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Lecture 1: Date: 30Th July Duration 1Hr Lecture 2: Date: 3Rd Aug. Duration Lecture 1: Date: 30 th July Duration 1hr Definition of continuous and discontinuous phase transitions; Critical exponents; Notion of symmetry breaking; Examples of models with continuous symmetry (classical XY model); Transverse Ising Models and idea of quantum phase transitions. Lecture 2: Date: 3 rd Aug. Duration 1hr Discussion on XY model and transverse Ising model; Recapitulation of Weiss mean field theory m=tanh(β Jqm + β h ) ; central limit theorem suggests that it should be exact in the limit of infinite number of nearest neighbors. We then look at an Ising model with infinite-range interaction where all the spins interact with each other = −J − given by the Hamiltonian H∑ sshsi j ∑ i ; where J is scaled by N so N i, j i that the energy is extensive. In the limit N → ∞ , (infinite range interaction) using the saddle point integration, we have shown that mean field theory is exact. Lecture 3: Date: 5 th Aug. Duration 2hrs Recapitulation of the saddle point result; (i) One can show using the expansion of the log(cosh..) term of the effective free energy that the Landau “free” energy expansion can be obtained. (ii) We also discussed Bragg-Williams theory of Ising model and its’ connection to the Landau theory (Chaikin, chapter 4, page 146). (iii) We also address the question whether a thermodynamic limit does always exist? (Goldenfeld, page 26). (iv) Finally, we have introduced the concept of variational mean field theory using the Bogoliubov inequality ( F≤ F + H − H for the 0 0 0 variational free energy; minimizing this we get the mean field Ising results. (Chaikin, page 198) Lecture 4: Date: 11 th Aug. Duration 1hr Recapitulation of the variational theorem. Discussion on the stability and convexity properties of the free energy function G(T,p) and F(T,V) . The behavior of the free energy functions across discontinuous and continuous transitions. Comments on Calpyeron equation. (Stanley Chapter 2) Lecture 5: Date: 12 th Aug. Duration 2hr (i) Comments on thermodynamics of magnets with localized magnetic moments. (ii) The Gibbs function is concave function of h and T. (for a proof: see Goldenfeld page 38) (iii) Properties of G(h) and F(m) across discontinuous and continuous transitions. (iv) g(h) satisfies all the properties of a concave function. To have a phase transition, we need a point of discontinuous slope at h=0 . (This does not violate the concavity). (v) Definition of phase, phase boundary and co-dimension (v) Symmetries of Ising model: (i) ghJT(,,)= g ( − hJT ,,) (ii) Sublattice symmetry: g(0, JT , )= g (0, − JT , ) (vi) Impossibility of phase transition due to symmetry; Impossibility theorem. (vii) Spontaneous symmetry breaking: The impossibility theory assumes that g(h) is differentiable for h → 0. But g(h) can have a form =− +σ σ > gh() g (0) mhs || oh ( ); 1 which is concave but not differentiable at h=0 . This explains the existence of spontaneous magnetization. The thermodynamic limit is essential to get the non-analytic behavior which is in turn essential for the spontaneous magnetization. The spontaneous 1 ∂g magnetization is appropriately defined as m = lim lim (the s h→0 N→∞ N∂ h limits do not commute). This is known as spontaneous ( h=0 ) symmetry breaking when the Hamiltonian is still up-down symmetric but the statistical averages are not symmetric. (In the Landau picture, property of one well comes in the process; ergodcity breaking) (Discussion are form Goldenfeld; chapter 2 and Stanley: Chapter 2) Lecture 6: Date: 17 th Aug. Duration 1hr Recapitulation: Discussion of spontaneous symmetry breaking; it was shown using the Arhenius law that in the thermodynamic limit, the reversal time of a oppositely magnetized block diverges. This results to the breaking of ergodicity. Discussion of fluctuation-response theorem and using non-local susceptibility it can shown that χ()q= β G () q where G(q) is Fourier = transformed connected correlation function. χ(q ) q→0 χ is the uniform susceptibility which diverges at the critical point. We have taken an Ising model with a site-dependent field hi and > ferromagnetic nearest neighbor interaction J. For T T C , using weak field = + and linear response, we get Siβ h i∑ JS ijj . Using the j∈ i = kB T χ Fluctuation- response and Fourier transform, we get G( q ) 2 2 ; 1+ q ξ 1 where is the correlation length. We get the exponents =; = 0 . ξ ν2 η Lecture 7: Date: 19 th Aug. Duration 2hrs < Recapitulation: Evaluation of the correlation function G(q) for T T c . We now two equations = + = = Siβ h i∑ JS ijj and mtanhβ∑ Jij S j tanh β Jqm . We shall look j∈ i j∈ i δ = − + at the fluctuation in Si S i m and expand β hi∑ J ij S j j∈ i J1− m 2 2 ∼ β ( ) around β Jqm ( that very weak field), we get ξ 2 ; giving 1− 2dβ J (1 − m ) 1 again =; = 0 . ν2 η 1. Questions i) Exponents satisfy relation like α+2 β + γ = 2 ; why? ii) The relation 2−α = ν d is satisfied with mean field exponents for d=4; why? iii) The critical exponents depend on (i) dimensionality (ii) symmetry and (iii) Nature of fixed points : Universality (iv) The ratio of critical amplitudes is also universal. 