Constraint equation for the lowest Landau level in the fractional quantum Hall system.

Abstract

Listen

Translate article

We apply the singular gauge transformation to the constraint equation for the lowest Landau level to investigate the generic properties of the quasiparticles of the fractional quantum Hall system by using the collective-field-theory approach. It shows a transparent connection with Laughlin s wave functions. If we take an average over the wave functional for the constraint equation, the resulting equation can be interpreted as the vortex equation for the fractionally charged quasiparticles. By introducing a generalized duality transformation, p(density)-8(phase) transformation, we can obtain the fractional statistics as well as the hierarchy scheme from the constraint equation. At present we have a fairly good understanding of the fractional quantum Hall effect' (FQHE) based on Laughlin's wave-function approach. Stimulated by an investigation of the off-diagonal long-range order for the FQHE, Girvin, and Girvin and MacDonald and others proposed a Ginzburg-Landau-like effective-field theory of the FQHE. This is in fact an eff'ective-field theory of a (2+1)-dimensional Schrodinger-Bose wave field coupled with a Chem-Simons gauge field that exhibits vortex solutions with finite energy and fractional charge. Although these are appealing, their connection to Laughlin's wave-function approach is not entirely clear. In particular, it is still not very clear to what extent the lowest Landau level (LLL) constraint has been properly accounted for. At the lowest Landau level the kinetic-energy part of the electron Hamiltonian, which usually bears the gauge properties of the system, should be projected away with only the Coulomb interaction term left. Therefore it is reasonable to expect that a rich variety of interesting properties for the FQHE systems, which are independent of the concrete form of the interactions, can be explored via the application of the Chem-Simons gauge-fieldtheory approach to the constraint for the electrons in the LLL. In this paper, we express the condition that all the electrons are in the LLL as a constraint equation for the state functional and apply the collective-field-theory technique ' which in fact is a field-theoretic approach, to this equation. The general solution of this constraint equation shows a transparent connection to Laughlin's wave function. If we take a quantum average of the constraint equation, the resulting equation can be interpreted as a vortex equation for the fractionally charged quasiparticles that is derived directly from the microscopic theory. The Coulomb interaction does not appear explicitly in the equation. The vortex solutions in turn require singularities of the phase of the wave functional. In the particle coordinate representation these singularities are pre cisely the ones that appear in the singular exponent of the prefactor of Laughlin's quasiparticles wave function. Next we introduce a generalized duality transformation as a generalized Fourier transformation — p(electron density)-8(phase variable conjugate to the electron density) transformation — of the state functional. Then the transformed constraint equation not only makes the fractional charge and statistics of quasiparticles (vortices) explicit, but also provides a natural derivation for the hierarchy scheme. We introduce the Hamiltonian for the interacting Xelectron system in a strong uniform magnetic field B by subtracting the zero-point energy for the lowest Landau level NAeB/2moc as