A brief note on the G$_2$ Affleck-Kennedy-Lieb-Tasaki chain

Referee Report

The manuscript entitled “A brief note on the G2 Affleck–Kennedy–Lieb–Tasaki chain” is an excellent short paper on the valence bond solid (VBS) with G2 symmetry. The authors provide a concise and well-rounded introduction and review of the construction by Affleck–Kennedy–Lieb–Tasaki.

The fundamental representation of G2 is 7-dimensional. On each lattice site, two such representations are introduced. In the decomposition of their tensor product, the representations 1 ⊕ 7 ⊕ 14 ⊕ 27 appear, of which all except the 14-dimensional adjoint representation are projected out.

The resulting rank-3 tensor serves as the basic building block for the multi-spin state composed of 14-dimensional adjoint representations: neighbouring tensors are connected by contracting their adjacent 7-dimensional components into singlets. Locally, the product of two adjoint (14-dimensional) representations may contain 1 ⊕ 14 ⊕ 27 ⊕ 77 ⊕ 77′. However, due to the specific construction of the VBS state — starting from four 7-dimensional representations of which two are contracted to a singlet — at most 1 ⊕ 7 ⊕ 14 ⊕ 27 can appear, resulting in 1 ⊕ 14 ⊕ 27.

The parental Hamiltonian, designed to take this constructed VBS state as its ground state, is chosen to annihilate the 1 ⊕ 14 ⊕ 27 components while assigning finite positive energy to the 77 ⊕ 77′ components. The authors investigate the two-point correlation function via a transfer matrix method and find it to be non-vanishing only for nearest neighbours. However, a string order parameter can be defined, which turns out to be finite.

The paper is well written, very accessible, and the results are, to my knowledge, new. The main body is presented with minimal technical detail — the authors rely primarily on fundamental properties of tensor product decompositions and the eigenvalues of the quadratic Casimir operator. The actual computations begin with the evaluation of the correlation function and the string order parameter, with technical details deferred to the appendices.

I would, however, suggest that the authors comment on the uniqueness of the ground state (not just of the VBS state itself). Typically, one would need to prove that the excited states remain gapped, even in the thermodynamic limit, to confirm uniqueness — a technically demanding part of the original AKLT construction. It would be valuable to clarify whether the G2 symmetry affects this aspect or simplifies the situation in any way.

I recommend this manuscript for publication in SciPost.

discovered "typos":

can be regarded as the G2 generalisation of the famous AKLT construction. -> can be regarded as the G2 version/analog of the famous AKLT construction.

ie -> i.e.

parent Hamiltonian -> parental Hamiltonian

hidden antiferro-magnetic order exist. -> hidden antiferro-magnetic order exists.

subindex -> index or subscript