Semiclassical perspective on Landau levels and Hall conductivity in an anisotropic Cubic Dirac Semi-Metal and the peculiar case of star-shaped classical orbits
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arXiv
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| Natura: | Preprint |
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2024
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| author | Jellal, Ahmed Bahlouli, Hocine Vogl, Michael |
| author_facet | Jellal, Ahmed Bahlouli, Hocine Vogl, Michael |
| contents | We study an anisotropic cubic Dirac semi-metal subjected to a constant magnetic field. In the case of an isotropic dispersion in the $x$-$y$ 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 $k_z=0$) has a degeneracy that is three times that of other states. This degeneracy manifests in the Hall conductivity as a step at zero chemical potential that is 3/2 the size of other steps. Moreover, as $n\to\infty$ we find energies $E_n\propto n^{3/2}$, which means the $n$-th step as a function of chemical potential roughly occurs at a value $μ\propto n^{3/2}$. We propose that these exciting features could be used to identify cubic Dirac semi-metals experimentally. Subsequently, we analyze the anisotropic case $v_{y}=λv_{x}$ with $λ\neq 1$. First, we consider a perturbative treatment around $λ\approx 1$ and find that energies $E_n\propto n^{3/2}$ still holds as $n\to\infty$. To gain further insight into the Landau level structure for a maximum anisotropy, we turn to a semi-classical 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 semi-classical formula for the energy, which again has the form as above. Our findings suggest that both isotropic and anisotropic cubic Dirac semi-metals should leave similar experimental imprints. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17902 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semiclassical perspective on Landau levels and Hall conductivity in an anisotropic Cubic Dirac Semi-Metal and the peculiar case of star-shaped classical orbits Jellal, Ahmed Bahlouli, Hocine Vogl, Michael Mesoscale and Nanoscale Physics We study an anisotropic cubic Dirac semi-metal subjected to a constant magnetic field. In the case of an isotropic dispersion in the $x$-$y$ 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 $k_z=0$) has a degeneracy that is three times that of other states. This degeneracy manifests in the Hall conductivity as a step at zero chemical potential that is 3/2 the size of other steps. Moreover, as $n\to\infty$ we find energies $E_n\propto n^{3/2}$, which means the $n$-th step as a function of chemical potential roughly occurs at a value $μ\propto n^{3/2}$. We propose that these exciting features could be used to identify cubic Dirac semi-metals experimentally. Subsequently, we analyze the anisotropic case $v_{y}=λv_{x}$ with $λ\neq 1$. First, we consider a perturbative treatment around $λ\approx 1$ and find that energies $E_n\propto n^{3/2}$ still holds as $n\to\infty$. To gain further insight into the Landau level structure for a maximum anisotropy, we turn to a semi-classical 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 semi-classical formula for the energy, which again has the form as above. Our findings suggest that both isotropic and anisotropic cubic Dirac semi-metals should leave similar experimental imprints. |
| title | Semiclassical perspective on Landau levels and Hall conductivity in an anisotropic Cubic Dirac Semi-Metal and the peculiar case of star-shaped classical orbits |
| topic | Mesoscale and Nanoscale Physics |
| url | https://arxiv.org/abs/2404.17902 |