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Composite-particles (Boson, Fermion) Theory of Fractional Quantum Hall Effect

A quantum statistical theory is developed for a fractional quantum Hall effects in terms of composite bosons (fermions) each of which contains a conduction electron and an odd (even) number of fluxons. The cause of the QHE is by assumption the phonon exchange attraction between the conduction electron ("electron", "hole") and fluxons (quanta of magnetic fluxes). We postulate that c-fermions with \emph{any} even number of fluxons have an effective charge (magnitude) equal to the electron charge $e$. The density of c-fermions with $m$ fluxons, $n_ϕ^{(m)}$, is connected with the electron density $n_{\mathrm e}$ by $n_ϕ^{(m)}=n_{\mathrm e}/m$, which implies a more difficult formation for higher $m$, generating correct values $me^2/h$ for the Hall conductivity $σ_{\mathrm H}\equiv j/E_{\mathrm H}$. For condensed c-bosons the density of c-bosons-with-$m$ fluxons, $n_ϕ^{(m)}$, is connected with the boson density $n_0$ by $n_ϕ^{(m)}=n_0/m$. This yields $σ_{\mathrm H}=m\,e^2/h$ for the magnetoconductivity, the value observed of the QHE at filling factor $ν=1/m$ ($m=$odd numbers). Laughlin's theory and results about the fractional charge are not borrowed in the present work.