Non-equilibrium DC noise in a Luttinger liquid with impurity

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by  P. Fendley

Non-equilibrium DC noise in a Luttinger liquid with impurity
P. Fendley and H. Saleur
Department of Physics, University of Southern California, Los Angeles CA 90089-0484
(January 1996, cond-mat/9601117)
We compute exactly the non-equilibrium DC noise in a Luttinger liquid with an impurity and
an applied voltage. By generalizing Landauer transport theory for Fermi liquids to interacting,
integrable systems, we relate this noise to the density fluctuations of quasiparticles. We then show
howtocomputethesefluctuationsusingtheBetheansatz. Thenon-trivialdensitycorrelationsfrom
the interactions result in a substantial part of the non-equilibrium noise. The final result for the
noise is a scaling function of thevoltage, temperature and impuritycoupling. It may eventually be
observable in tunnelingbetween edges of a fractional quantumHall effect device.
6
9
9
1
I. INTRODUCTION approach has it been possible to compute exactly the

n non-equilibrium noise in the presence of interactions.
a
J Thestudyoffluctuationsinsystemsofnon-interacting In a series of papers with A. Ludwig, we have devel-
particles is a well-understood problem. Of particular in- oped a description of a Luttinger liquid in terms of its
4
2 terest is the noise due to free electrons scattering off an interactingquasiparticles. As in our earlierwork,the re-
impurity, which has been discussed carefully in1–3. In sults rely crucially on the fact that the Luttinger model
1 these calculations, the only interactions are those by in- withimpurityisintegrable10,11. Thisformalismmakesit
v dividual particles with a barrier. The fact that the par- possible to do exactnon-perturbative calculations in the
7 ticles donotinteractwitheachotheris ofcoursecrucial. presenceof the impurity. We have computedthe current
1 When interactions are present and the system is out of and conductance through the impurity11,12, and the DC
1 equilibrium, it is not possible in general to go beyond noise at zero temperature13. At zero temperature, the
1
0 a perturbative analysis of the fluctuations. In particu- fluctuations are pure shot noise coming from the impu-
6 lar, the issue of possible singularities in the noise is very rity, so finding them does not require understanding the
9 difficult to resolve4. non-trivialquantumfluctuationsoftheinteractingquasi-
/ In this paper, we present an exact calculation of the particles. However,atnon-zerotemperature,thethermal
t
a non-equilibrium noise in a one-dimensional (interacting) noiseandthe shotnoisecannotbe separated,sothe full
m Luttinger liquid with a single impurity. This problem noise requires understanding these fluctuations. The ef-
- has attracted a great deal of attention because the edge fectofthesefluctuationsissubstantial;withoutincluding
d
of a fractional quantum Hall device should be a Lut- them one does not evenobtain the equilibrium Johnson-
n
o tinger liquid5–7. A point contact pinching the sample Nyquist formula when the impurity is not present. In
c plays the role of the impurity, with the gate voltage al- this paper, we calculate exactly the quasiparticle fluctu-
: lowing the impurity coupling to be varied. The noise ations, and use them to find the exact DC noise at any
v
i here is especially important because DC transport mea- temperature, voltage and impurity coupling.
X surements are not sensitive to the charge of the carriers, In sect. 2, we show how to compute density fluctu-
r while non-equilibriumnoise measurementsdo depend on ations of the quasiparticles in an integrable theory, and
a thischarge8. Weobtainthegeneralcurrent(andvoltage) applytheresultstofindthecurrentnoiseintheLuttinger
noise as a function of the applied voltage, temperature model without an impurity. In sect. 3, we couple an im-
and impurity coupling. We check in particular that the purity to the Luttinger liquid, and find the exactscaling
current noise corresponds to tunneling of uncorrelated curve. This involves a reformulation of Landauer’s ap-
electronsintheweak-backscatteringlimit,andtotunnel- proach suitable for interacting quasiparticles. In sect.
ing of uncorrelated Laughlin quasiparticles of fractional 4, we consider various limits, including the weak and
charge in the strong-backscatteringlimit. strong backscattering limit, and also derive some gen-
Inacriticaltheory,conformalinvarianceusuallyyields eralized fluctuation-dissipation results.
the correlators, from which the fluctuations follow. The
situationchangesdramaticallywhenanimpurityisintro-
II. DENSITY AND CURRENT FLUCTUATIONS
duced. This adds interactionsto the model and destroys
IN AN INTEGRABLE THEORY
conformal invariance. One can use the Keldysh formal-
ism to relate various types of fluctuations to each other,
andtoobtainresultstolowestorderinperturbationthe- In this section we first review known results for the
ory in the interaction strength8. Perturbed conformal quasiparticledensitiesinanintegrabletheory. Weextend
field theory yields other information9,4 (in particular it these calculations to compute the density fluctuations.
predicts a singularity at finite frequency), but in neither We then specialize to the case of interest, the Luttinger
1
liquid (in this section, without an impurity). For this to the Boltzmann weight of a configuration. This gener-
model, we find the current and charge fluctuations. alizationmayseemabitodd,butinanintegrabletheory
withdiagonalscattering,collisionschangeneithertheen-
ergy nor type of a particle. Therefore P (θ) is indepen-
i
A. General considerations dent oftime, evenwhen a currentis flowing. In the next
section we discuss how the densities change with time,
Weconsideranintegrabletheorywhosespaceofstates butintheveryspecialsituationwherethechangeresults
isdescribedentirelyintermsofquasiparticles. Inaninte- from a single impurity.
grabletheory,thescatteringiscompletelyelasticandfac- To determine the equilibrium values of the densities,
torizable, which means that the energy and momentum wefindtheconfigurationwhichminimizesthefreeenergy.
