We introduce the notion of entanglement measure for the universal classes of fractons as an entanglement between ocuppation-numbers of fractons in the lowest Landau levels and the rest of the many-body system of particles. This definition came as an entropy of the probability distribution {\it à la} Shannon. Fractons are charge-flux systems classified in universal classes of particles or quasiparticles labelled by a fractal or Hausdorff dimension defined within the interval
1<h<2 and associated with the fractal quantum curves of such objects. They carry rational or irrational values of spin and the spin-statistics connection takes place in this fractal approach to the fractional spin particles. We take into account the fractal von Neumann entropy associated with the fractal distribution function which each universal class of fractons satisfies. We consider the fractional quantum Hall effect-FQHE given that fractons can model Hall states. According to our formulation entanglement between occupaton-numbers in this context increases with the universality classes of the quantum Hall transitions considered as fractal sets of dual topological quantum numbers filling factors. We verify that the Hall states have stronger entanglement between ocuppation-numbers and so we can consider this resource for fracton quantum computing.
We develop the density-matrix renormalization group (DMRG) technique for numerically studying incompressible fractional quantum Hall (FQH) states on the sphere. We calculate accurate estimates for ground-state energies and excitation gaps at FQH filling fractions nu=1/3 and nu=5/2 for systems that are considerably larger than the largest ever studied by exact diagonalization. We establish, by carefully comparing with existing numerical results on smaller systems, that DMRG is a highly effective numerical tool for studying incompressible FQH states.
We theoretically examine entanglement in fractional quantum hall states, explicitly taking into account and emphasizing the quasi-two-dimensional nature of experimental quantum Hall systems. In particular, we study the entanglement entropy and the entanglement spectrum as a function of the finite layer thickness,
d, of the quasi-two-dimensional system for a number of filling fractions $\ensuremath{\nu}$ in the lowest and the second Landau levels: $\ensuremath{\nu}$ = 1/3, 7/3, 1/2, and 5/2. We observe that the entanglement measures are dependent on which Landau level the electrons fractionally occupy, and find that filling fractions 1/3 and 7/3, which are considered to be Laughlin states, weaken with
d in the lowest Landau level ($\ensuremath{\nu}=1/3$) and strengthen with
d in the second Landau level ($\ensuremath{\nu}=7/3$). For the enigmatic even-denominator $\ensuremath{\nu}=5/2$ state, we find that entanglement in the ground state is consistent with that of the non-Abelian Moore-Read Pfaffian state at an optimal thickness,
d. We also find that the single-layer $\ensuremath{\nu}=1/2$ system is not a fractional quantum Hall state consistent with the experimental observation. In general, our theoretical findings based on entanglement considerations are completely consistent with the results based on wave-function overlap calculations.
With the use of Haldane's spherical model, the fractional quantum Hall effect is studied by means of large-scale configuration-interaction calculations. The exact wave function and the ground-state energy are computed numerically up to ten electrons for filling \ensuremath{\nu}=(1/3), and up to seven electrons for filling \ensuremath{\nu}=(1/5). Quasiparticle, quasihole, and exciton energies are computed. The mutual relation between wave functions on the sphere and wave functions on the disk is fully analyzed. A wave function originally proposed by Halperin for the disk geometry is tested on the sphere; the numerical results up to six electrons for \ensuremath{\nu}=(2/5) and \ensuremath{\nu}=(2/9) are favorable.
While it is generally believed that a crystal state is favored at filling factors
ν<1/6 in the lowest Landau level, experiments show, at elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect (FQHE) at the standard sequences
ν=n/(6n±1) and
ν=n/(8n±1). To address this paradox, we investigate the nature of the state at filling factors
ν=1/7, 1/9, 1/11 and 2/13 by several methods and find that the competition between the crystal and liquid states is more subtle than previously thought. Exact diagonalization study in the spherical geometry shows behavior consistent with an FQHE liquid for system sizes available. Density matrix renormalization group study in the cylindrical geometry shows that the nature of the state depends on the particle number, with crystal favored for certain particle numbers but not all; these studies do not allow us to make conclusive statements regarding the thermodynamic nature of the state. A variational study shows that the composite-fermion crystal and the renormalized FQHE liquid are very close in energy. Based on these studies, it is not possible to rule out, definitively, an FQHE liquid with strong short-range crystalline correlations at these filling factors.