Helical edge magnetoplasmon in the quantum Hall effect regime

We present the microscopic treatment of edge magnetoplasmons (EMPs) for the regime of not-too-low temperatures defined by the condition $\hbar \omega_{c}\gg k_{B}T\gg \hbar v_{g}/2\ell_{0}$, where $v_{g}$ is the group velocity of the edge states, $\ell_{0}=\sqrt{\hbar /m^{\ast}\omega_{c}}$ is the magnetic length and $\omega_{c}$ is the cyclotron frequency. We find a weakly damped symmetric mode, named helical edge magnetoplasmon, which is localized at the edge states region for filling factors $\nu =1, 2$ and \textit{very strong dissipation} $\eta_{T}=\xi /k_{x}\ell_{T}\agt\ln (1/k_{x}\ell_{T})\gg 1$, where the characteristic length $\ell_{T}=k_{B}T\ell_{0}^{2}/\hbar v_{g}\gg \ell_{0}/2$ with $\xi $ being the ratio of the local transverse conductivity to the local Hall conductivity at the edge states and $k_{x}$ is the wave vector along the edge; here other EMP modes are strongly damped. The spatial structure of the helical edge magnetoplasmon, transverse to the edge, is strongly modified as the wave propagates along the edge. In the regime of \textit{weak dissipation}, $\eta_{T}\ll 1$, we obtain exactly the damping of the fundamental mode as a function of $k_{x}$. For $\nu=4$ and weak dissipation we find that the fundamental modes of $n=0$ and $n=1$ Landau levels (LLs) are strongly renormalized due to the Coulomb coupling. Renormalization of all these EMPs coming from a metal gate and air half-space is studied.