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Quasicrystals: A New Class of Ordered Structures
Abstract
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.
... Quasicrystals are non-periodic but long-range ordered systems found in a wide variety of physical systems including metallic alloys [1][2][3][4], photonic quasicrystals [5][6][7][8], ultra cold-atom systems [9][10][11] and twisted twodimensional (2D) materials. [12][13][14][15][16] Despite the increasing importance of quasicrystalline systems, the theoretical description of their physical properties is limited by the lack of the Bloch theorem. ...
... where (3) is the third Chern number calculated from arXiv:2201.04820v1 [cond-mat.mes-hall] 13 Jan 2022 the occupied states. ...
... The coefficient (3) in Eq. (10) is expressed as the third Chern number of the non-abelian Berry connection in the 6D Brillouin zone (BZ). Specifically, it is written as ...
We study the topological gap labeling of general 3D quasicrystals and we find that every gap in the spectrum is characterized by a set of the third Chern numbers. We show that a quasi-periodic structure has multiple Brillouin zones defined by redundant wavevectors, and the number of states below a gap is quantized as an integer linear combination of volumes of these Brillouin zones. The associated quantum numbers to characterize energy gaps can be expressed as third Chern numbers by considering a formal relationship between an adiabatic charge pumping under cyclic deformation of the quasi-periodic potential and a topological nonlinear electromagnetic response in 6D insulators.
... Shortly after Shechtman and his colleagues communicated the experimental discovery referred to above [1], Dov Levine and Paul J. Steinhardt (then at the University of Pennsylvania; now, at Princeton) suggested a theoretical model interpretation of Shechtman's discovery [16]. This was an important step ahead in understanding these heretofore unknown or, rather, unrecognized, structures, the more so as Shechtman's first attempts to understand them were not very successful [17]. ...
... Pattern 1725 is distinguished by "10 Fold ???". Reproduced by courtesy of and with permission from Dan Shechtman [16] and his response [18]. The magazine, The Chemical Intelligencer, in which this set of conversations appeared, stopped publication at the end of 2000. ...
... Alas, this was not the only time one could form the impression that Shechtman's and Mackay's contributions were downplayed. My other example goes back to January 1985, that is, only a few weeks after the appearance of Shechtman et al.'s paper [1] and the Levine and Steinhardt's paper [16]. The Science Section of the January 8, 1985, issue of The New York Times carried an article (p C2), titled "Theory of New Matter proposed." ...
On April 8, 1982, Dan Shechtman conducted an electron diffraction experiment on an aluminum/manganese alloy. The diffraction pattern showed tenfold symmetry although the rules of crystallography excluded such symmetry in extended structures. Alan L. Mackay had anticipated such structures, which fit his view of generalized crystallography. Shechtman persisted in claiming to have observed quasiperiodic structures despite denial of such structures even by Linus Pauling, the greatest authority in chemistry. When Shechtman’s claim was finally accepted, he was amply awarded for his contribution, including his Nobel Prize in 2011. The theoretical physicists Paul J. Steinhardt and Dove Levine coined the name quasicrystals, and advanced the field greatly by their models, but appeared to downplay somewhat the significance of prior predictions and of the experimental discovery.
... Thus solid state physics and Anderson localization was brought to optics [1][2][3][4] , leading to the development of photonic crystals and theories of controlling the flow of light through structured media. The discovery of quasicrystals [5][6][7] demonstrated that geometries with predictable long range order but no periodicity could play an important role in physics and materials science. This led to the development of photonic quasicrystals [8][9][10][11][12][13][14][15][16][17] , with the conceptual framework again provided by the analogy with quantum transport in solid-state physics. ...
... As the frequency changes and s(ω) sweeps across the complex plane, with s(0) = 0, the spectral measure μ, distribution of its eigenvalues, and localization properties of its eigenvectors, shown in Fig. 3, govern the frequency dependence of the phase and magnitude of ϵ * and the intensity and localization of E and D, shown in Fig. 4, according to the formulas in Eq. (6). Keeping these formulas with Im sðωÞ ( 1 in mind, we call resonant frequencies the values of ω where Re sðωÞ % λ j and the masses m j of μ are largest (shown in red in Fig. 3) and/or there's a large density of eigenvalues λ j with moderate to large values of m j . ...
From quasicrystalline alloys to twisted bilayer graphene, the study of material properties arising from quasiperiodic structure has driven advances in theory and applied science. Here we introduce a class of two-phase composites, structured by deterministic Moiré patterns, and we find that these composites display exotic behavior in their bulk electrical, magnetic, diffusive, thermal, and optical properties. With a slight change in the twist angle, the microstructure goes from periodic to quasiperiodic, and the transport properties switch from those of ordered to randomly disordered materials. This transition is apparent when we distill the relationship between classical transport coefficients and microgeometry into the spectral properties of an operator analogous to the Hamiltonian in quantum physics. We observe this order to disorder transition in terms of band gaps, field localization, and mobility edges analogous to Anderson transitions — even though there are no wave scattering or interference effects at play here.
... Two mentioned observations suggested that atoms in the material are structured in a non-periodic manner. The paper published in December 1984 coauthored by Levine and Steinhardt [LS84] named the phenomenon as a quasicrystallinity and the novel substance as a quasiperiodic crystal or quasicrystal. ...
... To represent the peculiar atomic structure of quasicrystals Levine and Steinhardt in [LS84] proposed the Penrose tilings [Pen74]. A tiling is a covering of a Euclidian plane by given geometric shapes without gaps and overlaps. ...
