classical-quantum state of the link at time $t+1$ is

$\displaystyle\widehat{\sigma}(t+1)$
$\displaystyle=\sum_{h^{t+1}}|h^{t+1}\rangle\langle
h^{t+1}|\otimes\widetilde{\sigma}(t+1;h^{t+1})$ (4.5)
$\displaystyle=\sum_{h^{t},a_{t},x_{t+1}}|h^{t},a_{t},x_{t+1}\rangle\langle
h^{t},a_{t},x_{t+1}|\otimes
d_{t}(h^{t})(a_{t})\mathcal{T}^{x_{t},a_{t},x_{t+1}}(\widetilde{\sigma}(t;h^{t})),$
(4.6)

and that

$\mathbb{E}[\widetilde{F}(t+1;\psi)]=\operatorname{Tr}[(|1\rangle\langle
1|_{X_{t+1}}\otimes\psi)\widehat{\sigma}(t+1)],$ (4.7)

using (2.59) we have two possibilities for every history $h^{t}$ (assuming
both actions are taken deterministically):

$\displaystyle a_{t}=0$
$\displaystyle\Rightarrow\mathbb{E}[\widetilde{F}(t+1;\psi)]=\langle\psi|\mathcal{T}^{X(t)(h^{t}),0,1}(\widetilde{\sigma}(t;h^{t}))|\psi\rangle$
$\displaystyle\qquad\qquad\quad\quad\quad\,=\Pr[H(t)=h^{t}]X(t)(h^{t})\langle\psi|\widehat{\mathcal{N}}(\rho(M(t)(h^{t})))|\psi\rangle$
$\displaystyle\qquad\qquad\quad\quad\quad\,=\Pr[H(t)=h^{t}]X(t)(h^{t})f_{M(t)(h^{t})+1}(\rho_{0}),$
(4.8) $\displaystyle a_{t}=1$
$\displaystyle\Rightarrow\mathbb{E}[\widetilde{F}(t+1;\psi)]=\Pr[H(t)=h^{t}]pf_{0}(\rho_{0})$
(4.9)

So the task is to determine which of the two quantities,
$X(t)(h^{t})f_{M(t)(h^{t})+1}(\rho_{0})$ and $pf_{0}(\rho_{0})$, is higher. If
the link is not active at time $t$, meaning that $X(t)(h^{t})=0$, then
requesting a link, i.e., selecting $a_{t}=1$, is higher than selecting
$a_{t}=0$ (since the latter leads to a value of
$\mathbb{E}[\widetilde{F}(t+1;\psi)]=0$ for $p>0$). On the other hand, if the
link is active at time $t$, then the task is to compare

$f_{M(t)(h^{t})+1}(\rho_{0})~{}(a_{t}=0)\quad\text{and}\quad
pf_{0}(\rho_{0})~{}(a_{t}=1)$ (4.10)

for every history $h^{t}$. Which of these two quantities is higher (and thus
which action is taken) depends on the success probability $p\in(0,1)$, the
noise model of the quantum memory, and on the target pure state $\psi$.

The policy discussed above is an example of a so-called “forward greedy”
policy: at each time step, we take the action that maximizes the expected
value of $\widetilde{F}$ at the next time step. As it turns out, however, this
policy is in general not optimal among all possible policies. In fact, as we
now show, the global optimal policy is a “reverse greedy” policy, which is
defined by first deterministically picking the action that maximizes the
expected value of $\widetilde{F}$ at the final time step, then proceeding
backwards in time, optimizing the actions recursively until we reach the first
time step.

######  Theorem 4.1 (Optimal finite-horizon policy for elementary link
generation).

For all $t\geq 1$, and for any pure state $\psi$,

$\max_{\pi}\mathbb{E}[\widetilde{F}(t+1;\psi)]=\sum_{x_{1}=0}^{1}\max_{a_{1}\in\\{0,1\\}}v_{1}^{*}(t;x_{1},a_{1}),$
(4.11)

where for all $1\leq j\leq t-2$,

$v_{j}^{*}(t;h^{j},a_{j})=\sum_{x_{j+1}=0}^{1}\max_{a_{j+1}\in\\{0,1\\}}v_{j+1}^{*}(t;h^{j+1},a_{j+1})\quad\forall~{}h^{j}\in\\{0,1\\}^{2j-1},a_{j}\in\\{0,1\\},$
(4.12)

and

$v_{t-1}^{*}(t;h^{t-1},a_{t-1})=\sum_{x_{t}=0}^{1}\max_{a_{t}\in\\{0,1\\}}\langle\psi|\widetilde{\sigma}^{\prime}(t+1;h^{t},a_{t},1)|\psi\rangle\quad\forall~{}h^{t-1}\in\\{0,1\\}^{2t-3},a_{t-1}\in\\{0,1\\},$
(4.13)

where $\widetilde{\sigma}^{\prime}(t+1;h^{t},a_{t},1)$ is defined in (4.4).
Furthermore, the optimal policy is deterministic and given by
$\pi^{*}=(d_{j}^{*}:1\leq j\leq t)$, where

$\displaystyle d_{j}^{*}(h^{j})$
$\displaystyle=\operatorname*{arg\,max}_{a_{j}\in\\{0,1\\}}v_{j}^{*}(t;h^{j},a_{j})\quad\forall~{}1\leq
j\leq t-1\quad\forall~{}h^{j}\in\\{0,1\\}^{2j-1},$ (4.14) $\displaystyle
d_{t}^{*}(h^{t})$
$\displaystyle=\operatorname*{arg\,max}_{a_{t}\in\\{0,1\\}}\langle\psi|\widetilde{\sigma}^{\prime}(t+1;h^{t},a_{t},1)|\psi\rangle\quad\forall~{}h^{t}\in\\{0,1\\}^{2t-1}.$
(4.15)

######  Remark 4.1.

The algorithm shown above for determining the optimal policy is often called
the “backward recursion” algorithm, and it is related to dynamic programming
because the algorithm solves the policy optimization problem over the entire
time duration via a sequence of simple optimization problems at each
individual time step.

