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 by whether it is in the majority or minority with its nearest neighbors. The spin updates its state using an ϵ-greedy algorithm. The parameter ϵ plays a 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 hyperscaling 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 a two-dimensional Ising model for lower learning rates.