††thanks: Corresponding<EMAIL_ADDRESS>
# Fragile topological dislocation modes

Gabriel Malavé  Facultad de Física, Pontificia Universidad Católica de Chile,
Vicuña Mackenna 4860, Santiago 8331150, Chile  Jorge Schifferli  Departamento
de Física, Universidad Técnica Federico Santa María, Casilla 110, Valparaíso,
Chile  Rodrigo Soto-Garrido  Facultad de Física, Pontificia Universidad
Católica de Chile, Vicuña Mackenna 4860, Santiago 8331150, Chile  Pedro A.
Orellana  Departamento de Física, Universidad Técnica Federico Santa María,
Casilla 110, Valparaíso, Chile  Vladimir Juričić  Departamento de Física,
Universidad Técnica Federico Santa María, Casilla 110, Valparaíso, Chile
Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes
Alfvéns väg 12, SE-106 91 Stockholm, Sweden

###### Abstract

We here introduce the concept of fragile dislocation modes, which are
localized only in a fraction of a topological phase, while otherwise leak into
the bulk continuum. We show that such dislocation modes are hosted in an
obstructed atomic topological phase in the two-dimensional Su-Schrieffer-
Heeger model, but only in a finite region with an indirect gap at high energy.
These dislocation modes are realized as chiral pairs at finite energies with
protection stemming from a combination of the chiral (unitary particle-hole)
and the point group (C4v) symmetries, but only when the indirect gap is open.
In this regime, we corroborate the stability of the defect modes by following
their localization and also by explicitly adding a weak chemical potential
disorder. Our findings, therefore, should be consequential for the
experimental observation of such modes in designer topological crystals and
classical metamaterials.

Figure 1: Two-dimensional (2D) Su-Schrieffer-Heeger (SSH) model with a single
dislocation. (a) The lattice featuring a single dislocation with the Burgers
vector ${\bf b}=2a{\bf e}_{x}$, with $a$ as the distance between nearest
neighbor sites. The sites within the unit cell are labeled as
$\alpha=1,...,4$, and with different colors. The dashed and solid lines denote
intra- and inter-unit-cell hoppings, $\tau$ and $t$, respectively [Eq. (1)].
(b) Band structure for the defect-free system along high-symmetry lines in the
Brillouin zone (periodic system) with the corresponding spectrum displayed in
Fig. S2 of the Supplementary Materials (SM). Notice an indirect bandgap
between the two positive (negative) energy bands, which closes for
$|t/\tau|=0.5$, as shown in the SM, Sec. S1. (c) The spectrum of the 2D SSH
model [Eq. (1)] with a single dislocation, and open boundaries. Notice two
chiral pairs of dislocation modes (marked in red), with energies $E=\pm 0.72$
and $E=\pm 1.13$. The corresponding LDOS is shown Fig. 2, implying that the
modes are localized. The system size is $40\times 40$ unit cells. In (b) and
(c), we set $t=0.3$ and $\tau=1$. Additional plots of the band structure are
given in Fig. S1 of the SM. Figure 2: Two chiral pairs of the fragile
dislocation bound states in the topological phase. Positive energy modes at
(a) $E=0.72$ and (c) $E=1.13$, respectively, transform under $A_{1}$ and
$B_{2}$ irreducible representations of the C4v group while their chiral
partners at (b) $E=-0.72$ and (d) $E=-1.13$, exchange the representation, i.e.
transform under $B_{2}$ and $A_{1}$ representations, respectively. Notice that
the modes are localized within a few unit cells in the defect core. Zoom-in of
each mode is shown in the inset. The values of the parameters are identical to
Fig. 1. The red (blue) color denotes the positive (negative) phase of the
wavefunction $\Psi_{i}$, while the circle’s area is proportional to the
amplitude $|\Psi_{i}|$ at the lattice site $i$. Sites with an amplitude value
less than $10^{-2}$ are left empty. Consult the Supplementary Materials for
additional details.

