# Topological disclination states and charge fractionalization
in a non-Hermitian lattice

Rimi Banerjee  School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore 637371, Singapore  Subhaskar Mandal
School of Physical and Mathematical Sciences, Nanyang Technological
University, Singapore 637371, Singapore  Yun Yong Terh  School of Physical and
Mathematical Sciences, Nanyang Technological University, Singapore 637371,
Singapore  Shuxin Lin  School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore 637371, Singapore  Gui-Geng Liu  School of
Physical and Mathematical Sciences, Nanyang Technological University,
Singapore 637371, Singapore  Baile Zhang<EMAIL_ADDRESS>School of
Physical and Mathematical Sciences, Nanyang Technological University,
Singapore 637371, Singapore  Centre for Disruptive Photonic Technologies,
Nanyang Technological University, Singapore, 637371, Singapore  Y. D. Chong
<EMAIL_ADDRESS>School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore 637371, Singapore  Centre for Disruptive
Photonic Technologies, Nanyang Technological University, Singapore, 637371,
Singapore

###### Abstract

We show that a non-Hermitian lattice with a disclination can host topological
disclination states that are induced by on-site gain and loss. The
disclination states are inherently non-Hermitian as they do not exist in the
limit of zero gain/loss. They arise from charge fractionalization in the non-
Hermitian lattice, which we establish using non-Hermitian Wilson loops
calculated with biorthogonal products. The model can be realized using an
array of optical resonators, with the emergence of the topological
disclination states manifesting as an abrupt shift in emission intensity and
frequency upon tuning the gain/loss level.

Introduction—In discrete lattices, dislocations and disclinations are
elementary defects that cannot be removed by local deformations, due to their
global topological structure [1, 2, 3, 4, 5, 6]. They are known to play an
important role in many condensed-matter phenomena, such as the melting of two-
dimensional solids [6]. In topological materials [7], such defects have a
special significance: they allow us to formulate “bulk-defect
correspondences”, generalizing the bulk-boundary correspondences that usually
serve as physical signatures for bulk band topology [8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 14, 18, 19, 20, 21, 22, 23, 24]. In certain higher-order
topological insulators (HOTIs) [25, 26, 27, 28, 29, 30, 31, 32, 33, 34], bulk-
defect correspondences may be used to probe topological properties that are
hard to access through boundary measurements [15, 16]. Aside from the
appearance of localized defect states, some bulk-defect correspondences
predict charge fractionalization, whereby a fraction of a unit charge
polarization is localized at the defect. This phenomenon, which results from
an incompatibility between the charge quantization principle and lattice
symmetries (a filling anomaly), underpins the topology of HOTIs [25, 26, 27].
Charge fractionalization and localized states have recently been observed
experimentally in 2D photonic lattices with disclinations [15, 16].

Theoretical analyses of topological materials, including most earlier studies
of bulk-defect correspondences, usually take Hermiticity as a starting
assumption. In recent years, however, there has been increasing interest in
non-Hermitian (energy non-conserving [35, 36, 37]) topological materials [38,
39, 40, 41, 42, 43, 44, 45, 46, 47]. Not only do such systems pose the
theoretically interesting challenge of formulating band topology without
Hermiticity, but they are also of practical interest: in the classical-wave
metamaterials commonly used to realize topological phases, loss and gain are
often non-negligible [48, 49, 50, 51, 52, 53, 54]. For example, topological
lasers, which are promising technological applications of topological states,
are inherently non-Hermitian (NH) [55, 56, 57, 58, 59, 60]. Research into NH
band topology has uncovered numerous surprises, such as NH topological phases
distinct from any Hermitian counterpart [38, 39, 61, 62, 47]. Recently, there
have been studies showing that gain and loss can induce boundary states in 1D
lattices [63, 64, 65] and corner states in 2D lattices [66, 67]. To our
knowledge, however, topological bulk-defect correspondences have not yet been
studied in the non-Hermitian regime.

