introduction to phase transitions in statistical physics ... · introduction to phase transitions...

Introduction to Phase Transitions inStatistical Physics and Field Theory

Motivation

Basic Concepts and Facts about Phase Transitions:

Phase Transitions in Fluids and Magnets

Thermodynamics and Statistical Mechanics

Order Parameter and Symmetry Breaking

Susceptibilities and Correlation Functions

Characteristics of Phase Transitions

Universality Classes

Important Models

IGS-Block course, LI, Oct 2006, J. Engels – p.1/??

Motivation

• Phase Transitions are of relevance everywhere in physics - starting with the the-ory of the early universe, in particle physics, in condensed matter physics, and ofcourse they are relevant in daily life.

• In the critical region of continuous transitions cooperative phenomena generatea large scale, the correlation length, although the fundamental interactions areshort range - most details of the microscopic structure become unimportant anda power law behaviour emerges. Such scaling laws occur also in the

clustering of galaxies, the distribution of earthquakes, the turbulence in fluids andplasmas, etc.

• Most interesting for particle physicists is: the connection between statistical fieldtheory of critical phenomena and quantum field theory, both froma theoretical point of view:

for the understanding of the role of renormalization and renormalizable field the-ories in particle physics, for the calculation of critical parameters,

and in a practical sense:

methods from statistical mechanics allow simulations of lattice gauge field theo-ries and lead to non-perturbative results


Phase Transitions in Fluids and Magnets

At a phase transition (PT) one observesa sudden change in the properties of asystem, e. g.

liquid⇐⇒ gasparamagnet⇐⇒ ferromagnetnormal conductor⇐⇒ superconductor

Phase diagram of a typical fluid:

Melting curve

T

p

p

Vapour

c

pressure curve

Tc

C

GasSublimation curve

LiquidSolid

Solid lines are phase boundaries

separating regions of stable phases

In crossing these lines:

Mass density ρ and energy density ε

change discontinuously, they jump

=⇒ 1. order phase transition

C is the critical end point

=⇒ 2. order phase transition


Vapour Pressure Curve

The vapour pressure curve is the liquid-gas coexistence line:

Order parameter

ρ

ρ

ρ

(T)

c

T Tc

C

liquid(T)

ρgas

"Order parameter" (OP): ρl − ρg

T = Tc : ρc = ρl(Tc) = ρg(Tc)

T > Tc : no distinction betweengas and liquid

At Tc: continuous transition, but Cv

diverges for ρ = ρc


Magnetic Phase Transitions

The phase diagram :

Analog to fluids:

T < Tc : 1. order PT

T = Tc : 2. order critical point

The order parameter :

T T

OP

c

M H = 0

inaccessible

Spontaneous Magnetization:

M(T < Tc,H = 0) 6= 0

M(T,H → 0+) = −M(T,H → 0−)

M(T ≥ Tc,H = 0) = 0


Thermodynamics and Statistical Mechanics

We consider only the equilibrium case. Thermodynamics assumes no particular model fora system, the variables are macroscopic quantities - for the system as a whole. There are

Extensive Variables: U =internal energy, V =volume, S =entropyNi =particle number of type i, ~M =total magnetic moment

Intensive Variables: T =temperature, p =pressure, ρ =density=mass/volumeµi =chemical potential of type i, ~H =external magnetic field

The magnetic field and the magnetic moment are in general vectors in N dimensions

~H = H~n, ~n = (n1, . . . , nN ), ~n2 = 1

A change in U (here without the internal energy of the field) is described by

dU = TdS − pdV − ~Md ~H

Mostly one treats either Fluids, where ~M = 0 or Magnetic systems with dV = 0.


Thermodynamic Potentials

Instead of using the internal energy U = U(S, V, ~H) one may use other potentials.Potential changes are achieved by Legendre transformations.

Thermodynamic Potentials Fluids Magnets

Internal energy U(S, V ) U(S, ~H)

dU = TdS − pdV dU = TdS − ~Md ~H

Helmholtz free energy F (T, V ) = U − TS F (T, ~H) = U − TS

dF = −SdT − pdV dF = −SdT − ~Md ~H

Gibbs free energy G(T, p) = F + pV G(T, ~M) = F + ~M ~H

dG = −SdT − V dp dG = −SdT + ~Hd ~M

A closed system has in equilibrium: at fixed S minimal U , at fixed U maximal Sand at fixed T minimal F .

