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1

Phases of the hard-plate lattice gas on a three-dimensional cubic lattice

Dipanjan Mandal,1, ∗ Geet Rakala,2, † Kedar Damle,3, ‡ Deepak Dhar,4, § and R. Rajesh5, 6, ¶

1Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom2Okinawa Institute of Scince and Technology, 1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa-ken, Japan

3Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai 400 005, India4Indian Institute of Science Education and Research,

Dr. Homi Bhabha Road, Pashan, Pune 411008, India5The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India

6Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India

(Dated: September 7, 2021)

We study the phase diagram of a system of 2 × 2 × 1 hard plates on the three dimensional cubiclattice, i.e. a lattice gas of plates that each cover a single face of the cubic lattice and touchthe four points of the corresponding square plaquette. We focus on the isotropic system, withequal fugacity for the three orientations of plates We show, using grand canonical Monte Carlosimulations, that the system undergoes two density-driven phase transitions with increasing densityof plates: first from a disordered fluid to a layered phase, and second from the layered phase toa sublattice ordered phase. In the layered phase, the system breaks up into occupied bilayersor equivalently slabs of thickness two along one spontaneously chosen cartesian direction, with ahigher density of plates occupying these bilayers relative to the density in the adjacent unoccupiedbilayers. In addition to breaking of lattice translation symmetry along one cartesian direction, thelayered phase additionally breaks the symmetry between the three types of plates, as two types ofplates, with normals perpendicular to the layering direction, have a higher density compared to thethird type. Also, the occupied bilayers of the layered phase have power-law columnar correlationswithin each bilayer, corresponding to power-law two-dimensional columnar order within the occupiedbilayers. In contrast, inter-bilayer correlations of the two-dimensional columnar order parametersdecay exponentially with the separation between the bilayers. In the sublattice ordered phase, thereis two-fold (Z2) breaking of lattice translation symmetry along all three cartesian directions. Wepresent evidence that the disordered to layered transition is continuous and consistent with thethree-dimensional O(3) universality class, while the layered to sublattice transition is discontinuous.

I. INTRODUCTION

Systems of particles interacting through only excludedvolume interaction may exist in different phases depend-ing on the shape and density of the particles. Thesefind applications in self-assembly [1–3] , effectiveness ofdrug delivery [4, 5], design of novel materials with specificoptical and chemical properties [6–8], design of molecu-lar logic gates [9–11], adsorption of gas on metallic sur-faces [12–14], etc. More generally, they are of interest assimple models of fluids [15] as well as being the simplestsystems to study critical behavior. Many shapes havebeen studied in the literature. These include differenttypes of polyhedra [2], colloidal superballs [3], rods [16]etc.Parallel to the study of models in the continuum, mod-

els of hard-core particles on lattices, known as hard corelattice gases (HCLGs) have also been studied. In lit-erature, many different geometrical shapes have beenstudied in two dimensional lattices, which include tri-angles [17], squares [18–23], dimers [24–27], Y-shaped

∗ dipanjan.mandal@warwick.ac.uk† geet.rakala@oist.jp‡ kedar@theory.tifr.res.in§ deepak@iiserpune.ac.in¶ rrajesh@imsc.res.in

particles [28], mixture of squares and dimers [29, 30],rods [31, 32], rectangles [33–36], discretised discs or thek-NN model [37–41], hexagons [42], etc., the last be-ing the only exactly solvable model. A variety of dif-ferent ordered phases may be observed including crys-talline, columnar or striped, nematic, power-law corre-lated phases, etc. Though many examples exist, it is notclear what the dependence between shapes of particlesand the emergent phases are, and in what order the dif-ferent phases appear with increasing density.

Comparatively less is known about HCLG models inthree dimensions. Detailed phase diagram that encom-passes all densities is known for only rods of shapek × 1 × 1 [43, 44] or 2 × 2 × 2 hard cubes [45]. Thenumerical study of HCLG models are constrained by dif-ficulties of equilibrating the system for densities close tothe maximal possible density, as the system gets stuck invery long-lived metastable systems. These difficulties aresubstantially reduced by using Monte Carlo algorithmsthat include cluster moves [29, 33, 34], which significantlydecrease the autocorrelation times.

The systems of plates or board-like particles in thecontinuum have been studied numerically [46–48]. Thephase diagram in the continuum is very rich, showingmultiple transitions with increasing particle densities,and varying aspect ratios. Different phases like smectic,biaxial smectic, uniaxial and biaxial nematic, columnarwith alignment along the long or short axis etc., arise. If

2

the orientations of the plates are restricted to the orthog-onal directions, then it is possible to obtain some rigorousresults regarding the nature of the phases, in particularfor a system of hard parallelepipeds of size 1 × kα × k,α ∈ [0, 1]. For plate like objects (1/2 < α < 1), it ispossible to show rigorously, for k ≫ 1, the existence of auniaxial nematic phase, where only minor axes of platesare aligned parallel to each other, and there is no trans-lational order [49]. However, the behavior of the corre-sponding lattice model, which is also natural from otherpoints of view [50, 51] which we summarize below, hasnot been studied away from full-packing.

