# Paper Introduction: Entanglement Entropy in Localized Systems.

In this article, we present an experimental study of entanglement entropy and correlation between artificially created localized systems. Localized systems are not in thermal equilibrium and the eigenstate thermalization hypothesis is not applicable. In this paper, the Aubry-Andre system is used. The Aubry-Andre system is a boson lattice system that can be easily turned into a localized system by adding defect terms. This Hamiltonian is

$H=-J\sum_{i}(a_i^\dagger a_{i+1}+h.c.)+\frac{U}{2}\sum_{i}n_i(n_i-1)+W\sum_{i}h_i n_i$

The result is where J is the tunneling effect coefficient, U is the Coulomb term, $h_i=cos(2\pi\beta+\phi),\beta\approx 1.618$ is the coefficient of the defect potential, and W is its strength. Also, $a_i^\dagger,a_i$ are the operators that create and annihilate bosons at site i, respectively, and $n_i=a_i^\dagger a_i$ is the number operator for site i. It has been confirmed that the maximum single-site entropy of this system alone becomes smaller as W is amplified, and its accumulation with time does not reach thermal equilibrium, and its value is smaller than that of the system with zero defect term. The number entropy, which is the usual quantum state entropy, and the configuration entropy, $S_c$ , which increases in proportion to the total amount of entanglements, were calculated and compared between the state of W=1.0, which is an integrable system with a site number of 8 on a one-dimensional lattice, and the state of W=8.9, which is a localized system. As a result, the number entropy increased more slowly in the localized system, while the configuration entropy started to increase after the number entropy converged, and the increase was also slower. Furthermore, it was confirmed that the difference in the size of the connected systems determines the maximum value of entropy, which reaches its maximum when the system size is half that of the connected systems, and becomes smaller the further away it is.

The result that a localized system, which does not reach thermal equilibrium, reaches thermal equilibrium when connected to another system may come as a shock to those who study quantum information and quantum computing hardware. If you are familiar with various physical phenomena, it may seem natural to you. However, it is still a topic that is expected to be developed in the future. For example, it is possible to find a way to suppress both entropies of the connection.