Hierarchical freezing in a lattice model

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by  Travis W. Byington

Hierarchical freezing in a lattice model
∗
Travis W. Byington and Joshua E. S. Socolar
Physics Department, Duke University, Durham, NC 27708
(Dated: January 30, 2012)
A certain two-dimensional lattice model with nearest and next-nearest neighbor interactions is
known to have a limit-periodic ground state. We show that during a slow quench from the high
temperature, disordered phase, the ground state emerges through an infinite sequence of phase
2 transitions. We define appropriate order parameters and show that the transitions are related
1 by renormalizations of the temperature scale. As the temperature is decreased, sublattices with
0 increasingly large lattice constants become ordered. A rapid quench results in glass-like state due
2 tokinetic barriers created bysimultaneous freezing on sublattices with different lattice constants.

n
a
J A recent result in tiling theory [1] presents a new op- black triangles with arbitrarily large side lengths. We
6 portunityforstudyingthe developmentoflong-rangeor- emphasize that the system is completely homogeneous,
2 der in a systemwith a non-periodic groundstate. A sin- being a lattice of identical units.
gle hexagonal prototile has been shown to force a limit-

]
periodic pattern (meaning a state made up of the union
h
c of an infinite number of periodic structures with lattice STAGGERED TETRAHEDRAL ORDER
e constantsofever-increasingsizes[2–4]). Theexistenceof
m
a local Hamiltonian with a limit-periodic ground state Thekeytoexplainingthebehaviorofthelatticemodel
- composed of a single unit repeated in different orien- is to focus on the patterns of (truncated) black triangles
t
a tations suggests that it may be possible for solid state of different sizes in Fig. 1(c). We refer to the smallest
t
s materials, colloidal systems, or perhaps even collections triangles as making up level 1, the next smallest as level
t. of macroscopic units to realize structures of this type. 2, etc., and note that the edge of a triangle at level n is
a
A similar type of complexity occurs in quasicrystals and a straight black stripe crossing k −1 tiles, where k ≡
m n n
hasimplicationsforelectronic,photonic,elastic,andfric- 2n−1. At each level we have a periodic arrangement of
-
d tionalproperties.[5]Evenin the absence ofany physical trianglescenteredontheverticesofahoneycomblattice.
n examples of limit-periodic phases, however, the statis- The level-1 pattern may form in four different ways.
o tical mechanical properties of the new tiling model are OneexampleisshowninFig.2,inwhichthetilesmarked
c remarkableandmayserveasa conceptuallyusefulinter- A do not contribute at all to the triangles. The other
[
polation between crystalline and glassy behavior. three are translations of this one, each with a different

2 In this paper, we study the spontaneous formation of non-contributing sublattice (B, C, or D). We associate
v
the limit-periodic structure. Working with a substan- with each tile j a “staggered tetrahedral spin” vector
2
tially more complex tiling model with square symmetry, σ =~e , where X indicates one of the four vertices of
4 1,j X
5 Mi¸ekisz showed that a series of partially ordered equi- areferencetetrahedron. (SeeFig.2.) Thespinofatileis
6 librium states exist at nonzero temperatures in a system determined both by the orientationof the diameter join-
0. with a limit-periodic ground state [6]. Here, we explic- ing its two black triangle corners and by the sublattice
1 itly define order parameters for the transitions, present that it belongs to, according to the map shown at right
1 numerical evidence for ordering at slow cooling rates in Fig. 2. For example, a tile with corners aligned verti-
1 through an infinite sequence of phase transitions, and callyandsittingontheD sublatticeisassignedσ =~e .
