We choose the following linearly independent set of $\chi$-indices that form a
basis: $1b_{1}:(0\,|\,-1,1,-1,0;1)$, $1b_{2}:(0\,|\,1,0,0,1;1)$,
$1c_{1}:(0\,|\,-1,0,-1,1;1)$, $1c_{2}:(0\,|\,0,1,1,0;1)$. By examining the
finite systems formed by the tilings of the $C_{3}$-symmetric unit cell in
Fig. 11(d) and (e), we see that the corner charges for $1b_{1}$, $1b_{2}$,
$1c_{1}$ and $1c_{2}$ are $1/3$, $1/3$, $0$ and $0$ respectively. This implies
that

$\left(\begin{array}[]{c}\frac{1}{3}\\\ \frac{1}{3}\\\ 0\\\
0\end{array}\right)=\left(\begin{array}[]{cccc}-1&1&-1&0\\\ 1&0&0&1\\\
-1&0&-1&1\\\ 0&1&1&0\end{array}\right)\left(\begin{array}[]{c}\lambda_{1}\\\
\lambda_{2}\\\ \lambda_{3}\\\ \lambda_{4}\end{array}\right),$ (71)

which gives $\lambda_{1,2,3,4}=\frac{1}{3},\frac{1}{3},-\frac{1}{3},0$.
Therefore,

$\displaystyle
Q_{\text{corner}}^{(3)}=\frac{1}{3}\left([K^{(3)}_{1}]+[K^{(3)}_{2}]-[K^{\prime(3)}_{1}]\right).$
(72)

###  E.3 $C_{4}$ symmetry

For the TR-symmetric case, $C_{4}$-symmetric lattices have three symmetry-
indicator invariants: $[X^{(4)}_{1}]$, $[M^{(4)}_{1}]$ and $[M^{(4)}_{2}]$.
The corner charge is given by a linear combination of these invariants

$Q_{\mathcal{T},\text{corner}}^{(4)}=\lambda_{1}[X^{(4)}_{1}]+\lambda_{2}[M^{(4)}_{1}]+\lambda_{3}[M^{(4)}_{2}].$
(73)

We choose the following linearly independent set of $\chi$-indices that form a
basis: $2c:(-1,-1,1;2)$, $1b_{1}:(-1,1,0;1)$, $1b_{2}:(2,0,0;2)$. By examining
the finite systems formed by the tilings of the $C_{4}$-symmetric unit cell in
Fig. 11(f) and (g), we see that the corner charges for $2c$, $1b_{1}$ and
$1b_{2}$ are $0$, $1/4$ and $1/2$ respectively (note that $1b_{2}:(2,0,0;2)$
is induced by two bands, each with Wannier centers at $1b$. Therefore, the net
corner charge is $(2\times 1/4)\text{ mod }1=1/2$). This implies that

$\left(\begin{array}[]{c}0\\\ \frac{1}{4}\\\
\frac{1}{2}\end{array}\right)=\left(\begin{array}[]{ccc}-1&-1&1\\\ -1&1&0\\\
2&0&0\end{array}\right)\left(\begin{array}[]{c}\lambda_{1}\\\ \lambda_{2}\\\
\lambda_{3}\end{array}\right),$ (74)

which gives $\lambda_{1,2,3}=\frac{1}{4},\frac{1}{2},\frac{3}{4}$. Therefore,

$\displaystyle
Q_{\mathcal{T},\text{corner}}^{(4)}=\frac{1}{4}\left([X^{(2)}_{1}]+2[M^{(4)}_{1}]+3[M^{(4)}_{2}]\right).$
(75)

For the TR-broken case, $C_{4}$-symmetric lattices have four symmetry-
indicator invariants: $[X^{(4)}_{1}]$, $[M^{(4)}_{1}]$, $[M^{(4)}_{2}]$ and
$[M^{(4)}_{4}]$. The corner charge is given by a linear combination of these
invariants

$Q_{\text{corner}}^{(4)}=\lambda_{1}[X^{(4)}_{1}]+\lambda_{2}[M^{(4)}_{1}]+\lambda_{3}[M^{(4)}_{2}]+\lambda_{4}[M^{(4)}_{4}].$
(76)

We choose the following linearly independent set of $\chi$-indices that form a
basis: $2c:(0\,|\,-1,-1,1,1;2)$, $1b_{1}:(0\,|\,-1,1,0,0;1)$,
$1b_{2}:(0\,|\,1,0,-1,1;1)$, $1b_{3}:(0\,|\,1,0,1,-1;1)$. By examining the
finite systems formed by the tilings of the $C_{4}$-symmetric unit cell in
Fig. 11(f) and (g), we see that the corner charges for $2c$, $1b_{1}$,
$1b_{2}$ and $1b_{3}$ are $0$, $1/4$, $1/4$ and $1/4$ respectively. This
implies that

$\left(\begin{array}[]{c}0\\\ \frac{1}{4}\\\ \frac{1}{4}\\\
\frac{1}{4}\end{array}\right)=\left(\begin{array}[]{cccc}-1&-1&1&1\\\
-1&1&0&0\\\ 1&0&-1&1\\\
1&0&1&-1\end{array}\right)\left(\begin{array}[]{c}\lambda_{1}\\\
\lambda_{2}\\\ \lambda_{3}\\\ \lambda_{4}\end{array}\right),$ (77)

which gives $\lambda_{1,2,3,4}=\frac{1}{4},\frac{1}{2},\frac{3}{8}$,
$\frac{3}{8}$. Therefore,