ofthequasiparticlesareindividually conserved,andthat This procedure is known as the thermodynamic Bethe
the multi-particle S matrix decomposes into a product ansatz16, and is discussed in the appendix. We define
of two-particle elements. For simplicity we require that the “pseudoenergies” ǫi by
the scattering is diagonal as well, which means that the
P 1
typesoftheparticlesdonotchangeinthecollision. This f = i , (3)
doesnotmeanthescatteringistrivial,becausetherecan i ni ≡ 1+eǫi−µi
be a phase shift; the S matrix element Sbulk(p ,p ) for
ij 1 2 whereǫ andthe chemicalpotentialsµ arescaledbythe
scatteringaparticleoftypeiwithoneoftypej isanon- i i
temperature to make them dimensionless. The result is
trivial function of the momenta. It will be convenient
the thermodynamic Bethe ansatz equations
to parametrize the momenta in terms of the rapidity θ
defined for gapless theories as p expθ. Then scale in-
variance requires that S matrix e∝lement is a function of ǫj(θ)= dθ′ Φjk(θ θ′)ln 1+e−ǫk(θ′)+µk(θ′) ,
− −
p1/p2, or equivalently, θ1 θ2. (In a massive theory one Xk Z h i
−
has p sinhθ and relativistic invariance requires also e (θ)
j
∝ + (4)
that S is a function of θ1 θ2; the results of the section T
−
apply to this case as well.)
The bar indicates thermal equilibrium values. The equi-
We study the system on a circle of large length L and
librium value F of the free energy per unit length is
at temperature T. Each quasiparticle has the energy e ;
i
in a gapless theory e = p expθ. We introduce
i i
the level densities n (θ) an±d th∝e filling fractions f (θ), F = T dθ e (θ)ln 1+e−ǫi(θ)+µi(θ) . (5)
i i i
−
defined so that Lni(θ)dθ is the number of allowed states Xi Z h i
foraquasiparticleoftypeiintherapidityrangebetween
The relations (4) and (5) still hold when the µ de-
θ and θ+dθ and f (θ) is the fraction of these which are i
i
occupied. Thus the density of occupied states per unit pends on θ. This makes it possible to compute correla-
tors by taking functional derivatives of the free energy
length P (θ) is given by
i
with respect to µ (θ), and then setting µ to the appro-
i i
Pi(θ)=ni(θ)fi(θ). priate value. This is because the partition function can
be written as the functional integral
Requiring that the wave functions be periodic in space
gives rise to the Bethe equations (setting h =1, not the
usual h¯ =1) Z = Pi(θ) e−F0L/T+L i dθµi(θ)Pi(θ),
" D #
n (θ)= dpi + dθ′ Φ (θ θ′)P (θ′), (1) Z Yi P R
i ij j
dθ − where the explicit form of F is given in the appendix
j Z 0
X (here,the exactformofthe integrationmeasureissome-
where the kernel is defined by what unclear, but this will not affect the following anal-
1 ∂ ysis). The equilibrium value FL/T is the saddle-point
Φij(θ)= 2iπ∂θ lnSibjulk(θ), valueoftheexponent. Taking−derivativeswithrespectto
µ (θ) brings down powers of P (θ) and gives expectation
i i
and all rapidity integrals in this paper run from to
values. For example, the equilibrium particle densities
−∞
. These Bethe equations hold for any allowed configu-
are given by
∞
ration,whetheritistheequilibriumconfigurationornot.
Notice that in a free theory the kernel vanishes and we 1 δZ 1 δF
obtain the free-particle relation ni(θ)=dpi/dθ. Pi(θ)= ZLδµ (θ) =−T δµ (θ). (6)
i i
We generalize the usual analysis by allowing for a
chemicalpotentialwhichdependsonθ,thuscontributing Ofcoursedoingthisfunctionalderivationisnotnecessary
for determining P ; a direct way is to relate it to ǫ using
i
exp L dθµ (θ)P (θ) (2) the definition (3) and the Bethe equations (1).
i i
" #
i Z
X
2
To find the current noise we will need the density- in the next section including an impurity. We find that
density correlation function, which is given by second- even though the current operator creates multi-particle
order functional derivatives: states,onlyone-particlestatescontributetotheDCfluc-
tuations.
1 δ2Z
P (θ)P (θ′)= The bulk Hamiltonian of the Luttinger model can be
i j ZL2δµi(θ)δµj(θ′) written in terms of the charge currents15
It is convenient to define the difference of an quantity π l
ahnavdeits equilibrium value, e.g. ∆Pi ≡ Pi−Pi. We then H0 = g Z0 dx jL2 +jR2 . (10)
(cid:2) (cid:3)
The overall constant g follows from the normalization
1 δ2F
∆P (θ)∆P (θ′)= (7) of the current; in the fractional quantum Hall problem
i j −LT δµi(θ)δµj(θ′) this is fixed by the charge of the electron and we have
g =ν when 1/ν is an odd integer5. Since in this system
In general, this equilibrium value is not diagonal: it can
the left and right movers are decoupled, we here treat
be non-zero even when θ = θ′ and i = j. The diag-
6 6 only the right movers. We study the model at non-zero
onal correlations come from the Fermi repulsion of the
temperatureT,whichcorrespondstoimaginarytimeson
particles, while the non-diagonalpart arises from the in-
a circle of circumference 1/T.
teractions. However, we show in the appendix that in
Without an impurity, the charge and current fluctua-
integrable systems there is at least one kind of diagonal
tions in a Luttinger liquid (or any other conformal field
fluctuation, even if interacting:
theory) can be found very straightforwardly from con-
formal field theory.9 The left- and right-moving current
∆f (θ)∆f (θ′) δ δ(θ θ′)
i j ∝ ij − operators jL(x+t) and jR(x t) are individually con-
−
served. To find the noise, we need to integrate the two-
It would be interesting to find a deep reason for this
pointfunctionofthecurrents. Conformalinvariancefixes
relation.
