# A Simple Factor in
Canonical Quantization yields
Affine Quantization
Even for Quantum Gravity

John R<EMAIL_ADDRESS>
Department of Physics and Department of Mathematics
University of Florida, Gainesville, FL 32611-8440

###### Abstract

Canonical quantization (CQ) is built around $[Q,P]=i\hbar 1\\!\\!1$, while
affine quantization (AQ) is built around $[Q,D]=i\hbar\,Q$, where
$D\equiv(PQ+QP)/2$. The basic CQ operators must fit $-\infty<P,Q<\infty$,
while the basic AQ operators can fit $-\infty<P<\infty$ and $0<Q<\infty$,
$-\infty<Q<0$, or even $-\infty<Q\neq 0<\infty$. AQ can also be the key to
quantum gravity, as our simple outline demonstrates.

##  1 Canonical Quantization to
Affine Quantization: CQ to AQ

The CQ equation $[Q,P]=QP-PQ=i\hbar 1\\!\\!1$ is multiplied by $Q$ to give
$\\{Q(QP-PQ)+(QP-PQ)Q\\}/2=\\{Q^{2}P+QPQ-QPQ-
PQ^{2}\\}/2=[Q,(QP+PQ)/2]\equiv[Q,D]=i\hbar\,Q$. This principal equation
requires that $Q\neq 0$, leading to the last expression which applies for
$Q>0$, $Q<0$, or both. From a classical promotion viewpoint for CQ it is
$-\infty<p\rightarrow P,q\rightarrow Q<\infty$, while for AQ it is
$-\infty<d\equiv pq\rightarrow D\equiv(PQ+QP)/2<\infty$, while $-\infty<q\neq
0\rightarrow Q\neq 0<\infty$. Note: $P^{\dagger}\neq P$, but
$P^{\dagger}Q=PQ$.

Not every set of classical variables lead to valid quantum operators, neither
for CQ or for AQ.The correct choice of classical variables to promote to
quantum operators for CQ are Cartesian coordinates, which creat a constant
vanishing curvature. The classical coordinates that creat a constant negative
curvature222A constant negative curvature is discussed in [1]. are the correct
coordinates to promote to quantum operators for AQ.333 Classical variables for
spin variables that form a constant positive curvature are the final set of
correct classical variables that promote to valid quantum operators. The
constant (+, 0, -1) curvatures are all addressed in [2], Sec. 1. Regarding CQ
and AQ operators, the principal difference is that CQ operators must span the
whole real line, while the AQ operator $Q$ strictly spans all positive
coordinates, all negative coordinates, or both, provided $Q\neq 0$.444Instead
of $-\infty<Q\neq 0<\infty$, we can also have $-\infty<Q\neq b<\infty$,
provided $b$ is finite.

Different sets of classical problems require valid CQ or AQ quantizations. For
example, a traditional harmonic oscillator, with $-\infty<p,q<\infty$, employs
CQ, while a half-harmonic oscillator, with $-\infty<p<\infty$, but
$0<q<\infty$, employs AQ [3]. Partial-harmonic oscillators, with
$-\infty<p<\infty$, while $-b<q<\infty$ and $0<b<\infty$, also employ AQ [4].

Examples from scalar field theories show that $\varphi^{12}_{3}$ leads to
“free” results for CQ while “non-free” results for AQ [5], and also “free”
results for $\varphi^{4}_{4}$ [6, 7, 8] while AQ leads to a “non-free” result
[9]. It becomes clear that CQ is not the only quantization procedure for
various classical systems! The author has even suggested that quantum field
theory might benefit by adding AQ to the usual procedures [10].

Regarding gravity, an affine result can offer a strictly positive metric,
i.e., $ds(x)^{2}=g_{ab}(x)\,dx^{a}\,dx^{b}>0$, provided
$dx^{c}\equiv\hskip-9.29993pt/\;\,0$. This ability offers a strong
contribution to the quantization of gravity, as sketched below.

##  2 The Essence of Quantum Gravity

The author has shown that an affine quantization is the natural procedure to
quantize Einstein’s gravity, e.g., [2, 11]. Here we offer an easy and
straightforward derivation that features the highlights of a natural quantum
gravity. Our procedure is based on the analysis of other quantum procedures
that lead toward a conventual Hamiltonian as part of a standard Schrödinger’s
representation followed by a suitable version of Schrödinger’s equation.

###  2.1 Classical gravity

The classical Hamiltonian, according to ADM [12], uses the metric
$g_{ab}(x)=g_{ba}(x)$ and its determinant $g(x)\equiv\det[g_{ab}(x)]$,555The
additional symbol $\\{g\\}$ stands for $\\{g_{ab}(\cdot)>0\\}$. the momentum
$\pi^{cd}(x)=\pi^{dc}(x)$, or more useful, the momentric666The name momentric
is taken from momentum and metric.
$\pi^{c}_{d}(x)\equiv\pi^{ce}(x)\,g_{de}(x)$, all of which leads to

$\displaystyle
H(\pi^{a}_{b},g_{cd})={\textstyle\int}\\{g(x)^{-1/2}[\pi^{a}_{b}(x)\pi^{b}_{a}(x)-\textstyle{\frac{1}{2}}\pi^{a}_{a}(x)\pi^{b}_{b}(x)]$
$\displaystyle\hskip 80.00012pt+g(x)^{1/2}(x)\,R(x)\\}\;d^{3}\\!x\;,$ (1)

where $R(x)$ is the Ricci scalar [13]. The Poisson brackets for the momentric
and metric fields are given by

$\displaystyle\\{\pi^{a}_{b}(x),\pi^{c}_{d}(x^{\prime})\\}=\textstyle{\frac{1}{2}}\,\delta^{3}(x,x^{\prime})\hskip
0.80002pt[\delta^{a}_{d}\hskip 0.80002pt\pi^{c}_{b}(x)-\delta^{c}_{b}\hskip
0.80002pt\pi^{a}_{d}(x)\hskip 0.80002pt]\;,$
$\displaystyle\hskip-1.99997pt\\{g_{ab}(x),\hskip
0.80002pt\pi^{c}_{d}(x^{\prime})\\}=\textstyle{\frac{1}{2}}\,\delta^{3}(x,x^{\prime})\hskip
0.80002pt[\delta^{c}_{a}g_{bd}(x)+\delta^{c}_{b}g_{ad}(x)\hskip 0.80002pt]\;,$
(2) $\displaystyle\hskip-3.00003pt\\{g_{ab}(x),\hskip
0.80002ptg_{cd}(x^{\prime})\\}=0\;.$

Observe that these Poisson brackets are valid even if we change $g_{ab}(x)$ to
$-g_{ab}(x)$, a property that allows us to retain only $g_{ab}(x)>0$. This is
not possible with the Poisson brackets for canonical variables.

