Helicity modulus of the quasi-one-dimensional XY model: Protection by an energy barrier

The helicity modulus of the quasi-one-dimensional classical XY model is examined in connection with liquid He confined in nanopores. The helicity modulus Upsilon(p) is suppressed with phase slippage, and its suppression is found to be well described with the scaling form obtained from the Tomonaga-Luttinger liquid theory in the one-dimensional limit. For phase slippage to occur, the system has to overcome an energy barrier, and the helicity modulus Upsilon(p) suppressed by phase slippage is not necessarily observed in experiments. The energy barrier for phase slippage is explicitly calculated. Using the result, the helicity modulus Upsilon(s) protected by the energy barrier is found to survive up to a higher temperature than Upsilon(p). The relation of the present results to experiments on liquid He in nanopores is also discussed.