# Non-Hermiticity induced Exceptional Points and Skin Effect in the Dice-
Haldane Model

Ronika Sarkar  Department of Physics, Indian Institute of Science, Bangalore
560012, India.  Arka Bandyopadhyay  Solid State and Structural Chemistry Unit,
Indian Institute of Science, Bangalore 560012, India.  Awadhesh Narayan
<EMAIL_ADDRESS>Solid State and Structural Chemistry Unit, Indian
Institute of Science, Bangalore 560012, India.

###### Abstract

The interplay of topology and non-Hermiticity has led to diverse, exciting
manifestations in a plethora of systems. In this work, we systematically
investigate the role of non-Hermiticity in the Chern insulating Haldane model
on a dice lattice. Due to the presence of a non-dispersive flat band, the
dice-Haldane model hosts a topologically rich phase diagram with the non-
trivial phases accommodating Chern numbers $\pm 2$. We introduce non-
Hermiticity into this model in two ways – through balanced non-Hermitian gain
and loss, and by non-reciprocal hopping in one direction. Both these types of
non-Hermiticity induce higher-order exceptional points of order three.
Remarkably, the exceptional points at high symmetry points occur at odd
integer values of the non-Hermiticity strength in the case of balanced gain
and loss, and at odd integer multiples of $1/\sqrt{2}$ for non-reciprocal
hopping. We substantiate the presence and the order of these higher-order
exceptional points using the phase rigidity and its scaling. Further, we
construct a phase diagram to identify and locate the occurrence of these
exceptional points in the parameter space. Non-Hermiticity has yet more
interesting consequences on a finite-sized lattice. Unlike for balanced gain
and loss, in the case of non-reciprocal hopping, the nearest-neighbour dice
lattice system under periodic boundary conditions accommodates a finite, non-
zero spectral area in the complex plane. This manifests as the non-Hermitian
skin effect when open boundary conditions are invoked. In the more general
case of the dice-Haldane lattice model, the non-Hermitian skin effect can be
caused by both gain and loss or non-reciprocity. Fascinatingly, the direction
of localization of the eigenstates depends on the nature and strength of the
non-Hermiticity. We establish the occurrence of the skin effect using the
local density of states, inverse participation ratio and the edge probability,
and demonstrate its robustness to disorder. Our results place the dice-Haldane
model as an exciting platform to explore non-Hermitian physics.

##  I Introduction

In condensed matter physics, most of the intricate phases of matter, including
magnetic and superconducting states, can be understood in the framework of the
celebrated Landau theory Sachdev (2011). However, the two-dimensional electron
gas at very low temperatures and under a strong transverse magnetic field
exhibits a quantized Hall conductance Klitzing et al. (1980) – such a
quantization is not subject to any spontaneous symmetry breaking.
Consequently, new concepts have been developed based on single-particle
dynamics in topological band theory to unravel the advent of the integer
quantum Hall effect. Haldane, in his seminal work, demonstrated that Dirac
points in honeycomb lattices, such as graphene are protected by both inversion
and time-reversal symmetry Haldane (1988); Sheng et al. (2006); Thonhauser and
Vanderbilt (2006); Hao et al. (2008); Bergholtz and Liu (2013); Haldane
(2017). Absence of any of these symmetries essentially leads to gapped spectra
with distinct topological nature. In particular, Semenoff mass assigns an
energy offset between the two sublattices of graphene and breaks the inversion
symmetry Semenoff (1984). These inversion symmetry broken systems give rise to
the normal or trivial insulators at half-filling. In contrast, a staggered
magnetic field that turns the second nearest neighbor hoppings complex also
breaks the time-reversal symmetry of the system without violating its
translational symmetry. These time-reversal symmetry broken Chern insulators
are at the heart of realizing quantized transverse Hall conductance in zero
external magnetic field condition, namely the quantum anomalous Hall effect.
In other words, the Haldane model is an elegant Chern insulator model on a
honeycomb lattice, that allows tunability between topologically trivial and
non-trivial phases by tuning the model parameters. The topological phase
diagram of the Haldane model has successfully been realized in experiments
using ultra cold fermionic atoms in optical lattices Jotzu et al. (2014).

Moreover, unlike graphene, some of the bipartite lattices possess an unequal
number of sublattices that offer an intriguing platform to realize perfectly
or compact localized states. These compact localized states exhibit non-
dispersive flat bands, i.e., the energy is independent of momentum in the
electronic band structure. The underlying mechanism behind such flat bands can
be well-explained in terms of destructive interference through various network
paths. For example, the bipartite dice lattice Sutherland (1986); Vidal et al.
(1998, 2001); Bercioux et al. (2009); Wang and Ran (2011); Möller and Cooper
(2012); Zhang and Jin (2020); Cheng et al. (2020); Wang et al. (2021a); Iurov
et al. (2022) is one of the first and most prominent examples where such flat
band physics was introduced. In a dice lattice, atoms are not only placed at
the vertices of hexagons but also at the centers. Therefore, the number of
sites with coordination number three is twice of those with coordination
number six. In contrast to usual honeycomb lattices, three-component fermions
invariably govern the low energy spectrum of the dice lattice. The two
dispersive bands form Dirac cones and touch each other at symmetry points $K$
and $K^{\prime}$ of the Brillouin zone (BZ), while the remaining one is flat
and lies at the Fermi level. The flat bands in the bipartite lattices occur
because of the chiral symmetry. In other words, the bipartite model systems
(such as dice) with a majority of one kind of sublattice invariably exhibit
chiral flat bands Sutherland (1986). The flat bands have been recently
experimentally realized in photonic crystals employing ultrafast laser
technology Mukherjee et al. (2015); Vicencio et al. (2015); Zong et al.
(2016); Mukherjee et al. (2018). Motivated by these interesting properties of
the dice lattice, the Haldane model has been extended in the form of a three-
band model with broken inversion and time-reversal symmetry Andrijauskas et
al. (2015). As expected, the topological phase diagram of the Haldane dice
lattice is richer compared to that of the graphene with more interesting
phases both within and outside the topologically non-trivial region Dey et al.
(2020).

Non-Hermitian physics Ashida et al. (2020); El-Ganainy et al. (2018); Kawabata
et al. (2019a); Rotter (2009); Martinez Alvarez et al. (2018); Bender (2007);
Ghatak and Das (2019), on the other hand, is a topic of growing widespread
interest. Non-Hermiticity finds applicability in various fields of photonics,
optics, and electronics, among others Cao and Wiersig (2015); Zhang et al.
(2018); Gupta et al. (2020); Zeng et al. (2022); Wang et al. (2021b); Ma and
Sheng (2016); Chou et al. (2011). Since these are open systems, the
corresponding non-Hermitian Hamiltonian can accommodate for the gain and loss
of particles or energy. Unusual properties, such as complex band spectra and
non-orthogonal eigenstates, are the outcomes of such non-Hermitian
Hamiltonians Bender and Boettcher (1998). In particular, non-Hermitian systems
can show a distinct class of spectral degeneracies known as exceptional points
(EPs) Bergholtz et al. (2021a); Miri and Alu (2019); Özdemir et al. (2019);
Huang et al. (2017); Krasnok et al. (2021); Banerjee and Narayan (2020, 2021);
Kawabata et al. (2019b); Hu et al. (2017); Chen et al. (2019); Minganti et al.
(2019); Alvarez et al. (2018), as well as exceptional contours Chowdhury et
al. (2021, 2022). At an EP the eigenvalues and the eigenvectors simultaneously
coalesce making the Hamiltonian defective, i.e., non-diagonalizable. The
number of eigenstates undergoing coalescence determines the order of the EP.
The study of EPs has gained immense interest in the field of photonic systems
Parto et al. (2021); Midya et al. (2018) and microwave cavities Dietz et al.
(2011); Dembowski et al. (2001a) among others, with interesting applications
such as uni-directional sensitivity Lin et al. (2011); Hodaei et al. (2017),
laser mode selectivity Feng et al. (2014); Hodaei et al. (2014) and opto-
mechanical energy transfer Xu et al. (2016). Non-Hermitian skin effect (NHSE),
on the other hand, is a feature unique to non-Hermitian systems where a
macroscopic fraction of eigenstates migrate to a boundary of the system as
soon as open boundary conditions (OBC) are imposed Yao and Wang (2018); Song
et al. (2019); Longhi (2019); Li et al. (2020a); Okuma et al. (2020); Kawabata
et al. (2020); Longhi (2020); Li et al. (2020b); Lu et al. (2021); Borgnia et
al. (2020). This extreme sensitivity of non-Hermitian systems to the boundary
conditions leads to an anomalous bulk-boundary correspondence Koch and Budich
(2020); Kunst et al. (2018); Xiao et al. (2020a); Edvardsson et al. (2019).
The NHSE has been experimentally observed recently in photonic systems Song et
al. (2020), electrical circuits Liu et al. (2021); Zou et al. (2021) and
acoustic topological insulators Zhang et al. (2021a, b) among others.

