Exact solutions to plaquette Ising models with free and periodic boundaries

Transcription

1 Exact solutions to plaquette Ising models with free and periodic boundaries Marco Mueller 1 W. Janke 1 and D. A. Johnston 2 1 Institut für theoretische Physik, Universität Leipzig, Germany 2 Department of Mathematics, School of Mathematical and Computer Sciences, Heriot-Watt University, Riccarton, Edinburgh, Scotland 24th of November, 2016 (Leipzig)

2 Motivation your first Monte Carlo simulation of spin-lattices was (is) erroneous almost surely compare to enumeration, exact solutions for finite lattices Exact solutions: 1d Ising model: {free, fixed, (anti)periodic}-boundary conditions 2d Ising model: {(anti)periodic, Brascamp-Kunz,... }-boundary conditions, no solution for free boundaries

3 Spin-Bond transformation: solving the 1d Ising chain for free boundary conditions: L 1 H = i i+1, Z 1d, free = ( ) L 1 exp β i i+1 {} spin-bond transformation: i {+1, 1}, 4 3 { 1, 2,... L } {τ 1, τ 2,... τ L } where τ 1 = 1 2, τ 2 = 2 3,..., τ L 1 = L 1 L and setting τ L = L, the mapping {} {τ} with an inverse relation of the form i = τ L τ L 1 τ L 2 τ i is one-to-one partition function factorises: Z 1d, free = ( ) L 1 L 1 exp β τ i = 2 {τ} τ i =±1 exp (βτ i ) = 2 L ch(β) L τ

4 Spin-Bond transformation: solving the 1d Ising chain (again) for periodic boundary conditions: L H = i i+1, i {+1, 1} τ spin-bond transformation: { 1, 2,... L } {τ 1, τ 2,... τ L } τ 1 = 1 2, τ 2 = 2 3,..., τ L = L L+1 = L 1, with an inverse relation of the form i = 1 τ 1 τ 2 τ 3 τ i 1, mapping is two-to-one and we have the constraint L L τ i = i 2 = 1

5 Spin-Bond transformation: solving the 1d Ising chain (again), cont d partition function: Z 1d, periodic = ( ) L exp β i i+1 {} ( ) ( L L ) = 2 exp β τ i δ τ i, 1 {τ} τ

6 Spin-Bond transformation: solving the 1d Ising chain (again), cont d partition function: Z 1d, periodic = ( ) L exp β i i+1 {} ( ) ( L L ) = 2 exp β τ i δ τ i, 1 {τ} = ( ) ) L L exp β τ i (1 + τ i {τ} τ

7 Spin-Bond transformation: solving the 1d Ising chain (again), cont d partition function: Z 1d, periodic = ( ) L exp β i i+1 {} ( ) ( L L ) = 2 exp β τ i δ τ i, 1 {τ} = ( ) ) L L exp β τ i (1 + τ i {τ} L = L exp (βτ i ) + τ i exp (βτ i ) τ i =±1 τ i =±1 τ = 2 L ch(β) L [ 1 + th(β) L].

8 Spin-Bond transformation, highlights solving the 1d Ising chain free periodic τ τ last spin, L remains untransformed is transformed cause of 2 summing over L two-to-one transformation additional constraint Z 2 L ch(β) L 1 2 L ch(β) [ L 1 + th(β) L]

9 Plaquette model (in 3d) H = [i,j,k,l] i j k l particular limit of 3d model of the gonihedric string H = 2κ i j + κ i j 1 κ 2 2 i,j i,j [i,j,k,l] i j k l D. A. Johnston, A. Lipowski, and R. P. K. C. Malmini, in Rugged Free Energy Landscape,s, Vol. 736 of Lecture Notes in Physics, Berlin Springer Verlag, edited by W. Janke (2008), pp

10 Anisotropic plaquette model H aniso ({}) = J x yz J y zx J z xy Jx =Jy =0 Haniso ({}) = J z L z z=1 [ ] 2d Jx =Jy =0 Zaniso = ( exp βh {} Jx =Jy =0 aniso ) ({}) = ( Z 2d, gonihedric ) Lz

