Field Theory for Disordered Conductors Vladimir Yudson Institute of Spectroscopy, RAS University of Florida, Gainesville, March 17, 2005 April 7, 2005

[E - H0 – U(r) ± iδ/2]G R(A) (r; r`) = δ( r – r` ) 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 no interaction 2  d  4 Efetov, Larkin, Khmelnitskii: fermionic σ-model Efetov: supersymmetric σ-model (for < g > ) 1980 1979 2005 2000 1990 Altshuler; Stone & Lee (mesoscopics) Muzykantskii & Khmelnitskii (secondary saddle-point; tails of P(t); ballistic effects) 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)

Content Field theory for P(ν) 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` ) + . . . = + Expansion in 1/kFl << 1 and , Ecτ ~ (l/L)2 << 1 = + … + GR(E+) GA(E)

Singular diagrams Diffuson q = p – p` = + + . . . GR(E+) GA(E) p p` + . . . GR(E+) GA(E) p p` q = p + p` Cooperon = + + . . . GR(E+) GA(E) p p`

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') 1/ = 2 Q(r) Hubbard-Stratonovich decoupling “Primary” saddle-point: Q2 = I; 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 σ-model + 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 )

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 σ-model + c1l2 (Q)4 + c2l2 (2 Q)2 + c3 2/D (Q)2 + … }  “extended” Anomalies (KL, KLY) RG transformation of additional scalar vertices Q

d = 2 RG transformation of additional vector vertices Q +Q  + Conformal structure: ±Q = xQ ± iyQ  d = 2 Growth of charges: zn ~ zn (0) exp [u (n2 – n)] u = ln(σ0/σ)  (1/g) ln(L/l) << 1

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 OR g/g < 0  = 1/( Ld) - mean level spacing

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

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 r* R Cutoff r* < r : (r*) = 0 r* ~ C l Solution: ( = 1, 2, 4) g(t) ~ (t)– 2g , 1 << t << (R/l)2 g(t) ~ exp[– g ln2(t/g)/4ln(R/l)] , t >> (R/l)2 Coincides with the RG result of AKL

Wanted: 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 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 r r1 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