# Low-energy dynamics of the Affleck-Kennedy-Lieb-Tasaki model in the one- and two-triplon basis

## TL;DR

This paper demonstrates the equivalence of magnon and triplon descriptions in the AKLT model, showing that multi-particle states' energies decompose into sums of single-particle energies, and constructs an effective Hamiltonian in this basis.

## Contribution

It proves the equivalence of multi-magnon and multi-triplon states and develops an effective Hamiltonian within the low-energy triplon basis for the AKLT model.

## Key findings

- Two-magnon and two-triplon energies are additive and identical.
- The low-energy Hilbert space is spanned by orthogonal one- and two-triplon states.
- The effective Hamiltonian accurately predicts the energy gap correction.

## Abstract

The elementary excitation in the antiferromagnetic spin-1 model known as the Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian has been described alternatively as magnons or kink-like solitons (triplons). The latter, which we call the triplon throughout this paper, has been proven equivalent descriptions of the same magnon excitation and not an independent branch of excited states. On the other hand, no careful examination of multi-magnon and multi-triplon equivalence was made in the past. In this paper we prove that two-magnon and two-triplon states are also identical descriptions of the same excited states, and furthermore that their energies break down as the sum of one-triplon energies exactly for the AKLT Hamiltonian. The statement holds despite the fact that the model is non-integrable. Such magnon/triplon dichotomy is conjectured to hold for arbitrary n-magnon and n-triplon states. The one- and two-triplon states form orthogonal sets that can be used to span the low-energy Hilbert space. We construct an effective version of the AKLT Hamiltonian within such subspace, and work out the correction to the one-triplon energy gap that finds excellent agreement with the known exact value.