# Exact Calculation of Entanglement in a 19-site 2D Spin System

## TL;DR

This paper presents an exact calculation of entanglement in a 19-site 2D transverse Ising model, revealing how entanglement can be tuned and identifying a quantum phase transition at a critical parameter value.

## Contribution

It introduces an exact computational method for entanglement in a 2D spin system and demonstrates control of entanglement via system parameters and impurities.

## Key findings

- Entanglement can be controlled by varying magnetic field parameters.
- A quantum phase transition occurs at λ ≈ 3.01.
- Impurities influence entanglement properties.

## Abstract

Using the Trace Minimization Algorithm, we carried out an exact calculation of entanglement in a 19-site two-dimensional transverse Ising model. This model consists of a set of localized spin-1/2 particles in a two dimensional triangular lattice coupled through exchange interaction J and subject to an external magnetic field of strength h. We demonstrate, for such a class of two-dimensional magnetic systems, that entanglement can be controlled and tuned by varying the parameter $\lambda$ = h/J in the Hamiltonian and by introducing impurities into the systems. Examining the derivative of the concurrence as a function of $\lambda$ shows that the system exhibits a quantum phase transition at about $\lambda_c$ = 3.01, a transition induced by quantum fluctuations at the absolute zero of temperature.