Majorana zero-modes in a dissipative Rashba nanowire

Arnob Kumar Ghosh ID $\star$ and Annica M. Black-Schaffer ID $\dagger$

Department of Physics and Astronomy, Uppsala University, Box 516, 75120
Uppsala, Sweden

$\star$<EMAIL_ADDRESS>, $\dagger$ annica.black-
<EMAIL_ADDRESS>
## Abstract

Condensed matter systems are continuously subjected to dissipation, which
often has adverse effects on quantum phenomena. We focus on the impact of
dissipation on a superconducting Rashba nanowire. We reveal that the system
can still host Majorana zero-modes (MZMs) with a finite lifetime in the
presence of dissipation. Most interestingly, dissipation can also generate two
kinds of dissipative boundary states: four robust zero-modes (RZMs) and two
MZMs, in the regime where the non-dissipative system is topologically trivial.
The MZMs appear via bulk gap closing and are topologically characterized by a
winding number. The RZMs are not associated with any bulk states and possess
no winding number, but their emergence is instead tied to exceptional points.
Further, we confirm the stability of the dissipation-induced RZMs and MZMs in
the presence of random disorder. Our study paves the way for both realizing
and stabilizing MZMs in an experimental setup, driven by dissipation.

###### Contents

  1. 1 Introduction
  2. 2 Model and method
    1. 2.1 Rashba NW-SC heterostructure
    2. 2.2 Lindblad master equation and dissipation
      1. 2.2.1 Lindblad spectrum
    3. 2.3 Winding number
  3. 3 Results: NW-SC heterostructure under uniform dissipation
    1. 3.1 Fate of MZMs under uniform dissipation
    2. 3.2 Emergence of MZMs and RZMs under dissipation
      1. 3.2.1 Stability against disorder
  4. 4 Conclusions and outlook
  5. A Obtaining $\mathcal{L}_{+}$
  6. B Computation of winding number
  7. C NW-SC heterostructure under single-site loss
    1. C.1 Fate of MZMs under single-site dissipation
      1. C.1.1 Dissipation in middle of NW
      2. C.1.2 Dissipation at end site of NW
    2. C.2 Non-topological system under single-site dissipation
      1. C.2.1 Dissipation at end site of NW

##  1 Introduction

The quest for Majorana fermions in topological superconductors has been
attracting an enormous amount of interest for the past two decades [1, 2, 3,
4, 5, 6], especially since they are proposed to be an essential ingredient of
topological quantum computers [7, 8, 9, 10, 11]. The pioneering paper by
Kitaev proposes that Majorana fermions can appear as Majorana zero-modes
(MZMs) in a spinless $p$-wave superconductor [1]. However, the lack of a
suitable candidate for a spinless $p$-wave superconductor barricades the
experimental realizability of this model. Nevertheless, Kitaev-like physics
has been shown to be possible in several different experimentally realizable
setups. One of the most popular routes is based on a one-dimensional (1D)
nanowire with Rashba spin-orbit coupling (SOC) in proximity to a conventional
spin-singlet $s$-wave superconductor and an applied magnetic field [12, 13, 3,
4, 14, 15, 6]. Numerous experiments have been performed based on this Rashba
nanowire-superconductor (NW-SC) heterostructure, primarily employing transport
measurements to find signature of the MZMs [14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26]. However, to date, a smoking-gun signature of the MZMs is
yet to be observed, especially as quantum dots-induced Andreev bound states or
the Kondo effect due to magnetic impurities present in the system have been
shown to easily mimic MZM signals in transport-based experiments [27, 28, 29,
30, 31, 32, 33, 34, 35].

In addition to the issues above for finding MZMs, topological systems are
typically described from the viewpoint that they are not coupled to the
environment, i.e. they are considered as entirely isolated systems. However,
in reality, they are never truly isolated; they constantly interact with their
surroundings as open systems [36, 37]. In an open system, the long-time
evolution is determined by the density matrix. Under the Markovian
approximation i.e., when the system is coupled to a memoryless bath [36, 38],
the evolution of the density matrix is governed by a Liouvillian superoperator
that takes the form of the Gorini-Kossakowski-Sudarshan-Lindblad master
equation, or simply the Lindblad master equation [39, 40]. Although the
application of the Lindblad master equation to topological systems has so far
been somewhat limited, it has still been used to study dissipation-engineered
preparation of some topological non-equilibrium steady states, which have also
been proposed for topological quantum computations [41, 38, 42, 43, 44, 45,
46, 47, 48]. The application of the Lindblad master equation is further not
limited to condensed matter systems but has been extensively employed e.g., in
quantum information [49], quantum optics [50], and atomic physics [51].

The past few years have also seen a huge interest in non-Hermitian (NH)
physics, which can encode some aspects of open systems. Especially the study
of NH topological systems has been blooming owing to their intriguing
properties that are absent in their Hermitian counterparts, such as the NH
skin effect [52, 53, 54, 55, 56], the emergence of exceptional points (EPs)
[57, 58, 59, 60, 61, 54, 62, 55, 56], breakdown of conventional bulk-boundary-
correspondence [63, 64, 54, 55], and ramification of symmetry operators [65].
NH open systems have also been shown to sometimes harbor boundary states that
are stabilized in the presence of dissipation [66, 67, 68, 69, 70]. In this
context, the Liouvillian superoperator that drives the dynamics of the density
matrix of the open system is an NH matrix, which then unravels the underlying
NH topology [71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81].

Furthermore, the interplay of NH physics and superconductivity has recently
exposed several intriguing phenomena to the above examples, such as the
generation of odd-frequency pairing [82], emergence of NH Bogoliubov Fermi
arcs [83], enhancement of superconductivity by EPs [84], and topological phase
transition with infinitesimal instability [85]. It has also been established
that MZMs can be obtained in a NH topological superconductor [86, 87, 88, 85,
89, 90, 91, 84], satisfying a modified NH Majonara condition [85]. Recently,
it has also been argued that non-Hermiticity can resolve the Majorana vs.
Andreev controversy [92]. Thus, non-Hermiticty provides a very interesting
avenue for topological superconductivity. However, most of the exploration of
topological superconductors within an NH setting is done by utilizing only
some specific forms of NH terms, often even just added ad-hoc. A few
counterexamples exist, such as creating an NH topological superconductor by
coupling the superconductor to a ferromagnetic lead [86, 82, 83] or allowing
dissipation in the system within the Kitaev model [67, 93].

In particular, to date, no systematic study exists of the effect of
dissipation, thereby creating an unexplored NH regime, on more experimentally
realizable models for topological superconductivity, such as Rashba NW-SC
heterostructures [12, 13, 3, 4, 14, 15, 6]. In fact several intriguing
questions exist: (a) What is the effect of dissipation on the Rashba NW-SC
heterostructure, and do we continue to obtain MZMs in such a dissipative
system? (b) Can dissipation even lead to the generation of topological
superconductivity and MZMs when starting from a Hermitian non-topological
regime? In this work, we answer both of these fascinating questions.

Figure 1: Schematic representation of NW-SC heterostructure setup encompassing
a 1D Rashba NW (orange dots connected by green lines) in close proximity to a
bulk conventional spin-singlet $s$-wave superconductor (violet) with an
applied in-plane magnetic field $B_{x}$. The system can host MZMs (cyan dots)
in the topological regime. The heterostructure is coupled to the environment
via dissipation. The loss due to the environment is encoded through the jump
operators: $L_{l}^{1},L_{l}^{2},\cdots,L_{l}^{N}$; with $N$ being the total
number of lattice sites.

In particular, we consider a 1D NW with Rashba SOC in direct proximity to a
conventional spin-singlet $s$-wave superconductor and with an applied in-plane
magnetic field, which is exposed explicitly to the surrounding environment,
see Fig. 1. We employ the Lindblad master equation to obtain an NH description
of this setup in terms of Lindblad spectra. First, we explore the fate of
MZMs, present in the Hermitian (closed system) limit, for increasing
dissipation. We show that the MZMs survive in the dissipative environment but
obtain a finite lifetime, see Fig. 2. Next, we demonstrate that dissipation
can even induce in-gap states localized at the edges of the system, where the
system in the Hermitian is non-topological and with no in-gap states. Here, we
obtain two kinds of zero-modes (ZMs): four robust ZMs (RZMs), whose origin we
trace back to the presence of EPs, and two ZMs mimicking MZMs, originating at
a topological phase transition where the bulk gap closes and a finite
topological winding number is achieved (see Figs. 3 and 4). Notably, we find
that both the RZMs and MZMs are robust against disorder (see Figs. 5 and 6).
Our study uncovers intriguing aspects of dissipative systems, where we observe
that dissipation goes hand-in-hand with NH physics and even renders the
generation of topologically protected MZMs.

