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

TL;DR

This study revisits the transfer-matrix method to analyze the first-order phase transition in the 2D Blume-Capel model, providing detailed phase coexistence data and insights into finite-size effects and interfacial tension at low temperatures.

Contribution

It offers a comprehensive transfer-matrix analysis with larger system sizes, revealing detailed phase coexistence and thermodynamic behavior in the 2D Blume-Capel model.

Findings

Phase coexistence curve matches Monte Carlo results

Exponential scaling of spectral gap with system size

Sharp transition signatures at low temperatures

Abstract

We investigate the first-order transition in the spin-1 two-dimensional Blume-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 the detailed phase coexistence curve which shows excellent quantitative agreement with the recent advanced Monte Carlo results. In the deep first-order area, we observe the exponential system-size scaling of the spectral gap of the transfer matrix from which linearly increasing interfacial tension is deduced with decreasing temperature. We find that the first-order signature at low temperatures is strongly pronounced with much suppressed finite-size influence in the examined thermodynamic properties of entropy, non-zero spin population, and specific heat. It turns out that the jump at the transition becomes increasingly sharp as it goes deep into the first-order area, which is in contrast to the Wang-Landau results where finite-size smoothing gets more severe at lower temperatures.