# Excitations in the Yang-Gaudin Bose gas

## TL;DR

This paper investigates the excitation spectrum of the Yang-Gaudin Bose gas, revealing significant finite-size effects, novel features like a $2k_F$ gap and roton minimum, and analyzing multi-spinon bound states both analytically and numerically.

## Contribution

It provides a detailed analysis of finite-size effects on excitations, including the $2k_F$ gap, roton minimum, and multi-spinon bound states, using both analytical and numerical methods.

## Key findings

- Pronounced finite-size effects in excitation dispersion relations.
- Persistence of $2k_F$ gap and roton minimum at finite density.
- Lorentzian form of $
abla$-string dressed energies near $ ext{lambda}=0$.

## Abstract

We study the excitation spectrum of two-component delta-function interacting bosons confined to a single spatial dimension, the Yang-Gaudin Bose gas. We show that there are pronounced finite-size effects in the dispersion relations of excitations, perhaps best illustrated by the spinon single particle dispersion which exhibits a gap at $2k_F$ and a finite-momentum roton minimum. Such features occur at energies far above the finite volume excitation gap, vanish slowly as $1/L$ for fixed spinon number, and can persist to the thermodynamic limit at fixed spinon density. Features such as the $2k_F$ gap also persist to multi-particle excitation continua. Our results show that excitations in the finite system can behave in a qualitatively different manner to analogous excitations in the thermodynamic limit. The Yang-Gaudin Bose gas is also host to multi-spinon bound states, known as $\Lambda$-strings. We study these excitations both in the thermodynamic limit under the string hypothesis and in finite size systems where string deviations are taken into account. In the zero-temperature limit we present a simple relation between the length $n$ $\Lambda$-string dressed energies $\epsilon_n(\lambda)$ and the dressed energy $\epsilon(k)$. We solve the Yang-Yang-Takahashi equations numerically and compare to the analytical solution obtained under the strong couple expansion, revealing that the length $n$ $\Lambda$-string dressed energy is Lorentzian over a wide range of real string centers $\lambda$ in the vicinity of $\lambda = 0$. We then examine the finite size effects present in the dispersion of the two-spinon bound states by numerically solving the Bethe ansatz equations with string deviations.