Abstract
The fractional quantum Hall effect has inspired searches for exotic emergent topological particles, such as fractionally charged excitations, composite fermions, abelian and nonabelian anyons and Majorana fermions. Fractionally charged skyrmions, which support both topological charge and topological vortexlike spin structure, have also been predicted to occur in the vicinity of 1/3 filling of the lowest Landau level. The fractional skyrmions, however, are anticipated to be exceedingly fragile, suppressed by very small Zeeman energies. Here we show that, slightly away from 1/3 filling, the smallest manifestations of the fractional skyrmion exist in the excitation spectrum for a broad range of Zeeman energies, and appear in resonant inelastic light scattering experiments as welldefined resonances slightly below the long wavelength spin wave mode. The spectroscopy of these exotic bound states serves as a sensitive tool for investigating the residual interaction between composite fermions, responsible for delicate new fractional quantum Hall states in this filling factor region.
Introduction
Skyrmions represent vortexlike spin structures in two dimensions, which are twodimensional stereographic projections of the spin hedgehog on a sphere. In a pioneering work, Sondhi et al.^{1} predicted integrally charged skyrmions for the quantum Hall ferromagnet near filling factor ν=1. These arise because the ν=1 integer quantum Hall effect state exhibits spontaneous ferromagnetism even in the absence of a Zeeman energy with the remarkable property that the addition or removal of a single electron causes a macroscopic number of spin flips^{1}. For nonzero Zeeman energies, the number of spin flips depends on the competition between the exchange and the Zeeman energies^{2}, that is, on the parameter , which characterizes the strength of the Zeeman splitting E_{Z}=gμ_{B}B. (Here ɛ is the dielectric constant of the background material, B is the perpendicular magnetic field, is the magnetic length, g is the Landé gfactor, and μ_{B} is the Bohr magneton. For the parameters of GaAs with the magnetic field specified in units of Tesla (T), we have .) The skyrmion physics is relevant for κ≲0.05 (B≲70 T for GaAs), and has been confirmed experimentally^{3,4}.
The fractional quantum Hall effect^{5} (FQHE) arises due to the formation of composite fermions, which are topological bound states of electrons and an even number (2p) of quantized vortices^{6}. Composite fermions experience an effective magnetic field and form Landaulike levels called Λ levels (ΛLs). Their filling factor ν* is related to the electron filling factor ν by the expression ν=ν*/(2pν*±1). The FQHE states at ν=n/(2n±1) are manifestations of integer quantum Hall effect of composite fermions and one may expect fractionally charged skyrmions close to composite fermion filling, ν*=1, which corresponds to the ν=1/3 state^{7}. Such fractional skyrmions (FSs) were already predicted in the work of Sondhi et al.^{1}, and their existence was subsequently verified in detailed microscopic calculations^{8,9,10}. These calculations indicated, however, that the fractional skyrmions are much more delicate than the integral skyrmions near ν=1, because the exchange interaction between composite fermions is much weaker than that between electrons. It was estimated that fractional skyrmions should occur only below κ≈0.009 (B≲2.5 T for GaAs). Consequently, for typical experimental parameters, when the filling factor is varied away from ν*=1 (or from ν=1/3), trivial quasiparticles, namely isolated composite fermion particles or holes, are produced rather than fractional skyrmions. The fractional skyrmions have been probed experimentally by suppressing g through the application of hydrostatic pressure^{11}. Certain E_{Z} dependencies of the excitations^{12,13} at ν=1/3 have also been interpreted in terms of skyrmion physics, but it is unclear how skyrmions may occur at the high κ values of these experiments, and an alternative explanation of the observations has been proposed^{14}. The binding energy of fractional skyrmions has not been measured so far, which would be important for a convincing observation.