In this paper, we demonstrate that a NH lattice containing a disclination can
host topological disclination states associated with fractional (1/2) charge.
Unlike the previously-studied disclination states of Hermitian lattices [8, 9,
10, 11, 12, 17, 14, 18, 19, 15, 16, 20, 21, 22, 23, 24], these NH disclination
states are induced solely by on-site gain and loss, and are non-existent in
the Hermitian limit. The model consists of a 2D graphene-like honeycomb
lattice with a disclination; the application of a specific pattern of
gain/loss (imaginary mass terms) generates a bulk gap in the real part of the
energy spectrum, in which the disclination states appear [68, 69, 70, 71].
Previously, gain/loss-induced topological states have been found in NH models
like the Takata-Notomi model, a 1D lattice hosting NH midgap boundary states
[63], as well as a NH 2D HOTI with gain/loss-induced corner states [66]. NH
topological disclination states, however, have not yet been identified. (There
have been some studies of how lattice defects affect the non-Hermitian skin
effect, which is a separate issue [72, 73, 74, 75].) This work also
establishes NH charge fractionalization at a lattice defect, a phenomenon
previously limited to Hermitian models [14, 15, 16]. The charge
fractionalization is established using Wilson loops derived from biorthogonal
products, and also using a NH formulation of the density of states. We will
also discuss an alternative model that features NH disclination states
unaccompanied by fractional charge.

Figure 1: (a) Schematic of a 2D NH lattice containing a disclination, with
nearest-neighbor hopping $J$ (black links) and on-site gain $i\gamma$ (red
circles) or loss $-i\gamma$ (gray circles). (b) $\mathrm{Re}(E)$ spectrum for
the finite lattice in (a), but expanded to 630 sites. Marker colors indicate
the degree of localization to the bulk/edge/corners/disclination (see
Supplemental Materials [76]). Horizontal dashes indicate the bulk gap obtained
from Fig. 2(a), in which lie the edge, corner, and disclination states. The
energies labelled $1$–$5$ correspond to disclination states. (c) Plot of
$\mathrm{Re}(E)$ versus $\gamma$, with the same marker color scheme as in (b).
The shaded green area corresponds to the bulk gap, and vertical magenta dashes
indicate the reference gain/loss level $\gamma=1.3$. (d) Intensity
distributions for the disclination states in (b). (e) Complex energy spectrum
for the disclination-bearing lattice of (b) and (d). In all subplots, the
hopping is $J=1$. Figure 2: (a) Schematic of a 2D NH honeycomb lattice with an
18-site unit cell (yellow region). From this, Fig. 1(a) can be generated by
cutting out a $\pi/3$ sector (blue region). (b) Real bulk band diagram for
$\gamma=0$ (gray dashes) and $\gamma=1.3$ (purple lines). (c) $\mathrm{Re}(E)$
spectrum for a 762-site hexagonal sample with $\gamma=1.3$. Marker colors
indicate the degree of localization to the bulk/edge/corners [76]. In the bulk
gap (horizontal dashes), there are edge and corner states, with the latter
pinned to $\mathrm{Re}(E)=0$. Inset: intensity ($|\psi|^{2}$) distributions
for representative corner and edge states. (d) Wilson loops for the lowest
nine bands at $\gamma=1.3$. Upper inset: the unit cell for this calculation.
Lower inset: layout of Brillouin zones and reciprocal lattice vectors
$b_{1,2}$. (e) Positions of Wannier centers (black and blue stars) near the
disclination. Those adjacent to the disclination core (blue stars) yield a
fractional charge of $1/2~{}(\mathrm{mod}~{}1)$. In all subplots, $J=1$.

Finally, we study the prospects for realizing such a NH lattice using pumped
optical resonators. In such a setting, the gain/loss-induced topological
disclination mode has a distinctive experimental signature in the form of an
increase in emission intensity and abrupt shift in peak frequency as the
gain/loss level is tuned. The unique properties of NH topological disclination
states, which are gain/loss-induced and yet possess a level of robustness due
to their topological origins, may eventually be useful for designing novel
lasers and related devices.

