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arXiv:hep-th/9202010 · International Journal of Modern Physics B

TOWARD A CONFORMAL FIELD THEORY FOR THE QUANTUM HALL EFFECT

GREG NAGAO

Published Oct 10, 1992• 0 citations
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Abstract

An effective Hamiltonian for the study of the quantum Hall effect is proposed. This Hamiltonian, which includes a "current-current" interaction has the form of a Hamiltonian for a conformal field theory in the large N limit. An order parameter is constructed from which the Hamiltonian may be derived. This order parameter may be viewed as either a collective coordinate for a system of N charged particles in a strong magnetic field; or as a field of spins associated with the cyclotron motion of these particles.

DOI
10.1142/s0217979292002462

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An effective Hamiltonian for the study of the quantum Hall effect is proposed. This Hamiltonian, which includes a "current-current" interaction has the form of a Hamiltonian for a conformal field theory in the large N limit. An order parameter is constructed from which the Hamiltonian may be derived. This order parameter may be viewed as either a collective coordinate for a system of N charged particles in a strong magnetic field; or as a field of spins associated with the cyclotron motion of these particles.
The QHE is studied in the context of a CFT. An effective field of NNN ``spins" associated with the cyclotron motion of particles is taken as an order parameter from which an effective Hamiltonian may be defined. This effective Hamiltonian describes the COM motion of the NNN particles (with coupling κ0κ_0κ0​) together with a current-current interaction (of strength κ1κ_1κ1​). Such a system gives rise to a CFT in the large NNN limit when κ0=κ1κ_0 = κ_1κ0​=κ1​. The Laughlin wavefunction is derived from this CFT as an N′N'N′-point correlation function of winding state vertex operators.
By using the Girvin–MacDonald effective Lagrangian for the Hall fluid system at filling ν = 1 we are able to evaluate the ground-state energy for a cylinder of infinite length. That is done by using the topological properties of the edge vacuum in the thermodynamic limit, Ne → ∞, ν = 1 fixed, From the vacuum energy we extract the value of the Casimir energy for both filling ν = 1 and ν = 1/m. The results obtained fix the central charge c of the underlying 2D Conformal Field Theory at c = 1. This correspondence is further analyzed by using the bosonic realization of the Kac–Moody algebra on the edge vacuum. Then the off-diagonal long-range order of the effective theory can be seen as an equivalent description of the topological order typical of the 2D CFT.
We extend the construction of the effective conformal field theory for the Jain hierarchical fillings proposed in Ref. 1 to the description of a quantum Hall fluid at nonstandard fillings [Formula: see text]. The chiral primary fields are found by using a procedure which induces twisted boundary conditions on the m scalar fields; they appear as composite operators of a charged and neutral component. The neutral modes describe parafermions and contribute to the ground state wave function with a generalized Pfaffian term. Correlators of Ne electrons in the presence of quasi-hole excitations are explicitly given for m=2.
The authors consider here the behavior of quantum Hall edges away from the Luttinger liquid fixed point that describes the low energy, large system limit. Using the links between quantum Hall wave functions and conformal field theories, they construct effective Hamiltonians and constrain their forms by considering bulk symmetries. In examining the effect of bulk interactions on this edge, they find remarkable simplifications to these effective theories that allow a very accurate description of the low-energy physics of quantum Hall edges relatively far away from the Luttinger liquid fixed point, and that apply to small systems and higher energies.
We extend the quantum Hall matrix model to include couplings to external electric and magnetic fields. The associated current suffers from matrix ordering ambiguities even in the classical model. We calculate the linear response at low momenta---this is unambigously defined. In particular, we obtain the correct fractional quantum Hall conductivity, and the expected density modulations in response to a weak and slowly varying magnetic field. In the quantum theory the ordering ambiguities are more severe, but we are able to construct a good density operator, satisfying the pertinent lowest Landau level commutator algebra, for the special case of $2\ifmmode\times\else\texttimes\fi{}2$ matrices. Our results support the conjecture that the quantum Hall matrix models do describe important aspects of the dynamics of electrons in the lowest Landau level.
The magnetic properties of a two-dimensional interacting electron system in a strong magnetic field are considered in the quantum Hall regime, where the motion of the electrons is restricted to the lowest Landau level. We apply an approach based on the many-particle quantum-field theory which we developed recently for the spin-polarized system in the fractional quantum Hall regime, and which treats the problem of the degeneracy of a partially filled Landau level properly. Via a gauge transformation of the fermion Green's function, we achieve a dressing of the propagator with multiple collective excitations. Our theory allows a calculation of the magnetic polarization M(T)M(T)M(T) for higher temperatures TTT, where $M(T)\ensuremath{\lesssim}0.4$ is not too large. For filling factor $\ensuremath{\nu}=1$ we obtain a magnetic polarization M(T)M(T)M(T) which agrees quite well with the data of a NMR experiment by Barrett [Phys. Rev. Lett. 74, 5112 (1995)].