Interpretable and unsupervised phase classification

Fully automated classification methods that provide direct physical insights into phase diagrams are of current interest. Interpretable, i.e., fully explainable, methods are desired for which we understand why they yield a given phase classification. Ideally, phase classification methods should also be unsupervised. That is, they should not require prior labeling or knowledge of the phases of matter to be characterized. Here, we demonstrate an unsupervised machine-learning method for phase classification, which is rendered interpretable via an analytical derivation of the functional relationship between its optimal predictions and the input data. Based on these findings, we propose and apply an alternative, physically-motivated, data-driven scheme, which relies on the difference between mean input features. This mean-based method does not rely on any predictive model and is thus computationally cheap and directly explainable. As an example, we consider the physically rich ground-state phase diagram of the spinless Falicov-Kimball model.

Figure 1

Our workflow to predict a phase diagram with indicators $I$ for phase transitions. Here, we illustrate the procedure for a two-dimensional parameter space. The parameter space is sampled on a grid, which yields a set of points $\left\{{\mathit{p}}_i\right\}$ of fixed system parameters. At each such point ${\mathit{p}}_i$ a set of samples $\left\{{\mathit{S}}_i\right\}$ is generated. Based on these samples, a scalar indicator for phase transitions $I\left({\mathit{p}}_i\right)$ is calculated. This indicator highlights the boundaries (red) between phases (grey). Different unsupervised phase classification schemes are established via different indicators.

Figure 2

(a) Sketch of the ground-state phase diagram of the spinless FKM. Red-dashed lines highlight the boundaries of the phases with ($\underline{1}$) segregated, ($\underline{2}$) diagonal, and ($\underline{3}$) axial orderings. For each ordering ($\underline{1}$)–($\underline{3}$), an example of a typical ground-state configuration ${\mathit{w}}_0$ ($L=20$) is shown on top. Here, the absence ($w_{0,i}=0$) and presence ($w_{0,i}=1$) of an $f$ particle at lattice site $i$ is denoted by a white or black square, respectively. [(b), (c)] ${\nabla }_{\mathit{p}}·\mathit{\delta }\mathit{p}$ [Eq. (4)] based on the predictions of a DNN trained using $\left|\mathcal{F}\right|$ as input in the (b) noise-free and (c) noisy case. The color scale in (b) and (c) was cut off at –1 and –2, respectively, for better visualization. There were very few distinct points in parameter space with a divergence signal ${\nabla }_{\mathit{p}}·\mathit{\delta }\mathit{p}$ below these cut-off values. (d) Illustration of the correlation functions that measure square (${\kappa }_n^{\mathrm{sq}}$), axial (${\kappa }_n^{\mathrm{ax}}$), and diagonal (${\kappa }_n^{\mathrm{di}}$) correlation at a distance $n$ from the origin [cf. Eq. (14)]. Blue squares denote the lattice sites marked by the corresponding stencil, where red denotes the origin. [(e), (f)] Correlation indicator $\mathrm{\Delta }\overline{\kappa }$ [Eq. (15)] in the (e) noise-free and (f) noisy case. Both ${\nabla }_{\mathit{p}}·\mathit{\delta }\mathit{p}$ and $\mathrm{\Delta }\overline{\kappa }$ serve as indicators for phase transitions (Fig. 1). The dashed line in (c) and (f) marks the cut along $\rho =63/400\approx 0.16$ analyzed in Fig. 3. Representative configurations ($\underline{1}$)–($\underline{9}$) for some of the largest predicted regions of stability (connected regions marked in blue by the indicator), i.e., phases, are shown on the top. These regions connect configurations of the same character. We checked this manually and selected a representative example configuration for some of the regions as a guidance for the reader. This post-process labeling is necessary as the system's phases are, in principle, not known beforehand. A finite-size scaling of the FKM phase diagram with square lattices of linear size $L=10$ and $L=16$ is provided in Appendix pp5.

