Phase transitions in a system of hard rectangles on the square lattice

0.58 MB·

by  Joyjit Kundu

Phase Transitions in a System of Hard Rectangles on the Square Lattice
Joyjit Kundu and R. Rajesh
∗ †
The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India
(Dated: May 19, 2014)
4
1 Thephasediagramofasystemofmonodispersedhardrectanglesofsizem×mkonasquarelattice
0 is numerically determined for m=2,3 and aspect ratio k=1,2,...,7. We show theexistence of a
2 disorderedphase,anematicphasewithorientationalorder,acolumnarphasewithorientationaland
partial translational order, and a solid-like phase with sublattice order, but no orientational order.
y
Theasymptoticbehaviorofthephaseboundariesforlargek aredeterminedusingacombinationof
a
M entropic arguments and a Bethe approximation. This allows us to generalize the phase diagram to
largermandk,showingthatfork≥7,thesystemundergoesthreeentropydrivenphasetransitions
withincreasingdensity. Thenatureofthedifferentphasetransitionsareestablishedandthecritical
6
exponentsfor thecontinuous transitions are determined using finitesize scaling.
1

] PACSnumbers: 64.60.De,64.60.Cn,05.50.+q
h
c
e I. INTRODUCTION densitycolumnarphase. Thetransitioniscontinuousfor
m
m = 2 [24–28], and first order for m = 3 [25]. When
- Thestudyofentropy-drivenphasetransitionsinasys- m , keeping k fixed, the lattice model is equiva-
t → ∞
a tem of long hard rods has a long history dating back to lent to the model of oriented rectangles in two dimen-
st Onsager’s demonstration [1] that the three dimensional sionalcontinuum,alsoknownastheZwanzigmodel[29].
t. systemundergoesatransitionfromanisotropicphaseto For oriented lines in the continuum (k →∞), a nematic
a an orientationally ordered nematic phase as the density phase exists at high density [22]. The only theoretical
m
of the rods is increased [1–5]. Further increase in den- results that exist are when m = 1 and k = 2 (dimers),
- sity may result in a smectic phase that partially breaks for which no nematic phase exists [30–33], k 1, when
d ≫
translational symmetry, and a solid phase [6, 7]. In two the existence of the nematic phase may be proved rigor-
n
o dimensional continuum space, the continuous rotational ously[19],andanexactsolutionforarbitraryk onatree
c symmetry remains unbroken but the system undergoes like lattice [5, 23].
[ a Kosterlitz-Thouless type transition from a low-density Less is known for other values of m and k. Simula-

3 phase with exponential decay of orientational correla- tions of rectangles of size 2 5 did not detect any phase
×
v tions to a high-density phase having quasi long range transitionwithincreasingdensity[34],whilethoseofpar-
0 order [8–14]. Experimental realizations include tobacco allelepipeds on cubic lattice show layered and columnar
9 mosaic virus [15], liquid crystals [3], carbon nanotube phases, but no nematic phase [35]. In general, numeri-
5 gels [16], and brownian squares [17]. The phenomenol- calstudies oflargerectanglesareconstrainedbythe fact
5
ogy is, however, much less clear when the orientations that it is difficult to equilibrate the system at high den-
.
1 are discrete, and the positions are either on a lattice or sities using Monte Carlo algorithms with localmoves, as
0 inthecontinuum,wheneventheexistenceofthenematic thesystemgetsjammedandrequirescorrelatedmovesof
4
phase has been convincingly seen in simulations only re- several particles to access different configurations.
1
cently [18, 19].
: Inadditiontobeingthelatticeversionofthehardrods
v Consider hard rectangles of size m mk on a two-
problem, the study of lattice models of hard rectangles
i dimensionalsquarelattice whereeachre×ctangleoccupies
X is useful in understanding the phase transitions in ad-
m (mk) lattice sites along the short (long) axis. The
r sorbed monolayers on crystal surfaces. The (100) and
a limiting cases when either the aspect ratio k = 1 or (110) planes of fcc crystals have square and rectangular
m=1 are better studied. When m=1 and k 7, there
symmetryandmaybetreatedwithlatticestatisticsifthe
≥
are, remarkably, two entropy-driven transitions: from a
adsorbate-adsorbateinteractionisnegligiblewithrespect
low-density isotropic phase to an intermediate density
to the periodic variation of the corrugation potential of
nematic phase, and from the nematic phase to a high-
the underlying substrate [36]. For example, the critical
density disordered phase [18, 20]. While the first transi-
behavior of a monolayer of chlorine (Cl) on Ag(100) is
tion is in the Ising universality class [21, 22], the second
well reproduced by the hard square model (k = 1) and
transitioncouldbenon-Ising[20],anditisnotyetunder-
the highdensity c(2 2)structureof the Cl-adlayermay
stood whether the high density phase is a re-entrantlow ×
be mapped to the high density phase of the hard square
densityphaseoranewphase[20,23]. Whenk =1(hard
problem [37]. Structures such as p(2 2), c(2 2) and
squares), the system undergoes a transition into a high × ×
(2 1)areorderedstructures. Latticegasmodelwithre-
×
pulsiveinteractionuptoforthnearestneighborhasbeen
used to study the phase behavior of selenium adsorbed
∗ [email protected] on Ni(100) [38]. The results in this paper show that dif-
† [email protected] ferentphaseswithorientationalandpositionalordermay
2
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Before evaporation After evaporation
shaped particles. The study of these models also have
(a)
been of continued interest in Statistical Physics as min-
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four distinct phases at different densities: isotropic, ne- Before Flip After Flip
(b)
matic, columnar, and sublattice phases. These phases,
suitableorderparameterstocharacterizethem,andother
FIG.1. (Color online) An illustration of theMonte Carlo al-
thermodynamic quantities are defined in Sec. III. From
gorithm. (a) Configurations before andafter theevaporation
extensive large scale simulations, we determine the rich
of horizontal rectangles with head on a particular row (de-
phase diagram for m = 2,3, and k = 1,...,7. The noted by an arrow). Sites denoted by cross symbols cannot
phasediagramform=2 isdiscussedinSec.IV.We find beoccupiedbyhorizontalrectanglesinthenewconfiguration.
