two-component normalized state

Topological charge without KR degeneracy
which defines S 3 . Then the following three real quantities, n 1 , n 2 and n 3 , are defined as a CP 1 representation of n i (i = 1, 2, 3): where σ a = σ a and the projection, P(x) = † , into the subspace spanned by (x) is introduced.
Since TrP = † = 1, the projection is expanded as P = 1 2 E 2 + P i σ i . The coefficients are given as P i = Tr P 1 2 σ i = 1 2 n i . Now we have rewritten P = 1 2 (E 2 + n i σ i ) = 1 2 (E 2 + H C (n)) and H C = n · σ = 2P − E 2 . Then H 2 C = 4P − 4P + E 2 = E 2 = n i σ i n j σ j = n i n i + i< j n i n j {σ i , σ j } = |n| 2 E 2 . This implies |n| 2 = 1. Therefore, the state can be considered as a positive energy eigenstate of H C by identifying n = R. It makes a CP 1 representation of the SO(3) nonlinear σ -model.

Topological charge with KR degeneracy
Tr T 3 p Tr Since TrTp = T ψ † ψ = 2, the projection is expanded as p = 1 2 E 2 + p a a . The coefficients are given as p a = Tr T ( a p)/4 = n a /2. Now we have rewritten p = 1 2 (E 2 + n a a ) = 1 2 (E 2 + H H (n)) and H H = n · = 2 p − E 2 . Then H 2 H = 4 p − 4 p + E 2 = E 2 = n i i n j j = n i n i E 2 + i< j n i n j { i , j } = |n| 2 E 2 . This implies |n| 2 = 1. Therefore, the state ψ can be considered as a positive energy KR multiplet of H H by identifying n = Q. This establishes the relation for the HP 1 representation of the SO(5) nonlinear σ -model.
Again using this decomposition of the five vectors n, let us discuss the topological charge Q H following [30,59,60]: J σ τ ω = N −1 µνλκρσ τ ω abcde n a ∂ µ n b ∂ ν n c ∂ λ n d ∂ ρ n e , Q H = dx 1 dx 2 dx 3 dx 4 J 567 = N −1 dx 1 dx 2 dx 3 dx 4 µνλρ567 abcde n a ∂ µ n b ∂ ν n c ∂ λ n d ∂ ρ n e This is again the direct relation between the second Chern number of the HP 1 model and the topological charge of the SO(5) nonlinear σ -model.