11 Two dimensional plaquette model: free boundaries in y-direction Spin-bond-transformation in y-direction, τ x,y = x,y x,y+1, with the condition τ x,ly = x,ly partition function factorises: Z 2d, gonihedric, free = exp β {} = exp β {τ} L x 1 x=1 L x 1 x=1 L y 1 y=1 L y 1 y=1 x,y x,y+1 x+1,y x+1,y+1 τ x,y τ x+1,y τ = 2 Lx ( Z 1d, Ising ) Ly 1 the factor 2 Lx comes from the L x sums over τ x,ly = x,lx = ±1 which do not appear in the exponent

12 Two dimensional plaquette model: mixed boundary conditions 2 Lx Ly (Lx 1)(Ly 1) ch (β) 2 Lx Ly ch (β) Lx (Ly 1) ( 1 + th (β) Lx ) Ly 1 τ

13 Two dimensional plaquette model: periodic boundaries Consider periodic boundary conditions in y-direction: x,ly +1 = x,1, here also in x-direction Lx +1,y = 1,y Spin-bond-transformation in y-direction, τ x,y = x,y x,y+1 is two-to-one and imposes L x constraints y τx,y = 1 Z 2d, gonihedric, periodic = 2 Lx exp β {τ} L x x=1 y=1 L y τ x,y τ x+1,y L x x=1 L y δ τ x,y, 1 y=1 the funny trick of rewriting the δ-constraints leads to complicated products we go straight to the high-temperature representation

14 Two dimensional plaquette model: periodic boundaries high-temperature representation Z 2d, gonihedric, periodic L x L y L x L y = 2 Lx exp β τ x,y τ x+1,y δ τ x,y, 1 {τ} x=1 y=1 L y = 2 Lx ch(β) Lx Ly {τ} L x y=1 x=1 x=1 y=1 ( ) 1 + th (β) τx,y τ x+1,y L x x=1 L y δ τ x,y, 1 y=1 similar to counting loops in the 2d Ising model, but simpler: only coupling in x-direction

15 Two dimensional plaquette model: periodic boundaries L y 2 Lx ch(β) Lx Ly {τ} L x y=1 x=1 ( ) 1 + th (β) τx,y τ x+1,y L x x=1 L y δ τ x,y, 1 y=1

16 Two dimensional plaquette model: periodic boundaries L y 2 Lx ch(β) Lx Ly {τ} L x y=1 x=1 ( ) 1 + th (β) τx,y τ x+1,y L x x=1 L y δ τ x,y, 1 y=1 ( ) 1 L x 2 Lx Ly ch(β) Lx Ly 2 v=0 ( Lx v ) (th(β) v + th(β) Lx v ) L y

17 Two dimensional plaquette model solving the 2d plaquette model free-free periodic-free periodic-periodic τ top line, x,ly not transformed transformed cause of 2 Lx summing over top row two-to-one additional constraint Z 2d, gonihedric, free = 2 Lx Ly (Lx 1)(Ly 1) ch (β) Z 2d, gonihedric, mixed = 2 Lx Ly ch (β) Lx (Ly 1) ( 1 + th (β) Lx ) Ly 1 ( ) 1 L x ( Lx ) Z 2d, gonihedric, periodic = 2 Lx Ly ch(β) (th(β) ) Lx Ly v + th(β) Lx v L y 2 v v=0

18 Anisotropic plaquette model (again) - fuki-nuke H fuki-nuke ({}) = J x yz J y zx

19 Three dimensional plaquette model: free boundaries in z-direction Spin-bond-transformation in z-direction τ x,y,z = x,y,z x,y,z+1 in a cuboidal L L L z, for one-to-one correspondence: equality on one plane τ x,y,lz = x,y,lz partition function factorises: H fuki-nuke ({τ}) = L L L z 1 x=1 y=1 z=1 ( τx,y,zτ x+1,y,z + τ x,y,zτ x,y+1,z ) Z fuki-nuke = {τ} exp ( βh fuki-nuke ({τ})) = 2 L2 ( Z 2d Ising ) Lz 1 the factor 2 L2 comes from the L L sums over τ x,y,lz = x,y,lz = ±1 which do not appear in the exponent free energy contributions 1 βf fuki-nuke lim L L 2 ln Z fuki-nuke = βf 2d Ising ln 2 + βf 2d Ising L z L z

