# First-order Quantum Phase Transitions and Localization in the 2D Haldane
Model with Non-Hermitian Quasicrystal Boundaries

Xianqi Tong  Department of Physics, Beijing Normal University, Beijing 100000,
People’s Republic of China  Su-Peng Kou<EMAIL_ADDRESS>Department of
Physics, Beijing Normal University, Beijing 100000, People’s Republic of China

(August 28, 2024)

###### Abstract

The non-Hermitian extension of quasicrystals (QC) are highly tunable system
for exploring novel material phases. While extended-localized phase
transitions have been observed in one dimension, quantum phase transition in
higher dimensions and various system sizes remain unexplored. Here, we show
the discovery of a new critical phase and imaginary zeros induced first-order
quantum phase transition within the two-dimensional (2D) Haldane model with a
quasicrystal potential on the upper boundary. Initially, we illustrate a phase
diagram that evolves with the amplitude and phase of the quasiperiodic
potential, which is divided into three distinct phases by two critical
boundaries: phase (I) with extended wave functions, PT-restore phase (II) with
localized wave functions, and a critical phase (III) with multifunctional wave
functions. To describe the wavefunctions in these distinct phases, we
introduce a low-energy approximation theory and an effective two-chain model.
Additionally, we uncover a first-order structural phase transition induced
(FOSPT) by imaginary zeros. As we increase the size of the potential boundary,
we observe the critical phase splitting into regions in proportion to the
growing number of potential zeros. Importantly, these observations are
consistent with groundstate fidelity and energy gap calculations. Our research
enhances the comprehension of phase diagrams associated with high-dimensional
quasicrystal potentials, offering valuable contributions to the exploration of
unique phases and quantum phase transition.

††preprint: APS/123-QED

##  I Introductioon

The exploration of open systems, characterized by non-Hermitian quantum
systems, has unraveled intriguing phenomena absent in their Hermitian
counterparts [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Notable examples
include PT symmetry and exceptional points [4, 5, 6, 7, 8, 9, 10, 11], non-
Bloch bulk-boundary correspondence [11, 12], and non-Hermitian skin effects
[11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Many of these phenomena are
related to parity-time (PT) symmetric Hamiltonians. These Hamiltonians
typically exhibit two phases as parameters vary: the PT-symmetric phase with
real eigenvalues and the PT-breaking phase with complex eigenvalues [2, 3, 4,
5]. These phenomena have been experimentally observed in open systems [23, 24,
25, 26, 27, 28, 29, 30], with promising applications in precision
measurements, nonreciprocal quantum devices, and topological transport. The
higher-order non-trivial interplay between the non-Hermitian skin effect and
the topological effect has led to the concept of a hybrid skin-topological
effect [31, 32, 33, 34, 35, 36].

Quasicrystals (QC) in closed quantum systems exhibit a plethora of fascinating
properties [37, 38, 39, 40, 41, 42, 43, 44, 45]. For instance, in the one-
dimensional (1D) Aubry-André-Harper (AAH) model, the introduction of finite
quasiperiodic strength leads to a transition from a metallic (extended) state
to an Anderson insulator (localized) [46, 47, 48, 49]. Critical phases are
vital for understanding the transitions from localized to extended states,
showing a range of fascinating phenomena including dynamical evolutions [50,
51, 52], critical spectral behavior [53, 54, 55, 56], and the multifractal
nature of wave functions [57, 58, 59, 60]. Expanding upon the AAH model,
variations incorporating different forms of quasiperiodic disorder and
interactions give rise to exotic phases, including critically localized states
[61, 62, 63] and many-body localization [64, 65, 66]. Recent research on non-
Hermitian extensions of the one-dimensional AAH model has uncovered such a
multicritical point marking the transition from localized to extended states,
accompanied by PT symmetry breaking and topological phase transitions [67, 68,
69]. However, the investigation of the interplay between the two-dimensional
(2D) chiral topological edge modes and the non-Hermitian quasicrystal
dissipation edge has not been previously explored.

In this context, our study not only reveals a complex phase diagram, but also
establishes a profound relationship between the size of the non-Hermitian
quasicrystal, the presence of imaginary zeros, and the increasing occurrence
of first-order structural phase transitions. In the phase diagram, there are
three distinct phases separated by two-phase boundaries: the extended phase
(I), the localized phase (III), and the critical phase (II), as illustrated in
Fig. 1. Additionally, we have also discovered an increasing number of phase
transitions (NPT) with the enlargement of the non-Hermitian quasicrystal size.
We explain these first-order structure phase transitions (FOSPT) in the
picture of phase splitting driven by imaginary zeros. Firstly, the quasi-
periodic modulated potential contains certain points where the potential
becomes zero. As the system parameters increase, a FOSPT occurs [70, 71].
Secondly, as the size of the system grows, the number of points with zero
quasiperiodic imaginary potential increases. These zero points divide the
imaginary potential into distinct domains, each having different positions for
undergoing phase transitions as parameters vary. Consequently, the NPT
increases with the size of the system. We have found that the NPT is
equivalent to the number of zero points, which also matches the count of non-
Hermitian domains. This phenomenon is unique to non-Hermitian systems and is
absent in their Hermitian counterparts.

In Sec. II, we establish the phase diagram by the inverse participation rate
and provide an interpretation in terms of the effective low-energy non-
Hermitian model. In Sec. III, we explore the relationship between the first-
order structure phase transition and the dimensions of the system. Section IV
is devoted to our conclusion.

