Emergence of limit-periodic order in tiling models

A two-dimensional (2D) lattice model defined on a triangular lattice with nearest- and next-nearest-neighbor interactions based on the Taylor-Socolar monotile is known to have a limit-periodic ground state. The system reaches that state during a slow quench through an infinite sequence of phase transitions. We study the model as a function of the strength of the next-nearest-neighbor interactions and introduce closely related 3D models with only nearest-neighbor interactions that exhibit limit-periodic phases. For models with no next-nearest-neighbor interactions of the Taylor-Socolar type, there is a large degenerate class of ground states, including crystalline patterns and limit-periodic ones, but a slow quench still yields the limit-periodic state. For the Taylor-Socolar lattic model, we present calculations of the diffraction pattern for a particular decoration of the tile that permits exact expressions for the amplitudes and identify domain walls that slow the relaxation times in the ordered phases. For one of the 3D models, we show that the phase transitions are first order, with equilibrium structures that can be more complex than in the 2D case, and we include a proof of aperiodicity for a geometrically simple tile with only nearest-neighbor matching rules.

Figure 1

(A) The Taylor-Socolar model prototile and matching rules. Discontinuities in the black and thick gray (purple) stripes have energetic penalties ${\epsilon }_1$ and ${\epsilon }_2$, respectively. (B) The black stripe model prototile and matching rules. Discontinuities in the black stripes have energetic penalties ${\epsilon }_1$. The model is equivalent to the Taylor-Socolar model with ${\epsilon }_2=0$. (C) The zonohedral model. [(a), (b), and (c)] Different views of the prototile. Each black bar reaches from one top face of the zonohedron to one of the bottom faces. Panel (c) shows the view down the 111 axis. [(d) and (e)] Two views of three tiles in separate layers. The black bars connecting through the shared faces of the tiles form a helix. Viewed along the 111 axis, the helix is seen to correspond to a small triangle in the Taylor-Socolar tiling. (D) The cubic model. The views shown correspond to those in (C).

Figure 2

(a) Level-1 ordering in the Taylor-Socolar model. A subset comprising three quarters of the tiles is shown. For each tile, the black and thick gray (purple) corner decorations are included, but not the long stripes. Each tile shown may be in any of the four orientations corresponding to the possible positions of the long black and thick gray (purple) stripes. The tiles lying on sublattice $A$ do not contribute any of the decorations in the pattern shown here. (b) Level-2 ordering in the Taylor-Socolar model. Three quarters of the tiles on sublattice $A$ of panel (a) participate in the formation of black and purple triangles. The light-colored tiles and decorations display the level-1 order. Double stripes indicate the possible locations of black and thick gray (purple) stripes on tiles that contribute corners to the level-1 triangles.

Figure 3

The spins used to define the order parameter for the level-1 transition. See text for explanation.

Figure 4

(a) The order parameters ${\varphi }_n$ vs $T$ from a quench for the case ${\epsilon }_2=1$, with quench parameters $\ell =64$, $T_0=2.0$, $T_f=0.0$, $\Delta T=0.01$, and $\tau =12\times {10}^5$ MCS. (b) The order parameters ${\varphi }_n$ vs $T$ from a quench for the case ${\epsilon }_2=0.5$, with the same quench parameters as in (a). (c) Data collapse obtained from the scaling theory for the data from (a). Deviations of the level-4 points from the others are finite-size effects due to the relatively small number of level-4 triangles in the system. (d) Solid circles show the dependence of $T_{c;1}$ on ${\epsilon }_2$, for ${\epsilon }_2\le 1$, obtained from simulations of slow quenches. For each value of ${\epsilon }_2$, $T_{c;1}$ is approximated as the highest temperature for which ${\varphi }_1>0.1$. Open circles show the dependence of $T_{c;n}$ on ${\epsilon }_2$, for ${\epsilon }_2\le {\epsilon }_1=1$, obtained from Eq. (8). The lines in this figure connect parameters for systems that exhibit equivalent behavior. As one follows a line downward and to the right, the open circles indicate transitions of levels $n=2$ through $n=5$.

Figure 5

Matched and mismatched corner configurations for level-3 edges.

