Ordering through learning in two-dimensional Ising spins

Abstract

We study two-dimensional Ising spins, evolving through reinforcement learning using their state, action, and reward. The state of a spin is defined as whether it is in the majority or minority with its nearest neighbours. The spin updates its state using an ε-greedy algorithm. The parameter ε plays the role equivalent to the temperature in the Ising model. We find a phase transition from long-ranged ordered to a disordered state as we tune ε from small to large values. In analogy with the phase transition in the Ising model, we calculate the critical ε and the three critical exponents β, γ, ν of magnetization, susceptibility, and correlation length, respectively. A hyper-scaling relation dν = 2β + γ is obtained between the three exponents. The system is studied for different learning rates. The exponents approach the exact values for two-dimensional Ising model for lower learning rates.