Dipole-dipole interactions mediated by a photonic flat band

Abstract

Flat bands (FBs) are energy bands with zero group velocity, which in electronic systems were shown to favor strongly correlated phenomena. Indeed, a FB can be spanned with a basis of strictly localized states, the so called compact localized states (CLSs), which are yet generally non-orthogonal. Here, we study emergent dipole-dipole interactions between emitters dispersively coupled to the photonic analogue of a FB, a setup within reach in state-of the-art experimental platforms. We show that the strength of such photon-mediated interactions decays exponentially with distance with a characteristic localization length which, unlike typical behaviours with standard bands, saturates to a finite value as the emitter's energy approaches the FB. Remarkably, we find that the localization length grows with the overlap between CLSs according to an analytically-derived universal scaling law valid for a large class of FBs both in 1D and 2D. Using giant atoms (non-local atom-field coupling) allows to tailor interaction potentials having the same shape of a CLS or a superposition of a few of these.

Full text

Dipole-dipole interactions mediated by a photonic flat band
Enrico Di Benedetto 1, Alejandro González-Tudela 2, and Francesco Ciccarello 1,3
1Universit`adegli Studi di Palermo, Dipartimento di Fisica e Chimica-Emilio Segrè, Via Archirafi 36, 90123 Palermo (Italy)
2CSIC – Instituto de Fisica Fundamental, C. de Serrano 113, 28006 Madrid (Spain)
3NEST, Istituto Nanoscienze-CNR, Piazza S. Silvestro 12, 56127 Pisa (Italy)
Flat bands (FBs) are energy bands with
zero group velocity, which in electronic sys-
tems were shown to favor strongly correlated
phenomena. Indeed, a FB can be spanned
with a basis of strictly localized states, the so
called compact localized states (CLSs), which
are yet generally non-orthogonal. Here, we
study emergent dipole-dipole interactions be-
tween emitters dispersively coupled to the
photonic analogue of a FB, a setup within
reach in state-of the-art experimental plat-
forms. We show that the strength of such
photon-mediated interactions decays exponen-
tially with distance with a characteristic local-
ization length which, unlike typical behaviours
with standard bands, saturates to a finite value
as the emitter’s energy approaches the FB. Re-
markably, we find that the localization length
grows with the overlap between CLSs accord-
ing to an analytically-derived universal scaling
law valid for a large class of FBs both in 1D
and 2D. Using giant atoms (non-local atom-
field coupling) allows to tailor interaction po-
tentials having the same shape of a CLS or a
superposition of a few of these.
1 Introduction
The coherent interaction between quantum emitters
and engineered low-dimensional photonic environ-
ments is a hot research area of modern quantum op-
tics, especially in the emerging framework of waveg-
uide QED [1–5]. A core motivation is unveiling qual-
itatively new paradigms of atom-photon interaction,
well beyond those occurring in standard electromag-
netic environments (like free space or low-loss cavi-
ties). These can potentially be exploited to implement
cutting-edge quantum information processing tasks or
to favor observation of new quantum many-body phe-
nomena. This relies on the fact that photonic baths
with tailored properties and dimensionality, e.g. pho-
tonic versions of electronic tight-binding models, can
Enrico Di Benedetto : enrico.dibenedetto@unipa.it
Alejandro González-Tudela : Correspondence email address:
a.gonzalez.tudela@csic.es
Francesco Ciccarello : Correspondence email address:
francesco.ciccarello@unipa.it
today be implemented in a variety of experimental
scenarios. This includes photonic crystals in the opti-
cal domain [6–9], superconducting circuits in the mi-
crowaves [10–13] and matter-wave emulators [14,15],
allowing the study even of lattices with non-trivial
properties such as occurrence of topological phases
(even in 2D) [16,17] or able to emulate curved spaces
[18].
A key phenomenon to appreciate the effect of an en-
gineered photonic bath are photon-mediated disper-
sive interactions. For instance, while a high-finesse
cavity can mediate an effective all-to-all interaction
between atoms in the dispersive regime [19,20], re-
placing the cavity with an engineered lattice of cou-
pled cavities can result in interactions with a non-
trivial shape of the interatomic potential when the
atoms are tuned in a photonic bandgap [21,22].
The resulting interaction range can be controlled via
modulation the detuning from the band edge, which
has been experimentally confirmed in circuit QED
[23,16,13,24] and predicted to be a resource for ef-
ficient quantum simulations [25–27] and implementa-
tion of hybrid quantum-classical algorithms [28]. Such
effective coherent interactions between atoms can be
understood as being mediated by atom-photon bound
states (BSs) formed by a single quantum emitter [29–
33,11,34–36,15,37,38], whose corresponding pho-
tonic wavefunction (typically exponentially localized
around the atom) in fact shapes the spatial profile
of the effective interatomic potential [21,39,40]. De-
pending on the bath structure, this can exhibit an un-
conventional or even exotic dependence on the emit-
ters’ positions which can be accompanied by topolog-
ical protection [41–47].
Remarkably, due to destructive interference mech-
anisms, some lattices can host a special type of bands
having in fact the same spectrum as a (one-mode)
cavity. Such a band is called flat band (FB) in that
its dispersion law is flat. Accordingly, a FB has zero
width so that its spectrum features only one frequency
just like a perfect cavity mode. Such kind of bands
are well-known to show up in certain natural and arti-
ficial lattice structures [48,49], a prominent example
occurring in the celebrated quantum Hall effect [50].
