Finite-temperature liquid-quasicrystal transition in a lattice model

We consider a tiling model of the two-dimensional square lattice, where each site is tiled with one of the 16 Wang tiles. The ground states of this model are all quasiperiodic. The systems undergoes a disorder to quasiperiodicity phase transition at finite temperature. Introducing a proper order parameter, we study the system at criticality and extract the critical exponents characterizing the transition. The exponents obtained are consistent with hyperscaling.

Figure 1

The 16 Ammann tiles. The four rows correspond to the four types of tiles (see text).

Figure 2

Binder cumulant of the tiling model. The collapse of crossings for various lattice sizes signals the location of the critical point.

Figure 3

Order parameter measurements performed at $T_1$ and $T_2$. Power-law behavior is fitted, $L=10$ is excluded from the analysis. Fitted exponents are $0.28$ and $0.33$.

Figure 4

Susceptibility $\chi$ fitted to a power-law behavior at $T_1$ and $T_2$. Fitted exponents are $1.44$ and $1.36$.

Figure 5

Measurements of $Q^{\prime }$ for $T_1=0.4211$. Fitted exponent is $0.99$.

Figure 6

Specific heat measurements at $T_1$ and $T_2$. Data are well fitted by a logarithmic function. For comparison, best power-law fits are presented in the inset, yielding exponents $0.23$ and $0.25$.