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A quasi-crystal is the natural extension of the notion of a crystal to structures with quasi-periodic, rather than periodic, translational order. Two and three-dimensional quasi-crystals are here classified by their symmetry under rotation, and it is shown that many disallowed crystals symmetries are allowed quasi-crystal symmetries. The diffraction pattern of an ideal quasi-crystal is analytically computed, and it is shown that the recently observed electron-diffraction pattern of an Al-Mn alloy is closely related to that of an icosahedral quasi-crystal.

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... Notwithstanding this, the remarkable sharpness of the obtained electron diffraction patterns clearly indicated a high coherency of the electrons spatial interference, comparable to the one usually encountered in classical periodic crystals [2]. The nature of the underlying long-range order was explained by invoking the mathematical notion of quasiperiodic (QP) functions [3], thereby widening the concept of ordered arrangement of matter from that based in periodic distribution of atoms through the space to that corresponding to QP ones [4]. Accordingly, these new materials were dubbed quasiperiodic crystals, or quasicrystals (QCs) for short. ...

... 3 With the notable exception of A. Mackay, see below. 4 Alan L. Mackay was awarded the 2010 Oliver E. Buckley Condensed Matter Prize from the American Physical Society for his "pioneering contributions to the theory of quasicrystals, including the prediction of their diffraction pattern". 5 BaTiO 3 is one of the best investigated perovskite oxide systems which is also widely used in thin film applications and oxide heterostructures. ...

... To this end, I recommend a careful reading of the insightful thoughts discussed by R. Lifshitz some time ago [31]. In particular, it may be pertinent to take a closer look at the originally proposed QC notion: according to D. Levine and P. Steinhardt "A quasicrystal is the natural extension of the notion of a crystal to structures with quasiperiodic, rather than periodic, translational order" [4]. Therefore, we may better define QCs as long-range ordered solids whose diffraction pattern exhibits symmetries that are incompatible with translational symmetry, with the proviso that this long-range order stems from the underlying quasiperiodic distribution of atoms throughout the space. ...

Four decades have elapsed since the first quasiperiodic crystal was discovered in the Al-Mn alloy system, and many progresses have been made during this time interval in the science of quasicrystals (QCs). Notwithstanding this, a significant number of open questions still remain regarding both fundamental and technological aspects. For instance: What are QCs good for? How can we improve the current provisional QC defi-nition? What is the role of the underlying quasiperiodic order and the characteristic inflation symmetry of these compounds in the emergence of their unusual physicochemical properties? or, What is the nature of chemical bonding in QCs formed in such different sorts of materials as alloys, oxides, or organic polymers? In this essay I will discuss these, and other closely related, issues from an interdisciplinary perspective and will comment on appealing prospectives in the field for the years to come.

... Notwithstanding this, the remarkable sharpness of the obtained electron diffraction patterns clearly indicated a high coherency of the electrons spatial interference, comparable to the one usually encountered in classical periodic crystals [2]. The nature of the underlying long-range order was explained by invoking the mathematical notion of quasiperiodic (QP) functions [3], thereby widening the concept of ordered arrangement of matter from that based in periodic distribution of atoms through the space to that corresponding to QP ones [4]. Accordingly, these new materials were dubbed quasiperiodic crystals, or quasicrystals (QCs) for short. ...

... 3 With the notable exception of A. Mackay, see below. 4 Alan L. Mackay was awarded the 2010 Oliver E. Buckley Condensed Matter Prize from the American Physical Society for his "pioneering contributions to the theory of quasicrystals, including the prediction of their diffraction pattern". is relatively small, so that the overall structure can be confidently described in terms of a perturbed averaged periodic structure. ...

... To this end, I recommend a careful reading of the insightful thoughts discussed by R. Lifshitz some time ago [28]. In particular, it may be pertinent to take a closer look at the originally proposed QC notion: according to D. Levine and P. Steinhardt "A quasicrystal is the natural extension of the notion of a crystal to structures with quasiperiodic, rather than periodic, translational order" [4]. Therefore, we may better define QCs as long-range ordered solids whose diffraction pattern exhibits symmetries that are incompatible with translational symmetry, with the proviso that this long-range order stems from the underlying quasiperiodic distribution of atoms throughout the space. ...

Four decades have elapsed since the first quasiperiodic crystal was discovered in the Al-Mn alloy system, and many progresses have been made during this time interval in the science of quasicrystals (QCs). Notwithstanding this, a significant number of open questions still remain regarding both fundamental and technological aspects. For instance: What are QCs good for? How can we improve the current provisional QC defi-nition? What is the role of the underlying quasiperiodic order and the characteristic inflation symmetry of these compounds in the emergence of their unusual physicochemical properties? or, What is the nature of chemical bonding in QCs formed in such different sorts of materials as alloys, oxides, or organic polymers? In this essay I will discuss these, and other closely related, issues from an interdisciplinary perspective and will comment on appealing prospectives in the field for the years to come.

... But Steinhardt and his student Dov Levine found a way to arrange pentagons in a nearly or quasiperiodic fashion without gaps, inspired by tilings in islamic art. They also demonstrated that the three dimensional equivalent, a quasiperiodic arrangement of atoms with five fold (icosahedral) symmetry, is possible and predicted that this should lead to an X-ray diffraction pattern as shown in Figure 3.29a (Levine and Steinhardt, 1984). Soon after, their hypothesis was verified when Dan Shechtman discovered the first compound with the predicted diffraction pattern in an artificially prepared Al-Mn alloy (Shechtman et al., 1984;Fig. ...

... But how was such a proof to be provided, given that almost no material was available for further investigations? (a) Theoretical X-ray diffraction pattern of five fold symmetry predicted by Levine and Steinhardt (1984) for a crystal with a quasiperiodic arrangement of atoms. Figure from Levine and Steinhardt (1984). ...

... (a) Theoretical X-ray diffraction pattern of five fold symmetry predicted by Levine and Steinhardt (1984) for a crystal with a quasiperiodic arrangement of atoms. Figure from Levine and Steinhardt (1984). (b) X-ray diffraction pattern observed by Shechtman (1984) for an Al-Mn alloy. ...

