Aharonov-Bohm effect in the non-abelian quantum Hall fluid

Transcription

1 PHYSICAL REVIEW B 73, Aharonov-Bohm effect in the non-abelian quantum Hall fluid Lachezar S. Georgiev Michael R. Geller Institute for Nuclear Research Nuclear Energy, 7 Tsarigradsko Chaussee, 784 Sofia, Bulgaria Department of Physics Astronomy, University of Georgia, Athens, Georgia , USA Received 7 January 006; published 4 May 006 The =5/ fractional quantum Hall effect state has attracted great interest recently, both as an arena to explore the physics of non-abelian quasiparticle excitations as a possible architecture for topological quantum information processing. Here we use a conformal field theoretic description of the Moore-Read state to provide clear tunneling signatures of this state in an Aharonov-Bohm geometry. While not probing statistics directly, the measurements proposed here would provide a first, experimentally tractable step towards a full characterization of the 5/ state. DOI: 0.03/PhysRevB PACS numbers: f, 7.0.Pm, Lx The quantum Hall fluid has become a paradigm of strongly correlated quantum systems., A combination of two-dimensional confinement strong magnetic field leads to rich phenomena driven by electron-electron interaction disorder. As such, the use of traditional theoretical techniques such as many-body perturbation theory has had only limited success, in an approach pioneered in 983 by Laughlin, 3 some of the most important advances have been made by correctly guessing the many-particle wave function. The states described by Laughlin, their generalizations,, have Hall conductances xy given in units of e /h by fractions with odd denominators only. The charged excitations, which have fractional charge 4 6 statistics, 7 are Abelian anyons. In 987, however, evidence for an even-denominator quantized Hall state at =5/ was discovered in the first-excited Lau level, 8 the state is now routinely observed in ultrahigh-mobility systems. 9 Motivated in part by this surprising result, Moore Read MR introduced the ground-state trial wave function MR z,z,...,z N =Pf z i z j ij z i z j for N even electrons in a partially occupied Lau level with complex coordinates z i, where the first term is the Pfaffian the stard Gaussian factor is suppressed. Exact diagonalization studies 3,4 indicate that the exact ground state at =5/ is close to. Moore Read also constructed excited-state wave functions. By identifying with a two-dimensional chiral conformal field theory CFT correlation function MR z,z,...,z N = z z z N of charge- fermion fields :e i : M, where is a u boson 5 M a neutral Majorana fermion, MR proposed CFT-based excited states of the form qh qh n z z N. 3 Here qh :e i/ 8 : is the fundamental charge-/4 quasihole field of the CFT, with the chiral spin field of the critical Ising model. The most spectacular prediction of Moore Read is that the quasiparticle excitations of the 5/ state are non- Abelian: An excited state of n quasiholes has degeneracy n, the braiding of their world lines generates elements from the orthogonal group SOn acting on the degenerate multiplet. This property follows from the correlation function 3 also from an alternative picture of the state as a BCS condensate of l= pairs of composite fermions, 6 which supports exotic non-abelian vortex excitations. 7,8 In addition to its intrinsic interest as a system to explore non-abelian quantum mechanics, Das Sarma et al. 9 have proposed to use a pair of antidots at =5/ to construct a topological quantum NOT gate, building on an intriguing idea by Kitaev to use the transformations generated by non- Abelian anyon braiding for fault-tolerant quantum computation. 