In finite volumes: ensemble average = time average


Thermodynamic Limit

We would like to work in the thermodynamic limit, i. e. in the limit where

V →∞ , Ni →∞ , at fixed intensive variables

Reasons :i) For finite systems the thermodynamic quantities are always analytic functions of

their variables. Singular behaviour as required for phase transitions does notoccur. Because of the finite volume the correlation length is finite.

ii) Spontaneous symmetry breaking exists only in the thermodynamic limit. In finitesystems the breaking is explicit by an external field or by the boundary conditions.

iii) In the thermodynamic limit the different ensembles are equivalent.

iv) In the thermodynamic limit there is no boundary (surface) dependence.

=⇒ The usual extensive variables become infinite in the thermodynamic limit. Wetherefore use densities : ε = U/V energy density

~M = ~M/V magnetizations = S/V entropy densityf = F/V = ε− Ts free energy density


Thermodynamic Relations

Fluids (T, V ) Magnets (T,H), N = 1

df = −sdT +(

∂f∂V

)

TdV df = −sdT −MdH

s = −(

∂f∂T

)

Vs = −

(

∂f∂T

)

H

ε = f + Ts =(

∂βf∂β

)

Vε = f + Ts =

(

∂βf∂β

)

H

p = −f − V(

∂f∂V

)

T= −

(

∂V f∂V

)

TM = −

(

∂f∂H

)

T

Cv =(

∂ε∂T

)

VCH =

(

∂ε∂T

)

H

κT = −1V

(

∂V∂p

)

Tχ =

(

∂M∂H

)

T

Equation of state

Specific heat

Susceptibility

We use k = 1 as unit, so that β = 1/T


Statistical Mechanics

In contrast to thermodynamics it provides a link between microscopic and macroscopicphysics, because it starts with a Hamiltonian H, which describes the microscopicinteraction.

Canonical ensemble, the partition function:

Z = Tr e−βH =∑

α

e−βEα

Eα = Eigenvalue of H in state α

U = 〈E〉 = 1

Z

∑

α

Eαe−βEα = −∂ lnZ

∂β= 〈H〉

F = −T lnZ , f = −TVlnZ

Z = e−βF


Statistical Mechanics

The fluctuation in the energy is given by

(∆U)2 = 〈(E − 〈E〉)2〉 = 〈E2〉 − 〈E〉2 = ∂2 lnZ

∂β2

More general, the average of a quantity X can be calculated by introducing a sourceterm XY in the Hamiltonian:

H = H0 −XY

and taking the derivative with respect to Y :

1

β

∂ lnZ

∂Y

∣

∣

∣

Y=0=1

βZ

∂

∂Y

(

∑

α

e−β(Eα−XαY )

)

Y=0

= 〈X〉 = −∂F∂Y

∣

∣

∣

Y=0

and1

β

∂〈X〉∂Y

∣

∣

∣

Y=0= 〈X2〉 − 〈X〉2


Order Parameter and Symmetry Breaking

Symmetry breaking for a ferromagnet, ~H = H~n, ~H,~n N -dim vectors

T < Tc: ferromagnetic phase, the spins are aligned, M measures the degree ofordering − the ordered state is not symmetric under rotationH 6= 0 =⇒ ~M aligns in ~H-direction, for ~H → − ~H, ~M jumps to − ~M

− explicit symmetry breakingH = 0 =⇒ direction of ~M is spontaneously chosen by the system

− spontaneous symmetry breaking

T > Tc: paramagnetic phase,H 6= 0 =⇒ ~M ∼ ~H, for ~H → − ~H,~M changes continuously to − ~M

− explicit symmetry breakingH = 0 =⇒ ~M = 0

− symmetry of state

T = Tc: critical line, like paramagnetic phaseH = 0 =⇒ ∂M/∂H is infinite

M

c

H

T > Tc

T < T T = Tc

N=1


The Order Parameter

Definition of the order parameter:

An (ideal) order parameter for a phase transition is a variable φ with

〈φ〉 ≡ 0 in Phase I

〈φ〉 6= 0 in Phase II

Here, 〈〉 means the "thermal average" over a long period in equilibrium at constanttemperature = the time average. Further remarks:

φ = φ(~x) is usually a local, fluctuating variable, which is not always an observable.

φ can be a scalar, a vector etc., real or complex.

The name "order parameter" is confusing, because it is also used for 〈φ〉, e. g. whenthe "order parameter" is shown in a plot. Better: φ(~x) is the order parameter field orlocal order parameter, 〈φ(~x)〉 is the order parameter.