it to be
There is a simple alternative formulation of (7) in
terms of the ǫi. We discuss this in the appendix . The π2T2g
result is written in terms of a function Kij defined by hjR(x,t)jR(0)i= [sinh(2π2T(x t))]2 (11)
−
Kij(θ,θ′) Φij(θ θ′) (1+eǫi(θ)−µi(θ))δijδ(θ θ′). and likewise for the left movers (The strange normaliza-
≡ − − −
tion in (11) arises because of our choice h = 1 instead
(8)
of the more common ¯h = 1. Also, we are ignoring sub-
Then we find that tleties coming from the chiral anomaly which vanish in
theinfinite-volumelimit.14)TheDCcurrentfluctuations
∆P (θ)∆P (θ′)= T ∂ + ∂ K−1(θ,θ′), (9) are then given simply by
i j L ∂θ ∂θ′ ij
(cid:18) (cid:19) 1
(∆I)2 lim dt eiωt j (x,t),j (0,0) + (L R)
where the inverse is defined via ≡ω→0 2h{ R R }i →
Z
=2gT, (12)
dθ′′ K (θ,θ′′)K−1(θ′′,θ′)=δ δ(θ θ′).
ik kj ij − where we added the contribution of left movers. This is
k Z
X
the well-known Johnson-Nykqvist formula (for spinless
Unfortunately,thisreformulation,whileratherelegant,is electrons), because the conductance G of this Luttinger
also rather useless because we cannot find a closed-form system is equal to g (see5,7 for example).
expression for K−1 explicitly. The problem in inverting We now rederive this result for the current noise from
(8)is thatK isnotafunctionofthe differenceθ θ′ but the quasiparticles. We have discussed the quasiparticles
−
ofboththevariablesindividually. Observefrom(9)that, for this system at length in11,12. The values of g where
as expected from general considerations, fluctuations of the scattering is diagonal are g = 1/t, t an integer. We
densities per unit length are of order O(1/L). thereforerestricttothesevalues,althoughweexpectthat
the final results will extend straightforwardly to general
coupling. Therearetdifferenttypesofquasiparticles;the
B. Current noise in a Luttinger liquid kink (labeled by +), the antikink ( ), and the breathers
−
(1...t 2). Since there is no scale in the theory, these
−
We now use these general results to compute the cur- right-movingparticlesaremassless,andhavethe disper-
rent noise in a particular system, the Luttinger liquid. sion relation e = p = m eθ, where m = m = m/2
i i i + −
We first find the noise using conformal field theory, and and m = msin[πj/(2(t 1))]. The overall scale m is
j
−
then rederive it using the quasiparticle picture. This arbitraryandcancelsoutofanyphysicalquantities. The
leads to a result which will be crucial to our analysis explicit formofthe kernels Φ in the Luttinger liquid is
jk
3
discussed for example in12, where we also give a simpler of µ = µ = V/(2T) and µ = 0 for j = 1...t
+ − j
− −
form of the thermodynamic Bethe ansatz equations (4). 2. Charge fluctuations can now be obtained by taking
Thegeneralfluctuationsdependonspaceandtime. To derivatives with respect to V, the rapidity-independent
computethemusingthe quasiparticles,onewouldliketo chemical potential. Using (13) with (6) and (7) gives
be able to recover expressions such as (11) based on the
action of current operators on multiparticle states. Un- ∂F
Q(V)= 2L
fortunatelythe theory,althoughitappears“almostfree” − ∂V
fromthe Bethe-ansatzpointofview,is trulyinteracting, ∂2F ∂Q
and physical observables act on the quasiparticle basis (∆Q)2 = 4LT =2T (14)
− ∂V2 ∂V
in a very complicated fashion. Typically, operators like
thecurrentortheenergydensityhavenon-vanishingma- We canfindQ(V)directly fromthe TBA.By comparing
trix elements (so-calledform factors17) between all pairs the derivative of (4) with (1), one finds that when µ is
of states with identical number of kinks minus number independent of θ
of antikinks; for instance, j (x,t) acting on the vacuum
R
can create an arbitrary number of breathers and pairs ∂ǫi(θ,V)
n (θ,V)=T . (15)
of kinks and antikinks. (Operators that are simple in i ∂θ
the quasiparticle basis, such as the density of kinks, are
Using the definitions of n and f gives
highly non-localin terms of the originalphysicalcurrent i i
j , and presumably of no physicalinterest.) This makes
R ∂ǫ (θ,V) 1
the computation of the space- and time-dependent fluc- dθ P (θ)=T dθ ±
tuations rather difficult, although not impossible.18 Z ± Z ∂θ 1+eǫ±(θ,V)−µ±
While local operators are quite complicated, their in-
The integrandis a total derivative,and from(13) we see
tegral over space can become much simpler. A typ-
that
ical example is the stress-energy tensor, whose inte-
gral gives rise to the energy and momentum opera- 1+e−ǫ−+(V/2T) ǫ(∞)
tors, both of which act diagonally on the multiparti- Q(V)=LT ln .
1+e−ǫ+−(V/2T)
cle states. Another example is the U(1) charge, Q = (cid:20) (cid:21)ǫ(−∞)
j dx. With the convention that the original physi-
R
Weseefrom(4)thatǫ( )= ,anditisstraightforward
cRal electrons have Coulomb charge 1, the breathers have to show that12 ∞ ∞
no charge, but the kink has charge q = +1 and the
antikink has charge q = 1. In our quasiparticle ba- sinh[(t 1)V/(2Tt)]
sis, Q acts diagonally, and−is a sum of one particle op- exp[ǫ±( )]= − .
−∞ sinh[V/(2Tt)]
erators: Qθ ,...θ = ( q ) θ ,...θ =
| 1 niq1...qn i | 1 niq1,...,qn
L (P P )dθ θ ,...θ . Thispropertyholds
+− − | 1 niq1,P...,qn This gives Q(V)=VL/t and using (14) gives
becausethesine-GordonHamiltonian(andthusitsmass-
(cid:0) R (cid:1)
lesslimit)andthechargeoperatorcommute,sotheycan (∆Q)2 =2LTg (16)
therefore be simultaneously diagonalized. It can also be
painstakingly checked using general formulas for matrix for any V. Using (∆I)2 = 1(∆Q)2 reproduces the de-
elements of the local current operator.17 L
sired result (12).
