# Quantized classical response from spectral winding topology

Linhu Li<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of
Quantum Metrology and Sensing $\&$ School of Physics and Astronomy, Sun Yat-
Sen University (Zhuhai Campus), Zhuhai 519082, China  Sen Mu  Department of
Physics, National University of Singapore, Singapore 117551, Republic of
Singapore  Ching Hua Lee<EMAIL_ADDRESS>Department of Physics, National
University of Singapore, Singapore 117551, Republic of Singapore  Jiangbin
Gong<EMAIL_ADDRESS>Department of Physics, National University of
Singapore, Singapore 117551, Republic of Singapore

###### Abstract

Topologically quantized response is one of the focal points of contemporary
condensed matter physics. While it directly results in quantized response
coefficients in quantum systems, there has been no notion of quantized
response in classical systems thus far. This is because quantized response has
always been connected to topology via linear response theory that assumes a
quantum mechanical ground state. Yet, classical systems can carry arbitrarily
amounts of energy in each mode, even while possessing the same number of
measurable edge modes as their topological winding. In this work, we discover
the totally new paradigm of _quantized classical response_ , which is based on
the spectral winding number in the complex spectral plane, rather than the
winding of eigenstates in momentum space. Such quantized response is classical
insofar as it applies to phenomenological non-Hermitian setting, arises from
fundamental mathematical properties of the Green’s function, and shows up in
steady-state response, without invoking a conventional linear response theory.
Specifically, the ratio of the change in one quantity depicting signal
amplification to the variation in one imaginary flux-like parameter is found
to display fascinating plateaus, with their quantized values given by the
spectral winding numbers as the topological invariants.

##  I Introduction

Topological quantization has captivated a generation of physicists since the
discovery of the quantum Hall effect Klitzing  _et al._ (1980). In the more
recent years, its standing as a novel quantum phenomenon was further
strengthened by the observation of discretely varying Hall conductances in the
quantum anomalous Hall Haldane (1988); Liu  _et al._ (2016) and quantum spin
Hall effects Kane and Mele (2005). Rigorously established through linear
response theory, the link between topological winding numbers and quantized
conductivity has indeed earned a place as a classic result of quantum
condensed matter physics. Indeed quantized Hall response and nontrivial Chern
topology are now widely regarded to be almost synonymous.

Yet, the concept of quantized response has so far eluded classical systems.
Without a quantum mechanical ground state, classical systems are not amendable
to conventional linear response theory, which expresses quantized responses as
perturbations upon a ground state. While classical metamaterials like photonic
crystals, acoustic structures and electrical circuits can possess an integer
number of topologically protected boundary modes Huber (2016); Nash  _et al._
(2015); Lu  _et al._ (2014); Imhof  _et al._ (2018), their response behaviors
are not based on the number of accessible channels, but on analog solutions of
differential equations that are by no means quantized.

In this work, we introduce the paradigm of quantized response based on the
winding of the spectrum in the complex energy plane, rather than the winding
of eigenstates in momentum space. For a long time, the complex spectral
winding number has been exploited in predicting the non-Hermitian pumping
under the open boundary conditions (OBCs) [known as the non-Hermitian skin
effect (NHSE)] Yao and Wang (2018); Okuma  _et al._ (2020); Zhang  _et al._
(2020a); Lee and Thomale (2019), leading to the breaking of bulk-boundary
correspondence and various anomalous topological phenomena Xiong (2018); Shen
_et al._ (2018); Kawabata  _et al._ (2019); Yao and Wang (2018); Lee and
Thomale (2019); Kunst  _et al._ (2018); Yao  _et al._ (2018); Yin  _et al._
(2018); Jiang  _et al._ (2018); Li  _et al._ (2019a); Song  _et al._ (2019a,
b); Okuma and Sato (2019); Mu  _et al._ (2020); Jiang  _et al._ (2019); Longhi
(2019); Jin and Song (2019); Lee  _et al._ (2019); Li  _et al._ (2020a);
Helbig  _et al._ (2020); Xiao  _et al._ (2020); Okuma  _et al._ (2020); Zhang
_et al._ (2020a); Lee  _et al._ (2020a); Li  _et al._ (2020b); Yang  _et al._
(2020); Yi and Yang (2020). However, to date no directly measurable quantity
has been associated with the spectral winding. Notably, spectral winding as a
topological feature is not directly related to quantum physics and is hence a
_classical_ concept. Furthermore, the notion of classical response can be also
a property of the Green’s function alone. While nontrivial complex energy
winding is in principle well-defined for quantum systems, it is most
physically relevant in classical settings like mechanical, photonic, plasmonic
and electrical systems where non-Hermiticity does not present significant
measurement difficulties Ghatak  _et al._ (2020); Xiao  _et al._ (2020);
Helbig  _et al._ (2020). Indeed, electrical circuits are governed by circuit
Laplacians whose complex eigenvalues merely indicate phase shifts or steady-
state impedances, rather than ephemeral excitations.

Specifically, what we find quantized is the response of the logarithm of the
Green’s function components with respect to an imaginary flux-like local
parameter that continuously adjust the system boundary conditions, forming
some quantum-Hall-like plateaus quantized according to the spectral winding
number. This discovery was inspired by the observation that deforming a non-
Hermitian system from periodic to open boundaries (PBCs to OBCs) Xiong (2018);
Lee and Thomale (2019); Lee  _et al._ (2019, 2020a); Mu  _et al._ (2020),
which is closely related to complex flux insertion, always reduces the
spectral winding number one by one till it reaches zero. In a classical setup
subject to a steady-state drive e.g. a circuit lattice with an input current,
the quantized quantity can be the logarithmic impedance experienced by the
response field, e.g. the voltage. This intriguing result is rooted in the way
non-Bloch eigenstates explore the interior of spectral loops, and can be
understood with an unexpected type of non-Hermitian pumping that is scale-free
Li  _et al._ .

##  II Results

###  II.1 Classical vs. Quantum response

In conventional literature, quantized response usually refers to quantized
linear response in a quantum setting, where an occupied quantum state is
driven by a time-dependent perturbation. In this work, however, our focus is
to highlight that it is still possible to extract a quantized quantity from
the response of a purely classical system, where there is no notion of an
occupied ground state. In the following, we shall first clarify the
distinction between quantum and classical response.

For concreteness, first consider the quantum setting described by a
Hamiltonian $\hat{H}(t)=\hat{H}_{0}+\epsilon(t)\hat{H}^{\prime}$, where
$\hat{H}_{0}$ is the equilibrium Hamiltonian, $\hat{H}^{\prime}$ the perturbed
operator and $\epsilon(t)$ a time-dependent factor that controls the
perturbation strength. The linear response of a chosen operator $\hat{\phi}$
(which has zero expectation when $\epsilon(t)=0$) is its expectation
$\hat{\phi}(t)$ after $\epsilon(t)$ is switched on. In frequency space, we
write

$\langle\hat{\phi}\rangle(\omega)=D(\omega)\epsilon(\omega),$ (1)

where

$\displaystyle D(\omega)$ $\displaystyle=$
$\displaystyle-i\int_{0}^{\infty}\langle[\hat{\phi}(t),\hat{H}^{\prime}(0)]\rangle\,dt$
(2) $\displaystyle=$ $\displaystyle-
iZ^{-1}\int_{0}^{\infty}\text{Tr}\left(e^{-\beta
H_{0}}[\hat{\phi}(t),\hat{H}^{\prime}(0)]\right)\,dt$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\sum_{nm}\frac{\langle n|\hat{\phi}|m\rangle\langle
m|\hat{H}^{\prime}|n\rangle(e^{-\beta E_{n}}-e^{-\beta
E_{m}})}{\omega+i\delta-(E_{m}-E_{n})}$

where $Z=\text{Tr}\left(e^{-\beta\hat{H}_{0}}\right)$, $\beta$ is the inverse
temperature and the trace is taken over all eigenstates $|n\rangle$ satisfying
$\hat{H}_{0}|n\rangle=E_{n}|n\rangle$. Very importantly, it has been assumed
that the system is in a ground state where the eigenstates $|n\rangle$ are
occupied according to the Boltzmann probability distribution $\propto
e^{-\beta E_{n}}$. By expressing the trace as a momentum-space integral which
shall correspond to a topological invariant, a topologically quantized
response coefficient can be obtained, for instance, with the DC limit of
$D(\omega)/i\omega$ in the case of Hall response.

