††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Experimental observation of exceptional bound states in a classical circuit
network

Deyuan Zou  Key Laboratory of Advanced Optoelectronic Quantum Architecture and
Measurements of Ministry of Education, Beijing Key Laboratory of Nanophotonics
and Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of
Technology, Beijing 100081, China.  Tian Chen  Key Laboratory of Advanced
Optoelectronic Quantum Architecture and Measurements of Ministry of Education,
Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems,
School of Physics, Beijing Institute of Technology, Beijing 100081, China.
Haiyu Meng<EMAIL_ADDRESS>School of Physics and Optoelectronics, Xiangtan
University, Xiangtan 411105, China  Department of Physics, National University
of Singapore, Singapore 117542  Yee Sin Ang<EMAIL_ADDRESS>Science,
Mathematics and Technology, Singapore University of Technology and Design,
Singapore 487372  Xiangdong Zhang<EMAIL_ADDRESS>Key Laboratory of
Advanced Optoelectronic Quantum Architecture and Measurements of Ministry of
Education, Beijing Key Laboratory of Nanophotonics and Ultrafine
Optoelectronic Systems, School of Physics, Beijing Institute of Technology,
Beijing 100081, China.  Ching Hua Lee<EMAIL_ADDRESS>Department of
Physics, National University of Singapore, Singapore 117542  Joint School of
National University of Singapore and Tianjin University, International Campus
of Tianjin University, Binhai New City, Fuzhou 350207, China

Exceptional bound (EB) states represent an unique new class of robust bound
states protected by the defectiveness of non-Hermitian exceptional points.
Conceptually distinct from the more well-known topological states and non-
Hermitian skin states, they were recently discovered [1] as a novel source of
negative entanglement entropy in the quantum entanglement context. Yet, EB
states have been physically elusive, being originally interpreted as negative
probability eigenstates of the propagator of non-Hermitian Fermi gases. In
this work, we show that EB states are in fact far more ubiquitous, also
arising robustly in broad classes of systems whether classical or quantum.
This hinges crucially on a newly-discovered spectral flow that rigorously
justifies the EB nature of small candidate lattice systems. As a highlight, we
present their first experimental realization through an electrical circuit,
where they manifest as prominent stable resonant voltage profiles. Our work
brings a hitherto elusive but fundamentally distinctive quantum phenomenon
into the realm of classical metamaterials, and provides a novel pathway for
the engineering of robust modes in otherwise sensitive systems.

In 2020, it was first realized that negative entanglement entropy can
mysteriously arise in a non-Hermitian free fermion lattice [1]. First
attributed to non-unitary bc-ghost conformal field theory (CFT) by Ryu et al
[2], it was subsequently realized [1] that it generically occurs in the
presence of exceptional points, which are branch points in the complex
bandstructure [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. While
exceptional points have been been extensively theoretically investigated and
experimentally observed in a variety of classical metamaterials [19, 20], what
was not previously known is that its defectiveness i.e. incompleteness of
eigenbasis also has profound impact on its quantum information aspects.
Specifically, the defectiveness allows the free fermion propagator to become
divergently asymmetric, leading to a non-cancellation of particle hoppings
across an entanglement cut, reminiscent of Hawking radiation near the surface
of a black hole [21, 22]. This non-conservation of probability leads to
negative entanglement.

A very pertinent question is: can such esoteric negative entanglement EB state
behavior manifest in realistic experiments? In this work, we theoretically and
experimentally provide an affirmative “yes”. As a special type of mathematical
eigenstate, EB states are fundamentally not restricted to any particular
physical setup. In particular, even though EB states are originally defined as
negative probability eigenstates of a fermion propagator [1], they can
equivalently exist as physical eigenstates of network Laplacians with an
identical mathematical form.

Key to the viability of EB states in classical networks is their remarkable
robustness. It is commonly believed that highly non-local operators, such as
the parent propagators of EB states, cannot be physically realized as a
lattice model due to the prohibitively large number of effective hoppings. In
this work, we discover a new form of adiabatic spectral continuity of negative
probability eigenvalues, reminiscent of the (completely physically different)
spectral flow due to topological pumping [23, 24, 25, 26], that ensures that
EB states continue to survive almost undisturbed even when the parent
propagator matrix is heavily truncated. Indeed, EB states remain prominently
well-gapped in very small lattice networks down to the order of
$\mathcal{O}(10^{1})$ sites, greatly facilitating their physical realization
in relatively simple classical networks such as electrical circuits without
the need for fine-tuning.

As the highlight, we experimentally implemented a defective free-fermion
propagator matrix in an electrical circuit network, and observed resonant
voltage profiles that exhibits excellent agreement with those of theoretically
predicted EB states. While the success of this experimental demonstration is
aided by the versatility of circuit connections, it more crucially hinges on
the abovementioned newly-discovered spectral flow that enables the rigorous
preservation of EB states in sufficiently small lattices with manageable
numbers of non-local couplings.

## Results

###  .1 Exceptional bound (EB) states as a novel type of robustness

Robust bound states are highly touted for their applications in topological
waveguides, non-Hermitian sensing or BIC systems [27, 28, 29, 30, 31, 32].
Recent literature features two conventional routes for generating robust bound
states - (i) topology and (ii) the non-Hermitian skin effect (NHSE). In the
former, robust boundary-localized states are protected by bulk topological
invariants such as the Chern or $\mathbb{Z}_{2}$ winding numbers [33, 34, 35,
36]. In the latter, “skin” states accumulate robustly at system boundaries due
to the directional amplification from asymmetric hoppings [37, 38, 39, 40, 41,
42].

Figure 1: Origin of EB states. In the traditional quantum entanglement
context, gapless points (Top) are characterized by their bounded occupied
Fermi sea projector $\bar{P}$ (Middle) whose eigenvalues $\bar{p}$ indicates
the occupancy of the lower band (orange). While usual gapless points that are
not geometrically defective i.e. Dirac points (Left column) possess only
$\bar{P}$ eigenvalues within $[0,1]$ (Bottom Left), defective exceptional
points (Right column) also exhibit special isolated EB eigenvalues
$\bar{p}_{\text{EB}}$, $1-\bar{p}_{\text{EB}}$ far outside of $[0,1]$ (Bottom
Right). The enigmatic negative occupation $\bar{p}_{\text{EB}}<0$ stems from
the non-locality of the 2-point functions that form the matrix elements of
$\bar{P}$ (Middle Right), and can be detected as pronounced impedance
resonances if $\bar{P}$, which is highly non-local, were to be implemented as
the Laplacian of an electric circuit.

By contrast, in this work, we theoretically and experimentally characterize a
new type of robustness from so-called exceptional bound (EB) states, whose
robustness rely neither from the NHSE nor nontrivial topology. Originally
interpreted as enigmatic free-fermion eigenstates with negative probability
[1], EB states arise from a totally new mechanism based on the geometric
defectiveness of exceptional points, where the eigenstates are not all
linearly independent. While a $N\times N$-component Hermitian Hamiltonian is
guaranteed full rank i.e. possess $N$ orthogonal eigenstates, a $N\times N$
non-Hermitian Hamiltonian can possess anywhere from one to $N$ linearly-
independent eigenstates.