I started my journey in science by studying noble gases implanted by the solar wind in dust grains on the surface of the Moon, and with many colleagues I have studied solar wind implanted noble gases in natural and artificial samples throughout my career, the latter exposed primarily by the Genesis space mission. Major questions are what noble gases in the solar wind can tell us about the present and the past Sun, and how they can contribute to understanding the formation and history of the planets and their building blocks, represented, for example, by meteorites. Since my early years as a postdoc, I have also been interested in noble gases (and radioactive nuclides) produced in meteorites and other extraterrestrial samples by interactions with energetic elementary particles from galactic cosmic radiation (and the Sun). These so called “cosmogenic” nuclides allow us to study the transport of meteorites to Earth, and the dynamics of the top surface layers (“regoliths”) on the Moon, asteroids, and comets. Cosmogenic noble gases are also crucial for studying even more exotic topics such as the history of tiny presolar grains that formed in the cooling envelopes of earlier generations of stars towards the end of their lives and were eventually incorporated into the meteoritic matter where they are found today. Cosmogenic noble gases in some tiny phases in meteorites are also likely tracers of our highly active Sun at a very early stage in its history. A few years later, I started my third major research topic in cosmochemistry, the study of primordial noble gases in meteorites and other extraterrestrial samples. These noble gases were incorporated into meteorites or their precursors in the early solar system or even in a presolar environment. I also participated in studies by colleagues of isotopic anomalies of other elements important in cosmochemistry, my expertise being mainly in aspects of the influence of cosmic rays on these elements. Although working in an Earth Science institution, it took quite a while before I started to also study noble gases (and radionuclides) in terrestrial samples. This is described in the second part of this contribution. A major focus was on cosmogenic noble gases and radionuclides produced in samples near the Earth’s surface. Although production rates of cosmogenic nuclides on Earth are several orders of magnitude lower than in space, making their analysis more challenging, they have become an important tool in geomorphology. Because stable noble gas nuclides are particularly well suited to the study of ancient landscapes, much of our work focused on areas with arid climates, such as Antarctica and the Andes in Chile, in collaboration with geoscience colleagues. We also participated in the large multinational CRONUS collaboration, funded by the European Union, a community effort to improve our knowledge of nuclide production rates at the Earth’s surface. In another major collaboration with external colleagues we are involved in noble gas analyses of water samples, ranging from lakes to aquifers to tiny inclusions in stalagmites. This research focuses on studying lake and groundwater dynamics, including contributions of mantle-derived noble gases such as in volcanic lakes. Atmospheric noble gases dissolved in suitable samples are also palaeotemperature indicators, supplementing information from other proxies such as oxygen isotopes.

... Notwithstanding this, the remarkable sharpness of the obtained electron diffraction patterns clearly indicated a high coherency of the electrons' spatial interference, comparable to those usually encountered in classical periodic crystals [2]. The nature of the underlying long-range order was explained by invoking the mathematical notion of quasiperiodic (QP) functions [3], thereby widening the concept of the ordered arrangement of matter from that based on the periodic distribution of atoms through space to that corresponding to QP arrangements [4]. Accordingly, these new materials were dubbed quasiperiodic crystals, or quasicrystals (QCs) for short. ...

... To this end, I recommend a careful reading of the insightful thoughts discussed by R. Lifshitz some time ago [31]. In particular, it may be pertinent to take a closer look at the originally proposed QC notion: according to D. Levine and P. Steinhardt, "A quasicrystal is the natural extension of the notion of a crystal to structures with quasiperiodic, rather than periodic, translational order" [4]. Therefore, we may better define QCs as long-range-ordered solids whose diffraction pattern exhibits symmetries that are incompatible with translational symmetry, with the proviso that this long-range order stems from the underlying quasiperiodic distribution of atoms throughout the space. ...

Four decades have elapsed since the first quasiperiodic crystal was discovered in the Al–Mn alloy system, and much progress has been made during this time on the science of quasicrystals (QCs). Notwithstanding this, a significant number of open questions still remain regarding both fundamental and technological aspects. For instance: What are QCs good for? How can we improve the current provisional QC definition? What is the role of the underlying quasiperiodic order and the characteristic inflation symmetry of these compounds in the emergence of their unusual physicochemical properties? What is the nature of chemical bonding in QCs formed in different sorts of materials such as alloys, oxides, or organic polymers? Herein these and other closely related issues are discussed from an interdisciplinary perspective as well as prospective future work in the field in the years to come.

... Quasicrystals (QCs), generally defined as quasiperiodic solids characterized by unconventional crystallographic rules (Levine and Steinhardt 1984), find applications in industry due to their physical properties, light absorption, reduced adhesion and friction, thermal insulation, coating on mechanical devices, etc. (Dubois 2012(Dubois , 2023. The investigation of the physical and chemical properties of QCs appears also relevant in the field of planetary geology despite their rare occurrence in nature. ...

... A couple of years after the experimental finding of icosahedral Al 86 Mn 14 by Shechtman et al. (1984) and the theoretical prediction by Levine and Steinhardt (1984), the first HP investigation of i-QC up to 14 GPa by monitoring changes in resistivity of i-Al 78 Mn 22 and i-Al 86 Mn 14 was performed by Parthasarathy et al. (1986). The authors reported a QCto-crystalline transition (plus the occurrence of pure Al) at 5 and 9.2 GPa, respectively. ...

We summarize the results of studies on quasicrystals (QCs) at extreme conditions over the last 4 decades with particular emphasis for compositions falling in the Al-based ternary system as the closest to those of quasicrystals discovered in nature, such as icosahedrite and decagonite. We show that, in contrast with what thought in the past, both pressure and temperature act to stabilize QCs, for which a clear phase transition to either crystalline approximants or amorphous material has been limited to very few compositions only. Such stabilization is proved by the compressibility behavior of QCs that resembles that of the pure constituent metals. Additional remarks come from the experimental observation of QC formation at high pressure and temperature in both static and dynamic experiments. These results seem, in conclusion, to suggest that the occurrence of QCs in nature might be more a rule rather than an exception.

... In 1984 Levine and Steinhardt (1984) hypothesized the existence of a novel type of material, a middle ground between the crystalline and glassy states; a material with theoretically impossible characteristics, and which they initially dubbed impossible crystal and later quasicrystal, short for "quasiperiodic crystal". ...

... Some researchers think that quasicrystals, being synthesized under highly controlled conditions, are very delicate, metastable materials too complicated to be stable phases of matter (Henley 1991). Others believe that quasicrystals are robust and energetically stable phases like ordinary crystals (Levine and Steinhardt 1984). Who is right? ...

By their very nature, discoveries are often unexpected and thus unpredicted. To a considerable extent, discoveries in the geological realm are disconnected from those in the laboratory sciences. In unusual situations, spectacular advances in cognate sciences result in geological or mineralogical discoveries. Such is the case with fullerenes and quasicrystals, whose histories will be briefly explored in the following pages.KeywordsFullerenesQuasicrystalsDiscoveryIcosahedriteSolar system

... The same year of the Mackay's paper there was the Shechtman's discovery (Fig. 2b), but it took him two years to get it published because of the community disbelief. Almost in the same months of 1984 Dov Levine and Paul J. Steinhardt (Levine and Steinhardt 1984) hypothesized the existence of a novel type of material with theoretically impossible characteristics (Fig. 2c), and which they dubbed quasicrystals, short for "quasiperiodic crystals". Few years later, Tsai et al. (1987) discovered a new quasicrystal alloy (Al 63 Cu 24 Fe 13 ), which should perhaps be viewed as the first bona fide synthetic quasicrystal (Fig. 2d). ...

... The topical issue on quasicrystals of the Rendiconti Lincei stems from an interdisciplinary conference resulting from mineralogical-crystallographic research, held at the Accademia dei Lincei in Rome on November 18, 2022, Levine and Steinhardt (1984); d experimental electron diffraction pattern reported by Tsai et al. (1987) for a synthetic Al-Cu-Fe alloy which was organized in the frame of the 2022 International Year of Mineralogy with the aim at providing a discussion on the mathematics, geometry, structure, natural occurrence, applications and physical behavior of quasicrystals. The conference, with internationally renowned speakers, was attended by almost 80 participants, who contributed to making of this event an occasion to discuss the controversial issues and outlooks of quasicrystals while the ever-growing list of experiments continues to daily present new and challenging questions. ...