This makes the computation of the DC noise much
Forcompleteness,we canuse this computationto give
easierthantheACnoise. ThepointisthatDCproperties
a quick proof of the diagonal action of Q. For any mul-
are obtained by integrating over all times, and since we
tiparticle state which we denote by , characterized
deal with chiral objects (objects which are functions of |Mi
densities P , one has
i
either x t or x + t), this is the same as integrating
−
over space. Thus DC current fluctuations are related to
dxj (x,t) dxj (x,t′) = X
charge fluctuations: for instance, as immediately seen hM| R R |Mi
from (12), one has (∆I)2 = 1(∆Q)2 (we set the Fermi Z Z
velocity equal to unity). L + L2 dθdθ′(P+(θ) P−(θ))(P+(θ′) P−(θ′)).
− −
Charge fluctuations can straightforwardly be com- Z
puted using the formalism of the previous section. For Here X represents the other contributions to the two-
a given configuration characterized by densities Pi, the point function:
total charge of the right movers is
2
X = dxj (x) ′ ei(EM−EM′ )t, (17)
Q=L (P+−P−)dθ. (13) M′ (cid:12)hM|Z R |Mi(cid:12)
Z X(cid:12) (cid:12)
(cid:12) (cid:12)
TheeffectofafiniteDCvoltagetoaddtothefreeenergy and EM is th(cid:12)e energy of the mult(cid:12)iparticle state . It
|Mi
a term VQ/2. This correspondsto chemical potentials is convenient to define
−
4
D(θ,θ′) ∆(P P )(θ)∆(P P )(θ′) (18) impurity combined with the S matrix elements for tun-
+ − + −
≡ − −
neling through the impurity. The former are given in
We then have the preceding section, while the S matrix elements are
known.10 The analysis is similar to one given for free
1
(∆I)2 =L D(θ,θ′)dθdθ′+ X electrons3; this is possible because of the simple action
L
Z Z of Q on multiparticle states explained the previous sec-
1 1
= (∆Q)2+ X tion.
L L As previously discussed11,12, the transport properties
1 of a Luttinger liquid with a backscattering impurity can
=2gT + X, (19)
L be mapped to a Luttinger system of right movers alone.
The backscatteredchargeQ Q due to left-righttun-
wherethesecondequalityfollowsfromthedefinitionofQ L− R
neling in the original problem is proportional to the
in(13),thelastequalityfollowsfrom(16),andX hasthe
charge Q in the new problem. It is not conserved be-
same expression as (17) but with a sum on intermediate
cause of the impurity scattering.
states restricted to E = E′ . However, the conformal
M M To start, we discuss the one-point function of the cur-
fieldtheoryresult(12)saysthat(∆I)2 =2gT,soX =0.
rent in the presence of the impurity, quickly rederiving
Since each contribution to X is a square modulus, and earlier results.11,12 There are two types of asymptotic
therefore positive, this means that each of these contri-
states; the in states injected by the reservoir from the
butions must vanish individually. Therefore,the compli-
left, and the out states, after scattering through the im-
cated matrix elements of the local current all disappear
purity. The out state must have the same momentum
after spatial integration, and the charge operator,which
and energy because the integrability allows only elastic
is all we need to compute DC current fluctuations, acts
scattering, so a kink can scatter only into a kink or an-
diagonallyonmultiparticlestates,andisasimplesumof
tikink of the same rapidity. The S matrix element for a
oneparticle operators. This factwillprovecrucialinthe
particle of rapidity θ and type i to scatter off the impu-
next section, where we include an impurity.
rity into a particle of type j is denoted by S (θ). These
ij
We also note that if one were to neglect the non- arewrittenoutin10;allwewillneedforthenoisearethe
diagonal density fluctuations, one does not obtain the
unitarity relation
Johnson-Nyquist formula even in this simple situation
with no impurity. This can easily be checked, because if S† S =δ (21)
ik kj ij
weweretotreattheparticlesasfreebutwithanon-Fermi
k
distribution function, then K−1(θ,θ′)= f δ δ(θ θ′). X
ij − i ij − and the magnitude
Using this to find D(θ,θ′) yields
1
(θ) S (θ)2 = (22)
sinh(gV/2T)cosh((1 g)V/2T) T ≡| ++ | 1+e2(g−1)(θ−θB)/g
∆I2approx =2T − . (20)
sinh(V/2T)
where θ parametrizes the impurity strength and is re-
B
lated to λ.12 In particular, it follows from the unitarity
Except at g = 1/2 (where the Luttinger liquid is equiv-
relation that S 2 =1 .
alent to a free fermion), this contradicts not only the | +−| −T
Theeffectoftheimpurityisthattheasymptoticstates
Johnson-Nyquistformulabutalsothesimplephysicalre-
θ and θ arenotindividualeigenstatesoftheHamil-
sult that when there is no impurity, the left movers and | i+ | i−
toniananymore. The single-particleeigenstatesarenow
right movers are not coupled, and the noise should not
depend on the voltage. θ 1 θ in+S (θ)θ out+S (θ)θ out
|| i+ ≡ √2 | i+ ++ | i+ +− | i−
1 (cid:8) (cid:9)
III. THE EFFECT OF AN IMPURITY ON THE ||θi− ≡ √2 |θii−n+S−−(θ)|θio−ut+S−+(θ)|θio+ut . (23)
NOISE
(cid:8) (cid:9)
The unitarity of the S matrix yields θ θ = δ .
i j ij
h || || i
Describing multiparticle eigenstates is a little more com-
In this section we add a single impurity to the Lut-
plicated; for our purpose it will be sufficient to think of
tinger liquid. This couples the left and right movers and
them as tensor products of the states (23)18.