Introduction.—The hallmark feature of topological phases are topological
electronic bands characterized by the bulk topological invariants [1, 2, 3, 4,
5], which thus cannot be localized on the real-space lattice [6], resulting in
the hallmark bulk-boundary correspondence [7, 8]. In this case, topological
lattice dislocation defects can host special topologically and symmetry-
protected localized modes, providing a common mechanism for the bulk probing
of topological phases covering a wide range of first- [9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22], higher-order phases [23], as well as the
dynamical Floquet [24] and non-Hermitian topological systems [25, 26, 27]. The
experimental observations of such dislocation modes were reported in both
quantum materials [28, 29] and classical metamaterials [30, 31, 32, 33]. This
defines the so-called bulk-dislocation correspondence, with an apparent
“duality” being at play: topological electronic wavefunctions in the bulk
therefore lacking Wannier representation in the real space yield localized
modes on the dislocation defects, provided that the band inversion is at a
finite momentum in the Brillouin zone (BZ) [9, 10, 11, 12, 14, 23]. On the
other hand, the absence of the localized dislocation modes does not imply
topological triviality of the phase, with the paradigmatic example being the
$\mathbb{Z}_{2}$ topological insulator featuring the band-inversion at
$\Gamma-$point [11].

Obstructed atomic topological phases possess Wannier representation, but at a
Wyckoff position away from the lattice site [34], with one-dimensional (1D)
Su-Schrieffer-Heeger (SSH) model as possibly the simplest example [35]. As a
consequence of this topological obstruction, they can feature charge
fractionalization [36, 37] and nontrivial disclination responses [38]. Given
the existence of this class of topological phases, a natural question arises
regarding dislocation response therein, since the phase admits a Wannier
representation, and therefore may obstruct the existence of the dislocation
modes, while its topological nature, which can be expressed through the band
representations at high-symmetry points (HSPs) [39, 34], is expected to yield
localized dislocation modes, for instance, when the band inversion is at a
finite momentum. As we show here, the answer is a superposition of the two
previously discussed possibilities: the dislocation modes become fragile.

Key results.—We here establish the existence of the fragile topological
dislocation modes, characterized as being localized, but only in a part of the
topological phase with an indirect bandgap at high energy, and otherwise are
delocalized. To this end, by performing numerical analysis of two-dimensional
(2D) SSH model, featuring a paradigmatic obstructed topological phase with
vanishing Berry connection [40], we find that a single dislocation can bind
several pairs of modes at finite energies throughout this phase (Fig. 1),
which are localized (Fig. 2) due to the dislocation-flux correspondence [23],
and symmetry protected against hybridization with each other. Crucially, our
numerical analysis of the evolution of the modes’ localization throughout the
topological phase demonstrates their stability (Fig. 3), but only in a finite
region of the topological phase, with the modes dissolving into the bulk when
a high-energy indirect gap closes, and the system becomes gapless, while, at
the same time, remaining topological. In addition, when localized, the
dislocation modes are robust with respect to a weak chemical-potential
disorder, see Fig. 4. Finally, the possible experimental relevance of these
fragile dislocation modes in metamaterials and the designer materials
platforms is outlined, particularly in a silicon-based designer lattice [41],
and a silicon dielectric metasurface platform [42].

2D SSH model..—We start with the Bloch Hamiltonian of the 2D SSH model on a
square lattice,
$\hat{H}=\sum_{\mathbf{k}}{{\Psi}}^{\dagger}_{\mathbf{k}}\mathbb{H}_{\mathbf{k}}{{\Psi}}_{\mathbf{k}}$,
which can be written in a block-off-diagonal form as [40]

$\displaystyle~{}\mathbb{H}_{\bf k}=\begin{pmatrix}0&\mathbb{Q}\\\
\mathbb{Q}^{\dagger}&0\end{pmatrix},\mathbb{Q}=\begin{pmatrix}{t+\tau
e^{2iak_{x}}}&{t+\tau e^{2iak_{y}}}\\\ {t+\tau e^{-2iak_{y}}}&{t+\tau
e^{-2iak_{x}}}\end{pmatrix},$ (1)

