# Distance between two manifolds, topological phase transitions and scaling
laws

ZhaoXiang Fang  ZhaoXiang Fang and Ming Gomg contribute equally School of
Physical Science and Technology, Xinjiang University, Urumqi, 830046, China
Ming Gong  Key Lab of Quantum Information, Chinese Academy of Sciences, School
of physics, University of Science and Technology of China, Hefei, 230026,
China  Synergetic Innovation Center of Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei, 230026, China
Guang-Can Guo  Key Lab of Quantum Information, Chinese Academy of Sciences,
School of physics, University of Science and Technology of China, Hefei,
230026, P.R. China  Synergetic Innovation Center of Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei, 230026,
P.R. China  Yongxu Fu<EMAIL_ADDRESS>International Center for Quantum
Materials, School of Physics, Peking University, Beijing, 100871, China  Long
Xiong<EMAIL_ADDRESS>International Center for Quantum Materials,
School of Physics, Peking University, Beijing, 100871, China

###### Abstract

Topological phases are generally characterized by topological invariants
denoted by integer numbers. However, different topological systems often
require different topological invariants to measure, such as geometric phases,
topological orders, winding numbers, etc. Moreover, geometric phases and its
associated definitions usually fail at critical points. Therefore, it’s
challenging to predict what would occur during the transformation between two
different topological phases. To address these issues, in this work, we
propose a general definition based on fidelity and trace distance from quantum
information theory: manifold distance. This definition does not rely on the
berry connection of the manifolds but rather on the information of the two
manifolds – their ground state wave functions. Thus, it can measure different
topological systems (including traditional band topology models, non-Hermitian
systems, and topological order models, etc.) and exhibit some universal laws
during the transformation between two topological phases. Our research
demonstrates that when the properties of two manifolds are identical, the
distance and associated higher-order derivatives between them can smoothly
transition to each other. However, for two different topological manifolds,
the higher-order derivatives exhibit various divergent behaviors near the
critical points. For subsequent studies, we expect the method to be
generalized to real-space or non-lattice models, in order to facilitate the
study of a wider range of physical platforms such as open systems and many-
body localization.

In the literature of topology and geometry, two manifolds are topologically
equivalent when and only when they can be smoothly deformed to each other Lee
(2010). The equivalent manifolds are uniformly characterized by one
topological index (invariant), such as the Chern number Lee (2010)
corresponding to the famous Thouless-Kohmoto-Nightingale-den Nijs (TKNN)
number derived from the linear response theory in the theory of topological
matters Hatsugai (1993); Kane and Mele (2005). This pioneering breakthrough
propelled the establishment of the topological classification of matters with
various symmetries in all dimensions Qi and Zhang (2011); Hasan and Moore
(2011); Qi and Zhang (2011); Schnyder  _et al._ (2009, 2008). Regardless of
the Hermiticity of the topological matters, theoretical explorations of non-
Hermitian quantum systems have significantly expanded the scope of condensed
matter physics in the past decade Ashida  _et al._ (2020); Bergholtz  _et al._
(2021); Gong  _et al._ (2018); Kawabata  _et al._ (2019a); Shen  _et al._
(2018); Yao and Wang (2018); Song  _et al._ (2019); Lieu (2018); Bagarello
_et al._ (2016); Lee (2016); Kunst  _et al._ (2018); Yao  _et al._ (2018); Lee
and Thomale (2019); Yokomizo and Murakami (2019); Okuma  _et al._ (2020);
Borgnia  _et al._ (2020); Yang  _et al._ (2020a); Zhang  _et al._ (2020a); Xue
_et al._ (2021); Guo  _et al._ (2021); Xue  _et al._ (2022); Fu and Zhang
(2023a), rapidly encompassing higher-order non-Hermitian systems Liu  _et al._
(2019); Lee  _et al._ (2019); Edvardsson  _et al._ (2019); Kawabata  _et al._
(2020); Okugawa  _et al._ (2020); Fu  _et al._ (2021); Yu  _et al._ (2021);
Palacios  _et al._ (2021); Li  _et al._ (2022), exceptional points Kawabata
_et al._ (2019b); Yokomizo and Murakami (2020); Yang  _et al._ (2020b); Zhang
_et al._ (2020b); Xue  _et al._ (2020); Yang  _et al._ (2021); Denner  _et
al._ (2021); Fu and Wan (2022); Mandal and Bergholtz (2021); Delplace  _et
al._ (2021); Liu  _et al._ (2021); Stålhammar and Bergholtz (2021); Ghorashi
_et al._ (2021a, b), and scale-free localization Li  _et al._ (2021, 2023);
Guo  _et al._ (2023); Fu and Zhang (2023b); Molignini  _et al._ (2023). In
open boundary conditions (OBSs), the non-Hermitian skin effect (NHSE) is a
remarkable feature that predicts an extensive number of eigenstates localized
at the edges as well as the breakdown of the Bloch band theory Lee (2016); Yao
and Wang (2018); Song  _et al._ (2019); Yokomizo and Murakami (2019); Zhang
_et al._ (2020a); Yang  _et al._ (2020a); Okuma  _et al._ (2020). A
comprehensive consequence is the difference of the topological transition
points between OBCs and periodic boundary conditions (PBCs).

However, a subtle question has never been addressed in either Hermitian or
non-Hermitian systems: what quantitative contexts will happen during the
deformation of the transition of two manifolds (topological phases)? In the
theory of continuous phase transitions, we aim to identify a set of parameters
that can universally manifest divergent behavior at the phase boundaries of
the system, which motivates us to ascertain the existence of the divergent
behavior at the topological phase boundaries.

Note that in quantum information science, there are two common ways to measure
the similarity between two pieces of information: trace distance and fidelity
(in the case of pure states, these are completely equivalent) Jozsa (1994);
Nandi  _et al._ (2018); Liang  _et al._ (2019); Gu (2010); Banchi  _et al._
(2015); Li (2012); Rastegin (2007, 2007); Zhang  _et al._ (2019); de Miranda
and Micklitz (2023); Liang  _et al._ (2019); Brito  _et al._ (2018); Rana  _et
al._ (2016). Based on this concept, various new definitions emerge for
different physical systems, such as the fidelity rate to characterize the
quantum phase transition of the ground state Gu (2010); Banchi  _et al._
(2015), the trace distance quantum discord to measure the quantum correlation
Rana  _et al._ (2016), and the minimum trace distance (the distribution
between test and the set of local correlations) to quantify the non-locality
of Bell-type inequalities Brito  _et al._ (2018). These generalized concepts
are built upon the significant distinctions in ”distances” between different
phases of matter, such as quantum magnetic and antiferromagnetic phases Gu
(2010), or the clear boundary in ”distances” between different systems, such
as quantum and classical systems Rana  _et al._ (2016). Thus, the measurement
of the quantum state serves as an inspiration for our investigation of the
topological phase boundaries Wiseman and Milburn (2009); Zeng  _et al._
(2019).

In this work, we establish a formulation for exploring the divergent behaviors
during the deformation of two manifolds (topological phases). We introduce the
concept of manifold distance (MD) with the aim of efficiently and directly
detecting the boundaries of topological phases. We discover that the manifold
distance and associated higher-order derivatives between two topologically
equivalent manifolds (topological phases) can smoothly transition to each
other, while the higher-order derivatives of the manifold distance exhibit
various divergent behaviors with universal scaling laws near the critical
points between two topologically equivalent manifolds (topological phases).
Our formulation of manifold distance is not only applicable to traditional
Hermitian systems but also applies to the topological transition points of
non-Hermitian systems under both OBCs and PBCs. Moreover, we also apply our
formulation to the (many-body) continuous systems, such as p-wave
superconductors and, Kitaev toric model (with topological order). Our research
paves an avenue that identifies the rigorous behaviors at the boundaries of
topological phases.

Thus, in this work, we are interested in exploring what happens during the
deformation of two manifolds. We introduce the concept of ”manifold distance”
with the aim of efficiently and directly detecting the boundaries of
topological phases. The pictorial illustration of this process is shown in
Fig. 1. Let us consider two manifolds $\mathcal{M}$ and
$\mathcal{M}^{\prime}$. The parameters in these two manifolds are mapped to
each other via a smooth function,

$f:{\bf k}\rightarrow{\bf k}^{\prime}=f({\bf k}).$ (1)

Then, we would show that various mapping functions, which represent the
distance between different ground state wavefunctions, exhibit divergent
behavior at critical points. Additionally, different systems exhibit distinct
scaling laws.

Firstly, we utilize manifold distance in traditional quantum models and then
generalize it to non-Hermitian models, with particular attention to the open
boundary case (OBCs). Theoretical explorations of non-Hermitian quantum
systems have significantly expanded the scope of condensed matter physics in
the past decade Ashida  _et al._ (2020); Bergholtz  _et al._ (2021); Gong  _et
al._ (2018); Kawabata  _et al._ (2019a); Shen  _et al._ (2018); Yao and Wang
(2018); Song  _et al._ (2019); Lieu (2018); Bagarello  _et al._ (2016); Lee
(2016); Kunst  _et al._ (2018); Yao  _et al._ (2018); Lee and Thomale (2019);
Yokomizo and Murakami (2019); Okuma  _et al._ (2020); Borgnia  _et al._
(2020); Yang  _et al._ (2020a); Zhang  _et al._ (2020a); Xue  _et al._ (2021);
Guo  _et al._ (2021); Xue  _et al._ (2022); Fu and Zhang (2023a), rapidly
encompassing higher-order non-Hermitian systems Liu  _et al._ (2019); Lee  _et
al._ (2019); Edvardsson  _et al._ (2019); Kawabata  _et al._ (2020); Okugawa
_et al._ (2020); Fu  _et al._ (2021); Yu  _et al._ (2021); Palacios  _et al._
(2021); Li  _et al._ (2022), exceptional points Kawabata  _et al._ (2019b);
Yokomizo and Murakami (2020); Yang  _et al._ (2020b); Zhang  _et al._ (2020b);
Xue  _et al._ (2020); Yang  _et al._ (2021); Denner  _et al._ (2021); Fu and
Wan (2022); Mandal and Bergholtz (2021); Delplace  _et al._ (2021); Liu  _et
al._ (2021); Stålhammar and Bergholtz (2021); Ghorashi  _et al._ (2021a, b),
and scale-free localization Li  _et al._ (2021, 2023); Guo  _et al._ (2023);
Fu and Zhang (2023b); Molignini  _et al._ (2023). In open boundary conditions,
the non-Hermitian skin effect (NHSE) is a remarkable feature that predicts an
extensive number of eigenstates localized at the edges under as well as the
breakdown of the Bloch band theory Lee (2016); Yao and Wang (2018); Song  _et
al._ (2019); Yokomizo and Murakami (2019); Zhang  _et al._ (2020a); Yang  _et
al._ (2020a); Okuma  _et al._ (2020). A comprehensive consequence is the
difference of the topological transition points between OBCs and periodic
boundary conditions (PBCs). Our formulation of manifold distance is not only
applicable to traditional Hermitian multiband systems, but also generalized to
characterize the topological transition points of non-Hermitian systems under
both OBCs and PBCs. Finally, we discuss the non-Brillouin zone systems, such
as p-wave superconductors, Kitaev toric model (with topological order),
confirming that these concepts still applicable in these systems.

Manifold distance.-The most natural choice of definition is based on trace
distance Jozsa (1994). For pure state, trace distance and fidelity are exactly
equivalent to each other Jozsa (1994); Liang  _et al._ (2019). Then we define
two different distances as

$d_{1}=1-|\langle\psi_{k}|\phi_{k^{\prime}}\rangle|^{2},\quad
d_{2}=\sqrt{1-|\langle\psi_{k}|\phi_{k^{\prime}}\rangle|^{2}}.$ (2)

Obviously, $d_{i}\geq 0$ and $d_{1}\leq d_{2}$.

We are interested in the pure state, while for the mixed state, the same
definition is still applicable: the overlap between the two distinct regions
should be defined by the distance between the two density matrices Jozsa
(1994). Thus, to measure the distance between the two manifolds, we define

Figure 1: Adiabatic deformation between two manifolds. (a) In this work, the
parameters in the two base spaces $\mathcal{M}$ and $\mathcal{M}^{\prime}$,
denoted as $k$ and $k^{\prime}$ respectively, can be connected by an arbitrary
smooth function $f$: $k\in\mathcal{M}\rightarrow
k^{\prime}\in\mathcal{M}^{\prime}$. (b) For most of the non-hermitian models
in this paper, these phase regions can be labeled from left to right as
I,II,III,IV, and the black dots represent the phase transition critical points
of these systems. (c) The integrand of the manifold distance $d_{2}$ in eq.
(2). At the phase transition point, the real part of $d_{2}$ must have a
singularity at some $k$ within the Brillouin zone, while away from the phase
transition point, $d_{2}$ is smooth throughout the entire Brillouin zone. This
is what causes the derivative of the manifold distance to diverge at the phase
boundary. The imaginary part of $d_{2}$ is always zero over the entire
integration range, or in some non-Hermitian models, it is an even function,
and therefore, the imaginary part does not contribute to manifold distance.
(d) If we replace the momentum k with a different form, then the manifold
distance still reflects the critical behavior of the phase boundary, only the
divergence coefficient is different. More specific examples can be found in
the supplementary materials Sup  in section I.

$\mathcal{D}_{1}=\int d{\bf k}d_{1}({\bf k},{\bf
k}^{\prime}),\quad\mathcal{D}_{2}=\int d{\bf k}d_{2}({\bf k},{\bf
k}^{\prime}).$ (3)

The manifold distance satisfies the strict definition of distance in geometry
and quantum information. However, the distance $\mathcal{D}_{i}=0$ means that
for each ${\bf k}$ and ${\bf k^{\prime}}$, the two wave-functions are
identical.

Figure 2: The non-Hermitian systems of manifolds distance $D$ and its second-
order derivatives $D^{\prime\prime}$. In fig. (a) $\sim$ (d), the model
represents a one-dimensional topological model in eq. (7), illustrating the
divergence of $D^{\prime\prime}$ in (a) and (b). As the coefficient
$\gamma_{2}$ of the non-Hermitian term decreases, the divergent behavior of
the manifold distance would gradually returns to the Hermitian case. Here,
fig. (a) $\sim$ (d) correspond to regions (I)$\sim$(IV) of the phase diagram
shown in fig. (1), respectively. Fig. (e) and (f) depict the divergence of
$D^{\prime\prime}$ for a two-dimensional topological model, corresponding to
regions (I) and (II) of the phase diagram in fig. (1).

