Collective Effects. Equilibrium and Nonequilibrium Physics

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1 1 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Collective Effects in Equilibrium and Nonequilibrium Physics Website: Caltech Mirror: Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March oday s Lecture he simplest magnet: the Ising model Calculate a simple phase transition from first principles Discuss the behavior near this second order phase transition Clarify the idea of broken symmetry Introduce Landau theory

2 2 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Equilibrium Statistical Mechanics Isolated system (microcanonical ensemble) Each accessible (micro)state equally probable hermodynamic potential: entropy S = k B ln n Probability of macroscopic configuration C: P(C) e S(C)/k B System in contact with heat bath at temperature (canonical ensemble) Probability of microstate n proportional to e β E n, with β = (k B ) 1 hermodynamic potential: free energy F = k B ln n e β E n Calculate the partition function Z = n e β E n Probability of macroscopic configuration C: P(C) e S(C)/k Be β E(C) = e β F(C) Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Ising Model d-dimensional lattice of N spins s i =±1 Hamiltonian H = 1 2 J s i s i+δ µ i i nn δ s i B

3 3 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Partition Function Canonical partition function Z = {s i } e β H{s i }. he enumeration of all configurations cannot be done for d 3, and although possible in d = 2, it is extremely hard there (a problem solved by Onsager). Ising solved the model in one dimension. We will use an approximate solution technique known as mean field theory. Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Free Spins in a Field H 0 = b i s i writing b for µb. his is easy to deal with, since the Hamiltonian is the sum over independent spins. Average spin on each site is s i = eβb e βb e βb = tanh(βb). + e βb he partition function is the product of single-spin partition functions Z 0 = [e βb + e βb ] N

4 4 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Mean Field heory In the mean field approximation we suppose that the ith spin sees an effective field b ef f which is the sum of the external field and the interaction from the neighbors calculated as if each neighboring spin were fixed at its ensemble average value b eff = b + J δ s i+δ. We now look for a self consistent solution where each s i takes on the same value s which is given by the result for noninteracting spins First look at b = 0 with ε = 2dβ Js. s = tanh[βb eff ] = tanh[β(b + 2dJs)]. s = tanh(2dβ Js) or ε/2dβ J = tanh ε Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Self Consistency ε/2dβ J = tanh ε > c ε/2dβj < c tanh ε ε For > c = 2dJ/k B the only solution is ε = 0 For < c two new solutions develop (equal in magnitude but opposite signs) with ε growing continuously below c.

5 5 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March s c Near c we can get the behavior by expanding tanh ε in small ε: ε = 2dβ J tanh ε with ε = 2dβ Js, k B c = 2dJ becomes ε = c (ε 1 3 ε3 ) or ε 2 = 3( c 1) giving to lowest order in small (1 / c ) s =± ( ) c 1/2 3 c Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Order Parameter Exponent Focus on the power law temperature dependence near c. Introduce the small reduced temperature deviation t = ( c )/ c. We had s =± ( c 3 c ) 1/2 Write this for small t < 0as: s t β order parameter exponent β = 1/2

6 6 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Susceptibility Exponent he spin susceptibility is χ = ds/db b=0 : so that (writing s = ds/db) s = tanh[β(b + 2Jds)] s = sech 2 [β(b + 2Jds)](β + c s ). Just above c, setting b = s = 0 χ = 1 k B c ( c c ) 1 giving a diverging susceptibility as approaches c from above χ t γ susceptibility exponent γ = 1 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Magnetization Exponent Exactly at c there is a nonlinear dependence s(b) of s on b: s = tanh[β c (b + 2Jds)] (β c b + s) 1 3 (β cb + s) 3 +. he s terms cancel, so we must retain the s 3 term. he linear term in b survives, so we can ignore terms in b 2, bs etc. his gives so that ( ) 3b 1/3 s( = c, b) sgnb +. k B c s b 1/δ sgnb magnetization exponent δ = 3.

