# Unique quantum impurity states driven by a vortex in topological
superconductors

Wei Su  Department of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China  College of Physics and Electronic Engineering, Center
for Computational Sciences, Sichuan Normal University, Chengdu 610068, China
Beijing Computational Science Research Center, Beijing 100084, China  Rui Wang
<EMAIL_ADDRESS>National Laboratory of Solid State Microstructures and
Department of Physics, Nanjing University, Nanjing 210093, China
Collaborative Innovation Center for Advanced Microstructures, Nanjing 210093,
China  Changfeng Chen  Department of Physics and Astronomy, University of
Nevada, Las Vegas, Nevada 89154, USA  Xiaoqun Wang<EMAIL_ADDRESS>Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China  Collaborative Innovation Center for Advanced Microstructures,
Nanjing 210093, China  Tsung-Dao Lee Institute, Shanghai Jiao Tong University,
Shanghai 200240, China  Beijing Computational Science Research Center, Beijing
100084, China

###### Abstract

The interplay of magnetic impurity and vortex in a topological superconductor
is of fundamental interest with major implications for implementing quantum
computation. There are multiple degrees of freedom interacting with the
impurity in the system, including the Majorana zero mode (MZM), the Caroli de
Gennes Matricon (CdGM) states, and the electron bulk states that form Cooper
pairs, which makes the impurity pinned vortex state elusive to date in
topological superconductors. Here, we present an accurate solution of the
problem, based on a generalized mapping scheme and the density-matrix
renormalization group (DMRG) method. We identify in-gap states that are
distinct from the established Yu-Shiba-Rusinov (YSR) states. The newly found
in-gap physics is driven by three prominent mechanisms: (i) the coupling of
impurity and bulk states leading to competition between Kondo screening and
Cooper pairing, resulting in a singlet-doublet quantum phase transition, (ii)
the coupling of impurity and CdGM states introducing an effective Zeeman
splitting to the doublet state, and (iii) the coupling of impurity and MZM
turning the singlet-doublet transition into a crossover. These mechanisms
cooperatively produce a unique spin-resolved local density of states. Despite
the MZM, a robust nearly zero-energy peak is generated, with comparable height
but opposite spin polarization as that of the MZM. These results signify novel
emergent physics and offer insights to elucidate intriguing experimental
phenomena.

###### pacs:

72.15.Qm,75.20.Hr,74.20.-z

Introduction.– MZMs are self-conjugate particles obeying non-Abelian
statistics, which in recent years have attracted great interest for their
promise in implementing topological quantum computation [1, 2]. Two-
dimensional topological superconductors (TSCs) are widely believed as the
suitable platform to host MZMs. Recent experimental progress suggests that
such TSCs can be realized either in fabricated heterostructures between
topological insulators and normal superconductors (SCs), or in certain
materials that simultaneously hosts nontrivial band topology and intrinsic
superconductivity, such as iron-based SCs,
$\mathrm{FeTe}_{0.55}\mathrm{Se}_{0.45}$. In both cases, the Cooper pairs in
the SC states proximate into the topological surface state with nontrivial
spin textures, resulting in an effective $p+ip$ pairing symmetry that are
conducive to hosting MZMs in the vortex regions. For clean TSCs in an external
magnetic field, the zero-bias peaks (ZBPs) detected by STM were taken as the
signature of MZMs [3, 4, 5, 6, 7, 8].

Impurities play important yet contrasting roles in influencing and probing
pertinent physics in SCs. On the one hand, magnetic scatterings of the SC bath
generate localized in-gap states, known as YSR states [9, 10], which offer
insights into the nature of the SC states. Studies of both s-wave SC and TSCs
without vortex reveal a quantum phase transition (QPT) between a singlet and
doublet state, which is driven by competing Kondo scale and SC gap [11, 12].
These states are experimentally accessible by STM measurements [13, 14] and
can be used to assess the underlying SC pairing symmetries. Moreover, MZMs can
be produced and manipulated without an external magnetic field [15, 16]. It
has been proposed [15] that a quantum anomalous vortex state can be induced by
a classical magnetic impurity via the Elliot-Yafet spin-orbital coupling, and
this scenario was supported by the STM observation of robust ZBPs near
interstitial Fe adatoms [17, 18, 19]. On the other hand, interactions of
impurities with other relevant degrees of freedom may complicate the physical
processes by generating features that could mask the origin of the ZBPs, thus
requiring in-depth elucidation. Recent spin-resolved STM studies found equal
weights between the opposite spin components of the ZBPs, raising the puzzling
prospects that the ZBPs are of YSR-type, rather than the previously expected
MZMs [20]. Currently, interpretation of the measured STM spectra in systems
with the coexistence of vortex and impurity are largely based on separate
physics of impurity induced bound states and vortex induced bound states.
However, the interaction between these distinct degrees of freedom may
qualitatively change the benchmark properties. Therefore, understanding the
interplay between magnetic impurity and vortex is crucial to clarifying the
fundamental physics underlying the experimental observations.

In this Letter, we study the generic problem of a vortex pinned by a quantum
magnetic impurity, as illustrated in Fig.1(a), with the aim to unveil the
salient effects of the quantum impurity on TSCs in the presence of the vortex.
In stark contrast to the vortex free case, we find that for TSCs with a vortex
where the MZM emerges, the previously established QPT between the singlet and
doublet states disappears and is replaced by a crossover between a resonant
scattering (RS) and a local moment (LM) regime. Moreover, both the impurity
and the MZM generate low-energy peaks in the local density of states (LDOS)
that possess opposite spin polarization. These results indicate that the
interplay between the MZMs and Kondo effect can lead to ZBPs with comparable
weights in distinct spin components, opening a new avenue for viable
interpretation of the intriguing and puzzling experimental STM observations
[20]. These findings showcase emergent novel physics stemming from the
interplay between MZMs and Kondo impurities, advancing fundamental
understanding of the intricate physical processes and underlying mechanisms
associated with the magnetic impurity pinned vortex in TSCs.

Impurity-vortex model in TSC.– We consider a quantum magnetic impurity in 2D
TSC with a magnetic flux pinned at the impurity as shown in Fig.1(a). This
system is modeled by the Hamiltonian
$H=H_{\textrm{imp}}+H_{\textrm{hyb}}+H_{0}$, where $H_{\textrm{imp}}$
describes an Anderson impurity

$H_{\textrm{imp}}=\sum_{\sigma}\epsilon_{f}\hat{f}_{\sigma}^{\dagger}\hat{f}_{\sigma}+U\hat{n}_{f\uparrow}\hat{n}_{f\downarrow},$
(1)

with $\hat{f}^{\dagger}_{\sigma}$ ($\hat{f}_{\sigma}$) being the creation
(annihilation) operator for spin $\sigma$ of the impurity and
$\hat{n}_{f\sigma}=\hat{f}^{\dagger}_{\sigma}\hat{f}_{\sigma}$ the number
operator. $\varepsilon_{f}$ is the energy of the local orbital and $U$ the on-
site Hubbard interaction. We consider here the symmetric Anderson model with
$\epsilon_{f}=-U/2$ and generalization to the asymmetric case is
straightforward.

The local impurity is hybridized with the bath via

$H_{\textrm{hyb}}=\sum_{\sigma}\int
d\mathbf{r}V(r)\left[\hat{f}_{\sigma}^{\dagger}\hat{c}_{\mathbf{r},\sigma}+h.c.\right],$
(2)

where $\hat{c}^{\dagger}_{\mathbf{r},\sigma}$ ($\hat{c}_{\mathbf{r},\sigma}$)
is the creation (annihilation) operator of the SC bath at $\mathbf{r}$. We
assume an isotropic coupling $V(\mathbf{r})=V(r)=V_{0}e^{-r/r_{0}}$ and set
$r_{0}=0.7\xi$ [15], where $\xi$ is the superconducting coherence length and
used as the length unit.