Figure 6

Inset: ${\varphi }_n$ vs $T$ for the black stripe model, ${\epsilon }_2=0$. The data are from simulated quenching on a rhombic domain of side length 64, with parameters $T_o=0.6$, $T_f=0.0$, $\Delta T=0.01$, and $\tau =12\times {10}^5$ MCS. Full panel: Data collapse from the scaling theory for the black stripe model applied to the data from the inset.

Figure 7

Image of one enantiomorph of the cubic prototile, which is an alternate representation of the zonohedral tile. The axis arrows turn from dark to light where they intersect the faces of the cube. (a) The arrow indicates the 111 axis of the cube, defined to be the $c$ axis. (b) A projection of the tile onto the plane perpendicular to the $c$ axis.

Figure 8

(a) A projection of a section of one of the limit-periodic states onto the plane perpendicular to the $c$ axis of the cubic lattice. The gray (red, green, and blue) triangles are formed from purple bars that were recolored according to their layer for ease of viewing. Level-1 helices correspond to the smallest black triangles, level-2 helices to the next largest, and so on. Similarly, level-1 triangles are the smallest colored triangles, level 2 the next largest and so on. (b) Image of nine layers of the 3D structure that gives the projection of the black bars in (a). The gray (colored) bars were left out for clarity.

Figure 9

Illustration of the structure of level-2 helices. Letters (a)–(f) label the parts of the helix, with (a), (c), and (e) being corners and (b), (d), and (f) being edges. The symbols R and L labels right- and left-handed helices, respectively.

Figure 10

[(a) and (b)] Representations of a level-1 helix. Cubes of the same color belong to separate columns. [(c) and (d)] Representation of a level-2 helix with level-1 designs left out for clarity. Bars belonging to a given cube are shown in the same color. Cubes containing bars of different colors do not interact with each other.

Figure 11

Helices of one chirality are colored gray (red), while the others are colored black (blue). (a) Level-1 lattice. (b) A section of a fully ordered structure.

Figure 12

A 2D projection of a region of an ordered level-1 structure. The different sublattices shown in different shades of gray contain only the cubes with layer indices in ${\ell }_2^0$. Bars not contributing to the level-1 structure are omitted for clarity.

Figure 13

(a) 2D projection of the two level-2 subsets ordered on the same sites. (b) 2D projection of the two level-2 subsets ordered with the centers of their honeycomb lattice on different sites, one depicted in black and the other in gray (orange). In both (a) and (b), designs not contributing to the level-1 or level-2 structures have been left out for clarity.

Figure 14

Order parameters of each subset of helices for the first three levels of the zonohedral model with ${\epsilon }_2={\epsilon }_1=1$. There is one order parameter for level 1, two for level 2, and four for level 3. The system is cooled from $T=2$ to $T=0$, in increments of $\Delta T=0.02$ with $\tau =1\times {10}^5$ MCS, and then heated in the same manner. Simulations are performed on a rhombic lattice of size $16\times 16\times 24$. During the cooling process, the order parameter of a level-$n$ subset is found only after level $(n-1)$ is ordered. The order parameters do not go to zero at high temperatures because of finite-size effects.

Figure 15

(a) Energy per site during a slow quench and subsequent heating. The phase transitions correspond to those of levels 1, 2, and 3, in order of decreasing $T$. See the end of Sec. 4b for the details of the simulation. (b) Free energy of the disordered, level-1 ordered, level-2 ordered, and level-3 ordered phases. The dot indicates the free energy of the simplest periodic competing phase.

Figure 16

Section of one of the periodic ground states. The largest helices belong to level 3.

Figure 17

Matched and mismatched corner configurations for the black bonds of level 3.

