radiation with MO transport has been explored. Finally, we have studied Faraday rotation which should be possible to be realized in the experiments.
A Chebyshev expansion
The first kind Chebyshev polynomials can be defined as,Tn(x)=cos(ncos−1(x)) in the range [−1, 1]. The recursion relations, T0(x)=1, T1(x)= xand Tn+1(x)=2xTn(x)−Tn−1(x) and the orthogonality relation,
are satisfied by these polynomials. The expansion of the Dirac delta in terms of Chebyshev polynomials, can be written as,
∆n()= 2Tn() π√
Also the Green‘s function can be expressed in terms of theTn(x) as,
The functiongσ,λrepresents both the retarded and the advanced Green’s function in the limit λ→0+, whereλis the finite broadening parameter. g+,0+ andg−,0+ are the retarded and the advanced Green’s function respectively. Hence, the Dirac deltas and Green’s function are combinations of a polynomial ofH0 (the unperturbed Hamiltonian) and a coefficient which are functions of the frequency and the energy parameters. The trace in the conductivity can TH-2574_166121018
be written as a trace over a product of polynomials and ˆhoperators. TheΓmatrix needed in the expression of conductivity is written as,
Γnα11···,···,αnmm= Tr N
1+δn10 · · ·h˜αmTnm(H0) 1+δnm0
whereNis the number of unit cells. The upper indices can be used for any number of indices:
α1=α11α21· · ·α1N1 and ˜hα1 =(i¯h)N1hˆα1. Here the coefficients of the Chebyshev expansion can be written similarly in a matrix form as,
Λnm(ω)=¯h Z ∞
+∆n()gmA(/h¯−ω), (A.0.8) where f()=(1+eβ(−µ))−1 is the Fermi-Dirac distribution function, where βis the inverse temperature andµis the chemical potential. Hence the first-order conductivity becomes,
σαβ(ω)= −ie2 Ωch¯2ω
whereΩcis the volume of the unit cell.
Also, the density of states can be found in terms of Chebyshev polynomials using Eq. (A.0.2) as,
NTrδ(−H0)= 1 π√
whereN=dim ˆH0is the Hilbert space dimension,µndenotes the Chebyshev moments. The Chebyshev momentsµnis,
µn= 1 N
2 TrTn(H0). (A.0.11)
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