###  2.2 Quantum variables

Passing to operator commutations by a promotion of the set of Poisson brackets
to operator commutations leads, as established in [11], to

$\displaystyle[{\hat{\pi}}^{a}_{b}(x),\hskip
0.80002pt{\hat{\pi}}^{c}_{d}(x^{\prime})]=i\hskip
0.80002pt\textstyle{\frac{1}{2}}\,\hbar\,\delta^{3}(x,x^{\prime})\hskip
0.80002pt[\delta^{a}_{d}\hskip
0.80002pt{\hat{\pi}}^{c}_{b}(x)-\delta^{c}_{b}\hskip
0.80002pt{\hat{\pi}}^{a}_{d}(x)\hskip 0.80002pt]\;,$
$\displaystyle\hskip-1.00006pt[{\hat{g}}_{ab}(x),\hskip
0.80002pt{\hat{\pi}}^{c}_{d}(x^{\prime})]=i\hskip
0.80002pt\textstyle{\frac{1}{2}}\,\hbar\,\delta^{3}(x,x^{\prime})\hskip
0.80002pt[\delta^{c}_{a}{\hat{g}}_{bd}(x)+\delta^{c}_{b}{\hat{g}}_{ad}(x)\hskip
0.80002pt]\;,$ (3) $\displaystyle\hskip-1.99997pt[{\hat{g}}_{ab}(x),\hskip
0.80002pt{\hat{g}}_{cd}(x^{\prime})]=0\;.$

As with the Poisson brackets, these commutators are valid if we change
${\hat{g}}_{ab}(x)$ to $-{\hat{g}}_{ab}(x)$. For the momentric and metric
operators, we again find that we can retain only ${\hat{g}}_{ab}(x)>0$.
Moreover, with suitable regularization, it follows that
${\hat{\pi}}^{a}_{b}(x)\,{\hat{g}}(x)^{-1/2}=0$, which is proved in [11].

###  2.3 Schrödinger’s representation and equation

Schrödinger’s representation chooses ${\hat{g}}_{ab}(x)=g_{ab}(x)$ and

$\displaystyle{\hat{\pi}}^{c}_{d}(x)=-i\textstyle{\frac{1}{2}}\hbar[g_{de}(x)(\partial/\partial
g_{ce}(x))+(\partial/\partial g_{ce}(x))g_{de}(x)]\;,$ (4)

which is well evaluated using a suitable regularization [11]. Finally, we are
led to Schrödinger’s equation

$\displaystyle\hskip-20.00003pti\hbar\,\partial\;\Psi(\\{g\\},t)/\partial
t=\\{\\!\\!\\{{\textstyle\int}\\{[{\hat{\pi}}^{a}_{b}(x)\,g(x)^{-1/2}\,{\hat{\pi}}^{b}_{a}(x)-\textstyle{\frac{1}{2}}{\hat{\pi}}^{a}_{a}(x)\,g(x)^{-1/2}\,{\hat{\pi}}^{b}_{b}(x)]$
$\displaystyle\hskip
100.00015pt+g(x)^{1/2}\,R(x)\\}\;d^{3}\\!x\\}\\!\\!\\}\;\Psi(\\{g\\},t)\;.$
(5)

###  2.4 Potential wave functions

The expression ${\hat{\pi}}^{a}_{b}(x)\,g(x)^{-1/2}=0$ points to a form of
wave function given by $\Psi(\\{g\\})=W(\\{g\\})\Pi_{y}\,g(y)^{-1/2}$. The
wave functions for the Hamiltonian are continuous in the metrics that make up
$\\{g\\}$, which is natural for a Hamiltonian that contains spatial
derivatives of the metric fields and enforces continuity within the Ricci
scalar, $R(x)$.

The normalization of such wave functions may be given by

$\displaystyle{\textstyle\int}|\Psi(\\{g\\})|^{2}{\cal{D}}\\{g\\}={\textstyle\int}|W(\\{g\\})|^{2}\Pi_{y}\,g(y)^{-1}\>{\cal{D}}\\{g\\}=1\;,$
(6)

while a matrix expression for an operator $A$ may be given by

$\displaystyle{\textstyle\int}\Psi^{\prime}(\\{g^{\prime}\\})^{*}\,A(\\{g^{\prime}\\},\\{g\\})\,\Psi(\\{g\\})\;{\cal{D}}\\{g^{\prime}\\}\,{\cal{D}}\\{g\\}$
$\displaystyle\hskip
20.00003pt={\textstyle\int}\Pi_{y^{\prime}}g^{\prime}(y^{\prime})^{-1/2}A(\\{g^{\prime}\\},\\{g\\})\/\Pi_{y}g(y)^{-1/2}\;{\cal{D}}\\{g^{\prime}\\}\,{\cal{D}}\\{g\\}\;.$
(7)

It may be noticed that the factors $\Pi_{y}\,g(y)^{-1/2}$ act somewhat like
‘coherent states’, e.g., [14], in these equations.

##  3 Conclusion

The forgoing study is focussed on the gravity Hamiltonian, which is the most
difficult aspect of quantum gravity. Additional topics, such as fulfilling
constraints, are examined in [11], along with a somewhat different viewpoint
in [2]. An earlier paper [15] is especially focussed on establishing a path
through the multiple constraints that quantum gravity faces.

Contributions by other researchers using affine procedures to quantize gravity
are welcome to add to the present story.

## References

  * [1] ”constant negative curvature“; Scholarpedia: Hyperbolic dynamics
  * [2] J. Klauder, “Using Coherent States to Make Physically Correct Classical-to-Quantum Procedures That Help Resolve Nonrenomalizable Fields Including Einstein’s Gravity”, Journal of High Energy Physics, Gravitation and Cosmology 7, (1019-1026 (2021); DOI:10.4236/jhepgc.2021.73060.
  * [3] L. Gouba, “Affine Quantization on the Half Line”, Journal of High Energy Physics, Gravitation and Cosmology 7, 352-365 (2021); DOI:10.4236/jhepgc.2021.71019.
  * [4] C. Handy and J. Klauder, “Proof that Half-Harmonic Operators become Full-Harmonic Oscillators after the Wall Slides Away”, arXiv:2108.00289.
  * [5] R. Fantoni, “Monte Carlo Evaluation of the Continuum Limit of $\varphi^{12}_{3}$”, J. Stat. Mech. (2021) 083102; arXiv:2011.09862.
  * [6] B. Freedman, P. Smolensky, D. Weingarten, “Monte Carlo Evaluation of the Continuum Limit of $\varphi^{4}_{4}$ and $\varphi^{4}_{3}$”, Phys. Lett. B 113, 481 (1982).
  * [7] M. Aizenman, “Proof of the Triviality. of $\varphi^{4}_{d}$ Field Theory and Some Mean-Field Features of Ising Models for $d>4$”, Phys. Rev. Lett.47, 1-4, E-886 (1981).
  * [8] J. Fröhlich, “On the Triviality of $\lambda\varphi^{4}_{d}$ Theories and the Approach to the Critical Point in $d\geq 4$ Dimensions”, Nuclear Physics B 200, 281-296 (1982).
  * [9] R. Fantoni and J. Klauder, “Affine Quantization of $\varphi^{4}_{4}$ Succeeds While Canonical Quantization Fails”, Phys. Rev. D 103, 076013 (2021).
  * [10] J. Klauder, “Evidence for Expanding Quantum Field Theory“,Journal of High Energy Physics, Gravitation and Cosmology 7, 1157-1160 (2021); DOI:10.4236/jhepgc.2021.73067.
  * [11] J. Klauder, “Using Affine Quantization to Analyze Nonrenotmalizable Scalar Fields and the Quantization of Einstein’s Gravity”, Journal of High Energy Physics, 9Gravitation and Cosmology 6, 802-816 (2020); DOI:10.4236/jhepgc.2020.64053.
  * [12] R. Arnowitt, S. Deser, and C. Misner, “The Dynamics of General Relativity”, Gravitation: An Introduction to Current Research, Ed. L. Witten, (Wiley & Sons, New York, 1962), p. 227; arXiv:gr-qc/0405109.
  * [13] “Ricci scalar”: Wikipedia; https;//wikipedia Ricci scalar
  * [14] J. Klauder and B.-S. Skagerstam, (1985) “Coherent States: Applications in Physics and Mathematical Physics”, World Scientific, Singapore; DOI:/org/10.1142/0096
  * [15] J. Klauder, “Fundamentals of Quantum Gravity”, arXiv:gr-qc/0612168v1.