##  II model and phase diagram

Figure 1: (a)-(e) xPBC/yOBC. (a) Schematic of the 2D Haldane model with
quasicrystal imaginary potential at the upper boundary. Quasiperiodic and zero
potentials are denoted by orange/yellow spheres against blue/green
backgrounds, respectively. Black solid lines represent NN hopping, black
dotted lines represent NNN hopping, and black dashed lines indicate
intermediate hidden layers ($L_{y}-1$). (b) The inverse participation ratio as
a function of $V$ and $h$, revealing three phases separated by two critical
lines: phase (i@) with extended wave functions, PT-restored phase (iii@)
featuring spatially localized wave functions, and critical phase (ii@) with
fractional wave functions. Parameters: $L_{x}=20$, $L_{y}=20$. (c)-(e) The
density $|\psi|^{2}$ as functions of $x$ at three distinct points,
$h=0.3,1.3,2.3$, with $V=1$, following Eq. (7). (c) IPR=0.11, (d) IPR=0.25,
(e) IPR=0.75.

The foundation of our study lies in the Hamiltonian, which exhibits different
forms under varying conditions:

$H=\begin{cases}H_{\text{AAH}},&L_{y}=1\\\ \text{two-chains},&L_{y}=2,\\\
H_{\text{Haldane}}+H_{\text{AAH}}^{\text{edge}},&L_{y}\rightarrow\infty,\end{cases}$
(1)

with two critical points of longitudinal dimensions $L_{y}=1,2$. When
$L_{y}=1$, the system returns back to the 1D non-Hermitian AAH model which
plays a key role in this paper

$H_{\text{edge}}^{\text{AAH}}=\sum_{n}V\cos(2\pi\alpha
n+ih)c_{n}^{\dagger}c_{n},$ (2)

where $c_{n}^{\dagger}$ and $c_{n}$ are the creation and annihilation
operators for a particle at the $n$-th site. $V$ and $h$ are the amplitude and
imaginary phase of the potential, and $\alpha$ is an irrational number for a
QC. Throughout this paper, we set $V=1$ as the energy unit.

Since $\alpha$ is an irrational number, it can be approximated by a sequence
of rational numbers $p_{n}/q_{n}$, where $p_{n}$, $q_{n}$ are prime numbers
and $p_{n}$, $q_{n}\rightarrow\infty$ as $n\rightarrow\infty$. In numerical
simulations, it is common practice to consider a finite (yet arbitrarily
large) number of sites $L=q_{n}$ on a ring with periodic boundary conditions,
where the occupation amplitudes are $\psi_{n+L}=\psi_{n}$.

When $L_{y}=2$, the system is no more than one-dimension, but a two-chain
system with a 1D AAH chain coupled to a hopping-only chain [56, 72]. Here we
focus on the non-Hermitian Haldane model with quasiperiodic complex potential
$H^{\text{AAH}}_{\text{edge}}$ on the upper boundary [Fig. 1(a)], where height
and circumference are $L_{y}$ and $L_{x}$. In the limits
$L_{y}\rightarrow\infty$, the Hamiltonian
$H=H_{\text{Haldane}}+H_{\text{edge}}^{\text{AAH}}$ and the Haldane model is
an important model for describing the topological insulator [73, 74],

$H_{\text{Haldane}}=t_{1}\sum_{\langle
nm\rangle}c_{n}^{\dagger}c_{m}+t_{2}\sum_{\langle\langle
nm\rangle\rangle}e^{i\phi_{nm}}c_{n}^{\dagger}c_{m},$ (3)

where the nearest-neighbor (NN) couplings are denoted by $t_{1}=1$, and the
next-nearest-neighbor (NNN) coupling coefficients are $t_{2}e^{i\phi_{nm}}$
with amplitude $t_{2}$ and phase $\phi_{nm}$. The symbols $\left\langle
n,m\right\rangle$ and $\left\langle\left\langle n,m\right\rangle\right\rangle$
denote the NN and NNN hopping, shown in Fig. 1(a) as black solid and black
dotted lines, respectively. The complex phase $e^{i\phi_{nm}}$ accounts for
the NNN hopping, and we set the positive phase direction to be clockwise
($|\phi_{nm}=\frac{\pi}{2}|$). Below we consider both x direction as a
periodic boundary condition (PBC) and y direction as an open boundary
condition (OBC), i.e., a cylindrical geometry.

Our focus is on revealing the critical phase emerging in the non-Hermitian
quasiperiodic boundary, which offers valuable insights into the boundary
effects in dissipative systems. To determine the phase diagram of the
Hamiltonian (1) under $L_{y}\rightarrow\infty$ condition, we compute the
inverse of the participation ratio (IPR)

$\displaystyle\text{IPR}_{n}=\frac{\sum_{m}\left|\braket{\psi_{n,m}^{R}}{\psi_{n,m}^{R}}\right|^{4}}{\left|\sum_{m}\braket{\psi_{n,m}^{R}}{\psi_{n,m}^{R}}\right|^{2}},$
(4)

as a function of $V$ and $h$, shown in Fig. 1(b). Here, $\ket{\psi_{n,m}^{R}}$
represents the right eigenstate of the $H$ corresponding to the energy
eigenvalue $E_{n}$, and $m=1,\dots,2L_{x}$. Specifically, when $n$ corresponds
to the ground state (denoted as $g$), $\ket{\psi_{g}}$ represents the ground
state, and $\text{IPR}_{g}$ quantifies the localization of the ground state.
Phase (I) with delocalized states has an $\text{IPR}_{g}\simeq 1/L\simeq 0$,
the PT-restore phase (III) with fully localized states, on the other hand, has
an $\text{IPR}_{g}\simeq 1$, and the critical phase (II) with fractional
states fall in between, with IPRg values ranging from 0 to 1, indicating an
intermediate level of localization, as shown in Fig. 1(b).

In contrast to the one-dimensional case, where only a transition from the
extended phase to the localized phase is observed [67, 68, 69], and distinct
from the mobility edge resulting from the coupling of two chains [60], our
scenario gives rise to a unique critical phase. Within this context, FOSPT
occurs in the ground state [see Appendix A]. In the following Appendix B, we
will present a more comprehensive analysis of the complete phase diagram and
offer analytically derived phase boundaries using an effective model.

