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

TL;DR

This paper tackles holographic physics within a discrete hyperbolic lattice by studying the (2+1)-D AdS Ising model using MPS/MPO methods. The authors deploy DMRG to obtain ground states up to ~232 spins and TEBD with MPOs to explore dynamics, focusing on bulk phase structure, boundary correlators, and entropy scaling. They find a bulk ordered/disordered transition, power-law boundary correlations in the disordered phase, and logarithmic boundary entanglement at criticality (with central charge $c\approx 1$) while the full system shows volume-law entanglement; OTOCs indicate scrambling behavior. The results illustrate both the potential and the current limitations of tensor-network approaches for discrete holographic models, highlighting the need for larger bond dimensions or quantum algorithms to access more detailed boundary CFT features.

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). Our spatial lattices correspond to regular tessellations of hyperbolic space with coordination number seven. We find the ground state of this model using the Density Matrix Renormalization Group (DMRG) algorithm which allowed us to probe lattices that range in size up to 232 sites. We explore the bulk phase diagram of the theory 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 the anti-de Sitter background. By tracing out the bulk indices, we are able to compute the density matrix for the boundary theory. At the critical point, we find the entanglement entropy exhibits the logarithmic dependence of boundary length expected for a one-dimensional CFT 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 highly connected systems. We also measure Out-of-time-Ordered-Correlators (OTOCs) to explore the scrambling behavior of the theory.

Figures (15)

• Figure 1: A summary of our MPS construction adapted to hyperbolic tessellations.
• Figure 2: Bulk phase transition for 12x12 square lattice flat space Ising model with $m = 0.001$
• Figure 3: Energy of the ground state v/s bond-dimension $\chi$ for $J_{zz}/J_x = 0.31$ for three layer lattice
• Figure 4: Bulk phase transition for Ising model on five-layer tesselated $AdS_3$ with $m = 0.001$.
• Figure 5: Magnetization for Ising model on five-layer tesselated $AdS_3$ with $m = 0.001$.