We first define the EB states in a platform-independent manner. Unlike
topological or skin states, EB states are not eigenstates of the Hamiltonian,
but are instead special eigenstates of $\bar{P}$, the occupied Fermi sea
projector [Fig. 1] restricted to a bounded region $[x_{L},x_{R}]$ [43, 44]:

$\displaystyle\bar{P}$ $\displaystyle=$
$\displaystyle\sum_{x,x^{\prime}\in[x_{L},x_{R}]}|x^{\prime}\rangle\langle
c^{\dagger}_{x^{\prime}}c_{x}\rangle\langle x|$ $\displaystyle=$
$\displaystyle\sum_{x,x^{\prime}\in[x_{L},x_{R}]}\left(\sum_{\mu\in\text{occ.}}\sum_{k}e^{ik(x^{\prime}-x)}\mu_{R}(k)\mu^{*}_{L}(k)\right)|x^{\prime}\rangle\langle
x|.$

Specifically, EB states, which arise only when an exceptional point is
present, are the eigenstates of $\bar{P}$ corresponding to eigenvalues far
outside of $[0,1]$ (yellow in Fig. 1 Bottom), the spectral range of an usual
Hermitian projector. As numerically computed in Fig. 2a, they possess
distinctive spatial profiles different from the exponential profiles of skin
or topological states. Here $c^{\dagger}_{x}$ ($c_{x}$) creates (annihilates)
a fermion at site $x$ and $\langle c^{\dagger}_{x^{\prime}}c_{x}\rangle$ is
its propagator within the occupied bands $\mu$, which are represented by right
and left eigenstates defined by $H(k)\mu_{R}(k)=\epsilon_{\mu}(k)\mu_{R}(k)$
and $H^{\dagger}(k)\mu_{L}(k)=\epsilon^{*}_{\mu}(k)\mu_{L}(k)$. For a non-
Hermitian Hamiltonian $H\neq H^{\dagger}$, the left and right eigenstates are
not simply complex conjugates of each other; they are necessarily defined as
such so that they form a biorthonormal basis satisfying
$\mu^{\prime}_{L}\cdot\mu^{*}_{R}=\delta_{\mu^{\prime}\mu}$. Physically, this
biorthogonal definition of the propagator $\langle
c^{\dagger}_{x^{\prime}}c_{x}\rangle$ preserves the probabilistic
interpretation of quantum mechanics under non-Hermiticity [45]. But
paradoxically, it would also give rise to negative probability EB eigenstates
as we shall shortly see.

We next review the notion of geometric defectiveness, such as to understand
its fundamental role in ensuring EB robustness. Consider a gapless non-
Hermitian Hamiltonian $H(k)$ with a $n$-fold degenerate eigenenergy
$\epsilon_{0}$ (exceptional point) at $k=k_{0}$. If it were Hermitian, there
would be $n$ linearly-independent eigenstates at $k_{0}$, all with eigenenergy
$\epsilon_{0}$. However, since $H$ is non-Hermitian, the degenerate subspace
at $k_{0}$ in general decomposes into a sum of Jordan blocks, namely
$H(k_{0})=H_{\text{non-deg}}\oplus H_{\text{deg}}$ where $H_{\text{non-deg}}$
is gapped and $H_{\text{deg}}$ is degenerate and can be brought to the form
$H_{\text{JB}}^{1}\oplus H_{\text{JB}}^{2}\oplus...\oplus H_{\text{JB}}^{j}$
via a similarity transform. Each Jordan block $H_{\text{JB}}^{i}$ (with
$i=1,...,j$) is $n_{i}\times n_{i}$ such that $\sum_{i=1}^{j}n_{i}=n$, and
takes the form

$H_{\text{JB}}^{i}=\left(\begin{matrix}\epsilon_{0}&1&0&\ldots&0\\\
0&\epsilon_{0}&1&\ldots&0\\\ 0&0&\epsilon_{0}&\ddots&0\\\
\vdots&\vdots&\vdots&\ddots&1\\\ 0&0&0&0&\epsilon_{0}\end{matrix}\right).$ (2)

Despite being $n_{i}\times n_{i}$, it possesses only a _single_ nonvanishing
eigenstate $(1,0,0,\ldots,0)^{T}$, and is thus of rank $1$. As such, the
$n\times n$ $H_{\text{deg}}$ and thus $H(k_{0})$ has only $j\leq N$
independent eigenstates i.e. has geometric multiplicity $j$. When $j<N$,
$H(k_{0})$ is known as being geometrically _defective_.

EB states precisely originate from the presence of special points $k_{0}$
known as exceptional points, where the Hamiltonian $H$ is geometrically
defective. This causes the corresponding $\bar{P}$ to become singular, as
explained below. From Eq. 2, it can be seen that within each Jordan block of
the degenerate subspace, the left and right eigenstates take the forms
$\mu_{L}=(0,...,1)^{T}$ and $\mu_{R}=(1,...,0)^{T}$, which are perfectly
orthogonal. As such, the existence of nontrivial Jordan blocks i.e. geometric
defectiveness makes the biorthogonal basis and thus the restricted Fermi sea
projector $\bar{P}$ ill-defined. In particular, as $k_{0}$ is approached,
$\bar{P}$ becomes more singular with increasingly divergent matrix elements
(propagators) $\langle c^{\dagger}_{x^{\prime}}c_{x}\rangle$. To concretely
see this, we specialize to the simplest exceptional point ansatz with double
degeneracy ($n=2$) in a single Jordan block, and consider perturbations around
$k=k_{0}=0$:

$\displaystyle H_{n=2}(k)$ $\displaystyle=\left(\begin{matrix}0&a_{0}+h(k)\\\
h(k)&0\end{matrix}\right)\sim\left(\begin{matrix}0&a_{0}\\\
\frac{1}{2}k^{2B}&0\end{matrix}\right)$ (3)

with $h(k)=\frac{1}{2}(2(1-\cos k))^{B}$, which reduces to $H^{1}_{\text{JB}}$
with $n_{1}=2$ and $\epsilon_{0}=0$ at $k=0$ upon rescaling. Choosing the
negative eigenenergy band to be occupied, we obtain

$\displaystyle\langle c^{\dagger}_{x^{\prime}}c_{x}\rangle_{n=2}$
$\displaystyle=\frac{1}{2}\sum_{k}e^{ik(x^{\prime}-x)}\left(\begin{matrix}1&-\sqrt{\frac{a_{0}+h(k)}{h(k)}}\\\
-\sqrt{\frac{h(k)}{a_{0}+h(k)}}&1\end{matrix}\right)$
$\displaystyle\sim\frac{1}{2}\sum_{k}e^{ik(x^{\prime}-x)}\left(\begin{matrix}1&-k^{-B}\sqrt{2a_{0}}\\\
-k^{B}/\sqrt{2a_{0}}&1\end{matrix}\right),$ (4)

which is divergently asymmetric due to the $\sim k^{-B}$ contributions near
$k=0$. Note that in the absence of $x_{L},x_{R}$ boundaries, the momentum $k$
is diagonal and $\bar{P}$ would still have eigenvalues $\bar{p}=0$ and $1$
only, as expected for a Fermi sea projector. But with these $x_{L},x_{R}$
boundary cutoffs, translation invariance is broken and the divergent $k^{-B}$
contributions from Eq. 4 dominate, yielding eigenvalues $\bar{p}$ that can be
prominently outside of $[0,1]$.

These eigenvalues outside of $[0,1]$ are known as the EB eigenvalues (labeled
as $\bar{p}_{\text{EB}}$), and their corresponding eigenstates dubbed EB
states [Fig. 1]. For a lattice system with $L$ unit cells, the magnitude of
$\bar{p}_{\text{EB}}$ scales like $L^{B-1}$, since the largest contributions
to $\langle c^{\dagger}_{x^{\prime}}c_{x}\rangle_{n=2}$ come from $k^{-B}$ at
the momentum points $k=\sim\pm\pi/L$ nearest to $k=0$. This is more rigorously
proven in the Methods. For every EB eigenvalue $\bar{p}_{\text{EB}}$, by
construction $1-\bar{p}_{\text{EB}}$ is also an eigenvalue of $\bar{P}$.
Arising due to the boundaries of $[x_{L},x_{R}]$, the number of EB states do
not scale with $L$: This is evident in Fig. 2a, which shows the $\bar{P}$
eigenvalues for $H_{n=2}$ (Eq. 3) with $B=3$, numerically plotted as a
function of boundary position $[x_{L},x_{R}]=[1,x_{cut}]$. Most eigenvalues
(dark blue) remain within $[0,1]$, with the exception of isolated pairs of EB
eigenvalues $\bar{p}_{\text{EB}}$ corresponding to EB eigenstates
$\psi_{1}^{\pm},\psi_{2}^{\pm}$ (pale blue).