For a 2D chiral topological insulator, chiral modes can only exist on the
boundary of topological materials. In the continuous limit, the effective
Hamiltonian in the low-energy is described as $H_{chiral}=v_{f}k$, where
$v_{f}=\frac{\partial H_{chiral}}{\partial k}_{k=k_{f}}$ is fermi velocity,
and $k$ is the vector of the chiral modes. In the long-wavelength and low-
frequency regimes, excitations are restricted to one-direction propagation and
protected by non-trivial bulk topology.

Then, we consider the effects of dissipation which the boundary potential has
a non-zero imaginary part, and the effective low-energy Hamiltonian [34, 35,
36] reads as

$H_{chiral}=v_{f}k+iV$ (5)

where the imaginary part of eigenvalues is dependent on the on-site
dissipation potential $V$. The Schrödinger equation of the dissipation chiral
modes is

$\left[-iv_{f}\frac{d}{dx}+iV(x)\right]\psi(x)=(\epsilon_{r}+i\epsilon_{i})\psi(x),$
(6)

where $\epsilon_{i}$ is the imaginary part of the eigenenergy.

Then, we get the solution of Eq. (6)

$\psi(x)=\frac{1}{\sqrt{C}}\exp\left(i\frac{\epsilon_{r}}{v_{f}}x\right)\exp\left(\int_{0}^{L_{x}}dx^{\prime}\frac{V\left(x^{\prime}\right)-\epsilon_{i}}{v_{f}}\right),$
(7)

where $1/\sqrt{C}$ is the normalization factor and the integration region (0,
$L_{x}$) is on the dissipation boundary.

Since the 2D Haldane model is periodic in x, the periodic boundary condition
gives $\psi(L_{x})=\psi(0)$. Then, we have

$i\frac{\epsilon_{r}}{v_{f}}L_{x}-\frac{\epsilon_{i}}{v_{f}}L_{x}+\frac{1}{v_{f}}\int_{0}^{L_{x}}dx^{\prime}V\left(x^{\prime}\right)=2i\pi
n,$ (8)

where $n\in\mathbb{Z}$. Then, the Eq. (8) can be reduced to

$\displaystyle\epsilon_{r}$ $\displaystyle=\frac{1}{v_{f}}\frac{2\pi
n}{L_{x}},$ (9) $\displaystyle\epsilon_{i}$
$\displaystyle=\frac{1}{L_{x}}\int_{0}^{L_{x}}dx^{\prime}V\left(x^{\prime}\right)=\widetilde{V},$
(10)

where the imaginary part of eigenenergy $\epsilon_{i}$ is the average value of
imaginary potential $\widetilde{V}$.

The first two components (7), similar to plane waves $exp(ikx)$, are uniformly
distributed throughout the entire space. However, when the sign of $v_{f}$ is
fixed, the third component (7) introduces exponential growth or decay is
possible, depending on the sign of $sgn(V\left(x\right)-\epsilon_{i})$ (either
1 or -1). For example, at position $x_{c}$, a change in the sign of the
imaginary potential occurs like a step function, specifically when
$V(x<x_{c})<0$ and $V(x>x_{c})>0$. If $v_{f}<0$, the edge state will exhibit a
peak at $x_{c}$. Conversely, if $V(x<x_{c})>0$ and $V(x>x_{c})<0$, and
$v_{f}>0$, the edge wave function will also display a peak at $x_{c}$. These
are exactly the black circles in Figs. 1(c)-(e), which are the positive
imaginary potentials.

In the specific case of our study, the dissipative potential takes the form of
a quasi-periodic pattern, as shown in Eq. (2). Substituting Eq. (2) into Eq.
(10), it can be reduced to

$\epsilon_{i}=-\frac{V\sin^{2}(\pi\alpha L_{x})}{\pi\alpha L_{x}}\sinh(h).$
(11)

Consequently, according to the above discussion, there will be a peak
approximating a period, due to the quasi-periodic nature of the imaginary
potential, as depicted in Figs. 1(c)-(e). To better visualize the phase
diagram [Fig. 1(b)] in different phases, we selected three positions along the
$V=1$ line in Fig. 1(b): $h=0.3,1.3$, 2.3. Among these positions, two critical
points are evident: the extended-critical transition point at $h_{1}=0.97$ and
the critical-localized phase transition at $h_{2}=1.41$.

The red and black circles represent positive and negative on-site potentials,
respectively, with a system size of $L_{x}=10$. As shown in Fig. 1(c), when
the imaginary phase $h=0.3$, the overall density of the wave function exhibits
minor fluctuations along the x-direction, corresponding to IPR$=0.11\approx
1/L_{x}$. At this point, the quasi-periodic potential is weak enough to be
seen as a perturbation, $V$ can be seen as part of the elliptical complex
energy spectrum, as shown in Appendix C. As the imaginary potential gradually
increases $(h=1.3)$, certain randomly distributed positions experience higher
density, while others exhibit decreased density. This leads to an intermediate
value IPR$=0.25$, indicating a wave function reminiscent of a fractal-like
state, as shown in Fig. 1(d). Finally, for a larger imaginary potential
$h=2.3$, the wave function localizes randomly at any position, with the
maximum occupation $|\psi|^{2}_{max}\approx 0.9$ is observed at $L_{x}=1$
(IPR=0.75), and the sharp peaks occur precisely at the transitions from black
(positive) to red (negative), as shown in Fig. 1(e).

##  III first-order structure phase transition

Figure 2: (a) The transverse dimension $L_{x}$ is a function of $h_{1}/h_{2}$.
(b) The longitudinal dimension $L_{y}$ is a function of $h_{1}/h_{2}$. (c) The
ground fidelity $F_{g}$ plotted against $h$ with $L_{x}=21,34,55,89,144,233$.
Inset: The NPT as a function of $L_{x}/L_{y}$. The red line represents
$L_{x}$, and the fit yields $\text{NPT}=0.05L_{x}+0.98$. The green line
corresponds to $L_{y}$.

