Current Address: ]Department of Applied Physics, Stanford University,
Stanford, CA 94305, USA.

# Are symmetry protected topological phases immune to dephasing?

Siddhant Midha  Department of Electrical Engineering, Indian Institute of
Technology Bombay, Powai, Mumbai-400076, India  Koustav Jana  [  Department of
Electrical Engineering, Indian Institute of Technology Bombay, Powai,
Mumbai-400076, India  Bhaskaran Muralidharan<EMAIL_ADDRESS>Department of
Electrical Engineering, Indian Institute of Technology Bombay, Powai,
Mumbai-400076, India

###### Abstract

Harnessing topological phases with their dissipationless edge-channels coupled
with the effective engineering of quantum phase transitions is a spinal aspect
of topological electronics. The accompanying symmetry protection leads to
different kinds of topological edge-channels which include, for instance, the
quantum spin Hall phase, and the spin quantum anomalous Hall phase. To model
realistic devices, it is important to ratify the robustness of the
dissipationless edge-channels, which should typically exhibit a perfect
quantum of conductance, against various disorder and dephasing. This work is
hence devoted to a computational exploration of topological robustness against
various forms of dephasing. For this, we employ phenomenological dephasing
models under the Keldysh non-equilibrium Green’s function formalism using a
model topological device setup on a 2D-Xene platform. Concurrently, we also
explicitly add disorder via impurity potentials in the channel and averaging
over hundreds of configurations. To describe the extent of robustness, we
quantify the decay of the conductance quantum with increasing disorder under
different conditions. Our analysis shows that these topological phases are
robust to experimentally relevant regimes of momentum dephasing and random
disorder potentials. We note that Rashba mixing worsens the performance of the
QSH phase and point out a mechanism for the same. Further, we observe that the
quantum spin Hall phase break downs due to spin dephasing, but the spin
quantum anomalous Hall phase remains robust. The spin quantum anomalous Hall
phase shows stark robustness under all the dephasing regimes, and shows
promise for realistic device structures for topological electronics
applications.

##  I Introduction

An exciting facet of research in modern condensed matter physics deals with
the manifestation of topology in the electronic states. Topological insulators
(TIs) are quantum states of matter, having a gapped bulk like a normal
insulator, and hosting protected states at the insulator-vacuum boundary [1,
2, 3]. These states are theorized to be robust against disorder as a result of
the accompanying symmetry protection [4]. There has been significant interest
in employing materials manifesting topological phases for applications in
electronics featuring these dissipation-less edge states [5, 6, 7, 8].
Particularly, 2D-Xene monolayers and monolayer transition metal
dichalcogenides (TMDs) are prominent candidates for 2D TIs [9, 10, 11], with
applications in topological electronics [5].
Although theorized to be robust against backscattering, there is a need for
detailed investigations of the effects of disorder in these topological phases
if they are to be used for any topological electronics applications. Previous
works have studied such effects in various regimes. Gap-opening mechanisms
facilitated via inclusion of electron-electron interaction in a many-body
picture has been reviewed in 2D TIs [12, 13, 14, 15] and 3D TIs [16, 17].
Interestingly, the combination of electron-electron interactions and momentum-
dependent spin polarization leads to a mean-field hamiltonian similar to a
Zeeman coupling [12], dubbed a ‘local magnetization’ [17]. The number of
Kramer’s pairs operational in the helical state also impacts the stability, as
outlined in [13, 15]. Effects of magnetic impurities have been studied in [18,
19]. A general analysis of the effects on incoherent electromagnetic noise on
the edge states is presented in [20].
In this work, we employ the non-equilibrium Green’s function (NEGF) formalism
[21, 22] to study the effects of disorder and dephasing in 2D Xene-devices
that are capable of hosting quantum spin Hall (QSH) and spin quantum anomalous
Hall (SQAH) phases. The NEGF formalism provides a phenomenological
computational framework for introducing different kinds of dephasing within
tight-binding models [23, 24, 25, 26]. Particularly, NEGF provides the
possibility of introducing momentum dephasing, which relaxes momentum as well
as phase, and spin dephasing, only relaxes spin, independently of each other
[22]. We study the effects of momentum and spin relaxation and analyze in
depth how the topological phase responds to dephasing via looking at the
conductance quantization in the presence of disorder. We also compare the
methods of NEGF enabled momentum dephasing and explicit addition of random
potentials followed by sample averaging. The effect of Rashba spin-mixing on
the robustness is studied, and a comparison of the robustness of the QSH and
the SQAH phases is provided. We facilitate the analysis with plots of the
bandstructures and wavefunction densities.
The rest of the paper is organized as follows: In Sec. II, we elaborate on the
structure of the channel, the topological phases and the device under study,
and outline the NEGF method for conducting the transport calculations. Next,
we discuss in detail the effects of momentum and spin dephasing on the QSH and
SQAH phases under various regimes in Sec. III. We discuss the implications of
the results and outline future work in Sec. IV.