We next discuss the interpretation of the EB eigenvalues
$\bar{p}_{\text{EB}}$. Since $\bar{P}$ projects onto the occupied bands within
$[x_{L},x_{R}]$, its eigenvalues $\bar{p}$ can be interpreted as the
occupation probability of their corresponding eigenstates. Indeed, for free
fermions, the departure of $\bar{p}$ from $0$ or $1$ represents the amount of
quantum entanglement across the boundaries of $[x_{L},x_{R}]$, as quantified
by the entanglement entropy
$S=-\sum_{\bar{p}}\left[\bar{p}\ln\bar{p}+(1-\bar{p})\ln(1-\bar{p})\right]$
[46], which is greatly contributed by values of $\bar{p}$ away from $0$ or
$1$. For gapped systems, all eigenvalues $\bar{p}$ lie exponentially close to
$0$ or $1$, leading to very little entanglement; for usual gapless systems
without geometric defectiveness, $\bar{p}$ become distributed within $[0,1]$
[Fig. 1 Bottom Left] with logarithmic spacings, leading to
$S\sim\frac{1}{3}\log L$, where $L$ is the system size [47, 48]. However, at
geometrically defective critical points, one or more pairs of eigenvalues
$\bar{p}_{\text{EB}}\sim L^{B}$ lie prominently outside of $[0,1]$. Literally
interpreted, the EB eigenvalue $1-\bar{p}_{\text{EB}}<0$ thus represents
negative probability: this is possible because the band occupancy is
characterized by a biorthogonal projector
$\sum_{k,\mu\in\text{occ.}}|\mu_{R}(k)\rangle\langle\mu_{L}(k)|$ involving
left/right eigenstates (bra/kets) that are generically different.
Alternatively, we can also understand why we can have
$1-\bar{p}_{\text{EB}}<0$, or equivalently $\bar{p}_{\text{EB}}>1$, through
the divergently asymmetric 2-point function $\langle
c^{\dagger}_{x^{\prime}}c_{x}\rangle$. By having strongly asymmetric
“hoppings” across the boundary of the cut region $[x_{L},x_{R}]$, there can be
a net gain of state amplitude into the region , resulting in
$\bar{p}_{\text{EB}}>1$, somewhat reminiscent of Hawking radiation at the
event horizon of a black hole [21, 22].

Figure 2: Robustness of EB states and their spectral flow.  (a) Flow of
occupation probability eigenvalues $\bar{p}$ for our canonical model
$H_{n=2}$, $B=3$ as the un-truncated region $[x_{L},x_{R}]=[1,x_{cut}]$ is
varied. While most $\bar{p}$ cluster around $0$ and $1$, two distinct branches
of EB states $\psi_{1}^{\pm}$ and $\psi_{2}^{\pm}$ exhibit distinct robust
spectral flows away from them, with the $x_{cut}\leftrightarrow L-x_{cut}$
symmetry arising from the equivalent roles of the truncated vs. un-truncated
regions. As $x_{cut}$ decreases, the EB eigenstates (insets) survive almost
unchanged, testimony to the adiabatic continuity between EB states defined on
very small and large $x_{cut}$ subsystems. (b,c) Distribution of the complex
$\bar{p}$ eigenvalues from 50 instances of random geometric disorder in the
$\bar{P}$ matrix elements with $x_{cut}=L-1$, with real and imaginary
fractional perturbations taken uniformly from $[-\delta,\delta]$. Unlike the
non-EB eigenvalues (central cloud), EB eigenvalues (red dashed) remain
relatively unperturbed and well-gapped, even in (c) where the largest matrix
elements scale like $L^{B-1}\sim\mathcal{O}(10^{3})$. We have fixed $a_{0}=1$
and $L=50$ unit cells in all panels.

###  .2 Exceptional bound (EB) states as physical eigenstates

Even though EB states possess interesting and unusual properties, to directly
observe them, they have to be eigenstates of a physical observable, and not
just the truncated projector $\bar{P}$. Hence, EB states have hitherto been
considered physically elusive, since $\bar{P}$ contains extremely non-local
matrix elements, and cannot be approximated by experimentally realistic
Hamiltonians or network Laplacians.

What has not been realized, however, is that EB states can survive robustly
even if $\bar{P}$ is extensively truncated, such that only a small proportion
of the non-local matrix elements remain. Below, we establish this important
fact, and furthermore show that EB states are relatively insensitive to system
disorder compared to all other eigenstates. These two discoveries make it
possible to observe EB states in rather small systems governed by
experimentally realistic Hamiltonians or network Laplacians that
mathematically simulate the heavily truncated $\bar{P}$, _without_ the need
for constructing the geometrically defective parent systems.

####  .2.1 EB spectral flow and their robustness against spatial truncation

EB spectra exhibits a novel type of spectral flow that gives EB states its
unusual robustness. This is illustrated in Fig. 2a, which shows how the
eigenvalues of $\bar{P}$ evolves as the position of its truncation cut
$x_{cut}=x_{R}$ is varied from $1$ to $L=50$ (we fix $x_{L}=1$). For
definiteness, we have used $H_{n=2}$ (Eq. 3) with exceptional point dispersion
$\sim k^{2B}=k^{6}$, although other choices of $B$ generically give rise to
qualitatively similar behavior [Fig. 5 of Methods].

Saliently, while almost all eigenvalues $\bar{p}$ are infinitesimally closely
locked to $0$ or $1$ (red dashed), there exist two pairs of eigensolutions
($\psi_{1}^{\pm}$ and $\psi^{\pm}_{2}$) that evolve away from $[0,1]$,
crossing each other at $x_{cut}=L/2=25$ in a perfectly symmetrical manner.
This “EB spectral flow” is reminiscent of the spectral flow of topological
states as flux is threaded (see Refs. [49, 23, 24, 25, 26]), even though there
cannot be any topological pumping in such a critical system.

The existence of this spectral flow is very important for the practical
physical demonstration of EB states, because it implies that an EB state which
is well-defined at large truncated system sizes $x_{cut}$, when the lattice
approximates a continuum, is adiabatically connected to a qualitatively
similar EB state at very small truncated systems with
$x_{cut}\sim\mathcal{O}(1)$. As such, bone-fide EB state phenomena can be
demonstrated by truncating $\bar{P}$ to just a few unit cells, at which their
conceptual origin becomes otherwise murky. Indeed, the $\psi_{1}^{\pm}$ EB
state in the nearly-continuum case of $x_{cut}=49$ unit cells (top right)
remains almost qualitatively unchanged as $x_{cut}$ gets smaller, keeping its
characteristic sine-like profile (derived in the Methods) even down to
$x_{cut}=4$ (bottom left). Its adiabatic continuity can be justified through
the large spectral gap separating the EB states from the other states. The
$\psi_{2}^{\pm}$ branch represents a different orthogonal eigensolution that
is odd about $x_{cut}=L/2$. For $x_{cut}=L$ (green), we simply have PBCs and
$\bar{p}$ reduces to exactly $0$ or $1$.

In the following, we sketch the origin of the robustness of this EB spectral
flow; the interested reader is invited to refer to the Methods for a more
complete proof. Unlike topological spectral flow, which is protected by
quantized Wannier center transport [49, 23, 24, 25, 26], EB spectral flow is
not protected by any topological invariant. Yet, the adiabatic continuity and
crisscrossing flows of the EB spectrum with $x_{cut}$ is equally inevitable
due to the divergently long-ranged nature of the 2-point functions that form
the matrix elements of $\bar{P}$. In the Methods, it was shown that the
largest 2-point functions decay slower than a power-law (for $B>1$), and
importantly also causes the EB states to exhibit approximately similar spatial
profiles. Due to their very slow decay, the extent of entanglement, as
captured by the departure of $\bar{p}_{\text{EB}}^{2}-\bar{p}_{\text{EB}}$
from zero, must therefore scale linearly with the size of the non-truncated
region $x_{cut}$ (Eq. 16). This spectral flow behavior originates from the
long-ranged 2-point functions, and cannot be destroyed unless the parent
Hamiltonian is no longer geometrically defective. As such, the EB states from
a heavily truncated $\bar{P}$ with very small number $x_{cut}$ of unit cells
must still be adiabatically connected to EB states defined on a large lattice.

####  .2.2 Robustness of EB states against hopping disorder

Besides surviving robustly even as the hopping model $\bar{P}$ is truncated
down to very small sizes, EB states are also remarkably robust to random
hopping disorder when $\bar{P}$ is taken as a physical Hamiltonian or
Laplacian. A key feature in the hoppings of $\bar{P}$ is that it ranges over
many orders of magnitude – as derived in the Methods, the largest hoppings
scale like $L^{B-1}$ (for $B>1$), while the smallest do not scale with $L$ at
all. This dramatic contrast between strong and weak hoppings is what gives
rise to the illusion of “negative probability” through a Hawking radiation-
like argument; yet, it also brings about the very real prospect of
experimental disorder in the few strongest hoppings drowning out all other
subtle signatures.