FBs are currently investigated in condensed matter
and photonics [51] because, due to their macroscopi-
cally large degeneracy and effective quenching of ki-
netic energy, they can favor emergence of many-body
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arXiv:2405.20382v3 [quant-ph] 18 Mar 2025

effects, highly-correlated phases of condensed matter
[52] and non-linearity [53]. Moreover, FBs are ex-
tremely sensitive to disorder [54]. Typically, a FB
arises when the lattice structure is such that, for each
unit cell, one can construct a compact stationary state
(i.e. strictly localized on a few sites), which exactly
decouples from the rest of the lattice by destructive
interference. As a hallmark, these states, called Com-
pact Localised States (CLSs) [55–57] in most cases are
non-orthogonal and can be used to construct a basis
of localized states spanning the FB space that is al-
ternative to the canonical basis of Bloch states (these
being in contrast orthogonal and unbound).
Given the above framework, here we tackle the
question as to what kind of photon-mediated interac-
tions are expected when atoms are dispersively cou-
pled to a photonic FB, and in particular whether
they are like those in the vicinity of a standard band
(i.e. with finite width) or instead like typical ones in
cavity QED. The question is non-trivial as a FB has
the same dimensionality of a standard band but, in
contrast, zero bandwidth. On the other hand, a FB
has a spectrum comprising only one frequency like a
one-mode cavity but contains a thermodynamically
large number of modes.
With these motivations, this paper presents a gen-
eral study of atom-photon interactions in the pres-
ence of photonic FBs 1, by focusing in particular
on photon-mediated interactions whose associated po-
tential shape is inherited (as for standard bands) from
the shape of atom-photon BSs. In contrast to the edge
of a standard band, it turns out that in the vicinity of
a FB the shape of the BS is insensitive to the detun-
ing, while the interaction range of photon-mediated
interactions generally remains finite even when the
atom energetically approaches the FB.
Importantly, we connect the atom-photon BS to the
form of the photonic CLSs characteristic of the consid-
ered band by demonstrating that the overlap between
CLSs provides the mechanism which enables atoms in
different cells to mutually interact. We derive an ana-
lytical and exact general relationship between the BS
localization length and the non-orthogonality of CLSs
valid for a large class of 1D and 2D lattices. Our the-
ory describes occurrence of atom-photon bound states
and dipole-dipole photon-mediated interactions in the
vicinity of a FB in the most general case, including
when CLSs lack orthogonality.
This work is organized as follows. In Section 2, we
introduce the model and notation and review some
basic concepts concerning atom-photon bound states
and photon-mediated interactions. In Section 3, we
present a case study illustrating some peculiar fea-
tures of BSs in the presence of FBs: this anticipates
some of the main original results of this work and
at the same time allows the reader to first familiar-
1Atoms coupled to photonic FBs in specific models appeared
in recent studies [58–60].
ize with the physics of flatbands. In Section 4, we
review the concept of Compact Localised States and
introduce some 1D models used in this paper. This is
then used in the following Sections 5and 6in order
to discuss bound states seeded by a FB, which repre-
sents a central result of our investigation. Finally, the
case of a giant atom coupled to a FB system and the
ensuing photon-mediated interactions is addressed in
Section mediated interactions is addressed in Section
7. Finally, in Section 8we draw our conclusions.
2 Setup and review of photon-
mediated interactions
Here, we introduce the general formalism we will work
with and review some basic notions and important
examples of photon-mediated interactions. We start
from atom-photon BSs, which are necessary in order
to appreciate the physics in the presence of FBs to be
presented later.
2.1 General model
We consider a quantum emitter modeled as a two-
level system whose pseudo-spin ladder operator is
σ=|g⟩⟨e|, with |g⟩and |e⟩being respectively the
ground and excited states, whose energy difference is
ω0. The atom is locally coupled under the rotating-
wave approximation to a photonic bath modeled as a
set of single-mode coupled cavities or resonators each
labeled by index x(which is generally intended as a
set of indexes). Accordingly, the total Hamiltonian
reads
H=ω0σ†σ+HB+ga†
x0σ+ H.c.,(1)
where x0labels the cavity which the emitter is coupled
to with strength g, assumed to weak in a sense that
will be specified better in the following sections. The
free Hamiltonian of the photonic bath HBreads
HB=X
x
ωxa†
xax+X
x=x′
Jxx′a†
x′ax+ H.c.,(2)
where ωxis the bare frequency of the xth cavity, ax
(a†
x) the associated creation (destruction) bosonic lad-
der operator while Jxx′denotes the photon hopping
rate between cavities xand x′. The Fock state where
cavity xhas one photon (with all the remaining ones
having zero photons) will be denoted as |x⟩=a†
x|vac⟩,
where |vac⟩is the vacuum state of the field.
While many properties discussed in this paper
require solely that the bath Bpossesses a FB,
which can happen even if the photonic bath is not
translationally-invariant, all of the examples that we
will discuss concern photonic lattices. In these cases,
Bis a D-dimensional lattice so that index xin the
above equations should be intended as the pair of in-
dexes (n, ν), where nstands for Dintegers labeling a
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primitive unit cells around the Bravais lattice vector
rn, while νlabels the sublattices. All lengths have to
be intended in units of the lattice constant. The nor-
mal frequencies of B(in the thermodynamic limit)
comprise a series of bands, labeled by index mand
with corresponding dispersion law ωm(k), where the
D-dimensional wave vector klies in the first Brillouin
zone (to make notation lighter, we will write kin place
of kwhenever possible). Accordingly, the bath Hamil-
tonian can be written in diagonal form as
HB=X
k,m
ωm(k)Ψ†
k,mΨk,m ,(3)
where Ψk,m and Ψ†
k,m are ladder operators associated
with the mth band.
