Coherence and interactions in diffusive systems. Lecture 4. Diffusion + e-e interations

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1 Cohrnc and inracions in diffusiv sysms G. Monambaux cur 4 iffusion + - inraions nsiy of sas anomaly phasing du o lcron-lcron inracions - inracion andau Frmi liquid picur iffusion slows down lcrons ( ) d / ( ) ρ( ) nsiy of sas anomaly nar F Corrcion o conduciviy phasing by - inracions ( )?? F

2 ifim of quasiparicl andau Frmi goldn rul ( ) W ω dω W : marix lmn of h inracion ω < ' < < ω iffusiv conducors : h ypical marix lmn of h inracion is nrgy dpndn (Alshulr-Aronov) h ffciv - inracion is proporional o h im spn in h inracion rgion : / ω W ( ω) P( ) d P () = d ( 4π ) / P() rurn probabiliy W ( ω) ω d / ( ) d / 3 nsiy of sas anomaly Harr Fock (xchang) 4

3 nsiy of sas anomaly δρ( ) involvs wo lcrons wih nrgy diffrnc hy say in phas during a im / δρ( ) ρ probabiliy o hav loops of im < / δρ( ) λ ρ ρ λ d / F vf V P() d 5 nsiy of sas anomaly δρ( ) ρ d / / λf v λ F ρ d λρ Pd ( ) = P( ) V g δρ( ) ln l l l d = d = d = 3 ln C + 3/ / = 6

4 nsiy of sas anomaly unnl conducanc anomaly unnl conducanc δg ( V) δρ( = V ) G ρ hick film (3d) hin film (d) Wir (d) Imry, Ovadyahu Saclay group 7 Corrcion o h conduciviy f δσ( ) = σ( ) d σ ( ) = ρ( ) Anomaly in ρ( ) corrcion δσ ( ) δσ ( ) ln l l l d = d = ln d = 3 C + 3/ k = k 8

5 Summary of quanum corrcions δ X X ln l X l l X d d d = = = 3 Corrcions of ordr /g X =, =, =, ω = ω = = = V k 9 phasing by - inracions mpraur dpndnc of h phas cohrnc lngh W.. in a quasi- wir ( )?? * W ( ) /3 icini,olan,ishop,98

6 phasing by - inracions Wak-localizaion corrcion : ( ) / d Δ g = s P ( ) ( ) Δ g = s P( ) iφ() d quasi- wir phasing : - inracion Phas cohrnc im /3 Alshulr,Aronov,Khmlniskii ( ) /3 : a qualiaiv drivaion quasi-d wir () iφ()? () - inracion = lcric flucuaing ponial Flucuaing phas iφ() Φ () Φ( ) = ( ) ( ) () = V(( r ), ) d Φ ( ) = V( r, ) V( r, ) V( r, ) V( r, ) d d Φ () d V

7 d Φ () V V k R k d yquis horm r σ S d Φ () k d S σ r iffusion r k 3/ Φ () σs 3/ σ S k /3 /3 yquis im (Aronov, Alshulr, Khmlniskii) Φ ( ) iφ() ( / ) 3/ 3 ( ) /3 : a daild drivaion quasi-d wir i () Φ Φ( ) = ( ) ( ) r () () ) Phas flucuaions origina from ponial flucuaions () = V( r, ) d ) Characriz ponial flucuaions yquis [ () ( )] ( ω) = V V k R kr r' [ V( r) V( r')] ( ω) = σ S k σ S kr r' [ V( r, ) V( r', )][ V( r, ') V( r', ')] = δ( ') σ S = 4

8 ( ) /3 : a daild drivaion quasi-d wir k () r( ) r( ) d S Φ = σ = r( ) r( ) d 3/ 3) Characrisic im = 3/ σ S k yquis im 3/ 4) Gaussian flucuaions of lcromagnic ponial () iφ() = Φ C hrmal flucuaions rajcoris 3/ C < r( ) r( ) d > C 5 ( ) /3 : a daild drivaion 5) Propry of a rownian bridg < 3/ r( ) r( ) d > C = r( ) d 3/ < > C Com al. iφ() = < 3/ r( ) d > C 6

9 ( ) /3 : a daild drivaion 6) Pah ingral formulaion Δ+ U( r) P( r, r', ) = δ ( rr') δ ( ) r() = r' r [( ) + U( r)] d 4 Prr (, ',) = r {} r() = r 7 ( ) /3 : a daild drivaion 6) Pah ingral formulaion r( ) d 3/ < > = C r() = r r [( ) + r( ) ] d 4 r() = r r {} P(,,) r r 3/ Prr (,,) P(, rr,) Φ i () = is soluion of : Δ+ r P(, r r',) δ ( r r') δ () 3/ = 8

10 ( ) /3 : a daild drivaion 7) aplac ransform iφ() γ P(,, rrγ) P (,,) rr d = is soluion of : γ Δ+ r P( r, r', γ) δ( r r') 3/ = 8) imnsionlss diffrnial quaion r = x γ + x P( x, x', γ ) δ ( x x') = x 9 ( ) /3 : a daild drivaion 9) Solv diffrnial quaion γ + x P( x, x', γ ) δ ( x x') = x Αι ( γ ) Pxx (,, γ ) = Αι' ( γ ) d Δ g = s P = s P x x iφ() γ ( ) (,, γ ) Δ g = s Αι ( γ ) Αι' ( γ )

11 ( ) /3 : a daild drivaion Δ g = s Αι ( γ ) Αι' ( γ ) Αι ( x) Αι' ( x) /+ x Assuming xponnial rlaxaion, w had obaind / γ Δ g = s P( ) d Δ g = s + γ / Conclusion: xponnial rlaxaion wih = Δg s + γ / is a vry good approximaion phasing by - inracions is vry wll dscribd by an xponnial rlaxaion / γ Δ g = s P( ) d σ S = = k /3 σs σs k = = = g( ) = k g ( )

12 iφ() = f π 4 3/.8.6 iφ() / G.M., E. Akkrmans, 3 /3 Saclay group 3 ( ) = A + /3 3 4

13 Cohrnc and inracions in diffusiv sysms G. Monambaux End of lcur 3 Spcral drminan 5 Unifid dscripion of msoscopic quaniis Wak-localizaion Avrag prsisn currn Conducanc flucuaions Varianc of prsisn currn 6

14 h spcral drminan Δg P() γ d = = ln S( γ ) γ + E γ n n Δ ψ = E n ψ En P () = n γ = = S( γ) = ( γ + En) n Spcral drminan : conains all informaions abou diffusion M. Pascaud, G.M., PR 8, 45 (999) 7 Unifid dscripion of msoscopic quaniis Wak-localizaion Avrag currn Conducanc flucuaions Currn flucuaions 8

15 Spcral drminan on a nwork M. Pascaud, G.M., PR 8, 45 (999) γ = nods bonds α β S( γ ) = sinh d M αβ Solv diffusion quaion on ach bond Currn consrvaion a h nods M M θ αβ αα αβ = coh β iθ αβ = sinh β 4π = A dl i α α β αβ cf: suprconducing nworks (Alxandr, Gnns) scaring on graphs (C. xir, G.M.) ik 9 Exampls S( γ ) = (cosh / cos 4 πϕ) γ = / ϕ AAS sinh / Δg cosh / - cos 4πϕ m m / Δg S ϕ ϕ ( γ ) = sinh / + (cosh / cos4 πϕ ) = (cosh / ϕ cos4 πϕ) ff sam form as for an isolad ring wih ff Δ m / ff g m ϕ ff ϕ ff Δg m m / Δg m m ( / ) + / 3