Fortunately, EB states turn out to be far more resilent towards hopping
disorder than non-EB states. To simulate component disorder in actual
experiments, the disorder is introduced by multiplying each hopping by a
random factor of $1+r_{R}+ir_{I}$, where $r_{R}$ and $r_{I}$ are real random
variables uniformly selected from $[-\delta,\delta]$. In electrical circuit
realizations, for instance, complex disorder can stem from stray capacitances
or inductances that lead to random phase shifts. Spectral plots with only real
or imaginary disorder can be found in the Supplement.

As shown in Figs. 2b and c for $H_{n=2}$ with $B=2$ and $B=3$ respectively,
the $\bar{P}$ eigenvalues from 50 independent instances of geometric hopping
disorder form dispersed spectral “clouds”. At zero disorder, the eigenvalues
$\bar{p}$ (black dots) are either infinitesimally close to $0$ or $1$ (non-EB
states), or take on isolated values outside of $[0,1]$ (EB states). With
nonzero fractional hopping disorder $\delta$, the non-EB eigenvalues quickly
become very dispersed, purportedly because very small fractional uncertainties
in the largest hoppings can be of large absolute magnitudes. Indeed, when the
disorder fraction reaches 5$\%$ ($\delta=0.05$) for the $B=2$ case [Figs 2b],
the $0$ and $1$ non-EB eigenvalues become so perturbed that their density
clouds cannot be separated - alternatively, we say that with even 5$\%$
complex disorder, the notion of occupancy becomes ill-defined. For the $B=3$
case [Figs 2c] with more strongly divergent hoppings in $\bar{P}$, this
threshold is lowered to about 0.5$\%$ ($\delta=0.005$), hardly a generous
uncertainty allowance for realistic experimental components [50, 51].

Yet, from Figs. 2b and c, it is evident that the EB eigenvalues (red dashed)
remain well-separated from non-EB eigenvalues even at $\delta=0.05$, when the
notion of non-EB state occupancy is not even well-defined. The fractional
spread in the EB eigenvalue “cloud” remains at approximately $\delta$ of the
unperturbed $\bar{p}_{\text{EB}}$, seemingly impervious to the disorder from
the largest hoppings. All in all, we have established the feasibility of
realizing EB states in lattices with significant geometric disorder, even in
regimes where it is hopeless to observe non-EB spectral signatures.

###  .3 Experimental observation of EB states

In the parent system $H$ which contains an exceptional point, the truncated
band projector $\bar{P}$ is not a physical observable and its EB eigenstates
would be experimentally elusive. However, by constructing an alternative
system described by a Hamiltonian or network Laplacian that is mathematically
identical to $\bar{P}$, we can directly observe EB eigenstates as unique and
isolated resonant peaks. Furthermore, instead of having a large number $L$ of
unit cells as in the parent system, $\bar{P}$ only needs to have
$x_{cut}\sim\mathcal{O}(1)$ unit cells to exhibit well-defined EB states.

###  .4 Description of experimental setup

We experimentally observed the EB states of our $H_{n=2}$ Hamiltonian by
implementing its $\bar{P}$ via an electrical LC circuit network [Fig. 3a,b],
with the non-Hermiticity introduced via operational amplifiers (op-amps)
encapsulated in negative impedance converters with current inversion (INICs,
see Methods). The steady state in electrical circuits is governed by
Kirchhoff’s law, which can be written in the matrix form
$\mathbb{I}=J\mathbb{V}$, where $\mathbb{I}=(I_{1},I_{2},...)^{T}$ represents
the input currents at nodes labeled $1,2,$ etc and
$\mathbb{V}=(V_{1},V_{2},...)^{T}$ represent their corresponding electrical
potentials. The matrix $J$, known as the circuit Laplacian, encodes the
component connectivity of the circuit, and can be designed to accurately
approximate the form of our matrix $\bar{P}$ as desired.

Figure 3: Experimental measurement of EB states. (a,b) Our LC circuit, which
implements Eq. 5 via the Laplacian given by Eqs. 18 and 19, consists of 6
nodes connected and grounded (circled 1 to 6) by various combinations of
capacitors and inductors. INICs containing non-Hermitian op-amps implement the
requisite asymmetric couplings. (c) Simulated and measured voltage profiles as
a function of AC frequency $\omega$, with resonances appearing at the
frequencies corresponding to all four eigenvalue clusters
$\bar{p}^{\prime}\approx-1,0,1,2$ (Eq. 6). However, only the voltage amplitude
profiles at the EB eigenvalues $\bar{p}^{\prime}\approx-1,2$ (colored “EB”)
resemble that of their corresponding eigenstates. Very good agreement (Right)
exist between the theoretically simulated $\bar{P}^{\prime}$ matrix elements
and their reconstructed values from node-resolved impedance measurements. (d)
Both the simulated (red) and measured (yellow) voltage amplitudes exhibit
resonances at the four frequencies corresponding to $\bar{P}^{\prime}$
eigenvalues, although the measured peaks are smoothed out by parasitic
resistances. The peak positions agree to within 1% (2-3 kHz). (e) Excellent
quantitative agreement between the simulated EB state amplitude profile $|V|$,
which is identical for both $\bar{p}^{\prime}_{\text{EB}}\approx-1,2$, and
their measured counterparts at slightly shifted $\bar{p}^{\prime}_{\text{EB}}$
frequencies.

In our experiment, we chose $x_{cut}=3$ and design our circuit [Fig. 3a] such
that the eigenvalue equation $\bar{P}\mathbb{V}=\bar{p}\mathbb{V}$ is
approximately satisfied when strong topolectrical resonances [50] occur. In
such scenarios, nonzero voltage potential profiles
($\mathbb{V}\neq\mathbb{0}$) can exist even with negligible external source
currents ($\mathbb{I}=0$), so the eigenvectors $\mathbb{V}$ are mapped into
the kernel of the circuit Laplacian $J$, i.e. $J\mathbb{V}=\mathbb{0}$. With
details left to the Methods, the circuit thus designed and fabricated is
illustrated in Figs. 3a,b, with 6 nodes (labeled $1$ to $6$) connected by
various combinations of capacitors, inductors and op-amps. It gives rise to
the following simplified eigenvalue equation

$\frac{1}{C_{0}}\left(\begin{matrix}\mathbb{C}_{13}&\mathbb{C}_{24}&\mathbb{0}\\\
\mathbb{C}_{24}&\mathbb{C}_{13}&\mathbb{C}_{24}\\\
\mathbb{0}&\mathbb{C}_{24}&\mathbb{C}_{13}\end{matrix}\right)\mathbb{V}=\bar{P}^{\prime}\,\mathbb{V}=\bar{p}^{\prime}\,\mathbb{V}$
(5)

where $\bar{P}^{\prime}$ is a matrix that closely approximates to the exact
$\bar{P}$ for parameters $a_{0}=1$, $B=7$, $L=4$ (see Supplement), as far as
commercially available circuit components and their uncertainty tolerances
allow. As elaborated in the Methods, its eigenvalue is related to the circuit
parameters via

$\bar{p}^{\prime}=\frac{1}{\omega^{2}LC_{0}}-\frac{C_{1}+2C_{2}+C_{3}+2C_{4}}{C_{0}},$
(6)

with $C_{0}=1$nF setting the capacitance scale. We have
$\mathbb{C}_{13}=C_{5}\,\mathbb{I}-C_{1}\sigma_{x}-i\sigma_{y}C_{3}$ and
$\mathbb{C}_{24}=-C_{2}\sigma_{x}-i\sigma_{y}C_{4}$, where
$\sigma_{x},\sigma_{y}$ are the Pauli matrices and $C_{1}=4.55$nF,
$C_{2}=2.87$nF, $C_{3}=4.3$nF, $C_{4}=3.04$nF and $C_{5}=0.50$nF are
capacitors in the circuit. $L=10\mu$H are grounded inductors introduced such
that $p^{\prime}$ can be freely tuned by adjusting the AC frequency $\omega$
(Eq. 6). To minimize component disorder, we have chosen to include only 5
unique capacitance parameters corresponding commercially available capacitors,
truncating the other matrix elements to zero; with them, Eq. 5 gives EB
eigenvalues $\bar{p}^{\prime}_{\text{EB}}=1.987$ and $-0.987$, which are
already extremely close to the exact EB eigenvalues
$\bar{p}_{\text{EB}}=2.007$ and $-1.007$.

