# Example 2 : Ferromagnetic Transition in MnSi or NixPd1-x

FM-PM transition as a function of

# 1. Generalized Nonlinear Sigma Model

The metal-insulator transition of noninteracting electrons is described by a matrix nonlinear sigma model ( Wegner 1979 )

Q is related to a matrix of single-electron degrees of freedom:

Interactions can be described by adding a term ( Finkelstein 1983 )

Properties:

• Describes diffusive phase (disordered Fermi liquid) and its instability

• Sectors of the Q-matrix/diffusive modes:
• particle-hole channel ()
• particle-particle channel ()
• spin-singlet ()
• spin-triplet ()

• Lower critical dimension dc+ = 2

• Critical behavior can be studied in -expansion about d = 2

# 2. Universality Classes

• External symmetry breakers (magnetic field, magnetic impurities, spin-orbit scattering) give some of the diffusive modes a mass:

## all

• The soft (diffusive) modes drive the transition (at least near d = 2)
Universality classes are characterized by the number of diffusive modes

• The experimental community is coming around on this point ( Itoh 2002 )

# 1. The Origin of the Myth: Low-Order RG

• Low-order perturbation theory
Kt is enhanced by disorder ( Altshuler & Aronov 1979 )

• RG to one-loop order
Kt diverges at a finite scale ( Finkelstein 1984 ):

• 2-loop RG does not cure this (some valiant attempts notwithstanding).

• Possible interpretations:

• Fundamental flaw of model
• Local moment formation ( Finkelstein )
• Artifact of low-order perturbation theory ( TRK & DB )

• Need to go beyond finite-loop order to find out

# 3. Solution of Integral Equations: Ferromagnetic Transition

Solution of integral equations

Transition to ferromagnetic state where Ds -> 0, D -> 0 ( TRK & DB 1991 ):

• Numerical solution:

# G

• Analytic solution (t=Gc-G):

• Power-law critical behavior with log corrections
• This is the exact critical behavior in d=3
• Nature of the transition a priori not obvious

# 4. Order Parameter Theory for the Magnetic Transition

• Results have been confirmed by a theory for the magnetization coupled to fermionic soft modes ( TRK's talk )

• This confirms the identification of the instability as a ferromagnetic transition

# 5. Confirmation By Other Approaches

The interpretation of Kt scaling to infinity as a FM transition has also been confirmed by two other groups:

• A sophisticated saddle-point solution of the Nonlinear Sigma Model finds a ferromagnetic state ( Chamon & Mucciolo 2000 )

• An effective action for the electronic magnetic moment finds that a ferromagnetic state minimizes the free energy ( Nayak and Yang 2002 )

# 6. Metal-Insulator Transition

• The model also contains a metal-insulator transition at a disorder larger than the PM-FM transition. This has been analyzed at 2-loop order ( TRK & DB 1992 ).

• Flow\phase diagram:

## (c) d=3 (b) 2<d<3 (a) d=2

### G

• Transitions explicitly described in d>2:
PMM-PMI, PMM-FMM, FMM-FMI

• Behavior in d=2 not clear