15th International Conference on High Magnetic Fields in Semiconductor Physics, Oxford, UK, 5 - 9 August 2002We review the main results of the anyon exciton model in light of recent criticism by Wojs and Quinn. We show that the appearance of fractionally charged anyon ions at the bottom of their numerically calculated excitation spectra is an artefact caused by finite-size effects in a spherical geometry

Parfitt, D.G.W.

Portnoi, M.E.

Open Research Exeter

arXiv:cond-mat/0306414v1  [cond-mat.mes-hall]  16 Jun 2003Anyon exciton revisitedD G W Parfitt and M E PortnoiSchool of Physics, University of Exeter, Exeter EX4 4QL, UKAbstract. We review the main results of the anyon exciton model in light of recent criticismby Wo´js and Quinn. We show that the appearance of fractionally charged anyon ions at thebottom of their numerically calculated excitation spectra is an artefact caused by finite-sizeeffects in a spherical geometry.1. Review of anyon exciton modelThe anyon exciton model [1, 2] provides a full classification of the multiple-branch spectraof a four-particle anyon exciton, a neutral composite particle consisting of a valence hole andthree anyons with charge −e/3. This model is only applicable for large enough separationbetween a photoexcited hole and a two-dimensional electron gas at exact fractional fillingfactor, when the Coulomb field from the hole cannot destroy the incompressible quantumliquid.As a neutral particle, the anyon exciton possesses an in-plane momentum k. At k = 0 itis also described by the angular momentum Lz = −L, where L is the degree of the polynomialsymmetric with respect to interchange of anyon coordinates. There is an additional quantumnumber which enumerates different polynomials of the same degree. An important parameterof the problem is the separation h between electron and hole confinement planes, measuredin units of magnetic length l. Two examples of numerically calculated energy spectra forh = 2 and 3 are shown in Fig. 1. For h = 2, the minimum of the spectrum occurs fornon-zero k, which means that the exciton in the lowest energy state possesses a non-zerodipole moment. The negative dispersion in Fig. 1a arises because of the mutual repulsion ofL = 2 and L = 3 branches. For higher values of h, the ground state is always at k = 0,as in Fig. 1b, and the lowest-branch dispersion is always positive. The value of L at k = 0for the lowest branch increases with increasing h, and thus direct optical transitions fromthe ground state are forbidden. However, for k 6= 0 the wavefunction is a mixture of stateswith different L values, and magnetoroton-assisted transitions become possible. Fig. 2, whichshows the negative charge distribution around a valence hole for two different values of k,implies a simple qualitative picture of such a transition. A quasihole appears as a result ofrecombination of the hole with two negatively-charged anyons which are close to it. Thequasihole and a split-off anyon form a magnetoroton. Notably, for the particular value ofh = 2 used in Figs. 1a and 2, the asymmetric k 6= 0 distribution has a lower energy than thesymmetric one. Nevertheless, we wish to emphasise that the exciton remains neutral, contraryto the statements in Ref. [3].Such features of the anyon exciton model as multiple-branch spectra, dark groundstates, and absence of L = 1 states coincide with the results of exact diagonalisation fora few-electron system in the spherical geometry [3, 4]. Similarities between the electrondensity distribution and pair correlation functions for low-lying states in planar and sphericalgeometries are discussed in Ref. [2]. We wish to emphasise that such a comparison is onlyAnyon exciton revisited 2Figure 1. Anyon exciton dispersion ǫ(k) for two values of electron-hole separation h.Numbers show L values; h is in units of magnetic length l.Figure 2. Electron density distribution in an anyon exciton for different values of the excitonin-plane momentum k. The distance h between the hole and the incompressible electron liquidis equal to two magnetic lengths. The hole is at the origin; the x-axis is chosen in the directionof the exciton dipole moment.valid in the region of intermediate electron-hole separation, h ≈ 2, where both approachesare applicable.The presence of a ‘tight’ (L = 0) state in the upper part of the exciton energy spectrum,together with a magnetoroton-assisted transition from the lower part of the spectrum, mayexplain the double-line feature observed in the intrinsic spectroscopy of incompressiblequantum liquids [5].Anyon exciton revisited 