This work is concerned with minimal fractional skyrmions, namely the skyrmions for which a composite fermion (CF) particle or hole is dressed by a single additional spinflip exciton (SFE). We show theoretically that such skyrmions exist in the excitation spectrum just below the Zeeman energy for a broad range of κ at filling factors slightly away from ν=1/3. These are accessible in resonant inelastic light scattering (RILS), because a photoexcited SFE that can bind, for ν>1/3 (ν<1/3), with a preexisting composite fermion particle (hole) to produce a negatively (positively) charged fractional skyrmion, denoted by FS^{−} (FS^{+}). We identify certain modes observed in RILS experiments^{15,16} with the minimal fractional skyrmions, supporting this identification by a detailed analysis of the experimental data, which shows qualitative and quantitative agreement between theory and experiment. In particular, the measured binding energies of the positively and negatively charged fractional skyrmions are seen to be in excellent agreement with the calculated binding energies.
Results
Theory
Employing a combination of exact and composite fermion diagonalization methods, we evaluate the binding energy of the minimal fractional skyrmion, that is, the amount by which it lies below the Zeeman energy, and estimate corrections due to finite quantum well thickness. We consider filling factors close to ν=1/3, where the density of composite fermion particles or holes is dilute and it suffices to consider a single composite fermion particle or hole. The composite fermion hole resides in the spinup lowest ΛL (0↑ ΛL) for all κ (Fig. 1a), whereas the composite fermion particle can reside either in spindown lowest ΛL (0↓ ΛL) for small κ (Fig. 1c) producing a partially polarized state, or in spinup second ΛL (1↑ ΛL) for large κ (Fig. 1e) producing a fully spinpolarized state.
We use the spherical geometry^{17}, in which N electrons move on the surface of a sphere, exposed to a radial magnetic field that produces a flux of 2Qφ_{0} through the surface of the sphere, where 2Q is a positive integer and φ_{0}=hc/e is a unit flux quantum. The distance between the electrons is defined as the chord distance on the sphere; whether the chord or the arc distance is chosen is unimportant because we evaluate the thermodynamic limit of the energy. The ν=1/3 state occurs at 2Q=3N−3 and has the spin quantum number S=N/2. A single composite fermion hole occurs at 2Q=3N−2 with the spin of the bare CF particle (Fig. 1a) also given by S=N/2. The composite fermion particle at 2Q=3N−4 can go either into 0↓ ΛL (Fig. 1c), with spin S=N/2−1, or into 1↑ ΛL (Fig. 1e), with spin S=N/2. The red dashes in Fig. 1g–i are obtained by exact diagonalization in these spin sectors. We can also construct explicit wave functions for these states in the composite fermion theory, which for the CF hole has the form where is the known wave function of a single hole at ν=1, u=cos(θ/2)exp(iφ/2) and v=sin(θ/2)exp(−iφ/2) are spinor coordinates, θ and φ are the polar and azimuthal angles on the sphere, and represents projection into the lowest Landau level (LLL). Wave functions for spinconserving and spinreversed composite fermion particles are constructed analogously. The red dots in g, h and i of Fig. 1 are the Coulomb energies of these wave functions.
To consider the fractional skyrmions, we next consider states containing an additional SFE, shown in Fig. 1b,d,f and consider the sector ΔS=−1, where ΔS is measured relative to the ground state. We show the spectra in these spin sectors in Fig. 1 in g, h and i. The black dashes show the exact Coulomb spectra obtained by numerical diagonalization. The black dots are the spectra obtained by the method of composite fermion diagonalization^{18}. For the latter, we first construct a basis of all states in the relevant spin sector at 2Q*=2Q−2(N−1) (which is the effective flux experienced by composite fermions), denoted by , where α labels different basis functions. We then composite fermionize this basis to obtain the correlated composite fermion basis at 2Q, given by . We finally diagonalize the Coulomb interaction in this basis to obtain eigen energies and eigen functions.