Throughout this work, we will focus on the single-
excitation sector, i.e. the subspace spanned by the set
{|e⟩|vac⟩,|g⟩|x⟩} (with xrunning over all cavities).
For the sake of simplicity, we will adopt a lighter no-
tation in what follows and replace
|e⟩|vac⟩→|e⟩,|g⟩|x⟩→|x⟩.
Hence, from now on |e⟩will be the state where the ex-
citation lies on the atom and the field has no photons,
while |x⟩is the state where the atom in the ground
state |g⟩and a single photon at cavity xhas one pho-
ton (with each of the remaining cavities in the vacuum
state). In particular, |x0⟩is the state where a single
photon lies at cavity x0(the one directly coupled to
the emitter).
2.2 Atom-photon bound states
Within the single-excitation sector, an atom-photon
bound state (BS) |ΨBS⟩is a normalized dressed state
such that H|ΨBS⟩=ωBS |ΨBS⟩, where ωBS is a real
solution of the pole equation
ωBS =ω0+g2⟨x0|GB(ωBS)|x0⟩,
while the wavefunction (up to a normalization factor)
reads [29]
|ΨBS⟩∝|e⟩+|ψBS ⟩with |ψBS⟩=g GB(ωBS )|x0⟩.
(4)
Here, GB(ω)is the bath resolvent or Green’s function
in the single-excitation subspace [29], whose general
definition is [cf. Eq. (3)]
GB(ω) = X
k,m
|Ψk,m⟩⟨Ψk,m|
ω−ωm(k).(5)
When Bis not a lattice, index (k, m)is just replaced
by the index(es) labeling the eigenmodes of the field.
Typically, a BS occurs when the atom is coupled off-
resonantly to B(as in all examples to be discussed in
this paper where ω0will be tuned within a bandgap of
bath B). In this case, to leading order in the coupling
strength we have
ωBS =ω0,|ψBS⟩=g GB(ω0)|x0⟩.(6)
Wavefunction |ψBS⟩describes a single photon local-
ized around the atom’s location x0.
The simplest, and in some respects trivial, example
of an atom-photon BS occurs in a single-mode cavity,
i.e. when HB=ωca†
x0ax0. In this case, the bath
Green’s function simply reads GB(ω) = |x0⟩⟨x0|/(ω−
ωc)so that the BS reduces to |ΨBS⟩=|e⟩+g
ω0−ωc|x0⟩
2.
Another paradigmatic instance of BS occurs when
Bis a 1D lattice of coupled cavities described by the
Hamiltonian HB=JPna†
xnaxn+1 +H.c.with J > 0.
The energy spectrum (in the thermodynamic limit)
consists of a single band in the interval [−2J, 2J]with
dispersion law ω(k) = 2Jcos k. The corresponding
Green’s function (for |ω|>2J, i.e. out of band) reads
[61,62]
⟨xn|GB(ω)|xm⟩=1
NX
k
eik(xn−xm)
ω−2Jcos k=
→(−1)|xn−xm|
2√Jδ exp−|xn−xm|
λ,
(7)
where the last equality holds in the thermodynamical
limit (N→ ∞) and for δ=ω−2J > 0(above the
upper band edge) and δ≪J. From here, we see that
the BS is exponentially localized with
λ=rJ
δ.(8)
Thus, the photon is exponentially localized around
the atom over a region of characteristic length λ(lo-
calization length). Remarkably, this scales as ∼δ−1
/2,
entailing that, as the atomic frequency approaches
the band edge (i.e. for δ→0+), the BS gets more
and more delocalized. Later on, we will see that in
the presence of a FB a very different behavior occurs
with the BS localization length saturating to a con-
stant value as δ→0+.
2.3 Effective Hamiltonian with many emitters
In the presence of many identical emitters, each in-
dexed by j, the Hamiltonian (1)is naturally general-
ized as
H=ω0X
j
σ†
jσj+HB+gX
j
(a†
xjσj+ H.c.),(9)
with xjlabeling the cavity which the jth atom is cou-
pled.
2This is one of the two dressed states in the single-excitation
sector, specifically the one with a dominant atomic component.
The other state instead has energy ≃ωcand is mostly photonic.
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When ω0lies within a photonic bandgap and Bis
in the vacuum state, the field degrees of freedom can
be adiabatically eliminated and the dynamics of the
emitters is fully described by the effective many-body
Hamiltonian [21,22,40]
Heff =X
ij Kij σ†
iσj+ H.c.,(10)
where
Kij =g⟨xi|ψBS,j ⟩.(11)
Here, |ψBS,j ⟩is the BS seeded by the jth atom, whose
expression is analogous to Eq. (6)with xjin place of
x0. Thus, when atoms are dispersively coupled to
the bath B, they undergo a coherent mutual interac-
tion described by the effective interatomic potential
Kij . Significantly, Eq. (11)shows that the interaction
strength between a pair of atoms is simply propor-
tional to the BS seeded by one emitter (as if this were
alone) on the cavity where the other emitter sits in.
Thus, the BS wavefunction of a single emitter in fact
embodies the spatial shape of the photon-mediated
interaction potential in the presence of many emitters
in the dispersive regime.
Based on the discussion in Section 2.2, for a set
of atoms all coupled to a single cavity we get Kij =
g2/(ω0−ωc)irrespective of iand j, meaning that the
emitters undergo an all-to-all interaction.
Instead, owing to the exponential localization of BS
(8), a set of atoms dispersively coupled to a homoge-
neous cavity array (see Section 2.2) undergo an effec-
tive short-range interaction described by
Kij =(−1)|xi−xj|
√Jδ exp−|xi−xj|
λ,(12)
where the BS localization length λ[see Eq. (8)] can
now be seen as the characteristic interaction range
depending on ω0.