While the implementation of the positive LC components is straightforward,
negative capacitors $-C_{3}$ and $-C_{4}$ are also present, as explicitly seen
from their asymmetric appearances in the $i\sigma_{y}$ terms of
$\mathbb{C}_{13}$ and $\mathbb{C}_{24}$ (Eq. 6). The implementation of
negative capacitive feedback requires more involved op-amp circuitry (see
Methods) which are decidedly non-Hermitian, drawing power from external
voltage sources. Note that this coupling asymmetry _cannot_ give rise to the
NHSE because the PBC spectrum of the projector consists of isolated spectral
points $0$ and $1$ by construction, and do not possess the spectral winding
necessary for the NHSE .

###  .5 Experimental results

To conclusively establish the observation of EB phenomena in our circuit, we
performed the (i) measurement of EB spectra as topolectrical resonances; (ii)
spatially mapping of the EB eigenstate from the electrical potential profile;
and (iii) reconstruction of the circuit Laplacian $J$ corresponding to
$\bar{P}^{\prime}$.

EB resonances in our circuit should be detectable via two-point impedance
measurements between any two nodes. Given any two nodes $i$ and $j$, the
impedance $Z_{ij}$ between them is given compactly by [50, 52]

$\displaystyle Z_{ij}$ $\displaystyle=$
$\displaystyle\sum_{\mu}\frac{\left(\langle\psi_{\mu}^{L}(i)|-\langle\psi_{\mu}^{L}(j)|\right)\left(|\psi_{\mu}^{R}(i)\rangle-|\psi_{\mu}^{R}(j)\rangle\right)}{j_{\mu}}\qquad$
(7)

where $j_{\mu}$ is the $\mu$-th eigenvalue of the circuit Laplacian $J$, with
$|\psi_{\mu}^{L}\rangle,|\psi_{\mu}^{R}\rangle$ its corresponding left and
right eigenstates. Since we have designed our circuit such that all
$\bar{P}^{\prime}$ eigenstates $\mathbb{V}$, particularly the EB eigenstates,
satisfy $J\mathbb{V}=0$, they are precisely the same voltage profiles
corresponding to $j_{\mu}=0$ i.e. divergent impedance $Z_{ij}$. In other
words, when the AC frequency $\omega$ is adjusted such that $\bar{p}^{\prime}$
satisfy the eigenvalue equation Eq. 5, the two-point impedance between any two
nodes should exhibit a resonance. Substituting the EB eigenvalues of
$\bar{p}^{\prime}_{\text{EB}}=1.987\approx 2$ and $-0.987\approx-1$ and non-EB
eigenvalues $\bar{p}^{\prime}=0.991,0.906,0.094$ and $0.009$ into Eq. 6, we
theoretical predict two resonant EB frequencies of 334kHz and 359kHz
corresponding to $\bar{p}^{\prime}\approx 2$ and $-1$ respectively, and four
non-EB resonant frequencies 342.0kHz, 342.6kHz, 349.3kHz and 350 kHz.

As shown in Fig. 3c, these predictions are indeed in very close agreement with
the simulated and measured voltage profiles $\mathbb{V}=(V_{1},...,V_{6})^{T}$
at the nodes, when an AC current is driven through the circuit. For
definiteness, we have chosen to connect node 2 and the ground with a AC signal
of amplitude 2V, even though the $\bar{p}^{\prime}$ eigenvalues corresponds to
topolectrical resonances $j_{\mu}=0$ which should always cause a divergent
voltage response, regardless of the input current configuration (Eq. 7). In
both the simulations and experimental measurements, pronounced resonances were
detected at the said EB and non-EB frequencies $\omega$. This is more
precisely plotted in Fig. 3d, which shows the qualitative agreement between
the theoretically simulated resonance peaks (red) and the measured peaks
(yellow), which are situated at $\omega=$336.6kHz
($\bar{p}^{\prime}_{\text{EB}}=1.7$), 343.2kHz ($\bar{p}^{\prime}=0.83$),
349.6kHz ($\bar{p}^{\prime}=0.05$) and 356.9kHz
($\bar{p}^{\prime}_{\text{EB}}=0.8$). Note that in terms of the absolute
frequency $\omega$, the measured and theoretically computed peaks only differ
by less than $1\%$, which is arguably excellent considering the effects of
component uncertainties and parasitic resistances. While the latter have also
broadened the measured peaks, the EB peaks remain well-resolved.

The EB state profiles can be directly read from the electrical potential
profiles $\mathbb{V}$ at the resonant EB frequencies, since EB states are
well-isolated and not susceptible to mixing. This is evidently shown in Fig.
3c, for both the simulated (red) and measured (yellow) voltage amplitude
profiles at the EB frequencies
$\bar{p}^{\prime}=\bar{p}^{\prime}_{\text{EB}}\approx 2$ or $-1$. Their
excellent agreement can be more explicitly seen in Fig. 3e, which compares the
normalized EB amplitude profiles (which are mathematically identical for
$\bar{p}^{\prime}_{\text{EB}}$ and $1-\bar{p}^{\prime}_{\text{EB}}$) between
experiment and theory. By contrast, the non-EB resonant profiles at
$\bar{p}^{\prime}\approx 0$ or $1$ are very different between the simulations
and measurements, because both of these eigenvalues are doubly degenerate and
thus possess no unique eigenstate. Note that we can only record the voltage
amplitude $|V|$ profile, and not $V$, since its sign cannot be unambiguously
determined from the oscilloscope.

To further confirm that we have experimentally realized the occupied band
projector $\bar{P}^{\prime}$ as a linearly transformed circuit Laplacian (see
Eq. 20 in the Methods), we performed node-to-ground impedance measurements to
reconstruct the individual Laplacian matrix elements. By running a current
$I_{x}=I_{0}\delta_{xi}$ from an arbitrary node $i$ to the ground and
measuring the electrical voltage potentials $\mathbb{V}$ at all the nodes, one
obtains the $i$-th column of the pseudoinverse $J^{-1}$ of the Laplacian viz.
$\mathbb{V}=J^{-1}\mathbb{I}$. This allows $J^{-1}$ and hence $J$ to be
measured, from which $\bar{P}^{\prime}$ can be recovered via Eq. 20. Comparing
the right panels of Fig. 3c, we see that most of the matrix elements of
$\bar{P}^{\prime}$ can indeed be accurately experimentally recovered.

###  .6 Discussion and outlook

Originally defined as enigmatic negative probability eigenstates of the free
fermion entanglement Hamiltonian, which is not a physical operator, EB states
lived in the domain of mathematical abstraction, being not observable in
principle. Yet, by successfully realizing their parent bounded free fermion
projector $\bar{P}$ in terms of the Laplacian of a _classical_ electrical
circuit, we have managed to experimentally observe these elusive EB states as
prominent resonance profiles.

The feasibility of such a circuit implementation relied not just on the
versatility of electrical circuit connectivity, but also to two hitherto
undiscovered contributing avenues of EB robustness: (i) a novel spectral flow
that links rigorously defined EB states in the continuum limit with their much
simpler discrete counterparts in a much smaller system [Fig. 2b], and (ii)
their far superior robustness against coupling disorder compared to non-EB
states [Fig. 2c]. This heightened robustness, in particular, could find
applications in the engineering of relative stable modes in systems that are
otherwise intrinsically very sensitive.

While reminiscent of the more familiar topological Wannier flow or non-
Hermitian skin states in some ways, EB states are decidedly a distinct new
phenomenon in classical settings. Despite exhibiting nontrivial EB
“polarization” in $\bar{p}$ as the entanglement region is varied, this
spectral flow is inherited from the long-ranged profile of its gapless 2-point
functions [Fig. 4], and not gapped topological invariant. And, despite the
presence of asymmetric couplings, EB systems are completely impervious to the
NHSE because their PBC ($x_{cut}=L$) spectra consists of isolated points that
cannot possibly collapse any further to accommodate the NHSE.

##  I Acknowledgements

CHL and HYM acknowledge the National Research Foundation Singapore grant under
its QEP2.0 programme (NRF2021-QEP2-02-P09), and the Ministry of education
(MOE) Tier-II grant (Award No. MOE-T2EP50222-0003).

##  II Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  III Author Contributions

DZ and TC performed the experiments and designed the circuits under the
supervision of XZ. HM and YSA improved the results and participated in the
manuscript writing. CHL proposed the initial idea, performed the computations,
wrote the manuscript and provided the overall supervision for the project. The
manuscript reflects the contributions of all authors.

##  IV Competing Interests

The authors declare no competing interests.

