Coulomb blockade of anyons
Coulomb blockade of anyons
Dmitri V. Averin and James A. Nesteroff
Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794-3800
(Dated: August 23, 2021)
Coulomb interaction turns anyonic quasiparticles of a primary quantum Hall liquid with filling
factor ν = 1/(2m + 1) into hard-core anyons. We have developed a model of coherent transport of
such quasiparticles in systems of multiple antidots by extending the Wigner-Jordan description of
1D abelian anyons to tunneling problems. We show that the anyonic exchange statistics manifests
itself in tunneling conductance even in the absence of quasiparticle exchanges. In particular, it can
be seen as a non-vanishing resonant peak associated with quasiparticle tunneling through a line of
three antidots.
PACS numbers: 73.43.-f, 05.30.Pr, 71.10.Pm, 03.67.Lx
Quasiparticles of two-dimensional (2D) electron liq-
uids in the regime of the Fractional Quantum Hall effect
(FQHE) have unusual properties of fractional charge [1]
and fractional exchange statistics [2, 3]. The fractional
quasiparticle charge was observed in experiments on an-
tidot tunneling [4] and shot-noise measurements [5, 6].
The situation with fractional statistics is so far less cer-
tain even in the case of the abelian statistics, which is the
subject of this work. Although the recent experiments [7]
demonstrating unusual flux periodicity of conductance
of a quasiparticle interferometer can be interpreted as a
manifestation of the fractional statistics [8, 9], this inter-
pretation is not universally accepted [10, 11]. There is a
number of theoretical proposals (see, e.g., [12, 13]) sug-
gesting tunnel structures where the statistics manifests
itself through noise properties. Partly due to complex-
ity of noise measurements, such experiments have not
been performed successfully up to now. In this work,
we show that coherent quasiparticle dynamics in multi-
antidot structures should provide clear signatures of the
exchange statistics in dc transport. Most notably, in tun-
neling through a line of three antidots, fractional statis-
tics leads to a non-vanishing peak of the tunnel conduc-
tance which would vanish for integer statistics.
These effects rely on the ability of quantum anti-
dots to localize individual quasiparticles of the QH liq-
FIG. 1: Tunneling of anyonic quasiparticles between oppo-
site edges of an FQHE liquid through quasi-1D triple-antidot
systems: (a) loop, (b) open interval. Quasiparticles tunnel
between the edges and the antidots with rates Γ1,2. The an-
tidots are coupled coherently by tunnel amplitudes ∆.
uids [4, 14, 15]. The resulting transport phenomena in
antidots are very similar to those associated with the
Coulomb blockade [16] in tunneling of individual elec-
trons in dots. For instance, similarly to a quantum dot
[19], the linear conductance of one antidot shows periodic
oscillations with each period corresponding to the addi-
tion of one quasiparticle [4, 14, 15, 17, 18]. Recently, we
have developed a theory of such Coulomb-blockade-type
tunneling for a double-antidot system [20], where quasi-
particle exchange statistics does not affect the transport.
The goal of this work is to extend this theory to antidot
structures where the statistics does affect the conduc-
tance. The two simplest structures with this property
consist of three antidots and have quasi-1D geometries
with either periodic or open boundary conditions (Fig. 1).
A technical issue that needed to be resolved to calculate
the tunnel conductance is that the anyonic field opera-
tors defined through the Wigner-Jordan transformation
[21, 22, 23, 24], are not fully sufficient in the situations of
tunneling. As we show below, to obtain correct matrix
elements for anyon tunneling, one needs to keep track
of the appropriate boundary conditions of the wavefunc-
tions which are not accounted for in the field operators.
Specifically, we consider the antidots coupled by tun-
neling among themselves and to two opposite edges of
the quantum Hall liquid (Fig. 1). The edges play the role
of the quasiparticle reservoirs with the transport voltage
V applied between them. We assume that the antidot-
edge coupling is weak and can be treated as a pertur-
bation. Quasiparticle transport through the antidots is
governed then by the kinetic equation similar to that for
Coulomb-blockade transport through quantum dots with
a discrete energy spectrum [25]. Coherent quasiparticle
dynamics requires that the relaxation rate Γd created by
direct Coulomb antidot-edge coupling is weak. This con-
dition should be satisfied if the edge-state confinement
is sufficiently strong [20]. The requirement on the con-
finement is less stringent in the case of the antidot line
(Fig. 1b), in which antidot quasiparticles move along the
edge, suppressing the antidot-edge coupling at low fre-
quencies. We also assume that all quasiparticle energies
http://arxiv.org/abs/0704.0439v2
on the antidots, tunnel amplitudes ∆, temperature T ,
Coulomb interaction energies U between quasiparticles
on different antidots, are much smaller than the energy
gap ∆∗ for excitations on each antidot. This condition
ensures that the state of each antidot is characterized
completely by the occupation number n of its relevant
quantized energy level. In any given range of the back-
gate voltage or magnetic field (which produces the over-
all shift of the antidot energies - see, e.g., [4, 14, 15]),
there can be at most one quasiparticle on each antidot,
n = 0, 1. This “hard-core” property of the quasipar-
ticles means that they behave as fermions in terms of
their occupation factors, despite the anyonic exchange
statistics. All these assumptions can be summarized as:
Γd,Γj ≪ ∆, U, T ≪ ∆∗.
