

Consider, e.g., a ferromagnet

Example: MnSi ( Pfleiderer et al 1997 )
Near the transition line (t > 0), one observes critical behavior :
correlation length 

relaxation time 

order parameter 

OP susceptibilty 
Critical behavior at the CPT is very well understood
Question: What is the critical behavior near the QCP ? 
_{} sets an energy scale
Crossover at _{} : 
Quantum vs. classical stat. mech.:
partition fct. 

classically 
statics + dynamics 

QM 
statics + dynamics 
LandauGinzburgWilson (LGW) theory:
classically 

QM 
Hypothesis (based on the above observation) ( Hertz 1976 ):
QPT in ddimensions related to CPT in (d+z)dimensions! 
Landau theory (valid classically for d > d_{c}^{+} = 4):
Equation of state: 

Continuous transition (if u > 0) at t=0 with meanfield critical exponents
Gaussian approximation for OP fluctuations:
(OrnsteinZernike plus electron dynamics) 
_{} 

Hertz actually derived the Landau theory from a microscopic model.
This theory and its generalizations ( Millis 1993 )
amounts to
It predicts
meanfield exponents
T_{c} ~ t^{3/4} (via a DIV)
Indication that the above conclusion cannot be correct:
Resolution of problem:
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 )
MnSi ( Pfleiderer et al. 1997 , 2001 )
lowT transition T_{c}(t) meanfield like
nonFermi liquid not superconducting

UGe_{2} (e.g., Saxena et al 2000 , Huxley et al 2001 , Pfleiderer & Huxley 2002 )
lowT transition
nonFermi liquid superconducting phase

ZrZn_{2} ( Pfleiderer et al 2001 ) and URhGe ( Aoki et al 2001 )
lowT transition Superconducting phase

Ni_{x}Pd_{1x} ( Nicklas et al 1999 )
lowT transition meanfield exponents no superconductivity

URu_{2x}Re_{x}Si_{2} ( Bauer et al 2002 )
lowT transition
exponents strongly
nonFermi liquid no superconductivity

Summary of Experiments:
Clean systems
1st order TCP at T > 0
2nd order meanfield exponents
Coexistence of superconductivity and ferromagnetism in sufficiently clean
samples
( different talk )
Disordered systems
Transition 2nd order with nonmean field critical behavior
NonFermi liquid behavior in whole regions in the PM phase
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 meanfield like ? In the disordered case, what causes the strange exponents ? 
Critical behavior determined by soft or massless modes
Keep all soft modes explicitly!
2 fields: OP fluctuations _{} (soft at t _{} 0, any T)
ph fluctuations _{} (soft at T = 0, any t)
Action:  




free fermions 


 paramagnon propagator 
 fermion propagator 
This is Landau/Gaussian theory, cf. Sec. I.2
2nd order transition with Landau exponents
Two time scales: 1) critical _{}
2) fermionic _{}
MFA for OP (_{}) + Gaussian approximation for fermions:
m > 0 coupling between OP and fermions
magnetic Goldstone modes appear instead
Integrate out fermions,  , etc. 
Contribution to equation of state
Generalized meanfield equation of state (d=3):

Effects of nonzero temperature (T) and disorder (G) :
T>0 gives ph exitations a mass
ln m > ln (m+T) tricritical point 
G>0 changes fermion dispersion relation
m^{d} > m^{d/2} G>0 changes sign of coefficient Generalized meanfield equation of state (d=3)
transition second order with

Phase diagram:
In agreement with experiments on MnSi, UGe_{2}, URuReSi
Cannot explain experiments on ZrZn_{2}, NiPd
Consider full theory (keep OP fluctuations)
Consider fixed points, then analyze stability
Consider clean case, disordered case proceeds analogously
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 = z_{M}
Unstable (c_{2} relevant) for d < 3 if z = z_{q}
This happens if fermion loops are involved, e.g., 
c_{2} 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
Need to keep loops systematically
e.g., u =  
u has negative correction
in perturbation theory, u(b > _{}) < 0 always
fluctuationinduced first order transition
Maps on classic fluctuationinduced first order transition in superconductors
( Halperin, Lubensky, Ma 1974 )
This is the generalized meanfield 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 t_{1} sufficiently large destroyed by critical fluctuations if t_{1} is small 
Explicit calculation Second order case has
meanfield exponents with log corrections in d = 3 (marginal operator c_{2})
nonmean 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 meanfield theory with log corrections
Correctly predicts exponents observed in URuReSi !
LowT_{c} itinerant ferromagnets are remarkably complex and interesting.
The T = 0 transition can be 1st order for generic reasons: A fluctuationinduced 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 fluctuationinduced 2nd order transition. This transition is in a new universality class.
Sufficiently strong nonmagnetic 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)
NonFermi liquid behavior (???)