Figure 1: a) Top and side views of the Xene nanoribbon: Top view shows the
honeycomb lattice structure, atoms from both the sublattices are depicted in
red and blue colors. Side view of shows the buckled nature of the honeycomb
lattice. The buckling height is given by $2l$. An out-of-plane electric field
$E_{z}$ applied perpendicular to the nanoribbon induces a potential difference
between the two sublattices, given as $V(A)=\Delta_{z}$ and
$V(B)=-\Delta_{z}$, where $\Delta_{z}=lE_{z}$ (b) Phase diagram showing the
possible topological phases of the device under the considered hamiltonian as
a function of the spin-orbit coupling strength $\lambda_{SO}$, sublattice
potential $\Delta_{z}=lE_{z}$, and the antiferromagnetic coupling strength
$M_{AF}$ [11] (c) Device structure under consideration: The channel, hosting
the topological phases, is connected to the source (left, $L$) and drain
(right, $R$) contacts. Under the NEGF model, the contacts are modelled with
self energies $\Sigma_{L/R}$, and dephasing is modelled through the
phenomenological NEGF dephasing probem, modelled as $\Sigma_{D}$, which
includes (i) spin dephasing, and (ii) momentum dephasing, shown as arrows.

##  II Formalism and Setup

###  II.1 Device Structure and Hamiltonian

We investigate the QSH and SQAH phases in the 2D buckled hexagonal lattice
structure as shown in Fig. 1(a). The top view of the hexagonal lattice shows
the $A$ and $B$ sublattice points in red and blue respectively. The side view
shows the buckled nature of the lattice with buckling height $2l$. The
electric field $E_{z}$ applied perpendicular to the lattice creates a
staggered sublattice potential $\Delta_{z}=lE_{z}$, with the net potential
difference between the sublattice points being $2\Delta_{z}$. The low energy
Bloch hamiltonian, is given as,

$\mathcal{H}(\mathbf{k})=\underbrace{\hbar
v_{f}(k_{x}\tau_{z}\sigma_{x}+k_{y}\sigma_{y})}_{\text{Nearest neighbour
hopping}}+\underbrace{\lambda_{SO}\sigma_{z}\tau_{z}s_{z}}_{\text{Spin-orbit
coupling}}+\\\
\underbrace{\lambda_{R}(\sigma_{x}s_{y}\tau_{z}-\sigma_{y}s_{z})}_{\text{Rashba
spin-mixing}}+\underbrace{\Delta_{z}\sigma_{z}}_{\text{Staggered sublattice
potential}}+\\\ \underbrace{M_{AF}\sigma_{z}s_{z}}_{\text{Antiferromagnetic
interaction}}$ (1)

