Physics of a Limit-Periodic Structureby

Zongjin Qian

Department of PhysicsDuke University

Date:Approved:

Joshua Socolar, Supervisor

Henry Greenside

Stephen Teitsworth

Thesis submitted in fulfillment of the requirements for Graduation with Distinctionin the degree of Bachelor of Science in the Department of Physics

in the Undergraduate School of Duke University2013

Copyright © 2013 by Zongjin QianAll rights reserved except the rights granted by the

Creative Commons Attribution-Noncommercial Licence

Abstract

This study concerns the properties of physical systems that can spontaneously form

non-periodic structures. It focuses on the Socolar-Taylor tiling model in which a

single space-filling prototile forces a limit-periodic pattern–a state made up of the

union of infinite levels of periodic structures with ever-increasing sizes–through local

interaction rules. A two-dimensional lattice model, which possesses the Socolar-

Taylor tiling as its ground state is constructed. It is known that during a slow

quench from an initial high-temperature, disordered phase, the ground state of the

model emerges through an infinite sequence of phase transitions. As temperature is

decreased, sublattices with periodic structures of increasing lattice constants become

ordered. In this study, we construct a theory based on one-dimensional Ising model to

explain the time scales required for equilibrium to be reached at a given temperature

by sublattices of increasing lattice constants. We observe a discrepancy in the scaling

behavior predicted by our theory and obtained from simulation of the tiling model,

which is likely due to finite size effect of the tiling model. We find that during a rapid

quench, the energy barriers created by competing domain walls cause the system to

fall out of equilibrium. Two types of domain wall with different physical structures

and energy costs are found in the system. The associated energetics of each type

of domain wall is discussed, and a particular type of domain wall is identified as

responsible for slowing down the ordering of the tiling system.

iii

Contents

Abstract iii

List of Tables vi

List of Figures vii

Acknowledgements ix

1 Introduction 1

1.1 Quasicrystals: Systems With Ordered Non-Periodic Structures . . . . 3

1.2 Tiling Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Behavior in a Rapid Quench . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Questions of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Previous Findings 11

2.1 Order Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Slow Quench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Scaling Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Relaxation Time 18

3.1 Scaling Relation for Relaxation Time . . . . . . . . . . . . . . . . . . 20

3.2 tn{tn´1 by MC Simulation of Tiling Model . . . . . . . . . . . . . . . 21

3.3 Theory Prediction of tI;n{tI;n´1 . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 Theory Calculation for τ2{τ1 . . . . . . . . . . . . . . . . . . . 28

iv

3.3.2 τn{τn´1 Ratios by MC Simulation of 1D Ising Systems . . . . . 35

4 Domain Dynamics 38

4.1 Domain Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Energy Costs of Infinite Domain Walls . . . . . . . . . . . . . . . . . 45

4.3 Domain Wall Energetics . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Conclusion 58

Bibliography 60

v

List of Tables

3.1 The ratios between the relaxation constants of different levels of struc-ture of the tiling model compared with theoretical predictions. . . . . 35

vi

List of Figures

1.1 A section of a perfect rhombic Penrose tiling. . . . . . . . . . . . . . 2

1.2 The prototile and its mirror image with color matching rules. . . . . . 7

2.1 The rules for assigning staggered tetrahedral spin vector. . . . . . . . 12

2.2 Order vs. temperature during a slow quench. . . . . . . . . . . . . . . 15

2.3 Scaling collapse of Fig. 2.2 data. . . . . . . . . . . . . . . . . . . . . . 17

3.1 The initial configuration for MC simulation of level 3. . . . . . . . . . 23

3.2 Φn vs. T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Φn vs. scaled T for n “ 1, 2, 3. . . . . . . . . . . . . . . . . . . . . . . 25

3.4 A diagram illustrating Ising systems at level 3. . . . . . . . . . . . . . 27

3.5 The configurations for a level 1 Ising System. . . . . . . . . . . . . . . 28

3.6 The configurations for a level 2 Ising System. . . . . . . . . . . . . . . 30

4.1 Domain walls present after a rapid quench. . . . . . . . . . . . . . . . 40

4.2 Configurations of sitting at T “ 0.6 for 69 ˆ 104 mcs. . . . . . . . . . 42

4.3 The first type of domain wall. . . . . . . . . . . . . . . . . . . . . . . 43

4.4 The second type of domain wall. . . . . . . . . . . . . . . . . . . . . . 44

4.5 Number of black stripe mismatches vs y for horizontal domain walls. 46

4.6 Number of color stripe mismatches vs y for horizontal domain walls. . 46

4.7 A horizontal domain wall of type II spanning a 64 ˆ 64 lattice withminimum mismatches. . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.8 Portion of a horizontal domain wall of type II with minimummismatches. 48

vii

4.9 Number of black stripe mismatches vs x for slanted domain walls. . . 49

4.10 Number of color stripe mismatches vs x for slanted domain walls. . . 50

4.11 A slanted domain wall of type I spanning a 64 ˆ 64 lattice with lownumber of mismatches. . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.12 Portion of a slanted domain wall of type I with low number of mis-matches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.13 A configuration that gets stuck in ordering due the presence of aslanted domain wall of type I. . . . . . . . . . . . . . . . . . . . . . . 53

4.14 Portion of a slanted domain wall of type I from the configuration inFig. 4.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.15 Portion of a slanted domain wall of type I with no mismatches in blackstripe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.16 Configurations of a sudden quench to T “ 0.6 and sitting for 5 ˆ 105

mcs followed by sitting at T “ 1.0 for 5 ˆ 104 mcs. . . . . . . . . . . 56

viii

Acknowledgements

I would like to acknowledge and extend my heartfelt gratitude to the following per-

sons who have made the completion of this thesis possible. My deepest gratitude goes

to Dr. Joshua Socolar, for giving me the opportunity to do research with him, and for

being abundantly helpful and offering invaluable assistance, support and guidance

along the way. Special thanks to Dr. Henry Greenside, for being a wonderful DUS,

and for his vital encouragement and support throughout my undergraduate study.

I would like to convey thanks to the department of physics for providing excellent

computing facilities, and to all faculty members and staff who extended help and

inspiration in one way or another. I wish to express my love and gratitude to my

beloved families and friends, for their understanding and endless love, through the

duration of my studies.

This work has been funded by a summer research fellowship from Triangle Materials

Research Science and Engineering Center (MRSEC).

ix

1

Introduction

This thesis addresses the properties of materials that can spontaneously form or-

dered non-periodic structures. Such structures are realized in several real physical

systems, most notably, in quasicrystals. Quasicrystals, short for quasiperiodic crys-

tals, are very different from the usual crystals as their structures can exhibit certain

symmetries such as the five-fold symmetry which are strictly forbidden in normal

crystalline structures. These unusual symmetries explain many of quasicrystals’ elec-

tronic, elastic and frictional properties. Since the date of their discovery, hundreds of

quasicrystalline samples have been made in laboratories, and technologically interest-

ing applications such as quasicrystalline coating on non-sticking, corrosion-resistant

cookware have been made. Today the structures of quasicrystals, their physical prop-

erties, and their spontaneous formation remain subjects of intense study.

In this thesis, I study the spontaneous formation of ordered non-periodic structures

by focusing on a recently discovered tiling model [1]. The idea of thinking ordered

non-periodic structures such as those displayed in quasicrystals as tiling models is

not novel. In fact, when the theoretical concept of quasicrystals was first conceived,

1

the initial inspiration was the two-dimensional Penrose tiling model [2] shown in

Fig. 1.1. Penrose had identified a pair of tiles that can fill the two-dimensional space

non-periodically, forming a self-similar patter full of five-fold symmetries. Several

theorists independently proposed the idea that the structures exhibited in Penrose

tiling may have analogies in solids [3–5].

Figure 1.1: A section of a perfect rhombic Penrose tiling. [6]

In this thesis, we focus on the Socolar-Taylor tiling which presents a single space-

filling prototile that forces a limit-periodic pattern (a state made up of the union of

infinite levels of periodic structures with ever-increasing sizes [7]) through local in-

teraction rules. This model is the first known example that a single two-dimensional

prototile can fill the entire two-dimensional Euclidean space with no overlap to form

a non-periodic pattern. Because of its simplicity, it has the significance of providing

a template for materials that can self-assemble into non-periodic structures using

just a single building block. It has been suggested that it may be possible for solid

state materials, colloidal systems, and collections of macroscopic units to achieve

structures displayed in the tiling model [8]. The allowed global tiling in the model

2

can be mapped to the states of a statistical mechanics model. Studying the ordering

dynamics of such a model in forming the limit-periodic phase would allow us to gain

insights into the unusual and remarkable properties of the tiling structures if they

were realized in real physical systems.

1.1 Quasicrystals: Systems With Ordered Non-Periodic Structures

For centuries, rigorous mathematical theorems dictated that crystals are ordered

solids with periodic structures that can be characterized by long-range translational

and rotational symmetries. It was commonly thought that solids come in two forms,

ordered and disordered. Ordered implies periodicity, while disordered solids describe

amorphous materials such as glass. It was well established by the classic theorems

of Schoenflies and Fyodorov proven in the 19th century that five-, seven- and any

higher-fold symmetry in two dimensions and icosahedral symmetry in three dimen-

sions are incompatible with periodicity and thus forbidden in crystals. Thus it is

not surprising that when Dan Shechtman et al. [9] announced the discovery of an

aluminum–manganese alloy with the forbidden icosahedral symmetry–the most fa-

mous forbidden symmetry in crystals–in 1984, the overall reaction of the scientific

community was strong skepticism. In a remarkable coincidence, the theoretical con-

cept of an ordered phase of matter that escapes periodicity and possesses long-range

icosahedral symmetry was proposed around the same time by Dov Levine and Paul

Steinhardt [5]. As an example, a three-dimensional quasicrystal can be constructed

using polyhedral units analogous to the Penrose tiles. The new hypothetical ordered

phase of matter was named quasicrystal, short for quasiperiodic crystal. The newly

proposed concept of quasicrystal accounts for the diffraction pattern of the type that

Shechtman observed. Furthermore, quasiperiodicity opens up an infinite possibility

of ordered solids with forbidden symmetries to crystals.