The remainder of this work is organized as follows. In Section 2, we discuss
the model Hamiltonian, introduce the form of dissipation, and review the
method that we employ. Section 3 presents the main results. In particular, we
discuss the fate of MZMs under uniform dissipation that are present already in
the Hermitian topological regime in Section 3.1, while we explore the
generation of RZMs and MZMs, as well as their stability against disorder in
Section 3.2. Finally, we conclude the work with the conclusions and outlook in
Section 4. In Appendix A and Appendix B, we present more details about the
method, while in Appendix C, we discuss the effect of single-site dissipation
on our system, which supplements our results in the main text.

##  2 Model and method

In this section, we discuss the details of the system, its Hamiltonian, the
nature of the dissipation, and the methods to handle this system.

###  2.1 Rashba NW-SC heterostructure

We consider a 1D NW with Rashba SOC placed on top of a conventional spin-
singlet $s$-wave superconductor with an applied in-plane magnetic field. The
schematic representation of this setup is depicted in Fig. 1. The system under
consideration can be described by the following Hamiltonian [12, 13, 3, 4, 14,
15, 6]

$\displaystyle H=$
$\displaystyle-\mu\sum_{i=1,\alpha}^{N}c^{\dagger}_{i\alpha}c_{i\alpha}+t_{\rm
h}\sum_{i=1,\alpha}^{N-1}c^{\dagger}_{i\alpha}c_{i+1\alpha}-i\lambda_{\rm
R}\sum_{i=1,\alpha\beta}^{N-1}c^{\dagger}_{i\alpha}(\sigma_{z})_{\alpha\beta}c_{i+1\beta}+B_{x}\sum_{i=1,\alpha\beta}^{N}c^{\dagger}_{i\alpha}(\sigma_{x})_{\alpha\beta}c_{i\beta}$
$\displaystyle+\Delta\sum_{i=1}^{N}c^{\dagger}_{i\uparrow}c^{\dagger}_{i\downarrow}+{\rm
h.c.}\ ,$ (1)

where $\mu$, $t_{\rm h}$, $\lambda_{\rm R}$, $B_{x}$, and $\Delta$ represent
the chemical potential, hopping amplitude, strength of the Rashba SOC,
magnitude of the in-plane magnetic field, and the proximity-induced $s$-wave
superconducting order parameter, respectively. Here, $c_{i\alpha}$
($c_{i\alpha}^{\dagger}$) stands for the electron creation (annihilation)
operator at lattice site $i$ with spin $\alpha=\uparrow,\downarrow$, and $N$
denotes the number of lattice sites in the 1D wire. The Pauli matrices
$\boldsymbol{\mathrm{\sigma}}$ acts on the spin degrees of freedom. For a NW
of infinite length and thus translationally invariant, we obtain a momentum-
space Bloch form of the Hamiltonian in Eq. (1) in the Bogoliubov-de Gennes
(BdG) basis
$\Psi_{k}=\left\\{c_{k\uparrow},c_{k\downarrow},-c^{\dagger}_{-k\downarrow},c^{\dagger}_{-k\uparrow}\right\\}$
reading $H_{\rm BdG}=\frac{1}{2}\sum_{k}\Psi_{k}^{\dagger}H(k)\Psi_{k}$, with
$H(k)$

$\displaystyle H(k)=(-\mu+2t_{\rm h}\cos k)\tau_{z}\sigma_{0}+2\lambda_{\rm
R}\sin k\tau_{z}\sigma_{z}+B_{x}\tau_{0}\sigma_{x}+\Delta\tau_{x}\sigma_{0}\
,$ (2)

where the Pauli matrices $\boldsymbol{\mathrm{\tau}}$ acts on the particle-
hole (PH) degrees of freedom. The Hamiltonian $H(k)$ respects both PH symmetry
$\mathcal{C}=\tau_{y}\sigma_{y}K$: $\mathcal{C}^{-1}H(k)\mathcal{C}=-H(-k)$
with $K$ being the complex-conjugation operator and chiral symmetry
$S=\tau_{y}\sigma_{z}$: $SH(k)S=-H(k)$. However, $H(k)$ breaks the time-
reversal symmetry $\mathcal{T}=\tau_{0}\sigma_{y}K$:
$\mathcal{T}^{-1}H(k)\mathcal{T}\neq H(-k)$ due to the presence of the non-
zero magnetic field $B_{x}$. We obtain the critical magnetic field needed a
topological phase transition by investigating the bulk gap-closing. At $k=0$
and $\pi$, the Hamiltonian $H(k)$, Eq. (2), exhibits gap-closing for
$B_{x}=\lvert B_{x,c1}\rvert=\sqrt{\left(\mu-2t_{\rm
h}\right)^{2}+\Delta^{2}}$ and $B_{x}=\lvert
B_{x,c2}\rvert=\sqrt{\left(\mu+2t_{\rm h}\right)^{2}+\Delta^{2}}$,
respectively [12, 94]. The system hosts two MZMs localized at the wire
endpoints (one MZM per endpoint) in the topological phase existing for $\lvert
B_{x,c1}\rvert<B_{x}<\lvert B_{x,c2}\rvert$. Throughout this work, we for
simplicity choose $t_{\rm h}=\mu=\Delta=1.0$ and $\lambda_{\rm R}=0.5$, unless
mentioned otherwise. Our observations do not explicitly depend on this
specific choice of parameters and will remain qualitatively the same for
different choices of parameters.

###  2.2 Lindblad master equation and dissipation

We consider that the system in Fig. 1 is coupled to an outside environment or
bath, and thereby subject to dissipation. We employ the Markovian
approximation, such that the environment is memoryless. This approximation is
valid when the dynamics of the environment are faster than the system. In this
case the evolution of the density matrix $\rho(t)$ follows the Lindblad master
equation [36, 38, 39, 40]:

$\displaystyle\frac{\partial\rho(t)}{\partial t}$
$\displaystyle=-i\left[H,\rho(t)\right]-\frac{1}{2}\sum_{m}\left(\left\\{L^{\dagger}_{m}L_{m},\rho(t)\right\\}-2L^{\dagger}_{m}\rho(t)L^{\dagger}_{m}\right)\
,$ (3a) $\displaystyle\equiv\mathcal{L}\rho(t)\ ,$ (3b)

where $\mathcal{L}$ is the Liouvillian superoperator and $L_{m}$’s are the
Lindblad jump operators. The effect of the bath-system coupling is encoded in
these jump operators [36, 38]. The first term on the right-hand side of Eq.
(3a) resembles the Liouville-von Neumann equation of motion and represents the
unitary evolution of the system in the absence of dissipation. The second term
is divided into two parts: the first part describes the continuous loss of
energy to the environment, and the second part denotes quantum jumps [73].
Without the quantum jumps, i.e. in the semi-classical limit, we obtain non-
unitary evolution of the density matrix $\rho(t)$, governed by an effective NH
Hamiltonian $H_{\rm eff}=H-\frac{i}{2}\sum_{m}L^{\dagger}_{m}L_{m}$, which
also represents the short-time dynamics of the system. In our work we go
beyond such an effective NH Hamiltonian description and use the full
formalism, including the effects of the quantum jumps. This adds substantial
complexity to solving the problem, and effects have also been observed in
experiments [95, 96, 97, 98, 99].

The jump operators $L_{m}$ can represent both onsite and bond losses or gains.
For simplicity, we consider a simple form of onsite loss, which can become
experimentally feasible. Although we may choose the form of the jump operators
differently, our work establishes that even this simple choice of jump
operators leads to interesting findings. The chosen jump operator at lattice
site $i$ reads

$\displaystyle L_{l}^{i}=$
$\displaystyle\sqrt{\gamma}\left(c_{i\uparrow}+c_{i\downarrow}\right),$ (4)

where the index $l$ stands for loss and $\gamma\geq 0$ encapsulates the
strength of the loss to the environment. In the main text, we consider uniform
loss i.e., $L_{l}^{i}\neq 0\,\,\forall i$, while in Appendix C, we provide
complementary results for single site loss.