3 Bound state near a photonic flat
band: case study
A major difference between the two paradigmatic
baths considered so far, i.e. a cavity versus a coupled-
cavity array, is that the latter can host propagating
photons whose speed is proportional to the slope of
the dispersion law ω(k). Notice that the presence of a
non-trivial dispersion law ω(k)is essential for the oc-
currence of photon-mediated interactions with finite
and ω0-dependent interaction range λ[cf. Eqs. (8)and
(12)].
In the remainder of this paper, we will deal with dis-
persive physics near a flat band (FB). A FB is a special
photonic band whose dispersion law is k-independent,
i.e.
ω(k) = constant.(13)
Thus, unlike a standard band, a FB has zero band-
width, a feature shared with a (perfect) cavity. Dif-
ferently from a cavity and analogously to a standard
band, however, the FB energy has a thermodynami-
cally large degeneracy (matching the number of lattice
cells N)3.
In order to introduce some typical properties of
photon-mediated interactions around a FB, in this
section we will consider the case study of the 1D saw-
tooth photonic lattice in Fig. 1(a) [63,64], arguably
the simplest yet non-trivial system where a FB shows
up. This is a bipartite lattice, where the nth cell con-
sists of a pair of cavities labeled anand bn. Each
cavity bnis coupled to its nearest neighbours bn±1
with photon hopping rate J > 0and with rate J√2
to cavities an−1and an4. All cavities have the same
bare frequency which we set to zero. The two bands
of Bhave dispersion laws (see Appendix A.1)
ωFB =−2J, ωd(k)=2J(1 + cos k),(14)
As shown in Fig. 1(b), the band labeled by subscript
dis a standard dispersive band of width 4J, below
which a FB of energy −2Jstands out (the energy
separation of this from the edge of band dis 2J).
We now discuss the atom-photon BS formed by
an emitter dispersively coupled to the 1D sawtooth
lattice in the two different regimes where the effect
of either the FB or the dispersive band is negligi-
ble. We start by tuning ω0very far from the FB
(so that this can be fully neglected) but relatively
near the lower edge of the dispersive band so as to
fulfill g≪δd≪δFB with δdand δFB respectively
the detuning from the lower edge of band dand from
the FB [see Fig. 1(b)]. The spatial profile of the BS
photonic wavefunction is plotted in Fig. 1(c)-(e) for
decreasing values of δd. Similarly to Eq. (8)(homoge-
neous array), the BS is exponentially localized around
the atom with a localization length which diverges as
δd→0+.
We next consider the regime g≪δFB ≪δd[see
Fig. 1(b)] in a way that the atom is significantly (al-
though dispersively) coupled only to the FB with the
contribution of the dispersive band now negligible. As
shown in Fig. 1(f)-(h), instead of diverging, the BS
localization length now saturates to a finite value as
δFB →0+. Correspondingly, the BS wavefunction
no longer changes when δFB is small enough, which
shows that this asymptotic BS is insensitive to the
atom frequency.
The simple instance just discussed suggests that
atom-photon BSs in the vicinity of a photonic FB,
3If bath Bis not a lattice, a FB occurs when there exists a
normal frequency with a thermodynamically large degeneracy.
An instance of such a situation are certain kinds of hyperbolic
systems, where there exists a FB gapped from the rest of the
energy spectrum [18].
4Strictly speaking, this is a special case of the sawtooth
lattice in which the ratio of the two hopping rates is constrained
so as to ensure the emergence of a FB.
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-0.5
0
0.5
1
-0.5
0
0.5
1
40 50 60
-0.5
0
0.5
1
40 50 60
Cell
Nearest-neighbour
overlap
(a) Sawtooth lattice
Compact
Localised
State
BS amplitude
(c)
(d)
(e)
(f)
Cell indexCell index
(g)
(h)
(b)
Energy
Flat band
Figure 1: Atom-photon bound state in the photonic sawtooth lattice. (a) Sketch of the sawtooth lattice, showing in particular
a primitive unit cell (red dashed box) and the values of hopping rates between nearest-neighbour cavities (the bare frequency
of each cavity is set to zero). Two overlapping CLSs are shown. (b) Frequency spectrum of the photonic lattice, comprising
a dispersive band between ω= 0 and ω= 4Jand additionally a FB at ωFB =−2J. We call δd(δFB) the detuning of
the atomic frequency ω0from the dispersive band (from the FB). (c)-(h) BS photonic wavefunction |ψBS⟩[see Eq. (4)] for
an atom coupled dispersively to the dispersive band [(c)-(e)] and to the FB [(f )-(h))] for decreasing values of δd(δFB). In
each case, we plot only the amplitude on sublattice an(the behaviour on sublattice bnis qualitatively similar) rescaled to the
maximum value and considered N= 100 unit cells under periodic boundary conditions with the atom coupled to cavity a50
with strength g= 0.001J.
along with the ensued photon-mediated interactions,
have a quite different nature compared to standard
photonic bands. While the fact that the BS remains
localized is somewhat reminiscent of the behavior in
a standard one-mode cavity, it is natural to wonder
what the BS localization length depends on. It will
turn out that this depends on the way in which the
emitter is coupled to the lattice as well as some intrin-
sic properties of the FB, in particular the orthogonal-
ity of the so called Compact Localized States (CLSs),
a key concept in FB theory. For this reason, the next
section is fully devoted to a review of CLSs, whose
main properties will be illustrated through variegated
examples. This will provide us with the necessary
theoretical basis to formulate general properties of
BSs in Section 5, including more general types of FBs
(e.g. occurring in 2D lattices).
