The Fractional Quantum Hall Effect

The fractional quantum Hall effect (FQHE) in the second orbital Landau level at even-denominator filling factor of 5/2 remains enigmatic and motivates our work. We theoretically consider the effect of the quasi-two-dimensional (2D) nature of the experimental fractional quantum Hall system on a number of FQH states (filling factors: 1/3, 1/5, and 1/2) in the lowest, second, and third orbital Landau levels (LLL, SLL, and TLL, respectively) by calculating the wave-function overlap, as a function of quasi-2D layer thickness, between the exact ground state of a model Hamiltonian and the consensus variational ansatz wave functions, i.e., the Laughlin wave function for 1/3 and 1/5 and the Moore-Read Pfaffian wave function for 1/2. Using large numerical overlap as a stability (or FQHE robustness) criterion, we find that the FQHE does not occur in the TLL (for any quasi-2D layer thickness), is the most robust for zero thickness (strict 2D limit) in the LLL for 1/3 and 1/5 and for 11/5 in the SLL, and is the most robust at finite thickness (4--5 magnetic lengths) in the SLL for the mysterious even-denominator 5/2 state and the presumably more conventional 7/3 state. We do not find any FQHE at 1/2 in the LLL for any thickness for the quasi-2D models considered in our work. Furthermore, we examine the orbital effects of a nonzero in-plane (parallel) magnetic field, finding that its application effectively reduces the quasi-2D layer thickness and, therefore, could destroy the FQHE at 5/2 and 7/3, while it enhances the FQHE at 11/5, in the SLL. The in-plane field also enhances the LLL FQHE states by making the quasi-2D system more purely 2D. The in-plane field effects could thus be qualitatively different in the LLL and the SLL by virtue of magneto-orbital coupling through the finite-thickness effect. Using exact diagonalization on the torus geometry, we show the appearance of the characteristic threefold topological degeneracy expected for the Pfaffian state. This signature is enhanced by nonzero thickness, corroborating our findings from overlap calculations. Our results have ramifications for wave-function engineering, opening the possibility of creating an optimal experimental system where the 5/2 FQHE state is more likely described by the Moore-Read Pfaffian state with obvious applications in the burgeoning field of fault-tolerant topological quantum computing.

It is generally agreed that there is large positional fluctuation in the ground state in the fractional quantized Hall effect. This is illustrated by a series of correlated Gaussian trial wave functions, one of which possesses an energy comparable to the Laughlin wavefunction as well as a gap in the excitation spectrum. The multiparticle exchange integral of this wave function is found to increase as the number of particles exchanged is increased. The shear modulus for this wave function as well as that of the Laughlin type wave function is calculated and is found to be finite. The effect of exchange is illustrated by considering the fractional quantized Hall effect in narrow channels. It is found that for narrow channels a gap exist in the excitation spectrum for even denominator filling factors. As the channel width is increased, this gap goes to zero and then becomes finite again.

The study of the electronic properties of quasi two dimensional systems has been a very exciting area of condensed matter physics during the last quarter of the 20th century. Among the most interesting discoveries in this area are the incompressible states showing integral and fractional quantum Hall effects. Incompressible quantum liquid states of the integral quantum Hall effect result from an energy gap in the single particle spectrum. The incompressibility of the fractional quantum Hall effect is completely the result of electron–electron interactions in a highly degenerate fractionally filled Landau level. Since the quantum Hall effect involves electrons moving on a two dimensional surface in the presence of a perpendicular magnetic field, we begin with a description of this problem.

We demonstrate that formulating the composite-fermion theory of the fractional quantum Hall (FQH) effect in terms of quaternions greatly expands its reach and opens the door into many interesting issues that were previously not amenable to quantitative theoretical investigation. As an illustration, we explore the possibility of a nematic or a charge-density wave instability of the composite-fermion Fermi sea at half-filled Landau level and of the nearby FQH states by looking for a gap closing instability of the neutral magneto-roton excitation. Our quaternion formulation of the FQH effect has been inspired by mathematical developments in the theoretical analyses of gravitational wave modes and cosmic microwave background radiation, where an important role is played by spin-weighted spherical harmonics that are nothing but monopole harmonics appearing in the spherical geometry for the FQH effect.

The recent discovery of the fractional quantum Hall effect (FQHE) at the even—denominator filling fraction v=5/2 represents yet another surprise in the area of two-dimensional electron physics. Previous to the work of WILLETT, et al.[1] the FQHE was strongly believed to occur solely at odd-denominator rational fractional fillings (v=p/q,q-odd) of the Landau levels. This belief was generated by both a wealth of experimental data and strong theoretical arguments[2] going as deep as the Pauli exclusion principle. How then does this new observation fit in with our overall understanding of the FQHE and 2D correlation effects in general? Here we present data[3] strongly suggesting that the underlying many-body ground state for the v=5/2 FQHE is not spin polarized, in contrast to the canonical v=1/3 and 2/3 incompressible quantum liquid states. Further, the apparent absence of even-denominator states in the lowest Landau level v<2 points to important differences between electron correlations in the first and second Landau levels. The purpose of the present paper is to review the existing experimental data[1,3] on the v=5/2 FQHE and show how this startling new result can, in principle at least, be accommodated by a generalized picture of the FQHE.