Figure 18

(a) Free energy of levels 2 and 3 scaled onto that of level 1 according to Eq. (38) in the full system (fluctuations are not artificially suppressed). Level-2 and level-3 curves have been scaled according to Eq. (38). The scaled transition temperatures for the different levels, marked by the heavy $+$s, are shown in the inset of panel (b). (b) Free energy of levels 2 and 3 scaled onto that of level 1 according to Eq. (38) when levels with indices less than $n$ are held fixed, showing perfect agreement with the scaling theory. The vertical dashed line through the intersection of the curves obtained by integrating down from high $T$ and integrating up from low $T$ marks the transition temperature. Inset: Critical temperatures of the levels-2 and -3 transitions scaled onto that of level 1. The diamonds are determined from simulations in which levels with indices less than $n$ are fixed by a conjugate field as in (b). The disks correspond to the systems with all equilibrium fluctuations present.

Figure 19

Inset: ${\varphi }_n$ vs time for levels 1, 2, and 3. The level-1 structure is relaxed on a $32\times 32$ lattice at $T_1=1.0$; level 2 is relaxed on a $64\times 64$ lattice at $T_2=0.603$; level 3 is relaxed on a $128\times 128$ lattice at $T_3=0.427$. Full panel: Data at long times from the runs shown in the inset with times scaled using ${\tau }_2/{\tau }_1=13.8$ and ${\tau }_3/{\tau }_2=10.7$.

Figure 20

Distinct ordered regions in a configuration of a $64\times 64$ lattice equilibrated for $5.4\times {10}^5$ Monte Carlo steps per tile after a rapid quench to $T=0.6$. Tiles are colored according to the sublattice ($A$, $B$, $C$, or $D$) that specifies the level-1 structure. [See Fig. 2.] The gray in the upper left (green) and dark gray (blue) regions are each single domains connected through the periodic boundary conditions.

Figure 21

Close views of two of the domain walls formed during a sudden quench of a $64\times 64$ lattice to the temperature $T=0.6$. The top panel shows a domain wall that roughens relatively easily due to the existence of multiple mismatches along the boundary. The lower panel shows a domain wall that becomes frozen due to the lack of black stripe mismatches in the level-1 and level-2 triangles on both sides of the boundary.

Figure 22

Mass decorations used for computation of the diffraction patterns. The tile on the right is the decoration associated with the tile orientation shown on the left. Each disk is taken to be a point mass, all of equal weight. The black disks indicate the positions of the point masses used for the ${\epsilon }_2=0$ case. The central tile shows the vectors used in Eq. (44).

Figure 23

(a) The mass density for computation of the diffraction pattern for the ${\epsilon }_2=0$ case (the black stripe model). Black dots represent point masses of equal mass. Gray lines are guides to the eye. The point masses touching the dashed (blue) lines form the level-2 unit cell. (b) The total computed diffraction pattern for levels 1 through 6 of the ${\epsilon }_2=0$ mass decoration. The largest wavevectors shown correspond to the basis vectors of the reciprocal lattice associated with the undecorated hexagonal tiling.

Figure 24

(a) A region of the mass density for computation of the diffraction pattern for the ${\epsilon }_2\ne 0$ mass decoration. Black and light gray (light purple) disks represent point particles of equal mass $m$. Darker gray (darker purple) disks represent particles of mass $2m$. (b) The total computed diffraction pattern for levels 1 through 6 of the ${\epsilon }_2\ne 0$ mass decoration. The largest wave vectors shown correspond to the basis vectors of the reciprocal lattice associated with the undecorated hexagonal tiling.

Figure 25

The thick gray (purple) chiral triangle, or a smaller or larger version of it, must occur in a layer of a periodic pattern. Note that each long edge ends at a corner that turns away from another long edge. The black stripes shown are then forced as a pyramid of layers is formed. The tile at the top of the pyramid cannot match all of its black stripes to its neighbors.

Figure 26

Proof of Lemma 2. If each edge in the red triangle (R) is shorter than one half the length of the black edge (S), then a blue edge (B) that is longer than each red one must exist.

Figure 27

Proof of Lemma 4. All possible choices of red lines (R) and corresponding black lines lead to the formation of a chiral triangle of the type shown in Fig. 25.

Figure 28

Proof of Lemma 5. (a) Base layer [thick gray, (purple)] required by Lemma 5 and forced layer (outlined white) above it. (b) Outlined white layer of (a) and forced layer [outlined gray (red)] above it. (c) Outlined gray (red) layer of (b) and forced layer (outlined white) above it.