# Interplay of Disorder and Point-Gap Topology: Chiral Modes, Localization and
Non-Hermitian Anderson Skin Effect in One Dimension

Ronika Sarkar<EMAIL_ADDRESS>Department of Physics, Indian Institute
of Science, Bangalore 560012, India.  Suraj S. Hegde  suraj.hegde@tu-
dresden.de Institute of Theoretical Physics and Wurzburg-Dresden Cluster of
Excellence ct.qmat, Technische Universitat Dresden, 01069 Dresden, Germany.
Awadhesh Narayan<EMAIL_ADDRESS>Solid State and Structural Chemistry
Unit, Indian Institute of Science, Bangalore 560012, India.

###### Abstract

Symmetry-protected spectral topology in extended non-Hermitian quantum systems
has interesting manifestations such as dynamically anomalous chiral currents
and skin effect. In this work, we study the interplay between symmetries and
disorder in a paradigmatic model for spectral topology - the non-reciprocal
Su-Schrieffer-Heeger model. We consider the effect of on-site perturbations
(both real and purely imaginary) that explicitly break the sub-lattice
symmetry. Such symmetry-breaking terms can retain a non-trivial spectral
topology but lead to a different symmetry class. We numerically study the
effect of disorder in on-site and non-reciprocal hopping terms. Using a real-
space winding number that is self-averaging and quantized, we investigate the
impact of disorder on the spectral topology and associated anomalous chiral
modes under periodic boundary conditions. We discover a remarkable robustness
of chiral current and its self-averaging nature under disorder. The value of
the chiral current retains the clean system value, is independent of disorder
strength and is tracked completely by the real-space winding number for class
A which has no symmetries, and class AIII, which has a sub-lattice symmetry.
In class $D^{\dagger}$, which has $PT$-symmetric on-site gain and loss terms,
we find that the disorder-averaged current is not robust while the winding
number is robust. We study the localization physics using the inverse
participation ratio and local density of states. As the disorder strength is
increased, a mobility-edge phase with a finite winding appears. The abrupt
vanishing of the winding number marks a transition from a partially localized
to a fully localized phase. Under open boundary conditions, we similarly
observe a series of transitions through skin effect–partial skin effect–no
skin effect phases. Further, we study the non-Hermitian Anderson skin effect
(NHASE) for different symmetry classes, where the system without skin effect
develops a disorder-driven skin effect at intermediate disorder values.
Remarkably while NHASE is present for different classes, the real-space
winding number shows a direct correspondence with it only when all symmetries
are broken.

##  I Introduction

Non-Hermitian systems, both in classical and quantum settings, have gained
interest because of their physical relevance and conceptual novelty, Martinez
Alvarez et al. (2018a); Bender (2007); Ashida et al. (2020); Ghatak and Das
(2019); Feng et al. (2017); Helbig et al. (2020); Hofmann et al. (2020); Li
and Lee (2022). Non-Hermitian concepts find a wide range of applicability in
fields such as optics, photonics, acoustics, mechanical metamaterials, and
biological open systems, to name a few – rapid experimental work is being done
to explore non-Hermitian aspects in these platforms Feng et al. (2017); Ma and
Sheng (2016); Chou et al. (2011); Midya et al. (2018); Ghatak et al. (2020).
Non-Hermitian lattice models have gained much importance in condensed-matter
settings, having introduced many interesting ideas such as complex eigen-
spectra, non-Hermitian skin effect Yao and Wang (2018); Yuce (2020); Song et
al. (2020); Okuma et al. (2020); Li et al. (2020a); Song et al. (2019a);
Kawabata et al. (2020); Liu et al. (2021); Longhi (2020); Li et al. (2020b);
Lu et al. (2021); Zhu et al. (2020); Okuma and Sato (2022); Borgnia et al.
(2020); Roccati (2021); Li et al. (2021), complex energy gaps, broken bulk-
boundary correspondence Kunst et al. (2018); Koch and Budich (2020);
Edvardsson et al. (2019); Cao et al. (2021); Xiao et al. (2020); Jin and Song
(2019), exceptional points Bergholtz et al. (2021); Miri and Alu (2019);
Özdemir et al. (2019); Huang et al. (2017); Krasnok et al. (2021); Kawabata et
al. (2019a); Hu et al. (2017); Chen et al. (2019a); Minganti et al. (2019); Xu
et al. (2017) among others. In contrast to the band topology in Hermitian
systems, the complex energy spectrum of non-Hermitian Hamiltonians is known to
have a notion of topological winding, which we term ‘spectral topology’. Skin
effect and chiral modes, among others, are manifestations of this spectral
topology. Much like band topology, it is protected by anti-unitary symmetries
and has been thoroughly classified under Bernhard-Sinclair classification
scheme Kawabata et al. (2019b); Lieu (2018a); Herviou et al. (2019a); Zhou and
Lee (2019); Magnea (2008). The spectral topology and the accompanying features
such as the chiral modes and currents have already been probed in experiments
using photonic lattices Wang et al. (2021); Ke et al. (2020); Pan et al.
(2018).

On the other hand, disorder has always played a vital role in condensed matter
systems, ranging from the seminal works on Anderson localization Anderson
(1958); Lee and Ramakrishnan (1985); Kramer and Mackinnon (1993); Phillips
(1993); Segev et al. (2013) to several works on disorder-driven phase
transitions in topological systems Claes and Hughes (2020); Schubert et al.
(2012); Sacksteder et al. (2015); Wu et al. (2016); Prodan (2011); Huang and
Shklovskii (2020); Rudner and Levitov (2009); Jiang and Lee (2022). The notion
of non-Hermiticity and its interplay with disorder has been a recent subject
of wide ranging interest. The effect of disorder on the inherently non-
Hermitian Hatano Nelson model Hatano and Nelson (1996); Kawabata and Ryu
(2021); Heinrichs (2001); Hatano and Obuse (2021); McDonald et al. (2022);
Longhi (2017) has been studied recently, where disorder has been introduced in
the hopping to obtain interesting physics, such as disorder driven phase
transitions from an extended phase to a bulk localized phase Longhi (2021) and
an non-Hermitian Anderson skin effect, where disorder gives rise to a skin
effect Claes and Hughes (2021), and more. Some studies of the non-Hermitian
Su-Schrieffer-Heeger (SSH) model Su et al. (1979); Lieu (2018b); Zhu et al.
(2014); Wu et al. (2021); Yuce and Ramezani (2019); Han et al. (2021); Zeuner
et al. (2015) have also been performed, where disorder has been introduced in
the hopping to find a non-Hermitian topological Anderson insulating phase
Zhang et al. (2020a).

In this paper, we undertake a comprehensive study of the interplay of disorder
and symmetry classes in the non-Hermitian SSH model. The point-gap topology in
the non-reciprocal SSH model is protected by the sub-lattice symmetry and
belongs to Class AIII. We consider a more general model with on-site terms
that breaks the sub-lattice symmetry. The resulting system has no symmetry and
belongs to Class A, which also has a corresponding point gap topology. In
contrast to the Hatano-Nelson model, which also belongs to the same class, we
find that in our case, for real on-site terms, there is an extended region in
the phase diagram with trivial winding (see Section III.4).

We then incorporate disorder, both real and imaginary, to the above mentioned
model and numerically study its effects. For convenience of the reader, we
summarize our key findings below:

  1. 1.

We introduce disorder in the on-site potential and the non-reciprocal hopping.
For periodic boundary conditions (PBC), we calculate the real-space winding
number and the current corresponding to the anomalous chiral mode associated
with the spectral topology Lee et al. (2019); Bessho and Sato (2021); Zhang et
al. (2020b). Both the winding number and the chiral current are self-
averaging. The value of the chiral current retains its clean system value and
is remarkably independent of disorder strength, becoming zero discontinuously
when the winding number becomes trivial. This is true for Classes AIII and A.
However, in Class D† the chiral current is zero on disorder averaging and
shows no equivalence to the winding number.

  2. 2.

Next, considering the inverse participation ratio (IPR) and local density of
states (LDOS), we show two disorder-induced transitions in the system under
PBC. As the disorder strength is tuned, we encounter a mobility-edge phase
where a fraction of states are localised, though the complex spectrum has a
finite winding. With the disorder-induced jump in the winding number from 1 to
0, the system undergoes a transition from a mobility edge phase to a fully
localised phase. The mobility edge in Class A and AIII occurs with respect to
real part of the energy, $\mathchar 572\relax(E)$, while for Class D† it
occurs as a function of the imaginary part, $\mathchar 573\relax(E)$.

  3. 3.

Under open boundary conditions (OBC), in all three classes A, AIII and D†,
upon adding disorder, we again find two disorder-induced transitions
characterised by the LDOS. Being a point-gapped topological system, in the
clean case, the skin effect is always present. However as the disorder
strength is increased, we find a mixed-phase with a fraction of states
localised in the bulk. At a critical disorder strength, the skin effect is
fully destroyed, with states now localised in the bulk. The real-space winding
number computed deep within the bulk also shows a transition to a trivial
value at this point. The winding number is a good indicator to track the
complete localization of states in the system.

  4. 4.

We discover a non-Hermitian Anderson skin effect (NHASE) for all the classes
under OBC when we start introducing disorder around the critical line
$\delta_{c}=\pm|1+t_{1}|$( of the non-disordered system). The winding number
shows a direct correspondence to the NHASE, only when all symmetries of the
system are broken.