In the fractional quantum Hall effect, the elementary excitations are quasi-particles with fractional charges as predicted by theory and demonstrated by noise and interference experiments. We observe Coulomb blockade of fractional charges in the measured magneto-conductance of a 1.4-micron-wide quantum dot. Interaction-driven edge reconstruction separates the dot into concentric compressible regions with fractionally charged excitations and incompressible regions acting as tunnel barriers for quasi-particles. Our data show the formation of incompressible regions of filling factors 2/3 and 1/3. Comparing data at fractional filling factors to filling factor 2, we extract the fractional quasi-particle charge e*/e = 0.32 ± 0.03 and 0.35 ± 0.05. Our investigations extend and complement quantum Hall Fabry-Pérot interference experiments investigating the nature of anyonic fractional quasi-particles.

A computational scheme is suggested to estimate neutral excitation energies in the fractional quantum Hall effect (FQHE) states. The FQHE states are systematized according to the Farey-number hierarchy structure. We show that besides the widely known Laughlin–Jain hierarchy of fractional states, there exist other “dark” hierarchies. Although hardly observed in the highest mobility samples, they can significantly affect the thermodynamics and spectral characteristics of the FQHE states. The known problems in the interpretation of the FQHE’s experimental results are explained in terms of the coexistence of two fundamentally different transformations of the electron system, one of which is a neutral excitation in the FQHE state, whereas the other is a transition between two FQHE ground states, one of which represents the Laughlin–Jain FQHE hierarchy and the other a state of “dark” hierarchies.

We have studied the energy gap in the 1/3-filling fractional quantum Hall effect state in GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As heterojunctions with a carrier concentration tuned by a gate voltage. We have measured in situ (i) a fractional signature in a capacitance between a two-dimensional electron system and a metal gate, and (ii) the activation energy of magnetoresistance. From the capacitance data, using recently proposed assumptions, we have evaluated the chemical potential discontinuity in the 1/3-filling state. It turned out to be in good agreement with a value expected from the measured activation energy under the assumption of the fractional quasiparticle charge ${\mathit{e}}^{\mathrm{*}}$=\ifmmode\pm\else\textpm\fi{}e/3. This result suggests that the approach used can be employed to determine absolute values of both the energy gap and the fractional quasiparticle charge.

The fractional quantum Hall effect (FQHE) arises from strong correlations between electrons when they are confined to two dimensions and exposed to a strong magnetic field. The underlying physics is the formation of topological particles called composite fermions (CFs), electron-vortex bound states whose integer quantum Hall effect explains a large majority of the observed FQHE states. In recent years, the focus has shifted to the more exotic states that originate from a weak residual interaction between composite fermions. These include chiral p-wave paired states of composite fermions at certain even denominator fractions, unconventional FQHE of composite fermions, and a series of CF crystals at low fillings. Aside from these states, we also review the FQHE in multicomponent systems, which has attracted renewed attention because of the observation of well-developed FQHE in several multivalley systems, such as graphene and AlAs quantum wells.

Duality and the fractional quantum Hall effect

Unitary polynomials in normal matrix models and wave functions for the fractional quantum Hall effects

The prominent “1/3” effect observed in the Hall effect plateaus of two-dimensional electron gas (2DEG) systems has been postulated to indicating 1/3 fractional charge quasiparticle excitations arising from electron-electron interactions. Tunneling shot-noise experiments on 2DEF exhibiting fractional quantum Hall effect (FQHE) shows evidence for tunnelling of particles with eand e/3 charges for a constant band mass. A “1/3” effect in the hydrogen molecule is seen in as much as its internuclear distance,dH-H = D− + D+, with ¦D+/D−¦ = 1/3. This is examined in terms of electron-electron interactions involving electron-and hole quasiparticles, (e-)and(h+), equivalent to those observed in FQHE shot -noise experiments. The(e/m) ratio of the (e−) and(h+) quasiparticles is kept at 1: −3. Instead of a 2DEG, these particles are treated as being in flat Bohr orbits. A treatment in the language of charge-flux tube composites for the hydrogen atom as well as the hydrogen molecule is attempted. Such treatment gives important insights into changes in chemical potential and bond energy on crossing a phase boundary during the atom-bond transition as well as on models for FQHE itself.

The spin of the quasi-particle in the fractional quantum Hall effect

Nonabelions in the fractional quantum hall effect

We numerically study the fractional quantum Hall effect at filling factors $\nu=12/5$ and 13/5 (the particle-hole conjugate of 12/5) in high-quality two-dimensional GaAs heterostructures via exact diagonalization including finite well width and Landau level mixing. We find that Landau level mixing suppresses $\nu=13/5$ fractional quantum Hall effect relative to $\nu=12/5$. By contrast, we find both $\nu=2/5$ and (its particle-hole conjugate) $\nu=3/5$ fractional quantum Hall effects in the lowest Landau level to be robust under Landau level mixing and finite well-width corrections. Our results provide a possible explanation for the experimental absence of the 13/5 fractional quantum Hall state as caused by Landau level mixing effects.