# Dislocation as a bulk probe of higher-order topological insulators

Bitan Roy<EMAIL_ADDRESS>Department of Physics, Lehigh University,
Bethlehem, Pennsylvania, 18015, USA  Vladimir Juričić<EMAIL_ADDRESS>Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes
Alfvéns väg 12, SE-106 91 Stockholm, Sweden  Departamento de Física,
Universidad Técnica Federico Santa María, Casilla 110, Valparaíso, Chile

###### Abstract

Topological materials occupy the central stage in the modern condensed matter
physics because of their robust metallic edge or surface states protected by
the topological invariant, characterizing the electronic band structure in the
bulk. Higher-order topological (HOT) states extend this usual bulk-boundary
correspondence, so they host the modes localized at lower-dimensional
boundaries, such as corners and hinges. Here we theoretically demonstrate that
dislocations, ubiquitous defects in crystalline materials, can probe higher-
order topology, recently realized in various platforms. We uncover that HOT
insulators respond to dislocations through symmetry protected finite-energy
in-gap electronic modes, localized at the defect core, which originate from an
interplay between the orientation of the HOT mass domain wall and the Burgers
vector of the dislocation. As such, these modes become gapless only when the
Burgers vector points toward lower-dimensional gapless boundaries. Our
findings are consequential for the systematic probing of the extended bulk-
boundary correspondence in a broad range of HOT crystals, and photonic and
phononic or mechanical metamaterials through the bulk topological lattice
defects.

##  I Introduction

The nontrivial topological invariant characterizing the bulk electronic band
structure gives rise to robust edge or surface modes, manifesting the hallmark
of a topological material - the bulk-boundary correspondence hasan-kane-
rmp2010 ; qi-zhang-rmp2011 . As such, these boundary modes have been so far
almost exclusively used to experimentally detect nontrivial electronic
topology, both in gapped molenkamp-science2007 ; hsieh-nature2008 ; xia-
natphys2009 ; hsieh-science2010 ; chen-science2010 ; dziawa-natmat2012  and
gapless xu-science2015 ; lv-prx2015  systems. Equally important, but much less
explored, is the direct probing of topological states in the bulk without
invoking the boundary modes, through their response to topological lattice
defects, such as _dislocations_ ran-natphys2009 ; teo-prb2010 ; juricic-
prl2012 ; asahi-prb2012 ; slager-natphys2013 ; hughes-prb2014 ; slager-prb2014
; you-prb2016 ; nag-roy-2020 ; panigrahi-2021 . Moreover, the topological
defect modes are more pristine, being immune to contamination by the
interfaces and independent of the surface termination. In fact, in the context
of experimental probing of topology in the quantum materials this aspect has
started to gain prominence only recently hamasaki-apl2017 ; nayak-sciadv2019 .

In $D-$dimensional $n$th order topological states benalcazar-science2017 ;
benalcazar-prb2017 ; song-prl2017 ; langbehn-prl2017 ; schindler-sciadv2018 ,
bulk probing of the electronic band topology should play an important role,
because the extended bulk-boundary correspondence is realized through gapless
modes on the lower, $(D-n)-$dimensional boundaries, characterized by
codimension $d_{c}=n$, such as hinges ($d_{c}=D-1$) and corners ($d_{c}=D$)
schindler-natphys2018 ; serra-nature2018 ; peterson-nature2018 ; imhof-
natphys2018 ; mittal-natphot2019 ; zhang-natphys2019 ; ni-natmat2019 ;
kempkes-natmat2019 . Their robustness originates from the combination of
spatial symmetries, such as discrete rotations, and non-spatial ones, such as
the reversal of time. Importantly, these protected modes on lower-dimensional
boundaries may be thought of as inherited from the parent, first-order
topological state (with $n=1$) upon partially gapping out its edge or surface
modes ($d_{c}=1$), which can, in turn, yield a hierarchical ladder of HOT
states calugaru-prb2019 ; nag-juricic-roy-prb2021 . This is accomplished by a
suitable _domain wall_ mass which changes sign across corners [see Fig. 1(a)]
or hinges, thus localizing topological modes at these lower-dimensional
boundaries [see Figs. 1(b), 2, 3(a) and 3(c)]. The reduced dimensionality of
the boundary may, however, hinder the experimental detection of the gapless
modes, and therefore HOT states require other means to directly probe the bulk
electronic topology.

As we demonstrate here, dislocations can serve as bulk probes of HOT
insulators through the binding of special topologically protected electronic
modes, see Figs. 1 and 3. To formulate the mechanism, we recall the Volterra
construction: in a two-dimensional (2D) lattice a dislocation can be created
by removing a line ending at the dislocation center (Volterra cut), and
reconnecting the sites across this cut so that the translational symmetry is
restored away from the defect center (core), see Fig. 1(a). Therefore, any
closed loop around the dislocation center features a missing translation by
the Burgers vector ${\bf b}$, which topologically characterizes the defect. As
such, a dislocation provides global frustration to the underlying crystalline
order, which translates into a nontrivial effect on the electrons hopping on
the lattice. Namely, an electron with a momentum ${\bf K}$ when encircling the
dislocation picks up a phase equal to ${\rm exp}[{i\Phi_{\rm dis}}]$, with
$\Phi_{\rm dis}={\bf K}\cdot{\bf b}\,({\rm mod}\,2\pi)$. In particular, for
topological states with the band inversion momentum at ${\bf K}_{\rm inv}$,
the hopping phase is $\Phi_{\rm dis}={\bf K}_{\rm inv}\cdot{\bf b}\,({\rm
mod}\,2\pi)$ ran-natphys2009 .

Figure 1: Dislocation defect on a square lattice. (a) The defect is obtained
through the Volterra cut-and-glue procedure by removing a line of atoms ending
at the center of the lattice (orange) and reconnecting the edges across this
Volterra cut, which are denoted by $+$ and $-$, right and left from the
center, respectively. The corresponding Burgers vector is ${\bf b}=-a{\bf
e}_{x}$. The HOT mass domain walls along (across) which it vanishes (changes
sign) are represented by the red and blue dashed lines for $\theta=0$ and
$\pi/2$, respectively, as given by Eq. (1). (b) Local density of states (LDoS)
for the dislocation mode localized at the defect core together with the four
corner modes in a second-order translationally-active HOT insulator with the
band inversion at the $M$ point of the BZ. Here, we set $\theta=0$ in the HOT
mass, so that the defect modes are at finite energies, while the corner modes
are at zero energy. (c) LDoS for the zero-energy dislocation modes in the
periodic system with a dislocation-antidislocation pair for $\theta=\pi/2$.
Here, we set $t=2B=1$, $m=3$ and $\Delta=0.20$ [see Eq. (2)]. Any site with
LDoS less than $10^{-3}$ is left empty. See also Appendix D.

To set the stage, recall that a _translationally-active_ first-order
topological insulator features at least one band inversion at a finite
(non-$\Gamma$) momentum in the Brillouin zone (BZ) slager-natphys2013
yielding gapless edge states. When $\Phi_{\rm dis}=\pi$ in a translationally-
active topological insulator, after encircling a dislocation, the electrons
pick up a hopping phase equal to $\exp({i\pi})=-1$ across the Volterra cut,
see Fig. 1(a). In turn, to resolve the frustration in the hopping introduced
by the defect through the nontrivial phase factor, a Kramers pair of zero-
energy states gets localized at the dislocation core juricic-prl2012 .

Figure 2: Spectral features of the dislocation modes in a 2D translationally-
active HOT insulator for $t=2B=1$ and $m=3$ [see Eqs. (1) and (2)]. (a) The
scaling of the spectral gap ($\delta E$) among four states localized at the
core of the dislocation with $\theta$ measuring the orientation of the four-
fold symmetry breaking mass domain wall [see Eq. (1)]. Note that these modes
become zero-energy states as $\theta\to\pi/2$, i.e. when the Burgers vector
becomes parallel to one of the four-fold symmetry breaking axes [see Fig.
1(a)]. The various choices of the amplitude of the $C_{4}$ symmetry breaking
mass ($\Delta$) are quoted in the figure. For small $\Delta$, the spectral gap
($\delta E$) scales _linearly_ with (b) $\cos\theta$ and (c) $\Delta$ [see Eq.
(4)]. Consult Appendix D for additional details.

In contrast, a second-order topological insulator features _gapped_ edge
states that stem from the mass domain wall in the bulk [Figs. (1)(a) and 4].
The domain wall mass gaps out the edge states but only _partially_ , in turn
producing the topological corner modes through the Jackiw-Rebbi mechanism
jackiw-rebbi . Now, when a dislocation is inserted, the defect modes, as we
show, still survive [Figs. 1(b),(c)], but, are moved away from zero energy,
since the edge states across the Volterra cut are gapped [Fig. 2(a)].
Importantly, when the orientation of the Burgers vector (${\bf b}$) is
parallel to the direction of the domain wall of the HOT mass, the defect modes
are pinned at zero energy. Hierarchy of the HOT states in this way directly
translates into the _spectral flow_ of the dislocation modes, detectable in
the tunneling spectroscopy measurements, for instance. The same mechanism is
analogously operative for three-dimensional (3D) second-order translationally-
active insulators: an edge dislocation hosts gapped modes which become gapless
only when the Burgers vector is parallel to the HOT mass domain wall.
Furthermore, a screw dislocation hosts gapless propagating modes only when it
is orthogonal to a gapless surface or equivalently parallel to the
corresponding surface normal [Fig. 3 and Fig. 5]. Finally, we emphasize that
the _composite_ $C_{4}{\mathcal{T}}$, ${\mathcal{P}}{\mathcal{T}}$ and
$C_{4}{\mathcal{P}}$ symmetries protecting the zero-energy corner (hinge)
modes, also protect both finite- and zero-energy dislocation modes, displayed
in Figs. 1(b), 3(a), and 3(c). Here $C_{4}$, ${\mathcal{T}}$ and
${\mathcal{P}}$ represent discrete four-fold rotational, time-reversal and
parity symmetries, respectively. For details consult Appendix E, where it is
also shown that this protection mechanism for dislocation modes extends to
$C_{4n}$ rotational symmetry breaking HOT insulators in two and three
dimensions, with $n>1$. Furthermore, in Appendix F we show that the defect
modes are protected also in the case of $C_{4n+2}$ rotational symmetry
breaking HOT insulators, where $n\geq 1$.

