††thanks: These two authors contributed equally.††thanks: These two authors
contributed equally.

# Geometric Response and Disclination-Induced Skin Effects in Non-Hermitian
Systems

Xiao-Qi Sun  Penghao Zhu  Taylor L. Hughes  Department of Physics and
Institute for Condensed Matter Theory, University of Illinois at Urbana-
Champaign, Urbana, Illinois 61801, USA

###### Abstract

We study the geometric response of three-dimensional non-Hermitian crystalline
systems with nontrivial point-gap topology. For systems with fourfold rotation
symmetry, we show that in the presence of disclination lines with a total
Frank angle which is an integer multiple of $2\pi$, there can be nontrivial
one-dimensional point-gap topology along the direction of the disclination
lines. This results in disclination-induced non-Hermitian skin effects. By
doubling a non-Hermitian Hamiltonian to a Hermitian three-dimensional chiral
topological insulator, we show that the disclination-induced skin modes are
zero modes of the effective surface Dirac fermion(s) in the presence of a
pseudomagnetic flux induced by disclinations. Furthermore, we find that our
results have a field theoretic description, and the corresponding geometric
response actions (e.g., the Euclidean Wen-Zee action) enrich the topological
field theory of non-Hermitian systems.

_Introduction.–_ Non-Hermitian Hamiltonians provide a natural formalism to
describe wave phenomena in the presence of loss and gain, which are ubiquitous
in both classical Schomerus (2013); Zhen  _et al._ (2015); Longhi  _et al._
(2015); Zeuner  _et al._ (2015); Lu  _et al._ (2015); Poli  _et al._ (2015);
Weimann  _et al._ (2016); Takata and Notomi (2018); Zhou  _et al._ (2018);
Cerjan  _et al._ (2019); Ghatak  _et al._ (2020); Weidemann  _et al._ (2020);
Helbig  _et al._ (2020a); Hofmann  _et al._ (2020); Wang  _et al._ (2021);
Palacios  _et al._ (2021) and quantum Choi  _et al._ (2010); Xiao  _et al._
(2017); Zhan  _et al._ (2017); Kozii and Fu (2017); Xu  _et al._ (2017); Shen
and Fu (2018); Papaj  _et al._ (2019); Zyuzin and Zyuzin (2018); Xiao  _et
al._ (2020); Nagai  _et al._ (2020) systems. Recently, there has been a
growing interest in the interplay between non-Hermiticity and topology. The
synergy of these two concepts has produced fruitful results in non-Hermitian
crytalline systems, such as new transport and dynamical features Rudner and
Levitov (2009); McDonald  _et al._ (2018); Song  _et al._ (2019a); Lee  _et
al._ (2019a); Wanjura  _et al._ (2020); Xue  _et al._ (2021); Li  _et al._
(2020a); Schomerus (2020); Yi and Yang (2020); Lee  _et al._ (2020); Helbig
_et al._ (2020b); Pará  _et al._ (2021), new forms of bulk-boundary
correspondence Hatano and Nelson (1996); Lee (2016); Leykam  _et al._ (2017);
Yao  _et al._ (2018); Yao and Wang (2018); Martinez Alvarez  _et al._ (2018);
Kunst  _et al._ (2018); Zhang  _et al._ (2021); Xiong (2018); Yokomizo and
Murakami (2019); Zirnstein  _et al._ (2019); Song  _et al._ (2019b); Imura and
Takane (2019); Jin and Song (2019); Yang  _et al._ (2020a), and non-Hermitian
analogy of topological insulators Shen  _et al._ (2018); Hirsbrunner  _et al._
(2019); Xi  _et al._ (2021); Lieu  _et al._ (2020); Tonielli  _et al._ (2020);
Altland  _et al._ (2020) and semimetals Cerjan  _et al._ (2018); Carlström and
Bergholtz (2018); Okugawa and Yokoyama (2019); Carlström  _et al._ (2019);
Moors  _et al._ (2019); Yoshida  _et al._ (2019); McClarty and Rau (2019);
Kawabata  _et al._ (2019a); Budich  _et al._ (2019); Yang and Hu (2019); Wang
_et al._ (2019); Yoshida  _et al._ (2020); Sun  _et al._ (2020); Yang  _et
al._ (2020b, 2019); Hu and Zhao (2021).

One of the most remarkable consequences of non-Hermiticity is the new class of
topological systems Gong  _et al._ (2018); Zhou and Lee (2019); Bergholtz  _et
al._ (2019); Kawabata  _et al._ (2019b); Liu  _et al._ (2019a); Liu and Chen
(2019); Wojcik  _et al._ (2020); Denner  _et al._ (2020) without Hermitian
analogs. These intrinsically non-Hermitian topological systems have
topological invariants associated with a point gap Gong  _et al._ (2018) at a
reference energy in the complex energy plane. In one spatial dimension, the
nontrivial point-gap topology produces the celebrated non-Hermitian skin
effect (NHSE) Hatano and Nelson (1996); Yao and Wang (2018), which generates
an extensive number of states localized at the boundaries of a system Lee and
Thomale (2019); Lee  _et al._ (2019b); Liu  _et al._ (2019b); Okuma  _et al._
(2020); Zhang  _et al._ (2020); Borgnia  _et al._ (2020); Kawabata  _et al._
(2020); Okugawa  _et al._ (2020); Ma and Hughes (2020); Fu  _et al._ (2021).
Recently, the magnetic-field-induced NHSE in three-dimensional (3D) non-
Hermitian Weyl semimetals has been studied, which originates from a 3D
nontrivial point-gap topology Bessho and Sato (2020). In addition to
electromagnetic response, there have been extensive studies of the _geometric_
response of Hermitian topological systems both in the continuum limit Avron
_et al._ (1995); Wen and Zee (1992); Read (2009); Hughes  _et al._ (2011);
Bradlyn  _et al._ (2012); Hughes  _et al._ (2013); Abanov and Gromov (2014);
Gromov and Abanov (2014); Gromov  _et al._ (2015); Biswas and Son (2016) and
at lattice level Teo and Hughes (2013); Benalcazar  _et al._ (2014);
Shapourian  _et al._ (2015); Li  _et al._ (2020b); Rao and Bradlyn (2020);
Peterson  _et al._ (2021); Liu  _et al._ (2021, 2019c). However, the
understanding of the interplay between geometry and non-Hermitian point-gap
topology is still preliminary.