Generalized to the Non-Hermitian.-Typically, non-Hermiticity is achieved by
introducing NH hoppings and/or with NH gain/loss terms Moiseyev (2011); Liu
_et al._ (2020); Ashida  _et al._ (2020); Bagarello  _et al._ (2016); Kawabata
_et al._ (2019a). In addition, some of the topological invariants of the
Hermitian Hamiltonians become generalized when the Hermiticity condition is
removed Bergholtz  _et al._ (2021); Gong  _et al._ (2018); Shen  _et al._
(2018); Yao and Wang (2018); Song  _et al._ (2019); Lieu (2018). Therefore, it
is reasonable to extend the concept of manifold distance to the non-Hermitian
case. Consider the eigenvalue equation of Non-Hermitian $H_{k}$:

$\displaystyle H_{k}\ket{\varphi_{n}}=E_{n}\ket{\varphi_{n}},\quad
H_{k}^{\dagger}\ket{\phi_{n}}=E_{n}^{{}^{\prime}*}\ket{\phi_{n}}.$ (4)

Here we have four choices

$d_{1}=1-|_{1}\langle a_{k}|b_{k^{\prime}}\rangle_{2}|^{2},\quad
d_{2}=\sqrt{1-|_{1}\langle a_{k}|b_{k^{\prime}}\rangle_{2}|^{2}}.$ (5)

where $a_{k}$ and $b_{k}$ correspond to $\psi_{k}$ and $\phi_{i}$.

Therefore, regardless of the chosen definition of manifold distance, the
behavior of the phase boundary is effectively captured, with only differences
in numerical values.

Certainly, it’s noteworthy that if the eigen-wave functions of $H$ and $H_{d}$
do not conform to the following form:

$\displaystyle(x+iy,u+iy),\quad(x-iy,u-iy)$ (6)

Divergence may arise due to the normalization of wave-function. Therefore, for
non-Hermitian systems, it is preferable to choose forms such as
$\langle\psi_{k}|\psi_{k^{\prime}}\rangle$ or
$\langle\phi_{k}|\phi_{k^{\prime}}\rangle$ to avoid this issue.

Hamiltonians.- We examine the divergent properties of phase boundaries through
one- and two-dimensional topological models Kitaev (2001); Greiter  _et al._
(2014); Leijnse and Flensberg (2012); Günter  _et al._ (2005); Leijnse and
Flensberg (2012); Greiter  _et al._ (2014); Chhajed (2020); Thakurathi  _et
al._ (2014); Sato and Fujimoto (2016); Ren  _et al._ (2016). The Hamiltonians
we consider are shown as follows:

$H_{1}(k)=\begin{pmatrix}\epsilon_{k}&\beta_{k}\\\
\beta_{k}&-\epsilon_{k}\end{pmatrix},\quad H_{2}({\bf
k})=\begin{pmatrix}\epsilon_{\bf k}&\beta_{\bf k}\\\ \beta_{\bf
k}^{*}&-\epsilon_{{\bf k}}\end{pmatrix}.$ (7)

In $H_{1}$, we choose $\epsilon=-2t\cos(k)+i\gamma$ and
$\beta_{k}=\alpha\sin(k)$; while in $H_{2}$, we choose $\epsilon_{\bf
k}=-2t(\cos(k_{x})+\cos(k_{y}))-\mu+i\gamma$ and $\beta_{\bf
k}=\alpha(\sin(k_{x})+i\sin(k_{y}))$, with $\mu$ being the chemical potential
in both cases and $t$ is the hopping between neighboring sites. The parameter
$\alpha$ maybe regarded as spin-orbit coupling or superconducting pairing
strength and $\gamma$ is the non-Hermitian term coefficients. These two models
are relevant to a large number of topological phases. For example, $H_{1}$ can
be regarded as the Kitaev toy model by approximating a spinless fermion to a
p-wave superconductor; it may also be regarded as the fermionized model of the
transverse Ising model. Similarly, $H_{2}$ can be regarded as a two-
dimensional superconducting model or the anomalous quantum Hall system.

In the following, let us first consider $H_{1}$. We assume the parameters for
$\mathcal{M}$ is $\mu$, $t$ and $\alpha$, while for $\mathcal{M}^{\prime}$
their corresponding parameters are $\mu^{\prime}$, $t^{\prime}$ and
$\alpha^{\prime}$. According to the expression of $d_{a}$, we derived that the
singular point happens at $k=0$ with
$\mu=2t\sqrt{1-\frac{\gamma^{2}}{\alpha^{2}}}$ or $k=\pi$ at
$\mu=-2t\sqrt{1-\frac{\gamma^{2}}{\alpha^{2}}}$. Also for $H_{2}$, the
singular point happens at

$\displaystyle\mu=\pm 2t(1+\sqrt{1-\frac{\gamma^{2}}{\alpha^{2}}})$ (8)

Transition from non-Hermitian to Hermitian.- As for the above Non-Hermitian
Hamiltonian $H_{1}$ and $H_{2}$, when $\gamma\to 0$, they both reback to
Hermitian systems. This transition phenomenon can be demonstrated using the
manifold distance. For instance, the divergence behavior of $D^{\prime\prime}$
at the phase boundary would reverts from non-Hermitian to Hermitian systems,
as shown in Fig. (2).

Specifically, consider a simplified 1D Hamiltonian

$\displaystyle H_{k}=\left(\begin{array}[]{cc}-\mu+i\gamma&\alpha k\\\ \alpha
k&\mu-i\gamma\\\ \end{array}\right),$ (11)

Figure 3: Manifold distance for systems with OBC. Figure (a) shows the
generalized Brillouin zone, where the first two subplots depict scenarios
without next-nearest-neighbor interactions, and the GBZ is circular. The
latter two diagrams include next-nearest-neighbor interactions, and the GBZ is
formed by two curved segments. Here, we have chosen parameters at the phase
transition point
$t=\sqrt{(t^{\prime})^{2}+(\frac{\gamma}{2})^{2}},t=t^{\prime}+(\frac{\gamma}{2})$
and its vicinity. It is evident that the variation in the image of GBZ cannot
capture the system’s phase transition process. For figures (b) $\sim$ (g), the
left column depicts the OBC case, while the right column corresponds to the
PBC case. The red and blue curves represent scenarios with and without next-
nearest-neighbor interactions in the system, respectively. As we see, in case
of $t_{3}=0$, the derivative of manifold distance diverges at the phase
boundary points in eq. (20) (PBC) or eq. (22) (OBC), that the phase
transitions points are $t\approx 1.20$ and $t\approx 1.67$, respectively,
which is consistent with the result of reference Yao and Wang (2018). However,
as $t_{3}\neq 0$, although we currently lack an analytical expression for the
phase boundary, we can numerically obtain it through the singularity of
manifold distance, here we show that the phase transition points are $t\approx
1.56$ for OBCs, and these results also consistent with reference Yao and Wang
(2018), which can be seen in the numerical spectra of real-space Hamiltonian
of an open chain. More specific examples can be found in the supplementary
materials Sup  in section II.

numerical calculations revealed that the divergence coefficients are mainly
affected by a certain set of parameters, so we assumed the first set of
parameters to be constants while concealing the subscripts in the follow-up.
Since the simplified Hamiltonian does not have a Brillouin zone, we only need
to consider integration intervals that cover the singularities of
$D^{\prime\prime}$ (here $D^{\prime\prime}$ is the partial derivative of $D$
respect to the chemical potential). When the non-Hermitian term $i\gamma$ is
large, we have

$\displaystyle
D^{{}^{\prime\prime}}\approx-\frac{\sqrt{\gamma}}{2\alpha\gamma}\frac{1}{\sqrt{\mu}},\gamma\gg\mu.$
(12)

As for $\gamma\to 0$,

$\displaystyle
D^{{}^{\prime\prime}}\propto\frac{1}{\mu\sqrt{\alpha^{2}+\mu^{2}}}\approx\frac{1}{\mu\alpha},$
(13)

it is the property of Hermitian system. It can be seen that when the non-
Hermitian term $i\gamma$ gradually disappears from larger values: the
divergence will gradually change from $\frac{1}{\sqrt{\mu}}$ to
$\frac{1}{\mu}$, and there would be a superposition of divergence behaviors in
this process. The coefficients for more models can be found in section IX of
the supplementary materials Sup .

Finally, although the divergence behavior of $D^{\prime\prime}$($D^{\prime}$)
is affected by several parameters, its divergence coefficient tends to depend
on only a few physical parameters for Hermitian case.

As for hermitian model defined in eq. 7, our numerical fitting confirm that

$\displaystyle
D^{{}^{\prime}}\propto\frac{1}{\sqrt{2}\alpha_{2}}\ln(|\mu_{2}-\mu_{2}^{*}|),\quad\text{for
1D}$ $\displaystyle
D^{{}^{\prime\prime}}\propto\frac{2}{\alpha^{2}}ln(|\mu_{2}-\mu_{2}^{*}|),\quad\text{for
2D}.$ (14)

and the non-Hermitian systems devergence behavior as follow

$\displaystyle
D^{{}^{\prime\prime}}\propto\frac{C_{1}}{\mu_{2}-\mu_{2}^{*}}+\frac{C_{2}}{\sqrt{|\mu_{2}-\mu_{2}^{*}|}},\quad\text{for
1D}$ (15) $\displaystyle D^{{}^{\prime\prime}}\propto
C_{1}ln(|\mu_{2}-\mu_{2}^{*}|)+\frac{C_{2}}{\sqrt{|\mu_{2}-\mu_{2}^{*}|}},\quad\text{for
2D}.$

More specific examples can be found in sections III and IV of the
supplementary materials Sup .

Open boundary for non-Hermitian systems.- The bulk-boundary correspondence was
developed for Hermitian case, due to gain and loss, some systems are
intrinsically non-Hermitian, which still holds the usual bulk-boundary
correspondence. However, for non-Hermitian systems, the open-boundary spectrum
is noticeably different from the periodic boundary, that the momentum space
Hamiltonian $H(k)$ cannot determine the zero modes. General the zeros modes or
phase boundary can be seen in the numerical spectra of real-space Hamiltonian
with open boundary, here we present a universally applicable approach that
directly employs manifold distance to determine phase boundaries in momentum
space. As a specific illustrative example, we consider the 1d PT-symmetry non-
Hermiant SSH model Yao and Wang (2018)

$\displaystyle H_{k}=\left(\begin{array}[]{cc}0&\beta(k)\\\ \beta^{*}(k)&0\\\
\end{array}\right),$ (18)

with

$\displaystyle\beta(k)=\frac{\gamma}{2}+t+(t^{{}^{\prime}}+t_{3})\cos(k)-i(t^{{}^{\prime}}-t_{3})\sin(k),$
(19)
$\displaystyle\beta^{*}(k)=-\frac{\gamma}{2}+t+(t^{{}^{\prime}}+t_{3})\cos(k)+i(t^{{}^{\prime}}-t_{3})\sin(k)$

We show a shortcut, which is applicable only to the $t_{3}=0$ case. For PBCs,
the phase critical lines are

$t=t^{\prime}\pm(\frac{\gamma}{2});\quad t=-t^{\prime}\pm(\frac{\gamma}{2}),$
(20)

and the devergence behavior as follow

$\displaystyle
D^{{}^{\prime\prime}}\propto\frac{a}{\sqrt{|t_{2}-t_{2}^{*}|}}+b\ln(|t_{2}-t_{2}^{*}|)$
(21)

Figure 4: Figure (a) depicts a schematic of the Kitaev p-wave superconducting
model, where under certain conditions, Majorana fermions may emerge at the
ends of the chain, leading the system into topological superconducting phase.
Figure (b) illustrates the Kitaev toric model, the honeycomb lattice can be
deformed to a brick-wall lattice without any change in the topology. Fig. (c)
$\sim$ (e) show the maifold distance for p-wave SC, which the phase transition
occurs at $\mu=0$, where $\mu$ represents the chemical potential. Fig. (f)
$\sim$ (h) show the maifold distance for Kitaev toric model. As we know, in
the ground state of this system, there exist two distinct physical phases: a
gapped phase, its excitations is Abelian anyons; and a gapless phase, which
features excitations of non-Abelian anyons. For the parameter space satisfies
the condition $J_{x}+J_{y}+J_{z}=1$, then the model exhibits these two
distinct phases. Select $J_{x}$ as an independent variable and ensure that
$J_{z}=\frac{1}{3}$ and $J_{x}+J_{y}+J_{z}=1$ are always satisfied, then we
have two phase transition points with the boundary between gapped and gapless
phase. One occurs at $J_{x}=\frac{1}{6}$, and the other occurs at
$J_{x}=\frac{1}{2}$, which is consistent with the properties of the derivative
of $D$. More specific examples can be found in the supplementary materials Sup
.

However, in non-Hermitian systems, the open-boundary spectra quite different
from periodic-boundary ones, which seems to indicate a complete breakdown of
bulk-boundary correspondence, and its transition points

$\displaystyle t=\pm\sqrt{\pm(t^{\prime})^{2}+(\frac{\gamma}{2})^{2}};\quad
t=\pm\sqrt{-(t^{\prime})^{2}+(\frac{\gamma}{2})^{2}},$ (22)

Nevertheless, the derivative of maifold distance $D$ can also manifest the
phase transition, whether the system has PBC or OBC, as show in fig.3.
However, for OBC, some modifications are necessary for manifold distance.

i) The integration region for manifold distance should be the generalized
Brillouin zone; more precisely, it should include the ”singularities” of GBZ;

ii) Correspondingly, it is necessary to extend the momentum $k$ to its complex
form, i.e.,

$\displaystyle k\to k-i\ln r$ (23)

which is equivalent to replacing the Bloch phase factor $e^{ik}$ by
$\beta\equiv re^{ik}$ in OBC.

Similarly, we could consider the case of systems with next-nearest neighbors.
As show in fig.4, although there is no analytical expression for the phase
boundary, we could numerically obtain it through the singularity of the
manifold distance.

In fact, for non-lattice systems, it is necessary to truncate the integration
domain, as long as these ”singular points” are contained within it. In this
way, its derivatives also exhibit singular properties near phase boundary,
such as topological order systems in fig.4.

To conclude, we defined manifold distance over two manifolds, and we have
shown that its higher-order derivatives could exhibit some scaling laws at the
critical points when crossing the topological phase boundary. We have
determined some of the divergence behaviors in one- and two-dimension model,
and proved that this approach can be extended to non-Hermitian models with
open boundary conditions (OBCs). For future research, we aim to extend this
concept to mixed states and apply it to open systems to observe the effects of
gain and loss on various physical experimental platforms, that
$D=\int_{GBZ}\operatorname{Tr}\left[\sqrt{\rho^{1/2}\sigma\rho^{1/2}}\right]d\bf{k}$.
We expect similar conclusions for open systems. Moreover, we aspire to extend
these definitions to broader domains, such as real space or quasi-crystal
system.

