Twisted boundary conditions of quantum spin chains near the Gaussian fixed points

Duality transformation, which relates a high-temperature phase to a low-temperature one, is used exactly to determine the critical point for several models (2D Ising, Potts, Ashkin-Teller, 8-vertex), as the self dual condition. By changing boundary condition, numerically we can determine the self-dual(critical) point of the Ashkin-Teller(or Gaussian) model. This is the first explicit application of the duality to the numerical calculation, with the use of boundary conditions.