Zero-Bias Anomaly in Disordered Wires

Abstract

We calculate the low-energy tunneling density of states $\ensuremath{\nu}(\ensuremath{\epsilon},T)$ of an $N$-channel disordered wire, taking into account the electron-electron interaction nonperturbatively. The finite scattering rate $1/\ensuremath{\tau}$ results in a crossover from the Luttinger liquid behavior at higher energies, $\ensuremath{\nu}\ensuremath{\propto}{\ensuremath{\epsilon}}^{\ensuremath{\alpha}}$, to the exponential dependence $\ensuremath{\nu}(\ensuremath{\epsilon},T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0)\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}{\ensuremath{\epsilon}}^{*}/\ensuremath{\epsilon})$ at low energies, where ${\ensuremath{\epsilon}}^{*}\ensuremath{\propto}1/(N\ensuremath{\tau})$. At finite temperature $T$, the tunneling density of states depends on the energy through the dimensionless variable $\ensuremath{\epsilon}/\sqrt{{\ensuremath{\epsilon}}^{*}T}$. At the Fermi level $\ensuremath{\nu}(\ensuremath{\epsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0,T)\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\sqrt{{\ensuremath{\epsilon}}^{*}/T})$.