Spin-Peierls transition of the dimer phase of the $J_1-J_2$ model: Energy cusp and CuGeO$_3$ thermodynamics

The spin-Peierls transition is modeled in the dimer phase of the spin-$1/2$ chain with exchanges $J_1$, $J_2 = alpha J_1$ between first and second neighbors. The degenerate ground state generates an energy cusp that qualitatively changes the dimerization $delta(T)$ compared to Peierls systems with nondegenerate ground states. The parameters $J_1 = 160$ K, $alpha = 0.35$ plus a lattice stiffness account for the magnetic susceptibility of CuGeO$_3$, its specific heat anomaly, and the $T$ dependence of the lowest gap.