In a homogeneous system 〈φ(~x)〉 is independent of ~x

〈φ(~x)〉 = 〈φ〉


The Order Parameter

The Hamiltonian is a function of φ:

H = H(φ(~x))

In many cases: H is invariant under certain transformations of φ, if ~H = 0 , e. g.

reversal φ(~x)→ −φ(~x) Ising model

change of phase φ(~x)→ eiαφ(~x) He4, fluid to superfluid

rotation ~φ(~x)→ A~φ(~x) Ferromagnet

. . .

Magnet: ~φ(~x) corresponds to the local magnetization and

~M = 〈~φ(~x)〉 = 〈~ϕ〉 where ~ϕ =1

V

∫

ddx~φ(~x)

~ϕ is the volume average of the local magnetization and ~M its thermalaverage, a global variable, the total magnetization of the magnet.


The Order of a Phase Transition

A phase transition occurs, if in the thermodynamic limit the free energy density f (or otherthermodynamic potentials) is non-analytic as a function of its parameters (T,H, etc. ).

A definition of order: A phase transition is of n′ th order, if the thermodynamic potentialhas (n− 1) continuous derivatives, but the n′ th derivativeis discontinuous or divergent.

Modern Classification (M.E. Fisher)

• Discontinuous or first order phase transition:one or more of the first derivatives of f has a finite discontinuity; the orderparameter is the first derivative of its conjugate variable =⇒ it jumps at Tc.

• Continuous (or second order) phase transition:the first derivatives of f are continuous, but one higher derivative is singular -includes all higher order transitions =⇒ the order parameter is continuous at Tc.

• Crossover:is no real phase transition, but there is a rapid change in the behaviour of thesystem =⇒ in (finite volume) simulations difficult to distinguish from true phasetransitions.


Susceptibilities and Correlation Functions

General Magnet: H = H0 − ~H∫

ddx~φ(~x)

N -dimensional vectors: ~H = H~n, ~φ(~x), ~ϕ

On a lattice: ~ϕ = 1Nx

∑

~x~φ(~x)

d-dimensional vector: ~x

H = H0 − V ~ϕ ~H Xa = V ϕa , Ya = Ha

〈V ϕa〉 = −∂F

∂Haor Ma = −

∂f

∂Ha= 〈ϕa〉

Tensor of susceptibilities:

χab = −∂2f

∂Ha∂Hb=∂Ma

∂Hb=∂〈ϕa〉∂Hb


The Susceptibilities

χab =∂〈ϕa〉∂Hb

=∂

∂Hb

(

1

Z

∑

α

ϕαae−β(Eα−V ~ϕα ~H)

)

=1

Z

∑

α

ϕαaβV ϕ

αb e−β(... ) − 1

Z2

(

∑

α

ϕαae−β(... )

)(

∑

α

βV ϕαb e−β(... )

)

χab = βV[

〈ϕaϕb〉 − 〈ϕa〉〈ϕb〉]

Total magnetic susceptibility ∼ fluctuation of the length of ~ϕ :

χtot =∑

a

χaa = Tr χ

χtot = βV[

〈~ϕ 2〉 − 〈~ϕ 〉2]


The Susceptibilities

Assumption: H0 is invariant under O(N)-rotations of ~φ(~x)

Then ~M aligns with ~H: ~M =M~n

~ϕ = ϕ‖~n+ ~ϕ⊥ , ~ϕ⊥~n = 0 , =⇒ M = 〈ϕ‖〉 , ~M⊥ = 〈~ϕ⊥〉 = 0

χtot = χL + (N − 1)χT with χL =∂M

∂H, χT =

M

H

Longitudinal susceptibility: χL = βV(

〈ϕ2‖〉 − 〈ϕ‖〉2)

↪→ Fluctuation of the longitudinal field component = fluctuation of M

Transverse susceptibility: χT = βV 〈~ϕ 2⊥〉/(N − 1)↪→ Fluctuation of a transverse field component

χab = χL nanb + χT (δab − nanb)


The Correlation Functions

Correlation functions (2-point connected) of the components of ~φ(~x) :

Gab(~x1, ~x2) = 〈φa(~x1)φb(~x2)〉 − 〈φa(~x1)〉〈φb(~x2)〉= 〈 (φa(~x1)− 〈φa(~x1)〉)(φb(~x2)− 〈φb(~x2) 〉

