# Exceptional Points and Braiding Topology in Non-Hermitian Systems with long-
range coupling

S. M. Rafi-Ul-Islam<EMAIL_ADDRESS>Department of Electrical and
Computer Engineering, National University of Singapore, Singapore 117583,
Republic of Singapore  Zhuo Bin Siu<EMAIL_ADDRESS>Department of
Electrical and Computer Engineering, National University of Singapore,
Singapore 117583, Republic of Singapore  Md. Saddam Hossain Razo
<EMAIL_ADDRESS>Department of Electrical and Computer Engineering, National
University of Singapore, Singapore 117583, Republic of Singapore  Haydar Sahin
<EMAIL_ADDRESS>Department of Electrical and Computer Engineering,
National University of Singapore, Singapore 117583, Republic of Singapore
Institute of High Performance Computing, A*STAR, Singapore 138632, Republic of
Singapore  Mansoor B.A. Jalil<EMAIL_ADDRESS>Department of Electrical and
Computer Engineering, National University of Singapore, Singapore 117583,
Republic of Singapore

###### Abstract

We present a study of complex energy braiding in a 1D non-Hermitian system
with $n$th order long range asymmetrical coupling. Our work highlights the
emergence of novel topological phenomena in such systems beyond the
conventional nearest-neighbor interaction. The modified SSH model displays $n$
distinct knots and links combinations in the complex energy-momentum space
under periodic boundary conditions (PBC), which can be controlled by varying
the coupling strengths. A topological invariant, namely the braiding index, is
introduced to characterize the different complex energy braiding profiles,
which depends on the zeros and poles of the characteristic polynomials.
Furthermore, we demonstrate that the non-Hermitian skin effect can be
localized at one or both ends, signifying conventional or bipolar
localization, depending on the sign of the braiding index. Phase transitions
between different braiding phases with the same (opposite) sign of the
topological invariant occur at Type-1 (Type-2) exceptional points, with Type-1
(Type-2) phase transitions accompanied by single (multiple) exceptional
points. We propose an experimental set-up to realize the various braiding
schemes based on the RLC circuit framework, which provides an accessible
avenue for implementation without recourse to high-dimensional momentum space
required in most other platforms.

##  I Introduction

The study of non-Hermitian systems has a long history, dating back to the
early days of quantum mechanics [1, 2, 3, 4, 5, 6]. However, it has gained
renewed interest in recent years, and has been analyzed in various contexts,
ranging from optics [7, 8, 9], photonics [10, 11, 12, 13], metamaterials [14,
15, 16, 17], quantum field theory [18, 19, 20], topolectrical circuits [21,
22, 23, 24, 25, 26, 27, 28, 24, 23, 29, 30, 31, 32, 33, 34, 35] to condensed
matter physics [36, 37, 38, 39, 5, 40, 41, 42]. Non-Hermitian systems are
usually characterized by non-Bloch theory [6, 43], and exhibit novel and
exotic physical phenomena that are not present in Hermitian systems. One of
the key features of non-Hermitian systems is the presence of complex eigen
energies [44, 45, 46], which can have a profound impact on the system’s
behavior. In particular, the concept of energy braiding [47, 48, 49, 50, 51,
52, 53, 54], i.e., the criss-crossing of two eigen energy braids in complex
energy space, has provided key insights into the understanding of non-
Hermitian properties, such as (i) the non-Hermitian skin effect [55, 56, 57,
58, 59, 60, 61, 62, 63, 64], which localizes the bulk states to the edges of
the system under open boundary conditions, and (ii) exceptional points [21,
65, 66, 67, 68], which are special points in the complex energy space where
two eigenvalues and their corresponding eigenvectors coalesce.

Complex energy braiding opens up a new avenue for exploring novel topological
phases and their corresponding phase transitions. This is owing to the
complex-valued eigenvalues of non-Hermitian systems confers extra degree of
freedom in their energy-momentum spectra [69, 70, 71, 72, 73]. Energy braiding
in non-Hermitian systems has been realized in a variety of platforms, both
theoretically and experimentally. On the theoretical side, the concept of
complex energy braiding has been studied using various models such as the
Kitaev chain [74, 75, 76, 77, 78], and the topological superconductor [79,
80]. Experimentally, energy braiding has been realized in a variety of
platforms such as photonic systems [48, 81, 82], metamaterial systems [83],
and electronic circuits [77, 84]. In photonic systems, complex energy braiding
has been observed in wave guide arrays and resonator arrays with asymmetrical
couplings [48, 85]. A single-band system with long-range coupling was
experimentally realized in an optical resonator system [85]. In mechanical
systems, the braiding of energy eigenvalues has been demonstrated in the
motion of coupled pendulums [86], while in electronic circuits, it has been
realized using LC (Inductor-Capacitor) circuits. Finally, one-band [87] and
two-band systems [53] with long-range coupling have been experimentally
realized in acoustic systems.

Despite the above plethora of studies on complex admittance energy braiding in
non-Hermitian systems, several key questions remain unanswered:

  1. 1.

Is it possible to extend the braiding concepts to multiple knots and links in
a generalized 1D system?

  2. 2.

Is there a correlation between the localization of the non-Hermitian skin
effect and the handedness of the energy braiding?

  3. 3.

Can generalized braiding circuits provide a general approach to trace the
locus of the exceptional points which is not tied to any specific model?

  4. 4.

Can an extended topological braiding index be defined to describe non-
Hermitian systems with arbitrary long-distance coupling?

In this letter, we aim to address the above-mentioned questions by reference
to the complex energy braiding of a model 1D non-Hermitian system which
incorporates long-range unidirectional coupling. Our model which is based on
the bipartite modified non-Hermitian SSH model [88], allows for an arbitrary
number of braids or knots to be engineered in the energy spectrum. Depending
on the degree of long-range coupling in the model, the eigen energy strands of
in the complex energy spectra can tangle with one another and form an
arbitrary number of loops. In general, an $n$th-order asymmetrical inter-chain
coupling allows for $n$ different configurations of complex energy braiding.
The braid configurations are characterized by a new topological invariant, the
knot/braiding index, which depends on the number of zeros and poles of the
characteristic polynomial of the system’s Hamiltonian. Interestingly, the sign
of the braiding index would determine the localization of the non-Hermitian
skin effect (NHSE) at either or both ends of the 1D chain. This is unlike the
conventional NHSE which is governed by the relative coupling strength within
the chain, and is restricted to unipolar (localized at one end) as opposed to
bipolar localization. Furthermore, distinct topological phases with different
numbers of knots can be transformed from one to the other via exceptional
points, which can be classified as Type-1 or Type-2 depending on the sign of
the braiding index across the phase boundary. Finally, we propose an
experimental framework for realizing these braiding schemes in an $RLC$
circuit, which offers a simple accessible route to the realization of braiding
in 1D non-Hermitian systems.

##  II Model with arbitrary number of knots

To realize various complex energy-momenta knots via 1D non-reciprocal lattice,
we consider a bipartite model with tunable long-range coupling with two types
of sublattice sites (A and B) per unit cell. The model Hamiltonian is given by

$H_{1}(k)=\begin{pmatrix}0&C_{\mathrm{AB},0}+C_{\mathrm{AB},-m}e^{-ikm}\\\
C_{\mathrm{AB},0}+C_{\mathrm{BA},n}e^{ikn}&0\end{pmatrix}.$ (1)

Eq. (1) represents a two-band model with long-range unidirectional couplings
of $n$ unit cells towards the right and $m$ unit cells towards the left. For a
given value of momentum $k$, the corresponding eigenvalues of $H_{1}(k)$ are
given by

$E_{\pm}(k)=\pm\sqrt{(C_{\mathrm{AB},0}+C_{\mathrm{AB},-m}e^{-ikm})(C_{\mathrm{AB},0}+C_{\mathrm{BA},n}e^{ikn})}.$
(2)

As the momentum $k$ varies from 0 to $2\pi$ within the Brillouin zone, the
trajectories of the two complex eigen energies may form knots in the 3D
$(\mathrm{Re}E,\mathrm{Im}E,k)$ complex energy–momentum space. If we assume,
without loss of generality that $m>n$, then the two eigen energy bands may
tangle with each other for up to $m$ times (e.g., if we set $m=3$, the energy
bands may tangle for zero, one, two, or three times) depending on the coupling
parameters. Fig. 1 shows exemplary knot systems realized by the Hamiltonian in
Eq. 1 with $m=3$ and $n=1$ in which the bands tangle once (Fig. 1a), twice
(Fig. 1b), and three times (Fig. 1c). These correspond to the Unknot, Hopf
link, and trefoil configurations, respectively. We can systematically describe
these configurations via the Artins notation [89, 52],where a complex braiding
with $p$ knots follows a sequence of band crossings, each of which can be
described by the braiding word $\tau_{1}^{\pm p}$. Here, $\tau_{1}^{1}$
($\tau_{1}^{-1}$) denotes the case where the trajectory of the first energy
strand crosses that of the second energy strand from the left (right) in the
projection of the 3D trajectory onto the $\mathrm{Im}E=\infty$ plane. The
systems in Fig. 1a – c can thus be described by the braid words
$\tau^{1}_{1}$, $\tau^{-2}_{1}$, and $\tau^{-3}_{1}$, respectively.

