phase transition: Because inelastic neutron scattering is proportional to Im χ$_{⊥}$ (+ , k , ω ), low-energy spin-waves with momentum near ¯ h k$_{cSDW}$ will dominate such measurements near the transition between the cSDW state and hidden magnetic order (see Fig. 3). Study of the expressions for the frequencies Ω$_{±}$ that are listed below Eq. (10) yields that the dominant spin-wave excitations centered at the cSDW wave numbers k$_{cSDW}$ disperse up from the groundstate energy as [ v 2 $_{l}$( k$_{l}$ − π/a ) $^{2}$+ v 2 $_{t}$k 2 t +Δ 2 $_{cSDW}$] 1 / 2 in general. (See Fig. 2.) Here k$_{l}$ and k$_{t}$ are the components of k that are respectively parallel and perpendicular to k$_{cSDW}$ . Near criticality, Δ$_{cSDW}$ → 0, the cSDW order is short range, with correlations lengths ξ ( l ) cSDW = v$_{l}$/ Δ$_{cSDW}$ and ξ ( t ) cSDW = v$_{t}$/ Δ$_{cSDW}$ in the longitudinal and transverse directions. At criticality, Δ$_{cSDW}$ = 0, the longitudinal velocity v$_{l}$ coincides with the spin-wave velocity v$_{0}$ for hidden magnetic order (13), while the transverse velocity v$_{t}$ is equal to the previous divided by the anisotropy parameter The latter coincides with the anisotropy of low-energy spinwaves that is predicted by the conventional J$_{1}$ - J$_{2}$ model, in which case J$_{1(2)}$ = [ J ‖ 1(2) + J ⊥ $_{1(2)}$] / 2 by Hund’s rule. Exact Diagonalization. The above semi-classical analysis indicates that large enough off- diagonal frustration can induce a quantum phase transition into hidden magnetic order that is unfrustrated, but that violates Hund’s rule. We shall now confirm this by obtaining the exact low-energy spectrum of states for the J$_{0}$ - J$_{1}$ - J$_{2}$ model (1) with two spin-1/2 moments per site over a 4 by 4 square lattice with periodic boundary conditions. The model Hamilto- nian operator (1) acts on a Hilbert space that is restricted to have 16 up spins and 16 down spins. Next, translational invariance is exploited in order to bring H into block-diagonal form. Each block of the Hamiltonian is labeled by the allowed momentum quantum num- bers: ( k$_{x}$a, k$_{y}$a ) = (0 , 0), ( π, 0), ( π, π ), ( π/ 2 , 0), ( π/ 2 , π/ 2) and ( π, π/ 2), plus their symmetric counterparts. Spin-flip symmetry is also exploited to further block-diagonalize the Hamil- tonian at such momenta into two blocks that are respectively even and odd under it. The combination of translation and spin-flip symmetries reduces the dimension of each block to a little under 19,000,000 states. Next, we construct the Hamiltonian operator (1) over each of these subspaces. Every term in Hamiltonian (1) represents a Heisenberg spin exchange that 12 v$_{l}$ v$_{t}$ = [ 2( J ‖ 2 + J ⊥ 2 ) + J ‖ 1 + J ⊥ 1 2( J ‖ 2 + J ⊥ 2 ) − J ‖ 1 − J ⊥ 1 ] 1 / 2 . (16) A (+ , k$_{cSDW}$ ) = πs [(4 J ⊥ 2 − J$_{0}$$_{c}$ ) / ( J$_{0}$ − J$_{0}$$_{c}$ )] 1 / $^{2}$. (15)