# Introduction to Phase Transitions in Statistical Physics ... ?· Introduction to Phase Transitions in…

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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

IGS-Block course, LI, Oct 2006, J. Engels p.2/??

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 and energy density

change discontinuously, they jump

= 1. order phase transition

C is the critical end point

= 2. order phase transition

IGS-Block course, LI, Oct 2006, J. Engels p.3/??

Vapour Pressure Curve

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

Order parameter

(T)

c

T Tc

C

liquid(T)

gas

"Order parameter" (OP): l g

T = Tc : c = l(Tc) = g(Tc)

T > Tc : no distinction betweengas and liquid

At Tc: continuous transition, but Cvdiverges for = c

IGS-Block course, LI, Oct 2006, J. Engels p.4/??

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

inacce

ssible

Spontaneous Magnetization:

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

M(T,H 0+) = M(T,H 0)

M(T Tc,H = 0) = 0

IGS-Block course, LI, Oct 2006, J. Engels p.5/??

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, =density=mass/volumei =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.

IGS-Block course, LI, Oct 2006, J. Engels p.6/??

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 TSdF = SdT pdV dF = SdT ~Md ~H

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

IGS-Block course, LI, Oct 2006, J. Engels p.7/??

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 : = U/V energy density

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

IGS-Block course, LI, Oct 2006, J. Engels p.8/??

Thermodynamic Relations

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

df = sdT +(

fV

)

TdV df = sdT MdH

s = (

fT

)

Vs =

(

fT

)

H

= f + Ts =(

f

)

V = f + Ts =

(

f

)

H

p = f V(

fV

)

T=

(

V fV

)

TM =

(

fH

)

T

Cv =(

T

)

VCH =

(

T

)

H

T = 1V

(

Vp

)

T =

(

MH

)

T

Equation of state

Specific heat

Susceptibility

We use k = 1 as unit, so that = 1/T

IGS-Block course, LI, Oct 2006, J. Engels p.9/??

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 eH =

eE

E = Eigenvalue of H in state

U = E = 1Z

EeE = lnZ

= H

F = T lnZ , f = TVlnZ

Z = eF

IGS-Block course, LI, Oct 2006, J. Engels p.10/??

Statistical Mechanics

The fluctuation in the energy is given by

(U)2 = (E E)2 = E2 E2 = 2 lnZ

2

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

lnZ

Y

Y=0=1

Z

Y

(

e(EXY )

)

Y=0

= X = FY

Y=0

and1

XY

Y=0= X2 X2

IGS-Block course, LI, Oct 2006, J. Engels p.11/??

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

IGS-Block course, LI, Oct 2006, J. Engels p.12/??

The Order Parameter

Definition of the order parameter:

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

0 in Phase I 6= 0 in Phase II

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

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

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

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

In a homogeneous system (~x) is independent of ~x

(~x) =

IGS-Block course, LI, Oct 2006, J. Engels p.13/??

The Order Parameter

The Hamiltonian is a function of :

H = H((~x))

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

reversal (~x) (~x) Ising model

change of phase (~x) ei(~x) He4, fluid to superfluid

rotation ~(~x) A~(~x) Ferromagnet

. . .

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

~M = ~(~x) = ~ where ~ =1

V

ddx~(~x)

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

IGS-Block course, LI, Oct 2006, J. Engels p.14/

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