Phase transitions in the Aristoff–Radin lattice model of quasicrystals: rigorous existence, order, and the temperature–chemical-potential phase diagram

The Aristoff–Radin model is a family of two-dimensional grand-canonical lattice gases built from Ammann's 16 aperiodic Wang tiles: non-overlapping square tiles of integer edge width $w\geq 1$ carry colored edges, and two non-overlapping tiles interact — with energy $\pm(3w-d)/(2w)$, where $d$ is the $\ell^1$-distance between their centers, negative when the facing edge colors match and positive otherwise — precisely when one of the two tiles overlaps one of the four $w\times w$ squares sharing an edge with the other; equivalently, for center offset $(\Delta x,\Delta y)$, when $\min(|\Delta x|,|\Delta y|)<w\leq\max(|\Delta x|,|\Delta y|)<2w$, so that $w\leq d<3w$ (diagonal near-neighbors such as offset $(w,w)$ do not interact). Monte Carlo simulation along the single isotherm $\beta=1.5$ found evidence of a discontinuous (first-order) transition between a low-density disordered phase and a high-density quasicrystalline phase for $w=2$ and $w=3$, and of an apparently continuous transition for $w=1$; no transition in the model has ever been proven, and the rest of the phase diagram is unexplored. The open problem is to determine the phase structure of this family: prove analytically the existence and the order of the transitions — including the columnar phase expected at high temperature and chemical potential — and locate the phase boundaries in the temperature–chemical-potential plane for all tile widths ($w>3$ was never simulated), locating thermodynamic-limit transition points and latent heats. The rigorous core is a recognized open problem of statistical mechanics: quasicrystalline Gibbs states for finite-range lattice gases have been constructed only in four spatial dimensions (Miekisz–Taati 2025); two and three dimensions remain open.

Background

Quasicrystals, discovered in 1984 in an Al–Mn alloy (Shechtman, Blech, Gratias and Cahn), are solids whose diffraction patterns show sharp Bragg peaks at rotational symmetries incompatible with any periodic crystal. Explaining why such aperiodically ordered structures are thermodynamically stable at positive temperature — in the language of equilibrium statistical mechanics, why a Hamiltonian with finite-range interactions should possess Gibbs states (equilibrium ensembles in the infinite-volume limit) with quasicrystalline long-range order — is a recognized open problem of mathematical statistical physics.

A natural discrete laboratory is provided by aperiodic tile sets. A Wang tile is a unit square whose four edges carry colors from a finite set; a finite set of such tiles is called aperiodic if it can tile the plane, but only in non-periodic ways. Ammann's set of 16 tiles (edge colors drawn from six colors, restricted to 16 four-color patterns) is a classical small example, and its tilings possess the long-range aperiodic order characteristic of quasicrystals. Leuzzi and Parisi (2000) introduced a lattice gas on the square lattice $\mathbb{Z}^2$ in which every site is occupied by exactly one such tile and nearest-neighbor tile pairs interact with energy $0$ or $-1$ according to whether the shared edge colors match. At fixed density of one tile per site, simulations (Leuzzi–Parisi 2000; Koch–Radin 2010) show a continuous (higher-order) transition at positive temperature between a disordered phase and a phase with quasicrystalline order, and Rotman and Eisenberg (2011) extracted critical exponents consistent with hyperscaling. All of these are simulation results; none is proven.

