Helicity modulus in the two-dimensional Hubbard model.

Abstract

The helicity modulus, which is the stiffness associated with a twisted order parameter, for the two-dimensional Hubbard model is calculated for the equivalent cases of (i) attractive on-site interaction (negative U) with arbitrary strength, arbitrary electron density, and zero magnetic field and (ii) repulsive on-site interaction (positive U) with arbitrary strength, at half-filling and in an arbitrary magnetic field. An explicit formula for the helicity modulus is derived using the Bogoliubov-Hartree-Fock approximation. An improved value for the helicity modulus is obtained by performing variational Monte Carlo calculations using a Gutzwiller projected trial wave function. To within a small correction term the helicity modulus is found to be given by -1/8 the average kinetic energy. The variational Monte Carlo calculation is found to increase the value of the helicity modulus by a small amount (about 5% for intermediate values of the interaction strength) compared to the results from the Bogoliubov-Hartree-Fock approximation. In the case of attractive interaction, from a comparison with the Kosterlitz-Thouless relation between critical temperature and helicity modulus, the critical temperature for a Kosterlitz-Thouless transition is calculated and a phase diagram is obtained. An optimal critical temperature is found for an intermediate value of U. We discuss connections of our results with results in the literature on the Hubbard model using the random-phase approximation and quantum Monte Carlo calculations.