By contrast, in the classical settings that we shall focus on i.e. photonic
waveguides, acoustic lattices and electronic circuit, the system do not settle
into a ground state, and Eq. 2 is inapplicable. The classical response
corresponds to the flow of arbitrary amounts of optical, phononic or electric
current, and not the modified (quantized) occupancy of quantum mechanical
eigenstates. Consider an external coherent drive
$\vec{\epsilon}=(\epsilon_{1},\epsilon_{2},...\epsilon_{L})$ with different
amplitudes applied to each of the $L$ sites of a lattice Xue  _et al._ (2020).
For a steady-state drive with a fixed frequency $\omega$,
$\vec{\epsilon}(t)=\vec{\epsilon}(\omega){\rm exp}(-i\omega t)$, and the
resultant classical response field at the same frequency can be written as
$\vec{\phi}(t)=\vec{\phi}(\omega){\rm exp}(-i\omega t)$ with the response
field amplitude given by

$\displaystyle\vec{\phi}(\omega)=G(\omega,\gamma)\vec{\epsilon}(\omega),~{}G(\omega,\gamma)=\frac{1}{\omega+i\gamma-H},$
(3)

analogous to Eq. 2 which is exclusively for quantum settings. Here $G$ is the
Green’s function matrix and $\gamma$ represents an overall gain/loss in the
system. For a signal entering the system from a single site $x$,
$\vec{\epsilon}$ only possesses one nonzero component $\epsilon_{x}$, and the
induced field at another site $y$ is $\phi_{y}=G_{yx}\epsilon_{x}$. In
particular, the directional signal amplification of a signal entering one end
of a 1D chain and measured at the other end is described by the two matrix
elements $G_{1N}$ and $G_{N1}$ Wanjura  _et al._ (2020); Xue  _et al._ (2020).

###  II.2 Motivation for quantized Green’s function response

In the spectral representation, the Green’s function Eq. 3 takes the form

$\displaystyle G=\sum_{n}\frac{1}{\omega+i\gamma-
E_{n}}|\Psi_{n}^{R}\rangle\langle\Psi^{L}_{n}|,$ (4)

where $|\Psi^{L/R}_{n}\rangle$ are the left/right eigenstates corresponding to
the $n$-th eigenenergy $E_{n}$. Its matrix elements $G_{xy}$ can be computed
by evaluating $|\Psi^{L/R}_{n}\rangle$ at $x$ and $y$. Eq. 4 is valid in both
classical and quantum settings, since the Hamiltonian is just the operator
that describes time evolution, and is well-defined regardless of whether
position and momentum commute.

Ordinarily, we do not expect the matrix elements of $G$ (or functions of them)
to respond to an external influence $\beta$ in a quantized manner, since there
is no reason why the derivatives of $(\omega-E_{n})^{-1}$ and the eigenstates
should conspire to add up to an integer. However, when translation symmetry is
broken, the eigenstates can potentially become exponentially localized like
$\sim e^{\kappa x}$, such that the matrix elements of $G$ are dominated by the
largest spatial decay rate $\kappa=\kappa_{\text{max}}$, with $G_{xy}\sim
e^{\kappa_{\text{max}}(y-x)}$. In such special scenarios, the response of $\ln
G_{xy}$ for a fixed interval $x-y$ is _wholly_ dependent on how
$\kappa_{\text{max}}$ varies with the external influence. In particular, if
$\kappa_{\text{max}}$ were to vary with an external parameter in a quantized
manner, so will the response quantity $\ln G_{xy}$.

In this work, we discover that $\ln G_{xy}$ indeed possess such a quantized
response if the external influence $\beta$ were to be an impurity parameter
tuning the boundary conditions, which coincides with tuning an imaginary flux
when the latter is sufficiently weak Lee and Thomale (2019). This quantized
quantity is furthermore equal to the winding number of the energy spectrum in
the complex energy plane. In the following sections, we shall elaborate on
these rather surprising findings, and show the classical quantized response
can be measured. While this quantization applies to both classical and quantum
systems, we shall call it the quantized _classical_ response to distinguish it
from the topological Hall response that exclusively exists in quantum systems.

###  II.3 Point-gap topology and PBC-OBC spectral evolution

To elucidate the role of spectral winding and motivate a suitable notion of
classical response, we consider a generic one-band non-Hermitian system with
$N$ lattice sites, described by the following tight-binding Hamiltonian

$\displaystyle
H=\sum_{x=1}^{N}\sum_{j=-r}^{l}t_{j}\hat{c}^{\dagger}_{j}\hat{c}_{x+j},$ (5)

with $t_{j}$ the hopping amplitude across $|j|$ lattice sites, $\hat{c}_{x}$
the annihilation operator of a (quasi-)particle at the $x$th lattice site,
$\hat{c}_{x+N}=\hat{c}_{x}$ representing the PBC, and $r$ ($l$) the maximal
range of the hopping toward right (left) direction. The associated momentum-
space Hamiltonian is given by

$H(z)=\sum_{j=-r}^{l}t_{j}z^{j}={P_{r+l}(z)}/{z^{r}},$ (6)

with $k$ the quasi-momentum, $z:=e^{ik}$, and $P_{r+l}(z)$ a $(r+l)$th-order
polynomial of $z$. For any $t_{j}\neq t_{-j}^{*}$ (assuming $t_{j}=0$ if $j>l$
or $j<-r$), the Hamiltonian becomes non-Hermitian and possesses a point-gap
topology, characterized by a nonzero spectral winding number w.r.t. a
reference energy $E_{r}$ enclosed by the PBC spectrum Gong  _et al._ (2018);
Okuma  _et al._ (2020); Zhang  _et al._ (2020a),

$\nu(E_{r})=\oint_{\mathcal{C}}\frac{dz}{2\pi}\frac{d}{dz}{\rm
arg}\det[H(z)-E_{r}],$ (7)

with $\mathcal{C}$ being the Brillouin zone (BZ), i.e. $k$ varying from $0$ to
$2\pi$. Simply put, $\nu(E_{r})$ gives the number of times the PBC spectrum
winds around $E_{r}$, as illustrated by a representative example in Fig. 1(a),
corresponding to the Hamiltonian of Eq. 5 with $r=l=2$ . As a side note,
$\nu(E_{r})$ also reflects the degeneracy of eigenmodes at $E_{r}$ when the
system is placed under semi-infinite boundary conditions (SIBC) Okuma  _et
al._ (2020).

Figure 1:  (a) PBC (cyan loops) and OBC (gray dots) spectra of the Hamiltonian
in Eq. (5) with $r=l=2$. The spectral winding number for each different regime
is indicated on the panel. (b) The quantity $I_{\beta}(E_{r})$ (Eq. 8) vs
$\beta$ for the three cases marked in (a). It diverges exactly $\nu$ times
when $E_{r}$ sits in a region of spectral winding $\nu$. Parameters are set at
$t_{1}=1$, $t_{-1}=0.5$, $t_{2}=2$, and $t_{-2}=0$, with $N=100$.

Unlike the loop-like PBC spectrum depicting a nontrivial point-gap topology,
the OBC spectrum must not cover any finite area in the complex plane, and so
generically must take the form of curves within the PBC spectral loops Xiong
(2018); Lee and Thomale (2019), e.g. the Y-shape lines formed by the gray dots
in Fig. 1(a). That is, any reference energy $E_{r}$ inside a PBC spectral loop
is enclosed $\nu(E_{r})$ times by the PBC spectrum as $k$ varies from 0 to
$2\pi$, but the same $E_{r}$ cannot be enclosed by the OBC spectrum. The
important qualitative insight is hence the following: if we continuously
deform the system from PBC to OBC, the evolving spectrum gradually collapses
from the PBC loop spectrum to the OBC line spectrum, and is therefore expected
to pass $E_{r}$ for $\nu(E_{r})$ times.

To further appreciate this understanding, we consider the real-space
Hamiltonian of Eq. (5) with the following substitution only at the system’s
boundary, i.e., $t_{j}\rightarrow e^{-\beta}t_{j}$, if $x+j>N$ or $x+j<1$.
This introduces an additional scaling factor, or an impurity, to the boundary
couplings of a 1D chain with $N$ unit cells. If one now continuously varies
$\beta$ from $0$ to infinity, a PBC-OBC spectral evolution can be examined in
detail.

