(Fractional) quantum Hall effect (1): Why is it interesting?

The quantum Hall effect was discovered in 1980 and was awarded the Nobel Prize in Physics just five years later. This exotic quantum effect opened a new field of physics entangled with topology, a big part of mathematics. Soon after this discovery, fractional quantum Hall (FQH) effect and high-temperature superconductors were found, in 1982 and 1986, respectively, highlighting the intricacies of strongly correlated electron systems, which can lead to completely unexpected physical properties. All have their own Nobel Prizes. This series of scientific explorations marks the last crazy time of innovations in physics in the late 20th century.

Fig. 1. A simple setup for the Hall effect. When the applied magnetic field is strong enough at low temperatures, the whole world of quantum phenomena opens.

Quantum Hall effect

While it is easy to see that high-temperature superconductors are important, it may be harder for laypeople to understand why the quantum Hall effect is so critical to science and technology. The quantization of the Hall resistance of a "dirty" device is shockingly robust and precise, being 25 812.807... Ω, so one per billion precision. A system with ~10^23 atoms with all the defects and impurities gives an exact number in a macroscopic measurement that only high-energy experiments with fundamental particles could achieve!

Soon after this experimental discovery, a theory was formalized to explain this phenomenon in the framework of Landau levels. Landau levels are the quantized energy levels in which electrons can occupy when a 2D electron gas experiences a strong magnetic field in the perpendicular direction. The Landau levels are filled with integer numbers of electrons; hence, the quantizations are scaled by integer factors: 0, 1, 2, 3,.... At the lowest filling factor (the ratio between the number of electrons and flux quanta), ν = 0 to 1, every electron has its own flux quantum, so they are in the lowest Landau level. The collective motions of electrons in this 2D plane can generate chiral edge currents on the edges of the devices. These currents suffer from no backscattering (sounds like a superconductor); therefore, they are of tremendous technological interest.

Fig. 2. Quantum Hall effect with chiral edge currents. These currents do not have backscatterings. Therefore, they have many potential applications in technology, including quantum information.

Fractional quantum Hall effect

The quantum Hall effect (QHE) is already interesting; the FQH effect is even more. To look back, the theory of the quantum Hall effect relies on the weakly interacting electron gas; thus, the phenomenon happens in dilute 2D electron systems. Then advances in semiconducting technology allowed scientists to produce a much higher level of charge mobilities than doped Si, such as in GaAs film. In this system, electrons are dense and they interact strongly. And now comes the interplay between the electron correlation and the Landau energy.

Once the electron correlation (Coulomb interaction) is strong enough, we cannot neglect it. Electrons now no longer ignore each other to sit in their own Landau levels. They start to push each other away. The results are the gaps in the energy spectrum at unexpected positions, at fractional filling factors. Therefore, we observed quantized Hall resistance at these fractional locations. The electrons now seem to acquire fractional charges if we talk about a similar picture in the integer QHE.

Fig. 3. The zoo of FQHE states, here, all states found are odd-denominator states. Credit figure: Stormer, Physica B: Condensed Matter, 177(14), 401-408 (1992).

The FQHE is still an active area in condensed matter theory, as many questions are yet to be answered, due to the complexity of the problem. Normally, when a standard system is subject to some complicated add-ons, perturbation theories are exploited, as long as the added terms are weak compared to the original Hamiltonian. In FQHE, this method doesn't work since the Coulomb interaction is comparable to Landau energies. Additionally, the degeneracies of the ground states of QHE further quench any hopes in this method.

This obstacle did not stop theorists from devising new ways to approach the problem. Robert Laughlin came up with an educated guess of trial wavefunctions, which he derived from the known wavefunction of the integer QHE in an annulus geometry. In more detail, he attaches a number of magnetic flux quanta to an electron and rewrites the wave function in the similar fashion. In this picture, the flux quanta help electrons better avoid each other, thus minimizing the Coulomb interactions. The flux quanta play the role of the charge drainage, effectively reducing the electron charge. This theory first successfully describes the FQHE at filling factor ν = 1/3 as each electrons are attached with 2 flux quanta, while the last one is responsible for the Landau quantization.

Fig. 4. Fractional quantum Hall effect at filling factor ν = 1/3 with 2 flux quanta attached per electron. The newly formed composite fermions are now at the new filling factor ν* = 1.

The success of this theory is the explanation of the FQHE at odd-denominator filling factors (most of the states), which appear a lot as a "zoo" (that's how physicists call them). This comes from the composite fermion theory, in which electrons need to be accompanied by an even number of quantum fluxes to form a new composite fermion; and there's one to create Landau levels for these composite fermions to experience their new QHE. So, there're even + 1 = odd number of flux quanta needed, hence odd denominators.

Fractional statistics

With fractional charges in 2D, the statistics of these particles are not ordinary. They are no longer normal fermions or bosons. Indeed, now they are called anyons as they can acquire any phase when two particles exchange positions, depending on their fractional charges. The special cases are non-Abelian anyons (which I wrote about their usefulness in quantum computation in an earlier post). They not only acquire a phase but also transform into other degenerate peers (QHE is highly degenerate). Transformations of a quantum state involve matrix operations, and matrix multiplication is non-commutative. There comes the name "non-Abelian". These states indeed are predicted to involve even-denominator FQHE, which I will talk about in the next post.