[PPT] - Physics in Hyperbolic Lattices Seung Ki Baek 1 in collaboration with PowerPoint Presentation

SLIDE 1

Physics in Hyperbolic Lattices

Seung Ki Baek1

in collaboration with

Beom Jun Kim1, Petter Minnhagen2, Hiroyuki Shima3 and So Do Yi1

1Sungkyunkwan University, Suwon, Korea 2Umeå University, Umeå, Sweden 3Hokkaido 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

SLIDE 2

Outline

1

Introduction

2

Percolation

3

Geometric XY spin

4

Diffusion

5

Summary

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 2 / 37

SLIDE 3

Outline

1

Introduction

2

Percolation

3

Geometric XY spin

4

Diffusion

5

Summary

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 3 / 37

SLIDE 4

Negatively Curved Surface

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(Upper) Positive and (below) negative curvature Image from Thurston et al. (1984)

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 4 / 37

SLIDE 5

Negatively Curved Surface - Continued

(Left) Photo courtesy of C. Gunn (2004) (Center) Image courtesy of N. Park (2003, c ✌APS) (Right) Hyperbolic soccerball from Wikepedia Extended indefinitely with keeping curvature

cf. Positive curvature ✘ Radius1

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 5 / 37

SLIDE 6

Poincaré Disk Representation

“Circle Limit III” by Escher (1959) Mapping of the negatively curved surface onto a unit disk Angles are preserved; Distances are not.

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 6 / 37

SLIDE 7

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

SLIDE 8

Peculiarities of Hyperbolic Lattices

Infinite dimensionality: (Average path length) ✴ log N

30 60 90 120 150 100 101 102 103 104 105 106 ℓ N 2D square lattice heptagonal lattice WS network 100 101 102 100 102 104 ℓ 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

SLIDE 9

Peculiarities of Hyperbolic Lattices

Infinite dimensionality: (Average path length) ✴ log N

30 60 90 120 150 100 101 102 103 104 105 106 ℓ N 2D square lattice heptagonal lattice WS network 100 101 102 100 102 104 ℓ 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

SLIDE 10

Peculiarities of Hyperbolic Lattices - Continued

Nonvanishing surface-volume ratio

■ N✭r✮ ✴ er ❂

✮ ❅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

pposed 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

SLIDE 11

Peculiarities of Hyperbolic Lattices - Continued

Nonvanishing surface-volume ratio

■ N✭r✮ ✴ er ❂

✮ ❅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

pposed 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

SLIDE 12

Peculiarities of Hyperbolic Lattices - Continued

Nonvanishing surface-volume ratio

■ N✭r✮ ✴ er ❂

✮ ❅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

pposed 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

SLIDE 13

Outline

1

Introduction

2

Percolation

3

Geometric XY spin

4

Diffusion

5

Summary

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 10 / 37

SLIDE 14

Percolation in Square Lattices, {4,4}

http://mathworld.wolfram.com Occupation probability p ✷ ❬0❀ 1❪

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 11 / 37

SLIDE 15

Scaling Exponents of Bond Percolation in ❢4❀ 4❣

10-6 10-4 10-2 100 100 101 102 103 P(s) s p=0.3 0.4 0.5 0.6 0.7 0.5 1 0.2 0.4 0.6 0.8 1 s1/N p L=32 64 128 256 512 Lβ/νs1/N (p-pc)L1/ν 1 0.2 0.4 0.6 0.8 1 S (103) p L=25 50 100 200 S N-γ/dν (p-pc)N1/dν

P✭s✮: Prob. distribution

f cluster sizes s

P✭s✮ ✘ s✜ at p ❂ pc Largest cluster size, s1, divided by N s1❂N ✘ ✭p pc✮☞ Average cluster size S ✑

P✵

s s2P✭s✮

P✵

s sP✭s✮

✘ ❥p pc❥✌

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 12 / 37

SLIDE 16

Other Quantities in ❢4❀ 4❣

10 20 b L-κ (a) L=51 101 201 401

1

1 (p-pc)L1/ν 0.2 0.4 0.6 0.8 0.48 0.50 0.52 s2/s1 p (b)

