Entropic crystallization of geometrically frustrated magnets on 1/1 approximant Tsai-type quasicrystal

Abstract

We have studied the antiferromagnetic Ising model on the icosahedral bcc lattice, as a model system of 1/1 approximant Tsai-type quasicrystals. We addressed thermal equilibrium properties of this system with Markov-chain Monte Carlo simulation supplemented with the parallel tempering technique to accelerate the relaxation dynamics. As a result, we found a second-order phase transition takes place to the magnetic ordered phase with ${\mathbb Z_3}\times {\mathbb Z_2}$ symmetry breaking. Despite the ordering, the low-temperature phase keeps macroscopic degeneracy as identified by finite residual entropy, $\mathcal{S}\sim0.1767/{\rm spin}$. Remarkably, the existence of residual entropy turns out to play a major role in the formation of magnetic order. Generation of domain wall is suppressed, as it reduces the residual entropy locally stored in icosahedra, beyond the gain of configurational entropy due to domain wall patterns. Magnetic order arises out of this competition as entropic crystallization, which manifest universal mechanism of strongly frustrated systems with large geometrical units.

Figures (5)

• Figure 1: (Color online) The icosahedral bcc network of magnetic ions on the 1/1 quasicrystal approximant.
• Figure 2: (Color online) (a) Structure of the connecting triangle. (b) Numbering of sites within an icosahedron. Rectangles on each plane are shown. (c) The spin configuration of the P1 state of the XY sector is shown (see Table \ref{['tab:table3']}). $\sigma=+1(-1)$ is shown with outward (inward) arrow. (d) The spin configuration of the S1 state of the XY sector is shown (see Table \ref{['tab:table3']}). The spin configuration is obtained by exchanging spins at sites $4$ and $8$ from the P1 state as shown with orange circles. (e) The positions of the 8 connecting triangles within an icosahedron are shown on the vertices of a cube. The spin minority representation of the P1 state is shown. (f) Two connecting triangles belonging to two neighboring icosahedra. The top triangle has a minority spin $\sigma=-1$ on the $zx$ plane, as colored in blue. Due to the ferromagnetic $J_2$, the $zx$ spin of the bottom triangle takes $\sigma=+1$, as colored in red.
• Figure 3: (Color online) Temperature dependence of (a) the energy, (b) the entropy per spin, (c) the specific heat, and (d) the order parameter, for $L=14\leftrightarrow N_{\rm spin}=65856$ spins. The values per spin are shown. In (a), in addition to the total energy, the intra- and inter-icosahedral energies are shown.
• Figure 4: (Color online) System size scaling of the phase transition. System size goes from $L=6\leftrightarrow N_{\rm spin}=5184$ spins to $L=20\leftrightarrow N_{\rm spin}=192000$ spins by steps of $2$. The system is well thermalised for $T\geq 10^{-2}$, but fails to thermalise for smaller temperatures. The entropy is therefore calculated down to the smallest reliable temperature for $T>10^{-2}$.
• Figure 5: (a) Single icosahedral ground state with the reference spin circled in red having $V_j=5$. The other spins are fully constrained by the triangle rule. (b) Single icosahedral ground state with the reference spin having $V_j=3$. Note that it is connected to (a) by flipping the two spins circled in green (see Sec. \ref{['sec:Sector']}).