I. A Simple Field Theory


    Consider quantum - theory coupled to a diffusive mode :

    Gaussian part


    interaction


    Gaussian propagators:

    Notes:

      Physically, this action (happens to) describe a disordered itinerant quantum ferromagnet
      The diffusive field is bilinear in the electron fields

      Two time scales (at least in the Gaussian theory):

      Diffusive time scale

      ~ 1/ ~ p2/H

      Critical time scale

      ~ 1/ ~ p4/C1

      (at t = 0)


      NB: ~ H p2/C1 = w(p) -> 0 for p -> 0


    Dangerous irrelevant variables (DIVs):

      Commonly, variables scaling to zero invalidate naive scaling since scaling functions singularly depends on them

      Example: Classical - theory

      In this field theory, = w acts as a DIV at a more fundamental level, as the RG flow equations
      singularly depend on it. This invalidates the loop expansion !




    II. Perturbation Theory


    One-loop corrections due to C2 :

      renormalizes a

      renormalizes H

      Both are logarithmically divergent in d = 4

      t, C1, C2, have no singular corrections


    Power counting

      t, C1, C2, have no singular corrections to any order

      Self-consistent one-loop theory is exact


    Options to determine critical behavior:

      Solve integral equations ( DB & TRK 1992 , DB et al 2001 ) Non-power-law critical behavior

      RG analysis: Structure of perturbation theory suggests standard expansion


    Questions:

      How can the RG manage to not produce power laws ?

      Why does one worry about renormalizability in the context of the field-theoretic formulation of the RG, but usually not in the context of the Wilsonian momentum-shell formulation ?

    Answers:

      The two questions are intimately related

      A well-defined renormalizability property is necessary to produce power-law critical behavior, even in the momentum-shell formulation



    III. Renormalization


      1. Power counting


      Choose scale dimensions:

        Length and frequency/temperature:

        [L] = -1

        [] = [T] = z

        Note: Work with a unique z, even though there are two time scales ( De Dominicis & Peliti 1978 );
        second time scale is implicit in the scale dimension of H
        (This is a matter of taste, one can as well work with two z's)

        Fields

        [] = [] = (d-2)/2

        a is dimensionless

        is not renormalized does not carry a field renormalization

        a is renormalized does carry a field renormalization
        critical exponent defined by

      Define dimensionless coupling constants

      and derive flow equations


      2. Exact flow equations


      t is not renormalized

      correlation length exponent


      C1 is not renormalized

      dynamical exponent


      C2 is not renormalized
      c2 is formally irrelevant !


      Remaining task: Determine , and the flow equation for h, from loop expansion


      3. Loop expansion


      Zero-loop order:

        c2 is irrelevant Gaussian fixed point with = 0 , z = 4 ( Hertz 1976 )

        NB: C2 and H are both irrelevant with scale dimension -2


      One-loop order:

        with U = C2/a H

        NB: Loop expansion parameter is U, not C2 !

        u-flow equation:

        FP value

        Seemingly controlled - expansion leads to power-law critical behavior at one-loop order
        Nothing is obviously wrong

        Gaussian FP unstable due to H being "dangerously irrelevant", compensating the irrelevance of C2
        This is a consequence of the diffusive time scale !


      Two-loop order:

        H() depends on : ( Remember: This represents ! )

        Write

        It is natural to introduce a running coupling constant :

        NB : Perturbation theory, and hence the flow eqs, depend singularly on w !

        Breakdown of loop (= ) expansion !


      Alternative interpretation:

        Eliminate w (or never introduce it) scale-dependent u-flow equation

        Renormalizability property:

          Power-law critical behavior

          requires

          u-flow equation to be scale independent

          requires

          special relation between coefficients of U2 log2 terms and U log terms

        This relation holds, e.g., in - theory, or in a theory with only one time scale

        Here, it is violated due to the presence of a second time scale


      Conclusion:

        - expansion breaks down due to combination of two irrelevant variables, c2 -> 0 , h -> 0 :

          Zero-loop term is ~ (c2)0
          One-loop term is ~ c2/h Gaussian FP invalidated
          Two-loop term is ~ (c2/h)2 log h Breakdown of loop expansion

        The fixed points at both zero- and one-loop order do not approximate the true critical behavior in any sense

        This first becomes apparent at two-loop order

        Correct critical behavior can be found by either

          Performing the RG procedure to all orders
          or
          Solving the integral equations

          Power laws times log-log-normal behavior ( DB & TRK 1992 , DB et al 2001 )
          E.g., the correlation length is given by


IV. Summary


Multiple time scales can lead to breakdown of the - expansion

Renormalizability property is necessary for power-law critical behavior; this can be violated by multiple time scales

Relevant for disordered quantum ferromagnets (see above), clean quantum ferromagnets (to be worked out),
but not, e.g., for classical fluids

Integrating out the fermions (i.e., the modes with the additional time scale) obscures all of this, so don't do that!