The square lattice Ising model on the rectangle

Transcription

1 The square lattice Ising model on the rectangle Fred Hucht, Theoretische Physik, Universität Duisburg-Essen, Duisburg Introduction Connection to Casimir forces Free energy contributions Analytical finite-size scaling limit Hyperbolic parametrization Exact mapping onto effective spin model Conclusions & Outlook Collaboration: Hendrik Hobrecht (see next talk), Felix M. Schmidt (UDE) AH, The square lattice Ising model on the rectangle I: finite systems, J. Phys. A: Math. Theor. 50, (207), arxiv: AH, The square lattice Ising model on the rectangle II: finite-size scaling limit, J. Phys. A: Math. Theor. 50, (207), arxiv: CompPhys 207, Leipzig

2 The 2d Ising model Exactly solved only if translationally invariant (TI) in at least one direction (yet) 2

3 The 2d Ising model Exactly solved only if translationally invariant (TI) in at least one direction (yet) Onsager (944): torus TI in both directions [] [] L. Onsager, Phys. Rev. 65, 7 (944) 2

4 The 2d Ising model Exactly solved only if translationally invariant (TI) in at least one direction (yet) Onsager (944): torus TI in both directions [] McCoy & Wu (973): cylinder TI in one direction [2] other direction homogeneous surface free energy other direction randomly distributed Griffith phase [] L. Onsager, Phys. Rev. 65, 7 (944) [2] B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model (973) 2

5 The 2d Ising model Exactly solved only if translationally invariant (TI) in at least one direction (yet) Onsager (944): torus TI in both directions [] McCoy & Wu (973): cylinder TI in one direction [2] other direction homogeneous surface free energy other direction randomly distributed Griffith phase exact results for arbitrary geometry at criticality from CFT [many] [] L. Onsager, Phys. Rev. 65, 7 (944) [2] B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model (973) 2

6 The 2d Ising model Exactly solved only if translationally invariant (TI) in at least one direction (yet) Onsager (944): torus TI in both directions [] McCoy & Wu (973): cylinder TI in one direction [2] other direction homogeneous surface free energy other direction randomly distributed Griffith phase exact results for arbitrary geometry at criticality from CFT [many] fc corner free energy fc in infinite system below [3,4] and above [5] Tc known exactly [] L. Onsager, Phys. Rev. 65, 7 (944) [2] B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model (973) [3] E.Vernier and J. L. Jacobsen, J. Phys. A: Math. Theor. 45, (202) [4] R. J. Baxter, J. Phys. A: Math. Theor. 50, 0400 (207) [5] AH, J. Phys. A: Math. Theor. 50, (207), arxiv:

7 The 2d Ising model Exactly solved only if translationally invariant (TI) in at least one direction (yet) Onsager (944): torus TI in both directions [] McCoy & Wu (973): cylinder TI in one direction [2] other direction homogeneous surface free energy other direction randomly distributed Griffith phase exact results for arbitrary geometry at criticality from CFT [many] fc corner free energy fc in infinite system below [3,4] and above [5] Tc known exactly Problem: exact solution on open rectangle not known (yet) [] L. Onsager, Phys. Rev. 65, 7 (944) [2] B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model (973) [3] E.Vernier and J. L. Jacobsen, J. Phys. A: Math. Theor. 45, (202) [4] R. J. Baxter, J. Phys. A: Math. Theor. 50, 0400 (207) [5] AH, J. Phys. A: Math. Theor. 50, (207), arxiv: ? 2

8 Inhomogeneous 2d Ising model on the cylinder partition function Z =Trexp LX MX K $`,m `,m `+,m + K l`,m `,m `,m+ `= m= `,m = ± K,M K $ L,m =0 allow all K $,l `,m different! M l`,m =0 special case K : open rectangle m Kl,m Kl,m K, σl,m K, KL-, l L 3

9 Critical Casimir forces 4

10 Critical Casimir forces (without corners yet) universal finite-size effect near criticality depends only on universality class, boundary conditions, shape t = T T c reduced residual free energy for cylinder geometry F res (t, L?,L k ) F (t, L?,L k ) Vf b (t) A f a s (t)+f b s (t) Hendrik B. G. Casimir ( ) F C a L? pbc pbc Michael E. Fisher (*93) F C A = L d k b critical Casimir force per area F C (t, L?,L k F res (t, L?,L k ) Pierre-Gilles de Gennes ( ) [8] M. E. Fisher and P.-G. de Gennes, C. R. Acad. Sci. Paris Ser. B 287, 209 (978) 5

11 Universal scaling functions both Fres and F C follow universal finite-size scaling functions F res (t, L?,L k ) ' (x, ) with scaling variables L d? F C (t, L?,L k ) ' #(x, ) x t L? + / ' t>0 L? b (t) / L? /L k 6

12 Universal scaling functions both Fres and F C with scaling variables follow universal finite-size scaling functions L d? F C (t, L?,L k ) ' #(x, ) x t F res (t, L?,L k ) ' (x, ) L? + L? /L k / ' t>0 L? b (t) / Experiment: critical thinning of 4 He films near the λ transition Garcia & Chan, Phys. Rev. Lett. 83, 87 (999) J Hx, 0L â â â â 6 x BKT x no free parameters! HaL Theory: MC simulation of XY model with open BCs AH, Phys. Rev. Lett. 99, 8530 (2007) 6

