# Boring (?) first-order phase transitions

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1 Boring (?) first-order phase transitions Des Johnston Edinburgh, June 2014 Johnston First Order 1/34

2 Plan of talk First and Second Order Transitions Finite size scaling (FSS) at first order transitions Non-standard FSS Johnston First Order 2/34

3 First and Second Order Transitions (Ehrenfest) First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable. Second-order transitions are continuous in the first derivative but exhibit discontinuity in a second derivative of the free energy. Johnston First Order 3/34

4 First and Second Order Transitions ( modern ) First-order phase transitions are those that involve a latent heat. Second-order transitions are also called continuous phase transitions. They are characterized by a divergent susceptibility, an infinite correlation length, and a power-law decay of correlations near criticality. Johnston First Order 4/34

5 First Order Transitions - Piccies First order - melting First order - field driven Johnston First Order 5/34

6 Second Order Transition H = ij σ i σ j ; Z(β) = {σ} exp( βh) Johnston First Order 6/34

7 First and Second Order Transitions - Piccies First order - discontinuities in magnetization, energy (latent heat) Second order - divergences in specific heat, susceptibility Johnston First Order 7/34

8 Looking at transitions Cook up a (lattice) model Identify order parameter, measure/calculate things Look for transition Continuous - extract critical exponents Finite Size Scaling (FSS) Johnston First Order 8/34

9 Define a model: q-state Potts Hamiltonian H q = ij δ σi,σ j Ising : H = σ i σ j ij Evaluate a partition function Z(β) = {σ} exp( βh q ) Derivatives of free energy give observables (energy, magnetization..) F (β) = ln Z(β) Johnston First Order 9/34

10 Measure 1001 Different Observables Order parameter M = (q max{n i } N)/(q 1) Per-site quantities denoted by e = E/N and m = M/N u(β) = E /N, C(β) = β 2 N[ e 2 e 2 ], B(β) = [1 e4 3 e 2 2 ]. m(β) = m, χ(β) = β N[ m 2 m 2 ], U 2 (β) = [1 m2 3 m 2 ]. Johnston First Order 10/34

11 Continuous Transitions - Critical exponents (Continuous) Phase transitions characterized by critical exponents Define t = T T c /T c Then in general, ξ t ν, M t β, C t α, χ t γ Can be rephrased in terms of the linear size of a system L or N 1/d ξ N 1/d, M N β/νd, C N α/νd, χ N γ/νd Johnston First Order 11/34

12 Scaling Another exponent... ψ(0)ψ(r) r d+2 η Scaling relations α = 2 νd ; α + 2β + γ = 2 Two independent exponents Johnston First Order 12/34

13 Now, first order... Formally νd = 1, α = 1 i.e. Volume scaling C N ; ξ N Squeezing a delta function onto a lattice Johnston First Order 13/34

14 What does a first order system look like (at PT) I? Johnston First Order 14/34

15 What does a first order system look like (at PT) II? E Hysteresis COLD HOT β Johnston First Order 15/34

16 What does a first order system look like (at PT) III? Phase coexistence P(E) e-05 1e-06 1e E Johnston First Order 16/34

17 Heuristic two-phase model A fraction W o in q ordered phase(s), energy ê o A fraction W d = 1 W o in disordered phase, energy ê d The hat = quantities evaluated at β Neglect fluctuations within the phases Johnston First Order 17/34

18 Energy moments Energy moments become e n = W o ê n o + (1 W o )ê n d And the specific heat then reads: C V (β, L) = L d β 2 ( e 2 e 2) = L d β 2 W o (1 W o ) ê 2 Max of C max V = L d (β ê/2) 2 at W o = W d = 0.5 Volume scaling Johnston First Order 18/34

19 FSS: Specific Heat Probability of being in any of the states p o e βld ˆf o and p d e βld ˆf d Time spent in the ordered states qp o Expand around β Solve for specific heat peak W o /W d qe Ld β ˆf o /e βld ˆf d 0 = ln q + L d ê(β β ) +... β Cmax V (L) = β ln q L d ê +... Johnston First Order 19/34