As shown in Fig. 1(a), by considering only $t_{\pm 1}$ and $t_{\pm 2}$, we
already have a representative and intriguing model whose spectral winding
number can be either 1 or 2 in the topologically nontrivial regime. Next we
consider the tuning of the boundary coupling via $t_{\pm 1}\rightarrow
e^{-\beta}t_{\pm 1}$ for $x=N$ and $t_{\pm 2}\rightarrow e^{-\beta}t_{\pm 2}$
for $x=N,N-1$. Let the $n$-th right eigenvalue of this model system with such
PBC-OBC interpolations be $E_{n}(\beta)$. Then the spectral evolution is all
captured by $E_{n}(\beta)$ vs $\beta$. To motivate the connection with the
Green’s function, and to capture incidences when $E_{n}(\beta)$ comes close to
$E_{r}$, we also define a quantity

$I_{\beta}(E_{r})=\sum_{n}|1/(E_{n}(\beta)-E_{r})|,$ (8)

i.e. the absolute sum of the inverse energy spacings between the evolving
spectrum $E_{n}(\beta)$ and a reference energy $E_{r}$. For an actual system
always of finite size, $E_{n}(\beta)$ is discretized, but it can still be made
to be very close to the PBC reference eigenvalue $E_{r}$ for a sufficiently
large system. As such, the quantity $I_{\beta}(E_{r})$ can be a diagnosis tool
to examine how many times $E_{n}(\beta)$ visits (the proximity of) $E_{r}$ as
$\beta$ is tuned. Moreover, the OBC limit can be essentially reached once
$\beta$ is beyond a critical value $\beta=\beta_{\rm OBC}\sim{N\eta}$ with
$\eta$ the effective localizing length of the eigenstates Koch and Budich
(2020); Li  _et al._ . With these understandings, one infers that as $\beta$
varies from $0$ to $\beta_{\rm OBC}$, $I_{\beta}(E_{r})$ is expected to
display high peaks whenever the complex spectral evolution passes through
$E_{r}$. As explained above, the total number of such local peaks then
reflects the spectral winding number $\nu(E_{r})$. In Fig. 1(b) we illustrate
$I_{\beta}(E_{r})$ as a function of $\beta$ for several $E_{r}$ denoted by the
stars of different colors in Fig. 1(a), corresponding to different spectral
winding numbers $\nu(E_{r})$. It is indeed observed that the number of peaks
of $I_{\beta}(E_{r})$ directly reflects the spectral winding number
$\nu(E_{r})$. In the Methods section, we offer more insights based on the so-
called generalized Brillouin zone (GBZ), to better understand why $\nu(E_{r})$
can be captured by the number of singularities encountered throughout the
complex spectral evolution.

###  II.4 Quantized response in signal amplification

While the previously defined quantity $I_{\beta}(E_{r})$ is useful in
diagnosing the spectral winding, it is not directly measurable. Below, we show
how it inspires another analogous quantity that is directly associated with
signal amplification. In particular, the quantity introduced below displays
quantized plateaus that precisely match spectral winding numbers, making it
possible to distinguish between one nontrivial point-gap topology from
another. This is a true advance as compared with earlier interesting attempts
where signal amplification was only used to probe NHSE under OBC Wanjura  _et
al._ (2020); Xue  _et al._ (2020); Yi and Yang (2020).

For our system with the PBC-OBC interpolation parameter $\beta$
($0<\beta<\beta_{\rm OBC}$), what enters into the expression of the Green’s
function $G$ in Eq. 4 is $1/[E_{r}-E_{n}(\beta)]$ again (as in
$I_{\beta}(E_{r})$) and hence the Green’s function should be able to capture
the complex spectral evolution. More importantly, it is found that the
associated eigenmodes under PBC-OBC interpolation pile up at its boundary in a
manner qualitatively different from that of NHSE Li  _et al._ . With further
derivation in the Method section, we find

$\displaystyle|G_{1N}|\propto e^{\beta},~{}\frac{d\ln|G_{1N}|}{d\beta}=1$ (9)

for a one-dimensional system with only the nearest-neighbor couplings,
provided $t_{1}>t_{-1}$ and $E_{r}$ falls within the loop-spectrum of the
system at a given $\beta$ (See Supplemental Materials). For $t_{-1}>t_{1}$,
one can analogously obtain $\frac{d\ln|G_{N1}|}{d\beta}=1$, corresponding to a
winding number $\nu(E_{r})=-1$. We emphasize that the variation in Eq. 9 can
be directly measured in a steady-state response experiment. In an electrical
circuit setting, this task can be done via impedance measurements, as will be
elaborated later.

For more general cases with couplings across $r$ ($l$) lattice sites to the
right (left), the system can be effectively understood as $m={\rm Max}[r,l]$
different sub-chains, with $t_{\pm m}$ viewed as the nearest neighbor coupling
on each sub-chain, and the rest understood as inter-sub-chain or intra-chain
longer-range couplings. For example, lattice sites $j$, $j+m$, $j+2m$,
$\cdots$, with each pair of neighbors coupled by $t_{\pm m}$, form the $j$-th
sub-chain, for $j=0,1,2,\cdots,m-1$. The above picture based on $m$ sub-chains
becomes most useful if $t_{\pm m}$ is much larger than all other hopping
parameters. Under this assumption these sub-chains are nearly decoupled from
each other in the bulk, and their respective edges are connected through the
couplings modified by $e^{-\beta}$. This then suggests a unit-cell structure
with $m$ sublattices, which greatly simplifies our consideration of signal
amplification. Indeed, the overall amplification can be captured by the
$m\times m$ block at the top-right (bottom-left) corners of the overall
Green’s function $G$, denoted as $G_{\leftarrow,m\times m}$
($G_{\rightarrow,m\times m}$), corresponding to measuring the output at the
first (last) $m$ sites of a signal entering from the last (first) $m$ sites.
Since each sub-chain yields an amplification factor proportional to
$e^{\beta}$, one qualitatively expects that $|G_{\leftarrow,m\times
m}|\equiv\det[G_{\leftarrow,m\times m}]\propto e^{m\beta}$ or
$|G_{\rightarrow,m\times m}|\equiv\det[G_{\rightarrow,m\times m}]\propto
e^{m\beta}$, depending on the amplification direction. This indicates that

$\nu_{\leftarrow,m}\equiv d\ln|G_{\leftarrow,m\times m}|/d\beta$ (10)

or analogously $\nu_{\leftarrow,m}\equiv d\ln|G_{\leftarrow,m\times
m}|/d\beta$ is quantized at $m$. On the other hand, the PBC spectrum of the
system winds $m$ times around the origin of the complex plane, a fact
obviously true since the associated momentum-space Hamiltonian is dominated by
the terms $e^{\pm imk}$. As theoretically shown in Supplementary Material sup
, this correspondence between quantized response and spectral winding indeed
holds, if $E_{r}$ falls inside the PBC spectrum loop with the spectral winding
number $\nu(E_{r})$ given by $\pm m$, where the $\pm$ sign is to be determined
by the actual direction of amplification.

Figure 2 presents computational results for $\ln\left|G_{\leftarrow,2\times
2}\right|$ and its derivative with respect to $\beta$ as functions of $\beta$,
denoted $\nu_{\leftarrow,2}$, again for the same system as that in Fig. 1. We
also compare these results with $I(E_{r})$ defined previously for a chosen
reference energy point $E_{r}=\omega+i\gamma$, with $\nu(E_{r})=2$. It is
observed that $\nu_{\leftarrow,2}$ as a measurable physical response shows
three clear plateaus quantized at $\nu_{\leftarrow,2}=2,1,0$. Echoing with the
jumps between these plateaus during the spectral evolution, $I_{\beta}(E_{r})$
shows local peaks whenever the spectral evolution passes through $E_{r}$. As
shown in Fig. 2(c), these transitions during the spectral evolution as a
result of increasing $\beta$ match precisely with the $\beta$ values for which
the complex spectrum touches the reference energy point and hence the spectral
winding number is about to jump. That is, the transitions between these
quantized plateaus have a clear topological origin and are hence identified as
topological transitions. As a side remark, observing all the plateaus in our
example here happens to require a broad regime of $\beta$ and hence cases with
very weak boundary coupling. This is however not a concern because the first
plateau at small $\beta$ is the one to reflect the topology under PBC.

Figure 2:  (a) Amplification ratio $|G_{\leftarrow,2\times 2}|$ and (b)
$\nu_{\leftarrow,2}$, the derivative of $\ln|G_{\leftarrow,2\times 2}|$ over
$\beta$, as functions of $\beta$, along with $I_{\beta}(E_{r})$, whose peaks
match well with the jumps between different plateaus. (c) Spectrum at
different values of $\beta$, corresponding to the five dashed lines in (b),
respectively. Red stars indicate the reference energy $E_{r}=\omega+i\gamma$,
with $\omega=-0.96$ and $\gamma=1$. Insets zoom in on the regime around
$E_{r}$ to give a clearer view of the relation between the shown spectrum and
$E_{r}$. Parameters are set at $t_{1}=1$, $t_{-1}=0.5$, $t_{2}=2$, $t_{-2}=0$,
and $N=100$.

