Extracting critical exponents by finite-size scaling with convolutional neural networks

Machine learning has been successfully applied to identify phases and phase transitions in condensed matter systems. However, quantitative characterization of the critical fluctuations near phase transitions is lacking. In this paper, we propose a finite-size scaling approach based on a convolutional neural network and analyze the critical behavior of a quantum Hall plateau transition. The localization length critical exponent learned by the neural network is consistent with the value obtained by conventional approaches. We show that the general-purposed method can be used to extract critical exponents in models with drastically different physics and input data, such as the two-dimensional Ising model and four-state Potts model.

Figure 1

Illustration of the perceptron model for calculating IPR and the scheme to detect the characteristic energy scale and its physics motivation. (a) The IPR can be expressed as a perceptron whose inputs are the square of the local electron density and whose weights $W_i\mathrm{s}$ are all set to be unity. A step-function activation can be introduced to roughly distinguish the localized states from the conducting ones. (b) We input labels for the eigenstates within $-E_m<E<E_m$ as conducting states and vary $E_m$ to observe the performance of the CNN. (c) The labeling scheme is motivated by the fact that the localization length $\xi \sim |E-E_c{|}^{-\nu }$ as energy $E$ approaches the critical point $E_c$. The states with $\xi$ greater than the system size $L$ are conducting; otherwise, they are localized.

Figure 2

CNN output $y_1$ of 2500 test states as a function of energy $E$ for (a) $E_m=0.065$, (b) $E_m=0.125$, and (c) $E_m=0.205$. The color scheme is such that the CNN correctly identifies the green points as conducting states and purple points as localized states, but misidentifies the red points. The horizontal dashed lines in (b) and (c) indicate $y_1=y_2$. (d) The scatter plot of the IPR for $E_m=0.205$. The color of a point is green if the state is classified as a conducting state with over 80% probability, purple as a localized state with over 80% probability, or red if neither is true. The horizontal line is a guide to eye as the separation of the conducting and localized states according to the CNN output. In all panels, the vertical lines mark $E_m$.

Figure 3

The characteristic V-shape of the performance curve $P\left(E_m\right)$. As $E_m$ increases, conducting states identified by the CNN emerge and, eventually, occupy the whole band, as indicated by the varying density of conducting (localized) states in the insets.

Figure 4

The performance curves for systems of size $L=9$-24. The minima of the curves can be extrapolated to $P\left(E_m\right)=1$ at $E_m=0$ in the thermodynamic limit. The inset shows that the locations of the minima for various $L$ can be fit by a power law $E_m^{\min }\left(L\right)\sim L^{-1/\nu }$ with exponent $\nu =2.22\pm 0.04$.

Figure 5

The performance curves for (a) the 2D Ising model and (c) the four-state Potts model on a square lattice. The minima of the curves $T_m^{\min }\left(L\right)-T_c\sim L^{-1/\nu }$ can be used to extract critical exponent (b) $\nu =0.985\pm 0.002$ in the Ising model and (d) $\nu =0.69\pm 0.01$ in the Potts model.

Figure 6

Architecture of the CNN model used in the present paper.

Figure 7

The performance curve $P\left(E_m\right)$ and its low-energy approximation $P_{<}\left(E_m\right)$ and high-energy approximation $P_{>}\left(E_m\right)$ for systems of size L = 12. The minima of $P\left(E_m\right)$ can be estimated by letting $P_{<}\left(E_m\right)=P_{>}\left(E_m\right)$. The shape factor $\zeta =0.7$ is used for calculating $P_{>}\left(E_m\right)$. In the inset, we illustrate the fluctuations of the CNN output as a function of energy. The input label of a state is determined by its energy while the classification of the state is by the CNN output $y_1$ (and $y_2)$. For states with fluctuating features, the classification cannot agree with the input labels in the red regions, leading to an imperfect performance.

Figure 8

(a) Shifted mean $y(E;E_m)$ and (b) its variance $\delta y(E;E_m)$ of the CNN output $y_1$ for $E_m=0.105$, 0.225, and 0.345. The lattice size is $L=12$.

Figure 9

The performance curve obtained by different CNN network layers for the Hofstadter model.

Figure 10

Comparison of four commonly used activation functions: ReLU, Leaky ReLU, ELU, and sigmoid functions.

Figure 11

The performance curves obtained with different nonlinear activation functions for the Hofstadter lattice model. The inset amplifies the curves near their minima.

Figure 12

The performance curves obtained with different numbers of filters for the Hofstadter model. The inset amplifies the curves near their minima.