In this paper, we systematically study the interplay between the effect of
non-Hermiticity and different kinds of hopping terms in the dice lattice
model. Later, we also study the role of disorder Anderson (1958); Lee and
Ramakrishnan (1985); Kramer and MacKinnon (1993); Phillips (1993); Schubert et
al. (2012); Segev et al. (2013); Sacksteder et al. (2015); Wu et al. (2016);
Claes and Hughes (2020); Sarkar et al. (2022) in the context of the non-
trivial effects brought about by non-Hermiticity. We start with only the
nearest neighbour hopping, then subsequently allow complex next-nearest
neighbour hopping terms similar to the Haldane model, and finally, introduce
the inversion breaking mass terms. For each case, we tune-up a non-Hermitian
balanced gain and loss and investigate the changes in the eigenspectra and
characterize the EPs which arise. We discover that third order EPs arise at
odd integer values of the non-Hermiticity strength in each case. Their
occurrence can be characterized using the phase rigidity which vanishes at the
EP. Further, the scaling of phase rigidity with respect to the non-Hermiticity
strength helps determine the order of the EP. When a non-reciprocal hopping is
introduced instead of gain and loss, we find that third order EPs occur at odd
integer multiples of $1/\sqrt{2}$. We also elucidate the complete phase
diagram to determine the regions where such higher-order EPs can be found in
the parameter space. Non-reciprocal hopping has interesting consequences when
we consider a finite sized dice-Haldane nanoribbon. For the dice lattice with
only nearest neighbour coupling, under periodic boundary conditions (PBC), the
spectrum under non-reciprocal hopping accommodates a finite, non-zero spectral
area in the complex plane unlike the gain and loss case where the complex
spectrum has an arc-like structure. This finite spectral area results in the
occurrence of the NHSE when OBC are imposed on the lattice with non-reciprocal
hopping. However, for the dice-Haldane model both gain and loss and
nonreciprocal hopping exhibits finite spectral area under PBC and hence
displays NHSE under OBC. The direction of the localization can be controlled
by the choice of the non-Hermiticity and its strength. We characterize the
NHSE using the local density of states, inverse participation ratio (IPR) and
the edge probability. This NHSE turns out to be fairly robust to disorder
owing to its topological protection. However, at sufficiently large disorder
strengths there is a complete destruction of NHSE accompanied by the bulk
localization of all the eigenstates.

##  II Dice-Haldane Lattice Model

Figure 1:  Illustrating the model, its Brillouin zone and the Hermitian phase
diagram. (a) Schematic of the non-Hermitian dice-Haldane lattice model. A, B
and C are the three sublattice sites. Hopping potential $t$ is the nearest
neighbour hopping between sublattices A-B and B-C. Lattice sites A (shown in
green) in general can have a Semenoff mass $+m$ and a non-Hermitian gain
$+i\delta$ while lattice sites C (orange) can have a Semenoff mass $-m$ and
non-Hermitian loss $-i\delta$. Panel (b) illustrates the next-nearest
neighbour Haldane-type hopping with strength $t_{2}$ and the flux enclosed by
these hopping potentials is $\phi$. (c) The BZ of the dice-Haldane lattice
showing the high symmetry points $M,K^{\prime},K$ and $\Gamma$. (d) The phase
diagram of the Hermitian dice-Haldane model where the region enclosed within
the curves $\pm m^{*}\sin\phi$ has a non-trivial Chern number. Outside the
curve lies the topologically trivial region. Further, we can find three
additional phases – AG: all-gapped (shown in blue), VG: valence-gapped (shown
in violet) and CG: conduction-gapped (shown in orange). Here $m^{*}$ is
expressed in units of $t_{2}$.

In this section, we introduce the Haldane model applied to the dice lattice.
The dice lattice can be viewed as a honeycomb lattice with an additional atom
at the centre of each hexagonal plaque, as depicted in Fig. 1(a). Therefore,
each unit cell of the dice lattice possesses three basis atoms denoted by A,
B, and C in our work. Among these three lattice sites, A and C are equivalent
with coordination number three, while B lattice sites have a coordination
number of six. There are two main schemes to obtain dice lattice – using cold
atoms confined in optical lattices Bercioux et al. (2009) and growing a
trilayer structure of cubic lattices viz. SrTiO3-SrIrO3-SrTiO3 along
crystallographic (111) direction Wang and Ran (2011). Under the tight-binding
framework, this model allows the nearest neighbour (nn) hopping ($t$) between
the sites A-B and B-C. Further, in the spirit of the Haldane model, we
consider a complex next-nearest neighbour (nnn) hopping ($t_{2}$) among the A
and C lattice sites, such that there is a non-zero flux enclosed by the path
formed by the nnn hopping terms. Hence, $t_{2}\rightarrow t_{2}e^{\pm i\phi}$
where, $\phi$, $+$ and $-$ indicate staggered flux and the sign of the phase
for counterclockwise and clockwise hopping about B lattice sites,
respectively. Additionally, a Semenoff mass $+m$ on A and $-m$ on C brings us
to the full dice-Haldane lattice model. The complete lattice model and the
corresponding BZ are schematically represented in Fig. 1(b) and (c)
respectively for convenience (note that for the Hermitian case: $\delta=0$).
The full Hamiltonian of the lattice model can be expressed as follows

$H=H_{nn}+H_{nnn}+H_{m},$ (1)

where $H_{nn}$ corresponds to the contribution from only nearest-neighbour
hopping. In our calculations, we fix $t=1/\sqrt{2}$, which sets the energy
scale of our system. $H_{nnn}$ corresponds to the contribution from next-
nearest neighbour hopping parameters and $H_{m}$ to the asymmetric Semenoff
mass terms on A and C sites. The expressions for the same are given by

$\begin{split}H_{nn}&=t\sum_{\langle
i,j\rangle}(c_{A,i}^{\dagger}c_{B,j}+c_{B,i}^{\dagger}c_{C,j}+h.c.),\\\
H_{nnn}&=t_{2}e^{\pm i\phi}\sum_{\langle\langle
i,j\rangle\rangle}(c_{A,i}^{\dagger}c_{A,j}+c_{C,i}^{\dagger}c_{C,j}+h.c.),\\\
H_{m}&=m\sum_{i}(c_{A,i}^{\dagger}c_{A,i}-c_{C,i}^{\dagger}c_{C,i}),\end{split}$
(2)

where, $c_{i}^{\dagger}$ and $c_{i}$ represent creation and annihilation
operator at $i-$th lattice site respectively and $h.c.$ indicates the
Hermitian conjugate partner of the given expression. Moreover,
$\langle.\rangle$ and $\langle\langle.\rangle\rangle$ denote nearest
neighbours and next nearest neighbours, respectively. It is needless to
mention that $H_{nnn}$ breaks the time-reversal (TR) symmetry in the model
without the requirement of a net external magnetic flux, while $H_{m}$ breaks
the inversion symmetry.

The low-energy electronic description of this dice model can be expressed as a
Dirac-Weyl Hamiltonian with pseudospin equal to one. Both the time-reversal
symmetry and inversion symmetry broken phases open a gap in the energy
spectra. However, these two gapped states are topologically distinct
Bandyopadhyay et al. (2020); Bandyopadhyay and Jana (2020), classified by
different Chern numbers. In other words, the Hermitian dice-Haldane model Dey
et al. (2020) harbours a richer phase diagram than the conventional Haldane
model, because the former gives rise to more phases both within and outside
the topological region of the usual phase diagram accommodating Chern numbers
$\pm 2$, as shown in Fig. 1(d). Particularly, the topologically non-trivial
region is bounded by the relation $m=\pm m^{*}\sin\phi$ and has a Chern number
$\pm 2$. Due to the bulk-boundary correspondence, two edge modes will appear
when open-boundary conditions are invoked. There are additional phases that
arise due to the dice lattice structure and its flat band – the all-gapped
phase (AG), where all three bands are gapped with no overlap, the valence-
gapped phase (VG), where the conduction band and flat band touch each other
while the valence band remains gapped, and lastly the conduction-gapped phase
(CG) where the conduction band is gapped while the valence and flat bands have
some overlap. The electronic band structures of the Hermitian dice-Haldane
model in the different phases AG, VG and CG can be found in Appendix. A.

##  III Effect of non-Hermitian Gain and Loss

Having acquainted ourselves with the Hermitian model, we now systematically
invoke non-Hermitian gain and loss into the dice-Haldane model and study its
interplay with $H_{nn}$, $H_{nnn}$ and $H_{m}$. Physically, non-Hermitian
balanced gain and loss can be thought of as a source attached to one unit cell
atom and an equally strong sink attached to another atom. Here, we allow
sublattice A sites to posses a gain $+i\delta$ while sublattice C sites posses
a loss $-i\delta$. Hence, the non-Hermiticity added to the Hamiltonian can be
described by considering an additional term to Eqn. 1, which is of the form

$H_{\delta}=i\delta\sum_{i}(c_{A,i}^{\dagger}c_{A,i}-c_{C,i}^{\dagger}c_{C,i}).$
(3)

Here, $\delta$ denotes the strength of non-Hermiticity. A point to note is
that when we turn off $t_{2}$ and $m$, the Hamiltonian $H$ has parity-time
($PT$) symmetry. Any non-zero value of $t_{2}$ or $m$ breaks the $PT$
symmetry.