Self-assembly is the process in which the components of a system, whether molecules, polymers, or macroscopic particles, are organized into ordered structures as a result of local interactions between the components themselves, without exterior guidance. In this paper, we speak about the self-assembly of aperiodic tilings. Aperiodic tilings serve as a mathematical model for quasicrystals - crystals that do not have any translational symmetry. Because of the specific atomic arrangement of these crystals, the question of how they grow remains open. In this paper, we state the theorem regarding purely local and deterministic growth of Golden-Octagonal tilings. Showing, contrary to the popular belief, that local growth of aperiodic tilings is possible.
... In this paper, we systematically clarify the effect of phase shifts on the typical multiple-Q spin textures, twodimensional (2D) SkLs and three-dimensional (3D) HLs, focusing on their topological properties and the emergent magnetic fields. We first establish a generic framework to deal with the phase shift by introducing the hyperspace with an additional dimension corresponding to the phase degree of freedom, inspired by the description of the phason degree of freedom in quasicrystals [66][67][68][69]. In the hyperspace representation, the 2D SkLs composed of the three spin density waves with the phase degree of freedom are mapped to 3D HLs in which the Dirac strings connecting the hedgehogs and antihedgehogs correspond to the skymion and antiskyrmion cores in the original 2D SkLs. ...
... Consequently, the hyperspace representation enables us to treat the phase degree of freedom as additional coordinates in the hyperspace. We note that the situation is analogous to the hyperspace introduced to understand the structures of quasiperiodic crystals, where the number of translation vectors are in general larger than the system dimension and the quasycrystals are obtained by a "slice" of a periodic structure in the hyperspace with additional dimensions spanned by the same number of vectors [66][67][68][69]. In the quasicrystals, the additional variables in the hyperspace are called phasons [74][75][76][77], which also supports the analogy. ...
A periodic array of topological spin textures, such as skyrmions and hedgehogs, is called the multiple-$Q$ spin texture, as it is represented by a superposition of multiple spin density waves. Depending on the way of superposition, not only the magnetic but also the topological properties are modified, leading to a variety of quantum transport and optical phenomena caused by the emergent electromagnetic fields through the Berry phase. Among others, the phase degree of freedom of the superposed waves is potentially important for such modifications, but its effect has not been fully elucidated thus far. Here we perform systematic theoretical analyses of magnetic and topological properties of the multiple-$Q$ spin textures with the phase degree of freedom. By introducing a hyperspace with an additional dimension corresponding to the phase degree of freedom, we establish a generic framework to deal with the phase shift in the multiple-$Q$ spin textures. Applying the framework to the two-dimensional 3$Q$ spin textures, we clarify the complete topological phase diagram while changing the phase and magnetization, which depends on the types of the superposed waves. We also study the three-dimensional 4$Q$ spin textures and clarify even richer topological phase diagrams. In particular, we find novel topological phase transitions associated with the previously unidentified Dirac strings on which the hedgehogs and antihedgehogs cause pair creation and fusion. Moreover, we demonstrate that phase shifts are caused by an external magnetic field in both 3$Q$ and 4$Q$ cases by analyzing the numerical data in the previous studies. Our results illuminate the topological aspects of the skyrmion and hedgehog lattices with the phase degree of freedom, which would be extended to other multiple-$Q$ textures and useful for the exploration of topologically nontrivial magnetic phases and exotic quantum phenomena.
... The nature of such a state will be discussed elsewhere. Quasicrystal was the first of such state with long-range correlation without periodicity, which has a periodic crystalline lattice in six-dimensions (Levine and Steinhardt, 1984). The ideal glass forms a crystal in infinite dimensions. ...
The conventional approach to elucidate the atomic structure of liquid and glass is to start with local structural units made of several atoms, and to use them as building blocks to form a global structure, the bottom-up approach. We propose to add an alternative top-down approach in which we start with a global high-temperature gas state and then apply interatomic potentials to all atoms at once. This causes collective density wave instability in all directions with the same wavelength. These two driving forces, local and global, are in competition and are mutually frustrated. The final structure is determined through the compromise of frustration between these two, which creates the medium-range-order. This even-handed approach on global and local potential energy landscapes explains the distinct natures of short-range order and medium-range order, and strong temperature dependence of various properties of liquid.
... To be precise, we mean a tiling generated by only a finite selection of different tiles, with each type of tile having a fixed arrangement of atoms within (at least 1). Any crystal can be described in this a way using only one tile by considering its underlying Bravais lattice [6,Chapter 4], but the definition above includes quasicrystals [39,26]. For the approach of quasicrystals by tilings see for example [21,22,32]. ...
For a certain class of discrete metric spaces, we provide a formula for the density of states. This formula involves Dixmier traces and is proven using recent advances in operator theory. Various examples are given of metric spaces for which this formula holds, including crystals, quasicrystals and the infinite cluster resulting from super-critical bond percolation on $\mathbb{Z}^d$.
... The quasicrystals (QCs) are ordered structures lacking periodic translational symmetry [1,2]. Different from the crystals which possess 2-, 3-, 4-, 6-fold rotational symmetry, the quasicrystals may have 5-, 8-, 10-, 12-, 18-fold rotational symmetry. ...
The self-assembly of two-dimensional dodecagonal quasicrystal (DDQC) from patchy particles are investigated by Brownian dynamics simulations. The patchy particle has a five-fold rotational symmetry pattern described by the spherical harmonics $Y_{55}$. From the formation of the DDQC obtained by an annealing process, we find the following mechanism. The early stage of the dynamics is dominated by hexagonal structures. Then, nucleation of dodecagonal motifs appears by particle rearrangement, and finally the motifs expand whole system. The transition from the hexagonal structure into the dodecagonal motif is made by the collective rational motion of the particles. The DDQC consists of clusters of dodecagonal motifs, which can be classified into several packing structures. By the analyses of the DDQC under fixed temperature, we find the fluctuations are characterised by changes in the network of the dodecagonal motifs. Finally we compare the DDQC assembled from the patchy particle system and isotropic particle system. The two systems both share a similar mechanism of the formation and fluctuation of DDQC.