In this Letter, we consider the geometric response of 3D non-Hermitian
crystalline systems having nontrivial point-gap topology. We show that
disclination lines in rotationally invariant systems can support one-
dimensional (1D) point-gap topology along the direction of the disclination
lines, hence leading to a corresponding NHSE. By mapping the non-Hermitian
problem to a 3D chiral topological insulator, we show that the disclination
skin modes are zero modes of the effective surface Dirac fermions subjected to
a pseudomagnetic field induced by the curvature singularities at the
disclinations. Furthermore, we show that we can describe this phenomenon with
the inclusion of geometric terms in the non-Hermitian field theory approach
Kawabata  _et al._ (2021).

_Point-gap topology and NHSE.–_ We shall first review point-gap topology and
its relation to the NHSE in 1D and the magnetic-field-induced NHSE in 3D. For
a non-Hermitian Bloch Hamiltonian $H$, one can define a point gap if its
spectrum does not cross a reference energy $E\in\mathbb{C}$, i.e.,
$\det(H\,{-}\,E)\neq 0$. This means that $\det(H\,{-}\,E)$ is a non-zero
complex number and can have a winding number in the 1D Brillouin zone (BZ)
protected by the point gap:

$W_{1}(E)=-\int_{0}^{2\pi}\frac{dk}{2\pi}\frac{\partial}{\partial
k}\arg\left[\det\left(H(k)-E\mathbb{1}\right)\right].$ (1)

$W_{1}(E)$ has been shown to count the number of eigenstates at energy $E$
that are localized at a semi-infinite boundary Okuma  _et al._ (2020).
Interestingly, $W_{1}(E)$ can be nonzero for a continuous region of energy $E$
in the complex energy plane (bounded by gap-closing points), and therefore
indicates an extensive number of eigenstates localized at the boundary. This
phenomenon is known as the NHSE. Furthermore, for a Hamiltonian with a
nontrivial $W_{1}(E)$, the long-lived excitations are chiral, and feature
anomalous dynamics absent in Hermitian systems Lee  _et al._ (2019a). To
illustrate this property, one simple example is a one-band model
$H(k)\,{=}\,e^{-ik}$ with nontrivial winding $+1$ for each reference energy
inside the unit circle. The most amplified mode is at $k\,{=}\,\,{-}\,\pi/2$
with chirality $\chi\,{=}\,\operatorname{sgn}\left[\text{Re}\left(\partial
H/\partial k\right)\right]\,{=}\,\,{+}\,1$. The NHSE and anomalous dynamics
are crucial features of 1D point-gap topology.

Similar to $W_{1}(E)$, one can define a quantized winding number $W_{3}(E)$
for a Bloch Hamiltonian $H({\bf{k}})$ in the 3D BZ at a reference energy $E$

$\displaystyle
W_{3}(E)=-\int_{\text{BZ}}\frac{d^{3}k}{24\pi^{3}}\epsilon^{ijk}\operatorname{tr}\Big{[}(\widetilde{H}^{-1}\partial_{k_{i}}\widetilde{H})$
(2)
$\displaystyle\times(\widetilde{H}^{-1}\partial_{k_{j}}\widetilde{H})(\widetilde{H}^{-1}\partial_{k_{k}}\widetilde{H})\Big{]},$

where $\widetilde{H}({\bf{k}})\equiv H({\bf{k}})-\mathbb{1}E$. The
generalization of the anomalous dynamics is straightforward: the long-lived
excitation is a 3D Weyl fermion. In a uniform magnetic field the Weyl node
exhibits chiral 1D Landau levels dispersing along the field direction. These
anomalous 1D chiral modes can be shown Bessho and Sato (2020) to be precisely
ascribed to the nontrivial 1D winding $W_{1}(E)=W_{3}(E)\Phi/2\pi$ with $\Phi$
being the total magnetic flux, and hence a magnetic-field-induced NHSE.

_Disclination-induced NHSE.–_ Beyond the NHSE associated with electromagnetic
fields, we can explore geometric probes of intrinsically non-Hermitian
topology. We focus on lattice disclinations that heuristically serve as
sources of geometric curvature. A disclination is classified by a Frank angle
and a Burgers vector equivalence class, which capture the amount of rotation
and translation accumulated by a vector parallel transported along a loop
encircling the disclination respectively. An example is given in Fig. 1, and
comprehensive discussions about the classification can be found in Refs.
Benalcazar  _et al._ (2014); Li  _et al._ (2020b). Due to the Frank angle and
the Burgers vector (equivalence class), a wave packet will generically obtain
a Berry phase when it adiabatically encircles the disclination, and thus the
disclinations can effectively generate a pseudomagnetic field. In the
following, we will demonstrate that the pseudomagnetic flux induced by
disclinations can lead to a NHSE in systems having nontrivial $W_{3}(E)$, and
this effect is captured by a nontrivial effective 1D winding number $W_{1}(E)$
along the direction of the disclination line.