###### Proof.

We start by using the definition of $\mathbb{E}[\widetilde{F}(t+1;\psi)]$ and
the expression in (4.3) to write

$\displaystyle\mathbb{E}[\widetilde{F}(t+1;\psi)]$
$\displaystyle=\sum_{h^{t+1}:x_{t+1}=1}\langle\psi|\widetilde{\sigma}(t+1;h^{t+1})|\psi\rangle$
(4.16)
$\displaystyle=\sum_{h^{t},a_{t}}^{1}\left(\prod_{j=1}^{t}\operatorname{Tr}[\pi(j;h_{j}^{t+1})|a_{j}\rangle\langle
a_{j}|]\right)\langle\psi|\widetilde{\sigma}^{\prime}(t+1;h^{t},a_{t},1)|\psi\rangle$
(4.17)
$\displaystyle=\sum_{x_{1},a_{1}=0}^{1}\operatorname{Tr}[\pi(1;x_{1})|a_{1}\rangle\langle
a_{1}|]v_{1}(t;x_{1},a_{1})$ (4.18)
$\displaystyle=\sum_{x_{1}=0}^{1}\operatorname{Tr}\left[\left(\sum_{a_{1}=0}^{1}|a_{1}\rangle\langle
a_{1}|v_{1}(t;x_{1},a_{1})\right)\pi(1;x_{1})\right],$ (4.19)

where we have let

$v_{1}(t;x_{1},a_{1})\coloneqq\sum_{\begin{subarray}{c}x_{2},\dotsc,x_{t}=0\\\
a_{2},\dotsc,a_{t}=0\end{subarray}}^{1}\left(\prod_{j=2}^{t}\operatorname{Tr}[\pi(j;h_{j}^{t+1})|a_{j}\rangle\langle
a_{j}|]\right)\langle\psi|\widetilde{\sigma}^{\prime}(t+1;h^{t},a_{t},1)|\psi\rangle.$
(4.20)

Notice that $v_{1}(t;x_{1},a_{1})$ does not depend on the decision variables
in $\pi(1;x_{1})$ for the first time step. Therefore,

$\max_{\pi}\mathbb{E}[\widetilde{F}(t+1;\psi)]=\sum_{x_{1}=0}^{1}\max_{\pi(1;x_{1})}\operatorname{Tr}\left[\left(\sum_{a_{1}=0}^{1}|a_{1}\rangle\langle
a_{1}|v_{1}^{*}(t;x_{1},a_{1})\right)\pi(1;x_{1})\right],$ (4.21)

where

$\displaystyle v_{1}^{*}(t;x_{1},a_{1})$
$\displaystyle\coloneqq\max_{\\{\pi(j;h_{j}^{t+1}):2\leq j\leq
t\\}}v_{1}(t;x_{1},a_{1})$ (4.22)
$\displaystyle=\max_{\\{\pi(j;h_{j}^{t+1}):2\leq j\leq
t\\}}\sum_{\begin{subarray}{c}x_{2},\dotsc,x_{t}=0\\\
a_{2},\dotsc,a_{t}=0\end{subarray}}^{1}\left(\prod_{j=2}^{t}\operatorname{Tr}[\pi(j;h_{j}^{t+1})|a_{j}\rangle\langle
a_{j}|]\right)\langle\psi|\widetilde{\sigma}^{\prime}(t+1;h^{t},a_{t},1)|\psi\rangle.$
(4.23)

The optimization in (4.21) is of the form $\max_{\rho:\rho\geq
0,\operatorname{Tr}[\rho]=1}\operatorname{Tr}[\rho H]$ for some Hermitian
operator $H$, and it is known (see, e.g., [176]) that this is equal to the
maximum eigenvalue $\lambda_{\max}(H)$ of $H$. The operator $H$ here is simply
$\sum_{a_{1}=0}^{1}|a_{1}\rangle\langle a_{1}|v_{1}^{*}(t;x_{1},a_{1})$, which
is diagonal in the basis $\\{|a_{1}\rangle\\}_{a_{1}=0}^{1}$; therefore, its
maximum eigenvalue is $\max_{a_{1}\in\\{0,1\\}}v_{1}^{*}(t;x_{1},a_{1})$.
Consequently,

$\max_{\pi}\mathbb{E}[\widetilde{F}(t+1;\psi)]=\sum_{x_{1}=0}^{1}\max_{a_{1}\in\\{0,1\\}}v_{1}^{*}(t;x_{1},a_{1}).$
(4.24)

