# Challenges in detecting topological superconducting transitions via
supercurrent
and phase probes in planar Josephson junctions

Pankaj Sharma  Department of Physics, Indian Institute of Technology Roorkee,
Roorkee 247667, India  Narayan Mohanta  Department of Physics, Indian
Institute of Technology Roorkee, Roorkee 247667, India
<EMAIL_ADDRESS>
###### Abstract

Topological superconductors harbor, at their boundaries and vortex cores,
zero-energy Majorana bound states, which can be the building blocks in fault-
tolerant topological quantum computing. Planar Josephson junctions host such
topological superconducting phases, highly tunable by external magnetic field
or phase difference between the superconducting leads. Despite many
theoretical and experimental studies, the signatures of the transition to a
topological superconducting phase, based on minima in the critical
supercurrent $I_{c}$ flowing across the junction, $0$-$\pi$ transition in the
ground state junction phase and their anisotropic magnetic-field response have
remained unsettled. Using rigorous numerical calculations with several
experimentally-relevant parameter settings, we show that $I_{c}$ and
$\varphi_{{}_{\rm GS}}$ cannot indicate unambiguously topological transition
in any realistic planar junctions. Furthermore, the anisotropic variations of
$I_{c}$ and $\varphi_{{}_{\rm GS}}$ with in-plane magnetic field appear in
junctions that are undoubtedly in trivial superconducting phase, raising
concerns on the effectiveness of these probes in identifying topological
transitions in planar junctions. We discuss possible strategies to confirm a
topological superconducting phase in these platforms.

##  I Introduction

Conclusive identification of a topological superconducting phase hosting zero-
energy Majorana bound states (MBS), is currently one of the central problems
in condensed matter physics. The MBS, because of their non-Abelian statistics,
are promising for building qubits for fault-tolerant topological quantum
computing Kitaev (2001, 2003); Nayak  _et al._ (2008); Alicea  _et al._
(2011); Alicea (2012). Extensive efforts in the past years in various
geometries with strong Rashba spin-orbit coupling (RSOC), including
semiconductor-superconductor nanowires Sau  _et al._ (2010); Oreg  _et al._
(2010); Lutchyn  _et al._ (2010); Mourik  _et al._ (2012); Das  _et al._
(2012); Deng  _et al._ (2016); Nichele  _et al._ (2017), magnetic adatoms-
superconductor interfaces Nadj-Perge  _et al._ (2013, 2014); Li  _et al._
(2016); Feldman  _et al._ (2017); Mohanta  _et al._ (2018), topological
insulator-superconductor interfaces Fu and Kane (2008); Chiu  _et al._ (2011);
Xu  _et al._ (2015) and oxide heterostructures Mohanta and Taraphder (2014);
Kuerten  _et al._ (2017), revealed zero-bias conductance peak (ZBCP) signature
to support the presence of the MBS. However, the challenge has been the
disentanglement of the MBS from other low-energy quasiparticles, primarily
Andreev bound states and impurity-induced states Kells  _et al._ (2012); Prada
_et al._ (2012); Stanescu  _et al._ (2012); Rainis  _et al._ (2013); Cayao
_et al._ (2015); Liu  _et al._ (2017); Suominen  _et al._ (2017); Vuik  _et
al._ (2019); Pan and Das Sarma (2020); Das Sarma and Pan (2021). Therefore, to
achieve the sought-after braiding of the MBS, and to overcome the inherent
instabilities in one-dimensional nanowire networks, two-dimensional platforms
such as the planar Josephson junctions were introduced Hell  _et al._ (2017a);
Pientka  _et al._ (2017); Shabani  _et al._ (2016); Nichele  _et al._ (2020);
Wu  _et al._ (2020); Alidoust  _et al._ (2021); Oshima  _et al._ (2022).
A planar Josephson junction consists of a two-dimensional electron gas (2DEG),
proximity-coupled to two superconducting leads (shown schematically in Fig.
1(a)). It has the additional advantage that the induced topological
superconductivity can be easily controlled by the phase difference between the
superconducting leads (tunable externally by applying a perpendicular magnetic
field). Transition to a topological superconducting phase in these junctions,
revealed by the emergence of the zero-energy MBS, appears at a critical value
of an in-plane magnetic field, applied along the length of the non-
superconducting channel when the phase difference between the superconducting
leads is close to $\pi$ Pientka  _et al._ (2017); Hell  _et al._ (2017b). This
topological superconducting transition has been suggested to be accompanied in
general by a minimum in the critical supercurrent $I_{c}$ and a jump from
nearly $0$ to nearly $\pi$ in the ground state phase $\varphi_{{}_{\rm GS}}$
Pientka  _et al._ (2017). Following this theoretical proposal, successive
experiments reported the ZBCP signature of the MBS within a range of magnetic
fields, predicted by the minima in $I_{c}$ Ren  _et al._ (2019); Fornieri  _et
al._ (2019). Further theoretical work suggested that for narrow
superconducting leads ($W_{\rm SC}\lesssim\xi$, where $\xi$ is the
superconducting coherence length), as used in the experiments, the minima in
$I_{c}$ do not necessarily indicate topological phase transitions Setiawan
_et al._ (2019). However, a subsequent study, with joint experimental and
theoretical efforts, concluded a $\pi$ phase jump in the junction phase and a
simultaneous minimum in $I_{c}$ at a critical magnetic field to be a signature
of topological superconducting transition Dartiailh  _et al._ (2021a). The
conclusions were further supported by the evidence of anisotropic response of
$I_{c}$ and $\varphi_{{}_{\rm GS}}$ with respect to the in-plane magnetic
field. Despite intensive analysis to exclude possible non-topological origins,
it remains unclear whether the correspondence between these critical fields
can be generalized for different junction dimensions and realistic parameters
such as RSOC strength. Moreover, these conflicting results created confusion
regarding the viability of these quantities ($I_{c}$ and $\varphi_{{}_{\rm
GS}}$) as reliable indicators of topological superconducting transition in
these planar Josephson junctions.

Figure 1: (a) A schematic of the planar Josephson junction, created by placing
two superconducting leads on top of a 2DEG based on a semiconducting quantum
well. Red markers show the regions in the metallic channel where MBS are
localized in the topological phase. (b) Energy eigenvalues versus their index
$n$ at $B\\!=\\!1.5$ T, $\varphi\\!=\\!\pi$ and $\mu\\!=\\!-0.4$ meV, showing
two eigenvalues at zero energy (in red), protected from the other states by an
energy gap. (c) Local density of state profile (normalized) for the lowest
positive energy eigenstate, revealing the localization of the two MBS at the
ends of the metallic channel (shown by the grey lines). (d), (e) Profiles of
charge density of states at $B\\!=\\!1.5$ T (MBS present) and at $B\\!=\\!0.5$
T (MBS absent), showing partial charge neutrality in the presence of MBS
(colorbar scales show two orders of magnitude reduction in the charge density
in (d)). The junction parameters are $L_{x}\\!=\\!0.5~{}\mu$m,
$L_{y}\\!=\\!2~{}\mu$m and $W\\!=\\!0.04~{}\mu$m.

In this paper, we show that the observables $I_{c}$, $\varphi_{{}_{\rm GS}}$,
and their anisotropic magnetic-field response cannot predict a topological
superconducting transition in any realistic planar Josephson junction,
irrespective of $W_{\rm SC}\lesssim\xi$ or $W_{\rm SC}>\xi$. We confirm these
findings using the presence (absence) of the zero-energy MBS as the direct
indicator of the topological (trivial) phase, in our numerical calculations
which were performed using various experimentally-relevant parameter settings
and device geometries. Under realistic conditions, $\varphi_{{}_{\rm GS}}$
exhibits a smooth monotonic variation with in-plane magnetic field rather than
a sharp jump. Even though, the critical fields for the minima in $I_{c}$ may
coincidentally match with the critical field for the topological
superconducting transition, we could not establish a general correspondence
between the two critical fields. We also find that the anisotropy in $I_{c}$
and $\varphi_{{}_{\rm GS}}$ with respect to the in-plane magnetic field cannot
conclusively distinguish a topological superconducting phase from a trivial
one. We discuss possible braiding realization of MBS as a prerequisite to
conclude topological superconductivity in a multi-terminal planar junction.

##  II Theoretical Model

The planar Josephson junction devices used in the experiments Ren  _et al._
(2019); Fornieri  _et al._ (2019); Dartiailh  _et al._ (2021a), are made of a
2DEG, created at a InAs/HgTe quantum well, and Al superconducting leads. To
describe such a 2DEG with proximity-induced superconductivity underneath the
superconducting leads, we use the below Hamiltonian

$\displaystyle\mathcal{H}$
$\displaystyle=\sum_{i,\sigma}(4t-\mu)c^{\dagger}_{i\sigma}c_{i\sigma}-t\sum_{\langle
ij\rangle,\sigma}(c^{\dagger}_{i\sigma}c_{j\sigma}+{\rm H.c.})$
$\displaystyle+\sum_{i}(\Delta_{i}c^{\dagger}_{i\uparrow}c^{\dagger}_{i\downarrow}+{\rm
H.c.})-\dfrac{g^{*}\mu_{{}_{\rm
B}}B}{2}\sum_{i,\sigma,\sigma^{\prime}}(\sigma_{y})_{\sigma\sigma^{\prime}}c^{\dagger}_{i\sigma}c_{i\sigma^{\prime}}$
$\displaystyle-\dfrac{\mathbf{i}\alpha}{2a}\sum_{\langle
ij\rangle,\sigma\sigma^{\prime}}(\bm{\sigma}\times\mathbf{d}_{ij})^{z}_{\sigma\sigma^{\prime}}~{}c^{\dagger}_{i\sigma}c_{j\sigma^{\prime}},$
(1)

