Introduction to Phase Transitions in Statistical Physics and Field Theory

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1 Introduction to Phase Transitions in Statistical 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/??

2 Motivation Phase Transitions are of relevance everywhere in physics - starting with the theory of the early universe, in particle physics, in condensed matter physics, and of course they are relevant in daily life. In the critical region of continuous transitions cooperative phenomena generate a large scale, the correlation length, although the fundamental interactions are short range - most details of the microscopic structure become unimportant and a power law behaviour emerges. Such scaling laws occur also in the clustering of galaxies, the distribution of earthquakes, the turbulence in fluids and plasmas, etc. Most interesting for particle physicists is: the connection between statistical field theory of critical phenomena and quantum field theory, both from a theoretical point of view: for the understanding of the role of renormalization and renormalizable field theories in particle physics, for the calculation of critical parameters, and in a practical sense: methods from statistical mechanics allow simulations of lattice gauge field theories and lead to non-perturbative results IGS-Block course, LI, Oct 2006, J. Engels p.2/??

3 Phase Transitions in Fluids and Magnets At a phase transition (PT) one observes a sudden change in the properties of a system, e. g. liquid gas paramagnet ferromagnet normal conductor superconductor Phase diagram of a typical fluid: p Melting curve Solid lines are phase boundaries p c Solid Liquid C Vapour pressure curve separating regions of stable phases In crossing these lines: Mass density ρ and energy density ɛ change discontinuously, they jump = 1. order phase transition Gas C is the critical end point Sublimation curve = 2. order phase transition T c T IGS-Block course, LI, Oct 2006, J. Engels p.3/??

4 Vapour Pressure Curve The vapour pressure curve is the liquid-gas coexistence line: ρ ρ liquid (T) "Order parameter" (OP): ρ l ρ g T = T c : ρ c = ρ l (T c ) = ρ g (T c ) ρ c Order parameter C T > T c : no distinction between gas and liquid ρ gas (T) Tc T At T c : continuous transition, but C v diverges for ρ = ρ c IGS-Block course, LI, Oct 2006, J. Engels p.4/??

5 Magnetic Phase Transitions The phase diagram : The order parameter : M H = 0 inaccessible OP Tc T Spontaneous Magnetization: Analog to fluids: T < T c : 1. order PT T = T c : 2. order critical point M(T < T c, H = 0) 0 M(T, H 0 + ) = M(T, H 0 ) M(T T c, H = 0) = 0 IGS-Block course, LI, Oct 2006, J. Engels p.5/??

6 Thermodynamics and Statistical Mechanics We consider only the equilibrium case. Thermodynamics assumes no particular model for a system, the variables are macroscopic quantities - for the system as a whole. There are Extensive Variables: Intensive Variables: U =internal energy, V =volume, S =entropy N i =particle number of type i, M =total magnetic moment T =temperature, p =pressure, ρ =density=mass/volume µ i =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 = (n 1,..., n N ), n 2 = 1 A change in U (here without the internal energy of the field) is described by du = T ds 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/??

7 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 = T ds pdv du = T ds Md H Helmholtz free energy F (T, V ) = U T S F (T, H) = U T S 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 S and at fixed T minimal F. In finite volumes: ensemble average = time average IGS-Block course, LI, Oct 2006, J. Engels p.7/??

8 Thermodynamic Limit We would like to work in the thermodynamic limit, i. e. in the limit where V, N i, 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 not occur. Because of the finite volume the correlation length is finite. ii) Spontaneous symmetry breaking exists only in the thermodynamic limit. In finite systems 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. We therefore use densities : ɛ = U/V energy density M = M/V magnetization s = S/V entropy density f = F/V = ɛ T s free energy density IGS-Block course, LI, Oct 2006, J. Engels p.8/??

