Scalar and vector Keldysh models in the time domain

0.3 MB·English

by  M. N. Kiselev

Scalar and vector Keldysh models in the time domain
M.N. Kiselev1 and K. Kikoin2
1International Center for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy
2School of Physics and Astronomy, Tel-Aviv University 69978, Israel
(Dated: January 15, 2009)
The exactly solvable Keldysh model of disordered electron system in a random scattering field
with extremely long correlation length is converted to the time-dependent model with extremely
9 longrelaxation. Thedynamicalproblemissolvedfortheensembleoftwo-levelsystems(TLS)with
0 fluctuatingwell depths havingthe discrete Z2 symmetry. It is shown also that the symmetric TLS
0 with fluctuating barrier transparency may be described in terms of theplanar Keldysh model with
2 dime-dependentrandomplanarrotationsinxyplanehavingcontinuousSO(2)symmetry. Thecase
of simultaneous fluctuations of the well depth and barrier transparency is subject to non-abelian
n
algebra. Application of this model to description of dynamic fluctuations in quantum dots and
a
optical lattices is discussed.
J

5
PACSnumbers: 73.21.La,73.23.Hk,85.35.Gv
1

]
The model with infinite correlation range of fluctuat- resultsin2DGaussianaveragingfortheGFindynamical
l
al ing fields V(r) proposed by L.V. Keldysh1 is one of few random field.
h exactlysolvableproblemsinthetheoryofdisorderedelec- Leaving for the last section the discussion of real sys-
- tron systems. The approximation tems, where TDKM arises as a description of generic
s
e disorder, we start with a toy model of an ensemble of
m D(r r′)= V(r)V(r′) =W2 (1) non-interacting two-level systems (TLS) in a randomly
− h i
fluctuatingenvironment. InstandardrealizationofTLS,
. makesidenticalalldiagramsforthe electronGreenfunc-
t
a tion (GF) in the order V2n. As a result summation of namely a double-valley well, particles are distinguished
m diagrammaticseries in the ”crosstechnique”2 reduces to not only by conventional quantum numbers but also by
their position in the well characterized by the index
- the problemofcalculationofcombinatoriccoefficientA
d n j = l,r of the left (l) or right (r) valley. The barrier
(number of pairwise coupling of scattering vertices). In
n betweenthevalleysischaracterizedbythetunnelingma-
fact A = (2n 1)!! is the total number of identical
o n
− trix element ∆ . The Hamiltonian of isolated TLS has
c diagrams in the order 2n. The perturbation series is 0
[ summed exactly1,3, and one deals with averaging of en- the form

1 semble of samples with constant V but the magnitude H(0) = ε n +Un2 ∆ (c†c +H.c.). (2)
of this field randomly changes from realization to real- TLS j j j − 0 l r
v
j
6 ization. Inmomentum space the correlationfunction (1) X(cid:0) (cid:1)
4 transforms into D(q) = (2π)3W2δ(q). The electron GF Here n = c†c is the particle occupation number, ε is
2 inKeldyshmodelisaveragedwith Gaussiandistribution the discjretejenjergy level in the valley j and U is thejin-
2 function characterized by the variance W2. teraction parameter for two particles in the same valley.
.
1 This model is not widely used in current literature The condition U ∆ is usually assumed. We consider
0 because it is difficult to propose an experimental de- spinless particles,≫having in mind that the theory can be
9 vice, where the conjecture (1) could be realized (see, applied both to interacting bosons and fermions (elec-
0 however,4). In the present paper we discuss a realiza- trons)withfrozenspindegreesoffreedom. Tobespecific
:
v tion of Keldysh model in time domain. In this case the we discuss tunneling electrons as an example.