The rest of the paper is organized as follows. In Sec. II, we discuss the
universal tight-binding model for the HOT insulators in $d=2$ and $d=3$.
Section III is devoted to the numerical results for the dislocation modes on a
square lattice. In Sec. IV we present a general argument for the existence of
the dislocation modes in a HOT insulator, and in Sec. V we show the numerical
results for the dislocation modes in 3D HOT insulators. We discuss the results
in Sec. VI and highlight their possible realizations in HOT crystals and
metamaterials. Additional technical details are relegated to the Appendices.

##  II Tight-binding model

To show the outlined general mechanism, we take the minimal, but the universal
tight-binding model describing a second-order topological insulator in $d=2$
and $d=3$ trifunovic-prx2019 ; calugaru-prb2019 , with the Hamiltonian
$H=\sum_{{\bf k}}\Psi_{\bf k}^{\dagger}\hat{h}\Psi_{\bf k}$, where
$\hat{h}=\hat{h}_{0}+\hat{h}_{\Delta}$, and

$\displaystyle\hat{h}_{0}$ $\displaystyle={\bf d}({\bf
k})\cdot{\bm{\Gamma}}+M({\bf k})\Gamma_{d+1},$ $\displaystyle\hat{h}_{\Delta}$
$\displaystyle=\Delta\;\Gamma_{d+2}\;\big{\\{}\cos\theta\left[\cos(k_{x}a)-\cos(k_{y}a)\right]$
$\displaystyle+\sin\theta\left[\sin(k_{x}a)\sin(k_{y}a)\right]\big{\\}}\equiv\Delta({\bf
k},\theta)\;\Gamma_{d+2}.$ (1)

Here $({\bm{\Gamma}},\Gamma_{d+1},\Gamma_{d+2})$ are the mutually
anticommuting four-component $\Gamma$ matrices, ${\bf k}$ is the momentum, and
$a$ is the lattice spacing. The above Hamiltonian breaks C4 rotational
symmetry about the $z-$axis, generated by $R_{4}=\exp(i\pi\Gamma_{12}/4)$,
where $\Gamma_{12}=[\Gamma_{1},\Gamma_{2}]/(2i)$, the time-reversal and the
parity symmetries with (representation-dependent) operators ${\mathcal{T}}$
and ${\mathcal{P}}$, respectively, but preserves their products
$C_{4}{\mathcal{T}}$, ${\mathcal{T}}{\mathcal{P}}$ and $C_{4}{\mathcal{P}}$.
Under four-fold rotation $k_{x}\to-k_{y}$ and $k_{y}\to k_{x}$. It should be
noted that with the above form of the generator of rotations, following the
Lie group, the HOT mass _always_ breaks discrete rotational symmetry.

In both $d=2$ and $d=3$, we take the form factors $d_{i}({\bf k})=t\sin
k_{i}a$, while the first-order mass is given by

$\displaystyle~{}M({\bf k})$
$\displaystyle=m-2B\bigg{[}d-\sum^{d}_{j=1}\cos(k_{j}a)\bigg{]}.$ (2)

We consider translationally-active $M$ phase in $d=2$ with a band inversion at
$M=(\pi/a,\pi/a)$ point in the BZ, which is realized in the parameter range
$4<m/B<8$. In $d=3$ we take translationally-active $R$ phase with the band
inversion at $R$ point in the BZ, $R=(\pi/a,\pi/a,\pi/a)$, for $8<m/B<12$. The
resulting first-order topological insulator supports edge and surface states,
both with $d_{c}=1$, respectively, in $d=2$ and $d=3$. Furthermore,
$\hat{h}_{\Delta}$ acts as a mass term for the topological edge (surface)
states, and leaves only the corners (hinges) gapless, yielding a second-order
topological insulator with corner (hinge) modes with $d_{c}=2$ [Figs. 1(b),
3(a), 3(c)]. To be concrete, in Eq. (1) we fix this Wilson-Dirac mass term so
that it changes sign under the $C_{4}$ rotation, transforming $k_{x}\to-
k_{y}$, $k_{y}\to k_{x}$. As such, this mass term for each value of the
parameter $0\leq\theta\leq\pi/2$ necessarily features a line across which it
changes sign. In particular, for $\theta=\pi/2$ the domain wall lies along the
principal axes, $k_{x}=0,k_{y}=0$, while for $\theta=0$ it is located along
the diagonals $k_{y}=\pm k_{x}$. See Fig. 1(a) for the HOT mass domain walls
in the real space.

##  III 2D Lattice dislocation modes

We perform numerical analysis of the translationally-active HOT $M$ phase,
hosting a band inversion at the $M=(\pi/a,\pi/a)$ point in the BZ. The
implementation of the model, given by Eq. (1), was carried out in the real
space on a square lattice hosting a dislocation defect with the Burgers vector
${\bf b}=a{\bf e}_{x}$, oriented in the lattice $x-$direction, as shown in
Fig. 1(a). See also Appendix D for details. In an open system, the dislocation
modes and the corner states coexist in the HOT $M$ phase, explicitly showing
that the defect can probe the extended bulk-boundary correspondence, see Fig.
1(b). Furthermore, we find that dislocations bind the modes in a lattice
without boundaries (periodic system), further corroborating their role as a
pure bulk probe of higher-order electronic topology, see Fig. 1(c). The
hybridization effects in both cases can be neglected, as the defect modes are
localized within a few lattice sites around its center, which is much shorter
than both the system size and the separation between the defects.

Figure 3: Dislocation defect modes in a 3D HOT insulator on a cubic lattice.
LDoS for the _closest_ to zero energy modes localized at (a) an edge
dislocation directed in the $z$-direction with the Burgers vector ${\bf
b}=a{\bf e}_{x}$ together with the zero-energy hinge modes, (b) an edge
dislocation-antidislocation pair directed in the $z$-direction with the
Burgers vectors ${\bf b}=\pm a{\bf e}_{x}$, (c) a screw dislocation with the
Burgers vector ${\bf b}=a{\bf e}_{z}$ together with the zero-energy hinge
modes, and (d) a screw dislocation-antidislocation pair with the Burgers
vectors ${\bf b}=\pm a{\bf e}_{z}$. Here (a) and (c) [(b) and (d)] are open
[periodic] systems. The edge dislocation modes are gapped [see Figs.
5(a),(b),(c)], while screw dislocation modes are gapless within the numerical
accuracy [${\mathcal{O}(10^{-6})}$], as explicitly shown in Fig. 5(f). In all
the cases the system is in a second-order topological $R$ phase with the band
inversion at the $R=(1,1,1)\pi/a$ point of the BZ. The orientation of the HOT
mass domain wall is fixed to be along the diagonals $k_{y}=\pm k_{x}$, i.e.
$\theta=0$ in Eq. (1). Throughout we set $t=B=1$, $m=10$, and $\Delta=0.40$
[see Eqs. (1) and (2)]. Details of the numerical analysis are presented in
Appendix D.

Most importantly, for any choice of the domain wall orientation ($\theta$), we
find that the defects feature mid-gap bound states at _finite_ energies, see
Fig. 2(a). As the domain wall orientation approaches the direction of the
Burgers vector ($\theta\to\pi/2$), the spectral gap ($\delta E$) between the
dislocation modes decreases. Eventually, when the two directions coincide
($\theta=\pi/2$), the modes become degenerate zero energy states. For the
spectral flow of the dislocation modes, see Fig. 6. Also notice that finite
energy dislocation modes are particle-hole partners, while they become
eigenmodes of the particle-hole operator when pinned at zero energy. See
Appendix C for details. The scaling of the gap with the domain wall
orientation for various amplitudes of the Wilson-Dirac mass ($\Delta$) is
displayed in Fig. 2(a), showing that $\delta E\to 0$ as $\theta\to\pi/2$. We
next present a general argument supporting this observation.

##  IV Dislocation modes: a general argument

To this end, we recall that in the parent first-order topological insulator
before reconnecting the edges across the Volterra cut, each edge (1) features
a Kramers pair of zero energy (due to a unitary particle-hole symmetry, see
Appendix C) helical modes, and (2) is perpendicular to the Burgers vector
${\bf b}$. The zero energy states at each of the edges are then the
eigenstates of the matrix $A_{\bf b}=i\Gamma_{\bf b}\Gamma_{3}$, where
$\Gamma_{\bf b}={\bm{\Gamma}}\cdot{\bf b}/|{\bf b}|$, and $\Gamma_{3}$ is the
mass matrix for the first-order topological insulator in $d=2$ [see Eq. (1)
and (2)], as explicitly shown in the Appendix A. The dislocation defect, which
is created by the Volterra construction, with the associated hopping $\pi$
phase factor, gives rise to the level repulsion among four zero modes at the
_pasted_ edges across the Volterra cut. However, a Kramers pair of modes
$|\Psi_{0}\rangle$ still remains pinned at zero energy and gets localized in
the defect core ran-natphys2009 .

