Digamma-Function Representation of the Ground-State Energy in Antiferromagnetic Heisenberg $\mathrm{XXX}_s$ Spin Chains

TL;DR

This paper presents a unified digamma-function based formula for the ground-state energy of antiferromagnetic Heisenberg XXX_s spin chains, generalizing known series for different spin representations.

Contribution

It introduces a novel, unified analytical expression for the ground-state energy density applicable to arbitrary spin-s in the XXX chain.

Findings

Reproduces Takhtajan-Babujian series for integer and half-integer spins

Provides a compact digamma-function representation

Enhances understanding of spin chain ground-state energies

Abstract

The antiferromagnetic Heisenberg spin chain remains a central framework for exploring exactly solvable models within quantum integrable systems. For the isotropic XXX chain, the ground-state energy per site of the spin-1/2 system is famously given by ln2. Extending the classic formulations to arbitrary spin-s, Takhtajan and Babujian derived two separate finite-series expressions for integer and half-integer spin representations. Current work introduces a unified analytical expression for the ground-state energy density in terms of the digamma function. This compact formulation reproduces both Takhtajan Babujian series.