4 Compact localized states
As for any band, the eigenspace made up by all pho-
tonic states where a single photon occupies a given
FB can be naturally spanned by the Bloch station-
ary states {|Ψk,FB⟩} with HB|Ψk,FB⟩=ωFB |Ψk,FB ⟩,
where kruns over the first Brillouin zone.
Accordingly, the projector onto the FB eigenspace
in the Bloch states basis {|Ψk,FB⟩} reads
PFB =X
k|Ψk,FB⟩⟨Ψk,FB|.(15)
By definition, when PFB is applied to a generic sin-
gle photon state it returns its projection onto the FB
eigenspace. Notice that, being translationally invari-
ant (see Bloch theorem [65]), each basis state |Ψk,FB⟩
is necessarily unbound, i.e. its wavefunction has sup-
port on the entire lattice. Yet, since photons in a FB
have zero group velocity [cf. Eq. (13)], it is natural to
expect the FB eigenspace to admit an alternative ba-
sis of localized (i.e. bound) states {|ϕn⟩}, one for each
unit cell indexed by n(when Bis a lattice). Such a
basis indeed exists and its elements are called Com-
pact Localised States (CLSs) [55] (an exception occurs
when the FB touches a dispersive band, a pathological
case that we will address later on). Note that a CLS
is compact in the sense that, typically, it is strictly
localized only on a finite, usually small, number of
neighbouring unit cells. There exists a general way to
express CLSs in the basis of Bloch states {|Ψk,FB⟩}
for a D-dimensional latttice, which reads [55]
|ϕn⟩=1
ND/2X
k∈BZ pf(k)e−ik·rn|Ψk,FB⟩,(16)
with Nthe number of cells and where f(k)≥0is a
suitable function (recall that rnis a Bravais lattice
identifying a unit cell).
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The idea behind CLSs is somewhat similar to stan-
dard Wannier states [66] in that a CLS can be ex-
pressed as a suitable superposition of unbound Bloch
states that yields a state localized around a lattice
cell. Unlike a Wannier state, however, a CLS is itself
an eigenstate of the bath Hamiltonian, i.e. HB|ϕn⟩=
ωFB |ϕn⟩, since the Bloch states entering the expan-
sion all have the same energy ωFB (reflecting the high
FB degeneracy).
4.1 Class Uand overlap parameters αd
Note that, while there is a one-to-one correspondence
between CLSs and unit cells, a CLS |ϕn⟩is generally
spread over U≥1cells, where the positive integer U
is the so called class of the CLS set [67]. In the special
case U= 1, each |ϕn⟩is entirely localized within the
nth cell so that the CLSs do not overlap in space and
are thus orthogonal, i.e. ⟨ϕn|ϕn′⟩=δnn′(an instance
is the double-comb lattice of Fig. 2). For U≥2, in-
stead, CLSs necessarily overlap in space with one an-
other, which remarkably causes them to be generally
non-orthogonal.
It is easy to understand that, for a 1D (generally
multipartite) lattice, CLSs of class U= 2 are such
that each CLS overlaps only its two nearest-neighbor
CLSs, being orthogonal to the all the remaining ones
(an instance is the sawtooth lattice of Fig. 1 which
we will discuss shortly). In such a case, the overlap
between a pair of CLSs thus reads
⟨ϕn|ϕn′⟩= 1 + α(δn,n′+1 +δn,n′−1),(17)
where α=⟨ϕn|ϕn±1⟩is an overlap parameter.
For 2D lattices, it is easier to think of class Uas
a 2D vector of integer components, describing how
many cells the CLS is spread over along either direc-
tion. For instance, if U= (2,2) the CLS will overlap
only its nearest-neighbors along each direction (an in-
stance being the checkerboard lattice of Fig. 5 to be
discussed later on). In this case, the overlap between
a pair of CLSs takes the form
⟨ϕn|ϕn′⟩= 1 + αx(δnx,n′
x+1 +δnx,n′
x−1)
+αy(δny,n′
y+1 +δny,n′
y−1),(18)
which now features two overlap parameters, αxand
αy, corresponding to the ˆexand ˆeydirections, respec-
tively. Likewise, for a generic D-dimensional model
with U= (2,2,...,2) (i.e., only nearest-neighbor
overlaps between CLSs along any ˆeddirection), the
overlap reads
⟨ϕn|ϕn′⟩= 1 +
D
X
d=1
αd(δn,n′+ˆ
ed+δn,n′−ˆ
ed),(19)
with αdthe overlap along the dth direction. Thus,
in general, there are D(generally) different overlap
parameters αd, as many as the spatial dimensions of
the lattice. A more general expression for CLSs of
class U > 2can be found in Appendix B, Eq. (62).
Notice that, owing to the degeneracy of the FB en-
ergy ωFB, the set of CLSs spanning the FB eigenspace
for a given lattice is not unique. We call minimal the
set of CLSs having the lowest class U. In the remain-
der, we will only consider minimal CLSs.
4.2 Instances of CLSs in 1D lattices
In order to make the reader familiarize with CLSs
and their properties (especially non-orthogonality),
we present next some examples of lattices exhibit-
ing FBs [see Fig. 2 and Fig. 3]. In this section, we
will only discuss 1D models, meaning that here the
cell index nis an integer and the wavevector ka real
number (see Appendix A for more details on those
models).
4.2.1 Double-comb lattice
The double-comb lattice is a tripartite lattice, which
comprises three sublattices a, b and c[see Fig. 2(a)].