The crucial observation is that the HOT mass matrix $\Gamma_{4}$ _commutes_
with the dislocation or edge-mode matrix $A_{\bf b}$, $[\Gamma_{4},A_{\bf
b}]=0$. The HOT mass matrix therefore reduces in the eigen-subspaces of
$A_{\bf b}$, introducing the level repulsion between the two zero modes and
thus symmetrically splits them about the zero energy. Furthermore, this
implies that the modes do not change the form, i.e. they remain localized
around at the defect, after introducing the HOT mass. The energy splitting
reads as

$\delta E=\langle\Psi_{0}|\Delta[{\bf K}_{\rm inv}-i{\bf b}({\bf
b}\cdot{\nabla})]|\Psi_{0}\rangle.$ (3)

Unless $\Delta[{\bf K}_{\rm inv}-i{\bf b}({\bf b}\cdot{\nabla})]=0$, i.e. when
the HOT mass vanishes in the direction of the Burgers vector, the energy
splitting is non-zero ($\delta E\neq 0$). Therefore, the interplay between the
orientation of the HOT mass domain wall and the Burgers vector pins the
dislocation modes precisely at zero energy when the two directions are
_parallel_ [see Fig. 1(a)]. The mechanism for the splitting of the dislocation
modes is also operative for gapping out the edges, ultimately yielding the
corner modes, implying that a dislocation can directly probe higher-order
bulk-boundary correspondence.

This mechanism captures the existence of the localized pair of dislocation
modes in the $M$ phase, split by the energy gap

$~{}\delta E=2\Delta\cos\theta{\tilde{E}},$ (4)

where

${\tilde{E}}\sim\left|\int
dx\left(\Psi_{0}^{(1,2)}(x)\right)^{\dagger}\partial_{x}^{2}\Psi_{0}^{(1,2)}(x)\right|,$

and $\Psi_{0}^{(1,2)}(x)$ are the zero-energy dislocation states in the first-
order phase (see Appendix A). The obtained energy gap ($\delta E$) implies
that the modes are pinned at zero energy only when the Burgers vector is
parallel with the mass domain wall ($\theta=\pi/2$), see Fig. 2(a).
Furthermore, for small $\Delta$, $\delta E$ scales linearly with $\cos\theta$
[see Fig. 2(b)] and $\Delta$ [see Fig. 2(c)]. Finally, a dislocation does not
feature any bound states either when $\Phi_{\rm dis}=0$ (as in the $\Gamma$
phase) or in the trivial phase.

##  V Dislocations in 3D HOT insulator

The above arguments can be straightforwardly extended to 3D second-order
topological insulators. Notice first that a 3D edge dislocation can be
obtained from its 2D analogue by translating it along an out-of-plane lattice
vector chaikin-book . Therefore, an edge dislocation in a 3D second-order
topological insulator should in general feature _gapped_ modes, which,
however, become _gapless_ for the Burgers vector parallel to the HOT mass
domain wall direction, analogously to the 2D case [see Fig. 5(a)].

A screw dislocation, being a true 3D defect, features the Burgers vector ${\bf
b}$ parallel to its orientation. An electron encircling the dislocation defect
once, skips a lattice distance $|{\bf b}|$ along the defect relative to the
perfect crystal chaikin-book . The “${\bf K}\cdot{\bf b}$” rule implies that
an electron then picks up a phase $\Phi_{\rm dis}={\bf K}_{\rm inv}\cdot{\bf
b}$. When this phase is nontrivial in a translationally-active first-order
insulator, the screw dislocation hosts gapless propagating modes ran-
natphys2009 . On the other hand, in a translationally-active HOT insulator
some of the surfaces are gapped, and the screw dislocation hosts gapless
propagating modes _only_ when it is oriented perpendicular to gapless
surfaces. See Appendix B. In particular, in a 3D second-order topological
insulator, with a single mass domain wall [see Eq. (1)], a screw dislocation
parallel to it hosts gapless modes, since the defect then pierces gapless $xy$
surfaces. Otherwise, a dislocation perpendicular to gapped $xz$ or $yz$
surfaces features gapped one-dimensional modes. See Fig. 5(f).

We numerically confirm this scenario in a 3D second-order topological
insulator described by the tight-binding model, exemplifying the $R-$phase on
the cubic lattice (see Eq. (1) and Appendix D). Both single edge dislocation
with Burgers vectors ${\bf b}=a{\bf e}_{x}$ extending in the $z-$direction in
an open system, which is also the propagation direction of the zero-energy
hinge modes, and a edge dislocation-antidislocation pair extending in the same
direction with Burgers vectors ${\bf b}=\pm a{\bf e}_{x}$ in a periodic system
indeed yield finite energy states when $\theta\neq\pi/2$, as shown in Fig.
3(a) and Fig. 3(b), respectively. For $\theta=\pi/2$ the dislocation modes
become gapless (same as in 2D). See Figs. 5(a),(b),(c). On the other hand, for
a single screw dislocation with ${\bf b}=a{\bf e}_{z}$ in a open system
(coexisting with the zero-energy hinge modes) and a screw dislocation-
antidislocation pair with ${\bf b}=\pm a{\bf e}_{z}$ in a periodic system, we
obtain _gapless_ dislocation modes for any HOT mass domain wall orientation
$\theta$ in Eq. (1), as displayed in Figs. 3(c) and 3(d). See also Fig. 5(f)
where vanishing of the gap is explicitly shown. The propagating modes are
gapless in this case because the defect pierces the gapless $xy$ surfaces, for
any $\theta$. The dislocation modes are localized within a few lattice sites
at the defect core, as can be seen from their LDoS in a plane perpendicular to
the dislocation direction displayed in Figs. 5(d) and 5(e). The screw
dislocation modes also inherit the $C_{4}$ symmetry preserved by the defect.

##  VI Discussion and conclusion

Our findings are experimentally consequential for probing HOT insulators,
apart from the crystalline systems, also in metamaterials. The paradigmatic
model of the 2D second-order topological insulator, the Benalcazar-Bernevig-
Hughes (BBH) model benalcazar-science2017 , equivalent to the minimal lattice
model in Eq. (1) roy-prr2019 , has been realized in the lattice of microwave
resonators peterson-nature2018 . A dislocation defect in this setup should be
created by a local hopping modification through $\pi$ phase factors across a
line of missing sites ending at the dislocation center, analogously to the
case of a translationally-active first-order topological insulator grinberg-
arxiv2019 , and a disclination peterson-arxiv2020 . In the BBH photonic
lattice, where the sign of the hopping also can be locally manipulated mittal-
natphot2019 , it should be therefore possible to introduce the dislocation
defects and observe the defect modes, as in first-order 2D topological
photonic crystals noh-natphot2018 ; li-natcomm2018 , and for a disclination
defect liu-arxiv2020 . Finally, the artificial lattices can host HOT phases,
as recently shown for Kagome lattice kempkes-natmat2019 , and we expect that
because of their tunability, our theoretical predictions can also be directly
tested in these platforms.

Most of the proposed and experimentally studied 3D HOT crystalline materials
turn out to be of the translationally-active type. For example, elemental Bi
exhibits a double band inversion at the $C_{3}$-symmetric $T$-point in the BZ
and supports mixed electronic topology manifesting through coexisting gapless
hinge and Dirac surface modes schindler-natphys2018 ; nayak-sciadv2019 ; hsu-
pnas2019 . Our general mechanism thus implies that an edge dislocation with
the Burgers vector in the $(111)$ direction parallel to the HOT mass domain
wall (see, Fig. 1c in Ref. schindler-natphys2018 ), so that also ${\Phi_{\rm
dis}}=\pi$, features gapless modes, protected by $C_{3}$ and time-reversal
symmetries. A screw dislocation oriented in the same direction should host
one-dimensional gapless (gapped) states if the $(111)$ surface is gapless
(gapped). Similarly, for a recently proposed HOT insulator in Zr(TiH2)2, with
band-inversion away from the $\Gamma$ point and the gapless modes along all
the edges in the cubic geometry zhang-nature2019 , we predict gapless (gapped)
modes in the core of an edge (a screw) dislocation with the Burgers vector
along a principal crystal axis. Finally, the candidate HOT insulators Bi4X4,
with X=Br,I, zhang-nature2019 ; vergniory-nature2019 ; tang-nature2019 ; yoon-
arxiv2020  feature band inversions at $R$ and $M=(\pi/a,\pi/a,0)$ points in
the BZ, and hence the dislocations should host the one-dimensional modes,
following the above general rule.

Here we demonstrated that dislocations can be instrumental in probing higher-
order electronic topology in insulators as a consequence of the subtle
interplay between the geometry of the HOT Wilson-Dirac mass in the momentum
space and real-space lattice topological defects. We furthermore demonstrate
the protection of the dislocation modes in the case of $C_{2n}$ rotational
symmetry breaking HOT insulators in both two and three dimensions (see
Appendices E and F), which pertains to physically relevant $C_{4}$ and $C_{6}$
symmetric crystals. On the other hand, when the order of the rotation is _odd_
, i.e. for $C_{2n+1}$ rotations, there is no higher-order Wilson-Dirac mass
term that changes sign under such rotation. Therefore, we cannot find second-
order topological mass. Further analysis of this case is left for future
investigation. Recently, it has been shown that also partial dislocations with
a Burgers vector which is a fraction of a primitive lattice vector can host
gapless propagating modes in 3D HOT insulators Queiroz-PRL2019 . In addition,
disclination can also host topological modes in 3D HOT insulators Geier-2021 .
Our findings motivate future investigation of the response to the dislocations
in HOT insulators on different crystalline lattices, such as, for instance,
Kagome lattice ezawa-PRL2018 . Finally, we expect that our mechanism will be a
useful guide for the experimental detection of the HOT phases in diverse
platforms, and consequential also for HOT semimetals calugaru-prb2019 ;
hughes-HOTDSM-PRB2018  and superconductors.