In both the exact and the composite fermion spectra, we find that a fractional skyrmion bound state (highlighted in yellow in panels Fig. 1g,h) is produced when a composite fermion particle in the 0↓ ΛL or a composite fermion hole in the 0↑ ΛL is dressed by the SFE. No such bound state is produced for the fully spin polarized state at ν>1/3, that is, for a composite fermion particle in the 1↑ ΛL. The exact density profiles of the fractional skyrmions are seen to be qualitatively different, and much smoother, than those of the composite fermion particle or composite fermion hole (Figs 2 and 3), which is what results in the lowering of the Coulomb energy. Despite the remarkably different structures, they all carry a precise fractional charge of magnitude e/3. Figure 4 shows the thermodynamic extrapolation of the binding energy of the fractional skyrmions, denoted , obtained from exact diagonalization results for finite systems. (The energy of FS is given by .) The thermodynamic limits for the binding energies are determined to be and for a system with zero thickness and no Landau level mixing.
The interpretation of the fractional skyrmions as bound states of three composite fermions (see panels b and d of Fig. 1) is confirmed by: the close agreement between the energies of the exact and the composite fermion wave functions (that is, the dashes and the dots in Fig. 1); by a comparison of the density profiles of the exact and composite fermion wave functions shown in Fig. 5; and the high overlap of ∼0.99 between the exact and the composite fermion wave functions for N=12.
For an accurate quantitative comparison with the experiment, we have estimated corrections due to finite transverse width of the quantum well wave function. We first use a local density approximation^{19} to obtain the transverse wave function ξ(z). The effective twodimensional interaction is given by: , where z_{1} and z_{2} are the coordinates perpendicular to the plane containing the electrons. At short distances this interaction is softer than the Coulomb interaction. For the fractional skyrmions, the change in the energy due to finite width is shown in the inset of Fig. 4. We use the composite fermion theory for obtaining the corrections due to finite width, because it is possible to go to larger systems in CFD than in exact diagonalization; the use of the composite fermion theory is justified given the above result showing the accuracy of the composite fermion theory. The finite size variations preclude a clean extrapolation to the thermodynamic limit 1/N→0, but it is clear that the binding energies for the fractional skyrmions are reduced only by a small amount. We take the average of all points in the inset of Fig. 4 as a measure of the reduction in the fractional skyrmion binding energy due to finite width, which gives for the positively and negatively charged fractional skyrmions energy reductions of 0.0013±0.0005 and 0.0010±0.0001 , respectively (with the error given by the standard deviation). We apply this correction to the binding energies obtained from exact diagonalization.
Experiment
The experimental results presented in this work are from RILS on a highquality GaAs single quantum well of width 33 nm, electron density n=5.5 × 10^{10} cm^{−2} and lowtemperature mobility μ=7.2 × 10^{6} cm^{2} Vs^{−1}. The magnetic field perpendicular to the sample is B=B_{Total}cosθ, where θ is the tilt of the sample with respect to the direction of the total magnetic field B_{Total}. The filling factor and magnetic length depend on the perpendicular field B, whereas the Zeeman energy on the total field B_{Total}, and thus tilting can be used to vary the parameter κ (κ is defined as the ratio of the Zeeman to Coulomb energy). Measurements were taken at two tilt angles, θ=30°±2° and θ=50°±2°, which correspond to κ≈0.018 and κ≈0.023. RILS spectra were obtained by tuning the incident laser photon energy E_{laser} to be close to the fundamental optical gap of GaAs to enhance the lightscattering crosssection. To identify all modes, it is important to scan over a range of energies of the incoming laser photon, because modes are picked out by resonant Raman scattering most prominently in a narrow range of parameters where the resonance condition is best satisfied. As seen in Fig. 6a,b, three modes can be identified for 30° tilt, whereas only two are seen for 50° tilt. In many cases, the number of modes and their energies are evident without fitting (see, for example, Fig. 2 in ref. 15 and Fig. 3 in ref. 16). In general, a detailed line shape analysis is necessary to determine the number and the energies of the observed modes. To this end, we determine the least number of Lorentzians that provide a reasonably good fit to the observed Raman line shapes, with the centres of Lorentzians giving the energies of the modes. Some representative fits for 30° tilt are shown in Fig. 7, where at least three Lorentzians are needed for a good fit to the observed line shape. For 50° tilt, a fit with two Lorentzians is found to be satisfactory.