In this section, we will first consider $H_{nn}+H_{nnn}$ and introduce
$\delta$, and thereafter, we will study the effect of non-Hermitian gain and
loss considering the full dice-Haldane Hamiltonian
$H=H_{nn}+H_{nnn}+H_{m}+H_{\delta}$. For each of these cases, we will
highlight the exotic physics arising at the high-symmetry points $M,K$ and
$\Gamma$, extensively discussing the occurrence of higher-order EPs at integer
$\delta$ values.

EPs occur when not only two or more eigenvalues become degenerate but also
their corresponding eigenvectors Heiss (2012, 2004). This leads to a collapse
of the Hilbert space into a lower-dimensional Hilbert space. The collapse of
two eigenvectors leads to a second order EP (EP2). A third order EP (EP3)
occurs on the collapse of three eigenvectors and so on. The coalescence of
eigenvectors can be characterized by the phase rigidity, which is a measure of
the bi-orthogonality of the eigenfunctions. It is given by Martinez Alvarez et
al. (2018); Rotter (2009); Alvarez et al. (2018)

$r_{\alpha}=\frac{\langle\phi_{\alpha}|\psi_{\alpha}\rangle}{\langle\psi_{\alpha}|\psi_{\alpha}\rangle},$
(4)

where $\psi_{\alpha}$ is the $\alpha$-th right eigenvector of $H$ while
$\phi_{\alpha}$ is the corresponding left eigenvector of $H$, i.e
$H|\psi_{\alpha}\rangle=\lambda_{\alpha}|\psi_{\alpha}\rangle$ and
$\langle\phi_{\alpha}|H=\lambda_{\alpha}\langle\phi_{\alpha}|.$ For a
Hermitian system $r_{\alpha}$ is always equal to unity as the right and left
eigenvectors are the same. For non-Hermitian systems, near an EP,
$r_{\alpha}\rightarrow 0$ for the states that coalesce.

Further, to determine the order of the EP, we can perform a scaling analysis
of the phase rigidity Zhang et al. (2020); Jaiswal et al. (2021). Here, the
Hamiltonian depends on two parameters ($k_{x}$, $\delta$). The scaling of
phase rigidity follows $|r|\sim|\delta-\delta_{EP}|^{\nu}$ for an $N$-th order
EP, where $\delta_{EP}$ is the value of $\delta$ for which an EP occurs. It is
noteworthy that when an anisotropic EP is approached from two orthogonal
directions in parameter space the scaling exponent $\nu$ can take values
$(N-1)$ or $(N-1)/2$ Xiao et al. (2019); Ding et al. (2018). In our case, we
fix $k_{x}$ and investigate the scaling of the phase rigidity with respect to
varying non-Hermiticity $\delta$ close to the EP. Hence, $\nu$ here, is given
by $(N-1)/2$ where $N$ is the order of the EP. In particular, we can plot
$\log|r|$ vs $\log|\delta-\delta_{EP}|$ to obtain $N$ from the slope. For
example, an EP2 will have a slope of 1/2, while an EP3 will have a slope of 1.

Figure 2:  Spectra with next-nearest-neighbour hopping and non-Hermiticity.
The $Re(E)$ (upper panel) and $Im(E)$ spectra (lower panel) have been shown
for different values of non-Hermiticity strength $\delta$. In (a) and (b)
$\delta=1.0$ showing an EP at the $M$ point. In (c) and (d) $\delta=2.0$, this
induces an EP at $k_{x}=1/3$. In (e) and (f) we find an EP at the $\Gamma$
point for $\delta=3.0$. For all plots $t=1/\sqrt{2}$, $t_{2}=0.06t$ and
$\phi=\pi/2$.

In addition to the nearest neighbour hopping $t$, first, we include the next-
nearest neighbour Haldane type hopping, which breaks time-reversal symmetry of
the system. In the presence of a balanced gain and loss term, the Hamiltonian
can be written as $H_{nn}+H_{nnn}+H_{\delta}$. The time-reversal symmetry
breaking induces a non-trivial band gap in the Hermitian system (discussed in
Appendix A). In other words, the degeneracy at $K$ and $K^{{}^{\prime}}$
points is lifted by non-vanishing $t_{2}$ (let, $\phi=\pi/2$). In the non-
Hermitian case, even a small value of $\delta$ produces a complex energy
spectrum as expected. Similar to the nearest neighbour case, finite imaginary
parts of the spectra first appear around $K$ and $K^{{}^{\prime}}$ points. On
the other hand, the conduction and valence bands of the real part of the
eigenspectra come closer with increasing $\delta$ and finally meet again at
$M$ point for $\delta=1.0$. These observations are illustrated in Fig. 2(a)
and (b). Further, the degeneracy of $Re(E)$ at $M$ point is found to be robust
to the values of $\delta$, for $\delta\geq 1$. However, the degeneracy for
$Im(E)$ is lifted beyond $\delta=1$. With further increase in $\delta$, we
have found another set of degeneracies of $Re(E)$ at the halfway point between
$\Gamma\rightarrow K/K^{{}^{\prime}}$ for $\delta=2$. Beyond this value of
$\delta$, the degeneracy of $Im(E)$ at the same ($\pm 1/3,0$) point is removed
[Fig. 2(c) and (d)]. It is important to note that the above mentioned
degeneracy of $Re(E)$ is robust for $\delta\geq 2$ similar to what happens at
the $M$ point. Finally, at $\delta\geq 3$ degeneracy at $\Gamma$ point appears
and disappears for $Re(E)$ and $Im(E)$, respectively, as presented in Fig.
2(e) and (f). Similar to the previous cases, the degeneracy of $Re(E)$ at
$\Gamma$ is robust after this critical point. There occur critical values of
$\delta$ at which the degeneracy in $Re(E)$ and $Im(E)$ exist simultaneously
at particular $k_{x}$ values. Now, we explore the possibility of these points
being EPs and their corresponding order at these critical $\delta$ values,
which are interestingly exact integers. The phase rigidity and its
corresponding scaling at $M$ and $\Gamma$ points are shown in Fig. 3. From the
zero value of $r_{\alpha}$ and the corresponding scaling giving a slope of
one, it is clear that all these points are indeed higher-order EPs of order
three.

Figure 3:  Phase rigidity for next-nearest-neighbour hopping with non-
Hermiticity. The upper panels show the phase rigidity, $r_{\alpha}$, as a
function of non-Hermiticity, $\delta$, while the lower panels show the scaling
of the corresponding $r_{\alpha}$. Panels (a) and (b) correspond to the $M$
point, where an EP is induced at $\delta=1$. Panels (c) and (d) correspond to
the $\Gamma$ point, where another EP is induced at $\delta=3.0$. The scaling
of $r_{\alpha}$ gives a slope of one in both cases implying the both the EPs
are of order three. The different colours in the plots correspond to different
eigenstates. The $r_{\alpha}$ for the two dispersive bands overlap. Here,
$t=1/\sqrt{2}$, $t_{2}=0.06t$ and $\phi=\pi/2$.

At this point, it is worth exploring the underlying reason behind the
emergence of EPs at high symmetry points due to odd integer values of
$\delta$. However, it is possible to get EPs away from high symmetry points at
even values of $\delta$. For this purpose, we have analytically calculated the
energy band dispersion of the dice lattice described by $H_{nn}+H_{\delta}$.
It is evident that the Hamiltonian will give rise to three energy bands
including the non-dispersive flat band at zero energy. The dispersive bands,
on the other hand, have the following expression.

$E_{\pm}(k_{x},\delta)=\pm\sqrt{3-\delta^{2}+4\cos(\pi k_{x})+2\cos(2\pi
k_{x})}.$ (5)

When these bands collapse with the flat band it gives rise to EPs at specific
strengths of non-Hermiticity $\delta$ given as below,

$\delta_{EP}=\pm\sqrt{3+4\cos(\pi k_{x})+2\cos(2\pi k_{x})}.$ (6)

Hence, from the solutions of the above equation we see that EP3 arises at
$\delta_{EP}=\pm 3$ and $\delta_{EP}=\pm 1$ for the $\Gamma$ ($k_{x}=0$) and
$M$ ($k_{x}=1$) points respectively.

Motivated by our above findings, we next add the inversion breaking Semenoff
mass term for setting up the complete dice-Haldane lattice model with the
Hamiltonian $H=H_{nn}+H_{nnn}+H_{m}$. We note that the mass term ($+m$ on $A$
lattice sites and $-m$ on $C$ lattice sites) leads to a critical value of $m$
in the units of $t_{2}$ ($m=m^{*}=0.16$), where a gap-closing occurs at the
$K$ point while the $K^{\prime}$ point remains gapped. Away from this critical
$m$ value, the bands become gapped again. In particular, for $m<m^{*}$, one
lies in the non-trivial topological region of the phase diagram whereas, for
$m>m^{*}$ topologically trivial spectra are obtained. The $m=m^{*}=0.16$ point
corresponds to the semi-metallic phase associated with band gap closing only
at $K$ but not at $K^{\prime}$. Corresponding band diagrams have been detailed
in Appendix. A.

Figure 4:  Non-Hermiticity in the complete dice-Haldane lattice model. In the
topologically non-trivial region $m<m^{*}$, non-Hermiticity strength $\delta$
induces an EP of order three close to the $M$ point in the spectrum. Panel (a)
shows the real part of energy spectrum, panel (b) shows the imaginary part of
the energy spectrum, and panel (c) shows the scaling of the phase rigidity
around $\delta_{EP}=0.94$ at $k_{x}=1.07$. The different colours here
correspond to different eigenstates. The $r_{\alpha}$ for the two dispersive
bands overlap. Here, $t=1/\sqrt{2},t_{2}=0.06t,\phi=\pi/2$ and $m=0.06$.

