The view of TK-SVM on the phase hierarchy in the classical kagome Heisenberg antiferromagnet

We illustrate how the tensorial kernel support vector machine (TK-SVM) can probe the hidden multipolar orders and emergent local constraint in the classical kagome Heisenberg antiferromagnet. We show that TK-SVM learns the finite-temperature phase diagram in an unsupervised way. Moreover, in virtue of its strong interpretability, it identifies the tensorial quadrupolar and octupolar orders, which define a biaxial $D_{3h}$ spin nematic, and the local constraint that underlies the selection of coplanar states. We then discuss the disorder hierarchy of the phases, which can be inferred from both the analytical order parameters and a SVM bias parameter. For completeness we mention that the machine also picks up the leading $\sqrt{3} \times \sqrt{3}$ correlations in the dipolar channel at very low temperature, which are however weak compared to the quadrupolar and octupolar orders. Our work shows how TK-SVM can facilitate and speed up the analysis of classical frustrated magnets.
Comment: 13 pages, 10 figures, 3 tables; invited for JPCM Special Issue on machine learning; v2: added discussions on dipolar orders