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We study the response of a two-dimensional classical XY model to a finite(non-infinitesimal) twist of the boundary conditions. We use Monte Carlosimulations to evaluate the free energy difference between periodic andtwisted-periodic boundary conditions and find deviations from the expectedquadratic dependence on the twist angle. Consequently, the helicity modulus(spin-stiffness) shows a non-trivial dependence on the twist angle. We showthat the deviation from the expected behavior arises because of a degeneracydue to the chirality of the spin-waves which leads to an additional entropycontribution in the quasi-long-range ordered phase. We give an improvedprescription for the numerical evaluation of the helicity modulus and resolvesome open questions related to anti-periodic boundary conditions. We alsodiscuss applications to discrete spin systems and some experimental scenarioswhere boundary conditions with finite twist are necessary.

Can helicity modulus be defined for boundary conditions with finite twist?