Aristoff and Radin (2011) — the source of this problem — extended the model in two ways. First, tiles became squares of integer edge length $w\geq 1$ that may not overlap (a hard core; for $w\geq 2$ this is an extended hard core like that of the hard-square lattice gas, whose densest configurations form a degenerate 'columnar' phase in which parallel rows or columns of squares slide independently). Second, the density became a free thermodynamic variable through a grand-canonical ensemble: a configuration $C$ of tiles in a finite volume is weighted by $\exp[-\beta(\mathcal{H}(C)-\mu\mathcal{N}(C))]$, where $\beta$ is the inverse temperature, $\mu$ the chemical potential, $\mathcal{N}(C)$ the number of tiles, and $\mathcal{H}$ the total pair energy. The pair interaction acts only between non-overlapping tiles one of which overlaps, in its interior, one of the four $w\times w$ squares sharing an edge with the other; in terms of the offset $(\Delta x,\Delta y)$ of the tile centers this is the condition $\min(|\Delta x|,|\Delta y|)<w\leq\max(|\Delta x|,|\Delta y|)<2w$, so the $\ell^1$-distance $d=|\Delta x|+|\Delta y|$ of an interacting pair satisfies $w\leq d<3w$ (diagonal near-neighbors, e.g. at offset $(w,w)$, do not interact). Its magnitude is $(3w-d)/(2w)$, negative when the facing edge colors match and positive when they do not. Perfect Ammann tilings are ground states; matching violations cost energy; the hard core and the entropy of the disordered phase compete with this ordering tendency. The order parameter $\chi$ is the normalized volume $|V_{j^*}|/|V|$ of the largest thick outer shell $V_{j^*}$ in which at least $80\%$ of the tile sites have a nearby tile whose 'long/short' edge pattern agrees with the boundary perfect tiling, with boundary conditions given by a fixed shell of tiles from a perfect tiling.

Along the single simulated isotherm $\beta=1.5$, Monte Carlo simulation found, for $w=2$ (near $\mu\approx 1.4$) and $w=3$ (near $\mu\approx 2.0$), developing jump discontinuities in the average volume fraction and in the energy per unit volume, a heat capacity developing a delta-function peak, and a discontinuity in $\chi$ — evidence for a first-order transition between a low-density disordered and a high-density quasicrystalline phase that breaks both spatial symmetry (a square sublattice is preferentially populated) and particle-type symmetry. For $w=1$ (near $\mu\approx 0.8$) the transition appears continuous, as in the fixed-density ancestor; introducing vacancies at $w=1$ did not change this character. By analogy with hard squares — where simulation indicates a continuous transition to the columnar phase for $w=2$ near volume fraction $0.92$ — the authors expect a richer diagram at higher temperatures: a columnar phase at high temperature and chemical potential, and a quasicrystalline-to-columnar transition upon heating at fixed high density.

The rigorous status is stark. For finite-range lattice models, aperiodic order at positive temperature has been proven only in one dimension (with long-range or specially stabilized interactions, and in one-dimensional thermodynamic-formalism 'quasi-crystals in subshifts' of Bruin–Leplaideur), and — very recently — in a four-dimensional construction (Miekisz–Taati 2025) whose two extra dimensions implement an error-correcting stacking of Ammann tilings. van Enter's 2014 review states plainly that proving the existence of a properly quasicrystalline state for a finite-range model remains open, and Miekisz–Taati note that the corresponding three-dimensional (and a fortiori two-dimensional) question remains a fundamental problem in statistical physics. There is even a conjecture that finite-range models in one and two dimensions have only finitely many extremal Gibbs states — which would forbid two-dimensional quasicrystalline Gibbs states — a conjecture the Aristoff–Radin simulations dispute. Proving the existence and order of the transitions in this concrete two-dimensional family is therefore simultaneously a well-posed question about a specific model and the concrete instance of a general open problem.