3

However the idea of a quasicrystal as a real physical phase of matter was not read-

ily accepted initially. First, X-ray studies [10] revealed that the diffraction peaks of

Shechtman’s icosahedral aluminum–manganese phase are not truly point-like, as pre-

dicted by the quasicrystal theory. Instead the diffraction peaks have experimentally

resolved finite widths. In addition, Shechtman’s icosahedral aluminum–manganese

phase was very unstable. Annealing could not sharpen the diffraction peaks and leads

to crystallization. Most critics also thought that the quasicrystal phase is physically

impossible. It was argued that in a quasiperiodic structure, no two atoms or clusters

could occupy identical positions, so they cannot self-organize into a perfect quasicrys-

talline structure. However in 1987, the synthesis of the first bona fide quasicrystal

sample in laboratory by Tsai et al [11] firmly established quasicrystal as a real phys-

ical phase of matter. Tsai et al. discovered a quasicrystal phase (Al63Cu24Fe13)

that exhibits sharp diffraction peaks and long-range icosahedral symmetry. Later

on, hundreds of high-quality quasicrystal samples with different forbidden symme-

tries have been identified.

Until very recently, all known samples of quasicrystals have been synthetic. It has

long remained a question of debate that whether quasicrystals could be metastable

states of matter and many contend that quasicrystals are too complicated to be sta-

ble and that all ground states of matter are crystalline. However, Luca Bindi and

Paul Steinhardt [12] recently found a natural quasicrystal in a meteorite whose com-

position is measured to be Al63Cu24Fe13, the same composition as the first bona fide

quasicrystal sample discovered by Tsai et al. The discovery of a natural quasicrys-

tal gives strong support for the argument that quasicrystals can be as energetically

stable as crystals, and can be formed under natural conditions, as predicted by the

original quasicrystal theory.

4

1.2 Tiling Model

The theory of tiling concerns using non-overlapping copies of a set of tiles to cover

the entire space of the Euclidean plane or some other geometric settings. It is not

hard to find a set of tiles that admits a periodic pattern characterized by transla-

tional and rotational symmetry. It is also not hard to find a set of tiles that admits

a nonperiodic pattern, but often the tiles employed could also be used to construct

periodic patterns. However, it is much more difficult to find a set of tile-types or

“prototiles” that can only be used to construct a nonperiodic pattern. Such sets are

called “aperiodic”. There are few known examples of aperiodic tilings, perhaps the

most famous one is the Penrose tiling [2]. In 2010, Joshua Socolar and Joan Tay-

lor [1] discovered a single prototile that covers the entire Euclidean plane only in a

nonperiodic way with a set of matching conditions. This is the first known aperiodic

tiling set with matching conditions made of just a single prototile in two-dimensional

Euclidean space. Historically, the discovery of metallic alloys which shares the essen-

tial structure of the Penrose tiling [9,11] proved that ordered nonperiodic structures

could form spontaneously. In material physics, the tiles may represent large building

blocks or clusters of atoms, and the matching rules which determine how tiles fit

together may represent the energetics of a physical system [7]. Hence a key moti-

vation for Socolar and Taylor’s search for a single prototile is the hope that their

tiling model could help to predict the properties of materials that can self-assemble

into ordered nonperiodic structures using just the same cluster of atoms or a uniform

building block.

The Socolar-Taylor tiling model presents a single space-filling prototile which is a

regular hexagon decorated with markings. A version of the prototile and its mirror

5

image is shown in Fig. 1.2(a). Each tile has a set of black stripe and colored stripe

decorations, and can be placed in one of its twelve possible orientations obtainable

by rotations of π{3 and reflection. Notice that the set of colored stripes are the set

of black stripes rotated by π{2 and scaled by a factor of?3. The model specifies a

set of rules to determine how nearest and next-nearest tiles should be oriented based

on the markings on the tiles. The rules for placing the tiles are as follows:

1. Nearest tiles must form continuous black stripes;

2. Next-nearest tiles that sit at the opposite ends of a tile edge must form con-

tinuous colored stripes.

The rules are illustrated in Fig. 1.2(b). A Hamiltonian can be defined for the tiling

model based on the“matching rules” for placing the adjacent tiles and next-nearest

neighbor tiles on a close-packed lattice. For the nearest neighbor pairs, energy zero

is assigned if the black stripes are continuous across the boundary of the tiles and

energy �1 otherwise. For the next-nearest neighbor pairs, energy zero is assigned if

the colored stripes at the closest corners are connected and energy �2 otherwise. It

has been shown that the ground state is a zero-energy structure displaying an infinite

hierarchy of triangular lattices with ever increasing lattice constant [8]. A portion of

the infinite ground state is shown in Fig 1.2(c). Each level of periodic sublattice is

scaled by a factor of 2 compared to its previous level. The ground state is however

non-periodic as there is no single sublattice with the largest lattice constant. The

non-periodicity is a direct consequence of the local “matching rules”. The rules en-

forces a simple periodic lattice at the smallest scale using a subset of the markings

of the tiles. The rules then effectively operate in the same manner at larger scales

to generate the same periodic lattices with arbitrarily large lattice constants [8].

A closer look at the infinite ground state of the model reveals that the black triangular

6

(a)

(b)

(c)

Figure 1.2: The prototile and its mirror image with color matching rules. (a) Thetwo tiles are reflected about a vertical axis. (b) For zero energy, the nearest neighbortiles must form continuous black stripes and the next-nearest neighbor tiles mustform continuous color stripes. The arrow indicates where the color stripes from twonext-nearest neighbor tiles are joined to form a continuous stripe. (c) A portion ofthe infinite zero-energy ground state.

sublattices of increasing lattice constants play a key role in explaining the behavior

of the tiling model. The sublattice made of the smallest triangles shall be referred as

the level 1 structure from now on, the sublattice made of the next smallest triangles

as the level 2 structure, etc. The edge of a black triangle at level n crosses kn ´ 1

tiles, where kn ” 2n´1. At each level, the black triangles form a periodic pattern in

which the lines connecting the centers of all the triangles form a honeycomb lattice.

Further, each sublattice is an exact replica of the previous one, scaled by a factor by 2.

7

1.3 Behavior in a Rapid Quench

A quenching procedure is a process in which one starts a system in an initial disor-

dered, high-temperature phase, and lower the temperature to monitor how the system

evolves toward the final state. For the tiling model we study, a quenching procedure

is very useful in investigating how the ordered limit-periodic ground state is formed

dynamically. Some preliminary evidence has been found that a rapid quench of the

tiling system leads to disordered states [8]. It has been shown that when the tiling

models orders during a quench, there is a hierarchy of ordering for sublattices of

increasing lattice constants, i.e. as the temperature is lowered, the level 1 structure

orders before the level 2 structure, which orders before the level 3 structure, etc.

However, if the quenching rate is too high, the level 1 could not become reasonably

well formed before the level 2 structure starts to order. As a result neither the level

1 nor the level 2 structure would become well ordered. Ref. [8] suggests that a rapid

quench would lead the system to some disordered final states due to the competition

between two or more levels of structures. The kinetic barriers created by a level of

structure when it fails to order would cause simultaneous frustration of the ordering

of the subsequent levels of structure. However, the reason that the system falls out of

equilibrium and ends up in some disordered phase is far from clear from any previous

findings.

1.4 Questions of Study

Previous work [8] has focused on the study of the equilibrium properties of the tiling

model, where the goal is to characterize the ordered phases that arise below the

critical temperatures in the tiling system. The usefulness of studying the tiling model

in the equilibrium framework is limited in understanding the physical properties

of the limit-periodic structures exhibited in the tiling model, especially if one is

8

interested in finding in nature or synthesizing materials that can spontaneously form

limit-periodic structures. A natural question to ask is how slowly one has to lower the

temperature of the system during a quench in a order to have all levels of structure

fully ordered so that the limit-periodic ground state could be reached? This is the

first question addressed in this thesis. The answer to this question tells the minimum

time scale required to achieve a certain number of perfectly ordered levels of structure

in the tiling model. We also know that a rapid quench results in disordered phases.

However, the origin of such disordered states are poorly understood. A second set

of questions addressed by this thesis are: what causes the system to fall out of

equilibrium if the temperature is lowered too fast? Once the system gets into a

disordered phase, how high is the energy barrier to get out and go to an equilibrium

state. The answer to these questions give insights on the nonequilibrium dynamics

of the tiling model.

1.5 Outline

An outline of the remainder of this chapter is as follows. Chapter 2 discusses in

detail some important previous findings on the properties of the tiling model. The

definition of an order parameter to quantify the ordering of the system is presented.

The behavior of the system in a slow quench is discussed and a scaling law that

maps the behavior of the tiling system from one level of sublattice to another is

introduced. In Chapter 3, we construct a theory based on one-dimensional Ising

model to find a scaling relationship on the time scales required for equilibrium to be

reached at a given temperature by sublattices of increasing lattice constants. The

scaling relationship is found through numerical simulation of the tiling model and

predicted using a theoretical model. Results suggest that there is a discrepancy be-

tween our theoretical prediction and the actual time scales. Chapter 4 discusses the

nonequilibrium dynamics of the tiling model. The formation of competing domains is

9

identified as the cause for the system’s ordering getting frozen during a rapid quench.

The two types of domain walls present in the system are examined. This is followed

by analyzing the domain wall energetics associated with each type of domain wall.

Finally, we identify a special type of domain wall as responsible for slowing down the

system’s recovery from a disordered state to an equilibrium state.

10

2

Previous Findings

2.1 Order Parameter

Defining a proper order parameter is an important first step towards understanding

the intricate equilibrium and dynamical properties of the tiling model. As we lower

the temperature of the tiling system slowly, the level 1 triangles form following a

second-order phase transition. As the temperature of the tiling system is further

lowered, subsequent levels of structure form following a hierarchy of phase transi-

tions. A family of vector order parameters Φn which characterizes the ordering of

the tiling system has been found [8]. There is one order parameter associated with

each level of structure. The level n order parameter saturates to 1 when the level

n triangles are perfectly formed. The definition of an order parameter for the tiling

system is described here.

For the level 1 structure, each tile can be assigned to one of four different sublattices,

denoted as A, B, C, D as shown in Fig. 2.1. Notice that a quarter of the tiles did not

contribute to the formation of black triangles in the level 1 structure. Each of the

11

four different sublattices could constitute the non-contributing sublattice at level 1.

