# Impurity flat band states in the diamond chain

A. M. Marques<EMAIL_ADDRESS>Department of physics $\&$ i3N,
University of Aveiro, 3810-193 Aveiro, Portugal  D. Viedma  Departament de
Física, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain  V.
Ahufinger  Departament de Física, Universitat Autònoma de Barcelona, E-08193
Bellaterra, Spain  R. G. Dias  Department of physics $\&$ i3N, University of
Aveiro, 3810-193 Aveiro, Portugal

###### Abstract

Flat band (FB) systems, i.e., models hosting dispersionless bands in their
energy spectrum, have attracted great interest recently. The eigenstates of
these FBs can be written as compact localized states (CLSs), meaning states
having finite support only over a small, non-extensive portion of the lattice.
When adjacent CLSs of a FB have spatial overlap, the subspace they span is, in
general, non-orthogonal. Here, we consider precisely this scenario, in the
form of a diamond chain with a finite magnetic flux per plaquette, which has a
gapped midspectrum FB. After developing the general framework for projecting
arbitrary operators onto a non-orthogonal subspace, we address the case of an
open chain with small local impurities in the midchain plaquette. The
framework is then applied to derive analytical expressions for the states of
the FB affected by the impurities. The results are illustrated for three
representative cases: two equal impurities, two symmetric impurities, and a
single impurity. We also address the effects of diagonal disorder on the
impurity states. Focusing on the equal impurities case, we show how these
states effectively feel an averaged disorder determined by their spatial
extension, which can be controlled by the flux. Due to this averaging effect,
these states exhibit enhanced robustness to disorder. For the single-impurity
case, we further reveal its exotic non-trivial topological phase, without
precedent in the literature on Hermitian models, characterized by a half-
integer winding number. By mapping the system onto an equivalent two-
dimensional (2D) tight-binding lattice decorated with a central impurity line,
the half-integer bulk invariants are shown to have a correspondence with the
appearance of a single in-gap edge state when open boundary conditions (OBC)
are imposed. Numerical simulations are performed to benchmark all analytical
predictions, showing a near-perfect agreement for flux values away from the
gap closing point. Our results lay the foundation for further inquiries on the
physics of non-orthogonal bases, and are expected to be of particular
relevance to the study of FB related phenomena.

##  I Introduction

About a decade ago, a renewed interest in FB systems [1] gave way to a new
cycle of studies devoted to the topic (a more detailed chronology can be found
in [2]), and it has no signs of slowing down. In this new wave of studies, the
focus has gradually shifted towards the CLSs that span every FB. For one-
dimensional (1D) models, it has been shown that these CLSs form an orthogonal
set when the FB projector is strictly local, and a non-orthogonal one
otherwise [3, 4]. Based on the form of the CLSs, FB generators were
constructed for 1D [5, 6] and higher-dimensional [7, 8, 9, 10, 11] models.

An interesting class of FB systems is the (self-explanatory) class of all-
bands-flat (ABF) models [12, 13], where transport is completely suppressed,
since any local excitation in an ABF model can be written as a linear
combination of CLSs belonging to the different FBs, leading to the phenomenon
of particle confinement known as Aharonov-Bohm (AB) caging [14]. The diamond
chain with a $\pi$-flux per plaquette [15, 16, 17, 18, 19, 20], along with its
$2^{n}$-root versions [21, 22], is a prime example of an ABF model exhibiting
AB caging that has been thoroughly investigated. In this and other related ABF
models, the effects of weak perturbations have been studied by projecting the
perturbation operator onto one of the FBs. For example, this method was
employed to study weak disorder in the ABF Creutz ladder [23, 24], and weak
interactions in both this model [25] and the diamond chain [26], shown to be
able to induce dynamical effects [27] in two-body states where it would
otherwise be absent, due to the infinite effective mass of the states of a FB.
Conversely, strong interactions can drive the appearance of effective ABF
many-body subspaces, in models with a dispersive single-particle energy
spectrum [28].

Here, we address the case of a diamond chain with a finite flux per plaquette,
and introduce a perturbation in the form of weak local potentials in the
midchain plaquette, as schematically illustrated in Fig. 1(a). By projecting
this impurity perturbation onto the gapped zero-energy FB, only two partially
overlapping CLSs have finite weight at the impurity sites. However, due to the
non-orthogonal nature of adjacent CLSs, we show that the analytical treatment
of the system requires one to reexpress the problem in terms of its dual basis
states. This greatly simplifies the problem, by reducing it to the solutions
of a decoupled dimer in a non-orthogonal dual subspace. We then derive the
analytical solutions for the impurity states that are lifted from the FB as a
function of the magnetic flux, focusing on the cases of two equal impurities,
two symmetric impurities, and a single-impurity, where near-perfect agreement
with numerical results is observed. Finally, for the single-impurity case, we
further demonstrate how it corresponds to a non-trivial topological phase
characterized by half-integer topological invariants. Upon mapping this system
onto an equivalent 2D lattice decorated with a central line of defects, we
provide indirect evidence of a refined bulk-boundary correspondence by
relating the half-integer bulk invariants with the presence of a single in-gap
edge state in the spectrum of the mapped lattice.

The rest of the paper is organized as follows. In Sec. II, we provide a brief
overview on the theory of non-orthogonal bases, with an emphasis on how to
project an operator onto such bases, and how to orthonormalize them by means
of the Löwdin transformation [29]. In Sec. III, we introduce the model
considered in this study, namely the diamond chain threaded by a finite flux
per plaquette, in its clean version, deriving its solutions and the form of
the overlap matrix for the CLSs spanning the zero-energy FB. In Sec. IV, we
perturb the model by including weak local impurities in the top and bottom
sites of the midchain plaquette, deriving the general analytical expression
for the impurity states that come off the FB, using the techniques described
in Sec. II. After that, we analyze in greater detail the particular cases of
equal impurities, symmetric impurities, and single-impurity. For the latter,
we also study its peculiar topological features, whose physical manifestation
requires the introduction of an equivalent 2D lattice for its observation.
Lastly, we present our conclusions in Sec. V.

##  II Basics of non-orthogonal bases

In this section, we will review some basic results of non-orthogonal bases,
particularly in what concerns the definition of projection operators and the
procedure for diagonalizing a Hamiltonian written in such a basis. We will
heavily rely in the theory outlined in [30], as well in the summarized results
provided in the appendix of [31]. Let us start by considering a non-orthogonal
basis $\\{\ket{i}\\}$, not necessarily normalized. The elements of the overlap
matrix $\mathbf{S}$ of this basis are written as $S_{ij}=\innerproduct{i}{j}$,
with $\mathbf{S}$ further assumed to be a real symmetric matrix
$S_{ij}=S_{ji}$. Evidently, $\mathbf{S}$ reduces to the identity $\mathbf{I}$
for an orthonormal basis. One can define the dual basis $\\{\ket{i^{*}}\\}$,
whose states are defined by the property
$\innerproduct{i}{j^{*}}=\delta_{ij}$, with $\delta_{ij}$ the Kronecker delta.
The transformation of states between the direct and dual bases is given by

$\ket{i}=\sum\limits_{j}S_{ij}\ket{j^{*}},\ \ \ \
\ket{i^{*}}=\sum\limits_{j}S_{ij}^{-1}\ket{j},$ (1)

where the usual slight abuse of language of writing $(\mathbf{S}^{-1})_{ij}$
as $S_{ij}^{-1}$ is used here and throughout the text for simplicity. The
identity operator can therefore be expressed in several different forms, as

$\hat{I}=\sum\limits_{i}\ket{i}\bra{i^{*}}=\sum\limits_{ij}\ket{i}S_{ij}^{-1}\bra{j},$
(2)

which is widely used, e.g., in systems of coupled coherent states [32, 33]
that form an overcomplete non-orthogonal basis, or alternatively as

$\hat{I}=\sum\limits_{i}\ket{i^{*}}\bra{i}=\sum\limits_{ij}\ket{i^{*}}S_{ij}\bra{j^{*}}.$
(3)

From $\hat{I}^{2}=\hat{I}$, one can derive the following useful identity,

$\sum\limits_{k}S_{ik}^{-1}S_{kj}=\delta_{ij}.$ (4)

###  II.1 Projection of operators onto a non-orthogonal subspace

It is convenient, for later use, to consider the case of a basis of size $N+M$
composed of two orthogonal subspaces,
$\\{\ket{i}\\}=\\{\ket{\text{Loc},n}\\}\cup\\{\ket{\text{Ext},m}\\}$, with
$n=1,2,\dots,N$ and $m=1,2,\dots,M$, and where, anticipating the notation that
will be used later on, $\\{\ket{\text{Loc},n}\\}$ stands for the non-
orthogonal subspace of localized states and $\\{\ket{\text{Ext},m}\\}$ for the
orthonormal subspace of extended states. The total overlap matrix reads as
$\mathbf{S}_{\text{tot}}=\mathbf{S}\oplus\mathbf{I}_{M}$, where
$\mathbf{I}_{M}$ is the identity matrix of size $M$. The projection operator
onto the non-orthogonal subspace is written as

