Non-parametric learning critical behavior in Ising partition functions: PCA entropy and intrinsic dimension

TL;DR

The paper tackles how to extract critical behavior from data-driven, non-parametric analyses of partition-function configurations in Ising models. It introduces two complementary tools: intrinsic dimension estimators (TWO-NN and PCA-based) and PCA entropy derived from the covariance spectrum. The key findings show that 3D volume complicates intrinsic-dimension estimation, but PCA entropy tracks the thermodynamic entropy and enables Tc extraction with <1% error via finite-size scaling (Tc^{2D}=2.266, Tc^{3D}=4.518}). This approach is computationally efficient and broadly applicable to many-body systems, including potential extensions to quantum path integrals and experimental data.

Abstract

We provide and critically analyze a framework to learn critical behavior in classical partition functions through the application of non-parametric methods to data sets of thermal configurations. We illustrate our approach in phase transitions in 2D and 3D Ising models. First, we extend previous studies on the intrinsic dimension of 2D partition function data sets, by exploring the effect of volume in 3D Ising data. We find that as opposed to 2D systems for which this quantity has been successfully used in unsupervised characterizations of critical phenomena, in the 3D case its estimation is far more challenging. To circumvent this limitation, we then use the principal component analysis (PCA) entropy, a "Shannon entropy" of the normalized spectrum of the covariance matrix. We find a striking qualitative similarity to the thermodynamic entropy, which the PCA entropy approaches asymptotically. The latter allows us to extract -- through a conventional finite-size scaling analysis with modest lattice sizes -- the critical temperature with less than $1\%$ error for both 2D and 3D models while being computationally efficient. The PCA entropy can readily be applied to characterize correlations and critical phenomena in a huge variety of many-body problems and suggests a (direct) link between easy-to-compute quantities and entropies.

Figures (10)

• Figure 1: TWO-NN $I_d$ estimation for the 3D Ising model. (a) Empirical cumulative distribution of data for $L=20$ and temperature $4.30$ and $4.70$, where the dashed line shows the linear fit used to estimate $I_d$. With the scale used in this plot, the linear fit above corresponds to Pareto distribution. (b) $I_{d}$ as the function of $T$ for different $L$. While we expect the $I_d$ to increase at high temperatures, and to drop as $T\to 0$, close to the transition point, this quantity features a local minimum, which becomes more apparent as the system size is increased.
• Figure 2: PCA-based $I_d$ estimation for 3D Ising model. (a) $I_{d}^{PCA}$ as a function of $T$ for different system sizes with cutoff $\epsilon = 0.10$. For such an ad hoc cutoff, $I_{d}^{PCA}$ abruptly drops to 1 below $T_c \approx 4.51$, while it rises above the transition. (b) $I_{d}^{PCA}$ for $L=32$, with varying cutoff $\epsilon$ [see Eq. \ref{['eq:Id_pca']}]. For sufficiently large values of $\epsilon$, $I_{d}^{PCA}$ does not drop to 1 just below the transition point. However, a signature of the transition can be observed as a visible change in the slope around $T_c \approx 4.51$.
• Figure 3: Comparison of $S_{PCA}$, for different sample sizes $N_r$, with the exact thermodynamic entropy per spin of the 2D Ising model with $L=48$ as a function of temperature. Both entropies have been normalized such that their maximum possible value is 1 [see Eq. \ref{['pca_entropy']}].
• Figure 4: $S_{PCA}$ as a function of temperature for different system sizes $L=32-80$, for 2D Ising model. These plots exhibit a clear crossing point in the vicinity of the transition point, suggestive of a finite-size scaling analysis.
• Figure 5: (a) Plot of $\delta S_{PCA}/\delta T$ as a function of temperature for the 2D Ising model. The location of the flex in $S_{PCA}$ is revealed by the peak in its derivative, occurring at $T^{\ast}(L)$. Solid lines show a smoothing curve of the data obtained via a standard smoothing spline function. (b) Linear finite-size scaling of the temperature where we get the maxima $T^{\ast}(L)$. This linear fit yields $T_c^{2D} = 2.266 \pm 0.061$.