# Probing topological phase transitions via quantum reflection in the graphene
family materials

P. P. Abrantes<EMAIL_ADDRESS>Instituto de Física, Universidade
Federal do Rio de Janeiro, 21941-972, RJ, Brazil  Tarik P. Cysne
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal Fluminense,
24210-346, RJ, Brazil  D. Szilard  Instituto de Física, Universidade Federal
do Rio de Janeiro, 21941-972, RJ, Brazil  F. S. S. Rosa  Instituto de Física,
Universidade Federal do Rio de Janeiro, 21941-972, RJ, Brazil  F. A. Pinheiro
Instituto de Física, Universidade Federal do Rio de Janeiro, 21941-972, RJ,
Brazil  C. Farina  Instituto de Física, Universidade Federal do Rio de
Janeiro, 21941-972, RJ, Brazil

###### Abstract

We theoretically investigate the quantum reflection of different atoms by two-
dimensional (2D) materials of the graphene family (silicene, germanene, and
stanene), subjected to an external electric field and circularly polarized
light. By using Lifshitz theory to compute the Casimir-Polder potential, which
ensures that our predictions apply to all regimes of atom-2D surface
distances, we demonstrate that the quantum reflection probability exhibits
distinctive, unambiguous signatures of topological phase transitions that
occur in 2D materials. We also show that the quantum reflection probability
can be highly tunable by these external agents, depending on the atom-surface
combination, reaching a variation of 40% for Rubidium in the presence of a
stanene sheet. Our findings attest that not only dispersive forces play a
crucial role in quantum reflection, but also that the topological phase
transitions of the graphene family materials can be comprehensively and
efficiently probed via atom-surface interactions at the nanoscale.

##  I Introduction

Two-dimensional (2D) materials constitute a remarkable material platform to
investigate fluctuation-induced phenomena, at the interdisciplinary frontiers
between condensed matter, atomic and molecular physics, and materials science
[1, 2, 3, 4]. It was theoretically predicted that dispersive forces [e.g.
Casimir and Casimir-Polder (CP) interactions] in graphene-based systems can be
efficiently controlled by external means, such as the application of magnetic
fields [6, 5, 7], strain engineering [8], carriers doping [9, 10, 11], and
suitable stacking in multilayer 2D systems [12, 13]. Other quantum vacuum-
related effects (as, for instance, CP torque in anisotropic molecules [14] and
quantum friction [15, 16]) have also been explored in 2D systems. Significant
experimental progress in the investigation of such phenomena in 2D materials
has also been made [17, 18, 19, 20], specially in the case of graphene. As a
result, it is clear that 2D materials play a pivotal role in the current and
future understanding of Casimir interactions and related physical phenomena
[21].

Among different 2D materials, the so-called graphene family materials
(silicene, germanene, and stanene - the 2D allotropes of Si, Ge, and Sn,
respectively) single out for exhibiting fascinating physical phenomena and a
broad range of applications [22]. They were synthesized only recently [23, 24,
25] and, in contrast to graphene, which has a spin-orbit coupling (SOC) around
a few $\mu$eV, the intrinsic SOC in these materials is higher [26, 27, 28], a
direct effect of their larger atomic number when compared to carbon. Their
intrinsic SOC, ranging from $2$ to $50$ meV, leads to a robust quantum spin
Hall insulator (QSHI) phase characterized by a $\mathbb{Z}_{2}$-topological
invariant [29]. These materials possess a buckled honeycomb lattice and, as a
result, topological phase transitions occur due to the characteristic response
of their electronic structures under the application of external electric bias
[30].

Quantum reflection (QR) consists of the reflection of incident quantum
particles under the influence of a potential that decreases monotonically in
the direction of the particle motion, despite the absence of any turning
points. It finds practical applications in the field of atom-optics [31, 32,
33, 34, 35, 36], such as in the design of atomic mirrors [37, 38, 39, 40],
atomic traps [41, 42], and diffraction gratings [43, 44] and it is also
related to the study of interactions involving Bose-Einstein condensates [45,
46, 47]. Furthermore, QR has been employed as an efficient method to probe
dispersive interactions between atoms and surfaces [49, 48, 44, 50, 51, 52]
(see Ref. [53] for application in atomic collisions). The understanding of
such interactions involving surfaces of complex materials [1, 54, 55] is
crucial for the description of many relevant phenomena in chemistry [56] and
applied physics [57, 52] of current interest.

In this paper, we investigate the QR of different atomic species by
representative 2D materials of the graphene family. Topological phase
transitions are induced by applying an external electric field and by
irradiating circularly polarized light in the 2D sheets. We show that QR
probability can be significantly modified by changing the intensities of these
external agents, specially in the case of materials with high intrinsic SOC
and more massive atoms. Our results demonstrate that topological phase
transitions clearly show up in the behavior of QR as a function of the
external electric field, suggesting that QR could be used as an alternative
and effective optical method to probe topological phase transitions in 2D
materials and other topologically non-trivial systems.

This paper is organized as follows. In the next section, we present the
methodology employed to evaluate the QR probabilities. Section III comprises
our results for the QR probability of a rubidium atom by different materials
of the graphene family and Sec. IV is dedicated to our final comments and
conclusions. Appendixes A and B contain important information on the
mathematical description of materials and atoms studied in this work, whereas
Appendix C presents results for QR of a sodium atom.

##  II Methodology

Figure 1(a) depicts the physical situation of interest. An atomic beam moves
towards a suspended sheet of a 2D graphene family material (silicene,
germanene, or stanene) with normal incidence. Each atom with energy $E$
interacts with the attractive potential generated by the 2D surface and, due
to the quantum particles wave nature, it has a non-zero probability to be
reflected by this attractive potential. An electric field $E_{z}$ is
perpendicularly applied to the 2D material which is also irradiated by a
circularly polarized light, characterized by the parameter $\Lambda$ (see
Appendix B for the precise definition of this parameter). Figure 1(b)
illustrates the topological phase diagram in the space of dimensionless
parameters $e\ell E_{z}/\lambda_{\textrm{SO}}$ and
$\Lambda/\lambda_{\textrm{SO}}$, where $\lambda_{\textrm{SO}}$ is the spin-
orbit coupling, $e$ is the modulus of electron charge and $\ell$ is the
material buckling. A description of this diagram can be found in Appendix B.
For further details, see Ref. [30].

Figure 1: (a) Atomic specimen being reflected by a suspended 2D material of
the graphene family. The system is under the influence of a static electric
field $E_{z}$ and it is shined by a circularly polarized light, characterized
by the parameter $\Lambda$. (b) Topological phase diagram of the 2D graphene
family materials. Horizontal dashed gray lines show the paths used in this
work to explore this diagram. The acronyms of each topological phase means:
band insulator (BI; $C=0$), quantum spin Hall insulator (QSHI; $C=0$, indexed
by the non-trivial $\mathbb{Z}_{2}$-index [29]), polarized-spin quantum Hall
insulator (PS-QHI; $C\leavevmode\nobreak\ =\leavevmode\nobreak\
-\leavevmode\nobreak\ 1$) and anomalous quantum Hall insulator (AQHI; $C=-2$),
with $C$ standing for the corresponding Chern numbers.