Here, we explore how the extended-critical and the critical-localized phase
transition points $h_{1},h_{2}$ are also affected by the dimensions
$L_{x},L_{y}$. The $log(L_{x})$ as a function of $h_{1}$ is shown in Fig.
2(a), where we can see $h_{1}\approx 0.97$ is almost a constant and
$log(L_{x})=kh_{2}+c$ is a linear function, where $k$ and $c$ are constants.
As the size of the system increases, the alternation of positive and negative
imaginary potentials becomes more pronounced. It is due to the effect of
positive imaginary potential gain that the wave function (Eq. (7)) tends to
disperse more to locations marked by positive imaginary potentials as the
system dimension expands. This inherent tendency leads to an increase in the
localization transition points of the wave function as the system dimensions
($L_{x}$) increase. In turn, $h_{1},h_{2}$ is shown in Fig. 2(b) as $L_{y}$
varies, when $h_{1}$, $h_{2}$ are all essentially constants.

We also find another interesting phenomenon that the critical phase fragments
into multiple parts, which cannot happen in the Hermitian system. We have
plotted the fidelity of the ground state $F_{g}$ as a function of $h$ for
different sizes $L_{x}=21,34,55,89,144,233$ in Fig. 2(c). It can be seen that
the NPT increases as $L_{x}$ increases. To know the relationship between NPT
and system size, we plot NPT as a function of $L_{x}/L_{y}$, as shown in the
subplot of Fig. 2 (c). It can be seen that when $L_{y}$ increases, the NPT is
kept constant because there is no increase in the size of $V(x)$, as shown by
the green line. However, the red line represents the NPT vs $L_{x}$. The
result given by the linear fit is NPT $=0.05L_{x}+0.98$. 0.05 means that for
every 20 lattices, there is a zero point increase in the imaginary potential,
and 0.98 means that there is a phase transition from a delocalized phase to a
localized phase at the beginning. Then for the $V(x)$ as in Eq. (2), there
will be sites where $Im(V(x))\approx 0$, which is where the FOSPT occurs with
parameters change. As the $L_{x}$ increases, the number of zero imaginary
potential points increases. These zeros partition the potential into distinct
domains, each hosting different phase transition points as the parameters
vary. This is the reason why the NPT increases as the $length(V(x))$
increases.

Figure 3: Energy spectrum of the Haldane model with different imaginary
potential impurities. (a) The energy spectra with two nearest-neighbor
imaginary impurities for $\gamma=1.600$, 3.356, 3.500, 6.000 from left to
right. (b) The energy spectra with two next nearest-neighbor imaginary
impurities for $\gamma=1.600$, 2.000, 3.400, 6.000 from left to right. The
parameters are chosen as $L_{x}=20$, $L_{y}=20$.

In this following, we aim to establish the connection between the FOSPT and
the imaginary potential. Fig. 3 illustrates the complex energy spectrum of two
scenarios: one with nearest-neighbor imaginary potential [Fig. 3(a)], defined
as

$H_{NN}=H_{Hermitian}+iV_{1}+iV_{2}/2,$ (12)

and another with next-nearest-neighbor imaginary potential [Fig. 3(b)] given
by

$H_{NNN}=H_{Hermitian}+iV_{1}+iV_{3}/2,$ (13)

where $H_{Hermitian}$ is an arbitrary Hermitian matrix, we consider the 2D
Haldane model $H_{Hermitian}=H_{Haldane}$ ( the simplest PT-symmetric matrix
is presented in Appendix D). In $V_{i}$, “$i=1,2,3$” represent the on-site
potential of any position on the boundary and are increasing in order. The
color of points represents the density of edge $|\psi_{edge}(h)|^{2}$.

In the first plot with $V=1.600$, points close to the x-axis depict
topological boundary states influenced by the imaginary potentials. Their
energy spectra form a semicircle with the $x$-axis, exhibiting a skinning
effect, as shown in Fig. 3(a). The two points with the largest imaginary part
correspond to two non-topological boundary states that are progressively
confined towards the boundary due to the impact of the imaginary potential.
The colors in the figure represent the density at the boundary, revealing that
the topological boundary states primarily occupy the boundary, while states
outside the energy gap are bulk states, except for the non-topological
boundary states. At $V=V_{c}=3.356$, a critical phase transition occurs the
two non-topological boundary states are degenerate. If the imaginary potential
is slightly larger, i.e., $V=3.500$, the real part of these two states remains
the same, but the imaginary part differs, resulting in a PT phase transition.
With further increases in $V$, the energy of the non-topological boundary
states increases in tandem with the imaginary potential, and the imaginary
parts of the topological boundary states become nearly zero, restoring PT
symmetry.

We also consider the case of next-nearest-neighbor imaginary potentials for
$V=1.600,2.000,3.400,6.000$. The energy spectra are more or less the same at
the beginning, and the energy of the two non-topological boundary states
undergoes a degenerate at $V=V_{c1}=2.000$, which is also the first critical
point. However, when $V=V_{c2}=3.400$, two additional non-topological boundary
states emerge near the topological boundary states. These two non-topological
boundary states undergo a second PT phase transition. As in Fig. 3(b), after
the occurrence of two PT phase transitions, there are only non-topological
boundary states localized at the two imaginary potential points, as in Fig.
3(b). This observation from the simplest case of two imaginary potentials can
be extended to scenarios involving multiple imaginary potentials, where PT
phase transitions occur whenever non-adjacent imaginary potentials are
present, see Fig. 2(c).

