Spin-1/2 string correlations and singlet-triplet gaps of frustrated ladders with ferromagnetic legs and alternate ferromagnetic and antiferromagnetic rungs

The frustrated ladder with alternate ferromagnetic exchange -J(F) and antiferromagnetic exchange J(A) to first neighbors and ferromagnetic exchange -J(L) to second neighbors is studied by exact diagonalization and density matrix renormalization group calculations in systems of 2N spins-1/2 with periodic boundary conditions. The ground state is a singlet (S = 0) and the singlet-triplet gap epsilon(T) is finite for the exchanges considered. Spin-1/2 string correlation functions g(1)(N) and g(2)(N) are defined for an even number N of consecutive spins in systems with two spins per unit cell; the ladder has string order g(2)(infinity)>0 and g(1)(infinity)=0. The minimum N-& lowast; of g(2)(N) is related to the range of ground-state spin correlations. Convergence to g(2)(infinity) is from below, and g1(N) decreases exponentially for N >= N-& lowast;. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of J(F), J(L), and J(A) reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the J(1)-J(2) model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of the ladder is a bond-order wave, a Kekul & eacute; VB diagram at J(L)=J(F)/2 <= J(A), that is reversed on interchanging -J(F) and J(A). Inversion symmetry is spontaneously broken in the dimer phase of the J(1)-J(2) model where the Kekul & eacute; diagrams are the doubly degenerate ground states at J(2)/J(1 )= 1/2.