allowsbackscattering. Animpurityatx=0corresponds
to adding a term5,7 Although Q acts diagonally on the asymptotic states,
itactsinamorecomplicatedfashionontheeigenstatesof
H =λcos[φ (0) φ (0)] the problem with impurity: this time, Q and the Hamil-
B L R
− tonian do not commute and cannot be simultaneously
to the Hamiltonian; we have written the theory in terms diagonalized. First one has
of a boson so that j = ∂ φ (x,t)/2π and j =
L x L R 1
∂xφR(x,t)/2π. Basically, the−idea is that we express the +hθ||Q||θi+ = 2 1+|S++(θ)|2−|S+−(θ)|2
noise in terms of properties of the system far from the
= ((cid:0)θ) (cid:1) (24)
T
5
and Q acts again as a sum of one-particle operators, albeit
non-diagonally. For each rapidity, there are two possible
−hθ||Q||θi− =−T(θ). (25) states ||θi+ and ||θi−, and in this subspace Q acts as
a two-by-two matrix. This matrix can alternatively be
The charge acts on multi-particle states by multiplying
representedbyintroducingfictitiousfermioncreationand
by the appropriate densities. Since we will be interested
annihilation operators as
intheleadingcontributionstothecurrent,wecanneglect
the effect of the boundary when quantizing the allowed
a†(θ)a (θ) a† (θ)a (θ) S (θ)2 (28)
rapidities. Therefore the densities are exactly those dis- + + − − − | +− |
cussed in the previous section. Since the impurity prob- h a†(θ)a (θ)S∗ S i a† (θ)a (θ)S∗ S
lemisagainmappedonpurelyrightmovers,therelation − + − ++ +−− − + +− ++
between DC fluctuations of the current and fluctuations We can thus represent the DC current as an integral
of the charge holds as before. It follows that over rapidities with (29) as the integrand. This formally
coincides with the free-electron calculation in3. There
I(V)= P (θ,V) P (θ,V) (θ)dθ. (26) are however major differences since populations at dif-
+ −
− T
Z ferent rapidities are correlated. For completeness, let us
(cid:2) (cid:3)
This is the central result of11,12, where it was however briefly rederive (27) in this approach. Since we neglect
the (O(1/L)) effect of the impurity on the densities, av-
derived quite differently.
erages have to be computed using the asymptotic states
The current noise follows in a similar fashion. The
and the Bethe equations for the bulk system. One then
central result of this paper is that
checks that for any such states, a†(θ)a (θ) = 0 and
h + − i
(∆I)2 =L D(θ,θ′)T(θ)T(θ′)dθdθ′ htaer†+m(θ)oaf+((2θ7′))af−o(llθo′w)as†−r(iθg)hit=aw0ayu.nlFesosrθth=e sθe′.conTdhetefirrmst,
Z Z
we need also to observe that a†(θ)a (θ)a (θ)a† (θ) =
+ P+(θ)(1−f−(θ))+P−(θ)(1−f+(θ)) a†(θ)a (θ) a (θ)a† (θ) +Oh(1+/L),+andth−eP (−1 fi )
Z (cid:2) (θ)[1 (θ)]dθ, ((cid:3)27) ahn+d P (+1 ifh )−follo−w forithe coincident term.+ − −
×T −T − − +
We finally discuss how to compute (27) numerically.
where we remind the reader that D(θ,θ′) is the density-
The second term is straightforward; the densities follow
densitycorrelatordefinedin(18). Again,fluctuationsare
fromtheTBAequations(4)andtherelation(15). There
computed without the impurity, its effect being negligi-
is a trick to make the numerics even easier; as indicated
ble at leading order in L. The first term in (27) is the
in sect. 4.6 below, this piece can be simply written as a
equivalent of (19), with the terms arising from (24)
T derivative of the current. The first term is much more
and (25). The second term is an additional contribution difficult. ExtractingD(θ,θ′) from (9) is difficult because
at coincident rapidities. It occurs because Q is not di-
itwouldrequireinvertinglargematricesandtakingtheir
agonal on the eigenstates. Since Q is diagonal on the
derivatives numerically. Taking functional derivatives to
asymptotic states, the only additional contribution due determineD(θ,θ′)directlyfrom(7)isalsodifficult. How-
to intermediate states is proportional to
ever,ifwechoosetherapidity-dependentchemicalpoten-
tials to be
θ Q θ θ Q θ = (θ)(1 (θ)),
+ −− +
h || || i h || || i T −T
V
and another with + . This correction does not µ (θ)= µ (θ)= +xS (θ)2,
+ − ++
↔ − − 2T | |
occur if the multiparticle state has a double occu-
pancy at a particular rapidity θ (a state of the form this defines a partition function Z(x). Then
...,θ,θ... ), because then the intermediate
...,+,−,...
|| i
state ||...,θ,θ...i...,−,−,... has two particles in the same L D(θ,θ′) (θ) (θ′)dθdθ′ = 1 ∂2F (29)
state, which is not allowedby the Bethe equations. This
T T −T ∂x2(cid:12)
gives rise to the additional densities in the second term Z Z (cid:12)x=0
(cid:12)
of (27), whose equilibrium value is Using this method, we calculated the noise e(cid:12)xplicitly at
(cid:12)
g =1/3. Somecurvesaregiveninfig.1. Amorephysical
P (θ)(1 f (θ))+P (θ)(1 f (θ)).