$\displaystyle
Q_{\text{corner}}^{(4)}=\frac{1}{4}\left([X^{(2)}_{1}]+2[M^{(4)}_{1}]+\frac{3}{2}[M^{(4)}_{2}]+\frac{3}{2}[M^{(4)}_{4}]\right).$
(78)

###  E.4 $C_{6}$ symmetry

For the TR-symmetric case, $C_{6}$-symmetric lattices have two symmetry-
indicator invariants: $[M^{(2)}_{1}]$ and $[K^{(3)}_{1}]$. The corner charge
is given by a linear combination of these invariants

$Q_{\mathcal{T},\text{corner}}^{(6)}=\lambda_{1}[M^{(2)}_{1}]+\lambda_{2}[K^{(3)}_{1}].$
(79)

We choose the following linearly independent set of $\chi$-indices that form a
basis: $2b:(0,2;4)$, $3c:(2,0;3)$. By examining the finite systems formed by
the tilings of the $C_{6}$-symmetric unit cells in Fig. 11(h) and (i), we see
that the corner charges for $2b$ and $3c$ are $1/3$ and $1/2$ respectively
(note that $2b:(0,2;4)$ is induced by four bands, with Wannier centers of each
pair at $2b$. The net corner charge is therefore ($2\times 2/3)\text{ mod
}1=1/3$). This implies that

$\left(\begin{array}[]{c}\frac{1}{3}\\\
\frac{1}{2}\end{array}\right)=\left(\begin{array}[]{cc}0&2\\\
2&0\end{array}\right)\left(\begin{array}[]{c}\lambda_{1}\\\
\lambda_{2}\end{array}\right),$ (80)

which gives $\lambda_{1,2}=\frac{1}{4},\frac{1}{6}$. Therefore,

$Q_{\mathcal{T},\text{corner}}^{(6)}=\frac{1}{4}[M^{(2)}_{1}]+\frac{1}{6}[K^{(3)}_{1}].$
(81)

For the TR-broken case, $C_{6}$-symmetric lattices have three symmetry-
indicator invariants: $[M^{(2)}_{1}]$, $[K^{(3)}_{1}]$ and $[K^{(3)}_{2}]$.
The corner charge is given by a linear combination of these invariants

$Q_{\text{corner}}^{(6)}=\lambda_{1}[M^{(2)}_{1}]+\lambda_{2}[K^{(3)}_{1}]+\lambda_{3}[K^{(3)}_{2}].$
(82)

We choose the following linearly independent set of $\chi$-indices that form a
basis: $2b_{1}:(0\,|\,0,1,-2;2)$, $2b_{2}:(0\,|\,0,1,1;2)$,
$3c:(0\,|\,2,0,0;3)$. By examining the finite systems formed by the tilings of
the $C_{6}$-symmetric unit cells in Fig. 11(h) and (i), we see that the corner
charges for $2b_{1}$, $2b_{2}$ and $3c$ are $2/3$, $2/3$ and $1/2$
respectively. This implies that

$\left(\begin{array}[]{c}\frac{2}{3}\\\ \frac{2}{3}\\\ \frac{1}{2}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}0&1&-2\\\ 0&1&1\\\
2&0&0\end{array}\right)\left(\begin{array}[]{c}\lambda_{1}\\\ \lambda_{2}\\\
\lambda_{3}\end{array}\right),$ (83)

which gives $\lambda_{1,2,3}=\frac{1}{4},\frac{2}{3},0$. Therefore,

$Q_{\text{corner}}^{(6)}=\frac{1}{4}[M^{(2)}_{1}]+\frac{2}{3}[K^{(3)}_{1}].$
(84)

Figure 12: All possible atomic limits that have (a) one and (b) two Wannier
centers with $C_{4}$\- (red) and $C_{2}$\- (blue) symmetric unit cells that
correspond to the same infinite lattice. All possible atomic limits that have
(d) one, (e) two, and (f) three Wannier centers with $C_{6}$\- (yellow),
$C_{3}$\- (green), and $C_{2}$\- (blue) symmetric unit cells that correspond
to the same infinite lattice. Note that some atomic limits with reduced
symmetry do not have Wannier centers fixed to maximal Wyckoff positions but
are instead movable (marked with a $\times$).

##  Appendix F Effect of unit cell choices on boundary states

When considering 2D PhCs with OAL bands, a particular choice of unit cell can
affect the relevant symmetries for the topological classification of the bulk
and the presence of boundary states in a finite tiling of that unit cell. For
example, consider all possible unit cell choices shown in Fig. 12(a). The unit
cells marked in red are $C_{4}$ symmetric and correspond to Wannier centers
located at $1a$ or $1b$ positions. The same infinite structure is also
consistent with unit cells that have reduced symmetry, in this case, $C_{2}$
symmetry, marked in blue. These correspond to Wannier centers located at the
$1a$, $1b$, $1c$, or $1d$ positions. A finite tiling of any of these unit
cells will result in edge or corner states depending on the dipole moment and
corner charge of their respective Wannier center configurations. This analysis
is performed diagrammatically for all possible Wannier center configurations
in Fig. 12.

Furthermore, when a choice of the unit cell reduces the symmetry of the
system, the new symmetry-reduced invariants may be found using the following
relations: Under TRS, the $C_{2}$ invariants of a $C_{4}$-symmetric PhC obey
$[X_{1}^{(2)}]=[Y_{1}^{(2)}]$ and $[M_{1}^{(2)}]=-2[M_{2}^{(4)}]$, and the
$C_{3}$ invariants of a $C_{6}$-symmetric PhC obey
$[K^{(3)}_{1}]=[K^{(3)}_{2}]$. Under broken TRS, the $C_{2}$ invariants of a
$C_{4}$-symmetric PhC obey $[X_{1}^{(2)}]=[Y_{1}^{(2)}]$ and
$[M_{1}^{(2)}]=-2[M_{2}^{(4)}]$, and the $C_{3}$ invariants of a
$C_{6}$-symmetric PhC obey $[K^{(3)}_{1}]=[K^{\prime(3)}_{1}]$ and
$[K^{(3)}_{2}]=[K^{\prime(3)}_{2}]$.

