T

# A Clinically Inspired Approach for Melanoma classification

###### Abstract

Melanoma is a leading cause of deaths due to skin cancer deaths and hence,
early and effective diagnosis of melanoma is of interest. Current approaches
for automated diagnosis of melanoma either use pattern recognition or
analytical recognition like ABCDE (asymmetry, border, color, diameter and
evolving) criterion. In practice however, a differential approach wherein
outliers (ugly duckling) are detected and used to evaluate nevi/lesions.
Incorporation of differential recognition in Computer Aided Diagnosis (CAD)
systems has not been explored but can be beneficial as it can provide a
clinical justification for the derived decision. We present a method for
identifying and quantifying ugly ducklings by performing Intra-Patient
Comparative Analysis (IPCA) of neighboring nevi. This is then incorporated in
a CAD system design for melanoma detection. This design ensures flexibility to
handle cases where IPCA is not possible. Our experiments on a public dataset
show that the outlier information helps boost the sensitivity of detection by
at least 4.1% and specificity by 4.0% to 8.9%, depending on the use of a
strong (EfficientNet) or moderately strong (VGG and ResNet) classifier.

Index Terms—  Melanoma, Skin cancer, flexible outlier detection, Ugly
Ducklings

##  1 Introduction

Melanoma is an aggressive and lethal form of skin cancer contributing to a
majority of skin cancer fatalities. Its early detection is crucial to increase
the chances of survival by a large margin. Since patients can have several
lesions in some specific locations diagnosis is a time-intensive task for
melanoma lesions which can lead to misclassifications, motivating the need for
a CAD.

Typically, clinicans perform three major tasks while evaluating a skin lesion:
overall examination of the lesion i.e. Lesion Focussed Analysis (LFA) lesion-
level evaluation by analytical methods such as 7-Point Checklist [1], ABCDE
approach [2] and an IPCA in a region, to identify any Ugly Ducklings. In CAD
development, deep learning solutions have been proposed for diagnosing
melanoma by performing only LFA [3], [4] as well as a combination of LFA and
7-point checklist [5].

When numerous nevi (moles) are present in a region, it has been observed that
melanoma lesions deviate from the rest and exhibit very different properties.
Hence, detecting outliers or _Ugly Ducklings_ has been argued [6] [7], to
improve the effectiveness of screening and early detection. Public awareness
campaigns about melanoma also advise individuals with nevi to perform a self-
exam to identify the ugly ducklings and report the same to medical experts.

Since a differential recognition has not been explored in CAD development for
melanoma detection, there are 2 questions of interest. First, is on whether
outlier detection can be beneficial to melanoma classification by a CAD
system. It is possible that either multiple nevi are not present in a patient
or a CAD has no access to patient-level information to identify contextual
lesions. This leads to the second question, namely on how to design a CAD
system that is flexible in handling a variable input, namely, a singleton set,
or a contextual set of lesions. In this paper, we answer these questions with
the following contributions.

  * •

A solution for outlier detection based on a pretrained CNN. The output of this
module is an outlier score for each of the contextual lesions. Such a score
can aid clinicians to decide which of the lesions are to be excised and help
reduce the excision rate of benign lesions while adding explanation for the
inference.

  * •

A modular design for a CAD system for melanoma detection. The design includes
a solution for the integration of outlier detection module with a module for
melanoma classification which can be a pre-existing system or a new design.
This integration permits the functioning of the overall system even in the
absence of contextual lesions information.

(a) Contextual Lesions from the torso of a patient

(b) Cosine Distance matrix $\mathcal{M}_{12\times 12}$

Fig. 1: Ugly Ducklings and Cosine Distance. Fig (a): 12 lesions of a patient
on the torso, along with target and outlier score; 1 indicates an ugly
duckling. Fig (b): the cosine distance map of each lesion with other
neighbouring/contextual lesions. The map shows the 7th and 9th lesions having
high cosine distances, indicating that they are very dissimilar and hence they
are labelled as ugly ducklings.

##  2 Integration of IPCA in learning Models

We propose a CAD system that incorporates IPCA (to identify ugly ducklings)
and uses its results for deriving the final diagnosis of melanoma.

###  2.1 Identifying Ugly Ducklings

The task at hand is to identify the ugly ducklings (or outliers) among a set
of contextual lesions (i.e. from one region) for a patient. Let $R$ be the
region of a patient with lesions $L_{i}$, ($i\in[1,N],i\in\mathbb{Z}$). Given
a set of lesions for $R$, for every $L_{i}$, the corresponding contextual
lesions are nothing but $L_{j},\forall j\in[1,N],j\neq i$. In order to
determine if a lesion is an outlier or not, we propose to use a score based on
the cosine distance between $L_{i}$ and the contextual lesion $L_{j}$ to
characterise $L_{i}$. The details of the proposed methods are presented next.

Features of $L_{i}$ are extracted using a CNN pretrained on a melanoma dataset
(The ISIC 2019 [8, 9, 10] was used in our work). Next, a global average
pooling is applied on these feature maps to obtain a feature vector
$\boldsymbol{g_{i}}$.

We use the cosine distance of a lesion from all contextual lesions to define a
matrix $\mathcal{M}(i,j);i,j\;\in[1,N]$ as

$\displaystyle\mathcal{M}(i,j)$
$\displaystyle=1-\frac{\boldsymbol{g_{i}}\cdot\boldsymbol{g_{j}}}{\|\boldsymbol{g_{i}}\|\times\|\boldsymbol{g_{j}}\|}$
(1)

When a lesion, say $L_{3}$, is an outlier among $L_{j};j\in[1,N]$ the choice
of Cosine distance ensures that $\mathcal{M}(3,j)$ has a higher value than if
it was similar to the other lesions in the set.

Given a set of contextual lesions, the average value of the cosine distances
is a good way to characterise any lesion in the set. Thus, given a lesion
$L_{i}$ its outlier score is defined as

$\mathcal{O}_{i}=\frac{\sum_{j=1,j\neq i}^{j=N}\mathcal{M}(i,j)}{N-1}$ (2)

From the above definition, it is seen that $\mathcal{O}_{i}\in[0,1]$; in
practice, $\mathcal{O}_{i}=0$ is possible only when all the contextual lesions
are duplicates. The robustness of any outlier detection will depend on the
sample size $N$ as too small an $N$ can lead to misleading results. Hence,
empirically, the required $N$ was taken to be 6. When $N\leq 5$, IPCA is not
possible, and we treat $L_{i}$ as an outlier with $\mathcal{O}_{i}=1$. This
definition will ensure that when the system does not have access to a
patient’s other neighbouring lesions, every lesion should be suspected and
decision should be based only on morphological analysis or LFA.

Finally, the outliers are identified using an analysis based on the Inter
Quartile Range ($IQR$) which is the difference between first ($Q_{1}$) and
third quartiles ($Q_{3}$) of $\mathcal{O}_{i}$. A lesion $L_{i}$ is an outlier
if,

$\mathcal{O}_{i}\geq Q_{3}+k\times IQR$ (3)

where, the value of ${k}$ determines the tolerance of the outlier detector.
Higher the value of ${k}$, lower is the tolerance. Fig 1. shows some sample
contextual leasions and a visualisation of the $\mathcal{O}$ matrix as an
image.

###  2.2 Proposed CAD system

We now turn to the CAD system design. A schematic of the proposed design is
shown in Fig. 2. The top pipeline is a melanoma classifier whose decision is
to be moderated based on the outlier score. This is done as follows.

Assume a CNN based classifier $f$ with embeddings, $f(L_{i})$ derived after a
Global Average Pooling. These are passed through a fully connected network
(FCN), $h$ to obtain $h(L_{i})$. The outlier scores, $\mathcal{O}_{i}$ are
used to weigh these features by performing
$h^{\prime}(L_{i})=\mathcal{O}_{i}\cdot h(f(L_{i}))$, so that the ugly
ducklings are learnt with more importance. In order to obtain the final
decision, a binary classification is done based on $h^{\prime}(L_{i})$ to
predict labels, $p_{i}$ using a FCN $m$, such that
$p_{i}=m(h^{\prime}(L_{i}))$, as shown in Fig. 2.

Fig. 2: Proposed CAD system for melanoma detection

There are two ways to use the outlier scores, $\mathcal{O}_{i}$: (a) The score
can be injected into the network to give an increased weight to ugly ducklings
and penalise misclassification, or, (b) it can be directly included in the
loss function. Method (a) ensures that the outlier scores is used for
inference, whereas (b) guarantees its role only during the learning phase. We
chose (a).

The objective of the system is to minimize loss ($\mathcal{L}$) between the
ground truth distribution of lesion labels, $y_{i}$ and $p_{i}$ for lesion
$L_{i}$ by finding optimal network weights $f_{w}^{*},h_{w}^{*},m_{w}^{*}$ for
networks $f,h,m$ such that:

$f_{w}^{*},h_{w}^{*},m_{w}^{*}=\operatorname*{arg\,min}_{f_{w},h_{w},m_{w}}\;\mathcal{L}(y_{i},p_{i})$
(4)

Consider the networks, $f,h,m$. If the inputs are $x_{f},x_{h},x_{m}$ and
network weights are $f_{w},h_{w},m_{w}$ respectively, then by chain rule,

$\displaystyle\frac{\partial\mathcal{L}(y_{i},p_{i})}{\partial m_{w}}$
$\displaystyle=\frac{\partial\mathcal{L}(y_{i},p_{i})}{\partial
p_{i}}\cdot\frac{\partial p_{i}}{\partial m_{w}}$ (5)
$\displaystyle\frac{\partial\mathcal{L}(y_{i},p_{i})}{\partial h_{w}}$
$\displaystyle=\frac{\partial\mathcal{L}(y_{i},p_{i})}{\partial
p_{i}}\cdot\frac{\partial p_{i}}{\partial x_{m}}\cdot\frac{\partial
x_{m}}{\partial h_{w}}$
$\displaystyle\frac{\partial\mathcal{L}(y_{i},p_{i})}{\partial f_{w}}$
$\displaystyle=\frac{\partial\mathcal{L}(y_{i},p_{i})}{\partial
p_{i}}\cdot\frac{\partial p_{i}}{\partial x_{m}}\cdot\frac{\partial
x_{m}}{\partial x_{h}}\cdot\frac{\partial x_{h}}{\partial f_{w}}$

Substituting $p_{i}=m(x_{m})$, $x_{m}=\mathcal{O}_{i}\cdot h(x_{h})$,
$x_{h}=f(x_{f})$ and $x_{f}=L_{i}$, we have,

$\displaystyle\frac{\partial x_{m}}{\partial h_{w}}$
$\displaystyle=\mathcal{O}_{i}\cdot\frac{\partial h(x_{h})}{\partial h_{w}}$
(6) $\displaystyle\frac{\partial\mathcal{L}(y_{i},p_{i})}{\partial h_{w}}$
$\displaystyle=\mathcal{O}_{i}\cdot\frac{\partial\mathcal{L}(y_{i},p_{i})}{\partial
p_{i}}\cdot\frac{\partial p_{i}}{\partial x_{m}}\cdot\frac{\partial
h(x_{h})}{\partial h_{w}}$
$\displaystyle\frac{\partial\mathcal{L}(y_{i},p_{i})}{\partial f_{w}}$
$\displaystyle=\mathcal{O}_{i}\cdot\frac{\partial\mathcal{L}(y_{i},p_{i})}{\partial
p_{i}}\cdot\frac{\partial p_{i}}{\partial x_{m}}\cdot\frac{\partial
h(x_{h})}{\partial x_{h}}\cdot\frac{\partial x_{h}}{\partial f_{w}}$

Thus, from equation 6 we can see that $\mathcal{O}_{i}$ scales the gradients
of the networks. The higher value of 6 for an outlier, therefore will ensure
that the ugly ducklings are penalised more for misclassification, yielding the
optimal network weights $f^{*}_{w},h^{*}_{w},m^{*}_{w}$ according to the
objective function in equation 4.

