Generalized antiferromagnetic Heisenberg spin ladders

We utilize the path-integral technique to derive the non-linear Sigma model (NL sigmaM) for generalized antiferromagnetic spin-ladder systems on the s quare lattice with diagonal (next-nearest neighbor) interactions in additio n to the nearest neighbor interaction. The model Hamiltonian is: H = Sigma (n)(a=1)' Sigma (1) J(a)S(a(i))(.)S(a)(i+1) + J ' S-a,a+1(a)(i)S-.(a+1)(i) + K(a,a+1)Sa(i) (.) Sa+1(i+1) + Ma,a+1Sa(i+1) (.) S-a(i+1) (.) S-a(i). The topological term of the NL sigmaM is absent for the spin-s ladder with an e ven number of legs and is equal to 2 pis for the ladder with an odd number of legs. The spin wave velocity is s[Sigma (a)(J(a) - M-a,M-a+1 - K-a,K-a+1 )/Sigma L-b,c(b,c)-1](1/2) where L-a,L-b = 4J(a) + J ' (a,a+1) + J ' (a-1) - M-a,M-a+1- M-a,M-a-1-K-a,K-a+1-K-a,K-a-1 when a = b, and L-a,L-b = J ' (a ,b) + K-a,K-b + M-a,M-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 ladd er with spin (s) over tilde and s, in which diagonal interactions occur onl y in the even (or odd) cells. The Berry phase is found to be dependent on t he coupling constants. The expressions of the spin-wave velocity and spin g ap are also given for even-leg ladders. The operator approach to the genera lized spin-ladder problem is also presented. Finally we address the finite- size NL sigmaM treatment of this system. (C) 2001 Elsevier Science B.V. All rights reserved.