Physics in Hyperbolic Lattices Seung Ki Baek 1 in collaboration with Beom Jun Kim 1 , Petter Minnhagen 2 , Hiroyuki Shima 3 and So Do Yi 1 1 Sungkyunkwan University, Suwon, Korea 2 Umeå University, Umeå, Sweden 3 Hokkaido University, Sapporo, Japan The 3rd KIAS Conference on Statistical Physics Nonequilibrium Statistical Physics of Complex Systems 1 July, 2008 S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 1 / 37
Outline Introduction 1 Percolation 2 Geometric XY spin 3 Diffusion 4 Summary 5 S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 2 / 37
Outline Introduction 1 Percolation 2 Geometric XY spin 3 Diffusion 4 Summary 5 S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 3 / 37
Negatively Curved Surface 0 0 -50 -50 -100 -100 -150 -150 -200 -200 10 5 -10 0 -5 0 -5 5 10 -10 100 100 50 50 0 0 -50 -50 -100 -100 10 5 -10 0 -5 0 -5 5 -10 10 Image from Thurston et al. (1984) (Upper) Positive and (below) negative curvature S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 4 / 37
Negatively Curved Surface - Continued (Left) Photo courtesy of C. Gunn (2004) ✌ APS) (Center) Image courtesy of N. Park (2003, c (Right) Hyperbolic soccerball from Wikepedia Extended indefinitely with keeping curvature cf. Positive curvature ✘ Radius � 1 S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 5 / 37
Poincaré Disk Representation Mapping of the negatively curved surface onto a unit disk Angles are preserved; Distances are not. “Circle Limit III” by Escher (1959) S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 6 / 37
Hyperbolic Tessellation Schläfli symbol: ❢ m ❀ n ❣ ❂ ❢ 7 ❀ 3 ❣ n regular m -gons meet at each vertex. Negative curvature for every ❢ m ❀ n ❣ such that ✭ m � 2 ✮✭ n � 2 ✮ ❃ 4 cf. ❢ 4 ❀ 4 ❣ : square lattices S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 7 / 37
Peculiarities of Hyperbolic Lattices Infinite dimensionality: (Average path length) ✴ log N 150 2D square lattice heptagonal lattice 120 WS network 10 2 90 10 1 ℓ ℓ 60 10 0 10 0 10 2 10 4 30 N 0 10 0 10 1 10 2 10 3 10 4 10 5 10 6 N Mean-field behaviors for Ising [Ueda et al. (2007)] and XY spins [Gendiar et al. (2008)] by imposing traslational invariance S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 8 / 37
Peculiarities of Hyperbolic Lattices Infinite dimensionality: (Average path length) ✴ log N 150 2D square lattice heptagonal lattice 120 WS network 10 2 90 10 1 ℓ ℓ 60 10 0 10 0 10 2 10 4 30 N 0 10 0 10 1 10 2 10 3 10 4 10 5 10 6 N Mean-field behaviors for Ising [Ueda et al. (2007)] and XY spins [Gendiar et al. (2008)] by imposing traslational invariance S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 8 / 37