We will next explore the effect of the non-Hermitian gain and loss in both the
topologically non-trivial and trivial phases. For this purpose, we first chose
a value of $m$ ($m=0.06$) that satisfies the $m<m^{*}$ criterion for being
topologically non-trivial. Further, we introduce and systematically vary
$\delta$ to investigate its effect on the complex energy band structure. We
find that the sole variation of non-Hermiticity strength $\delta$ can bring
about a gap-closed real energy spectrum. This gap closing takes place close to
$M$ point for $\delta\sim 1.0$ as presented in Fig. 4(a). Here, the imaginary
spectrum is also triply degenerate [Fig. 4(b)], which subsequently gaps out.
Hence, we find a third order EP at $k_{x}=1.07$ for $\delta_{EP}=0.94$. The
scaling of the phase rigidity around this EP is shown in Fig. 4(c), confirming
its nature. Furthermore, we choose $m$ ($m>m^{*}$) such that we start from the
topologically trivial phase and then invoke non-Hermiticity. In this condition
the $Re(E)$ spectra never undergo band closing, and thus EPs cannot emerge,
even for arbitrarily large values of $\delta$. Therefore, we have discovered
that inversion symmetry breaking in the dice Haldane model offers an EP near
the $M$ point only in the topologically non-trivial case. On the other hand,
an EP at $\Gamma$ point can be obtained primarily in the inversion-symmetric
conditions, i.e., $m=0$. For better understanding, the complete phase diagram
for the emergence of EPs at the $\Gamma$ and $M$ points in the parameter space
of the dice-Haldane model is presented in Fig. 5(a) and (b), respectively. The
regions in the parameter space where the phase rigidity approaches zero are
the regions where EPs can be found. We observe extended regions with phase
rigidity values very close to zero. This indicates that fine tuning of
parameters $\delta$ and $m/t_{2}$ is not required to obtain these exceptional
regions expanding the possibilities for reaching low values of phase rigidity
Li et al. (2022). The occurrence of EPs also signify a topological phase
transition as these are regions of band-gap closing. It is important to note
that these phase transitions are possible only at low values of $m$
($m<m^{*}$). This follows from our prior observation that gap closings cannot
occur solely due to non-Hermiticity unless we are in the topologically non-
trivial region of the Hermitian dice-Haldane model. In contrast to the
conventional Haldane model, the topological phase transition is now driven by
a complex mass term. In particular, the edge states that usually occur in the
Hermitian topological phase of Haldane model are also observed in the presence
of gain and loss ($\delta$). However, the topological protection of the edge
states in the $\mathcal{PT}$ symmetry broken phase holds only up to a critical
value of $\delta$. This phase transition is associated with passing through a
third-order EP. On the other hand, if we choose the value of mass $m$ outside
the topologically non-trivial region of the Hermitian case, there is no
possible value of $\delta$ that will manifest in protected edge states and
will trace the system back into the topological region. In other words, to
obtain EPs at any finite, non-zero value of $\delta$, we require the Hermitian
system to be placed initially within the topological region. Tuning $\delta$
can bring about a topological phase transition enabling the occurrence of EPs
in the non-Hermitian model.

Figure 5:  Phase diagram showing the occurrence of EPs as a function of the
model parameters. Panel (a) shows the phase rigidity at the $\Gamma$ point
while panel (b) corresponds to the $M$ point. Zero values of phase rigidity
imply the existence of third-order EPs, at the corresponding parameter values.
Note the extended region in the parameter space with phase rigidity values
very close to zero. Here, $t=1/\sqrt{2}$ and $\phi=\pi/2$.

Now, we will invoke a different class of non-Hermiticity – non-reciprocal
hopping along one direction. In particular, we have assigned $t+\gamma$
($t-\gamma$) to the hopping parameters $C\rightarrow B$ ($B\rightarrow C$) and
$B\rightarrow A$ ($A\rightarrow B$) in the vertical direction in Fig. 1(a).
Even in the absence of the non-Hermitian gain and loss we observe that non-
reciprocal hopping solely can induce EPs in the system at specific strengths
of non-Hermiticity $\gamma$. The deformation of the electronic bands under
this non-reciprocity is qualitatively similar to our previous results with
non-Hermitian gain and loss. However, the critical values of $\gamma$ at which
the EPs occur differ from the critical $\delta$ we found in the above
discussion. For example, in the case with only nearest neighbour hopping EPs
are induced at critical $\gamma$ values of

$\gamma_{EP}=\pm\sqrt{\frac{3}{2}+2\cos(\pi k_{x})+\cos(2\pi k_{x})}.$ (7)

Therefore, from Eqn. 6 it is clear that $\gamma_{EP}$ values are related to
the corresponding $\delta_{EP}$ values by the relation
$\gamma_{EP}=\delta_{EP}/\sqrt{2}$. Numerically, we have verified the
occurrence of EP3 at $\gamma_{EP}=\pm\frac{3}{\sqrt{2}}$ and
$\gamma_{EP}=\pm\frac{1}{\sqrt{2}}$ for the $\Gamma$ and $M$ points
respectively. It is interesting to study this non-reciprocal hopping in the
context of the dice-Haldane nanoribbon to see the effects of OBC [see Section
IV(B)].

##  IV Finite-size effects: Dice-Haldane nanoribbon

Having understood the effect of non-Hermiticity in the $k$-space model of the
dice lattice sheet, we next move on to the study of another physically
important case of the dice-Haldane nanoribbon extended along one direction
(say $x$). The schematic of the finite size nanoribbon considered here is
shown in Fig. 6. Here $L_{x}$ and $L_{y}$ are the dimensions of the nanoribbon
in the $x$ and $y$ directions, respectively. We particularlt focus on a real
space model of the nanoribbon, where both $L_{x}$ and $L_{y}$ are finite. The
real space model with open-boundary conditions in both directions leads to
interesting consequences when we invoke non-Hermiticity as we will discuss
shortly.

Figure 6:  Schematic of the dice-Haldane nanoribbon with non-reciprocal
hopping. The hopping and on-site parameters remain the same as in the dice-
Haldane sheet case. An additional kind of non-Hermiticity $\gamma$ has been
introduced in the nanoribbon, namely a non-reciprocal hopping. This favours
nearest-neighbour hopping from right to left along the $x$ direction rather
than from left to right.

Considering the Hermitian dice-Haldane model, we can expect that in the
topologically non-trivial phase, each edge of the lattice will exhibit two
chiral edge states since the Chern number is $\pm 2$. These edge modes lie in
the band gap and connect two bulk bands. In contrast to the honeycomb lattice,
the spectrum exhibits two unidirectional chiral states per edge for $m<m^{*}$
that cross over from the bulk states near the Fermi level. However, for
$m>m^{*}$, the edge states near the flat band are counter-propagating at a
given edge. Hence, no net current will flow through it. Consequently, the bulk
states remain gapped out, and the bulk boundary correspondence continues to
hold. Corresponding figures and a discussion can be found in Appendix. C.

It is important to note that these edge states are quite robust to both real
and complex on-site disorder, i.e., despite some disorder-induced distortion
in the shape of the bands, the edge states persist up to large values of
disorder. We have checked this for a disorder of the form $\Delta_{j}$ on each
lattice site, where $\Delta_{j}$ is allowed to be real or imaginary,
corresponding to real and imaginary on-site disorder, respectively. Here, $j$
denotes the lattice site and $\Delta_{j}=\Delta\omega_{j}$ where
$\omega_{j}\in[-1,1]$. The disordered Hamiltonian has the form
$H=H_{nn}+H_{nnn}+H_{m}+H_{dis}$ where,

$H_{dis}=\sum_{j}\Delta_{j}c_{j}^{\dagger}c_{j}.$ (8)

We will study the effect of this complex random on-site disorder in more
detail when we introduce non-Hermiticity in the finite nanoribbon geometry. We
address the interplay of non-Hermiticity and disorder in Section. IV.2.

###  IV.1 Nanoribbon with non-Hermitian gain and loss

We first consider the effect of non-Hermitian balanced gain and loss in the
dice-Haldane nanoribbon. The Hamiltonian under such considerations is given by
$H=H_{nn}+H_{nnn}+H_{m}+H_{\delta}$. It is worth noting that the topological
edge states found in the Hermitian regime, forming conducting channels between
the conduction and valence bands are robust even in the presence of non-
Hermiticity. For a range of increasing values of $\delta$, up to a system
dependent critical value $\delta_{c}$, the topological edge states can be
clearly discerned from the energy band diagram. A detailed discussion can be
found in Appendix. C.

