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Collective Excitations of Composite Fermions Across Multiple Lambda LETTERS PUBLISHED ONLINE: 10 MAY 2009 | DOI: 10.1038/NPHYS1275 Collective excitations of composite fermions across multiple 3 levels Dwipesh Majumder1, Sudhansu S. Mandal1 and Jainendra K. Jain2* The fractional quantum Hall state1 is a quintessential system kinetic energy levels in this reduced magnetic field, called 3 for the study of collective quantum behaviour. In such levels, and their filling factor ν∗ is related to the electron filling a system, the collective behaviour results in the creation factor ν through the relation ν D ν∗=(2pν∗ C 1). In particular, at of so-called composite fermions, quasi-particles formed by ν D n=(2pn C 1), the ground state consists of n filled 3 levels. In electrons attached to magnetic flux quanta. Recently, a the composite-fermion theory, the lowest-energy neutral excitation new collective mode was unexpectedly observed in Raman is a particle–hole pair, or an exciton, of composite fermions, scattering experiments2 on such a system as it was found to wherein a single composite fermion from the topmost occupied split off from the familiar ‘fundamental’ long-wavelength mode 3 level is excited into the lowest unoccupied 3 level (Fig. 1b on increase of the wave vector. Here, we present results from shows the fundamental composite-fermion exciton at ν D 2=5). extensive theoretical calculations that make a compelling case The validity of this description has been confirmed for fractions of that this mode corresponds to an excitation of a composite the form ν D n=(2pn C 1) by comparison to exact diagonalization fermion across two 3 levels—effective kinetic energy levels results as well as to experiment11,12. This physical explanation for resembling Landau levels for such particles. In addition to the neutral collective excitations is distinct from the single-mode explaining why this excitation merges with the fundamental approximation, and, in particular, suggests the possibility of extra mode in the long-wavelength limit, our theory also provides collective modes, in which a composite fermion is excited across a good quantitative account of the amount of splitting, and two or more 3 levels, as shown schematically in Fig. 1c, in complete makes several experimentally verifiable predictions. analogy to the collective modes of an integral quantum Hall Unlike the well-known quantum phenomena of superfluidity state13. However, it is far from obvious that the composite-fermion and superconductivity, the fractional quantum Hall effect1 does collective modes across different 3 levels should merge in the long- not entail any Bose–Einstein condensation but occurs as a wavelength limit. In fact, a model that takes composite fermions as result of the formation of topological electron–vortex bound non-interacting produces collective modes spaced by the effective states called composite fermions3. Transport4, light scattering5,6 cyclotron energy in the long-wavelength limit, as also found for and phonon scattering7,8 have been extensively used during the the dispersions obtained in the composite-fermion Chern–Simons past quarter of a century to probe its numerous excitations. approach14,15; if correct, this would make such physics irrelevant to Of particular significance is the neutral collective mode, which the new collective mode discovered in ref. 2. For a more definitive was first studied theoretically at Landau-level filling ν D 1=3 test, however, the composite-fermion exciton-mode spectrum must in a single-mode approximation9, wherein, following Feynman's be evaluated in a microscopic approach that includes effects of theory of the phonon–roton mode of helium superfluid, the inter-composite-fermion interactions. excitation is modelled as a density wave. The neutral collective Exact diagonalization studies of the FQHE state do not by mode was detected by Raman scattering5, with the observations themselves provide an understanding of the underlying physics, generally consistent with the predictions of the single-mode and are not useful in the present context, because, as seen below, approximation in the long-wavelength limit. More recently, systems as large as 200 particles are required for investigating however, Hirjibehedin et al.2 have discovered that this mode is not the experimentally relevant wave vectors; the Fock space increases a single mode, as believed earlier, but splits into two as the wave exponentially with the number of particles, and at present exact vector is increased. By definition, the single-mode approximation diagonalization is possible