With this motivation, here we use such a cluster algo-rithm to study the phase diagram of a system of 2×2×1hard plates on the three dimensional cubic lattice, i.e. alattice gas of plates that each cover a single face of thecubic lattice and touch the four points of the correspond-ing square plaquette. We focus on the isotropic system,with equal fugacity for the three orientations of plates,so that “µ plates” (with normal along the µ axis) haveequal fugacity for all µ (µ = x, y, z) (see Ref. [50] forthe anisotropic case). We show, using grand canonicalMonte Carlo simulations, that the system undergoes twodensity-driven phase transitions with increasing densityof plates: first from a disordered fluid to a spontaneouslylayered phase, and second from the layered phase to asublattice ordered phase.

In the layered phase, the system breaks up into oc-cupied bilayers, i.e. slabs of thickness two along onespontaneously chosen cartesian direction, with a some-what higher density of plates occupying these bilayersrelative to the density in the adjacent unoccupied bi-layers. Thus, this phase involves a two-fold symmetrybreaking of lattice translation symmetry along a singlespontaneously chosen cartesian direction. Additionallythe symmetry between the three types of plates is spon-taneously broken, as two types of plates, with normalsperpendicular to the layering direction, have a higherdensity compared to the third type. Intriguingly, theoccupied bilayers of the layered phase have power-lawcolumnar correlations within each bilayer, correspondingto power-law two-dimensional columnar order within theoccupied bilayers. In contrast, inter-bilayer correlationsof the two-dimensional columnar order parameters decayexponentially with the separation between the bilayers.

In the sublattice ordered phase, there is two-fold (Z2)breaking of lattice translation symmetry along all threecartesian directions. In this phase, one of the eight sub-lattices of the cubic lattice are preferentially occupiedthough each type of particle breaks translational sym-metry along only two directions (see Sec. III for moreprecise definitions). The disordered to layered transi-tion occurs at density ρDL ≈ 0.941. From finite sizescaling, we show that this transition is continuous andconsistent with the O(3) universality class perturbed bycubic anisotropy. The transition from layered to sublat-tice phase occurs at density ρLS ≈ 0.974. We show thatthis second transition is discontinuous. We note that the

fully packed system of 2× 2× 1 hard plates on the cubiclattice also has a very rich phase diagram as a functionof anisotropy in the fugacity of the three orientation ofplates. This is discussed in a parallel work [50].

As mentioned above and in Ref. [50], there is a com-plementary point of view that makes this vacancy-drivenphysics of hard plates particularly interesting on thecubic lattice. The basic point can be understood bythinking of the constraints on the position and mobil-ity of individual vacancies in this system close to thefull-packing limit, and contrasting these with the corre-sponding constraints (or lack thereof) in systems of k-mers (k > 2) [31, 32, 52] or dimers (k = 2) [24–27].

Consider removing a single dimer from a fully-packeddimer model on the bipartite square or cubic lattice. Thisintroduces two vacancies, one on the A sublattice andthe other on the B sublattice of the bipartite lattice.As the dimers move around while obeying the hard-coreconstraint on their positions, the two vacancies can sep-arate from each other and move individually via hopsto next-nearest-neighbor sites. In other words, the onlyconstraint on them is that the two vacancies must oc-cupy opposite sublattices. Turning to long rigid rods oflength k with k > 2, the situation is not very different:Consider the k vacancies, created by the removal of a sin-gle rod from the fully-packed system. Apart from someconstraints on the sublattices of sites that can be simul-taneously occupied by these k vacancies, these vacanciescan move around and separate from one another.

This should be contrasted with the constraints facedby the four vacancies that are created when a single hardplate is removed from the fully-packed system on the cu-bic lattice. These vacancies are only free to move astwo nearest neighbor pairs, and that too only in direc-tions perpendicular to the pairing axis. In bipartite dimermodels, each dimer can be thought of as a dipole, andthe fully-packed limit is understood in terms of a coarse-grained height action that describes the potential fieldin a system of fluctuating dipoles. This provides a nat-ural description of the Coulomb correlations of bipartitedimer models [29, 53–57]. Isolated vacancies correspondto charged monopoles in this description. Any nonzerodensity of vacancies then corresponds to a nonzero den-sity of free charges, which introduces a finite correlationlength and destroys the Coulomb liquid phase. Althoughless is known, the effect of a small density of vacancies onfully-packed k-mers is expected to be quite similar, sincethe full-packing limit again admits a multi-componentheight description and isolated vacancies now correspondto vector charges [31, 32]. In contrast, since vacancies inthe fully-packed plate system can only move in pairs,there is no “charge” associated with them. Instead, va-cancies give rise to dipolar defects in the coarse-grainedeffective field theory [50]. Thus, our results, particularlythe transition to the spontaneously layered phase andthe critical correlations of the occupied bilayers, can beviewed as being a direct consequence of this restrictedmobility of vacancy defects in the hard plate lattice gas;

3

this point of view is particularly appropriate since thetransition to the critical layered phase occurs at a verysmall vacancy density of ρcritvac = 0.026.