The rest of the paper is organized as follows: In Section II, we describe the
non-Hermitian SSH model and discuss its symmetries under different on-site
terms. In Section III, we briefly summarize some features of the clean system
without disorder and present the real space formalism for winding number.
Here, we investigate the effects of adding a disorder-free on-site potential
and note the changes it causes in the phase diagram. Finally, in Section IV,
we present the numerical results on introducing disorder, using real space
calculations of the winding number,chiral current, edge density, IPR and LDOS.
Section V concludes by summarizing our results and providing an outlook.

##  II The system

###  II.1 Our model

Figure 1:  Illustrating the model. Schematic of the non-Hermitian SSH lattice
model with hopping and on-site disorder. $A$ and $B$ correspond to the two
sublattices of the model while $j$ is the lattice site index. $t_{1}$ and
$t_{2}$ are the intra and inter cell hopping strengths, respectively.
$\delta_{1,2}$ are the non-Hermiticity parameters making the intracell hopping
non-reciprocal. $\Delta_{1}$ and $\Delta_{2}$ are the on-site potentials added
to $A$ and $B$ sublattices, respectively.

We consider the generalized non-Hermitian SSH model, which has been widely
investigated for its point gap topology Zhu et al. (2014); He and Chien
(2020); Chen et al. (2019b); Mochizuki et al. (2020). Our generalized
Hamiltonian has the form

$\displaystyle H$
$\displaystyle=\sum_{j}[(t_{1}+\delta_{1})c_{j,A}^{\dagger}c_{j,B}+(t_{1}-\delta_{2})c_{j,B}^{\dagger}c_{j,A}$
(1)
$\displaystyle+t_{2}(c_{j,B}^{\dagger}c_{j+1,A}+c_{j+1,A}^{\dagger}c_{j,B})$
$\displaystyle+\Delta_{1}c_{j,A}^{\dagger}c_{j,A}+\Delta_{2}c_{j,B}^{\dagger}c_{j,B}].$

Here, $c^{\dagger}$ and $c$ are the creation and annihilation operators, while
$t_{1}$ and $t_{2}$ are the intra and inter cell hopping strengths,
respectively (See Fig.1). The summation over $j$ runs over all sites up to
$n$, which is the total number of lattice sites. Special cases of our
Hamiltonian are the chiral symmetric ($\Delta_{1}=\Delta_{2}=0$) and $PT$
symmetric ($\Delta_{1}=-\Delta_{2}=i\mu$) versions of this model, and have
been studied in Lieu (2018b); Herviou et al. (2019b). Here we consider,
$\delta_{1,2}=\delta+\delta^{R}_{1/2}$, where $\delta^{R}_{1}$ and
$\delta^{R}_{2}$ are the disorder terms added to the non-Hermiticity parameter
$\delta$ – this can also be thought of as disorder in the forward and backward
hopping terms, respectively. Similarly,
$\Delta_{1,2}=\Delta_{0}+\Delta^{R}_{1,2}$, where $\Delta_{0}$ is the
disorder-free on-site energy, while $\Delta^{R}_{1}$ and $\Delta^{R}_{2}$ are
the disorder terms added to the on-site potential at $A$ and $B$ sublattices,
respectively. For our study,
$\delta^{R}_{1,2}=\delta^{R}\omega_{\delta_{1,2}}$, where $\delta^{R}$ is the
strength of disorder and $\omega_{\delta_{i}}$ is chosen randomly from the
disorder window $[-1,1]$. Similarly,
$\Delta^{R}_{1,2}=\Delta^{R}\omega_{\Delta_{1,2}}$, where $\Delta^{R}$ is the
disorder strength and $\omega_{\Delta_{i}}\in[-1,1]$. We next discuss the
symmetries of our model in the absence of disorder.

###  II.2 Symmetries

Table 1:  Symmetry classes of the SSH model. Classification of the symmetry classes of the SSH model upon including an on-site potential. Here, SLS refers to sublattice symmetry and $PT$ is parity-time symmetry. The symmetry classes are in accordance with References Kawabata et al., 2019b; Herviou et al., 2019a. On-site potential | Antiunitary symmetry | Symmetry class
---|---|---
$\Delta_{1}=\Delta_{2}=0$ | SLS , $\sigma_{z}H\sigma_{z}^{-1}=-H$ | AIII
$\Delta_{1}\neq\Delta_{2}\neq 0$ | None | A
$\Delta_{1}=i\mu,\Delta_{2}=-i\mu$ |  $PT$, $\sigma_{x}H\sigma_{x}^{-1}=H^{*}$ | $D^{\dagger}$

The non-Hermitian system described by the Hamiltonian in Eq. 1 with
$\Delta_{1,2}=0$ possesses a sublattice symmetry (SLS) given by Kawabata et
al. (2019b) $\sigma_{z}H\sigma_{z}^{-1}=-H$ (here $\sigma_{x,y,z}$ are the
Pauli spin matrices). Therefore, it belongs to Class AIII. SLS can be broken
by on-site terms. If we consider the non-Hermitian SSH model in momentum ($k$)
space, our Hamiltonian with on-site energies $\Delta_{1,2}$ has the matrix
form:

$H(k)=\begin{pmatrix}\Delta_{1}&t_{2}e^{-ik}+t_{1}-\delta_{2}\\\
t_{2}e^{ik}+t_{1}+\delta_{1}&\Delta_{2}\end{pmatrix}.$ (2)

For sublattice symmetry we must have $\sigma_{z}H(k)\sigma_{z}=-H(k)$.
However, for our Hamiltonian,

$\begin{split}\sigma_{z}H(k)\sigma_{z}&=\begin{pmatrix}\Delta_{1}&-t_{2}e^{-ik}-t_{1}+\delta_{2}\\\
-t_{2}e^{ik}-t_{1}-\delta_{1}&\Delta_{2}\end{pmatrix}\\\
&\neq-H(k)\end{split}$ (3)

Hence, any non-zero on-site term breaks the sublattice symmetry of the system.
Further, on-site energy can be introduced in several ways. Different choices
of the on-site terms and the symmetry classes they correspond to are presented
in Table 1. For the present study, we focus on the following types of on-site
terms and corresponding disorder: (a) symmetric on-site disorder:
$\Delta^{R}_{1}=\Delta^{R}_{2}$ in Eq. 1. Such an onsite term explicitly
breaks SLS and places the system in Class A. The symmetric case satisfies the
following constraint: $\sigma_{x}H^{T}\sigma_{x}^{-1}=H$, which however is not
an antiunitary symmetry. (b) random on-site disorder:
$\Delta^{R}_{1}\neq\Delta^{R}_{2}$, which also belongs to Class A. (c)
$PT$-symmetric on-site disorder: $\Delta^{R}_{1}=i\mu$ and
$\Delta^{R}_{2}=-i\mu$. This establishes the symmetry
$\sigma_{x}H\sigma_{x}^{-1}=H^{*}$ and belongs to Class D† Herviou et al.
(2019a).

Our calculations show that the sub-lattice symmetric case with only hopping
disorder (Class AIII; $\delta^{R}_{1,2}\neq 0,\Delta^{R}_{1,2}=0$), gives
qualitatively similar results as the symmetric on-site disorder case (Class A;
$\delta^{R}_{1,2}=0,\Delta^{R}_{1}=\Delta^{R}_{2}\neq 0$) [for a discussion,
see Section IV.3]. Later, we will consider the case of random on-site disorder
and $PT$-symmetric disorder to highlight new features of the disordered non-
Hermitian SSH system [Section IV.4, IV.6].

##  III Discussions for the Clean System

We first briefly revisit the clean system and discuss some of its important
properties. In this case, the disorder parameters in Eq. 1 are set to zero to
obtain the usual non-Hermitian SSH model, i.e.
$\delta^{R}_{1,2}=\Delta^{R}_{1,2}=0$.

###  III.1 Energy Spectrum of Clean System

The usual complex spectra of the non-Hermitian SSH model without disorder can
be classified into four phases Herviou et al. (2019a); Ghatak and Das (2019) –
non-Hermitian topological phase (NH Top), Hermitian topological phase (H Top),
non-Hermitian trivial phase (NH Triv) and Hermitian trivial phase (H Triv).
The different phases can be characterized by the spectral winding number
around different base energies. In $k$ space, the spectral winding number is
given by Claes and Hughes (2021)

$W=\int_{-\pi}^{\pi}\frac{dk}{2\pi
i}\partial_{k}\>\textup{ln}\>[\textup{det}(H(k)-E_{b})],$ (4)

where $E_{b}$ is the base energy about which the winding number is calculated
and $H(k)$ is the Bloch Hamiltonian of the system in momentum space. The
winding number is non-trivial in the NH Top phase with $E_{b}=0$, since at
this point the spectrum shows a point-gap (see Fig. 2a). With non-trivial
winding around some complex base energy, there usually occurs a non-zero
chiral current in the system which we will investigate in Section IV.3.