... Compared with the conventional quasicrystals where all of the atoms are intrinsically located within a quasiperiodic order [36,37], graphene quasicrystal is viewed as an extrinsic quasicrystal (i.e., engineered quasicrystals) because its quasiperiodicity arises from the interlayer coupling between two graphene monolayers. Thus, figuring out the origin of quasicrystalline order requires a deep understanding of how the interlayer states interact with each other. ...
The incommensurate 30 • twisted bilayer graphene (BG) possesses both relativistic Dirac fermions and quasiperiodicity with 12-fold rotational symmetry arising from the interlayer interaction [Ahn et al., Science 361, 782 (2018) and Yao et al., Proc. Natl. Acad. Sci. USA 115, 6928 (2018)]. Understanding how the interlayer states interact with each other is of vital importance for identifying and subsequently engineering the quasicrystalline order in the layered structure. Herein, via symmetry and group representation theory we unravel the interlayer hybridization selection rules governing the interlayer coupling in both untwisted and twisted BG systems. Compared with the only allowed equivalent hybridization in D 6h untwisted BG, D 6 twisted BG permits equivalent and mixed hybridizations, and D 6d graphene quasicrystal allows both equivalent and nonequivalent hybridizations. The energy-dependent hybridization strengths in graphene quasicrystal and D 6 twisted BG show two remarkable characteristics: (i) near the Fermi level the weak hybridization owing to the relatively large energy difference between Dirac bands from top and bottom layers, and (ii) in high-energy regions the electron-hole asymmetry of hybridization strength with stronger interlayer coupling for holes, which arises from the non-nearest-neighbor interlayer hoppings and the wave-function phase difference between pairing states. These hybridization-generated band structures and their hybridization strength characteristics are verified by the calculated optical conductivity spectra. Our theoretical study paves a way for revealing the interlayer hybridization in van der Waals layered systems.
... It should also be noted that the patterns in Figure 2 happen to bear resemblance to quasicrystals [34], which are related to Penrose tilings [35]. Thus, we can infer that GRIDS will likely also contain patterns that bear resemblance to 4D-quasicrystals, which might lead to further insights in the future. ...
The causal set program and the Wolfram physics project leave open the problem of
how a graph that is a (3+1)-dimensional Minkowski spacetime according to its simple geodesic distances could be generated solely from simple deterministic rules. This paper provides a solution by describing simple rules that characterize discrete Lorentz boosts between 4D lattice graphs, which combine further to form Wigner rotations that produce isotropy and lead to the emergence of the continuous Lorentz group and the (3+1)-dimensional Minkowski spacetime. On such graphs, the speed of light, the proper time interval, as well as the proper length are all shown to be highly accurate.
... Later, Bindi et al. demonstrated that quasicrystals can also originate naturally, in the presence of extreme conditions such as collisions between asteroids [21]. Their properties have already been the subject of extensive theoretical investigation [22], motivated in part by the discovery of aperiodic tilings that can cover the plane without being bounded by the symmetries of classical crystallography, such as the Penrose tiling [23]. At finite temperature, the thermodynamic features of quasicrystals can be established in terms of the interplay between different length and energy scales pertaining to the inter-particle potentials [24]. ...
In this work, we explore the relevant methodology for the investigation of interacting systems with contact interactions, and we introduce a class of zonal estimators for path-integral Monte Carlo methods, designed to provide physical information about limited regions of inhomogeneous systems. We demonstrate the usefulness of zonal estimators by their application to a system of trapped bosons in a quasiperiodic potential in two dimensions, focusing on finite temperature properties across a wide range of values of the potential. Finally, we comment on the generalization of such estimators to local fluctuations of the particle numbers and to magnetic ordering in multi-component systems, spin systems, and systems with nonlocal interactions.
... Later, Bindi et al. demonstrated that quasicrystals can also originate naturally, in the presence of extreme conditions such as collisions between asteroids [19]. Their properties have already been the subject of extensive theoretical investigation [20], motivated in part by the discovery of aperiodic tilings that can cover the plane without being bounded by the symmetries of classical crystallography, such as the Penrose tiling [21]. At finite temperature, the thermodynamic features of quasicrystals can be established in terms of the interplay between different length and energy scales pertaining to the inter-particle potentials [22]. ...
In this work, we explore the relevant methodology for the investigation of interacting systems with contact interactions, and we introduce a class of zonal estimators for Path-integral Monte Carlo methods, designed to provide physical information about limited regions of inhomogeneous systems. We demonstrate the usefulness of zonal estimators by their application to a system of trapped bosons in a quasiperiodic potential in two dimensions, focusing on finite temperature properties across a wide range of values of the potential. Finally, we comment on the generalization of such estimators to local fluctuations of the particle numbers and to magnetic ordering in multi-component systems, spin systems, and systems with nonlocal interactions.
... Their configurations in physical space lack translational periodicity, but long-range order as well as high-order rotational symmetries are present [13]. The unique symmetries of quasicrystals are revealed by their exotic sharp Bragg diffraction patterns, first experimentally observed by Shechtman et al. [14] and theoretically reported in the pioneering work of Levine and Steinhardt [15]. Investigation of properties arising from such unique symmetries has resulted in novel applications such as lasing [16,17], superior sensing and imaging [18], guiding and bending of waves [19], super-focusing [13], superconductivity [20] and topological wave transport [21,22]. ...