These considerations are important for PhC design since a second “cladding”
material is often required to confine the boundary states of the topological
“core”. With a different choice of unit cell made for the cladding material,
both the core and cladding can have identical band structures and therefore
conveniently overlapping bandgaps while having different topological
invariants.


resemblance with electrodynamics. One can further the semblance by observing
that the holonomy $\delta\ket{\phi}$ is nothing but the flux of the local
“magnetic field" (which is similar to the curvature/“gauge field" $F_{\mu\nu}$
in quantum mechanics) through an infinitesimal loop on $\mathcal{B}$.

We note that the gauge field $F_{\mu\nu}$ is not generically gauge-invariant
except for the Abelian case where it transforms under a gauge transformation
as,

$F_{\mu\nu}\rightarrow UF_{\mu\nu}U^{\dagger}.$ (266)

To construct a gauge-invariant quantity, let us recall the definition of
parallel transport of a vector $\ket{\phi}$ on a curved manifold via Eq.
(259). The general solution of the Eq. (260) can be written in terms of a
Dyson series,

$\vec{\Phi}_{||}({\lambda}_{f})=\mathbb{P}_{||}({\lambda}_{f},{\lambda_{i}})\vec{\Phi}({\lambda}_{i}),$
(267)

where $\vec{\Phi}_{||}(\vec{\lambda}_{f})$ is $\vec{\Phi}(\vec{\lambda}_{i})$
after being transported from the point $\vec{\lambda}_{i}$ to
$\vec{\lambda}_{j}$ on $\mathcal{B}$ and,

$\mathbb{P}_{||}(\vec{\lambda}_{f},\vec{\lambda_{i}})=\mathcal{P}e^{i\int_{\vec{\lambda}_{i}}^{\vec{\lambda}_{f}}A^{\mu}d\lambda_{\mu}},$
(268)

is the parallel propagator which transports a vector parallely along a curve
$C$; $\mathcal{P}$ being the path-ordering operator on $C$. The parallel
propagator is sometimes also known as a Wilson line. From Eqs. (256) and
(268), it is apparent that the parallel propagator also suffers from the
ambiguity of gauge freedom. Under a gauge transformation, the propagator
transforms as,

$\mathbb{P}_{||}(\vec{\lambda}_{f},\vec{\lambda_{i}})\rightarrow
U(\vec{\lambda}_{f})\mathbb{P}_{||}(\vec{\lambda}_{f},\vec{\lambda_{i}})U^{\dagger}(\vec{\lambda}_{i}).$
(269)

In a similar methodology employed in Sec. A.1, a gauge invariant can be
constructed by considering a parallel transport along a closed curve
($\vec{\lambda}_{i}=\vec{\lambda}_{f}=\vec{\lambda}$), which is realized by
the parallel propagator,

$\mathbb{P}_{o}=\mathcal{P}e^{i\oint A^{\mu}d\lambda_{\mu}},$ (270)

Under a gauge transformation, the closed loop propagator transforms as,

$\mathbb{P}_{o}\rightarrow
U(\vec{\lambda})\mathbb{P}_{o}U^{\dagger}(\vec{\lambda}).$ (271)

It can now be seen that the quantity,

$W={\rm Tr}\left(\mathcal{P}e^{i\oint A^{\mu}d\lambda_{\mu}}\right),$ (272)

must be gauge invariant under a generic ${\rm U(N)}$ gauge transformation due
to the cyclic property of trace. This gauge invariant quantity is popularly
known as the Wilson loopwang19  holonomy on the parameter manifold
$\mathcal{B}$. A more useful form is the gauge-invariant action/index which is
defined as,

$\mathcal{C}=\frac{1}{i}\ln{\rm Tr}\left(\mathcal{P}e^{i\oint
A^{\mu}d\lambda_{\mu}}\right)=c_{\phi}.$ (273)

We have thus constructed a gauge invariant index $c_{\phi}$ that characterizes
states $\ket{\phi({\lambda})}\in\mathcal{V}_{{\lambda}}$ on the generic
parameter manifold $\mathcal{B}$. However, in practice, the parameter manifold
is often discrete as opposed to a continuous one. To exemplify, consider a
many-body Hamiltonian describing non-interacting particles moving on a
periodic lattice. This Hamiltonian then respects a discrete translational
symmetry and not a continuous one. Utilizing this symmetry, one might recast
the complete description in Fourier space parametrized by conserved
quasimomenta $(\\{k_{i}\\}_{i=1}^{i=d})$ which now form a $d$-dimensional
discrete parameter space. The discrete space approaches a continuum manifold
in the thermodynamic limit of the system. It is thus important to extend the
formalism, discussed thus far in the context of continuous manifold, to
situations in which the parameter space is discrete.

