Estimating correlations and entanglement in the two-dimensional Heisenberg model in the strong-rung-coupling limit

We consider the isotropic Heisenberg model in a magnetic field in the strong-rung-coupling limit on a two-dimensional (2D) rectangular zig-zag lattice of arbitrary size, and determine the one-dimensional (1D) effective model representing the low-energy manifold of the 2D model up to second order in perturbation theory. We consider a number of Hermitian operators defined on the Hilbert space of the 2D model, and systematically work out their action on the low-energy manifold, which are operators on the Hilbert space of the 1D effective model. For a class of operators among them, we demonstrate that the expectation values computed in the low-energy manifold of the 2D model can be mimicked by the expectation values of the corresponding operators in the 1D effective model even beyond the perturbation regime of the system parameters. We further argue that quantitatively estimating partial trace-based measures of entanglement in the 2D model may be done in the same fashion only in the perturbation regime. Our results and approach are expected to be useful in investigating observables and entanglement in the 2D models with large system sizes due to the advantage of using the effective 1D model with a smaller Hilbert space as a proxy.