Symmetry and ergodicity breaking, mean-field study of the Ising model Rafael Mottl, ETH Zurich Advisor: Dr. Ingo Kirsch Topics Introduction Formalism The model system: Ising model Solutions in one and more dimensions Symmetry and ergodicity breaking Conclusion Introduction phase A phase B Tc temperature T phase transition order parameter Critical exponents Tc T Formalism Statistical mechanical basics sample region Ω i) certain dimension d ii) volume V (Ω) iii) number of particles/sites N(Ω) iv) boundary conditions Hamiltonian defined on the sample region H − Ω = K Θ k T n n B n Depending on the degrees of freedom Coupling constants Partition function 1 −βHΩ({Kn},{Θn}) β = ZΩ[{Kn}] = T r exp kBT Free energy FΩ[{Kn}] = FΩ[K] = −kBT logZΩ[{Kn}] Free energy per site FΩ[K] fb[K] = lim N(Ω)→∞ N(Ω) i) Non-trivial existence of limit ii) Independent of Ω iii) N(Ω) lim = const N(Ω)→∞ V (Ω) Phases and phase boundaries Supp.: -fb[K] exists - there exist D coulping constants: {K1,...,KD} -fb[K] is analytic almost everywhere - non-analyticities of f b [ { K n } ] are points, lines, planes, hyperplanes in the phase diagram Dimension of these singular loci: Ds = 0, 1, 2,... Codimension C for each type of singular loci: C = D − Ds Phase : region of analyticity of fb[K] Phase boundaries : loci of codimension C = 1 Types of phase transitions fb[K] is everywhere continuous. Two types of phase transitions: ∂fb[K] a) Discontinuity across the phase boundary of ∂Ki first-order phase transition b) All derivatives of the free energy per site are continuous across the phase boundaries Continuous phase transition Overview −βHΩ(K,{Θn}) ZΩ[K] = Tr exp FΩ[K] = −kBTlogZΩ[K] FΩ[K] fb[K] = lim N(Ω)→∞ N(Ω) ∂ ∂fb[K] ǫin[K] = (βfb[K]) M[K] = − ∂β ∂H internal energy magnetization ∂M[K] ∂ǫin[K] C[K] = χT [K] = ∂T ∂H heat capacity magnetic susceptibility Critical exponents How do the measurable quantities change in the neighbourhood of a critical point? T − TC Critical temperature TC t = TC heat capacity C ∼|t|−α Order parameter (t < 0) M ∼|t|β for H ≡ 0 susceptibility χ ∼|t|−γ Equation of state M ∼ H1/δ Correlation length - Length scale over which the fluctuations of the microscopic degrees of freedom are significantly correlated with each other (Cardy: Scaling and Renormalization in Statistical Physics) - Strong dependence on temperature near a phase transition diverging to infinity at the transition itself Critical exponents for the correlation function t → 0 Γ(r) −→ r−pe−r/ξ −ν Correlation length ξ ∼|t| Power-law decay (t=0) p = d − 2 + η The model system: Ising model - attempt to simulate a domain in a ferromagnetic material - Spontaneous magnetization in absence of an external magnetic field - Critical temperature: Curie temperature Importance of Ising model - Equivalent models (lattice gas, binary alloy) - The