Xi analog of infinite spatial correlation is the long mem- We start with the singly occupied TLS, where the
ory effect, which can be realized in many physical situa- constraint N = n = 1 is imposed on the Hamilto-
r i i
a tions(seebelow). Thestructureofperturbationseriesin nian, introduce pseudospin operators σ+ = c†c ,σ− =
time-dependent Keldysh model (TDKM) is the same as P l r
c†c ,2σ =n n , and reduce (2) to
in original one, and the long characteristic times of dy- r l z l− r
namical correlation play the same part as infinite range H(0) = δ σ ∆ σ µ (N 1). (3)
spatialcorrelationofstatic randompotentials. Since the TLS − 0 z − 0 x− 0 −
timeaxisistheonlycoordinateinthisproblem,itseffec- inthepseudospinsubspace. Heretheasymmetryparam-
tive dimension is ”0+1”. Moreover, the TDKM admits eter δ = ε ε play the role of effective ”magnetic”
0 r l
naturalgeneralizationoforiginalKeldyshmodel. Wewill field, the Lagr−ange parameter µ controls constraint.
0
showthatthedynamicalfluctuationsintimedomainmay The scalar fluctuation field is introduced as random
be bothofscalarandof vectorcharacter. The kinematic fluctuationsofTLSasymmetry,namelyasatime depen-
constraint existing in the vector TDKM results in elim- dent field δ (t)=δ +h (t) determined by its moments
ρ 0 ρ
ination of essential part of diagrams in cross technique,
butthesummationofperturbationseriesisstillexact. It h (t)=0, h (t)h (t+τ)=D(τ). (4)
ρ ρ ρ
2
Here the overline stands for the ensemble average. Thus accordancewith (4) andmaking the Fourier transforma-
wereducedtheoriginalmodeltotheeffectivespinHamil- tion by means of (5), we come to the series
tonian in magnetic field with random time-dependent
component. The problem can be reformulated as a ∞
GR (ε)=g (ε) 1+ A ζ2ng2n(ε) (8)
study of propagation of fermions along the time axis in j,s j n j
" #
the presence of time-dependent random scalar potential nX=1
δ (t)(n n )/2,and the crosstechnique may be usedin
caρlculatri−on olf the propagators5. Here gl(ε)=(ε+iη)−1 and gr =(ε+δ0+iη)−1 are the
barepropagators,A =(2n 1)!!istheabovementioned
n
The analog of Keldysh conjecture in time domain is −
combinatoric coefficient (see Fig. 1, where several first
a slowly varying random field exp( γt). A very long
∼ − irreducible diagrams are shown).
relaxation time τ 1/γ with small γ is presumed, so
rel
∼
that the noise correlationfunction is given by
2ζ2γ
D(ω)= lim =2πζ2δ(ω) (5)
γ→0ω2+γ2
FIG. 1: Irreducible Feynman diagrams for scalar TDQM.
(the noise correlation function (5) is normalized in such
Solid and wavy lines stand for gi(ε) and D(ω).
awaythatthecorrespondingverticesaredimensionless).
In this limit the averaged spin propagator describes the
Like in the real space Keldysh model1, the series (8)
ensemble of states with a field δ = const in a given
maybesummedbymeansoftheintegralrepresentation7
state, but this constant is random in each realization.
for(2n 1)!!. Thenchangingtheorderofsummationand
The ”planar-type” TDKM may be derived in a similar
−
integration (Borel summation), one comes to the follow-
way. For this sake one should introduce a random com-
ing equation for the left valley GF
ponentinthetunneling matrixelement,∆=∆ +∆ (t)
0 ρ
with ∆ρ(t) = 0 and make similar conjecture (5) about GR (ε)= 1 ∞ e−z2/2ζ2 dz (9)
the correlation function F(τ) = ∆ρ(t)∆ρ(t+τ), namely l,s ζ√2π ε z+iη
approximate its Fourier transform by Z−∞ −
Remarkably, the single electron GF in this model has
F(ω)= lim 4ξ2γ =4πξ2δ(ω) (6) no poles, singularities or branch cuts. Similar procedure
γ→0ω2+γ2 may be applied to the Green function GR. As a result
r
of this Gaussian averagingthe ”magnetization” σ is re-
z
Thus,wetreatourtoyHamiltonian(3)inthefollowing ducedandthecorrespondingresponsetotransversalfield
way. First we study the limiting case δ0 ∆0, were the is modified accordingly.