A key strength of the proposed design is that any existing CAD system for
melanoma detection can be plugged in place of the melanoma classifier.

##  3 Experiments

###  3.1 Dataset

Experiments were done using the ISIC 2020 Challenge Dataset [11]. This dataset
provides patient-level contextual information and has 33,126 dermoscopic
images of 2056 patients from Europe, North America and Australia with an
average of 16 lesions per patient and 584 confirmed melanomas.

###  3.2 Dataset Preprocessing and Augmentation

All lesion images were resized to a fixed size ($512\times 512\times 3$) and
normalized using ImageNet statistics. Augmentation was applied via affine
transformations (random shifts, shear, zoom, rotation, horizontal and vertical
flips), random change of saturation, contrast, and brightness. This was done
on the train, test, and validation sets. In all our experiments, test time
augmentations with 22 steps was done on both test and validation sets during
inference.

###  3.3 Implementation Details

Computing outlier scores: In ISIC 2020 dataset, not every patient has multiple
images/lesions that are sufficient enough to find ugly ducklings. Hence, only
patients who have more than 5 contextual lesion images were considered for
outlier detection assessment. The features of each of these lesions belonging
to a patient in a given location was extracted by using an EfficientNet B6
network [12] pretrained on the ISIC 2019 dataset. These features were then
used to compute outlier scores as given by equation 2. Empirical results
showed that the outlier detector performed better when $k$ in equation 2 is
set to 1.

Different variants of the proposed system in Fig. 2 was created by changing
the classifier module : EfficientNet B6 Noisy-Student (NS) [13], EfficientNet
B7 NS . All networks were trained on TPU (Tensor Processing Units) with 8 TPU
v3 cores and 128 GiB memory. In the interest of computation time, the
outlier/ugly duckling labels were precomputed to serve as an additional
information to CNN classifier as shown in Fig 2. A group 5-fold split was
applied on patient ID to ensure that train and validation sets have no patient
in common. This prevented data leakage.

The Rectified Adam optimizer [14] was used with a Focal-Loss based cross
entropy loss [15] function with an initial learning rate as 3$e$-3 which is
reduced on plateau. Each fold was trained for 25 epochs and early stopping was
used to prevent any overfitting.

Model |  Without |  |
---|---|---|---
Ugly Ducklings |  With |  |
Ugly Ducklings |  |  |  |
| Sen. | Spec. | Sen. | Spec.
EfficientNet |  |  |  |
-B6 Noisy Student | 0.89 | 0.82 | 0.91 | 0.84
EfficientNet |  |  |  |
-B7 Noisy Student | 0.84 | 0.81 | 0.89 | 0.85
ResNet 101 | 0.91 | 0.74 | 0.88 | 0.82
VGG 19 | 0.87 | 0.72 | 0.87 | 0.77
Table 1: Sensitivity & Specificity of models without and with Ugly ducklings
information.

###  3.4 Results and Discussion

First we report on the outlier detection performance. When the cosine distance
based scores are treated as class scores, the attained performance was 0.82
sensitivity and 0.81 specificity which is comparable to benchmarked deep
learning models.

The sensitivity and specificity values of the proposed CAD system variants
(see section 3) were calculated at the knee point of the ROC curve and are
presented in Table 1. The inclusion of ugly duckling information is seen to
lead to a consistent boost in specificity and sensitivity. While the boost is
by moderate amounts for variants with strong classifier models like
EfficientNets, it is more significant for variants with weaker networks like
VGG [16], ResNet [17]. A 3% decrease in sensitivity is more than offset by a
boost of 10.8 % in specificity.

Overall, our experimental results support the emphasis of the clinical studies
on the importance of the use of ugly ducklings in early detection of melanoma.
While the boost in performance depends on the type of classifier used, the key
advantage of our design is that the output of the system is a prediction label
together with an outlier score which, in the case of contextual lesions,
serves as a much needed confidence measure to clinicians.

The current benchmark on ISIC 2020 dataset is 0.949 Area under the ROC (AUC)
score, which is recorded in a Kaggle competition, launched jointly by SIIM
(Society for Imaging Informatics in Medicine) and ISIC (International Skin
Imaging Colloboration). According to [18], the top scoring team ensembled
multiple EfficientNet models (B3-B7), SE-ResNeXt-101 and ResNeXt-101 with
various input resolutions ranging from 384$\times$384 to 1024$\times$1024
along with meta data. Only patient-level information, in the form meta-data,
was used for the diagnosis of melanoma, rather than the patient’s contextual
images. In contrast, our approach, when implemented with an simple ensemble of
weighted average of scores of only two EfficientNet B6 and B7 Noisy student
models on 512x512 images, achieved an AUC of 0.940 and 0.935 for inference
with and without integration of ugly ducklings information, respectively.
These results are promising and indicate the potential for improvements with
more extensive ensembling of various models.

##  4 Conclusion

We presented an approach based on the use of contextual information in
clinical practice to assess melanoma. Specifically we are identifying outliers
and using them jointly with visual features to assess melanoma classification.

The key strength of the proposed approach is that it provides a clinically
meaningful explanation for a decision by including an outlier score. The
proposed design permits a flexible integration of outlier detector with any
existing architecture. It needs to be emphasised that the design includes both
an outlier detector and a classifier to flexibly handle a range of scenarios
and cases: cases where information about contextual lesions is available or
not as well as cases with isolated lesions.

The experimental results indicate that the inclusion of Ugly duckling method
in CAD systems is beneficial both in terms of detection performance boost and
in providing a supportive clinical explanation when apt.

## References

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# Chiral metals and entrapped insulators in a one-dimensional topological non-
Hermitian system

Ayan Banerjee<EMAIL_ADDRESS>Solid State and Structural Chemistry
Unit, Indian Institute of Science, Bangalore 560012, India  Suraj S. Hegde
<EMAIL_ADDRESS>Institut für Theoretische physik, Technische
Universität Dresden, 01069 Dresden, Germany  Max-Planck Institute for the
Physics of Complex Systems, Nöthnitzer straße 38, Dresden 01187, Germany
Adhip Agarwala<EMAIL_ADDRESS>Max-Planck Institute for the Physics of
Complex Systems, Nöthnitzer straße 38, Dresden 01187, Germany  International
Centre for Theoretical Sciences, Tata Institute of Fundamental Research,
Bengaluru 560089, India  Awadhesh Narayan<EMAIL_ADDRESS>Solid State and
Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012,
India

###### Abstract

In this work we study many-body ‘steady states’ that arise in the non-
Hermitian generalisation of the non-interacting Su-Schrieffer-Heeger model at
a finite density of fermions. We find that the hitherto known phase diagrams
for this system, derived from the single-particle gap closings, in fact
correspond to distinct non-equilibrium phases, which either carry finite
currents or are dynamical insulators where particles are entrapped. Each of
these have distinct quasi-particle excitations and steady state correlations
and entanglement properties. Looking at finite-sized systems, we further
modulate the boundary to uncover the topological features in such steady
states – in particular the emergence of leaky boundary modes. Using a variety
of analytical and numerical methods we develop a theoretical understanding of
the various phases and their transitions, and uncover the rich interplay of
non-equilibrium many-body physics, quantum entanglement and topology in a
simple looking, yet a rich model system.

##  I Introduction

Quantum matter and its novel manifestations are often characterized by an
intricate interplay of quantum fluctuations, ideas of band topology and
entanglement Wen_RMP_2017; Ludwig_PS_2015; Chiu_RMP_2016; Hasan_RMP_2010;
Qi_RMP_2011. A rather new addition to this group has been the notion of non-
Hermiticity, which has its own illustrious history of being visited time and
again, in context of dynamical systems, open systems or parity-time ($PT$)
symmetric systems, which have been realized in various physical platforms
alvarez2018topological; torres2019perspective; ashida2020non;
bergholtz2021exceptional; Carlstrom20; Huber20; Lee14; Lee14a; Lieu20; Wei17;
vanCaspel19; Dangel18; Panda20; Lieu_PRA-2019; ghatak2019new;
guo2021entanglement; Wang_2021. While one strategy has been to investigate the
role of non-Hermiticity that arises in the effective Hamiltonian of some
degrees of freedom (due to coupling to a bath or leads etc.), another has been
to sideline the above question and explicitly model the non-Hermitian
Hamiltonians and investigate their properties.

