Nonmonotonic External Field Dependence of the Magnetization in a Finite Ising Model: Theory and MC Simulation

Abstract

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 $h/M^{\delta }=f(h{L}^{\beta \delta /\nu },t/h^{1/\beta \delta })$ where $t=(T-T_c)/T_c$ is the reduced temperature, $h$ is the external field and $L$ is the size of system. Below $T_c$ and at $T_c$ the theory predicts a nonmonotonic dependence of $f(x,y)$ with respect to $x\equiv h{L}^{\beta \delta /\nu }$ 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 $T_c$.

arXiv commentLaTex, 12 pages

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