Method for estimating spin-spin interactions from magnetization curves

We develop a method to estimate the spin-spin interactions in the Hamiltonian from the observed magnetization curve by machine learning based on Bayesian inference. In our method, plausible spin-spin interactions are determined by maximizing the posterior distribution, which is the conditional probability of the spin-spin interactions in the Hamiltonian for a given magnetization curve with observation noise. The conditional probability is obtained with the Markov chain Monte Carlo simulations combined with an exchange Monte Carlo method. The efficiency of our method is tested using synthetic magnetization curve data, and the results show that spin-spin interactions are estimated with a high accuracy. In particular, the relevant terms of the spin-spin interactions are successfully selected from the redundant interaction candidates by the $l_1$ regularization in the prior distribution.

Figure 1

Schematic research flows for forward modeling and Bayes modeling. $P\left(A\right)$ is the probability distribution of $A$, and $P\left(A\right|B)$ is the conditional probability of $A$ given $B$.

Figure 2

Cross validation for $S=4$. We evaluate ${\mathrm{\Delta }}^{\left(s\right)}\left(\lambda \right)$ in each case. The average of ${\mathrm{\Delta }}^{\left(s\right)}\left(\lambda \right)$ over the four cases is used to estimate ${\lambda }^{*}$.

Figure 3

Lattice structure and types of spin-spin interactions in the model Hamiltonian used to generate the synthetic magnetization curve. Numbers indicate the types of lattice points.

Figure 4

(a) Magnetization curve ${\left\{m(H_l,{\mathbf{x}}^{\prime })\right\}}_{l\in D}$ for $L=160$ at the zero temperature in the theoretical model defined by Eq. (21) with model parameter ${\mathbf{x}}^{\prime }=(1,4,5,6,0,0,0,0.1)$. (b) Observed magnetization curve ${\left\{m^{\mathrm{ex}}\left(H_l\right)\right\}}_{l\in D}$ represented by ${\left\{m(H_l,{\mathbf{x}}^{\prime })\right\}}_{l\in D}$ with added Gaussian noise of the mean zero and standard deviation of 0.004.

Figure 5

The $\lambda$ dependence of estimated spin-spin interactions $J_k$ and $b$ using the prior distributions of (a) type I, (b) type II, and (c) type III, respectively. The results are the averages over the four sets of training data. Dashed lines indicate the input spin-spin interactions $J_k$ and $b$. The shaded area in (c) indicates that the estimated spin-spin interactions are roughly the same as the values used in the inputted magnetization curve.

Figure 6

The $\lambda$ dependence of the prediction error ${\mathrm{\Delta }}_{\mathrm{av}}\left(\lambda \right)$ obtained by cross validation for each type of prior distribution. ${\mathrm{\Delta }}_{\mathrm{av}}\left(\lambda \right)$ takes a minimum for $\lambda =0.004$ in type III.

Figure 7

Estimated magnetization curve ${\left\{m(H_l,{\mathbf{x}}^{*})\right\}}_{l\in D}$ for $L=160$ when $\lambda =0.004$ in type III and inputted observed magnetization curve ${\left\{m^{\mathrm{ex}}\left(H_l\right)\right\}}_{l\in D}$. Table 1 shows the estimated spin-spin interactions ${\mathbf{x}}^{*}$.