Under these conditions, the antidot tunneling is domi-
nated by the antidot energies. The quasi-1D geometry of
the antidot systems we consider makes it possible to in-
troduce the quasiparticle “coordinate” x numbering suc-
cessive antidots; e.g., x = −1, 0, 1 for systems in Fig. 1.
The quasiparticle Hamiltonian can be the written as
[ǫxnx − (∆xξ†x+1ξx + h.c.)] +
Ux,ynxny , (1)
where ǫx are the energies of the relevant localized states
on the antidots (taken relative to the common chemical
potential of the edges at V = 0), ∆x is the tunnel cou-
pling between them, Ux,y is the quasiparticle Coulomb
repulsion, and nx ≡ ξ†xξx. The quasiparticle operators
ξ†x, ξx in (1) can be viewed as the Klein factors left in the
standard operators for the edge-state quasiparticles when
all the edge magneto-plasmon modes are suppressed by
the gap ∆∗. Characteristics of such Klein factors depend
on the geometry of a specific tunneling problem; non-
trivial examples can be found in [12, 13, 26, 27]. In the
Hamiltonian (1), ξx describe the hard-core anyons with
exchange statistics πν. Wigner-Jordan transformation
expresses them through the Fermi operators cx [21]:
ξx = e
iπ(ν−1)
nzcx , ξyξx = ξxξye
iπνsgn(x−y), (2)
with similar relations for ξ†.
Anyonic statistics creates an effective interaction be-
tween the quasiparticles which can be understood as the
Aharonov-Bohm (AB) interaction between a flux tube
“attached” to one of the particles and the charge carried
by another. In general, this interaction can be masked by
the direct Coulomb interaction Ux,y. In the antidot loop
(Fig. 1a), however, Ux,y is constant, Ux,y = U , and the
interaction term in (1) reduces to Un(n− 1)/2, with n =
x nx – the total number of the quasiparticles on the an-
tidots. In this case, the Coulomb interaction contributes
to the energy separation between the group of states with
different n, but does not affect the level structure for
given n. The hard-core property of quasiparticles limits n
to the interval [0, 3]. For n = 0 and n = 3, the system has
the “empty” and “completely filled” state with respective
energies E0 = 0 , E3 =
x ǫx+3U . The spectrum E1k of
the three n = 1 states |1k〉 =
x φk(x)ξ
x|0〉, is obtained
as usual from (1). In the uniform case ǫx = ǫ, ∆x = ∆,
with an external AB phase ϕ, one has φk(x) = e
ikx/L1/2
E1k = ǫ−∆cos k , k = (2πm+ ϕ)/L , (3)
where m = 0, 1, 2, and the loop length is L = 3.
Anyonic statistics can be seen in the n = 2 states,
|2l〉 = (1/
xy ψl(x, y)ξ
x|0〉. The fermion-anyon
relation (2) suggests that the stationary two-quasiparticle
wavefunctions should coincide up to the exchange phase
with that for free fermions:
ψl(x, y) =
eiπ(1−ν)sgn(x−y)/2√
φq(x) φq(y)
φp(x) φp(y)
. (4)
Here φs are the single-particle eigenstates of the Hamilto-
nian (1). (The states (4) are numbered with the index l of
the third “unoccupied” eigenstate of (1) complementary
to the two occupied ones q, p.) The boundary conditions
for the φs are affected by the exchange phase in Eq. (4).
To find them, we temporarily assume for clarity that co-
ordinates x, y are continuous and lie in the interval [0, L].
Subsequent discretization does not change anything sub-
stantive in this discussion. The 1D hard-core particles
are impenetrable and can be exchanged only by moving
one of them, say x, around the loop from x = y + 0 to
x = y − 0 (Fig. 2a). Since the loop is imbedded in the
underlying 2D system, such an exchange means that the
wavefunction acquires the phase factor eiπν , in which the
sign of ν is fixed by the properties of the 2D system, e.g.
