# Entanglement entropy of the long-range Dyson hierarchical model

## TL;DR

This paper investigates the entanglement entropy of a long-range quantum spin chain with hierarchical interactions, revealing an unexpected area law at criticality, contrasting with typical long-range models.

## Contribution

It introduces a real-space renormalisation group approach to analyze entanglement in a hierarchical long-range model, uncovering novel area law behavior at criticality.

## Key findings

- Entanglement entropy follows an area law at criticality.
- Analytical results for large and small decay exponents $\sigma$.
- Numerical validation of recursive equations.

## Abstract

We study the ground state entanglement entropy of the quantum Dyson hierarchical spin chain in which the interaction decays algebraically with the distance as $r^{-1-\sigma}$. We exploit the real-space renormalisation group solution which gives the ground-state wave function in the form of a tree tensor network and provides a manageable recursive expression for the reduced density matrix of the renormalised ground state. Surprisingly, we find that at criticality the entanglement entropy obeys an area law, as opposite to the logarithmic scaling of short-range critical systems and of other non-hierarchical long-range models. We provide also some analytical results in the limit of large and small $\sigma$ that are tested against the numerical solution of the recursive equations.