where $t\\!=\\!\hbar^{2}/2m^{*}a^{2}$ is the hopping energy, $m^{*}$ is the
effective mass of the electrons, $a$ is unit spacing of the considered square
lattice grid, $i$ and $j$ represent lattice site indices, $\sigma$ and
$\sigma^{\prime}$ are indices for spins ($\uparrow$, $\downarrow$), $\mu$ is
the chemical potential, $\Delta_{i}$ is the superconducting pairing
(conventional $s$ wave) amplitude at site $i$, $\mu_{{}_{\rm B}}$ is the Bohr
magneton, $g^{*}$ is the effective g-factor of the electrons, $B$ is the
magnetic field applied along the length of the metallic channel ($y$
direction, in the chosen axes description, as shown in Fig. 1(a)),
$\bm{\sigma}$ represents the Pauli matrices, $\alpha$ is the RSOC strength,
$\mathbf{i}$ is the imaginary number, and $\mathbf{d}_{ij}$ denotes the unit
vector from site $i$ to $j$. The pairing amplitude is zero in the metallic
channel and of constant magnitude on the two superconducting regions, but it
has a phase difference of $\varphi$ between the superconducting regions. We
use open boundary conditions and the following parameters $m^{*}\\!=\\!0.026$
$m_{0}$ ($m_{0}$ being the rest mass of electrons), $g^{*}\\!=\\!10$,
$\Delta\\!=\\!0.2$ meV, and $\alpha\\!=\\!30$ meV-nm, typical for Al/InAs
systems Smith III and Fang (1987); Mayer  _et al._ (2019); Carrad  _et al._
(2020). We use the lattice grid spacing $a\\!=\\!10$ nm in our calculations
throughout. The eigenvalues and eigenvectors of the Hamiltonian (1) were
obtained by diagonalizing it using the unitary transformation
${c}_{i\sigma}=\sum_{n}u^{n}_{i\sigma}{\gamma}_{n}+v^{n*}_{i\sigma}\gamma^{\dagger}_{n}$,
where $u^{n}_{i\sigma}$ ($v^{n}_{i\sigma}$) represents quasi-particle (quasi-
hole) amplitudes, and ${\gamma}_{n}$ (${\gamma}_{n}^{\dagger}$) represents
fermionic annihilation (creation) operator of the Bogoliubov-de Gennes quasi-
particles corresponding to the $n^{\rm th}$ eigenstate.

Figure 2: Variation with the applied magnetic field $B$ of the critical
supercurrent $I_{c}$ (normalized) and the ground state phase $\varphi_{{}_{\rm
GS}}$ (top row) and quasiparticle energy spectrum (bottom row) for three
different cases: (a) a long planar Josephson junction
($L_{x}\\!=\\!0.5~{}\mu$m, $L_{y}\\!=\\!2~{}\mu$m and $W\\!=\\!0.04~{}\mu$m)
with RSOC strength $\alpha\\!=\\!30$ meV-nm, (b) the same junction as in (a)
but with $\alpha\\!=\\!0$, (c) a relatively shorter junction
($L_{x}\\!=\\!0.3~{}\mu$m, $L_{y}\\!=\\!0.15~{}\mu$m and $W\\!=\\!0.1~{}\mu$m)
with $\alpha\\!=\\!0$. The chemical potential in (a) and (b) is
$\mu\\!=\\!-0.4$ meV and that in (c) is $\mu\\!=\\!0.5$ meV; the phase
difference between the superconducting leads in all three cases is
$\varphi\\!=\\!\pi$. MBS do not appear in the quasiparticle energy spectra in
(b) and (c), but $I_{c}$ exhibits zeros at some critical fields and
simultaneously sharp transitions in $\varphi_{{}_{\rm GS}}$. In case (a), MBS
appear (shown by the red lines near zero-energy within the greyed range of
$B$), $I_{c}$ shows a minimum at around $B\\!=\\!2.5$ T and $\varphi_{{}_{\rm
GS}}$ increases gradually from zero toward $\pi$ with increasing $B$; however,
the critical field for the minimum in $I_{c}$ is different from the critical
field at which MBS emerge. Figure 3: Magnetic field-variation of critical
supercurrent $I_{c}$ (normalized) and ground state phase $\varphi_{{}_{\rm
GS}}$ (first and third rows), and corresponding quasiparticle energy spectrum
(second and fourth rows) of a planar Josephson junction of dimension
$L_{x}\\!=\\!0.4~{}\mu$m, $L_{y}\\!=\\!1.6~{}\mu$m and $W\\!=\\!0.08~{}\mu$m,
at different RSOC strengths $\alpha$ (in meV-nm unit). Other parameters used
are $\varphi\\!=\\!\pi$ and $\mu\\!=\\!-0.4$ meV. The cyan vertical lines show
quantitative agreement of the critical field for the first minimum in $I_{c}$
with the critical field for the emergence of zero-energy MBS.

##  III Results

###  III.1 Appearance of MBS

Zero-energy MBS appear at the two ends of the middle metallic channel of the
planar Josephson junction when the phase difference $\varphi$ between the
superconducting leads is nearly $\pi$, and the length of the channel $L_{y}$
is sufficiently large, so that the overlap between the two MBS appearing at
its ends is minimized. Other essential requirements are (i) RSOC strength
needs to be greater than a critical value, which we found to be nearly $20$
meV-nm, (ii) magnetic field strength $B$ and chemical potential $\mu$ need to
be chosen suitably within a range. Quasi-particle energy spectrum for our
considered geometry ($L_{y}\\!=\\!2~{}\mu$m, transverse length
$L_{x}\\!=\\!0.5~{}\mu$m, channel width $W\\!=\\!0.04~{}\mu$m and
superconducting lead width $W_{\rm SC}\\!=\\!230$ nm (less than the coherence
length $\xi\\!=\\!\hbar v_{F}/(\pi\Delta)\\!\simeq\\!885~{}nm$), shown in Fig.
1(b), reveals presence of a pair of topologically-protected zero-energy MBS at
$\varphi\\!=\\!\pi$, $B\\!=\\!1.5$ T, and $\mu\\!=\\!-0.4$ meV. These zero-
energy states are localized near the metallic channel ends, as shown by the
local density of states profile in Fig. 1(c), obtained via
$\rho_{i}=\sum_{\sigma}(|u_{i\sigma}|^{2}+|v_{i\sigma}|^{2})$ for the lowest
positive-energy eigenstate. The local charge density of states
$\rho_{ci}=\sum_{\sigma}(|u_{i\sigma}|^{2}-|v_{i\sigma}|^{2})$, at two
magnetic fields, $B\\!=\\!1.5$ T (with MBS) and $B\\!=\\!0.5$ T (without MBS)
as shown in Figs. 1(d) and 1(e) respectively, reveals two orders of magnitude
reduction in its value, indicating the realization of a partial charge
neutrality, supporting further the appearance of the charge-neutral MBS in the
junction. Charge density profile in the presence of MBS shows a density-wave-
like pattern, a feature that generically appears in all geometries including
the nanowire and planar Josephson junction ones, and is a manifestation of
oscillatory MBS wave functions Mohanta  _et al._ (2021, 2019); sup . We use
these direct confirmations of MBS to identify topological superconducting
phase in our geometries.

###  III.2 Critical supercurrent and ground state phase

The observables of our main interest are the critical supercurrent $I_{c}$ and
the ground state phase $\varphi_{{}_{\rm GS}}$, which were predicted to be
natural diagnostics for topological transitions in these Josephson junctions
Pientka  _et al._ (2017). We calculate the critical supercurrent using
$I_{c}\\!=\\!{\rm max}\\{I(\varphi)\\}$, where the supercurrent as a function
of the phase difference $\varphi$ is given by the below thermodynamic relation
Beenakker (1992a, b)

$\displaystyle I(\varphi)=\dfrac{2e}{\hbar}\dfrac{d{\cal F}}{d\varphi},$ (2)

and the free energy ${\cal F}$ of the junctions is computed using
quasiparticle energies $E_{n}$ using the relation

$\displaystyle{\cal
F}=-2k_{{}_{B}}T\sum_{E_{n}>0}\ln\Bigl{[}2\cosh\Bigl{(}\dfrac{E_{n}}{2k_{B}T}\Bigr{)}\Bigr{]}$
(3)

where $k_{{}_{B}}$ is the Boltzmann constant and $T$ is the temperature. We
use $k_{{}_{B}}T\\!=\\!0.43\Delta$, for the results presented here and explore
temperature effect in the Supplemental Material sup . The ground state phase
$\varphi_{{}_{\rm GS}}$ is determined as the phase difference that minimizes
${\cal F}$. We obtain $\varphi_{{}_{\rm GS}}$ within $0$ and $\pi$, by mapping
values appearing above $\pi$ to this range.

In Fig. 2(a) (top panel), we show field variations of $I_{c}$ and
$\varphi_{{}_{\rm GS}}$ for a long junction of dimension and parameters (with
finite RSOC) as in Fig. 1. We find that $I_{c}$ exhibits a minimum at
$B\\!\approx\\!2.5$ T, accompanied by a gradual increase in $\varphi_{{}_{\rm
GS}}$ from $0$ toward $\pi$ with increasing magnetic field strength $B$.
Field-variation of quasiparticle energy spectrum (below panel in Fig. 2(a)),
reveals the emergence of zero-energy MBS within the range $1.1~{}{\rm
T}\\!\lesssim\\!B\\!\lesssim\\!1.7~{}{\rm T}$. Evidently, beyond
$B\\!\approx\\!2.5$ T, the junction is not in topological superconducting
phase since MBS are absent. The critical magnetic field $B\\!\approx\\!2.5$ T,
given by the first minimum in $I_{c}$, is therefore not consistent with the
topological phase boundaries. On the other hand, $\varphi_{{}_{\rm GS}}$ does
not reveal any sharp transitions in this case with typical RSOC strengths,
available in the discussed 2DEGs, and hence it is also not a good indicator
for topological superconducting transitions. Fig. 2(b) depicts $B$-variation
of $I_{c}$, $\varphi_{{}_{\rm GS}}$ (top panel) and of quasiparticle energy
spectrum (bottom panel) for the same Josephson junction as in Fig. 2(a) but
with zero RSOC. In this case, $I_{c}$ exhibits a zero and $\varphi_{{}_{\rm
GS}}$ exhibits a sharp jump from $0$ to $\pi$ at a critical field $B\approx
2.75$ T; however, the junction is in trivial phase since MBS do not appear for
the entire range of $B$ considered here. We confirm our findings by
calculating $I_{c}$ and $\varphi_{{}_{\rm GS}}$ also for relatively-shorter
junctions; results for a short junction of dimension $L_{x}\\!=\\!0.3~{}\mu$m,
$L_{y}\\!=\\!0.15~{}\mu$m and $W\\!=\\!0.1~{}\mu$m at zero RSOC are shown in
Fig. 2(c). Multiple ranges of $B$, defined by $\varphi_{{}_{\rm
GS}}\\!\approx\\!\pi$, can be identified clearly in this case, while the
junction remains in trivial superconducting phase. Because of the mean-field
nature of the adopted formalism and constant pairing gap $\Delta$, the
critical fields are overestimated i.e. larger than the actual critical fields
found in experiments.