$\displaystyle\hat{P}_{\text{Loc}}$ $\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{N}\ket{\text{Loc},j}\bra{\text{Loc},j^{*}}$
(5) $\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{N}\ket{\text{Loc},j}\sum\limits_{i=1}^{N+M}S_{ij}^{-1}\bra{i}$
$\displaystyle=$
$\displaystyle\sum\limits_{i,j=1}^{N}\ket{\text{Loc},j}S_{ij}^{-1}\bra{\text{Loc},i},$

where $\ket{\text{Loc},j^{*}}$ belongs to the dual localized subspace and the
orthogonality between subspaces was used in the last line. Correspondingly,

$\displaystyle\hat{P}_{\text{Loc}}^{\dagger}$ $\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{N}\ket{\text{Loc},j^{*}}\bra{\text{Loc},j}$
(6) $\displaystyle=$
$\displaystyle\sum\limits_{i,j=1}^{N}\ket{\text{Loc},i}S_{ij}^{-1}\bra{\text{Loc},j}=\hat{P}_{\text{Loc}}.$

Given an arbitrary operator $\hat{A}$, its projection onto the non-orthogonal
subspace can be developed as

$\displaystyle\hat{A}_{\text{Loc}}$ $\displaystyle=$
$\displaystyle\hat{P}_{\text{Loc}}^{\dagger}\hat{A}\hat{P}_{\text{Loc}}$ (7)
$\displaystyle=$
$\displaystyle\sum\limits_{i,j=1}^{N}\ket{\text{Loc},i^{*}}A_{ij}\bra{\text{Loc},j^{*}},$

where $A_{ij}=\bra{\text{Loc},i}\hat{A}\ket{\text{Loc},j}$.

###  II.2 Symmetric orthonormalization

Given a non-orthogonal basis, one can always find an orthonormalized basis
through the Gram-Schmidt method [34], which relies on a recursive state-by-
state orthogonalization procedure. However, this method can be very
computationally demanding and prone to numerical instabilities. A more
efficient method consists of using the Löwdin transformation [29, 35] to
perform a symmetrical orthogonalization of a non-orthogonal basis. With this
transformation, and from a non-orthogonal basis spanning $\\{\ket{i}\\}$, with
$i=1,2,\dots,N$, one can find the orthogonal basis states with the help of the
overlap matrix as

$\ket{\tilde{i}}=\sum\limits_{j}S_{ij}^{-\frac{1}{2}}\ket{j},$ (8)

where, again, we use the simplified notation
$S_{ij}^{-\frac{1}{2}}\equiv(\mathbf{S}^{-\frac{1}{2}})_{ij}$. The secular
equation, in this basis, is computed through the effective Hamiltonian
$\tilde{\mathbf{H}}$,

$\displaystyle\tilde{\mathbf{H}}\tilde{\pmb{\psi}_{j}}$ $\displaystyle=$
$\displaystyle E_{j}\tilde{\pmb{\psi}_{j}},\ \ j=1,2,\dots,N,$ (9)
$\displaystyle\tilde{\mathbf{H}}$ $\displaystyle=$
$\displaystyle\mathbf{S}^{-\frac{1}{2}}\mathbf{H}\mathbf{S}^{-\frac{1}{2}},$
(10)

where $\mathbf{H}$ is the Hamiltonian matrix written in the starting non-
orthogonal $\\{\ket{i}\\}$ basis, also called the core Hamiltonian [36], and
$\tilde{\pmb{\psi}_{j}}$ is the $j^{\text{th}}$ eigenvector with components
spanning the orthogonal $\\{\ket{\tilde{i}}\\}$ basis.

##  III Diamond chain with flux

The specific system we will consider from now on will be that of Fig. 1(a),
consisting of a diamond chain with a $\phi=2\pi\frac{\Phi}{\Phi_{0}}$ reduced
magnetic flux per plaquette, with $\Phi$ the magnetic flux and $\Phi_{0}$ the
magnetic flux quantum. Setting the lattice constant to unit, the bulk
Hamiltonian of the clean system is given, in the
$\\{\ket{A(k)},\ket{B(k)},\ket{C(k)}\\}$ basis, with $k$ the quasimomentum, by

$\displaystyle H(k)$ $\displaystyle=$ $\displaystyle-t\begin{pmatrix}0&h\\\
h^{\dagger}&0\end{pmatrix},$ (11) $\displaystyle h^{\dagger}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}e^{i\frac{\phi}{4}}+e^{i(k-\frac{\phi}{4})}\\\
e^{-i\frac{\phi}{4}}+e^{i(k+\frac{\phi}{4})}\end{pmatrix}$ (12)

where $t$ is the magnitude of the hopping term. Diagonalization of this
Hamiltonian yields three energy bands,

$\displaystyle E_{0}$ $\displaystyle=$ $\displaystyle 0,$ (13) $\displaystyle
E_{\pm}(k)$ $\displaystyle=$ $\displaystyle\pm 2t\sqrt{1+\cos
k\cos\frac{\phi}{2}},$ (14)

where $E_{\pm}(k)$ are dispersive for all $\phi$ except for $\phi=\pi$, where
they become flat [see Fig. 1(b)]. An energy gap between the zero-energy FB
$E_{0}$ and $E_{\pm}(k)$ opens at $k=\pi$ for $\phi\neq 0$, with a value of
$E_{\text{gap}}=2t\sqrt{1+\cos\frac{\phi}{2}}$.

The eigenstates corresponding to $E_{0}$ can be written as CLSs, which for
finite flux span over two plaquettes [18], as shown in Fig. 1(a). Consecutive
normalized CLSs of the zero-energy FB are written, in the
$\\{\ket{B_{j}},\ket{C_{j}},\ket{B_{j+1}},\ket{C_{j+1}},\ket{B_{j+2}},\ket{C_{j+2}}\\}$
basis, where $\ket{\zeta_{l}}$ stands for the state at site $\zeta=B,C$ of
unit cell $l$ (note that $A$ sites are absent from this basis, as the CLSs
have no weight on them), as

$\displaystyle\ket{\text{Loc},j}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}1,-e^{-i\frac{\phi}{2}},e^{-i\frac{\phi}{2}},-1,0,0\big{)}^{T},$
(15) $\displaystyle\ket{\text{Loc},j+1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}0,0,1,-e^{-i\frac{\phi}{2}},e^{-i\frac{\phi}{2}},-1\big{)}^{T},$
(16)

which overlap in the two middle sites [see Fig. 1(a)]. By taking advantage of
this overlap, it was already shown that projecting spatially modulated onsite
interactions on the CLSs can induce topological behavior in a two-body system
[26]. Here, we will focus on the single-particle case.

Figure 1: (a) Diamond chain with $\phi$ flux per plaquette, distributed
uniformly by the four hopping terms of each plaquette, with positive phase
accumulated in the clockwise direction. Green shaded region represents the
$q$th unit cell with sites $A_{q}$, $B_{q}$ and $C_{q}$, while blue and orange
shaded regions represent the spatial width of adjacent CLSs
$\ket{\text{Loc},q+3}$ and $\ket{\text{Loc},q+4}$, respectively, which overlap
in the middle $B$ and $C$ sites. Onsite impurity potentials
$\epsilon_{{}_{B}}$ and $\epsilon_{{}_{C}}$ are placed at the orange $B_{q+1}$
and light blue $C_{q+1}$ sites, respectively. (b) Bulk energy spectrum, in
units of $t$, of the diamond chain for three representative values of the
$\phi$ flux per plaquette. The zero-energy flat band is common to all $\phi$
values.

The elements of the overlap matrix within the non-orthogonal subspace of CLSs
of the zero-energy FB, $S_{ij}=\innerproduct{\text{Loc},i}{\text{Loc},j}$, are
given, using (15)-(16), by

$\begin{cases}S_{jj}=1,\\\ S_{jj+1}=S_{j+1j}=\frac{1}{2}\cos\frac{\phi}{2},\\\
S_{ij}=0,\absolutevalue{i-j}>1,\end{cases}$ (17)

yielding a tridiagonal real and symmetric matrix,

$\mathbf{S}=\begin{pmatrix}1&a&\\\ a&1&a\\\ &a&1&\ddots\\\ &&\ddots&\ddots\\\
&&&&1&a&\\\ &&&&a&1&a\\\ &&&&&a&1\end{pmatrix},$ (18)

where the entries not shown are zeros and $a:=\frac{1}{2}\cos\frac{\phi}{2}$.
Notice how $a=0$ for $\phi=\pi$, such that $\mathbf{S}$ reduces to
$\mathbf{I}$, that is, the CLSs form an orthogonal basis, even though adjacent
CLSs still have spatial overlap [26]. The formula for the elements of the
inverse of a tridiagonal matrix is known [37], and reduces, in our case of
interest (that of $\mathbf{S}^{-1}$), to

$S_{nm}^{-1}=\begin{cases}(-1)^{n+m}\frac{1}{a}\frac{U_{n-1}(\frac{1}{2a})U_{N-m}(\frac{1}{2a})}{U_{N}(\frac{1}{2a})},\
\ n\leq m,\\\
(-1)^{n+m}\frac{1}{a}\frac{U_{m-1}(\frac{1}{2a})U_{N-n}(\frac{1}{2a})}{U_{N}(\frac{1}{2a})},\
\ n>m,\end{cases}$ (19)

where $U_{n}(x)$ is the Chebyshev polynomial of the second kind, defined as

$U_{n}(x)=\frac{\sinh\big{[}(n+1)\theta\big{]}}{\sinh\theta},\ \
\absolutevalue{x=\cosh\theta}>1,$ (20)

whose argument, for the case of (19), reads as

$x=\cosh\theta=\frac{1}{2a}=\sec\frac{\phi}{2},$ (21)

Substituting (20) and (21) in (19) leads to

$S_{nm}^{-1}=(-1)^{n+m}2\coth\theta\frac{\sinh\big{[}n\theta\big{]}\sinh\big{[}(N-m+1)\theta\big{]}}{\sinh\big{[}(N+1)\theta\big{]}},$
(22)

for $n\leq m$, and

$S_{nm}^{-1}=(-1)^{n+m}2\coth\theta\frac{\sinh\big{[}m\theta\big{]}\sinh\big{[}(N-n+1)\theta\big{]}}{\sinh\big{[}(N+1)\theta\big{]}},$
(23)

for $n>m$, such that $S_{ij}^{-1}=S_{ji}^{-1}$. Note that the tridiagonal and
repeating structure of $\mathbf{S}$ in (18) is formally equivalent to that of
an open linear chain with uniform couplings and a constant energy shift. The
analytical solution of these types of linear 1D models has been recently
derived [38] and, therefore, one could alternatively find $\mathbf{S}^{-1}$
through the known eigenvalues and eigenvectors of $\mathbf{S}$.