the direction of magnetic field in the case of FQHE liq-
uid. Next, if the second particle is moved similarly, from
y = x+ 0 to y = x− 0, the wavefunction changes in the
same way, for a total factor ei2πν . Equation (4) shows
that only one of these changes can agree with the 1D form
of the exchange phase. As a result, the wavefunction (4)
satisfies different boundary conditions in x and y:
ψl(L, y) = ψl(0, y)e
iϕ, ψl(x, L) = ψl(x, 0)e
i(ϕ+2πν). (5)
Conditions (5) on the wavefunction (4) mean that the
single-particle functions φ in (4) satisfy the boundary
condition that correspond to the effective AB phase
ϕ′ = ϕ+ π− πν, i.e. the addition of an extra quasiparti-
cle to the loop changed the AB phase by π − πν, where
−πν comes from the exchange statistics and π from the
hard-core condition. This gives the energies of the two-
quasiparticle states (4) as U + E1q + E1p, where, if the
loop is uniform, the single-particle energies are given by
Eq. (3) with ϕ → ϕ′. In this case,
k E1k = 0, and
the energies E2l of the two-quasiparticle states can be
written as:
E2l = 2ǫ+ U −∆cos l , l = (2πm′ + ϕ− πν)/3 , (6)
(a) (b)
FIG. 2: Exchanges of hard-core anyons on a 1D loop: (a) real
exchanges by transfer along the loop embedded in a 2D sys-
tem; (b) formal exchanges describing the assumed boundary
conditions (5) of the wavefunction.
where m′ = 0, 1, 2.
One of the consequences of this discussion is that the
sign of ν in the 1D exchange phases of Eqs. (2) and (4)
can be chosen arbitrarily for a given fixed sign of the
2D exchange phase. Reversing this sign only exchanges
the character of the boundary conditions (5) between x
and y. This fact has simple interpretation. Although
the 1D hard-core anyons can not be exchanged directly,
formally, coordinates x and y in Eq. (4) are independent
and one needs to define how they move past each other
at the point x = y. Depending on whether the x-particle
moves around y from below or (as in Fig. 2b) from above,
its trajectory does or does not encircle the flux carried
by the y particle, and the boundary condition for x is
or is not affected by the statistical phase. The choice
made for x immediately implies the opposite choice for
y (Fig. 2b), accounting for different boundary conditions
(5). This interpretation shows that in calculation of any
matrix elements, the participating wavefunctions should
be taken to have the same boundary conditions. While
this requirement is natural for processes with the same
number of anyons, it is less evident for tunneling that
changes the number of anyons in the system. Indeed, the
most basic, tunnel-Hamiltonian, description of tunneling
into the point z of the system leads to the states
ξ†z |1k〉 = (1/
ψk(x, y)ξ
x|0〉 , (7)
ψk(x, y) = [φk(x)δy,z − eiπ(1−ν)sgn(x−y)δx,zφk(y)]/
One can see that Eq. (7) automatically implies specific
choice of the boundary conditions which corresponds to
the tunneling anyon not being encircled by anyons al-
ready in the system. This means that in the calculation
of the tunnel matrix elements with the states (4), one
should always pair the coordinate of the tunneling anyon
with the discontinuous one in (5). Then, the tunnel ma-
trix elements are obtained as
〈2l|ξ†z|1k〉 =
ψ∗l (x, z)φk(x) . (8)
For instance, in the case of uniform loop with states (3)
and (6), we get up to an irrelevant phase factor
〈2l|ξ†z|1k〉 = (2/3) cos[(k − l)/2] . (9)
Specific anyonic interaction between quasiparticles can
be seen in the fact that the matrix elements (9) do not
vanish for any pair of indices k, l. In the fermionic case
ν = 1, one of the elements (9) always vanishes for any
given k, since the two-particle state after tunneling neces-
sarily has one particle in the original single-particle state.
By contrast, the tunneling anyon can shift existing par-
ticle out of its state.
The matrix elements involving empty or fully occu-
pied states coincide with those for fermions. Taken to-
gether with Eqs. (8) and (9) for transitions between the
partially filled states, they determine the rates Γj(E) =
γjfν(E)|〈ξ†z〉|2 of tunneling between the jth edge and the
antidots, where γj is the overall magnitude of the tun-
neling rate, and
fν(E) = (2πT/ωc)
ν−1|Γ(ν/2+iE/2πT )|2e−E/2T /2πΓ(ν)
is its energy dependence associated with the Luttinger-
liquid correlations in the edges [28]. Here Γ(z) is the
gamma-function and ωc is the cut-off energy of the edge
excitations. The rates Γj(E) can be used in the stan-
dard kinetic equation to calculate the conductance of the
antidot system [20]. Anyonic statistics of quasiparticles
affects the position and amplitude of the conductance
peaks through the shift of the energy levels by quasiparti-
cle tunneling (described, e.g., by Eq. (6)) and through the
kinetic effects caused by the anyonic features in the ma-
trix elements (8). In the case of the antidot loop (Fig. 1a),
however, effects of statistics are masked by the external
AB flux ϕ through the loop. Since the area of practical
antidots is much larger than the internal area of the loop,
ϕ is essentially random and can not be controlled by ex-
ternal magnetic field on the relevant scale of one period
of conductance oscillations. Below, we present the results
for conductance for the similar case of a line of antidots
(Fig. 1b), the conductance of which is insensitive to the
AB flux, and shows effects of fractional statistics in the
tunneling matrix elements.
