Non-equilibrium noise in the (non-) Abelian fractional quantum Hall effect

20 Pages·2014·0.38 MB·English

Non-equilibrium noise in the (non-)Abelian fractional quantum Hall effect
O. Smits
∗
Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Rd, Dublin, Ireland and
School of Mathematics, Trinity College, Dublin 2, Ireland
J. K. Slingerland
Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Rd, Dublin, Ireland and
Department of Mathematical Physics, National University of Ireland, Maynooth, Ireland
4
1
S. H. Simon
0
Rudolf Peierls Centre for Theoretical Physics, Oxford, OX1 3NP, UK and
2
Department of Mathematical Physics, National University of Ireland, Maynooth, Ireland

n (Dated: January 21, 2014)
a
J WeanalysethenoiseoftheedgecurrentofagenericfractionalquantumHallstateinatunnelling
point contact system. We show that the non-symmetrized noise in the edge current for thesystem
8
out-of-equilibriumiscompletelydeterminedbythenoiseinthetunnellingcurrentandtheNyquist-
1
Johnson(equilibrium)noiseoftheedgecurrent. Simplyput,thenoiseinthetunnellingcurrentdoes
] not simply add up the equilibrium noise of the edge current. A correction term arises associated
l with the correlation between the tunnelling current and the edge current. We show, using a non-
l
a equilibrium Ward identity, that this correction term is determined by the anti-symmetric part of
h thenoiseinthetunnellingcurrent. Thisleadstoanon-equilibriumfluctuation-dissipation theorem
-
andrelatedexpressionsfortheexcessandshotnoiseofthenoiseintheedgecurrent. Ourapproach
s
e makes use of simple properties of the edge, such as charge conservation and chirality, and applies
m to generic constructions of the edge theory which includes edges of non-Abelian states and edges
withmultiplechargedchannels. Twoimportanttoolswemakeuseofarethenon-equilibriumKubo
.
t formula and thenon-equilibrium Ward identity. We discuss these identities in the appendix.
a
m
- I. INTRODUCTION
d
4 2
n
o The fractional quantum Hall effect1 is an example of
c
a topological phase of matter2. At each plateaux the
[
electrical Hall resistance is quantized and the collective

1 behaviour of the electrons is said to be topologically
v
1 ordered3. Characteristic features of these phases are a S 1 3 D
topological quantum field theory as the low energy de-
8
5 scription, the presence of a bulk energy gap, a robust-
4 nessofthe low-energytheoryagainstlocalperturbations
Figure 1. Sketch of a point contact. A current is injected at
. and quasiparticle excitations known as anyons4–6. Non-
1 the source (S), flows along the edge and is collected by the
0 Abelian anyons in particular obey a very rich general- drain(D).Atthepointcontactquasiparticlestunnelbetween
4 ization of exchange statistics, and the ν =5/2 state has the edges and a backscattering current forms flowing from
1 been put forward as a candidate for the realization of the lower to the upper edge. The probes 1 through 4 can be
: these quasiparticles7–9. Although much effort has been usedtomeasurethelocaledgecurrentandthecorresponding
v
i put into studying this and other candidates phases the noise.
X experimental discovery of a non-Abelian anyon is as of
r yetanopenquestion. Thestakesarehighasnon-Abelian
a
anyonscouldleadtotherealizationofatopologicalquan- edges. This results in a tunnelling currentwhich is char-
tum computer10–12. acterized by the specific edge theory and the underlying
The edge of a fractional quantum Hall state is re- topological order. Because of this both the tunnelling
sponsible for the transportproperties of the system13,14. currentanditsfluctuations(alsoknownasthenoise)can
Edge states are chiral and topologically protected, and be used to identify the topological order of the system.
backscatteringofchargecanonlyoccurbetweenopposite The expressionfor the noise and in particular its rela-
edges. Tunnellingexperimentsinthefractionalquantum tiontothetunnellingcurrent(or,equivalently,thetrans-
Hall effect make use of this property and probe the low- mission)has been studied perturbatively for generaland
energy states of the system through use of a tunnelling specific quantum Hall states18–32. For special cases such
point contact15–17. A tunnelling point contact acts as a asthe integer quantumHalleffect33–35 andthe Laughlin
constrictionwhichforcesopposite edgestogetherandin- series36–38 there are also non-perturbative results. The
duces tunnelling of (charged) quasiparticles between the simplestexampleofaperturbativeapproachistheSchot-
2
tky relation39, which arises in the low temperature and 4
weak tunnelling limit. It relates the shot noise and tun- jL
nellingcurrentthroughSIB(0)=e∗IB whichcanbeused jL−IB
to measure the quasiparticle charge. However, a univer- IB
sal expression relating the noise and the current non-
perturbatively is still an open question. jR jR−IB
Experiments that measure shot noise40–58 do not ac- S 3 D
tually measure the noise in the tunnelling current di-
rectly, but instead look at the noise in the outgoing
Figure 2. Sketch of a point contact. An edge current j
edge currents. To clarify, consider Figure 1 which shows R
is injected at the source S, and is partially reflected by the
a schematic of the experimental setup of a tunnelling
point contact resulting in a tunnelling current I . The edge
B
point contact. A current is injected at the source (S). current after the point contact is therefore j −I (on the
R B
It flows along the edge and is partially reflected at the basis of charge conservation and the chiral structure of the
point contact. The dotted line represents the tunnelling edge). Thenoisemeasuredatprobe3correspondstoEq.(2).
current. This current and the corresponding noise are In this work this relations are derived at the operator level.
not measured directly, but instead end up in the out-
going branches of the edge currents. A probe located
Supposewenowmeasurethenoiseoftheedgecurrentat
at position 3 or 4 then measures the local edge current
probe 3. This noise is roughly givenby the squareof the
and corresponding fluctuations (this probe can also be
operator or
incorporated with the drain – here we use a simplistic
picture).
