Friedel oscillations in one-dimensional fermion systems with long-range Coulomb interaction

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Using a bosonization and variational Gaussian wave-functional method we analytically calculate the Friedel oscillations induced by a single impurity in one-dimensional electron systems with long-range Coulomb interaction. It is shown that the Friedel oscillations are drastically different from those in electron systems with short-range interaction. First, in the long-range case the oscillatory part of the Friedel oscillations is not periodic as that in the short-range case. Second, in the long-range case the envelope of the Friedel oscillations decays as $\mathrm{exp}[\ensuremath{-}\mathrm{const}(\mathrm{ln}{x)}^{1/2}]$ far away from the impurity, which is much slower than the power law in the short-range case.