Disordered Electron Systems II

Roberto Raimondi

•Perturbative thermodynamics

•Renormalized Fermi liquid

•RG equation at one-loop

•Beyond one-loop

Workshop on Disorder and Interactions Savoyan Castle, Rackeve, Hungary

4-6 april 2006

Thanks to C. Di CastroC. Castellani

Main features of non-interacting case

i. Physics: interference of trajectoriesii. Ladder and crossed diagrams only in response functions

No change in single-particle properties

Physical meaning:•Interference between impurity and self-consistent potential•Due to disorder also Hartree potential is disordered

Interaction: DOS diagram

How it works?

Poles dominate integral

Log from power counting

Large momentum transfer

Exchange?

Small momentum transfer

Altshuler, Aronov, Lee 1980

Neglect crossing for simplicity

Also thermodynamics singular

First order correction

To compute the spin susceptibility B-dependence needed

Via Zeeman coupling diffuson ladder changes

Altshuler,, Aronov, Zyuzin 1983

Perturbative Conductivity

These sum to zero

Hartree diagrams not shown

WL: localizing

EEI: depends on whichScattering is stronger

Only direct ladders involved!

Additional RG couplings

Altshuler Aronov 1979Altshuler Aronov Lee 1980 Altshuler Khmelnitskii Larkin 1980

Effective Hamiltonian

Related to Landau quasi-particle scattering amplitudes

Spin channels

Singlet

Triplet

Landau Fermi-liquid assumption: all singular behavior comes from particle-hole bubble, i.e., screening of quasiparticles

Finkelstein 1983

How to build the renormalized theorySkeleton structure

Irreducible vertex for cutting a ladder

“wave function”

Frequency dressing

diffusion

Dynamic infinite resummationStatic part

Scattering amplitude

Renormalized ladder

Spin response: triplet channel

Charge response: singlet channel

Castellani, Di Castro, Lee, Ma 1984

Wave function DOSWard identities

Responsefunction

Spin

Infinite resummation

Castellani, Di Castro, Lee, Ma, Sorella, Tabet 1986

Ladder self-energy

Different log-divergent integrals

More diagrams•Hartree•P-H exchange

One-ladder Two-ladders Three-ladders

DOS Castellani, Di Castro 1986

Meaning of the different log-integrals

Screened Coulomb interaction Different length scales•Dynamical Diffusion length•Mean free path•Screening length

Three regimes of screened interaction

Extra singularity due to LR

Felt over a diffusive trajectory

Not relevant region

I.

II.

III.

Potential in II almost uniform

Absorbed into a gauge factor

Drops in gauge-invariant quantities

Explains cancellation in

Extra singularity only in Finkelstein 1983, Kopietz 1998

The last step: replace in the perturbative calculationsof specific-heat, susceptibility, conductivity

Effectivecouplings Drops out

With Coulomb long range forces

Dynamical amplitude

Dress magnetic field with Fermi-liquid screening

RG equations

Local moment formation?

Strong coupling runaway due to spin fluctuations

at

Castellani, Di Castro, Lee, Ma 1984Finkelstein 1983,1984Castellani, Di Castro, Lee, Ma, Sorella, Tabet 1984

Critical line

Perfect metal

As in 2D local moment?

Effective equation

Approaching the critical line

Finite!

Scaling law

Magnetic field

No contribution from triplet with

As in non-interacting case

Magnetic impurities and spin-orbit

No contribution from all triplet channels, then no

If pure WL effects are included (Cooperon ladder)

Magnetic field only controls approach to C.P. Katsumoto et al 1987

One-loop Two-loop

In d=2 a MIT

Metallic side NFL as in one-loop

Non-magnetic case beyond one-loop

Belitz and Kirkpatrick 1990,1992

In d=3 a MIT

Metallic side is FL

Only diagrams relevant for

Extend to N valleys

Useful limit for

N=2 for silicon

Two-loop for

Different physics

Thermodynamics close to MIT

Metal

Insulator

Separatrices for MIT

No magnetic instability, qualitative agreement with Prus et al 2003,

Kravchenko et al, 2006

Punnoose and Finkelstein 2005

Castellani: JCBL February 2006

Kravchenko et al 2004

Prus, Yaish, Reznikov, Sivan, Pudalov 2003

Experiments in 2D (cf. Pudalov’s lecture)

•Enhancement•Exclusion of Stoner instability

New method for thermodynamic M

Conclusions

•With magnetic couplings, good agreement •General case: strong coupling run-away •In 3D enhanced thermodynamics seen in the exps

Only selective limits with different physics•Large exchange: MIT in 3D and 2D, 2D metal with MI •Large number of valleys: MIT in 2D, perfect metal, weaker MI

One-loop

Two-loop

Theory provides a reasonable scenario, but more work needed