# Faraday rotation and transmittance as markers of topological phase
transitions in 2D materials

Manuel Calixto  Corresponding author<EMAIL_ADDRESS>Department of Applied
Mathematics, University of Granada, Fuentenueva s/n, 18071 Granada, Spain
Institute Carlos I for Theoretical for Theoretical and Computational Physics
(iC1), Fuentenueva s/n, 18071 Granada, Spain  Alberto Mayorgas
<EMAIL_ADDRESS>Department of Applied Mathematics, University of Granada,
Fuentenueva s/n, 18071 Granada, Spain  Nicolás A. Cordero<EMAIL_ADDRESS>Department of Physics, University of Burgos, E-09001 Burgos, Spain
International Research Center in Critical Raw Materials for Advanced
Industrial Technologies (ICCRAM), University of Burgos, E-09001 Burgos, Spain
Institute Carlos I for Theoretical for Theoretical and Computational Physics
(iC1), Fuentenueva s/n, 18071 Granada, Spain  Elvira Romera<EMAIL_ADDRESS>Department of Atomic, Molecular and Nuclear Physics, University of Granada,
Fuentenueva s/n, 18071 Granada, Spain  Institute Carlos I for Theoretical for
Theoretical and Computational Physics (iC1), Fuentenueva s/n, 18071 Granada,
Spain  Octavio Castaños<EMAIL_ADDRESS>Institute of Nuclear
Sciences, National Autonomous University of Mexico, Apdo. Postal 70-543,
04510, CDMX, Mexico

###### Abstract

We analyze the magneto-optical conductivity (and related magnitudes like
transmittance and Faraday rotation of the irradiated polarized light) of some
elemental two-dimensional Dirac materials of group IV (graphene analogues,
buckled honeycomb lattices, like silicene, germanene, stannane, etc.), group V
(phosphorene), and zincblende heterostructures (like HgTe/CdTe quantum wells)
near the Dirac and gamma points, under out-of-plane magnetic and electric
fields, to characterize topological-band insulator phase transitions and their
critical points. We provide plots of the Faraday angle and transmittance as a
function of the polarized light frequency, for different external electric and
magnetic fields, chemical potential, HgTe layer thickness and temperature, to
tune the material magneto-optical properties. We have shown that
absortance/transmittance acquires extremal values at the critical point, where
the Faraday angle changes sign, thus providing fine markers of the topological
phase transition. In the case of non-topological materials as phosphorene, a
minimum of the transmittance is also observed due to the energy gap closing by
an external electric field.

###### pacs:

03.65.Vf, 03.65.Pm,

##  I Introduction

Two-dimensional (2D) materials have been extensively studied in recent years
(and are expected to be one of the crucial research topics in future years)
especially because of their remarkable electronic and magneto-optical
properties which make them hopeful candidates for next generation
optoelectronic devices. Graphene is the archetype of a 2D nanomaterial with
exceptional high tensile strength, electrical conductivity, transparency, etc.
In spite of being the thinnest one, it exhibits a giant Faraday rotation
($\Theta_{\mathrm{F}}\sim 6^{\circ}$) on polarized light in single- and
multilayer arrangements [1, 2, 3, 4, 5, 6] with experimental confirmation [7].
Faraday rotation is a fundamental magneto-optical phenomenon used in various
optical control, laser technology and magnetic field sensing techniques.

Magneto-optical properties of other buckled honeycomb lattices, like silicene
[8], have been studied in [9, 10, 11], together with other monolayer
transition metal dichalcogenides [12] and anisotropic versions like
phosphorene [13]. Magneto-optical measurements also provide signatures of the
topological phase transition (TPT; see [14, 15, 16] for standard textbooks on
the subject) in inverted HgTe/CdTe quantum wells (QW), distinguishing quantum
Hall (QH) from quantum spin Hall (QSH) phases [17], where one can tune the
band structure by fabricating QWs with different thicknesses $\lambda$. A
universal value of the Faraday rotation angle, close to the fine structure
constant, has been experimentally observed in thin HgTe QW with critical
thickness [18].

To determine experimentally the Faraday rotation effect in Dirac materials it
is convenient to consider: (1) A transverse-magnetic-polarized wave incident
from the left onto a single topological insulator sandwiched by dielectric
layers, which yields an enhancement of the Faraday rotation with an angle
larger than $700$ mrad and with a transmission higher than $90\%$ [19]. (2) A
graphene sheet sandwiched by one-dimensional topological photonic crystals
also an enhancement of the Faraday rotation can be achieved with high
transmittance [20]. (3) In thin films of 3D topological insulators [21] or by
considering thin films of Floquet topological insulators where giant Faraday
and Ker rotations have been observed under the action of a perpendicular
magnetic field or in a non-resonant optical field [22]. The inverse Faraday
effect (IFE) has been studied in Dirac materials in 2D and 3D, and these
studies have concluded that IFE is stronger than in conventional
semiconductors. Then the Dirac materials can be potentially useful for the
optical control of magnetization in optoelectronic devices [23].

Information theoretic measures also provide signatures of the TPT in silicene
[24, 25, 26, 27, 28] and HgTe/CdTe QWs [29], as an alternative to the usual
topological (Chern) numbers. They also account for semimetalic behavior of
phosphorene [30, 31] under perpendicular electric fields.

In this paper we perform a comparative study of the magneto-optical properties
of several 2D Dirac materials, looking for TPT signatures when the band
structure is tuned by applying external fields or by changing the material
characteristics. For this purpose, we focus on transmittance and Faraday
rotation near the critical point of the topological phase transition for
topological materials such as silicene and HgTe quantum wells. We found that,
for these materials, transmittance attains an absolute minimum
$\mathcal{T}_{0}$ at the critical TPT point for a certain value $\Omega_{0}$
of the normal incident polarized light frequency. This minimal behavior does
not depend on the chosen values of magnetic field, chemical potential and
temperature, although the location of $\Omega_{0}$ varies with them. An
inflection point of the Faraday angle is observed at each peak of the
transmittance, coinciding in frequency. As a novel perspective, we study that
non-topological materials, such as phosphorene, also exhibit an extremal value
of the transmittance when the energy gap is closed by an external electric
field.

The organization of the article is as follows. In Sec. II we discuss the
structure of time independent Bloch Hamiltonians for general two-band 2D-Dirac
material models, their Chern numbers and their minimal coupling to an external
perpendicular magnetic field. We particularize to graphene analogues
(silicene, germanene, etc.) in Sec. II.1, zincblende heterostructures
(HgTe/CdTe quantum wells) in Sec. II.2 and anisotropic materials like
phosphorene in II.3, calculating their energy spectrum and Hamiltonian
eigenstates (Landau levels) and describing their topological phases (when they
exist). In Sec. III we recall the Kubo-Greenwood formula for the magneto-
optical conductivity tensor $\bm{\sigma}$ of a 2D electron system in a
perpendicular magnetic field $B$ and an oscillating electric field of
frequency $\Omega$. In particular, we are interested in analyzing the
transmittance and Faraday rotation of linearly polarized light of frequency
$\Omega$ for normal incidence on the 2D material. Magneto-optical properties
of graphene analogues, zincblende heterostructures and phosphorene are
analyzed in Sections III.1, III.2 and III.3, respectively. For topological
insulator materials, we find that the critical point is generally
characterized by a minimum transmittance $\mathcal{T}_{0}$ at a given light
frequency $\Omega_{0}$, where the Faraday angle changes sign. The effect of
anisotropies is also discussed in phosphorene in Section III.3. Finally, Sec.
IV is devoted to conclusions.

##  II Some two-band 2D-Dirac material models

The time independent Bloch Hamiltonian of a two-band 2D insulator has the
general form

$H(\bm{k})=\epsilon_{0}(\bm{k})\tau_{0}+\bm{d}(\bm{k})\cdot\bm{\tau},$ (1)

where $\bm{\tau}=(\tau_{x},\tau_{y},\tau_{z})$ is the Pauli matrix vector,
$\tau_{0}$ denotes the $2\times 2$ identity matrix and $\bm{d}(\bm{k})$
parameterizes an effective spin-orbit coupling near the center $\Gamma$ or the
Dirac valleys $K$ and $K^{\prime}$ of the first Brillouin zone (FBZ), with
$\bm{k}=(k_{x},k_{y})$ the two-dimensional wavevector. The energy of the two
bands is $\epsilon_{\pm}(\bm{k})=\epsilon_{0}(\bm{k})\pm|\bm{d}(\bm{k})|$.

To distinguish between band insulator and topological insulator phases, one
can use the TKNN (Thouless-Kohmoto-Nightingale-Nijs) formula [32] providing
the Chern-Pontryagin number (related to the quantum spin Hall conductance and
the Berry phase [33])

$\mathcal{C}=\frac{1}{2\pi}\int\int_{\mathrm{FBZ}}d^{2}\bm{k}\left(\frac{\partial\hat{\bm{d}}(\bm{k})}{\partial
k_{x}}\times\frac{\partial\hat{\bm{d}}(\bm{k})}{\partial
k_{y}}\right)\cdot\hat{\bm{d}}(\bm{k}),$ (2)

with $\hat{\bm{d}}=\bm{d}/|\bm{d}|$, which counts the number of times (winding
number) the unit vector $\hat{\bm{d}}(\bm{k})$ wraps around the unit sphere as
$\bm{k}$ wraps around the entire FBZ. The Chern number $\mathcal{C}$ usually
depends on the sign of some material and (external) control parameters in the
Hamiltonian $H$ (see later for some examples), taking different values in
different phases. We shall see that magneto-optical conductivity measures also
capture the topological phase transition.

We shall consider the interaction with a perpendicular magnetic field
$\bm{B}=(0,0,B)$. Promoting the wavevector $\bm{k}$ to the momentum operator
$\bm{k}\to\bm{p}/\hbar=-\mathrm{i}\bm{\nabla}$, this interaction is introduced
through the usual minimal coupling, $\bm{p}\to\bm{P}=\bm{p}+e\bm{A}$ with
$\bm{A}=(A_{x},A_{y})=(-By,0)$ the electromagnetic potential (in the Landau
gauge) and $e$ the elementary charge (in absolute value). After Peierls’
substitution, which results in

$k_{x}\to P_{x}/\hbar=\frac{a^{\dagger}+a}{\sqrt{2}\ell_{B}},\quad k_{y}\to
P_{y}/\hbar=\frac{a^{\dagger}-a}{\mathrm{i}\sqrt{2}\ell_{B}},$ (3)

the Hamiltonian (1) can be eventually written in terms of creation
$a^{\dagger}$ and annihilation

${a}=\frac{\ell_{B}}{\sqrt{2}\hbar}(P_{x}-\mathrm{i}P_{y})=\frac{-1}{\sqrt{2}\ell_{B}}(y-y_{0}+\mathrm{i}\ell_{B}^{2}{p}_{y}/\hbar),$
(4)

operators, where $\ell_{B}=\sqrt{\hbar/(eB)}$ is the magnetic length and
$y_{0}=\ell_{B}^{2}k_{x}$ is the coordinate of the conserved center of the
cyclotron orbit.

Let us review some relevant physical examples.

###  II.1 Graphene analogues: silicene, germanene, etc

Silicene, germanene, and other transition metal dichalcogenides (of the Xene
type) exhibit an intrinsic non-zero spin-orbit coupling
$H_{\mathrm{so}}=-\frac{1}{2}s\xi\Delta_{\mathrm{so}}\tau_{z}$ ($s=\pm 1$ is
the spin of the electron and $\xi=\pm 1$ refer to the Dirac valleys $K$ and
$K^{\prime}$) due to second neighbors hopping terms in the tight binding model
[34]. Spin-orbit interaction $H_{\mathrm{so}}$ combined with and external
perpendicular electric field coupling
$H_{\Delta_{z}}=\frac{1}{2}\Delta_{z}\tau_{z}$, gives
$\bm{d}(\bm{k})=(v\hbar\xi k_{x},v\hbar k_{y},\Delta_{s\xi})$, where
$\Delta_{s\xi}=(\Delta_{z}-s\xi\Delta_{\mathrm{so}})/2$ results in a tunable
(Dirac mass) gap (see e.g. [35, 36, 37, 38]). In Table 1 we show a comparative
of spin-orbit coupling and Fermi velocity values for several 2D materials.

|  $\Delta_{\mathrm{so}}$ (meV) |  $l$ (Å) |  $v$ ($10^{5}$m/s)
---|---|---|---
Si | 4.2 | 0.22 | 4.2
Ge | 11.8 | 0.34 | 8.8
Sn | 36.0 | 0.42 | 9.7
Pb | 207.3 | 0.44 | –
Table 1:  Approximate values of model parameters $\Delta_{\mathrm{so}}$ (spin-
orbit coupling), $l$ (interlattice distance) and $v$ (Fermi velocity) for two
dimensional Si, Ge, Sn and Pb sheets. These data have been obtained from
first-principles computations in [38] ($\Delta_{\mathrm{so}}$ and $l$) and
[39, 40] ($v$).