only non-trivial example of a phase transition that can be worked out be mathematical rigor in statistical mechanics - Compare computer simulations of the model with exact solution Characterization of the Ising model i) Periodic latticeΩ in d dimensions ii) Lattice containsN(Ω) fixed points called lattice sited iii) For each site: classical spin variable S i = +/- 1 (i = 1, …, N) in a definite direction (degrees of freedom) iv) Most general Hamiltonian −HΩ = HiSi + JijSiSj + KijkSiSj Sk + ... i∈Ω i,j i,j,k v) Number of possible configurations: 2N(Ω) T r ≡ ··· ≡ S =±1 S =±1 S =±1 1 2 N(Ω) {Si=±1} Assumptions i) two-spin coupling only Kijk = 0,... ii) External magnetic field spatially constant Hi ≡ H iii) Nearest-neighbour interactions only → i,j <ij> iv) Isotropic interactions Jij ≡ J v) Hypercubic lattice z = 2d vi) Ferromagnetic material J > 0 −HΩ = H Si + J SiSj i∈Ω <i,j> β(H ∈ Si+J SiSj ) ZΩ[H, T, J] = exp i Ω <i,j> {Si=±1} Arguments for phase transition in d = 1,2 dimensions with H = 0 FΩ[K] = Ein[K] − TSΩ[K] = −J SiSj − T kBlog(♯(states)) <i,j> d = 1 A B N → ∞ FN,B − FN,A = 2J − kBT log(N − 1) → −∞ d = 2 A B n bonds FN,B − FN,A = [2J − log(z − 1)kBT ]n 2J TC = kBlog(z − 1) Long range order short range order - nearest neighbour interaction: short range interaction - What about: J Jij = σ |ri − rj| Answer: σ < 1 thermodynamic limit does not exist 1 σ 2 long range order persist for 0 < T < T C 2 σ short-range interaction: no ferromagnetic state for T > 0 Solutions in one and more dimensions - ad hoc methods H = 0 - recursion method d = 1 - transfer matrix method H = 0 (Kramers, Wannier 1941) - low temperature expansion d = 2 H = 0 - Onsager solution (1944) d = 1, 2, 3 H = 0 - mean-field method (Weiss) d = 1 H = 0 Transfer matrix method Reduce the problem of calculating the partition function to the problem of finding the eigenvalues of a matrix Assumption: SN+1 = S1 SN+1 = S1 SN S2 −βHΩ(H,J,{Si}) S ZΩ[H, J] = T r exp 3 HΩ(H, J, {Si}) = −H Si − J SiSj i∈Ω <i,j> h := βH K := βJ h Si+K SiSi+1 ZΩ[h, K] = ... e i i S1 SN eh+K − λ e−K det ≡ 0 Calculation: e−K e−h+K − λ K 2 −4K ⇒ λ1,2 = e (cosh(h) ± sinh(h) + e ) 2 −4K ⇒ fb(h, K, T ) = −JkBTlog(cosh(h) + sinh(h) + e ) Spatial correlation in one dimension (T > 0) Definition: two point correlation function d = 1 H = 0 G(i, j) := (Si − Si)(Sj − Sj) = SiSj − SiSj j G(i, i + j) = SiSi+j = (tanh(K)) G(i, i + j) = e−jlog(coth(K)) ≡ e−j/ξ 1 ⇒ ξ = =∼ 1/2 eJ/(kB T ) log(coth)K)) Onsager solution d = 2 H = 0 Notation and boundary conditions: Columns → 12 3 ... n n + 1 ≡ 1 µ = {s , s , . , s } 1 α 1 2 n αth row 2 sn+1 = s1 , µn+1 = µ1 s 3 2 .. N = n Row ← n Elements of the Hamiltonian: n + 1 ≡ 1 n ′ ′ E(µ, µ ) = −ǫ sksk k=1 n n E(µ) =$ −ǫ sksk+1 − H sk $ 2 k=1 k=1 $ $ Partition function: n −β [E(µα,µα+1)+E(µα)] ZΩ[H, T ] = ... e α=1 µ µ 1 