Figure 1:  Complex energy braiding with arbitrary number of knots in a 1D two-
band modified SSH chain with long-range coupling From left to right : complex
energy plane projection, 3D complex energy–$k$ braiding, and schematic
representation of braiding for $m=3$, $n=1$ and $C_{\mathrm{AB},0}=1$ in (a)
an Unknot system with one knot and $C_{\mathrm{AB},-m}=0.22$,
$C_{\mathrm{BA},n}=-1.5$, (b) Hopf link system with two knots and
$C_{\mathrm{AB},-m}=1.4$, $C_{\mathrm{BA},n}=1.6$, and (c) trefoil system with
three knots and $C_{\mathrm{AB},0}=1.4$, $C_{\mathrm{BA},n}=-0.4$. The dotted
line and circles denote the crossing points of the yellow-blue line with a
line slightly displaced above the $\mathrm{Re}E$ axis mentioned in the text.
The yellow-blue and gray-red lines in the left and middle plots represent the
eigen energies of the two distinct bands with darker colors (i.e., red and
blue) representing larger values of $k$ for $0\leq k<2\pi$.

Furthermore, the different types of knot configurations can be characterized
by an integer topological number known as the braiding index $\xi$, which is
defined as [48]

$\xi\equiv\int\frac{\mathrm{d}k}{2\pi
i}\partial_{k}\mathrm{log}\left|H(k)-\frac{1}{2}\mathrm{Tr}H(k)\mathbf{I}_{2}\right|$
(3)

where $\mathbf{I}_{2}$ is the two-by-two identity matrix.

The magnitude of $\xi$ indicates the number of times that the eigen energy
strands of $H(k)$ wind or braid around each other in
$(\mathrm{Re}(E),\mathrm{Im}(E),k)$ space, while its sign denotes the net
handedness of the braiding. Using the fact that the determinant of a square
matrix is equal to the product of its eigenvalues, introducing
$\tilde{H}(k)\equiv H(k)-\frac{1}{2}\mathrm{Tr}H(k)\mathbf{I}_{2}$, and
denoting the two eigenvalues of $\tilde{H}(k)$ as $\tilde{E}_{\pm}(k)$, $\xi$
can be rewritten as

$\xi=\int\frac{\mathrm{d}k}{2\pi}\
\partial_{k}\left(\mathrm{Arg}\tilde{E}_{+}(k)+\mathrm{Arg}\tilde{E}_{-}(k)\right).$
(4)

Because the Hamiltonian Eq. 1 is traceless, $\xi$ for this Hamiltonian can be
interpreted as counting the number of times that its eigenvalues wind around
the complex energy origin. For example, the leftmost plot in Fig. 1a shows
that for this Unknot system, the two energy strands join together to form a
single loop that winds in the counter-clockwise direction as $k$ increases (as
is evident from the color coding of the energy strings where darker colors
correspond to larger $k$ values), and hence corresponds to a braiding index of
+1. For the Hopf link in Fig. 1b, the gray-red and yellow-blue bands each
winds around the origin once in a clockwise manner, and hence collectively
give rise to an overall braiding index of -2. For the trefoil in Fig. 1c, the
gray-red and yellow-blue bands each winds around the origin for one and a half
times in a counter-clockwise manner. (The fact that each band winds around the
origin for one and a half times can be seen from, e.g., the yellow-blue line
crossing a line slightly displaced above the real energy axis only once on the
negative side but twice on the positive side, with the crossing points
depicted by open circles in the left panel of Fig. 1c. ) The two bands
therefore collectively give rise to a braiding index of -3. Interestingly, for
two-band systems, each of the two energy strings begins and ends at the same
value of complex energy at $k=0$ and $k=2\pi$, respectively, when $\xi$ is
even, but when $\xi$ is odd, the start-point of one band coincides with the
end-point of the other, and vice versa (so that the trajectory of the two
bands collectively form a closed loop).

Figure 2:  Braiding index phase diagram  (a) - (c) Schematic illustration of
distribution of $\beta$s at which $E=0$ on complex $\beta$ plane at (a)
$\xi=0$, (b) $\xi=1$, and (c) $\xi=-2$ for a system described by Eq. (1) for
$m=3$ and $n=1$. The yellow circle denotes the value of $\beta$ due to the
$n=1$ term and the blue circles the values of $\beta$ due to the $m=3$ term.
(d) $\xi$ phase diagram of system in (a) with $C_{\mathrm{AB},0}=1$. (e)
Schematic illustration of distribution of $\beta$s at which $E=0$ on complex
$\beta$ plane on transition line between $\xi=-2$ and $\xi=1$. (f) Complex
energy projection and (g) $(\mathrm{Re}(E),\mathrm{Im}(E),k)$-space trajectory
of energy strings at transition in (e). The black circles in (g) denote the
string-touching points at $E=0$.

Beyond counting the number of times that the eigen energy strands wind around
the origin on the complex energy plane, the braiding index has an alternative
mathematical interpretation. Eq. (3) can be recast into the form of a contour
integral over the complex unit circle in $\beta\equiv\exp(ik)$ space:

$\xi=\frac{1}{2\pi
i}\oint_{|\beta|=1}\frac{\partial_{k}\tilde{E_{+}}(\beta)}{\tilde{E_{+}}(\beta)}.$
(5)

Eq. (5) can be evaluated using the argument principle as $\xi=N-P$, where
$N(P)$ is the number of zeros (poles) of $E_{+}$ enclosed inside the contour
integral. From Eq. 2,
$\tilde{E}_{+}=\sqrt{(C_{\mathrm{AB},0}+C_{\mathrm{AB},-m}\beta^{-m})(C_{\mathrm{AB},0}+C_{\mathrm{BA},n}\beta^{n})}$.
There are therefore $m$ poles of $E_{+}$ at $\beta=0$ within the complex unit
circle due to the $\beta^{-m}$ term. The zeros of $\beta$ occur when
$\beta^{m}=-C_{\mathrm{AB},-m}/C_{\mathrm{AB},0}$, the solutions of which are
given by the following equation

$\beta=\beta_{1,f}=|C_{\mathrm{AB},-m}/C_{\mathrm{AB},0}|^{1/m}e^{(i(\mathrm{Arg}(C_{\mathrm{AB},-m})-\mathrm{Arg}(C_{\mathrm{AB},0})+(2f+1)\pi/m))},$
(6)

where, $f\in(1,...,m)$, and when
$\beta^{n}=-C_{\mathrm{AB},0}/C_{\mathrm{BA},n}$, i.e., the solutions of which
are given by

$\beta=\beta_{2,g}=|C_{\mathrm{AB},0}/C_{\mathrm{BA},n}|^{1/n}e^{(i(\mathrm{Arg}(C_{\mathrm{AB},0})-\mathrm{Arg}(C_{\mathrm{BA},n})+(2g+1)\pi/n))},$
(7)

where, $g\in(1,...,n)$. Depending on the relative magnitudes of
$C_{\mathrm{AB},0}$, $C_{\mathrm{AB},-m}$, and $C_{\mathrm{BA},n}$, the $m$
($n$) values of $\beta_{1,f}$ ($\beta_{2,g}$) may fall inside, on, or outside
the complex unit circle. Neglecting for now the cases where the $\beta_{1,f}$s
or $\beta_{2,g}$s lie exactly on the complex unit circle, the possible values
of $\xi$ are thus $-m$, when $|C_{\mathrm{AB},-m}/C_{\mathrm{AB},0}|>1$,
$|C_{\mathrm{AB},0}/C_{\mathrm{BA},n}|>1$ so that the $\beta_{1,f}$s and
$\beta_{2,g}$s all lie outside the unit circle; 0, when
$|C_{\mathrm{AB},-m}/C_{\mathrm{AB},0}|<1$,
$|C_{\mathrm{AB},0}/C_{\mathrm{BA},n}|>1$ so that the $m$ $\beta_{1,f}$s all
lie within the unit circle while the $n$ $\beta_{2,g}$s all lie outside (Fig.
2a); $n$, when $|C_{\mathrm{AB},-m}/C_{\mathrm{AB},0}|<1$,
$|C_{\mathrm{AB},0}/C_{\mathrm{BA},n}|<1$, so that the $m$ $\beta_{1,f}$s and
$n$ $\beta_{2,g}$s all lie inside the unit circle (Fig. 2b) ; and $(n-m)$, so
that the $n$ $\beta_{2,g}$s all lie within the unit circle while the $m$
$\beta_{1,f}$s all lie outside (Fig. 2c). This dependence of $\xi$ on the
relative magnitudes of $C_{\mathrm{AB},0}$, $C_{2L}$, and $C_{\mathrm{BA},n}$
results in the phase diagram shown in Fig. 2d where the boundaries between
different values of the braiding indices are defined by the equations:
$C_{\mathrm{AB},-m}=\pm C_{\mathrm{AB},0}$ and $C_{\mathrm{BA},n}=\pm
C_{\mathrm{AB},0}$.

