Quantum Ferromagnetic Transitions in Itinerant Electron Systems *
    (with emphasis on the order of the transition)

    I. Introduction
    II. Experiments
    III. Soft-Mode Field Theory
    IV. Summary and Conclusion
    Dietrich Belitz

    University of Oregon

    		* In collaboration with
    T.R. Kirkpatrick
    T. Vojta
    S. Sessions
    M.T. Mercaldo
    R. Narayanan

    I. INTRODUCTION

    Consider a ferromagnet with
    schematic phase diagram

    Example: MnSi ( Pfleiderer et al 1997 )

    Critical behavior at the CPT is very well understood

    Question: What is the critical behavior near the QCP ?

    Hypothesis ( Hertz 1976 ):

    QPT in d-dimensions related to CPT in (d+z)-dimensions!

    Landau theory (valid classically for d > dc+ = 4):

      Equation of state:

      Continuous transition (if u > 0) at t=0 with mean-field critical exponents

      Gaussian approximation for OP fluctuations:

      (Ornstein-Zernike plus electron dynamics)

      dc+ = 1, MF critical behavior in d = 3 ( Hertz 1976 )

      Predictions ( Hertz 1976, Millis 1993 )

        continuous transition

        mean-field exponents

        Tc ~ t3/4 (via a DIV)

    Landau theory not exact since it ignores soft modes in addition to the OP fluctuations:

      Particle-hole excitations (soft at T=0, any t)

      They couple to the OP fluctuations and invalidate LGW theory.
      This affects both disordered and clean systems. ( TRK & DB 1996 ; TRK, DB, et al 1997 ff )

    II. EXPERIMENTS

      A (subjective) selection of experiments on the ferromagnetic QPT:

      MnSi ( Pfleiderer et al. 1997 , 2001 )

        clean (?)

        low-T transition
        1st order

        Tc(t) mean-field like

      UGe2 (e.g., Saxena et al 2000 , Huxley et al 2001 , Pfleiderer & Huxley 2002 )

        clean

        low-T transition 1st order

        Superconducting phase

      ZrZn2 ( Pfleiderer et al 2001 ) and URhGe ( Aoki et al 2001 )

        clean

        low-T transition
        2nd order

        Superconducting phase

      NixPd1-x ( Nicklas et al 1999 )

        clean (?)

        low-T transition
        2nd order

        mean-field exponents

      URu2-xRexSi2 ( Bauer et al 2002 )

        disordered

        low-T transition
        2nd order

        exponents strongly
        non-mean field
        like (e.g., = 3/2 )

      Summary of Experiments:

      Clean systems

        Transition at T = 0 may be 1st or 2nd order

        1st order TCP at T > 0

        2nd order mean-field exponents

        Coexistence of superconductivity and ferromagnetism in sufficiently clean samples
        ( different talk )

      Disordered systems

        Transition 2nd order with non-mean field critical behavior

      Questions:

      In clean systems, why is the transition 1st order in some systems, 2nd order in others ?

      When it's second order, why are the exponents mean-field like ?

      In the disordered case, what causes the strange exponents ?

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

      1. Effective Action

      Critical behavior determined by soft or massless modes

      Keep all soft modes explicitly!

      2 fields: OP fluctuations (soft at t 0, any T)

      p-h fluctuations
      (soft at T = 0, any t)

      Action:

        Local, static
        LGW
        functional

        free fermions

        coupling

      2. Gaussian Approximation

      		paramagnon
      propagator

      		fermion
      propagator

        This is Landau/Gaussian theory, cf. Sec. I

        2nd order transition with Landau exponents

        Two time scales: 1) critical

        2) fermionic

      3. Renormalization-Group Analysis

        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 (and a crucial sign!)

        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

        Transition always 2nd order

        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 (???)