#  Effect of quintessence on the Nature of Kerr-newman blackhole shadow with
clouds of strings

Gowtham Sidharth M  School of Advanced Sciences,Vellore Institute of
Technology, Chennai - 600048, India  Sanjit Das  School of Advanced
Sciences,Vellore Institute of Technology, Chennai - 600048, India

###### Abstract

In this paper we have took reissner nordstrom blackhole with cloud of strings
and surrounds it with quintessence. we processed the metric through newman
janis algorithm to get its rotating counterpart. the blackhole in study now is
a roating charged blackhole with clouds of string surrounded by quintessence.
we studied its nature of effective potential and unstable photon orbits.
Finally we have plotted the blackhole shadow for various variable profiles.

Charged Particle deflection, dipole magnetic field, Gauss-Bonnet theorem,
gravitational lensing,Jacobi-Randers metric

##  I Introduction

From their early theoretical framework to the latest observerence of the
blackhole key-1 ; key-2 .It has remained the most enigmatic object in the
entire cosmos.General Relativity establishes that the space time at the near
vicinity of the black hole is so warpped that the photon orbits around the
blackhole in an unstable circular orbit. There are plenty of research has been
already done on both rotating and non-rotating caseskey-3 ; key-4 ; key-5 ;
key-6 ; key-7 ; key-8 ; key-9 ; key-10 ; key-11 ; key-12 ; key-13 .where the
shadow is a perfect circle in later case and distorted in the former due to
the frame-dragging effect. A blackhole with presence of charge has also been
studied. In addition to this there are black holes with an addisional dark
energy quintessenisal field which was first proposed by kesilev. the
quintessential field is present to explain the accleratated expansion of our
universe key-14 ; key-15 ; key-16 .

In this paper, Instead of assuming point particles,we consider the the
blackhole to be reside on on dimensional strings key-16 ; key-17 ; key-18 ;
key-19 ; key-20 ; key-21 ; key-22 ; key-23 ; key-24 ; key-25 ; key-26 .A
static spherically symmetric spacetime metric for this black hole, has been
studied in Ref. In our case we considered a Ressiner nordstrom blackhole in a
stingy universe with added quintessential field. using Newman Janis algorithm
we have arrived at a metric for rotating charge blackhole with string cloud in
pressece of quintessential fieldFin

This Section 2 we discuss about this Kerr-Newman-Kiselev blackhole with string
cloud, Section 3 discusses about the photon orbit and effective potential,we
have plotted the blackhole shadow in section 4 and summarized the paper in
section 5.

##  II Kerr - Newman - kiselev BH with string cloud

The metric of Reissner-Nordstrom surrounded by quintessence and cloud of
stringkey-28  is given as,

$\begin{split}ds^{2}&=-\left(1-b-\frac{2M}{r}+\frac{q^{2}}{r^{2}}-\frac{c}{r^{3\omega_{q}+1}}\right)dt^{2}\\\
&+\left(1-b-\frac{2M}{r}+\frac{q^{2}}{r^{2}}-\frac{c}{r^{3\omega_{q}+1}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}\end{split}$
(1)

Following newman janis algorithmkey-27 , we have our metric to be,

$\begin{split}ds^{2}&=-\left(1-\frac{2\rho
r}{\Sigma}\right)dt^{2}+\frac{\Sigma}{\Delta(r)}dr^{2}+\Sigma d\theta^{2}\\\
&-\frac{4a\rho
rsin^{2}\theta}{\Sigma}{dtd\phi}+sin^{2}\theta\left(r^{2}+a^{2}+\frac{2a^{2}\rho
rsin^{2}\theta}{\Sigma}\right)d\phi^{2}\end{split}$ (2)

Where,

$2\rho r\ =r^{2}+a^{2}-\Delta(r)$ (3)

$\Delta(r)=\ \ r^{2}\ +\ a^{2}\ -\ 2\ M\ r\ -\ q^{2}\ +\ c\ r^{1-3\omega}\ +b\
\ r^{2}$ (4)

$\Sigma=r^{2}+a^{2}\cos^{2}{\theta}$ (5)

In the above equations,q being the charge of the blackhole, M and c term
corresponds to mass and quintessence parameter,while a and b denotes the spin
and string cloud parameter. The metric reduces to kerr newman kiselev
blackhole when b = 0 and can be further reducedto kerr newman with b = c = 0
and to kerr with b=c=q=0 and finally to schwarschild by a=b=c=q=0.

