The energy spectrum is Γ Γ = [ E min , E max ] which, in this case, depends on the particular J i j used. If all J i j are 1 (the ferromagnetic Ising model), E min = 0 (e.g. all spins are 1.) and E max = 2 D N (half spins are up, half spins are down). Also notice that in this system, Γ Γ ∈ ∈ Z