The results presented in Fig. 2 are particularly stimulating. Indeed, therein
neither the next-nearest-neighbor coupling nor the nearest-neighbor coupling
is dominating. Yet quantized plateaus at $m=1$ and $m=2$ are still obtained.
Returning to our decoupled sub-chain picture above, this indicates that for
different reference energy points, the behavior of this system is
topologically equivalent to that of one single chain or that of two weakly
coupled sub-chains. With this perspective, we propose to examine
$\nu_{\leftarrow,m}$ vs. different choices of $m$ in order to fully map out
the phase boundaries, without any prior knowledge of spectral winding. We thus
proceed to find the maximal $\nu_{\leftarrow,m}$ by scanning $m$, for
sufficiently small $\beta$. The obtained value is then expected to yield
$\nu(E_{r})$ of the studied system under PBC. To this end we define
$\nu_{\leftarrow}={\rm Max}[\nu_{\leftarrow,1},\nu_{\leftarrow,2}]$, given
that our model system at most has effectively two sub-chains (one can
similarly define $\nu_{\rightarrow}$). We present in Fig. 3(a) our results of
$\nu_{\leftarrow}$ at $\beta=0$ for different $\omega$ and $\gamma$ , with the
phase boundaries identified there in excellent agreement with the actual PBC
spectrum shown in Fig. 1(a). The only subtlety is that we also obtain a
negative value $\nu_{\leftarrow}=-1$ in the topological trivial regime
(theoretically this result is also explained in Supplementary Material sup ).
However, in these regimes both $\nu_{\leftarrow}$ and $\nu_{\rightarrow}$ are
found to be negative, the amplification factor is far less than unity, and
hence there is actually no signal amplification after all. One may just
exploit this additional feature to locate regimes with zero spectral winding.

Figure 3: The topological phase diagram and simulation of impedance
measurement in a circuit realization. (a) The topological phase diagram, in
excellent agreement with that shown in Fig. 1 is mapped out from examining
physical response functions $\nu_{\leftarrow,m}$ with $m=1$ or $m=2$, for the
model depicted in the main text, with parameters set at $t_{1}=1$,
$t_{-1}=0.5$, $t_{2}=2$, $t_{-2}=0$, and $N=100$. At the phase boundaries,
$E_{r}=\omega+i\gamma$ is an eigenvalue of the system, leading to the
divergence of the Green’s function matrix. (b) Quantization of
$\frac{d\ln|Z_{m}|}{d\beta}$, which is directly obtainable from simulated
circuit measurement data of the impedance between the first $m$ and last $m$
nodes of the circuit lattice, here computed with $50$ nodes. $Z_{m}$ (Eq. 12
is the $m\times m$ matrix of impedances between the measured nodes, and
$\beta$ is the parameter controlling the strengths of the couplings connecting
the two ends. The complex admittance parameter $\Omega$ indicated on panel (b)
is connected with parameters $\omega$ and $\gamma$ plotted in panel (a) with
$\omega={\rm Re}(\Omega)$ and $\gamma={\rm Im}(\Omega)$. The first plateau
encountered when $\beta$ is increased from zero gives the nonzero topological
winding $\nu$ (green, yellow for $\nu=1,2$). When $\nu=0$, the logarithmic
gradients of both $Z_{1,2}$ are close to or smaller than $0$ (blue),
corresponding to the absence of signal amplification toward either direction.

###  II.5 Measurement of quantized response in electrical circuits

We next elaborate on how the quantized response can be directly extracted by
measuring the impedance in an electrical circuit setting. Instead of external
perturbations, a circuit is most naturally driven by a steady-state AC or DC
current, with voltage response given by Kirchhoff’s law
$\mathbb{I}=L\mathbb{V}$, where $L$ is the circuit Laplacian matrix and the
components of $\mathbb{I}$ and $\mathbb{V}$ are respectively the input
currents and voltages at each node. Suppose that the circuit is then grounded
by identical circuit components with complex admittances $-\Omega$. In this
case, the full (grounded) Laplacian becomes $J=L-\Omega\,\mathbb{I}$, and the
voltage distribution due to the input current are given by Lee  _et al._
(2018a, 2020b)

$\mathbb{V}_{i}=[J^{-1}]_{ij}\mathbb{I}_{j}=G_{ij}\mathbb{I}_{j}=\left[\sum_{n}\frac{|\Psi^{R}_{n}\rangle\langle\Psi_{n}^{L}|}{\Omega-
E_{n}}\right]_{ij}\mathbb{I}_{j}$ (11)

where the eigenvalues $E_{n}$ and L/R eigenstates $|\Psi^{L/R}_{n}\rangle$ are
that of the Laplacian $J$. Notably, the quantity in the square parentheses
agree exactly with our definition of the Green’s function (Eq. 4), with
$\Omega$ taking the role of $\omega+i\gamma$. We extract the physical response
through the impedance $Z_{ij}$ between the $i$-th and $j$-th nodes, which is
related to the Green’s function via $Z_{ij}=G_{ii}+G_{jj}-G_{ij}-G_{ji}$. By
varying the identical grounding admittances $\Omega$ via a combination of RLC
components with $\pm\pi/2$ relative phase shifts, we will be able to
effectively access the response from different regions of the complex spectral
plane.

Quantized classical response can be obtained from a circuit whose Laplacian
exhibits nontrivial spectral winding. This requires effectively asymmetric
couplings between nodes, which has already been demonstrated in existing
topolectrical experiments through combinations of capacitors, inductors and
INICs (negative impedance converter with current inversion) comprising
operation amplifiers Hofmann  _et al._ (2019); Helbig  _et al._ (2020); Ezawa
(2019a), as further elaborated in the Supplementary Material. The boundary
couplings can also be adjusted to effect the variation with $\beta$ through
tunable inductors connected in series with the asymmetric couplings Li  _et
al._ . Arbitrarily large spectral winding numbers can always be achieved by
coupling sufficiently distant nodes, which can be much more feasibly done in
electrical circuits compared to other platforms.

In analogy to $\nu_{\leftarrow,m}$ from Eq. 10, we can define
$Z_{m}=\text{det}\,Z_{ij}|_{i\leq m,N-j<m}$, the determinant of the $m\times
m$ matrix of impedances between the first $m$ nodes at one end with the last
$m$ at the other end of the circuit chain. Although it is not exactly
equivalent to $\nu_{\leftarrow,m}$, it is expected to vary with $\beta$ in a
similar manner, since the impedance $Z_{ij}$ is dominated by the component of
the Green’s function that produces the directional amplification. Therefore,
by keeping track of the effective $\beta(\omega)$ and $\Omega(\omega)$, the
logarithmic gradient of the impedance determinant $\frac{d\ln|Z_{m}|}{d\beta}$
will be expected to exhibit quantized jumps as $\Omega(\omega)$ crosses the
boundary between regions of different topological winding $\nu$. For
$m\leqslant 2$, which includes the model we had considered (Figs. 1, 2 and 3),
we explicitly have

$Z_{m=2}=Z_{1,N-1}Z_{2,N}-Z_{1,N}Z_{2,N-1},$ (12)

whose gradients are dominated by those of terms like $G_{1,N-1}G_{2,N}$ and
$G_{1,N}G_{2,N-1}$ in the presence of directional amplification (toward the
first lattice site). As demonstrated via the simulated measurements in Fig.
3(b), the gradient $\frac{d\ln|Z_{m=1,2}|}{d\beta}$ indeed exhibits plateaus
quantized at the winding number $\nu$ (blue, green, yellow for $\nu=0,1,2$
respectively) where $\Omega$ is tuned to. For higher topological winding, we
take the plateau that first occurs when $\beta$ is increased from $0$, i.e.
the plateau closest to periodic boundary conditions.

##  III Conclusions

In this work, we have introduced the new paradigm of quantized classical
response, where a quantized response coefficient can always be extracted from
how the Green’s function varies with the evolution of an imaginary flux-like
parameter. Being based on the topological winding properties of the Green’s
function in the complex energy plane, this quantization does not assume the
existence of any quantum mechanical ground state, and applies to all systems,
classical and quantum. Specifically, we show that the spectral winding number
is directly detectable as a steady-state response coefficient to changes in
the boundary condition. Such correspondence between spectral winding numbers
and quantized response is arguably broader in scope than in the case of
momentum-space topology, because spectral winding does not even require
translational invariance. Our results are relevant to a number of current
experimental platforms of non-Hermitian systems Hofmann  _et al._ (2020);
Helbig  _et al._ (2020); Xiao  _et al._ (2020); Wanjura  _et al._ (2020); Li
_et al._ (2020b); Coulais  _et al._ (2020). In the context of classical
electrical circuits, we have shown that a quantized response can be easily
extracted from extremely experimentally accessible impedance measurements.

###### Acknowledgements.

Acknowledgement.- We would like to thank Da-Jian Zhang for helpful
discussions. L. L. acknowledges funding support by the Key-Area Research and
Development Program of GuangDong Province under Grants No. 2019B030330001 and
the Startup Grant of Sun Yat-sen University (No. 71000-18841245). J.G.
acknowledges funding support by the Singapore NRF Grant No. NRF-NRFI2017-04
(WBS No. R-144-000-378-281). C.H.L acknowledges funding support by the
Singapore MOE Tier-1 start-up grant (WBS R-144-000-435-133).

