Anomalous Hall Resistance in Bilayer Quantum Hall Systems

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by  Z. F. Ezawa

Anomalous Hall Resistance in Bilayer Quantum Hall Systems
Z.F. Ezawa1, S. Suzuki1 and G. Tsitsishvili2
1Department of Physics, Tohoku University, Sendai, 980-8578 Japan
2Department of Theoretical Physics, A. Razmadze Mathematical Institute, Tbilisi, 380093 Georgia
We present a microscopic theory of the Hall current in the bilayer quantum Hall system on
7
the basis of noncommutative geometry. By analyzing the Heisenberg equation of motion and the
0
continuityequationofcharge,wedemonstratetheemergenceofthephasecurrentinasystemwhere
0
theinterlayerphasecoherencedevelopsspontaneously. Thephasecurrentarrangesitselftominimize
2
the total energy of the system, as induces certain anomalous behaviors in the Hall current in the

n counterflow geometry and also in the drag experiment. They explain the recent experimental data
a for anomalous Hall resistances due to Kellogg et al. [M. Kellogg, I.B. Spielman, J.P. Eisenstein,
J
L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 88 (2002) 126804; M. Kellogg, J.P. Eisenstein,

4 L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 93 (2004) 036801] and Tutuc et al. [E. Tutuc, M.
Shayegan and D.A. Huse,Phys. Rev. Lett. 93 (2004) 036802] at ν =1.
]
l
l
a
h I. INTRODUCTION There arises new phenomena associated with the in-
- terlayer phase coherence in the bilayer QH system1,2.
s Thebilayersystemhasthepseudospindegreeoffreedom,
e The emergence of the Hall plateau together with the
m where the electron in the front (back) layer is assigned
vanishing longitudinal resistance has been considered to
to carry the up (down) pseudospin. Provided the layer
. be the unique signal of the quantum Hall (QH) effect1,2.
t separation d is reasonably small, the interlayer phase
a This is certainly the case in the monolayer QH sys-
coherence13,14 emergesdue to the Coulombexchangein-
m tem. However, recent experiments3,4 have revealed an
teraction. Thesystemiscalledthe pseudospinQHferro-
- anomalous behavior of the Hall resistance in a coun-
d magnet. This is clearly seen by examining the coherence
terflow geometry in the bilayer QH system that both
n the longitudinal and Hall resistances vanish at the to- length ξϑ of the interlayer phase field ϑ(x), which is cal-
o culated as
tal bilayer filling factor ν = 1. Another anomalous
c
[ Hall resistance has been reportedin a dragexperiment5.
Though a suggestion6 has been made that the anoma-
1
lous phenomenon would occur owing to excitonic excita- πJd
v ξ =2ℓ s , (1.1)
3 tions (electron-hole pairsin opposite layers)in the coun- ϑ Bs∆SAS
6 terflow transport, there exists no theory demonstrating
0 these phenomena explicitly in a unified way.
1
The aim of this paper is to present a microscopic the-
0 whereJd isthepseudospinstiffnessand∆ isthetun-
ory of Hall currents to understand the mechanism of s SAS
7
0 the anomalousHallresistance3,4,5 discoveredexperimen- neling gap. It is observed that the interlayer phase co-
herence develops well for Jd ∆ .
/ tally. Intheordinarytheorythecurrentisdefinedasthe s ≫ SAS
t
a No¨ther current, which arises from the kinetic Hamilto- It has been argued in an effective theory1 that the
m nian. However, this is quite a nontrivial problem in the
phase current, ∂ ϑ(x), flows in the pseudospin QH
- QH system7 since the kinetic Hamiltonian is quenched ferromagnet. In∝thisipaper,wepresentamicroscopicfor-
d within each Landau level8,9. There is a Lagrangian
n approach10tothisproblem,butitseemsquitedifficultto mulation of the phase current, and show that the phase
o currentarrangesitselftominimizethetotalenergyofthe
gobeyondthe one-bodyformalismwithinthis approach.
c system and makes the Hall resistance vanish in a coun-
v: We propose a new formalism to elucidate the current in terflow geometry3,4. Furthermore, it explains also the
the QH system.
i anomalous Hall resistance in the drag experiment5.
X We start with a microscopic Hamiltonian11 describ-
r ing electrons in the lowest Landau level. The intriguing This paper is composed as follows. Section II is de-
a
feature is that the dynamics is determined not by the voted to a concise review of the microscopic formalism
kinetic Hamiltonian but by noncommutative geometry. of the QH system based on the noncommutative geome-
The noncommutative geometry12 means that the guid- try. InSectionIII weanalyzethe Heisenbergequationof
ingcenterX =(X,Y)issubjecttothenoncommutative motion in the QH system. In Section IV the formula is
relation, [X,Y] = iℓ2, with ℓ the magnetic length. derived for the current in the spin ferromagnet from the
− B B
We derive the formula for the electric current from the continuityequation. InSectionVwestudythecurrentin
Heisenberg equation of motion and the continuity equa- the pseudospinferromagnet. InSectionVI wedetermine
tion of charge based on the noncommutative relation. It the phase current by minimizing the total energy of the
agrees with the standard formula for the Hall current in system. InSectionVIIweinvestigatehowtheanomalous
the monolayer system. Hall resistance occurs in the pseudospin ferromagnet.
