Two-Dimensional Electron System in High Magnetic Fields: Wigner Crystal versus Composite-Fermion Liquid

The two-dimensional system of electrons in a high magnetic field offers an opportunity to investigate a phase transition from a quantum liquid into a Wigner solid. Recent experiments have revealed an incipient composite fermion liquid in a parameter range where theory and many experiments had previously suggested the Wigner crystal phase, thus calling into question our current understanding. This Letter shows how very small quantitative corrections ($<1\%$) in the energy due to the weak interaction between composite fermions can cause a fundamental change in the nature of the ground state, thus providing insight into the puzzling experimental results.

Figure 1

Examples of configurations of ${}^2\mathrm{C}\mathrm{F}\mathrm{s}$ at ${\nu }^{*}=1$ ($\nu =1/3$). Panel (a) shows pictorially the ground state of noninteracting composite fermions, while panel (b) shows two ${}^2\mathrm{C}\mathrm{F}$ particle-hole pairs. The analogous configurations at $\nu =1/7$, $1/9$, and $1/11$ will contain ${}^6\mathrm{C}\mathrm{F}\mathrm{s}$, ${}^8\mathrm{C}\mathrm{F}\mathrm{s}$, and ${}^{10}\mathrm{C}\mathrm{F}\mathrm{s}$, respectively.

Figure 2

The ground state energy per particle as a function of the number of composite fermions, $N$, for $\nu =1/7$, $1/9$, and $1/11$. $E^{(0)}$ is the energy of the noninteracting CF system, while $E^{(2)}$ and $E^{(3)}$ are successively better approximations including interaction effects. The cross on the $y$ axis shows the energy of the Wigner crystal, taken from Ref. 9, which has an uncertainty of $\sim 0.0005$ in the chosen units. The error bars denote the statistical uncertainty in the Monte Carlo, and the solid straight lines are the best chi-square fits for $E^{(3)}$. The number near each solid diamond equals $|⟨{\chi }^{(3)}|{\Psi }^{(0)}⟩{|}^2/(⟨{\chi }^{(3)}|{\chi }^{(3)}⟩⟨{\Psi }^{(0)}|{\Psi }^{(0)}⟩)$.