Non-standard finite-size scaling at first-order phase transitions Marco Mueller, Wolfhard Janke, Des Johnston Coventry MECO39, April 2014 Mueller, Janke, Johnston Non-Standard First Order 1/24
Plan of talk The Standard Story - FSS at first order transitions A Problem - 30 standard deviations of problem A Solution - Non-standard FSS Mueller, Janke, Johnston Non-Standard First Order 2/24
The Standard story FSS at a first order phase transition Potts model as an example Simple two state model Mueller, Janke, Johnston Non-Standard First Order 3/24
First and Second Order Transitions First order - discontinuities in magnetization, energy (latent heat) Second order - divergences in specific heat, susceptibility Mueller, Janke, Johnston Non-Standard First Order 4/24
The q-state Potts model Hamiltonian H q = ij δ σi,σ j Evaluate a partition function, derivatives give observables Z(β) = {σ} exp( βh q ) Two dimensions ferromagnetic, q = 2, 3, 4 continuous d 3 (and mean field), q = 2 continuous otherwise first order Mueller, Janke, Johnston Non-Standard First Order 5/24
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 Mueller, Janke, Johnston Non-Standard First Order 6/24
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 Mueller, Janke, Johnston Non-Standard First Order 7/24
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 +... Mueller, Janke, Johnston Non-Standard First Order 8/24
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 +... 1 L d FSS Mueller, Janke, Johnston Non-Standard First Order 9/24
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 ) Mueller, Janke, Johnston Non-Standard First Order 10/24
A Problem... A 3D plaquette Ising model Its dual A (big) critical temperature discrepancy Mueller, Janke, Johnston Non-Standard First Order 11/24
A 3D Plaquette Ising action 3D cubic, spins on vertices H = 1 2 σ i σ j σ k σ l [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] Mueller, Janke, Johnston Non-Standard First Order 12/24
And the dual... Dual, by hand Z(β) = cosh (β) [1 + tanh (β) (σ i σ j σ k σ l )] {σ} [ijkl] which can be written as Z(β) = [2 cosh (β)] 3L3 [tanh (β)] n(s) Matchbox spins {S} Mueller, Janke, Johnston Non-Standard First Order 13/24
And the dual... II 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, 2 2 2 ij x ij y ij z Mueller, Janke, Johnston Non-Standard First Order 14/24
The Problem Original model: L = 8, 9,..., 26, 27, periodic bc, 1/V fits β = 0.549994(30) Dual model: L = 8, 10,..., 22, 24, periodic bc, 1/V fits βdual = 1.31029(19) β = 0.55317(11) Estimates are about 30 error bars apart. Mueller, Janke, Johnston Non-Standard First Order 15/24
A Solution... Degeneracy Modified FSS Mueller, Janke, Johnston Non-Standard First Order 16/24
Groundstates: Plaquette Persists into low temperature phase: degeneracy 2 3L Mueller, Janke, Johnston Non-Standard First Order 17/24
Groundstates: Dual ++ ++ ++ ++ + + + + (a) ++ ++ ++ (b) ++ ++ ++ ++ ++ ++ + + ++ ++ ++ (c) + ++ ++ + (d) ++ ++ σ τ στ Dual degeneracy Mueller, Janke, Johnston Non-Standard First Order 18/24
Ground state Mueller, Janke, Johnston Non-Standard First Order 19/24
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 +... Mueller, Janke, Johnston Non-Standard First Order 20/24
FSS Plaquette Hamiltonian fits Dual Hamiltonian fits 0.550 0.548 0.546 0.544 β 0.542 0.540 0.538 β Cmax V = 0.551221(11) β Bmin = 0.551347(7) β eqw = 0.551221(11) 10 5 10 20 p 10 35 10 50 10 65 10 80 1.50 1.25 1.00 0.75 0.50 e L = 13 β eqh = 0.551288(14) 0.536 0.001 0.002 0.003 0.004 0.005 0.006 0.007 L 2 1.310 1.305 1.300 β 1.295 1.290 1.285 β Cmax V = 1.31330(15) β Bmin = 1.31333(12) β eqw = 1.31330(15) 10 0 10 10 10 20 p 10 30 10 40 10 50 10 60 1.50 1.25 1.00 0.75 0.50 e L = 12 β eqh = 1.31318(16) 1.280 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 L 2 Mueller, Janke, Johnston Non-Standard First Order 21/24
Quality of fits L max L max 6 11 16 21 26 6 11 16 21 26 β C max V β +a/l 3 β +a/l 2 4 9 14 19 24 L min L max L max 6 11 16 21 26 6 11 16 21 26 β Bmin β +a/l 3 β +a/l 2 4 9 14 19 24 L min Q 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Forcing a fit to 1/L 3 gives much poorer quality Mueller, Janke, Johnston Non-Standard First Order 22/24
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,... Mueller, Janke, Johnston Non-Standard First Order 23/24
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, 14757 (1993) M. Mueller, W. Janke and D. A. Johnston, Non-Canonical Finite-Size Scaling at First-Order Phase Transitions [arxiv:1312.5984] Mueller, Janke, Johnston Non-Standard First Order 24/24