The Chern number (2) turns out to be

$\mathcal{C}_{s\xi}=\xi\,\mathrm{sign}(\Delta_{s\xi}),$ (5)

where we have integrated on the whole plane, as corresponds to the FBZ in the
continuum limit (zero lattice constant). Therefore, the topological phase is
determined by the sign of the Dirac mass at each valley $\xi$. More precisely,
there is a TPT from a topological insulator (TI,
$|\Delta_{z}|<\Delta_{\mathrm{so}}$) to a band insulator (BI,
$|\Delta_{z}|>\Delta_{\mathrm{so}}$), at a charge neutrality point (CNP)
$\Delta_{z}^{(0)}=s\xi\Delta_{\mathrm{so}}$, where there is a gap cancellation
between the perpendicular electric field and the spin-orbit coupling.

Using the general prescription (3), the minimal coupling with a perpendicular
magnetic field $B$ then results in a different Hamiltonian $H_{\xi}$ for each
valley $\xi=\pm 1$

$H_{1}=\left(\begin{array}[]{cc}\Delta_{s,1}&{\hbar\omega}a\\\
{\hbar\omega}a^{\dagger}&-\Delta_{s,1}\end{array}\right),\;H_{-1}=\left(\begin{array}[]{cc}\Delta_{s,-1}&-{\hbar\omega}a^{\dagger}\\\
-{\hbar\omega}a&-\Delta_{s,-1}\end{array}\right),$ (6)

where $\omega=\sqrt{2}v/\ell_{B}$ denotes the cyclotron frequency. The
eigenvalues of both Hamiltonians are simply:

$E_{n}^{s\xi}=\left\\{\begin{array}[]{l}\mathrm{sgn}(n)\sqrt{|n|\hbar^{2}\omega^{2}+\Delta_{s\xi}^{2}},\quad
n\neq 0,\\\ -\xi\Delta_{s\xi},\quad n=0,\end{array}\right.$ (7)

and the corresponding eigenstates are written in terms of Fock states
$||n|\rangle$, for Landau level (LL) index $n=0,\pm 1,\pm 2,\dots$ [valence
$(-)$ and conduction $(+)$ states], as spinors

$|\bm{n}\rangle_{s\xi}=\left(\begin{array}[]{c}A_{n}^{s\xi}\left\lvert|n|-\frac{\xi+1}{2}\right\rangle\\\
B_{n}^{s\xi}\left\lvert|n|+\frac{\xi-1}{2}\right\rangle\end{array}\right),$
(8)

with coefficients (see [41, 42, 9, 43] for similar results)

$\displaystyle A_{n}^{s\xi}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}\frac{\mathrm{sgn}(n)}{\sqrt{2}}\sqrt{1+\mathrm{sgn}(n)\cos\theta_{n}^{s\xi}},\quad
n\neq 0,\\\ (1-\xi)/2,\quad n=0,\end{array}\right.$ (11) $\displaystyle
B_{n}^{s\xi}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}\frac{\xi}{\sqrt{2}}\sqrt{1-\mathrm{sgn}(n)\cos\theta_{n}^{s\xi}},\quad
n\neq 0,\\\ (1+\xi)/2,\quad n=0,\end{array}\right.$ (16)

where
$\theta_{n}^{s\xi}=\arctan\left(\hbar\omega\sqrt{|n|}/\Delta_{s\xi}\right)$,
that is, $\cos\theta_{n}^{s\xi}=\Delta_{s\xi}/|E_{n}^{s\xi}|$. Note that
$A_{n}^{s\xi}$ and $B_{n}^{s\xi}$ can eventually be written as
$\cos(\theta_{n}^{s\xi}/2)$ or $\sin(\theta_{n}^{s\xi}/2)$, depending on
$\mathrm{sgn}(n)$.

In Figure 1 we plot the low energy spectra of silicene, given by (7), as a
function of the external electric field $\Delta_{z}$, together with the charge
neutrality (critical) points $\Delta_{z}^{(0)}=\pm|\Delta_{\mathrm{so}}|$
(marked by vertical dashed lines) at which the TPT takes place.

Figure 1: Low energy spectra of silicene as a function of the external
electric potential $\Delta_{z}$ (in $\Delta_{\text{so}}$ units) for $B=0.05$
T. Landau levels $n=\pm 1,\pm 2$ and $\pm 3$ [valence $(-)$ and conduction
$(+)$], at valley $\xi=1$, are represented by thin solid lines, blue for
$s=-1$ and red for $s=1$ (for the other valley we simply have
$E_{n}^{s,-\xi}=E_{n}^{-s,\xi}$). The edge states $n=0$ are represented by
thick lines at both valleys: solid at $\xi=1$ and dashed at $\xi=-1$. Vertical
dashed gray lines indicate the charge neutrality points separating band
insulator ($|\Delta_{z}|>\Delta_{\mathrm{so}}$) from topological insulator (
$|\Delta_{z}|<\Delta_{\mathrm{so}}$) phases.

###  II.2 HgTe/CdTe quantum wells

In [44, 45, 46, 47] it was shown that quantum spin Hall effect can be realized
in mercury telluride-cadmium telluride semiconductor quantum wells. Similar
effects were also predicted in Type-II semiconductor quantum wells made from
InAs/GaSb/AlSb [48]. The surface states in these 3D topological insulators can
be described by a 2D modified effective Dirac Hamiltonian

$H=\left(\begin{array}[]{cc}H_{+}&0\\\
0&H_{-}\end{array}\right),\,H_{s}(\bm{k})=\epsilon_{0}(\bm{k})\tau_{0}+\bm{d}_{s}(\bm{k})\cdot\bm{\tau},$
(17)

where $s=\pm 1$ is the spin and $H_{-}(\bm{k})=H_{+}^{*}(-\bm{k})$
(temporarily reversed). The expansion of $H_{s}(\bm{k})$ about the center
$\Gamma$ of the first Brillouin zone gives [45]

$\epsilon_{0}(\bm{k})=\gamma-\delta\bm{k}^{2},\quad\bm{d}_{s}(\bm{k})=(\alpha
sk_{x},\alpha k_{y},\mu-\beta\bm{k}^{2}),$ (18)

where $\alpha,\beta,\gamma,\delta$ and $\mu$ are expansion parameters that
depend on the heterostructure (the HgTe layer thickness $\lambda$). The most
important one is the mass or gap parameter $\mu$, which changes sign at a
critical HgTe layer thickness $\lambda_{c}$ when going from the normal
($\lambda<\lambda_{c}$ or $\mu/\beta<0$) to the inverted
($\lambda>\lambda_{c}$ or $\mu/\beta>0$) regime [49]. Typical values of these
parameters for different HgTe layer thickness (below and above $\lambda_{c}$)
can be found in [49] and in Table 2 ($\gamma$ can be neglected).

$\lambda$(nm) |  $\alpha$(meV$\cdot$nm) |  $\beta$(meV$\cdot$nm2) |  $\delta$(meV$\cdot$nm2) |  $\mu$(meV)
---|---|---|---|---
5.5 | 387 | -480 | -306 | 9
6.1 | 378 | -553 | -378 | -0.15
7.0 | 365 | -686 | -512 | -10
Table 2:  Material parameters for HgTe/CdTe quantum wells with different HgTe
layer thicknesses $\lambda$ [49].

The energy of the two bands is

$\epsilon_{\pm}(\bm{k})=\epsilon_{0}(\bm{k})\pm\sqrt{\alpha^{2}\bm{k}^{2}+(\mu-\beta\bm{k}^{2})^{2}}.$
(19)

The TKNN formula (2) for $\bm{d}_{s}(\bm{k})$ provides the Chern number

$\mathcal{C}_{s}=s[\mathrm{sign}(\mu)+\mathrm{sign}(\beta)],$ (20)

where we have integrated on the whole plane, as corresponds to the continuum
limit. According to Table 2, $\beta$ does not change sing and, therefore, the
topological phase transition occurs when $\mu$ changes sign, as already
mentioned. In reference [49], the normal and inverted regimes are equivalently
given by the sign of $\mu/\beta$.

Using again the general prescription (3), the minimal coupling with a
perpendicular magnetic field $B$ now results in

$\displaystyle H_{+}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\gamma+\mu-\frac{(\delta+\beta)(2N+1)}{\ell_{B}^{2}}&\frac{\sqrt{2}\alpha}{\ell_{B}}a\\\
\frac{\sqrt{2}\alpha}{\ell_{B}}a^{\dagger}&\gamma-\mu-\frac{(\delta-\beta)(2N+1)}{\ell_{B}^{2}}\end{array}\right),$
(23) $\displaystyle H_{-}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\gamma+\mu-\frac{(\delta+\beta)(2N+1)}{\ell_{B}^{2}}&-\frac{\sqrt{2}\alpha}{\ell_{B}}a^{\dagger}\\\
-\frac{\sqrt{2}\alpha}{\ell_{B}}a&\gamma-\mu-\frac{(\delta-\beta)(2N+1)}{\ell_{B}^{2}}\end{array}\right),$
(26)

with $N=a^{\dagger}a$. A Zeeman term contribution

$H_{s}^{Z}=-\frac{s}{2}B\mu_{\mathrm{B}}\left(g_{\mathrm{e}}\frac{\tau_{0}+\tau_{z}}{2}+g_{\mathrm{h}}\frac{\tau_{0}-\tau_{z}}{2}\right)$
(28)

can also be added to the Hamiltonian, with $\mu_{\mathrm{B}}\simeq 0.058$
meV/T the Bohr magneton and $g_{\mathrm{e,h}}$ the effective (out-of-plane)
$g$-factors for electrons and holes (conduction and valence bands).

Using (Fock state) eigenvectors $||n|\rangle$ of the (Landau level) number
operator $N=a^{\dagger}a$, one can analytically obtain the eigenspectrum

$\displaystyle E_{n}^{s}=$
$\displaystyle\,\gamma-\tfrac{2\delta|n|-s\beta}{\ell_{B}^{2}}-s\tfrac{g_{\mathrm{e}}+g_{\mathrm{h}}}{4}B\mu_{\mathrm{B}}$
(29)
$\displaystyle+\mathrm{sgn}(n)\sqrt{\tfrac{2\alpha^{2}|n|}{\ell_{B}^{2}}+\left(\mu-\tfrac{2\beta|n|-s\delta}{\ell_{B}^{2}}-s\tfrac{g_{\mathrm{e}}-g_{\mathrm{h}}}{4}B\mu_{\mathrm{B}}\right)^{2}},$

for LL index $n=\pm 1,\pm 2,\pm 3,\dots$ [valence $(-)$ and conduction $(+)$]
, and

$E_{0}^{s}=\gamma-s\mu-\frac{\delta-s\beta}{\ell_{B}^{2}}-B\mu_{\mathrm{B}}\left(\frac{s+1}{4}g_{\mathrm{h}}+\frac{s-1}{4}g_{\mathrm{e}}\right),$
(30)

for the edge states $n=0$, $s=\pm 1$. These eigenvalues coincide with those in
[50, 51, 17] for the identification $s=\\{-1,1\\}=\\{\uparrow,\downarrow\\}$.