Figure 1: $C_{4}$-symmetric lattice with a disclination. Starting from the red
circle, the parallel transport around the loop provides net rotation of
$-\pi/2$ and net transportation of $-3\mathbf{e}_{1}$. The Frank angle of this
disclination is $\pi/2$. The equivalence class of Burgers vector is the parity
of the sum of the Burgers vector components: $-3\bmod 2=1$.

_Hermitian description.–_ In order to see how the pseudomagnetic flux induced
by disclinations leads to a NHSE in lattice models, it is convenient to
introduce a doubled and Hermitianized Hamiltonian with chiral symmetry
Feinberg and Zee (1997); Gong  _et al._ (2018):

$\mathcal{H}({\bf{k}})=\left(\begin{array}[]{cc}0&H({\bf{k}})^{\dagger}-E^{*}\\\
H({\bf{k}})-E&0\end{array}\right),$ (3)

where $H({\bf{k}})$ is the non-Hermitian Hamiltonian we want to study, and $E$
is the reference energy on which we focus. There are two main features of this
approach: (i) The topological winding number $W_{3}(E)$ of $H({\bf{k}})$
equals the chiral winding number of $\mathcal{H}({\bf{k}})$. Hence, the bulk-
boundary correspondence of (Hermitian) chiral symmetric insulators indicates
that a nonzero $W_{3}(E)$ implies the existence of $|W_{3}(E)|$ protected
surface Dirac cones (SDCs). (ii) The existence of an _exact_ zero mode of
$\mathcal{H}$ implies the existence of an eigenstate at energy $E$ ($E^{*}$)
for $H$ ($H^{\dagger}$) depending on its chirality.

Figure 2: (a) A tabulation of
$(\operatorname{sgn}\left[\gamma-\text{Im}E\right],\chi_{i})$ at the eight
Weyl points of Eq. (4) when $1<\operatorname{Im}E<3$. (b) the locations of
Dirac cone(s) on the surface of the Hermitian Hamiltonian in Eq. (3) when
$1<\operatorname{Im}E<3$ (red), $-1<\operatorname{Im}E<1$ (yellow), and
$-3<\operatorname{Im}E<-1$ (blue).

For an explicit illustration, let us focus on a concrete non-Hermitian Weyl
semimetal model with a Bloch Hamiltonian:

$H({\bf{k}})=t\sin k_{x}\sigma_{x}+t\sin k_{y}\sigma_{y}+t\sin
k_{z}\sigma_{z}+i\gamma({\bf{k}}),$ (4)

where $\gamma({\bf{k}})=\cos k_{x}+\cos k_{y}+\cos k_{z}$. This model has a
$C_{4z}$ rotational symmetry represented by:

$\displaystyle
C_{4z}=e^{-i\frac{\pi}{4}\sigma_{z}}e^{-i\pi\mathcal{L}_{z}/2},$ (5)
$\displaystyle
C_{4z}H(k_{x},k_{y},k_{z})C_{4z}^{\dagger}=H(k_{y},-k_{x},k_{z}),$

where the Pauli matrices represent the spin degree of freedom, and
$\mathcal{L}_{z}$ is the angular momentum of the orbital that we put on each
site, which takes values $-1,0,1,2$ $\operatorname{mod}4,$ for
$C_{4z}$-symmetric systems. We will see below that the orbital rotation phase
in Eq. (5) is important for the geometric response. For this model, the 3D
winding number $W_{3}(E)$ has a simplified formula Sun  _et al._ (2018);
Higashikawa  _et al._ (2019); Bessho and Sato (2020):

$W_{3}(E)=\sum_{i}\frac{1}{2}\text{sgn}\left[\gamma({\bf{Q}}_{i})-\text{Im}E\right]\chi_{i},$
(6)

where ${\bf{Q}}_{i}$ are the momenta of the eight Weyl points in the Hermitian
limit, and $\chi_{i}\,{=}\,\pm 1$ are their corresponding chiralities. The
positions of the Weyl points in momentum space are shown in Fig. 2 (a), as
well as their corresponding
$(\operatorname{sgn}\left[\gamma-\text{Im}E\right],\chi_{i})$. This model
therefore has a nontrivial 3D point gap winding number of $W_{3}(E)\,{=}\,1$
for $1\,{<}\,|\operatorname{Im}E|\,{<}\,3$, and $W_{3}(E)\,{=}\,-2$ for
$|\operatorname{Im}E|\,{<}\,1$.

Now we consider (the Hermitian) $\mathcal{H}$ in a semi-infinite bulk geometry
terminated with a surface normal $\hat{z}$ and its two-dimensional (2D)
reduced Brillouin zone (rBZ) [see Fig. 2(b)]. We start with the simple case
where $1\,{<}\,\operatorname{Im}E\,{<}\,3$ and $W_{3}(E)\,{=}\,1$. The low-
energy effective Hamiltonian on the top $C_{4z}$-symmetric surface termination
is a rotationally symmetric SDC at $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\Gamma}$ as shown in Fig. 2(b) :

$\mathcal{H}_{\text{eff}}=v(k_{x}\tau_{x}+k_{y}\tau_{y}),$ (7)

where the Pauli matrices $\tau_{i}$ are the effective degrees of freedom, and
the chiral symmetry is represented by $\tau_{z}$.