0 Unfortunately, the braiding matrices generated by 3 are not computationally universal. But by generalizing the pairing present in MR to clusters of k particles, Read Rezayi have proposed a hierarchy of ground excited states CFT correlators of parafermion currents reducing to 3 when k=. These parafermion states are already computationally universal at k =3. Computation with non-abelian quasiparticles will be incredibly challenging experimentally. Demonstrating that the actual 5/ state is in the universality class of demonstrating that the quasiparticles are indeed non-abelian are necessary first steps. Direct interferometric thermodynamic probes of the non-abelian statistics have been proposed recently.,3 Here we propose an even simpler test of the MR state, in a similar antidot geometry. Although the tunneling measurement proposed below does not directly probe the non-abelian nature of the excitations, it can distinguish between the tunneling of ordinary electrons that of a bosonic charge- excitation we call, which is allowed by the CFT which has lower scaling dimension. And the existence of two inequivalent charge- excitations is itself a fingerprint of the non-abelian nature of the fundamental quasihole qh. Experimental observation of the electron tunneling channels would give confidence in the MR state, the powerful CFT approach,, by extension, the k parafermion hierarchy necessary for universal topological quantum computation. The geometry we consider is illustrated in Fig.. The 098-0/006/730/ The American Physical Society

2 L. S. GEORGIEV AND M. R. GELLER PHYSICAL REVIEW B 73, TABLE I. Some chiral conformal fields their quantum numbers. c 0 are the scaling dimensions in the charged neutral sectors, c + 0 is the total CFT dimension, mod is the statistics angle. FIG.. Color online. Antidot inside a =5/ Hall bar with weak tunneling points at x x, threaded by an Aharonov-Bohm AB flux. =/3 realization of this system was considered by one of us previously, 4 a dual configuration, the double pointcontact interferometer, was studied previously by Chamon et al. 5 In the strong-antidot-coupling regime pictured, the edge states, indicated by the arrows, are strongly reflected by the antidot. This regime can be realized experimentally in two physically distinct ways: i The Hall fluid can be pinched off near the x i, leaving large tunneling barriers or vacuum regions. Only electrons can tunnel through these vacuum regions. ii The system can begin in the weak-antidotcoupling regime, where current from the upper edge tunnels through the antidot acting as a macroscopic impurity to the lower one, the temperature is then lowered. The weaktunneling regime, which permits quasiparticle tunneling, is unstable at low temperatures, as in the =/3 quantum point contact, 6 the system then flows to the stable strongantidot-coupling fixed point, which can be described by an effective weak-tunneling theory. 7 In the =/3 case, this effective theory contains electron tunneling only is identical to that of case i. The most dramatic illustration of this fact comes from the exact Bethe ansatz solution of the chiral Luttinger liquid model for a =/3 point contact containing only charge-/ 3 quasiparticles, which nonetheless describes electronlike tunneling far off resonance. 8 In principle, the effective weak-tunneling theory for case ii can be different than that of i, we will consider this possibility here. Because there is no exact solution available for the =5/ quasiparticle tunneling model, we will guess the form of the effective theory. Guided by the reasonable condition that any charge excitation of the CFT preserving the stability of the fixed point can potentially tunnel the charge requirement introduced to recover the =/3 result, we need to consider a cluster of at least four fundamental quasiholes qh. According to the fusion rule =+ M, 9,30 the product qh qh qh qh =+ yields two distinct tunneling objects, the electron/hole a charge- boson, :e i :, whose quantum numbers are summarized in Table I. There are also excitations less relevant than the electron that we do not need to consider. We emphasize that is an allowed excitation of the Pfaffian state satisfying the parity rule. 9,30 The scaling dimension of is, if it tunnels, it will dominate electron tunneling in the low-temperature limit, leaving a clear experimental signature. We turn now to a calculation of the source-drain current 4 Field Charge c 0 / Quasihole qh particle 0 0 Majorana M 0 0 Electron IV,T, = I 0 V,T + I AB V,Tcos + as a function of voltage V temperature T, which according to our analysis can be decomposed into flux-independent period- oscillatory AB components. Here is the mean electrochemical potential of the contacts, v/l is the noninteracting level spacing on the antidot with edge velocity v circumference L, is the AB flux in units of h/e. The Hamiltonian in the strong-antidot-coupling regime is H = H L + H R + H, with H = i B i + * i B i. i=, The Hamiltonians for the uncoupled right- left-moving edge states of length L sys, H R = v 0 c L sysl 4 H L = v L sysl 0 c 4, are given by the zero modes of the CFT stress tensors Tz T z, L 0 dz i ztz L 0 dz i z T z, which satisfy the Virasoro algebra with central charge c=3/ Refs Then, L 0 = J 0 + J n J n + n= n= n M n+/ M n /, similarly for L 0. The Laurent-mode expansion of the Majorana field with antiperiodic boundary conditions on the cylinder z=e ix/l sys is M z = n / nz M z n M with n / = dz i zn M z. The u current Jzi z has mode expansion