####  2.2.1 Lindblad spectrum

To incorporate the full effects of openness within the Lindblad formalism,
including the effects of quantum jumps, we employ third quantization. The
details of the third quantization method are introduced in Refs. [100, 101,
102]. For the sake of completeness, we provide the main steps for obtaining
the matrix form of the Liouvillian $\mathcal{L}$ here while the technical
details are found in Appendix A. To proceed, we represent all the complex
fermions in terms of real Majorana fermions by employing the transformation

$\displaystyle c_{i\uparrow}=\frac{\left(w_{Ai}+iw_{Bi}\right)}{2}\quad{\rm
and}\quad c_{i\downarrow}=\frac{\left(w_{Ci}+iw_{Di}\right)}{2}\ ,$ (5)

where $w_{\alpha i}$ represent real Majorana fermions, satisfying the
fermionic anti-commutation relation as $\left\\{w_{\alpha i},w_{\beta
i}\right\\}=2\delta_{\alpha,\beta}\delta_{i,j}$; with $\alpha,\beta=A,B,C,D$
and $i,j=1,2,\cdots,N$. We recast the Hamiltonian Eq. (1) and the jump
operator, Eq. (4) in the Majorana basis as

$\displaystyle H_{\rm M}=$ $\displaystyle\sum_{a,b}^{4N}w_{a}H_{a,b}w_{b}\ ,$
(6) $\displaystyle L_{l}=$ $\displaystyle\sum_{b}^{4N}l_{l,b}w_{b}\ ,$ (7)

where $H_{\rm M}$ is a $(4N\times 4N)$ matrix and $L_{l}$ is $(4N\times 1)$
column matrix. Here, we define the vector
$\underline{w}=\left\\{w_{A1},w_{B1},w_{C1},w_{D1},\cdots,w_{Ai},w_{Bi},w_{Ci},w_{Di},\cdots\right\\}^{T}$,
with $T$ being the transpose operation. The next step is to consider a Fock-
space $\mathcal{K}$, also known as Liouville-Fock space. The density matrix is
part of that space $\mathcal{K}$, $\rho(t)\in\mathcal{K}$, and is a
$(16N^{2}\times 1)$-vector in this Liouville-Fock space (while $\rho(t)$ is a
$(4N\times 4N)$ matrix in the normal Hilbert space). We consider the vector
space $\mathcal{K}$ spanned by the polynomial $P_{\underline{\alpha}}$, such
that

$\displaystyle P_{\underline{\alpha}}=\prod_{a=1}^{4N}w_{a}^{\alpha_{a}},$ (8)

where $\alpha_{a}=0,1$ and denotes the occupation of the Majorana fermion
states. Thus, the inner product is defined as $\braket{x}{y}=2^{-4N}{\rm
tr}\left[x^{\dagger}y\right]$. We also define the adjoint fermion annihilation
and creation operator as

$\displaystyle\phi_{a}\ket{P_{\underline{\alpha}}}=\delta_{\alpha_{a},1}\ket{w_{a}P_{\underline{\alpha}}}\quad{\rm
and}\quad\phi_{a}^{\dagger}\ket{P_{\underline{\alpha}}}=\delta_{\alpha_{a},0}\ket{w_{a}P_{\underline{\alpha}}}\
.$ (9)

Since, $w_{a}^{2}=1$ $\phi_{a}$ annihilates the Majorana fermion $w_{a}$,
while $\phi_{a}^{\dagger}$ creates the Majorana fermion $w_{a}$. These
operators follow the fermion anti-commutation relation
$\left\\{\phi_{a},\phi_{b}^{\dagger}\right\\}=\delta_{a,b}$.

Assuming an even number of fermionic degrees of freedom in the system, we can
now represent the Liouvillian that governs the dynamics of the dissipative NW-
SC heterostructure by $\mathcal{L}_{+}$. We provide the steps to obtain
$\mathcal{L}_{+}$ in Appendix A and here we only report the expression:

$\displaystyle\mathcal{L}_{+}=\frac{i}{2}\begin{pmatrix}\underline{\phi}^{\dagger}\cdot&\underline{\phi}\cdot\end{pmatrix}\begin{pmatrix}-X^{\dagger}&-iY\\\
0&X\end{pmatrix}\begin{pmatrix}\underline{\phi}^{\dagger}\\\
\underline{\phi}\end{pmatrix}-A_{0}\ ,$ (10)

where we define $X=4H_{\rm M}+i(M+M^{T})$, $Y=2(M-M^{T})$, and
$A_{0}=\frac{1}{2}{\rm Tr}[X]$, with $M_{jk}=l^{T}_{j,l}l^{*}_{l,k}$. Here,
$\underline{\phi}^{\dagger}$ and $\underline{\phi}$ represent the vector form
of the adjoint creation and annihilation operators, respectively. Since
$\mathcal{L}_{+}$ takes an upper triangular form, the spectrum of the
Liouvillian, which is also called the Lindblad spectrum, is completely
determined by that of $X$, i.e. it is the operator $X$ that dictates the
dynamics of the system [78]. Thus, we obtain

$\displaystyle\mathcal{L}_{+}=i\sum_{a}E_{a}\chi_{a}^{\prime}\chi_{a}\ ,$ (11)

where $E_{a}$ are the eigenvalues of $X$ and $\chi_{a}^{\prime}$ and
$\chi_{a}$ are the normal-master modes satisfying
$\left\\{\chi_{a}^{\prime},\chi_{b}\right\\}=\delta_{a,b}$ [100, 101, 102]. It
is also known that the topological properties of $\mathcal{L}_{+}$ are
mimicked by the matrix $X$ [72]. Thus, we can focus fully on $X$, as it
dictates both the dynamical and topological properties of the system. Owing to
the form of the loss in Eq. (4), we can finally obtain the matrix $X$ for a
dissipative NW-SC heterostructure, which in momentum-space reads

$\displaystyle X(k)=$ $\displaystyle 4H_{\rm M}(k)+i\left(M+M^{T}\right)$
$\displaystyle=$ $\displaystyle(-\mu+2t_{\rm h}\cos
k)\tilde{\tau}_{0}\tilde{\sigma}_{y}+2\lambda_{\rm R}\sin
k\tilde{\tau}_{z}\tilde{\sigma}_{0}-B_{x}\tilde{\tau}_{x}\tilde{\sigma}_{y}+\Delta\tilde{\tau}_{y}\tilde{\sigma}_{x}+\frac{i\gamma}{2}\left(\tilde{\tau}_{0}\tilde{\sigma}_{0}+\tilde{\tau}_{x}\tilde{\sigma}_{0}\right)\
,$ (12)

where $\boldsymbol{\mathrm{\tilde{\tau}}}$ and
$\boldsymbol{\mathrm{\tilde{\sigma}}}$ are newly defined Pauli matrices.

Further, we note that one of the differences between $X$ and the NH
Hamiltonians so far usually considered in the literature [54, 55] is that the
imaginary part of the eigenvalue of the matrix $X$, i.e., the Lindblad
spectrum ${\rm Im}~{}E$, is always greater than zero, which ensures that the
density matrix decays over time, following
$\rho(t)=\exp\left[i\mathcal{L}_{+}t\right]\rho(0)$. This, in turn, also
provides a $10$-fold classification of the Lindbladians [72] in contrast to
all NH Hamiltonians, which exhibit a $38$-fold classification instead [65].

###  2.3 Winding number

To investigate the topological properties of a dissipative NW-SC
heterostructure, we compute a winding number based on the matrix $X$. The
matrix $X(k)$ satisfies pseudo-anti-Hermiticity symmetry ($\Gamma$): $\Gamma
X^{\dagger}(k)\Gamma=-X(k)$; with $\Gamma=\tilde{\tau}_{x}\tilde{\sigma}_{x}$
[103]. We employ this pseudo-anti-Hermiticity symmetry to define the winding
number $\nu$ as [104]

$\displaystyle\nu=\frac{i}{2\pi}\int_{\rm BZ}dk\ {\rm
tr}\left[q(k)^{-1}\partial_{k}q(k)\right]\ .$ (13)

We provide the detailed procedure to obtain $q(k)$ and $\nu$ in Appendix B.
Our results establish that we do not observe any skin effects and that the
conventional Hermitian bulk-boundary correspondence is intact. Thus, we can
always employ this momentum space topological invariant to obtain the
topological phase boundaries of the system.

##  3 Results: NW-SC heterostructure under uniform dissipation

This section is devoted to our main results. We consider two cases: first,
when the isolated (Hermitian) Hamiltonian of the Rashba NW-SC heterostructure
is topological, and then when the isolated system is topologically trivial. In
both cases, we mainly focus on the Lindblad spectra i.e., the eigenvalue
spectrum of the matrix $X$ as described in Section 2.2.1. Throughout this
section, we usually employ open boundary condition (OBC) when computing the
Lindblad spectra and the local density of states (LDOS), unless mentioned
otherwise. We further find that the boundary modes always have their real
energy equal to zero, which makes us term them ZMs, akin to those in the
Hermitian system.