Acknowledgments.- This work is supported by …

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Supplemental Online Material for “Distance between two manifolds, topological
phase transitions and scaling laws”

###### Contents

  1. A Manifold distance
  2. B PT-symmetry 1d non-Hermiant SSH model
    1. B.1 Divergence behavior of phase boundary
    2. B.2 Different definitions of $F$
    3. B.3 Fitting the phase boundary behavior
    4. B.4 Hermitian case
    5. B.5 Divergence coefficient
    6. B.6 Open boundary conditions
  3. C 1D Kitaev toy model
    1. C.1 Hermitian model
    2. C.2 Divergence coefficient
    3. C.3 1D non-Hermitian Kitaev toy model
  4. D 2D topological superconductor model
    1. D.1 Hermitian model
    2. D.2 Divergence coefficient
    3. D.3 2D non-Hermitian topological superconductor model
  5. E 2d P-wave SC
  6. F 3D spin-orbit Coupled Degenerate Fermi Gases
  7. G Manifold Distance of Simplified Hamiltonian
    1. G.1 Hermitian case
    2. G.2 Non-Hermitian case
  8. H The Kitaev honeycomb model
  9. I Summary of the Critical Divergence Behavior of $D$ at the Phase Boundary

##  Appendix A Manifold distance

The ability to distinguish between quantum states is crucial for various
quantum operations. Typically, fidelity and trace distance are employed for
this purpose.

Fidelity is a measure used to quantify the similarity between two quantum
states, with values ranging from 0 to 1. A fidelity value of 0 indicates
complete dissimilarity, while a value of 1 signifies perfect identity.
Fidelity is particularly relevant when the system is in a pure state. For pure
states, fidelity is given by:

$\displaystyle F(|\psi\rangle,|\phi\rangle)=|<\psi|\phi>\mid.$ (24)

For mixed states, it is given by:

$\displaystyle
F(\rho,\sigma)=\operatorname{Tr}\left[\sqrt{\rho^{1/2}\sigma\rho^{1/2}}\right]$
(25)

Trace distance, on the other hand, quantifies the distance between two quantum
states and is defined as:

$\displaystyle
D\left(\rho_{1},\rho_{2}\right)=\frac{1}{2}\left\|\rho_{1}-\rho_{2}\right\|_{1}$
(26)

where $\|M\|_{1}=\sqrt{M^{\dagger}M}$ represents the Schatten-1 norm.

The relationship between trace distance and fidelity is given by:

$\displaystyle 1-F\left(\rho_{1},\rho_{2}\right)\leq
D\left(\rho_{1},\rho_{2}\right)\leq\sqrt{1-F^{2}\left(\rho_{1},\rho_{2}\right)},$
(27)

with equality holding if and only if both quantum states are pure states.

As an illustration in a two-energy band system, consider two pure states
$|*\rangle\phi$ and $|*\rangle\psi$. Assuming the existence of two orthogonal
states $|*\rangle 0$ and $|*\rangle 1$, such that:

$\displaystyle\ket*{\phi}=\ket*{0};\ket*{\psi}=\cos(\theta)\ket*{0}+\sin(\theta)\ket*{1},$
(28)

then the density matrices $\rho$ and $\sigma$ are given by:

$\displaystyle\rho=\bra*{\phi}\ket*{\phi};\sigma=\bra*{\psi}\ket*{\psi},$ (29)

For a two-dimensional matrix, assuming the orthogonal states to be:

$\displaystyle\ket*{0}=\left(\begin{array}[]{cc}1\\\ 0\\\
\end{array}\right);\ket*{1}=\left(\begin{array}[]{cc}0\\\ 1\\\
\end{array}\right),$ (34)

we have

$\displaystyle\sqrt{1-|\bra*{\phi}\ket*{\psi}|^{2}}=|\sin(\theta)|,$ (35)

and

$\displaystyle\frac{1}{2}Tr|\ket*{\phi}\bra*{\phi}-\ket*{\psi}\bra*{\psi}|$
$\displaystyle=\frac{1}{2}Tr|\left(\begin{array}[]{cc}1-\cos^{2}(\theta)&-\cos(\theta)\sin(\theta)\\\
-\cos(\theta)\sin(\theta)&-\sin^{2}(\theta)\\\ \end{array}\right)|$ (38)
$\displaystyle=|\sin(\theta)|,$ (39)

Finally, we have

$\displaystyle\frac{1}{2}Tr|\rho-\sigma|=\sqrt{1-|\bra*{\phi}\ket*{\psi}|^{2}}.$
(40)

##  Appendix B PT-symmetry 1d non-Hermiant SSH model

The Su-Schrieffer-Heeger model (SSH model) is a simple low-dimensional
topological system initially used to study polyacetylene organic molecules (in
which carbon atoms are arranged alternately with carbon atoms through single
and double bonds), which is a one-dimensional composition of spinless Fermi
lattice, where only interactions between nearest-neighbor lattices are taken
into account, and the interactions are characterized by alternating strong and
weak couplings. This property is also characteristic of many organic dimeric
compounds. Here we do not consider spin effects, but only the contribution of
leaps between electrons. In this section, we would demonstrate the equivalence
of various definitions of manifold distance in describing phase boundaries
(both for Hermitian and certain non-Hermitian systems), and in subsequent
sections we would use only one of these definitions Su  _et al._ (1980); Zhu
_et al._ (2014); Lieu (2018); Yao and Wang (2018). The Hamiltonian is given
by:

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}0&\frac{\gamma}{2}+t+(t^{{}^{\prime}}+t_{c})\cos(k)-i(t^{{}^{\prime}}-t_{c})\sin(k)\\\
-\frac{\gamma}{2}+t+(t^{{}^{\prime}}+t_{c})\cos(k)+i(t^{{}^{\prime}}-t_{c})\sin(k)&0\\\
\end{array}\right),$ (43)

where $t_{c}$ characterizes the next-nearest-neighbor interaction. We simplify
the Hamiltonian by neglecting $t_{c}$:

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}0&\frac{\gamma}{2}+t+t^{{}^{\prime}}\cos(k)-it^{{}^{\prime}}\sin(k)\\\
-\frac{\gamma}{2}+t+t^{{}^{\prime}}\cos(k)+it^{{}^{\prime}}\sin(k)&0\\\
\end{array}\right),$ (46)

The phase boundary of the system with periodic boundary conditions (PBCs)
satisfies:

$\displaystyle t=t^{{}^{\prime}}\pm(\frac{\gamma}{2});\quad
t=-t^{{}^{\prime}}\pm(\frac{\gamma}{2}).$ (47)

###  B.1 Divergence behavior of phase boundary

Consider

$\displaystyle F=1-|\bra*{\phi_{1}}\ket*{\psi_{2}}|^{2},$ (48)

the manifold distance $D$ and its derivative as

$\displaystyle D=\int_{BZ}Fdk;\quad D^{{}^{\prime}}=\int_{BZ}\frac{\partial
F}{\partial t_{2}}dk;\quad
D^{{}^{\prime\prime}}=\int_{BZ}\frac{\partial^{2}F}{\partial t_{2}^{2}}dk,$
(49)

Parameters | $\gamma_{1}$ | $\gamma_{2}$ | $t_{1}$ | $t_{1}^{{}^{\prime}}$ | $t_{2}^{{}^{\prime}}$ | Critical point 1 | Critical point 2 | Critical point 3 | Critical point 4
---|---|---|---|---|---|---|---|---|---
1 | 1.3 | 0.4 | 0.7 | 0.3 | 0.6 | -0.8 | -0.4 | 0.4 | 0.8
2 | 1.3 | 0.4 | 0.7 | 0.3 | 0.7 | -0.9 | -0.5 | 0.5 | 0.9
3 | 1.3 | 0.4 | 0.7 | 0.3 | 0.8 | -1.0 | -0.6 | 0.6 | 1.0
Table 1: Three sets of parameters used to study the phase boundary divergence
behavior.

The divergence behavior of $D$ and its derivatives near the phase boundary is
illustrated as Fig. 5.

Figure 5: As depicted in the figure, there are no divergence points other than
the phase transition point. The manifold distance $D$ undergoes a continuous
and smooth transition at the critical point, $D^{{}^{\prime}}$ is continuous
but not smooth through the phase transition point; and $D^{{}^{\prime\prime}}$
diverges in the critical points.

Similar conclusions are obtained for different choices of $k$ when shifted by
a constant $c$. This indicates that $D^{\prime\prime}$ effectively
characterizes the phase boundary regardless of the specific form of $k$, as
long as the momentum is within the Brillouin zone. We select three parameter
sets from table (2), each assigned a different $k^{\prime}$ for
$c=0,0.6,1.2,1.8$.

Parameters | $\gamma_{1}$ | $\gamma_{2}$ | $t_{1}$ | $t_{1}^{{}^{\prime}}$ | $t_{2}^{{}^{\prime}}$ | Critical point 1 | Critical point 2 | Critical point 3 | Critical point 4
---|---|---|---|---|---|---|---|---|---
1 | 2.3 | -0.5 | -0.7 | 0.3 | -0.2 | -0.45 | -0.05 | 0.05 | 0.45
2 | 0.3 | -0.7 | 1.7 | -2.44 | -1.1 | -1.45 | -0.75 | 0.75 | 1.45
3 | -1.87 | -0.8 | -5.7 | -7.7 | 0.2 | -0.6 | -0.2 | 0.2 | 0.6
Table 2: Parameters used for comparing
$D,D^{{}^{\prime}},D^{{}^{\prime\prime}}$ behavior under the condition
$k^{{}^{\prime}}=k+c$. Figure 6: As shown in the figure, when different form
of $k$ and $k^{{}^{\prime}}$ are chosen, the function curve $D$ and
$D^{{}^{\prime}}$ would be different, but $D^{{}^{\prime\prime}}$ also diverge
at the phase transition points.

###  B.2 Different definitions of $F$

Consider the four definitions of $F$ as follow

$\displaystyle F_{1}=1-|\bra*{\phi_{1}}\ket*{\phi_{2}}|^{2};\quad
F_{2}=1-|\bra*{\psi_{1}}\ket*{\psi_{2}}|^{2};\quad
F_{3}=1-|\bra*{\psi_{1}}\ket*{\phi_{2}}|^{2};\quad
F_{4}=1-|\bra*{\phi_{1}}\ket*{\psi_{2}}|^{2},$ (50)

the

$\displaystyle H\ket*{\phi}=E\ket*{\phi};\quad
H^{\dagger}\ket*{\psi}=E^{*}\ket*{\psi},$ (51)

Assume that

$\displaystyle\ket*{\phi_{1}}=(x_{1}+iy_{1},u_{1}+iv_{1})^{T};\quad\ket*{\phi_{2}}=(x_{2}+iy_{2},u_{2}+iv_{2})^{T};$
(52)
$\displaystyle\ket*{\psi_{1}}=(x_{1}-iy_{1},u_{1}-iv_{1})^{T};\quad\ket*{\psi_{2}}=(x_{2}-iy_{2},u_{2}-iv_{2})^{T},$
$\displaystyle\bra*{\phi_{1}}=(x_{1}-iy_{1},u_{1}-iv_{1});\quad\bra*{\phi_{2}}=(x_{2}-iy_{2},u_{2}-iv_{2});$
(53)
$\displaystyle\bra*{\psi_{1}}=(x_{1}+iy_{1},u_{1}+iv_{1});\quad\bra*{\psi_{2}}=(x_{2}+iy_{2},u_{2}+iv_{2}),$

then

$\displaystyle|\bra*{\phi_{1}}\ket*{\phi_{2}}|^{2}$
$\displaystyle=\bra*{\phi_{1}}\ket*{\phi_{2}}\bra*{\phi_{2}}\ket*{\phi_{1}}$
(54)
$\displaystyle=\\{(x_{1}+iy_{1})(x_{2}-iy_{2})+(u_{1}+iv_{1})(u_{2}-iv_{2})\\}\\{(x_{1}-iy_{1})(x_{2}+iy_{2})+(u_{1}-iv_{1})(u_{2}+iv_{2})\\},$

$\displaystyle|\bra*{\psi_{1}}\ket*{\psi_{2}}|^{2}$
$\displaystyle=\bra*{\psi_{1}}\ket*{\psi_{2}}\bra*{\psi_{2}}\ket*{\psi_{1}}$
(55)
$\displaystyle=\\{(x_{1}+iy_{1})(x_{2}-iy_{2})+(u_{1}+iv_{1})(u_{2}-iv_{2})\\}\\{(x_{1}-iy_{1})(x_{2}+iy_{2})+(u_{1}-iv_{1})(u_{2}+iv_{2})\\},$

$\displaystyle|\bra*{\psi_{1}}\ket*{\phi_{2}}|^{2}$
$\displaystyle=\bra*{\psi_{1}}\ket*{\phi_{2}}\bra*{\phi_{2}}\ket*{\psi_{1}}$
(56)
$\displaystyle=\\{(x_{1}-iy_{1})(x_{2}-iy_{2})+(u_{1}-iv_{1})(u_{2}-iv_{2})\\}\\{(x_{1}+iy_{1})(x_{2}+iy_{2})+(u_{1}+iv_{1})(u_{2}+iv_{2})\\},$

$\displaystyle|\bra*{\phi_{1}}\ket*{\psi_{2}}|^{2}$
$\displaystyle=\bra*{\phi_{1}}\ket*{\psi_{2}}\bra*{\psi_{2}}\ket*{\phi_{1}}$
(57)
$\displaystyle=\\{(x_{1}-iy_{1})(x_{2}-iy_{2})+(u_{1}-iv_{1})(u_{2}-iv_{2})\\}\\{(x_{1}+iy_{1})(x_{2}+iy_{2})+(u_{1}+iv_{1})(u_{2}+iv_{2})\\},$

therefore

$\displaystyle|\bra*{\phi_{1}}\ket*{\phi_{2}}|^{2}=|\bra*{\psi_{1}}\ket*{\psi_{2}}|^{2};\quad|\bra*{\psi_{1}}\ket*{\phi_{2}}|^{2}=|\bra*{\phi_{1}}\ket*{\psi_{2}}|^{2},$
(58)

Finally, we have proved that these four definitions are actually equivalent.

$\displaystyle F_{1}=F_{2};\quad F_{3}=F_{4}.$ (59)

Noteworthily, if the eigen-wave function of $H$ and $H_{d}$ are not of the
following form

$\displaystyle(x+iy,u+iy),\quad(x-iy,u-iy)$ (60)

The divergence may be arise due to the normalization of wave-function.
Therefore, for non-Hermitian systems, it is preferable to choose the forms of
$F_{1}$ or $F_{2}$ to avoid this issue.

###  B.3 Fitting the phase boundary behavior

The $D^{{}^{\prime\prime}}$ is divergent near the phase boundary, its
divergence behavior and the fitting function are shown below. It is found that
all phase boundaries exhibit the superposition divergence law

$\displaystyle
D^{{}^{\prime\prime}}\propto\frac{a}{\sqrt{|t_{2}-t_{2}^{*}|}}+bln(|t_{2}-t_{2}^{*}|)$
(61)

The parameters of table (B.3) are chosen to fit the phase boundary.

