Quantum Hall Effect under Rotation and Mass of the Laughlin Quasiparticles

4 Pages·0.13 MB·English

by  Bo Zhao

Quantum Hall Effect under Rotation and Mass of the Laughlin Quasiparticles
Bo Zhao and Zeng-Bing Chen∗
Department of Modern Physics, University of Science and Techlogy of China, Hefei, Anhui 230027, China
3
WeconsiderthequantumHalleffectinducedbymagneticfieldandrotation, whichcandrivethe
0
Hall samples into the quantum Hall regime and induce fractional excitations. Both the mass and
0
thechargeoftheLaughlinquasiparticlesarepredictedtobefractionally quantized. Theobservable
2
effectsinducedbyrotationarediscussed. BasedontheusualHallsamplesunderrotation,wepropose

n an experimental setup for detecting the macroscopic quantization phenomena and the fractional
a mass of theLaughlin quasiparticles.
J

3 PACSnumbers: 73.43.Cd
2

] In condensed matter, two-dimensional (2D) electron in QH samples using a twofold U(1) gauge invariance.
l
l gases are of particular interest due to the discovery of Thechargeofthecurrentcarriers(theLaughlinquasi-
a
h fractional quantum Hall (QH) effect by Tsui, Stormer, particles) in fractionalQH effect is fractionized as νe for
- andGossard[1]. Itisoneofthe spectacularmacroscopic simple Hall droplet, as is now well known. Now a fas-
s
e quantization phenomens and has been investigated for cinating question arises: Is the mass of the fractionally
m two decades both theoretically and experimentally [2]. chargedquasiparticlesinafractionalQHliquidalsofrac-
One of the exotic aspects is the Laughlin quasiparti- tionized? Tothebestofourknowledge,thisquestionhas
.
at cles of fractional charge e (l odd for fundamental frac- notbeeninvestigateduptodate. Itsanswerisnotasob-
l
m tional filling fraction) [12] which has been demonstrated vious as in a superconductor where the current carriers
- recently by several beautiful experiments [17]. Similar are the Cooper pairs with mass mCP being duplation of
d ideas have been explored in another strongly correlated electron’smassm ,namely,m =2m . Inusualquan-
e CP e
n
system–dilute cold neutral atoms in Bose-Einstein con- tum Hall effect an excitation can be induced by piercing
o
densates (BECs) in recent years [3, 4, 5, 6]. The rapidly a magnetic flux Φ = 2π~. However,there is no physical
c e e
[ rotating BECs, which can be treated as 2D interact- principle to determine the mass of the excitations.

ing atoms, will lie in the lowest landau level (LLL) and Motivated by our previous work [6], in this paper we
1
v the nondegenerate ground state is of the form of the fa- dealwithsuchkindofproblemsfromamicroscopicview.
9 mous Laughlin state [3, 4]. In our previous paper [6], Namely,weareconcernedwithQHeffectunderrotation.
3 we predicted some macroscopicquantizationphenomena Ofparticularinterestisthe properties(e.g.,the mass)of
4 which are the “atomic twin of the electronic brother” thefractionalexcitations. ByusingtheusualQHsamples
1
on the base of the similarity between these two strongly under rotation, we propose an experiment to measure
0
correlated many-body systems. The quantized “mass- the fractionalmass of the Laughlinquasiparticles,which
3
0 conductance” σ(m) and the quasiparticles of fractional could be tested under current experimental conditions.