Solving $\Delta(r)$ = 0 will give us the event horizon , the plot $\Delta(r)$
vs r below gives us the comparison for different blackholes.

Figure 1: Horizon plots for different blackholes.

##  III Photon Orbits

###  III.1 Unstable Circular Orbit

There is a particular unstable orbit in which a photon revolves around the
blackhole,depending on the conditions of impact parameter the would leave this
orbit to either fell into the blackhole or escape to infinity. The equation of
such orbit can calculated using hamilton-Jacobi Variable separation
methodkey-29 . Hamilton - Jacobi equation in its general form is given as,

$\frac{\partial S}{\partial\lambda}=-\frac{1}{2}g^{{\mu\nu}}\frac{\partial
S}{\partial x^{\mu}}\frac{\partial S}{\partial x^{\nu}}$ (6)

Where $\lambda$ is the affine parameter and S is the action thats defined as,

$S\ =\frac{1}{2}m^{2}\ \lambda\ -\ Et\ +\ L\varphi\ +\ S_{r}\ (r)\ +\
S_{\theta}\ (\theta)$ (7)

where m is the rest mass, E and L are energy and Angular momentum which are
the constants of motion.

Solving the Hamilton jacobi equation will yield,

$\frac{\partial S_{r}}{\partial r}=\frac{\sqrt{R(r)}}{\Delta(r)}$ (8)

$\frac{\partial S_{\theta}}{\partial\theta}=\sqrt{\Theta(\theta)}$ (9)

Where ,

$R(r)=(aL\ -\ (a^{2}\ +\ r^{2}\ )E)^{2}\ -\ (K\ +\ (aE\ -\ L)^{2}\
)\mathrm{\Delta}$ (10) $\Theta(\theta)=K\
-\left(\frac{L^{2}}{\text{si}n^{2}\theta}\ -\ a^{2}\text{\ E}\
^{2}\right)Cos\theta$ (11)

where K is the the seperation constant. The trajectory of a photon is
calculated with two impact parameter,

$\xi=\frac{L}{E}$ $\eta=\frac{K}{E^{2}}$

Rewritting R(r) and $\Theta(\theta)$ in terms of impact parameters

$R_{p}(r)=(a\ \xi\ -\ (a^{2}\ +\ r^{2}\ ))2-(\eta\ +\ (a-\xi)^{2}$ (12)
$\Theta_{p}(\theta)\ =\ \eta\ -\left(\frac{\xi^{2}}{\text{si}n^{2}\theta}-\
a^{2}\ \right)cos^{2}\theta$ (13)

The equation of geodesic motion can be written the form

$\Sigma\ t^{\prime}=\frac{((r^{2}+\ a^{2}\ )E\ -\ aL)(r^{2}\ +\ a^{2}\
)}{\Delta(r)}a(aESin^{2}\ (\theta\ )\ -\ L)$ (14) $\Sigma\ r^{\prime}\
=\sqrt{R(r)}$ (15) $\Sigma\ \theta^{\prime}=\sqrt{\Theta(\theta)}$ (16)
$\Sigma\ \varphi^{\prime}=\frac{a((r^{2}\ +\ a^{2}\ )E\ -\
aL)}{\Delta(r)}-\frac{\text{aESi}n^{2}\ (\theta\ )\ -\ L}{\text{Si}n^{2}\
(\theta\ )}$ (17)

##  IV Effective Potential

The relationship between r’ and effective potential is given as,

$V_{\text{eff}}=\frac{E^{2}-1}{2}-\frac{1}{2}r^{\prime}$ (18)

Effective potential is the maximum potential energy for a photon to have zero
radial velocity and zero radial acceleration and there by determining the
photons’s stability.

