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Physica E 6 (2000) 72–74
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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
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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.).
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