Figure 2:  Non-Hermitian topological phase with disorder belonging to
different symmetry classes. (a) The complex space energy spectrum in the
undisordered NH Top phase in the presence of real on-site terms, which has a
point gap at $E=0$. In (b), we have added real hopping disorder to the clean
system with disorder strength $\delta^{R}=2.0$. This shows the NH Top spectrum
in class AIII. Panel (c) corresponds to class A, where we have added random
real on-site disorder to each sublattice with disorder strength
$\Delta^{R}=2.0$. In (d), we have included imaginary on-site disorder of the
form $\pm i\Delta^{R}$ to sublattices A and B, leading to class D†. Despite
disorder, the NH Top phase under all the symmetry classes shows a point-gap
topology up to a considerable disorder strength. Here we choose $n=200$,
$t_{1}=1.1$, $\delta=1.2$. The chosen parameters correspond to the NH Top
phase of the disorder-free system.

###  III.2 Winding Number in Real Space

Let us next summarize the real space formalism for the winding number. First,
we define a base energy, $E_{b}$, around which we wish to calculate the
winding. We then construct the matrix $H-E_{b}I$ and perform a singular value
decomposition to get the form $H-E_{b}I=MSN^{\dagger}$, where $S$ is a
diagonal matrix with eigenvalues along the diagonal. Then, we define
$Q=MN^{\dagger}$ and $P=NSN^{\dagger}$, such that $H-E_{b}I=QP$. Now we have
obtained the polar decomposed form of the Hamiltonian and we can write Claes
and Hughes (2021); Song et al. (2019b)

$W=\frac{1}{L^{\prime}}\mathrm{Tr^{\prime}}(Q^{\dagger}[Q,X]).$ (5)

Here, $X$ is the position operator denoting the position of each lattice site.
Before taking the trace, we need to eliminate a sufficient number of lattice
sites from both ends of the one-dimensional lattice so that the winding number
is determined solely by the bulk properties of the system. So, we define a
cut-off length, $l$, from both ends of the one-dimensional lattice. The
effective bulk length is then $L^{\prime}=L-2l$ ($L$ is the total length of
the lattice). Here, $\mathrm{Tr^{\prime}}$, therefore, denotes that the trace
has been taken only over this bulk length. This formulation gives a winding
number which is quantized and robust, and allows to definitively characterize
the different phases and identify phase transitions which may be parameter-
driven or disorder-driven. For our calculations we fix $l=0.2L$.

###  III.3 Chiral Modes and Non-Hermitian Skin Effect

The spectrum of the system under PBC in the non-Hermitian topological phase,
shown in Fig. 2, illustrates that the spectrum accommodates point gaps. In
such a point-gapped spectrum, the complex energies for which its real part of
the spectrum $\mathchar 572\relax(E)$ is gap-less correspondingly have both
positive and negative imaginary energy parts. These eigenvalues correspond to
dynamically anomalous chiral modes of the system Lee et al. (2019); Bessho and
Sato (2021); Zhang et al. (2020b). Even though gapless modes come in pairs of
opposite chiralities as in a Hermitian system, the non-Hermiticity stabilises
only one of these modes. The chirality of the modes are determined by the
spectral winding and only one of them survive the longest on evolving through
time. In order to study the long-time dynamics of the system, it is sufficient
to study this chiral mode Banerjee et al. (2021). For example, in the NH Top
phase (Fig. 2a), the chiral mode is located at $\mathchar 572\relax(E)=0$ at
the top of the spectrum (shown in red). This chiral mode is responsible for a
chiral current, which is unidirectional in nature and is an observable of the
system. We develop the formalism to calculate the chiral current in Section
IV.3.

The non-Hermitian skin effect is another unique property of non-Hermitian
systems where, as long as the spectrum accommodates a point-gap topology under
PBC, all the eigenstates under OBC migrate to an edge determined by the
direction of the persistent current Jiang et al. (2020); Zhang et al. (2020b);
Martinez Alvarez et al. (2018b); Lee and Thomale (2019); Okuma et al. (2020).
From the spectral topology shown in Fig. 2, we see that the non-Hermitian
phase under different symmetry classes will show a skin effect when we cut
open the system under PBC and switch to OBC. Interestingly, the non-Hermitian
skin effect can also occur starting from trivial winding and introducing
disorder, which we will elucidate later in Section IV.6.

###  III.4 Introduction of Real Symmetric On-site Energy

Next, in order to illustrate the effect on of onsite terms on non-Hermitian
topological phases we consider the simplest case of
$\Delta_{1}=\Delta_{2}=\Delta_{o}$. Fig. 3 shows the evolution of the phase
diagram of the system with increasing values of such an on-site energy
$\Delta_{o}$. Fig. 3a shows the well-studied phase diagram Herviou et al.
(2019b) of the non-Hermitian SSH model with $\Delta_{o}=0$, where the non-
trivial winding about $E_{b}=0$ corresponds to the non-Hermitian topological
phase. The $t_{1}=t_{2},\delta=0$ is the gapless point in between the two
gapped Hermitian SSH topological phases. On addition of even a small non-
Hermiticity $\delta$, the spectrum becomes complex and there is an anisotropy
introduced in the right and left movers, effectively resulting in a chiral
mode characteristic of the NH Top phase. On introducing finite $\Delta_{0}$ at
$\delta=0$ (which shifts the ‘chemical potential’ such that it is within a
band) one has a window of gapless points around $t_{1}=t_{2}$. Now introducing
non-Hermiticity $\delta$ over this gapless window leads to the NH Top phase as
seen in Fig.3 (b and c). For larger values of $\Delta_{o}$ the gapless window
occurs only at larger values of $t_{1}$ as seen in 3(d)

Figure 3: Phase diagram with real symmetric on-site potential. Winding number
as a function of parameters $t_{1}$ and $\delta$ for different values of on-
site energy $\Delta_{o}$. The critical lines of the regular NH SSH model can
be seen in (a) where $\delta_{c}=\pm|1\pm t_{1}|$ shows a transition in
winding number. As the on-site energy increases (b)-(d) the NH Top region
moves towards higher $t_{1}$ values invading the former H Triv region. Here,
$t_{2}=1$, $n=100$ and base energy $E_{b}=0$.  Figure 4:  Phase diagram with
imaginary anti-symmetric on-site potential, $\Delta_{1,2}=\pm i\mu$. (a) The
winding number as a function of $\delta$ and $t_{1}$ for $\mu=0.5$. (b) The
winding number as a function of $\mu$ and $t_{1}$ for $\delta=0.5$. The region
with non-trivial winding number is the non-Hermitian topological phase which
is extended due to the presence of $\mu$. Here $n=100$.

###  III.5 Introduction of Imaginary Antisymmetric On-site Energy

Here, we look at the changes in the phase diagram on introducing an imaginary
antisymmetric on-site energy, $\Delta_{1}=+i\mu$ and $\Delta_{2}=-i\mu$, such
that the system retains $PT$ symmetry and belongs to the D† class. The phase
diagram is shown in Fig. 4 and is in agreement with those reported in
Reference Herviou et al., 2019a. Fig. 4a shows the winding number as a
function of $\delta$ and $t_{1}$ when $\mu=0.5$ and Fig. 4b shows the winding
number as a function of $\mu$ and $t_{1}$ when the non-Hermiticity parameter
$\delta=0.5$. The winding number has been calculated with respect to base
energy $E_{b}=0.0$. Hence non-trivial winding number corresponds to the non-
Hermitian topological phase. In both cases, we can see an extended region of
the same due to the introduction of non-zero $\mu$.

Figure 5:  Characterizing the disorder-induced phases under PBC. (a) The
winding number, $W$, as a function of disorder. (b) The number of localized
states (with IPR $\geq$ 0.4) as a function of disorder, demarcating three
regions corresponding to the different phases. (c) IPR as a function of
$\mathchar 572\relax(E)$ for the three regions, where I, II and III correspond
to $\Delta^{R}=$ 0.0, 1.0, 4.5, respectively. The inset shows the zoomed in
plot for Region I (in brown). (d) LDOS for Region I ($\Delta^{R}=0.0$) which
is the extended or delocalized phase, (e) Region II ($\Delta^{R}=1.0$), which
is the mobility-edge phase, and (f) Region III ($\Delta^{R}=4.5$), the bulk-
localized phase. All IPR plots have been disorder-averaged. Winding number is
self-averaging. LDOS plots (shown only up to the 40th lattice site) are for a
representative disorder configuration. Here, number of lattice sites $n=400$,
$t_{1}=1.0$, $t_{2}=1.0$, $\delta=0.3$, $\Delta_{o}=0.0$, symmetric on-site-
disorder strength $\Delta^{R}$ varies from 0 to 5 in steps of 0.05.

##  IV Introducing Disorder

Next, we introduce disorder in the system through the hopping and the on-site
energy terms. In this section, we study how disorder affects various
properties such as the chiral current, winding number and density of states.
We compare the effect of hopping and on-site disorder and also investigate
disorder-driven topological transitions as well as localization transitions.

###  IV.1 Energy spectra under disorder in the different symmetry classes

We find that the point-gap topology of the different phases of the clean
system persists even in the presence of disorder for all classes A, AIII and
D†, up to considerable disorder strengths. This hints at a persistent non-
trivial winding number and a non-zero chiral current even in the presence of
disorder which we elaborate in the following sub-sections. In Fig. 2, we show
the non-Hermitian topological phase of the clean system and the changes in the
complex space spectra as we introduce disorder transitioning the system
between different symmetry classes. Fig. 2(b) shows class AIII, where disorder
has been introduced in the hopping terms, while Fig. 2(c) shows class A, where
a random real on-site energy has been added to each sublattice. Fig. 2(d)
corresponds to class D†, where imaginary disordered on-site energy of the form
$\pm i\Delta^{R}$ has been introduced to sublattices A and B, respectively. In
all three disordered cases the spectrum retains its point-gap topology until
the critical disorder strength.

