Post on 25-Feb-2016
description
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