With this article, we briefly retrace the history of quasicrystals and introduce the Topical collection on “ Quasicrystals: State of the art and outlooks ”, consisting of a number of review articles published in the frame of a conference held at the Accademia dei Lincei in November 2022.

... This is, however, by far not the most general structural behaviour and many more complex but still geometrically appealing physical systems do not possess any global symmetry. One prominent example are quasicrystals [6][7][8] who generically do not possess any global symmetry but are governed by a plethora of local symmetries arranged in a quasiperiodic manner [9,10]. Here the notion of a local symmetry refers to the situation * Peter.Schmelcher@physnet.uni-hamburg.de ...

... Individual impurities add to this spectrum eigenstates which are exponentially localized [27,28]. The situation becomes richer in terms of symmetries in the case of quasicrystals with their longrange aperiodic order [6][7][8]. Indeed, the iterative action of a given substitution rule underlying aperiodic lattices lead to a plethora of e.g. local reflection symmetries or, more precisely, they lead to a quasiperiodic recurrence of reflection symmetries [9] which manifests itself in the corresponding return map. ...

Symmetries are known to dictate important physical properties and can be used as a design principle in particular in wave physics, including wave structures and the resulting propagation dynamics. Local symmetries, in the sense of a symmetry that holds only in a finite domain of space, can be either the result of a self-organization process or a structural ingredient into a synthetically prepared physical system. Applying local symmetry operations to extend a given finite chain we show that the resulting one-dimensional lattice consists of a transient followed by a subsequent periodic behaviour. Due to the fact that, by construction, the implanted local symmetries strongly overlap the resulting lattice possesses a dense skeleton of such symmetries. We proof this behaviour on the basis of a class of local symmetry operations allowing us to conclude upon the 'asymptotic' properties such as the final period, decomposition of the unit-cell and the length and decomposition of the transient. As an example case, we explore the corresponding tight-binding Hamiltonians. Their energy eigenvalue spectra and eigenstates are analyzed in some detail, showing in particular the strong variability of the localization properties of the eigenstates due to the presence of a plethora of local symmetries.

... The towering tall buildings, solid bridges, and almost all daily infrastructures around us are based on understanding the structures' mechanical properties. In recent years, with the development of metamaterials [1][2][3][4][5] with abnormal mechanical properties, including negative Poisson ratio [6][7][8][9], negative compressibility [10], negative thermal expansivity [11], statical non-reciprocity [12], vanishing shear modulus [13], ultra stiffness [14,15], as well as the indepth study on jammed particles [16,17], granular materials [18], quasi-crystals [19][20][21], the investigation of the multi-body system's elasticity and vibrational behavior has received immense attention from the academic community. In the analysis of the mechanical response of a multi-body system, we can usually assume the ''individual bodies'' as the mass points and regard the interactions between them as the harmonic potentials. ...

The multi-body system's elasticity and vibrational behavior is an old topic. The conventional method to study this issue is to transform a multi-body system to an abstracted network model consisting of nodes connected by central-force springs. In recent years, abnormal mechanical responses, including negative Poisson ratio, vanishing shear modulus, non-trivial topological characteristics, are discovered in such network systems making this topic be with lasting charms. In this paper, we investigate the vibrational modes of a series of distorted square lattice networks. The distorted degree for this network is only characterized by one parameter, the twisting angle. By studying their phonon spectrums, we find a band gap between the two lowest modes of the distorted square lattice. The gap width is linearly correlated to , and the gap will be closed when reaches a specific value. Our results may find applications in device protection and energy harvesting.

... This included icosahedral Al-Cu-Fe quasicrystals (Bindi et al., 2009(Bindi et al., , 2011 and decagonal Al-Ni-Fe quasicrystals (Bindi et al., 2015a(Bindi et al., , 2015b, representing the first natural occurrence of these quasiperiodic materials. This class of matter has properties intermediate between crystalline and amorphous materials, being characterized by atomic structures that are ordered but aperiodic and, therefore, lack translational symmetry while retaining higher order symmetries (Levine & Steinhardt, 1984;Shechtman et al., 1984). ...

A recently described micrometeorite from the Nubian desert (Sudan) contains an exotic Al‐Cu‐Fe assemblage closely resembling that observed in the Khatyrka chondrite (Suttle et al., 2019; Science Reports 9:12426). We here extend previous investigations of the geochemical, mineralogical, and petrographic characteristics of the Sudan spherule by measuring oxygen isotope ratios in the silicate components and by nano‐scale transmission electron microscopy study of a focused ion beam foil that samples the contact between Al‐Cu alloys and silicates. O‐isotope work indicates an affinity to either OC or CR chondrites, while ruling out a CO or CM precursor. When combined with petrographic evidence we conclude that a CR chondrite parentage is the most likely origin for this micrometeorite. SEM and TEM studies reveal that the Al‐Cu alloys mainly consist of Al metal, stolperite (CuAl), and khatyrkite (CuAl 2 ) together with inclusions in stolperite of a new nanometric, still unknown Al‐Cu phase with a likely nominal Cu 3 Al 2 stoichiometry. At the interface between the alloy assemblage and the surrounding silicate, there is a thin layer (200 nm) of almost pure MgAl 2 O 4 spinel along with well‐defined and almost perfectly spherical metallic droplets, predominantly iron in composition. The study yields additional evidence that Al‐Cu alloys, the likely precursors to quasicrystals in Khatyrka, occur naturally. Moreover, it implies the existence of multiple pathways leading to the association in reduced form of these two elements, one highly lithophile and the other strongly chalcophile.

... Its quasicrystalline system departs from conventional lattice structure and suggests new modes of space configuration while achieving perfect packing. Its unique structure has been explored in art and science due to novel material property and aesthetic qualities (Levine and Steinhardt 1984;Bursill and Ju Lin 1985;Aranda\Lasch 2018). ...

... The concept of bond-orientational order is crucial for understanding crystals, 1-7 quasi-crystals, 8 glasses, [9][10][11][12][13] and even morphogenesis in living systems. 14 Bond Order Parameters (BOPs) have been developed to quantify this order, initially in 2D crystallization, 15,16 and later extended to 3D structures. ...

Bond-orientational order in DNA-assembled nanoparticles lattices is explored with the help of recently introduced Symmetry-specific Bond Order Parameters (SymBOPs). This approach provides a more sensitive analysis of local order than traditional scalar BOPs, facilitating the identification of coherent domains at the single bond level. The present study expands the method initially developed for assemblies of anisotropic particles to the isotropic ones or cases where particle orientation information is unavailable. The SymBOP analysis was applied to experiments on DNA-frame-based assembly of nanoparticle lattices. It proved highly sensitive in identifying coherent crystalline domains with different orientations, as well as detecting topological defects, such as dislocations. Furthermore, the analysis distinguishes individual sublattices within a single crystalline domain, such as pair of interpenetrating FCC lattices within a cubic diamond. The results underscore the versatility and robustness of SymBOPs in characterizing ordering phenomena, making them valuable tools for investigating structural properties in various systems.

... Unlike the periodic structures that feature translational symmetries or 1-, 2-, 3-, 4-, and 6-fold rotational symmetries, this new structure has a 5-fold rotational symmetries. Later, researchers coined this new long-range ordered structure "quasicrystals" [23]. Since the discovery of the 5-fold quasicrystals, more different structures with 5-, 6-, 8-, 12-, and 20-fold symmetries have emerged in various metallic alloys [33,36]. ...

