Strong Coupling Expansions for Antiferromagnetic Heisenberg S=1/2 Ladders

The properties of antiferromagnetic Heisenberg $S=\frac{1}{2}$ ladders with 2, 3, and 4 chains are expanded in the ratio of the intra- and interchain coupling constants. A simple mapping procedure is introduced to relate the 4 and 2-chain ladders which holds down to moderate values of the expansion parameters. A second order calculation of the spin gap to the lowest triplet excitation in the 2- and 4-chain ladders is found to be quite accurate even at the isotropic point where the couplings are equal. Similar expansions and mapping procedures are presented for the 3-chain ladders which are in the same universality class as single chains.