II. Generalized MeanField Theory


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 ( Hertz 1976 ):
QPT in ddimensions related to CPT in (d+z)dimensions! 
Caution: Hypothesis plausible, but in general NOT correct (see below)
Order parameter: _{} average magnetization (MFA)
(Free) energy density: 
^{} 
Equation of state: 

Landau theory predicts continuous transition at t=0 with meanfield critical exponents
Gaussian approximation for OP fluctuations:
(OrnsteinZernike plus electron dynamics) 
_{} 
Classical MFT valid for 4 < d suggests QM MFT valid for 4 < d + z = d + 3 or d > 1

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)
The resolution of these problems turns out to affect both disordered and clean systems
( TRK & DB 1996 ; TRK, DB, et al 1997 ff )
Phase transition physics is determined by soft modes ( = massless particles) in the theory.
These are,
particlehole excitations (at T=0, any t) _{}
ph excitations contribute to f_{0} _{}
FM phase
broken symmetry
some ph excitations acquire a mass
contribution to f:
_{}
Note: ph excitations soft for all t
They are the Goldstone modes of a spontaneously broken symmetry
`` Generic Scale Invariance ''
Conclusion: Landau theory misses modemode coupling contribution to the 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 diagrams:
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 as predicted by
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 generalized meanfield critical behavior
NonFermi liquid behavior in whole regions in the PM phase
Keep all soft modes explicitly!
2 fields: OP fluctuations _{}
ph fluctuations _{}
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:
Nonanalyticities due to fermion loops,  , etc. 
Predicts first order transition (always!)
T>0 gives fermions a mass TCP at T>0 in agreement with exps on MnSi, UGe_{2} 
Cannot explain experiments on ZrZn_{2}, NiPd
Assign scale dimensions
which may be either z_{M} or z_{q} ! 
and look for fixed points
Discuss clean case, quote results for disordered case
Look for fixed point with
[G] = [H] = 0 (Fermi liquid)
[a] = [c_{1}] = 0 (Landau theory)
Check stability:
[t] = 2 > 0 _{} = 1/2
[u] =  (d  1) < 0 for d>1
[c_{2}] = (4  d  z)/2
[c_{2}] = (3  d)/2 if z = z_{q}
NB: 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 
Instability of first order transition leads to nonmean field, fluctuationinduced second order
transition. Explicit calculation
Second order case in d = 3 has meanfield exponents with log corrections
(due to the marginal operator c_{2} ); nonmean field exponents in d < 3
Consistent with experiments on clean systems !
Explicit calculation for disordered case
Critical behavior as described by generalized meanfield theory with log corrections
Agrees with experiment on NiPd !
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 (???)