Pokrovsky, VL

Pryadko, LP

Physical Review Letters

English

Texas A&M University

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SU-ITP-93-14
Quasi-Fermi Distribution and Resonant Tunneling of
Quasiparticles with Fractional Charges
V.L. Pokrovsky
Physics Dept., Texas A&M University, College Stat., TX 77843∗
and
L.D. Landau Institute for Theoretical Physics,
V-334, Kosygina 2, Moscow, GSP-1, 117940, Russia
L.P. Pryadko
Physics Dept., Stanford University, Stanford, CA 94305†
(May 18, 1993)
Abstract
We study the resonant tunneling of quasiparticles through an impurity be-
tween the edges of a Fractional Quantum Hall sample. We show that the
one-particle momentum distribution of fractionally charged edge quasiparti-
cles has a quasi-Fermi character. The density of states near the quasi-Fermi
energy at zero temperature is singular due to the statistical interaction of
quasiparticles. Another effect of this interaction is a new selection rule for
the resonant tunneling of fractionally charged quasiparticles: the resonance
is suppressed unless an integer number of electrons occupies the impurity. It
allows a new explanation of the scaling behavior observed in the mesoscopic
fluctuations of the conductivity in the FQHE.
1
The question regarding the extent to which the fractionally charged quasiparticles pro-
posed by Laughlin [1] are real and whether they can be observed individually, was recently
resolved by experiments [2]. In these experiments FQHE samples with constriction were
studied, in order to observe essentially one-particle tunneling processes of the quasiparticles.
In particular, the frequencies of the mesoscopic fluctuations of the longitudinal resistance in
the FQHE with ν = 1/3 were compared to those in the Integer QHE.
Previously, Jain and Kivelson [3] suggested that the the resonant tunneling of electrons
from one edge to another through an impurity could cause an enhancement of the dissipative
resistance in the IQHE samples with a narrow constriction. Kivelson and Pokrovsky [4]
proposed an analogous mechanism for the fractionally charged quasiparticles in the FQHE.
Their model implied simple scaling laws for the periods of the mesoscopic oscillations in
the vicinity of the state with filling factor ν ≡ p/q : ∆B ∝ q at fixed gate voltage VG, and
∆VG ∝ p at fixed magnetic field B. Both predictions have been confirmed experimentally [2].
Nevertheless, the theoretical understanding of this scaling can not be considered as satis-
factory. In particular, Kivelson [5] derived quasiclassical quantization rules for a multi-anyon
bound state at the impurity allowing for the statistical interaction; his scaling relations are
different from the observed ones. P. Lee [6] supported Kivelson’s result from the position
of the theory of edge quasiparticles. He accounted for the obvious discrepancy with the
experiment with the Coulomb blockade.
Another problem in understanding these experiments is that the mechanism of resonant
tunneling usually implicates the existence of a Fermi level for excitations. It clearly exists
for the case of the IQHE but is much less obvious for the FQHE. Recently Haldane [7]
defined the generalized Pauli principle for anyons. This principle, however, does not imply
the existence of the distinct Fermi level required to explain the resonances in tunneling.
The purpose of this work is to elucidate these general questions and give a new explana-
tion for the experimental result.
Let us start with the quasiclassical quantization rule derived by Kivelson. It reads
2
Φ = mφ∗0 +Nφ0, (1)
where Φ is the total magnetic flux through an area A surrounded by the trajectory of the
quasiparticle, φ0 = hc/e is the flux quantum for an electron, φ
∗
0 = qφ0 is the flux quantum
for a quasiparticle (anyon) with charge e∗ = e/q, N is the number of quasiparticles captured
by the impurity, and m is the angular momentum of the tunneling quasiparticle. The first
term in the r.h.s. of equation (1) is required by the gauge invariance, while the second one
simply shows that each quasiparticle is bound with one flux quantum. The same quantum
spectrum arises in the exact solution for a system of N anyons in a quadratic potential [8].
At a given gate voltage VG, the area A enclosed by the trajectory, corresponding to the
Fermi level, is the same for any quantized value of Hall conductivity. This is true because
the Laughlin liquid is incompressible. Therefore, the intervals of the magnetic field between
