13.1. Review of Geometric Phases

13.1.1. Berry Phase

Adiabatic Principle

In quantum mechanics, adiabatic principle states that given a system Hamiltonian \(H(\xi(t))\) which depends on a parameter \(\xi(t)\) and varies slowly on time, the system will stay on the instantaneous eigenstate if it starts with such an eigenstate,

\[\ket{\Psi_n(t)} = \exp\left( - \frac{i}{\hbar} \int_0^t E_n(t') dt' \right)\ket{\Psi_n(0)},\]

where \(E_n(t)\) is the instanteneous eigen value, i.e.,

\[H(t) \ket{\Psi_n(t)} = E_n(t) \ket{\Psi_n(t)}.\]

The adiabatic approximation is, however, not always right. In general, we can always assume a general state to be

\[\ket{\Psi_n(t)} = c_n(t) \exp\left( - \frac{i}{\hbar} \int_0^t E_n(t') dt' \right)\ket{\Psi_n(0)},\]

as this form is pluged back to the Schrodinger equation, we find it has a form of phase

\[c_n(t) = c_n(0) e^{i\gamma_n(t)},\]

where

\[\gamma_n = i \int_0^t \bra{\Psi_n(t')} \frac{d}{dt'} \ket{\Psi_n(t')} dt'.\]

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 \(H(\xi(t))\). And a loop integral becomes tricky since poles are to be considered.

13.1.2. 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

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