Semiclassical perspective on Landau levels and Hall conductivity in an anisotropic cubic Dirac semimetal and the peculiar case of star-shaped classical orbits

We study an anisotropic cubic Dirac semimetal subjected to a constant magnetic field. In the case of an isotropic dispersion in the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>x</a:mi><a:mtext>−</a:mtext><a:mi>y</a:mi></a:mrow></a:math> plane, with parameters <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>v</b:mi><b:mi>x</b:mi></b:msub><b:mo>=</b:mo><b:msub><b:mi>v</b:mi><b:mi>y</b:mi></b:msub></b:mrow></b:math>, it is possible to find exact Landau levels, indexed by the quantum number <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>n</c:mi></c:math>, using the typical ladder operator approach. Interestingly, we find that the lowest energy level (the zero-energy state in the case of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msub><d:mi>k</d:mi><d:mi>z</d:mi></d:msub><d:mo>=</d:mo><d:mn>0</d:mn></d:mrow></d:math>) 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>n</e:mi><e:mo>→</e:mo><e:mi>∞</e:mi></e:mrow></e:math>, we find energies <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:msub><f:mi>E</f:mi><f:mi>n</f:mi></f:msub><f:mo>∝</f:mo><f:msup><f:mi>n</f:mi><f:mrow><f:mn>3</f:mn><f:mo>/</f:mo><f:mn>2</f:mn></f:mrow></f:msup></f:mrow></f:math>, which means the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>n</g:mi><g:mi>th</g:mi></g:mrow></g:math> step as a function of the chemical potential roughly occurs at a value <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>μ</h:mi><h:mo>∝</h:mo><h:msup><h:mi>n</h:mi><h:mrow><h:mn>3</h:mn><h:mo>/</h:mo><h:mn>2</h:mn></h:mrow></h:msup></h:mrow></h:math>. We propose that these exciting features could be used to experimentally identify cubic Dirac semimetals. Subsequently, we analyze the anisotropic case <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:msub><i:mi>v</i:mi><i:mi>y</i:mi></i:msub><i:mo>=</i:mo><i:mi>λ</i:mi><i:msub><i:mi>v</i:mi><i:mi>x</i:mi></i:msub></i:mrow></i:math>, with <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>λ</j:mi><j:mo>≠</j:mo><j:mn>1</j:mn></j:mrow></j:math>. First, we consider a perturbative treatment around <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mi>λ</k:mi><k:mo>≈</k:mo><k:mn>1</k:mn></k:mrow></k:math> and find that energies <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mrow><l:msub><l:mi>E</l:mi><l:mi>n</l:mi></l:msub><l:mo>∝</l:mo><l:msup><l:mi>n</l:mi><l:mrow><l:mn>3</l:mn><l:mo>/</l:mo><l:mn>2</l:mn></l:mrow></l:msup></l:mrow></l:math> still hold as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>n</m:mi><m:mo>→</m:mo><m:mi>∞</m:mi></m:mrow></m:math>. 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.