Figure 2: $V_{eff}$ vs r for varying q with M = 1; e = 1; k = 4; L = 2.1; a =
1; c = b = 0.1;. Figure 3: $V_{eff}$ vs r for varying c with M = 1; e = 1; k =
4; L = 2.1; a = 1; b = 0.1;q=0.4. Figure 4:  $V_{eff}$vs r for varying b with
M = 1; e = 1; k = 4; L = 2.1; a = 1; c = 0.1; q=0.4.

From figure 2 - 4 , we have plotted effective potential against r for various
profiles such as by varying charge 1, quintessence parameter c and string
parameter b. In fig 2, the maxima of the effective potential decreases with
increase in charge, we can also notice that the peak is shifted toward the
left as charge is being increased. In Fig 3, the maxima increases with
increase in quintessense value and peak is shifted left. Similarly in fig 4,
the maxima increases with string parameter b and peak shifting towards to left
can be seen here too.

##  V Shadows

For a photon to have spherical orbit,it should have null radial velocity and
null radial acceleration,which means that,

$R_{p}(r)=\ 0$ $\frac{\text{d\ }R_{p}}{\text{d\ r}}\ =\ 0$

With this we can rewrite the impact parameter as,

$\xi=\frac{-4r\mathrm{\Delta}(r)\ +\ a^{2}\ \mathrm{\Delta}^{\prime}(r)\ +\
r^{2}\ \mathrm{\Delta}^{\prime}(r)}{a\ \mathrm{\Delta}^{\prime}(r)}$ (19)
$\eta=\frac{r^{2}\ (16a^{2}\ \mathrm{\Delta}(r)\ -\ 16\mathrm{\Delta}^{2}(r)\
+\ 8r\mathrm{\Delta}(r)\ \mathrm{\Delta}^{\prime}(r)\ -\ r^{2}\
\mathrm{\Delta}^{\prime 2}(r)\ )}{a^{2}\mathrm{\Delta}^{\prime 2}(r)}$ (20)

The Shape of the black hole depends on the celestial coordinates $\alpha$ and
$\beta$ , which is given as

$\alpha=-{r_{0}}^{2}Sin\theta\frac{d\phi}{dr}$ (21) $\beta=\
-{{r_{o}}^{2}\frac{d\theta}{\text{dr}}}$ (22)

Where $\alpha$ and $\beta$ has limit $r_{o}$ goes to infinity,using the
geodesic equation the celestial coordinates can be written in terms of impact
parameter.

$\alpha=\frac{-\xi}{sin(\theta)}$ (23)

$\beta=\sqrt{\eta+a^{2}Cos^{2}(\theta)-\xi^{2}cot^{2}(\theta)}$ (24)

The shadow can we obtained by plotting $\alpha$ vs $\beta$ in the equatorial
plane.

Figure 5: Shadow plot fot KNK - cloud string black hole for variation in spin
parameter(left) snd charge(right) Figure 6:  Shadow plot fot KNK - cloud
string black hole for variation in quintessense (left) snd string
parameter(right).

##  VI conclusion

In this paper we have took reissner nordstrom blackhole with cloud of strings
and surrounds it with quintessence. we processed the metric through newman
janis algorithm to get its rotating counterpart. the blackhole in study now is
a roating charged blackhole with clouds of string surrounded by quintessence.
we studied its nature of effective potential and unstable photon orbits. we
have observed that the effective potential maxima is directly proportional to
the increase in charge,quintessence and string parameter and the peak seems to
incline towards the left.

Finally, we have plotted the blackhole shadow for various variable
profiles.from figure 5, As expected the shadow shape starts to distort as we
go for higher spin values. it also seems to a distorted circles for higher
values of charge but upon closer observation,the circles are in deed
concentric proving that only size of the shadow decreases with increase in
charge. From figure 6, It is evident that the size of the shadow decreases
with increase in quintessence and string parameters.