Let us now consider the case where the $m$ values of $\beta_{1,f}$s (or $n$
values of $\beta_{2,g}$s) fall on the unit circle on the complex $\beta$ plane
(Fig. 2e), for which the value of $C_{\mathrm{AB},-m}$ ($C_{\mathrm{BA},n}$)
must necessarily fall on one of the phase boundaries between different values
of $\xi$. This is because the $\beta_{1,f}$s and/or $\beta_{2,g}$s have to
pass through the unit circle, for the number of $\beta$s solutions to
$E_{+}=0$ that lie within the unit circle to change. The correspondence
between the complex unit circle on the $\beta$ plane to real values of $k$,
i.e., $\beta\equiv\exp(ik)$ means that there are $m$, $n$, or $(m+n)$ values
of $k$ corresponding to zero energy on the PBC spectrum when only
$|C_{\mathrm{AB},-m}/C_{\mathrm{AB},0}|=1$, only
$|C_{\mathrm{AB},0}/C_{\mathrm{BA},n}|=1$, or
$|C_{\mathrm{AB},-m}/C_{\mathrm{AB},0}|=|C_{\mathrm{AB},0}/C_{\mathrm{BA},n}|=1$,
respectively. Figs. 2f and 2g illustrate this for the specific case of $m=3$.
For the particular model in Eq. (1), the zeros of the $E_{+}$ also happen to
be exceptional points because these zeros occur only when $H_{1}(k)$ is rank-
deficient and all the matrix elements in the left or right columns are zero.

Figure 3:  Braiding number and NHSE (a) Distribution of $\xi_{r}(E)$ on the
complex energy plane and PBC spectrum for the Hopf link system of Fig. 1b with
$m=3$, $n=1$, $C_{\mathrm{AB},0}=1$, $C_{\mathrm{AB},-m}=1.4$, and
$C_{\mathrm{BA},n}=1.6$. The two energy bands are indicated by the blue and
red lines, and the arrows on the lines indicate the direction of increasing
$k$. The blue dot A and red dot B denote values of $E$ for which $\xi_{r}(E)$
takes the values of -2 and -1, respectively. (b) Distribution of the number of
distinct eigenstates with $|\beta|>1$ on the complex energy plane for the
system in (a). (c) PBC spectra for the Hopf link system in (a), and the OBC
spectra and spatial density distribution for its eigenstates in a finite
system with 40 nodes. Darker dots indicate a higher density distribution. The
lines comprising faint blue (green) dots denote eigenstates localized towards
the right (left) with larger (smaller) node indices. (The green dots in (c)
are located at $E=0$.) (d) Variation of proportion of OBC eigenstates in
finite systems comprising 40 nodes localized nearer the left edge with
$C_{\mathrm{AB},-m}$ and $C_{\mathrm{BA},n}$. The dotted white lines denote
the partitions between phase regions with different values of the braiding
index $\xi$, which are indicated at the top left corner of the regions. The
values of $(C_{\mathrm{AB},-m},C_{\mathrm{BA},n})=(1.4,1.6)$ plotted in (a) –
(c), and $(C_{\mathrm{AB},-m},C_{\mathrm{BA},n})=(0.8,1.1)$ plotted in (e) and
(f) are indicated. (e) Distribution of $\xi_{r}(E)$ on the complex energy
plane for a system with with $m=3$, $n=1$, $C_{\mathrm{AB},0}=1$,
$C_{\mathrm{AB},-m}=0.8$, and $C_{\mathrm{BA},n}=1.1$. (f) PBC spectrum, and
OBC spectrum and eigenstate spatial distribution for a finite system
containing 40 nodes for the system in (e).

##  III GBZ and NHSE variation in complex braiding configurations

We comment briefly on the relationship between the braiding index and the non-
Hermitian skin effect (NHSE). In 1D systems, a winding number (or braiding
index) $\xi_{r}(E)$ [90] may be defined with respect to a reference energy $E$
as

$\xi_{r}(E)=\int\frac{\mathrm{d}k}{2\pi
i}\partial_{k}\mathrm{log}\left|H(k)-E\right|.$ (8)

The braiding index $\xi$ in Eq. (3) may be interpreted as a special case of
the winding number $\xi_{r}(E)$ where $E$ is set to
$\frac{1}{2}\mathrm{Tr}(H)$, which for the model in Eq. (1) is equal to 0. The
winding number $\xi_{r}(E)$ may be visually determined from the PBC spectrum
of a system provided that the individual bands can be separately identified
and the direction of increasing $k$ indicated.

For example, in Fig. 3a the winding number at point A. $\xi_{r}(i)=-2$ because
the bands indicated by the red and blue lines each winds around point A in a
complete circle along the clockwise direction. In contrast, the winding number
at point B, $\xi_{r}(B)=-1$ because point B falls completely outside the loop
enclosed by the blue band and is enclosed only the red band, which winds
around the point in the counter-clockwise direction.

The magnitude of the winding number $\xi_{r}(E)$ is associated with the number
of bulk NHSE states that exist in a semi-infinite system at $E$, as explained
shortly, while the sign of the winding number denotes whether it is a semi-
infinite system that extends from 0 to $+\infty$ (positive $\xi_{r}(E)$) or a
semi-infinite system that extends from $-\infty$ to 0 that hosts such bulk
NHSE states ([91]) The braiding index
$\xi=\xi_{r}\left(-\frac{1}{2}\mathrm{Tr}(H)\right)$ therefore indicates the
number and localization of NHSE states that exist at
$E=-\frac{1}{2}\mathrm{Tr}(H)$ in the semi-infinite system.

To further elucidate the relationship between the winding number and semi-
infinite eigenstates, we examine these eigenstates more carefully. We consider
first the eigenstates for a semi-infinite system that extends from $x=-\infty$
to $x=0$. We introduce the surrogate Hamiltonian $H_{1}(\beta)$ [58] to the
Hamiltonian in Eq. (1) as

$H_{1}(\beta)=\begin{pmatrix}0&C_{\mathrm{AB},0}+C_{\mathrm{AB},-m}\beta^{-m}\\\
C_{\mathrm{AB},0}+C_{\mathrm{BA},n}\beta^{n}&0\end{pmatrix},$ (9)

where $\beta\equiv\exp(ik)$ and $k$ is now allowed to be complex. If an
eigenstate of the semi-infinite system at a given energy $E$ exists, the
eigenstate can be generically written as a linear combination

$\psi(x)=\sum^{a=n+1}_{a=1,|\beta_{a;E}|\geq 1}|\phi_{a;E}\rangle
c_{a}\beta_{a;E}^{x}$ (10)

where $\beta=\beta_{a;E}$ is the $a$th value of $\beta$ for which one of the
eigenvalues of $H_{1}(\beta=\beta_{a;E})$ is $E$ and $|\phi_{a}\rangle_{E}$ is
the corresponding eigenvector. The summation of $a$ from 1 to $n+1$ in Eq.
(10) is due to the following considerations: The coupling between the B node
of a unit cell and the A node of the $n$th neighbor to its right represented
by the $\beta^{n}$ term in Eq. (9) requires the A node component of $\psi(x)$,
i.e. $\langle A|\psi(x)$ to satisfy the boundary conditions $\langle
A|\psi(1)=\langle A|\psi(2)=...=\langle A|\psi(n)=0$ beyond the rightmost unit
cell of the semi-infinite system at $x=0$. A linear combination of $n+1$
linearly independent eigenvectors is therefore required in Eq. (10) to ensure
that these $n$ constraints are always satisfied. Moreover, the corresponding
$\beta$ values for these eigenvectors are required to satisfy
$|\beta_{a;E}|\geq 1$ to ensure that $\psi(x)$ remains bounded as
$x\rightarrow-\infty$.

Whether such right-localized eigenstates exist in a semi-infinite system
extending from $x=-\infty$ to $x=0$ for a given $E$ therefore hinges on
whether there are at least $n+1$ linearly independent eigenvectors of
$H_{1}(\beta)$ satisfying $|\beta_{a;E}|\geq 1$ at that given $E$. The
characteristic equation for $|H_{1}(\beta)-E\mathbf{I}_{2}|=0$ can be cast
into the form of a $(m+n)$th order polynomial
$E^{2}\beta^{m}-(C_{\mathrm{AB},0}\beta^{m}+C_{\mathrm{AB},-m})(C_{\mathrm{AB},0}+C_{\mathrm{BA},n}\beta^{n})=0$,
which in general has $(m+n)$ roots for the $\beta$s, corresponding to $(m+n)$
linearly independent eigenvectors.

Now recall that the braiding index is $\xi=N-P$ where $N$ is the number of
zeros of $E_{+}$ and $P=m$ is the number of poles of $E_{+}$ that lie within
$|\beta|<1$ on the complex $\beta$ plane.As before, for simplicity, we first
set aside the cases where any of the $\beta$ values lies exactly on the unit
circle. The zeros of $E_{+}$ on the complex $\beta$ plane, are merely the
$\beta_{a;E=0}$ values corresponding to $E=0$. The number of $\beta_{a;E=0}$s
that lie within the complex unit circle on the $\beta$ plane (excluding the
$\beta=0$ term), i.e. $N$, is therefore $m+\xi$ because $\xi=N-m$. Conversely,
the number of $\beta$s that lie outside the complex unit circle on the $\beta$
plane is $m+n-(m+\xi)=n-\xi$. In particular, $\xi=-1$ implies that there are
$n+1$ values of $\beta$ that lie outside the complex plane with $n+1$
corresponding linearly independent eigenvectors. This is exactly the number of
linearly independent eignvectors with $|\beta|\geq 1$ required to constitute a
right-localized semi-infinite NHSE eigenstate. A more negative value of $\xi$
implies that there is an excess number of $|\beta|\geq 1$ eigenstates that may
be used to construct a right-localized semi-infinite NHSE eigenstate. The fact
that only $(n+1)$ such $|\beta|\geq 1$ eigenstates are required out of the
$n+|\xi|$ available ones implies that we can obtain $C^{n+|\xi|}_{n+1}$
linearly independent right-localized NHSE eigenstates in a semi-infinite
system extending from $x=-\infty$ to $x=0$ at $E=0$. Here, $C^{a}_{b}$ denotes
the number of different combinations of picking $a$ items out of a set of $b$
items. (Note that this differs from the interpretation in [91] where $|\xi|$
itself is interpreted to be the number of degenerate semi-infinite NHSE
eigenstates.) In a similar manner, a positive value of $\xi$ implies that
there are $C^{m+\xi}_{m+1}$ linearly independent left-localized NHSE
eigenstates in a semi-infinite system extending from $x=0$ to $x=\infty$ at
$E=0$.