##  IV Methods

###  IV.1 Insights based on generalized Brillouin zone

Here we offer more insights based on the so-called generalized Brillouin zone
(GBZ), to better understand why $\nu(E_{r})$ can be captured by the complex
spectral evolution. According to the non-Bloch band theory, the OBC spectrum
can be described by the PBC one in a GBZ, using a complex deformation of the
quasi-momentum $k\rightarrow k+i\kappa_{\rm OBC}(k)$ Yao and Wang (2018); Lee
and Thomale (2019); Lee  _et al._ (2018b, 2020a); Yokomizo and Murakami
(2019). The PBC-OBC spectral evolution can then be effectively described by
$k\rightarrow k+i\kappa(k)$ with $\kappa(k)$ varying from $0$ to $\kappa_{\rm
OBC}(k)$, with $\kappa_{\rm OBC}(k)$ having the minimal magnitude to yield the
OBC spectrum Yokomizo and Murakami (2019); Lee and Thomale (2019). The PBC-OBC
spectral evolution can hence be understood as arising from tuning $\kappa(k)$
and hence deforming the BZ to the GBZ, as shown in Fig. 4. Moreover, an
winding number can be defined as,

$\nu(E_{r})=\oint_{\rm GBZ}\frac{dz}{2\pi}\frac{d}{dz}{\rm
arg}\det[H(z)-E_{r}],$ (13)

analogous to that of Eq. (7), with integration in the GBZ instead of the BZ.
This winding number must be zero when the OBC spectrum is reached because,
again, the OBC spectrum cannot enclose any finite area Okuma  _et al._ (2020);
Zhang  _et al._ (2020a). Following the Cauchy principle, the spectral winding
number is found to be $\nu(E_{r})=N_{\rm zero}-N_{\rm pole}$, where $N_{\rm
zero}$ and $N_{\rm pole}$ are the counting of zeros and poles enclosed by the
integration path (BZ or GBZ) weighted by their respective orders. The
conclusion is hence as simple as follows. If we continuously tune $\kappa(k)$,
the PBC-OBC spectral evolution must pass through different zeros of
$[\frac{P_{r+l}(z)}{z^{r}}-E_{r}]$ [colored dots in Fig. 4] for a total of
$\nu(E_{r})$ times, such that the spectral winding number reduces from
$\nu(E_{r})$ to $0$ eventually when the integration path approaches the GBZ.
Thus, during the complex spectral evolution, the spectrum under $k\rightarrow
k+i\kappa(k)$ must pass the reference energy $E_{r}$ for a total of
$\nu(E_{r})$ times, constituting a rather formal argument to justify our
treatment in the main text.

Figure 4: A typical example of the Brillouin zone (BZ) and the generalized
Brillouin zone (GBZ). The BZ and GBZ are given by cyan and gray dots
respectively, in the complex plane of $z:=e^{ik}e^{-\kappa(k)}$. Here
$\kappa(k)$ represents a complex deformation of the momentum $k$. The black
dash loop indicates an evolving GBZ with $\kappa(k)$ between $0$ and
$\kappa_{\rm OBC}(k)$, the value that gives the spectrum under the open
boundary condition. Blue, red, and yellow dots are the zeros of $H(z)-E_{r}$,
where $H(z)$ is the PBC Hamiltonian of the system and $E_{r}$ is the chosen
reference energy for calculating the winding number. The system is chosen as
the same as that in Fig. 1 in the main text, i.e. $H(z)=2z^{2}+z+1/2z$.

###  IV.2 Directional signal amplification versus the PBC-OBC spectral
evolution

####  IV.2.1 Cases with only nearest-neighbor coupling

As mentioned in the main text, under an external drive
$\vec{\epsilon}(t)=\vec{\epsilon}(\omega){\rm exp}(-i\omega t)$ and an overall
on-site gain/loss parameter $\gamma$, the resultant response field
$\vec{\phi}(t)$ can be written as $\vec{\phi}(t)=\vec{\phi}(\omega){\rm
exp}(-i\omega t)$, with

$\displaystyle\vec{\phi}(\omega)=G(\omega,\gamma)\vec{\epsilon}(\omega),~{}G(\omega,\gamma)=\frac{1}{E_{r}-H},$
(14)

where $E_{r}=\omega+i\gamma$, $G$ is the Green’s function matrix. The
amplification factors for a signal toward the left and the right are described
by the matrix elements $G_{1N}$ and $G_{N1}$, respectively. For a non-
Hermitian $H$, the Green’s function matrix can be expressed in the spectral
representation Zirnstein  _et al._ ; Xue  _et al._ (2020)

$\displaystyle
G(\omega,\gamma)=\frac{1}{E_{r}-H}=\sum_{n}\frac{1}{E_{r}-E_{n}^{R}}|\Psi^{R}_{n}\rangle\langle\Psi^{L}_{n}|,$
(15)

with $|\Psi^{R}_{n}\rangle$ the $n$th right eigenstate of $H$ with eigenenergy
$E_{n}^{R}$, and $\langle\Psi^{L}_{n}|$ the corresponding left eigenstate.

To be more explicit, consider the Hatano-Nelson model under the PBC-OBC
interpolation, described by the following Hamiltonian

$H_{\beta}=\sum_{x=1}^{N-1}(t_{1}\hat{c}^{\dagger}_{x}\hat{c}_{x+1}+t_{-1}\hat{c}^{\dagger}_{x+1}\hat{c}_{x})+e^{-\beta}(t_{1}\hat{c}^{\dagger}_{N}\hat{c}_{1}+t_{-1}\hat{c}^{\dagger}_{1}\hat{c}_{N}),$
(16)

also with the assumption $t_{1}>t_{-1}$ without loss of generality. Let the
$n$-th right eigenstate be
$|\Psi_{n}^{R}\rangle=\sum_{x=1}^{\infty}\psi_{x,n}^{R}\hat{c}^{\dagger}_{x}|0\rangle$,
with $|0\rangle$ the vacuum state. Using the eigenvalue-eigenstate equation
$H_{\beta}|\Psi_{n}^{R}\rangle=E_{n}^{R}|\Psi_{n}^{R}\rangle$, one obtains

$\displaystyle
t_{1}\psi^{R}_{x+1,n}+t_{-1}\psi^{R}_{x-1,n}=E_{n}^{R}\psi^{R}_{x,n}~{}$ (17)

for $x\in[2,N-1]$, and

$\displaystyle e^{-\beta}t_{1}\psi^{R}_{1,n}+t_{-1}\psi^{R}_{N-1,n}$
$\displaystyle=$ $\displaystyle E_{n}^{R}\psi^{R}_{N,n},$ (18) $\displaystyle
t_{1}\psi^{R}_{2,n}+e^{-\beta}t_{-1}\psi^{R}_{N,n}$ $\displaystyle=$
$\displaystyle E_{n}^{R}\psi^{R}_{1,n}.$ (19)

If we further assume $t_{1}\gg t_{-1}$ and the following exponentially
decaying eigensolutions:

$\displaystyle\psi_{x,n}^{R}=C_{n}e^{-M_{n}^{R}(x-1)}$ (20)

with $C_{n}$ the normalization constant and $M_{n}^{R}>0$, then one obtains

$\displaystyle
t_{1}e^{-M_{n}^{R}}=E_{n}^{R},M_{n}^{R}=\frac{\beta-i2n\pi}{N}.$ (21)

Note that here $|E_{n}^{R}/t_{1}|$ is determined by the ratio $\beta/N$. That
is, the eigenvalues $E_{n}^{R}$ will be distributed on a circle on the complex
plane, whose radius depends only on $t_{1}$ as well as the ratio $\beta/N$.

Likewise, the left eigenstates under the same assumptions satisfy
$H^{\dagger}_{\beta}|\Psi_{n}^{L}\rangle=E_{n}^{L}|\Psi_{n}^{L}\rangle$ and
$E_{n}^{L}=(E_{n}^{R})^{*}$, and they are found to be

$\displaystyle\psi_{x,n}^{L}$ $\displaystyle=$ $\displaystyle
C_{n}^{*}e^{-M_{n}^{L}(N-x)},t_{1}e^{-M_{n}^{L}}=E_{n}^{L},$ $\displaystyle
M_{n}^{L}$ $\displaystyle=$ $\displaystyle\frac{\beta+i2n\pi}{N}.$ (22)

From the biothorgonal condition $\langle\Psi_{n}^{L}|\Psi_{n}^{R}\rangle=1$,
we then obtain the normalization constant

$\displaystyle C_{n}=\frac{e^{-i\frac{N-1}{N}\pi
n}}{\sqrt{Ne^{-\beta(N-1)/N}}}.$ (23)

Figure 5: (a) Spectra under PBC (cyan loop), OBC (gray dots), and (black
circles) the PBC-OBC interpolation through $\beta$. The red star indicates the
reference energy $E_{r}=\omega+i\gamma$ for further results in (b) and (c).
Here $E_{r}$ falls right on the spectrum of $H_{\beta}$, as the value of
$\beta=\beta_{c}\approx 6.2$ is read out from (b) and (c). (b) and (c) show
the amplification ratios $|G_{1N}|$ and $|G_{N1}|$ [(b2) and (c2)] and the
derivatives of their logarithms over $\beta$ [(b3) and (c3)] as functions of
$\beta$, where the quantity $I_{\beta}(E_{r})$ indicates with its peaks when
$E_{r}$ is passed through by the $H_{\beta}$ spectrum. The parameters are
$t_{1}=1$ and $t_{-1}=0.5$.