##  IV Impurity states

In this section, we will study the case, depicted in Fig. 1(a), where
impurities affect the local potentials at the top and bottom sites of a single
plaquette in the middle of an open diamond chain with $N+1$ unit cells (such
that one can construct $N$ zero-energy CLSs). The corresponding impurity
operator has the general form

$\hat{V}=\epsilon_{{}_{B}}\ket{B_{q+1}}\bra{B_{q+1}}+\epsilon_{{}_{C}}\ket{C_{q+1}}\bra{C_{q+1}},$
(24)

where $\epsilon_{{}_{B}}$ and $\epsilon_{{}_{C}}$ are local impurity
potentials, left as free parameters, and $q=\frac{N}{2}$, with $N$ assumed
even. We consider a finite flux per plaquette, $\phi\neq 0$, to have a finite
$E_{\text{gap}}$ between the flat and dispersive bands, and also that
$\epsilon_{{}_{B}},\epsilon_{{}_{C}}\ll E_{\text{gap}}$, such that higher-
order terms mediated by the dispersive bands can be neglected. Through (7),
the projected impurity operator onto the non-orthogonal FB subspace is written
as

$\displaystyle\hat{V}_{{}_{FB}}$ $\displaystyle=$
$\displaystyle\hat{P}^{\dagger}\hat{V}\hat{P}$ (25) $\displaystyle=$
$\displaystyle\sum\limits_{n,m=q}^{q+1}\ket{\text{Loc},n^{*}}V_{nm}\bra{\text{Loc},m^{*}},$

where $V_{nm}=\bra{\text{Loc},n}\hat{V}\ket{\text{Loc},m}$ and the sums run
only from $q$ to $q+1$ since, from the form of the CLSs in (15)-(16) and of
$\hat{V}$ in (24), only the CLSs with these indices are affected by the
impurity potentials. One can readily evaluate all the relevant $V_{nm}$
elements, which reduce to the $\\{V_{qq},V_{qq+1},V_{q+1q},V_{q+1q+1}\\}$ set,
to obtain

$\displaystyle\hat{V}_{{}_{FB}}$ $\displaystyle=$ $\displaystyle
v\sum\limits_{n=q}^{q+1}\ket{\text{Loc},n^{*}}\bra{\text{Loc},n^{*}}+$ (26)
$\displaystyle+$
$\displaystyle\Big{[}w\ket{\text{Loc},q+1^{*}}\bra{\text{Loc},q^{*}}+\text{H.c.}\Big{]},$

with $v=\frac{\epsilon_{{}_{B}}+\epsilon_{{}_{C}}}{4}$ acting as a local
potential on the dual CLSs at $q$ and $q+1$, and
$w=\big{(}\epsilon_{{}_{B}}e^{-i\frac{\phi}{2}}+\epsilon_{{}_{C}}e^{i\frac{\phi}{2}}\big{)}/4$
a complex hopping term between the states.

At this point the reader might suspect of a circular reasoning being employed
here. After all, the impurities placed at the midchain plaquette in (24) only
affect two adjacent CLSs, with indices $q$ and $q+1$, which are part of the
non-orthogonal set of CLSs. When we translate the system into the dual basis,
we arrive at the effective dimer model of (26) which, again, only involves the
two dual CLSs $q$ and $q+1$ that, in turn, are themselves also a part of the
non-orthogonal set of dual CLSs. It appears that one can go back and forth
between the original and dual basis indefinitely, without moving the problem
forward.

What, then, is to be gained by this procedure? We can always restrict
ourselves, in either basis, to the two states set
$\\{\ket{\text{Loc},q^{(*)}},\ket{\text{Loc},q+1^{(*)}}\\}$, and assume that
the rest of the localized states $\\{\ket{\text{Loc},l^{(*)}}\\}$, where
$l\neq q,q+1$, are orthogonalized first with respect to the former two, and
then between themselves. However, and crucially, orthogonalization of this
residual set, in the original basis, produces states of varying spatial
extension (without losing their localized nature [39]), i.e., that spread to
more than two consecutive plaquettes. Due to this, several states in the
orthogonalized set $\\{\ket{\text{Loc},l}\\}$ will have a finite weight at the
$q+1$ plaquette where the impurities are located, such that, from (24), we
will have $V_{nm}\neq 0$ at least for some $n,m\neq q,q+1$, and the effective
model is not restricted to two states in the original basis. On the other
hand, the impurity operator of the dual basis, given in (26), is not spatially
defined, but instead directly couples the $q$ and $q+1$ dual CLSs. As such,
regardless of the particular spatial profile of the dual states in the
orthogonalized $\\{\ket{\text{Loc},l^{*}}\\}$ set, all effective impurity
elements involving these states, computed from (26), will vanish, given that
they are assumed orthogonal to the
$\\{\ket{\text{Loc},q^{*}},\ket{\text{Loc},q+1^{*}}\\}$ states appearing in
$\hat{V}_{{}_{FB}}$. Therefore, working in the dual basis greatly simplifies
the problem, now reduced to the diagonalization of a dimer model defined in a
two-state dual basis.

Figure 2: Analytical energy curves (light blue and yellow curves) obtained
from diagonalization of the effective two-state system in (35), as a function
of the magnetic flux $\phi$ for (a) symmetric impurities
$\epsilon_{{}_{B}}=-\epsilon_{{}_{C}}=\epsilon$, (b) equal impurities
$\epsilon_{{}_{B}}=\epsilon_{{}_{C}}=\epsilon$, and (c) a single impurity,
$\epsilon_{{}_{B}}=\epsilon$ and $\epsilon_{{}_{C}}=0$. Blue and red dots
correspond to the numerically obtained energies of the two states of the FB
that are affected by the impurity, for an open diamond chain with 53
plaquettes pierced by a $\phi$ flux, with the impurities placed at the $B$ and
$C$ sites of the midchain plaquette in (a) and (b), and at its $B$ site in
(c). Note that the overlap between analytical curves and numerical dots
renders the former invisible for most of the spectrum in all three cases.
(d)-(f) Probability amplitudes of the eigenstate marked by a blue circle in
the corresponding cases above. Only the central part of the chain is shown.
Golden bars (red dots) correspond to numerical (analytical) results. The left
insets plot the same results in a semi-logarithmic scale and for a smaller
portion of the chain. Sites without red dots are $A$ sites where the CLSs
forming the analytical subspace have zero weight. The right insets depict the
impurities at the midchain plaquette, with green (purple) color indicating an
$\epsilon$ ($-\epsilon$) value for the impurity at that site. The blue circle
(red square) in (c) also corresponds to the state shown on the top (bottom) of
Fig. 5.

