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Computing critical exponents in 3D Ising model via pattern recognition/deep learning approach

Timothy A. Burt

TL;DR

This study computed three critical exponents for the 3D Ising model with Metropolis Algorithm with Metropolis Algorithm using Finite-Size Scaling Analysis on six cube length scales, and performed a supervised Deep Learning approach (3D Convolutional Neural Network or CNN) to train a neural network on specific conformations of spin states.

Abstract

In this study, we computed three critical exponents ($α, β, γ$) for the 3D Ising model with Metropolis Algorithm using Finite-Size Scaling Analysis on six cube length scales (L=20,30,40,60,80,90), and performed a supervised Deep Learning (DL) approach (3D Convolutional Neural Network or CNN) to train a neural network on specific conformations of spin states. We find one can effectively reduce the information in thermodynamic ensemble-averaged quantities vs. reduced temperature t (magnetization per spin $<m>(t)$, specific heat per spin $<c>(t)$, magnetic susceptibility per spin $<χ>(t)$) to \textit{six} latent classes. We also demonstrate our CNN on a subset of L=20 conformations and achieve a train/test accuracy of 0.92 and 0.6875, respectively. However, more work remains to be done to quantify the feasibility of computing critical exponents from the output class labels (binned $m, c, χ$) from this approach and interpreting the results from DL models trained on systems in Condensed Matter Physics in general.

Computing critical exponents in 3D Ising model via pattern recognition/deep learning approach

TL;DR

This study computed three critical exponents for the 3D Ising model with Metropolis Algorithm with Metropolis Algorithm using Finite-Size Scaling Analysis on six cube length scales, and performed a supervised Deep Learning approach (3D Convolutional Neural Network or CNN) to train a neural network on specific conformations of spin states.

Abstract

In this study, we computed three critical exponents () for the 3D Ising model with Metropolis Algorithm using Finite-Size Scaling Analysis on six cube length scales (L=20,30,40,60,80,90), and performed a supervised Deep Learning (DL) approach (3D Convolutional Neural Network or CNN) to train a neural network on specific conformations of spin states. We find one can effectively reduce the information in thermodynamic ensemble-averaged quantities vs. reduced temperature t (magnetization per spin , specific heat per spin , magnetic susceptibility per spin ) to \textit{six} latent classes. We also demonstrate our CNN on a subset of L=20 conformations and achieve a train/test accuracy of 0.92 and 0.6875, respectively. However, more work remains to be done to quantify the feasibility of computing critical exponents from the output class labels (binned ) from this approach and interpreting the results from DL models trained on systems in Condensed Matter Physics in general.

Paper Structure

This paper contains 11 sections, 4 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Results/Discussion
  4. Simulation and finite-size scaling analysis
  5. Equilibration times
  6. <c>(T), <m>(T), χ(T) results
  7. Critical exponents α, β, γ results from FSSA
  8. Using Deep Learning on spin states to predict different phases of matter
  9. Class label definitions and interpretation
  10. DL implementation and results
  11. Conclusions

Figures (7)

  • Figure 1: Equilibration times for L=20 subset. Magnetization per spin $<\tau_m>$ left, energy per spin $<\tau_e>$ right. Two independent runs were concatenated together to give the ensemble mean/std. err.
  • Figure 2: From left to right -- Ensemble-averaged $<m>(T), <c>(T), <\chi>(T)$ for L=20. Error bars are $\pm\sigma_{\mu}$. A bootstrapping technique was used to generate fluctuation statistics $(c, \chi)$.
  • Figure 3: Top left to right -- Uncollapsed results for $c, m, \chi$ vs. reduced temperature $t\equiv \frac{T-T_C}{T_C}$ on L=20,40,80 set (L=30,60,90 not shown). Bottom left to right -- Corresponding data collapsed scaling functions for $\tilde{c}, \tilde{m}, \tilde{\chi}$ after FSSA was performed, using the determined critical exponents.
  • Figure 4: Top -- Combined results for $m, c, \chi$ vs. t for L=30,60,90. Bottom -- Box & whisker plot depicting each quantity split into each of the six classes and its resulting statistics.
  • Figure 5: Left -- Class 0 (Far $T_C$ paramagnetic phase) L=20 conformation example, before padding. Red voxels represent spin up (1), empty voxels represent spin down (-1). Right -- Training accuracy vs. epochs on the L=20 miniset.
  • ...and 2 more figures