###  3.1 Fate of MZMs under uniform dissipation

Figure 2: (a) Real part of the Lindblad spectra ${\rm Re}~{}E$ as a function
of dissipation strength $\gamma$. (b) ${\rm Re}~{}E$ of the Lindblad spectrum
as a function of the state index $m$, at the black dashed line in (a). Inset
shows zoomed-in eigenvalue spectrum close to ${\rm Re}~{}E=0$. The ${\rm
Re}~{}E$ is weighted by the ${\rm Im}~{}E$ value, see color bar in (a,b). (c)
Local density of states (LDOS) computed for ${\rm Re}~{}E=0$ states as a
function of lattice sites $x$ at the black dashed line in (a). Here
$B_{x}=2.0>\lvert B_{x,c1}\rvert$ and we use $100$ lattice sites.

In a realistic experimental scenario, a setup is always in contact with the
environment. Thus, it is worth examining the fate of MZMs in a topological 1D
Rashba NW-SC heterostructure. To this end, we consider the isolated setup,
i.e. system Eq. (1) with no dissipation, to reside in the topological regime:
$\lvert B_{x,c1}\rvert<B_{x}<\lvert B_{x,c2}\rvert$ [12, 94]. We then add
uniform dissipation and we plot the real part of the eigenvalue spectra ${\rm
Re~{}}E$ of the matrix $X$ as a function of the dissipation strength $\gamma$
in Fig. 2(a). The ${\rm Re~{}}E$ are further colored corresponding to their
imaginary part ${\rm Im~{}}E$. As seen from the color bar of Fig. 2(a), ${\rm
Im~{}}E\geq 0$ always, ensuring the decay of the density matrix over time.
Notably, the MZMs present in the Hermitian limit, $\gamma=0$, continue to
exist as ${\rm Re~{}}E=0$ states, but they now carry also a finite imaginary
part, which increases with increasing $\gamma$. This finite imaginary part can
be understood as the lifetime of the MZMs in a dissipative system.
Furthermore, we find that the gap protecting the MZMs from all other states
diminishes with $\gamma$, especially for large dissipation, as clearly
depicted in Fig. 2(a).

We next choose a small realistic value of the dissipation strength (black
dashed line in Fig. 2(a)) and examine the Lindblad spectrum and the LDOS. In
Fig. 2(b), we plot the real part of the Lindblad spectrum ${\rm Re~{}}E$ as a
function of the state index $m$. The color again represents the imaginary part
of the eigenvalue ${\rm Re~{}}E$. In the inset, we show the states closest to
${\rm Re~{}}E=0$, which corroborates the presence of two MZMs. The LDOS
associated with these two MZMs is illustrated in Fig. 2(c). We clearly see
that the MZMs are localized at the end of the NW, even for the dissipative
system. We further confirm the topological properties of the system by
computing the winding number $\nu$ using Eq. (13). We always obtain $\nu=1$
when MZMs are present in the system. These results show that the MZMs present
in the non-dissipative, or Hermitian, topological regime of a 1D Rashba NW-SC
heterostructure survive in a dissipative background. In Appendix C, we
establish the same results for single-site dissipation.

###  3.2 Emergence of MZMs and RZMs under dissipation

Having established that the MZMs in the Hermitian Rashba NW-SC heterostructure
survive under dissipation, we next investigate the effect of dissipation in
the topologically trivial limit and particularly seek whether we can obtain
non-trivial topology with boundary states via dissipation only. To achieve
this, we set $B_{x}<\lvert B_{x,c1}\rvert$, such that the isolated NW-SC
heterostructure does not exhibit any zero-energy end states and where the bulk
is always trivially gapped with $\nu=0$. Then, we couple this system with the
environment and illustrate the results in Figs. 3, 4, and 5. We uncover the
generation of two kinds of ZMs: four robust ZMs (RZMs), which we will show are
not connected to the bulk topology, and two MZMs, which emerge due to the
closing of the bulk gap and finite bulk winding number.

Figure 3: (a) Real part of the Lindblad spectra ${\rm Re}~{}E$ as a function
of dissipation strength $\gamma$, with color bar representing the imaginary
part of the eigenvalue. (b) Eigenvalues of the four middle states: $(2N-1)$,
$2N$, $(2N+1)$, and $(2N+2)$, i.e. closest to ${\rm Re}~{}E=0$. Right axis
corresponds to the scalar product $\lvert\braket{\Psi_{i}}{\Psi_{j}}\rvert$ of
the corresponding eigenstates. Orange dashed line indicates the position of
EPs. (c) Real part of the Lindblad spectra employing OBC (blue dots) and PBC
(green dots) as a function of $\gamma$ and winding number $\nu$ on the right
axis (red line). The winding number exhibits a jump from $0$ to $1$ when there
is a gap closing in the PBC spectrum. (d,e) Winding number $\nu$ in the
$B_{x}\mathchar 45\relax\gamma$ plane for fixed $\mu=1.0$ in (d) and in the
$\mu\mathchar 45\relax\gamma$ plane for fixed $B_{x}=1.2$ (e). Color stars
correspond to the dashed vertical lines in (a,c). Here $B_{x}=1.2<\lvert
B_{x,c1}\rvert$ in panels (a,b,c).

First, we investigate the phase diagram as a function of $\gamma$. We plot
${\rm Re}~{}E$ as a function of dissipation strength $\gamma$ in Fig. 3(a) and
observe that there are no states at ${\rm Re}~{}E=0$ when $\gamma=0$ but with
the increase in $\gamma$, we observe the appearance of states at or near ${\rm
Re}~{}E=0$, see yellow and gray regions in Fig. 3(a). In the yellow and gray
regions, we obtain four RZMs and two MZMs at ${\rm Re}~{}E=0$, respectively.
The yellow region emerges without any closing of the bulk gap, see the left
part of the yellow region in Fig. 3(a) and also the eigenvalue spectra for the
system obeying periodic boundary condition (PBC) in Fig. 3(c). Nevertheless,
we observe that the left edge of the yellow region in Fig. 3(a) corresponds to
EPs. In Fig. 3(b), we illustrate this formation of two second-order EPs: the
$2N$\- and $\left(2N+2\right)$-th, and the $\left(2N-1\right)$\- and
$\left(2N+1\right)$-th states coalesce pairwise with each other, forming two
second-order EPs at the orange dashed line (corresponding to the left edge of
the yellow region). Notice that the real, imaginary, and absolute eigenvalues
all merge at the orange dashed line in Fig. 3(b). At this point, the scalar
product $\lvert\braket{\Psi_{2N}}{\Psi_{2N+2}}\rvert$ (magenta solid line) and
$\lvert\braket{\Psi_{2N-1}}{\Psi_{2N+1}}\rvert$ (cyan dashed line) also
becomes unity, thus forming two parallel set of states, another signature of
EPs. EPs are known to be topological [57, 58, 59, 60, 61, 54, 62, 55, 56] and
we here find that EPs can additionally also induce ZMs. This finding of EP-
generated ZMs, here called RZMs due to their disorder stability, as found
below, is one of the unexpected findings of this work. We further note that
the RMZs appearing in the yellow region cannot possess a finite winding
number, since they are not connected to any bulk states. Interestingly,
similar features have recently been observed in the Su-Schrieffer-Heeger
model, where the emergence of ZMs was also not found to be associated with a
bulk gap closing [66, 105].

Figure 4: (a) [(c)] Real part of the Lindblad spectrum ${\rm Re}~{}E$ as a
function of state index $m$ for $\gamma=1.5$ [$\gamma=5.0$]. Modes close to
${\rm Re}~{}E=0$ are shown in the inset. Color bar represents ${\rm Im}~{}E$.
(b,d) LDOS computed for the ${\rm Re}~{}E=0$ states in (a,c). Here
$B_{x}=1.2<\lvert B_{x,c1}\rvert$.