The attractive potential between incident atomic specimens and a given 2D
suspended material of the graphene family is determined by the Casimir-Polder
(CP) potential. At low temperatures, the CP potential between a neutral but
polarizable atom and a flat surface can be calculated using the Lifshitz
formula [14, 58, 59] (see also Refs. [61, 60] for a thorough derivation of the
Lifshitz formula in materials with finite Hall conductivity), according to

$\displaystyle U(z)$
$\displaystyle=\frac{\hbar}{\epsilon_{0}c^{2}}\int_{0}^{\infty}\\!\frac{d\xi}{2\pi}\xi^{2}\,\alpha(i\xi)\int\\!\\!\frac{d^{2}{\bm{k}}}{(2\pi)^{2}}\frac{e^{-2\kappa
z}}{2\kappa}$ (1)
$\displaystyle\times\left[r^{ss}({\bm{k}},i\xi)-\left(1+\frac{2c^{2}k^{2}}{\xi^{2}}\right)r^{pp}({\bm{k}},i\xi)\right]\,,$
(3)

where $\kappa=\sqrt{\xi^{2}/c^{2}+k^{2}}$, $\alpha(i\xi)$ denotes the atomic
electric polarizability as a function of imaginary frequencies $i\xi$, and
$r^{ss}({\bm{k}},i\xi)$ and $r^{pp}({\bm{k}},i\xi)$ stand for the diagonal
reflection coefficients associated to the 2D material. As usual, $s$ and $p$
mean the transverse electric and transverse magnetic polarizations,
respectively. Note that, in contrast to configurations involving only
macroscopic objects, the reflection coefficients mixing the polarizations $s$
and $p$ do not contribute to Eq. (3) when one considers the interaction
between an isotropic atom and an macroscopic object with Hall conductivity
[61, 62]. The reflection coefficients, the model for the atomic
polarizability, and other parameters related to each atom considered in this
work can be found in Appendix A. For the sake of simplicity, we shall neglect
thermal corrections and the roughness of the 2D material in this present work
[63, 64, 65].

Considering an atom of mass $m$ and energy $E$ under the influence of any
potential $U(z)$, the time-independent Schrödinger equation reads

$\displaystyle\frac{\partial^{2}\psi(z)}{\partial
z^{2}}+\frac{p^{2}(z)}{\hbar^{2}}\psi(z)=0\,,$ (4)

with

$p(z)=\sqrt{2m[E-U(z)]}\,.$ (5)

The WKB solutions are good approximations when the atom is located far from
the monolayer, when compared to the length scale associated to the CP
interaction, to wit, $c/\xi_{l}$ (see Table 1 in Appendix A). In such a case,
a solution of the form [66]

$\displaystyle\psi(z)=\frac{c_{+}(z)}{\sqrt{|p(z)|}}e^{i\phi(z)}+\frac{c_{-}(z)}{\sqrt{|p(z)|}}e^{-i\phi(z)}\,,$
(6)

is a very convenient one, where $\phi(z)$ is written as

$\displaystyle\phi(z)=\int^{z}_{z_{0}}\\!dz^{\prime}\,\frac{p(z^{\prime})}{\hbar}\,.$
(7)

The previous ansatz is very suitable, since it transforms the second order
Schrödinger equation into a set of two coupled first-order differential
equations for the coefficients $c_{+}(z)$ and $c_{-}(z)$. In fact,
substituting Eqs. (6) and (7) into Eq. (4), it can be shown that [67]

$\displaystyle\frac{\partial c_{+}(z)}{\partial z}$ $\displaystyle=$
$\displaystyle e^{-2i\phi(z)}\frac{c_{-}(z)}{2p(z)}\frac{\partial
p(z)}{\partial z}\,,$ (8) $\displaystyle\frac{\partial c_{-}(z)}{\partial z}$
$\displaystyle=$ $\displaystyle
e^{2i\phi(z)}\frac{c_{+}(z)}{2p(z)}\frac{\partial p(z)}{\partial z}\,.$ (10)

It is reasonable to assume that any atom that reaches the surface of the
material will not be reflected, but adsorbed to it, leading to the boundary
conditions $c_{+}(0)=0$ and $c_{-}(0)=1$. The quantum reflection probability
is then defined as [68, 69, 67, 38, 70]

$\displaystyle
R=\lim_{z\rightarrow\infty}\Bigg{|}\frac{c_{+}(z)}{c_{-}(z)}\Bigg{|}^{2}\,.$
(11)

Information about the efficiency of a given potential $U(z)$ to give rise to
QR can be extracted from the so-called badlands function [68, 69, 67, 38, 70]

$\displaystyle
Q(z)=\frac{\hbar^{2}}{2p^{2}(z)}\left[\frac{\phi^{\prime\prime\prime}(z)}{\phi^{\prime}(z)}-\frac{3}{2}\left(\frac{\phi^{\prime\prime}(z)}{\phi^{\prime}(z)}\right)^{2}\right]\,,$
(12)

with $p(z)$ and $\phi(z)$ given by (5) and (7) and the primes indicate
derivatives with respect to $z$. It is common in the literature to associate
the highest probabilities of QR occurrence to regions of highest values of
$Q(z)$ [68, 69, 67, 38, 70]. However, for very low energies (very high de
Broglie wavelengths), this is not the case and QR must be thought of as a
quite non-local effect so that the correlation between the QR probability and
the badlands function seems to be not valid anymore. In other words, the
unsuitability of this function only occurs at the threshold of the QR
($E\rightarrow 0$) [71]. Nevertheless, as will be discussed in the next
section, this work is concerned with the intermediate energy regime, where
tuning the QR probability by applying external fields is more easily achieved,
therefore, the usual interpretation based on badlands functions still holds.
For a given energy, the badlands function exhibits a peak whose maximum
occurs, by assumption, at position $z_{m}$. To solve Eqs. (8) and (10)
numerically, we need to choose points $z_{i}$ and $z_{f}$ of the space, such
that $z_{i}\ll z_{m}\ll z_{f}$. The boundary conditions are applied at
$z_{i}$, i.e., $c_{+}(z_{i})=0$ and $c_{-}(z_{i})=1$, and the limit of Eq.
(11) is taken at $z_{f}$, i.e., $R=|c_{+}(z_{f})/c_{-}(z_{f})|^{2}$. In this
sense, $z_{i}$ and $z_{f}$ can be understood as convergence parameters and
their typical values are around a few Angstroms and a fraction of centimeters,
respectively. For further details on the methods of solving coupled
differential equations in the QR problem, we refer the reader to Refs. [68,
69, 67, 38, 70].

We computed the QR probability for two atomic species of experimental
relevance (Rb [72] and Na [45, 46]), and for three graphene family materials
already synthesized (stanene [23], germanene [24], and silicene [25]) in
vacuum. For a given atomic specie, with parameters specified in Table 1, and a
given reflective material, whose parameters are presented in Table 2, the QR
probability is a function of the energy of the incident particles beam, the
applied electric field, and the intensity of the circularly polarized laser.
By modifying these external agents, the 2D materials undergo different
topological phase transitions that drastically affect their optical
conductivities (see Appendix B) and, consequently, the atomic reflection
probability. We found that the effect of external perturbations in QR is more
pronounced for heaviest atoms. In what follows, we present our results for QR
of a Rb atom by the graphene family materials. Similar results for a Na atom
can be found in Appendix C.

