Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2013 / Paper 1 / 3 / e / Solution

Differentiate the uniform effective action:To reproduce the printed coefficient, define the spin-summed density of states per site byThe saddle-point approximation then givesThe uniform spin field and magnetization obey $M=U(n_{\uparrow }-n_{\downarrow })=2U⟨S_z⟩$ for occupations per site. If a one-spin density of states were used instead, the factor $1/2$ would be absent. The spin field is an energy, rather than directly a magnetic moment.