that all transitions except the isotropic-columnar tran- (b) An example of the flip move for rectangles of size 2×6.
sition for k = 6 are continuous. The critical exponents A rotatable or flippable plaquette of size 6×6, consisting of
and universality classes of the continuous transitions are three aligned rectangles, is shown by the dashed line. After
determined. Section V contains the details about the theflip move, thehorizontal rectangles become vertical.
phase diagram and the nature of the phase transitions
for m = 3. In Sec. VI, we use a Bethe approximation
and estimates of entropies for the different phases to de-
moves that was introduced in Refs. [20, 51] to study the
termine the phase boundaries for large k, allowing us
problem of hardrods (m=1). We describe the algorithm
to generalize the phase diagram to arbitrary m and k.
below. Givenavalidconfigurationofrectangles,inasin-
In particular, it allows us to take the continuum limit
gle move, a row or a column is chosen at random. If a
m , thus obtaining predictions for a system of ori-
→ ∞ rowischosen,then allhorizontalrectangleswhose heads
ented rectangles in the continuum. Sec. VII contains a
(bottom,leftcorner)lieinthatrowareremoved,leaving
summary and a discussion of possible extensions of the
the rectangles with heads in other rows untouched. The
problem.
emptiedrownowconsistsoftwokindsofsites: forbidden
sites that cannot be occupied with horizontal rectangles
duetothepresenceofverticalrectanglesinthesamerow
II. THE MODEL AND MONTE CARLO orduetorectangleswithheadsintheneighboring(m 1)
ALGORITHM −
rows,and sites that may be occupied by horizontalrect-
angles in a valid configuration. An example illustrating
We define the model on a square lattice of size L L the forbidden sites is shown in Fig. 1(a). It is clear that
×
with periodic boundary conditions. Consider a mono- the sites that may be occupied are divided into intervals
dispersed system of rectangles of size m mk such that ofcontiguous empty sites. The problemof occupationof
×
theaspectratioisk. Arectanglecanbeeitherhorizontal theemptiedrowwithanewconfigurationnowreducesto
orvertical. Ahorizontal(vertical)rectangleoccupiesmk the problem of occupying the empty intervals. However,
sitesalongthex(y)-axisandmsitesalongthey(x)-axis. theemptyintervalsmaybeoccupiedindependentofeach
No two rectangles may overlap. An activity z = eµ is other,astheoccupationofoneisnotaffectedbythecon-
associated with each rectangle, where µ is the chemical figurationofrectangles in the remaining ones. Thus, the
potential. re-occupation of the emptied row reduces to a problem
We study this system using constant µ grand canon- of occupying a one dimensional interval with rods. This
ical Monte Carlo simulations. The algorithm that we problem is easily solvable and the equilibrium probabili-
implementisanadaptationofthealgorithmwithcluster ties of each new configuration may be easily calculated.
3
WerefertoRefs.[20,51]forthecalculationoftheseprob- horizontal (vertical), then their heads, or bottom left
abilities. If a column is chosen instead of a row, then a corners, preferably lie in rows (columns) that are sep-
similar operationis performed for the verticalrectangles aratedbym. Thus,itbreaksthetranslationalsymmetry
whose heads lie in that column. in the direction perpendicular to the orientationbut not
In additionto the aboveevaporation-depositionmove, parallel to the orientation. Clearly, there are 2m sym-
we find that the autocorrelation time is reduced con- metric C phases. In this phase the rectangles can slide
siderably by introducing a flip move. In this move, a muchmorealongone lattice direction. The fourthphase
site (i,j) is picked at random. If it is occupied by the is the crystalline sublattice (S) phase with no orienta-
head of a horizontal rectangle, then we check whether tional order [see Fig. 2(f)]. We divide the square lattice
(i,j+m),(i,j+2m),...,(i,j+[k 1]m) sites are occu- into m2 sublattices by assigning to a site (i,j) a label (i
−
pied by the heads of horizontal rectangles. If that is the mod m)+m (j mod m). The sublattice labeling for
×
case, we call this set of k aligned rectangles a rotatable the casem=2 is showninFig.2(a). Inthe S phase,the
plaquette of horizontal rectangles. In the flip move such heads of the rectangles preferablyoccupy one sublattice,
a rotatable plaquette of size mk mk, containing k hor- breakingtranslationalsymmetryinboth the directions.
×
izontalrectangles,is replacedby a similar plaquette of k
From the symmetry of the system we would expect
verticalrectangles. Anexampleoftheflipmoveisshown
up to 7 phases. The orientational symmetry could be
inFig.1(b). If(i,j)is occupiedby the headofa vertical
presentorbrokenwhilethetranslationalsymmetrycould
rectangleanda rotatableplaquette ofverticalrectangles
beunbroken,brokenalongonlyoneorbothx-andy-di-
is present, then it is replacedby a plaquette of k aligned
rections. If the orientational symmetry is broken, then
horizontal rectangles. A Monte Carlo move corresponds
thetranslationalsymmetrycouldbebrokeneitherparal-
to 2L evaporation–deposition moves and L2 flip moves.
lelorperpendicular tothe preferredorientations. Outof
Itiseasytocheckthatthealgorithmisergodicandobeys
the 7 possibilities, we do not observe (i) a phase with no
detailed balance.
