In a previously asked question, the answer seems to be talking about the half-integer spin and the spin-statistics theorem. But I want to consider the Fermi statistics in terms of an adiabatic process of the exchanging particles and the Berry phase associated with that.
To make my question concrete, suppose two Fermions are trapped in harmonic potential $V(x-X_1)$ and $V(x-X_2)$ and the ground state wavefunction for each trapped particle is given by $\langle x|X_1\rangle = \psi(x-X_1), \langle x|X_2\rangle=\psi(x-X_2)$. We always assume $\psi(x-X)$ is sufficiently localized around $X$ compared to $|X_1-X_2|$ so that $\langle X_1|X_2\rangle = 0$. The wavefunction for the two Fermions is given by $|\Psi(X_1, X_2)\rangle=(1/\sqrt{2})(|X_1\rangle|X_2\rangle - |X_2\rangle|X_1\rangle)$ so that it satisfies the Fermi statistics.
Then we can slowly change the parameters $X_1, X_2$ so that after some time the two particles are exchanged. At the end of this process, we should get a wavefunction $|\Phi\rangle$ proportional to $|\Psi(X_2, X_1)\rangle=-|\Psi(X_1, X_2)\rangle$.
I expected that the Berry phase calculation of this process would give a phase of $\pi$, corresponding to Fermi statistics. However, since the single-particle groundstate wavefunction of the harmonic oscillator is always real, the Berry connection for this process is always zero, and thus there is no Berry phase.
This seems to suggest that the Fermi statistics cannot be associated with the Berry phase, although people talk a lot about the anyon statistics and the Berry phase.
Can the Fermi statistics be understood in terms of the Berry phase? If so, which part of my argument above is wrong?
Edited(2024/08/29): my question seemed unclear, so let me clarify. My question is the following:
"When we move two fermions at $X_1, X_2$ slowly and exchange their position so that they will be at $X_2, X_1$, the wavefunction should be the same as before exchanging except a phase of $\pi$ due to the Fermi statistics. Can we understand this phase in terms of the Berry phase, given that the wavefucntion is antisymmetric with respect to the exchange of particle indices?"
My original question is a more detailed description of how this exchange is achieved. I wanted to calculate the Berry phase of this exchange process, but I couldn't get the Berry phase of $\pi$.
