# Analytical results on the Heisenberg spin chain in a magnetic field

## TL;DR

This paper develops a recursive, algebraic method to analytically compute the ground state magnetization and free energy series of the Heisenberg spin chain in a magnetic field, with convergence properties studied numerically.

## Contribution

It introduces a closed-form recursive approach to solve Bethe equations for the Heisenberg chain, enabling analytical series expansions of physical quantities.

## Key findings

- Series for magnetization and free energy converge in their respective domains.
- Recursion formulas derived from Bethe equations allow algebraic computation of coefficients.
- Method applies to models with complex Bethe root distributions.

## Abstract

We obtain the ground state magnetization of the Heisenberg and XXZ spin chains in a magnetic field $h$ as a series in $\sqrt{h_c-h}$, where $h_c$ is the smallest field for which the ground state is fully polarized. All the coefficients of the series can be computed in closed form through a recurrence formula that involves only algebraic manipulations. The radius of convergence of the series in the full range $0<h\leq h_c$ is studied numerically. To that end we express the free energy at mean magnetization per site $-1/2\leq \langle \sigma^z_i\rangle\leq 1/2$ as a series in $1/2-\langle \sigma^z_i\rangle$ whose coefficients can be similarly recursively computed in closed form. This series converges for all $0\leq \langle \sigma^z_i\rangle\leq 1/2$. The recurrence is nothing but the Bethe equations when their roots are written as a double series in their corresponding Bethe number and in $1/2-\langle \sigma^z_i\rangle$. It can also be used to derive the corrections in finite size, that correspond to the spectrum of a free compactified boson whose radius can be expanded as a similar series. The method presumably applies to a large class of models: it also successfully applies to a case where the Bethe roots lie on a curve in the complex plane.