Hamiltonian theory of gaps, masses, and polarization in quantum Hall states

Abstract

In two short papers I had described an extension, to all length scales, of the Hamiltonian theory of composite fermions ͑CF͒ that Murthy and I developed for the infrared, and applied it to compute finite-temperature quantities for quantum Hall fractions. I furnish details of the extended theory and apply it to Jain fractions ϭp/(2psϩ1). The explicit operator description in terms of the CF allows one to answer quantitative and qualitative issues, some of which cannot even be posed otherwise. I compute activation gaps for several potentials, exhibit their particle-hole symmetry, the profiles of charge density in states with a quasiparticle or hole ͑all in closed form͒, and compare to results from trial wave functions and exact diagonalization. The Hartree-Fock approximation is used, since much of the nonperturbative physics is built-in at tree level. I compare the gaps to experiment, and comment on the rough equality of normalized masses near half-and quarter-filling. I compute the critical fields at which the Hall system will jump from one quantized value of polarization to another, and the polarization and relaxation rates for half-filling as a function of temperature and propose a Korringa-like law. After providing some plausibility arguments, I explore the possibility of describing several magnetic phenomena in dirty systems with an effective potential, by extracting a free parameter describing the potential from one data point and then using it to predict all the others from that sample. This works to the accuracy typical of this theory ͑10-20 %͒. I explain why the CF behaves like a free particle in some magnetic experiments when it is not, what exactly the CF is made of, what one means by its dipole moment, and how the comparison of theory to experiment must be modified to fit the peculiarities of the quantized Hall problem.

Quantitative theory of composite fermions in Bose-Fermi mixtures at $ν=1$

“…In order to represent the Hamiltonian in terms of the CFs and CBs introduced earlier, we employ the density operators defined in Eqs. ( 15), ( 16), (31), and (32) which satisfy the GMP algebra. The corresponding Hamiltonian is…”

Collective excitations of quantum Hall states under tilted magnetic field

“…It provides a microscopic way to derive the CF picture, where the energy scale is correctly set to the Coulomb interaction instead of the kinetic energy. It is also useful in the calculation of the charge gap [27], the energy cost to create a wellseparated quasiparticle/quasihole pair, which is equivalent to add up the energy of creating a hole in the filled CF LL and the energy of creating a CF in the empty CF LL. In that case, a preferred density ρ p = ρ e − c 2 χ is employed in the interaction Hamiltonian Eq.…”