Persistent homology as an order parameter for classical spins

Classical spin models can feature rich phase diagrams that are difficult to characterize without knowing the order parameter. Phases can have strong correlations while lacking magnetic ordering, such as e.g. nematic ordering which breaks spin rotational symmetry. We show that persistent homology – a relatively new field in applied mathematics – can be used as a universal framework to construct a phase diagram [1]. Finally, we demonstrate this new approach using the XXZ model on the pyrochlore lattice and automatically identify all six phases, including hidden phases such as the spin-ice phase.
[1] Olsthoorn B., Hellsvik J., Balatsky A. V. (2020). Finding hidden order in spin models with persistent homology–arXiv:2009.05141 (accepted in PRResearch, appearing soon).