October 2, 1996

Statistical Mechanics of Aperiodic Tilings

Steven A. Janowsky^{∗} and Hans Koch^{†} Department of Mathematics University of Texas Austin, TX 78712

Abstract We consider a statistical mechanical spin system where the ground states are nonperiodic tilings of the plane. Numerical simulations indicate that there is a phase transition between an ordered and a disordered state.

1 Introduction

Quasicrystals are a form of matter that exhibit long range order, but are nonperiodic. One attempt to model the properties of quasicrystals has been through the use of aperiodic tiling models such as the Penrose tiles. More accurately, one is really using the tiling models to describe the quasicrystal at zero temperature—at positive temperatures the quasicrystals will have defects that are unaccounted for in the tiling models.

We attempt to remedy this defect by studying tilings at finite temperature. We do this via the standard language of equilibrium statistical mechanics by first translating the tiling model into an equivalent spin system.

2 The Model

2.1 Tiling Model

Our model [1] consists of 16 (Wang-style) prototiles, which are perhaps best thought as being squares of the same size that can be translated (with fixed orientation) to any lattice site. There are six different types of edges, and two tiles can be placed next to each other only if their adjacent edges are of the same type. The different type of prototiles are listed in figure 1. We see, for example, that to the right of the first kind of tile we can place only the second, third or fourth kind of tile.

∗janowsky@math.utexas.edu

†koch@math.utexas.edu

2 1 2 1  6 4 1 1  5 4 1 2  3 6 1 2  3 5 2 2  1 1 6 4  1 2 5 4  1 2 3 6
2 2 3 5  5 3 5 3  6 3 5 5  5 5 6 3  6 5 6 5  4 5 6 6  6 6 4 5  4 6 4 6

Figure 1: The 16 different prototiles.

One can also view these prototiles as being rectangular, with edges 1 and 2 having length 1 and the remaining edges having length (1 + \sqrt{5})/2. Then the first prototile is a small square, the next 4 are horizontal rectangles, 6–9 are vertical rectangles, and the remaining 7 are large squares. The “all squares” view is more convenient for treating the model as a lattice spin system while the rectangular view is more convenient for determining the properties of a perfect tiling of the plane, i.e. the ground state.

With these prototiles, it is possible to create a tiling of the entire plane (where the edge-matching condition is preserved), and this can only be done in a nonperiodic fashion [1]. Perhaps more commonly studied as an example of nonperiodic tiling are the (pair of) Penrose tiles []; we need this more complicated example in order to be able to consider a tiling as a spin configuration on the two-dimensional square lattice. Our current model with sixteen prototiles was chosen because until recently it was the smallest known set of aperiodic Wang prototiles; recently a set of 13 prototiles was discovered [] but the set of sixteen may still be preferable in some respects since the horizontal and vertical directions are equivalent, unlike the set of thirteen.

2.2 Spin Model

We can treat the tiling model as a spin model. At each site (x, y) of the square lattice we have a spin σ(x, y) that takes values from 1 to 16, corresponding to the 16 different prototiles. Each spin σ(x, y) is assigned a local energy h(x, y) that is equal to the number of neighboring spins (tiles) that have an edge mis-match: This energy ranges from 0 for a “perfect fit” to 4 for a “complete mis-match”. Then we define the total energy in a volume Λ as the sum of the local energies,

H_{Λ} = ∑_{(x,y)∈Λ} h(x, y) ,

with appropriate boundary conditions—see figure 2.

If we choose reasonable boundary conditions, the ground states of this model are simply subsets of the possible tilings of the plane produced by the tiling model. Such states exhibit a peculiar type of long range order (quasicrystalline in nature). A key question is whether or not this order persists at finite temperature.

2 1 2 1 6 4 1 1 5 4 1 2 3 5 2 2
2 2 3 5 5 3 5 3 6 3 5 5 5 3 5 3

Figure 2: With free boundary conditions, this example of a spin configuration would have energy 3.

The difficulty in answering this question is a consequence of the rather bizzare nature of the Hamiltonian for our spin system. It is highly anisotropic and satisfies no obvious symmetry conditions. There are numerous ground states: in an n × n volume there are O(n^{2}) ground states. Standard techniques for determining the phase structure of a system, such as Peierls’ arguments or correlation inequalities do not seem to be applicable.

At zero temperature the system has long range order. At high temperature the system is disordered. There are four basic phase diagrams that seem possible for this sytem:

1. There is a phase transition from long range order to disorder at finite temperature.

2. There is a phase transition from long range order to disorder at zero temperature.

3. There is a phase transition from long range order to some intermediate state at finite temperature, and then a transition to disorder at a higher temperature.

4. There is a phase transition from long range order to some intermediate state at zero temperature, and then a transition to disorder at a finite temperature.

Of course more complicated phase diagrams are possible—for example models with an infinite number of transitions are possible [].

