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arXiv:1912.10327 · Physical review. B./Physical review. B

Deformations of the boundary theory of the square-lattice AKLT model

John M. Martyn, Kohtaro Kato, Ángelo Lucia

Published Jul 10, 2020• 1 citation
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Abstract

It remains an open problem to prove the existence of a spectral gap in the 2D square-lattice Affleck-Kennedy-Lieb-Tasaki (AKLT) model. Here, the authors express the boundary state of the 2D AKLT model in terms of a classical loop model. By simulating the loop model and numerically analyzing the AKLT boundary state, the authors evidence a spectral gap in the square-lattice AKLT model. In addition, the authors vary the parameters of the loop model and study its three distinct phases.

DOI
10.1103/physrevb.102.035121

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It remains an open problem to prove the existence of a spectral gap in the 2D square-lattice Affleck-Kennedy-Lieb-Tasaki (AKLT) model. Here, the authors express the boundary state of the 2D AKLT model in terms of a classical loop model. By simulating the loop model and numerically analyzing the AKLT boundary state, the authors evidence a spectral gap in the square-lattice AKLT model. In addition, the authors vary the parameters of the loop model and study its three distinct phases.
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.
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 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 S=1 Affleck-Kennedy-Lieb-Tasaki (AKLT) quantum spin chain was the first rigorous example of an isotropic spin system in the Haldane phase. The conjecture that the S=3/2 AKLT model on the hexagonal lattice is also in a gapped phase has remained open, despite being a fundamental problem of ongoing relevance to condensed-matter physics and quantum information theory. Here we confirm this conjecture by demonstrating the size-independent lower bound Δ>0.006 on the spectral gap of the hexagonal model with periodic boundary conditions in the thermodynamic limit. Our approach consists of two steps combining mathematical physics and high-precision computational physics. We first prove a mathematical finite-size criterion which gives an analytical, size-independent bound on the spectral gap if the gap of a particular cut-out subsystem of 36 spins exceeds a certain threshold value. Then we verify the finite-size criterion numerically by performing state-of-the-art DMRG calculations on the subsystem.
Using exact diagonalization and tensor network techniques, we compute the gap for the Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian in one and two spatial dimensions. Tensor network methods are used to extract physical properties directly in the thermodynamic limit, and we support these results using finite-size scalings from exact diagonalization. Studying the AKLT Hamiltonian perturbed by an external field, we show how to obtain an accurate value of the gap of the original AKLT Hamiltonian from the field value at which the ground state verifies e0<0{e}_{0}<0e0​<0, which is a quantum critical point. With the tensor network renormalization group methods we provide direct evidence of a finite gap in the thermodynamic limit for the AKLT models in the one-dimensional chain and two-dimensional hexagonal and square lattices. This method can be applied generally to Hamiltonians with rotational symmetry, and we also show results beyond the AKLT model.