1

1 (p-pc)L1/ν

B: Number of boundary points b: Number of boundary points connected to the middle s2❂s1: Second largest

vs. Largest

Only one threshold, pc ❂ 0✿5

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 13 / 37

SLIDE 17

Bond Percolation in ❢3❀ 7❣

0.5 1 s1/N (a) l=6 8 10 ∞ 0.5 1 0.0 0.2 0.4 0.6 0.8 1.0 S (103) p (b) l=6 8 10 10-6 10-3 100 101 102 103 P(s) s (c) p=0.2 0.3 0.4 0.5 0.6

pu ✘ 0✿4 A broad peak in S P✭s✮ ✘ s✜✭p✮ for a wide range of p

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 14 / 37

SLIDE 18

Two Thresholds pc and pu in ❢3❀ 7❣

100 102 b (a) l=5 6 7 8 0.2 0.4 0.6 0.8 b/B (b) l=5 6 7 8 ∞ Nκ-1 1 p=0.3 0.4 0.2 0.4 0.6 0.8 0.1 0.2 0.3 0.4 0.5 s2/s1 p (c) l=4 6 8 10

Unbounded clusters first form at p ❂ pc ✘ 0✿2. b❂B becomes finite at p ❂ pu ✘ 0✿37 as N ✦ ✶. A unique unbounded cluster at p ❂ pu

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 15 / 37

SLIDE 19

Tree Approximation at p ✷ ✭pc❀ pu✮

Approximation to a z-ary tree, ❢✶❀ z ✰ 1❣ pc ❂ 1❂z and pu ❂ 1 P✭s✮ ✘ s✜ with ✜ ✘ 2 ✰ log✭1❂p✮

log✭zp✮ between pc and pu:

✜ ✦ ✚ ✶ if p ✦ pc 2 if p ✦ pu

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 16 / 37

SLIDE 20

Cluster size distribution at p ✷ ✭pu❀ 1✮

A C B D O Schematic view of a hyperbolic lattice (upto the dotted line)

n 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

SLIDE 21

Notes on Percolation

Two thresholds in a hyperbolic lattice ❢m❀ n❣

■ pc: Unbounded clusters begin to form.

Measured by the connections to the middle point Estimated by pc ✘ 1❂✭n 1✮

■ pu: A single cluster overwhelms the others.

Measured by the ratio s2❂s1 Estimated by pu ✘ m❂✭m ✰ n✮

Cluster size distribution, P✭s✮

■ p ✷ ✭0❀ pc✮: P✭s✮ ✘ exp✭s✮ ■ p ✷ ✭pc❀ pu✮: P✭s✮ ✘ s✜ with ✜ ✙ 2 ✰

log✭1❂p✮ log❬✭n1✮p❪

■ p ✷ ✭pu❀ 1✮: P✭s✮ ✘ exp❬✑✭p✮ log s❪ ❂ s✑✭p✮ ■ Thereby S peaks broadly around pu. S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 18 / 37

SLIDE 22

Notes on Percolation

Two thresholds in a hyperbolic lattice ❢m❀ n❣

■ pc: Unbounded clusters begin to form.

Measured by the connections to the middle point Estimated by pc ✘ 1❂✭n 1✮

■ pu: A single cluster overwhelms the others.

Measured by the ratio s2❂s1 Estimated by pu ✘ m❂✭m ✰ n✮

Cluster size distribution, P✭s✮

■ p ✷ ✭0❀ pc✮: P✭s✮ ✘ exp✭s✮ ■ p ✷ ✭pc❀ pu✮: P✭s✮ ✘ s✜ with ✜ ✙ 2 ✰

log✭1❂p✮ log❬✭n1✮p❪

■ p ✷ ✭pu❀ 1✮: P✭s✮ ✘ exp❬✑✭p✮ log s❪ ❂ s✑✭p✮ ■ Thereby S peaks broadly around pu. S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 18 / 37

SLIDE 23

Notes on Percolation

Two thresholds in a hyperbolic lattice ❢m❀ n❣

■ pc: Unbounded clusters begin to form.