2
II. NONCOMMUTATIVE GEOMETRY whereλA arethegeneratingmatricesofSU(N).Wesum-
marize the electron density and the isospin density into
A planar electron performs cyclotron motion in mag- the density matrix as
netic field, B = (0,0, B ). The electron coordi-
− ⊥ 1
nate x = (x,y) is decomposed into the guiding center D = δ ρ+(λ ) I . (2.8)
µν µν A µν A
N
X = (X,Y) and the relative coordinate R = (R ,R ),
x y
x = X +R, where Rx = Py/eB⊥ and Ry = Px/eB⊥ It is given by
−
with P = (P ,P ) the covariant momentum. The com-
x y
mutation relations are [X,Y] = −iℓ2B, [Px,Py]= i~2/ℓ2B Dµν(p)=e−ℓ2Bp2/4Dˆµν(p) (2.9)
and [X,P ] = [X,P ] = [Y,P ] = [Y,P ] = 0, with
x y x y
ℓ2 = ~/eB . They imply that the guiding center and in the momentum space, together with
B ⊥
the relative coordinate are independent variables.
1
The kinetic Hamiltonian, Dˆ (p)= me−ipX n c†(m)c (n). (2.10)
µν 2π h | | i ν µ
mn
P2 1 1 X
HK = 2M = 2M(Px−iPy)(Px+iPy)+ 2~ωc, (2.1) We callDˆµν(p)the baredensity. Thedifference between
D and Dˆ is negligible for sufficiently smooth field
creates Landau levels with gap energy ~ω = ~eB /M. µν µν
c ⊥ configurations.
When it is large enough, excitations across Landau lev-
In the succeeding analysis of the dynamics of the
els aresuppressedatsufficiently low temperature. It is a
SU(N) QH system, the W (N) algebra satisfied by the
∞
goodapproximationtoprohibitallsuchexcitationsbyre-
bare density,
quiringelectronconfinementtotheLowestLandaulevel.
Weexplorethephysicsofelectronsconfinedtothelow- 2π[Dˆµν(p),Dˆστ(q)]=δµτe+2iℓ2Bp∧qDˆσν(p+q)
est Landau level, where the electron position is specified
solely by the guiding center X = (X,Y), whose X and δσνe−2iℓ2Bp∧qDˆµτ(p+q),
−
Y components are noncommutative, (2.11)
[X,Y]= iℓ2. (2.2) playsthebasicrole. Wehavealreadyderivedthisrelation
− B based on the noncommutative relation (2.2) and the an-
We introduce the operators ticommutation relation (2.6) in our previous works11,15:
See (3.19) in the first reference of Ref.11.
1 1
b= (X iY), b† = (X +iY), (2.3)
√2ℓ − √2ℓ
B B
III. EQUATIONS OF MOTION
obeying [b,b†]=1, and define the Fock states
1 The Heisenberg equation of motion determines the
n = (b†)n 0 , b0 =0, (2.4) quantum mechanical system. It is given by
| i √n! | i | i
d
where n = 0,1,2, . The QH system provides us with i~ Dˆ (p)=[Dˆ (p),H] (3.1)
··· dt µν µν
an ideal 2-dimensional world with the built-in noncom-
mutative geometry. for electrons in the lowest Landau level. In ordinary
We assume that the electron carries the SU(N) index. physics the dynamics arises from the kinetic Hamilto-
Forinstance,itcarriesthespinSU(2)indexinthemono- nian. However, this is not the case here, since the ki-
layersystem,andthespin-pseudospinSU(4)indexinthe netic Hamiltonian (2.1) commutes with Dˆ (p). In-
µν
bilayer system. The SU(N) electron field Ψ(x) has N deed, H contains only the relative coordinate R =
K
components. It is given by ( P /eB ,P /eB ), while Dˆ (p) contains only the
y ⊥ x ⊥ µν
−
guiding center X = (X,Y). Nontrivial dynamics can
ψµ(x)= xn cµ(n) (2.5) arise because the guiding center has the noncommuta-
h | i
n tive coordinates obeying (2.2). It is remarkable that the
X
dynamicsarisesfromtheverynatureofnoncommutative
for electrons in the lowest Landau level, where c (n) is
µ geometry in the QH system.
the annihilation operator acting on the Fock state n ,
| i ThetotalHamiltonianH consistsoftheCoulombterm
c (n),c†(m) =δ δ . (2.6) HC and the rest term Hrest,
{ µ ν } mn µν
The physical variables are the electron density ρ(x) and H =HC+Hrest, (3.2)
the isospin field I (x),
A whereH standsfortheZeemanterminthemonolayer
rest
1 system and additionally the tunneling and bias terms in
ρ(x)=Ψ†(x)Ψ(x), IA(x)= Ψ†(x)λAΨ(x), (2.7) the bilayer system. All of them are represented in terms
2
3
of the bare densities. Hence, we areable to calculate the IV. CURRENTS IN SPIN QH FERROMAGNET
Heisenbergequationofmotion(3.1)basedontheW (N)
∞
algebra (2.11). InthemonolayerQHsystemthephysicalvariablesare
What is observed experimentally is the classical field the electron density ρ(x) and the spin field S (x),
a
(x), which is the expectation value of Dˆ (x) by a
µν µν
D 1
Fock state, ρ(x)=Ψ†(x)Ψ(x), S (x)= Ψ†(x)τ Ψ(x) (4.1)
a a
2
µν(x)= Dˆµν(x) . (3.3) with (2.5) for Ψ(x). We denote the corresponding clas-
D h i
sical field as
WeconsidertheclassofFockstateswhichcanbewritten
as ̺(x)= ρˆ(x) , (x)= Sˆ(x) . (4.2)
h i S h i
Here τ are the Pauli matrices for the spin space.