In this paper, we present numerical and experimental evidence of directional wave behavior, i.e. beaming and diffraction, along high-order rotational symmetries of quasicrystalline elastic metamaterial plates. These structures are obtained by growing pillars on an elastic plate following a particular rotational symmetry arrangement, such as 8-fold and 10-fold rotational symmetries, as enforced by a design procedure in reciprocal space. We estimate the dispersion properties of the waves propagating in the plates through Fourier transformation of transient wave-fields. The procedure identifies, both numerically and experimentally, the existence of anisotropic bands characterized by high energy density at isolated regions in reciprocal space that follow their higher order rotational symmetry. Specific directional behavior is showcased at the identified frequency bands, such as wave beaming and diffraction. This work expands the wave directionality phenomena beyond the symmetries of periodic configurations (e.g., 4-fold and 6-fold), and opens new possibilities for applications involving the unusual high-order wave features of the quasicrystals such as superior guiding, focusing, sensing and imaging.
... The breaking of translation symmetry can be achieved in two different ways: either the localization of atoms becomes random or it preserves a quasi-periodic pattern. Quasicrystals are precisely crystals, in which the position of atoms is not arbitrary but the translation symmetry is broken [57][58][59]. In order to understand the elastic description of such crystals we consider a one-dimensional line of atoms parameterized by two sublattices a and b. ...
There has been a surge of interest in effective non-Lorentzian theories of excitations with restricted mobility, known as fractons. Examples include defects in elastic materials, vortex lattices or spin liquids. In the effective theory novel coordinate-dependent symmetries emerge that shape the properties of fractons. In this review we will discuss these symmetries, cover the effective description of gapless fractons via elastic duality, and discuss their hydrodynamics.
... Quasicrystals first observed by Shechtman in 1982 [7] are aperiodic systems with symmetry elements (like 5-, 10-fold rotational axes) of the diffraction pattern that are incompatible with translational symmetry [8]. Aperiodic symmetry also occurs locally in the atomic structure, making the structural and diffraction description of these materials much harder compared to periodic crystals [9]. ...
Quasicrystals have attracted a growing interest in material science because of their unique properties and applications. Proper determination of the atomic structure is important in designing a useful application of these materials, for which a difficult phase problem of the structure factor must be solved. Diffraction patterns of quasicrystals consist of a periodic series of peaks, which can be reduced to a single envelope. Knowing the distribution of the diffraction image into series, it is possible to recover information about the phase of the structure factor without using time-consuming iterative methods. By the inverse Fourier transform, the structure factor can be obtained (enclosed in the shape of the average unit cell, or atomic surface) directly from the diffraction patterns. The method based on envelope function analysis was discussed in detail for a model 1D (Fibonacci chain) and 2D (Penrose tiling) quasicrystal. First attempts to apply this technique to a real Al-Cu-Rh decagonal quasicrystal were also made.
... The rapidly solidified Al 86 Mn 14 alloys (Shechtman et al. 1984) [1] have demonstrated the possibility of perfect order, but non-periodic atomic arrangements exhibiting an icosahedral symmetry (m 35) [1]. This experimental observation was explained by Levine and Steinhardt (1984) [6] based on the argument that an aperiodic lattice with icosahedral bond orientation order is possible, and the mathematics of diffraction does not necessarily require periodicity. In a similar time frame, unaware of the concept of 'quasicrystal, ' Penrose (1974) [7] had already developed a scheme of non-periodic filling of space by a finite number of tiles, which was later used by Mackay in 1981 [8] to build the three-dimensional analog tiles and to emphasize their role in crystallography. ...
Quasicrystals (QCs) are intermetallic materials with long-range ordering but with lack of periodicity. They have attracted much interest due to their interesting structural complexity, unusual physical properties, and varied potential applications. The last four decades of research have demonstrated the existence of different forms of QC composed of several metallic and non-metallic systems, which have already been exploited in several applications. Recently, with the experimental realization of 2D (atomically thin) metals, the potential applications of these structures have significantly increased (such as inflexible electronics, optoelectronics, electrocatalysis, strain sensors, nano-generators, innovative nano-electromechanical systems, and biomedical applications). As a result, high-quality 2D metals and alloys with engineered and tunable properties are in great demand. This review summarizes the recent advances in the synthesis of 2D single and few layered metals and alloys using quasicrystals. These structures present a large number of active sites for hydrogen evolution process catalysis and other functional properties. In this review, we also highlighted the possibility of using QC to synthesize other 2D metals and to explore their physical and chemical properties.
... A periodic array of alternating dielectrics and plasmas called plasma photonic crystals (PPCs) has been the subject of much interest [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] owing to its optical properties. Besides crystals, Levine and Steinhardt [20] introduced quasicrystals -materials with properties that are ordered in space but do not possess an exact periodicity. Photonic quasicrystals have been studied extensively [21][22][23][24][25][26][27][28]. ...
We demonstrate theoretically and numerically that a warm fluid model of a plasma supports space-time quasicrystalline structures. These structures are highly nonlinear, two-phase, ion acoustic waves that are excited autoresonantly when the plasma is driven by two small amplitude chirped-frequency ponderomotive drives. The waves exhibit density excursions that substantially exceed the equilibrium plasma density. Remarkably, these extremely nonlinear waves persist even when the small amplitude drives are turned off. We derive the weakly nonlinear analytical theory by applying Whitham's averaged variational principle to the Lagrangian formulation of the fluid equations. The resulting system of coupled weakly nonlinear equations is shown to be in good agreement with fully nonlinear simulations of the warm fluid model. The analytical conditions and thresholds required for autoresonant excitation to occur are derived and compared to simulations. The weakly nonlinear theory guides and informs numerical study of how the two-phase quasicrystalline structure "melts" into a single phase traveling wave when one drive is below a threshold. These nonlinear structures may have applications to plasma photonics for extremely intense laser pulses, which are limited by the smallness of density perturbations of linear waves.
... Quasicrystals (QCs) are long-range-ordered solids that exhibit self-similar diffraction patterns incompatible with translational symmetry (Shechtman et al., 1984;Levine & Steinhardt, 1984). The first dodecagonal quasicrystal (DDQC) was found in small particles of an Ni-Cr alloy (Ishimasa et al., 1985). ...