Measured by the connections to the middle point Estimated by pc ✘ 1❂✭n 1✮

■ pu: A single cluster overwhelms the others.

Measured by the ratio s2❂s1 Estimated by pu ✘ m❂✭m ✰ n✮

Cluster size distribution, P✭s✮

■ p ✷ ✭0❀ pc✮: P✭s✮ ✘ exp✭s✮ ■ p ✷ ✭pc❀ pu✮: P✭s✮ ✘ s✜ with ✜ ✙ 2 ✰

log✭1❂p✮ log❬✭n1✮p❪

■ p ✷ ✭pu❀ 1✮: P✭s✮ ✘ exp❬✑✭p✮ log s❪ ❂ s✑✭p✮ ■ Thereby S peaks broadly around pu. S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 18 / 37

SLIDE 24

Notes on Percolation

Two thresholds in a hyperbolic lattice ❢m❀ n❣

■ pc: Unbounded clusters begin to form.

Measured by the connections to the middle point Estimated by pc ✘ 1❂✭n 1✮

■ pu: A single cluster overwhelms the others.

Measured by the ratio s2❂s1 Estimated by pu ✘ m❂✭m ✰ n✮

Cluster size distribution, P✭s✮

■ p ✷ ✭0❀ pc✮: P✭s✮ ✘ exp✭s✮ ■ p ✷ ✭pc❀ pu✮: P✭s✮ ✘ s✜ with ✜ ✙ 2 ✰

log✭1❂p✮ log❬✭n1✮p❪

■ p ✷ ✭pu❀ 1✮: P✭s✮ ✘ exp❬✑✭p✮ log s❪ ❂ s✑✭p✮ ■ Thereby S peaks broadly around pu. S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 18 / 37

SLIDE 25

Notes on Percolation

Two thresholds in a hyperbolic lattice ❢m❀ n❣

■ pc: Unbounded clusters begin to form.

Measured by the connections to the middle point Estimated by pc ✘ 1❂✭n 1✮

■ pu: A single cluster overwhelms the others.

Measured by the ratio s2❂s1 Estimated by pu ✘ m❂✭m ✰ n✮

Cluster size distribution, P✭s✮

■ p ✷ ✭0❀ pc✮: P✭s✮ ✘ exp✭s✮ ■ p ✷ ✭pc❀ pu✮: P✭s✮ ✘ s✜ with ✜ ✙ 2 ✰

log✭1❂p✮ log❬✭n1✮p❪

■ p ✷ ✭pu❀ 1✮: P✭s✮ ✘ exp❬✑✭p✮ log s❪ ❂ s✑✭p✮ ■ Thereby S peaks broadly around pu. S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 18 / 37

SLIDE 26

Notes on Percolation

Two thresholds in a hyperbolic lattice ❢m❀ n❣

■ pc: Unbounded clusters begin to form.

Measured by the connections to the middle point Estimated by pc ✘ 1❂✭n 1✮

■ pu: A single cluster overwhelms the others.

Measured by the ratio s2❂s1 Estimated by pu ✘ m❂✭m ✰ n✮

Cluster size distribution, P✭s✮

■ p ✷ ✭0❀ pc✮: P✭s✮ ✘ exp✭s✮ ■ p ✷ ✭pc❀ pu✮: P✭s✮ ✘ s✜ with ✜ ✙ 2 ✰

log✭1❂p✮ log❬✭n1✮p❪

■ p ✷ ✭pu❀ 1✮: P✭s✮ ✘ exp❬✑✭p✮ log s❪ ❂ s✑✭p✮ ■ Thereby S peaks broadly around pu. S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 18 / 37

SLIDE 27

Notes on Percolation

Two thresholds in a hyperbolic lattice ❢m❀ n❣

■ pc: Unbounded clusters begin to form.