Parameters | $\gamma_{1}$ | $\gamma_{2}$ | $t_{1}$ | $t_{1}^{{}^{\prime}}$ | $t_{2}^{{}^{\prime}}$ | Critical point 1 | Critical point 2 | Critical point 3 | Critical point 4
---|---|---|---|---|---|---|---|---|---
1 | 1.3 | 0.4 | 0.7 | 0.3 | 0.6 | -0.8 | -0.4 | 0.4 | 0.8
Figure 7: Fitting of the phase boundary near the four phase transition points.
Horizontal coordinates $x=t_{2}-t_{2}^{*}$, where $t_{2}^{*}$ represents each
critical point. The fit curve for Fig. (a)-(h) are:
$\frac{0.00784}{\sqrt{|t_{2}-t_{2}^{*}|}}+5.87313\ln(|t_{2}-t_{2}^{*}|)-0.91928$,
$\frac{2.03932}{\sqrt{|t_{2}-t_{2}^{*}|}}+6.84244\ln(|t_{2}-t_{2}^{*}|)+5.89555$,
$\frac{-2.01754}{\sqrt{|t_{2}-t_{2}^{*}|}}-6.14734\ln(|t_{2}-t_{2}^{*}|)-7.21072$,
$\frac{-0.00274}{\sqrt{|t_{2}-t_{2}^{*}|}}-5.62399\ln(|t_{2}-t_{2}^{*}|)-3.12874$,
$\frac{-2.09107}{\sqrt{|t_{2}-t_{2}^{*}|}}+4.52992\ln(|t_{2}-t_{2}^{*}|)+20.2738$,
$\frac{0.004}{\sqrt{|t_{2}-t_{2}^{*}|}}-4.89576\ln(|t_{2}-t_{2}^{*}|)-3.26324$,
$\frac{-2.11250}{\sqrt{|t_{2}-t_{2}^{*}|}}-6.24336\ln(|t_{2}-t_{2}^{*}|)-12.04630$.

###  B.4 Hermitian case

When $\gamma\to 0$, the model reverts to the Hermitian case, and the
Hamiltonian is given by:

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}0&t+t^{{}^{\prime}}\cos(k)-it^{{}^{\prime}}\sin(k)\\\
t+t^{{}^{\prime}}\cos(k)+it^{{}^{\prime}}\sin(k)&0\\\ \end{array}\right),$
(64)

Compared to the non-Hermitian case, there are only two phase transition points
and the phase boundary is

$\displaystyle t=\pm t^{{}^{\prime}},$ (65)

The behavior of $D^{{}^{\prime}}$ and the parameter table are shown as follow

Parameters | $t_{1}$ | $t_{1}^{{}^{\prime}}$ | $t_{2}^{{}^{\prime}}$ | Critical point 1 | Critical point 2
---|---|---|---|---|---
1 | -0.7 | 0.3 | -0.2 | -0.2 | -0.2
2 | 1.7 | -2.44 | -1.1 | -1.1 | 1.1
3 | -5.7 | -7.7 | 0.2 | -0.2 | 0.2
Figure 8: ”$D^{\prime}$ exhibits logarithmic divergence, the Horizontal
coordinates $x=\ln\left(\left|t_{2}-t_{2}^{\prime}\right|\right)$.

###  B.5 Divergence coefficient

Now let’s determine the logarithmic divergence coefficient for $D^{\prime}$.
Suppose $D^{\prime}\propto$ $k_{i}x+b$, where
$x=\ln\left(t_{2}-t_{2}^{\prime}\right)$, and we observed that $k_{i}$ is
mainly influenced by $t_{2}$.

Parameters | $t_{1}$ | $t_{1}^{{}^{\prime}}$ | $t_{2}^{{}^{\prime}}$ | critical point 1 | critical point 2
---|---|---|---|---|---
1 | -1.0 $\sim$ 1.0 | 0.3 | -0.2 | -0.2 | -0.2
2 | -0.7 | -1.0$\sim$1.0 | -0.2 | -0.2 | -0.2
3 | -0.7 | 0.3 | -1.0$\sim$1.0 | -$t_{2}^{{}^{\prime}}$ | $t_{2}^{{}^{\prime}}$
Figure 9: As shown in this figure, while keeping the other parameters fixed
and varying only one set of parameters, the trend of $k_{i}$ with respect to
parameter changes is depicted. It can be observed that the divergence
coefficient $k_{i}$ is primarily influenced by the parameter $t_{2}^{\prime}$,
while the other parameters only cause slight variations or alter its sign.

According to the above analysis, it can be concluded that the divergence
coefficient is mainly influenced by $t_{2}^{\prime}$, and the singular
behavior of $D^{\prime}$ is primarily caused by the divergence of $F^{\prime}$
at $k=\pi$. Therefore, we obtain the simplified expression for $F$ at $k=\pi$:

$\displaystyle
F=\frac{1}{4}(2+\frac{2(t_{1}t_{2}+t_{1}^{{}^{\prime}}t_{2}^{{}^{\prime}}+(t_{2}t_{1}^{{}^{\prime}}+t_{1}t_{2}^{{}^{\prime}})\cos(k))}{(e^{ik}t_{1}+t_{1}^{{}^{\prime}})(e^{ik}t_{2}+t_{2}^{{}^{\prime}})}),$
(66)

Integrate the expression over the interval $(0,2\pi)$ and selecting the
divergent terms:

$\displaystyle\frac{(t_{2}+t_{2}^{{}^{\prime}})(-t_{2}+t_{2}^{{}^{\prime}})\ln(-t_{2}+t_{2}^{{}^{\prime}})}{2(t_{2}^{{}^{\prime}})^{2}},$
(67)

let

$\displaystyle x=-t_{2}+t_{2}^{{}^{\prime}},$ (68)

then

$\displaystyle\frac{x\ln(x)}{t_{2}^{{}^{\prime}}},$ (69)

Taking the derivative with respect to $t_{2}$, we have:

$\displaystyle\frac{\ln(x)}{t_{2}^{{}^{\prime}}}.$ (70)

In the end, we obtain the divergence coefficient, namely

$\displaystyle
D^{{}^{\prime}}\propto\frac{1}{t_{2}^{{}^{\prime}}}\ln(|t_{2}-t_{2}^{{}^{\prime}}|).$
(71)

###  B.6 Open boundary conditions

In quantum mechanics, while Hermitian Hamiltonians are conventionally
associated with real eigenvalues, Hermiticity is not a necessary condition. If
a system possesses combined spatial inversion and time-reversal symmetry (PT
symmetry), even if the Hamiltonian is non-Hermitian, there exists a parameter
space (PT-symmetry unbroken region) where energy eigenvalues are guaranteed to
be real Magorrian  _et al._ (1998); Bender and Boettcher (1998). When the
system transitions from the PT-symmetry unbroken region to the PT-symmetry
broken region, it must pass through exceptional points, presenting a unique
phenomenon in non-Hermitian systems.

Experimentally, PT-symmetric non-Hermitian quantum mechanics can be realized
in classical wave systems, such as optical systems, by introducing gain and
loss.

In the realm of non-Hermitian SSH models, a remarkable phenomenon arises
regarding the localization of eigenstates. Apart from localized zero-energy
boundary states, the ”bulk” wavefunction of the system also exhibits
localization, departing from the conventional Bloch wavefunctions Fu and Wan
(2022); Zhang  _et al._ (2020a). We consider the trial wavefunction in real
space as
$|\psi\rangle=\sum_{n}\beta^{n}\left(\phi_{A}|n,A\rangle+\phi_{B}|n,B\rangle\right)$.
By applying the eigenvalue equation $\hat{H}_{k}|\psi\rangle=E|\psi\rangle$,
we obtain the bulk equations

$\displaystyle{\left[\left(t+\gamma\right)+t^{\prime}\beta^{-1}\right]\phi_{B}=E\phi_{A},}$
$\displaystyle{\left[\left(t-\gamma\right)+t^{\prime}\beta\right]\phi_{A}=E\phi_{B}.}$

Hence,

$\left[\left(t-\gamma\right)+t^{\prime}\beta\right]\left[\left(t+\gamma\right)+t^{\prime}\beta^{-1}\right]=E^{2},$

which yields two solutions $\beta_{1,2}(E)$, satisfying
$\beta_{1}\beta_{2}=\left(t-\gamma\right)/\left(t+\gamma\right)$. The general
bulk eigenstates are expressed as linear combinations of the two $\beta$
solutions, i.e.,

$\displaystyle|\Psi\rangle=\left|\psi_{1}\right\rangle+\left|\psi_{2}\right\rangle$
$\displaystyle\left|\psi_{j}\right\rangle=\sum_{n}\beta_{j}^{n}\left(\phi_{A}^{(j)}|n,A\rangle+\phi_{B}^{(j)}|n,B\rangle\right),j=1,2$

Introducing open boundary conditions, we obtain boundary equations

$\displaystyle\left(t+\gamma\right)\psi_{1,B}-E\psi_{1,A}=0,$
$\displaystyle\left(t-\gamma\right)\psi_{N.A}-E\psi_{N,B}=0,$

where
$\psi_{n,q}=\beta_{1}^{n}\phi_{q}^{(1)}+\beta_{2}^{n}\phi_{q}^{(2)},q=A,B,n=1,2,\cdots,N$.
Combining bulk and boundary equations, we obtain

$\beta_{1}^{N+1}\left(t-\gamma+t^{\prime}\beta_{2}\right)=\beta_{2}^{N+1}\left(t-\gamma+t^{\prime}\beta_{1}\right).$

In the limit of large $N$ or in the thermodynamic limit, the condition
satisfied by the above equation is $|\beta_{1}|=|\beta_{2}|$, along with the
combined condition
$\beta_{1}\beta_{2}=\left(t-\gamma\right)/\left(t+\gamma\right)$. Thus, the
condition for the existence of bulk eigenstates is given by

$|\beta_{1}|=|\beta_{2}|\equiv
r=\sqrt{\left|\frac{t-\gamma}{t+\gamma}\right|}.$

When $r<1$, all bulk wavefunctions are localized at the left boundary of the
system, which is known as the non-Hermitian skin effect. The occurrence of the
non-Hermitian skin effect renders the Bloch band theory ineffective.
Consequently, we obtain the bulk state spectrum with open boundaries in the
generalized Brillouin zone as

$E_{\pm}(k)=\pm\sqrt{t^{2}+(t^{\prime})^{2}-\gamma^{2}+t^{\prime}\sqrt{t^{2}-\gamma^{2}}\left[\operatorname{sgn}\left(t+\gamma\right)e^{ik}+\operatorname{sgn}\left(t-\gamma\right)e^{-ik}\right]}$

$\beta$ solutions corresponding to zero-energy boundary states are
$\beta_{1,2}^{E\rightarrow
0}=-\frac{t-\gamma}{t^{\prime}},-\frac{t^{\prime}}{t+\gamma}$. The critical
point satisfies the condition

$\left|\beta_{1,2}^{E\rightarrow 0}\right|=r$

This equation yields the critical point
$t=\pm\sqrt{(t^{\prime})^{2}+\gamma^{2}}$.

##  Appendix C 1D Kitaev toy model

###  C.1 Hermitian model

In 2001, Kitaev proposed a one-dimensional theoretical model capable of
realizing Majorana fermionic states Kitaev (2001). In the trivial phase, all
Majorana fermions are confined to the same lattice and form pairs. This
implies that all Majorana fermions are bound within the lattice, resulting in
ordinary fermions in 1d chains. In contrast, the topological phase, each end
of the one-dimensional chain is left with an unpaired Majorana fermion, while
the others form pairs. Therefore, in the topological phase, there is the
prospect of finding Majorana fermions at the ends of the chain. The existence
of Majorana zero modes is a characteristic feature of topological
superconductors. Majorana fermions, originally elusive in particle physics,
have now been realized in condensed matter systems. In this section, we would
consider the conventional one-dimensional Kitaev model, obtained by directly
Fourier transforming a lattice system with p-wave superconducting pairing.
This Hamiltonian satisfies particle-hole symmetry.

Let $\mu_{2}^{*}$ and $-\mu_{2}^{*}$ represent the positive and negative
critical points of the system, respectively; $-\mu_{2}^{*}-\epsilon$,
$-\mu_{2}^{*}+\epsilon$, $\mu_{2}^{*}-\epsilon$ and $\mu_{2}^{*}+\epsilon$
represent four distinct neighborhoods of critical points. For convenience,
these regions are labeled from left to right as Region I, II, III and IV. The
illustration is provided as follow.

The Hamiltonian and phase boundary equation are given by:

$\displaystyle H_{k}=\left(\begin{array}[]{cc}-2t\cos(k)-\mu&\alpha\sin(k)\\\
\alpha\sin(k)&2t\cos(k)+\mu\\\ \end{array}\right)$ (74)

$\displaystyle 4t^{2}\cos^{2}(k)+\mu^{2}+4t\mu\cos(k)+\alpha^{2}\sin^{2}(k)=0$
(75)

Choosing $k=0,\pi$ to substitute into the above equation, we have

$\displaystyle\mu=\pm 2t$ (76)

The parameters of table (C.1) are chosen to fit the phase boundary.

Parameters | $\alpha_{1}$ | $\alpha_{2}$ | $t_{1}$ | $t_{2}$ | $\mu_{1}$ | $\mu_{2}^{*}=\pm 2t_{2}$
---|---|---|---|---|---|---
1 | 1.5 | -1.0 | 1.4 | 0.5 | -1.1 | $\pm 1.0$
2 | 1.5 | 1.0 | 2.7 | 0.4 | 1.1 | $\pm 0.8$
3 | 1.2 | 1.0 | 1.5 | 0.3 | 1.1 | $\pm 0.6$
4 | 1.5 | 1.0 | 2.7 | 0.6 | 1.4 | $\pm 1.2$
5 | 1.5 | 1.0 | 2.7 | 0.45 | 1.3 | $\pm 0.9$
6 | 1.5 | -1.0 | 1.7 | -0.35 | 1.1 | $\pm 0.7$
Figure 10: Fitting of $D^{{}^{\prime}}$ near the region I and II, here
$\mu_{2}-\mu_{2}^{*}$. Figure 11: Fitting of $D^{{}^{\prime\prime}}$ near the
region I and II, here $\mu_{2}-\mu_{2}^{*}$. Figure 12: Fitting of
$D^{{}^{\prime}}$ near the region III and IV, here $\mu_{2}^{*}-\mu_{2}$.
Figure 13: Fitting of $D^{{}^{\prime\prime}}$ near the region III and IV, here
$\mu_{2}^{*}-\mu_{2}$. Figure 14: Fig. (a)$\sim$(d) show the logarithmically
divergence near the region (I)-(IV).

###  C.2 Divergence coefficient

The divergence of $D^{\prime}$ originates from the singularity of the
integrated function $F^{\prime}$. At the singularity, $F^{\prime}$ must be
replaceable by a simple expression, given by:

$\displaystyle F_{sim}=\sqrt{\frac{w^{2}}{w^{2}+\alpha_{2}^{2}d_{k}^{2}}}$
(77) $\displaystyle
w=-2t_{2}+\mu_{2}+\sqrt{\alpha_{2}^{2}d_{k}^{2}-(2t_{2}-\mu_{2})(2t_{2}(-1+d_{k}^{2})+\mu_{2})};\quad
d_{k}=k-\pi$ (78)

Finally we have the divergence coefficient of $D^{{}^{\prime}}$:

$\displaystyle D^{{}^{\prime}}=k_{i}\ln(|\mu_{2}-\mu_{2}^{*}|),\quad
k_{i}=\frac{1}{\sqrt{2}\alpha_{2}}.$ (79)

It can be inferred that $|\alpha_{2}|$ has the greatest impact on the slope,
while the influence of other parameters is relatively small. When one of the
parameters is changed while keeping the others fixed, the properties of the
divergence coefficient are depicted in Figure 15.

