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

# Kinked linear response from non-Hermitian pumping

Fang Qin<EMAIL_ADDRESS>Department of Physics, National University of
Singapore, Singapore 117551, Singapore  Ruizhe Shen<EMAIL_ADDRESS>Department of Physics, National University of Singapore, Singapore 117551,
Singapore  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  Ching
Hua Lee<EMAIL_ADDRESS>Department of Physics, National University of
Singapore, Singapore 117551, Singapore  Joint School of National University of
Singapore and Tianjin University, International Campus of Tianjin University,
Binhai New City, Fuzhou 350207, China

###### Abstract

Non-Hermiticity is known to give rise to modified topological bulk-boundary
correspondences, which predict the presence of topological boundary modes
through appropriately modified topological invariants. Yet, little is
currently known about how non-Hermiticity affects the precise linear response
of wavepackets beyond their overall spectral flow. In this work, we discover
that, generically, non-Hermiticity gives rise to abrupt and prominent kinks in
the semi-classical wavepacket trajectories of quantum gases, despite the
absence of sudden physical impulses. This physically stems from a hitherto
underappreciated intrinsic non-locality from non-Hermitian pumping, even if
all physical couplings are local, thereby resulting in enigmatic singularities
in the band structure that lead to discontinuous band geometry and Berry
curvature. For a concrete experimental demonstration, we propose an ultracold
atomic setup in a two-dimensional optical lattice with laser-induced loss,
such that response kinks can be observed without fine-tuning in the physical
atomic cloud dynamics. Our results showcase unique non-monotonic behavior from
non-Hermitian pumping beyond the non-Hermitian skin effect and suggest new
avenues for investigating non-Hermitian dynamics on ultracold atomic
platforms.

Introduction.– Linear response theory, as epitomized by the Kubo formula for
Hall conductivity, plays a ubiquitous role in connecting experimental response
signatures with bulk band geometry and topology Halperin (1982); Thouless  _et
al._ (1982); Niu  _et al._ (1985); Von Klitzing (1986); Hatsugai (1993);
Albrecht  _et al._ (2001); Bernevig and Zhang (2006); Hu  _et al._ (2016);
Zilberberg  _et al._ (2018); Galeski  _et al._ (2020); Qin  _et al._ (2020,
2022a, 2022b, 2023a). First investigated in ferromagnetic settings by Karplus
and Luttinger Karplus and Luttinger (1954), Hall responses are now known to
strikingly affect thermoelectric Xiao  _et al._ (2010); Jiang  _et al._
(2022); Xu  _et al._ (2022); Zhang  _et al._ (2019); Qin  _et al._ (2014) and
magneto-transport Xiao  _et al._ (2010); Tsui  _et al._ (1982) properties.
Beyond quantized conductivity, much transport behavior can be derived from the
semiclassical dynamics of Bloch electrons under external fields Chang and Niu
(1996); Gosselin  _et al._ (2006); Bliokh  _et al._ (2007); Lee  _et al._
(2018); Gao (2019); Tuloup  _et al._ (2020); Lapa and Hughes (2019).

While the behavior of linear response is already well-understood in Hermitian
settings, non-Hermiticity introduces a host of new subtleties. For instance,
the presence of gain/loss can lead to non-quantized Hall conductivity from
finite electron lifetimes Wang and Wang (2019); Wang  _et al._ (2022). A non-
Hermitian linear response theory is developed for Hermitian systems with a
non-Hermitian probe Pan  _et al._ (2020). The inequivalence of the left and
right eigenstates of a biorthogonal basis also complicates Kubo’s celebrated
formula Lau and Clerk (2018); Geier and Hauke (2022); Sticlet  _et al._ (2022)
and associated Green functions Borgnia  _et al._ (2020); Li  _et al._ (2021);
Zirnstein  _et al._ (2021); Hashemi  _et al._ (2022).

In this work, we discover that, generically, asymmetric non-Hermitian
couplings can give rise to intriguing response kinks in an atomic gas. These
kinks are surprising because they seemingly correspond to divergently large
semi-classical impulses, even though no such physical impulses exist. We trace
their existence to an emergent form of non-locality (even though the physical
system is completely local) caused by directed non-Hermitian pumping, which is
already known for producing boundary-localized skin modes Yao and Wang (2018);
Song  _et al._ (2019); Lee  _et al._ (2019); Lee and Thomale (2019); Longhi
(2019); Li  _et al._ (2020a); Lee  _et al._ (2020); Qin  _et al._ (2020);
Helbig  _et al._ (2020); Okuma  _et al._ (2020); Li  _et al._ (2020b); Lee
(2021); Shen and Lee (2022); Jiang and Lee (2022a); Yang  _et al._ (2022);
Rafi-Ul-Islam  _et al._ (2022); Zhang  _et al._ (2022a); Qin  _et al._ (2023b,
c); Jiang and Lee (2022b); Yokomizo and Murakami (2020a); Li  _et al._ (2023).
Mathematically, the emergent non-locality can be encoded in the non-
analyticity of the general Brillouin zone (GBZ), where non-Hermitian pumping
causes the effective lattice momentum $k\rightarrow k+i\kappa(k)$ to be
generically complex-deformed by an envelope factor $\kappa(k)$ that contains
cusp singularities in all but the simplest models. As such, all quantities
that contain momentum-space gradients, such as the band metric and Berry
curvature, become discontinuous. Thus kinks appear in their associated linear
responses, with the very large “virtual” impulses encoding the emergent non-
locality.

A caveat is that these response kinks tacitly hinge on the notion of band
occupancy and thus cannot appear in metamaterial platforms such as non-
Hermitian photonic lattices Regensburger  _et al._ (2012); Feng  _et al._
(2017); Midya  _et al._ (2018); Sun  _et al._ (2023) and topolectrical
circuits Helbig  _et al._ (2020); Hofmann  _et al._ (2020); Li  _et al._
(2019a); Ezawa (2019); Liu  _et al._ (2021); Zhang  _et al._ (2022b); Shang
_et al._ (2022); Zhang  _et al._ (2023); Zhu  _et al._ (2023); Hohmann  _et
al._ (2023), whose great success are restricted to single-particle phenomena
111But see Refs. Lee (2021); Di Liberto  _et al._ (2016); Gorlach and Poddubny
(2017); Olekhno  _et al._ (2020); Shen and Lee (2022); Koh  _et al._ (2022);
Poddubny (2023) for schemes to simulate few-particle phenomena.. As such,
building on rapid parallel developments in ultracold atomic systems Li  _et
al._ (2019b); Lapp  _et al._ (2019); Ren  _et al._ (2022); Liang  _et al._
(2022); Gou  _et al._ (2020); Qin  _et al._ (2023b), we formulate a detailed
experimental proposal for observing these kinks in the trajectories of atomic
clouds.

Non-analytic complex momentum deformation from non-Hermitian pumping.– We
first review how asymmetric couplings can deform the effective lattice
momentum non-analytically into the complex plane. A generic two-dimensional
(2D), non-interacting lattice Hamiltonian can be written as

$H=\sum_{x,y}\left(\sum_{ji;\alpha\beta}t^{\alpha\beta}_{ji}c^{\alpha\dagger}_{j+x,i+y}c^{\beta}_{x,y}\right),$
(1)

where $x,y$ and $\alpha,\beta$ respectively index the unit cell position and
sublattices. The hopping amplitude from unit cell $(x,y)$ and sublattice
$\beta$ to unit cell $(x+j,y+i)$ and sublattice $\alpha$ is given by
$t^{\alpha\beta}_{ji}$. Since a realistic lattice is finite and bounded, we
shall consider open boundary conditions (OBCs) with widths $L_{x}$ and
$L_{y}$, such that $t^{\alpha\beta}_{ji}=0$ whenever $(x,y)$ or $(x+j,y+i)$ is
not within $[1,L_{x}]\times[1,L_{y}]$. In the presence of asymmetric couplings
($|t^{\alpha\beta}_{ji}|\neq|t^{\beta\alpha}_{ij}|$), states are amplified in
the direction of stronger hopping, leading to the net effect of non-Hermitian
pumping where all states eventually accumulate against the boundaries Yao and
Wang (2018); Lee and Thomale (2019); Yokomizo and Murakami (2019); Chen  _et
al._ (2019); Li  _et al._ (2019c); Zhang  _et al._ (2020); Yang  _et al._
(2020a); Okuma  _et al._ (2020); Li  _et al._ (2020b); Qin  _et al._ (2023c,
b); Shen and Lee (2022); Jiang and Lee (2022a); Lin  _et al._ (2023).

To illustrate how complex-deformed momentum arises, we consider the square
lattice in two dimensions for simplicity 222For more complicated lattices,
boundaries of different orientations may have to be separately treated Li and
Yi (2023); Kawabata  _et al._ (2020b); Zhang  _et al._ (2022a); Li  _et al._
(2022); Lei  _et al._ (2023); Manna and Roy (2023), and the effective BZ may
even be dimensionally reduced when the lattice cannot be decomposed into the
sum of 1D lattices Jiang and Lee (2022a)., such that non-Hermitian pumping in
either dimension can be separately treated. For a 1D subchain
$H_{\text{1D}}(y)=\sum_{x}\sum_{j;\alpha\beta}t^{\alpha\beta}_{j}c^{\alpha\dagger}_{j+x,y}c^{\beta}_{x,y}$,
the periodic boundary condition (PBC) spectrum is simply given by the
eigenvalues of the matrix symbol $H_{\text{1D}}(z)$, where

$H_{\text{1D}}^{\alpha\beta}(z)=\sum_{j}t^{\alpha\beta}_{j0}z^{j}$ (2)

and $z=e^{ik_{x}}$, $k_{x}\in\\{2\pi/L_{x},4\pi/L_{x},\cdots\\}$. However,
under OBCs, the accumulated boundary “skin” states drastically break
translation invariance, and the correct bulk description can only be obtained
by complex deforming $k_{x}\rightarrow\bar{k}_{x}=k_{x}+i\kappa(k_{x})$ with
an appropriate $\kappa(k_{x})$ such that approximate translation invariance
(i.e., “bulk-boundary correspondence” Yokomizo and Murakami (2019); Kawabata
_et al._ (2020a); Yokomizo and Murakami (2020a); Yi and Yang (2020); Yokomizo
and Murakami (2020b); Yang  _et al._ (2020a); Yokomizo and Murakami (2021);
Deng and Yi (2019); Lee (2022)) is restored Kunst  _et al._ (2018); Xiong
(2018); Yang  _et al._ (2020a); Xiao  _et al._ (2020). The deformation
$\kappa(k_{x})=-i\log|z|$ is determined from the requirement that all OBC
wavefunctions vanish at both ends of $[1,L_{x}]$ Yao and Wang (2018); Lee and
Thomale (2019); Lee  _et al._ (2019); Li  _et al._ (2020b); Ashida  _et al._
(2020); Okuma  _et al._ (2020); Ding  _et al._ (2022); Zhang  _et al._
(2022c); Jiang and Lee (2022a); Lin  _et al._ (2023). This is equivalent to
the condition in the dispersion equation

$\text{Det}\left[H_{\text{1D}}(z)-E_{\text{OBC}}\,\mathbb{I}\right]=0,$ (3)

which possesses doubly degenerate solutions $z,z^{\prime}$ with
$|z|=|z^{\prime}|$ at OBC eigenenergies $E_{\text{OBC}}$. This is because the
smallest solution branch of $\kappa(k_{x})=-i\log|z|$ controls the dominant
decay rate of eigensolutions $\psi_{k_{x}}$ viz.
$|\psi_{k_{x}}(x)|\sim|e^{i\bar{k}_{x}x}|=e^{-\kappa(k_{x})x}$, and a
superposition of two eigensolutions that decay at identical rates is needed to
satisfy OBCs at both left and right boundaries.

Figure 1: Discontinuous bands and their Berry curvature from complex momentum
singularities. For our illustrative Hamiltonian $H_{\text{Toy}}$ (Eq. (4))
with $\alpha=1$, $\beta=2$, $A=3B$, the effective momentum $\bar{k}_{x}$ (Eq.
(5)) acquires an imaginary contribution $\text{Im}[\bar{k}_{x}]=\kappa(k_{x})$
(red) that is non-analytic at $k_{x}=\pi/3,\pi,5\pi/3$, wherever different
solution branches (blue) intersect. In (b), likewise, the real part of energy
bands (red for the upper band and blue for the lower band) of our cold atom
Hamiltonian $\bar{\cal H}$ defined by Eq. (10) exhibits discontinuities at
$k_{y}=\pm 0.5\pi$ under $\hat{x}$-OBCs and $\hat{y}$-PBCs. (c) The Berry
curvature Re$(\bar{\Omega}_{xy})$ of $\bar{\cal H}$ also exhibits
discontinuities at the same locations $k_{y}=\pm 0.5\pi$, even though it still
integrates to an integer multiple of $2\pi$. Parameters for (b) and (c) are
$\phi_{0}=\pi/4$, $\Omega_{r}=2$, $\Omega_{0}=1$, $\gamma=1.2$, $t_{y}=0.5$,
and $t_{x}=1$.

