II. A Model for Interacting Disordered Electrons III. Ferromagnetic Transition in the Generic Universality Class


Most obvious examples in disordered interacting electron systems:
Conductivity and dielectric susceptibility: 
Infrared absorption coefficient: 



FMPM transition as a function of
hydrostatic pressure (MnSi)

or composition (Ni_{x}Pd_{1x})




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

Q is related to a matrix of singleelectron degrees of freedom:
Interactions can be described by adding a term ( Finkelstein 1983 )

Properties:
Lower critical dimension d_{c}^{} = 2
QPTs can be studied in _{}expansion about d = 2
Sectors of the Qmatrix/diffusive modes:
The sectors can be probed separately:
The soft (diffusive) modes are expected to drive the QPTs (at least near d = 2)
Universality classes are characterized by the number of diffusive modes
Symmetry 
Magnetic 
Magnetic impurities 
Spinorbit scattering 
None 
Universality class 
MF 
MI 
SO 
Generic 
Soft modes 
ph 
ph 
ph and pp 
all 
Experimental evidence now supports this picture ( Itoh 2002 )
Loop expansion (= expansion in _{} = d2) yields
Critical observables (inter alia):
Exponents are known to first order in _{}
This problem was (qualitatively) solved 20 years ago, and the analysis is conventional!
The generic universality class has given rise to confusion
Loworder perturbation theory K_{t} is enhanced by disorder ( Altshuler & Aronov 1979 )
RG to oneloop order K_{t} diverges at a finite scale ( Finkelstein 1984 ):
2loop RG does not cure this (some valiant attempts notwithstanding).
Possible interpretations:
Fundamental flaw of model
Local moment formation ( Finkelstein )
Artifact of loworder perturbation theory ( TRK & DB )
Need to go beyond finiteloop order to find out
One can control perturbation theory to all orders by using 1/K_{t} as a small parameter.
Coupled integral equations ( TRK & DB 1990 ) for
Transition to ferromagnetic state where D_{s} > 0, D > 0 ( TRK & DB 1991 ):
Numerical solution (d=3):
D_{s}, D 

G 
Analytic solution (t=G_{c}G, d=3):
Comments:
Results have been confirmed by a theory for the magnetization coupled to fermionic soft modes ( DB, TRK, Mercaldo, Sessions 2001 )
Keep soft ph excitations Coupled field theory
Gaussian theory yields FM transition with powerlaw critical behavior
Dangerously irrelevant quartic terms Same integral equations as above
the identification of the instability as a ferromagnetic transition
An earlier version of this theory ( TRK & DB 1996 ), which had integrated out the ph excitations, yielded the same result except for the log corrections to scaling.
The interpretation of K_{t} scaling to infinity as a FM transition has also been confirmed by two other groups:
An effective action for the electronic magnetic moment finds that a ferromagnetic state minimizes the free energy ( Nayak and Yang 2003 talk G6.003)
The model also contains a metalinsulator transition at a disorder larger than the PMFM transition. This has been analyzed at 2loop order ( TRK & DB 1992 ).
Flow\phase diagram:
K_{t}/(1+K_{t}) 
(c) d=3 

G/(1+G) 
G 
Transitions explicitly described in d>2:
PMMPMI, PMMFMM, FMMFMI
Behavior in d=2 not clear
Nonlinear sigma model yields physically sensible results for all universality classes
Describes MI transition in d>2 for all but the generic universality class
Generic universality class displays FM transition
Runaway flow is artifact of loworder perturbative treatment, gives way to FM transition . This has been confirmed by at least three independent approaches.
FM transition followed by MI transition in d = 2 + _{}