/ mass replace, respectively, the electron-conductanceσ(e) First, we consider a 2D rotating QH sample (see Fig.
t
a and fractionally charged quasiparticles in the usual QH 1) under a high magnetic field −Bzˆ, which is perpendic-
m
effect. But the requirements for observing similar QH ulartothe2Dplane. Assumethatthesampleisconfined
- physics in BECs are demanding due to the difficulty in by a harmonic potential with trapping frequency ω and
d 0
the current BEC experiments to deposit a large enough then rotated, along the zˆ-direction, at an angular fre-
n
o angular momentum to reach the QH regime. quency ω = ω0. In the rotating frame the 2D electrons
c are described by [11]
v: Essentially, the phenomena predicted in Ref. [6] in
i rapidly-rotating stongly-correlated cold atoms are in- Htot = Hi+ V(ri−rj). (1)
X X X
duced by the noninertial force on massive particles in i i<j
r
the rotating frame. Thus, these effects should also occur
a Here H = (pi−mωz×ri+eAi)2 is the single-body Hamilto-
in other strongly correlatedmany-body systems, even in i 2m
nian;V(r −r )desbcribesthetwo-bodyinteraction,which
the electron gases. The detection of effects induced by i j
rotationinmagnetizableandmetalshasahistoryofmore in this case is the Columb interaction e2 between
|ri−rj|
than one hundred year [7]. The famous experiment on electrons as is familiar in QH effect. The Hamiltonian
thisrespectwasdonebyBarnettwhomeasuredthemag- (1) can be naturally realized in a rotating quantum-dot
netic fieldinducedby rotationin1915[8]. Recentexper- system under the same conditions specified above.
iments on the rotating effect were to measure the mass Now let us first consider the single-body Hamiltonian.
ofthecurrentcarriers(Cooperpairs)insuperconductors As is done in our previous paper [6], one can deal with
[9]. Recently,Fischeretal. [10]havestudiedtherotation the rotation and the magnetic field in a uniform way,
2
namely
q∗A∗ =mωz×r+eA, (2)
where A=−1Bz×r (with Bb = ∇×A and B∗ = ∇×
2
A∗) and q∗ is the effective charge. Then single-body
b
Hamiltonian is simplified to be
(p−q∗A∗)2 [p−(mω+ eB)z×r]2
H = = 2 , (3)
2m 2m
b
which is completely analogous to the Hamiltonian with
only effective magnetic field B∗. So when the mag- FIG. 1: Schematic view of the rotating sample under high
netic field B or/and the rotating frequency ω is large magnetic field B.
enough, the system can be studied only in the lowest
landau level (LLL). In LLL the single-body states read
ψ (η) = N ηlexp[−|η|2/(4a )2] (l = 0,1,2,...) in terms
l l 0 will induce not only the Hall electrostatic field but also
of the 2D complex coordinate η = x+iy ≡ a η. Here
0 thegravitational-likeaccelerationfield. Thetwoinduced
N =[πl!(2a )l+1]−1/2 is the normalizationconstant and
l 0 fields are caused by the two-body interaction. Then the
a = ~/(2mω+eB) is the effective magnetic length.
0 equivalentparticlecurrentinducedbythechangeofmag-
Wepcan also define the filling fraction as usual netic field B (piercing a magnetic flux Φ = 2π~) or ro-
e e
2πn~ 2πn~ tating velocity ω (piercing a vortex Φ = 2π~) can be
ν = = . (4) m m
q∗B∗ 2mω+eB described in a uniform way
Inthiscasethemany-bodysystemwillshowsomemacro- dN ν dΦ
j = = , (8)
scopic quantum phenomena due to the repulsive two- dt 2π~ dt
body interaction as in the electron or atom QH effect.
where Φ = m Ω × dr−e A·dr=N 2π~ is the to-
Its nondegenerate ground state is the Laughlin state v
tal fluxes and jH is also theHequivalent electron-number
Ψl =NlY(ηi−ηj)le−14Pi|ηi|2, (5) cTuhrirseenqt.uatSioonfimlleinangsftrhaacttieointhneratpuierarclliyngraeamdsagνnet=icflNNuνx.
i<j Φ = 2π~ or piercing a vortex Φ = 2π~ in a disk or
e e m m
whichwasfirstputforwardbyLaughlininaclassicpaper cylinder sample will induce a particle current as long as
in 1983 [12] to explain the fundamental filling fraction the sample is in Hall regime.
