The electron correlations near the half-filling of the lowest and excited Landau levels (LL's) are studied using numerical diagonalization. It is shown that in the low-lying states electrons avoid pair states with relative angular momenta
R corresponding to positive anharmonicity of the interaction pseudopotential
V(R). In the lowest LL, the superharmonic behavior of
V(R) causes Laughlin correlations (avoiding pairs with
R=1) and the Laughlin-Jain series of incompressible ground states. In the first excited LL,
V(R) is harmonic at short range and a different series of incompressible states results. Similar correlations occur in the paired Moore-Read $\ensuremath{\nu}=\frac{5}{2}$ state and in the $\ensuremath{\nu}=\frac{7}{3}$ and
38 states, all having small total parentage from
R=1 and 3 and large parentage from
R=5. The $\ensuremath{\nu}=\frac{7}{3}$ and
38 states are different from Laughlin $\ensuremath{\nu}=\frac{1}{3}$ and
32 states and, in finite systems, occur at a different LL degeneracy (flux). The series of Laughlin-correlated states of electron pairs at $\ensuremath{\nu}{=2+2/(q}_{2}+2)=\frac{8}{3},$
25, 512, and
37 is proposed, although only in the $\ensuremath{\nu}=\frac{5}{2}$ state pairing has been confirmed numerically. In the second excited LL,
V(R) is subharmonic at short range and (near the half-filling) the electrons group into spatially separated larger $\ensuremath{\nu}=1$ droplets to minimize the number of strongly repulsive pair states at
R=3 and 5.
We have investigated the behavior and stability of a Wigner crystal of rotating dipolar fermions in two dimensions. Using an ansatz wave function for the ground state of rotating two-dimensional dipolar fermions, which occupy only partially the lowest Landau level, we study the correlation energy, elastic moduli and collective modes of Wigner crystals in the lowest Landau level. We then calculate the mean square of the displacement vector of Wigner crystals. The critical filling factor, below which the crystalline state is expected, is evaluated at absolute zero by use of the Lindeman's criterion. We find that the particle (hole) crystal is locally stable for filling factor is less than 1/15 (between filling factors 14/15 and 1), where the stable regime of the crystal is much narrower than the result from Baranov, Fehrmann and Lewenstein, [Phys. Rev. Lett. 100, 200402 (2008)].
We propose a ground-state trial wave function for a two-dimensional Wigner crystal in a strong perpendicular magnetic field. The wave function includes Laughlin-Jastrow correlations between electron pairs, and may be interpreted as a crystal state of composite fermions or composite bosons. Treating the power
m of the Laughlin-Jastrow factor as a variational parameter, we use quantum Monte Carlo simulations to compute the energy of these new states. We find that our wave functions have lower energy than existing crystalline wave functions in the lowest Landau level. Our results are consistent with experimental observations of the filling factor at which the transition between the fractional quantum Hall liquid and the Wigner crystal occurs for electron systems. Exchange contributions to the wave functions are estimated quantitatively and shown to be negligible for sufficiently small filling factors.
We generalize a path integral Monte Carlo formula to investigate the phase boundary between the fractional quantum Hall liquid and the Wigner crystal for a two-dimensional electron (fermion) system in the presence of strong magnetic fields and at low temperatures. At very low temperature, the state of the system is found to be a solid at filling factors $\ensuremath{\nu}=1∕7$,
1∕9, and
1∕11, while the state remains liquidlike at $\ensuremath{\nu}=1∕3$ and
1∕5. The temperature-density phase diagrams are obtained at low filling factors. At $\ensuremath{\nu}=1∕7$ and a fixed electron density, the calculated melting temperature is $\ensuremath{\sim}220\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$, in good agreement with some experimental results. The nature of the phase transition is suggested to be continuous.
The interplay between strongly correlated liquid and crystal phases for two-dimensional electrons exposed to a high transverse magnetic field is of fundamental interest. Through the non-perturbative fixed phase diffusion Monte Carlo method, we determine the phase diagram of the Wigner crystal in the
ν−κ plane, where
ν is the filling factor and
κ is the strength of Landau level mixing. The phase boundary is seen to exhibit a striking
ν dependence, with the states away from the magic filling factors
ν=n/(2pn+1) being much more susceptible to crystallization due to Landau level mixing than those at
ν=n/(2pn+1). Our results explain the qualitative difference between the experimental behaviors observed in n-doped and p-doped GaAs quantum wells, and, in particular, the existence of an insulating state for
ν<1/3 and also for
1/3<ν<2/5 in low density p-doped systems. We predict that in the vicinity of
ν=1/5 and
ν=2/9, increasing LL mixing causes a transition not into an ordinary electron Wigner crystal but rather into a strongly correlated crystal of composite fermions carrying two vortices.
A low-disorder, two-dimensional electron system (2DES) subjected to a large perpendicular magnetic field and cooled to very low temperatures provides a rich platform for studies of many-body quantum phases. The magnetic field quenches the electrons' kinetic energy and quantizes the energy into a set of Landau levels, allowing the Coulomb interaction to dominate. In excited Landau levels, the fine interplay between short- and long-range interactions stabilizes bubble phases, Wigner crystals with more than one electron per unit cell. Here, we present the screening properties of bubble phases, probed via a simple capacitance technique where the 2DES is placed between a top and a bottom gate and the electric field penetrating through the 2DES is measured. The bubbles formed at very low temperatures screen the electric field poorly as they are pinned by the residual disorder potential, allowing a large electric field to reach the top gate. As the temperature is increased, the penetrating electric field decreases and, surprisingly, exhibits a pronounced minimum at a temperature that appears to coincide with the melting temperature of the bubble phase. We deduce a quantitative phase diagram, as a function of Landau level filling factor (\ensuremath{\nu}) and temperature, for the transition from the bubble to liquid phases for $4\ensuremath{\le}\ensuremath{\nu}\ensuremath{\le}5$.