≫
interdot tunneling is consideredas a perturbation to the Next, we formulate the ”planar” TDKM for the bare
longitudinal termaffectedbystochastizationinthescalar HamiltonianH(0) (2) withimpenetrable barrier∆ 0
TDKM. This problem may be solved by means of the andsymmetricTvLaSlleys,ε =ε , isolatedfromeacho0t→her.
l r
standard technique1,3,6 generalized for the time-domain The transverse random perturbation is introduced by
case. Then we turn to the case of transversal random
tunneling potential and solve this problem by means of H (t)=∆ (t)σ++∆∗(t)σ− (10)
ρ ρ ρ
correspondingly modified ”planar” TDKM.
We start with the scalar TDKM and treat the term so that the inter-valley tunneling is stochastisized by
H(0) =δ (n n )/2 asa zeroorderapproximationwith meansofaveragingintime-domainwithcorrelationfunc-
k 0 r− l
electronoccupyingthelevelε inthegroundstate. With- tion(6). The tunnel matrix element∆ is transformedas
l
outstochasticperturbationtheroleofthetunnelingterm ∆ρ ∆ρei(ϕr−ϕl) under the gauge transformation cj
tHo⊥th=e g−ro∆u0nσdxsitsatine waditmhitxhinegcoarrcehsaprognedtinragnesnfeerrgeyxclietvoenl cjeU→iϕnjl,ikaendthweescparleasrumTDeK∆Mρ ,tothbeenaoctoicmeapblelexpvaarritabolfe.di→a-
shifts, ε ε ∆2/δ for the ground and excited grams in the perturbation series disappears due to the
states, rle,rsp→ectivle,rly∓. T0he0time-dependent perturbation kinematical restrictions σ+σ+ = σ−σ− = 0 (see also9).
stochastisizes this simple picture. Onlythe diagramswith pseudospinoperatorsorderedas
...σ+σ−σ+σ−...surviveinthe expansionforthe GFof
Let us introduce the retarded propagators for the
planar model
scalar TDKM
∞
GRj,s(t−t′)=hcj(t)c†j(t′)iR =−ih[cj(t)c†j(t′)]+i. (7) GRj,p(ε)=gj(ε)+ Bn(√2ξ)2ngj2n+1(ε). (11)
n=1
and consider their evolution on the time axis under the X
influence of random component h (t) (”random longitu- The vertices in the cross technique are now ”colored” in
j
dinal magnetic field” in pseudospin notation), first as- accordance with two terms entering the random poten-
suming γ 0, ∆ /γ 0. After averaging GR(t t′) in tial. Theverticeswithdifferentcolorshavetobeordered
→ 0 → j −
3
(Figs. 1, 2) and corresponding skeleton diagrams2,3 are
still valid. However, the important reservation should
be made: the self energy cannot be treated as a renor-
malizationofbarepolebecausethe bareanddressedGF
are connected by non-local integral operators [see Eqs.
FIG. 2: First non-vanishing vertex correction to the Green (9),(13)]. Nevertheless, the ordinary differential equa-
function self-energies in the planar Keldysh model. Black tion connecting the GF and its derivative over energy
and white sites correspond to two terms in the Hamiltonian can be obtained for both versions of TDKM. This equa-
(??); transversal pseudospin correlation functions F(ω) (6) tion was found for the scalar Keldysh model in3. Here
are represented bydashed lines. we derive this equation without appealing to the Ward
identity andthen generalizethe derivationprocedure for
the planar model.
