Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be
Fractional quantum Hall quasiparticles are famous for having fractional electric charge. Recent experiments report that the quasiparticle’s effective electric charge determined through tunneling current noise measurements can depend on the system parameters such as temperature or bias voltage. Sev...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1284532018-01-10T03:03:00Z Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be Snizhko, K. Квантовые эффекты в полупpоводниках и диэлектриках Fractional quantum Hall quasiparticles are famous for having fractional electric charge. Recent experiments report that the quasiparticle’s effective electric charge determined through tunneling current noise measurements can depend on the system parameters such as temperature or bias voltage. Several works proposed to understand this as a signature for edge theory properties changing with energy scale. I consider two of such experiments and show that in one of them the apparent dependence of the electric charge on a system parameter is likely to be an artefact of experimental data analysis. Conversely, in the second experiment the dependence cannot be explained in such a way. 2016 Article Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be / K. Snizhko // Физика низких температур. — 2016. — Т. 42, № 1. — С. 79–88. — Бібліогр.: 37 назв. — англ. 0132-6414 PACS: 73.43.–f http://dspace.nbuv.gov.ua/handle/123456789/128453 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Квантовые эффекты в полупpоводниках и диэлектриках Квантовые эффекты в полупpоводниках и диэлектриках |
| spellingShingle |
Квантовые эффекты в полупpоводниках и диэлектриках Квантовые эффекты в полупpоводниках и диэлектриках Snizhko, K. Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be Физика низких температур |
| description |
Fractional quantum Hall quasiparticles are famous for having fractional electric charge. Recent experiments
report that the quasiparticle’s effective electric charge determined through tunneling current noise measurements
can depend on the system parameters such as temperature or bias voltage. Several works proposed to understand
this as a signature for edge theory properties changing with energy scale. I consider two of such experiments and
show that in one of them the apparent dependence of the electric charge on a system parameter is likely to be an
artefact of experimental data analysis. Conversely, in the second experiment the dependence cannot be explained
in such a way. |
| format |
Article |
| author |
Snizhko, K. |
| author_facet |
Snizhko, K. |
| author_sort |
Snizhko, K. |
| title |
Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be |
| title_short |
Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be |
| title_full |
Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be |
| title_fullStr |
Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be |
| title_full_unstemmed |
Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be |
| title_sort |
tunneling current noise in the fractional quantum hall effect: when the effective charge is not what it appears to be |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2016 |
| topic_facet |
Квантовые эффекты в полупpоводниках и диэлектриках |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/128453 |
| citation_txt |
Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be / K. Snizhko // Физика низких температур. — 2016. — Т. 42, № 1. — С. 79–88. — Бібліогр.: 37 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT snizhkok tunnelingcurrentnoiseinthefractionalquantumhalleffectwhentheeffectivechargeisnotwhatitappearstobe |
| first_indexed |
2025-07-09T09:07:23Z |
| last_indexed |
2025-07-09T09:07:23Z |
| _version_ |
1837159726547206144 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1, pp. 79–88
Tunneling current noise in the fractional quantum Hall effect:
when the effective charge is not what it appears to be
Kyrylo Snizhko
Faculty of Physics, Taras Shevchenko National University of Kyiv, Kyiv 03022, Ukraine
Shortly after leaving Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
E-mail: snezhkok@gmail.com
Received July 24, 2015, published online November 23, 2015
Fractional quantum Hall quasiparticles are famous for having fractional electric charge. Recent experiments
report that the quasiparticle’s effective electric charge determined through tunneling current noise measurements
can depend on the system parameters such as temperature or bias voltage. Several works proposed to understand
this as a signature for edge theory properties changing with energy scale. I consider two of such experiments and
show that in one of them the apparent dependence of the electric charge on a system parameter is likely to be an
artefact of experimental data analysis. Conversely, in the second experiment the dependence cannot be explained
in such a way.
PACS: 73.43.–f Fractional quantum Hall effect.
Keywords: quantum Hall effect, tunneling experiments, effective charge, neutral mode.
1. Introduction
The fractional quantum Hall effect (FQHE) is a field of
intensive research nowadays, with one of the main reasons
for that is its supporting of quasiparticle excitations with
unusual properties. Quasiparticles in the FQHE have the
electric charge which is a fraction of the electron charge, and
are predicted to have other unusual properties such as
anyonic or even non-Abelian statistics. The quasiparticles
obeying the non-Abelian statistics would potentially allow
for performing topologically protected quantum computa-
tions (TPQC) (i.e., quantum computations in which qubits
are protected from decoherence by “topological order” of
the system) [1]. Therefore, finding the properties of
quasiparticles in different FQHE states is an important task.
Measuring tunneling current noise is a powerful method
for finding the properties of quasiparticle excitations in the
FQHE, in particular the tunneling quasiparticle electric
charge. In the regime of weak tunneling of quasiparticles
the tunneling current shot noise is proportional to the tun-
neling current itself, the proportionality coefficient, called
the Fano factor, is the tunneling quasiparticle charge [2]. If
several quasiparticles contribute to the tunneling processes,
then the Fano factor is some average of the quasiparticles'
charges. It is in this way that the first confirmation was
given for the fractional charge of quasiparticles in the
FQHE with filling factor = 1/3ν [3,4].