##  III Results and discussions

Figure 2 exposes the results for QR of a Rb atom by a stanene sheet under the
influence of electric field and circularly polarized light. We consider that
the energy of the incident Rb atom is $E=10^{-4}$ neV (see next discussion).
In Fig. 2(a), we show the QR probability as a function of the applied electric
field for five values of the laser parameter $\Lambda$. Each curve corresponds
to a path symbolized by a horizontal dashed gray line in Fig. 1(b). Let us
start with the analysis of the blue curve, with
$\Lambda/\lambda_{\textrm{SO}}=0$, meaning that the circularly polarized light
is absent [path I in Fig. 1(b)] and the electric field varying from $e\ell
E_{z}/\lambda_{\textrm{SO}}=0$ to $e\ell E_{z}/\lambda_{\textrm{SO}}=3$. The
key feature that we highlight is the non-monotonic behavior of the QR
probability. It starts decreasing from $R\approx 0.48$ until it reaches the
minimum value of $R\approx 0.39$ at the critical point of the topological
phase transition $e\ell E_{z}/\lambda_{\textrm{SO}}=1$. At this point, the
electronic spectrum of stanene becomes gapless and this point separates a
quantum spin-Hall insulator (QSHI) phase from a trivial band insulator (BI)
one. If we continue to increase the intensity of the electric field going into
the deep trivial insulator phase, the QR probability increases and reaches
$R\approx 0.55$ for $e\ell E_{z}/\lambda_{\textrm{SO}}=3$. It is worth
mentioning that both topological phases discussed above are indexed by zero
Chern number, but this is not synonymous of topological triviality. In fact,
it is possible to attribute other topological invariant to the quantum spin
Hall insulator phase called $\mathbb{Z}_{2}$ invariant, which is not trivial
[29]. But this $\mathbb{Z}_{2}$ invariant does not have a direct connection
with charge conductivity as it is the case of Chern number (see Appendix B).

Figure 2: Results for QR of a Rb atom with energy $E=10^{-4}$ neV by a
suspended sheet of pristine stanene ($\lambda_{\textrm{SO}}=50$ meV, chemical
potential $\mu=0$ and inverse of the scattering time
$\Gamma=10^{-4}\lambda_{\textrm{SO}}/\hbar$, as described in Appendix B). (a)
QR probability as a function of the electric field for different values of the
laser parameter. (b) Percentual modification in the QR probability caused by
the electric field.

Now, we consider the presence of circularly polarized light, turning on the
laser parameter to a constant value of
$\Lambda/\lambda_{\textrm{SO}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.5$,
represented by the red curve in Fig. 2(a). In such a configuration, the cusp
of the later case splits into others, stamping the existence of two
topological phase transitions. This is in agreement with the topological
phases crossed by path II in Fig. 1(b), in which there is an emergence of a
third topological phase called polarized-spin quantum Hall insulator (PS-QHI),
between $e\ell E_{z}/\lambda_{\textrm{SO}}=0.5$ and $e\ell
E_{z}/\lambda_{\textrm{SO}}=1.5$, that separates the aforementioned QSHI and
BI phases. Analogously, the phase diagram of Fig. 1(b) can be fully explored
by the paths III, IV, and V associated with
$\Lambda/\lambda_{\textrm{SO}}=1,1.5,2$, respectively, and their results for
the QR probability are also presented in Fig. 2(a). A remarkable
characteristic of all curves is the presence of a cusp exactly at the
topological phase transition points. This suggests at least two routes to
explore these results. Firstly, one can take advantage of these phase
transitions to control and tune the QR probability, adjusting these external
agents (the circularly polarized laser and the electric field) to enhance it
or diminish it according to what may be more convenient. Secondly, these
results may provide a great opportunity of using the QR as a simple optical
tool to probe these topological phase transitions experimentally, since the QR
probability changes its behavior whenever the combination of electric field
and laser intensities hit a point of the phase transition diagram.

Figure 3: (a) The probability $R$ of a Rb atom suffering QR from a stanene
sheet as a function of its incident energy $E$. (b) Relative modification in
the CP energy between the Rb atom and the 2D surface caused by the applied
electric field as a function of the distance $z$. (c) The plot of $Q(z)$ as a
function of the distance $z$ for the incident energy $E=10^{-4}$ neV chosen in
the intermediate energy regime between the classical and the quantum limit of
QR [see panel (a)]. In all plots, $\Gamma=10^{-4}\lambda_{\textrm{SO}}/\hbar$,
$\mu=0$, and $\Lambda=0$.

To achieve a deeper and more complete understanding of the phenomenology
involved in the QR physics by these topological materials, we present a
detailed discussion of the case where $\Lambda/\lambda_{\textrm{SO}}=0$ [path
I depicted in Fig. 1(b)]. In Fig. 3 we show the QR as a function of the
incident energy [panel (a)], the normalized CP energy as a function of
distance [panel (b)], and $Q(z)$ as a function of $z$ for an incident energy
$E=10^{-4}$ neV [panel (c)]. Each curve denotes a distinct value of $e\ell
E_{z}/\lambda_{\textrm{SO}}\leq 1$, judiciously chosen to explore the region
in which stanene crosses the QSHI phase up to the critical point at $e\ell
E_{z}/\lambda_{\textrm{SO}}=1$. In Fig. 4 we present similar results but for
electric field values for which stanene goes from this critical point to the
BI phase ($e\ell E_{z}/\lambda_{\textrm{SO}}\geq 1$).

Figure 4: (a) The probability $R$ of a Rb atom suffering QR from a stanene
sheet as a function of its incident energy $E$. (b) Relative modification in
the CP energy between the Rb atom and the 2D surface caused by the applied
electric field as a function of the distance $z$. (c) The plot of $Q(z)$ as a
function of the distance $z$ for the incident energy $E=10^{-4}$ neV chosen in
the intermediate energy regime between the classical and the quantum limit of
QR [see panel (a)]. In all plots, $\Gamma=10^{-4}\lambda_{\textrm{SO}}/\hbar$,
$\mu=0$, and $\Lambda=0$.

Both Figs. 3 and 4 clarify the cusps in QR probability associated to the
topological phase transitions. From Fig. 3, it should be noticed that the QR
probability presents three regimes. (i) The low energy limit, in which the
wavelike nature of the quantum particle stands out and $R\rightarrow 1$ as
$E\rightarrow 0$. (ii) The high energy limit, in which $R\rightarrow 0$ as
$E\rightarrow\infty$ and the particle behaves classically. (iii) The
intermediate regime lying in between the classical and the quantum ones where
$0<R<1$, in which a choice of incident energy is advantageous, since the QR
probability can be substantially altered by combined effects of external
agents [5, 7]. In this intermediate regime, Fig. 3(a) points out that the QR
probability decreases with the electric field in the QSHI-phase for most range
of energies of the incident particle. This feature can be explained with the
aid of Figs. 3(b) and (c), that show the effect of electric field on the CP
energy and on function $Q(z)$, respectively, for a particle with energy
$E=10^{-4}$ neV. The peak of the $Q(z)$ is located around $z\approx 1$ $\mu$m
for all values of $e\ell E_{z}$ in the QSHI phase, meaning that this is the
distance regime where the QR is most salient. In this same topological phase,
the result of the electric field on the CP energy is to enhance its intensity
as the system approaches the critical point [3(b)]. This effect is noticeable
for distances $z\geq 0.1$ $\mu$m only, but that contemplates the peak of
$Q(z)$. As a consequence, the height of the peak of $Q(z)$ decreases and the
outcome is a decrease of the QR probability when the electric field is varied
from $e\ell E_{z}/\lambda_{\textrm{SO}}=0$ to $1$ [3(c)]. An opposite behavior
can be identified when we consider the BI phase presented in Fig. 4. As can be
verified in panel (a), the QR probability increases with the intensity of the
electric field in intermediate energy regimes. In addition, panel (b) reveals
that the CP energy decreases with the electric field at long distances, which
causes the height of the peak of $Q(z)$ to increase in panel (c). The final
result translates into an enhancement of the QR probability with the increase
of the electric field intensity in the BI phase. Ultimately, the results
presented in Figs. 3 and 4 can be traced back to the cusp at the topological
critical point in the curve of $\Lambda=0$ in Fig. 2(a). This same reasoning
can be borrowed to every results of the main text in order to understand all
structures of cusps.