7 7 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Internal Energy With a little more effort we can calculate the internal energy U and other thermodynamic potentials. We will do this in zero magnetic field only. In the mean field approximation U is simply given by Nd bonds each with energy Js 2 for < c : ( ) U = NdJs 2 c 3NdJ. For > c the energy is zero in mean field theory (an indication of the limitations of this theory, since clearly there will be some lowering of energy from the correlation of nearest neighbors). c Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Free Energy For noninteracting spins in field b we had F = k B ln Z 0 with Z 0 = [e βb + e βb ] N. Evaluate (?) the free energy for the interacting spins replacing b by b eff = 2Jds. his turns out not to be quite right, so call the expression F I (I for independent) F I = Nk B ln [e (c/ )s (c/ + e )s] replacing 2dJ/k B by c. his is not quite correct, because we have double counted the interaction energy. So we need to subtract off a term U to correct for this [ F = F I U = Nk B ln e (c/ )s + e (c/ )s] + NdJs 2. Near c expand this in small s F I = Nk B ln 2 Nk B c 2 [ ( c ) s ( ) 3 c s ] 4.

8 8 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Specific Heat Exponent F = Nk B ln 2 NJd [ ( c ) s ( ) 3 c s ] 4 Minimize F with respect to s gives, as before ± ( ) 1/2 3 c s = c for < c 0 for c and the reduction in F below c for nonzero s δf = 3 ( ) 2 NdJ c 2 +. c he power law dependence of δf near c is used to define the specific heat exponent δf t 2 α specific heat exponent α = 0 Specific heat is C = d 2 F/d 2 is zero above c, and jumps to 3Nk B /2at c Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Summary b c

9 9 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March b c s c Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March b c C δf c

10 10 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March b c χ c Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March b c s b

11 11 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March b c s b Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March When is mean field theory exact? Mean field theory is a useful first approach giving a qualitative prediction of the behavior at phase transitions. It becomes exact when a large number of neighbors participate in the interaction with each spin: in high enough spatial dimension d; for long range interactions. In other cases it is only an approximate theory, and fluctuations are important.

12 12 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Fluctuations May Destroy the Order One dimension L Energy cost to flip a cluster of length L is 4J Probability of flipping cluster e 4β J No ordering at any nonzero temperature (the problem Ising solved) Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Fluctutations in 2d Ising Model L Energy to flip cluster of size L grows as roughly 8JL Ordering occurs at nonzero transition temperature

13 13 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Fluctuations Change Exponents Exponents for the Ising model Quantity Dependence MF 2d 3d Order parameter s t β, t < 0 β = 1 2 β = 1 8 β = 0.33 Susceptibility χ t γ γ = 1 γ = 7 4 γ = 1.25 Free energy δf t 2 α α = 0 α = 0 α = 0.12 Order parameter at c s b 1/δ sgnb δ = 3 δ = 15 δ = 4.8 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March General Remarks A new state grows continuously out of the previous one: for c the two states become quantitatively the same. For < c equally good but macroscopically different states exist. his is a broken symmetry the thermodynamic states do not have the full symmetry of the Hamiltonian. Instead the different thermodynamic states below c are related by this symmetry operation. he thermodynamic potentials F, U, S...are continuous at c but not necessarily smooth (analytic). Fluctuations involving admixtures of the other states become important as c, so that mean field theory will not in general be a good approximation near c. hermodynamic potentials show power law behavior in 1 / c near c. he derivatives of the potentials (specific heat, susceptibility etc.) similarly show power laws, and will diverge at c if the power is negative. he exponents are different than the values calculated in mean field theory, and are usually no longer rationals. It is not possible to classify phase transitions into higher orders (second, third etc.) according to which derivative of the free energy is discontinuous (Ehrenfest).

14 14 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Landau heory of Second Order Phase ransitions Landau theory formalizes these ideas for any second order phase transition. Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Landau heory of Second Order Phase ransitions Second order phase transitions occur when a new state of reduced symmetry develops continuously from the disordered (high temperature) phase. he ordered phase has a lower symmetry than the Hamiltonian spontaneously broken symmetry. here will therefore be a number (sometimes infinite) of equivalent symmetry related states (e.g., equal free energy). hese are macroscopically different, and so thermal fluctuations will not connect one to another in the thermodynamic limit. o describe the ordered state we introduce a macroscopic order parameter ψ that describes the character and strength of the broken symmetry. he order parameter ψ is an additional thermodynamic variable