Translation invariance: Gab(~x1, ~x2) = Gab(~x) , ~x = ~x2 − ~x1Rotation invariance: Gab(~x) = Gab(r) , r = |~x|

=⇒ Gab(~x) = 〈φa(0)φb(~x)〉 − 〈φa(0)〉〈φb(~x)〉∫

ddxGab(~x) = V [ 〈φa(0)ϕb〉 − 〈φa(0)〉〈ϕb〉 ]

= V [ 〈ϕaϕb〉 − 〈ϕa〉〈ϕb〉 ] = χab/β

FD-Theorem: χab = β

∫

ddxGab(~x)


The Correlation Functions

Longitudinal and transverse correlation functions :

GL(~x) = 〈φ‖(0)φ‖(~x)〉 − 〈ϕ‖〉2

GT (~x) =1

N − 1 〈φ⊥(0)φ⊥(~x)〉

Like for χ: Gab = GL nanb +GT (δab − nanb)

Gtot =∑

aGaa = GL + (N − 1)GT = 〈~φ(0)~φ(~x)〉 − 〈~ϕ 〉2

In general: For larger distancesr = |~x| the fields become more andmore uncorrelated and

G(~x) ∼ e−r/ξ

rτ

ξ or ξL,T is the correlation lengthIf ξ →∞ a long range order develops and G(~x) has a power behaviour.


Characteristics of First Order Transitions

There is a latent heat ` 6= 0 :

` = limT→T+

c

ε(T )− limT→T−c

ε(T ) 6= 0 , ` =

∫ T+c

T−c

Cv(T )dT , Cv = CH

The order parameter M jumps at Tc because :

limT→T−c

M 6= 0 , limT→T+

c

M = 0

The correlation length ξ stays finite at Tc and :

limT→T+

c

ξ − limT→T−c

ξ 6= 0 ,

The fluctuations / susceptibilities at Tc grow ∼ V :

χ = βV(

〈ϕ2‖〉 −M2

)

= βV⟨

(ϕ‖ −M)2⟩

∼ V β(∆M)2 ∼ V

Cv =β2

V

(

〈E2〉 − 〈E〉2)

= β2V 〈(E

V− ε)2〉 ∼ V β2(ε+ − ε−)

2 ∼ V


Characteristics of Continuous Transitions

At T = Tc :

• There is no latent heat

• The order parameter vanishes

• The correlation length diverges, i.e. thefluctuations of the order parameter fieldgrow infinitely, a long range order isestablished

` = 0

M(Tc) = 0

limT→Tc

ξ →∞

Close to Tc :

• The critical behaviour is described by power laws withcritical exponents.The exponents are the same for T → T+c and T → T−c .


Characteristics of Continuous Transitions

Reduced temperature: t = (T − Tc)/TcThe following laws are valid in the thermodynamic limit

Singular behaviour of an observable F (t) near Tc or t = 0 for H = 0

F (t) = A|t|λ(1 + b|t|λ1 + . . . ) , λ1 > 0

λ = critical exponent

λ = limt→0

ln |F (t)|ln |t|

λ can, in principle, be different for t→ 0+ and t→ 0−, but is not!

A = critical amplitude, is different for t→ 0+ and t→ 0− !

(1 + b|t|λ1 + . . . ) = corrections to scaling, b is different for t→ 0±


The Scaling Laws

Leading terms of the scaling laws, thermodynamic limit

high T phaselow T phase

critical line

t > 0, H = 0, high T phase

M = 0

Cv = Cns +A+

αt−α

N = 1 :

χ = C+t−γ

ξ = ξ+t−ν

N > 1 :

χL = C+t−γ = χT

ξL = ξ+t−ν = ξT

• t = 0, H = 0, critical point

r large : G(r) ∼ 1

rd−2+η

η measures deviations frompure Gaussian behaviour


The Scaling Laws

t < 0, H = 0, low T phase

M = B(−t)β

Cv = Cns +A−

α(−t)−α

N = 1 :

χ = C−(−t)−γ

ξ = ξ−(−t)−ν

N > 1 :

χL,T , ξL,T diverge

Goldstone effect

t = 0, H > 0, critical isotherm

M = BcH1/δ ⇔ H = DcMδ

Cv = Cns +Ac

αcH−αc , αc =

α

βδ

N = 1 :

χ = CcH1/δ−1 , Cc =Bc

δ

ξ = ξcH−νc , νc =ν

βδ

N > 1 :

χL = CcH1/δ−1 , χT = BcH1/δ−1

ξL,T = ξcL,TH−νc


The Scaling Laws

Further facts and comments

• Specific heat:

Cns is a background term coming from the non-singular part of f ;the amplitudes A± are positive ; α < 1 because ` = 0 :

lim∆T→0

∫ Tc+∆T

Tc

t−αdT = Tc lim∆T→0

t1−α

1− α∣

∣

∆T/T

0= 0 only if α < 1

α < 0, α = 0 (special case) are possible.