Fig. 2.1 shows an example in which the tiles sitting on sublattice A do not contribute

to the the level 1 black triangles. At level 2, if the level 1 structure is perfectly or-

dered, the corners of the level 2 black triangles must come from the non-contributing

sublattice of level 1. The system size for level 2 is thus effectively 1{4 of that of level

1. Note that the level 1 non-contributing sublattice is an exact copy of the original

lattice only that the distance between tiles are scaled by a factor of 2. At level 2,

the four sublattices can again be assigned for the level 2 tiles. A non-contributing

sublattice can also be defined and it gives the tiles to form the level 3 structure. Such

construction can be repeated ad infinitum so that the tiles of the level n structure

come from the non-contributing sublattice of level n ´ 1.

Figure 2.1: The pattern of the level 1 triangles formed when the non-contributingtiles are from those of the A sublattice and the rules for assigning spin vectors fortiles of different sublattices. For explanation of the dashed lines and grey bar, seetext. Adapted with permission from “Hierarchical Freezing in a Lattice Model” byT. W. Byington and J. E. S. Socolar, Physical Review Letters, vol. 108 (2012), pp.045701.

It has been shown the an order parameter can be explicitly defined for each level of

structure [8] by associating a “staggered tetrahedral spin” vector with each tile. The

rules for assigning the spin vector for each tile is illustrated by Fig. 2.1. See also

12

Ref. [8] for details. Each tile j at level n is assigned a spin vector σjn “ �eX , where

�eX is a unit vector pointing in one of the four corners of a reference tetrahedron

labeled “X”. The spin for each tile is determined by both its sublattice and the

orientation of the line connecting the two black triangle corners. Specifying the spin

vector for a tile does not completely determined the orientation of the tile. There

are four compatible orientations associated with a spin vector as shown in Fig. 2.1.

For a tile that has spin vector �eA, there are two possible locations of its long black

stripe as indicated by the two dashed grey lines, and there are two possible locations

of its long colored stripe as indicated by the grey diagonal arrow. When a perfectly

ordered level 1 structure has sublattice A as its non-contributing sublattice, all the

tiles on sublattice B, C, and D have spin vector �eA while the tiles on sublattice A

does not contribute to the ordering so they could have any of the other three spin

vectors. The situation is exactly the same for any subsequent level of structure. The

average spin for level n is simply �σ�n “ 1Nn

řj σ

jn, where Nn is the number of tiles

at level n and is determined by the relation Nn “ Nn´1{4 as discussed before. The

order parameter, Φn for level n can now be defined very easily. For each level of

structure, It is given by the maximum of the projection of the average spin along the

four tetrahedral spin vectors: Φn ” maxX

r�eX ¨ �σ�ns. The order parameter Φn goes to

its maximum value 1 when level n triangles become perfectly ordered. The tiles on

the non-contributing sublattice X could be in any orientations.

2.2 Slow Quench

The study of how the tiling system evolves from an initial random configuration at

finite temperatures reveals some remarkable properties of the model. Long range or-

der emerges in the tiling model during a slow quench from high to low temperatures

13

as a consequence of the local interactions of the tiles. This is an example of the phe-

nomenon known as “Emergence”, the hallmark of many self-organizing systems [13].

In many complex systems, there are multiple agents interacting dynamically in many

ways, following local updating rules and paying little attention to any higher level

instructions. In our tiling model, the nearest pairs and next-nearest pairs of tiles

play the role of the dynamically interacting agents. The systems become “emergent”

when the local interactions results in discernible macroscopic behaviors. As Murray

Gell-Mann once said: surface complexity arising out of deep simplicity.

Ref. [8] shows that during a slow quench from a high temperature, disordered phase,

the limit-periodic ground state emerges through an infinite sequence of second-order

phase transitions. Further, there is an hierarchy of phase transitions. As temperature

is decreased, sublattices with periodic structures of increasing lattice constant become

ordered. These results are reproduced here in this thesis as a confirmation and are

used to illustrate the physical properties of the tiling model. In our study, we look at

the special case �1 “ �2 “ � “ 1. Fig. 2.2 shows the behavior of Φn for n “ 1, 2, 3, 4 for

a slow quench. The simulations are done on a 64 ˆ 64 rhombus lattice with periodic

boundary conditions using the standard Metropolis algorithm. In each Monte Carlo

Step (mcs), each tile in the lattice has an equal probability of being chosen and N

random tile selections are made, where N “ 64 ˆ 64 is the total number of tiles

in the rhombus domain. For each tile selection, one of the twelve possible states is

randomly chosen as a possible move. The tile is then changed into the proposed state

according to the distribution:

ωiÑj “ min�1, e´∆EiÑj{T(

(2.1)

If the chosen move lowers the energy of the system, it is accepted with certainty,

14

otherwise, the move is accepted with a probability of e´∆EiÑj{T , where ∆EiÑj is the

change in energy of the proposed flip. The temperature T is lowered by δT “ 0.01

after every r “ 1.2 ˆ 106 mcs, where r controls the quenching rate. Fig. 2.2 shows a

clear sequence of phase transitions during a slow quench. Each subsequent level of

structure becomes well ordered only if its previous level is perfectly ordered.

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���������������������������������������������������������������������������������������������������������������������������������������������������������������

��

�

��

�������������������������������������

� n�1� n�2� n�3� n�4

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

Temperature T�Ε�kB�

�n

Figure 2.2: Behavior of the order parameter Φn vs. the temperature T during aslow quench. Simulations were performed on a 64ˆ64 rhombic domain for �1 “ �2 “ 1and r “ 1.2 ˆ 106.

2.3 Scaling Theory

The tiling model follows a hierarchy of phase transitions to become fully ordered

during a slow quench. For the level n structure to become fully ordered, it requires

the level n ´ 1 structure to be almost perfectly ordered before level n transition

occurs. A consequence is that assuming level n ´ 1 is perfectly ordered, the phase

15

transition of the level n is identical to that of the level n ´ 1 up to rescaling of the

temperature. Ref. [8] finds a relation between Tn and T1 such that the behavior of

the system at level n at temperature Tn is identical to that at level 1 at temperature

T1. It is given as:

tanh

ˆ�

2T1

˙“

„tanh

ˆ�

2Tn

˙kn

(2.2)

where kn ” 2n´1, as defined before. Or, equivalently, for all n,

tanh

ˆ�

2Tn

˙“

„tanh

ˆ�

2Tn`1

˙2

(2.3)

One immediate usefulness of this formula is that one can deduce the level n phase

transition temperature Tc;n for any n, given the level 1 transition temperature Tc;1.

As shown by Fig. 2.3, an excellent data collapse for n “ 1, 2, 3, 4 is obtained using

Eq.(2.2) by plotting ΦnpTnq as a function of T1pTnq. Note the system size for the

level 4 structure is only 8 ˆ 8, and the deviation of the level 4 data points for high

temperatures is due to the finite size effect. The deviated points represent projec-

tions of onto different tetrahedral spin vectors.

16

�������������������������������������������������

�

�

����������������������

�������������������������������������������������������������������������������������������������������������������������������

����������������

�

�������

����������������������������������������������������������������������

������

�

�

����

����������������������������������������������

��

�

��

������������������������������������ � n�1

� n�2� n�3� n�4

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

Temperature T�Ε�kB�

�n

Figure 2.3: Behavior of the order parameter Φn as a function of the rescaledtemperature T1pTnq during a slow quench. The deviation of the level 4 data pointsis due to the finite size of the system.

17

3

Relaxation Time

This chapter addresses the first question raised at the end of Sec. 1.4. The question

asks how slowly the temperature should be lowered during a slow quench in order for

the system to reach the limit-periodic ground state? It is a fact that the system goes

through a sequence of phase transitions before it reaches the ground state. Further,

the order with which the phase transitions happen follow a strict hierarchy. The level

n ´ 1 structures must be sufficiently ordered before the level n structures starts to

order in order for both levels to reach equilibrium. Based on these facts, a quenching

procedure that guarantees the well ordering of all levels of structure so that the

ground state emerges as the final state is as follows:

1. Start the system at a high temperature T ą Tc;1, where the tiles are completely

disordered; here Tc;1 is the transition temperature for the level 1 structure.

2. Instantaneously cool the system to a lower temperature T1 satisfying Tc;2 ăT1 ă Tc;1; the level 1 structure starts to equilibrate but the level 2 structure

remains disordered as T1 is above the level 2 transition temperature, Tc;2.

3. Wait enough time until the level 1 structure becomes fully ordered; At this

18

point, the level 2 structure remains completely disorientated.

4. Instantaneously cool the system to a lower temperature T “ T2pT1q, whichis the rescaled temperature of T1 given by the Rescaling Theory as stated

in Eq.(2.2). T2 satisfies Tc;3 ă T2 ă Tc;2 as a consequence of the Rescaling

Theory; the level 2 structure starts to equilibrate but the level 3 structure

remains disordered.

5. Repeat step 2 and 3 ad infinitum, each time instantaneously cool the system

to the temperature T “ TnpT1q, where Tn satisfies Tc;n`1 ă Tn ă Tc;n. wait for

the level n structure to reach equilibrium before lowering the temperature to

T “ Tn`1pT1).

Note Tn satisfying Tc;n`1 ă Tn ă Tc;n is a direct consequence of the Rescaling Theory

which says the physics of the level n structure at the T1 rescaled temperature, Tn is

exactly the same as that of the level 1 structure at T1; Since T1 satisfies Tc;2 ă T1 ăTc;1, Tn too should be in between the transition temperature of its corresponding

level of structure and the level above. The above procedure would produce a final

state of the limit-periodic phase. Given an infinite system, the waiting time required

to generate the desired final state would be infinite since there is no largest sublattice

in the tiling model which means the procedure has to be carried on forever. However

it is still meaningful to ask how long one has to wait to have a certain number,

say k levels of structure fully ordered. The minimum waiting time, tmin is given by

tmin “ řkn“1 tn, where tn is the minimum time required for level n structure to reach

equilibrium at temperature Tn, given that all previous n ´ 1 levels have been fully

ordered following our quenching procedure. Now the question we originally asked

could be sharpened as the following: What is the minimum time scale required

for levels of structure of increasing lattice constants to reach equilibrium at the

19

temperatures tTnukn“1? Answering this question effectively breaks the total dynamics

of the tiling model into just the individual dynamics of the different levels of structure.

