Learning Degenerate Manifolds of Frustrated Magnets with Boltzmann Machines

TL;DR

This work shows that Restricted Boltzmann Machines can serve as effective generative tools for learning the statistical structure of frustrated classical magnets, including highly degenerate ground-state manifolds and emergent constraint-driven correlations. By benchmarking on the 1D ANNNI model at its multiphase point and on kagome spin ice, the authors demonstrate that RBMs can reproduce nontrivial correlation functions and symmetry-breaking phenomena, provided appropriate bias fields are used to capture emergent orders. The study highlights the RBM’s capacity to internalize constraint hierarchies and to provide compact probabilistic representations of complex spin ensembles, offering a complementary approach to traditional Monte Carlo sampling for exploring frustrated systems. These results open avenues for applying generative ML models to broader classes of spin liquids, Coulomb phases, and gauge-constrained materials, with potential extensions to deeper architectures and larger systems.

Abstract

We show that Restricted Boltzmann Machines (RBMs) provide a flexible generative framework for modeling spin configurations in disordered yet strongly correlated phases of frustrated magnets. As a benchmark, we first demonstrate that an RBM can learn the zero-temperature ground-state manifold of the one-dimensional ANNNI model at its multiphase point, accurately reproducing its characteristic oscillatory and exponentially decaying correlations. We then apply RBMs to kagome spin ice and show that they successfully learn the local ice rules and short-range correlations of the extensively degenerate ice-I manifold. Correlation functions computed from RBM-generated configurations closely match those from direct Monte Carlo simulations. For the partially ordered ice-II phase -- featuring long-range charge order and broken time-reversal symmetry -- accurate modeling requires RBMs with uniform-sign bias fields, mirroring the underlying symmetry breaking. These results highlight the utility of RBMs as generative models for learning constrained and highly frustrated magnetic states.

Figures (10)

• Figure 1: Schematic diagram of a restricted Boltzmann machine (RBM) in an Ising-spin representation. The visible spins $\bm{\sigma}$ (green) are coupled to the hidden spins $\bm{\tau}$ (yellow) through the weight matrix $\bm W$. The local fields $\mathbf{b}$ and $\mathbf{b}'$ appearing in the RBM Hamiltonian are indicated by external boxes connected to the visible and hidden layers, respectively.
• Figure 2: (a) Competing interactions in the 1D ANNNI chain: ferromagnetic nearest-neighbor bonds $J_1$ and antiferromagnetic next-nearest-neighbor bonds $J_2$. (b) Ferromagnetic ground state for $\kappa = J_2 / J_1 < 1/2$, and (c) the anti-phase $\langle 2, 2\rangle$ state for $\kappa > 1/2$.
• Figure 3: Plot of $\varepsilon_W$ during training of the ice-II phase RBM. Early oscillatory behavior indicates rapid adjustment of model parameters, while decay of $\varepsilon_W$ at later iterations indicates convergence of the gradient descent procedure.
• Figure 4: Comparison of the spin-spin correlation function $C(r)$ obtained from Markov-chain Monte Carlo sampling and from RBM-generated configurations for the ANNNI chain at the multiphase point $\kappa = 1/2$ and $T = 0$.
• Figure 5: (a) Ice-rule configurations (2-in/1-out and 1-in/2-out) for three spins on a triangle unit of the kagome lattice, each carrying an emergent magnetic charge of $Q = \pm 1$ unit. (b) Elementary excitations consist of defect triangles with 3-in or 3-out spin configurations, corresponding to magnetic charges of $Q = \pm 3$. (c) Example of an ice-I state. Small red and blue circles mark triangles that satisfy the ice-rule constraint and carry magnetic charges $Q=+1$ and $-1$, respectively, while large red and blue circles denote defect triangles with $Q=\pm 3$ that violate the constraint. The inset illustrates a single triangular unit cell with the three easy-axis directions $\hat{\mathbf e}_{1,2,3}$ of the kagome lattice. (d) Example of the charge-ordered ice-II phase, in which the $\pm 1$ magnetic charges form a staggered pattern that breaks the $Z_2$ symmetry relating up- and down-triangles.