Figure 7:  Spectra and LDOS with non-Hermitian balanced gain and loss. Panels
(a) and (b) show the complex energy spectrum for the torus (PBC) geometry of
the nanoribbon and panels (c) and (d) show the respective LDOS for the same
system under OBC. Panels (a) and (c) correspond to the dice lattice
($t_{2}=0$) with non-Hermitian balanced gain and loss. The energy spectrum in
(a) does not enclose a finite, non-zero area. This implies that under OBC
there will be no occurrence of NHSE. The corresponding LDOS of the system
under OBC shown in (c) demonstrates the absence of skin effect. Panels (b) and
(d) correspond to the full dice-Haldane lattice with gain and loss. Here,
$t_{2}=0.06t$. The spectrum shown in (b) encloses a finite non-zero spectral
area indicating the possibility of an NHSE under OBC, which is established by
the LDOS shown in (d). In this case, the localization of states occur at both
the top and bottom edges of the nanoribbon. Here, $t=1/\sqrt{2},\phi=\pi/2$,
$m=0$ and gain and loss strength $\delta=2.0$.

One of the striking features of non-Hermitian systems has been the discovery
of NHSE and it is interesting to understand whether our proposed system
exhibits this feature. It may be noted that these fascinating phenomena unique
to non-Hermitian systems, such as non-Bloch EPs and skin effects, can be well
explained in terms of the generalized Brillouin zone (GBZ) formalism Song et
al. (2019); Yang et al. (2020). We note that it has been established that a
two-dimensional system under OBC can exhibit NHSE if and only if under PBC the
complex eigenspectrum encloses a finite non-zero spectral area Zhang et al.
(2022). We consider a periodic version of the system, i.e., a dice-Haldane
torus and invoke balanced gain and loss. When we consider the dice lattice
with only nearest neighbour interactions under PBC, the complex eigenspectrum
of the system has an arc-like structure and does not enclose any finite
spectral area in the complex plane, as shown in Fig. 7(a). This indicates the
absence of an NHSE when we invoke OBC. In order to probe this, we calculate
the local density of states (LDOS) for the system under OBC, i.e., the finite
nanoribbon. To calculate the LDOS, we evaluate
$\sum_{\alpha}|\psi_{\alpha}(x_{i})|^{2}$ at each lattice site ($x_{i}$) which
gives us LDOS($x_{i}$). The plot of the corresponding LDOS for the dice
lattice in Fig. 7(c) verifies that NHSE is indeed absent and the states are
distributed over the lattice. However, when we consider the full dice-Haldane
periodic system ($t_{2}\neq 0$) with balanced gain and loss, the complex
spectrum does enclose a finite area [shown in Fig. 7(b)]. This translates to
the occurence of NHSE when OBC is invoked. Remarkably, the localization of the
eigenstates occurs at both the top and bottom edges of the nanoribbon which
can be seen fom the plot of the LDOS in Fig.7(d). So, non-Hermitian gain and
loss is able to cause an NHSE in the dice-Haldane model and not in the dice
model which accounts for only nearest neighbour hopping. The nature of the
skin effect occurring only at the top and bottom edges of the dice-Haldane
system [as shown in Fig. 7(d)], can be explained further through the winding
number and the complex energy spectra by imposing OBC in one direction while
retaining PBC in the other. A detailed discussion and corresponding figures
can be found in Appendix. D. For all cases of our computations the dice-
Haldane nanoribbon has 72 $\times$ 36 sites ($n=2592$), unless stated
otherwise.

###  IV.2 Nanoribbon with non-reciprocal hopping

Figure 8:  Effect of disorder on the spectra and LDOS for the dice-Haldane
lattice with non-reciprocal hopping. Panels (a), (b) and (c) show the complex
spectra of the dice-Haldane model under PBC with non-reciprocal hopping.
Panels (d), (e) and (f) show the corresponding LDOS for the same system under
OBC. Panels (a) and (d) show the disorder-free system. Here, the complex
eigenspectrum [shown in (a)] encloses a non-zero spectral area which
translates to a skin effect under OBC. The corresponding LDOS [shown in (d)]
depicts an NHSE, with states accumulating close to $x=1$, which gradually
decreases as we move rightward. Panels (b) and (e) correspond to disorder
strength $\Delta=1$. Here, the spectral area [shown in (b)] is still non-zero
and finite, implying an existence of NHSE, which is established by the LDOS in
(e). For panels (c) and (f) where $\Delta=10$, the spectral area has
disappeared indicating the destruction of NHSE. The corresponding LDOS shown
in (f) attributes to the same where the skin effect has completely disappeared
due to localization in the bulk. Here, $t=1/\sqrt{2},t_{2}=0.06t,\phi=\pi/2$,
$m=0$ and non-reciprocity strength $\gamma=2.0$.

As we analyzed previously for the periodic dice-Haldane sheet, in this
section, we consider the dice-Haldane nanoribbon and study the effect of non-
reciprocal hopping. In particular, non-reciprocal nearest-neighbour hopping is
introduced only along the $x$ direction. We have the hopping values $t-\gamma$
along $+x$ and $t+\gamma$ along $-x$ directions, i.e., we have a biased
hopping strength that favours hopping from right to left rather than from left
to right. We invoke this non-reciprocal hopping throughout the bulk of the
nanoribbon [See Fig. 6]. We study the spectrum and LDOS of the system with
this type of non-Hermiticity and find strikingly different behaviour than in
the case of balanced gain and loss. Unlike in the prior case, when we
introduce a non-reciprocal hopping, i.e., $\gamma\neq 0$, the PBC spectrum
covers a finite area in the complex plane even in case of the dice model with
$t_{2}=0$, which implies an NHSE under OBC. In fact, the effect of non-
reciprocal hopping on the dice model is qualitatively the same as its effect
on the dice-Haldane lattice. We will study the latter in detail and also
discuss the effect of disorder for this case. The spectrum of the dice-Haldane
model under PBC with non-reciprocal hopping accommodates a finite spectral
area in the complex plane shown in Fig. 8(a). This indicates the possibility
of an NHSE when we impose OBC in the system. Next, we investigate the effect
of disorder on this spectral area where the nature of disorder has been taken
according to Eqn. 8. When the disorder strength is taken to be $\Delta=1$, the
complex spectrum undergoes some disortion but still accommodates a finite area
[Fig. 8(b)]. However, at a large disorder strength ($\Delta=10$) shown in Fig.
8(c) the spectral area disappears. This indicates that under OBC, the NHSE
will gradually get destroyed due to the introduction of disorder. Next, we
investigate the behaviour of the LDOS to verify the above findings and to
visualise the occurrence and subsequent disappearance of the NHSE under
disorder.

In Fig. 8(d) the LDOS can be seen to be higher around small values of $x$ and
decreases as we go to higher $x$. This implies a maximal concentration of
eigenstates near the left edge of the system. Thus, non-reciprocal hopping,
when invoked throughout the bulk of the nanoribbon, causes an NHSE. It is
important to note that the NHSE in this case is different from that caused by
gain and loss. Here, the localization of the eigenstates is at one edge (left)
of the lattice and is also directionally different than in the previous case
where NHSE occurred at the top and bottom edges. In the former case of gain
and loss, the inclusion of the Haldane next-nearest neighbour hopping is
essential for realizing the skin effect. Here, the staggered magnetic flux in
the presence of non-Hermitian gain and loss introduces chiral edge currents in
two different directions for the two distinct sublattices A and C Cai et al.
(2019). Consequently, the eigenmodes are localized at the top and bottom edges
of the ribbon under OBC. In contrast, the non-reciprocal hopping along one
direction offers a directionally-biased propagation of eigenstates, causing
localization at the left edge. We find that the LDOS does not vary
continuously from high to low as we move along $+x$ but shows regions of high
value followed by those of lower value. This feature is due to the missing
hopping terms between sublattices A and C, which inhibits the complete flow of
the eigenstates leftwards. This can be pictured from the schematic in Fig. 6,
which suggests that the accumulation of states will be greater on sublattice C
and gradually decrease towards the following A lattice site. Now, we look at
the effect of disorder on the NHSE. Upon increasing the value of disorder
strength, there occurs a localization of the eigenstates, as shown in Fig. 8(e
and f). In Fig. 8(e) corresponding to $\Delta=1$ the concentration of
eigenstates at the left edge has decreased, signifying a partial destruction
of the NHSE. Finally, at large values of disorder ($\Delta=10$), a complete
localization of the eigenfunctions is found, causing a low value of the LDOS
over all $x$, implying the complete destruction of the NHSE [Fig. 8 (f)].

Figure 9:  IPR and edge probability with non-reciprocal bulk. The IPR has been
plotted for all eigenstates as a function of $Re(E)$ and the corresponding
colour denotes its probability density at the left edge of the nanoribbon.
Panel (a) corresponds to $\Delta=0$, where the IPR is very high for the states
localized at the edge and hence establishes the occurrence of NHSE. In panel
(b) the IPR values have diminished, implying the reduction of skin effect due
to disorder. Here, disorder strength $\Delta=1$. Panel (c) corresponding to
$\Delta=10$, shows very high values of IPR but very low edge population
implying the occurrence of disorder-induced bulk localization and complete
destruction of NHSE. A zoomed in plot is shown in the inset. Here, the IPR and
edge probability have been disorder averaged over 1000 configurations. The
number of hexagonal layers in the $y$ direction has been taken to be 35. We
have set $t=1/\sqrt{2},t_{2}=0.06t,\phi=\pi/2,m=0,\delta=0,\gamma=2$.