###### Acknowledgements.

B.R. was supported by the Startup Grant from Lehigh University. V.J.
acknowledges support of the Swedish Research Council (VR 2019-04735).  Data
and code availability The data that support the plots within this paper and
other findings of this study are available from the authors upon reasonable
request.

Figure 4: Local density of states (LDoS) for four closest to the zero energy
modes on (a) a regular square lattice (solid black box) and (b) an oblique
square lattice (solid black box), cut from a regular square lattice (dashed
black box). While the corners in (a) are placed along the body diagonal
directions, they are placed along the principle axes in (b). The corresponding
values of the parameter $\theta$ appearing in the second-order Wilson-Dirac
mass are quoted in each figure [see Eq. (1)]. Consequently, in panel (a) the
corner modes appear when $\theta=0$, for which the HOT Wilson-Dirac mass
changes its sign across the body diagonals, while the near zero energy modes
occupy all the edges when $\theta=\pi/2$. On the other hand, in oblique square
lattice in (b), the corner states appear when $\theta=\pi/2$ for which the
second-order Wilson-Dirac mass changes its sign across the principle axes. The
closet to zero energy states in this setup occupy all the edges when
$\theta=0$.

##  Appendix A Dislocation modes in a 2D second-order $M$ phase: Continuum
model

In this Appendix we derive the form of the localized dislocation modes from
the continuum model for the two-dimensional (2D) $M$ phase with a the Burgers
vector ${\bf b}=a{\bf e}_{x}$. We start with the lattice Hamiltonian [Eqs. (1)
and (2) in $d=2$ with specifically chosen form factors] $H=\sum_{{\bf
k}}\Psi_{\bf k}^{\dagger}\hat{h}\Psi_{\bf k}$, where
$\hat{h}=\hat{h}_{0}+\hat{h}_{\Delta}$, and

$\displaystyle~{}\hat{h}_{0}$
$\displaystyle=t\left[\sin(k_{x}a)\Gamma_{1}+\sin(k_{y}a)\Gamma_{2}\right]+\left\\{m-2B\left[2-\cos(k_{x}a)-\cos(k_{y}a)\right]\right\\}\Gamma_{3},$
$\displaystyle\hat{h}_{\Delta}$
$\displaystyle=\Delta\big{\\{}\cos\theta\left[\cos(k_{x}a)-\cos(k_{y}a)\right]+\sin\theta\left[\sin(k_{x}a)\sin(k_{y}a)\right]\big{\\}}\Gamma_{4}=\Delta({\bf
k},\theta)\Gamma_{4}.$ (5)

Here, $\Psi_{\bf k}$ is a four-component spinor, the exact form of which does
not affect the following discussion, while ${\bf k}$ is the momentum, and $a$
is the lattice spacing. The mutually anticommuting four-component $\Gamma$
matrices satisfy the Clifford algebra
$\\{\Gamma_{j},\Gamma_{k}\\}=2\delta_{jk}$ for $j,k=1,\cdots,5$. The following
discussion only rests on this anicommuting Clifford algebra, not on the exact
representations of the $\Gamma$ matrices.

This model for the regime of parameters $0<m/B<8$ describes a 2D first-order
topological insulator and a HOT insulator (second-order) for $\Delta=0$ and
finite $\Delta$, respectively. Furthermore, when $4<m/B<8$, the model features
the band inversion at the $M=(\pi/a,\pi/a)$ point (the $M$ phase), while for
$0<m/B<4$, the band inversion is at the $\Gamma=(0,0)$ point (the $\Gamma$
phase) of the BZ. Notice that $\\{\hat{h}_{0},\hat{h}_{\Delta}\\}=0$, and
therefore $\hat{h}_{\Delta}$ acts as a mass term for the topological edge
states of $\hat{h}_{0}$. This mass term changes sign under the $C_{4}$
rotation and, as such, assumes the profile of a discrete symmetry breaking
Wilson-Dirac mass, the exact form of which depends on the parameter
$\theta\in[0,\pi/2]$. In particular, for $\theta=\pi/2$ the domain wall lies
along the diagonals $k_{y}=\pm k_{x}$, while for $\theta=0$ it is located
along the principal axes, $k_{x}=0,k_{y}=0$.

We now comment on the structure of the corner modes for $\theta=0$ and
$\pi/2$. Let us consider a 2D square lattice of linear dimension $L$ in each
direction, such that four corners are at $(\pm L/2,\pm L/2)$. Four corner
modes are then sharply localized around these corners for $\theta=0$, and with
increasing $\theta$ they become more delocalized. By contrast, if the crystal
is cut in such a way that four corners are located at $(\pm L/2,0)$ and
$(0,\pm L/2)$, the corner modes are most prominently localized when
$\theta=\pi/2$ and gradually delocalize as $\theta$ is ramped down to zero.
However, irrespective of the sharpness of the corner modes, the system always
describes a 2D second-order topological insulator for any $\theta$. These
outcomes are shown in Fig. 4. A similar structure also appears for the hinge
modes for 3D second-order topological insulator, discussed in Appendix B.

The continuum Hamiltonian is obtained by expanding the above lattice
Hamiltonian close to the bandgap closing at the $M$ point, $k_{x}=\pi+q_{x}$,
$k_{y}=\pi+q_{y}$, with $\Delta=0$, which in the real space (${\bf
q}\rightarrow-i{\nabla}$) reads as

${\hat{h}}_{\rm
M}=it\Gamma_{1}\partial_{x}+it\Gamma_{2}\partial_{y}-\Gamma_{3}\left[{\tilde{m}}+{\tilde{B}}(\partial^{2}_{x}+\partial^{2}_{y})\right].$
(6)

Here, ${\tilde{m}}=8B-m>0$, ${\tilde{B}}=B>0$, and we set $a=1$.

The two edges along the lines $x_{\pm}=\pm a$ [Fig. 1(a)], before the
dislocation is introduced through the Volterra construction, feature
topological gapless edge states, resulting from the corresponding zero modes
of the edge Hamiltonian $H_{\rm
edge}=(it\Gamma_{1}\partial_{x}-({\tilde{m}}+{\tilde{B}}\partial^{2}_{x})\Gamma_{3})\otimes\mu_{3}$,
where the vector of Pauli matrices ${\bm{\mu}}$ acts in the space of the two
edges. When dislocation introduces a $\pi$ hopping phase, the reconnection of
the edges across the Volterra cut is modeled by a hopping Hamiltonian between
them in the form $H_{\rm D}=t\,{{\rm sgn}(x)}\Gamma_{1}\otimes\mu_{1}$, where
${{\rm sgn}(x)}$ is the “sign” function. This term, through the sign change of
the hopping across the cut, takes into account that the (low-energy) electrons
acquire a $\pi$ phase when encircling the defect. Its form ensures that when
the phase factor is trivial, the connected edge modes are trivially gapped
out, as in the $\Gamma$ phase.

We then look for the zero energy modes of the Hamiltonian $H_{\rm edge}+H_{\rm
D}$, or explicitly

$\left\\{[it\Gamma_{1}\partial_{x}-({\tilde{m}}+{\tilde{B}}\partial^{2}_{x})\Gamma_{3}]\otimes\mu_{3}+t\,{{\rm
sgn}(x)}\Gamma_{1}\otimes\mu_{1}\right\\}\Psi_{0}(x)=0.$ (7)

After multiplying this equation from the left by $-i\Gamma_{1}\otimes\mu_{3}$,
we obtain

$\left[it\partial_{x}+i\Gamma_{1}\Gamma_{3}({\tilde{m}}+{\tilde{B}}\partial^{2}_{x})+t\,{{\rm
sgn}(x)}\mu_{2}\right]\Psi_{0}(x)=0.$ (8)

The form of the above equation implies that we can take the following ansatz
for the solution

$\Psi_{0}(x)=f(x)\chi_{\sigma}\otimes\varphi_{\rho},$ (9)

where

$i\Gamma_{1}\Gamma_{3}\chi_{\sigma}=\sigma\chi_{\sigma},\,\,\mu_{2}\varphi_{\rho}=\rho\varphi_{\rho},$
(10)

which for an exponentially localized solution at both edges
$f(x)\sim\exp(-\lambda|x|)$, yields

$-t\lambda+({\tilde{m}}+{\tilde{B}}\lambda^{2})\sigma{\rm sgn}(x)+t\rho=0.$
(11)

Choosing $\sigma{\rm sgn}(x)=+1,\rho=+1$, and considering the regime
$t^{2}>4{\tilde{B}}({\tilde{m}}+t)$, we obtain two characteristic inverse
localization lengths

$\lambda_{1,2}=\frac{t\pm\sqrt{t^{2}-4{\tilde{B}}(t+{\tilde{m}})}}{2{\tilde{B}}}>0.$
(12)

Notice that the spinors $\chi_{\sigma}$ are doubly degenerate, because of the
anticommuting property of the four-component Hermitian $\Gamma$ matrices. This
form of the localization length together with the continuity condition across
the reconnected edges, $\Psi_{0}(x\to 0)=0$, yields a pair of the zero energy
dislocation modes given by

$\Psi_{0}^{(1,2)}(x)={\mathcal{C}}\chi^{(1,2)}_{{\rm
sgn}(x)}\otimes\varphi_{+1}\left({\rm e}^{-\lambda_{1}|x|}-{\rm
e}^{-\lambda_{2}|x|}\right)\equiv{\mathcal{C}}\chi^{(1,2)}_{{\rm
sgn}(x)}\otimes\varphi_{+1}\psi(x),$ (13)

mentioned in the main text.

