Theory of Nonlinear Luttinger Liquids

We developed a generalization of the Luttinger liquid theory which allowed us to consider threshold singularities in the momentum-resolved dynamic response functions at arbitrary momenta ({\it i.e.}, far away from the Fermi points). The main difficulty the new theory overcomes is the accounting for a generic non-linear dispersion relation of quantum particles which form the liquid. We derive an effective ``quantum impurity'' Hamiltonian which adequately describes the dynamics of the system at the near-threshold energies. The phenomenological theory for the constants of such Hamiltonian is built; it expresses the constants in terms of other measurable properties (energy spectra of the excitations) of the liquid. One of the most important dynamic correlation functions we consider is the momentum-resolved electron spectral function at arbitrary momenta. The spectral function is directly measurable in tunneling experiments. It is singular at the spectrum of the lowest-energy excitation branch. In the absence of spin polarization, this is the branch of spinon excitations. The derivation of the phenomenological relations for the threshold exponent uses the $SU(2)$ and Galilean invariance of the electron liquid. We also consider in detail the case of single-species fermions, which adequately describes the fully spin-polarized electron gas [1]. The theory of threshold exponents is valid at arbitrary wave vectors $k$, including the vicinities of Fermi points $\pm k_F$. There, the exponents approach universal values [2] which depend only on the Luttinger liquid parameter $K$. Remarkably, the found exponents differ from the predictions of the conventional linear Luttinger liquid theory. The deviations from that theory though are confined to the region close to the threshold; while being wide away from the Fermi points, the width of that region scales as $|k\pm k_F|^3$ at $k\to\pm k_F$ in the absence of spin polarization, and as $(k\pm k_F)^2$ for polarized electrons. \\[4pt] [1] A. Imambekov, L.I. Glazman, Phys. Rev. Lett., {\bf 102}, 126405 (2009)\\[0pt] [2] A. Imambekov, L.I. Glazman, Science, {\bf 323}, 228 (2009)