S (j I )2 =j2 +I2 (j I +I j )
Thissetupthenbegsthequestion: howisthenoiseat, 3 ∼ R− B R B − R B B R
S +S ∆S . (2)
say,probenumber3relatedtothenoiseinthetunnelling ∼ bg IB −
current? In this work we derive such a relation based on
InthesecondlineweidentifyI2 withthenoiseinthetun-
generalgrounds. Weuseconservationofchargecombined B
nelling current S and S a type of background noise
with the chiral structure of the edge. Any charge tun- IB bg
(the noise of the edge current in the absence of a point
nelling from the upper to the lower edge will end up at
contact). There also appears a third term ∆S, which
probe number 3 due to the chiralstructure. Inthis work
representsthecouplingofthetunnellingcurrentwiththe
we study the exact expression relating the noise in the
equilibriumedgecurrent. This extratermarisesbecause
outgoing current to the noise in the tunnelling current.
the noise is not linear in the edge current.
Thisquestionhasbeenstudiedseveraltimesbefore,both
In this work we reproduce this argument at the op-
non-perturbatively18,24,34,36,59andperturbatively20,31,60.
erator level, which is also what distinguishes our ap-
Wewillgiveasummaryofourapproachandourresults
proach from previous work18,24,60. We analyse the non-
in the next section. What is important to keep in mind
symmetrized noise in the outgoing edge current S for
is that the expressionwhich relates the noise in the edge 3
generic quantum Hall systems. For that we use two new
currenttothenoiseinthetunnellingcurrentisnot linear.
tools, which we have developed in this work. The first
The fluctuations of the tunnelling current do not simply
isthenon-equilibrium Kubo formula. ThisNE-Kubofor-
add to the fluctuations in the edge current. The relation
mula formally extends the expression for linear response
between the noise in the edge current and the noise in
theory to all orders of perturbation theory, and from it
thetunnellingcurrentisalsoknownasanon-equilibrium
we obtain the current equation (Kirchoff’s law) Eq. (1)
fluctuation-dissipation theorem.
at the operator level. Using the same logic we obtain a
formula of the form Eq. (2).
The secondtool we developis a non-equilibrium Ward
II. SUMMARY AND OVERVIEW OF THIS
identity. A Ward identity is an identity imposed on the
WORK
correlation function, due to the presence of a symme-
try in the theory. In this work the symmetry is as-
Let us present an intuitive picture of the derivation in sociated with charge conservation (and j is the asso-
R
this work which relates the noise in the outgoing edge ciated conserved current), which leads to a well-known
current to the noise in the tunnelling current. We start Ward identity61. We have extended this identity to
with the simplified Figure 2. A current is injected at S correlation functions evaluated in the non-equilibrium
into the lower edge carried by the (right moving) edge system. The non-equilibrium Ward identity is used to
current jR. This chiral current is partially reflected by simplify the expression for the correction term ∆S
the point contact, where a tunnelling current IB tunnels (jRIB + IBjR). This results in the anti-symmetrize∼d
to the upper edge and ends up in the left moving edge noise of I , i.e. this correction term is proportional to
B
current. On the basis of charge conservation the edge ∆S (S (ω) S ( ω)).
current that is measured by probe 3 equals Th∼e finIBal res−ultIiBs a−n expression for the noise in the
edge current related to the noise in the tunnelling cur-
j =j I . (1)
3 R B rent, see Eq. (80). Therefore to compute the noise in
−
3
the edge current, we only need to determine the expres- weemphasizethatourmainanalysisisquitegeneraland
sionforthe noiseinthe tunnelling currentwhichisoften does not require all technical details associated with the
easier to obtain and for which more work has been per- edgetheory. Therequiredinputforthetreatmentonthe
formed. Related to this is an expression for the excess noiseis(1)theedgecurrentoperatorj (2)thequasipar-
R
noise Eq. (84), a non-equilibrium fluctuation-dissipation ticle operatorψ whichis usedto representquasiparticle
†
theorem Eq. (91) and an expression for the shot-noise tunnelling, and (3) the non-equal time commutation re-
limit Eq.(92). Our mainworkfocusesonanedge witha lations of the edge current and quasiparticle operator,
single chargedchannel (described by a chiralboson) and Eq. (25). These relations combined with some basic as-
possibly one or multiple neutral channels. In addition sumptions,suchastranslationalinvarianceandchirality,
we show how the results extend to quantum Hall edges are enough input for our main work which is treated in
withmultiplechargedmodes,possiblycounterpropagat- SectionIVandbeyond. Soalthoughthediscussioninthis
ing. Finally, we also look at similar expressions for the sectionissomewhattechnicalandbrief,itisonlyneeded
noise at the remaining probes of Figure 1, and the noise to motivate the origin of the edge current operator and
ofcombinationsoftheseprobes(i.e. thenoiseinthevolt- to describe the general idea of the edge theory.
age difference of probe 3 and 4). All results are valid to Our treatment of the edge theory is very similar to
all orders of perturbation theory. our previous work in Ref. 62. A quantum Hall system
An important simplification that we assume is that is a topologicallyorderedsystem, inwhich chiralgapless
away from the point contact the edge is described by a states develop at the edge as a consequence of anomaly
collectionoffree anddecoupled channels,eachdescribed cancellation13,14,63. In the long wavelength limit the ef-
by a chiralconformalfieldtheory in the long wavelength fective edge theory is a chiralconformal field theory and
limit. Interaction effects and disorder, which can for in- it comes equipped with a set of quasiparticle operators
stance cause equilibration of the edge currents after the and fusion rules. Non-Abelian states7,8,64 are character-
point contact, are beyond the scope of this work. We izedbythepresenceofquasiparticleswithmultiplefusion
also note that this paper looks at the (non-equilibrium) channels.