We can also discuss the degree of tunability of the QR probability. To this
end, we define the quantity

$\displaystyle\delta_{\Lambda}(E_{z})=\left[\frac{R_{\Lambda}\left(E_{z}\right)-R_{\Lambda}\left(E_{z}=0\right)}{R_{\Lambda}\left(E_{z}=0\right)}\right]\times
100\%\,,$ (13)

that furnishes an estimate of the QR relative variation induced by the applied
electric field for a given laser intensity. Figure 2(b) shows the degree of
tunability of the QR probability for the same parameters employed in Fig.
2(a), revealing an amplitude of variation that comfortably lies in the
measurement precision of QR experiments [72, 45, 46, 73, 74, 75, 36].

Figures 5 and 6 show similar results, but for QR of a Rb atom by the other two
materials (germanene and silicene, respectively), indicating that QR by
germanene is also quite sensitive. Unfortunately, due to the lower spin-orbit
coupling of silicene compared with the ones of germanene and stanene, the
control of QR by the electric field is not so intense in this material, as can
be seen contrasting Fig. 6(b) with Figs. 2(b) and 5(b). Despite that, the
noticeable presence of cusp when the material crosses a given topological
phase transition persists in the case of silicene and their optical detection
could also remain possible. From the perspective of material synthesis, it is
noteworthy to mention that it should be easier to fabricate germanene and
silicene with currently available methods [23, 24, 25].

Figure 5: Results for QR of a Rb atom with energy $E=10^{-4}$ neV by a
suspended sheet of pristine germanene ($\lambda_{\textrm{SO}}=20$ meV, $\mu=0$
and $\Gamma=10^{-4}\lambda_{\textrm{SO}}/\hbar$). (a) QR probability as a
function of the electric field for different values of the laser parameter.
(b) Percentual modification in the QR probability caused by the electric
field. Figure 6: Results for QR of a Rb atom with energy $E=10^{-4}$ neV by a
suspended sheet of pristine silicene ($\lambda_{\textrm{SO}}=2$ meV, $\mu=0$
and $\Gamma=10^{-4}\lambda_{\textrm{SO}}/\hbar$). (a) QR probability as a
function of the electric field for different values of the laser parameter.
(b) Percentual modification in the QR probability caused by the electric
field.

As a final comment, we briefly discuss the last cousin of the graphene family,
the plumbene, synthesized in 2019 [76]. Although its spin-orbit coupling is
almost twice as strong as that one of stanene, density-functional calculations
have shown that plumbene is a band insulator (BI) at neutrality ($\mu=0$)
[77]. The induction of a quantum spin-Hall insulator phase can be performed by
electron doping, originating a physics like the one described here [77]. In
such a case, we expect that the effects of electric field and circularly
polarized laser should be much more noticeable, as well as the control of QR
probability. However, the simplified Dirac Hamiltonian described in Eq. (18)
and the related topological phase diagram of Fig. 1(b) may not be applicable
for plumbene, and more detailed modeling would be necessary.

##  IV Final Remarks and Conclusions

We have investigated the quantum reflection (QR) of atoms by graphene family
materials (stanene, germanene, and silicene). In our setup, we have considered
two atomic species of experimental importance (Na and Rb) and the system was
subject to a perpendicular electric field and a circularly polarized light.
These external agents may induce several topological phase transitions and the
fundamental motivation was to understand how they may affect atom-surface
interactions. We have demonstrated that, whenever the 2D material undergoes a
given topological phase transition, it leaves a clear signature in the QR
probability. Therefore, our findings reveal that one can look at them from two
different perspectives. The first one involves the use of these materials as a
promising platform to manipulate and tune the QR probability through an
external agent. The second one is to take advantage of the QR as a simple
optical tool to probe these topological phase transitions experimentally.
Bearing all this in mind, we also determined the sensitivity of the QR
probability due to variations of the electric field and laser intensities,
verifying that they lie within the scope of typical experimental precision in
many situations. A higher degree of control is achieved in the case of Rb-
stanene, in which the modification on the QR probability may reach 40%. This
can be mainly associated with two aspects: (i) The heaviest mass of the Rb
atom, which makes the QR to be dominated by the longer distances regime of the
CP energy (a region most affected by external agents), and (ii) the higher
spin-orbit coupling of stanene, making the effect of perturbations more
pronounced. Altogether we expect that these results allow for an alternative
way to control QR in increasingly smaller scales, as well as contribute to a
better understanding of the rich physics in Casimir effect and other related
phenomena present in these materials.

## Acknowledgements

C.F. and F.S.S.R. acknowledge Conselho Nacional de Desenvolvimento Científico
e Tecnológico (CNPq) for financial support (Grant numbers 310365/2018-0 9 and
309622/2018-2). F.S.S.R. (Grant number E26/203.300/2017) and P.P.A.
acknowledge Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro
(FAPERJ). T.P.C acknowledges Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior (CAPES). D.S. and F.A.P also thank the funding agencies.

##  Appendix A Reflection coefficients and atomic polarizabilities

The reflection coefficients of a 2D material sheet with finite longitudinal
and transverse optical conductivities, needed in Eq. (3) to evaluate the
Casimir-Polder energy, can be obtained by solving Maxwell’s equations with the
appropriate boundary conditions [6, 5, 7, 78]. These formulas can be written
as

$\displaystyle r^{ss}({\bm{k}},i\xi)$
$\displaystyle=\frac{2\sigma_{xx}(i\xi)Z^{\textrm{H}}+\eta_{0}^{2}[\sigma_{xx}^{2}(i\xi)+\sigma_{xy}^{2}(i\xi)]}{-\Phi({\bm{k}},i\xi)}\,,$
(14) $\displaystyle r^{pp}({\bm{k}},i\xi)$
$\displaystyle=\frac{2\sigma_{xx}(i\xi)Z^{\textrm{E}}+\eta_{0}^{2}[\sigma_{xx}^{2}(i\xi)+\sigma_{xy}^{2}(i\xi)]}{\Phi({\bm{k}},i\xi)}\,,$
(15) $\displaystyle\Phi({\bm{k}},i\xi)$
$\displaystyle=[2+Z^{\textrm{H}}\sigma_{xx}(i\xi)][2+Z^{\textrm{E}}\sigma_{xx}(i\xi)]$
$\displaystyle+[\eta_{0}\sigma_{xy}(i\xi)]^{2}\,,$ (16)

where $Z^{\textrm{H}}=\xi\mu_{0}/\kappa$,
$Z^{\textrm{E}}=\kappa/(\xi\epsilon_{0})$, and
$\eta_{0}^{2}=\mu_{0}/\epsilon_{0}$. Additionally, $\sigma_{xx}(i\xi)$ and
$\sigma_{xy}(i\xi)$ denote the longitudinal and transverse conductivities of
the 2D material, respectively, as a function of imaginary frequencies $i\xi$.