With preparations above the matrix element of the Green’s function $G$ can
then be found as follows:

$\displaystyle G_{1N}$ $\displaystyle=$
$\displaystyle\sum_{n}\frac{1}{E_{r}-E_{n}^{R}}\psi_{1,n}^{R}(\psi_{N,n}^{L})^{*}$
(24) $\displaystyle=$
$\displaystyle\sum_{n}\frac{1}{E_{r}-t_{1}e^{-\frac{\beta-i2n\pi}{N}}}\frac{1}{N(e^{-\frac{\beta-i2n\pi}{N}})^{N-1}}$
$\displaystyle=$
$\displaystyle\sum_{n}\frac{e^{\beta}e^{-\frac{\beta-i2n\pi}{N}}}{E_{r}-t_{1}e^{-\frac{\beta-i2n\pi}{N}}}\frac{1}{N}.$
(25)

We may now attempt to rewrite the discrete sum in Eq. (25) in terms of a loop
integral of a complex variable $z$, by defining $k_{n}:=2n\pi/N$ and
$z:=e^{-\beta/N}e^{ik_{n}}$. To that end one must implicitly assume that $N$
under consideration is sufficiently large so as to use an integral to replace
the discrete sum. With this in mind, for a given $\beta$, $\beta/N$ is assumed
to be vanishingly small, and hence essentially we are working in the regime of
$|z|\rightarrow 1$. Under these conditions, the sum in Eq. (25) can then be
evaluated by the following integral

$\displaystyle G_{1N}=\oint_{|z|=e^{-\beta/N}}\frac{1}{2\pi
i}\frac{e^{\beta}}{E_{r}-t_{1}z}dz,$ (26)

which is found to be

$\displaystyle G_{1N}=-\frac{e^{\beta}}{t_{1}},$ (27)

if $z_{0}\equiv E_{r}/t_{1}$ satisfies $|z_{0}|<e^{-\beta/N}$, i.e. the pole
of the integrand falls within the the integral loop. This condition leads to

$\displaystyle|E_{r}|<t_{1}e^{-\beta/N}=|E^{R}_{n}|,$ (28)

meaning that the reference energy $E_{r}$ falls within the loop-spectrum of
$H_{\beta}$. The above detailed theoretical considerations indicate that, so
long as $E_{r}$ is enclosed by the loop spectrum of $H_{\beta}$, we have

$\displaystyle\frac{d\ln|G_{1N}|}{d\beta}=1,$ (29)

which is just the claim in the main text regarding how to use a quantized
physical response to detect the spectral winding number $\nu(E_{r})$, as
computationally verified in Fig. 5(b).

Next we investigate what happens if we let $\beta$ exceed $\beta_{c}$, the
critical $\beta$ value for which a reference energy point $E_{r}$ falls
exactly on the loop spectrum of $H_{\beta}$ (as shown in Fig. 5(a)). Let us
first recall the result from Eq. (21), which indicates that the radius of the
loop spectrum of $H_{\beta}$ scales with $\beta/N$. Specifically, for a given
lattice size $N$ and a given reference point $E_{r}$ under investigation, one
immediately obtains that

$\displaystyle\beta_{c}=-N\ln|E_{r}/t_{1}|,$ (30)

which is clearly proportional to $N$. As such, to probe the regime of
$\beta\geq\beta_{c}$ , $\beta$ should at least linearly increase with $N$ as
well. This being the case, we can no longer approximate the discrete sum in
Eq. (25) as a loop integral with $N\rightarrow\infty$ in mind, simply because
it has the factor $e^{\beta}$ in its numerator, which diverges with $N$
because in the regime of interest $\beta$ diverges with an increasing $N$.

Based on the discussions above, in the regime of $\beta\geq\beta_{c}$, what is
under investigation becomes the evaluation of the same discrete sum, but with
$\beta$ scaling proportionally with $N$, and hence both becoming sufficiently
large in the event of using a loop integral to replace this discrete sum.
Though we still use the same complex variable $z=e^{-\beta/N}e^{ik_{n}}$ to
invoke a possible loop integral, we see that $|z|$ is now far from unity. To
reflect this, we now use the alternative expression of the discrete sum in Eq.
(24) and then rewriting it as the following integral, with the integrand
having one pole of order $N$ at $z=0$, and another $1$st-order pole at
$z=E_{r}/t_{1}$,

$\displaystyle G_{1N}$ $\displaystyle=$
$\displaystyle\oint_{|z|=e^{-\beta/N}}\frac{1}{2\pi
i}\frac{1}{z^{N}(E_{r}-t_{1}z)}dz.$ (31)

Assuming that $|z_{0}|=|E_{r}/t_{1}|>e^{-\beta/N}$, namely, the reference
energy $E_{r}$ is outside the loop, so $z=0$ is the only pole of the integrand
enclosed by the integration path, we have

$\displaystyle G_{1N}=\frac{t_{1}^{N-1}}{E_{r}^{N}},$ (32)

a value independent of $\beta$, which is again consistent with the
computational results in Fig. 5(b2). Further, using the previous expression of
$\beta_{c}$ from Eq. (30), we find that in this case
$G_{1N}=\frac{t_{1}^{N-1}}{E_{r}^{N}}=e^{\beta_{c}}/t_{1}$. Interestingly,
though this magnitude $e^{\beta_{c}}/t_{1}$ exponentially larger than
$e^{\beta}/t_{1}$ obtained earlier for $\beta<\beta_{c}$, this amplification
factor is saturated and no longer depends on $\beta$ in the regime of
$\beta>\beta_{c}$.

As a side note, one might wonder why we cannot also use the loop integral in
Eq. (31) to treat the first case, namely, a fixed $\beta$ in the regime of
$\beta<\beta_{c}$ but with $N$ approaching sufficiently large values. As said
earlier, in this case we essentially perform the summation under the condition
of $|z|=1$. Under this condition we always have $z^{N}=1$ and hence the
expression in Eq. (31) is no longer useful.

In the same fashion, we can now proceed to examine $G_{N1}$, which depicts how
the signal is amplified or suppressed in the other direction. The matrix
element $G_{N1}$ is found to be the following,

$\displaystyle G_{N1}$ $\displaystyle=$
$\displaystyle\sum_{n}\frac{1}{E_{r}-E_{n}^{R}}\psi_{N,n}^{R}(\psi_{1,n}^{L})^{*}$
(33) $\displaystyle=$
$\displaystyle\sum_{n}\frac{1}{E_{r}-t_{1}e^{-\frac{\beta-i2n\pi}{N}}}\frac{(e^{-\frac{\beta-i2n\pi}{N}})^{N-1}}{N}$
$\displaystyle=$
$\displaystyle\sum_{n}\frac{e^{-\beta}e^{\frac{\beta-i2n\pi}{N}}}{E_{r}-t_{1}e^{-\frac{\beta-i2n\pi}{N}}}\frac{1}{N}.$
(34)

Rewriting Eq. (34) in terms of a loop integral, we have

$\displaystyle G_{N1}$ $\displaystyle=$
$\displaystyle\oint_{|z|=e^{-\beta/N}}\frac{1}{2\pi
i}\frac{e^{-\beta}}{z^{2}(E_{r}-t_{1}z)}dz.$ (35)

The integrand has a 1st-order pole at $z_{0}=E_{r}/t_{1}$, and a second-order
pole at $z_{1}=0$. Therefore we have

$\displaystyle G_{N1}=\frac{e^{-\beta}t_{1}}{E_{r}^{2}}$ (36)

if $|E_{r}|>t_{1}e^{-\beta/N}$ (reference energy falls outside the loop of
integral), and

$\displaystyle G_{N1}=0$ (37)

if $|E_{r}|<t_{1}e^{-\beta/N}$ (reference energy $E_{r}$ falls inside the loop
of integral). Here because of the factor $e^{-\beta}$ in Eqs. (34) and Eq.
(35), the replacement of the discrete sum by the loop integral is always valid
by assuming a sufficiently large $N$, i.e., regardless of whether $\beta$ is
assumed to be fixed or assumed to scale linearly with $N$. Thus, results
obtained above for both $\beta>\beta_{c}$ and $\beta<\beta_{c}$ are valid,
which are indeed consistent with our numerical results. Note also that for
fixed $\beta$, our numerical results for finite systems give a small but
nonzero $G_{N1}$ when $\beta<\beta_{c}$ (e.g. $|G_{N1}|\approx e^{-5}$ in Fig.
5(c2)), which vanishes when further increasing $N$ (not shown).