Next, we focus on the gray region of Fig. 3(a), which occurs after a clear
bulk gap closing at the interface between the yellow and the gray region. In
this phase, we obtain two zero-energy modes, designated as MZMs below (see
Fig. 4(c)). The bulk gap closing is clearly seen in the PBC eigenvalue spectra
(green dots) in Fig. 3(c). Since the MZMs emerging in this phase are connected
to the bulk states, we can compute the winding number $\nu$, thereby employing
the bulk states to unravel the bulk topology. We depict the winding number
$\nu$ as a function of $\gamma$ on the right axis of Fig. 3(c). The winding
number jumps from $0$ to $1$ at the bulk gap closing point, thus exhibiting a
non-zero value as soon as the MZMs are present in the system. We thus find a
topological phase transition exhibiting MZMs when the bulk gap closes. This
non-trivial phase with $\nu=1$ is clearly driven by dissipation and is not
present in the closed system. This is the second unexpected finding of this
work. To map out the phase diagram, we also plot the winding number $\nu$ in
the $B_{x}\mathchar 45\relax\gamma$ and $\mu\mathchar 45\relax\gamma$ planes
in Fig. 3(d) and (e), respectively. These phase diagrams show that dissipation
can drive a trivial NW-SC heterostructure with $\nu=0$ in the Hermitian limit
($\gamma=0$) to a topologically non-trivial phase with $\nu=1$ with
dissipation-induced MZMs. We observe a visible increase in the parameter space
for $\gamma\neq 0$ compared to $\gamma=0$, for example, as indicated by the
colored stars corresponding to the vertical dashed lines in Fig. 3(a,c). This
means we can obtain MZMs for a smaller value of the magnetic field
$B_{x}\leq\lvert B_{x,c1}\rvert$ for a fixed $\mu$, if we allow for
dissipation and we have dissipation-induced MZMs.

Having understood the emergence of RZMs and MZMs from the phase diagram, we
now further investigate their properties. In particular, we show the real part
of the Lindblad spectrum ${\rm Re}~{}E$ corresponding to the blue dashed line
in Figs. 3(a,c) (or equivalently blue stars in Figs. 3(d,e)), as a function of
the state index $m$ in Fig. 4(a). In the inset of Fig. 4(a), we demonstrate
the eigenvalues close to ${\rm Re}~{}E=0$, which corroborates the presence of
four RZMs. However, they have different imaginary parts, divided into two
sets: orange and blue. This can be interpreted as different lifetimes of the
RZMs. To study their localization properties, we plot the LDOS computed at
${\rm Re}~{}E=0$ in Fig. 4(b), which indicates that the RZMs are localized at
the two ends of the system. This is similar to any topological boundary state,
despite the RZMs not appearing in a topologically non-trivial phase, as
indicated by the zero winding number.

In Fig. 4(c), we depict ${\rm Re}~{}E$ as a function of the state index $m$
corresponding to the black dashed line in Figs. 3(a,c) (or equivalently black
stars in Figs. 3(d,e)). We observe from the inset that there are now two
states at ${\rm Re}~{}E=0$, the two MZMs already identified Fig. 2(b). Here,
both the ${\rm Re}~{}E=0$ MZMs incur the same value of imaginary part i.e.,
they have the same lifetime. We illustrate the LDOS associated with the MZMs
at ${\rm Re}~{}E=0$ in Fig. 4(d), showing how they are primarily localized to
the end points of the wire. We also note that the peak height of the two MZMs
are equal in Fig. 4(d), in contrast with the RZMs in Fig. 4(b), where the peak
heights are not the same in both the ends. Taken together, these results
illustrate that dissipation alone can lead to the generation of ZMs, both
topologically non-trivial MZMs, and topologically trivial RZMs, in an
otherwise trivial system with no in-gap states.

####  3.2.1 Stability against disorder

Figure 5: Disorder-averaged real part of the Lindblad spectra ${\rm Re}~{}E$
as a function of dissipation strength $\gamma$ for $w=0.5$ (a) and $w=1.0$
(b). Color bar represents ${\rm Im}~{}E$. Here $B_{x}=1.2<\lvert
B_{x,c1}\rvert$, and we use $500$ lattice sites and take an average over $50$
disorder configurations.

Having demonstrated the emergence of both RZMs and the MZMs in a dissipative
NW with no topology in the closed Hermitian limit, we check the stability of
these states in the presence of disorder. This is a particularly valid
question for the RZMs, since they do not enjoy any protection from the bulk
topology, but elucidating disorder stability also for MZMs is of experimental
relevance. To this end, we add an onsite, Anderson-type, non-magnetic random
disorder potential to the Hamiltonian of the NW:
$-\sum_{i\alpha}V_{i}c^{\dagger}_{i\alpha}c_{i\alpha}$. Here $V_{i}$ is
uniformly distributed in the range $[-w/2,w/2]$, with $w$ the strength of the
disorder potential. We consider an average over $50$ random configurations of
the disorder potential in our numerical computation.

We show the disorder-averaged real part of the Lindblad spectra ${\rm Re}~{}E$
as a function of $\gamma$ in Figs. 5(a) and (b) for $w=0.5$ and $1.0$,
respectively. Although disorder-averaged eigenvalues are not directly
observable, these spectra still capture the overall effects of disorder, and
especially we get to know when disorder-induced (near) ZMs appear as a
function of $\gamma$, which signals disorder destroying the clean limit
results. For the smaller disorder strength $w=0.5$ in Fig. 5(a), we do not
observe any significant difference compared to the clean case in Fig. 3(a),
except that the point separating two phases with the RZMs and the MZMs,
respectively, is somewhat broadened in the disordered system. However, the
bulk gap in terms of its real energy is reduced in the disorder system. For
the comparatively strong disorder strength $w=1.0$ in Fig. 5(b) (note $t=1$),
we note that a dramatic reduction and even destruction of the real energy bulk
gap, such that no bulk gap longer protects the MZMs. Interestingly, the RZMs
seem to be more resilient compared to the MZMs, as they remain well separated
from all other states even at these strong disorder levels.

Figure 6: (a) Disorder-averaged real part of the Lindblad spectrum ${\rm
Re}~{}E$ as a function of state index $m$ and (b) the LDOS computed for ${\rm
Re~{}E=0}$ for disorder strength $w=0.5$. In the inset of panels (a) and (b),
we repeat (a) and (b) for $w=1.0$. Here $\gamma=1.5$. In panels (c) and (d),
we repeat (a) and (b) for $\gamma=5.0$. Here $B_{x}=1.2<\lvert
B_{x,c1}\rvert$.

To gain more insights into the stability of the RZMs and the MZMs, we select
two fixed dissipation strengths, dashed blue and black lines in Fig. 5, and
investigate the localization properties of the in-gap states in the presence
of disorder. In particular, in Fig. 6(a), we show the disorder-averaged ${\rm
Re}~{}E$ as a function of state index $m$ for the states close to ${\rm
Re}~{}E=0$ when the system hosts RZMs and at $w=0.5$ (blue dashed line in Fig.
5(a)). In the inset, we instead use disorder strength $w=1.0$ (blue dashed
line in Fig. 5(b)). We observe that the RZMs are separated from all other bulk
states in terms of their real energy and continue to appear at ${\rm
Re}~{}E=0$ even for a very disordered NW. In Fig. 5(b), we depict the LDOS
associated with the ${\rm Re}~{}E=0$ states. The end states continue to be
well separated from each other in space. The main conclusion from Fig. 5(b)
(with $w=0.5$) and its inset (with $w=1.0$) is that with an increase in the
strength of the disorder, the RZMs appear to be largely unaltered, only their
peak height becomes slightly more asymmetric. Due to this robustness, we name
these four ZMs, RZMs.

On the other hand, for the MZMs, the diminishing of the gap separating the
zero-energy states and the bulk states is more abrupt with the increase in
disorder. We depict the disorder-averaged ${\rm Re}~{}E$ as a function of the
state index $m$ for the states close to ${\rm Re}~{}E=0$ in Fig. 5(c) for
$w=0.5$, while using $w=1.0$ in the inset. For $w=1.0$, the gap in the real
energy is essentially nonexistent. In Fig. 5(d), we investigate the LDOS
computed at ${\rm Re}~{}E=0$ for $w=0.5$, while $w=1.0$ in the inset. The
leaking of the MZMs into the bulk is clearly visible for both disorder
strengths. This is expected as the MZMs are energetically not very separated
from the bulk states anymore and shows how the destruction of the bulk gap
directly reduced the MZM localization. Here, we also notice that the peak
height at both NW ends is not the same anymore. It is also interesting to
compare the stability towards the disorder of these dissipation-induced MZMs
to their Hermitian topological counterpart, having the same gap in the real
part of the eigenvalue. We find that the MZMs appearing in the Hermitian
topological case are overall more robust compared to the dissipation-induced
ones. This phenomenon can be understood from the fact that the diminishing of
the bulk gap with the disorder is much less in the Hermitian counterpart. In
particular, we can still obtain MZMs separated from the other states in the
Hermitian case for $w=1$.