The corresponding eigenvectors are

$|\bm{n}\rangle_{s}=\left(\begin{array}[]{l}A_{n}^{s}\left\lvert|n|-\frac{s+1}{2}\right\rangle\\\
B_{n}^{s}\left\lvert|n|+\frac{s-1}{2}\right\rangle\end{array}\right),$ (31)

with coefficients

$\displaystyle A_{n}^{s}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}\frac{\mathrm{sgn}(n)}{\sqrt{2}}\sqrt{1+\mathrm{sgn}(n)\cos\vartheta_{n}^{s}},\quad
n\neq 0,\\\ (1-s)/2,\quad n=0,\end{array}\right.$ (34) $\displaystyle
B_{n}^{s}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}\frac{s}{\sqrt{2}}\sqrt{1-\mathrm{sgn}(n)\cos\vartheta_{n}^{s}},\quad
n\neq 0,\\\ (1+s)/2,\quad n=0,\end{array}\right.$ (39)

where

$\vartheta_{n}^{s}=\arctan\left(\frac{\sqrt{2|n|}\,\alpha/\ell_{B}}{\mu-\frac{2\beta|n|-s\delta}{\ell_{B}^{2}}-s\frac{g_{\mathrm{e}}-g_{\mathrm{h}}}{4}B\mu_{\mathrm{B}}}\right).$
(40)

As for the graphene analogues in (16), the coefficients $A_{n}^{s}$ and
$B_{n}^{s}$ can eventually be written as sine and cosine of half angle,
depending on $\mathrm{sgn}(n)$.

According to (30), the band inversion for edge states occurs when

$E_{0}^{+}=E_{0}^{-}\Rightarrow
B_{\mathrm{inv}}=\frac{\mu}{e\beta/\hbar-\mu_{\mathrm{B}}(g_{\mathrm{e}}+g_{\mathrm{h}})/4},$
(41)

which gives the critical magnetic field $B_{c}$ which separates the QSH and QH
regimes [51]. For example, for the material parameters in Table 2
corresponding to a QW thickness $\lambda=7.0$ nm and $g$-factors
$g_{\mathrm{e}}=22.7,g_{\mathrm{h}}=-1.21$, one obtains
$B_{\mathrm{inv}}\simeq 7.4$ T. See also Figure 2 for a graphical
representation of this band inversion.

From now on we shall discard Zeeman coupling for the sake of convenience since
our main conclusions remain qualitatively equivalent. We address the
interested reader to the Supplemental Material [52] where we reproduce some
results of Reference [17] for non-zero Zeeman coupling and contrast with the
zero Zeeman coupling case.

We shall use a linear fit

$\displaystyle\mu(\lambda)$ $\displaystyle=$ $\displaystyle
77.31\,-12.53\lambda,$ $\displaystyle\alpha(\lambda)$ $\displaystyle=$
$\displaystyle 467.49-14.65\lambda,$ $\displaystyle\beta(\lambda)$
$\displaystyle=$ $\displaystyle 283.58-138.16\lambda,$
$\displaystyle\delta(\lambda)$ $\displaystyle=$ $\displaystyle
458.46-138.25\lambda,$ (42)

of the material parameters in Table 2 as a function of the HgTe layer
thickness $\lambda$ (dimensionless units and $\lambda$ in nm units). In all
cases the coefficient of determination is $R^{2}>0.99$. Looking at
$\mu(\lambda)$ in (42), we can obtain an estimation of the critical HgTe
thickness at which the topological phase transition occurs as

$\mu=0\Rightarrow\lambda_{c}=6.17~{}\mathrm{nm}.$ (43)

In Figure 2 we plot the low energy spectra given by (29) and (30) as a
function of the HgTe layer thickness $\lambda$, where we have extrapolated the
linear fit (42) to the interval [4 nm, 8 nm]. When neglecting Zeeman coupling,
the band inversion for edge states (41) occurs for $B=\hbar\mu/(e\beta)$
which, using the linear fit (42), provides a relation

$\lambda_{\mathrm{inv}}(B)=\frac{368.31-2.05B}{59.7-B}$ (44)

between the applied magnetic field $B$ (in Tesla) and the HgTe layer thickness
$\lambda_{\mathrm{inv}}(B)$ (in nanometers) at which the band inversion
$E_{0}^{+}=E_{0}^{-}$ takes place. Note that
$\lambda_{\mathrm{inv}}(B)\simeq\lambda_{c}=6.17$ nm for low $B\ll 1$ T, and
that $E_{0}^{+}=E_{0}^{-}\simeq 0$ meV at this point as shows Figure 2.

Figure 2: Low-energy spectra $E_{n}^{s}$ of a HgTe/CdTe quantum well as a
function of the HgTe layer thickness $\lambda$ for $B=0.5$ T. Landau levels
$n=\pm 1,\pm 2,\pm 3$ [valence $(-)$ and conduction $(+)$] are represented by
thin solid lines, blue for spin $s=-1$ and red for $s=1$. Edge states ($n=0$)
are represented by thick lines. A vertical dashed black line indicates the
HgTe thickness $\lambda_{\mathrm{inv}}(0.5)=6.20$ nm$\>\simeq\lambda_{c}$
where the band inversion for edge states occurs for $B=0.5$ T according to
(44).

###  II.3 Phosphorene as an anisotropic material

The physics of phosphorene has been extensively studied [53, 54, 55, 56, 57,
58, 59, 60, 61, 62, 63, 64, 65, 66, 67]. There are several approaches to the
low energy Hamiltonian of phosphorene in the literature. Rudenko et al. [68]
and Ezawa [69] propose a four-band and five-neighbors tight-binding model
later simplified to two-bands [69]. Several approximations of this two-band
model have been used in [70, 71, 72, 13]. We shall choose for our study the
Hamiltonian

${H}=\left(\begin{array}[]{cc}E_{\mathrm{c}}+\alpha_{x}k_{x}^{2}+\alpha_{y}k_{y}^{2}&\gamma
k_{x}\\\ \gamma
k_{x}&E_{\mathrm{v}}-\beta_{x}k_{x}^{2}-\beta_{y}k_{y}^{2}\end{array}\right).$
(45)

proposed by Zhou and collaborators [13]. This corresponds to a Bloch
Hamiltonian (1) with

$\displaystyle\epsilon_{0}(\bm{k})$ $\displaystyle=$
$\displaystyle\frac{E_{\mathrm{c}}+E_{\mathrm{v}}+(\alpha_{x}-\beta_{x})k_{x}^{2}+(\alpha_{y}-\beta_{y})k_{y}^{2}}{2},$
(46) $\displaystyle\bm{d}(\bm{k})$ $\displaystyle=$ $\displaystyle\left(\gamma
k_{x},0,\frac{E_{\mathrm{c}}-E_{\mathrm{v}}+(\alpha_{x}+\beta_{x})k_{x}^{2}+(\alpha_{y}+\beta_{y})k_{y}^{2}}{2}\right),$

The Hamiltonian (45) provides a trivial Chern number (2), even in the presence
of a tunable perpendicular constant electric field (see below), which means
that monolayer phosphorene does not have a topological phase _per se_. It has
been shown that topological transitions can be induced in phosphorene when
rapidly driven by in-plane time-periodic laser fields [73]; these are called
in general “Floquet topological insulators” (see e.g. [74, 75, 76]), but we
shall not consider this possibility here. Although phosphorene is not a
topological material, we will see in Sec. III.3 that the critical magneto-
optical properties (e.g., minimum transmittance) observed for silicene and
HgTe QWs are still valid in phosphorene when closing the energy gap through an
external electric field. Another possibility to modify the energy gap could be
by applying strain [60, 70] (see later in Sec. III.3).

The material parameters of phosphorene can be written in terms of conduction
(c) and valence (v) effective masses as (see [13] for more information)

$\alpha_{x,y}=\frac{\hbar^{2}}{2m_{\textrm{c}x,\textrm{c}y}},\quad\beta_{x,y}=\frac{\hbar^{2}}{2m_{\textrm{v}x,\textrm{v}y}},$
(47)

with

$\begin{array}[]{ll}m_{\textrm{c}x}=0.793m_{\textrm{e}},&m_{\textrm{c}y}=0.848m_{\textrm{e}},\\\
m_{\textrm{v}x}=1.363m_{\textrm{e}},&m_{\textrm{v}y}=1.142m_{\textrm{e}},\end{array}$
(48)

and $m_{\textrm{e}}$ is the free electron mass. Conduction and valence band
edge energies are $E_{\mathrm{c}}=0.34$ eV and $E_{\mathrm{v}}=-1.18$ eV, so
that the energy gap is $E_{\textrm{g}}=E_{\mathrm{c}}-E_{\mathrm{v}}=1.52$ eV.
The interband coupling parameter is $\gamma=-0.523$ eV$\cdot$nm.

When coupling to an external perpendicular magnetic field, the anisotropic
character of phosphorene slightly modifies Peierls’ substitution (3), which
now adopts the following form

$k_{x}\to\frac{P_{x}}{\hbar}=\frac{a^{\dagger}+a}{\sqrt{2}\alpha_{yx}\ell_{B}},\quad
k_{y}\to\frac{P_{y}}{\hbar}=\frac{\alpha_{yx}(a^{\dagger}-a)}{\mathrm{i}\sqrt{2}\ell_{B}},$
(49)

with $\alpha_{yx}=\left(\frac{m_{\mathrm{c}y}}{m_{\mathrm{c}x}}\right)^{1/4}$.
Therefore, applying this prescription to (45), the final Hamiltonian can be
written as

$\displaystyle H=$
$\displaystyle\,\hbar\omega_{\gamma}({a}+{a}^{\dagger})\tau_{x}+\left[E_{\mathrm{c}}+\hbar\omega_{\mathrm{c}}({a}^{\dagger}{a}+1/2)\right]\frac{\tau_{0}+\tau_{z}}{2}$
(50)
$\displaystyle+\left[E_{\mathrm{v}}-\hbar\omega_{\mathrm{v}}({a}^{\dagger}{a}+1/2)-\hbar\omega^{\prime}({a}^{2}+{a}^{\dagger}{}^{2})\right]\frac{\tau_{0}-\tau_{z}}{2},$

in terms of the annihilation (and creation ${a}^{\dagger}$) operator

${a}=\sqrt{\frac{m_{\mathrm{c}y}\omega_{c}}{2\hbar}}\left(y-y_{0}+i\frac{\hat{p}_{y}}{m_{\mathrm{c}y}\omega_{\mathrm{c}}}\right),$
(51)

in analogy to (4), where some effective frequencies have been defined as

$\begin{array}[]{ll}\omega_{\mathrm{c}}=\frac{eB}{\sqrt{m_{\mathrm{c}x}m_{\mathrm{c}y}}},&\omega_{\gamma}=\frac{\gamma}{\sqrt{2}\hbar\alpha_{yx}\ell_{B}},\\\
\omega_{\mathrm{v}}=(r_{x}+r_{y})\omega_{\mathrm{c}},&\omega^{\prime}=(r_{x}-r_{y})\omega_{\mathrm{c}}/2,\end{array}$
(52)

with

$r_{x}=\frac{m_{\mathrm{c}x}}{2m_{\mathrm{v}x}},\,r_{y}=\frac{m_{\mathrm{c}y}}{2m_{\mathrm{v}y}}.$

As we did for silicene, we shall also consider here the application of a
perpendicular electric field to the phosphorene sheet in the usual form [77]
$\hat{H}_{\Delta}=\Delta_{z}\tau_{z}$, with $\Delta_{z}$ the on-site electric
potential. Unlike for silicene and HgTe QWs, the diagonalization of the
phosphorene Hamiltonian (50) has to be done numerically [30].