Since the two-state dual basis
$\\{\ket{\text{Loc},q^{*}},\ket{\text{Loc},q+1^{*}}\\}$ is itself non-
orthogonal, we first need to orhogonalize it through the Löwdin transformation
[29, 35], as outlined in Sec. II.2. The first step requires us to find the
form of $\mathbf{S}_{*}^{-\frac{1}{2}}$, where $\mathbf{S}_{*}$ is the overlap
matrix written in this basis. Using (1), it is straightforward to show that
the elements of $\mathbf{S}_{*}$ can be expressed as

$\displaystyle S_{*,nm}$ $\displaystyle=$
$\displaystyle\innerproduct{\text{Loc},n^{*}}{\text{Loc},m^{*}}$ (27)
$\displaystyle=$
$\displaystyle\sum\limits_{i,j}S_{ni}^{-1}\innerproduct{\text{Loc},i}{\text{Loc},j}S_{jm}^{-1}$
$\displaystyle=$
$\displaystyle\sum\limits_{j}\Big{(}\sum\limits_{i}S_{ni}^{-1}S_{ij}\Big{)}S_{jm}^{-1}$
$\displaystyle=$ $\displaystyle\sum\limits_{j}\delta_{nj}S_{jm}^{-1}$
$\displaystyle=$ $\displaystyle S_{nm}^{-1},$

where (4) was used in the fourth line. Using $q=\frac{N}{2}$ and assuming the
thermodynamic limit ($N\to\infty$), such that
$\sinh\big{[}x(N)\big{]}\to\frac{e^{x(N)}}{2}$ in (22)-(23), we obtain the
following expressions for some elements of the inverse overlap matrix,

$\begin{cases}S_{q+\Delta q+\Delta}^{-1}=\coth\theta\\\
S_{qq+\Delta}^{-1}=(-1)^{\Delta}\coth\theta e^{-\Delta\theta}\end{cases},\ \
\Delta=0,1,2,\dots,$ (28)

with $S_{qq+\Delta}^{-1}=S_{q+\Delta q}^{-1}$. With the formulas of (28), we
find the form of $\mathbf{S}_{*}^{-\frac{1}{2}}$,

$\displaystyle\mathbf{S}_{*}^{-\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}S_{qq}^{-1}&S_{qq+1}^{-1}\\\
S_{qq+1}^{-1}&S_{q+1q+1}^{-1}\end{pmatrix}^{-\frac{1}{2}}$ (29)
$\displaystyle=$ $\displaystyle\begin{pmatrix}\coth\theta&-\coth\theta
e^{-\theta}\\\ -\coth\theta
e^{-\theta}&\coth\theta\end{pmatrix}^{-\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\coth\theta}}\begin{pmatrix}\alpha_{-}+\alpha_{+}&\alpha_{-}-\alpha_{+}\\\
\alpha_{-}-\alpha_{+}&\alpha_{-}+\alpha_{+}\end{pmatrix},$

with $\alpha_{\pm}=\frac{1}{\sqrt{1\pm e^{-\theta}}}$. Combining (8) and (29),
we write the orthogonal two-state basis
$\\{\ket{\tilde{\varphi}_{q}},\ket{\tilde{\varphi}_{q+1}}\\}$ as

$\displaystyle\ket{\tilde{\varphi}_{q}}$ $\displaystyle=$
$\displaystyle\sum\limits_{j=q}^{q+1}S^{-\frac{1}{2}}_{*,qj}\ket{\text{Loc},j^{*}}$
(30) $\displaystyle=$
$\displaystyle\sum\limits_{j=q}^{q+1}\sum\limits_{l}S^{-\frac{1}{2}}_{*,qj}S^{-1}_{jl}\ket{\text{Loc},l},$
$\displaystyle\ket{\tilde{\varphi}_{q+1}}$ $\displaystyle=$
$\displaystyle\sum\limits_{j=q}^{q+1}S^{-\frac{1}{2}}_{*,q+1j}\ket{\text{Loc},j^{*}}$
(31) $\displaystyle=$
$\displaystyle\sum\limits_{j=q}^{q+1}\sum\limits_{l}S^{-\frac{1}{2}}_{*,q+1j}S^{-1}_{jl}\ket{\text{Loc},l}.$

In turn, the impurity matrix is constructed from (26) as

$\mathbf{V}_{{}_{FB}}=\begin{pmatrix}V_{*,qq}&V_{*,qq+1}\\\
V_{*,q+1q}&V_{*,q+1q+1}\end{pmatrix},$ (32)

with $V_{*,nm}=\bra{\text{Loc},n^{*}}\hat{V}_{{}_{FB}}\ket{\text{Loc},m^{*}}$
and

$\displaystyle V_{*,qq}$ $\displaystyle=$ $\displaystyle
V_{*,q+1q+1}=v\big{[}(S_{qq}^{-1})^{2}+(S_{qq+1}^{-1})^{2}\big{]}+(w+\bar{w})S_{qq}^{-1}S_{qq+1}^{-1},$
(33) $\displaystyle V_{*,qq+1}$ $\displaystyle=$ $\displaystyle
V_{*,q+1q}^{*}=\bar{w}(S_{qq}^{-1})^{2}+w(S_{qq+1}^{-1})^{2}+2vS_{qq}^{-1}S_{qq+1}^{-1},$
(34)

where $\bar{w}$ is the complex conjugate of $w$. According to (10), the
effective impurity matrix is computed from (28), (29) and (32) and yields

$\displaystyle\mathbf{\tilde{V}}_{{}_{FB}}$ $\displaystyle=$
$\displaystyle\mathbf{S}_{*}^{-\frac{1}{2}}\mathbf{V}_{{}_{FB}}\mathbf{S}_{*}^{-\frac{1}{2}}$
(35) $\displaystyle=$
$\displaystyle\begin{pmatrix}\frac{\epsilon_{{}_{B}}+\epsilon_{{}_{C}}}{4}&\frac{\epsilon_{{}_{B}}+\epsilon_{{}_{C}}}{4}e^{-\theta}+i\frac{\epsilon_{{}_{B}}-\epsilon_{{}_{C}}}{4}\sqrt{1-e^{-2\theta}}\\\
\frac{\epsilon_{{}_{B}}+\epsilon_{{}_{C}}}{4}e^{-\theta}-i\frac{\epsilon_{{}_{B}}-\epsilon_{{}_{C}}}{4}\sqrt{1-e^{-2\theta}}&\frac{\epsilon_{{}_{B}}+\epsilon_{{}_{C}}}{4}\end{pmatrix}.$

Diagonalization of this matrix yields, through (21), the flux-dependent
energies $E_{\pm}=E_{\pm}(\phi)$ (which will be referred hereafter as the
“energy bands”, with the flux taking the place of a synthetic momentum) of the
two states $\ket{\tilde{\varphi}_{\pm}}=\ket{\tilde{\varphi}_{\pm}(\phi)}$
within the zero-energy FB that are affected by the impurities at the $q+1$
plaquette, which are determined as

$\begin{pmatrix}\ket{\tilde{\varphi}_{+}}\\\
\ket{\tilde{\varphi}_{-}}\end{pmatrix}=\mathbf{U}\begin{pmatrix}\ket{\tilde{\varphi}_{q}}\\\
\ket{\tilde{\varphi}_{q+1}}\end{pmatrix},$ (36)

where $\mathbf{U}$ is the unitary matrix with eigenvectors as its rows, with
elements $[U]_{ij}\equiv U_{ij}$, where $i,j=q,q+1$. Using (30)-(31), we can
express $\ket{\tilde{\varphi}_{\pm}}$ in terms of the original CLSs,

$\displaystyle\ket{\tilde{\varphi}_{+}}$ $\displaystyle=$
$\displaystyle\sum\limits_{j=q}^{q+1}\sum\limits_{l}\Big{[}U_{qq}S^{-\frac{1}{2}}_{*,qj}+U_{qq+1}S^{-\frac{1}{2}}_{*,q+1j}\Big{]}S^{-1}_{jl}\ket{\text{Loc},l},$
(37) $\displaystyle\ket{\tilde{\varphi}_{-}}$ $\displaystyle=$
$\displaystyle\sum\limits_{j=q}^{q+1}\sum\limits_{l}\Big{[}U_{q+1q}S^{-\frac{1}{2}}_{*,qj}+U_{q+1q+1}S^{-\frac{1}{2}}_{*,q+1j}\Big{]}S^{-1}_{jl}\ket{\text{Loc},l}.$
(38)

Let us address separately three representative choices for the
$\epsilon_{{}_{B}},\epsilon_{{}_{C}}$ local impurities shown in the right
insets of Fig. 2(d)-(f) and analyze their solutions.

###  IV.1 Symmetric impurities
($\epsilon_{{}_{B}}=-\epsilon_{{}_{C}}=\epsilon$)

Figure 3: Snapshots of the amplitudes of the higher (lower) impurity state at
the top (bottom) panel at different flux values, indicated on top, for a
partial section of an open diamond chain with 53 complete plaquettes, with
equal impurities $\epsilon=0.1$ in the middle one. The energy of both states
at each flux value is given in Fig. 2(b).

For symmetric impurities, $\epsilon_{{}_{B}}=-\epsilon_{{}_{C}}=\epsilon$,
where the impurity strength $\epsilon$ is given in units of the hopping
parameter $t$, the effective impurity matrix in (35) reduces to

$\mathbf{\tilde{V}}_{{}_{FB}}=-\frac{\epsilon}{2}\sqrt{1-e^{-2\theta}}\sigma_{y},$
(39)

with $\sigma_{\mu}$ ($\mu=x,y,z$) the $\mu$ Pauli matrix. The respective
eigenvalues and eigenvectors read as

$\displaystyle E_{\pm}$ $\displaystyle=$
$\displaystyle\pm\frac{\epsilon}{2}\sqrt{1-e^{-2\theta}},$ (40)
$\displaystyle\begin{pmatrix}\ket{\tilde{\varphi}_{+}}\\\
\ket{\tilde{\varphi}_{-}}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}i&1\\\
-i&1\end{pmatrix}\begin{pmatrix}\ket{\tilde{\varphi}_{q}}\\\
\ket{\tilde{\varphi}_{q+1}}\end{pmatrix}.$ (41)

In Fig. 2(a), we compare the analytical energies of (40) with numerical
results as a function of the magnetic flux, for a chain with 53 plaquettes and
symmetric local impurities at the top and bottom sites of the midchain
plaquette [depicted in the right inset of Fig. 2(d)]. Two states with
symmetric energies are lifted from the flat band for a finite magnetic flux
per plaquette. Even though the perturbation is already quite large,
$\epsilon=0.1$, that is, away for all $\phi$ from the $\epsilon\ll
E_{\text{gap}}$ condition assumed in the projection of the impurity operator
onto the flat band in (25), there is a nearly perfect match between analytical
and numerical curves for the entire spectrum.