1 C
v: present a scaling theory of the transition hierarchy. We Notethatspecifyingσ doesnotcompletelyspecifythe
1,j
i also show that rapid quenching produces a state with orientation of tile j. There are four consistent choices,
X
frozen disorder resulting from the competition between corresponding to the two possible locations of the long
r
a two or more levels of the hierarchy. black stripe and two possible orientations of the flag bar
Our model Hamiltonian is based on rules for placing perpendiculartothatstripe. Notealsothatforanygiven
hexagonaltiles ofthe type shownin Fig.1(a)on a close- tile, σ1 can take only three of the four possible values.
packed lattice. Each tile may be placed in any of the We define the average total spin σ ≡ 1 σ ,
1,tot N j 1,j
twelveorientationsobtainablebyrotationsbyπ/3andre- whereN isthenumberoftilesinthesystem. Thesystem
P
flection. Foreachnearestneighborpair,weassignenergy exhibitstetrahedralsymmetryinthe followingsense: for
zero if the black stripe is continuous across their shared each configuration with a given σ , there is another
1,tot
boundaryandǫ otherwise. Foreachnext-nearestneigh- withidenticalenergyhavingσ′ relatedtoσ byan
1 1,tot 1,tot
borpair,theenergyiszeroiftheflagsintheirclosestcor- operation in the 24-element tetrahedral group T . The
d
nerspointinthesamedirectionandǫ otherwise. Ref.[1] mappingfromoperationsonthelatticetoelementsofT
2 d
shows that for positive ǫ and ǫ , the ground state of an is given in Table I.
1 2
infinite system is a zero-energy structure that contains Considernowthe level-2triangles. Ifthe level-1struc-
2
Lattice operation Td operation on φ δT = 0.01 after every Nτ steps, where τ controls the
quench rate.
Rotation by2π/3 → Rotation by 2π/3
For the rapid quench (see Fig. 3), φ reaches its sat-
about centerof X about eX 1
uration value but φ does not. The ordering on level
2
2 ceases to increase once the level-3 transition temper-
Reflection through edge → Reflection through
ature is reached, at which point interactions associated
shared byX and Y (eX,eY) plane
with level 3 prevent further ordering of level 2. For the
slow quench, on the other hand, Fig. 4 shows a sequence
Translations taking → Rotations by π of clear transitions. The number of transitions that can
X sublattice to Y about eX+eY be observedislimited bythe systemsize. Atlevel5(not
shown), the edge length of a black triangle is 15 tiles, so
Rotation by2π/3 → Rotary inversion our 64×64 system is only 4×4 at level 5.
followed byreflection
Because the symmetry of the order parameter is the
same as that of a 4-state Potts model, we expect the
transition to fall in the same universality class, which
TABLEI.Symmetryoperationsforthetotalstaggeredtetra-
has an order parameter exponent β = 1/12. [7] Prelimi-
hedral spin. The left column specifies an operation on the
nary numerical investigations are consistent with a very
2D tiling pattern, where X,Y ∈{A,B,C,D} each represent
atileinthecorrespondingsublatticeofFig.2. Therightcol- sharp second order transition, having not revealed any
umnspecifies3Doperations ontheorderparameterinterms hysteresis in a cooling and heating cycle. None of the
of the tetrahedral star of vectors eX, where X is the label analysis below depends on the critical behavior or even
shown on Fig. 2. on the order of the transition.
SCALING THEORY
ture is perfectly ordered,the corners of the level-2 trian-
gles must come from the non-contributing sublattice at
The systemis consideredfully orderedatleveln when
level 1. That sublattice (e.g., all of the A’s in Fig. 2)
all of the level-n triangle corners occur in the pattern
is a precise copy of the original lattice (with distances
of Fig. 1(c). In the ordered pattern, at each level, flags
scaled by a factor of 2), so we can define a new stag-
in the tiles surroundinga non-contributingtile also form
gered tetrahedral spin σ ≡ 1 σ , where the
2,tot N/4 j 2,j triangular structures of the type shown at left in Fig. 2.
sum runs only over the sublattice of interest. The con-
P The long black stripes and the flag bars perpendicular
structioncanberepeatedadinfinitum, withσ being
n,tot to them, however, need not be in their ground state ori-
well-defined if and only if the ordering on level n−1 is
entations. Their role is only to mediate the interactions
sufficiently strong to unambiguously specify which sub-
between the triangle corners. We will see below that if
lattice will order at level n.
thetrianglecornersatalllevelsm<nareassumedtobe
perfectly ordered and immovable, the level-n dynamics
are identical to the level-1 dynamics at a rescaled tem-
FRUSTRATION AND ORDER IN QUENCHES perature (and possibly a rescaled ǫ ). We discuss first
2
the special case ǫ =ǫ ≡ǫ.