As before, the quasiparticle Hamiltonian is given by
Eq. (1). In this geometry, the interaction energy U1 ≡
U1,0 = U0,−1 between the nearest-neighbor antidots is in
general different from the interaction U2 ≡ U1,−1 between
the quasiparticles at the ends. The localization energies
on the antidots can be written as ǫj = ǫ+xδ+2λ|x|. We
consider first the unbiased line, δ = 0. At low tempera-
tures, T ≪ ∆, U , only the ground states of n quasipar-
ticles with energies En participate in transport: E0 = 0,
E1 = ǫ + λ − ω, E2 = 2ǫ + 3λ − ω̄ + (Ua + Ub)/2, and
E3 = 3ǫ+ 2Ua + Ub + 4λ, where ω = (∆
2 + λ
2)1/2
and ω̄ is given by the same expression with λ replaced by
λ̄ = λ − (U1 − U2)/2. In this regime, the linear conduc-
tanceG consists of three peaks, with each peak associated
with addition of one more quasiparticle to the antidots,
(eν)2
γ1 + γ2
anfν(En+1 − En)
1 + exp[−(En+1 − En)/T ]
, (10)
where an ≡ |〈n + 1|ξ†0|n〉|2. The amplitudes a0, a2 are
effectively single-particle, and thus, independent of the
−10 −5 0 5
 = 2.0∆
 = 1.5∆
T = 0.3∆
−10 −5 0 5
−2 0 2
FIG. 3: Linear conductance G of the antidot line in a ν = 1/3
FQHE liquid (Fig. 1b) as a function of the common anti-
dot energy ǫ relative to the edges. In contrast to electrons
(ν = 1, left inset), tunneling of quasiparticles with fractional
exchange statistics produces non-vanishing conductance peak
associated with transition between the ground states of one
and two quasiparticles. The maximum of this peak is shown in
the right inset (ν = 1/3 – solid, ν = 1 – dashed line) as a func-
tion of the bias δ. The curves are plotted for ∆1 = ∆2, λ = 0,
γ1 = γ2; conductance is normalized to G0 = (eν)
2Γ1(0)/∆1.
exchange statistics: a0 = (ω + λ)/2ω, and a2 = (ω̄ −
λ̄)/2ω̄. By contrast, the amplitude a1 of the transition
from one to two quasiparticles is multi-particle, and is
found from Eqs. (4) and (8) to be strongly statistics-
dependent,
(ω + λ)ω(ω̄ − λ̄)ω̄
cos2(πν/2) . (11)
In particular, a1 vanishes for electron tunneling (ν = 1),
but is non-vanishing in the case of fractional statistics,
e.g., for ν = 1/3, when cos2(πν/2) = 3/4. This is illus-
trated in Fig. 3 which shows the conductance G obtained
by direct solution of the full kinetic equation for tunnel-
ing through the antidots. Qualitatively, the vanishing
amplitude a1 for electrons can be understood as a re-
sult of destructive interference between the two terms in
the two-particle wavefunction which correspond to dif-
ferent ordering of the added/existing electron on the an-
tidot line. The opposite signs of these two terms lead
to vanishing overlap with the single-particle state in the
tunnel matrix element. Fractional statistics of quasipar-
ticles makes this destructive interference incomplete. Fi-
nite bias δ 6= 0 along the line suppresses this interference
making the effect of the statistics smaller. One can still
distinguish the fractional statistics by looking at the de-
pendence of the amplitude of the middle peak of conduc-
tance on the bias δ shown in the right inset in Fig. 3.
In conclusion, we have developed a model of coherent
transport of anyonic quasiparticles in systems of mul-
tiple antidots. In antidot loops, addition of individual
quasiparticles shifts the quasiparticle energy spectrum by
adding statistical flux to the loop. In the case without
loops, energy levels are insensitive to quasiparticle statis-
tics, but the statistics still manifests itself in the quasi-
particle tunneling rates and hence dc tunnel conductance
of the antidot system.
The authors would like to thank F.E. Camino, V.J.
Goldman, J.K. Jain, V.E. Korepin, Yu.V. Nazarov, O.I.
Patu, V.V. Ponomarenko, and J.J.M. Verbaarschot for
discussions. This work was supported in part by NSF
grant # DMR-0325551 and by ARO grant # DAAD19-
03-1-0126.
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http://arxiv.org/abs/cond-mat/0702614