## Methods

###  IV.1 Approximate analytic solution for EB eigenvalues and their spectral
flow

Our starting point is the simplest possible defective Hamiltonian ansatz [1]

$H_{n=2}(k)=\left(\begin{matrix}0&a_{0}+h(k)\\\ h(k)&0\end{matrix}\right)$ (8)

with $h(k)=\frac{1}{2}\left(2(1-\cos k)\right)^{B}\sim k^{2B}$, such that the
matrix symbol of the projector $P$ is given by Eq. 4 of the main text:

$P(k)=\left(\begin{matrix}\mathbb{I}&-U\\\ -D&\mathbb{I}\end{matrix}\right)$
(9)

where $U(k)=D(k)^{-1}=\sqrt{\frac{a_{0}+h(k)}{h(k)}}$. EB states arise when
$P$, or equivalently $U=D^{-1}$, becomes real-space truncated into $\bar{P}$
(or equivalently $\bar{U}\neq\bar{D}^{-1}$). Essentially, this means that
$\bar{P}$ is no longer a projector, and EB states encode the extent by which
$\bar{P}^{2}\neq\bar{P}$. At this point, it can be observed that the operator

$\Lambda=4(\bar{P}^{2}-\bar{P})=\left(\begin{matrix}\bar{U}\bar{D}-\mathbb{I}&0\\\
0&\bar{D}\bar{U}-\mathbb{I}\end{matrix}\right)$ (10)

not only measures how much $\bar{P}$ fails to be a projector, but also allows
its eigensolutions to be separately solved in a two smaller decoupled
subspaces. EB eigenvalues $\bar{p}_{\text{EB}}$ directly correspond to special
eigenvalues
$\bar{\lambda}_{\text{EB}}=4\bar{p}_{\text{EB}}(\bar{p}_{\text{EB}}-1)$ of
$\Lambda$ that are far from zero.

To proceed, we examine Eq. 10 in real space, and borrow a few observations
about the spatial decay behavior of $U$ and $D$ that would be shown in the
next subsubsection:

  * •

$U_{x}$ diverges with $L$, with $U_{x}\sim L^{B-1}$ for $B\geq 2$, and
$U_{x}\sim\log L$ for $B=1$.

  * •

$U_{x}$ decays slowly in space, logarithmically for $B=1$, linearly for $B=2$,
and asymptotically quadratically for $B>2$.

  * •

$D_{x}$ does not diverge, and decays rapidly.

where $U_{x}$ and $D_{x}$ are the real-space matrices of $U$ and $D$, obtained
via Fourier transformation. Putting these observations together, it is found
that $\bar{U}\bar{D}-\mathbb{I}$ must be dominated by its first slowly-
decaying column, since the subsequent columns are attenuated by the rapid
decay of $D_{x}$ coefficients. Denoting an EB eigenstate as
$\psi=(\psi_{\uparrow},\psi_{\downarrow})^{T}$, such that
$(\bar{U}\bar{D}-\mathbb{I})|\psi_{\uparrow}\rangle=\bar{\lambda}_{\text{EB}}|\psi_{\uparrow}\rangle$
(and analogously for $|\psi_{\downarrow}\rangle$), the above considerations
imply that $|\psi_{\uparrow}\rangle\approx U_{x}$ up to normalization, as
verified in Fig. 4. With some algebra, it can also be shown that the EB
eigenvalues occur in pairs
$p_{\text{EB}}^{\pm}=\frac{1}{2}\left(1\pm\sqrt{1+\bar{\lambda}_{\text{EB}}}\right)$,
where

$\bar{\lambda}_{\text{EB}}\approx-\frac{\sum_{x,x^{\prime},x^{\prime\prime}=0}D_{x}U_{-x^{\prime}}D_{x^{\prime}+x^{\prime\prime}}U_{-(x+x^{\prime\prime})}}{\sum_{x=0}D_{x}U_{-x}},$
(11)

the summation taken over the un-truncated region $[x_{L},x_{R}]$.

As an immediate corollary, for $B\geq 2$ such that $U_{x}\sim L^{B-1}$, we
also asymptotically have
$\bar{p}_{\text{EB}}\sim\sqrt{\bar{\lambda}_{\text{EB}}}\sim L^{(B-1)/2}$,
since the numerator of $\bar{\lambda}_{\text{EB}}$ is quadratic in the
$U_{x}$, while the denominator is linear in them.

####  IV.1.1 Asymptotic behavior of the two-point function

Here, we provide some details on the asymptotic scaling properties of the two-
point function, which are the matrix elements of $\bar{P}$. From Eq. 9, the
two-point function between the $\uparrow,\downarrow$ sublattices are given
asymmetrically by Fourier coefficients $-U_{x}$ and $-D_{x}$. We have

$\displaystyle U_{x}$ $\displaystyle=$ $\displaystyle-2\langle
c^{\dagger}_{x,\uparrow}c_{0,\downarrow}\rangle$ (12) $\displaystyle\approx$
$\displaystyle\sqrt{a_{0}}\int_{\pi/L}^{L}\frac{e^{ikx}+e^{-ikx}dk}{(2\sqrt{1-\cos
k})^{B}}$

which simplifies to

$\displaystyle U_{x}|_{B=1}$ $\displaystyle\sim$ $\displaystyle
2\sqrt{a_{0}}\log\frac{L}{\pi x}-\gamma,$ (13) $\displaystyle U_{x}|_{B=2}$
$\displaystyle\sim$ $\displaystyle 2\sqrt{a_{0}}\left(\frac{L}{x}-\pi
x\right)$ (14) $\displaystyle U_{x}|_{B>2}$ $\displaystyle\sim$ $\displaystyle
4\sqrt{a_{0}}\left(\frac{L}{\pi}\right)^{B-1}\left(1-\frac{\pi^{2}x^{2}}{2L^{2}}\right).$
(15)

For $B=1$, $\gamma=\int_{1}^{\infty}(\lfloor k\rfloor^{-1}-k^{-1})dk\approx
0.577$ is the Euler-Mascheroni constant. For $B\geq 2$, we see from Eq. 12
that lattice momentum regularization sets the dominant contribution to be
$U_{x}\sim L^{B-1}$, which diverges with $L$ (this result has nothing to do
with the cut position $x_{cut}$). By taking the next order approximation with
$x<L$, we obtain asymptotically quadratic scaling with $x$ for all $B>2$,
except the $B=2$ case exhibiting a special linear scaling. Beyond $x>L/2$, the
scaling behavior takes on its antisymmetric “dual” value under $x\rightarrow
L-x$, as shown in Fig. 4, which confirms that the above spatial behavior of
$U_{x}$ also approximately applies to that of their corresponding EB states.

(a)  (b)

(c)  (d)

Figure 4: The similar spatial profiles of 2-point function $U_{x}$ and the
leading EB state. Shown in (a)-(d) are the spatial profiles of the 2-point
function $U_{x}=-2\langle c^{\dagger}_{x,\uparrow}c_{0,\downarrow}\rangle$
(dotted curves), juxtaposed against the largest-eigenvalue EB eigenstates
$\psi_{\uparrow}(x)$ of $\bar{P}$ (solid curves). Evidently,
$\psi_{\uparrow}(x)\propto U_{x}$, as expected, since $U_{x}$ forms a
dominating column in the matrix $\bar{P}$. For both $B=3,4$ (and in fact all
higher $B$), the profiles are quadratic at small or large $x$.

The Fourier coefficients $D_{x}$, however, do not diverge since there is no
divergent singularity in $D(k)=U(k)^{-1}$. As such, they remain small and
decay exponentially with $x$.

####  IV.1.2 Spectral flow of $\bar{p}_{\text{EB}}$ with respect to $x_{cut}$

In principle, Eq. 11 already allows one to make a reasonable accurate
prediction of the value of $\bar{\lambda}_{EB}$ and hence
$\bar{p}_{\text{EB}}$. However, given the universality of the decay behavior
of $U_{x}$ and $D_{x}$, not just for our illustrative model but also for
generic exceptional point dispersions, further conclusions are possible.