Parameters | $\alpha_{1}$ | $\alpha_{2}$ | $t_{1}$ | $t_{2}$ | $\mu_{1}$
---|---|---|---|---|---
1 | -1.73 | 0.63 | 0.9 | -0.85 | 2.1
Figure 15: Using the parameters from Table C.2, numerical calculations were
performed to obtain the slope $k_{i}$, where the phase transition point is
$\mu_{2}^{*}=-2t_{2}$.

###  C.3 1D non-Hermitian Kitaev toy model

The non-Hermitian Kitaev toy model as

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}-2t\cos(k)-\mu+i\gamma&\alpha\sin(k)\\\
\alpha\sin(k)&2t\cos(k)+\mu-i\gamma\\\ \end{array}\right)$ (82)

The phase boundary equation is:

$\displaystyle\mu^{2}+4t\mu\cos(k)+4t^{2}\cos^{2}(k)+\alpha^{2}\sin^{2}(k)-\gamma^{2}+i[-2\gamma
u-4\gamma t\cos(k)]=0$ (83)

Solving this equation, we have:

$\displaystyle\mu=\pm
2t\sqrt{1-\frac{\gamma^{2}}{\alpha^{2}}},\quad\gamma^{2}\leq\alpha^{2}$ (84)

The parameters of table (C.3) are chosen to fit the phase boundary.

Parameters | $\alpha_{1}$ | $\alpha_{2}$ | $t_{1}$ | $t_{2}$ | $\gamma_{1}$ | $\gamma_{2}$ | $\mu_{1}$ | $\mu_{2}^{*}=\pm 2t_{2}(\sqrt{1-\gamma_{2}/\alpha_{2}})$
---|---|---|---|---|---|---|---|---
1 | 1.5 | -1.7 | 2.7 | 0.5 | 1.4 | 0.8 | 1.1 | $\pm 0.88$
2 | 1.5 | -1.7 | 2.7 | 0.5 | 0.014 | 0.0045 | 1.1 | $\pm 0.99996$
3 | 1.5 | -1.7 | 2.7 | 0.5 | 0.0014 | 0.0008 | 1.1 | $\pm 1$
4 | 1.5 | -1.7 | 2.7 | 0.5 | 0.0014 | 0.00065 | 1.1 | $\pm 1$
5 | 1.5 | -1.7 | 2.7 | 0.5 | 0.0014 | 0.00055 | 1.1 | $\pm 1$
6 | 1.5 | -1.7 | 2.7 | 0.5 | 0.0014 | 0.00045 | 1.1 | $\pm 1$
Figure 16: As shown in this figure, $D^{{}^{\prime\prime}}$ are always
diverges in region (I), and only when $\gamma_{2}$ is small enough, it would
exhibit divergent behavior in region (II). Figure 17: As shown in this figure,
$D^{{}^{\prime\prime}}$ are always diverges in region (IV), and only when
$\gamma_{2}$ is small enough, it would exhibit divergent behavior in region
(III).

(1). When $\gamma$ is large enough($\gamma_{2}>10^{-6}$), the divergence in
region (I) and (IV) is given by:

$\displaystyle
D^{{}^{\prime\prime}}\propto\frac{C_{1}}{\mu_{2}-\mu_{2}^{*}}+\frac{C_{2}}{\sqrt{|\mu_{2}-\mu_{2}^{*}|}},$
(85)

regions (II) and (III) do not exhibit divergent behavior.

(2). When $\gamma$ is small enough($\gamma_{2}<10^{-6}$), the divergence in
region (I) to (IV) is given by:

$\displaystyle D^{{}^{\prime\prime}}\propto\frac{C_{1}}{\mu_{2}-\mu_{2}^{*}}$
(86)

Next, we show that the non-Hermitian model reverts to Hermitian case. As shown
in Fig. 18 to 20, when $\gamma\to 0$, the divergence behavior of the phase
boundary also reback to the Hermitian case.

Figure 18: The left panel shows the divergence behavior of the region (I),
where $x=\frac{1}{\mu_{2}-\mu_{2}^{*}}$. When $\gamma_{2}$ takes a larger
value, the divergence behavior is
$\frac{C_{1}}{\mu_{2}-\mu_{2}^{*}}+\frac{C_{2}}{\sqrt{|\mu_{2}-\mu_{2}^{*}|}}$,
as $\gamma_{2}$ decreases, the curve becomes straight gradually, so it is
proved that the divergence behavior has returned to the traditional 1d Kitaev
toy model. the right figure shows the divergence behavior ofregion (IV), the
same as left figure. Figure 19: The left panel shows the divergence behavior
of the region (II), where $x=\frac{1}{\mu_{2}^{*}-\mu_{2}}$. When $\gamma_{2}$
takes a larger value, the divergence behavior is
$\frac{C_{1}}{\mu_{2}-\mu_{2}^{*}}+\frac{C_{2}}{\sqrt{|\mu_{2}-\mu_{2}^{*}|}}$,
as $\gamma_{2}$ decreases, the curve becomes straight gradually, so it is
proved that the divergence behavior has returned to the traditional 1d Kitaev
toy model. the right figure shows the divergence behavior ofregion (III), the
same as left figure. Figure 20: The left and right figure show the
$D^{{}^{\prime\prime}}$ in region (II) and (III). As $\gamma$ decreases, the
divergence behavior gradually becomes evident, eventually reverting to the
Hermitian case.

##  Appendix D 2D topological superconductor model

###  D.1 Hermitian model

The 2D topological superconductor Hamiltonian show as

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}-2t(\cos(k_{x})+\cos(k_{y}))-\mu&\alpha(\sin(k_{x})+i\sin(k_{y}))\\\
\alpha(\sin(k_{x})-i\sin(k_{y}))&2t(\cos(k_{x})+\cos(k_{y}))+\mu\\\
\end{array}\right)$ (89)

the phase boundary satisfy:

$\displaystyle(-2t(\cos(k_{x})+\cos(k_{y}))-\mu)^{2}+\alpha|\sin(k_{x})+i\sin(k_{y})|^{2}=0$
(90)

that

$\displaystyle\mu^{2}+4t\mu\cos(k_{x})+4t^{2}\cos^{2}(k_{x})+4t\mu\cos(k_{y})+8t^{2}\cos(k_{x})\cos(k_{y})+4t^{2}\cos^{2}(k_{y})+\alpha^{2}\sin^{2}(k_{x})+\alpha^{2}\sin^{2}(k_{y})=0$
(91)

Let $k_{x},k_{y}$ take the values $0,\pi$ respectively, then we have the phase
boundaries of the system as:

$\displaystyle\mu=\pm 4t$ (92)

The parameters of table (D.1) are chosen to fit the phase boundary.

Parameters | $\alpha_{1}$ | $\alpha_{2}$ | $t_{1}$ | $t_{2}$ | $\mu_{1}$ | $\mu_{2}^{*}=\pm 4t_{2}$
---|---|---|---|---|---|---
1 | 1.5 | -1.0 | 1.7 | 0.45 | 1.1 | $\pm 1.8$
2 | 1.5 | -1.0 | 1.7 | -0.35 | 1.1 | $\pm 1.4$
3 | 1.5 | -1.0 | 1.7 | -0.55 | 1.1 | $\pm 2.2$
Figure 21: Fig. (a)$\sim$(f) show the logarithmically divergence in the region
(I)-(II), here $x=\mu-\mu^{*}$. Figure 22: Fig. (a)$\sim$(f) show the
logarithmically divergence in the region (III)-(IV), here $x=\mu-\mu^{*}$.
Figure 23: Fig. (a)$\sim$(f) show the logarithmically divergence in the region
(I)-(IV), here $X=\ln(|\mu-\mu^{*}|)$.

For 2D topological superconductor model, we have:

$\displaystyle D^{{}^{\prime\prime}}\propto Cln(|\mu_{2}-\mu_{2}^{*}|)$ (93)

###  D.2 Divergence coefficient

Similar to the one-dimensional model, the divergence coefficient is only
related to the parameter $\alpha_{2}$, while other parameters have a minor
impact on the divergence coefficient. We construct the following approximate
model:

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}\frac{k^{2}}{2m}-\mu&\alpha(\sin(k_{x})+i\sin(k_{y}))\\\
\alpha(\sin(k_{x})-i\sin(k_{y}))&\mu-\frac{k^{2}}{2m}\\\ \end{array}\right)$
(96)

Since the divergence coefficient is independent of the parameters in the first
group $\alpha_{1},\mu_{1},t_{1}$, assume $\psi_{1}=(1,0)$, then we have the
expression of $F$

$\displaystyle
F=\frac{\sqrt{1+\frac{k^{2}-2m\mu}{\sqrt{4\alpha^{2}k^{2}m^{2}+(k^{2}-2m\mu)^{2}}}}}{\sqrt{2}},$
(97)

note that

$\displaystyle\iint dk_{x}dk_{y}=2\pi\int dk$ (98)

so

$\displaystyle D=\int_{0}^{\Lambda}2\pi kFdk;\quad
D^{{}^{\prime\prime}}=\int_{0}^{\Lambda}2\pi
k\frac{\partial^{2}F}{\partial\mu^{2}}dk$ (99)

let $k\to 0$ and extract the divergent term from the above expression
$D^{{}^{\prime\prime}}$,

$\displaystyle\frac{m\sqrt{m}\alpha(8\alpha^{2}m^{2})\ln(\mu^{2})}{8(\alpha^{2}m-\mu)^{7/2}}$
(100)

When the parameters are close to the phase transition point, we can set
$\mu\to 0$, that

$\displaystyle\frac{2\ln(\mu)}{\alpha^{2}}$ (101)

Finally we have

$\displaystyle
D^{{}^{\prime\prime}}\propto\frac{2}{\alpha^{2}}ln(|\mu_{2}-\mu_{2}^{*}|)$
(102)

The figure below shows a comparison between the numerical results and the
fitting for the divergence coefficient $k_{i}$.

###  D.3 2D non-Hermitian topological superconductor model

The 2D non-hermitian topological superconductor model Hamiltonian and its
phase boundary are given by:

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}-2t(\cos(k_{x})+\cos(k_{y}))-\mu+i\gamma&\alpha(\sin(k_{x})+i\sin(k_{y}))\\\
\alpha(\sin(k_{x})-i\sin(k_{y}))&2t(\cos(k_{x})+\cos(k_{y}))+\mu-i\gamma\\\
\end{array}\right)$ (105) $\displaystyle\mu=\pm
2t(1+\sqrt{1-\frac{\gamma^{2}}{\alpha^{2}}}),\quad\gamma^{2}\leq\alpha^{2}$
(106)

The parameters of table (3) are chosen to fit the phase boundary.

Parameters | $\alpha_{1}$ | $\alpha_{2}$ | $t_{1}$ | $t_{2}$ | $\gamma_{1}$ | $\gamma_{2}$ | $\mu_{1}$ | $\mu_{2}^{*}=\pm 2t_{2}(1+\sqrt{1-\gamma_{2}/\alpha_{2}})$
---|---|---|---|---|---|---|---|---
1 | 1.5 | 1.0 | 2.7 | 0.25 | 0.4 | 0.4 | 1.1 | $\pm 0.118$
2 | 1.5 | 1.0 | 2.7 | 0.25 | 0.4 | 0.2 | 1.1 | $\pm 0.118$
3 | 1.5 | 1.0 | 2.7 | 0.25 | 0.04 | 0.05 | 1.1 | $\pm 0.118$
4 | 1.5 | 1.0 | 2.7 | 0.25 | 0.04 | 0.02 | 1.1 | $\pm 0.118$
5 | 1.5 | 1.0 | 2.7 | 0.25 | 0.004 | 0.008 | 1.1 | $\pm 0.118$
6 | 1.5 | 1.0 | 2.7 | 0.25 | 0.004 | 0.002 | 1.1 | $\pm 0.118$
Table 3: The parameter tables used in Fig. 24$\sim$LABEL:tr-17 is designed to
highlight the transitional behavior of the phase boundary neighborhoods from
the non-Hermitian 2D model to the conventional 2D Hermitian model as $\gamma$
gradually decreases. In the table, parameters other than
$\gamma_{1},\gamma_{2}$ remain unchanged, and both of them decrease
simultaneously. Figure 24: As shown in the left panel with
$x=\mu_{2}-\mu_{2}^{*}$, the fitting results for region (I) with Tab. 3 are
presented. It is evident from the plots that the superimposed
divergence($C_{1}ln(|\mu_{2}-\mu_{2}^{*}|)+C_{2}/\sqrt{|\mu_{2}-\mu_{2}^{*}|}$)provides
an excellent fit. The right panel displays the fitting results for region
(IV), showing similar divergence behavior as observed in region (I). Figure
25: As shown in the left panel with $x=\mu_{2}-\mu_{2}^{*}$, the fitting
results for region (I) with Tab. 3 are presented. Only logarithmic divergence
behavior ($Cln(|\mu_{2}-\mu_{2}^{*}|)$) is observed. The right panel displays
the fitting results for region (III), demonstrating similar divergence
behavior as observed in region (II).

Now, we summarize the conclusions regarding the divergence behavior of the 2D
topological superconductor model:

1\. If the non-Hermitian contributions to the system are significant, i.e.,
when $\gamma_{2}>10^{-3}$, the divergence behavior in region (I) and (IV) as
follow

$\displaystyle D^{{}^{\prime\prime}}\propto
C_{1}ln(|\mu_{2}-\mu_{2}^{*}|)+C_{2}/\sqrt{|\mu_{2}-\mu_{2}^{*}|}$ (107)

2\. If the non-Hermitian contributions to the system can be neglected, i.e.,
when $\gamma_{2}<10^{-3}$, the divergence behavior in region (I) and (IV) as
follow

$\displaystyle D^{{}^{\prime\prime}}\propto C_{1}ln(|\mu_{2}-\mu_{2}^{*}|)$
(108)

The divergence behavior gradually transitions to the Hermitian case.

3\. Regardless of the value of $\gamma$, both region (II) and (III) exhibit
logarithmic divergence, remaining consistent with the Hermitian case.

##  Appendix E 2d P-wave SC

Topological superconductors are a class of materials characterized by the
presence of gapless edge states and the existence of superconducting pairing.
In 2000, N. Read and D. Green constructed a two-dimensional p-wave
superconducting model based on the BCS theory of superconductivity Read and
Green (2000). This model involves spin-triplet pairing with orbital angular
momentum equal to 1, resulting in two distinct phases — the strong pairing
phase and the weak pairing phase. The weak pairing phase exhibits edge states
due to the presence of Majorana fermions at the system’s edge. In 2008, Fu and
Kane proposed the construction of topological superconductors by utilizing the
interaction between surface states of three-dimensional topological insulators
and s-wave superconductivity Fu and Kane (2008). In the Nambu basis, the
Hamiltonian can be expressed as a $4\times 4$ matrix. Essentially, this model
remains effectively two-dimensional. According to Fu and Kane’s work, J.D.
Sau, R.M. Lutchyn, and others utilized semiconductor thin films coupled to
s-wave superconductors to achieve p-wave pairing Lutchyn  _et al._ (2010).
Subsequently, an increasing number of materials have been employed to couple
with conventional s-wave superconductors to realize Majorana zero-energy
states. In this section, we investigate the 2D p-wave superconductors manifold
distance and subsequently extend our researches to spin-orbit coupling case.

