Entanglement blossom in a simplex matryoshka

TL;DR

This paper generalizes the rainbow chain to higher dimensions, revealing complex entanglement structures and interpolating between models relevant to black hole physics and holography.

Contribution

It introduces a higher-dimensional, frustration-free entanglement model based on simplices, with an analytically diagonalizable Hamiltonian using Young operators.

Findings

Analytical diagonalization of the effective Hamiltonian.

Model interpolates between SYK and XX spin chain.

Introduces curvature via disinclination defects in the lattice.

Abstract

Exotic entanglement entropy scaling properties usually come with interesting entanglement structures in real space and novel metrics of the spacetime lattice. One prominent example is the rainbow chain where lattice sites symmetric about the center form entangled Bell pairs due to an effective long-range coupling from the strong inhomogeneity of the coupling strength. This manuscript generalizes the rainbow chain to higher dimensional space on lattices with Hausdorff dimension one and enlarged local Hilbert space keeping the Hamiltonian frustration free. The effective Hamiltonian from the Schrieffer-Wolf transformation is given by a stacking of layers of $k$-simplices with $0$-dimensional (fully-connected) antiferromagnetic Hamiltonians, which can be diagonalized analytically with Young operators. The original lattice can be obtained from proliferating disinclination defects in a regular $k$-dimensional cubical lattice, which introduces curvature at the center of the lattice. The model interpolates between the SYK model and the free-fermionic XX spin chain, and hence might be potentially useful in understanding black hole physics and holography.