only for 10–12 particles for the filling cannot accommodate a doublet. A hydrodynamic approach has factors of interest here. Our quantitative investigations below been proposed10 to account for the experimental observation, exploit accurate trial wavefunctions for composite fermions3. The but does not take into account the microscopic physics of the standard spherical geometry is used in our calculations, which fractional quantum Hall effect (FQHE), does not naturally explain considers electrons moving on the surface of a sphere, subjected the merging of the two modes in the long-wavelength limit and also to a radial magnetic field. The magnetic field can be thought greatly underestimates the splitting. to emanate from a `magnetic monopole' of strength Q at the We show here that this new mode finds a natural explanation centre, which produces a total magnetic flux of 2Qφ0 through within the composite-fermion theory3. Composite fermions are the surface of the sphere. This maps into a system of composite bound states of electrons and an even number (2p) of quantized fermions at an effective flux Q∗ D Q − N C 1, with Q chosen vortices. Because of the Berry phases produced by the bound so that the state at Q∗ is an integral quantum Hall state at vortices, composite fermions effectively experience a much reduced filling ν∗ D n. The wavefunction 9CF-g for the FQHE ground ∗ magnetic field B D B − 2pρφ0 (B is the external magnetic field, state at ν D n=(2n C 1) is obtained by composite-fermionizing the ∗ g ρ is the two-dimensional density and φ0 D hc=e is called the ν D n integral quantum Hall state 8 . To model neutral collective flux quantum). Composite fermions form their own Landau-like excitations, we first construct wavefunctions of the excitons of the 1Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India, 2104 Davey Laboratory, Physics Department, Pennsylvania State University, University Park, Pennsylvania 16802, USA. *e-mail: [email protected]. NATURE PHYSICS j VOL 5 j JUNE 2009 j www.nature.com/naturephysics 403 © 2009 Macmillan Publishers Limited. All rights reserved. LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS1275 Λ abc 2 1 0 2/5 ground state 1 2 composite-fermion exciton 0 2 composite-fermion exciton Figure 1 j Schematic diagram of composite-fermion excitons. Each composite fermion is shown as an electron carrying vortices represented by arrows. a, Representation of the ground state at ν D 2=5 as two filled 3 levels. b,c, 1 ! 2 (b) and 0 ! 2 (c) composite-fermion excitons. a b 0.20 0.18 0.18 0.16 0.16 0.14 ) ) l l ε ε 0.12 / / 0.14 2 2 e e ( ( Δ Δ 0.12 0.10 0.10 0.08 0.08 0.06 0.06 0.04 0 0.5 1.0 1.5 2.0 2.5 3.0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ql ql c d 0.010 0.03 N = 100 N = 50 0.008 0.02 0.006 0.004 0.01 Spectral weight Spectral Spectral weight Spectral 0.002 0 0 0 0.5 1.0 1.5 2.0 2.5 3.0 0 0.1 0.2 0.3 0.4 0.5 ql (ql)4 Figure 2 j Dispersions of several composite-fermion excitons and their spectral weights. a, The three lowest composite-fermion exciton modes at ν D 1=3, obtained from 0 ! 1, 0 ! 2 and 0 ! 3 excitons. The error bar at the end of each curve represents the typical statistical error in the energy 2/` determinedp by the Monte Carlo method. The energies are quoted in units of e , where is the dielectric constant of the background semiconductor and ` D hcN =eB is the magnetic length. b, The three lowest composite-fermion exciton modes at ν D 2=5, obtained from 1 ! 2, 0 ! 2 and 1 ! 3 excitons. c, Spectral weights for the three modes at ν D 1=3 for N D 100. The curves from top to bottom correspond to the three modes shown in a respectively from bottom to top. d, The spectral weight for the lowest-energy composite-fermion exciton at ν D 1=3 at small q` for N D 50 and N D 100. All results in this CF-ex figure are for the Coulomb eigenstates χλ . f8ex g integral quantum Hall state, denoted by λ,L , where L is the total Even though our immediate interest is in understanding the orbital angular momentum of the exciton and λ labels different splitting of the collective mode at ν D 1=3, we consider, for excitons of the type shown in Fig. 1. We composite-fermionize completeness, the three lowest collective modes at two filling f9CF-exg ! ! ! ν D = ! this basis to obtain λ,L , which gives a set of basis functions factors: 0 1, 0 2 and 0 3 modes at 1 3, and 1 2, for composite-fermion excitons. We orthonormalize this basis and 1 ! 3 and 0 ! 2 modes at ν D 2=5. We have studied systems diagonalize the Coulomb Hamiltonian to obtain the energies of with 50, 100 and 200 particles at both ν D 1=3 and ν D 2=5.
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