Figure 6:  Disorder induced phases under OBC. (a) The density of states at the
edge ($\rho_{edge}$) which drops down from close to unity as the skin effect
disappears. This helps distinguish between Regions I, II and III. The inset
shows the change in winding number with disorder which distinguishes between
Regions II and III. The LDOS for (b) Region I ($\Delta^{R}=0.0$), which is the
skin phase, (c) Region II ($\Delta^{R}=1.5$) which is the mixed phase, and (d)
Region III ($\Delta^{R}=4.5$), which is the bulk-localized phase. Here number
of lattice sites, $n=400$, $t_{1}=1.0$, $t_{2}=1.0$, $\delta=0.3$,
$\Delta_{o}=0.0$ and symmetric on-site-disorder strength
$\Delta_{1}=\Delta_{2}=\Delta^{R}$ varies from 0 to 5 in steps of 0.05.

###  IV.2 Disorder induced localization in Class A and AIII

The interplay of localization and winding number has been studied widely in
the context of disorder and quasiperiodic systems Shnerb and Nelson (1998,
1998); Zhai et al. (2020); Jiang et al. (2019); Zeng and Xu (2020). From our
computations, we discover three disorder-induced phases in the system under
both PBC and OBC. We characterise these using various measures, such as the
winding number, inverse participation ratio (IPR), number of localised states
($N_{Loc}$), the local density of states (LDOS) and the density of states at
the edge ($\rho_{edge}$). We briefly describe these quantities in the
following: 1) The measure of localization (of a state $\alpha$ with
eigenvector $\psi_{\alpha}$) is calculated using the IPR, which is given by
Kramer and Mackinnon (1993)
$I_{\alpha}=\frac{\sum_{i}|\psi_{\alpha}(x_{i})|^{4}}{(\sum_{i}|\psi_{\alpha}(x_{i})|^{2})^{2}}$.
$I_{\alpha}$ close to unity implies that the state is very localized, while a
very low IPR is indicative of a delocalized state. 2) To evaluate the LDOS, we
calculate $\sum_{\alpha}|\psi_{\alpha}(x_{i})|^{2}$ at each lattice site
($x_{i}$). 3) We characterize a state as localized if its IPR $\geq 0.4$. The
number of such states gives us $N_{Loc}$ which we use to categorize the
observed phases. 4) $\rho_{edge}$ gives us the density of states at the edge
of the system, where the edge has been quantified as the first two lattice
sites.

In Fig. 5, we present the characterization of the three disorder-induced
phases under PBC. We have named them in a manner similar to the disordered
phases found in the Hatano-Nelson model Longhi (2021). While the winding
number is non-trivial, we find the extended (or delocalized) phase (Region I)
and the mobility-edge phase (Region II). Region I is characterised by all
states being delocalized, hence $N_{Loc}$ is zero (Fig. 5b). The IPR is very
low for all the states (Fig. 5c) and the LDOS is similar at all lattice sites
(Fig. 5d). Region II is characterized by an intermediate, non-zero $N_{Loc}$
(Fig. 5b), high IPR for some states beyond a certain $\mathchar 572\relax(E)$
and low IPR for other states (Fig. 5c). The LDOS fluctuates over the lattice
sites (Fig. 5e). In this phase, extended and localized states coexist Yuce and
Ramezani (2022). Next, as soon as winding number becomes trivial, all the
states in the system localize due to high disorder. In this scenario,
$N_{Loc}\sim n$ (Fig. 5a,b). The IPR of all states starts to rise (Fig. 5c),
and the LDOS shows localization of all states in the bulk (Region III) (Fig.
5f) .

Next, we present the characterization of the disorder-induced phases found
under OBC in Fig. 6. Fig. 6b shows the LDOS for the skin phase, in which all
states are localised at the edge of the lattice. Fig.6c shows the mixed phase,
where a fraction of states show skin effect, while the remaining fraction of
states move into the bulk. With a further increase in disorder strength, the
system transitions to the bulk-localized phase (Region III), where all the
states are localized in the bulk, as can be seen from the LDOS in Fig. 6d. The
IPR is always relatively high as the states are either localized at the edge
or localized in the bulk of the system. The three phases can be distinguished
with the help of $\rho_{edge}$ and the winding number (Fig. 6a). $\rho_{edge}$
shows a sharp fall as we enter the mixed phase from the skin phase and
subsequently goes to zero in the bulk-localized phase, accompanied by a
winding number change from 1 to 0, demarcating Regions II and III.

It is important to note that under both PBC and OBC, a change in winding
number coincides with the transition from a partially localized to a fully
localized phase. We find that as long as there is a finite non-zero winding,
the system does not transition to a fully localized phase. The chosen
parameters in the above discussion correspond to class A. We note that class
AIII shows the same qualitative features.

###  IV.3 Chiral Current

Figure 7: Correspondence between chiral current and winding number. (a),(c)
The winding number, $W$, as a function of disorder for under PBC. (b),(d) The
chiral current as a function of different kinds of disorder in the same
system. (a) and (b) correspond to the effect of introducing a symmetric on-
site disorder strength (class A) with $\Delta_{1}=\Delta_{2}=\Delta^{R}$.
Panels (c) and (d) show the effect of hopping disorder (class AIII). We find
that the system is more robust to on-site disorder rather than disorder in
hopping and that there is a clear equivalence between winding number and
chiral current with disorder. Here $n=400$, $t_{1}=1.0$, $t_{2}=1.0$,
$\delta=0.3$, and $\Delta_{o}=0.0$.

As mentioned in Sec. III.3, the dynamically anomalous chiral mode is on the
characteristics of the spectral topology. In our analysis of the disordered
system we use the current due to the chiral mode, $J_{C}$. The current
operator is given by Banerjee et al. (2021); Asbóth et al. (2016)

$J_{C}=\frac{iL}{2}[\langle c_{j}^{\dagger}c_{j+1}\rangle-\langle
c_{j+1}^{\dagger}c_{j}\rangle],$ (6)

where the expectation value is taken with respect to the chiral mode, i.e.,
the eigenvector with the maximum positive imaginary energy at the real-gapless
point. While the negative imaginary counter part is also a chiral mode, it
decays over long times. Here, $j\in[1,L-1]$. If $j$ is taken to be odd,
$J_{C}$ corresponds to the intracell current, while an even $j$ yields the
intercell current. We note that $J_{C}$ is independent of the value of $j$ as
the same chiral current flows through the entire lattice under PBC.

From Fig. 7 we can see that the winding number and the chiral current show a
correspondence for both on-site disorder (Fig. 7a,b) and hopping disorder
(Fig. 7c,d). These correspond to classes A and AIII, respectively. Regions I,
II and III are the same disorder-induced phases found under PBC in Section
IV.2. The chiral current directly corresponds with the winding number and is
thus self-averaging i.e without averaging over different disorder
configurations, the current approaches a unique mean value for large enough
systems. First, let us consider the value of the chiral current in the
disorder free case.In momentum space, the Hamiltonian of the disorder-free
system can be written as $H(k)=\boldsymbol{d(k)}.\boldsymbol{\sigma}$, where
$|\boldsymbol{d(k)}|=\pm\sqrt{t_{1}^{2}+t_{2}^{2}-\delta^{2}+2t_{1}t_{2}\cos
k-2it_{2}\delta\sin k}=\pm\epsilon(k)$. Then, the current in the system can be
obtained as $j(k)\sim\nabla_{k}\epsilon(k)$. We obtain

$j(k)=\frac{i}{2}(-e^{-ik}/s+e^{ik}s),$ (7)

where, $s=-\sqrt{(t_{1}+t_{2}e^{-ik}-\delta)/(t_{1}+t_{2}e^{ik}+\delta)}$. We
identify the chiral mode at a momentum $k_{C}$. The chiral current is then
$J_{C}=j(k_{C})$. The chiral mode in the non-Hermitian topological phase of
the nH-SSH model is found at $k_{C}=\pi$ and the value of the current depends
on the system parameters we choose.

Now remarkably, on adding disorder we find that the magnitude of the current
is very robust even for substantial disorder strengths up to a critical value
where it goes to zero. This is one of the important observations of our work.
The symmetry protected anomalous chiral current is self-averaging and is
extremely robust to disorder. We also note that our numerical result is
consistent with recent field theoretical analysisKawabata et al. (2021), which
showed that in the continuum limit the response current at a real-gapless
point is exactly given by the winding.

It is important to note here that we obtain the same qualitative observations
irrespective of whether we introduce a disorder in hopping
($\delta^{R}_{1,2}$; class AIII) or an on-site disorder ($\Delta^{R}_{1,2}$;
class A). However, the non-trivial phase characterized by a non-vanishing
chiral current and a finite winding number is more robust to on-site disorder
rather than hopping disorder. The hopping disorder has the higher probability
of impeding the flow of chiral mode compared to the onsite terms. The chiral
current behaves differently in symmetry class D† as will be discussed in the
following subsection.