To explicitly see that such $\kappa(k_{x})$ solutions generically possess
cusps, we examine the simplest class of asymmetric hopping models with two
different nonzero hopping amplitudes $A,B$, one $\alpha$ site towards the left
and the other $\beta$ site towards the right:

$H_{\text{Toy}}(z)=Az^{\alpha}+\frac{B}{z^{\beta}},$ (4)

where $z=e^{ik_{x}}$ under PBCs, and
$z=e^{i\bar{k}_{x}}=e^{-\kappa(k_{x})}e^{ik_{x}}$ under OBCs. Since the unit
cell is trivial, Eq. (3) reduces to $E_{\text{OBC}}=Az^{\alpha}+Bz^{-\beta}$,
which is invariant under $z\rightarrow ze^{2\pi i/(\alpha+\beta)}$,
$E_{\text{OBC}}\rightarrow E_{\text{OBC}}e^{2\pi i\alpha/(\alpha+\beta)}$. As
such, $z=z(k_{x})$ and hence $\kappa(k_{x})$ must have a period of
$2\pi/(\alpha+\beta)$ i.e. $\kappa\left(k_{x}+\frac{2\pi
m}{\alpha+\beta}\right)=\kappa(k_{x})$ with $k_{x}\in[0,2\pi/(\alpha+\beta)]$,
$m$ labeling the branch. It can be verified by direct substitution that with
the ansatz (Fig. 1(a)) Jiang and Lee (2022a)

$\kappa(k_{x})=\frac{1}{\alpha+\beta}\log\left(\frac{A\sin(\alpha
k_{x})}{B\sin(\beta k_{x})}\right),$ (5)

the red branch with the smallest value of $\kappa(k_{x})$, and hence the
slowest spatial decay gives $E_{\text{OBC}}(k_{x})=e^{\frac{2\pi
im}{\alpha+\beta}}\sin[(\alpha+\beta)k_{x}]\times\sqrt[(\alpha+\beta)]{A^{\beta}B^{\alpha}/[\sin^{\alpha}(\alpha
k_{x})\,\sin^{\beta}(\beta k_{x})]}$, which collapses into a line segment in
the complex energy plane. This implies that any $E_{\text{OBC}}$ must be
visited by two $k_{x}$ values, hence satisfying the requirement of double
degeneracy.

Importantly, $\kappa(k_{x})$ as given by Eq. (5) generically exhibits kinks
where different branches of the smallest $\kappa(k_{x})$ join (except when
$\alpha=\beta=1$), since that is where the phase of $E_{\text{OBC}}$ jumps
discontinuously Li  _et al._ (2020b). In generic models, there would be
multiple solution branches of $\kappa(k_{x})$, leading to inevitable jumps in
the definition of the complex momentum. Physically, these singularities arise
from the competition between the non-local pumping from asymmetric couplings
of different ranges.

Discontinuous Berry curvature.– We next discuss how non-analyticities in the
bulk momentum description can give rise to jumps in the band geometry and
topology. For a square lattice, the effective 2D deformed momentum is
$\bar{\mathbb{k}}=(k_{x}+i\kappa_{x}(\mathbb{k}),k_{y}+i\kappa_{y}(\mathbb{k}))$,
with $\kappa_{x},\kappa_{y}$ in general dependent on both momentum components
$\mathbb{k}=(k_{x},k_{y})$. Consider a two-band 2D Hamiltonian

$\displaystyle{\cal H}(\mathbb{k})={\cal
H}(k_{x},k_{y})=d_{x}\sigma_{x}+d_{y}\sigma_{y}+d_{z}\sigma_{z}+d_{0}\mathbb{I},$
(6)

with the Pauli matrices $\sigma_{x,y,z}$ in pseudospin space. Under OBCs in
both $x,y$ directions, the complex-deformed biorthogonal Berry curvature,
computed with the occupied left and right momentum eigenstates
$\langle\psi_{L}|$ and $|\psi_{R}\rangle$, is

$\displaystyle\bar{\Omega}_{xy}(\mathbb{k})$
$\displaystyle=\text{Im}\left[\langle\partial_{k_{x}}\psi_{L}(\bar{\mathbb{k}})|\partial_{k_{y}}\psi_{R}(\bar{\mathbb{k}})\rangle\right]$
(7) $\displaystyle=\frac{i}{2}\left[(\partial_{k_{x}}D)(\partial_{k_{y}}{\rm
In}R)-(\partial_{k_{y}}D)(\partial_{k_{x}}{\rm In}R)\right],$ (8)

with
$D=d_{z}(\bar{\mathbb{k}})/\sqrt{d_{x}^{2}(\bar{\mathbb{k}})+d_{y}^{2}(\bar{\mathbb{k}})+d_{z}^{2}(\bar{\mathbb{k}})}$
and
$R=\sqrt{[d_{x}(\bar{\mathbb{k}})-id_{y}(\bar{\mathbb{k}})]/[d_{x}(\bar{\mathbb{k}})+id_{y}(\bar{\mathbb{k}})]}$
Sup . Do note that while the eigenstates $\langle\psi_{L}|$ and
$|\psi_{R}\rangle$ are evaluated at complex-deformed momenta
$\bar{\mathbb{k}}$, their derivatives $\partial_{k_{x}},\partial_{k_{y}}$ are
taken with respect to the physical momenta $\mathbb{k}$, different from
existing definitions Shen  _et al._ (2018); Liang and Huang (2013); Xu  _et
al._ (2017); Yokomizo and Murakami (2019); Lieu (2018). This is because while
$\bar{\mathbb{k}}$ compensates for the broken translation invariance from
boundary-accumulated pumped states, the derivatives
$\partial_{k_{x}},\partial_{k_{y}}$ must be taken with respect to the physical
momenta $\mathbb{k}$ in order to capture the response from conjugate physical
fields.

The kinks from $\kappa_{x}(\mathbb{k})$ and $\kappa_{y}(\mathbb{k})$
correspond to discontinuities in their gradients, and hence also the Berry
curvature Sup . Take, for instance, the contribution

$\displaystyle\frac{dD({\bar{k}_{x},\bar{k}_{y}})}{dk_{x}}=i\left[\frac{\partial
D({\bar{\mathbb{k}}})}{\partial\bar{k}_{x}}\frac{d\kappa_{x}(\mathbb{k})}{dk_{x}}\\!+\\!\frac{\partial
D({\bar{\mathbb{k}}})}{\partial\bar{k}_{y}}\frac{d\kappa_{y}(\mathbb{k})}{dk_{x}}\right].$
(9)

Even though $\partial D(\bar{\mathbb{k}})/\partial\bar{k}_{\mu}$, $\mu=x,y$
are in general continuous for gapped bands, they are multiplied by
$d\kappa_{\mu}(\mathbb{k})/dk_{x}$ gradients, which are discontinuous at the
kinks.

For concreteness, we specialize Eq. (6) to a model that can be feasibly
realized in an ultracold atomic lattice, as discussed later and in Section SI
of Sup :

$\displaystyle d_{x}=\Omega_{r}+2\Omega_{0}\sin\phi_{0}\sin k_{x},$ (10)
$\displaystyle d_{y}=2\Omega_{0}\cos\phi_{0}\sin k_{x},$ $\displaystyle
d_{z}=i\gamma-2t_{x}\cos k_{x}-2t_{y}\cos k_{y},$

where $t_{x}=\Omega_{0}$, $i\gamma$ originates from onsite loss under constant
energy shift Li  _et al._ (2019b); Lapp  _et al._ (2019); Ren  _et al._
(2022), and reciprocal couplings are between like pseudospins of neighboring
unit cells.

To derive its kinked spectral deformation, we first diagonalize ${\cal
H}(k_{x},k_{y})$ via $\text{Det}({\cal H}-E\,\mathbb{I})=0$ to obtain the
dispersion relation

$E^{2}(k_{x},k_{y})=t_{+}z+t_{-}z^{-1}+t_{0},$ (11)

where $z=e^{ik_{x}}$,
$t_{+}=-2it_{x}(\gamma+\Omega_{r}\sin\phi_{0}+2it_{y}\cos k_{y})$,
$t_{-}=-2it_{x}(\gamma-\Omega_{r}\sin\phi_{0}+2it_{y}\cos k_{y})$,
$t_{0}=\Omega_{r}^{2}-\gamma^{2}+4t_{x}^{2}+4t_{y}^{2}\cos^{2}k_{y}-4i\gamma
t_{y}\cos k_{y}$. Up to a constant $t_{0}$, Eq. (11) reduces to Eq. (4) with
$\alpha=\beta=1$, such that $\kappa_{x}$ is independent of $k_{x}$. As such,
under x-OBCs and y-PBCs (cylinder), the effective Hamiltonian is ${\cal
H}(\bar{k}_{x},k_{y})={\cal H}(k_{x}+i\kappa_{x}({\bf k}),k_{y})={\cal
H}(k_{x}+i\kappa_{x}(k_{y}),k_{y})$ where

$\displaystyle\kappa_{x}(k_{y})\\!=\\!\ln\sqrt{\frac{t_{+}}{t_{-}}}\\!=\\!\ln\sqrt{\frac{\gamma\\!+\\!\Omega_{r}\sin\phi_{0}\\!+\\!2it_{y}\cos
k_{y}}{\gamma\\!-\\!\Omega_{r}\sin\phi_{0}\\!+\\!2it_{y}\cos k_{y}}}.$ (12)

Band singularities occur at non-analytic $k_{y}=\pm\frac{\pi}{2}$ where
$\text{Im}[\kappa_{x}(k_{y})]$ for ${\cal H}(\bar{k}_{x},k_{y})$ vanishes.
Indeed, the real band surfaces $\text{Re}[E_{\text{OBC}}]$ fracture along
these lines (Fig. 1(b)), and the real Berry curvature
$\text{Re}[\bar{\Omega}_{xy}(\mathbb{k})]$ likewise exhibits discontinuities
(Fig. 1(c)). Due to the emergent non-locality, these discontinuities are
expected to appear also in other quantities characterizing the OBC band
geometry or topology Marzari and Vanderbilt (1997), such as the Fubini-Study
metric Fubini (1905); Wikipedia (2022); Neupert  _et al._ (2013); Lee and Qi
(2014); Lee  _et al._ (2017); Mera and Ozawa (2021).

Figure 2: Kinked semiclassical responses from Berry curvature discontinuities.
(a) Schematics of detecting kinked responses through semiclassical dynamics.
Two atomic clouds (green, blue) with zero initial momenta are respectively
subject to forces ${\bf F}_{\pm}=F_{0}{\bf e}_{y}\pm F_{x}{\bf e}_{x}$, and
consequently move along trajectories (dashed) related by reflection symmetry
about $x=0$. The Berry curvature response can be extracted from their
difference in $x$-center-of-mass $\Delta\overline{\mathbf{r}}_{x}$ (Eq. (14)).
(b) Due to non-Hermitian pumping, kinks appear in the
$\Delta\overline{\mathbf{r}}_{x}$ (solid black) of our cold-atom model Eq.
(10). They coincide with the Berry curvature discontinuities (dashed)
encountered during the evolution, which are driven by the force parameters
$F_{0}=1$, $F_{x}=1.5$. (c) Origin of the response kinks, as seen from the two
atomic cloud trajectories (green, blue) ${{\mathbf{k}}}_{\pm}(t)$. Evidently,
Berry curvature discontinuities (beige/brown interfaces) are encountered at
$F_{0}t=\frac{\pi}{2},\frac{3\pi}{2},...$ (blue and green dots), and they
cause kinks to appear in $\Delta\overline{\mathbf{r}}_{x}$, as shown in (b).
Parameters are $\phi_{0}=\pi/4$, $\Omega_{r}=2$, $\Omega_{0}=1$, $\gamma=1.2$,
$t_{y}=0.2$, and $t_{x}=1$.

Response kinks from discontinuous Berry curvature.– We next demonstrate that
the Berry curvature discontinuities (Fig. 1(c)) are not just mathematical
artifacts but are in fact experimentally detectable via semiclassical
wavepacket dynamics. In Hermitian systems, a wavepacket can always be
decomposed into real momentum eigenstates obeying well-known semiclassical
equations of motion. Such a picture remains valid in the presence of non-
Hermitian pumping if the canonical momentum is replaced by the complex
$\kappa$-deformed momentum. The velocity of a wavepacket in a generic bounded
2D non-Hermitian system evolves as Longhi (2009, 2015); Xu  _et al._ (2017):

$\displaystyle\overline{\mathbf{v}}(t)\\!=\\!\frac{d\bar{\bf
r}}{dt}\\!=\\!\frac{1}{\hbar}\frac{\partial{\rm Re}[E(\bar{\bf
k})]}{\partial{\bf k}}\\!-\\!\frac{1}{\hbar}({\bf F}\times{\bf e}_{z}){\rm
Re}[\bar{\Omega}_{xy}(\bar{\bf k})],$ (13)

where $\bar{\bf r}$ describes its center of mass, and ${\bf
F}(t)=(F_{x}(t),F_{y}(t))$ is an externally applied force. The first term
$\partial{\rm Re}[E(\bar{\bf k})]/\partial{\bf k}$ is the usual canonical
velocity, with $E(\bar{\bf k})$ being that of the lower occupied band, but
evaluated at the effective momentum $\bar{\bf k}$. The second term involving
${\rm Re}[\bar{\Omega}_{xy}(\bar{\bf k})]$ is the anomalous velocity
contribution 333${\rm Im}[\bar{\Omega}_{xy}(\bar{\bf k})]$ only contributes
additional growth or decay to the wavepacket and is often small enough to be
neglected or eliminated by post-selection, as in our cold-atom setup
elaborated later Sup .