For the sake of simplicity, let us consider the case of a two dimensional
parameter space. An extension to higher dimensions can be subsequently made in
a straight forward manner. Assuming a square lattice, we label each point on
the discrete parameter space by the coordinates
$\vec{\lambda}\in\\{na\hat{\mu},ma\hat{\nu}\\}$ where $a$ is the distance
between neighboring lattice points and the numbers $\\{n,m\\}$ take integral
values. The discrete space then reaches a continuum manifold in the limit
$a\rightarrow 0$. We now proceed to construct Wilson loops on this discrete
parameter space. It is evident that the Wilson line originating from the point
$\vec{\lambda}$ and connecting the next consecutive lattice point in the
direction $\hat{\mu}$, for $a\ll 1$, can be approximated as,

$\mathcal{U}_{\mu}({\lambda})=\mathbb{P}_{||}({\lambda})\simeq
e^{iA^{\mu}({\lambda})a}.$ (274)

The group-valued quantities $\mathcal{U}_{\mu}$ on the lattice is also known
as a link variable simply because it lives on the “links" connecting lattice
points while the quantum states $\ket{\phi({\lambda})}$ reside on the lattice
points itself. Using the link variables, we can therefore generate a parallel
transformation on a square loop on the lattice such that the enclosed area is
a minimum. The rules to do so can be inferred immediately: (i) to parallely
transport a vector from $\vec{\lambda}$ to

Figure 26: Schematic picture of the lattice Wilson loop on a discrete square
lattice denoting the link variables and path traversed around the closed loop
enclosing an area of $a^{2}$, $a$ being the lattice parameter.

$(\vec{\lambda}+a\hat{\mu})$, one must operate with
$\mathcal{U}_{\mu}(\vec{\lambda})$, and (ii) to induce the reverse transport
from $(\vec{\lambda}+a\hat{\mu})$ to $\vec{\lambda}$, one must similarly
operate with $\mathcal{U}^{\dagger}_{\mu}(\vec{\lambda}+a\hat{\mu})$. Hence,
for the infinitesimal square loop over a plaquette of minimum area ($=a^{2}$)
on the discrete lattice, the logarithm of the gauge invariant lattice Wilson
loop $W_{L}$ (see Fig. 26)can be written asfukui05 ,

$W_{L}=\frac{1}{i}\ln{\rm
Tr}\left[\mathcal{U}_{\nu}^{\dagger}(\vec{\lambda})\mathcal{U}^{\dagger}_{\mu}(\vec{\lambda}+a\hat{\nu})\mathcal{U}_{\nu}(\vec{\lambda}+a\hat{\mu})\mathcal{U}_{\mu}(\vec{\lambda})\right].$
(275)

Substituting Eq. (274) for the link variables in Eq. (275) and retaining terms
up to the lowest order in $a$, one obtains,

$W_{L}\simeq\frac{1}{i}\ln{\rm Tr}\left(e^{ia^{2}F_{\mu\nu}+....}\right),$
(276)

where we have retained terms up to quadratic in order $a$. This expression can
further be simplified for $a\ll 1$ as,

$\displaystyle W_{L}$ $\displaystyle\simeq$ $\displaystyle a^{2}{\rm
Tr}F_{\mu\nu}={\rm Tr}\left(F_{\mu\nu}\right)d\lambda^{\mu}d\lambda^{\nu},$
(277)

where we have evaluated the logarithm in its principal branch, i.e.,

$\left|{\rm
Tr}\left(F_{\mu\nu}\right)d\lambda^{\mu}d\lambda^{\nu}\right|<\pi.$ (278)

In the continuum limit $a\rightarrow 0$, Eq. (277) approaches an exact
equality. It is also to be noted that the Wilson loop around an infinitesimal
plaquette is simply the flux of the gauge field through the plaquette.
Therefore, it is natural to expect that if the Wilson loop encloses a finite
region $S$, the total flux through the loop equals the sum of the fluxes
through all infinitesimal plaquettes $dS\in S$ within the loop. We can
therefore write the discretized version of the index $\mathcal{C}$ defined in
Eq. (273) as,

$\displaystyle\mathcal{C}(C)$ $\displaystyle\simeq$
$\displaystyle\sum\limits_{dS\in S}W_{L}(dS)$ (279) $\displaystyle=$
$\displaystyle\sum\limits_{dS\in S}{\rm
Tr}\left(F_{\mu\nu}\right)\Delta^{\mu\nu}+2\pi\eta(dS),$

where $\Delta^{\mu\nu}$ is the area element of each plaquette. The integer
valued field $\eta(dS)$ defined over each plaquette restricts the complex
logarithm in Eq. (275) in the principal branch, thus satisfying Eq. (278). In
the continuum limit, the relation becomes an exact equality and is also known
as the Stokes’ theorem,

$\mathcal{C}(C)=\frac{1}{i}\ln{\rm
Tr}\left(\mathcal{P}e^{i\oint_{C}A^{\mu}d\lambda_{\mu}}\right)=\int_{S}{\rm
Tr}\left(F_{\mu\nu}\right)d\sigma^{\mu\nu},$ (280)

where all logarithms are evaluated in the principal branch.

###  A.3 Topological invariants

####  A.3.1 Chern invariants

We are now fully equipped with the necessary abstractions and ready to probe
the topological phases of physical quantum systems. Consider a non-interacting
fermionic many body system described by the Hamiltonian $H$ on a periodic
lattice. As discussed previously, one can utilize the discrete spatial
translation symmetry to rewrite the Hamiltonian in terms of
decoupled/conserved momentum modes $\vec{k}$ in Fourier space,

$H=\bigoplus_{\vec{k}}H(\vec{k}),$ (281)

where the vector $\vec{k}$ parametrizes the underlying manifold. For example,
in $d=2$, we have $\vec{k}\equiv\\{k_{1},k_{2}\\}$ where the coordinates
$k_{1}$ and $k_{2}$ specify position along two basis directions in reciprocal
space. As we have already seen in Sec. A.1, for one-dimensional systems, the
Berry phase $\Phi_{B}$ (for singly filled bands i.e., $N=1$) characterizes the
topological classification of the eigenstates of $H(k)$. In this section, we
focus primarily on the topological classification of two dimensional ($d=2$)
systems and construct the relevant topological invariants which characterize
such systems.