Some of more recent experiments [5–10] that studied
more complicated FQHE states report that the “effective
charge” determined from tunneling current noise depends on
external parameters such as temperature [6,8], bias voltage
across the tunneling contact [9], other system parameters
[10]. Three distinct mechanisms proposed recently can
contribute to evolution of the Fano factor. Two of them
assume that the FQHE edges behave differently at different
energy scales: either due to energy cutoffs of edge
transport channels [11–13] or due to edge reconstruction
[14]. The third one, considered in Ref. 15, assumes de-
pendence of quasiparticle tunneling amplitudes on experi-
mental parameters, which can change relative importance
of different quasiparticles' contributions.
However, there is a subtlety regarding the data analysis
in experimental works. Namely, experimentalists [3–10,16]
tend to use for analysis a formula that is not based on a real-
istic FQHE model but is a generalization of a formula which
can be derived for free electrons. As it was analyzed in
Ref. 17 in the case of = 1/3,ν the formula used by experi-
mentalists can agree well with the exact theory under certain
conditions, but deviates from the exact theory otherwise.
This can give rise to misinterpretations of experimental
data. In particular, the "effective charge" obtained this way
is not necessarily the same as the Fano factor.
In this work I analyze the results of Ref. 10 regarding
= 2/3ν and of Ref. 6 regarding = 2/5.ν I show that in the
former case the data can be explained within the minimal
© Kyrylo Snizhko, 2016
Kyrylo Snizhko
= 2/3ν FQHE edge model [18,19] without additional
structure such as energy cutoffs or edge reconstruction.
Therefore, the effective charge dependence on external
parameters appears to be a data analysis artefact in this
case. In the case of the data of Ref. 6 regarding = 2/5,ν
the charge dependence on the system temperature cannot
be explained in this simple way.
The paper structure is as follows. In Sec. 2 I introduce a
general scheme of the experiments I discuss. Then in
Sec. 3 I describe a theoretical model that can be used to
analyze such experiments. This model is not easy to treat,
therefore in Sec. 4 I discuss the three existing approaches
to analyzing the model and the experiments: the one based
on perturbative treatment of tunneling processes (Sec. 4.1),
the one based on exactly solving the model in the cases
when this can be done (Sec. 4.2), and the one typically
used by experimentalists — the phenomenological ap-
proach (Sec. 4.3). Finally, Secs. 5 and 6 present original
results of analyzing the data regarding = 2/3ν and 2/5,
respectively. Some concluding remarks are made in Sec. 7.
2. Tunneling experiments in the FQHE:
a typical scheme
The typical scheme of the experiments I am going to
discuss is presented in Fig. 1.
There are two FQHE edges (upper and lower) along
which transport of electric charge and of heat can occur,
the rest of the sample is insulating. Each edge contains at
least one “charged mode” — the channel, excitations in
which carry electric charge and are responsible for charge
transport. The transport channels are chiral, i.e., excitations
in a channel can propagate in one direction only. If there
are several charged modes in an edge, I assume that all of
them flow in one direction. Apart from the charged modes
the edges can support “neutral modes”. These are transport
channels that do not carry electric charge. They can, how-
ever, transport heat, spin etc. The neutral modes can be
absent at all, there can be one or several of them. Neutral
modes are also chiral. Some of the neutral modes can flow
in the same direction as the charged ones, some — in the
opposite direction.
The upper and lower edges come close together at the
quantum point contact (QPC) where tunneling of
quasiparticles between the edges can take place. Apart
from the QPC, the edges are separated and do not interact
with each other.
Experimental equipment is connected to the system
through four Ohmic contacts (yellow rectangles). Ground 1
contact is grounded. Source S is used to inject electric cur-
rent sI into the lower edge. Voltage probe is used to meas-
ure the current I flowing into it, and its noise. If no tunnel-
ing takes place at the QPC, then = sI I and the noise of I
is just the Johnson–Nyquist noise. However, if there is tun-
neling at the QPC, then both I and its noise carry infor-
mation about the tunneling processes. Finally, Source N is
used to inject current nI into the system. As one can see
from the scheme, the electric current itself does not flow into
the system. However, its injection can excite the neutral
modes of the upper edge, and if some of them flow opposite
to the charged mode, they can influence the tunneling pro-
cesses at the QPC.
3. Tunneling experiments in the FQHE: the model
In this section I briefly outline the standard model for ana-
lyzing the experiments described in the previous section.
The model contains three distinct ingredients: single
edge model (to describe each of the two edges), tunneling
processes model, and a model for interaction of the Ohmic
contacts with the edge.
Here I consider the case of Abelian edge theories. The
non-Abelian ones can be considered similarly, but I do not
analyze them in this paper. For a general discussion of how
the FQHE edge theories are constructed see Ref. 20.
A single Abelian edge can be described in terms of N
bosonic fields iϕ with = 1, . . ., ,i N one for each edge
mode. The action for the fields is*
21= ( ( ) ),
4 m x m t m m x m
m
S dxdt v−χ ∂ ϕ ∂ ϕ − ∂ ϕ
π ∑∫ (1)
Fig. 1. (Color online) A typical experiment scheme. Two FQHE
edges form a quantum point contact (QPC) at which quasiparticles
can tunnel between the edges. The Ohmic contact Ground 1 is
grounded. Source N and Source S are used to inject some electric
current into the system. Measurement of the electric current and its
noise is performed at Voltage probe. T0 is the system and its envi-
ronment temperature when currents Is, In are not injected.
