Zero-Bias Anomaly in Disordered Wires

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