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Physica E 6 (2000) 72–74 www.elsevier.nl/locate/physe Strong bulk-edge coupling in the compressible half-lled quantum Hall state Milica V. Milovanovica; b , Efrat Shimshonib; ∗ a Department b Department of Physics, The Technion, Haifa 32000, Israel of Mathematics-Physics, Oranim-Haifa University, Tivon 36006, Israel Abstract We study bulk and edge correlations in the compressible half-lled state, using a modied version of the plasma analogy. The corresponding plasma has anomalously weak screening properties, and as a consequence we nd that the correlations along the edge do not decay algebraically as in the Laughlin (incompressible) case, while the bulk correlations decay in the same way. The results suggest that due to the strong coupling between charged modes on the edge and the neutral fermions in the bulk, reected by the weak screening in the plasma analogue, the (attractive) correlation hole is not well dened on the edge. Hence, the system there can be modeled as a free Fermi gas of electrons. We nally comment on a possible scenario, in which the Laughlin-like dynamical edge correlations may nevertheless be realized. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Compressible half-lled state; Quantum Hall eect 1. Introduction Laughlin’s theory of the fractional quantum Hall eect (QHE) [1,2] was given in terms of wave functions of the ground state and quasihole excitation. Using a plasma analogy to calculate the static many-body correlators, which characterize these wave functions, he was able to advance a very successful physical picture of the electron system. The wave functions, describing the incompressible states, contain the Laughlin–Jastrow factor, which leads to a special, later introduced, Girvin–MacDonald (GM) ∗ Corresponding author. correlations in the bulk [3], and Wen’s correlations on the edge [4 – 6]. The Laughlin–Jastrow factor is everpresent in QHE states — it exists even in the compressible half-lled state [7], for which an explicit wave function has been proposed by Read and Rezayi (RR) [8,9]. The question arises whether its manifestations, in terms of the above-mentioned correlations, survive in more general quantum Hall states, and in particular in the compressible states. These correlations provide important information on the status of “Bose condensation” in the state, and on the nature of charged excitations. Experimentally, these correlations are in principle accessible by tunneling measurements. Indeed, recent 1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 0 5 0 - 8 M.V. Milovanovic, E. Shimshoni = Physica E 6 (2000) 72–74 edge-tunneling experiments by Grayson et al. [10] prompted the question whether the Luttinger liquid picture [3], which is characterized by Wen’s correlations, is valid for general QHE systems, including the compressible states. A number of theoretical works [11–13] have attempted to explain the puzzling results of Ref. [10], in terms of charged excitations on the edge that are eectively decoupled from the bulk (see also Refs. [14 –16]). To test these ideas from a microscopic point of view, we focus here on the correlators which characterize the compressible QHE system at lling factor = 12 [17]. We assume that the ground state is well described by the RR wave function [5] (1) RR = PLLL det [exp(iki Rj )]L ; i; j where PLLL stands for the projector to the lowest Landau level (LLL), and L is the Laughlin wave function with m = 2. The physical picture underlying this state is that of neutral dipoles (electrons bound to correlation holes), that have Fermi statistics [18–25]. The GM and Wen’s correlations (appropriately redened for the RR state) are then derived [17] employing the analogy with an anomalous, weakly screening plasma. This implies a dramatic modication of the behavior of charged edge excitations with respect to the incompressible states. Below we sketch our derivation and the principal results. (2) 0 where Ve (z − z ) is the interaction between two point-like charges (of charge m=2 each) immersed in a plasma of charge m particles. A diagrammatic expansion of this interaction [17] yields (in q-space) Ve (|q|) = −2m2 =|q|2 ; 1 + (2m2 =|q|2 )n (z; z 0 ) ∼ |z − z 0 |−m=2 : (4) We now consider the parallel of the GM correlator in the 12 state (Eq. (1)). The approach mentioned above yields a diagrammatic expansion, in which the vertex is modied by the Fermi correlations