* In this section I put e = ħ = kB = 1 unless the opposite is stated explicitly. Here e is the elementary charge, ħ is the Planck constant,
kB is the Boltzmann constant.
80 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1
Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be
where = 1mχ ± determine chiralities of the modes (plus
for counterclockwise-movers and minus for clockwise-
movers*), and > 0mv are the modes' propagation veloci-
ties. Without loss of generality, I put = 1mχ + for
= 1, . . ., lm N and = 1mχ − for = 1, . . .,lm N N+ ( lN is
thus the number of counterclockwise-moving modes).
The electric current J α 0(J is the electric charge den-
sity, 1J is the electric current flowing along the edge) has
the form
1= ,
2 m m
m
J qα αβ
βε ∂ ϕ
π ∑ (2)
where the symbol αβε denotes the fully antisymmetric
tensor with ,α β taking values t and x (or 0 and 1, respec-
tively) and 01= = 1.txε ε The numbers iq should satisfy
the constraint [21,22]
2 = ,m m
m
qχ ν∑ (3)
where ν is the filling factor.
As it was mentioned in the previous section, I assume that
all the modes that carry electric charge flow in one direction.
Formally this means that = 0mq for = 1, . . ., ,lm N N+ i.e.,
only counterclockwise-propagating modes can carry electric
charge.
The quantized fields mϕ obey the commutation relations
,[ ( , ), ( , )] = sgn ( ) ,m m m m m mx t x t i X X′ ′ϕ ϕ − π − δ′ ′ ′ (4)
where = .m m mX x v t−χ +
It is convenient to introduce local quasiparticle operators
2 /2
( , ) = : exp ( , ) : .
2
gm
m m m
m
LV x t i g x t
− ϕ π
∑
∑g (5)
These operators can be not used when describing transport
along a single edge, but are important for tunneling pro-
cesses. Here L is the edge length, : ... : stands for the nor-
mal ordering, 1= ( ,..., ),Ng gg and mg ∈ are the
quasiparticle quantum numbers. The quasiparticles' quan-
tum numbers are quantized, i.e., the set of allowed vectors
g is discrete. The quasiparticle's two most important quan-
tum numbers, the electric charge Q and the scaling dimen-
sion ,δ are equal to
= ,m m m
m
Q q gχ∑ (6)
21= .
2 m
m
gδ ∑ (7)
Having constructed a single edge theory, one can de-
scribe tunneling between the two edges at the QPC as hop-
ping of local quasiparticles from one edge to another. The
Hamiltonian for such processes is [2,21,23,24]
( )† ( )= (0, ) (0, ) h.c.u l
TH V t V tη +∑ g g g
g
, (8)
here the superscripts ( ), ( )u l label quantities relating to the
upper and the lower edges, respectively, ηg are the tunnel-
ing amplitudes; for simplicity I have put the position of the
QPC to the origin of coordinates. In the limit of large edge
length ( )L → ∞ the dominant contribution to the tunneling
processes comes from the quasiparticles with the smallest
scaling dimension δ **. In the following I label such
quasiparticle types by = 1, . . ., ,i n with the quasiparticle
electric charges being iQ (in the units of the elementary
charge e), their common scaling dimension being = ,iδ δ
and the full set of quantum numbers being .ig
The final component is a model for interaction between
the Ohmic contacts and the edge. For this work I use the
following set of assumptions regarding the interaction.
I assume that when an edge mode flows into an Ohmic
contact all the excitations are absorbed by the latter, and
the state of inflowing modes does not influence the state of
modes that flow away from the contact. I also assume that
an edge mode emitted by an Ohmic contact, when no cur-
rent is injected into it, is in thermal equilibrium with the
contact and its environment. When an electric current is
injected through an Ohmic contact, the only change to the
state of the charged mode(s) emanating from the contact is
the change of their chemical potentials, so that they carry
the injected current. The influence of current injection on
the neutral modes should be a matter of separate investiga-
tion. For this work I assume that the neutral modes that
propagate in the same direction as the charged mode(s) are
not influenced by current injection at all, while the
counterpropagating neutral modes get heated due to this.
Therefore, if counterflowing neutral modes are present in
the edge, the temperature of the upper edge near the QPC
is 0= ( ) ,nT I Tλ with ( ) (0) = 1.nIλ ≥ λ Details of this
heating for = 2/3ν were investigated in Ref. 15.
Before reviewing the existing approaches to solving the
model outlined above, I define the observables that are
measured in the experiments I consider below.
Current I flowing into Voltage probe contact (see
Fig. 1) is equal to 1J component of the current ,J α de-
fined in Eq. (2), taken at some point to the right of the QPC
along the lower edge. I denote the operator of this current
as ˆ( ).I t Then, the average current flowing into Voltage
* In this section I put e = ħ = kB = 1 unless the opposite is stated explicitly. Here e is the elementary charge, ħ is the Planck con-
stant, kB is the Boltzmann constant.