• η < 2 , because of the FD-theorem, at Tc :

χ = β

∫

ddxG(~x) ∼∫ R

R0

drrd−11

rd−2+η∼ R2−η

2− η

For R→∞ the expression must diverge =⇒ 2− η > 0


The Scaling Laws

• Hyperscaling laws:

The six main critical exponents α, β, γ, δ, η, ν are not independent,because of 4 (hyper)scaling laws

α+ 2β + γ = 2 2− α = β(δ − 1)(2− η)ν = γ dν = 2− α

The laws were first derived as inequalities from thermodynamics.

• For d > 4 the critical exponents assume the mean field values(for the Landau-Ginzburg model also at d = 4 )

γ = 1 , ν =1

2, η = 0 , α = 0 , β =

1

2, δ = 3

the hyperscaling relation 2− α = dν does not hold for d ≥ 5 .


Scale Freedom

• The power laws are called scaling laws, because they are scalefree

Homogeneous functions of one variable satisfy

f(bx) = g(b)f(x)

=⇒ g(b) = λp , p = the degree of homogenuity

Power law functions are homogeneous :

f(x) = Axθ , f(bx) = A(bx)θ = bθf(x)

=⇒ a change in the units of x can be compensated by a changein the units of f(x) - the function looks on all scales alike - that isscale freedom.



Remarkable: Systems with physically different kinds of continuous tran-sitions can be assigned to one Universality Class (UC).

Systems with short range, isotropic couplings with the same space dimension d,

and same symmetry and dimension N of the order parameter belong usually to

the same UC.

Members of the same UC have :− different values of Tc , but

− the same critical exponents

− different critical amplitudes, but universal

amplitude ratios / combinations, as e. g.

C+/C− , A+/A− , Rχ = C+DcBδ−1

− the same scaling functions, e. g. the

coexistence curve ρ/ρc(T/Tc) for fluids:

Guggenheim 1945



Historically:

The equations between the critical exponents, and the scaling laws aresatisfied automatically, if f̂ = βf(t,H) and G(r, t)

are generalized homogeneous functions of their arguments

f̂(bpx, bqy) = bf̂(x, y)

Equation of state H =M δψH(t/M1/β) Widom(1965)

Correl. function G(r, t) = ψG(r/ξ)/rd−2+η Kadanoff(1966)

Here, ψH and ψG are universal apart from metric factors.

=⇒ A theory, which explains the fact that all different members of a UC share itsproperties, must identify the important or relevant parameters of a system.

Idea of Renormalization Group Theory: Systematic reduction of thenumber of degrees of freedom by successive scale transformations


Important Models

The universality classes of these models cover many important systems of StatisticalPhysics and Particle Physics. The knowledge of the universal parameters of these simplesystems serves to classify phase transitions.

In the rest of the lecture we include a factor β = 1/T in f ,H and therefore in the

couplings, e. g. in the magnetic field ~H, that is we use reduced variables. However, we

use the old names

βH → H , f̂ = βf → f , β ~H → ~H

Due to ~H-rescaling the susceptibility looses a factor, whereas the magnetization

remains the same

Ma = −∂f

∂Ha, χab =

∂Ma

∂Hb

Mnewa =Mold

a , χnewab = χold

ab /β =

∫

ddxGab(~x)

U = 〈E〉 = 1

Z

∑

α

Eαe−βEα = −∂ lnZ

∂β= 〈H〉

F = −T lnZ , f = −TVlnZ

Z = e−βF


Lattice Models

Lattice models: Spin variables ~si = ~φ(~xi) are defined on the sites i of aregular d−dimensional lattice (usually hypercubic).

