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


    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






    pressure curve



    GasSublimation curve


    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 Tc




    "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



    M H = 0



    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 +(



    TdV df = sdT MdH

    s = (



    Vs =





    = f + Ts =(



    V = f + Ts =





    p = f V(





    V fV


    TM =





    Cv =(



    VCH =





    T = 1V




    T =





    Equation of state

    Specific heat


    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 =


    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


    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 :








    e(EXY )



    = X = FY




    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




    T > Tc

    T < T T = Tc


    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



    ~ 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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