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}\frac{k^{2}}{2m}-\mu&\alpha(k_{x}+ik_{y})\\\
\alpha(k_{x}-ik_{y})&-(\frac{k^{2}}{2m}-\mu)\\\ \end{array}\right).$ (111)

Its phase boundary satisfy

$\displaystyle(\frac{k^{2}}{2m}-\mu)^{2}+\alpha^{2}(k_{x}^{2}+k_{y}^{2})^{2}=0.$
(112)

The critical points are:

$\displaystyle\mu=0.$ (113)

The definition of manifold distance differs from previous systems as there is
no Brillouin zone. Consequently, we need to impose momentum cutoffs in our
analysis. Although the integral
$D=\iint_{-\infty}^{\infty}Fd_{kx}d_{ky}=2\pi\iint_{0}^{\infty}dk$ may exhibit
divergences, the cutoff $\Lambda$ ensures $D$ remains continuously smooth
across the phase transition point, its derivatives also exhibit singular
properties.

The parameters of table (E) are chosen to fit the phase boundary.

Parameters | m | $\alpha_{1}$ | $\alpha_{2}$ | $\mu_{1}$
---|---|---|---|---
1 | 0.5 | 1.5 | 1 | 0.7
2 | 0.5 | 1.5 | 1 | 2.1
3 | 0.5 | 1.7 | 1 | 1.4
4 | 0.5 | 1.5 | -1 | 1.6
5 | 0.5 | 1.5 | -1 | 1.9
6 | 0.5 | 1.5 | -1 | 2.2
Figure 26: As shown in the figure, utilizing the parameters from Tab. E, where
$x=\mu_{2}-\mu^{*}_{2}=\mu_{2}$, $D^{{}^{\prime\prime}}$ exhibits logarithmic
divergence on both sides of the phase transition point $\mu^{*}$. Figure 27:
As shown in the figure, let $x=ln(\mu_{2}-\mu_{2}^{*})$,
$D^{{}^{\prime\prime}}$ exhibits logarithmic divergence on both sides of the
phase transition point $\mu^{*}$.

The expression for $F_{k}$ is:

$\displaystyle
F_{k}=\frac{k}{\sqrt{2}}\sqrt{1+\frac{-k^{4}-4m^{2}\mu_{1}\mu_{2}+2mk^{2}(\mu_{1}+\mu_{2}-2m\alpha_{1}\alpha_{2})}{\sqrt{4\alpha_{1}^{2}k^{2}m^{2}+(k^{2}-2m\mu_{1})^{2}}\sqrt{4\alpha_{2}^{2}k^{2}m^{2}+(k^{2}-2m\mu_{2})^{2}}}}$
(114)

Due to the divergence of $F_{k}^{{}^{\prime\prime}}$ at the phase transition
point, which occurs at $k=0$, let $k\to 0,\mu_{2}\to 0$, we obtain:

$\displaystyle
F_{k}^{{}^{\prime\prime}}=\frac{-\alpha_{2}^{2}k^{3}(\alpha_{2}^{2}k^{2}\mu_{1}+6\mu_{2}(\mu_{1}\mu_{2}-|\mu_{1}|\sqrt{\alpha_{2}^{2}k^{2}+\mu_{2}^{2}}))}{4\sqrt{2}\mu_{1}(\alpha_{2}^{2}k^{2}+\mu_{2}^{2})^{3}(1-\frac{\mu_{1}\mu_{2}}{|\mu_{1}|\sqrt{\alpha_{2}^{2}k^{2}+\mu_{2}^{2}}})^{\frac{3}{2}}}$
(115)

$\displaystyle D^{{}^{\prime\prime}}=\int F_{k}^{{}^{\prime\prime}}dk$ (116)

let

$\displaystyle\mu_{2}=\pm\text{e}^{-x}$ (117)

where the positive and negative signs correspond to $\mu_{2}>0$ and
$\mu_{2}<0$ respectively. Then we have

(1). $\mu_{2}>0$

$\displaystyle
D^{{}^{\prime\prime}}(x)\propto-\frac{x}{4\sqrt{2}\alpha_{2}^{2}}$ (118)

(2). $\mu_{2}<0$

$\displaystyle
D^{{}^{\prime\prime}}(x)\propto\frac{x}{4\sqrt{2}\alpha_{2}^{2}}$ (119)

##  Appendix F 3D spin-orbit Coupled Degenerate Fermi Gases

Similarly, manifold distance can also be employed to a four-level system. For
the four-level system, only the intermediate two bands (with the opening and
closing of energy gap) exhibit divergence in $D^{{}^{\prime}}$ or
$D^{{}^{\prime\prime}}$ at the phase boundary, while the other two bands show
no divergence phenomenon.

We consider a 3D degenerate Fermi Gases introduced with a Rashba-type SOC in
the $xy$ plane and a perpendicular Zeeman field along the $z$ direction. In
experiment, 2D degenerate Fermi gases can be realized using a one-dimensional
deep optical lattice, and the Rashba SOC with Zeeman field can be realized
using the adiabatic atoms Zwierlein  _et al._ (2005); Gong  _et al._ (2011,
2012); Xiong  _et al._ (2023). The Hamiltonian for this system can be written
as $\left(\hbar=K_{B}=1\right)$:

$\displaystyle\quad H=H_{0}+H_{\mathrm{int}}.$ (120)

where the single-particle Hamiltonian is:

$\displaystyle
H_{0}=\sum_{\mathbf{k}\gamma\gamma^{\prime}}c_{\mathbf{k}\gamma}^{\dagger}\left[\xi_{\mathbf{k}}I+\alpha\left(k_{y}\sigma_{x}-k_{x}\sigma_{y}\right)+\Gamma\sigma_{z}\right]_{\gamma\gamma^{\prime}}c_{\mathbf{k}\gamma^{\prime}},$
(121)

with $\gamma=\uparrow,\downarrow$. Here $\mu$ is the chemical potential,
$\epsilon_{k}=\frac{k_{x}^{2}+k_{y}^{2}}{2m}$ is the free particle energy,
$\xi_{k}=\epsilon_{k}-\mu$ is the reduced particle energy, $\Gamma$ is the
strength of the Zeeman field, $\alpha$ is the Rashba SOC strength and
$\pi=\alpha(-ik_{x}+k_{y})$, $I$ is the $2\times 2$ unit matrix, $\sigma_{i}$
is the Pauli matrix, and $c_{\mathbf{k}\gamma}$ is the annihilation operator.

In the mean-field approximation, the $s$-wave pair potential has the following
form:

$\displaystyle\Delta=g\sum_{\mathbf{k}}\left\langle
c_{\mathbf{k}\downarrow}c_{-\mathbf{k}\uparrow}\right\rangle$ (122)

and the interaction term is obtained as:

$\displaystyle H_{\text{int
}}=-\Delta^{2}/g+\Delta\sum_{\mathbf{k}}\left(c_{\mathbf{k}\downarrow}c_{-\mathbf{k}\uparrow}+c_{-\mathbf{k}\uparrow}^{\dagger}c_{\mathbf{k}\downarrow}^{\dagger}\right)$
(123)

Here, we ignore the constant term; under the Nambu spinor basis, we can obtain
such an expression:

$\displaystyle\Psi_{\mathbf{k}}=\left(c_{\mathbf{k}\uparrow},c_{\mathbf{k}\downarrow},c_{-\mathbf{k}\downarrow}^{\dagger},-c_{-\mathbf{k}\uparrow}^{\dagger}\right)^{T}$
(124)

the Hamiltonian is
$H=\sum_{\mathrm{k}}\Psi_{\mathrm{k}}^{\dagger}H_{\mathrm{k}}\Psi_{\mathrm{k}}$,
where the Hamiltonian $H_{k}$ is:

$\displaystyle
H_{k}=\left(\begin{array}[]{cccc}\xi_{k}+\Gamma&\pi^{\dagger}&\Delta&0\\\
\pi&\xi_{k}-\Gamma&0&\Delta\\\ \Delta&0&-\xi_{k}+\Gamma&-\pi^{\dagger}\\\
0&\Delta&-\pi&-\xi_{k}-\Gamma\end{array}\right),$ (129)

The quasiparticle excitation energy $\Lambda_{k}$ and its relation between the
eigenvale equation of $H_{k}$

$\displaystyle\Psi_{k}^{\dagger}H_{k}\Psi_{k}=\Psi_{k}^{\dagger}UU^{\dagger}H_{k}UU^{\dagger}\Psi_{k}=\beta_{k}^{\dagger}\Lambda_{k}\beta_{k},$
(130)
$\displaystyle\Lambda_{k}=\left(\begin{array}[]{cccc}\sqrt{E_{f}+2E_{0}}&0&0&0\\\
0&\sqrt{E_{f}-2E_{0}}&0&0\\\ 0&0&-\sqrt{E_{f}-2E_{0}}&0\\\
0&0&0&-\sqrt{E_{f}+2E_{0}}\end{array}\right),$ (135)

where

$\displaystyle
E_{f}=k^{2}\alpha^{2}+\Gamma^{2}+\Delta^{2}+\xi_{k}^{2},E_{0}=\sqrt{\Gamma^{2}\Delta^{2}+\xi_{k}^{2}(k^{2}\alpha^{2}+\Gamma^{2})},$
(136)

When $\alpha\to 0,\Gamma\to 0$ and $E_{0}=0$, the system would be explained by
the standard BCS theory.

The parameters of table (E) are chosen to fit the phase boundary.

Parameters | $m$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Delta_{1}$ | $\Delta_{2}$ | $\mu_{1}$ | $\alpha_{1}$ | $\alpha_{2}$ | critical points$\pm\sqrt{\Gamma_{2}^{2}-\Delta_{2}^{2}}$
---|---|---|---|---|---|---|---|---|---
1 | 0.5 | 0.5 | 1.0 | 0.4 | 0.8 | 0.2 | 0.9 | 1.4 | $\pm 0.6$
2 | 0.5 | 0.75 | 1.0 | 0.3 | 0.6 | 1.4 | 0.95 | 1.2 | $\pm 0.8$
3 | 0.5 | 0.75 | 0.5 | 0.3 | 0.3 | 2.4 | 0.35 | 0.6 | $\pm 0.4$
4 | 0.5 | 0.4 | 0.5 | 0.73 | 0.4 | 0.85 | 0.65 | 0.8 | $\pm 0.3$
5 | 0.5 | 0.4 | $\sqrt{1.3}$ | 0.73 | $\sqrt{0.3}$ | 1.85 | 0.69 | 0.4 | $\pm 1.0$
6 | 0.5 | 0.4 | $\sqrt{1.88}$ | 0.73 | $\sqrt{0.44}$ | 1.85 | 0.69 | 0.4 | $\pm 1.2$
Figure 28: The figure depicts a schematic representation of the manifold
distance on the left side of the phase transition point
$-\sqrt{\Gamma_{2}^{2}-\Delta_{2}^{2}}$, where $D^{\prime\prime}$ exhibits
logarithmic divergence at the phase boundary, where $x=$
$\mu_{2}-\mu_{2}^{*}$. Figure 29: The figure depicts a schematic
representation of the manifold distance on the right side of the phase
transition point $\sqrt{\Gamma_{2}^{2}-\Delta_{2}^{2}}$, where
$D^{\prime\prime}$ exhibits logarithmic divergence at the phase boundary,
where $x=$ $\mu_{2}-\mu_{2}^{*}$. Figure 30: As shown in the figure, let
$x=ln(\mu_{2}-\mu_{2}^{*})$, $D^{{}^{\prime\prime}}$ exhibits logarithmic
divergence on both sides of the phase transition point $\mu^{*}$.

##  Appendix G Manifold Distance of Simplified Hamiltonian

In this section, we attempt to explain why the derivative of the manifold
distance diverges at the phase boundary and how the non-Hermitian divergence
transitions back to the Hermitian scenario.

###  G.1 Hermitian case

Consider a simplified Hamiltonian to investigate the divergence of manifold
distance derivative in the Hermitian case,

$\displaystyle H_{k}=\left(\begin{array}[]{cc}-\mu&\alpha k\\\ \alpha k&\mu\\\
\end{array}\right),$ (139)

here

$\displaystyle F=\sqrt{1-|\bra*{\phi_{1}}\ket*{\psi_{2}}|^{2}},$ (140)

The previous numerical calculations indicate that the divergence coefficient
is primarily influenced by a specific set of parameters. Therefore, we assume
the first set of parameters to be constant, while omitting the subscripts.

$\displaystyle F=\sqrt{1-|\bra*{\phi}\ket*{\psi}|^{2}},$ (141)
$\displaystyle(\varphi_{1},\varphi_{2})=(a,b);\quad(\varphi_{1}^{*},\varphi_{2}^{*})=(a^{*},b^{*}),$
(142)

then the analytical expression for $F$ is given by:

$\displaystyle F=\frac{1}{\sqrt{2}}\sqrt{1+\frac{\alpha
k(a^{*}b+ab^{*})+\mu(-|a|^{2}+|b|^{2})}{\sqrt{\alpha^{2}k^{2}+\mu^{2}}}},$
(143)

let $a=1,b=0$

$\displaystyle
F=\frac{1}{\sqrt{2}}\sqrt{1-\frac{\mu}{\sqrt{\alpha^{2}k^{2}+\mu^{2}}}},$
(144)

and $k\to q/\alpha$

$\displaystyle D=\int_{-1}^{1}Fdk\to\frac{1}{\alpha}\int_{-1}^{1}Fdq,$ (145)

$\displaystyle F=\frac{1}{\sqrt{2}}\sqrt{1-\frac{\mu}{\sqrt{q^{2}+\mu^{2}}}},$
(146)

set $q\to\mu k$ again

$\displaystyle\frac{1}{\alpha}\int_{-1}^{1}Fdq\to
2\frac{\mu}{\alpha}\int_{0}^{1/(\alpha\mu)}Fdk,\quad
F=\frac{1}{\sqrt{2}}\sqrt{1-\frac{\mu}{\sqrt{\mu^{2}k^{2}+\mu^{2}}}},$ (147)

then $D$ is given by:

$\displaystyle
D=\int_{0}^{\frac{1}{\alpha\mu}}\frac{\mu\sqrt{2-\frac{2\mu}{\sqrt{(1+k^{2})\mu^{2}}}}q}{\alpha},$
(148)

Perform a second-order Taylor expansion of the integrand of $F$ around
$k=\infty$,

$\displaystyle\frac{\sqrt{2}\mu}{\alpha}-(\frac{\mu}{\sqrt{2}\alpha})\frac{1}{k}-(\frac{\mu}{4\sqrt{2}\alpha})\frac{1}{k^{2}},$
(149)

After integrating this expression, we obtain:

$\displaystyle
D=\frac{\mu\ln(|\mu|)}{\sqrt{2}\alpha}+\frac{8+\alpha^{2}\mu^{2}}{4\sqrt{2}\alpha^{2}}+\frac{\mu\ln(|\alpha|)}{\sqrt{2}\alpha},$
(150) $\displaystyle
D^{{}^{\prime}}=\frac{2+\alpha|\mu|+2\ln\alpha}{2\sqrt{2}\alpha}+\frac{\ln(\mu)}{\sqrt{2}\alpha},$
(151) $\displaystyle
D^{{}^{\prime\prime}}=\frac{1}{2\sqrt{2}}+\frac{|\mu|}{\sqrt{2}\alpha},$ (152)

Figure 31: Graphs of the analytical expressions with $\alpha=1.2$.

