1. For d=3,2 from ε<< ε F it follows that ετ >> e-e h, i.e.,

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1 Quasiparticle decay rate at T = 0 in a clean Fermi Liquid. ω +ω Fermi Sea τ e e ( ) F ( ) log( ) Conclusions:. For d=3, from << F it follows tat τ >> e-e, i.e., tat te qusiparticles are well determined and te Fermi-liquid approac is applicable.. For d= τ e-e is of te order of, i.e., tat te Fermi-liquid approac is not valid for d systems of interacting fermions. Luttinger liquids F F d d d = = 3 =

2 Quasiparticle relaxation rate in 0D case T=0 ω + ω Offdiagonal matrix element M δ ( ω,, ) << g δ

3 Quasiparticle relaxation rate in 0D case T=0 Fermi Golden Rule M τ δ e-e 0< ω< ω< < 0 ( ω,, ) τ e-e L 0D case: D = δ ET L < L, i.e., < E T M δ (L)/g Eac gives ( U.Sivan, Y.Imry & A.Aronov,994 ) δ

4 Quasiparticle relaxation rate in disordered conductors Fermi Golden Rule M τ δ e-e 0< ω< ω< < 0 ( ω,, ) T=0 0D case: L < L, i.e., < E T D τ L = d>0 case: L > L, i.e., > E T e-e τe-e δ δ ET At L L te rate is of te order of te mean level spacing δ. It sould not cange, wen we keep increasing te system size, i.e. decreasing te Touless energy L

5 Quasiparticle relaxation rate in disordered conductors Fermi Golden Rule M τ δ e-e 0< ω< ω< < 0 ( ω,, ) 0D case: L < L, i.e., < E T D L = τ e-e T=0 δ ET d>0 case: L > L, i.e., > E T τ δ e-e L τ << A. Scmid 973 e-e g L B.A. & A.Aronov 979

6 Matrix elements at large,,ω >> E T, energies M τ δ e-e 0< ω< ω< < 0 ( ω,, ) τe-e g L << M,, ( ω) 3 + d L Lω d δ δ ω ω D M ( ω,, ) δ ( L) ω 0

7 Quasiparticle relaxation rate in disordered conductors M τ δ e-e 0< ω< ω< < 0 (, T ) ( ω,, ) M ( n ) n ( n ) τ δ ω + ω + ω e-e ω ( ω,, ) T>0 T=0 T>0 n = exp T Fermi distribution function a) T=0 -no problems: ω and converges ω ω b) T>0 -a problem: T and diverges! M ω ω,, D Abraams, Anderson, Lee & Ramakrisnan 98 + d d

8 T>0 -a problem: /τ e-e diverges T τ ω e-e (, T) ω ( n ) ω d/ d D B.A., A.Aronov & D.E. Kmelnitskii (983): Divergence of is not a catastrope: /τ e-e as no pysical meaning E.g., for energy relaxation of ot electrons processes wit small energy transfer ω are irrelevant. τ g L Q: Is it te energy relaxation rate tat determines te applicability of te Fermi liquid approac?

9 T>0 -a problem: /τ e-e diverges T τ ω e-e (, T) ω ( n ) ω d/ d D B.A., A.Aronov & D.E. Kmelnitskii (983): Divergence of is not a catastrope: /τ e-e as no pysical meaning E.g., for energy relaxation of ot electrons processes wit small energy transfer ω are irrelevant. τ g L Pase relaxation: in a time t after a collision δ (π ω t) / processes wit energy transfer ω smaller tan /τ are irrelevant.? τ

10 Wat is Depasing? Wat is Depasing?. Suppose tat originally a system(an electron) was in a pure quantum state. It means tat it could be described by a wave function wit a given pase.. External perturbations can transfer te system to a different quantum state. Suc a transition is caracterized by its amplitude, wic as a modulus and a pase. 3. Te pase of te amplitude can be measured by comparing it wit te pase of anoter amplitude of te same transition. Example: Fabri-Perrot interferometer beam splitter mirror

11 4. Usually we can not control all of te perturbations. As a result, even for fixed initial and final states, te pase of te transition amplitude as a random component. 5. We call tis contribution to te pase, δ, random if it canges from measurement to measurement in an uncontrollable way. 6. It usually also depends on te duration of te experiment, t: δ = δ(t) 7. Wen te time t is large enoug, δ exceeds π, and interference gets averaged out. 8. Definitions: δ( τ ) π τ pase coerence time; /τ depasing rate

12 Wy is Depasing rate important? Imagine tat we need to measure te energy of a quantum system, wic interacts wit an environment and can excange energy wit it. Let te typical energy transferred between our system an te environment in time t be δ(t). Te total uncertainty of an ideal measurement is environment () t δ ; t t 0 t! δ } Δ () t δ () t ( t ) δ ( t ) + t quantum uncertainty Tere sould be an optimal measurement time t=t*, wic minimizes Δ(t) : Δ(t*)= Δ min t* τ! * * t * Δ min τ

13 δ Wy is Depasing rate important? t* * * t * Δ ( t ) δφ ( t ) min τ τ It is depasing rate tat determines te accuracy at wic te energy of te quantum state can be measured in principle.

