Addendum: “Landau quantized dynamics and spectra for group-VI dichalcogenides, including a model quantum wire” [AIP Advances 7, 065316 (2017)]

This addendum presents some additional material to clarify and extend the results of the original paper¹ (it also corrects a couple of typographical errors).

Further to the discussion of the off-diagonal elements of the retarded Green’s function, it should be noted that the results stated for $G21′,G12′$ in Eqns. (28, 29, ff.) are restricted to the K-point alone (γ_ν = γ; 1_ν = +1). More generally, for both K(1_ν = +1) and K′(1_ν = −1) jointly the corresponding statement reads as

and it must be borne in mind that in Eq. (24) D_n± also depends on 1_ν through $ϵn±2$⁠,

Alternatively expressed, the results for K, K′ separately are given by

where we use the notation

with $ϵn2≡(2n+2)γ2eB$⁠. Clearly, the off-diagonal elements are just interchanged with the interchange of K and K′.

Moreover, in momentum representation the off-diagonal elements maybe determined by Eq. (26) as $ω±G2112′(p,ω)=[(γpx±iγνpy)+(iγeB2∂∂py±γνeB2∂∂px)]G1122′(p,ω)$⁠.

In connection with our discussion of the spectral weight matrix A, it is worthwhile to point out that the sum rule of Eq. (58) is characteristic of the spectral weight of the single particle Green’s function (which is more often written in momentum representation). It should be emphasized that since the spectral weight matrix $Ar,r′,T$ satisfies the homogeneous counterpart of the Green’s function equation, it has an analytically similar solution to that of the retarded matrix Green’s function for T > 0; and it is devoid of the Heaviside unit step function η₊(T) which causes the retarded Green’s function to vanish for T < 0, whereas the spectral weight remains finite for T < 0. (For example, a relation for A(R, ω) corresponding to the integral representation of $G1122′(R,ω)$in Eq. (22) would involve a time-τ -integral over the range −∞ ≤ τ ≤ ∞, instead of 0 ≤ τ ≤ ∞ for $G1122′(R,ω)$⁠.) Expressed in operator terms (without specification of representation), the fact that A(T) satisfies the homogenous equation for the retarded Green’s function in the interval [ −∞ < T < ∞] without η₊(T) means that A(T) = exp(−iHT). This may be verified from another point of view using the relations involving the retarded Green’s function G(T) ≡ G_ret(T) given by G(T) = −iη₊(T) exp(−iHT) and A(ω) = −2ImG(ω) (Ref. 13, Eq. (7.5.12); ω → ω + i0⁺), so that $A(ω)=2Imi∫0∞dTexp(iT(ω−H))=−2Im(ω+i0+−H)−1=2πδ(ω−H)$⁠.

The full Fourier-time transform therefore confirms that $A(T)=∫−∞∞dω2πexp(−iωT)A(ω)=exp(−iHT)$⁠.

Incidentally, the spectral weight matrix A employed here is related to the more familiar spectral weight function A as A = Trace A, which includes all contributing states (Ref. 13, Chapt. 7.2).

Furthermore, the appearance of Dirac δ(ω−…)-functions in Eq. (65) to the exclusion of energy denominators that appear in Eq. (23) for the retarded Green’s function elements $G1122′$is a consequence of the time- Fourier transform integration range for A being (−∞ → ∞), whereas the corresponding integration range for the retarded Green’s function G is (0 → ∞); and the imaginary part emerges from the Dirac prescription 1/(ω + i0⁺) = P(1/ω) − πiδ(ω), in conformance with the insert above (for the end of the paragraph of Eq. (58)).

In addition, a “+” sign was inadvertently omitted on the left side of Eq. (20), which is also missing a pair of brackets, calling for the replacement of Eq. (20) by

Also, in Eqns. (62), (63), m and $M$ should be replaced by script capital $M±$⁠. These proofreading corrections do not affect the validity of any of the results of the article.