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

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/

Introductory remarks

• 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

ψH = S(0,−∞)ψH0 ψi (t) = S(t , 0)ψH

• From state vectors to operators

AH(t) = S−1(t , 0)Ai (t)S(t , 0)

• By introducing S ≡ S(∞,−∞) the key formula is

〈ψH|T (AH(t1)BH(t2))|ψH〉 = 〈ψH0 |S−1T (Ai (t1)Bi (t2)S)|ψH0〉

i.e. we have transformed average over the interacting state in anaverage over the free one

The Keldysh Time Contour• The problem with the previous formula is that S is time ordered, while

S−1 is anti-time ordered• The problem is solved by invoking the adiabatic theorem. The state at

t =∞, adiabatically evolved from the state at t = −∞ acquires just adivergent phase factor (Gell-Mann and Low (1951))

ψH0 = exp(iα)ψH0 S = exp(iα)

• The unpleasent S−1 factor can be moved into the denominator

〈ψH |T (AH(t1)BH(t2))|ψH〉 =〈ψH0 |T (Ai (t1)Bi (t2)S)|ψH0〉

〈ψH0 |S|ψH0〉• There is an alternative strategy: the Keldysh contourCK = (−∞,∞)

S(∞,−∞), which includes both time-ordered and

antitime-ordered factors via the ordering operator TK . The S-matrixbecomes

SK = TK exp

−iZCK

dt HIi (t)

!• The key formula for perturbation theory becomes

〈ψH |T (AH(t1)BH(t2))|ψH〉 = 〈ψH0 |TK (Ai (t1)Bi (t2)SK )|ψH0〉• For a stationary state both strategies can be used. For a non-equilibrium

state one must use the Keldysh strategy.

The Keldysh Green function

The beauty of the Keldysh time path is that the definition of the Greenfunction is like the equilibrium case

G(x1, x2) = −i〈TK Ψ(x1)Ψ†(x2)〉, xi = (xi , ti )

The price to pay is that there are actually several Green functions to calculate.

It is convenient to introduce a matrixstructure in the so-called Keldysh space toindicate which Keldysh-contour branch thetime arguments belong to

G(x1, x2)→ G =

„G11 G12

G21 G22

«

Time− ordered G11(x1, x2) = −i〈T Ψ(x1)Ψ†(x2)〉Lesser(Densitymatrix) G12(x1, x2) = +i〈Ψ†(x2)Ψ(x1)〉

Greater G21(x1, x2) = −i〈Ψ(x1)Ψ†(x2)〉Antitime− ordered G22(x1, x2) = −i〈T Ψ(x1)Ψ†(x2)〉

The Green functions are not all independent and obey the causality condition

G11 + G22 = G12 + G21

The Keldysh rotation

• Due to the fact that the components of G are not independent, it isconvenient to transform the matrix G as

G→ 12

„1 −11 1

«„1 00 −1

«G„

1 1−1 1

«• In this new representation the Green’s function is

G =

„GR GK

GZ GA

«=

12

„G11 −G22 −G12 + G21 G11 + G22 + G12 + G21

G11 + G22 −G12 −G21 G11 −G22 + G12 −G21

«• The rotated components are

GR(x1, x2) = −iΘ(t1 − t2)〈Ψ(x1)Ψ†(x2) + Ψ†(x2)Ψ(x1)〉GA(x1, x2) = +iΘ(t2 − t1)〈Ψ(x1)Ψ†(x2) + Ψ†(x2)Ψ(x1)〉GK (x1, x2) = −i〈Ψ(x1)Ψ†(x2)−Ψ†(x2)Ψ(x1)〉GZ (x1, x2) = 0

• 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

ψcl =1√2

(ψ1 + ψ2)

ψqu = − 1√2

(ψ1 − ψ2)