Note that the Hamiltonian (50) preserves the parity $\pi(n,s)=e^{i\pi n_{s}}$
of the state $|\bm{n}\rangle_{s}$, with $n_{s}=n+(s+1)/2$ (see e.g. [30]).
This means that the matrix elements
${}_{s}\langle\bm{n}|H|\bm{n}^{\prime}\rangle_{s^{\prime}}\propto\delta_{\pi(n,s),\pi(n^{\prime},s^{\prime})}$
are zero between states of different parity. Therefore, this parity symmetry
helps in the diagonalization process and any (non-degenerate) eigenstate of
$H$ has a definite parity. The Hamiltonian eigenstates can now be written as

$|\psi_{l}\rangle=\sum_{n,s}c^{(l)}_{n,s}|\bm{n}\rangle_{s},$ (53)

where $l\in\mathbb{Z}$ denotes the LL index ($l>0$ for conduction and $l\leq
0$ for valence band). The sum $\sum_{n,s}$ is constrained to $\pi(n,s)=\pm 1$,
depending on the even $(+)$ and odd $(-)$ parity of $k$. The coefficients
$c^{(k)}_{n,s}$ are obtained by numerical diagonalization of the Hamiltonian
matrix, which is truncated to $n\leq N$, with $N$ large enough to achieve
convergent results for given values of the magnetic and electric fields. In
particular, we have used Fock states with $N\leq 1000$ to achieve convergence
(with error tolerance $\leq 10^{-15}$ eV) for $B=0.5$ T in the six first
Hamiltonian eigenvalues in the range $-1.55\leq\Delta_{z}\leq-1.49$ eV. The
resulting spectrum, as a function of the electric field potential
$\Delta_{z}$, can be seen in Figure 3 for a magnetic field of $B=0.5$ T
(higher magnetic fields need less Fock states to achieve convergence). The
vertical dashed line gives the point $\Delta_{z}=-1.520$ eV at which the
electric potential equals minus the energy gap
$E_{\textrm{g}}=E_{\mathrm{c}}-E_{\mathrm{v}}=1.52$ eV of phosphorene. This is
not really a critical point in the same sense as
$\Delta_{z}^{(0)}=\Delta_{\mathrm{so}}=4.2$ meV for silicene and
$\lambda_{c}=6.17$ nm for HgTe QWs, since phosphorene as such (as already
said) does not display a topological phase. However, we will see in Section
III.3 that the phosphorene transmittance still presents a minimum at
$\Delta_{z}^{(0)}=-1.523$ eV, which closes the energy gap
$E_{\textrm{g}}=1.52$ eV at low magnetic fields.

It is also interesting to note that the LLs of phosphorene are degenerated in
pairs for an electric potential below $\Delta_{z}\simeq-1.53$ eV. Namely, we
obtain numerically that $|E_{l}^{\text{even}}-E_{l+1}^{\text{odd}}|\leq
10^{-4}$ eV for all $\Delta_{z}<-1.53$ eV and $l=-6,-4,-2,0,2,4$ as it shows
the left hand side of Figure 3. This energy degeneracy will influence the
conductivity as well.

Figure 3: Low energy spectra $E_{l}$ of phosphorene as function of the
electric field potential $\Delta_{z}$ for thirteen Hamiltonian eigenstates
$l=-6,\dots,0,\dots,6$ and a magnetic field $B=0.5$ T. Valence and conduction
band states of even (odd) parity $l=\pm 2,\pm 4,\pm 6$ ($n=\pm 1,\pm 3,\pm 5$)
are represented in red (blue) color. The edge state $E_{0}^{\text{even}}$ is
represented by a thick black line. The vertical dashed black line is the point
$\Delta_{z}=-E_{\textrm{g}}=-1.520$ eV at which the electric potential equals
the energy gap of phosphorene.

##  III Magneto-optical conductivity

The magneto-optical conductivity tensor $\bm{\sigma}$ of a 2D electron system
in a perpendicular magnetic field $B$ and an oscillating electric field of
frequency $\Omega$, can be obtained from Kubo-Greenwood formula [32, 78, 79]
in the Landau-level representation:

$\sigma_{ij}(\Omega,B)=\frac{\mathrm{i}\hbar}{2\pi\ell_{B}^{2}}\sum_{\bm{n},\bm{m}}\frac{f_{m}-f_{n}}{E_{n}-E_{m}}\frac{\langle\bm{m}|j_{i}|\bm{n}\rangle\langle\bm{n}|j_{j}|\bm{m}>}{\hbar\Omega+E_{m}-E_{n}+i\eta},$
(54)

where

$\bm{j}=\frac{\mathrm{i}e}{\hbar}[H,\bm{r}]=\frac{e}{\hbar}\nabla_{\bm{k}}H$
(55)

is the current operator, with $\bm{r}=(x,y)$ and
$\nabla_{\bm{k}}=(\partial_{k_{x}},\partial_{k_{y}})$ [the minimal coupling
prescription (3) is understood under external electromagnetic fields], and
$f_{n}=1/(1+\exp[({E}_{n}-\mu_{\mathrm{F}})/(k_{B}T)])$ is the Fermi
distribution function at temperature $T$ and chemical potential
$\mu_{\mathrm{F}}$. In the zero temperature limit, the Fermi function $f_{n}$
is replaced by the Heaviside step function $\Theta(\mu_{\mathrm{F}}-E_{n})$,
which enforces the Pauli exclusion principle for optical transitions (they are
allowed between occupied and unoccupied states). The parameter $\eta$ is a
small residual scattering rate of charge carriers and, although the exact
shape of $\sigma_{ij}$ would depend on the details of the scattering
mechanisms, using a constant $\eta$ gives a good, qualitative description of
the essential mechanisms relevant for magneto-optical experiments. In
$\sum_{\bm{n}}$ of eq. (54) it is also implicit the sum over spin $s$ and
valley $\xi$, besides the LL index $n$ (for graphene, there is a twofold spin
and valley degeneracy, so that the extra sum just contributes with a
degeneracy factor $g=4$). We shall measure $\sigma_{ij}$ in units of the
conductance quantum $\sigma_{0}=e^{2}/h=38.8$ $\mu$S [78] and renormalize the
currents as $\bar{j}=j/(e/\hbar)=\nabla_{\bm{k}}H$, so that

$\frac{\sigma_{ij}(\Omega,B)}{\sigma_{0}}=\frac{\mathrm{i}}{\ell_{B}^{2}}\sum_{\bm{n},\bm{m}}\frac{f_{m}-f_{n}}{E_{n}-E_{m}}\frac{\langle\bm{m}|\bar{j}_{i}|\bm{n}\rangle\langle\bm{n}|\bar{j}_{j}|\bm{m}>}{\hbar\Omega+E_{m}-E_{n}+i\eta},$
(56)

We shall analyze the transmittance and Faraday rotation of linearly polarized
light of frequency $\Omega$ for normal incidence on the 2D material, where the
electric fields of incident ($\bm{E}^{i}$) and transmitted ($\bm{E}^{t}$)
waves are related through the conductivity tensor $\bm{\sigma}$ by the formula
[80, 81, 82]

$\bm{E}^{t}=\left(\bm{I}+\tfrac{1}{2}Z_{0}\bm{\sigma}\right)^{-1}\cdot\bm{E}^{i},$
(57)

where $Z_{0}=2\alpha/\sigma_{0}$ is the vacuum impedance ($\alpha=1/137$ is
the fine-structure constant) and $\bm{I}$ denotes the $2\times 2$ identity
matrix. We also assume that the incident field is linearly polarized in the
$x$ axis, that is $\bm{E}^{i}=(E_{x}^{i},0)$. From here, the transmittance
$\mathcal{T}$ and the Faraday rotation angle $\Theta_{\mathrm{F}}$ (in
degrees) are [2, 82]

$\displaystyle\mathcal{T}=$
$\displaystyle\,\frac{1}{2}(|t_{+}|^{2}+|t_{-}|^{2})\simeq
1-Z_{0}\mathrm{Re}(\sigma_{xx})\,,$ (58) $\displaystyle\Theta_{\mathrm{F}}=$
$\displaystyle\,\frac{1}{2}(\arg(t_{+})+\arg(t_{-}))\simeq\frac{180}{2\pi}Z_{0}\mathrm{Re}(\sigma_{xy})\,,$
(59)

where $t_{\pm}=E^{t}_{\pm}/|\bm{E}^{i}|$ are the transmission amplitudes in
the circular polarization basis [83, 84] or chiral basis [85],
$\bm{E}^{t}_{\pm}=E^{t}_{x}\pm\mathrm{i}E^{t}_{y}$. $\mathrm{Re}(\sigma_{ij})$
means the real part of $\sigma_{ij}$ and $\arg(t_{\pm})$ the complex argument.
We have also provided the approximate expressions in the limit of weak
absorption for isotropic materials. Note that, in this case, according to
(59), the absorption peaks of $\mathrm{Re}[\sigma_{xx}(\Omega)]$ shown in
Figure 6, correspond to dips of the transmittance $\mathcal{T}$. Silicene and
HgTe QWs have both longitudinal conductivities equal
$\sigma_{xx}=\sigma_{yy}$, but this symmetry is broken for anisotropic
materials like phosphorene [10, 85] (see later on Section III.3). Therefore,
in phosphorene, we cannot apply the approximation in eq.(59) and we have to
use the strict equality.

In the circular polarization (right- and left-handed $\pm$) basis, the
conductivity is given by $\sigma_{\pm}=\sigma_{xx}\pm\mathrm{i}\sigma_{xy}$,
and the absorptive part is therefore
$\mathrm{Re}(\sigma_{\pm})=\mathrm{Re}(\sigma_{xx})\mp\mathrm{Im}(\sigma_{xy})$.
In the Supplemental Material [52] we provide extra plots for the silicene
conductivity under circular polarization which reproduce the results of [9].

###  III.1 Magneto-optical properties of graphene analogues

The current operator (55) for this case is
$\bm{j}=(j_{x},j_{y})=ev(\xi\tau_{x},\tau_{y})$. The matrix elements

$\begin{array}[]{l}\langle\bm{m}|\tau_{x}|\bm{n}\rangle_{s\xi}=A_{m}^{s\xi}B_{n}^{s\xi}\delta_{|m|-\xi,|n|}+A_{n}^{s\xi}B_{m}^{s\xi}\delta_{|m|+\xi,|n|},\\\
\langle\bm{m}|\tau_{y}|\bm{n}\rangle_{s\xi}=-\mathrm{i}A_{m}^{s\xi}B_{n}^{s\xi}\delta_{|m|-\xi,|n|}+\mathrm{i}A_{n}^{s\xi}B_{m}^{s\xi}\delta_{|m|+\xi,|n|},\end{array}$
(60)

provide the familiar selection rules $|n|=|m|\pm 1$ for LL transitions.
Plugging (60) into the general expression (56) we obtain the magneto-optical
conductivity for graphene analogues. In Figure 4 we plot the real and
imaginary parts of the conductivity tensor components $\sigma_{ij}$ (in
$\sigma_{0}=e^{2}/h$ units) of silicene as a function of the polarized light
frequency $\Omega$ at three different electric potentials
$\Delta_{z}=0.5\Delta_{\text{so}},\Delta_{\text{so}},1.5\Delta_{\text{so}}$
around the critical point $\Delta_{z}^{(0)}=\Delta_{\mathrm{so}}$, for a
magnetic field $B=0.05$ T and some representative values of the chemical
potential $\mu_{\mathrm{F}}=2.1$ meV, temperature $T=1$ K and scattering rate
$\eta=0.1$ meV. For $\hbar\Omega\in[0,20]$ meV, we achieve convergence with
100 LLs, that is, restricting the sum in (56) as
$\sum_{n=-\infty}^{\infty}\to\sum_{n=-100}^{100}$. More explicitly, for the
parameters mentioned above,

$\left|\sum_{n=-100}^{n=100}\sigma_{ij}-\sum_{n=-99}^{n=99}\sigma_{ij}\right|/\sigma_{0}\leq\begin{cases}10^{-5}\quad\text{if}\>\sigma_{ij}=\mathrm{Re}(\sigma_{xx})\,,\\\
10^{-15}\quad\text{if}\>\sigma_{ij}=\mathrm{Re}(\sigma_{xy})\,,\\\
10^{-3}\quad\text{if}\>\sigma_{ij}=\mathrm{Im}(\sigma_{xx})\,,\\\
10^{-14}\quad\text{if}\>\sigma_{ij}=\mathrm{Im}(\sigma_{xy})\,.\\\
\end{cases}$ (61)

Figure 4: Real and imaginary parts of the longitudinal $\sigma_{xx}$ and
transverse Hall $\sigma_{xy}$ magneto-optical conductivities in a silicene
monolayer under three different electric potentials
$\Delta_{z}=0.5\Delta_{\text{so}},\Delta_{\text{so}},1.5\Delta_{\text{so}}$,
as a function of the polarized light frequency $\Omega$ and in
$\sigma_{0}=e^{2}/h$ units. We set the conductivity parameters as
$\mu_{\mathrm{F}}=2.1$ meV, $B=0.05$ T, $T=1$ K and $\eta=0.1$ meV.