9 Thermodynamic Relations Fluids (T, V ) Magnets (T, H), N = 1 df = sdt + ) s = ( f T ɛ = f + T s = p = f V ) C v = ( ɛ T κ T = 1 V ( f V ) V ( βf β ( ) f V V ( V p ) T T dv ) V ( = V f T V ) T df = sdt MdH ( ) s = f T H ( ) ɛ = f + T s = βf β ( ) M = f H T ( ) C H = ɛ T H ) χ = ( M H T H Equation of state Specific heat Susceptibility We use k = 1 as unit, so that β = 1/T IGS-Block course, LI, Oct 2006, J. Engels p.9/??

10 Statistical Mechanics In contrast to thermodynamics it provides a link between microscopic and macroscopic physics, because it starts with a Hamiltonian H, which describes the microscopic interaction. Canonical ensemble, the partition function: Z = Tr e βh = α e βe α E α = Eigenvalue of H in state α U = E = 1 Z E α e βe α α = ln Z β = H F = T ln Z, f = T V ln Z Z = e βf IGS-Block course, LI, Oct 2006, J. Engels p.10/??

11 Statistical Mechanics The fluctuation in the energy is given by ( U) 2 = (E E ) 2 = E 2 E 2 = 2 ln Z β 2 More general, the average of a quantity X can be calculated by introducing a source term XY in the Hamiltonian: H = H 0 XY and taking the derivative with respect to Y : 1 β ln Z Y Y =0 = 1 βz Y ( α e β(e α X α Y ) ) Y =0 = X = F Y Y =0 and 1 β X Y Y =0 = X 2 X 2 IGS-Block course, LI, Oct 2006, J. Engels p.11/??

12 Order Parameter and Symmetry Breaking Symmetry breaking for a ferromagnet, H = H n, H, n N-dim vectors T < T c : ferromagnetic phase, the spins are aligned, M measures the degree of ordering the ordered state is not symmetric under rotation H 0 = M aligns in H-direction, for H H, M jumps to M explicit symmetry breaking H = 0 = direction of M is spontaneously chosen by the system spontaneous symmetry breaking T > T c : paramagnetic phase, H 0 = M H, for H H, M changes continuously to M explicit symmetry breaking H = 0 = M = 0 symmetry of state M T < T c T = T c T > T c H T = T c : critical line, like paramagnetic phase H = 0 = M/ H is infinite N=1 IGS-Block course, LI, Oct 2006, J. Engels p.12/??

13 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 0 in Phase II Here, means the "thermal average" over a long period in equilibrium at constant temperature = 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. when the "order parameter" is shown in a plot. Better: φ( x) is the order parameter field or local order parameter, φ( x) is the order parameter. In a homogeneous system φ( x) is independent of x φ( x) = φ IGS-Block course, LI, Oct 2006, J. Engels p.13/??

14 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) e iα φ( x) He 4, fluid to superfluid rotation φ( x) Aφ( x) Ferromagnet Magnet:... φ( x) corresponds to the local magnetization and M = φ( x) = ϕ where ϕ = 1 V d d x φ( x) ϕ is the volume average of the local magnetization and M its thermal average, a global variable, the total magnetization of the magnet. IGS-Block course, LI, Oct 2006, J. Engels p.14/??

15 The Order of a Phase Transition A phase transition occurs, if in the thermodynamic limit the free energy density f (or other thermodynamic 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 potential has (n 1) continuous derivatives, but the n th derivative is 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 order parameter is the first derivative of its conjugate variable = it jumps at T c. 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 T c. Crossover: is no real phase transition, but there is a rapid change in the behaviour of the system = in (finite volume) simulations difficult to distinguish from true phase transitions. IGS-Block course, LI, Oct 2006, J. Engels p.15/??

16 Susceptibilities and Correlation Functions General Magnet: H = H 0 H d d x φ( x) N-dimensional vectors: H = H n, φ( x), ϕ d-dimensional vector: x On a lattice: ϕ = 1 N x x φ( x) H = H 0 V ϕ H X a = V ϕ a, Y a = H a V ϕ a = F H a or M a = f H a = ϕ a Tensor of susceptibilities: χ ab = 2 f H a H b = M a H b = ϕ a H b IGS-Block course, LI, Oct 2006, J. Engels p.16/??

