Searching magnetic states using an unsupervised machine learning algorithm with the Heisenberg model

Magnetism is a canonical example of a spontaneous symmetry breaking. The symmetry of a magnetic state below the Curie temperature is spontaneously broken even though the Hamiltonian is invariant under symmetry. Recently, machine learning algorithms have been successfully utilized to study topics in physics. We applied unsupervised machine learning algorithms to find the magnetic ground states of the Heisenberg model. A fully connected neural network was used to generate the spin configuration from randomly coded features, and the magnetic energy was selected as the cost to be minimized during the machine learning process. We found that ground states solved by the unsupervised learning process are consistent with the theoretical solution. Also, we compared the results with those from traditional computational methods and found that the machine learning algorithm provides an efficient method to solve the magnetic state numerically.

Figure 1

The process scheme of our algorithm containing a fully connected neural network layer. $n_b$ is the batch size, and $n$ is the size of the input vector. The square grid model was used with lateral sizes $L_x$ and $L_y$, and $m$ is $L_x\times L_y$. The color wheel in the spin configuration box indicates in-plane spin directions, and white-black is for the out-of-plane spin directions.

Figure 2

(a) The spin configurations generated by our algorithm. The three figures in a column are spin configurations generated by different inputs after ${10}^0,{10}^1,{10}^2,{10}^3,{10}^4$, and ${10}^5$ training iteration. (b)–(d) The spin configurations and their Fourier transformations after ${10}^5$ iterations with (b) $n=8$, (c) $n=16$, and (d) $n=64$.

Figure 3

(a) The log-log graph of the simulated energy ${\langle \varepsilon \rangle }_{n_b}$, (b) the graph of short and long order parameters calculated from the $n=32$ result, and (c) the log-log graphs for the magnitude of weight and bias. A dashed black line at $I_c$ divides the graphs into two stages, LFL and GFL. Inset figures in (a) are the representative spin configurations generated at iteration ${10}^0$, ${10}^1$, ${10}^2$, $3\times {10}^2$, $5\times {10}^2$, ${10}^3$, ${10}^4$, and ${10}^5$ with $n=32$.

Figure 4

The generated spin configuration ${\vec{\mathit{S}}}_{\mathrm{config}}$ and vector maps, which show the information contained in weight and bias, ${\mathbf{W}}_{\mathrm{maps}}$ and ${\mathbf{B}}_{\mathrm{map}}$. Iteration progressed from the top to the bottom row. The numbers of ${\mathbf{B}}_{\mathrm{map}}$ and ${\mathbf{W}}_{\mathrm{maps}}$ were 1 and $n$. We sampled eight weight vector maps among $n$ maps to briefly show the training process of $\mathbf{W}$.

Figure 5

(a) Log-log graph of the simulated energy values, ${\varepsilon }_{\mathrm{sim}}$, from the greedy method, simulated annealing, and our algorithm. The simulated annealing process was performed with different cooling speeds, fast cooling (MC-I) and slow cooling (MC-II). The inset graph shows the magnified region in the last 200 s, and the result of $n=32$ (yellow line) reached the lowest energy value. (b) Spin configurations generated from (i) greedy, (ii) MC-I, (iii) MC-II, and machine learning (iv) $n=$ 8, (v) 16, and (vi) 32 each after a 1-h simulation.

Figure 6

Simulated spin configurations of (a) a skyrmion lattice, (b) a magnetic vortex, (c) and two-length scale chiral structures generated in a bilayer system (top and bottom layer).