Each peak on the plot of the conductivity $\mathrm{Re}(\sigma_{xx})$ against
$\hbar\Omega$ represents an electron transition between two LLs $n,m$
connected by the selection rules $|n|=|m|\pm 1$ and generally arranged above
and below the Fermi level $\mu_{\mathrm{F}}$; this latter constrain comes from
the Fermi functions factor $(f_{m}-f_{n})$ of the Kubo formula (54), which
becomes a step function at low temperatures. For more information, see the
Supplemental Material [52] where we illustrate these electron transitions by
arrows in the energy spectrum in an animated gif. The value of $\hbar\Omega$
where a peak of the conductivity occurs coincides with the energy difference
$(E_{n}-E_{m})$ of the LL transition $n\to m$. This is clear by looking at the
denominator of the Kubo formula. For example, the two main peaks of
$\mathrm{Re}(\sigma_{xx})$ at low frequencies $\hbar\Omega\in[2,6]$ meV in
Figure 4 correspond to the transitions $0\to 1$ for spin and valley $s=\xi=1$
and $s=\xi=-1$ (purple and green arrows in the animated gif of [52]). The
other conductivity peaks located at higher frequencies correspond to electron
transitions between higher LLs and different spin/valley combinations
according to (60). When the external electric field $\Delta_{z}$ is such that
the energy differences of the two main peaks are the same, that is, when
$E_{1}^{++}-E_{0}^{++}$ is equal to $E_{1}^{--}-E_{0}^{--}$, both peaks merge
into a bigger one. Using the silicene spectrum energy equation (7), we find
that this condition is fulfilled at the critical point
$\Delta_{z}=\Delta_{\text{so}}$ for any value of the magnetic field $B$. This
result implies that we can extract information of the TPT occurring at
$\Delta_{z}^{(0)}=\Delta_{\text{so}}$ by looking at the conductivity
$\mathrm{Re}(\sigma_{xx})$ plot for different values of $\Delta_{z}$.

To be more specific, in Figure 5 we represent the behavior of the two
observables given in (59), that is, the Faraday angle $\Theta_{\mathrm{F}}$
and the transmittance $\mathcal{T}$, as a function of the polarized light
frequency $\Omega$ around the critical point
$\Delta_{z}^{(0)}=\Delta_{\mathrm{so}}=4.2$ meV. We focus on the frequency
interval $\hbar\Omega\in[2,6]$ meV where the main peaks (transition $0\to 1$)
in Figure 4 are located. We find an absolute minimum of the transmittance
$\mathcal{T}_{0}=0.704$ at the critical point
$\Delta_{z}^{(0)}=\Delta_{\text{so}}$ and $\hbar\Omega=4.06$ meV. This
“minimal” behavior does not depend on the particular values of magnetic field,
chemical potential and temperature, which only change the actual value of
$\mathcal{T}_{0}$ and $\hbar\Omega$ of the peak. Actually, the minimum peaks
in the transmittance plot are related to the maximum peaks of the absortance
$\mathrm{Re}(\sigma_{xx})$, according to equation (59). The Faraday angle at
the critical point (black curve in Figure 5) changes sign at the minimum
transmittance point $\hbar\Omega=4.06$ meV, a behavior that can also be
extrapolated to other 2D materials (se later for HgTe QWs and phosphorene). In
fact, each peak of the transmittance in Figure 5 coincides in frequency with
an inflection point of the Faraday angle, where it attains a value of 0
degrees.

Changing the chemical potential $\mu_{\mathrm{F}}$ locks/unlocks other
electronic transitions, so we would see different peaks in the conductivity
and transmittance plots (see e.g., [10]). Increasing the scattering rate
$\eta$ smoothes the peaks in the transmittance, so it would be more difficult
to distinguish when they overlap. We have choosen values of $\eta$
approximately an order of magnitude below the frequency of the conductivity
peaks, for which the resolution is fine.

Figure 5: Transmittance $\mathcal{T}$ and Faraday angle $\Theta_{\mathrm{F}}$
(in degrees) in a silicene monolayer as a function of the incident polarized
light frequency $\Omega$, and for different electric fields below and above
the critical (black line) electric field
$\Delta_{z}^{(0)}=\Delta_{\text{so}}=4.2\,$meV. $\mathcal{T}$ and
$\Theta_{\mathrm{F}}$ are symmetric about $\Delta_{z}^{(0)}$. We set the
conductivity parameters as $\mu_{\mathrm{F}}=2.1$ meV, $B=0.05$ T, $T=1$ K and
$\eta=0.1$ meV.

For completeness, in the Supplemental Material [52] we show several contour
plots of the Faraday angle using different cross sections in the
$\\{\hbar\Omega,\Delta_{z},B,T,\mu_{\mathrm{F}}\\}$ parameter space.

###  III.2 Magneto-optical properties of zincblende heterostructures

From the Hamiltonian (17), the current operator (55) for zincblende
heterostructures is

$\displaystyle j_{x}^{s}$ $\displaystyle=$
$\displaystyle\frac{e}{\hbar}\left(s\alpha\tau_{x}-2k_{x}(\beta\tau_{z}+\delta\tau_{0})\right),$
$\displaystyle j_{y}^{s}$ $\displaystyle=$
$\displaystyle\frac{e}{\hbar}\left(\alpha\tau_{y}-2k_{y}(\beta\tau_{z}+\delta\tau_{0})\right),$
(62)

which, after minimal coupling according to the general prescription (3),
results in

$\displaystyle j_{x}^{s}$ $\displaystyle=$
$\displaystyle\frac{e}{\hbar}\left(s\alpha\tau_{x}-\sqrt{2}\frac{a^{\dagger}+a}{\ell_{B}}(\beta\tau_{z}+\delta\tau_{0})\right),$
$\displaystyle j_{y}^{s}$ $\displaystyle=$
$\displaystyle\frac{e}{\hbar}\left(\alpha\tau_{y}+\mathrm{i}\sqrt{2}\frac{a^{\dagger}-a}{\ell_{B}}(\beta\tau_{z}+\delta\tau_{0})\right).$
(63)

Note that, in fact, $j_{y}^{s}$ does not depend on $s$. The current matrix
elements for this case are

$\displaystyle\langle\bm{m}|j_{x}^{s}|\bm{n}\rangle_{s}=$
$\displaystyle\,\frac{es\alpha}{\hbar}\Xi_{m,n}^{s,+}-\frac{\sqrt{2}e}{\hbar\ell_{B}}\Phi_{m,n}^{s,+}\,,$
$\displaystyle\langle\bm{m}|j_{y}^{s}|\bm{n}\rangle_{s}=$
$\displaystyle\,-\mathrm{i}\frac{e\alpha}{\hbar}\Xi_{m,n}^{s,-}+\mathrm{i}\frac{\sqrt{2}e}{\hbar\ell_{B}}\Phi_{m,n}^{s,-}\,,$
(64)

where

$\displaystyle\Xi_{m,n}^{s,\pm}=$
$\displaystyle\,(A_{m}^{s}B_{n}^{s}\delta_{|m|-s,|n|}\pm
A_{n}^{s}B_{m}^{s}\delta_{|m|+s,|n|})\,,$ (65)
$\displaystyle\Phi_{m,n}^{s,\pm}=$
$\displaystyle\,((\delta+\beta)A_{m}^{s}A_{n}^{s}+(\delta-\beta)B_{m}^{s}B_{n}^{s})$
$\displaystyle{\scriptstyle\times\left(\sqrt{|n|+1+\tfrac{s-1}{2}}\delta_{|m|-1,|n|}\right.\left.\pm\sqrt{|n|-\tfrac{s+1}{2}}\delta_{|m|+1,|n|}\right)}\,.$

Despite the more involved structure of the current than for silicene, the
corresponding matrix elements maintain the same familiar selection rules
$|n|=|m|\pm 1$ for LL transitions.

Inserting the matrix elements (60) into the general expression (56) we obtain
the magneto-optical conductivity for general zincblende heterostructures. In
Figure 6 we plot the real and imaginary parts of the conductivity tensor
components $\sigma_{ij}$ (in $\sigma_{0}=e^{2}/h$ units) of a HgTe QW as a
function of the polarized light frequency $\Omega$ at three different HgTe
layer thicknesses $\lambda=5.50\,\text{nm}<\lambda_{c}$,
$\lambda=6.17\,\text{nm}=\lambda_{c}$, and
$\lambda=7.00\,\text{nm}>\lambda_{c}$, a magnetic field $B=0.5$ T and some
representative values of the chemical potential $\mu_{\mathrm{F}}=12.5$ meV,
temperature $T=1$ K and scattering rate $\eta=0.5$ meV. For
$\hbar\Omega\in[0,60]$ meV, we achieve convergence with 100 LLs, that is,
restricting the sum in (56) as
$\sum_{n=-\infty}^{\infty}\to\sum_{n=-100}^{100}$. More explicitly, for the
parameters mentioned above,

$\left|\sum_{n=-100}^{n=100}\sigma_{ij}-\sum_{n=-99}^{n=99}\sigma_{ij}\right|/\sigma_{0}\leq\begin{cases}10^{-5}\quad\text{if}\>\sigma_{ij}=\mathrm{Re}(\sigma_{xx})\,,\\\
10^{-4}\quad\text{if}\>\sigma_{ij}=\mathrm{Re}(\sigma_{xy})\,,\\\
10^{-3}\quad\text{if}\>\sigma_{ij}=\mathrm{Im}(\sigma_{xx})\,,\\\
10^{-7}\quad\text{if}\>\sigma_{ij}=\mathrm{Im}(\sigma_{xy})\,.\\\ \end{cases}$
(66)

Figure 6: Real and imaginary parts of the longitudinal $\sigma_{xx}$ and
transverse Hall $\sigma_{xy}$ magneto-optical conductivities in a bulk HgTe QW
of thickness $\lambda=5.50,6.17,7.00$ nm, as a function of the polarized light
frequency $\Omega$ and in $\sigma_{0}=e^{2}/h$ units. We set the conductivity
parameters as $\mu_{\mathrm{F}}=12.5$ meV, $B=0.5$ T, $T=1$ K and $\eta=0.5$
meV.

Similar to silicene, we can see in Figure 6 that there are multiple peaks in
the absorptive components $\mathrm{Re}(\sigma_{xx})$ and
$\mathrm{Im}(\sigma_{xy})$, corresponding to transitions between occupied and
unoccupied LLs obeying the selection rules $|n|=|m|\pm 1$. At lower
frequencies $\hbar\Omega\in[0,30]$ meV, inside each curve of Figure 6, we find
the main peaks corresponding to the transitions $0\to 1$ for spin $s=1$ and
$s=-1$. Both peaks merge approximately at $\lambda\simeq\lambda_{c}=6.17$ nm.
This is because the energy differences $E_{1}^{+}-E_{0}^{+}$ and
$E_{1}^{-}-E_{0}^{-}$ are similar when $\lambda\simeq\lambda_{c}$ for low
magnetic fields $B\ll 1$ T, according to equations (29,30). In order to extend
this result to higher values of the magnetic field, we insert the parameter
fits (42) into the equation $E_{1}^{+}-E_{0}^{+}=E_{1}^{-}-E_{0}^{-}$, and
solve it numerically for $\lambda^{*}=\lambda^{*}(B)$, obtaining the values
represented by blue dots in Figure 7. These values fit the equation

$\lambda_{\text{fit}}^{*}(B)=\frac{218.4-17.3B}{35.4-2.8B}\,,$ (67)

which is represented as an orange curve in Figure 7. Consequently, only for
small magnetic fields, we can infer the critical thickness $\lambda_{c}$ where
the TPT in HgTe QW occurs from the conductivity $\mathrm{Re}({\sigma_{xx}})$
plot, that is, $\lambda^{*}\simeq\lambda_{c}=6.17$ nm for $B\ll 1$ T.