where $\sigma_{i}$, $\tau_{i}$, and $s_{i}$ denote the pauli matrices in the
space of the sublattice points $A$-$B$, the valley index $K$-$K^{\prime}$ and
the spin $\uparrow$ \- $\downarrow$ for $i\in\\{x,y,z\\}$. The quantity
$\hbar$ is the reduced Planck’s constant and $v_{f}$ is the Fermi velocity
given as $v_{f}=3ta_{0}/2$, with $t$ being the nearest neighbour hopping and
$a_{0}$ the side length of the hexagonal lattice. Moreover, $\lambda_{SO}$ is
the strenght of the spin-orbit coupling, $\lambda_{R}$ denotes the Rashba spin
mixing interaction, and $\Delta_{z}$ encapsulates the perpendicular electric
field as discussed before. Lastly, the antiferromagnetic interaction, which
can be realized via proximity coupling is quantified using the $M_{AF}$ term.
The tight-binding hamiltonian used for numerical calculations can be found in
[10, Appendix A].
The interplay of the parameters $(\lambda_{SO},\Delta_{z},M_{AF})$ results in
the realization of various topological phases in the channel, which are
adiabatically disconnected from each other. Figure 1(b) elucidates these
phases diagramatically. Notably, in the absence of antiferromagnetic
interaction ($M_{AF}=0$): for $\Delta_{z}<\lambda_{SO}$ the channel is in the
QSH state with zero total Chern number and a non-zero spin Chern number,
indicating helical edge states. With antiferromagnetic interaction
($M_{AF}\neq 0$): for $\Delta_{z}>M_{AF}-\lambda_{SO}$ the channel is in the
SQAH phase with a non-zero total Chern number, possessing spin-polarized
chiral edge states.

###  II.2 Transport Calculations

We consider the following device structure as outlined in Fig. 1(c): The
channel is $L$ atoms long and $W$ atoms wide, and is sandwiched between the
source and the drain terminals. All the proximity effects and electric fields
are present in the channel region only. That is, the electric field, causing
the sublattice potential $\Delta_{z}$ and the Rashba spin-mixing
$\lambda_{R}$, along with the antiferromagnetic interaction $M_{AF}$ is
considered zero outside the $L\times W$ channel region for the calculations
below.
The NEGF method, is a means of attaching contacts to the Schrödinger equation
for performing quantum transport calculations [21, 22]. Moreover, the NEGF
method also enables the modelling of phase-breaking non-coherent processes
withing the self-consistent Born approximation. We use the NEGF method along
with the Landauer-Buttiker formalism for our numerical simulations. In the
regime of coherent transport, the Retarded Green’s function $G_{R}$ for the
channel is defined as,

$G^{R}(E)=((E+\iota\eta)I-H-\Sigma_{L}-\Sigma_{R})^{-1}$ (2)

where, $\eta$ is a small damping parameter, $H$ is the channel (tight-binding)
hamiltonian, $I$ is the identity matrix, $\Sigma_{L(R)}$ are the self energies
describing the left $(L)$ and the right $(R)$ contacts. The advanced Green’s
Function is defined as $G^{A}=(G^{R})^{\dagger}$. Using the contact self-
energies, the broadening matrices are defined as

$\Gamma_{L/R}=\iota(\Sigma_{L/R}-\Sigma^{\dagger}_{L/R})$ (3)

We get the spectral function, $A$ as

$A(E)=\iota[G^{R}(E)-G^{A}(E)]$ (4)

The diagonal elements of the spectral function represent the local density of
states. The electron correlation function matrix is given as

$G^{n}(E)=G^{R}(E)[\Sigma^{in}_{L}+\Sigma^{in}_{R}]G^{A}(E)$ (5)

where, the in-scattering matrices are given as
$\Sigma^{in}_{L/R}(E)=f_{L/R}(E)\Gamma_{L/R}(E)$ using the fermi functions
$f_{L/R}$ of the contacts. The transmission from the left to the right contact
at the energy $E$ is computed as,

$T(E)=Tr(\Gamma_{L}G^{R}\Gamma_{R}G^{n})$ (6)

The conductance is given as $G(E)=e^{2}/h\times T(E)$.

Figure 2: Band structure diagrams for the (a) QSH phase with
$\lambda_{SO}=0.15eV$, $\Delta_{z}=1meV$ and $\lambda_{R}=0$; the (b) SQAH
phase with $\lambda_{SO}=0.15eV$, $M_{AF}=0.201eV$ $\Delta_{z}=0.2eV$, and
$\lambda_{R}=0$. The bandgap for both the phases is $0.298eV$. Figure 3:
Effect of momentum dephasing on the QSH phase: We employ the NEGF dephasing
model with verying dephasing strengths $D$ add IID disorder potentials of
variance $\sigma^{2}$ in the channel, with the relation $D_{0}=\sigma^{2}$ (a)
Conductance for the cases $\lambda_{R}=0$ (blue, dotted) and
$\lambda_{R}=15meV$ (red, filled) for various cases of disorder and dephasing;
(b) Wavefunction density over the channel for a pristine QSH phase (c)
Wavefunction density over the channel for momentum dephasing with
$D_{0}=0.1t^{2}$ Figure 4: Effect of momentum dephasing on the SQAH phase: We
employ the NEGF dephasing model with verying dephasing strengths $D_{0}$ and
add IID disorder potentials of variance $\sigma^{2}$ in the channel, with the
relation $D_{0}=\sigma^{2}$ (a) Conductance for the cases $\lambda_{R}=0$
(blue, dotted) and $\lambda_{R}=15meV$ (red, filled) for various cases of
disorder and dephasing; (b) Wavefunction density over the channel for a
pristine SQAH phase (c) Wavefunction density over the channel for momentum
dephasing with $D_{0}=0.1t^{2}$