consequent bound states of a quasiparticle are ∆Bq = φ
∗
0/A if the number of quasiparticles
N is fixed. The scaling, consistent with these intervals, was observed experimentally.
However, during the tunneling the number of quasiparticles N coupled with the impurity
changes by one. It corresponds to the change of the flux Φ by a single flux quantum φ0 instead
of φ∗0 = qφ0. Corresponding periods ∆B1 = φ0/A have not been observed experimentally.
The solution to this puzzle lies in the fractional statistics of quasiparticles. Consider
the situation where N quasiparticles are initially bound to the impurity, and the tunneling
quasiparticle arrives at an orbit enclosing all of them. In the quasiclassical approximation,
the wave function of this quasiparticle will gain a phase factor z = exp(i2piN/q) after
each complete revolution around the quantized orbit. More accurately, it is multiplied by
z(1 − γ/2), where γ is the total probability of tunneling from the impurity to either left or
right edge. The total tunneling amplitude contains a series
tLR =
∑
k
zk(1− γ/2)k =
1
1− z(1− γ/2)
. (2)
Usually resonant enhancement of the tunneling happens when all the amplitudes, corre-
sponding to different numbers of revolutions in (2) are coherent, i.e. z = 1. This is obviously
3
the case for the usual Fermi quasiparticles (ν = 1). For a fractional value of ν, the contribu-
tions of q consequent revolutions almost cancel each other. Thus the resonant tunneling is
suppressed unless N/q is an integer. In other words, the tunneling of an anyon is resonantly
enhanced only if an integer number of electrons are already bound to the impurity. This
simple selection rule restores the scaling suggested in ref. [4] and agrees with experiment.
The scaling of the oscillation intervals on the gate voltage ∆VG [4] is also easily repro-
duced. Indeed, at a fixed magnetic field B the change ∆VG corresponding to a new resonance
is determined by the change of the area
∆A = ∆Φ/B, (3)
where ∆Φ is the change of the flux through the trajectory. As we have already established,
∆Φ = qφ0 for the resonant tunneling at ν = p/q. On the other hand, the value of the
magnetic field Bν , corresponding to the filling factor ν, is approximately 1/ν times B1. As
a result we obtain ∆Aν = p∆A1 and ∆V
ν
G = p∆V
1
G
These intuitive and semi-classical arguments are supported by direct calculations in the
framework of Wen’s theory of edge excitations [9]. Simultaneously, we find the distribution
of edge quasiparticles over momenta to confirm the conjecture of its Fermi-like character [4].
All calculations have been performed for special values of ν = 1/q, where q is an odd integer.
In Wen’s theory the operator creating a quasiparticle
ψ†(x, t) = :eiφ(x,t) : (4)
at the point x of the edge is associated with the chiral Bose field φ(x, t) of an edge magneto-
plasmon. This field obeys the commutation relationship
[φx,t, φx′,t] = −ipiν sign(x− x
′) (5)
and is related to the charge density ρ = e/2pi ∂φ/∂x at the edge. The permutation relations
of the anyon operators (4) are
ψ†(x, t)ψ†(x′, t) = eipiν sign(x−x
′)ψ†(x′, t)ψ†(x, t). (6)
4
To find the distribution of edge quasiparticles over momenta, it is necessary to calcu-
late the Fourier-transformation G˜p of the simultaneous correlation function G(x − x
′) =
〈ψ†(x, t)ψ(x′, t)〉. We have performed this calculation explicitly with the following result:
G˜p can be represented as the product
G˜(p′) = g
T
(p′)
1
exp(βp′v) + 1
, (7)
where the momentum p′ = p − pF , pF is a Fermi momentum, and v is the drift velocity
along the edge for both the chiral field and the anyons. The second factor is the usual
Fermi-distribution, while the first one can be treated as the temperature-dependent density
of states; it is an even function of p′. At T = 0 the density of states g
T
(p′) has a singularity ∝
|p′|ν−1 and diverges at the Fermi-level. The singularity is smeared out at a finite temperature.
Details of the calculations will be published elsewhere.
To investigate the tunneling, we modelled the impurity as a void in the incompressible
quantum Hall liquid, its edge being an additional environment for the edge quasiparticles.
The perimeter Li of this edge was assumed to be small enough to neglect the probability
of thermal excitation of states with non-zero angular momentum m. The outer edge, on the
contrary, was assumed to be in the thermodynamical limit; it serves as a thermostat. The