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# Instability of QBC systems to Topological Anderson Insulating phases

Nicolau Sobrosa1, Miguel Gonçalves2, Eduardo V. Castro1,3 1Centro de Física
das Universidades do Minho e Porto, Departamento de Física e Astronomia,
Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal
2CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais,
1049-001 Lisboa, Portugal  3Beijing Computational Science Research Center,
Beijing 100084, China

###### Abstract

Here we study the instabilities of a quadratic band crossing system to Chern
insulating states and uncorrelated disorder. We determined the phase diagram
in the plane of topological mass versus disorder strength, characterizing the
system with respect to spectral, localization and topological properties. In
the clean limit, the system has two gapped Chern insulating phases with Chern
numbers $C=\pm 2$, and a trivial phase with $C=0$. For finite disorder, the
quadratic band crossing points are unstable to emergent gapless Chern
insulating phases with $C=\pm 1$, not present in the clean limit. These phases
occupy a considerable region of the phase diagram for intermediate disorder
and show features of topological Anderson insulators: it is possible to reach
them through disorder-driven transitions from trivial phases.

##  I Introduction

Topological insulators are a remarkable state of electronic matter. They show
quantized responses that are proportional to topological invariants and, as a
consequence, typically very robust to perturbations and system’s details
(Hasan and Kane, 2010; Qi and Zhang, 2011; with Taylor L. Hughes, 2013; Chiu
et al., 2016). Topological band insulators, as paradigmatic examples of
systems with non-trivial topology, have been extensively studied and are
fairly well understood (with Taylor L. Hughes, 2013). Non-trivial topology
however also manifests in systems with broken translational invariance that
are not described by topological band theory. Among these systems, disordered
topological insulators are a popular sub-group (Wu et al., 2016).

Topological phases are robust to disorder regarding that no symmetry
protecting the topological properties is broken (Altland and Zirnbauer, 1997;
Xiao et al., 2010). In quantum Hall insulators, disorder even plays a
fundamental role for the observation of a quantized Hall conductance. In the
case of Chern insulators where time-reversal symmetry is broken, disorder
localizes every eigenstate except at specific energies (Kramer and MacKinnon,
1993; Onoda and Nagaosa, 2003; Onoda et al., 2007; Castro et al., 2015, 2016).
The extended eigenstates at these energies carry finite Chern numbers and are
therefore responsible for a quantized Hall response as long as the Fermi level
lies between them (the localized states cannot change this response).
Topological phase transitions in disordered Chern insulators occur when the
extended states merge and annihilate at the Fermi level (through the so-called
“levitation and annihilation” mechanism), becoming localized (Onoda et al.,
2007).

Quite surprisingly, in some systems topological phase transitions between
trivial and topological phases can occur by increasing disorder. These
disorder-driven topological phases are now known as topological Anderson
insulators (TAI) (Li et al., 2009; Groth et al., 2009). This phase has been
observed in many models, including paradigmatic models of topological
insulators in two-dimensions (2D) such as the Kane-Mele model (Kane and Mele,
2005; Orth et al., 2016) and the disordered Haldane model (Song et al., 2012;
García et al., 2015; Gonçalves et al., ).

Systems with quadratic band crossings (QBC) in 2D are also very interesting
because, contrary to conventional band degeneracy points, they are associated
with a finite Berry phase of $\pm 2\pi$. Due to the finite density of states
at QBC, these systems are unstable to interactions (Sun et al., 2009):
originating nematic phases with two Dirac cones, each carrying half of the
QBC’s Berry phase; or gap openings that may give rise to topological
insulating phases precisely due to the non-trivial Berry phase of the QBC
(Uebelacker and Honerkamp, 2011; Murray and Vafek, 2014; Ray et al., 2018;
Zeng et al., 2018). The fate of interaction induced topological insulating
phases in the presence of disorder has been examined in Refs. (Wang et al.,
2017; Dong et al., 2020) within the one-loop RG approach. The suppression of
topological phases under increasing disorder and possible transition to
trivial phases was predicted.

In this work we study the interplay between the instability of a QBC system to
a Chern insulating state and to disorder of the Anderson type. The full phase
diagram in the plane of the gap opening coupling parameter $B$ and disorder
strength $W$ is shown in Fig. 1. For null disorder, QBC ponits (QBCP) occur
for $B=-2,0$, otherwise, the system is a gapped trivial insulator or a Chern
insulator with Chern number $C=\pm 2$. These phases are also present for
finite disorder, but new gapless Chern insulating phases also emerge. Besides
gapless and gapped phases with $C=\pm 2$, new gapless topological phases with
$C=\pm 1$, not present in the clean limit, arise. In fact, the most important
result of our work is that the QBCP are unstable to the formation of these
phases, for any finite disorder. Finally, the existence of TAI phenomena is
also clear: it is possible to undergo a transition between the trivial phase
and the disorder-induced topological phases with $C=\pm 1$ by increasing
disorder.