where $\Psi_{\bf k}=(c_{1,{\bf k}},c_{2,{\bf k}},c_{3,{\bf k}},c_{4,{\bf
k}})^{\top}$ represents the annihilation operator for the four sites in the
unit cell, as shown in Fig. 1(a), and $c_{\alpha,{\bf k}}$, $\alpha=1,...,4$
is the operator annihilating the state at the site $\alpha$ within the unit
cell, with the momentum ${\bf k}$, and $a$ as the nearest-neighbor distance.
Here, parameter $t$ ($\tau$) represents the inter- (intra-)unit-cell hopping
amplitude, and we fix $\tau=1$ in the following. See Sec. S1 of the
Supplemental Materials (SM) for additional details. The band structure in the
topological phase for the defect-free system is shown in Fig. 1(b), with high-
energy bands separated by an indirect bandgap from the low-energy ones for
$|t/\tau|<0.5$. Importantly, this Hamiltonian features $C_{4v}$ point-group,
and the chiral (unitary particle-hole) symmetry, $\\{\Pi,\mathbb{H}_{\bf
k}\\}=0$, generated by $\Pi=\sigma_{3}\otimes\sigma_{0}$, with $\sigma_{0}$ as
the $2\times 2$ identity matrix and $\sigma_{i}$ ($i=1,2,3$) the usual Pauli
matrices. The latter symmetry implies that the eigenstates of the Hamiltonian
(1) forming a chiral pair $(\Psi_{\bf k},\Pi\Psi_{{\bf k}})$ are with the same
absolute value but the opposite signs of energy, $\mathbb{H}_{\bf
k}\Pi^{n}\Psi_{\bf k}=(-1)^{n}E_{\bf k}\Pi^{n}\Psi_{\bf k}$ ($n=0,1$), and the
chiral operator exchanges the irreducible representations (irreps) of the
$C_{4v}$ group, $A_{1}$ into $B_{2}$, and vice versa.

Since the model features both space inversion, represented by
$\mathcal{I}_{S}=\sigma_{0}\otimes\sigma_{1}$ and time-reversal symmetry,
$\mathcal{I}_{T}=K$, with $K$ as the complex conjugation, the Berry connection
vanishes identically [40]. As such, topological phases of the 2D SSH model
[Eq. (1)] are distinguished by the relative difference of the irreps
pertaining to high-energy electronic bands, at the HSPs $M$ and $X(Y)$ with
respect to the $\Gamma$ point, which may be viewed as a double band-inversion
[43]. Namely, in a topologically trivial phase, with the Wyckoff position
centered at the site of the unit cell, the irreps at the HSPs are equal. In
the topological phase, on the other hand, the irreps of $C_{4v}$ ($C_{2v}$)
point group at the $M$ ($X$ and $Y$) point(s) differ with respect to the
$\Gamma$ point, moving the Wyckoff position to the center of the unit cell,
away from any atomic position, consequently yielding four zero-energy corner
modes, implying nontrivial edge Zak phases and charge fractionalization [36].
Importantly, only the zero energy states of the two closest-to-zero energy
bulk bands touching at the $M$ and $\Gamma$ points transform under the 2D $E$
representation of $C_{4v}$ group, while finite-energy modes transform under
$A_{1}$ and $B_{2}$ irreps. The two phases are separated by a phase
transition, at $|t|=1$ in the Hamiltonian (1), when the gap between the bulk
and the edge states closes, and the system becomes a featureless metal.

Fragile dislocation modes.—We now include a dislocation defect in the lattice,
as shown in Fig. 1(a), with the Burgers vector ${\bf b}=2a{\bf e}_{x}$, equal
to a primitive lattice vector. Since the lowest energy metallic bands touch at
the $M$ point, the dislocation acts as an effective $\pi$ flux, $\Phi_{\rm
dis}={{\bf K}_{M}}\cdot{\bf b}=\pi$ [9, 10, 11]. Furthermore, the Burgers
vector is orthogonal to the gapped edges, and the dislocation therefore binds
finite energy modes [23], which, as we show in the following, remain stable
until the indirect bandgap closes, and they dissolve into the bulk continuum.

To this end, we first implement the 2D SSH model, given by Eq. (1), on a real
space lattice with a single dislocation and open boundary conditions, see Fig.
1(a), and numerically diagonalize it. The numerical calculations were
performed using the Kwant code [44]. In the obtained spectrum, displayed in
Fig. 1(c), we show two chiral pairs of modes at finite energies, representing
two localized dislocation modes. The corresponding local density of states
(LDOS) is displayed in Fig. 2, and shows that the modes within the chiral pair
transform under $A_{1}$ and $B_{2}$ representations of the $C_{4v}$ group (see
also Sec. S2 and Fig. S3 in the SM). Furthermore, among these states, the two
with the same sign of energy transform under different 1D irreps, rendering
them mutually orthogonal. The modes are buried within the edge continuum,
separated by an indirect gap from the bulk modes [Fig. 1(c) and Fig. S4 in the
SM], and as such protected from the mixing with the bulk modes, while the
hybridization with edge modes is avoided due to the geometry (bulk-edge
separation). However, as the indirect gap is starting to close, the two bulk
bands approach the energy of the dislocation modes and they eventually
hybridize at the gap closing, $t=0.5$, since the dislocation and (some of) the
bulk modes become degenerate and transform under the same irreps, $A_{1}$ and
$B_{2}$.