• The connection with the various Green functions

GK → 〈ψclψcl〉GR → 〈ψclψqu〉GA → 〈ψquψcl〉GZ → 〈ψquψqu〉

A first example: the Fermi gas at equilibrium

• 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

gK (p, ε; x, t) = gR(p, ε; x, t)F0(ε)− F0(ε)gR(p, ε; x, t) = 2(1− 2f (ε))

is integrable due to antisymmetry of the hyperbolic tangent and

gK (p, η = 0; x, t) = 0

• The Eilenberger equation for gK with φ = 0, A = 0 and Σ = 0 reads

(∂t + vF p · ∂x)gK = 0

• Then by averaging over p and integrating over ε

∂teN0

2

Z ∞−∞

dε〈gK 〉+ ∂x ·eN0

2

Z ∞−∞

dε〈vF pgK 〉 = 0

which yields the continuity equation for charge and current

δρ(x, t) =eN0

2

Z ∞−∞

dε〈gK (p, ε; x, t)〉 j =eN0

2

Z ∞−∞

dε〈vF pgK 〉

Continuity equation in the presence of potentials

Let us insert the potentials φ and A

(∂t − e∂tφ(x, t)∂ε + vF p · ∂x + evF p · ∂tA(x, t)∂ε)gK = 0

Again by averaging over p and integrating over ε

∂tδρ(x, t) + ∂x · j(x, t) = 0

with

δρ(x, t) =eN0

2

Z ∞−∞

dε〈gK (p, ε; x, t)〉 − 2e2N0φ(x, t)

j(x, t) =eN0

2

Z ∞−∞

dε〈vF pgK 〉

since Z ∞−∞

dε∂εgK (p, ε; x, t) = gK (p, ε; x, t)|∞−∞ = gKeq(p, ε; x, t)|∞−∞ = 4

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))

D(x1, x2) = −i〈TKφ(x1)φ(x2)〉

Boson on the Keldysh contour

D11(x1, x2) = −i〈Tφ(x1)φ(x2)〉D12(x1, x2) = −i〈φ(x2)φ(x1)〉D21(x1, x2) = −i〈φ(x1)φ(x2)〉D22(x1, x2) = −i〈Tφ(x1)φ(x2)〉

As in the case of Fermions,also Boson Green functionsobey the condition

D11 +D22 = D12 +D21

Boson-fermion interaction matrix structure

In Keldysh space Ψ†φΨ ≡ Ψ†„φ1 00 −φ2

«Ψ

φ1(x) t ∈ (−∞,∞)φ2(x) t ∈ (∞,−∞)

By making the Keldysh rotation as for fermions

DK (x1, x2) = −i〈nφ(x1), φ(x2)

o〉

DR(x1, x2) = −iΘ(t1 − t2)〈[φ(x1), φ(x2)]〉DA(x1, x2) = −iΘ(t2 − t1)〈[φ(x1), φ(x2)]〉

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

〈δ(g∂xg)K 〉φ = ∂x〈g(2)K 〉φ +12〈hg(1)K (∂xg(1)Z )2F + 2F (∂xg(1)Z )g(1)K

i〉φ

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

δjQC = N0DZ ∞−∞

dω2π

Z ∞−∞

dεX

q

(F (ε, x)∂xF (ε− ω, x))2D2q2DR(q, ω)

(Dq2 − iω)3

δσ = − e2

2π2~ln„

1T τ

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Summary of the fourth lecture

• 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

5. Thermal transport and Coulomb blockade

1. Definition of heat current

2. Quantum corrections and W-F law

3. Transport through a tunnel junction

4. Coulomb Blockade

Some history

Pre-history• Wiedemann-Franz• Boltzmann-Drude-Sommerfeld

κ

σT= L0 =

π2k2B

3e2

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

gi (η, t) = eiϕi (t+η/2)g0,i (η, t)e−iϕi (t−η/2) ϕi (t) =

Zdtφi (t)

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

(−2iε∓ D∂2x )F R,A(ε, x) = ∓2i∆(x)(1 + (1/2)(F R,A(ε, x))2)

• At first limit to linear term in ∆ and F

F R,A(ε, x) = ±Z

d x′PR,Aε (x− x′)∆(x′)

• 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

Andreev scattering in pictures

Blonder, Tinkham, Klapwijk (1982), Shelankov (1980)

NIS structure

S I NLeft Right

x0 L

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

Thank you for listening!