II. MODEL AND ALGORITHM

Consider a L×L×L cubic lattice with periodic bound-ary along the three orthogonal directions. The latticesites may be empty or occupied by 2 × 2 × 1 plates,each of which covers one square face of the cubic latticeand touches the four sites of the corresponding plaquette.Three types of plates are possible depending on the ori-entation of the normal to the plate, i.e., x, y and z platescorresponding to plates lying in the yz, zx and xy planesrespectively. The plates interact through excluded vol-ume interaction i.e., no two particles may overlap. Weassociate activity sp and s0 to each plate and vacancyrespectively. These are normalized through

s1/4p + s0 = 1, (1)

where the power 1/4 accounts for the volume of a plate.We study the system using grand canonical Monte

Carlo simulations. Conventional Monte Carlo simula-tions involving local evaporation, deposition, diffusion,and rotation moves are inefficient in equilibrating suchsystems especially when the packing fraction approachesfull packing. These difficulties may be over come by al-gorithms that include cluster moves. In particular, analgorithm that updates strips of sites of size proportionalto L using transfer matrices has been particularly use-ful for hard core lattice gas models [29, 32, 38, 58]. Webriefly describe the algorithm and give details of its im-plementation for the system of hard plates. The imple-mentation and terminology closely follow that followedin Ref. [29] where the phase diagram was obtained for amixture of dimers and squares on a square lattice at allpacking densities.Let a tube be defined as a 2× 2× L subsystem of the

lattice, made up of L plaquettes of size 2× 2× 1. Choosea tube at random in any one of the three orthogonal di-rections. Remove all the plates that are completely con-tained within the tube. There may be some protrudingplates that are not fully contained within the tube, buttouch sites of this tube. These plates are left undisturbed.Due to these protrusions, the shape of the tube (after re-moval of fully contained plates) is complicated and can becharacterized by assigning different morphologies to eachsection depending on the protrusion. There are are 16such morphologies possible for each section and they arelisted in Fig. 1(a). In order to provide a visual depictionthat is easier to read, we use a space-filling conventionfor depicting the protruding plates. In this space fillingconvention, each site of the original cubic lattice maps toa unit cube of the dual cubic lattice, and each plate isa space-filling object that occupies a 2 × 2 × 1 slab con-sisting of 4 adjacent elementary cubes of the dual lattice.

1

9

2 3 4 5 6 7 8

10 11 12 13 14 15 16

1 2 3 4 5 6 7 8

Morphologies

States

(a)

(b)

FIG. 1. Schematic diagram of (a) sixteen possible morpholo-gies and (b) eight possible states, that are used to constructthe transfer matrix. To represent different states we havetaken the projection of particles in xy-plane. Black representblocked site and brown, red, green respectively represents pro-jection of y, x and z-particles.

Note that this alternate description is behind the com-monly used terminology, also used here, which refers tothe hard plates as 2× 2× 1 cuboids.

The aim is to refill the tube with a new configuration ofplates that are fully contained within the tube, but withthe correct equilibrium probability. The probability ofthis new configuration may be calculated using transfermatrices. Any 2× 2× 1 section with a given morphologymay be filled by plates in utmost eight different ways.The possible states for a section is listed in Fig. 1(b).Among the sixteen possible morphologies, there are fif-teen morphologies with partially blocked sites. The re-maining one morphology [morphology-16 as shown inFig. 1 (a)] represents a complete blockage in the cho-sen tube. We have to thus calculate 152 = 225 differenttransfer matrices of size 8×8. Let Tm1,m2

be the transfermatrix where the system is transferring from morphologym2 to morphologym1. The matrix element may be writ-ten as

Tm1,m2(i, j) = cm1,m2

(i, j)WpW0, (2)

where cm1,m2(i, j) is the compatibility factor, Wp is the

weight associated with the particle that sits on morphol-ogy m1 and W0 is the weight of vacancies present onmorphology m2 after depositing particle on morphologym1. The compatibility factor cm1,m2

(i, j) is 1 if the statesi and j are compatible on morphologies m1 and m2, oth-erwise it equals zero. The weights associated with theparticles and vacancies may be written as

Wp = sns

p , ns = 0, 1, 2, (3)

W0 = sn0

0 , n0 = 0, 1, 2, 3, 4. (4)

Examples of few transfer matrices are given in

4

Eqs. (5)–(7).

T1,1 =

s40 s20 s20 s20 s20 1 1 1sps

20 0 sp 0 0 0 0 0

sps20 sp 0 0 0 0 0 0

sps20 0 0 0 sp 0 0 0

sps20 0 0 sp 0 0 0 0

s2p 0 0 0 0 0 0 0s2p 0 0 0 0 0 0 0sps

40 sps

20 sps

20 sps

20 sps

20 sp sp sp

(5)

T1,3 =

s30 0 s0 0 s0 0 0 00 0 0 0 0 0 0 0

sps0 0 0 0 0 0 0 00 0 0 0 0 0 0 0

sps0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

sps30 0 sps0 0 sps0 0 0 0

(6)

T3,1 =

s40 s20 s20 s20 s20 1 1 10 0 0 0 0 0 0 0

sps20 sp 0 0 0 0 0 0

0 0 0 0 0 0 0 0sps

20 0 0 sp 0 0 0 0

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

(7)

The partition function of a closed 2× 2×L tube withmorphology m1, . . . ,mL may be written as

Zc =∑

i

〈i|TmL,m1Tm1,m2

. . . TmL−1,mL|i〉, (8)

where |i〉 is the state vector of state i. The partition func-tion for the open tube of length X < L may be writtenas

Zo = 〈Lm1|Tm1,m2

Tm2,m3. . . TmX−1,mX

|RmX〉, (9)

where 〈Lm1| and |RmX

〉 are respectively left and rightvectors that may be written as

Lm1(n) = T16,m1

(1, n), (10)

RmX(n) = TmX ,16(n, 1). (11)

Calculating the partition function, we occupy each sec-tion, section by section, according to the calculated prob-abilities.To speed up equilibration as well as to reduce autocor-

relation times, we also implement a flip move in which apair of adjacent parallel plates of same type is replacedby another pair of adjacent parallel plates whose type ischosen randomly.Tubes that are separated from each other may be up-

dated simultaneously, thus allowing for parallelization.The results in the paper are obtained from a parallelizedversion of the above algorithm.