Measured by the connections to the middle point Estimated by pc ✘ 1❂✭n 1✮

■ pu: A single cluster overwhelms the others.

Measured by the ratio s2❂s1 Estimated by pu ✘ m❂✭m ✰ n✮

Cluster size distribution, P✭s✮

■ p ✷ ✭0❀ pc✮: P✭s✮ ✘ exp✭s✮ ■ p ✷ ✭pc❀ pu✮: P✭s✮ ✘ s✜ with ✜ ✙ 2 ✰

log✭1❂p✮ log❬✭n1✮p❪

■ p ✷ ✭pu❀ 1✮: P✭s✮ ✘ exp❬✑✭p✮ log s❪ ❂ s✑✭p✮ ■ Thereby S peaks broadly around pu. S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 18 / 37

SLIDE 28

Notes on Percolation

Two thresholds in a hyperbolic lattice ❢m❀ n❣

■ pc: Unbounded clusters begin to form.

Measured by the connections to the middle point Estimated by pc ✘ 1❂✭n 1✮

■ pu: A single cluster overwhelms the others.

Measured by the ratio s2❂s1 Estimated by pu ✘ m❂✭m ✰ n✮

Cluster size distribution, P✭s✮

■ p ✷ ✭0❀ pc✮: P✭s✮ ✘ exp✭s✮ ■ p ✷ ✭pc❀ pu✮: P✭s✮ ✘ s✜ with ✜ ✙ 2 ✰

log✭1❂p✮ log❬✭n1✮p❪

■ p ✷ ✭pu❀ 1✮: P✭s✮ ✘ exp❬✑✭p✮ log s❪ ❂ s✑✭p✮ ■ Thereby S peaks broadly around pu. S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 18 / 37

SLIDE 29

Outline

1

Introduction

2

Percolation

3

Geometric XY spin

4

Diffusion

5

Summary

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 19 / 37

SLIDE 30

Two Kinds of Parallelism

i j (a) i j

ij

ψ (b)

Internal phase vs. Geometric angle ✁ij ❂ ✒i ✒j ✥ij H ❂ J P cos ✁ij

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 20 / 37

SLIDE 31

Geometric Frustration

C A B O (Solid) Poincaré disk (Dotted) Geodesics ⑦ SA ❦ ⑦ SB and ⑦ SA ❦ ⑦ SC, but ⑦ SB ✻❦ ⑦ SC

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 21 / 37

SLIDE 32

Spin-Glass Susceptibility

For replicas ☛ and ☞, ✤SG ❂ N ❉ q2❊ ❂ N ✜ S✭✜❀ t0✮❀ S✭✜❀ t0✮ ✑ ✷ ✻ ✹

✜

❳

t❂1

☞ ☞ ☞ ☞ ☞ ☞ 1 N

N

❳

j❂1

ei❬✣☛

j ✭t0✰t✮✣☞ j ✭t0✰t✮❪

☞ ☞ ☞ ☞ ☞ ☞

2✸

✼ ✺

1 2 3 0.1 0.2 0.3 t0 (107) T S(τ;0) MC steps t0

Equilibration over t0, Average over ✜ S✭✜❀ t0✮ ✴ ✜ after equilibrated

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 22 / 37

SLIDE 33

Exchange Monte-Carlo Results

100 200 0.1 0.2 0.3 χSG T (a) N=315 (l=4) 847 (l=5) 2240 (l=6) 5887 (l=7) 102 10-4 10-3 10-2 χSG N-1 (b) β=10 11 12 13 14 15 16

✤SG diverges at low T. ☞: inverse temperature Zero-temperature glass transition: ✤SG ✴ N at T ❂ 0 by definition

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 23 / 37

SLIDE 34

Spin Configurations

High T Low T Red: High-energy bonds Blue: Low-energy bonds

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 24 / 37

SLIDE 35

Notes on Geometric XY Spins

Interacting along geodesic lines on the space

■ cf. Magnetic order-disorder transition

f internal XY spins with Tc ❂ 0

Geometric frustration induces glassy nature without disorder.