Since the other 2-point function $D_{x}$ exhibits rapid decay, the dominant
contributions in Eq. 11 must contain terms with minimal range of $D_{x}$, i.e.
$x=0$ and $x^{\prime}=-x^{\prime\prime}$ for the numerator and $x=0$ for the
denominator. In particular, for relatively small $x_{cut}/L$, that allows us
to further approximate Eq. 11 by

$\displaystyle\bar{\lambda}_{\text{EB}}$ $\displaystyle\sim$
$\displaystyle-\frac{D_{0}}{U_{0}}\sum_{x^{\prime\prime}}^{x_{cut}}U_{x^{\prime\prime}}U_{-x^{\prime\prime}}+\mathcal{O}\left(e^{-2/B}\right)$
(16) $\displaystyle\approx$ $\displaystyle U_{0}D_{0}\,x_{cut}$
$\displaystyle\sim$ $\displaystyle\left(\frac{L}{\pi}\right)^{B-1}\times
x_{cut}$

where we have assumed that $B>1$. In the first line, we have only kept the
$D_{x=0}$ term, and neglected all subleading terms. In the second line, we
note that $U_{x}$ changes slowly with $x^{\prime\prime}$, especially at small
$x^{\prime\prime}$, and the sum should thus approximately scale like the
number of terms $x_{cut}$. In practice, this constant summand approximation
works well even for $x_{cut}$ comparable to $L$.

In summary, we expect to have an eigensolution
$\bar{\lambda}_{\text{EB}}=4\bar{p}_{\text{EB}}(1-\bar{p}_{\text{EB}})$ which
scales approximately linearly with the size of the un-truncated region
$x_{cut}$, which is defined by $[x_{L},x_{R}]=[1,x_{cut}]$. And adiabatic
continuity is expected due to the “smooth” linear dependence on $x_{cut}$. By
construction, this argument pertains to only _one_ pair of EB eigenvalues:
there can be other EB spectral branches that behave very differently.

To complete the justification of the EB spectral flow, we note that since
swapping the truncated and un-truncated regions should not physically affect
their entanglement, an linearly increasing $\bar{\lambda}_{EB}$ (i.e. square-
root increasing $\bar{p}_{\text{EB}}$) solution must be accompanied by another
identical but decreasing solution with $x_{cut}$ and $L-x_{cut}$ swapped.
Indeed, this is what is numerically observed, as in Figs. 2, 5 and plots
within the Supplement.

Interestingly, because of this “reflection” symmetry, for the smallest
possible truncation with $x_{cut}=1$ unit cell, the two sole eigenvalues must
be equivalent to the largest $\bar{p}_{\text{EB}}$ at $x_{cut}=L-1$. This is a
vivid example of how EB behavior in the largest un-truncated systems is “dual”
to the smallest one unit-cell $\bar{P}$ system.

(a)  (b)  (c)  (d)  (e)

Figure 5: Effect of changing the order $2B$ of the exceptional point. The
$\bar{p}$ spectrum of our canonical model $H_{n=2}$ as the truncation position
$x_{cut}$ is varied. From (a) to (e), we have $B=1,2,3,4,7$ respectively, all
computed with $L=40$ and $a_{0}=1$. The values of $\bar{p}_{\text{EB}}$
increases exponentially with $B$, as predicted by Eq. 16, even though they
qualitatively unchanged. Saliently, the EB branches increase in a slightly
concave manner for all cases (together with their mirror images), which is in
good numerical agreement with the theoretical prediction
$\bar{p}_{\text{EB}}\propto
1\pm\sqrt{1+\bar{\lambda}_{\text{EB}}}\sim\left(\frac{L}{\pi}\right)^{B/2}\sqrt{x_{cut}}$
from Eq. 16, with $L/\pi=12.73$.

###  IV.2 Details on the circuit observation of EB states

####  IV.2.1 Truncated projector from circuit Laplacian

In our experiment, we chose the parameters $x_{cut}=3$, $L=4$, $a_{0}=14$ and
$B=7$ such as to obtain a $\bar{P}$ with EB eigenvalues that are approximately
$2$ and $-1$, in addition to two pairs of non-EB eigenvalues close to $0$ and
$1$. Explicitly, $\bar{P}=$

$\left(\begin{array}[]{cccccc}0.5&-8.8529&0&-5.9055&0&0\\\
-0.2566&0.5&0.1712&0&0&0\\\ 0&-5.9055&0.5&-8.8529&0&-5.9055\\\
0.1712&0&-0.2566&0.5&0.1712&0\\\ 0&0&0&-5.9055&0.5&-8.8529\\\
0&0&0.1712&0&-0.2566&0.5\\\ \end{array}\right)$ (17)

with EB eigenvalues $\bar{p}_{\text{EB}}=2.0073,-1.0073$ and doubly degenerate
non-EB eigenvalues $\bar{p}=0,1$. Note that with the chosen parameters, some
off-diagonal matrix elements are also negligibly small. Next, we attempt to
design a circuit with not more than four unique capacitance coupling values,
such that its circuit Laplacian $J$ has all these $\bar{P}$ eigenvectors in
its kernel to an excellent approximation. By considering commericially
available capacitor values, our designed Laplacian is $J=J_{g}+J_{c}$,
comprising a grounded part

$J_{g}=\left[\frac{1}{i\omega L}+i\omega C_{5}\right]\mathbb{I}_{6\times
6}+i\omega\left(\begin{array}[]{cccccc}C_{2}+C_{4}&0&0&0&0&0\\\
0&C_{2}+2C_{3}+3C_{4}&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&3C_{3}+4C_{4}&0&0\\\
0&0&0&0&C_{2}+C_{4}&0\\\ 0&0&0&0&0&C_{2}+2C_{3}+3C_{4}\end{array}\right)$ (18)

and a inter-node coupling part exclusively involving capacitors:

$J_{c}=\left(\begin{array}[]{cccccc}C_{1}+C_{2}+C_{3}+C_{4}&-C_{3}\\!-\\!C_{1}&0&-C_{4}\\!-\\!C_{2}&0&0\\\
C_{3}\\!-\\!C_{1}&C_{1}+C_{2}\\!-\\!C_{3}\\!-\\!C_{4}&C_{4}\\!-\\!C_{2}&0&0&0\\\
0&-C_{2}\\!-\\!C_{4}&C_{1}+2C_{2}+C_{3}+2C_{4}&-C_{1}\\!-\\!C_{3}&0&-C_{2}\\!-\\!C_{4}\\\
C_{4}\\!-\\!C_{2}&0&C_{3}\\!-\\!C_{1}&C_{1}+2C_{2}\\!-\\!C_{3}\\!-\\!2C_{4}&C_{4}\\!-\\!C_{2}&0\\\
0&0&0&-C_{2}\\!-\\!C_{4}&C_{1}+C_{2}+C_{3}+C_{4}&-C_{1}\\!-\\!C_{3}\\\
0&0&C_{4}\\!-\\!C_{2}&0&C_{3}\\!-\\!C_{1}&C_{1}+C_{2}\\!-\\!C_{3}\\!-\\!C_{4}\end{array}\right)$
(19)

where $C_{1}=4.55$nF, $C_{2}=2.87$nF, $C_{3}=4.3$nF, $C_{4}=3.04$nF and
$C_{5}=0.50$nF. The inductor $L=10\mu$H introduces a relative factor of
$\omega^{-2}$ that allows the AC frequency $\omega$ to be effectively used as
a tuning knob for shifting the eigenvalues.

For states $\mathbb{V}$ in the kernel of $J$, $J\mathbb{V}=\mathbb{0}$ is
designed to recover the eigenvalue equation for $\bar{P}$. Due to limitations
on the commercially available capacitors, this would only be approximate: we
call the approximately obtained truncated projector $\bar{P}^{\prime}$. From
the above, $(J_{c}+J_{g})\mathbb{V}=\mathbb{0}$ can be re-arranged as

$\frac{1}{C_{0}}\left[\frac{1}{\omega^{2}L}-(C_{1}+2C_{2}+C_{3}+2C_{4})\right]\mathbb{V}=-\left(\begin{matrix}-C_{5}&C_{1}+C_{3}&0&C_{2}+C_{4}&0&0\\\
C_{1}-C_{3}&-C_{5}&C_{2}-C_{4}&0&0&0\\\
0&C_{2}+C_{4}&-C_{5}&C_{1}+C_{3}&0&C_{2}+C_{4}\\\
C_{2}-C_{4}&0&C_{1}-C_{3}&-C_{5}&C_{2}-C_{4}&0\\\
0&0&0&C_{2}+C_{4}&-C_{5}&C_{1}+C_{3}\\\
0&0&C_{2}-C_{4}&0&C_{1}-C_{3}&-C_{5}\end{matrix}\right)\mathbb{V}$ (20)

or simply $\bar{p}^{\prime}\mathbb{V}=\bar{P}^{\prime}\mathbb{V}$, where
$\bar{p}^{\prime}=\frac{1}{C_{0}}\left[\frac{1}{\omega^{2}L}-(C_{1}+2C_{2}+C_{3}+2C_{4})\right]$
and $\bar{P}^{\prime}$ is the matrix on the RHS (Eqs. 5 and 6). $C_{0}=1$nF
normalizes the capacitances such that the non-EB eigenvalues are very close to
$0$ and $1$. Numerically,

$\bar{P}^{\prime}=\left(\begin{array}[]{cccccc}0.5&-8.85&0&-5.91&0&0\\\
-0.25&0.5&0.17&0&0&0\\\ 0&-5.91&0.5&-8.85&0&-5.91\\\
0.17&0&-0.25&0.5&0.17&0\\\ 0&0&0&-5.91&0.5&-8.85\\\ 0&0&0.17&0&-0.25&0.5\\\
\end{array}\right)$ (21)

which differs very little from the exact $\bar{P}$ given by Eq. 17 above.