Figure 7: Numerical solutions $\lambda^{*}$ (in nm, blue dots) of the equation
$E_{1}^{+}-E_{0}^{+}=E_{1}^{-}-E_{0}^{-}$ (energies (29,30) of HgTe QW) for 50
different values of the external magnetic field $B$. In orange, non-linear fit
(67) of the numerical values.

The behavior of the Faraday angle and the transmittance as a function of the
polarized light frequency $\Omega$ around the critical HgTe layer thickness
$\lambda_{c}=6.17$ nm (at which the material parameter $\mu$ changes
sign/Chern number) is shown in Figure 8. As for silicene, we focus on the
lower frequencies $\hbar\Omega\in[0,30]$ meV where the main peaks are located,
and find again a minimum of the transmittance, this time
$\mathcal{T}_{0}=0.78$, at the critical point $\lambda_{c}$ and
$\hbar\Omega=15.0$ meV. For this material, the “minimal” behavior does depend
on the particular values of magnetic field, as we saw in equation (67).
However, for small magnetic fields like $B=0.5$ T in Figure 8, the minimum of
the transmittance still takes place at $\lambda^{*}\simeq\lambda_{c}=6.17$ nm.
The Faraday angle at the critical point (black curve in Figure 8) changes sign
at the minimum transmittance frequency $\hbar\Omega=15.0$ meV, a behavior
shared with silicene.

Figure 8: Transmittance $\mathcal{T}$ and Faraday angle $\Theta_{\mathrm{F}}$
(in degrees) in a bulk HgTe QW as a function of the polarized light frequency
$\Omega$, and for thickness $\lambda<\lambda_{c},\lambda=\lambda_{c}$ and
$\lambda>\lambda_{c}$, with $\lambda_{c}=6.17$ nm (black line). We set the
conductivity parameters as $\mu_{\mathrm{F}}=12.5$ meV, $B=0.5$ T, $T=1$ K and
$\eta=0.5$ meV.

For completeness, in the Supplemental Material [52] we show several contour
plots of the Faraday angle using different cross sections in the
$\\{\hbar\Omega,\lambda,B,T,\mu_{\mathrm{F}}\\}$ parameter space.

###  III.3 Magneto-optical properties of phosphorene and effect of
anisotropies

From the phosphorene Hamiltonian (45), the current operator (55) is

$\displaystyle j_{x}^{s}$ $\displaystyle=$
$\displaystyle\frac{e}{\hbar}\left(\gamma\tau_{x}+k_{x}(\tau_{0}(\alpha_{x}-\beta_{x})+\tau_{z}(\alpha_{x}+\beta_{x}))\right),$
$\displaystyle j_{y}^{s}$ $\displaystyle=$
$\displaystyle\frac{e}{\hbar}k_{y}\left(\tau_{0}(\alpha_{y}-\beta_{y})+\tau_{z}(\alpha_{y}+\beta_{y})\right),$
(68)

which, after minimal coupling, according to prescription (49), results in

$\displaystyle j_{x}^{s}$ $\displaystyle=$
$\displaystyle\frac{e}{\hbar}\left(\gamma\tau_{x}+\frac{a^{\dagger}+a}{\sqrt{2}\alpha_{yx}\ell_{B}}(\tau_{0}(\alpha_{x}-\beta_{x})+\tau_{z}(\alpha_{x}+\beta_{x}))\right),$
$\displaystyle j_{y}^{s}$ $\displaystyle=$
$\displaystyle\frac{e}{\hbar}\frac{\alpha_{yx}(a^{\dagger}-a)}{\mathrm{i}\sqrt{2}\ell_{B}}\left(\tau_{0}(\alpha_{y}-\beta_{y})+\tau_{z}(\alpha_{y}+\beta_{y})\right).$
(69)

Plugging these matrix elements into the general expression (56) we obtain the
magneto-optical conductivity for phosphorene. Note that, unlike silicene and
HgTe QW, there is now a large asymmetry between $\sigma_{xx}$ and
$\sigma_{yy}$ (about one order of magnitude difference), as evidenced by
Figure 9. This asymmetry was already highlighted by [71], where tunable
optical properties of multilayer black phosphorus thin films were studied for
$B=0$. In Figure 9 we plot the real and imaginary parts of the conductivity
tensor components $\sigma_{ij}$ (in $\sigma_{0}=e^{2}/h$ units) of phosphorene
as a function of the polarized light frequency $\Omega$, for some values of
the electric potential around $\Delta_{z}^{(0)}=-E_{\textrm{g}}=-1.52$ eV
(closing the energy gap), a magnetic field of $B=0.5$ T, like in Figure 3, and
some representative values of the chemical potential $\mu_{\mathrm{F}}=-0.417$
eV, temperature $T=1$ K and scattering rate $\eta=0.2$ meV. We are using the
same threshold of $N=1000$ Fock states that we used to find convergence in the
first 6 Hamiltonian eigenstates of the numerical diagonalization in Figure 3.
This convergence is ensured for $\hbar\Omega\in[0,20]$ meV. The anisotropic
character of phosphorene also implies that the current $j_{y}^{s}$ is
significantly lower than $j_{x}^{s}$ [the Hamiltonian (45) is of second order
in $k_{y}$]. This makes transversal components of the conductivity
significantly lower than longitudinal components. This is why we have disposed
Figure 9 in a slightly different manner from Figures 4 for silicene and 6 for
HgTe QW, which display a more isotropic structure.

Due to the parity symmetry of the Hamiltonian (50), only the electronic
transitions between LLs of different parities are allowed [30]. The main peak
(smaller frequency) of the conductivity $\mathrm{Re}(\sigma_{xx})$ in Figure 9
corresponds to the electronic transitions $E_{0}^{\text{even}}\to
E_{3}^{\text{odd}}$ and $E_{1}^{\text{odd}}\to E_{2}^{\text{even}}$, which
have approximately the same energy difference for all $\Delta_{z}<-1.53$ eV
with a tolerance $\leq 10^{-14}$ eV. That is, $E_{0}^{\text{even}}$ and
$E_{1}^{\text{odd}}$, and $E_{2}^{\text{even}}$ and $E_{3}^{\text{odd}}$, are
degenerate for all $\Delta_{z}<-1.53$ eV as the spectrum in Figure 3 shows.
When the degeneration is broken around the electric potential
$\Delta_{z}\simeq-1.53$ eV, the main conductivity $\mathrm{Re}(\sigma_{xx})$
peak splits into two as we can see in Figure 9.

Figure 9: Real and imaginary parts of the longitudinal
$\sigma_{xx},\sigma_{yy}$ and transverse Hall $\sigma_{xy}$ magneto-optical
conductivities in a phosphorene monolayer, as a function of the polarized
light frequency $\Omega$ and in $\sigma_{0}=e^{2}/h$ units. Phosphorene is
under a perpendicular electric field potential
$\Delta_{z}^{(0)}=-E_{\textrm{g}}=-1.52$ eV closing the energy gap in Figure
3. The $y$-axis ticks have different values in each subplot as the
conductivities $\sigma_{xy}$ and $\sigma_{yy}$ attain smaller values than
$\sigma_{xx}$ (phosphorene anisotropy). We set the conductivity parameters as
$\mu_{\mathrm{F}}=-0.417$ meV, $B=0.5$ T, $T=1$ K and $\eta=0.2$ meV.

The anisotropic character of phosphorene also affects the Faraday angle, which
attains much lower values (in absolute value) than for silicene or HgTe QWs.
Indeed, in Figure 10 we plot Faraday angle and transmittance as a function of
the polarized light frequency $\Omega$ for different electric field potentials
$-1.535\leq\Delta_{z}\leq-1.519$ eV. Like for silicene and HgTe QWs, we find a
minimal behavior in the transmittance of phosphorene $\mathcal{T}_{0}=0.50$
for a polarized light frequency $\hbar\Omega=2.6$ meV at electric field
potential $\Delta_{z}^{(0)}=-1.523$ eV, which is close to minus the energy gap
$-E_{\textrm{g}}=-1.52$ eV. Note that this value of the minimal transmittance
of phosphorene is much smaller than for silicene and HgTe QWs; actually, the
assumption of low absortance in formula (59) is no longer valid here and we
have used the exact expressions for $\mathcal{T}$ and $\Theta_{\mathrm{F}}$ in
(59). Moreover, unlike for graphene analogues and HgTe QWs, this minimum of
the transmittance does not seem to be related to the union of two conductivity
peaks into a bigger one; rather, it is simply related to the energy gap
closure. Actually, the critical electric potential $\Delta_{z}^{(0)}$ where
the transmittance of phosphorene reaches a minimum depends on the magnetic
field $B$ chosen, as Figure 11 shows. We perform a non-linear fit of the
numerical values of $\Delta_{z}^{(0)}(B)$ and obtain the equation ($B$ in
dimensionless units)

$\left(\Delta_{z}^{(0)}\right)_{\text{fit}}(B)=\frac{-77.4-3.5B}{50.9+2.2B}\,\textrm{eV}\,,$
(70)

which is represented as a orange curve in Figure 11. For small magnetic
fields, we can deduce that the critical electric field potential is similar to
minus the energy gap $-E_{\textrm{g}}$ of phosphorene, that is
$\Delta_{z}^{(0)}(B)\simeq-E_{\textrm{g}}=-1.52$ eV for $B\ll 1$ T. We have
also checked numerically that the critical electric potentials
$\Delta_{z}^{(0)}(B)$ are independent of the parameters $\mu_{\mathrm{F}}$ and
$\eta$ for a fixed magnetic field $B$. However, we set different values of
$\mu_{\mathrm{F}}$ for small fields $B\leq 2$ T (see caption of Figure 11), in
order to avoid blocking the electric transition $E_{1}^{\text{odd}}\to
E_{2}^{\text{even}}$ of the main peak of the transmittance. We also increment
$N$ as $B$ decreases in order to achieve convergence in the diagonalization.

Additionally, Figure 10 shows how one peak of the transmittance splits into
two around $\Delta_{z}\simeq-1.53$ eV (blue lines), since the LL
$E_{0}^{\text{even}}$ breaks its degeneration approximately for
$\Delta_{z}>-1.53$ eV (see Figure 3). For $\Delta_{z}=\Delta_{z}^{(0)}=-1.523$
eV (thick black line), the big peak on the left in Figure 10 corresponds to
the electronic transition $E_{1}^{\text{odd}}\to E_{2}^{\text{even}}$, and
moves toward smaller values of $\hbar\Omega$ when increasing $\Delta_{z}$. The
other small peak in the black line corresponds to the electronic transition
$E_{0}^{\text{even}}\to E_{3}^{\text{odd}}$, which moves toward bigger values
of $\hbar\Omega$ when increasing $\Delta_{z}$. The Faraday angle also presents
inflection points at the frequencies where the peaks of the transmittance are
located.

Figure 10: Transmittance $\mathcal{T}$ and Faraday angle $\Theta_{\mathrm{F}}$
(in degrees) in a phosphorene monolayer as a function of the polarized light
frequency $\Omega$, and for electric fields $-1.535<\Delta_{z}<-1.519$ eV
around the minus energy gap $-E_{\textrm{g}}=-1.52$ eV. The black line
corresponds to the electric potential $\Delta_{z}^{(0)}=-1.523$ eV$\,\simeq-
E_{\textrm{g}}$ where the transmittance attains a minimum of
$\mathcal{T}_{0}=0.5$ at $\hbar\Omega=2.5$ meV. We set the conductivity
parameters as $\mu_{\mathrm{F}}=-0.417$ meV, $B=0.5$ T, $T=1$ K and $\eta=0.2$
meV. Figure 11: Electric field potential at which phosphorene transmittance
reaches a minimum, as a function of different magnetic fields. In orange, non-
linear fit (70) of the numerical values. In general, we set the conductivity
parameters $\mu_{\mathrm{F}}=-0.41$ eV, $T=1$ K, $\eta=1$ meV, and use $N=300$
Fock state in the numerical diagonalization, for all $B\geq 3$ T. For smaller
magnetic fields $B=0.1,0.5,1,2$ T, we set
$\mu_{\mathrm{F}}=-0.419,-0.418,-0.416,-0.416$ eV respectively. In the case of
$B=0.1$ T we also set $\eta=1$ meV and $N=500$ Fock states to achieve energy
diagonalization convergence.