To isolate the Berry curvature response and extract signatures of its
discontinuities, we compare the semiclassical trajectories of two initially
stationary wavepackets subject to forces ${\bf F}_{\pm}=F_{y}(t){\bf e}_{y}\pm
F_{x}{\bf e}_{x}$ with opposite ${\bf e}_{x}$ components, as sketched in Fig.
2(a). These wavepackets acquire different effective momenta Price and Cooper
(2012) $\bar{\bf k}_{\pm}(t)={\bf k}_{\pm}(t)+i\kappa_{x}({\bf k}_{\pm}(t))$,
where ${\mathbf{k}}_{\pm}(t)=\hbar^{-1}\int_{0}^{t}{\bf F}_{\pm}dt^{\prime}$,
and thus propagate along two different trajectories. If we further assume an
even energy dispersion, such that $\frac{\partial{\rm
Re}[E(\overline{\mathbf{k}}_{+}(t^{\prime}))]}{\partial
k_{x}}+\frac{\partial{\rm
Re}[E(\overline{\mathbf{k}}_{-}(t^{\prime}))]}{\partial k_{x}}=0$ Sup , only
the Berry curvature contribution survives, and the two trajectories are
separated by an $x$-displacement 444We do not consider the $y$-displacement
since the two trajectories are designed to cancel off the non-Berry curvature
contributions in the $x$-displacement. of

$\displaystyle\Delta\overline{\mathbf{r}}_{x}(t)$
$\displaystyle\\!=\\!\int_{0}^{t}(\overline{\mathbf{v}}_{x}(t^{\prime})|_{{\bf
F}_{+}}+\overline{\mathbf{v}}_{x}(t^{\prime})|_{{\bf F}_{-}})dt^{\prime}$ (14)
$\displaystyle\\!=\\!-\frac{1}{\hbar}\\!\int_{0}^{t}\\!F_{y}(t^{\prime}{})\operatorname{Re}\\!\left[\bar{\Omega}_{xy}\left(\overline{\mathbf{k}}_{-}\left(t^{\prime}\right)\right)\\!+\\!\bar{\Omega}_{xy}\left(\overline{\mathbf{k}}_{+}\left(t^{\prime}\right)\right)\right]\\!dt^{\prime}.$

Plotted in Fig. 2(b) is the evolution of the center-of-mass difference
$\Delta\overline{\mathbf{r}}_{x}(t)$ (solid black) for our model Eq. (10): it
exhibits sharp kinks when discontinuities are encountered in the underlying
Berry curvatures ${\rm Re}[\bar{\Omega}_{xy}(\bar{\bf k}_{\pm}(t))]$
(green,blue) along the trajectories. This is concretely illustrated in Fig.
2(c), which shows the momentum-space trajectories ${\bf k}_{\pm}(t)$ (green,
blue) on the ${\rm Re}[\bar{\Omega}_{xy}({\bf k}_{\pm})]$ landscape from Fig.
1(c). With constant $F_{x}$ and $F_{y}(t)=F_{0}$ such that ${\bf k}_{\pm}(t)$
traces out the paths $k_{x}=\pm k_{y}$, Berry curvature discontinuities are
encountered by both trajectories whenever $k_{y}=F_{0}t=(n+1/2)\pi$. These
discontinuities exert sudden impulses that give rise to abrupt kinks in the
real-space trajectory, as captured by $\Delta\overline{\mathbf{r}}_{x}$. These
enigmatic kinks, which arise solely from the non-locality of non-Hermitian
pumping and not physical impulses, can be detected through the motion of atoms
in our designed experimental setup below.

Figure 3:  (a) Cold Rydberg atoms (cyan) are trapped in an optical lattice
with effective hoppings $t_{x},t_{y}$ created by the lasers described in
(b,c). They move according to Eq. (13), subject to a force ${\bf F}$ with $x$
and $y$-components respectively, arising from tilting and shaking the optical
lattice. (b) To realize our Hamiltonian in Eq. (10), various Raman lasers
generate effective onsite (blue) and nearest-neighbor (red) spin-orbit
couplings (SOCs) between $|\uparrow\rangle$ and $|\downarrow\rangle$ ground
states, mediated by excited states $|e_{2\downarrow}\rangle$ and
$|e_{1\uparrow}\rangle$. Other lasers partake in generating the optical
lattice potential (red, purple) and laser-induced loss (yellow) Sup . (c) 3D
configuration of the lasers in (b), with polarization directions indicated in
green. (d) Simulated $x$-displacement differences
$\Delta\overline{\mathbf{r}}_{x}(t)$ between two atomic clouds under opposite
$F_{x}$ tilts, which generically exhibit prominent response kinks and plateaus
without fine-tuning of parameters, such as the shaking frequency (shown here
for $\omega=0.82,0.83$ and $0.85t_{x}/\hbar$ where $t_{x}/\hbar=2\pi\times
900$ Hz). (e) Origin of response kinks: Momentum-space trajectories of
$\mathbf{k}_{\pm}(t)=\frac{1}{\hbar}\int_{0}^{t}{\bf
F}_{\pm}(t^{\prime})dt^{\prime}$ (purple,blue) for the
$\omega=0.85t_{x}/\hbar$ case (solid green solid curve) in (d). They
inevitably pass through regions of strongly discontinuous Berry curvature
(blue or red) at momenta $k_{y}a_{y}=\pm\pi/2$, which are encountered at times
$t=0.04$, $0.29$, and $0.36$ ms in this illustrative case. Other parameters
are $\phi_{0}=\pi/4$ Atala  _et al._ (2013), $\Omega_{r}=\gamma=t_{x}$,
$t_{y}=4.5t_{x}$, and $a_{x}=a_{y}=767$ nm, with external force amplitudes
being $F_{x}a_{x}/t_{x}=1$ and $F_{0}a_{y}/t_{x}=3.5$.

Detecting kinked responses with cold atoms.– We propose a cold atom setup
(Fig. 3(a,b,c)) for realizing the model Eq. (10) with predicted response
kinks. It requires atoms with two ground states (labeled as
$|\uparrow\rangle,|\downarrow\rangle$) and five suitably located excited
states (Fig. 3(b)), such as 87Rb whose detailed energy levels are elaborated
in the Supplement Sup . Several Raman lasers connect the two
$|\uparrow\rangle,|\downarrow\rangle$ ground states through various excited
states $|e_{1\uparrow}\rangle,|e_{2\downarrow}\rangle$. Laser fields ${\bf
E}_{1x}$ (red) and ${\bf E}_{3y}$ (purple) respectively generate the optical
lattice potential in the ${\bf e}_{x}$ and ${\bf e}_{y}$ directions, which
also determine the nearest-neighbor hopping amplitudes $t_{x},t_{y}$ in the
effective tight-binding description Sup . Pairs of Raman laser fields ${\bf
E}_{2x},{\bf E}^{\prime}_{2x}$ (blue) and ${\bf E}_{1x},{\bf E}_{1z}$ (red)
respectively induce effective on-site and nearest-neighbor spin-orbit coupling
terms $\Omega_{r}$ and $\Omega_{0}$, mediated by excited states
$|e_{2\downarrow}\rangle$ and $|e_{1\uparrow}\rangle$. A reciprocity-breaking
phase $\phi_{0}$ can be introduced through Ramam laser beams ${\bf E}_{1x}$
and ${\bf E}_{1z}$. Finally, non-Hermiticity is introduced through laser-
induced loss with rate $\gamma$ by depopulating atoms from the ground state
$|\downarrow\rangle$ to another excited state Li  _et al._ (2019b); Lapp  _et
al._ (2019); Ren  _et al._ (2022); Zhou  _et al._ (2022); Li and Yi (2023,
2022); Shen  _et al._ (2023). All in all, as detailed in Section SI of Sup ,
we arrive at the effective tight-binding Hamiltonian given by Eq. (10) in
momentum space, with the additional restriction $\Omega_{0}=t_{x}$ so that the
Berry curvature is discontinuous. Related experimental setups have
demonstrated considerable freedom in tuning $t_{x}/\hbar$, $t_{y}/\hbar$,
$\Omega_{0}/\hbar$, $\Omega_{r}/\hbar$, and $\gamma/\hbar$ within the range of
$2\pi\times[60,1000]$ Hz Fölling  _et al._ (2007); Lin  _et al._ (2011); Zhang
_et al._ (2012); Wang  _et al._ (2012); Cheuk  _et al._ (2012); Qu  _et al._
(2013); Atala  _et al._ (2013); Huang  _et al._ (2016); Meng  _et al._ (2016);
Wu  _et al._ (2016); Sun  _et al._ (2018); Li  _et al._ (2019b); Lapp  _et
al._ (2019); Liang  _et al._ (2022); Ren  _et al._ (2022); Zhou  _et al._
(2022). The required $x$-direction OBCs can be realized by lifting the optical
potential along the edges Zhou  _et al._ (2022).

To observe the response kinks, one can initialize Anderson  _et al._ (1995);
Davis  _et al._ (1995); Bradley  _et al._ (1995); Griffin  _et al._ (1996);
Fried  _et al._ (1998); Dalfovo  _et al._ (1999); Robert  _et al._ (2001);
Pereira Dos Santos  _et al._ (2001); Takasu  _et al._ (2003); Griesmaier  _et
al._ (2005); Pitaevskii and Stringari (2003); Pethick and Smith (2008); Regal
_et al._ (2004) two independent groups of atoms with zero momenta, and drive
the two atom clouds Aidelsburger  _et al._ (2015); Mancini  _et al._ (2015);
Goldman  _et al._ (2016); Cooper  _et al._ (2019) with different periodic
force fields ${\bf F}_{\pm}(t)=F_{0}\cos(\omega t){\bf e}_{y}\pm F_{x}{\bf
e}_{x}$. We choose an oscillatory $F_{y}(t)=F_{0}\cos(\omega t)$, which can be
implemented through shaking Parker  _et al._ (2013); Jotzu  _et al._ (2014);
Zhang and Zhou (2014); Zheng and Zhai (2014), to confine the atomic clouds to
a small sample region. The $F_{x}$ force component can be realized by tilting
the optical platform.

Through density snapshots Aidelsburger  _et al._ (2015); Goldman  _et al._
(2016), response kinks can be detected in the difference in the $x$-center-of-
mass $\Delta\overline{\mathbf{r}}_{x}(t)$ of the two clouds, as plotted in
Fig. 3(d) for an experimentally realistic scenario characterized by
$t_{x}/\hbar=2\pi\times 900$ Hz, forces $F_{x}=t_{x}/a_{x}$ and
$F_{0}=3.5t_{x}/a_{y}$, and lattice constants $a_{x}=a_{y}=767$ nm Su  _et
al._ (2023); Yang  _et al._ (2020b, 2017). $F_{x}$ can be gravitationally
introduced through a tilt angle [Fig. 3(a)] of
$\arcsin\left[t_{x}/(mga_{x})\right]$, which is $\approx 33^{\circ}$ for 87Rb
atoms of mass $m=1.45667\times 10^{-25}$ kg.

Importantly, since the response kinks arise inevitably whenever the semi-
classical trajectories cross Berry curvature discontinuities, they are
observable without fine-tuning, as evident in the approximately similar
responses from slightly different shaking frequencies $\omega$ [Fig. 3(d)], or
different initial atomic cloud momenta Sup . The kinks (abrupt jumps) in
$\Delta\overline{\mathbf{r}}_{x}(t)$ occur at times $t=0.04$, $0.29$ and
$0.36$ms, whenever $k_{y}(t)a_{y}$ crosses $\pm\pi/2$ and
$k_{x}(t)a_{x}/\pi=F_{x}a_{x}t/(\pi\hbar)=0.07$, $0.52$, and $0.65$, where
Berry curvature discontinuities (red arrows in Fig. 3(e)) occur. With response
kinks or plateaus occurring within the timescale of 1 ms, decoherence will not
be significant Li  _et al._ (2019b); Liang  _et al._ (2022).

Discussion. In this work, we uncovered intriguing non-Hermitian kinked
responses beyond the non-Hermitian skin effect, as captured by discontinuities
in the Berry curvature, or in fact, any band geometric quantity containing
momentum derivatives of the band eigenstates. The advantage of our
experimental proposal is the capacity to measure the signature of kinked
responses as prominent plateaus in short-time dynamics, which is tailored to
the state-of-the-art cold atom experimental platforms. Our proposal lays the
groundwork for future experimental demonstrations of other non-Hermitian
transport phenomena with ultracold atomic platforms.

Curiously, even though the discontinuous Berry curvature is phenomenologically
manifested in the semi-classical response of wavepackets, it does not affect
the Hall response of the entire band. Indeed, the Chern number $\bar{{\cal
C}}=-\frac{1}{2\pi}\int{\rm Re}(\bar{\Omega}_{xy})dk_{x}dk_{y}$ can be shown
to surprisingly remain quantized despite the violent deformations of the
Brillouin zone.

###### Acknowledgements.

Acknowledgments. We acknowledge helpful discussions with Haowei Li and Lihong
Zhou. Fang Qin is supported by the QEP2.0 grant from the Singapore National
Research Foundation (Grant No. NRF2021-QEP2-02-P09). All data and code of this
work are available from the corresponding authors upon reasonable request.

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## Supplementary Material for “Kinked linear response from non-Hermitian
pumping”

In this supplement, we provide the following:

SI. Elaboration of our proposal for realizing our non-Hermitian discontinuous
Berry curvature model in an optical lattice.

SII. Details of the singular behavior in the biorthogonal Berry curvature
induced by non-Hermitian pumping.