S =eiW S , (3.4) a
| i | 0i We study the electric current in the QH state. The
current is introduced originally to guarantee the charge
where W is an arbitrary element of the W (N) algebra
∞ conservation. This is the case also in the noncommuta-
which represents a general linear combination of the op-
tive plane,
erators c†(m)c (n). The state S is assumed to be of
the form1ν1 µ | 0i e dρˆ(x)=∂ Jˆ(x), (4.3)
i i
− dt
|S0i= c†µ(n) νµ(n)|0i, (3.5) where eρˆ(x) is the chargedensity and Jˆi(x) is the cur-
Yµ,n(cid:2) (cid:3) rentin−the lowestLandaulevel. The physically observed
current is the classical current given by
where ν (n) takes the value either 0 or 1 specifying
µ
whethertheisospinstateµatthestate n isoccupiedor (x)= Jˆ(x) . (4.4)
| i Ji h i i
not, respectively. Though it may not include all states,
it certainly contains all integer QH states, by which we Taking the expectation value of (4.3) andusing the clas-
meanthegroundstateaswellasallquasiparticleexcited sical equation of motion (3.7), we have
states at ν =integer.
d
Theclassicalfieldsatisfiedtheclassicalequationofmo- ∂i i(x)= e ̺(x)= e[̺(x), ]PB. (4.5)
J − dt − H
tion. Itisconstructedby takingthe expectationvalueof
the Heisenberg equation of motion (3.1), The formula for the current i(x) is obtained from this
J
continuity equation by integrating it.
d Inthe monolayerQH systemthe Hamiltonianconsists
i~ µν(x)= Dˆµν(p)H HDˆµν(p) . (3.6) of the Coulombterm, the Zeeman term and the electric-
dtD h i−h i
field term,
We can verify11 for the class of states (3.4) that
H =H +H +H . (4.6)
C Z E
d We introduce the scalar potential ϕ(x) to produce an
(x)=[ (x), ] , (3.7)
dtDµν Dµν HPB electric field,
where = H istheclassicalHamiltonian,providedthe Ei(x)= ∂iϕ(x), (4.7)
H h i −
classical density is endowed with the Poissonstructure
in the Hamiltonian H . The classical Hamiltonian is
E
given by = H , which reads11
2πi~[ µν(p), στ(q)]PB =δµτe+2iℓ2Bp∧q σν(p+q) H h i
D D D
δσνe−2iℓ2Bp∧q µτ(p+q). H=HD+HX+HZ+HE, (4.8)
− D
(3.8) where
The classical Coulomb energy consists of the direct and D =π d2qVD(q)̺( q)̺(q), (4.9a)
H −
exchange energies11, C = D+ X, and we obtain Z
H H H
= π d2pV (p) ( p) (p)
X X a a
= + + , (3.9) H − S − S
H HD HX Hrest Z ha=Xxyz
1
from the total Hamiltonian (3.2). Here, and + ̺( p)̺(p) , (4.9b)
HD Hrest 4 −
havethe same expressionsas HC andHrest, respectively, = 2π∆ (0). i (4.9c)
withthereplacementofDˆ by : Theexchangeterm HZ − ZSz
µν µν
HterXmissianntehwe ftoelrlmow.inWgesegcivtieonexs.pDlicit expressions of these HE =−e d2qe−q2ℓ2B/4ϕ(−q)̺(q), (4.9d)
Z
4
with This is zero because the relation
VD(q)=4πeε2q e−ℓ2Bq2/2, (4.10) Z d2qf(−q)g(k+q)sin(k∧q)
| |
VX(p)=√24ππeε2ℓBI0(ℓ2Bp2/4)e−ℓ2Bp2/4. (4.11) =−Z d2qf(k+q)g(−q)sin(k∧q) (4.15)
holds for any two functions f and g. Hence, there is no
Here, I (x) is the modified Bessel function. contribution from .
0 X
H
It is straightforward to calculate the Poisson bracket We finally examine the contribution from E by eval-
(4.5) with (4.8), uating (4.12d). Expanding sin[ℓ2(k q)/2], wHe obtain
B ∧
[̺(k),HD]PB Ji(k)=−ie22πℓ~2Bεij d2qe−k2ℓ2B/4qjϕ(q)̺(k−q)
2 k q Z
= d2qV (q) ̺(q),̺(k q) sin ℓ2 ∧ , 1 k q 2
~ D { − } B 2 1 ℓ2 ∧ + . (4.16)
Z (cid:18) (cid:19) × − 3! B 2 ···
(4.12a)
h (cid:16) (cid:17) i
In a constant electric field E such that
[̺(k), ] j
X PB
H
1 k q q ϕ(q)=2πiE δ(q), (4.17)
= d2qV (q)̺( q)̺(k+q)sin ℓ2 ∧ i j
− 2~ X − B 2
Z (cid:18) (cid:19) we find
2 k q
− ~Z d2qVX(q)Sa(−q)Sa(k+q)sin(cid:18)ℓ2B ∧2 (4(cid:19).12b) Ji(k)= e~2ℓ2BεijEje−k2ℓ2B/4̺(k−q). (4.18)
[̺(k), ] =0, (4.12c) On the incompressible state (4.13) it yields
Z PB
H
[̺(k), ]
HE PB e2ℓ2
= e d2qe−q2ℓ2B/4ϕ(q)̺(k q)sin ℓ2 k∧q . Ji(x)= ~BεijEjρ0. (4.19)
− 2π~ − B 2
Z (cid:18) (cid:19) This is the standard formula for the Hall current.
(4.12d)
We havedemonstratedthat the CoulombandZeeman
interactions do not affect the Hall current, as expected.
WeevaluatethemonthegroundstateintheQHsystem,
However, this is not a trivial result since the exchange
where the classical density is given by
Coulomb interaction yields rather complicated formulas
in the midstream of calculations. As we have verified,
̺(k)=2πρ δ2(k) (4.13)
0 they vanish in the spin QH ferromagnet. On the other
hand, as we shall see in the succeeding sections, the ex-
with ρ0 the total electron density. This is known as the change Coulomb interaction produces the phase current
incompressibility condition1, implying that the QH sys- in the pseudospin QH ferromagnet.
tem is an incompressible liquid.
