Edge exponent in the dynamic spin structure factor of the Yang-Gaudin model

The dynamic spin structure factor $\mathcal{S}\left(k,\omega \right)$ of a system of spin-1/2 bosons is investigated at arbitrary strength of the interparticle repulsion. As a function of $\omega$ it is shown to exhibit a power-law singularity at the threshold frequency defined by the energy of a magnon at given $k$. The power-law exponent is found exactly using a combination of the Bethe ansatz solution and an effective-field theory approach.

Figure 1

Excitation spectrum in the model (3). (a) and (b) show the dispersion curve ${\omega }_{-}\left(k\right)$ [Eq. (12)] of the magnon (solid line) and the exact lower bound of the particle-hole continuum (dashed line) for two values of $\gamma =g/{\rho }_0$. (c) shows the $\gamma$-dependent exact lower bound $E_p$ of the particle-hole continuum at $k=k_F\equiv \pi {\rho }_0$. (d) shows the $\gamma$-dependent magnon energy $E_m={\omega }_{-}$ at $k=k_F$.

Figure 2

(Color online) The function $\alpha \left(k\right)$ on the right hand side of Eq. (22) is plotted for different values of the dimensionless coupling constant $\gamma$. The values of the Luttinger parameter $K$ are indicated for each curve and correspond in increasing order to $\gamma =\infty$, 1.65, 0.56, 0.238 and 0.109, respectively.