Using the NEGF method, one can simply add non-coherent phase-breaking
processes using a ficticious self energy $\Sigma_{D}$ and in-scattering matrix
$\Sigma^{in}_{D}$ [23]. In this work, we concern ourselves with two models:

  1. 1.

Momentum relaxation: We have,

$\displaystyle\Sigma_{D}(i,j)$ $\displaystyle=\bar{D}(i,j)G^{R}(i,j)\equiv
g_{1}(G^{R})$ (7) $\displaystyle\Sigma^{in}_{D}(i,j)$
$\displaystyle=\bar{D}(i,j)G^{n}(i,j)\equiv g_{1}(G^{n})$ (8)

with $\bar{D}(i,j)=D_{0}\times\delta_{i,j}$. This choice of
$(\Sigma_{D},\Sigma^{in}_{D})$ relaxes both phase and momentum. Moreover, this
can be related to random impurity potentials through the equation

$\bar{D}(i,j)\sim\langle U_{D}(i)U^{*}_{D}(j)\rangle$ (9)

where $U_{D}$ denotes the impurity potential. In the simulations, we use the
NEGF dephasing model as well as explicitly add random potentials.

  2. 2.

Spin relaxation: We have,

$\displaystyle[\Sigma_{D}](i,j)$
$\displaystyle=D_{0}\sum_{k\in\\{x,y,z\\}}s_{k}[G^{R}](i,j)s_{k}\equiv
g_{2}(G^{R})$ (10) $\displaystyle[\Sigma^{in}_{D}](i,j)$
$\displaystyle=D_{0}\sum_{k\in\\{x,y,z\\}}s_{k}[G^{n}](i,j)s_{k}\equiv
g_{2}(G^{n})$ (11)

where $[G^{R}](i,j)$ and $[G^{n}](i,j)$ denote $2\times 2$ spin sub-blocks.
This only relaxes the spin and conserves the momentum.

Including both coherent and phase breaking processes, we summarise the
calculations with the following four equations,

$\displaystyle G^{R}$
$\displaystyle=((E+\iota\eta)I-H-\Sigma_{L}-\Sigma_{R}-\Sigma_{D})^{-1}$ (12)
$\displaystyle G^{n}$
$\displaystyle=G^{R}(E)[\Sigma^{in}_{L}+\Sigma^{in}_{R}+\Sigma^{in}_{D}]G^{A}(E)$
(13) $\displaystyle\Sigma_{D}$ $\displaystyle=g(G^{R})$ (14)
$\displaystyle\Sigma^{in}_{D}$ $\displaystyle=g(G^{n})$ (15)

where $g\in\\{g_{1},g_{2}\\}$ depends on the dephasing model under study.
Numerically, these equations are solved self consistently under a stochastic
approximation scheme.
We use the KWANT [27] package in python as the playground for all the
numerical simulations. The lattice structure is defined using KWANT, which
enables access to the self-energies of the contacts. Thereafter, the
subsequent analysis is done using the NEGF method outlined above.

##  III Results and Discussion

In this section, we report the results of our numerical simulations. A Xene
nanoribbon of length $L=30$ atoms and width $W=13$ atoms is used for the
analysis, unless otherwise stated. Moreover, the parameter values used for the
QSH and SQAH phases are summarized in Table 1. Unless explicitly stated, the
parameters in this table are used for the calculations. We first analyze the
effects of momentum dephasing and random potential disorder, and then move on
to spin dephasing.