relation between the noise of the edge currents and the The edge contains a U(1) symmetry due to the cou-
tunnelling current. We do not determine the expression pling with the electromagnetic field. For instance, the
for the noise or its relation to the tunnelling current. edgetheoryoftheAbelianLaughlinstateatν =1/(2m+
In Section III we start with a summary of a generic 1) is described by a uˆ(1) chiral current algebra, also
quantum Hall edge. We focus on the definition of the known as the chiral boson or chiral Luttinger liquid14.
edgecurrentoperatorinthechiralbosonmodel,thecon- More complicated edge theories are constructed by com-
struction of a quasiparticle operator and the non-equal bining neutral degrees of freedom with one or multiple
timecommutationrelationsofthetheory,andextendthis chiral bosons. These neutral degrees of freedom do not
construction to edges with multiple charged channels. coupletotheelectromagneticfieldandareresponsiblefor
In Section IV we discuss the model of a point con- the non-Abelian nature of the corresponding trial state.
tactandinSectionVwe summarizethe non-equilibrium Inthisworkweassumethequasiparticleoperatorsatthe
formalism. For this formalism we assume that, initially, edge obey the following decomposition65
the point contact is absent and the system is at equi-
= uˆ(1) uˆ(1) . (3)
librium and finite temperature. We also discuss the Aedge Wn⊗ ⊗···⊗
non-equilibrium Kubo formula, which is proven in Ap- Heretheuˆ(1)’scorrespondtothedifferentchargedchan-
pendix A. In Section VI we apply the NE-Kubo formula nels of the edge. Since we are interested mostly in the
totheedgecurrentoperatorwhichresultsinanoperator- properties of the charged channels we describe all neu-
version of Kirchoff’s law. traldegreesoffreedomcollectivelythrough . Wefirst
n
W
The main results regarding the noise are obtained in discuss the case of a single charged channel, and expand
SectionVII.Thismakesuseofthenon-equilibriumWard this to the multichannel case at the end of this section.
identitytosimplifytheexpressionforthecorrectionterm
∆S. We obtain expressions for the non-symmetrized
noise,theexcessnoiseandtheshotnoise(allofthenoise A. The charged channel in the absence of
tunnelling
in the edge current) and generalize these expression to
themultichannelcase. Finally,SectionVIIIdiscussesex-
pressions for the noise in related quantities. We discuss We consider the charged channel13,66–68 on the lower
our findings in the Section X edge described by a chiral boson ϕ subject to a voltage
biasU inthegaugea =0. Thefieldiscompactifiedand
x
the action is given by
III. THE EDGE OF A FRACTIONAL 1
QUANTUM HALL STATE SR = ηR∂tϕR∂xϕR vc(∂xϕR)2 dtdx
4π − −
ZΣR
(cid:2) (cid:3)
√ν
In this section we discuss the edge theory of a generic + U [∂ ϕ ] dtdx . (4)
R x R
2π
fractional quantum Hall state. Before we come to this ZΣR
4
Throughout this work we mostly focus on a right mov- B. Edge current operator
ing edge current boson, which in the single channel case
corresponds to a single right moving chiral boson. It is The charge density operator is the zeroth component
coupled to the potential UR and moves along the edge of a conserved edge current (ρ(x,t),j(x,t)). To obtain
ΣR. We can easily switch to a left moving boson by re- the edge current density operator we use the continuity
placingR L. ThechiralityηR =1iswrittenexplicitly equation
→
(and η = 1). Finally v is the edge velocity. Quan-
L c
−
tization of this action is performed in e.g. Ref. 68. The ∂tρR(x)+∂xjR(x)=0 . (12)
non-local commutation relations are
In terms of the bosonic field the continuity equation
[ϕR(x),ϕR(x′)]=iηRπsgn(x x′) (5) reads ∂x(√2πν∂tϕ(x)+j(x))=0, which determines the
− edge current in terms of ∂ ϕ up to an x-independent
t
with sgn(x) = +1, 0, 1 for the regions x>0, x=0 and term. This term is set to zero by demanding that the
x<0. Heisenberg’s equation of motion results in current operator produces the usual Hall relation. The
edge current operator is
( η ∂ v ∂ )ϕ = √νU . (6)
− R t− c x R − R √ν
j (x)= ∂ ϕ (x) . (13)
R t R
−2π
Using the equations of motion we can extend the com-
mutation relations to non-equal time Usingtheequationsofmotion(6)wehavethealternative
form in terms of the charge density operator
[ϕ (x,t),ϕ (0,0)]=iη πsgn(x η v t)
R R R − R c ν
[∂xϕR(x,t),ϕR(0,0)]=iηR2πδ(x−ηRvct)) . (7) jR(x)=ηRvcρR(x)−ηR2πUR (14)
The left and right moving bosons commute. The de- HerewerecallthatbyreplacingR Lweobtaintheleft
→
pendency on the combination x v t reflects the chiral moving chiral boson. The total current running through
c
nature of the systemand we assu±me the system is trans- the system is given by
lational invariant. The charge density along the edge is Iˆ(x)=j (x)+j (x) . (15)
identified with the operator 0 R L
This total current operator is non-local as it adds the
√ν edge current densities on opposite edges. In a more gen-
ρ (x)= ∂ ϕ (x) . (8)
R x R
2π eralsettingthetotalcurrentoperatorisobtainedbytak-
ing the (bulk + edge) current density operator and in-
Thecorrespondingconservedchargeistheelectriccharge
upperedge
tegrating along a cross section J(x,y)dy, see
operator loweredge
e.g. Ref. 69. This reduces to Eq. (15) when the continu-
R
√ν ity equation Eq. (12) holds.
R = ∂xϕR(x) dx . (9) Wehavedefinedthevacuumsuchthatitischargeneu-
Q 2π
ZΣR tral. This implies the vanishing of the one-point correla-
tor ρ (x) = ∂ ϕ (x) =0 and we find for the current
The total edge Hamiltonian includes the contribution of h R i h x R i
densities on the edge
the neutral channel, which we discuss in Section IIIC.