####  IV.2.2 Experimental design and procedure

The experimental circuit is designed using the PADS circuit design software,
which assists in selecting the PCB configuration, stackup layout, internal
layers and grounding. To prevent unintended interference between different PCB
layers, a grounding layer is placed in the gap between any two layers.
Moreover, all PCB traces are designed to have a relatively large width (0.5mm)
to reduce the parasitic inductance, and the spacings between electronic
components are also made sufficient large (1.0mm) to avoid spurious inductive
coupling. A relay is used to simultaneously switch on or off the input
voltages across all nodes when required. The op-amps used are of the model
LT1363, and are powered through the DC power supply UNI-T UTP1306S.

To ensure accurate experimental implementation of our theoretically designed
circuit Laplacian (Eqs. 18, 19), the component uncertainties have to be
minimized and their stability optimized. Capacitors of requisite capacitances
$C_{1}=4.55$nF, $C_{2}=2.87$nF, $C_{3}=4.3$nF, $C_{4}=3.04$nF and
$C_{5}=0.50$nF with 0.5% error tolerance are obtained by selecting from a
large sample of commercially available 3.0nF and 4.7nF capacitors with 10%
error tolerance; C0G ceramic capacitors are chosen for their stability.

The inductors contribute a major source of dissipation that can smooth out the
desired impedance resonance peaks. To minimize that loss, we select only the
inductors with the highest Q values at the operational frequency with
impedances of no more than 200m$\Omega$, as measured through a WK6500B
impedance analyzer.

To measure the resonances and hence map out the EB state profiles, an input AC
signal is input at an arbitrary node, and a digital storage oscilloscope
(Agilent Technologies Infiniivision DSO7104B) measures the voltage signal
$\mathbb{V}$ at each node. The input signal, which is chosen to be at node 2,
is generated by an arbitrary waveform generator (ROGOL DG1022Z) with amplitude
2V, with frequency swept from 310kHz to 400kHz to accommodate all expected
resonances ($\bar{P}^{\prime}$ eigenvalues).

To map out the matrix elements of the circuit Laplacian $J$, the same
arbitrary waveform generator is used to successively input a signal with
amplitude 2V and frequency 350kHz into each node, one node at a time. For a
chosen input node $i$, the current is $I_{j}\propto\delta_{ij}$, and the
relation $\mathbb{V}=J^{-1}\mathbb{I}$ allows the $i$-th column of $J^{-1}$ to
be read by measuring the resultant voltages $\mathbb{V}$ on all the nodes. By
iterating over $i$, $J^{-1}$ can be reconstructed, thereby yielding the
Laplacian and its corresponding $\bar{P}^{\prime}$ through matrix inversion.

####  IV.2.3 Implementation of asymmetric capacitive couplings through INICs

Asymmetric terms in the circuit Laplacian are implemented via negative
impedance converter through current inversion (INICs), which simulates
negative capacitance in one direction, while maintaining positive capacitance
in the other direction. As shown in Fig. 6, the main ingredient of an INIC is
an operational amplifier (op-amp); for experimental implementation we used the
model LT1363. In the negative feedback configuration of the op-amp, the
potentials at its positive and negative terminals are almost equal at $V_{1}$,
with negligible current flowing into them. As such, we can write down the
currents $I_{1}$ and $I_{2}$ as

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{Z_{+}}(V_{1}-V_{0})$ $\displaystyle I_{2}$
$\displaystyle=$ $\displaystyle\frac{1}{Z_{-}}(V_{1}-V_{0})=i\omega
C_{3}(V_{2}-V_{1})$ (22)

where $Z_{+},Z_{-}$ are the total impedances of the positive and negative
feedback loops. In general, $Z_{+}$ and $Z_{-}$ can take arbitrary values,
depending on the components in these feedback loops, but for our application,
it suffices to set them to be identical. Letting the feedback loops to consist
of a parallel configuration of a resistor $R_{a}$ and a capacitor $C_{a}$,
such as to avoid resonances at high frequencies, we have
$Z_{+}=Z_{-}=(1/R_{a}+1/(i\omega C_{a}))^{-1}$. By writing the current-voltage
relations in matrix form, we have

$\displaystyle\left(\begin{matrix}I_{1}\\\ I_{2}\end{matrix}\right)$
$\displaystyle=$ $\displaystyle i\omega
C_{3}\left(\begin{matrix}-Z_{-}/Z_{+}&Z_{-}/Z_{+}\\\
-1&1\end{matrix}\right)\left(\begin{matrix}V_{1}\\\ V_{2}\end{matrix}\right)$
(23) $\displaystyle=$ $\displaystyle i\omega C_{3}\left(\begin{matrix}-1&1\\\
-1&1\end{matrix}\right)\left(\begin{matrix}V_{1}\\\ V_{2}\end{matrix}\right),$

which manifestly contains an asymmetric Laplacian matrix. In particular,
$I_{1}=I_{2}$ instead of the usual case of $I_{1}=-I_{2}$, where the current
flow is conserved. All in all, the net effect is that this INIC behaves like
an usual capacitor $C_{3}$ from the right, but behaves like a negative
capacitor $-C_{3}$ from the left. Combined with other circuit elements, the
INIC allows for the possibility of arbitrarily asymmetric circuit components.

Figure 6: Anatomy of an INIC An INIC component in our circuit behaves like an
usual capacitor $C_{3}$ from the right ($V_{2}$), but behaves like a negative
capacitor $-C_{3}$ from the left ($V_{1}$). It consists of an op-amp connected
to two identical feedback loops, which implement the non-reciprocal non-
Hermiticity by allowing for the non-conservation of current.

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Supplementary materials

##  S1 Further properties of exceptional bound (EB) spectra

In this supplementary section, we provide further plots of various aspects of
EB spectra.

(a)  (b)  (c)

Figure S1: Effect of the magnitude of exceptional point asymmetry $a_{0}$.
Here we show how the EB spectrum $\bar{p}$ is affected by $a_{0}$ in the
parent defective Hamiltonian
$H_{n=2}(k)=h(k)\sigma_{x}+a_{0}(\sigma_{x}+i\sigma_{y})$,
$h(k)=\frac{1}{2}(2(1-\cos k))^{B}$, $B=3$. Depicted are (a) $a_{0}=0.01$, (b)
$a_{0}=1$ and (c) $a_{0}=100$, all computed in a system with $L=40$ unit
cells. As evident, the spectrum does not qualitatively change much, even
though $a_{0}$ ranges over four orders of magnitude.

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure S2: Robustness of the EB spectrum with respect to real and imaginary
disorder. Distribution of eigenvalues $\bar{p}$ from 30 instances of random
geometric disorder in the $\bar{P}$ matrix elements corresponding to
$H_{n=2}$, with $L=40$, $x_{cut}=39$ and fractional disorder between $-1\%$ to
$1\%$. The left column plots (a,c,e,g) depict real disorder while the right
column plots (b,d,f,g) depict imaginary disorder. Each row depicts a different
value of $B$: (a,b) $B=1$, (c,d) $B=2$, (e,f) $B=3$ and (g,h) $B=7$. The EB
eigenvalues (far left or right) remain relatively unperturbed and well-
resolved compared to the non-EB eigenvalues (central clouds), stays
real/imaginary upon real/imaginary disorder, and in general also move further
from $[0,1]$ with increasing $B$.