Nonmonotonic external field dependence of the magnetization in a finite Ising model: Theory and MC simulation

Using $\varphi^4$ field theory and Monte Carlo (MC) simulation we investigate the finite-size effects of the magnetization M for the three-dimensional Ising model in a finite cubic geometry with periodic boundary conditions. The field theory with infinite cutoff gives a scaling form of the equation of state $h/M^\delta = f(hL^{\beta\delta/\nu}, t/h^{1/\beta\delta})$ where $t=(T-T_{\rm c})/T_{\rm c}$ is the reduced temperature, h is the external field and L is the size of system. Below Tc and at Tc the theory predicts a nonmonotonic dependence of f(x,y) with respect to $x \equiv hL^{\beta\delta/\nu}$ at fixed $y \equiv t/h^{1/\beta \delta}$ and a crossover from nonmonotonic to monotonic behaviour when y is further increased. These results are confirmed by MC simulation. The scaling function f(x,y) obtained from the field theory is in good quantitative agreement with the finite-size MC data. Good agreement is also found for the bulk value $f(\infty,0)$ at Tc