A four-dimensional (4D) model is presented of the first oxide dodecagonal quasicrystal found in a Ba–Ti–O ultra-thin film on a Pt(111) single-crystal substrate. The 4D model, with a 4D dodecagonal lattice constant a d = 8.39 Å, was derived by considering a tile decoration model of dodecagonal Niizeki–Gähler tiling composed of squares, triangles and 30° rhombuses. The model consists of four kinds of occupation domain, and 4D positional vectors defining the shape of each occupation domain are given. Moreover, the atomic arrangement of two Ba–Ti–O periodic approximants, the sigma-phase approximant and a 25.6 Å approximant were derived from the 4D model by the introduction of linear phason strains.
... For example, PC exhibits the so-called slow light effect at the edge of the photonic bandgap (PBG), which is a property of great practical interest as it can be leveraged to construct state of the art miniaturized optical devices like amplifiers, lasers, detectors, sensors, wave-length modulators, among other applications [3]. Parallel to these developments in the field of standard PCs, many works have been done on aperiodic photonic structures, specifically quasicrystals [4], since their discovery by Dan Shechtman in 1984 [5]. Aperiodic systems are constructed from specific deterministic rules and present long-range order with discrete spectral contributions [6], as well as rotational symmetry limited to a few-folds of the periodic lattice [7]. ...
The study of photonic crystals, artificial materials whose dielectric properties can be tailored according to the stacking pattern of its constituents, remains an attractive research area. Very recently, the propagation of light waves in periodic and quasiperiodic graphene embedded dielectric multilayers have also been considered. The presence of graphene between consecutive layers induces the emergence of the so-called graphene induced photonic bandgap. In this article, we employ a transfer-matrix treatment to study the effects of (i) random arrangement, (ii) length of the multilayers and (iii) chemical potential on the robustness of the graphene induced photonic bandgap (GIPBG) in photonic crystals composed of random dielectric multilayers with graphene embedded. The photonic crystals considered here are composed of two materials, namely, silicon dioxide (layer A=SiO2) and titanium dioxide (layer B=TiO2). Our numerical results quantitatively show that the random arrangement and length of the multilayers cause minor changes in the bandgap edge of the low frequency GIPBG, i.e., the low frequency GIPBG is robust regarding composition and length of the structures. On the other hand, the chemical potential strongly affects the robustness of the low frequency GIPBG. In fact, the low frequency GIPBG is strongly dependent on the properties of graphene, reinforcing that it is possible to adjust its width and bandgap edge by tuning the chemical potential via a gate voltage.
... 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 (CSDF) model whose ground state is a crystal where each unit cell is a fractal.
... In this frame quasicrystals were regarded as a somewhat intermediate structure between an ordered and a disordered system Shechtman et al. (1984). This notion however was later challenged in light of the fact that the so-called quasicrystals exhibit long range order, orientational symmetries and a discrete diffraction patterns Levine and Steinhardt (1984), all properties that are shared with ordered matter. This suggests that quasicrystals should be interpreted as an extension of the notion of crystals. ...
This thesis deals with a variety of topics that are relevant for the theory of ultracold atoms, with a focus on the many interaction regimes that can be obtained in these systems. The topics are presented in increasing order of complexity with regards to these interaction regimes. In the first chapter we are prompted by an experiment on a realization of an interacting Aubry-Andr´e Hamiltonian driven by a periodic modulation, where a localisation-delocalisation transition is observed. We will model an analogue non interacting system and show that it reproduces an equivalent phase diagram. Moreover we are able to provide a physical explanation for the critical amplitude and the critical frequency for the delocalisation transition. Next, we will consider a model of N atoms which are non interacting with the addition of a single light impurity that interacts with them. We will propose a simplified model based on the Born-Oppenheimer approximation and a polaron-like picture through which we are able to estimate the energy of the system and to address the question of the
existence of stable clusters bound by an impurity in the large N limit. In the rest of the thesis the pivot will be on the Andreev-Bashkin effect which describes the drag that each component of a mixture of two superfluids exerts on the other, as a result of their mutual interactions. We will first propose a microscopic theory based on linear response theory that describes the drag, and derive its implications on the nature of the excited states in a superfluid mixture. Then we will compute the effect of the drag on the spin speed of sound and the spin dipole mode which can in principle observed in experiments. Analytical results for the case of a weakly interacting mixture are presented as a benchmark for our methods. Finally we will focus on the Andreev-Bashkin effect in a Bose-Hubbard Hamiltonian in a one dimensional ring lattice. We will show that the effect is enhanced for attractive intraspecies interactions, more so close to the transition to paired superfluidity. A discussion on the correction brought by the drag in the low energy Luttinger theory for the model is also presented.
... The work suggests that the NV center spin system (assisted by dynamical decoupling) may provide a versatile platform to realize the counterpart of real-space lattice, i.e., the Floquet lattice. With incommensurate frequencies, it is possible to further explore the quasicrystals with infinite lattice [52][53][54]. With commensurate frequencies, the Floquet lattice has the periodic boundary conditions such that the lattice contracts to a cylinder [12]. ...
The features of topological physics can manifest in a variety of physical systems in distinct ways. Periodically driven systems, with the advantage of high flexibility and controllability, provide a versatile platform to simulate many topological phenomena and may lead to novel phenomena that can not be observed in the absence of driving. Here we investigate the influence of realistic experimental noise on the realization of a two-level system under a two-frequency drive that induces topologically nontrivial band structure in the two-dimensional Floquet space. We propose a dynamical decoupling scheme that sustains the topological phase transition overcoming the influence of dephasing. Therefore, the proposal would facilitate the observation of topological frequency conversion in the solid state spin system, e.g. NV center in diamond.
