  The calculation of response functions in correlated electronic systems is one
of the most important problems in the condensed matter physics. To obtain a
physical response function, preserving both the Ward-Takahashi identity and the
fluctuation-dissipation theorem are crucial. Here we propose a self-consistent
many body method within the GW framework to calculate the response functions
based on the fluctuation-dissipation theorem, which also satisfies the
Ward-Takahashi identity. The validity of this methodology is demonstrated on
the two-dimensional one-band Hubbard model, where both the Ward-Takahashi
identity and fluctuation-dissipation theorem are verified numerically.
Moreover, comparing to the accurate spin susceptibility of the determinantal
Monte Carlo approach, the results obtained from our method are quite
satisfactory and the computational cost are greatly reduced.

Strongly Correlated Electrons