Field Theory for Disordered Conductors

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Field Theory for Disordered Conductors
Vladimir Yudson Institute of Spectroscopy,
RAS

• University of Florida, Gainesville, March 17,
  2005
• 
  April 7, 2005

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Electrons in a random potential
E - H0 U(r) id/2G R(A) (r r) d( r r )
Physical quantities Global DOS ?(E) L?d ?m
d( E Em) i/(2pLd )TrGR(E) -
GA(E) Conductance g gxx 1/(2
L2)TrvxGR -GAvxGR -GA (E EF )

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Gang of 4 Scaling hypothesis Gorkov, Larkin,
Khmelnitskii (renormalizability) Wegner
(bosonic) nonlinear s-model
no interaction
2 ? d ? 4
2005
2000
1990
Kravtsov Lerner Kravtsov, Lerner, Yudson
(anomalies in the extended s-model (for ltgn gt
)) Altshuler, Kravtsov, Lerner log-normal tails
of distribution functions P(g), P(?), P(t?)
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Content

• Averaged quantities and nonlinear s-model
• Extended s-model, averaged moments, and
  log-normal tails of P(g), P(?), and P(t?)
• Secondary saddle-point approach and log-normal
  tails of P(t?)
• Hypothetical relationship between P(?) and P(t?)
• Field theory for P(?)

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Averaged quantities - diagrammatics
G0-1(E) - U id/2G R(A) (r r) d( r r )


Expansion in 1/kFl ltlt 1
and ??, Ect (l/L)2 ltlt 1
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Singular diagrams
Diffuson
q p p
q p p
Cooperon
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Field theory and nonlinear s-model
Basic object Greens function
Primary representation
Primary variable local field ?(r)
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Averaging over disorder
ltU(r)U(r') ? ?(r - r')
1/? 2???
Hubbard-Stratonovich decoupling
Primary saddle-point Q2 I Q V ?z V-1
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Moments ltgngt, lt?ngt (modifications ? ? ?i Q ?
Qij i,j 1, , n )
Hydrodynamic expansion ( ??, l/L ltlt 1 )
Str ln ?(??d D/4) Str (?Q)2 2i?/D ?Q
? usual s-model
c1l2 (?Q)4 c2l2 (?2 Q)2 c3? ?2/D (?Q)2
? extended
Perturbation theory Parametrization Q (1
iP)?z (1 iP)-1 P?z ?z P Q ?z
(1 2iP 2P2 2iP3 2P4 . . . )
Str ln ? ??d StrD(?P)2 i?P2 P4

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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
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One-parameter scaling hypothesis
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Moments ltgngt, lt?ngt (modifications ? ? ?i Q ?
Qij i,j 1, , n )
Hydrodynamic expansion ( ??, l/L ltlt 1 )
Str ln ?(??d D/4) Str (?Q)2 2?/D ?Q
? usual s-model
c1l2 (?Q)4 c2l2 (?2 Q)2 c3? ?2/D (?Q)2
? extended
Anomalies (KL, KLY)
RG transformation of additional scalar vertices
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RG transformation of additional vector vertices
Conformal structure ?Q ?xQ i?yQ ?
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AKL Growth of cumulant moments lt(?g/g)n gtc
lt(??/?)n gtc g1-n (l/L)2(n1) exp u (n2
n),
n gt n0 u-1
ln (L/l) g
g22n , n lt n0
Log-normal asymptotics of distribution functions
P(x) exp (1/4u) ln2(x/??) , x ??/? gt
0 OR ?g/g lt 0
? 1/(? Ld) - mean level spacing
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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?
P(? ) ? P(t? )
Anomalous contribution to DOS ?? 1/?E t?
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Muzykantskii Khmelnitskii Secondary
saddle-point approach
ltg(t)gt g0exp( t/?) ? d?/2? exp(-i?t) ? DQ
exp (- SQ) , SQ ??/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?/?
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d2 , disk geometry (Mirlin)

• ? (r) - singular at r ? 0
• (breakdown of the diffusion approximation!)

Requirement ??(r) lt 1/l
Cutoff r lt r ??(r) 0
r C l
Solution
(? 1, 2, 4)
g(t) (t?) 2??g , 1 ltlt t? ltlt (R/l)2
g(t) exp ??g ln2(t/?g)/4ln(R/l) , t? gtgt
(R/l)2
Coincides with the RG result of AKL
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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
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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 s-model ) Non-perturbative solutions
(secondary saddle point) Is all this possible
for distribution functions?
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Results
Effective field theory for the characteristic
function
Free action
Source action
k diag1, -1
Q(r) Qr1,r2(r) Q2(r) I
C 1 4?2?2-1
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Derivation
Primary representation
- bi-local primary superfields
K diag?, I
Explicitly
?z i?y
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Integrating over
Finally
Averaging over disorder and H-S decoupling of
by a matrix field
As a result
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Primary saddle-point approximation
Saddle-point
Slow functional
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Correspondence with the perturbation theory
ln P(s) ? s (s?/2?)2 ?q?0 (Dq2)?2
lt??/?gt 0 lt (??/?)2 gtc ?2/(2?2) ?q?0
(Dq2)?2
Non-trivial accidental cancellation of two-loop
contributions to lt (??/?)3 gtc
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Beyond the perturbation theory
Secondary saddle-point equation
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Wigner representation
In the considered hydrodynamical approximation
(non-extended s-model)
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Long-tail asymptotics of
Inverse transformation
Parametrization of the bosonic sector

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Boson action
Self-consistency equation
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Saddle-point equation for boson action
MK equation, for comparison
?2? (r) ?2sinh ?(r) 0 ?2 i?/D
Formal solution
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Saddle-point action leads to
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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 s-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