Now we introduce disclination lines parallel to the $z$ direction, which
correspond to curvature singularities in the continuous effective Dirac model
[e.g. Eq. (7)]. The curvature is represented by the coupling to a spin
connection $\omega_{i}$ via the rotation generators. The Frank angle of
disclination lines can be expressed in terms of $\omega_{i}$ as
$\Theta_{z}\,{=}\,\int(\partial_{x}\omega_{y}\,{-}\,\partial_{y}\omega_{x})dxdy$.
In addition to the coupling to $\tau_{z}$, the spin connection also couples to
$\mathcal{L}_{z}\mathbb{1}$, which adds an effective gauge flux
$\mathcal{L}_{z}\Theta_{z}$ to the Dirac fermion Sup . According to the index
theorem Nakahara (2003), this flux $\mathcal{L}_{z}\Theta_{z}$, will lead to
robust zero modes on the top surface with a total number of
$\nu\,{=}\,|\mathcal{L}_{z}\Theta_{z}/(2\pi)|$. Furthermore, the zero modes
are eigenmodes of $\tau_{z}$ with eigenvalue (chirality)
$\tau\,{=}\,\text{sgn}~{}{\mathcal{L}_{z}\Theta_{z}}$ Sup . For
$\mathcal{L}_{z}\Theta_{z}\,{>}\,0$, the eigenmodes will have $\tau\,{=}\,+1$
and correspond to skin modes of $H$ on the top surface and at energy $E$. By a
similar argument for the bottom surface, one can derive that for
$\mathcal{L}_{z}\Theta_{z}\,{<}\,0$, there are zero modes of $\mathcal{H}$
corresponding to skin modes of $H$ on the bottom surface and at energy $E$. It
is known Okuma  _et al._ (2020) that the number of zero modes of $\mathcal{H}$
with $\tau\,{=}\,+1$ equals $|W_{1}(E)|$, and that
$\text{sgn}[W_{1}(E)]\,{=}\,+/-$ indicates that the zero modes with
$\tau\,{=}\,+1$ are on the top/bottom surface. Hence, we can conclude that
when $1\,{<}\,\operatorname{Im}E<3$ and $W_{3}(E)\,{=}\,1$, the disclination
induces a winding number of $W_{1}(E)\,{=}\,\mathcal{L}_{z}\Theta_{z}/(2\pi)$.
We note that the effective flux is contributed _only_ by the orbital rotation
generator, which commutes with $\tau_{i}$. From here on, we focus only on this
part to compute the effective flux for a Dirac fermion in order to determine
$W_{1}(E)$.

Let us proceed to discuss cases where the SDCs are at momenta away from
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\Gamma}$. Since disclinations are classified by a Frank angle $\Theta_{z}$
and a Burgers vector (equivalence class) of ${\bf{b}}\,{=}\,(b_{x},b_{y})$,
for a SDC with nonzero momentum ${\bf{Q}}_{\perp}$, the Burgers vector will
also contribute an effective flux of $-{\bf{Q}}_{\perp}\cdot{\bf{b}}$. With
this consideration in mind, we analyze other topologically nontrivial regimes
of our model. First, when $-3\,{<}\,\operatorname{Im}E\,{<}-1$, there is a
single SDC at $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{M}$
[see Fig. 2 (b)], and the effective flux it feels is Sup

$\Phi^{\text{eff}}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{M})=\mathcal{L}_{z}\Theta_{z}-(\pi,\pi)\cdot{\bf{b}}.$ (8)

From the index theorem for 2D Dirac fermions, as well as the relation between
$W_{1}$ and the number of surface zero modes, it is straightforward to see
that the effective flux in Eq. (8) leads to a winding number of
$W_{1}(E)\,{=}\,\Phi^{\text{eff}}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{M})/(2\pi)$. Second, when $-1\,{<}\,\operatorname{Im}E\,{<}\,1$, there is
a pair of SDCs at $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{X}$
and $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Y}$
[see Fig.2 (b)]. Since a $C_{4z}$ rotation takes one SDC to another, and the
translation phase obtained by each SDC is different, these two SDCs together
form an irreducible representation of the space group. A subtlety arises that
we cannot _individually_ define the effective flux for each SDC, and we need
to consider the (possibly non-Abelian) effective flux of both. In doing so, we
write the $C_{4z}$ rotation operator for these two SDCs acting on the
_orbital_ degrees of freedom:

$C_{4z}=\sigma_{x}^{v}e^{-i\frac{\pi}{2}\mathcal{L}_{z}}=\exp\left(-i\frac{\pi}{2}\mathcal{L}_{z}\sigma_{x}^{v}\right),$
(9)

where the superscript $v$ indicates the operator is in the valley space
spanned by $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{X}$
and $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Y}$,
and $\sigma_{x}^{v}$ exchanges the two valleys. From Eq. (9), we can see
$\mathcal{L}_{z}\sigma_{x}^{v}$ is the orbital rotation generator, which leads
to a non-Abelian flux for SDCs at the two valleys. The effective flux can be
written as $\Phi_{r}\,{=}\,\mathcal{L}_{z}\Theta_{z}\sigma_{x}^{v}$, where
$\Theta_{z}$ is a multiple of $\pi/2$ in the $C_{4z}$-symmetric case. In
addition, the translation phase of SDCs at the two valleys contributes another
matrix flux:

$\Phi_{t}=-\pi\begin{pmatrix}b_{x}&0\\\ 0&b_{y}\end{pmatrix}.$ (10)

Combining these two contributions, we define the total non-Abelian effective
flux felt by SDCs at two valleys to be

$e^{i\Phi^{\text{eff}}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{X}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{Y})}=e^{i\Phi_{t}}e^{i\Phi_{r}}.$ (11)

Now applying the index theorem again Sup , we find a winding number of
$W_{1}(E)\,{=}\,-\text{tr}[\Phi^{\text{eff}}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{X}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{Y})]/(2\pi)$, where the minus sign comes from the chiral winding number of
$-1$ of the SDCs. Notice that the analysis from the index theorem is for the
effective Dirac Hamiltonian in the continuum limit, however, we will show that
this indeed captures the disclination-induced skin effect in our lattice model
[cf. Eq. (4)].

Figure 3: Schematic illustration of a lattice construction with four
disclinations, each of which has a Frank angle of $\pi/2$. (a) is the gluing
procedure Sup , and (b) is a completed lattice with disclinations.

