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 fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.
Recent experiments have shown that, counterintuitively, applying a magnetic field to a Wigner crystal can induce quantum Hall effects. In this Letter, we use variational Monte Carlo to show that a magnetic field can melt zero-field Wigner crystals into integer quantum Hall liquids. This melting originates from quantum oscillations in the liquid's ground state energy, which develops downward cusps at integer filling factors due to incompressibility. Our calculations establish a range of densities in which this quantum melting transition occurs.
The nature of the state at low Landau-level filling factors has been a long-standing puzzle in the field of the fractional quantum Hall effect (FQHE). While theoretical calculations suggest that a crystal is favored at filling factors $\ensuremath{\nu}\ensuremath{\lesssim}\frac{1}{6}$, experiments show, at somewhat elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect at $\ensuremath{\nu}=\frac{1}{7}, \frac{2}{11}, \frac{2}{13}, \frac{3}{17}, \frac{3}{19}, \frac{1}{9}, \frac{2}{15}$, and
172, which belong to the standard sequences $\ensuremath{\nu}=n/(6n\ifmmode\pm\else\textpm\fi{}1)$ and $n/(8n\ifmmode\pm\else\textpm\fi{}1)$. To address this paradox, we investigate the nature of some of the low-$\ensuremath{\nu}$ states, specifically $\ensuremath{\nu}=\frac{1}{7}, \frac{2}{13}$, and
91, by variational Monte Carlo, density matrix renormalization group, and exact diagonalization methods. We conclude that in the thermodynamic limit, these are likely to be incompressible fractional quantum Hall liquids, albeit with strong short-range crystalline correlations. This suggests a natural explanation for the experimentally observed behavior and a rich phase diagram that admits, in the low-disorder limit, a multitude of crystal-FQHE liquid transitions as the filling factor is reduced.
Trial wave functions which, in the thermodynamic limit, describe states relevant to the fractional quantum Hall effect are explicitly constructed for up to eight electrons. Estimates are given for the ground-state energies and pair-correlation functions and for the low-lying excitation energies for states associated with the filling factors \ensuremath{\nu}=(1/3), (2/7), and (2/5). The results help explain why the filling factors \ensuremath{\nu}=${\ensuremath{\nu}}_{m}$=m/(2m+1), m=1,2,..., and \ensuremath{\nu}=1-${\ensuremath{\nu}}_{m}$ show strong quantum Hall effects.