##  SI Effective non-Hermitian model for discontinuous Berry curvature and its
realization in an ultracold atomic setup

In the main text, to explore the kinked response from the non-Hermitian
pumping Yao and Wang (2018); Okuma  _et al._ (2020); Li  _et al._ (2020b, a),
we require a lattice model that exhibits discontinuous Berry curvature. Here,
we show how such an effective lattice model can result from a cold-atom
optical lattice setup described by the following Hamiltonian density

$\displaystyle\mathcal{H}(x,y)\\!=\\!\frac{\hat{\bf
p}^{2}}{2m}\\!+\\!V(x,y)\\!+\\![({\rm A}(x)\\!+\\!{\rm
B}(x))\sigma_{+}\\!+\\!{\rm h.c.}]\\!+\\!i\gamma\sigma_{z},$ (S1)

in a two-component basis
$(\left|\uparrow\right\rangle\\!,\\!\left|\downarrow\right\rangle)^{T}$, taken
from appropriate atomic states described below. Here $\hat{\bf
p}^{2}=\hat{p}_{x}^{2}+\hat{p}_{y}^{2}$ is the squared momentum operator,
${\rm A}(x)$ and ${\rm B}(x)$ are the spin-orbit couplings (SOCs) induced by
two sets of Raman lasers that generate the on-site and nearest-neighbor spin
flip, $\gamma$ is the effective decay rate of the spin-dependent on-site loss
achieved by laser, and $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$ with the
Pauli matrices $\sigma_{j}$ ($j=x,y,z$). The optical lattice potential is of
the form

$V(x,y)=-V_{x}\cos^{2}(\pi x/a_{x}+\phi_{1-})-V_{y}\cos^{2}(\pi
y/a_{y}+\phi_{3-}),$ (S2)

where $V_{x}$ and $V_{y}$ are the lattice depths, $a_{x}$ and $a_{y}$ are
lattice constants Wu  _et al._ (2016), and phases $\phi_{1-}$ and $\phi_{3-}$
are determined by the electric fields ${\bf E}_{1x}$ and ${\bf E}_{3y}$ that
will be described later in subsection D.

Below, we elaborate on the details of each aspect of the above experimental
setup.

###  SI.1 Raman-transition scheme for 87Rb

Figure S1: (a) Energy level structures for the relevant Raman transitions in
87Rb. Red (${\bf E_{1x}},{\bf E_{1z}}$) and Blue (${\bf E_{2x}},{\bf
E^{\prime}{}_{2x}}$) arrows denote laser transitions for effective spin-orbit
coupling (SOC) between the ground states, mediated by excited states. Red
(${\bf E_{1x}}$) and purple (${\bf E_{3y}}$) arrows generate the $x$\- and
$y$-direction optical lattice potentials. Another laser (yellow) gives rise to
depopulation losses from the $|\downarrow\rangle$ ground state. The chosen
hyperfine ground states are $|\downarrow\rangle=|F=1,m_{F}=-1\rangle$ and
$|\uparrow\rangle=|F=1,m_{F}=0\rangle$ for 52S1/2, and the chosen excited
states are $|e_{1\uparrow}\rangle=|F^{\prime}{}=1,m^{\prime}_{F}=0\rangle$,
$|e_{1\downarrow}\rangle=|F^{\prime}{}=1,m^{\prime}_{F}=-1\rangle$,
$|e_{3\uparrow}\rangle=|F^{\prime}{}=2,m^{\prime}_{F}=0\rangle$,
$|e_{3\downarrow}\rangle=|F^{\prime}{}=2,m^{\prime}_{F}=-1\rangle$ for 52P1/2,
and $|e_{2\downarrow}\rangle=|F^{\prime}{}=1,m^{\prime}_{F}=-1\rangle$ for
52P3/2. (b) Raman transitions giving rise to the nearest $x$-neighbor SOC with
amplitude $M_{0}\sin(k_{1}x)e^{i\phi_{0}}$ with
$M_{0}=-\hbar\Omega_{1x}\Omega_{1z}/\Delta_{0}$, which is mediated by the
$|e_{1\uparrow}\rangle=|F^{\prime}{}=1,m^{\prime}_{F}=0\rangle$ excited state.
(c) Raman transitions that give rise to the on-site effective SOC with
amplitude $M_{r}e^{i(2k_{2}x+\phi_{2})}$ with
$M_{r}=-\hbar\Omega_{2x}\Omega^{\prime}_{2x}/\Delta_{r}$, which is mediated by
the $|e_{2\downarrow}\rangle=|F^{\prime}{}=1,m^{\prime}_{F}=-1\rangle$ excited
state. (d) Laser-induced depopulation loss for the
$i\gamma_{\downarrow}\sigma_{z}$ term. We couple one of the ground states
$|\downarrow\rangle$ with an excited state
$|F^{\prime}{}=2,m^{\prime}{}_{F}=-2\rangle$ of 52P1/2. The yellow dashed
arrow line represents the laser-induced loss from Eq. (S9).

To engineer the model Eq. (S1) in a cold atom system, we propose to use the
energy levels of 87Rb atoms as a concrete realization Wu  _et al._ (2016); Lin
_et al._ (2011); Dumitrescu and Endlich (2007); Steck (2001); Grimm  _et al._
(2000). As shown in the Raman transition diagram in Fig. S1(a), there are two
hyperfine ground states
$\left|\uparrow\right\rangle=\left|F=1,m_{F}=0\right\rangle$ and
$\left|\downarrow\right\rangle=\left|F=1,m_{F}=-1\right\rangle$ from the
52S1/2 energy manifold. We can also make use of five additional hyperfine
excited states: four in the 52P1/2 manifold:
$\left|F^{\prime}=1,m^{\prime}_{F}=0\right\rangle$,
$\left|F^{\prime}=1,m^{\prime}_{F}=-1\right\rangle$,
$\left|F^{\prime}=2,m^{\prime}_{F}=0\right\rangle$,
$\left|F^{\prime}=2,m^{\prime}_{F}=-1\right\rangle$, and
$\left|F^{\prime}=1,m^{\prime}_{F}=-1\right\rangle$ in the 52P3/2 manifold.

###  SI.2 Two types of spin-orbit couplings

Here, we discuss the two types of spin-orbit couplings (SOCs) in our setup:

  1. 1.

Type 1: Effective nearest $x$-neighbor SOC $M_{0}$ (shown again in Fig. S1(b))

This is generated by two Raman lasers: a standing wave along the $x$ direction
with the optical-field vector ${\bf E}_{1x}$ (which also generates the optical
lattice $V_{x}$ along the $x$ direction), and a plane wave propagating along
the $z$ direction with the optical-field vector ${\bf E}_{1z}$, as given by

$\displaystyle{\bf E}_{1x}$
$\displaystyle\\!=\\!E_{1x}e^{i\phi_{1+}}[e^{i(k_{1}x+\phi_{1-})}\\!+\\!e^{-i(k_{1}x+\phi_{1-})}]{\bf
e}_{z}\\!=\\!2E_{1x}e^{i\phi_{1+}}\cos(k_{1}x+\phi_{1-}){\bf e}_{z},$ (S3)
$\displaystyle{\bf E}_{1z}$
$\displaystyle\\!=\\!2E_{1z}e^{ik_{1}z+i\phi_{1z}}{\bf e}_{x},$ (S4)

where ${\bf e}_{z}$ and ${\bf e}_{x}$ are the directions of the polarization,
$\phi_{1\pm}=(\phi_{1x}\pm\phi^{\prime}_{1x})/2$ with $\phi_{1x}$ and
$\phi^{\prime}_{1x}$ the phases of incident and reflected lights,
respectively. By fixing $\phi_{1-}=-\pi/2$ and choosing a frame such that
$z=0$, we have

$\displaystyle{\bf E}_{1x}$
$\displaystyle\\!=\\!2E_{1x}e^{i\phi_{1+}}\sin(k_{1}x){\bf e}_{z},$ (S5)
$\displaystyle{\bf E}_{1z}$ $\displaystyle\\!=\\!2E_{1z}e^{i\phi_{1z}}{\bf
e}_{x}.$ (S6)

The optical fields ${\bf E}_{1x}$ and ${\bf E}_{1z}$ give rise to transitions
between the ground states $|\uparrow\rangle,|\downarrow\rangle$ and the
excited state 52P${}_{1/2}|1,0\rangle$ (Fig. S1(a,b)), thereby leading to
effective SOC between pseudospins $|\uparrow\rangle$ and $|\downarrow\rangle$
mediated by this excited state. By adiabatic elimination, it can be shown that
Qin  _et al._ (2023b); Scully and Zubairy (1997); Peng and Li (1998) the SOC
amplitude is given by $M_{0}\sin(k_{1}x)e^{i\phi_{0}}$, where
$\phi_{0}=\phi_{1+}-\phi_{1z}$ and
$M_{0}=-\hbar\Omega_{1x}\Omega_{1z}/\Delta_{0}$ with
$|\Delta_{0}|\gg|\Omega_{1x}|,|\Omega_{1z}|$ . Here, $\Omega_{1x}$ and
$\Omega_{1z}$ are the Rabi frequencies for the respective transitions shown in
Fig. S1(b) and $\Delta_{0}$ is the single-photon detuning of the excited
state.

In general, the Rabi frequency $\Omega$ associated with the transition between
two arbitrary states $|a\rangle,|b\rangle$ is given by $\Omega={\bf d}_{\rm
eff}\cdot{\bf E}/\hbar$, where ${\bf E}$ is the electric field that drives the
transition and ${\bf d}_{\rm eff}\equiv\langle b|\hat{\bf
d}|a\rangle=-e\langle b|\hat{\bf r}|a\rangle$ is the effective dipole matrix
element with $\hat{\bf r}=(\hat{x},\hat{y},\hat{z})$ Wu  _et al._ (2016).

  2. 2.

Type 2: Effective on-site SOC $M_{r}$ (shown again in Fig. S1(c))

An additional SOC channel, which is on-site, is mediated by the excited state
52P${}_{3/2}|1,-1\rangle$ and produced by two plane-wave Raman lasers with
opposite wave vectors along the $x$ direction as

$\displaystyle{\bf E}_{2x}$
$\displaystyle\\!=\\!2E_{2x}e^{ik_{2}x+i\phi_{2x}}{\bf e}_{z},$ (S7)
$\displaystyle{\bf E}^{\prime}_{2x}$
$\displaystyle\\!=\\!2E^{\prime}_{2x}e^{-ik_{2}x+i\phi^{\prime}_{2x}}{\bf
e}_{y},$ (S8)

where ${\bf e}_{x}$ and ${\bf e}_{y}$ are the directions of the polarization.
As before, it can be shown with adiabatic elimination that the effective SOC
amplitude between pseudospins $|\uparrow\rangle,|\downarrow\rangle$ is
$M_{r}e^{i(2k_{2}x+\phi_{2})}$ with
$M_{r}=-\hbar\Omega_{2x}\Omega^{\prime}_{2x}/\Delta_{r}$,
$|\Delta_{r}|\gg|\Omega_{2x}|,|\Omega^{\prime}_{2x}|$ and
$\phi_{2}=\phi_{2x}-\phi^{\prime}_{2x}$. Here, $\Omega_{2x}$ and
$\Omega^{\prime}_{2x}$ are the respective Rabi frequencies and $\Delta_{r}$ is
the single-photon detuning of the excited state (illustrated in Fig. S1(c)).

Experimentally, the ratio $k_{2}/k_{1}$ can be conveniently tuned by adjusting
the propagating direction of ${\bf E}^{\prime}_{2x}$ relative to ${\bf
E}_{1x}$ Zhou  _et al._ (2022).

###  SI.3 Laser-induced loss

Besides SOCs, optical driving can give rise to laser-induced loss (in addition
to spontaneous decay) when the laser excites atoms to another state other than
the ground state pseudospins of interest. As shown in Fig. S1(d), a laser-
induced loss term can be generated by coupling the hyperfine ground state
$|\downarrow\rangle=|F=1,m_{F}=-1\rangle$ for 52S1/2 to an excited state such
as $|F^{\prime}{}=2,m^{\prime}{}_{F}=-2\rangle$ for 52P1/2 Li  _et al._
(2019b); Lapp  _et al._ (2019); Ren  _et al._ (2022). We can write it as

$\displaystyle\hat{H}_{\mathrm{loss}}=-i\gamma_{\downarrow}\left|\downarrow\right\rangle\left\langle\downarrow\right|=i\gamma\sigma_{z}-i\gamma(\left|\uparrow\right\rangle\left\langle\uparrow\right|+\left|\downarrow\right\rangle\left\langle\downarrow\right|),$
(S9)

where $\gamma_{\downarrow}=2\gamma$ is the effective decay rate. Here the
overall loss term
$-i\gamma(\left|\uparrow\right\rangle\left\langle\uparrow\right|+\left|\downarrow\right\rangle\left\langle\downarrow\right|)$
can be compensated by post-selection on $|\uparrow\rangle$ and
$|\downarrow\rangle$ during experimental measurements Zhou  _et al._ (2022);
Shen  _et al._ (2023). In our proposed experiment, $10^{5}$ atoms are
initialized in the system Liang  _et al._ (2022), and about $10^{4}$ atoms out
of them should survive through our proposed short-time dynamics. Thus, we
shall henceforth drop this overall loss term and keep only the
$i\gamma\sigma_{z}$ term.