The optimal decision at the first time step is therefore given by
$\pi^{*}(1;x_{1})\coloneqq|d_{1}^{*}(x_{1})\rangle\langle d_{1}^{*}(x_{1})|$,
where the optimal decision function $d_{1}^{*}$ is deterministic and given by

$d_{1}^{*}(x_{1})\coloneqq\operatorname*{arg\,max}_{a_{1}\in\\{0,1\\}}v_{1}^{*}(t;x_{1},a_{1})\quad\forall~{}x_{1}\in\\{0,1\\}.$
(4.25)

Now, in order to determine $v_{1}^{*}(t;x_{1},a_{1})$, we can write
$v_{1}(t;x_{1},a_{1})$ as follows:

$\displaystyle v_{1}(t;x_{1},a_{1})$
$\displaystyle=\sum_{x_{2},a_{2}=0}^{1}\operatorname{Tr}[\pi(2;h_{2}^{t+1})|a_{2}\rangle\langle
a_{2}|]v_{2}(t;h_{2}^{t+1},a_{2})$ (4.26)
$\displaystyle=\sum_{x_{2}=0}^{1}\operatorname{Tr}\left[\left(\sum_{a_{2}=0}^{1}|a_{2}\rangle\langle
a_{2}|v_{2}(t;h_{2}^{t+1},a_{2})\right)\pi(2;h_{2}^{t+1})\right],$ (4.27)

where

$v_{2}(t;h_{2}^{t+1},a_{2})\coloneqq\sum_{\begin{subarray}{c}x_{3},\dotsc,x_{t}=0\\\
a_{3},\dotsc,a_{t}=0\end{subarray}}^{1}\left(\prod_{j=3}^{t}\operatorname{Tr}[\pi(j;h_{j}^{t+1})|a_{j}\rangle\langle
a_{j}|]\right)\langle\psi|\widetilde{\sigma}^{\prime}(t+1;h^{t},a_{t},1)|\psi\rangle.$
(4.28)

Then, observing that $v_{2}(t;h_{2}^{t+1},a_{2})$ does not depend on the
decision variables in $\pi(2;h_{2}^{t+1})$, we apply the same reasoning as
above to conclude that

$v_{1}^{*}(t;x_{1},a_{1})=\sum_{x_{2}=0}^{1}\max_{a_{2}\in\\{0,1\\}}v_{2}^{*}(t;h_{2}^{t+1},a_{2}).$
(4.29)

Then, in a similar manner to above, we find that the optimal decision is given
by $\pi^{*}(2;h_{2}^{t+1})\coloneqq|d_{2}^{*}(h_{2}^{t+1})\rangle\langle
d_{2}^{*}(h_{2}^{t+1})|$, where the optimal decision function $d_{2}^{*}$ is
again deterministic and given by

$d_{2}^{*}(h_{2}^{t+1})\coloneqq\operatorname*{arg\,max}_{a_{2}\in\\{0,1\\}}v_{2}^{*}(t;h_{2}^{t+1},a_{2})\quad\forall~{}h_{2}^{t+1}\in\\{0,1\\}^{3}.$
(4.30)

In general, we have that

$\displaystyle v_{j}(t;h_{j}^{t+1},a_{j})$
$\displaystyle\coloneqq\sum_{\begin{subarray}{c}x_{j+1},\dotsc,x_{t}=0\\\
a_{j+1},\dotsc,a_{t}=0\end{subarray}}^{1}\left(\prod_{k=j+1}^{t}\operatorname{Tr}[\pi(k;h_{k}^{t+1})|a_{k}\rangle\langle
a_{k}|]\right)\langle\psi|\widetilde{\sigma}(t+1;h^{t},a_{t},1)|\psi\rangle$
(4.31)
$\displaystyle=\sum_{x_{j+1},a_{j+1}=0}^{1}\operatorname{Tr}[\pi(j+1;h_{j+1}^{t+1})|a_{j+1}\rangle\langle
a_{j+1}|]v_{j+1}(t;h_{j+1}^{t+1},a_{j+1})$ (4.32)
$\displaystyle=\sum_{x_{j+1}=0}^{1}\operatorname{Tr}\left[\left(\sum_{a_{j+1}=0}^{1}|a_{j+1}\rangle\langle
a_{j+1}|v_{j+1}(t;h_{j+1}^{t+1},a_{j+1})\right)\pi(j+1;h_{j+1}^{t+1})\right]$
(4.33)

for all $1\leq j\leq t-2$. We recursively optimize each $v_{j}$, leading to

$\displaystyle v_{j}^{*}(t;h_{j}^{t+1},a_{j})$
$\displaystyle\coloneqq\max_{\\{\pi(k;h_{k}^{t+1}):j+1\leq k\leq
t\\}}v_{j}(t;h_{j}^{t+1},a_{j})$ (4.34)
$\displaystyle=\sum_{x_{j+1}=0}^{1}\max_{a_{j+1}\in\\{0,1\\}}v_{j+1}^{*}(t;h_{j+1}^{t+1},a_{j+1}).$
(4.35)

Then, the optimal action $d_{j}^{*}(h_{j}^{t+1})$ to be performed
deterministically at the $j^{\text{th}}$ time step is given by