0 1

4 5

2 3

6 7even-xz plane odd-xz plane

z0 0

0 0

x

1 1

4 5 4 5

1 1

4 5 4 5

z

7

7

7

7

x

2 3 2 3

6 6

2 3 2 3

6 6

FIG. 2. Division of the full lattice into eight sublattices0, 1, . . . , 7, depending on whether each coordinate is odd oreven. The arrows show the orientation of the three axes x, yand z.

For each value of activity, we ensure that equilibra-tion has been achieved by starting the simulations withconfigurations that correspond to different phases, andensuring that the final equilibrium state is independentof the initial state.

III. DIFFERENT PHASES OF SYSTEM

A. Order parameter and disordered phase

We observe three different phases in our simulations asdensity is varied. To characterize them, it is convenientto divide the full lattice into eight sublattices depend-ing on whether the x, y, and z coordinates of a site areeven (0) or odd (1), as shown in Fig. 2. A lattice site(x, y, z) belongs to the sublattice constructed out of thebinary number zyx where each coordinate is modulo two.Except for plates that cover a plaquette on an edge thatwraps around the periodic direction, we assign each plateto the site with least x, y and z coordinates (of the foursites touched by it). For plates on wrapping plaquettes,this definition is of course modified in the obvious way toremain consistent with the treatment of bulk plates. Foreach plate, the corner that touches this chosen site is the“head” of the plate.

To characterize the phases quantitatively, we definesublattice densities ρji as the volume fraction of plates oftype j = x, y, z whose heads are in sublattice i = 0, . . . , 7.We also define three particle densities ρj , eight sublatticedensities ρi, and total density ρ as

ρj =

7∑

i=0

ρji , j = x, y, z,

ρi =∑

j=x,y,z

ρji , i = 0, . . . , 7, (12)

ρ =

7∑

i=0

ρi.

To quantify the breaking of translational invariancein the different directions, it is convenient to define the

5

quantities

ℓx =1

L3

∑

x,y,z

φ(x, y, z)(−1)x,

ℓy =1

L3

∑

x,y,z

φ(x, y, z)(−1)y, (13)

ℓz =1

L3

∑

x,y,z

φ(x, y, z)(−1)z,

where φ(x, y, z) is 1 if the site is occupied by the head ofa plate and zero otherwise. The square of the layeringorder parameter, which characterizes the layered phase,may be defined as

Λ2 = ℓ2x + ℓ2y + ℓ2z. (14)

The columnar vector ~C with components (cx, cy, cz) maybe written as

cx =1

L3

∑

x,y,z

(−1)y+zφ(x, y, z),

cy =1

L3

∑

x,y,z

(−1)x+zφ(x, y, z), (15)

cz =1

L3

∑

x,y,z

(−1)x+yφ(x, y, z).

The square of the columnar order parameter may be de-fined as

Γ2 = c2x + c2y + c2z. (16)

We also define the square of the order parameter ω tocharacterize the sublattice phase

ω2 = ℓ2xℓ2yℓ

2z. (17)

To capture the breaking of particle number symmetry,we define a nematic order parameter Π as

Π2 =(

ρz −ρy

2−ρx

2

)2+

3

4

(

ρy − ρx)2. (18)

When Π is non-zero, particle symmetry is broken.In a sublattice ordered phase, we expect ω2, Λ2 and

Γ2 to all tend to nonzero values in the thermodynamiclimit. In contrast, in the layered phase, we expect ω2 totend to zero as 1/L6 and Γ2 to tend to zero as 1/L3 inthe thermodynamic limit, while Λ2 tends to a nonzerolimit.Variation of the square of different order parameters

Λ2, Γ2 and ω2 as a function of L−1 are shown in Fig. 3(a-c) for sp = 0.300, sp = 0.360 and sp = 0.420. The quan-tity Λ2 decays to zero as L−3 for sp = 0.300 and takesnon-zero values for sp = 0.360 and sp = 0.420. Thequantity Γ2 decays to zero as L−3 for both sp = 0.300and sp = 0.360, and takes non-zero value for sp = 0.420.Similarly ω2 also decays to zero for both sp = 0.300 andsp = 0.360, but the decay obey different power laws,

which are L−9 and L−6 respectively. For sp = 0.420,ω2 takes non-zero values. Observing the decay of differ-ent order parameters and their saturation, we may saythat the system is in disordered, layered and sublatticephase for sp = 0.300, sp = 0.360 and sp = 0.420 respec-tively. This establishes the presence of the three phasesdescribed in our introductory discussion. For a bird’s eyeview of the phase diagram as a function of plate fugac-ity, we plot the fugacity dependence of the various orderparameters in Fig. 4. We clearly observe a layered phase(Λ2 6= 0, Γ2 = 0, ω2 = 0) and a sublattice phase (Λ2 6= 0,Γ2 6= 0, ω2 6= 0). The variation of Π2 as a function ofsp is also shown in Fig. 4. Π2 is zero in both disorderedand sublattice phase, and takes nonzero values only inthe layered phase, which indicates asymmetric densitiesof three types of particles in the layered phase.The characterization of the disordered phase is

straightforward: at low densities, the plates form a dis-ordered fluid, with their heads uniformly distributed, i.e.each of the sublattice densities are equal for the threedifferent types of plates, i.e.,

ρi =ρ

8, i = 0, 1, . . . , 7,

ρj =ρ

3, j = x, y, z, (19)

Π2 = 0.

