Collective Effects. Equilibrium and Nonequilibrium Physics

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1 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March 2006 Collective Effects in Equilibrium and Nonequilibrium Physics Website: Caltech Mirror: Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Today s Lecture Magnets with continuous symmetry (XY model) Landau free energy and the Mexican hat Angle/phase variables Goldstone s theorem Generalized rigidity Spin wave excitations and the Mermin-Wagner theorem Topological defects Kosterlitz-Thouless transition

2 2 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March XY Model s i θ i d-dimensional lattice of N spins s i with s i =freetorotateinaplane Hamiltonian H = 2 J s i s i+δ µ i i nn δ s i B Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Mean Field Theory We could go through mean field theory is before. I will leave you to show that for free XY spins with H = i s i b the mean spin is given by s i =L XY (βb) with the function L XY defined in terms of Bessel functions L XY (x) = I (x) I 0 (x) x 2 x3 6 + and that there is a phase transition to an ordered phase, s i =swith nonzero s, below a transition temperature T c given by k B T c = dj.

3 3 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Landau Free Energy Near T c we expand the free energy gained from a nonzero s as a Taylor expansion δ f = f (s, T ) f o (T ) α(t )s s + 2 β(t )(s s)2 + γ(t ) s s or expanding α, β, γ around their values at T c δ f = a(t T c )s s + 2 b(s s)2 + γ s s Note that the gradient term is a separate dot product of the and s vectors s s = i,α i s α i s α Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Mexican Hat Free Energy δ f = a(t T c )s s + 2b(s s)2 Free Energy s x s y Rotation of s everywhere through any angle leads to an equivalent state (same free energy, etc.)

4 4 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Angle remains a thermodynamic variable far below T c In Landau theory near T c the complete order parameter s(r) is a thermodynamic variable because the free energy changes involving s are small, and so the relaxation dynamics will be slow. Far below T c the magnitude s will relax quickly to the minimum of the Mexican hat, on a microscopic time scale. The angle variable will relax slowly: For a uniform rotation (the same change in everywhere) there is no relaxation For a rotation (r) that varies slowly in space the relaxation will be slow The orientation of the order parameter remains a new thermodynamic variable for any temperature in the ordered phase Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Goldstones Theorem For any continuous symmetry that is broken there is a low energy collective degree of freedom, with an energy that goes to zero as the wave length becomes infinite or For any continuous symmetry that is broken there is a low frequency collective mode, with a frequency that goes to zero as the wave number of the mode goes to zero Developed in the context of particle physics where the low frequency mode becomes zero mass boson. The search for the Higgs Boson coming from the supposed breaking of the electro-weak gauge symmetry is one of the great challenges in current experiments at the supercolliders.

5 5 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Phase Transition with an Angle Symmetry Easy plane ferromagnets and antiferromagnets Superfluidity: is the phase of the Bose condensed wave function Superconductivity: is the phase of the electron pair wave function Nematic liquid crystals in thin films or electric field: is the orientation of the long molecules in the plane And to some degree: Solids: is the phase of the density wave ρ e i(q r+ ) giving the translational position of the atoms Smectic liquid crystal: gives the translation of the planes of atoms Rayleigh-Bénard convection: gives the translation of the convection rolls Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Generalized Rigidity Far below T c we are left with the free energy dependence on the order parameter angle (r) δ f = 2 K ( )2 There is a new long range rigidity or stiffness in the system: Θ(r) Boundary dependent free energy scales as L rather than e L/ξ as in the absence of broken symmetry. We may use this result to rigorously define the stiffness constant K. L

6 6 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March 2006 This rigidity coming from broken symmetry is of course vital in the world around us: Elasticity of solids: shear modulus Superfluid and superconducting: superfluid density Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Spin-Wave Excitations Small perturbations from a uniform state: (r) = 0 + δ (r) Free energy of spin-wave excitations is F = 2 K d d x( δ ) 2 Introduce mode expansion δ (r) = q δ q e iq r (δ q = δ q ) so that F = 2 K d d x qq (q q )δ q δ q e iq r e iq r This gives (with the system volume ) F = 2 K q q 2 δ q 2 energy of spin wave distortion at wave vector q proportional to q 2 quadratic effective Hamiltonian: a Gaussian model : P(δ q ) e β Kq2 δ q 2 /2

7 7 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Mermin-Wagner-Hohenberg Theorem There is no long range order for a system with a broken continuous symmetry in two dimensions or less at any nonzero temperature. Arises because of the divergence of the thermally excited long wavelength (Goldstone) modes. Illustrative calculation: assume that long range order exists with order parameter s 0, and look for corrections to this coming from the thermal excitation of the Goldstone modes. Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March d = 2 Illustrative Calculation Free energy of spin-wave excitations is F = 2 K q q 2 δ q 2 Equipartition theorem tell us that δ q 2 = k B T Kq 2 Now lets correct the mean spin for these fluctuations ) s =s 0 cos s 0 ( 2 δ 2 with δ (r) 2 = δ q e iq r δ q e iq r = δ q 2 q q q = qc (2π) 2 2πqdq k B T 0 Kq 2 = k qc BT dq 2π K 0 q