in the fractional QH effect. Here N is an unimportant Theexoticfractionalexcitations,whicharethecurrent
l
normalization constant, and l > 0 is an even number carriers, are of particular interest in electron QH effect
for bosons or an odd number for fermions. Then in this [12] and in atom QH effect [6]. What is the relation
statethefillingfractionwillbequantizedasusualν = 1. between these two kinds of excitations? The answer to
l
The “atomic Hall-conductance” in atomic QH effect [6] this question can be approached in the present case. As
andtheusualquantizedHall-conductanceinelectronQH usual, the fractional charge should read Q∗ = q∗. In
l
effect [2] is naturally unified as orderto see clearlyits meaning we addthe externalfield
E∗, which is induced by piercing flux Φ [6] and reads
e2 m2
+ =l. (6)
σ(e)2π~ σ(m)2π~ Q∗E∗ = eE+mg = eE+ mg. (9)
l l l
This means that neither of the two Hall conductances is
quantized, but their combination is quantized. This relation means that the quasiparticles will respond
The equivalent electron-number current (the real cur- to the electric field as e and acceleration field as m.
l l
rent carriers are not the electron but the quasiparticles, This fact implies that the quasiparticles are of both the
as will be discussed below) is fractional charge and the fractional mass. Thus, pierc-
ing a magnetic flux Φ = 2π~ and/or piercing a vortex
ν e e
j = (eE +mg ), (7) Φ = 2π~ in the rotating Hall sample will induce exci-
x 2π~ y y m m
tations from the unique vacuum state (i.e., the Laughlin
where E and g are induced Hall electric field and state) and lead to the same Laughlin quasiparticle state
y y
gravitational-like acceleration field, respectively. Equa- [6, 12]. Consequently, the excitations possess simultane-
tion (7) is identical to the twofold U(1) gauge invari- ously fractional charge and fractional mass. This result
ant current obtained in Ref. [10]. It means that when is necessarily the property of the nondegenerate ground
one inputs a particle current in the sample, the current state and does not depend on the origin of the ground
3
is
H = H′+ V(r −r ), (11)
tot i i j
X X
i i<j
where H′ = (pi+eAi)2. Laughlin’s gedanken experiment
i 2m
is to pierce a magnetic flux Φ = 2π~, which will induce
e e
a fractional charge transfer in the Hall sample, as has
already been demonstrated by experiments [14, 15, 16].
Based on the similar sample one can also measure the
mass of the fractionallychargedquasiparticles whichare
the current carriers in QH sample.
LetusconsiderthecaseofpiercingavortexΦ = 2π~
m m
FIG. 2: Schematic view of the Corbino sample to verify the in the sample by rotating the fractional Hall Corbino
fractional mass of the Laughlin quasiparticles. Rotating the
disk. Due to Faraday’s law [6, 10], this process will in-
sampletoinduceg willexcitethequasiparticlesandinducea
duceanaccelerationfieldandexcitetheelectrongas. As
voltage in the capacitance. The voltage plateau will demon-
described above it will induce a radialHall mass current
strate the fractional mass.
