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

TL;DR

This paper investigates the complex-analytic properties of the finite-volume one-point function in the critical Ising chain, revealing a natural boundary and connections to Lambert series and Borel resummation.

Contribution

It demonstrates the natural boundary of the analytically continued one-point function and links its singularities to Lambert-type divisor sums, providing new insights into its asymptotics.

Findings

The one-point function has a natural boundary along the negative real axis.

Its singular behavior matches a Lambert-type series for the odd-divisor-squared sum.

Large-$N$ asymptotics of the form factor are also derived and analyzed.

Abstract

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.