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arXiv:cond-mat/0702274 · Physical Review B

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

Anı́bal Iucci, Gregory A. Fiete, Thierry Giamarchi

Published May 21, 2007• 29 citations
arXivHTMLPDF

Abstract

We obtain a closed-form analytical expression for the zero-temperature Fourier transform of the $2{k}_{F}$ 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.

DOI
10.1103/physrevb.75.205116

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We derive an analytic expression for the zero temperature Fourier transform of the density-density correlation function of a multicomponent Luttinger liquid with different velocities. By employing Schwinger identity and a generalized Feynman identity exact integral expressions are derived, and approximate analytical forms are given for frequencies close to each component singularity. We find power-like singularities and compute the corresponding exponents. Numerical results are shown for the case of three components.
We perform a rigorous computation of the spin and charge density correlations of the 1d repulsive Hubbard model at weak coupling, focusing on the properties of the Fourier transform at momentum 0 and ±2 p F , if p F is the Fermi momentum. We prove that the interaction changes the singularity at ±2 p F (the discontinuity in the derivative becomes a power law singularity) while the singularity at 0 is essentially unchanged. Our results show that the logarithmic divergences at zero momentum recently found in [7], which would be in contrast with Luttinger liquid behaviour, are indeed spurious.
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We compute the spectral function rho (q, omega ) of the one-dimensional Luttinger model. We discuss the distinct influences of charge-spin separation and of the anomalous dimensions of the fermion operators and their evolution with correlation strength. Charge-spin separation shows up in finite spectral weight at frequencies between v sigma q and v rho q, where v sigma and v rho are the velocities of spin and charge fluctuations, while spectral weight above v rho q and below -v rho q is generated by the hybridization of the Fermi surface at +or-k F by interactions. There are non-universal power-law singularities at these special frequencies. We discuss the consistency of recent photoemission experiments on low-dimensional conductors with a Luttinger-liquid picture, which then would suggest very strong long-range interactions. It is pointed out that many-particle correlation functions in principle exhibit similar singularities, but they probe different and complementary aspects of the Fermi-surface interactions.
We develop a nonperturbative zero-temperature theory for the dynamic response functions of interacting one-dimensional spin-1/2 fermions. In contrast to the conventional Luttinger liquid theory, we take into account the nonlinearity of the fermion dispersion exactly. We calculate the power-law singularities of the spectral function and the charge- and spin-density structure factors for arbitrary momenta and interaction strengths. The exponents characterizing the singularities are functions of momenta and differ significantly from the predictions of the linear Luttinger liquid theory. We generalize the notion of the spin-charge separation to the nonlinear spectrum. This generalization leads to phenomenological relations between threshold exponents and the threshold energy.
We provide analytic expressions for the Green's functions in position-frequency space as well as for the tunneling density of states of various Luttinger liquids at zero temperature: the standard spinless and spinful Luttinger liquids, the helical Luttinger liquid at the edge of a topological insulator, and the Luttinger liquid that appears either together with an ordering transition of nuclear spins in a one-dimensional conductor or in spin-orbit split quantum wires in an external magnetic field. The latter system is often used to mimic a helical Luttinger liquid, yet we show here that it exhibits significantly different response functions and, to discriminate, we call it the spiral Luttinger liquid. We give fully analytic results for the tunneling density of state of all the Luttinger liquids as well as for most of the Green's functions. The remaining Green's functions are expressed by simple convolution integrals between analytic results.