Cohrnc and inracions in diffusiv sysms G. Monambaux Cours 4 iffusion + - inraions nsiy of sas anomaly phasing du o lcron-lcron inracions
Why ar h flucuaions univrsal and wak localizaion is no? ΔG G cl g P() d δg G d P () g cl / dx x d / P () = 4π d / dx / x x= / Univrsal if d < Univrsal if d < 4 d / /
- inracion Landau Frmi liquid picur iffusion slows down lcrons nsiy of sas anomaly nar ( ) d / ( ) ρ( ) F Corrcion o conduciviy phasing by - inracions Lφ ( )?? F 3
Lifim of quasiparicl Landau 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 avrag im spn in h inracion rgion : W / ω ( ω) P( ) d P () = d ( 4π ) / P() rurn probabiliy W ( ω) ω d / ( ) d / > E c 4
nsiy of sas anomaly Harr Fock (xchang) 5
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 6
nsiy of sas anomaly δρ( ) ρ λ d Pd ( ) P( ) g d / / λf vf ρ λ ρ = V δρ( ) L ln l L l L l d = d = d = 3 ln C + 3/ / φ L φ L = 7
nsiy of sas anomaly unnl conducanc anomaly unnl conducanc δgv ( ) δρ( = V) G ρ hick film (3d) hin film (d) Wir (d) Imry, Ovadyahu Saclay group 8
Corrcion o h conduciviy f δσ( ) = σ( ) d σ ( ) = ρ( ) Anomaly in ρ( ) corrcion δσ ( ) δσ ( ) L ln l L l L l L d = d = ln d = 3 C + 3/ k = L = k 9
Summary of quanum corrcions δ X LX l L L LX ln l l L X d = d = d = 3 Corrcions of ordr /g LX = L, Lφ = φ, L =, Lω = ω L = = L = V k
phasing by - inracions mpraur dpndnc of h phas cohrnc lngh W.L. in a quasi- wir Lφ ( )?? * W Lφ ( ) φ L φ /3 Licini,olan,ishop,98
.L. Alshulr, A.G. Aronov,.E. Khmlniskii, J. Phys. C 5, 7367 (98) Effcs of lcron-lcron collisions wih small nrgy ransfrs on quanum localizaion
phasing by - inracions Wak-localizaion corrcion : φ( ) / φ Δ g = s P( ) d φ( ) Δ g = s P( ) iφ() d quasi- wir phasing : - inracion Phas cohrnc im φ /3 Alshulr,Aronov,Khmlniskii 3
φ ( ) /3 : a qualiaiv drivaion quasi-d wir φ() iφ()? φ () - inracion = lcric flucuaing ponial Flucuaing phas iφ() Φ () Φ( ) = φ( ) φ ( ) φ() = V(( r ), ) d Φ ( ) = V( r, ) V( r, ) V( r, ) V( r, ) d d Φ () d V 4
yquis horm d Φ () V V k R k d 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/ 5
φ ( ) /3 : a daild drivaion quasi-d wir i () Φ Φ( ) = φ( ) φ ( ) r φ() φ() ) Phas flucuaions origina from ponial flucuaions φ() = V( r, ) d ) Characriz ponial flucuaions yquis [ () ( )] ( ω)= V V L k R kr r' [ V( r) V( r')] ( ω) = σ S kl σ S kr r' [ V( r, ) V( r', )][ V( r, ') V( r', ')] = δ( ') σ S = 6
φ ( ) /3 : a daild drivaion quasi-d wir k () r( ) r( ) d S Φ = σ 3/ = r( ) r( ) d 3) Characrisic im = 3/ σ S k yquis im 3/ 4) Gaussian flucuaions of lcromagnic ponial () iφ() = Φ C hrmal flucuaions rajcoris C < 3/ r( ) r( ) d > C 7
φ ( ) /3 : a daild drivaion 5) Propry of a rownian moion < 3/ r( ) r( ) d > C = r( ) d 3/ < > C iφ() = < 3/ r( ) d > C 8
φ ( ) /3 : a daild drivaion 6) Pah ingral formulaion Δ+ U( r) P( r, r', ) = δ ( r r') δ ( ) r() = r' r [( ) + U( r)] d 4 Prr (, ',) = r {} r() = r 9
φ ( ) /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/ =
φ ( ) /3 : a daild drivaion 7) Laplac 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
φ ( ) /3 : a daild drivaion 9) Solv diffrnial quaion γ + x P( x, x', γ ) δ ( x x') = x Pxx (,, γ ) Αι ( γ ) = Αι' ( γ ) d g s P( ) s P( x, x, γ ) L iφ() γ Δ = = Δ g = s L Αι ( γ ) Αι' ( γ )
φ ( ) /3 : a daild drivaion Δ g = s L Αι ( γ ) Αι' ( γ ) Αι ( x) Αι' ( x) /+ x Assuming xponnial rlaxaion, w had obaind / γ φ Δ g = s P( ) d Δ g = s + γ L φ / Conclusion: Δg s + γ L / xponnial rlaxaion wih φ = is a vry good approximaion 3
phasing by - inracions is vry wll dscribd by an xponnial rlaxaion φ Δ g = s P( ) / γ d φ σ S = = k /3 k σs σs = = = L g( L ) = k gl ( ) 4
iφ() = f π 4 3/.8.6 iφ() /.4. 3 4 5 6 7 G.M., E. Akkrmans, Phys. Rv. L. 95, 643 (5) 5
/3 lcron-lcron Saclay group 3 lcron-phonon φ ( ) = A + /3 3 6
Gomric ffcs on dphasing du o - inracion d Φ () d V d Φ () d k S σ r 7
d Φ () V V k R r σ S d yquis horm k d Φ () d k S σ r phasing dpnds on h naur of h diffusiv rajcoris iffusiv Ergodic r r L 3/ 3/ σ S Φ k () Φ () kl σ S c /3 σ S σ /3 k c S kl Ludwig, Mirlin xir,g.m. yquis im AAK his im probs h gomry of h sysm 8