The rest of the analysis straightforwardly follows from the fact that
$[\Gamma_{4},i\Gamma_{1}\Gamma_{3}]=0$ implying that $\Gamma_{4}$ reduces in
the eigen-subspaces of $i\Gamma_{1}\Gamma_{3}$. Therefore, a perturbation
proportional to $\Gamma_{4}$ symmetrically splits the dislocation modes about
the zero energy. Explicitly, the HOT perturbation

$\hat{h}_{\Delta}=\Delta\big{\\{}\cos\theta\left[\cos(k_{x}a)-\cos(k_{y}a)\right]+\sin\theta\sin(k_{x}a)\sin(k_{y}a)\big{\\}}\Gamma_{4},$
(14)

after the expansion about the $M$ point at ${\bf K}_{M}=(\pi/a,\pi/a)$,
$k_{x}=\pi/a+q_{x}$, $k_{y}=\pi/a$, with $q_{x}=-i\partial_{x}$, takes the
form

$\hat{h}_{\Delta,M}=-\frac{1}{2}\Delta\cos\theta\partial_{x}^{2}.$ (15)

Therefore the splitting of the modes is given by

$\delta E=2\Delta\cos\theta{\tilde{E}},$ (16)

where

${\tilde{E}}=\frac{1}{2}\left|\int
dx\left(\Psi^{(1,2)}_{0}(x)\right)^{\dagger}\partial_{x}^{2}\Psi^{(1,2)}_{0}(x)\right|,$
(17)

as announced in the main text. Finally, we note that in the above derivation
of the dislocation modes in $d=2$ and also in $d=3$ (see Appendix B), we only
used the Clifford algebra of the $\Gamma$ matrices,
$\\{\Gamma_{i},\Gamma_{j}\\}=2\delta_{ij}$, for $i,j=1,...,5$, and the
existence of the modes is therefore independent of the $\Gamma-$matrix
representation. Since $[\Gamma_{4},\Gamma_{13}]=0$, even though introduction
of the HOT mass generically yields finite energy dislocation modes unless
$\theta=\pi/2$, they cannot be mixed with rest of the bulk states, and
therefore remain robust and protected. We further substantiate this claim by
demonstrating the symmetry protection of the dislocation modes in Appendix E
in the case of $C_{4n}$ rotational symmetry, while an analogous analysis for
$C_{4n+2}$ rotational symmetry is presented in Appendix F.

Figure 5: (a) Scaling of the spectral gap ($\delta E$) among four states
localized at the core of the dislocations with the orientation of the $C_{4}$
symmetry breaking Wilson-Dirac mass $\theta$ [see Eq. (18)] in a cubic system
of linear dimension $L=16$ in each direction in the presence of an edge
dislocation-antidislocation pair with the Burgers vector ${\bf b}=\pm a{\bf
e}_{x}$. It shows that when the Burger vector is parallel to one of the
$C_{4}$ symmetry breaking axis, i.e, $\theta=\pi/2$, edge dislocation modes
become gapless. The linear scaling of the spectral gap $\delta E$ with (b)
$\cos\theta$ and (c) $\Delta$, for small $\Delta$, are in agreement with our
theoretical predictions from Appendix B. Localization of the dislocation modes
in the presence of (d) edge dislocation-antidislocation pair with the Burgers
vector ${\bf b}=\pm a{\bf e}_{x}$, and (e) screw dislocation-antidislocation
pair with the Burgers vector ${\bf b}=\pm a{\bf e}_{z}$, on a specific $xy$
plane along the $z$ direction ($z=6$) in a cubic system with linear dimension
$L=12$ in each direction. The direction of the HOT mass domain wall is fixed
to be $\theta=0$ [Eq. (18)]. It shows that these modes are always highly
localized around the core of dislocations. (f) Scaling of the spectral gap
($\delta E$) among the states localized at the dislocation core in a system of
linear dimension $L=16$ in each direction, in the presence of a screw
dislocation-antidislocation pair, when the Burgers vector is ${\bf b}=\pm
a{\bf e}_{z}$ (red) and ${\bf b}=\pm a{\bf e}_{y}$ (blue). Respectively in
these two cases the screw dislocation pierces surfaces hosting gapless and
gapped modes, and concomitantly the dislocation modes are also gapless and
gapped.

##  Appendix B 3D edge and screw dislocation modes: The continuum model

In this Appendix we start with the lattice model for a three-dimensional
second-order topological insulator on a cubic lattice, analogous to Eq. (A),
but with [Eqs. (1) and (2) in $d=3$]

$\displaystyle h_{0}$
$\displaystyle=t\left[\sin(k_{x}a)\Gamma_{1}+\sin(k_{y}a)\Gamma_{2}+\sin(k_{z}a)\Gamma_{3}\right]+\left\\{m-2B\left[3-\cos(k_{x}a)-\cos(k_{y}a)-\cos(k_{z}a)\right]\right\\}\Gamma_{4},$
$\displaystyle\hat{h}_{\Delta}$
$\displaystyle=\Delta\big{\\{}\cos\theta\left[\cos(k_{x}a)-\cos(k_{y}a)\right]+\sin\theta\left[\sin(k_{x}a)\sin(k_{y}a)\right]\big{\\}}\Gamma_{5}=\Delta({\bf
k},\theta)\Gamma_{5}.$ (18)

We now consider the $R$ phase, with the band inversion at the
$R=(\pi/a,\pi/a,\pi/a)$ point in the BZ, realized for the values of the
parameters $8<m<12$, $t=1$ and $B=1$, and expand the above Hamiltonian about
the $R$ point for $\Delta=0$ to obtain

$h_{0,R}(k_{x},k_{y},k_{z})=t\left[-k_{x}\Gamma_{1}-k_{y}\Gamma_{2}-k_{z}\Gamma_{3}\right]+(-{\tilde{m}}+Bk^{2})\Gamma_{4},$
(19)

where ${\tilde{m}}=12B-m>0$ in the topological $R$ phase. We take $B=1$, and
$k_{i}=\pi/a+q_{i}$, for $i=x,y,z$.

A 3D edge dislocation is obtained by translating its 2D counterpart along a
particular lattice direction representing the defect line. Therefore, the
conclusions obtained above in the 2D case for a Burgers vector along the
$C_{4}$ symmetry breaking $x$ or $y$ direction directly apply to the 3D case:
an edge dislocation should in general feature gapped modes which become
gapless when the Burgers vector is parallel to the domain wall
($\theta=\pi/2$), consistent with our numerical findings, see Fig. 5(a)-(c).
Also, when the Burgers vector is oriented along the $C_{4}$ symmetry axis,
therefore piercing a gapless surface (in this case the $xy$ surface), the
defect should feature gapless modes.

A screw dislocation with the Burgers vector ${\bf b}=a{\bf e}_{z}$ can be
introduced by the Volterra construction as follows. We first choose a slip
half-plane for $y=0$, $x>0$, whose neighbor half-plane $y=a{\bf e}_{y}$, $x>0$
is displaced by the Burgers vector ${\bf b}=a{\bf e}_{z}$ relative to the slip
plane. Translational symmetry is then restored by reconnecting the bonds
between the slip and the neighbor plane everywhere, except close to the
dislocation line along the $z$-direction. An electron sliding down through the
dislocation picks up a phase $\Phi_{\rm dis}={\bf K}_{\rm inv}\cdot{\bf
b}=\pi$ upon encircling it once.

The Hamiltonian for the slip plane and its neighbor before the reconnection
reads

$h_{xz}(k_{x},k_{z})=h_{0,R}(k_{x},\pi/a,k_{z})\otimes\mu_{3},$ (20)

where the vector of Pauli matrices ${\bm{\mu}}$ acts in the space of the two
surfaces. The $\pi$ phase factor is then introduced by modifying the sign of
the hopping in the $y$-direction across the domain wall at $x=0$, because the
slip plane is orthogonal to the $y$-axis, yielding

$h_{\rm 3D-dis}=t\,{\rm sgn}(x)\;\Gamma_{1}\otimes\mu_{1}.$ (21)

The form of this Hamiltonian is chosen so that when the phase factor is
trivial, the surface zero modes are gapped out, as in the $\Gamma$ phase,
realized for $t=B=1$ and $0<m<4$ in Eq. (18). We now look for the zero modes
of the surface Hamiltonian using that the translational symmetry is preserved
in the $z$-direction and is explicitly broken in the $x$-direction. Hence,
$k_{x}=\pi/a-i\partial_{x}$, $k_{z}=\pi/a$, and

$\left\\{[it\Gamma_{1}\partial_{x}-({\tilde{m}}+{\tilde{B}}\partial^{2}_{x})\Gamma_{4}]\otimes\mu_{3}+t\,{{\rm
sgn}(x)}\Gamma_{1}\otimes\mu_{1}\right\\}\Psi_{0,{\rm 3D}}(x)=0,$ (22)

which is identical to the Eq. (8), up to the change
$\Gamma_{3}\rightarrow\Gamma_{4}$. The form of the pair of zero modes is
therefore given by Eq. (13), with the difference that here
$i\Gamma_{1}\Gamma_{4}\chi_{\sigma}=\sigma\chi_{\sigma}$. Gapless propagating
modes along the dislocation line are obtained by “translating” the zero modes
along the dislocation direction $\Psi^{(1,2)}_{\rm
3D}(k_{z})\sim\Psi^{(1,2)}_{0,{\rm 3D}}(x)e^{ik_{z}z}$. Notice that this
solution explicitly breaks $C_{4}$ symmetry which is a consequence of the
specific Volterra cut not preserving this symmetry. Namely, there are two
equivalent slip planes, $x-z$ and $y-z$, each of them individually breaking
$C_{4}$ symmetry. On the lattice, this symmetry is certainly preserved away
from the dislocation line and therefore the zero mode solution in the
continuum inherits it, so that, after a proper regularization the spatially
dependent part should obey $\Psi_{\rm 3D}(x,y)=\Psi_{\rm 3D}(-y,x)$. This
feature, as we show below is inherited by the HOT dislocation modes, and as we
also find in our numerical analysis, see Fig. 5(e) for the plot of the local
density of states (LDoS) of the closest to zero energy modes on a particular
$x-y$ plane.

