Electrodynamical properties of gapless edge excitations in the fractional quantum Hall states.

We propose a scheme to construct the edge states for the Fractional Quantum Hall (FQH) states obtained by the Jain’s scheme. The low energy effective theory of the edge excitations is obtained. We calculate the number of the branches and the electron propagators of the edge states for various FQH states. We demonstrate that two FQH states with the same filling fraction may have different topological orders and may support different edge excitations. PACS numbers: 73.20.Dx, 73.20.Mf, 73.40.Kp * Published in Mod. Phys. Lett. B 5 39 (1991) Recently the edge excitations in the Fractional Quantum Hall (FQH) states have attracted a lot of attentions. The dynamical properties of the edge excitations in FQH states have been studied in Ref. 1,2,3,4,5 with help of the Kac-Moody (K-M) algebra. It is shown that the electrons on the edge of a FQH states form a new kind of quantum liquid state4 which cannot be described by the Fermi liquid theory. DC transport properties of the edge channels were discussed in Ref. 6,7,8. The experimental and the theoretical results about the edge excitations in the Integral Quantum Hall (IQH) states can be found in Ref. 9 and Ref. 10. In Ref. 4 it is argued that the QH states with filling fraction ν 6= 1/m 11,12 must have more than one branch of the edge excitations. In this letter we will discuss in detail the edge excitations in these generic QH states. Some aspects of the edge excitations of the hierarchy QH states have been studied in Ref. 8. Here we will give a different approach which leads to a more detailed picture about the edge excitations. The properties of the edge excitations are not only determined by the filling fraction, but also determined by the internal structures of the FQH wave function. Such internal structures may be called the topological orders in the FQH states13,14. The filling fraction alone does not provide enough information for the edge excitations. To determine the properties of the edge excitations, we must also specify the topological orders in the FQH states. In this letter we will specify the topological orders by specifying a particular way to construct the FQH wave functions. We are going to use the Jain’s sheme proposed in Ref. 12. More general characterizations of the topological orders can be found in Ref. 13,14. Let us first review Jain’s construction using a FQH state with filling fraction ν = 5 as an example. His trick is to view the electrons as bound states of a charge e 5 fermion and a charge 4e 5 boson. The filling fraction ν and the electron density ne is related by ν = 1 e hc B ne. Since the e 5 fermions and the 4e 5 have the same densities as the electrons nf = nb = ne, the e 5 fermions have a filling fraction νf = 5 e hc B nf = 2 and the 4e 5 bosons have a filling fraction νb = 5 4e hc B nb = 1 2 . We start from the limit in which the fermions and bosons have no interactions and there is a hardcore repulsive interaction between the bosons. In this limit the bosons form a νb = 1 2 FQH state described by the Laughlin’s wave function Ψ1/2(z b i ) = [ ∏ i<j (zb i − zb j)]exp(− 1 4 ∑ i 4e 5 B h̄c |zb i |2) (1) and the fermions form a νf = 2 IQH state described by wave function Ψ2(z f i ). Here z b i and z f i are coordinates of the bosons and the fermions. Now let us bind the fermion and the boson together. This is equivalent to suppress the nb−nf fluctuations (i.e., nb−nf = 0). In this case the low lying excitations are described by the wave functions of the bound states, or equivalently, the wave function of the electrons. The above picture leads to a trial wave function for the ground state Ψ(zi) = Ψ1/2(z b i )Ψ2(z f i )|zb i =z i =zi (2) where zi are coordinates of the electrons and zb i = z f i is the projection to the nf − nb = 0 sector. (2) is exactly the wave function constructed by Jain for the ν = 5 FQH state. 2 The quasi particle excitations with electric charge 5 e can be shown to have a statistics θ = 5πm 2 in such a state.15 In our notation Ψν is the QH wave function with a filling fraction ν for the fictitious particles. The charge of of the fictitious particles are chosen such that Ψ1/2 and Ψ2 have the same particle density despite they have different filling fractions. The essence of the above construction is the following. We first decompose the electrons into fictitious particles. Those fictitious particles have simple filling fractions such that one can easily write down their wave functions. By introducing the fictitious particles, we introduce some unphysical degrees of freedom. We need to make a projection to project away all the unphysical degrees of freedom and to obtain the correct physics for the electrons. In particular the ground state wave function is obtained by doing such a projection (see (2)). The Jain’s scheme12 is very convenient for the construction of the edge excitations. This is because the projection can be done at the effective theory level. Let us first discuss the edge excitations in the ν = n mn+1 FQH state (m is an even integer). The electrons are decomposed into fermions with a charge q1 = ν ne and bosons with a charge q2 = νme. The electron wave function of the FQH state is given by Ψ(zi) = Ψn(zi)Ψ1/m(zi) (3) Let us assume that the FQH state has a disk-like geometry. The boundary of the disk is parameterized by x. In Ref. 1,4 It is shown that the edge excitations are described by the U(1) K-M algebras. Those edge excitations can be regarded as surface waves propagating along the boundary of the incompressible QH fluid.2,5 The surface wave can be described by the “edge density” ρ(x) = neh(x) where h(x) is the displacement of the edge. Note ρ is an one dimensional density with dimension 1/[length]. Before the projection, the charge q2 bosons form a νb = 1 m FQH state and support a single branch of edge excitations. Those excitations are described by the the following K-M algebra1,3,4 [ρ0,k, ρ0,k′] = 1 m k 2π δk+k′ (4) where ρ0 is the edge density of the bosons. The boson creation operator on the edge is given by4 ψ0 =: eimφ0 : where ∂xφ0 = 2πρ0. ψ0 carries an electric charge q2. The charge q1 fermions form a νf = n IQH states which support n branches of edge excitations.10 Those edge excitations are described by [ρi,k, ρi′,k′ ] = k 2π δk+k′δi,i′, i, i ′ = 1, 2, . . . , n (5) The fermion creation operators on the edge are given by ψi =: eiφi : with ∂xφi = 2πρi, i = 1, . . . , n. They carry an electric charge q1. ρi and ψi are the edge density and the fermions of the ith Landau level. The coupling between the edge densities and the external electric potential is given by (q2ρ0 + n ∑