A Pseudogap in the Single-Particle Density of States of a Tomonaga-Luttinger Liquid

We study a single-particle density of states (DOS) in a model of Tomonaga-Luttinger liquid with small (large) values of the Luttinger parameter $K_{\text{rc}}<0.17$, ($K_{\text{rc}}>5.82$), in the charge sector. We explain that such values of $K_{\text{rc}}$ may be achieved by electron-phonon interactions without generating the spin gap. We suggest that electron-phonon interactions can be partially incorporated into the Tomonaga-Luttinger liquid scheme by introducing frequency dependent $K_{\text{rc}}(\omega )$. We demonstrate that when the low-frequency asymptotic value $K_{\text{rc}}(0)<0.17$ or $K_{\text{rc}}(0)>5.82$, the single-particle DOS has a pseudogap behavior in frequency where $\rho (\omega )\mathrm{/}\omega$ vanishes at small frequencies. The DOS exhibits a peak the position of which scales as $[K(0) + K^{-1}(0)]\omega_0/2$where$\omega_0$ is the characteristic phonon frequency.