Recent content by taishizhiqiu

@protonsarecool Thanks for explaining ##[a, \eta]_+ = 0##. However, I think it is not the correct way to just write ##\eta## out of the overlap in, for example, ##\bra{1} \eta \ket{1} = \bra{1} \ket{0} - \bra{1} \ket{\eta} = - (- \eta) \bra{1} \ket{1} = \eta ##. I am not sure I can draw a...

Thanks for the reply. ##a## is the annihilation operator of the state: ##a|1\rangle=|0\rangle##, ##\langle 0 | a = \langle 1 |## (I have already edited the original post for clarification). It is important to notice that Grassmann numbers are supposed to anticommute with annihilation and...

Most textbooks on fermionic path integral only briefly introduce Grassmann numbers. However, I want a more systematic treatment to feel comfortable about this approach. For illustration, I have several examples here. Example 1: Consider a system with only one state, how to calculate ##\langle...

I don't understand this. Can you show me an example?

OK, I finally learned spontaneous symmetry breaking. However, I am stilled confused. Supposing two local minimum in ferromagnets, time reversal symmetry will transform one into another, both of which are ground states of the Hamiltonian. Thus, it seems that ##T## is actually commutable with...

Thanks for the suggestion.

Oh, yeah. I am sorry, ##\psi_{nk}##s do live in the same Hilbert space. However, in berry connection, we only use ##u_{nk}## and they do not satisfy PBC.

Hey, they satisfy different boundary conditions: ##\psi_{k}(x+R)=e^{ikR}\psi(k)##

So, in the original question, I said wave functions (##\psi_k##)can be identified at opposite edges of brillouin zone. However, since ##\psi_k## in different points of brillouin zone do not live in the same Hilbert space, I don't know whether we can define berry phase using ##\psi_k##.

Oh, I am sorry. You are right, but the question remains.

What's the difference between your equation and mine? They look the same!

Recently, topological concepts are popular in solid state physics, and berry connection and berry curvature are introduced in band theory. The integration of berry curvature, i.e. chern number, is quantized because Brillouin zone is a torus. However, I cannot justify the argument that...

I am told that in ferromagnets, time reversal symmetry is broken. However, I don't know any hamiltonian terms in solid that can break time reversal symmetry. So is there a hamiltonian term I don't know or is there any subtlety in ferromagnets?

Why x is not a single-valued operator will lead to breakdown of the claim in my original post? After all, we can define x to be between ##-L/2## and ##L/2##.

According to bloch theorem, wave function in crystals should be like ##\psi_k(x)=e^{ikx}u_k(x)##, where ##u_k(x+a)=u_k(x)## and ##a## is lattice constant. So ##\langle u(k)|\partial_k|u(k)\rangle## should be something like ##\int u^*_k(x)\partial_k u_k(x)dx##, although it doesn't make sense...