Exact Exponents of Edge Singularities in Dynamic Correlation Functions of 1D Bose Gas

The spectral function and dynamic structure factor of bosons interacting by contact repulsion and confined to one dimension exhibit power-law singularities along the dispersion curves of the collective modes. We find the corresponding exponents exactly, by relating them to the known Bethe ansatz solution of the Lieb-Liniger model. Remarkably, the Luttinger liquid theory predictions for the exponents fail even at low energies, once the immediate vicinities of the edges are considered.

Figure 1

(a) Dynamic structure factor (DSF) $S(k,\omega )$ and (b) spectral function $A(k,\omega )$. Shaded areas indicate the regions where they are nonvanishing. Lieb’s particle mode ${\epsilon }_1(k)$ and hole excitation mode ${\epsilon }_2(k)$ are indicated. For spectral function $A(k,\omega )$ region with $\omega >0(\omega <0)$ corresponds to the particle (hole) part of the spectrum. Notations of $\mu$ indicate which exponents of Eqs. (16, 17, 18) should be used in Eq. (4).

Figure 2

Spectral function $A(k,\omega )$ for $\omega >0$ near $k=0$. Exact exponents $\overline{{\mu }_{+}}=1/\sqrt{K}-1/(2K)$, $\underline{{\mu }_{+}}=-1/\sqrt{K}-1/(2K)$ are valid only in the immediate vicinities of ${\epsilon }_1$ and ${\epsilon }_2$ marked by dashed lines. Luttinger liquid behavior given by Eqs. (27) is valid only sufficiently far away from $\epsilon =vk$, for $\omega -vk\gg {\epsilon }_1-{\epsilon }_2$.