Figure 1: Phase diagram in the $B-W$ plane. The Chern number was calculated
for a system with $35\times 35$ lattice points and was averaged for $100$
disorder realizations. The gray areas correspond to the gapped regions
obtained through calculations of the DOS. The error bars for the Chern number
were obtaining by fixing the disorder intensity and varying $B$. The red boxes
indicate the results for the critical points calculated with the TMM. The size
of the bar is determined in such a way that outside them the system is clearly
localized ($\Lambda_{M}$ decreases with M, see Fig. 5).

##  II Model and methods

We study a QBCP system realized in the square lattice with two orbitals per
site. The model considers first-neighbor hoppings between the same orbitals
and second-neighbors hoppings connecting different orbitals. The Hamiltonian
for the disorder-free model can be written in the reciprocal space as

$H_{0}=\sum_{\mathbf{k}}\bm{\Psi_{k}^{\dagger}}\mathcal{H}_{k}\bm{\Psi_{k}}\,,$
(1)

where $\bm{\Psi_{k}}=\left(\mid 1,\mathbf{k}\rangle,\mid
2,\mathbf{k}\rangle\right)$ is the two component spinor of the two orbitals
and

$\mathcal{H}_{k}=\mathbf{h}\cdot\bm{\sigma}\,,$ (2)

with $\bm{\sigma}$ the Pauli vector and the vector $\mathbf{h}$ given by:

$\displaystyle h_{x}$ $\displaystyle=2t_{x}\sin k_{x}\sin k_{y}$
$\displaystyle h_{y}$ $\displaystyle=0$ $\displaystyle h_{z}$
$\displaystyle=t_{z}\left(\cos k_{x}-\cos k_{y}\right).$ (3)

In the following, we set $t_{x}=t_{z}=t$ and $t$ to unity.

This model has two QBCP: at $\Gamma=\left(0,0\right)$ and
$M=\left(\pm\pi,\pm\pi\right)$. The term $h_{y}$ is necessarily zero for a
QBCP to exist. It is possible to break time-reversal symmetry by adding a
finite $h_{y}$, that also opens a gap. For a constant $h_{y}$, the system is a
trivial insulator. Similarly to the Haldane Model (Haldane, 1988) we add a
$k-$dependent component which allows us to tune independently the gap at each
QBCP. We used a simple choice that is dependent of a unique parameter, $B$:

$h_{y}=1+\frac{B+1}{2}\left(\cos k_{x}+\cos k_{y}\right).$ (4)

In real-space, the constant term corresponds to a intracell complex hopping
between different orbitals and the term with $k-$modulation to a nearest
neighbor complex hopping between different orbitals. As the modification in
Eq. 4 does not change $h_{x}$ and $h_{z}$, a QBCP still exists when $h_{y}=0$
as seen before. For $B=0$, there is a QBCP at $M$ and for $B=-2$, there is a
QBCP at $\Gamma$.

The disorder potential is easily added through:

$H=H_{0}+\sum_{i}\xi_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i}\>,$ (5)

where $\xi_{i}$ are site-dependent potentials that follow the uniform
distribution,

$P_{W}\left(\xi_{i}\right)=\frac{1}{W}\Theta\left(\left|\xi_{i}\right|-\frac{W}{2}\right)\,,$
(6)

where $W$ defines the disorder strength.

