Hamiltonian theories of the fractional quantum Hall effect

This paper reviews progress on the fractional quantum Hall effect (FQHE) based on what we term Hamiltonian theories, i.e., theories that proceed from the microscopic electronic Hamiltonian to the final solution via a sequence of transformations and approximations, in either the Hamiltonian or path-integral approach, as compared with theories based on exact diagonalization or trial wave functions. The authors focus on the Chern-Simons approach, in which electrons are converted to Chern-Simons fermions or bosons that carry along flux tubes, and on their own extended Hamiltonian theory, in which electrons are paired with pseudovortices to form composite fermions whose properties are a lot closer to the ultimate low-energy quasiparticles. The article addresses a variety of qualitative and quantitative questions: In what sense do electrons really bind to vortices? What is the internal structure of the composite fermion and what does it mean? What exactly is the dipole picture? What degree of freedom carries the Hall current when the quasiparticles are localized or neutral or both? How exactly is the kinetic energy quenched in the lowest Landau level and resurrected by interactions? How well does the composite-fermion picture work at and near $\nu =1/2?$ Is the system compressible at $\nu =1/2?$ If so, how can composite fermions be dipolar at $\nu =1/2\mathrm{}$ and still be compressible? How is compressibility demonstrated experimentally? How does the charge of the excitation get renormalized from that of the electron to that of the composite fermion in an operator treatment? Why do composite fermions sometimes appear to be free when they are not? How does one compute (approximate) transport gaps, zero-temperature magnetic transitions, the temperature-dependent polarizations of gapped and gapless states, the NMR relaxation rate ${1/T}_1$ in gapless states, and gaps in inhomogeneous states? It is seen that though the Chern-Simons and extended Hamiltonian approaches agree whenever a comparison is possible, results that are transparent in one approach are typically opaque in the other, making them truly complementary.