Due to quasicrystals having long-range orientational order but without translational symmetry, traditional numerical methods usually suffer when applied as is. In the past decade, the projection method has emerged as a prominent solver for quasiperiodic problems. Transforming them into a higher dimensional but periodic ones, the projection method facilitates the application of the fast Fourier transform. However, the computational complexity inevitably becomes high which significantly impedes e.g. the generation of the phase diagram since a high-fidelity simulation of a problem whose dimension is doubled must be performed for numerous times. To address the computational challenge of quasiperiodic problems based on the projection method, this paper proposes a multi-component multi-state reduced basis method (MCMS-RBM). Featuring multiple components with each providing reduction functionality for one branch of the problem induced by one part of the parameter domain, the MCMS-RBM does not resort to the parameter domain configurations (e.g. phase diagrams) a priori. It enriches each component in a greedy fashion via a phase-transition guided exploration of the multiple states inherent to the problem. Adopting the empirical interpolation method, the resulting online-efficient method vastly accelerates the generation of a delicate phase diagram to a matter of minutes for a parametrized two-turn-four dimensional Lifshitz-Petrich model with two length scales. Moreover, it furnishes surrogate and equally accurate field variables anywhere in the parameter domain.

... Quasicrystals are unique classes of materials that exhibit a regular atomic structure with non-repeating patterns, forming a contrast between the disordered arrangement of glasses and the fully periodic arrangement of crystals. They can be characterized by the presence of long-range order in terms of their translational and orientational symmetries, but the patterns do not repeat at regular intervals [20][21][22]. Due to this unique arrangement of the sites, they exhibit rare features such * akif.keskiner@bilkent.edu.tr † onur.erten@asu.edu ...

We develop an exactly solvable model with Kitaev-type interactions and study its phase diagram on the dual lattice of the quasicrystalline Ammann-Beenker lattice. Our construction is based on the $\Gamma$-matrix generalization of the Kitaev model and utilizes the cut-and-project correspondence between the four-dimensional simple cubic lattice and the Ammann-Beenker lattice to designate four types of bonds. We obtain a rich phase diagram with gapped (chiral and abelian) and gapless spin liquid phases via Monte Carlo simulations and variational analysis. We show that the ground state can be further tuned by the inclusion of an onsite term that selects 21 different vison configurations while maintaining the integrability of the model. Our results highlight the rich physics at the intersection of quasicrystals and quantum magnetism.

... It is a crystal in six-dimensions projected to three-dimensions with an irrational projection angle. 57 The DW state is a crystal in N-dimensions projected with nearly random phase factors because the Bragg sphere can be considered to be made of an infinite number of Bragg points covering the sphere. Below, we show that the DW state, described by the pseudo-density function ρpp(r), is close to the structurally coherent glass state of liquid except for details of the SRO. ...

The atomic pair-distribution function of simple liquid and glass shows exponentially decaying oscillations beyond the first peak, representing the medium-range order (MRO). The structural coherence length that characterizes the exponential decay increases with decreasing temperature and freezes at the glass transition. Conventionally, the structure of liquid and glass is elucidated by focusing on a center atom and its neighboring atom shell characterized by the short-range order (SRO) and describing the global structure in terms of overlapping local clusters of atoms as building units. However, this local bottom-up approach fails to explain the strong drive to form the MRO, which is different in nature from the SRO. We propose to add an alternative top-down approach based upon the density wave theory. In this approach, one starts with a high-density gas state and seeks to minimize the global potential energy in reciprocal space through density waves using the pseudopotential. The local bottom-up and global top-down driving forces are not mutually compatible, and the competition and compromise between them result in a final structure with the MRO. This even-handed approach provides a more intuitive explanation of the structure of simple liquid and glass.

... The discovery of quasicrystals (Shechtman et al., 1984) brought to light a new family of structures that exhibit long-range non-periodic order. The atoms in this new state of matter are not arranged in a periodic manner similar to traditional crystals, but exhibit a complicated long-range translational order that is not periodic (Levine & Steinhardt, 1984;Socolar et al., 1985;Levine & Steinhardt, 1986;Ishii & Fujiwara, 2008;Al Ajlouni, 2011). The discovery of these forbidden symmetries have inspired a substantial theoretical research concerning systems that are situated between perfectly periodic and chaotically random. ...

To cite this article: Rima Ajlouni (2023): Derived from the traditional principles of Islamic geometry, a methodology for generating non-periodic long-range sequences in one-dimension for 8-fold, 10-fold, and 12-fold rotational symmetries, Journal of Mathematics and the Arts,

... The sharp diffraction spots indicated the existence of a long-range translational order, and the ten-fold symmetric diffraction pattern was incompatible with a periodic order, thus verifying that the phase could not be crystalline. Soon after this discovery, a new classification scheme for solids was proposed, with the aforementioned phase denoted a quasicrystal (QC) [9][10][11] . Essentially, QCs have a quasiperiodic long-range translational order that is compatible with crystallographically disallowed rotational symmetries such as five-, ten-, and twelve-fold symmetries. ...

van der Waals (vdW) layered transition-metal chalcogenides are attracting significant attention owing to their fascinating physical properties. This group of materials consists of abundant members with various elements, having a variety of different structures. However, all vdW layered materials studied to date have been limited to crystalline materials, and the physical properties of vdW layered quasicrystals have not yet been reported. Here, we report on the discovery of superconductivity in a vdW layered quasicrystal of Ta1.6Te. The electrical resistivity, magnetic susceptibility, and specific heat of the Ta1.6Te quasicrystal fabricated by reaction sintering, unambiguously validated the occurrence of bulk superconductivity at a transition temperature of ~1 K. This discovery can pioneer new research on assessing the physical properties of vdW layered quasicrystals as well as two-dimensional quasicrystals; moreover, it paves the way toward new frontiers of superconductivity in thermodynamically stable quasicrystals, which has been the predominant challenge facing condensed matter physics since the discovery of quasicrystals almost four decades ago.

... While no material has yet been discovered with the symmetries of the Hat it would seem likely that nature would realize such an elegant construction, just as Penrose tilings were discovered to describe the surfaces of icosahedral quasicrystals [92][93][94][95]. A promising solid-state platform is the engineered adsorption of atoms to constrain scattering of surface states. ...

The discovery of the Hat, an aperiodic monotile, has revealed novel mathematical aspects of aperiodic tilings. However, the physics of particles propagating in such a setting remains unexplored. In this work we study spectral and transport properties of a tight-binding model defined on the Hat. We find that (i) the spectral function displays striking similarities to that of graphene, including six-fold symmetry and Dirac-like features; (ii) unlike graphene, the monotile spectral function is chiral, differing for its two enantiomers; (iii) the spectrum has a macroscopic number of degenerate states at zero energy; (iv) when the magnetic flux per plaquette ($\phi$) is half of the flux quantum, zero-modes are found localized around the reflected `anti-hats'; and (v) its Hofstadter spectrum is periodic in $\phi$, unlike other quasicrystals. Our work serves as a basis to study wave and electron propagation in possible experimental realizations of the Hat, which we suggest.