3.1 Scaling Relation for Relaxation Time

To answer the above question, the key is the sublattice displayed in each level of

structure. Each level is made of the black and color triangles. The black triangles

in the level n structure has an edge that crosses kn ´ 1 tiles, where kn ” 2n´1 as

discussed in Sec. 1.2. The set of color triangles associated with the level n struc-

ture is just the set of black triangles rotated by π{2 and scaled by a factor of?3.

Referring to the configuration shown in Fig. 1.2, the level n structure is bound to

become fully ordered once the corners of the level n black triangles are locked in. At

that stage, the middle tiles on an edge of the black triangle can be any of the four

possible orientations corresponding to two locations of the long black stripe and two

locations of the long colored stripe. Their role is to mediate the interactions until

one corner of the triangle develops a correlation with the other corner. Once this

correlation is developed, the level n structure will always come to equilibrium in a

short time scale compared to the time required to develop correlation between the

corners of the level n triangles (The system does not exhibit hysteresis [8]).

If every level m ă n is perfectly ordered, it is expected that the level n dynamics is

identical to the level 1 dynamics in every aspect except the time required for correla-

tion to develop between the corners of the black triangles. The correlation time for a

certain level of structure is then determined by the edge length of its black triangles.

At level n, each edge of the triangle crosses kn ´1 tiles, with black stripes joining the

nearest neighbor tiles. There are altogether kn possible mismatches in black stripes

in each edge, hence kn nearest neighbor bonds between two corner tiles. Each edge

can effectively be treated as a 1D Ising system of length kn ` 1. Given that every

20

level m ă n is perfectly locked in, the minimum time tn required for equilibrium to

be reached by the level n structure at Tn is essentially determined by the time it

takes for two end spins in the Ising system to correlate to each other. As the length

of the Ising system scales with the number of level of structure, a scaling relation

is expected to exist for the correlation time of end spins. Determining the scaling

relation for the correlation time of Ising systems of increasing length would then

provide us with a scaling relation for the sequence of minimum time, ttnukn“1.

More precisely, if the size of each level of structure, i.e. the number of black triangles,

is controlled to be the same, then the only difference between the different levels of

structure is really just the edge length of the black triangles. We make the following

hypothesis: given that the system sizes for both levels are the same and all lower levels

of structure are perfectly ordered, the ratio between the minimum time required for

level n and n´1 to reach equilibrium, tn{tn´1, is determined by the ratio between the

time it takes to reach equilibrium by the 1D Ising systems corresponding to the two

levels of structure, tI;n{tI;n´1. To investigate the truth of this hypothesis, we carry

out theory calculations and numerical simulations for the Ising systems to obtain the

ratio, tI;n{tI;n´1, and compare it with the ratio, tn{tn´1 obtained from the simulation

using the tiling model.

3.2 tn{tn´1 by MC Simulation of Tiling Model

To test our hypothesis about the correspondence between the scaling behavior of the

different levels of tiling structure and that of 1D Ising systems, we first carry out

Monte Carlo simulation of the tiling model using the quenching procedure discussed

at the beginning of Chapter 3. The procedure guarantees the well ordering of all

levels of structure of the tiling model so that the limit-periodic ground state emerges

in the end. It converts the dynamics of the tiling model at each temperature in tTnu

21

to just the dynamics of the level n structure as all previous n ´ 1 are already locked

in when the level n structure starts to order. Monte Carlo Simulation of the tiling

model following the quenching procedure is used to find the ratio, tn{tn´1, between

the minimum time required for level n and n ´ 1 to reach equilibrium at the tem-

perature Tn and Tn´1 respectively. We define the number of mcs run as T . For

sufficiently large T , the order parameter, Φn for the level n structure approaches its

equilibrium value exponentially. The relaxation constant τn could be extracted from

the curve of Φn as a function of T . As the time scale of the transient behavior of the

curve for ΦnpT q is very small compared to the time scaled required for Φn to reach

its equilibrium value exponentially, we could neglect the initial part of the curve for

ΦnpT q where the transient behavior occurs. Hence taking the ratio τn{τn´1 gives a

satisfactory estimate of the ratio tn{tn´1.

According to the quenching procedure, the method of simulation of the tiling model

is as follows: Let T1 “ 1.0. For the level n structure, level 1 through level n ´ 1

structures are set to be perfectly ordered, while the level n structure is set to be

completely random; The tiling system is then equilibrated at the temperature Tn,

the rescaled temperature of T1 given by the Rescaling Law in Eq. (2.2).

A snapshot of a typical starting configuration for the level 3 structure is showcased in

Fig. 3.1. Note that the level 1 and 2 structures are perfectly in place. It is intended

that the level 3 structure is initially random, however in order to lock in the level

1 and 2 structures, the long black stripes of the middle tiles of level 3 are initially

always parallel to the potential edges of the level 3 black triangles. During the MC

simulation of the level 3’s relaxation to equilibrium, the level 3 middle tiles would go

to orientations such that the long black stripes are nonparallel to the potential edges

of the level 3 black triangles with negligible probability due to high energy cost of

22

those moves. The only allowed positions of the long black stripes are as illustrated

in Fig. 2.1 by the dashed grey lines, corresponding to the states |Ó� and |Ò� in our

analogy of 1D Ising systems.

Figure 3.1: A portion of the initial configuration used for the MC simulation ofthe level 3 structure. The level 1 and 2 structures are perfectly locked in, while thelevel 3 structure is random.

MC simulations of relaxation to equilibrium are done for the level 1, 2 and 3 struc-

tures. Curves of Φn as a function of mcs, T for n “ 1, 2, 3 are shown in Fig. 3.2. The

tiling system sizes used for the relaxation of level 1, 2 and 3 are 32ˆ32, 64ˆ64, and

128 ˆ 128 respectively such that the number of tiles that need to be ordered dur-

ing the relaxation of each level is the same. The level 1 structure is equilibrated at

T1 “ 1.0; The level 2 and 3 structures are equilibrated at the rescaled temperatures

of T1 at T2 “ 0.603335 and T3 “ 0.427097 respectively. Together these conditions

ensure the difference in the relaxation time required for each level is expected to

23

be only due to the different time required for the corners of the black triangles at

different levels to develop correlations with each other.

�

�

�

���������������������������������������

�������������������������������������������������������������������������������������������������������������

�

����������������������������������������������

������������������������������

������������

������������������

������������������������

�����������������������

� n�1� n�2� n�3

0 5000 10000 15000 20000 250000.0

0.2

0.4

0.6

0.8

1.0

�

�n

Figure 3.2: The plot of Φn vs. T for n “ 1, 2, 3. The tiling system sizes used forthe relaxation of level 1, 2, and 3 are 32 ˆ 32, 64 ˆ 64, and 128 ˆ 128 respectively.The temperatures of equilibration for level 1, 2, and 3 are T1 “ 1.0, T2 “ 0.603335and T3 “ 0.427097 respectively.

We neglect the transient behavior of ΦnpT q and compare the curves for ΦnpT q for

Φn ě 0.8. The relaxation constant τn is extracted for each curve and their ratios are

listed in the first row of Tab. 3.1. In Fig. 3.3, the curve for Φ1pT q has been scaled in

T by the factor of τ2{τ1 ˆ τ3{τ2 “ 13.8 ˆ 10.7. The curve for Φ2pT q has been scaled in

T by the factor of τ3{τ2 “ 10.7. The scaled curves of Φ1pT q, Φ2pT q and Φ3pT q showvery good agreement with each other, indicating a good estimate of the ratio τ2{τ1and τ3{τ2. As discussed earlier, the ratio τ2{τ1 and τ3{τ2 approximates the ratio t2{t1and t3{t2 respectively.

24

�

�

������������������������������������������

����������������������

����������������������������

�

�������������������������������������������������������

������������������������������������

�������������������������������������

�������������������������������������������������������������������

���������������������������������������

����������������������������������������������������������������

� n�1� n�2� n�3

0 400000 800000 12000000.80

0.85

0.90

0.95

1.00

�

�n

Figure 3.3: The plot of Φn vs. scaled T for n “ 1, 2, 3. The ratio between therelaxation constants for level 1 and 2 gives τ2{τ1 “ 13.8, while the ratio betweenthe relaxation constants for level 2 and 3 gives τ3{τ2 “ 10.7. The three curves showgood agreement with each other after the mcs T for level 1 is scaled by the factorτ2{τ1 ˆ τ3{τ2, and the mcs T for level 2 is scaled by the factor τ3{τ2.

3.3 Theory Prediction of tI;n{tI;n´1

The next step of testing our hypothesis is to find tI;n{tI;n´1, the ratio between the

time required for 1D Ising systems that correspond to two subsequent levels of struc-

ture in the tiling model to reach equilibrium. The ratio tI;2{tI;1 is found through a

direct theoretical calculation, while the ratio between higher levels of Ising systems

are obtained through MC simulation of the 1D Ising systems using the Metropolis

Algorithm, the same algorithm we used in the simulation of the tiling model. The

settings of the problem we need to solve in order to obtain the desired ratios are as

follows: In each level of structure, let the total number of 1D Ising systems, s to be

the same since we want to control the system size for each level to be the same. At

level n, the length of the Ising systems is kn ` 1. For each Ising system, the first

25

spin is fixed in the up orientation with the state |Ò�, and the rest are initialized to

random orientations. Spin updating rules follow the standard Metropolis Algorithm

except that only two out of the three spins (the first spin is fixed) are allowed to

be flipped. One Monte Carlo Step (mcs) is defined as potentially updating s ˆ kn

spins-the number of movable spins-among s 1D Ising systems asynchronously. For

each pair of spin bond, energy is 0 if the pair of spins is aligned, and 1 if the pair

is anti-aligned. An Ising system is ordered if the first and last spins are aligned, i.e.

both in the |Ò� state. When level n structure reaches equilibrium, an equilibrium

fraction of the Ising systems will be in the ordered configurations. The exact value

of the fraction is determined from Boltzmann statistics by taking the ratio between

the Boltzmann factor for the ordered configuration and the sum of the Boltzmann

factors for all the possible configurations of the Ising system. For level n, If we

plot the fraction of Ising systems that are ordered as a function of T , it is expected

that for sufficiently large T , the Ising systems approaches the equilibrium fraction

exponentially. The the relaxation time constant, τn can then be extracted from the

exponential curve. τn determines the rate of approach to equilibrium for the level

n Ising systems, hence we can approximate the ratio tI;n{tI;n´1 by taking the ratio

τn{τn´1 as discussed before.