A. O(N)−invariant spin models (non-linear σ−models)

Heisenberg Hamiltonian

H = −J∑

<ij>

~si~sj − ~H∑

i

~si

< ij > − summation over nearest neighbour sites,

~si = N −component unit vector at site i, ~s 2i = 1,

like ~H, J is β−rescaled and therefore J ∼ 1/T ,

Partition Function

Z =

∫

∏

i

dNsiδ(~s2i − 1)e−H


Lattice Models

N = 1: the Ising model, si = ±1 , scalar spin variablesN = 2: the XY model, ~si = (cos θi, sin θi)N = 3: the Heisenberg model (of a real 3d ferromagnet)N = 4: the O(4) model

d = 1: the Ising model has no phase transition

d = 2: the Ising model has a 2. order phase transition atJc = ln(1 +

√2)/2, the Z(2)−symmetry, which breaks at

Jc is discrete (model solved by Onsager 1944);

N ≥ 2: no real symmetry breaking phase transition(Mermin- Wagner- Colemann theorem): instead aKosterlitz-Thouless transition

d = 3: 2. order phase transition

d ≥ 4: 2. order phase transition with mean field exponents


Lattice Models

B. The q−state Potts model

HamiltonianH = −J

∑

<ij>

δσiσj

si is a scalar with q states σi = 1, 2, . . . , q ;δ the Kronecker symbol

Partition Function

Z =∏

i

q∑

σi=1

e−H

d = 2: 2. order phase transitions for q ≤ 4 , 1. order for q ≥ 5d = 3: 2. order phase transitions for q = 2 , 1. order for q ≥ 3


Lattice Models

C. More general Hamiltonians with spins ~φi of unbound length

H = −J∑

<ij>

~φi~φj +∑

i

V (~φ 2i )− ~H∑

i

~φi

where∫ ∞

−∞ebx

2−V (x)dx <∞ for all real b

Special case: the φ4−Hamiltonian

H = −J∑

<ij>

~φi~φj +∑

i

[λ(~φ 2i − 1)2 + ~φ 2i ]− ~H∑

i

~φi

It is the lattice discretization of the Landau-Ginzburg-Wilson

Hamiltonian of continuum theory.


Continuum Models

C. The Landau-Ginzburg-Wilson Hamiltonian

H =∫

ddx

{

1

2|∇~φ(x)|2 + r

2~φ(x) 2 +

u

4!(~φ(x) 2)2 − ~H~φ(x)

}

Translation from lattice

~φi → ~φ(x)J−1/2 , r =2− 4d

J− 2d , u =

4!λ

J2, ~H → ~HJ1/2

Partition Function: Z =

∫

[dNφ(x)]e−H

The Gaussian Model is a special case of the LGW-Hamiltonian,where u = 0 =⇒ the components of ~φ(x) do not interact, it issufficient to consider a single component, the theory corresponds toa free, Euclidean theory of a particle with mass m = r1/2 .


Books, Reviews and Reports

1. J. Zinn Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press 2002.

2. J.J. Binney, N.J. Dowrick, A.J. Fisher and M.E.J. Newman, The Theory of CriticalPhenomena, Clarendon Press 2002.

3. J. Cardy, Scaling and Renormalization in Statistical Physics, Clarendon Press 2002.

4. J.M. Yeomans, Statistical Mechanics of Phase Transitions, Clarendon Press 2000.

5. H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, ClarendonPress 1971.

6. A.Z. Patashinskii and V.I. Pokrovskii, Fluctuation Theory of Phase Transitions,Pergamon, Oxford 1979.

7. A. Pelissetto and E. Vicari, Critical Phenomena and Renormalization-Group Theory,Physics Reports 368 (2002) 549.

8. K.G. Wilson and J. Kogut, The Renormalization Group and the ε-Expansion, Phys.Rep. 12 (1974) 75.

9. M.E. Fisher, Rev. Mod. Phys. 46 (1974) 597; Rev. Mod. Phys. 70 (1998) 653.

10. K.G. Wilson, Rev. Mod. Phys. 47 (1975) 773; Rev. Mod. Phys. 55 (1983) 583.

11. Phase Transitions and Critical Phenomena, C. Domb and M.S. Green (eds.) 1972-76,Vols. 1-6; C. Domb and J.L. Lebowitz (eds) 1983-2001, Vols. 7-19, Academic Press,London.

12. J. Cardy, ed., Finite Size Scaling, Elsevier 1988, Amsterdam.IGS-Block course, LI, Oct 2006, J. Engels – p.37/??

introduction to phase transitions in statistical physics ... · introduction to phase transitions...

Documents