We first examine the Poisson bracket (4.12b) for .
D
H
Substituting the incompressibility condition into (4.12a) V. CURRENTS IN PSEUDOSPIN QH
it is trivial to see that [̺(k), ] = 0. Hence, there FERROMAGNET
D PB
H
is no contribution to the Hall current from the direct
Coulomb term D. We proceed to study the electric currents in the bi-
H
We next examine the Poisson bracket (4.12b) for . layer system. Though the actual system has the spin-
X
The term involving ̺( k′)̺(k+k′)sin[ℓ2(k q)/2] vHan- pseudospin SU(4) structure, we consider the spin-frozen
− B ∧
ishes because of the incompressibility condition. The re- system since the spin does not affect the current: See
maining term involves S ( k′)S (k+k′). Since we are AppendixB.Theelectronfield(2.5)hastwocomponents
z z
−
concernedaboutahomogeneousflowofelectrons,taking ψα(x) correspondingto the front(α=f) and back(α=b)
the nontrivial lowestorder term in the derivative expan- layers. The physical variables are the electron densities
sion of potential V (k), we find ρα(x) in the two layers,and the pseudospin field P (x),
X a
1
1 ρα(x)=ψα†(x)ψα(x), P (x)= Ψ†(x)π Ψ(x)
d2qVX(k′)Sz(−q)Sz(k+q)sin 2ℓ2Bk∧q a 2 a (5.1)
Z (cid:18) (cid:19)
1 with (2.5), where π are the Pauli matrices for the pseu-
V (0) d2qS ( q)S (k+q)sin ℓ2k q a
≃ X z − z 2 B ∧ dospin space. We use notations
Z (cid:18) (cid:19)
=0. (4.14) ̺α(x)= ρˆα(x) , (x)= Pˆ (x) (5.2)
a a
h i P h i
5
for the classical variables. We note that the capacitance energy is given by
TheHamiltonianH consistsoftheCoulombterm,the
tunnelingterm,thegatetermandtheelectric-fieldterm, ǫcap =4(ǫ−D−ǫ−X). (5.10)
H =H +H +H +H , (5.3) The scalar potentials ϕf(x) and ϕb(x) are introduced to
C T gate E
produce the electric fields
which are explicitly given by (A1) in Appendix A. The
gate term has been introduced to make a density imbal- Ef = ∂ ϕf, Eb = ∂ ϕb (5.11)
ance between the two layers. The average density reads i − i i − i
in the front and back layers within the Hamiltonian
1+σ 1 σ
ρf = 0ρ , ρb = − 0ρ (5.4) (5.6e).
0 2 0 0 2 0
We investigate the classical equation of motion (3.7)
in each layer. We call σ0 the imbalance parameter. to derive the classical current Jiα(x) in each layer. It is
As we show in Appendix A, the classical Hamiltonian defined so that the charge conserves locally,
is rearrangedinto
H d̺f 1
e =∂ f(x) (x), (5.12a)
H = D+ X+ T+ bias+ E, (5.5) − dt iJi − dJz
H≡h i H H H H H
d̺b 1
where and arethe directandexchangeCoulomb e =∂ b(x)+ (x), (5.12b)
HD HX − dt iJi dJz
energy terms,
where (x) is the tunneling current between the two
z
=π d2pV+(p)̺( p)̺(p) layers.J
HD D −
Z The tunneling term T contributes only to the tun-
H
+4π d2pV−(p) ( p) (p) 8πǫ−σ (0) neling current z(x), and it is given by
D Pz − Pz − D 0Pz J
Z
(5.6a) 1 e d
(x)= [̺f ̺b] =e[ , ] . (5.13)
z z T PB
dJ 2dt − P H
= π d2pVd(p) ( p) (p) (cid:12)HT
HX − X Pa − Pa (cid:12)
aX=x,yZ The current in the layer α=(cid:12)(cid:12)f, b is given by
1
π d2pV (p) ( p) (p)+ ̺( p)̺(p)
−+8πZǫ−Xσ0PzX(0),(cid:20)Pz − Pz 4 − (5(cid:21).6b) ∂iJiα(x)=−e dd̺tα(cid:12)(cid:12)HD+HX+Hbias+HE, (5.14)
(cid:12)
and HT and Hbias are the tunneling and bias terms, wanhdic[h̺αc,onsis]ts.ofW[e̺αe,xHprDe(cid:12)]sPsB̺,α[̺inα,tHerXm]PsBo,f ̺[̺αan,Hdbias]aPsB
E PB z
= 2π∆ P (0), (5.6c) H P
T SAS x
H −
σ 1 1
Hbias =−2π 1 0 σ2∆SASPz(0), (5.6d) ̺f = 2̺+Pz, ̺b = 2̺−Pz. (5.15)
− 0
while the electric-fieldpterm reads Note that ̺α =ρα0 with (5.4) in the ground state.