###  SI.4 Optical lattice potential

As a 2D system, an optical lattice potential along both the $x$ and $y$
directions is needed. In the $x$-direction, the optical lattice potential
$V(x)$ is generated from the electric field ${\bf E}_{1x}$ [Eq. (S3)], which
also serves to generate the nearest-neighbor SOC. We have
$V(x)=-V_{x}\cos^{2}(k_{1}x+\phi_{1-})$ with lattice depth $V_{x}$, lattice
constant $a_{x}=\pi/k_{1}$, and the same phase offset $\phi_{1-}$ as in ${\bf
E}_{1x}$ [Eq. (S3)].

The optical lattice potential along the $y$ direction is generated by another
electric field

$\displaystyle{\bf E}_{3y}$
$\displaystyle\\!=\\!E_{3y}(e^{ik_{3}y+i\phi_{3y}}\\!+\\!e^{-ik_{3}y+i\phi^{\prime}_{3y}}){\bf
e}_{x}\\!=\\!E_{3y}e^{i\phi_{3+}}[e^{i(k_{3}y+\phi_{3-})}\\!+\\!e^{-i(k_{3}y+\phi_{3-})}]{\bf
e}_{x}\\!=\\!2E_{3y}e^{i\phi_{3+}}\cos(k_{3}y+\phi_{3-}){\bf e}_{x},$ (S10)

where ${\bf e}_{x}$ is the direction of the polarization,
$\phi_{3\pm}=(\phi_{3y}\pm\phi^{\prime}_{3y})/2$ with $\phi_{3y}$ and
$\phi^{\prime}_{3y}$ the phases of incident and reflected lights,
respectively. It generates the $y$-lattice potential
$V(y)=-V_{y}\cos^{2}(k_{3}y+\phi_{3-})$ with lattice depth $V_{y}$ and lattice
constant $a_{y}=\pi/k_{3}$ along $y$ direction, and a phase offset $\phi_{3-}$
from ${\bf E}_{3y}$.

In all, the optical lattice potential then takes the form Wu  _et al._ (2016)

$\displaystyle
V(x,y)=-V_{x}\cos^{2}(k_{1}x+\phi_{1-})-V_{y}\cos^{2}(k_{3}y+\phi_{3-}).$
(S11)

Its optical depths are given by
$V_{x}=-\hbar|\Omega_{1x}|^{2}/\Delta_{1\sigma}$
($\sigma=\uparrow,\downarrow$) and
$V_{y}=-\hbar|\Omega_{3y}|^{2}/\Delta_{2\sigma}$ with the detunings
$\Delta_{1\sigma}$, $\Delta_{2\sigma}$ and Rabi frequencies
$\Omega_{1x},\Omega_{3y}$ satisfying $|\Delta_{1\sigma}|\gg|\Omega_{1x}|$, and
$|\Delta_{2\sigma}|\gg|\Omega_{3y}|$.

###  SI.5 Effective cold-atom model

Based on the above prescription for our proposed setup, the Hamiltonian
density for our two-component ($\left|\uparrow\right\rangle$ and
$\left|\downarrow\right\rangle$) atomic gas in a two-dimensional optical
lattice (along $x$ and $y$) is described by

$\displaystyle~{}~{}{\cal H}(x,y)=\frac{\hat{\bf
p}^{2}}{2m}\\!+\\!V(x,y)\\!+\\![{\rm A}(x)\sigma_{+}\\!+\\!{\rm
h.c.}]\\!+\\![{\rm B}(x)\sigma_{+}\\!+\\!{\rm h.c.}]\\!+\\!i\gamma\sigma_{z}$
$\displaystyle\\!=\\!\frac{\hat{\bf
p}^{2}}{2m}\\!+\\!V(x,y)\\!+\\!M_{0}\sin(k_{1}x)(e^{i\phi_{0}}\sigma_{+}\\!+\\!{\rm
h.c.})\\!+\\!M_{r}(e^{i\phi_{r}}\sigma_{+}\\!+\\!{\rm
h.c.})\\!+\\!i\gamma\sigma_{z}$ $\displaystyle\\!=\\!\frac{\hat{\bf
p}^{2}}{2m}\\!+\\!V(x,y)\\!+\\!M_{0}\sin(k_{1}x)(e^{i\phi_{0}}\sigma_{+}\\!+\\!{\rm
h.c.})\\!+\\!M_{r}(e^{i(2k_{2}x+\phi_{2})}\sigma_{+}\\!+\\!{\rm
h.c.})\\!+\\!i\gamma\sigma_{z}$ $\displaystyle\\!=\\!\frac{\hat{\bf
p}^{2}}{2m}\\!+\\!V(x,y)\\!+\\!M_{0}\sin(k_{1}x)(e^{i\phi_{0}}\sigma_{+}\\!+\\!e^{-i\phi_{0}}\sigma_{-})\\!+\\!M_{r}(e^{i(2k_{2}x+\phi_{2})}\sigma_{+}\\!+\\!e^{-i(2k_{2}x+\phi_{2})}\sigma_{-})\\!+\\!i\gamma\sigma_{z}$
$\displaystyle\\!=\\!\frac{\hat{\bf
p}^{2}}{2m}\\!+\\!V(x,y)\\!+\\!\frac{M_{0}\sin(k_{1}x)}{2}[(e^{i\phi_{0}}\\!+\\!e^{-i\phi_{0}})\sigma_{x}\\!+\\!i(e^{i\phi_{0}}\\!-\\!e^{-i\phi_{0}})\sigma_{y}]\\!+\\!i\gamma\sigma_{z}$
$\displaystyle~{}~{}\\!+\\!\frac{M_{r}}{2}[(e^{i(2k_{2}x+\phi_{2})}\\!+\\!e^{-i(2k_{2}x+\phi_{2})})\sigma_{x}\\!+\\!i(e^{i(2k_{2}x+\phi_{2})}\\!-\\!e^{-i(2k_{r}x+\phi_{2})})\sigma_{y}]$
$\displaystyle\\!=\\!\frac{\hat{\bf
p}^{2}}{2m}\\!+\\!V(x,y)\\!+\\!M_{0}\sin(k_{1}x)(\cos\phi_{0}\sigma_{x}\\!-\\!\sin\phi_{0}\sigma_{y})\\!+\\!M_{r}[\cos(2k_{2}x+\phi_{2})\sigma_{x}\\!-\\!\sin(2k_{2}x+\phi_{2})\sigma_{y}]\\!+\\!i\gamma\sigma_{z}$
$\displaystyle\\!=\\!\frac{\hat{\bf
p}^{2}}{2m}\\!+\\!V(x,y)\\!+\\![M_{0}\sin(k_{1}x)\cos\phi_{0}+M_{r}\cos(2k_{2}x+\phi_{2})]\sigma_{x}\\!-\\![M_{0}\sin(k_{1}x)\sin\phi_{0}\\!+\\!M_{r}\sin(2k_{r}x+\phi_{2})]\sigma_{y}\\!+\\!i\gamma\sigma_{z},$
(S12)

where the SOC terms are found to be ${\rm
A}(x)=M_{0}\sin(k_{1}x)e^{i\phi_{0}}$ and ${\rm B}(x)=M_{r}e^{i\phi_{r}}$ with
$\phi_{r}=2k_{2}x+\phi_{2}$. Here $\hat{\bf
p}^{2}=\hat{p}_{x}^{2}+\hat{p}_{y}^{2}$ is the squared momentum operator,
$V(x,y)=-V_{x}\cos^{2}(k_{1}x+\phi_{1-})-V_{y}\cos^{2}(k_{3}y+\phi_{3-})$ is
the optical lattice potential and $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$
with $\sigma_{j}$ ($j=x,y,z$) being Pauli matrices. The lattice constants
along $x$ and $y$ directions are $a_{x}=\pi/k_{1}$ and $a_{y}=\pi/k_{3}$,
respectively.

####  SI.5.1 Tight-binding model

To obtain the effective tight-binding model from the above Hamiltonian
density, we expand the field operator
$\hat{\psi}_{\sigma}(x,y)=\sum_{j_{x},j_{y}}W_{0}(x-a_{x}j_{x},y-a_{y}j_{y})\hat{c}_{j_{x},j_{y},\sigma}$,
where $\sigma=\uparrow,\downarrow$, $a_{x}$ and $a_{y}$ are the lattice
constants along $x$ and $y$ respectively, $W_{0}(x)$ is a maximally localized
Wannier function Marzari and Vanderbilt (1997); Lee and Qi (2014); Lee and Ye
(2015) of the lowest-band ($n=0$), and $j_{x}/j_{y}$ is the index of lattice
sites along $x/y$. In this way, the second-quantized single-particle
Hamiltonian can be reduced to the tight-banding model, with the following
parameters:

The nearest-neighbor-hopping terms are

$\displaystyle t_{x}$ $\displaystyle=-\int dx\int
dyW_{0}^{*}(x,y)\left[\frac{\hat{p}_{x}^{2}}{2m}\\!-\\!V_{x}\cos^{2}(k_{1}x+\phi_{1-})\right]W_{0}(x-a_{x},y),$
(S13) $\displaystyle t_{y}$ $\displaystyle=-\int dx\int
dyW_{0}^{*}(x,y)\left[\frac{\hat{p}_{y}^{2}}{2m}\\!-\\!V_{y}\cos^{2}(k_{3}y+\phi_{3-})\right]W_{0}(x,y-a_{y}).$
(S14)

The on-site spin-flip-hopping terms are

$\displaystyle t_{\uparrow\downarrow}^{j_{x}}$ $\displaystyle=\int dx\int
dyW_{0}^{*}(x-x_{j_{x}},y)M_{r}e^{i(2k_{2}x+\phi_{2})}W_{0}(x-x_{j_{x}},y)\equiv
e^{i\phi_{r}j_{x}}\Omega_{r},$ (S15) $\displaystyle
t_{\downarrow\uparrow}^{j_{x}}$ $\displaystyle=\int dx\int
dyW_{0}^{*}(x-x_{j_{x}},y)M_{r}e^{-i(2k_{2}x+\phi_{2})}W_{0}(x-x_{j_{x}},y)\equiv
e^{-i\phi_{r}j_{x}}\Omega_{r},$ (S16)

where we use $\phi_{r}=2k_{2}a_{x}+\phi_{2}=2\pi k_{2}/k_{1}+\phi_{2}$, and
$\Omega_{r}=M_{r}\int dx\int
dyW_{0}^{*}(x,y)M_{r}e^{i(2k_{2}x+\phi_{2})}W_{0}(x,y)$,
$M_{r}=-\hbar\Omega_{2x}\Omega^{\prime}_{2x}/\Delta_{r}$.

The nearest-neighbor spin-flip terms are

$\displaystyle t_{\uparrow\downarrow}^{j_{x},j_{x}+1}$ $\displaystyle=\int
dx\int
dyW_{0}^{*}(x-x_{j_{x}},y)[M_{0}\sin(k_{1}x)e^{i\phi_{0}}]W_{0}(x-x_{j_{x}+1},y)$
$\displaystyle=\int dx\int
dyW_{0}^{*}(x,y)[M_{0}\sin(k_{1}(x+x_{j_{x}}))e^{i\phi_{0}}]W_{0}(x-a_{x},y)$
$\displaystyle=e^{i\phi_{0}}(-1)^{j_{x}}\int dx\int
dyW_{0}^{*}(x,y)[M_{0}\sin(k_{1}x)]W_{0}(x-a_{x},y)\equiv
e^{i\phi_{0}}(-1)^{j_{x}}\Omega_{0},$ (S17) $\displaystyle
t_{\uparrow\downarrow}^{j_{x},j_{x}-1}$ $\displaystyle=\int dx\int
dyW_{0}^{*}(x-x_{j_{x}},y)[M_{0}\sin(k_{1}x)e^{i\phi_{0}}]W_{0}(x-x_{j_{x}-1},y)$
$\displaystyle=\int dx\int
dyW_{0}^{*}(x,y)[M_{0}\sin(k_{1}(x+x_{j_{x}}))e^{i\phi_{0}}]W_{0}(x+a_{x},y)$
$\displaystyle=e^{i\phi_{0}}(-1)^{j_{x}}\int dx\int
dyW_{0}^{*}(x,y)[M_{0}\sin(k_{1}x)]W_{0}(x+a_{x},y)$
$\displaystyle=e^{i\phi_{0}}(-1)^{j_{x}}\int dx\int
dyW_{0}^{*}(x-a_{x},y)[M_{0}\sin(k_{1}(x-a_{x}))]W_{0}(x,y)$
$\displaystyle=-e^{i\phi_{0}}(-1)^{j_{x}}\int dx\int
dyW_{0}^{*}(x-a_{x},y)[M_{0}\sin(k_{1}x)]W_{0}(x,y)\equiv-e^{i\phi_{0}}(-1)^{j_{x}}\Omega_{0}=-t_{\uparrow\downarrow}^{j_{x},j_{x}+1},$
(S18) $\displaystyle t_{\downarrow\uparrow}^{j_{x}+1,j_{x}}$
$\displaystyle=\int
dxW_{0}^{*}(x-x_{j_{x}+1},y)[M_{0}\sin(k_{1}x)e^{i\phi_{0}}]W_{0}(x-x_{j_{x}},y)$
$\displaystyle=\int dx\int
dyW_{0}^{*}(x,y)[M_{0}\sin(k_{1}(x+x_{j_{x}+1}))e^{i\phi_{0}}]W_{0}(x+a_{x},y)$
$\displaystyle=e^{i\phi_{0}}(-1)^{j_{x}+1}\int dx\int
dyW_{0}^{*}(x,y)[M_{0}\sin(k_{1}x)]W_{0}(x+a_{x},y)$
$\displaystyle=-e^{i\phi_{0}}(-1)^{j_{x}}\int dx\int
dyW_{0}^{*}(x-a_{x},y)[M_{0}\sin(k_{1}(x-a_{x}))]W_{0}(x,y)$
$\displaystyle=e^{i\phi_{0}}(-1)^{j_{x}}\int dx\int
dyW_{0}^{*}(x-a_{x},y)[M_{0}\sin(k_{1}x)]W_{0}(x,y)\equiv
e^{i\phi_{0}}(-1)^{j_{x}}\Omega_{0}=t_{\uparrow\downarrow}^{j_{x},j_{x}+1},$
(S19)

where $\Omega_{0}=\int dx\int
dyW_{0}^{*}(x,y)[M_{0}\sin(k_{1}x)]W_{0}(x-a_{x},y)=\int dx\int
dyW_{0}^{*}(x-a_{x},y)[M_{0}\sin(k_{1}x)]W_{0}(x,y)$,
$M_{0}=-\hbar\Omega_{1x}\Omega_{1z}/\Delta_{0}$, and we have used the
identities