Using the electric charge operators, the grand canonical ν
j (x,t) = η U (16)
Hamiltonian K0 of the total system is given by h R i − R2π R
The expectation values are with respect to the
K =H +H U U (10)
0 0,c 0,n− RQR− LQL equilibrium Hamiltonian at finite temperature, i.e.
H = vc dx (∂ ϕ )2+(∂ ϕ )2 (11) =Tr e−βK0 . For the total current we obtain
0,c x R x L h···i ···
4π the familiar Hallrelationbetween voltageandcurrentin
Z (cid:2) (cid:3)
(cid:2) (cid:3) the absence of backscattering
Here H and H are the Hamiltonians of the charged
0,c 0,n
ν
and neutral channels of both left and right movers. I = Iˆ = j + j = (U U ) . (17)
max 0 L R L R
The combinationH U followsfromthe action h i h i h i 2π −
0,c,R R R
− Q
Eq. (4), and we set H0,c =H0,c,L+H0,c,R. in units where ~ =e =1. Throughout this work Imax is
Eq. (10) is of the form of a grand canonical Hamil- calledtheequilibriumcurrentwhichreferstothecurrent
tonian K =H µNˆ, with a generalization of the running through the system in the absence of tunnelling
0 0
number operator−Nˆ. AlthoughQthe edges are held at dif- between edges. We define V = U U as the source-
L R
−
ferent chemical potentials we still refer to this system drain voltage.
and the corresponding Hamiltonian as the equilibrium A tunnelling point contact induces backscattering of
system. When we include the point contact we refer to charge and this modifies the Hall relation (17). Con-
the system as out of equilibrium. cretely a so-calledbackscatteringcurrentflows alongthe
5
point contact from one edge to the other. On the basis The exponential and σ operator correspond to the
of charge conservation we expect that this modifies the charged and neutral channel, respectively. The neutral
Hall relation to channel itself is also chiral (i.e. we have a left- and
right moving version σ ), but we will not write this
R/L
I =Imax IB (18) explicitly. The charged operator is normal ordered and
−h i
we assume the operator is properly normalized, see e.g
In this work we will prove this relation on the operator Ref.67. Bothoperatorsarecharacterizedbytheirconfor-
level and we study its effect on the noise in the edge maldimension71h andh . Inparticular,forthecharged
n c
current. part we have h = Q2.
For later purposes we therefore require the au- c 2ν
For each quasiparticle operator we also have a conju-
tocorrelator of the current which determines the
gate operator which has opposite charge and equal con-
equilibrium noise of the edge current. We set formal dimension70,71
∆j (x,t)=j (x,t) j (x,t) . Since ∂ ϕ =0 we
R R R x R
have ∆jR(x,t)=−−vch√2πν∂xϕRi. This hgivesifor the ψR(x,t)∝σ¯R(x,t)eiηR√QνϕR(x,t) . (22)
autocorrelator67
Here σ¯ is the unique operator in the conformalfield the-
S (t)= ∆j (0,t)∆j (0,0) ory which fuses to the identity with σ
jR h R R i
ν (πkBT)2 σ σ¯ =1+... . (23)
= (19) ×
(2π)2sin πk T(δ+it) 2
B If the right hand side contains multiple fusion products,
(cid:0) (cid:1) then the quasiparticle is non-Abelian. In some cases,
withδ aUVregulator. ThecorrespondingFouriertrans-
form is22 such as the Moore-Read state7 the neutral part of the
quasiparticle operator is self-dual meaning σ =σ¯.
ThequasiparticleoperatorcarriesachargeQmeasured
S (ω)=ωN(ω)G . (20)
jR in units of e = 1. This follows from the commutation
where N(ω) = coth( ω )+ 1 and G = ν is half of relation with the electric charge operator
2kBT 4π
the total conductivity of the system (the other half is
attributed to the left moving edge). [QR,ψR†(x,t)]=QψR†(x,t) . (24)
Finally, there is also the commutation relation between
the edge current and the quasiparticle operator at non-
C. Neutral channel and quasiparticles
equal times. Using Eq. (7) we obtain
The neutralchanneldescribesedge degreesoffreedom [jR(x,t),ψR†(0,0)]=ηRvcQψR†(0,0)δ(x−ηRvct) . (25)
whichdonotcoupletothe externalvoltagebias. Similar
toourpreviouswork62wedonotspecifytheexactnature
of the neutralpart, and only demand that the decompo- D. Generalization to multiple charged channels
sition(3)holds. Inthecaseofnon-Abelianstatesitisthe
neutral channel which is responsible for the non-Abelian The single chiral boson model is only sufficient to ex-
nature of the quasiparticle. plaintheLaughlinseriesatfillingfractionν =1/(2M+1)
In this work we are interested in the properties of the with M a positive integer. This construction can be ex-
edge current operator. This operator completely decou- tendedthroughuseofneutralchannels,whichallowsfora
ples from the neutral channel. So although the neutral diverse range of filling fractions. An alternative method
channel plays an import role in specifying the topologi- is to consider multiple copies of chiral bosons, each of
cal order of the system, it does not explicitly enter the which couples to the electromagnetic field. Both con-
remaining analysis of this work. structions are needed to account for the wide variety of
With that in mind we now give a short overview of observed filling fractions.
how the neutral channel enters the description of the We follow here the treatment of Ref. 66 and Ref. 14.
quasiparticles. The neutral channel is described in the We assume the bosons are decoupled from each other.
