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It depends on what symmetry class you are interested in. Spin Chern numbers are defined when you have U(1) spin rotation around a fixed axis. But for TI, there is only time-revesal symmetry (together …

I don't know what is a "p+ip insulator". There is indeed p+ip superconductor, which belongs to the class D and characterized by an integer invariant, the Chern number.

It is not true that if $\mathbf{k}\neq -\mathbf{k}$ the matrix element $\langle u_i(\mathbf{k}|T|u_j(\mathbf{k})\rangle$ vanishes. Remember that $u(\mathbf{k})$ are Bloch wavefunctions, which are eige …

Here is a derivation: Let me focus on the sum in the expression of $\sigma_{xy}$. First notice that $\langle \alpha|(\partial_{k_x}|\beta\rangle)=-(\partial_{k_x}\langle \alpha|)|\beta\rangle$ because …

The statement cited is misleading. If the system is bosonic, and no symmetry is considered, then indeed all gapped Hermitian Hamiltonians are trivial. With symmetry there can be distinct symmetry-prot …

There are similar topological invariants for band structures in one dimension, but an important difference is that these invariants always require some symmetry in the band Hamiltonians, for example p …

Q1: Consider the example of a Haldane phase protected by time-reversal symmetry. There is a spin-1/2 on the edge, and to polarize it (so the system is gapped and the ground state is unique) one must b …

In either cases the bulk is superconducting. I don't understand why you ask whether it is p-wave or Cooper pair. In this context, "p-wave" always means p-wave pairing, so always a superconductor to be …

The difference is obvious: In the second figure, the blue and red line connect the valence and conduction bands. These are actually surface states. So regardless of where your chemical potential lies, …

When the Chern numbers of the two bands go from $(1,-1)$ to $(-1,1)$, the Chern number of the occupied band changes sign. Even though the number of chiral edge modes is the same, they have opposite ch …

If we define $\mathcal{T}=i\sigma_y K$ where $K$ is complex conjugation, i.e. $\mathcal{T}\psi_\uparrow \mathcal{T}^{-1}=\psi_\downarrow, \mathcal{T}\psi_\downarrow \mathcal{T}^{-1}=-\psi_\uparrow$, …

To have time-reversal symmetry, there needs to be a unitary operator $U$ such that $ U\sigma^x U^{-1}=-\sigma^x, U\sigma^y U^{-1}= \sigma^y, U\sigma^z U^{-1}=\sigma^z$. These follow from $T=UK$, and I …

There is no fundamental difference between the signatures found in the two works (Kouwenhoven and Yazdani). Both are tunneling spectroscopy, which roughly measures whether there is a zero-energy singl …

It seems that you thought $\Phi_0$ is a trivial SPT while $\Phi$ is nontrivial. This does not make sense without defining the symmetry transformation. The fact that $\Phi=U\Phi_0$ means both are produ …

This paper https://arxiv.org/abs/1904.05491 proves that there can not be any net energy current in equilibrium state on a lattice system, which implies that left-moving and right-moving modes must bal …