We carried out a complete study of the phase diagram of the model in Eq. 5,
characterizing spectral, topological and localization properties. The density
of states (DOS) was calculated for finite systems containing more than
$10^{6}$ sites using the Kite quantum transport software that has a very
efficient implementation of the kernel polynomial method (KPM) (João et al.,
2020). The topological phase diagram was obtained by computing the Chern
number through the Coupling Matrix Method introduced in Ref. (Fukui et al.,
2005). The localization properties were characterized through the Transfer
Matrix Method (TMM) (Mackinnon and Kramer, 1981; MacKinnon and Kramer, 1983;
Hoffmann and Schreiber, 2012), that also allowed to cross-check the Chern
number results. This method considers a finite system with a large
longitudinal width $L$ and a transverse length $M$ which we varied in order to
find the localization length in terms of $M$, $\lambda_{M}$. With the goal of
obtaining the localization properties we studied the normalized localization
length, $\Lambda_{M}=\lambda_{M}/M$, in the following way: if $\Lambda_{M}$
decreases with $M$ the states are localized in the thermodynamic limit which
correspond to a insulating behaviour; on the other hand, if $\Lambda_{M}$
increases with $M$ the states are extended. Finally, a constant $\Lambda_{M}$
is characteristic of critical states that appear at transition points between
different phases. We choose $L$ in order to obtain $\lambda_{M}$ with less
than $1\%$ error. It is clear that the behaviour of $\Lambda_{M}$ can capture
topological phase transitions at finite disorder. As mentioned in the
introduction, the spectrum of finite-disorder Chern insulators consists of
localized states, except at specific energies where extended critical states
live. Topological phase transitions occur when these states cross (and merge
at) the Fermi level. Therefore, for a disordered Chern insulator,
$\Lambda_{M}$ should always decrease with $M$ except at the topological phase
transitions, where it becomes $M$-independent.

##  III Results

###  III.1 Topological Properties

In this section we present details on the topological phase diagram of Fig. 1.
The different colors indicate different Chern Numbers and the black lines
represent the topological transitions. For null disorder the system undergoes
a transition from $C=\pm 2$ to $C=0$ at the points where a QBCP appears. For
finite disorder, the phases with $C=0,\pm 2$ survive and a new gapless phase
with $C=\pm 1$ appears. The latter phases are TAI as it is possible to reach
them by increasing disorder from a topologically trivial phase, at fixed $B$.
For large enough disorder, all topological phases are suppressed, in agreement
with one-loop RG calculations (Wang et al., 2017).

Figure 2: Chern number results obtained at fixed $W$ and variable $B$. The
results were averaged over 120 disorder realizations. Each color depicts three
examples of disorder intensities used. The full black curve represents the
topological phases of the system calculated for null disorder where it is very
clear that no $C=\pm 1$ phases exist.

To obtain the transition lines, $W$ was fixed at some value and $B$ was varied
continuously. For each disorder strength we performed an average over $120$
different disorder configurations with $35\times 35$ sites. Some examples of
the obtained curves are represented in the Fig. 2 including the values for the
clean limit. The continuous variation of the chern number is expected to
disappear in the thermodynamic limit, where transitions between different
Chern numbers should become sharp. The errors that appear in Fig. 1 are
determined by the curves of the Fig. 2. The error bars were considered to be
the range over which the Chern number was more than $5$% away from its integer
value.

It is not clear from the phase diagram in Fig. 1 if the finite-disorder Chern
insulating phases with $C=\pm 1$ exist for any infinitesimal disorder. To
understand if this is the case, we fixed $B=0$ (QBC system for $W=0$) and
computed the Chern number as a function of $W$ for different system sizes. The
results are shown in Fig. 3, where it is clear that the $C=-1$ phase occurs
for disorder strengths that approach $W=0$ as the system size is increased.
These results suggest that in the thermodynamic limit the $B=0$ QBC point is
unstable to the formation of the $C=-1$ phase for any infinitesimal disorder.
The same is expected for the QBC point at $B=-2$ as suggested by the symmetry
of the phase diagram around $B=-1$.

Figure 3: Topological phases as function of disorder for $B=0$ (QBC at $W=0$)
and variable system sizes with $N$ unit cells in the direction of each lattice
vector.

###  III.2 Gapped/Gapless Regions

Figure 4:  Examples of DOS calculated for the points A,B,C marked in Fig. 1.
The inset shows a zoom around the Fermi level which corresponds to $E=0$ at
half-filling. $20$ disorder realizations were always performed and a system of
$1024\times 1024$ sites was considered.