Figure 3

Analysis of the transition from nonsegregated to segregated orderings occurring at $U\approx 2$ along the line scan [dashed line in Figs. 2 and 2] from $U_{\min }=1$ to $U_{\max }=8$ at fixed $\rho =63/400\approx 0.16$. [(a), (d)] Predictions $\widehat{U}$ and corresponding divergence $\partial \delta U/\partial U$ of a DNN trained on the line scan with $\left|\mathcal{F}\right|$ as input, as well as the indicator $\mathrm{\Delta }\overline{x}$ [Eq. (16)] based on (b), (e) the entire set of correlation functions $\mathit{\kappa }$ (points) or a subset thereof denoted as ${\mathit{\kappa }}_{1,2-6,7-10}$ (dashed lines), and (c), (f) $\left|\mathcal{F}\right|$. Here, ${\mathit{\kappa }}_{1,2-6,7-10}$ denotes the set of correlation functions measuring square (sq), axial (ax), and diagonal (di) correlations over 1, 2–6, and 7–10 lattice sites, respectively. $\langle \mathrm{\Delta }\overline{x}\rangle$ denotes the average difference signal over the entire line scan and is subtracted to account for noise arising due to finite sample statistics. The inset in panel (b) shows the average input $\mathit{\kappa }$ for the nonsegregated phase (I) and segregated phase (II) obtained from averaging over all inputs for $U$ ranging from $U=1.0$ to $U=1.2$ and $U=2.6$ to $U=8.0$, respectively. The results shown in panels (c) and (f) are obtained using $n_{\mathrm{trafo}}=20$ for offline data augmentation at which we find the indicator to converge. The degree of red in (a)–(c) denotes an increasingly positive value of the respective indicator for phase transitions (Fig. 1); (d)–(f) configurations visualized using the same color scale as for the points in (a)–(c), respectively (see Fig. 5 in Appendix pp1 for all configurations).

Figure 4

Indicator $\mathrm{\Delta }\overline{x}$ [Eq. (16)] of the mean-based method based on $\left|\mathcal{F}\right|$ as input on the two-dimensional parameter space of the FKM in the (a) noise-free and (b) noisy case. For offline data augmentation we use $n_{\mathrm{trafo}}=20$ at which we find the indicator to converge.

Figure 5

Ground-state configurations ($L=20$) along the line scan from $U_{\min }=1$ to $U_{\max }=8$ at fixed $\rho =63/400\approx 0.16$ [cf. dashed line in Figs. 2 and 2 in the main text]. At each value of $U$, we show the 10 configurations arising from independent simulated annealing runs that are considered in the noisy case.

Figure 6

Vector field $\mathit{\delta }\mathit{p}=\widehat{\mathit{p}}-\mathit{p}$ obtained using a DNN trained with $\left|\mathcal{F}\right|$ as input for the FKM phase diagram in (a) the noise-free and (b) noisy case [see Figs. 2 and 2 in the main text for the vector-field divergence signal, respectively].

Figure 7

Vector-field divergence ${\nabla }_{\mathit{p}}·\mathit{\delta }\mathit{p}$ as a function of $U$ at at fixed $\rho =63/400\approx 0.16$ obtained analytically based on Eqs. (12) and (13), as well as numerically using a DNN trained with $\left|\mathcal{F}\right|$ as input for the two-dimensional ground-state phase diagram in the noise-free case [see Fig. 2 in the main text for full phase diagram].

Figure 8

Representative filters for computing the order parameter for diagonal ($F_{\mathrm{di}}$) or axial ($F_{\mathrm{ax}}$) orderings. Here, black denotes 1 and white denotes –1.

Figure 9

Values of order parameters [(a), (d)] $O_{\mathrm{sg}}$, [(b), (e)] $O_{\mathrm{ax}}$, and [(c), (f)] $O_{\mathrm{di}}$ on the two-dimensional parameter space of the FKM in the (a)–(c) noise-free and (d)–(f) noisy case.

Figure 10

Indicator $\mathrm{\Delta }\overline{x}$ [Eq. (16) in the main text] of the mean-based method based on $\mathit{\kappa }$ as input on the two-dimensional parameter space of the FKM in the noise-free case for two-dimensional square lattices of linear size (a) $L=10$, (b) $L=16$, and (c) $L=20$. The density of heavy particles $\rho$ ranges from $1/L^2$ to half-filling $(\mathrm{\Delta }\rho =1/L^2)$ and $U$ ranges from 1 to $8(\mathrm{\Delta }U=0.2)$. This allows for an assessment of finite-size effects on the FKM phase diagram. In particular, we observe that the phase diagram is converging with increasing lattice size but still displays some finite-size effects at $L=20$.

Figure 11

Parameter-generic indicator $\mathrm{\Delta }{\overline{x}}_{\mathrm{tot}}$ [Eq. (17) in the main text] of the mean-based method based on $\mathit{\kappa }$ as input on the two-dimensional parameter space of the FKM in the (a) noise-free and (b) noisy case.