1 2
We define an order parameter for each level of struc- Thepartitionfunctionforthelevel-nsystemcanbeex-
ture: φ ≡max[e ·σ ],whereX ∈{A,B,C,D}and pressed as follows. A configuration of the system can be
n X n,tot
e isaunitvectorinthedirectionlabeled“X”inFig.2. specified by giving, for every edge of each hexagon, the
X
We take the magnitude σ of each spin to be 3/2 so that location of the black stripe meeting that edge and, for
themaximumφ isunityforalln. Themaximumφ oc- everyvertex,theorientationoftheflagatthatvertex. A
n 1
cursifandonlyifthelatticeofsmalltrianglesisperfectly configurationisallowedifandonlyifthespecificationfor
ordered, regardless of the orientations of the tiles on the everyhexagoncorrespondstosomeorientationofthetile
X sublattice. Figs.3and4showthebehaviorofφ with decorationofFig.1(a),independentofwhetherneighbor-
n
n = 1,2,3,4 for a rapid quench and for a slow quench. ing tiles match properly. At level n, the contribution to
The simulations were done on rhombic domain with pe- the partition function from a given configuration of the
riodic boundary conditions. Because the ground state is triangle corners is a product of contributions from all of
nonperiodic,theremustbesomemismatchesatalltimes. thenearestandnext-nearestneighborbondsbetweenthe
Fordomainsofsize2n×2n,thesmallestpossiblenumber corner tiles. Each of these bonds consists of k −1 tiles,
n
ofmismatchesis4. Thesimulationsemployedastandard with black stripes joining nearest neighbors of the level-
Metropolisalgorithminwhicharandomtile is chosenat nsublatticeandflagbarsjoiningnext-nearestneighbors.
each step and one of the twelve orientations of that tile Fig. 2 shows an example of each type of bond for the
is chosen randomly as a possible move. T is lowered by n=2 case. The dashed grey lines show the two possible
3
locations of the stripe on a triangle edge joining the cor- The analysis is more complicated for the generic case
nersthatwouldbepresentontheAsublattice. Thegrey ǫ < ǫ , but the essential phenomenology appears to be
2 1
diagonal shows the two possibe orientations of the flag the same. Let T (ǫ ,ǫ ) be the level-n transition tem-
c;n 1 2
bar that forms a triangle edge for next-nearest neighbor perature. Assumingthatdeviationsdiscussedinthepre-
interactions between two A tiles. vious paragraph can still be neglected, T must be an
c;n
The full partition function for the level-n system is increasing function of each of its arguments. Thus we
musthaveǫ /ǫ <T (ǫ ,ǫ )/T (ǫ ,ǫ )<1,wherethe
2 1 c;n 1 2 c;n 1 1
firstinequalityfollowsfromthe factthatasimple rescal-
Z (T)= ζ , (1)
n n,b ing of all energies gives T (ǫ ,ǫ )=(ǫ /ǫ )T (ǫ ,ǫ ).
