A minimal tensor network beyond free fermions

We would like to thank all three referees for their reports and their valuable comments and suggestions. We are delighted to see that the referees appreciate our work and recommend its publication. Before we address the more detailed comments, we would like to discuss the main concerns raised by the referees ('weaknesses').

Regarding the concern of referee 2, that there might be limited interest in our results, we would like to mention that since the initial submission of our manuscript, we had the opportunity to present our results at conferences and were met with genuine interest from the 'strongly correlated electrons/tensor network methods'-community, mostly due to the applicability of the duality mapping to more general statistical mechanics systems. We thus think that our case study of this minimal model will be appreciated by colleagues working in (frustrated) magnetism and statistical mechanics.

Referee 1 remarked that a numerical study of the phase diagram would be desirable. In fact, we have in the meantime started to numerically investigate the phase diagram of our model and have obtained promising first results using iPEPS methods. However, the focus of the current manuscript is on establishing the duality mapping between spin systems and fermionic systems beyond the Gaussian limit and on demonstrating the power of this method by constructing the phase diagram without relying on numerics. Thus, we would like to keep an in-depth numerical study of the phase diagram separate and present it elsewhere.

Referee 3 remarked that our work is restricted to a specific model while also pointing out that the idea of the duality are not limited to the specific model and that it would be interesting to see it applied to more general spin models. We fully agree with the referee and -- given the positive feedback we received so far -- expect that the method might be picked up by other researchers for other spin models of interest.

Apart from these high-level remarks referees 1 and 2 raised some further points, while referee 3 was entirely satisfied with the manuscript.

Referee 1 remarked that an appendix explaining the mapping between spins and fermions would be helpful and that it is not entirely clear which aspects of the phase diagram are calculated and which are 'guessed'. We have added a paragraph at the end of the phase diagram discussion that clarifies this point. To summarise, the topology of the phase diagram, as well as the approximate location of the Lifshitz point, and the exact location of the transition points along the a=0 and b=0 line are calculated/inferred. The shape of the transition lines was 'guessed' using preliminary numerical studies. Regarding the fermion-spin mapping, we would like to point out that we have already provided an appendix on that topic in another publication (Topological dualities via tensor networks,C. Wille, J. Eisert, A. Altland, Phys. Rev. Research 6, 013302, 2024, DOI: https://doi.org/10.1103/PhysRevResearch.6.013302) that we refer to in the current manuscript. The referee remarked that this appendix is not sufficient as well. We agree with the referee, that the appendix alone is not sufficient, as it only explains details on the consistent ordering of fermionic modes, while the main procedure of the mapping itself also relies on the general understanding of formulating a fermionic tensor network as a Grassmann integral and the procedure to identify tensor indices with Grassmann variables. Both of these concepts are explained in Section II.B of the mentioned reference, which we previously failed to point out. We have now included an explicit reference to that section and hope that this will provide the necessary background information for those who want to understand the working mechanisms behind the mapping rather than just a summary of how it is applied.

Referee 2 had three minor comments that we all addressed as will be clear from the list of changes below.

Changes addressing comments by referee 1
- include reference to sec II B of Ref [16] in second paragraph of section 2
- include sentence at the end of sec 3.2 specifying which parts of the phase diagram are calculated and which are 'guessed'

Changes recommended by referee 2
- modified caption of Fig. 1 to include statement that b controls the strength of nonlinearity
- defined the acronym ANNNI towards the end of the introduction
- rephrased the sentence "By introducing local non-linearities into the free tensor network, specified by a single parameter, a, with the non-linearity strength modulated by another parameter, b, we have set up a rather simple two-parameter model." to "Starting from a free tensor network specified by a single parameter $a$ and introducing local non-linearities modulated by a second parameter $b$ we have set up a rather simple two-parameter model beyond the free fermion limit. " at the bottom of page 11 to avoid ambiguity

Other changes
-fixed typo beta ^2-> beta in inline inequality in sec. 3.2
-changed 'tricritical point' to 'multicritical point' throughout (The Lifshitz point in our 2-parameter model is a multicritical point and also a triple point. However, it not a tricritical point.)