17 The Susceptibilities χ ab = ϕ a H b = H b = 1 Z α ( 1 Z ) ϕ α a e β(e α V ϕ α H) α ϕ α a βv ϕ α b e β(... ) 1 Z 2 ( ) ( ϕ α a e β(... ) α α βv ϕ α b e β(... ) ) χ ab = βv [ ϕ a ϕ b ϕ a ϕ b ] Total magnetic susceptibility fluctuation of the length of ϕ : χ tot = a χ aa = Tr χ χ tot = βv [ ϕ 2 ϕ 2] IGS-Block course, LI, Oct 2006, J. Engels p.17/??

18 The Susceptibilities Assumption: H 0 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 χ tot = χ L + (N 1)χ T with χ L = M H, χ T = M H Longitudinal susceptibility: χ L = βv ( ϕ 2 ϕ 2 ) Fluctuation of the longitudinal field component = fluctuation of M Transverse susceptibility: χ T = βv ϕ 2 /(N 1) Fluctuation of a transverse field component χ ab = χ L n a n b + χ T (δ ab n a n b ) IGS-Block course, LI, Oct 2006, J. Engels p.18/??

19 The Correlation Functions Correlation functions (2-point connected) of the components of φ( x) : G ab ( x 1, x 2 ) = φ a ( x 1 )φ b ( x 2 ) φ a ( x 1 ) φ b ( x 2 ) = (φ a ( x 1 ) φ a ( x 1 ) )(φ b ( x 2 ) φ b ( x 2 ) Translation invariance: G ab ( x 1, x 2 ) = G ab ( x), x = x 2 x 1 Rotation invariance: G ab ( x) = G ab (r), r = x = G ab ( x) = φ a (0)φ b ( x) φ a (0) φ b ( x) d d xg ab ( x) = V [ φ a (0)ϕ b φ a (0) ϕ b ] FD-Theorem: = V [ ϕ a ϕ b ϕ a ϕ b ] = χ ab /β χ ab = β d d xg ab ( x) IGS-Block course, LI, Oct 2006, J. Engels p.19/??

20 The Correlation Functions Longitudinal and transverse correlation functions : G L ( x) = φ (0)φ ( x) ϕ 2 G T ( x) = 1 N 1 φ (0)φ ( x) Like for χ: G ab = G L n a n b + G T (δ ab n a n b ) G tot = a G aa = G L + (N 1)G T = φ(0) φ( x) ϕ 2 In general: For larger distances r = x the fields become more and more uncorrelated and G( x) e r/ξ r τ ξ or ξ L,T is the correlation length If ξ a long range order develops and G( x) has a power behaviour. IGS-Block course, LI, Oct 2006, J. Engels p.20/??

21 Characteristics of First Order Transitions There is a latent heat l 0 : l = lim T T + c ɛ(t ) lim T T c ɛ(t ) 0, l = T + c T c C v (T )dt, C v = C H The order parameter M jumps at T c because : lim T T c M 0, lim T T + c M = 0 The correlation length ξ stays finite at T c and : lim T T + c ξ lim T Tc ξ 0, The fluctuations / susceptibilities at T c grow V : χ = βv ( ϕ 2 M 2) = βv (ϕ M) 2 V β( M) 2 V C v = β2 V ( E 2 E 2) = β 2 V ( E V ɛ)2 V β 2 (ɛ + ɛ ) 2 V IGS-Block course, LI, Oct 2006, J. Engels p.21/??

22 Characteristics of Continuous Transitions At T = T c : There is no latent heat The order parameter vanishes The correlation length diverges, i.e. the fluctuations of the order parameter field grow infinitely, a long range order is established l = 0 M(T c ) = 0 lim ξ T T c Close to T c : The critical behaviour is described by power laws with critical exponents. The exponents are the same for T T + c and T T c. IGS-Block course, LI, Oct 2006, J. Engels p.22/??

23 Characteristics of Continuous Transitions Reduced temperature: t = (T T c )/T c The following laws are valid in the thermodynamic limit Singular behaviour of an observable F (t) near T c or t = 0 for H = 0 F (t) = A t λ (1 + b t λ ), λ 1 > 0 λ = critical exponent λ = lim t 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 λ ) = corrections to scaling, b is different for t 0 ± IGS-Block course, LI, Oct 2006, J. Engels p.23/??

