# Large-$S$ limit of the large-$N$ theory for the triangular antiferromagnet

## TL;DR

This paper shows that incorporating Gaussian corrections in a large-$N$ Schwinger boson approach to the triangular antiferromagnet reproduces classical spin wave results in the large-$S$ limit, highlighting the importance of spinon interactions.

## Contribution

It demonstrates that Gaussian ($1/N$) corrections are essential to recover semiclassical spin wave theory within a large-$N$ framework for the triangular antiferromagnet.

## Key findings

- Gaussian corrections cancel saddle-point contributions
- Magnons are spinon bound states from RPA poles
- Magnon dispersion differs from saddle-point spinons

## Abstract

Large-$S$ and large-$N$ theories (spin value $S$ and spinor component number $N$) are complementary, and sometimes conflicting, approaches to quantum magnetism. While large-$S$ spin-wave theory captures the correct semiclassical behavior, large-$N$ theories, on the other hand, emphasize the quantumness of spin fluctuations. In order to evaluate the possibility of the non-trivial recovery of the semiclassical magnetic excitations within a large-$N$ approach, we compute the large-$S$ limit of the dynamic spin structure of the triangular lattice Heisenberg antiferromagnet within a Schwinger boson spin representation. We demonstrate that, only after the incorporation of Gaussian ($1/N$) corrections to the saddle-point ($N=\infty$) approximation, we are able to exactly reproduce the linear spin wave theory results in the large-$S$ limit. The key observation is that the effect of $1/N$ corrections is to cancel out exactly the main contribution of the saddle-point solution; while the collective modes (magnons) consist of two spinon bound states arising from the poles of the RPA propagator. This result implies that it is essential to consider the interaction of the spinons with the emergent gauge fields and that the magnon dispersion relation should not be identified with that of the saddle-point spinons.