To consolidate the above arguments, we next study the behaviour of the IPR
Kramer and MacKinnon (1993) and the probability density of the eigenstates at
the edge after averaging both the quantities over multiple disorder
configurations. We averaged over 1000 disorder configurations. The IPR for the
$\alpha$-th eigenstate, $I_{\alpha}$, is defined as

$I_{\alpha}=\frac{\sum_{r}\absolutevalue{\psi_{\alpha}(r)}^{4}}{\left(\sum_{r}\absolutevalue{\psi_{\alpha}(r)}^{2}\right)^{2}}.$
(9)

For localized states, IPR is close to 1 while for extended states IPR is very
low. We further define the edge probability, $P_{\alpha}$, of state
$\psi_{\alpha}$ as

$P_{\alpha}=\frac{\sum_{x_{i}=1}^{x_{E}}\absolutevalue{\psi_{\alpha}(x_{i})}^{2}}{\sum_{x_{i}}\absolutevalue{\psi_{\alpha}(x_{i})}^{2}},$
(10)

where, $x_{E}$ represents the width of the edge, which we take as the first
five lattice sites from the left end of the ribbon ($x_{E}=5$). In the case of
the Hermitian dice-Haldane nanoribbon the dispersive bands are delocalized
with very low IPR and there is no skin effect as expected. In presence of non-
reciprocity in the bulk ($\gamma=2$) we observe IPR $\rightarrow$ 1 for the
eigenstates localized at the edge. This can be seen from Fig. 9(a) where the
edge probability is denoted by the colour bar showing that the states with
high $P_{\alpha}$ correspond to high IPR. Hence, for the disorder-free case
with non-reciprocal bulk we can confirm the occurrence of NHSE. Next, we look
at the effect of disorder on the skin effect – the presence of disorder
essentially reduces the edge localization in the system. This can be discerned
from the diminished values of the IPR in Fig. 9(b), presented for disorder
strength $\Delta=1$. Yet, the higher values of IPR are predominantly
contributed by the eigenstates near the edge. Further, for large disorder
strength ($\Delta=10$), shown in Fig. 9(c), the IPR is uniformly high,
although $P_{\alpha}$ is very low for all the eigenstates. This corresponds to
the disorder induced bulk localization and hence the complete destruction of
NHSE.

Finally, we also note that when we invoke non-reciprocal hopping only along
the upper and lower edges of the system and not in the bulk, surprisingly,
here too there is the occurrence of the NHSE which is similarly destroyed at
large values of disorder.

##  V Summary and discussion

In this work, we have systematically studied the effect of non-Hermiticity in
the Chern insulating dice-Haldane lattice.

We introduced non-Hermiticity in this model in two ways: (i) using balanced
gain and loss terms and (ii) setting non-reciprocal hopping parameters.
Introducing non-Hermiticity through any of the above means invariably causes
higher-order exceptional points. Our analytical description revealed that the
exceptional points at high symmetry points emerge at odd integer values of the
gain and loss non-Hermiticity strength and at $1/\sqrt{2}$ times the previous
values in the case of non-reciprocal hopping. Further, we showed that the
dice-Haldane lattice consisting of complex next-nearest neighbour hopping and
the Semenoff mass offers a rich topological phase diagram. The robustness of
the topological edge states was critically examined with non-Hermiticity and
complex disorder. Moreover, we discover that, unlike the gain and loss case,
the non-reciprocal hopping triggers a fascinating non-Hermitian skin effect
under OBC for the dice lattice with only nearest neighbour couplings. However,
both kinds of non-Hermiticity can cause NHSE in the more general dice-Haldane
nanoribbon. Remarkably, the NHSE caused by gain and loss generates
localization at the top and bottom edges while non-reciprocity results in an
NHSE at the left edge of the nanoribbon. The directionality of localization of
maximal eigenstates can hence be tuned using the nature of non-Hermiticity and
its strength. The skin effect is protected by a finite spectral area in the
complex plane under PBC in real space. Furthermore, the LDOS, IPR, and edge
probability calculations also demonstrate the occurrence of the skin effect
and its robustness to the disorder.

Our study is fundamental to understanding the tunability of the dice-Haldane
model under the influence of non-Hermiticity, especially in the context of EPs
which have been experimentally realized in microwave cavity resonators
Dembowski et al. (2001b) and coupled electronic circuits Stehmann et al.
(2004). Specifically, EPs of order three have been realized in coupled
acoustic cavity resonators Ding et al. (2016a) and optical cavity systems
Hodaei et al. (2017). It would be interesting to engineer already fabricated
dice lattices to introduce non-Hermitian gain and loss or non-reciprocal
hopping to obtain these higher-order EPs.

The key ingredients for attaining a dice lattice in cold atomic systems are
three pairs of counterpropagating lasers placed at an angle of $120^{\circ}$
with respect to each other Bercioux et al. (2009); Möller and Cooper (2012).
This laser setup essentially divides the 2D plane into six equivalent parts.
Further, the interference causes standing waves that give rise to the required
potential traps of the optical lattice. In particular, a dice lattice with
lattice constant $a_{0}$ can be constructed by using six linearly polarized
laser beams of wavelength $\lambda=3a_{0}/2$. Another plausible pathway for
fabricating dice lattices is the use of coupled resonators Zhang and Jin
(2020). The prescription is the incorporation of additional resonators at the
center of hexagonal rings of the honeycomb lattice. The ring-shaped primary
resonators of the dice lattice are effectively connected with each other via
auxiliary resonators placed in between. The Haldane model has been
experimentally realized in optical lattices using ultracold atoms Jotzu et al.
(2014). Notably, the time-reversal symmetry can be broken through complex
next-nearest-neighbor tunnelling induced by circular modulation of the lattice
position in time. Additionally, the deformation of lattice geometry by
applying unidirectional in-plane force with the help of a magnetic field
gradient provides an energy offset and breaks the inversion symmetry.

Non-Hermiticity has been successfully engineered into several optical
lattices, acoustic systems and topoelectrical circuits Ding et al. (2016b,
2018); Zhang and Franz (2020); Dong et al. (2021). In particular, the
inclusion of non-Hermiticity in the three-site Lieb lattice using coupled
optical waveguides Xiao et al. (2020b) can be feasibly extended to our lattice
system. It has been established that optical lattices fabricated using
femtosecond-direct-laser-writing can adduce non-Hermitian gain and loss
through periodic ‘breaks’ in the waveguides, which lead to loss of radiation
modes. This loss can be tuned using the length of the breaks.Cerjan et al.
(2019). This method of engineering gain and loss has also been successfully
realized in a graphene-like honeycomb lattice Rechtsman et al. (2013).
Further, it has been proposed that atomic loss in ultracold atomic gas systems
can be generated using a resonant optical beam to kick the weakly trapped
atoms or by using a radio frequency to excite the atom to an irrelevant state,
thereby simulating lossXu et al. (2017). On-site gain and loss can be
effectively mapped onto a non-Hermiticity-controlled coupling between
neighbouring atoms. A synthetic imaginary gauge field engineered strategically
can make these couplings asymmetric Longhi et al. (2015). Such complex gauge
potentials causing non-reciprocal hopping can be implemented using a non-
Hermitian anti-resonance ring Midya et al. (2018). Due to the directional
coupling, the photons become attenuated or amplified depending on their
direction of travel. Furthermore, two-dimensional non-Hermitian systems with
gain and loss or non-reciprocity have been proposed in classical
topoelectrical circuits Zhang and Franz (2020); Dong et al. (2021), where the
non-Hermiticity can be ingeniously engineered using combinations of
resistances and LC-tanks. Information about the eigenenergies can be extracted
from the electrical response, admittance and impedance resonances Lee et al.
(2018). In light of the above rapid experimental advances, we believe our
theoretical findings can be experimentally tested in the near future.

## ACKNOWLEDGEMENTS

R.S. acknowledges Indian Institute of Science for financial support. A.B.
thanks the IoE postdoctoral fellowship for support. R.S. and A.B. would also
like to thank Ayan Banerjee for several useful discussions. A.N. acknowledges
support from the start-up grant (SG/MHRD-19-0001) of the Indian Institute of
Science and DST-SERB (project number SRG/2020/000153).