! c;n 2 2 2 1 c;n 1 1
configurations bonds
X Y
where b ∈ {odd,even} represents the state of a given
DISCUSSION
bond (mismatched or matched) in the given configura-
tion. Each edge is effectively a 1D Ising system with k
n
possible mismatches (see Fig. 5(a)), which gives The ground state of the tiling model is discussed in
detailinRef. [1]. The complexityofthe structureis best
ζ = 1 1+e−ǫ/T kn − 1−e−ǫ/T kn ; revealedbythepatternoftileparities(thegreyandwhite
n,odd
2 in Fig. 1(c)). We have shown here that the model also
(cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
ζ = 1 1+e−ǫ/T kn + 1−e−ǫ/T kn . (2) has remarkable properties at finite temperatures. First,
n,even 2 itexhibitsahierarchyofdistinctthermodynamicphases,
(cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21) with each successive one corresponding to formation of
Now because the configuration sums are identical for
an ordered lattice with a lattice constant twice as large
all levels, the behavior of the system at level n at some
as the last. Second, each transition is unusual in that
temperatureT willbeidenticaltothatforlevel1attem-
n the lowtemperature phaseleavesonequarterofthe tiles
peratureT ifζ (T )=αζ (T )andζ (T )=
1 n,odd n 1,odd 1 n,even n as “rattlers” with undetermined orientations. Finally,
αζ (T ) for some normalization constant α. Elimi-
1,even 1 the kinetics of ordering become frustrated if the quench
natingαfromthese equationsyields,aftersomealgebra,
rateisfastenoughthatthetemperaturedropsbelowT
c;n
ǫ ǫ kn before φn−1 has reached a sufficiently high value. The
tanh = tanh (3) lattereffectisreminiscentofglassformation,wherelower
2T 2T
(cid:18) 1(cid:19) (cid:20) (cid:18) n(cid:19)(cid:21) energy states can be reached through slower quenching
butanynonzeroquenchrateeventuallyleadstotrapping
or, equivalently, for all n,
ina nonequilibriumconfiguration.[8] The presentmodel
ǫ ǫ 2 shows that this can happen in the context of a series of
tanh = tanh . (4)
transitions that each establish true long-range order.
2T 2T
(cid:18) n(cid:19) (cid:20) (cid:18) n+1(cid:19)(cid:21) Many questions remain concerning the precise nature
From Eq. (3), the transition temperature for large n is of the staggered tetrahedral phase transition, the pos-
sibility of reaching the ground state via growth from a
1 n
T = +O . (5) small seed, the geometry and energetics of topological
c;n nlog2−log log coth( ǫ ) 22n defects in this system, and the scaling of the time re-
2Tc;1 (cid:16) (cid:17)
quired for ordering at each level.
h h ii
OnemightworrythatEq.(5)willbreakdownbecause Todate,weknowofnoexamplesofspontaneousemer-
φ maynotbefullysaturatedatT . Nevertheless,as gence of limit-periodic order in a real system. The
n c;n+1
shown in Fig. 5, we obtain an excellent data collapse for present work suggests that such systems may exist and
severallevelsby plotting φ (T ) as a function ofT (T ). exhibitnovelproperties. Thehexagonaltilingconsistsof
n n 1 n
(Note: The×’sonthehightemperaturesideofthetran- a single unit that may be realized as a cluster of atoms,
sitionrepresentfinite-sizefluctuationsandcorrespondto asurfacepatternonacolloidalparticle,orevenamicro-
projections onto different tetrahedral vectors.) The col- machinedbrick. Itisalsopossiblethatsuchsystemshave
lapse shows that the residual disorder after the transi- already been observed but not recognized because they
tion at level n has a very small effect on the transition present as “poor” crystals in diffraction studies. (See
at level n + 1. For n = 1, the data for Fig. 4 show Ref.[9]foradiscussionofdiffractionfromalimit-periodic
clearly that φ is very close to unity (≈ 0.999) at the structure.) Systems of interest may be two-dimensional
1
critical temperature for φ . The scaling argument takes monolayersonaflatsubstrate,surfacereconstructionson
2
thisvaluetobeexactlyunity,inwhichcaseEq.(4)leads crystals, or bulk phases. See Refs. [1] for a 3D tile that
to φ (T )=φ (T ) for all n. If the small deviation enforcesthepresentstructurethroughitsshapealoneby
n c;n+1 1 c;2
from unity causes the scaling to break down at large n, allowing direct contact between next-nearest neighbors.
it appears that the difficulty will only emerge at lattice Mi¸ekisz’s study [10] based on an extension of Robin-
sizes too large to be probed computationally. son’s set of six Wang tiles [11] indicates that a limit-
4
periodicgroundstatewithsquaresymmetryshouldshow (W. H. Freeman & Co., New York,NY,USA,1986).
a similar sequence of transitions. See [12] and [13] for [3] J. E. S. Socolar and J. M. Taylor, The Math-
2Dand3Dexamplesoflimit-periodic tilingswithsquare ematical Intelligencer 34 (2012), online at
http://www.springerlink.com/content/v06145n476l13xp0/.
symmetry composed of only two types of tiles, which
[4] M. Baake, R. Moody, and M. Schlottmann, Journal of
maybemoreamenabletothetypeofanalysisperformed
Physics A: Mathematical and General 31, 5755 (1998),
above.
