Field Theory for Disordered
Conductors
Vladimir Yudson
Institute of Spectroscopy, RAS
University of Florida, Gainesville, March 17, 2005
April 7, 2005
Electrons in a random potential
[E - H0 – U(r) ± iδ/2]G R(A) (r; r`) = δ( r – r` )
Physical quantities:
Global DOS:
ν(E) = Ld m δ( E – Em) = i/(2πLd )Tr{[GR(E) - GA(E)]}
Conductance:
g = gxx = 1/(2 L2)Tr{vx[GR -GA]vx[GR -GA]};
(E = EF )
“Gang of 4”: Scaling hypothesis
Gor’kov, Larkin, Khmelnitskii
(renormalizability)
Wegner: (bosonic) nonlinear σ-model
1979
no interaction
2d
4
Efetov, Larkin, Khmelnitskii: fermionic σ-model
Efetov: supersymmetric σ-model (for < g > )
1980
1990
Altshuler; Stone & Lee
(mesoscopics)
Kravtsov & Lerner; Kravtsov, Lerner, Yudson:
(anomalies in the extended σ-model (for <gn > ))
Altshuler, Kravtsov, Lerner: log-normal tails of
distribution functions P(g), P(ν), P(t)
2005
2000
Muzykantskii & Khmelnitskii
(secondary saddle-point;
tails of P(t); ballistic effects)
Content
• Averaged quantities and nonlinear σ-model
• Extended σ-model, averaged moments, and log-normal
tails of P(g), P(ν), and P(t)
• Secondary saddle-point approach and log-normal tails of
P(t)
• Hypothetical relationship between P(ν) and P(t)
• Field theory for P(ν)
Averaged quantities - diagrammatics
[G0-1(E) - U ± iδ/2]G R(A) (r; r`) = δ( r – r` )
 U (r )U (r ' )   (r  r ' )
=
+
Expansion in 1/kFl << 1
l  vF
1

+
and , Ecτ ~ (l/L)2 << 1
 2 ( E F )
GR(E+)
=
GA(E)
...
+…+
Singular diagrams
1
D ( q,  ) 
 i  Dq 2
Diffuson
p
GR(E+)
=
p`
+
+
q = p + p`
GR(E+)
=
p`
+ ...
GA(E)
Cooperon
p
q = p – p`
GA(E)
+
+
+ ...
Field theory and nonlinear σ-model
Basic object: Green’s function
Primary representation:
Primary variable: local field Ψ(r)
Price for the absence of the denominator:
Replica trick: {Ψi ; i = 1, …, n  0 }
OR supersymmetry: Ψ = (S,
)
OR ….
Averaging over disorder
<U(r)U(r') =  (r r')
Hubbard-Stratonovich decoupling
“Primary” saddle-point: Q2 = I;
1/ =
2
Q(r)
Q = V z V-1
symmetry breaking term
Moments <gn>, <n> (modifications: Ψ  Ψi ; Q  Qij ; i,j = 1, …, n
)
“Hydrodynamic” expansion ( , l/L << 1 ) :
Str ln {…} = (d D/4) Str{ (Q)2 + 2i/D Q +  usual σmodel
+ c1l2 (Q)4 + c2l2 (2 Q)2 + c3 2/D (Q)2 + … } 
“extended”
Perturbation theory
Parametrization: Q = (1– iP)Λz (1 – iP)-1 ;
PΛz = – Λz P
Q = Λz (1 + 2iP – 2P2 – 2iP3 + 2P4 + . . . )
Str ln {…} =  d Str{D(P)2 – iP2 + # P4 + …}
RG transformation of the usual nonlinear sigma model
Separation of “fast” and “slow” degrees of freedom:
Q2 = I;
Q = V z V-1 → Vslow Vfast z Vfast-1 Vslow-1
Qfast
RG equation ( g = 2D )
d ln g
a b
  ( g )  d  2   2  ...
d ln L
g g
One-parameter scaling hypothesis
Moments <gn>, <n> (modifications: Ψ  Ψi ; Q  Qij ; i,j = 1, …, n
)
“Hydrodynamic” expansion ( , l/L << 1 ) :
Str ln {…} = (d D/4) Str{ (Q)2 + 2/D Q +  usual σmodel
+ c1l2 (Q)4 + c2l2 (2 Q)2 + c3 2/D (Q)2 + … } 
“extended”
Anomalies (KL, KLY)
RG transformation of additional scalar vertices
Q
RG transformation of additional vector vertices
Q
+

