Residual entropy and spin gap in a one-dimensional frustrated antiferromagnet

We show that the low-energy sector of the $S=1/2\mathrm{}$, antiferromagnetic Heisenberg model on a one-dimensional lattice of coupled tetrahedra consists of $2^N$ replica of the spectrum of the dimerized Heisenberg chain, where N is the number of tetrahedra. This provides a proof of the following properties: (i) there is a residual ground-state entropy per spin equal to $\mathrm{}(1/4)\mathrm{}\ln 2$; (ii) there is a singlet-triplet gap as long as the coupling between the tetrahedra is smaller than the internal one. These properties are compared to available results on the pyrochlore lattice.