A plethora of work, in recent years, has been dedicated to this latter theme
and, in particular, the role of non-Hermiticity in lattice models has been
widely investigated alvarez2018topological; ghatak2019new;
torres2019perspective; ashida2020non; bergholtz2021exceptional. Most notable
among this class of works has been the investigations into the role of
topology vis-a-vis non-Hermiticity. While building a catalog of such results,
here, would be an impossible task; we point out the underlying themes which
have been in discussion in the literature (see References
alvarez2018topological; ghatak2019new; torres2019perspective; ashida2020non;
bergholtz2021exceptional for recent reviews). Introduction of non-Hermiticity,
in general, in the Hamiltonian matrix results in complex eigenvalues, which
leads to the idea of a band diagram being generalized to a complex energy
plane, where the notion of a band gap is modified to prohibition of touching a
base energy in the complex spectrum Gong18. Thus a new notion of ‘a
topological phase’ was introduced in these systems, which involves winding of
the single particle energies over the entire complex spectra Gong18;
Kawabata19; lee2019anatomy. It is important to note, however, that such a
winding is inherently distinct from the case where a half-filled Hermitian
system is called “topological”, where it is often a response of the many-
particle state in equilibrium that can be written in terms of such topological
quantities which quantify the behaviour of single particle wavefunctions over
the Brillouin zone Resta_RMP_1994; Watanabe_PRX_2018; Thouless_PRL_1982.
Absence of this latter connection is a result of a rather incomplete
understanding between the many-body state of a non-Hermitian non-interacting
system and its single particle complex eigenspectrum, as we discuss below.

While the object of interest in Hermitian lattice tight-binding models is the
ground state and its low energy excitations, in a non-Hermitian system, the
appearance of complex eigen-spectra requires us to restructure the framework
for characterising the phases of matter. For instance, given a finite density
of particles (say, half-filling of bands) in a non-interacting equilibrium
system, a gapped single particle spectrum implies that the many-body state is
characterized by correlations, which are often topological invariants of the
lowest filled band. Such a direct mapping between the single-particle states
and the many-body state is absent for non-Hermitian systems, given the system
may not reach an equilibrium state but rather a non-equilibrium steady state
at long times. Therefore, the method of conventional energy minimization to
determine a many-body state is not directly applicable to a non-Hermitian
system. In fact, not unlike the classical non-reciprocal transitions, it is
possible that the description of the many-body state is eminent only within a
set of dynamical equations rather than an underlying free energy Fruchart2021.

In this work we study such non-equilibrium steady states that arise in a
concrete setting of a microscopic non-Hermitian lattice model at a finite
density of fermions. The prescription to form such steady states following
References Lieu_PRA-2019; LieuPRB19; Panda20; Carlstrom20 is the following.
Given a non-interacting Hamiltonian which has a set of single particle
eigenvalues given by $E_{i}=E^{R}_{i}+\mathbbm{i}E^{I}_{i}$
($i=\\{1,\ldots,L\\}$), a many-body state constructed out of $N_{p}$ particles
is a direct product of such single particle states, with a corresponding many
body energy. Note that we use $\mathbbm{i}$ for the square root of unity,
while reserving $i$ for the site index. There are ${}^{L}C_{N_{p}}$ number of
such many-body states defined by $|\Psi_{j}\rangle$ with energy
$\mathcal{E}_{j}=\mathcal{E}^{R}+\mathbbm{i}\mathcal{E}^{I}$. Each of these
states evolves in time with a generalized time-evolution operator. A general
many-body wave-function in this Hilbert space will, therefore, evolve as

$|\Psi(t)\rangle=\sum_{j}\psi_{j}|\Psi_{j}(t)\rangle=\sum_{j}\psi_{j}e^{-\mathbbm{i}(\mathcal{E}_{j})t}|\Psi_{j}(t=0)\rangle.$
(1)

So, any generic many-body state at late times ($t\rightarrow\infty$) will
eventually evolve into the many-body eigenstate with the largest value of
$\mathcal{E}^{I}$. We define this direct product state as the steady state of
the system.

The model we focus on in this work is the generalization of the paradigmatic
Su-Schrieffer-Heeger (SSH) Hamiltonian but with non-reciprocal hoppings to
model the effect of non-Hermiticity (see Fig. 1 for a schematic). In
particular, here, we would discuss the physics of this system in light of the
many-body steady states it realizes – the correlations, low energy
excitations, the entanglement signatures and the effect of tuning the boundary
conditions. Given that this is a non-interacting system, not unexpectedly, all
the properties can be determined from the single-particle eigenspectrum –
however, as we will show below, this steady states interpretation and the
intervening phase transitions provide insights into the nature of distinct
non-equilibrium phases and their transitions in non-Hermitian quantum systems.

The choice of nH-SSH (non-Hermitian SSH) model to illustrate this physics is
most natural given the immense body of work that has been dedicated towards
this particular model Lee_PRB_2020; Herviou_PRA_2019; Lieu_PRB_2018;
Jin_PRB_2019; yao2018non; kunst2018biorthogonal; liu2019topological;
song2019non; LieuPRB19; KentaPRB19; Schomerus13; MartinezPRB18; YinPRA18;
YaoPRL18; StJeanNature17; Albrecht_Nature_2016; KunstPRB2019; Dangel18;
borgnia2020non; masuda2021relationship; Zirnstein_PRL_2021;
vyas2021topological; he2020non; Silberstein_PRB_2020. In particular, we follow
the notations introduced in Reference Herviou_PRA_2019, where a phase diagram
for the nH-SSH model is presented in terms of the single-particle gap closings
both for periodic and open boundary conditions, showing that the bulk-edge
correspondence does not translate, in a straight forward manner, to non-
Hermitian systemshelbig2020generalized; hofmann2020reciprocal;
weidemann2020topological. Interestingly, in Reference YaoPRL18, an
interpretation for appearance of boundary modes and related non-Hermitian skin
effect was understood within the framework of a generalized Brillouin zone,
while in Reference kunst2018biorthogonal, such a correspondence was seen with
bulk bi-orthogonal polarization. In this work, we revisit this problem in
terms of the many-body steady states the system realizes. We find four
distinct non-equillbirium phases with characteristic currents, low-energy
excitations, correlations and entanglement signatures. We provide a
comprehensive understanding of these non-equilibrium phases and the nature of
the transitions between them. We further discuss the effects of weakening the
boundary term, such that topological boundary modes appear in these steady
states. We further point out various connections between this many-body
perspective and the single-particle dispersion framework discussed previously
in the literature Herviou_PRA_2019.

Figure 1: Schematic of the non-Hermitian SSH model. The non-Hermitian SSH
model with non-reciprocal hoppings. Intra-unit cell hopping along the right
and left are $t_{1}\pm\gamma/2$, giving rise to non-Hermiticity. Inter-unit
cell hopping is fixed at $t_{2}$. Figure 2: Phase diagram of the non-Hermitian
SSH model. The non-equilibrium phases that arise in the non-Hermitian SSH
model at half-filling [see eqn. (2)], shows four distinct phases. These
include two central charge, $c=2$, chiral-Fermi sea (cFS) phases (labelled A
and B), one $c=1$ cFS phase, and a non-Hermitian chiral insulating state with
$c=0$, termed entrapped insulator (EI).

The central finding of this work is presented in Fig. 2, where we show that by
varying the parameters characterizing the nH-SSH model – (a) $t_{1}/t_{2}$:
the dimerization scale and (b) $\gamma/t_{2}$: the non-reciprocal intra-unit-
cell hopping (see Fig. 1), one can access four distinct non-equillbrium phases
(i) a chiral metal with central charge $2$ (A) [$c=2$ cFS-A], (ii) a chiral
metal with central charge $1$ [$c=1$ cFS], (iii) another chiral metal with
central charge $2$ [c$=2$ cFS-B], and (iv) an entrapped insulator EI [$c=0$].

In the rest of the paper we discuss the physics underlying these phases. In
Section II we introduce the Hamiltonian and discuss its symmetries. In Section
III we present the methodology and introduce various diagnostic tools to
characterize these phases. In Section IV we explore the emergence of these
four distinct phases and their properties and connections to spectral
topology, as well as their low energy theories. In Section V we examine the
effect of tuning the boundary conditions from periodic to open and the
realization of leaky boundary modes. Finally, in Section VI we summarize our
results and present an outlook.

##  II Non-Hermitian SSH model

###  II.1 Model

Figure 3: Nature of complex eigenspectrum. The real and imaginary parts of the
single particle energy eigenvalues are shown in (a) $c=2$ cFS-A ($t_{1}=0.5$
and $\gamma=0.2$ ) with two spectral loops separated by a real line gap, (b)
$c=1$ cFS, ($t_{1}=1.05$ and $\gamma=0.5$) which has a single loop, and in (c)
entrapped insulator phase ($t_{1}=0.1$ and $\gamma=2.5$) which has two loops
but separated by an imaginary line gap. The purple diamonds denote the filled
states at half-filling while red (blue) stars show growing (decaying) chiral
modes. The dotted lines (crosses) indicate the line gap (point gaps). Here,
$t_{1}$ and $\gamma$ are scaled in terms of $t_{2}$. We stick to this
convention unless otherwise specified.

The Hamiltonian of interest for the nH-SSH model is given by Herviou_PRA_2019
(also called “chiral nH-SSH” model, see for e.g. Lieu_PRB_2018),

$\displaystyle H=-$
$\displaystyle\sum_{i}[t_{1}(c^{\dagger}_{i,A}c_{i,B}+h.c.)+t_{2}(c^{\dagger}_{i+1,A}c_{i,B}+h.c.)]$
$\displaystyle+$
$\displaystyle\sum_{i}\frac{\gamma}{2}(c^{\dagger}_{i,B}c_{i,A}-c^{\dagger}_{i,A}c_{i,B}),$
(2)

where $c^{\dagger}_{i,\alpha}(c_{i,\alpha})$ is the fermionic creation
(annihilation) operator at site $i$ for sublattice $\alpha=A,B$ (see Fig. 1).
The intra- and inter-unit cell hopping amplitudes are given by $t_{1}$ and
$t_{2}$, respectively, and $\gamma$ introduces a non-reciprocity only in the
intra-unit cell hopping, thereby introducing non-Hermiticity in the system. We
consider $t_{1},t_{2},\gamma$ to be real valued. Moving to $k$-space by
Fourier transforming the operators [using
$c^{\dagger}_{k,A}=\frac{1}{\sqrt{L}}\sum_{i}\exp(-\mathbbm{i}ki)c^{\dagger}_{i,A}$],
we get

$\begin{split}H(k)=\mbox{\boldmath$d$}(k)\cdot\mbox{\boldmath$\sigma$}=\begin{bmatrix}0&t_{2}e^{-\mathbbm{i}k}+t_{1}-\gamma/2\\\
t_{2}e^{\mathbbm{i}k}+t_{1}+\gamma/2&0\end{bmatrix},\end{split}$ (3)

where
$\mbox{\boldmath$d$}(k)\equiv(d_{1},d_{2},d_{3})=(t_{1}+t_{2}\cos{k},t_{2}\sin{k}-\mathbbm{i}\gamma/2,0)$
and $\sigma$ is the triad of Pauli matrices. The eigenvalues of $H(k)$ are
given by $E_{k}=\pm d(k)$, where

$d(k)=\pm\sqrt{t_{1}^{2}+t_{2}^{2}-\gamma^{2}/4+2t_{1}t_{2}\cos{k}-\mathbbm{i}t_{2}\gamma\sin{k}}.$
(4)

The right eigenvectors (corresponding to $\pm d(k)$ energy) are of the form

$|\psi^{\pm}\rangle=\dfrac{1}{\sqrt{1+|a|^{2}}}\begin{pmatrix}a\\\ \pm 1\\\
\end{pmatrix},$ (5)

where

$a=-\sqrt{\frac{t_{1}+t_{2}e^{-\mathbbm{i}k}-\gamma/2}{t_{1}+t_{2}e^{\mathbbm{i}k}+\gamma/2}}.$
(6)

The Hamiltonian has a sub-lattice symmetry ${\cal
S}:c_{iA}\rightarrow-c^{\dagger}_{iA},c_{iB}\rightarrow
c^{\dagger}_{iB},({\cal S}H^{\dagger}{\cal S}^{-1}=H)$,
$\sigma_{z}H(k)\sigma_{z}^{-1}=-H(k)$ Kawabata19.