+
+
Q

 +
Conformal structure: ±Q = xQ ± iyQ

Growth of charges: zn ~ zn (0) exp [u (n2 – n)]
u = ln(σ0/σ)  (1/g) ln(L/l) << 1
d=2
AKL: Growth of cumulant moments:
<(g/g)n >c ~ <(/)n >c ~ g1-n (l/L)2(n–1) exp [u (n2 – n)],
n > n0 ~ u-1 ln (L/l) ~ g ;
~ g2–2n ,
n < n0
Log-normal asymptotics of distribution functions:
P(x) ~ exp[– (1/4u) ln2(x/) ],
x = / > 0
 = 1/( Ld) - mean level spacing
OR
g/g < 0
AKL: Long-time current relaxation - distribution of relaxation times
Anomalous contribution to ()  log-normal asymptotics of (t)
(t) ~ exp[– (1/4u) ln2(t/g) ] ~  exp[– t/t ] P(t) dt
Distribution of relaxation times:
P(t ) ~ exp[– (1/4u) ln2(t /)
]
Relationship between P( ) and P(t ) (Kravtsov; Mirlin)
Energy width of a “quasi-localized” state: E ~ 1/ t
Anomalous contribution to DOS:  ~ 1/E ~ t
P( )  P(t )
Muzykantskii & Khmelnitskii: Secondary saddle-point approach
<g(t)> = g0exp(– t/) +  d/2 exp(-it)  DQ […] exp (S[Q]) ,
S[Q] = /4  dr Str{ D(Q)2 + 2iQ }
Parametrization: Q =VHV-1 ; H = H(, F)
Saddle-point equation for (r): 2 + 2sinh  = 0 ; 2 =
i/D
Boundary conditions: |leads = 0 ; n|insulator = 0
Integration over   self-consistency equation:
 dr/Ld [cosh  – 1] =
t/
d=2 , disk geometry (Mirlin):
=  (r) - singular at r  0
(breakdown of the diffusion approximation!)
Requirement: |(r)| <
1/l
Cutoff r* < r : (r*) =
0
r* ~ C l
( = 1, 2,
4)
g(t) ~ (t)– 2g , 1 << t << (R/l)2
r*
Solution:
g(t) ~ exp[– g ln2(t/g)/4ln(R/l)] , t >> (R/l)2
Coincides with the RG result of AKL
R
However, no alternative way for P() and P(g)
(no field theory for P() and P(g) !)
Wanted:
Field theory for distribution function
Requirements:
disorder averaging
slow functional description
RG analysis
non-perturbative solutions
Standard routine:
Primary field-theoretical representation:
fields
Averaging over disorder
Hubbard-Stratonovich decoupling, new field Q(r)
Primary saddle point approximation ( Q2 = I )
Slow Q-functional (nonlinear σ-model )
Non-perturbative solutions (secondary saddle point)
Is all this possible for distribution functions?
Results:
Effective field theory for the characteristic function
“Free action”:
“Source action”:
k = diag{1, -1}
Q(r) = { Qr1,r2(r) }
Q2(r) = I
C = [1 + 422]-1
Derivation
Primary representation:
- bi-local primary superfields
K = diag{, I}
Explicitly:
= z + iy
Integrating over
Finally:
Averaging over disorder and H-S decoupling of
by a matrix field
As a result
“Primary” saddle-point approximation
Saddle-point:
Slow functional:
Correspondence with the perturbation theory
ln P(s) =  s + (s/2)2 q0 (Dq2)2 +
…
</> = 0 ; < (/)2 >c = 2/(22) q0
(Dq2)2
Non-trivial “accidental” cancellation of
two-loop contributions to < (/)3 >c
Beyond the perturbation theory
Secondary saddle-point equation:
- a sub-manifold of the saddle-point manifold
smooth dependence on (r1 + r2)/2
small difference |r1 – r2| ~ l
r1
r
r2
Wigner representation:
k
In the considered hydrodynamical approximation
(non-extended σ-model):
Long-tail asymptotics of
Inverse transformation:
Parametrization of the bosonic sector:
Boson action:
Self-consistency equation
Saddle-point equation for boson action
MK equation, for comparison:
2 (r) + 2sinh (r) = 0 ; 2 = i/D
Formal solution
Green’s function of Laplace operator
Self-consistency at r = r`
Trivial solution:
Non-trivial solution for small
Saddle-point action leads to:
:
Conclusions
A working field-theoretical representation has been
developed for DOS distribution functions
The basic element – integration over bi-local primary
superfields
Slow functional (in the spirit of a non-extended nonlinear
σ-model) has been derived
Non-perturbative secondary saddle-point approach has led
to a log-normal asymptotics of distribution functions
The formalism opens a way to study the complete statistics
of fluctuations in disordered conductors