Collective excitations in the ferro- and antiferromagnetic states of the two-dimensional repulsive Hubbard model

Collective excitations in antiferromagnetic and ferromagnetic states of the two-dimensional repulsive Hubbard model are investigated both in the phase transition vicinity (Ginsburg–Landau region $|T-T_c|\ll T_c$) and in the low-temperature region $T\ll T_c$ using the functional integration region.
The collective excitation spectrum consists of one branch at $T>T_c$ and one more branch (Goldstone branch) arises at $T<T_c$. The Goldstone branch at $T\ll T_c$ in the antiferromagnetic state is calculated including the anisotropic correction term to the linear dispersion law. The value of the non-phonon branch at the zero momentum is equal to the double gap of the fermion energy $2\Delta$. All the spectrum branches in the ferromagnetic state vanish at zero momentum ($\lim_{k\to0}E(k)=0$). Bibliography: 6 titles.