Analyticity, asymptotics and natural boundary for a one-point function of the finite-volume critical Ising chain

This note reports the following observation: the finite-volume expectation value of the spin operator (the one-point function) between the $\mathbb{Z}_2$-even and odd ground states in the critical periodic Ising chain, when continued as a complex-analytic function of the system length $N$ through the Borel resummation of its large-$N$ expansion, has a natural boundary of analyticity along the negative real axis. The singular behavior near the negative real axis, after an exponential map, is the same as that of a Lambert-type series for the odd-divisor-squared sum near the unit circle $|z|=1$. The same divisor sum also governs the strengths of the Borel discontinuities of the one-point function's factorially-divergent large-$N$ asymptotics. We also report the all-order large-$N$ asymptotics of the leg function for the finite-volume spin-operator form factor, and the similarities to certain known quantities in the literature.