To explore the role of RSOC and variation in the width of the metallic channel
in the agreement of the critical fields, we perform similar analyses at
different RSOC strengths for a junction with a relatively wider metallic
channel (dimension $L_{x}\\!=\\!0.4~{}\mu$m, $L_{y}\\!=\\!1.6~{}\mu$m,
$W\\!=\\!0.08~{}\mu$m and $W_{\rm SC}\\!=\\!160$ nm $<\xi$). From the results,
shown in Fig. 3, we note the following: (i) $I_{c}$ at the first minimum and
at the first maximum increases with increasing RSOC strength $\alpha$ till
$\alpha\\!\approx\\!30$ meV-nm, above which it decreases again; such a non-
monotonic field dependence exists for shorter junctions also sup , (ii) sharp
transitions in $\varphi_{{}_{\rm GS}}$ appear only for small values of
$\alpha$ for which the Josephson junction is in trivial phase always; the
gradual increase in $\varphi_{{}_{\rm GS}}$ with $B$ was also found in Ref.
Dartiailh  _et al._ (2021a), (iii) the second critical field for the second
minimum of $I_{c}$ or drop in $\varphi_{{}_{\rm GS}}$ increases with
increasing $\alpha$. Therefore, for typical $\alpha$ values, available in the
discussed 2DEGs, the second critical field will be beyond the usually-
realizable values. The near-zero-energy MBS start to appear above
$\alpha\\!\approx\\!20$ meV-nm with a well-defined energy gap, and the
critical field for the first minimum in $I_{c}$ appears near the critical
field for the topological superconducting transition. Coincidentally, the
critical fields are in better agreement for the case of $\alpha\\!=\\!30$ meV-
nm for this junction, than the result presented in Fig. 2(a) for a junction
with a narrower metallic channel. This comparison indicates that the agreement
is better for junctions with a wider metallic region. However, there is no
general correspondence among these critical fields, particularly for the long
junctions considered in the experiments Ren  _et al._ (2019); Fornieri  _et
al._ (2019); Dartiailh  _et al._ (2021a). With increasing $\alpha$, the
critical field for the emergence of MBS decreases, compatible with RSOC-driven
topological superconducting transitions.

We also investigate the in-plane magnetic field anisotropy in $I_{c}$, a
signature used in Ref. [Dartiailh  _et al._ , 2021a] in support of topological
phase transition. We rotate the in-plane magnetic field angle $\theta$
(measured with respect the $+y$-direction, along the length of the channel)
from $\theta\\!=\\!0$ to $\theta\\!=\\!\pi$, and plot $I_{c}$,
$\varphi_{{}_{\rm GS}}$ and the energy spectrum in Fig. 4. We show two cases:
(a) $B\\!=\\!1.5$ T (topological phase at $\theta\\!=\\!0$ as shown in Fig. 3)
(b) $B\\!=\\!4$ T (non-topological phase at $\theta\\!=\\!0$ as shown in Fig.
3), at a fixed RSOC strength $\alpha\\!=\\!30$ meV-nm. It is evident that the
magnetic anisotropy in $I_{c}$ and $\varphi_{{}_{\rm GS}}$ exists in both
cases. There is no noticeable difference in the field variation of $I_{c}$ and
$\varphi_{{}_{\rm GS}}$ (see also Fig.S3 in Ref. sup, ). Hence, further
evidence such as the demonstration of the non-Abelian characters of the MBS is
required, in addition to the signatures in $I_{c}$ and $\varphi_{{}_{\rm
GS}}$, in order to unambiguously detect a topological superconducting
transition in these planar junctions.

Figure 4: Variation of $I_{c}$, $\varphi_{{}_{\rm GS}}$ and the quasi-particle
energy spectrum with in-plane magnetic field angle $\theta$ at
$\varphi\\!=\\!\pi$ ($\mu$ in meV unit, $\alpha$ in meV-nm unit and $B$ in T)
for two cases: (a) topological phase at $\theta\\!=\\!0$, and (b) non-
topological phase at $\theta\\!=\\!0$. The system’s dimensions are the same as
in Fig 3. Parameters are shown in the insets of the top panels.

##  IV Discussion and outlook

To generalize our findings, we performed similar calculations of $I_{c}$ and
$\varphi_{{}_{\rm GS}}$, as presented above for various other scenarios: (i)
planar junctions with wider superconducting leads i.e. $W_{\rm SC}\\!>\\!\xi$
and wider channels, (ii) magnetic field applied only in the metallic channel
region, (iii) non-uniform $g$ factor sup . Based on these extensive analyses
of different parameters and device geometries, we conclude that, with
realizable device dimensions and realistic parameters, $I_{c}$ and
$\varphi_{{}_{\rm GS}}$ cannot identify topological superconducting
transitions in planar Josephson junctions.

Other observables used in previous experiments include (i) fractional ac
Josephson effect and missing Shapiro steps due to $4\pi$ periodic current-
phase relation Kwon  _et al._ (2004); San-Jose  _et al._ (2012); Sau and
Setiawan (2017); Rokhinson  _et al._ (2012); Wiedenmann  _et al._ (2016);
Deacon  _et al._ (2017), (ii) ZBCP signature Ren  _et al._ (2019); Fornieri
_et al._ (2019); Banerjee  _et al._ (2023a, b), (iii) bulk gap closing
signature Banerjee  _et al._ (2023a, b), (iv) local and non-local transport
spectroscopy Banerjee  _et al._ (2023a), and (v) negative conductance
curvature at zero bias $(\partial^{2}G/\partial V^{2})|_{V=0}$ Fornieri  _et
al._ (2019). These signatures likely cannot confirm topological
superconductivity separately due to the presence of non-Majorana states
Dartiailh  _et al._ (2021b); Rosenbach  _et al._ (2021), but together may
constitute a procedure for the detection of the topological superconductivity.

To look forward, it is intuitively appealing to explore more complicated
geometries such as multi-terminal planar Josephson junctions Riwar  _et al._
(2016); Pankratova  _et al._ (2020) for realizing Majorana fusion and non-
Abelian Majorana braiding Flensberg (2011); van Heck  _et al._ (2012); Hyart
_et al._ (2013); Hell  _et al._ (2017b); Stern and Berg (2019); Pandey  _et
al._ (2023). These complicated geometries may come with new challenges, one of
them clearly being the issue of fixing the direction of the in-plane magnetic
field which is required to be along the metallic channel length. This
particular problem can be overcome by placing underneath the Josephson
junction a chiral magnetic texture such as a skyrmion crystal which can
provide a gauge field and create a two-dimensional topological
superconductivity in the entire 2DEG Desjardins  _et al._ (2019); Mohanta  _et
al._ (2021). Nonetheless, these planar Josephson junctions provide a versatile
two-dimensional platform, capable of manipulating MBS with more efficient
control knobs Laeven  _et al._ (2020); Alidoust  _et al._ (2018); Scharf  _et
al._ (2019); Moehle  _et al._ (2021); Paudel  _et al._ (2021), and it is
possible to realize even more exotic states Klinovaja and Loss (2014);
Vinkler-Aviv  _et al._ (2017).

## Acknowledgements

PS was supported by Ministry of Education, Government of India via a research
fellowship. NM acknowledges support of initiation grant (No.
IlTR/SRIC/2116/FIG) from IIT Roorkee. Numerical calculations were performed at
the computing resources of PARAM Ganga at IIT Roorkee, provided by National
Supercomputing Mission, implemented by C-DAC, and supported by the Ministry of
Electronics and Information Technology and Department of Science and
Technology, Government of India.

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Supplementary Information: "Challenges in detecting topological
superconducting transitions via supercurrent and phase probes in planar
Josephson junctions"

Pankaj Sharma and Narayan Mohanta

Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667,
India

Figure S1: Magnetic field variation of critical supercurrent ($I_{c}$) and
ground state phase ($\varphi_{{}_{\rm GS}}$) of planar Josephson junctions of
different dimensions and different RSOC strengths ($\alpha$ in units of meV-
nm) and fixed chemical potential ($\mu=5$ meV-nm): (a) wide channel and narrow
superconducting leads ($W>\xi$, $W_{\rm SC}\ll\xi$, $W$ and $W_{\rm SC}$ in
units of $nm$), (b) narrow channel and wide superconducting leads ($W\ll\xi$,
$W_{\rm SC}\gtrsim\xi$), (c) narrow channel and wider superconducting leads
($W\ll\xi$, $W_{\rm SC}>\xi$) with $W_{\rm SC}$ approximately twice of the
superconducting coherence length $\xi$.  Figure S2: Magnetic field variation
of critical supercurrent ($I_{c}$) and ground state phase ($\varphi_{{}_{\rm
GS}}$) of planar Josephson junctions of different dimensions: (a),(b) when the
field is applied only in the metallic channel, or (c) when the $g$ factor is
different in different regions. In panel (a), the junction dimensions and
other parameters that are not specified are the same as in Fig.2(c) of the
main text ($W$ and $W_{\rm SC}$ in nm unit). In panel (b), the dimensions and
parameters are the same as in Fig. S1(c). In panel (c), the parameters used
are $g\\!=\\!20$ in the superconducting leads and $g\\!=\\!50$ in the metallic
channel. Figure S3: Magnetic field variation of critical supercurrent
($I_{c}$) and ground state phase ($\varphi_{{}_{\rm GS}}$) at different field
angles $\theta$. The first two columns are for $\theta\\!=\\!10^{\circ}$ and
$\theta\\!=\\!20^{\circ}$ at a fixed RSOC strength $\alpha\\!=\\!30$ meV-nm
which supports MBS at $\theta\\!=\\!0^{\circ}$ (as shown in Fig. 3 of the main
text). The third and fourth columns are for $\theta\\!=\\!10^{\circ}$ and
$\theta\\!=\\!20^{\circ}$ at a fixed RSOC strength $\alpha\\!=\\!5$ meV-nm
which does not support MBS at $\theta\\!=\\!0^{\circ}$ (as shown in Fig. 3 of
the main text). All other parameters are the same as in Fig. 3 of the main
text. Figure S4:  (a) Critical supercurrent ($I_{c}$) and ground state phase
($\varphi_{{}_{\rm GS}}$) with varying in-plane magnetic field ($B$) in a long
planar Josephson junction with a narrow channel (same as the one used in Fig
2(a) in the main paper) at a temperature $k_{B}T\\!\simeq\\!1.29\Delta$;
$I_{c}$ shows a minimum around $B\\!=\\!3.5$ T, while the ground state phase
($\varphi_{{}_{\rm GS}}$) gradually increases from nearly zero to nearly
0.75$\pi$. (b) Temperature variation of $I_{c}$ at a fixed magnetic field
$B\\!=\\!2.75$ T, showing a monotonic decrease in $I_{c}$ with increasing
temperature.