###  II.2 Single particle eigenvalues and spectral topology

The nH-SSH model has been extensively studied. Here we briefly summarize the
main results (see for e.g.,Herviou_PRA_2019; Lieu_PRA-2019). The single
particle spectrum shows gap closings (for absolute values of energies) for
lines $\gamma=2(t_{1}\pm 1)$ and $\gamma=2(1-t_{1})$ dividing the phase
diagram into four regions (see Fig. 2): (i) $\gamma<2(1-t_{1})$ – the complex
spectrum when plotted in the complex plane $\\{Re[E],Im[E]\\}$ comprises of
two spectral lobes separated in real energy by a line gap Kawabata19;
YinPRA18; ghatak2019new [see Fig. 3(a)], (ii) $t_{1}>1+\gamma/2$ – spectral
topology is again similar to (i) [see Fig. 3(a)], (iii)
$\gamma>2(|1-t_{1}|),\gamma<2(1+t_{1})$ – the spectral topology is that of a
single loop in the complex plane [see Fig. 3(b)], and (iv) $\gamma>2(1+t_{1})$
– the complex spectrum has a two lobe structure with the lobes vertically
displaced along the imaginary axis [see Fig. 3(c)].

The spectral topology can be characterized by the winding number, $w$,
Kawabata19

$w=\int_{-\pi}^{\pi}\frac{dk}{2\pi\mathbbm{i}}\partial_{k}(\ln(\text{det}[H(k)-E_{B}])),$
(7)

where $E_{B}$ is some base energy. Immediately, the four regions [(i)-(iv)]
are characterized by winding numbers $(1,1,1/2,0)$ Kawabata19; YinPRA18;
ghatak2019new respectively. Interestingly, these winding numbers have a one-
to-one correspondence with non-decaying chiral modes, as we briefly discuss
next.

Figure 4: Single particle dispersion and currents of $c=2$ chiral metals. The
real (orange-green lines) and imaginary part (cyan-blue lines) of the
eigenspectrum and single particle current [see eqn. (13)] as a function of $k$
for various parameters are shown. In (a-b) $t_{1}=0.01,\gamma=0.2$ [$c=2$
cFS-A] (c-d) $t_{1}=2.1,\gamma=0.75$ [$c=2$ cFS-B]. For half-filling, the
occupied $k$ points are further highlighted by black diamonds.

Each of the single particle states are associated with a current determined by
the group velocity ($\sim\frac{dRe[E_{k}]}{dk}$) and a lifetime given by the
inverse of imaginary part of the energy eigenvalue $Im[E_{k}]$. This
immediately shows the existence of chiral modes which decay/grow in the single
particle spectrum (see Fig. 3). In an equilibrium system, such modes are
stationary states at equal energy which guarantees that any equilibrium state
cannot have a finite current (this follows from the celebrated Nielsen-
Ninomiya theorem NIELSEN1981219). However in a non-equilibrium setting, such
as here, these chiral modes have imaginary part of energy eigenvalue of
opposite sign, which means that at long times, only one can be populated
leading to steady states which can carry a finite current bessho2020. In fact,
the existence of such chiral modes and the finite winding numbers are inter-
related, symmetry protected and guaranteed via the properties of the spectral
topology Kawabata19; LeePRL2019; yoshida2021correlation. Having discussed the
properties of the nH-SSH model using its single-particle spectrum and its
properties, we next delve into the discussion of the many-body states obtained
here at a finite filling.

Figure 5: Single particle dispersions and currents of $c=1$ cFS and $c=0$ EI.
The real (orange-green lines) and imaginary part (cyan-blue lines) of the
eigenspectrum and single particle current [see eqn. (13)] as a function of $k$
for various parameters are shown. In (a-b) $t_{1}=1.05,\gamma=0.5$ [$c=1$
cFS], and (c-d) $t_{1}=0.01,\gamma=3.5$ [EI]. For half-filling, the occupied
$k$ points are further highlighted by black diamonds.

##  III Methodology

In this section we briefly discuss our method for construction of steady
states at finite filling, and define quantities of interest such as current
and correlation functions in these steady states.

A quadratic non-Hermitian Hamiltonian of the form ${\cal
H}=\Psi^{\dagger}H\Psi$ is diagonalized using $H=U_{R}\Lambda U^{-1}_{R}$
where

$\displaystyle\tilde{c}^{\dagger}_{i}=\sum_{j=1}^{L}(U_{R})_{ji}c^{\dagger}_{j}\qquad\tilde{c}_{i}=\sum_{j=1}^{L}(U^{-1}_{R})_{ij}c_{j}$
(8)

and $c^{\dagger}_{j}(c_{j})$ are the fermionic creation (annihilation)
operators defined at site $j$ such that ${\cal
H}=\sum_{i}E_{i}\tilde{c}^{\dagger}_{i}\tilde{c}_{i}$ where
$E_{i}=E^{R}_{i}+\mathbbm{i}E^{I}_{i}$, and $E^{R}_{i}$ and $E^{I}_{i}$ are
the real and imaginary parts respectively of every eigenvalue $E_{i}$. These
operators satisfy the conventional fermionic algebra
$\\{\tilde{c}^{\dagger}_{i},\tilde{c}_{j}\\}=\\{c^{\dagger}_{k},c_{l}\\}=\delta_{ij}$
and
$\\{\tilde{c}^{\dagger}_{i},\tilde{c}^{\dagger}_{j}\\}=\\{c^{\dagger}_{k},c^{\dagger}_{l}\\}=0$,
such that any particle number state with $N_{p}$ particles

$|\Psi_{\alpha}\rangle=\tilde{c}^{\dagger}_{i_{1}}\tilde{c}^{\dagger}_{i_{2}}\ldots\tilde{c}^{\dagger}_{i_{N_{p}-1}}\tilde{c}^{\dagger}_{N_{p}}|\Omega\rangle,$
(9)

has a well-defined many-body energy ${\cal
H}|\Psi_{\alpha}\rangle=\Big{(}\sum_{j=1}^{N_{p}}E_{i_{j}}\equiv\mathcal{E}_{\alpha}\Big{)}|\Psi_{\alpha}\rangle$,
that evolves in time as

$|\Psi_{\alpha}(t)\rangle=e^{-\mathbbm{i}(\mathcal{E}_{\alpha})t}|\Psi_{\alpha}(t=0)\rangle.$
(10)

Given the non-unitary nature of evolution, such a many-body state needs to be
renormalized during the time evolution Faisal_JPB_1981. The state which grows
the fastest is given by maximum value of $\mathcal{E}^{I}_{\alpha}$.
Therefore, at long times, the steady state of the system is defined as the one
which has the largest value of $\mathcal{E}^{I}_{\alpha}$. Any other many-body
state ($|\Psi_{\alpha^{\prime}}\rangle$) with energy
$\mathcal{E}^{R}_{\alpha^{\prime}}+\mathbbm{i}\mathcal{E}^{I}_{\alpha^{\prime}}$
has a real gap
$\Delta^{R}_{\alpha\alpha^{\prime}}=\mathcal{E}^{R}_{\alpha^{\prime}}-\mathcal{E}^{R}_{\alpha}$
and similarly an imaginary gap as
$\Delta^{I}_{\alpha\alpha^{\prime}}=\mathcal{E}^{I}_{\alpha^{\prime}}-\mathcal{E}^{I}_{\alpha}$.
The latter also defines the lifetime of the $\alpha^{\prime}$ state
$\equiv\frac{1}{\Delta^{I}_{\alpha\alpha^{\prime}}}$ for the excitation to
decay into the steady state. Applying this for the nH-SSH model, we calculate
the following quantities:

##### Current response:

The current in any system is the response of the Hamiltonian to an external
gauge field. Threading a small flux $\phi$ [$t_{2}\rightarrow
t_{2}e^{\mathbbm{i}\frac{\phi}{L}}$] through a one-dimensional ring [see eqn.
(2)] leads to the current operator as

$J=\lim_{\phi\rightarrow 0}-L\frac{\partial
H(\phi)}{\partial\phi}=-\mathbbm{i}\sum_{i}\left(t_{2}c^{\dagger}_{i+1,A}c_{i,B}-t_{2}c^{\dagger}_{i,B}c_{i+1,A}\right).$
(11)

When Fourier transformed (using
$c^{\dagger}_{k,A}=\frac{1}{\sqrt{L}}\sum_{i}\exp(-\mathbbm{i}ki)c^{\dagger}_{i,A}$),
the operator becomes

$J=-\sum_{k}\mathbbm{i}e^{\mathbbm{i}k}c^{\dagger}_{k,A}c_{k,B}+\mathbbm{i}e^{-\mathbbm{i}k}c^{\dagger}_{k,B}c_{k,A}.$
(12)

When evaluated on a single particle state $|\psi^{\pm}\rangle$ [see eqn. (5)],
one obtains

$j_{k\pm}=\pm\frac{\mathbbm{i}}{2L}(-e^{-\mathbbm{i}k}/a+e^{\mathbbm{i}k}a).$
(13)

For a filled state, depending on the occupied states one can, therefore,
straightforwardly calculate the total current which is an integral over the
corresponding region of the Brillouin zone.

