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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 $x\text{\u2212}y$ plane, with parameters ${v}_{x}={v}_{y}$, it is possible to find exact Landau levels, indexed by the quantum number $n$, using the typical ladder operator approach. Interestingly, we find that the lowest energy level (the zero-energy state in the case of ${k}_{z}=0$) 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 $n\to \infty $, we find energies ${E}_{n}\propto {n}^{3/2}$, which means the $n\mathrm{th}$ step as a function of the chemical potential roughly occurs at a value $\mu \propto {n}^{3/2}$. We propose that these exciting features could be used to experimentally identify cubic Dirac semimetals. Subsequently, we analyze the anisotropic case ${v}_{y}=\lambda {v}_{x}$, with $\lambda \ne 1$. First, we consider a perturbative treatment around $\lambda \approx 1$ and find that energies ${E}_{n}\propto {n}^{3/2}$ still hold as $n\to \infty $. 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

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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 Jellal^{1,2}, Hocine Bahlouli^{3}, and Michael Vogl^{3,4,*}

^{1}Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb Doukkali University, P.O. Box 20, 24000 El Jadida, Morocco^{2}Canadian Quantum Research Center, 204-3002 32 Avenue Vernon, British Columbia V1T 2L7, Canada^{3}Physics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia^{4}Interdisciplinary 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 ${\sigma}_{xy}^{\left(r\right)}$ as a function of dimensionless chemical potential ${\mu}^{\left(r\right)}=\mu /{\omega}_{c}$. The left panel shows the Hall conductivity for our problem of a cubic Dirac semimetal ($\beta =50/{\omega}_{c}$), and the right panel shows the case of graphene ($\beta =100/{\omega}_{c}$) as a comparison.

###### Figure 2

Plot of the dimensionless Hall conductivity ${\sigma}_{xy}^{\left(r\right)}$ as a function of the dimensionless chemical potential ${\mu}^{\left(r\right)}=\mu /{\omega}_{c}$ at inverse temperature $\beta =50/{\omega}_{c}$, with “mass” term $m=-3$.

###### Figure 3

Plot of the potential landscape $V\left(x\right)$ (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 ${S}_{\alpha}$.

###### 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 $m$ (a dimensionless version of momentum ${k}_{z}$). The right plot shows the relative error between the exact and approximate energies for $m=0$.

###### Figure 6

Plot of phase space trajectories in terms of unitless momentum ${\stackrel{\u0303}{p}}_{i}={p}_{i}{\left(eB\right)}^{-1/2}$, unitless energy $\epsilon =E{v}^{-1}{\left(eB\right)}^{-3/2}$, and position $\stackrel{\u0303}{x}=x{\left(eB\right)}^{1/2}$ (recall that $v$ is not a velocity and we set $\hslash =1$). In both cases, we set ${p}_{y}=0$ because this term leads to only a shift of the trajectory center along the $x$ axis. The red curve is for $\epsilon =1$, and the blue one is for $\epsilon =0.02$.

###### 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 ${\stackrel{\u0303}{p}}_{x}=\sqrt{{x}^{2}/3+\epsilon /\left(6x\right)}$, and the orange curve is given by ${\stackrel{\u0303}{p}}_{x}=\sqrt{{x}^{2}/3-\epsilon /\left(6x\right)}$.

###### Figure 8

The relative error between exact and approximate energies. The blue line serves to guide the eye.