To evaluate the effectiveness of machine learning in systems with competing interactions, we developed a self-learning quantum Monte Carlo (SLQMC) method to simulate the phase transition in the classical Holstein spin-fermion model. In SLQMC, machine-learning techniques are employed to approximate the free energy, thereby bypassing the need for exact diagonalization and significantly reducing computational cost. We assess the performance of SLQMC using both linear regression and neural network models. Our results show that both models are capable of capturing the phase transition from the antiferromagnetic (AFM) state to the charge-density-wave (CDW) state. However, the sampling efficiency decreases near the AFM-CDW phase transition, which is attributed to the increased mean-squared error of the machine-learning model. Additionally, the sampling efficiency decreases with increasing lattice size. This suppression is due to the increased root-mean-squared error as the machine-learning model is applied to a large lattice and the finite-size effect, wherein the energy gap between the ground state and low-energy excited states decreases as the lattice grows. Our findings highlight the necessity of highly accurate machine-learning models to simulate theoretical models with complex, competing microscopic interactions on a large lattice.

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• Charge density waves
• Magnetic order
• Magnetic phase transitions
• Phase diagrams
• Strongly correlated systems
• Monte Carlo methods