Model—We begin with the 2D NH lattice shown in Fig. 1(a), which has real
nearest-neighbour hopping $J$ and bears a disclination at its core. Each site
is assigned an imaginary mass corresponding to gain ($i\gamma$, red circles)
or loss ($-i\gamma$, gray circles), where $\gamma\geq 0$ is the gain/loss
level [35]. The Hamiltonian is

$H=\sum_{n}i\gamma_{n}a^{\dagger}_{n}a_{n}+J\sum_{\langle
nn^{\prime}\rangle}\left(a^{\dagger}_{n^{\prime}}a_{n}+\mathrm{h.c.}\right),$
(1)

where $\gamma_{n}=\pm\gamma$, $a_{n}^{\dagger}$ and $a_{n}$ are the creation
and annihilation operators on site $n$, and $\langle nn^{\prime}\rangle$
denotes nearest neighbors. For $\gamma=0$, this reduces to a Hermitian
graphene lattice with a disclination [10], which has no bulk gap and no edge,
corner, or disclination states [76].

For nonzero $\gamma$, Fig. 1(b) shows the real part of the spectrum. The
lattice is based on Fig. 1(a) but expanded to 630 sites, with $\gamma=1.3$ and
$J=1$. There are several eigenenergies near the center of the gap, marked in
blue and red, which turn out to be edge and corner states localized to the
outer boundary of the sample [76].

We also see five eigenenergies, two doublets and a singlet, marked in yellow.
These turn out to be gain/loss-induced disclination states. As shown in Fig.
1(c), they emerge into the bulk gap at nonzero values of $\gamma$. Their
intensity distributions are plotted in Fig. 1(d), showing that they are
localized to the vicinity of the disclination core. The corresponding phase
plots, presented in the Supplemental Materials [76], reveal the singlet state
to have all five sites at the disclination core in-phase, whereas the doublets
have more complicated phase distributions. The distribution of eigenenergies
in the complex $E$ plane is shown in Fig. 1(e). Later, we show that these NH
disclination states are also associated with a fractional charge of $1/2$.

To understand these results, we consider the related disclination-free NH
honeycomb lattice shown in Fig. 2(a). It has $C_{6}$ rotational symmetry, with
a unit cell (yellow hexagon) of 18 sites. From this lattice, we can generate
the disclination lattice of Fig. 1(a) through a Volterra (“cut-and-glue”)
process, by removing a $\pi/3$ sector (blue-shaded region) and reconnecting
the seams. With this choice of gain/loss distribution, there is no seam or
discontinuity after the Volterra process (there are other gain/loss
distributions that are simpler and have smaller unit cells, but lack such a
property).

Fig. 2(b) shows the real part of the bulk band diagram for the NH lattice of
Fig. 2(a), with hopping $J=1$. In the Hermitian limit (i.e., $\gamma=0$), the
eigenenergies are real and gapless (gray dashes); this is simply the band
diagram for graphene, with the original $K$ and $K^{\prime}$ points folded
onto $\Gamma$. For $\gamma\neq 0$, a real line gap—i.e., a gap in the real
part of the energy, $\mathrm{Re}(E)$ [47]—opens up around $\mathrm{Re}(E)=0$
(purple lines). In Fig. 2(c), we plot $\mathrm{Re}(E)$ for a finite hexagonal
sample with 762 sites. This spectrum exhibits the same real line gap as the
bulk system, but with additional edge and corner states occupying the bulk
gap. The intensity distributions for two exemplary states are plotted in the
inset of Fig. 2(c). Evidently, the lattice behaves much like a NH version of a
HOTI [25], with the nonzero gain and loss generating a real line gap as well
as in-gap boundary states [66, 76].

Topological characterization—In Hermitian HOTIs, the topological states are
tied to the deeper phenomenon of robust charge fractionalization [14, 15, 16].
Here, we argue that the disclination in the present NH model induces a
fractional charge of 1/2.

For the periodic lattice of Fig. 2(a), we numerically calculate Wilson loops
involving all nine bands situated below the real line gap. We first adjust the
shape of the unit cell to avoid the ambiguity of certain sites falling on the
edges of the unit cell [Fig. 2(d), upper inset]. Next, we discretize the first
Brillouin zone along the reciprocal lattice vectors $b_{1}$ and $b_{2}$ [Fig.
2(d), lower inset]. Starting from a given $k$, the Wilson loop matrices are
[66, 67]

$\displaystyle W_{j}=F_{j}(k+N\Delta k_{j})\cdots F_{j}(k+\Delta
k_{j})F_{j}(k),$ (2)

where $N$ is the number of discretization points in the $b_{j}$ direction,
$\Delta k_{j}$ is the wavevector step, and $F_{j}(k)$ is a $9\times 9$ matrix
of biorthogonal products:

