Metal-Insulator Transitions
Ferromagnetic Transitions

II. A Model for Interacting Disordered Electrons

Generalized Nonlinear Sigma-Model
Universality Classes
Metal-Insulator Transitions

III. Ferromagnetic Transition in the Generic Universality Class

Low-Order RG
Perturbation Theory to All Orders
Ferromagnetic Transition
Order Parameter Theory
Confirmation by Other Approaches
Metal-Insulator Transition

Magnetic Properties of Disordered
Interacting Electrons ^*

D. Belitz⁺
University of Oregon

Abstract: The magnetic properties of disordered interacting electron systems are discussed. It is shown that the runaway flow in d>2 that has given rise to confusion in the past is an artifact of a one-loop approximation. The perturbation theory can be controlled and resummed to all orders, and for small disorder near d=2 a ferromagnetic transition emerges. This has been confirmed by an explicit order parameter theory for the magnetic transition.

^* With T.R. Kirkpatrick, M.T. Mercaldo, S.L. Sessions

⁺ belitz@greatwhite.uoregon.edu

I. QUANTUM PHASE TRANSITIONS

Quantum phase transitions occur at T=0 as a function of some non-thermal control parameter.
Most obvious examples in disordered interacting electron systems:

Metal-insulator transitions
Ferromagnetic transitions

Example 1 : Metal-Insulator Transition in Si:P ( Rosenbaum et al. 1983 )

Conductivity and dielectric susceptibility:

Infrared absorption coefficient:

Example 2 : Ferromagnetic Transition in MnSi or Ni_xPd_1-x

FM-PM transition as a function of

hydrostatic pressure (MnSi)
( Pfleiderer at al. 1997 ):

or composition (Ni_xPd_1-x)
( Nicklas et al. 1999 ):

II. A MODEL FOR INTERACTING DISORDERED ELECTRONS

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:

Lower critical dimension d_c^- = 2

QPTs can be studied in -expansion about d = 2

2. Universality Classes

Sectors of the Q-matrix/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 breaker	Magnetic field	Magnetic impurities	Spin-orbit scattering	None
Universality class	MF	MI	SO	Generic
Soft modes	p-h singlet + 1 triplet	p-h singlet	p-h and p-p singlet	all

Experimental evidence now supports this picture ( Itoh 2002 )

3. Metal-Insulator Transitions in Systems with Spin-Flip Mechanisms

Loop expansion (= expansion in = d-2) yields

The universality classes MF, MI, and SO all display a metal-insulator transition at one-loop order ( Finkelstein 1983,1984 )
Critical observables (inter alia):

Correlation length

Conductivity

Density of states

Specific heat

Exponents are known to first order in

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!
The generic universality class has given rise to confusion

III. FERROMAGNETIC TRANSITION in the GENERIC UNIVERSALITY CLASS

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

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

RG to one-loop order K_t 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

2. Perturbation Theory To All Orders

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),

3. Solution of Integral Equations: Ferromagnetic Transition

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:

Power-law critical behavior with log corrections
This is the exact critical behavior in d=3
Nature of the transition a priori not obvious (initial interpretation thought there was no long-range order)

4. Order Parameter Theory for the Magnetic Transition

Results have been confirmed by a theory for the magnetization coupled to fermionic soft modes ( DB, TRK, Mercaldo, Sessions 2001 )

spin-triplet interaction

Keep soft p-h excitations Coupled field theory

Gaussian theory yields FM transition with power-law 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 p-h excitations, yielded the same result except for the log corrections to scaling.

5. Confirmation By Other Approaches

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

Chamon & Mucciolo 2000

An effective action for the electronic magnetic moment finds that a ferromagnetic state minimizes the free energy ( Nayak and Yang 2003 talk G6.003)

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:

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:
PMM-PMI, PMM-FMM, FMM-FMI

Behavior in d=2 not clear

IV. SUMMARY AND CONCLUSION

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 low-order 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 +

Conductivity and dielectric susceptibility:							Infrared absorption coefficient:

hydrostatic pressure (MnSi) ( Pfleiderer at al. 1997 ):									or composition (Ni_xPd_1-x) ( Nicklas et al. 1999 ):

Example 1 : Metal-Insulator Transition in Si:P ( Rosenbaum et al. 1983 )

Conductivity and dielectric susceptibility:

Infrared absorption coefficient:

Example 2 : Ferromagnetic Transition in MnSi or NixPd1-x

hydrostatic pressure (MnSi) ( Pfleiderer at al. 1997 ):

or composition (NixPd1-x) ( Nicklas et al. 1999 ):

Symmetry breaker

Magnetic field

Magnetic impurities

Spin-orbit scattering

None

Universality class

MF

MI

SO

Generic

Soft modes

p-h singlet + 1 triplet

p-h singlet

p-h and p-p singlet

all

Kt(l->) -> One can control perturbation theory to all orders by using 1/Kt as a small parameter. Coupled integral equations ( TRK & DB 1990 ) for the spin diffusivity Ds (~ 1/Kt), and the heat diffusivity D (~ 1/H),

Ds, D

G

Kt/(1+Kt)

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

G/(1+G)

G

Example 2 : Ferromagnetic Transition in MnSi or Ni_xPd_1-x

hydrostatic pressure (MnSi)
( Pfleiderer at al. 1997 ):

or composition (Ni_xPd_1-x)
( Nicklas et al. 1999 ):

Symmetry
breaker

Magnetic
field

p-h
singlet + 1 triplet

p-h
singlet

p-h and p-p
singlet

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),

D_s, D

K_t/(1+K_t)

(c) d=3

(b) 2<d<3

(a) d=2