Landau levels for a Weyl pair in a monolayer medium and thermal quantities


  • Abdullah Guvendi Erzurum Technical University
  • Abdelmalek Boumali Erzurum Technical University



Landau levels; graphene; Weyl fermions, charge carriers; many-body system; thermal properties


In this paper, we consider a Weyl pair under the effect of an external uniform magnetic field in a monolayer medium without considering any charge-charge interaction between the particles. Choosing the interaction of the particles with the magnetic field in the symmetric gauge we seek for an analytical solution of the corresponding form of a one-time fully-covariant two-body Dirac equation derived from quantum electrodynamics via the action principle. As it is usual with two-body problems, we separate the relative motion and center of mass motion coordinates. Assuming the center of mass is at rest, we derive a matrix equation in terms of the relative motion coordinates without considering any group theoretical method. This equation gives a wave equation in exactly soluble form and accordingly we obtain the spinor components and complete energy eigen-states (in closed form) for such a spinless composite structure. Our results not only give exact Landau levels for such a Weyl pair in a monolayer medium but also show the considered system behaves as a two-dimensional harmonic oscillator. Furthermore, our findings give exactly the excited states of a Weyl particle under the effect of uniform external magnetic field in a monolayer graphene sheet and there is no imprint to distinguish these modes from each other. This means that the performed experiments based on Landau levels for a monolayer graphene sheet may actually involve many-body effects. Our results provide a suitable basis to analyze the associated thermal quantities and accordingly we discuss the thermal properties by determining free energy, total energy, entropy and specific heat for the composite system in question.


G. Breit, The effect of retardation on the interaction of two electrons, Physical Review 34 (1929) 553,

E. E. Salpeter, and H. A. Bethe, A relativistic equation for bound-state problems, Physical Review 84 (1951) 1232,

A. O. Barut and S. Komy, Derivation of Nonperturbative Relativistic Two-Body Equations from the Action Principle in Quantumelectrodynamics, Fortschritte der Physik/Progress of Physics 33 (1985) 309,

A. Guvendi and Y. Sucu, An interacting fermion-antifermion pair in the spacetime background generated by static cosmic string, Physics Letters B 811 (2020) 135960,

A. Guvendi, S. Zare and H. Hassanabadi, Exact solution for a fermion-antifermion system with Cornell type nonminimal coupling in the topological defect-generated spacetime, Physics of the Dark Universe 38 (2022) 101133,

A. O. Barut and N. Unal, A new approach to bound-state quantum electrodynamics: I. Theory, Physica A: Statistical Mechanics and its Applications 142 (1987) 467,

M. Moshinsky and G. Loyola, Barut equation for the particleantiparticle system with a Dirac oscillator interaction, Foundations of physics 23 (1993) 197,

S. Deser, R. Jackiw, and G. Hooft, Three-dimensional Einstein gravity: dynamics of flat space, Annals of Physics 152 (1984) 220,

E. Witten, 2+1 dimensional gravity as an exactly soluble system, Nuclear Physics B 311 (1988) 46,

M. Banados, C. Teitelboim and J. Zanelli, Black hole in threedimensional spacetime, Physical Review Letters 69 (1992) 1849,

K. S. Novoselov et al., Electric field effect in atomically thin carbon films, Science 306 (2004) 666,

A. K. Geim and K. S. Novoselov, The rise of graphene, Nature materials 6 (2007) 183,

M. O. Goerbig, Electronic properties of graphene in a strong magnetic field, Reviews of Modern Physics 83 (2011) 1193,

Y. Sucu and N. Unal, Exact solution of Dirac equation in 2+1 ¨ dimensional gravity, Journal of mathematical physics 48 (2007) 052503,

A. Guvendi, Dynamics of a composite system in a point source-induced space-time, International Journal of modern Physics A 36 (2021) 2150144,

B. P. Mandal and S. Verma, Dirac oscillator in an external magnetic field, Physics Letters A 374 (2010) 1021,

A. Guvendi, R. Sahin, and Y. Sucu, Exact solution of an exciton energy for a monolayer medium, Scientific reports 9 (2019) 1,

A. Guvendi, Relativistic Landau levels for a fermionantifermion pair interacting through Dirac oscillator interaction, The European Physical Journal C 81 (2021) 1,

G. B. Arfken, H. J. Weber and F. E. Harris, Mathematical Methods for Physicists, Seventh Edition: A Comprehensive Guide, (Academic Press, Cambridge, 2012),

A. Guvendi and H. Hassanabadi, Relativistic vector bosons with non-minimal coupling in the spinning cosmic string spacetime, Few-Body Systems 62 (2021) 1,