$d_{j}^{*}(h_{j}^{t+1})=\operatorname*{arg\,max}_{a_{j}\in\\{0,1\\}}v_{j}^{*}(t;h_{j}^{t+1},a_{j}),\quad\forall~{}1\leq
j\leq t-2\quad\forall~{}h_{j}^{t+1}\in\\{0,1\\}^{2j-1}.$ (4.36)

For time step $t-1$, we have

$v_{t-1}(t;h_{t-1}^{t+1},a_{t-1})\coloneqq\sum_{x_{t}=0}^{1}\operatorname{Tr}\left[\left(\sum_{a_{t}=0}^{1}|a_{t}\rangle\langle
a_{t}|\langle\psi|\widetilde{\sigma}^{\prime}(t+1;h^{t},a_{t},1)|\psi\rangle\right)\pi(t;h_{t}^{t+1})\right],$
(4.37)

so that the optimal action at time $t-1$, to be performed deterministically,
is given by

$d_{t-1}^{*}(h_{t-1}^{t+1})=\operatorname*{arg\,max}_{a_{t-1}\in\\{0,1\\}}v_{t-1}^{*}(t;h_{t-1}^{t+1},a_{t-1})\quad\forall~{}h_{t-1}^{t+1}\in\\{0,1\\}^{2t-3},$
(4.38)

where

$v_{t-1}^{*}(t;h_{t-1}^{t+1},a_{t-1})\coloneqq\sum_{x_{t}=0}^{1}\max_{a_{t}\in\\{0,1\\}}\langle\psi|\widetilde{\sigma}^{\prime}(t+1;h^{t},a_{t},1)|\psi\rangle.$
(4.39)

For the final action $a_{t}$ at time $t$, we have

$d_{t}^{*}(h^{t})=\operatorname*{arg\,max}_{a_{t}\in\\{0,1\\}}\langle\psi|\widetilde{\sigma}^{\prime}(t+1;h^{t},a_{t},1)|\psi\rangle\quad\forall~{}h^{t}\in\\{0,1\\}^{2t-1}.$
(4.40)

This completes the proof. ∎

######  Remark 4.2.

Observe that the backward recursion algorithm presented in Theorem 4.1 is
exponentially slow in the horizon time because of the fact that the number of
histories grows exponentially in time—the number of histories up to time $t$
is $2^{2t-1}$. Therefore, given a finite horizon time $t$, the functions
$v_{j}^{*}$ in (4.12), which are used to determine the optimal action at the
$j^{\text{th}}$ time step for every history up to the $j^{\text{th}}$ time
step, for $1\leq j\leq t$, have an exponentially increasing number of values.
One way to avoid this issue is to use reinforcement learning to discover
(near-optimal) policies. Another path, which is an interesting direction for
future work, is to determine whether a speed-up can be obtained by trying to
find a quantum algorithm for calculating the optimal policy in a quantum
decision process.

##  5 Summary and outlook

In this work, we have cast elementary link generation in a quantum network
within the framework of quantum partially observable Markov decision
processes, as defined in [85]. At each stage in the elementary link generation
procedure, an agent decides whether to request a new quantum state or to keep
the quantum state currently in the quantum memories at the nodes. We defined
the transition maps acting on the quantum state of any elementary link
corresponding to these two actions, and we determined expressions for the
expected quantum state and expected fidelity of any elementary link at any
time for any policy and any noise model for the quantum memories. We then
looked at the special memory cutoff policy. Using the framework of decision
processes allowed us to determine both the finite-horizon and infinite-horizon
behavior of the link under the memory cutoff policy, in terms of the expected
quantum state, expected fidelity, and expected rates. We then showed how
dynamic programming can be used to obtain optimal policies in the finite-
horizon setting.

We view this work as the starting point for applying reinforcement learning to
develop practical protocols for entanglement distribution in quantum networks,
since decision processes provide the theoretical underpinnings of
reinforcement learning. As such, one direction for future work is to use the
tools developed in this work to perform reinforcement learning of (near-
optimal) policies for elementary link generation. This would be especially
useful when developing policies for long horizon times, since the dynamic
programming technique presented here is exponentially slow in the horizon time
(see Remark 4.2). Another direction for future work is to go beyond the
elementary link level and use quantum decision processes, and thus
reinforcement learning by extension, as a general technique to develop
policies for full-fledged entanglement distribution protocols in quantum
networks that incorporate entanglement distillation and joining measurements.
Such a development would involve multiple interacting agents (for the
different sets of nodes) and multiple environments (for the different edges),
and it would build on prior works [116, 18, 44, 47, 126] that examine quantum
network protocols beyond the elementary link level. One simple example of how
to extend the results of this work in this direction is to use the policies
for elementary link generation determined via the techniques developed in this
work as “sub-policies” on top of which policies for routing can be used in
order to generate long-distance entanglement in a network. Finally, it would
be interesting to apply the mathematical framework developed here to analyze
the all-photonic quantum repeater architecture [36], in which photonic graph
states are used at the elementary link level.

## Acknowledgments

This work is dedicated to the memory of Jonathan P. Dowling.

The author acknowledges support from the National Science Foundation and from
the National Science and Engineering Research Council of Canada Postgraduate
Scholarship.

