I. Introduction

  1. QPT Concepts
  2. Landau Theory

II. Generalized Mean-Field Theory

  1. Breakdown of Landau Theory
  2. Generalized Mean-Field Theory

III. Experiments

IV. Soft-Mode Field Theory

  1. Effective Action
  2. Gaussian Approximation
  3. Generalized Mean-Field Theory
  4. RG Analysis

V. Summary and Conclusion

Quantum Ferromagnetic Transitions
in Itinerant Electron Systems*


D. Belitz+
University of Oregon

Abstract: After a brief introduction to phase transitions in general, and quantum phase transitions in particular, I will discuss the quantum or zero-temperature transition in bulk (3-d) itinerant ferromagnets. Conventional theory concludes that this transition should be of second order with mean-field critical exponents. I will first show that this conclusion is incorrect due to the presence of soft particle-hole excitations that couple to the order parameter. A generalized mean-field theory that takes these modes into account predicts a first-order transition in clean systems, and a second-order one with non-mean field critical exponents in systems with quenched disorder. Finally, a full RG analysis that includes the order parameter fluctuations shows that under certain conditions the first-order transition in clean systems can become unstable with respect to a fluctuation-induced second-order transition. I will discuss these conclusions in the light of recent experiments.


* With T.R. Kirkpatrick, T. Vojta, S.L. Sessions, M.T. Mercaldo, R. Narayanan
+ belitz@greatwhite.uoregon.edu



I. INTRODUCTION




II. GENERALIZED MEAN-FIELD THEORY




III. EXPERIMENTS




IV. SOFT-MODE FIELD THEORY (DB et al 2001a , 2001b , TRK & DB 2002 )




    3. Generalized Mean-Field Theory


    MFA for OP () + Gaussian approximation for fermions:


      Integrate out fermions generalized MF equation of state

      Nonanalyticities due to fermion loops, , etc.


      Predicts first order transition (always!)


      T>0 gives fermions a mass

      TCP at T>0 in agreement with exps on MnSi, UGe2

      Cannot explain experiments on ZrZn2, NiPd



    4. Renormalization-Group Analysis


    Assign scale dimensions

    which may be either zM or zq !

    and look for fixed points


    Discuss clean case, quote results for disordered case



      a) Tree level


      Look for fixed point with

      [G] = [H] = 0 (Fermi liquid)

      [a] = [c1] = 0 (Landau theory)


      Check stability:

      [t] = 2 > 0 = 1/2

      [u] = - (d - 1) < 0 for d>1

      [c2] = (4 - d - z)/2

        [c2] = - (d - 1)/2 if z = zM

        [c2] = (3 - d)/2 if z = zq

        NB: This happens if fermion loops are involved, e.g.,

      c2 relevant with respect to Landau FP for d < 3 !

      Landau FP unstable due to fermion loops


      Applies to disordered case as well, only the numbers change



      a) Loop level


      Need to keep loops systematically

      e.g., u =


      u has negative correction

      in perturbation theory, u(b -> ) < 0 always

      fluctuation-induced first order transition

      Maps on classic fluctuation-induced first order transition in superconductors
      ( Halperin, Lubensky, Ma 1974 )

      This is the generalized mean-field theory in RG language


      Go beyond perturbation theory: H(b -> ) ->

      Depending on initial conditions, u(b -> ) < 0 or > 0

      Transition may be first or second order !


      Physical interpretation :

      H ~ specific heat coefficient
      Critical behavior of specific heat couples back to u

      Energy scales involved: Correlation energy vs Fermi energy or bandwidth

      First vs second order depends on ratio of energy scales;
      strong correlation favors first order

      First order transition
      stable if t1 sufficiently large
      destroyed by critical fluctuations if t1 is small


      Instability of first order transition leads to non-mean field, fluctuation-induced second order
      transition. Explicit calculation

      Second order case in d = 3 has mean-field exponents with log corrections
      (due to the marginal operator c2 ); non-mean field exponents in d < 3

      Consistent with experiments on clean systems !


      Explicit calculation for disordered case

      Critical behavior as described by generalized mean-field theory with log corrections

      Agrees with experiment on NiPd !



    V. SUMMARY AND CONCLUSION


    Low-Tc itinerant ferromagnets are remarkably complex and interesting.


    The T = 0 transition can be 1st order for generic reasons: A fluctuation-induced 1st order transition preempts the continuous Landau transition.


    Crucial for this mechanism are soft fermionic modes and the resulting two time scales in the problem.


    If the 1st order transition is too close to the second order one it is trying to preempt, it becomes unstable with respect to a fluctuation-induced 2nd order transition. This transition is in a new universality class.


    Sufficiently strong non-magnetic disorder causes the transition to always be 2nd order. This constitutes yet another universality class, which is exactly solved.


    These results explain, or are consistent with, all experiments to date.


    Properties that are not understood (at least not completely):

      Ferromagnetic superconducting phase (partially understood)

      Non-Fermi liquid behavior (???)