Non-Universal Fractional Quantum Hall States in a Quantum Wire

The ground state as well as low-lying excitations in a 2D electron system in strong magnetic fields and a parabolic potential is investigated by the variational Monte Calro method. Trial wave functions analogous to the Laughlin state are used with the power-law exponent as the variational parameter. Finite size scaling of the excitation energy shows that the correlation function at long distance is characterized by a non-universal exponent in sharp contrast to the standard Laughlin state. The Laughlin-type state becomes unstable depending on strength of the confining potential. Typeset using REVTEX 1 The fractional quantum Hall (FQH) effect occurs in two-dimensional electron systems under strong magnetic field [1]. The ground state is described as an incompressible quantum liquid with an energy gap [2]. On the edge of the FQH state, however, gapless excitations exist. Hence the edge excitations play an important role in low-energy response of the FQH system. In a large system, the edge excitations are described as the chiral TomonagaLuttinger (TL) liquid. It has been discussed that the long-distance behavior in the edge state is universal in the sense that the exponent is determined by the filling alone and is independent of other details of the system [3]. In a mesoscopic system where the length scale can be made comparable to the radius of the cyclotron motion, the FQH state should be affected by the finite dimension of the system. In this paper we study the FQH state in a quantum wire which can be fabricated as a conducting channel in the semiconductor heterojunction. If the channel is wide enough, the chiral nature of the edge excitation should remain. In a narrow channel, on the contrary, two edges are not independent objects. Then the universality of the asymptotic behavior might be lost. The unique property of the quantum wire is that one can control the strength of interaction between the edges, and hence the chiral nature. This property is not shared with more popular geometries like a disk or a semi-infinite plane. The quantum wire in a strong magnetic field has been studied theoretically by Yoshioka with use of the exact diagonalization method [4]. He has shown that a FQH state is realized under suitable condition with respect to the strength of a confinement potential and the Coulomb repulsion. The exact diagonalization can deal with only small systems containing several electrons. As an alternative we use in this paper the variational Monte Calro (VMC) method [5,6] to investigate much larger systems with O(10) electrons. We aim at deriving explicitly the non-universal property of the FQH state in the quantum wire, and show how the property depends on the size of the system. For this purpose we investigate not only the ground state but low-lying excitations for the Laughlin-type wave function [2]. Consider a two-dimensional electron system under a strong magnetic field and a parabolic potential in the y-direction. There is no potential in the x-direction for which we impose the 2 periodic boundary condition with length Lx. The magnetic field B is applied antiparallel to the z-direction. Then the one-electron Hamiltonian in the Landau gauge is written as H0 = 1 2me [ (px + e c By) + py ] + 1 2 meω 2 0y 2 , (1) where me is the electron mass and ω0 is the strength of confinement potential. The Hamiltonian is rewritten for later convenience as H0 = − 1 2 h̄Ω [ 1 μ (λ∂x + i y λ ) + μλ∂ y ] − h̄ 2 2meλ ω 0 Ω2 λ∂ x . (2) We have introduced μ = l/λ = √ 1 + (ω0/ωc) and Ω = √ ω2 c + ω 2 0, where l = √ h̄c/eB and ωc = eB/mec. Ω and λ give the effective cyclotron frequency and the Larmor radius in the presence of the parabolic potential. The eigenstate φn,k and the eigenvalue En,k of this Hamiltonian are obtained as