3 AHARONOV-BOHM EFFECT IN THE NON-ABELIAN¼ Jz = J n z n, nz with J n = dz i zn Jz. Here J 0 / J 0/ are the usual g=/ Luttinger-liquid number currents N R N L. The operators B i L x i R x i, i =,, entering Eq. 5 are tunneling operators acting at points x i in Fig.. The fields L R appearing in B depend on the tunneling object. The tunneling amplitudes at x x are = e i/+ = e i/+, where, with no loss of generality, can be taken to be real. The bare amplitude depends on microscopic details type of tunneling object. The tunneling current IV,T, can be calculated by linear response theory along the lines of Refs. 4 5, leading to I 0 V,T =4e Im X = ev I AB V,T =4e Im X = ev, where X ij is the Fourier transform of the response function, X ij t itb i t,b j 0, with B i te ih 0t B i e ih 0t, the thermal average A = trae H 0 tre H 0 is taken with respect to the Hamiltonian H 0 H R + H L R N R L N L. The finite-temperature correlation function X ij t is computed in three steps: i First, it is split into products of finitetemperature correlation functions with chirality of the form ± x,t ± x,t. ii Then these one-dimensional thermal correlation functions are obtained as zerotemperature correlation functions after Wick rotation to imaginary time on a cylinder with circumference L T v/k B T. iii Finally, we map the cylinder to the complex plane by the conformal transformation z ± =e iv±x/l T where, for primary fields with scaling dimension, ± z ± z = z z for z z. 6 Under the conformal map a chiral primary field transforms as ± x, / dz i div±x ± z, by going back to real time we obtain ±i/l T ± x,t ± 0 = sh x vt ± i/l T, 7 where is a positive infinitesimal required by 6. Finally, the desired response function is X ij t = t 4 L T Imsh x i x j + vt + i/l T sh x i x j vt i/l T. 8 When is an integer, which will be the case here, the transform X ij has an infinite number of poles of order can be obtained by residue summation. Note that the local response function which determines the direct current I 0 could be obtained as the limit X = lim x x 0 X. We expect that both the electrons particles will contribute to the observed tunneling current, as will the filled Lau level. If only electrons tunnel, they will contribute I el 0 = e el 64 T h 40 T 0 4 ev ev 3 + ev PHYSICAL REVIEW B 73, T T I el AB = e el T/T 0 3 h sh 3 T/T 0 ev + T T 0 3cth T T 0sin ev +6 ev T T 0 cth T T 0cos ev, 0 which is identical to that of the g =/3 chiral Luttinger liquid. 4,5 Here T 0 v/k B L is a temperature scale associated with the level spacing of the antidot edge state of velocity v circumference L. Above T 0, the thermal length L T becomes smaller than L the AB oscillations become washed out. /vl is a dimensionless tunneling amplitude. Each of the two filled Lau levels also contributes a Fermiliquid FL current I FL 0 =e /h FL V I FL AB = e h FL T/T 0 ev sht/t 0 sin. The contribution from the channel alone is I 0 = e 4 T h 3 T 0 V+ 4 ev k B T 3 I AB = e T/T 0 h sh T/T 0 T T 0cth T ev T 0sin ev ev cos 4. The oscillations of the AB currents as functions of the volt

