Influence of Generic Scale Invariance on Quantum Critical Behavior:

II. Generalized MeanField Theory


UGe_{2}, ZrZn_{2}, (MnSi) 
(pressure tuned, clean) 

URu_{2x}Re_{x}Si_{2} 
(concentration tuned, disordered) 

Ni_{x}Pd_{1x} 
(concentration tuned, clean?) 

Clean materials all show tricritical point , with 2nd order transition at high T, and 1st order transition at low T:








T=0 1st order transition persists at nonzero magnetic field, ends at quantum critical point

This behavior of clean systems appears to be universal
URuReSi shows 2nd order transition with nonmeanfield exponents (e.g., _{} = 3/2) ( Bauer et al 2002 )
NiPd shows 2nd order transition with meanfield exponents ( Nicklas et al 1999 )
MnSi shows NFL behavior in PM phase ( Pfleiderer et al 2001 , 2004 )
Consider Landau theory with order parameter m = average magnetization
(Free) energy density: 
Equation of state: 

Landau theory predicts
first order transition if u < 0.
Sandeman et al 2003 , Shick et al 2004
Band structure in UGe_{2} leads to u < 0
Origin of 1st order transition at T = 0
Caveats:  Not universal 
TCP needs to be put in by hand 
Hertz 1976
Meanfield theory correctly describes T = 0 transition for all d > 1 in clean systems,
and for all d > 0 in disordered ones.
Breakdown of Landau theory for d _{}
3 (clean), and d _{} 4
(dirty), respectively
( TRK & DB 1996 ;
TRK, DB, et al 1997 ff )
Phase transition physics is determined by all of the soft modes in the problem.
These are,
particlehole excitations (at T=0, any t) _{}
ph excitations contribute to _{ } _{}
FM phase
broken symmetry
some ph excitations acquire a mass
contribution to f:
_{}
Note: 
particlehole excitations soft for all values of t 
They are the Goldstone modes of a spontaneously broken symmetry ( Wegner 1979 ) 

Generic Scale Invariance 

It is impossible to construct a local field theory in terms
of the OP only.


This is not revolutionary.
Analogous effects occur at other phase transitions,
and in particle physics,

Conclusion: Landau theory misses modemode coupling contribution to the equation of state:
Generalized meanfield equation of state (d=3):

This is generic!
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:
Effect of nonzero magnetic field (h):
Generalized meanfield theory produces the experimentally

Keep all soft modes explicitly!
two fields: 
OP fluctuations 

ph fluctuations 
Action:  









The effective action has been analyzed at various levels:
Hertz theory (fixed point unstable with respect to c_{2})
b) Generalized MeanField Theory:
+ Gaussian approximation for fermions:
, etc. 
Yields generalized meanfield equation of state (see above)
Predicts tricritical point , and correct h  dependence
c) RG Analysis:
Two time scales: 
Critical time scale 
z = 3 (clean), or z = 4 (disordered) 
fermionic time scale 
z = 1 (clean), or z = 2 (disordered) 
Upper critical dimension 
d_{c}^{+} = 3 
(clean) 
d_{c}^{+} = 4 
(disordered) 
Disordered case:
Conventional d_{c}^{+}  _{} expansion does not work! ( DB, TRK, JR 2004 , condmat/0406350 )
Subdominant time scale leads to a dangerous irrelevant variable entering the flow equations !
Oneloop analysis yields misleading results!
Resummation to all orders (disordered case)
Perturbatively exact solution
Critical behavior as described by generalized meanfield
theory with log corrections
to powerlaw scaling
( TRK & DB 1992 , DB et al 2001 )
Clean case:
Conclusion probably not reliable (see above) Needs to be revisited!
LowT_{c} itinerant ferromagnets are remarkably complex and interesting.
The T = 0 transition is 1st order for generic reasons: A fluctuationinduced 1st order transition preempts the continuous Landau transition.
Crucial for this mechanism: Fermionic modes and the resulting two time scales in the problem.
Theory explains
magnetic field dependence of phase diagram
Sufficiently strong nonmagnetic disorder drives transition 2nd order.
New universality class, which is solved exactly. Agrees with experimental results on URuReSi.
Properties that are not understood (at least not completely):
Ferromagnetic superconducting phase (partially understood)
NiPd (awaits clarification of clean case)
NonFermi liquid behavior in MnSi (???)