1. Workshop on Disorder and Interactions Savoyan Castle, Rackeve, Hungary Disordered Electron Systems II • Perturbative thermodynamics • Renormalized Fermi liquid • RG equation at one-loop • Beyond one-loop Roberto Raimondi Thanks to C. Di Castro C. Castellani 4-6 april 2006

2. Main features of non-interacting case Physics: interference of trajectories Ladder and crossed diagrams only in response functions No change in single-particle properties Interaction: DOS diagram Physical meaning: • Interference between impurity and self-consistent potential • Due to disorder also Hartree potential is disordered

3. Large momentum transfer How it works? Altshuler, Aronov, Lee 1980 Poles dominate integral Exchange? Small momentum transfer Log from power counting Neglect crossing for simplicity

4. Also thermodynamics singular First order correction To compute the spin susceptibility B-dependence needed Via Zeeman coupling diffuson ladder changes Altshuler,, Aronov, Zyuzin 1983

5. Perturbative Conductivity These sum to zero Hartree diagrams not shown Only direct ladders involved! Altshuler Aronov 1979 Altshuler Aronov Lee 1980 Altshuler Khmelnitskii Larkin 1980 WL: localizing EEI: depends on which Scattering is stronger Additional RG couplings

6. Effective Hamiltonian Finkelstein 1983 Spin channels Singlet Triplet Related to Landau quasi-particle scattering amplitudes Landau Fermi-liquid assumption: all singular behavior comes from particle-hole bubble, i.e., screening of quasiparticles

7. How to build the renormalized theory Skeleton structure Castellani, Di Castro, Lee, Ma 1984 Static part Dynamic infinite resummation Irreducible vertex for cutting a ladder Scattering amplitude “wave function” Renormalized ladder Frequency dressing diffusion Charge response: singlet channel Spin response: triplet channel

8. Response function Infinite resummation Ward identities Wave function DOS Spin Castellani, Di Castro, Lee, Ma, Sorella, Tabet 1986

9. Ladder self-energy • More diagrams • Hartree • P-H exchange Different log-divergent integrals Two-ladders Three-ladders One-ladder DOS Castellani, Di Castro 1986

10. Meaning of the different log-integrals • Different length scales • Dynamical Diffusion length • Mean free path • Screening length Screened Coulomb interaction Three regimes of screened interaction Felt over a diffusive trajectory I. Extra singularity due to LR II. Not relevant region III. Drops in gauge-invariant quantities Potential in II almost uniform Explains cancellation in Absorbed into a gauge factor Extra singularity only in Finkelstein 1983, Kopietz 1998

11. The last step: replace in the perturbative calculations of specific-heat, susceptibility, conductivity Dynamical amplitude Dress magnetic field with Fermi-liquid screening Effective couplings Drops out With Coulomb long range forces

12. Castellani, Di Castro, Lee, Ma 1984 Finkelstein 1983,1984 Castellani, Di Castro, Lee, Ma, Sorella, Tabet 1984 RG equations Strong coupling runaway due to spin fluctuations at Local moment formation?

13. Effective equation Critical line Perfect metal Approaching the critical line Finite! Scaling law As in 2D local moment?

14. Magnetic field No contribution from triplet with As in non-interacting case

15. 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

16. Non-magnetic case beyond one-loop Only diagrams relevant for Two-loop One-loop In d=3 a MIT Metallic side is FL In d=2 a MIT Metallic side NFL as in one-loop Belitz and Kirkpatrick 1990,1992

17. N=2 for silicon Extend to N valleys for Useful limit

18. Two-loop for Punnoose and Finkelstein 2005 Different physics Insulator Thermodynamics close to MIT Separatrices for MIT Metal No magnetic instability, qualitative agreement with Prus et al 2003, Kravchenko et al, 2006 Castellani: JCBL February 2006

19. New method for thermodynamic M Experiments in 2D (cf. Pudalov’s lecture) • Enhancement • Exclusion of Stoner instability Prus, Yaish, Reznikov, Sivan, Pudalov 2003 Kravchenko et al 2004

20. Conclusions • With magnetic couplings, good agreement • General case: strong coupling run-away • In 3D enhanced thermodynamics seen in the exps One-loop 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 Two-loop Theory provides a reasonable scenario, but more work needed