The quasiclassical Keldysh Green function Lectures delivered at the Universit` a di Camerino Roberto Raimondi Dipartimento di Fisica, Universit ` a di Roma Tre http://www.fis.uniroma3.it/raimondi 13-20 May 2009 c Questa opera ` e pubblicata sotto una Licenza Creative Commons. http://creativecommons.org/licenses/by-n/c-nd/2.5/it/
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The quasiclassical Keldysh Green functionLectures delivered at the Universita di Camerino
Roberto Raimondi
Dipartimento di Fisica, Universita di Roma Trehttp://www.fis.uniroma3.it/raimondi
• The Keldysh technique is the way to a quantum kinetic equation, but theresulting equations are in general very complicated and difficult to solve
• The quasiclassical approximation, which works well for degenarateFermi systems, is a concrete example, where, starting from a quantumformulation, one makes a link to the phenomenologicl Boltzmannequation for quasiparticles
• The cases which I include in these lectures show that is not onlypossible to give a microscopic justification to the Boltzmann equation,but also to derive systematically correction terms
• The quasiclassical approximation, furthermore, even though does nothave the elegance of the functional integral methods, provides very oftena much clearer physical picture
• Even at equilibrium, when diagrammatic methods are sufficient, thequasiclassical approximation has the advantage of a more compactderivation
I hope to convince you! Let us get started.
References
1. L.V. Keldysh, Sov. Phys. JETP 20, 1018 (1965).Where all began
2. L. P. Pitaevskii, E.M. Lifshitz, Physical Kinetics: Volume 10.I find this relatively short account of the basics of the Keldysh technique one ofthe best. You just learn exactly what is necessary to know in pure Landau style.
3. J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986).This is a classic reference full of hystorical and technical information. It gives therules of the game
4. P. Schwab and R. Raimondi, Annalen der Physik 12, 471 (2003).This deals mainly with quantum corrections.
5. A. Kamenev and A. Levchenko, ArXiv:0901.3586v2 (Les Houches).A modern and self-contained account based on functional integrals. The authorsreally know the subject.
Program of the lectures
1. The Keldysh technique
2. The quasiclassical approximation
3. Weak localization
4. Quantum interaction correction
5. Thermal transport and Coulomb Blockade
6. Superconductivity and Andreev scattering
1. The Keldysh technique
1. The Keldysh time path
2. The example of the Fermi gas
3. Dyson equation
4. Perturbation theory
5. Disorder and self-consistent Born approximation
Let us recall the standard perturbation theory• Split the Hamiltonian into free and interacting parts
H = H0 + HI
• Go to the interaction picture , which in terms of Schrodinger andHeisenberg pictures reads
ψi (t) = eiH0tψS(t)
ψi (t) = eiH0te−iHtψH(t)
Note that at t = 0 all pictures coincide• The time evolution in the interaction picture is given by the S-matrix
ψi (t1) = S(t1, t2)ψi (t2)
• The S-matrix is naturally time ordered
S(t1, t2) = T exp„−iZ t1
t2
dt HIi (t)«
• The problem is how to connect the time evolution of the S-matrix in theinteraction picture with the relation between the free and interactingstates of the Heisenberg picture
Trick of the adiabatic turning on
• In the far past and future there is no interaction HI → e−ε|t|HI
• Then the interaction picture state vector in the far past and future and att = 0 is related to the free and interacting Heisenberg state vector
ψi (±∞) = ψH0 ψi (0) = ψH
• To have a connection at all times one needs the S-matrix
• The vanishing of GZ is not automatic if the evolution on the two timepaths is not the same. This may happen in the presence of quantumfluctuations. We will come back to this when treating interactions.
The meaning of the rotation
• The rotation defines classical and quantum fields
• One can compute explicitly all the Green functions starting from
ψ(x, t) =1√V
Xp
eip·xe−iξpt ξp =p2
2m− µ
• The retarded and advanced Green function provide information aboutthe energy spectrum
GR,A(p, ε) =ˆε− ξp ± i0+˜−1
,
• The Keldysh Green function tells about the distribution function
GK (p, ε) =“
GR(p, ε)F (ε)− F (ε)GA(p, ε)”, F (ε) = tanh
“ ε
2T
”This is basically the the fluctuation-dissipation theorem
• Observables are given in terms of the lesser Green function
G12 =12
(GK −GR + GA) = −f (ε)(GR −GA)
where f (ε) is the Fermi function
Dyson equations• From the definition, the derivation of the Dyson equation is like in the
equilibrium case. For future use, we show both left- and right-sideequation of motion
(G−10 − Σ)G = δ(x1 − x2) G(G−1
0 − Σ) = δ(x1 − x2)
• The differential operator is a matrix in Keldysh space
G−10 (x1, x2) =
»i∂t1 −
12m
(−i∇x1 + eA(x1))2 + eφ(x1) + µ
–1δ(x1 − x2)
• While the electromagnetic potentials φ(x),A(x) are shown explicitly(with e > 0), disorder scattering and electron-electron interactions arecontained in the self-energy
• The self-energy has a triangular structure as the Green function
Σ =
„ΣR ΣK
0 ΣA
«• The Keldysh Green function can always be expressed in terms of the
retarded and advenced Green functions and Keldysh self-energyh(GR,A
0 )−1 − ΣR,Ai
GR,A = 1 GK = GRΣK GA
Perturbation expansion in an external potential
• Let U be one-body external potential. The zero and first order terms ofthe perturbation expansion are
G =
„GR
0 GK0
0 GA0
«+
„GR
0 GK0
0 GA0
«„U 00 U
«„GR
0 GK0
0 GA0
«+ . . .