[6]
m2
mj =ν g . (12)
r 2π~ φ
state. So what we got above should be correct even in
usual nonrotating Hall sample under high magnetic field Notethatthecurrentcarriersarequasiparticles. Assum-
B, namely, the fractional excitations with charge e in ing e∗ and m∗ are the charge and mass of the quasi-
l
electron QH effect possess the fractional mass m. paticles, respectively, one can get the electric current
It seems that the electron-conductance willl not be ej =ν e∗ m2g with g = Φ. . The charge transferred
r m∗2π~ φ φ 2πR
quantizedasusualduetotherelationI = ν (eV+mΦ) through this process is
x 2π~
(Φ is the induced gravational-like potential [6], I is the
x
e∗ m2
equivalentelectroncurrent)ifweaddsourceanddrainon Q=2πR ej dt=2πR dtν g
therotatingQHsample. Butthisisnotthecase. Infact, Z r Z m∗2π~ φ
thequantumHallvoltageorthegravitational-likepoten- e∗ m2 . e∗ m2 2π~
=ν dtΦ=ν
tial are all caused by the two-body interaction, which is m∗2π~Z m∗2π~ m
nowtheColumbinteraction. Thegravational-likeforceis m
=νe∗ . (13)
alsoinducedby the two-bodyinteractionandbehavesas m∗
another electrostatic field eV′ ≡ mΦ. What is actually
According to the above results that e∗ = e and m∗ = m
measured is the total Hall voltage V = V +V′. Then l l
tot we will get Q = νe. This transferred charge is equal
the quantization relation in usual QH effect is formally to piercing a flux Φ = 2π~. While the fractional charge
preserved e
ofthe Laughlinquasiparticleshas been demonstratedby
severalexperiments through measuring the shot noise in
σ(em) ≡ eIx =ν e2 . (10) tunnelingeffect[17],thefractionalmassisunnoticedand
Vtot 2π~ thus, not determined up to date. So this process opens
a door to determine the fractional mass of the Laughlin
Now let us consider how to measure the effects in- quasiparticles.
duced by rotation, in particular, the fractional mass of For this purpose we propose a way to measure the
the quasiparticles. This issue is currently difficult for proportion me∗∗ between the charge and the mass of the
BECsforwhichthecurrentexperimentsarefarfromthe Laughlin quasiparticles to demonstrate the fractional
Hall regime. We turn to other samples, such as the elec- mass. The experimental device is depicted in Fig. 2.
tron Hall sample, to find a way to measure these effects This device has been used in QH effect to test Laugh-
for there have been some experiments to measure the lin’sgedankenexperiment[15,16]. TheCorbinodiskhas
rotating effects and the mass of Cooper pairs in super- reached the fractional quantum Hall regime under high
conductors[9]. Inthefollowingwewillconsideraspecial magnetic field B perpendicular to the 2D plane. Then
case and propose a method to measure the physical ef- rotating the sample to excite the quasiparticles will in-
fects induced by rotation and the fractional mass of the duce charge to transfer from one edge to the other. The
Laughlin quasiparticles. charge transferred in this process is
First we consider a usual Hall Corbino disk sample [2] e∗ m2
withoutrotation. TheHamiltoniandescribingthesystem Q= m∗πR2ν2π~2△ω, (14)
4
whereRistheradiusoftheCorbinodisk,△ωistherota- (Springer-Verlag, New York,1990).
tion angular velocity. This transferredchargecan be de- [3] N. K. Wilkin, J. M. F. Gunn, and R. A. Smith, Phys.
tected by measuring the voltage in capacitance, V = Q. Rev. Lett. 80, 2265 (1998); N. R. Cooper and N. K.
C Wilkin, Phys. Rev. B 60, R16279 (1999); N. K. Wilkin
Now the rotation only plays the role of exciting the QH
andJ.M.F.Gunn,Phys.Rev.Lett. 84,6(2000); N.R.
sample. Since the accuracy of the typical measurements
Cooper , N. K. Wilkin, and J. M. F. Gunn, Phys. Rev.
inQHeffectishigh,therotationfrequencydoesnotneed Lett. 87, 120405 (2001).
to be too large, say, about 103 ∼ 104s−1, is enough for [4] B. Paredes, P.Fedichev,J. I. Cirac, and P.Zoller, Phys.
current experimental conditions [8, 9, 10] to observe the Rev. Lett. 87, 010402 (2001); B. Paredes, P. Zoller, and
predicted effects. The methods to observe similar phe- J. I.Cirac, Phys.Rev.A 66, 033609 (2002).
nomenon in QH effect by changing magnetic field in this [5] T.-L.Ho,Phys.Rev.Lett.87,060403(2001);E.J.Muller
and T.-L. Ho, Phys.Rev.Lett. 88, 180403 (2002); T.-L.