Overall, we obtain that the gradient of $\ln|G_{N1}|$ with respect to $\beta$
is again quantized, with

$\displaystyle\frac{d\ln|G_{N1}|}{d\beta}=-1,$ (38)

if $E_{r}$ is NOT enclosed by the loop spectrum of $H_{\beta}$, corresponding
to $\beta>\beta_{c}$ [Fig. 5(c)]. If we further increase $\beta$ the system
shall approach the OBC limit when $\beta=\beta_{\rm OBC}\approx\alpha N$ with
$\alpha=\ln(\sqrt{t_{1}/t_{-1}})$, where the spectrum falls on the same lines
as the OBC spectrum Kunst  _et al._ (2018); Koch and Budich (2020).
Nevertheless, this limit is still not exactly like the OBCs, as the two
boundaries are still weakly connected. For example, due to the (weak) boundary
couplings, a flux threading cannot be gauged away and can lead to fluctuation
of eigenergies, unlike in real OBC cases. Indeed, from our numerical results,
we indeed see that $G_{N1}$ keeps decreasing after $\beta$ exceeds $\beta_{\rm
OBC}$, and becomes a constant when $\beta\gtrsim 2\beta_{\rm OBC}-\beta_{c}$
[Fig. 5(c3)].

####  IV.2.2 Cases with only $m$th-nearest neighbor coupling

Consider now a 1D non-Hermitian chain with only the $m$th-nearest neighbor
couplings:

$\displaystyle H_{\beta}$ $\displaystyle=$
$\displaystyle\sum_{x=1}^{N-m}(t_{m}\hat{c}^{\dagger}_{x}\hat{c}_{x+m}+t_{-m}\hat{c}^{\dagger}_{x+m}\hat{c}_{x})$
$\displaystyle+e^{-\beta}\sum_{x=N-m+1}^{N}(t_{m}\hat{c}^{\dagger}_{x}\hat{c}_{x+m-N}+t_{-m}\hat{c}^{\dagger}_{x+m-N}\hat{c}_{x}).$

Here we first assume $N/m$ is an integer, thus the system is decoupled into
$m$ identical 1D sub-chains, and the eigenstates satisfy

$\displaystyle t_{m}\psi^{R}_{x+m,n}+t_{-m}\psi^{R}_{x-m,n}$ $\displaystyle=$
$\displaystyle E_{n}^{R}\psi^{R}_{x,n}$ (40)

for $x\in[2,N-m]$, and

$\displaystyle e^{-\beta}t_{m}\psi^{R}_{s,n}+t_{-m}\psi^{R}_{N-2m+s,n}$
$\displaystyle=$ $\displaystyle E_{n}^{R}\psi^{R}_{N-m+s,n},$ (41)
$\displaystyle t_{m}\psi^{R}_{m+s,n}+e^{-\beta}t_{-m}\psi^{R}_{N-m+s,n}$
$\displaystyle=$ $\displaystyle E_{n}^{R}\psi^{R}_{s,n},$ (42)

with $s=1,2,...,m$ labelling different sub-chains. As in the previous
discussion for the case of $m=1$, we assume $t_{m}\gg t_{-m}$ to obtain some
simple analytical results. Here we replace the labels $x$ and $n$ with $x_{s}$
and $n_{s}$ for each sub-chain. We take the ansatz

$\displaystyle\psi^{R}_{x_{s},n_{s}}=C_{n_{s}}e^{-M_{n_{s}}^{R}(x_{s}-1)},$
(43)

with $n_{s}$ only takeing values from $1$ to $N_{m}=N/m$, given that each sub-
chain contains only $N_{m}$ lattice sites, and $x_{s}=(x-s+m)/m$ ranging from
$1$ to $N_{m}$, with $x$ being $s,m+s,2m+s,...,N-m+s$, and more importantly,

$\displaystyle
t_{m}e^{-M_{n_{s}}^{R}}=E_{n_{s}}^{R},M_{n_{s}}^{R}=\frac{\beta-i2n_{s}\pi}{N_{m}}.$
(44)

Similarly, the left eigenstates are given by

$\displaystyle\psi_{x_{s},n_{s}}^{L}$ $\displaystyle=$ $\displaystyle
C_{n_{s}}^{*}e^{-M_{n_{s}}^{L}(N_{m}-x_{s})},t_{m}e^{-M_{n_{s}}^{L}}=E_{n_{s}}^{L},$
$\displaystyle M_{n_{s}}^{L}$ $\displaystyle=$
$\displaystyle\frac{\beta+i2{n_{s}}\pi}{N_{m}}.$ (45)

Again, from the biothorgonal condition
$\langle\Psi_{n}^{L}|\Psi_{n}^{R}\rangle=1$, we have the normalization
constants

$\displaystyle
C_{n_{s}}=\frac{e^{-i\frac{N_{m}-1}{N_{m}}\pi{n_{s}}}}{\sqrt{N_{m}e^{-\beta(N_{m}-1)/N_{m}}}}.$
(46)

Note that in the Green’s function matrix, the element $G_{1N}$ shall always be
zero as each sub-chain is decoupled from the others. In this case, the
directional amplification of each sub-chain corresponds to the element
$G_{s(N-m+s)}$, with

$\displaystyle G_{s(N-m+s)}$ $\displaystyle=$
$\displaystyle\sum_{n_{s}}\frac{1}{E_{r}-E_{n_{s}}^{R}}\psi_{x_{s}=1,{n_{s}}}^{R}(\psi_{x_{s}=N_{m},{n_{s}}}^{L})^{*}$
(48) $\displaystyle=$
$\displaystyle\sum_{n_{s}}\frac{1}{E_{r}-t_{m}e^{-\frac{\beta-i2n_{s}\pi}{N_{m}}}}\frac{1}{N_{m}(e^{-\frac{\beta-i2n_{s}\pi}{N_{m}}})^{N_{m}-1}}$
$\displaystyle=$
$\displaystyle\sum_{n_{s}}\frac{e^{\beta}e^{-\frac{\beta-i2n_{s}\pi}{N_{m}}}}{E_{r}-t_{m}e^{-\frac{\beta-i2n_{s}\pi}{N_{m}}}}\frac{1}{N_{m}}.$

As $n_{s}$ takes value from $1$ to $N_{m}=N/m$, here we need to define
$k_{s}=i2mn_{s}\pi/N$, so that the summation can be replaced by an integral
with $k_{s}$ varying from $0$ to $2\pi$. Similar to the case with only
nearest-neighbor coupling, we then have

$\displaystyle\frac{d\ln|G_{s(N-m+s)}|}{d\beta}=1$ (49)

for each sub-chain, when $E_{r}$ is enclosed by the loop-like spectrum of each
sub-chain. This result indicates that each sub-chain has its own spectral
winding number $\nu_{s}(E_{r})=1$, but for the original 1D chain with $m$th-
nearest neighbor couplings, the element $G_{1N}$ and $G_{N1}$ are zero as
sites $1$ and $N$ belong to different decoupled sub-chains.

On the other hand, the $m$-sub-chain picture here also indicates an effective
unit-cell structure with $m$ sublattices, even though the sublattices are
physically equivalent on the lattice. Thus the directional amplification of
the overall system comprised by these sub-chains/sublattices shall be
described by the combination of that of each sub-chain, corresponding to the
corner blocks of the overall Green function matrix,

$\displaystyle G_{\leftarrow,m\times
m}=\left(\begin{array}[]{cccc}G_{1(N-m+1)}&G_{1(N-m+2)}&\cdots&G_{1N}\\\
G_{2(N-m+1)}&G_{2(N-m+2)}&\cdots&G_{2N}\\\ \vdots&\vdots&\vdots&\vdots\\\
G_{m(N-m+1)}&G_{m(N-m+2)}&\cdots&G_{mN}\\\ \end{array}\right),$ (54)
$\displaystyle G_{\rightarrow,m\times
m}=\left(\begin{array}[]{cccc}G_{(N-m+1)1}&G_{(N-m+1)2}&\cdots&G_{(N-m+1)m}\\\
G_{(N-m+2)1}&G_{(N-m+2)2}&\cdots&G_{(N-m+2)m}\\\
\vdots&\vdots&\vdots&\vdots\\\ G_{N1}&G_{N2}&\cdots&G_{Nm}\\\
\end{array}\right),$ (59)

for signals moving toward the left- and the right side, respectively. In the
above schematic scenario where the sub-chains are fully decoupled from each
other, only the diagonal elements of the above two matrices are nonzero, i.e.
$(G_{\leftarrow,m\times m})_{ab}\propto\delta_{ab}e^{\beta}$ when $E_{r}$ is
enclosed by the spectrum on the complex plane, and $(G_{\rightarrow,m\times
m})_{ab}\propto\delta_{ab}e^{-\beta}$ when $E_{r}$ is NOT enclosed by the
spectrum on the complex plane. This being the case, we arrive at

$\displaystyle\det[G_{\leftarrow,m\times m}]\propto
e^{m\beta},~{}\det[G_{\rightarrow,m\times m}]\propto e^{-m\beta},$ (60)

in the above two cases repectively. Thus we have

$\displaystyle\nu_{\leftarrow,m}:=\frac{d\ln|G_{\leftarrow,m\times
m}|}{d\beta}=m,$ (61)

corresponding to the spectral winding number $\nu(E_{r})=m$ for the case with
only the $m$th-nearest neighbor couplings. Furthermore, this conclusion shall
also be valid when $N/m$ is not an integer. In such cases, the system still
possesses the $m$-sub-chain picture, only that the two ends of one sub-chain
are connected to those of other sub-chains now. This conclusion is also
verified by our numerical calculations. As also mentioned in the main text,
the physical analysis here is also valid if the system has many different
types of coupling coexisting, but is still dominated by one type of coupling.
The topological robustness of the response can then retain the quantization.