The analogy between the settings of the problem for the Ising systems and for the

tiling model is apparent. The various conditions in the Ising systems are intended to

reproduce those in the tiling system. Fig. 3.4 gives an illustration of the Ising systems

at level 3. The bonds between pairs of spins in the Ising system represent the bonds

created by the black stripes joining the nearest neighbor tiles along the edges of the

black triangles. An aligned pair of spins represents a pair of nearest neighbor tiles

with matched black stripes joining them. Besides, the energy assignments for the

Ising spin bonds are in accordance with those stated in the local “matching rules”

26

for the tiles. Further, the definition for one mcs for the Ising systems matches that

for the tiling model. In every mcs, on average there is one attempt to update every

spin, whereas in the tiling model, on average there is also one attempt to update

every tile in one mcs.

Mismatched corners

Matched corners

Level 3

Figure 3.4: A diagram illustrating Ising systems at level 3. The Ising system at thetop has anti-aligned end spins, corresponding to an unordered edge of black triangle,while the Ising system at the bottom has aligned end spins, corresponding to anordered edge of black triangle. The arrows indicate the spin states represented bythe end and middle tiles in their respective orientations. Each anti-aligned pair ofspin bond represents a mismatch in black stripes of neighboring tiles. The fractionα is greater than β as the end tiles have more energetically favorable positions to goto.

To complete the analogy, one more parameter needs to be introduced to the 1D Ising

systems: Suppose a spin was in the state |Ò� originally. In the standard Metropolis

Algorithm for the 1D Ising model, when the spin is chosen to be updated during

a mcs, it is tested whether it should be flipped to the opposite state |Ó�. Suppose

that there is only a probability of ρ that state |Ó� is generated to be tested for the

spin chosen. With probability 1´ ρ, the spin would just stay in its original state |Ò�without going through the Metropolis updating procedure. This extra parameter ρ is

introduced because when a tile on an edge of a black triangle is chosen to be updated,

as discussed in Sec. 3.2, among the 12 possible orientations it could be tested with,

27

only a fraction of the orientations correspond to a flip of the allowed positions of the

long black stripe which are parallel to the edges of the black triangles. Furthermore,

this fraction is different for the middle and end tiles on an edge of black triangle

because the end tiles have more allowed positions of the black stripe than the middle

tiles. The middle tiles participate in the formation of black triangles of the previous

levels, hence only have two energetically favorable positions of the long black stripe,

corresponding to spin states |Ò� and |Ó�. We let ρ “ α for the end spins in the 1D

Ising systems, and let ρ “ β for the middle spins, where α and β are constants.

3.3.1 Theory Calculation for τ2{τ1

For the simple case of level 1 and level 2 Ising systems , the relaxation time con-

stant, τ can be solved exactly. For higher level structures, MC simulations using the

settings stated above for the problem of Ising systems are used to find τ .

At level 1, each 1D Ising system has 2 spins, corresponding to two tiles on each edge

of the level 1 black triangles, with the leftmost spin fixed in the state |Ò�. Fig. 3.5

shows each configuration of the Ising system and how each configuration could be

updated after one spin has been flipped. The numbers in the parentheses are the

corresponding energies of the configurations.

(1)(0) (0)(1)Figure 3.5: On the left side are the possible configurations of a level 1 Ising system,and on the right side are the possible configurations after one spin has been flipped. Inthe case of level 1, only the second spin could be flipped since the first is always fixed.The numbers in the parenthesis are the corresponding energies of the configurations.

Define at time step t (after 1 ˆ t spins have been updated), the number of Ising

systems with the ordered configuration |ÒÒ� as nt, its fraction among a total of s

28

Ising systems as qt. Then nt`1 is related to nt by:

nt`1 “ αp pnt ´ 1q qt ` αp1 ´ pq nt qt ` p1 ´ αq nt qt

` p1 ´ αqnt p1 ´ qtq ` α pnt ` 1q p1 ´ qtq

“ nt ´ α rpp ` 1qqt ´ 1s

(3.1)

Where p “ e´1{T1 is the Boltzmann factor which indicates the probability that a

move is accepted if it increases the energy of the Ising system by 1 at temperature

T1. Divide both sides by s, the total number of Ising systems, then:

qt`1 “ qt ´ α

srpp ` 1qqt ´ 1s (3.2)

At equilibrium, let q˚ be the equilibrium value of qt, then:

q˚ “ q˚ ´ α

srpp ` 1qq˚ ´ 1s (3.3)

Solve for q˚ to get:

q˚ “ 1

p ` 1(3.4)

Let qt “ q˚ ` �t, qt`1 “ q˚ ` �t`1, where �t and �t`1 are the deviations from the

equilibrium fraction at time t and t ` 1 respectively. It follows from Eq.(3.2) and

Eq.(3.4) that:

�t`1 “ �t”1 ´ α

spp ` 1q

ı(3.5)

One mcs is defined as potentially updating all the movable spins in the Ising systems.

At level 1, the total number of movable spins is s. It follows that:

�T `1 “ �T”1 ´ α

spp ` 1q

ıs(3.6)

29

Take the limit where s Ñ 8 as the size of level 1 becomes infinitely large, then:

limsÑ8

�T `1 “ �T e´αpp`1q (3.7)

Applying the above relation recursively, one arrives at:

�T “ �0e´T αpp`1q (3.8)

Where �0 is the deviation from the equilibrium fraction q˚ at time 0. The equation

for �T has the form of an exponential function, so the relaxation constant for level 1

can now be extracted:

τ1 “ 1

αpp ` 1q (3.9)

Now for level 2, each 1D Ising system has 3 spins, corresponding to three tiles on

each edge of the level 2 black triangles, with the leftmost spin fixed in the state

|Ò�. Fig. 3.6 shows all possible configurations of the Ising system and how each

configuration could be updated after one spin has been flipped.

(1)(0) (0)(1)(2) (1)

(1)(2) (2)(1)(0) (1)

Figure 3.6: On the left side are the possible configurations of a level 2 Ising system,and on the right side are the possible configurations after one spin has been flipped.The first spin is fixed in the up orientation as always. The numbers in the parenthesisare the corresponding energies of the configurations.

30

Define at time step t (after 2 ˆ t spins have been updated), the number of Ising

systems with configurations |ÒÒÒ�, |ÒÓÒ�, |ÒÒÓ� as n1,t, n2,t, n3,t respectively. Their

fractions among a total of s Ising systems as q1,t, q2,t, q3,t respectively. n1,t`1, n2,t`1,

n3,t`1 can be related to n1,t, n2,t, n3,t respectively:

n1,t`1 “ 1

2

`2n1,t ´ q1,tpαp1 ` βp2q ` βq2,t ` αq3,t

˘` 1

2αp1 pn1,t ´ 1q q1,t

` 1

2αp1 ´ p1qn1,tq1,t ` 1

2βp2 pn1,t ´ 1q q1,t ` 1

2βp1 ´ p2qn1,tq1,t ` 1

2p1 ´ αqn1,tq1,t

` 1

2p1 ´ βqn1,tq1,t ` 1

2p1 ´ βqn1,tq2,t ` 1

2β pn1,t ` 1q q2,t ` n1,tq2,t

2` 1

2α pn1,t ` 1q q3,t

` n1,t`1 ´ q1,t ´ q2,t ´ q3,t

˘` 1

2p1 ´ αqn1,tq3,t ` n1,tq3,t

2

“ 1

2

“2n1,t ´ q1,tpαp1 ` βp2q ` βq2,t ` αq3,t

‰

(3.10)

n2,t`1 “ 1

2βp1 ´ p2qn2,tq1,t ` 1

2βp2 pn2,t ` 1q q1,t ` 1

2p1 ´ βqn2,tq1,t ` n2,tq1,t

2

` 1

2α pn2,t ´ 1q q2,t ` 1

2β pn2,t ´ 1q q2,t ` 1

2p1 ´ βqn2,tq2,t

` 1

2αp1 pn2,t ` 1q

`1 ´ q1,t ´ q2,t ´ q3,t

˘` 1

2p1 ´ αqn2,tq2,t ` n2,tq3,t

` 1

2αp1 ´ p1qn2,t

`1 ´ q1,t ´ q2,t ´ q3,t

˘` 1

2p1 ´ αqn2,t

`1 ´ q1,t ´ q2,t ´ q3,t

˘

` 1

2n2,t

`1 ´ q1,t ´ q2,t ´ q3,t

˘

“ 1

2

“2n2,t ` αp1 ` q1,tpβp2 ´ αp1q ´ αp1q2,t ´ αp1q3,t ´ αq2,t ´ βq2,t

‰

(3.11)

31

n3,t`1 “ 1

2αp1 ´ p1qn3,tq1,t ` 1

2αp1 pn3,t ` 1q q1,t ` 1

2p1 ´ αqn3,tq1,t ` n3,tq1,t

2

` n3,tq2,t ` 1

2α pn3,t ´ 1q q3,t ` 1

2p1 ´ αqn3,tq3,t ` 1

2β pn3,t ´ 1q q3,t

` 1

2β pn3,t ` 1q

`1 ´ q1,t ´ q2,t ´ q3,t

˘` 1

2n3,t

`1 ´ q1,t ´ q2,t ´ q3,t

˘

` 1

2p1 ´ βqn3,tq3,t ` 1

2p1 ´ βqn3,t

`1 ´ q1,t ´ q2,t ´ q3,t

˘

“ 1

2

“2n3,t ` β ` q1,tpαp1 ´ βq ´ βq2,t ´ αq3,t ´ 2βq3,t

‰

(3.12)

Where p1 “ e´1{T2 and p2 “ e´2{T2 are the Boltzmann factors which indicate the

probabilities that a move is accepted if it increases the energy of the Ising system by

