Probing hidden spin order with interpretable machine learning

Abstract

The search of unconventional magnetic and nonmagnetic states is a major topic in the study of frustrated magnetism. Canonical examples of those states include various spin liquids and spin nematics. However, discerning their existence and the correct characterization is usually challenging. Here we introduce a machine-learning protocol that can identify general nematic order and their order parameter from seemingly featureless spin configurations, thus providing comprehensive insight on the presence or absence of hidden orders. We demonstrate the capabilities of our method by extracting the analytical form of nematic order parameter tensors up to rank 6. This may prove useful in the search for novel spin states and for ruling out spurious spin liquid candidates.The statistical learning of phases is nowadays an active field of research [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Despite the enormous recent progress, learning or classifying intricate phases in manybody systems remains a daunting task. Many recent algorithmic advances are tried and tested in only the simplest of models, and their applicability to more complex situations remains an open question. The ability to interpret results to gain physical insight has been identified as one of the key challenges in the application of machine learning techniques to the domain of physics. Still, recent approaches struggle and this is only exacerbated when going beyond those simple models. However, those situations can also be arenas for machine learning methods to demonstrate their features and prove their worth, in comparison to-or complementary to-traditional methods.One such arena may be found in frustrated spin and spin-orbital-coupled systems [18]. These systems have rich phase diagrams, supporting various spin nematic (multipolar ordered) [19][20][21][22][23][24][25][26][27][28] and spin liquid phases [29][30][31][32][33][34][35]. However, to distinguish these two types of phases is often tricky, since both of them are invisible to conventional magnetic measurements. Indeed, there have been steady reports of "hidden" multipolar orders from a magnetically disordering state [36][37][38][39][40][41][42][43][44][45][46][47][48]. Moreover, identifying the right characterization of a spin-nematic order can also be a nontrivial task. For instance, in the low temperature phase of the classical Heisenberg-Kagomé antiferromagnet, a hidden quadrupolar order was found first [36], followed by the realization of an additional octupolar order [37] and its optimal order parameter [39,40].The aforementioned multipolar orders are only the simplest ones admitted by the subgroup structure of O(3). There are indeed myriads of more complicated multipolar orders where even the abstract classification of their order parameters has only been accomplished two years ago [49][50][51]. Along with the diverse interactions and lattice geometries in frustrated systems, identifying or ruling out certain orders becomes a difficult task for traditional methods, as there is n...

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