longwavelengthlimitbysomechiralconformalfieldthe- The action of the right moving edge is given by
orywhichcomesequippedwith aconsistentsetoffusion
rules70 andsomeHamiltonianHn. ThisHamiltonianen- SR = 1 ηi∂tϕi∂xϕi vi(∂xϕi)2 dtdx
ters the definition of the grand canonical Hamiltonian 4π i ZΣR − −
K , see Eq. (10). In addition there is some characteris- X (cid:2) (cid:3)
0 1
tic neutral edge velocity vn, and in general vn 6= vc. A + 4πUR κi ∂xϕi dtdx . (26)
general quasiparticle operator is of the form i Z
X
Each chiral boson ϕ has its own edge velocity v , a chi-
ψR†(x,t)∝σR(x,t) e−iηR√QνϕR(x,t) (21) rality ηi and a coupiling parameter κi > 0. Theiindex i
6
refers to the i’th chiral boson of the right-moving edge. This restriction is in fact a consequence of anomaly
The left moving edge consists of a similar set of bosons, cancellation63, so we assume that it holds. Unlike the
but with opposite chiralities i.e. ηL = ηR, etc. We will single-channel case the conductivity does not uniquely
i − i
alwaysworkwith the rightmovingcurrentunless explic- specifythecouplingsκ (recallthatinsinglechannelcase
i
itly statedotherwise. Itispossibletohaveκ =0,which we simply have κ =√ν). To fully specify the topolog-
i 1
corresponds to a chiral boson which does not couple to ical order we also need to define the electron operators
the electromagneticfield. Suchabosonalreadyfalls into of the theory,which in turn determines the quasiparticle
the category of neutral channels, so we assume κ >0. content. Werefertotheliteratureforfurtherdiscussions
i
It is possible to formulate the edge theory in terms on this classification scheme.
of coupled chiral bosons, which is usually done through A generic quasiparticle operator is of the form
use ofa K-matrix3,15. Starting fromthis formulationwe
can always switch to a different basis of fields through a ψR†(x,t)∝σR(x,t)e−iPiηiqiϕi(x,t) (34)
linear transformation, which results in an action of the
which is defined by the q ’s. The electric charge Q of
formEq.(26). Thereforethere is no lossofgeneralityby i
the quasiparticle is determined using the commutation
assuming decoupled chiral bosons.
relation with the charge operator
For each boson we have the equation of motion
1
(−ηi∂t−vc∂x)ϕi =−κiUR . (27) QψR† =[QR,ψR†]= 2π κi [∂xϕi(x),ψR†] dx . (35)
i Z
X
Since the channels are decoupled we can apply the same
It follows that the charge is given by
argument as before to obtain the edge current operator
foreachchannelseparately. Thechargedensity,itscorre-
Q= κ q . (36)
sponding conserved charge and the edge current density i i
i
operator of the i’th channel are X
In addition the conformaldimension for the i’th channel
κi κi is h = qi2 and so the total conformal dimension equals
ρi = ∂xϕi , i = ∂xϕi dx , (28) i 2
2π Q 2π h=h +h with
ZΣR n c
κ κ2
j = i∂ ϕ =η v ρ η i U . (29) q2
i −2π t i i i i− i2π R hc = i . (37)
2
i
Likewise, the commutation relations also decouple X
Finally, the non-equal time commutation relations be-
[∂ ϕ (x,t),ϕ (0,0)]=iη 2πδ(x η v t)δ . (30) tween the current and the quasiparticle is given by
x i j i i i ij
−
The totalcharge density, electric chargeand edge cur- [jR(x,t),ψR†(y,t′)]=
rentoftherightmovingedgeisthesumoftheseoperators
ηiviκiqiδ(x−y−ηivi(t−t′)) ψR†(y,t′) . (38)
ρR = ρi , R = i , jR = ji . (31) (cid:16)Xi (cid:17)
Q Q
Xi Xi Xi The generic form of the quasiparticle operator (34) in-
volves all the channels of the edge theory, although this
A similar definition applies to the left moving edge.
mixing does not always occur.
The total current operator is again the sum j (x)+
R An example of a state which is described by multiple
j (x), Eq. (15). To obtain the current-voltage relation
L charged chiral bosons is the Moore-Read trial state7,8,72
(17) we assume that each channel is in chemical equilib-
of the ν = 5 plateau73,74. Here we deal with a half-filled
rium,meaningthedensitymatrixisoftheforme−βK0/Z 2
LandaulevelontopoftwofullyfilledLandaulevels. The
andthechargedensityofeachchannelvanishes ρ =0.
h ii edge theory consists of two chiral bosons with couplings
The expectation value of the right-moving edge current
κ =κ =1, a third chiral boson with κ = 1 and a
is 1 2 3 √2
neutralchanneldescribedbythechiralIsingmodel. This
hjR(x,t)i=−21πUR ηiκ2i (32) ccoormrepslpetoenlydsdteocoaucpolendduacntdivihtayvoeftνhe=sa52m. Aelclhcihraalnitnye.lsTahree
i
X quasiparticleoperatorsdonotmixdifferentchiralbosons,
and similarly for the left-moving edge current. For a so for each quasiparticle the sum appearing in Eq. (34)
right moving edge we require η κ2 >0, while for a consists of only one term.
i i i
left moving edge it is negative. The usual conductivity Asecondexampleisahierarchialtrialstate75,76 ofthe
relation Eq. (17) is obtained p(cid:0)rPovided w(cid:1)e have ν = 2 plateau. The trial state is formed through con-
5
densation of quasiparticles in the ν = 1 state. The cor-
3
η κ2 =ν . (33) responding edge14 consists of two (co-propagating) chi-
i i
Xi ral bosons with couplings κ1 = √13 and κ2 = √115, which
7
jL,bg asissketchedinFigure3. Thetunnellingoperatoristhe
jL,tot operator which tunnels a quasiparticle from the lower to
jL o the upper edge and it is defined as
V =ψL†(x=0)ψR(x=0) . (39)
j
R Here ψ is the quasiparticle operator defined in Eq. (21)
j
R,tot and Eq. (34). The quasiparticle is characterized by its
j
R,bg o quasiparticlechargeQandconformaldimensionsh and
n
h . In the multichannel case we assume the couplings
c
Figure 3. The point contact induces tunnelling between the
κ and individual charges q are known. The tunnelling
i i
two edges. Tunnelling occurs between the inner channels of
Hamiltonian is the tunnelling operator together with a
the edges. We decompose the total edge current (j )
R/L,tot tunnelling coupling constant
into channels which are partially reflected (j ), and which
R/L
are fully transmitted (j ).