The above arguments that $\xi=\xi_{r}(E=0)$ is correlated to the number of
semi-infinite NHSE eigenstates at $E=0$ can be readily extended to the case of
arbitary $E$. Fig. 3b shows $n(|\beta|>1)$, i.e., the number of $\beta$s for
which $|\beta|>1$ corresponding to each value of $E$ in the parameter set in
Fig. 3a, for which $m=3$ and $n=1$. It can be readily verified by a comparison
between Fig. 3a and 3b that the relation between the negative values of $\xi$
in Fig. 3a to $n(|\beta|\geq 1)$ in Fig. 3b is given by $n(|\beta|\geq
1)=\xi_{r}+m$ as discussed above. Further, the PBC spectrum coincides with the
boundary lines between different values of $\xi_{r}(E)$. This is because
changes in $\xi_{r}(E)$ correspond to changes in the number of $\beta$s within
the complex unit circle on the $\beta$ plane, which are necessarily
accompanied by $\beta_{a,E}$s crossing the complex unit circle as $E$ is
varied. A $\beta_{a,E}$ values that lies on the complex unit circle
corresponds to a real value of $k=-i\mathrm{ln}(\beta_{a,E})$ in Eq. (1) and
therefore lies on the PBC spectrum.

Let us label the $\beta_{a,E}$s such that
$|\beta_{1,E}|\leq|\beta_{2,E}|\leq...|\beta_{n+m,E}|$. Then, the general
Brillouin zone (GBZ) of Eq. (1) for a system with open boundary conditions
(OBC) in the thermodynamic limit is given by the loci of $E$ for which
$|\beta_{m,E}|=|\beta_{m+1,E}|$. Recall from the above discussion that the
number of $\beta_{a;E}$s that lie within the complex unit circle on the
$\beta$ plane at energy $E$ is $m+\xi_{r}(E)$. For a value of $E$ that lies on
the GBZ, a positive finite value of $\xi_{r}(E)$ therefore implies that
$|\beta_{m,E}|=|\beta_{m+1,E}|\leq 1$, which corresponds to a left-localized
OBC NHSE state. Conversely, a negative value of $\xi_{r}(E)$ in turn
corresponds to a right-localized OBC NHSE state. Although strictly speaking,
the GBZ condition $|\beta_{m,E}|=|\beta_{m+1,E}|$ applies only in the
thermodynamic limit, the NHSE localization direction of a sufficiently long
finite-sized chain follows that of the chain in the thermodynamic limit. In
particular, a positive (negative) value of the braiding index $\xi=\xi_{r}(0)$
implies that OBC NHSE states within the central area around $E=0$ bounded by
the PBC spectrum are localized at the left (right) edge of the system. This is
illustrated in Fig. 3c, which shows the eigenenergy and spatial density
distribution for the eigenstates of a 40-node long finite system described by
the Hamiltonian with the parameter set of Fig. 3a, b. As predicted by the
negative signs of $\xi_{r}(E)$ shown in Fig. 3a, most of the eigenstates are
localized towards the right edge (large $N$) of the system, except for two
topological zero modes (TZMs) that are localized towards the left edge (these
TZMs are not described by the GBZ theory).

Apart from the TZMs, the NHSE localization direction may differ from that
predicted by $\xi$ outside the central area on the complex energy plane around
$E=0$ bounded by the PBC spectrum. Fig. 3d shows $f_{\mathrm{L}}$, the
fraction of the 40 eigenstates of a 40 node-long system described by Eq. (1)
localized closer to the left edge, as a function of $C_{\mathrm{AB},-m}$ and
$C_{\mathrm{BA},n}$. A $f_{\mathrm{L}}$ value of 1 indicates that all of the
eigenstates are localized closer to the left edge while a proportion of 0
indicates that all of the eigenstates are localized closer to the right edge.
In the $\xi_{r}(E)=-2$ regions at the four corners of the plot in Fig. 3d,
there are regions at which $f_{\mathrm{L}}$ deviates very slightly from 0.
These slight deviations are due to TZMs that are localized at the left edge
opposite to that of the NHSE eigenstates, as shown in Fig. 3c. Apart from the
$\xi_{r}(E)=-3$ regions where all the eigenstates are localized to the right,
bipolar localization, in which the NHSE eigenstates at different eigenenergies
are localized at different edges of the system, occurs throughout much of the
$C_{\mathrm{AB},-m}$–$C_{\mathrm{BA},n}$ parameter space. One example at
$(C_{\mathrm{AB},-m},C_{\mathrm{BA},n})=(0.8,1.1)$ is shown in Fig. 3e and f
where Fig. 3f shows that the eigenstates in the central complex energy region
around $E=0$ are localized to the left, consistent with $\xi_{r}(E)=1$,
whereas the more numerous eigenstates in the complex $E$ side-lobes are
localized towards the right, consistent with $\xi_{r}(E)=-1$ in these lobes,
as shown in Fig. 3e.

##  IV Continuous jumps in braiding index via intra-unit cell coupling and
onsite coupling

Figure 4:  Continuous jumps in $\xi$ via intra-unit cell coupling  (a) $\xi$
phase diagram of system with surrogate Hamiltonian given by Eq. (11) with
$C_{\mathrm{AB},0}=C_{i}=1$, $m=3$, and $n=1$. The $\xi=-1$ regions, which are
absent in Fig. 3d, are highlighted in white. (b) (Left) Variation of complex
$\beta$s for which $E=0$ with $C_{\mathrm{AB},-m}$ at $C_{\mathrm{BA},n}=2$,
as denoted by the white dotted line in (a). The values of $\beta$ for which
$|\beta|>1$ ($|\beta|<1$) are marked in black (blue). (Right) Projection of
graph on left onto complex $\beta$ plane. The points with lighter (darker)
colors correspond to smaller (larger) values of $C_{\mathrm{AB},-m}$. (c),
(d), (e) show the evolution of the PBC spectrum with $C_{\mathrm{AB},-m}$ in
the vicinity of the $\xi=0$ to $\xi=-2$ transition denoted by the white arrow
in (a). The black arrows in (c) and (e) show the movement direction of the
energy bands as $C_{\mathrm{AB},-m}$ is increased across the transition.

In the model in Eq. 1, there is a constraint in the braiding index $\xi$ in
that it changes only in steps of $\pm m$ or $\pm n$ . This is because the $m$
($n$) roots of $C_{\mathrm{AB},0}+C_{\mathrm{AB},-m}\exp(-ikm)=0$
($C_{\mathrm{AB},0}+C_{\mathrm{BA},n}\exp(ikn)=0$) in Eq. (2) all have the
same absolute values. We can remove this constraint by modifying the
Hamiltonian with the introduction of a finite on-site potential, i.e.,

$H_{2}(\beta)=\begin{pmatrix}-C_{i}&C_{\mathrm{AB},0}+C_{\mathrm{AB},-m}\beta^{-m}\\\
C_{\mathrm{AB},0}+C_{\mathrm{BA},n}\beta^{n}&C_{i}\end{pmatrix},$ (11)

in which case the eigenenrgies $E$ now satisfy

$C_{i}^{2}-(C_{\mathrm{AB},0}+C_{\mathrm{BA},n}\beta^{n})(C_{\mathrm{AB},0}+C_{\mathrm{AB},-m}\beta^{-m})=E^{2}.$
(12)

This results in a phase diagram for which all values of $\xi$ across the
entire range of $\xi=-m$ to $\xi=n$ appear, as shown in Fig. 4a. Due to the
offset $C_{i}^{2}$ in Eq. 12, the $(m+n)$ values of $\beta$ that satisfy the
$E=0$ condition will no longer have the same absolute values in general (Fig.
4b), thus removing the restriction on the change in $\xi$ to steps of $\pm m$
or $\pm n$ as was the case when $C_{i}=0$. As mentioned previously, a change
in $\xi$ will be accompanied by the appearance of a zero eigenenergy on the
PBC spectrum since $\xi$ varies with the number of $\beta$ values at which the
energy eigenvalues are 0 and $|\beta|<1$. From Eq. 12, it can be seen that
$E=0$ corresponds to an eigenenergy at which the two energy bands coincide
with each other (Fig. 3d). As $C_{\mathrm{AB},-m}$ in Fig. 4)c–e is varied
towards the boundary between two $\xi$ values (along the dotted line shown in
Fig. 4a), the two bands begin to approach each other along the real energy
axis (Fig. 4c). They then touch each other at $E=0$ at the transition point
corresponding to the $\xi$ boundary line (Fig. 4d), and then drift apart along
the imaginary energy axis (Fig. 4e) as $C_{AB,-m}$ is increased further. As
the energy bands touch at the transition point, they swap partners, which in
turn results in a change in the connectivity between the energy bands and
accordingly, the number of times the two bands wind around $E=0$. Consider the
series of figures Fig. 4c–e. Before the transition (Fig. 4c),
$\mathrm{Re}(E^{2})$ is less than 0 at the minimum value of $|E|$, which
results in the two bands being separated by a finite gap along the real energy
axis; $\mathrm{Re}(E^{2})$ becomes 0 at the transition point, which results in
the bands touching (Fig. 4d); and finally $\mathrm{Re}(E^{2})$ becomes
positive after the transition point in Fig. 4e, which results in the two bands
being separated by a finite gap along the imaginary energy axis.