4 L. S. GEORGIEV AND M. R. GELLER PHYSICAL REVIEW B 73, FIG.. Color online The Aharonov-Bohm current I AB V,T for = tunneling as a function of the applied voltage V compared to =3/ electron tunneling reduced by a factor of 00 the chiral Fermi-liquid current =/ multiplied by a factor of 0 at T=T 0. age for the three channels at temperature T=T 0 are shown in Fig.. An experiment will observe these two or three transport channels in parallel, the contribution of each determined by the bare amplitudes, el, FL. However, the channel will always dominate the electron channel in the low-energy limit. The linear conductance GdI/dV V 0 for these channels takes the form G,T = G 0 T + cos + G AB T, where G 0 G AB are the direct AB conductances, respectively. For the channel they read G 0 T = h e 4 T 3 T 0, G AB T = e h4 T/T 0 sh T/T 0 T T 0cth T T 0, while for the electron channel we obtain G 0 el T = e h4 el G el AB T = e h4 el T/T 0 3 sh 3 T/T 0 +3 T T 0cth T T T T 04 T T 0, cth T T 0 FIG. 3. Color online. Aharonov-Bohm conductances for electron tunneling at =5/. The Fermi-liquid conductance is also shown. G FL 0 T = e h FL, G FL AB T = e T/T 0 h FL sht/t 0 are plotted in Fig. 3. Both the electron contributions display a pronounced maximum as a function of temperature. In the T 0 limit the contribution, varying as T, dominates the electron contribution. The temperature dependence of G in the low- high-temperature regimes is summarized in Table II. There is also an interesting zerotemperature nonlinear regime k B TeVk B T 0, where both the direct AB currents vary as I V 3 I el V 5, independent of temperature. Finally, we also note that the limit of a single =5/ quantum point contact in the stable, strong-tunneling regime follows from our results by letting T 0 ; the electron contribution in this case agrees with that TABLE II. Asymptotic tunneling conductance. T 0 is a crossover temperature defined in Eq.. evk B Tk B T 0 evk B T 0 k B T Fermi liquid G 0 const G 0 const = G AB const G AB Te T/T 0 particle G 0 T G 0 T = G AB T G AB T 3 e T/T 0 Electron G 0 T 4 G 0 T 4 = 3 G AB T 4 G AB T 5 e 3T/T 0 These conductances are compared to the Fermi-liquid ones

5 AHARONOV-BOHM EFFECT IN THE NON-ABELIAN¼ for tunneling between a FL =5/ edge state. 3 In conclusion, we have used the CFT picture of Moore Read to calculate the AB tunneling spectrum in a =5/ antidot geometry. Observing the electron possibly transport channels will give evidence in support of the non-abelian nature of the 5/ state. PHYSICAL REVIEW B 73, We thank Ady Stern for useful discussions. L.S.G. has been partially supported by the FP5-EUCLID Network Program of the EC under Contract No. HPRN-CT by the Bulgarian National Council for Scientific Research under Contract No. F-406. M.R.G. was supported by the NSF under Grant Nos. DMR CMS The Quantum Hall Effect, nd ed., edited by R. E. Prange S. M. Girvin Springer-Verlag, Berlin, 990. Perspectives in Quantum Hall Effects, edited by S. Das Sarma A. Pinczuk Wiley, New York, R. B. Laughlin, Phys. Rev. Lett. 50, V. J. Goldman B. Su, Science 67, L. Saminadayar, D. C. Glattli, Y. Jin, B. Etienne, Phys. Rev. Lett. 79, R. de Picciotto, M. Reznikov, M. Meiblum, V. Umansky, G. Bunin, D. Mahalu, Nature London 389, F. E. Camino, W. Zhou, V. J. Goldman, Phys. Rev. B 7, R. Willett, J. P. Eisenstein, H. L. Stormer, D. C. Tsui, A. C. Gossard, J. H. English, Phys. Rev. Lett. 59, W. Pan, J. S. Xia, V. Shvarts, D. E. Adams, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, K. W. West, Phys. Rev. Lett. 83, J. P. Eisenstein, K. B. Cooper, L. N. Pfeiffer, K. W. West, Phys. Rev. Lett. 88, J. S. Xia, W. Pan, C. L. Vincente, E. D. Adams, N. S. Sullivan, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, K. W. West, Phys. Rev. Lett. 93, G. Moore N. Read, Nucl. Phys. B 360, R. H. Morf, Phys. Rev. Lett. 80, E. H. Rezayi F. D. M. Haldane, Phys. Rev. Lett. 84, This is normalized according to zz= lnz z. 6 N. Read D. Green, Phys. Rev. B 6, D. A. Ivanov, Phys. Rev. Lett. 86, A. Stern, F. von Oppen, E. Mariani, Phys. Rev. B 70, S. Das Sarma, M. Freedman, C. Nayak, Phys. Rev. Lett. 94, A. Y. Kitaev, Ann. Phys. N.Y. 303, 003. N. Read E. Rezayi, Phys. Rev. B 59, A. Stern B. I. Halperin, Phys. Rev. Lett. 96, P. Bonderson, A. Kitaev, K. Shtengel, Phys. Rev. Lett. 96, M. R. Geller D. Loss, Phys. Rev. B 56, C. de C. Chamon, D. E. Freed, S. A. Kivelson, S. L. Sondhi, X. G. Wen, Phys. Rev. B 55, K. Moon, H. Yi, C. L. Kane, S. M. Girvin, M. P. A. Fisher, Phys. Rev. Lett. 7, This assumes, of course, that there is a stable strong-tunneling fixed point here that there are no other stable fixed points. 8 P. Fendley, A. W. W. Ludwig, H. Saleur, Phys. Rev. Lett. 74, A. Cappelli, L. S. Georgiev, I. T. Todorov, Commun. Math. Phys. 05, L. S. Georgiev, Nucl. Phys. B 65, M. Milovanovic N. Read, Phys. Rev. B 53, N. Read E. Rezayi, Phys. Rev. B 54,