• Replace GK0 with the equilibrium form GK
0 = GR0 F0 − F0GA
0
GK = (GR0 + GR
0 UGR0 )F0 −GR
0 (UF0 − F0U)GA0 − F0(GA
0 + GA0 UGA
0 ) + . . .
• By observing the partial resummations
GR = GR0 +GR
0 UGR0 +. . . , GA = GA
0 +GA0 UGA
0 +. . . , F = F0−[U,F0]+. . .
One obtains that the equilibrium relation of the Fermi gas is valid in general
GK = GRF − FGA
The standard model of disorder• We consider in the Hamiltonian a termZ
dxψ†(x)U(x)ψ(x) =Xp,p′
c†pU(p− p′)cp
• The lowest order Born approximation corresponds to the self-energy
• The average over disorder realizations is indicated by a dashed line anddefines the type of disorder model
• The white noise model
U(x)U(x′) = u20δ(x− x′)
where u0 is the scattering amplitude
Self-consistent Born approximation
• Let us evaluate the lowest order self-energy
ΣR = δ(x− x′)u20
Xp
Z ∞−∞
dε2π
e−iε(t−t′)
ε− ξp + i0+
≈ −iδ(x− x′)2πu20N0
Z ∞−∞
dξpe−i(ξp−i0+)(t−t′) µ ∼ ∞
= −iδ(x− x′)
2τδ(t − t ′) τ−1 = 2πu2
0N0
• Actually a self-consistent solution is obtained since what matters is theposition of the pole of the Green function in the complex plane, but notits distance from the real axis
ΣR = −iδ(x− x′)2πN0τ
Xp
e−i(ξp−i/2τ)(t−t′)
• The important physical assumption is that the distance of the pole fromthe real axis is small compared to the Fermi energy
εF τ � 1 or pF l � 1
The form of the self-energy
• The retarded Green function has the form
GR = (ε− ξp + i/2τ)−1 GA = (GR)∗
• The self-consistent self-energy can be generalized in the Keldysh space
Σ =δ(x− x′)2πN0τ
G(x, t ; x, t ′)
• The above self-energy can be inserted into the Dyson equation. Thisform treats the scattering from different impurities incoherently. We willcome back to this.
Summary of the first lecture
• In this lecture we have seen how the perturbation theory based on theKeldysh technique works
• In particular, we have studied the important case of the perturbationexpansion for an external potential
• We have seen how the impurity technique leads to a functional form forthe self-energy
• In the next lecture, we will introduce the so-called quasiclassicalapproximation, which will allow us to derive effective kinetic equations
2. The Quasiclassical approximation
1. The Eilenberger equation
2. Meaning of ξ-integration
3. Observables
4. Gauge invariance and Drude formula
The quasiclassical approximation
• The idea is that in order to describe the response to an externaldisturbance, it is necessary to keep all the information at the microscopiclevel
• The basic assumption is that external disturbances have length scalesmuch larger than the Fermi wave length
• The arguments of the Green function can be divided in center-of-massand relative coordinates
x =x1 + x2
2, r = x1 − x2, t =
t1 + t22
, η = t1 − t2
• The Wigner form corresponds to Fourier transform with respect to therelative coordinates
G(p, η; x, t) =
Zdre−ip·rG
“x +
r2, t +
η
2; x− r
2, t − η
2
”
The gradient expansion
• Suppose to consider a convolution (1 ≡ x1 and so for 2 and 3)
(A · B)(1, 2) =
Zd3 A(1, 3)B(3, 2)
≡Z
d3 A„
1 + 32
, 1− 3«
B„
2 + 32
, 3− 2«
=
Zd3 A
„1 + 2
2+
3− 22
, 1− 3«
B„
1 + 22
+3− 1
2, 3− 2
«≈
Zd3 A
„1 + 2
2, 1− 3
«B„
1 + 22
, 3− 2«
+ . . .
• By making the Fourier transform with respect to the relative coordinate ofthe free arguments 1− 2
(A · B)p = A(x,p)B(x,p) + . . .
• All other terms can be obtained by a gradient expansion
(AB)(p, x) = e−i(∂Ax ∂
Bp−∂
Ap∂
Bx )/2A(p, x)B(p, x)
Quasiclassical equation of motion (G. Eilenberger, Z. Phys. 214, 195 (1968) )
• Subtract LH and RH Dyson equations from one another so that the deltafunctions cancel. Then Fourier transform and keep the lowest order termin the gradient expansion»
i∂t + i1m
p · ∂x
–G(p, x) =
hΣ(p, x), G(p, x)
i• The electromagnetic potential enter through the covariant derivatives
∂tG = [∂t − ieφ(x, t + η/2) + ieφ(x, t − η/2)] G(p, η; x, t)∂xG = [∂x + ieA(x, t + η/2)− ieA(x, t − η/2)] G(p, η; x, t)
• Define the quasiclassical Green function
g(p, η; x, t) =iπ
ZdξG(p, η; x, t), ξ = p2/2m − µ
The Eilenberger equationh∂t + vF p · ∂x
ig(p, η, x, t) = −i
ˆΣ(p, x), g(p, η, x, t)
˜, gg = 1
The normalization condition is necessary since the equation is homogeneous
Eilenberger decomposition
Meaning of the ξ-integration
The momentum integration can be divided into an energy integration, whichcorresponds to the ξ-integration and an integration over the Fermi surface
GR(x1, x2) =X
p
eip·(x1−x2)
ε− ξ + i0+, r = x1 − x2
=
Zdp⊥dp‖(2π)2
eip‖r
ε− vF (p‖ − pF )− p2⊥
2m + i0+
= −iei(pF +ε/vF )r
vF
Zdp⊥2π
e−ip2⊥r/2pF
= −
s2πipF r
m2π
ei(pF +ε/vF )r
= GR0 (r, ε = 0)gR(x1, x2)
At large r = |x1 − x2| integral dominated by the extrema in the exponentialand the Green function is factorized in rapidly and slowly varying terms
Shelankov analysis (A. L. Shelankov, J. Low Temp. Phys. 60, 29 (1985))
• The definition of the quasiclassical Green function appears to be definedby a non-convergent integral
• The slowly varying part is, however, well defined due to the exponential
gR(x1, x2) =i
2π
Zdξ
eiξr/vF
ε− ξ + i0+= eiεr/vF
• According to Shelankov, in general, we may write
gR(x1, x2) =i
2π
Zdξeiξr/vF GR(p, x), p = pr
• This leads to a well-defined quasiclassical Green function
gR(p, η; x, t) = limr→0
iπ
Zdξ cos
„ξrvF
«GR(p, η; x, t)
• In the Fermi gas, for instance,
gR(p, η; x, t) = δ(η) = −gA(p, η; x, t)
Connection to observables and quasiclassical Green function
• The observables are obtained by evaluating the Green function atequal-time and for doing so it is important to integrate energy first andmomentum after
ρ(x, t) = (−e)(−i)X
p
Z ∞−∞
dε2π
G12(p, ε)
• The momentum integration is divided into ξ-integration and angleintegration X
p
· · · = 〈Z
dξ . . . 〉, 〈. . . 〉 =
Zdp . . .