##  4 Conclusions and outlook

To summarize, in this work, we study the effect of dissipation on a 1D Rashba
NW placed in direct proximity to a conventional spin-singlet $s$-wave
superconductor. The effect of dissipation, including quantum jumps, is
encompassed in the Liouvillian superoperator. We utilize third quantization to
obtain an NH description of the system. We show that the MZMs that emerge in
the topological regime of the isolated system, remain even in the dissipative
regime. However, the MZMs now have a finite lifetime due to dissipation. We
further discover that it is easy to engineer two kinds of ZMs starting from a
non-topological system entirely due to dissipation: four RZMs and two MZMs.
The RZMs are located at the two ends of the NW, however, they are not
associated with any bulk states or a non-trivial bulk topology but rather
originate from EPs induced by the dissipation. In contrast, the MZMs are
linked to the bulk states, manifested both by a topological phase transition
where the bulk gap closes and a finite bulk winding number, computed based on
pseudo-anti-Hermiticity symmetry. We show that both the MZMs and the RZMs are
robust against random disorder, but notably, the RZMs are more robust against
the disorder compared to the MZMs.

For the dissipation-driven MZMs generated even when the isolated system is
topologically trivial, an exciting future avenue is to showcase how the
properties of the setup and the MZMs can be successfully utilized within
quantum computation [41, 38, 42, 43, 44, 45, 46, 47, 48], thereby opening for
more flexible and environment-tolerant quantum computing platforms. In the
case of dissipation-generated RZMs via EPs, they unravel the intricate
behavior of NH systems, while also shedding light on the interplay between the
NH terms and end states, beyond a NH skin effect. It would here be interesting
to further investigate the topological properties of these RZMs, tied to the
EPs. In particular, if it is possible to gain more control over the emergence
of these modes a priori, with the knowledge of the isolated system’s
Hamiltonian and the specific form of the dissipation. A next step could be to
find a possibility to seek EP-generated boundary states in higher dimensions.
Interestingly, the MZMs and the RZMs never overlap in parameter space. Still,
having them exist side-by-side at only slightly different dissipation
strengths raises questions on how to yet again distinguish MZMs from other
spurious zero-energy modes. In fact, our EP-generated RZMs might be relevant
to consider in current experiments seemingly not being able to detect MZMs.

Furthermore, we always obtain boundary modes with a finite lifetime, which
will decay over time. However, in a dissipative setup, it has been
demonstrated that one can also obtain boundary modes that can have a zero or a
very tiny imaginary part, implying that these states are not as affected as
the remaining bulk states [69, 70]. Thus, only the boundary states survive in
the long time limit. A fascinating direction would be to search for such more
immune MZMs, possibly even having a lifetime approaching infinity, in the
presence of dissipation.

Finally, in our study, we only demonstrate the generation of MZMs by
considering a static system. However, it is known that MZMs can also be
generated from static topologically trivial systems by employing periodic
drives [94, 67]. Thus, it would be intriguing to investigate the interplay of
periodic drive and dissipation to generate MZMs. In addition, we may ask
whether we can get also RZMs in a driven dissipative system and it would be
intriguing if one could even get RZMs at the Floquet zone boundary.

## Acknowledgements

We acknowledge fruitful discussions with Rodrigo Arouca. We acknowledge
financial support from the Knut and Alice Wallenberg Foundation, through the
Wallenberg Academy Fellows program KAW 2019.0309 and project grant KAW
2019.0068. Part of the computations were performed at the Uppsala
Multidisciplinary Center for Advanced Computational Science (UPPMAX) provided
by the National Academic Infrastructure for Supercomputing in Sweden (NAISS),
partially funded by the Swedish Research Council through grant agreements no.
2022-06725 and no. 2018-05973.

##  Appendix A Obtaining $\mathcal{L}_{+}$

In this Appendix A we discuss the procedure to obtain the matrix form of
$\mathcal{L}_{+}$ [Eq. (10)] using the adjoint fermion representation [100,
101, 102]. Following the operations of the adjoint fermions on the basis
states $\ket{P_{\underline{\alpha}}}$, see Eq. (9), we identify the following
relations

$\displaystyle\ket{w_{a}P_{\underline{\alpha}}}=$
$\displaystyle\left(\phi_{a}^{\dagger}+\phi_{a}\right)\ket{P_{\underline{\alpha}}}\
,$ $\displaystyle\ket{P_{\underline{\alpha}}w_{a}}=$
$\displaystyle\mathcal{P}_{\rm
F}\left(\phi_{a}^{\dagger}-\phi_{a}\right)\ket{P_{\underline{\alpha}}},$ (A.1)

where the last equation is obtained employing the anti-commutation properties
of the Majorana operators $w_{a}$. Here $\mathcal{P}_{\rm F}$ is the fermion
parity operator defined as $\mathcal{P}_{\rm F}=(-1)^{\sum_{a}\alpha_{a}}$
$=\exp\left(i\pi\mathcal{N}\right)$, with $\mathcal{N}$ being the number
operator. Employing Eqs. (A.1) and the adjoint fermion anti-commutation
relations, we obtain the following identities, which are useful to find the
quadratic representation of the Liouvillian:

$\displaystyle\ket{w_{a}w_{b}P_{\underline{\alpha}}}=$
$\displaystyle\left(\phi_{a}^{\dagger}\phi_{b}^{\dagger}+\phi_{a}^{\dagger}\phi_{b}+\phi_{a}\phi_{b}^{\dagger}+\phi_{a}\phi_{b}\right)\ket{P_{\underline{\alpha}}}\
,$ (A.2a) $\displaystyle\ket{P_{\underline{\alpha}}w_{a}w_{b}}=$
$\displaystyle\left(\phi_{a}^{\dagger}\phi_{b}^{\dagger}-\phi_{a}^{\dagger}\phi_{b}-\phi_{a}\phi_{b}^{\dagger}+\phi_{a}\phi_{b}+2\delta_{a,b}\right)\ket{P_{\underline{\alpha}}}\
,$ (A.2b) $\displaystyle\ket{w_{a}P_{\underline{\alpha}}w_{b}}=$
$\displaystyle\mathcal{P}_{\rm
F}\left(\phi_{a}^{\dagger}\phi_{b}^{\dagger}-\phi_{a}^{\dagger}\phi_{b}+\phi_{a}\phi_{b}^{\dagger}-\phi_{a}\phi_{b}\right)\ket{P_{\underline{\alpha}}}\
.$ (A.2c)

The density matrix $\rho(t)$ can now be represented by the basis element
$\ket{P_{\underline{\alpha}}}$. The unitary part of the Liouvillian
$\mathcal{L}$ reads as

$\displaystyle\mathcal{L}_{0}\rho:=$ $\displaystyle-i\left[H,\rho(t)\right]$
$\displaystyle=-i\sum_{a,b}H_{a,b}\left(\ket{w_{a}w_{b}P_{\underline{\alpha}}}-\ket{P_{\underline{\alpha}}w_{a}w_{b}}\right)$
$\displaystyle=-4i\sum_{a,b}\phi_{a}^{\dagger}H_{a,b}\phi_{b}\quad\left({\rm
using~{}Eqs.~{}(\ref{idt1}),~{}(\ref{idt2}),~{}and}~{}H_{\rm M}^{ba}=-H_{\rm
M}^{ab}\right)$ $\displaystyle=-4i\underline{\phi}^{\dagger}\cdot H_{\rm
M}\underline{\phi}\,,$ (A.3)

