Quantum field theory and composite fermions in the fractional quantum Hall effect

Abstract

We derive a composite fermion model for the fractional quantum Hall effect from relativistic quantum electrodynamics. With simple arguments it is shown that a special Chern Simons transformation of the Dirac electrons in four spacetime dimensions leads in the low energy limit to a single particle Hamiltonian for composite fermions in three dimensions with correction terms such as Rashba-or Dresselhaus-spin-orbit coupling and zitterbewegung. Furthermore we provide a mechanism to quantum-mechanically project the quantum fields defined in the four dimensional Minkowski space to three dimensions. This leads to a relativistic field theory and especially a composite fermion field theory in three dimension. This projection map can be combined with the projection onto a Landau level or composite fermion Landau level respectively. This results in a quasi relativistic quantum field theory on a noncommutative plane. The phenomenological models resulting from this approach are discussed and allow a systematical exploration of the effects of the spin and the condensation in a Landau level. We expect from the relativistic approach corrections in terms of spin-orbit coupling effects. From the projection onto Landau levels we expect a modification of the dispersion relation and a modified composite fermion mass. Furthermore, the BRST quantization for Chern Simons theories with compact gauge group is reviewed and the phenomenological consequences within a composite fermion model with spin are discussed. The connection to Wess Zumino Witten theories is recalled and a possible link between the corresponding central charge of the related affine Lie algebra and the composite fermion filling factor is pointed out.

Composite Fermions QED Lagrangian Density in Fractional Formulation

“…Composite Fermions are a theoretical concept in condensed matter physics that explains the behavior of electrons when subjected to a strong magnetic field [1][2][3][4]. These electrons can form composite particles with unique physical properties, such as those seen in fractional quantum Hall states [5][6][7].…”

Determining the filling factors of fractional quantum Hall states using knot theory

“…Moreover the theory of classical vortices has been formulated in terms of knots [28,37]. Some progress with quantum vortices such as thouse in superconductivity can be inflicted from a Chern-Simons theory [38][39][40][41]. Amazingly path integrals of a Chern-Simons theory has explicitly shown the tipical skein relations of Jones polynomials which are invariants of knots [42][43][44].…”