I. Introduction

  1. QPT Concepts
  2. Landau Theory

II. Experiments

III. Soft-Mode Field Theory

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

IV. Conclusion

Quantum Ferromagnetic Transitions
(and p-Wave Superconductivity)
in Itinerant Electron Systems*


D. Belitz+
University of Oregon

Abstract: This talk gives an overview of the quantum phase transition in itinerant ferromagnets. It is shown that this problem is much more interesting from a theoretical point of view than previous results suggested. In systems without quenched disorder, the upper critical dimension is dc+ = 3, and in d = 3 the transition can be either of first order or of second order, depending on microscopic details. Both the first-order transition and the second-order one are fluctuation-induced. In the second order case, the critical behavior in d = 3 is mean field-like with logarithmic corrections to scaling. In the presence of sufficiently strong quenched disorder, the transition is always of second order. The upper critical dimension is dc+ = 6, and the critical behavior in d=3 has been determined exactly. These results are compared with various recent experiments.


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



I. INTRODUCTION




    4. Renormalization-Group Analysis


    Consider full theory (keep OP fluctuations)

    Consider fixed points, then analyze stability

    Consider clean case, disordered case proceeds analogously



      a) Tree level


      Look for fixed point with Gaussian propagators (Hertz's fixed point)


      Such a fixed point exists. Stability analysis

      Stable for d > 1 as long as z = zM

      Unstable (c2 relevant) for d < 3 if z = zq

        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

      Explicit calculation Second order case has

      mean-field exponents with log corrections in d = 3 (marginal operator c2)
      non-mean field exponents in d < 3

      Correctly predicts / is consistent with experiments on clean systems !


      Explicit calculation for disordered case

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

      Correctly predicts exponents observed in URuReSi !



    IV. 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 (???)