The total current, $J$, can be equivalently expressed as follows

$J=\int_{0}^{2\pi}n(E_{k})\langle j_{k}\rangle dk=\oint n(E_{k})dE_{k},$ (14)

where the integral is over the spectral loop in the complex energy plane and
$n(E_{k})$ is an effective distribution function which defines which states
are occupied Zhang_2020. Therefore the way the steady states are formed here
is to consider a distribution function of the kind

$n(E_{k})=\frac{1}{1+\exp(-\beta(\text{Im}[E_{k}]-\mu))},$ (15)

where $\beta=1/T$ and $\mu$ is the chemical potential which is set to zero for
the half-filled case.

##### Correlation functions:

Given the occupancy of the single-particle states, the correlation functions
are given by

$\displaystyle C^{B,A}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{L}\sum_{k}\frac{1}{2a}e^{-\mathbbm{i}kr}(n_{k+}-n_{k-}),$
(16) $\displaystyle C^{A,B}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{L}\sum_{k}\frac{a}{2}e^{-\mathbbm{i}kr}(n_{k+}-n_{k-}),$
(17) $\displaystyle C^{A,A}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{L}\sum_{k}\frac{1}{2}e^{-\mathbbm{i}kr}(n_{k+}+n_{k-}),$
(18)

where $n_{k\pm}$ are the occupations of the $\pm$ bands.

##### Entanglement:

Having calculated the correlators, it is straightforward to evaluate the
entanglement entropy, $S_{l}$, of a sub-system of length ($l$)
Peschel_JPA_2003. The eigenvalues of the subsystem limited correlator matrix
($\equiv e_{i}$) are related to the subsystem entanglement entropy via

$S_{l}=-\sum_{i}\Big{(}e_{i}\log(e_{i})+(1-e_{i})\log(1-e_{i})\Big{)}.$ (19)

Figure 6: Nature of $c=2$ cFS-A. (a) The behavior of chiral current, for three
different values of $t_{1}$, as a function of the non-Hermitian co-efficient
$\gamma$ [see eqn. (21)]. (b) The correlator, $C_{B,A}(r)$, as a function of
$r$ at $\gamma=0.5$ for different values of $t_{1}$. The numerical values from
real space calculation are the points ($L=300$). The continuous lines are the
expressions shown in eqn. (23) for appropriate parameters. (c) Entanglement
entropy for three different values of $t_{1}$ as a function of $l/L$ for
$L=300$.

Having discussed the various quantities of interest, we next investigate the
various non-equilibrium phases realized in the nH-SSH model.

##  IV Non-equilibrium phases and their transitions

The non-equilibrium phase diagram as a function of $t_{1}/t_{2}$ and
$\gamma/t_{2}$, that we obtain, is shown in Fig. 2. We discover four distinct
non-equilibrium phases which we describe below. Before introducing the non-
Hermiticity parameter $\gamma$, let us briefly recall the phases which arise
as a function of $t_{1}$ at $\gamma=0$.

#####  $\gamma=0$ limit:

At $\gamma=0$ for both $t_{1}<t_{2}$ and $t_{1}>t_{2}$ one has gapped phases,
which have a bond dimer like order. The fermion-fermion correlator decays
exponentially with a functional form $C_{i,j}\sim e^{-r/\xi}$, where $\xi$ is
the correlation length $(\xi^{-1}=\dfrac{1}{2}|\ln{\frac{t_{1}}{t_{2}}}|)$
calabrese2009entanglement. At $t_{1}=t_{2}$, there exists a gapless point,
which corresponds to a free Fermi sea that has two linearly dispersing Dirac
fermions at $k=\frac{\pi}{2}$ and $\frac{3\pi}{2}$ (for a single site unit
cell, the same points change to $k=0,2\pi$ for a two site unit cell due to
Brillouin zone folding), rendering it a critical theory with central charge
$c=1$. We now discuss the effects of including non-Hermiticity and the phases
that emerge.

###  IV.1 $c=2$ chiral metal (CFS-A)

The first non-trivial phase arises when we have both $t_{1}<t_{2}$ and
$\gamma<t_{2}$. The Hermitian system at ($t_{1}\sim 0,\gamma=0$) exhibits two
nearly flat bands, one of which is fully filled in equilibrium. Even a
perturbatively small $\gamma$ immediately leads to a finite imaginary part of
the energy eigenvalues [see Fig. 4(a)], which makes the equilibrium state
unstable and takes the many-body state to a unique steady state which has a
finite current. It is in this sense that the non-Hermitian limit of the system
is not perturbatively connected to the Hermitian physics. Not unlike the
Hatano-Nelson model, where the system stabilizes a steady state with a
Galilean boosted Fermi sea; here the Hermitian band insulator splits into two
Fermi seas occupying the two bands (orange and green) respectively from
$\pi<k<2\pi$ (orange band) and from $0<k<\pi$ (green) [see Fig. 4(a)].

Figure 7: Steady state and the nature of excitations in the chiral FS (A)
phase. A schematic showing the steady state as well as excitations in the
cFS-A phase (see section. IV.1). The spectral loops of width $\Delta_{t}$
(top) are separated by a line gap $\Delta_{bg}$ for a system of total
bandwidth $\Delta_{bw}$. At half-filling six steady states exist (cFS-A-(5))
which are separated in real energies due to particle hole excitations (shown
using arrows). The six steady states are shown below.

Nature of the steady state: It is interesting to note that exactly at
$\gamma=0$, the system has only the lower band filled. Inclusion of any finite
$\gamma$ provides a time scale $\tau_{k,n}=\frac{1}{E^{I}_{k,n}}$ to each of
the states after which the state grows (decays) given $E^{I}_{k,n}>0$
($E^{I}_{k,n}<0$). Interestingly, every band chooses both signs for
$E^{I}_{k,n}$, therefore while the occupation of the lower band from $0<k<\pi$
is stable, at long times, the system moves to a steady state where the
particles occupy the higher (real) energy band between $0<k<\pi$.

An illustrative way of seeing this is to plot the single particle spectrum in
the complex plane which shows two closed spectral loops [see Fig. 7 (top
panel)]. These loops have a finite winding with a width $\Delta_{t}$ along the
real axis. In the $\gamma=0$ limit this corresponds to the dispersing width of
each of the bands. The loops are further separated in real energy by a “band”
gap of $\Delta_{bg}$ (also called line gap). The complete band width of the
spectrum along real axis is $\Delta_{bw}$.

The existence of the spectral loops is symmetry protected, which guarantees
existence of a state with complex energy $-E$ given a state with energy $E$.
Given the existence of four eigenvalues that have a zero imaginary component,
at half filling, the system has six many-body states which have the same total
imaginary component, while different values of the real component. Therefore,
both the spectral winding and the number of steady states in the system are
intricately related. The non-equilibrium steady state (NESS) corresponds to
the minimum total real energy and the maximum total imaginary component
characterized by a split Fermi sea on both the lower and the upper band (see
Fig. 7) and four Fermi points.

Excitations: The NESS can excite into other five steady states (cFS(1)-(5))
via particle-hole excitations, which leads to states that have the same total
imaginary component, however having a gap in real energies determined by
$\Delta_{t}$ and $\Delta_{bw}$ [see Fig. 7 (bottom)]. These excitations do not
decay and can have finite amplitudes at long times. Each of these multiple
steady states have indistinguishable current and real space correlations.
However, these steady states themselves are gapless and are dominated by
particle-hole excitations within the spectral loops which, in turn, are
characterized by point gaps in the complex energy plane. Given the continuum
of excitations within each loop, these excitations live longer, closer they
are to the Fermi points.

Current: The occupied bands in the system lead to a finite current for the
NESS, given that the single particle states are current carrying states and
contribute additively to the many-body state. This is in contrast with any
Hermitian system where current contributions from various single particle
states cancel pairwise, leading to an equilibrium state with zero current. For
instance, here both the $+$ band and the $-$ bands have a positive
contribution to the current [see Fig. 4(b)]. It is easy to integrate
$j_{k\alpha}$ to find the total current in the system, even when $\gamma$ is
perturbatively small. This finite current

$j_{o}=\frac{2}{\pi}t_{1},$ (20)

is characteristic of the NESS. Non-zero $\gamma$ leads to a quadratic increase
[see Fig. 6 (a)]

$J_{|\gamma|>\epsilon,\epsilon\rightarrow
0}\sim\mathop{\mathrm{sign}}(\gamma)\frac{2t_{1}}{\pi}(1+\frac{3}{25}\gamma^{2})$
(21)

In fact the same behavior is expected for all the six steady states in the
system, given the current is determined collectively by the bulk of the single
particle states.

Let us stress that the origin of this finite current in the NESS is purely
topological in nature. A finite winding of $H(k)$ around any reference point
in the complex spectral plane ensures the existence of a finite additive
current from various single particle states, as seen in eqn. (14) Zhang_2020.
In the cFS-A phase, the current contributions stem from the winding of the two
spectral loops. While the spectral topology of the Hamiltonian gives rise to
finite current, the notion of steady state filling ensures a purely real total
current through cancellation of all imaginary parts [see Fig.4(b)].

Correlations: Having discussed the current carrying nature of this state, it
is natural to pose the question regarding how the fermions are mutually
correlated. It is interesting to note that even while the fermions at the same
sublattice are un-correlated, i.e.,

$C^{A,A}(r)=C^{B,B}(r)=\frac{1}{2}\delta(r),$ (22)

the fermions on two different sublattices are power law correlated (to leading
order dependence on $\gamma$)

$C^{A,B}(r)\sim\frac{\gamma}{r}\sin(\frac{\Delta k_{F}r}{2}),$ (23)

with characteristic oscillations governed by a finite density of fermions in
each band ($\Delta k_{F}=\pi$). This $\gamma$ dependent behavior is in
addition to a $\gamma$ independent power law with similar oscillations. This
is in direct contrast with the Hermitian limit of a band insulator where each
of these correlators fall exponentially with distance [see Fig. 6(b)]. It is
also important to note that generally for a non-Hermitian system $\langle
c^{\dagger}_{i}c_{j}\rangle\neq[\langle
c^{\dagger}_{j}c_{i}\rangle]^{\dagger}$. The above features, therefore,
qualifies that this non-Hermitian phase is in fact a chiral-metal.

