Effect of a charged impurity on the fractional quantum Hall effect: Exact numerical treatment of finite systems

We propose a model of an anyon exciton consisting of a hole and several anyons, and apply it to the spectroscopy of an incompressible quantum liquid. Fractionalization of the electron charge makes properties of such entities quite different from those of usual magnetoexcitons. The model describes a number of properties established by few-particle simulations, including an abrupt change in emission vs electron-hole asymmetry of the system. The attractive field of a hole may eliminate the hard core constraint for anyons. The effect of exciton-magnetoroton coupling is discussed. PACS numbers: 73.20.Dx, Phenomena in two-dimensional (2D) electron systems related to the fractional quantum Hall effect (FQHE) [1,2] and Wigner crystallization were originally discovered by means of magnetotransport. Later on the formation of an incompressible quantum liquid (IQL) [3], underlying the FQHE, and related phenomena have been investigated by optical experiments, which have become a powerful tool in the field [4]. These findings stimulated theoretical activity on the optical spectroscopy of the FQHE [5][6][7][8]. It has been shown that a hidden symmetry, which is inherent in 2D systems with charge symmetric electron-hole interaction (V ee = Vhh = -Veh), results in exact cancellation of the effect of the electron background on optical spectra. Optical spectra of symmetric systems are trivial, since they coincide with the spectra of free magnetoexcitons, and are insensitive to electron phase transitions. Therefore, the spectroscopy of chargeasymmetric systems acquires a special importance. Gap widths for IQLs determined from cusp strengths [5(a)] in extrinsic optical emission spectra were reported [4(a)] for such systems.It is one of the most remarkable properties of IQLs that their elementary excitations carry fractional charge [3], and are anyons [9,10], i.e., obey fractional statistics [11,12]. The theory [5(c),8] predicts that fractional charges should manifest themselves by dramatic changes in the position and the intensity of the emission band vs the asymmetry parameter. The ratio h/l, where h is a distance between electron and hole confinement planes, and / = (ch/eH) 1 / 2 is the magnetic length, may be chosen as such a parameter. Numerical simulations for few-electron systems are accessible only for small values of h/l < 1. For the opposite limit case of strongly asymmetric systems, h/l > 1, the approach based on the anyon concept seems to be most promising. An exciton, appearing against a background of an electron IQL, is a neutral entity consisting of a valence hole and several anyons, e.g., if the filling factor v = 1/3, the charge of anyons e* = -e/3, their statistical charge a = -1/3 (for comparison, a -0 for boso»ns, and a = 1 for fermions) [9], and the number of anyons N = 3. If h ^> /, the mean separation between anyons in an exciton is about h, which is larger than the anyon size, I. Therefore, anyons are well defined particles, anyon-anyon and anyon-hole interactions follow a Coulomb law in the leading approximati...

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“…AE-MR coupling manifests itself also in the oscillatory behavior of d\,{p) ( Fig. 1) (similarly to the screening of charged impurities [20,22]). …”

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confidence: 81%

“…3. The separation between an anyon and ion increases with K, and an AE produces a strong Coulomb field acting on the IQL, which is known to become unstable when an external charge about e* approaches it [22]. A simple idea that the AE creates a virtual MR and makes a bound state with it describes the behavior at h « 0 very well.…”

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confidence: 99%

Anyon excitons

Rashba

Portnoi

1993

Phys. Rev. Lett.

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“…Thus the Hall conductance of the pure system is the average Chern number of these q states, fractionally quantized at H = pe 2 / qh. However, numerical calculations show that impurities lift the degeneracy in a finite system, 29 implying that the Hall conductance would only be quantized to fractional values under superfluous conditions. 30 Wen and Niu 31 proposed that states of a system exhibiting the FQHE are topologically degenerate on a torus ͑in general, on high-genus Riemann surfaces͒ in the thermodynamic limit.…”

Section: Introductionmentioning

confidence: 99%

Mobility gap in fractional quantum Hall liquids: Effects of disorder and layer thickness

et al. 2005

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We study the behavior of two-dimensional electron gas in the fractional quantum Hall regime in the presence of finite layer thickness and correlated disordered potential. Generalizing the Chern number calculation to many-body systems, we determine the mobility gaps of fractional quantum Hall states based on the distribution of Chern numbers in a microscopic model. We find excellent agreement between experimentally measured activation gaps and our calculated mobility gaps, when combining the effects of both disordered potential and layer thickness. We clarify the difference between mobility gap and spectral gap of fractional quantum Hall states and explain the disorder-driven collapse of the gap and the subsequent transitions from the fractional quantum Hall states to the insulator.

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“…͑There does exist numerical work demonstrating the effect of disorder; see, for example, Ref. 12.͒ My approach has been to take experimental points and fit them to the theory with the ZDS potential and ask what is needed. This is done solely to obtain a feeling for its size and also compare it to the values computed for the pure system with no LL mixing, using say, the local-density approximation.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian theory of gaps, masses, and polarization in quantum Hall states

Shankar

2001

Phys. Rev. B

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In two short papers I had described an extension, to all length scales, of the Hamiltonian theory of composite fermions ͑CF͒ that Murthy and I developed for the infrared, and applied it to compute finite-temperature quantities for quantum Hall fractions. I furnish details of the extended theory and apply it to Jain fractions ϭp/(2psϩ1). The explicit operator description in terms of the CF allows one to answer quantitative and qualitative issues, some of which cannot even be posed otherwise. I compute activation gaps for several potentials, exhibit their particle-hole symmetry, the profiles of charge density in states with a quasiparticle or hole ͑all in closed form͒, and compare to results from trial wave functions and exact diagonalization. The Hartree-Fock approximation is used, since much of the nonperturbative physics is built-in at tree level. I compare the gaps to experiment, and comment on the rough equality of normalized masses near half-and quarter-filling. I compute the critical fields at which the Hall system will jump from one quantized value of polarization to another, and the polarization and relaxation rates for half-filling as a function of temperature and propose a Korringa-like law. After providing some plausibility arguments, I explore the possibility of describing several magnetic phenomena in dirty systems with an effective potential, by extracting a free parameter describing the potential from one data point and then using it to predict all the others from that sample. This works to the accuracy typical of this theory ͑10-20 %͒. I explain why the CF behaves like a free particle in some magnetic experiments when it is not, what exactly the CF is made of, what one means by its dipole moment, and how the comparison of theory to experiment must be modified to fit the peculiarities of the quantized Hall problem.

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Effect of a charged impurity on the fractional quantum Hall effect: Exact numerical treatment of finite systems

Cited by 66 publications

References 14 publications

Anyon excitons

Anyon excitons

Mobility gap in fractional quantum Hall liquids: Effects of disorder and layer thickness

Hamiltonian theory of gaps, masses, and polarization in quantum Hall states

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