I. Different junction dimensions:
In Fig S1, we present the variations of the critical supercurrent ($I_{c}$)
and the ground state phase ($\varphi_{{}_{\rm GS}}$) for different dimensions
of planar Josephson junctions. We consider both narrow superconducting leads
i.e. $W_{\rm SC}\\!\lesssim\\!\xi$, where $\xi$ is the superconducting
coherence length (found out to be about 885 nm for the considered junctions),
and wide superconducting leads i.e. $W_{\rm SC}\\!>\\!\xi$. For these
considered junctions, a topological phase does not appear, as the junctions do
not support the Majorana bound states (MBS). The description in Ref. [Setiawan
_et al._ , 2019] for the limit $W_{\rm SC}\\!\lesssim\\!\xi$ suggested that
$I_{c}$ does not unambiguously reveal a topological superconducting
transition. Here, we find that for both these limits, i.e. for junctions of
any dimensions, both $I_{c}$ and $\varphi_{{}_{\rm GS}}$ do not indicate a
topological phase transition.

II. Non-uniform magnetic field and $g$ factor:
In Fig S2, we show the field variation of $I_{c}$ and $\varphi_{{}_{\rm GS}}$
when the external field is applied only in the metallic channel and also when
the planar Josephson junction has different $g$-factors in the metallic
channel and superconducting leads. We find that the first minimum in $I_{c}$
occurs at a larger value of the magnetic field when the field is applied only
in the middle channel; this is true for both narrow and wide leads as shown in
Fig. S2(a) and Fig. S2(b), respectively, without the trace of any topological
transition. No distinct behavior is found for the case when the $g$ factor in
the superconducting leads is less than that in the metallic region, as shown
in Fig. S2(c).

III. Rotation of in-plane magnetic field:
In Fig S3, the in-plane field variation at different field angles $\theta$
(measured with respect to the $+y$ direction) is shown for two cases: (i)
$\alpha\\!=\\!30$ meV-nm (topological phase at $\theta\\!=\\!0$ as in Fig. [3]
of the main text), (ii) $\alpha\\!=\\!5$ meV-nm (non-topological phase at
$\theta\\!=\\!0$ as in Fig. [3] of the main text). We find that the case (i)
at $\theta\\!=\\!10^{\circ}$ still supports zero-energy MBS. At
$\theta\\!=\\!20^{\circ}$ the MBS disappear. But both panels (first and second
columns of Fig. S3) show minima in $I_{c}$ at a critical magnetic field, and
corresponding gradual increase in $\varphi_{{}_{\rm GS}}$ from nearly $0$ to
nearly $\pi$ for both $\theta\\!=\\!10^{\circ}$ and $\theta\\!=\\!20^{\circ}$.
The non-topological case (ii) (third and fourth columns of Fig. S3) also show
minima in $I_{c}$ and $0-\pi$ jumps in $\varphi_{{}_{\rm GS}}$ without showing
any zero energy MBS at different tilt angles of the applied field. Even if the
applied field is tilted, the minima in $I_{c}$ occurs at around the same
magnitude of the in-plane field for topological and non-topological cases at
$\theta\\!=\\!0$. The above analysis suggests that the tilting of the in-plane
field away from the channel-length direction destroys any existing topological
superconducting phase, but the minima in $I_{c}$ and $0-\pi$ jumps in
$\varphi_{{}_{\rm GS}}$ prevail for topological as well as non-topological
cases.

IV. Temperature variation:
In Fig S4(a), the behavior of $I_{c}$ and $\varphi_{{}_{\rm GS}}$ is presented
for a long junction ($L_{y}$ large) with a narrow channel ($W_{\rm SC}$
small), at a temperature, given by $k_{{}_{B}}T\\!\simeq\\!1.29\Delta$,
approximately three times the value used in the results in the main paper. The
critical supercurrent exhibits its first minimum around a magnetic field value
of $3.5$ T, whereas, in Fig. 2(a) of the main paper, this minimum was observed
around $2.5$ T. This shift to a higher magnetic field strength for the first
minimum in $I_{c}$ is because of a higher temperature of the junction. On the
other hand, $\varphi_{{}_{\rm GS}}$ gradually increases from nearly zero and
saturates at a value close to $\pi$ at this elevated temperature also.
However, the deviation of this saturation value of $\varphi_{{}_{\rm GS}}$
from $\pi$ increases with increasing temperature. These calculations indicate
that the first minimum in $I_{c}$ occurs at a higher magnetic field strength,
and its value at the higher critical field decreases as the temperature
increases. In Fig. S4(b), we show temperature variation of $I_{c}$ for the
considered long junction with a narrow channel. As temperature increases,
$I_{c}$ decreases monotonically, revealing a shoulder-like feature at around 3
K. Temperature variation of $I_{c}$, observed experimentally in a planar S-N-S
geometry Golikova  _et al._ (2013), also shows a monotonic decrease. In our
case, $I_{c}$ does not approach zero even for temperatures exceeding $10$ K,
contrary to what is reported in this experimental observations. This deviation
is because of a constant pairing gap used in our simulations. Interestingly, a
similar shoulder-like feature was also seen in the experiments Golikova  _et
al._ (2013).

## References

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# A new family of analytical anisotropic solutions by gravitational decoupling

Milko Estrada<EMAIL_ADDRESS>Departamento de Física, Facultad de
ciencias básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile.
Instituto de Matemática, Física y Estadística, Universidad de las Américas,
Manuel Montt 948, Providencia,Santiago, Chile.  Francisco Tello-Ortiz
<EMAIL_ADDRESS>Departamento de Física, Facultad de ciencias básicas,
Universidad de Antofagasta, Casilla 170, Antofagasta, Chile.

###### Abstract

This work is focused in the study of analytic anisotropic solutions to
Einstein’s field equations, describing spherically symmetric and static
configurations by way of the gravitational decoupling through the method of
Minimal Geometric Deformation (MGD). For this we apply MGD to Heintzmann’s
solution obtaining two new analytic and well behaved anisotropic solutions, in
which all their parameters such as the effective density, the effective radial
and tangential pressure, as well as radial and tangential sound speed, fulfill
each of the requirements for the physical acceptability available in the
literature.

anisotropy,

##  I Introduction

In recent years has emerged a great interest in the anisotropic solutions of
Einstein’s equations. This is so because the anisotropic scenarios describe
some relevant astrophysical phenomena. Harko and Mak Harko2  showed some
examples of this: nuclear matter may be anisotropic in certain high density
ranges, or from the point of view of the Newtonian gravity, spherical galaxies
can have anisotropic matter distribution. Harko and Mak in the references
Harko2 ; Harko1  argue that the interior of a star must fulfill the general
physical conditions that describe a well behaved isotropic or anisotropic
solution . Due to the later arguments, several models of anisotropic compact
objects and its criteria have been studied in the literature. anisotropico1 ;
anisotropico2 ; anisotropico3 ; anisotropico4 ; anisotropico5 ; anisotropico51
; anisotropico52 .

According to the above mentioned and other arguments, in recent years there
has risen a great interest in finding new anisotropic analytic solutions of
Einstein equations, but this is not an easy task due the highly nonlinear
behaviour of these equations (some examples of how to generate anisotropic
solutions are exposed at references anisotropico6 ; anisotropico7 ;
anisotropico8 ). To address this problem, J. Ovalle Ovalle1 , proposed a
method called Minimal Geometric Deformation (MGD), which leads to finding new
analytic anisotropic solutions of Einstein equations for spherically symmetric
and static configurations. MGD was initially employed for the study of
solutions at the braneworld scenarios Ovalle3 ; Ovalle4  and then this was
spread to the study of black hole solutions Ovalle5 ; Ovalle6  (other
applications can be seen on references Ovalle7 ; Ovalle8 ; Ovalle9 ; Ovalle10
).

In this method the isotropic energy momentum tensor $\bar{T}_{\mu\nu}$ is
deformed by an additional source $\Theta_{\mu\nu}$ whose coupling is
proportional to the constant $\alpha$, and causes anisotropic effects on the
self-gravitating system. This additional source can contain new fields, like
scalar, vector and tensor fields Ovalle2 . However in this work the source
$\Theta_{\mu\nu}$ will represents a generic gravitational source. Then the
energy momentum tensor reads:

$T_{\mu\nu}\mapsto\bar{T}_{\mu\nu}+\alpha\Theta_{\mu\nu},$ (1)

with the corresponding conservation equation:

$\nabla_{\nu}T^{\mu\nu}=0.$ (2)

Ovalle at reference Ovalle2  argue “To summarise, the MGD-decoupling amounts
to the following procedure: given two gravitational sources A and B, standard
Einstein’s equations are first solved for A, and then a simpler set of quasi-
Einstein equations are solved for B. Finally, the two solutions can be
combined in order to derive the complete solution for the total system ”. The
remarkable of this method is that: one isotropic solution A is deformed and it
produces a combined anisotropic solution $A\cup B$ that preserves spherical
symmetry. Due to the above explanation it is very interesting to apply MGD to
an isotropic well behaved and spherically symmetric solution and then analyze
the behavior of the new anisotropic solution (but now with an anisotropic
criterion). Regarding this, in reference Lake  one hundred twenty seven
isotropic solutions were analyzed where only $9$ of them were well behaved
from physical point of view.

One interesting isotropic spherically symmetric solution is the Heintzmann’s
space time Heint . In reference Lake  it was shown that this solution is well
behaved for an arbitrary election of the constants (specifically the special
case in which the constants $a$, $c$ and $A$ have magnitude equal to 1 on
equation (25)). In this work we will show that this solution is still well
behaved for values of mass and radii of physical interest described below.
These values yield to values of constants $a$ and $c$ that are no longer
restricted to be equal to one. Heintzmann’s solution has been extended to an
anisotropic charged case in the references Heint1 ; Heint2  also showing that
it is a well behaved solution.