15 15 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Free Energy Expansion Since the order parameter grows continuously from zero at the transition temperature, Landau suggested an expansion of the free energy: aylor expansion in order parameter ψ (i.e., analytic) he free energy must be invariant under all symmetry operations of the Hamiltonian, and the terms in the expansion are restricted by these symmetry considerations. he order parameter may take on different values in different parts of the system ψ(r), and so we introduce the free energy density f F = d d xf(ψ, ) and expand the free energy density f in the local order parameter ψ(r) for small ψ. Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Expansion for the Ising Model For the Ising ferromagnet, use ψ = m( r), with m( r) the magnetization per unit volume averaged over some reasonably macroscopic volume. he free energy is invariant under spin inversion, and so the aylor expansion contains only even powers of m f (m, ) f o ( ) + α( )m β( )m4 +γ( ) m m. he last term in the expansion gives a free energy cost for a nonuniform m( r). A positive γ ensures that the spatially uniform state gives the lowest value of the free energy. Higher order terms could be retained, but are not usually necessary for the important behavior near c.

16 16 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Minimum Free Energy f (m, ) f o ( ) α( )m β( )m4 + γ( ) m m. If fluctuations are small, the state that minimizes the free energy will be the physically realized state. his is not always the case, and Landau s theory corresponds to a mean field theory that ignores these fluctuations. For the Ising magnet the minimum of F is given by a uniform m( r) = m satisfying α m + β m 3 = 0. giving ± α/β for α<0 m =. 0 for α>0 Identify α = 0 as where the temperature passes through c, and expand near here α( ) a( c ) + β( ) b + γ( ) γ + Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Exponents etc. f f o α( )m β( )m + γ( )( m) 2 a( c )m bm4 + γ( m) 2 and the value m = m minimizing f ( a ) 1/2 m (c ) 1/2 b for < c. Evaluating f at m gives f f 0 a2 ( c ) 2 2b. hese are the same results for the exponents we found directly from mean field theory.

17 17 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March f f o = a( c )m bm4 f > c = c < c m Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Coupling to Magnetic Field f (m,, B) = f 0 ( ) + a( c )m bm4 + γ( m) 2 mb. he magnetic field couples directly to the order parameter and is a symmetry breaking field. Minimizing f with respect to m gives a susceptibility diverging at c, e.g., for > c χ = m B = 1 B=0 2a ( c) 1, and at = c for small B ( ) 1 1/3 m = B 1/3. 2b Again the exponents are the same as we found in from mean field theory.

18 18 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March First Order ransitions We can also encounter first order broken-symmetry transitions. For example the liquid solid transition is described by the strength of density waves ρe i q r. In three dimensions we need three density waves with wave vectors that prescribe the reciprocal lattice; we will suppose all three components have the same magnitude ρ. Since the density perturbation is added to the uniform density of the liquid, there is no symmetry under changing sign of the density wave, and so the free energy expansion now may have a cubic term (suppose uniform ρ for simplicity). f f 0 = a( c )ρ 2 + cρ bρ4 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March f f 0 = a( c )ρ 2 + cρ bρ4 solid f >> m > m = m c << m = c << m ρ

19 19 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March At high temperatures there is a single minimum at ρ = 0 corresponding to the liquid phase. As the temperature is lowered a second minimum develops, but at a free energy that remains higher than the liquid. At m the new free energy becomes equal: this is the melting temperature where the liquid and solid have the same free energy. Below m the solid has the lower free energy. here is a jump in ρ at m, not a continuous variation as at a second order transition. he temperature c at which the minimum in the free energy at ρ = 0 disappears (i.e. the liquid state no longer exists) is not of great physical significance. he Landau expansion technique may not be accurate at a first order transition: the jump in ρ at the transition means there is no guarantee that the truncated expansion in ρ is a good approximation. Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Another way a first order transition can occur is if the coefficient of the quartic term turns out to be negative. hen we must extend the expansion to sixth order to get finite results (otherwise the free energy is unbounded below): f f 0 = a( c )m b m cm6.

20 20 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March f f 0 = a( c )m b m cm6 f = f = c m Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March Next Lecture Magnets with continuous symmetry (XY model) Spin waves and the Mermin-Wagner theorem opological defects Kosterlitz-houless transition