##  V  Impact of bidirectional asymmetric long range coupling on braiding
profile

In the models introduced so far, the inter-unit cell A to B sublattice site
coupling $\langle\mathrm{A}|H(\beta)|\mathrm{B}\rangle$ occurs only to the
left ( $C_{\mathrm{AB},-m}\beta^{-m}$) while the B to A sublattice coupling
$\langle\mathrm{B}|H(\beta)|\mathrm{A}\rangle$ occurs only to the right
($C_{\mathrm{AB},0}\beta^{n}$) term. This implies that the maximum value of
$|\xi|$ that can occur is $\mathrm{max}([m,n])$, i.e., the largest distance of
the inter-unit cell coupling either along the left or right directions. We can
further generalize the model to extend the range of possible $\xi$ values and
attain a larger maximum value of $|\xi|$ for a given inter-cell coupling
distance by incorporating inter-unit cell bidirectional coupling of both
directions in both the A to B sublattice sites and vice versa. The
corresponding Hamiltonian in the presence of this bidirectional coupling is
given by

$H_{3}(\beta)=\begin{pmatrix}-C_{i}&\sum_{a=1}^{m_{\mathrm{AB}}}C_{\mathrm{AB};-a}\beta^{-a}+\sum_{b=0}^{n_{\mathrm{AB}}}C_{\mathrm{AB};b}\beta^{b}\\\
\sum_{c=1}^{m_{\mathrm{BA}}}C_{\mathrm{BA};-c}\beta^{-c}+\sum_{d=0}^{n_{\mathrm{BA}}}C_{\mathrm{BA};d}\beta^{d}&C_{i}\end{pmatrix}.$
(13)

Figure 5:  Extended values of $\xi$ in the presence of bi-directional coupling
(a) $\xi$ phase diagram of system with surrogate Hamiltonian on the
$C_{\mathrm{AB};3}$–$C_{\mathrm{BA};1}$ plane at
$C_{\mathrm{AB};0}=C_{\mathrm{BA};0}=C_{i}=1$, and
$C_{\mathrm{AB};2}=C_{\mathrm{BA};-3}=3$. (b), (c) PBC spectra of system in
(a) at (b) $(C_{\mathrm{AB};3},C_{\mathrm{BA};1}=(4.5,1.5)$ with $\xi=-6$ and
(c) $(C_{\mathrm{AB};3},C_{\mathrm{BA};1}=(1.5,4.5)$ with $\xi=3$ .

Fig. 5 shows an exemplary phase diagram for a system represented by Eq. (13)
where $m_{\mathrm{AB}}=m_{\mathrm{BA}}=3$, $n_{\mathrm{AB}}=2$, and
$n_{\mathrm{BA}}=1$ and two PBC spectra from the two extreme (positive and
negative) values of $\xi=n_{\mathrm{AB}}+n_{\mathrm{BA}}=3$ and
$\xi=-m_{\mathrm{AB}}-m_{\mathrm{BA}}=-6$ obtained when the $\beta$ terms with
the largest positive and negative exponents in the upper right element of Eq.
(13) are multplied with their counterparts in the lower left element. The
presence of coupling along both the left and right directions in the two off-
diagonal terms of $H$ implies that the exponents of the $\beta$ terms sum up
in the characteristic equation for the eigenenergy , which is given by

$E^{2}-C_{i}^{2}+\left(\sum_{a=1}^{m_{\mathrm{AB}}}C_{\mathrm{AB};-a}\beta^{-a}+\sum_{b=0}^{n_{\mathrm{AB}}}C_{\mathrm{AB};b}\beta^{b}\right)\left(\sum_{c=1}^{m_{\mathrm{BA}}}C_{\mathrm{BA};-c}\beta^{-c}+\sum_{d=0}^{n_{\mathrm{BA}}}C_{\mathrm{BA};d}\beta^{d}\right)=0.$
(14)

In the characteristic equation, the exponents of $\beta$ now range from
$-(m_{\mathrm{AB}}+m_{\mathrm{BA}})$ to $n_{\mathrm{AB}}+n_{\mathrm{BA}}$.
Correspondingly, for the example depicted in Fig. 5, the values of $\xi$ range
from $-(m_{\mathrm{AB}}+m_{\mathrm{BA}})=-6$ when all of the
($m_{\mathrm{AB}}+m_{\mathrm{BA}}+n_{\mathrm{AB}}+n_{\mathrm{BA}}$) $|\beta|$
values at $E=0$ fall outside the unit circle on the complex $\beta$ plane
(Fig. 5b) to $n_{\mathrm{AB}}+n_{\mathrm{BA}}=3$ when all of the $|\beta|$
values fall within the unit circle. Compared to the earlier models with
unidirectional couplings depicted in Figs. 2 to 4 for which $m=3$ resulted in
the maximum value of $|\xi|$ being only 3,the presence of bidirectional
coupling results in the doubling of the maximum value of $|\xi|$ to 6 even
though the furthest inter-unit cell coupling is still maintained to the third
neighbor.

##  VI Topological phase transition through exceptional curves

Two topologically distinct phases associated with different braiding index
cannot be continuously transformed into one another without transiting through
a vanishing bandgap state, at which point the braiding index changes its
value. This transition signifies the presence of exceptional points (EPs)
lying on the boundary between the two distinct phases. Specifically, all
points lying on the phase boundaries between two distinguished knot
configurations(see phase diagram of Fig. 2d and 4a) constitute EPs which are
characterized by vanishing bandgap with real solution of $k$.

To illustrate the phase transition via EPs, let us consider Eq. 1 and set
$n=1$ without loss of generality. The modified Laplacian then reads

$H(k,m)=(C_{\mathrm{AB},0}+C_{\mathrm{AB},-m}e^{-imk})\sigma^{+}+(C_{\mathrm{AB},0}+C_{\mathrm{BA},n}e^{ik})\sigma^{-}$
(15)

where, $\sigma^{\pm}=\frac{1}{2}(\sigma_{x}\pm i\sigma_{y})$.Note that the
band-gap of the two band non-Hermitian model can vanish in two possible ways.
First, if the transition points lie at the phase boundary between two opposite
handedness (i.e., the sign of the braiding index changes from positive sign to
negative sign or vice versa) configurations, the two strings with undercross
($\tau_{1}^{m}$) configuration transforms into an overcross ($\tau_{1}^{-m}$)
configuration or vice versa, this sort of phase transition is referred as
Type-2 phase transition which is accompanied by $m$ number of EPs (here we
vary $k_{x}$ from $-\pi$ to $\pi$ for simplicity). On the other hand, a second
type of phase transition occurs when two strings change its braiding index
number at the phase transition points while maintaining the same sign, so that
the handedness of the braiding and the NHSE localization remains at the same
edge. This is the so-called Type-1 phase transition, where an undercross
(overcross) configuration remains undercross (overcross) in the braiding
configuration. Type-1 transition is accompanied by a single EP.

Figure 6: Exceptional points evolution in the various braided non-Hermitian
bands  (a) Phase diagram and the phase boundaries with various types and
number of EPs in the two band system defined in Eq. 15 with second order long
range coupling (i.e. $m=2$). Type-1 and Type-2 EPs are defined at the phase
boundaries between different knot index with same sign and opposite sign,
respectively and shown in the vertical and horizontal arrows, respectively.
(b-c) Various type EPs in the complex energy-momentum space. Type-1 transition
is accompanied by single EP located at $k=0$ in (b). Type-2 transition is
accompanied by two EPs located at $k=\pm\pi/2$ in (c).

As an illustrative example, we consider a system with second-order long range
coupling (i.e. setting $m=2$ in Eq. 15), and plot the corresponding phase
diagram in Fig. 6. Clearly, the phase boundaries coincide with the
$C_{\mathrm{AB},-m}/C_{\mathrm{AB},0}=\pm 1$ and
$C_{\mathrm{BA},n}/C_{\mathrm{AB},0}=\pm 1$ lines. Interestingly, the band
diagrams corresponding to points lying on the lines PQ and RS (corresponding
to $C_{\mathrm{AB},-m}=\pm C_{\mathrm{AB},0}$ ) host two exceptional points
with Type-2 transition as there are only two real values of $k$ (i.e.,
$k=\pm\pi/2$ for RS line, see Fig. 6c and $k=0,\pi$ for PQ line) which result
in zero bandgap of Eq. 15. On the contrary, the band diagrams corresponding to
lines AB and CD (corresponding to $C_{\mathrm{BA},n}=\pm C_{\mathrm{AB},0}$)
represents the Type-1 phase transition with a single exceptional point as
there exist only a single real value of $k$ (i.e., $k=0$ for AB line, see Fig.
6b and $k=\pi$ for EF line) which satisfies the zero bandgap condition in Eq.
15. Thus a transition from unlink ($|\xi|=0$) to Hopf link with $|\xi|=2$
(unknot with $|\xi|=1$) phase constitutes a Type-2 (Type-1) transition (see
Fig. 6a).