3 Results

We attempted to determine which of the above scenarios is correct through the use of numerical simulations. In figure 3 we show the energy per tile and its variance (the specific heat) for a 257×257 system. For the energy we show two curves: the lower curve is for increasing temperature and the upper curve for decreasing temperature. These two curves do not match because certain types of defects are effectively frozen into the system.

The data is certainly consistent with a phase transition at T ≈ 0.43, but it is not convincing. Finite-size scaling provides little help; the size 257 system’s data is only very slighly sharper than the data for a system of size 129.

0.000 0.001 0.002 Specific Heat 0.003 0.004 0.005 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8 Specific heat 0.0 0.2 0.4 0.6 Average Energy 0.8 1.0 1.2  Decreasing Temperature Increasing Temperature

Figure 3: Average energy per tile and specific heat as a function of temperature.

This very slow divergence of the specific heat makes it more likely that the transition (if it exists) is of the 4th type mentioned: a Kosterliz-Thouless transition would produce a logarithmically divergent specific heat, consistent with what was observed numerically.

Supporting this result are data for the correlations in this model. Let η_{small}(x, y) = ∑_{k=1}^{5} δ_{k,σ(x,y)}, and let η_{large}(x, y) = ∑_{k=6}^{16} δ_{k,σ(x,y)}. The variable η_{small}(x, y) is equal to 1 if the tile at (x, y) has left and right sides of type 1 or 2, and zero otherwise; the variable η_{large}(x, y) is complementary. We choose these particular spin values since if we treat the tiles as rectangles, η_{small}(x, y) corresponds to a tile of height 1 and η_{large}(x, y) corresponds to a tile of height (1 + \sqrt{5})/2.

In figure 4 we plot the (unnormalized) correlation function

G(x) = ∑_{y}⟨η_{small}(0, y)η_{small}(x, y) + η_{large}(0, y)η_{large}(x, y)⟩

for a system with fixed ground-state boundary conditions at x = 0. At zero temperature, G(x) = 1, while at infinite temperature, G(x) = 73/128.

The results at high temperature clearly show exponential decay of the correlation function. The low temperature results are somewhat different. While there does not appear to be long range order (at least for this particular correlation function) the correlations do decay slowly, indicating a possibly infinite correlation length.

150000 200000 250000 300000 350000 400000 450000 500000 0  50  100  150  200  250 T = .40 T = .41 T = .42 T = .43 T = .44 T = .45 T = .47

Figure 4: correlation function for various temperatures

Lastly, we examined the Fourier transform of our system. In particular, let

ϕ(x, y) = δ_{σ(x,y),1} − ∑_{(x,y)∈Λ} δ_{σ(x,y),1}/|Λ|

be the variable describing the density of type-1 tiles, and consider the maximum value

S(Λ) = max_{(k_{x},k_{y})} ⟨| ∑_{(x,y)∈Λ} e^{ik_{x}x}ik_{y}yϕ(x, y)|⟩ .

The scaling of S(Λ) with |Λ| indicates to some extent the type of order that the system posseses. A completely ordered system behaves like S(Λ) ∼ |Λ|; in general one would expect S(Λ) ∼ |Λ|^{α}. In figure 5 we plot the values of S(Λ) for Λ a 256 × 256 square for different temperatures; on the same plot are the values of the exponent α that one would obtain from the ratio

α = \frac{log[S(Λ)/S(1)]}{log |Λ|},

where S(1) refers to a single site system.

We see that there is an apparent transition.

4 Discussion

Tiling models such as the one studied here can be viewed as a map between the real numbers (on the unit circle) and tiling configurations. Each real angle produces a certain subset of ground states.

30 100 300 1000 3000 10000  1.0 0.9 0.8 0.7 0.6 0.5 0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8 S(Λ)  scaling exponent temperature ⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄ ∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘

Figure 5: S(Λ) for various temperatures. The ⋄’s represent increasing temperature, the ∘’s decreasing temperature, showing the observed hysteresis. Corresponding scaling exponents are also shown.

In a finite volume, the ground states are described by rational approximations to the real numbers. Ground states that are “nearby” have “nearby” real numbers.

We can imagine performing some type of block-spin renormalization group on this model. Two neighboring blocks with angles θ_{1} and θ_{2} will have energies that differ at least by |θ_{1} − θ_{2}|L_{B}, where L_{B} is the size of the block. If this was the only contribution to the energy, we would effectively have a rotator model undergoing a Kosterlitz-Thouless transition.

Unfortunately the angle description is incomplete, and it is not known if these contributions are dominant or if there is a more important contribution.

References

[1] Branko Grünbaum and G. C. Shepard, Tilings and Patterns, W. H. Freeman, 1987.

[2] Jacek Mi ekisz, “A Microscopic Model with Quasicrystalline Properties,” J. Stat. Phys. 58, 1137–1149 (1990).