If these zeros are replaced by finite imaginary potentials, we find that the
NPT returns to the situation in Fig. 1(b) and does not vary with $L_{x}$, as
shown by the green line. Moreover, when the length of the imaginary potential
is constant, i.e. length($V(x)$) = 20, we plot the NPT as a function of
$L_{y}$, again still with the green line. This means the phase transition
depends only on the imaginary potential change.

##  IV Conclusion

We uncover the rich phase diagram of the two-dimensional Haldane model with
edge quasi-periodic dissipation. Overall, the fractional dimension, the
largest imaginary part of the eigenvalues, the scaling exponent, and the
ground state fidelity provide valuable insights into the localization
properties and phase transitions in the system. The system exhibits extended,
critical, and localized phases, with the criticality appearing between the
extended-critical and critical-localized transitions. In the low-energy
approximation, we show phase transitions of the wave function in different
phases. We then projected the original Hamiltonian onto the boundary subspace
and obtained an effective two-chain model which yielded a phase diagram
similar to that of the original Hamiltonian, effectively capturing its phase
transition properties.

We also analyzed the effect of transverse and longitudinal dimensions on the
phase transition. The results show that the longitudinal dimensions do not
affect the phase transition, but the critical point from the critical phase to
the localized phase becomes larger with increasing transverse dimensions,
which is due to the gain effect of the positive imaginary potential, which
weakens the localization of the wave function. Then we also find the
phenomenon of critical phase tearing, which is unique in non-Hermitian
systems, due to the first-order structural phase transition of the system
caused by the zeros of the quasi-periodic imaginary potential.

###### Acknowledgements.

We are grateful to Gao Xianlong, Yiling Zhang, Xin-Ran Ma, Qian Du, Yufei Zhu
for valuable suggestions on the manuscript. This work is supported by NSFC
Grants No. 11974053 and No. 12174030.

##  Appendix A FIRST-ORDER PHASE TRANSITION

Figure 4: (a) The logarithm of energy gap $log(\Delta_{g})$ as a function of
$h$. (b) The first-order derivative of the ground state energy $dE_{g}/dh$ as
a function of $h$. Both $log(\Delta_{g})$ and $dE_{g}/dh$ exhibit
discontinuities at $h_{1}$ and $h_{2}$. The parameters are chosen as
$L_{x}=20$, $L_{y}=20$.

The quantum phase transition in the Hermitian system refers to the
nonanalyticity of avoided level-crossing or actual level-crossing [75].
However, the spectra of the non-Hermitian system are generally complex. There
are some line gaps and point gaps that have been found in the non-Hermitian AA
systems [6]. When the parameters are changed, the line gaps may be experienced
many times close to reopening processes if there is more than one zero point.
Traditionally, quantum phase transition is characterized by singularities of
the ground state energy and the energy gap between the ground state and the
first excited state. First-order quantum phase transition is identified by
abrupt changes in the first derivative of the energy.

In Fig. 4 (a), we plot the logarithm of the energy gap $log(\Delta_{g})$ as a
function of $h$, where

$\displaystyle\Delta_{g}=E_{f}-E_{g}.$ (14)

$E_{f}$ is the first excited state energy and $E_{g}$ is the ground state
energy. There are two discontinuous points $h_{1}$ and $h_{2}$, at which the
gap is closed. It also corresponds to the phase transition in Fig. 1(b).

To determine the phase transition type, we calculated the ground state energy
$E_{g}$ and its first derivative $dEg/dh$ as a function of $h$. In Fig. 4(b),
the black dots represent the ground state energy, while the red dots represent
its first derivative. Although $E_{g}$ is continuous, the discontinuity of
$dEg/dh$ at $h_{1}$ and $h_{2}$ shows the first-order nature of quantum phase
transition, similar to in Fig. 4(a).

##  Appendix B FRACTION DIMENSION, PT-SYMMETRY BREAKING, AND FIDELITY

Figure 5: (a)-(c). The fractal dimensions $\tau$ as a function of the real
part of eigenenergies $Re(E)$ for three different $h=0.2,1.4,2.6$. In (d), the
largest value of $|Im(E)|$ is a function of $h$. (e) The minimal scaling
exponent as a function of $h$. (f) The fidelity of ground states $F_{g}$
versus $h$. Abrupt changes at $h_{1}$ and $h_{2}$ are present in (d)-(f).
Inset: Phase diagram of the effective two-chain model $H_{eff}$. The
parameters are chosen the same as Fig. 3.

The localization behavior of the system’s wave function is a crucial
observable that requires precise measurements. Wave functions are commonly
characterized by their fractal dimension, quantified by the IPR, which follows
a scaling relation of

$\displaystyle\text{IPR}\sim(L_{x})^{-\tau},$ (15)

where $\tau$ represents the fractal dimension (FD). The FD provides a valuable
perspective to understand how states expand and fluctuate as the system size
increases. Similar to IPR, When $\lim_{L_{x}\rightarrow\infty}\tau=1$, it
indicates an extended wave function. Conversely, if the wave function is
localized with peaks only at a few lattice points and negligible amplitudes
elsewhere, it implies $\lim_{L_{x}\rightarrow\infty}\tau=0$. Fractal wave
functions exhibit FD values within the range of
$0<\lim_{L_{x}\rightarrow\infty}\tau<1$.

In Figs. 5(a-c), we compare the fractal dimensions of the extended, critical,
and localized phases. Specifically, we examine three points along the $V=1$
line in Fig. 1(b): $h=0.2,1.4,$ and $2.6$. The two black dashed lines in all
plots correspond to $Re(E)=\pm 1$ positions, between which states represent
topological boundary states when $h$ is small. In Fig. 5(a), the majority of
states are concentrated at the top, indicating extended states with $\tau=1$.
Fig. 5(b) shows a decreasing trend in both topological and non-topological
boundary states, with $\tau$ values ranging from $0.1$ to $0.6$. These non-
topological edge states refer to those influenced by the presence of an
imaginary potential, which tends to localize at the boundary. Furthermore, it
is worth noting that at $\tau=0.6$, we observe the separation between extended
states and fractal states localized at the dissipative boundary.