_Numerical results.–_ To find a quantized $W_{1}(E)$ and the corresponding
NHSE, we need a total pseudomagnetic flux that is time-reversal odd and an
integer multiple of $2\pi$. However, there is a subtlety on the lattice that a
flux of $n\pi$ passing through a single lattice plaquette is time-reversal
even and incompatible with the chiral anomalous dynamics and the corresponding
NHSE. Thus, for our lattice calculations this motivates us to consider orbital
angular momentum $\mathcal{L}_{z}\,{=}\,\pm 1$ and introduce four
disclinations 111Here we use open boundary conditions in $x$ and $y$
directions. We also discuss the case of eight disclinations with total Frank
angle $4\pi$ in the Supplemental Material Sup , which enables us to compactify
$x$ and $y$ directions to a 2-sphere. The numerical results agree with our
Hermitian description and suggest that the appearance of disclination-induced
NHSE along $z$ direction does not depend on boundary physics in $x$ and $y$
directions in our model., each with a Frank angle of
$\Theta^{s}_{z}\,{=}\,\pm\pi/2$, where the superscript $s$ is used to
distinguish the Frank angle of a single disclination and the total Frank angle
$\Theta_{z}$ in the system. For illustration purposes, let us fix
$\mathcal{L}_{z}\,{=}\,1,$ and focus on the case where each of the four
disclinations has a Frank angle of $\Theta_{z}^{s}\,{=}\,\pi/2$ in the
following. We discuss other cases, e.g.,
$(\Theta_{z}^{s}\,{=}\,\pi/2,\,\mathcal{L}_{z}\,{=}-1)$, and
$(\Theta_{z}^{s}\,{=}-\pi/2,\,\mathcal{L}_{z}\,{=}\pm 1)$, etc., in the
Supplemental Material Sup .

For our numerical lattice calculations we construct four disclinations with a
Frank angle of $\Theta_{z}^{s}\,{=}\,\pi/2$ by a process shown in Fig. 3.
Then, we add the hopping terms determined by the Hamiltonian in Eq. (4) on the
disclinated lattice to derive a Hamiltonian $H_{\text{dis}}(k_{z})$. We note
that we are using disclinations with “plaquette-type” cores Teo and Hughes
(2013), which implies a Burgers vector _class_ ${\bf{b}}^{s}\,{=}\,a\hat{x},$
where $a$ is the lattice constant and the superscript $s$ indicates the
Burgers vector is for a single disclination. Note that because of the $C_{4z}$
symmetry we could equivalently say that ${\bf{b}}^{s}\,{=}\,a\hat{y},$ hence a
Burgers vector class. Let us first focus on the simplest regime when
$1\,{<}\,\operatorname{Im}E\,{<}\,3$ and $W_{3}(E)=1$. To show the nontrivial
1D point gap winding number $W_{1}(E)=1$ [cf. Eq. (1)] of
$H_{\text{dis}}(k_{z})$, we plot
$\,{-}\,\operatorname{arg}\det\left(H_{\text{dis}}(k_{z})-E\mathbb{1}\right)$
at $E=2i$ in Fig. 4(a). Figure 4(b) shows the spectrum of $H_{\text{dis}}$
under periodic boundary conditions along the $z$ direction, where a loop
circling the $1\,{<}\,\operatorname{Im}E\,{<}\,3$ region can be seen. The skin
modes on the top surface, which are qualitatively captured by
$W_{1}(E)\,{=}\,1,$ are indicated by blue dots in the spectrum of open
boundary conditions in the $z$ direction, as shown in Fig. 4(c). We find ten
skin modes in the region of $1\,{<}\,\operatorname{Im}E\,{<}\,3,$ which is
consistent with the extensive NHSE for a 1D line in a 3D system where
$N_{z}\,{=}\,10$. In Fig. 4(d), we show an exponentially decaying wave
function of a representative skin mode [circled in Fig. 4(c)]. Hence, when
$1\,{<}\,\operatorname{Im}E\,{<}\,3$, $\mathcal{L}_{z}\,{=}\,\,{+}\,1$, and
there is pseudomagnetic flux $2\pi$, we have shown numerically that
$W_{1}(E)\,{=}\,1$ and there are corresponding skin modes, which is consistent
with our Hermitian description above 222Numerical results for all other cases
with $1\,{<}\,\operatorname{Im}E\,{<}\,3$ (i.e.,
$\mathcal{L}_{z}\,{=}\,\,{-}\,1$ and pseudomagnetic flux $2\pi$, and
$\mathcal{L}_{z}=\pm 1$ and pseudomagnetic flux $\,{-}\,2\pi$) are shown in
the Supplemental Material, and are also consistent with our Hermitian
description. .

Figure 4: Numerical calculations for the non-Hermitian Weyl semimetal
described by Eq. (4) on a lattice with four disclinations as shown in Fig. 3,
and $t=1/2$ is used in all calculations. The lattice has 10 unit cells along
$z$ direction, and 200 unit cells in the $x$-$y$ plane at each $z$. (a) shows
the nontrivial one-dimensional point gap winding number $W_{1}(E)$ at $E=2i$.
(b) and (c) show the energy spectra under periodic and open boundary condition
along $z$ direction, where the blue dots in (c) are skin modes on the top
surface. (d) shows the wave function along $z$ direction for the state
indicated by a red circle in (c).