8 8 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Result ) s s 0 ( 2 δ 2 with δ 2 = k qc BT 2π K 0 dq q Integral diverges because of the denominator as q 0! Divergence occurs for d 2 and is logarithmic for d = 2. No long range order for d 2 For rigorous proofs see Mermin and Wagner: Phys. Rev. Lett. 7, 33 (966) Hohenberg: Phys. Rev. 58, 383 (967) Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Correlation Function C( r r ) = cos[ (r) (r )] = exp[i( (r) (r ) A general result says for a Gaussian model (i.e., a model with a quadratic effective Hamiltonian such as given by our free energy) exp[i( (r) (r )] { } = exp 2 [ (r) (r )] 2. The result only depends on r r by the translational invariance, and so we may take r = 0 for convenience. Then introducing the Fourier mode representation [ (r) (0)] 2 = qq δ q δ q (e iq r )(e iq r ) = q δ q 2 2( cos q r)

9 9 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March We can roughly see how this comes as follows. Introduce the modes C(r) = exp i δ q (e iq r ) q Now expand the exponential, and use the fact that averages of odd powers of δ q are zero C(r) = Re[ 2 δ q δ q (e iq r )](e iq r )] + ] qq = δ q 2 ( cos q r) + q = exp δ q 2 ( cos q r) q (it can be shown that the higher order terms work for the last expression) Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March C(r) = exp[ q δ q 2 ( cos q r)] The sum appearing in the exponential is almost the same one we did before, since for large q r the cos(q r) term oscillates rapidly and may be replaced by its average value of zero. However for small q < r the ( cos q r) factor becomes proportional to q 2, and the logarithmic divergence is cutoff: q δ q 2 ( cos q r) k BT 2π K qc r dq q = k BT 2π K ln q cr Note that inaccuracies in our evaluation of the integral can be absorbed into the definition of the constant q c. We get the important result of power law decay of correlations with an exponent that depends on temperature C(r) r η with η(t ) = k BT 2π K

10 0 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Topological Defects Topological invariant (winding number) dl = 2π Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Topological Defects Topological invariant (winding number) dl = 2π

11 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Topological Defects Topological invariant (winding number) dl = 2π Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Topological Defects Topological invariant (winding number) dl = 2π

12 2 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Topological Defects Topological invariant (winding number) dl = 2π Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Topological Defects Plus-minus pair gives winding number 0 dl = 0 2π

13 3 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Topological Defects Plus-minus pair gives winding number 0 dl = 0 2π Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Topological Defects - + Plus-minus pair gives winding number 0 dl = 0 2π

14 4 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Importance of Topological Defects Random arrangement of defects (of both signs) eliminates long range order and gives exponential decay of correlations May eliminate the generalized rigidity Dislocations in crystals Vortices in superfluids and superconductors Provide a mechanism for dynamics in an ordered state Dominate dynamics in quenches from disordered initial conditions (coarsening, cf. cosmic strings, the Kibble mechanism) May be important in phase transition (e.g., 2d XY model) Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Vortex Energetics in 2d Energy of a vortex E = 2 K d 2 x( ) 2 = 2 K 2πr drr 2 = π K ln(l/ξ) + c Entropy of a vortex Free energy of a vortex S = k B ln(l 2 /ξ 2 ) + c F = (π K 2k B T ) ln(l/ξ) + c < 0 for T >πk/2k B

15 5 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Kosterlitz-Thouless Transition - 2d XY Model At low temperatures there is a phase with a nonzero spin stiffness constant K but no long range order: correlations decay as a power law that depends on temperature C(r) r η with η(t ) = k BT 2π K (T ) In the low temperature phase there are bound pairs of plus-minus vortices, but no free vortices At the Kosterlitz-Thouless temperature T KT given by k B T KT = π K/2 there is a vortex unbinding transition leading to free vortices ForT > T KT there is exponential decay of correlations AtT = T KT the stiffness constant K jumps discontinuously to zero The ratio K (T KT )/k B T KT = 2/π and is universal Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Dynamics Have not yet specified the dynamics. Different dynamics is appropriate for different physical systems, depending on what other thermodynamic variables there are, and how the order parameter couples to them. Some examples follow:

16 6 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Thin Film Nematic describes the orientation of long molecules in the plane, and relaxes to minimize the free energy. Most easily expressed in complex notation (s x, s y ) S = S e i. Free energy becomes F = d d x [a(t T c ) S b S 4 + γ S 2] Relaxational dynamics is Ṡ = τ r δf δs τ r Ṡ = a(t c T )S b S 2 S + γ 2 S Far below T c the magnitude S relaxes rapidly to the value S 0 at the bottom of the Mexican hat almost everywhere, leaving the low frequency phase dynamics = γ 2 τ r (or = δf τ δ with F = 2 d d xk( ) 2 ) i.e., relaxation by diffusion. Vortex dynamics may also be important. Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Easy Plane Magnet Spins can point in any direction, but anisotropy favors alignment in XY plane. S z The XY and Z components of the spin have different properties: S S z = s iz is a conserved quantity i in S = s i is the XY order parameter i in S z and are canonically conjugate variables in the Hamiltonian equations Ṡ z = δf δ, where the free energy (away from T c ) F = d d x = δf δs z [ ] 2 K ( )2 + S2 z 2χ S zb z plays the role of the effective Hamiltonian (b z is µb z with B z the external field in the z direction, χ is the z-spin susceptibility, and I ve chosen the simple case of B = 0)

17 7 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March F = d d x [ 2 K ( )2 + 2 S2 z /χ S zb z ] Ṡ z = δf δ, = δf δs z S z S For b z = 0 and small deviations δ from the uniform state δṡ z = K 2 δ δ = χ δs z leading to propagating spin waves e i(q r ωt) with the dispersion relation ω = cq and the propagation speed c = K/χ Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March Next Lecture Superfluids and superconductors What are superfluidity and superconductivity? Description in terms of a macroscopic phase Josephson effect Supercurrents that flow for ever Four sounds