Spin- 12 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 -JF and antiferromagnetic exchange JA to first neighbors and ferromagnetic exchange -JL 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 εT is finite for the exchanges considered. Spin-1/2 string correlation functions g₁(N) and g₂(N) are defined for an even number N of consecutive spins in systems with two spins per unit cell; the ladder has string order g₂(∞)>0 and g₁(∞)=0. The minimum N* of g₂(N) is related to the range of ground-state spin correlations. Convergence to g₂(∞) is from below, and g₁(N) decreases exponentially for N≥N*. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of JF, JL, and JA reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the J₁-J₂ 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\'e VB diagram at JL=JF/2≤JA, that is reversed on interchanging -JF and JA. Inversion symmetry is spontaneously broken in the dimer phase of the J₁-J₂ model where the Kekul\'e diagrams are the doubly degenerate ground states at J₂/J₁=1/2.