* Clockwise-movers are left-movers at the lower edge and right-movers at the upper edge. Correspondingly, the counterclockwise-
movers are the right-movers at the lower edge and left-movers at the upper edge.
** One can see this from L–δ factor in Eq. (5). This statement is also confirmed by Monte Carlo simulations [25].
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1 81
Kyrylo Snizhko
probe is ˆ= ( ) .I I t〈 〉 It is also convenient to introduce oper-
ator ˆ( ) = ( ) .I t I t Iδ −
If there is no tunneling at the QPC, then I is equal to
the current sI injected at Source S. As soon as there is
tunneling, some part of the quasiparticles will not reach
Voltage probe, with =s TI I I− being the tunneling cur-
rent. Define the following quantities:
• transmission rate = / st I I ;
• tunneling (or reflection) rate = / = 1T sr I I t− ;
• measured current noise*
1( ) = exp( ) { (0), ( )} ,
2
S d i I I
∞
−∞
ω τ ωτ 〈 δ δ τ 〉∫
(9)
where { } denotes the anti-commutator.
In what follows I only use the zero-frequency noise
( = 0).S ω It is also convenient to talk about the excess
noise
Nyquist 0( = 0) = (0) (0) = (0) ,
2
S S S S Tν
ω − −
π
(10)
where 0T is the system temperature when no currents are
injected.
4. Three approaches to theoretical description
of tunneling experiments
4.1. Perturbative treatment of tunneling
The model described in the previous section is hard to
solve. Exact solutions are available only in exceptional
cases. Therefore, the most generally applicable approach is
to treat the tunneling Hamiltonian (8) as a small perturba-
tion. Then in the lowest nontrivial order of perturbation
theory one obtains the following results [15,28]**:
4 1
0
4 1
4 ( )
= ,B
i i
is
e k T
r G
I
δ−
δ+
π
κ∑
(11)
2 4 1
0
4 1
4 ( )
(0) = ,B
i i
i
e k T
S F
δ−
δ+
π
κ∑
(12)
2
2 2
0
sin
= sin 2
(sinh ) (sinh )
i i s
i
Q Q j t
G dt
t t
∞ δ
δ δ
λ
πδ
λ∫ , (13)
02= cos 2 sin 2 ,TT T
i i iF F Fπδ − πδ
π
(14)
21 4
2
2 20
cos
= lim 1 4 (sinh ) (sinh )
TT i s
i i
Q j t
F Q dt
t t
∞ δ− δ
δ δε→+ ε
λε
+ − δ λ
∫ , (15)
2 2
0
2 2
0
cos
=
(sinh ) (sinh )
T i i s
i
Q t Q j t
F dt
t t
∞ δ
δ δ
λ
λ∫ , (16)
0 0
0
= , = ,s
s B
I ej I k T
I h
ν π (17)
where 0T is the equilibrium system temperature, sI is the
current injected into Source S, = ( )nIλ λ is related to the
upper edge heating due to injection of current nI (the upper
edge temperature at near the QPC is 0= ),T Tλ = 2h π is
the Planck constant, ν is the filling factor,
22( )2= | | ,
igmi i m
m
v−κ η ∏g and i enumerates different qua-
siparticles participating in tunneling. I remind the reader that
iQ are the electric charges of the quasiparticles and δ is
their common scaling dimension. The formulas (13), (15),
(16) are correct for < 1/2,δ for 1/2δ ≥ they should be
modified. However, typically the quasiparticles contributing
to the tunneling processes are predicted to have < 1/2.δ
If one applies the formulas above to analyze experi-
mental data, one often finds a significant disagreement be-
tween the theory and experiment already for the tunneling
rate r (see, e.g., [16,29,30] and references therein). This is
believed to be the result of nonuniversal physical processes
in the system which can lead to (a) renormalization of the
scaling dimension δ [31–34] and/or (b) tunneling ampli-
tudes iη
g
depending on various external parameters such as
the injected currents ,sI ,nI system temperature T0 [15].
Both effects are likely to be relevant in realistic situations.
The tunneling amplitudes should be exponentially sensitive
to the distance between the edges in the QPC since they de-
scribe tunneling of quasiparticles under a barrier. The dis-
tance between the electrostatically confined edges is in turn
sensitive to the edges' electrostatic potentials, which change
in the course of a tunneling experiment. The scaling dimen-
sion renormalization is also likely to be relevant in experi-
ments. For example, the mechanism of renormalization due
to 1/f noise, proposed in Ref. 34, is extremely robust: even
vanishingly small interaction of the FQHE edge with the 1/f
* One must be cautious when comparing formulas and data for noise from different articles since there are two conventions regard-
ing the definition of the noise spectral density. While some authors (see, e.g., [26]) use the same definition as I do, others (see,
e.g., [10,27]) adopt the definition which is twice as large as the one used here.
** Here and in the rest of the paper I restore the elementary charge e, the Planck constant ħ, and the Boltzmann constant kB, which I
had put to 1 in Sec. 3.
82 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1
Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be
noise can produce a finite renormalization of the scaling
dimension. Moreover, this mechanism (unlike the ones of
Refs. 31–33) is equally applicable for any type of the FQHE
edge: with or without counterflowing modes.