Figure 8: Steady state current in cFS-A phase. Steady state current, $J$, as a
function of density, $\rho$, at $t_{1}=0.25$ in the cFS-A phase. Note the
behavior characteristic of exclusion processes with a maximum at half-filling.
Here we set $\gamma=0.1$.

Entanglement: We next focus on the entanglement properties of this phase,
where a part of the system under consideration in real space ($l$) is traced
over, and the entanglement of the remaining part is calculated. Given the non-
interacting system, we use the prescription given by Peschel and Calabrese
calabrese2009entanglement; chang2020entanglement; guo2021entanglement and
extract the central charge of the system using (for periodic boundary
conditions)

$S=\frac{c}{3}\ln{[\frac{L}{\pi}\sin{(\dfrac{\pi l}{L}})]},$ (24)

where we find that $c=2$ [see Fig. 6(c)]. This is consistent with the fact
that the system behaves like two disjoint Fermi surfaces (corresponding to two
bands) with a Fermi volume of $\pi$, and therefore gapless low energy
excitations on four Fermi points, where each potentially contributes
$c=\frac{1}{2}$ to the central charge.

Away from half-filling: All of the discussion above have made use of the fact
that we keep the non-Hermitian system at a finite density of fermions which is
fixed to ${\it half}$. This implicitly assumes that even while single particle
states themselves can decay, the system pumps in new particles to maintain a
finite filling. However, a finite current is rooted in the spectral topology
and the filling only changes the contribution of single-particle states to the
total current. Away from half-filling the nature of the metallic state does
not change, since both the bands still get partially but equally filled, the
four Fermi points still remain, and correlations are still power law, albeit
with a different wavelength of oscillation, which, in turn, corresponds to the
changing Fermi volume. However the value of the steady state current is
density dependent and follows a behavior characteristic of exclusion processes
$\sim\rho(1-\rho)$ (see for e.g. Derrida2007) as seen in Fig. 8. In this way
it lends credence to the ideas of non-Hermitian systems modelling either open
systems or boundaries of higher dimensional symmetry protected topological
systems LeePRL2019, where the boundary states are maintained at a finite
filling even the while the system is connected to the leads.

Figure 9: Many body properties of $c=1$ cFS. (a) Chiral current for two
different values of $t_{1}$, as a function of the non-Hermitian co-efficient
$\gamma$ [see Eq.25] (b) The correlator, $C_{B,A}(r)$, as a function of $r$
for $t_{1}=1.0$ for different values of $\gamma$. The numerical values from
real space calculation are the points ($L=300$). The continuous lines are the
expressions shown in eqn. (26) for appropriate parameters. (c) The central
charge variation is shown as a function $\gamma$ ($t_{1}=0.5$) for two system
sizes. The non-analytic behaviour in the current and the central charge arise
at $t_{1}=0.5$ due to cFS-A to $c=1$ cFS transition.

###  IV.2 $c=1$ chiral metal (cFS)

There exists another distinct chiral metallic phase in this system with two
Fermi points. As we show below this phase is smoothly connected to the Hatano-
Nelson steady state, which is realized in the celebrated single band non-
reciprocal hopping Hamiltonian hatano1996localization. The ground state of
$t_{1}=t_{2},\gamma=0$ point is a half-filled Fermi sea with two Fermi points.
Any non-reciprocal hopping immediately shifts the Fermi sea to a finite-
momentum current carrying steady state still with two Fermi points, and
gapless excitations. While at $\gamma=0$ this phase is unstable to
dimerization, a finite $\gamma$ opens up a window where even upon introduction
of dimerization ($t_{1}\neq t_{2}$) the chiral metallic phase continues to
remain stable. We next discuss the properties of this phase.

Nature of steady state and excitations: At $\gamma=0$ the complex spectrum is
a single continuous line which spans the total bandwidth. The equilibrium
state at half-filling is given by occupation of all single particle states
with $Re[E]<0$. Introduction of a non-zero $\gamma$ adds an imaginary
component to the single particle eigenvalues which leads to single loop in the
complex spectrum (see Fig.3). The existence of two eigenvalues with zero
imaginary component leads to two steady states which are separated in real
energy by their band width $\Delta_{bw}$. Each of these states however has a
set of gapless excitations, which decay with time.

Current and correlations: In this phase, both the steady states carry a finite
current (see Fig. 9), where the current is given by

$J\sim\mathop{\mathrm{sign}}{(\gamma)}\frac{2t_{1}}{\pi}[1-\frac{\gamma^{2}}{16}].$
(25)

The current comes from the finite winding of a single loop in the complex
spectral plane. As is expected of a gapless state the fermionic correlations
decay in a power law fashion and the existence of a single Fermi surface with
a Fermi volume of $2\pi$ leads to no characteristic oscillations in the inter-
sublattice correlator such that even while
$C^{A,A}(r)=C^{B,B}(r)=\frac{1}{2}\delta(r)$, one finds (the leading order
dependence on $\gamma$)as

$C^{A,B}(r)\sim\frac{\gamma}{r}.$ (26)

which again is in addition to an $\gamma$ independent power-law part.

Entanglement and density dependence: Given two Fermi points, we expectedly
find $c=1$ from the subsystem entanglement scaling (see Fig. 9),
characterizing the phase as a $c=1$ chiral metal. Going away from half-filling
again leads to a change in the current density, additionally resulting in
oscillations in the correlators characteristic of the reduced Fermi volume
(not shown).

###  IV.3 Entrapped band insulator

Figure 10: Many body properties of $c=0$ EI. (a) The correlator $C_{B,A}(r)$
in logarithmic scale as a function of $r$ for different values of $\gamma$.
The numerical values from real space calculation are the points ($L=300$). The
continuous lines are the expressions shown in eqn. (27) for appropriate
parameters. (b) Entanglement entropy for different parameter values of
$\gamma$, as a function of $l/L$. Here, we have set $t_{1}=0.1$.

A novel non-equilibrium phase of matter which is realized here is that of an
entrapped insulator (EI) which gets stabilized for
$\gamma/t_{2}>2(t_{1}/t_{2}+1)$ region of the phase diagram (see Fig. 2). One
illustrative limit to start from is the $t_{2}=0$ ($t_{1}\rightarrow\infty$)
and $\gamma\neq 0$ where the chain breaks into a sequence of dimer states,
albeit with strong non-reciprocal coupling within each dimer. At exactly half-
filling, each unit cell can host a single electron stabilizing a non-Hermitian
insulating state. Given every particle is dynamically trapped in a unit cell
with a finite intra-unit cell current, we call this new state an EI. However,
the inter-unit cell current remains zero. It is this phase which continues to
remain stable for large values of $\gamma$ and a window of $t_{1}$ and $t_{2}$
regions.

Nature of the steady state: The complex spectrum again shows a two-lobe
structure with a line-gap in this phase, however the line gap now exists in
the imaginary eigenvalue direction (see Fig. 3). At half-filling the upper
lobe is completely filled creating a gap to excitations which are short-lived
with a finite lifetime.

Current and correlations: Given the gapped character of the system this steady
state has zero current. This is also seen through the fact that the spectral
loop in this case has zero winding. The inter-sublattice fermion correlator
decays exponentially with a form given by

$C^{A,B}(r)\sim\exp(-r/\xi),$ (27)

where $\xi\approx 1/\gamma$ (see Fig. 10).

Entanglement: The bipartite entanglement entropy as a function of subsystem
size shows characteristic oscillations with periodicity of two sites
signalling a dimer order with a value $\sim\log(2)$ when the partition is
intra-unit cell, and $\sim 0$ when the partition is inter-unit cell (see Fig.
10). One finds the inter-unit cell entropy to be negative for for larger
values of $t_{1}$, as has been reported in other non-Hermitian systems
chang2020entanglement, however its interpretation in terms of information is
not clear at present and merits further investigation.

###  IV.4 $c=2$ chiral metal (cFS-B)

We now briefly discuss another chiral metallic phase which appears when
$t_{1}\gg t_{2}$. While many of the essential properties of this phase are
similar to cFS-A, one distinctive feature is the orbital character of the
steady state which we discuss next.

At $\gamma=0$ the $t_{1}>t_{2}$ and $t_{1}<t_{2}$ phases are both insulating
with a band gap, however the orbital character of the bands are distinct
Asboth_springer_2016. For instance for $t_{1}<t_{2}$ the half-filled
equilibrium state has a distinct polarization compared to $t_{1}>t_{2}$ which
is reflected in a winding number that changes discontinuously at $t_{1}=t_{2}$
Resta_book_2007; Resta_RMP_1994; Watanabe_PRX_2018. Non-zero $\gamma$ does not
change the orbital character of the single particle bands in the $\gamma=0$
limit. For instance, decomposing the band wavefunctions ($n=0,1$) into the
bonding (antibonding) contributions ($c^{\dagger}_{nk}=\psi^{n}_{\alpha
k}c^{\dagger}_{k\alpha}+\psi^{n}_{\beta k}c^{\dagger}_{k\beta}$ ) where
$c^{\dagger}_{k\alpha}=\frac{1}{\sqrt{2}}\Big{(}c^{\dagger}_{kA}+c^{\dagger}_{kB}\Big{)}$
and
$c^{\dagger}_{k\beta}=\frac{1}{\sqrt{2}}\Big{(}c^{\dagger}_{kA}-c^{\dagger}_{kB}\Big{)}$,
shows the characteristic twist of the bands when $t_{1}<t_{2}$ (when
$\gamma\neq 0$) when $|\psi^{n}_{\alpha k}|^{2}$ is plotted for both the bands
(see Fig. 11). Introduction of non-Hermitian instability due to $\gamma$ leads
to half-filling of both these bands (therefore maximizing the total imaginary
component) for the steady state.

Moreover, the current in the system is dominated by the lower hopping scale
which is $t_{2}=1$ in the regime when $t_{1}>t_{2}$. Therefore the current in
cFS-B starts at $2/\pi$ when $\gamma\rightarrow 0$ and is independent of the
value of $t_{1}$ (See Fig. 12).

Figure 11: Distinction between two chiral phases cFS-A and cFS-B through
orbital band character. The real part of the dispersion is shown with
$|\psi^{n}_{\alpha k}|^{2}$ contribution in color [see Section IV.4] with (a)
$t_{1}=0.5$ and $\gamma=0.5$, and (b) $t_{1}=1.5$ and $\gamma=0.5$. Note the
difference in the orbital character around $k=\pi$ for the two cases. Figure
12: Scaling of chiral current. Chiral current as a function of $t_{1}$ for
different values of non-Hermitian coefficient $\gamma$. The current in the
system saturates to $2/\pi$ as a function of $t_{1}$ for ($t_{1}\geq 1.0$).
The current is dominated by the lowest of the two hopping scales
$t_{1},t_{2}$.