Next, consider the regime when $-3\,{<}\,\operatorname{Im}E\,{<}-1$. Since
each of the four disclinations in Fig. 3 has Burgers vector class
${\bf{b}}^{s}\,{=}\,a\hat{x}$, the effective flux [cf. Eq. (8)] contributed
from each disclination is $-\pi/2$. In total, there is a
$\Phi^{\text{eff}}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{M})\,{=}\,-2\pi$ pseudomagnetic flux, and we expect a nontrivial 1D point-
gap winding of $W_{1}(E)\,{=}\,-1$, which is confirmed by our numerical
calculation shown in Fig. 5 (a). Finally, consider the regime when
$-1\,{<}\,\operatorname{Im}E\,{<}\,1$. By substituting
${\bf{b}}^{s}\,{=}\,a\hat{x}$ and $\Theta^{s}_{z}\,{=}\,\pi/2$ into $\Phi_{t}$
and $\Phi_{r}$ defined above, one can calculate
$e^{i\Phi^{\text{eff}}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{X}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{Y})}\,{=}\,\sigma_{y}^{v}$, of which the two eigenmodes feel time-reversal
even flux ($\Phi_{1}\,{=}\,0$ and $\Phi_{2}\,{=}\,\pi$) on each disclination,
and thus we expect a trivial 1D point-gap winding of $W_{1}(E)\,{=}\,0$. This
is confirmed by our numerical calculation shown in Fig. 5(b).

Figure 5: The 1D point-gap winding number $W_{1}(E)$ on a lattice with four
disclinations as shown in Fig. 3 for (a) $E=-2i$ and (b) $E=0i$.

_Conclusions and discussions.–_ We studied the geometric response of a
$C_{4z}$-symmetric 3D non-Hermitian lattice model with nontrivial point-gap
topology characterized by $W_{3}$. We found that disclinations can induce
NHSEs along the disclination lines. Interestingly, we note that our results
can be described by the framework of the Euclidean field theory for non-
Hermitian systems recently proposed in Ref. Kawabata  _et al._ (2021). The
disclination-induced NHSE can be captured by a Euclidean Wen-Zee term encoding
the Frank angle contributionWen and Zee (1992):

$S_{\text{WZ}}=\frac{\mathcal{L}_{z}W_{3}(E)}{2\pi}\int
d^{3}{\bf{x}}\epsilon^{ijk}A_{i}\partial_{j}\omega_{k},$ (12)

and a term containing the mixing of the $U(1)$ gauge field and the
frame/translation gauge field for the Burgers vector equivalence class. These
geometric response actions enrich the types of topological field theory
responses for non-Hermitian systems. Interested readers are referred to the
Supplementary Material for more details Sup .

It is also important to note that our model is expected to be realizable in a
variety of platforms Schomerus (2013); Zhen  _et al._ (2015); Longhi  _et al._
(2015); Zeuner  _et al._ (2015); Lu  _et al._ (2015); Poli  _et al._ (2015);
Weimann  _et al._ (2016); Takata and Notomi (2018); Zhou  _et al._ (2018);
Cerjan  _et al._ (2019); Ghatak  _et al._ (2020); Weidemann  _et al._ (2020);
Helbig  _et al._ (2020a); Hofmann  _et al._ (2020); Wang  _et al._ (2021);
Palacios  _et al._ (2021) previously used to study non-Hermitian physics.
Besides directly observing the boundary localized skin modes, one can also
probe the disclination-induced skin effect using the chiral propagation of
wave packets created near disclinations in the bulk. Furthermore, our model
study serves as a first concrete example to theoretically understand geometric
response in non-Hermitian topological systems, which will motivate the pursuit
of the non-Hermitian analogy of more geometric responses such as Hall
viscosity in future research.

###### Acknowledgements.

_Acknowledgments.–_ X.-Q. S. acknowledges support from the Gordon and Betty
Moore Foundations EPiQS Initiative through Grant GBMF8691. P. Z. and T. L. H.
thank the US Office of Naval Research (ONR) Multidisciplinary University
Research Initiative (MURI) grant N00014-20- 1-2325 on Robust Photonic
Materials with High-Order Topological Protection for support.

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  * Note (1) Here we use open boundary conditions in $x$ and $y$ directions. We also discuss the case of eight disclinations with total Frank angle $4\pi$ in the Supplemental Material Sup , which enables us to compactify $x$ and $y$ directions to a 2-sphere. The numerical results agree with our Hermitian description and suggest that the appearance of disclination-induced NHSE along $z$ direction does not depend on boundary physics in $x$ and $y$ directions in our model.
  * Note (2) Numerical results for all other cases with $1\tmspace+{.1667em}{<}\tmspace+{.1667em}\operatorname{Im}E\tmspace+{.1667em}{<}\tmspace+{.1667em}3$ (i.e., $\mathcal{L}_{z}\tmspace+{.1667em}{=}\tmspace+{.1667em}\tmspace+{.1667em}{-}\tmspace+{.1667em}1$ and pseudomagnetic flux $2\pi$, and $\mathcal{L}_{z}=\pm 1$ and pseudomagnetic flux $\tmspace+{.1667em}{-}\tmspace+{.1667em}2\pi$) are shown in the Supplemental Material, and are also consistent with our Hermitian description.

##  I Supplemental Material for: Geometric response and disclination-induced
skin effects in non-Hermitian systems

##  II Effective flux from index theorem

In this section, we justify the index theorem for SDCs used in the main text.
In our case, we have rotation (around $z$ direction) generator
$\mathcal{L}_{z}\mathbb{1}+\tau_{z}/2$ for a SDC (at $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\Gamma}$). When we introduce disclinations, the spin connection
$\omega_{\mu}$ with $\mu=1,2$ will enter the Dirac operator, which can be
written as

$\not{D}_{SDC}=e^{\mu}_{a}\gamma^{a}\left[\mathbb{1}\partial_{\mu}-i\omega_{\mu}(\mathcal{L}_{z}\mathbb{1}+\tau_{z}/2)\right]=e^{\mu}_{a}\gamma^{a}(\mathbb{1}\partial_{\mu}-i\mathcal{L}_{z}\omega_{\mu}\mathbb{1}-i\omega_{\mu}\tau_{z}/2),$
(S1)

where $e^{\mu}_{a}$ is the frame field and $\gamma^{1,2}=\tau_{x,y}$. If we
compare Eq. (S1) with the most general Dirac operator coupled to a U(1) gauge
field and to the background geometry:

$\not{D}=e^{\mu}_{a}\gamma^{a}(\mathbb{1}\partial_{\mu}-iA_{\mu}\mathbb{1}-\frac{i}{4}\omega^{ab}_{\mu}\gamma_{ab}),$
(S2)

where $\omega_{\mu}^{ab}$ is the spin connection. We can see $\omega_{\mu}$ in
Eq. (S1) represents the spin connection component
$\omega_{\mu}^{12}=-\omega^{21}_{\mu}$, and correspondingly
$\gamma^{12}=-\gamma^{21}=\tau_{z}$. If we further consider
$\mathcal{L}_{z}\omega_{\mu}$ as an effective Abelian gauge field
$A^{\text{eff}}_{\mu}$, it is straightforward to see that Eq. (S1) is a
special case of Eq. (S2) on a manifold $M$ described by the fixed
$\omega_{\mu}^{ab}$. Then we can apply the index theorem for the general Dirac
operator (see Chapter 12 of Ref. Nakahara (2003)) to Eq. (S1):

$\nu_{+}-\nu_{-}=\frac{1}{2\pi}\int_{M}\partial_{x}A^{\text{eff}}_{y}-\partial_{y}A^{\text{eff}}_{x}=\frac{\mathcal{L}_{z}}{2\pi}\int_{M}\partial_{x}\omega_{y}-\partial_{y}\omega_{x}=\frac{\mathcal{L}_{z}\Theta_{z}}{2\pi},$
(S3)

where $\nu_{+}$ ($\nu_{-}$) is the number of zero modes with chirality
$\tau=+1$ ($-1$). There are a total of $\nu\equiv|\nu_{+}-\nu_{-}|$ number of
robust zero modes. In the case of $\nu_{+}-\nu_{-}>0$, the $\nu$ robust zero
modes have chirality $\tau=+1$, which corresponds to skin modes. We note that
the RHS of Eq. (S3) only has contribution from the flux of the (effective)
abelian gauge field, and has no contribution from the coupling of spin
connection to $\gamma_{ab}$, which is true for Dirac operators in 2D Nakahara
(2003). We also note that the above discussion is for a SDC with chiral
winding number $+1$, for SDCs with chiral winding number $-1$ there will be an
extra minus sign on the RHS of Eq. (S3). To apply the index theorem in the
main text, we should consider that the top SDC has chiral winding number
$\text{sgn}[W_{3}(E)]$ and the bottom SDC have the opposite chiral winding
number in our model. With similar analysis, we can also find the effective
Abelian/non-Abelian gauge flux and use the correct index theorem in other
cases.

Before proceeding to other cases, we detour to show that the NHSE derived from
the index theorem in the Hermitianized Hamiltonian discussed above also has a
field theoretic description, which is of help to gain better understanding of
the disclination induced NHSE. In Ref. Kawabata  _et al._ (2021), Kawabata _et
al._ proposes a Euclidean topological field theory for non-Hermitian systems
with nontrivial point-gap topology. They formally introduce a Euclidean action
coupling to a spatial gauge field for each reference energy $E$

$S_{E}[\boldsymbol{A},\psi_{E}]=\int\psi_{E}^{\dagger}[H(-i\partial-\boldsymbol{A})-E]\psi_{E}d^{2n+1}x.$
(S4)

Then, the effective action upon integrating out fermions is the $(2n+1)$-D
Chern-Simons action with coefficient $W_{2n+1}(E)$. For example, the Chern-
Simons action in 1D:

$S_{E}[A_{x}]=W_{1}(E)\int dxA_{x},$ (S5)

leads to a chiral current

$j_{x}=\frac{\delta S_{E}[A_{x}]}{\delta A_{x}}=W_{1}(E),$ (S6)

which can capture the chiral anomalous dynamics and the NHSE in 1D. Similarly,
the Chern-Simons action in 3D, $i.e.$,

$S_{E}[\boldsymbol{A}]=\frac{W_{3}(E)}{4\pi}\int
d^{3}\epsilon_{ijk}A_{i}\partial_{j}A_{k}$ (S7)

can capture the magnetic field induced NHSE by a dimension reduction:

$\frac{W_{3}(E)}{4\pi}\int
d^{3}\epsilon_{ijk}A_{i}\partial_{j}A_{k}=\frac{W_{3}(E)}{2\pi}\int_{S}B_{z}dxdy\int
dzA_{z}\equiv W_{1,z}(E)\int dzA_{z}\ \Rightarrow\
W_{1,z}(E)=\frac{W_{3}(E)}{2\pi}\int_{S}B_{z}dxdy.$ (S8)

Motivated by Eqs. (S7) and (S8), it is straightforward to see the NHSE due to
the winding number $\mathcal{L}_{z}\Theta_{z}/(2\pi)$ can be captured by a
Euclidean Wen-Zee term:

$S_{\text{WZ}}=\frac{\mathcal{L}_{z}W_{3}(E)}{2\pi}\int
d^{3}{\bf{x}}\epsilon^{ijk}A_{i}\partial_{j}\omega_{k}.$ (S9)

When have a disclination line along $z$ direction, Eq. (S9) leads to

$W_{1}(E)=\frac{\mathcal{L}_{z}W_{3}(E)}{2\pi}\int_{S}\Omega_{z}dxdy=\frac{\mathcal{L}_{z}W_{3}(E)\Theta_{z}}{2\pi},$
(S10)

where $\Omega_{z}\equiv\partial_{x}\omega_{y}-\partial_{y}\omega_{x}$.