The presence of translational invariance allows us to impose periodic boundary
conditions, which implies that all physical observables are uniquely defined
within the first BZ in the reciprocal space. The underlying parameter-manifold
is therefore compact, i.e., the manifold closes in on itself. For example, the
BZ in one dimension, $k\in[0,2\pi)$, has the geometry of a closed ring (known
as a $\mathcal{S}^{1}$). Similarly, in $d=2$, the BZ assumes the form of a
two-dimensional torus (denoted as a $T^{2}$) as both $k_{1},k_{2}\in[0,2\pi)$.

Let us first consider the Abelian case of $N=1$ where only one single-particle
eigenstate of the Hamiltonian $H(\vec{k})$ is occupied for all $\vec{k}\in
BZ$. Any simple closed loop $\Gamma$ on the surface of the two-dimensional BZ
torus can be contracted to a point on the surface itself. The holonomy of an
eigenstate $\ket{\psi}=\ket{n(k_{1},k_{2})}$, defined in Eq. (273), can be
evaluated as,

$\mathcal{C}(\Gamma)=\frac{1}{i}\ln\left(e^{i\oint_{\Gamma}A^{\mu}dk_{\mu}}\right),$
(282)

where we have omitted the path ordering operator $\mathcal{P}$ as the gauge
group is Abelian. The loop $\Gamma$ then divides the surface of the parameter-
space into two distinct sections. Although the area enclosed by the loop in
the two sections are different, the line integral in Eq. (282) is identical to
the surface integral (see Eq. (280)) on either of the sections. For our
purpose, we evaluate the surface integral over the surface $S$ which expands
in area as the loop $\Gamma$ shrinks in perimeter. We thus arrive at,

$\mathcal{C}(\Gamma)=\frac{1}{i}\ln\left(e^{i\oint_{\Gamma}A^{\mu}dk_{\mu}}\right)=\int_{S}\left(F_{\mu\nu}\right)d\sigma^{\mu\nu},$
(283)

However, note that as the loop $\Gamma$ shrinks to a point, the line integral
in Eq. (283) vanishes and the surface $S$ starts covering the complete BZ.
This apparently suggests that the net gauge flux through the total BZ must
always vanish. However, this only holds true for topologically trivial
systems. It is seen that there also exists topologically non-trivial systems,
for which it becomes impossible to uniquely define a consistent gauge for all
single-particle states residing on the complete BZ. For example, consider the
family of quantum systems in which the ground state in a given basis can be
represented by points on a Bloch sphere,

$\ket{\psi(\theta(\vec{k}),\phi(\vec{k}))}\equiv\begin{bmatrix}f(\theta(\vec{k}),\phi(\vec{k}))\\\
g(\theta(\vec{k}),\phi(\vec{k}))\\\
\end{bmatrix}=\begin{bmatrix}e^{i\phi(\vec{k})}\sin\frac{\theta(\vec{k})}{2}\\\
\cos\frac{\theta(\vec{k})}{2}\\\ \end{bmatrix},$ (284)

where $\theta(\vec{k})\in[0,\pi]$, $\phi(\vec{k})\in[0,2\pi]$ and we have
chosen the gauge in which the function $g$ satisfies,

${\rm Im}[g(\theta(\vec{k}),\phi(\vec{k}))]=0.$ (285)

The gauge-fixing rule in Eq. (285) is well behaved at all points except the
south pole $\theta=\pi$, where the Eq. (285) is ill-defined. On the other
hand, changing the gauge condition to ${\rm
Im}[f(\theta(\vec{k}),\phi(\vec{k}))]=0$ does not resolve the issue as it is
then ill-defined at $\theta=0$. If the ground state
$\ket{\psi(\theta(\vec{k}),\phi(\vec{k}))}$ is topologically non-trivial, it
is found to be mapped to both the poles of the Bloch sphere for two different
values of $\vec{k}$ within the BZ. Thus evaluating the integral in Eq. (283)
necessitates choosing at least two different gauges on disjoint sections,
$S_{1}$ and $S_{2}$, of the BZs such that the gauge-fixing condition is well-
defined in their respective sections. Thus, now the total gauge flux through
the compact BZ,

$\mathcal{C}=\oint_{BZ}F_{\mu\nu}d\sigma^{\mu\nu}=\left(\int_{S_{1}}d\sigma^{\mu\nu}_{1}-\int_{S_{2}}d\sigma^{\mu\nu}_{2}\right)\leavevmode\nobreak\
F_{\mu\nu}.$ (286)

Also, the connection under two different gauges in $S_{1}$ and $S_{2}$ must be
related by a simple ${\rm U(1)}$ gauge transformation,

$\displaystyle\ket{\psi}_{1}$ $\displaystyle=$ $\displaystyle
e^{-i\gamma}\ket{\psi}_{2},$ $\displaystyle A^{\mu}_{1}$ $\displaystyle=$
$\displaystyle A^{\mu}_{2}-\partial_{\mu}\gamma.$ (287)

Therefore, from the Stokes’ theorem separately applied on both the sections,
the total flux reduces to,

$\mathcal{C}=\oint_{\Gamma^{*}}\partial_{\mu}\gamma
dk^{\mu}=2\pi\nu\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{\rm where}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\nu\in\mathcal{Z},$ (288)

where $\Gamma^{*}$ is the boundary between the disjoint sections. The second
equality in the above equation follows from the requirement of single-
valuedness of the vectors $\ket{\psi(\vec{k})}$ on the boundary loop
$\Gamma^{*}$ and we thus obtain an integer quantized gauge-invariant
topological index. The gauge fixing is however not an issue while evaluating
the total flux as the gauge field $F_{\mu\nu}$ is in itself completely gauge
invariant. We have therefore the gauge invariant integral quantity, also
popularly known as the Chern number of a statetong16 ,

