Quantum Phase Transitions
in
Interacting Disordered Electron Systems
D. Belitz
University of Oregon
Quantum phase transitions occur at T=0 as
a function of some nonthermal control parameter.
Example 1 : MetalInsulator Transition in Si:P
( Rosenbaum et al. 1983 )
Conductivity and dielectric susceptibility:

Infrared absorption coefficient:



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:
 Describes diffusive phase (disordered Fermi liquid) and its instability
 Sectors of the Qmatrix/diffusive modes:
 particlehole channel (_{})
 particleparticle channel (_{})
 spinsinglet (_{})
 spintriplet (_{})
 Lower critical dimension d_{c}^{+} = 2
 Critical behavior can be studied in _{}expansion
about d = 2
 External symmetry breakers (magnetic field, magnetic impurities, spinorbit scattering)
give some of the diffusive modes a mass:
Symmetry breaker

Magnetic field

Magnetic impurities

Spinorbit scattering

None

Universality class

MF

MI

SO

Generic

Soft modes

ph singlet + 1 triplet

ph singlet

ph and pp singlet

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 )
Loop expansion (= expansion in _{} = d2)
 The universality classes MF, MI, and SO all display a metalinsulator transition
at oneloop order ( Finkelstein 1983,1984 )
 Critical observables (inter alia):
 Correlation length _{}
 Conductivity _{}
 Density of states _{}
 Specific heat _{}
 Exponents are known to first order in _{}:

MI

MF

SO

_{}

1/_{} + O(1)

1/_{} + O(1)

1/_{} + O(1)

s

1 + O(_{})

1 + O(_{})

1 + O(_{})

_{}

1/_{} + O(1)

1/2(1ln2)_{}

2/_{} + O(1)

_{}

1  _{}/4 + O(_{})^{2}

1

1  _{}/2 + O(_{})^{2}

 Results were confirmed and interpreted by other methods ( Castellani et al. 1984  1987 )
 This problem was (qualitatively) solved 20 years ago, and the
analysis is conventional!
(However, no known method for quantitatively calculating exponents in d=3)

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
K_{t}(l>_{}) > _{}
One can
control perturbation theory to all orders by using
1/K_{t} as a small parameter.
Coupled integral equations ( TRK & DB 1990 )
for
 the spin diffusivity D_{s} (~ 1/K_{t}), and
 the heat diffusivity D (~ 1/H),
Solution of integral equations
Transition to ferromagnetic state
where D_{s} > 0, D > 0 ( TRK & DB 1991 ):
 Numerical solution:
D_{s}, D



G

 Analytic solution (t=G_{c}G):
 Comments:
 Powerlaw critical behavior with log corrections
 This is the exact critical behavior in d=3
 Nature of the transition a priori not obvious
 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
The interpretation of K_{t} scaling to infinity as a FM
transition has also been confirmed by two other groups:
 A sophisticated saddlepoint 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 )
 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
(b) 2<d<3 (a) d=2


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, followed by
MI transition
Runaway flow is artifact of loworder perturbative treatment,
gives way to FM transition . This has been confirmed by at least
three independent approaches.