Gabor Frames for Quasicrystals, KK-theory, and Twisted Gap Labeling

We study the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal $\Lambda,$ and the $K$-theory of the twisted groupoid $C^*$-algebra $\mathcal{A}_\sigma$ arising from a quasicrystal. In particular, we construct a finitely generated projective module \mathcal{H}_\L over $\mathcal{A}_\sigma$ related to time-frequency analysis, and any multiwindow Gabor frame for $\Lambda$ can be used to construct an idempotent in $M_N(\mathcal{A}_\sigma)$ representing \mathcal{H}_\L in $K_0(\mathcal{A}_\sigma).$ We show for lattice subsets in dimension two, this element corresponds to the Bott element in $K_0(\mathcal{A}_\sigma),$ allowing us to prove a twisted version of Bellissard's gap labeling theorem

There are integral heptagons, no three points on a line, no four on a circle

We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erd\H{o}s.Comment: 4 pages, 1 figur

Nematicity, magnetism and superconductivity in FeSe

Iron-based superconductors are well known for their complex interplay between structure, magnetism and superconductivity. FeSe offers a particularly fascinating example. This material has been intensely discussed because of its extended nematic phase, whose relationship with magnetism is not obvious. Superconductivity in FeSe is highly tunable, with the superconducting transition temperature, $T_\mathrm{c}$, ranging from 8 K in bulk single crystals at ambient pressure to almost 40 K under pressure or in intercalated systems, and to even higher temperatures in thin films. In this topical review, we present an overview of nematicity, magnetism and superconductivity, and discuss the interplay of these phases in FeSe. We focus on bulk FeSe and the effects of physical pressure and chemical substitutions as tuning parameters. The experimental results are discussed in the context of the well-studied iron-pnictide superconductors and interpretations from theoretical approaches are presented.Comment: Topical Review submitted to Journal of Physics: Condensed Matte

Order-Parameter Symmetries of Domain Walls in Ferroelectrics and Ferroelastics

The symmetry of boundaries between ferroelectric, ferroelastic and antiphase domains is a key element for a theoretical understanding of their properties. Here, we derive this symmetry from their organic relation to the symmetry of the primary transition order parameters. The domain wall symmetries are shown to coincide with directions of the order-parameter n-dimensional vector space, corresponding to sum of the vectors associated with adjacent domain states. This property is illustrated by the determination of the domain wall maximal symmetries in BaTiO3, LaAlO3, SrTiO3 and Gd2(MoO4)3. Besides, the domain pattern in YMnO3 is interpreted as resulting from an annihilation-creation process, the annihilation of the antiphase domain walls creating six ferroelectric domain walls merging at a single point.Comment: 5 pages, 3 figure

First-principles study of PbTiO3_3 under uniaxial strains and stresses

The behavior of PbTiO$_3$ under uniaxial strains and stresses is investigated from first-principles calculations within density functional theory. We show that irrespectively of the uniaxial mechanical constraint applied, the system keeps a purely ferroelectric ground-state, with the polarization aligned either along the constraint direction ($FE_z$ phase) or along one of the pseudo-cubic axis perpendicular to it ($FE_x$ phase). This contrasts with the cases of isotropic or biaxial mechanical constraints for which novel phases combining ferroelectic and antiferrodistortive motions have been previously reported. Under uniaxial strain, PbTiO$_3$ switched from a $FE_x$ ground state under compressive strain to $FE_z$ ground-state under tensile strain, beyond a critical strain $\eta_{zz}^c \approx +1$\%. Under uniaxial stress, PbTiO$_3$ exhibits either a $FE_x$ ground state under compression ($\sigma_{zz} < 0$) or a $FE_z$ ground state under tension ($\sigma_{zz} > 0$). Here, however, an abrupt jump of the structural parameters is also predicted under both compressive and tensile stresses at critical values $\sigma_{zz} \approx$ $+2$ GPa and $- 8$ GPa. This behavior appears similar to that predicted under negative isotropic pressure and might reveal practically useful to enhance the piezoelectric response in nanodevices.Comment: Submitted, 9 pages, 9 figure