Estimating entanglement in 2D Heisenberg model in the strong rung-coupling limit

TL;DR

This paper develops a method to estimate entanglement in a 2D Heisenberg model by mapping it to an effective 1D XXZ model in the strong rung-coupling limit, enabling simplified analysis of complex quantum states.

Contribution

It introduces a systematic approach to approximate 2D entanglement properties using a 1D XXZ model, including operator mapping and numerical validation.

Findings

Effective 1D description emerges in the strong rung-coupling limit.

The 1D XXZ model accurately estimates entanglement in the 2D model.

Partial trace entanglement measures are well-approximated by the 1D model.

Abstract

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.