Efimov effect in Dirac semi-metals

The Efimov effect is defined as a quantum state with discrete scaling symmetry and a universal scaling factor. It has attracted considerable interests from nuclear to atomic physics communities. In a Dirac semi-metal, when an electron interacts with a static impurity through a Coulombic interaction, the same kinetic scaling and the interaction energy results in the Efimov effect. However, even when the Fermi energy lies exactly at the Dirac point, the vacuum polarization of the electron-hole pair fluctuation can still screen the Coulombic interaction, which leads to deviations from the scaling symmetry and eventually breaks down of the Efimov effect. This energy distortion of the Efimov states due to vacuum polarization is a relativistic electron analogy of the Lamb shift for the hydrogen atom. Motivated by the recent experimental observations in two- and three-dimensional Dirac semi-metals, we herein investigate this many-body correction to the Efimov effect and the conditions that allow some of the Efimov-like quasi-bound states to be observed in these condensed matter experiments.