Figure 1: (a) Schematic plot of a magnetic flux pinned by a quantum magnetic
impurity in a 2D TSC. (b, c) Effective star models for the TSC without or with
a vortex, respectively. $\gamma_{n,m}$ denotes the Bogoliubov quasi-particles,
where $n$ denotes the energy level and $m$ represents for the orbital angular
momentum (OAM). The quasi-particles $\gamma^{\prime}_{n,1}$ are related to
$\gamma_{n,1}$ via the particle-hole transformation. (d) The effective chain
model for the TSC without and with a vortex.

We consider a 2D TSC with effective $p+ip$ symmetry induced by spin-momentum
locking, i.e,

$H_{0}=\int\mathrm{d}\mathbf{r}\;\sum_{\sigma,\sigma^{\prime}}\hat{c}^{\dagger}_{\mathbf{r},\sigma}\left[v_{F}\left[\left(\bm{\sigma}\times\mathbf{p}\right)\cdot\mathbf{z}\right]_{\sigma,\sigma^{\prime}}-\mu\delta_{\sigma,\sigma^{\prime}}\right]\hat{c}_{\mathbf{r},\sigma^{\prime}}\\\
+\int\mathrm{d}\mathbf{r}\;\Delta(\mathbf{r})(\hat{c}^{\dagger}_{\mathbf{r,\uparrow}}\hat{c}^{\dagger}_{\mathbf{r,\downarrow}}+h.c.).$
(3)

Eq.(1)-Eq.(3) describe the Anderson impurity in extrinsic TSCs, e.g., the
topological surface state in proximity to a normal $s$-wave SC or the surface
TSC of $\mathrm{FeTe}_{0.55}\mathrm{Se}_{0.45}$[5, 15].

The magnetic flux pinned at the impurity generates the vortex state in TSCs,
resulting in a winding phase in the gap function
$\Delta(\mathbf{r})=\Delta(r)e^{i\nu\theta}$, with $\mathbf{r}=(r,\theta)$ and
the winding number $\nu$ being an integer. We take $\Delta(r)=\Delta_{0}$ for
$\nu=0$ and $\Delta(r)=r\Delta_{0}/\sqrt{r^{2}+\xi^{2}}$ for $\nu=1$[21], such
that the characteristic length of the vortex is of the order of SC coherence
length $\xi$. Although $\Delta(r)$ is not self-consistently determined here,
it will not qualitatively affect the impurity states, as the specific form of
$\Delta(r)$ only generates minor modifications to the CdGM vortex core states
[22, 23]. In the following, we adopt the parameter values of
$v_{F}=\sqrt{2\pi}V_{0}=1$, $\mu=0.2$ and $\Delta_{0}=0.1$ without losing
generality in probing the main intended physics.

Mapping to star and chain models.– The incorporation of a vortex into the
quantum impurity problem is highly nontrivial. To take into account the
contributions from the MZM and CdGM states in addition to the bulk electrons,
we generalize the conventional mapping scheme in NRG calculations.
Specifically, we adopt a disk geometry of the bath as shown in Fig.1(a). The
resulting vortex-impurity model has rotational symmetry, allowing an expansion
with respect to 2D spherical functions with the characteristic quantum number
$m$, i.e., the orbital angular momentum (OAM). By diagonalizing $H_{0}$
through Bogoliubov transformation in real space, the original impurity model
is mapped into the star models as shown in Fig.1(b) and Fig.1(c) for $\nu=0$
and $\nu=1$, respectively, with the impurity directly coupled to Bogoliubov
quasi-particles[24].

The OAM index $m$ plays a key role in determining the structure of the star
model. First, for the vortex-free TSC ($\nu=0$), the impurity is coupled to
the states with OAM $m=0$ and $m=-1$ [11], as shown in Fig.1(b). These two
orbitals form the pseudo-spin degree of freedom and screen the impurity. In
the presence of a vortex ($\nu=1$), however, the situation is totally
different, as shown in Fig.1(c). The spin-down impurity state couples to the
Bogoliubov particles and CdGM modes via hopping only, whereas the spin-up
state couples to the Bogoliubov particles and the MZM via both hopping and
pairing. The different coupling mechanisms can lead to distinct impurity
states with the presence of a vortex.

We further tridiagonalize the star models by the block Lanczos approach [25,
26, 27, 24] to generate 1D chains with only nearest neighbor interactions,
where the impurity is located at the boundary, as shown in Fig.1(d). We
preserve the MZM-impurity and CdGM-impurity couplings in this procedure to
keep track of their distinct impacts on the system. The chain length depends
on the truncation of the Bessel basis $j_{\mathrm{max}}$. We use
$j_{\mathrm{max}}=100$ in our calculations. The resulting chain is of the
length $400$ ($200$) for the spin-down (spin-up) environment.

The star models in Fig.1(b)-(c) and chain models in Fig.1(d) exhibit different
symmetries for TSC without or with vortex. The real spin in the TSC is not
conserved because of the spin-momentum locking. However, the pseudo-spin
$S_{z}$, which comprises $m=0$ and $m=-1$, is a conserved quality for $\nu=0$.
However, the vortex further breaks the pseudo-spin symmetry, leaving only two
conserved quantities, namely the total number $N_{\downarrow}$ of the quasi-
particles coupled to $f_{\downarrow}$, and the number parity $P_{\uparrow}$ of
the quasi-particles coupled to $f_{\uparrow}$.

Figure 2: (a) Calculated energy gap $\Delta E_{i}$ as a function of $U$ for
$\nu=1$, which indicates the occurrence of a number of in-gap states. The
states shown are all doubly degenerate for even and odd $P_{\uparrow}$, as a
result of the MZM. For comparison, the result for $\nu=0$ case is shown by the
yellow curve, which displays a singlet-doublet QPT. The occupation number (b)
and the magnetic momentum (c) obtained at the impurity versus $U$ for $\nu=1$.
The same legends are used in (a-c) to denote the states in different symmetry
sectors. “es” denotes the first excited state in the corresponding sector. (d)
shows the entanglement entropy for the impurity, and the mutual information
between the impurity and MZM/CdGM.

Unique in-gap states driven by MZM and CdGM.– We use DMRG to target all in-gap
states for the chain models in Fig.1(d) [24]. In Fig.2(a) we show energy gaps
defined by $\Delta E_{i}=E_{i}-E_{0}$, where $E_{i}$ and $E_{0}$ are the
energies of the $i$-th excited states and ground state, respectively. For the
vortex-free case $\nu=0$, $\Delta E_{1}$ firstly decreases to zero and then
increases with raising $U$ [28, 11, 29], as shown by the yellow curve in
Fig.2(a). The critical point $U_{c}$ indicates a QPT, separating the singlet
($U<U_{c}$) and doublet ($U>U_{c}$) ground state [11].

We now focus on TSC with a vortex. Utilizing the conserved quantities
$P_{\uparrow}$ and $N_{\downarrow}$, all the low-energy states can be
classified into different sectors. In each sector, we calculate the low-energy
spectrum, the average occupation number at the impurity
$\langle\hat{n}_{f\uparrow}+\hat{n}_{f\downarrow}\rangle$ and the impurity
moment $\langle\hat{n}_{f\uparrow}-\hat{n}_{f\downarrow}\rangle$, which are
shown in Fig.2(a), (b) and (c), respectively. As shown in Fig.2(a), the
previously established singlet-doublet transition for the vortex free case
vanishes. Instead, a number of in-gap states take place, without showing any
signature of a QPT.