Under a strong imaginary potential at $h=2.6$ in Fig. 5(c), we discover that
all non-topological boundary states move to the bottom with $\tau=0$, and
their count precisely matches the number of topological boundary states, which
is $L_{x}$. However, the topological boundary states return to the top of the
plot, indicating their return to extended states unaffected by the imaginary
potential.

In fact, the impact of the imaginary potential on the system’s topological
states is observed in the critical phase, as depicted in Figs. 5(c). A
captivating question arises: How does the variation of the parameter influence
the imaginary part of the topological boundary states? Fig. 5(d) illustrates
the behavior of the largest imaginary part $|Im(E)|$ vs $h$. Remarkably, a
sudden surge from zero occurs at $h_{1}\approx 0.97$, followed by an abrupt
decline to zero at $h_{2}\approx 1.41$. Remarkably, when $h<0.97$ (PT-
symmetric) and $h>1.41$ (PT-restored), $max(Im(E_{e}))$ remains close to zero,
in agreement with Figs. 5(a) and (c). However, it is in the intermediate
region that we observe the most pronounced impact of the imaginary potential,
resulting in non-zero values of $Im(E_{e})$. This observation verifies our
earlier conjecture and highlights the intricate interplay between the
parameter variation and the imaginary part of the topological boundary states,
see Eq. (6).

One calculation similar to the fractal dimension is the scaling exponent,
which can be obtained from the on-site probabilities of any wave function
$\psi_{n}$. The on-site probability is restricted to the boundary. According
to the fractal theorem, the scaling of the maximum on-site probability is
expressed as

$\displaystyle max(p_{n,\text{edge}})\sim(2L_{x})^{-\beta_{n}^{\text{edge}}},$
(16)

where $p_{n,edge}=|\psi_{n,\text{edge}}|^{2}$ and
$\text{edge}=1,\dots,2L_{x}$. To determine the extended, critical, and
localized wave functions, we only need to investigate the minimum value of the
exponent $\beta_{min}^{\text{edge}}$.

Referring to Fig. 5(e), we focus on the boundary scaling exponent of the
ground state. Strikingly, we observe a precipitous decline in
$\beta_{min}^{edge}$ precisely at the critical points $h_{1}$ and $h_{2}$.
When $h<h_{1}$, the system manifests an extended phase, thereby approximating
$\beta_{min}^{edge}$ to 1. Conversely, for $h>h_{2}$, the system assumes phase
(III), whereby the boundary wave functions localize, yielding
$\beta_{min}^{edge}\approx 0$. Within the critical phase, $\beta_{min}^{edge}$
spans the interval (0.1, 0.25), providing evidence of a fractal nature of the
ground state.

We have carried out various calculations, some of which are based on knowledge
of the physical properties of the system. However, in Fig. 5(f), we
investigate the behavior of the fidelity $F_{g}$ vs $h$. Fidelity has a
distinct advantage in that it does not require prior familiarity with the
order parameters or symmetries of the system. Typically, we can expect that as
the ground state structure undergoes sharp changes, the fidelity will abruptly
decrease near the critical points of the system. We focus on the boundary
subspace and explore the boundary fidelity, quantified as:

$\displaystyle F_{g}(h,\delta
h)=\left|\braket{\psi_{g,\text{edge}}(h)}{\psi_{g,\text{edge}}(h+\delta
h)}\right|,$ (17)

where $\delta h$ is a small quantity,
$\ket{\psi_{g,\text{edge}}(h)}=\sum_{edge}\ket{\psi_{edge}}\braket{\psi_{edge}}{\psi_{g}(h)}$
and $\ket{\psi_{g}(h)}$ satisfies the eigenvalue equation
$H(h)\ket{\psi_{g}(h)}=E_{g}\ket{\psi_{g}(h)}$. At the critical points near
$h_{1}$ and $h_{2}$, the overlap of the ground states $F_{g}$ undergoes a
dramatic decrease, decreasing from $1$ to $0.76$ and from $1$ to $0$,
respectively.

Based on the above description, the physical properties of the original
Haldane model can essentially be captured by the characteristics of its
boundaries. Therefore, we introduce the concept of the boundary effective
Hamiltonian $H_{\text{edge}}$ to further comprehend the expanded-critical and
critical-localized phase transitions depicted in Fig. 1(b). Directly, we
project the $H$ onto the boundary subspace using the boundary projection
operator $P_{\text{edge}}$. The effective edge Hamiltonian is then given by:

$\displaystyle H_{\text{eff}}$ $\displaystyle=P_{edge}\leavevmode\nobreak\
H\leavevmode\nobreak\ P_{edge}$
$\displaystyle=H_{\text{AA}}+H_{\text{free}}+H_{c},$ (18)

where
$H_{\text{AA}}=\sum_{m}\left(a_{m}^{\dagger}a_{m+1}+\text{h.c.}\right)+V\cos(2\pi\alpha
m+ih)a_{m}^{\dagger}a_{m}$ represents the non-Hermitian Aubry-André-Harper
model, $H_{\text{free}}=\sum_{m}b_{m+1}^{\dagger}b_{m}+\text{h.c.}$ is the
free chain with only the nearest-neighbor hopping term, and
$H_{c}=a_{m}^{\dagger}b_{m}+\text{h.c.}$ represents their coupling, as seen in
Eq. (1) with $L_{y}=2$. The $H_{\text{eff}}$ is reduced to a two-chain model

$\displaystyle H_{\text{eff}}=$
$\displaystyle\sum_{j=1}^{2}\sum_{n}\left[V_{j,m}c_{j,m}^{\dagger}c_{j,m}+t\left(c_{j,m}^{\dagger}c_{j,m+1}+\text{
H.c. }\right)\right]$ (19)
$\displaystyle+\lambda\sum_{m=odd}\left(c_{1,m}^{\dagger}c_{2,m}+\text{ H.c.
}\right).$

where $V_{1,m}=V\cos(2\pi\alpha m+ih)$ when $j=1$ and when $j=2$, $V_{2,m}=0$.