WeneedtocalculatethePoissonbracketsfor̺and
z
P
E = e d2qe−q2ℓ2B/4 ϕf( q)̺f(q)+ϕb( q)̺b(q) . with various Hamiltonians. When we estimate them on
H − − − thestatesatisfyingtheincompressibilitycondition(4.13),
Z
(cid:2) (5.6(cid:3)e) many terms vanish precisely by the same reasons as in
Various Coulomb potentials are defined by the monolayer system. We now demonstrate that there
exists a new contribution from the exchange interaction
V =V++V−, Vd =V+ V−, (5.7)
X X X X X − X X to the current, as is the novel feature in the pseu-
H
dospin QH ferromagnet. Let us explicitly write down
and
only those parts in the Poisson brackets that yield non-
VD±(p)=8πeε2|p|(cid:16)1±e−|p|d(cid:17)e−21ℓ2Bp2, (5.8a) vPaonTisihsshoerninegbirscaocanktenrteibw[utti(eokrnm)s,tionvt]ohlev,icnugrrceonst(.12ℓ2Bk ∧q) in the
V±(p)=ℓ2B d2ke−iℓ2Bp∧kV±(k), (5.8b) Pz HX PB
X π D
Z
[ (k), ]
with the interlayer separation d. We have also defined Pz HX PB
ǫ ℓ2
ǫ− = 1ρ d2xV−(x), ǫ− = 1ρ d2xV−(x). =− ~ab d2qVXd(q)Pa(−q)Pb(k+q)cos 2Bk∧q
D 2 0 D X 4 0 X Z (cid:18) (cid:19)
Z Z + , (5.16)
(5.9)
···
6
wherethe index a runsoverxandy. Makingthe deriva-
tive expansion of Vd(q) and cos(1ℓ2k q), we find
X 2 B ∧
4eJd
f(X)(x)= b(X)(x)= s(∂ ∂ ),
Ji −Ji ~ρ2 iPx·Py− iPy·Px
0
(5.17)
where
Jsd =Js − π2ℓd + 1+ ℓd22 ed2/2ℓ2Berfc d/√2ℓB
r B B
h (cid:16) (cid:17) (cid:16) (5.(cid:17)18i)
together with
1 e2
J = (5.19) FIG. 1: (a) Hall currents are injected to the front and back
s
16√2π4πεℓB layers independently. The Hall and diagonal resistances are
measuredonlyinoneofthelayers. (b)Asimplifiedpictureis
is the pseudospin stiffness.
given to represent the same measurement, where the symbol
The electric field yields a nonzero contribution as in V in a circle indicates that the measurement is done on this
the monolayer case, layer.
[̺α(k), ]
E PB
H
= e d2qe−41ℓ2Bqϕα(q)̺α(k q)sin ℓ2 k∧q , We haveshownthatthe phasecurrentarisesinthe pres-
− 2π~ 0 − B 2 ence of the interlayer phase coherence.
Z (cid:18) (cid:19) We should mention that the emergence of the phase
as corresponds to the monolayer formula (4.19). Hence, current, ∂ ϑ(x), in the pseudospin QH ferromagnet
i
we obtain the standard formula for the Hall current in hasalread∝ybeenpointedoutinaneffectivetheory1based
each layer, on an intuitive and phenomenological reasoning. In this
paperwehavepresentedamicroscopicformulationofthe
α(E)(x)= e2ℓ2Bε E ρα, (5.20) phase current.
Ji ~ ij j 0
on the incompressible ground state.
VI. DIAGONAL AND HALL RESISTANCES
Finally,the Poissonbracketwiththe tunneling Hamil-
tonian is exactly calculable,
Let us first review the Hall current in the monolayer
1 system with homogeneous electron density ρ . The elec-
[P (k), ] = ∆ P (k), (5.21) 0
y HT PB ~ SAS z tricfieldE drivestheHallcurrentintothedirectionper-
pendicular to it,
which yields
e2ℓ2ρ
Jz(x)=−e~d∆SASPy(x) (5.22) Ji = ~B 0εijEj. (6.1)
We apply the electric field so that the current flows into
to the tunneling current.
the x direction, as implies E = 0. Hence the diagonal
We parametrize the classicalpseudospin field in terms x
resistance vanishes,
of the interlayerphase field ϑ(x) andthe imbalance field
σ(x),
E
x
R =0, (6.2)
xx
P (x)= 1ρ 1 σ2(x)cosϑ(x), ≡ Jx
x
2 0 − and the Hall resistance is given by
1 p
P (x)= ρ 1 σ2(x)sinϑ(x),
y −2 0 − Ey ~ 2π~
R = = . (6.3)
Pz(x)= 1ρ σ(px). (5.23) xy ≡ Jx e2ℓ2Bρ0 νe2
2 0
ThesignalsoftheQHeffectconsistofthedissipationless
TheHallcurrentisgivenbythesumof(5.17)and(5.20),
current(6.2) and the development of the Hall plateauat
the magic filling factor ν.
eJd e2ℓ2
f(x)= s (1 σ2)∂ ϑ+ Bε Ef̺f, (5.24a) What occurs in actual systems is as follows. We feed
Ji ~ − i ~ ij j the current into the x direction. Due to the Lorentz
x
b(x)= eJsd(1 σ2)∂ ϑ+ e2ℓ2Bε Eb̺b. (5.24b) force electroJns accumulate at the edge of the sample,
Ji − ~ − i ~ ij j which generates such an electric field E that makes
y
7
the given amount of current flow into the x direc- the equation of motion also for the electric field. Equiv-
x
J
tion [Fig.1]. The relation between the current and the alently it is necessary to minimize the Coulomb energy
electric field is fixed kinematically by the formula (6.1) stored in the volume between the two layers.
in the monolayer system. The energy due to the electric field E(x,z) between
We proceed to study the QH current in the imbal- the two layers is given by the sum of the Maxwell term
anced bilayer system at σ , where the electron densities and the source term,
0
are given by (5.4) in the ground state. We assume the
ε
sampleparameter∆SAS =0sothatthereisnotunneling HE = d2xdzE2(x,z) e d2xdzϕ(x,z)ρ(x,z),
currentbetweenthetwolayers. AsweshowinAppendix 2 −
Z Z
(6.8)
A, the interlayer phase field ϑ(x) is gapless in the limit
where
∆ =0,butthe imbalancefieldσ(x)hasthe gapǫ .