The impurity ground state can be understood at the two limits, i.e., $U=0$ and
$U\to\infty$. For $U=0$, the charge fluctuations become dominant, as we find
in Fig.2(c) that $\langle\hat{n}_{f\uparrow}-\hat{n}_{f\downarrow}\rangle\sim
0$ for $U=0$, which implies that the ground state is located in the resonant
scattering (RS) regime [28, 30]. For $U\to\infty$, the large Hubbard $U$
ensures the single-occupancy condition at the impurity, as electron number
$\langle\hat{n}_{f\uparrow}+\hat{n}_{f\downarrow}\rangle$ approaches 1 for
large $U$, as shown by Fig.2(b). Moreover, as indicated by Fig.2(c), the local
moment $\langle\hat{n}_{f\uparrow}-\hat{n}_{f\downarrow}\rangle$ saturates to
$\pm 1$ for large $U$, showing that the impurity remains effectively
unscreened and a doublet ground state emerge at this limit. With increasing
$U$ from 0 to $\infty$, there is no signature of any QPTs, implying a
crossover from the RS to the LM regime.

To further validate the nature of ground state, we also calculate the
entanglement entropies (EE) $S_{A}$, which measures the entanglement between
subsystem A and its environment. The black curve in Fig.2(d) is the result of
$S_{\textrm{Imp}}$. $S_{\textrm{Imp}}\rightarrow 2\ln 2$ is reached for
$U\rightarrow 0$, indicating that not only the spin but also the charge
degrees of freedom are entangled with the bath. $S_{\textrm{Imp}}$ decreases
down to vanishingly small values with increasing $U$, reflecting a LM regime
with effectively free local moment. We also calculated the mutual
information[31] between the impurity and MZM or CdGM state. The mutual
information is defined as
$I_{\textrm{A}:\textrm{B}}=S_{\textrm{A}}+S_{\textrm{B}}-S_{\textrm{AB}}$,
which measures the entanglement between two subsystems A and B. As shown in
Fig.2(d), both $I_{\textrm{Imp}:\textrm{MZM}}$ and
$I_{\textrm{Imp}:\textrm{CdGM}}$ vanish at large $U$, which indicates that the
correlations between the impurity and the in-gap modes are indeed negligible
in the LM regime.

Figure 3: The in-gap states as a function of $U$ for different couplings to
the impurity. (a) shows the spectrum with turning off the impurity-MZM and
impurity-CdGM coupling, and (b) shows the results with turning off the
impurity-MZM coupling. (c) The zoom-in spectrum near the critical point with
gradually turning on the impurity-MZM coupling controlled by the factor $a$.
After completely turning on the impurity-MZM coupling with $a=1$, the green
(red, blue) curve in Fig.2(a) will be reproduced by the continuous evolution
of the green (red, blue) curve in (b) and (c), while the dotted curve in (a)
or (b) will be pushed into the continuum.

To examine the crossover behavior in the intermediate $U$ regime, we divide
the system into four parts: the impurity, the CdGM state, the MZM state, and
the bulk states consisting of the Bogoliubov particles with $E>\Delta_{0}$.
Then, we evaluate the low-energy spectrum with artificially tuning the
impurity-CdGM and impurity-MZM coupling. First, we turn off both the impurity-
CdGM and impurity-MZM coupling. The spectrum shown in Fig.3(a) well reproduces
the vortex-free case in Fig.2(a), displaying a singlet-doublet QPT. Then, we
turn on the impurity-CdGM coupling. Interestingly, although the feature of QPT
is still preserved, the CdGM induces a significant splitting of the doublet
state, as shown by Fig.3(b). In fact, through a Schrieffer-Wolff
transformation [32], we show that the impurity-CdGM coupling essentially
contributes to an effective Zeeman field to the impurity [24].

We then gradually turn on the impurity-MZM coupling controlled by a factor $a$
in front of the coupling. Fig.3(c) shows the spectrum around the critical
point $U_{c}$ with raising $a$ up to $0.04$. As long as $a$ is nonzero, the
gapless point of the QPT is turned into an avoided level crossing, reproducing
the previously observed crossover. The avoided level crossing is a direct
result of the model symmetry. With $a=0$, the parity of impurity-bulk system
$P^{\textrm{Bulk+Imp}}_{\uparrow}$ is conserved. The even and odd parity
sectors are degenerate at the critical point. However, such a degeneracy is
broken as long as the MZM-impurity coupling is turned on, which in turn gaps
out the critical point [24].

Figure 4: Spin-resolved in-gap LDOS $\rho_{f}^{\sigma}$ for different values
of $U$. The left corner builds up the one-to-one correspondence between the
LDOS peaks and the level transition processes (both are labeled by the numbers
from 1 to 5). $(P_{\uparrow},N_{\downarrow})$ denotes the symmetry sectors
introduced in Fig.2. Dotted and solid levels with the same color are
degenerate in energy, which are shifted intentionally for the purpose of
clarity.

Spin-resolved LDOS.– We now calculate the spin-resolved local density of
states (LDOS) at the impurity site under zero temperature, which is crucial to
understanding the STM experiments [13, 33, 34, 33]. In the Lehmann
representation

$\rho_{f}^{\sigma}(\omega)=\sum_{m}\left[\left|\langle\psi_{m}|f_{\sigma}^{\dagger}|\psi_{0}\rangle\right|^{2}\delta(\omega+E_{0}-E_{m})\right.\\\
+\left.\left|\langle\psi_{0}|f_{\sigma}^{\dagger}|\psi_{m}\rangle\right|^{2}\delta(\omega+E_{m}-E_{0})\right],$
(4)

where $E_{0}$ ($E_{m}$) is the energy of the ground state $|\psi_{0}\rangle$
($m$-th excited state $|\psi_{m}\rangle$). The LDOS is then obtained with
taking into account all the in-gap states as shown by Fig.2(a), whose energies
satisfy $E_{m}-E_{0}<\Delta$. Besides, a broadening factor $b=0.001$ is chosen
in the calculation to show the peaks more clearly.

As shown by Fig.4, the LDOS is highly spin sensitive, which is expected since
the TSC bath is effectively spin polarized. For $\rho_{f}^{\uparrow}$, there
always takes place a robust zero-energy peak [35, 36], as a result of the MZM,
which is only coupled to $f_{\uparrow}$ as indicated by Fig.1(d). Besides, two
more peaks arise in $\rho_{f}^{\uparrow}$ at higher energy; one comes from the
hole component ($\omega<0$) and the other from the electron component
($\omega>0$). With increasing $U$, both peaks gradually move toward
$\omega=0$. Besides, the hole component peak disappears around $U\sim 1$,
while the electron component saturates at the energy of the CdGM, i.e.,
$E_{\textrm{CdGM}}$ for $U\gtrsim 1$, as indicated by Fig.4.

Remarkably, distinct LDOS feature is found for $\rho_{f}^{\downarrow}$. A
significant peak (labeled by “2”) emerges, which is originated from the
transition between the ground state the first excited state,as indicated by
Fig.2(a). Hence, it is a direct consequence of the RS-LM crossover. With
increasing $U$, the peak continuously evolves and approaches to $\omega=0$ at
large $U$. Furthermore, for $U\gtrsim 1.8$, the peak enjoys comparable height
but opposite spin-polarization as that of the MZM peak, signifying an
interesting LDOS feature relevant to experiments. Last, we also clarify the
origin of all the LDOS peaks by showing their one-to-one correspondence with
the level transition processes, as indicated by Fig.4.

It should be noted that the occurrence of two LDOS peaks with comparable
weight and opposite spin-polarization near zero energy has been observed in
recent experiment in TSC [13]. Although it was explained by YSR states, our
findings show new possibilities stemming from the novel crossover associated
with the impurity-pinned vortex state.