However, in this work I assume that no scaling dimen-
sion renormalization happens, but the tunneling amplitudes
can depend on the system parameters*.
For such a case, it has been argued in [15,28] that in-
stead of considering r and (0)S separately, it is advanta-
geous to consider their ratio (noise to tunneling rate ratio,
NtTRR)
(0)( ) = = .
i i
i
s s
i i
i
F
SX I eI
r G
κ
κ
∑
∑
(18)
In the large-Is limit one obtains
| | ( ) 1( ) = =
i i
i
s j I ss n i i
i
F
X I eI
Gλ ≥
κ
κ
∑
∑
4 1
0 04= | | ( , ),
i i
i
s
i i
i
Q
e I O I I
Q
δ+
δ
κ
+ λ
κ
∑
∑
(19)
or equivalently
* *
| | ( ) 1(0) = | | = | |,j I s Ts n
S Q er I Q e Iλ ≥
(20)
4 1
*
4= .
i i
i
i i
i
Q
Q
Q
δ+
δ
κ
κ
∑
∑
(21)
Therefore, in the regime of weak quasiparticle tunneling,
the large-Is asymptote of the ratio of the measured excess
noise and the tunneling current is equal to some average of
the quasiparticle charges *Q (the coefficient *Q e is often
called the Fano factor). This well-known result is correct
not just for the model I consider here, but is quite robust
against nonuniversal processes that may influence the
physics at the QPC [31].
The average (or effective) charge *Q may be not a con-
stant but a function of sI as the tunneling amplitudes iηg
contained in iκ may depend on sI strongly. However, in
the cases I consider in this paper this does not happen. If
all the quasiparticles participating in tunneling processes
have the same charge =iQ Q (as it is for the model of
= 2/5ν I consider), then * =Q Q independently of the
tunneling amplitudes' dependence on the current. In the
case of = 2/3ν not all the iQ are equal. However, when
the ratios /i jκ κ are constant (as it has been shown [15]
for the data I consider below), then again *Q does not de-
pend on .sI
A more accurate large-Is asymptotic expression for the
NtTRR, obtained in [28],
| | ( ) 1( ) =s j Is n
X I λ ≥
4 1
2 2 2
0 0
04
2 8= | | , ,
i i
i
s
s si i
i
Q
I I
e I eI O
I IQ
δ+
δ
κ
λ− δ
+ + πκ
∑
∑
(22)
may be useful in some cases. Its subleading term contains
information about the scaling dimension of the tunneling
quasiparticles.
To conclude the section, the main statements I would
like the reader to take from it are as follows. The pertur-
bative treatment of tunneling processes allows one to ob-
tain results for experimental observables in the limit of
weak quasiparticle tunneling. From large-Is asymptote of
the ratio of the excess noise and the tunneling rate one can
obtain some average of the tunneling quasiparticles' char-
ges *,Q which is often called the effective charge.
4.2. Exact solutions
The cases for which the model of tunneling experiments
in QHE can be solved exactly are scarce: only two cases
are known to me. One is the case of = 1ν integer QHE
(IQHE), the other is the Laughlin sequence of states
= 1/(2 1),kν + .k ∈ I briefly discuss these two cases
below.
In the case of = 1ν IQHE the simplest edge theory is a
theory of free chiral electrons. It can be rewritten into the
model of a single free chiral boson of the type described in
Sec. 3 through the standard bozonization technique [35].
* I must acknowledge here, that an unknown dependence of the tunneling amplitudes leads to a huge ambiguity: for example, one
can fit any dependence of r on Is. The question regarding possible dependences of the tunneling amplitudes deserves a study. For
example, in the case of ν = 2/3 considered in Ref. 15 the dependence turns out not to be generic. The tunneling amplitudes for the
quasiparticles there depend significantly on Is and In, but in the same way for different quasiparticles, i.e., the ratios κi/κj are con-
stant. An explanation for such a restriction is unknown to me. In contrast to this, the approach with scaling dimensions being
renormalized, uses a few unknown fitting parameters, but no unknown functions. In some cases the latter approach allows one to
describe the data for both tunneling rate and noise very well using a finite number of fitting parameters [11–13]. Whether it is pos-
sible to describe the case of ν = 2/3 considered in Ref. 15 within this approach is a matter of future investigation.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1 83
Kyrylo Snizhko
The edge contains a charged mode and no neutral modes,
so the injection of nI does not influence the observable
quantities. The quasiparticle with the smallest scaling di-
mension (i.e., the one giving dominant contribution to the
tunneling processes) in this case is the electron. Therefore,
the problem of tunneling is a problem of free electrons that
can be scattered back by a δ-function barrier. Such a model
can be solved exactly [26,27,36]. The results are as fol-
lows. The tunneling rate = constr , i.e., r does not depend
on .sI The excess noise has the form
0
2(0) = (1 ) coth ,
2
s
s
j
S r r eI eI
π − − π
(23)
sj and 0I are defined in Eq. (17). In the limit 0r → this
expression can be reproduced by treating tunneling
perturbatively as in Sec. 4.1, with the electron as the tun-
neling quasiparticle (the electric charge = 1,Q the scaling
dimension = 1/2).δ
The simplest models for the FQHE edge at
= 1/(2 1),kν + k ∈ also contain a single charged mode.