B. Layered phase

With increasing density, we observe that the systemundergoes a transition into a layered phase. In this phase,the system breaks up into weakly interacting slabs ofwidth 2 along one of the three directions. In the other twodirections, translational invariance is maintained. Also,the symmetry among the three types of plates is broken,and the density of plates whose normal is in the directionof layering is suppressed compared to the other two typesof plates.The time evolution of the sublattice densities, when the

system is in a layered phase, with layering in x-direction,is shown in Fig. 5. Fig. 5(a) compares the densities ofthe three types of plates. It is clear that the density ofx-plates is suppressed compared to y and z-plates, whenthe layering is in the x-direction, i.e., ρy ≈ ρz ≫ ρx. Atthe same time, Fig. 5(b)–(d) show that while x-plates oc-cupy all sublattices equally, y and z-plates preferentiallyoccupy planes with odd x (in this case), correspondingto ρ1, ρ3, ρ5, and ρ7.

C. Sublattice phase

At higher densities including full packing, we observea sublattice phase. In this phase, particle symmetry is re-stored, but translational invariance is broken in all threedirections, as in a solid-like phase.

6

(a) (b) (c)

10-4

10-3

10-2

10-1

1/150 1/120 1/100 1/80

L-3

⟨Λ2⟩

L-1

sp=0.300sp=0.360sp=0.420

10-6

10-5

10-4

10-3

10-2

1/150 1/120 1/100 1/80

L-3

L-3

⟨Γ2⟩

L-1

sp=0.300sp=0.360sp=0.420

10-12

10-10

10-8

10-6

10-4

1/150 1/120 1/100 1/80

L-9

L-6

⟨ω2⟩

L-1

sp=0.300sp=0.360sp=0.420

FIG. 3. Variation of the square of the (a) layered order parameter 〈Λ2〉, (b) columnar order parameter 〈Γ2〉 and (c) sublatticeorder parameter 〈ω2〉 as a function of L−1 for different values of sp.

The time evolution of the sublattice densities, when thesystem is in a sublattice phase is shown in Fig. 6. Outof the eight sublattices, one of them is occupied pref-erentially. At the same time, there is solid-like sublat-tice ordering as can be seen seen from Fig. 6(a). Thesublattice densities for each type of plate are shown inFig. 6(b)–(d). For each type of the plates, two sublat-tices are preferred, as in a columnar phase. The preferredsublattice densities are [ρ2, ρ3], [ρ1, ρ3] and [ρ3, ρ7] for x,y and z-plates respectively. Time profile of total sub-lattice density ρi breaks into four labels [see Fig. 6(a)].The top and bottom labels are ρ3 and ρ4 respectively.Two intermediate labels are degenerate with three den-sities in each label. Higher intermediate label has densi-ties ρ1, ρ2, ρ7 and lower intermediate label has densitiesρ0, ρ5, ρ6 respectively. The pattern of the labels may beunderstood from the right panel of Fig. 2 where the sub-lattice division is shown schematically. The labels aredivided depending on the lowest distance between thesublattice-3 (most occupied) and other sublattices. Thedensity decreases with increasing the distance betweensublattices.

One could imagine the sublattice phase as follows.Consider a collection of 2×2×2 cubes that are arrangedin a periodic manner to favor one sublattice. If the cubesare now replaced by a pair of plates of the same kind(each cube in three ways), then the phase that is ob-tained is similar to the sublattice phase that we see inthe system of hard plates. Unlike the layered phase, thedensities of the three types of plates are equal.

IV. PHASE TRANSITIONS

We now study the nature of the disordered-layered andlayered-columnar phase transitions.

1. Disordered to layered phase transition

To capture the symmetry breaking in the differenttransitions, we consider the squared order parameter Λ2

as defined in Eq. (14). In the disordered phase Λ2 ≈ 0,while in a pure layered phase Λ2 ≈ 1. The variation ofΛ2 with activity sp for different system size is shown inFig. 4(a). We clearly observe a transition from disorderedphase to layered phase with increasing particle density.

We now study the disordered-layered transition usingthe squared order parameter Λ2. We define the Bindercumulant UΛ associated with Λ2

UΛ = 1−9

15

〈Λ4〉

〈Λ2〉2. (20)

We study the critical behavior using finite size scaling.Near the critical point, according to finite-size scalingtheory, the singular part of the Binder cumulant obey

UΛ(ǫ, L) ≃ fΛ(ǫL1/ν), (21)

where ǫ = sp− sc is the deviation from the critical point,ν is the critical exponent, and fΛ is the scaling functions.