In Fig. 2(d), we plot the spatial profile of the eigenstate corresponding to
the encircled positive energy at $\phi=\frac{\pi}{2}$ in Fig. 2(a). The
analytical amplitude at each site, depicted as red dots, was computed by
applying the general expression in (37), with the $\mathbf{U}$ matrix for the
symmetric impurities case given in (41), and agrees very well with the
numerical one, represented by the golden bar at the corresponding site. The
inset shows the same results but in a semi-logarithmic scale. Sites without
red dots correspond to A sites where the localized states have zero weight,
according to (15)-(16). The negligible numerical weight at these sites comes
from higher-order terms that couple the perturbed states with the top and
bottom dispersive bands.

###  IV.2 Equal impurities ($\epsilon_{{}_{B}}=\epsilon_{{}_{C}}=\epsilon$)

For equal impurities, $\epsilon_{{}_{B}}=\epsilon_{{}_{C}}=\epsilon$, the
effective impurity matrix in (35) reduces to

$\mathbf{\tilde{V}}_{{}_{FB}}=\frac{\epsilon}{2}\big{(}\sigma_{0}+e^{-\theta}\sigma_{x}\big{)},$
(42)

where $\sigma_{0}$ is the $2\times 2$ identity matrix. The respective
eigenvalues and eigenvectors read as

$\displaystyle E_{\pm}$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{2}\big{(}1\pm e^{-\theta}\big{)},$ (43)
$\displaystyle\begin{pmatrix}\ket{\tilde{\varphi}_{+}}\\\
\ket{\tilde{\varphi}_{-}}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}\begin{pmatrix}\ket{\tilde{\varphi}_{q}}\\\
\ket{\tilde{\varphi}_{q+1}}\end{pmatrix}.$ (44)

In Fig. 2(b), we compare the analytical energies of (43) with numerical
results as a function of the magnetic flux for the impurity distribution
depicted in the right inset of Fig. 2(e). Two positive energy states are
lifted from the FB for a finite magnetic flux per plaquette, and become
degenerate with $E=0.5\epsilon$ at $\phi=\pi$. Since $E_{\text{gap}}\to 0$
when $\phi\to 0\ (\text{mod}\ 2\pi)$, near the edges of the spectrum higher-
order couplings to the dispersive bands can no longer be safely neglected, and
the analytical treatment breaks down in that region. However, this region is
still very small for $\epsilon=0.1$, and shrinks even further with decreasing
$\epsilon$. The spatial amplitudes of the eigenstate with the energy marked by
the blue circle in Fig. 2(b) are depicted in Fig. 2(e). As for the symmetric
case above, a very good agreement is found between the numerical and
analytical results.

####  IV.2.1 Disorder

The localization length of the impurity states is controlled by the flux, as
illustrated in Fig. 3. Having established numerically the validity of the
analytical solutions considerably away from the $\epsilon\ll 1$ limit assumed
in their derivation (the impurity strength is $\epsilon=0.1$ in Fig. 2, and
even higher values still yield good agreement with numerical results), one can
study the effect of small disorder on the impurity states. More specifically,
we will consider diagonal disorder on an open diamond chain with 101
plaquettes threaded by a $\phi$ flux, by setting the onsite potential at each
site to a random value taken from a uniform distribution in the interval
$[-\frac{\delta}{2},\frac{\delta}{2}]$, with $\delta=0.02$ the disorder
strength, with a standard deviation given by
$\sigma_{\text{ud}}=\frac{\delta}{\sqrt{12}}\approx 5.8\times 10^{-3}$. For
the sites at the midchain plaquette hosting the impurities, the disorder is
added on top of $\epsilon=0.1$. We perform 20000 disorder realizations for
each flux value and track the energies of the impurity states, from where we
construct their respective histograms, as exemplified for the higher one at
$\phi=\frac{9\pi}{20}$ in the inset of Fig. 4(b). Remarkably, the underlying
uniform distribution of the disorder generates a normal distribution for the
energies of the impurity states, whose mean approaches their disorder-free
values given in Fig. 2(b) with increasing disorder realizations. This result
is consistent with the central limit theorem (CLT), i.e., for each realization
the impurity states effectively feel an averaged disorder related to their
localization length, that is, to their spatial extension.

Figure 4: (a) Standard deviation, as a function of the flux, of the energy of
both impurity states for an open diamond chain with 101 plaquettes, with equal
impurities $\epsilon=0.1$ in the middle one, taken from 20000 realizations of
diagonal disorder at each point, with disorder strength $\delta=0.02$. (b)
Effective number of sites involved in the disorder averaging effect, computed
from the corresponding standard deviation values in (a) in accordance with the
CLT. The inset shows the energy histogram for the encircled higher-energy
state at $\phi=0.45\pi$, with the dashed orange curve corresponding to the
Gaussian fit from which the standard deviation is extracted.

In Fig. 4(a), we plot the standard deviation of the Gaussian fit of the energy
distributions of both impurity states as a function of the flux . The flux was
taken in $\frac{\pi}{20}$ increments and restricted to the
$[\frac{3\pi}{20},\frac{15\pi}{20}]$ interval, since near $\phi=0,\pi$ the
energy separation of the states is of the order of $\delta$ [see numerical
results in Fig. 2(b) in these regions], which leads to a reordering of the
states and to an overlap of their corresponding energy distributions,
preventing their independent analysis. According to the CLT, the standard
deviation of the normal distribution is related to the sample size, which in
our case corresponds to the effective number of sites $N_{e}$ over which the
disorder is averaged for each impurity state, through
$\sigma=\frac{\sigma_{\text{ud}}}{\sqrt{N_{e}}}$, with $\sigma_{\text{ud}}$
given above. With this formula, and from the $\sigma$ values given in Fig.
4(a), we plot in Fig. 4(b) the flux dependence of $N_{e}$ for both impurity
states. Interestingly, it can be seen that they follow opposite tendencies
with increasing flux: while $N_{e}$ decreases exponentially for the lower
state, it increases monotonously for the higher state.

The evolution of the spatial profile of the impurity states with flux, shown
in Fig. 3, can be used to explain their different response to disorder. The
lower state becomes globally more and more localized until it becomes a CLS at
$\phi=\pi$, which agrees with its decreasing $N_{e}$ in Fig. 4(b), since the
disorder is being averaged over fewer sites, therefore generating a higher
effective onsite disorder on the lower state. The higher state, on the other
hand, has dominant contributions from the impurity sites at lower fluxes (see
central peaks in the top panel of Fig. 3 with growing weight for decreasing
flux). The weights at these sites are in an anti-symmetric configuration that
becomes the full eigenstate (a CLS) at $\phi=0$, while they are in a symmetric
configuration for the lower state. At zero flux, applying the same rotation to
the $B$ and $C$ sites at each plaquette has been shown to decouple the anti-
symmetric state, while the symmetric one connects with adjacent $A$ sites to
form a uniform chain [40]. As flux is increased, the anti-symmetric
configuration of impurity sites couples to the uniform chain with growing
strength [41] and, as with the lower state, the higher state develops a
spatially extended profile that becomes increasingly localized (see the
evolution of the components at sites other than the central impurity ones in
the top panel of Fig. 3). Conversely, the peaks at the impurity sites, which
are the main contributors to the number of effective sites $N_{e}$ of the
averaged disorder, continuously decrease with increasing flux, that is, spread
their weight to the other sites. This effect, which tends to increase $N_{e}$,
overpowers the localization effect of the other components, which tends to
decrease $N_{e}$ (as for the lower state), leading to the net increase of
$N_{e}$ with flux seen in Fig. 4(b).

While the normal distribution of the sample means expected from CLT is only
exact, strictly speaking, in the limit of infinite sample size,
$N_{e}\to\infty$, we have checked numerically that a Gaussian fit is in very
good agreement with the energy histogram of all points in Fig. 4(b), including
for the higher state, as exemplified in the inset, where the effective number
of sites responsible for the averaged disorder is small, ranging from
$N_{e}\approx 2.5$ to $N_{e}\approx 5$.

The averaging of diagonal disorder on the impurity states, conjectured to be
the mechanism behind the enhanced robustness to disorder displayed by certain
domain wall states with similar localization properties [42], is
quantitatively confirmed here by the numerical results of Fig. 4. For the
equal impurities case studied in this section, this enhanced robustness to
disorder is greater for the lower impurity state at low flux values where,
according to Fig. 4(b), $N_{e}$ is maximized. This is demonstrated in a follow
up paper, where the response to disorder of effective systems of coupled
impurity states is analyzed in greater detail [43].