The phase diagram is almost unchanged, as shown in the subplot of Fig. 5(f).
In the subplot of Fig. 5(f), two critical lines can also be seen dividing the
whole phase diagram into three regions: extended, localized, and critical
phase. This phase diagram is almost the same as Fig. 1(c), except that the
critical line from the extended phase to the critical phase is not
particularly obvious, and the extended phase is also reduced. This is due to
size effects, and in the case of $L_{y}\rightarrow\infty$ the subplot of Fig.
5(f) will change back to Fig. 1(c).

##  Appendix C DISSIPATION IN WEAK DISSIPATION CASE

Figure 6: The phenomena in the regime of small imaginary potentials under
gain/loss domain wall condition. (a) A comparison between numerical results
(yellow dots) and theoretical predictions (blue line) for the complex energy
spectrum of topological states. Inset: Energy spectra in whole complex energy
space. (b) The topological protected edge state on gain/loss boundaries (red
line) and the wave function for low-energy approximation. The parameters are
chosen as $L_{x}=20$, $L_{y}=20$, $h=0.2$ and $\gamma=0.2$.

Energy bands that encircle a point gap with nonzero winding numbers under PBC
will exhibit non-Hermitian skin effects under OBC. All eigenmodes are
localized at the boundary. However, in the above cases, periodic boundary
conditions ensure unidirectional chiral currents. Therefore, we introduce OBC
at the gain-loss boundary, creating a dissipation domain wall. We define the
Hamiltonian for the domain as

$\displaystyle H_{domain}=H+\text{domain},$ (20)

where $H$ is the Hamiltonian in Eq. (1) under $L_{y}\rightarrow\infty$, and
$\text{domain}=H_{G}(x)+H_{L}(x)$, with subscripts $G$ and $L$ representing
the ’“gain” and “loss” domains, respectively. In each domain, gain ($x<0$) or
loss ($x>0$) generates a constant on-site imaginary potential $\pm i\gamma$.
This results in a purely imaginary shift in the spectra of $H$.

Fig. 6(a) shows the complex energy spectrum of this Hamiltonian $H_{domain}$
in the $|Re(E)|<1$ regime under weakly dissipative conditions. where the
vertical axis represents $\left|Im(E)\right|$. It corresponds to the region
marked by the red dashed lines in the entire complex energy spectrum of the
Hamiltonian, as shown in the subplots of Fig. 6(a). While the colors indicate
the probability density of the eigenstates at the boundaries
$density_{edge}=|\psi_{edge}|^{2}$. The complex energy satisfy half-ellipse
equation, i.e.,
$\frac{\operatorname{Re}(E)^{2}}{a^{2}}+\frac{|\operatorname{Im}(E)|^{2}}{b^{2}}=1$.

In the limit of $h\rightarrow 0$, the quasiperiodic dissipation potential in
Eq. (2) undergoes a Taylor expansion

$\displaystyle\underset{h\rightarrow 0}{\operatorname{lim}}V_{n}$
$\displaystyle=\underset{h\rightarrow 0}{lim}\cos(2\pi\alpha n+ih)$
$\displaystyle\approx\cos(2\pi\alpha n)+i\sin(2\pi\alpha n)h+O(h^{2}).$ (21)

The imaginary potential perturbs the edge modes, leading to a correction in
their eigenenergies, given by

$\displaystyle E^{1}$
$\displaystyle=\braket{\psi^{0}_{e}}{H_{edge}}{\psi^{0}_{e}}$
$\displaystyle=\braket{\psi^{0}_{e}}{\cos(2\pi\alpha n)+i\sin(2\pi\alpha
n)h}{\psi^{0}_{e}}$
$\displaystyle=\sum_{n=0}^{L_{x}}\left|\psi_{n}^{0}\right|^{2}\cos(2\pi\alpha
n)+ih\sum_{n=0}^{L_{x}}\left|\psi_{n}^{0}\right|^{2}\sin(2\pi\alpha n)$
$\displaystyle=\operatorname{Re}(E^{1})+i\operatorname{Im}(E^{1}),$ (22)

where $\ket{\psi_{e}^{0}}$ is an eigenfunction of $H$ satisfying
$H\ket{\psi_{e}^{0}}=E^{0}\ket{\psi_{e}^{0}}$,
$\operatorname{Re}(E^{1})=\sum_{n=0}^{L_{x}}\left|\psi_{n}^{0}\right|^{2}\cos(2\pi\alpha
n)$ represents the real part of the correction to the eigenvalues, while
$\operatorname{Im}(E^{1})=h\sum_{n=0}^{L_{x}}\left|\psi_{n}^{0}\right|^{2}\sin(2\pi\alpha
n)$ denotes the imaginary part.