R/L,bg HT =Γ +Γ∗ † . (40)
V V
It is treated as a perturbation to the grand canonical
brings the conductivity to ν = 2. A simplified descrip- Hamiltonian K , Eq. (10).
5 0
tionassumesthedistancebetweenthetwochargedchan- Inthepresenceofavoltagebiasanetcurrentofquasi-
nelsislargeandthechiralbosonscanbetreatedascom- particles tunnels from one edge to the other, resulting
pletely decoupled. Each quasiparticle operator is then in a tunnelling current. This is called the tunnelling or
associated with strictly one chiral boson. backscattering current I . It is defined as the rate of
B
Inpractice the distance between the channels is small, changeofthechargedifferencebetweentheedges. Using
theCoulombinteractionneedstobetakenintoaccount14 Heisenberg’s equation of motion we obtain
and the channels no longer decouple (although the cur-
rents still commute). In this case it is possible to diago- Iˆ e d( )= ie[ ,K +H ] .
B L R L R 0 T
nalize the interaction term through a linear transforma- ≡ 2dt Q −Q − 2 Q −Q
tion of the fields. The new fields are, again, completely (41)
decoupled. In this new basis the quasiparticle and elec-
Charge is conserved in the equilibrium system, so
tron operators are constructed from multiple fields, and
[K , ]=0. The tunnelling operators are defined in
in particular the sum appearing in (34) contains both 0 QR/L
terms of quasiparticle operators with charge Q, and so
chiral bosons of the new basis.
[ , ]= [ , ]=Q . This also applies to the case
Wefinalizethisdiscussionbynotingthatitiscurrently QL V −QR V V
of multiple chargedchannels. We have
not completely clear if the case of counter propagating
charge modes arises in the quantum Hall effect, as they Iˆ = iQe Γ Γ . (42)
B ∗ †
have never been experimentally verified. One explana- − V − V
tion for this is that counter propagating modes are un- (cid:0) (cid:1)
stableinthepresenceofdisorder. InRef.77itwasfound B. Background current and multichannel case
that for the ν =2/3 state disorder induces tunnelling of
chargebetweenthe counterpropagatingmodes. This re-
The point contact induces tunnelling of quasiparticles
sults in a different effective edge theory that consists of
between the innermost channels of the left- and right
asinglechargedmodeandacounterpropagatingneutral
moving edge. In particular it does not always involve
mode. In this work we do not consider such dynamical
all edge channels. An example is the Moore-Read state
effects which alter the edge theory away from the point
for the ν = 5/2 plateau. In this case the outer chan-
tot
contact. We simply assume the different channels com-
nelscorrespondtothefullyfilledLandaulevelswhichare
pletely decouple, and allow for the possibility of counter
fullytransmitted. Tunnellingoccursonlybetweenthein-
propagatingmodes. Arecentexperiment78 suggeststhat
ner channels described by the chiral Ising model times a
counter propagatingneutralmodes are in factpresentin
chiral boson.
multiple states, including the ν =5/2 state.
We therefore decompose the edge current into two
pieces: the channels which are fully transmitted and
not involved in the tunnelling process (called the back-
IV. TUNNELLING POINT CONTACT
ground current), and the channels which are partially
reflected (called the reflected current). This decomposi-
A. Tunnelling Hamiltonian and tunnelling current tion is sketched in Figure 3. The corresponding current
operators are denoted j for the background current
R,bg
We consider a quantum Hall bar at filling fraction ν and jR for the reflected current. The total edge current
withtwodisconnectededges19,20,79. Apointcontactacts operator is written as
as a restriction forcing opposite edges together thereby
j =j +j (43)
inducing tunnelling of quasiparticles between the edges R,tot R R,bg
8
Inadditionthereflectedcurrentandbackgroundcurrent andS (t,t)=1. FollowingRef.80wefactorizethe time
K
can also consist of multiple channels. Note also that the evolution operator as (t,t )=e iK0(t t0) (t,t ).
K 0 − − 0
S U
conductivity splits accordingly FromEq.(49)itfollowsthattheunitaryoperator (t,t )
0
U
satisfies the equation of motion
ν =ν+ν . (44)
tot bg
i∂ (t,t )=H (t) (t,t ) (50)
The decomposition (43) is reflected in the definition of tU 0 T U 0
the tunnelling Hamiltonian and the tunnelling current, HT(t) eiK0tHTe−iK0t . (51)
≡
Eq. (40)and (41). The perturbation H commutes with
T
the current operatorsof the channels not involvedin the Here HT(t) is in an interaction-like picture with its time
tunnelling, i.e. [ R,bg,HT]=0 and so evolution dictated by the unperturbed Hamiltonian K0.
