Phase diagram of fractional quantum Hall effect of composite fermions in multicomponent systems

While the integer quantum Hall effect of composite fermions manifests as the prominent fractional quantum Hall effect (FQHE) of electrons, the FQHE of composite fermions produces further, more delicate states, arising from a weak residual interaction between composite fermions. We study the spin phase diagram of these states, motivated by the recent experimental observation by Liu and co-workers [Phys. Rev. Lett. 113, 246803 (2014) and private communication] of several spin-polarization transitions at 4/5, 5/7, 6/5, 9/7, 7/9, 8/11, and 10/13 in GaAs systems. We show that the FQHE of composite fermions is much more prevalent in multicomponent systems, and consider the feasibility of such states for systems with $\mathcal{N}$ components for an SU($\mathcal{N}$) symmetric interaction. Our results apply to GaAs quantum wells, wherein electrons have two components, to AlAs quantum wells and graphene, wherein electrons have four components (two spins and two valleys), and to an H-terminated Si(111) surface, which can have six components. The aim of this paper is to provide a fairly comprehensive list of possible incompressible fractional quantum Hall states of composite fermions, their SU($\mathcal{N}$) spin content, their energies, and their phase diagram as a function of the generalized “Zeeman” energy. We obtain results at three levels of approximation: from ground-state wave functions of the composite fermion theory, from composite fermion diagonalization, and, whenever possible, from exact diagonalization. Effects of finite quantum well thickness and Landau-level mixing are neglected in this study. We compare our theoretical results with the experiments of Liu and co-workers [Phys. Rev. Lett. 113, 246803 (2014) and private communication] as well as of Yeh et al., [Phys. Rev. Lett. 82, 592 (1999)] for a two-component system.

Figure 1

Some examples illustrate the notation used in Eq. (10). The green solid dots represent electrons while the red solid dots with $2p$ arrows are composite fermions carrying $2p$ vortices. The overline notation is used to indicate a particle-hole transformation while ${\left[\right]}_{2p}$ and ${\left[\right]}_{-2p}$ denote composite fermionization with parallel and reverse flux attachment, respectively. Note that even though the two fully polarized $\nu =4/5$ states shown above, namely ${[1+{\left[1\right]}_2]}_{-2}$ and $\overline{{\left[1\right]}_4}$, appear different, they represent the same state, as explained in the text.

Figure 2

Extrapolation of the ground-state energy to the thermodynamic limit, assuming zero thickness. The density correction has been applied.

Figure 3

Extrapolation of the ground-state energy to the thermodynamic limit, assuming zero thickness. The density correction has been applied. Among the fractions that allow several multicomponent states, the data are convincing only at $\nu =7/19$ and $\nu =10/27$; even here it is difficult to conclude anything beyond error bars (see Table 15).

Figure 4

Phase diagram of the CF FQHE states in the filling factor regions $1/3<\nu <2/5$ (upper panel) and $2/3>\nu >1$ (lower panel). The various possible states are shown for several fractions, along with the theoretical critical Zeeman energies where transitions between them are expected. (The “?” symbol is used to represent transitions that are expected to occur but for which the critical Zeeman energies have not been estimated.) The spin contents and polarizations of the states can be found in the main text, but lowest state at even-numerator fractions is spin singlet and the highest state is fully spin polarized. The states in the top (bottom) panel are obtained from states of CF in the filling factor range $1<{\nu }^{*}<2$ by parallel (reverse) flux attachment. The dots in the upper panel are obtained with the help of CF diagonalization, whereas those in the lower panel are estimated from exact diagonalization studies. In both cases the thermodynamic energies are obtained by extrapolation of finite system results to obtain the critical Zeeman energies.

Figure 5

Critical Zeeman energies for transitions at 4/5, 9/7, 5/7, 7/9, and 10/13 as a function of the effective transverse width $\lambda$ (see text for definition) in units of the magnetic length $\ell$. The results are taken from Liu et al. [9] and Yeh et al. [8], as indicated on the figure. The experiment of Yeh et al. employed heterostructures, which correspond to very small transverse widths. Theoretical values at zero width are encircled in black. They are in good agreement with zero width limits of the experiments.

Figure 6

Energy spectrum of the 4/11 spin singlet FQHE state, where composite fermions form a spin singlet 4/3 FQHE state. The spectra shown by dots (different colors representing different spin quantum numbers $S$) were obtained by CF diagonalization. The dashes in the top two panels show the exact spectrum obtained by exact diagonalization in the full Hilbert space. The spherical geometry is used for the calculation. $L$ is the total orbital angular momentum, $2Q$ is the flux through the sphere in units of the flux quantum, and $N$ is the total number of electrons. The bottom panel shows the two lowest-energy collective modes, namely the charge exciton and the spin-flip exciton.