$\nu=\frac{1}{2\pi}\oint_{BZ}F_{\mu\nu}d\sigma^{\mu\nu}.$ (289)

From the discussion following Eq. (244) and that the Chern number can be
represented in terms of a holonomy on the manifold generated by the gauge
potential $A_{\mu}$ (see Eq. (280)), it follows that the Chern number must
remain invariant under adiabatic transformations on the state
$\ket{\psi(k_{1},k_{2})}$ over which it is being evaluated. Thus, in analogy
with the Berry phase, the Chern number induces a topological classification of
quantum states on a two dimensional parameter-manifold. For the generic case
when $N$ single-particle states are occupied, it follows from the discussion
in Sec. A.2 that the appropriate gauge-invariant quantity (see Eq. (272))
classifying the topological phases of the system is the ${\rm U(N)}$ Chern
number defined bytong16 ; hughes08 ,

$\tilde{\nu}=\frac{1}{2\pi}\oint_{BZ}{\rm
Tr}\left(F_{\mu\nu}\right)d\sigma^{\mu\nu}.$ (290)

For a finite-size system, the parameter space or the BZ is discrete. As
discussed in Sec. A.2, one can construct Wilson loops on the discrete $d=2$
lattice with the lattice parameter chosen as $a=2\pi/L\ll 1$, where $L$ is the
linear dimension of the system. To calculate the total flux through the
discrete parameter lattice, we simply sum over the discrete Wilson loops (see
Eq. (279)) over all the infinitesimal square plaquettes $dS$ of area $a^{2}$
making up the BZ,

$\nu_{L}=\frac{1}{2\pi}\sum\limits_{dS\in BZ}W_{L}(dS),$ (291)

where $W_{L}$ is given by Eq. (275). Focusing on the simple case of $N=1$, the
discrete Wilson loop is found to assume the form (see Eq. (279)),

$\displaystyle W_{L}(dS)$ $\displaystyle=$ $\displaystyle
F_{\mu\nu}a^{2}+2\pi\eta(dS),$ $\displaystyle F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\delta_{\mu}A_{\nu}-\delta_{\nu}A_{\mu},$ (292)

where $\delta_{\mu}$ are finite difference operators on the discrete lattice.
Thus, on summing over all such plaquettes on the compact BZ, one finds,

$\nu_{L}=\sum\limits_{dS\in BZ}\eta(dS).$ (293)

The lattice Chern number $\nu_{L}$ is thus also integer quantized despite the
parameter space being discrete in a finite-size system. Note that the lattice
Chern number is physically meaningful only if it approaches the Chern number
defined in the thermodynamic limit as the system size is increased. However,
since $\nu_{L}$ is always an integer, it cannot approach $\nu$ continuously as
$L\rightarrow\infty$ and hence, there must be a critical size $L_{c}$ for
which the lattice chern number exactly reaches the continuum Chern number. To
this end, we note that the lattice Chern number can only change if the
condition in Eq. (278) is violated for at least one plaquette within the BZ.
In other words, $\nu_{L}$ can only change when

$\left|F_{\mu\nu}a^{2}\right|=\left|\frac{4\pi^{2}F_{\mu\nu}}{L^{2}}\right|\sim\pi.$
(294)

In this case, one needs to choose an altogether different integer field
$\eta(dS)$ so as to restore the condition in Eq. (278). We can now perform an
order of magnitude analysis of the dependence of the lattice Chern number on
the spatial dimension $L$. From Eq. (291) one might estimate,

$F_{\mu\nu}\sim\frac{2\pi\nu_{L}}{4\pi^{2}},$ (295)

where $4\pi^{2}$ is the total area of the BZ. Substituting in Eq.(294), we
obtain an order of magnitude dependence of the lattice Chern number with the
lattice dimension,

$L\sim\sqrt{2|\nu_{L}|}.$ (296)

This shows that as the system size $L$ increases and the lattice becomes
finer, the lattice Chern number monotonically transits between different
quantized values until at $L=L_{c}$, it reaches the exact Chern number defined
in the continuum limit. Any subsequent increase in $L$ does not alter the
lattice Chern number and hencefukui05 ,

$\nu_{L}=\nu\leavevmode\nobreak\ \leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ \leavevmode\nobreak\ L\gtrsim
L_{c}=\sqrt{2|\nu|}.$ (297)

Since $L_{C}$ is a finite length, this method of finding the Chern number for
the continuum theory is far more advantageous numerically as one does not have
to go to the thermodynamic limit of system size as required in Eq. (289).
Here, it is also worth noting that the discrete lattice arises in finite-size
systems which remain gapped even at the QCP, which exists in the continuum
limit of the system and is necessarily gapless. Finally, we recall that the
topological phases of a thermodynamically large system are separated by
gapless QCPs, at which the adiabatic conditions break down. However, given
that no truly gapless point exists in finite-size systems, it seems possible
to connect phases characterized by distinct values of the lattice Chern number
without any breakdown of adiabaticity. This is particularly remarkable as the
lattice Chern number mimics the continuum Chern number above the critical size
$L_{c}$.

