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Multipole symmetries are of interest in multiple contexts, from the study offracton phases, to nonergodic quantum dynamics, to the exploration of newhydrodynamic universality classes. However, prior explorations have focused oncontinuum systems or hypercubic lattices. In this work, we systematicallyexplore multipole symmetries on arbitrary crystal lattices. We explain how,given a crystal structure (specified by a space group and the occupied Wyckoffpositions), one may systematically construct all consistent multipole groups.We focus on two-dimensional crystal structures for simplicity, although ourmethods are general and extend straightforwardly to three dimensions. Weclassify the possible multipole groups on all two-dimensional Bravais lattices,and on the kagome and breathing kagome crystal structures to illustrate theprocedure on general crystal lattices. Using Wyckoff positions, we provide anin-principle classification of all possible multipole groups in any spacegroup. We explain how, given a valid multipole group, one may construct aneffective Hamiltonian and a low-energy field theory. We then explore thephysical consequences, beginning by generalizing certain results originallyobtained on hypercubic lattices to arbitrary crystal structures. Next, weidentify two seemingly novel phenomena, including an emergent, robust subsystemsymmetry on the triangular lattice, and an exact multipolar symmetry on thebreathing kagome lattice that does not include conservation of charge(monopole), but instead conserves a vector charge. This makes clear that thereis new physics to be found by exploring the consequences of multipolarsymmetries on arbitrary lattices, and this work provides the map for theexploration thereof, as well as guiding the search for emergent multipolarsymmetries and the attendant exotic phenomena in real materials based onnonhypercubic lattices.

Multipole groups and fracton phenomena on arbitrary crystalline lattices