Peculiarities of Hyperbolic Lattices - Continued Nonvanishing surface-volume ratio ■ N ✭ r ✮ ✴ e r ❂ ✮ ❅ N ❅ r ❂ N ✦ const. ■ Deviation from MF Ising [Shima et al. (2006)] ■ Zero temperature transition for XY [Baek et al. (2007)] ■ Thermodynamic limit? Intrinsic length scale: curvature “ ✿ ✿ ✿ the one thing in it which is opposed to our conceptions is that, there must exist in space a linear magnitude, determined for itself.” ■ No freedom in choosing a lattice constant without changing curvature for any given ❢ m ❀ n ❣ ■ Real-space scaling transformation prevented S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 9 / 37
Peculiarities of Hyperbolic Lattices - Continued Nonvanishing surface-volume ratio ■ N ✭ r ✮ ✴ e r ❂ ✮ ❅ N ❅ r ❂ N ✦ const. ■ Deviation from MF Ising [Shima et al. (2006)] ■ Zero temperature transition for XY [Baek et al. (2007)] ■ Thermodynamic limit? Intrinsic length scale: curvature “ ✿ ✿ ✿ the one thing in it which is opposed to our conceptions is that, there must exist in space a linear magnitude, determined for itself.” ■ No freedom in choosing a lattice constant without changing curvature for any given ❢ m ❀ n ❣ ■ Real-space scaling transformation prevented S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 9 / 37
Peculiarities of Hyperbolic Lattices - Continued Nonvanishing surface-volume ratio ■ N ✭ r ✮ ✴ e r ❂ ✮ ❅ N ❅ r ❂ N ✦ const. ■ Deviation from MF Ising [Shima et al. (2006)] ■ Zero temperature transition for XY [Baek et al. (2007)] ■ Thermodynamic limit? Intrinsic length scale: curvature “ ✿ ✿ ✿ the one thing in it which is opposed to our conceptions is that, there must exist in space a linear magnitude, determined for itself.” ■ No freedom in choosing a lattice constant without changing curvature for any given ❢ m ❀ n ❣ ■ Real-space scaling transformation prevented S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 9 / 37
Outline Introduction 1 Percolation 2 Geometric XY spin 3 Diffusion 4 Summary 5 S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 10 / 37
Percolation in Square Lattices, {4,4} http://mathworld.wolfram.com Occupation probability p ✷ ❬ 0 ❀ 1 ❪ S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 11 / 37
Scaling Exponents of Bond Percolation in ❢ 4 ❀ 4 ❣ P ✭ s ✮ : Prob. distribution 10 0 p=0.3 0.4 10 -2 of cluster sizes s 0.5 P(s) P ✭ s ✮ ✘ s � ✜ at p ❂ p c 0.6 10 -4 0.7 10 -6 10 0 10 1 10 2 10 3 s 1 Largest cluster size, s 1 , L β / ν s 1 /N L=32 divided by N s 1 /N 0.5 64 s 1 ❂ N ✘ ✭ p � p c ✮ ☞ (p-p c )L 1/ ν 128 256 512 0 0 0.2 0.4 0.6 0.8 1 p Average cluster size P ✵ L=25 s s 2 P ✭ s ✮ 50 S N - γ /d ν S ✑ P ✵ S (10 3 ) 1 100 s sP ✭ s ✮ 200 ✘ ❥ p � p c ❥ � ✌ (p-p c )N 1/d ν 0 0 0.2 0.4 0.6 0.8 1 p S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 12 / 37
Other Quantities in ❢ 4 ❀ 4 ❣ B : Number of boundary L=51 points (a) 20 101 201 b L - κ b : Number of boundary 401 -1 0 1 10 (p-p c )L 1/ ν points connected to the middle 0 (b) s 2 ❂ s 1 : Second largest 0.8 s 2 /s 1 0.6 vs. Largest -1 0 1 0.4 (p-p c )L 1/ ν 0.2 Only one threshold, p c ❂ 0 ✿ 5 0.48 0.50 0.52 p S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 13 / 37