###  IV.3 Single impurity ($\epsilon_{{}_{B}}=\epsilon,\epsilon_{{}_{C}}=0$)

Finally, let us address the case where a single impurity,
$\epsilon_{{}_{B}}=\epsilon$, is placed at the top site of the $q+1$
plaquette. The effective impurity matrix in (35) reduces now to

$\displaystyle\mathbf{\tilde{V}}_{{}_{FB}}$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{4}\big{(}\sigma_{0}+e^{-\theta}\sigma_{x}-\sqrt{1-e^{-2\theta}}\sigma_{y}\big{)}.$
(45)

The respective eigenvalues and eigenvectors read as

$\displaystyle E_{\pm}$ $\displaystyle=$ $\displaystyle\frac{\epsilon}{4}(1\pm
1),$ (46) $\displaystyle\begin{pmatrix}\ket{\tilde{\varphi}_{+}}\\\
\ket{\tilde{\varphi}_{-}}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}1&e^{-i\vartheta}\\\
1&-e^{-i\vartheta}\end{pmatrix}\begin{pmatrix}\ket{\tilde{\varphi}_{q}}\\\
\ket{\tilde{\varphi}_{q+1}}\end{pmatrix},\ $ (47) $\displaystyle\cos\vartheta$
$\displaystyle:=$ $\displaystyle
e^{-\theta}=\text{Exp}\Big{[}-\cosh^{-1}\big{[}\sec\frac{\phi}{2}\big{]}\Big{]},$
(48)

where the formula for $\theta$ given in (21) was used in the last line. Both
energy bands are flat and, in particular, we have $E_{-}=0$, as shown in Fig.
2(c), meaning that, when a single impurity is present, the system can always
rotate the subspace of dual localized states in order to generate a state,
namely $\ket{\tilde{\varphi}_{-}}$ in (47), with zero weight at the impurity
site, $\innerproduct{B_{q+1}}{\tilde{\varphi}_{-}}=0$, such that it remains
degenerate with the other FB states. This is illustrated in Fig. 5, where a
partial profile of the $\ket{\tilde{\varphi}_{+}}$ and
$\ket{\tilde{\varphi}_{-}}$ eigenstates, computed through (47) for
$\phi=\frac{\pi}{2}$, is depicted [the complete profile of
$\ket{\tilde{\varphi}_{+}}$ is shown in Fig. 2(f), which agrees very well with
numerics]. It can be seen that the weight of the state at the impurity site is
maximum for $\ket{\tilde{\varphi}_{+}}$ and zero for
$\ket{\tilde{\varphi}_{-}}$.

Figure 5: Partial profile of the $\ket{\tilde{\varphi}_{+}}$ and
$\ket{\tilde{\varphi}_{-}}$ eigenstates with energies marked with a blue
circle and a red square in Fig. 2(c), respectively, and where the radius of
each circle represents the amplitude of the wavefunction at the respective
site and the color represents its phase, coded by the color bar at the right.
The arrow indicates the $B_{q+1}$ site where the
$\epsilon_{{}_{B}}=\epsilon=0.1$ impurity is placed, and where
$\ket{\tilde{\varphi}_{+}}$ ($\ket{\tilde{\varphi}_{-}}$) has maximum (zero)
amplitude.

####  IV.3.1 Topological characterization

The topological characterization of the single-impurity case reveals
interesting new features. Let us first summarize the usual approach to the
topological characterization of gapped 1D two-band models. The relevant
topological invariant for characterizing the energy gap is the Zak phase of
the lower band, which can be expressed as

$\gamma=i\oint
dR\bra{\tilde{\varphi}(R)_{-}}\frac{d}{dR}\ket{\tilde{\varphi}(R)_{-}},$ (49)

where the integral traces a closed path along the dimensional parameter $R$.
For the textbook Su-Schrieffer-Heeger (SSH) two-band model [44], consisting of
a chain with staggered hopping terms $t_{1}$ and $t_{2}$, with the former
(latter) corresponding to the intracell (intercell) hopping parameter [45],
the lowest energy bulk eigenstate has the same form of
$\ket{\tilde{\varphi}_{-}}$ in (47) when written in its site state basis,
$\ket{\tilde{\varphi}_{-}(k)}=1/\sqrt{2}(1,-e^{-i\vartheta(k)})^{T}$, where
$k$ is the momentum and $\cos\vartheta(k)=\frac{t_{1}}{t_{2}\sin k}+\cot k$
[46, 38]. Inserting $\ket{\tilde{\varphi}_{-}(k)}$ in (49), where $R=k$ and
the integral goes over the Brillouin zone (BZ), we get

$\gamma=\frac{1}{2}\int_{\vartheta(0)}^{\vartheta(2\pi)}d\vartheta(k)=\frac{\Delta\vartheta}{2},$
(50)

with $\Delta\vartheta=\vartheta(2\pi)-\vartheta(0)$ yielding the total change
in the $\vartheta(k)$ phase around the BZ.

Figure 6: Phase $\vartheta$ of the lowest energy eigenstate as a function of
the magnetic flux $\phi$ (momentum $k$) for the single-impurity model (SSH
model). The color of each curve matches the color of the corresponding
horizontal axis. For the SSH model, the ratio $t_{1}/t_{2}=2$
($t_{1}/t_{2}=0.5$) was used for the topologically (non-)trivial case.

From the profile of $\vartheta(k)$ shown in Fig. 6, which has been
experimentally confirmed recently in different platforms [47, 48], one readily
obtains $\gamma=0(\pi)$ for the (non-)trivial phase. According to the bulk-
boundary correspondence, an open SSH chain will have zero edge states in the
trivial $\gamma=0$ phase, and two edge states in the non-trivial $\gamma=\pi$
phase, exponentially localized at opposite ends of the chain (up to
hybridization effects stemming from finite size corrections).

Let us turn our attention to the single-impurity case, with the gapped
effective two-band spectrum given by the solid curves in Fig. 2(c). It is
clear that the relevant parameter of integration is now, not the momentum $k$,
but rather the reduced magnetic flux $\phi$, as it changes by one flux
quantum, that is, from $\phi=0$ to $\phi=2\pi$. By inserting
$\ket{\tilde{\varphi}_{-}}=\ket{\tilde{\varphi}_{-}(\phi)}$, given in (47), in
(49) leads to (50) with $k\to\phi$. Therefore, the Zak phase depends on the
total variation of $\vartheta(\phi)$ in (48) over one flux quantum. The blue
solid curve in Fig. 6 shows that $\Delta\vartheta=\pi$, leading to a Zak phase
$\pi$-quantized to a half-integer value, $\gamma=\frac{\pi}{2}$.
Alternatively, after rewriting the effective impurity matrix in (45) as

$\mathbf{\tilde{V}}_{{}_{FB}}=\frac{\epsilon}{4}\sigma_{0}+\begin{pmatrix}0&h(\phi)\\\
h^{\ast}(\phi)&0\end{pmatrix},$ (51)

where $h(\phi)=\frac{\epsilon}{4}e^{i\vartheta(\phi)}$, one can compute the
winding number [45, 49] as

$W=\frac{1}{2\pi i}\int_{0}^{2\pi}d\phi\frac{d}{d\phi}\log
h(\phi)=\frac{\Delta\vartheta}{2\pi}=\frac{\gamma}{\pi}=\frac{1}{2},$ (52)

which similarly produces a half-integer value, meaning that $\phi$ needs to
change adiabatically by two full cycles to produce one complete winding of
$h(\phi)$ around the origin in the complex plane. Furthermore, from Fig. 6 and
the form of the eigenvectors in (47), it can be seen that they are
$4\pi$-periodic in $\phi$ [since both energy bands are flat, see (46), the
$4\pi$-periodicity is not explicitly manifested there], and also that the
eigenvectors exchange in multiples of $2\pi$ as

$\ket{\tilde{\varphi}_{\pm}(\phi)}=\ket{\tilde{\varphi}_{\mp}(\phi+2\pi)}=\ket{\tilde{\varphi}_{\pm}(\phi+4\pi)}.$
(53)

This behavior is remarkably similar to that observed in chiral-symmetric non-
Hermitian 1D models where, due to the fact that each exceptional point (EP)
[50] carries a topological charge of $\pm 1/2$, the absolute value of the
winding number, $|W|$, corresponds to half the number of times each EP is
encircled as the momentum $k$ traverses the BZ (assuming the wrapping around
all EPs follows the same circulation, such that all individual contributions
to $W$ have the same sign). In particular, if a single EP is encircled once
then one has $|W|=1/2$ [51, 52, 53] and, analogously, the eigenstates of both
bands (assuming a two-band model) are also exchanged after traversing the BZ
[51, 54]. This non-Hermitian $|W|=1/2$ case has been associated with the
presence of a single edge state under open boundary conditions (OBC), and each
$1/2$ increment in $|W|$ has been further associated with the appearance of an
extra edge state [55]. Concerning the effective model for the single-impurity
case studied in this section, it corresponds, to the best of our knowledge, to
the first example of a Hermitian model with a half-integer winding number.

Regarding the Zak phase, on the other hand, the other known example of a 1D
Hermitian model where it yields a half-integer value at its energy gaps is the
diamond chain with $\pi$-flux per plaquette [19, 21] [where $W$ is not well-
defined since the off-diagonal block of the Hamiltonian in its chiral
representation, corresponding here to the scalar $h(\phi)$ in (51), is not a
square matrix]. The $\gamma=\frac{\pi}{2}$ value for the finite-energy bands
of this model is a consequence of having a noncentered inversion-axis within
the unit cell [56], such that the inversion operator picks up a $k$-dependence
over one of the sublattices. Accordingly, a modified bulk-boundary
correspondence is manifested by this system, one that matches a half-quantized
Zak phase at a given energy gap with the presence of a single edge state at
that gap, when open boundaries are considered [19, 21, 16, 17].