###  IV.4 Imaginary On-site Disorder : Class D†

Figure 8: Disorder-induced properties under PBC in Class D†. (a) The Winding
number $W$ as a function of PT-symmetric disorder for the system under PBC.
(b) The chiral current as a function of the same disorder which varies from 0
to 5 in steps of 0.05. The chiral current randomly fluctuates between -1 and
1, which when disorder averaged becomes trivially zero (shown in the inset).
Panel (c) shows the number of localised states for the same disorder, while
panel (d) shows the IPR at $\Delta^{R}=1.0$ (Region II), which has a mobility
edge with respect to the imaginary part of the eigenvalues. The figure shows
that for Class D†, winding no. tracks the complete localization in the system
demarcating Regions II and III. $N_{Loc}$ helps demarcate between the extended
phase (Region I), intermediate mobility edge (Region II) and the bulk
localised phase (Region III). The chiral current does not show equivalence to
the winding number anymore; rather, it fluctuates between 1 and -1, averaging
out to 0 on taking a disorder average. The IPR, contrary to Class A and AIII,
shows a mobility edge as a function of $\mathchar 573\relax(E)$. Here, no. of
lattice sites $n=400,t_{1}=1.0,t_{2}=1.0,\delta=0.3,\Delta_{o}=0.0$. Figure 9:
Winding number and edge density under OBC in Class D†. (a) The winding number
$W$ as a function of PT-symmetric disorder for the system under OBC. (b) The
density of states at the edge under the same disorder which varies from 0 to 5
in steps of 0.05. The winding number change from 1 to 0 tracks the complete
localization of the states into the bulk. The edge density $\rho_{edge}$ shows
skin states in Region I, an intermediary mixed phase with partial localization
and partial skin states (Region II) and finally Region III where all the
states are localised in the bulk. Here, no. of lattice sites
$n=400,t_{1}=1.0,t_{2}=1.0,\delta=0.3,\Delta_{o}=0.0$.

In this section, we discuss the effects of introducing an on-site disorder of
the form $\Delta_{1}=i\Delta^{R}\omega_{\Delta_{1}}$ and
$\Delta_{2}=-i\Delta^{R}\omega_{\Delta_{2}}$, such that
$\omega_{\Delta_{1}}=\omega_{\Delta_{2}}$. This configuration of disorder
leads us to symmetry class D† which, unlike the other classes A and AIII
(discussed earlier), includes anti-symmetric imaginary on-site terms chosen
randomly from a window of disorder of strength $\Delta^{R}$. Similar to the
previous discussion, here too, we obtain three disorder-induced phases under
both PBC and OBC. However, as the disorder is now purely imaginary, different
features are found compared to classes A and AIII, which we highlight here.

Fig. 8 shows the disorder-induced phases under PBC. As before, Region I is the
extended phase where the number of localized states, $N_{Loc}=0$ (Fig. 8c),
Region II is the mobility-edge phase with an intermediate number of localized
states, and Region III is the bulk-localized phase where all the states are
localized in the bulk ($N_{Loc}\sim n$). We find that the winding number (see
Fig. 8a) tracks the localization physics, showing a sharp transition from 1 to
0 as soon as all the states in the system get localized. Thus, it is effective
in demarcating between Regions II and III. Interestingly, the chiral current
(shown in Fig. 8b) does not show an equivalence to the winding number anymore,
unlike in classes A and AIII. The chiral current fluctuates between 1 and -1
at small values of disorder and at sufficiently large values it stabilises to
0. On disorder-averaging the chiral current over 100 configurations of
disorder, we found that $J_{C}$ is identically zero over the entire disorder
region (shown in the inset of Fig. 8b). Thus, in class $D^{\dagger}$, the
chiral current is not robust to disorder nor equivalent to the winding number.
Even though the current can be finite for a given disorder configuration,
random fluctuations in the equal gain and loss terms destroys the current when
disorder averaged. Turning to localization physics, the mobility edge phase in
Region II, is distinct from the mobility edge found for classes A and AIII.
Here, the mobility edge occurs with respect to $\mathchar 573\relax(E)$ rather
than $\mathchar 572\relax(E)$, as for the other symmetry classes. This is
understandable since the disorder we add here is purely imaginary, contrary to
the other symmetry classes. The mobility edge phase is well characterized by
the IPR as a function of $\mathchar 573\relax(E)$, as shown in Fig. 8d.

To characterize the phases under OBC, we use the winding number and the edge
density of states ($\rho_{edge}$), as reliable indicators. Fig. 9 shows the
disorder-induced phases under OBC for class D†. Here, Region I is the skin
phase, where $\rho_{edge}\sim 1$, i.e., all the states are localized at the
skin (Fig. 9b). Region II is the mixed phase with an intermediate edge
density, indicating a mixture of skin states and bulk states. Region III is
the bulk localized phase, where $\rho_{edge}\sim 0$, indicating no skin
states. Here too, the winding number tracks the complete localization of the
system, clearly demarcating Regions II and III.

###  IV.5 Disorder around phase boundaries under PBC

Figure 10:  Winding number as a function of disorder from a critical line
under PBC. Plot of winding number as function of on-site symmetric disorder
and hopping disorder showing a $0\rightarrow 1\rightarrow 0$ transition when
we sit on the critical line $\delta_{c}=|1+t_{1}|$. Here, parameter values
$t_{1}=1.1,\delta_{c}=2.1,\Delta_{o}=0.0$ and $n=100$.

After investigating the effect of disorder in various classes, next we
consider the effect of disorder on a system sitting on the critical lines in
the phase diagram of the non-reciprocal SSH model without onsite terms . In
the phases with a non-zero winding number $W$, on the introduction of either
hopping ($\delta^{R}$) or symmetric on-site disorder ($\Delta^{R}$) or both,
$W$ changes from 1 to 0 at a certain parameter-dependent critical disorder
strength. Interestingly, starting from the set of critical lines:
$\delta_{c}=\pm|1+t_{1}|$, demarcating the NH Triv and NH Top phases, $W$
shows a $0\rightarrow 1\rightarrow 0$ transition. This behavior persists
irrespective of the different disorder configurations and even for choice of
parameters slightly displaced from the critical line. Fig. 10 shows the
winding number as a function of disorder strengths $\delta^{R}$ and
$\Delta^{R}$, when one is positioned on the critical line
$\delta_{c}=|1+t_{1}|$ under PBC and introduces disorder. As we have
established that there exists a correspondence between the winding number and
the chiral current for the concerned symmetry classes A and AIII, such a phase
diagram indicates that we can start from a system with no chiral current and
tune it using either hopping or on-site disorder (or both) to obtain a non-
zero chiral current. At the critical lines, $\delta_{c}=\pm|1-t_{1}|$, which
lie between the NH Top and H Triv phases, $W$ shows a $1\rightarrow 0$
transition.

###  IV.6 Non-Hermitian Anderson skin effect

Figure 11:  The NHASE under OBC. (a) The winding number as a function of
hopping disorder strength $\delta^{R}$ and random on-site disorder strength
$\Delta^{R}$ which vary from 0 to 4 in steps of 0.05. (b) The density of
states at the edge of the lattice as a function of both $\delta^{R}$ and
$\Delta^{R}$. A system with no skin effect initially develops a skin effect
under the effect of disorder which subsequently disappears. Here,
$\Delta_{1,2}=\Delta^{R}\omega_{\Delta_{1,2}}$ such that
$\omega_{\Delta_{1}}\neq\omega_{\Delta_{2}}\implies\Delta_{1}\neq\Delta_{2}$.
When there is no symmetry in the system, the figure shows that the winding
number and density of edge states show the same variation with disorder. Here,
$n=100,t_{2}=1.0,t_{1}=1.1,\delta=2.1,\Delta_{o}=0.0$. Figure 12:  The winding
number and NHASE under OBC in for imaginary disorder. (a) The Winding number
as a function of disorder strength $\delta^{R}$ for hopping and disorder
strength $\Delta^{R}$ for imaginary on-site terms. Both vary from 0 to 4 in
steps of 0.05. (b) The density of states at the edge of the lattice as a
function of both $\delta^{R}$ and $\Delta^{R}$. Here, $\Delta_{1,2}=\pm
i\Delta^{R}\omega_{\Delta_{1,2}}$ such that
$\omega_{\Delta_{1}}\neq\omega_{\Delta_{2}}$. Here too, the winding no.
corresponds to the occurrence of anomalous edge states. The parameter values
here are: $n=100,t_{2}=1.0,t_{1}=1.1,\delta=2.1,\Delta_{o}=0.0$.