... The discovery of quasicrystals in the early 1980s [1] marked a paradigm shift in crystallography and solidstate physics, prompting investigations into their structural and electronic properties [2][3][4][5][6][7]. Quasicrystals are usually synthetized in the laboratory after fast solidification of certain alloys [1,8], but have also been observed at their natural state in meterorites [9,10] and residues of nuclear blasts [11]. ...

Quasicrystals, a fascinating class of materials with long-range but nonperiodic order, have revolutionized our understanding of solid-state physics due to their unique properties at the crossroads of long-range-ordered and disordered systems. Since their discovery, they continue to spark broad interest for their structural and electronic properties. The quantum simulation of quasicrystals in synthetic quantum matter systems offers a unique playground to investigate these systems with unprecedented control parameters. Here, we investigate the localization properties and spectral structure of quantum particles in 2D quasicrystalline optical potentials. While states are generally localized at low energy and extended at high energy, we find alternating localized and critical states at intermediate energies. Moreover, we identify a complex succession of gaps in the energy spectrum. We show that the most prominent gap arises from strongly localized ring states, with the gap width determined by the energy splitting between states with different quantized winding numbers. In addition, we find that these gaps are stable for quasicrystals with different rotational symmetries and potential depths, provided that other localized states do not enter the gap generated by the ring states. Our findings shed light on the unique properties of quantum quasicrystals and have implications for their many-body counterparts.

... Figure 8b was a selected area diffraction pattern (SADP) that was obtained from the particle marked "X" in Figure 8a. This SADP displays the characteristic five-fold symmetry expected for the [000001] zone axis of the i-phase [19,[37][38][39][40][41]. Thus, the bright particles observed in BSE SEM images, such as Figure 4b, were mostly retained in the i-phase. ...

Quasicrystalline Al93Fe3Cr2Ti2 (at.%) gas-atomized powders, which exhibit a metastable composite microstructure, were used to produce coatings by cold spray additive manufacturing processing (CSAM) using different processing parameters. The metastable composite microstructure provides the Al93Fe3Cr2Ti2 alloy with excellent mechanical properties. At the same time, the metastability of its microstructure, achieved by the high cooling rates of the gas atomization process, limits the processability of the Al93Fe3Cr2Ti2 powder. The purpose of this study was to investigate the effect of process parameters on the CSAM of quasicrystalline Al93Fe3Cr2Ti2 powder. The powder was sieved and classified to a size range of −75 µm. Using N2 carrier gas combined with different temperatures, pressures, nozzle apertures, and deposition substrate conditions, cold-sprayed coatings were produced. The porosity and thickness of the coatings were evaluated by image analyses. By SEM, XRD, DSC, and TEM, the microstructure was identified, and by Vickers microhardness, the mechanical properties of the coatings were investigated. Dense (≤0.50% porosity) and thick (~185.0 µm) coatings were obtained when the highest pressure (4.8 MPa), highest temperature (475 °C), and lowest nozzle aperture (A) were used in combination with an unblasted substrate. The SEM, XRD, and DSC data showed that the composite powder’s microstructure was retained in all coatings with no decomposition of the metastable i-phase into equilibrium crystalline phases. Supporting these microstructural results, all coatings presented a high and similar hardness of about 267 ± 8 HV. This study suggests that the CSAM process could, therefore. produce metastable quasicrystalline Al93Fe3Cr2Ti2 coatings with a composite microstructure and high hardness.

... This discovery was initially met with resistance: the existence of structures in which atoms can be arranged in spatial structures which lack long-range periodicity, while still preserving sufficient long-range order to generate discrete Bragg peaks clashed with the elegant picture of crystals as consisting of a repeating unit cell. Nonetheless, eventually materials with this property, which were called quasi-crystals (QCs) [2], changed the way in which scientists interpret the crystal state, by disentangling the concept of order from the concept of periodicity, to the point where the very definition of crystals had to be changed to include aperiodic structures [3]. ...

Quasi-crystals are aperiodic structures that present crystallographic properties which are not compatible with that of a single unit cell. Their revolutionary discovery in a metallic alloy, less than three decades ago, has required a full reconsideration of what we defined as a crystal structure. Surprisingly, quasi-crystalline structures have been discovered also at much larger length scales in different microscopic systems for which the size of elementary building blocks ranges between the nanometric to the micrometric scale. Here, we report the first experimental observation of spontaneous quasi-crystal self-assembly at the millimetric scale. This result is obtained in a fully athermal system of macroscopic spherical grains vibrated on a substrate. Starting from a liquid-like disordered phase, the grains begin to locally arrange into three types of squared and triangular tiles that eventually align, forming 8-fold symmetric quasi-crystal that has been predicted in simulation but not yet observed experimentally in non-atomic systems. These results are not only the proof of a novel route to spontaneously assemble quasi-crystals but are of fundamental interest for the connection between equilibrium and non-equilibrium statistical physics.

... Extended author information available on the last page of the article et al. 1984). Just a few weeks afterwards, a rationale for the aperiodic structure of this material was published by Levine and Steinhardt (1984) who by the way coined the name "quasicrystals" (QC) to designate this specific and unprecedented type of order in solids. Quasicrystals comprise nowadays quite different types of materials since they were found in metallic alloys, polymers and oxides (Schirber 2007;Cartwright 2013). ...

The discovery of quasicrystals by Shechtman et al . in 1982–84 has revolutionised our understanding of crystals and order in solids. Shechtman was awarded a Nobel Prize in Chemistry in 2011 to recognize the importance of this breakthrough. Soon after the initial publication, a patent was filed by the author to secure the potential application of these new materials to the fabrication of low-stick surfaces adapted to the industrial production of cooking utensils. Quite a few more patents followed, covering several areas of technological relevance such as low friction, thermal insulation, solar light absorption, etc. The first application failed, although it reached market. Few others never developed to this stage, but also a (very) small number can now be considered as commercially successful. This is especially the case of polymers reinforced with a quasicrystal powder that are especially adapted to additive manufacturing or 3D printing. Also very advanced is the use of a blend of quasicrystalline and complex intermetallic powders to mark and authenticate an object in a way that cannot be counterfeit. The present article reviews the state of the art and outlines the physics behind few technological breakthroughs that are based on quasicrystalline alloys in the areas of mechanical engineering and solid–solid or solid–liquid adhesion. For the sake of brevity, applications in the areas of catalysis, solar and thermo-electric devices are only shortly evoked.
Graphical abstract

... One such concept is the moire crystals in twisted multilayer materials [1], leading to crystals with extremely large unit cells. A separate class of order is quasicrystals [2,3]. Another generalization that attracted interest for a long time is the class of order where a classical field demonstrates crystallization coexisting with the spontaneous breaking of additional symmetries. ...

We propose a generalization of the crystalline order: the ground-state fractal crystal. We demonstrate that by deriving a simple continuous-space-discrete-field model whose ground state is a crystal where each unit cell is a fractal.

... In early 1984, the discovery of quasiperiodic crystals (also called quasicrystals) in aluminum-magnesium alloys, achieved by Shechtman and his co-workers, led to the formation of a new type of solid different from the other well-known amorphous and periodic crystals based on basic atomic or molecular structures [1] . The two-dimensional (2D) diffraction pattern of this quasicrystal was found to present a long-range order feature (but with no periodicity), fully countering the law of the existing ordered crystal, and was demonstrated to match Penrose and Mackay's mathematical model in which a mosaic could be laid with a few rhombic tiles [2][3][4] . Figures 1(a) and 1(b) show the common five-fold Penrose lattice tiling and the corresponding numeric diffraction pattern [5] . ...

