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The spin$-1/2$ chain with ferromagnetic exchange $J_1 < 0$ between firstneighbors and antiferromagnetic $J_2 > 0$ between second neighbors supports twospin-Peierls (SP) instabilities depending on the frustration $\alpha =J_2/\vert J_1\vert$. Instead of chain dimerization with two spins per unitcell, $J_1-J_2$ models with $\alpha > 0.65$ and linear spin-phonon coupling areunconditionally unstable to sublattice dimerization with four spins per unitcell. Unequal $J_1$ to neighbors to the right and left extends the model togapped ($\gamma > 0$) chains with conditional SP transitions at $T_{SP}$ todimerized sublattices and a weaker specific heat $C(T)$ anomaly. The spinsusceptibility $\chi(T)$ and $C(T)$ are obtained in the thermodynamic limit bya combination of exact diagonalization of small systems with $\alpha > 0.65$and density matrix renormalization group (DMRG) calculations of systems up to$N \sim 100$ spins. Both $J_1-J_2$ and $\gamma > 0$ models accountquantitatively for $\chi(T)$ and $C(T)$ in the paramagnetic phase of$\beta$-TeVO$_{4}$ for $T > 8$ K, but lower $T$ indicates a gapped chaininstead of a $J_1-J_2$ model as previously thought. The same parameters and$T_{SP} = 4.6$ K generate a $C(T)/T$ anomaly that reproduces the anomaly at the$4.6$ K transition of $\beta$-TeVO$_{4}$, but not the weak $\chi(T)$ signature.

physical review. b./physical review. b

Spin Peierls transition of $J_{1}-J_{2}$ and extended models with  ferromagnetic $J_{1}$.Sublattice dimerization and thermodynamics of zigzag  chains in $\beta$-TeVO$_{4}$