####  A.3.2 Bott invariant

So far we have focused on topological invariants which classify
translationally invariant systems. However, it turns out that it is possible
to define a topological invariant which does not depend on the translational
invariance of the system, yet which remarkably reduces to the conventional
Chern number with periodic boundary conditions. Consider a two-dimensional
insulating lattice system with the lattice coordinates being
$\\{x_{i},y_{i}\\}$ and described by the Hamiltonian $H$. It is then
straightforward to see that the Bott index defined in Eq. (133) of Sec. V.1
(also see Ref. rigol15, ),

$\nu_{B}=\frac{1}{2\pi}{\rm Im}\left[{\rm
Tr}\ln\left(\tilde{T}_{X}\tilde{T}_{Y}\tilde{T}_{X}^{\dagger}\tilde{T}_{Y}^{\dagger}\right)\right]$
(298)

is a real number which is independent of the boundary conditions imposed on
the lattice. This quantity is known as the Bott index. Using $C^{*}$ algebras,
it can be rigorously shown that the quantity $\nu_{B}$ is indeed an integer
quantized topological invariant which can also topologically classify systems
with broken translational symmetry as well as disordered quantum systems.
Nevertheless, a simpler understanding of the Bott index can be made by
inspecting it with periodic boundary conditions, which simplifies the
analysis.

As in preceding discussions, we describe the lattice in the momentum space
$\\{k_{x},k_{y}\\}$ such that in one BZ, $k_{x},k_{y}\in[0,2\pi)$ with
periodic boundary conditions. In the momentum space, the action of the
operator $T_{X(Y)}$ is simply to generate translations in momentum direction
$k_{x(y)}$ by an amount $\Delta k_{x(y)}=2\pi/L$. To see this, we note that
the position operators $X$ and $Y$ have the representations
$i\partial_{k_{x}}$ and $i\partial_{k_{y}}$, respectively, in reciprocal
space. Consequently, the action of the operators $T_{X},T_{Y}$ when expanded
in a Taylor series gives,

$\displaystyle T_{X(Y)}\ket{\psi_{n}(k)}$ $\displaystyle=$
$\displaystyle\ket{\psi(k)}-\partial_{k_{x(y)}}\ket{\psi_{n}(k)}\Delta
k_{x(y)}$ (299)
$\displaystyle+\frac{1}{2}\partial_{k_{x(y)}}^{2}\ket{\psi_{n}(k)}\Delta
k_{x(y)}^{2}-\dots$ $\displaystyle=$ $\displaystyle\ket{\psi_{n}(k-\Delta
k_{x(y)})},$

Hence, it follows that

$\displaystyle\braket{\psi_{m}({k})}{T_{X(Y)}}{\psi_{n}({k}^{\prime})}$
$\displaystyle=$
$\displaystyle\braket{\psi_{m}({k})}{\psi_{n}({k}^{\prime})}\delta_{{k^{\prime}},k+\Delta
k_{x(y)}}$ (300) $\displaystyle=$
$\displaystyle\delta_{m,n}\delta_{{k^{\prime}},k+\Delta k_{x(y)}}.$

From Eq.(298), it is then easy to see that the Bott index with periodic
boundary conditions reduces to,

$\nu_{B}^{n}=\\\ \frac{1}{2\pi}\sum_{k\in BZ}{\rm
Im}\left[\ln\Big{(}\left(T_{X}\Big{)}^{nn}\Big{(}{T}_{Y}\Big{)}^{nn}\left({T}_{X}^{\dagger}\right)^{nn}\left({T}_{Y}^{\dagger}\right)^{nn}\right)\right].$
(301)

It is thus clear that for each $k$, the argument of the logarithms in the
above equations traces out a Wilson loop (see Eq. (275)) around a plaquette
with $k$ residing at one corner of the plaquette. Thus, the summation is thus
carried out over all such plaquettes covering the complete BZ. We therefore
conclude that the Bott index is indeed a gauge invariant and quantized
topological invariant which is equivalent to the lattice Chern number
$\nu_{L}$ with periodic boundary conditions. We can then expect that, similar
to the lattice Chern number, one does not need to cross a gapless QCP in the
thermodynamic limit to connect two regions separated by inequivalent Bott
index or lattice Chern numbers. The is significant in the context of the
dynamical preparation of topologically non-trivial quantum states in real
physical systems as discussed in Sec. V.1.

##  Appendix B A perturbative expansion of the steady state Majorana
correlations

In Sec. IV.3.2, we discussed the dynamical defects generated in the two-point
Majorana correlations of one-dimensional Kitaev-like systems when driven
adiabatically in the presence of a Markovian bath. While the lowering of
defects with increasing ramp duration $\tau$ (in the small $\tau$ limit) is a
result of approach towards the adiabatic limit of ramping protocol, the linear
rise following the optimal $\tau_{0}$ needs further scrutiny.

To this end, we perturbatively expand (see Ref. souvik20, ) the solution of
the dynamical equations of motion for the two-point Majorana correlations
where we make use of the condition that $\tau\ll\tau_{B}$ or $\kappa\tau\ll 1$
($\kappa\sim 1/\tau_{B}$) to identify $\kappa\tau$ as the small parameter in
the perturbation.