... We interchangeably call this problem lattice Fourier interpolation, or semi-continuous signal reconstruction, as lattice frequencies can be viewed as interpolating between discrete and continuous domains. Solving this problem is our key tool en-route to faster sparse-recovery, but it is also interesting on its own right; One concrete motivation for this problem is modeling and analyzing Quasicrystals [SBGC84,LS84,LS86,MM10]. Formally, a discrete measure on R n is a Fourier quasicrystal if its Fourier transform is also a discrete measure. ...
We revisit the classical problem of band-limited signal reconstruction -- a variant of the *Set Query* problem -- which asks to efficiently reconstruct (a subset of) a $d$-dimensional Fourier-sparse signal ($\|\hat{x}(t)\|_0 \leq k$), from minimum noisy samples of $x(t)$ in the time domain. We present a unified framework for this problem, by developing a theory of sparse Fourier transforms over *lattices*, which can be viewed as a "semi-continuous" version of SFT, in-between discrete and continuous domains. Using this framework, we obtain the following results: $\bullet$ *High-dimensional Fourier sparse recovery* We present a sample-optimal discrete Fourier Set-Query algorithm with $O(k^{\omega+1})$ reconstruction time in one dimension, independent of the signal's length ($n$) and $\ell_\infty$-norm ($R^* \approx \|\hat{x}\|_\infty$). This complements the state-of-art algorithm of [Kap17], whose reconstruction time is $\tilde{O}(k \log^2 n \log R^*)$, and is limited to low-dimensions. By contrast, our algorithm works for arbitrary $d$ dimensions, mitigating the $\exp(d)$ blowup in decoding time to merely linear in $d$. $\bullet$ *High-accuracy Fourier interpolation* We design a polynomial-time $(1+ \sqrt{2} +\epsilon)$-approximation algorithm for continuous Fourier interpolation. This bypasses a barrier of all previous algorithms [PS15, CKPS16] which only achieve $>100$ approximation for this problem. Our algorithm relies on several new ideas of independent interest in signal estimation, including high-sensitivity frequency estimation and new error analysis with sharper noise control. $\bullet$ *Fourier-sparse interpolation with optimal output sparsity* We give a $k$-Fourier-sparse interpolation algorithm with optimal output signal sparsity, improving on the approximation ratio, sample complexity and runtime of prior works [CKPS16, CP19].
... Quasicrystals are aperiodic structures characterized by long-range order and lack of translational symmetry [1,2]. The order can be revealed in the Fourier spectrum of the structure that has a countable set of Fourier components [3][4][5][6]. ...
We report on the evolution of the spin-wave spectrum under induced structural disorder in one-dimensional magnonic quasicrystal. We study theoretically a system composed of ferromagnetic strips arranged in a Fibonacci sequence for several stages of disorder in the form of phasonic defects, where different rearrangements are introduced. By transition from the quasiperiodic order towards disorder, we show a gradual degradation of the fine fractal spectra of spin-waves, and the evolution of the band gaps. The analysis shows the mode localization evolution, which is an inevitable property of the quasicrystals. In particular, significant enhancement of the spin-wave mode localization and disappearing of the van Hove singularities at the band gap edges with increasing number of defects can be indication of the phasonic defects existing in the structure. This work fills gap in the knowledge of defected magnonic Fibonacci quasicrystals and opens the way to the study of the phasonic defect in two-dimensional magnonic quasicrystals.
... However, many experimentally relevant systems have correlated-but-aperiodic [25] (or nearly aperiodic) spatial inhomogeneity. Examples include quasicrystals [26], incommensurate optical lattices [27], and moiré materials [28][29][30]. Each case provides experimental evidence for correlated phases [31][32][33], such as quantum criticality without tuning in the Yb-Al-Au heavy-fermion quasicrystal [34][35][36][37] and observation of insulating phases at integer fillings of the moiré unit cell in magic-angle twisted bilayer graphene [38][39][40]. ...
The Anderson model for a magnetic impurity in a one-dimensional quasicrystal is studied using the numerical renormalization group (NRG). The main focus is elucidating the physics at the critical point of the Aubry-Andre (AA) Hamiltonian, which exhibits a fractal spectrum with multifractal wave functions, leading to an AA Anderson (AAA) impurity model with an energy-dependent hybridization function defined through the multifractal local density of states at the impurity site. We first study a class of Anderson impurity models with uniform fractal hybridization functions that the NRG can solve to arbitrarily low temperatures. Below a Kondo scale $T_K$, these models approach a fractal strong-coupling fixed point where impurity thermodynamic properties oscillate with $\log_b T$ about negative average values determined by the fractal dimension of the spectrum. The fractal dimension also enters into a power-law dependence of $T_K$ on the Kondo exchange coupling $J_K$. To treat the AAA model, we combine the NRG with the kernel polynomial method (KPM) to form an efficient approach that can treat hosts without translational symmetry down to a temperature scale set by the KPM expansion order. The aforementioned fractal strong-coupling fixed point is reached by the critical AAA model in a simplified treatment that neglects the wave-function contribution to the hybridization. The temperature-averaged properties are those expected for the numerically determined fractal dimension of $0.5$. At the AA critical point, impurity thermodynamic properties become negative and oscillatory. Under sample-averaging, the mean and median Kondo temperatures exhibit power-law dependences on $J_K$ with exponents characteristic of different fractal dimensions. We attribute these signatures to the impurity probing a distribution of fractal strong-coupling fixed points with decreasing temperature.