orientational order but partial translational order, (ii) a
Weimplementaparallelizedversionofthe abovealgo-
phasewithorientationalorderandcompletetranslational
rithm. In the evaporation-deposition move, we simulta-
order (iii) a smectic like phase in which orientationalor-
neously update all rows that are separated by m. Once
deris presentandtranslationalsymmetry parallelto the
allrowsareupdatedinthismanner,the columnsareup-
orientation is broken.
dated. We also parallelize the flip move. The lattice is
dividedintoL2/(m2k2)blocksofsizemk mk. Theflip- To distinguish among the four different phases we de-
×
ping of each of these blocks is independent of the other fine the following order parameters:
and may therefore be flipped simultaneously. We flip a
rotatableplaquettewithprobability1/2. Theparalleliza-
tionandefficiency ofthe algorithmallowsus to simulate
N N
large systems (up to L = 810) at high densities (up to Q1 =m2kh h− vi, (1a)
N
0.99).
We check for equilibration by starting the simulations Q =m2kh| jm=20−1nje2mπi2j|i, (1b)
with two different initial configurationsand making sure 2 P N
tthiaaltctohnedifitnioanl.eOqunielibcroinufimgusrtaattieonisiisnadefpuellnydneenmtaotfitchsetaintei-, Q =m2kh| mj=−01rje2πmij|−| mj=−01cje2πmij|i, (1c)
3 P N P
where all rectangles are either horizontal or vertical and
n n n +n
the other is a random configuration where rectangles of Q =m2kh 0− 1− 2 3i, (1d)
4
bothverticalandhorizontalorientationsaredepositedat N
random.
whereN andN arethe totalnumberofhorizontaland
h v
vertical rectangles respectively, n is the number of rect-
III. THE DIFFERENT PHASES i
angles whose heads are in sublattice i = 0,...,m2 1,
−
r are the number of rectangles whose heads are in row
Snapshots of the different phases that we observe in j
(j mod m), and c are the number of rectangles whose
simulations are shown in Fig. 2. First is the low density j
heads arein column (j mod m). All four orderparame-
isotropic (I) phase in which the rectangles have neither
ters are zero in the I phase. Q is non-zero in the N and
orientational nor translational order [see Fig. 2(c)]. Sec- 1
C phases, Q is non-zero in the C and S phases, Q is
ond is the nematic (N) phase in which the rectangles 2 3
non-zeroonly in the C phase, andQ is non-zeroonly in
have orientational order but no translational order [see 4
the S phase. Q in Eq. (1d) has been defined for m=2.
Fig. 2(d)] . In this phase, the mean number of horizon- 4
Its generalization to m 3 is straightforward.
tal rectangles is different from that of vertical rectan- ≥
gles. The third phase is the columnar (C) phase, having We now define the thermodynamic quantities that are
orientational order and partial translational order [see useful to characterize the transitions between the differ-
Fig. 2(e)]. In this phase, if majority of rectangles are ent phases. Q ’s second moment χ , compressibility κ
i i
4
SL0 (H) ν defined by Q ( ǫ)β, ǫ < 0, χ ǫ−γ, κ ǫ−α,
2 3 2 3 SL1 (H) and ξ ǫ−ν, wh∼ere−ξ is the correlat∼ion| |length,∼ǫ| | 0,
∼| | | |→
SL2 (H) and Q representsany of the orderparameters. Only two
0 1 0 1
SL3 (H) exponents areindependent, others being relatedto them
SL0 (V) through scaling relations.
2 3 2 3
SL1 (V) The critical exponents α, β, γ and ν are obtained by
SL2 (V) finite sizescalingofthe differentquantitiesnearthe crit-
0 1 0 1
SL3 (V) ical point:
(a) (b) U f (ǫL1/ν), (3a)
u
≃
Q L β/νf (ǫL1/ν), (3b)
− q
≃
χ Lγ/νf (ǫL1/ν), (3c)
χ
≃
κ Lα/νf (ǫL1/ν), (3d)
κ
≃
where f , f , f , and f are scaling functions.
u q χ κ
IV. PHASE DIAGRAM AND CRITICAL
BEHAVIOR FOR m=2
(c) (d) In this section, we discuss the phase diagram for the
case m = 2 and aspect ratio k = 1,2,...7. The critical
exponents characterizing the different continuous transi-
tions are determined numerically.
A. Phase Diagram
Thephasediagramobtainedfromsimulationsform=
2 and integer k are shown in Fig. 3. The low density
phase is an I phase for all k. The case k =1 is different
fromotherk. Inthiscase,theproblemreducestoahard
(e) (f)
square problem and orientational order is not possible
as there is no distinction between horizontal and verti-
FIG. 2. (Color online) Snapshots of different phases (at dif- cal rectangles. The hard square system undergoes only
ferent densities) in a system of 2×14 hard rectangles. (a) onetransitionwithincreasingdensity,itbeingacontinu-
The four sub lattices when m = 2. (b) The color scheme:
ous transitionfrom the I phase to a C phase [24, 44–46].