In order to study the existence of a spectral gap at the Fermi level ($E=0$ at
half-filling) we computed the DOS using the kernel polynomial method
implemented in Kite (João et al., 2020). In Fig. 1 is present the
gapped/gapless transitions. The system was considered gapped when the DOS at
the Fermi Level was below a certain threshold, $\rho_{\text{cut}}$, that was
determined by exact diagonalization of the Hamiltonian in Eq. (5). In Appendix
C we show the results that support our choice. The error in the transition
from gapped to gapless corresponds to a variation of $\pm 25\%$ of that
criterion. This error corresponds to the thickness of the gapped-gapless
transition lines in Fig. 1. The small error shows that variations in the
criterion do not affect significantly the results, especially in the regions
of the topological phase transitions.

###  III.3 Localization

Figure 5: Normalized localization length $\Lambda_{M}$ as funtion of system
transversal size $M$.

We studied the normalized localization length $\Lambda_{M}$ obtained through
the TMM, at the Fermi level ($E=0$). $\Lambda_{M}$ always decreases with $M$,
except at specific points. This means that, as expected, all states at the
Fermi level are localized except when a topological phase transition occurs
accompanied by the merging of extended critical states carrying opposite Chern
numbers. Example results are shown in Fig. 5. The results in Fig. 5(a) are for
$W=5$ and a range of $B$ containing the $C=2$/$C=1$ and $C=1$/$C=0$
transitions. Fig. 5(b) corresponds to a cut at $W=8$, for a range of $B$
containing all the topological phase transitions. The phase transition points
were considered to be those with constant $\Lambda_{M}$. The uncertainty in
these points was considered to be the range of $B$ over which there isn’t an
undoubtful decrease of $\Lambda_{M}$ with $M$. This uncertainty is represented
in Fig. 1 by the shaded red bars, being perfectly bounded by the error bars in
the Chern number.

##  IV Discussion

The phase diagram of the QBC system here unveiled shows many features that are
characteristic of well-known disordered topological insulators (Onoda and
Nagaosa, 2003; Onoda et al., 2007; Castro et al., 2015, 2016; Li et al., 2009;
Groth et al., 2009; Orth et al., 2016; Song et al., 2012; García et al., 2015;
Gonçalves et al., ; Liu et al., 2020; Stützer et al., 2018; Song et al., 2015;
Su et al., 2016; Kuno, 2019; Hsu and Chen, 2020; Wu and Liu, 2021), such as
the existence of robust finite-disorder gapped and gapless topological
insulating phases. However, there is a feature that distinguishes it from the
previously explored models: the existence of new disorder-driven topological
phases with $C=\pm 1$, absent in the clean limit. Our results suggest that the
existence of a QBCP is a key ingredient for the formation of these phases. In
particular, the finite-size scaling analysis in Fig. 3 is a strong indication
that they are instabilities of the QBCPs: in the thermodynamic limit, any
infinitesimal amount of disorder should drive the QBC system to one of these
phases. Furthermore, from the phase diagram in Fig. 1, it is clear that for
low disorder the new topological phases are located around the clean-limit
QBCP.

The new phases with $C=\pm 1$ are also TAI since they can be reached by
increasing disorder from a trivial phase. TAI phenomena were by now observed
in a multitude of disordered topological insulators. However, there is an
important difference from the conventional TAI in the present case: the $C=\pm
1$ TAI phases have a Chern number that does not exist in any zero-disorder
topological phase of the model. Therefore, these TAI phases do not evolve
smoothly from the clean-limit phases as disorder is increased, in contrast to
conventional TAIs.

A possible variation of the QBC system is to split the QBCP into two Dirac
cones with a suitable perturbation (see SM). Even in this case, a finite
disorder gives rise to the $C=\pm 1$ phases which could be an argument against
the importance of the QBCP for their existence. Nonetheless, the topological
information carried by the split Dirac cones and the QBCP is the same since no
gap is opened in the splitting process. The topological properties of the new
system can then be traced back to the possibility of creating a QBCP without
closing the gap.