###  IV.5 Low energy theories and nature of phase transitions

Figure 13: Schematic of extended phase diagram. A schematic of the extended
phase diagram of the system showing different non-equilibrium phases[see eqn.
(2)] including the $c=2$ cFS, $c=1$ cFS and the EI phase.

Having the discussed the character of the steady states and their low energy
excitations, we now discuss the nature of these phase transitions between
various chiral metals and entrapped insulating states (Fig. 13). We note that
a conventional energy minimization to find the ground state or the excitations
is not sufficient to describe the steady states. Therefore, one needs to go
beyond conventional paradigm to describe such phase transitions, as we will
see next.

####  IV.5.1 Hatano-Nelson Model and $c=1$ cFS

The Hatano-Nelson Hamiltonian is given by hatano1996localization

$H_{HN}=\sum_{i}-t(c^{\dagger}_{i}c_{i+1}+h.c.)+\gamma(c^{\dagger}_{i}c_{i+1}-h.c.),$
(28)

which leads to a single particle dispersion given by
$\epsilon(k)=-2t\cos(k)+2\mathbbm{i}\gamma\sin(k)$. While for the equilibrium
Hermitian system ($\gamma=0$), the ground state is given by
$0<k<\frac{\pi}{2},\frac{3\pi}{2}<k<2\pi$, which has an entanglement scaling
$c=1$ with low energy states that disperse linearly $H(k)\sim
v_{F}|k|\Big{(}c^{\dagger R}_{k}c^{R}_{k}+c^{\dagger L}_{k}c^{L}_{k}\Big{)},$
where $R,L$ specify the right and left moving modes.

In presence of any $\gamma$ the single particle eigenvalues get a positive
imaginary coefficient for $\pi<k<2\pi$ leading to a steady state where
$\pi<k<2\pi$ are occupied.

Figure 14: Hatano-Nelson model at half-filling. In the absence of non-
Hermiticity ($\gamma=0$) the equilibrium system is given by two Fermi points
with linearized excitations. In presence of finite $\gamma$ the shifted Fermi
sea has quadratic excitations at band minima and band edges (blue), and short-
lived linear excitations (green).

The low energy excitations is therefore quadratically dispersing fermions,
albeit with a lifetime that is proportional to $\frac{1}{k}$,

$H_{HN}(k)\sim\Big{[}(-2+k^{2})t-\mathbbm{i}2\gamma k\Big{]}c^{\dagger
R}_{k}c^{R}_{k}+\Big{[}(2-k^{2})-\mathbbm{i}2\gamma k\Big{]}c^{\dagger
L}_{k}c^{L}_{k}.$ (29)

Along with excitations, another branch of short-lived excitations exist where
the linearly dispersing fermions can scatter from the right (left) moving
branch to the other left (right) branch with an energy $|(k-k_{o})|$
($k_{o}=\pi$) which are linear in momentum but with a finite lifetime (see
Fig. 14). These excitations carry current in the opposite direction but
eventually decay, thereby not changing the total steady state current of the
many-body state. Interestingly the sub-system scaling of the entanglement
entropy continues to show $c=1$. It is this feature which is characteristic of
the chiral Fermi sea and is shared with the $c=1$ chiral Fermi surface which
occurs in the complete phase diagram of Fig. 2 for the nH SSH model.

In a reduced Brillouin zone for a two site unit cell, as is appropriate for
our system, the above picture continues to hold, however, with both the
effective mass of fermions and lifetimes being $\gamma$ and $t_{1}$ dependent.

$H(k)\sim\Big{[}(-\epsilon+\frac{k^{2}}{m})+\mathbbm{i}\frac{k}{\tau}\Big{]}c^{\dagger
R}_{k}c^{R}_{k}+\Big{[}(\epsilon-\frac{k^{2}}{m})+\mathbbm{i}\frac{k}{\tau}\Big{]}c^{\dagger
L}_{k}c^{L}_{k},$ (30)

where
$\epsilon=2+t_{1}-\frac{\gamma^{2}}{16},m^{-1}=(1/4)+\frac{t_{1}}{8}-\frac{\gamma^{2}}{128}$
and $\tau=\frac{8}{(2-t_{1})\gamma}$ about $\gamma=0,t_{1}=1$. Interestingly,
the short-lived linear excitations now are also at small momentum given the
zone-folding with a velocity scale given by $\sim t_{1}$ and a
$\tau\propto\frac{1}{\gamma}$.

####  IV.5.2 $c=2$ cFS to $c=1$ cFS

The $c=2$ cFS is essentially two copies of the above Hatano-Nelson phase,
however, made out of bonding and anti-bonding orbitals of the dimerized
system, which are at two different values of real energies. The natural point
to investigate the $c=2$ FS is to start from $t_{1}=0,\gamma=0$ point, where
the Hamiltonian is exactly diagonalized by the following transformation

$\displaystyle c_{i\alpha}=\frac{1}{\sqrt{2}}(c_{i+1,A}+c_{iB}),\quad
c_{i\beta}=\frac{1}{\sqrt{2}}(c_{i+1,A}-c_{iB}).$ (31)

A perturbative $\gamma$ when projected to just the $\alpha$ and $\beta$ bands
to leading order gives the following non-Hermitian Hamiltonian

$\displaystyle H_{\alpha}=\sum_{i}-(c^{\dagger}_{\alpha i}c_{\alpha
i})+\frac{\gamma}{4}(c^{\dagger}_{\alpha i}c_{\alpha,i+1}-h.c.),$ (32)
$\displaystyle H_{\beta}=\sum_{i}+(c^{\dagger}_{\beta i}c_{\beta
i})+\frac{\gamma}{4}(c^{\dagger}_{\beta i}c_{\beta,i+1}-h.c.),$ (33)

where we have ignored the terms which mix $\alpha,\beta$ sectors. The above
effective Hamiltonian essentially leads to two Galilean-boosted Fermi seas,
which however, are separated in real energies (see Fig. 7). A change of sign
of $\gamma$ leads to a different $c=2$ cFS, where the currents are in the
opposite direction. A similar interpretation is valid for large $t_{2}$, where
$t_{1}\rightarrow 0$ and another set of dimers can be formed with $zero$
polarization, $c_{i\alpha}=\frac{1}{\sqrt{2}}(c_{i,A}+c_{iB})$ and
$c_{i\beta}=\frac{1}{\sqrt{2}}(c_{i,A}-c_{iB})$, with the effective non-
Hermitian model given by eqn. (33) with a different orbital content. Again for
$\gamma<0$ the current direction changes. Therefore one realizes four
different kinds of $c=2$ cFS (see Fig. 13) in the extended regime of $\gamma$
and $t_{1}$. Just like the discussion of the single Hatano-Nelson model in the
last subsection, here for each $c=2$ cFS, one obtains two copies of quadratic
and linear excitation branches.

The transition from such a $c=2$ cFS to a $c=1$ cFS is essentially a Lifshitz
transition characterized by exceptional point physics at $k=\pi$. The
eigenvalues of the single particle Hamiltonian at $k=\pi$ is given by [see
eqn. (3)] $E_{k}=\pm\sqrt{(t_{1}-t_{2})^{2}-\frac{\gamma^{2}}{4}}$, which
shows that while the real spectrum is gapped as soon as $t_{1}\neq t_{2}$ for
$\gamma=0$, there exists an extended regime for $\gamma>2|(t_{1}-t_{2})|$
where the real eigenvalues are gapless, while the imaginary eigenvalues show a
gap. Therefore, starting at any $t_{1},t_{2}$ ($\gamma\rightarrow 0$) where a
cFS-2 phase exists $\Delta_{bg}\sim|t_{2}-t_{1}|$ (see Fig. 7) with
quadractically dispersing fermions. Upon tuning the non-Hermiticity parameter,
$\gamma$, the branches merge at a critical value of
$\lambda_{c}=2|(t_{1}-t_{c})|$ close to which
$\Delta_{bg}\sim\sqrt{\gamma_{c}-\gamma}$ – showing the characteristic non-
analyticity of a phase transition governed by exceptional point physics.
Interestingly both the real and imaginary eigenvalues of energy scale as
$\sim\sqrt{k}$ reflecting the non-analyticity of this low energy theory at the
exceptional points aron2020landau.

####  IV.5.3 Entrapped insulator to $c=2$ cFS

To understand the entrapped insulator, it is illustrative to discuss the large
$\gamma$ limit, where a different band insulator can be obtained with
eigenfunctions
$\frac{1}{\sqrt{2}}\Big{(}\pm\mathbbm{i}c^{\dagger}_{iA}+c^{\dagger}_{iB}\Big{)}$
– this has a finite intra-unit cell current but no inter-cellular current. The
transition from the $c=2$ cFS ($t_{1}<t_{2},\gamma\rightarrow 0$) to an
insulating steady state is again by an exceptional point, where the imaginary
gap scales as $\sqrt{\gamma}$ near the phase transition.

Figure 15: Phase diagram under open boundary conditions. Four phases arise in
the non-Hermitian SSH model [see eqn. (8)] under open boundary conditions. Of
these (i) entrapped insulator (EI) and (ii) $c=2$ metal, are non-equilibrium
phases in their steady states, while (iii) topological band insulator and (iv)
trivial band insulator are in equilibrium.  Figure 16: Entanglement entropy
for $c=2$ metal. The entanglement entropy for $c=2$ metal under open boundary
conditions as a function of the bond length $l/L$ for $L=160$. The numerical
data can be fit to $\frac{c}{6}\ln{[\frac{L}{\pi}\sin{(\dfrac{\pi l}{L}})]}$,
where, $c/6=0.337$ (solid purple line). Here we have set $t_{1}=0.1$,
$\gamma=0.75$. Figure 17: Single particle complex spectra under open boundary
conditions. (a) $c=2$ metal ( $t_{1}=0.5$, $\gamma=1.72$) (b) Entrapped
insulator( $t_{1}=0.5$, $\gamma=3.0$) (c) Topological band insulator (
$t_{1}=0.5$, $\gamma=0.5$) (d) Trivial insulator phase ( $t_{1}=2.0$,
$\gamma=0.5$)

##  V Tuning boundary conditions

Having discussed the properties and phases of our system under periodic
boundary conditions, we next study the system under open boundary conditions.
For the Hermitian case, we know that an open system contains boundary modes
which are robust to disorder. It is natural question to pose if such modes can
be obtained in a non-Hermitian system and how they effect the steady states.

