A new nonsingular mapping between bosons and fermions is introduced in connection with the fractional quantum Hall effect. The mapping connects a fermion system of filling ${\ensuremath{\nu}}_{\mathit{f}}$ to a boson system of filling ${\ensuremath{\nu}}_{\mathit{b}}$ through the relation ${\ensuremath{\nu}}_{\mathit{f}}^{\mathrm{\ensuremath{-}}1}$=${\ensuremath{\nu}}_{\mathit{b}}^{\mathrm{\ensuremath{-}}1}$+1. The ground state of the boson system is shown to map accurately to that of the fermion system if and only if they are both incompressible quantum Hall states. The accuracy of the mapping is shown to be related to the validity of the mean-field theory. The relevance of this mapping to off-diagonal long-range order is discussed by numerically studying the binding of wave-function zeros to particle positions.
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.
We demonstrate the existence of a novel type of off-diagonal long-range order in the fractional-quantum-Hall-effect (FQHE) ground state. This is revealed for the case of fractional filling factor \ensuremath{\nu}=1/m by application of Wilczek's ``anyon'' gauge transformation to attach m quantized flux tubes to each particle. The binding of the zeros of the wave function to the particles in the FQHE is a (2+1)-dimensional analog of oblique confinement in which a condensation occurs, not of ordinary particles, but rather of composite objects consisting of particles and gauge flux tubes.
Quantum Hall states at filling fraction $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5/2$ are examined by numerical diagonalization. Spin-polarized and spin-unpolarized states of systems with $N\ensuremath{\le}18$ electrons are studied, neglecting effects of Landau level mixing. We find that the ground state is spin polarized. It is incompressible and has a large overlap with paired states like the Pfaffian. For a given sample, the energy gap is about 11 times smaller than at $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3$. Evidence is presented of phase transitions to compressible states, driven by the interaction strength at short distance. A reinterpretation of experiments is suggested.
A natural generalization of the Heisenberg uncertainty principle inequality holding for non-Hermitian operators is presented and applied to the fractional quantum Hall effect (FQHE). This inequality was used in a previous paper to prove the absence of long-range order in the ground state of several one-dimensional (1D) systems with continuous-group symmetries. In this paper, we use it to rule out the occurrence of Bose-Einstein condensation in the bosonic representation of the FQHE wave function proposed by Girvin and MacDonald. We show that the absence of off-diagonal long-range order in this two-dimensional (2D) problem is directly connected with the
q2 behavior of the static structure function S(q) at small momenta.
The single-particle spectral function for an incompressible fractional quantum Hall state in the presence of a scalar short-ranged attractive impurity potential is calculated via exact diagonalization within the spherical geometry. In contrast to the noninteracting case, where only a single bound state below the lowest Landau level forms, electron–electron interactions strongly renormalize the impurity potential, effectively giving it a finite range, which can support many quasi-bound states (long-lived resonances). Averaging the spectral weights of the quasi-bound states and extrapolating to the thermodynamic limit, for filling factor ν = 1/3 we find evidence consistent with localized fractionally charged e /3 quasi-particles. For ν = 2/5, the results are slightly more ambiguous, due to finite size effects and possible bunching of Laughlin-quasi-particles.