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 Castro C. Castellani

Main features of non-interacting case i.Physics: interference of trajectories ii.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 which Scattering is stronger Only direct ladders involved! Additional RG couplings Altshuler Aronov 1979 Altshuler 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 theory Skeleton structure Irreducible vertex for cutting a ladder “wave function” Frequency dressing diffusion Dynamic infinite resummation Static part Scattering amplitude Renormalized ladder Spin response: triplet channel Charge response: singlet channel Castellani, Di Castro, Lee, Ma 1984

Wave functionDOS Ward identities Response function 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-ladderTwo-laddersThree-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 calculations of specific-heat, susceptibility, conductivity Effective couplings 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 1984 Finkelstein 1983,1984 Castellani, 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