+ − − − − + waytoexpresstheimpuritycouplingisasθB ln[TB/T],
≡
Of course,using the usual arguments (see the appendix) where TB is a crossover parameter analogous to the
thefluctuationsinthistermaresuppressedbyafactorof Kondo temperature. (This definition corresponds to set-
1/L,sowecanreplaceitwiththeproductoftheindivid- tingthe arbitraryscalem=T.) Thenoise(∆I)2 is then
ual equilibrium values, as in (27). Since fluctuations of afunctionofthreevariablesV,T andT . Since(∆I)2/T
B
densitiesperunitlengthareoforderO(1/L),bothterms isdimensionless,itisafunctionoftwovariables,sayV/T
in (27) are of order O(1). andT /T. Infig.1eachcurveisafunctionofT /T ata
B B
Q acts diagonally on multiparticle states as a sum of fixedvalueofV/T. WenoticethatasV/T isincreased,a
one particle operators. In the presence of the impurity, peak develops, similarly to the conductance12. However,
6
this peak appears at a lower value of V/T than does the Nyquist formula holds. We first observe that due to the
peak in the conductance. symmetry of the kernels Φ = Φ , the Bethe equations
ij ji
(1) and the TBA equations (4) yield
1
V/T=0 n =n n ǫ =ǫ ǫ
+ − + −
V/T=2 ≡ ≡
V/T=6
V/T=10 atany voltage. Similarly,when µ (θ)= µ (θ) µ(θ),
+ −
− ≡
we have
0.8
P [µ(θ)]=P [ µ(θ)],
+ −
−
from which it follows that
0.6
T n[µ(θ)]=n[ µ(θ)] ǫ[µ(θ)]=ǫ[ µ(θ)]. (31)
e/ − −
s
oi
n Weseefromsection2thatD(θ,θ′)canbe writtenasthe
0.4 functional derivative
δ(P (θ) P (θ))
D(θ,θ′) =L + − − . (32)
V=0 δµ(θ′) µ(θ)=0
0.2 (cid:12) (cid:12)
(cid:12) (cid:12)
Because P = nf and n and ǫ are even(cid:12) in µ, the only
i i
non-vanishing term in the functional derivative at µ=0
arises from the direct dependence of the filling fractions
0 (3) on µ. Hence, at vanishing voltage,
-3 -2 -1 0 1 2 3
ln (T_B/T)
2
FIG.1. The noise (∆I)2/T as a function of impurity cou- D(θ,θ′) = n f (1 f )δ(θ θ′). (33)
pling for four different values of V/T, where the coupling V=0 L − −
g=1/3. (cid:12)
Therefore, the(cid:12)charge fluctuations are diagonal and free
intheabsenceofanappliedvoltage. Thisfactalsofollows
easily from the symmetry of the TBA equations under
IV. VARIOUS LIMITS
+ when V =0, combined with (9). One then finds
↔−
We now consider various limits of (27) to check our
(∆I)2 =2 nf(1 f) dθ =2TG, (34)
expressionsandto investigatefurther physicalaspectsof V=0 − T
the noise in the interacting system. A first trivial limit Z
(cid:12)
is of course the case with no impurity, where =1 and where we us(cid:12)ed the linear-response result for the conduc-
the results of section 3 reduce to those of sectTion 2. tance G.11 Thus in limit V =0 we recover the Johnson-
Nyquist formula. This is an extremely non-trivial check
on our result, because the conductance here depends on
A. T =0 the impurity coupling and temperature.
Atvanishingtemperature,therearenofluctuations,so
D(θ,θ′) = 0; the noise is pure shot noise. For positive C. Free fermions
voltage, f =0, so the noise reads simply
−
When g = 1/2, the Luttinger liquid is equivalent to
(∆I)2 = P (θ) (θ) (θ)2 dθ, (30) free fermions. The bulk S matrices are unity, so the
+
T −T kernels Φ = 0 and the pseudoenergy Tǫ (θ) = m eθ.
Z h i ij i i
Then K (θ,θ′) = (1/f )δ δ(θ θ′) is easy to invert,
inagreementwith13. Thereitisalsoshownthatasimple ij − i ij −
yielding
fluctuation-dissipation-type relation relates the noise to
thecurrent. Unfortunately,wehavenotbeenabletofind 1
a finite-temperature analog of this relation. D(θ,θ′)= L P+(1−f+)+P−(1−f−) δ(θ−θ′)
(cid:2) (cid:3)
and using n =n n, we have
+ −
≡
B. V =0
(∆I)2 = n f (1 f )+f (1 f )
+ − − − − + T
WhenthevoltageV =0,weshowthatthechargefluc-
Z
tuationsD(θ,θ′)arediagonalandfindthattheJohnson- (cid:8)(cid:2)(f f )2 2 dθ (cid:3)
− +− − T
(cid:9)
7
in agreement with the known results2,3. the same arguments of section 3 with the the transmis-
This limit illustrates the importance of the fluctua- sion amplitude replaced with the reflection amplitude
T
tions. We note that if one were to use this formula for 1 . Therefore
−T
g =1/2asanapproximationforthecorrectformula(27),
on6e obtains an answer larger than the correct one, even (∆v)2 =L D(θ,θ′)(1 (θ)(1 (θ′))dθdθ′
−T −T
if we use the appropriate interacting distribution func- Z Z
tionsderivedfromtheTBA.Forexample,as 1,one
+ P (θ)(1 f (θ))+P (θ)(1 f (θ))
obtains (20) instead of the correct 2gT. TheTd→ifference + − − − − +
Z
betweenthetwoissubstantialforV =0,especiallywhen (cid:2) (θ)(1 (θ))dθ, ((cid:3)36)
6 ×T −T
g is small.
Bysimplealgebra,andusing(35),wederivethefollowing
identity
D. Strong-backscattering limit (∆I)2 =g2(∆v)2+2T(2G g). (37)
−
From (22) we see that for θ << θ , the transmission This is the zero-frequency limit of the general relation
B
amplitude 0. Thus the limit θ is the strong between current and voltage fluctuations, equation (3)
B
backscatterTing→limit, where the impur→ity∞splits the sys- of8.
tem in two. In the expression (27) for the noise, in this
limit we can neglect 2 compared to , except for very
energetic particles wiTth θ θ . HowTever, for θ F. Weak-backscattering limit
B
≈ → ∞
we have ǫ and f 0, and the contribution of this
→∞ →
region to the integral is negligible. We can therefore ap- The weak-backscattering limit is given by θ .
B
→ −∞
proximate (27) in this limit by neglecting the 2 terms This is not as easy to treat as the strong backscattering
T
entirely: limit, because the fluctuations for very-low-energyparti-
cles are not negligible, preventing a naive expansion in
1 . We therefore have not succeeded in checking the
(∆I)2 n f (1 f )+f (1 f ) dθ.