The dynamical equations of motion for the two-point Majorana correlations can
be expressed in terms of a $2L\times 2L$ dimensional covariance matrix $C(t)$
defined as $C_{i,j}(t)={\rm Tr}(a_{i}a_{j}\rho(t))-\delta_{i,j}$, which
satisfies,

$\dot{C}(t)=-X(t)C(t)-C(t)X^{T}(t)+iY(t),$ (302)

where $X(t)=4(i\mathcal{H}(t)+\mathrm{Re}[M(t)])$,
$Y(t)=4(\mathrm{Im}[M(t)]-\mathrm{Im}[M^{T}(t)])$ and
$M(t)=\sum_{i}l_{i}\otimes l_{i}^{*}$. The matrix $\mathcal{H}(t)$ corresponds
to single particle Hamiltonian in Majorana basis while $M(t)$ encodes all the
bath information and is therefore time independent in our case. Substituting
the ansatz $C(t)=Q(t)C(0)Q^{T}(t)-iP(t)Q^{T}$ in the above equation, where
$P(T)$ and $Q(t)$ are two real matrices, results in two simpler equations
prosen11

$\displaystyle\dot{Q}(t)=-X(t)Q(t),$ (303a)
$\displaystyle\dot{P}(t)=-X(t)P(t)-Y(t)Q^{-T}(t),$ (303b)

with $Q(0)=\mathcal{I}$ and $P(0)=\mathbb{O}$.

The uniform and time independent coupling of the bath with the system in our
case ($\kappa_{i}=\kappa$) allows us to rewrite $M(t)=M=\kappa\widetilde{M}$
and $Y(t)=Y=\kappa\widetilde{Y}$ where all elements of matrix $\widetilde{M}$
and $\widetilde{Y}$, are dimensionless. Assuming natural units, we substitute
$t=\tilde{t}\tau$, $\widetilde{\mathcal{H}}=\mathcal{H}\tau$, and
$\tilde{\kappa}=\kappa\tau$ to arrive at the non-dimensionalized version of
the above equations,

$\displaystyle\dot{Q}(\tilde{t})=-4\left(i\widetilde{\mathcal{H}}(\tilde{t})+\widetilde{\kappa}Re[\widetilde{M}]\right)Q(\tilde{t}),$
(304a)
$\displaystyle\dot{P}(\tilde{t})=-4\left(i\widetilde{\mathcal{H}}+\widetilde{\kappa}Re[\widetilde{M}]\right)P(\tilde{t})-\widetilde{\kappa}\widetilde{Y}Q^{-T}(\tilde{t}).$
(304b)

Solving the above pair of matrix equations perturbatively to first order in
$\widetilde{\kappa}$, we obtain

$\displaystyle
C(\tilde{t})=V(\tilde{t})C(0)V^{T}(\tilde{t})-\widetilde{\kappa}V(\tilde{t})\big{[}C(0)\Lambda^{T}(\tilde{t})+\Lambda(\tilde{t})C(0)-\Gamma(\tilde{t})\big{]}V^{T}(\tilde{t})+\mathcal{O}(\widetilde{\kappa}^{2}),$
(305)

where

$\displaystyle
V(\tilde{t})=\mathcal{T}e^{-4i\int_{0}^{\tilde{t}}\widetilde{\mathcal{H}}(\tilde{t}^{\prime})d\tilde{t}^{\prime}},$
(306a)
$\displaystyle\Lambda(\tilde{t})=4\int_{0}^{\tilde{t}}V^{T}(\tilde{t}^{\prime})Re[\widetilde{M}]V(\tilde{t}^{\prime})d\tilde{t}^{\prime},$
(306b)
$\displaystyle\Gamma(\tilde{t})=i\int_{0}^{\tilde{t}}V^{T}(\tilde{t}^{\prime})\widetilde{Y}V(\tilde{t}^{\prime})d\tilde{t}^{\prime}.$
(306c)

The end of the ramp protocol corresponds to $\tilde{t}=1$ in the rescaled
units; the covariance matrix at the end of the ramp is therefore obtained as

$\displaystyle
C(1)=V(1)C(0)V^{T}(1)-\widetilde{\kappa}K+\mathcal{O}(\widetilde{\kappa}^{2}),$
(307)

where
$K=V(1)\big{[}C(0)\Lambda^{T}(1)+\Lambda(1)C(0)-\Gamma(1)\big{]}V^{T}(1)$ is a
constant matrix with dimensionless elements. Next, we note that the first term
in the above equation $V(1)C(0)V^{T}(1)$ correspond to an unitary time-
evolution of the covariance matrix generated by the Hamiltonian
$\widetilde{\mathcal{H}}$. Further, this term captures the full evolution of
the covariance matrix in the absence of bath ($\widetilde{\kappa}=0$). We,
therefore identify this term as the time-evolved covariance matrix $C^{U}(1)$
in the absence of any dissipative channels. Returning to the original unscaled
units, we finally obtain,

$\displaystyle C(\tau)=C^{U}(\tau)-\kappa\tau
K+\mathcal{O}(\kappa^{2}\tau^{2}).$ (308)

Let us now consider the Majorana edge correlation
$\mathrm{Tr}(\rho(\tau)\theta)=iC_{1,2L}(\tau)$ at the end of the ramp. As we
are interested in the adiabatic limit $L\ll\tau$, we assume that the unitary
ramp (in absence of dissipation) induces negligible excitations on its own,
i.e., $iC^{U}_{1,2L}\simeq\bra{\psi_{f}^{0}}\theta\ket{\psi_{f}^{0}}$. Using
Eq. (308), the defect $\chi(\tau)$ is obtained as,

$\displaystyle\chi(\tau)=\kappa\tau K_{1,2L}+\mathcal{O}(\kappa^{2}\tau^{2}).$
(309)

Hence, we conclude that in the limit $L\ll\tau\ll\tau_{B}$, the defect at the
end of the ramp scales as $\chi(\tau)\sim\kappa\tau$ in the leading order of
perturbation. Further, we note that the above scaling holds true for all
pairwise Majorana correlations or elements of the covariance matrix.

Finally, we note that the perturbation theory, in essence, extracts the effect
of a weak environmental effect (perturbation). The defect generated because of
non-adiabatic effects arising from a unitary finite ramp duration can not be
captured within this framework.

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