32. Classical limit in planar and spherical geometryWhen the separation h exceeds two magnetic lengths (which is a requirement for the anyonexciton model to be valid), the ground state at k = 0 is formed by the states with angularmomentum obeying the superselection rule L = 3N , where N ≥ 2. All these states are hard-core states, i.e. their wavefunctions are zero if any two of the anyon coordinates coincide. Thismeans that the three anyons in the exciton form an equilateral triangle; this is in completeagreement with the classical picture. In fact, for large values of separation, the result of aquantum mechanical calculation for the ground state energy is the same as that obtained by asimple classical approach. This is true for quasielectrons obeying both anyonic and bosonicstatistics [2].It is natural to compare the results of classical calculations for anyon excitons and ionsin both planar (Fig. 3) and spherical (Fig. 4) geometries. These calculations are basedon minimising the classical potential energy of the system, and the hole and anyons areconsidered as point charges. Fig. 3a shows the ‘classical’ k = 0 anyon exciton, the energy ofwhich is given byE =13√3r− 1(h2 + r2)1/2, (1)where r is the distance from each anyon to the centre of negative charge. Here, and in whatfollows, charge is measured in units of electron charge e. Minimising this energy with respectto r givesEmin = −(23)3/2 1h≈ −0.544h. (2)This result coincides exactly with quantum mechanical calculations in the limit of large h [2].A similar calculation for an ion (Fig. 3b) gives a potential energyE =118r− 23(h2 + r2)1/2, (3)which, when minimised, leads toEmin = −(122/3 − 1)3/218h≈ −0.485h. (4)Therefore, as expected, the neutral exciton is more energetically favourable than a positively-charged ion.For a spherical geometry, with the valence hole at the north pole of the sphere, the anyonexciton corresponds to the configuration shown in Fig. 4a, whereas for the ion, one of theanyons goes to the south pole of the sphere (Fig. 4b), which corresponds to infinity on theplane. Note that the electron-hole separation is taken into account by introducing an effectiveinteraction V = −(r2c + h2)−1/2/3, where rc is taken as the anyon-hole chord length. Theexpression for the energy (with the radius of the sphere taken as unity) isE =√39(1− d2)1/2− 1(h2 − 2d + 2)1/2, (5)where d is the distance between the centre of negative charge and the centre of the sphere.Minimising this expression in the limit of large electron-hole separation h, we obtainEmin =13√3− 1h+1h3− 32h5+ O(1h6). (6)Anyon exciton revisited 4h h−1/3−1/3−1/3+1 +1(a) Anyon exciton (b) Anyon ion−1/3−1/3r rFigure 3. ‘Classical’ anyon exciton (a) and ion (b) in the planar geometry, where h is theseparation between the electron and hole confinement planes and r is the distance betweeneach anyon and the centre of negative charge. Note that charges are scaled by the electroncharge e.d d−1/3−1/3+1 +1−1/3−1/3−1/3(a) Anyon exciton (b) Anyon ion−1/3Figure 4. ‘Classical’ anyon exciton (a) and ion (b) in the spherical geometry, where d is thedistance between the centre of negative charge and the centre of the sphere.A similar calculation for the configuration in Fig. 4b gives a potential energyE =√327(1 + d)1/2(1− 3d)1/2+2√327(1 + d)1/2− 23(h2 − 3d + 1)1/2− 13(h2 + 4)1/2, (7)which, when minimised, yieldsEmin =13√3− 1h+1h3− 94h5+ O(1h6). (8)Anyon exciton revisited 5Comparing Eqs. 6 and 8, one can see that for large electron-hole separation in a sphericalgeometry, the positively-charged ion is energetically more favourable than the neutral exciton,which is not the case in the more realistic planar geometry.Introducing realistic form factors [3], which reduce anyon-anyon repulsion at largedistances, is unlikely to push one of the anyons to infinity and make the ion energeticallyfavourable in the planar geometry.In conclusion, we believe that the appearance of fractionally-charged anyon ions at thebottom of the numerically calculated excitation spectra [3] is an artefact caused by finite-sizeeffects in the spherical geometry.References[1] Rashba E I and Portnoi M E 1993 Phys. Rev. Lett. 70 3315-3318[2] Portnoi M E and Rashba E I 1996 Phys. Rev. B 54 13791-13806[3] Wo´js A and Quinn J J 2001 Phys. Rev. B 63 0453031-04530313; ibid. 63 0453041-04530411; Solid StateCommun. 118 225-229[4] Apalkov V M and Rashba E I 1995 Solid State Commun. 95 421-424[5] Plentz F, Heiman D, Pinczuk A, Pfeiffer L N, and West K W 1996 Surf. Sci. 361/362 30-33, and referencestherein

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