• The quasiclassical Green function at equal-time interchanges the correctorder of integration
g12(p, η = 0; x, t) =
Z ∞−∞
dε2π〈g12(p, ε; x, t)〉
• Already, at equilibrium, for the Fermi gas case, this leads to problemssince
g12(p, ε; x, t) = −2f (ε), f Fermi function
which is not integrable
Continuity equation
• At equilibrium the Keldysh component is well defined since
The expression for the density shows clearly how the quassiclassical Greenfunction captures the low energy part
Impurity scattering and kinetic equation
• Disorder in the Born approximation just shifts the pole in the complexplane for the retarded and advanced Green functions, which then do notchange gR = −gA = 1
• The RHS of the Keldysh component of the Eilenberger equation yields
[Σ, g]K = ΣRgK + ΣK gA − gRΣK − gK ΣA
= − iτ
gK − 2ΣK
= − iτ
gK − 21
2πN0τ
Xp
GK (p, ε; x, t)
= − iτ
(gK − 〈gK )
which has the form of the collision integral with the scattering-in andscattering-out terms well-konwn in the Boltzmann’s equation within therelaxation-time approximation
Drude conductivity and gauge invariance: vector gauge
• A = −tE
(∂t + vF p · ∂x − evF p · E∂ε)gK =1τ
(〈gK 〉 − gK )
• By looking for a uniform solution, we take the p-wave component of bothsides
〈pgK 〉 =evF τ
dE∂εgK
• To linear order in the electric field
〈pgK 〉 =evF τ
dE∂εgK
eq
• By integration over the energy ε
j =eN0vF
2〈pgK 〉 =
eN0vF
24evF τ
2E = σDE
Drude conductivity and gauge invariance: scalar gauge
• φ(x) = −E · x Apparently the scalar potential drops out
(∂t + vF p · ∂x)gK =1τ
(〈gK 〉 − gK )
• However〈pgK 〉 = −vF τ
2∂x〈gK 〉
• We must use the condition of charge uniformity
∂xδρ(x, t) = ∂x
„eN0
2
Z ∞−∞
dε〈gK (p, ε; x, t)〉 − 2e2N0φ(x, t)«
= 0
The bottom line is the minimal substitution also in this case
∂x → ∂x − eE∂ε
Summary of the second lecture
• In this lecture we have derived the Eilenberger equation for thequasiclassical Green function
• We have applied this equation to derive the semiclassical theory ofelectrical transport
• In the next lectures we are going to apply the Eilenberger equation todifferent situations to obtain corrections to the semiclassical level
3. Weak Localization
1. Crossed diagrams
2. Backscattering
3. Corrections to conductivity
Some hystorical remarks
• Anderson localization was invented in 1958, half a century ago andperhaps would deserve an entirely dedicated lecture course
• Here, I concentrate on the phenomenon of weak localization, which is aperturbative effect on the metallic behavior
• My emphasis will be on the advantage of the quasiclassical method inorder to capture the physical origin of the phenomenon
Quantum interference and crossed diagrams
The Born approximationcorresponds to the Rainbowdiagrams. Interference betweenscattering events at different sites isnot allowed
The maximally crossed diagramsdescribe the interference betweentime-reversed paths (G. Bergmann ,Phys. Rep., 107, 1 (1984).)
Corrections to the self-energy
Let us consider a term for the Keldysh Green function
GRUGR . . .GRUGK UGA . . .GAUGA
Note that an equal number of GR and GA precede and follow GK
We take the disorder average selecting the diagrams with maximum crossing
!!K =
Disorder average yields momentumconservation at each impurity line
Q = p + p′
p′ = Q− pp′1 = Q− p1
p′2 = Q− p2
p′3 = Q− p3
The Cooperon channel
The so-called Cooperon is the sum of alldiagrams with an arbitrary number ofcrossed impurity lines (J. S. Langer and T.Neal, Phys. Rev. Lett. 16, 984 (1966).)
