# Bell correlations at Ising quantum critical points

## TL;DR

This paper demonstrates that Bell inequalities based on two-body correlations are violated near the quantum-critical point of the Ising model with power-law interactions, revealing non-local quantum behavior at criticality.

## Contribution

It introduces a permutationally invariant Bell inequality applicable to the Ising model with power-law interactions and shows its violation at quantum critical points using analytical and numerical methods.

## Key findings

- Bell inequality violation occurs near the quantum-critical point.
- Maximum violation is observed for infinite-range interactions ($=0$).
- Quantum-critical correlations lead to squeezing of collective-spin fluctuations.

## Abstract

When a collection of distant observers share an entangled quantum state, the statistical correlations among their measurements may violate a many-body Bell inequality, demonstrating a non-local behavior. Focusing on the Ising model in a transverse-field with power-law ($1/r^\alpha$) ferromagnetic interactions, we show that a permutationally invariant Bell inequality based on two-body correlations is violated in the vicinity of the quantum-critical point. This observation, obtained via analytical spin-wave calculations and numerical density-matrix renormalization group computations, is traced back to the squeezing of collective-spin fluctuations generated by quantum-critical correlations. We observe a maximal violation for infinite-range interactions ($\alpha=0$), namely when interactions and correlations are themselves permutationally invariant.