Excited state correlations of the finite Heisenberg chain

TL;DR

This paper extends the known factorized correlation formulas from the ground state to excited states in finite Heisenberg chains, using Bethe roots, and validates the approach with numerical data.

Contribution

It proposes a conjecture that the factorized correlation formulas for ground states can be applied to excited states with the exact Bethe roots, supported by numerical validation.

Findings

Perfect agreement with numerical data for both XXZ and XXX chains.

Derived a new formula for the nearest-neighbour correlator in the XXX chain.

Conjecture applies broadly to excited states, linking existing theory with TBA-like descriptions.

Abstract

We consider short range correlations in excited states of the finite XXZ and XXX Heisenberg spin chains. We conjecture that the known results for the factorized ground state correlations can be applied to the excited states too, if the so-called physical part of the construction is changed appropriately. For the ground state we derive simple algebraic expressions for the physical part; the formulas only use the ground state Bethe roots as an input. We conjecture that the same formulas can be applied to the excited states as well, if the exact Bethe roots of the excited states are used instead. In the XXZ chain the results are expected to be valid for all states (except certain singular cases where regularization is needed), whereas in the XXX case they only apply to singlet states or group invariant operators. Our conjectures are tested against numerical data from exact diagonalization and coordinate Bethe Ansatz calculations, and perfect agreement is found in all cases. In the XXX case we also derive a new result for the nearest-neighbour correlator $\{\sigma_1^z\sigma_2^z\}$, which is valid for non-singlet states as well. Our results build a bridge between the known theory of factorized correlations, and the recently conjectured TBA-like description for the building blocks of the construction.