in the wave function. The eective interaction in the fake plasma then becomes −2m2 =|q|2 ; 1 + (2m2 =|q|2 )s0 (q) 3 kf |q| s0 (q) = : (5) 4 2 In the coordinate space, at large distances Ve ∼ 1=r, i.e. it is still long ranged and only partially screened. Nevertheless, Eq. (2) implies asymptotically the same algebraic decay of the GM correlations as in the Laughlin states. This signies the presence of some kind of coherent order even in the compressible state, and correspondingly the existence of weakly localized vortices (correlation holes). The weak screening properties of the plasma analogue corresponding to the 12 state have a far more signicant eect on the edge excitations. To see this, note that in the Laughlin 1=m-states, within the plasma analogy the unit-charge correlator on the edge of a conned droplet can be expressed as [4 – 6] Ve (|q|) = (6) 0 The plasma analogy enables the calculation of correlators and expectation values in LLL quantum many-body states, by mapping them onto a classical statistical mechanics problem — the two-dimensional Coulomb gas, at an “inverse temperature” = 2=m. Thus, for example, the GM correlator corresponding to the Laughlin state 1=m can be written as (z; z 0 ) = n|z − z 0 |−m=2 exp{Ve (z − z 0 )}; namely, a screened Coulomb interaction. As a result, the GM correlator Eq. (2) acquires the familiar asymptotic form Gc(L) (z; z 0 ) ∼ exp{Vim (x)}; 2. Principal results 73 (3) where x = |z − z |, and Vim (x) is the interaction of a conned plasma with a point-like charge m placed at a distance x from its boundary. Due to the perfect screening in the plasma, it is dominated by an interaction of this external charge with its image: Vim (x) = −m ln(x): (7) The resulting correlator (which here can be identied with the electron propagator) then exhibits an algebraic decay Gc(L) ∼ 1=xm , characteristic of a Luttinger liquid. In contrast, the anomalous plasma corresponding to the compressible 12 state is not suciently well screened, and the electrostatic parallel of the image charge term Eq. (7) is found to be (to leading order) independent of x [17]. As a consequence, the correlator Gc is nearly constant, indicating that charged excitations on the edge are not well-dened. This is a 74 M.V. Milovanovic, E. Shimshoni = Physica E 6 (2000) 72–74 signature of a strong coupling between the edge and the bulk. To nd out the electron correlator, we note that the electron can be viewed as a composite of a charged object (described by Gc ) and a neutral component, which introduces the Fermi statistics. The combined correlator is sin(kxF x) ; (8) x where g depends on the boundary conditions. Namely, the electron correlations on the edge are as if the system was a free (two-dimensional) Fermi gas of electrons. Ge (x) ∼ g 3. Concluding remarks Our results suggest that at = 12 , due to strong coupling between charged modes on the edge and the neutral Fermions (dipoles) in the bulk, the (attractive) correlation hole is not well dened on the edge. Hence, the system can be modeled as a free Fermi gas of electrons. Indeed, an experimental evidence for the enhanced bulk-edge coupling at = 12 is provided by measurements of non-local resistance (Rnl ) [26,27], which monitor the relative dominance of edge transport: the observed pattern of Rnl () imitates Rxx () for nearly every 61, except for a sever suppression of the former around = 12 . Finally, while our results contradict the validity of an eective one-dimensional description of the edge excitations on the static level, the dynamics, in a certain energy range, may decouple the edge and bulk so as to recover the Laughlin-like behavior apparent in Ref. [10] (see also Ref. [13]). Acknowledgements We acknowledge useful discussions with A. Auerbach, J. Feinberg, V. Goldman J.H. Han, A. MacDonald, R. Rajaraman, S. Sondhi, A. Stern and X.-G. Wen, and especially with N. Read. This work was partly supported by the United States–Israel Binational Science Foundation (BSF), and the Israeli Academy of Sciences (M.V.M.). References [1] R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. [2] R.B. Laughlin, in: R.E. Prange, S.M. Girvin (Eds.), The Quantum Hall Eect, Springer, New York, 1990. [3] S.M. Girvin, A.H. MacDonald, Phys. Rev. Lett. 58 (1987) 1252. [4] X.-G. Wen, Phys. Rev. B 41 (1990) 12 838. [5] X.-G. Wen, Phys. Rev. B 43 (1991) 11 025. [6] X.-G. Wen, Int. J. Mod. Phys. 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