# Numerical Investigation of Spin Excitations in a Doped Spin Chain

## TL;DR

This study investigates how doping affects spin excitations in a one-dimensional Hubbard model, revealing excitation shifts and emphasizing the importance of higher-order terms for accurate low-energy spin models, with insights supported by mean-field theory.

## Contribution

It demonstrates the significance of 3-site and higher-order terms in accurately modeling doped 1D Hubbard systems and employs a combined numerical and analytical approach.

## Key findings

- Spin excitations harden with electron doping and soften with hole doping.
- Higher-order terms are essential for quantitative agreement with experimental spectra.
- Mean-field theory explains the doping-dependent shifts and bandwidth changes.

## Abstract

We study the doping evolution of spin excitations in a 1D Hubbard model and its downfolded spin Hamiltonians, by using exact diagonalization combined with cluster perturbation theory. In all models, we observe hardening (softening) of spin excitations upon electron (hole) doping, which are reminiscent of recent experiments on 2D cuprate materials. We also find that the 3-site and even higher-order terms are crucial for the low-energy effective spin models to reproduce the magnetic spectra of doped Hubbard systems at a quantitative level. To interpret the numerical results, we further employ a strong coupling slave-boson mean-field theory. The mean-field theory provides an intuitive understanding of the overall compact support of dynamic spin structure factors, including the shift of zero-energy modes and change of spin excitation bandwidth with doping. Our results can serve as predictive benchmarks for future inelastic x-ray or neutron scattering experiments on doped 1D antiferromagnetic Mott insulators.