Review of Geometric Phases ==================================== Berry Phase ------------------ .. admonition:: Adiabatic Principle :class: note In quantum mechanics, adiabatic principle states that given a system Hamiltonian :math:H(\xi(t)) which depends on a parameter :math:\xi(t) and varies slowly on time, the system will stay on the instantaneous eigenstate if it starts with such an eigenstate, .. math:: \ket{\Psi_n(t)} = \exp\left( - \frac{i}{\hbar} \int_0^t E_n(t') dt' \right)\ket{\Psi_n(0)}, where :math:E_n(t) is the instanteneous eigen value, i.e., .. math:: H(t) \ket{\Psi_n(t)} = E_n(t) \ket{\Psi_n(t)}. The adiabatic approximation is, however, not always right. In general, we can always assume a general state to be .. math:: \ket{\Psi_n(t)} = c_n(t) \exp\left( - \frac{i}{\hbar} \int_0^t E_n(t') dt' \right)\ket{\Psi_n(0)}, as this form is pluged back to the Schrodinger equation, we find it has a form of phase .. math:: c_n(t) = c_n(0) e^{i\gamma_n(t)}, where .. math:: \gamma_n = i \int_0^t \bra{\Psi_n(t')} \frac{d}{dt'} \ket{\Psi_n(t')} dt'. This phase is named after Berry thus Berry Phase. What is special about Berry phase is that it is not always be removed even it looks like a global phase. The reason is that this integral becomes a loop integral when we are dealing with a periodic behavior of Hamiltonian :math:H(\xi(t)). And a loop integral becomes tricky since poles are to be considered. Refs & Notes ------------------ 1. Mehta, P. (2009). Topological phase in two flavor neutrino oscillations. Physical Review D, 79(9), 096013. doi:10.1103/PhysRevD.79.096013