Quantum Hall Ferromagnetic States and Spin-Orbit Interactions in the Fractional Regime

The competition between the Zeeman energy and the Rashba and Dresselhaus spin-orbit couplings is studied for fractional quantum Hall states by including correlation effects. A transition of the direction of the spin polarization is predicted at specific values of the Zeeman energy. We show that these values can be expressed in terms of the pair-correlation function, and thus provide information about the microscopic ground state. We examine the particular examples of the Laughlin wave functions and the $5/2$-Pfaffian state. We also include effects of the nuclear bath.

Figure 1

Values of $g_c$ (negative dots) and $\Delta g$ (positive dots), as obtained from Eqs. (4, 5) for states at $\nu =1/M$ (and their particle-hole conjugates), and $\nu =1/2$. The density is chosen such that $B=5/\nu$ (in $T$), $\alpha =0$, and other parameters are given in the main text. The dashed line is the noninteracting value of $g_c$ ($\Delta g=0$ for noninteracting electrons). Inset: general plot of the polarization angle ${\theta }_m$ [see Eq. (3)] as a function of the $g$ factor.

Figure 2

Same as Fig. 1, in the range $2<\nu <3$. At $\nu =5/2$ both composite Fermi sea (CFS) and Pfaffian (Pf) results are displayed. The density is such that $B=12.5/\nu$ (in $T$) and $\alpha =0$. The noninteracting $g_c$ is also plotted (dashed line).

Figure 3

Plot of the values of $g_c$ (negative) and $\Delta g$ (positive) from Eqs. (4, 5) at $\nu =5/2$, $B>5\mathrm{T}$ and other parameters as in the text. The solid and dashed lines correspond to the Pfaffian state and CF sea, respectively [$f_{1,2}$ as in Eqs. (10, 11)]. The dotted line is the linear noninteracting contribution to $g_c$ ($f_1=0$). The dots mark the $B$-field values of 17, 36.

Figure 4

Nuclear shift of the hyperfined modified electron $g$ factor (see text) as a function of temperature $T$ at different values of $B$. In the inset, the average nuclear polarization $⟨I⟩=\sum _i{x}_i⟨{\widehat{I}}_z{⟩}_i$ is also shown.