In the fields of light manipulation and localization, quasiperiodic photonic crystals, or photonic quasicrystals (PQs), are causing an upsurge in research because of their rotational symmetry and long-range orientation of transverse lattice arrays, as they lack translational symmetry. It allows for the optimization of well-established light propagation properties and has introduced new guiding features. Therefore, as a class, quasiperiodic photonic crystal fibers, or photonic quasi-crystal fibers (PQFs), are considered to add flexibility and richness to the optical properties of fibers and are expected to offer significant potential applications to optical fiber fields. In this review, the fundamental concept, working mechanisms, and invention history of PQFs are explained. Recent progress in optical property improvement and its novel applications in fields such as dispersion control, polarization-maintenance, supercontinuum generation, orbital angular momentum transmission , plasmon-based sensors and filters, and high nonlinearity and topological mode transmission, are then reviewed in detail. Bandgap-type air-guiding PQFs supporting low attenuation propagation and regulation of photonic density states of quasiperiodic cladding and in which light guidance is achieved by coherent Bragg scattering are also summarized. Finally, current challenges encountered in the guiding mechanisms and practical preparation techniques, as well as the prospects and research trends of PQFs, are also presented.

... The forcing of nonperiodic structure, the substitution symmetry, and the emergence of the golden ratio are all reminiscent of the well-known Penrose tiles [2], which have served as a paradigmatic example of quasicrystalline structure, providing significant insights into the properties of physical quasicrystals. [3][4][5] Smith et al. have shown that the hat tile shape (or matching rules for the metatiles) forces some form of nonperiodic structure. The purpose of the present paper is to provide a characterization of that structure, showing that these tilings are quasicrystalline but possess some novel symmetry properties. ...

We show that the tilings of the plane with the Smith hat aperiodic monotile (and its mirror image) are quasicrystals with hexagonal (C6) rotational symmetry. Although this symmetry is compatible with periodicity, the tilings are quasiperiodic with an incommensurate ratio characterizing the quasiperiodicity that stays locked to the golden mean as the tile parameters are continuously varied. Smith et al. [arXiv:2303.10798 (2023)] have shown that the hat tiling can be constructed as a decoration of a substitution tiling employing a set of four "metatiles." We analyze a modification of the metatiles that yields a set of "Key tiles," constructing a continuous family of Key tiles that contains the family corresponding to the Smith metatiles. The Key tilings can be constructed as projections of a subset of 6-dimensional hypercubic lattice points onto the two-dimensional tiling plane, and when projected onto a certain 4-dimensional subspace this subset uniformly fills four equilateral triangles. We use this feature to analytically compute the diffraction pattern of a set of unit masses placed at the tiling vertices, thereby establishing the quasiperiodic nature of the tiling. We comment on several unusual features of the family of Key tilings and hat decorations and show the tile rearrangements associated with two tilings that differ by an infinitesimal phason shift.

... In this context, there has been significant interest in exploring the existence of topological states in quasicrystals (QCs). While they lack translational periodicity, QCs have long-range order [25,26] and form a particularly interesting class of aperiodic lattices since they may exhibit spatial symmetries which are forbidden in periodic crystals, such as 5, 7, 8 and 10-fold rotational symmetries in the plane, and e.g. icosahedral symmetries in 3D [27]. ...

We investigate the existence of interface states induced by broken inversion symmetries in two-dimensional quasicrystal lattices. We introduce a 10-fold rotationally symmetric quasicrystal lattice whose inversion symmetry is broken through a mass dimerization that produces two 5-fold symmetric sub-lattices. By considering resonator scatterers attached to an elastic plate, we illustrate the emergence of bands of interface states that accompany a band inversion of the quasicrystal spectrum as a function of the dimerization parameter. These bands are filled by modes which are localized along domain-wall interfaces separating regions of opposite inversion symmetry. These features draw parallels to the dynamic behavior of topological interface states in the context of the valley Hall effect, which has been so far limited to periodic lattices. We numerically and experimentally demonstrate wave-guiding in a quasicrystal lattice featuring a zig-zag interface with sharp turns of 36 degrees, which goes beyond the limitation of 60 degrees associated with 6-fold symmetric (i.e., honeycomb) periodic lattices. Our results provide new opportunities for symmetry-based quasicrystalline topological waveguides that do not require time-reversal symmetry breaking, and that allow for higher freedom in the design of their waveguiding trajectories by leveraging higher-order rotational symmetries.

... In the 'cut and project' method the tiles are typically represented by different shapes or lines. This geometric representation of tilings is useful in modelling physical quasi-crystalline materials as certain sequences are highly ordered and aperiodic, which is similar to the structures of these quasi-crystals [2] [12][16] [13]. Thus the point-sets provided from these tilings can provide interesting diffraction and symmetry properties on quasi-crystals [13]. ...

In this paper, two generating function representations of the Fibonacci Substitution Tiling are derived and proven to converge on the interval -1<x<1. A sequence of signs for the Fibonacci Substitution is established along with a conjecture that the interval of convergence has an infinite number of zeroes.

... The discovery of quasicrystals in 1982 by Dan Schechtman marked a paradigm shift within the field of crystallography since, prior to his discovery, order and repeatability was associated exclusively to periodicity [19,20]. It took Dan Shechtman two years to convince his colleagues about the existence of quasicrystals, which was eventually recognized by the Nobel Prize in Chemistry in 2011 [21]. ...

This paper compares the mechanical properties of a class of lattice metamaterials with aesthetically-pleasing patterns that are governed by the mathematics of aperiodic order. They are built up of ordered planar rod networks and exhibit higher non-crystallographic rotational symmetries. However, they lack the translational symmetry associated with periodic lattice metamaterials. We present schematics illustrating their development based on pattern-unique mathematical substitution rules and exploit a numerical framework from previous work to demonstrate that they exhibit fascinating near-isotropic properties. The lattice structures are compared to the well-known hexagonal lattice with respect to their elastic anisotropy measure and the proximity of their bulk and shear moduli to Hashin–Shtrikman-Walpole limits. The study lends insight into the cost-constrained benefits of introducing additional connectivities between aperiodically-ordered point sets. The results show that aperiodic lattices have the potential to yield superior mechanical properties to periodic ones subject to the mechanical rigidity of the underlying shapes that constitute the pattern. The inherent ‘near-isotropy’ associated with these aperiodic structures, even with uniform strut thicknesses and at low fractional densities, and the ordered and varied orientation of their lattice struts present them as promising mesoscale architecture for solving complex multi-axially loaded structural optimization problems, providing inspiration for this study.

We theoretically study the real-space distribution of the supercurrent that flows under a uniform vector potential in a two-dimensional quasiperiodic structure. This is done by considering the attractive Hubbard model on the quasiperiodic Ammann-Beenker structure and studying the superconducting phase within the Bogoliubov-de Gennes mean-field theory. Decomposing the local supercurrent into the paramagnetic and diamagnetic components, we numerically investigate their dependencies on average electron density, temperature, and the angle of the applied vector potential. We find that the diamagnetic current locally violates the current conservation law, necessitating compensation from the paramagnetic current, even at zero temperature. The paramagnetic current shows exotic behaviors in the quasiperiodic structure, such as local currents, which are oriented transversally or reversely to that of the applied vector potential.

