Spin triangle chain with ferromagnetic and antiferromagnetic interactions on the transition line

We investigate spin-$\frac{1}{2}$ anisotropic model of a linear chain of triangles with competing ferro- and antiferromagnetic interactions and ferromagneic Heisenberg interactions between triangles. For a certain ratio between interactions the one-magnon excitation band is dispersionless leading to an existence of localized-magnon states which form macroscopically degenerated ground states. The spectrum of excitations has a specific structure depending on the value of the triangle-triangle interaction. Such specific structure determines the low-temperature thermodynamics and, in particular, the temperature dependence of a specific heat. In the limit of strong anisotropy of interactions the spectrum has a multi-scale structure which consists of subsets rank-ordered on small parameter. Each subset is responsible for the appearance of the peak in the temperature dependence of the specific heat.