Conclusion and discussion.–  We have employed the DMRG method to study
magnetic impurity pinned vortex in TSC. The results reveal fundamentally
distinct behaviors of the in-gap states in TSC, which are driven by three
physical processes in cooperation with each other. First, the impurity-bulk
coupling generates the competition between Kondo screening and Cooper pairing.
Second, the impurity-CdGM coupling leads to an effective Zeeman splitting for
the impurity. Third, the impurity-MZM coupling produces a crossover between
the RS and LM regime with increasing the Hubbard $U$. The crossover behavior
as well as the in-gap states are expected to be general properties of the
impurity-pinned vortex state in superconductors due to the following reasons.
First, the crossover is the result of the model symmetries at the critical
point, and it should occur as long as the impurity-MZM coupling breaks the
conservation of total spin (pseudospin). Second, since the TSC bath is
polarized by the vortex, the in-gap states are robust against spin-dependent
perturbation, such as the external magnetic field or asymmetric parameters in
the Anderson model. Remarkably, over a wide parameter range, we show that, in
addition to the MZM, the crossover generates another LDOS peak near zero-
energy, which exhibits the comparable weight but opposite spin polarization as
that of MZM. The LDOS and the in-gap physics found here are originated from
the mutual effects from the CdGM and MZM in TSC, and are clearly different in
nature from all known impurity states in superconductors. Our results clarify
the long-standing impurity problem in TSCs with a vortex, which could provide
further guidance to the experimental studies of TSCs.

###### Acknowledgements.

We are grateful to Lin Li and Shang Xin for fruitful discussions. This work
was supported by the Natural Science Foundation of China (No. 11904245) and
Postdoctoral Science Foundation of China (No. 2021M690330), and was also
supported by the High Performance Computing Center of Sichuan Normal
University, China.

Supplementary Material for: Unique quantum impurity states driven by a vortex
in topological superconductors

##  Appendix A An outline of the methods

The incorporation of an additional vortex into the quantum impurity problem is
highly nontrivial at least in the following two senses. First, the DOS of the
bath is complicated by the CdGM vortex core states and MZM, thus the
conventional numerical renormalization group (NRG) method cannot be directly
applied. Second, the interplay between the quantum magnetic impurity and the
MZMs can sharply alter the ground state of impurity state, resulting in novel
physical features available by STM measurements. We adopted the numerical
scheme specified by the following steps, using which we can accurately extract
the effect of the bulk states as well as the in-gap bound states on the
impurity physics.

  1. 1.

Diagonalization of the TSC via Bogoliubov transformation in real space. The
impurity is then explicitly coupled to the Bogoliubov quasi-particles, which
is referred to the star model in the main text. In this way we can directly
keep track of the impurity-MZM and impurity-CdGM coupling (see Sec.B for
details.).

  2. 2.

Tridiagonalization of the star model into a chain with only the nearest
neighbor interactions through block Lanczos. The process can preserve the form
of the impurity-MZM and impurity-CdGM couplings (see Sec.C for details.).

  3. 3.

Application of the DMRG solver to obtain all the in-gap states of the chain
model (see Sec.D for details on implementation and assurance of convergence).

No approximation is made within the above scheme, except for (1) a truncation
of the Bessel functions in step 1 and (2) truncation of basis in DMRG. The
error introduced by both truncations are controllable by examining the
computational convergence.

##  Appendix B  Bogoliubov transformation

In Nambu basis
$\Psi(\mathbf{r})=(c_{\mathbf{r},\uparrow},c_{\mathbf{r},\downarrow},c^{\dagger}_{\mathbf{r},\downarrow},-c^{\dagger}_{\mathbf{r},\uparrow})^{T}$,
$H_{0}=\int\mathrm{d}\mathbf{r}\;\Psi^{\dagger}(\mathbf{r})\mathcal{H}_{0}\Psi(\mathbf{r})$.
In terms of Nambu spinor
$\Phi_{n}(\mathbf{r})\equiv(u_{n\uparrow},u_{n\downarrow},v_{n\downarrow},-v_{n\uparrow})^{T}$,
the Bogolubov-de Gennes (BdG) equation is
$\mathcal{H}_{0}\Psi_{n}(\mathbf{r})=E_{n}\Psi_{n}(\mathbf{r})$, with

$\mathcal{H}_{0}=\left(\begin{array}[]{cc}H&\Delta(\mathbf{r})\\\
\Delta^{\ast}(\mathbf{r})&-\sigma_{y}H^{\ast}\sigma_{y}\end{array}\right).$
(BS1)

The Hamiltonian $\mathcal{H}_{0}$ can be diagonalized by Bogoliubov
transformation,

$\displaystyle c_{\sigma}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\sum_{n}\left[u_{n\sigma}(\mathbf{r})\gamma_{n}+v_{n\sigma}^{\ast}(\mathbf{r})\gamma_{n}^{\dagger}\right],$
(BS2) $\displaystyle c^{\dagger}_{\sigma}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\sum_{n}\left[u^{\ast}_{n\sigma}(\mathbf{r})\gamma^{\dagger}_{n}+v_{n\sigma}(\mathbf{r})\gamma_{n}\right].$
(BS3)

Note that particle-hole symmetry ensures that if
$\Phi_{n}=\left[u_{n,\uparrow},u_{n,\downarrow},v_{n,\downarrow},-v_{n\uparrow}\right]^{T}$
is an eigenstate with energy $E_{n}$, then
$\left[-v^{\ast}_{n,\uparrow},-v^{\ast}_{n,\downarrow},-u^{\ast}_{n,\downarrow},u^{\ast}_{n\uparrow}\right]^{T}$
is also an eigenstate with energy $-E_{n}$. After transformation,

$H_{0}=\sum_{n}E_{n}\gamma_{n}^{\dagger}\gamma_{n}+E_{g},$ (BS4)

where $E_{g}$ is the ground state energy and, the summation is over the
excitation states with positive energy.

The inverse transformation is given by

$\gamma_{n}^{\dagger}=\int\mathrm{d}\mathbf{r}\;\sum_{\sigma}\left[u_{n\sigma}(\mathbf{r})c_{\sigma}^{\dagger}(\mathbf{r})+v_{n\sigma}(\mathbf{r})c_{\sigma}(\mathbf{r})\right],$
(BS5a)
$\gamma_{n}=\int\mathrm{d}\mathbf{r}\;\sum_{\sigma}\left[u^{\ast}_{n\sigma}(\mathbf{r})c_{\sigma}(\mathbf{r})+v^{\ast}_{n\sigma}(\mathbf{r})c^{\dagger}_{\sigma}(\mathbf{r})\right],$
(BS5b)

with $\gamma_{n}^{\dagger}$ and $\gamma_{n}$ being the creation and
destruction operators of Bogoliubov quasi-particles. They satisfy the
fermionic anti-commutation relation

$\displaystyle\\{\gamma_{n},\gamma_{n^{\prime}}\\}=\\{\gamma_{n}^{\dagger},\gamma_{n^{\prime}}^{\dagger}\\}=0,\
\ \\{\gamma_{n},\gamma_{n^{\prime}}^{\dagger}\\}=\delta_{nn^{\prime}}.$ (BS6)

Figure BS1: Single particle spectrum for $\nu=1$ without impurity. $m$ is the
OAM index.

For the TSC, the helical Dirac surface state carries an extra Berry phase as
$\partial_{x}\pm\partial_{y}=\exp(\pm
i\theta)\left(\partial_{r}\pm\frac{i}{r}\partial_{\theta}\right)$[37, 15]. The
BdG equation thus can be solved by assuming the following wave function

$\Phi^{\textrm{TSC}}_{nm}(r,\theta)=e^{im\theta}\left[u_{nm\uparrow}(r),u_{nm+1\downarrow}(r)e^{i\theta},\right.\\\
\left.v_{nm-\nu\downarrow}(r)e^{-i\nu\theta},-v_{nm-\nu+1\uparrow}(r)e^{-i(\nu-1)\theta}\right]^{T},$
(BS7)

where $m=0,\pm 1,\pm 2,\cdots$ are integers. The particle-hole symmetry is
defined between OAM $m$ and $-m+\nu-1$. For $\mu=0$, there exists another
particle-hole symmetry within each OAM $m$. This symmetry is inherited from
the particle-hole symmetry of the surface state at the Dirac point.