In order to describe the atomic polarizabilities in the Casimir-Polder
potential [Eq. (3)], we employed a Lorentz oscillator model with a single
resonance, given by

$\displaystyle\alpha_{l}(i\xi)=\frac{\alpha_{l}(0)}{1+\frac{\xi^{2}}{\xi_{l}^{2}}}\,.$
(17)

$l$ |  |  $\alpha_{l}(0)$ (a.u.) |  |  $\hbar\xi_{l}$ (eV) |  | $m(\times 10^{-27}\text{ Kg})$
---|---|---|---|---|---|---
Na |  | 162.6 |  | 2.13 |  | 38.17
Rb |  | 318.6 |  | 1.68 |  | 141.92
Table 1: Data for Na and Rb atoms. This table contains parameters of the
single Lorentz-oscillator model needed in Eq. (17) [12] and their masses. ($1$
a.u. $=1.648\times 10^{-41}$ C2m2J-1).

The different fitted parameters for each atomic specimen analyzed here are
well-known from the literature, see Table 1. For materials with higher spin-
orbit coupling, such as stanene, stronger electric fields are necessary to
explore the entire topological phase diagram of Fig. 1 (b). In turn, these
electric fields may cause a significant modification in the atomic resonance
frequency $\xi_{l}$ via the Stark effect. For instance, in the case of QR of
Rb atom by stanene, this correction can be of the order of 25$\%$ of the
values presented in Table 1. Nevertheless, such modification only implies in a
minor numerical correction in the short-distance regime of the CP energy and,
consequently, does not affect our results on the QR probability. Therefore, in
the numerical results presented in this work, we neglected the effects of the
applied electric field and circularly polarized light on the atom, due to the
dominant impact of these external agents in the optical conductivity of the
reflective 2D surface (see Appendix B).

##  Appendix B Optical conductivities

The low energy description of buckled materials of the graphene family under
the influence of external perpendicular electric field ($E_{z}$) and
circularly polarized laser beam ($\Lambda$) is given by the Hamiltonian [30],

$\displaystyle H_{s}^{\eta}=\hbar v_{F}(\eta
k_{x}\tau_{x}+k_{y}\tau_{y})+\frac{\Delta^{\eta}_{s}}{2}\tau_{z}-\mu\,,$ (18)

where

$\displaystyle\Delta_{s}^{\eta}=\eta s\lambda_{\textrm{SO}}-e\ell
E_{z}-\eta\Lambda\,.$ (19)

In these equations, $\eta=\pm 1$ is the quantum number related to $K$ and
$K^{\prime}$ valleys of the Brillouin zone of the honeycomb material,
$\tau_{x,y,z}$ are Pauli matrices related to the sublattice degree of freedom,
$s=\pm 1$ for electron spins $\uparrow$ and $\downarrow$, and $k_{x(y)}$ are
the electron momentum relative to $K$ ($K^{\prime}$) valleys. Also,
$v_{F}=\sqrt{3}at/2$ is the Fermi velocity, $a$ is the lattice parameter, $t$
is the hopping parameter, and $\mu$ is the chemical potential of the material,
which we set $\mu=0$ in the numerical calculations of this work (see Refs. [9,
10, 11] for details about the impact of the chemical potential in the Casimir-
Polder interaction in Dirac materials). In Eq. (19), $\lambda_{\textrm{SO}}$
is the intrinsic spin-orbit coupling, $e$ is the modulus of electron charge,
$\ell E_{z}$ is the sublattice potential generated by the external electric
field $E_{z}$, and the parameter $\Lambda=\pm 8\pi\alpha
v_{F}^{2}I_{0}/\omega_{0}^{3}$ is associated to the circularly polarized light
of intensity $I_{0}$ and frequency $\omega_{0}$ irradiated in the material.
Furthermore, $\alpha$ is the fine structure constant and the $+$ ($-$) sign
denotes the left (right) circular polarization [79]. Material parameters can
be found in Table 2.

Material |  |  $\lambda_{\textrm{SO}}$ (meV) |  |  $t$ (eV) |  |  $\ell$ (Å) |  |  $a$ (Å)
---|---|---|---|---|---|---|---|---
Silicene |  | 2 |  | 1.6 |  | 0.11 |  | 3.86
Germanene |  | 20 |  | 1.3 |  | 0.16 |  | 4.02
Stanene |  | 50 |  | 1.3 |  | 0.20 |  | 4.70
Table 2: Spin-orbit coupling ($\lambda_{\textrm{SO}}$), hopping parameter
($t$), buckling ($\ell$) and lattice parameter ($a$) of each graphene family
material used in this work. Parameters were taken from references [26, 27, 28,
22]. Note: There exists some discordance in the literature for
$\lambda_{\textrm{SO}}$ in the case of stanene. Figure 7: Evolution of the
electronic spectrum of the Hamiltonian of Eqs. (18) and (19), along the path I
of Fig. 1 (b), i. e., $\Lambda/\lambda_{\rm SO}=0$, and (a) $e\ell
E_{z}/\lambda_{\rm SO}=0$, (b) $e\ell E_{z}/\lambda_{\rm SO}=0.5$, (c) $e\ell
E_{z}/\lambda_{\rm SO}=1$, (d) $e\ell E_{z}/\lambda_{\rm SO}=1.5$, and (e)
$e\ell E_{z}/\lambda_{\rm SO}=2$. The left panels show the spectrum for the
valley $K$ ($\eta=+1$) and the right panels show the spectrum for the valley
$K^{\prime}$ ($\eta=-1$). The black-dashed line depicts the spectrum for the
spin-$\uparrow$ sector ($s=+1$), and the red-dotted line depicts the spectrum
for the spin-$\downarrow$ sector ($s=-1$). The topological critical point is
represented in panel (c), where the electronic spectrum becomes gapless, and
it separates the QSHI phase [panels (a) and (b)] and the BI phase [panels (d)
and (e)].

In Fig. 7, we analyze the evolution of the electronic spectrum of the
Hamiltonian shown in Eqs. (18) and (19), exploring the path I
($\Lambda/\lambda_{\rm SO}=0$) of Fig. 1 (b). For the cases presented in
panels 7 (a) and 7 (b), where $e\ell E_{z}/\lambda_{\rm SO}=0$ and $e\ell
E_{z}/\lambda_{\rm SO}=0.5$, respectively, the 2D material exhibits a QSHI
phase, indexed by a non-trivial $\mathbb{Z}_{2}$-topological invariant, if the
chemical potential $\mu$ lies inside the gap of the electronic spectrum. This
topological index remains invariant over small changes in the Hamiltonian, as
long as the electronic gap remains open and still contains the chemical
potential [29]. However, as the electric field is enhanced, the topological
gap becomes increasingly small. When the field reaches the value corresponding
to $e\ell E_{z}/\lambda_{\rm SO}=1$, shown in panel 7 (c), the spin-$\uparrow$
($\downarrow$) sector of the electronic spectrum becomes gapless at the valley
$K$ ($K^{\prime}$), and the system touches the topological critical point. If
one continues to increase the electric field intensity, the electronic gap
reopens but now exhibiting a trivial-insulator (or BI) behavior, as shown in
panels 7 (d) and 7 (e). The two insulating phases presented in Fig. 7 are
characterized by a Chern number $C=0$.