To calculate the derivative dG(ε)/dε for the scalar
inalternatingway,andthecorrelationlinesconnectonly
TDKM, we start with expansion (8) (index s is omit-
the vertices of opposite color (see Fig. 2.)
ted below for the sake of brevity). It is convenient to
As a result of the above kinematic restrictions, the
count the energy off the position of chemical potential
combinatoric coefficient B = n!. Then we use the inte-
n µ [seeEq. (3)]inthe middlebetweenthelevelsε . So,
gralrepresentation10 for n!andtransformthe series(11) 0 l,r
we shift the energies ε ε = ε δ /2 for the left and
into → ± ∓ 0
right GR, respectively.
∞ The same procedure, which leads to Eq. (9) for GR
GRj,p(ǫ)=gi(ǫ) 1+2 tdt t√2ξgj(ǫ)]2n e−t2 gives the following equation for its derivative:
( Xn Z0 h i ) z2
Herewesubstitutedt2 forthe variablez. Thenchanging dG g2(ε ) ∞ − 1 z3
= α e 2ζ2 1+ dz.
tGhRe(ǫo)rdinertootfhseuimntmegartaioln and integration, we transform dε −ζ√2π Z−∞ (cid:20) ζ2εα−z+iη(cid:21)
j
(α = ). Calculating the integral and substituting
GR (ǫ)= ∞2tdt gj(ǫ) e−t2 (12) g2(εα)=∓ε−α2 we come to the differential equation
j,p 1 2t2ξ2g2(ǫ)
Z0 − j dG(ε )
ζ2 α =1 ε G(ε ) (15)
α α
Takingintoaccounttheexplicitformofthefreepropaga- dε −
tor g (ǫ), we change the integration variable once more,
j similartothatobtainedfortherealspacescalarKeldysh
t=u/√2ξ, and transform (12) into model3,6.
Generalizationofthisprocedureforthesymmetricpla-
GRj,p(ǫ)= ∞ u2dξu2 ǫ u1+iη + ǫ+u1+iη e−u2/2ξ2 ndaiffremreondtiealti(nδg0 =the0)seisriemso(r1e1c)uomvebrertshoemeen.erWgye.sTtahretnwtihthe
Z0 (cid:18) − (cid:19) analog of Eq. (12) for the derivative has the form
(13)
Nucoowsφw,ey =inturosdinucφe, stoheth”actaurte=sian”x2c+ooyr2diannadtesd,xdxy == dG = g2(ε) 1+ ∞2tdt2(2t2−1)t2ξ2g2(ε)e−t2
dε − 1 2t2ξ2g2(ε)
ududφ. The angle independent integral (13) may be (cid:20) Z0 − (cid:21)
p
rewritten as The subsequent variable change which gave Eq. (13) for
the GF gives for its derivative the following equation
1 +∞ dxe−x2/2ξ2 +∞ dye−y2/2ξ2
GR (ǫ)=
j,p 2Z−∞ ξ√2π Z−∞ ξ√2π dG = g2(ε) 1+ 1 J4 J2 (16)
dε − 2 ξ4 − ξ2
1 1 (cid:20) (cid:18) (cid:19)(cid:21)
+ . (14)
where
"ǫ x2+y2+iη ǫ+ x2+y2+iη#
−
∞ z2
This resultpis a natural generpalization of the one- J = dzznexp g(ε z)+( 1)n+1g(ε+z)
n −2ξ2 − −
dimensional Gaussian averaging (9) characteristic for Z0 (cid:18) (cid:19)
(cid:2) (cid:3)
the scalar TDKM to the two-dimensional Gaussian av- Aftersomemanipulations,theseintegralsarerepresented
eraging of planar random field with purely transversal via the GF for the planar model (13):
(xy) fluctuations. Only the modulus of random field
J =2εξ2G 2ξ2, J = 4ξ4+ε2J (17)
r = x2+y2 is averaged, whereas the angular variable 2 − 4 − 2
remainsirrelevantduetothein-planeisotropyofthesys-
Substituting these integrals in Eq. (16), we come even-
p
tem. Like in the scalar model, the averaged GF has no
tually to the differential equation
singularities.