We now show that the data near the disordered-layeredtransition are consistent with the O(3) universality classwith cubic anisotropy, wherein the values of the criticalexponents are known to be ν = 0.704, β = 0.362, andγ = 1.389 [59, 60]. The variation of UΛ with sp fordifferent system sizes is shown in Fig. 7(a). The datafor different system sizes cross each other at the criticalpoint sDL

p ≈ 0.323. The corresponding critical density

is ρDL ≈ 0.940. The data for the Binder cumulant fordifferent L collapse onto one curve when the variablesu and ss are scaled as in Eq. (21) with ν = 0.704, asshown in Fig. 7(b). From the excellent data collapse,we conclude that the transition, most likely, belongs tothe universality class of the three dimensional Heisenbergmodel with cubic anisotropy [59, 60].

7

(a)

(b)

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.30 0.32 0.34 0.36 0.38 0.40 0.42

⟨Λ2⟩

20⟨Γ2⟩

sp

L=80L=100L=120L=150

0

1

2

3

4

5

6

0.30 0.32 0.34 0.36 0.38 0.40 0.42

⟨Π2⟩

sp

×10-4

L=80L=100L=120L=150

FIG. 4. Variation of the square of the (a) translational orderparameters 〈Λ2〉, 〈Γ2〉, and (b) nematic order parameter 〈Π2〉with activity of plate sp. The data are for for system sizesL = 80, 100, 120, 150.

2. Layered to sublattice phase transition

In this section, we study the nature of the second tran-sition from layered to sublattice phase. Suitable orderparameters are Γ2 and Π2 as defined in Eq. (14) andEq. (18) respectively. The associated Binder cumulantsmay be defined as

UΓ = 1−1

2

〈Γ4〉

〈Γ2〉2, (22)

UΠ = 1−1

2

〈Π4〉

〈Π2〉2, (23)

We show that the transition is discontinuous in nature.The variation of 〈Γ2〉 and 〈Π2〉 with sp, for different

system sizes, is shown in Fig. 4(a) and (b) respectively.We observe that both order parameters have a sharp vari-ation across the transition point, and the data for dif-ferent system sizes intersect each other with the curvesbecoming steeper with increasing system size. These aresignatures of a discontinuous transition. We also plot theprobability distribution of course grained total density ρ

0.30

0.31

0.32

0.33

0.34

1×106

2×106

3×106

4×106

�z, �

y

�x�

x,

�

y,

�

z

time

0.00

0.02

0.04

0.06

0.08

0.10

1×106

2×106

3×106

4×106

i=0, 1, 2, 3, 4, 5, 6, 7

�

ix

time

0.00

0.02

0.04

0.06

0.08

0.10

1×106

2×106

3×106

4×106

i=1, 3, 5, 7

i=0, 2, 4, 6

�

iy

time

0.00

0.02

0.04

0.06

0.08

0.10

1×106

2×106

3×106

4×106

i=1, 3, 5, 7

i=0, 2, 4, 6

�

iz

time

(a) (b)

(c) (d)

FIG. 5. The temporal evolution of different thermodynamicquantities is shown for an equilibrated layered phase at ac-tivity sp = 0.380 and for system size L = 120. (a) The threeplate densities ρx, ρy, ρz. The eight sublattice densities ρi for(b) x-plates, (c) y-plates, and (d) z-plates, where the sub-scripts i = 0, . . . , 7 denote the different sublattices and thesuperscripts x, y, z denote the different types of plates.

(a) (b)

(c) (d)

0.0

0.1

0.2

0.3

1×106

2×106

3×106

4×106

i=3

i=1, 2, 7

i=0, 5, 6

i=4

i

time

0.00

0.02

0.04

0.06

0.08

0.10

1×106

2×106

3×106

4×106

i=4, 5

i=0, 1, 6, 7

i=2, 3

ix

time

0.00

0.02

0.04

0.06

0.08

0.10

1×106

2×106

3×106

4×106

i=4, 6

i=0, 2, 5, 7

i=1, 3

�

iy

time

0.00

0.02

0.04

0.06

0.08

0.10

1×106

2×106

3×106

4×106

i=0, 4

i=1, 2, 5, 6

i=3, 7

�

iz

time

FIG. 6. The temporal evolution of different thermodynamicquantities is shown for an equilibrated sublattice phase at ac-tivity sp = 0.460 and for system size L = 120. The eightsublattice densities (a) ρi summing over three types of par-ticles, for individual particle type (b) x-plates, (c) y-platesand (d) z-plates, where the subscripts i = 0, . . . , 7 denote thedifferent sublattices and the superscripts x, y, z denote thedifferent types of plates.

8

(a) (b)

0.0

0.1

0.2

0.3

0.4

-20 -10 0 10 20

U

εL1/ν

L=100L=120L=150L=200

0.0

0.1

0.2

0.3

0.4

0.320 0.322 0.324 0.326 0.328

UΛ

sp

L=100L=120L=150L=200

FIG. 7. Data for Binder cumulant UΛ near the disordered-layered transition. (a) UΛ for different system sizes intersectclose to sDL

p ≈ 0.323. (b) UΛ for different system sizes collapseonto a curve when the parameter are scaled as in Eq. (21) withexponent ν = 0.704.

and order parameters Γ2, Π2 as shown in Fig. 8(a), (b)and (c) respectively. We do the course graining of blocksize 51 in the time series of different quantities to reducethe noise. We see the typical double peaked distributionfor ρ and order parameters as the activity crosses thecritical activity, showing coexistence. The jump in thedensity across the transition is quite small (of the order10−4) and therefore quite difficult to detect in simula-tions. More evidence of the discontinuous nature maybe found by examining Binder cumulants. The variationof Binder cumulant UΓ and UΠ are shown in Fig. 9(a)and (b) respectively, for different system sizes. We seeevidence for both cumulants being non-monotonic andnegative near the transition, a signature of a first ordertransition. We conclude, within our numerical simula-tions, that the layered-sublattice transition is discontin-uous.

