Spinon excitation spectra of the $J_1$-$J_2$ chain from analytical calculations in the dimer basis and exact diagonalization

TL;DR

This paper analytically and numerically investigates the excitation spectra of the frustrated spin-1/2 Heisenberg chain, revealing new dispersion relations, gap openings, and incommensurability phenomena at specific coupling ratios.

Contribution

It introduces a tractable analytical approach using the dimer basis and exact diagonalization to study spinon excitations in the J1-J2 chain, recovering and extending previous results.

Findings

Gap opens at J2=0.25J1

Incommensurability appears at J2=9/17J1

Analytical results support the independent spinons picture

Abstract

The excitation spectrum of the frustrated spin-$1/2$ Heisenberg chain is reexamined using variational and exact diagonalization calculations. We show that the overlap matrix of the short-range resonating valence bond states basis can be inverted which yields tractable equations for single and two spinons excitations. Older results are recovered and new ones, such as the bond-state dispersion relation and its size with momentum at the Majumdar-Ghosh point are found. In particular, this approach yields a gap opening at $J_2=0.25J_1$ and an onset of incommensurability in the dispersion relation at $J_2=9/17J_1$ [as in S. Brehmer \emph{et al.}, J. Phys.: Condens. Matter \textbf{10}, 1103 (1998)]. These analytical results provide a good support for the understanding of exact diagonalization spectra, assuming an independent spinons picture.