Abstract
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Highlights
If we focus on a single bandgap, topological physical systems[1-6] are usually classified by Abelian groups, with the prime example being the ten-fold classification[7,8] of Hermitian topological insulators and superconductors
Non-Abelian topological charges in four-band models Here, for simplicity we focus on a four-band PT symmetric system
We will see their difference more explicitly by extending the 1D Hamiltonian onto a 2D plane (Figs. 1d and e); The second category consists of six conjugacy classes which can be distinguished using single band Zak phase arguments, regarding which two of the four bands have Zak phases of π; In the last category, all eigenstates flip their sign after k runs across the 1D FBZ
Summary
We briefly recall some facts about rotations in the four-dimension[23-25]. For each rotation R, there is at least one pair of orthogonal 2-planes (the 2-planes are dubbed as invariant planes) - A and B which are invariant under the rotation R and span the four-dimensional space, i.e. for any a⃗ ∈ A and bŒ⃗ ∈ B we have a⃗ ⊥ bŒ⃗, Ra⃗ ∈ A and RbŒ⃗ ∈ B. When the two rotation angles satisfy |α| = |β|, the rotation R is called isoclinic rotation, where there are infinitely many pairs of orthogonal 2planes. Isoclinic rotations with αβ > 0 are denoted as left-isoclinic; those with αβ < 0 as right-isoclinic. In some ideal cases, +⁄− q&")% consists of purely left/right-isoclinic rotations, where there are infinitely many pairs of orthogonal 2-planes. The calculation is made by a surjective homomorphism, ρ: q ∈ SU(2) → R)P ∈ SO(3), whose kernel is {1, −1} indicting SU(2) is a double cover of SO(3). Left/right isoclinic rotations are represented by left/right multiplication of unit quaternions. A −b −c −d u RX%(r) = }bc da −ad −cbƒ ©yxa d −c b a z (S11) This is a left quaternion multiplication of r by qQ.
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