Quantum Hall physics

A two-dimensional electron gas in a strong magnetic field has discrete energy levels (Landau levels) separated by the cyclotron energy. In a homogeneous system, each level has a macroscopic degeneracy that is proportional to the area of the system. This means that the system is highly sensitive to perturbations such as Coulomb interactions and disorder. As a result, exotic grounds states emerge from the manifold of degenerate states in the lowest Landau level.

In a non-interacting system, the degeneracy of the Landau levels can be lifted by disorder and the Zeeman effect. Each level splits into two spin-polarized levels, each of which is broadened by disorder into a band of states ("Landau bands"). These states are localized in space except at one energy at the centre of each spin-split Landau band. The Hall conductance is quantized in units of e²/h (integer quantum Hall effect) when the Fermi level is in the localized part of the band. A jump from one quantum Hall plateau to another occurs when the Fermi level crosses the special delocalized state at the centre of each band. This sharply-defined location of the delocalized state in the spectrum is responsible for the robustness of the integer quantum Hall effect to disorder. In fact, this phenomenon now provides the metrological standard for the measurement of the electronic charge.

[ Further reading on IQHE: Nobel Prize 1985. ]

Analogous to Hund's rule in atomic physics, Coulomb interactions give rise to ferromagnetic exchange between the electron spins. This gives us a quantum Hall ferromagnet at integer fillings, even in the absence of Zeeman coupling to polarize the electron spins. This becomes important in systems where the g-factor for the Zeeman coupling is weak. The important low-energy excitations are topological spin textures (or skyrmions). Each skyrmion has a large spin quantum number but carries only unit electrical charge. The current challenge is to understand how these magnetic correlations might affect the quantum Hall physics in these systems.

Disorder is an important ingredient in understanding experimental studies of the quantum Hall ferromagnets. We have made progress in understanding the phase diagram as a function of disorder and electron density. We model the system as a Heisenberg ferromagnet, together with a disordered potential nucleates skyrmions and antiskyrmions. We find that strong disorder destroys the ferromagnetic state. This gives rise to a spin glass where the local magnetic moments are frozen into random directions so that the system has zero net magnetization. We have proposed a scaling picture for the structure of the phase diagram. This has been supported by a Monte Carlo simulation of finite-size systems.

• S. Rapsch, J.T. Chalker and D.K.K. Lee, "Spin textures, screening and excitations in dirty quantum Hall ferromagnets",  Phys. Rev. Lett.  88, 036801 (2002). 
• D.K.K. Lee, S. Rapsch and J.T. Chalker, "Dirty quantum Hall ferromagnets and the quantum Hall spin glasses",  Phys. Rev. B 67 195322 (2003). 

Even in the absence of Coulomb interactsion, disorder itself radically modified the wavefunctions of the states in each Landau band. The classic integer quantum Hall effect arises from spin-polarized bands. However, if the disorder broadening of the band becomes comparable to the Zeeman splitting, spin-orbit scattering between these spin-degenerate Landau bands becomes important.

One can ask whether the delocalized states from each spin state survive as two separate states, or whether only one delocalized state remains after strong spin mixing. Surprisingly, the network model I have studied in relation to localization in random magnetic fields gives us an answer to this problem as well. We have found that the system does have a pair of delocalization transitions. They are now positioned away from the band centre, but they remain in the same universality class as the spin-split case. 

• D.K.K. Lee and J.T. Chalker, "A unified model for two localization problems - electron states in spin-degenerate Landau levels and in a random magnetic field",  Phys. Rev. Lett.  72, 1510 (1994). 
• D.K.K. Lee, " Degenerate Landau bands with interband disorder: a semiclassical picture",  Phys. Rev. B.  50, 7734 (1994). 

This prediction of two separate critical points is consistent with the general belief that the Hall conductance jumps by single units of e²/h in the quantum Hall transitions described by single-particle localization theory. However, larger jumps in the Hall conductance have been observed experimentally, notably at weaker fields. In these weak-field systems, it is important to consider take into magnetic correlations among the electron spins which might not be completely polarized by the external magnetic field. This brings us back to the physics of the quantum Hall ferromagnet (see above). We need to understand how the magnetic correlations affect the transport properties and the localization transition in the quantum Hall system.