Therefore, we see that anisotropies affect the values of the Faraday angle and
transmittance. There are mechanical ways of introducing anisotropies in 2D
materials by subjecting them to strain (like for strained [86] or rippled [87]
graphene). This kind of anisotropies can be treated by replacing the scalar
Fermi velocity $v$ by a $2\times 2$ symmetric tensor $\bm{v}$ (see e.g. [82]).
Namely, for graphene, the Hamiltonian (1) vector $\bm{d}$ components
$d_{j}=\hbar vk_{j}$ are replaced by $d_{j}=\hbar k_{i}v_{ij},i=1,2,d_{3}=0$.
Actually, for uniformly strained graphene with strain tensor
$\bm{\varepsilon}$, the Fermi velocity tensor is (up to first order)
$\bm{v}=v(\tau_{0}-\beta\bm{\varepsilon})$ (see e.g. [88, 82]), where
$\beta\sim 2$. The relation between the isotropic $\bm{\sigma}^{0}$ and the
anisotropic $\bm{\sigma}$ magneto-optical conductivity tensors is simply
$\bm{\sigma}(\Omega,B)=\bm{v}\bm{\sigma}^{0}(\Omega,\mathcal{B})\bm{v}/\det(\bm{v})$,
with $\mathcal{B}=B\det({\bm{v}})/v^{2}$ an effective magnetic field.
Interesting discussions on how measurements of dichroism and transparency for
two different light polarization directions can be used to determine the
magnitude and direction of strain can be found in [81]. Also, photoelastic
effects in graphene [86], strain-modulated anisotropies in silicene [89, 90],
etc. The band gap $E_{\textrm{g}}=E_{\mathrm{v}}-E_{\mathrm{c}}$ of
phosphorene can be furthermore modulated by strain and by the number of layers
in a stack [60, 70].

##  IV Conclusions

We have studied magneto-optical properties of different 2D materials, focusing
on transmittance and Faraday rotation near the critical point of the
topological phase transition for topological insulators like silicene and HgTe
quantum wells. We have seen that, in all topological 2D materials analyzed,
transmittance attains an absolute minimum $\mathcal{T}_{0}$ at the critical
TPT point for a certain value $\Omega_{0}$ of the normal incident polarized
light frequency. This is a universal behavior for graphene analogues, that is,
the minimal behavior of the transmittance does not depend on the chosen values
of magnetic field, chemical potential and temperature, although the location
of $\Omega_{0}$ varies with them. In addition, we have found that each peak of
the transmittance coincides in frequency with an inflection point of the
Faraday angle, for a fixed selection of the electric field, magnetic field,
chemical potential and temperature parameters.

This extremal universal behavior is shared with other topological 2D materials
like HgTe quantum wells as long as the applied magnetic field remains small
enough $B\ll 1$ T. In HgTe quantum wells we have verified that there is a
minimum of the transmittance $\mathcal{T}_{0}$ at the critical HgTe layer
thickness at a given frequency $\Omega^{\prime}_{0}$ (for this material this
minimal behavior depends on the magnetic field) and the Faraday angle at the
critical point changes sign at the minimum transmittance frequency
$\Omega^{\prime}_{0}$.

For other non-topological anisotropic materials like phosphorene, this minimal
behavior of the transmittance still remains when the energy gap is closed, the
Faraday angle being much smaller (in absolute value) than in silicene and HgTe
QWs. In this case the critical electric potential where the transmittance
reaches a minimum depends on the magnetic field.

Therefore, these extremal properties of transmittance/absortance and chirality
change of Faraday angle at the critical point turn out to provide sharp
markers of either the topological phase transition or the energy gap closure.

## Acknowledgments

We thank the support of the Spanish MICINN through the project
PGC2018-097831-B-I00 and Junta de Andalucía through the projects
FEDER/UJA-1381026 and FQM-381. AM thanks the Spanish MIU for the FPU19/06376
predoctoral fellowship. OC is on sabbatical leave at Granada University,
Spain, since the 1st of September 2023. OC thanks support from the program
PASPA from DGAPA-UNAM.