Furthermore, in this work we will analyze the minimal geometric deformation of
isotropic Heintzmann’s solution and we will get two new anisotropic solutions.
The matching conditions are obtained with the Schwarzschild exterior solution
and then we studied the physical admissibility of these new solutions. For
this, we will use realistic values of radii and mass that correspond to the
starts 4U 1538-52, RXJ 1856-37 and Vela X-1 anisotropico51 ; anisotropico52 ,
again these values of mass and radii will lead to values of the constants $a$
and $c$ that are no longer restricted to be equal to one.

This work is organized as follows: section II presents the Einstein equations
for the energy momentum tensor (1), in section III we explain in more details
the MGD method, in section IV we study the physical acceptability of the
isotropic Heintzmann’s solution for typical values of radius and mass of some
compact objects. Section V is devoted to the application of the MGD method to
the Heintzmann’s solution, showing two new anisotropic physically acceptable
solutions, finally section VI summarize the essentials of this work and
exposes some conclusions.

##  II Einstein’s field equations for multiple sources

Starting with the standard Einstein’s equations:

$G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=-\kappa^{2}T_{\mu\nu},$
(3)

where $T_{\mu\nu}$ is given by (1) and $\bar{T}_{\mu\nu}$ corresponds to a
perfect fluid:

$\bar{T}_{\mu\nu}=(\bar{\rho}+\bar{p})\,u_{\mu}\,u_{\nu}-\bar{p}\,g_{\mu\nu},$
(4)

being $u^{\mu}$ the four-velocity, $\bar{\rho}$ and $\bar{p}$ the density and
the isotropic pressure respectively.

In Schwarzschild coordinates the spherically symmetric line element reads:

$ds^{2}=e^{\nu(r)}\,dt^{2}-e^{\lambda(r)}\,dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right),$
(5)

where $\nu=\nu(r)$ and $\lambda=\lambda(r)$ are purely radial functions and
$r$ ranging from $r=0$ (the object center) to $r=R$ (the object surface). The
line element (5) satisfy the Einstein’s equations (3), which leads to:

$\displaystyle\kappa^{2}\left(\bar{\rho}+\alpha\,\Theta_{0}^{0}\right)$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{1}{r^{2}}-e^{-\lambda}\left(\frac{1}{r^{2}}-\frac{\lambda^{\prime}}{r}\right)\
,$ (6) $\displaystyle\kappa^{2}\left(\bar{p}-\alpha\,\Theta_{1}^{1}\right)$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\frac{1}{r^{2}}+e^{-\lambda}\left(\frac{1}{r^{2}}+\frac{\nu^{\prime}}{r}\right)\
,$ (7) $\displaystyle\kappa^{2}\left(\bar{p}-\alpha\,\Theta_{2}^{2}\right)$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{e^{-\lambda}}{4}\left(2\nu^{\prime\prime}+\nu^{\prime
2}-\lambda^{\prime}\nu^{\prime}+2\frac{\nu^{\prime}-\lambda^{\prime}}{r}\right),$
(8)

The conservation equation (2), which is a linear combination of Eqs. (6)-(8),
yields:

$\bar{p}^{\prime}+\frac{\nu^{\prime}}{2}(\bar{\rho}+\bar{p})-\alpha(\Theta^{1}_{1})^{\prime}+\frac{\nu^{\prime}}{2}\alpha(\Theta^{0}_{0}-\Theta^{1}_{1})+\frac{2\alpha}{r}(\Theta^{2}_{2}-\Theta^{1}_{1})=0.$
(9)

Here the prime means differentiation respect to $r$. So, the perfect fluid is
recovered in the limit $\alpha\to 0$.

In order to simplify the above system, we identify the effective density, the
effective radial and tangential pressures as:

$\displaystyle\rho=\bar{\rho}+\alpha\Theta^{0}_{0}$ (10) $\displaystyle
p_{r}=\bar{p}-\alpha\Theta^{1}_{1}$ (11) $\displaystyle
p_{t}=\bar{p}-\alpha\Theta^{2}_{2},$ (12)

where in the extra fluid $\Theta^{1}_{1}\neq\Theta^{2}_{2}=\Theta^{3}_{3}$. So
the total anisotropy introduced by the generic source $\Theta_{\mu\nu}$ is
given by:

$\Pi\equiv p_{t}-p_{r}=\alpha\left(\Theta_{1}^{1}-\Theta_{2}^{2}\right)\ $
(13)

##  III Minimal Geometric Deformation Method

This method for spherically symmetric distributions was proposed in references
Ovalle1 ; Ovalle2 . We will begin by considering a solution to Eqs. (6)-(8)
with $\alpha=0$, namely, a GR perfect fluid solution
$\\{\eta,\mu,\bar{\rho},\bar{p}\\}$, where $\eta$ and $\mu$ are the
corresponding metric functions:

$ds^{2}=e^{\eta(r)}dt^{2}-\mu(r)^{-1}dr^{2}-r^{2}d\Omega^{2}.$ (14)

Turning on the parameter $\alpha$ we can see the effects of the source
$\Theta_{\mu\nu}$ on the perfect fluid solution
$\\{\eta,\mu\,\bar{\rho},\bar{p}\\}$. These effects can be encoded in the
geometric deformation undergone by the perfect fluid geometry $\\{\eta,\mu\\}$
in equation (14) as follows:

$\mu(r)\mapsto e^{-\lambda(r)}=\mu(r)+\alpha f^{*}(r)$ (15)
$\eta(r)\mapsto\nu(r)=\eta(r),$ (16)

it means that only the radial component of the line element (14) is deformed,
where $f^{*}(r)$ is the corresponding deformation to the radial part. Upon
replacing equations (15) and (16) in the Einstein equations (6)-(8), the
system splits into two sets of equations :

  1. 1.

The standard Einstein equations for a perfect fluid (with $\alpha=0$), where
$\eta(r)=\nu(r)$:

$\displaystyle\kappa^{2}\bar{\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}-\frac{\mu}{r^{2}}-\frac{\mu^{\prime}}{r}$ (17)
$\displaystyle\kappa^{2}\bar{p}$ $\displaystyle=$
$\displaystyle-\frac{1}{r^{2}}+\mu\left(\frac{1}{r^{2}}+\frac{\nu^{\prime}}{r}\right)$
(18) $\displaystyle\kappa^{2}\bar{p}$ $\displaystyle=$
$\displaystyle\frac{\mu}{4}\left(2\nu^{\prime\prime}+\nu^{\prime
2}+2\frac{\nu^{\prime}}{r}\right)+\frac{\mu^{\prime}}{4}\left(\nu^{\prime}+\frac{2}{r}\right),$
(19)

along with the conservation equation (2) with $\alpha=0$, namely
$\nabla_{\nu}\bar{T}^{\mu\nu}$ , yielding:

$\bar{p}^{\prime}+\frac{\nu^{\prime}}{2}\left(\bar{\rho}+\bar{p}\right)=0,$
(20)

which is a linear combination of equations (17)-(19).

  2. 2.

The terms of order $\alpha$ give rise to the following quasi-Einstein
equations Ovalle1 ; Ovalle2  , which includes the source $\Theta_{\mu\nu}$:

$\displaystyle\kappa^{2}\Theta^{0}_{0}$ $\displaystyle=$
$\displaystyle-\frac{f^{*}}{r^{2}}-\frac{f^{*\prime}}{r}$ (21)
$\displaystyle\kappa^{2}\Theta^{1}_{1}$ $\displaystyle=$
$\displaystyle-f^{*}\left(\frac{1}{r^{2}}+\frac{\nu^{\prime}}{r}\right)$ (22)
$\displaystyle\kappa^{2}\Theta^{2}_{2}$ $\displaystyle=$
$\displaystyle-\frac{f^{*}}{4}\left(2\nu^{\prime\prime}+\nu^{\prime
2}+2\frac{\nu^{\prime}}{r}\right)-\frac{f^{*\prime}}{4}\left(\nu^{\prime}+\frac{2}{r}\right).$
(23)

The conservation equation (2) then yields to $\nabla_{\nu}\Theta^{\mu\nu}=0$,
which explicitly reads:

$\left(\Theta^{1}_{1}\right)^{\prime}-\frac{\nu^{\prime}}{2}\left(\Theta^{0}_{0}-\Theta^{1}_{1}\right)-\frac{2}{r}\left(\Theta^{2}_{2}-\Theta^{1}_{1}\right)=0.$
(24)

It is worth to stress that equations (20) and (24) imply that there is no
exchange of energy momentum between the perfect fluid and the extra source
$\Theta^{\mu}_{\nu}$. So only there is purely gravitational interaction.

##  IV Heintzmann’s solution

In this section we will analyze if the isotropic Heintzmann’s solution Heint
is still well behaved for our physical interest parameters of mass and radii,
these correspond to the starts 4U 1538-52, RXJ 1856-37 and Vela X-1
anisotropico51 ; anisotropico52 . In reference Lake  this solution is called
as Heint IIa, The line element is:

$ds^{2}=A^{2}\left(1+ar^{2}\right)^{3}dt^{2}-\left(1-\frac{3ar^{2}}{2}\frac{1+c\left(1+4ar^{2}\right)^{-1/2}}{1+ar^{2}}\right)^{-1}dr^{2}-r^{2}d\Omega^{2},$
(25)

where $a$, $c$ and $A$ are constant parameters. The pressure $\bar{p}$ and the
energy density $\bar{\rho}$ are:

$\displaystyle\bar{p}(r)$ $\displaystyle=$
$\displaystyle-\frac{3a\left[\left(3ar^{2}-3\right)\left(1+4ar^{2}\right)^{1/2}+c\left(1+7ar^{2}\right)\right]}{2\kappa^{2}\left(4ar^{2}+1\right)^{1/2}\left(1+ar^{2}\right)^{2}}$
(26) $\displaystyle\bar{\rho}(r)$ $\displaystyle=$
$\displaystyle\frac{3a\left[\left(4a^{2}r^{4}+13ar^{2}+3\right)\left(1+4ar^{2}\right)^{1/2}+3c+9acr^{2}\right]}{2\kappa^{2}\left(4ar^{2}+1\right)^{1/2}\left(1+ar^{2}\right)^{2}}.$
(27)