≈ + − − − − + T we−akT backscattering limit analytically. Some steps can
Z
(cid:2) (cid:3) be accomplished however. We can evaluate the behavior
By simple algebra one finds
of the second term in (27) using the identity
V g ∂
f+(1 f−)+f+(1 f−)=coth (f+ f−) (1 )= .
− − 2T − T −T 2(g 1)∂θ T
(cid:18) (cid:19) − B
Hence Therefore
V V n f (1 f )+f (1 f ) (1 )dθ
(∆I)2 ≈coth 2T n (f+−f−)T =coth 2T I. Z + − − + − + T −T
(cid:18) (cid:19)Z (cid:18) (cid:19) (cid:2)g V (cid:3) ∂
= coth( ) n (f f )
This is the noise coming from the tunneling of uncorre- 2(g 1) 2T +− − ∂θ T
lated electrons, a result expected on physical grounds8. − Z B
g V ∂I
= coth( ) . (38)
2(g 1) 2T ∂θ
B
−
E. Generalized-fluctuation dissipation results
In the weak backscattering limit, the current can be ex-
panded in a power series I = I +c(V,g,T)e2(1−g)θB
max
By the same arguments as in section 2, the conduc- (where I =gV), so
max
tance G=dI/dV can be written as
∂I
=2(1 g)(I I )
G= L D(θ,θ′) (θ)dθdθ′. (35) ∂θB − − max
2T T
Z and
The voltage difference v between the two channels is
(∆I)2 gcoth(V/(2T))(I I)+2T(2G g)
given by8 ≈ max− −
+ D(θ,θ′)(1 (θ))(1 (θ′))dθdθ′.
1 −T −T
v = (Imax I), Z Z
g −
On the other hand, in this limit physical arguments say
that the noise comes from the uncorrelated tunneling of
where I is current with no impurity. Since the noise
max
in v is proportional to the noise in the reflected (as op- Laughlin quasiparticles of charge g, yielding8
posed to transmitted) current, it is therefore given by
(∆I)2 gcoth(gV/2T)(I I)+2T(2G g).
max
≈ − −
8
This will hold if and only if 0 f 1. Thenumberofstatesintheintervalbetween
i
≤ ≤
θ and θ+dθ is then
L D(θ,θ′)(1 (θ))(1 (θ′))dθdθ′
(n (θ)dθ)!
−T −T i
Z Z
(P (θ)dθ)!([n (θ) P (θ)]dθ)!.
gV V i i − i
g coth( ) coth (I I). (39)
max
≈ 2T − 2T − Stirling’s formula gives the entropy
(cid:20) (cid:18) (cid:19)(cid:21)
This can be checked explicitly in the case g = 1/2, but
S = dθ[n lnn P lnP (n P )ln(n P )].
in general we need to resort to a numerical check. For i i− i i− i− i i− i
g = 1/3 we computed this integral by expressing it as Xi Z
the derivative of Z(x), as discussed at the end of sec- A given configuration is completely characterizedby the
tion 3. We found Z(x) by numerically solving the TBA knowledge of the densities P (θ) , or by the knowledge
i
equations,andtook the derivativesnumerically. We find of the pseudoenergies ǫ (θ{) defi}ned in (3). These sets
i
that (39) is indeed satisfied for any voltage as one takes areindividuallycomple{tebec}ausetheBetheequation(1)
θB →∞. gives the ni in terms of the Pi, and so knowing the ǫi
determines the P andvice versa. Either set canbe used
i
as convenient variables for the functional integration. In
V. CONCLUSION whatfollowswekeepdependentvariablesinintermediate
computations for notational simplicity.
Wehavepresentedinthispaperanexactcomputation The various quantities are obtained by a functional
of the non-equilibrium noise in a Luttinger liquid with integration with the Boltzmann weight e−FL/T, where
a single impurity. The approach is somewhat similar to F = E TS. As discussed in sect. 2, the full action
−
theoneforfreeelectronsbecausethemodelisintegrable. includes a chemical potential, and it can be written as
However, the quasiparticles of this integrable model are
notfree,andthecomputationrequiresasakeyingredient F = (e µ ) P dθ T P ln(1+eǫi−µi)
i i i i
the equilibrium fluctuations of densities, which are not − −
i Z i Z
diagonal and are quite complicated. X +eǫi−µiln(1+e−ǫi+µXi) dθ, (cid:2)
The computation of AC properties appears more dif-
ficult. In addition to the correlations between densities where we generalize the customar(cid:3)y situation by allowing
at different rapidities induced by the Bethe equations, rapidity-dependent chemical potentials µ (θ).
i
another key ingredient to take into account is that lo- The equilibrium values ǫ or P minimize the action
i i
cal physical operators are infinite sums of multiparticle (A1) subject to the constraint of the Bethe equations
operators with complicated matrix elements (form fac- (1). The first-order variation gives
tors). We hope however that a computation similar to18
is possible.
∆F = e ∆P P eǫi−µiln(1+e−ǫi+µi)∆ǫ
i i i i
−
i Z i Z
X X
WewouldliketothankA.Ludwigforpreviouscollabo- T ln(1+eǫi−µi)+eǫi−µiln(1+e−ǫi+µi) ∆P .
i
−
rationsandinterestingconversations. Thisworkwassup- i Z
X (cid:2) (cid:3)
ported by the Packard Foundation, the National Young
Recall that the ∆P are given in terms of ∆ǫ via the
Investigatorprogram(NSF-PHY-9357207)andtheDOE i i
Betheequations. Requiring∆F =0givestheTBAequa-
(DE-FG03-84ER40168).
tions (4) with the equilibrium free energy (5).