Figure 1g–l show energies of experimental modes observed in RILS at two tilt angles between magnetic field direction and the planenormal. For 50° tilt the results are taken from Gallais et al.^{15}. For 30° tilt, the experimental points shown in Fig. 1k are deduced from the RILS spectra of Dujovne et al.^{16}, but the detailed line shape analysis performed here gives more accurate energies than those quoted in that work.
Discussion
We identify the mode just below the Zeeman energy with the minimal fractional skyrmion. This identification is supported by several observations. At exactly ν=1/3, no modes are observed below E_{Z}, as expected. For v≲1/3, a single subE_{Z} mode is observed^{15}. The excellent quantitative agreement between theory and experiment seen in Fig. 1j confirms its identification with the positively charged fractional skyrmion. We next consider v≳1/3. At 30° tilt, the energy of the mode slightly below E_{Z} is in excellent agreement with the calculated energy of the the negatively charged fractional skyrmion (including finite thickness correction). We attribute the absence of this mode at 50° tilt to transition from a partially polarized ground state into a fully spinpolarized ground state as κ is raised from 0.018 to 0.023 by increasing the tilt. This is consistent with calculations^{20} that have shown that the ground state at v≳1/3 has a transition from partially spinpolarized state to a fully spinpolarized state at κ≈0.020. We note that unlike their integral counterparts, the positively and negatively charged fractional skyrmions are not related by particlehole symmetry, as indicated by their different binding energies.
A discussion of various approximations is in order. Landau level mixing and disorder, not included in our calculations, are likely to provide small corrections. Calculations^{21} have shown that Landau level mixing is a minor effect, especially because finite width weakens the short range part of the interaction that is primarily responsible for causing admixture with higher Landau levels. Disorder is also known significantly to diminish the energy of a charged excitation. However, we are concerned here with the change in the energy of a chargeneutral SFE due to its binding with an already present composite fermion particle or hole, which we expect to be less sensitive to disorder^{21}. We have also neglected interaction between the fractional skyrmion and other composite fermion particles or holes, which is valid only close to ν=1/3 where the density of composite fermion particles or holes is very small. The ν dependence of the energy of fractional skyrmion indicates its renormalization due to the presence of other composite fermion particles or holes in its vicinity. The much weaker dependence of the measured dispersion of negatively charged fractional skyrmion indicates a weaker interaction between composite fermion particles. The ν dependence of the fractional skyrmion energy can in principle allow a further investigation into the intercomposite fermion interactions, although we have not pursued that here.
The excitation spectrum also contains skyrmions binding K>2 SFEs, with energy . These lie in the continuum above E_{Z} for experimental parameters considered here. We expect that these couple weakly to RILS because they require the incident photon to excite K≥2 SFEs, a higher order scattering process.
For completeness, we have considered excitations other than the fractional skyrmions. Figure 8 shows schematically various elementary excitations of the states corresponding to a, c and e of Fig. 1. We have performed an exhaustive study of all of these excitations by the standard methods of the composite fermion theory^{21,22,23}, as well as from an extrapolation of the results available from exact diagonalization. The Coulomb contributions to their energy are given in Table 1 for both zero width and the width of 33 nm (which is close to the quantum well width in the experiment). The mode shown in Fig. 8a can be viewed as the mode in Fig. 8d plus a spin wave; given that the Coulomb energy of the spin wave goes to zero at small wave vector according to Larmor’s theorem, these two have the same Coulomb energies (although they have different Zeeman contributions). For the modes shown in Fig. 8c,f,i,j (Fig. 8d) we need to add (subtract) the E_{Z} before comparing with the experiments.
