nonlocal interactions
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2022 ◽  
Vol 43 (1) ◽  
pp. 1-20
Author(s):  
Binying Wang ◽  
Jinxing Liu ◽  
A. K. Soh ◽  
Naigang Liang

AbstractWe have proposed an “exact” strain gradient (SG) continuum model to properly predict the dispersive characteristics of diatomic lattice metamaterials with local and nonlocal interactions. The key enhancement is proposing a wavelength-dependent Taylor expansion to obtain a satisfactory accuracy when the wavelength gets close to the lattice spacing. Such a wavelength-dependent Taylor expansion is applied to the displacement field of the diatomic lattice, resulting in a novel SG model. For various kinds of diatomic lattices, the dispersion diagrams given by the proposed SG model always agree well with those given by the discrete model throughout the first Brillouin zone, manifesting the robustness of the present model. Based on this SG model, we have conducted the following discussions. (I) Both mass and stiffness ratios affect the band gap structures of diatomic lattice metamaterials, which is very helpful for the design of metamaterials. (II) The increase in the SG order can enhance the model performance if the modified Taylor expansion is adopted. Without doing so, the higher-order continuum model can suffer from a stronger instability issue and does not necessarily have a better accuracy. The proposed SG continuum model with the eighth-order truncation is found to be enough to capture the dispersion behaviors all over the first Brillouin zone. (III) The effects of the nonlocal interactions are analyzed. The nonlocal interactions reduce the workable range of the well-known long-wave approximation, causing more local extrema in the dispersive diagrams. The present model can serve as a satisfactory continuum theory when the wavelength gets close to the lattice spacing, i.e., when the long-wave approximation is no longer valid. For the convenience of band gap designs, we have also provided the design space from which one can easily obtain the proper mass and stiffness ratios corresponding to a requested band gap width.


Author(s):  
José A. Carrillo ◽  
Young-Pil Choi

AbstractWe rigorously derive pressureless Euler-type equations with nonlocal dissipative terms in velocity and aggregation equations with nonlocal velocity fields from Newton-type particle descriptions of swarming models with alignment interactions. Crucially, we make use of a discrete version of a modulated kinetic energy together with the bounded Lipschitz distance for measures in order to control terms in its time derivative due to the nonlocal interactions.


2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Elmer Guardado-Sanchez ◽  
Benjamin M. Spar ◽  
Peter Schauss ◽  
Ron Belyansky ◽  
Jeremy T. Young ◽  
...  

Author(s):  
Dibya Phuyal ◽  
Soham Mukherjee ◽  
Swarup K. Panda ◽  
Gabriel J. Man ◽  
Konstantin Simonov ◽  
...  

2021 ◽  
Vol 82 (6) ◽  
Author(s):  
Shin-Ichiro Ei ◽  
Hiroshi Ishii ◽  
Makoto Sato ◽  
Yoshitaro Tanaka ◽  
Miaoxing Wang ◽  
...  

2021 ◽  
Vol 10 (2) ◽  
Author(s):  
Chao-Ming Jian ◽  
Yichen Xu ◽  
Xiao-Chuan Wu ◽  
Cenke Xu

One dimensional (1d) interacting systems with local Hamiltonians can be studied with various well-developed analytical methods. Recently novel 1d physics was found numerically in systems with either spatially nonlocal interactions, or at the 1d boundary of 2d quantum critical points, and the critical fluctuation in the bulk also yields effective nonlocal interactions at the boundary. This work studies the edge states at the 1d boundary of 2d strongly interacting symmetry protected topological (SPT) states, when the bulk is driven to a disorder-order phase transition. We will take the 2d Affleck-Kennedy-Lieb-Tasaki (AKLT) state as an example, which is a SPT state protected by the SO(3) spin symmetry and spatial translation. We found that the original (1+1)d boundary conformal field theory of the AKLT state is unstable due to coupling to the boundary avatar of the bulk quantum critical fluctuations. When the bulk is fixed at the quantum critical point, within the accuracy of our expansion method, we find that by tuning one parameter at the boundary, there is a generic direct transition between the long range antiferromagnetic Néel order and the valence bond solid (VBS) order. This transition is very similar to the Néel-VBS transition recently found in numerical simulation of a spin-1/2 chain with nonlocal spatial interactions. Connections between our analytical studies and recent numerical results concerning the edge states of the 2d AKLT-like state at a bulk quantum phase transition will also be discussed.


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