With the field theoretic description in mind, let us come back to the
discussion about index theorem in other cases. When the SDC at
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{M}$,
we add an extra flux related to translation, $-(\pi,\pi)\cdot{\bf{b}}$, to the
effective flux. This is because for SDCs with non-zero momenta
${\bf{Q}}_{\perp}=(Q_{x},Q_{y})$, there will be another effective gauge field
$-Q_{\perp,a}e_{\mu}^{a}$ besides $\omega_{\mu}\mathcal{L}_{z}$ and the extra
effective gauge flux can be computed as

$-Q_{\perp,a}\oint
dx^{\mu}e^{a}_{\mu}=-Q_{\perp,a}b^{a}=-{\bf{Q}}_{\perp}\cdot{\bf{b}}.$ (S11)

Like the effective flux from $\omega_{\mu}\mathcal{L}_{z}$ can be captured by
a bulk Wen-Zee term in Eq. (S9), we tentatively propose a term at low energy
to capture the translation phase:

$S_{t}=\frac{1}{2\pi}\int
d^{3}{\bf{x}}\epsilon^{ijk}A_{i}Q_{\perp,a}\partial_{j}e^{a}_{k},$ (S12)

the details of which will be left to future research.

When there are a pair of SDCs at $X$ and $Y$ of the same winding number, we
have a 2D Dirac operator under a non-Abelian effective gauge field related to
orbital angular momentum, $\mathcal{L}_{z}\sigma_{x}^{v}\otimes\mathbb{1}$,
and a non-Abelian effective gauge field related to translation,
$\begin{pmatrix}(\pi,0)\cdot\mathbf{e}_{\mu}&0\\\
0&(0,\pi)\cdot\mathbf{e}_{\mu}&\end{pmatrix}\otimes\mathbb{1}$. Then, we need
to consider the non-Abelian effective flux $\Phi^{\text{eff}}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{X}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Y})$
contributed by both and we have

$\nu_{+}-\nu_{-}=\pm\text{sgn}[{W_{3}(E)}]\frac{\text{tr}\left[\Phi^{\text{eff}}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{X}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{Y})\right]}{2\pi},$ (S13)

for the top/bottom surface as used in the main text.

Figure S1: Numerical calculations for a non-Hermtian Weyl semimetal described
by Eq. (8) in a lattice where there are four disclinations with Frank angle
$\pi/2$ as shown in (a). (b), (c) and (d) are for $\mathcal{L}_{z}=1$. (e),
(f) and (g) are for $\mathcal{L}_{z}=-1$. Figure S2: Numerical calculations
for a non-Hermtian Weyl semimetal described by Eq. (8) in a lattice where
there are four disclinations with Frank angle $-\pi/2$ as shown in (a). (b),
(c) and (d) are for $\mathcal{L}_{z}=1$. (e), (f) and (g) are for
$\mathcal{L}_{z}=-1$.

##  III Supplemental information for numerical calculations

Here, we first describe more details about how we perform the numerical
calculations. When gluing the open edges at $\phi$ and $\phi+\pi/2$ shown in
Fig. 2 (a) in the main text, we add a phase $\exp(-i\mathcal{L}_{z}\pi/2)$.
This is because by rotating one edge anti-clockwisely by $\pi/2$ to glue it
with another edge, the states on that edge will obtain a phase
$\exp(-i\mathcal{L}_{z}\pi/2)$. Meanwhile, we also rotate $\sigma_{x}$ to
$\sigma_{y}$, and rotate $\sigma_{y}$ to $-\sigma_{x}$ when gluing. This part
is given by the rotation of the internal degrees of freedom (or spin), which
is generated by $\sigma_{z}$.

Next, we present numerical results for $\mathcal{L}_{z}=\pm 1$ in the lattice
where there are four disclinations, each of them has $\Theta^{s}_{z}=\pm\pi/2$
. In summary, the main conclusions we can draw from these results are: (i) the
1D point gap winding number for the $1<\operatorname{Im}E<3$ case can be
correctly predicted by the Wen-Zee term in Eq. (5). However, (ii) the correct
prediction of 1D point gap winding number for the $-3<\operatorname{Im}E<-1$
case and the $-1<\operatorname{Im}E<1$ case requires a combination of rotation
and translation lattice information, and can be done through including an
extra effective flux as shown in the last section.

Last, to show that the appearance of disclination-induced skin effect does not
depend on the physics of boundaries in $x-y$ directions, we compactify the
$x-y$ plane at each $z$ by gluing two lattices with four disclinations, each
with Frank angle $\pi/2$, as shown in Fig. S3(a), and then calculate the 1D
point gap winding number. Since we have eight “plaquette-type” disclinations
in the lattice, following the index theorem discussed above, we expect to have
$W_{1}(E)=2$ when $1<\operatorname{Im}E<3$, $W_{1}(E)=0$ when
$-1<\operatorname{Im}E<1$, and $W_{1}(E)=-2$ when $-3<\operatorname{Im}E<-1$,
which are confirmed by our numerical results shown in Fig. S3(b-d).

Figure S3: (a) illustrates the construction of a lattice with compactified
$x-y$ planes and eight disclinations, each with Frank angle $\pi/2$. We take
two copies of the disclinated lattice and glue their edges with dotted lines
having the same color (pink, green, blue, and brown), then we get the lattice
in the rightmost picture, of which the $x-y$ plane at each $z$ is compatified
as the surface of a cuboid (that is homeomorphic to a sphere). The eight
disclination cores are indicated by the red dots on the right. (b), (c) and
(d) shows the 1D point gap winding number for $E=2i$, $E=0$, and $E=-2i$,
respectively.