Generalized antiferromagnetic Heisenberg spin ladders

We utilize the path-integral technique to derive the non-linear Sigma model (NLσM) for generalized antiferromagnetic spin-ladder systems on the square lattice with diagonal (next-nearest neighbor) interactions in addition to the nearest neighbor interaction. The model Hamiltonian is: H =∑ a=1 n l ∑ i J a S a (i)· S a (i+1)+J a,a+1 ′ S a (i)· S a+1 (i)+K a,a+1 S a (i)· S a+1 (i+1)+M a,a+1 S a (i+1)· S a (i) . The topological term of the NLσM is absent for the spin-s ladder with an even number of legs and is equal to 2πs for the ladder with an odd number of legs. The spin wave velocity is s[∑a(Ja−Ma,a+1−Ka,a+1)/∑b,cLb,c−1]1/2 where La,b=4Ja+Ja,a+1′+Ja,a−1′−Ma,a+1−Ma,a−1−Ka,a+1−Ka,a−1 when a=b, and La,b=Ja,b′+Ka,b+Ma,b, when |a−b|=1. The spin gaps are predicted for spin ladders with an even number of chains. We also consider a two-leg ladder with spin s and s, in which diagonal interactions occur only in the even (or odd) cells. The Berry phase is found to be dependent on the coupling constants. The expressions of the spin-wave velocity and spin gap are also given for even-leg ladders. The operator approach to the generalized spin-ladder problem is also presented. Finally we address the finite-size NLσM treatment of this system.