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A Python program for calculating the metrics necessary to perform information-theory based symmetry classifications and quantifications of transmission electron diffraction spot patterns is introduced. It is the first of its kind, in that it implements objectivity into crystallographic symmetry classifications and quantifications of approximate zone axis patterns from crystals. The equations by which the program operates as well as the required inputs are given. The results of the program's analysis of an experimental transmission electron diffraction spot pattern from a crystal with a pseudo-hexagonal lattice metric and a rectangular-centered Bravias lattice is used as an example. The program will eventually be appended to allow analysis of the other hierarchical translational pseudo-symmetry and Bravais lattice type combinations. Crystallographic Rsym values of traditional classifications into projected point symmetry groups are provided alongside information-theoretic results of the new program's analysis for comparison purposes.
We consider a bosonic $s$ and $p$ orbital system in a face-centered cubic (FCC) optical lattice, and predict a fluctuation-induced instability towards the orbital analogue of Palmer-Chalker state, which is originally proposed in an electronic spin system. For bosons loaded in the FCC optical lattice, the single-particle spectrum has four degenerate band minima with their crystal momenta forming a tetrahedron in Brillouin zone. In the weakly interacting regime, the ensuing many-particle ground state, at the classical level, underlies a four-sublattice tetrahedral supercell of spontaneously generated $p$-orbital angular momenta through the Bravias-Bloch duality between real and momentum space, and is macroscopically degenerate originating from the geometric frustration. The fluctuations on top of the classical ground state lift its degeneracy and select the Palmer-Chalker ordering of $p$-orbital angular momenta as the quantum ground state through order-by-disorder mechanism. These findings raise the exciting possibility of simulating the Palmer-Chalker state with its orbital counterpart in ultracold atomic gases.
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