Spinful composite fermions in a negative effective field

In this paper, we study fractional quantum Hall composite fermion wave functions at filling fractions $\nu =2/3,$ 3/5, and $4/7$. At each of these filling fractions, there are several possible wave functions with different spin polarizations, depending on how many spin-up or spin-down composite fermion Landau levels are occupied. We calculate the energy of the possible composite fermion wave functions and we predict transitions between ground states of different spin polarizations as the ratio of Zeeman energy to Coulomb energy is varied. Previously, several experiments have observed such transitions between states of differing spin polarization and we make direct comparison of our predictions to these experiments. For more detailed comparison between theory and experiment, we also include finite-thickness effects in our calculations. We find reasonable qualitative agreement between the experiments and composite fermion theory. Finally, we consider composite fermion states at filling factors $\nu =2+2/3$, $2+3$/5, and $2+4/7$. The latter two cases we predict to be spin polarized even at zero Zeeman energy.

Figure 1

A Summary of the qualitative predictions of CF theory (modified from a similar figure in Ref. 9). The figure shows the filling of effective LLs with spin-up and spin-down composite fermions. Different cases are labeled by their quantum numbers $n_{\uparrow }$ and $n_{\downarrow }$, the number of filled spin-up and spin-down LLs, e.g., (1:1) denotes $n_{\uparrow }=1$ and $n_{\downarrow }=1$. The degree of spin polarization ${\gamma }_e$ is calculated using Eq. (2).

Figure 2

Extrapolations to the thermodynamic limit for the Coulomb energies per electron of various CF trial ground-state wave functions in the LLL. Only the linear extrapolations in $1/N$ are shown. Error bars are smaller than the data markers and so are not plotted.

Figure 3

Predicted and measured values for the critical Zeeman energy per electron from Table plotted against the reciprocal number of filled effective LLs, $1/n$. Transitions are labeled by their degree of spin polarization ${\gamma }_e$ [defined in Eq. (2)]. In the text we explain how the theoretical predictions of the Critical Zeeman per electron are calculated. Experimental values are taken from Kukushkin et al. (Ref. 3) and from Du et al. (Ref. 7). We explain in the text how the experimental values are deduced. Circled sets of points correspond to the same transition. For the series of ${}^2{\mathit{CF}}_{-\mathit{n}}\to {}^2{\mathit{CF}}_{(-\mathit{n}-1,-1)}$ transitions (i.e., those transitions for which higher value of ${\gamma }_e$ is 1) there is a trend for the critical Zeeman energy to increase as $n$ increases. In all cases theory tends to underestimate the critical Zeeman energy.

Figure 4

Graphs showing the impact of the finite thickness correction on our results. The energy is given in units of $e^2/\epsilon l_0$ and the thickness parameter is given in units of $l_0$. To obtain these curves, weighted extrapolations to the thermodynamic limit are calculated for the sets of data points at finite $N$ for 501 discrete values of the parameter $d$ between 0 and 5 and then the curves are interpolated. We use the extrapolation scheme that gives minimum uncertainty in the extrapolated value (i.e., possibly with the smallest system size removed from the extrapolation). (a) Interaction energy per electron associated with a selection of the CF ground states in the thermodynamic limit due to a modified Coulomb potential $V_{\mathit{FH}}(r,d)$ given in Eq. (13), plotted as a continuous function of the thickness parameter $d$. The associated error bars for these extrapolations are not drawn as they are too small to see on the plot. (b) Theoretical predictions for the critical Zeeman energy per electron due to a finite-thickness potential $V_{\mathit{FH}}(r,d)$ given in Eq. (13), plotted as a continuous function of the thickness parameter $d$ for selected CF states. The critical Zeeman energy is calculated using Eq. (12). Representative error bars for a selected subset of the data are drawn at intervals on the plot. Note that the data have been smoothed between $d=0.2$ and 0.3 in order to remove a numerical artifact.

Figure 5

Predicted values for the critical Zeeman per electron modified by the inclusion of finite thickness effects (data from Table ) and measured values for the critical Zeeman energy per electron from Table plotted against the reciprocal number of filled effective LLs, $1/n$. Transitions are labeled by their degree of spin polarization ${\gamma }_e$ [defined in Eq. (2)]. In the text we explain how the theoretical predictions of the critical Zeeman energy per electron are calculated, taking into account a finite thickness correction to the potential appropriate to the conditions of the different experimental systems considered. Experimental values are taken from Kukushkin et al. (Ref. 3) and from Du et al. (Ref. 7). We explain in the text how the experimental values are deduced. Circled sets of points correspond to the same transition. As in Fig. 3, the theory somewhat underestimates the critical Zeeman energy.