###  III.1 Momentum Dephasing & Disorder

Parameter | Value | Units
---|---|---
$\lambda_{SO}$ | $0.15$ | $eV$
$\Delta_{z}$ | $1$ | $meV$
$\lambda_{R}$ | $\\{0,15\\}$ | $meV$
$M_{AF}$ | $0$ | $meV$

Parameter | Value | Units
---|---|---
$\lambda_{SO}$ | $0.15$ | $eV$
$\Delta_{z}$ | $0.2$ | $eV$
$\lambda_{R}$ | $\\{0,15\\}$ | $meV$
$M_{AF}$ | $0.201$ | $eV$

Table 1: Parameter values for the numerical calculations performed for the QSH
case (left table; with $\Delta_{z}<\lambda_{SO}$) and the SQAH case (right
table; with $\Delta_{z}>M_{AF}-\lambda_{SO}$). These parameters are used
unless otherwise stated. Further, a hopping value of $t=1eV$ is used.

We study the effects of disorder-induced momentum dephasing on the QSH and the
SQAH phases. We add momentum depahasing using the NEGF model discussed in Sec.
II as well as random impurity potentials to the channel and average over
hundreds of configurations. This acheives the purpose of studying the effect
of dephasing on the topological phases, as well as stressing that the NEGF
dephasing model is a viable alternative to potential averaging for studying
disorder in tight-binding systems, where the latter can be computationally
expensive. We add potential disorder under the independent and identically
distributed (IID) regime, viz., the potential at each site is independent.
Thus, we have $\langle U_{D}(i)U^{*}_{D}(j)\rangle=\langle
U_{D}^{2}\rangle\delta_{i,j}$. Two models for the distribution are used: (i)
Gaussian, $U\sim\mathcal{N}(0,\sigma^{2})$ and (ii) Uniform
$U\sim\text{unif}[-W,W]$; where, $\mathcal{N}(\mu,\sigma^{2})$ denotes a
Gaussian distribution with mean $\mu$ and variance $\sigma^{2}$,
$\text{unif}[-W,W]$ denotes the uniform distribution with support on $[-W,W]$,
and $X\sim p(\cdot)$ denotes a random variable with $X$ with the distribution
$p(\cdot)$. Since the magnitude of $D_{0}$ in the dephasing model depends on
the variance of these distributions, we maintain,

$\sigma^{2}=\frac{W^{2}}{12}\text{ and }\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ D_{0}=\sigma^{2},$ (16)

for consistency of the strength of disorder across the two potential-averaging
and the dephasing method. Note that the variance of a $\text{unif}[-W,W]$
random variable is $W^{2}/12$.

Figure 5: Conductance for the cases $\lambda_{R}=0$ (blue, dotted) and
$\lambda_{R}=15meV$ (red, filled) under momentum dephasing with
$D_{0}=0.01t^{2}$ as a function of the width $W$, measured as the number of
atoms, for the (a) QSH case and the (b) SQAH case

The effect of momentum dephasing and disorder potentials on the conductance of
the QSH phase is outlined in Fig. 3(a). We plot the conductance as a function
of increasing disorder, quantified by the $\sigma$. Moreover, for the
dephasing curves, the relation given in (16) is followed. We also show the
wavefunction plots of the clean QSH phase as well as QSH under dephasing with
$D_{0}=0.1t^{2}$. We note that, for $\lambda_{R}=0$, the QSH phase is starkly
immune to disorder and dephasing with a worst-case decrease in conductance of
$\sim 0.05\%$. The three models agree for QSH with $\lambda_{R}=0$. If we turn
on the Rashba interaction, $\lambda_{R}\neq 0$, we note a worst case
conductance drop of $0.3\%$. While still a miniscule decrease, it is to be
noted that it is nearly a order higher than the previously noted decrease in
conductance without Rashba mixing. Moreover, we note that for high disorder,
the impurity potentials show worse conductance than the dephasing model.
Further, the wavefunction density plots for the clean QSH channel and QSH
under dephasing of strength $D_{0}=0.1t^{2}$, both with $\lambda_{R}=0$ are
shown in Fig. 3(b) and (c) respectively. In these, one observes a uniform
discolouration under disorder, and the decay of the edge states into the bulk
is visible. The plot for the dephased case with $\lambda_{R}\neq 0$ is
similar, and not shown.
The conductance response for the SQAH case is shown in Fig. 4(a). We observe
that for both the $\lambda_{R}$ cases the three models agree reasonably. The
fact that conductance for the $\lambda_{R}\neq 0$ case is lower than that for
$\lambda_{R}=0$ at high disorder is much less pronounced than for the
conductance response of QSH. The wavefunction plots are qualitatively the same
as QSH, shown in Fig. 4(b) and (c), and show discolouration under disorder,
but show a lesser decay into the bulk. The worst case conductance drop is
$0.06\%$, which is similar to the QSH case without Rashba mixing.
It is to be noted that the wavefunction plots only show the density for
currents flowing from the left contact to the right contact, and hence, only
one edge is illuminated for SQAH and both for QSH.

