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Semiclassical perspective on Landau levels and Hall conductivity in an anisotropic cubic Dirac semimetal and the peculiar case of star-shaped classical orbits
Ahmed Jellal, Hocine Bahlouli, and Michael Vogl
Phys. Rev. B 109, 235434 – Published 26 June 2024
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Abstract
We study an anisotropic cubic Dirac semimetal subjected to a constant magnetic field. In the case of an isotropic dispersion in the plane, with parameters , it is possible to find exact Landau levels, indexed by the quantum number , using the typical ladder operator approach. Interestingly, we find that the lowest energy level (the zero-energy state in the case of ) has a degeneracy that is 3 times that of other states. This degeneracy manifests in the Hall conductivity as a step at a zero chemical potential 3/2 the size of other steps. Moreover, as , we find energies , which means the step as a function of the chemical potential roughly occurs at a value . We propose that these exciting features could be used to experimentally identify cubic Dirac semimetals. Subsequently, we analyze the anisotropic case , with . First, we consider a perturbative treatment around and find that energies still hold as . To gain further insight into the Landau level structure for a maximum anisotropy, we turn to a semiclassical treatment that reveals interesting star-shaped orbits in phase space that close at infinity. This property is a manifestation of weakly localized states. Despite being infinite in length, these orbits enclose a finite phase space volume and permit finding a simple semiclassical formula for the energy, which has the same form as above. Our findings suggest that both isotropic and anisotropic cubic Dirac semimetals should leave similar experimental imprints.
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- Received 27 April 2024
- Revised 11 June 2024
- Accepted 13 June 2024
DOI:https://doi.org/10.1103/PhysRevB.109.235434
![Semiclassical perspective on Landau levels and Hall conductivity in an anisotropic cubic Dirac semimetal and the peculiar case of star-shaped classical orbits (10) Semiclassical perspective on Landau levels and Hall conductivity in an anisotropic cubic Dirac semimetal and the peculiar case of star-shaped classical orbits (10)](https://i0.wp.com/cdn.journals.aps.org/files/icons/creativecommons.png)
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Published by the American Physical Society
Physics Subject Headings (PhySH)
- Research Areas
Density of statesElectrical conductivityElectrical propertiesHall effectLandau levels
Condensed Matter, Materials & Applied Physics
Authors & Affiliations
Ahmed Jellal1,2, Hocine Bahlouli3, and Michael Vogl3,4,*
- 1Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb Doukkali University, P.O. Box 20, 24000 El Jadida, Morocco
- 2Canadian Quantum Research Center, 204-3002 32 Avenue Vernon, British Columbia V1T 2L7, Canada
- 3Physics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
- 4Interdisciplinary Research Center for Intelligent Secure Systems, KFUPM, Dhahran 31261, Saudi Arabia
- *Contact author: ssss133@googlemail.com
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Images
Figure 1
Plot of the dimensionless Hall conductivity as a function of dimensionless chemical potential . The left panel shows the Hall conductivity for our problem of a cubic Dirac semimetal (), and the right panel shows the case of graphene () as a comparison.
Figure 2
Plot of the dimensionless Hall conductivity as a function of the dimensionless chemical potential at inverse temperature , with “mass” term .
Figure 3
Plot of the potential landscape (thick blue line) and energy (dashed line). Two distinct potential pots are visible. Each will have its own associated classical periodic orbits with an action .
Figure 4
Phase space curve of an electron in a cubic semimetal subjected to a constant magnetic field.
Figure 5
The left plot shows Landau levels as a function of the mass parameter (a dimensionless version of momentum ). The right plot shows the relative error between the exact and approximate energies for .
Figure 6
Plot of phase space trajectories in terms of unitless momentum , unitless energy , and position (recall that is not a velocity and we set ). In both cases, we set because this term leads to only a shift of the trajectory center along the axis. The red curve is for , and the blue one is for .
Figure 7
Plot of one sector of the star orbit. Marked in blue is the area one needs to compute. The blue curve is given by , and the orange curve is given by .
Figure 8
The relative error between exact and approximate energies. The blue line serves to guide the eye.