However, solving such models exactly requires the use of
the Bethe ansatz technique. The details for the solution
together with answers can be found in Refs. 23, 24, 37.
Analytic answers for this case are only available for zero-
temperature 0( = 0)T . Therefore, the use of this exact solu-
tion is not easy and requires a certain skill level in using
Bethe ansatz.
4.3. Phenomenological approach
A third, phenomenological, approach to treating the
experiment model is often used in experimental papers
[3–10,16]. Essentially it does not use any solution of the
model but generalizes the answer of Eq. (23) for = 1:ν
*
* 0
2(0) = (1 ) coth ,
2
s
s
jeS r r e I eI
e
π
− − π
(24)
where sj and 0I are defined in Eq. (17), and *e is a phe-
nomenological parameter.
The advantages of this approach are the simplicity of
Eq. (24) and the correct leading asymptotic behavior:
=0(0) = 0,Is
S (25)
and for 1r
* *| | 1(0) = | | = | | .j s Ts
S e r I e I
(26)
The last equation is in accord with Eq. (20) for ** = .e eQ
Formula (24) was compared against the exact solution
for the simplest model of = 1/3ν FQHE in Ref. 17, where
a good agreement was found.
However, there are several disadvantages to using for-
mula (24). As it has been pointed out in Ref. 12, the value
of *e extracted from real experimental data with the help
of formula (24) depends strongly on the range of Is consid-
ered. Second, the formula cannot be derived from a model
for the FQHE. Indeed, the result (23) is derived for the
model of noninteracting electrons. If one replaces the
charge of particles e by *,e then the edge conductance
would be 2 2*( ) / / .e h e h≠ ν E.g., for the Laughlin series of
states * = ,e eν so that 2 2 2*( ) / = / .e h e hν Third, the for-
mula does not catch correctly the subleading terms in the
large-Is asymptotic behavior of NtTRR (22) that carry in-
formation about the tunneling quasiparticles' scaling di-
mension. Finally, it does not have a natural way to include
the influence of ,nI since such an effect is impossible in
the simplest model of = 1ν IQHE.
From the above one can see that formula (24) is a good
interpolation formula, but not more. In particular, one should
be careful when trying to apply it to the experiments in
which injection of current nI plays an important role.
5. = 2/3 :ν Neutral mode heating and behavior
of the effective charge
The = 2/3ν edge has been predicted to support a single
charged mode and a single counterpropagating neutral
mode [18,19]. The experiment of the type described in
Sec. 2 that was reported in Ref. 10 was able to confirm
qualitatively the existence of a counterflowing neutral
mode. I and my co-authors analyzed the experiment data
quantitatively [15] and found a good quantitative agree-
ment between the data and the theory described in Sec. 4.1.
However, Ref. 10 reported a dependence of the effective
charge *e on the injected current nI (see Fig. 3b of
Ref. 10), while in our theory the effective charge does not
have such a dependence. In this section I explain the origin
of this apparent discrepancy.
The = 2/3ν edge can be described with a model that
contains a single charged mode and a single neutral mode
that propagates opposite to the charged one. Then, according
to what has been discussed in Sec. 3, the upper edge gets
heated upon injection of current ,nI so that the upper edge
temperature near the QPC is equal to 0( ) ,nI Tλ while the
lower edge temperature is always equal to 0.T There are
three quasiparticles that give contribution to the tunneling
processes, their charges are 1 2= = 1/3Q Q and 3 = 2/3,Q
their scaling dimension is = 1/3.δ Using the formulas of
Sec. 4.1 one can compare the theory with the experimental
data for the tunneling rate 0.2r ≈ presented in Fig. 3a of
Ref. 10. In Ref. 15, as a result of such comparison, it was
found that the experimental data can be described well with
( ) = 1 | | ,a
n nI C Iλ + where C = 5.05(13)nA–a and a =
= 0.54(5) and parameter 3 1 2= /( )κ κ κ + κ being independ-
ent of sI and nI and equal to = 0.39.κ The temperature
T0 = 10 mK is taken to be the same for all values of nI .
From the formulas of Sec. 4.1 one can see that the effective
charge *Q is then equal to * 0.5Q ≈ independently of .nI
84 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1
Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be
In Ref. 10 the experimental data is analyzed with the help
of Eq. (24) using *e and 0T as fitting parameters. The in-
terpretation connected to this is that for = 0nI the tempera-
ture 0T is equal to the environment temperature, but injec-
tion of 0nI ≠ heats up the whole system (or at least both
edges of the QPC) leading to a different value of 0T in
Eq. (24). The resulting agreement with the experimental data
is also good, however, the behavior of the effective charge
*e is different. One would expect ** = /(1 ) 0.63e Q e r e− ≈
independently of ,nI but Fig. 3b of Ref. 10 says that the
effective charge *e varies from * 2 /3e e≈ to * 0.4e e≈ as a
function of .nI
In order to explain this I fit the perturbative formula for
measured current noise*
2
pert 0(0) = ( ) 2 ,s B
eS rX I k T
h
ν
+ (27)
where ( )sX I is defined in Eq. (18), with the phenomeno-
logical formula
2*
*phen 0 0
2(0) = (1 ) coth 2 .
