Electronic properties in a two-dimensional disordered electron liquid: Spin-valley interplay

We report a detailed study of the influence of the spin and valley splittings on such physical observables of the two-dimensional disordered electron liquid as resistivity and spin and valley susceptibilities. We explain qualitatively the nonmonotonic dependence of the resistivity on temperature in the presence of a parallel magnetic field. In the presence of either spin or valley splitting we predict a temperature dependence of the resistivity with two maximum points.

Figure 1

Projection of the RG flow in the three-dimensional parameter space $\left({\sigma }_{xx},{\gamma }_v,{\gamma }_s\right)$ onto the $\left({\gamma }_v,{\gamma }_s\right)$ plane for the $\text{SU}\left(2\right)\times \text{SU}\left(2\right)$ symmetry case [Eqs. (63, 64, 65)]. Dots denote the line at which $1+6f\left({\gamma }_v\right)+f\left({\gamma }_s\right)=0$. The dashed line indicates the line ${\gamma }_v={\gamma }_s$ (see text).

Figure 2

Schematic dependence of the resistance $\rho =1/\left(\pi {\sigma }_{xx}\right)$ on $\xi$. Curves $a$, $b$, and $c$ correspond to the flow lines $a$, $b$, and $c$ in Fig. 1 (see text).

Figure 3

Projection of the RG flow in the three-dimensional parameter space $\left({\sigma }_{xx},{\gamma }_v,{\gamma }_s\right)$ onto the $\left({\gamma }_v,{\gamma }_s\right)$ plane for the completely symmetry-broken case [Eqs. (86, 87, 88)]. Dots denote the line at which $1+2f\left({\gamma }_v\right)+f\left({\gamma }_s\right)=0$ (see text).

Figure 4

Schematic dependence of the resistance $\rho =1/\left(\pi {\sigma }_{xx}\right)$ on $\xi$ along the flow line $a$ in Fig. 3 (see text).

Figure 5

Schematic $\rho \left(T\right)$ dependence in the case of zero parallel magnetic field (see text).

Figure 6

Schematic $\rho \left(T\right)$ dependence in the presence of both spin and valley splitting in the case ${\Delta }_s<{\Delta }_v$. For the opposite case, the behavior will be similar (see text).

Figure 7

Schematic $\rho \left(T\right)$ dependence in the presence of a strong parallel magnetic field: ${\Delta }_v,T_{\text{max}}^{\left(I\right)}<{\Delta }_s$ (see text).

Figure 8

Schematic dependence of the ration ${\chi }_v/{\chi }_s$ on temperature: (a) for ${\Delta }_s<{\Delta }_v$; (b) for ${\Delta }_s={\Delta }_v$; (c) for ${\Delta }_s>{\Delta }_v$. The temperature scales $T_{a,c}\equiv {\Delta }_s$.