Aperiodic spin chains at the boundary of hyperbolic tilings

TL;DR

This paper investigates a boundary spin chain model with aperiodic couplings inspired by hyperbolic tilings, revealing power-law decay of correlations, a logarithmic scaling of entanglement entropy, and no phase transition in mutual information.

Contribution

It introduces a novel boundary spin chain model reflecting hyperbolic tiling inflation rules and analyzes its properties using strong-disorder RG techniques.

Findings

Two-point functions decay as a power-law with exponent one.

Entanglement entropy scales as the logarithm of N, the number of degrees of freedom.

Mutual information shows no phase transition at finite distances.

Abstract

In view of making progress towards establishing a holographic duality for theories defined on a discrete tiling of the hyperbolic plane, we consider a recently proposed boundary spin chain Hamiltonian with aperiodic couplings that are chosen such as to reflect the inflation rule, i.e. the construction principle, of the bulk tiling. As a remnant of conformal symmetry, the spin degrees of freedom are arranged in multiplets of the dihedral group under which the bulk lattice is invariant. For the boundary Hamiltonian, we use strong-disorder RG techniques and evaluate correlation functions, the entanglement entropy and mutual information for the case that the ground state is in an aperiodic singlet phase. We find that two-point functions decay as a power-law with exponent equal to one. Furthermore, we consider the case that the spin variables transform in the fundamental representation of $SO(N)$, leading to a gapless system, and find that the effective central charge obtained from the entanglement entropy scales as $\ln N$, reflecting the number of local degrees of freedom. We also determine the dependence of this central charge on the parameters specifying the bulk tiling. Moreover, we obtain an analytical expression for the mutual information, according to which there is no phase transition at any finite value of the distance between the two intervals involved.