14 T>0 -a problem: /τ e-e diverges T τ ω e-e (, T) ω ( n ) ω d/ d D B.A., A.Aronov & D.E. Kmelnitskii (983): Divergence of is not a catastrope: /τ e-e as no pysical meaning E.g., for energy relaxation of ot electrons processes wit small energy transfer ω are irrelevant. Pase relaxation: in a time t after a collision δ (π ω t) / processes wit energy transfer ω smaller tan /τ are irrelevant. (, T) τ T τ ω ω> τ g L ( n ) ω d/ d D

15 e-e interaction Electric noise E α Fluctuation- dissipation teorem: Electric noise - randomly time and space - α dependent electric field α E r, t E ( k., ω ) Correlation function of tis field is completely determined by te conductivity σ k,ω : E β = ω cot ω α β, ω k σ ω αβ αβ k (, k) T k (, k) ω σ k T Noise intensity increases wit te temperature, T, and wit resistance g

16 E α E β = ω cot α β, ω k σ ω αβ αβ ω k (, k ) T k (, k ) ω σ k T ( L) - Touless conductance def. e R( L) R( L) - resistance of te sample wit lengt (d) area (d) g { } L τ T g L L D τ - depasing lengt D - diffusion constant of te electrons

17 T τ Tis is an equation! g L

18 τ T g L Tis is an equation! g( L) L d L τ were is te number of dimensions: d= for wires; d= for films, L T ( ) 4 d τ T 4 ( ) T d = d T 3 d =

19 T L τ g L D τ Fermi liquid is valid (one particle excitations are well Tτ ( T) defined), provided tat >

20 T L τ g L D τ Fermi liquid is valid (one particle excitations are well Tτ ( T) defined), provided tat >. In a purelyd cain, g,and, terefore, Fermi liquid teory is never valid.

21 Magnetoresistance Φ O No magnetic field = O Wit magnetic field H = π Φ/Φ 0 Φ = HS - magnetic flux Φ = c/e - troug te loop 0 flux quantum

22 Weak Localization, Magnetoresistance in Metallic Wires I 3. K.7 K 0 mk B ΔR/R L ~ 0.5 mm mk <L φ =(Dτ φ ) L = / D φ τ φ B (T) V d case; strong spin-orbital coupling Δ R A = R + R e L L 3 L H L H = eh A area of te wire cross-section

23 Can we always reliably extract te inelastic depasing rate from te experiment Weak localization: NO - everyting tat violates T-invariance will destroy te constructive interference EXAMPLE: random quenced magnetic field B? Mesoscopic fluctuations: YES - Even strong magnetic field will not eliminate tese fluctuations. It will only reduce teir amplitude by factor. A But Slow diffusion of te impurities will look as depasing in mesoscopic fluctuations measurements

24 Magnetic Impurities - before - after T-invariance is clearly violated, terefore we ave depasing Mesoscopic fluctuations Magnetic impurities cause depasing only troug effective interaction between te electrons. T 0 Eiter Kondo scattering or quencing due to te RKKY excange. In bot cases no elastic depasing

25 Inelastic depasing rate /τ can be separated at least in principle ω ω oter electrons ponons magnons two level systems

26 THE EXPERIMENTAL CONTROVERSY Moanty, Jariwala and Webb, PRL 78, 3366 (997) T -3 T -/3 Saturation of τ : Artifact of measurement? Real effect in samples?

27 Zero-point Oscillations Collision between te quantum particle and a armonic oscillator e - energy counted from te Fermi level ( E ) n = ω n +. T > ω. T << ω > ω; n > 0 Te particle and te oscillator can excange energy n= n= << ω; n = 0 No energy excange between te oscillator and te particle n=0 Inelastic scattering depasing Pure elastic scattering No depasing

28 e Zero-point Oscillations Collision between te quantum particle and a armonic oscillator - energy counted from te Fermi level ( E ) n = ω n + T << ω << ω; n = 0 No energy excange between te oscillator and te particle Pure elastic scattering No depasing ω Zero-point oscillations

29 Caos in Nuclei Delocalization? Witout interactions between fermions energy of eac of te particles is conserved, i.e., tere are as many integrals of motion as tere are excited particles. Fermi Sea For a finite Fermi Gas one sould expect Poisson situation for te eigenstates of te wole system.

30 Caos in Nuclei Delocalization? generations Fermi Sea.... Delocalization in Fock space Expansion of a typical eigenstate of te many-body system in te basis of states wit given number of excitations involves a large number of terms

31 Quasiparticle relaxation rate in 0D case ω + ω τ e-e δ ET ( U.Sivan, Y.Imry & A.Aronov,994 ) Becomes incorrect as < E δ soon as provided tat T = δ g δ ( B.A, Y.Gefen. A Kamenev & L.Levitov,994 )