Identifying interesting relationships between pairs of variables in large data sets is increasingly important. Here, we present a measure of dependence for two-variable relationships: the maximal information coefficient (MIC). MIC captures a wide range of associations both functional and not, and for functional relationships provides a score that roughly equals the coefficient of determination (R2) of the data relative to the regression function. MIC belongs to a larger class of maximal information-based nonparametric exploration (MINE) statistics for identifying and classifying relationships. We apply MIC and MINE to data sets in global health, gene expression, major-league baseball, and the human gut microbiota and identify known and novel relationships.

Detecting Novel Associations in Large Data Sets

Motivated by a recent experimental observation of superconductivity in the Al-Zn-Mg quasicrystal, we study the low-temperature behavior of electrons moving in the quasiperiodic potential of the Ammann--Beenker tiling in the presence of a local attraction. We employ the Bogoliubov--de Gennes approach for approximants of different sizes and determine the local pairing amplitude ${\mathrm{\ensuremath{\Delta}}}_{i}$ as well its spatial average, ${\mathrm{\ensuremath{\Delta}}}_{0}$, the superconducting order parameter. Due to the lack of periodicity of the octagonal tiling, the resulting superconducting state is inhomogeneous, but we find no evidence of the superconductivity islands, as observed in disordered systems, with ${\mathrm{\ensuremath{\Delta}}}_{i}\ensuremath{\rightarrow}0$ at ${T}_{c}$ for all sites. In the weak-coupling regime, we find that the superconducting order parameter depends appreciably on the approximant size only if the Fermi energy sits at a pseudogap in the noninteracting density of states, with ${\mathrm{\ensuremath{\Delta}}}_{0}$ decreasing as the system size increases. These results are in line with the experimental observations for the Al-Zn-Mg quasicrystal, and they suggest that, despite their electronic structure, quasicrystals are prone to display conventional BCS-like superconductivity.

Conventional superconductivity in quasicrystals

Eight years since the discovery of the binary antiferromagnetic (AFM) quasicrystal approximants $\mathrm{C}{\mathrm{d}}_{6}R$ [R. Tamura et al., Phys. Rev. B 82, 220201 (2010)], we report the observation of an AFM transition in quasicrystal approximants Au-Al-R ($R=\mathrm{Gd}$ and Tb), which represent examples of AFM in ternary quasicrystal approximants. Magnetic susceptibility, magnetization, and specific heat coherently show that the Au-Al-R ($R=\mathrm{Gd}$ and Tb) approximants undergo an AFM transition at ${T}_{\mathrm{N}}=9.4$ and 11.8 K, respectively. The AFM transitions are found near the phase boundary of the wide single-phase region where the chemical disorder is substantially low. This finding of ternary AFM approximants has far deeper implications compared with that of the binary AFM approximants: It now indicates the possibility to synthesize new AFM approximants in many other systems by the appropriate replacement of Cd with two other elements, and thus enables exploration of various magnetic orders over AFM spin icosahedra in solids, in particular, their evolution as a function of the electron-per-atom (e/a) ratio. Moreover, the finding provides us a missing route to synthesize an AFM quasicrystal by similar composition tuning.

Antiferromagnetic order is possible in ternary quasicrystal approximants

Quasicrystals (QCs), first discovered in 1984, generally do not exhibit long-range magnetic order. Here, we report on long-range magnetic order in the real icosahedral quasicrystals (i QCs) Au-Ga-Gd and Au-Ga-Tb. The Au65Ga20Gd15i QC exhibits a ferromagnetic transition at TC = 23 K, manifested as a sharp anomaly in both magnetic susceptibility and specific heat measurements, along with an appearance of magnetic Bragg peak below TC. This is the first observation of long-range magnetic order in a real quasicrystal, in contrast to the spin-glass-like behaviors observed for the other magnetic quasicrystals found to date. Moreover, when Gd is replaced by Tb, i.e., for the Au65Ga20Tb15i QC, a ferromagnetic behavior is still retained with TC = 16 K. Although the sharp anomaly in the specific heat observed for the Au65Ga20Gd15i QC becomes broadened upon Tb substitution, neutron diffraction experiments clearly show marked development of magnetic Bragg peaks just below TC, indicating long-range magnetic order for the Au65Ga20Tb15i QC also. Our findings can contribute to the further investigation of exotic magnetic orders formed on real quasiperiodic lattices with unprecedented highest global symmetry, i.e., icosahedral symmetry.

Experimental Observation of Long-Range Magnetic Order in Icosahedral Quasicrystals.

We study magnetic properties in the half-filled Hubbard model on the Ammann-Beenker tiling. First, we focus on the domain structure with locally eightfold rotational symmetry to examine the strictly localized confined states for the tight-binding model. We count the number of vertices and confined states in the larger domains generated by the deflation operations systematically. Then, the fraction of the confined states, which plays an important role for magnetic properties in the weak coupling limit, is obtained as $p=1/2{\ensuremath{\tau}}^{2}$, where $\ensuremath{\tau}(=1+\sqrt{2})$ is the silver ratio. It is also found that the wave functions for confined states are densely distributed in the system and thereby the introduction of the Coulomb interactions immediately induces the finite staggered magnetizations. Increasing the Coulomb interactions, the spatial distribution of the magnetizations continuously changes to those of the Heisenberg model. We discuss crossover behavior in the perpendicular space representation and reveal the superlattice structure in the spatial distribution of the staggered magnetizations.

Superlattice structure in the antiferromagnetically ordered state in the Hubbard model on the Ammann-Beenker tiling

Composition-driven spin glass to ferromagnetic transition in the quasicrystal approximant Au-Al-Gd

A neutron powder diffraction experiment has been performed on the quasicrystal approximant ${\mathrm{Au}}_{72}{\mathrm{Al}}_{14}{\mathrm{Tb}}_{14}$, a body-centered-cubic crystal of icosahedral spin clusters. The long-range antiferromagnetic order was confirmed at the transition temperature ${T}_{\mathrm{N}}=10.4$ K. The magnetic structure consists of noncoplanar whirling spins on the icosahedral clusters, arranged in an antiferroic manner. A simple icosahedral spin-cluster model with uniaxial anisotropy accounts well the whirling spin order as well as the in-field metamagnetic transition, indicating that the icosahedral symmetry is essential.

Asuka Ishikawa

Papers

Antiferromagnetic order is possible in ternary quasicrystal approximants

Experimental Observation of Long-Range Magnetic Order in Icosahedral Quasicrystals.

Composition-driven spin glass to ferromagnetic transition in the quasicrystal approximant Au-Al-Gd

Whirling spin order in the quasicrystal approximant Au72Al14Tb14

Magnetism of tsai-type quasicrystal approximants