eightcolorscorrespondingtotwoorientations,horizontal(H)
This transition belongs to the Ashkin Teller universality
and vertical (V),and heads of rectangles beingon oneof the
class(seeRefs.[25–28]forrecentnumericalstudies). For
four sublattices, denoted by SL0 to SL3. (c) The isotropic
k = 2,3, we find that the system undergoes one contin-
phase where all 8 colors are present. (d) The nematic phase,
dominatedby4colorscorrespondingto4sublatticesandone uous transition directly from the I phase to a crystalline
orientation. (e) The columnar phase, dominated by 2 colors S phase. On the other hand, the system with k =4,5,6
correspondingtotwosublatticesandoneorientation. (f)The may exist in I, C, or S phases. With increasing density,
sublatticephase,dominatedby2colorscorrespondingtoone the system undergoes two phase transitions: first from
sublattice and 2 orientations. the I to a C phase which could be continuous or first or-
der,andsecond,fromtheCtoaSphasewhichiscontin-
uous. For k =7, we observethree continuoustransitions
and the Binder cumulant U are defined as
i
with increasing density: first from the I to the N phase,
χi =hQ2iiL2, (2a) second into the C phase and third into the S phase. By
κ=[ ρ2 ρ 2]L2, (2b) confirmingtheexistenceoftheNandCphasesfork =8,
h i−h i
we expect the phase behavior for k 8 to be similar to
U =1 hQ4ii . (2c) that for k =7. ≥
i − 3 Q2 2
h ii The system undergoes more than one transition only
The transitions are accompanied by the singular be- for k k = 4. We now present some supporting
min
≥
havioroftheabovethermodynamicquantitiesatthecor- evidence for this claim. In Fig. 4 (a) we show the prob-
responding critical densities. Let ǫ=(µ µ )/µ , where ability distribution of the nematic order parameter Q ,
c c 1
−
µ is the critical chemical potential. The singular be- when k = 4, for different values of µ and fixed L, close
c
havior is characterizedby the critical exponents α, β, γ, totheI-Candthe C-Stransitions. Forlowervaluesofµ,
5
8 12
10 (a) L=120 (b)
N L=240
9 L=360
7 8
C )1 8 µ=6.30 )1
Q µ=6.45 Q
6 P( 7 µ=6.50 P(
µ=6.60 4
6 µ=6.75
µ=6.85
5 5 µ=7.00
0
k -0.1 -0.05 0 0.05 0.1 -0.4 -0.2 0 0.2 0.4
Q Q
4 1 1
I HD FIG.4. (Coloronline)(a)Theprobabilitydistributionofthe
3 orderparameterQ1 fork=4atdifferentµvalues,whenL=
416. (b) The same for different system sizes when µ=6.55.
2
B. Critical behavior for the isotropic-sublattice
1
(I-S) phase transition
0.7 0.75 0.8 0.85 0.9 0.95 1
ρ The system of rectangles with m = 2 undergoes a di-
rect I-S transition for k = 2,3. At this transition, the
translationalsymmetrygetsbrokenalongbothx- andy-
FIG. 3. (Color online) Phase diagram for rectangles of size directionbuttherotationalsymmetryremainspreserved.
2×2k. I, N, C and HD denote isotropic, nematic, columnar We study this transition using the order parameter Q
2
and high density phases respectively. The HD phase is a C [see Eq. (1b)]. Q is non-zero in the S phase and zero
2
phase for k = 1 and a S phase for k > 1. The data points in the I phase. In this case Q and Q remains zero for
1 3
are from simulation, while the continuous lines and shaded all values of µ. The data collapse of U , Q , and χ for
2 2 2
portions are guides to the eye. The shaded portion denotes
different values of L near the I-S transition are shown in
regions of phase coexistence.
Fig.5fork=2andinFig.6fork =3. Fromthecrossing
of the Binder cumulant data for different L, we estimate
the critical chemical potential µ 5.33 (ρ 0.930)for
c c
≈ ≈
k = 2 and µ 6.04 (ρ 0.925) for k = 3. The order
c c
≈ ≈
parameter increases continuously with µ from zero as µ
c
iscrossed,makingthetransitioncontinuous. SincetheS
the distribution is peaked around zero corresponding to phase has a four fold symmetry due to the four possible
the Iphase. With increasingµ, thedistributionbecomes sublattices,weexpectthetransitiontobeintheAshkin-
flat and two symmetric maxima appear at Q = 0 (Q Teller universality class. Indeed, we find a good collapse
1 3
6
also becomes nonzero simultaneously), corresponding to with β/ν = 1/8 and γ/ν = 7/4. Numerically, we find
a C phase. On increasing µ further, the two maxima ν =1.18 0.06 for k =2 and ν =1.23 0.07 for k =3.
± ±
continuously merge into a single peak at Q =0, corre- ν being larger than 1, we do not observe any divergence
1
sponding to the S phase (Q also becomes zero and Q in κ. This transition could have also been studied using
3 4
becomes nonzero). Fig. 4 (b) shows the distribution of the order parameter Q .
4
Q forthreedifferentsystemsizesatafixedvalueofµfor
1
whichP(Q )hastwosymmetricmaximaatQ =0. The
1 1
6
two peaks become sharper and narrower with increasing
C. Critical behavior of the isotropic-columnar
L. We find the similar behavior for P(Q ) also. From
3 phase (I-C) transition
the above,we conclude that the C phase exists for k =4
albeit for a very narrow range of µ. For k = 2 and 3
The I-C transition is seen for k = 4,5,6. When for
wedo notobservethe existenceofa columnarphase and
k = 4, µ for the I-C and the C-S transitions are close
find that the probability distributions of Q and Q are c
1 3
to each other, making k =4 unsuitable for studying the
peaked around zero for all µ. Hence, we conclude that
criticalbehavior. We, therefore,study the I-C transition
k =4.
min
fork =5(2 10rectangles)andk =6(2 12rectangles).
