  Prediction of the Curie temperature is of significant importance for the
design of ferromagnetic materials. Even though the Curie temperature has been
estimated using the Heisenberg model, magnetic exchange coupling parameters
widely used is thus far based on first-principles calculations at zero
temperature. In the explicit consideration of temperature effects, it is
important to minimise the total free energy, because the magnetic and phonon
free energies correlate with each other. Here, we propose a first-principles
thermodynamic approach to minimise the total free energy considering both the
influences of magnetism on phonons and the feedback effect from phonons to
magnetism. By applying our scheme to bcc Fe, we find a significant reduction of
the Curie temperature due to the feedback effect. This result inevitably
enforces us to change our convention as follows: we should use exchange
coupling constants for the disordered local moment state, not for the
ferromagnetic state, in the prediction of the Curie temperature. Our results
not only change the fundamental understanding of finite-temperature magnetism
but also provide a general framework to predict the Curie temperature more
accurately.