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Topological defects in a double-mirror quadrupole insulator displace diverging
charge

Isidora Araya Day 1 $\star$, Anton R. Akhmerov 2 and Dániel Varjas 3

1 QuTech and Kavli Institute of Nanoscience, Delft University of Technology,
Delft 2600 GA, The Netherlands

2 Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box
4056, 2600 GA Delft, The Netherlands

3 Department of Physics, Stockholm University, AlbaNova University Center, 106
91 Stockholm, Sweden

⋆<EMAIL_ADDRESS>
$16$ February 2022

## Abstract

We show that topological defects in quadrupole insulators do not host
quantized fractional charges, contrary to what their Wannier representation
indicates. In particular, we test the charge quantization hypothesis based on
the Wannier representation of a disclination and a parametric defect. Since
disclinations necessarily strain the lattice and parametric defects require
closed curves in parameter space, both defects break four-fold rotation
symmetry, even away from their origin. The Wannier representation of the
defects is thus determined by local reflection symmetries. Contrary to the
hypothesis, we find that the local charge density decays as $\sim 1/r^{2}$
with distance, leading to a diverging defect charge. Because topological
defects are incompatible with four-fold rotation symmetry, we conclude that
defect charge quantization is protected by sublattice symmetry, and not higher
order topology.

See also: online presentation recording

A topological quadrupole insulator hosts quantized half-integer corner charges
[1, 2, 3, 4, 5], with the Benalcazar-Bernevig-Hughes (BBH) model being a
canonical example of this phase (see Fig. 1(a)). In the bulk a quantized
quadrupole moment serves as a topological invariant, providing the quadrupole
insulator with a bulk-corner correspondence [6, 7]. While the precise
definition of the bulk quadrupole moment is a subtle issue [8, 9], the bulk-
corner correspondence is guaranteed by the combination of the four-fold
rotation symmetry and two anti-commuting reflection symmetries.

Figure 1: (a) The BBH model has four orbitals per unit cell, connected via
$\gamma_{x,y}$ and $\lambda_{x,y}$ hoppings. The dashed hopping lines have an
extra phase of $-1$ with respect to the solid lines, threading a $\pi$ flux
through each plaquette. (b) The phase diagram shows that the bulk gap closes
at the grey dots, when $\lvert\gamma_{x}/\lambda_{x}\rvert=1$ and
$\lvert\gamma_{y}/\lambda_{y}\rvert=1$. The orange line indicates the set of
parameters that preserve four-fold rotation symmetry and the dashed purple
curve is a circular gapped path centered in $(\gamma_{x},\gamma_{y})=(1,1)$
with radius $\gamma_{r}=1$. (c) A disclination is created by effectively
removing one orbital from a unit cell. We expect both defects to host
localized fractional charges. (d) A parametric defect is inserted in the BBH
model using a gapped path in the phase diagram. The parametrization of the
hoppings shifts the strongly bounded unit cells around the defect, creating a
charge deficiency at it. The hopping strengths are given by the loop shown in
(b).

Disclinations in quadrupole insulators were proposed to serve as bulk probes
of topology, emulating the behavior of the corners by trapping quantized
fractional charges [10]. Furthermore, Refs. [10, 11] demonstrated that if a
band insulator admits a real space Wannier representation— _i. e._ has a
vanishing Chern number—then the non-uniform distribution of the Wannier
centers establishes the corner and defect charges quantization. The topology
of a quadrupole insulator was recently probed via disclination defects,
successfully finding defect bound states in microwave metamaterials [12].

An alternative pathway to creating topological defects, so far not explored in
the context of quadrupole insulators, is to make the Hamiltonian parameters
position-dependent [13, 14] by making the Hamiltonian $H(\bm{k},\bm{r})$
gradually vary with position. In conventional topological phases, as long as
$H(\bm{k},\bm{r})$ remains gapped far away from the defect, the presence of a
defect bound state is protected by the defect invariant. Breaking the four-
fold rotation symmetry while preserving both reflection symmetries allows us
to introduce non-contractible loops in the space of the Hamiltonians of a
quadrupole insulator [15], see Fig. 1(b). Therefore, similarly to the
quantized disclination charge, one may expect that the charge of a topological
parametric defect is quantized in a quadrupole insulator.

The topological arguments presented in earlier works, as well as the Wannier
center considerations above suggest that the defect charge quantization is a
direct consequence of topology. There is, however, no rigorous proof that the
defect charge quantization is indeed a consequence of the quantized quadrupole
moment. Our goal, therefore, is to put the charge quantization hypothesis to a
test and rigorously identify the conditions under which the defect charge is
quantized.

To test the hypothesis we use the BBH tight-binding model: a two-dimensional
lattice with four orbitals per unit cell. Figure 1(a) shows the model, which
corresponds to a quadrupole insulator that admits a Wannier representation,
and whose Bloch Hamiltonian is:

$\displaystyle\mathcal{H}(\bm{k})=$
$\displaystyle[\gamma_{x}+\lambda_{x}\cos(k_{x})]\Gamma_{4}+\lambda_{x}\sin(k_{x})\Gamma_{3}+$
$\displaystyle[\gamma_{y}+\lambda_{y}\cos(k_{y})]\Gamma_{2}+\lambda_{y}\sin(k_{y})\Gamma_{1},$
(1)

where $\gamma_{x,y}$ and $\lambda_{x,y}$ account for intra-cell and inter-cell
hopping amplitudes, respectively. Here $\Gamma_{k}=-\sigma_{2}\otimes\tau_{k}$
for $k\in\\{1,2,3\\}$, and $\Gamma_{4}=\sigma_{1}\otimes\tau_{0}$, with
$\sigma_{i}$ and $\tau_{i}$ Pauli matrices acting on the orbital degrees of
freedom.

