# Future prospects

radiation with MO transport has been explored. Finally, we have studied Faraday rotation which should be possible to be realized in the experiments.

### Appendices

TH-2574_166121018

## 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)−Tn1(x) and the orthogonality relation,

Z 1

1Tn(x)Tm(x) dx

√1−x2nm1+δn0

2 (A.0.1)

are satisfied by these polynomials. The expansion of the Dirac delta in terms of Chebyshev polynomials, can be written as,

δ(−H0)=

X

n=0

n()Tn(H0)

1+δn0, (A.0.2)

where

n()= 2Tn() π√

1−2. (A.0.3)

Also the Green‘s function can be expressed in terms of theTn(x) as,

gσ,λ(,H0)=h¯

X

n=0

gσ,λn ()Tn(H0)

1+δn0, (A.0.4)

where

gσ,λn ()=−2σieniσcos1(+iσλ)

p1−(+iσλ)2. (A.0.5)

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

"

α1Tn1(H0)

1+δn10 · · ·h˜αmTnm(H0) 1+δnm0

#

, (A.0.6)

whereNis the number of unit cells. The upper indices can be used for any number of indices:

α111α21· · ·α1N1 and ˜hα1 =(i¯h)N1α1. Here the coefficients of the Chebyshev expansion can be written similarly in a matrix form as,

Λn=Z

−∞

df()∆n() (A.0.7)

and

Λnm(ω)=¯h Z

−∞

df()

gRn(/h¯+ω)∆m()

+∆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)= −ie2c2ω

"

X

n

ΓnαβΛn+X

nm

Λnm(ω)Γnmα,β

#

(A.0.9)

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,

ρ()= 1

NTrδ(−H0)= 1 π√

1−2

X

0

µnTn(), (A.0.10)

whereN=dim ˆH0is the Hilbert space dimension,µndenotes the Chebyshev moments. The Chebyshev momentsµnis,

µn= 1 N

1+δn,0

2 TrTn(H0). (A.0.11)

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