Figure 6: Spectral flow of the dislocation modes (red) in a system of linear
dimension $L=28$ and with the periodic boundary in both directions in a two-
dimensional second-order topological insulator with a pair of edge
dislocation-antidislocation, for $t=2B=1.0$, $m=3.0$ and $\Delta=0.20$, as a
function of the relative angle ($\theta$) between the Burgers vector ${\bf
b}=\pm a{\bf e}_{x}$ and the HOT mass domain wall (see Fig. 1(a) of the main
text). The bulk states are shown in black. As the HOT mass domain wall becomes
parallel to the Burgers vector ($\theta=\pi/2$), the dislocation modes become
zero-energy states. Note that for any value of $\theta$ the dislocation modes
(at zero energy or finite energies) are always well separated from the bulk
states, and they are particle-hole partners of each other, due to the unitary
particle-hole symmetry generated by $\Theta=\Gamma_{5}$ (see Appendix C), and
thus do not mix with the bulk states. Hence, they are protected by the bulk
topology and cannot be removed from the system (by mixing with the bulk
states, for example). Here, $n$ is the index for the energy eigenvalues
($E_{n}$). In Appendix E, we also show that the composite symmetries of HOT
insulators forbid addition of any new perturbation in the system, leaving the
dislocation modes at finite and zero energy symmetry protected.

We now introduce the HOT mass term proportional to $\Gamma_{5}$, as given by
the Hamiltonian $h_{\Delta}$ in Eq. (18), that may gap out otherwise gapless
propagating modes along the dislocation since
$[\Gamma_{5},i\Gamma_{1}\Gamma_{4}]=0$, analogous as in the case in two
dimensions. The splitting depends also on the form factor $\Delta({\bf
k},\theta)$, which after expanding about the band inversion $R$ point becomes

$\Delta(x,y)=\frac{\Delta}{2}\left\\{\cos\theta(\partial_{y}^{2}-\partial_{x}^{2})+2\sin\theta\partial_{x}\partial_{y}\right\\},$
(23)

which then yields for the energy gap

$\delta E\sim\int
dxdy\Psi^{*}(x,y)\\{\cos\theta(\partial_{y}^{2}-\partial_{x}^{2})+2\sin\theta\partial_{x}\partial_{y}\\}\Psi(x,y)=0,$
(24)

because of the $C_{4}$ symmetry [both $(\partial_{x}^{2}-\partial_{y}^{2})$
and $\partial_{x}\partial_{y}$ are odd, while $\Psi(x,y)$ is even under the
$C_{4}$]. The $x-y$ surface therefore yields gapless modes for the screw
dislocation piercing it, as also expected from the fact that it is gapless.
This is precisely what we find in the numerical analysis, see Fig. 5(f) (red
dots). The plot of the LDoS of the closest to zero energy modes shows
explicitly the $C_{4}$ symmetry, see Fig. 5(e). The result can also be
appreciated from the fact that in the continuum limit, featuring the full
$U(1)$ rotational symmetry in the $x-y$ plane, the dislocation represents a
$\pi$ magnetic flux tube, which is known to carry the gapless mode when
piercing a gapless surface ran-natphys2009 ; slager-prb2014 . On the other
hand, when the screw dislocation is oriented in the $x$ or $y$ direction,
piercing therefore a gapped surface, it carries gapped modes. This can be
directly seen by recalling that the dislocation zero modes have to break the
$C_{4}$ symmetry, and thus the spatial part of the zero energy solution for
the dislocation with Burgers vector in the $y$ direction is $\Psi(x)$, as
given in Eq. (13). The energy splitting is then

$\delta E\sim\Delta\cos\theta\int dx\psi(x)\partial_{x}^{2}\psi(x)\neq 0,$
(25)

for any $\theta\neq\pi/2$. This is consistent with the numerically extracted
scaling of $\delta E$ versus the HOT mass parameter $\Delta$, shown in Fig.
5(f) (blue dots). Therefore, as long as the screw dislocation pierces the
gapped surfaces, the modes along the dislocation line are also gapped with the
gap scaling with the size of the surface gap ($\Delta$). The dislocation
modes, however, remain within the bulk bandgap, determined by the first-order
topological mass proportional to the $\Gamma_{4}$ matrix in Eq. (18).

Operator | Symmetry
---|---
${\mathcal{T}}$ | ${\mathcal{P}}$ | $C_{4}$ | ${\mathcal{P}}{\mathcal{T}}$ | $C_{4}{\mathcal{P}}$ | $C_{4}{\mathcal{T}}$
$\hat{h}_{0}$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
$\hat{h}_{\Delta}$ | $\times$ | $\times$ | $\times$ | $\checkmark$ | $\checkmark$ | $\checkmark$
Table 1: Symmetry transformations of the first-order Hamiltonian
($\hat{h}_{0}$) and the second-order mass term ($\hat{h}_{\Delta}$), given by
Eq. (1) in the main text. The representation of the $\Gamma$ matrices is given
in Eq. (26). Time-reversal (${\mathcal{T}}$), parity (${\mathcal{P}}$) and
$C_{4}$ rotational (${\mathcal{R}}_{4}$) symmetries are, respectively,
represented by the operators
${\mathcal{T}}=\tau_{2}\otimes\sigma_{1}{\mathcal{K}}$,
${\mathcal{P}}=\tau_{1}\otimes\sigma_{3}$, and
${\mathcal{R}}_{4}=\exp\left(i\frac{\pi}{4}\tau_{0}\otimes\sigma_{3}\right)$.
For a scalar (pseudoscalar) $M$ under an operation $X$ it holds
$XMX^{\dagger}=X$ ($XMX^{\dagger}=-X$). Scalar and pseudoscalar operators are
denoted by $\checkmark$ and $\times$ in the table. Here $\checkmark(\times)$
also indicates whether an operator is even (odd) under a specific symmetry.

##  Appendix C Unitary and anti-unitary particle-hole symmetry

In this Appendix we discuss the unitary and anti-unitary particle-hole
symmetry of both first-order and second-order topological insulators that
respectively protects zero-energy edge, surface, corner and hinge modes, and
both zero and finite energy dislocation modes. For this analysis we choose the
following representation of the $\Gamma$ matrices satisfying the anticommuting
Clifford algebra

$~{}\Gamma_{1}=\tau_{1}\otimes\sigma_{1},\Gamma_{2}=\tau_{1}\otimes\sigma_{2},\Gamma_{3}=\tau_{1}\otimes\sigma_{3},\Gamma_{4}=\tau_{3}\otimes\sigma_{0},\Gamma_{5}=\tau_{2}\otimes\sigma_{0},$
(26)

in the Hamiltonian for the second-order HOT insulator in $d=2$ and $d=3$,
given by Eq. (1) in the main text.

Next we discuss the particle-hole or the spectral symmetry of both the first-
order and second-order topological insulators in $d=2$ and $d=3$. It is
defined in terms of an operator, say $\Theta$, which _anticommutes_ with a
generic Hamitlonian $\hat{h}_{\rm gen}$, i.e., $\\{\hat{h}_{\rm
gen},\Theta\\}=0$. If such operator ($\Theta$) exists, then all the
eigenstates of the $\hat{h}_{\rm gen}$ at positive and negative energies $\pm
E_{n}$, denoted by $|\pm E_{n}\rangle$, are related to each other according to
$\Theta|\pm E_{n}\rangle=|\mp E_{n}\rangle$. This statement follows from the
relation

$\displaystyle\hat{h}_{\rm gen}|\pm E_{n}\rangle=\pm E_{n}|\pm
E_{n}\rangle\Rightarrow\Theta\left(\hat{h}_{\rm gen}|\pm E_{n}\rangle\right)$
$\displaystyle=$ $\displaystyle\pm E_{n}\left(\Theta|\pm
E_{n}\rangle\right)\Rightarrow-\hat{h}_{\rm gen}\left(\Theta|\pm
E_{n}\rangle\right)=\pm E_{n}\left(\Theta|\pm E_{n}\rangle\right)$ (27)
$\displaystyle\Rightarrow$ $\displaystyle\hat{h}_{\rm gen}\left(\Theta|\pm
E_{n}\rangle\right)=\mp E_{n}\left(\Theta|\pm E_{n}\rangle\right),$

and hence $\Theta|\pm E_{n}\rangle=|\mp E_{n}\rangle$. Otherwise, the
particle-hole symmetry generator $\Theta$ can be either unitary or antiunitary
roy-prr2019 . The above definition of the particle-hole or spectral symmetry
then shows that if there exist any state at precise zero energy, which can
only be achieved in the true thermodynamic limit, then it must be an
eigenstate of $\Theta$ with eigenvalue $+1$ or $-1$. However, in any finite
system, all the states are at finite energies, and the states which reside at
zero energy in the thermodynamic limit are also placed at finite but extremely
small (typically $\sim 10^{-6}-10^{-8}$) energies. Next we show that such
particle-hole or spectral symmetry operator always exists for the universal
models of both first-order and second-order topological insulators in $d=2$
and $d=3$ [see Eq. (1)], and thereby protects both zero energy topological
boundary modes (such as the edge, surface, corner and hinge modes), as well as
the dislocation modes (both zero and finite energy ones).