$\displaystyle\sin(k_{1}(x+x_{j_{x}}))\\!=\\!\sin(k_{1}(x+a_{x}j_{x}))\\!=\\!\sin(k_{1}x+\pi
j_{x})\\!=\\!(-1)^{j_{x}}\sin(k_{1}x),$ (S20)
$\displaystyle\sin(k_{1}(x+x_{j_{x}+1}))\\!=\\!\sin(k_{1}(x+a_{x}(j_{x}+1)))\\!=\\!\sin(k_{1}x+\pi(j_{x}+1))\\!=\\!(-1)^{j_{x}+1}\sin(k_{1}x))\\!=\\!-(-1)^{j_{x}}\sin(k_{1}x),$
(S21)
$\displaystyle\sin(k_{1}(x-a_{x}))\\!=\\!\sin(k_{1}x-\pi)\\!=\\!-\sin(k_{1}x).$
(S22)

The above overlap integrals allow us to write down the effective Hamiltonian
in the following tight-binding form:

$\displaystyle\hat{H}_{\rm tb}$
$\displaystyle=-t_{x}\sum_{j_{x},j_{y}}(\hat{c}_{j_{x},j_{y},\uparrow}^{\dagger}\hat{c}_{j_{x}+1,j_{y},\uparrow}+\hat{c}_{j_{x},j_{y},\downarrow}^{\dagger}\hat{c}_{j_{x}+1,j_{y},\downarrow}+{\rm
h.c.})-t_{y}\sum_{j_{x},j_{y}}(\hat{c}_{j_{x},j_{y},\uparrow}^{\dagger}\hat{c}_{j_{x},j_{y}+1,\uparrow}+\hat{c}_{j_{x},j_{y},\downarrow}^{\dagger}\hat{c}_{j_{x},j_{y}+1,\downarrow}+{\rm
h.c.})$
$\displaystyle~{}~{}+\Omega_{0}\sum_{j_{x},j_{y}}[e^{i\phi_{0}}(-1)^{j_{x}}(\hat{c}_{j_{x},j_{y},\uparrow}^{\dagger}\hat{c}_{j_{x}+1,j_{y},\downarrow}+\hat{c}_{j_{x}+1,j_{y},\uparrow}^{\dagger}\hat{c}_{j_{x},j_{y},\downarrow})+{\rm
h.c.}]$
$\displaystyle~{}~{}+\Omega_{r}\sum_{j_{x},j_{y}}(e^{i\phi_{r}j_{x}}\hat{c}_{j_{x},j_{y},\uparrow}^{\dagger}\hat{c}_{j_{x},j_{y},\downarrow}+{\rm
h.c.})+i\gamma\sum_{j_{x},j_{y}}(\hat{c}_{j_{x},j_{y},\uparrow}^{\dagger}\hat{c}_{j_{x},j_{y},\uparrow}-\hat{c}_{j_{x},j_{y},\downarrow}^{\dagger}\hat{c}_{j_{x},j_{y},\downarrow}).$
(S23)

Redefining the spin-down operator $\hat{c}_{j_{x},j_{y},\downarrow}\rightarrow
e^{i\pi(x_{j_{x}}/a_{x}+y_{j_{y}}/a_{y})}\hat{c}_{j_{x},j_{y},\downarrow}=(-1)^{j_{x}+j_{y}}\hat{c}_{j_{x},j_{y},\downarrow}$
Wu  _et al._ (2016); Liu  _et al._ (2013, 2014); Pan  _et al._ (2015, 2016);
Zhou  _et al._ (2022); Wang  _et al._ (2018); Sun  _et al._ (2018); Liang  _et
al._ (2023) with $x_{j_{x}}=a_{x}j_{x}$, $y_{j_{y}}=a_{y}j_{y}$, and
$\phi_{r}=\pi$ (i.e., performing a gauge transformation), we recast the
Hamiltonian into

$\displaystyle\hat{H}_{\rm tb}$
$\displaystyle=-t_{x}\sum_{j_{x},j_{y}}(\hat{c}_{j_{x},j_{y},\uparrow}^{\dagger}\hat{c}_{j_{x}+1,j_{y},\uparrow}-\hat{c}_{j_{x},j_{y},\downarrow}^{\dagger}\hat{c}_{j_{x}+1,j_{y},\downarrow}+{\rm
h.c.})-t_{y}\sum_{j_{x},j_{y}}(\hat{c}_{j_{x},j_{y},\uparrow}^{\dagger}\hat{c}_{j_{x},j_{y}+1,\uparrow}-\hat{c}_{j_{x},j_{y},\downarrow}^{\dagger}\hat{c}_{j_{x},j_{y}+1,\downarrow}+{\rm
h.c.})$
$\displaystyle~{}~{}+\Omega_{0}\sum_{j_{x},j_{y}}[-e^{i\phi_{0}}(\hat{c}_{j_{x},j_{y},\uparrow}^{\dagger}\hat{c}_{j_{x}+1,j_{y},\downarrow}-\hat{c}_{j_{x}+1,j_{y},\uparrow}^{\dagger}\hat{c}_{j_{x},j_{y},\downarrow})+{\rm
h.c.}]$
$\displaystyle~{}~{}+\Omega_{r}\sum_{j_{x},j_{y}}(\hat{c}_{j_{x},j_{y},\uparrow}^{\dagger}\hat{c}_{j_{x},j_{y},\downarrow}+{\rm
h.c.})+i\gamma\sum_{j_{x},j_{y}}(\hat{c}_{j_{x},j_{y},\uparrow}^{\dagger}\hat{c}_{j_{x},j_{y},\uparrow}-\hat{c}_{j_{x},j_{y},\downarrow}^{\dagger}\hat{c}_{j_{x},j_{y},\downarrow}).$
(S24)

####  SI.5.2 Effective Hamiltonian in momentum space

Performing a 2D Fourier transformation, one obtains

$\displaystyle\hat{c}_{j_{x},j_{y},\sigma}$
$\displaystyle=\sum_{k_{x},k_{y}}e^{i(k_{x}a_{x}j_{x}+k_{y}a_{y}j_{y})}\hat{c}_{k_{x},k_{y},\sigma},$
(S25) $\displaystyle\hat{c}_{j_{x},j_{y},\sigma}^{\dagger}$
$\displaystyle=\sum_{k_{x},k_{y}}e^{-i(k_{x}a_{x}j_{x}+k_{y}a_{y}j_{y})}\hat{c}_{k_{x},k_{y},\sigma}^{\dagger}.$
(S26)

Therefore, our effective Hamiltonian in momentum space is given by

$\displaystyle\hat{H}_{\rm eff}$
$\displaystyle=\sum_{k_{x},k_{y}}\left[-t_{x}e^{ik_{x}a_{x}}\hat{c}_{k_{x},k_{y},\uparrow}^{\dagger}\hat{c}_{k_{x},k_{y},\uparrow}+t_{x}e^{ik_{x}a_{x}}\hat{c}_{k_{x},k_{y},\downarrow}^{\dagger}\hat{c}_{k_{x},k_{y},\downarrow}-t_{y}e^{ik_{y}a_{y}}\hat{c}_{k_{x},k_{y},\uparrow}^{\dagger}\hat{c}_{k_{x},k_{y},\uparrow}+t_{y}e^{ik_{y}a_{y}}\hat{c}_{k_{x},k_{y},\downarrow}^{\dagger}\hat{c}_{k_{x},k_{y},\downarrow}\right.$
$\displaystyle\left.~{}~{}-\Omega_{0}e^{i\phi_{0}}(e^{ik_{x}a_{x}}-e^{-ik_{x}a_{x}})\hat{c}_{k_{x},k_{y},\uparrow}^{\dagger}\hat{c}_{k_{x},k_{y},\downarrow}+\Omega_{r}\hat{c}_{k_{x},k_{y},\uparrow}^{\dagger}\hat{c}_{k_{x},k_{y},\downarrow}\right]+{\rm
h.c.}+i\gamma\sum_{k_{x},k_{y}}(\hat{c}_{k_{x},k_{y},\uparrow}^{\dagger}\hat{c}_{k_{x},k_{y},\uparrow}-\hat{c}_{k_{x},k_{y},\downarrow}^{\dagger}\hat{c}_{k_{x},k_{y},\downarrow})$
$\displaystyle=\sum_{k_{x},k_{y}}\left[(-t_{x}e^{ik_{x}a_{x}}-t_{y}e^{ik_{y}a_{y}}+{\rm
h.c}+i\gamma)\hat{c}_{k_{x},k_{y},\uparrow}^{\dagger}\hat{c}_{k_{x},k_{y},\uparrow}+(t_{x}e^{ik_{x}a_{x}}+t_{y}e^{ik_{y}a_{y}}+{\rm
h.c}-i\gamma)\hat{c}_{k_{x},k_{y},\downarrow}^{\dagger}\hat{c}_{k_{x},k_{y},\downarrow}\right.$
$\displaystyle\left.~{}~{}+[\Omega_{r}-i2\Omega_{0}e^{i\phi_{0}}\sin(k_{x}a_{x})]\hat{c}_{k_{x},k_{y},\uparrow}^{\dagger}\hat{c}_{k_{x},k_{y},\downarrow}+[\Omega_{r}+i2\Omega_{0}e^{-i\phi_{0}}\sin(k_{x}a_{x})]\hat{c}_{k_{x},k_{y},\downarrow}^{\dagger}\hat{c}_{k_{x},k_{y},\uparrow}\right]$
(S27)
$\displaystyle\\!=\\!\sum_{k_{x},k_{y}}(\hat{c}_{k_{x},k_{y},\uparrow}^{\dagger},\hat{c}_{k_{x},k_{y},\downarrow}^{\dagger})$
$\displaystyle\times\\!\begin{pmatrix}\\!-t_{x}(e^{ik_{x}a_{x}}+e^{-ik_{x}a_{x}})-t_{y}(e^{ik_{y}a_{y}}+e^{-ik_{y}a_{y}})+i\gamma&\Omega_{r}-i2\Omega_{0}e^{i\phi_{0}}\sin(k_{x}a_{x})\\\
\Omega_{r}+i2\Omega_{0}e^{-i\phi_{0}}\sin(k_{x}a_{x})&t_{x}(e^{ik_{x}a_{x}}+e^{-ik_{x}a_{x}})+t_{y}(e^{ik_{y}a_{y}}+e^{-ik_{y}a_{y}})-i\gamma\\!\end{pmatrix}\\!\\!\begin{pmatrix}\\!\hat{c}_{k_{x},k_{y},\uparrow}\\!\\\
\\!\hat{c}_{k_{x},k_{y},\downarrow}\\!\end{pmatrix}\\!$
$\displaystyle\\!=\\!\sum_{k_{x},k_{y}}(\hat{c}_{k_{x},k_{y},\uparrow}^{\dagger},\hat{c}_{k_{x},k_{y},\downarrow}^{\dagger})$
$\displaystyle\times\\!\begin{pmatrix}\\!-2t_{x}\cos(k_{x}a_{x})-2t_{y}\cos(k_{y}a_{y})+i\gamma&\Omega_{r}+2\Omega_{0}\sin\phi_{0}\sin(k_{x}a_{x})-i2\Omega_{0}\cos\phi_{0}\sin(k_{x}a_{x})\\\
\Omega_{r}+2\Omega_{0}\sin\phi_{0}\sin(k_{x}a_{x})+i2\Omega_{0}\cos\phi_{0}\sin(k_{x}a_{x})&2t_{x}\cos(k_{x}a_{x})+2t_{y}\cos(k_{y}a_{y})-i\gamma\\!\end{pmatrix}\\!\\!\begin{pmatrix}\\!\hat{c}_{k_{x},k_{y},\uparrow}\\!\\\
\\!\hat{c}_{k_{x},k_{y},\downarrow}\\!\end{pmatrix}\\!$
$\displaystyle\\!=\\!\sum_{k_{x},k_{y}}(\hat{c}_{k_{x},k_{y},\uparrow}^{\dagger},\hat{c}_{k_{x},k_{y},\downarrow}^{\dagger})\\!\begin{pmatrix}\\!i\gamma\\!-\\!2t_{x}\cos(k_{x}a_{x})\\!-\\!2t_{y}\cos(k_{y}a_{y})&\Omega_{r}-i2\Omega_{0}e^{i\phi_{0}}\sin(k_{x}a_{x})\\\
\Omega_{r}+i2\Omega_{0}e^{-i\phi_{0}}\sin(k_{x}a_{x})&\\!-\\!i\gamma\\!+\\!2t_{x}\cos(k_{x}a_{x})\\!+\\!2t_{y}\cos(k_{y}a_{y})\\!\end{pmatrix}\\!\\!\begin{pmatrix}\\!\hat{c}_{k_{x},k_{y},\uparrow}\\!\\\
\\!\hat{c}_{k_{x},k_{y},\downarrow}\\!\end{pmatrix}\\!,$ (S28)

where we have used

$\displaystyle\sum_{k^{\prime}_{x},k^{\prime}_{y}}\sum_{k_{x},k_{y}}e^{i(k_{x}-k^{\prime}_{x})a_{x}j_{x}}e^{i(k_{y}-k^{\prime}_{y})a_{y}j_{y}}=\sum_{k^{\prime}_{x},k^{\prime}_{y}}\delta_{k_{x},k^{\prime}_{x}}\delta_{k_{y},k^{\prime}_{y}}\rightarrow\sum_{k_{x},k_{y}}.$
(S29)