Bond Percolation in ❢ 3 ❀ 7 ❣ p u ✘ 0 ✿ 4 1 l=6 (a) A broad peak in S 8 s 1 /N 0.5 10 P ✭ s ✮ ✘ s � ✜ ✭ p ✮ ∞ 0 for a wide range of p l=6 S (10 3 ) 1 (b) 8 0.5 10 0 0.0 0.2 0.4 0.6 0.8 1.0 p 10 0 p=0.2 0.3 0.4 P(s) 10 -3 0.5 0.6 (c) 10 -6 10 1 10 2 10 3 s S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 14 / 37
Two Thresholds p c and p u in ❢ 3 ❀ 7 ❣ 10 2 Unbounded clusters first l=5 (a) form at p ❂ p c ✘ 0 ✿ 2. 6 b 7 10 0 8 1 b ❂ B becomes finite at l=5 p=0.3 0.8 6 0.4 p ❂ p u ✘ 0 ✿ 37 7 0.6 b/B 8 0 as N ✦ ✶ . ∞ 0.4 N κ -1 0 0.2 (b) l=4 0.8 A unique unbounded 6 s 2 /s 1 0.6 8 cluster at p ❂ p u 0.4 10 (c) 0.2 0.1 0.2 0.3 0.4 0.5 p S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 15 / 37
Tree Approximation at p ✷ ✭ p c ❀ p u ✮ Approximation to a z -ary tree, ❢✶ ❀ z ✰ 1 ❣ p c ❂ 1 ❂ z and p u ❂ 1 P ✭ s ✮ ✘ s � ✜ with ✜ ✘ 2 ✰ log ✭ 1 ❂ p ✮ log ✭ zp ✮ between p c and p u : ✚ ✶ if p ✦ p c ✜ ✦ if p ✦ p u 2 S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 16 / 37
Cluster size distribution at p ✷ ✭ p u ❀ 1 ✮ A B D C O Schematic view of a hyperbolic lattice (upto the dotted line) on the Poincaré disk (solid) P ✭ s ✮ is dominated by the surface of an s -sized cluster P ✭ s ✮ ✘ exp ❬ � ✑ ✭ p ✮ ✂ ✭ surface to cut ✮❪ ✘ exp ❬ � ✑ ✭ p ✮ ✂ ✭ OB ✰ OC ✮❪ ✘ exp ❬ � ✑ ✭ p ✮ log s ❪ ✘ s � ✑ ✭ p ✮ S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 17 / 37
Notes on Percolation Two thresholds in a hyperbolic lattice ❢ m ❀ n ❣ ■ p c : Unbounded clusters begin to form. Measured by the connections to the middle point Estimated by p c ✘ 1 ❂ ✭ n � 1 ✮ ■ p u : A single cluster overwhelms the others. Measured by the ratio s 2 ❂ s 1 Estimated by p u ✘ m ❂ ✭ m ✰ n ✮ Cluster size distribution, P ✭ s ✮ ■ p ✷ ✭ 0 ❀ p c ✮ : P ✭ s ✮ ✘ exp ✭ � s ✮ log ✭ 1 ❂ p ✮ ■ p ✷ ✭ p c ❀ p u ✮ : P ✭ s ✮ ✘ s � ✜ with ✜ ✙ 2 ✰ log ❬✭ n � 1 ✮ p ❪ ■ p ✷ ✭ p u ❀ 1 ✮ : P ✭ s ✮ ✘ exp ❬ � ✑ ✭ p ✮ log s ❪ ❂ s � ✑ ✭ p ✮ ■ Thereby S peaks broadly around p u . S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 18 / 37
Notes on Percolation Two thresholds in a hyperbolic lattice ❢ m ❀ n ❣ ■ p c : Unbounded clusters begin to form. Measured by the connections to the middle point Estimated by p c ✘ 1 ❂ ✭ n � 1 ✮ ■ p u : A single cluster overwhelms the others. Measured by the ratio s 2 ❂ s 1 Estimated by p u ✘ m ❂ ✭ m ✰ n ✮ Cluster size distribution, P ✭ s ✮ ■ p ✷ ✭ 0 ❀ p c ✮ : P ✭ s ✮ ✘ exp ✭ � s ✮ log ✭ 1 ❂ p ✮ ■ p ✷ ✭ p c ❀ p u ✮ : P ✭ s ✮ ✘ s � ✜ with ✜ ✙ 2 ✰ log ❬✭ n � 1 ✮ p ❪ ■ p ✷ ✭ p u ❀ 1 ✮ : P ✭ s ✮ ✘ exp ❬ � ✑ ✭ p ✮ log s ❪ ❂ s � ✑ ✭ p ✮ ■ Thereby S peaks broadly around p u . S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 18 / 37
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