2
s
s B
je eS r r e I eI k T
e h
π ν
− − + π
(28)
In other words, I repeat the analysis done in Ref. 10, using
the theory of Ref. 15 instead of experimental data.
A typical resulting fit is shown in Fig. 2. The range of
sI values corresponds to the experimental data range in
Ref. 10. As one can see, the fit of perturbative formula (27)
by phenomenological formula (28) is very good. However,
the perturbative theory only starts leveling off to its large-
sI asymptote (22) at | | 1sI ≈ nA. Therefore, one can hard-
ly expect the fitted *e to correspond to the proper effective
charge. Indeed, the fitted value of *fitted 0.48e e≈ differs
from the expected ** = /(1 ) 0.63 .e Q e r e− ≈ The system
temperature fitted
0 19T ≈ mK given by the fit is also differ-
ent from both the lower edge temperature 0 = 10 mKT and
the upper edge temperature near the QPC 0( ) = 80 mK.nI Tλ
I repeat the fitting procedure for different values of
( )nIλ . The resulting dependence of the fitted effective
charge is shown in Fig. 3. One can see that the dependence
of the fitted effective charge *fittede closely follows the data
reported in Ref. 10. At the same time the Fano factor (21)
stays constant since 3 1 2/( )κ = κ κ + κ is independent of nI .
This suggests that the true origin of the reported effective
charge dependence on nI is the use of phenomenological
formula (28) for the analysis of experimental data, and is not
related to the nonuniversal behavior of the tunneling ampli-
tudes, nor to complications in the = 2/3ν edge theory (for
example, like the ones proposed in Refs. 12, 14**.
* A careful reader has noticed that the Nyquist noise 2νe2kBT0/h here is two times greater than the Nyquist noise subtracted in
Eq. (10). The Nyquist noise subtracted in Eq. (10) is the Nyquist noise of a single edge to the left of Voltage probe contact in
Fig. 1. However, in a real experiment there is a similar set of edge transport channels to the right of Voltage probe. While these are
not influenced by the injection of Is and In, they still contribute to the experimentally measured noise through the Nyquist noise.
That is the origin of the Nyquist noise “doubling” here.
** Though, the more elaborate models proposed in Refs. 12, 14 may be necessary to explain other observed effects. Consult the
works themselves for more details.
30
25
20
15
10
5
0
–1.0 –0.5 0 0.5 1.0
Is, nA
S(
0)
, 1
0
A
·s
–3
0
2
Nyquist noise
Fig. 2. (Color online) ν = 2/3 measured current noise: a fit of
the perturbative theory by the phenomenological formula. The
green points are generated with the help of perturbative theory
for T0 = 10 mK and λ(In) = 8 (In ≈ 1.8 nA). The red curve is a fit
of these points by phenomenological formula (28). The dashed cyan
curve is the large-Is asymptote of the perturbative theory.
0.7
0.6
0.5
0.4
0.3
0 1 2 3 4 5
e*
, e
In, nA
Fig. 3. (Color online) ν = 2/3 measured current noise: depend-
ence of the effective charge *e on current In. The blue points and
the blue solid line show the data of Fig. 3b of Ref. 10. The green
points and the green dashed line show the dependence of *fittede
on In obtained by fitting the perturbative theory with phenomeno-
logical formula (28). The error bars are due to uncertainty in the
function λ(In). The black horizontal lines correspond to
* = 2 /3e e and * = /3.e e
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1 85
Kyrylo Snizhko
6. = 2/5 :ν System temperature dependence
of the effective charge
One of the other cases when an unexpected dependence
of the effective charge on external parameters was reported
concerns = 2/5ν [6]. Namely, the effective charge was
reported to depend on the system temperature 0.T In this
section I apply the same methodology as in the previous
section to this case. I find that the effective charge depend-
ence on temperature cannot be explained the same way.
The simplest model for = 2/5ν edge contains two edge
channels: the charged one and the neutral one. However,
unlike in the case of = 2/3,ν the neutral mode here propa-
gates in the same direction as the charged mode. Therefore,
the injection of nI should not have any effect on the
measured current noise whatsoever. This was confirmed in
Ref. 10. Therefore, I expect that both edges have the same
temperature 0T . There are two quasiparticles which con-
tribute most to the tunneling processes, their electric
charges are 1 2= = 2/5,Q Q their scaling dimension is
= 1/5.δ Therefore, the parameters iκ drop out of the
perturbative expression for the NtTRR.
Figures 2b and 2c of Ref. 6 give some data regarding
the measured current noise behavior for 0.02.r ≈ The data
of Fig. 2c of Ref. 6 presents data on the behavior of the
effective charge *e as a function of temperature 0T ob-
tained with the help of phenomenological formula (28).
I repeat the analysis of the previous section for this
case, fitting perturbative formula (27) (using the parame-
ters corresponding to the experimental ones) with phenom-
enological formula (28). A typical resulting fit is shown in
Fig. 4. The same data with the Nyquist noise subtracted are
shown in Fig. 5. As one can see, the fit is very good. How-
ever, the fitted values of the effective charge and the sys-
tem temperature are slightly overestimated: *fitted 0.46e e≈
and fitted
0 71T ≈ mK.