SAS cap
Consequently,theexcitationofσ(x)issuppressedatsuf- ϕf ϕb
ficiently low temperature. Hence we set σ(x)=σ0 in all Ex =−∂xϕ, Ey =∂yϕ, Ez =− −d . (6.9)
formulas.
We imagine the electric fields Ef and Eb driving the Notethat(6.8)is equivalenttothe ”surfaceterm”(5.6e)
j j
Hall currents to flow into the x direction [Fig.1]. As we inthe equationofmotion(3.7)sincethe electrondensity
havearguedintheprevioussection,thebasicformulafor ρ(x,z) is nonzero only on the two layers.
the current is (5.24), or When there are constant fields Ef and Eb into the
y y
y direction on the layers, the field E between the two
z
e e2ℓ2 layers is given by
f = (1 σ2)Jd∂ ϑ+ Bε Efρf, (6.4a)
Ji ~ − 0 s i ~ ij j 0
Jib =− ~e(1−σ02)Jsd∂iϑ+ e2~ℓ2BεijEjbρb0, (6.4b) Ez =−ϕf−dϕb = Eyf −d Eyby. (6.10)
We carry out the integration over z and then over the
in the imbalance configuration at σ .
0 plane in (6.8),
Sinceoursystemisassumedtobehomogeneousinthe
y direction, the variables depend only on x. Thus, εd εd
H = w d2x (Ef)2+(Eb)2 + d2xE2
E 2 y y 2 z
Ef =Eb =0. ∂ ϑ=0, (6.5) Z Z
x x y εd L2 (cid:0) ε(cid:1)L4
= w (Ef)2+(Eb)2 + (Ef Eb)2, (6.11)
and 2 y y 24d y− y
(cid:0) (cid:1)
Ef Eb where dw is the thickness of the layer, and L is the size
Rf x =0, Rb x =0, (6.6) of the sample. Note that there is no contribution from
xx ≡ f xx ≡ b
Jx Jx the source term due to the parity,
for f =0 and b =0. The Hall current is given by
Jx 6 Jx 6 d2xdzϕ(x,z)ρ(x,z) L/2 dyy =0. (6.12)
f =e(1 σ2)Jd∂ ϑ+ e2ℓ2Bρ0(1+σ )Ef, (6.7a) Z ∝Z−L/2
Jx ~ − 0 s x 2~ 0 y
The energy density is given by
e e2ℓ2ρ
b = (1 σ2)Jd∂ ϑ+ B 0(1 σ )Ef. (6.7b)
Jx − ~ − 0 s x 2~ − 0 y = εdw (Ef)2+(Eb)2 + εL2(Ef Eb)2. (6.13)
HE 2 y y 24d y − y
Consequently the relation between the current and the
(cid:0) (cid:1)
electric field is not fixed kinematically in the presence of The important observation is that the second term di-
the interlayer phase difference ϑ(x). verges in the large limit of the sample size L. The order
Any set of Ef, Eb and ϑ seems to yield the given ofthesamplesizeisL 1mm,whilethetypicalsizepa-
y y ≃
amounts of currents f and b provided they satisfy rameterisℓB d dB 10nm,asimpliesL/ℓB 106.
Jx Jx Itis a goodap≃prox≃imati≃onto take the limit L ≃. We
(6.7). In the actual system the unique set of them is
→∞
then find
realized: It is the one that minimizes the energy of the
system. It is a dynamical problem how Ef and Eb are
y y Ef =Eb (6.14)
determined in the bilayer system. y y
A bilayer system consists of the two layers and the to make the energy density finite.
volume between them. The dynamics of electrons is de- We rewrite (6.7) as
scribed by the Hamiltonian (5.5), which is defined on
the two planes. The tunneling termhas been introduced 2~ f e
Ef = Jx (1 σ )Jd∂ ϑ , (6.15a)
just to guarantee the charge conservation. We have so y e2ℓ2ρ 1+σ − ~ − 0 s x
far neglectedthe electric fieldin the volume betweenthe B 0 (cid:20) 0 (cid:21)
2~ b e
two layers, since it does not contribute to the equation Eb = Jx + (1+σ )Jd∂ ϑ . (6.15b)
of motion (3.7) for electrons. We now need to analyze y e2ℓ2Bρ0 (cid:20)1−σ0 ~ 0 s x (cid:21)
8
FIG.2: (a)Thesameamountofcurrentflowsonbothlayers
in thesame direction. (b) The same amount of current flows
on both layers in the opposite directions. (c) and (d) The
current flows only on the front layer. In these experiments
the diagonal and Hall resistances are measured at one of the
layers indicated by V in a circle.
FIG.3: Byfeedingdifferentcurrentsintothetwolayers,the
Hall resitance Rxy (solid curve) and the longitudinal resis-
The condition Ef =Eb requires tance Rxx (dotted curve) are measured on one of the layers
y y (indicated by V in a circle). (a) In the counterflow exper-
iment the opposite amounts of currents are fed to the two
Eyf −Eyb = e2ℓ22~ρ 1+Jxfσ − 1Jxbσ − 2e~Jsd∂xϑ =0, tlaaykeerns,fwrohmereRRef.xf4x.=(bR)xfIyn=th0eadnroamgaelxopuesrlyimaetnνt=the1.cDurarteantaries
B 0 (cid:18) 0 − 0 (cid:19)(6.16) fed only to the front layer, where Rxfy takes anomalously the
or same value at ν = 1 and ν = 2. It is also remarkable that
Rf = R¯b at ν = 1, where R¯b ≡ Eb/Jf. Data are taken
xy xy xy y x
~ f b from Ref.5.