V. CORRELATIONS IN LAYERED PHASE

In this section we characterize the correlations in thelayered phase. To examine the intra-bilayer and inter-bilayer correlations, we define the in-plane columnar or-der parameter (ℓx(z), ℓy(z)) of a layer z as

ℓx(z) =1

L3

L−1∑

x,y=0

(−1)xφ(x, y, z),

ℓy(z) =1

L3

L−1∑

x,y=0

(−1)yφ(x, y, z), (24)

where φ(x, y, z) = 1, if the site is occupied by the head ofa plate, and zero otherwise. The inter-bilayer correlationG(p, L) for two bilayers separated by a distance p in thelayering direction is defined as

G(p, L) =1

L

L−1∑

z′=0

[ℓx(z′)ℓx(z

′+p)+ℓy(z′)ℓy(z

′+p)], (25)

The variation of the normalized correlation functionGn(p, L) = G(p, L)/G(0, L) for the layered phase isshown in Fig. 10(a) for different systems sizes. It is clearthat it decays exponentially with p. We conclude thatin the layered phase, the interaction between the bilayersis weak and decays rapidly with inter-bilayer distance.The variation of G(0, L) and G(4, L) as a function of Lis shown in Fig. 10(b) and (c) respectively. For large L,we see that these approach the behavior

G(p, L) ≈ AL−2. (26)

Intrigued by this power-law behavior, we note that ifviewed along the layering axis, each occupied bilayermaps to a lattice gas of hard squares and dimers on asquare lattice. In this mapping, the plates with nor-mals perpendicular to the layering axis correspond todimers, and pairs of plates with normals along the layer-ing axis correspond to hard squares. Plates that straddletwo neighboring occupied bilayers and couple them cor-respond to a pair of correlated dipolar defects in the twodimensional representation of both bilayers. This moti-vates the definition of a local two-dimensional columnarorder parameter field ψ exactly as in Ref. [29]. To under-stand this power-law behavior better, we have measuredthe connected two-point function C(r, 0) of ψ within a bi-layer, and studied the L dependence ofC(L/4, 0). Withinthe layered phase, we find that C(L/4) displays power-law behavior C(L/4) ∼ 1/Lη, with a η > 2 that dependson the plate fugacity (see Fig. 10(d)).This appearance of critical correlations in the layered

phase at nonzero vacancy density is quite surprising atfirst sight. We understand it as follows: As already notedin the Introduction, and discussed in parallel work [50],vacancies in the hard plate system near full packing haverestricted mobility. Indeed, they can only move in pairsthat can be thought of as dipoles. Additionally, as ar-gued in parallel work [50], when the system is in a spon-taneously layered phase, two neighboring occupied bilay-ers can only be straddled by a pair of parallel adjacentplates in the thermodynamic limit (more precisely, otherconfigurations are entropically subdominant in the ther-modynamic limit). Taken together, these two featuresof the system imply that in the equivalent two dimen-sional hard square and dimer gas describing an occupiedbilayer, there are no isolated “charged” defects. Indeed,the defects that dominate in the thermodynamic limitof the equivalent two-dimensional system have a dipolarcharacter, and the coupling between neighboring occu-pied bilayers is quadrupolar in nature as described inRef. [50]. The dipolar character of the defects in theequivalent two-dimensional system of hard squares anddimers implies that two-dimensional power-law colum-nar order can survive in the presence of these defects.On the other hand, the quadrupolar nature of the cou-pling between neighboring occupied bilayers [50] implies,by an argument entirely analogous to that of Ref. [50],that inter-bilayer couplings do not lock these power-lawordered bilayers together into a long-range ordered state.

9

(a)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.9736 0.9738 0.9740 0.9742

P(�

)

�

× 104

sp=0.3920sp=0.3925sp=0.3930

0

3

6

9

12

15

0.0×100

1.5×10-3

3.0×10-3

P(Γ

2)

Γ2

×102

sp=0.3910sp=0.3925sp=0.3940

0

2

4

6

8

10

0.0×100

3.0×10-4

6.0×10-4

P(Π

2)

Π2

×103

sp=0.3910sp=0.3925sp=0.3940

(b) (c)

FIG. 8. Plot of probability distribution of (a) total density ρ, (b) Γ2 and (c) Π2 near layered to sublattice transition for L = 150.

0.0

0.1

0.2

0.3

0.4

0.5

0.370 0.380 0.390 0.400

U

�

sp

L=100L=120L=150

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.380 0.390 0.400

UΠ

sp

L=100L=120L=150(a) (b)

FIG. 9. Variation of the Binder cumulant (a) UΓ and (b) UΠ

as a function of sp for different L.

