Holographic analysis of boundary correlation functions for the hyperbolic-lattice Ising model

TL;DR

This paper explores the boundary correlation functions of the hyperbolic-lattice Ising model using holographic principles and CTMRG, revealing power-law decay with oscillations and insights into scaling dimensions influenced by lattice curvature.

Contribution

It introduces a holographic analysis of boundary correlations in the hyperbolic-lattice Ising model, linking geometric properties to correlation scaling and examining CTMRG cutoff effects.

Findings

Boundary correlation functions decay as a power law with oscillations.

Bulk correlations decay exponentially.

Long-distance boundary correlations are well captured even with small bond dimension.

Abstract

We analyze boundary spin correlation functions of the hyperbolic-lattice Ising model from the holographic point of view. Using the corner-transfer-matrix renormalization group (CTMRG) method, we demonstrate that the boundary correlation function exhibits power-law decay with quasi-periodic oscillation, while the bulk correlation function always decays exponentially. On the basis of the geometric relation between the bulk correlation path and distance along the outer edge boundary, we find that scaling dimensions for the boundary correlation function can be well explained by the combination of the bulk correlation length and background curvatures inherent to the hyperbolic lattice. We also investigate the cutoff effect of the bond dimension in CTMRG, revealing that the long-distance behavior of the boundary spin correlation is accurately described even with a small bond dimension. In contrast, the sort-distance behavior rapidly loses its accuracy.