Starting at or around any critical point on the line $\delta_{c}=1+t_{1}$,
under OBC, where there is no skin effect, we find that by introducing disorder
in hopping or on-site energy (or both), a skin effect develops and
subsequently disappears at high disorder strengths. This disorder-induced
appearance of the skin effect has been dubbed the non-Hermitian Anderson skin
effect (NHASE) Zhang et al. (2020a), and was also found recently in the
Hatano-Nelson model by Claes and Hughes Claes and Hughes (2021), upon
introducing disorder in the hopping potential

Remarkably, the NHASE in our system occurs irrespective of the presence or
absence of symmetries such as SLS, $PT$ or particle-hole symmetry, i.e., the
NHASE can be seen for classes A, AIII as well as D†. This anomalous behaviour
as a function of disorder is presented in Fig. 11(b), which shows that a skin
effect occurs at intermediate disorder values and vanishes at large disorder.
Here, $\delta^{R}$ and $\Delta^{R}$ are the disorder strengths of hopping and
on-site real disorder respectively. The same observations hold if we chose
$\Delta_{1}=\Delta_{2}$ (class A with a constraint) or $\Delta_{1,2}=\pm
i\Delta^{R}\omega$ (class D†) and $\Delta_{1,2}=0$ which corresponds to class
AIII. The behaviour of the edge densities in Fig. 11(b) and Fig.12b are
illustrative of this, though their relation to the winding number warrants
special attention.

In Fig. 11(a) and 12(a), we can see that the winding number shows a similar
behavior as the corresponding edge density. This direct correspondence between
the winding number and the NHASE occurs only if we break all the symmetries in
the system by taking a random on-site disorder, such that
$\Delta^{R}_{1}\neq\Delta^{R}_{2}$, where $\Delta^{R}$ could in general be
real or imaginary. We show that starting from a system with zero winding
number, as we introduce a random disorder in either hopping or on-site energy
or both, the system acquires a non-trivial winding at a critical disorder
strength, and as soon as the winding number changes to 1, the NHASE develops
in the system. Fig. 11 shows this correspondence between the winding number
and the density of states at the edge of the lattice, as we introduce hopping
or on-site disorder. The density of states at the edge is the number of states
localized at the edge as a fraction of the total number of eigenstates of the
system.
It is important to note that our findings demonstrate that the origin of skin
effect in non-Hermitian systems is the point-gap spectral topology under PBC
Kawabata et al. (2019b), irrespective of having trivial or non-trivial
winding. The occurrence of a non-zero current in the system is a consequence
of non-trivial winding Kawabata et al. (2021). On the other hand, only when we
break SLS ($\sigma_{z}H\sigma_{z}^{-1}=-H$), $PT$ symmetry
($\sigma_{x}H\sigma_{x}^{-1}=H^{*}$), can we expect a direct relationship
between NHASE and winding, but the NHASE is nevertheless present irrespective
of presence or absence of symmetries. The winding number, which is effective
in tracking the localization physics in all the symmetry classes A, AIII and
D† is unable to track the Anderson skin effect (NHASE) when there is any
underlying symmetry or constraint present in the system.

##  V Summary and Conclusions

In view of studying the interplay between symmetry-protected spectral topology
in non-Hermitian systems and disorder, we have considered the addition of both
real and imaginary on-site potentials to the non-Hermitian SSH model. We have
systematically investigated the effect of different kinds of disorder in
different symmetry classes by numerically computing the real-space winding
number, localization characteristics and the chiral current associated
associated with dynamically anomalous chiral mode. We found a correspondence
between the chiral current and winding number in classes A and AIII. The
chiral current is self-averaging under disorder and is remarkably robust,
retaining its clean-system-value up to significant disorder strengths and
dropping to zero only beyond a critical disorder strength. This robustness of
the chiral current may be of value in technological applications. Localization
transitions due to disorder have been probed under both PBC and OBC, where the
system exhibits three distinct phases. We found that a non-trivial winding
persists up to significant disorder strengths, resisting a complete
localization of all the states. We have also demonstrated the occurrence of
the NHASE in all three symmetry classes (A, AIII and D†), where starting from
the disorder-free system, which has no skin effect, the system develops a non-
Hermitian skin effect on introducing disorder. One of our striking
observations is that the correspondence between the real-space winding number
and the NHASE holds only when all symmetries are broken.

We note that the non-Hermitian effective Hamiltonian is valid for short time-
scales before the jump processes occur and the steady state dynamics of non-
Hermitian systems requires the consideration of jump operators and a
Lindbladian approach. One can incorporate non-Hermitian gain and loss as well
as non-reciprocal hopping into the Su-Schrieffer Heeger model using these jump
operators as has been discovered in Ref. Song et al., 2019c. For systems
connected to a suitably generic bath, where the Hamiltonian is quadratic in
Fermi operators, the system dynamics can be expressed by a “quadratic
Lindbladian” whose diagonalization gives the complex eigenvalues of the system
Lieu et al. (2020). In such a case, following the Lindbladian formalism one
can infer the eigenspectrum for a set of parameter values and hence possibly
calculate the winding number which will determine the topological phase.
Disorder can be incorporated into the model as a fluctuation in these
parameters. A deeper understanding of the effect of disorder, its resilience
and the time-dynamics is an interesting direction for future work.

Our formalism is general in terms of the Hamiltonian, winding number, chiral
current and the other indicators we use. Hence, we expect our analysis of
disorder, non-Hermitian skin effect and the single mode (chiral) current to
hold for bosonic cases as well. When considering bosonic or fermionic
statistics particularly, one would have to be careful in the way one
constructs the many body state. The filling factor and interaction terms would
be crucial quantities in the differing physical manifestation of bosons and
fermions. Consequences of a bosonic or fermionic system with disorder is a
very interesting topic for future studies.

There have already been breakthroughs in experimental realization and probing
of non-Hermitian physics in optical photonic and mechanical systems Lin et al.
(2021); Ding et al. (2016); Liu et al. (2020); Cerjan et al. (2019); Qin et
al. (2021); Weidemann et al. (2020); Wu et al. (2020); Zhang et al. (2021).
Our results could be corroborated in these settings. On the other hand,
further theoretical analyses is required towards understanding the disorder-
induced topological transitions, criticality and localization in non-Hermitian
systems. This could possibly done under the field theoretical framework of
recent works Kawabata et al. (2021); Moustaj et al. (2021). The self-averaging
nature of quantities such as the localisation length are understood well in
terms of transfer matrices in disordered Hermitian systems. The self-averaging
nature of the winding number and the chiral current needs to be addressed in
non-Hermitian systems. Many-body effects in non-Hermitian systems have gained
significant attention very recently Banerjee et al. (2021); Hamazaki et al.
(2019); Zhai et al. (2020); Kawabata et al. (2022); Denner et al. (2021). It
would be interesting to study the interplay of symmetry classes, disorder and
many-body effects. Another important direction would be studying the different
symmetry classes and disorder directly in open quantum systems.

## ACKNOWLEDGEMENTS

RS would like to thank D. Sen for several useful discussions, and acknowledge
S. Chakraborty, A. Banerjee and A. Bandyopadhyay for their help. We thank A.
Agarwala and A. Banerjee for a related collaboration. We would especially like
to thank Tobias Meng for useful comments on the draft. SSH acknowledges
funding from the Deutsche Forschungsgemeinschaft and the National Research
Fund Luxembourg via the project TOPREL (ME 4844/3-1 and C20/MS/14764976). SSH
would also like to acknowledge funding from the Deutsche
Forschungsgemeinschaft via the Emmy Noether Programme ME4844/1-1 (project id
327807255), the Collaborative Research Center SFB 1143 (project id 247310070),
and the Cluster of Excellence on Complexity and Topology in Quantum Matter
ct.qmat (EXC 2147, project id 390858490). AN acknowledges support from the
start-up grant (SG/MHRD-19-0001) of the Indian Institute of Science and from
DST-SERB (project number SRG/2020/000153).