C(Q, !) =
!! "
2
! +"
2
The correction to the self-energy is a convolution with respect to therelative momentum
δΣK (p, ε; q, ω) =X
Q
C(Q, ω)GK (Q− p, ε; q, ω)
The momentum Q describes the effect of multiple scattering
Diffusion pole and backscatteringBy using the time-reversalinvariance one line can bereversed turning the crosseddiagrams into a geometricseries of ladder diagrams. Theladder is a particle-particlescattering channel as in theCooper mechanism forsuperconductivity
The building block
ηRA =X
p
GR(p, ε+ω
2)GA(Q− p, ε− ω
2) = 2πN0τ
h1 + iωτ − DQ2τ + . . .
i
C(Q, ω) =1
2πN0τηRA 1
2πN0τ+ · · · =
12πN0τ 2
»1
DQ2 − iω− 1–
The diffusion pole selects processes with p′ ≈ −p
A technical detail
Xp
GR(p, ε)GA(p, ε) =
Z ∞−µ
dξN(ξ)
(ε− ξ + i/2τ)(ε− ξ − i/2τ)
≈ N0
Z ∞−∞
dξ1
(ε− ξ + i/2τ)(ε− ξ − i/2τ)
= 2πN0τ
All integrals involving a number of retarded and advanced Green function canbe made in the same way by using the residue Cauchy theorem. It isimportant to note that if all the Green functions are retarded or advanced theintegral vanishes
Quasiclassical procedure
The ξ-integration yields the final form to be used in the Eilenbergerequation
1π
ZdξhδΣ, G
iK=
1π
Zdξ“δΣRGK + δΣK GA −GRδΣK −GK δΣA
”
By using
GK = GRF − FGA =12
“GRgK − gK GA
”Correction to the scattering-in term of the kinetic equation
δσk = −1π
ZdξhGR“−p, ε+
ω
2
”− GA
“−p, ε−
ω
2
”i2 12
gK (−p, ε,q, ω)X
Q
C(Q, ω)
(S. Hershfield and V. Ambegaokar, Phys. Rev. B 34, 2147 (1986).)But the story is not complete yet, since we dot know about diagrams whenthe numbers of GR and GK before and after GK is not the same
The arising of the Hikami boxIn addition to this diagram,we must consider thatobatined by the interchangeof R and A
p′ = Q− p + q/2p′1 = Q− p1 + q/2p′2 = Q− p2
p′3 = Q− p3
Correction to the scattering-out term of the kinetic equation
δΣK =1
2πN0τ
Xp1
GR“
Q− p1 +q2, ε+
ω
2
”GR“
Q− p +q2, ε+
ω
2
”× GK (p1,q, ε, ω)
XQ
C(Q, ω)
Kinetic equation with quantum corrections
(∂t + vF p · ∂x)gK (p, η, x, t) = −1τ
hgK (p, η, x, t)− 〈gK (p, η, x, t)〉
i+
1τ
Z ∞−∞
dt ′α(t − t ′)
×hgK (−p, η, x, t ′)− 〈gK (p, η, x, t ′)〉
i
α(t − t ′) = 2τ 2X
Q
Z ∞−∞
dω2π
e−iω(t−t′)C(Q, ω)
• The correction to the in- and out- terms maintain charge conservation• The non local-in-time response in the collision integral corresponds to
the memory effect of a self-retracing trajectory
Weak localization correction
j =
»1−
Z ∞−∞
dt ′α(t − t ′)–σDE
α2D(t − t ′) =Θ(t − t ′)(2π)2N0D
1t − t ′
• Renormalization of conductivity• In 2D logarithmic corrections
Z ∞−∞
dt ′α(t − t ′)⇒Z t−τ
−τφ
dt ′1
(2π)2N0D1
t − t ′=
1(2π)2N0D
lnτφτ
The controlling parameter is the conductance measured in units of thequantum of conductance
σ = σD −2e2
h1
2πlnτφτ
Scaling theory and MIT
• τ time above which diffusive motion set in• τφ maximum time over which phase memory lasts• in general τφ ∼ T−p
• in finite system at low temperatures τφ → L2/D
Dimensionless conductance
Effective perturbation theory
g =σDLd−2
2e2/h
σ = σD
„1− 1
π
1g
lnLl
«• L. P. Gorkov, A. I. Larkin, and D. E. Khmelnitskii, Pis’ma Zh. Eksp. Teor. fiz. 30,
248 (1979) [JETP Lett. 30, 228 (1979)].
• E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys.Rev. Lett. 42, 673 (1979).
Summary of the third lecture
• In this lecture we have applied the quasiclassical Green functionapproach to weak localization
• We have derived corrections term to the kinetic equation associated tothe backscattering mechanism originating from the interference ofself-retracing trajectories
• In the next lecture we are going to illustrate a different quantumcorrection originating from the electron-electron interaction
4. Quantum Interaction Corrections
1. Electron-electron interaction
2. Density of states
3. Electrical conductivity
Diffusive approximation in general
• Let us recall the Eilenberger equation for the quasiclassical Greenfunction
[∂t + vF p · ∂x] g = − 12τ
[〈g〉, g] , gg = 1
• Diffusive motion makes the Green function almost isotropic and allowsexpansion in s-wave and p-wave
g = gs + p · gp + . . . , gsgs = 1,n
gs, gp
o= 0
One obtains the Usadel diffusive equation (K. D. Usadel, Phys. Rev. Lett. 25,507 (1970).)
