Cluster Dynamics Stay Fast-Until Tricriticality

Abstract

Cluster Monte Carlo algorithms are widely regarded as the most effective route to overcoming critical slowing down in lattice spin systems. Whether this acceleration persists in the presence of vacancies and multicritical fluctuations, however, remains unresolved. We address this question through a systematic dynamic-scaling study of hybrid cluster-local update schemes in the two-dimensional Blume-Capel model, which exhibits a line of continuous Ising-like transitions terminating at a tricritical point. Along the entire critical line, hybrid dynamics retain the near-optimal efficiency of pure cluster updates despite the presence of annealed vacancies. Strikingly, this acceleration collapses precisely at tricriticality, where the dynamic critical exponent reverts to the local-update value. We trace this breakdown to the correlated percolation of vacancies, whose emergent system-spanning geometry obstructs nonlocal relaxation in the spin sector. Our results identify a fundamental geometric limitation of cluster acceleration at tricriticality and establish vacancy percolation as the mechanism controlling dynamic universality in hybrid Monte Carlo dynamics.