In general, Type-2 transition host $m$ number of EPs, for a system with long-
range coupling of the order of $m$, since there are $m$ real solutions of $k$
corresponding to zero bandgap. On the other hand, the Type-1 phase transition
hosts only a single EP, regardless of the degree of the long-range coupling
(i.e. $m$). We summarized the locations of $m$ Type-2 EPs and single Type-1 EP
in the $k$ space for several example of long-range order in Table 1 (note that
for simplicity we consider the values of $k$ within the range of $-\pi$ to
$\pi$).

Table 1:  Location and number of various Type of EPs for different order of long range coupling for the 1D non-Hermitian system defined in Eq. 15. Type-1 and Type-2 EPs emerge when phase boundary lie between two distinct phases with two knot index of same sign and opposite sign, respectively. Type-1 phase transition hosts only single EPs irrespective to the choice of long range order while the number of EPs lying on the Type-2 phase boundary is equal to the order of the long range coupling. We used second order to sixth order long range coupling here. Note that we vary $k$ from $-\pi$ to $\pi$ for simplicity.  Order of long range coupling |  Type-1 transition Transition through AB line |  Type-1 transition Transition through EF line |  Type-2 transition Transition through PQ line |  Type-2 transition Transition through RS line
---|---|---|---|---
$m=2$ | $k=0$ | $k=\pi$ | $k=0,\pi$ | $k=\pm\frac{\pi}{2}$
$m=3$ | $k=0$ | $k=\pi$ | $k=0,\pm\frac{2\pi}{3}$ | $k=\pm\frac{\pi}{3},\pi$
$m=4$ | $k=0$ | $k=\pi$ | $k=0,\pm\frac{2\pi}{4},\pi$ | $k=\pm\frac{\pi}{4},\pm(\pi-\frac{\pi}{4})$
$m=5$ | $k=0$ | $k=\pi$ | $k=0,\pm\frac{2\pi}{5},\pm(\pi-\frac{\pi}{5})$ | $k=\pm\frac{\pi}{5},\pm(\pi-\frac{\pi}{5}),\pi$
$m=6$ | $k=0$ | $k=\pi$ | $k=0,\pm\frac{2\pi}{6},\pm(\pi-\frac{2\pi}{6}),\pi$ | $k=\pm\frac{\pi}{6},\pm(\pi-\frac{\pi}{6}),\pm\frac{\pi}{2}$

##  VII Proposal for Topological Circuit Realization

The intricate energy braiding described by the model in Eq. 1 can be
implemented in a practical electrical circuit using an RLC configuration, as
depicted in Fig. 7, for the specific parameter values of $m=2$ and $n=1$ under
PBC configuration. For the corresponding OBC setup, the terminal couplings
should be replaced with appropriate capacitors to ensure consistent total node
admittance across all nodes (refer to Appendix A for detailed information).
The behavior of this system is governed by Kirchhoff’s law, as expressed in
Eq. (16).

$I_{i}(\omega)=\sum_{j}J_{ij}(\omega)V_{j}(\omega)$ (16)

Here, $I_{i}(\omega)$ represents the current flowing into the $i^{th}$ node,
$J_{ij}$ denotes the circuit admittance between the $i^{th}$ and $j^{th}$
nodes, and $V_{j}(\omega)$ is the voltage at the $j^{th}$ node. The circuit
Laplacian, $J_{ij}(\omega)$, is given by:

$J_{ij}(\omega)=i\omega\left[\sum_{m}C_{ij,m}-\sum_{n}\frac{1}{\omega^{2}L_{i,n}}\right]$
(17)

Where $C_{ij,m}$ represents $m^{th}$ capacitor connected between node $i$ and
$j$, and $L_{i,n}$ are the grounding $n^{th}$ inductor connected to node $i$.
Applying Fourier transformation yields the $k-$space representation of the
circuit Laplacian, as seen in Eq. (18):

$J(k)=i\omega\begin{pmatrix}C_{AB,0}+C_{AB,n}-\frac{1}{\omega^{2}L_{a}}&-C_{AB,0}-C_{AB,-m}e^{-i\,m\,k}\\\
-C_{AB,0}-C_{AB,n}e^{i\,n\,k}&C_{AB,0}+C_{AB,-m}-\frac{1}{\omega^{2}L_{b}}\end{pmatrix},$
(18)

Sub-lattice node A and B are grounded through inductors $L_{a}$ and $L_{b}$
respectively. $C_{AB,0}$ is a bidirectional capacitor between node A and B,
$C_{AB,n}$ is an unidirectional capacitor from node A towards B and
$C_{AB,-m}$ is an unidirectional capacitor from node B towards A (Fig. 7).

To realize any complex braiding phase, the system needs to be driven at the
resonant frequency
$\omega_{r}=\frac{1}{\sqrt{L_{a}(C_{AB,0}+C_{AB,n})}}=\frac{1}{\sqrt{L_{b}(C_{AB,0}+C_{AB,-m})}}$.

The real-space matrix formation of $J(k)$ for both PBC and OBC can be found in
Appendix A, supplemented by detailed derivations and circuit diagrams.
Current-inversion negative impedance converters (INIC) topology, incorporating
LT1363 Op-Amps, are employed to realize unidirectional capacitors
$C_{AB,m},C_{AB,n}$ (as shown in Fig 7(c)). To stabilize the Op-Amp output, a
small capacitance ($C_{a}$) in the pico Farad range and a low-resistance
($R_{a}$) less than 1 k$\Omega$ are connected in parallel. Since achieving a
perfect unidirectional capacitance is challenging in practice, $C_{a}$ and
$R_{a}$ are meticulously tuned and tested to attain the best possible
unidirectional capacitance for the capacitors used in this study. Detailed
results of the INIC optimization can be found in Appendix B.

Figure 7: (a) Circuit realization of the non-Hermitian braiding system with
long-range coupling in periodic boundary conditions. There are 10 unit cells
(one representative unit cell is highlighted with a light yellow rectangle)
with A (green dot) and B (red dot) sub-lattice sites. Every node is grounded
via an inductor as shown in (b). A and B sub-lattice nodes are connected with
a bi-directional capacitor $C_{AB,0}$ within each unit cell. For this
demonstration, we selected long-range coupling $m=2$ and $n=1$. So, each A
node of the unit cell is connected to the B node of the second unit cell to
its left via a unidirectional capacitor $C_{AB,m}$ and similarly, the B node
of the cell is connected to the A node of the first unit cell to its right via
another unidirectional capacitor $C_{AB,n}$. All these long-range couplings
are realized through a uni-directional capacitor designed with the current-
inversion negative impedance converters (INIC) as shown in (c). $C_{a}$ and
$R_{a}$ are incorporated into the INIC to stabilize the outputs of the LT1363.
This will generate unidirectional capacitance of $C_{i}$ from node B to A.

To illustrate the practical implementation of our proposed braiding scheme,
four distinct braiding phases—referred to as phases 1 through 4 in Fig 6—have
been selected for execution in the electrical circuit set-up. In the circuit
implementation of the phases, we dynamically adjust the circuit parameters to
demonstrate the versatility and adaptability of our proposed approach. More
specifically, as the circuit is exposed to complex admittance eigen values
with both positive and negative amplitudes, it is vulnerable to non-causal
instabilities[31]. To prevent the system going unstable, we grounded each of
the nodes with additional resistor $R_{0}$ to shift up the whole spectrum to
have complex eigen values with no negative imaginary parts. This shifting,
however, has no effect in realizing the braiding topologies this study is
concerned about. Moreover, this ensures converged transient circuit simulation
results as well. Careful numerical analysis was carried out to get the optimum
value of $R_{0}$ which was chosen as $20\Omega$. Also, parasitic resistance of
$100m\Omega$ was added to each inductors to mimic more real-world system.
Table 2 lists all the chosen parameters for the four different phases.

Table 2: Circuit Parameters to realize the four different braiding phases Phase | $C_{AB,0}$ | $C_{AB,m}$ | $C_{AB,n}$ | $L_{a}$ | $L_{b}$
---|---|---|---|---|---
1 | 4.7 nF | 0.94 nF | 2 nF | 94.52 $\mu$H | 112.28 $\mu$H
2 | 4.7 nF | 0.94 nF | 20 nF | 25.64 $\mu$H | 112.28 $\mu$H
3 | 4.7 nF | 20 nF | 2 nF | 94.52 $\mu$H | 25.64 $\mu$H
4 | 4.7 nF | 20 nF | 20 nF | 25.64 $\mu$H | 25.64 $\mu$H

Interestingly, the complex energy admittance spectrum can be reconstructed
from the circuit by applying current at each node and measuring the node
voltages. This allows the construction of the circuit Green’s function.
Subsequently, the inversion of this Green’s function provides the resulting
circuit Laplacian. From the constructed circuit Laplacian, eigenvalues and
eigen modes corresponding to the admittance spectrum and node voltage
distribution can be calculated. Fig. 8 shows the complex admittance spectrum
corresponding to the circuit of Fig. 7. The numerical results (red dots) are
compared with LTSpice simulation results (blue dots), revealing that the
electrical circuit setup can almost perfectly realize all the distinct phases
of the complex braiding system. Although there are slight discrepancies
between the numerical results and the circuit simulations, these mainly arise
from the inclusion of parasitic resistance in the inductors and the imperfect
unidirectional capacitance realized through the INIC, as discussed in detail
in Appendices C-E. The necessary precautions for choosing parameters related
to the INIC are thoroughly covered in these appendices to ensure the transient
analysis of the circuit converges without causing the OpAmps to saturate.
Furthermore, the effects of parasitics and disorder on the braiding
configurations are discussed in Appendix D.

