Macroscopic Degeneracy and FSS at 1st Order Phase Transitions

Transcription

1 Macroscopic Degeneracy and FSS at 1st Order Phase Transitions Marco Mueller (hero), Wolfhard Janke (good guy), Des Johnston (villain) Krakow, Oct 2015 Mueller, Janke, Johnston Degeneracy/FSS 1/22

2 Plan of talk Standard 1st order FSS A Problem with 1st order FSS for a plaquette 3D Ising model A solution Mueller, Janke, Johnston Degeneracy/FSS 2/22

3 First and Second Order Transitions First order - discontinuities in magnetization, energy (latent heat) Second order - divergences in specific heat, susceptibility Mueller, Janke, Johnston Degeneracy/FSS 3/22

4 The q-state Potts model Hamiltonian H q = ij δ σi,σ j Evaluate the partition function, derivatives give observables Z(β) = {σ} exp( βh q ) Mueller, Janke, Johnston Degeneracy/FSS 4/22

5 1st Order FSS: Heuristic two-phase model A fraction W o in q ordered phase(s), energy e o A fraction W d = 1 W o in disordered phase, energy e d Ignore transits Mueller, Janke, Johnston Degeneracy/FSS 5/22

6 1st Order FSS: Energy moments Energy moments become e n = W o e n o + (1 W o )e 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 ) e 2 Max of C max V = L d (β e/2) 2 at W o = W d = 0.5 Volume scaling Mueller, Janke, Johnston Degeneracy/FSS 6/22

7 1st Order FSS: Specific Heat peak shift Probability of being in any of the states W o q exp( βl d f o ), W d q exp( βl d f d ) Take logs, expand around β ln(w o /W d ) = ln q + βl d (f d f o ) = ln q + L d e(β β ) Solve for specific heat peak W o = W d, ln(w o /W d ) = 0 β Cmax V (L) = β ln q L d e +... Mueller, Janke, Johnston Degeneracy/FSS 7/22

8 1st Order FSS: summary Peaks grow as L d Critical points shift as 1/L d Except Fixed boundaries (1/L leading term) Z(β) = [ e β(ld f d +L d 1 f o) + qe β(ld f o+l d 1 f d ) ] [1 +...] Mueller, Janke, Johnston Degeneracy/FSS 8/22

9 A 3D Plaquette Ising model 3D cubic lattice, spins on vertices H = σ i σ j σ k σ l One parameter family of Gonihedric Ising models (Savvidy, Wegner) H κ = 2κ i,j σ i σ j + κ 2 i,j κ = 0 strong 1st order transition σ i σ j 1 κ 2 σ i σ j σ k σ l Mueller, Janke, Johnston Degeneracy/FSS 9/22

10 The Gonihedric action Savvidy Gonihedric action S = ij X µ (i) X µ (j) θ ij θ ij = π α ij X i Gonia: angle Hedra: face α ij X j Mueller, Janke, Johnston Degeneracy/FSS 10/22

11 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 Standard duality relation tanh β = e 2β Mueller, Janke, Johnston Degeneracy/FSS 11/22

12 The Problem High precision multicanonical simulation, determine critical point(s) L = , periodic bc, 1/L 3 fits - a nice exercise for a PhD student (Marco) Original model: β = (30) Dual model: βdual = (19) β = (11) Estimates are about 30 error bars apart Mueller, Janke, Johnston Degeneracy/FSS 12/22

13 The one slide about simulation methods Multicanonical histograms Mueller, Janke, Johnston Degeneracy/FSS 13/22

14 The Solution Blame the student Incorrect, try again What is special about plaquette model? Mueller, Janke, Johnston Degeneracy/FSS 14/22

15 Groundstates: Plaquette Persists into low temperature phase: degeneracy 2 3L Mueller, Janke, Johnston Degeneracy/FSS 15/22

16 Groundstates: Dual (a) (b) (c) (d) σ τ στ Dual degeneracy Mueller, Janke, Johnston Degeneracy/FSS 16/22

17 Ground state Mueller, Janke, Johnston Degeneracy/FSS 17/22

18 1st Order FSS with Exponential Degeneracy Normally q is constant Suppose instead q e L becomes β Cmax V (L) = β ln q L d e +... β Cmax V (L) = β 1 L d 1 e +... Mueller, Janke, Johnston Degeneracy/FSS 18/22

19 FSS Plaquette Hamiltonian fits Dual Hamiltonian fits β β Cmax V = (11) β Bmin = (7) β eqw = (11) p e L = 13 β eqh = (14) L β β Cmax V = (15) β Bmin = (12) β eqw = (15) p e L = 12 β eqh = (16) L 2 Mueller, Janke, Johnston Degeneracy/FSS 19/22

20 Quality of fits L max L max β C max V β +a/l 3 β +a/l L min L max L max β Bmin β +a/l 3 β +a/l L min Q Forcing a fit to 1/L 3 gives much poorer quality Mueller, Janke, Johnston Degeneracy/FSS 20/22

21 Conclusions Standard 1st order FSS: 1/L 3 corrections in 3D Fixed BC: 1/L (surface tension) Exponential degeneracy: 1/L 2 in 3D Quantum case? Mueller, Janke, Johnston Degeneracy/FSS 21/22

22 References G.K. Savvidy and F.J. Wegner, Nucl. Phys. B 413, 605 (1994). Y. Hashizume and M. Suzuki, Int. J. Mod. Phys. B 25 (2011) 73. M. Mueller, W. Janke and D. A. Johnston, Phys. Rev. Lett. 112 (2014) M. Mueller, D. A. Johnston and W. Janke, Nucl. Phys. B 888 (2014) 214; Nucl. Phys. B 894 (2015) 1. Mueller, Janke, Johnston Degeneracy/FSS 22/22