Comparison between numerical fitting and analytical results in Tab .4 and Tab
.5.

$\mu<0$

Parameters | $\alpha=1.2$ | $\alpha=2.0$ | $\alpha=0.7$
---|---|---|---
Analytical | $0.3536+\frac{0.5893}{\mu}$ | $0.3536+\frac{0.3536}{\mu}$ | $0.3536+\frac{1.010}{\mu}$
fitting | $0.2429+\frac{0.5893}{\mu}$ | $0.088+\frac{0.3536}{\mu}$ | $0.708+\frac{1.010}{\mu}$
Table 4: Comparison between analytical expression and numerical fitting for
$D^{{}^{\prime\prime}}$, where $F=\sqrt{1-|\bra*{\phi}\ket*{\psi}|^{2}}$.

$\mu>0$

Parameters | $\alpha=1.2$ | $\alpha=2.0$ | $\alpha=0.7$
---|---|---|---
Analytical | $0.3536+\frac{0.5893}{\mu}$ | $0.3536+\frac{0.3536}{\mu}$ | $0.3536+\frac{1.010}{\mu}$
fitting | $0.2481+\frac{0.5893}{\mu}$ | $0.089+\frac{0.3536}{\mu}$ | $0.735+\frac{1.010}{\mu}$
Table 5: Comparison between analytical expression and numerical fitting for
$D^{{}^{\prime\prime}}$, where $F=\sqrt{1-|\bra*{\phi}\ket*{\psi}|^{2}}$.

Due to the numerous approximations made in the derivation process, there may
be discrepancies between the fitting constants and the analytical expression.
However, the agreement in the divergent coefficient is excellent.

Similarly, when $F$ is defined as follows:

$\displaystyle F$
$\displaystyle=1-|\bra*{\phi}\ket*{\psi}|^{2}=\frac{1}{2}-\frac{\mu}{2\sqrt{\alpha^{2}k^{2}+\mu^{2}}},$
(153) $\displaystyle
D=\int_{-1}^{1}Fdk=\frac{1}{\alpha}(\alpha+\mu\ln(|\alpha|)+\mu\ln(|\mu|)-\mu\ln(\alpha^{2}+\alpha\sqrt{\alpha^{2}+\mu^{2}})),$
(154) $\displaystyle
D^{{}^{\prime}}=\frac{1}{\sqrt{\alpha^{2}+\mu^{2}}}+\frac{1}{\alpha}(\ln|\alpha|-\mu\ln(|\alpha|(\alpha+\sqrt{\alpha^{2}+\mu^{2}}))),$
(155) $\displaystyle D^{{}^{\prime\prime}}=\frac{1}{\alpha},$ (156)

Figure 32: Graphs of the analytical expressions with $\alpha=1.2$.

Comparison between numerical fitting and analytical results in Tab .4 and Tab
.5.

$\mu<0$

Parameters | $\alpha=1.2$ | $\alpha=2.0$ | $\alpha=0.7$
---|---|---|---
Analytical | $\frac{0.8333}{\mu}$ | $\frac{0.5000}{\mu}$ | $\frac{1.4286}{\mu}$
fitting | $\frac{0.8333}{\mu}$ | $\frac{0.5000}{\mu}$ | $\frac{1.4286}{\mu}$
Table 6: Comparison between analytical expression and numerical fitting for
$D^{{}^{\prime\prime}}$, where $F=1-|\bra*{\phi}\ket*{\psi}|^{2}$.

$\mu>0$

Parameters | $\alpha=1.2$ | $\alpha=2.0$ | $\alpha=0.7$
---|---|---|---
Analytical | $\frac{0.8333}{\mu}$ | $\frac{0.5000}{\mu}$ | $\frac{1.4286}{\mu}$
fitting | $\frac{0.8333}{\mu}$ | $\frac{0.5000}{\mu}$ | $\frac{1.4286}{\mu}$
Table 7: Comparison between analytical expression and numerical fitting for
$D^{{}^{\prime\prime}}$, where $F=1-|\bra*{\phi}\ket*{\psi}|^{2}$.

###  G.2 Non-Hermitian case

Adding an imaginary part to the simplified Hermitian Hamiltonian,

$\displaystyle H_{k}=\left(\begin{array}[]{cc}-\mu+i\gamma&\alpha k\\\ \alpha
k&\mu-i\gamma\\\ \end{array}\right),$ (159) $\displaystyle
D=2\int_{\gamma/\alpha}^{1}Fdk,\quad
F=\frac{1}{2}(1-\frac{2\mu}{\sqrt{\alpha^{2}k^{2}-(\gamma-i\mu)^{2}}+\sqrt{\alpha^{2}k^{2}-(\gamma+i\mu)^{2}}}),$
(160)

Similarly, directly integrating, we obtain the expression:

$\displaystyle D$
$\displaystyle=\frac{1}{4\alpha\gamma}\\{\alpha(4r-i(-\sqrt{\alpha^{2}-(\gamma-i\mu)^{2}}+\sqrt{\alpha^{2}-(\gamma+i\mu)^{2}}))+i\gamma(4i\gamma+\sqrt{\mu(-2i\gamma+\mu)})$
$\displaystyle+i(\gamma-i\mu)^{2}\ln(\frac{\gamma+\sqrt{\mu(2i\gamma+\mu)}}{\alpha+\sqrt{\alpha^{2}-(\gamma-i\mu)^{2}}})+i(\gamma+i\mu)^{2}\ln(\frac{\alpha+\sqrt{\alpha^{2}-(\gamma+i\mu)^{2}}}{\gamma+\sqrt{\mu(-2i\gamma+\mu)}})\\},$
(161) $\displaystyle D^{{}^{\prime}}$
$\displaystyle=-\frac{1}{2\alpha\gamma}\\{(\gamma-i\mu)\ln(\frac{(\alpha+\sqrt{\alpha^{2}-(\gamma-i\mu)^{2}})}{(\gamma+\sqrt{\mu(2i\gamma+\mu)})})+(\gamma+i\mu)\ln(\frac{\alpha+\sqrt{\alpha^{2}-(\gamma+i\mu)^{2}}}{\gamma+\sqrt{\mu(-2i\gamma+\mu)}})\\},$
(162)

$\displaystyle D^{{}^{\prime\prime}}$
$\displaystyle=-\frac{i}{2\alpha\sqrt{\mu}}(\frac{1}{-2i\gamma+\mu}-\frac{1}{2i\gamma+\mu})-\frac{i}{2\gamma}(\frac{1}{\sqrt{\alpha^{2}-(\gamma-i\mu)^{2}}}-\frac{1}{\sqrt{\alpha^{2}-(\gamma+i\mu)^{2}}})$
$\displaystyle-\frac{1}{2\alpha\gamma}i\ln(\frac{(\alpha+\sqrt{\alpha^{2}-(\gamma+i\mu)^{2}})(\gamma+\sqrt{\mu(2i\gamma+\mu)})}{(\alpha+\sqrt{\alpha^{2}-(\gamma-i\mu)^{2}})(\gamma+\sqrt{\mu(-2i\gamma+\mu)})}),$
(163)

Figure 33: Graphs of the analytical expressions with $\alpha=2.0,\gamma=1.2$.

Comparison between numerical fitting and analytical results in Tab .4 and Tab
.5.

$\mu<0$

Parameters | $\alpha=1.2$ |  $\alpha=2.0$,$\gamma=1.2$ |  $\alpha=2.4$,$\gamma=1.8$
---|---|---|---
Analytical | $-0.0273-\frac{0.4393}{\sqrt{-\mu}}$ | $-0.0114+\frac{0.2283}{\sqrt{-\mu}}$ | $-0.0050-\frac{0.15530}{\sqrt{-\mu}}$
fitting | $-0.0241-\frac{0.4391}{\sqrt{-\mu}}$ | $-0.010-\frac{0.2282}{\sqrt{-\mu}}$ | $-0.0044-\frac{0.15525}{\sqrt{-\mu}}$
Table 8: Comparison between analytical expression and numerical fitting for
$D^{{}^{\prime\prime}}$, where $F=1-|\bra*{\phi}\ket*{\psi}|^{2}$.

$\mu>0$

Parameters | $\alpha=1.2$ |  $\alpha=2.0$,$\gamma=1.2$ |  $\alpha=2.4$,$\gamma=1.8$
---|---|---|---
Analytical | $0.0273+\frac{0.4391}{\sqrt{\mu}}$ | $0.0113+\frac{0.2282}{\sqrt{\mu}}$ | $0.005+\frac{0.15528}{\sqrt{\mu}}$
fitting | $0.0242+\frac{0.4391}{\sqrt{\mu}}$ | $0.010+\frac{0.2282}{\sqrt{\mu}}$ | $0.0044+\frac{0.15525}{\sqrt{\mu}}$
Table 9: Comparison between analytical expression and numerical fitting for
$D^{{}^{\prime\prime}}$, where $F=1-|\bra*{\phi}\ket*{\psi}|^{2}$.

Simplification of the expression for different values of $\gamma$:

(1) When $\gamma$ is large, we can perform a third-order Taylor expansion of
$D$,$D^{{}^{\prime}}$,$D^{{}^{\prime\prime}}$ around $\mu=0$, yielding:

$\displaystyle
D=-\frac{7\mu^{5/2}}{30\alpha\gamma^{3/2}}-\frac{2\mu^{3/2}}{3\alpha\sqrt{\gamma}}+\frac{\gamma}{2\alpha}-\frac{\mu(\ln(\alpha\sqrt{-\gamma^{2}}))-\ln(\alpha\gamma)}{\alpha},$
(164) $\displaystyle
D^{{}^{\prime}}=-\frac{7\mu^{3/2}}{12\alpha\gamma^{3/2}}-\frac{\sqrt{\mu}}{\alpha\sqrt{\gamma}}-\frac{(\ln(\alpha\sqrt{-\gamma^{2}}))-\ln(\alpha\gamma)}{\alpha},$
(165) $\displaystyle
D^{{}^{\prime\prime}}=-\frac{\sqrt{\gamma}}{2\alpha\gamma}\frac{1}{\sqrt{\mu}}-\frac{7}{16\alpha\gamma^{3/2}}\sqrt{\mu}\approx-\frac{\sqrt{\gamma}}{2\alpha\gamma}\frac{1}{\sqrt{\mu}},$
(166)

(2) When $\gamma$ is sufficiently small, the non-Hermitian term can be treated
as a perturbation, we can directly perform a zeroth-order Taylor expansion of
$D$ at $\gamma=0$, resulting in the expression:

$\displaystyle D$
$\displaystyle=1+\frac{\mu}{\alpha}\ln(\frac{|\mu|}{\alpha+\sqrt{\alpha^{2}+\mu^{2}}}),$
(167)

$\displaystyle D^{{}^{\prime}}$
$\displaystyle=\frac{1}{\sqrt{\alpha^{2}+\mu^{2}}}+\frac{\ln(\frac{|\mu|}{\alpha+\sqrt{\alpha^{2}+\mu^{2}}})}{\alpha}\approx\frac{1}{|\alpha|}+\frac{1}{\alpha}(\ln(|\mu|)),$
(168)

$\displaystyle D^{{}^{\prime\prime}}$ $\displaystyle=\frac{1}{\alpha\mu},$
(169)

As we see, when $\gamma\to 0$, the divergent behavior reback to the Hermitian
system.

It can be inferred that as the non-Hermitian parameter $\gamma$ gradually
decreases from a large value to 0, the divergent behavior would transition
from $\frac{1}{\sqrt{\mu}}$ to $\frac{1}{\mu}$, with the presence of
superimposed divergent behaviors during the process.

It is evident that in the process of decreasing the non-Hermitian parameter
$\gamma$ from a relatively large value to zero, the divergent behavior
transitions from $\frac{1}{\sqrt{\mu}}$ to $\frac{1}{\mu}$, and there is
overlapping divergence behavior during this process.

Alternatively, these results can be directly inferred from the analytical
expression of $D^{{}^{\prime\prime}}$.

$\displaystyle D^{{}^{\prime\prime}}$
$\displaystyle=-\frac{i}{2\alpha\sqrt{\mu}}(\frac{1}{-2i\gamma+\mu}-\frac{1}{2i\gamma+\mu})-\frac{i}{2\gamma}(\frac{1}{\sqrt{\alpha^{2}-(\gamma-i\mu)^{2}}}-\frac{1}{\sqrt{\alpha^{2}-(\gamma+i\mu)^{2}}})$
$\displaystyle-\frac{1}{2\alpha\gamma}i\ln(\frac{(\alpha+\sqrt{\alpha^{2}-(\gamma+i\mu)^{2}})(\gamma+\sqrt{\mu(2i\gamma+\mu)})}{(\alpha+\sqrt{\alpha^{2}-(\gamma-i\mu)^{2}})(\gamma+\sqrt{\mu(-2i\gamma+\mu)})}),$
(170)

(1) The first term reflecting the $\frac{1}{\sqrt{\mu}}$ divergence, and this
term does not exist when $\gamma=0$;

(2) For the second term, a first-order approximation is obtained by Taylor
expanding around $\gamma=0$

$\displaystyle\frac{|\mu|}{\alpha\mu^{3}}\gamma,$ (171)

when $\gamma=0$, this term also vanishes.

(3) The third term, when expanded to zeroth order by a Taylor series around
$\gamma=0$, yields.

$\displaystyle\frac{1}{\mu\sqrt{\alpha^{2}+\mu^{2}}}\approx\frac{1}{\mu\alpha},$
(172)

##  Appendix H The Kitaev honeycomb model

Figure 34: Graphs of the Kitaev honeycomb model and its ”brick-wall”
structure.