13 Open rectangle (with corners): free energy contributions & analytic finite-size scaling limit 7

14 Free energy contributions on the rectangle we find from decomposition compare with decomposition of residuals (!) F res strip(l, M) = log det( + Y) F (L, M) =Lf b,s (M)+F s,c (M)+F res strip(l, M) F (L, M) =LMf b + Lf $ s + Mf l s + f c + F res (L, M) F res (L, M) =Lf res b,s (M)+F res s,c (M)+F res strip(l, M) Transfer matrix propagation direction $ 4 f c 2 f $ s F (L, M) F strip (L, M) 4 f c f b,s (M) 2 F s,c(m) F (L, M) 2 F s,c(m) sketch of contributions M 2 f l s f b 2 f l s 4 f c 2 f $ s 4 f c L 8

15 Finite-size scaling limit scaling variables x =2 M = L M reduced temperature = z/z c scaling functions Casimir potential Casimir force F res (L, M) ' (x, ) F(L, M) ' M 2 #(x, ) our model Casimir potential F res (L, M) =Lf res b,s (M)+F res s,c (M)+F res strip(l, M) ) (x, ) = (oo) (x)+ s,c (x)+ (x, ) Casimir force #(x, &) = (oo) (x)+ (x, ) #(x,.) = # (oo) (x) x 0 s,c(x) (x, ) (x, ) (oo) (x) = Z p! x2 +! d! log + 2 x p 2 0 x2 +! 2 + x e 2p x 2 +! 2 # (oo) (x) = Z 0 d! p x 2 +! 2 + p! x2 +! 2 + x p x2 +! 2 x e2p x 2 +! 2 9

16 Universal finite-size scaling functions Casimir potential [8] (x, ) = (oo) (x)+ s,c (x)+ (x, ) log. divergence from corner free energy s,c (x) = 8 log x 3 4 at x = 0 we confirm CFT predictions, e.g., log 2 x x + reg. Θ(x,ρ) Casimir potential Θ s,c (x) 8 Ψ(x,) ρ = ρ = 2 ρ = 4 ρ = 8 ρ = 6 ρ = x (0, ) = log (i ) 0.6 Casimir force -xθ s,c (x) ψ(x,) Casimir amplitude x = 0 Casimir potential! C( ) = 4 log (i ) Casimir force changes sign with x and ρ (0, ) = ϑδ(xδ,ρ) ρ = ρ = 2 ρ = ρ = 5/6 ρ = 2/3 ρ = /2 ρ = /4 ρ = /8 ρ = /6 ρ = 0 [8] AH, J. Phys. A: Math. Theor. 50, (207), arxiv: x δ 0

17 Hyperbolic Parametrization (isotropic) FSS limit z 7! z c x 2M L 7! M ' 7! 7! M! M M of Onsager dispersion ω plane for x = 2 cosh = t + z + t z cos ' gives π ω2 ω4 π = p x characteristic polynomial π 2 ω π 2 P ( ) cos + x sin ω3 ω5 complex integrals in Φ plane: branch cuts solution: hyperbolic parametrization = x sinh!, = x cosh! 0 - π π 2 separation of odd & even zeroes!

18 Calculation of s,c (x). s,c (x) was calculated numerically in [8] using a scaling relation new result: direct calculation of regularized infinite double product M 2 log Y µ,+ µ= µ odd MY =2 even 2( µ,+,+ ) : zeroes of a certain TM characteristic polynomial double product single product s,c (x) result after hyperbolic parametrization: 2 2 Z 0 dt Z t 0 dw log(cosh 2 t cosh 2 w) S(t)S(w) cosh t cosh w second contribution simpler (single Cauchy integral) x -log 2 [8] AH, J. Phys. A: Math. Theor. 50, (207), arxiv:

19 Effective spin model effective Hamiltonian (sµ = {0,}, b ) H e = MX MX K µ s µ s + L µ< = µ= h X M ˆµs µ + b µ= µs µ i K,2 two sublattices odd/even ( µ=± ) with equal sub. magnetization long-range couplings Kµν, intra-sub FM, inter-sub AFM 4 M magnetic moments ˆµ > 0, magnetic field L 5 free energies: F res strip(l, M) =F e (L, M) critical Casimir force magnetization in magnetic field L F strip (L, M) F strip(l, res M) log Tr e H eff = m e (L) model might be used as starting point for further analysis 3

20 Conclusions & Outlook big step towards exact solution of 2d Ising model on (open) rectangle problem reduced from 4LM 4LM to M/2 M/2 FSS functions calculated analytically Θ(x,ρ) Casimir potential Θ s,c (x) 8 Ψ(x,) ρ = ρ = 2 ρ = 4 ρ = 8 ρ = 6 ρ = corner free energy log divergence of Casimir potential Casimir force changes sign 3 2 K, x hyperbolic parametrization lead to great simplifications (see also next talk by Hendrik) residual matrix can be mapped exactly onto effective spin model with M spins 4 5 M ϑδ(xδ,ρ) Casimir force -xθ s,c (x) ψ(x,) ρ = ρ = 2 ρ = ρ = 5/6 ρ = 2/3 ρ = /2 ρ = /4 ρ = /8-0.4 ρ = /6 ρ = x δ 4

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