Combinatorial problems arising in puzzles, origami, and (meta)material design have rare sets of solutions, which define complex and sharply delineated boundaries in configuration space. These... Show moreCombinatorial problems arising in puzzles, origami, and (meta)material design have rare sets of solutions, which define complex and sharply delineated boundaries in configuration space. These boundaries are difficult to capture with conventional statistical and numerical methods. Here we show that convolutional neural networks can learn to recognize these boundaries for combinatorial mechanical metamaterials, down to finest detail, despite using heavily undersampled training sets, and can successfully generalize. This suggests that the network infers the underlying combinatorial rules from the sparse training set, opening up new possibilities for complex design of (meta)materials. Show less
We formulate a hydrodynamic theory of p-atic liquid crystals, namely, two-dimensional anisotropic fluids endowed with generic p-fold rotational symmetry. Our approach, based on an order parameter... Show moreWe formulate a hydrodynamic theory of p-atic liquid crystals, namely, two-dimensional anisotropic fluids endowed with generic p-fold rotational symmetry. Our approach, based on an order parameter tensor that directly embodies the discrete rotational symmetry of p-atic phases, allows us to unveil several unknown aspects of flowing p-atics, that previous theories, characterized by O(2) rotational symmetry, could not account for. This includes the onset of long-ranged orientational order in the presence of a simple shear flow of arbitrary shear rate, as opposed to the standard quasi-long-ranged order of two-dimensional liquid crystals, and the possibility of flow alignment at large shear rates. Show less
Chagnet, N.; Chapman, S.; Boer, J. de; Zukowski, C. 2022
We study circuit complexity for conformal field theory states in an arbitrary number of dimensions. Our circuits start from a primary state and move along a unitary representation of the Lorentzian... Show moreWe study circuit complexity for conformal field theory states in an arbitrary number of dimensions. Our circuits start from a primary state and move along a unitary representation of the Lorentzian conformal group. Different choices of distance functions can be understood in terms of the geometry of coadjoint orbits of the conformal group. We explicitly relate our circuits to timelike geodesics in anti-de Sitter space and the complexity metric to distances between these geodesics. We extend our method to circuits in other symmetry groups using a group theoretic generalization of the notion of coherent states. Show less
We study the phase behavior of a quasi-two-dimensional cholesteric liquid crystal shell. We characterize the topological phases arising close to the isotropic-cholesteric transition and show that... Show moreWe study the phase behavior of a quasi-two-dimensional cholesteric liquid crystal shell. We characterize the topological phases arising close to the isotropic-cholesteric transition and show that they differ in a fundamental way from those observed on a flat geometry. For spherical shells, we discover two types of quasi-two-dimensional topological phases: finite quasicrystals and amorphous structures, both made up of mixtures of polygonal tessellations of half-skynnions. These structures generically emerge instead of regular double twist lattices because of geometric frustration, which disallows a regular hexagonal tiling of curved space. For toroidal shells, the variations in the local curvature of the surface stabilizes heterogeneous phases where cholesteric patterns coexist with hexagonal lattices of half-skyrmions. Quasicrystals and amorphous and heterogeneous structures could be sought experimentally by self-assembling cholesteric shells on the surface of emulsion droplets. Show less
Reusch, S.; Stein, R.; Kowalski, M.; Velzen, S. van; Franckowiak, A.; Lunardini, C.; ... ; Zimmerman, E. 2022