∂ ϑ= Jx Jx . (6.17)
x 2eJd 1+σ − 1 σ
s (cid:18) 0 − 0(cid:19)
and
Substituting (6.17) into (6.15) we obtain
~ Rf Eyf =4π~, Rb Eyb =4π~. (7.1a)
Eyf =Eyb = e2ℓ2Bρ0 Jxf+Jxb . (6.18) xy ≡ Jxf νe2 xy ≡ Jxb νe2
(cid:0) (cid:1)
This is the standard result of the bilayer QH current.3,4
We conclude that the Hall resistance is given by
Ef 2π~ b
Rf y = 1+ Jx , (6.19a) B. Counterflow Experiment (b)
xy ≡ f νe2 f
Jx (cid:18) Jx(cid:19)
Eb 2π~ f
Rb y = 1+ Jx (6.19b) Thecounterflowexperiment[Fig.2(b)]ismostinterest-
xy ≡ b νe2 b ing. Since f = b, we obtain from (6.17) that
Jx (cid:18) Jx (cid:19) Jx −Jx
in each layer. We note that both the diagonal and Hall ~ f
resistances are independent of the imbalance parameter ϑ= Jxx+constant. (7.2)
eJd
σ . s
0
and
VII. ANOMALOUS BILAYER HALL Ef Eb
Rf y =0, Rb y =0. (7.3a)
CURRENTS xy ≡ b xy ≡ b
Jx Jx
WeapplytheseformulastoanalyzetypicalbilayerQH The result is remarkablesince it is againstthe naivepic-
currents[Fig.2],andcomparethe results withthe exper- ture of the QH effect. Recall that the essential signal of
imental data3,4,5. Though the experiments were carried the QH effect is considered to be the development of the
out at the balanced point (σ =0), our results are valid plateau. The vanishing of the Hall resistance in the QH
0
also for imbalanced configurations (σ =0). regime is a new phenomenon. This anomalous behavior
0
6 has been observedexperimentally by Kellogg et al.3 and
Tutuc et al.4 at ν =1, as illustrated in Fig.3(a).
A. Experiment (a)
C. Drag Current (c) & (d)
The same amounts ofcurrentare fed to the twolayers
in the experiment [Fig.2(a)]. Since f = b, we obtain
Jx Jx
from (6.17) that The drag experiment [Figs.2(c) and (d)] is also very
interesting, where the current is fed only to the front
ϑ=constant, layer. The Hall resistance is measured in the front layer
9
[Fig.(c)]andalsointhebacklayer[Figs.(d)]. Since b = contains the phase current in the bilayer QH ferromag-
Jx
0, we obtain from (6.17) that net. It is a dynamical problem how the phase current
flows. We have shown that it flows in such a way that
~ f theHallcurrentbehavesanomalouslyasdiscoveredinre-
ϑ= Jx x+constant. (7.4)
2eJd cent experiments3,4,5. Furthermore, the anomalous Hall
s
resistance is unchanged even if the density imbalance is
In the drag experiment the definition (6.19) for Rb be- made between the two layers. These experimental data
xy
comesmeaninglesssince b =0. Weadoptthedefinition provide us with another proofof the interlayerphase co-
Ji herence spontaneously developed in the bilayer system.
Eb
R¯b y (7.5)
xy ≡ f
Jx
APPENDIX A: SU(2) EFFECTIVE
with the use of f. Then, we find HAMILTONIAN
Jx
Ef 2π~ Eb 2π~
Rf y = , R¯b y = . (7.6a) In this appendix we derive the classical Hamiltonian
xy ≡ f νe2 xy ≡ f νe2 (5.6)fromthefield-theoreticalHamiltonian(5.5),orH =
Jx Jx
H++H−+H +H +H . Eachtermisgiveninterms
In particular, we have C C T gate E
of the electron density ρ(x) and the pseudospin density
P (x) as follows,
Ef 2π~ a
Rf =R¯b = y = at ν =1. (7.7)
xy xy f e2 1
Jx H+ = d2xd2y V+(x y)ρ(x)ρ(y), (A1a)
C 2 −
On the other hand, if there is no interlayer coherence, Z
the QH current in the front layer is H− =2 d2xd2y V−(x y)P (x)P (y), (A1b)
C − z z
Z
νe2
Jif = 4π~εijEjf (7.8) HT =−∆SAS d2xPx(x), (A1c)
Z
at the balance point, and H = ∆ d2xP (x), (A1d)
gate bias z
−
Z
Ef 4π~
Rf = y = . (7.9) H = e d2x ϕf(x)ρf(x)+ϕb(x)ρb(x) , (A1e)
xy f νe2 E −
Jx Z
(cid:2) (cid:3)
In particular we have where ∆ is the bias parameter to take care of the
bias
charge imbalance made by the gate voltages in (A1d).
Rf = Eyf = 2π~ at ν =2. (7.10) The Coulomb potentials are
xy f e2
Jx e2 1 1
V±(x)= . (A2)
It is prominent that from (7.7) and (7.10) the Hall resis-
8πǫ x ± x2+d2
tance is the same at ν =1 and ν =2. These theoretical (cid:18)| | | | (cid:19)
resultsexplainthedragexperimentaldataduetoKellogg p
We take the expectation value by the Fock state (3.4).
et al.5, as illustrated in Fig.3(b).