■ Increasing spin-glass susceptibility at low T ■ Increasing equilibration time at low T

Transition temperature estimated as Tg ✙ 0

■ Spins are not completely frozen at any T ❃ 0.

Degenerate ground states

■ Domain structures S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 25 / 37

SLIDE 36

Notes on Geometric XY Spins

Interacting along geodesic lines on the space

■ cf. Magnetic order-disorder transition

f internal XY spins with Tc ❂ 0

Geometric frustration induces glassy nature without disorder.

■ Increasing spin-glass susceptibility at low T ■ Increasing equilibration time at low T

Transition temperature estimated as Tg ✙ 0

■ Spins are not completely frozen at any T ❃ 0.

Degenerate ground states

■ Domain structures S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 25 / 37

SLIDE 37

Notes on Geometric XY Spins

Interacting along geodesic lines on the space

■ cf. Magnetic order-disorder transition

f internal XY spins with Tc ❂ 0

Geometric frustration induces glassy nature without disorder.

■ Increasing spin-glass susceptibility at low T ■ Increasing equilibration time at low T

Transition temperature estimated as Tg ✙ 0

■ Spins are not completely frozen at any T ❃ 0.

Degenerate ground states

■ Domain structures S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 25 / 37

SLIDE 38

Notes on Geometric XY Spins

Interacting along geodesic lines on the space

■ cf. Magnetic order-disorder transition

f internal XY spins with Tc ❂ 0

Geometric frustration induces glassy nature without disorder.

■ Increasing spin-glass susceptibility at low T ■ Increasing equilibration time at low T

Transition temperature estimated as Tg ✙ 0

■ Spins are not completely frozen at any T ❃ 0.

Degenerate ground states

■ Domain structures S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 25 / 37

SLIDE 39

Outline

1

Introduction

2

Percolation

3

Geometric XY spin

4

Diffusion

5

Summary

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 26 / 37

SLIDE 40

Diffusion in Euclidean Spaces

Diffusion equation ❅✣ ❅t ❂ Dr2✣✭⑦ r❀ t✮ Displacement ✴ ♣ t In d ✔ 2 dimension, a diffusing particle returns back. If d ❃ 2, it has nonzero probability of escape.

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 27 / 37

SLIDE 41

Random Walk on a Heptagonal Lattice, ❢7❀ 3❣

Walk between sites = Reflection of a triangle ✭z1❀ z2❀ z3✮ ✦ ✭z1❀ z2❀ z✄

3✮ by z✄ 3 ❂ w ✰ r 2 ✖ z3✖ w with

w ❂ z1z2✭ ✖

z1 ✖ z2✮✭z1z2✮ ✖ z1z2z1 ✖ z2

and r ❂ ❥w z1❥ ❂ ❥w z2❥

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 28 / 37

SLIDE 42

Numerical Result

10 20 100 200 300 〈 dh 〉 n precision=10-3 10-5 10-7 10-10 10-12

Displacement ✴ t, rather than ♣ t

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 29 / 37

SLIDE 43

Continuum Limit

Laplace-Beltrami operator in hyperbolic polar coordinates ✹ ❂ ❅2 ❅r 2 ✰ cosh r sinh r ❅ ❅r ✰ 1 sinh2 r ❅2 ❅✒2 Solving ❅t✣ ❂ ✹✣ [Monthus et al. (1996)] ✣✭r❀ ✒❀ t✮ ✴ t3❂2et❂4 ❩ ✶ ❩ 2✙ u✭r ✵❀ ✒✵❀ 0✮I✭t❀ ✚✮ sinh r ✵d✒✵dr ✵ with ✚ ❂ distance✭✭r❀ ✒✮❀ ✭r ✵❀ ✒✵✮✮ and I✭t❀ ✚✮ ❂ ❩ ✶

✚

se s2

4t ds

♣cosh s cosh ✚

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 30 / 37

SLIDE 44

Numerical Integration

0.5 1 5 10 φ(r) sinh(r) r t=0.1 t=0.5 t=1.0 t=1.5 2 4 6 2 4 〈 r 〉 t

Probability distribution (Inset: Expected distance)

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 31 / 37

SLIDE 45

Localized Electron on Heptagonal Lattice

10 5 1 10 20 〈 l 〉 t L=4 6 8 10 12 14

Starting on the first level, the wavefunction extends to upper levels.