Although the GF lost its pole structure, the stan- dG(ε) ξ2
ξ2 =1 εG(ε) 1 (18)
dard Feynman rules for construction of irreducible parts dε − − ε2
(cid:18) (cid:19)
4
which is obviously the generalization of Eq. (15). These DoS DoS
s p
two equation may be rewritten in a unified way: 0.2 0.5
0.5 1.0
2.0 1.5
d
ε G 1 = ζ2G2 G−1
α α,s− α,sdε α,s a) b)
1 d
εG 1 = ξ2G2 εG−1 (19) -0.5 0 0.5 -1 0 1
p− p εdε p
(cid:20) (cid:21)
(cid:0) (cid:1)
FIG. 4: Density of states in scalar (left) and planar (right)
The solutions ofthese equations satisfying the boundary
model (all unitsare arbitrary).
conditionG(ǫ )=ǫ−1isgivenby(9,13). Itisworth
→∞
notingthatthedifferentialoperatorintherighthandside
of the secondequation is nothing but div in polar coor-
ε doublehumpGaussianstructure. Theinformationabout
dinates. Thisformreflectseffectivetwo-dimensionalityof
position of electron in right or left valley is completely
Gaussian averaging in the planar TDKM. Now one may
lost at ζ/δ > 1. In the planar model for symmetric
0
introduce vertex parts using the analogy with the Ward
TLS instead two-peak structure due to avoided crossing
identities
characteristicforcoherenttunneling,wegetapseudogap
around zero energy due to stochastic tunneling, which
d 1 d
Γs = dεG−s1, Γp = εdε(εG−p1). (20) survives at any variance ξ.
Let us allow now the fluctuations of both longitudinal
These vertices, together with equations (19) will be use- and transverse components which corresponds to simul-
ful for calculation of response functions of our TLS (see taneous fluctuations of the well depth and barrier trans-
below). Likeintheselfenergyparts(Figs. 1,2),thepla- parency for the case of symmetric TLS (δ0 = 0). We
nar TDKM lacks most of diagrams of scalar model due solve this problem by means of path integral formalism.
to the kinematic restrictions: the sites in the vertices of The Lagrangianandcorrespondingactionaredefinedon
triangle are of the same color, black and white sites al- the Keldysh contour K (see e.g.11)
ternate,anddashedlinesconnectsitesofoppositecolors.
Firstnonvanishingverticesforbothmodels areshownin (t)= c¯ i∂ c H, S = (t)dt. (23)
j t j K
Fig. 3 L j=l,r − ZKL
X
Here c¯ ,c are Grassmann variables describing the elec-
j j
tron. The time-dependent gauge transformationc (t)
j
→
cj(t)eiϕj(t), converts the fluctuation of the well depth to
thefluctuationofthephase ofthetunnelmatrixelement
under the choice
t
FIG.3: Firstnon-vanishingvertexdiagramsfortheΓinscalar ϕ (t)= h (t′)dt′. (24)
j j
(left) and planar (right) TDKM. Z−∞
We therefore identify the longitudinal and transverse
The density of states (DoS) in stochastisized TLS is noisewith phasefluctuations ofthe barriertransparency
given by the imaginary parts of GF (9), (13) for scalar and fluctuations of the modulus of the tunnel matrix el-
and planar TDKM, respectively: ement, respectively and unify them in the path integral
description.
2 ε2+δ2 εδ
ν (ε)= exp 0 cosh 0 , (21) The ensemble averaging
s ζ√2π − 2ζ2 ζ2
(cid:18) (cid:19) (cid:18) (cid:19)
... = dh P(h ) d∆∗d∆ P (∆∗,∆ )...