Each cavity cnof the central sublattice is coupled to
the nearest-neighbour cavities cn±1with photon hop-
ping rate Jand with rate tto cavities anand bn(up-
per and lower sublattices, respectively). Cavities an
and bnhave the same bare frequency ωc, while the
one of cnis set to zero. The spectrum of B, which is
plotted in a representative case in Fig. 2(b), features
a FB at energy ωFB =ωcand two dispersive bands.
The occurrence of such FB is easy to predict since the
antisymmetric state
|ϕn⟩=1
√2|an⟩−|bn⟩(20)
clearly decouples from state |cn⟩(hence the rest of the
lattice) due to destructive interference and is thereby
an eigenstate of the bath Hamiltonian HBwith energy
ωc. Evidently, there exists one such state for each cell
n, explaining the origin of the FB at energy ωFB =ωc.
As each |ϕn⟩is strictly localized within a unit cell [see
Fig. 2(a)], states {|ϕn⟩} form a set of orthogonal CLSs
of class U= 1 based on the previous definition.
4.2.2 Sawtooth lattice
We already introduced the sawtooth lattice in Section
3and Fig. 1. Somewhat similarly to the double-comb
lattice, the FB at ωFB =−2Jalso arises through
destructive interference. Indeed, the superposition
|ϕn⟩=1
2|an⟩+|an−1⟩ − √2|bn⟩(21)
decouples from sites bn±1, hence the rest of the lattice.
Since we can build up one such state for each cell, the
set {|ϕn⟩} form a basis spanning the FB eigenspace.
Unlike the double-comb lattice, however, it is clear
that each state |ϕn⟩is not localized within a single
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Compact
Localised
State
Compact
Localised
State
(a) (c)
Double-comb lattice Stub lattice
Energy
(b)
(d)
Figure 2: Instances of one-dimensional lattices with FB. (a)
Double-comb lattice (the frequency of cavity cnis zero, while
the one of cavities anand bnis ωc). (b) Field spectrum of
the double-comb lattice showing a FB at energy ω=ωc. (c)
Stub lattice (the frequency of all cavities is set to zero). (d)
Field spectrum of the stub lattice, showing a zero-energy FB.
In (a) and (c) we highlight the primitive unit cell (red dashed
line) and possible choices of CLSs. Notice that CLSs are not
overlapping in the double-comb lattice in panel (a), but they
do overlap in the stub lattice (c).
unit cell and is not orthogonal to the two CLSs |ϕn±1⟩
[cf. Fig. 1(a)]. Indeed, it is easy to verify that Eq. (17)
holds in this case with α= 1/4, meaning that this set
is of class U= 2. Therefore, these CLSs form a non-
orthogonal basis of the FB eigenspace.
4.2.3 Stub lattice
The stub lattice (or 1D Lieb lattice) [63,68,69] is
the tripartite lattice sketched in Fig. 2(c), where each
cavity bnis coupled to cavities cn±1with rate Jand
side-coupled to cavity anwith rate J√∆where ∆≥0
is a dimensionless parameter. The spectrum of HBis
symmetric around ω= 0, at which energy a FB arises
(ωFB = 0). The gap separating the FB from each
dispersive band is proportional to J√∆[see Fig. 2(c)].
Similarly to the previous lattices and CLSs, the ori-
gin of the zero-energy FB can be understood by noting
that state
|ϕn⟩=1
√2+∆ |an⟩+|an+1⟩ − √∆|cn⟩(22)
decouples from the rest of the chain. Like in the saw-
tooth lattice, the non-orthogonal set {|ϕn⟩} form a
CLS of class U= 2 since Eq. (17)holds also in the
present case with
α=1
2+∆.(23)
Notice that ∆controls both the overlap between CLSs
and the energy gap between the FB and each dis-
persive band (such a tunability is not possible in the
sawtooth model). For ∆→0+we get α→1/2−and
zero band gap. For growing ∆, the non-orthogonality
parameter αgets smaller and smaller [indeed |ϕn⟩is
more and more localized around cavity cn, cf. (22)]
while the gap gets larger and larger.
Note that the stub lattice enjoys chiral symmetry
[70] which guarantees a zero-energy FB to exist [71]
even in the presence of disorder provided that it does
not break chiral symmetry 5.
4.2.4 1D Kagomè lattice
The 1D Kagomè model sketched in Fig. 3(a) is a lat-
tice with five sublattices, representing the 1D version
of the popular 2D Kagomè model [72]. In this model
each cavity is coupled to its nearest neighbour with
rate −J, except for pairs (an, bn) and (dn, en) which
are coupled with rate J. This system differs from
the previous instances in that there exists a FB (of
frequency 2J) which touches the edge of a dispersive
band [see Fig. 3(b)].
Even in this lattice, one can construct CLSs of class
U= 2 having the form [see Fig. 3(a)]
|ϕn⟩=1
√6h|cn⟩+|cn+1⟩−|an⟩−|bn⟩−|dn⟩−|en⟩i,
(24)
which fulfill Eq. (17)with α=⟨ϕn|ϕn±1⟩= 1/6.
5 Bound states in a FB: general prop-
erties
Armed with the notion of CLSs, we are now ready
to establish general properties of atom-photon BSs.
This and the following two sections contain the main
results of our work.
5.1 Atom-photon bound state as a superposi-
tion of compact localized states
We recall that the BS shape is generally dictated by
the field’s bare Green’s function [see Eq. (4)]. As in
the examples discussed in Section 3, here we consider
the regime where the emitter is dispersively coupled
to one specific band (not necessarily a FB), which
we will call ˜m, in such a way that the effects of all
the other bands can be neglected. This in particular
happens when, as in Section 3, there is a finite energy
gap between band ˜mand all the other bands (no band
crossing/touching). Another circumstance where this
regime holds (as we will see later on) is when the
contribution of band ˜mto the density of states (at
energies close to ω0) is dominant compared to that
from the other bands.