We introduce a disclination by removing the quarter of the lattice that is
spanned by $\theta\in[3\pi/2,2\pi)$, changing the positions of the remaining
sites according to $\theta\rightarrow 4/3\theta$, while keeping
$r\equiv|\bm{r}|$ constant (see Fig. 1(c)). We keep the lattice isotropic by
choosing $\gamma_{x}=\gamma_{y}=\gamma$ and $\lambda_{x}=\lambda_{y}=\lambda$.
If the disclination center lies between the strongly coupled sites, the
disclination removes an orbital from one of the strongly coupled clusters. The
Wannier centers are located at the center of these clusters, such that the
resulting strongly bonded triangle has $3/2$ electrons at half-filling, and
therefore a half-integer charge. An alternative way to come to the same
conclusion is by observing that each of the three corners must have a charge
$1/2$ because the quadrupole insulator is in its topological phase. The
remaining half integer charge must then be bound to the disclination
center—the only remaining lattice defect.

Both the argument above and the proof of Ref. [10] ignore the effect of
lattice distortion around the defect center. To study the effect of strain on
the tight-binding model, we modulate the hopping amplitudes by changing them
proportionally to the bond length:

$\displaystyle\gamma^{\prime}(\bm{r}_{1},\bm{r}_{2})=\gamma(1+\alpha_{\gamma}(\lvert\bm{r}_{1}-\bm{r}_{2}\rvert-a))$
(2)
$\displaystyle\lambda^{\prime}(\bm{r}_{1},\bm{r}_{2})=\lambda(1+\alpha_{\lambda}(\lvert\bm{r}_{1}-\bm{r}_{2}\rvert-a)),$
(3)

where $\gamma^{\prime}$ and $\lambda^{\prime}$ are hopping amplitudes coupling
the orbitals that would be connected by a hopping $\gamma$ or $\lambda$
without a disclination, $a=1/2$ is the bond rest length, and $\bm{r}_{1}$,
$\bm{r}_{2}$ are the coordinates of the two orbitals. The factors
$\alpha_{\gamma}$ and $\alpha_{\lambda}$ are the electron-lattice couplings.
To keep the system in the topological phase, we require that far away from the
disclination the ratio $\lvert\gamma/\lambda\rvert<1$. Because the
disclination necessarily introduces strain, without fine-tuning that ensures
$\alpha_{\gamma}=\alpha_{\lambda}=0$, the disclination necessarily breaks the
four-fold rotation symmetry even far away from its center.

A parametric defect is an alternative way to create a quantized fractional
charge. We create this defect in the BBH model at an arbitrary lattice
coordinate $\bm{r}_{0}$. We fix $\lambda_{x}=\lambda_{y}=1$ and vary the
hopping strengths $\gamma_{x}$ and $\gamma_{y}$ with displacement $\bm{r}$
from the defect. For simplicity we consider a defect where $\gamma_{x}$ and
$\gamma_{y}$ only depend on the angle $\theta$ between $\bm{r}$ and the
$x$-axis. We require that as the angle $\theta$ of $\bm{r}$ in polar
coordinates advances by $2\pi$, the vector $(\gamma_{x},\gamma_{y})$ encloses
a loop around the point $(1,1)$. As an example, Fig. 1(d) shows the hopping
strengths corresponding to the circular loop in the parameter space shown in
purple in Fig. 1(b). The center of the defect has an odd-sized cluster of
strongly coupled sites, which contains a half-integer number of electrons at
half-filling, similarly to a disclination. Tracking the positions of the
Wannier centers within the unit cell [15] as a function of $\bm{r}$, shown in
Fig. 2, demonstrates the presence of the defect charge.

Figure 2:  The parametric defect interpolates between the trivial (top right
quadrant) and topological (bottom left quadrant) phases of the BBH model (grey
dots connected by lines) by smoothly modulating the hoppings and preserving
local mirror symmetries. While in the trivial phase the Wannier centers (blue
circles) are localized at the center of the unit cell, they are localized at
the corners in the topological phase. In all unit cells the Wannier centers
come in pairs invariant under local mirror symmetries. The square cells, each
centered in between pairs of nearest Wannier centers, tile the space away from
the defect. The unoccupied quarter unit cell at $\bm{r}=0$ corresponds to the
missing fractional charge $e/2$.