##  Appendix A Hermitian dice-Haldane band diagrams

Figure 10:  Energy spectra of the Hermitian model. The band structure of the
Hermitian dice-Haldane model along high symmetry points $M-K^{\prime}-\Gamma-
K-M$. The upper panels show the three-dimensional spectrum as a function of
$k_{x}$ and $k_{y}$, while the lower panels show the corresponding two-
dimensional plots at $k_{y}=0$. Panels (a) and (b) show the effect of only
nearest-neighbour hopping with $t_{2}=m=0$. Here, all three bands are gapless
at the $K$ and $K^{\prime}$ points. In (c) and (d) next-nearest neighbour
hopping has been included. Here, $t_{2}=0.1,\phi=\pi/2$ while $m=0$. In (e)
and (f) $m=0.3$ while $t_{2}=0$ shows the effect of the Semenoff mass term.
Both $t_{2}$ and $m$ open up a gap in the spectrum. For all plots the value of
nearest neighbour hopping is chosen to be $t=1/\sqrt{2}$. Figure 11:  Effect
of the Semenoff mass in the Hermitian model with both $t$ and $t_{2}.$ In (a)
$m=0.06$, in (b) $m=0.16$ and in (c) $m=0.30$. An arbitrarily small Semenoff
mass opens up a gap in the spectrum which subsequently closes in (b)
indicating a semi-metallic phase. On increasing the value of $m$ further, all
bands become gapped again. This critical value of $m$ is $m^{*}=0.16$ which
separates the topologically non-trivial and trivial regions (a) and (c),
respectively. For all plots $t=1/\sqrt{2},t_{2}=0.06t$ and $\phi=\pi/2$.
Figure 12:  Energy spectra of the different phases in the Hermitian model. The
band structure of the Hermitian dice-Haldane model along high symmetry points
$M-K^{\prime}-\Gamma-K-M$. The upper panels show the three-dimensional
spectrum as a function of $k_{x}$ and $k_{y}$, while the lower panels show the
corresponding two-dimensional plots at $k_{y}=0$. Panels (a) and (b) show the
all-gapped (AG) phase with $\phi=\pi/2$ and $m=0$. Here, all three bands are
gapped for all values of $k_{x}$ and $k_{y}$. Panels (c) and (d) show the
valence-gapped (VG) phase where the conduction and flat bands are gapless
while the valence band remains gapped. Here, $\phi=0$ and $m=0.15$. In (e) and
(f) $m=0.15$ and $\phi=\pi$ which shows the conduction-gapped (CG) phase,
where the conduction band is gapped while the valence and the flat bands are
gapless. For all plots the values of $t=1/\sqrt{2}$ and $t_{2}=0.06t$.

We perform a Fourier transform of the Hamiltonian given in Eqn. 1 and Eqn. 2
in order to obtain the energy band diagram in the two-dimensional momentum
space. To encompass all the relevant physics of the model, we choose a
symmetry path $M(1,0)-K^{{}^{\prime}}(-2/3,0)-\Gamma(0,0)-K(2/3,0)-M(1,0)$ in
the BZ, which includes all the high-symmetry points. Note that the coordinates
of the symmetry points are given in units of $2\pi/a$, where $a$ is the
lattice constant of the dice lattice. Later, we will systematically invoke
non-Hermiticity in our model and study the physics around these high-symmetry
points. The dice lattice with only nearest neighbour interactions
($H_{nnn}=H_{m}=0$), exhibit two Dirac-like dispersive bands while a
dispersion-less flat band lies at the Fermi level as shown in Fig. 10 (a) and
(b). The Dirac points lie at the symmetry points $K$ and $K^{{}^{\prime}}$ of
the BZ. Next, the complex next nearest neighbour hopping term ($H_{m}=0$)
splits the Dirac cones as shown in Fig. 10 (c) and (d) resulting in non-
trivial topological band structures. On the contrary, only Semenoff mass term
($H_{nnn}=0$) induces a trivial or normal band gap in the system as given in
Fig. 10 (e) and (f). It is worth noting that both $H_{nnn}$ and $H_{m}$ open
up a gap in the energy spectrum even for arbitrarily small values of $t_{2}$
and $m$. The competing nature of $H_{nnn}$ and $H_{m}$ leads to the rich
topological phase diagram of the dice-Haldane model.

In particular, Fig. 11 illustrates the effect of the Semenoff mass on the
electronic band structure of the non-trivial dice-Haldane model. Even an
arbitrarily small non-zero value of $m$ opens up a band gap at the $K$ and
$K^{\prime}$ points of the BZ while maintaining topologically non-trivial
features [Fig. 11(a)]. With a further increase in $m$, for a critical value
$m=m^{*}$, the $K^{\prime}$ point becomes triply degenerate while the $K$
point develops no such band touching [Fig. 11(b)]. Beyond this point, for
$m>m^{*}$, the system becomes topologically trivial and the energy bands are
completely non-degenerate for all higher values of $m$ [Fig. 11(c)]. It is
interesting that the different phases of the Hermitian dice-Haldane model
exhibit qualitatively different energy band structures. The AG phase features
three non-degenerate bands as shown in Fig. 12 (a) and (b). The upper panel in
Fig. 12 shows the three-dimensional band structure as a function of $k_{x}$
and $k_{y}$, while the lower panel is a two-dimensional plot of the dispersion
relations along line ($M-K^{{}^{\prime}}-\Gamma-K-M$) joining the high
symmetry points of the BZ. The VG phase is shown in Fig. 12 (c) and (d) where
the valence band remains gapped while the flat band and conduction band are
degenerate at some points in the BZ. Fig. 12 (e) and (f) show the CG phase
where the conduction band is non-degenerate while the valence and flat bands
become gapless.

##  Appendix B Effect of gain and loss in the dice lattice model

Figure 13:  Spectra with nearest-neighbour hopping and non-Hermiticity. The
$Re(E)$ (upper panels) and $Im(E)$ spectra (lower panels) have been shown for
different values of non-Hermiticity strength $\delta$. In (a) and (b)
$\delta=0.5$ showing that the non-Hermitian gain and loss instantly makes the
$K$ and $K^{\prime}$ points of $Re(E)$ degenerate while the degeneracy is
lifted in $Im(E)$. In (c) and (d) $\delta=1.0$, this induces an EP at the $M$
point where both the $Re(E)$ and $Im(E)$ are simultaneously degenerate.
Similarly in (e) and (f) we find an EP at the $\Gamma$ point for $\delta=3.0$.
For all plots we have set $t=1/\sqrt{2}$. Figure 14:  Phase rigidity for
nearest-neighbour hopping with non-Hermiticity. Panel (a) shows the phase
rigidity of the eigenfunctions $r_{\alpha}$ as a function of non-Hermiticity
$\delta$, while panel (b) shows the scaling of the corresponding $r_{\alpha}$
around the concerned EP. Here, the plots have been shown for the $\Gamma$
point where an EP is induced at $\delta=3.0$. At this point,
$r_{\alpha}\rightarrow 0$ shown in (a). Panel (b) shows the logarithmic
scaling of $r_{\alpha}$ around $\delta_{EP}=3.0$ gives a slope of unity
implying it is an EP of order three. The different colours in both the plots
correspond to different eigenstates. The $r_{\alpha}$ for the two dispersive
bands overlap. For both plots we have set $t=1/\sqrt{2}$.

We consider the nearest neighbour hopping in the Hamiltonian, while keeping
$t_{2}$ and $m$ switched off. We introduce non-Hermiticity to the system and
study its effects as we vary the strength of gain and loss, $\delta$. In this
condition, the Hamiltonian has the form $H=H_{nn}+H_{\delta}$, where $\delta$
is tuned methodically. The band diagram for the Hermitian case, i.e., at
$\delta=0$, has been previously shown in Fig. 10(a) and (b). As we invoke gain
and loss, a complex dispersion relation appears even for arbitrarily small
values of $\delta$. Such a dispersion relation results in a complex eigenvalue
spectrum owing to the non-Hermitian nature of the Hamiltonian. Particularly,
the real part of the spectra in this condition become a single-sheeted
hyperboloid around the $K$ point, near which the imaginary part of the spectra
has a non-vanishing contribution as shown in Fig. 13(a) and (b). Here, the
appearance of complex energy spectra is exciting because the $PT$ operator
still commutes with the Hamiltonian. Therefore, the only explanation is that
the Hamiltonian and $PT$ symmetry operators do not possess the same set of
eigenvectors underpinned by the anti-linear nature of the $T$ operator.
Similar features have also been evinced for graphene in the presence of non-
Hermitian gain and lossKremer et al. (2019). This can be explicitly shown for
the dice lattice where the $\hat{P}$ operator maps A $\leftrightarrow$ C
lattice sites and is given by the following matrix:

$\hat{P}=\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\end{pmatrix},$ (11)

and $\hat{T}$ is the antiunitary complex conjugation operator for our spinless
system. It can be shown that the commutation relation $[\hat{P}\hat{T},H(k)]$
is invariably zero for arbitrary values of $\delta$. On the other hand, the
eigenvectors of $\hat{P}\hat{T}$ and $H$ operators, particularly shown below
for the $\Gamma$ point, are clearly distinct.

$\hat{P}\hat{T}\rightarrow\begin{bmatrix}-1\\\ 0\\\
1\end{bmatrix},\begin{bmatrix}1\\\ 0\\\ 1\end{bmatrix},\begin{bmatrix}0\\\
1\\\ 0\end{bmatrix}.$ (12)

$H\rightarrow\begin{bmatrix}-1\\\ a\\\
1\end{bmatrix},\begin{bmatrix}a\eta_{-}\\\ \eta_{-}\\\
1\end{bmatrix},\begin{bmatrix}a\eta_{+}\\\ \eta_{+}\\\ 1\end{bmatrix},$ (13)

where $\eta_{\pm}=a\pm\sqrt{2+a^{2}}$ and $a=\frac{\sqrt{2}}{3}i\delta$.
Hence, despite $PT$ symmetry of the system, the eigenstates correspond to the
broken-$PT$ phase.

