Critical behavior of the helicity modulus for the classical Heisenberg model

Abstract

The critical scaling of the helicity modulus of the classical O(3) $3d$ Heisenberg ferromagnet is studied directly. Monte Carlo methods that impose either an antiperiodic boundary condition or a finite twist of definite handedness across otherwise periodic boundaries in one lattice direction are used to measure scale-dependent enthalpy variations in a simple cubic lattice at the ferromagnetic critical temperature. Finite-size scaling is then used to determine the critical exponents ${v}_{E}$ and ${v}_{F}$ for helicity and, by evaluating three independent hyperscaling-linked pairs of $\ensuremath{\nu}$ and $\ensuremath{\alpha}$, to test hyperscaling for this model. It is observed that antiperiodic boundary conditions in particular constrain the lattice to have a nonzero topological charge, establishing a connection between topological charge and helicity in the model.