This previous analysis of the electronic spectrum can be repeated to
understand all topological phase transitions crossed by paths II, III, IV, and
V of Fig. 1 (b). For instance, by changing the parameters along path II
($\Lambda/\lambda_{\rm SO}=0.5$), the 2D material undergoes three distinct
topological phases, that are separated by two topological critical points. The
first one occurs for $e\ell E_{z}/\lambda_{\rm SO}=0.5$, where the
spin-$\uparrow$ sector of the spectrum becomes gapless at valley $K$, while
the spectrum remains gapped at valley $K^{\prime}$, and separates the QSHI
from the PS-QHI phase, indexed by a non-trivial Chern number $C=-1$. The
second critical point occurs for $e\ell E_{z}/\lambda_{\rm SO}=1.5$, where the
spin-$\downarrow$ sector of the electronic spectrum becomes gapless at valley
$K^{\prime}$, while it remains gapped at valley $K$, and separates the PS-QHI
from the BI phase. Lastly, to analyze the phase transition between the AQHI
phase, indexed by a Chern number $C=-2$, and the PS-QHI phase, we consider
path IV ($\Lambda/\lambda_{\rm SO}=1.5$). At the critical point $e\ell
E_{z}/\lambda_{\rm SO}=0.5$, that separates these two topological phases, the
spin-$\uparrow$ sector of the electronic spectrum becomes gapless at valley
$K^{\prime}$ and remains gapped at valley $K$. Furthermore, extrapolating the
path IV until $e\ell E_{z}/\lambda_{\rm SO}=2.5$, there is another critical
point to be crossed, separating the PS-QHI and the BI phases, where the
spin-$\downarrow$ sector of the spectrum becomes gapless at valley
$K^{\prime}$, while it remains gapped at valley $K$. This analysis exhausts
the different types of phase transitions presented in Fig. 1 (b), that occur
whenever the material hits the points of the parameter space for which the
electronic spectrum becomes gapless. Among the topological invariants
described above, the Chern number has particular importance in the context of
this work, due to its connection with the DC limit of the Hall conductivity
(see next paragraph) that appears in the reflection coefficients of the 2D
material. The non-zero Chern numbers are related to the existence of localized
edge-states in finite samples [80], responsible for the transport of the Hall
conductivity that is robust against disorder and weak interactions, a
characteristic aspect of non-trivial topological systems [29, 80].

Longitudinal and transverse optical conductivities for graphene family
materials described by Hamiltonian (18) can be obtained by the usual Kubo
formalism in the linear response regime [81, 82] and closed analytical
expressions were obtained in Refs. [21, 79]. In the low temperature regime,
the spin and valley resolved longitudinal conductivity is given by

$\displaystyle\frac{\sigma^{\eta,s}_{xx}(\omega)}{\sigma_{0}/2\pi}$
$\displaystyle=\frac{4\mu^{2}-|\Delta_{s}^{\eta}|^{2}}{2\hbar\mu\Omega}\Theta(2\mu-|\Delta_{s}^{\eta}|)$
$\displaystyle+\left[1-\frac{|\Delta_{s}^{\eta}|^{2}}{\hbar^{2}\Omega^{2}}\right]\tan^{-1}\\!\left[\frac{\hbar\Omega}{M}\right]+\frac{|\Delta_{s}^{\eta}|^{2}}{\hbar\Omega
M}$ (20)

and the transverse (Hall) conductivity is found to be

$\frac{\sigma^{\eta,s}_{xy}(\omega)}{\sigma_{0}/2\pi}=\frac{2\eta\Delta_{s}^{\eta}}{\hbar\Omega}\tan^{-1}\\!\left[\frac{\hbar\Omega}{M}\right]\,.$
(21)

In addition, $\sigma^{\eta,s}_{yy}(\omega)=\sigma^{\eta,s}_{xx}(\omega)$ and
$\sigma^{\eta,s}_{xy}(\omega)=-\sigma^{\eta,s}_{yx}(\omega)$. The total
conductivities are obtained from
$\sigma_{ij}(\omega)=\sum_{s,\eta}\leavevmode\nobreak\
\sigma^{\eta,s}_{ij}(\omega)$. In previous equations, $\Theta$ is the
Heaviside function, $\sigma_{0}=e^{2}/(4\pi)$,
$M=\text{max}(|\Delta_{s}^{\eta}|,2|\mu|)$, and $\Omega=-i\omega+\Gamma$, with
$\Gamma=1/(2\tau)$ and $\tau$ being the scattering time that accounts for
effects of impurities [9]. It is worth mentioning that we assume the local
approximation ($|\vec{q}|\rightarrow 0$) in Eqs. (20) and (21). This
approximation describes very well the dispersive forces in graphene [83] and
should also apply to the other graphene family materials of Table 2 since they
have very similar Fermi velocities.

In Fig. 8 we plot the real and imaginary parts of the longitudinal and
transverse conductivities for different combinations of parameters
$\\{\Lambda/\lambda_{\textrm{SO}},e\ell E_{z}/\lambda_{\textrm{SO}}\\}$, so
that we capture some representative points of the topological phase diagram
modeled by Eq. (18). From panels (b) and (d), it can be noticed the relation
between the static limit of the transverse conductivity and the Chern number
of the topological phase: $\sigma_{xy}(\omega\rightarrow 0)=[e^{2}/(2h)]C$
[21, 79, 80]. Equations (20) and (21) in combination with Eqs. (14-17) are
used in Eq. (3) to compute the CP potential between the incident atoms and the
2D graphene family material.

Figure 8: Panels (a) and (b): Real (solid) and imaginary (dashed) parts of
$\sigma_{xx}(\omega)$ and $\sigma_{xy}(\omega)$ for some combinations of
parameters $\\{e\ell
E_{z}/\lambda_{\textrm{SO}},\Lambda/\lambda_{\textrm{SO}}\\}$. A: $\\{0,0\\}$,
B: $\\{0,1\\}$ and C: $\\{0,2\\}$, as indicated in the topological phase
diagram of panel (f). Panels (c) and (d): the same as before but for A:
$\\{0,0\\}$, D: $\\{1,0.5\\}$ and E: $\\{2,0\\}$, as in panel (f). We set
$\Gamma=10^{-4}\lambda_{\textrm{SO}}/\hbar$ and $\mu=0$ in all cases. Panels
(e) and (f): Topological phase diagram of the graphene family materials
described by the Hamiltonian (18). Panel (e) shows the phase diagram with the
acronyms and their respective Chern numbers, while panel (f) sketches the set
of parameters used in panels (a)-(d). A detailed discussion of this phase
diagram and the optical conductivities can be found in Refs. [30, 21, 79].
Figure 9: Results for a Na atom with incident energy $E=10^{-3}$ neV. QR
probability as a function of the applied electric field by a suspended sheet
of pristine (a) stanene, (c) germanene and (e) silicene. Percentual
modification in the QR probability due to the electric field by a suspended
sheet of pristine (b) stanene, (d) germanene and (f) silicene. In all cases
different curves refer to different values of the laser parameter.

##  Appendix C Results for Na atom

Figure 9 shows results similar to those presented before, but for QR of a Na
atom, which is also relevant in experimental setups [45, 46]. In this case, we
chose an incident energy $E=10^{-3}$ neV and the outcomes are qualitatively
the same as for Rb. The QR probability shows a cusp whenever the graphene
family material undergoes a topological phase transition induced by $E_{z}$
for a fixed $\Lambda$. The degree of tunability of QR probability is also more
evident using materials with higher spin-orbit coupling as the reflective
material. As we already pointed out, if we compare the degree of control,
given by Eq. (13), for the QR probability of Rb and Na by the same graphene
family surface, we find that Rb allows a greater control by $E_{z}$ than Na.
This is due to the fact that Rb is more massive than Na, resulting in a lower
energy regime for which the QR of Rb is more tunable than the QR of Na [7].
Consequently, by comparing the QR of Rb and Na at the intermediate energy
regime, the former is dominated by longer distances (bigger $z_{m}$) between
the atom and the reflective material as compared to the latter, where the CP
energy exhibits a higher degree of control with $E_{z}$.

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