1 and 2 respectively at temperature T2. Divide both sides of Eq.(3.10)–(3.12) by s,

the total number of Ising systems, then:

q1,t`1 “ q1,t ´ 1

2s

“q1,tpαp1 ` βp2q ` βq2,t ` αq3,t

‰(3.13)

q2,t`1 “ q2,t ` 1

2s

“αp1 ` q1,tpβp2 ´ αp1q ´ αp1q2,t ´ αp1q3,t ´ αq2,t ´ βq2,t

‰(3.14)

q3,t`1 “ q3,t ` 1

2s

“β ` q1,tpαp1 ´ βq ´ βq2,t ´ αq3,t ´ 2βq3,t

‰(3.15)

By following the same procedure as in the calculation for level 1, we standardize

q1,t, q2,t and q3,t around their equilibrium values q˚1 , q

˚2 and q˚

3 respectively. Let �1,t,

�2,t and �3,t be the deviations from the equilibrium fractions at time t respectively,

then three recursive relations can be obtained for the deviations from the equilibrium

fractions. Written in matrix form:

»

–�1,t`1

�2,t`1

�3,t`1

fi

fl “

»

——–

2s´p1α´p2β2s

β2s

α2s

p2β´p1α2s

2s´p1α´α´β2s ´p1α

2s

p1α´β2s ´ β

2s2s´α´2β

2s

fi

ffiffifl

»

–�1,t�2,t�3,t

fi

fl (3.16)

Or:

32

�t`1 “ A�t (3.17)

WhereA is the 3 by 3 matrix in Eq.(3.16). One mcs is defined as potentially updating

all the movable spins in the Ising systems. At level 2, the total number of movable

spins is s ˆ k2 “ 2s. It follows that:

�T `1 “ A2s�T (3.18)

Let �0 be the deviation from the equilibrium fraction vector q˚ at time 0. Applying

Eq.(3.18) recursively to arrive at:

�T “ pA2sqτ�0 (3.19)

Suppose the eigenvalues and eigenvectors of A are tλiu3i“1 and tviu3i“1 respectively.

�0 can be written as a linear combination of the eigenvectors of A,

�0 “3ÿ

i“1

civi (3.20)

Where tciu3i“1 are real constants. Eq.(3.19) becomes:

�T “3ÿ

i“1

cipλ2si qτvi (3.21)

For illustrative purpose, let us set α “ 1 and β “ 1 and proceed with the calculation.

It follows:

A “

»

——–

2s´p1´p22s

12s

12s

p2´p12s

2s´p1´22s ´p1

2s

p1´12s ´ 1

2s2s´32s

fi

ffiffifl (3.22)

The eigenvalues of A can be solved:

33

λ1 “ 1 ´ 1

s

λ2,3 “ 1 ´ 1

4s

ˆ2p1 ` p2 ` 3 ˘

bp22 ´ 2p2 ` 5

˙ (3.23)

Raise the eigenvalues to the power of 2s and take the limit where s Ñ 8:

limsÑ8

λ2s1 “ e´2

limsÑ8

λ2s2,3 “ e

´ 12

´2p1`p2`3˘

?p22´2p2`5

¯ (3.24)

For sufficiently large number of mcs, or T , the term with λ2s3 dominates in Eq. (3.21):

�T » c3pλ2s3 qT v3

“ c3e´ 1

2

´2p1`p2`3´

?p22´2p2`5

¯T v3

(3.25)

An Ising system is ordered if the first spin and last spin are aligned, i.e. in one of

the configurations of |ÒÒÒ�, |ÒÒÓ�. We denote the total fraction deviation from the

equilibrium fraction of ordered systems, q˚ after T mcs as �tot;T . Then for sufficiently

large T :

�tot;T “ c3pv31 ` v32qe´ 12

´2p1`p2`3´

?p22´2p2`5

¯T (3.26)

From Eq.(3.26), the relaxation time constant τ2 for level 2 in the case that both

α “ 1 and β “ 1 can be extracted:

τ2 “ 2

2p1 ` p2 ` 3 ´ap22 ´ 2p2 ` 5

(3.27)

Now the value of τ2{τ1, which approximates tI;2{tI;1, can be calculated using the above

calculations for τ1 and τ2. Set T1 “ 1.0 and T2 “ 0.603335, which is the rescaled

temperature of T1 for the level 2 Ising systems. For each edge of the black triangles,

the number of tile orientations that correspond to a flip of the allowed positions of

34

the long black stripe is in fact 2 out of 12 for the middle tiles, and 12 out of 12

for the end tiles. This gives the fraction α “ 1 and β “ 16 . However, the situation

is more complicated for the middle tiles. The long color stripe of each middle tile,

which is perpendicular to the long black stripe, could have mismatches with the the

long color stripes directly above and below it. Depending on the positions of the

long color stripes above and below each middle tile, 1 of the 2 allowed orientations of

the middle tile could be energetically unfavorable, and thus forbidden. This means

the fraction β could be as low as 112 and as high as 1

6 . Setting β “ 112 , we obtain a

theoretical ratio of τ2{τ1 “ 14.464; Setting β “ 19 , τ2{τ1 “ 11.033; Setting β “ 1

6 ,

τ2{τ1 “ 7.618.

3.3.2 τn{τn´1 Ratios by MC Simulation of 1D Ising Systems

The τn{τn´1 ratios for the higher levels are obtained from MC simulations of the 1D

Ising systems using the settings as stated at the beginning of Sec. 3.3. We set the

temperatures of the Ising systems for level 1, 2, and 3 as T1 “ 1.0, T2 “ 0.603335,

and T3 “ 0.427097 to match the temperatures used for equilibration of the corre-

sponding levels of structure of the tiling model. The results for α “ 1 and β “ 112

are summarized in Tab. 3.1 for level 1, 2, and 3.

τ2{τ1 τ3{τ2Simulation of Tiling Model 13.8 10.7Simulation of Ising Model 14.45 4.10Direct Calculation (Ising) 14.46 -

Table 3.1: The ratios between the relaxation constants for the level 1, 2, and 3structure of the tiling model obtained from MC simulation of the tiling model arecompared with the ratios predicted by simulation of 1D Ising systems as well as fromdirect calculations.

The ratios between the relaxation constants from the MC simulations of the tiling

model are also listed in Tab. 3.1 for comparison. For τ2{τ1, the predicted value from

35

theory calculation and simulation of 1D Ising systems are in good agreement with

each other. Further, they agree reasonably well with the value from simulation of the

tiling model. However, there is an obvious discrepancy between the predicted value

of τ3{τ2 and that from simulation of the tiling model. We have tried using different

values of α and β unsuccessfully. In fact, setting β to any value higher than 1{12 de-

creases the predicted value of τ3{τ2, thus widens the discrepancy. This disagreement

is still not well understood at the current stage of research. To resolve the discrep-

ancy, a better understanding is needed for how the long color stripes of the middle

tiles on each edge of black triangles interact with their next-nearest neighbors above

and below them. We have also hypothesized that different values of β may have

to be set for different middle tiles along the edge of a black triangle. However, an

analysis which examines how often each middle tile is flipped during MC simulations

of the tiling model does not support this hypothesis. An ongoing effort to resolve

the discrepancy involves removing the color stripes of the tiling model altogether,

i.e. setting �2 “ 0, so that the color stripes no longer play a role in the dynamics of

the tiling model. Preliminary evidence suggests that without the color stripes, the

tiling model would still be able to relax to its limit-periodic ground state. Finding

the τn{τn´1 ratios in the case �2 “ 0 would test the validity of using 1D Ising systems

to model the dynamics of the tiling model.

The most likely explanation of the observed discrepancy in the scaling behavior is

due to the finite-size effect in MC simulations of the tiling model. Our theory based

on 1D Ising model assumes the collective physics effects of ordering aside from the

the development of correlation between the corners of the triangles are the same

for different levels of structure. This assumption is valid if the system size is large

enough. The collective physics effects of each level’s ordering include the domain

dynamics. Due to the limited sizes of the system we used for MC simulations of

the tiling model, the domain dynamics may not be the same for level 1, 2 and 3,

thus the scaling behavior of the time required for equilibration by different levels

of structure is affected by the finite-size effect. We shall return to this discussion

36

in the concluding paragraph after we study the domain dynamics in the next chapter.

37

4

Domain Dynamics

Some preliminary evidence has been found that a rapid quench of the tiling system

leads to disordered states [8]. If the rate of dropping the temperature is too high so

that the temperature drops below the level n transition temperature Tc;n before Φn´1

can reach a sufficiently high value, then neither level n nor level n ´ 1 could become

well ordered. The result is that the ordering the system gets frozen and the system

ends up in some disordered configuration. Intuitively, this could be understood if we

look at the non-contributing sublattice for each level of structure. When the level 1

structure fails to order during a rapid quench, the value of its order parameter, Φ1

would be far from its maximum value 1. When the temperature drops to T2;c so that

the level 2 structure starts to order, level 2 would be ignorant about which is the

non-contributing sublattice for level 1 and hence on which sublattice to establish the

level 2 structure. As a result, level 2 would also fail to become fully ordered. Any

subsequent level would face the same difficulty of choosing a sublattice to establish

its order, and hence the system as a whole would fail in getting ordered. The behav-

ior of the system during a rapid quench is reminiscent of glass formation, where any

nonzero quench rate would eventually lead to the system getting trapped in some

38

nonequilibrium states [14].

Several questions remain poorly understood from any previous findings. First, why

does the system falls out of equilibrium during a rapid quench? More precisely, what

are the kinetic barriers created by a certain level of structure when it fails to or-

der that cause the simultaneous frustration of the ordering of the subsequent levels?

Second, when the system goes into a disordered phase during a rapid quench, does

the system get stuck there? Could the system return to equilibrium if we relax the

system at some finite temperature for a reasonable amount of time? If the answer is

yes, then what is the energy cost of removing the kinetic barriers created by that level

of structure when it fails to order. These are the questions addressed by this chapter.

4.1 Domain Walls

A closer look at the configuration of the tiling system resulted from a rapid quench

reveals that domain walls form when the ordering of a certain level of structure gets

frustrated. Fig. 4.1 shows that several domains form after a size 32 ˆ 32 system

goes through a rapid quench in which the initial temperature, T “ 2.0 is lowered by

δT “ 0.01 after every r “ 10 mcs.