$F_{j}^{mn}(k)=\big{\langle}u_{m}^{L}(k+\Delta
k_{j})\big{|}u_{n}^{R}(k)\big{\rangle}.$ (3)

Here, $m,$ $n$ index bands below the line gap at $\mathrm{Re}(E)=0$, and
$u^{L/R}(k)$ denotes left/right eigenvectors of the Bloch Hamiltonian, which
satisfy the biorthogonality relation
$\bra{u_{m}^{L}(k)}\ket{u_{n}^{R}(k)}=\delta_{mn}$ [35].

From the eigenvalues $\lambda_{j}$ of $W_{j}$, we retrieve the phases
$\nu_{j}=-i\ln(\lambda_{j})$. Fig. 2(d) plots $\mathrm{Re}(\nu_{2})$ as the
initial $k$ point varies along the $b_{1}$ direction. Throughout this
trajectory, $\mathrm{Im}(\nu_{2})$ is close to zero. The plot for
$\mathrm{Re}(\nu_{1})$, as $k$ varies along $b_{2}$, behaves very similarly
and is thus omitted. The Wilson loop exhibits no overall winding, but crosses
$\pm\pi$ an odd number of times. This is the same behavior observed in the
Hermitian Wu-Hu model, using standard rather than biorthogonal inner products
to calculate $W_{j}$ matrices [77, 78, 79, 80]. That Hermitian model is known
to belong to the 2D Stiefel-Whitney class and supports a HOTI phase with
Wannier centers at the $3c$ Wyckoff position [80, 81, 82].

Informed by these results, we take a close-up view of the disclination-bearing
lattice from Fig. 1(a). As shown in Fig. 2(e), the Wannier centers (black and
blue stars) lie at the boundaries of the unit cells, contributing
$1/2~{}(\mathrm{mod}~{}1)$ spectral charge to each adjacent cell [14, 15].
Since the boundary of the disclination core passes through an odd number of
Wannier centers (blue stars), the disclination carries a fractional charge of
$1/2~{}(\mathrm{mod}~{}1)$. In Hermitian lattices, a similar argument for
charge fractionalization at disclinations [14] has been verified in
experiments on microwave metamaterials and photonic crystals [16, 15].

Another way to identify disclination charge is to examine the local density of
states [16, 15, 27]. For each site $j$, we define the expected density

$\langle\Pi_{jm}\rangle=\bra{\psi^{L}_{m}}\Pi_{j}\ket{\psi^{R}_{m}},$ (4)

where $\Pi_{j}=\ket{j}\bra{j}$ is a local projection operator and
$|\psi^{L/R}_{m}\rangle$ are the biorthogonal $m$-th left/right eigenstates of
the full lattice. For our NH model, $\langle\Pi_{jm}\rangle$ is complex-valued
[83, 84, 47]. We define the disclination charge as

$\displaystyle
Q_{d}=\sum_{mj}\big{|}\langle\Pi_{jm}\rangle\big{|}\;(\text{mod~{}1}),$ (5)

with $j$ summed over sites near the disclination core [76], and $m$ summed
over all states below the bulk gap—i.e., $\mathrm{Re}(E)$ below the bottom
dashes in Fig. 1(b). For the lattice of Fig. 2(e), we obtain $Q_{d}=0.53$,
close to the predicted value of $0.5$. This result is compatible with the
previous Wannier center analysis.

Figure 3: (a)–(b) An alternative NH lattice, similar to Figs. 1(a) and 2(a),
but with a different gain/loss pattern. (c) Wilson loop for the lowest nine
bands, with $J=1$ and $\gamma=0.75$. (d) Close-up view of the disclination-
bearing lattice, with the Wannier centers marked by black stars. Figure 4: (a)
Time domain simulation results for a disclination-bearing lattice [Fig. 1(a)]
driven by a monochromatic excitation on three sites at the disclination core
(green stars, inset), with $F=0$ on all other sites. The total intensity on
the five core sites (stars, inset) is plotted against the excitation frequency
$\omega_{0}$ for different $\gamma$, with $J=1$. (b) Plot of peak frequency
versus $\gamma$. The shaded green area indicates the bulk gap of the undriven
lattice. (c) Steady-state intensity profile for two values of $\gamma$. In the
large-$\gamma$ regime, the intensity is strongly localized to the disclination
core.