Problem Statement

Consider the Aristoff–Radin family of grand-canonical lattice gases on $\mathbb{Z}^2$. A configuration is a set of pairwise non-overlapping square tiles of integer edge length $w\geq 1$ centered on the lattice, each tile carrying one of the 16 edge-coloring types of Ammann's aperiodic Wang-tile set (16 square tiles whose edges are colored from six colors and which tile the plane, but only non-periodically). Overlapping tiles are forbidden (infinite energy). A pair of non-overlapping tiles interacts if and only if one of them overlaps, with nonempty interior, one of the four $w\times w$ squares that share an edge with the other; writing $(\Delta x,\Delta y)$ for the offset of the two tile centers, this is the condition $\min(|\Delta x|,|\Delta y|)<w\leq\max(|\Delta x|,|\Delta y|)<2w$, and in particular not every non-overlapping pair with $\ell^1$-distance $d=|\Delta x|+|\Delta y|$ below $3w$ interacts — diagonal pairs such as offset $(w,w)$ have zero interaction. For an interacting pair the energy is $H=\pm(3w-d)/(2w)$: negative when the two facing edge colors match (e.g. the north edge of the lower tile and the south edge of the upper tile) and positive when they do not. The grand-canonical Gibbs measure at inverse temperature $\beta$ and chemical potential $\mu$ weights a configuration $C$ by $\exp[-\beta(\mathcal{H}(C)-\mu\mathcal{N}(C))]$, with $\mathcal{N}(C)$ the number of tiles. (The original simulations used square volumes of side a multiple of $w$ with fixed boundary shells taken from perfect tilings, and the order parameter $\chi$, defined in the source by $\chi(C)=|V_{j^*}|/|V|$: a tile 'agrees' with the boundary perfect tiling $T$ if its long/short edge pattern (colors 1, 2 'short'; 3, 4, 5, 6 'long') equals that of the tile of $T$ at the nearby site $(wi,wj)$, and $j^*$ is the largest $j$ such that at least $80\%$ of the sites in the thick outer shell $V_j=V\setminus[-j,j]^2$ have a nearby agreeing tile.)

What is known: along the single simulated isotherm $\beta=1.5$, Monte Carlo evidence of a first-order transition between a low-density disordered phase and a high-density quasicrystalline phase (one with the long-range aperiodic order of a perfect Ammann tiling) for $w=2$ (near $\mu\approx 1.4$) and $w=3$ (near $\mu\approx 2.0$), and of an apparently continuous transition for $w=1$ (near $\mu\approx 0.8$). No other isotherm and no width $w>3$ has been simulated, no finite-size-scaling extrapolation to the thermodynamic limit has been completed, and no transition in the model has been proven.

Determine the phase structure of this model family, in two linked parts serving one common objective — the existence, order, and location of the phase transitions of the Aristoff–Radin quasicrystal lattice gas in its natural $(\beta,\mu,w)$ parameter space:

(1) Existence and order. Prove — or disprove, with equal rigor — that in the thermodynamic limit the model exhibits phase transition(s), and establish their order (continuous versus discontinuous): the discontinuous disordered-to-quasicrystalline transition at low temperature for $w=2$ and $w=3$; the apparently continuous transition for $w=1$; and the transitions expected at higher temperature involving the columnar phase, in particular a quasicrystalline-to-columnar transition at fixed high chemical potential upon heating. The enumerated transitions are independent sub-claims, each resolvable in either direction; the same determination is sought for every tile width $w$, including $w>3$, so that a rigorous proof or disproof of any one of them settles that sub-claim, while a complete resolution of part (1) requires settling all of them across the full width range. A proof of first-order character should identify the coexisting phases (including the broken spatial sublattice and particle-type symmetries) and, where possible, the latent heat (the discontinuity in density and energy per volume at coexistence).

(2) Location and extrapolation. For each transition whose existence is established or left standing by part (1), determine the loci of the phase boundaries in the temperature–chemical-potential plane as functions of the tile width $w$ — separating the disordered, columnar, and quasicrystalline phases, and including widths $w>3$ — locating the thermodynamic-limit transition points and latent heats underlying the observed size-dependent developing discontinuities and heat-capacity peaks. A transition rigorously ruled out in part (1) contributes no boundary to locate; should part (1) be resolved entirely in the negative, the phase structure is settled and this location demand is discharged in full.

As a secondary, explicitly optional extension noted in the source, one may ask whether replacing the soft matching interaction by an equivalent hard (infinite) attraction/repulsion — bringing the model closer to the rigorously studied multitype Widom–Rowlinson lattice gases — changes this phase structure. This extension is not required for the completion of the core objective above.

References

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