Boltzmann transport theory for many-body localization

We investigate a many-body localization transition based on Boltzmann transport theory. Introducing weak-localization corrections into a Boltzmann equation, Hershfield and Ambegaokar rederived the Wolfle-Vollhardt self-consistent equation for the diffusion coefficient [Phys. Rev. B 34, 2147 (1986)]. We generalize this Boltzmann equation framework, introducing electron-electron interactions into the Hershfield-Ambegaokar Boltzmann transport theory based on the study of Zala-Narozhny-Aleiner [Phys. Rev. B 64, 214204 (2001)]. Here, not only Altshuler-Aronov corrections but also dephasing effects are taken into account. As a result, we obtain a self-consistent equation for the diffusion coefficient in terms of the disorder strength and temperature, which extends the Wolfle-Vollhardt self-consistent equation in the presence of electron correlations. Solving our self-consistent equation numerically, we find a many-body localization insulator-metal transition, where a metallic phase appears from dephasing effects dominantly instead of renormalization effects at high temperatures. Although this mechanism is consistent with that of recent seminal papers [Ann. Phys. (NY) 321, 1126 (2006); Phys. Rev. Lett. 95, 206603 (2005)], we find that our three-dimensional metal-insulator transition belongs to the first-order transition, which differs from the Anderson metal-insulator transition described by the Wolfle-Vollhardt self-consistent theory. We speculate that a bimodal distribution function for the diffusion coefficient is responsible for this first-order phase transition.