We begin with the first-order topological insulators. Notice that for such
systems in both $d=2$ and $d=3$, we can always find a unitary particle-hole
symmetry operator, namely $\Theta=\Gamma_{5}$, irrespective of the
representation of the $\Gamma$ matrices. We note that with the representation
of the $\Gamma$ matrices, shown in Eq. (26), we also also find one antiunitary
particle-hole symmetry generator, namely
$\Theta=(\tau_{2}\sigma_{2}){\mathcal{K}}$, for 3D first-order topological
insulator, where ${\mathcal{K}}$ is the complex conjugation and
$\Theta^{2}=+1$. Such antiunitary particle-hole operator will play an
important role for 3D HOT insulator. For now, one can immediately appreciate
that the unitary particle-hole operator $\Gamma_{5}$, pins the boundary edge
and surface modes at zero-energy and prevents them from mixing with the bulk
states. In addition, it also forbids the gapless dislocation modes in both 2D
and 3D first-order topological insulators from mixing with the bulk mode.
Next, we proceed to HOT insulators.

For a 2D HOT insulator, we can still find $\Theta=\Gamma_{5}$. It protects the
four zero-energy corner states [see Fig. 1(b)]. Furthermore, given that the
in-gap, but finite- or zero-energy dislocation modes are always well separated
from the bulk states, see Fig. 6, and appear in pairs, such unitary particle-
hole operator also prevents the finite-energy (for any $\theta\neq\pi/2$) and
the zero-energy (for $\theta=\pi/2$) dislocation modes from mixing with the
bulk states. Thus neither the zero nor the finite energy dislocation modes can
be removed from the system, and they are protected by the particle-hole
symmetry. Finally, for a 3D HOT insulator, we cannot find any unitary
particle-hole operator, since the corresponding Hamiltonian operator exhausts
all five mutually anticommuting four-component $\Gamma$ matrices.
Nevertheless, the particle-hole symmetry is then generated by an antiunitary
operator $\Theta=(\tau_{2}\sigma_{2}){\mathcal{K}}$. Then the above discussion
for the 2D HOT insulator immediately generalizes to 3D HOT insulator, and one
can immediately conclude that (1) the four zero-energy hinge modes, and (2)
both finite- and zero-energy dislocation modes are protected by the antiuntary
particle-hole symmetry. Finally, we note that the existence of such an
antiunitary particle-hole symmetry generator is also independent of the
explicit representation of the $\Gamma$ matrices roy-prr2019 . In Appendix E,
we show that composite symmetries of both 2D and 3D HOT insulators forbid
addition of any new term to their corresponding universal Hamiltonian, shown
in Eqs. (A) and (18), respectively. Therefore, both finite and zero energy
dislocation modes are symmetry protected.

Operator | Symmetry
---|---
Compact form | Explicit form | ${\mathcal{T}}$ | ${\mathcal{P}}$ | $C_{4}$ | ${\mathcal{P}}{\mathcal{T}}$ | $C_{4}{\mathcal{P}}$ | $C_{4}{\mathcal{T}}$
$\Gamma_{0}$ | $\tau_{0}\sigma_{0}$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
$\Gamma_{3}$ | $\tau_{1}\sigma_{3}$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
$\Gamma_{4}$ | $\tau_{3}\sigma_{0}$ | $\times$ | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\times$
$\Gamma_{5}$ | $\tau_{2}\sigma_{0}$ | $\times$ | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\times$
$\Gamma_{12}$ | $-\tau_{0}\sigma_{3}$ | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$ | $\times$
$\Gamma_{34}$ | $\tau_{2}\sigma_{3}$ | ✓ | $\times$ | $\checkmark$ | $\times$ | $\times$ | $\checkmark$
$\Gamma_{35}$ | $-\tau_{3}\sigma_{3}$ | ✓ | $\times$ | $\checkmark$ | $\times$ | $\times$ | $\checkmark$
$\Gamma_{45}$ | $\tau_{1}\sigma_{0}$ | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$ | $\times$
$Y^{1}=(\Gamma_{1},\Gamma_{2})$ | $(\tau_{1}\sigma_{1},\tau_{1}\sigma_{2})$ | $\times$ | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\times$
$Y^{2}=(\Gamma_{23},\Gamma_{13})$ | $(\tau_{0}\sigma_{1},\tau_{0}\sigma_{2})$ | $\checkmark$ | $\times$ | $\checkmark$ | $\times$ | $\times$ | $\checkmark$
$Y^{3}=(\Gamma_{14},\Gamma_{24})$ | $(\tau_{2}\sigma_{1},\tau_{2}\sigma_{2})$ | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$ | $\times$
$Y^{4}=(\Gamma_{15},\Gamma_{25})$ | $(-\tau_{3}\sigma_{1},-\tau_{3}\sigma_{2})$ | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$ | $\times$
Table 2: Symmetry transformation of the identity matrix ($\Gamma_{0}$), five
$\Gamma$ matrices in Eq. (26) and ten commutators
$\Gamma_{jk}=[\Gamma_{j},\Gamma_{k}]/(2i)$ under time-reversal
(${\mathcal{T}}$), parity (${\mathcal{P}}$) and $C_{4}$ rotational
(${\mathcal{R}}_{4}$) symmetries, which are, respectively, represented by the
operators ${\mathcal{T}}=\tau_{2}\otimes\sigma_{1}{\mathcal{K}}$,
${\mathcal{P}}=\tau_{1}\otimes\sigma_{3}$, and
${\mathcal{R}}_{4}=\exp\left(i\frac{\pi}{4}\tau_{0}\otimes\sigma_{3}\right)$.
Scalar and vector operators are denoted by $\checkmark$, while $\times$
denotes pseudoscalar and pseudovector operators. Here $\checkmark(\times)$
also indicates whether an operator is even (odd) under a specific symmetry.

##  Appendix D Details of the numerical analysis and additional results

This Appendix is devoted to highlight the parameter values used in Figs. 1 and
2 of the main text, and discuss additional numerical results for three-
dimensional second-order topological insulators, hosting one-dimensional hinge
modes along the $z$ direction, in the presence of edge and screw dislocation-
antidislocation pair, shown in Fig. 5.

For two-dimensional topological insulators, we always choose $t=1$, $B=1/2$,
and $m=3$ [see Eq. (A)], so that the system is in the $M$ phase. To realize a
second-order topological insulator, we set $\Delta=0.20$ in Figs. 1(b) and
1(c). For Figs. 2(a)-(c), we consider a dislocation-antidislocation pair in a
periodic system with linear dimension $L=28$ in each direction. The locations
of the dislocations in these figures are identical to the ones in Fig. 1(c).

For three-dimensional topological insulators, we always set $t=B=1$ and $m=10$
[see Eq. (18)], so that the system is in the $R$ phase. To realize a second-
order topological insulator, we set $\Delta=0.40$ in Fig. 3.

We now discuss additional numerical results for 3D second-order topological
insulators in the presence of dislocations, displayed in Fig. 5. In Figs.
5(a)-(c) we show the scaling of the spectral gap ($\delta E$) among four
states localized in the dislocation core in the presence of an edge
dislocation-antidislocation pair with Burgers vectors ${\bf b}=\pm a{\bf
e}_{x}$. It shows that only when the Burgers vector is parallel to one of the
$C_{4}$ symmetry breaking axis ($\theta=\pi/2$) these modes become gapless
[Fig. 5(a)]. Otherwise, the spectral gap scales linearly with $\cos\theta$ and
$\Delta$, as shown in Figs. 5(b) and (c), respectively, when $\Delta$ is
small, in agreement with our scaling argument presented in Appendix B.

Figs. 5(d) and (e) show the localization (through LDoS) of the dislocation
modes around the core of edge and screw dislocation-antidislocation pair,
respectively, in the $xy$ plane for a specific $z=6$. These two figures are
projections of Figs. 3(b) and (d) of the main text, respectively, for a
specific $z$.

Fig. 5(f) shows the scaling of the spectral gap ($\delta E$) between the
states localized in the dislocation core in the presence of a screw
dislocation-antidislocation pair for two specific orientations of the
corresponding Burgers vectors, namely, when ${\bf b}=\pm a{\bf e}_{z}$ (red)
and ${\bf b}=\pm a{\bf e}_{y}$ (blue). For ${\bf b}=\pm a{\bf e}_{z}$ the
screw dislocation pair pierces the $xy$ surfaces that host gapless modes,
while for ${\bf b}=\pm a{\bf e}_{y}$ it pierces the $xz$ planes which, on the
other hand, accommodate gapped modes. Concomitantly, for these two
orientations of the screw dislocation-antidislocation pair the modes localized
in its core are respectively gapless and gapped, in agreement with our
analytical arguments from Appendix B.

