Influence of lattice geometry on the tricritical point

Determine how different geometric structures influence the location and properties of the tricritical point in Blume–Capel models, including the anisotropic square-lattice J_x–J_y Blume–Capel model.

Background

The paper places the J_x–J_y Blume–Capel model within broader studies of how dimensionality, lattice geometry, and approximation methods affect critical and tricritical behavior. Because exact solutions are available only in restricted cases and existing estimates vary with the underlying geometry and analytical approach, the general impact of geometric structure on the tricritical point remains unresolved. The authors investigate one specifically designed anisotropic square-lattice realization using Monte Carlo simulations, but do not establish a general characterization across different geometries.

References

This implies that the influence of different geometric structures on the tricritical point remains incompletely understood.

— Phase diagram and transition properties of the Blume-Capel model: A Monte Carlo study  (2610.07567 - Li et al., 6 Oct 2026) in Section 1, Introduction

Several questions remain open. The critical exponents along the second-order branch, the expected logarithmic divergence of the specific heat, and the immediate neighborhood of the tricritical point, where the Ising-based extrapolation of Table~\ref{tab:secondorder} becomes unreliable, all require considerably larger lattices: at $T=0.38$ the condition $L\gtrsim6\xi$ already demands $L\approx150$.

— High-resolution mapping of the phase boundary of the honeycomb-lattice Blume--Capel ferromagnet  (2610.11068 - Filho et al., 8 Oct 2026) in Section “Summary and outlook”