The Bogoliubov transformation that diagonalizes $\mathcal{H}_{0}$ of the TSC
is given by

$c_{\uparrow}(\mathbf{r})=\frac{1}{\sqrt{2\pi}}\sum_{nm}\left[e^{im\theta}u_{nm\uparrow}(r)\gamma_{nm}\right.\\\
\left.+e^{-im\theta}v^{\ast}_{nm-\nu+1\uparrow}(r)e^{i(\nu-1)\theta}\gamma_{nm}^{\dagger}\right],$
(BS8)
$c_{\downarrow}(\mathbf{r})=\frac{1}{\sqrt{2\pi}}\sum_{nm}\left[e^{im\theta}u_{nm+1\downarrow}(r)e^{i\theta}\gamma_{nm}\right.\\\
\left.+e^{-im\theta}v^{\ast}_{nm-\nu\downarrow}(r)e^{i\nu\theta}\gamma_{nm}^{\dagger}\right].$
(BS9)

In practice, the radial functions $u(r)$ and $v(r)$ are solved by expanding
with the Bessel functions

$\phi_{m,j}(r)=\frac{\sqrt{2}}{RJ_{m+1}(\beta_{m,j})}J_{m}\left(\beta_{m,j}\frac{r}{R}\right).$
(BS10)

These Bessel functions form a complete set of basis on a disc with radius $R$
and $j=1,2,\cdots,j_{\textrm{max}}$. $j_{\textrm{max}}$ should be $\infty$ in
principle, but a truncation is used in practice. The BdG equation is then
solved in the basis of

$(u_{m,j,\uparrow},u_{m+1,j,\downarrow},v_{m-\nu,j,\downarrow},-v_{m-\nu+1,j,\uparrow})_{j=1\cdots
j_{\textrm{max}}}.$

Diagonalizing the corresponding $4j_{\textrm{max}}\times 4j_{\textrm{max}}$
Hamiltonian matrices numerically, we get the Bogoliubov quasi-particle
spectrum in Fig.BS1. $m$ being an integer for the TSC with a vortex is crucial
for supporting the MZM in the OAM $m=0$[37]. In addition to the MZM, there are
CdGM states[38], carrying integer quantum numbers of $m=\pm 1,\pm 2,\cdots$.

After transformation, the hybridization between the impurity and TSC is

$H^{\textrm{TSC}}_{\textrm{hyb}}=\sum_{\sigma}\int\mathrm{d}\mathbf{r}\;V(r)\hat{f}_{\sigma}^{\dagger}\hat{c}_{\mathbf{r},\sigma}+h.c.\\\
=\sqrt{2\pi}\sum_{n}\int\mathrm{d}r\;rV(r)f^{\dagger}_{\uparrow}\left[u_{n0\uparrow}(r)\gamma_{n0}+v^{\ast}_{n0\uparrow}(r)\gamma_{n\nu-1}^{\dagger}\right]\\\
+\sqrt{2\pi}\sum_{n}\int\mathrm{d}r\;rV(r)f^{\dagger}_{\downarrow}\left[u_{n0\downarrow}(r)\gamma_{n,-1}+v^{\ast}_{n0\downarrow}(r)\gamma_{n,\nu}^{\dagger}\right]\\\
+h.c..$ (BS11)

It is seen that both $f_{\uparrow}$ and $f_{\downarrow}$ couple to $m=0,-1$
channels for $\nu=0$, while $f_{\uparrow}$($f_{\downarrow}$) couples to the
$m=0$($m=\pm 1$) channel(s) for $\nu=1$.

Eq.(BS11) combined with Eq.(BS4) and $H_{\textrm{imp}}$ defines the star
models, where the impurity directly couples to Bogoliubov quasi-particles. For
TSC with $\nu=1$, a particle-hole transformation can be performed for
$\gamma^{\dagger}_{n,1}\to\gamma^{\prime}_{n,1}$. The pairings between the
impurity and $\gamma^{\dagger}_{n,1}$ are transformed into hopping terms
between the impurity and $\gamma^{\prime}_{n,1}$, correspondingly. Note that
the pairing persists in the coupling between $f_{\uparrow}$ and
$\gamma_{n,0}$.

##  Appendix C  Block Lanczos and the chain models

We use block Lanczos to transform the star models to chain models that contain
only nearest interactions[25, 26, 27], which are in a more suitable form for
numerical calculations. Using the coupling coefficient in Eq.(BS11) as the
initial vector and tridiagonalize the bulk Hamiltonian Eq.(BS4), the resulting
chain model for the vortex-free TSC is obtained as

$H_{\textrm{chain}}^{\nu=0}=H_{\textrm{imp}}+\left(\sum_{\sigma}t_{0\sigma}f^{\dagger}_{\sigma}b_{1\sigma}+\Delta^{\prime}_{0}f_{\downarrow}b_{1\uparrow}+h.c.\right)+H_{\textrm{env}}^{\nu=0}$
(CS12)

where

$H^{\nu=0}_{\textrm{env}}=\sum_{\sigma,i=1}^{2j_{\textrm{max}}}\varepsilon_{i\sigma}b^{\dagger}_{i\sigma}b_{i\sigma}+\sum_{i=1}^{2j_{\textrm{max}}}\left(\Delta_{i}b^{\dagger}_{i\uparrow}b^{\dagger}_{i\downarrow}+h.c.\right)\\\
+\sum_{i=1}^{2j_{\textrm{max}}-1}\left(\sum_{\sigma}t_{i\sigma}b^{\dagger}_{i\sigma}b_{i+1\sigma}+\Delta^{\prime}_{i}b_{i\downarrow}b_{i+1\uparrow}+h.c.\right)$
(CS13)

For $\nu=1$, it is desirable to leave the form of the impurity-MZM
($\gamma_{0,0}$) and impurity-CdGM ($\gamma_{0,-1}$) coupling intact. This can
be achieved by making the coupling terms between $f_{\sigma}$ and
$\gamma_{0,0(-1)}$ in Eq.(BS11) invariant, while using other coupling terms as
the initial vector to tridiagonalize the bulk Hamiltonian,
$H_{0}^{\prime}=\sum_{n,m=0,\pm
1}^{\prime}E_{n,m}\gamma_{n,m}^{\dagger}\gamma_{n,m}$ (Here $\sum_{n,m=0,\pm
1}^{\prime}$ excludes terms with $(n,m)=(0,0)$ and $(0,-1)$). This procedure
leads to the following chain model:

$H_{\textrm{chain}}^{\nu=1}=H_{\textrm{imp}}+H_{\textrm{chain}}^{\nu=1,\uparrow}+H_{\textrm{chain}}^{\nu=1,\downarrow}\\\
+E_{0,0}\gamma_{0,0}^{\dagger}\gamma_{0,0}+E_{0,-1}\gamma_{0,-1}^{\dagger}\gamma_{0,-1}\\\
+[f^{\dagger}_{\uparrow}(U_{00\uparrow}\gamma_{0,0}+V_{00\uparrow}\gamma_{0,0}^{\dagger})+U_{00\downarrow}f^{\dagger}_{\uparrow}\gamma_{0,-1}+h.c.]$
(CS14)

where $U_{nm\sigma}=\sqrt{2\pi}\int\mathrm{d}r\;rV(r)u_{nm\sigma}(r)$,
$V_{nm\sigma}=\sqrt{2\pi}\int\mathrm{d}r\;rV(r)v^{\ast}_{nm\sigma}(r)$, and

