Deriving the order parameters of a spin-glass model using principal component analysis

We investigate the relationship between the order parameters of spin models and principal component analysis (PCA). PCA is applied to the spin configurations generated from the Boltzmann distribution for the cases of uniformly interacting and randomly interacting spin models, and the datasets obtained for various specific temperatures are analyzed. In the case of the uniformly interacting spin model, the first principal component is found to be equivalent to the magnetization in the ordered phase, which is the order parameter. For the randomly interacting spin model, we apply the analysis to datasets generated by the Sherrington-Kirkpatrick model. When PCA is performed under the assumption that the Hadamard product of the spin configuration is taken as a new dataset, it is found that the first principal component coincides with the spin-glass order parameter. By analytically treating the limit in which the number of datasets is infinite, it is shown that the first principal component agrees with the order parameter.

Figure 1

Phase diagram of the Sherrington-Kirkpatrick model [31, 32, 33]. The quantity $\beta$ represents the inverse temperature. Each phase is characterized by zero or positive values of $m$ and $q_{\mathrm{EA}}$. The red $+$ and the blue $\times$ signs indicate points where we perform PCA. A detailed explanation is given in the next section.

Figure 2

Weight vectors ${\mathit{w}}_1$ (left column) and ${\mathit{w}}_2$ (right column) indicated by density plots. The quantity $\beta J$ is fixed at 0.40, 0.45, and 0.50 in the first, second, and third rows, respectively. There is a phase-transition point between the first and the second row.

Figure 3

First principal component vs second principal component of the Ising model data at each temperature. The horizontal and vertical axes indicate the first and second principal components, respectively. Gray open circles, red squares, and blue triangles represent data points for $\beta J=0.40,0.45,$ and 0.50, respectively.

Figure 4

PCA results for dataset $\left\{{\mathit{S}}^{\left(\mu \right)}\right\}$ across the transition point between the paramagnetic and ferromagnetic phases. The subfigures in the left and middle columns show the weight vectors ${\mathit{w}}_1$ and ${\mathit{w}}_2$, respectively. The subfigures in the right column show the first principal component vs the second principal component. The quantity $\beta J$ is fixed at 0.6, 0.7, and 0.8 in the first, second, and third rows, respectively. The first, second, and third rows correspond to (a), (b), and (c) in Fig. 1, respectively. The transition between the paramagnetic and ferromagnetic phases occurs between the first and second rows.

Figure 5

PCA results for dataset $\left\{{\mathit{Q}}^{(\mu ,{\mu }^{\prime })}\right\}$ across the transition between the paramagnetic and ferromagnetic phases. The subfigures in the left and middle columns show the weight vectors ${\mathit{w}}_1^{\prime }$ and ${\mathit{w}}_2^{\prime }$, respectively. The subfigures in the right column show the first principal component vs the second principal component. The quantity $\beta J$ is fixed at 0.6, 0.7, and 0.8 in the first, second, and third rows, respectively. The first, second, and third rows correspond to (a), (b), and (c) in Fig. 1, respectively. The transition between the paramagnetic and ferromagnetic phases occurs between the first and second rows.

Figure 6

PCA results for dataset $\left\{{\mathit{S}}^{\left(\mu \right)}\right\}$ across the transition between the paramagnetic and spin-glass phases. The subfigures in the left column show the weight vector ${\mathit{w}}_1$. The subfigures in the right column show the first principal component vs the second principal component. The quantity $\beta J$ is fixed at 0.9, 1.1, and 1.2 in the first, second, and third rows, respectively. The first, second, and third rows correspond to (d), (e), and (f) in Fig. 1, respectively. The transition between the paramagnetic and spin-glass phases occurs between the first and second rows.

Figure 7

PCA results for dataset $\left\{{\mathit{Q}}^{(\mu ,{\mu }^{\prime })}\right\}$ across the transition between the paramagnetic and spin-glass phases. The subfigures in the left and middle columns show the weight vectors ${\mathit{w}}_1^{\prime }$ and ${\mathit{w}}_2^{\prime }$, respectively. The subfigures in the right column show the first principal component vs the second principal component. The quantity $\beta J$ is fixed at 0.9, 1.1, and 1.2 in the first, second, and third rows, respectively. The first, second, and third rows correspond to (d), (e), and (f) in Fig. 1, respectively. The transition between the paramagnetic and spin-glass phases occurs between the first and second rows.