24 The Scaling Laws Leading terms of the scaling laws, thermodynamic limit t > 0, H = 0, high T phase M = 0 C v = C ns + A+ α t α N = 1 : N > 1 : low T phase high T phase χ = C + t γ χ L = C + t γ = χ T critical line ξ = ξ + t ν ξ L = ξ + t ν = ξ T t = 0, H = 0, critical point r large : G(r) 1 r d 2+η η measures deviations from pure Gaussian behaviour IGS-Block course, LI, Oct 2006, J. Engels p.24/??

25 The Scaling Laws t < 0, H = 0, low T phase M = B( t) β C v = C ns + A α ( t) α t = 0, H > 0, critical isotherm M = B c H 1/δ H = D c M δ N = 1 : χ = C ( t) γ ξ = ξ ( t) ν N > 1 : χ L,T, ξ L,T diverge Goldstone effect C v = C ns + A c α c H α c, α c = α βδ N = 1 : χ = C c H 1/δ 1, C c = Bc δ ξ = ξ c H ν c, ν c = ν βδ N > 1 : χ L = C c H 1/δ 1, χ T = B c H 1/δ 1 ξ L,T = ξ c L,T H ν c IGS-Block course, LI, Oct 2006, J. Engels p.25/??

26 The Scaling Laws Further facts and comments Specific heat: C ns is a background term coming from the non-singular part of f ; the amplitudes A ± are positive ; α < 1 because l = 0 : Tc + T lim t α dt = T c lim T 0 T T 0 c t 1 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 T c : χ = β d d xg( x) R drr d 1 1 R2 η R 0 rd 2+η 2 η For R the expression must diverge = 2 η > 0 IGS-Block course, LI, Oct 2006, J. Engels p.26/??

27 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. IGS-Block course, LI, Oct 2006, J. Engels p.27/??

28 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) = λ p, p = the degree of homogenuity Power law functions are homogeneous : f(x) = Ax θ, f(bx) = A(bx) θ = b θ f(x) = a change in the units of x can be compensated by a change in the units of f(x) - the function looks on all scales alike - that is scale freedom. IGS-Block course, LI, Oct 2006, J. Engels p.28/??

29 Universality Classes Remarkable: Systems with physically different kinds of continuous transitions 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 T c, but the same critical exponents different critical amplitudes, but universal amplitude ratios / combinations, as e. g. C + /C, A + /A, R χ = C + D c B δ 1 the same scaling functions, e. g. the coexistence curve ρ/ρ c (T/T c ) for fluids: Guggenheim 1945 IGS-Block course, LI, Oct 2006, J. Engels p.29/??

30 Universality Classes Historically: The equations between the critical exponents, and the scaling laws are satisfied automatically, if ˆf = βf(t, H) and G(r, t) are generalized homogeneous functions of their arguments ˆf(b p x, b q y) = b ˆf(x, y) Equation of state H = M δ ψ H (t/m 1/β ) Widom(1965) Correl. function G(r, t) = ψ G (r/ξ)/r d 2+η Kadanoff(1966) Here, ψ H and ψ G are universal apart from metric factors. = A theory, which explains the fact that all different members of a UC share its properties, must identify the important or relevant parameters of a system. Idea of Renormalization Group Theory: Systematic reduction of the number of degrees of freedom by successive scale transformations IGS-Block course, LI, Oct 2006, J. Engels p.30/??

31 Important Models The universality classes of these models cover many important systems of Statistical Physics and Particle Physics. The knowledge of the universal parameters of these simple systems 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, ˆf = βf f, β H H Due to H-rescaling the susceptibility looses a factor, whereas the magnetization remains the same M new a M a = f H a, = Ma old, χ new ab χ ab = M a H b d d xg ab ( x) = χ old ab /β = IGS-Block course, LI, Oct 2006, J. Engels p.31/??