###  IV.1 Coupling with Schwarzschild exterior solution

We will present the conditions for the coupling of isotropic solution (25)
with the Schwarzschild vacuum exterior solution:

$ds^{2}=\left(1-\frac{2\bar{M}}{r}\right)dt^{2}-\left(1-\frac{2\bar{M}}{r}\right)^{-1}dr^{2}-r^{2}d\Omega^{2}.$
(28)

For this, we will take into account the Israel-Darmois matching conditions. At
the stellar surface $\Sigma$ defined by $r=R$ these conditions give Israel :

$\left[{G}_{\mu\nu}r^{\nu}\right]_{\Sigma}=0,$ (29)

where $r_{\nu}$ is an unit radial vector and $\left[F\right]_{\Sigma}\equiv
F\left(r\rightarrow R^{+}\right)-F\left(r\rightarrow R^{-}\right)$, for any
function $F=F(r)$. Using equation (29) and the general Einstein equations, we
find that:

$\left[\bar{T}_{\mu\nu}r^{\nu}\right]_{\Sigma}=0,$ (30)

which leads to:

$\left[\bar{p}\right]_{\Sigma}=0,$ (31)

because in the external vacuum Schwarzchild solution the pressure is zero,
then the last equation yields:

$\bar{p}(R)=0.$ (32)

Using the equation (26), we find the value of the constant $c$ from equation
(32):

$c=\frac{3\left(1+4aR^{2}\right)^{1/2}\left(1-aR^{2}\right)}{1+7aR^{2}}.$ (33)

The second condition, also given by the Israel-Darmois conditions, says that
the line element must be continuous i.e. there must be no jumps in the metric:

$\left[ds^{2}\right]_{\Sigma}=0,$ (34)

so,

$g_{tt}^{\,\,-}(R)=g_{tt}^{\,\,+}(R),$ (35)

and:

$g_{rr}^{\,\,-}(R)=g_{rr}^{\,\,+}(R),$ (36)

where $g_{tt}$ and $g_{rr}$ are the temporal and radial components of the
metric. Then equating (25) with (28) and using the conditions (35) and (36) we
get:

$A^{2}\left(1+aR^{2}\right)^{3}=1-\frac{2\bar{M}}{R},$ (37)
$1-\frac{3aR^{2}}{2}\frac{1+c\left(1+4aR^{2}\right)^{-1/2}}{1+aR^{2}}=1-\frac{2\bar{M}}{R}.$
(38)

We will take $\bar{M}$ and $R$ as free parameters. Inserting the parameter $c$
from equation (33) into the equation (38), we find:

$a=\frac{\bar{M}}{R^{2}(3R-7\bar{M})},$ (39)

then with this value for $a$ we obtain the value for $c$ from equation (33)
and for $A$ from equation (37).

The criteria for the physical admissibility are well known (as example see
reference Lake ), and could represent a particular case of the conditions of
subsection V.1 when $p_{r}=p_{t}$ (excluding the energy momentum tensor
conditions). In this work we will choose the following values of mass and
radius: $\bar{M}=0,87$ solar mass and $R=7,866$ KM with a compactness factor
$u=\bar{M}/R=0,16$; $\bar{M}=0,9041$ solar mass and $R=6$ KM with a
compactness factor $u=\bar{M}/R=0,22$ and $\bar{M}=1,77$ solar mass and
$R=9,56$ KM with a compactness factor $u=\bar{M}/R=0,27$ (these values are
typical of compact objects. Indeed these correspond to the starts 4U 1538-52,
RXJ 1856-37 and Vela X-1, respectively anisotropico51 ; anisotropico52 ). With
these values we obtain the constants $a$, $c$ and $A$ from equations (39),
(33) and (37). In the figure (1) we observe that in the three cases the
density and pressure are positives and decreasing, the pressure vanishes at
the boundary and the square of light velocity is less than 1 (for simplicity
in the graphic analysis we take $\kappa=1$). Therefore Heintzmann’s solution
is well behaved under our physical assumptions. Throughout the text $u$
represents in all figures the ratio mass-radius.

Figure 1: The panel $a)$ shows the isotropic pressure which vanishes at the
boundary $r=R$. Panel $b)$ shows that the sound speed velocity obeys causality
condition $v_{s}\leq 1$. Panel $c)$ exhibits the monotonically decreasing
behaviour of the density with increasing $r$. Finally panel $d)$ shows the
decreasing pressure-density ratio $\bar{p}/\bar{\rho}$. In the graphics the
pressure and density are divided by the value of $a$. These plots correspond
to solution of section IV

.

##  V Applying MGD to Heintzmann’s solution

In this section we will apply the MGD method to Heintzmann’s solution and we
will obtain two new analytic solutions. Furthermore, we will test if these new
solutions are well behaved from physical point of view, to do this we will use
realistic values of mass and radii that correspond to the starts 4U 1538-52,
RXJ 1856-37 and Vela X-1 anisotropico51 ; anisotropico52 .

From equation (25) we have:

$\displaystyle e^{\nu(r)}$ $\displaystyle=$ $\displaystyle
A^{2}\left(1+ar^{2}\right)^{3}$ (40) $\displaystyle\mu(r)$ $\displaystyle=$
$\displaystyle
1-\frac{3ar^{2}}{2}\frac{1+c\left(1+4ar^{2}\right)^{-1/2}}{1+ar^{2}}.$ (41)

Now we apply the MGD method to Heintzmann’s solution (25). Following equations
(15)-(16) we get:

$\displaystyle e^{\nu(r)}$ $\displaystyle=$ $\displaystyle
A^{2}\left(1+ar^{2}\right)^{3}$ (42) $\displaystyle e^{-\lambda(r)}$
$\displaystyle=$ $\displaystyle
1-\frac{3ar^{2}}{2}\frac{1+c\left(1+ar^{2}\right)^{-1/2}}{1+ar^{2}}+\alpha
f^{*}(r),$ (43)

where $f^{*}(r)$ is determined imposing a mimic constraint on the density or
pressure. Respect this, two choices leading to physically acceptable solutions
are following mimic constraints Ovalle10 ; Ovalle2 :

$\Theta^{0}_{0}(r)=\bar{\rho}(r),$ (44)

or

$\Theta^{1}_{1}(r)=\bar{p}(r).$ (45)

As soon as any mimic constraint (44) or (45) is imposed, $f^{*}(r)$ can be
determined . Respect to the constant parameters $a$, $c$ and $A$, they are
determined matching the expressions (42) and (43) with the corresponding
Schwarszchild’s exterior solution.

$ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$
(46)

The first matching condition is given by equations (29) and (11), obtaining:

$p_{r}(R)=0,$ (47)

so:

$\bar{p}(R)-\alpha\Theta^{1}_{1}(R)=0.$ (48)

The second matching condition is given by equation (34), yielding to:

$\displaystyle A^{2}\left(1+aR^{2}\right)^{3}$ $\displaystyle=1-\frac{2M}{R},$
(49)

and,

$\displaystyle\left(1-\frac{3aR^{2}}{2}\frac{1+c\left(1+4aR^{2}\right)^{-1/2}}{1+aR^{2}}\right)+\alpha
f^{*}$ $\displaystyle=1-\frac{2M}{R}.$ (50)

Equating equations (49) and (50) we obtain:

$A^{2}\left(1+aR^{2}\right)^{3}=\left(1-\frac{3aR^{2}}{2}\frac{1+c\left(1+4aR^{2}\right)^{-1/2}}{1+aR^{2}}\right)+\alpha
f^{*}$ (51)

###  V.1 Admissibility of the solution

On the other hand when one considers an anisotropic matter distribution the
basic requirements change slightly, these are given by references Harko1 ;
Harko2 ; Herrera1 :

  1. 1.

the density and pressure $p_{r}$ should be positive inside the star.

  2. 2.

the gradients $d\rho/dr$ , $dp_{r}/dr$ and $dp_{t}/dr$ should be negative.

  3. 3.

inside the static configuration the speed of sound should be less than the
speed of light, i.e. $0\leq dp_{r}/d\rho\leq 1$ and $0\leq dp_{t}/d\rho\leq
1$.

  4. 4.

a physically reasonable energy-momentum tensor has to obey the conditions
$\rho-p_{r}-2p_{r}\geq 0$ and $\rho+p_{r}+2p_{t}\geq 0$.

  5. 5.

the radial pressure must vanish but the tangential pressure may not vanish at
the boundary $r=R$ of the sphere. However, the radial pressure is equal to the
tangential pressure at the center of the fluid sphere.

In this article we will analyze the above criteria. However, some of these
requirements are still in discussion,for example in Maartens  it has been
claimed that a solution which disrupts the condition $3$ could still be
allowed.

Other authors argued that the speed of sound must decreases with increasing
$r$. But the latest assumption could be not valid for anisotropic fluid, since
the speed of sound at the isotropic case could be directly related to the
material rigidity. In facts it is expected that stable configurations have a
greater rigidity inside and that it decrease with increasing $r$, i.e. the
sound speed should be decreasing as increasing $r$. But the latest assumption
is not mandatory for anisotropic fluids Herrera2 .

Respect to item 1, some authors have argued that if ordinary matter is
present, described by a high density state equation, in which anisotropies
naturally appear, one should impose the condition of the positivity of all
pressures. However, for more exotic types of matter (dark energy, or as in our
case: scalar or vectorial fields, or generic gravitational sources, etc.) the
tangential pressure may be negative inside the star, and it may not vanish on
the surface. Exotic objects like compact gravastars or compact boson stars
could have negative pressures Harko3  .

###  V.2 Solution I: mimic constraint for pressure

We will use the contraint (45), so from equation (11) we get the following
value for the effective radial pressure:

$p_{r}=(1-\alpha)\bar{p}(r),$ (52)

where $\bar{p}(r)$ is given by equation (26).

Equating equations (18) and (22) we obtain the following expression for
$f^{*}(r)$:

$\displaystyle
f^{*}(r)=\frac{3ar^{2}}{2}\frac{1+c\left(1+4ar^{2}\right)^{-1/2}}{1+ar^{2}}+\frac{ar^{2}+1}{7ar^{2}+1}-1.$
(53)

Hence from equation (43) the deformed radial metric component reads:

$e^{-\lambda(r)}=\left(1-\alpha\right)\mu(r)+\alpha\left(\frac{ar^{2}+1}{7ar^{2}+1}\right),$
(54)

where $\mu(r)$ corresponds to equation (41). The interior metric functions
given by Eqs. (42) and (54) represent the Heintzmann’s solution minimally
deformed by the generic anisotropic source $\Theta_{\mu\nu}$. We can see that
the limit $\alpha\rightarrow 0$ in Eq. (54) leads to the standard Heintzmann’s
solution for perfect fluid.

