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We investigate a many-body localization transition based on a Boltzmanntransport theory. Introducing weak localization corrections into a Boltzmannequation, Hershfield and Ambegaokar re-derived the Wolfle-Vollhardtself-consistent equation for the diffusion coefficient [Phys. Rev. B {\bf 34},2147 (1986)]. We generalize this Boltzmann equation framework, introducingelectron-electron interactions into the Hershfield-Ambegaokar Boltzmanntransport theory based on the study of Zala-Narozhny-Aleiner [Phys. Rev. B {\bf64}, 214204 (2001)]. Here, not only Altshuler-Aronov corrections but alsodephasing effects are taken into account. As a result, we obtain aself-consistent equation for the diffusion coefficient in terms of the disorderstrength and temperature, which extends the Wolfle-Vollhardt self-consistentequation in the presence of electron correlations. Solving our self-consistentequation numerically, we find a many-body localization insulator-metaltransition, where a metallic phase appears from dephasing effects dominantlyinstead of renormalization effects at high temperatures. Although thismechanism is consistent with that of recent seminal papers [Ann. Phys. (N. Y).{\bf 321}, 1126 (2006); Phys. Rev. Lett. {\bf 95}, 206603 (2005)], we find thatour three-dimensional metal-insulator transition belongs to the first ordertransition, which differs from the Anderson metal-insulator transitiondescribed by the Wolfle-Vollhardt self-consistent theory. We speculate that abimodal distribution function for the diffusion coefficient is responsible forthis first order phase transition.

Boltzmann transport theory for many body localization