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We consider the antiferromagnetic Heisenberg and the repulsive Hubbard modelon two $N$-site one-dimensional lattices, which support dispersionlessone-particle states corresponding to localized states on triangular trappingcells. We calculate the degeneracy of the ground states in the subspaces with$n\le n_{\max}$, $n_{\max}\propto N$ magnons or electrons as well as thecontribution of these states (independent localized states) to thermodynamicquantities. Moreover, we discuss another class of low-lying eigenstates(so-called interacting localized states) and calculate their contribution tothe partition function. We also discuss the effect of extra interactions, whichlift the degeneracy present due to the chirality of the localized states ontriangles. The localized states set an extra low-energy scale in the system andlead to a nonzero residual ground-state entropy and to one (or more) additionallow-temperature peak(s) in the specific heat. Low-energy degrees of freedom inthe presence of perturbations removing degeneracy owing to the chirality can bedescribed in terms of an effective (pseudo)spin-1/2 transverse $XX$ chain.

Localized states on triangular traps and low-temperature properties of the antiferromagnetic Heisenberg and repulsive Hubbard models