Furthermore, in the basis
$\hat{\Psi}_{k_{x},k_{y}}=(\hat{c}_{k_{x},k_{y},\uparrow},\hat{c}_{k_{x},k_{y},\downarrow})^{T}$,
our model Hamiltonian in momentum space can be written as

$\displaystyle\hat{\cal H}_{\rm eff}(k_{x},k_{y})$
$\displaystyle=\begin{pmatrix}i\gamma-2t_{x}\cos(k_{x}a_{x})-2t_{y}\cos(k_{y}a_{y})&\Omega_{r}-i2\Omega_{0}e^{i\phi_{0}}\sin(k_{x}a_{x})\\\
\Omega_{r}+i2\Omega_{0}e^{-i\phi_{0}}\sin(k_{x}a_{x})&-i\gamma+2t_{x}\cos(k_{x}a_{x})+2t_{y}\cos(k_{y}a_{y})\end{pmatrix}$
(S30) $\displaystyle=\begin{pmatrix}i\gamma-
t_{x}(z+z^{-1})-2t_{y}\cos(k_{y}a_{y})&\Omega_{r}-\Omega_{0}e^{i\phi_{0}}(z-z^{-1})\\\
\Omega_{r}+\Omega_{0}e^{-i\phi_{0}}(z-z^{-1})&-i\gamma+t_{x}(z+z^{-1})+2t_{y}\cos(k_{y}a_{y})\end{pmatrix}$
$\displaystyle=d_{x}\sigma_{x}+d_{y}\sigma_{y}+d_{z}\sigma_{z},$ (S31)

where $z=e^{ik_{x}a_{x}}$, and we have used
$\cos(k_{x}a_{x})=\frac{e^{ik_{x}a_{x}}+e^{-ik_{x}a_{x}}}{2}$,
$\sin(k_{x}a_{x})=\frac{e^{ik_{x}a_{x}}-e^{-ik_{x}a_{x}}}{2i}$,

$\displaystyle d_{x}$
$\displaystyle=\Omega_{r}+2\Omega_{0}\sin\phi_{0}\sin(k_{x}a_{x})=\Omega_{r}-i\Omega_{0}\sin\phi_{0}(z-z^{-1}),$
(S32) $\displaystyle d_{y}$
$\displaystyle=2\Omega_{0}\cos\phi_{0}\sin(k_{x}a_{x})=-i\Omega_{0}\cos\phi_{0}(z-z^{-1}),$
$\displaystyle d_{z}$
$\displaystyle=i\gamma-2t_{x}\cos(k_{x}a_{x})-2t_{y}\cos(k_{y}a_{y})=i\gamma-
t_{x}(z+z^{-1})-2t_{y}\cos(k_{y}a_{y}).$

Therefore, the eigenenergies take the form of

$\displaystyle E^{2}(k_{x},k_{y})$
$\displaystyle=d_{x}^{2}+d_{y}^{2}+d_{z}^{2}$ (S33)
$\displaystyle=[\Omega_{r}-i\Omega_{0}\sin\phi_{0}(z-z^{-1})]^{2}+[-i\Omega_{0}\cos\phi_{0}(z-z^{-1})]^{2}+[i\gamma-
t_{x}(z+z^{-1})-2t_{y}\cos(k_{y}a_{y})]^{2}$
$\displaystyle=t_{x}^{2}(z+z^{-1})^{2}-\Omega_{0}^{2}(z-z^{-1})^{2}-i2\Omega_{r}\Omega_{0}\sin\phi_{0}(z-z^{-1})-2t_{x}(i\gamma-2t_{y}\cos(k_{y}a_{y}))(z+z^{-1})$
$\displaystyle~{}~{}+\Omega_{r}^{2}-\gamma^{2}+4t_{y}^{2}\cos^{2}(k_{y}a_{y})-i4\gamma
t_{y}\cos(k_{y}a_{y}).$ (S34)

By enforcing $t_{x}=\Omega_{0}$ (through adjustment of the physical
parameters) in order to obtain our desired model, we have

$\displaystyle E^{2}(k_{x},k_{y})$
$\displaystyle=-2t_{x}(i\Omega_{r}\sin\phi_{0}+i\gamma-2t_{y}\cos
k_{y})z-2t_{x}(-i\Omega_{r}\sin\phi_{0}+i\gamma-2t_{y}\cos k_{y})z^{-1}$
$\displaystyle~{}+\Omega_{r}^{2}-\gamma^{2}+4t_{x}^{2}+4t_{y}^{2}\cos^{2}(k_{y}a_{y})-i4\gamma
t_{y}\cos(k_{y}a_{y}),$ (S35) $\displaystyle=t_{+}z+t_{-}z^{-1}+t_{0},$ (S36)

where

$\displaystyle t_{+}$
$\displaystyle=-2t_{x}(i\Omega_{r}\sin\phi_{0}+i\gamma-2t_{y}\cos(k_{y}a_{y}))=-i2t_{x}(\Omega_{r}\sin\phi_{0}+\gamma+i2t_{y}\cos(k_{y}a_{y})),$
(S37) $\displaystyle t_{-}$
$\displaystyle=-2t_{x}(-i\Omega_{r}\sin\phi_{0}+i\gamma-2t_{y}\cos(k_{y}a_{y}))=-i2t_{x}(-\Omega_{r}\sin\phi_{0}+\gamma+i2t_{y}\cos(k_{y}a_{y})),$
(S38) $\displaystyle t_{0}$
$\displaystyle=\Omega_{r}^{2}-\gamma^{2}+4t_{x}^{2}+4t_{y}^{2}\cos^{2}(k_{y}a_{y})-i4\gamma
t_{y}\cos(k_{y}a_{y}).$ (S39)

If we define the dimensionless wave numbers as $k_{x}\leftarrow k_{x}a_{x}$
and $k_{y}\leftarrow k_{y}a_{y}$ for simplicity, we can rewrite Eq. (S32) as

$\displaystyle d_{x}=\Omega_{r}+2\Omega_{0}\sin\phi_{0}\sin k_{x},$ (S40)
$\displaystyle d_{y}=2\Omega_{0}\cos\phi_{0}\sin k_{x},$ $\displaystyle
d_{z}=i\gamma-2t_{x}\cos k_{x}-2t_{y}\cos k_{y},$

which is consistent with Eq. (10) of the main text.

##  SII Band singularities from non-Hermitian pumping

###  SII.1 Complex deformation from non-Hermitian pumping

We consider cylindrical boundary conditions for our Hamiltonian of Eq. (S31)
with OBCs in the $\hat{x}$-direction and PBCs in the $\hat{y}$-direction. Here
we set the dimensionless crystal momenta as $k_{x}\leftarrow k_{x}a_{x}$ and
$k_{y}\leftarrow k_{y}a_{y}$ for simplicity, and rewrite Eq. (S30) as follows:

$\hat{\cal H}_{\rm
eff}(k_{x},k_{y})=\begin{pmatrix}i\gamma-2t_{x}\cos(k_{x})-2t_{y}\cos(k_{y})&\Omega_{r}-i2\Omega_{0}e^{i\phi_{0}}\sin(k_{x})\\\
\Omega_{r}+i2\Omega_{0}e^{-i\phi_{0}}\sin(k_{x})&-i\gamma+2t_{x}\cos(k_{x})+2t_{y}\cos(k_{y})\end{pmatrix}.$
(S41)

As discussed in the main text, non-Hermitian pumping in the $x$-OBC direction
leads to the complex deformation $k_{x}\rightarrow k_{x}+i\kappa_{x}({\bf k})$
Yao and Wang (2018); Okuma  _et al._ (2020); Li  _et al._ (2020b, a)

$\displaystyle\kappa_{x}(k_{y})=\ln\sqrt{\frac{t_{+}}{t_{-}}}=\ln\sqrt{\frac{\gamma+\Omega_{r}\sin\phi_{0}+i2t_{y}\cos(k_{y})}{\gamma-\Omega_{r}\sin\phi_{0}+i2t_{y}\cos(k_{y})}}.$
(S42)

By specializing to $\phi_{0}=\pi/4$,

$\displaystyle\kappa_{x}(k_{y})=\ln\sqrt{\frac{\gamma+2it_{y}\cos\left(k_{y}\right)+\frac{\Omega_{r}}{\sqrt{2}}}{\gamma+2it_{y}\cos\left(k_{y}\right)-\frac{\Omega_{r}}{\sqrt{2}}}}.$
(S43)

Under the complex deformation $k_{x}\rightarrow k_{x}+i\kappa_{x}({\bf k})$,
with parameters specialized to $t_{x}=\Omega_{0}$ and $\phi_{0}=\pi/4$, the
eigenenergies of Eq. (S41) take the form

$\displaystyle\bar{E}^{2}(k_{x}+i\kappa_{x}(k_{y}),k_{y})=\Omega_{r}^{2}-\gamma^{2}+4t_{x}^{2}+4t_{y}^{2}\cos^{2}{k_{y}}-4i\gamma{t_{y}}\cos{k_{y}}$
$\displaystyle-4i\gamma
t_{x}\cos\left(k_{x}+\frac{1}{2}i\log\left(\frac{\gamma+2it_{y}\cos
k_{y}+\frac{\Omega_{r}}{\sqrt{2}}}{\gamma+2it_{y}\cos
k_{y}-\frac{\Omega_{r}}{\sqrt{2}}}\right)\right)+4t_{x}t_{y}\cos\left(k_{x}+\frac{1}{2}i\log\left(\frac{\gamma+2it_{y}\cos
k_{y}+\frac{\Omega_{r}}{\sqrt{2}}}{\gamma+2it_{y}\cos
k_{y}-\frac{\Omega_{r}}{\sqrt{2}}}\right)-k_{y}\right)$
$\displaystyle+4t_{x}t_{y}\cos\left(k_{x}+\frac{1}{2}i\log\left(\frac{\gamma+2it_{y}\cos
k_{y}+\frac{\Omega_{r}}{\sqrt{2}}}{\gamma+2it_{y}\cos
k_{y}-\frac{\Omega_{r}}{\sqrt{2}}}\right)+k_{y}\right)+2\sqrt{2}t_{x}\Omega_{r}\sin\left(k_{x}+\frac{1}{2}i\log\left(\frac{\gamma+2it_{y}\cos
k_{y}+\frac{\Omega_{r}}{\sqrt{2}}}{\gamma+2it_{y}\cos
k_{y}-\frac{\Omega_{r}}{\sqrt{2}}}\right)\right).$ (S44)

To express this as a Hatano-Nelson model along the $x$-direction with
$k_{y}$-valued coefficients, we make use of

$\displaystyle\sin\left(k_{x}+\frac{1}{2}i\log\left(\frac{\gamma+2it_{y}\cos
k_{y}+\frac{\Omega_{r}}{\sqrt{2}}}{\gamma+2it_{y}\cos
k_{y}-\frac{\Omega_{r}}{\sqrt{2}}}\right)\right)=\frac{e^{ik_{x}}}{2i\sqrt{\frac{\gamma+2it_{y}\cos
k_{y}+\frac{\Omega_{r}}{\sqrt{2}}}{\gamma+2it_{y}\cos
k_{y}-\frac{\Omega_{r}}{\sqrt{2}}}}}-\frac{e^{-ik_{x}}}{2i}\sqrt{\frac{\gamma+2it_{y}\cos
k_{y}+\frac{\Omega_{r}}{\sqrt{2}}}{\gamma+2it_{y}\cos
k_{y}-\frac{\Omega_{r}}{\sqrt{2}}}},$ (S45)
$\displaystyle\cos\left(k_{x}+\frac{1}{2}i\log\left(\frac{\gamma+2it_{y}\cos
k_{y}+\frac{\Omega_{r}}{\sqrt{2}}}{\gamma+2it_{y}\cos
k_{y}-\frac{\Omega_{r}}{\sqrt{2}}}\right)\right)=\frac{e^{ik_{x}}}{2\sqrt{\frac{\gamma+2it_{y}\cos
k_{y}+\frac{\Omega_{r}}{\sqrt{2}}}{\gamma+2it_{y}\cos
k_{y}-\frac{\Omega_{r}}{\sqrt{2}}}}}+\frac{e^{-ik_{x}}}{2}\sqrt{\frac{\gamma+2it_{y}\cos
k_{y}+\frac{\Omega_{r}}{\sqrt{2}}}{\gamma+2it_{y}\cos
k_{y}-\frac{\Omega_{r}}{\sqrt{2}}}}.$

With that, Eq. (S44) simplifies to

$\bar{E}^{2}(k_{x},k_{y})=\sqrt{t_{+}t_{-}}(e^{ik_{x}}+e^{-ik_{x}})+t_{0},$
(S46)

with

$\displaystyle t_{+}$
$\displaystyle=-2it_{x}\left(\frac{\Omega_{r}}{\sqrt{2}}+\gamma+2it_{y}\cos
k_{y}\right),$ (S47) $\displaystyle t_{-}$
$\displaystyle=-2it_{x}\left(-\frac{\Omega_{r}}{\sqrt{2}}+\gamma+2it_{y}\cos
k_{y}\right),$ (S48) $\displaystyle t_{0}$
$\displaystyle=\Omega_{r}^{2}-\gamma^{2}+4t_{x}^{2}+4t_{y}^{2}\cos^{2}k_{y}-i4\gamma
t_{y}\cos k_{y}.$ (S49)

This energy dispersion is manifestly even along the $\hat{k}_{x}$ and
$\hat{k}_{y}$ directions, as required by our response kink setup in the text.

