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1. Snaking without subcriticality: grain boundaries as non-topological defects

   https://doi.org/10.1093/imamat/hxab032  

   Subramanian, Priya; Archer, Andrew J; Knobloch, Edgar; Rucklidge, Alastair M (July 2021, IMA Journal of Applied Mathematics)

   null (Ed.)

   Abstract Non-topological defects in spatial patterns such as grain boundaries in crystalline materials arise from local variations of the pattern properties such as amplitude, wavelength and orientation. Such non-topological defects may be treated as spatially localized structures, i.e. as fronts connecting distinct periodic states. Using the two-dimensional quadratic-cubic Swift–Hohenberg equation, we obtain fully nonlinear equilibria containing grain boundaries that separate a patch of hexagons with one orientation (the grain) from an identical hexagonal state with a different orientation (the background). These grain boundaries take the form of closed curves with multiple penta-hepta defects that arise from local orientation mismatches between the two competing hexagonal structures. Multiple isolas occurring robustly over a wide range of parameters are obtained even in the absence of a unique Maxwell point, underlining the importance of retaining pinning when analysing patterns with defects, an effect omitted from the commonly used amplitude-phase description. Similar results are obtained for quasiperiodic structures in a two-scale phase-field model. 

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2. Universality and quantum criticality in quasiperiodic spin chains

   https://doi.org/10.1038/s41467-020-15760-5  

   Agrawal, Utkarsh; Gopalakrishnan, Sarang; Vasseur, Romain (May 2020, Nature Communications)

   Abstract Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems and disordered ones as well. Quasiperiodic criticality was previously understood only in the special limit where the couplings follow discrete quasiperiodic sequences. Here we consider generic quasiperiodic modulations; we find, remarkably, that for a wide class of spin chains, generic quasiperiodic modulations flow to discrete sequences under a real-space renormalization-group transformation. These discrete sequences are therefore fixed points of a functional renormalization group. This observation allows for an asymptotically exact treatment of the critical points. We use this approach to analyze the quasiperiodic Heisenberg, Ising, and Potts spin chains, as well as a phenomenological model for the quasiperiodic many-body localization transition. 

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3. Effect of electromechanical coupling on locally resonant quasiperiodic metamaterials

   https://doi.org/10.1063/5.0119914  

   LeGrande, Joshua; Bukhari, Mohammad; Barry, Oumar (January 2023, AIP Advances)

   Electromechanical metamaterials have been the focus of many recent studies for use in simultaneous energy harvesting and vibration control. Metamaterials with quasiperiodic patterns possess many useful topological properties that make them a good candidate for study. However, it is currently unknown what effect electromechanical coupling may have on the topological bandgaps and localized edge modes of a quasiperiodic metamaterial. In this paper, we study a quasiperiodic metamaterial with electromechanical resonators to investigate the effect on its bandgaps and localized vibration modes. We derive here the analytical dispersion surfaces of the proposed metamaterial. A semi-infinite system is also simulated numerically to validate the analytical results and show the band structure for different quasiperiodic patterns, load resistors, and electromechanical coupling coefficients. The topological nature of the bandgaps is detailed through an estimation of the integrated density of states. Furthermore, the presence of topological edge modes is determined through numerical simulation of the energy harvested from the system. The results indicate that quasiperiodic metamaterials with electromechanical resonators can be used for effective energy harvesting without changes in the bandgap topology for weak electromechanical coupling. 

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4. Fast numerical solutions to direct and inverse scattering for bi-anisotropic periodic Maxwell's equations

   Nguyen, Dinh-Liem; Truong, Trung (January 2023, AMS Contemporary Mathematics)

   Nguyen, Dinh-Liem; Nguyen, Loc; Nguyen, Thi-Phong (Ed.)

   This paper is concerned with the numerical solution to the direct and inverse electromagnetic scattering problem for bi-anisotropic periodic structures. The direct problem can be reformulated as an integro-di erential equation. We study the existence and uniqueness of solution to the latter equation and analyze a spectral Galerkin method to solve it. This spectral method is based on a periodization technique which allows us to avoid the evaluation of the quasiperiodic Green's tensor and to use the fast Fourier transform in the numerical implementation of the method. For the inverse problem, we study the orthogonality sampling method to reconstruct the periodic structures from scattering data generated by only two incident fields. The sampling method is fast, simple to implement, regularization free, and very robust against noise in the data. Numerical examples for both direct and inverse problems are presented to examine the efficiency of the numerical solvers. 

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5. Multiple scale method applied to homogenization of irrational metamaterials

   https://doi.org/10.1109/Metamaterials49557.2020.9284990  

   Guenneau, S.; Zolla, F.; Cherkaev, E.; Wellander, N. (September 2020, 2020 Fourteenth International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials))

   null (Ed.)

   We adapt the multiple scale method introduced over 40 years ago for the homogenization of periodic structures [1], to the quasiperiodic (cut-and-projection) setting. We make use of partial differential operators (gradient, divergence and curl) acting on periodic functions of m variables in a higher-dimensional space that are projected onto operators acting on quasiperiodic functions in the n-dimensional physical space (m>n). We replace heterogeneous quasiperiodic structures, coined irrational metamaterials in [2], by homogeneous media with anisotropic permittivity and permeability tensors, obtained from the solution of annex problems of electrostatic type in a periodic cell in higher dimensional space. This approach is valid when the wavelength is much larger than the period of the higher dimensional elementary cell. 

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