Loop ensembles in Stochastic Series Expansion of Two-Dimensional Heisenberg Antiferromagnets

The Stochastic Series Expansion (SSE) method along with resummation over the spin or flavor values maps the partition function of a quantum antiferromagnet to a closely-packed loop gas model in one higher dimension. Earlier work by Nahum et al. has shown that certain closely-packed three-dimensional loop gas models exhibit phases dominated by macroscopic loops, wherein the corresponding joint distribution of loop lengths is Poisson-Dirichlet. On grounds of universality, the same is expected of the ensemble of loops obtained in (2+1)-dimensional quantum antiferromagnets, albeit the loops emerge from a different microscopic origin. We sample the SSE loop ensemble for SU(N) antiferromagnets on a square lattice using Monte Carlo and study how the joint distribution varies with the degree of representation N and inverse temperature $\beta$. We observe that, for low temperatures and small N($\leq$ 4), the distribution indeed shows characteristics of Poisson-Dirichlet behaviour when antiferromagnetic correlations dominate the system.