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Savoyan Castle, Rackeve, Hungary. Workshop on Disorder and Interactions . Disordered Electron Systems I. Introduction Scaling theory Microscopic theory Non-interacting case. Roberto Raimondi. Thanks to C. Di Castro C. Castellani. 4-6 april 2006.

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disordered electron systems i

Savoyan Castle, Rackeve, Hungary

Workshop on Disorder and Interactions

DisorderedElectron Systems I.

  • Introduction
  • Scaling theory
  • Microscopic theory
  • Non-interacting case

Roberto Raimondi

Thanks to C. Di Castro

C. Castellani

4-6 april 2006


Key problem: metal-insulator transition (MIT)

  • MIT from interplay of disorder and interaction
  • Metallic side in terms of Fermi liquid
  • Aim: describe MIT as continuous phase transition
  • Tasks:identify couplings and critical modes

Key physics:quantum interference corrections

G. Bergman Phys. Rep. 107, 1 (1984)

P.A. Lee and T.V. Ramakrishnan Rev. Mod. Phys. 57, 287 (1985)

B.L. Altshuler and A.G. Aronov in Electron-electron Interactions in Disordered Systems,

Eds. M.Pollak and A.L. Efros North-Holland, Amsterdam (1984) p.1

A.M. Finkelstein Sov. Sci. Rev.14, 1 (1990)

D. Belitz and T.R. Kirkpatrick Rev. Mod. Phys. Rep. 66, 261 (1994)

C. Di Castro and R. Raimondi in The Electron Liquid Paradigm in Condensed Matter Physics

Proceedings of the Inter. School of Physics E. Fermi,

Eds. G.F. Giuliani and G. Vignale IOP Press 20041. Cond-mat/0402203


Semiclassical theory: Drude-Boltzmann-Sommerfeld

Random walk of step

Diffusive motion

Response function and Einstein’s relation

Fermi gas case:


Quantum corrections: self-intersecting trajectories

Return probability

Self-intersection probability

Summing all times

Task for microscopic theory:

Diffusion modes as critical modes

Inverse conductivity as expansion parameter


Scaling theory

Thouless’s argument

Edwards and Thouless 1972

Control parameter: dimensionless conductance


Scaling hypothesis:

Depends on g only

Fixed point:

Critical exponent:

Abrahams, Anderson, Licciardello, Ramakrishnana 1979


Power behavior of physical quantities

Correlation length

Scaling law

Metallic side expansion

Time reversal invariance

B-field or magnetic impurities


Basic tool: linear response theory

Castellani, Di Castro, Forgacs, Tabet 1983

Real space

Fourier space

Charge conservation

Gauge invariance



Response functions and Ward identities

Bare vertex

Dressed vertex

Ward identity


Check: free case

Consequences of W.i.

Dynamic part


Phenomenological theory obeys all !


Microscopic theory: Green function

Task: recover semiclassical approach as the zeroth order in

Disorder expected effect

Finite lifetime

Quasi-particle pole

Disorder model: Gaussian random variable


Self-consistent Born approximation

Key approximation:

Self-consistent solution, only position of the pole matters

Abrikosov, Gorkov, Dzyaloshinski


Microscopic theory: response functions

“Rainbow” for

“Ladder” for

W. I.

Langer, Neal 1976

Recover the semiclassical result!


How to go beyond and keep interference processes

Role of crossed diagrams

Expansion parameter

Maximally crossed diagrams

Enhanced backscattering due to time-reversed paths


Correction to response function

Ladder self-energy

Weak localization correction

Gorkov, Larkin, Khmelnitskii 1979


What about B?

Crossed diagrams in real space

B enters via

a “mass” in the diffusion propagator


Magnetoresistance and dephasing time

Crossover when

Measure of


Spin effects: magnetic impurities and spin-orbit coupling


Singlet and Triplet channels





  • Dolan Osheroff PRL ‘79
  • Giordano et al PRL’79

WL seen in films and wires



  • Dynes, Geballe, Hull, Garno PRB 83

Thomas et al PRB ‘82 GeSb

  • Hertel et al PRL ‘83 Nb Si
  • Rhode Micklitz al PRB ‘87 BiKr

Compensated Smc and alloys



Si-P critical exponent puzzle

  • Rosenbaum et al PRL ‘80, PRB ‘83
  • Stupp et al PRL ‘93
  • Shafarman et al PRB ‘89 Si As
  • Dai et al PRB ‘93 Si B

Uncompensated SiP

Si As n-doped, Si B p-doped


Anomalous B-dependence of critical exponent

CuMn Magnetic impurities ?

AlGaAs Si

Okuma et al ‘87

Katsumoto et al JPSJ ‘87

  • Dai et al et al PRB ‘93 Si P

Si Au Strong Spin Orbit

Nishida et al SSP ‘84


Unexpected anomalies

Singularity in DOS

  • McMillan Mochel PRL ‘81 Ge Au
  • Hertel et al PRL ‘83 Nb Si

Low-T enhancement of specific heat

  • Kobayashi et al SSC ‘79 Si P
  • Thomas et al PRB ‘81 Si P
  • Paalanen et al PRL ‘88 Si P
  • Lakner et al PRL ‘89 Si P

Low-T enhancement of spin susceptibility

  • Ikeata et al SSC ‘85
  • Paalanen et al PRL ‘86
  • Alloul Dellouve PRL ‘87
  • Hirsch et al PRL ‘92
  • Schlager et al EPL ‘97

Key issue: how e-e interaction changes the game?


Last but no least: 2D MIT in Si-MOSFETs and heterostructures

Kravchenko and SarachikRep. Progr. Phys.67, 1 (2004)

Quantum effects

Key parameter:

  • Unexpected with non-interacting theory
  • Strong magnetoresistance in parallel field
  • Open issue whether there is a MIT



End of part I.

  • Program for next lecture
  • Explore perturbative effects of interaction
  • Landau Fermi-liquid formulation
  • Renormalizability of response function
  • RG equations