While the defect charge quantization hypothesis relies only on the spatial
symmetry arguments, the BBH model Eq. (1) is minimal and therefore it has
additional symmetries. In particular, the sublattice symmetry of the BBH model
by itself leads to charge quantization. To see this we consider the local
charge density in a unit cell

$\displaystyle\rho_{ij}=\sum_{E_{l}<E_{F}}\sum_{\alpha}\lvert\Psi_{\alpha
l}^{ij}\rvert^{2},$ (4)

where $E_{l}\leq E_{F}\equiv 0$ labels the occupied energies with the Fermi
level fixed in the bulk gap and $\alpha\in\\{1,2,3,4\\}$ labels the four
orbitals of the unit cell. The indices $i,j$ are the lattice coordinates of a
unit cell, where the defect origin is at $i=0,j=0$, and $\Psi_{\alpha
l}^{ij}=\braket{i,j,\alpha\lvert\Psi_{l}}$. We express the total number of
states in a unit cell through the contributions of eigenstates at different
energies:

$4=\sum_{E_{l}<0}\sum_{\alpha=1}^{4}\lvert\Psi_{\alpha
l}^{ij}\rvert^{2}+\sum_{E_{l}=0}\sum_{\alpha=1}^{4}\lvert\Psi_{\alpha
l}^{ij}\rvert^{2}+\sum_{E_{l}>0}\sum_{\alpha=1}^{4}\lvert\Psi_{\alpha
l}^{ij}\rvert^{2}\\\ =2\sum_{E_{l}<0}\sum_{\alpha=1}^{4}\lvert\Psi_{\alpha
l}^{ij}\rvert^{2}+\sum_{E_{l}=0}\sum_{\alpha=1}^{4}\lvert\Psi_{\alpha
l}^{ij}\rvert^{2}=2\rho_{ij}+\sum_{E_{l}=0}\sum_{\alpha=1}^{4}\lvert\Psi_{\alpha
l}^{ij}\rvert^{2},$ (5)

where we use the sublattice symmetry in the second equality. Finally, the
total defect charge $q_{\textrm{tot}}$ integrated over a square-shaped area
surrounding the defect is the charge excess with respect to the uniform charge
density $\rho_{ij}=2$:

$\displaystyle q_{\textrm{tot}}(R)$
$\displaystyle=\sum_{i,j=-R/2}^{R/2}\Big{(}\rho_{ij}-2\Big{)}.$ (6)

Substituting Eq. (5) into Eq. (6) we obtain

$\displaystyle q_{\textrm{tot}}(R)$
$\displaystyle=-\frac{1}{2}\sum_{i,j=-R/2}^{R/2}\sum_{E_{l}=0}\sum_{\alpha=1}^{4}\lvert\Psi_{\alpha
l}^{ij}\rvert^{2}.$ (7)

Therefore in presence of the sublattice symmetry, the total defect charge is
$N_{0}/2$, where $N_{0}$ is the number of zero-energy modes localized at the
defect. The number of the zero-energy modes, in turn, is defined by the second
winding number of the Hamiltonian around the defect, and protected already by
the sublattice symmetry alone [14]. Furthermore, because of the bulk gap, the
zero-modes are exponentially localized.

Sublattice symmetry also guarantees the disclination charge quantization,
despite the center of the disclination locally breaking the sublattice
symmetry. To demonstrate this, we use perturbation theory and consider
$3/4\textsuperscript{th}$ of the BBH lattice the unperturbed Hamiltonian. This
Hamiltonian inherits sublattice symmetry from the BBH model and therefore it
hosts fractional and exponentially localized charges at its six corners. To
obtain a disclination, we add a perturbation that consists on the hopping
terms that couple orbitals from neighboring unit cells on the position-
transformed lattice, such that the lattice is glued. Using that the system is
gapped, we conclude that the Fermi level Green’s function, and therefore
charge density, changes by an amount that decays exponentially with distance
from the disclination’s origin. Because sublattice symmetry applies near the
sample corners, we conclude there is a $1/2$ charge that may only be located
near the disclination.

To distinguish the topological properties of the quadrupole insulator from the
effect of the sublattice symmetry, we add hoppings with magnitude $\delta$
connecting sites from the same sublattice in the neighboring unit cells. In
order to preserve the anti-commuting reflection and the four-fold rotation
symmetries of the original model, we choose the value of all these hoppings to
be the same. We change the sign of the additional hoppings when gluing the
different sides of the wedge at the disclination in order for the cut to be
gauge-compatible with the rest of the lattice.

Figure 3:  Local charge density at half-filling in the defects. A defect
displaces both negative (blue) and positive (red) excess charge in the absence
of sublattice symmetry, resulting in a divergent defect charge. (a-b) The
parametric defect and the disclination with sublattice symmetry. (c-d) The
parametric defect and the disclination without sublattice symmetry.

We numerically study the defect charge inserting the defects at the center of
a $L^{2}=50\times 50$ square system. We create the parametric defect using a
circular path of radius $\gamma_{r}=1$ centered at
$\gamma_{x}/\lambda_{x}=\gamma_{y}/\lambda_{y}=1$ (see Fig. 1(b)), and the
disclination using $\gamma=\lambda/2=0.5$ and
$\alpha_{\gamma}=\alpha_{\lambda}=\alpha_{\delta}=1$. To obtain the defect
charges we integrate the charge density distributions shown in Fig. 3(a-d).
Our results in Fig. 4(a) confirm that the total defect charge converges to
$1/2$ in presence of sublattice symmetry, and demonstrate an apparent
convergence to a non-quantized value otherwise, unless the electron-lattice
coupling is neglected, in agreement with Ref. [10]. This lack of quantization
is explained by considering the absolute charge deviation:

$\displaystyle q(R)$
$\displaystyle=\sum_{i,j=-R/2}^{R/2}\Big{\lvert}\rho_{ij}-2\Big{\rvert}.$ (8)

While the absolute charge deviation converges to a finite value in presence of
sublattice symmetry, it diverges when the sublattice symmetry and the local
four-fold rotation symmetry are broken, as shown in Fig. 4(b). According to
our expectations, the defect charge convergence is exponential (Fig. 4(c)). On
the other hand, the divergence matches a $\sim 1/r^{2}$ decay of the local
charge density, as confirmed by $\Delta q(R)=q(R+1)-q(R)\propto 1/R$ in Fig.
4(d). The lack of absolute convergence in combination with conditional
convergence means that depending on the summation order, the defect charge may
assume an arbitrary value according to the Riemann rearrangement theorem.
While the absolute convergence of charge deviation is required, it is also a
weaker condition on charge quantization than the vanishing charge variance
studied in Ref. [16].