The level 1 tiles colored in red have spin vector �eA, and they form a domain for which

the non-contributing sublattice for level 1 is sublattice A. The tiles in sublattice A

in the red domain could have any of the other three spin vectors, however they are

colored in dark red to emphasize that they are part of the red domain. Similarly, the

purple, green and yellow domains have sublattice B, C, D as their non-contributing

sublattices respectively. The color stripes are omitted in Fig. 4.1 to reduce visual

distraction, but they do participate in forming the domain walls. The four domains

39

Figure 4.1: Four domain walls are present after a rapid quench from T “ 2.0.Temperature is lowered by δT “ 0.01 after every r “ 10 mcs. The red, purple,green and yellow domains have sublattice A, B, C and D as their non-contributingsublattice respectively.

compete with each other as each wants its non-contributing sublattice to be the

non-contributing sublattice for the entire level 1 structure, and there is no single

dominating domain during the rapid quench. Clearly the energy barriers created by

the domain walls prevented the system from getting further ordered. Even though

the domains are only colored for level 1 here, they could form in any level of struc-

ture. Level 2 domains could develop within each of the four level 1 domains shown

in Fig. 4.1, and level 3 domains could develop within the domains of level 2, etc.

The domain walls created when a level of structure fails to order prevent the system

from getting further ordered during a rapid quench, but would the system go back

40

to an equilibrium state if we relax the system at a finite non-zero temperature for

a reasonable amount of time such that a single domain emerges by conquering the

others (We require the sitting time to be reasonable, as if one waits for an astronom-

ical amount of time, there is always a small chance that the system equilibrates)?

Fig. 4.2 illustrates such an example. However, as we will see later, this is not al-

ways the case. Whether the system could equilibrate by sitting at a finite non-zero

temperature for a reasonable amount of time depends on the temperature and the

type of domain walls present in the system. An example in which the system gets

stuck in a configuration, and the ordering of the system freezes is discussed in Sec. 4.3.

Fig. 4.2(a) shows a configuration of a size 64ˆ64 system after a sudden quench from

an initial high temperature to T “ 0.6, followed by sitting at that temperature for

5 ˆ 105 mcs. While Fig 4.2(b–f) show the subsequent configurations of the system

after further sitting at T “ 0.6. As the sitting temperature T “ 0.6 is below the

level 2 transition temperature, T2;c and above the level 3 transition temperature,

T3;c, the highest level of structure that can be formed within each domain is level 2.

An examination of these configurations reveals that there are two types of domain

walls present in the tiling model. A type I domain wall is shown in Fig. 4.3, which

is a zoomed in view of the domain wall between the yellow and green domains in

the lower left part of the configuration in Fig. 4.2(a). A type I domain wall is made

of tiles of alternating colors arranged along a straight line. The black triangles from

two different domains are separated by the long black stripes of the tiles along the

domain wall. Long correlation in the long black stripes often develops such that it

is possible to have no mismatches in the black stripes along the domain wall.

A type II domain wall is shown in Fig.4.4, which is a zoomed in view of the domain

41

II

II

II

II

IIII

(a) T “ 50 ˆ 104 (b) T “ 54 ˆ 104

(c) T “ 66 ˆ 104 (d) T “ 67 ˆ 104

(e) T “ 68 ˆ 104 (f) T “ 69 ˆ 104

Figure 4.2: The configurations of a size 64 ˆ 64 system after a rapid quench toT “ 0.6 and sitting at that temperature for 69ˆ104 mcs. The black and color stripesare omitted here. The roman numerals indicate the types of domain wall. (a) Theconfiguration after being suddenly quenched to T “ 0.6 and sitting for 50ˆ 104 mcs.(b–f) Subsequent configurations after furthering sitting at T “ 0.6.

42

Figure 4.3: The type I domain wall shown from a zoomed in view of the domainwall between the yellow and green domains in the lower left part of the configurationin Fig. 4.2(a).

walls between the yellow and purple domains in the lower right part of the config-

uration in Fig. 4.2(a). Along the type II domain wall, the black triangles from two

different domains get intercepted by each other. In many cases, the incomplete level

1 black triangles from one domain are connected to the incomplete black triangles of

higher levels from the other domain. In this case, the highest level of black triangles

formed is level 2, so an entanglement between the level 1 and 2 incomplete black

triangles across two different domains is observed.

Observing how the domain walls in the configurations in Fig. 4.2 have evolved over

a period of 69ˆ104 mcs, it is found that the type I domain walls are much harder to

move around than the type II domain walls. The middle yellow domain is bounded

by two domain walls of type I as indicated by the labels “ I” in Fig 4.2(a). The rest

43

Figure 4.4: The second type of domain wall shown from a zoomed in view of thedomain walls between the yellow and purple domains in the lower right part of theconfiguration in Fig. 4.2(b).

of the domain walls are of type II as indicated by labels “II”. The yellow domain has

expanded through moving its type II domain walls while its type I domain walls have

changed in length but maintained their shapes and positions. Apparently, the type

I domain wall is intrinsically harder to move than the type II domain wall. The part

of the green domain pointed by an arrow in Fig. 4.2(a) is bounded by one domain

wall of type I and one of type II. It has shrunk over time despite the fact that its

type I domain wall remains stable. However the blue domain on the other side of

its type II domain wall gets taken by the yellow domain, which then pushed the

type II domain wall of the green domain inward. The invasion of the yellow domain

subsequently consumes the type I domain wall of the green domain which served as

a defense. Sitting at T “ 0.6 for more time, it is observed that the yellow domain

eventually conquers the other domains through the expansion of its type I domain

44

walls.

4.2 Energy Costs of Infinite Domain Walls

The energetics associated with the two types of domain wall are different, and this

accounts for the difference in the difficulty of move the two types of domain wall.

Here we construct flat domain walls that span the entire system in different direc-

tions, and examine the energy costs of moving the domain walls around. The type

of domain wall arisen in each direction and the corresponding number of mismatches

along the domain walls give insights into the energy barriers created by the domain

walls and their associated dynamics.

First, two flat domain walls are constructed in the horizontal direction to span a size

64 ˆ 64 system in a rhombus domain. The system is made of two domains, a red

and a yellow one. Each domain is in the limit-periodic ground state, i.e. all levels of

structure are well ordered. Note that under the periodic boundary conditions, there

are at least two mismatches present in the system [8]. In fact one mismatch in black

stripe is each present in the red and yellow domain. Define a coordinate system for

the rhombus domain such that the x-axis is parallel to the horizontal direction, and

the y-axis is parallel to the slanted direction. The origin is set at the bottom right

corner of the domain (the direction of axes and origin are marked in Fig. 4.7). The

lower domain wall is fixed at y “ 15 and the upper wall is initially set at y “ 40.

The upper domain is subsequently moved upward to the position y “ 60 in steps of

∆y “ 1. The number of mismatches in both the black and color stripes on the upper

domain wall are plotted as a function of the y position respectively in Fig. 4.5 and

Fig. 4.6.

45

�

�

�

��

�

�

�

�

�

�

�

�

�

�

�

�

�

�

� �

40 45 50 55 6040

60

80

100

120

y

Num

.ofblack

stripemismatches

Figure 4.5: Number of black stripe mismatches vs y for horizontal domain wallsseparating perfectly ordered red and yellow domain.

�

� � �

�

� � �

�

� � �

�

� � �

�

� � �

�

40 45 50 55 60

130

140

150

160

170

180

190

y

Num

.ofcolorstripemismatches

Figure 4.6: Number of color stripe mismatches vs y for horizontal domain wallsseparating perfectly ordered red and yellow domain.

46

The minimum mismatches in black tripe on the upper horizontal domain wall is 62

and, is obtained when y “ 50 and 54; While the minimum number of mismatches in

color stripe on the upper horizontal domain wall is 127 when y “ 54 and 58. There

are other y values at which the number of mismatches in black and color stripes are

just off by 1 or 2 from the minimum numbers. It is clear that periodic behavior

is present in the number of mismatches in both the black and color stripes when

the y value is changed. For the black stripe, the number of mismatches remains

relatively high for three steps of ∆y “ 1 increment and reaches around the minimum

on the fourth step of increment; While for the color stripe, the number of mismatches

remains stable around the minimum for three steps of ∆y “ 1 increment and shoots

to a high value on the fourth step of increment. The resultant horizontal domain

walls are found to be of type II.

y

x

Figure 4.7: A configuration showing two domains separated by a horizontal upperdomain wall of type II at y “ 54 with minimum mismatches. The black and colorstripes are omitted here.

A configuration that corresponds to the upper domain wall having the minimum

47

Figure 4.8: Zoomed in view of a portion of the upper horizontal domain wall oftype II in Fig. 4.7. Following the blue path, a mismatch in black stripe is connectedto another mismatch in black stripe along the domain wall. Following the greenpath, a mismatch in color stripe is connected to another mismatch in color stripealong the domain wall.

number of mismatches in both the black and color stripe decorations at y “ 54 is

shown in Fig. 4.7. A closer view of the upper domain wall is shown in Fig. 4.8. Ex-

amining the mismatches in the black stripe decoration reveals that each mismatch in

black stripe along the domain wall is always connected to another mismatch in black

stripe along the domain wall. One such example is highlighted by the blue path in

Fig. 4.8. The total number of mismatches in black stripe along the upper domain is

64, that is 1 mismatch in black stripe per tile. For the mismatches in color stripe, it

is also the case that each mismatch must have a companion along the domain wall,

connected by a path of incomplete colored triangle. One example is highlighted by

48

the green path in Fig. 4.8. There is a total of 128 mismatches in color stripes along

the upper domain wall, that is 2 mismatches in color stripe every tile.

Next, two flat domain walls are constructed in the slanted direction to separate a

red and a yellow domain. Each domain is in the limit-periodic ground state. The

left slanted domain wall starts at x “ 42 and rises at an angle of 60˝ with respect to

the x-axis to span the entire 64ˆ64 rhombus lattice. The right domain wall initially

starts at x “ 6, and the starting position is subsequently moved to x “ 36 in steps of

∆x “ 1. The number of mismatches in both the black and color stripes on the right

slanted domain wall are plotted as a function of the starting x position respectively

in Fig. 4.9 and Fig. 4.10. The resultant slanted domain walls are of type I.