× ×
The critical behavior is best studied using the order
The N phase exists only for k 7. This is also true parameter Q [see Eq. (1c)]. Q is non-zero only in the
3 3
≥
form=1[18]. Toseethis,noticethattheI-Ctransition Cphase. First,wepresentthecriticalbehaviorfork =5.
for k = 6 is first order (see Fig. 3). If a nematic phase The simulation data for different system sizes are shown
existsfork =6,thenthefirsttransitionwouldhavebeen in Fig. 7. From the crossing of the Binder cumulant
continuous and in the Ising universality class [21]. curves, we obtain µ 4.98 (ρ 0.876). The transi-
c c
≈ ≈
6
0.7
0.6
(a) 0.6 (b)
0.6 0.6 (b)
(a) 0.4
0.4
2 2 3 3
U 0.5 U 0.5 U 0.2 U
0.2
L=120 L=120 L=400 L=400
0.4 L=240 L=240 0 L=600 L=600
L=360 0.4 L=360 L=800 0 L=800
-0.2
4 4.5 5 5.5 6 6.5 -2 -1 0 1 2 3 4 4.8 4.9 5 5.1 1 2 3 4 5
µ ln |ε L1/ν| µ ln |ε L1/ν|
2
L=120 3 L=120
1.6 L=240 L=240 1.6 (c) 3 (d)
L=360 (c) L=360 (d)
βν/QL2 01..82 γν-/χ L2 1.5 βν/Q| L3 01..82 γν-/χ L3 1.5
| L=400 L=400
0.4 0.4 L=600 L=600
L=800 L=800
0 0 0
-2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 1 2 3 4 5 1 2 3 4 5
ln |ε L1/ν| ln |ε L1/ν| ln |ε L1/ν| ln |ε L1/ν|
FIG. 5. (Color online) Data collapse for different L near the FIG. 7. (Color online) The data for different L near the I-C
I-S transition for rectangles of size 2×4 (m=2, k=2). We transition collapse when scaled with exponents β/ν = 1/8,
find µc ≈ 5.33 (ρc ≈ 0.93). The exponents are β/ν = 1/8, γ/ν = 7/4, and ν = 0.82. We find µc ≈ 4.98 (ρc ≈ 0.876).
γ/ν =7/4 and ν ≈1.18. Data are for rectangles of size 2×10.
class. The data for different L collapse with β/ν = 1/8,
0.6 (a) 0.6 (b)
γ/ν = 7/4 and ν = 0.82 0.06 (see Fig. 7), confirming
±
the same. Unlike the I-S transition for k = 2,3, ν < 1
U2 U2 0.5
0.5 and lies between the Ising and q=4 Potts points. At the
L=120 L=120 I-C transition partial breaking of translational symme-
0.4
L=240 L=240 try and complete breaking of rotationalsymmetry occur
0.4 L=360 L=360
simultaneously.
5.2 5.6 6 6.4 -2 0 2 4
µ L0/ν ln |ε L1/ν| The I-C transition for k = 4 is also continuous [see
1.6 Fig.4(a)],andisthereforeexpectedtobeintheAshkin-
2 Teller universality class. However, there is no reason to
(c) (d)
1.2 expect that ν will be the same as that for k =5.
ν ν 1.5
β/Q L2 0.8 γ-/χ L2 1 FigF.or8(ka)=s6h,owthsetIh-Ce ttirmanespitrioofinleisosfurdpernissiitnyglnyefiarrstthoerdIe-Cr.
L=120 L=120 transition. ρ alternates between two well defined densi-
0.4 L=240 0.5 L=240
ties, one corresponding to the I phase and the other to
L=360 L=360
0 the C phase. This is also seen in the probability distri-
-2 0 2 4 -2 0 2 4
ln |ε L1/ν| ln |ε L1/ν| butionfordensity[seeFig.8(b)]. NeartheI-Ctransition
it shows two peaks corresponding to the I and the C
FIG. 6. (Color online) Data collapse for different L near the phases. Thus, at µ = µIcC, the density has a disconti-
I-S transition for rectangles of size 2×6 (m=2, k=3). We nuity, which is shown by the shaded region in the phase
find µc ≈ 6.04 (ρc ≈ 0.925). The exponents are β/ν = 1/8, diagram(see Fig. 3). The probability distribution of the
γ/ν =7/4 and ν ≈1.23 orderparameterQ showssimilarbehavior[seeFig.8(c)].
3
. Nearµ=µIC the distributionshowsthreepeaks: oneat
c
Q = 0 corresponding to the I phase and the other two
3
at Q = 0, corresponding to the C phase. At µIC the
3 6 c
tion is found to be continuous. There are four possible three peaks become of equal height and the order pa-
columnar states: majority of heads are either in even rameter Q jumps from zero to a non-zero value. These
3
or odd rows (when horizontal orientation is preferred), peakssharpenwithincreasingsystemsize[seeFig.8(d)].
or in even or odd columns (when vertical orientation is These are typical signatures of a first order transition.
preferred). Due to this four fold symmetry, we expect Hence, we conclude that the I-C transition may be con-
the I-C transitionto be in the Ashkin-Teller universality tinuous or first order depending on k.
7
(a) µ=3.02 120 (b) µµ==33..0012 0.6
0.82 µ=3.03 0.6
ρ 0.81 ρP() 80 U1 0.4 (a) U1 0.4 (b)
L=448 L=448
40
L=560 0.2 L=560
0.8 0.2 L=672 L=672
L=784 L=784
0 0
3e+07 4e+07 5e+07 0.8 0.81 0.82 1.6 1.7 1.8 1.9 -2 -1 0 1 2 3 4 5
t ρ µ ln |ε L1/ν|
µ=3.01 4 L=240
6 µµ==33..0023 (c) 3 (d) L=576 2 3.6 LL==454680
1.6 L=672
P(Q)3 4 P(Q)3 2 βν/Q| L1 01..82 L=448 (c) γν-/χ L1 2.4 L=784 (d)
2 1 | L=560 1.2
0.4 L=672
L=784
0 0
-0.8 -0.4 0 0.4 0.8 -0.8 -0.4 0 0.4 0.8 0 0
-2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5
Q3 Q3 ln |ε L1/ν| ln |ε L1/ν|
FIG. 8. (Color online) Data for the density ρ and the order
FIG. 9. (Color online) The data for different L near the I-
parameter Q3 for rectangles of size 2×12. (a) Equilibrium
N transition collapse when scaled with the Ising exponents
time profile of ρ near the I-C transition, for µ = 3.02 and
L = 720. Probability distribution, near the I-C transition, β/ν = 1/8, γ/ν = 7/4, ν = 1 and µc ≈ 1.77. The critical
of (b) ρ for different values of µ when L = 576, (c) Q3 for density ρc ≈0.746. Data are for rectangles of size 2×14.
