Title: Efimov Effect in the Dirac Semi-metals,
Abstract: Efimov effect refers to quantum states with discrete scaling symmetry and a
universal scaling factor, and has attracted considerable interests from nuclear
to atomic physics communities. In a Dirac semi-metal, when an electron
interacts with a static impurity though a Coulomb interaction, the same scaling
of the kinetic and interaction energies also gives rise to such a Efimov
effect. However, even when the Fermi energy exactly lies at the Dirac point,
the vacuum polarization of electron-hole pair fluctuation can still screen the
Coulomb interaction, which leads to derivation from this scaling symmetry and
eventually breakdown of the Efimov effect. This distortion of the Efimov bound
state energy due to vacuum polarization is a relativistic electron analogy of
the Lamb shift for the hydrogen atom. Motivated by recent experimental
observations in two- and three-dimensional Dirac semi-metals, in this paper we
investigate this many-body correction to the Efimov effect, and answer the
question that under what condition a good number of Efimov-like bound states
can still be observed in these condensed matter experiments.