## References

  * Crassee  _et al._ [2011] I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, Giant faraday rotation in single- and multilayer graphene, Nature Physics 7, 48 (2011).
  * Ferreira  _et al._ [2011] A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids, Phys. Rev. B 84, 235410 (2011).
  * Fialkovsky and Vassilevich [2012] I. Fialkovsky and D. V. Vassilevich, Faraday rotation in graphene, The European Physical Journal B 85, 384 (2012).
  * Tymchenko  _et al._ [2013] M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, Faraday rotation due to excitation of magnetoplasmons in graphene microribbons, ACS Nano 7, 9780 (2013).
  * Yang  _et al._ [2014] F. Yang, X. Xu, and R.-B. Liu, Giant faraday rotation induced by the berry phase in bilayer graphene under strong terahertz fields, New Journal of Physics 16, 043014 (2014).
  * Lin  _et al._ [2017] C.-Y. Lin, T.-N. Do, Y.-K. Huang, and M.-F. Lin,  _Optical Properties of Graphene in Magnetic and Electric Fields_, 2053-2563 (IOP Publishing, 2017).
  * Shimano  _et al._ [2013] R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, Quantum faraday and kerr rotations in graphene, Nature Communications 4, 1841 (2013).
  * Spencer and [Eds.] M. J. Spencer and T. M. (Eds.),  _Silicene: Structure, Properties and Applications,_ (Springer Series in Materials Science vol. 235, 2016).
  * Tabert and Nicol [2013a] C. J. Tabert and E. J. Nicol, Magneto-optical conductivity of silicene and other buckled honeycomb lattices, Phys. Rev. B 88, 085434 (2013a).
  * Shah and Anwar [2019] M. Shah and M. S. Anwar, Magneto-optical effects in the landau level manifold of 2d lattices with spin-orbit interaction, Opt. Express 27, 23217 (2019).
  * Nualpijit and Soodchomshom [2022] P. Nualpijit and B. Soodchomshom, Enhanced faraday rotation and control of pure spin-valley optical conductivity by topological phase in silicene, Physica E: Low-dimensional Systems and Nanostructures 137, 115011 (2022).
  * Hien  _et al._ [2020] N. D. Hien, C. V. Nguyen, N. N. Hieu, S. S. Kubakaddi, C. A. Duque, M. E. Mora-Ramos, L. Dinh, T. N. Bich, and H. V. Phuc, Magneto-optical transport properties of monolayer transition metal dichalcogenides, Phys. Rev. B 101, 045424 (2020).
  * Zhou  _et al._ [2015] X. Y. Zhou, R. Zhang, J. P. Sun, Y. L. Zou, D. Zhang, W. K. Lou, F. Cheng, G. H. Zhou, F. Zhai, and K. Chang, Landau levels and magneto-transport property of monolayer phosphorene, Scientific Reports 5, 12295 (2015).
  * Shen [2012] S.-Q. Shen,  _Topological Insulators: Dirac Equation in Condensed Matters_ (Springer-Verlag Berlin Heidelberg, 2012).
  * Bernevig [2013] B. A. Bernevig,  _Topological Insulators and Topological Superconductors_ (Princeton University Press, 2013).
  * J. K. Asboth and A. Palyi [2016] L. O. J. K. Asboth and A. S. A. Palyi,  _A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions_ (Springer International Publishing Switzerland, 2016).
  * Scharf  _et al._ [2015] B. Scharf, A. Matos-Abiague, I. Žutić, and J. Fabian, Probing topological transitions in HgTe/CdTe quantum wells by magneto-optical measurements, Phys. Rev. B 91, 235433 (2015).
  * Shuvaev  _et al._ [2016] A. Shuvaev, V. Dziom, Z. D. Kvon, N. N. Mikhailov, and A. Pimenov, Universal faraday rotation in hgte wells with critical thickness, Phys. Rev. Lett. 117, 117401 (2016).
  * Ardakani and Zare [2021] A. G. Ardakani and Z. Zare, Strong faraday rotation in a topological insulator single layer using dielectric multilayered structures, J. Opt. Soc. Am. B 38, 2562 (2021).
  * Liang  _et al._ [2020] Y. Liang, Y. Xiang, and X. Dai, Enhancement of graphene faraday rotation in the one-dimensional topological photonic crystals, Opt. Express 28, 24560 (2020).
  * Shao  _et al._ [2017] Y. Shao, K. W. Post, J.-S. Wu, S. Dai, A. J. Frenzel, A. R. Richardella, J. S. Lee, N. Samarth, M. M. Fogler, A. V. Balatsky, D. E. Kharzeev, and D. N. Basov, Faraday rotation due to surface states in the topological insulator (bi1–xsbx)2te3, Nano Letters 17, 980 (2017).
  * Shah  _et al._ [2023] M. Shah, M. Q. Mehmood, Y. S. Ang, M. Zubair, and Y. Massoud, Magneto-optical conductivity and giant faraday-kerr rotation in floquet topological insulators, Phys. Rev. B 107, 235115 (2023).
  * Tokman  _et al._ [2020] I. D. Tokman, Q. Chen, I. A. Shereshevsky, V. I. Pozdnyakova, I. Oladyshkin, M. Tokman, and A. Belyanin, Inverse faraday effect in graphene and weyl semimetals, Phys. Rev. B 101, 174429 (2020).
  * Calixto and Romera [2015a] M. Calixto and E. Romera, Identifying topological-band insulator transitions in silicene and other 2d gapped dirac materials by means of rényi-wehrl entropy, EPL (Europhysics Letters) 109, 40003 (2015a).
  * Romera and Calixto [2015a] E. Romera and M. Calixto, Uncertainty relations and topological-band insulator transitions in 2d gapped dirac materials, Journal of Physics: Condensed Matter 27, 175003 (2015a).
  * Calixto and Romera [2015b] M. Calixto and E. Romera, Inverse participation ratio and localization in topological insulator phase transitions, Journal of Statistical Mechanics: Theory and Experiment 2015, P06029 (2015b).
  * Romera and Calixto [2015b] E. Romera and M. Calixto, Band inversion at critical magnetic fields in a silicene quantum dot, EPL (Europhysics Letters) 111, 37006 (2015b).
  * Romera  _et al._ [2018] E. Romera, M. Calixto, and J. Bolívar, Information measures and topological-band insulator transitions in 2d-dirac materials under external circularly polarized lasers, and static electric and magnetic fields, Physica A: Statistical Mechanics and its Applications 511, 174 (2018).
  * Calixto  _et al._ [2022] M. Calixto, N. A. Cordero, E. Romera, and O. Castaños, Signatures of topological phase transitions in higher landau levels of hgte/cdte quantum wells from an information theory perspective, Physica A: Statistical Mechanics and its Applications 605, 128057 (2022).
  * Castaños  _et al._ [2019] O. Castaños, E. Romera, and M. Calixto, Information theoretic analysis of landau levels in monolayer phosphorene under magnetic and electric fields, Materials Research Express 6, 106316 (2019).
  * Calixto  _et al._ [2021] M. Calixto, E. Romera, and O. Castaños, Analogies between the topological insulator phase of 2d dirac materials and the superradiant phase of atom-field systems, International Journal of Quantum Chemistry 121, e26464 (2021), https://onlinelibrary.wiley.com/doi/pdf/10.1002/qua.26464 .
  * Thouless  _et al._ [1982] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982).
  * Xiao  _et al._ [2010] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  * Kane and Mele [2005] C. L. Kane and E. J. Mele, Quantum spin hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005).
  * Drummond  _et al._ [2012] N. D. Drummond, V. Zólyomi, and V. I. Fal’ko, Electrically tunable band gap in silicene, Phys. Rev. B 85, 075423 (2012).
  * Liu  _et al._ [2011a] C.-C. Liu, W. Feng, and Y. Yao, Quantum spin hall effect in silicene and two-dimensional germanium, Phys. Rev. Lett. 107, 076802 (2011a).
  * Liu  _et al._ [2011b] C.-C. Liu, H. Jiang, and Y. Yao, Low-energy effective hamiltonian involving spin-orbit coupling in silicene and two-dimensional germanium and tin, Phys. Rev. B 84, 195430 (2011b).
  * Tsai  _et al._ [2013] W.-F. Tsai, C.-Y. Huang, T.-R. Chang, H. Lin, H.-T. Jeng, and A. Bansil, Gated silicene as a tunable source of nearly 100% spin-polarized electrons, Nature Communications 4, 1500 (2013).
  * Trivedi  _et al._ [2014] S. Trivedi, A. Srivastava, and R. Kurchania, Silicene and Germanene: A First Principle Study of Electronic Structure and Effect of Hydrogenation-Passivation, Journal of Computational and Theoretical Nanoscience 11, 781 (2014).
  * van den Broek  _et al._ [2014] B. van den Broek, M. Houssa, E. Scalise, G. Pourtois, V. V. Afanas‘ev, and A. Stesmans, Two-dimensional hexagonal tin: ab initio geometry, stability, electronic structure and functionalization, 2D Materials 1, 021004 (2014).
  * Stille  _et al._ [2012] L. Stille, C. J. Tabert, and E. J. Nicol, Optical signatures of the tunable band gap and valley-spin coupling in silicene, Phys. Rev. B 86, 195405 (2012).
  * Tabert and Nicol [2013b] C. J. Tabert and E. J. Nicol, Valley-spin polarization in the magneto-optical response of silicene and other similar 2d crystals, Phys. Rev. Lett. 110, 197402 (2013b).
  * Tahir and Schwingenschlögl [2013] M. Tahir and U. Schwingenschlögl, Valley polarized quantum hall effect and topological insulator phase transitions in silicene, Scientific Reports 3, 1075 (2013).
  * Novik  _et al._ [2005] E. G. Novik, A. Pfeuffer-Jeschke, T. Jungwirth, V. Latussek, C. R. Becker, G. Landwehr, H. Buhmann, and L. W. Molenkamp, Band structure of semimagnetic ${\mathrm{hg}}_{1-y}{\mathrm{mn}}_{y}\mathrm{Te}$ quantum wells, Phys. Rev. B 72, 035321 (2005).
  * Bernevig  _et al._ [2006] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin hall effect and topological phase transition in HgTe quantum wells, Science 314, 1757 (2006), https://www.science.org/doi/pdf/10.1126/science.1133734 .
  * König  _et al._ [2007] M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Quantum spin hall insulator state in HgTe quantum wells, Science 318, 766 (2007), https://www.science.org/doi/pdf/10.1126/science.1148047 .
  * König  _et al._ [2008] M. König, H. Buhmann, L. W. Molenkamp, T. Hughes, C.-X. Liu, X.-L. Qi, and S.-C. Zhang, The quantum spin hall effect: Theory and experiment, Journal of the Physical Society of Japan 77, 031007 (2008), https://doi.org/10.1143/JPSJ.77.031007 .
  * Liu  _et al._ [2008] C. Liu, T. L. Hughes, X.-L. Qi, K. Wang, and S.-C. Zhang, Quantum spin hall effect in inverted type-ii semiconductors, Phys. Rev. Lett. 100, 236601 (2008).
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  * Büttner  _et al._ [2011] B. Büttner, C. X. Liu, G. Tkachov, E. G. Novik, C. Brüne, H. Buhmann, E. M. Hankiewicz, P. Recher, B. Trauzettel, S. C. Zhang, and L. W. Molenkamp, Single valley dirac fermions in zero-gap HgTe quantum wells, Nature Physics 7, 418 (2011).
  * Scharf  _et al._ [2012] B. Scharf, A. Matos-Abiague, and J. Fabian, Magnetic properties of HgTe quantum wells, Phys. Rev. B 86, 075418 (2012).
  * [52] For details, see the Supplemental Material at, URL_will_be_inserted_by_publisher.
  * Corbridge [2013] D. Corbridge,  _Phosphorus: Chemistry, Biochemistry and Technology 6th edn_ (CRC Press, 2013).
  * Bridgman [1914] P. W. Bridgman, Two new modifications of phosphorus.,  _Journal of the American Chemical Society_ , Journal of the American Chemical Society 36, 1344 (1914).
  * Bridgman [1916] P. W. Bridgman, Further note on black phosphorus.,  _Journal of the American Chemical Society_ , Journal of the American Chemical Society 38, 609 (1916).
  * Zhu and Tománek [2014] Z. Zhu and D. Tománek, Semiconducting layered blue phosphorus: A computational study, Phys. Rev. Lett. 112, 176802 (2014).
  * Guo  _et al._ [2014] H. Guo, N. Lu, J. Dai, X. Wu, and X. C. Zeng, Phosphorene nanoribbons, phosphorus nanotubes, and van der waals multilayers,  _The Journal of Physical Chemistry C_ , The Journal of Physical Chemistry C 118, 14051 (2014).
  * Guan  _et al._ [2014] J. Guan, Z. Zhu, and D. Tománek, Phase coexistence and metal-insulator transition in few-layer phosphorene: A computational study, Phys. Rev. Lett. 113, 046804 (2014).
  * Liu  _et al._ [2014] H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, Phosphorene: An unexplored 2d semiconductor with a high hole mobility,  _ACS Nano_ , ACS Nano 8, 4033 (2014).
  * Carvalho  _et al._ [2016] A. Carvalho, M. Wang, X. Zhu, A. S. Rodin, H. Su, and A. H. Castro Neto, Phosphorene: from theory to applications, Nature Reviews Materials 1, 16061 (2016).
  * Wan  _et al._ [2014] R. Wan, X. Cao, and J. Guo, Simulation of phosphorene schottky-barrier transistors, Applied Physics Letters 105, 163511 (2014), https://doi.org/10.1063/1.4900410 .
  * Liu  _et al._ [2015] H. Liu, Y. Du, Y. Deng, and P. D. Ye, Semiconducting black phosphorus: synthesis, transport properties and electronic applications, Chem. Soc. Rev. 44, 2732 (2015).
  * Akhtar  _et al._ [2017] M. Akhtar, G. Anderson, R. Zhao, A. Alruqi, J. E. Mroczkowska, G. Sumanasekera, and J. B. Jasinski, Recent advances in synthesis, properties, and applications of phosphorene, npj 2D Materials and Applications 1, 5 (2017).
  * Li  _et al._ [2014] L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, Black phosphorus field-effect transistors, Nature Nanotechnology 9, 372 (2014).
  * Chen  _et al._ [2017] P. Chen, N. Li, X. Chen, W.-J. Ong, and X. Zhao, The rising star of 2d black phosphorus beyond graphene: synthesis, properties and electronic applications, 2D Materials 5, 014002 (2017).
  * Ling  _et al._ [2015] X. Ling, H. Wang, S. Huang, F. Xia, and M. S. Dresselhaus, The renaissance of black phosphorus, Proceedings of the National Academy of Sciences 112, 4523 (2015), https://www.pnas.org/content/112/15/4523.full.pdf .
  * Xu  _et al._ [2016] R. Xu, S. Zhang, F. Wang, J. Yang, Z. Wang, J. Pei, Y. W. Myint, B. Xing, Z. Yu, L. Fu, Q. Qin, and Y. Lu, Extraordinarily bound quasi-one-dimensional trions in two-dimensional phosphorene atomic semiconductors,  _ACS Nano_ , ACS Nano 10, 2046 (2016).
  * Rudenko and Katsnelson [2014] A. N. Rudenko and M. I. Katsnelson, Quasiparticle band structure and tight-binding model for single- and bilayer black phosphorus, Phys. Rev. B 89, 201408 (2014).
  * Ezawa [2014] M. Ezawa, Topological origin of quasi-flat edge band in phosphorene, New Journal of Physics 16, 115004 (2014).
  * Rodin  _et al._ [2014] A. S. Rodin, A. Carvalho, and A. H. Castro Neto, Strain-induced gap modification in black phosphorus, Phys. Rev. Lett. 112, 176801 (2014).
  * Low  _et al._ [2014] T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, Tunable optical properties of multilayer black phosphorus thin films, Phys. Rev. B 90, 075434 (2014).
  * Ezawa [2015] M. Ezawa, Highly anisotropic physics in phosphorene, Journal of Physics: Conference Series 603, 012006 (2015).
  * Dutreix  _et al._ [2016] C. Dutreix, E. A. Stepanov, and M. I. Katsnelson, Laser-induced topological transitions in phosphorene with inversion symmetry, Phys. Rev. B 93, 241404 (2016).
  * Lindner  _et al._ [2011] N. H. Lindner, G. Refael, and V. Galitski, Floquet topological insulator in semiconductor quantum wells, Nature Physics 7, 490 (2011).
  * Kitagawa  _et al._ [2011] T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, Transport properties of nonequilibrium systems under the application of light: Photoinduced quantum hall insulators without landau levels, Phys. Rev. B 84, 235108 (2011).
  * Cayssol  _et al._ [2013] J. Cayssol, B. Dóra, F. Simon, and R. Moessner, Floquet topological insulators, Physica Status Solidi (RRL) – Rapid Research Letters 7, 101 (2013).
  * Chen  _et al._ [2020] G.-H. Chen, Y.-N. Chen, Y.-W. Zhou, Y.-L. Sun, and E.-J. Ye, Strain and electric field tunable electronic transport in armchair phosphorene nanodevice with normal-metal electrodes, AIP Advances 10, 105012 (2020), https://doi.org/10.1063/5.0021775 .
  * Mahan [2000] G. D. Mahan,  _Many-Particle Physics, 3rd edition_ (Kluwer Academic/Plenum Publishers, New York, 2000).
  * Allen [2006] P. Allen, Chapter 6 electron transport, in  _Conceptual Foundations of Materials_, Contemporary Concepts of Condensed Matter Science, Vol. 2, edited by S. G. Louie and M. L. Cohen (Elsevier, 2006) pp. 165–218.
  * Stauber  _et al._ [2008] T. Stauber, N. M. R. Peres, and A. K. Geim, Optical conductivity of graphene in the visible region of the spectrum, Phys. Rev. B 78, 085432 (2008).
  * Oliva-Leyva and Naumis [2015] M. Oliva-Leyva and G. G. Naumis, Tunable dichroism and optical absorption of graphene by strain engineering, 2D Materials 2, 025001 (2015).
  * Oliva-Leyva and Wang [2017] M. Oliva-Leyva and C. Wang, Magneto-optical conductivity of anisotropic two-dimensional Dirac–Weyl materials, Annals of Physics 384, 61 (2017).
  * Chiu  _et al._ [1976] K. Chiu, T. Lee, and J. Quinn, Infrared magneto-transmittance of a two-dimensional electron gas, Surface Science 58, 182 (1976).
  * O’Connell and Wallace [1982] R. F. O’Connell and G. Wallace, Ellipticity and faraday rotation due to a two-dimensional electron gas in a metal-oxide-semiconductor system, Phys. Rev. B 26, 2231 (1982).
  * Chakraborty  _et al._ [2023] A. Chakraborty, G. Bian, and G. Vignale, Frequency-dependent faraday and kerr rotation in anisotropic nonsymmorphic dirac semimetals in a magnetic field (2023), arXiv:2302.05385 [cond-mat.mes-hall] .
  * Pereira  _et al._ [2011] V. M. Pereira, R. M. Ribeiro, N. M. R. Peres, and A. H. C. Neto, Optical properties of strained graphene, Europhysics Letters 92, 67001 (2011).
  * Schiefele  _et al._ [2016] J. Schiefele, L. Martin-Moreno, and F. Guinea, Faraday effect in rippled graphene: Magneto-optics and random gauge fields, Phys. Rev. B 94, 035401 (2016).
  * Pellegrino  _et al._ [2011] F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, Linear response correlation functions in strained graphene, Phys. Rev. B 84, 195407 (2011).
  * Farokhnezhad  _et al._ [2017] M. Farokhnezhad, M. Esmaeilzadeh, and K. Shakouri, Strain-modulated anisotropy of quantum transport properties in single-layer silicene: Spin and valley filtering, Phys. Rev. B 96, 205416 (2017).
  * Siu and Jalil [2021] Z. B. Siu and M. B. A. Jalil, Effective hamiltonian for silicene under arbitrary strain from multi-orbital basis, Scientific Reports 11, 7575 (2021).