1 ε2 h inoise ρ l ρ ρ ρ tr ρ ρ
νp(ε)= ξ2|ε|exp −2ξ2 . (22) Z Z
(cid:18) (cid:19) isdonewiththehelpofprobabilitydistributionfunctions
for longitudinal and transverse fluctuations
Inthe scalarmodelν (ε)is a superpositionoftwoGaus-
s
sians centeredaroundε andε , respectively. In the pla-
l r 1
narmodel ν (ε) is representedby a single Gaussianwith P = exp dtdt′h (t)D−1(t t′)h (t′)
p l ρ ρ
ζ√2π − −
a dip ”burnt” around zero energy (Fig. 4). (cid:18) ZK (cid:19)
Switching on the tunneling term H in scalar model
⊥
without randomfield, we come to the picture of two lev- 1
P = exp dtdt′∆∗(t)F−1(t t′)∆ (t′) .
els mutually repulsed due to coherent interdot tunnel- tr 2πξ2 − ρ − ρ
ing and broadened due to incoherenttime-dependent in- (cid:18) ZK (cid:19)
tradotfluctuations. The spectrumis still gapfulatsmall TheGF’scanbecalculatedbymeansofgeneratingfunc-
enough ratio ζ/δ , If the ζ . δ , the DoS merges into tionalcorrespondingto KeldyshactionS inastandard
0 0 K
5
way12,13. Inthe”infinitememory”limit(5),(6)weeasily
express the GF of the electron in the symmetric double
well potential
GR (ǫ)= 1 ∞ dze−z2/2ζ2 FIG.5: Diagramsforbareloop(transversesusceptibility)and
j,v ζξ2(2π)3/2 loop with dressed vertex (longitudinal susceptibility).
Z−∞
dw∗dwe−|w|2/2ξ2 ǫ±z (25)
(ǫ+iη)2 z2 w2
Z − −| | i dtexp(iωt) σz(t)σz(0) R,whichisrepresentedbytwo
h i
loops with vertex corrections (Fig. 5). Here both solid
Noticing that the GFs do not depend on the well index
R
lines correspond either to j = l or to j = r. We confine
j andperforming the integrationoveranglesinspherical
ourselveswith calculationof static susceptibility, ω 0.
coordinate system we get →
In order to work at finite temperatures we turn to Mat-
GR(ǫ)= 1 ∞dρρexp ρ2 subaraGreenfunctionsfunctionsG(iǫn)(similarcalcula-
v 2ξ −2ξ2 tioncanbe doneontheKeldyshcontourintherealtime
Z0 (cid:18) (cid:19) path integral formalism) and susceptibility
erf ρ ξ2−ζ2
2ξ2ζ2 1 1
+ . (26) χ(iω )=T (iω +iǫ ) (iǫ )Γ(iǫ ,iω ) (30)
(cid:16)ξq2 ζ2 (cid:17) ǫ ρ+iη ǫ+ρ+iη m Gj m n Gj n n m
− (cid:18) − (cid:19) Xn,j
The Eqps (25, 26) generalize (9) and (14) for the Thenthe firstofWardidentities (20) providesus with
case of anisotropic vector Keldysh model. The three- the exact equation for the vertex
dimensionalGaussianaveragingin(25)standsforvector
characterof the randomfield distributed onan ellipsoid. ζ2(GR )2ΓR (ǫ,0)=ǫGR 1, (31)
j,s j,s j,s−
Typical size of semi-axes is defined by the variances of
giving an access to the exact evaluation of χ(0). Com-
longitudinal and transverse noises. Like in the scalar an
bining (30) with (31) and analytical continuation of (9),
planarmodels,theaveragedGFhasnosingularities. The
we find
angle φ dependence is absent in (25) due to the in-plane
isotropy of the model Ptr =Ptr(|∆|2) which is preserved ∞ ydye−y2/2 (2y αδ0)ζ
here. The limits of strong easy axis ξ 0 and easy plane χ (0)= n − (32)
→ k − √2πζ F 2T
ζ 0 anisotropycorrespondto scalarandplanarmodels, α=∓Z−∞ (cid:18) (cid:19)
→ X
correspondingly.
