Galois Solvability of Finite-Size Bethe Solutions in the Heisenberg Chain

TL;DR

This paper investigates the algebraic complexity of finite-size solutions in the Heisenberg chain, revealing that Bethe roots and ground state wavefunctions become Galois unsolvable as system size increases.

Contribution

It analyzes the algebraic structure of exact ground states and Bethe roots, showing their increasing complexity and potential Galois unsolvability in larger chains.

Findings

Bethe roots become Galois unsolvable for chains of 8 or more sites.

Ground state wavefunction coefficients become unsolvable for 10 or more sites.

Algebraic complexity grows rapidly with system size, affecting explicit solvability.

Abstract

The spin-1/2 Heisenberg antiferromagnetic chain is the canonical example of an integrable quantum many-body model. Despite its exact solvability, explicit finite-size solutions are typically only accessible via numerical evaluation of the Bethe ansatz equations. Here, we analyse the algebraic structure of the exact, symbolic ground states for chains up to ten sites using the coordinate Bethe ansatz. We show that both the ground state wavefunction and the Bethe-roots rapidly develop algebraic complexity with respect to system size, but at different rates. The Bethe-roots appear to become Galois unsolvable for chains of eight or more sites, whereas the ground state wavefunction coefficients and energy appear to become unsolvable for ten or more sites. This demonstrates a lack of explicit analytic tractability in a quantum integrable model due to algebraic complexity.