while the non-unitary part can be represented as

$\displaystyle\mathcal{D}\left[L_{m}\right]\rho:=$
$\displaystyle-\frac{1}{2}\sum_{m}\left(\left\\{L^{\dagger}_{m}L_{m},\rho(t)\right\\}-2L^{\dagger}_{m}\rho(t)L^{\dagger}_{m}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{m,a,b}l_{m,a}l_{m,b}^{*}\left(\ket{w_{a}w_{b}P_{\underline{\alpha}}}+\ket{P_{\underline{\alpha}}w_{a}w_{b}}-2\ket{w_{a}P_{\underline{\alpha}}w_{b}}\right)$
$\displaystyle=$ $\displaystyle\sum_{m,a,b}\frac{1+\mathcal{P}_{\rm
F}}{2}\left[\phi_{a}^{\dagger}\left(l_{m,a}l_{m,b}^{*}-l_{m,b}l_{m,b}^{*}\right)\phi_{b}^{\dagger}-\phi_{a}^{\dagger}\left(l_{m,a}l_{m,b}^{*}+l_{m,b}l_{m,b}^{*}\right)\phi_{b}\right]$
$\displaystyle+\sum_{m,a,b}\frac{1-\mathcal{P}_{\rm
F}}{2}\left[\phi_{a}\left(l_{m,a}l_{m,b}^{*}-l_{m,b}l_{m,b}^{*}\right)\phi_{b}-\phi_{a}\left(l_{m,a}l_{m,b}^{*}+l_{m,b}l_{m,b}^{*}\right)\phi_{b}^{\dagger}\right]$
$\displaystyle\qquad\qquad\left({\rm
using~{}Eqs.~{}(\ref{idt1}),~{}(\ref{idt2}),~{}and~{}(\ref{idt3})}\right)$
$\displaystyle=$ $\displaystyle\sum_{a,b}\frac{1+\mathcal{P}_{\rm
F}}{2}\left[\phi_{a}^{\dagger}\left(M_{a,b}-M_{b,a}\right)\phi_{b}^{\dagger}-\phi_{a}^{\dagger}\left(M_{a,b}+M_{b,a}\right)\phi_{b}\right]$
$\displaystyle+\sum_{a,b}\frac{1-\mathcal{P}_{\rm
F}}{2}\left[\phi_{a}\left(M_{a,b}-M_{b,a}\right)\phi_{b}-\phi_{a}\left(M_{a,b}+M_{b,a}\right)\phi_{b}^{\dagger}\right]$
$\displaystyle\qquad\qquad\left({\rm
defining~{}}M_{a,b}=\sum_{m}l_{m,a}l_{m,b}^{*}\right)$ $\displaystyle=$
$\displaystyle\frac{1+\mathcal{P}_{\rm
F}}{2}\left[\underline{\phi}^{\dagger}\cdot\left(M-M^{T}\right)\underline{\phi}^{\dagger}-\underline{\phi}^{\dagger}\cdot\left(M+M^{T}\right)\underline{\phi}\right]$
$\displaystyle+\frac{1-\mathcal{P}_{\rm
F}}{2}\left[\underline{\phi}\cdot\left(M-M^{T}\right)\underline{\phi}-\underline{\phi}\cdot\left(M+M^{T}\right)\underline{\phi}^{\dagger}\right]\
.$ (A.4)

Thus, the full Liouvillian reads as

$\displaystyle\mathcal{L}=$ $\displaystyle-4i\underline{\phi}^{\dagger}\cdot
H_{\rm M}\underline{\phi}+\frac{1+\mathcal{P}_{\rm
F}}{2}\left[\underline{\phi}^{\dagger}\cdot\left(M-M^{T}\right)\underline{\phi}^{\dagger}-\underline{\phi}^{\dagger}\cdot\left(M+M^{T}\right)\underline{\phi}\right]$
$\displaystyle+\frac{1-\mathcal{P}_{\rm
F}}{2}\left[\underline{\phi}\cdot\left(M-M^{T}\right)\underline{\phi}-\underline{\phi}\cdot\left(M+M^{T}\right)\underline{\phi}^{\dagger}\right]\
.$ (A.5)

Now, considering the physical situation where have an even number of fermionic
operators, we can set $\mathcal{P}_{\rm F}=+1$. In this case, the Liouvillian
$\mathcal{L}$, Eq. (A.5) boils down to $\mathcal{L}_{+}$, Eq. (10). This
concludes the derivation of $\mathcal{L}_{+}$.

##  Appendix B Computation of winding number

In this Appendix B, we provide the steps to obtain the definition of the
winding number $\nu$ in Eq. (13). The main ingredient to compute the winding
number is the pseudo-anti-Hermiticity symmetry $\Gamma$. The matrix $X$ being
NH means left and right eigenstates are different:

$\displaystyle X(k)\ket{\Psi_{{\rm R},n}(k)}=E_{n}(k)\ket{\Psi_{{\rm
R},n}(k)}\quad{\rm and}\quad X^{\dagger}(k)\ket{\Psi_{{\rm
L},n}(k)}=E_{n}^{*}(k)\ket{\Psi_{{\rm L},n}(k)}\ .$ (B.1)

The pseudo-anti-Hermiticity symmetry demands that $\ket{\Psi_{{\rm
L},-n}(k)}=\Gamma\ket{\Psi_{{\rm R},n}(k)}$ such that $E_{-n}^{*}(k)=$
$-E_{n}(k)$ [103]. Thus, we define an operator $Q(k)$ as

$\displaystyle Q(k)=$
$\displaystyle\frac{1}{2}\Bigg{(}\sum_{n>0}\ket{\Psi_{{\rm
L},n}(k)}\bra{\Psi_{{\rm R},n}(k)}-\sum_{n<0}\ket{\Psi_{{\rm
L},n}(k)}\bra{\Psi_{{\rm R},n}(k)}$ $\displaystyle+\sum_{n>0}\ket{\Psi_{{\rm
R},n}(k)}\bra{\Psi_{{\rm L},n}(k)}-\sum_{n<0}\ket{\Psi_{{\rm
R},n}(k)}\bra{\Psi_{{\rm L},n}(k)}\Bigg{)}\ .$ (B.2)

Here, $Q(k)$ is a Hermitian matrix satisfying $\Gamma Q(k)\Gamma=-Q(k)$ [103].
Thus, $\Gamma$ appears as an emergent chiral symmetry for $Q(k)$, although
$X(k)$ explicitly breaks it. Employing the basis in which $\Gamma$ is diagonal
with $U_{\Gamma}\Gamma U_{\Gamma}^{-1}={\rm diag}\left(1,1,-1,-1\right)$, we
can represent $Q(k)$ as

$\displaystyle U_{\Gamma}Q(k)U_{\Gamma}^{-1}=\begin{pmatrix}0&q(k)\\\
q^{*}(k)&0\end{pmatrix}\ .$ (B.3)

Using $q(k)$, we define the topological invariant $\nu$ for the system in Eq.
(13). We can also compute this topological invariant in real space for a
system obeying PBC [104].

##  Appendix C NW-SC heterostructure under single-site loss

In the main text, we consider the case where the whole NW is subjected to
dissipation. However, we can also easily envision a setup where only a few or
even a single lattice site is exposed to the environment and encapsulates the
effect of dissipation. In this appendix, we report the results of the extreme
other limit from the main text, that of single-site dissipation. Overall, we
find similar results as in the main text, except that we also often obtain
states at the actual dissipation site. As a consequence, we report these
results as an appendix, as they primarily complement the main text results.
Akin to the main text, we divide our discussions into two parts: the effect of
dissipation on a topological and non-topological isolated NW-SC
heterostructure.

Overall, we discuss the two limiting cases: loss at one of the middle sites of
the chain, here at $n=50$, and loss at the very end site of the chain, $n=1$,
considering a NW with $100$ lattice sites with $n$ representing the lattice
site subjected to loss. The loss at lattice site $n$ can be encapsulated via
the jump operator

$\displaystyle L_{l}^{n}=$
$\displaystyle\sqrt{\gamma}\left(c_{n\uparrow}+c_{n\downarrow}\right)\ .$
(C.1)

Following this jump operator $L_{l}^{n}$, we construct the corresponding
Liouvillian $\mathcal{L}_{+}$ and the matrix $X$, just as the main text.

###  C.1 Fate of MZMs under single-site dissipation

We first consider the loss at a single site of a topologically non-trivial NW-
SC heterostructure with MZMs in the Hermitian (isolated) system regime.
Overall, we find that the lattice site at which the loss is implemented can
significantly change its behavior, as detailed below.

####  C.1.1 Dissipation in middle of NW

We first showcase the results for a system encompassing loss at one of the
middle sites ($n=50$) of the chain in Fig. 7. We plot the real part of the
Lindblad spectra ${\rm Re~{}}E$ as a function of the amplitude of the loss
$\gamma$ for a system obeying OBC in Fig. 7(a). We observe the presence of the
MZMs at ${\rm Re~{}}E=0$ for $\gamma=0$, as expected. However, as we increase
the strength of the dissipation, we notice that two other states come closer
to the ${\rm Re~{}}E=0$ (around $\gamma\sim 4$) and then remain there with
further increase in $\gamma$. To investigate these states more, we depict the
real and imaginary parts of the eigenvalues of the four middle states,
$\left(2N-1\right)$, $2N$, $\left(2N+1\right)$, and $\left(2N+2\right)$, in
Fig. 7(b). Both the real (blue dots) and imaginary (green dots) parts of the
eigenvalues of the MZMs stay at zero for all the values of $\gamma$. While for
the $\left(2N-1\right)$-th and $\left(2N+2\right)$-th states, the real parts
decrease to zero around $\gamma=4$ (red dots), although they still have finite
imaginary part (magenta dots).