Here n (x) is the Fermi distribution function. The
We notice that the DoS for vector TDKM is also F
asymptotic behavior of static susceptibility χ(0) is
characterized by a pseudogap. The energy dependence
ν (ε) ε2 for small energies should be contrasted with
v 1/T, T (ζ,δ )
ν (ε)∼const for scalar ν (ε) ε for planar model behav- χ(0) ≫ 0 (33)
s ∼ p ∼ ∼ 1/ζ, ζ (T,δ0)
ior. Thisdependencereflectsthefactthattheprobability (cid:26) ≫
toremainclosetoinitiallevelpositiondecreaseswithin- There is no vertex correctionto transversesusceptibility
crease of effective dimensionality of the problem due to χ (0) = σ ,σ . It is given by the bare loop formed
⊥ + − R
h i
stochastizationofthecomplextunnelingmatrixelement. by the left and right GF. In case of symmetric TQD
One should specially mention the degenerate case of (δ = 0) and big variance ζ ∆ this function is as
0 0
≫
theisotropicvectorTDKM,characterizedbyξ=ζ=λand smooth as χ (0) with changing ζ and T. Its asymptotic
k
describingrotationofpseudospinonaBlochsphere. Per- behavior is given by the same Eq. (33). The physical
forming simple algebra we get an equation for GF sense of these results in obvious: the information about
positionofelectroninagivenwellislostatstrongenough
1 d
εG 1=λ2G2 ε2G−1 (27) stochastization ζ T.
v− v ε2dε v Next, we calcul≫ate the susceptibility for planar model
(cid:20) (cid:21)
(cid:0) (cid:1) in case of symmetric TLS with δ = 0. In this case
and the Ward Identity 0
χ (0) differs from χ (0) by factor 2, so that it is enough
⊥ k
1 d to calculate the latter one. Now we appealto the second
Γv = ε2dε(ε2G−v1). (28) equation from (19) with Γp defined in (20). Then like
in previous case the calculation of χ (0) is reduced to
k
TheDoSfortheisotropicmodelisgivenbytheexpression finding the combination iǫ (iǫ ) 1. Straightforward
n p n
G −
computation gives
1 ε2
ν (ε)= ε2exp . (29)
v λ3√2π (cid:18)−2λ2(cid:19) χ (0)= 1 ∞y2dye−y2/2tanh yξ . (34)
k ξ 2T
Our next task is calculation of response func- Z0 (cid:18) (cid:19)
tions. In scalar TDKM the longitudinal suscepti- Thebehaviorofχ asafunctionofT andξisclosetothat
k
bility is given by the correlation function χ (ω) = for scalar model, including the asymptotic dependence
k
6
(33). Theequationsforstaticsusceptibilitiesinisotropic matrixelement. Inthe planarmodelone dealsonly with
vector TDKM can be easily found with help of (25, 26) the planar (xy) rotations, so the relevant symmetry is
SO(2)withstillabelianalgebra. Thephasefluctuationin
χ(0)= 1 ∞y3dye−y2/2tanh yλ . (35) thatcasearediscreteZ2 (φ=0,π toprovidethecondition
λ√2π Z0 (cid:18)2T(cid:19) ∆¯ρ=0), while modulus fluctuations determine effective
2d behavior. The symmetry of isotropic vector TDKM
These susceptibilities are characterized by the same is SU(2) and corresponding algebra is non-abelian.
