Quantum Hall ferromagnets

Quantum Hall (QH) systems continuously provide us with fascinating phenomena both physically and mathematically. They have received renewed interest owing to the discovery of quantum coherence associated with the spin and layer degrees of freedom. They have also proved to be ideal systems to play with noncommutative geometry. When an electron is confined within the lowest Landau level, its position is described solely by the guiding center, whose X and Y coordinates do not commute with one another. Hence, the QH system is formulated as a dynamical system in the noncommutative plane. We construct the microscopic theory of the QH system based on noncommutative geometry. Although the microscopic theory is necessary to derive some key formulae, it is intuitively clear to use the composite-boson theory to understand the mechanism how quantum coherence develops spontaneously. In the spontaneously broken phase of the spin SU(2) symmetry, there arises a topological soliton flipping several spins coherently. It is the quasiparticle (charged excitation) called a skyrmion. Skyrmions have been experimentally observed both in integer and fractional QH systems. More remarkable is the bilayer QH system, where the layer degree of freedom acts as the pseudospin. Due to the parallelism between the spin and the pseudospin, in the spontaneously broken phase of the pseudospin SU(2) symmetry, the Goldstone mode is the pseudospin wave, and the quasiparticle is a topological soliton to be identified with the pseudospin skyrmion. A new feature is the phase current, which induces anomalous behavior of the Hall resistance in a counterflow geometry. Another new feature is the tunnelling current, which demonstrates the Josephson-like phenomena. Furthermore, the parallel magnetic field penetrates between the two layers, and forms a soliton lattice beyond the commensurate-incommensurate phase-transition point. There are experimental indications for the phase current, the dc Josephson current, the pseudospin skyrmion and the soliton lattice.