Probing hidden spin order with interpretable machine learning

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.

Figure 1

The square root of the decision function, $\sqrt{d\left(\mathbf{x}\right)}$, trained at different ranks for the tetrahedral order. The true order parameter curve is shown for comparison, and $\mathrm{d}\left(\mathbf{x}\right)$ has been rescaled linearly, such that their endpoints match up. Insufficient tensor ranks do not result in a meaningful order parameter.

Figure 2

The coefficient matrix $C_{\mu \nu }$ for the tetrahedral order learnt using the rank-3 kernel. (a) Full $C_{\mu \nu }$ matrix, where the multi-indices $\mu ,\nu =({\alpha }_1,{\alpha }_2,{\alpha }_3;a_1,a_2,a_3)$ are lexicographically ordered. Each block is assigned coordinates $[{\alpha }_1{\alpha }_2{\alpha }_3;{\alpha }_1^{\prime }{\alpha }_2^{\prime }{\alpha }_3^{\prime }]$. Nontrivial blocks have mutually exclusive color indices. (b) Details of the $[\mathrm{lmn};\mathrm{lmn}]$ block, in comparison with (c) where trivial self-contractions have been removed.

Figure 3

Block structure of $C_{\mu \nu }$ for (a) the octahedral and (b) the dodecahedral order, learnt using the rank-4 kernel. Each pixel corresponds to a block of $C_{\mu \nu }$, identified by coordinates $[{\alpha }_1{\alpha }_2{\alpha }_3{\alpha }_4;{\alpha }_1^{\prime }{\alpha }_2^{\prime }{\alpha }_3^{\prime }{\alpha }_4^{\prime }]$. The value of each pixel is given by the squared Frobenius norm of the corresponding block.

Figure 4

Deviation $\delta$ for the tetrahedral order as a function of the number of training samples. The insets show excerpts of the coefficient tensor for selected points.

Figure 5

Deviation $\delta$ for the tetrahedral order against different discriminatory temperatures $T_{\mathrm{disc}}$ used to classify training samples.