(a) (b)

(c) (d)

0.1

0.2

0.5

1.0

0 20 40 60 80 100

sp=0.380

Gn(p

,L)

p

L=120L=140L=168L=192

2

3

4

6

9

120 144 168 192

L-2G

(0,L

)

L

×10-4

2

3

4

6

9

120 144 168 192

L-2

G(4

,L)

L

×10-4

0.001

0.003

0.010

80 120 160 200

L-2.51

sp=0.365

C(L

/4,0

)

L

FIG. 10. Variation of (a) normalized correlation functionGn(p,L) and (d) exponent η(p) as a function of p. Plot of un-normalized correlation function (b) G(0, L) and (c) G(4, L).

0.014

0.016

0.018

0.02

80 100 120 140

L-0.5

�

|∆s|⟩L

-1

L

sp=0.375sp=0.385

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 0.2 0.4 0.6 0.8 1.0

⟨|∆

|⟩

kL-1

L=80L=100L=120L=140(a) (b)

FIG. 11. (a) Variation of the absolute value of total vacancycharge |∆| in a k × L rectangular box on a occupied layerwith kL−1, for different L in a layered phase with sp = 0.375.(b) Power-law scaling of the saturation charge |∆s| with L fordifferent sp in the layered phase.

Evidence in favor of this coarse-grained description canbe obtained by monitoring the total “charge” in eachoccupied bilayer. This is defined as:

∆ =∑

x,y

(−1)x+yδσ,1, (27)

where the vacancy field σ is 1 if the site is empty, other-wise σ is 0. The variation of average absolute charge asa function of kL−1 for different L and fixed sp = 0.375is shown in Fig. 11(a). The absolute charge decays sym-metrically from the saturation value ∆s because of theperiodic boundary. The scaling of |∆sL

−1| with L isshown in Fig. 11(b). This perimeter-law scaling clearlyshows that there are no free charges in the system.

VI. SUMMARY AND DISCUSSION

In this paper we studied the phases and phase transi-tions in a system of 2 × 2 × 1 hard plates on the threedimensional cubic lattice using Monte Carlo simulations.

10

disordered layered sublattice

FIG. 12. Schematic phase diagram of 2 × 2 × 1 hard platesmodel. The red dot represents a continuous transition and theblue dots and dotted line represent the coexistence regime ina first order transition.

Three types of plates are possible depending on their ori-entation. We have shown that the system undergoes twoentropy driven phase transitions with increasing the den-sity of particles: first from disordered to layered and sec-ond from layered to sublattice. A schematic phase dia-gram is shown in Fig. 12. In the fully packed limit, thesystem has sublattice order. We show that the disorderedto layered transition is continuous and the critical behav-ior is consistent with the universality class of the threedimensional Heisenberg model with cubic anisotropy. Onthe other hand, we show that the transition from layeredto columnar phase is discontinuous.In the layered phase, the density of one type of plate is

suppressed, and translational symmetry is broken in onlyone direction. We showed that in-plane correlations aredecay as a power law. On the other hand, the correla-tions between different layers decrease exponentially withthe separation between layers. In the sublattice phase,the system has translational order in all three directions.Each type of particle has columnar order in the directionnormal to the plane of the particle. On the other hand, ifone looks at the densities of sublattices [see Fig. 6], thenone sublattice is preferentially occupied, as in crystalline-sublattice order.Although, numerous analytical [49, 61, 62], experimen-

tal [63, 64] and computer simulation [48, 65, 66] studiesindicate the presence of biaxial nematic phase, in whichthe system exhibits orientational order along all threeinternal axis of the particle, in the system of anisotropicplate like objects in three dimensions, there is long stand-ing debate regarding the existence of this phase. In thispaper we have not found any biaxial nematic phase inthe system of plates with side length k = 2. It would be

very interesting to check the existence and stability of bi-axial nematic phase for the lattice models of plates withlarger side length. The system of rectangular plates withdifferent aspect ratio having hard core and/or attractiveinteraction are also promising area for future study.

A two dimensional section of the system of hard platescorresponds to a problem of hard squares and dimers.This model, when the activities of dimers and squarescan be varied independently, has a very rich phase dia-gram including two lines of critical points meeting at apoint [29, 30]. Thus, one can expect that if the activi-ties of the three kinds of plates in three dimensions canbe independently varied, then a very rich phase diagramcan be expected, especially at full packing, where regionsof power-law correlated phases should exist. We will de-scribe the phases of the fully packed regime in anotherpaper [50].

ACKNOWLEDGMENTS

We thank K. Ramola and N. Vigneshwar for helpfuldiscussions. The simulations were carried out on the highperformance computing machines Nandadevi at the In-stitute of Mathematical Sciences, and the computationalfacilities provided by the University of Warwick ScientificComputing Research Technology Platform. Some of thiswork contributed to the Ph.D thesis of DM submittedto the Homi Bhabha National Institute (HBNI). GR wassupported by the TQM unit of Okinawa Institute of Sci-ence and Technology during the final stages of this work.KD was supported at the TIFR by DAE, India and inpart by a J.C. Bose Fellowship (JCB/2020/000047) ofSERB, DST India, and by the Infosys-ChandrasekharanRandom Geometry Center (TIFR). D.D.’s work waspartially supported by Grant No. DST-SR- S2/JCB-24/2005 of the Government of India.

Author contributions DM performed the computationswith assistance from GR. KD, RR, and DD conceived anddirected this work, and finalized the manuscript usingdetailed inputs from DM.

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