Can Helicity Modulus Be Defined For Boundary Conditions With Finite Twist?

We study the response of a one-dimensional quantum rotor Hamiltonian to a finite (non-infinitesimal) twist of the boundary conditions. After mapping the system to two-dimensional classical XY model, we use large-scale Monte Carlo simulations to evaluate the free energy difference between periodic and twisted-periodic boundary conditions and find deviations from the expected quadratic dependence on the twist angle. Consequently, the helicity modulus (spin-stiffness) shows a non-trivial dependence on the twist angle. We show that the deviation from the expected behavior arises because of a degeneracy due to the chirality of spin-waves in the ordered phase which leads to an additional entropy contribution of spin-waves of mixed helicities. We give an improved prescription for the numerical evaluation of the helicity modulus and resolve some open questions related to anti-PBC [1]. We also discuss applications to discrete spin systems [2] and some experimental scenarios [3] where boundary conditions with finite twist are necessary.

[1] R. G. Brown and M. Ciftan, Phys. Rev. B, 74, 224413 (2006).

[2] Y. Kumano, et. al. Physical Review B, 88, 10, 104427 (2013).

[3] R. E. Troncoso, A. Brataas, and A. Sudbø, Physical Rev Letters, 125, 23, 237204 (2020).