Q The time evolution operator is also known as the S-
U
[ R,tot,HT]=[ R,HT] . (45) matrix operator and it is given by Dyson’s series
Q Q
Wecanthereforetreatthebackgroundcurrentasanequi- t
librium system unaffected by the perturbation. (t,t0)= exp i HT(t′) dt′
U T −
Zt0
∞(cid:0) ( i)n n t (cid:1) n
=1+ − dt H (t ) (52)
V. NON-EQUILIBRIUM FORMALISM i T j
n! T
nX=1 hiY=1Zt0 i jY=1
A. Formalism
Here is the time-ordering operator and the exponen-
T
tiated form is an abbreviation for the corresponding ex-
The presence of a point contact together with an ap- pansion. Similarly, we set for an operator
pliedvoltagebiascouples the upper andloweredges and O
forcesthesystemoutofequilibriumandwerequireafor- (t)=eiK0t e iK0t . (53)
OK0 O −
malismthattakesthisintoaccount. Inanon-equilibrium
formalism80 the tunnelling Hamiltonian is treated as a By using the factorization of the unitary time evolution
perturbationofthegrandcanonicalHamiltonianK . Ini- operatorin(48)andtakingthelimitt weobtain
0 0
→−∞
tially at some time t<t the perturbationis absent and for an operator its expectation value
0
O
the systemisdescribedbyanequilibriumdensitymatrix
of the form OI(t)≡U†(t,−∞)OK0(t)U(t,−∞) (54)
w0 w(t0)=e−K0/kBT/Z . (46) hOI(t)i=Tr w0OI(t)
≡
We also denote as the expectation value with re- =Tr(cid:2)w0U†(t,−(cid:3) ∞)OK0(t)U(t,−∞) . (55)
h···i
spect to w0, Here I(t) is sti(cid:2)ll the Heisenberg representatio(cid:3)n (t),
K
O O
but with the time evolution operators factorized. The
Tr w . (47)
h·i≡ 0··· superscript I denotes that the tunnelling Hamiltonian is
This density matrix furthe(cid:2)r facto(cid:3)rizes as a product of switched on and the operator is taken in the Heisenberg
density matrices – one for each channel of the system. representation. TheeffectoftheperturbationHT iscom-
Atsometimet theperturbationisswitchedonadiabat- pletely captured by the time evolution operator . All
0 U
ically and the systemis slowlydriven awayfrom equilib- correlatorsare evaluated with respect to the equilibrium
rium. Eventually,aftertheperturbationisfullyswitched density matrix w0.
on(t t ) the systemis describedby a steady state. In As an example the expectation value of the tunnelling
0
≫
our approach we make use of the fact that (1) the ini- current is given by
tialstateisanequilibriumstateand(2)theunitarytime
evolution of the system is completely described by the IB =hIˆBI(t)i (56)
(known) perturbed Hamiltonian K =K0+HT. IˆI(t)= (t, )Iˆ (t) (t, ) . (57)
Concretely, when the system reaches a steady state B U† −∞ B U −∞
the expectation value of an operator is given by
Ifwewanttoexplicitlydeterminethiscorrelatorweneed
O
(t) =Tr w (t) where (t) is the Heisenberg
0 K K to resort to perturbation theory.
hO i O O
representation of the operator with respect to the
(cid:2) (cid:3) O
grand canonical Hamiltonian K,
B. A non-equilibrium Kubo formula
OK(t)=SK† (t,t0)OK(t0)SK(t,t0) . (48)
The unitary time evolution operator S (t,t ) solves the In the formalism presented here the effect of the tun-
K 0
Schr¨odinger equation, nelling perturbation is fully captured by the time evolu-
tionoperator (t,t ). Inlinearresponse theorythe time
0
i∂ (t,t )=K (t,t ) (49) U
t K 0 K 0 evolution operator Eq. (52) is expanded to lowest order
S S
9
in the tunnelling coupling constant, which leads to the Here jI and jI are the edge currents in the interaction
R L
Kubo formula, picture, Eq. (54). We focus initially on an edge with a
singlechargedchannelandcommentonthemultichannel
t
case at the end of the section.
OI(t)∼=OK0(t)−i [OK0(t),HT(t′)] dt′+... . We now apply the non-equilibrium Kubo formula
Z−∞ (58) Eq. (59). For this we need the commutator of the edge
currentandthetunnellingHamiltonian. Weusethecom-
The dots represent higher order contributions. We mutation relations of the edge current with the quasi-
present here an extension of the Kubo formula, which particle operator, Eq. (25), and the expression of the
includes the higher order contributions. This non- tunnelling Hamiltonian in terms of the quasiparticles,
equilibrium Kubo formula is given by81 HT =ΓψL†ψR+c.c.. This gives
OI(t)=OK0(t) [jR(x,t),HT(t′)]=−iηRvcIˆB(t′)δ(x−ηRvc(t−t′))
t [jL(x,t),HT(t′)]=iηLvcIˆB(t′)δ(x ηLvc(t t′)) (63)
i (t, )[ (t),H (t)] (t, ) dt (59) − −
− U† ′ −∞ OK0 T ′ U ′ −∞ ′ with η = +1 and η = 1. Plugging this into (59) for
Z−∞ jI aRnd performingLthe−integration over t results in
We emphasize that this expression is an operator iden- R/L ′
tity. Ref.81 obtains this formula for class of operators jI(x,t)=j (x,t) θ(x)IˆI(t x/v ) . (64)
which commute with the equilibrium Hamiltonian K . R R − B − c
0
The second term is the difference of the operator in a jLI(x,t)=jL(x,t)−θ(−x)IˆBI(t+x/vc) (65)
system in equilibrium and a system out of equilibrium,
Here θ(x) is the unit step function, and jI(x,t) and IˆI
B
δ I(t) I(t) (t) (60) are the edge currentand the tunnelling current operator
O ≡O −OK0 in the interaction representation, see Eq. (57).
t
= i (t, )[ (t),H (t)] (t, ) dt This operator has an intuitive meaning. It is a reflec-
− U† ′ −∞ OK0 T ′ U ′ −∞ ′ tionofboth chargeconservationandthe chiralstructure
Z−∞
of the edge current. Consider Eq. (64) for the rightmov-
This equationseparatesthe effectofthe perturbationon
ing current. For the region x < 0 the operator reduces
tthhee soypsetreamtoirsOforwcehdenouthteofpeeqrtuuilribbartiuiomn.is turned on and to jRI(x,t) = jR(x,t), meaning the current operator in
this region is not affected by the presence of the tun-
A proof of this relation is presented in appendix A.
nelling point contact. This is as expected, since the re-
In this proof we apply the expansion of the time evolu-
gion x < 0 is “upstream” of the point contact. For the
tion operator Eq. (52) to the operator in the interaction
regionx>0 the backscatteringcurrentI at a retarded
B
representation Eq. (54). Through some combinatorial
time (t x/v ) is subtracted. The backscattering cur-
c
manipulations of these expansions we recover the non- −
rentisthechargetransferredfromthelowertotheupper
equilibrium Kubo formula (59).
edgeandisthereforesubtractedfromthecurrentpastthe
point contact (it is also subtracted from the left moving
current because of the direction of total current). The
VI. EDGE CURRENT OPERATOR IN THE
identity resembles Kirchoff’s law as charge is conserved
NON-EQUILIBRIUM FORMALISM
along the point contact.