At this point, we will match the deformed interior metric given by (42) and
(54),with the exterior vacuum Schwarzschild solution (46). Using (47), and
since $\alpha\neq 0$ in equation (52) we arrive to:

$\bar{p}(R)=0,$ (55)

which, according to the Eq. (26) leads to:

$c=\frac{3\left(1+4aR^{2}\right)^{1/2}\left(1-aR^{2}\right)}{1+7aR^{2}}.$ (56)

The continuity of the metric is given by the equations (49) and (50):

$A^{2}\left(1+aR^{2}\right)^{3}=1-\frac{2M}{R},$ (57)

and

$\left(1-\alpha\right)\mu(r)+\alpha\left(\frac{ar^{2}+1}{7ar^{2}+1}\right)=1-\frac{2M}{R},$
(58)

Eqs. (56), (57) and (58) are the necessary and sufficient conditions to match
the interior solution with the exterior Schwarzschild space-time.

We will evaluate $c$ from equation (56) at equation (58) and we get the
following expression for $a$:

$a=\frac{M}{R^{2}(3R-7M)}.$ (59)

Then with above expression (59), we will obtain the value for $c$ from
equation (56) and for $A$ from equation (57).

On the other hand replacing $f^{*}(r)$ given by (53) in the equations (21) and
(23) we obtain $\Theta^{0}_{0}(r)$ and $\Theta^{2}_{2}(r)$:

$\displaystyle\Theta^{0}_{0}(r)=-\frac{9a}{2\kappa^{2}}\frac{\zeta(r)}{\left(ar^{2}+1\right)^{2}\left(7ar^{2}+1\right)^{2}\left(4ar^{2}+1\right)^{3/2}}$
(60)
$\displaystyle\Theta^{2}_{2}(r)=-\frac{3a}{2\kappa^{2}}\frac{\xi(r)}{\left(ar^{2}+1\right)^{2}\left(7ar^{2}+1\right)^{2}\left(4ar^{2}+1\right)^{1/2}},$
(61)

where the functions $\zeta(r)$ and $\xi(r)$ are:

$\displaystyle\zeta(r)=$
$\displaystyle\sqrt{1+4ar^{2}}\Big{(}-3-15ar^{2}+19a^{2}r^{4}+131a^{3}r^{6}+28a^{4}r^{8}\Big{)}+17car^{2}+91a^{2}r^{4}c$
(62) $\displaystyle+147a^{3}r^{6}c+c$ $\displaystyle\xi(r)=$
$\displaystyle\sqrt{1+4ar^{2}}\Big{(}189a^{3}r^{6}-45a^{2}r^{4}-21ar^{2}-3\Big{)}+343a^{3}r^{6}c+147a^{2}r^{4}c+21car^{2}+c.$
(63)

The above expressions allow to us find the effective density and the effective
tangential pressure from equations (10) and (12):

$\displaystyle{\rho}(r)$ $\displaystyle=$
$\displaystyle\bar{\rho}(r)-\frac{\alpha}{2\kappa^{2}}\frac{9a\zeta(r)}{\left(ar^{2}+1\right)^{2}\left(7ar^{2}+1\right)^{2}\left(4ar^{2}+1\right)^{3/2}}$
(64) $\displaystyle{p}_{t}(r)$ $\displaystyle=$
$\displaystyle\bar{p}(r)+\frac{\alpha}{2\kappa^{2}}\frac{3a\xi(r)}{\left(ar^{2}+1\right)^{2}\left(7ar^{2}+1\right)^{2}\left(4ar^{2}+1\right)^{1/2}}.$
(65)

From the physical admissibility analysis we will fix as free parameters the
constants $M$ and $R$ corresponding again to the same values of mass and
radius above declared in section IV. We evaluate at equation (59) and obtain
the value of $a$ and from equation (56) we obtain the value of $c$. In figures
(2), (3), (4) and (5) we see that all conditions from physical admissibility
of subsection (V.1) are satisfied.

Figure 2: The three panels display the radial (lower curve) and the tangential
(upper curve) pressures, exhibiting a monotonically decreasing behaviour with
the increasing $r$. All the above quantities are divided by $a$, i.e.
$P_{r}=p_{r}/a$ and $P_{t}=p_{t}/a$.These plots correspond to solution of
subsection V.2 with $\alpha=0.2$. Figure 3: Panel $a)$ shows the effective
radial pressure. Panel $b)$ shows the effective tangential pressure and panel
$c)$ shows the decreasing effective energy density. All these quantities are
normalized by $a$. These plots correspond to solution of subsection V.2 with
$\alpha=0.2$. Figure 4: Panels $a)$ and $b)$ show the squares of radial and
tangential velocities, respectively. These plots correspond to solution of
subsection V.2 with $\alpha=0.2$. Figure 5: Panel $a)$ and $b)$ show the
energy conditions $\rho+P_{r}+2P_{t}$ and $\rho-P_{r}-2P_{t}$ respectively.
These plots correspond to solution of subsection V.2 with $\alpha=0.2$.

###  V.3 Solution II: mimic constraint for density

An alternative choice leading to a physically acceptable solution is the
“mimic constraint” for the density (44), then from equation (10) we obtain:

$\rho=\bar{\rho}(1+\alpha),$ (66)

then equating the equations (17) and (21) we get $f^{*}(r)$, which reads:

$f^{*}(r)=\frac{B}{r}-\frac{3ar^{2}}{2}\frac{1+c\left(1+4ar^{2}\right)^{-1/2}}{1+ar^{2}},$
(67)

where the density $\bar{\rho}$ from Eq. (27) has been used. To avoid a
singular behaviour at the center $r=0$ we must impose $B=0$, then the equation
(15) yields:

$e^{-\lambda(r)}=\mu(r)-\alpha\left(\frac{3ar^{2}}{2}\frac{1+c\left(1+4ar^{2}\right)^{-1/2}}{1+ar^{2}}\right).$
(68)

Next evaluating $f^{*}(r)$ from equation (67) at equations (22) and (23) we
find the expressions for $\Theta^{1}_{1}(r)$ and $\Theta^{2}_{2}(r)$:

$\displaystyle\Theta^{1}_{1}=\frac{3a}{2\kappa^{2}}\frac{(\sqrt{1+4ar^{2}}+c)(1+7ar^{2})}{(1+ar^{2})^{2}\sqrt{1+4ar^{2}}}$
(69)
$\displaystyle\Theta^{2}_{2}=\frac{3a}{2\kappa^{2}}\frac{\sqrt{1+4ar^{2}}(9ar^{2}+1)+7ar^{2}c+c}{(1+ar^{2})^{2}\sqrt{1+4ar^{2}}}.$
(70)

With these expressions (69) and (70) the radial and tangential pressures (11)
and (12) are:

${p}_{r}(r)=\bar{p}(r)-\alpha\frac{3a}{2\kappa^{2}}\frac{(\sqrt{1+4ar^{2}}+c)(1+7ar^{2})}{(1+ar^{2})^{2}\sqrt{1+4ar^{2}}},$
(71)

and

${p}_{t}(r)=\bar{p}(r)-\alpha\frac{3a}{2\kappa^{2}}\frac{\left(\sqrt{1+4ar^{2}}(9ar^{2}+1)+\left(7ar^{2}+1\right)c\right)}{(1+ar^{2})^{2}\sqrt{1+4ar^{2}}}.$
(72)

The continuity of the effective radial pressure (47) along with the equation
(71), yield to:

$c=-\frac{\sqrt{1+4aR^{2}}(-3+3aR^{2}+\alpha+7\alpha
aR^{2})}{(1+\alpha)(1+7aR^{2})}.$ (73)

.

Using the expression given by (67), the matching conditions in (49) and (50)
lead to:

$A^{2}\left(1+aR^{2}\right)^{3}=1-\frac{2M}{R},$ (74)

and:

$1-\left(1+\alpha\right)\left(\frac{3ar^{2}}{2}\frac{1+c\left(1+4ar^{2}\right)^{-1/2}}{1+ar^{2}}\right)=1-\frac{2M}{R},$
(75)

Eqs. (73)-(75) are the necessary and sufficient conditions to match the
exterior Schwarzschild solution with the deformed interior solution.

Inserting $f^{*}(r)$ from equation (67) and $c$ from equation (73) at equation
(75) we again find the same value for $a$ from equation (59). Next with this
value for $a$ we will obtain the value for $c$ from equation (73) and A from
equation (74).

For testing the admissibility of our solution we again choose the same values
of mass and radius from subsections IV and V.2 but now $\alpha=0,3$, obtaining
$a$ from equation (59) and $c$ from equation (73). For these values figure (6)
shows the radial pressure ${p}_{r}(r)$ and tangential pressure ${p}_{t}(r)$
inside the spherical distribution, showing how the magnitude of the anisotropy
increases towards the surface.The tangential pressure is less than radial
pressure i.e there is an attractive force. Also we observe that the tangential
pressure becomes negative before the boundary. In figures (6),(7),(8) and (9)
we observe that all requirements of admissibility of subsection (V.1) are
fulfills .

With the same values of mass and radius also it is possible show a scenario
where the tangential pressure is greater than radial pressure and being both
positives ı.e there is a repulsive force, for example with $\alpha=-0,3$. In
figures (10), (11), (12) and (13) we see again, that all the admissibility
conditions are fulfills, therefore with $\alpha=-0,3$ the solution is also
well behaved.

Figure 6: The three panels display the tangential (lower curve) and the radial
(upper curve) pressures, exhibiting a monotonically decreasing behaviour with
the increasing $r$. All the above quantities are divided by the constant $a$,
i.e. $P_{r}=p_{r}/a$ and $P_{t}=p_{t}/a$. These plots correspond to solution
of subsection V.3 with $\alpha=0.3$. Figure 7: Panel $a)$ shows the effective
radial pressure. Panel $b)$ shows the effective tangential pressure and panel
$c)$ shows the decreasing effective energy density. All these quantities are
divided by the value of $a$. These plots correspond to solution of subsection
V.3 with $\alpha=0.3$. Figure 8: Panels $a)$ and $b)$ show the squares of
radial and tangential velocities respectively. These plots correspond to
solution of subsection V.3 with $\alpha=0.3$. Figure 9: Panel $a)$ and $b)$
show the energy conditions $\rho+P_{r}+2P_{t}$ and $\rho-P_{r}-2P_{t}$,
respectively. These plots correspond to solution of subsection V.3 with
$\alpha=0.3$. Figure 10: The three panels display the radial (lower curve) and
the tangential (upper curve) pressures, exhibiting a monotonically decreasing
behaviour with the increasing $r$. All the above quantities are divided by the
constant $a$, i.e. $P_{r}=p_{r}/a$ and $P_{t}=p_{t}/a$. These plots correspond
to solution of subsection V.3 with $\alpha=-0.3$. Figure 11: Panel $a)$ shows
the effective radial pressure. Panel $b)$ shows the effective tangential
pressure and panel $c)$ shows the decreasing effective energy density. All
these quantities are divided by the value of $a$. These plots correspond to
solution of subsection V.3 with $\alpha=-0.3$. Figure 12: Panels $a)$ and $b)$
show the squares of radial and tangential velocities, respectively. These
plots correspond to solution of subsection V.3 with $\alpha=-0.3$. Figure 13:
Panel $a)$ and $b)$ show the energy conditions $\rho+P_{r}+2P_{t}$ and $\rho-
P_{r}-2P_{t}$, respectively. These plots correspond to solution of subsection
V.3 with $\alpha=-0.3$.

###  V.4 Stability conditions

One important aspect in the study of compact objects in general relativity
describing anisotropic matter distributions is the stability of the model. In
this section we analyze the stability of the present model from two
perspectives. The first one is via the adiabatic index $\Gamma$ and the second
one is through the analysis of the square of the sound speeds using Abreu et.
al.Abreu  studies based on the Herrera’s cracking concept herrera5 .
It is well known that a spherically symmetric and static configuration
associated with a Newtonian isotropic matter distribution will collapse if the
adiabatic index is $\Gamma<4/3$. In distinction with the relativistic
anisotropic fluid spheres the collapsing condition becomes Herrera3 ; Herrera4

$\Gamma<\frac{4}{3}+\left[\frac{1}{3}\kappa\frac{\rho_{0}p_{r0}}{|p^{\prime}_{r0}|}r+\frac{4}{3}\frac{\left(p_{t0}-p_{r0}\right)}{|p^{\prime}_{r0}|r}\right]_{max}$
(76)

where $\rho_{0}$, $p_{r0}$ and $p_{t0}$ are the initial density, radial and
tangential pressure when the fluid is in static equilibrium. The second term
in the right hand side represents the relativistic corrections to the
Newtonian perfect fluid and the third term is the contribution due to
anisotropy. Heintzmann and Hillebrandt hillebrandt  showed that the stability
condition for a relativistic compact object is given by $\Gamma>4/3$. Since
the gravitational collapse takes place in the radial direction of the
configuration it is sufficient to calculate the adiabatic index $\Gamma_{r}$
in such direction. We can explicitly obtain the adiabatic index from, chan

$\Gamma_{r}=\frac{\rho+p_{r}}{p_{r}}\frac{dp_{r}}{d\rho}.$ (77)

Figure (14) shows that all our models satisfies the above criteria. Therefore
it is a stable model. On the other hand, Abreu et. al. analysis regarding the
square of the sound speeds within the stellar configuration, provides us a
criteria to determine the regions where the model is stable. So the model is
stable if $v^{2}_{r}-v^{2}_{t}$ remains between $0$ and $1$ and unstable if
$v^{2}_{r}-v^{2}_{t}$ remains between $-1$ and $0$. We can see that solutions
of figures 15(a) and 15(b) becomes potentially stable since they fulfill the
condition $0<v_{r}^{2}-v_{t}^{2}<1$, whereas the solutions of figure 15(c)
becomes unstable.

(a) Solutions of subsection V.2. (b) Solutions of subsection V.3 with
$\alpha=-0.3$ (c) Solutions of subsection V.3 with $\alpha=0.3$

Figure 14: Adiabatic index criterion.

(a) Solutions of subsection V.2. (b) Solutions of subsection V.3 with
$\alpha=-0.3$ (c) Solutions of subsection V.3 with $\alpha=0.3$

Figure 15: Abreu criterion $v_{r}^{2}-v_{t}^{2}$.

##  VI Conclusions and remarks

We showed that by imposing our matching conditions the isotropic Heintzmann’s
solution is well behaved using typical mass and radius values for some compact
objects such as 4U 1538-52, RXJ 1856-37 and Vela X-1, whose compactness factor
$u$ are $0,16$, $0,22$ and $0,27$, respectively. The choice of Heintzmann IIa
solution is merely arbitrary, because there is not evidence that the mentioned
compact stars are described fully by this solution.

Then, applying the Minimal Geometric Deformation method to Heintzmann’s
solution we have found two new analytic anisotropic and spherically symmetric
solutions of Einstein field equations. These two solutions are the results of
the decoupling of Einstein equations in a isotropic sector described by
$\bar{T}_{\mu\nu}$ and the sector described by the quasi-Einstein equations
with a source $\Theta_{\mu\nu}$. Due to the application of MGD the perfect and
the extra fluid are separately conserved. So, the combination of these two
sectors has only gravitational interaction, and does not has exchange of
energy momentum.

We have shown that these two new solutions are well behaved for the above
parameters of mass and radius, since they fulfill every admissibility criteria
of the subsection (V.1). To accomplish it we use $\alpha=0,2$ and $\alpha=0,3$
in the first and second constraint, respectively. Moreover considering
$\alpha=0,3$ in the second constraint, we obtained an attractive force where
the tangential pressure becomes negative before the boundary of the compact
object, but using $\alpha=-0,3$ a repulsive force appears. In all cases the
solutions are well behaved. The matching conditions are perfectly satisfied in
both solutions.

All the sources $\Theta_{\mu\nu}$ used in this works are generic gravitational
sources, therefore does not represent to a high density equation of state i.e.
does not impose that the tangential pressure is positive inside of the fluid
sphere.

In the figure (16) we observe how any perfect fluid solution can be
consistently extended to the anisotropic domain via the MGD approach. In our
case the seed well behaved Heintzmann’s solution $\\{\nu,\mu\\}$ is extended
to an anisotropic well behaved scenario $\\{\nu,\mu+\alpha\cdot f^{*}\\}$.

Figure 16: Minimal Geometric Deformation approach.

In this work we do not make a complete study about compact objects, however
our solutions would serve for future studies about these objects. Moreover,
working in CGS system table 1 shows the values of the central and surface
energy density and central pressure, corresponding to the isotropic
Heintzmann’s solution from section IV and both anisotropic extensions from
subsection V.2 (mimic constraint for pressure) and from subsection V.3 (mimic
constraint for density). We found values for central and surface density of
order of $10^{14}-10^{15}g/cm^{3}$ and central pressure of order of
$10^{34}-10^{35}$ $\mbox{dyne}/cm^{2}$. These are typical values for compact
objects, some examples can be found in references anisotropico51 ;
anisotropico52 ; Anisotropico9 .

Table 1: Energy density and central pressure in CGS system. Solution | $u$ | $\alpha$ | $a$ |  $c$ (dimension | Central den- | Surface den- | Central pressu-
---|---|---|---|---|---|---|---
|  |  | $(cm^{-2})$ | less | sity ($g/cm^{3}$) | sity ($g/cm^{3}$) | re $(\mbox{dyne}/cm^{2})$
section IV | $0,163$ | $0$ | $1,422\cdot 10^{-13}$ | $1,969$ | $1,018\cdot 10^{15}$ | $7,57\cdot 10^{14}$ | $1,06\cdot 10^{35}$
| $0,222$ | $0$ | $4,287\cdot 10^{-13}$ | $1,551$ | $2,634\cdot 10^{15}$ | $1,672\cdot 10^{15}$ | $4,49\cdot 10^{35}$
| $0,27$ | $0$ | $2,756\cdot 10^{-13}$ | $1,151$ | $1,43\cdot 10^{15}$ | $7,612\cdot 10^{14}$ | $3,683\cdot 10^{35}$
subsection V.2 | $0,163$ | $0,2$ | $1,422\cdot 10^{-13}$ | $1,969$ | $1,089\cdot 10^{15}$ | $7,324\cdot 10^{14}$ | $8,48\cdot 10^{34}$
| $0,222$ | $0,2$ | $4,287\cdot 10^{-13}$ | $1,551$ | $2,934\cdot 10^{15}$ | $1,598\cdot 10^{15}$ | $3,592\cdot 10^{35}$
| $0,27$ | $0,2$ | $2,756\cdot 10^{-13}$ | $1,151$ | $1,676\cdot 10^{15}$ | $7,195\cdot 10^{14}$ | $2,946\cdot 10^{35}$
subsection V.3 | $0,163$ | $0,3$ | $1,422\cdot 10^{-13}$ | $1,246$ | $1,001\cdot 10^{15}$ | $7,653\cdot 10^{14}$ | $5,551\cdot 10^{34}$
| $0,222$ | $0,3$ | $4,287\cdot 10^{-13}$ | $0,9$ | $2,552\cdot 10^{15}$ | $1,706\cdot 10^{15}$ | $2,371\cdot 10^{35}$
| $0,27$ | $0,3$ | $2,756\cdot 10^{-13}$ | $0,558$ | $1,348\cdot 10^{15}$ | $7,878\cdot 10^{14}$ | $1,966\cdot 10^{35}$
| $0,163$ | $-0,3$ | $1,422\cdot 10^{-13}$ | $3,311$ | $1.035\cdot 10^{15}$ | $7,489\cdot 10^{14}$ | $5,049\cdot 10^{34}$
| $0,222$ | $-0,3$ | $4,287\cdot 10^{-13}$ | $2,761$ | $2,722\cdot 10^{15}$ | $1,638\cdot 10^{15}$ | $2,119\cdot 10^{35}$
| $0,27$ | $-0,3$ | $2,756\cdot 10^{-13}$ | $2,251$ | $1,513\cdot 10^{15}$ | $7,345\cdot 10^{14}$ | $1,717\cdot 10^{35}$

###### Acknowledgements.

The work of ME was partially founded by FONDECYT Regular under Grant No.
1151107 . FTO thanks the financial support of the post-graduate program at the
Universidad de Antofagasta-Chile.

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