Figure 8: Admittance energy spectrum -comparison between numerical results and
LTSpice simulation of the circuit shown in Fig. 7. Red and blue dots represent
numerical and LTSpice simulation results, respectively. (a-d) show the OBC
admittance spectrum, while (e-h) show the PBC admittance spectrum. All these
results were obtained with 10-unit cells. In LTSpice, we added 500m$\Omega$
ESR(Effective Series Resistance) to each inductor. Other circuit parameters
used in realizing different phases are outlined in Table 2.

##  VIII Conclusion

In conclusion, we present a comprehensive analysis of complex energy braiding
in 1D non-Hermitian systems with a general $m$th order long-range asymmetrical
coupling. Our work provides a new perspective on the emergence of novel
topological phenomena induced by long-range coupling beyond the conventional
nearest-neighbor interaction. By varying the coupling strengths, we
demonstrate that the modified SSH model displays $m$ distinct knots and links
topologies in the complex energy-momentum space under periodic boundary
conditions (PBC). These topologies can be characterized by a new topological
invariant, the braiding index. We establish a correlation between the braiding
index and the zeros and poles of the characteristic polynomials, providing a
novel insight to the topology of complex energy braiding.

Additionally, our work shows a deep connection between the braiding topology
and eigen state localization due to the non-Hermitian skin effect (NHSE). We
find that the NHSE can be localized at one or both terminals of the system,
signifying the conventional or bipolar skin localization respectively,
depending on the sign of the braiding index. Furthermore, we find that
different complex energy braiding configurations with the same sign (opposite
sign) can be transformed continuously through Type-1 (Type-2) phase
transitions which traverse through single (multiple) exceptional points (EPs).

Finally, we propose a realistic experimental setup for realizing various
braiding schemes in an RLC circuit. This circuit-based platform to realize
braiding topology is readily implementable using basic circuit components and
avoids the requirement for high-dimensional momentum space unlike most other
proposed platforms [48, 54]. The proposed electrical circuit setup provides a
promising platform for exploring the rich physics of complex energy braiding
in non-Hermitian systems with non-reciprocal long-range coupling.

#### Acknowledgments

This work is supported by the Ministry of Education (MOE) Tier-II Grant
MOE-T2EP50121-0014 (NUS Grant No. A-8000086-01-00), and MOE Tier-I FRC Grant
(NUS Grant No. A-8000195-01-00).

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##  Appendix A  Circuit Laplacian Formation

In this section, we undertake a detailed derivation of the circuit Laplacian
specifically for the case where $m=2$ and $n=1$, considering a circuit with 10
unit cells, corresponding to 20 nodes. Our objective is to apply Kirchhoff’s
Current Law (KCL) as expressed in Eq. 16 to every node within the circuit
illustrated in Fig. 7.

Node 1: | $i\omega\left(C_{0}+C_{n}-\frac{1}{\omega^{2}L_{a}}+\frac{1}{i\omega R_{0}}\right)V_{1}-i\omega C_{0}V_{2}-i\omega C_{m}V_{18}=I_{1}$
---|---
Node 2: | $-i\omega C_{0}V_{1}+i\omega\left(C_{0}+C_{m}-\frac{1}{\omega^{2}L_{b}}+\frac{1}{i\omega R_{0}}\right)V_{2}-i\omega C_{n}V_{3}=I_{2}$
Node 3: | $i\omega\left(C_{0}+C_{n}-\frac{1}{\omega^{2}L_{a}}+\frac{1}{i\omega R_{0}}\right)V_{3}-i\omega C_{0}V_{4}-i\omega C_{m}V_{20}=I_{3}$
Node 4: | $-i\omega C_{0}V_{3}+i\omega\left(C_{0}+C_{m}-\frac{1}{\omega^{2}L_{b}}+\frac{1}{i\omega R_{0}}\right)V_{4}-i\omega C_{n}V_{5}=I_{4}$
$\vdots$ | $\vdots$
Node 19: | $-i\omega C_{n}V_{16}+i\omega\left(C_{0}+C_{n}-\frac{1}{\omega^{2}L_{a}}+\frac{1}{i\omega R_{0}}\right)V_{19}-i\omega C_{0}V_{20}=I_{19}$
Node 20: | $-i\omega C_{n}V_{1}-i\omega C_{0}V_{19}+i\omega\left(C_{0}+C_{m}-\frac{1}{\omega^{2}L_{b}}+\frac{1}{i\omega R_{0}}\right)V_{20}=I_{20}$

In matrix format, we can write

$i\omega\begin{pmatrix}d_{a}&-C_{0}&0&0&0&\cdots&{\color[rgb]{1,0,0}-C_{m}}&0&0\\\
-C_{0}&d_{b}&-C_{n}&0&0&\cdots&0&0&0\\\
0&0&d_{a}&-C_{0}&0&\cdots&0&0&{\color[rgb]{1,0,0}-C_{m}}\\\
0&0&-C_{0}&d_{b}&-C_{n}&\cdots&0&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\ddots&\cdots&\cdots\\\
0&0&0&0&-C_{n}&\cdots&0&d_{a}&-C_{0}\\\
{\color[rgb]{1,0,0}-C_{n}}&0&0&0&0&\cdots&0&-C_{0}&d_{b}\end{pmatrix}\begin{pmatrix}V_{1}\\\
V_{2}\\\ V_{3}\\\ V_{4}\\\ V_{5}\\\ \vdots\\\ V_{16}\\\ V_{17}\\\ V_{18}\\\
V_{19}\\\ V_{20}\end{pmatrix}=\begin{pmatrix}I_{1}\\\ I_{2}\\\ I_{3}\\\
I_{4}\\\ \vdots\\\ I_{16}\\\ I_{17}\\\ I_{18}\\\ I_{19}\\\
I_{20}\end{pmatrix}$ (19)

where $d_{a}=C_{0}+C_{n}-\frac{1}{\omega^{2}L_{a}}+\frac{1}{i\omega R_{0}}$
and $d_{b}=C_{0}+C_{m}-\frac{1}{\omega^{2}L_{b}}+\frac{1}{i\omega R_{0}}$ are
the diagonal admittances. The real-space representation of the model in Eq. 1
mirrors 19, with the exception of the diagonal terms. By exciting the circuit
at the resonant frequency
$\omega_{r}=\frac{1}{\sqrt{L_{a}(C_{AB,0}+C_{AB,n})}}=\frac{1}{\sqrt{L_{b}(C_{AB,0}+C_{AB,-m})}}$,
we can effectively eliminate these diagonal terms.

To realize the OBC system, the capacitors coupling the two ends must be
removed, meaning the terms highlighted in red in Eq. 19 will be replaced by
zero. Subsequently, when the Laplacian matrix is multiplied by the complex
frequency $i\omega$, the real spectrum of the Laplacian corresponds to the
imaginary spectrum of Eq. 1, and vice versa.

##  Appendix B Unidirectional Capacitor

Achieving unidirectional coupling in an electrical circuit involves the
utilization of a unidirectional capacitor, which can be effectively
implemented through a negative impedance converter with current inversion
(INIC). The circuit configuration is depicted in Fig. 9. Employing this INIC
setup enables the realization of a unidirectional capacitor.

To establish a unidirectional capacitance, denoted as $C$, between node 1 and
node 2, it is imperative to ensure that the capacitance from node 1 to 2 is
$C$, while the capacitance from node 2 to 1 is effectively zero. The INIC
setup, as illustrated in Fig. 9, facilitates the creation of this
unidirectional capacitance by setting $C_{1}=C_{2}=C/2$ in Fig. 9a. For the
purpose of stabilizing the Op-Amp output, a capacitor $C_{a}$ is added in
parallel with a resistor $R_{a}$. Together, these components form an impedance
denoted as $Z_{a}$. The unidirectionality of capacitive reactance, denoted as
$Z_{c}$, is achieved through this setup.

Figure 9: Implementation of asymmetric capacitive coupling between two nodes
using an INIC when the coupling direction is reversed. (a) Circuit setup for
achieving unequal capacitance of $C_{1}+C_{2}$ for coupling from node 1 to 2,
while the capacitance from node 2 to 1 is $C_{2}-C_{1}$. For
$C_{1}=C_{2}=C/2$, unidirectional coupling capacitance of $C$ from 1 to 2 is
achieved, while the capacitance from node 2 to 1 is approximately zero
($C_{2}-C_{1}\approx 0$). (b) The same setup with the sign of $C_{1}$ reversed
by reversing the polarity of the applied voltage to the OPamps. Hence, for
realizing unidirectional coupling by setting $C_{1}=C_{2}=C$, coupling from
node 1 to node 2 experiences almost zero capacitance ($C_{2}-C_{1}\approx 0$),
while coupling from node 2 to node 1 involves a coupling of $C$.