Alternative gain/loss pattern—Fig. 3(a)–(b) shows a lattice with an
alternative gain/loss distribution. Like the previous design, the periodic
lattice has 18 sites per unit cell, and the gain/loss pattern produces no seam
under the Volterra process. For $\gamma\neq 0$, this lattice exhibits a real
line gap, but no corner or disclination states (for details, see the
Supplemental Materials [76]). We thus interpret this as a NH analogue of a
HOTI in the trivial phase. Moreover, if we apply a disclination with a Frank
angle of $2\pi/3$, the resulting $C_{4}$-symmetric lattice does host a midgap
disclination state [76], consistent with Deng et al.’s finding that a Wu-Hu
lattice in its trivial phase hosts a disclination state for a $2\pi/3$
disclination but not a $\pi/3$ disclination [21].

It is also instructive to perform the Wilson loop calculation for this case.
As shown in Fig. 3(c), we obtain trivial windings reminiscent of the trivial
phase of the Hermitian Wu-Hu model with Wannier centers at the $1a$ Wyckoff
position [81, 82]. Fig. 3(d) shows the positions of the Wannier centers (black
stars), which lie at the centers of the unit cells and thus do not contribute
fractional charge to the disclination core. This is consistent with the
lattice’s aforementioned lack of gain/loss-induced disclination-bound states,
and further supported by the calculation yielding $Q_{d}=-0.08$.

Experimental signatures—The present model should be experimentally realizable
using classical-wave platforms, such as arrays of photonic resonators [65, 64,
67, 85, 86, 87]. The model’s inter-site hoppings are reciprocal, positive, and
nearest-neighbor, whereas the non-Hermiticity enters only as on-site gain and
loss. A possible complication is that inter-site hoppings in a real lattice
may be spatially inhomogeneous due to varying physical distances between
resonators. However, our numerical studies show that such inhomogeneities do
not substantially affect the disclination states [76].

As an example of how the NH disclination states could be identified in an
experiment, we insert the lattice of Fig. 1(a) into a driven Schrödinger
equation,

$\displaystyle i\frac{\partial\psi}{\partial
t}=H\psi-i\gamma\psi+Fe^{-i\omega_{0}t},$ (6)

where $\psi$ is the state vector, $H$ is the NH lattice Hamiltonian, and an
additional loss $\gamma$ is applied to all sites. This can describe a setup
where individual resonators have fixed loss and tunable gain, with the latter
tuned so that one sublattice has zero net gain/loss and the other has loss
$2\gamma$. Eq. (6) also includes an excitation of frequency $\omega_{0}$ and
spatially-dependent amplitude $F$, concentrated on three disclination core
sites as indicated in the inset of Fig. 4(a) (overlapping one of the
disclination states).

Fig. 4(a) shows the resulting total intensity on the five disclination core
sites, versus the excitation frequency $\omega_{0}$. As we tune $\gamma$ to
larger values, the intensity peak shifts closer to mid-gap and increases
greatly in magnitude. As shown in Fig. 4(b), the peak frequency exhibits an
abrupt jump, coinciding with the emergence of the disclination state into the
theoretically-predicted gain/loss-induced bulk gap. The steady-state intensity
profiles, plotted in Fig.4(c), show strong localization in the large-$\gamma$
regime.

Conclusion—We have shown that disclinations in 2D lattices can host non-
Hermitian disclination states, which emerge when a bulk gap is created by a
specific pattern of gain/loss on the lattice sites. The disclinations are
shown to carry fractional charge $1/2$, thereby extending recent results on
Hermitian topological disclination states into the non-Hermitian case. Because
these topological disclination states are induced by gain/loss tuning, they
might be used as the basis for novel lasers whose modes can be manipulated via
selective pumping.

This work was supported by the National Research Foundation (NRF), Singapore
under Competitive Research Programme NRF-CRP23-2019-0005 and NRF-
CRP23-2019-0007, and NRF Investigatorship NRF-NRFI08-2022-0001.

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