I. Quantum Phase Transitions

1. Metal-Insulator Transitions
2. Ferromagnetic Transitions

II. A Model for Interacting Disordered Electrons

1. Generalized nonlinear sigma model
2. Universality classes

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

IV. The Generic Universality Class

1. The origin of the myth: Low-order RG
2. Perturbation theory to all orders: Integral equations
3. Solution of the integral equations: Magnetic transition
4. Order parameter theory for the magnetic transition
5. Confirmation by other approaches
6. Metal-insulator transition

V. Summary and Conclusion

I. QUANTUM PHASE TRANSITIONS

Quantum phase transitions occur at T=0 as a function of some non-thermal control parameter.

---

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 (NixPd1-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:

• 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 d_c⁺ = 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:

  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

  ---

• 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 )

III. METAL-INSULATOR TRANSITIONS IN SYSTEMS WITH SPIN-FLIP MECHANISMS

---

Loop expansion (= expansion in = d-2)

• 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 :

  	MI	MF	SO
  	1/ + O(1)	1/ + O(1)	1/ + O(1)
  s	1 + O()	1 + O()	1 + O()
  	1/ + O(1)	1/2(1-ln2)	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)

---

IV. 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

---

Solution of integral equations

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

• Numerical solution:

  Ds, D
  	G

• Analytic solution (t=G_c-G):
  

• 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

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 K_t 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:

  Kt/(1+Kt)									(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

V. 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, followed by

MI 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.