Spin Ice and Kagome Magnets on Every Stacking

Two families of magnets are built from corner-sharing triangles and tetrahedra: spin ice, whose magnetic atoms sit on the pyrochlore network of DICTO Janus c/h Pyrochlore (#60), and the kagome antiferromagnets Mn3X, whose manganese atoms form kagome nets. Both are close-packed layers, so both can stack cubic, hexagonal or any mixture. We count exactly what changes with the share h of hexagonal layers. On the pyrochlore network of any stacking, three quarters of the sites lie in kagome layers and one quarter in triangular layers; per tetrahedron the boat-shaped six-loops number 3h/2 and the chair-shaped 2 − 3h/2. Yet the closed loops of the network do not see the stacking at all: on five stackings the numbers of closed non-backtracking loops of every length from 6 to 34 bonds are identical, while the shells of neighbours differ by up to 5%. We call this the Loop Equality (computed, conjecture for all lengths). It gives a counting reason why the frozen-in entropy of spin ice and of water ice is the same for cubic and hexagonal stacking, as recent tensor-network work finds, and predicts the same value for every stacking, the disordered ice of clouds included. In the kagome magnets each manganese atom lies on exactly 1 + 3(1 − h) kagome planes: four in cubic Mn3Ir, kagome in all directions, one in hexagonal Mn3Sn, kagome in layers only. Theorems 1 and 2 are proved in the paper and checked from explicit coordinates; the Loop Equality is computed through length 34 and stated as a conjecture for all lengths; sources are listed. The ideas and names are DICTO's. Computation and proof with Claude (Anthropic). Sources checked are listed in the paper. The study (scripts and results) is attached. Licence CC BY-SA 4.0. More at dictospheres.com.