5This zero-energy FB always exists in lattices with chiral
symmetry and an odd number of sublattices (Lieb’s Theorem
[71]). This is because chiral energy imposes the energy spec-
trum to be symmetric around ω(k) = 0 for any k. As the
total number of energy bands is odd, one of the bands must
necessarily be a zero-energy FB.
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Cell index
BS amplitude
(c)
(a)
Compact
Localised
State
1D Kagome lattice
Energy
Band touching
(b)
Figure 3: (a) Structure of the 1D Kagomè model, featuring five sublattices with hopping rates ±J. A possible choice of CLS
is higlighted in green. (b) Energy spectrum featuring four dispersive bands and a FB of energy ωFB = 2J. The FB touches
the upper edge of one of the dispersive bands. (c) Bound state seeded by an atom in the dispersive regime δFB ≫gwith
δFB =ω0−2Jthe detuning from the FB, which we report in the legend (we show only the BS wavefunction on sublattice
a). We consider 100 cells with the emitter coupled to cavity a50. In contrast e.g. to the case in Fig. 1 (f)-(h), the localization
length here does depend on δFB (which is due to the band touching as explained later on in Section 6).
In such regime, we can approximate the BS wave-
function as [cf. Eqs. (4),(5)and (6)]
ψBS(x) = g⟨x|GB(ω0)|x0⟩ ≃ gX
k
⟨x|Ψk, ˜m⟩⟨Ψk, ˜m|x0⟩
ω0−ω˜m(k)
(25)
(notice that only the contribution of band ˜mis re-
tained).
Wavefunction ψBS(x), including its shape and
width, will generally depend on the emitter frequency
ω0, hence on its detuning from the ˜mth band edge.
However, if band ˜mis a FB, in Eq. (25)we can re-
place ω˜m(k)with ωFB, which is k-independent, and
obtain the BS wavefunction (here xlabels a generic
lattice site, even beyond 1D)
ψBS(x)≃g
ω0−ωFB ⟨x|PFB|x0⟩,(26)
where we substituted the projector PFB on the FB
eigenspace [see Eq. (15)]. Evidently, the shape and
width of the BSs become independent of the detuning
of the atom from the FB in agreement with the be-
haviour emerging from Fig. 1(f)-(h). Notice that this
independence holds also for an atom dispersively cou-
pled to a standard cavity in which case however the
BS is trivial. A typical case where Eq. (26)holds is
when g≪δFB ≪δd, where gis the coupling strength
while δFB and δdare respectively the emitter’s detun-
ing from the FB and the edge of the nearest disper-
sive band (see e.g. Section 3), where it is understood
that there exists a finite gap separating the FB from
the dispersive bands. Interestingly, Eq. (26)can hold
even if this gap vanishes, e.g. when the FB touches
the edge of a dispersive band, provided that the FB
contribution to the density of states dominates over
those due to dispersive bands: we will see an example
of such a case in Section 6.
Having assessed the independence of the detuning,
we next characterize the structure and spatial range
of the BS (which will then also be those of photon-
mediated interactions, see Section 2.3). For this aim,
taking advantage of Eq. (16), it is convenient to re-
express the FB projector (15)in terms of the CLSs
basis as (see Appendix B for details)
PFB =X
n,n′
ξnn′|ϕn′⟩⟨ϕn|(27)
with
ξnn′=1
NDX
k
1
f(k)eik·(rn−rn′).(28)
Notice that the scalar product between two CLSs de-
pends on function f(k)as [cf. Eq. (16)]
⟨ϕn|ϕn′⟩=1
NDX
k
f(k)eik·(rn−rn′).(29)
Remarkably, unlike the Bloch-states expansion of
Eq. (15), the decomposition of Eq. (27)in terms
of CLSs is non-diagonal. This is a consequence of
the CLSs’ non-orthogonality discussed in the previ-
ous section. Indeed, for ⟨ϕn|ϕn′⟩=δnn′(orthogo-
nal CLSs) we have f(k)=1and hence ξnn′=δnn′
in a way that Eq. (27)reduces to a standard diago-
nal expansion in the CLS basis. On the other hand,
provided that only nearest-neighbour CLSs are over-
lapping (as in all the examples of this work), f(k)is
generally given by
f(k) = 1 + 2
D
X
d=1
αdcos kd(30)
where Dis the lattice dimension, kdthe dth compo-
nent of wave vector kand αdthe overlap between
a pair of nearest-neighbor CLSs lying along the dth
direction (see Section 4.1). Expression (30)can be
easily derived by comparing Eq.(19)and Eq. (29). It
is clear that since f(k)≥0we have Pd|αd| ≤ 1/2.
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Finally, plugging Eq. (27)into Eq. (26)yields
ψBS(x)≃g
ω0−ωFB X
n
wn(x0)ϕn(x),(31)
with
wn(x0) = X
n′
ξnn′ϕ∗
n′(x0)(32)
which expresses the BS wavefunction as a weighted su-
perposition of CLSs ϕn(x) = ⟨x|ϕn⟩where the weight
function wn(x0)is defined by Eq. (32)and depends on
the cavity x0to which the emitter is coupled to. In
particular, the weight function wn(x0)is proportional
to ϕ∗
n′(x0) = ⟨ϕn′|x0⟩, which is non-zero only if cavity
x0overlaps the CLS |ϕn′⟩. As a consequence, being
the CLSs strictly compact, the weight function wn
[cf. Eq. (32)] involves only a finite number of terms.