Figure 7: (a) Lieb lattice with a mid-length line of onsite impurity
potentials $\epsilon$ at every other site (red dots) along the $y$-direction.
All hopping terms are given by $t$. (b) Diamond chain obtained from Fourier
transforming the lattice in (a) along the periodic $y$-direction, using the
indicated supercell. The $y$-momentum $k_{y}$ becomes the equivalent of a flux
per plaquette. The dashed hoppings pick up a positive $k_{y}$ phase in the
clockwise direction within each plaquette.

From the examples mentioned above, the same pattern seems to emerge both for
Hermitian and non-Hermitian models, namely that a half-integer bulk
topological invariant is connected to the presence of a single edge state
under OBC. The question now becomes: given that the same half-integer
invariants (winding number and Zak phase) are observed for our single-impurity
case, is there a physical signature of this non-trivial behavior? Recall that
the “bulk” bands in Fig. 2(c) are defined over the reduced flux $\phi$, and
not over a momentum variable $k$. Therefore, the concept of OBC does not
directly apply here 111Note that the bulk-boundary correspondence relates
quantized bulk invariants defined for the momentum-space with the presence of
edge states under OBC in the complementary real-space, since position and
momentum are conjugate variables. The conjugate variable of the (reduced)
magnetic flux is the (reduced) electric charge, a central relation in the
study of quantum electrical circuits [78]. However, and for our purposes, it
is not clear what the meaning of OBC in a “charge-space” could be, if any, and
what would correspond to an edge state in this quantized charge-space.. To
circumvent this limitation, we adopt a strategy based on an exact mapping as
follows. First, we consider the modified Lieb lattice depicted in Fig. 7(a),
with a line of impurity sites at every other site along the $y$-direction (red
sites), positioned at the center of the $x$ coordinate. Then, assuming PBC
(OBC) in the $y$-direction ($x$-direction), we Fourier transform the lattice
along the $y$-direction (with lattice constant set to unity) using the
supercell represented in Fig. 7(a), obtaining the open diamond chain in Fig.
7(b). As a result, this chain becomes the same as the single-impurity case of
Fig. 1(a), with $k_{y}$ taking the place of $\phi$, and up to an
inconsequential gauge transformation that accumulates all the $k_{y}$ “flux”
in a single hopping term within each plaquette. Therefore, for a small value
of $\epsilon$ we expect the effective low-energy subspace to behave as in Fig.
2(c), with $\phi\to k_{y}$. The same half integer invariants $\gamma$ and $W$
in (50) and (52), respectively, are obtained when integrating now over the
Brillouin zone in $k_{y}$.

Figure 8: (a) Energy spectrum, with $t$ as the energy unit, of the impurity
decorated open Lieb lattice, with $N_{x}=41$ ($N_{y}=200$) unit cells in the
$x$ ($y$) direction and impurities $\epsilon=0.1$ placed along unit cell 21 in
$x$ according to the pattern in Fig. 7(a). (b) Zoomed energy spectrum of the
region delimited by the orange dashed box in (a). (c)-(e) Depiction of the
amplitude distribution of the eigenstate with index (c) $n=16201$, (d)
$n=16202$, and (e) $n=16396$, with energies given by the corresponding
encircled states in (b).

In order to test if this translates into the appearance of a gapped edge state
in real-space, we impose OBC along the $y$-direction also, i.e., we consider
the Lieb lattice in Fig. 7(a) with OBC for both directions. In Fig. 8(a), we
plot the energy spectrum for a chain $N_{x}=41$ ($N_{y}=200$) unit cells in
the $x$ ($y$) direction, with impurities $\epsilon=0.1$ placed along unit cell
21 in $x$ according to the pattern in Fig. 7(a). The overall features of the
spectrum in Fig. 8(a) are the expected ones for a Lieb lattice, namely a
gapless system of two continua of dispersive states above and below the
localized states of the zero-energy flat band. However, a closer look at the
states with energies around $0\leq E\lesssim\frac{\epsilon}{2}$ [notice how
these limits correspond to the energies of the flat bands of the effective
system in Fig. 2(c)], shown in the zoomed region of Fig. 8(b), reveals the
presence of an energy gap. As expected, an isolated edge state is seen to
appear within this gap, with the spatial profile depicted in Fig. 8(c).

The presence of an energy gap was not guaranteed a priori since, loosely
speaking, the spectrum sweeps all the $k_{y}$ values allowed by the OBC, and
the gap between the dispersive and flat bands closes as $k_{y}\to 0$ [58].
Moreover, contrary to the analytical results for the effective model, the
numerical results of Fig. 2(c) also show that the gap between the two bands
closes at the edges (see the descending blue dots tending towards the red ones
near $\phi\to k_{y}=0,2\pi$). But since this gap closing region is very sharp,
getting narrower as $\epsilon\to 0$, few solutions are expected to appear in
this region under OBC. Indeed, some states at the left edge of the top band in
Fig. 8(b) are bending towards the zero-energy band, while nonetheless keeping
close to the top flat band at $E=\frac{\epsilon}{2}=0.05$, such that the gap
remains open and the edge state fully gapped. In Fig. 8(d), we show the
spatial profile of the leftmost state of this subset. It can be seen that,
while consisting mostly of a bulk state along the impurity line, there is
still a finite weight on the region of the right edge state, which gradually
shrinks as the state index increases, eventually vanishing once the flat band
states at $E=0.05$ are reached [see the profile of the state in Fig. 8(e)]. We
attribute the right edge component for this subset of states to a perturbative
hybridization with the gapped edge state that further sharpens their energy
bending.

The results of this exact mapping provide another example where the half-
integer topological bulk invariants, $\gamma=\pi W=\frac{\pi}{2}$, have a
correspondence with the presence of a single gapped edge state under OBC. In
turn, this provides indirect evidence that the topologically non-trivial
nature of the single-impurity case studied in this section can be related to
specific physical manifestations. We note that this bulk-boundary
correspondence should also be observed upon switching the single impurity from
the $B$ site of the midchain plaquette, as in Fig. 7(b), to its $C$ site. In
terms of the Lieb lattice in Fig. 7(a), this corresponds to shifting the
impurities along the impurity line, exchanging its red and green sites. As we
detail in Appendix A, this shift leads to a physically distinct configuration
that produces small changes to the low-energy spectrum in Fig. 8(b). However,
as we show there, the validity of this mapping is not affected, as a single
in-gap state is still present.

##  V Conclusions

We studied a diamond chain with a gapped FB in its energy spectrum upon
introducing a finite magnetic flux per plaquette, whose states can be
described as CLSs spanning a non-orthogonal basis subspace. After outlining
the mathematical formalism required for the analytical treatment of such non-
orthogonal bases, we applied it to the general solutions of the impurity
states that are lifted from the FB by small local potentials placed at the
midchain plaquette. For the two equal, two symmetric and single-impurity
cases, the numerical results were shown to closely follow the analytical
predictions, except for flux values very close to the critical gap closing
point between flat and dispersive bands, corresponding to the expected
breakdown region of the theory.

Disorder studies were conducted for the case of equal impurities. By applying
small disorder following a uniform distribution to the diamond chain, the
energy histogram of both impurity states was seen to follow a normal
distribution for a wide range of flux values. These findings were connected to
the CLT, such that, from relating the standard deviations of the normal and
uniform distributions, an effective number of sites involved in a disorder
averaging effect [42] was computed for each impurity state. We discussed how
this effect translates into an enhanced robustness of the states to diagonal
disorder, which is one of the most common kinds of disorder present in real
systems.

A topological characterization of the impurity levels was carried out, where
the flux was interpreted as a synthetic momentum. While the equal and
symmetric impurity cases were characterized as topologically trivial, the
single-impurity case manifested a peculiar type of non-trivial topology,
namely one defined by half-integer invariants such as the Zak phase and
winding number. The latter corresponds, as far as we know, to the first of its
kind in the context of Hermitian models. Given that the invariants are
computed by considering the reduced magnetic flux as the “bulk” variable of
integration, and the variation of the flux by a flux quantum as the “Brillouin
zone”, the bulk-boundary correspondence cannot be directly checked. However,
an indirect way of circumventing this limitation was formulated, based on an
exact mapping between the diamond chain with a single impurity and a two-
dimensional Lieb lattice with a central line of impurities. The mapped lattice
displays the same low-energy effective impurity bulk bands, classified by the
same half-integer invariants. For fully open boundaries, its energy spectrum
reveals the presence of a single edge state within the gap of this low-energy
subspace, providing an indirect evidence of the unconventional bulk-boundary
correspondence associated with the topology of the single-impurity case.

Regarding the realization of the model studied here, shown in Fig. 1(a), we
highlight two platforms. One is electrical circuits [59], whose flexibility
and plasticity allows one to recreate most tight-binding models simply by
coupling nodes with capacitors or inductors in the desired geometry. By
incorporating negative impedance converters with current inversion (INICs)
[60, 61] in phase-control units, it has been shown [62] that arbitrary phase
factors can be associated to the couplings, enabling the generation of a
synthetic flux at each plaquette, as required by the target model of Fig.
1(a). The impurity potentials at the central nodes can be simulated by adding
extra capacitors, with small capacitance values, to the grounding schemes of
these nodes. Preliminary numerical studies should be undertaken to determine
the optimal parameters of the circuit since, on the one hand, the admittance
elements of the impurity nodes should be small, compared to the admittance gap
between flat and dispersive bands, and, on the other hand, they should be
sufficiently higher than the tolerance of the other components, so as to not
have its effects washed way by this inherent disorder. As such, very low
tolerance components should be considered. After feeding the circuit with an
alternated current with the eigenfrequency of the impurity states, standard
voltage and two-point impedance measurements can be used to detect them.