The Coulomb energy is decomposed into the direct and
exchange energies11,
VIII. CONCLUSION
H+ =π d2pV+(p)̺ˆ( p)̺ˆ(p)
h Ci D −
In this paper we have analyzed the dynamics of elec- Z
1
trons confined to the lowest Landau level based on non- π d2pV+(p) ( p) (p)+ ̺( p)̺(p)
− X Pz − Pz 4 −
commutativegeometry. Inordinaryphysicsthedynamics Z (cid:20) (cid:21)
arises from the kinetic Hamiltonian. In the QH system,
π d2pV+(p) ( p) (p), (A3a)
however, it arises from the very nature of noncommuta- − X Pa − Pa
a=x,yZ
tivegeometry,thatistheW (N)algebra(2.11)satisfied X
∞
by the bare density Dˆµν(p). hHC−i=4π d2pVD−(p)Pz(−p)Pz(p)
As an application we have derived the formula for the Z
electric current. We have found that the Coulomb in- π d2pV−(p) ( p) (p)+ 1̺( p)̺(p)
teraction yields quite complicated contributions through − X Pz − Pz 4 −
Z (cid:20) (cid:21)
the exchange term to the current. Nevertheless, we re-
produce the standardformula for the Hall currentin the +π d2pVX−(p)Pa(−p)Pa(p). (A3b)
monolayer QH ferromagnet. However, the Hall current a=x,yZ
X
10
Allothertermsaresimplyconvertedintothecorrespond- uptothesecondorderinσ(x)andϑ(x). Theircoherence
ing classical terms, lengths are
H = 2π∆ (0), (A4)
hhHgTaitei=−−2πS∆AbSiaPsPxz(0), (A5) ξϑ =2ℓBsπp∆1−SAσS02Jsd,
hHEi=−e d2qe−q2ℓ2B/4 ϕf(−q)̺f(q)+ϕb(−q)̺b(q) . ξ =2ℓ π[(1−σ02)Js+σ02Jsd] . (A15)
Z (cid:2) (A6(cid:3)) σ Bsǫcap(1−σ02)+∆SAS/ 1−σ02
We now analyze the ground-state condition. We substi- It is remarkable that the coherent lenpgth ξ becomes in-
ϑ
tute finitely large as ∆ 0, though ξ remains finite due
SAS σ
→
tothecapacitance-energyparameterǫ . Itfollowsfrom
1 cap
̺(x)=ρ , P (x)= ρ (√1 σ ,0,σ ) (A7) (A8) that the condition ∆ < ǫ is necessary to ob-
0 a 0 0 bias cap
2 0 − tain the gapless mode.
into the total energy H , and minimize it with
H ≡ h i
respect to σ . As a result we obtain the condition
0
APPENDIX B: SU(4) EFFECTIVE
σ
0 HAMILTONIAN
∆bias = ∆SAS+σ0ǫcap (A8)
1 σ2
− 0
We have ignored the spin degree of freedom when we
with (5.10). Eliminapting the parameter ∆ from the
bias haveanalyzedtheHallcurrentsinthebilayerQHsystem
total classical Hamiltonian
inSectionVI. Inthisappendix,includingallcomponents
= H+ + H− + H + H + H , (A9) of the SU(4) QH system, we derive the effective Hamil-
H h Ci h Ci h Ti h gatei h Ei tonian and justify this simplification.
we can rearrange it into There are 15 isospin components,
1 1
H=HD+HX+HT+Hbias+HE, (A10) Sa = 2Ψ†σaΨ(x), Pa = 2Ψ†πaΨ,
where various terms are given by (5.6) in text. 1
R = Ψ†σ π Ψ. (B1)
We proceedto analyzeasmallfluctuationofthe pseu- ab 2 a b
dospin field P (x) around the ground state (A7), which
a
We denote their classical fields as , and as in
describes the pseudospin wave. We parametrize Sa Pa Rab
(4.2) and (5.2). The classical Hamiltonian has already
P (x)= ρ0n†(x)π n(x), (A11) been derived11, where the Coulomb energy density is
a a
2
Hcl =πV+(p)̺( p)̺(p)+4πV−(p) ( p) (p)
where π is the Pauli matrix, and h Ci D − D Pz − Pz
a π
Vd(p)[ ( p) (p)+ ( p) (p)
n(x)= √12(cid:18)√√11+−σσ00 −√√11−+σσ00 (cid:19)(cid:18)p1−η(|xη)(x)|2 (cid:19) − 2 X Sa − +SaRab(−Pp)aR−ab(pP)a]
1 √1+σ0 √1−σ0 1 (A12) −πVX−(p)[Sa(−p)Sa(p)+Pz(−p)Pz(p)
≃ √2(cid:18)√1−σ0 −√1+σ0 (cid:19)(cid:18)η(x)(cid:19) +Raz(−p)Raz(p)]
π
in the linear approximation,with VX(p)̺( p)̺(p). (B2)
− 8 −
σ(x)+iϑ(x) σ(x) iϑ(x)
η(x)= , η†(x)= − . (A13) TheZeemanterm,thetunnelingtermandthebiasterms
2 2 are given by (4.9c), (5.6c) and (5.6d), respectively.
The ground state is given by
Thepseudospinfield(A11)isreducedtothegroundstate
(A7) for η(x) = 0. Substituting the pseudospin field 1 1
(A11) into (A10) together with this parametrization, we Sag = 2δaz, Pag = 2( 1−σ02δax+σ0δaz),
find 1 q
g = δ ( 1 σ2δ +σ δ ) (B3)
(1 σ2)J +σ2Jd Rab 2 az − 0 bx 0 bz
= − 0 s 0 s(∂ σ)2 q
k
H 2 in the imbalanced configuration at σ . We analyze a
0
ρ ∆ small fluctuation of the isospin field around the ground
+ 0 ǫ (1 σ2)+ SAS σ2
4 " cap − 0 1 σ2# state. We may parametrize
− 0
ρ ρ ρ
+ 12Jsd(∂kϑ)2+ ρ40 ∆1pS−ASσ02ϑ2 (A14) Sa = 20n†σan, Pa = 20n†πan, Rab = 20n†σaπ(bBn4,)
p