Figure S2: Plot of the real part of $\kappa_{x}(k_{y})$. We set the parameters
as $\phi_{0}=\pi/4$, $\Omega_{0}/t_{x}=1$, $t_{y}/t_{x}=0.2$,
$\Omega_{r}/t_{x}=2$, $\gamma/t_{x}=1.2$, with $t_{x}$ as the energy unit.
Divergences appear at $k_{y}=\pm\frac{\pi}{2}$, where the deformation
$\kappa_{x}(k_{y})$ vanishes.

###  SII.2 Discontinuities of the Berry curvature

In order to compute the biorthogonal Berry curvature in a non-Hermitian
system, with or without non-Hermitian pumping, note that the left (bra) and
right (ket) eigenstates are not identical:

$\displaystyle\hat{\cal H}_{\rm eff}|\psi_{R}\rangle=E|\psi_{R}\rangle,$ (S50)
$\displaystyle\hat{\cal H}_{\rm
eff}^{\dagger}|\psi_{L}\rangle=E^{*}|\psi_{L}\rangle.$ (S51)

For any two-band Hamiltonian, such as our Hamiltonian in Eq. (S31), it can be
written as $\bar{\cal H}_{\rm
eff}=[(d_{+}+d_{-})/2]\sigma_{x}+[i(d_{+}-d_{-})/2]\sigma_{y}+d_{z}\sigma_{z},$
with $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$, such that we can perform
the following parametrization on the Bloch sphere:

$\displaystyle d_{x}$
$\displaystyle=d\sin\theta\cos\phi=d\sqrt{1-D^{2}}\cos\phi,$ (S52)
$\displaystyle d_{y}$
$\displaystyle=d\sin\theta\sin\phi=d\sqrt{1-D^{2}}\sin\phi,$ (S53)
$\displaystyle d_{z}$ $\displaystyle=d\cos\theta=dD,$ (S54) $\displaystyle d$
$\displaystyle=\sqrt{d_{x}^{2}+d_{y}^{2}+d_{z}^{2}},$ (S55) $\displaystyle D$
$\displaystyle=\frac{d_{z}}{\sqrt{d_{x}^{2}+d_{y}^{2}+d_{z}^{2}}},$ (S56)
$\displaystyle R$
$\displaystyle=\sqrt{\frac{d_{+}}{d_{-}}}=\sqrt{\frac{d_{x}-id_{y}}{d_{x}+id_{y}}}=\sqrt{\frac{e^{-i\phi}}{e^{i\phi}}}=e^{-i\phi}.$
(S57)

Hence, our Hamiltonian can be expressed as

$\displaystyle\bar{\cal H}_{\rm eff}$
$\displaystyle=d_{x}\sigma_{x}+d_{y}\sigma_{y}+d_{z}\sigma_{z}=\begin{pmatrix}d_{z}&d_{x}-id_{y}\\\
d_{x}+id_{y}&-d_{z}\end{pmatrix}=\begin{pmatrix}dD&d\sqrt{1-D^{2}}(\cos\phi-i\sin\phi)\\\
d\sqrt{1-D^{2}}(\cos\phi+i\sin\phi)&-dD\end{pmatrix}$
$\displaystyle=d\begin{pmatrix}D&\sqrt{1-D^{2}}e^{-i\phi}\\\
\sqrt{1-D^{2}}e^{i\phi}&-D\end{pmatrix}=d\begin{pmatrix}D&\sqrt{1-D^{2}}R\\\
\sqrt{1-D^{2}}R^{-1}&-D\end{pmatrix}.$ (S58)

We have the left and right eigenstates as

$\displaystyle|\psi_{R}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\pm
R\sqrt{1\pm D}\\\ \sqrt{1\mp D}\end{pmatrix},$ (S59)
$\displaystyle|\psi_{L}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\pm\sqrt{1\pm
D^{*}}/R^{*}\\\ \sqrt{1\mp D^{*}}\end{pmatrix},$ (S60)
$\displaystyle\langle\psi_{R}|=\frac{1}{\sqrt{2}}\begin{pmatrix}\pm
R^{*}\sqrt{1\pm D^{*}}&\sqrt{1\mp D^{*}}\end{pmatrix},$ (S61)
$\displaystyle\langle\psi_{L}|=\frac{1}{\sqrt{2}}\begin{pmatrix}\pm\sqrt{1\pm
D}/R&\sqrt{1\mp D}\end{pmatrix}.$ (S62)

Here, there are two choices for each bra and ket, and in principle, we can
compute four possible inequivalent Berry curvatures, although all four can be
shown to integrate to the same Chern number Shen  _et al._ (2018). We shall
choose the biorthogonal option with $\langle\psi_{L}|$ bra and
$|\psi_{R}\rangle$ ket, which is the most physical in terms of the
probabilistic interpretation of quantum mechanics Brody (2013). Doing so, we
obtain the biorthogonal Berry curvature as

$\displaystyle\Omega_{xy}$
$\displaystyle=i\left(\langle\partial_{k_{x}}\psi_{L}|\partial_{k_{y}}\psi_{R}\rangle-\langle\partial_{k_{y}}\psi_{L}|\partial_{k_{x}}\psi_{R}\rangle\right)$
$\displaystyle=\frac{i}{2}\left[\begin{pmatrix}\partial_{k_{x}}(\sqrt{1+D}/R)&\partial_{k_{x}}\sqrt{1-D}\end{pmatrix}\begin{pmatrix}\partial_{k_{y}}(R\sqrt{1+D})\\\
\partial_{k_{y}}\sqrt{1-D}\end{pmatrix}-\begin{pmatrix}\partial_{k_{y}}(\sqrt{1+D}/R)&\partial_{k_{y}}\sqrt{1-D}\end{pmatrix}\begin{pmatrix}\partial_{k_{x}}(R\sqrt{1+D})\\\
\partial_{k_{x}}\sqrt{1-D}\end{pmatrix}\right]$
$\displaystyle=\frac{i}{2R}\left[(\partial_{k_{x}}D)(\partial_{k_{y}}R)-(\partial_{k_{y}}D)(\partial_{k_{x}}R)\right]=\frac{i}{2}\left[(\partial_{k_{x}}D)(\partial_{k_{y}}\ln
R)-(\partial_{k_{y}}D)(\partial_{k_{x}}\ln R)\right],$ (S63)

where

$\displaystyle\partial_{k_{y}}D(\bar{k}_{x},k_{y})=\frac{d(D(\bar{k}_{x},k_{y}))}{dk_{y}}=\frac{\partial(D(\bar{k}_{x},k_{y}))}{\partial
k_{y}}+i\frac{\partial(D(\bar{k}_{x},k_{y}))}{\partial\bar{k}_{x}}\frac{d\kappa_{x}(k_{y})}{dk_{y}}.$
(S64)

In our cylindrical geometry under $x$-OBCs, non-Hermitian pumping occurs in
the $x$ direction, and the complex flux $\kappa_{x}(k_{y})$ contributes to the
derivative of $D(\bar{k}_{x},k_{y})$ and ${\rm In}R(\bar{k}_{x},k_{y})$.

###  SII.3 Robustness of the semiclassical trajectories

Figure S3: The $x$-displacement in Eq. (S65) (solid green lines). The kinked
responses appear when the wavepacket momentum
$k_{y}(t)=\frac{1}{\hbar}\int_{0}^{t}F_{0}\cos(\omega t^{\prime})dt^{\prime}$
(blue curves) approaches $\pm\pi/2$ under the external fields $F_{x}=7$ and
$F_{0}=4$ with $\omega=2$, as marked by the black lines. The parameters are:
$t_{x}=2$, $\Omega_{r}=2$, $t_{y}=6$, and $\gamma=0.5,1.0,2.0$, and $2.5$ for
(a)-(d).

Here, we show that the semiclassical response kinks remain very prominent even
if the initial wavepacket exhibits uncertainty in its momentum distribution.
As discussed in the main text, the $x$-displacements of two initially
stationary wavepackets driven by forces ${\bf F}_{\pm}(t)=F_{0}\cos(\omega
t){\bf e}_{y}\pm F_{x}{\bf e}_{x}$ are given as follows Longhi (2009, 2015);
Xu  _et al._ (2017):

$\displaystyle\Delta\overline{\mathbf{r}}_{x}(t)$
$\displaystyle=\int_{0}^{t}(\overline{\mathbf{v}}_{x}(t^{\prime})|_{{\bf
F}_{+}}+\overline{\mathbf{v}}_{x}(t^{\prime})|_{{\bf F}_{-}})dt^{\prime}$
(S65) $\displaystyle=-\frac{1}{\hbar}\int_{0}^{t}\left[\frac{\partial
E(\overline{\mathbf{k}}_{+}(t^{\prime}))}{\partial k_{x}}+\frac{\partial
E(\overline{\mathbf{k}}_{-}(t^{\prime}))}{\partial
k_{x}}\right]dt^{\prime}-\frac{1}{\hbar}\\!\int_{0}^{t}[\mathbf{F}_{\pm}(t^{\prime})\\!\times\\!\mathbf{e}_{z}]\operatorname{Re}\\!\left[\bar{\Omega}_{xy}\left(\overline{\mathbf{k}}_{-}\left(t^{\prime}\right)\right)\\!+\\!\bar{\Omega}_{xy}\left(\overline{\mathbf{k}}_{+}\left(t^{\prime}\right)\right)\right]\\!dt^{\prime}$
$\displaystyle=-\frac{1}{\hbar}\\!\int_{0}^{t}F_{y}(t)\operatorname{Re}\\!\left[\bar{\Omega}_{xy}\left(\overline{\mathbf{k}}_{-}\left(t^{\prime}\right)\right)\\!+\\!\bar{\Omega}_{xy}\left(\overline{\mathbf{k}}_{+}\left(t^{\prime}\right)\right)\right]\\!dt^{\prime},$

where $\mathbf{v}_{x}$ is the velocity along $x$, $\bar{\Omega}_{xy}({\bf k})$
is the Berry curvature of model Eq. (S41), $\bar{\bf k}_{\pm}(t)={\bf
k}_{\pm}(t)+i\kappa_{x}({\bf k}_{\pm}(t))$, and
$E(\overline{\mathbf{k}}_{\pm}(t^{\prime}))$ denotes the lower band of model
Eq. (S41). Here, the even energy dispersion of the band allows all non-Berry
curvature contributions to cancel, leaving only the response purely from the
Berry curvature. Thus, Berry curvature discontinuities from singular points of
$\kappa_{x}(k_{y})$ lead to unique responses in the form of kinked
$x$-displacement, as elaborated in the main text. Such kinked responses are
manifested as the abrupt jumps (dark green) in Fig. S3 as the $k_{y}$ momentum
approaches $\pm\pi/2$.

To test the robustness of such jumps or kinks, we slightly perturb the initial
momentum distribution of the wave packet from zero. Importantly, they are
shown to be robust: under the initial $k_{y}$ shifts $\delta k_{y}=0.05$ and
$\delta k_{y}=0.1$ in the wavepacket, we observe a broadened distribution of
$\Delta\overline{\mathbf{r}}_{x}$ (light green). But importantly, the abrupt
jump is still very much protected, even though the time when the momentum
reaches the Berry curvature discontinuity varies a little.

Figure S4: The $x$-displacement in Eq. (S65) (solid green lines) and the
perturbative results under $\delta k_{y}=0.05$ (yellow curves) and $0.1$ (red
curves) shifts in the initial wavepacket. Manifestly, the strongly kinked
response (largely abrupt jumps) remains robust. The model is Eq. (S41), with
parameters $t_{x}=2$, $\Omega_{r}=2$, $\gamma=2$, and $t_{y}=6,7,8$, and $9$
for (a)-(d). The external fields are $F_{x}=7$ and $F_{0}=4$ with $\omega=2$.
The abrupt jumps occur precisely when the wavepacket momentum
$k_{y}(t)=\frac{1}{\hbar}\int_{0}^{t}F_{0}\cos(\omega t^{\prime})dt^{\prime}$
(blue curve) approaches $\pm\pi/2$, as marked by the black lines.