ThefactthatwesubtracttheoperatorIˆI fromj ata
Intheabsenceofthepointcontact,thecurrentthrough B R
retardedtimet x/v isamanifestationofthechiraland
c
the system is given by the usual quantum Hall relation −
causal structure. Chirality and translational symmetry
I = ν (U U ). In the presence of a point contact
max 2π L− R enforces all observables to be functions of the combina-
this Hall relation no longer holds. The point contact
tion t x/v . A similar argumentis used in Ref. 60 as a
c
induces a tunnelling current I , which is effectively a −
B derivationoftheedgecurrentoperatorforthesystemout
form of backscattering, since the edge currents of the
of equilibrium. The chiral structure takes into account
system are chiral. On the basis of charge conservation
the position of the point contact (at x = 0, hence the
R
we expect the current in the presence of a point contact
step function), the chirality of the edge (right-moving)
to be
and the finite velocity of the charged channel.
The totalcurrentoperatorin the interacting regime is
I =I I . (61)
0 max− B now
We now show that this relation is also satisfied at the IˆI(x,t)=j (x,t)+j (x,t) IˆI t x/v . (66)
level of the operators. For this we make use of the non- 0 R L − B −| | c
equilibriumKuboformula. Recallthatintheinteraction This indeed reproduces the current r(cid:0)elation Eq(cid:1). (61)
representation the total current operator is
I = j (x,t)+j (x,t) IˆI t x/v
0 h R L i−h B −| | c i
IˆI(x,t)=jI(x,t)+jI(y,t) . (62) =I I . (67)
0 R L max− B (cid:0) (cid:1)
10
A similar relation applies to the charge density opera- jI jI
L L
tors. When we apply the non-equilibrium Kubo formula
to these operators we find
1
ρI(x,t)=ρ (x,t) IˆI(t x/v )θ(x)
R R − v B − c
c
1
ρIL(x,t)=ρL(x,t)+ vcIˆBI(t+x/vc)θ(−x) (68) jRI jRI
NotethatthesignofI intheequationsaremerelyacon-
B Figure 4. The point contact viewed as a scattering source
sequenceofourconventions(directionofthe currentand
at x = x = 0 with the edges depicted as incoming and
R L
backscatteringcurrent,chargeofthetunnellingquasipar- outgoing edge currents. The arrows denote the direction of
ticle, etc.) thelocal electric current. The edge currents are taken in the
Let us remark on the more general case of multiple interaction picture.
charged channels. First note that the inclusion of back-
ground currents (see Section IVB) does not modify the
relation, since the background currents commute with tunnelling Hamiltonian on the edge current. In this sec-
the tunnelling Hamiltonian. This is intuitively clear, tion we analyse the noise in the edge currentin the non-
since the background currents are fully transmitted. equilibrium system. Using the identity Eq. (66) we can
In the more general case the additional charged chan- relate the noise in the edge current out of equilibrium
nels do not commute with the tunnelling Hamiltonian. to the noise in the tunnelling current. This results in
The total edge current is a sum of the background cur- a non-equilibrium fluctuation-dissipation theorem18 and
rents plus the reflected edge currents an expression for the excess noise in the edge current.
Put differently, we are studying the effects of the non-
jR,tot =jR+jR,bg , jR = ji . (69) equilibrium Kubo formula on autocorrelators and their
i Fourier transform.
X
Let us first recall some definitions22,82,83. Given an
Each channel is characterised by its own edge velocity
operator we set ∆ (t) = ˆ(t) ˆ and define the
vi and chirality ηi. The commutator of the edge current O O O −hOi
autocorrelatorS (t) as
operator with the tunnelling Hamiltonian becomes
O
S (t)= ∆ (t)∆ (0)
[jR,tot(x,t),HT(t′)]=[jR(x,t),HT(t′)]= O h O O i
κ q = (t) (0) 2 . (73)
iIˆ (t) i iη v δ(x η v (t t)) . (70) hO O i−hOi
B ′ i i i c ′
− Q − −
Xi The non-symmetrizednoise is the corresponding Fourier
and for completeness we also note the left moving edge transform
(with chiralities ηL)
i
S (ω)= eiωtS (t) dt . (74)
O O
[jL,tot(x,t),HT(t′)]=[jL(x,t),HT(t′)]= Z
κ q
iIˆ (t) i iηLv δ(x+ηLv (t t)) . (71) The symmetric and antisymmetric combinations of the
B ′ Q i i i c − ′ noise are denoted by
i
X
The charge of the quasiparticle in this case is given by 1
C (ω)= (S (ω)+S ( ω)) (75)
Q = iκiqi. The edge current operator in the interac- O 2 O O −
tion picture is given by 1
P R (ω)= (S (ω) S ( ω)) . (76)
O 2 O − O −
jI (x,t)=
R,tot
κ q The same notation is used in Ref. 18.
j (x,t) i i η θ(η x)IˆI(t η x/v ) . (72)
R,tot − Q i i B − i i
i
X(cid:0) (cid:1)
Thesummationreflectsthechiralstructureofeachchan- A. Noise in the outgoing edge current
nel separately and the current relation Eq. (61) is again
obtained. InthespiritofRef.20wethinkofthepointcontactas
a scattering source with the edges as two incoming and
two outgoing branches, see Figure 4. We focus on the
VII. NON-EQUILIBRIUM NOISE noise in the outgoing branch of the right-moving edge,
which corresponds to the noise in jI (x,t) for x > 0.
R,tot
The main result of the previous section is the oper- We first consider the case of a single reflected charged
ator identity Eq. (66) which captures the effect of the channel plus any number of background currents which