different values of µ when L = 576, and (d) Q3 for L = 240
(µ=2.99) and L=576 (µ=3.02).
orientationalsymmetrygetsbroken. Ifthenematicphase
consists of mostly horizontal (vertical) rectangles, then
D. Critical behavior of the isotropic-nematic phase thereisnopreferenceoverevenandoddrows(columns).
(I-N) transition In the columnar phase the system chooses either even
or odd rows (columns), once the orientationalsymmetry
is broken. Due to the two broken symmetry phases we
We find that the nematic phase exists only for k
≥ expectthistransitiontobeintheIsinguniversalityclass.
7. We study the I-N phase transition for k = 7 using
We indeed find gooddata collapsewhen U , Q andχ
the order parameter Q [see Eq. (1a)]. Q is non-zero 3 3 3
1 1 | |
for different system sizes are scaledwith Ising exponents
in the N and C phases and zero in the I phase. We
(see Fig. 10). The critical chemical potential or critical
confirmthattheorderedphaseisanNphasebychecking
density is obtained from the crossing point of the binder
that Q , which is non-zero only in the C phase, is zero.
3
cumulant U for different L. We find µ 1.92 (ρ
In the nematic phase the rectangles may choose either 3 c c
≈ ≈
0.766)forthistransition. Weexpectthecriticalbehavior
horizontal or vertical orientation. Thus, we expect the
to be same for k >7.
transition to be in the Ising universality class. When
m=1,thishasbeenverifiedusingextensiveMonteCarlo
simulations [21]. Here, we confirm the same for m = 2.
ThedataforU , Q andχ fordifferentLcollapseonto F. Critical behavior of the columnar-sublattice
1 1 1
| |
one curve when scaled with the two dimensional Ising phase (C-S) transition
exponents β/ν =1/8,γ/ν =7/4,andν =1 (see Fig.9).
We find µc 1.77 (ρc 0.746). We note that the value The C-S transition exists for k 4. This transitions
ofU2atthe≈pointwher≈ethecurvesfordifferentLcrossis is studied by choosing k = 5. W≥e characterize the C-
slightlysmallerthanthe Isingvalue0.614. This suggests S transition using the order parameter Q which is non
4
that larger system sizes are necessary for better collapse zero only in the S phase. In the C phase the system
of the data. choosesoneparticularorientationandeitherevenorodd
rowsorcolumns,dependingontheorientation. Thiscor-
responds to two sublattices being chosen among four of
E. Critical behavior of the nematic-columnar phase them. In the C-S transition the translational symme-
(N-C) transition try gets brokencompletely by choosing a particularsub-
lattice, but along with that the orientational symmetry
The N-C transition is also studied for k = 7, using gets restored. This transition is found to be continuous.
the orderparameterQ . Q is zeroin the nematic phase The data of U , Q and χ for different L near the C-S
3 3 4 4 4
| |
butnonzerointhecolumnarphase. AttheI-Ntransition transition collapse well when scaled with the exponents
8
L=300
0.6 0.6 0.6 L=400 0.6
(a) (b) L=600
0.4
0.4 0.4 0.4
3 3 4 (a) 4 (b)
U U 0.2 U U
0.2 L=448 L=448 0.2 0.2
L=560 L=560 L=300
0
L=672 L=672 L=400
0 L=784 L=784 0 0 L=600
-0.2
1.8 1.85 1.9 1.95 2 -1 0 1 2 3 4 8.8 9.2 9.6 10 0 1.5 3 4.5
µ ln |ε L1/ν| µ L0/ν ln |ε L1/ν|
L=448 L=300 L=300
1.6 3 L=560 1.2 L=400 1.5 L=400
(c) L=672 (d) L=600 L=600
βν/Q| L3 01..82 L=448 γν-/χ L3 1.5 L=784 βν/Q| L4 0.8 (c) γν-/χ L4 1 (d)
| L=560 | 0.5
0.4
0.4 L=672
L=784
0 0
-1 0 1 2 3 4 -1 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
ln |ε L1/ν| ln |ε L1/ν| ln |ε L1/ν| ln |ε L1/ν|
FIG. 10. (Color online) The data for different L near the FIG.11. (Coloronline)ThedatafordifferentLneartheC-S
N-Ctransition collapse whenscaled withtheIsingexponents transition collapse when scaled with exponents β/ν = 1/8,
β/ν = 1/8, γ/ν = 7/4, ν = 1 and µc ≈ 1.92. The critical γ/ν = 7/4, ν = 0.83 and µc ≈ 9.65. The critical density ρc
density ρc ≈0.766. Dataare for rectangles of size 2×14. ≈0.958. Data are for rectangles of size 2×10.
belonging to the Ashkin-Teller universality class. The
estimated critical exponents are β/ν = 1/8, γ/ν = 7/4 matches with that for m=1 [18].
and ν = 0.83 0.06 (see Fig. 11). Binder cumulants for
±
different system sizes cross at µ 9.65 (ρ 0.958).
c c
≈ ≈
We expect similar behavior for k=4 and 6 but possibly
with different ν. The C-S transition occurs at very high
density. With increasing k, the relaxation time becomes B. The isotropic-columnar phase (I-C) transition
increasingly large, making it difficult to obtain reliable
data for the C-S transition when k 6.
