The area law and real-space renormalization

TL;DR

This paper examines how real-space renormalization techniques handle quantum systems with area law correlations, highlighting the importance of central points in the system for accurate local results, supported by numerical examples.

Contribution

It analyzes the accuracy scaling of renormalization methods for gapped systems and introduces the significance of central points in achieving better local approximations.

Findings

Central points have more accurate local quantities.

Numerical results confirm the importance of system center.

Spatial anisotropy affects accuracy in renormalization.

Abstract

Real-space renormalization-group techniques for quantum systems can be divided into two basic categories - those capable of representing correlations following a simple boundary (or area) law, and those which are not. I discuss the scaling of the accuracy of gapped systems in the latter case and analyze the resultant spatial anisotropy. It is apparent that particular points in the system, that are somehow `central' in the renormalization, have local quantities that are much closer to the exact results in the thermodynamic limit than the system-wide average. Numerical results from the tree-tensor network and tensor renormalization-group approaches for the 2D transverse-field Ising model and 3D classical Ising model, respectively, clearly demonstrate this effect.