Non-relativistic representation of the Jackiw-Rebbi soliton

Authors

  • David Valenzuela Pontificia Universidad Catolica de Chile
  • Julio César Pérez Pedraza Universidad Michoacana de San Nicolás de Hidalgo
  • Alfredo Raya Universidad Michoacana de San Nicolás de Hidalgo

DOI:

https://doi.org/10.31349/RevMexFisE.20.020501

Keywords:

Jackiw-Rebbi model, Foldy-Wouthuysen transformation, topological insulators

Abstract

We consider the Foldy-Whouthuysen (FW) transformation of the Dirac equation coupled to a background soliton field which is equivalent to a position-dependent massm(x) such that at each limit x → ±∞, the mass to the left and to the right tends to a (possibly different) constant, with a sign difference at each side. We then build-up a third order unitarily transformed Schrödinger-like Hamiltonian as a counterpart of the corresponding to the well known Jackiw-Rebbi model. By further FW-transforming the Dirac spinor, we establish the relation between the non-relativistic and relativistic wave functions up to this order of approximation for generic position dependent mass profiles. For the economic choice m(x) = m0x/|x|, we find that these spinors are the same up to an overall constant.

References

R. Jackiw and C. Rebbi, Solitons with fermion number ½, Phys. Rev. D 13 (1976) 3398, https://doi.org/10.1103/PhysRevD.13.3398

A. Niemi and G. Semenoff, Fermion number fractionization in quantum field theory, Physics Reports 135 (1986) 99, https://doi.org/10.1016/0370-1573(86)90167-5

D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-Dimensional Magnetotransport in the Extreme Quantum Limit, Phys. Rev. Lett. 48 (1982) 1559, https://doi.org/10.1103/PhysRevLett.48.1559

D. K. Campbell and A. R. Bishop, Solitons in polyacetylene and relativistic-field-theory models, Phys. Rev. B 24 (1981) 4859, https://doi.org/10.1103/PhysRevB.24.4859.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Soliton excitations in polyacetylene, Phys. Rev. B 22 (1980) 2099, https://doi.org/10.1103/PhysRevB.22.2099

M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82 (2010) 3045, https://doi.org/10.1103/RevModPhys.82.3045

X. Yuan, M. Bowen, and P. S. Riseborough, The Dirac equation as a model of topological insulators, Philosophical Magazine 100 (2020) 1324, https://doi.org/10.1080/14786435.2020.1726523

D. G. Angelakis, P. Das, and C. Noh, Probing the topological properties of the Jackiw-Rebbi model with light, Scientific Reports 4 (2014) 6110, https://doi.org/10.1038/srep06110

T. X. Tran and F. Biancalana, Linear and nonlinear photonic Jackiw-Rebbi states in interfaced binary waveguide arrays, Phys. Rev. A 96 (2017) 013831, https://doi.org/10.1103/PhysRevA.96.013831

G. González, Dirac Equation and Optical Wave Propagation in One Dimension, physica status solidi (RRL) – Rapid Research Letters 12 (2018) 1700357, https://doi.org/10.1002/pssr.201700357

G. Gonzalez, Dirac equation in one dimensional transformation optics (2017), 10.48550/ARXIV.1707.06743, URL: https://arxiv.org/abs/1707.06743.

G. Gonzalez, et al., Electrostatic simulation of the Jackiw-Rebbi zero energy state, Rev. Mex. Fis. E 65 (2019) 30–33 https://doi.org/10.31349/RevMexFisE.65.30

R. Rubiano, Magnetostatic Analogy of the Zero Energy State of Jackiw-Rebbi (2019), 10.48550/ARXIV.1907.04479, URL: https://arxiv.org/abs/1907.04479.

F. Khosravi, T. Van Mechelen, and Z. Jacob, Dirac wire: Fermionic waveguides with longitudinal spin, Phys. Rev. B 100 (2019) 155105, https://doi.org/10.1103/PhysRevB.100.155105

Y. Nishida, L. Santos, and C. Chamon, Topological superconductors as nonrelativistic limits of Jackiw-Rossi and Jackiw-Rebbi models, Phys. Rev. B 82 (2010) 144513, https://doi.org/10.1103/PhysRevB.82.144513

L. L. Foldy and S. A. Wouthuysen, On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit, Phys. Rev. 78 (1950) 29, https://doi.org/10.1103/PhysRev.78.29

T. L. Li and K. J. Kuhn, Band-offset ratio dependence on the effective-mass Hamiltonian based on a modified profile of the GaAs-AlxGa1−xAs quantum well, Phys. Rev. B 47 (1993) 12760, https://doi.org/10.1103/PhysRevB.47.12760

F. S. A. Cavalcante, et al., Form of the quantum kinetic-energy operator with spatially varying effective mass, Phys. Rev. B 55 (1997) 1326, https://doi.org/10.1103/PhysRevB.55.1326

S. S.-Q. Shen, Topological Insulators, 1st ed. (Springer Berlin, Heidelberg, 2012), p. 225.

E. Ley-Koo, Recent progress in confined atoms and molecules: Superintegrability and symmetry breakings, Rev. Mex. Fís. 64 (2018) 326, https://doi.org/10.31349/RevMexFis.64.326

D. J. Griffiths, Introduction to electrodynamics, 2nd ed.

Downloads

Published

2023-06-28

How to Cite

[1]
D. Valenzuela, J. C. Pérez Pedraza, and A. . Raya, “Non-relativistic representation of the Jackiw-Rebbi soliton”, Rev. Mex. Fis. E, vol. 20, no. 2 Jul-Dec, pp. 020501 1–, Jun. 2023.

Issue

Vol. 20 No. 2 Jul-Dec (2023): Revista Mexicana de Física E

Section

05 Letters

License

Copyright (c) 2023 David Valenzuela, Julio César Pérez Pedraza, Alfredo Raya

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Authors retain copyright and grant the Revista Mexicana de Física E right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.