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

The spin$-1/2$ chain with ferromagnetic exchange $J_1 < 0$ between first neighbors and antiferromagnetic $J_2 > 0$ between second neighbors supports two spin-Peierls (SP) instabilities depending on the frustration $α= J_2/\vert J_1\vert$. Instead of chain dimerization with two spins per unit cell, $J_1-J_2$ models with $α> 0.65$ and linear spin-phonon coupling are unconditionally unstable to sublattice dimerization with four spins per unit cell. Unequal $J_1$ to neighbors to the right and left extends the model to gapped ($γ> 0$) chains with conditional SP transitions at $T_{SP}$ to dimerized sublattices and a weaker specific heat $C(T)$ anomaly. The spin susceptibility $χ(T)$ and $C(T)$ are obtained in the thermodynamic limit by a combination of exact diagonalization of small systems with $α> 0.65$ and density matrix renormalization group (DMRG) calculations of systems up to $N \sim 100$ spins. Both $J_1-J_2$ and $γ> 0$ models account quantitatively for $χ(T)$ and $C(T)$ in the paramagnetic phase of $β$-TeVO$_{4}$ for $T > 8$ K, but lower $T$ indicates a gapped chain instead 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 $β$-TeVO$_{4}$, but not the weak $χ(T)$ signature.