Repeating the fitting procedure for different values of
0 ,T I obtain the dependence of the effective charge on
temperature shown in Fig. 6. At the lowest considered
temperature 0 = 10T mK the approach can explain the
oberved effective charge slightly higher than the expected
* = 2 /5.e e However, at higher temperatures the effective
charge reported in Ref. 6 drops down, while the fitted
charge *fittede grows.
Therefore, in this case the effective charge dependence
on an external parameter cannot be explained as a peculiar-
ity of the phenomenological formula used for the data
analysis. In other words, the data of Ref. 6 regarding
= 2/5ν does not agree with the perturbative theory for the
simplest = 2/5ν edge model.
Finding models that can describe the data goes beyond
the present article. However, I would like to mention sev-
eral possibilities.
The authors of Refs. 11, 12 showed that the data can be
explained with the help of a more complicated model that
(a) introduces energy cutoffs to the edge modes, (b) takes
into account tunneling of the quasiparticles having the next
smallest scaling dimension, (c) introduces renormalization
of the scaling dimension due to nonuniversal processes
(and assumes that tunneling amplitudes do not depend on
sI and 0 ).T
While the model of Refs. 11, 12 allows one to achieve a
good quantitative agreement, it incorporates several as-
pects not usually considered. Therefore, it would be inter-
esting to check whether it is possible to describe the data
without some of the complications. For example, one could
investigate the influence of the quasiparticles having the
next smallest scaling dimension without introducing scal-
ing dimension renormalization and/or edge mode cutoffs.
Fig. 4. (Color online) ν = 2/5 measured current noise: a fit of the
perturbative theory by the phenomenological formula. The green
points are generated with the help of perturbative theory for T0 =
= 70 mK. The red curve is a fit of these points by phenomenolo-
gical formula (28).
Fig. 5. (Color online) ν = 2/5 measured current excess noise: a fit
of the perturbative theory by the phenomenological formula. The
green points are generated with the help of perturbative theory for
T0 = 70 mK. The red curve is the fitted curve from Fig. 4 less the
Nyquist noise. The dashed cyan curve is the large-Is asymptote of
the perturbative theory.
86 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1
Tunneling current noise in the fractional quantum Hall effect: when the effective charge is not what it appears to be
There may be other important effects. One can consider
a model, in which the injection of current sI excites the
copropagating neutral mode in = 2/5ν just as the injection
of current nI excites the counterpropagating neutral mode
in = 2/3.ν
Finally, one can investigate the influence of bulk dy-
namics on the experimental observables. Indeed, the bias
voltage corresponding to injection of the experimentally
used current Is = 3 nA onto the = 2/5ν edge is of the
order of 200 µV, which corresponds to energies (in the
units of temperature) about 2 K. With the typical bulk
gap in the FQHE systems on the order of (and typically
less than) 1 K one can expect that bulk dynamics is in-
volved at such voltages.
7. Conclusion
In this paper I compared two approaches to analyzing
tunneling current noise experiments in the FQHE: the ap-
proach based on the perturbative treatment of tunneling
processes in the model describing such experiments in the
FQHE and the approach that uses a phenomenological
generalization of the theory which describes such experi-
ments in = 1ν IQHE.
The analysis of Sec. 5 shows that using the phenomeno-
logical formula can lead to misinterpretation of the exper-
imental data, like the false dependence of the effective
charge on current .nI However, this does not always hap-
pen, as shows the case of Sec. 6.
While one should be cautious when interpreting the pa-
rameters, such as the effective charge *e or system tem-
perature 0 ,T obtained with the help of the phenomenologi-
cal theory, the formula itself shows a remarkable ability to
fit the proper theory well, as can be seen from Secs. 5, 6
and as was previously shown in Ref. 17. Therefore, one
can use the phenomenological formula as an efficient way
to encode a large set of experimental points into two num-
bers *e and 0.T However, one can then miss subtle effects
related to the scaling dimension of the quasiparticles par-
ticipating in tunneling.
Finally, I would like to emphasize that the phenomeno-
logical approach is used in most papers that analyze experi-
mental data of the tunneling current noise experiments in the
FQHE, including Refs. 3–10. Exceptions like Refs. 11–15
are rare. Therefore, reanalyzing the available experimental
data with the help of proper theory can be highly beneficial.
One reason is that false effects, such as in the case consid-
ered in Sec. 5, are possible. Another reason is that missed
effects are also possible.
Acknowledgments
I would like to thank Alessandro Braggio, Jianhui Wang,
Vadim Cheianov, and Oles Shtanko for useful discussions.
The research leading to these results has received funding
from the European Research Council under the European
Union's Seventh Framework Programme (FP7/2007-2013) /
ERC grant agreement No 279738 - NEDFOQ.
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88 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 1
1. Introduction
2. Tunneling experiments in the FQHE: a typical scheme
3. Tunneling experiments in the FQHE: the model
4. Three approaches to theoretical description of tunneling experiments
4.1. Perturbative treatment of tunneling
4.2. Exact solutions
4.3. Phenomenological approach
5. Neutral mode heating and behavior of the effective charge
6. System temperature dependence of the effective charge
7. Conclusion
Acknowledgments
|