Dynamical spin correlations in the Heisenberg ladder under a magnetic field and correlation functions in the SO(5) ladder

The zero-temperature dynamical spin-spin correlation functions are calculated for the spin-$\frac{1}{2}$ two-leg antiferromagnetic Heisenberg ladder in a magnetic field above the lower critical field $H_{c1\mathrm{}}$. The dynamical structure factors are calculated, which exhibit both massless modes and massive excitations. These modes appear in different sectors characterized by the parity in the rung direction and by the momentum in the direction of the chains. The structure factors have power-law singularities at the lower edges of their support. The results are also applicable to the spin-1 Heisenberg chain. The implications are briefly discussed for the various correlation functions and the $\pi$ resonance in the SO(5) symmetric ladder model.