samplearefamiliarintheusualelectronQHeffect,where
HoandE.J.Muller,Phys.Rev.Lett.89,050401(2002).
the quantized conductance is measured. This result is
[6] Z.-B. Chen, B. Zhao, and Y.-D. Zhang,
alsoavailableto integerQHeffectwhere ν =integerand
cond-mat/0211187.
the current carriers are electrons. [7] J. C. Maxwell, A Treatise on Electricity and Magnetism
In summary, we have exploited the quantum Hall ef- ( Dover,New York,1954 ).
fect induced by magnetic field and/or rotation. In this [8] S. J. Barnett, Phys.Rev. 6, 239 (1916).
case both the magnetic field and the rotation are avail- [9] A.F.Hildebrandt,Phys.Rev.Lett.12,190(1964); J.E.
able to reach the quantum Hall regime and can induce Zimmerman and J. E. Mercereau, Phys. Rev. Lett. 14,
887 (1965); J. Tate, S. B. Felch, and B. Cabrera, Phys.
the fractional excitations. Both the mass and charge of
Rev. B 42, 7885 (1990).
the quasiparticles, which are the current carriers in the
[10] U. R. Fischer and Schopohl, Europhys. Lett. 54, 502
Hall sample, are fractionally quantized. The observable
(2001); U. R. Fischer, C. H¨aussler, J. Oppenl¨ander, and
effects induced by rotation have been discussed. An ex- N. Schopohl, Phys.Rev B 64, 214509 (2001).
perimental proof of the macroscopic quantization phe- [11] This Hamiltonian also describles a rotating Hall sample
nomena and the fractional mass of the Laughlin quasi- except for a centrifugal term which can betreated as an
particles has been proposed. We hope that our predic- external U(1) field, as is done by Fischer et al. in [10].
[12] R. B. Laughlin, Phys.Rev.Lett. 50, 1395 (1983).
tion canbe verifiedin the near future by using the usual
[13] R. B. Laughlin, Phys.Rev.B 23, 5632 (1981).
quantum Hall samples under rotation.
[14] V. T. Dolgopolov, A. A. Shashkin, N. B. Zhitenev, S.
We thank Professor Yong-De Zhang for helpful dis- I. Dorozhkin, and K. von Klitzing, Phys. Rev. B 46,
cussion and U. P. Fischer for bringing Ref. [10] into our 12560 (1992); V. T. Dolgopolov, A A. Shashkin, G. V.
attention. ThisworkwassupportedbytheNationalNat- Kravchenko,S.I.Dorozhkin,andK.vonKlitzing,Phys.
ural Science Foundation of China, the Chinese Academy Rev. B 48, 8480 (1993).
of Sciences and the Natural Fundamental Research Pro- [15] V. T. Dolgopolov and A. A. Shashkin, Phys. Rev. Lett.
86, 5566 (2001).
gram (under Grant No. 2001CB309300).
[16] B. Jeanneret, B. D. Hall, H.-J. Bu¨hlmann, R. Houdr´e,
and M. Ilegems, Phys. Rev.B 51, 9572 (1995).
[17] R. de-Picciotto, M. Reznikov, M. Heiblum, V. Uman-
sky, G. Bunin & D. Mahalu, Nature (London) 389, 162
(1997);L.Saminadayar,D.C.Glarrli,Y.JinandB.Eri-
∗
Electronic address: [email protected] enne,Phys.Rev.Lett.79, 2526 (1997); M.Reznlkov,R.
[1] D.C.Tsui,H.L.Stormer,andA.C.Gossard,Phys.Rev. de-Picciotto, T. G. Griffiths, M. Heiblum and V.Uman-
Lett.48, 1559 (1982). sky,Nature (London) 399, 238 (1999).
[2] R.E.PrangeandS.M.Girvin,The QuantumHallEffect