Anyon statistics in a lattice Moore-Read state

I’m trying to understand this paper¹, in particular the remark after Eq. 26.

Let me rephrase the problem. According to the paper, one can write the Berry phase as
$$
theta_B=ioint_Gammafrac{1}{C}frac{partial C}{partial w_k} mathrm{d} w_k +c.c.=ioint_Gammafrac{partial ln C}{partial w_k} mathrm{d} w_k +c.c.
$$
where $Gamma$ is the closed path along which we move anyon $k$, whose position is given by a complex number $w_k=x_k+iy_k$ and $C$ is the normalization constant (I consider the simplest case of nondegenerate states, so I drop the index $alpha$). Here, $C=sqrt{sum_{n}Psi_{n}bar{Psi}_{n}}$ where $Psi_{n}$ depend on $w_k$ and not $bar{w}_k$ (the bar denotes complex conjugation). Likewise, $bar{Psi}_{n}$ depend on $bar{w}_k$ and not $w_k$. I’m fine with the derivation of the above equation. But now, according to the paper, if we show that $C$ is lattice-periodic in $w_k$, $bar{w}_k$ (which they show numerically, see Fig. 4b), then $theta_B$ phase is zero, and I do not understand why.

I know that the case considered in the paper is the lattice version of the situation considered in this paper². In this case, the system is continuous, and thus instead of the periodicity they need to show that $C$ is a constant. And then it is fairly obvious to me that the phase is zero, since all the derivatives vanish. But I don’t know how to extend it to the case when we have periodic $C$ instead of a constant.

In a naive interpretation of the lattice case, we can say that:
$$
oint_Gammafrac{partial ln C}{partial w_k} mathrm{d} w_k=ln C |_{end}-ln C |_{start}
$$
which vanishes, because the start and the end of the path coincide. This reasoning would be correct in the case of real integrals, but I think it does not work for complex contour integrals.

Consider the following counterexample: just one basis state and one anyon (thus I drop the indices) with $Psi=w$. Let us consider $Gamma$ parametrized by $w=e^{it}$, so that $C=sqrt{w bar{w}}=1$ everywhere on the path. A constant is clearly periodic on the path, so according to the naive reasoning the integral should yield 0. But, calculating explicitly
$$
oint_Gammafrac{1}{C}frac{partial C}{partial w} mathrm{d} w=
oint_Gammafrac{1}{2C^2}frac{partial C^2}{partial w} mathrm{d} w=
oint_Gammafrac{1}{2w bar{w}}frac{partial w bar{w}}{partial w} mathrm{d} w=
oint_Gammafrac{1}{2w} mathrm{d} w= pi i
$$
which is not zero. So the periodicity within the path is not enough to guarantee the vanishing of the result. But the lattice periodicity is a stronger claim (a periodicity in two directions). However, I don’t know how to use it to show that the phase vanishes.

Maybe this is a simple issue within the complex analysis. I am not very good in this subject, and all the books I have found focus on the holomorphic functions, while $C$ is clearly not holomorphic. If anyone can recommend a good book where these subjects are discussed, it would also be fine.

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1. S. Manna, J. Wildeboer, G. Sierra & A. E. B. Nielsen, "Non-Abelian quasiholes in lattice Moore-Read states and parent Hamiltonians", Phys. Rev. B 98, 165147 (2018), arXiv:1807.11222.
2. P. Bonderson, V. Gurarie & C. Nayak, "Plasma Analogy and Non-Abelian Statistics for Ising-type Quantum Hall States", Phys. Rev. B 83, 075303 (2011), arXiv:1008.5194.