Finally, the results obtained here are not restricted to disorder of the
Anderson type. We also obtained the phase diagram for binary disorder, being
qualitatively similar to the phase diagram for Anderson disorder (see SM).

##  V Conclusions

We have studied a model of a QBC system under gap-opening and disorder-
inducing couplings. A complete spectral, topological and localization analysis
was carried out in order to obtain a detailed phase diagram. We not only found
that the topological phases existing in the clean limit were robust to
disorder but also that new gapless topological phases were formed. Most
importantly, we found a new instability of the QBCP: a disorder-induced
instability to gapless topological phases with Chern numbers $C=\pm 1$, that
are absent in the clean limit.

An interesting question for future work is whether instabilities of the QBCPs
to electron-electron interactions can give rise to topological phases with
similar properties to the gapless topological insulators here uncovered. The
full phase diagram capturing the interplay between disorder and interactions
would then be a natural follow-up.

##  Appendix A Evolution of Phase Diagram with $B_{x}$ constant perturbation

To study the robustness of the QBCP we introduce the term
$\bm{b}\cdot\bm{\sigma}$ in the Hamiltonian written in the reciprocal space
where $\bm{b}$ can be a constant vector with three components,
$\bm{b}=$$\left(b_{x},b_{y},b_{z}\right)$. The $b_{x}$ perturbation lifts the
degeneracy of the QBCP, splitting it into two Dirac Cones. In fig. 6, we show
the value of the Chern number in the plane $B$ vs $b_{x}$ with increasing
disorder. For a small $b_{x}$ the phase diagram of the fig. 2 is practically
unchanged. It is also clear from the figure that the $C=\pm 1$ phases exist
even when $b_{x}\neq 0$.

Figure 6: Evolution of phase diagram in the $B-b_{x}$ plane. The black lines
correspond to the phase transition for null disorder corresponding to the
points where there are Dirac Cones or QBCPs in the spectrum. All points were
averaged with 100 disorder configurations for systems of size $15\times 15$.

##  Appendix B Qualitative Phase Diagram for Binary Disorder

In this section, we present the phase diagram for binary disorder. In this
case, the on-site potentials $\epsilon_{i}$ follow the following distribution:

$P_{V}(\varepsilon_{i})=\frac{1}{2}\left(\delta\left(\varepsilon_{i}\right)+\delta\left(V-\varepsilon_{i}\right)\right)$

Figure 7: Qualitative description of the phase diagram with Binary disorder.
All points were averaged with 200 disorder configurations with systems of size
$15\times 15$

The Chern number results are shown in fig. 7. Qualitatively, the phase diagram
is very similar to the one obtained for Anderson disorder in Fig. 1. In
particular, the disorder-induced topological phases are still present and the
reentrant TAI behaviour is also observed. However it is needed a smaller
degree of disorder to destroy all the non-trivial phases. Besides this,, for
large $B$ (absolute value), the $C=\pm 1$ phases are much narrower with binary
disorder which means that they are more robust to Anderson disorder.

##  Appendix C Gapped/gapless regions

In this section, we present details on the criterion used in the main text to
distinguish gapped and gapless regimes.

ss

Figure 8: Difference from the two states directly above and under the Fermi
Level, computed using exact diagonalization for different systems with $N$
sites.

Due to the finite resolution of kernel polynomial methods (KPM), it is
typically challenging to find the transition point between gapped and gapless
regimes. In particular, the DOS obtained through KPM may have a finite
spectral weight at energies within gaps. We therefore use finite-size scaling
results from exact diagonalization to find a suitable $\rho_{{\rm cut}}$ below
which the KPM DOS should be considered null. For the system to be gapless the
difference between the states immediately above and under the Fermi level must
converge to 0 in the thermodynamic limit. To define $\rho_{{\rm cut}}$, we set
the parameter $B=-1$. After fitting the finite-size scaling results to a cubic
function and extrapolating to $N\rightarrow\infty$, we observe that for a
disorder around $W\approx 4.2$ the system must be gapless. It was, then, just
a matter of computing the DOS with Kite (João et al., 2020) for $B=-1$ and
that critical disorder and then fix the $\rho(E=0)$ to be the $\rho_{cut}$.

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