Interaction-Induced Magnetoresistance: From the Diffusive to the Ballistic Regime

We study interaction-induced quantum correction $\delta {\sigma }_{\alpha \beta }$ to the conductivity tensor of electrons in two dimensions for arbitrary $T\tau$, where $T$ is the temperature and $\tau$ the transport mean free time. A general formula is derived, expressing $\delta {\sigma }_{\alpha \beta }$ in terms of classical propagators (“ballistic diffusons”). The formalism is used to calculate the interaction contribution to the magnetoresistance in a classically strong transverse field and smooth disorder in the whole range of temperatures from the diffusive ($T\tau \ll 1$) to the ballistic ($T\tau \gtrsim 1$) regime.

Figure 1

Diagrams for the interaction correction to ${\sigma }_{\alpha \beta }$. The wavy (dashed) lines denote the interaction (impurity scattering), the shaded blocks are impurity ladders, and the $+/-$ symbols denote the signs of the Matsubara frequencies. The diagrams obtained by a flip and/or by an exchange $+\leftrightarrow -$ should also be included.

Figure 2

Functions $G_0(T\tau )$ (a) and $G_F(T\tau )$ (b) determining the $T$ dependence of the exchange term for pointlike, Eq. (11), and Coulomb, Eq. (12), interaction, respectively.

Figure 3

Hartree contribution, $G_{\mathrm{H}}(T\tau )$, for (a) weak interaction, $\kappa /k_F=0.1,0.2,0.3,0.5$, and (b) strong interaction, $F_0=-0.3,-0.4,-0.5$ (from bottom to top); (c) schematic plot of MR $\delta {\rho }_{xx}(B)$ in different temperature regimes: (1) $T_1\ll {\tau }^{-1}$; (2) ${\tau }^{-1}\ll T_2\ll T_{\mathrm{H}}$; (3) $T_3\gg T_{\mathrm{H}}$.