Quantum Ising Model on $(2+1)-$Dimensional Anti$-$de Sitter Space using Tensor Networks

TL;DR

This paper investigates the quantum Ising model on (2+1)-dimensional anti-de Sitter space using tensor network methods, revealing phase transitions, holographic boundary correlations, and scrambling behavior.

Contribution

It applies tensor network techniques to a hyperbolic space quantum Ising model, analyzing phase structure, boundary correlations, entanglement scaling, and chaos indicators.

Findings

Identified disordered and ordered phases with a phase transition.

Observed power law boundary correlations consistent with holography.

Measured volume law entanglement and OTOCs indicating scrambling.

Abstract

We study the quantum Ising model on (2+1)-dimensional anti-de Sitter space using Matrix Product States (MPS) and Matrix Product Operators (MPOs). We explore the bulk phase diagram of the theory on regular tessellations of hyperbolic space with coordination number seven and find disordered and ordered phases separated by a phase transition. We find that the boundary-boundary spin correlation function exhibits power law scaling deep in the disordered phase of the Ising model consistent with holography. At the critical point, we find the boundary entanglement entropy scales logarithmically with subsystem size but away from this, we see a linear scaling. In comparison, the full system exhibits a volume law scaling, which is expected in chaotic and/or highly connected systems. We also measure Out of time Ordered Correlators (OTOCs) to explore the scrambling behavior of the theory.