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Abstract
We have studied the finite-size behavior at magnetic phase transitions by using extensive Monte Carlo simulations. For the second-order transition in the simple cubic Ising model, we have investigated the critical behavior by implementing the Wolff cluster flipping algorithm and data analysis with histogram reweighting in quadruple precision arithmetic. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, we have obtained the critical quantities with precision that exceeds all previous Monte Carlo estimates. For the first-order ``spin-flop" transition in the 3D anisotropic Heisenberg antiferromagnet in an external field, we have explored the finite-size behavior of the transition between the Ising-like antiferromagnetic state and the canted, $XY$-like state. Finite-size scaling for a first-order phase transition where a continuous symmetry is broken is developed using an approximation of Gaussian probability distributions with a phenomenological ``degeneracy" factor, $q$, included. Our theory yields $q = pi$, and it predicts that for large linear dimension $L$ the field dependence of all moments of the order parameters as well as the fourth-order cumulants exhibit universal intersections, where the values of these intersections can be expressed in terms of the factor $q$. The agreement between our theory and high-resolution multicanonical simulation data implies a heretofore unknown universality can be invoked for first-order phase transitions.