Transfer-matrix approach to the Blume-Capel model on the triangular lattice

TL;DR

This paper uses an advanced transfer-matrix method to precisely analyze the phase transition and tricritical point of the spin-1 Blume-Capel model on a triangular lattice, confirming theoretical predictions.

Contribution

It provides the most accurate location of the tricritical point for this model using transfer-matrix techniques and finite-size scaling, surpassing previous Monte Carlo estimates.

Findings

Accurate tricritical point location matching conformal field theory.

Observation of exponential spectral gap scaling in the first-order regime.

Validation of the transfer-matrix method's effectiveness for complex lattice models.

Abstract

We investigate the spin-$1$ Blume-Capel model on an infinite strip of the triangular lattice using the transfer-matrix method combined with a sparse-matrix factorization technique. Through finite-size scaling analysis of numerically exact spectra for strip widths up to $L = 19$, we accurately locate the tricritical point improving upon recent Monte Carlo estimates. In the first-order regime, we observe exponential scaling of the spectral gap, reflecting the linear growth of interfacial tension as the temperature decreases below the tricritical point. Finally, we validate our tricritical point estimate through precise agreement with conformal field theory predictions for the tricritical Ising universality class. Our results underscore the continued utility of the transfer-matrix approach for studying phase transitions in complex lattice models.