Gapless singlet modes in thekagoméstrips: A study through density-matrix-renormalization group and strong-coupling analysis

Recently, Azaria et al. [Phys. Rev. Lett. 81, 1694 (1998)] have studied strips of the kagom\'e lattice in the weak-coupling limit, where they consist of two spin-half chains on the outside weakly coupled to an array of half-integer spins in the middle. Using a number of mappings, they have arrived at the interesting result that in this system all spin excitations are gapped but there are gapless spinless modes. Here, we study these kagom\'e strips in the limit where the interchain couplings are comparable to the coupling to the middle spins by density-matrix-renormalization group and by a strong-coupling analysis. In the limit when the coupling to the middle-spin dominates, the five-spins of the unit-cell reduce to a single $S=3/2$ spin, and the overall system has well known gapless spin excitations. We study the phase transition from this phase to the weak-coupling phase. We also carry out a strong-coupling analysis away from the $S=3/2$ limit, where the five-spin blocks have four degenerate ground states with $S=1/2,$ which can be thought of as two spin and two pseudospin degrees of freedom. A numerical study of this strong coupling model also suggests a finite spin gap.