∂t gs − D∂x · gs∂xgs = 0
The generalized current is related to the gradient of the s-wave component
gp = −vF τ gs∂xgs
How to include electron-electron interactionHubbard-Stratonovich transformationInteraction mediated by a boson (photon)field φ (G. Zala, B. N. Narozhny, and I. L.Aleiner, Phys. Rev. B 64, 214204 (2001))
Notice the anticommutator in theKedysh component and thecommutator in the retarded andadvanced components
By means of the Keldysh rotation Ψ†φΨ⇒ Ψ†′φ′Ψ′
φ =
„φ1 00 −φ2
«⇒ φ′ =
12
„φ1 + φ2 φ1 − φ2
φ1 − φ2 φ1 + φ2
«≡ φ1′σ0 + φ2′σ1
• φ1′ classical component
• φ2′ quantum component
〈TK φ′(x1)φ′(x2)〉 =
i2
„DK (x1, x2) DR(x1, x2)
DA(x1, x2) 0
«
Interaction propagator in a diffusive (Fermi gas) metal
As for fermions, the Keldysh component is expressed in terms of thedistribution function, which at equilibrium has the standard form of the Bosestatistics
DK (q, ω) =hDR(q, ω)−DA(q, ω)
icoth
“ ω
2T
”The retarded and advanced part are given by the RPA resummation forstandard screening
DR(q, ω) =V (q)
1− V (q)χ(q, ω)χ(q, ω) = −2N0
Dq2
Dq2 − iω
I Dq2 > ω DR(q, ω) ≈ 2πe2
κ2D
II Dq2 < ω < Dqκ2D DR(q, ω) ≈ −2πe2 iωDq2
III Dqκ2D < ω DR(q, ω) ≈ 2πe2
q
Various regimes oftransferred momentum andfrequency
• I. Physics of diffusive modes and interference effects• II. Physics of coupling to modes of the e.m. environment
Eilenberger equation with interaction: three steps
• Expand in powers of φ
g = g(0) + g(1) + g(2) + . . . , g(0) =
„1 2F0 1
«• Solve up to quadratic order in φ
∂t gs − D∂x · gs∂xgs = ihφ, g
iφ =
„φ1 φ2
φ2 φ1
«• Average over the fluctuations of φ
Notice that for an external(classical) field φ2 = 0 due tocausality. Since for a quantum fieldφ2 6= 0, the Green function has anextra component gZ , which,however, must vanish afteraveraging
g =
„gR gK
gZ gA
«〈g〉φ =
„〈gR〉φ 〈gK 〉φ
0 〈gA〉φ
«
How it actually works in more detail
The normalization conditiong2 = 1 makes only twocomponents independent
gR = 1− 12
(g(1)Z + g(1)K g(1)Z ) + . . .
gA = −1 +12
(g(1)K + g(1)Z g(1)K ) + . . .
Up to second order in φ, g(1)Z and g(1)K are enough, because the currentoperator
〈(g∂xg)K 〉φ = 〈gR∂xgK + gK∂xgA〉φhas a correction of the form
After averaging, the Keldysh component must be related to retarded andadvanced components via the distribution function
〈g(2)K 〉φ = 〈g(2)R〉φF − F 〈g(2)A〉φ
First order equations
One obtains non homegenous linear differential equations
(∂t − D∂2x )g(1)K (η, x, t) = 2i
hφ1(t + η/2, x)− φ1(t − η/2, x)
iF (η, t)
(∂t + D∂2x )g(1)Z (η, x, t) = 2iδ(η)φ2(t , x)
The solution can be written in terms of the appropriate Green function for thediffusion operator
g(1)K (η, x, t) = 2iZ
dt ′x′Ltt′(x, x′)hφ1(t ′ +
η
2, x′)− φ1(t ′ − η
2, x′)
iF (η, t ′)
g(1)Z (η, x, t) = −2iδ(η)
Zdt ′x′Lt′t (x, x
′)φ2(t ′, x′)
The Green function for the diffusion operator corresponds to the ladderresummation of impurity lines in the diagrammatic approach
(∂t − D∂2x )Ltt′(x, x
′) = δ(t − t ′)δ(x− x′)
Density of states
δNN0
= 〈g(2)R(ε, x, t)〉φ = −iX
q
Z ∞−∞
dω2π
DR(q, ω)
(Dq2 − iω)2 tanh“ ε− ω
2T
”
• The diffusion poles come from g(1)Z and g(1)K
• The distribution function comes from g(1)K
• The interaction comes from 〈φ1φ2〉
Comparison with diagrammatic approach (B. L. Altshuler, A.G. Aronov A, and P.A. Lee, Phys. Rev. Lett., 44, 1288 (1980). )
(a) (b) (c)
(e) (f)(d)
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Degree of singularity of the corrections anf physical origin
• In 2D for short range interactions the correction is logarithmic
δN(ε)
N0= N0V R(0, 0)t ln |ετ |
and controlled by the parameter t = 1/(2π)2N0D ∝ 1/g as in the weaklocalization
• This is the famous zero-bias anomaly in DOS• However for long range interaction the correction is log-square!
δN(ε) = − t4
ln (|ε|τ) ln„|ε|
τD2κ42D
«This is due to the fact that the integral is dominated by regime II fortransferred momentum and frequency
Regime II corresponds to long distances where interference effects are nolonger effective. We will come back to this point.
