Incorporating non-local anyonic statistics into a graph decomposition

- Treats a difficult problem (non solvable interacting model)
- Mathematically involved

- The novelty/the objectives are not clearly given.
- For a novice, the steps are not precisely explained, the vocabulary is not simply defined.

The authors derive a perturbative series expansion of the toric code in the presence of a magnetic field h, that they use to calculate the ground state energy to order 10 and charge and flux gap to order 9, and to order 10 for hy=0. The terms are in powers of hx, hy and hz. After a review of the toric code model, and a setting of the notations in Sec. 1, they derive a pCUT expansion, for which the authors are specialists, allowing to construct an effective Hamiltonian that does not couple sectors with different quasi particle numbers and use it in a linked-cluster expansion using hypergraph decomposition.

This article is mathematically/algorithmically involved and results from several previous articles using the same technic (which is not a criticism in itself) and the same vocabulary. However, despite its length, the authors fail to make it self-consistent and easy to follow. I am convinced the calculations are correct and involved, but the authors should give more detailed explanation of the articulations between technical parts, and more careful definitions of the objects they use, with a thought for those who have not read all the previous articles. I list questions and remarks containing examples of this below.

On another side, the novelty of the calculations is not clearly stated. What was the initial question that has motivated this work (except applying a well-mastered technique to a new model). The phase diagram of this model is already known (ref 25). What information can be obtained from the series ? Do the gap close at some magnetic field, or does the series diverge before ?
Is is the first time such a method is applied to a model with anyons ? If I understand correctly, the way fractionalized excitations are taken into account when (hyper ?)graphs contain loops is inherent to the linked-cluster decomposition, and is not realized through a special adaptation of the method, but the authors elegantly evidence how it appears through the spin background ?

I think this article deserves publication in SciPost, after answering my comments.

Questions and remarks:

- Sec. 2.1: the ground state energy E_O^{tc} is said to be -N/2, but from Eq. (2), I would have expected -N, as N is the number of plaquettes and of stars.

- In Sec. 3, the matrix elements of T(m) of Eq. 15 are caculated. But I don't understand where it leads. Where are the C(m) evaluated ? And how are they related to H^{TCF} ? Is Sec. 3 only a preparation to Sec. 4, where the real calculations are done, and does it only explain the calculation of the T(m) matrix elements ? Then, the title of subsections 3.1 and 3.2 (GS and 1QP energy) are misleading.
The sequence of steps leading to the expansions, and their logic, should be explained more carefully.

- The last paragraph of p.7 is hard to folllow with the introduction of bonds $b^(\alpha)}$.
Would the notation $b_i^(\alpha)}$ not be more adapted, with a sum over $i$ and $\alpha$ replacing the sums over $b^(\alpha)$ ? (this notation with sites in index is already used in the caption of Fig 4, but not in the figure, and should be used in Table 1 for example)
It seems that "bond" designs two different objects when applied to a site $i$ and to an operator $b^(\alpha)$. $b^(\alpha)$ is a composite object containing a site $i$ and 0 or 2 vortice, and 0 or 2 plaquettes, also called 'bond'.

- What means 'ordered set' for such a b-bond ? Is it not enough to precise that $\tau$ first acts on site, then on flux sites and finally on charge sites ?

- p. 14: between Eqs 50 and 51, "note that the sum runs over all subclusters c of C and not over all distinct sub-clusters of C". What is the difference between the two expressions ?
In the sum of Eq. 50, the symbol should be "included and different from", as C is included in itself.

- p. 14: a definition of a cluster is required, and/or of H_{eff}.
Does a cluster include sites ? b-bonds ? plaquettes ? Same question for H^c_{eff}. Do a "cluster" with no plaquette and no star has H_{eff}=0 ? These notations are used all along p. 14, and only explained p. 17, with many references to other articles. An hypergraph and a König representation of an hypergraph could be better defined.

- After Eq. 56, what does a "local state" represent ? A cluster is thus not just a collection of bonds, but includes a spin-configuration ? or is it simply the x,y or z from the b-bonds ?

- Could the authors precise in which sense the bonds joins multiple sites not necessarily in a symmetric way ? (p.15)

- Is the cluster expansion realised on the initial Hamiltonian or on the pCUT Hamiltonian H_{eff} (which seems to be the case according to Eq. 51-53). Then again, the C(m) coefficients are required, but their obtention not detailed.

- H^{eff} -> H_{eff}