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 !

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