A further increase in $\delta$ extends the degeneracy of $Re(E)$ to $M$, while
the degeneracy at this point in $Im(E)$ is lifted as presented in Fig. 13(c)
and (d). Most interestingly, this critical transition at the $M$ point occurs
exactly at $\delta=1.0$. At this value of $\delta$, we find that the three
eigenvalues and eigenfunctions coalesce at the $M$ point, giving rise to a
third order EP. As we increase $\delta$ further, the branches of $Re(E)$
becomes degenerate while the degeneracy is lifted in $Im(E)$ at the
corresponding $k_{x}$ values. At $\delta=3.0$, we find the coalescence of the
three eigenvalues and eigenfunctions at the $\Gamma$ point [Fig. 13(e) and
(f)]. This is again a third order EP, however appearing at a different point
in the BZ. The phase rigidity and its scaling for this EP are shown in Fig.
14. Fig. 14(a) shows the variation of phase rigidity $r_{\alpha}$ of the
eigenvectors as a function of $\delta$. $r_{\alpha}\rightarrow 0$ as
$\delta\rightarrow\delta_{EP}=3.0$. The scaling of the phase rigidity has been
plotted on a logarithmic scale in Fig. 14(b) whose slope is one, denoting that
the higher order EP is indeed of order three.

##  Appendix C Non-Hermiticity and Edge States in the dice-Haldane nanoribbon

Figure 15:  Effect of gain and loss on the edge states of the nanoribbon.
Here, the nanoribbon has been considered to be finite in the $y$ direction
while we have imposed PBC along $x$. Panels (a) and (b) correspond to the
Hermitian nanoribbon. Panel (a) shows the real spectrum for $m=0.06$ and panel
(b) shows the spectrum for $m=0.3$. The spectra have been plotted along high
symmetry lines $X-\Gamma-X$. In the topologically non-trivial region we find
conducting edge states in the spectrum [shown in panel (a)], while in the
topologically trivial region [shown in panel (b)] no non-trivial edge states
are present. Panels (c) and (d) show the persistence of edge states in the
nanoribbon with non-Hermitian gain and loss while in the topologically non-
trivial region ($m=0.06$). The edge states are robust to $\delta$ up to a
critical value of $\delta_{c}\approx 1$, after which they disappear. (c) The
presence of edge states in the real energy spectra can be seen for
$\delta=0.5<\delta_{c}$. (d) The edge state is absent for
$\delta=1.5>\delta_{c}$, as expected. The corresponding imaginary spectra for
(c) and (d) have been shown in (e) and (f) respectively. The system with non-
Hermiticity has a finite non-zero $Im(E)$ yet the real part of the spectra can
still accommodate edge states. Here, the number of hexagonal layers in the $y$
direction has been taken to be 70. We have chosen $t=1/\sqrt{2}$,
$t_{2}=0.06t$, and $\phi=\pi/2$.

Topological zero energy modes which are found in the non-trivial region of the
Hermitian dice-Haldane model have been shown in Fig. 15 (a), characterised by
the linear zero-energy crossings from the conduction to the valence band. The
zero modes dissapear as one moves into the topologically trivial region of the
phase diagram, as shown in Fig. 15(b). Fig. 15(a) and (b) show the spectrum
close to the Fermi level ($E=0$) along the line joining high symmetry points:
$X-\Gamma-X$.

Sitting in the topologically non-trivial region, we have studied the energy
spectrum for a varying non-Hermiticity strength, $\delta$. We found that the
existing edge states are robust to values of $\delta$ up to $\delta_{c}=0.8$
for the energy scale of our system, after which the edge states cannot be
discerned from the bulk states. This can be seen in Fig. 15(c), where we still
decipher clear edge states for $\delta<\delta_{c}$ and subsequently the edge
states disappear for $\delta>\delta_{c}$ [Fig. 15(d)]. Fig. 15(e) and (f) show
the imaginary spectra corresponding to Fig. 15(c) and (d) respectively. We can
see that the spectra develops finite non-zero $Im(E)$ as a result of
introducing non-Hermiticity into the nanoribbon, yet the real spectra can
accommodate topological edge states upto $\delta_{c}$.

It is well established that the bulk boundary correspondence in a system
generally breaks down due to the presence of non-Hermiticity Helbig et al.
(2019); Xiong (2018); Lee (2016); Xiao et al. (2020c); Bergholtz et al.
(2021b). As a consequence, the Chern number of the momentum space bulk
Hamiltonian fails to predict the existence of topological edge states
correctly. In principle, a topological invariant may be calculated by taking
into account the GBZ formalism Yang et al. (2020). However, for a two-
dimensional, non-trivial model, it is a challenging task to deduce the GBZ. An
alternative approach has been explored in Ref. Reséndiz-Vázquez et al. (2020),
where a finite two-dimensional topological Haldane lattice has been
constructed with arbitrary edge types. In these finite two-dimensional
systems, edge states appear even if the eigenvalues are not entirely real due
to the presence of gain and loss. Moreover, in this $\mathcal{PT}$ symmetry
broken phase, the topological protection has been determined by the number of
edge states that remain within the dissipation- or amplification-free region.
The absence of back-scattering, hence the topological phase, has been
confirmed by calculating the time evolution of an edge state. As expected, the
number of real edge states essentially depends on the strength of gain and
loss ($\delta$), which drives the topological transition. We have used a
similar formalism for our dice-Haldane model to calculate the critical values
of $\delta$ ($\delta_{c}$) corresponding to the topological protection. The
numerical value of $\delta_{c}$ has been obtained from the maximum value of
gain or loss for which at least 5% of the edge states remain dissipation- or
amplification-free. We note that the value of $\delta_{c}$ for the dice-
Haldane system varies with system size similar to the conventional Haldane
model. For example, $\delta_{c}=0.8$ for a smaller lattice size, say $n=288$.
However, for the system with sufficiently large number of lattice sites
(checked for $n=1152$ and $n=5220$), the critical value of $\delta_{c}$
saturates to $\approx 1$ as depicted in the Fig. 16. Fig.15(c) and (d)
illustrate the effect of gain and loss strength on the real part of the dice-
Haldane nanoribbon band structure in momentum space. Here, under the time-
reversal symmetry broken condition, the Semenoff mass term is chosen as
$m=0.06$, ensuring the non-trivial topological phase of the Hermitian system
to start with. We observed that the non-Hermiticity in terms of gain and loss
above the critical value essentially destroys the topologically protected edge
states in the dice-Haldane lattice.

Figure 16:  Real and imaginary parts of energy for the dice-Haldane nanoribbon
with gain and loss. $Re(E)$ is shown in red and $Im(E)$ is shown in blue.
Panel (a) shows $Re(E)$ and $Im(E)$ for gain and loss strength $\delta=0.5$.
Panel (b) corresponds to $\delta=1.5$. The other parameters for the plots are
$t=1/\sqrt{2}$, $t_{2}=0.06t$, $m=0.0$, and $\phi=\pi/2$. The total number of
lattice sites $n=5220$. The number of edge states that remain in the
dissipation- or amplification-free region is reduced with increasing gain and
loss strength.

##  Appendix D Geometry dependent skin effect under non-Hermitian gain and
loss

The finite spectral area of the dice-Haldane torus (PBC in both directions) in
the complex plane cannot solely explain the appearance of skin effect only on
the top and bottom edges. The nature of skin effect shown in Fig. 7(b) and (d)
under non-Hermitian gain and loss resembles the geometry-dependent skin effect
mentioned in Ref. Zhang et al. (2022). To illustrate, we calculate the Zak
phase Zhang and Song (2019) and the corresponding winding number ($W$) of the
filled bands Banerjee et al. (2022) along different directions, supporting our
findings of the skin effect. Additionally, we elucidate how the complex energy
spectrum can also be used to predict NHSE in the top and bottom edges
($y$-direction) in Fig. 7(b) and (d) rather than along the length
($x$-direction), which can occur on imposing non-reciprocal hopping [Fig. 8].
For that purpose, we consider the following two cases –

Case1: We impose PBC along the $y$ direction while we have OBC along $x$. The
energy spectrum in the complex plane of this nanoribbon under gain and loss
shows a loop-like structure enclosing a finite spectral area. This implies a
skin effect when OBC is imposed along the $y$ direction, i.e., along the top
and bottom edges of the sheet. To confirm this observation, we have calculated
the Zak phase and the corresponding winding number. We find that the filled
bands contribute to $W=1$ [See: Fig. 17(a)] as per expectation.

Case 2: We consider the ribbon to be periodic in the $x$ direction and open
along $y$. The spectrum in this case has an arc-like structure in the complex
plane, which does not enclose a finite spectral area. This implies that on
imposing OBC in the $x$ direction, skin effect will not occur at the right or
left edges. In this geometry we find $W=0$ for the filled bands. [See: Fig.
17(b)].

Figure 17:  Complex energy spectra and winding number with gain and loss in
the dice-Haldane ribbon under two different orientations. Panel (a) shows the
complex energy spectrum of the ribbon with PBC along $y$ and OBC along $x$.
The spectrum encloses a finite spectral area which implies the existence of a
skin effect in the $y$ direction when one imposes OBC. The skin effect has
been confirmed by the winding number, $W=1$. Panel (b) shows the complex
spectrum of the ribbon with PBC along $x$ and OBC along $y$. The spectrum
shows an arc-like structure with no enclosed area. This signifies that no skin
effect will occur along the $x$ direction when OBC is imposed. The absence of
skin effect in this case has been confirmed by the winding number, $W=0$. For
the above plots the values of $t=1/\sqrt{2}$, $t_{2}=0.06t$, $m=0$,
$\delta=2.0$, and $\phi=\pi/2$.

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