20 Interlude: topology

21 Interlude: topology

22 Interlude: topology

23 Interlude: topology

24 Three dimensional plaquette model: periodic boundaries in z-direction Spin-bond-transformation in z-direction τ x,y,z = x,y,z x,y,z+1 is two-to-one and imposes L L constraints z τx,y,z = 1 L L H fuki-nuke ({τ}) = L z x=1 y=1 z=1 ( τx,y,zτ x+1,y,z + τ x,y,zτ x,y+1,z ), Z fuki-nuke = = L L 2 L2 exp ( βh fuki-nuke ({τ})) δ x=1 y=1 L z z=1 {τ} L L L z exp ( βh fuki-nuke ({τ})) 1 + {τ} x=1 y=1 z=1 τ x,y,z, 1 τ x,y,z

25 Three dimensional plaquette model: periodic boundaries in z-direction Z fuki-nuke = L L exp ( βh fuki-nuke ({τ})) 1 + {τ} x=1 y=1 L z x=1 y=1 z=1 τ x,y,z + O (ττ) = ( ) Lz L L ( ) Lz Z 2d Ising 1 + τ x,y Z2d Ising + O(ττ) assuming translational invariance in each layer (2d periodic Ising model) Z fuki-nuke = ( ) ) Lz Z 2d, Ising (1 + L 2 C Lz 1 + O (ττ) C 1 = τ 1,1 Z2d, Ising is the normalized one-point function (magnetization) O (ττ) = 1 2 ( L x 1 =1 L y 1 =1 L x 2 =1 L y 2 =1 ( τ x1,y 1 τ x2,y 2 Z2d Ising ) Lz 1 ) + O (τττ)

26 Three dimensional plaquette model: periodic boundaries in z-direction Zfuki-nuke = X exp ( β Hfuki-nuke ({τ })) 1 + Lz L X L Y X τx,y,z + O (τ τ ) x =1 y =1 z =1 {τ } = Z2d Ising Lz L X L X 1 + hτx,y iz2d Lz Ising + O(τ τ ) x =1 y =1 I assuming translational invariance in each layer (2d periodic Ising model) Zfuki-nuke = Z2d, Ising Lz 1 + L2 C1Lz + O (τ τ ) I C1 = hτ1,1 iz2d, Ising is the normalized one-point function (magnetization) I O (τ τ ) = PL PL 1 2 x 1 =1 y1 =1 PL x2 =1 PL y2 =1 hτx1,y1 τx2,y2 iz2d Lz Ising 1 + O (τ τ τ )

27 Fuki-Nuke: full-periodic ( ) Lz L L ( ) Lz Z2d Ising 1 + τ x,y Z2d Ising + O(ττ) x=1 y=1

28 Fuki-Nuke: full-periodic ( ) Lz L L ( ) Lz Z2d Ising 1 + τ x,y Z2d Ising + O(ττ) x=1 y=1 without the power L z in O(ττ) (high-temperature) susceptibility of the 2d Ising model, no closed-form expression too late, discovered in loop-matrix calculations already T. Jonsson and G. K. Savvidy, Phys. Lett. B 449 (1999) 253; T. Jonsson and G. K. Savvidy, Nucl. Phys. B 575 (2000) 661; G. K. Savvidy, J. High Energy Phys. 09 (2000) 44; G. K. Savvidy, Mod. Phys. Lett. B 29 (2015)

29 Conclusion identical spin-bond transformation can be treated explicitly for the 1d Ising and 2d plaquette models τ τ τ the 3d fuki-nuke model: explicit closed-form solution, as long as one boundary is free and 2d Ising model boundary is known the 3d fuki-nuke model: fully-periodic lattice creates sum over non-trivial n-point correlation functions the (full) 3d plaquette model: to be investigated (or maybe not)

30 Acknowledgements I Nikolay Izmailian, Roman Kotecký I George Savvidy I Loïc Turban I Wolfhard Janke, Des Johnston I CQT-group This work is, with full references (but less pictures), in press at Nucl. Phys. B, arxiv: supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre SFB/TRR 102 (project B04), the Deutsch-Französische Hochschule (DFH-UFA) through the Doctoral College L4 under Grant No. CDFA-02-07, and by the EU IRSES Network DIONICOS under Grant No