≥ The I-C transition exists when k 6. We study the
≤
this transition for k = 6, using the order parameter Q .
3
NowtherearesixpossiblechoicesfortheCphase: heads
V. PHASE DIAGRAM AND CRITICAL
are predominantly in one of the rows 0,1 or 2 (mod 3)
BEHAVIOR FOR m=3
with all the columns equally occupied (if horizontal ori-
entation is preferred) or in one of the columns 0,1 or 2
A. Phase diagram (mod 3) and all the rows are equally occupied (if ver-
tical orientation is preferred). Making an analogy with
Thephasediagramthatweobtainform=3,isshown the six state Potts model, we expect the I-C transition
in Fig. 12. When k = 1, the corresponding hard square tobefirstorder. Theprobabilitydistributionoftheden-
system has a single, first order transition from the I sity ρ and the order parameter Q for k = 3 near the
3
| |
phase into the C phase [25]. The shaded region between I-C transition is shown in Fig. 13. The distributions are
two points denotes a region of phase coexistence. For clearly double peaked at and near the transition point,
2 k 6, the system undergoes two first order transi- one corresponding to the I phase and the other to the
≤ ≤
tions with increasing density: first an I-C transition and C phase. We find that these peaks become sharper with
second a C-S transition. This is unlike the case m = 2, increasing system size. This is suggestive of a first order
where for k = 2 and 3 we find only one transition. For phasetransitionwithadiscontinuityinbothdensityand
k = 7, we find three transitions as in the m = 2 case. orderparameterasµcrossesµ . Thediscontinuityinthe
c
ThefirsttransitionfromItoNphaseiscontinuouswhile density is denoted by the shaded regions of Fig. 12. The
the second from N to C phase appears to be first order. chemical potential at which the I-C transition occurs is
Although we cannot obtain reliable data for the third given by, µ 3.93. Similar behavior is seen near the
∗IC ≈
transitionintotheSphase,weexpectittobefirstorder. I-Ctransitionforrectanglesofsize3 3kwithk =2,3,4
×
We note that the minimum value of k beyond which the and 5. We observe that the discontinuity in the density
nematicphaseexistsis 7forbothm=2andm=3,and increases with k.
9
2.5 (a) (b)
N 0.6
7
2
6 C P(Q)1 1 .15 U1 0.4 L=420
µ=2.72 L=630
5 0.5 µµ==22..7860 0.2 L=756
µ=2.84 0.614
0
-0.8 -0.4 0 0.4 0.8 2.7 2.8 2.9 3
k 4 Q1 µ
I H FIG. 14. (Color online) The I-N transition for rectangles of
3
D size 3×21. (a) Distribution of the order parameter Q1 near
the I-N transition. The data are for L = 420. (b) Binder
2 cumulantfordifferentsystemsizescrossesatµc ≈2.92(ρc≈
0.79). Valueof U at µc is ≈0.61.
1
Isinguniversalityclass. InFig.14(a),thedistributionof
0.75 0.8 0.85 0.9 0.95 1 theorderparameterQ1 nearthe I-Ntransitionisshown.
ρ
Thetwosymmetricpeaksofthedistributioncomecloser
with decreasingµ and merge to a single peak, this being
a signature of a continuous transition. The Binder cu-
FIG. 12. (Color online) Phase diagram for rectangles of size mulant U for different system sizes cross at µ 2.92
1 c
3×3k. HDdenoteshighdensity. TheHDphaseisaCphase (ρ 0.79)[seeFig.14(b)]. ThevalueofU atµ=≈µI N
for k =1 and a S phase for k>1. The data points are from c ≈ c−
is very close to the U value (0.61) for the Ising univer-
simulationwhilethecontinuouslineandshadedportionsarea c
sality class.
guidetotheeye. Theshadedportionsdenoteregionsofphase
coexistence. ExcepttheI-Ntransition, allthetransitionsare
found to be first order.
D. The nematic-columnar phase (N-C) transition
20
100 (a) µ=3.91 (b) µ=3.91 The N-C transition is studied for k = 7 using the or-
µ=3.93 µ=3.93
80 µ=3.94 15 µ=3.94 der parameter Q3. Contrary to our expectation that
the N-C transition should be in the q=3 Potts univer-
ρ) 60 Q|)3 10 sality class, we observe a first order transition. The
P( 40 P(| temporaldependence ofthe density nearthe N-C transi-
5 tion is shown in Fig. 15(a). Density jumps between two
20
well separated values corresponding to the two different
0 0
0.8 0.82 0.84 0.86 0 0.3 0.6 0.9 phasesnearthe coexistence. Fig 15(b) showsthe discon-
ρ |Q3| tinuity in the order parameter Q3 near the transition.
| |
P(Q ) shows two peaks of approximately equal height
3
| |
FIG.13. (Coloronline)Distributionof(a)thedensityρnear nearµ 3.12. However,wearelimitedinourabil-
∗N C
the I-C transition, (b) the order parameter |Q3| near the I- ity to obt−ain≈reliable data for 3 21 rectanglesfor larger
C transition. The data are for rectangles of size 3×18 and system sizes, and the observed×first order nature could
L=432.
be spurious.