$H^{\nu=1,\uparrow}_{\textrm{chain}}=\left(t_{0}f^{\dagger}_{\uparrow}b_{1\uparrow}+\Delta_{0}f_{\uparrow}b_{1\uparrow}+h.c.\right)+H^{,\nu=1,\uparrow}_{\textrm{env}},$
(CS15)
$H^{\nu=1,\uparrow}_{\textrm{env}}=\sum_{i=1}^{2j_{\textrm{max}}}\varepsilon_{i}b^{\dagger}_{i\uparrow}b_{i\uparrow}\\\
+\sum_{i=1}^{2j_{\textrm{max}}-1}\left(t_{i}b^{\dagger}_{i\uparrow}b_{i+1\uparrow}+\Delta_{i}b_{i\uparrow}b_{i+1\uparrow}+h.c.\right),$
(CS16)
$H^{\nu=1,\downarrow}_{\textrm{chain}}=\left(t_{0}f^{\dagger}_{\downarrow}b_{1\downarrow}+h.c.\right)+H^{\nu=1,\downarrow}_{\textrm{env}},$
(CS17)
$H^{\nu=1,\downarrow}_{\textrm{env}}=\sum_{i=1}^{4j_{\textrm{max}}}\varepsilon_{i}b^{\dagger}_{i\downarrow}b_{i\downarrow}+\sum_{i=1}^{4j_{\textrm{max}}-1}\left(t_{i}b^{\dagger}_{i\downarrow}b_{i+1\downarrow}+h.c.\right)$
(CS18)

##  Appendix D  Numerical details of DMRG calculations

The largest number of states kept in DMRG is $2,000$ for TSC without the
vortex, with the truncation error on the density matrix $\epsilon<10^{-10}$.
For the TSC with $\nu=1$, keeping about $300$ states is sufficient to reach
the same level of accuracy. For $U=0$, the resulting errors in energy are less
than $10^{-7}$ compared to the exact results. We used $R=100$ and
$j_{\textrm{max}}=100$ in our calculations, with the topological degeneracy
preserved to the order of $10^{-3}\Delta_{0}$. The convergence of
$j_{\textrm{max}}$ is carefully checked with $j_{\textrm{max}}=60\to 140$,
with energy differences being less than $10^{-3}\Delta_{0}$.

We mention here that although it is easier to obtain the ground state and the
low-energy excited states in the gapped TSC compared to the gapless thermal
baths, local minimums can still take place if one starts from random initial
states. To avoid the local minimums, we first obtain the low-energy states for
short chains (with the impurity), and then successively add into the system
more and more sites of the environment.

##  Appendix E Schrieffer-Wolff transformation

To reveal more clearly the effect of the CdGM mode, we perform the Schrieffer-
Wolff transformation $\mathcal{U}=e^{\mathcal{S}}$ to eliminate the coupling
term to linear order in $V$[32]. The unitarity requires
$\mathcal{U}^{-1}=\mathcal{U}^{\dagger}$, implying
$\mathcal{S}^{\dagger}=-\mathcal{S}$. The transformed Hamiltonian can be
expressed as follows:

$\displaystyle H_{\textrm{eff}}$ $\displaystyle=$
$\displaystyle\mathcal{U}H\mathcal{U}^{\dagger}=\sum_{0}^{\infty}\frac{1}{n!}[\mathcal{S},H]_{n}$
$\displaystyle\approx$
$\displaystyle\mathcal{H}_{0}+H_{\textrm{hyb}}+[\mathcal{S},\mathcal{H}_{0}]+[\mathcal{S},H_{\textrm{hyb}}]+\frac{1}{2}[[\mathcal{S},[\mathcal{S},\mathcal{H}_{0}]]$

with $\mathcal{H}_{0}\equiv H_{0}+H_{\textrm{imp}}$. By demanding

$H_{\textrm{hyb}}+[\mathcal{S},\mathcal{H}_{0}]=0,$ (ES20)

$H_{\textrm{hyb}}$ is then eliminated to linear order, and the transformed
Hamiltonian becomes

$H_{\textrm{eff}}=\mathcal{H}_{0}+\frac{1}{2}[\mathcal{S},H_{\textrm{hyb}}].$
(ES21)

Figure ES2: Strength of the effective Zeeman term and that contributed by the
CdGM mode as a function of $U$.

The generator for $\mathcal{S}=\mathcal{S}_{0}-\mathcal{S}_{0}^{\dagger}$
reads as

$\mathcal{S}_{0}=\sum_{n}\left\\{U^{\ast}_{n\uparrow}\left[\frac{1-n_{f\downarrow}}{E_{n,0}-\epsilon_{\uparrow}}+\frac{n_{f\downarrow}}{E_{n,0}-\epsilon_{\uparrow}-U}\right]\gamma_{n,0}^{\dagger}f_{\uparrow}\right.\\\
-V^{\ast}_{n\uparrow}\left[\frac{1-n_{f\downarrow}}{E_{n,0}+\epsilon_{\uparrow}}+\frac{n_{f\downarrow}}{E_{n,0}+\epsilon_{\uparrow}+U}\right]\gamma_{n,0}f_{\uparrow}\\\
+U^{\ast}_{n\downarrow}\left[\frac{1-n_{f\uparrow}}{E_{n,-1}-\epsilon_{\downarrow}}+\frac{n_{f\uparrow}}{E_{n,-1}-\epsilon_{\downarrow}-U}\right]\gamma_{n,-1}^{\dagger}f_{\downarrow}\\\
-\left.V^{\ast}_{n\downarrow}\left[\frac{1-n_{f\uparrow}}{E_{n,1}+\epsilon_{\downarrow}}+\frac{n_{f\uparrow}}{E_{n,1}+\epsilon_{\downarrow}+U}\right]\gamma_{n,1}f_{\downarrow}\right\\},\\\
$ (ES22)

which can be checked by using $[n,f]=-f$ and $[n,f^{\dagger}]=f^{\dagger}$.
The transformed Hamiltonian can be arranged into a concise form

$H_{\textrm{eff}}=\mathcal{H}_{0}^{\prime}+\mathcal{H}_{\textrm{zeeman}}+H_{\textrm{scattering}}\\\
+H_{\textrm{spin-flip}}+H_{\textrm{quasispin-flip}},$ (ES23)

where $\mathcal{H}_{0}^{\prime}$ is a renormalized $\mathcal{H}_{0}$. Note
that both $U$ and $\epsilon_{f}$ is renormalized to $U^{\prime}$ and
$\epsilon_{f}^{\prime}$, and the particle-hole symmetry is broken even for
$\epsilon_{f}=-U/2$ because of the polarized bath.

Importantly, we see from above that there emerges an effective Zeeman
splitting term
$\mathcal{H}_{\textrm{Zeeman}}=h_{z}(n_{f\uparrow}-n_{f\downarrow})$, with the
strength

$h_{z}=\sum_{n}\frac{|U_{n\downarrow}|^{2}}{E_{n,-1}-\epsilon_{f}}-\sum_{n}\frac{|U_{n\uparrow}|^{2}}{E_{n,0}-\epsilon_{f}}\\\
-\sum_{n}\frac{|V_{n\downarrow}|^{2}}{E_{n,1}+\epsilon_{f}+U}+\sum_{n}\frac{|V_{n\uparrow}|^{2}}{E_{n,0}+\epsilon_{f}+U}.$
(ES24)

The contribution of the impurity-MZM coupling to the effective Zeeman term is
$h_{z}^{\textrm{ZM}}=-|U_{0\uparrow}|^{2}\left[1/\epsilon_{f}+1/(\epsilon_{f}+U)\right]$
since $|U_{0\uparrow}|^{2}=|V_{0\uparrow}|^{2}$, which is zero in the
symmetric case $\epsilon_{f}=-U/2$. Fig.ES2 shows the strength of the Zeeman
term, along with the contribution of the CdGM state

$h^{\textrm{CdGM}}_{z}=\frac{|U_{0\downarrow}|^{2}}{E_{0,-1}-\epsilon_{f}}$
(ES25)

It is clearly shown by Fig.S2 that it is the CdGM state that mainly
contributes to the effective Zeeman term.