Precipitation of a stable quasicrystalline i-phase has been discovered in a Mg-6Zn-3Al (wt%, ZA63) magnesium alloy, possibly a first involving a stable i-phase. Dissolution of i-phase solidified at grain boundaries leads to precipitation in the Mg-matrix during cooling due to a solvus boundary observed by phase diagram calculation. Phase transformation, nucleation characteristics, morphology and interfaces with the matrix have been studied. Differential scanning calorimetry (DSC) and in-situ X-ray diffraction (XRD) showed dissolution and precipitation of i-phase during heating and cooling cycles, forming a hysteresis between about 250 and 325∘C. The quasilattice parameter aRof i-phase was calculated to be 5.17 Å at room temperature, close to the c-parameter 5.18 Å of the Mg-matrix. At dissolution/nucleation, the diffraction peaks of i-phase were sharper, (quasi)lattice constants aRand cMg become equal to each other (5.22 Å) and the lattice strain in the matrix was minimized. Precipitation occurred with a fivefold plane parallel to a matrix pyramidal {011¯1} plane, which gave rise to two possible orientation relationships (OR): 5f∥[0001] (OR3’) and 2f∥[0001] (OR1). Presence of both ORs in the same precipitates generated icosahedral twinning. The precipitates were faceted sharply on fivefold, twofold and threefold planes which were on basal, prismatic and pyramidal planes of the Mg-matrix. Results suggest the OR selection to be nucleation temperature dependent. Based on the crystallographic relationships, 3-dimensional tiling model of icosahedral lattice has been used to model lattice match with the matrix.
We study the effect of carrier doping to the Mott insulator on the Penrose tiling, aiming at clarifying the interplay between quasiperiodicity and strong electron correlations. We numerically solve the Hubbard model on the Penrose-tiling structure within a real-space dynamical mean-field theory, which can deal with a singular self-energy necessary to describe the Mott insulator and spatial inhomogeneity. We find that the strong correlation effect produces a charge distribution unreachable by a static mean-field approximation. In a small doping region, the spectrum shows a site-dependent gap just above the Fermi energy, which is generated by a singularly large self-energy emergent from the Mott physics and regarded as a real-space counterpart of the momentum-dependent pseudogap observed in a square-lattice Hubbard model.
A comprehensive framework of characterizing complex self-assembled structures with a set of orientational order parameters is presented. It is especially relevant in the context of using anisotropic building blocks with various symmetries. Two classes of tensor order parameters are associated with polyhedral nematic, and bond orientational order, respectively. For the latter, a variation of classical bond order parameters is introduced that takes advantage of the symmetry of constituent particles, and/or expected crystalline phases. These Symmetrized Bond Order Parameters (SymBOPs) can be averaged over an entire system, or assigned locally to an individual bond. By combining that with bond percolation procedure, one is able to identify coherent domains within a self-assembled structure. As a demonstration of the proposed framework, we apply it to a simulated hybrid system that combines isotropic and patchy particles with octahedral symmetry. Not only does the methodology allow one to identify individual crystalline domains, but it also detects coherent clusters of a peculiar compact amorphous structure that is not space-filling and lacks any long-range order.
Topological insulators constitute a novel state of matter with scatter-free edge states surrounding an insulating bulk. Conventional wisdom regards the insulating bulk as essential, since the invariants describing the topological properties of the system are defined therein. Here, we study fractal topological insulators based on exact fractals comprised exclusively of edge sites. We present experimental proof that, despite the lack of bulk bands, photonic lattices of helical waveguides support topologically protected chiral edge states. We show that light transport in our topological fractal system features increased velocities compared to the corresponding honeycomb lattice. By going beyond the confines of the bulk-boundary correspondence, our findings pave the way toward an expanded perception of topological insulators and open a new chapter of topological fractals.
Quasicrystals can enhance wave transmission and boost nonlinear phenomena due to rotational symmetry forbidden in conventional periodic crystals. As a projection of higher-dimensional quasicrystal lattice, quasiperiodic structures represent a novel platform that exhibits robust microwave-absorbing capacity in thin thickness and digs out the influence of structural configurations on electromagnetic wave absorption from local and global aspects. Due to the synergy effect of composites and structure, quasiperiodic metamaterials with Penrose tiling achieve 12.2 GHz in effective absorption bandwidth (reflection loss < −10 dB), from 5.78 GHz to 18.00 GHz with 2 mm thickness. It has obvious advantages in absorbing electromagnetic wave compared with corresponding periodic absorbers according to the characteristic of local anisotropy and higher order translational symmetry in Brillouin zone. By enhancing multiple scattering and nonlinear coupling effects, Penrose tiling provides new insights to design absorbers with outstanding electromagnetic performance and higher flexibility and tunability.
Two problems of a one-dimensional (1D) piezoelectric quasicrystal (QC) wedge are investigated, i.e., the two sides of the wedge subject to uniform tractions and the wedge apex subject to the concentrated force. By virtue of the Stroh formalism and Barnett-Lothe matrices, the analytical expressions of the displacements and stresses are derived, and the generalized solutions for the critical wedge angles are discussed. Numerical examples are given to present the mechanical behaviors of the wedge in each field. The results indicate that the effects of the uniform tractions and the concentrated force on the phonon field displacement are larger than those on the phason field.
Nanostructured superlattices are expected to play a significant role in the next generation of technological devices, specially due to their adjustable physical properties. In terms of heat transport, materials with low thermal conductivities can be useful in thermoelectric devices or heat shields, while materials with high thermal conductivities are fundamental for heat dissipation in miniaturized electronic devices. In general, transport properties are dominated by translational symmetry and the presence of unconventional symmetries might lead to unusual transport characteristics. In this work, we report our results from nonequilibrium molecular dynamics simulations to investigate phonon heat transport in periodic and quasiperiodic graphene-hBN superlattices. The periodic superlattices are built with alternating equal-sized domains of graphene and hBN, while the quasiperiodic case follows the Fibonacci sequence, which lies between periodic and disordered structures. Periodic superlattices can facilitate coherent phonon transport due to constructive interference at the boundaries between the materials. Nonetheless, it is possible to induce a crossover from a coherent to an incoherent transport regime by increasing the length of individual domains, thus adjusting the superlattice period. We also show that the quasiperiodicity can suppress coherent phonon transport in these superlattices. We attribute this behavior to the increased inhomogeneity in the distribution of interfaces, which increases for each Fibonacci generation, hindering coherent phonon transport in the superlattices. The suppression of coherent thermal transport enables a higher degree of control on heat conduction at the nanoscale, and shows potential for application in thermoelectric devices and heat management.
