Efimov effect in Dirac semi-metals

Abstract

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.

Keywords

Dirac semi-metal / Efimov effect / screening

References

[1] V. Efimov, Energy levels arising from resonant two-body forces in a three-body system, Phys. Lett. B 33(8), 563 (1970)

[2] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H. C. Nägerl, and R. Grimm, Evidence for Efimov quantum states in an ultracold gas of caesium atoms, Nature 440(7082), 315 (2006)

[3] B. Huang, L. A. Sidorenkov, R. Grimm, and J. M. Hutson, Observation of the second triatomic resonance in Efimov’s scenario, Phys. Rev. Lett. 112(19), 190401 (2014)

[4] R. Pires, J. Ulmanis, S. Häfner, M. Repp, A. Arias, E. D. Kuhnle, and M. Weidemüller, Observation of Efimov resonances in a mixture with extreme mass imbalance, Phys. Rev. Lett. 112(25), 250404 (2014)

[5] S. K. Tung, K. Jiménez-García, J. Johansen, C. V. Parker, and C. Chin, Geometric scaling of Efimov states in a ⁶Li-¹³³Cs mixture, Phys. Rev. Lett. 113(24), 240402 (2014)

[6] M. Kunitski, S. Zeller, J. Voigtsberger, A. Kalinin, L. P. H. Schmidt, M. Schoffler, A. Czasch, W. Schollkopf, R. E. Grisenti, T. Jahnke, D. Blume, and R. Dorner, Observation of the Efimov state of the helium trimer, Science 348(6234), 551 (2015)

[7] S. Deng, Z. Y. Shi, P. Diao, Q. Yu, H. Zhai, R. Qi, and H. Wu, Observation of the Efimovian expansion in scaleinvariant Fermi gases, Science 353(6297), 371 (2016)

[8] Z.-Y. Shi, R. Qi, H. Zhai, Z. Yu, Dynamic super Efimov effect,Phys. Rev. A 96, 050702(R) (2017)

[9] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81(1), 109 (2009)

[10] M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82(4), 3045 (2010)

[11] X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83(4), 1057 (2011)

[12] A. V. Shytov, M. I. Katsnelson, and L. S. Levitov, Atomic collapse and quasi-Rydberg states in graphene, Phys. Rev. Lett. 99(24), 246802 (2007)

[13] V. M. Pereira, J. Nilsson, and A. H. C. Neto, Coulomb impurity problem in graphene, Phys. Rev. Lett. 99(16), 166802 (2007)

[14] Y. Nishida, Vacuum polarization of graphene with a supercritical Coulomb impurity: Low-energy universality and discrete scale invariance, Phys. Rev. B 90(16), 165414 (2014)

[15] Y. Nishida, Renormalization group analysis of graphene with a supercritical Coulomb impurity, Phys. Rev. B 94(8), 085430 (2016)

[16] H. Wang, H. Liu, Y. Li, Y. Liu, J. Wang, J. Liu, Y. Wang, L. Li, J. Yan, D. Mandrus, X. C. Xie, and J. Wang, Discrete scale invariance and fermionic Efimov states in ultra-quantum ZrTe₅, arXiv: 1704.00995

[17] O. Ovdat, J. Mao, Y. Jiang, E. Y. Andrei, and E. Akkermans, Observing a scale anomaly and a universal quantum phase transition in graphene, Nat. Commun. 8(1), 507 (2017)

[18] A. V. Shytov, M. I. Katsnelson, and L. S. Levitov, Vacuum polarization and screening of supercritical impurities in graphene, Phys. Rev. Lett. 99(23), 236801 (2007)

[19] H. Isobe and N. Nagaosa, Theory of a quantum critical phenomenon in a topological insulator: (3+ 1)- dimensional quantum electrodynamics in solids, Phys. Rev. B 86(16), 165127 (2012)

[20] S. K. Jian and H. Yao, Correlated double-Weyl semimetals with Coulomb interactions: Possible applications to HgCr₂Se₄ and SrSi₂, Phys. Rev. B 92(4), 045121 (2015)

[21] E. Braaten and H. W. Hammer, Universality in fewbody systems with large scattering length, Phys. Rep. 428(5–6), 259 (2006)

[22] See Appendix A for detailed derivation.

RIGHTS & PERMISSIONS

2018 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature