# Anomalous Shiba states in topological iron-based superconductors

Areg Ghazaryan  Institute of Science and Technology Austria, Am Campus 1, 3400
Klosterneuberg, Austria  Ammar Kirmani  Physics Department, City College of
the City University of New York, NY 10031, U.S.A.  Rafael M. Fernandes  School
of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota
55455, US  Pouyan Ghaemi  Physics Department, City College of the City
University of New York, NY 10031, U.S.A.  Physics Program, Graduate Center of
City University of New York, NY 10031, U.S.A.

###### Abstract

We demonstrate the formation of robust zero energy modes close to magnetic
impurities in the iron-based superconductor FeSe1-xTex. We find that the
Zeeman field generated by the impurity favors a spin-triplet inter-orbital
pairing as opposed to the spin-singlet intra-orbital pairing prevalent in the
bulk. The preferred spin-triplet pairing preserves time-reversal symmetry and
is topological, as robust, topologically-protected zero modes emerge at the
boundary between regions with different pairing states. Moreover, the zero
modes form Kramers doublets that are insensitive to the direction of the spin
polarization or to the separation between impurities. We argue that our
theoretical results are consistent with recent experimental measurements on
FeSe1-xTex.

Introduction: Topological superconductivity (TSc) is a quantum state that has
been extensively explored Hasan and Kane (2010); Qi and Zhang (2011); Sato and
Ando (2017), particularly due to its application for realizing Majorana zero
modes (MZM) Alicea (2012); Flensberg  _et al._ (2021). In recent years, the
iron-based superconductor FeSe1-xTex (FST) emerged as a promising candidate
for TSc Wang  _et al._ (2015). It was theoretically predicted that the band
inversion in the electronic structure of FST can lead to localized MZM at the
end of vortex lines in the superconducting phase Xu  _et al._ (2016). One of
the appeals of superconducting FST is the comparable energy scales between the
superconducting gap and the Fermi energy, which leads to a wide spectral
resolution of the vortex zero mode Kreisel  _et al._ (2020). The experimental
observation of such vortex zero modes has led to extensive studies of FST as a
bulk TSc Zhang  _et al._ (2018); Wang  _et al._ (2018); Zhu  _et al._ (2020);
Machida  _et al._ (2019); Miao  _et al._ (2018). The interplay between
topology, superconductivity and magnetism has also been investigated in
relation to surface states and generating new superconducting order Li  _et
al._ (2021); Hu  _et al._ (2020); Zaki  _et al._ (2021); Youmans  _et al._
(2018); Dzero and Levchenko (2022); Li  _et al._ (2022). On the other hand,
several of the properties of FST, such as the existence of vortices with and
without zero modes in the same sample, have been puzzling Machida  _et al._
(2019); Wang  _et al._ (2020); Chiu  _et al._ (2020); Ghazaryan  _et al._
(2020); Hu and Zhang (2022).

Another type of in-gap states in superconductors are the Shiba states, which
form near magnetic impurities Yu (1965); Shiba (1968); Rusinov (1969). In
certain regimes in-gap states can also be formed for non-magnetic impurity Sau
and Demler (2013) even when the system respects time-reversal symmetry
Matsumoto  _et al._ (2009); Mashkoori  _et al._ (2019) In FST, in-gap states
have been observed near interstitial Fe atoms Yin  _et al._ (2015), which
behave as magnetic impurities Thampy  _et al._ (2012). Interestingly, several
of the properties of these in-gap states are different from those of
conventional Shiba states. First, Shiba states generally have a finite energy
unless the microscopic properties of the superconductor and the coupling
strength of the magnetic impurity with the itinerant electrons are fine tuned
Balatsky  _et al._ (2006a). Yet in FST the in-gap states at many of the
magnetic impurities are observed at zero energy Yin  _et al._ (2015); Fan  _et
al._ (2021). The other surprising property is the absence of a hybridization
gap in the Shiba-state energies of two nearby magnetic impurities Yin  _et
al._ (2015), which contrasts with the standard behavior seen in conventional
systems Ruby  _et al._ (2018). STM measurements have further shown that, while
the energy of the in-gap state is zero when the impurity is at the center of
the unit cell, it becomes finite when the impurity is pushed toward the edge
of the unit cell Fan  _et al._ (2021). Proposals such as anomalous quantum
vortices forming at magnetic impurities Jiang  _et al._ (2019) or effective
$\pi$-phase shifts at the impurity sites Song  _et al._ (2022); Björnson  _et
al._ (2017) have been put forward, but a comprehensive description of
properties of these states is still lacking.

In this Letter we present an alternative mechanism for formation of zero
energy states close to magnetic impurities in FST. The key property of FST
leading to the type of in-gap states discussed in this Letter is the possible
existence of multiple superconducting pairing instabilities energetically
close to each other. Near an impurity, the Zeeman field generated by it
strongly impacts this balance between different types of pairing. In
particular, by solving the linearized gap equations, we find that the pairing
state favored by the Zeeman field in this region is topologically distinct
from the pairing state in regions farther than a multiple lattice spacing from
the magnetic impurity. As a result, a pair of zero-energy states form in the
boundary region between the two types of pairing. The resulting zero-energy
states form bubbles surrounding the magnetic impurity, with a radius of the
order of multiple lattice spacing. Since these states arise from the
topological character of the superconducting state, their energy is generally
pinned at zero.

When two impurities approach each other, the regions around them where the
topological superconductivity is dominant merge. As a result, the zero modes
surround a larger area and do not become gapped. Furthermore, the type of
pairing that is preferred close to the impurity is determined by the
symmetries of the system. Therefore, the position of the impurity in the unit
cell can affect the type of pairing selected and, consequently, the
development of zero modes. Interestingly, the triplet superconducting pairing
that forms close to the magnetic impurity is time-reversal symmetric, and the
boundary states are Kramers pairs that are insensitive to the direction of the
spin-polarization of the magnetic impurity. The later property is consistent
with results from spin-polarized STM measurements Wang  _et al._ (2021).
Another feature of the magnetic-impurity induced in-gap state in FST is that
it becomes gapped once the magnetic impurity approaches a magnetic vortex Fan
_et al._ (2021). This is because the two zero modes enclosing the magnetic
impurity hybridize in the presence of a Zeeman field. Therefore, due the
vortex hosting a sizable magnetic field, it gaps out the zero modes.

Model Hamiltonian: FST is a member of the family of iron-based superconductors
Fernandes  _et al._ (2022) with non-symmorphic $P4/nmm$ space group symmetry
due to the buckling of the chalcogen atoms inside the $2$-Fe unit cell
Fernandes and Chubukov (2016); Cvetkovic and Vafek (2013); Lohani  _et al._
(2020). At the $\Gamma$ point, the $P4/nmm$ space group is isomorphic to the
$D_{4h}$ point group. The generators of the $D_{4h}$ group can be taken to be
$\pi_{x}$, $\pi_{z}$ and $\pi_{X}$ reflection planes. Here, $x$ and $y$
connect nearest-neighbor Fe atoms, whereas $X$ and $Y$, rotated by
$45^{\circ}$ with respect to $x$ and $y$, connect next-nearest-neighbor Fe
atoms. The FeSe layers are stacked along $z$. The band structure close to
$\Gamma$ point mainly contains $p_{z}$, $d_{xz}$ and $d_{yz}$ orbitals and the
most general effective Hamiltonian including spin-orbit coupling (SOC) can be
constructed using the method of invariants Inui  _et al._ (1990); Bir  _et
al._ (1974). The inversion-odd $p_{z}$ orbital is essential for realizing band
inversion along the $\mathrm{\Gamma}-Z$ direction and thus the nontrivial
topology of the band structure Xu  _et al._ (2016); Lohani  _et al._ (2020).
In the basis
$\psi_{\mathbf{k}}=\left(|d_{+}\uparrow\rangle,|d_{-}\uparrow\rangle,|p_{z}\uparrow\rangle,|d_{+}\downarrow\rangle,|d_{-}\downarrow\rangle,|p_{z}\downarrow\rangle\right)$,
where $d_{\pm}=\left(d_{yz}\pm id_{xz}\right)^{T}$, the Hamiltonian is given
by Lohani  _et al._ (2020)

$H\left(\mathbf{k}\right)=\sigma_{0}\otimes
H_{0}\left(\mathbf{k}\right)+H_{\mathrm{SOC}}\left(\mathbf{k}\right),$ (1)

where for small in-plane momentum:

$\displaystyle H_{0}\left(\mathbf{k}\right)$
$\displaystyle=\left(\begin{array}[]{ccc}M_{1}\left(\mathbf{k}\right)&\beta
k^{2}_{+}&\delta k_{-}\\\ \beta k^{2}_{-}&M_{1}\left(\mathbf{k}\right)&-\delta
k_{+}\\\ \delta k_{+}&-\delta
k_{-}&M_{2}\left(\mathbf{k}\right)\end{array}\right),$ (5)

with $k_{\pm}=k_{x}\pm ik_{y}$ and
$M_{n}\left(\mathbf{k}\right)=E_{n}+\left(k^{2}_{x}+k^{2}_{y}\right)/2m_{nx}+t_{nz}\left(1-\cos\left(k_{z}\right)\right)$.

The non-zero elements of the SOC Hamiltonian
$H_{\mathrm{SOC}}\left(\mathbf{k}\right)$ are
$H^{\mathrm{22}}_{\mathrm{SOC}}\left(\mathbf{k}\right)=H^{\mathrm{44}}_{\mathrm{SOC}}\left(\mathbf{k}\right)=-H^{\mathrm{11}}_{\mathrm{SOC}}\left(\mathbf{k}\right)=-H^{\mathrm{55}}_{\mathrm{SOC}}\left(\mathbf{k}\right)=\lambda_{1}$,
and
$H^{\mathrm{16}}_{\mathrm{SOC}}\left(\mathbf{k}\right)=H^{\mathrm{35}}_{\mathrm{SOC}}\left(\mathbf{k}\right)=\sqrt{2}\lambda_{3}\sin\left(k_{z}\right)$,
as well as the matrix elements related by hermiticity. The parameters $\beta$,
$\delta$, $E_{1}$, $E_{2}$, $m_{1}$, $m_{2}$, $t_{1}$, $t_{2}$, $\lambda_{1}$
and $\lambda_{3}$ were previously determined using density functional theory
Xu  _et al._ (2016).

Figure 1: (a) Phase boundary between different inter-orbital triplet
($\Delta_{2\pm}$) and intra-orbital spin-singlet ($\Delta_{1}$) pairing states
for $\mu=0\,\mathrm{eV}$ when the Zeeman field is along the $\hat{\mathbf{z}}$
or $\hat{\mathbf{x}}$ direction. Singlet (triplet) pairing emerges in the
light gray (light red) region, following the phase boundary for the Zeeman
field along the $\hat{\mathbf{z}}$ direction. (b) Band structure in the
$\Gamma-Z$ direction. The selected chemical potentials are those for which the
phase boundaries between $\Delta_{2+}$ and $\Delta_{1}$ are shown in (c). The
Zeeman field is in the $\hat{\mathbf{z}}$ direction for (c).

Symmetries and pairing states:  The types of pairing states in FST can be
classified according to irreducible representations of the $D_{4h}$ symmetry
group of the underlying lattice Sigrist and Ueda (1991). $D_{4h}$ has five
odd-parity and five even-parity irreducible representations. We only consider
pairing with zero center-of-mass momentum. In the even sector, the standard
pairing is intra-orbital spin-singlet that transforms as the $\mathrm{A_{1g}}$
representation and has the form $\Delta_{1}:\sigma_{0}\otimes\mathbf{1}_{3}$.
We use the standard BdG basis where the wave function has the form
$\Psi_{\mathbf{k}}=\left(\psi_{\mathbf{k}},i\sigma_{y}K\psi_{\mathbf{k}}\right)^{T}$.
Due to the presence of SOC, there is a spin-triplet inter-orbital pairing
involving the $d_{xz}$ and $d_{yz}$ orbitals that also transforms as
$\mathrm{A_{1g}}$, of the form $\sigma_{z}d_{xz}d_{yz}$. This pairing state,
favored by the Hund’s coupling Vafek and Chubukov (2017), does not pair the
$p_{z}$ orbitals. Therefore, when the chemical potential is close to the
$p$-$d$ band inversion point, $\sigma_{z}d_{xz}d_{yz}$ pairing state is not
energetically favorable.

On the other hand, there are four types of odd-parity pairing states that are
inter-orbital triplet and involve the $p_{z}$ orbital. They are
$\Delta_{2\pm}:\sigma_{x}p_{z}d_{xz}\pm\sigma_{y}p_{z}d_{yz}$, which
transforms as the $\mathrm{A_{1u}}$ ($+$) or $\mathrm{B_{1u}}$ ($-$)
irreducible representation; and
$\Delta_{3\pm}:\sigma_{y}p_{z}d_{xz}\mp\sigma_{x}p_{z}d_{yz}$, which
transforms as $\mathrm{A_{2u}}$ ($+$) or $\mathrm{B_{3u}}$ ($-$). In the
$d_{\pm}$ basis, these gap functions generate the Bogoliubov-de Gennes mean-
field pairing terms:

$\displaystyle H_{\Delta_{\\{2,3\\},+}}$
$\displaystyle=\Delta_{\\{2,3\\},+}i^{\\{0,1\\}}\tau_{x}\left(\begin{array}[]{cccccc}0&0&0&0&0&1\\\
0&0&0&0&0&0\\\ 0&0&0&0&-1&0\\\ 0&0&0&0&0&0\\\ 0&0&\mp 1&0&0&0\\\ \pm
1&0&0&0&0&0\\\ \end{array}\right),$ (12) $\displaystyle
H_{\Delta_{\\{2,3\\},-}}$
$\displaystyle=\Delta_{\\{2,3\\},-}i^{\\{0,1\\}}\tau_{x}\left(\begin{array}[]{cccccc}0&0&0&0&0&0\\\
0&0&0&0&0&1\\\ 0&0&0&-1&0&0\\\ 0&0&\mp 1&0&0&0\\\ 0&0&0&0&0&0\\\ 0&\pm
1&0&0&0&0\\\ \end{array}\right),$ (19)

where the upper (lower) signs refer to $\Delta_{2}$ ($\Delta_{3}$) and the
Pauli matrix $\tau_{j}$ acts in Nambu particle-hole space. The overall phase
was chosen such that both $\Delta_{2,\pm}$ and $\Delta_{3,\pm}$ respect time-
reversal symmetry.

From numerical analysis, we identify that all these pairing states, except
$\Delta_{2+}$, are nodal. The novel structure of the pairing gaps has
important implications for the topological character of the superconducting
state. Indeed, a time-reversal and odd-parity pairing is topological if it is
gapped and encloses an odd number of time-reversal invariant momenta Fu and
Berg (2010). The space group $P4/nmm$ has the non-symmorphic symmetry
operation inversion followed by half translation, which at the $\Gamma$ point
corresponds to the inversion symmetry operation of $D_{4h}$ Cvetkovic and
Vafek (2013). As a result, the triplet pairings of FST identified above,
$\Delta_{2\pm}$ and $\Delta_{3\pm}$, have odd parity. Since only $\Delta_{2+}$
has a full gap, it is the only one among those four that can be in a
topological superconducting state, as long as it encloses an odd number of
time-reversal invariant momenta.

Effect of the Zeeman field: To determine the dominant pairing instability as a
function of the Zeeman field we utilize the linearized gap equation Fischer
(2013). We start by defining the finite temperature superconducting
susceptibilities:

$\displaystyle\chi_{lm}$
$\displaystyle=-\frac{T}{N}\sum_{\omega_{n}\mathbf{p}}\mathrm{Tr}\left[\frac{H_{\Delta_{l}}^{\dagger}}{\Delta_{l}}G_{0}\left(i\omega_{n},\mathbf{p},\Delta_{Z}\right)\times\right.$
$\displaystyle\hskip
80.0pt\left.\frac{H_{\Delta_{m}}}{\Delta_{m}}G_{0}\left(-i\omega_{n},\mathbf{p},-\Delta_{Z}\right)\right],$
(20)

where $H_{\Delta_{l}}$ is the pairing Hamiltonian, $N$ is the number of
momentum points, and
$G_{0}\left(i\omega_{n},\mathbf{p},\Delta_{z}\right)=\sum_{j}\mathcal{P}_{j\mathbf{p}\Delta_{Z}}/\left(i\omega_{n}-\epsilon_{j\mathbf{p}\Delta_{z}}\right)$
is the normal state Green’s function. $j$ runs through the bands, $\omega_{n}$
is the Matsubara frequencies and $\mathcal{P}_{j,\mathbf{p}\Delta_{Z}}$ is the
projection operator onto band $j$ at momentum $\mathbf{p}$. The Zeeman-field
energy splitting is $\Delta_{Z}$. Finally, the subscripts $l$ and $m$ label
the five pairing states discussed above: the singlet $\Delta_{1}$ and the
triplets $\Delta_{2\pm}$, $\Delta_{3\pm}$. The linearized gap equations, which
determine the critical temperature $T_{c}$ for each pairing instability, take
the form

$\displaystyle-\frac{1}{g_{s}}-\chi_{11}$ $\displaystyle=0,$ (21)
$\displaystyle-\frac{2}{3g_{t}}-\chi_{tt}$ $\displaystyle=0,$ (22)

where $g_{l}$ are the superconducting coupling constants arising from the
microscopic interactions, with $s=1$ and $t=\pm 2,\,\pm 3$. In deriving these
expressions, we ignored mixing between different pairing channels, e.g.
$\Delta_{2+}$ and $\Delta_{2-}$, that is allowed by the Zeeman field. We will
justify this later.

Figure 2: (a) Schematic representation of the interstitial Fe atom and its
surrounding triplet pairing state, separated from the spin-singlet pairing
realized away from the atom. (b) Corresponding total probability distribution,
along the $\hat{\mathbf{z}}$ direction, for the zero-energy mode Kramers pair.
The system size is 600 lattice sites, divided into three regions of
singlet/triplet (topological)/singlet pairings with 200 sites each. The
chemical potential is $\mu=0\,\mathrm{meV}$, and $k_{x}=k_{y}=0$.

To proceed, we need to discuss the coupling constants $g_{l}$. We assume,
without specifying a mechanism, that there is attractive interaction in the
singlet channel, since this is the state realized in the bulk. As for the
triplet states, according to what we discussed above, only $\Delta_{2+}$ can
correspond to a topological pairing state, which can display zero-energy
modes. It is therefore crucial to identify the microscopic interactions that
favors $g_{2\pm}$ over $g_{3\pm}$. We start from a generalized Hubbard-
Kanamori interacting Hamiltonian Kanamori (1963); Georges  _et al._ (2013);
Graser  _et al._ (2009) for the $p_{z}$, $d_{xz}$, and $d_{yz}$ orbitals,
which includes anisotropic Hund’s terms $J_{1}$, $J_{2}$ and $J_{3}$. We find
a simple condition under which there is an effective attractive interaction
only for the $\Delta_{2\pm}$ states (see Supplementary Information):
$J_{1}>J_{2}$ and $J_{3}+J_{1}-J_{2}<V<J_{3}+J_{2}-J_{1}$, where $V$ is the
inter-orbital Hubbard repulsion. We assume this condition is satisfied, set
$g_{2\pm}=g_{t}$, and focus hereafter on the $\Delta_{2\pm}$ and $\Delta_{1}$
states only.

Fig. 1 (a) shows the phase boundaries for the different pairing states in the
parameter space of Zeeman splitting and the coupling constants ratio
$g_{t}/g_{s}$. The Zeeman field is taken in the $\hat{\mathbf{z}}$ or
$\hat{\mathbf{x}}$ (in-plane) direction. The phase boundaries between
$\Delta_{2+}$ and $\Delta_{2-}$ are quite separated, such that $\Delta_{2+}$
dominates for small values of the coupling constant ratio, regardless of the
field direction. This not only shows that $\Delta_{2+}$ is the favored pairing
state for larger Zeeman coupling, but also that the Zeeman-induced mixing with
$\Delta_{2-}$ should be small, which justifies dropping this term in the gap
equations. Fig. 1 (a) shows that phase boundary is almost insensitive to the
direction of the Zeeman field.

To show that $\Delta_{2+}$ pairing is robust with respect to changes in the
chemical potential, Fig. 1 (b,c) shows the small changes experienced by the
$\Delta_{2+}$-$\Delta_{1}$ phase boundary for four different values of $\mu$.
Therefore, upon increasing the Zeeman coupling, the pairing state will
transition from intra-orbital singlet $\Delta_{1}$ to inter-orbital triplet
$\Delta_{2+}$.

While the preferred pairing is robust with respect to the chemical potential
modification, its topological nature depends on the chemical potential. The
relevant time-reversal invariant momenta are at the $\Gamma$ and $Z$ points.
For $\mu>-0.577\,\mathrm{eV}$, the Fermi surface at the $Z$ point is filled
whereas for $\mu<0.07\,\mathrm{eV}$, all three bands at $\Gamma$ are empty.
Therefore, in this range of $\mu$, the $\Delta_{2+}$ pairing state is
topological. Consequently, in Fig. 1 (c,d), only two chemical potential values
$\mu=0\,\mathrm{eV}$ and $\mu=0.05\,\mathrm{eV}$ correspond to a topological
superconductor. The cases $\mu=0.075\,\mathrm{eV}$ and $\mu=0.1\,\mathrm{eV}$
are generally non-topological. For $\mu=0.075\,\mathrm{eV}$, the lowest band
at the $\Gamma$ point is quite close to the Fermi energy. As a result, a
topological phase transition can be induced by increasing the pairing
amplitude or further tuning the chemical potential. Since this requires
considerable tuning, difficult to achieve experimentally, we focus on values
of the chemical potential where $\Delta_{2+}$ is strictly topological.

When the chemical potential is in the band-inversion gap, the normal-state
itself is topological Xu  _et al._ (2016) (see Fig. 1 (c)). This is manifested
by Majorana bound states at the end of the vortex cores where they cross the
sample boundary Fu and Kane (2008); Wang  _et al._ (2018). In this case, the
bulk pairing state is intra-orbital spin-singlet $\Delta_{1}$ which is not
topological in the bulk, but it also induces topological superconductivity on
the surface Fu and Kane (2008). Since our interest is in the vicinity of bulk
impurities, we will not discuss surface effects.

Figure 3: (a) Schematic representation of bringing two interstitial Fe atoms
closer. (b) Dependence of the in-gap energy states on the effective two
impurity separation in $\hat{\mathbf{z}}$ direction. The two impurity state is
simulated by a five region sandwich with superconducting pairings
singlet/triplet/singlet/triplet/singlet; the distance between the boundaries
of the two triplet regions is $N_{S}$ sites. (c,d) Corresponding total
probability when two Fe atoms are farther apart and closer to each other,
respectively. For (c,d) the system is the same as in (b) with $N_{S}=25$ for
(c) and $N_{S}=40$ for (d). Chemical potential is $\mu=0\,\mathrm{meV}$ and
$k_{x}=k_{y}=0$.

Majorana modes: Having identified the dominant pairing state in the presence
of Zeeman coupling, we now examine the possible boundary mode localized
between regions with $\Delta_{2+}$ and spin-singlet $\Delta_{1}$ pairings.
Since for a considerable range of chemical potential values the $\Delta_{2+}$
pairing is topological, the presence of zero-energy edge modes is expected.
The $\Delta_{2+}$ pairing respects time-reversal symmetry, therefore the zero
modes are Kramers pairs with opposite spin.

Figs. 2 and 3 depict the structure of zero modes for single and double
magnetic impurities. Figs. 2 (a) and (b) show the case of a single Fe
impurity. The effective magnetic field generated by the magnetic impurity
modifies the superconducting order parameter in its surroundings, such that
$\Delta_{2+}$ pairing is realized. The boundary with the bulk singlet pairing
features zero energy modes. Therefore, scanning tunneling spectroscopy close
to the interstitial iron atom should detect a robust zero bias peak Yin  _et
al._ (2015); Fan  _et al._ (2021); Wang  _et al._ (2021). The observed zero
mode is different from usual Shiba state observed next to a magnetic impurity
Balatsky  _et al._ (2006b), whose energy depends on the amplitude of the
exchange interaction and requires special tuning to be fixed at zero energy.
In contrast, in our case, as long as the magnetic field is sufficiently large
to favor topological $\Delta_{2+}$ pairing, the zero-energy mode is robust.

Fig. 3 shows the evolution of the zero modes when the two impurities are
brought close to each other. As shown in Fig. 3 (b), while two of the zero
modes are gapped out, the ones located on the outer edge of the two-impurity
region remain robust. This is consistent with the experimental observation in
Ref. Yin  _et al._ (2015) and is different from the expected hybridization
between Shiba states when the two impurities are brought close to each other.
This shows that the corresponding zero modes form bubbles around the magnetic
impurities, which combine into a single bubble enclosing both atoms when the
impurities are close to each other.

Finally, we comment on the size of the bubble formed around the interstitial
magnetic impurity. Analyzing the Friedel oscillations from neutron scattering
data, Ref. Thampy  _et al._ (2012) used a five-orbital Hubbard model to
estimate that the nearest-neighbor spin exchange between the interstitial and
the surrounding Fe atoms is about 70 meV. This is an order of magnitude larger
than the Zeeman field required to observe the triplet pairing state and the
topological superconducting region around the magnetic impurity (see Fig. 1).
Therefore, despite the exchange interaction being short-ranged, we expect the
radius of the bubble to be in the range of a multiple lattice spacing,
consistent with experimental observations Yin  _et al._ (2015).

conclusions: We have shown that the peculiar features of the zero bias peak
experimentally observed at interstitial iron atoms in FST can be reconciled if
we consider the modification of the superconducting order parameter close to
the impurity. We have shown that the Zeeman field prefers inter-orbital
triplet pairing which, for a certain range of chemical potential values, is
topological. As a result, zero-energy modes naturally occur at the boundary
between inter-orbital triplet and intra-orbital singlet pairings. These modes
are robust and do not get modified by changes in the exchange interaction,
contrary to the conventional Shiba states. It has been experimentally observed
that the zero modes are not spin-polarized Wang  _et al._ (2021), a feature
that is also consistent with our theoretical model. The obtained triplet
pairing state near the impurity respects time-reversal symmetry, such that the
zero modes are always doubly-degenerate and have opposite spin due to the
Kramers theorem. This mechanism is also capable of explaining the robustness
of the zero modes when two impurities are brought close to each other, since
in this case only two out of four modes get hybridized.

###### Acknowledgements.

We thank Armin Rahmani, Andrey V. Chubukov, Jay D. Sau and Ruixing Zhang for
fruitful discussions. AK and PG are supported by NSF-DMR2037996. PG also
acknowledges support from NSF-DMR1824265. RMF was supported by the U. S.
Department of Energy, Office of Science, Basic Energy Sciences, Materials
Sciences and Engineering Division, under Award No. DE-SC0020045. Part of this
work was performed at the Aspen Center for Physics, which is supported by
National Science Foundation grant PHY-1607611.

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