Stability of the dislocation modes.—To establish the stability of the modes,
we further analyze their evolution by following the localization of the
closest-to-zero positive-energy mode throughout the topological phase, see
Fig. 3. Notice first that when the indirect gap is open, the spectral weight
of the dislocation mode is distributed within two unit cells around the defect
center [Fig. 3(a)-(d)], see also Fig. 2(a) for the real space profile of the
mode when $t=0.3$. Once the system enters the parameter regime close to the
indirect bulk gap closing ($t=0.5$), the maximum spectral weight decreases
significantly, as can be seen from Fig. 3(e). Upon entering the gapless
regime, the dislocation modes dissolve into the bulk, see Fig. 3(f), due to
the hybridization with the bulk modes. See also Fig. S5 in the SM for the
evolution of the LDOS of the dislocation modes in the topological phase.

The evolution of the dislocation and the corner modes throughout the
topological phase are qualitatively comparable, up to the closing of the
indirect gap, see Fig. S7 in the SM. Notice that the maximum spectral weight
of the corner modes also significantly drops as the system enters the gapless
regime. We emphasize that the gap between two modes scales with the indirect
gap, indicating that the dislocation can serve as a bulk probe of this
topological phase, but only in the regime with the indirect gap, see Fig. S6
in the SM.

Figure 3: Localization of the closest-to-zero positive-energy dislocation mode
in the topological phase. The local density of states (LDOS) of the mode is
shown for the following values of the intra-unit-cell hopping parameter $t$ in
the Hamiltonian (1): (a) $t=0$; (b) $t=0.1$; (c) $t=0.2$; (d) $t=0.3$; (e)
$t=0.4$; (f) $t=0.5$. We fix $\tau=1$, the system size is $40\times 40$ unit
cells, and the LDOS is shown only up to $10$ lattice sites away from the
dislocation center since the LDOS farther away is negligible. The distance is
given in units of the nearest-neighbor spacing, $a$. The LDOS profile of the
mode on the square lattice for $t=0.3$ is shown in Fig. 2(a). The analogous
plots for the corner modes are available in Sec. S4 of the Supplementary
Materials.

We furthermore analyze the stability of the modes with respect to the
chemical-potential disorder on average, preserving both chiral and $C_{4v}$
symmetries. We implement the random chemical potential so that at a particular
lattice point $i$, the chemical potential is given by:

$~{}\mu_{i}=\mu(1+\delta_{i}),$ (2)

where the random variable $\delta_{i}\in[-x,x]$ is uniformly distributed, with
$x$ as the variation, taken to be fixed. We clearly observe that the modes
remain localized for a weak chemical-potential disorder, see Fig. 4(a), while
in the strong-disorder regime, the modes delocalize into the bulk, as shown in
Fig. 4(b).

_Mechanism of the modes’ emergence and dissolution_. To explain the emergence
of dislocation modes, we invoke the chiral and $C_{4v}$ symmetries of the 2D
SSH model [Eq. (1)], which, in fact, _guarantee_ their existence, together
with dislocation-flux correspondence [23]. To this end, we first recall that
the dislocation modes are formed by the Volterra process connecting the two
edges across the removed line of unit cells ending at the dislocation center
[Fig. 1(a)]. The low-energy modes localized at the two sides of the “trench”
formed before the dislocation is constructed by connecting the two edges then
hybridize to yield well localized defect modes. The 2D SSH topological metal
has two zero-energy (finite-energy) modes at the lower (upper) corners of the
trench, as shown in Fig. S3 of the SM, for a representative value $t=0.3$. Two
pairs of the top trench modes transform under even ($A$) and odd ($B$)
representations of the $C_{2}$ group, which become one-dimensional $A_{1}$ and
$B_{2}$ irreps of the $C_{4v}$ point group after reconnecting the sides of the
trench in the Volterra process, and thereby restore the complete $C_{4v}$
symmetry, see also Sec. S2 of the SM. This irrep content is fixed by the
invariance of the modes under the (two) diagonal reflections in the unit cell.
After the edges are reconnected, all the dislocation modes are at finite
energy due to the hybridization and the $\pi-$flux localization, with the
chiral symmetry forcing the modes to organize in chiral pairs. Furthermore,
the chiral pairs, in general, emerge with absolute values of energy, say
$\varepsilon_{1}>\varepsilon_{0}$, where $\varepsilon_{1}$ ($\varepsilon_{0}$)
corresponds to the finite (zero) energy modes before the reconnection. If
positive energy dislocation modes transform under $B_{2}$ and $A_{1}$ irreps,
then their chiral partners at negative energies swap the irreps, i.e., they
transform under $A_{1}$ and $B_{2}$, and therefore, the modes are pairwise
protected, independently of the energy splitting between the modes (they can
even be degenerate and still orthogonal). Now, the crucial part is that the
modes in high-energy bulk bands close to the $M$ point transform under the
same $A_{1}$ and $B_{2}$ representation of the $C_{4v}$ group, and therefore
dislocation modes are protected from hybridization as long as the indirect gap
between the positive energy bands is finite. Once this gap closes, the
dislocation modes leak in the bulk, and therefore dissolve, since the spectral
gap protection is no longer operative, as can be explicitly observed in Fig.
S5 in the SM. Finally, even though the dislocation modes transform under the
same irreps as the edge modes, the mixing between them is suppressed by the
geometry, i.e. the bulk-edge separation.