�

�

��

�

�

��

�

�

�

��

�

���

��

�

�

�

� �

�

�

�

���

�

5 10 15 20 25 30 3520

40

60

80

100

x

Num

.ofblack

stripemismatches

Figure 4.9: Number of black stripe mismatches vs x for slanted domain wallsseparating perfectly ordered red and yellow domain.

The minimum mismatches in black stripe on the right slanted domain wall is 38 for

49

�

�

��

�

�

���

�

�

�

�

�

��

�

�

�

�

�

�� ��

�

�

��

�

�

5 10 15 20 25 30 35100

120

140

160

180

200

x

Num

.ofcolorstripemismatches

Figure 4.10: Number of color stripe mismatches vs x for slanted domain wallsseparating perfectly ordered red and yellow domain.

starting position x “ 31 and 34. The minimum number of mismatches is much lower

than that in the case of horizontal domain walls. This is because a long straight

line made of the long black stripes belonging to tiles of alternate colors forms part

of the domain wall. There could be no mismatches in black stripe along the straight

line separating the two domains as long correlation in black stripe develops. This

is illustrated by a zoomed in view of the domain wall in Fig. 4.12. Later we will

see that the minimum mismatches in black stripe could be even lower if the red

and yellow domains do not have all their levels of structure perfectly ordered. The

minimum number of mismatches in color stripe is 130 at starting position x “ 32.

Here, it is also the case that a mismatch in black stripe or in color stripe must have

a companion mismatch along the domain wall, as illustrated by the blue and green

paths in Fig. 4.12 respectively. The periodic behavior in the number of mismatches

as the starting x position changes is less obvious in the case of the slanted domain

walls. However, it is still clear that the number of mismatches in both the black

50

stripe and the color stripe reaches a relatively low value every few steps of increment

in the starting x position.

y

x�

Figure 4.11: A configuration showing two domains separated by a slanted rightdomain wall of type I at starting position x “ 31 with low number of mismatches.The black and color stripes are omitted here.

4.3 Domain Wall Energetics

Now we are ready to answer why the system’s ordering gets frozen during a rapid

quench. Domain walls often form during a rapid quench of the tiling system. We

have seen a case in Fig. 4.2 that the system goes into a configuration made of several

domains separated by domain walls of different types, and one domain eventually

conquers the others after long enough time. However, there are also cases in which the

system would just get stuck in a particular configuration, and it takes an astronomical

amount of time to move the domain walls around. Fig. 4.13 presents such an example.

The configuration is formed by a sudden quench of a 64 ˆ 64 system from an initial

high temperature with a random configuration to the temperature T “ 0.6. The

51

Figure 4.12: Zoomed in view of a portion of the right slanted domain wall of typeI in Fig. 4.11. Following the blue path, a mismatch in black stripe is connected toanother mismatch in black stripe along the domain wall. Following the green path, amismatch in color stripe is connected to another mismatch in color stripe along thedomain wall.

system is then equilibrated at this temperature for 5 ˆ 105 mcs. At T “ 0.6, the

level 2 structure is well ordered as is evident in Fig. 4.14. The resultant configuration

has two long slanted domain walls of type I separating a red and a yellow domain.

It is found that the domain walls can hardly be moved by keeping sitting at the

temperature T “ 0.6. As a result, the system gets stuck in this configuration and

could not reach equilibrium at T “ 0.6 within a reasonable amount of time.

52

Figure 4.13: A configuration that gets stuck in ordering due the presence of aslanted domain wall of type I that spans the entire 64ˆ64 system. The configurationis resulted from a sudden quench from an initial high temperature to T “ 0.6 andsitting for 5 ˆ 105 mcs. The black and color stripes are omitted here.

Figure 4.14: Zoomed in view of a domain wall of type I from the configuration inFig. 4.13.

53

Siting at temperatures even lower than T “ 0.6 makes it even harder to move the

stable domain walls of type I. This is because the chance of a tile going into an

orientation that does not lower the system’s energy and thus upsets the energetically

favorable domain walls becomes smaller and smaller at lower temperatures. It is

noteworthy that sitting at a temperature lower than T “ 0.6 allows higher levels

of structure to develop within each domain until the size of the triangular lattice

becomes too large to be accommodated by the domain sizes. As shown by Fig. 4.15,

there could be no mismatches in black stripes at all along the domain wall. All the

mismatches are in color stripe decorations. The domain wall of type I obtained in

this case is more energetically favorable than the one we constructed in Fig. 4.12

due to zero mismatches in black stripes. However, an examination of the entire

configuration that is partly shown in Fig. 4.15 reveals that even though all levels of

structure that can be fitted into each domain are ordered, not every black triangle

is perfectly formed. In other words, there are mismatches in black stripes within

each domain. While the configuration we constructed in Fig. 4.12 has all levels of

structure perfectly ordered within each domain, and mismatches in black stripes are

only along the domain wall.

The only way to disrupt the stable domain wall of type I is to increase the tempera-

ture of the system. A simulation of equilibrating the configuration shown in Fig. 4.13

at T “ 1.0 demonstrates the red domain could eventually conquer the yellow domain

in 5 ˆ 104 mcs. The subsequent configurations of the system are shown in Fig. 4.16.

The relative difficulty of moving the domain walls around and thus of one domain to

conquer the others is clearly determined by the energetics of the domain walls present

in the system. Due to the different nature of the two types of domain wall, the en-

ergetics associated with the two types of domain wall are also different. Clearly the

54

Figure 4.15: This is the same portion of the slanted domain wall of type I as inFig. 4.14 after dropping the temperature of the configuration in Fig. 4.13 to T “ 0.05and sitting for another 5 ˆ 104 mcs. There is no mismatches in black stripe alongthe domain wall.

domain walls of type I are energetically more favorable and represent local minima

on the free energy map of the tiling system. This is not surprising as a domain wall of

type I could minimize the number of mismatches by having no mismatches in black

stripes along the domain wall. To disrupt a domain wall of type I, the black triangles

on the two sides of the domain wall that are right next to the long black straight

line have to be broken. This would turn a long straight domain wall of type I into a

wiggly shape, setting the system free from the energy minima imposed by the domain

wall of type I. Part of a domain wall of type I could turn into a domain wall of type

II, which is much easier to move around and break through. However, the process

of breaking all the black triangles next to the domain wall could cost a significant

amount of energy. Thus a domain wall of type I could present an insurmountable

energy barrier. In the case discussed above, sitting at T “ 1.0 allows only the level 1

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(a) T “ 0 (b) T “ 1 ˆ 104

(c) T “ 2 ˆ 104 (d) T “ 3 ˆ 104

(e) T “ 4 ˆ 104 (f) T “ 5 ˆ 104

Figure 4.16: (a) A configuration of a size 64ˆ 64 system after a sudden quench toT “ 0.6 and sitting at that temperature for 5 ˆ 105 mcs. (b–f) Configurations aftera sudden increase of the temperature to T “ 1.0 and sitting for every 1 ˆ 104 mcs.The black and color stripes are omitted here.

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structure to develop. The energy of cost of breaking the level 1 triangles right next

to the domain wall of type I is not so high, plus the fact that the temperature is

high, the probability of having tiles along or right next to the domain wall go to ori-

entations that would disrupt the domain wall is high, determined by the Boltzmann

factor e´∆E{T , where ∆E is the change in energy of the move. Thus it is relatively

easy to break a type I domain wall. However as the temperature is decreased, more

and more levels of structure develope within each domain, and the energy cost of

breaking the higher levels of triangle right next to the domain wall becomes very high.

Combined with the effect of the low temperature, the Boltzmann factor e´∆E{T is too

low to allow a significant amount of tiles to go to orientations that would disrupt a

domain wall of type I. Hence, the type I domain walls become practically immovable.

Often a configuration resulted from a rapid quench would have a mixture of domain

walls of both types. As seen in examples shown in Fig. 4.2, one domain conquers an-

other most easily by breaking the domain walls of type II, whereas the type I domain

walls remain intact and stable unless they get consumed by domains expanding in

a direction that is parallel to the type I domain wall. It is found that for each pair

of colored domains, there exists a special direction in which the type I domain walls

could arise. In a large size system, many domains could be formed during a rapid

quench. The system tends to relax to a configuration in which the type I domain

walls establish themselves wherever possible in the special directions, while the type

II domain walls form in the other directions. At a given temperature, with more

type I domain walls present, it is more difficult to restore the system to equilibrium

as the type I domain walls slow down the ordering of the tiling system.

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5

Conclusion

The Socolar-Taylor tiling model uses a single space-filling prototile to force a limit-

periodic pattern through local matching rules. A two-dimensional lattice model

which possesses the this tiling as its ground state is constructed. During a slow

quench from an initial high-temperature, disordered phase, the ground state of the

model emerges through an infinite sequence of phase transitions following a strict

hierarchy. As temperature is decreased, sublattices with periodic structures of in-

creasing lattice constants become ordered. A theory based on one-dimensional Ising

model is constructed to explain the time scales required for equilibration at a given

temperature by sublattices of increasing lattice constants. Compared with the re-

sults from the simulation of the tiling mode, the theory successfully predicts the

ratio of the time scales required by the level 1 and 2 structures to reach equilib-

rium. However, discrepancies exist for higher levels of structure in predicting the

scaling behavior of the time scales to reach equilibrium. We see that the domain

dynamics determines the extreme slowness of the system’s relaxation to equilibrium

during thermal quenches. In building the theory based on Ising model to predict

the scaling behavior, we have assumed the domain dynamics is equivalent for all

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levels of structure. Even at level 2 and 3, it requires systems of very large sizes for

this assumption to be true, so that the finite size effect is negligible. However, the

sizes of the tiling model used for the level 2 and 3 simulation are only 64 ˆ 64 and

128ˆ128 respectively, which are not large enough to see the full domain dynamics of

the model. This is the most likely cause of the observed discrepancies in the scaling

behavior between the theory prediction and the results of simulation.

During a rapid quench, the energy barriers created by competing domain walls pre-

vent the system from getting further ordered and cause the system to fall out of

equilibrium. Two types of domain wall with different physical structures and energy

costs are found in the system. During a domain expansion, the associated energetics

makes the type I domain walls much harder to be broken through or moved around

than the type II domain walls. In a large size system, domain walls of type I are es-

tablished wherever possible in special directions during a rapid quench. The relative

difficulty of moving the type I domain walls slows down the relaxation of the system

to equilibrium.

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