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arXiv:cond-mat/9707251 · Physical Review Letters

Quantum Hall Fluids on the Haldane Sphere: A Diffusion Monte Carlo Study

V. Melik-Alaverdian, N. E. Bonesteel, Gerardo Ortíz

Published Dec 29, 1997• 40 citations
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Abstract

A generalized diffusion Monte Carlo method for solving the many-body Schr\"odinger equation on curved manifolds is introduced and used to perform a ``fixed-phase'' simulation of the fractional quantum Hall effect on the Haldane sphere. This new method is used to study the effect of Landau level mixing on the $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3$ energy gap and the relative stability of spin-polarized and spin-reversed quasielectron excitations.

DOI
10.1103/physrevlett.79.5286

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A generalized diffusion Monte Carlo method for solving the many-body Schr"odinger equation on curved manifolds is introduced and used to perform a ``fixed-phase'' simulation of the fractional quantum Hall effect on the Haldane sphere. This new method is used to study the effect of Landau level mixing on the \ensuremathν=1/3\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3\ensuremathν=1/3 energy gap and the relative stability of spin-polarized and spin-reversed quasielectron excitations.
The spin transitions in the fractional quantum Hall effect provide a direct measure of the tiny energy differences between differently spin-polarized states and thereby serve as an extremely sensitive test of the quantitative accuracy of the theory of the fractional quantum Hall effect, and, in particular, of the role of Landau-level mixing in lifting the particle-hole symmetry. We report on an accurate quantitative study of this physics, evaluating the effect of Landau-level mixing in a nonperturbative manner using a fixed-phase diffusion Monte Carlo method. We find excellent agreement between our calculated critical Zeeman energies and the experimentally measured values. In particular, we find, as also do experiments, that the critical Zeeman energies for fractional quantum Hall states at filling factors ν=2-n/(2n±1) are significantly higher than those for ν=n/(2n±1), a quantitative signature of the lifting of particle-hole symmetry due to Landau-level mixing.
Based on Haldane's spherical geometrical formalism of the two-dimensional quantum Hall fluids, the relation between the noncommutative geometry of S2 and the two-dimensional quantum Hall fluids is exhibited. A finite matrix model on the two-sphere is explicitly constucted as an effective description of the fractional quantum Hall fluids of finite extent, and the complete sets of physical quantum states of this matrix model are determined. We also describe how the low-lying excitations in the model are constructed in terms of the quasi-particle and quasi-hole excitations. It is shown that there exists a Haldane hierarchical structure in the two-dimensional quantum Hall fluid states of the matrix model. These hierarchical fluid states are generated by the parent fluid state by condensing the quasi-particle and quasi-hole excitations level by level.
Most analyses of the ground state wavefunctions of the QHE ignore electron spin and simply use anti-commuting scalars: either because strong magnetic fields split the spins and the ground state is completely spin polarised or because the ground state is effectively spin degenerate so the only effect of spin is to double the degeneracy of the ground state. A useful approach is that of the Haldane sphere where the system is put on a 2-dimensional sphere and a normal magnetic field is generated by placing a magnetic monopole at the centre of the sphere. The analysis of the fractional quantum Hall effect in this approach is somewhat subtle, as the ground state wavefunction of the unperturbed Hamiltonian is not unique in Haldane's approach. When the fermionic nature of the charged particles is fully incorporate into this picture the analysis is modified and we show that, incorporating Jain's composite fermion picture into the Haldane sphere leads to a unique unperturbed ground state for the fractional quantum Hall effect on a sphere. A mass gap then ensures stability under perturbations. An important tool in the analysis is the Atiyah-Singer index theorem.
We compare ground-state energies by variational Monte Carlo of the spin unpolarized Halperin 331 and the spin polarized Moore-Read (MR) Pfaffian fractional quantum Hall states at half filling of the lowest Landau level (LLL) and the second Landau level (SLL) as a function of small deviations around the Coulomb point via the finite-thickness effect and direct alterations to the the first two Haldane pseudopotentials. In the comparison we find that in the LLL, either the 331 state or the MR Pfaffian may be lower in energy depending on the deviations. In the SLL, however, the MR Pfaffian is consistently lower in energy except for large deviations. These results suggest that even under moderate deviations in the interaction potential (through various physical processes such as finite thickness, Landau-level mixing, etc.), the MR Pfaffian description is more energetically favorable than the Halperin 331 state in the half-filled SLL (i.e., $\ensuremath{\nu}=5/2$), consistent with recent experimental investigations.