Electrical Current
j = −eN0D2
Zdε(gs∂xgs)K = δjBorn + δjQC
(a)
(b)
(c)
(c)
(e)
(d)
• Diagrams (c), (d) and (e) correspond to δjBorn and eventually cancel• (a) and (b) correspond to δjQC and yield the interaction corrections
(B. L. Altshuler, A.G. Aronov A, and P. A. Lee, Phys. Rev. Lett., 44, 1288 (1980). )
Structure of the corrections
Note that δjBorn
• has the same structure as the density of states• gives a term proportional to the gradient of the density• and vanishes at uniform density as in a linear response calculation
δjBorn = −D∂xeN0
2
Z ∞−∞
dεδ〈g(2)K (ε, x, t)〉φ = −D∂xδρ(x, t)
δjQC
• has three ladders due to the internal derivative• the factor Dq2 and F∂xF make relevant regime I for the Coulomb
interaction• yileds the quantum interaction correction to the elctrical conductivity
• We have set the formalism for e-e interaction in disordered systems• Corrections to density of states and conductivity• Identification of two important regimes of effects
Classical period• Chester and Tellung (1960)• Langer (1962)
General validity based on:• Independent quasiparticles• Fermion statistics• Elastic scattering
Modern period• Castellani, Di Castro, Kotliar, Lee,
Strinati (1987,1988)• Livamov, Reizer, and Sergeev (1991)• Niven and Smith (2003)
With quantum corrections
Heat current at semiclassical level• Consider the energy density in terms of the quasiclassical Green
function
ρQ = πN0(−i∂η)gKs (η, x, t)|η=0 = −N0
Z ∞−∞
dε εgKs (ε, x, t)
• Check that it gives the correct value for the specific heat of the electrongas by using
gKs (η, x, t) = −2i
πP 1η
+ 2iπT 2
6η + . . .
so that
ρQ =2π2N0
6T 2
• Derive each term of the Eilenberger equation with respect to η, therelative time, set η = 0 and take the angle average
∂t (−i∂η)gKs (η, x, t)|η=0 + ∂x
vF
d(−i∂η)gK
p (η, x, t)|η=0 = 0
• One gets a continuity equation for the energy and an expression for theenergy current
jQ =πN0vF
d(−i∂η)gK
p (η, x, t)|η=0
A fourier transform
The semiclassical result
• The p-wave is related to the s-wave by the usual relation
gKp (η, x, t) = −l∂xgK
s (η, x, t)
• Space dependence via local equilibrium T (x)
−l∂xgKs (η, x, t) = −2iπl
T (x)
3η∂xT (x)
• By taking the derivative with respect to η
−i∂η
„−2iπl
T (x)
3η∂xT (x)
«= (−∂xT (x)) 2πl
T (x)
3
• The thermal current becomes
jQ = (−∂xT (x))π2
32N0vF l
dT (x) = (−∂xT (x))
π2
3e2 σDT
Quantum Corrections to Heat Current
The strategy is as for electrical current:• There are two terms
δjQ =N0D
2
Zdε εδ(gs∂xgs)K = δjaQ + δjbQ
• Instead of the charge vertex e, there appears an energy vertex ε• The first term has the same structure as the density of states
δjaQ = DN0∇TZ
dε εZ
dω2π
∂T (Fε−ω(x)Fε(x))× ImX
q
DR(ω,q)
(−iω + Dq2)2
• The second term instead is like the quantum corrections to the electricalconductivity
δjbQ = DN0∇TZ
dε εZ
dω2π
Fε(x)∂T Fε−ω(x)× 4d
ImX
q
Dq2DR(ω,q)
(−iω + Dq2)3
Conclusions about the Wiedemann-Franz law
• Previous literature shows some controversy• Castellani, Di Castro, Kotliar, Lee, Strinati (1987) RG analysis confirms W-F• Livanov, Reizer, Sergev (1991) finds extra terms that violate W-F• Niven and Smith (2003) also find extra terms• Apparently no explanation for the different results• Our analysis clarifies the issue
• The final result shows an extra term which violates the W-F law
κ =π2
3k2
BTe2
„σD + δσ +
12
e2
πhln(~Dκ2
2d/kBT )
«• The term that is in agreement with W-F comes from integration of
diffusive modes corresponding to energies T < ω < τ−1 which is regimeI
• The term violating W-F is due to energies ω < T in regime II and isbeyond the reach of RG analysis since the singularity is purely infrared
The Coulomb blockade theory
(Yu. V. Nazarov, Zh. Eksp. Teor. Fiz. 95, 975 (1989) [JETP Lett. 66, 214 (1989)]. )The physics associated to the violation of the W-F is also clearly seen whenconsidering the Coulomb interaction affecting transport through a tunneljunction
Left Right
0 x
In the Eilenberger equation we neglectspace dependence but for which side thequasiclassical Green function belongs to
∂t gi (η, t) = iehφi , gi
i, i = L,R
The solution can be expressed as a gauge transformation, since thefluctuating Hubbard-Stratonovich field depends on time only
If only one electrode were present the field φ could be eliminated by a gaugetransformation. This explains why the singularities in the density of statesdrop out from physical quantities
Tunneling current
• We need boundary conditions connecting the quasiclassical Greenfunctions on the two sides of the junction
• In general, for arbitrary tunnel transmittance this is a difficult problem• We confine to the tunneling limit within the Bardeen model with tunneling
amplitude TLR
ΣL = TLRGR(0, 0)T ∗LR ΣR = TRLGL(0, 0)T ∗RL
• The tunneling current is then the Keldysh component of the commutator
j =GT
8e
Z ∞−∞
dε〈[gL(ε, t), gR(ε, t)]K 〉φ
GT ∝ TLRT ∗LR tunneling conductance• In the absence of interaction standard tunneling theory
How interaction affects tunneling by changing the density of the states
• Gauge transformation: eliminate the interaction from one lead, say theleft one
• The average over the fluctuating field factorizes
j =GT
8e
Z ∞−∞
dε(gRL − gA
L )(gRR − gA
R)(FR − FL) FL,R = tanh„ε+ eVL,R
2T
«
What is left is the average over the φ field of the retarded Green function
〈gRR (t1, t2)〉φ = 〈
“eiϕ(t1)
”1i
„δ(t1 − t2) 2F (t1, t2)
0 −δ(t1 − t2)
«ij
“−eiϕ(t1)
”j1〉φ
The matrix structure is associated to the quantum component φ2
e±iϕ( t) = e±iϕ1(t)„
cosϕ2(t) ±i sinϕ2(t)±i sinϕ2(t) cosϕ2(t)
«〈eiφ〉φ = e−(1/2)〈φ2〉
P-theory
The effect of the interaction is condensed into a single function
J(t) = e2Z ∞−∞
dω2π
ImDR(ω)
»1− cos(ωt)
ω2 − isin(ωt)ω
–and in the Fourier transform of its exponential
P(ω) =
Z ∞−∞
dteiωteJ(t)
The correction to the density of states depends of the spectral density of theinteraction
δ〈gRR (ε)〉φ =
Z ∞−∞
dω2π
12
(FR(ε+ ω)− FR(ε− ω))P(ω)
G.-L. Ingold and Yu. V. Nazarov, in [91], chapter 2 Single Charge Tunneling, edited byH. Grabert and M. Devoret, NATO ASI Series B, Vol. 294, (Plenum Press, New York,1992).