Figure 6: Depiction of degradation caused by momentum relaxation in the
presence of Rashba spin mixing interaction in the QSH phase: The Fermi surface
of radius $k_{F}$ of the 2D sample along with the spin-texture caused by the
Rashba interaction is shown. A momentum relaxation in a spin-down-left-mover,
of order $\Delta k$, results in non-zero overlap with the spin-up-right-mover,
thereby facilitating spin-flipping. Figure 7: Effect of spin dephasing on
topological phases: (a) Conductance as a function of the spin dephasing
strength $D_{0}$ for the QSH (blue, dotted) and the SQAH (red, filled) phases
(b-c) Wavefunction density across the channel for (b) the SQAH phase and (c)
the QSH phase under spin dephasing of value $D_{0}=0.0014t^{2}$ Figure 8:
Interplay of the staggered potential $\Delta_{z}$ and spin dephasing in the
QSH phase: (a) Conductance as a function of the spin dephasing strength
$D_{0}$ for $\Delta_{z}=1meV$ (blue, dotted) and the $\Delta_{z}=0.1eV$ (red,
filled) cases (b-c) Wavefunction density across the channel for (b)
$\Delta_{z}=1meV$ and (c) $\Delta_{z}=0.1eV$ (d-e) Bandstructure diagram of
QSH with (d) $\Delta_{z}=1meV$ and (e) $\Delta_{z}=0.1eV$. The dots on
bandstructure diagram, for the same energy level (horizontal dotted line)
illustrate momentum transfer required for spin-flipping.

The different behaviours of Rashba spin-mixing in the two topological phases
can be understood with the fact that the QSH phase is a helical liquid with
counter-propagating spin-locked edge modes on both sides of the sample, and
the SQAH phase has chiral edge modes, with only one spin direction forming the
edge current. Introduction of the Rashba spin-mixing in an electronic systems
causes a spin-texture to be formed at the Fermi surface [28], as shown in Fig.
6. Now, consider the effect of momentum dephasing coupled with this spin-
texture, as depicted in Fig. 6. A stochastic change in momentum $\Delta k$ of
a left-mover with spin down, can cause it to move to a different momentum
state which is locked with another spin state. The latter now has the
possibility of having a non-zero overlap with the right-mover up-spin state,
hence facilitating backscattering. On the other hand, there is no such
possibility for backscattering in the SQAH phase, as the mode is chiral and
not helical. We elaborate more on this issue in the next subsection.
Now, we study the effect of conductance under worst-case disorder of magnitude
$D_{0}=0.01t^{2}$, which corresponds to $\sigma=0.1t$, with the width of the
channel $W$. The results are reported in Fig. 5. For both the QSH and SQAH
cases, one observes plateaus for $W>25$. The pronounced drop in the QSH case
owing to Rashba mixing can be seen clearly.

