Breakdown of the Perturbative Renormalization Group:
Ferromagnetic Quantum Criticality 
University of Oregon
Ted Kirkpatrick




Gaussian propagators:
Notes:
Two time scales (at least
in the Gaussian theory):
Diffusive time scale _{ } 
_{} ~ 1/_{} ~ p^{2}/H 

Critical time scale _{ } 
_{} ~ 1/_{} ~ p^{4}/C_{1} 
(at t = 0) 
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 !
_{} renormalizes H _{}
Both are logarithmically divergent in d = 4
t, C_{1}, C_{2}, _{} have no singular corrections
Power counting
Selfconsistent oneloop theory is exact
Options to determine critical behavior:
RG analysis: Structure of perturbation theory suggests standard _{} expansion
Questions:
Why does one worry about renormalizability in the context of the fieldtheoretic formulation of the RG, but usually not in the context of the Wilsonian momentumshell formulation ?
Answers:
A welldefined renormalizability property is necessary to produce powerlaw critical behavior, even in the momentumshell formulation
Choose scale dimensions:
[_{}] = [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
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
_{}

C_{1} is not renormalized
_{}

C_{2} is not renormalized
_{}
c_{2} is formally irrelevant !
Remaining task: Determine _{}, and the flow equation for h, from loop expansion
3. Loop expansion
Zeroloop order:
NB: C_{2} and H are both irrelevant with scale dimension 2
Oneloop order:
with U = C_{2}/a H
NB: Loop expansion parameter is U, not C_{2} !
uflow equation: _{ }
FP value
_{ }

Seemingly controlled _{} 
expansion leads to powerlaw critical behavior at oneloop order
Nothing is obviously wrong
Gaussian FP unstable due to H being "dangerously
irrelevant", compensating the irrelevance of C_{2}
This is a consequence of
the diffusive time scale !
Twoloop 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) scaledependent uflow equation
Renormalizability property:
Powerlaw critical behavior
requires
uflow equation to be scale independent
requires
special relation between coefficients of U^{2} log^{2} 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, c_{2} > 0 , h > 0 :
The fixed points at both zero and oneloop order do not approximate the true critical behavior in any sense
This first becomes apparent at twoloop order
Correct critical behavior can be found by either
Power laws times loglognormal behavior
( DB & TRK 1992 , DB et al 2001 )
E.g., the
correlation length is given by
Renormalizability property is necessary for powerlaw 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!