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.

Figure 1

Diffusion coefficient as a function of the disorder strength $\lambda =1/\left(\pi k_{F}l\right)$ and the external frequency $\omega \tau$ in the Wölfle-Vollhardt self-consistent theory for an Anderson metal-insulator transition. Here, $k_F$ is the Fermi momentum, $l$ ($\tau$) is the mean free path (time), and ${\lambda }_c=1/\sqrt{3\pi }$. Part (a) shows the diffusion coefficient as a function of disorder strength $\lambda /{\lambda }_c$ and external frequency $\omega \tau$. Part (b) shows the diffusion coefficient as a function of the disorder strength, where the thick red line denotes the dc diffusion coefficient and the thick blue (dashed blue) line denotes the real (imaginary) part of the ac diffusion coefficient at $\omega \tau =0.01$. Part (c) displays the real part of the ac diffusion coefficient as a function the external frequency, where the blue-thick (red-thick) line denotes it in the metallic, denoted by $M$ (insulating, denoted by $I$) phase of $\lambda /{\lambda }_c=0.7$ ($\lambda /{\lambda }_c=2$). Part (d) exhibits the imaginary part of the ac diffusion coefficient as a function the external frequency, where the blue-thick dashed (red-dashed) line expresses it in the metallic (insulating) phase of $\lambda /{\lambda }_c=0.7$ ($\lambda /{\lambda }_c=2$). For comparison, we also show the Drude conductivity as the thick black (dashed black) line.

Figure 2

The power-law behavior near the transition point, $\lambda ={\lambda }_c=1/\sqrt{3\pi }$ with $\omega \tau =0$. (a) The blue line is a log-log plot of $\mathrm{Re}\tilde{D}$ vs $\omega \tau$ with $\lambda ={\lambda }_c$. The red line is a fitting curve of the form ${\left(\omega \tau \right)}^x$, where $x\approx 0.330$. (b) The blue line is a log-log plot of $\mathrm{Re}\tilde{D}$ vs $\delta \lambda /{\lambda }_c$ with $\omega \tau =0$. The red line is a fitting curve of the form $\delta {\lambda }^x$ with $x\approx 0.998$. (c) The blue-dashed line is a log-log plot of $-\mathrm{Im}\tilde{D}$ vs $\omega \tau$ with $\lambda /{\lambda }_c=2$, which is in the insulating phase. The red-dashed line is a fitting curve of the form $\delta {\lambda }^x$ with $x\approx 1.000$.

Figure 3

(a) Log-linear plot of $\mathrm{Re}\tilde{D}$ vs $\delta \lambda /{\lambda }_c$ with three different values of $\omega \tau$, as shown in the inset. (b) Log-linear plot of $\mathrm{Re}\tilde{D}/{\left(\omega \tau \right)}^{y_D}$ vs $(\delta \lambda /{\lambda }_c)/{\left(\omega \tau \right)}^{y_{\delta \lambda }}$.

Figure 4

Electrical conductivity in terms of disorder strength $\lambda$. (a) The black line represents electric conductivity in the absence of electron correlations. The blue and red lines express the electric conductivity with interactions for $T\tau =0$ and 0.1, respectively. (b) We compare the non-RPA conductivity (red solid) with an RPA (red dashed) form for interactions at $T\tau =0.1$.

Figure 5

Diffusion coefficient as a function of the disorder strength $\lambda =1/\left(\pi k_{F}l\right)$ and reduced temperature $T\tau$ in our Boltzmann-transport self-consistent theory for a many-body localization transition. Part (a) expresses a three-dimensional plot of the diffusion coefficient as a function of both disorder strength and temperature. Note that there are two surfaces in a metallic phase. Part (b) displays the diffusion coefficient as a function of the disorder strength at a finite temperature $\tilde{T}=0.1$. The gray solid (gray dashed) line marks the larger (smaller) solution of Eq. (37). A deviation between the numerical analysis and the analytic approach originates from the fact that we neglect the low-temperature contribution given by ${\int }_0^{\tilde{T}}d\tilde{\epsilon }$ in the analytic approach. Part (c) shows the diffusion coefficient as a function of temperature in a many-body localized insulating phase of $\lambda =0.2$. The reason why we cut the gray solid (gray dashed) line is that the analytic expression Eq. (37) can be justified within the regime of $T\tau \ll 1/\left(2\pi \lambda \right)$.

Figure 6

Phase diagram of a many-body localization transition in the plane of the disorder strength and temperature.