20 FSS: Binder Cumulant Energetic Binder cumulant Use (again) B(β, L) = 1 e4 3 e 2 2 e n = W o ê n o + (1 W o )ê n d to get location of min: β Bmin (L) β Bmin (L) = β ln(qê2 o/ê 2 d ) L d ê L d FSS Johnston First Order 20/34

21 An Aside You can do this more carefully Pirogov-Sinai Theory (Borgs/Kotecký) Z(β) = ] [ ] [e βld f d + qe βld f o 1 + O(L d e L/L 0 ) Z(β) 2 ( qe βld (f d +f 0 βl )/2 d (f d f 0 ) cosh + 1 ) 2 2 ln q x = βld (f d f 0 ) ln q Ld ê(β β ) ln q +... Johnston First Order 21/34

22 A Bit Dull... Peaks grow like L d = V = N Critical temperatures shift like 1/L d No other numbers (?) Johnston First Order 22/34

23 Some other numbers - Fixed BC Fixed boundary conditions Z(β) = [ e β(ld f d +2dL d 1 f d ) + qe β(ld f o+2dl d 1 f o) ] [1 +...] x = Ld (f d f 0 ) + dl d 1 ( 2 f d f 0 ) ln q al d (β β ) + bl d (since f d = a 1 + ẽ d (β β ) +..., fo = a 2 + ẽ o (β β ) +...) β 1 L It is clear that there can now be O(1/L) corrections Johnston First Order 23/34

24 Fitting the data - 8-state Potts Estimated from peak in χ: β c (L) = β + a L +... Johnston First Order 24/34

25 Some other numbers - degeneracy A 3D plaquette Ising model Its dual A big critical temperature discrepancy (30σ) Johnston First Order 25/34

26 A 3D Plaquette Ising action 3D cubic, spins on vertices H = 1 σ i σ j σ k σ l 2 [i,j,k,l] NOT H = U ij U jk U kl U li, U ij = ±1 Z 2 Lattice Gauge [i,j,k,l] Johnston First Order 26/34

27 And the dual An anisotropically coupled Ashkin-Teller model H dual = 1 σ i σ j 1 τ i τ j 1 σ i σ j τ i τ j, ij x ij y ij z Johnston First Order 27/34

28 The Problem Original model: L = 8, 9,..., 26, 27, periodic bc, 1/L d fits β = (30) Dual model: L = 8, 10,..., 22, 24, periodic bc, 1/L d fits βdual = (19) β = (11) Estimates are about 30 error bars apart. Johnston First Order 28/34

29 A Solution... Degeneracy Modified FSS Johnston First Order 29/34

30 Groundstates: Plaquette Persists into low temperature phase: degeneracy 2 3L Johnston First Order 30/34

31 Ground state Johnston First Order 31/34

32 1st Order FSS with Exponential Degeneracy Normally q is constant Suppose instead q e L β Cmax V (L) = β ln q L d ê +... become β Bmin (L) = β ln(qê2 o/ê 2 d ) L d ê +... β Cmax V (L) = β 1 L d 1 ê +... β Bmin (L) = β ln(ê2 o/ê 2 d ) L d 1 ê +... Johnston First Order 32/34

33 Conclusions Standard 1st order FSS: 1/L 3 corrections in 3D Fixed BC: 1/L (surface tension) Exponential degeneracy: 1/L 2 in 3D Further applications may be higher-dimensional variants of the gonihedric model, ANNNI models, spin ice systems, orbital compass models,... Johnston First Order 33/34

34 References K. Binder, Rep. Prog. Phys. 50, 783 (1987) C. Borgs and R. Kotecký, Phys. Rev. Lett. 68, 1734 (1992) W. Janke, Phys. Rev. B 47, (1993) M. Mueller, W. Janke and D. A. Johnston, Phys. Rev. Lett. 112, (2014) Johnston First Order 34/34

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