  In this paper, we calculate entanglement in the isotropic Heisenberg model in
a magnetic field on a two-dimensional rectangular zig-zag lattice in the strong
rung-coupling limit, using the one-dimensional XXZ model as a proxy. Focusing
on the leading order in perturbation, for arbitrary size of the lattice, we
show how the one-dimensional effective description emerges. We point out
specific states in the low-energy sector of the two-dimensional model that are
well-approximated by the one-dimensional spin-1/2 XXZ model. We propose a
systematic approach for mapping matrix-elements of operators defined on the
two-dimensional model to their low-energy counterparts on the one-dimensional
XXZ model. We also show that partial trace-based description of entanglement in
the two-dimensional model can be satisfactorily approximated using the
one-dimensional XXZ model as a substitute. We further show numerically that the
one-dimensional XXZ model performs well in estimating entanglement quantified
using a measurement-based approach in the two-dimensional model for specific
choices of measured Hermitian operators.
