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We investigate the first-order transition in the spin-1 two-dimensionalBlume-Capel model in square lattices by revisiting the transfer-matrix method.With large strip widths increased up to the size of 18 sites, we construct thedetailed phase coexistence curve which shows excellent quantitative agreementwith the recent advanced Monte Carlo results. In the deep first-order area, weobserve the exponential system-size scaling of the spectral gap of the transfermatrix from which linearly increasing interfacial tension is deduced withdecreasing temperature. We find that the first-order signature at lowtemperatures is strongly pronounced with much suppressed finite-size influencein the examined thermodynamic properties of entropy, non-zero spin population,and specific heat. It turns out that the jump at the transition becomesincreasingly sharp as it goes deep into the first-order area, which is incontrast to the Wang-Landau results where finite-size smoothing gets moresevere at lower temperatures.

First-order transitions and thermodynamic properties in the 2D Blume-Capel model: the transfer-matrix method revisited