Generic Scale Invariance and Critical Behavior :


free energy is analytic function of order parameter: 

order parameter susceptibility has OrnsteinZernike form: 

Landau theory predicts
1st order transition if u < 0 powerlaw behavior for r , h 0 , e.g.
longrange correlations at r = 0 , e.g.
with a correlation length _{}
Long range due to soft
or massless modes (viz., orderparameter fluctuations)


Wilson 1970's LGW theory, and (Wilsonian) RG
Fieldtheoretic generalization of Landau theory:
with an ``action''
Free energy displays scaling:
Universality classes
Experiment: Scaling and universality are observed
Wilson & Fisher 1972 Exponents can be calculated in a d_{c}^{+}  _{} expansion
Ferrell et al 1967, Halperin & Hohenberg 1967 dynamical scaling of time correlation functions,
with z the dynamical critical exponent
3 independent exponents: 2 static ones plus z
thermodynamics can be calculated independent of the dynamics
In quantum mechanics, statics and dynamics couple:
Scaling of free energy generalizes to
with an LGW action
The functional form of _{} depends on the details of the problem
Imaginary time direction finite for T > 0 crossovers from quantum to classical scaling

Two scenarios:
Crucial points:
LGW theory will be a local field theory only if no soft modes have been integrated out !
Check for soft modes other than the order parameter fluctuations
This hold for both quantum and classical theories!
Critical soft modes occur at isolated critical points in the phase diagram
Modes can be soft in entire regions of the phase diagram Generic scale invariance
Consider two mechanisms for GSI in real space. GSI can also occur in time space longtime tails
Spontaneous breaking of a continuous symmetry soft modes in the entire broken symmetry phase
More generally: Modes soft at zero wavenumber (mechanical model has no wavenumber concept)
Examples:
Theoretical realization: Classical O(2)  symmetric_{} theory
action 
Simplest realization: U(1)
Examples:
Theoretical realization: scalar gauge theory
Gauge field eats the Goldstone mode, becomes massive in the process
Situation reversed from case without gauge symmetry ( Anderson 1963 , Higgs 1964 )
Expect GSI, if present, to influence critical behavior
Consider two examples in detail, mention a few others
Consider a superconductor
Order parameter is complex scalar field (amplitude + phase of Cooper pair)
Cooper pairs carry charge q = 2e Coupling to E&M vector potential (transverse photons)
Action given by the gauge theory given above
Extend to 3+1 dimensions A simple particle physics model: Scalar QED (scalar mesons + photons)
Treat _{} in meanfield approximation A can be integrated out exactly
meanfield free energy
with _{} and _{} closely related to r and u, and _{}
NB: F is not an analytic function of _{} !
For later reference: In d=4 one finds
Consequences for superconductors ( Halperin, Lubensky, Ma 1974 )
Consequences for particle physics ( Coleman & Weinberg 1973 )
Consider a nematic liquid crystal
Same conclusions as above. 1st order transition has been observed!
Effects orderparameter fluctuations:
Quality of meanfield approximation depends on ration of length scales (type I vs type II)
Experiment 1st order in type I materials, 2nd order in type II materials
Numerics + theory
1st order transition gives way to inverted XY universality class
(incompletely understood)
Soft modes:
Order parameter fluctuations _{}
Particlehole excitations contribute to the free energy
in the paramagnetic phase _{}
and in the ferromagnetic phase
broken symmetry
some ph excitations acquire a mass
contribution
_{}
Meanfield free energy in d = 3 same as for superconductor/liquid crystal in d = 4
Generalized meanfield equation of state (d=3):

1st order transition! ( DB, TRK, TV 1999 )
Experiments: Sufficiently clean
materials all show
tricritical
point , with 2nd order transition at high T,
and
1st order transition at low T:








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:
Experiment: URu_{2x}Re_{x}Si_{2} shows 2nd order transition with _{} = 3/2 ( Bauer et al 2002 )
Effect of nonzero magnetic field ( DB, TRK, J. Rollbühler 2004 ):

At QCPs Hertz theory is valid Meanfield quantum critical behavior
Experiment: Wing structure observed in MnSi ( Pfleiderer et al 2001 ) and ZrZn_{2} ( Uhlarz et al 2004 )
Generic soft modes can strongly influence critical behavior, and even destroy it, in both classical and quantum systems
Generic soft modes are ubiquitous in metals at T = 0
Quantum phase transitions in metals are i.g. not simple
Classical example : Superconductor/nematic liquid crystal.
GSI due to gauge field fluctuationinduced 1st order transition
QM example : Itinerant quantum ferromagnet.
GSI due to Goldstone mode
fluctuationinduced 1st order transition at h = 0
meanfield QCP at h _{} 0
Another example:
LTTs in classical fluids are an example of temporal GSI
Enhancement of LTTs at criticality Modification of critical dynamics