Figure 4: The defect charge diverges in the absence of sublattice symmetry.
(a) The total charge $q_{\textrm{tot}}$ integrates to $1/2$ only when
sublattice symmetry is present (dashed lines), otherwise $q_{\textrm{tot}}$
depends on the integration area (solid lines), demonstrating the lack of
absolute convergence. (b) In the presence of sublattice symmetry (dashed
lines), the charge deviation $q$ is quantized to $1/2$, otherwise it diverges
(solid lines). The divergence of the parametric defect’s charge is more
pronounced when the paths are asymmetric in parameter space. (c) The defect
charge deviation converges exponentially when $\delta=0$ (dashed lines), as
$-\log\lvert q(R+1)-q(R)\rvert\propto R$. (d) For $\delta\neq 0$ the local
charge density decays as $1/R^{2}$ as indicated by $q(R+1)-q(R)\propto 1/R$.

In order to confirm the total charge divergence we determine the asymptotic
behavior of the local charge density away from the defect. Sufficiently far
away from the defect in a large enough sample, the Hamiltonian’s position
dependence becomes small compared to the inverse energy gap while it changes
slowly in space. Thus, the local real space variations of the Hamiltonian
determine the charge density and its local response is captured by
perturbation theory that treats the position dependence as a perturbation. To
account for $\rho\propto 1/r^{2}$ found via numerical simulations, it is
enough to consider a general position-dependent perturbation expanded to
second order around $\bm{r}^{*}$ far away from the defect center $\bm{r}_{0}$,

$\displaystyle H^{\prime}(\delta\bm{r})=H^{\prime 0}+\delta\bm{r}_{i}H^{\prime
1}_{i}+\delta\bm{r}_{i}H^{\prime 2}_{ij}\delta\bm{r}_{j}+h.c.,$ (9)

where $H^{\prime 0}$, $H^{\prime 1}$ and $H^{\prime 2}$ are the zeroth, first
and second order components of the Hamiltonian perturbation, and
$\delta\bm{r}\equiv\bm{r}-\bm{r}^{*}$. By construction, $H^{\prime 0}$,
$H^{\prime 1}$, and $H^{\prime 2}$ are invariant under reflection. While
$H^{\prime 1}$ is reflection-symmetric, $\bm{r}_{i}H^{\prime 1}_{i}$ is odd
under reflection symmetries. The perturbative response of the local charge
density is invariant under reflection and it is a power series of $H^{\prime
1}$ and $H^{\prime 2}$. As a consequence of being odd under reflection, the
first order contribution of $H^{\prime 1}$ to the charge density vanishes,
while the contribution of $H^{\prime 2}$ and the second order contribution of
$H^{\prime 1}$ remain. Because both types of defects have a finite
$dH/d\theta\to\textrm{const}$ with $r\to\infty$, $H^{\prime 1}\sim 1/r$ and
$H^{\prime 2}\sim 1/r^{2}$. Therefore, we confirm that the charge density
universally decays as $1/r^{2}$ unless the Hamiltonian is fine-tuned or it has
additional symmetries.

In summary, we investigated charge quantization of defect bound states in
quadrupole insulators as it was predicted in Refs. [10, 11] and reported in
Ref. [12]. Specifically, we analyzed a disclination and a parametric defect in
the BBH model and found that the previously reported charge quantization is a
consequence of the sublattice symmetry, or of neglecting the effects of
lattice strain around the disclination. The importance of sublattice symmetry
was appreciated in the early studies of soliton-fermion bound states in one-
dimensional models [17, 18, 19]. This symmetry is not inherent to quadrupole
insulators, but without it the defect charge is not quantized, because the
absolute charge deviation diverges. Our findings demonstrate that topological
protection of the filled bands, despite being characterized by a quantized
topological invariant, is weaker than the protection of individual states in
topological insulators.

## Acknowledgements

We thank B. Seradjeh for drawing our attention to relevant literature. D. V.
is thankful to R. Queiroz for enlightening discussions.

## Data availability

The code used to produce the reported results is available on Zenodo [20].

### Author contributions

D. V. defined the project goal and formulated the hypothesis. I. A. D.
implemented the numerical checks with guidance from D. V. and A. R. A. All
authors interpreted the results and contributed to the project planning. I. A.
D. and A. R. A. wrote the manuscript with input form D. V.

### Funding information

This work was supported by the Netherlands Organization for Scientific
Research (NWO/OCW), as part of the Frontiers of Nanoscience program and an NWO
VIDI grant 016.Vidi.189.180. We acknowledge the support from QuTech Academy
through the Scholarship. D. V. was supported by the Swedish Research Council
(VR) and the Knut and Alice Wallenberg Foundation.

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