Three-dimensional higher-order topological semimetals in crystalline systems exhibit higher-order Fermi arcs on one-dimensional hinges, challenging the conventional bulk-boundary correspondence. However, the existence of higher-order Fermi arc states in aperiodic quasicrystalline systems remains uncertain. In this paper, we present the emergence of three-dimensional quasicrystalline second-order topological semimetal phases by vertically stacking two-dimensional quasicrystalline second-order topological insulators. These quasicrystalline topological semimetal phases are protected by rotational symmetries forbidden in crystals, and are characterized by topological hinge Fermi arcs connecting fourfold degenerate Dirac-like points in the spectrum. Our findings reveal an intriguing class of higher-order topological phases in quasicrystalline systems, shedding light on their unique properties.

Since the discovery of the quasicrystal, approximately 100 stable quasicrystals are identified. To date, the existence of quasicrystals is verified using transmission electron microscopy; however, this technique requires significantly more elaboration than rapid and automatic powder X‐ray diffraction. Therefore, to facilitate the search for novel quasicrystals, developing a rapid technique for phase‐identification from powder diffraction patterns is desirable. This paper reports the identification of a new Al–Si–Ru quasicrystal using deep learning technologies from multiphase powder patterns, from which it is difficult to discriminate the presence of quasicrystalline phases even for well‐trained human experts. Deep neural networks trained with artificially generated multiphase powder patterns determine the presence of quasicrystals with an accuracy >92% from actual powder patterns. Specifically, 440 powder patterns are screened using the trained classifier, from which the Al–Si–Ru quasicrystal is identified. This study demonstrates an excellent potential of deep learning to identify an unknown phase of a targeted structure from powder patterns even when existing in a multiphase sample.

We study the low-temperature phases of interacting bosons on a two-dimensional quasicrystalline lattice. By means of numerically exact path integral Monte Carlo simulations, we show that for sufficiently weak interactions the system is a homogeneous Bose-Einstein condensate that develops density modulations for increasing filling factor. The simultaneous occurrence of sizeable condensate fraction and density modulation can be interpreted as the analogous, in a quasicrystalline lattice, of supersolid phases occurring in conventional periodic lattices. For sufficiently large interaction strength and particle density, global condensation is lost and quantum exchanges are restricted to specific spatial regions. The emerging quantum phase is therefore a Bose glass, which here is stabilized in the absence of any source of disorder or quasidisorder, purely as a result of the interplay between quantum effects, particle interactions and quasicrystalline substrate. This finding clearly indicates that (quasi)disorder is not essential to observe Bose glass physics. Our results are of interest for ongoing experiments on (quasi)disorder-free quasicrystalline lattices.

Symmetries are known to dictate important physical properties and can be used as a design principle in particular in wave physics, including wave structures and the resulting propagation dynamics. Local symmetries, in the sense of a symmetry that holds only in a finite domain of space, can be either the result of a self-organization process or a structural ingredient into a synthetically prepared physical system. Applying local symmetry operations to extend a given finite chain we show that the resulting one-dimensional lattice consists of a transient followed by a subsequent periodic behavior. Due to the fact that, by construction, the implanted local symmetries strongly overlap the resulting lattice possesses a dense skeleton of such symmetries. We proof this behavior on the basis of a class of local symmetry operations allowing us to conclude upon the “asymptotic” properties such as the final period, decomposition of the unit cell and the length and appearance of the transient. As an example case, we explore the corresponding tight-binding Hamiltonians. Their energy eigenvalue spectra and eigenstates are analyzed in some detail, showing in particular the strong variability of the localization properties of the eigenstates due to the presence of a plethora of local symmetries.

Predicting quasicrystal structures is a multifaceted problem that can involve predicting a previously unknown phase, predicting the structure of an experimentally observed phase, or predicting the thermodynamic stability of a given structure. We survey the history and current state of these prediction efforts with a focus on methods that have improved our understanding of the structure and stability of known metallic quasicrystal phases. Advances in the structural modeling of quasicrystals, along with first principles total energy calculation and statistical mechanical methods that enable the calculation of quasicrystal thermodynamic stability, are illustrated by means of cited examples of recent work.

We develop an exactly solvable model with Kitaev-type interactions and study its phase diagram on the dual lattice of the quasicrystalline Ammann-Beenker lattice. Our construction is based on the Γ-matrix generalization of the Kitaev model and utilizes the cut-and-project correspondence between the four-dimensional simple cubic lattice and the Ammann-Beenker lattice to designate four types of bonds. We obtain a rich phase diagram with gapped (chiral and abelian) and gapless spin liquid phases via Monte Carlo simulations and variational analysis. We show that the ground state can be further tuned by the inclusion of an onsite term that selects 21 different vison configurations while maintaining the integrability of the model. Our results highlight the rich physics at the intersection of quasicrystals and quantum magnetism.

The incorporation of the quasicrystalline phase into the metal matrix offers a wide range of potential applications in particle-reinforced metal-matrix composites. The analytic solution of the piezoelectric quasicrystal (QC) microsphere considering the thermoelectric effect and surface effect contained in the elastic matrix is presented in this study. The governing equations for the QC microsphere in the matrix subject to the external electric loading are derived based on the nonlocal elastic theory, electro-elastic interface theory, and eigenvalue method. A comparison between the existing results and the finite-element simulation validates the present approach. Numerical examples reveal the effects of temperature variation, nonlocal parameters, surface properties, elastic coefficients, and phason coefficients on the phonon, phason, and electric fields. The results indicate that the QC microsphere enhances the mechanical properties of the matrix. The results are useful for the design and understanding of the characterization of QCs in micro-structures.

The structural, electronic, and magnetic properties of novel functional magnetic intermetallics, in particular some of the most prominent magnetic Heusler alloys, have been reviewed with emphasis on their complex magnetic behavior and nonergodic glassy features, in contrast to the well‐known properties of Ti ‐ Ni and Ni ‐ Al shape memory alloys. Calculational procedures have been listed throughout the text and the theoretical results have been compared with experimental findings where possible. The thermodynamic properties of a few systems ( Ni – Co – Mn – In ), which undergo a structural phase transformation of displacive nature in conjunction with a metamagnetic first‐order phase transition, have been discussed in detail. The importance of such magnetostructural phase transition together with the so‐called thermodynamic (kinetic) arrest phenomenon for the appearance of a giant magnetocaloric effect has been underlined. The list of references should allow bridging the gap between shape memory alloys such as Ti – Ni (including recent ab initio calculations that explain the rapid decrease of the martensitic transformation temperature with disorder) and the “exotic properties” of the magnetic intermetallics.

We study light propagation of electromagnetic waves in a medium with a homogeneous refractive index varying quasi-periodically in time. We study two types of time-varying quasi-crystals: Andrey Aubrey and Fibonacci sequence.

