Concurrence and entanglement entropy in a dimerized spin-1/2 two-leg ladder

We consider the isotropic spin-1/2 two-leg ladders with the dominant spatially modulated rung exchanges. We study the effect of a uniform magnetic field on the ground state phase diagram of the model using the perturbation theory and the numerical Lanczos method. The ground state phase diagram consists of two gapless Luttinger liquid (LL) and three gapped phases. Numerically, we calculate the concurrence between two spins and entanglement entropy between legs. Numerical experiment shows that in principle the gapless LL phases are different. In the first LL phase, only spins on rungs are entangled, but in the second LL phase the spins on legs are long-distance entangled. Therefore the concurrence between spins on legs can be considered as a function to distinguish the LL phases.