Energy Levels of Polarons in a Magnetic Field

Abstract

A variational method closely related to the intermediate coupling method of Lee, Low, and Pines is used to calculate the ground-state energy and low-lying excited states of the Fr\"ohlich Hamiltonian with a uniform time-independent magnetic field. The energy is calculated in a power series in $\frac{{\ensuremath{\omega}}_{c}}{\ensuremath{\omega}}$ to order ${(\frac{{\ensuremath{\omega}}_{c}}{\ensuremath{\omega}})}^{2}$, where ${\ensuremath{\omega}}_{c}$ is the cyclotron resonance frequency of the electron in the absence of electron-phonon interaction and $\ensuremath{\omega}$ is the frequency of the longitudinal optical phonons. It is shown that in the presence of electron-phonon interaction the energy of the $n\mathrm{th}$ magnetic level is no longer proportional to $n$ and that the effective mass for motion along the direction of the magnetic field is a function of $n$. The calculated variational energies approach the weak field result expected from the calculation of Lee, Low, and Pines (LLP) when $\frac{{\ensuremath{\omega}}_{c}}{\ensuremath{\omega}}\ensuremath{\rightarrow}0$, and in the weak coupling limit the ground-state energy becomes exact to order ${(\frac{{\ensuremath{\omega}}_{c}}{\ensuremath{\omega}})}^{2}$.