The AKLT models on the singly decorated diamond lattice and two degree-4 planar lattices are gapped
arXiv:2010.03137
Abstract
Recently various 2D AKLT models have been shown to be gapped, including the one on the hexagonal lattice. Here we report on a non-trivial 3D AKLT model which consists of spin-2 entities on the diamond lattice sites and one single spin-1 entity between every neighboring spin-2 site. Although the nonzero gap problem for the uniformly spin-2 AKLT models on the diamond and square lattices is still open, we are able to establish the existence of the gap for two planar lattices, which we call the inscribed square lattice and the triangle-octagon lattice, respectively. So far, these latter two models are the only two uniformly spin-2 AKLT models that have a provable nonzero gap above the ground state. We also discuss some attempts in proving the gap existence on both the square and kagome lattices. In addition, we show that if one can solve a finite-size problem of a weighted AKLT Hamiltonian and if the gap is larger than certain threshold, then the model on the square lattice is gapped in the thermodynamic limit. The threshold of the gap scales inversely with the linear size of the finite-size problem.
arXiv comment15 pages, 16 figures, 4 tables
References in corpus (12)
- Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
- Novel schemes for measurement-based quantum computation
- Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice is a Universal Quantum Computational Resource
- Measurement-based quantum computer in the gapped ground state of a two-body Hamiltonian
- Quantum computational capability of a 2D valence bond solid phase
- Demonstrating the AKLT spectral gap on 2D degree-3 lattices
- Spectral gaps of AKLT Hamiltonians using Tensor Network methods
- Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice
- Hybrid valence-bond states for universal quantum computation
- A class of two-dimensional AKLT models with a gap
- Quantum computational universality of spin-3/2 Affleck-Kennedy-Lieb-Tasaki states beyond the honeycomb lattice
- AKLT models on decorated square lattices are gapped