## References

  * Martinez Alvarez et al. (2018a) V. M. Martinez Alvarez, J. E. Barrios Vargas, M. Berdakin, and L. E. F. Foa Torres, The European Physical Journal Special Topics 227, 1295–1308 (2018a), ISSN 1951-6401, URL http://dx.doi.org/10.1140/epjst/e2018-800091-5.
  * Bender (2007) C. M. Bender, Reports on Progress in Physics 70, 947 (2007).
  * Ashida et al. (2020) Y. Ashida, Z. Gong, and M. Ueda, Advances in Physics 69, 249 (2020).
  * Ghatak and Das (2019) A. Ghatak and T. Das, Journal of Physics: Condensed Matter 31, 263001 (2019).
  * Feng et al. (2017) L. Feng, R. El-Ganainy, and L. Ge, Nature Photonics 11 (2017).
  * Helbig et al. (2020) T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, Nature Physics 16, 747 (2020), eprint 1907.11562.
  * Hofmann et al. (2020) T. Hofmann, T. Helbig, F. Schindler, N. Salgo, M. Brzeziń ska, M. Greiter, T. Kiessling, D. Wolf, A. Vollhardt, A. Kabaši, et al., Physical Review Research 2 (2020), URL https://doi.org/10.1103%2Fphysrevresearch.2.023265.
  * Li and Lee (2022) L. Li and C. H. Lee, Science Bulletin 67, 685 (2022), URL https://doi.org/10.1016%2Fj.scib.2022.01.017.
  * Ma and Sheng (2016) G. Ma and P. Sheng, Science Advances 2, e1501595 (2016).
  * Chou et al. (2011) T. Chou, K. Mallick, and R. K. P. Zia, Reports on Progress in Physics 74, 116601 (2011), ISSN 1361-6633, URL http://dx.doi.org/10.1088/0034-4885/74/11/116601.
  * Midya et al. (2018) B. Midya, H. Zhao, and L. Feng, Nature communications 9, 1 (2018).
  * Ghatak et al. (2020) A. Ghatak, M. Brandenbourger, J. van Wezel, and C. Coulais, Proceedings of the National Academy of Sciences 117, 29561 (2020), URL https://doi.org/10.1073%2Fpnas.2010580117.
  * Yao and Wang (2018) S. Yao and Z. Wang, Physical Review Letters 121 (2018), ISSN 1079-7114, URL http://dx.doi.org/10.1103/PhysRevLett.121.086803.
  * Yuce (2020) C. Yuce, Physics Letters A 384, 126094 (2020), ISSN 0375-9601, URL http://dx.doi.org/10.1016/j.physleta.2019.126094.
  * Song et al. (2020) Y. Song, W. Liu, L. Zheng, Y. Zhang, B. Wang, and P. Lu, Phys. Rev. Applied 14, 064076 (2020), URL https://link.aps.org/doi/10.1103/PhysRevApplied.14.064076.
  * Okuma et al. (2020) N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Phys. Rev. Lett. 124, 086801 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.124.086801.
  * Li et al. (2020a) L. Li, C. H. Lee, S. Mu, and J. Gong, Nature communications 11, 1 (2020a).
  * Song et al. (2019a) F. Song, S. Yao, and Z. Wang, Physical review letters 123, 170401 (2019a).
  * Kawabata et al. (2020) K. Kawabata, M. Sato, and K. Shiozaki, Physical Review B 102, 205118 (2020).
  * Liu et al. (2021) S. Liu, R. Shao, S. Ma, L. Zhang, O. You, H. Wu, Y. J. Xiang, T. J. Cui, and S. Zhang, Research 2021 (2021).
  * Longhi (2020) S. Longhi, Physical Review B 102, 201103 (2020).
  * Li et al. (2020b) L. Li, C. H. Lee, and J. Gong, Physical review letters 124, 250402 (2020b).
  * Lu et al. (2021) M. Lu, X.-X. Zhang, and M. Franz, Physical review letters 127, 256402 (2021).
  * Zhu et al. (2020) X. Zhu, H. Wang, S. K. Gupta, H. Zhang, B. Xie, M. Lu, and Y. Chen, Physical Review Research 2, 013280 (2020).
  * Okuma and Sato (2022) N. Okuma and M. Sato, _Non-hermitian topological phenomena: A review_ (2022), URL https://arxiv.org/abs/2205.10379.
  * Borgnia et al. (2020) D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Physical Review Letters 124 (2020), URL https://doi.org/10.1103%2Fphysrevlett.124.056802.
  * Roccati (2021) F. Roccati, Phys. Rev. A 104, 022215 (2021), URL https://link.aps.org/doi/10.1103/PhysRevA.104.022215.
  * Li et al. (2021) L. Li, C. H. Lee, and J. Gong, Communications Physics 4 (2021), URL https://doi.org/10.1038%2Fs42005-021-00547-x.
  * Kunst et al. (2018) F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Physical Review Letters 121 (2018), ISSN 1079-7114, URL http://dx.doi.org/10.1103/PhysRevLett.121.026808.
  * Koch and Budich (2020) R. Koch and J. C. Budich, The European Physical Journal D 74 (2020), ISSN 1434-6079, URL http://dx.doi.org/10.1140/epjd/e2020-100641-y.
  * Edvardsson et al. (2019) E. Edvardsson, F. K. Kunst, and E. J. Bergholtz, Physical Review B 99, 081302 (2019).
  * Cao et al. (2021) Y. Cao, Y. Li, and X. Yang, Physical Review B 103, 075126 (2021).
  * Xiao et al. (2020) L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Nature Physics 16, 761 (2020).
  * Jin and Song (2019) L. Jin and Z. Song, Physical Review B 99, 081103 (2019).
  * Bergholtz et al. (2021) E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Reviews of Modern Physics 93 (2021), ISSN 1539-0756, URL http://dx.doi.org/10.1103/RevModPhys.93.015005.
  * Miri and Alu (2019) M.-A. Miri and A. Alu, Science 363, eaar7709 (2019).
  * Özdemir et al. (2019) Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Nature materials 18, 783 (2019).
  * Huang et al. (2017) Y. Huang, Y. Shen, C. Min, S. Fan, and G. Veronis, Nanophotonics 6, 977 (2017).
  * Krasnok et al. (2021) A. Krasnok, N. Nefedkin, and A. Alu, arXiv preprint arXiv:2103.08135 (2021).
  * Kawabata et al. (2019a) K. Kawabata, T. Bessho, and M. Sato, Physical review letters 123, 066405 (2019a).
  * Hu et al. (2017) W. Hu, H. Wang, P. P. Shum, and Y. D. Chong, Physical Review B 95, 184306 (2017).
  * Chen et al. (2019a) C. Chen, L. Jin, and R.-B. Liu, New Journal of Physics 21, 083002 (2019a).
  * Minganti et al. (2019) F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, Physical Review A 100, 062131 (2019).
  * Xu et al. (2017) Y. Xu, S.-T. Wang, and L.-M. Duan, Phys. Rev. Lett. 118, 045701 (2017), URL https://link.aps.org/doi/10.1103/PhysRevLett.118.045701.
  * Kawabata et al. (2019b) K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Physical Review X 9 (2019b), ISSN 2160-3308, URL http://dx.doi.org/10.1103/PhysRevX.9.041015.
  * Lieu (2018a) S. Lieu, Phys. Rev. B 98, 115135 (2018a), URL https://link.aps.org/doi/10.1103/PhysRevB.98.115135.
  * Herviou et al. (2019a) L. Herviou, N. Regnault, and J. H. Bardarson, SciPost Physics 7 (2019a), ISSN 2542-4653, URL http://dx.doi.org/10.21468/SciPostPhys.7.5.069.
  * Zhou and Lee (2019) H. Zhou and J. Y. Lee, Physical Review B 99 (2019), URL https://doi.org/10.1103%2Fphysrevb.99.235112.
  * Magnea (2008) U. Magnea, Journal of Physics A: Mathematical and Theoretical 41, 045203 (2008), URL https://doi.org/10.1088%2F1751-8113%2F41%2F4%2F045203.
  * Wang et al. (2021) K. Wang, A. Dutt, K. Y. Yang, C. C. Wojcik, J. Vučković, and S. Fan, Science 371, 1240–1245 (2021), ISSN 1095-9203, URL http://dx.doi.org/10.1126/science.abf6568.
  * Ke et al. (2020) S. Ke, D. Zhao, J. Fu, Q. Liao, B. Wang, and P. Lu, IEEE Journal of Selected Topics in Quantum Electronics 26, 1 (2020).
  * Pan et al. (2018) M. Pan, H. Zhao, P. Miao, S. Longhi, and L. Feng, Nature communications 9, 1 (2018).
  * Anderson (1958) P. W. Anderson, Physical review 109, 1492 (1958).
  * Lee and Ramakrishnan (1985) P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985), URL https://link.aps.org/doi/10.1103/RevModPhys.57.287.
  * Kramer and Mackinnon (1993) B. K. Kramer and A. Mackinnon, Reports on Progress in Physics 56, 1469 (1993).
  * Phillips (1993) P. Phillips, Annual Review of Physical Chemistry 44, 115 (1993), URL https://doi.org/10.1146/annurev.pc.44.100193.000555.
  * Segev et al. (2013) M. Segev, Y. Silberberg, and D. N. Christodoulides, Nature Photonics 7, 197 (2013).
  * Claes and Hughes (2020) J. Claes and T. L. Hughes, Physical Review B 101 (2020), ISSN 2469-9969, URL http://dx.doi.org/10.1103/PhysRevB.101.224201.
  * Schubert et al. (2012) G. Schubert, H. Fehske, L. Fritz, and M. Vojta, Physical Review B 85 (2012), ISSN 1550-235X, URL http://dx.doi.org/10.1103/PhysRevB.85.201105.
  * Sacksteder et al. (2015) V. Sacksteder, T. Ohtsuki, and K. Kobayashi, Physical Review Applied 3 (2015), ISSN 2331-7019, URL http://dx.doi.org/10.1103/PhysRevApplied.3.064006.
  * Wu et al. (2016) B. Wu, J. Song, J. Zhou, and H. Jiang, Chinese Physics B 25, 117311 (2016).