Finally, we show the real space version of the tight-binding model, reported
in Eq. (1) of the main text in the momentum space, in the presence of a single
edge dislocation in open system on a 2D square lattice for $\theta=0$. By
performing the Fourier transform, we arrive at

$\displaystyle\hat{h}_{\rm real}$
$\displaystyle=\bigg{(}\bigg{[}\left\\{\left(\sum^{\ell_{x}-2}_{n_{x}=1}\sum^{\ell_{y}-1}_{n_{y}=1}+\sum^{L_{x}-1}_{n_{x}=\ell_{x}+1}\sum^{\ell_{y}-1}_{n_{y}=1}+\sum^{L_{x}-1}_{n_{x}=1}\sum^{L_{y}}_{n_{y}=\ell_{y}}\right)c^{\dagger}_{n_{x},n_{y}}c_{n_{x}+1,n_{y}}+\sum^{\ell_{y}}_{n_{y}=1}c^{\dagger}_{\ell_{x}-1,n_{y}}c_{\ell_{x}+1,n_{y}}\right\\}\otimes\frac{2B\Gamma_{3}+it\Gamma_{1}+\Delta\Gamma_{4}}{2}$
$\displaystyle+\left\\{\left(\sum^{\ell_{x}-1}_{n_{x}=1}\sum^{L_{y}-1}_{n_{y}=1}+\sum^{L_{x}}_{n_{x}=\ell_{x}+1}\sum^{L_{y}-1}_{n_{y}=1}\right)c^{\dagger}_{n_{x},n_{y}}c_{n_{x},n_{y}+1}+\sum^{L_{y}-1}_{n_{y}=\ell_{y}}c^{\dagger}_{\ell_{x},n_{y}}c_{\ell_{x},n_{y}+1}\right\\}\otimes\frac{2B\Gamma_{3}+it\Gamma_{2}-\Delta\Gamma_{4}}{2}\bigg{]}+H.c.\bigg{)}$
$\displaystyle+\sum^{L_{x}L_{y}-(\ell_{y}-1)}_{j=1}c^{\dagger}_{j,j}c_{j,j}\otimes(m-4B)\Gamma_{3}.$
(28)

In this construction, the dislocation center is located at
$(\ell_{x},\ell_{y})$ [see Fig. 1 of the main text]. Now this construction can
be generalized to introduce a pair of edge dislocation-antidislocation in a 2D
periodic system, as well as edge and screw dislocations in 3D.

##  Appendix E Symmetry protection of dislocation modes

In this Appendix we take the representation of the $\Gamma$ matrices as given
by Eq. (26) in the Hamiltonian for the second-order HOT insulator in $d=2$ and
$d=3$, given by Eq. (1) in the main text. Antiunitary time-reversal
(${\mathcal{T}}$), unitary parity (${\mathcal{P}}$) and $C_{4}$ rotational
(${\mathcal{R}}_{4}$) symmetries are respectively represented by the operators
${\mathcal{T}}=\tau_{2}\otimes\sigma_{1}{\mathcal{K}}$,
${\mathcal{P}}=\tau_{1}\otimes\sigma_{3}$, and
${\mathcal{R}}_{4}=\exp\left(i\frac{\pi}{4}\tau_{0}\otimes\sigma_{3}\right)$,
where ${\mathcal{K}}$ is the complex conjugation. Under time-reversal and
parity, momentum changes sign, ${\bf k}\rightarrow-{\bf k}$, while it is a
vector under the $C_{4}$ rotation: $(k_{x},k_{y})\rightarrow(-k_{y},k_{x})$.
The first-order topological Hamiltonian ($\hat{h}_{0}$) and the second-order
mass term ($\hat{h}_{\Delta}$) are both invariant _only_ under the composite
${\mathcal{P}}{\mathcal{T}}$, ${C}_{4}{\mathcal{T}}$, ${C}_{4}{\mathcal{P}}$
symmetries, see Table 1. We show that there are no additional symmetry allowed
terms that can be added to the HOT Hamiltonian besides the ones already
present and the trivial one, proportional to unity matrix, which is, of
course, always symmetry allowed in an insulating system, and it only causes an
overall shift of energy eigenvalues. Finally, we point out that this proof
pertains to any second-order topological insulator with $C_{4n}$ rotational
symmetry breaking second-order Wilson-Dirac mass because in this case for any
$n$ the HOT mass is _even_ under ${\bf k}\to-{\bf k}$.

As a first step, we classify all the sixteen four-dimensional Hermitian
matrices, constructed from five mutually anticommuting Hermitian $\Gamma$
matrices given by Eq. (26), according to the above three composite symmetries,
see Table 2. We see that only the matrix $\Gamma_{3}$, which represents the
first-order topological mass, is invariant under all the three composite or
product symmetry operations. One can also form scalars under the $C_{4}$
rotation in terms of the four vectors in the form,

$H_{V}^{j}=\Delta_{j}(\cos\chi Y^{j}_{1}+\sin\chi Y^{j}_{2}),$ (29)

where $j=1,2,3,4$ corresponds to four vectors $Y^{j}$ from the last four rows
in Table 2, and the angle $\chi\rightarrow\chi+\pi/2$ under the $C_{4}$.
However, as explicitly shown in Table 3, none of these Hamiltonians is
invariant under _both_ parity and time-reversal, and therefore they violate
either $C_{4}{\mathcal{T}}$, $C_{4}{\mathcal{P}}$, or
${\mathcal{P}}{\mathcal{T}}$. Notice that the ${\mathcal{P}}{\mathcal{T}}$
symmetry assures the double degeneracy of the electronic bands in the model,
since it is antiunitary and squares to $-1$ roy-prr2019 . Therefore, there are
no new symmetry allowed terms in the 2D HOT Hamiltonian given by Eq. (1) in
the main text besides the ones already present. This in turn implies that both
both finite and zero energy dislocation modes in a 2D second-order topological
insulator are symmetry protected by the same combinations of the spatial and
non-spatial symmetries as the bulk Hamiltonian.

In three spatial dimensions, we choose the first order Hamiltonian so that the
matrix $\Gamma_{5}$ multiplies the factor $t\sin(k_{z}a)$ (see Table 2), while
we keep the rest of the terms identical as in the Hamiltonian for the 2D
second-order topological insulator [a slightly different notation than the one
in Eq. (1)]. Therefore, requiring the same symmetries, the above results imply
that the same composite symmetries protect the dislocation modes in 3D as in
two spatial dimensions.

In Fig. 6, we show that in the presence of dislocations the energy spectra
possess two gaps. One is set by the first-order topological mass (among the
black colored bulk states) and the other one is determined by the HOT mass
$\Delta({\bf k},\theta)$ among the dislocation modes (red states). The latter
one in addition also depends on the parameter $\theta$, measuring the relative
orientation between the HOT mass domain walls and the Burgers vector. Only
when $\theta=\pi/2$, for which the HOT mass domain wall is parallel to the
Burgens vector, the dislocation modes become gapless. However, as we argued
above, these two distinct energy scales are symmetry protected and we cannot
add any symmetry allowed perturbation that can mix these two scales. The same
conclusion holds analogously in 3D. Thus both finite and zero energy
dislocation modes are symmetry protected.

##  Appendix F Symmetry protection for $C_{4n+2}$ symmetry breaking second-
order topological insulator

In this Appendix, we discuss the symmetry protection of the dislocation modes
when the rotational symmetry is of the form $C_{4n+2}$, where $n$ is an
integer, as for instance is the case for $C_{6}$ rotations. Specifically for
$C_{6}$ symmetry breaking HOT insulators the second-order Wilson-Dirac mass
takes the form

$\hat{h}_{\Delta}=\Delta\Gamma_{d+2}\left[\cos\theta\;\sin(k_{x}a)\left\\{3\cos(k_{y}a)-\cos(k_{x}a)+2\right\\}+\sin\theta\;\sin(k_{y}a)\left\\{3\cos(k_{x}a)-\cos(k_{y}a)+2\right\\}\right],$
(30)

in the $M$ phase [compare with Eq. (1)] roy-goswami-juricic . In this case,
the second-order Wilson-Dirac mass is _odd_ under ${\bf k}\to-{\bf k}$, and
the protection of the dislocation modes can be realized only through the
parity (${\mathcal{P}}$) and the time-reversal (${\mathcal{T}}$) symmetries
(no rotational or a composite symmetry is required now). Therefore, the
inspection of the Table 2, shows that the only $\Gamma$ matrices preserving
both parity and time-reversal symmetries are $\Gamma_{0}$ and $\Gamma_{3}$.
Then our argument from Appendix E can be extended straightforwardly to show
that dislocation modes (at both finite and zero energies) cannot be mixed with
the other bulk states and hence they are symmetry protected. Furthermore, this
argument straightforwardly generalizes to all $2D$ second-order topological
insulators for which the second order Dirac-Wilson mass is odd under the
transformation ${\bf k}\to-{\bf k}$. Finally, due to the form of the
Hamiltonian for 3D second-order topological insulators, this argument
straightforwardly applies also to this case.

Operator | Symmetry
---|---
${\mathcal{T}}$ | ${\mathcal{P}}$ | $C_{4}$ | ${\mathcal{P}}{\mathcal{T}}$ | $C_{4}{\mathcal{P}}$ | $C_{4}{\mathcal{T}}$
$H_{V}^{1}$ | $\times$ | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\times$
$H_{V}^{2}$ | $\checkmark$ | $\times$ | $\checkmark$ | $\times$ | $\times$ | $\checkmark$
$H_{V}^{3}$ | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$ | $\times$
$H_{V}^{4}$ | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$ | $\times$
Table 3: Symmetry transformations of the Hamiltonians defined in Eq. (29). The
notation is the same as in the Tables 1 and 2. Notice that none of the
Hamiltonians is invariant under all the three composite symmetries. Here
$\checkmark(\times)$ indicates whether an operator is even (odd) under a
specific symmetry.

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