The scattering term is given by

$H_{\textrm{scattering}}=\sum_{mn}W^{\downarrow}_{mn}n_{f\downarrow}\gamma_{m,0}^{\dagger}\gamma_{n,0}\\\
+\sum_{mn}\left(Z^{\downarrow}_{mn}n_{f\downarrow}\gamma_{m,0}^{\dagger}\gamma_{n,0}^{\dagger}+h.c.\right)\\\
+\sum_{mn}\sum_{l=\pm
1}W^{\uparrow}_{mn,l}n_{f\uparrow}\gamma_{m,l}^{\dagger}\gamma_{n,l}\\\
+\sum_{mn}\left(Z^{\uparrow}_{mn}n_{f\downarrow}\gamma_{m,-1}^{\dagger}\gamma_{n,1}^{\dagger}+h.c.\right)$
(ES26)

This term can be absorbed into a shift of the single-particle energy of the
TSC. Besides, the quasi spin-flip term is given by

$H_{\textrm{quasispin-flip}}=\\\
\sum_{mn}\left(K^{(1)}_{mn}\gamma_{m,0}\gamma_{n,1}f_{\uparrow}f_{\downarrow}+K^{(2)}_{mn}\gamma_{m,0}^{\dagger}\gamma_{n,-1}^{\dagger}f_{\uparrow}f_{\downarrow}+h.c.\right)\\\
+\sum_{mn}\left(L^{(1)}_{mn}\gamma_{m,-1}^{\dagger}\gamma_{n,0}f_{\uparrow}f_{\downarrow}+L^{(2)}_{mn}\gamma_{m,0}^{\dagger}\gamma_{n,1}f_{\uparrow}f_{\downarrow}+h.c.\right)$
(ES27)

This term is usually neglected since double occupation of the impurity site is
suppressed at $U\gg 0$.

Another other interesting term is the spin flipping term $H_{\textrm{spin-
flip}}$, which screens the magnetic impurity. This term is given by

$H_{\textrm{spin-flip}}=\\\
\sum_{mn}\left(J^{(1)}_{mn}\gamma_{m,0}^{\dagger}\gamma_{n,1}f^{\dagger}_{\uparrow}f_{\downarrow}+J^{(2)}_{mn}\gamma_{m,-1}^{\dagger}\gamma_{n,0}f^{\dagger}_{\uparrow}f_{\downarrow}+h.c.\right)\\\
+\sum_{mn}\left(T^{(1)}_{mn}\gamma_{m,-1}^{\dagger}\gamma_{n,0}^{\dagger}f^{\dagger}_{\uparrow}f_{\downarrow}+T^{(2)}_{mn}\gamma_{m,0}\gamma_{n,1}f^{\dagger}_{\uparrow}f_{\downarrow}+h.c.\right).$
(ES28)

##  Appendix F  Deviation from the vortex center

When the impurity is spatially separated from the vortex center by a distance
$\mathbf{r}_{0}$, more OAMs are involved in the hybridization. With enlarging
$\mathbf{r}_{0}$, one expects the impurity physics to be transformed into the
vortex-free case. We consider this question and show that the ground state can
indeed be adiabatically evolved into that of the vortex-free case. We assume
an isotropic coupling $V(\mathbf{r})=V(|\mathbf{r}-\mathbf{r}_{0}|)$ and
expand the interaction as $V(\mathbf{r})=\sum_{m}V_{m}(r)e^{im\theta}$
($V_{m}(r)=\frac{1}{2\pi}\int\mathrm{d}\theta\;V(\mathbf{r})e^{-im\theta}$),
then the hybridization is transformed into:

$\begin{split}H_{\textrm{hyb}}&=\sum_{\sigma}\int\mathrm{d}\mathbf{r}\;V(\mathbf{r})\hat{f}_{\sigma}^{\dagger}\hat{c}_{\mathbf{r},\sigma}+h.c.\\\
&=\sqrt{2\pi}\sum_{mn}\int\mathrm{d}r\;rV_{m}(r)\\\
&\times\left\\{f_{\uparrow}^{\dagger}\left[u_{n,-m\uparrow}(r)\gamma_{n,-m}+v^{\ast}_{n,m,\uparrow}\gamma_{n,m+\nu-1}^{\dagger}\right]\right.\\\
&+\left.f_{\downarrow}^{\dagger}\left[u_{n,-m\downarrow}(r)\gamma_{n,-m-1}+v^{\ast}_{nm\downarrow}\gamma_{n,m+\nu}^{\dagger}\right]\right\\}+h.c.\end{split}$
(FS29)

Figure FS3: Single particle in-gap spectrum as a function of $1/(r_{0}+1)$,
where $r_{0}$ is the distance between the impurity and the vortex center. At
$r_{0}\to\infty$, the impurity state is obtained from $\nu=0$, and the MZM and
CdGM states are obtained from the vortex state without impurity.

This is essentially a two-dimensional model, which is generally difficult to
solve except for $U=0$. For $U=0$, the single particle spectrum is show in
Fig.FS3 as a function of the distance between the impurity and the vortex
center. Here, the truncation of the maximum OAM is chosen as
$m_{\textrm{max}}=170$ for $r_{0}=24$. The data for $r_{0}\to\infty$ is
obtained from vortex free case and vortex state without impurity,
respectively. As we observe in Fig.S4, when the impurity is gradually
separated from the vortex center, the energies of the two polarized in-gap
impurity states evolve continuously and non-monotonically. Meanwhile, they
become closer and closer to the degenerate doublet in the vortex-free case.
The fluctuating behavior of the energy in Fig.S3 is mainly caused by the
spatial distribution of the CdGM and MZM states.

##  Appendix G Symmetry analysis accounting for the RS-LM crossover

Figure GS4: Schematic plot to show that at QPT the accident degeneracy is
lifted by the impurity-MZM coupling between
$P^{\textrm{Bulk+Imp}}_{\uparrow}=0$ and $1$ sectors, leaving only the
topological degeneracy.

In the main text we concluded that the RS-LM crossover is a general result
because of the symmetry at the critical point. We demonstrate this point more
clearly in this section. We intentionally introduced a controlling factor $a$
in front of the impurity-MZM coupling to keep track of its effect on the
transition point. In the non-coupling limit ($a=0$), a QPT occurs at $U\approx
0.15$ with the other parameters given in the main text. An infinitesimal $a$
is sufficient to turn the QPT into a crossover, manifesting as an avoided
level crossing, as we showed in the manuscript.

Here we present a symmetry analysis to account for the avoided level crossing.
For $a=0$ at the QPT, the ground state has quadruple degeneracy.
$N_{\downarrow}$ is the same for those four states, so it is irrelevant in the
following analysis. Taking the free MZM into account, the parity in the bulk-
impurity system ($P^{\textrm{Imp-Bulk}}_{\uparrow}$) and the parity of the MZM
($P^{\textrm{MZM}}$) are conserved separately. The total parity is then given
by $P^{\uparrow}=(P^{\textrm{Imp-Bulk}}_{\uparrow}+P^{\textrm{MZM}})$(mod 2).
The four degenerated states can be separated into two sectors as shown in
Fig.GS4. One is $(P^{\textrm{MZM}},P^{\textrm{Imp-
Bulk}}_{\uparrow})=\left\\{(0,0),(1,0)\right\\}$, and the other is
$(P^{\textrm{MZM}},P^{\textrm{Imp-
Bulk}}_{\uparrow})=\left\\{(0,1),(1,1)\right\\}$. Within each sector, the
remaining degeneracy is of topological origin, as a result of the presence of
MZM. However, the degeneracy of the states in between the two sector is not
topological but an accidental one. We find that the impurity-MZM coupling off-
diagonal entries that mix the $P^{\textrm{Bulk+Imp}}_{\uparrow}=0$ and $1$
sectors, which in turn lifts the accident degeneracy, leaving only the
topological degeneracy. This results in the avoided level crossing numerically
observed in the main text.