An analytical solution of the guided wave propagation in a multilayered two-dimensional decagonal quasicrystal plate with imperfect interfaces is derived. According to the elastodynamic equations of quasicrystals (QCs), the wave propagating problem in the plate is converted into a linear control system by employing the state-vector approach, from which the general solutions of the extended displacements and stresses can be obtained. These solutions along the thickness direction are utilized to derive the propagator matrix which connects the physical variables on the lower and upper interfaces of each layer. The special spring model, which describes the discontinuity of the physical quantities across the interface, is introduced into the propagator relationship of the multilayered structure. The total propagator matrix can be used to propagate the solutions in each interface and each layer about the multilayered plate. In addition, the traction-free boundary condition on the top and bottom surfaces of the laminate is considered to obtain the dispersion equation of wave propagation. Finally, typical numerical examples are presented to illustrate the marked influences of stacking sequence and interface coefficients on the dispersion curves and displacement mode shapes of the QC laminates.
Quasicrystals are considered as a new class of intermetallics with the long-range quasiperiodic ordering, and have attracted research interest over the last 3 decades due to their interesting structural complexities, unique physical/mechanical properties, and potentials for applications. Quasicrystals exhibit ultimate complexity with a lack of periodicity in the 3-dimensional space, but give rise to sharp diffraction patterns with rotational symmetry forbidden for periodic crystals. The crystal structure of quasicrystalline intermetallics can be understood in terms of appropriate decoration of 3-dimensional Penrose quasilattice using higher dimensional crystallography. Here, we have reviewed research outcomes in the area of quasicrystals, with an emphasis on the synthesis, structure, stability, as well as properties, and potential applications related to mostly icosahedral quasicrystals and other relevant complex intermetallics.
In this paper, we review the specific field that combines topological photonics and deep learning (DL). Recent progress of topological photonics has attracted enormous interest for its novel and exotic properties such as unidirectional propagation of electromagnetic waves and robust manipulation of photons. These phenomena are expected to meet the growing demands of next-generation nanophotonic devices. However, to model and engineer such highly-complex systems are challenging. Recently, DL, a subset of machine learning methods using neural network (NN) algorithms, has been introduced in the field of nanophotonics as an effective way to capture a complex nonlinear relationship between design parameters and their corresponding optical properties. In particular, among various fields of nanophotonics, DL applications to topological photonics empowered by NN models have shown astonishing results in capturing the global material properties of topological systems. This review presents fundamental concepts of topological photonics and the basics of DL applied to nanophotonics in parallel. Recent studies of DL applications to topological systems using NN models are discussed thereafter. The summary and outlook showing the potential of taking data-driven approaches in topological photonics research and general physics are also discussed. © 2022 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
A mode-I crack in a one-dimensional (1D) orthorhombic quasicrystal (QC) strip under in-plane phonon and phason stress loading is considered. Fourier transforms are applied to reduce the mixed boundary value problem of the mode-I crack to solving a system of simultaneous singular integral equations. Asymptotic expressions of the phonon and phason stresses and displacement fields near the crack tips have been obtained in an explicit form. The crack-tip singularities of the mode-I crack have been investigated and the intensity factors of the stresses in the phonon and phason fields are derived explicitly. The stress intensity factors (SIFs) and the hoop stress intensity factors (HSIFs) have been determined to investigate the effect of the geometric size and the crack kinking phenomenon. The effect of the thickness ratio of the cracked strip on the SIFs and energy release rates has been investigated. When the thickness of the cracked strip becomes infinite large, the results obtained for the crack problem can be reduced to the analytic solution for a mode-I crack in an infinite 1D orthorhombic QC media.
Quasicrystal materials have aroused extensive attentions of researchers due to their excellent properties. In this paper, for the first time, functionally graded two-dimensional piezoelectric QC wedges and spaces which subjected to line force, charge or dislocation are investigated. By virtue of equilibrium and constitutive equations, the real form expressions of stresses and displacements are directly derived without introducing complex vectors. All expressions can be degraded to the classical solutions of homogeneous materials when material parameters have no concern with angle. Before the numerical results are presented, the comparative study between solutions of homogeneous materials from Stroh formalism and solutions of homogeneous materials from our method is performed to verify the accuracy of the formulation and numerical procedure. The mechanical behaviors of QC wedges and spaces under different loads are analyzed carefully by numerical examples. The numerical results reveal that the variations of stress and displacement in each field have much to do with the wedge angle and boundary conditions; line force has little effect on lattice rearrangement. Moreover, the numerical results can be used as references in engineering application when analyzing functionally graded materials.
Partitions appear across physics like Bose-Einstein statistics, dimer coverings, Potts model, crystal growth, Vasiliev theory, quasicrystals, Penrose-like random tilings, and so on. Since the general analytical study of partitions is rather tough, so statistical study is immensely preferred. This paper derives an asymptotic formula to evaluate the m-th raw moment of the multiplicity of a part for a broad category of partitions given by a specific generating function. In addition to being a multifold refinement/generalization of results in the literature, this work allows to calculate useful statistical quantities like mean and variance of the number of appearance of a part which is of deeper significance to analysts. The obtained expression is numerically validated to be in close agreement with the actual values. This study instigates the development of a general paradigm where partition theory is exposed to more sophisticated statistical analysis, thus, opening new avenues of research in this field.
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.