For ν<1/3 only a single subE_{Z} mode (the positively charged fractional skyrmion) is expected from theory and only one such mode is observed experimentally. For ν>1/3 other subE_{Z} modes are possible. In this region, using the parameters of ref. 16, in addition to the fractional skyrmion, the mode shown in Fig. 8d lies below E_{Z} for the partially spinpolarized state, and the mode shown in Fig. 8i lies below the E_{Z} for the fully spinpolarized state. We assign the mode in panel l of Fig. 1 to the excitation shown in Fig. 8i. The Coulomb energy contribution to the measured energy, ∼−0.013 , is to be compared with the theoretical Coulomb energy (including finitewidth correction) of −0.018 . We further assign the mode in panel k of Fig. 1 indicated by green stars to the excitation shown in Fig. 8d. The Coulomb contribution to the energy of this mode is theoretically calculated to be 0.018 , whereas the measured one is 0.030 . We find this level of agreement acceptable, considering that both these numbers are differences between the energies of two single particle excitations, which are known to be sensitive to the effects of disorder.
RILS can also detect magnetoroton modes, which are particlehole excitations of composite fermions (to be distinguished from the minimal fractional skyrmions that are bound state of three composite fermions). The magnetoroton modes of the 1/3 or 2/5 states occur at energies much higher than E_{Z} for the parameters of the experiments in question. Additional low energy magnetorotons are in principle possible at incompressible FQHE states of composite fermions in the range 1/3<ν<2/5, such as 4/11, 5/13 and 3/8 (refs 24, 25, 26), which correspond to neutral spinconserving modes of the composite fermions in the minority spin sector^{27,28,29}. Several observations indicate that this physics is not relevant to the mode identified above as the negatively charged fractional skyrmion. First, the composite fermionFQHE states at 4/11, 5/13 and 3/8 are stabilized at temperatures below the minimum temperature (50 mK) of the experiments of Fig. 1. Second, the experimental mode identified as the fractional skyrmion does not require a fine tuning of ν. Third, the energy of the observed mode agrees with the theoretically calculated energy of the fractional skyrmion. Finally, the neutral magnetoroton modes of 4/11, 3/8 and 5/13 are expected to occur at ∼0.002 (refs 30, 31), which is much lower than the energy of the fractional skyrmion.
In conclusion, we have shown that the smallest version of the fractionally charged skyrmions can be created optically in the vicinity of ν=1/3 when a photoexcited neutral exciton forms a bound state with an already present charged composite fermion particle or hole. Furthermore, we provide strong evidence that the modes observed slightly below the longwavelength spinwave mode at the Zeeman energy are precisely these skyrmions. In particular, the measured binding energy of these skyrmions is in excellent agreement with the theory. The study of skyrmions for ν≈1/3 provides a sensitive probe into the intercomposite fermion interaction, and also sheds light on the spin polarization of the ground state.
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How to cite this article: Balram, A. C. et al. Fractionally charged skyrmions in fractional quantum Hall effect. Nat. Commun. 6:8981 doi: 10.1038/ncomms9981 (2015).
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Acknowledgements
We acknowledge financial support from the US Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DESC0005042 (A.C.B., J.K.J.), from the US National Science Foundation under grant DMR1306976 (A.P.), from the National Science Centre, Poland, under grant 2014/14/A/ST3/00654 (A.W.). We thank Research Computing and Cyberinfrastructure at Pennsylvania State University (supported in part through instrumentation funded by the National Science Foundation through grant OCI0821527), Wrocław Centre for Networking and Supercomputing and Academic Computer Centre CYFRONET, both parts of PLGrid Infrastructure, and the DFG excellence programme Nanosystems Initiative Munich (U.W.). We thank Indian Institute of Science Education and Research, Pune for their kind hospitality during the final stages of this work.
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A.P. and J.K.J. planned and organized this project; U.W. and A.P. analysed the experimental data; A.C.B. and A.W. performed theoretical calculations; all authors discussed the results and contributed to the writing of the manuscript.
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Balram, A., Wurstbauer, U., Wójs, A. et al. Fractionally charged skyrmions in fractional quantum Hall effect. Nat Commun 6, 8981 (2015). https://doi.org/10.1038/ncomms9981
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DOI: https://doi.org/10.1038/ncomms9981
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