Fourier transform of the $2k_F$ Luttinger liquid density correlation function with different spin and charge velocities

We obtain a closed-form analytical expression for the zero-temperature Fourier transform of the $2k_F$ component of the density-density correlation function in a Luttinger liquid with different spin and charge velocities. For frequencies near the spin and charge singularities, approximate analytical forms are given and compared with the exact result. We find power-law-like singularities leading to either divergence or cusps, depending on the values of the Luttinger parameters, and compute the corresponding exponents. Exact integral expressions and numerical results are given for the finite-temperature case as well. We show, in particular, how the temperature rounds the singularities in the correlation function.

Figure 1

(Color online) Real and imaginary parts of ${\chi }_{2k_F}(q,\omega )$ as a function of frequency (filled lines) for $q=0$, $v_{\sigma }∕v_{\rho }=0.4$, and (a) $K_{\rho }=1$, $K_{\sigma }=0.4$, (b) $K_{\rho }=0.4$, $K_{\sigma }=0.2$, (c) $K_{\rho }=0.4$, $K_{\sigma }=1$, and (d) $K_{\rho }=0.8$, $K_{\sigma }=1$. The last case corresponds to the repulsive Hubbard model away from half filling. The comparison with the approximate expressions from Eqs. (13, 14) are also shown (dashed lines). The charge energy is defined as $E_{\rho }=k_F{v}_{\rho }$.

Figure 2

(Color online) Imaginary parts of ${\chi }_{2k_F}(q,\omega )$ as a function of frequency for $q=0$, $v_{\sigma }∕v_{\rho }=0.1$, $K_{\rho }=0.5$, $K_{\sigma }=1$. In the inset, we observe the suppression of the correlation function under the effect of temperature (we show its behavior for fixed values of $\omega$).

Figure 3

(Color online) Imaginary parts of ${\chi }_{2k_F}(q,\omega )$ as a function of frequency for $q=k_F∕2$, $v_{\sigma }∕v_{\rho }=0.2$, $K_{\rho }=0.4$, $K_{\sigma }=0.2$. The four peak structure is the effect of taking $q\ne 0$.