C. Critical behavior of the isotropic-nematic phase E. The columnar-sublattice phase (C-S) transition
(I-N) transition
The C-S transition is studied by choosing k = 2. We
Asform=2,form=3wefindtheexistenceofthene- use the order parameter Q which is nonzeroonly in the
4
matic phase only for k 7. We study the I-N transition S phase. The probability distribution of the density ρ
≥
for k = 7 with the order parameter Q . It is expected and the order parameter Q for 3 6 rectangles near
1 4
| | ×
to be in the Ising universality class since there are two the C-S transitions is shown in Fig. 16. The distribu-
possible choices of the orientation: either horizontal or tions are againdouble peaked at and near the transition
vertical. We are unable to obtain good data collapse for point,makingtheC-Stransitionfirstorder. Thesepeaks
Q , χ and U as the relaxation time increases with becomesharperwithincreasingsystemsize. Thediscon-
1 1 1
| |
increasing m and k. Instead, we present some evidence tinuityinthedensityneartheC-Stransitionisverysmall
for the transition being continuous and belonging to the and can also be seen in the shaded portions of Fig. 12.
10
(a) µ=3.16 8 (b) µ=3.10 B. The nematic–columnar phase boundary
0.84
µ=3.12
6 µ=3.16
To obtain the asymptotic behavior of the N-C phase
ρ 0.83 Q|)3 4 boundary, we use an ad hoc Bethe approximation
P(| scheme for rods due to DiMarzio [53], adapted to other
2 shapes [54]. To estimate the phase boundary of the
0.82 nematic–columnar transition of m mk rectangles on
×
0 the square lattice with M = L L sites, we require
0 5e+07 1e+08 0 0.4 0.8 ×
the entropy as a function of the occupation densities
t |Q |
3
of the m types of rows/columns. The calculations be-
come much simpler, if we consider a fully oriented phase
FIG.15. (Coloronline)(a)Temporalvariationofdensitynear
with only horizontalrectangles. Now, the nematic phase
the N-C transition, (b) distribution of the order parameter
|Q3| near the N-C transition. The data are for rectangles of corresponds to the the phase where there is equal occu-
size 3×21 and L=756. pancyofeachofthemtypesofrows,whilethecolumnar
phase breaks this symmetry and preferentially occupies
one type of row. For this simplified model with only one
1000 (a) µ=9.30 5 (b) µ=9.30 orientation, we estimate the entropy within an ad hoc
µ=9.33 µ=9.33
800 µ=9.36 4 µ=9.36 Bethe approximation as detailed below. We present the
calculation for m = 2, classifying the rows as even and
ρ) 600 Q|)4 3 oddrows. Generalizationtohighervaluesofmisstraight
P( 400 P(| 2 forward.
200 1 LettherebeNe(No)numberofrectangleswhoseheads
(left bottom site of the rectangle) occupy even (odd)
0 0
0.978 0.98 0.982 0 0.2 0.4 0.6 0.8 rows. We firstplace the Ne rectangles one by one on the
ρ |Q4| even rows. Given that je rectangles have already been
placed,thenumberofwaysinwhichthe(j +1)th rectan-
e
FIG.16. (Coloronline)Distributionof(a)thedensityρnear glecanbeplacedmaybe estimatedasfollows. Thehead
theC-S transition and (e) theorder parameter |Q4| near the of the (j +1)th rectangle has to be placed on an empty
e
C-S transition. The data are for rectangles of size 3×6 and siteofanevenrow. WedenotethissitebyA(seeFig.17).
L=720.
ThesiteAcanbechosenatrandomin(M/2 2kj )ways,
e
−
M/2beingthenumberofsitesinevenrowsand2kj be-
e
ing the number of occupied sites in the evenrows by the
Weestimateµ 9.33. SimilarbehaviorneartheC-S
transitions is a∗Cls−oSo≈bserved for k >2, but the relaxation je rectangles. Wenowrequirethatthe2k−1consecutive
sites to the right of A are also empty. The probability
time increases with k. of this being true is [P (B A)]2k 1, where P (B A) is
x − x
| |
the conditionalprobability thatB (see Fig. 17) is empty
given that A is empty. In terms of M and j , P (B A)
e x
|
VI. ESTIMATION OF THE PHASE is given by
BOUNDARIES USING ANALYTICAL METHODS
M 2kj
P (B A)= 2 − e . (4)
x | M 2kj +j
In this section we obtain the asymptotic behavior of 2 − e e
the phase diagram for large k using theoretical argu-
To place the (j +1)th rectangle, we also require that
ments. e
the site C (see Fig. 17) and the 2k 1 consecutive sites
−
to the right of C are also empty. The probability of
this being true is P (C A) P (D B C)2k 1, where
y xy −
| × | ∩
A. The isotropic–nematic phase boundary Py(C A) is the conditional probability that C is empty
|
given A is empty, and P (D B C) is the conditional
xy
| ∩
probabilitythatD (seeFig.17)isemptygiventhatboth
The critical density for the I-N phase transition, for
BandC areempty. SitesCandDbelongstoanoddrow.
fixed m and k 1 may be determined by making an
≫ SincebothAandBareempty,CandDcanbe occupied
analogywith the continuumproblem. The limit k ,
→∞ onlybyrectangleswithheadsinthe sameoddrow. But,
keeping m fixed corresponds to the system of oriented
there are no such rectangles. Therefore, P (C A) = 1,
l5i2n]e.sTinhethceocnosntatinntuAum1 .isFeosrtitmhiastepdrotbolebmeρ≈c ≈6.0A[11/8k, 5[128]., ahnavdePbxeye(nDp|Blac∩edC,)th=e (1j. +Th1u)tsh, rgeivcteanntghleatmjyaeyrbe|ectpalnagcleeds
Thus, we expect ρI N A /k, where A is independent e
c− ≈ 1 1 in
of m. For k = 7, we observe only a weak dependence of
ρI N on m with the critical density being 0.745 0.005 M
(mc−=1),0.744 0.008(m=2)and0.787 0.010(m±=3). νje+1 =(cid:18) 2 −2kje(cid:19)×[Px(B|A)]2k−1 (5)
± ±