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We perform Monte Carlo simulations, combining both the Wang-Landau and theMetropolis algorithms, to investigate the phase diagrams of the Blume-Capelmodel on different types of nonregular lattices (Lieb lattice (LL), decoratedtriangular lattice (DTL), and decorated simple cubic lattice (DSC)). Thenonregular character of the lattices induces a double transition (reentrantbehavior) in the region of the phase diagram at which the nature of the phasetransition changes from first-order to second-order. A physical mechanismunderlying this reentrance is proposed. The large-scale Monte Carlo simulationsare performed with the finite-size scaling analysis to compute the criticalexponents and the critical Binder cumulant for three different values of theanisotropy $\Delta/J \in \big\{ 0, 1, 1.34 \textrm{ (for LL)}, 1.51 \textrm{(for DTL and DSC)} \big\}$, showing thus no deviation from the standard Isinguniversality class in two and three dimensions. We report also the location ofthe tricritical point to considerable precision: ($\Delta_t/J=1.3457(1)$; $k_BT_t/J=0.309(2)$), ($\Delta_t/J=1.5766(1)$; $k_B T_t/J=0.481(2)$), and($\Delta_t/J=1.5933(1)$; $k_B T_t/J=0.569(4)$) for LL, DTL, and DSC,respectively.

journal of statistical mechanics

Monte Carlo studies of the Blume-Capel model on nonregular two- and  three-dimensional lattices: Phase diagrams, tricriticality, and critical  exponents