The Kitaev honeycomb model is a two-dimensional spin model initially
introduced and studied by Kitaev. This model consists of three types of
nearest-neighbor interactions, categorized as $\mathrm{XX}$, $\mathrm{YY}$,
and $\mathrm{ZZ}$ interactions depending on their directional alignment Kitaev
(2006); Kitaev and Preskill (2006). The Hamiltonian is given by:

$\displaystyle H=-J_{x}\sum_{x\text{-bonds
}}\sigma_{R_{w}}^{x}\sigma_{R_{b}}^{x}-J_{y}\sum_{y\text{-bonds
}}\sigma_{R_{w}}^{y}\sigma_{R_{b}}^{y}-J_{z}\sum_{z\text{-bonds
}}\sigma_{R_{w}}^{z}\sigma_{R_{b}}^{z}$ (173)

Here, $\sigma_{i}^{\alpha}(\alpha=x,y,z)$ refers to the Pauli matrices at site
$i$, and $\langle
i,j\rangle_{\alpha\alpha}(\alpha=\mathrm{X},\mathrm{Y},\mathrm{Z})$ denotes
nearest-neighbor interactions of type $\alpha\alpha$ between sites $i$ and
$j$, with $J_{\alpha}$ representing the strength of the interaction.

The Hamiltonian can be transformed into a fermionic form using the Jordan-
Wigner transformation Mancini (2008). It’s noteworthy that the ”brick-wall”
structure is equivalent to the original two-dimensional Kitaev model
structure. The Jordan-Wigner transformation is given by

$\displaystyle\sigma_{m,n}^{+}$
$\displaystyle=2\left[\Pi_{n^{\prime}<n}\Pi_{m^{\prime}}\sigma_{m^{\prime},n^{\prime}}^{z}\right]\left[\prod_{m^{\prime}<m}\sigma_{m^{\prime},n^{\prime}}^{z}\right]c_{m,n}^{\dagger}$
(174) $\displaystyle\sigma_{m,n}^{z}$
$\displaystyle=2c_{m,n}^{\dagger}c_{m,n}-1$ (175)

Here, $c^{\dagger}$ and $c$ are fermionic creation and annihilation operators,
$\sigma^{+}=\sigma^{x}+i\sigma^{y}$ is twice the spin raising operator at a
specific position. Then we have

$\displaystyle H=$ $\displaystyle J_{x}\sum_{x\text{-bonds
}}\left(c^{\dagger}-c\right)_{w}\left(c^{\dagger}+c\right)_{b}$ (176)
$\displaystyle-J_{y}\sum_{y\text{-bonds
}}\left(c^{\dagger}+c\right)_{b}\left(c^{\dagger}-c\right)_{w}$ (177)
$\displaystyle-J_{z}\sum_{z\text{-bonds
}}\left(2c^{\dagger}c-1\right)_{b}\left(2c^{\dagger}c-1\right)_{w}$ (178)

Here, the subscripts $w$ and $b$ represent the white and black particles,
respectively. It’s noteworthy that in the Kitaev honeycomb model, there exists
a relationship that
$\alpha_{r}=\left(c-c^{\dagger}\right)_{b}\left(c+c^{\dagger}\right)_{w}$ is a
constant and its value $1^{[108]}$ for the ground state. Therefore, we can
diagonalize this model.

Introduction of two Majorana fermions for each site.

$\displaystyle A_{w}=\frac{\left(c-c^{\dagger}\right)_{w}}{i}\quad
B_{w}=\left(c+c^{\dagger}\right)_{w}$ (179) $\displaystyle
A_{b}=\left(c+c^{\dagger}\right)_{b}\quad
B_{b}=\frac{\left(c-c^{\dagger}\right)_{b}}{i}$ (180)

After the transformation the model becomes:

$\displaystyle H=-iJ_{x}\sum_{x\text{-bonds
}}A_{w}A_{b}+iJ_{y}\sum_{y\text{-bonds }}A_{b}A_{w}-iJ_{z}\sum_{z\text{-bonds
}}iA_{b}A_{w}$ (181)

In order to diagonalize the Hamiltonian, we introduce a fermion in each
$z$-bonds, that

$\displaystyle d_{r}=\frac{1}{2}\left(A_{w}+iA_{b}\right)\quad
d_{r}^{\dagger}=\frac{1}{2}\left(A_{w}-iA_{b}\right)$ (182)

where $A_{w}$ and $A_{b}$ refer to the Majorana fermion on the white site and
the black site, then resolve by Fourier transformation:

$\displaystyle
d_{r}=\frac{1}{\sqrt{\Omega}}\sum_{q}e^{i\boldsymbol{q}\boldsymbol{r}}d_{q}.$
(183)

then the Hamiltonian is

$\displaystyle\left(\begin{array}[]{cc}H_{k}=2\text{Jz}-2\text{Jx}\cos(\text{kx})-2\text{Jy}\cos(\text{ky})&i(\text{Jx}\sin(\text{kx})+\text{Jy}\sin(\text{ky}))\\\
-i(\text{Jx}\sin(\text{kx})+\text{Jy}\sin(\text{ky}))&-2\text{Jz}+2\text{Jx}\cos(\text{kx})+2\text{Jy}\cos(\text{ky})\\\
\end{array}\right)$ (186)

Figure 35: The phase diagram in the parameter space where
$J_{x}+J_{y}+J_{z}=1$. The regions labeled with ”Abelian” represent gapped
phases with Abelian anyon excitations, while the region labeled with ”Non-
Abelian” corresponds to a gapped phase with non-Abelian anyon excitations. The
phase transitions at values in the $J_{x}+J_{y}+J_{z}=1$ plane.

##  Appendix I Summary of the Critical Divergence Behavior of $D$ at the Phase
Boundary

a.The derivative of manifold distance $D$ exhibits divergence phenomena at the
phase boundaries;

b.For non-Hermitian systems, $D^{{}^{\prime\prime}}$ shows superimposed
divergence. As the non-Hermitian term gradually to $0$, the divergence
behavior of the phase boundary also reback to the Hermitian case.

c. Although this paper provides various definitions for manifold distance, but
it does not affect the occurrence of divergence in $D^{{}^{\prime}}$ or
$D^{{}^{\prime\prime}}$ at the phase boundaries.

Noteworthily, if the eigen-wave function of $H$ and $H_{d}$ are not of the
following form

$\displaystyle(x+iy,u+iy),\quad(x-iy,u-iy)$ (187)

it is preferable to choose the forms of $F_{1}$ or $F_{2}$.

d. When different momentum $k$ is chosen for parameter sets $1$ and $2$, with
$k^{{}^{\prime}}=k+c$ or $k^{{}^{\prime}}=f(k)$, although $D$ may exhibit
numerical differences, $D^{{}^{\prime}}$ or $D^{{}^{\prime\prime}}$ still
diverges at the phase boundaries, and the divergence behavior remains
unaffected.

1\. 1d Kitaev toy model

$\displaystyle H_{k}=\left(\begin{array}[]{cc}-2t\cos(k)-\mu&\alpha\sin(k)\\\
\alpha\sin(k)&2t\cos(k)+\mu\\\ \end{array}\right)$ (190)

phase boundary

$\displaystyle\mu=\pm 2t$ (191)

manifold distance

$\displaystyle
D^{{}^{\prime}}\propto\frac{1}{\sqrt{2}\alpha_{2}}\ln(|\mu_{2}-\mu_{2}^{*}|),\quad
D^{{}^{\prime\prime}}\propto\frac{1}{\sqrt{2}\alpha_{2}(\mu_{2}-\mu_{2}^{*})}$
(192)

2\. 1d Kitaev non-hermitian toy model

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}-2t\cos(k)-\mu+i\gamma&\alpha\sin(k)\\\
\alpha\sin(k)&2t\cos(k)+\mu-i\gamma\\\ \end{array}\right)$ (195)

phase boundary

$\displaystyle\mu=\pm 2t\sqrt{1-\frac{\gamma^{2}}{\alpha^{2}}}$ (196)

(1). When $\gamma$ is large enough($\gamma_{2}>10^{-6}$), the divergence
behavior in region (I) and (IV) as follow

$\displaystyle
D^{{}^{\prime\prime}}\propto\frac{C_{1}}{\mu_{2}-\mu_{2}^{*}}+\frac{C_{2}}{\sqrt{|\mu_{2}-\mu_{2}^{*}|}},$
(197)

(2). When $\gamma$ is small enough($\gamma_{2}<10^{-6}$), the divergence
behavior in region (I) and (IV) as follow

$\displaystyle D^{{}^{\prime\prime}}\propto\frac{C_{1}}{\mu_{2}-\mu_{2}^{*}}$
(198)

The divergence behavior gradually transitions to the Hermitian case.

3\. Regardless of the value of $\gamma$, both region (II) and (III) exhibit
logarithmic divergence, remaining consistent with the Hermitian case.

Now, we summarize the conclusions regarding the divergence behavior of 2D
topological superconductor model as follows:

1\. If the non-Hermitian contributions to the system are significant, i.e.,
when $\gamma_{2}>10^{-3}$, the divergence behavior in region (I) and (IV) as
follow

$\displaystyle D^{{}^{\prime\prime}}\propto
C_{1}ln(|\mu_{2}-\mu_{2}^{*}|)+C_{2}/\sqrt{|\mu_{2}-\mu_{2}^{*}|}$ (199)

and region (II) and (III) do not exhibit divergent behavior.

2\. If the non-Hermitian contributions to the system can be neglected, i.e.,
when $\gamma_{2}<10^{-3}$, the divergence behavior in region (I) and (IV) as
follow

$\displaystyle D^{{}^{\prime\prime}}\propto C_{1}ln(|\mu_{2}-\mu_{2}^{*}|)$
(200)

The divergence behavior gradually transitions to the Hermitian case.

3\. Regardless of the value of $\gamma$, both region (I) $\sim$ (III) exhibit
logarithmic divergence, remaining consistent with the Hermitian case.

3\. 2d p-wave SC model

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}-2t(\cos(k_{x})+\cos(k_{y}))-\mu&\alpha(\sin(k_{x})+i\sin(k_{y}))\\\
\alpha(\sin(k_{x})-i\sin(k_{y}))&2t(\cos(k_{x})+\cos(k_{y}))+\mu\\\
\end{array}\right)$ (203)

phase boundary

$\displaystyle\mu=\pm 4t$ (204)

manifold distance

$\displaystyle
D^{{}^{\prime\prime}}\propto\frac{2}{\alpha^{2}}ln(|\mu_{2}-\mu_{2}^{*}|)$
(205)

4\. 2d non-hermitian p-wave SC model

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}-2t(\cos(k_{x})+\cos(k_{y}))-\mu+i\gamma&\alpha(\sin(k_{x})+i\sin(k_{y}))\\\
\alpha(\sin(k_{x})-i\sin(k_{y}))&2t(\cos(k_{x})+\cos(k_{y}))+\mu-i\gamma\\\
\end{array}\right)$ (208)

phase boundary

$\displaystyle\mu=\pm 2t(1+\sqrt{1-\frac{\gamma^{2}}{\alpha^{2}}})$ (209)

1\. When $\gamma_{2}>10^{-3}$, the divergence behavior in region (I) and (IV)
as follow

$\displaystyle D^{{}^{\prime\prime}}\propto
C_{1}ln(|\mu_{2}-\mu_{2}^{*}|)+C_{2}/\sqrt{|\mu_{2}-\mu_{2}^{*}|}$ (210)

and region (II) and (III) do not exhibit divergent behavior.

2\. If the non-Hermitian contributions to the system can be neglected, i.e.,
when $\gamma_{2}<10^{-3}$, the divergence behavior in region (I) and (IV) as
follow

$\displaystyle D^{{}^{\prime\prime}}\propto C_{1}ln(|\mu_{2}-\mu_{2}^{*}|)$
(211)

3\. Regardless of the value of $\gamma$, both region (II) and (III) exhibit
logarithmic divergence, remaining consistent with the Hermitian case.

5\. 1d PT-symmetry non-Hermiant ssh model

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}0&\frac{\gamma}{2}+t+t^{{}^{\prime}}\cos(k)-it^{{}^{\prime}}\sin(k)\\\
-\frac{\gamma}{2}+t+t^{{}^{\prime}}\cos(k)+it^{{}^{\prime}}\sin(k)&0\\\
\end{array}\right),$ (214)

critical points

$\displaystyle t=t^{{}^{\prime}}\pm(\frac{\gamma}{2});\quad
t=-t^{{}^{\prime}}\pm(\frac{\gamma}{2}).$ (215)

manifold distance

$\displaystyle
D^{{}^{\prime\prime}}\propto\frac{a}{\sqrt{|t_{2}-t_{2}^{*}|}}+bln(|t_{2}-t_{2}^{*}|)$
(216)

6\. 1d PT-symmetry ssh model

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}0&t+t^{{}^{\prime}}\cos(k)-it^{{}^{\prime}}\sin(k)\\\
t+t^{{}^{\prime}}\cos(k)+it^{{}^{\prime}}\sin(k)&0\\\ \end{array}\right),$
(219)

critical points

$\displaystyle t=\pm t^{{}^{\prime}},$ (220)

manifold distance

$\displaystyle
D^{{}^{\prime}}\propto\frac{1}{t_{2}^{{}^{\prime}}}ln(|t_{2}-t_{2}^{*}|),\quad
D^{{}^{\prime\prime}}\propto\frac{1}{t_{2}^{{}^{\prime}}}\frac{1}{|t_{2}-t_{2}^{*}|}$
(221)

7\. P-wave superconductor model

$\displaystyle
H_{k}=\left(\begin{array}[]{cc}\frac{k^{2}}{2m}-\mu&\alpha(k_{x}+ik_{y})\\\
\alpha(k_{x}-ik_{y})&-(\frac{k^{2}}{2m}-\mu)\\\ \end{array}\right)$ (224)

critical points

$\displaystyle\mu=0$ (225)

manifold distance

$\displaystyle
D^{{}^{\prime\prime}}\propto\frac{\text{ln}(|\mu_{2}|)}{4\sqrt{2}\alpha_{2}^{2}}$
(226)

8\. SC with SOC

$\displaystyle
H_{k}=\left(\begin{array}[]{cccc}\xi_{k}+\Gamma&\pi^{\dagger}&\Delta&0\\\
\pi&\xi_{k}-\Gamma&0&\Delta\\\ \Delta&0&-\xi_{k}+\Gamma&-\pi^{\dagger}\\\
0&\Delta&-\pi&-\xi_{k}-\Gamma\end{array}\right)$ (231)

where, $\gamma=\uparrow,\downarrow$, $\mu$ is the chemical potential,
$\epsilon_{k}=\frac{k_{x}^{2}+k_{y}^{2}}{2m}$ is the free particle energy,
$\xi_{k}=\epsilon_{k}-\mu$ is the reduced particle energy, $\Gamma$ is the
strength of the Zeeman field, $\alpha$ is the Rashba SOC strength and
$\pi=\alpha(-ik_{x}+k_{y})$, $I$ is the $2\times 2$ unit matrix, $\sigma_{i}$
is the Pauli matrix, and $c_{\mathbf{k}\gamma}$ is the annihilation operator.

phase boundary

$\displaystyle\mu=\pm\sqrt{\Gamma^{2}-\Delta^{2}},$ (232)

manifold distance

$\displaystyle D^{{}^{\prime\prime}}\propto\ln(\mu_{2}-\mu_{2}^{*})$ (233)