2. The Vanderwaal gas has same set of critical exponents as the mean −' =1 ' − 2 field Ising. For a fluid system, Grr( )2 ρ ()() rr ρ ρ and ρ − + = dd rG r . The structure factor Sq()∝ Gq () ∼ q 2 η at = leads κT β ∫ ( ) T T c to critical opalescence. Divergence of S(q) for q → 0 implies the divergence of compressibility. 3. Discussion on a) Correction to Scaling b) Experimental determination of critical exponents (Goldenfeld, Chapter 4) 4. Introduction to Landau-Ginzburg functional. Lecture 8: Date: 21st Aug. Duration 1.5 hrs (i) Recapitulation: Landau-Ginzburg Free energy functional (for discussion on functional derivatives refer to Chaikin and Lubensky, Chapter 3, Appendix and Euler-Lgranges equation, see Goldsetin; last chapter on mechanics of continuous bodies) (ii) Critical exponents are the same as mean field exponents (All the mean field approaches lead to the same set of critical exponents) (iii) Fluctuation-response theorem helps in getting the correlation function which now has the mean field form 1 2 1 G( q ) = = ξ so that ν = and = 0 +2 + 2 2 2 η r12 um cq c()1+ q ξ 2 (iv) LG theory is a mean field theory that includes the spatial variation of the order parameter in a systematic way. (Landau-Ginzburg: Chaikin Chapter 4, Goldenfeld Chapter 5) Critique: (i) Mean field theory ignores fluctuations and hence over- estimates the critical temperature. Predicts a phase transition even for a one-dimensional Ising Model. (ii) Critical exponents are very much off the experimentally obtained values. (iii) Critical exponents are independent of dimension and also symmetery!! (iv) Spin models with continuous symmetry: XY, Heisenberg and n-vector models; within the LG theory the critical exponents. 1 2 (v) The parallel correlation functions G( q ) = = ξ || +2 + 2 2 2 r12 um cq c()1+ q ξ show the same behavior as the Ising (n=1) case where the transverse 1 1 correlation G( q ) = = for T < T which shows that the ⊥ r+4 um2 + cq 2 cq 2 c < transverse correlation length is always divergent for T T C . This is the consequence of the rotational symmetry of the spin space (for XY model n=2, the Landau functional looks a like the bottom of a bottle) . This also results in massless excitaions known as Goldstone Bosons. (For discussion, see Chaikin, Chapter 4 and Goldenfeld, Chapter 11). Lecture 9: Date: 25th Aug. Duration 1 hrs Recapitulation: Continuous symmetry breaking, transverse correlation, Goldstone Modes and Mermin-Wagner theorem. We learnt that the (logarithmic) divergence in the limit q → 0 in the transverse correlation in the position space yields the concept of the lower-critical dimension. In other words, continuous symmetry can not be spontaneously broken for d ≤ 2 . We can not get a superfluid, superconductor or spontaneous magnetization in a XY or Heisenberg Model at finite temperature if d ≤ 2 . If a symmetry is broken, or in the ordered phase, we have rigidity of the order parameter and topological defects. For a XY model, the order parameter field is conveniently written as mexp( iθ ( x ) ) where m is the magnitude and θ is the spatially varying transverse component. Lecture 10: Date: 27th Aug. Duration 1.5 hrs 1. Using the XY Hamiltonian, we calculate the fluctuations about the ordered phase. We show that in terms of fluctuations, one can rewrite the LG functional. If ϕ2 describes the fluctuation in the direction perpendicular to the order, then there is no quadratic (mass) term associated with ϕ2 and hence the correlation is power-law. (Goldstein, Chapter 11) 2. We studied the XY model in the presence of a weak magnetic field and expanding the Landau-Ginzburg Hamiltonian for weak field, showed that k Tm 2 the transverse correlation is given by G( q ) = B . This shows that ⊥ 2 + ρsq h1 m in presence of a field the correlation function decays exponentially with a 1/2 correlation length = ρs (Chaikin, Chapter 6). Using the scaling ξh h1 m ∼(2−d ) ∼ ν (d − 2) → ρs ξ t (vanishes as t 0 ), we derived some scaling relations. 3. Connection to superfluidity: Order parameter is the ground state = iθ ( x ) wave-function ψ (x ) ns e which has the same symmetry as the XY model and v= ∇θ; ∇× v = 0 . Superfluid can not participate in a sm s rotational motion which is contradicted by the rotating bucket 1 = experiment! Now, ∇θ ⋅dl = 2 n π if ∇θ = eÙθ ; ∇× v 0 everywhere ∫ 2π r s except r=0 .
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