Monte Carlo Study of the Yukawa Coupled Two Spin Ising Model

We consider a particular 4 state spin system composed of two Ising spins (~$s_x, \; \sigma_x$~) with independent hopping parameters $\kappa_1, \kappa_2$, coupled by a bilinear Yukawa term, $y s_x \sigma_x$. The Yukawa term is solely responsible for breaking the global $Z_2 \times Z_2$ symmetry down to $Z_2$. This model is intended as an illustration of general coupled Higgs system where scalars can arise both as composite and elementary excitations. For the Ising example in 2d, we give convincing numerical evidence of the universality of the two spin system with the one spin Ising model, by Monte Carlo simulations and finite scaling analysis . We also show that as we approach the phase transition, universality arises by a separation of low mass spin waves from an extra set of spin waves with an energy gap that diverges as the correlations length diverges