###  III.2 Spin Dephasing

We observed overall robustness of the topological phases under non-magnetic
impurities captured by momentum dephasing and disorder potentials. Now, we
will outline the effects of spin disorder on these phases using the NEGF spin
dephasing model ($g=g_{2}$) as discussed in Sec. II.
We show the effects of spin dephasing on the conductance of the QSH and SQAH
phases in Fig. 7(a). These are in stark contrast to the momentum dephasing
results. We observe that the QSH phase is not robust to spin dephasing, and
the QSH conductance drops to near zero for large dephasing values. Whereas,
the SQAH phase remains robust and shows minimal drop in conductance even for
large dephasing values. The wavefunction plots, shown in Fig. 7(b-c) for SQAH
and QSH respectively, support these findings, and show a decay of the QSH
wavefunction across the length of the channel. The SQAH wavefunction appears
to be faded, but is not decaying with the length. Note that spread of the
wavefunction into the bulk is absent here, as compared to momentum dephasing.
Another interesting point is the role of $\Delta_{z}$ in the robustness of QSH
to spin dephasing. For any value of $\Delta_{z}<\lambda_{SO}$ one can
theoretically predict a QSH phase. But, is there any merit to choosing a
particular value? We observe that, for higher $\Delta_{z}=0.1eV$ (less than
$\lambda_{SO}$ to ensure the topological phase) the decay is much lesser than
for a small $\Delta_{z}=1meV$. The conductance response in this case is shown
in Fig. 8(a). The wavefunction density plots, shown in Fig. 8 (b) and (c) for
$\Delta_{z}=1meV$ and $\Delta_{z}=0.1eV$ respectively, illustrate this effect.
We explain this through a careful look at the bandstructures for the two
cases. These are plotted in Fig. 8(d-e). Consider elastic scattering in spin
from the green dot state to the blue dot state, at the same energy for the two
cases of $\Delta_{z}$. For the $\Delta_{z}=1meV$, the spin bands are
overlapping and hence a small momentum transfer is required for the spin-
flipping. For the $\Delta_{z}=0.1eV$ case, the spin bands move horizontally
apart, and hence the spin flip requires a larger momentum transfer. Thus, for
the same amount of disorder the latter case is much more robust to spin-
flipping.
We end this section with a general note about the spin dephasing model, and
the results obtained. With a magnetic impurity in the QSH case, time-reversal
symmetry is broken and backscattering is possible between the counter-
propagating modes at each edge. While in the SQAH case the edge modes are
chiral – there is no direct possibility of backscattering even in the presence
of magnetic impurities. The behaviour of the edge modes thus leads to a clear
contrast in the response of these phases under magnetic disorder, as was seen
in Fig. 7. This is qualitatively similar to how the spin-texture at the Fermi
surface caused by Rashba mixing degrades the QSH case but not the SQAH case.
It has been shown that spin flipping scattering can be caused by non-magnetic
impurities [12] and particular forms of random Rashba spin-mixing interactions
[29]. Our spin dephasing results phenomenologically model a culmination of
such effects, which would concern the robustness of such disorders in real
devices.

##  IV Conclusions

This work was devoted to a computational exploration of topological robustness
against various forms of dephasing. For this, we employed phenomenological
dephasing models using the Keldysh non-equilibrium Green’s function technique
on a model topological device setup on a 2D-Xene platform. Concurrently, we
also explicitly added disorder via impurity potentials in the channel and
averaging over hundreds of configurations. To describe the extent of
robustness, we quantified the decay of the conductance quantum with increasing
disorder under different conditions. Our analysis showed that these
topological phases are robust to experimentally relevant regimes of momentum
dephasing and random disorder potentials. Moreover, we have pointed out a
mechanism which accounts for the relatively worse performance of the QSH phase
under momentum dephasing in the presence of Rashba interaction. We observed
that the QSH phase break downs in the presence of spin dephasing, but the SQAH
remains robust. Although, we showed that the QSH phase, under spin dephasing,
performs better when the magnitude of the perpendicular electric field is
higher; a mechanism for the same in terms of the bandstructures was outlined.
The SQAH phase showed stark robustness under all the dephasing regimes, and
can be potentially employed in realistic device structures for topological
electronics applications.

## Acknowledgements

The authors wish to acknowedge insightful discusions with Supriyo Datta. The
author BM wishes to acknowledge the support by the Science and Engineering
Research Board (SERB), Government of India, Grant No. CRG/2021/003102, and the
Ministry of Human Resource Development (MHRD), Government of India, Grant No.
STARS/APR2019/NS/226/FS under the STARS scheme. The authors SM and BM
acknowledge support of the Dhananjay Joshi Foundation under an Endowment to
IIT Bombay.

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