many-body quantum mechanical states at the impurity are well-defined in the limit of a
small tunneling coupling
Ht =
∫
dxdyt(x, y)ψ†(x)ψi(y) + h.c., (8)
where ψi(y) is the annihilation operator for the edge quasiparticles at the impurity. This
limit allows us to reduce the evolution equation of the density matrix at the impurity to a
set of kinetic equations, describing the evolution of probabilitiesWN = 〈PˆN〉 to have exactly
N quasiparticles at the impurity, where PˆN is the appropriate projection operator.
The transition amplitudes are connected with different equilibrium averages similar to
〈ψ†i (y)ψi(y
′)PˆN 〉. In the generic case they are periodic only at the q-fold boundary [10],
gaining the phase 2piN/q in each cycle. This phase, similar to the Berry phase, is the
5
exact consequence of the fractional statistics and does not depend on the distribution of
edge phonons. As usual, the broken symmetry leads to the selection rule for the allowed
transitions; namely, transitions N −→ N + 1 are suppressed unless N/q is integer. This
statement coincides with our conclusion extracted from the semi-classical model. In contrast
to the previous derivation, we made no assumptions about the geometrical properties of the
orbit of the tunneling quasiparticle.
The factorization (7) yields the Boltzmann distribution for the probabilities of differ-
ent many-particle states of the impurity in equilibrium. In the presence of the inter-edge
potential difference, however, the impurity population depends on the tunneling probabil-
ities. The current-voltage dependence (see Figure 1) is highly non-linear and asymmetric,
especially in the vicinity of the resonance.
In conclusion, we have found new selection rules for the resonant tunneling of quasi-
particles in the FQHE, arising from the broken symmetry specific to anyons. The equilib-
rium momentum distribution of the edge quasiparticles has quasi-Fermi properties with the
temperature-dependent density of states. This explains the appearance of resonant tunnel-
ing effects in the anyonic systems and, in particular, the scaling properties of the mesoscopic
pattern measured in the experiment [2].
Upon completion of this work, we received the preprint [11], where the RG equations for
resonant inter-edge tunneling are solved numerically in a different geometry. The authors
considered neither scaling properties of the resonant effects, nor the momentum distribution
of quasiparticles. Their main emphasis was the line shape.
V. L. P. is indebted to Steve Kivelson for numerous discussions and to J. S. Langer and
Institute of Theoretical Physics in Santa-Barbara for the hospitality extended to him at the
initial stage of this work. L. P. P. wishes to thank the Soros Foundation for partial financial
support under grant # S92.56, and Jared Levy for valuable comments on the manuscript.
6
REFERENCES
∗ Present address.
† E-mail address leonid@leland.stanford.edu.
[1] R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
[2] J.A. Simmons, H.P. Wei, L.W. Engel, D.C. Tsui and M. Shayegan, Phys. Rev. Lett.
63, 1731 (1989); J.A. Simmons, S.W. Hwang, D.C. Tsui et al, Phys. Rev. B44, 12933
(1991); S.W. Hwang, J.A. Simmons, D.C. Tsui and M. Shayegan, Surf. Sci. 263, 72
(1991).
[3] J.K. Jain and S.A. Kivelson, Phys. Rev. B37, 4111 (1988).
[4] S.A. Kivelson and V.L. Pokrovsky, Phys. Rev. B40, 1373 (1989).
[5] S.A. Kivelson, Phys. Rev. Lett. 65, 3369 (1990).
[6] P. Lee, Phys. Rev. Lett. 65, 2206 (1990).
[7] F.D.M. Haldane, Phys. Rev. Lett. 67, 937 (1991).
[8] G.V. Dunne, A. Lerda, and C.A. Trugenberger, Mod. Phys. Lett, A6, 2819 (1991); A.P.
Polychronakos, Phys. Lett. B264, 362 (1991); S. Chou, Phys. Rev. D44, 2533 (1991).
[9] X.-G. Wen, Int. J. Mod. Phys. B6, 1711 (1992).
[10] Our definition of the normal ordering is different from the conventional one in the
treatment of the zero momentum mode of the chiral field φ.
[11] K. Moon, H. Yi, C.L. Kane, S.M. Girvin and M.P.A. Fisher, preprint, April 1993.
7
FIGURES
FIG. 1. Non-linear resonant tunneling current I versus the inter-edge voltage V expressed in
the units of the temperature T at different values of the one-particle energy E of the bound state
at the impurity.
8


Quasi Fermi distribution and resonant tunneling of quasiparticles with fractional charges

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Quasi Fermi distribution and resonant tunneling of quasiparticles with fractional charges

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