Learning multiple order parameters with interpretable machines

Machine-learning techniques are evolving into a subsidiary tool for studying phase transitions in many-body systems. However, most studies are tied to situations involving only one phase transition and one order parameter. Systems that accommodate multiple phases of coexisting and competing orders, which are common in condensed matter physics, remain largely unexplored from a machine-learning perspective. In this paper, we investigate multiclassification of phases using support vector machines (SVMs) and apply a recently introduced kernel method for detecting hidden spin and orbital orders to learn multiple phases and their analytical order parameters. Our focus is on multipolar orders and their tensorial order parameters whose identification is difficult with traditional methods. The importance of interpretability is emphasized for physical applications of multiclassification. Furthermore, we discuss an intrinsic parameter of SVM, the bias, which allows for a special interpretation in the classification of phases, and its utility in diagnosing the existence of phase transitions. We show that it can be exploited as an efficient way to explore the topology of unknown phase diagrams where the supervision is entirely delegated to the machine.

Figure 1

Coefficient matrix $C_{\mu \nu }=C_{ab,a^{\prime }{b}^{\prime }}$ for the quadrupolar order with the rank-2 kernel. The pattern can be decomposed according to Eq. (30) and leads to the uniaxial nematic tensor $Q_{ab}$. 28K samples have been used for training.

Figure 2

Graphical representation of different training schemes. Samples are taken from temperatures corresponding to the shaded regions and labeled as “ordered” (blue) or “disordered” (red). The first three schemes use a continuous distribution of temperatures from $T_A\ll T_c$ to $T_B=\infty$ and differ by the discriminatory temperature $T_{\mathrm{disc}}$ used to assign the labels, where $T_{\mathrm{disc}}=T_c=0.52$ (a), $T_{\mathrm{disc}}=0.4<T_c$ (b), and $T_{\mathrm{disc}}=0.64>T_c$ (c) have been chosen, respectively. The last three schemes take samples at two discrete temperatures. Panels (a)–(d) are discussed in Sec. 5b, and (e) and (f) in Sec. 5c. The concrete temperatures read as $T_A=0.2,T_B=\infty ,T_D=0.25,T_E=2$.

Figure 3

The decision function, obtained from the earlier training with regularization $\nu =0.1$, is measured as an observable in Monte Carlo simulations at temperatures spanning the phase transition. The left panel shows the numerical values of the decision functions as obtained from SVM. When training the SVM with the rank-1 kernel (shown in gray), the quadrupolar order cannot be captured, resulting in an erratic decision function. The curves (a), (b), and (d) are trained with the rank-2 kernel and correspond to the eponymous training schemes (cf. Fig. 2). The zero crossing of the decision functions marks the decision boundary, whereas the constant value for $T>T_c$ is given by $-\rho$ (cf. Table 1). The right panel shows the same curves, but shifted up by $\rho$, and are seen to match the square of the uniaxial nematicity after being rescaled.

Figure 4

The phase diagram for the generalized quadrupolar order (cf. Ref. [45]). The points labeled $⓪$ through $③$ are used to generate training samples.

Figure 5

The coefficient matrix distinguishing the isotropic $\left[\text{O}\right(3\left)\right]$ and uniaxial ($D_{\infty h}$) phase of the generalized quadrupolar order. The left panel displays the full matrix $C_{\mu \nu }$ where the multi-indices $\mu$ and $\nu$ are sorted lexicographically and where the color indices are more significant. The resulting block structure is shown in the right panel. The block weights have been obtained by summing all matrix elements within each block over their component indices and normalizing to the top-right $\left(\mathrm{nn},\mathrm{nn}\right)$ block. The interpretation leads to an alternative expression of the nematic $Q_{ab}$ tensor, given in Eq. (34). The pattern exhibited within each block is identical to Fig. 1.

Figure 6

The block structures of the $C_{\mu \nu }$ matrix of the remaining two decision functions of the generalized quadrupolar order, (a) the uniaxial ($D_{\infty h}$) and biaxial ($D_{2h}$) phases, and (b) the isotropic $\left[\text{O}\right(3\left)\right]$ and biaxial ($D_{2h}$) phases. The latter is a superposition of the patterns exhibited by the other two decision functions, shown in Figs. 5 and 6. The block weights are not fixed, but rather the coefficients $p_1$ and $p_2$ vary with the relative strength of the two orderings.

Figure 7

The block structure of the $C_{\mu \nu }$ matrix learned by the rank-3 kernel, distinguishing the octupolar phase ($D_{3h}$) from the isotropic $\left[\text{O}\right(3\left)\right]$ and the uniaxial phase ($D_{\infty h}$). The color indices are sorted lexicographically and those of the nonvanishing blocks are explicitly given. The interpretation of this pattern gives the rank-3 octupolar ordering tensor in Eq. (37).

Figure 8

Internal structure of the $\left(\mathrm{lll},\mathrm{lll}\right)$ block of the coefficient matrix in Fig. 7. The axes are labeled by lexicographically sorted triplets of component indices ($xxx,xxy$, ...). This pattern corresponds to the expression (38) and in turn gives the one-spin form of the octupolar ordering tensor in Eq. (40).

Figure 9

Graph representation of the relation between points on a $10\times 10$ grid as inferred from the bias $\rho$ of the corresponding SVM classifiers ($\nu =0.1$). Any two vertices are connected by an edge if $\rho >{\rho }_c=2.5$. The choice of the optimal value of ${\rho }_c$ is illustrated in the text. The phase diagram which has been found based on the peak susceptibilities (cf. Fig. 4) is shown in red for reference. The phase boundaries are intersected by only few graph edges and this happens mostly where grid points are very close to the phase transition.

Figure 10

Upper panel: elements of the Fiedler vector corresponding to the graph obtained for a $23\times 23$ grid at a threshold value of ${\rho }_c=12$. Each element is rendered as a pixel at the location of its corresponding grid point in parameter space and color coded according to its value. The correct phase diagram (cf. Fig. 4) is shown in white for reference. The SVM result shows ambiguity when very close to phase boundaries, however, this tentative phase diagram suffices to guide subsequent trainings to extract order parameters. Lower panel: a histogram of the elements' values is shown next to the color scale used to encode them in the upper panel.