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# Algorithmic properties of $\mathbf{QK4.3}$ and $\mathbf{QS4.3}$

M. Rybakov  IITP RAS, HSE University, Tver State University  D. Shkatov
University of the Witwatersrand, Johannesburg

###### Abstract

We prove that predicate modal logics $\mathbf{QK4.3}$ and $\mathbf{QS4.3}$ are
undecidable—more precisely, $\Sigma^{0}_{1}$-complete–in languages with two
individual variables, one modandic predicate letter, and one proposition
letter.

It is natural to expect that predicate modal logics should be algorithmically
harder than the classical predicate logic $\mathbf{QCl}$, just as
propositional modal logics are, as a rule, algorithmically harder than the
classical propositional logic. Nevetheless, numerous predicate modal logics
are just as hard as $\mathbf{QCl}$, i.e., $\Sigma^{0}_{1}$-complete: some—such
as $\mathbf{QK}$, $\mathbf{T}$, $\mathbf{S4}$, and $\mathbf{S5}$—are
recursively axiomatizable over $\mathbf{QCl}$ [15, 12]; others are recursively
embeddable [27, 30] into $\mathbf{QCl}$ through the standard traslation. It,
however, turns out that $\Sigma^{0}_{1}$-complete predicate modal logics can
be distinguished from $\mathbf{QCl}$ by algorithmic properties of their
fragments: while the monadic fragment of $\mathbf{QCl}$ is decidable [18, 2],
the monadic fragments of most $\Sigma^{0}_{1}$-complete modal logics are not
[17]; while the two-variable fragment of $\mathbf{QCl}$ is decidable [21, 13],
the two-variable fragments of most $\Sigma^{0}_{1}$-complete modal predicate
logics are not [16, 29]. This leads to the study of the algorithmic properties
of the fragments of modal predicate logics.

The algorithmic properties of one-variable and two-variable fragments of
first-order modal logics are also of interest due to close links between those
fragments and, respectively, two-dimensional and three-dimensional
propositional modal logics [11, 9, 35, 36, 33].

The study of the algorithmic properties of fragments of predicate modal, and
related superintuitionistic, logics (see [17, 19, 20, 22, 8, 1, 10, 37, 16,
29, 32, 34]; for a summary of results, see [32, Introduction]) is much less
advanced than similar research for $\mathbf{QCl}$ [4].

In the present paper, we attempt to identify the minimal undecidable fragments
of the predicate counterparts $\mathbf{QK4.3}$ and $\mathbf{QS4.3}$ of the
well-know propositional modal logics $\mathbf{K4.3}$ and $\mathbf{S4.3}$. It
is known [6] that $\mathbf{QK4.3}$ and $\mathbf{QS4.3}$ are finitely
axiomatizable over $\mathbf{QCl}$ and, hence, they are
$\Sigma^{0}_{1}$-complete. The logics $\mathbf{QK4.3}$ and $\mathbf{QS4.3}$
are faithfully characterized using Kripke semantics with expanding domains
[15], [12, §3.1]: they are determined by all, respectively, strict and partial
linear orders [6]. A closely related logic $\mathbf{QK4.3.D.X}$ is determined
by the rationals with the natural strict order, viewed as a Kripke frame [6].

The main interest of the results presented here is due the techniques used:
the known techniques for proving lower bounds in predicate modal and
superintuitionistic logics in languages with a few variables and a few
predicate letters [29, 32, 34], being based on propositional-level techniques
[14, 5, 23, 24, 26, 25, 28] developed for logics of frames with unbounded
branching, are inapplicable to logics of linear frames (the only exception
being our earlier work [31], where the techniques used for establishing
results reported here originate).

Predicate modal languages are obtained by enriching the classical preciate
language with a unary modal connective $\Box$ (for more backgroud on predicate
modal logic, see [15, 7, 12]

To recall the definitions of the logics we study, we use the following
notation: if $\Gamma$ is a set of formulas and $\varphi$ is a formula,
$\Gamma\oplus\varphi$ denotes the closure of $\Gamma\cup\\{\varphi\\}$ under
modes ponens, generalization, necessitation, and predicate substitution. Then,

$\begin{array}[]{lcl}\mathbf{QK}&=&\parbox{28.33342pt}{$\mathbf{QCl}$}\leavevmode\nobreak\
\oplus\leavevmode\nobreak\ \Box(p\to q)\rightarrow(\Box p\to\Box q);\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\mathbf{QK4}&=&\parbox{28.33342pt}{$\mathbf{QK}$}\leavevmode\nobreak\
\oplus\leavevmode\nobreak\ \Box p\rightarrow\Box\Box p;\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\
\mathbf{QS4}&=&\parbox{28.33342pt}{$\mathbf{QK4}$}\leavevmode\nobreak\
\oplus\leavevmode\nobreak\ \Box p\rightarrow p;\vskip 3.0pt plus 1.0pt minus
1.0pt\\\
\mathbf{QS4.3}&=&\parbox{28.33342pt}{$\mathbf{QS4}$}\leavevmode\nobreak\
\oplus\leavevmode\nobreak\ \Box(\Box p\rightarrow q)\vee\Box(\Box q\rightarrow
p);\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\mathbf{QK4.3}&=&\parbox{28.33342pt}{$\mathbf{QK4}$}\leavevmode\nobreak\
\oplus\leavevmode\nobreak\ \Box(p\wedge\Box p\rightarrow
q)\vee\Box(q\wedge\Box q\rightarrow p);\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\mathbf{QK4.3.D.X}&=&\parbox{28.33342pt}{$\mathbf{QK4.3}$}\leavevmode\nobreak\
\oplus\leavevmode\nobreak\ \Diamond\top\leavevmode\nobreak\
\oplus\leavevmode\nobreak\ \Box\Box p\rightarrow\Box p.\end{array}$

######  Theorem 1.

Logics $\mathbf{QK4.3}$, $\mathbf{QS4.3}$, and $\mathbf{QK4.3.D.X}$ are
$\Sigma^{0}_{1}$-complete in languages containing one monadic predicate
letter, one proposition letter, and two individual variables.

Theorem 1 is proved as follows: we encode a $\Sigma^{0}_{1}$-hard tiling
problem [3] using predicate modal formulas with only two variables, and only
binary and unary predicate letters; then, we simulate binary letters with
monadic one; finally, we simulate the monadic letters with just one monadic
and one nullary letter; the latter reductions do not use more than two
individual variables.

This result can be extended to logics containing the Barcan formula
$\mathit{bf}=\forall x\,\Box P(x)\to\Box\forall x\,P(x)$:

######  Theorem 2.

Logics $\mathbf{QK4.3}\oplus\mathit{bf}$, $\mathbf{QS4.3}\oplus\mathit{bf}$,
and $\mathbf{QK4.3.D.X}\oplus\mathit{bf}$ are $\Sigma^{0}_{1}$-complete in the
language containing one monadic predicate letter, one proposition letter, and
two individual variables.

Furthermore, we obtain the following generalisation of Theorems 1 and 2:

######  Theorem 3.

Every logic in the intervals
$[\mathbf{QK4.3},\mathbf{QK4.3.D.X}\oplus\mathit{bf}]$ and
$[\mathbf{QS4.3},\mathbf{QS4.3}\oplus\mathit{bf}]$ are $\Sigma^{0}_{1}$-hard
in the language containing one monadic predicate letter, one proposition
letter, and two individual variables.

Our proofs of Theorems 1–3 rely on Kripke completeness of $\mathbf{QK4.3}$,
$\mathbf{QS4.3}$, and $\mathbf{QK4.3.D.X}$ and on soundness of
$\mathbf{QK4.3}\oplus\mathit{bf}$, $\mathbf{QS4.3}\oplus\mathit{bf}$, and
$\mathbf{QK4.3.D.X}\oplus\mathit{bf}$ with respect to, respectively,
$\mathbf{QK4.3}$-frames, $\mathbf{QS4.3}$-frames, and
$\mathbf{QK4.3.D.X}$-frames with constant domains.

This research has been supported by the Russian Science Foundation with grant
21–18–00195; it has been carried out at Tver State University.

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