n ′ Properties of the matrix P: - µ|P|µ′ := e−β[E(µ,µ )+E(µ)] - P : 2n × 2nmatrix n - ZΩ[H, T ] = T rP 1 1 ⇒ lim log(ZΩ[H = 0,T ]) = lim log(λmax) N→∞ N n→∞ n largest eigenvalue of P FΩ[0,T ] −kBT logZΩ[0,T ] 1 fb[H = 0,T ] = lim = lim = −kBT lim log(λmax) N(Ω)→∞ N(Ω) N(Ω)→∞ N(Ω) n→∞ n π 1 1 2 2 ⇒ βfb[0,T ] = − log (2 cosh 2βǫ) − dφ log 1 + 1 − κ sin φ 2π 0 2 κ = 2[cosh 2φ coth2φ]−1 Mean-field theory Arguments for mean-field theory: - Simplest treatment of an interacting statistical mechanical system - often gives a qualitatively correct picture of the phase diagram of a given model - „the“ mean-field theory Hamiltonian −HΩ = H Si + J SiSj i∈Ω <i,j> Partition function β(H ∈ Si) ZΩ[H, T, J = 0] = exp i Ω = {Si=±1} N = expβHSk = (eβH + e−βH )N = (2 cosh βH)N S =±1 k=1 k Assumption: d-dimensional hypercubic lattice z = 2d Postulate: effective field due to the magnetic moments of all other spins −HΩ = H Si + J SiSj = Si[H + J Sj] i∈Ω <i,j> i∈Ω j n.n. Sj = Sj + J (Sj − Sj =∼ Sj = 2dM j n.n. j n.n. j n.n. j n.n. mean field fluctuation in the mean field −HΩ = Si[H + J Sj] = Si[H + 2dJM] ≡ Heff Si i∈Ω j n.n. i∈Ω i∈Ω N N ⇒ ZΩ[H, T ] = (2 cosh βHeff ) = (2 cosh β[H + 2dJM]) Free energy per site N(Ω) −kBT logZΩ[H, T ] −kBT log(2 cosh β[H + 2dJM]) fb[H, T ] = lim = lim N(Ω)→∞ N(Ω) N(Ω)→∞ N(Ω) ⇒ fb[H, T] = −kBT log 2 cosh β[H + 2dJM] ∂f [K] H + 2dJM magnetization per site M[K] = − b = tanh ∂H kBT 2 f(M) T < TC 1 2dJ T = TC = kB M -2 -1 1 2 T > TC -1 Spontaneous magnetization: H = 0 -2 Critical exponents for mean-field theory of the Ising model tanh H + tanh Mτ H + 2dJM kB T M = tanh = H kBT 1 + tanh tanh Mτ T kB T τ = C T Expand for small H,M: H τ 3 = M(1 − τ) + M 3(τ − τ 2 + + ...) + ... kBT 3 2 T − TC 1 H = 0 M = 3 + ... ⇒ β = TC 2 H 3 τ = 1 = M + ... ⇒ δ = 3 kBT 3 1 2 2 τ M = 0 = χT (1 − τ) + 3M χT (τ − τ + ) + ... ⇒ γ = 1 kBT 3 Symmetry and ergodicity breaking The ergodic hypothesis dynamical degrees of freedom ηi(t) Any observable A({ηi(t)}) t 1 ′ ′ time average Ata := lim dt A({ηi(t )}) t→∞ t 0 expectation value Aeqm := dηiPeqm({ηi})A({ηi)}) i ! Ata = Aeqm Symmetry of the system at H = 0 −HΩ({Si}) = H Si + J SiSj = J SiSj i∈Ω <i,j> <i,j> −HΩ({−Si}) = J (−Si)(−Sj) = J SiSj = −HΩ({Si}) <i,j> <i,j> statistical mechanical probability of finding the system in the state {S i}: exp(−βHΩ({Si})) PΩ({Si}) = ZΩ[K] M ≡ Si = T r[PΩ({Si})Si] = 0 The restricted ensemble* divide phase space into components Γ = Γα α Assumptions on the properties of components : (a) Confinement: there is no chance for the phase point for moving from one component to another (b) Internal ergodicity: ergodic hypothesis for a particular component Remark : component decomposition depends on the external parameters of the system appearance, disappearance, bifurcation, merge of components * Palmer, Adv.Phys., (1982), Vol.