To compute the fluctuations, we expand this action
to second order around the saddle point values (denoted
APPENDIX A: DENSITY FLUCTUATIONS
with a bar). Writing ∆(2)P for the variation of P in-
FROM THE THERMODYNAMIC BETHE i i
duced by first and second order variations in ǫ we get
ANSATZ
Theenergyperunitlengthisgivensimplybysumming ∆(2)F/T = ei∆(2)Pi/T
up the individual energies e (θ) of the particles: i Z
i X
P eǫi−µiln(1+e−ǫi+µi)∆ǫ
i i
−
E = ei(θ)Pi(θ)dθ. Xi Z
Z
∆(2)P ln(1+eǫi−µi)+eǫi−µiln(1+e−ǫi+µi)
i
In order to compute the entropy, we utilize the fact −
i Z
that solutions of the Bethe equations allow only one X (cid:2) (cid:3)
1 (∆ǫ )2
quasiparticle per level, so that the filling fractions obey P eǫi−µi ln(1+e−ǫi+µi) i .
− i − 1+eǫi−µi 2
i Z (cid:20) (cid:21)
X
9
Thedefinitionofǫ(3)relatesthefluctuationsofn,P and Then it follows from the definition of K that
ǫ:
∂eǫk(α) ∂ ∂
δ(α α′)δ = + K (α,α′).
e−ǫi+µi∆(2)(ni Pi)=∆(2)Pi+Pi ∆ǫi+ 1(∆ǫi)2 . ∂α − km −(cid:18)∂α ∂α′(cid:19) km
− 2
(cid:20) (cid:21)
Using these two identities and the definition of the in-
Also, from the Bethe equations (1) we have verse of K yields the relation (9) relating the density
fluctuations to K−1.
∆(2)n = Φ ∆(2)P .
i ij j
∗
j
X
It follows that ∆(2)P depends only on the combination
i
∆ǫ + 1(∆ǫ )2. Because the first-order variation ∆F =
j 2 j
0, the linear terms in ∆ǫ and many others cancel out,
leaving only 1R.Landauer,PhysicaD38(1989)594;G.B.Lesovik,JETP
Lett.49(1989)594;M.Bu¨ttiker,Phys.Rev.Lett.65(1990)
1 P 2901.
∆(2)F = 2 1+e−iǫi+µi(∆ǫi)2. 2R.Landauer and Th. Martin, Physica B175 (1991) 167.
i Z 3M. Bu¨ttiker, Phys.Rev.B46 (1992) 12485.
X
4C.de C. Chamon, D.E. Freed and X.G. Wen, Phys. Rev.
(A similar result appears in the original paper by Yang
B51 (1995) 2363, cond-mat/9408064.
and Yang16 on bosons with repulsive delta interactions). 5X.G. Wen, Phys. Rev. B41 (1990) 12838; Phys. Rev. B43
Hence the fluctuations of ǫ are diagonal:
(1991) 11025.
6K.Moon,H.Yi,C.L.Kane,S.M.GirvinandM.P.A.Fisher,
1+e−ǫi+µi
∆ǫ (θ)∆ǫ (θ′) = δ δ(θ θ′). (A1) Phys.Rev.Lett. 71 (1993) 4381, cond-mat/9304010.
h i j i LPi ij − 7C.L. Kane and M.P.A. Fisher, Phys. Rev. B46 (1992)
15233.
In terms of the filling fractions fi we can recast this as 8C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 72 (1994)
724.
1
n (θ)∆f (θ)n (θ′)∆f (θ′) = n f (1 f )δ δ(θ θ′). 9X.G. Wen,Phys. Rev.B44 (1991) 5708.
h i i j j i L i i − i ij − 10S. Ghoshal and A.B. Zamolodchikov, Int. J. Mod. Phys.
This is the same result as one would have for a free A9(1994) 3841, hep-th/9306002.
11P. Fendley, A.W.W. Ludwig and H. Saleur, Phys. Rev.
theory: fluctuations restricted to the filling fractions are
Lett. 74 (1995) 3005, cond-mat/9408068.
free. The difference with a free theory is that we have
12P.Fendley,A.W.W.LudwigandH.Saleur,Phys.Rev.B52
correlated fluctuations ∆n ∆n as well as ∆n ∆f .
The density-densityhflucituatjioin D(θ,θ′) hcanibejoib- (1995)8934,cond-mat/9503172.Forareview,seeProceed-
ings of the 19th IUPAP International Conference on Sta-
tained from (A1) by expressing∆ǫ in terms of the ∆P .
i i tistical Physics (World Scientific) toappear.
The Bethe ansatz equations (1) yield 13P. Fendley, A.W.W. Ludwig and H. Saleur, Phys. Rev.
Lett. 75 (1995) 2196, cond-mat/9505031.
(1+eǫi−µi)∆Pi+eǫi−µiPi∆ǫi = Φij ∆Pj, 14seeforexample,P.FendleyandK.Intriligator,Phys.Lett.
∗
j B 319 (1993) 132.
X
15A. Luther and I. Peschel, Phys. Rev. B12 (1975) 3908;
which can be recast as
L.P.KadanoffandA.C.Brown,Ann.Phys.121(1979)318;
F.D.M. Haldane, J. Phys. C14 (1981) 2585.
eǫi−µiPi∆ǫi = Kij(θ,α)∆Pj(α)dα, 16C.N. Yangand C.P. Yang, J. Math. Phys. 10 (1969) 1115;
j Z Al.B. Zamolodchikov, Nucl.Phys. B342 (1991) 695.
X 17F.Smirnov,Form Factors inCompletely Integrable Models
whereKij isdefinedin(8). Pluggingthisinto(A1)gives of Quantum Field Theory, (World Scientific,1992).
anexplicitformofD(θ,θ′)intermsofanintegralofK−1: 18F. Lesage, H. Saleur, S. Skorik, submitted to Phys. Rev.
Lett., cond-mat/9512087.
∆P (θ)∆P (θ′)
i j
h i
1
= dα K−1(θ,α)K−1(θ′,α)eǫk(α)−µkn (α).
L ik jk k
k Z
X
This equation can be simplified. First note that when µ
is independent of α, (15) yields
∂eǫk(α)
eǫk(α)n (α)=T .
k
∂α
10