Resistor-Capacitor Model
q C
R
• When an electron tunnels through the junction, a charge is added to the(right) lead
• The charge creates an extra potential• After some time, the charge relaxes and the lead goes back to a charge
neutrality condition• The process may be represented in terms of an equivalent circuit with a
capacitor and a resistor in parallel
Resistor-Capacitor Model: continuedThe solution of the circuit equations
Q(t) = qe−t/τ
U(t) =qC
e−t/τ
where τ = RC
allows to obtain the effective interaction
∂tQ(t) = qδ(t)
U(t) =
Z ∞−∞
dt ′DR(t − t ′)Q(t ′)
which defines the screened interaction
By setting q = e and defining the charging energy Ec = e2/(2C)
J(t) = Ec
Z ∞−∞
dωπτ
1ω2 + τ−2
»1− cos(ωt)
ω2 − isin(ωt)ω
–In the limit of large τ
J(t) ≈ iEc t − EcTt2
and for T → 0
P(ω) = 2πδ(ω + Ec)
• Suppression of the density of statesand blocking of tunneling
• Resummation of the zero-biasanomaly in DOS
〈gRR (ε)〉φ = Θ(|ε| − Ec)
Summary of the fifth lecture
• Thermal transport: continutity equation and heat current• Two types of corrections to the heat current• Coulomb blockade
6. Superconductivity
1. General equation
2. Landau-Ginzburg limit
3. Andreev scattering and ZBA
Quasiclassical formulation
Eilenberger’s eq. ∂t
nσz , g
o− iε [σz , g]− D∂xg∂xg = i
ˆ∆, g
˜gg = 1
We consider the diffusive limit, but the clean limit can be also analyzed
Keldysh space σz =
„σz 00 σz
«g =
„gR gK
0 gA
«∆ =
„∆ 00 ∆
«The phase of the order parameter is fixed
Nambu spacegR,A = GR,Aσz + iF R,Aσy , gK = gR f − f gA f = f0σ0 + fz σz
The distribution function f0 ± fz refer to particles and holes, respectively
Self-consistency ∆ =gN0π
2gK (η = 0, x, t) ∆ = i∆σy
Larkin A I and Ovchinikov Y N in Non equilibrium superconductivity 1986 eds.Langeberg D N and Larkin A I (NorthHolland)
Uniform equilibrium case
Distribution functions
Normalization condition
f0 = tanh“ ε
2T
”fz = 0
(GR)2 − (F R)2 = 1
Explicit solution
GR,A(ε) = ± εp(ε+ i0+)2 −∆2
F R,A(ε) = ± sign(ε)∆p(ε+ i0+)2 −∆2
,
∆ =gN0π
2
Z ∞−∞
dε2π
tanh“ ε
2T
”(F R(ε)− F A(ε))
Direct evaluation from BCS theory
The same expression can of course be obtained by the ξ-integration of theBCS Green functionIn BCS we have
u2p =
12
„1± ξp
Ep
«v2
p = 1− u2p E2
p = ξ2p + ∆2
GR =iπ
Z ∞−∞
d ξp
u2
p
ε− Ep + i0++
v2p
ε+ Ep + i0+
!
=
Z ∞−∞
d ξp
“u2
pδ(ε− Ep) + v2pδ(ε+ Ep)
”=
Z ∞0
d ξpδ(ε− Ep)
=εp
(ε+ i0+)2 −∆2
Landau-Ginzburg Equation for the order parameter
• Consider the Eilenberger equation for the anomalous part F and expressG using the normalization condition
• The Green function P plays the role of propagator of the Cooper pair
(−2iε∓ D∂2x )PR,A
ε (x− x′) = −2iδ(x− x′)
• To solve the integral equation, expand the order parameter
∆(x′) = ∆(x′+x−x) = ∆(x)+(x′−x)·∂x∆(x′)+12
(x′−x)i (x′−x)j∂2ij ∆(x)+. . .
