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Collective phenomena :
from Phase Transitions to
Statistical Field Theory
Phase transitions and exponents
Example 1 : para/ferromagnetic phase transition
Fromhttp://en.wikipedia.org/wiki/Magnetic_domain
Spontaneous magnetization vanishes above some critical temperature, which depends on material.
Fromhttp://fr.wikipedia.org/wiki/Température_de_Curie
Example 2 : liquid crystals
Discovered by F. Reinitzer in 1888Systems of 'rigid rods', each characterized by its position and its orientation. Isotropic phase at high temperature and low densityThere exist other phases, described by G. Friedel (1922):
Fromhttp://en.wikipedia.org/wiki/Liquid_crystal
Example 3 : lambda transition in liquid helium
From http://ltl.tkk.fi/research/theory/helium.html
No viscosityHuge thermal conductivity
2.17 K
Fromhttp://www.yutopian.com/Yuan/TFM.html
Two-fluid model
Kammerlingh Onnes et la liquéfaction de l'hélium (in French):http://www.bibnum.education.fr/files/BRADU_HELIUM_LIQUIDE.pdf
Ising model: (1926)
Si
Sj
Spins Si = +1 or -1
At zero temperature: two uniform ground states Si = +1,∀ i or S
i =-1, ∀ i
Magnetization :
Sj
What happens at non-zero temperature ? Does M remain equal to zero or does it vanish as soon as T>0 ?
Answer : depends on space dimension (Peierls, 1936 ; Onsager, 1944)
neighbours
There is a non-zero magnetization at all temperatures T smaller than Tc
How is it possible ?
Dimension D=1 :M/N
+1
-1
Dimension D=2 :M/N
+1
-1
Tc
T T
Two types of behavior :
Magnetization fluctuates around 0Weak memory of initial condition
T < Tc
Magnetizaton fluctuates around +m(T) and -m(T)Memory of initial condition (up to T-dependent time)
A bit of phenomenology …Monte Carlo simulation of the Ising model in D=2 dimensions
(don't worry, we'll see later how it works)
Link: http://www.ibiblio.org/e-notes/Perc/ising640.htm
time
m
+m(T)
- m(T)
time
mm
0 T > T
c
From a « static » point of view:
● For any temperature T>0 and finite system size N, the overall magnetization is zero (as found in numerical simulations)
● One can break the spin reversal symmetry with a weak field h :
The previous calculation gives :
Therefore
What about ?
Free energy f at fixed magnetization :(we will do the explicit computation within mean-field theory later)
T > Tc
T < Tc
where
f
m
f
m
0
f
m
(h=0) (h>0)
Thus
Spontaneous magnetization :
Spin reversal symmetryis spontaneously broken
+1
Tc
T
mS(T)
Yes, but in a physical system, N is finite ...Yes again, but N is large ...
Let us look again at the magnetization vs. the external field h.
T > Tc
T < Tc
+1
-1
h
m(T,N=∞,h)
+ mS(T)
- mS(T)
+1
-1
h
m(T,N=∞,h)
Slope at origin :
N large, χ of the order of NTT
c
T > Tc
The magnetization m vanishes on average but there exists a local order:
T < Tc
From a mathematical point of view, the correlation length is now defined through :
Correlations between spins :
The correlation length ξ is the typical size of the + or - domains ; It increases as the temperature decreases.
The magnetization m is different from zero, e.g. m>0. There appear smalldomains of - spins in the + phase : Their size is ξ ; it decreases as the temperature decreases.
Question : How does ξ behave as a function of temperature in D=1 ? You'll know after the break...
+1
Tc
T
Critical exponents:
Characterize the behaviour of the properties of the system (observables ) at, or close to the critical temperature.
● Magnetization and β :
mS(T)
NB : Not a mere definition: one has to show that the functional dependence is correct ...
● Susceptibility and γ :
NB : there could be a priori two exponents, one for T < Tc and one for T > T
c
Slope at origin for magnetization:
TTc
● Specific heat and α :
NB : The specific heat is proportional to the variance of the energy (fluctuation-dissipation theorem), hence the energy has large fluctuations at the critical point
TTc
● Correlation length and ν :
TTc
at « large » distance R and T > Tc
++
+
+
+
++
+
-
-
(connected correlation for T< Tc)
ξ
++
+
+
+
+ ++
---
--
- -
- -
-
-
++-
+
+
+ + + +
++
+ +
+++
-
-
Experimental data: liquid crystals at the N-S transitions
Exponent values : γ ≈ 0.9 , ν ≈ 0.6(wave number q
0 = 1 nm-1)
● Correlations at the critical point and η :
R
à « large » distance R and for T = Tc
correlation
Exponential decay for T ≠ Tc
Power law decay at T = Tc
● Magnetization as a function of the field at the critical point and δ :
+1
T = Tc
h
mS(h)
Presence of a singularity since the susceptibility is infinite at the critical temperature
NB : it is possible to define many more exponents associated to other observables. Examples will be given to you in the tutorials.
Why are critical exponents interesting at all ?
They do not depend on the details of the model. For instance :
Ising model in D=2square lattice
Tc
α,β,γ,δ,η,ν
Ising model in D=2triangular lattice
Tc
α,β,γ,δ,η,ν≠=
Consequence : The exponents of the real physical systems are identical to the ones of the theoretical models !
Illustration : The λ transition in superfluid helium
H. Kleinert, Theory and satellite experiment for critical exponent α of λ-transition in superfluid helium, Physics Letters A 277 (2000) 205–211
[2] Ahlers, measure of heat capacity of helium 4 (1971)[9] Goldner et al., measure of the heat flux velocity (1992)[1] Lipa et al., experiment in spatial shuttle under micro-gravity (1992).
Why is there a linear relationship between α and ν ?
theory (XY),simulations
A few questions :
● Why are there identities linking critical exponents ? How many critical exponents are independent ?
● Why are critical exponents universal ?
● How can critical exponent be calculated in practice ?
Next week !
Next next week !
Next months (& year) !