###  B.1 Zero-Capacitance

In this subsection, we delve into the intricacies of achieving nearly
unidirectional capacitive coupling by approaching zero capacitance in one
direction. In Fig. 9(a), we make the assumption of the Op-Amp’s linear
operation, and due to its very high open-loop gain, we can consider the
voltages at its inverting and non-inverting terminals to be equal, i.e.,
$V_{+}=V_{-}$. This implies $V_{1}=V_{f}$. Additionally, owing to the Op-Amp’s
extremely high input impedance, the input currents at both inverting and non-
inverting terminals are approximately zero, i.e., $i_{+}=i_{-}\approx 0$.

Now, the current from node 1 to the output terminal of the Op-Amp is given by:

$i_{1}=\frac{V_{1}-V_{o}}{Z_{a}}$ (20)

Similarly, the current from the output of the Op-Amp to the inverting terminal
will be $i_{2}=\frac{V_{o}-V_{f}}{Z_{a}}=\frac{V_{o}-V_{1}}{Z_{a}}$ as
$V_{1}=V_{f}$. Since the current going to the Op-Amp through the inverting
terminal, $i_{-}$, is zero, the same $i_{2}$ will flow through $C_{1}$, i.e.,
$i_{2}=\frac{V_{1}-V_{2}}{Z_{C}}$.

$\displaystyle\frac{V_{o}-V_{1}}{Z_{a}}$ $\displaystyle=$
$\displaystyle\frac{V_{1}-V_{2}}{Z_{C_{1}}}$ $\displaystyle V_{o}$
$\displaystyle=$ $\displaystyle V_{1}+\frac{Z_{a}}{Z_{C_{1}}}(V_{1}-V_{2})$
(21)

where $Z_{C_{1}}=\frac{1}{j\omega C_{1}}$ is the capacitive reactance of
capacitor $C_{1}$ at frequency $\omega$.

From (20):

$\displaystyle i_{1}$ $\displaystyle=$
$\displaystyle\frac{V_{1}-V_{o}}{Z_{a}}$ $\displaystyle Z_{a}i_{1}$
$\displaystyle=$ $\displaystyle
V_{1}-V_{1}-\frac{Z_{a}}{Z_{C_{1}}}(V_{1}-V_{2})$
$\displaystyle\frac{V_{1}-V_{2}}{i_{1}}$ $\displaystyle=$ $\displaystyle-
Z_{C_{1}}$

This results in an equivalent negative capacitor. If another capacitor $C_{2}$
with impedance $Z_{C_{2}}$ is added in parallel, the total equivalent
capacitance will be approximately $C_{2}-C_{1}$. Choosing $C_{1}=C_{2}$ leads
to an equivalent capacitance of approximately zero.

###  B.2 Non-zero Capacitance

In the scenario where we reverse the polarity of the voltage applied to the
same circuit, as illustrated in Fig. (9)(b), positive capacitance can be
achieved. Utilizing the same analysis as before, we consider $i_{+}=i_{-}=0$
and $V_{f}=V_{2}$. Consequently, the current flowing through $C_{1}$ is given
by:

$\displaystyle i_{C}$ $\displaystyle=$
$\displaystyle\frac{V_{1}-V_{2}}{Z_{C_{1}}}$
$\displaystyle\frac{V_{1}-V_{2}}{i_{C}}$ $\displaystyle=$ $\displaystyle
Z_{C_{1}}$

In this case, we obtain a positive capacitive reactance $Z_{C_{1}}$.
Introducing a parallel capacitor $C_{2}$ results in a total capacitance of
$C_{1}+C_{2}$.

###  B.3 Testing Unidirectional Capacitance

To test the unidirectional capacitance, we setup a typical $RC$ low-pass
filter and used our unidirectional capacitor as a test capacitance. In Fig.
10(a) we tested the zero capacitance and by reversing the polarity of the
capacitor in Fig. 10(b) we test non-zero capacitance.

Figure 10:  Unidirectional Capacitor testing setup (a) zero-capacitance test
(b) non-zero capacitance test Figure 11: Low pass filter response using
unidirectional capacitor (a) $0.47nF$ (b) $1nF$ (c) $4.7nF$ (d) $10nF$

Fig. 11 shows LTSpice simulation results of testing our unidirectional
capacitors in a low pass filter configuration. From the voltage
gain($=\frac{V_{o}}{V_{s}}$), we calculated the 3dB frequency of a low pass
$RC$ filter as $f_{3dB}=\frac{1}{2\pi RC}$. We used $1k\Omega$ as a test
resistor. For zero (blue line) and non-zero(red line) configurations, we found
non-zero capacitances are 17, 23, 39 and 50 times higher than the zero
capacitances.

##  Appendix C Constraints on Knot Types Achievable in Our Setup

In our model, not all types of knot configurations can be realized; only knots
with an unknotting number of one can be achieved. The unknotting number, a
concept from knot theory, measures the minimum number of crossing changes
needed to transform a given knot into the simplest possible knot, known as the
unknot (a simple loop). In the context of our model with two bands in a 1D
setup containing long-range coupling, it is indeed true that only twist knots
with an unknotting number of one can be realized. For example, in the case of
a Hopf link or a trefoil knot configuration, only a single crossing needs to
be changed to transform the knot configuration into an unknot configuration.
Furthermore, it is true that knots with larger unknotting numbers (such as the
Cinquefoil knot with an unknotting number of two) are not possible in our
current two-band model, as each band must have a unique complex value at each
$k$-point.

##  Appendix D Robustness of the knot configurations against disorder

Our system demonstrates robustness against small parasitic effects. To verify
this, we varied the components within a 5% tolerance of Eq. 1 and plotted the
topological knot phase diagram in Fig.12a-c, which showed a profile almost
identical to the original one. Thus, the knot configuration is quite protected
against small parasitic variations. However, the situation changes when a
$\sigma_{z}$ term is included in the Hamiltonian. If we rewrite the
Hamiltonian in Eq. 1 with a mass term as

$H_{2}\left(k\right)=H_{1}\left(k\right)+C_{z}\sigma_{z};$ (22)

where, $\sigma_{z}$ is the third Pauli matrix and $C_{z}$ corresponds to the
mass-like term in the Hamiltonian, this term breaks the chiral symmetry of the
Hamiltonian. In the presence of a small mass term (i.e., $C_{z}\ll C_{i}$),
the braiding index phase diagram does not change significantly (see Fig.12d).
Therefore, the complex braiding/knot configuration remains protected. However,
with a larger mass-like term (i.e., $C_{z}\approx C_{i}$), the braiding index
phase diagram changes a lot and the region with zero knot index (unknotting
phase) expands in the parameter plane, as shown in Fig.12f, while retaining
other knot configurations with non-zero knot index in smaller range of
parameters. As a result, knot structures convert to an unknot configuration
across the wider parameter regime. Thus, knot structures are still protected
in the presence of a large chiral symmetry-breaking ${C_{z}\sigma}_{z}$ term
but at the different coupling parameters range.

Figure 12:  Braiding index phase diagram for the system described by Eq. 1 of
the main text with $m=3$ and$n=1$, showing the effects of various tolerances
in the coupling parameters: (a) 0% tolerance, (b) 2.5% tolerance, and (c) 5.0%
tolerance. (d-f) Braiding index phase diagram in the presence of a mass-like
term proportional to $\sigma_{z}$ at different $C_{z}$ values, (d) $C_{z}=0$,
(e) $C_{z}=0.2$, and (f) $C_{z}=1.2$. Common parameter used: $C_{AB,0}=1$.

##  Appendix E Stability of the Proposed Electric Circuit Network

In this section, we discuss some important points regarding the stability of
the proposed electric circuit network. Indeed, the use of active elements such
as operational amplifiers (OpAmps) in a negative impedance converter
configuration to achieve unidirectional couplings can introduce stability
concerns.

To address this, we conducted additional analyses beyond the steady-state AC
simulations originally performed with LTSpice. Specifically, we examined the
time dynamics of the system to assess the stability of the circuit. Our
findings indicate that the eigenvalues of the Laplacian with a negative
imaginary part can indeed lead to exponentially increasing time evolution of
the corresponding eigenstates, potentially causing instability. For systems
with periodic boundary conditions, this issue is particularly critical as even
slightly unstable modes can diverge due to the loop feedback of the boundary
conditions. To mitigate this risk, careful design of the OpAmp circuit is
essential, including the implementation of feedback mechanisms to ensure
stability.

Additionally, we have outlined the steps taken to ensure stability in the
possible experimental setup, including the design parameters and feedback
mechanisms used to counteract potential instabilities. To prevent the system
from going unstable due to non-causal instability, we added an additional
resistor $R_{0}$ to each node to shift the entire admittance spectrum upward
such that no complex eigenvalues have negative imaginary parts. This
adjustment does not alter the eigenstate profile, allowing us to still observe
the same braiding patterns. With these changes in the circuit design, time
domain analysis of the system converges, and the system becomes stable.

Furthermore, to achieve almost real-like outputs from circuit simulations, we
added a parasitic resistance of $100\ m\Omega$ to each inductor. While this
inclusion of parasitic resistance results in slight mismatches from the
numerical results, as shown in Fig. 8 of the main manuscript, the circuit now
provides almost real outputs that are stable.