The other strong possibility consists in implementing the diamond chain in
photonic lattices. Three different options can be considered: (i) photonic
waveguide arrays, where the synthetic flux can be generated by an appropriate
bending of the lattice along its longitudinal direction [15], while the
impurity potentials are controlled by the index of refraction of the
corresponding waveguides (which in turn depends on the difference in lasing
speed that is used to write them [63]); (ii) microring resonator arrays, where
the synthetic flux can be controlled by displacing or modifying the optical
path of the auxiliary rings used to effectively couple the main ones [64, 65,
66], and the impurities can be introduced by locally perturbing the resonance
condition of the main lattice rings [67] (through slightly modifying their
size, width or refractive index); (iii) arrays of coupled multiorbital
waveguides [68, 20, 69, 70] guiding staggered orbital angular momentum (OAM)
modes of order $l=0,\pm 1$ where, in analogy with systems of ultracold atoms
[71, 72, 73], the synthetic flux can be produced by geometrical manipulation
of the waveguide pattern of the excited OAM modes [74, 75], while local
potentials can again be introduced via changes to their radii or refractive
index.

Our work provides a conceptual frame within which future studies on systems
with non-orthogonal bases can be carried out. In particular, we discuss in
greater detail in a companion piece to this paper [43] how one can, using the
implementation platform (iii) above, and by decorating the diamond chain with
impurities placed at regular intervals, extract impurity subspaces realizing
different effective models from the FB, including topological ones hosting
edge states.

## Acknowledgments

A.M.M. and R.G.D. developed their work within the scope of the Portuguese
Institute for Nanostructures, Nanomodelling, and Nanofabrication (i3N)
Projects No. UIDB/50025/2020, No. UIDP/50025/2020, and No. LA/P/0037/2020,
financed by national funds through the Fundação para a Ciência e Tecnologia
(FCT) and the Ministério da Educação e Ciência (MEC) of Portugal. A.M.M.
acknowledges financial support from i3N through the work Contract No. CDL-
CTTRI-46-SGRH/2022. D.V. and V.A. acknowledge financial support from the
Spanish State Research Agency AEI (contract No.
PID2020-118153GBI00/AEI/10.13039/501100011033) and Generalitat de Catalunya
(Contract No. SGR2021-00138).

##  Appendix A Comments on the single-impurity case

The Hamiltonian of the diamond chain in Fig. 1(a) can be parametrized as
$H=H(\phi,\epsilon_{{}_{B}},\epsilon_{{}_{C}})$. For the general case, a
$y$-mirror quasi-symmetry, with the mirror axis going through all the spinal
$A$ sites, can be defined as

$M_{y}H(\phi,\epsilon_{{}_{B}},\epsilon_{{}_{C}})M_{y}^{-1}=H(-\phi,\epsilon_{{}_{C}},\epsilon_{{}_{B}}),$
(54)

where $M_{y}$ is the $y$-mirror operator.

Figure 9: (a) Energy spectrum, with $t$ as the energy unit, of the impurity
decorated open Lieb lattice, with $N_{x}=41$ ($N_{y}=200$) unit cells in the
$x$ ($y$) direction and impurities $\epsilon=0.1$ placed along unit cell 21 in
$x$ as in Fig. 7(a), but with every impurity shifted by one position along the
impurity line (corresponding to exchanging the red and green sites along this
line). (b) Zoomed energy spectrum of the region delimited by the orange dashed
box in (a). (c)-(e) Depiction of the amplitude distribution of the eigenstate
with index (c) $n=16201$, (d) $n=16202$, and (e) $n=16396$, with energies
given by the corresponding encircled states in (b).

Under this quasi-symmetry, the flux is reversed and the impurities exchange
their positions, such that it only becomes an exact symmetry for the equal
impurities case $\epsilon_{{}_{B}}=\epsilon_{{}_{C}}$. On the other hand,
under time-reversal, the system behaves as

$KH(\phi,\epsilon_{{}_{B}},\epsilon_{{}_{C}})K^{-1}=H(-\phi,\epsilon_{{}_{B}},\epsilon_{{}_{C}}),$
(55)

where $K$ is the conjugation operator obeying $KK^{-1}=1$. The combination of
$y$-mirror and time-reversal operations yields

$KM_{y}H(\phi,\epsilon_{{}_{B}},\epsilon_{{}_{C}})M_{y}^{-1}K^{-1}=H(\phi,\epsilon_{{}_{C}},\epsilon_{{}_{B}}).$
(56)

For a general eigenstate $\ket{\psi_{n}}$ of $H$, the Schrödinger equation has
the form

$H(\phi,\epsilon_{{}_{B}},\epsilon_{{}_{C}})\ket{\psi_{n}}=E_{n}\ket{\psi_{n}}.$
(57)

Applying $KM_{y}$ from the left to both sides of (57) leads to

$H(\phi,\epsilon_{{}_{C}},\epsilon_{{}_{B}})KM_{y}\ket{\psi_{n}}=E_{n}KM_{y}\ket{\psi_{n}},$
(58)

where (56) was used. By defining
$\ket{\psi_{n}^{\prime}}:=KM_{y}\ket{\psi_{n}}$, (58) becomes

$H(\phi,\epsilon_{{}_{C}},\epsilon_{{}_{B}})\ket{\psi_{n}^{\prime}}=E_{n}\ket{\psi_{n}^{\prime}},$
(59)

which, as could be expected, tells us that the energy spectrum is invariant
under an exchange of the impurities.

Through the mapping developed for the single-impurity case in Sec. IVC, we
arrived at the substitution $\phi\to k_{y}$. The corresponding combined quasi-
symmetry acting on the diamond chain depicted in Fig. 7(b)) becomes, from
(56), and up to a gauge transformation that redistributes the $k_{y}$ “flux”
equally between all hopping terms of each plaquette,

$KM_{y}H(k_{y},\epsilon,0)M_{y}^{-1}K^{-1}=H(k_{y},0,\epsilon).$ (60)

As shown in (59), $H(k_{y},\epsilon,0)$ and $H(k_{y},0,\epsilon)$ are
isospectral. However, when OBC are considered in the $y$-direction, and for an
integer number of supercells, this isospectrality is lost due to the fact that
placing the impurities at the $B$ sites [as in Fig. 7(a)] or at the $C$ sites
[corresponding to exchanging the red and green sites along the impurity line
in Fig. 7(a)] corresponds to two nonequivalent configurations. This can be
understood as follows. When the clean, impurity-free lattice in Fig. 7(a) with
PBC along the $y$-direction is considered, the $y$-mirror axis is shifted from
the center of the supercell [56, 76], which induces a $k_{y}$-dependence in
the $y$-mirror operator of the lattice [77]. More importantly, if OBC are now
imposed along the $y$-direction, the lattice loses the $y$-mirror symmetry,
since the top and bottom ends are different: from Fig. 7(a), it can be checked
that, for OBC along the $y$-direction and an integer number of supercells, the
top end is a horizontal line of sites, while the bottom end is composed of a
horizontal sequence of dangling sites, thus breaking $y$-mirror symmetry (it
can only be restored by adding extra sites from an incomplete supercell at one
of the edges). This entails that, when the impurity line is included, the
positioning of the impurities at the $B$ or $C$ sites leads to physically
distinct configurations.

The question we want to address here is the following: when changing the
impurities from the $B$ sites along the impurity line, as in Fig. 7(a), to the
$C$ sites (by exchanging the red and green sites along this line), do the
relevant qualitative features of the low-energy spectrum remain the same? In
other words, does one still observe an energy gap between two flat or nearly-
flat bands with a single in-gap edge state, as in Fig. 8(b), such that the
bulk-boundary correspondence is preserved, that is, is the system insensitive
to the choice of impurity positioning? In Fig. 9, we replicate the energy
spectrum of the same Lieb lattice with the spectrum of Fig. 8(a), apart from
changing the position of the impurities to the $C$ sites along the impurity
line, following the procedure explained above. As can be seen in the zoomed
low-energy spectrum of Fig. 9(b), a single in-gap state is found, albeit much
closer in energy to the top bulk band than in Fig. 8(b). The spatial profile
of this state is depicted in Fig. 8(c), confirming its edge localized nature.
The energy state immediately above this one, depicted in Fig. 9(d), is
confirmed to be a bulk state of the impurity line. Even though this first bulk
state of the top band is closer to the in-gap state than in Fig. 8(b), a
weaker hybridization with the edge state than for the state in Fig. 8(d) is
observed. Finally, the last state of the top band is shown in Fig. 9(e),
closely replicating the one in Fig. 8(e). In summary, while small quantitative
differences are observed between the two choices of impurity positioning, both
share the same qualitative features, most notably the presence of a single in-
gap state in the relevant low-energy sector of the spectrum.

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