Figure 7: (a) Real part of the Lindblad spectra ${\rm Re}~{}E$ as a function
of dissipation strength $\gamma$. (b) Eigenvalues for the four middle states.
Both the real and imaginary parts of eigenvalue for the $2N$\- and $(2N+1)$-th
states stay at zero, while the $(2N-1)$-th and $(2N-2)$-th state show some
transition. (c) Eigenvalues as a function of state index, employing OBC for
$\gamma=8$. The zoomed-in eigenvalues close to ${\rm Re}~{}E=0$ are shown in
inset I1, while inset I2 illustrates the eigenvalues for a system obeying PBC.
(d) LDOS for states with ${\rm Re}~{}E=0$. The LDOS for a system obeying PBC
is shown in the inset. Here $B_{x}=2.0>\lvert B_{x,c1}\rvert$.

Next, we consider a cut from Fig. 7(a) at $\gamma=8.0$ and plot the
corresponding real part of the Lindblad spectrum ${\rm Re~{}}E$ as a function
of the state index $m$ in Fig. 7(c), with the zoomed-in eigenvalue close to
${\rm Re~{}}E=0$ shown in inset I1. We identify the existence of four modes at
${\rm Re~{}}E=0$ as expected from Figs. 7(a,b). In order to identify whether
all the states are end modes, we further employ PBC, i.e. connect the two ends
of the chain, and plot the corresponding eigenvalue spectrum in inset I2. We
observe that then there exist still two modes at ${\rm Re~{}}E=0$ for a system
obeying PBC. Thus, two of the four states appearing at ${\rm Re~{}}E=0$ are
not end states, but actually exist in the bulk. To investigate more
specifically the localization properties of all ${\rm Re~{}}E=0$ modes, we
compute the LDOS at ${\rm Re~{}}E=0$ as a function of the lattice site $x$ in
Fig. 7(d). We observe three peaks: two at the ends of the chains, which we
associate with the MZMs, present already in the Hermitian limit, and one at
the site at which the loss is incorporated. We also demonstrate the LDOS
employing PBC in which only the ZMs at the loss site survive, see inset in
Fig. 7(d). Thus, we conclude that strong single-site loss in the interior of
the NW induces states at ${\rm Re~{}}E=0$, localized at the loss site. This
phenomenon is seemingly akin to that of defect-induced states [106, 107] or
the disorder-induced states in a Rashba NW [32, 35], but here driven purely by
dissipation.

####  C.1.2 Dissipation at end site of NW

Figure 8: (a) Real part of the Lindblad spectra ${\rm Re}~{}E$ as a function
of dissipation strength $\gamma$. (b) ${\rm Im~{}E}$ for the two middle states
as a function of $\gamma$. (c) Lindblad spectrum as a function of state index
$m$ for a fixed $\gamma=2$ and (d) LDOS computed for the ${\rm Re}~{}E=0$
states. Here $B_{x}=2.0>\lvert B_{x,c1}\rvert$.

Having demonstrated the effect of loss in the interior of the NW, we now
discuss the effect of the loss at one of the end sites (here $n=1$) of the NW.
In Fig. 8(a), we illustrate the real part of the Lindblad spectra ${\rm
Re~{}}E$ as a function of $\gamma$. The MZMs that are present in the Hermitian
(isolated) system continue to appear at ${\rm Re~{}}E=0$. However, when we
investigate the imaginary parts ${\rm In~{}}E$ in Fig. 8(b), we notice that
one of the MZMs incurs a finite imaginary part for non-zero $\gamma$. Thus,
one MZM suffers from having a finite lifetime. The increase in the imaginary
part of the MZMs is, however, not monotonic with respect to $\gamma$, as seen
in Fig. 8(b), with the imaginary part even starting to decrease for
$\gamma\sim 2$, before saturating to a finite value for large $\gamma$. This
is quite an interesting behavior as the MZM then becomes more resistant to
dissipation with higher dissipation. The other MZM always remains unaffected
by dissipation, which is expected, since it is located at the other end of the
NW, which is far away from the loss site.

Next, we consider a cut from Fig. 8(a) at $\gamma=2$ and plot the real part of
the Lindblad spectrum ${\rm Re~{}}E$ as a function of state index $m$ in Fig.
8(c). The associated imaginary parts of the eigenvalue spectrum ${\rm Im~{}}E$
are indicated by color. In the inset of Fig. 8(c) we showcase the eigenvalues
close to ${\rm Re~{}}E=0$. We clearly identify the existence of two MZMs, with
one getting a large finite imaginary part. To investigate the localization
property of these MZMs, we plot the LDOS of the ${\rm Re~{}}E$ states as a
function of lattice sites in Fig. 8(d). We observe that the MZMs are sharply
localized at the ends of the system. However, the peak height of the left MZM,
where loss is applied, is less compared to the right one. Also, the position
of the left peak is shifted by one lattice site to the right. We observe that
this shifting of the peak happens only after the imaginary part of one of the
MZMs reaches the maximum value around $\gamma=2$. Thus, the MZM
counterbalances the effect of the dissipation by shifting away from the
dissipative site. This explains why the imaginary eigenvalue part can decrease
despite increasing dissipation. This is an interesting finding, which also
signifies the robustness of MZM against dissipation. It would be intriguing to
analytically find an expression that has a dependence on the position of loss
on MZMs. However, we leave this for future studies.

Figure 9: (a) Real part of Lindblad spectra ${\rm Re}~{}E$ as a function of
$\gamma$ for a system having loss at one of the ends of the NW. (b)
Eigenvalues, real ${\rm Re}~{}E$ (red dots) and absolute ${\rm Abs}~{}E$ (blue
dots) for the two middle states. The scalar product of these two states
$\lvert\braket{\Psi_{i}}{\Psi_{j}}\rvert$ (green dots) on the right axis of
(b). (c) Eigenvalue as a function of the state index, employing OBC for
$\gamma=2$. The zoomed-in eigenvalues close to ${\rm Re~{}E}$ are shown in
inset I1, while inset I2 illustrates the eigenvalues for a system obeying PBC.
(d) LDOS corresponding to ${\rm Re~{}E=0}$. Here $B_{x}=1.2<\lvert
B_{x,c1}\rvert$.

###  C.2 Non-topological system under single-site dissipation

Having investigated the effect of single-site loss on a topological Rashba NW-
SC heterostructure, we finally discuss the effect of single-site loss in a
non-topological setup. We only show the results for loss at $n=1$, since, we
observe the most interesting behavior in this. Dissipation at the interior of
the NW does not change the physics substantially.

####  C.2.1 Dissipation at end site of NW

With the loss at one of the end sites of chain ($n=1$), we plot the real part
of the Lindblad spectra ${\rm Re}~{}E$ as a function of $\gamma$ in Fig. 9(a).
We identify the emergence of ZMs at ${\rm Re~{}}E=0$ for a range of $\gamma$.
However, akin to the RZMs in Fig. 3(a) in the main text, also here, these ZMs
are not associated with any bulk gap closing and can thus not have a
topological bulk winding associated with them either. To understand their
origin, we plot the eigenvalues, both real part ${\rm Re~{}}E$ (red) and
absolute value ${\rm Abs~{}}E$ (blue), of the two middle states: $2N$ and
$(2N+1)$ in Fig. 9(b). We observe that when the modes at ${\rm Re~{}}E=0$
appear or disappear, their absolute part ${\rm Abs~{}}E$ also merges. We also
illustrate the scalar product (green) of the $2N$-th and $(2N+1)$-th
eigenstate on the right axis of Fig. 9(b). One can see that these eigenstates
become parallel exactly when the corresponding absolute part of the
eigenvalues becomes degenerate. Thus, we have EPs at these two points, marked
with black dashed lines. We can thus relate the appearance/disappearance of
ZMs to EPs. This phenomenon is similar to the RZMs that we discuss in Fig.
3(a) in the main text. However, for uniform dissipation, we find that the RZMs
disappear with the increase in $\gamma$ due to the closing of the bulk gap,
while the disappearance of the ZMs for single-site loss is tied to an EP.

In Fig. 9(c), we show the real part of the Lindblad spectra ${\rm Re~{}}E$
with their imaginary parts ${\rm Im~{}}E$ weighted by color, as a function of
state index $m$. In inset I1, we depict the states close to ${\rm Re~{}}E=0$,
while in inset I2, we show the eigenvalue spectra for a system obeying PBC. We
observe the presence of two states at ${\rm Re~{}}E=0$ in the inset I1 of Fig.
9(c), but no states appearing at ${\rm Re~{}}E=0$ in the inset I2 of Fig.
9(c). Thus, these zero-energy states appear only when the ends of the chains
are open. We also demonstrate the LDOS associated with the ${\rm Re~{}}E=0$
states in Fig. 9(d). These ZMs are localized at only one end of the system,
where the loss is incorporated. This generation of ZMs by single-site loss is
interesting since they appear as end states and thereby produce a signature in
the zero-bias conductance peak mimicking an MZM.

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