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 32 / 37

SLIDE 46

Notes on Diffusion

Ballistic motion

■ No recurrence ■ Convergence in direction [Kendall (1984)]

Classical diffusion is as fast as quantum one.

cf. A quantum particle can easily get back to the center [Childs et
al. (2002)]

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 33 / 37

SLIDE 47

Notes on Diffusion

Ballistic motion

■ No recurrence ■ Convergence in direction [Kendall (1984)]

Classical diffusion is as fast as quantum one.

cf. A quantum particle can easily get back to the center [Childs et
al. (2002)]

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 33 / 37

SLIDE 48

Notes on Diffusion

Ballistic motion

■ No recurrence ■ Convergence in direction [Kendall (1984)]

Classical diffusion is as fast as quantum one.

cf. A quantum particle can easily get back to the center [Childs et
al. (2002)]

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 33 / 37

SLIDE 49

Outline

1

Introduction

2

Percolation

3

Geometric XY spin

4

Diffusion

5

Summary

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 34 / 37

SLIDE 50

Summary

New physics in infinite-dimensional structures

■ Two thresholds and power-laws in percolation ■ Structural glass with geometric XY spins ■ Ballistic diffusion comparable to a quantum particle

Geometric effects

■ Presence of a boundary ■ Frustration ■ Radial drift

Further questions

■ Periodic boundary? [Sausset et al. (2007)] ■ Other systems? S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 35 / 37

SLIDE 51

Summary

New physics in infinite-dimensional structures

■ Two thresholds and power-laws in percolation ■ Structural glass with geometric XY spins ■ Ballistic diffusion comparable to a quantum particle

Geometric effects

■ Presence of a boundary ■ Frustration ■ Radial drift

Further questions

■ Periodic boundary? [Sausset et al. (2007)] ■ Other systems? S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 35 / 37

SLIDE 52

Summary

New physics in infinite-dimensional structures

■ Two thresholds and power-laws in percolation ■ Structural glass with geometric XY spins ■ Ballistic diffusion comparable to a quantum particle

Geometric effects

■ Presence of a boundary ■ Frustration ■ Radial drift

Further questions

■ Periodic boundary? [Sausset et al. (2007)] ■ Other systems? S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 35 / 37

SLIDE 53

For Further Reading I

W. P

. Thurston and J. R. Weeks

Sci. Am. 120, 108 (July 1984).
C. Gunn

Visualizing Non-euclidean Geometry Proceeedings of Bolyai Bicentennial Conference, edited by Prekopa et al., Budapest, (2004), p. 121

N. Park, M. Yoon, S. Berber, J. Ihm, E. Osawa and D. Tománek
Phys. Rev. Lett. 91, 237204 (2003).
K. Udeda, R. Krcmar, A. Gendiar and T. Nishino
J. Phys. Soc. Jpn. 76, 084004 (2007).
A. Gendiar, R. Krcmar, K. Ueda and T. Nishino
Phys. Rev. E 77, 041123 (2008).

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 36 / 37

SLIDE 54

For Further Reading II

H. Shima and Y. Sakaniwa
J. Phys. A: Math. Theor. 39, 4921 (2006).
S. K. Baek, P

. Minnhagen and B. J. Kim EPL 79, 26002 (2007).

C. Monthus and C. Texier
J. Phys. A: Math. Theor. 29, 2399 (1996).
W. S. Kendall

Séinaire de probabilités (Strasbourg) 18, 70 (1984).

A. M. Childs, E. Farhi and S. Gutmann

Quantum Information Processing 1, 35 (2002).

S.K. Baek (SKKU) Hyperbolic Lattice NSPCS08 37 / 37