Seealso D.Frettl¨oh,PhDThesis, Universit¨atDortmund
WethankPatrickCharbonneau,TomLubensky,David (2002).
Huse, and Giulio Biroli for helpful conversations. This [5] C. Janot, Quasicrystals: a primer (Oxford University
work was supported by an Undergraduate Summer Fel- Press., New York,1997).
lowshipfromtheDukePhysicsDepartmentandtheNSF [6] J. Mie¸kisz, Phys.Lett. A 138, 415 (1989).
[7] F. Y.Wu,Reviews of Modern Physics 54, 235 (1982).
Research Triangle MRSEC (DMR-1121107).
[8] P. G. Debenedetti and F. H. Stillinger,
Nature 410, 259 (2001).
[9] M. Baake and U. Grimm,
Philosophical Magazine 91, 2661 (2011).
[10] J. Mie¸kisz, J. Stat. Phys. 58, 1137 (1990).
∗
[email protected] [11] R. Robinson, InventionesMathematicae 12, 177 (1971).
[1] J. E. S. Socolar and J. M. Taylor, Journal of Combina- [12] C.Goodman-Strauss,European Journal of Combinatorics 20, 375 (1999)
torial Theory: Series A 118, 2207 (2011). [13] C.Goodman-Strauss,European Journal of Combinatorics 20, 385 (1999)
[2] B. Gru¨nbaum and G. C. Shephard,Tilings and patterns
5
a
H L b
H L
c
H L
FIG. 1. The hexagonal prototile and its mirror image. (a) Thetwo tiles are related byreflection about a vertical line. (b) For
zeroenergy,adjacent tilesmustform continuousblackstripesandflagdecorations at oppositeendsofatileedge(asindicated
by thearrows) must point in the same direction. (c) A portion of an infinitetiling that has zero energy.
(cid:1)
(cid:2)
(cid:3)
(cid:1)
(cid:4)
FIG. 2. The sublattices employed in the definition of the order parameter and the mapping from tile orientations to spin
vectors. ThetilesoftheAsublatticearelabeled along withonetileeach oftheB,C,andD sublattices. Thefigureshowsthe
patternoflevel-1triangles formed whenthenon-contributingtilesarethoseoftheAsublattice. Forexplanation ofthecluster
shown at left, thedashed lines, and grey bar, see text.
6
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(cid:1)(cid:1)(cid:1)
Φn 0.5(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
0(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)
0 1 2
T
(cid:1) n(cid:2)1
(cid:1) n(cid:2)2
(cid:2) n(cid:2)3
FIG. 3. Right: The behavior of the order parameters for a rapid quench. Simulations were performed on a 64×64 rhombic
domain for ǫ1 =ǫ2 =1 andτ =120. Left: Theresult of arapid quenchon an 8×8lattice showing thehighdensity ofdefects
that persists at long times. There are nodefects, however, in thelevel-1 structure,consistent with the high value of φ1.
1(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
(cid:1)
n 0.5 (cid:4)
Φ (cid:4)
(cid:2)
(cid:1) (cid:1)
(cid:2)
(cid:4)
0 (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:1)(cid:4)(cid:2)
0 1 2
T
(cid:1) n(cid:2)1
(cid:1) n(cid:2)2
(cid:2) n(cid:2)3
(cid:4) n(cid:2)4
FIG. 4. Right: The behavior of the order parameters for a slow quench. Simulations were performed on a 64×64 rhombic
domain for ǫ1 =ǫ2 =1 and τ =12×105. Left: Theresult of a slow quenchon an 8×8 lattice showing theminimum possible
numberof defects.
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(cid:127)
n"3
n"4
n 0.5
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FIG.5. (a)Matchedandmismatchedcornerconfigurations,shownhereforlevel-3triangleedges. (b)ScalingcollapseofFig.4
data. The deviation of thelevel-4 points on theright is dueto thefinite size of thesystem.