Experimental feasibility.—The primary candidate for the realization of the
proposed fragile dislocation modes is the Si lattice, recently reported to
host the 2D SSH model in a rectangular geometry [41]. In particular, the
tunable lattice geometry of this platform should allow for a straightforward
implementation of a dislocation defect, thereby probing the defect modes via
scanning tunneling spectroscopy. Furthermore, designed metasurfaces, such as
the one recently reported in a silicon-based system [42], offer another
promising route for their realization, which furthermore may control the
energy splitting of the modes through the tunability of the hopping. Photonic
crystals [30, 45], acoustic crystals [46], mechanical resonator networks [31],
and topolectric circuits [47, 48, 49, 50], represent other possible platforms
for the realization of the fragile dislocation modes.

Discussion & outlook.—In this paper, we have demonstrated that dislocation
defects in an obstructed atomic topological phase can host fragile dislocation
modes: the defect modes which are stable, but only in a finite fraction of the
topological phase. Our results furthermore elucidate the role of the structure
of the spectrum within the same topological phase for the response of the
defects.

Our findings should be consequential for further understanding of the
obstructed phases in terms of the physical observables, and are expected to
corroborate future studies of their stability in terms of disorder and
interaction effects. Their generalization to higher-dimensional SSH models
[51], where crystalline, (anti-)unitary particle-hole and/or time-reversal
symmetries may provide a further protection to the fragile dislocation modes,
is an interesting prospect for future investigation, which may be relevant for
quantum materials, e.g., bismuth [52].

In closing, our work leaves an open question regarding the possibility of the
existence of dislocation, or more generally, lattice-defect bound states in
the continuum (BICs), _symmetry-protected_ against the hybridization with bulk
bands [53]. This issue is especially pertinent given that the zero-energy
corner modes in the 2D SSH model form symmetry-protected BICs [54], and were
experimentally observed [45, 55], while BICs are relevant in a variety of
topological condensed-matter systems [56, 57, 58, 59, 60, 61]. Our discovery
of fragile dislocation modes may encourage further research of the bulk-defect
correspondence in obstructed atomic topological phases, and, in particular,
provide new physical insights, e.g. in terms of experimental probes, into
topological classifications based on the symmetry content of the electronic
bands [39, 34].

Acknowledgments.—This work was supported by ANID/ACT192023 (G.M. and R.S-G),
ANID/ACT210100 (R.S.-G. and V.J.), the Swedish Research Council Grant No. VR
2019-04735 (V.J.), Fondecyt (Chile) Grants No. 1200399 (R.S.-G.), No. 1201876
(P.A.O. and J. S.) and No. 1230933 (V.J. and P.A.O.) Nordita is partially
supported by Nordforsk. V. J. is thankful to Bitan Roy for useful discussions.

Figure 4: Effect of the random chemical potential on the local density of
states corresponding to the dislocation modes in Fig. 2(a). We implement the
disorder [Eq. (2)] with a fixed variation $x=0.1$ and values of the chemical
potential (a) $\mu=0.3$ and (b) $\mu=3$. The color code is the same as in Fig.
2, sites with an amplitude value less than $10^{-2}$ are left empty, and zoom-
in of each mode is shown in the inset.

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