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Journal of Physics A Mathematical and General

Fractional quantum Hall effect on the 2-sphere: a quasispin analysis

Roger Sollie, Kaare Olaussen

Published Nov 7, 1990• 3 citations
Publisher

Abstract

The authors study a class of models for the fractional quantum Hall effect, where the system is mapped onto the surface of a sphere around a strong magnetic monopole. When the magnetic charge of the monopole is half-integer (in units of h(cross)/e), there exists an SU(2) quasispin group operating on the Fock space of these models. This group commutes with the group of rotations on the sphere, but not with the particle number operator N. This fact may be utilized to easily construct states of definite angular momentum at high particle numbers from states of the same angular momentum at lower particle numbers. Using the Wigner-Eckart theorem the matrix elements of the Hamiltonian between these states can also be constructed. For a large class of pair interactions this leads to a computational method of finding exact eigenstates and eigenvalues of the Hamiltonian.

DOI
10.1088/0305-4470/23/21/026

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The authors study a class of models for the fractional quantum Hall effect, where the system is mapped onto the surface of a sphere around a strong magnetic monopole. When the magnetic charge of the monopole is half-integer (in units of h(cross)/e), there exists an SU(2) quasispin group operating on the Fock space of these models. This group commutes with the group of rotations on the sphere, but not with the particle number operator N. This fact may be utilized to easily construct states of definite angular momentum at high particle numbers from states of the same angular momentum at lower particle numbers. Using the Wigner-Eckart theorem the matrix elements of the Hamiltonian between these states can also be constructed. For a large class of pair interactions this leads to a computational method of finding exact eigenstates and eigenvalues of the Hamiltonian.
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 study the behavior of an interacting two-dimensional charged particle system in a transverse magnetic field and give a simple picture of the features that lead to the fractional quantum Hall effect. In particular, we discuss how the interparticle interactions give rise to an energy-level substructure in the lowest Landau level and develop a relatively simple model that gives insight into the analytical form of the Laughlin wave function. We analyze the model by transforming to a reference frame that rotates at half the cyclotron angular frequency. In the rotating frame, the vector potential for the magnetic field is replaced by a quadratic scalar potential. It is then evident that the system of interacting particles is a bound state, even in the absence of a charge-neutralizing background. We solve exactly a many-particle system in the rotating frame using a harmonic model interaction. In addition to being a good example for demonstrating basic methods in quantum mechanics, we show explicitly that the Laughlin function is an eigenstate of this model system.
New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry ±1/4\pm1/4±1/4 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta NϕN_ϕNϕ​ is shown to be related to the number of electrons NNN by Nϕ=2N−(5−q)N_ϕ = 2N-(5-q)Nϕ​=2N−(5−q) with qqq being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at Nϕ=2N−3N_ϕ = 2N-3Nϕ​=2N−3. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.
A low-energy neutral quasiparticle in a fractional quantum Hall system appears in the latter’s energy spectrum on a sphere as a series of many-body excited states labeled by the angular momentum L L and whose energy is a smooth function of L L in the limit of large sphere radius. We argue that the signature of a nonvanishing spin (intrinsic angular momentum) s s of the quasiparticle is the absence, in this series, of states with total angular momentum less than s s . We reinterpret the missing of certain states, observed in an exact-diagonalization calculation of the spectrum of the \nu=7/3 ν = 7 / 3 FQH state in a wide quantum well as well as in many proposed wave functions for the excited states as a consequence of the spin-2 nature of the zero-momentum magnetoroton.
The Halperin $({m}^{\ensuremath{'}},m,n)$ fractional quantum Hall effects of two-component quantum particles are studied in topological checkerboard lattice models. Here for $m\ensuremath{\ne}{m}^{\ensuremath{'}}$, we demonstrate the emergence of fractional quantum hall effects with the associated K=(m+211m)\mathbf{K}=\left(\begin{array}{cc}m+2& 1\\ 1& m\end{array}\right)K=(m+21​1m​) matrix (even m=2m=2m=2 for boson and odd m=3m=3m=3 for fermion) in the presence of both strong intercomponent and intracomponent repulsions. Through exact diagonalization and density-matrix renormalization group calculations, we elucidate their topological fractionalizations, including (i) the $det|\mathbf{K}|=({m}^{2}+2m\ensuremath{-}1)$-fold ground-state degeneracies and (ii) fractionally quantized topological Chern number matrix $\mathbf{C}={\mathbf{K}}^{\ensuremath{-}1}$. Our flat band model provides a paradigmatic example of a microscopic Hamiltonian featuring fractional quantum Hall effect with partial spinpolarization.