Three-dimensional higher-order topological semimetals in crystalline systems exhibit higher-order Fermi arcs on one-dimensional hinges, challenging the conventional bulk-boundary correspondence. However, the existence of higher-order Fermi arc states in aperiodic quasicrystalline systems remains uncertain. In this work, we present the emergence of three-dimensional quasicrystalline second-order topological semimetal phases by vertically stacking two-dimensional quasicrystalline second-order topological insulators. These quasicrystalline topological semimetal phases are protected by rotational symmetries forbidden in crystals, and are characterized by topological hinge Fermi arcs connecting fourfold degenerate Dirac-like points in the spectrum. Our findings reveal an intriguing class of higher-order topological phases in quasicrystalline systems, shedding light on their unique properties.

Quantum simulation of quasicrystals in synthetic bosonic matter now paves the way for the exploration of these intriguing systems in wide parameter ranges. Yet thermal fluctuations in such systems compete with quantum coherence and significantly affect the zero-temperature quantum phases. Here we determine the thermodynamic phase diagram of interacting bosons in a two-dimensional, homogeneous quasicrystal potential. We find our results using quantum Monte Carlo simulations. Finite-size effects are carefully taken into account and the quantum phases are systematically distinguished from thermal phases. In particular, we demonstrate stabilization of a genuine Bose glass phase against the normal fluid in sizable parameter ranges. We interpret our results for strong interactions using a fermionization picture and discuss experimental relevance.

In general, the photonic crystal (PC) is a periodical optical structure, but there are some studies considering aperiodic structures. If we insert a defect layer into a one-dimensional periodic PC to break its translational symmetry order (TSO), some peaks, called defect modes, appear in the transmittance spectrum. The defect layer thickness governs the frequencies of these defect modes but almost does not affect the other part of the spectrum. The discovery of quasi-crystals tells us that not only the TSO but also other orders can produce Bragg diffraction. It is well known that triadic Cantor set (TCS) PCs, which lack TSO but have a self-similar symmetry order (SSO), still exhibit narrow transmission peaks. In this work, we try to break the SSO in TCS PCs and find the resulting optical phenomena, where single-negative materials and dielectrics are chosen as the constituents of PCs. The study method is the transfer matrix method, and the calculation results show that the background intensity of the transmittance spectrum rather than the frequency of peaks obviously periodically changes with the break of SSO. It follows that the SSO does have physical meaning, and not only the transmission peaks but also the background should be treated as a significant optical property.

A metallic solid (Al-14-at. pct.-Mn) with long-range orientational order, but with icosahedral point group symmetry, which is inconsistent with lattice translations, has been observed. Its diffraction spots are as sharp as those of crystals but cannot be indexed to any Bravais lattice. The solid is metastable and forms from the melt by a first-order transition.

Bond-orientational order in molecular-dynamics simulations of supercooled liquids and in models of metallic glasses is studied. Quadratic and third-order invariants formed from bond spherical harmonics allow quantitative measures of cluster symmetries in these systems. A state with short-range translational order, but extended correlations in the orientations of particle clusters, starts to develop about 10% below the equilibrium melting temperature in a supercooled Lennard-Jones liquid. The order is predominantly icosahedral, although there is also a cubic component which we attribute to the periodic boundary conditions. Results are obtained for liquids cooled in an icosahedral pair potential as well. Only a modest amount of orientational order appears in a relaxed Finney dense-random-packing model. In contrast, we find essentially perfect icosahedral bond correlations in alternative "amorphon" cluster models of glass structure.

Complex alloy structures, particularly those of transition metals, are ; considered as determined by the geometricnl requirements for sphere packing. A ; characteristic of the class of structures discussed is that tetrahedral groupings ; of atoms occur everywhere in the structure--alternatively stated, coordination ; polyhedra have only triangular faces. The topological and geometrical properties ; of such polyhedra are examined and rules and theorems regarding them are deduced. ; Justification is given for the prominence of four such polyhedra (for ; coordination numbers of 12, 14, 15, and 16) in actual structures. General ; principles regarding the combination of these polyhedra into full structures are ; deduced and necessary definitions are given for terms that facilitate the ; detailed discussion of this class of structures. (auth);

Supercooled liquids and metallic glasses can be viewed as defected states of bond orientational order. Surfaces of constant negative curvature contain an irreducible density of point disclinations in a hexatic order parameter. Analogous defect lines in an icosahedral order parameter appear in three-dimensional flat space. The Frank-Kasper phases are ordered networks of these lines, which, when disordered, provide an appealing model for structure in metallic glasses.

Three-dimensional bond orientational order is studied via computer simulations of 864 particles interacting through a Lennard-Jones pair potential. Long-range orientational fluctuations appear upon supercooling about ten percent below the equilibrium melting temperature. The fluctuations suggest a broken icosahedral symmetry with extended correlations in the orientations of local icosahedral packing units.

Recently developed scaling concepts in the theory of quasiperiodic dynamical systems are used to develop an exact renormalization group applicable to the discrete, quasiperiodic Schrödinger equation. To illustrate the power of the method, we calculate the universal scaling properties of the states and eigenvalue spectrum at and below the localization transition for an energy which corresponds to an integrated density of states of 1/2. The modulating potential has a frequency 1/2(√5-1) relative to the underlying lattice for the example we work out in greatest detail.

A defect description of liquids and metallic glasses is developed. In two dimensions, surfaces of constant negative curvature contain an irreducible density of point disclinations in a hexatic order parameter. Analogous defect lines in an icosahedral order parameter appear in three-dimensional flat space. Frustration in tetrahedral particle packings forces disclination lines into the medium in a way reminiscent of Abrikosov flux lines in a type-II superconductor and of uniformly frustrated spin-glasses. The defect density is determined by an isotropic curvature mismatch, and the resulting singular lines run in all directions. The Frank-Kasper phases of transition-metal alloys are ordered networks of these lines, which, when disordered, provide an appealing model for structure in metallic glasses.

The Penrose pattern is a tiling of two-dimensional and of three-dimensional space by identical tiles of two kinds (acute and obtuse rhombi with α = 72° and 144° in two dimensions and acute and obtuse rhombohedra with α = 63.43° and 116.57° in three dimensions). The two-dimensional pattern is a section through that in three dimensions. When joining (or recursion) rules are prescribed, the pattern is unique and non-periodic. It has local five-fold axes and thus represents a structure outside the formalism of classical crystallography and might be designated a quasi-lattice.

A theory of dislocation-mediated melting in two dimensions is described
in detail, with an emphasis on results for triangular lattices on
both smooth and periodic substrates. The transition from solid to
liquid on a smooth substrate takes place in two steps with increasing
temperatures. Dissociation of dislocation pairs first drives a transition
out of a low-temperature solid phase, with algebraic decay of translational
order and long-range orientational order. This transition is into
a "'liquid-crystal"' phase characterized by exponential decay of
translational order, but power-law decay of sixfold orientational
order. Dissociation of disclination pairs at a higher temperature
then produces an isotropic fluid. The behavior of the specific heat,
structure factor, and various elastic constants near these transitions
is worked out. We also discuss the applicability of our results to
melting on a periodic substrate. Dislocation unbinding should describe
melting of a "'floating"' (and, in general, incommensurate) adsorbate
solid into a high-temperature fluid phase. The orientation bias imposed
by the substrate can alter or eliminate the disclination-unbinding
transition, however. Transitions from a floating solid into a low-temperature
registered or partially registered phase can also be mapped onto
the dislocation-unbinding transition, but only at certain special
values of the coverage. Substrate reciprocallattice vectors play
the role of Burger's vectors in this case.