A tensor network study of the complete ground state phase diagram of the spin-1 bilinear-biquadratic Heisenberg model on the square lattice

Dear editor,

Thank you for forwarding the referee reports on the manuscript titled "A tensor network study of the complete ground state phase diagram of the spin-1 bilinear-biquadratic Heisenberg model on the square lattice".

Both referees are positive about the paper and recommend publication after we address the points discussed below.

The first referee's concern about the absence of the use of variational optimization will be discussed in our direct reply to the first referee. Additionally, the same reply contains an extensive pointwise discussion on the referee's excellent suggestions for improvements regarding the amount of jargon in the paper. The list of changes made in accordance to the referee's suggestions can be found below. The second referee's concern about how the two-dimensional AKLT state relates to the Haldane phase we find in the BBH model will be addressed in our direct reply to the second referee. In addition, we have added an extra paragraph to the paper clarifying the issue raised by the referee.

We hereby submit a revised version of our paper containing the changes listed below to SciPost.

Sincerely yours,
Ido Niesen and Philippe Corboz

List of changes:

1. Introduction:
- added a footnote explaining the term "classical ground state",
- extended a sentence to explain the meaning of spin-nematic states,
- rewrote the second paragraph to include a more explicit description of Hamiltonian and corresponding operators, and,
- added an explicit reference to the iPEPS phase diagram.

3. The model: rewrote most of this section to make it more accessible to non-experts. In particular, we
- moved the definition of the quadrupolar operators forward,
- added a figure explaining the meaning of the discs (quadrupolar states),
- defined the action of the time-reversal operator explicitly, and elaborated on the notion of spin-nematic states (in relation to time-reversal invariance),
- added a few sentences on the relation between spin-nematics and nematics in liquid crystals,
- inserted J(S) and J(Q) directly into Eq. 4,
- explicitly listed all SU(3)-symmetric points, and,
- added a sentence and a footnote on the Gell-Mann matrices and their relation to SU(3).

4. Previous studies:
- marked the SU(3) points with black dots in Fig. 3,
- added note on the color coding in Fig. 4, and,
- changed 'corner' to 'octant'.

5. Description of all phases:
- added a paragraph explaining that the Haldane phase we find does not correspond to the two-dimensional AKLT state.