asymptotic behavior as those for scalar/planar models. Onemaymentionseveralmorephysicalsystems,where
the scalar and/or planar TDQM is useful. One of such
The toy model of noninteracting TLS under dynami- models is the big quantum dot with charge fluctua-
cal stochastization demonstrate some generic properties tions accompanied by longitudinal and transversal spin
ofTDKM:(i)thelossofcharacteristicspinorpseudospin fluctuations14. The class of Gaussian ensembles corre-
behavior at variance exceeding temperature; (ii) the ef- sponds to infinite-range correlations in the charge and
fective two-dimensionality of Gaussian averaging in pla- spin sectors of the model. Both spin and charge inter-
nar TDKM as its main distinction from scalar model; actions contain stochastic component15, leaving a room
(iii) the effective three-dimensionality of vector TDKM. for original formulation of the Keldysh model. The
These features survive also in more realistic situations. gauge field theory, based on functional bosonization be-
Oneofpossibleapplicationsofthistheoryistheproblem ing formulated in the time-domain, opens a possibil-
of electron tunneling through double quantum dot in a ity of stochastic treatment of dynamic processes. As is
regime of strong Coulomb blockade, where the source of shown in14, the transverse spin correlation function for
stochastizationis a randomtime-dependent gate voltage anisotropic spin exchange contains both short-time and
applied to one of the valleys5. The case of N = 2 was long-time correlationparts. While the short-time (white
considered, where the starting Hamiltonian H(0) is that noise) correlations dominate away from the Stoner in-
of Eq. (2) with added spin index. In this case the scalar stability, the (infinitely) long-time correlations become
TDKMmaybe usedinthe limitofslowfluctuations(5), importantasoneapproachesthe regimeofstrongfluctu-
the double quantum dot looses its spin characteristicsat ations of the magnetization. The long-time part of the
lowT,andtheKondo-typezerobiasanomalyissmeared model is equivalent to the planar TDKM.
accordingly8. The important difference between spinful Another interesting object is the optical superlattice
andspinlessTLSmodels is intheir symmetry. The sym- consisting of biased double wells16. The bias is random,
metryofTLSwithN=2consideredin8isSO(5),whichis butthenumberofatomsinthesameinallTLSinthisex-
reduced to SO(3) for low-energypart of excitation spec- perimentalsetupdue to the ”interactionblockade”. One
trum, so the Lie algebra is non-abelian. However, it was may expect that the well population in these TLS could
possible to introduce scalar TDKM due to abelian char- be stochastized in accordance with Fig. 4, provided the
acter of time-dependent random gauge field. Keldysh-typefluctuations (5)or(6)with longrelaxation
ThesymmetryofthescalarTLSwithN=1isgivenby times were realized experimentally.
the discrete group Z with abelian algebra. The gauge We thank Boris Altshuler, Jan von Delft, Yuri
2
transformation allows to identify the fluctuations of the Galperin, Yuval Gefen, Jean Richert and Wilhelm Zw-
scalar model as U(1) fluctuations of the phase of tunnel erger for helpful discussions.
1 L.V. Keldysh, ”Semiconductors in strong electric field”, transformed under ”local” time-independent gauge trans-
D. Sci. Thesis, Lebedev Institute,Moscow, 1965. formation and therefore average to zero according to
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Methods of Quantum Field Theory in Statistical Physics, 10 n!=Γ(n+1)=R0∞dzzne−z.
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(1993). pany,Dordrecht,1983.
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87 M(2n.N−.K1)i!s!e=lev2,nK√1.πKΓi(kno+in,1a/n2)d=J.√R12iπchRe−∞rt∞,adrtXt2inve:−08t2023..2676; 1156 IBP..L6C.2Kh,eu1irn4le8at8n,6d,S(I2..0LT0.r0oA)t.lzekiyn,erM,a.ndFeBld.,L.UA.ltSschhunleorr,rbPehrygse.r,ReMv..
Physica Status Solidi (c) (in press)
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9 The correlators ∆(t)∆(t+τ) and ∆∗(t)∆∗(t+τ) are Lett. 101, 090404 (2008).