Exact Solution to the Moment Problem for the XY Chain

TL;DR

This paper provides an exact solution to the moment problem for the XY chain, revealing the analytic structure of the generating function and challenging existing extrapolation methods like CMX.

Contribution

It offers explicit forms for moments, the cumulant generating function, and the Resolvent Operator, and analyzes their complex analytic properties.

Findings

The generating function has a finite radius of convergence and a specific algebraic decay.

The Resolvent exhibits a branch cut and essential singularity near the ground state energy.

CMX extrapolation methods are shown to be flawed and inapplicable in this context.

Abstract

We present the exact solution to the moment problem for the spin-1/2 isotropic antiferromagnetic XY chain with explicit forms for the moments with respect to the Neel state, the cumulant generating function, and the Resolvent Operator. We verify the correctness of the Horn-Weinstein Theorems, but the analytic structure of the generating function <e^{-tH}> in the complex t-plane is quite different from that assumed by the "t"-Expansion and the Connected Moments Expansion due to the vanishing gap. This function has a finite radius of convergence about t=0, and for large t has a leading descending algebraic series E(t)-E_0 ~ At^{-2}. The Resolvent has a branch cut and essential singularity near the ground state energy of the form G(s)/s ~ B|s+1|^{-3/4} exp(C|s+1|^{1/2}). Consequently extrapolation strategies based on these assumptions are flawed and in practise we find that the CMX methods are pathological and cannot be applied, while numerical evidence for two of the "t"-expansion methods indicates a clear asymptotic convergence behaviour with truncation order.