More generally, we multiply a scaling factor, $\lambda$, to the boundary bond
in order to smoothly tune between an open and a periodic problem (see
koch2020bulk), and discuss its effect on the steady state physics. When
$\lambda=0$ (open boundary conditions), as has been reported in earlier
studies focusing on single particle spectrum (see, for e.g., Reference
Herviou_PRA_2019), one finds four distinct regions as shown in Fig. 15.
Briefly, for $\gamma>2t_{1}$ all eigenvalues are real, while for
$\gamma<2t_{1}$ eigenvalues can have both real and imaginary components.
Moreover, within the region $2\sqrt{t^{2}_{1}-1}<\gamma<2\sqrt{t^{2}_{1}+1}$,
one finds zero energy eigenvalues which smoothly connect to topological
boundary modes in the Hermitian limit ($\gamma=0,t_{1}<t_{2}$). Given that our
steady states are constructed by populating modes with the largest imaginary
component, such long-lived non-equilibrium states are only defined for
$\gamma<2t_{1}$, while for $\gamma>2t_{1}$ one obtains a conventional
equilibrium ground state which is obtained by minimizing the total real
energy. This apparent mismatch between the thermodynamic phase diagram for the
periodic and the open problem is, however, an artifact of extreme sensitivity
of the non-equillibrium phases to the $\lambda=0$ limit, and, as we show
below, any infinitesmal $\lambda$ leads to stable steady states albeit with
subtle effects arising due to topological boundary modes.

Four distinct phases can be identified in the open system as shown in Fig. 15
and Fig. 17. Out of these, two: (i) trivial band insulator and (ii)
topological band insulator are the equilibrium phases which are smoothly
connected to the Hermitian limit ($\gamma=0$). While in the thermodynamic
limit both these regions have a bulk band gap, the latter has quasi-degenerate
boundary modes at zero real energies. The other two phases (iii) $c=2$ metal
and (iv) entrapped insulator, are non-equilibrium steady states which arise by
populating the eigenmodes with largest imaginary components. We find that the
metallic phase has gapless excitations which can be long-lived, and has a zero
total current in the system (as is expected for an open system). Investigating
the bipartite entanglement leads to a central charge $c=2$ (see Fig. 16).
Interestingly, this metallic phase contains two zero energy modes which have
both real and imaginary components to be zero. This leads to quasi-degenerate
steady states in the system which can be excited via particle-hole
excitations. Having discussed the many-body phases as realized in the
completely open system, we now investigate their stability to weak
hybridization ($\lambda$) between the boundary sites.

We find that, in the thermodynamic limit, any perturbatively small $\lambda$
immediately leads to the non-equilibrium phases as realized in the periodic
system as shown in Fig. 2. However, as $\lambda$ is tuned, two phenomena
generically occur (a) boundary modes in regions of the metal and band
insulator give way to Fermi seas albeit with boundary modes either by changing
the gap between the steady states or by leaky boundary modes, and (b) the
steady state current gets immediately saturated to the value in the periodic
problem as soon as $\lambda\neq 0$. We therefore interpolate from the open
(see Fig. 15) to periodic system as a function of $\lambda$ (see Fig. 2) to
see how the steady state evolves. In the Hermitian limit $(\gamma=0)$ when
$t_{1}<t_{2}$ the boundary modes for the open system hybridize and gap out
(with the gap $\propto\lambda$), eventually merging with the bulk bands at
$\lambda=1$.

Figure 18: Instability of the topological band insulator region to cFS-A. (a)
The spectrum in the complex plane for different values of $\lambda$ for
$t_{1}=0.5,\gamma=0.5$. (b) The current, $J$, as a function of $\lambda$. The
inset shows the behaviour of $\Delta_{bg}$ by tuning $\lambda$. (c) A
schematic of the energy spectrum. Here $t_{1}=0.5$. Figure 19: Instability of
the $c=2$ metal region to $c=2$ cFS-A. (a) The complex spectrum for different
values of $\lambda$ for $t_{1}=0.12,\gamma=1.2$. (b) The current, $J$,
variation with $\lambda$ for two different representative values of $t_{1}$
and $\gamma$.

##### Instability to cFS-2 A:

We start from the open system in either the topological (or trivial) band
insulator and introduce a $\lambda$ to interpolate to the $c=2$ cFS. While the
spectrum is purely real for $\lambda=0$, any finite value of $\lambda$
introduces an imaginary part to the eigenvalues and moves the boundary modes
along the real line towards the two lobes [see the spectrum in Fig. 18 (a)].
This immediately leads to a finite steady state current as shown in Fig. 18
(b) for a few representative values. Therefore, we find that the open boundary
phase diagram is not stable, and immediately leads to a chiral metal phase.
However, there is a distinction as to how the $c=2$ cFS is approached either
from topological or trivial band insulator regimes. Note that the $c=2$ cFS
has six steady states with gaps given by $\Delta_{bg}$ and $\Delta_{t}$ (see
Fig. 7). When $\lambda$ is tuned such that the $c=2$ CFS approaches a
topological band insulator, two bulk modes move towards zero in the real
spectrum thereby tuning both $\Delta_{t}\rightarrow\Delta_{bw}/2$ and
$\Delta_{bg}\rightarrow 0$ [see Fig. 18 (c)], thus offering a tunability
between many-body two quasi-degenerate steady states by occupying or emptying
the boundary mode locally. It is in this sense that these steady states
realized in $c=2$ cFS-A are in fact tunable and are inherently distinct from
the $c=2$ cFS-B. Thus the role of single particle boundary modes here is to
provide an avenue for interesting quasi-degenerate steady states which are
boundary tunable. Similar physics is at play when $\lambda$ interpolates
between $c=2$ non-chiral metal realized in the open system to the $c=2$ chiral
FS. Again any finite value of $\lambda$ immediately leads to a chiral current,
and existence of boundary modes, which, in turn, lead to quasi degenerate
steady states (see Fig. 19).

Figure 20: Instability of the $c=2$ metal region to $c=1$ cFS. (a) Tuning of
the complex spectrum with varying $\lambda$. (b) Variation of the current,
$J$, with $\lambda$ at two representative choices of $t_{1}$ and $\gamma$.
(inset: variation of the lifetime $\tau_{g}$ of the boundary excitations as a
function of $\lambda$ for $t_{1}=0.5,\gamma=1.72$).

##### Instability to $c=1$ cFS:

We next discuss the instability to another phase from the topological band
insulator, where the system goes to the chiral $c=1$ FS. At $\lambda=0$, given
that we are starting at the topological band insulator, we have two zero
energy eigenmodes. However, unlike the instability to $c=2$ phase, here the
boundary modes move along the imaginary axis leading to a finite lifetime for
the boundary mode (see Fig. 20). The system immediately obtains a finite
current and two degenerate steady states. Each of them, however, has an
excitation over the boundary which has a finite lifetime. The way the value of
lifetime changes with $\lambda$ is shown in the inset of Fig. 20 (b). Unlike
the previous case, the lifetime decreases exponentially with increasing
$\lambda$ leading therefore to the leaky boundary modes.

The entrapped insulator phase in the open system gets smoothly connected to
the periodic system and is therefore stable to tunability of boundary
conditions as is expected for a steady state with no finite current. With this
we have discussed the stability of the all the phases we realized in the open
system and the role of boundary modes in such steady state physics.

##  VI Summary and discussion

In this work we revisited the nH-SSH model and the phases it realizes under
finite filling. We find that the single-particle phase diagram gets
reinterpreted in terms of distinct non-equilibrium phases of matter, which get
realized at long times. We further analysed each of the phases in terms of its
correlations, gap to excitations and entanglement features. Developing the low
energy theories of various phases and intervening transitions, we build a
qualitative understanding of the phases and its excitations in terms of the
spectral topology. We further tuned the boundary conditions to take the system
from an open to a periodic problem, and find that the nature of the
equilibrium phases in the open system are perturbatively unstable to a weak
coupling and leads to finite currents in the system.

An important question, that we have side-stepped here, is how such non-
Hermitian Hamiltonians get realized in an experimental setting and where this
is a valid effective description. While in open systems, where bath degrees of
freedom are systematically integrated out and the effective density matrix of
a system is evolved – one naturally uses the Lindbaldian approach which can in
general be non-Hermitian AltlandPRX2021; bergholtz2021exceptional. Here we
presume the existence of the nH-SSH system and that one can keep it at a
finite density. This is a particularly useful construct where one may consider
such non-Hermitian models and their steady states to be anomalous theories
which get realized on the boundaries of higher-dimensional topological phases
LeePRL2019.

Our work opens up several avenues to explore. One natural scope of expanding
our findings is to consider higher dimensions, where one expects such chiral
phases and entrapped insulating phases to be generic. An interesting direction
will be to investigate the role of disorder and impurities in such systems and
the meaning of resulting topological protection in such many-body states
obtained in the non-Hermitian phases. How an equilibrium state reaches such
steady states in time and the corresponding dynamics of the density matrix due
to introduction of non-Hermiticity is another interesting future scope.
Similarly the role of many-body physics and interactions is something that has
been studied only recentlyalsallom2021fate; Lee_PRB_2020; Liu2020; Mu20;
Shen21; zhang2021observation; Yoshida21 and is worth pursuing in future. In
conclusion, our different many-body perspective on the non-Hermitian phases
can lead to interesting insights where the interplay of topology, entanglement
and correlations conspire to produce novel phases of quantum matter.

###### Acknowledgements.

AB is supported by the Prime Minister’s Research Fellowship (PMRF). SSH
acknowledges MPI-PKS, Dresden for support during the project. AA acknowledges
enlightening discussions with Animesh Panda, Sumilan Banerjee, Vatsal Dwivedi,
Aabhaas Mallik, Abhisodh Prakash and a related collaboration with Amit
Chatterjee and Arghya Das. AA acknowledges partial financial support through
Max Planck partner group on strongly correlated systems at ICTS. AA
acknowledge the visitor’s program at MPI-PKS, Dresden for hospitality. AN is
supported by the start-up grant (SG/MHRD-19-0001) of the Indian Institute of
Science and by DST-SERB (project number SRG/2020/000153).

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