32 Lattice Models Lattice models: Spin variables s i = φ( x i ) are defined on the sites i of a regular d dimensional lattice (usually hypercubic). A. O(N) invariant spin models (non-linear σ models) Heisenberg Hamiltonian H = J <ij> s i s j H i s i < ij > summation over nearest neighbour sites, s i = N component unit vector at site i, s 2 i = 1, like H, J is β rescaled and therefore J 1/T, Partition Function Z = d N s i δ( s i 2 1)e H i IGS-Block course, LI, Oct 2006, J. Engels p.32/??

33 Lattice Models N = 1: the Ising model, s i = ±1, scalar spin variables N = 2: the XY model, s i = (cos θ i, sin θ i ) N = 3: the Heisenberg model (of a real 3d ferromagnet) N = 4: the O(4) model d = 1: d = 2: d = 3: d 4: the Ising model has no phase transition the Ising model has a 2. order phase transition at J c = ln(1 + 2)/2, the Z(2) symmetry, which breaks at J c is discrete (model solved by Onsager 1944); N 2: no real symmetry breaking phase transition (Mermin- Wagner- Colemann theorem): instead a Kosterlitz-Thouless transition 2. order phase transition 2. order phase transition with mean field exponents IGS-Block course, LI, Oct 2006, J. Engels p.33/??

34 Lattice Models B. The q state Potts model Hamiltonian H = J <ij> δ σi σ j s i is a scalar with q states σ i = 1, 2,..., q ; δ the Kronecker symbol Partition Function Z = i q σ i =1 e H d = 2: 2. order phase transitions for q 4, 1. order for q 5 d = 3: 2. order phase transitions for q = 2, 1. order for q 3 IGS-Block course, LI, Oct 2006, J. Engels p.34/??

35 Lattice Models C. More general Hamiltonians with spins φ i of unbound length H = J <ij> φ i φj + i V ( φ 2 i ) H i φ i where e bx2 V (x) dx < for all real b Special case: the φ 4 Hamiltonian H = J <ij> [λ( φ 2 i 1) 2 + φ 2 i ] H i φ i φj + i φ i It is the lattice discretization of the Landau-Ginzburg-Wilson Hamiltonian of continuum theory. IGS-Block course, LI, Oct 2006, J. Engels p.35/??

36 Continuum Models C. The Landau-Ginzburg-Wilson Hamiltonian H = d d x { 1 2 φ(x) 2 + r φ(x) u } 4! ( φ(x) 2 ) 2 H φ(x) Translation from lattice φ i φ(x)j 1/2, r = 2 4d J 2d, u = 4!λ J 2, H HJ 1/2 Partition Function: Z = [d N φ(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 is sufficient to consider a single component, the theory corresponds to a free, Euclidean theory of a particle with mass m = r 1/2. IGS-Block course, LI, Oct 2006, J. Engels p.36/??

37 Books, Reviews and Reports 1. J. Zinn Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press J.J. Binney, N.J. Dowrick, A.J. Fisher and M.E.J. Newman, The Theory of Critical Phenomena, Clarendon Press J. Cardy, Scaling and Renormalization in Statistical Physics, Clarendon Press J.M. Yeomans, Statistical Mechanics of Phase Transitions, Clarendon Press H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Clarendon Press A.Z. Patashinskii and V.I. Pokrovskii, Fluctuation Theory of Phase Transitions, Pergamon, Oxford A. Pelissetto and E. Vicari, Critical Phenomena and Renormalization-Group Theory, Physics Reports 368 (2002) K.G. Wilson and J. Kogut, The Renormalization Group and the ɛ-expansion, Phys. Rep. 12 (1974) M.E. Fisher, Rev. Mod. Phys. 46 (1974) 597; Rev. Mod. Phys. 70 (1998) K.G. Wilson, Rev. Mod. Phys. 47 (1975) 773; Rev. Mod. Phys. 55 (1983) Phase Transitions and Critical Phenomena, C. Domb and M.S. Green (eds.) , Vols. 1-6; C. Domb and J.L. Lebowitz (eds) , 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/??

Phase transitions and critical phenomena

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