• Plug F into the self-comsistency equation
gN0
»ln
TTc
+7ζR(3)
8π2T 2 ∆2(x)− πD8Tc
∂2x
–∆(x) = 0
A technical detail: the coefficient of the linear term
Z ∞−∞
dε2π
tanh“ ε
2T
”(F R(ε, x)− F A(ε, x))
= ∆(x)
Z ∞−∞
dε2π
tanh“ ε
2T
”(
Zd x′PR
ε (x− x′) +
Zd x′PA
ε (x− x′))
= ∆(x)
Z ∞−∞
dε2π
tanh“ ε
2T
”„ 1ε+ i0+
+1
ε− i0+
«≈ 4∆(x) ln
ωD
T
A technical detail: the coefficient of the derivative term
16∂2
x ∆(x)
Z ∞−∞
dε2π
tanh“ ε
2T
”(
Zd x′PR
ε (x′)x′2 +
Zd x′PA
ε (x′)x′2)
= iD2∂2
x ∆(x)
Z ∞−∞
dε2π
tanh“ ε
2T
”„ 1(ε+ i0+)2 −
1(ε− i0+)2
«= −i
D2∂2
x ∆(x)
Z ∞−∞
dε2π
tanh“ ε
2T
”∂ε
„1
(ε+ i0+)− 1
(ε− i0+)
«=
πD2T
A technical detail: the coefficient of the cubic term
Solve the cubic term by using the first order result
F R,A(3)(ε, x) = ±12
„Zdx′PR,A
ε (x′)«3
∆3(x) = ±12
∆3(x)
(ε± i0+)3
To Plug it into the self-consistency condition, integrate over energy
18
∆3(x)
Z ∞−∞
dε2π
tanh“ ε
2T
”„ 1(ε+ i0+)3 +
1(ε− i0−)3
«= − 1
π2T 2
∞Xn=0
1(2n + 1)3
= −7ζR(3)
8π2T 2
Tunneling current• The boundary condition in the tunneling limit as in the case of Coulomb
blockade (M. Y. Kuprianov and V. F. Lukichev, Sov. Phys. JETP 64, 139(1988).)
I =σ
egR∂xgR =
GT
2e[gR , gL],
• We allow for the phase of the order parameter
gR(A)i = (iF R(A)
i sin(φi ), iFR(A)i cos(φi ),G
R(A)i )
• The current has two components
j =GT
8e
Z ∞−∞
dε(IJ + IPI)
• The first term IJ is the Josephson current which is different from zeroonly the order parameter on the two sides of the junction has a differentphase
IJ = isin(φ1 − φ2)hf0R(F R
R − F AR )(F R
L + F AL ) + f0L(F R
R + F AR )(F R
L − F AL )i
• In the Josephson only the symmetrix part of the distribution functionenters
f0L,R =12
(tanh((ε+ eVL,R)/2T ) + tanh((ε− eVL,R)/2T ))
Clearly the Josephson current exists even at zero voltage drop
Andreev scattering A. F. Andreev, Sov. Phys.JETP 19, 1228 (1964).
• The second term is called quasiparticle and interference current
IPI =h(GR
L −GAL )(GR
R −GAR) + cos(φ1 − φ2)(F R
R + F AR )(F R
L + F AL )i
(fzL−fzR).
• It can be non zero only when a bias is applied
fzL,R =12
(tanh((ε+ eVL,R)/2T )− tanh((ε− eVL,R)/2T ))
• The terms with GR and GL yields the standard quasiparticle current• The terms with FL and FR yields Andreev scattering
Andreev scattering and proximity effect
These two phenomena are two aspects of the same thing: at the S-Ninterface one electron is scattered back as a hole and this makes theGorkov’s function F leaking in the normal side, i.e., yielding the proximityeffect
Experiments show anenhancement at low voltage(Kastalskii et al. (1991)) thatcannot be explained by thesimple BTK theory
j =GT
8e
Z ∞−∞
dεh(F R
R + F AR )(F R
L + F AL )i
(fzL − fzR).
F RL 6= 0
Superconductor at equilibrium
F RR (ε, x), fz(ε, x)
To be determined by solving the kinetic equations
Kinetics equations (Zaitsev (1990), Volkov and Klapwijk (1992))
• First solve the spectral problem
∂xgR(A)∂xgR(A) = iεhσz , gR(A)
i• Plug the result into the equation for the Kedysh component
∂x(gR∂xgK + gK∂xgA) = 0
• Express gK = gR f − f gA
∂x
h∂x f − gR(∂x f )gA
i− (gR∂xgR)(∂x f )− (∂x f )(gA∂xgA) = 0
• Finally obtain a diffusion-like equation with an energy andposition-dependent coefficient
∂x
h(1−GRGA − F RF A)∂xfz
i= 0
fz(x) = m(x)fz(L)− fz(0)
m(L)+ fz(0) m(x) =
Z x
0
dx ′
1−GR(x ′)GA(x ′)− F R(x ′)F A(x ′)
Zero bias anomaly (ZBA)
• On the normal side the diffusion equation with boundary condition (forL→∞) F (ε,∞) = 0 yields
F R(ε, x) = F F (ε, 0)ei√
2i|ε|/Dx ≡ F RR ei√
2i|ε|/Dx
• To connect the Gorkov’s function on the two sides we use the standardboundary conditions (current conservation) which are valid for a weakproximity effect
σSe
gRR∂RgR
R =GT
2e[gR
R , gRL ] ⇒ iF R
R =
sD
2i|ε|GT
2σSF R
L
One sees that the Gorkov function on the normal side is smaller by afactor GT with respect to that on the superconducting side but there issquare-root of the energy at the denominator
• The smallness of the factor G2T is compensated by the an effective
conductance of the normal side (at low T , LT � L)
GNIS ≈G2
T
GeffN
GeffN =
σSLT
LT =
rDT
Summary of all the lectures
• With superconductivity we have ended our tour through the applicationsof the quasiclassical Green function approach
• My aim has been that of showing the flexibility of the method and itsvaste range of applicability
• I think that in most cases is more powerful of the conventionaldiagrammatics, but its application is not straightforward. It requires somephysical understanding of the physical phenomenon under study
• A current field of application of the method is to systems with spin-orbitinteraction and spin-splitted Fermi surfaces