Supplementary Materials

##  V Cases with couplings across different ranges

In the main text, we have already witnessed an intriguing example where both
nearest-neighboring and next-nearest-neighbor couplings present in the system.
In that case, with respect to different reference energy points, the spectral
winding can have different nonzero values and this is also manifested as
different quantized responses. Benchmarking this with our decoupled sub-chain
picture, this indicates that there is competition in different regimes, and
the underlying topological robustness associated with either single-chain
physics or two coupled sub-chain physics still yields, remarkably, quantized
responses. The next question is then, if we add more and more complexity to a
non-Hermitian lattice system with many coexisting hopping length scales, can
we still observe quantization and hence a clear correspondence between
spectral winding and the signal amplification. Our answer is yes based on more
computational tests.

Figure S1: (a) PBC (blue loop) and OBC (black dots) spectra of a system with
many coexisting hopping length scales, and the PBC-OBC spectral evolution
between them (blue-purple curves), tuned by the parameter $\beta$. Four
reference energy values $E_{r}$ are chosen to inspect the possible
quantization of the gradient of the logarithm of directional amplification
with respect to $\beta$, each with a different winding number $\nu(E_{r})$, as
indicated by the four colored marks. (b)-(e) show the summed reverse energy
spacing $I_{\beta}(E_{r})=\sum_{n}|1/(E_{n}(\beta)-E_{r})|$, logarithm of the
amplification ratio for the block of $G$ associated with the winding number
$\nu(E_{r})$, as well as $\nu_{\leftarrow,m}$ and $\nu_{\rightarrow,m}$, as
functions of $\beta$. In (e) we have $\nu(E_{r})=0$, hence we only illustrate
amplification ratios $G_{\leftarrow,1\times 1}$ and $G_{\rightarrow,1\times
1}$, which are both less than unity, indicating no amplification for a signal
moving either toward the left or the right. Parameters are $t_{2}=2$,
$t_{-2}=1$, $t_{3}=0$, $t_{-3}=1$, and $t_{\pm m}=0$ for all other values of
$m$, with $N=300$ lattice sites.

Consider then a lattice model with nonzero terms
$\\{t_{-r},t_{-r+1},...,t_{l-1},t_{l}\\}$, and none of them is dominating over
the rest. We can still view this system as one comprised of $m$ sub-chains,
with $m\leqslant{\rm Max}[r,l]$. We can investigate if the same response
functions $\nu_{\leftarrow,m}$ or $\nu_{\rightarrow,m}$ can reflect the
spectral winding behaviors. To proceed specifically, consider the following
Hamiltonian as an example,

$\displaystyle
H=\sum_{x=1}^{N}\sum_{j=-r}^{l}t_{j}\hat{c}^{\dagger}_{j}\hat{c}_{x+j},$ (S1)

with $r=l=3$, and the boundary coupling tuned via $t_{j}\rightarrow
e^{-\beta}t_{j}$ when it connects sites at different ends of this 1D chain.
The rather complicated spectral winding behavior is shown in Fig. S1(a),
indicating winding numbers -1, 1, 2, and 0. As shown in Fig. S1(b)-(d), the
obtained $\nu_{\leftarrow,m}$ or $\nu_{\rightarrow,m}$ for $m=1$ or $m=2$
still shows relatively clear plateaus for cases with nonzero $\nu(E_{r})$,
with the transitions of these plateaus in excellent agreement with the
critical $\beta$ values for which the spectral winding numbers make jumps. In
panel (c3), the quantization in $\nu_{\leftarrow,1}$ is clearly seen. A
careful reader might notice that in panel (b3) and (d3), the plateaus of the
obtained response function $\nu_{\rightarrow,1}$ are so not clearly quantized.
To double check if this is merely a finite-size effect, we have increased the
size of the model system and then much better quantized plateaus are indeed
observed, as presented in Fig. S2. As to the case of $\nu(E_{r})=0$ labeled by
the yellow star in Fig. S1(a), no amplification is obtained for a signal
moving toward either the left- or the right-hand side, as indicated by the
always-negative $\ln|G_{\leftarrow,1\times 1}|$ and
$\ln|G_{\rightarrow,1\times 1}|$ in Fig. S1(e).

Figure S2: (a) $\nu_{\rightarrow,1}$ and (b) $\nu_{\leftarrow,2}$ as functions
of $\beta/N$ for the two cases in Fig. S1(b) and (d) respectively, with
different numbers of lattice $N=300$, $400$, and $600$. As our previous
analytical results suggest that the transition value $\beta_{c}$ is
proportional to $N$, here the variable $\beta$ is rescaled by a factor of
$1/N$, so as to map the transition points for systems with different sizes to
the same parameter. The plateaus are seen to be flatter (hence better
quantization) for larger systems. Other parameters are $t_{1}=1$ and
$t_{-1}=0.5$.

##  VI Illustrative details of circuit for probing the response of the Hatano-
Nelson model

To be concrete, we provide explicit details for the realization of the
simplest case of the circuit representing the Hatano-Nelson model with
topological winding $\nu=1$. It consists of a chain of nodes connected by
unbalanced couplings which simultaneously give rise to non-Hermiticity and
non-reciprocity. It is well-established that such couplings can be realized
with INICs (negative impedance converters with current inversion) Hofmann  _et
al._ (2019); Helbig  _et al._ (2020), which contains operation amplifiers that
break the reciprocity. To realize the tuning of the spectral reference point
$\Omega$ and end-to-end couplings $e^{-\beta(\omega)}$, we also include
tunable inductors and additional RLC elements as according to Ref. Li  _et
al._ . Its Laplacian, together with these tunable elements, takes the form

$\displaystyle J=$
$\displaystyle\biggl{[}e^{-\beta(\omega)}\left(e^{\alpha}|N\rangle\langle
0|+e^{-\alpha}|0\rangle\langle
N|\right)+\sum_{x=0,\pm}^{N-1}e^{\pm\alpha}|x\rangle\langle x\pm 1|$
$\displaystyle-\sum_{x=0}^{N}\left(2\cosh\alpha-\omega_{0}^{2}/\omega^{2}+\Omega(\omega)\right)|x\rangle\langle
x|\biggl{]}\times i\omega C$ (S2)

where $\alpha=\tanh^{-1}\frac{C_{1}}{C_{2}}$, $C=\sqrt{C_{2}^{2}-C_{1}^{2}}$,
$\omega_{0}^{-2}=l_{gr}C$ and
$\Omega=\frac{-C_{\Omega}+iR_{\Omega}/\omega}{C}$. $C_{1}$ and $C_{2}$ are
capacitors involved in internode couplings as according to Ref. Li  _et al._ ,
and $R_{\Omega}$, $l_{gr},C_{\Omega}$ are RLC elements that connect each node
to the ground. By varying the choice of $C_{\Omega}$ and $R_{\Omega}$,
$\Omega$ and hence different points of the complex eigenvalue plane can be
sampled. In this circuit, the $\nu=1$ and $\nu=0$ regions are separated by the
curve $\Omega=2\cosh(\alpha+ik)-2\cosh\alpha-\omega_{0}^{2}/\omega^{2}$. The
effective flux $\beta(\omega)$ can be adjusted by tuning the AC frequency
$\omega$, and is given by $\beta(\omega)=\ln[1-\omega^{2}l(C_{2}-C_{1})]$,
where $l$ is an inductor involved in the internode couplings which controls
the sensitivity of the tuning. Clearly, by including additional couplings
between more distant nodes Li  _et al._ (2019b); Lee  _et al._ (2020b); Ezawa
(2019b); Lu  _et al._ (2019); Zhang  _et al._ (2020b); Song  _et al._ (2020);
Stegmaier  _et al._ (2020), this circuit construction can be extended to
models with additional further couplings, such as that shown in the main text
with $t_{1}=1,t_{-1}=0.5,t_{2}=2,t_{-2}=0$.

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