Symmetry and Ergodicity Breaking, Mean-Field Study of the Ising Model

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 <ul style="display: flex;"><li style="flex:1">phase A </li><li style="flex:1">phase B </li></ul>Tc temperature T phase transition order parameter ꢀ Critical exponents TTc Formalism Statistical mechanical basics Ωsample region di) certain dimension ii) volume V (Ω) N(Ω) iii) number of particles/sites iv) boundary conditions Hamiltonian defined on the sample region ꢀHΩ −=KnΘn kBT nDepending on the degrees of freedom Coupling constants Partition function 1β = ZΩ[{Kn}] = Tr exp−βH ({K },{Θ }) Ω<ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li></ul>kBT FΩ[{Kn}] = FΩ[K] = −kBTlogZΩ[{Kn}] Free energy Free energy per site FΩ[K] fb[K] = lim N(Ω)→∞ N(Ω) i) Non-trivial existence of limit Ωii) Independent of iii) lim N(Ω) = const N(Ω)→∞ V (Ω) Phases and phase boundaries fb[K] <ul style="display: flex;"><li style="flex:1">Supp.: - </li><li style="flex:1">exists </li></ul>{K1, . . . , KD} - there exist D coulping constants: <ul style="display: flex;"><li style="flex:1">-</li><li style="flex:1">is analytic almost everywhere </li></ul>fb[K] f [{Kn}] are points, lines, planes, hyperplanes in the - non-analyticities of b phase diagram Ds = 0, 1, 2, . . . ꢀ Dimension of these singular loci: Codimension C for each type of singular loci: C = D − Ds fb[K] Phase: region of analyticity of 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 ZΩ[K] = Tr exp−βH (K,{Θ }) ΩnFΩ[K] = −kBTlogZΩ[K] FΩ[K] fb[K] = lim N(Ω)→∞ N(Ω) ∂ ∂fb[K] ǫin[K] = (βfb[K]) ∂β M[K] = − ∂H <ul style="display: flex;"><li style="flex:1">internal energy </li><li style="flex:1">magnetization </li></ul>∂M[K] ∂ǫin[K] χT [K] = C[K] = ∂H ∂T magnetic susceptibility heat capacity Critical exponents How do the measurable quantities change in the neighbourhood of a critical point? T − TC t = TC Critical temperature TC C ∼|t|−α heat capacity βfor H ≡ 0 M ∼|t| Order parameter (t < 0) susceptibility χ ∼|t|−γ M ∼ H1/δ Equation of state 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 ξ ∼|t|−ν Correlation length p = d − 2 + η Power-law decay (t=0) 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 lattice ii) Lattice contains in d dimensions Ωfixed points called lattice sited N(Ω) iii) For each site: classical spin variable Si = +/- 1 (i = 1, …, N) in a definite direction (degrees of freedom) iv) Most general Hamiltonian <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li></ul>−HΩ = <ul style="display: flex;"><li style="flex:1">HiSi + </li><li style="flex:1">JijSiSj + </li></ul>KijkSiSjSk + . . . i,j i∈Ω i,j,k 2N(Ω) v) Number of possible configurations: <ul style="display: flex;"><li style="flex:1">ꢀ ꢀ </li><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li></ul>Tr ≡ <ul style="display: flex;"><li style="flex:1">· · · </li><li style="flex:1">≡</li></ul>S1=±1 S2=±1 SN(Ω)=±1 {Si=±1} Assumptions Kijk = 0, . . . i) two-spin coupling only Hi ≡ H ii) External magnetic field spatially constant iii) Nearest-neighbour interactions only <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li></ul>→<ul style="display: flex;"><li style="flex:1">i,j </li><li style="flex:1"><ij> </li></ul>iv) Isotropic interactions v) Hypercubic lattice Jij ≡ J z = 2d J > 0 vi) Ferromagnetic material <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li></ul>−HΩ = H Si + J SiSj <i,j> i∈Ω ꢀ<ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li></ul>i∈Ω Si+J <i,j> SiSj) <ul style="display: flex;"><li style="flex:1">ZΩ[H, T, J] = </li><li style="flex:1">expβ(H </li></ul>{Si=±1} Arguments for phase transition in d = 1,2 dimensions with H = 0 ꢀFΩ[K] = Ein[K] − TSΩ[K] = −J SiSj − TkBlog(♯(states)) <i,j> A Bd = 1 N → ∞ FN,B − FN,A = 2J − kBTlog(N − 1) → −∞ d = 2 A Bn 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: JJij = σ|ri − rj| Answer: σ < 1 thermodynamic limit does not exist 1 ꢀ σ ꢀ 2 long range order persist for 0 < T < TC short-range interaction: no ferromagnetic state for T > 0 2 ꢀ σ Solutions in one and more dimensions - ad hoc methods - recursion method H = 0 d = 1 - transfer matrix method H = 0 (Kramers, Wannier 1941) - low temperature expansion <ul style="display: flex;"><li style="flex:1">H = 0 </li><li style="flex:1">d = 2 </li></ul>- Onsager solution (1944) - mean-field method (Weiss) d = 1, 2, 3 H = 0 <ul style="display: flex;"><li style="flex:1">d = 1 </li><li style="flex:1">H = 0 </li></ul>Transfer matrix method Reduce the problem of calculating the partition function to the problem of finding the eigenvalues of a matrix SN+1 = S1 SN+1 = S1 Assumption: SN S2 ZΩ[H, J] = Tr exp−βH (H,J,{S }) S3 Ωi<ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li></ul><ul style="display: flex;"><li style="flex:1">HΩ(H, J, {Si}) = −H </li><li style="flex:1">Si − J </li></ul>SiSj <i,j> i∈Ω <ul style="display: flex;"><li style="flex:1">h := βH </li><li style="flex:1">K := βJ </li></ul>ꢀ ꢀ <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li></ul><ul style="display: flex;"><li style="flex:1">i Si+K </li><li style="flex:1">i SiSi+1 </li></ul>ZΩ[h, K] = . . . eh S1 SN <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>eh+K − λ e−K e−h+K − λ det ≡ 0 Calculation: e−K ꢃ<ul style="display: flex;"><li style="flex:1">⇒ λ1,2 = eK(cosh(h) ± sinh(h)2 + e−4K </li><li style="flex:1">)</li></ul>ꢃ⇒ fb(h, K, T) = −JkBTlog(cosh(h) + sinh(h)2 + e−4K) Spatial correlation in one dimension (T > 0) <ul style="display: flex;"><li style="flex:1">H = 0 </li><li style="flex:1">d = 1 </li></ul>Definition: two point correlation function G(i, j) := ꢁ(Si − ꢁSiꢂ)(Sj − ꢁSjꢂ)ꢂ = ꢁSiSjꢂ − ꢁSiꢂꢁSjꢂ G(i, i + j) = ꢁSiSi+jꢂ = (tanh(K))j G(i, i + j) = e−jlog(coth(K)) ≡ e−j/ξ 1J/(kBT) ∼ ⇒ ξ = 1/2 e =log(coth)K)) <ul style="display: flex;"><li style="flex:1">H = 0 </li><li style="flex:1">d = 2 </li></ul>Onsager solution Columns → 3Notation and boundary conditions: ... n21n + 1 ≡ 1 12µα = {s1, s2, . . . , sn}αth row sn+1 = s1 , µn+1 = µ1 3N = n2 nElements of the Hamiltonian: n + 1 ≡ 1 nꢀE(µ, µ′) = −ǫ sks′k k=1 <ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li></ul><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li></ul>E(µ) = −ǫ sksk+1 − H sk $$2$$<ul style="display: flex;"><li style="flex:1">k=1 </li><li style="flex:1">k=1 </li></ul>Partition function: ꢀ ꢀ ꢀnα=1[E(µα,µα+1)+E(µα)] ZΩ[H, T] = . . . e−β µ1 µn −β[E(µ,µ′)+E(µ)] ꢁµ|P|µ′ꢂ := e <ul style="display: flex;"><li style="flex:1">Properties of the matrix P: </li><li style="flex:1">-</li></ul>--P : 2n × 2nmatrix ZΩ[H, T] = TrPn <ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li></ul>⇒ lim log(ZΩ[H = 0, T]) = lim log(λmax) <ul style="display: flex;"><li style="flex:1">N→∞ N </li><li style="flex:1">n→∞ n </li></ul>largest eigenvalue of P <ul style="display: flex;"><li style="flex:1">FΩ[0, T] </li><li style="flex:1">−kBTlogZΩ[0, T ] </li></ul>N(Ω) 1 = −kBT lim log(λmax n→∞ n <ul style="display: flex;"><li style="flex:1">fb[H = 0, T] = lim </li><li style="flex:1">= lim </li><li style="flex:1">)</li></ul>N(Ω)→∞ N(Ω) N(Ω)→∞ ꢄπꢅ<ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li></ul><ul style="display: flex;"><li style="flex:1">⇒ βfb[0, T] = − log (2 cosh 2βǫ) − </li><li style="flex:1">dφ log 1 + 1 − κ2sin2φ </li></ul>2π 20κ = 2[cosh 2φ coth2φ]−1

Symmetry and Ergodicity Breaking, Mean-Field Study of the Ising Model

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