Notably, this correction also depends on the $\left|\psi_{n}^{0}\right|^{2}$
distribution, as seen in Eq. (22). Additionally, the distribution of
topological edge states at the boundary is energy-dependent. As we shift from
the energy gap to higher or lower energy bands, the distribution of
topological edge states inside the material becomes increasingly significant.
Consequently, the energy spectrum follows an elliptic function dependence that
is governed by these edge states, expressed as:

$\displaystyle\frac{\left[\operatorname{Re}(E^{0})+\operatorname{Re}(E^{1})\right]^{2}}{a^{2}}+\frac{\left[\operatorname{Im}(E^{1})\right]^{2}}{b^{2}}=1.$
(23)

The discrepancy between the theory and the numerical results is attributed to
the higher order perturbation, as shown in Fig. 6(a). As the imaginary
potential becomes smaller, the perturbation theory aligns more closely with
the numerical outcomes

Due to the inclusion of gain/loss domain walls in the up boundary, the
effective chiral Hamiltonian in Eq. (5) is substituted with
$H_{chiral}=v_{f}k+iV+\text{domain}$. And the Schrödinger equation with OBC
can be rewritten as

$\displaystyle\left[-iv_{f}\frac{d}{dx}+\text{domain}+iV(x)\right]\psi(x)=(\epsilon_{r}+i\epsilon_{i})\psi(x).$
(24)

The solution of the wave function is given by

$\displaystyle\psi(x)=\frac{1}{\sqrt{C}}\exp\left(i\frac{\epsilon_{r}}{v_{f}}x\right)\exp\left(\int_{0}^{L_{x}}dx^{\prime}\frac{\text{domain}+V\left(x^{\prime}\right)-\epsilon_{i}}{v_{f}}\right).$
(25)

Here $\epsilon_{i}$ is still the average imaginary potential, see Eq. (10).
Since the domain walls have opposite signs, $\int_{0}^{L_{x}}\text{domain}=0$.

To validate our prediction, we performed numerical calculations (blue line),
which closely match our analytical results (red star-line), as shown in Fig.
6(b). Consequently, $H_{G}(H_{L})$ still exhibits localized topological edge
states due to the inherent topological properties of the photonic topological
insulator, in the presence of non-Hermitian effects. Additionally, the
localization length of the wave function in Fig. 6(b) is inversely
proportional to
$\frac{1}{\xi}\propto\text{domain}+V\left(x^{\prime}\right)-\epsilon_{i}$.

##  Appendix D PHASE TRANSITION IN IMAGINARY POTENTIAL

Figure 7: The energy spectrum of the simplest four lattice model. (a) The real
and imaginary parts of the energy spectrum of $H_{a}$ as a function of
$\gamma$. (b) The real and imaginary parts of the energy spectrum of $H_{b}$
as a function of $\gamma$.

We consider a four lattices model [Eq. (12)] where two non-adjacent lattices
are dissipative and the other two are not, and the Hamiltonian $H_{NNN}=H_{a}$

$H_{a}=\left(\begin{array}[]{cccc}i\gamma&t&0&0\\\ t&0&t&0\\\ 0&t&i\gamma&t\\\
0&0&t&0\\\ \end{array}\right),$ (26)

where the $t$ is the nearest hopping term and the $\gamma$ is the non-adjacent
on-site imaginary potential. $H_{a}$ is PT-symmetric, i.e.,
$PTH_{a}(PT)^{-1}=H_{a}$. The exceptional points are $\gamma=(\sqrt{5}\pm
1)t$, and the eigenvalues are

$\displaystyle\lambda_{1}=\frac{1}{2}\left(-\sqrt{-\gamma^{2}-2\left(\sqrt{5}-3\right)t^{2}}+i\gamma\right),$
$\displaystyle\lambda_{2}=\frac{1}{2}\left(\sqrt{-\gamma^{2}-2\left(\sqrt{5}-3\right)t^{2}}+i\gamma\right),$
$\displaystyle\lambda_{3}=\frac{1}{2}\left(-\sqrt{2\left(\sqrt{5}+3\right)t^{2}-\gamma^{2}}+i\gamma\right),$
$\displaystyle\lambda_{4}=\frac{1}{2}\left(\sqrt{2\left(\sqrt{5}+3\right)t^{2}-\gamma^{2}}+i\gamma\right).$
(27)

As the eigenvalues are symmetric, we only need to consider $\lambda_{i}>0$
($i=1,2,3,4$), see Fig.7 (a). The system hold $PT$ symmetry when
$\gamma<(\sqrt{5}-1)t$, where $Im(\lambda_{i})$ are all equal to each other.
When $(\sqrt{5}-1)t<\gamma<(\sqrt{5}+1)t$, the imaginary part
$Im(\lambda_{1},2)$ are different, so $\gamma=(\sqrt{5}-1)t$ is the first
critical point. Moreover, the $PT$-symmetry is broken again when
$\gamma>(\sqrt{5}+1)t$, that the real part of the four modes vanishes.

Conversely, the Hamiltonian with two adjacent imaginary potentials is

$H_{b}=\left(\begin{array}[]{cccc}i\gamma&t&0&0\\\ t&i\gamma&t&0\\\ 0&t&0&t\\\
0&0&t&0\\\ \end{array}\right),$ (28)

where $H_{b}$ is also PT-symmetric. Then the eigenvalues of $H_{b}$ is

$\displaystyle\lambda_{1}=\frac{1}{2}\left(-\sqrt{-\gamma^{2}-2t\sqrt{5t^{2}-4\gamma^{2}}+6t^{2}}+i\gamma\right),$
$\displaystyle\lambda_{2}=\frac{1}{2}\left(\sqrt{-\gamma^{2}-2t\sqrt{5t^{2}-4\gamma^{2}}+6t^{2}}+i\gamma\right),$
$\displaystyle\lambda_{3}=\frac{1}{2}\left(-\sqrt{-\gamma^{2}+2t\sqrt{5t^{2}-4\gamma^{2}}+6t^{2}}+i\gamma\right),$
$\displaystyle\lambda_{4}=\frac{1}{2}\left(\sqrt{-\gamma^{2}+2t\sqrt{5t^{2}-4\gamma^{2}}+6t^{2}}+i\gamma\right).$
(29)

However, there is only one exceptional point where
$Im(\lambda_{1})=Im(\lambda_{2})$ and $Im(\lambda_{3})=Im(\lambda_{4})$, as
shown in Fig. 7(b).

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