  * Prodan (2011) E. Prodan, Journal of Physics A: Mathematical and Theoretical 44, 113001 (2011).
  * Huang and Shklovskii (2020) Y. Huang and B. I. Shklovskii, Phys. Rev. B 101, 014204 (2020), URL https://link.aps.org/doi/10.1103/PhysRevB.101.014204.
  * Rudner and Levitov (2009) M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 102, 065703 (2009), URL https://link.aps.org/doi/10.1103/PhysRevLett.102.065703.
  * Jiang and Lee (2022) H. Jiang and C. H. Lee, Chinese Physics B 31, 050307 (2022), URL https://doi.org/10.1088%2F1674-1056%2Fac4a73.
  * Hatano and Nelson (1996) N. Hatano and D. R. Nelson, Physical Review Letters 77, 570–573 (1996), ISSN 1079-7114, URL http://dx.doi.org/10.1103/PhysRevLett.77.570.
  * Kawabata and Ryu (2021) K. Kawabata and S. Ryu, Physical review letters 126, 166801 (2021).
  * Heinrichs (2001) J. Heinrichs, Physical Review B 63, 165108 (2001).
  * Hatano and Obuse (2021) N. Hatano and H. Obuse, Annals of Physics 435, 168615 (2021).
  * McDonald et al. (2022) A. McDonald, R. Hanai, and A. A. Clerk, Physical Review B 105, 064302 (2022).
  * Longhi (2017) S. Longhi, Physical Review B 95, 014201 (2017).
  * Longhi (2021) S. Longhi, Physical Review B 103 (2021), ISSN 2469-9969, URL http://dx.doi.org/10.1103/PhysRevB.103.144202.
  * Claes and Hughes (2021) J. Claes and T. L. Hughes, Physical Review B 103 (2021), ISSN 2469-9969, URL http://dx.doi.org/10.1103/PhysRevB.103.L140201.
  * Su et al. (1979) W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979), URL https://link.aps.org/doi/10.1103/PhysRevLett.42.1698.
  * Lieu (2018b) S. Lieu, Physical Review B 97 (2018b), ISSN 2469-9969, URL http://dx.doi.org/10.1103/PhysRevB.97.045106.
  * Zhu et al. (2014) B. Zhu, R. Lü, and S. Chen, Physical Review A 89 (2014), ISSN 1094-1622, URL http://dx.doi.org/10.1103/PhysRevA.89.062102.
  * Wu et al. (2021) H. Wu, L. Jin, and Z. Song, Physical Review B 103, 235110 (2021).
  * Yuce and Ramezani (2019) C. Yuce and H. Ramezani, Physical Review A 100, 032102 (2019).
  * Han et al. (2021) Y. Han, J. Liu, and C. Liu, New Journal of Physics 23, 123029 (2021).
  * Zeuner et al. (2015) J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, Physical review letters 115, 040402 (2015).
  * Zhang et al. (2020a) D.-W. Zhang, L.-Z. Tang, L.-J. Lang, H. Yan, and S.-L. Zhu, Science China Physics, Mechanics & Astronomy 63 (2020a), ISSN 1869-1927, URL http://dx.doi.org/10.1007/s11433-020-1521-9.
  * Lee et al. (2019) J. Y. Lee, J. Ahn, H. Zhou, and A. Vishwanath, Physical Review Letters 123 (2019), ISSN 1079-7114, URL http://dx.doi.org/10.1103/PhysRevLett.123.206404.
  * Bessho and Sato (2021) T. Bessho and M. Sato, Phys. Rev. Lett. 127, 196404 (2021), URL https://link.aps.org/doi/10.1103/PhysRevLett.127.196404.
  * Zhang et al. (2020b) K. Zhang, Z. Yang, and C. Fang, Physical Review Letters 125 (2020b), ISSN 1079-7114, URL http://dx.doi.org/10.1103/PhysRevLett.125.126402.
  * He and Chien (2020) Y. He and C.-C. Chien, Journal of Physics: Condensed Matter 33, 085501 (2020), ISSN 1361-648X, URL http://dx.doi.org/10.1088/1361-648X/abc974.
  * Chen et al. (2019b) R. Chen, C.-Z. Chen, B. Zhou, and D.-H. Xu, Physical Review B 99 (2019b), ISSN 2469-9969, URL http://dx.doi.org/10.1103/PhysRevB.99.155431.
  * Mochizuki et al. (2020) K. Mochizuki, N. Hatano, J. Feinberg, and H. Obuse, Physical Review E 102 (2020), ISSN 2470-0053, URL http://dx.doi.org/10.1103/PhysRevE.102.012101.
  * Herviou et al. (2019b) L. Herviou, J. H. Bardarson, and N. Regnault, Physical Review A 99 (2019b), ISSN 2469-9934, URL http://dx.doi.org/10.1103/PhysRevA.99.052118.
  * Song et al. (2019b) F. Song, S. Yao, and Z. Wang, Physical Review Letters 123 (2019b), ISSN 1079-7114, URL http://dx.doi.org/10.1103/PhysRevLett.123.246801.
  * Banerjee et al. (2021) A. Banerjee, S. S. Hegde, A. Agarwala, and A. Narayan, _Chiral metals and entrapped insulators in a one-dimensional topological non-hermitian system_ (2021), eprint 2111.02223, URL https://arxiv.org/abs/2111.02223.
  * Jiang et al. (2020) H. Jiang, R. Lü, and S. Chen, The European Physical Journal B 93 (2020), ISSN 1434-6036, URL http://dx.doi.org/10.1140/epjb/e2020-10036-3.
  * Martinez Alvarez et al. (2018b) V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, Phys. Rev. B 97, 121401 (2018b), URL https://link.aps.org/doi/10.1103/PhysRevB.97.121401.
  * Lee and Thomale (2019) C. H. Lee and R. Thomale, Phys. Rev. B 99, 201103 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.99.201103.
  * Shnerb and Nelson (1998) N. M. Shnerb and D. R. Nelson, Phys. Rev. Lett. 80, 5172 (1998), URL https://link.aps.org/doi/10.1103/PhysRevLett.80.5172.
  * Zhai et al. (2020) L.-J. Zhai, S. Yin, and G.-Y. Huang, Phys. Rev. B 102, 064206 (2020), URL https://link.aps.org/doi/10.1103/PhysRevB.102.064206.
  * Jiang et al. (2019) H. Jiang, L.-J. Lang, C. Yang, S.-L. Zhu, and S. Chen, Phys. Rev. B 100, 054301 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.100.054301.
  * Zeng and Xu (2020) Q.-B. Zeng and Y. Xu, Phys. Rev. Research 2, 033052 (2020), URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.033052.
  * Yuce and Ramezani (2022) C. Yuce and H. Ramezani, arXiv preprint arXiv:2203.02129 (2022).
  * Asbóth et al. (2016) J. K. Asbóth, L. Oroszlány, and A. Pályi, Lecture Notes in Physics (2016), ISSN 1616-6361, URL http://dx.doi.org/10.1007/978-3-319-25607-8.
  * Kawabata et al. (2021) K. Kawabata, K. Shiozaki, and S. Ryu, Physical Review Letters 126 (2021), ISSN 1079-7114, URL http://dx.doi.org/10.1103/PhysRevLett.126.216405.
  * Song et al. (2019c) F. Song, S. Yao, and Z. Wang, Phys. Rev. Lett. 123, 170401 (2019c), URL https://link.aps.org/doi/10.1103/PhysRevLett.123.170401.
  * Lieu et al. (2020) S. Lieu, M. McGinley, and N. R. Cooper, Phys. Rev. Lett. 124, 040401 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.124.040401.
  * Lin et al. (2021) Q. Lin, T. Li, L. Xiao, K. Wang, W. Yi, and P. Xue, _Observation of non-hermitian topological anderson insulator in quantum dynamics_ (2021), URL https://arxiv.org/abs/2108.01097.
  * Ding et al. (2016) K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, Phys. Rev. X 6, 021007 (2016), URL https://link.aps.org/doi/10.1103/PhysRevX.6.021007.
  * Liu et al. (2020) S. Liu, S. Ma, C. Yang, L. Zhang, W. Gao, Y. J. Xiang, T. J. Cui, and S. Zhang, Phys. Rev. Applied 13, 014047 (2020), URL https://link.aps.org/doi/10.1103/PhysRevApplied.13.014047.
  * Cerjan et al. (2019) A. Cerjan, S. Huang, M. Wang, K. P. Chen, Y. Chong, and M. C. Rechtsman, Nature Photonics 13, 623 (2019), URL https://doi.org/10.1038%2Fs41566-019-0453-z.
  * Qin et al. (2021) G.-Q. Qin, R.-R. Xie, H. Zhang, Y.-Q. Hu, M. Wang, G.-Q. Li, H. Xu, F. Lei, D. Ruan, and G.-L. Long, Laser & Photonics Reviews p. 2000569 (2021), URL https://doi.org/10.1002%2Flpor.202000569.
  * Weidemann et al. (2020) S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Science 368, 311 (2020).
  * Wu et al. (2020) J. Wu, X. Huang, J. Lu, Y. Wu, W. Deng, F. Li, and Z. Liu, Phys. Rev. B 102, 104109 (2020), URL https://link.aps.org/doi/10.1103/PhysRevB.102.104109.
  * Zhang et al. (2021) X. Zhang, Y. Tian, J.-H. Jiang, M.-H. Lu, and Y.-F. Chen, Nature communications 12, 1 (2021).
  * Moustaj et al. (2021) A. Moustaj, L. Eek, and C. M. Smith, _Field theoretical study of disorder in non-hermitian topological models_ (2021), eprint 2107.14271.
  * Hamazaki et al. (2019) R. Hamazaki, K. Kawabata, and M. Ueda, Phys. Rev. Lett. 123, 090603 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.123.090603.
  * Kawabata et al. (2022) K. Kawabata, K. Shiozaki, and S. Ryu, Phys. Rev. B 105, 165137 (2022), URL https://link.aps.org/doi/10.1103/PhysRevB.105.165137.
  * Denner et al. (2021) M. M. Denner, A. Skurativska, F. Schindler, M. H. Fischer, R. Thomale, T. Bzdušek, and T. Neupert, Nature Communications 12 (2021), URL https://doi.org/10.1038%2Fs41467-021-25947-z.