Comment on “Chern-Simons theory and atypical Hall conductivity in the Varma phase”

Remarks on the Duffin-Kemmer-Petiau oscillator in an external magnetic field

“…For both cases, we add a term of the form a † i c j along with the Hamiltonian H 0 , resulting in the emergence of a bandgap energy. Following a procedure similar for graphene, it is possible to demonstrate that the dynamics of the charge carriers on the Lieb lattice in the low-energy regime exhibit a relativistic-like behavior [21], as described by the two-dimensional DKPlike Hamiltonian:…”

“…Therefore, the threecomponent spinor Φ = (Φ a , Φ b , Φ c ) T correctly describes spin-1 quasiparticles, similarly to that studied in this remark. Nevertheless, it is necessary to establish the equivalence between the Hamiltonian form (48) and the equation of motion (1), which is achieved using the constraint equation v F β i β 0 β 0 p i Φ = m 1 − β 0 β 0 Φ [21,33].…”

“…It is straightforward to demonstrate that the above system of equations allows us to reconstruct a new DKPO equation following the same structure as (11), considering only one of the two three-dimensional irreducible representations of the matrices β μ and a three-component spinor Φ = (Φ 1 , Φ 2 , Φ 3 ) T , as predicted by [8]. The 3 × 3 matrices are given by [21]…”

“…The DKPO arises as a kind of tensor coupling with a linear potential that leads to the harmonic-oscillator (a) E-mail: aobispo@utp.edu.pe (corresponding author) problem in the weak-coupling limit. Although the DKP equation and the DKPO are widely studied models in four-dimensional space, recently, a great deal of theoretical research has been done on these systems in (2 + 1)dimensional space-time, which are motivated by their potential applications in other areas of physics, for instance, in condensed-matter systems [19][20][21]. However, the DKP algebra in (2 + 1) dimensions only generates a set of 35 linearly independent matrices, leading to a trivial representation, a four-dimensional representation (spin-0 sector), and a twin three-dimensional representation (spin-1 sector).…”

“…Returning our attention to the system of equations ( 19)- (21), it is possible to combine the equations and after some algebraic manipulations we obtain an equation of motion for the Φ 1 component,…”