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arXiv:1911.01410 · Physical Review Letters

Demonstrating the Affleck-Kennedy-Lieb-Tasaki Spectral Gap on 2D Degree-3 Lattices

Nicholas Pomata, Tzu-Chieh Wei

Published Apr 28, 2020• 34 citations
arXivHTMLPDF

Abstract

We report results on solving a long outstanding problem-whether the two-dimensional spin-3/2 antiferromagnetic valence-bond model of Affleck, Kennedy, Lieb, and Tasaki (AKLT) possesses a nonzero gap above its ground state. We exploit a relation between the anticommutator and sum of two projectors and apply it to ground-space projectors on regions of the honeycomb lattice. After analytically reducing the complexity of the resultant problem, we are able to use a standard Lanczos method to establish the existence of a nonzero gap. This approach is also successfully applied to spin-3/2 AKLT models on other degree-3 semiregular tilings, namely, the square-octagon, star, and cross lattices, where the complexity is low enough that exact diagonalization can be used instead. In addition, we close the previously open cases of hybrid AKLT models on the singly decorated honeycomb and singly decorated square lattices.

DOI
10.1103/physrevlett.124.177203

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arXiv:1501.03512•Physical Review Letters•Jun 18, 2015•22 citations
We report results on solving a long outstanding problem-whether the two-dimensional spin-3/2 antiferromagnetic valence-bond model of Affleck, Kennedy, Lieb, and Tasaki (AKLT) possesses a nonzero gap above its ground state. We exploit a relation between the anticommutator and sum of two projectors and apply it to ground-space projectors on regions of the honeycomb lattice. After analytically reducing the complexity of the resultant problem, we are able to use a standard Lanczos method to establish the existence of a nonzero gap. This approach is also successfully applied to spin-3/2 AKLT models on other degree-3 semiregular tilings, namely, the square-octagon, star, and cross lattices, where the complexity is low enough that exact diagonalization can be used instead. In addition, we close the previously open cases of hybrid AKLT models on the singly decorated honeycomb and singly decorated square lattices.
The nonzero spectral gap of the original two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) models has remained unproven for more than three decades. Recently, Abdul-Rahman et al. (arXiv:1901.09297) provided an elegant approach and proved analytically the existence of a nonzero spectral gap for the AKLT models on the decorated honeycomb lattice (for the number nnn of spin-1 decorated sites on each original edge no less than 3). We perform calculations for the decorated square lattice and show that the corresponding AKLT models are gapped if $n\ensuremath{\ge}4$. Combining both results, we also show that a family of decorated hybrid AKLT models, whose underlying lattice is of mixed vertex degrees 3 and 4, are also gapped for $n\ensuremath{\ge}4$. We develop a numerical approach that extends beyond what was accessible previously. Our numerical results further improve the nonzero gap to $n\ensuremath{\ge}2$, including the establishment of the gap for n=2n=2n=2 in the decorated triangular and cubic lattices. The latter case is interesting, as this shows that the AKLT states on the decorated cubic lattices are not N\'eel ordered, in contrast to the state on the undecorated cubic lattice.
We study the spectral properties of a family of quantum antiferromagnets on two-dimensional (2D) lattices. This family of models is obtained by a deformation of the well-studied 2D quantum antiferromagnetic model of Affleck, Kennedy, Lieb, and Tasaki (AKLT); they are described by two-body, frustration-free Hamiltonians on a three-colorable lattice of spins. Although the existence of a spectral gap in the 2D AKLT model remains an open question, we rigorously prove the existence of a gap for a subset of this family of quantum antiferromagnets. Along with providing progress for the gap problem in AKLT-type antiferromagnets in 2D, this result has implications for the theory of quantum computation as it provides a family of two-body Hamiltonians for which the ground state is a resource for universal quantum computation and for which a spectral gap is proven to exist.
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.
The two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) model on a honeycomb lattice has been shown to be a universal resource for quantum computation. In this valence bond solid, however, the spin interactions involve higher powers of the Heisenberg coupling (S[over →](I)·S[over →](j))(n), making these states seemingly unrealistic on bipartite lattices, where one expects a simple antiferromagnetic order. We show that those interactions can be generated by orbital physics in multiorbital Mott insulators. We focus on t(2g) electrons on the honeycomb lattice and propose a physical realization of the spin-3/2 AKLT state. We find a phase transition from the AKLT to the Néel state on increasing Hund's rule coupling, which is confirmed by density matrix renormalization group simulations. An experimental signature of the AKLT state consists of protected, free S=1/2 spins on lattice vacancies, which may be detected in the spin susceptibility.