Landau Level Mixing and Solenoidal Terms in Lowest Landau Level Currents

We calculate the lowest Landau level (LLL) current by working in the full Hilbert space of a two dimensional electron system in a magnetic field and keeping all the non-vanishing terms in the high field limit. The answer a) is not represented by a simple LLL operator and b) differs from the current operator, recently derived by Martinez and Stone in a field theoretic LLL formalism, by solenoidal terms. Though that is consistent with the inevitable ambiguities of their Noether construction, we argue that the correct answer cannot arise naturally in the LLL formalism. Typeset using REVTEX 1 The microscopic theory of the quantum Hall effect (QHE) [1] is constructed around a large magnetic field (B) expansion which reflects the fact that the QHE is a strong field phenomenon. At integer QHE filling factors (ν) the strong field states are non-degenerate eigenstates of the kinetic energy—filled Landau levels—and further terms in the expansion, obtained by perturbing in the interaction, serve largely to renormalize excitation energies. At the fractional fillings the eigenstates of the kinetic energy are macroscopically degenerate and the lowest order solution requires degenerate perturbation theory in the interaction in a given Landau level. This is the step where the novel physics arises; inclusion of Landau level mixing is, as in the IQHE, qualitatively unimportant. It is worth emphasizing that the irrelevance of LL mixing is really an example of adiabatic continuity, for the ratio (e/ǫl)/h̄ωc (l = √ h̄c/eB is the magnetic length, h̄ωc = eB/mc is the cyclotron frequency calculated with the band effective mass and ǫ is the background dielectric constant) that, naively, controls the mixing is O(1) in the experiments and is not negligible [2]. The solution of the degenerate problem for ν < 1, and by extension for other fractions, is greatly facilitated by restricting the Hilbert space to the lowest Landau level (LLL) and working with projected operators [3]. This procedure is not without its subtleties for restricting the Hilbert space restricts the intermediate states that occur in evaluating the product of a string of operators and hence the product of the projections does not, in general, equal the projection of the product; the simplest instance of this effect is that the projected particle co-ordinates obey the equal time commutator [x, y] = il. This letter is concerned with another, unphysical, instance that was first noted by Girvin, MacDonald and Platzman [4]. If HL, ρL(r) and jL(r) are the projected Hamiltonian, density and current operators then, ∇ · jL(r) = 0 while ∂tρL(r) = i h̄ [HL, ρL(r)] 6= 0 (1) thus violating continuity. These authors noted that for smoothly varying potentials V (r), ∂tρL(r) could be rearranged into the divergence of the drift current jD(r) = (c/B )(B × ∇V (r))ρL(r). Subsequently, Sondhi and Kivelson [5] showed that the drift current could be obtained as a LL mixing contribution that survives at arbitrarily high fields. (Readers 2 unfamiliar with this issue may find it helpful to skim the concluding discussion at this point.) Recently, Martinez and Stone [6] used a field theory incorporating the LLL constraint to construct a LLL Noether current for the potential scattering problem that (automatically) satisfies continuity. Subsequently, Rajaraman [7] derived the same operator as well as a form valid for interacting systems by working directly with the projected equations of motion as in the usual derivation of the Schrödinger current. In this communication we take a different tack: we calculate the LLL current by working in the full Hilbert space of the system and keeping all the non-vanishing terms in the high field limit. We show that our answer, which is the correct physical current, differs from the the results in [6,7] by solenoidal terms that do not appear to arise naturally in the projected problem. Furthermore, the current in an arbitrary state, generically, has a (solenoidal) contribution that is not calculable in advance of diagonalizing the projected Hamiltonian and hence it is not possible to specify a useful form for a current operator in the LLL that reproduces the exact answer. We return to the implications of our results at the end. Projection: We begin by reformulating, following [5], the high field dynamics as a purely LLL problem; this is an ancient technique, we provide details solely in the interests of clarity [8]. The Hamiltonian is, H = 1 λ HK + V (2) where the kinetic term, HK = 1 2m ∫ dr ψ(r) [ h̄ i ∇− e c Ab(r) ]2 ψ(r) (3) includes the vector potential (Ab) of the uniform magnetic field while, V = ∫ dr U(r)ψ(r)ψ(r) + 1 2 ∫ drdr ψ(r)ψ(r)V (r − r)ψ(r)ψ(r) (4) includes the interactions as well as any one-body potentials. The parameter λ serves, formally, to organize the large field expansion which is then a small λ expansion. The restriction to the LLL is acheived by constructing a hermitian generator T such that, 3