M. Abramowitz and I. A. Stegum, Handbook of Mathematical Functions, (Dover Publications Inc., New York, 1965)

V. A. Miransky and I. A. Shovkovy, Quantum field theory in a magnetic field: From quantum chromodynamics to graphene and Dirac semimetals, Physics Reports 576 (2015) 1,

M. Castillo-Celeita and D. J. Fernandez, Dirac electron in graphene with magnetic fields arising from first-order intertwining operators, Journal of Physics A: Mathematical and Theoretical 53 (2020) 035302,

J. W. McClure, Diamagnetism of graphite, Physical Review 104 (1956) 666,

G. Li and E. Y. Andrei, Observation of Landau levels of Dirac fermions in graphite, Nature Physics 3 (2007) 623,

G. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, (1999)

V. Kac and P. Cheung, Quantum Calculus, Springer, (2001)

P. Erdos and S. S. Wagstaff, The fractional parts of the Bernoulli numbers, Illinois Journal of Mathematics 24 (1980) 104,

D. Elliot, The Euler-Maclaurin formula revisited, Journal of the Australian Mathematical Society: Series B, Applied Mathematics 40 (1998) E27,

A. Boumali and H. Hassanabadi, The thermal properties of a two-dimensional Dirac oscillator under an external magnetic field, The European Physical Journal Plus 128 (2013) 124, 31

A. Boumali, Thermodynamic properties of the graphene in a magnetic field via the two-dimensional Dirac oscillator, Physica Scripta 90 (2015) 045702,

M. H. Pacheco, R. V. Maluf, C.A.S. Almeida, and R. R. Landim, Three-dimensional Dirac oscillator in a thermal bath, Europhysics Letters 108 (2014) 10005,

Z. Jiang et al., Infrared Spectroscopy of Landau Levels of Graphene, Physical Review Letters 98 (2007) 197403,

A. Boumali, One-dimensional thermal properties of the Kemmer oscillator, Physica Scripta 76 (2007) 669,

V. A. Miransky, and I. A. Shovkovy, Quantum field theory in a magnetic field: From quantum chromodynamics to graphene and Dirac semimetals, Physics Reports 576 (2015) 1,

J. Peng, Z. Chen, B. Ding, and H. M. Cheng, Recent Advances for the Synthesis and Applications of 2-Dimensional Ternary Layered Materials, Research 6 (2023) 1,

K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. Castro, Neto, 2D materials and van der Waals heterostructures, Science 353 (2016) 9439,

V. Lukose, R. Shankar, and G. Baskaran, Novel electric field effects on Landau levels in graphene, Physical Review Letters 98 (2007) 116802,

N. Gu, M. Rudner, A. Young, P. Kim, and L. Levitov, Collapse of Landau levels in gated graphene structures, Physical Review Letters 106 (2011) 066601,

F. Guinea, M. I. Katsnelson, and A. K. Geim, Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering Nature Physics 6 (2010) 30,

Y. Betancur-Ocampo, E. Diaz-Bautista, and T. Stegmann, Valley-dependent time evolution of coherent electron states in tilted anisotropic Dirac materials, Physical Review B 105 (2022) 045401,

G. J. Iafrate, V. N. Sokolov, and J. B. Krieger, Quantum transport and the Wigner distribution function for Bloch electrons in spatially homogeneous electric and magnetic fields, Physical Review B 96 (2017) 144303,

E. Liu, J. van Baren, T. Taniguchi, K. Watanabe, Y. C. Chang, and C. H. Lui, Landau-Quantized Excitonic Absorption and Luminescence in a Monolayer Valley Semiconductor, Physical Review Letters 124 (2020) 097401, 44

F. J. Pena, and E. Munoz, Magnetostrain-driven quantum engine on a graphene flake, Physical Review E 91 (2015) 052152,

A. A. Greshnov, Room-temperature quantum Hall effect in graphene: the role of the two-dimensional nature of phonons, Journal of Physics: Conference Series 568 (2014) 052010, https://doi:10.1088/1742-6596/568/5/052010

G. Li, A. Luican-Mayer, D. Abanin, L. Levitov, and E. Y. Andrei, Evolution of Landau levels into edge states in graphene, Nature Communications 4 (2013) 1744, https://doi:10.1038/ncomms2767




How to Cite

A. Guvendi and A. Boumali, “Landau levels for a Weyl pair in a monolayer medium and thermal quantities”, Rev. Mex. Fís., vol. 69, no. 6 Nov-Dec, pp. 061701 1–, Nov. 2023.



17 Thermodynamics and Statistical Physics