Real-Space Treatment of Spin-Peierls Transitions: Gap Equation of the S = 1/2 Heisenberg Antiferromagnetic Chain

The spin-Peierls transition at T$_{SP}$ of s = 1/2 Heisenberg AF chains with linear spin-phonon (sp-ph) coupling is modeled by exact solution up to 18 spins. Spin correlations alter the gap relation between T$_{SP}$ and the singlet-triplet gap, E$_{ST}$, from the free-fermion or mean-field (mf) result by 30{\%}. Direct solution accounts for the spin-Peierls transition, magnetic susceptibility and magnetization of the prototypical TTF$^{+}$ (tetrathiafulvalene) spin chain in TTFCu(BDT). Strong sp-ph coupling is required for size convergence. Exact analysis of finite chains conserves total spin, while mf or bosonization approximations for spinless fermions in infinite chains only conserve the z component and have axial rather than spherical symmetry in spin space.