Expansion (31)is a central result of this work. No-
tice that for non-overlapping CLSs (class U= 1) we
have wn=ϕ∗
n(x0)(since ξnn′=δnn′as we saw previ-
ously), which entails that in this case the BS just coin-
cides the CLS overlapping the cavity (if any). Hence,
for U= 1, atoms sitting in different cells will just
not interact. This situation is reminescent of cavity
QED, where CLSs play the role of non-overlapping
cavity modes: atoms in isolated cavities have no way
to cross-talk. In this sense, FBs of class U= 1 show
a cavity-like behaviour. In contrast, for U≥2, CLSs
do overlap each other in a way that now ξnn′=δnn′
and thus the BS is a superposition of more than one
CLS. We see that the interaction between atoms lo-
cated in different cells is now possible and this is a
consequence of the CLSs’ overlap. This situation is
quite different from standard cavity QED. In partic-
ular, the CLSs cannot be interpreted as overlapping
cavity modes since, if so, these would in fact couple
the cavities affecting their spectrum non-trivially (in
contrast to the present FB).
In the following, we consider the recurrent case
where only nearest-neighbor CLSs overlap, which
happens for U= 2 for 1D lattices and U= (2,2)
for 2D ones (cf. Section 4.1) and show that the BS
is exponentially localized by deriving the localization
length as an explicit general function of the CLSs’
overlap. We point out however that the present the-
ory is valid even when overlapping CLSs are not lim-
ited to nearest-neighbor ones, which is illustrated in
Appendix E with an example of 1D tripartite lattice
showing up a FB of class U= 3.
5.2 1D lattices, U= 2
In this case, Eq. (30)reduces to f(k) = 1 + 2αcos k,
with |α|=|⟨ϕn|ϕn+1⟩| ≤ 1/2. Plugging this into
Eq. (28)and working out the resulting integral in the
thermodynamic limit (N→ ∞), one ends up with
(see Appendix C for the proof )
ξnn′=(−sgn α)|n−n′|
√1−4α2exp−|n−n′|
λ1D ,(33)
0.00 0.25 0.50
0.1
1
10
Figure 4: Universal scaling of the BS localization length λBS
as a function of the CLSs overlap |α|for 1D and 2D lat-
tices based on Eqs. (34)and (36), respectively (in the com-
mon case that only nearest-neighbour CLSs overlap). Here,
|α|= 1/2and |α|= 1/4are the maximum allowed values
of |α|, respectively in 1D and 2D, consistent with the hy-
pothesis that only nearest-neighbour CLSs overlap. In 2D,
the behaviour is the sum of two decaying exponentials with
localization lengths λ2D and λ′
2D [cf. Eq. (36)].
where sgn αdenotes the sign of αwhile
1
λ1D
=settsech(2|α|),(34)
being settsech(x)the inverse function of the hyper-
bolic secant. Therefore, ξnn′decays exponentially
with |n−n′|with a characteristic length λ1D which
is a growing function of the non-orthogonality coeffi-
cient α[see Fig. 4(b) showing that λ1D vanishes for
α= 0 and diverges for |α| → 1/2]. Based on a re-
sult derived in Appendix D, we get that the BS has
the same exponential shape and localization length as
ξnn′. For instance, in the case of the sawtooth lattice
we have α= 1/4, which replaced in Eq. (33)yields
λBS ≃0.759 in perfect agreement with the scaling
observed in Fig. 1(f-h).
5.3 2D lattices, U= (2,2)
For a 2D lattice and a square geometry (we focus on
this case as all our 2D examples have such structure),
Eq. (30)is given by f(kx, ky) = 1 + 2αxcos kx+
2αycos ky. For the sake of argument, we focus on
the isotropic case αx=αy=α, (a generalization to
αx=αyis straightforward but expressions get in-
volved). One can then show that (see Appendix C)
ξnn′=Aexp−|rn−rn′|
λ2D +Bexp−|rn−rn′|
λ′
2D ,
(35)
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where A, B are constants while
1
λ2D
=settsech
2α
1−2α
,1
λ′
2D
=settsech
2α
1+2α
(36)
(αis subject to the constraint |α| ≤ 1/4). Thus, in
this case the decay results from the superposition of
two exponential functions with localization length λ2D
and λ′
2D, respectively. As in 1D, both localization
lengths grow with |α|(see Fig. 4(b)]. Differently from
1D, however, when αapproaches its maximum value
1/4 only λ2D diverges while λ′
2D instead saturates to
the finite value λ′
2D(α→1/4) ≃0.567 (for α→ −1/4,
λ2D saturates to the same value while λ′
2D diverges).
This means that in this limiting case the weight
function ξnn′is a single exponential having localiza-
tion length equal to λ≃0.567. In 2D, thereby,
photon-mediated interactions are finite-ranged pro-
vided that the CLSs have non-zero overlap.
6 Flatband touching a dispersive band
The conclusions developed so far apply to lattices fea-
turing an energetically isolated FB, i.e. separated by a
finite gap from all the remaining bands [recall Eq. (26)
where only the contribution of the FB is retained].
This rules out in particular those lattices where a FB
arises on the edge of a dispersive band. In 1D, such
band touching happens for example in the stub lat-
tice for ∆=0(see Section 4.2.3 and Fig. 2) and in the
Kagomè model (see Section A.4 and Fig. 3). Indeed,
it turns out that in this case, an atom dispersively
coupled to the FB seeds a BS with features analo-
gous to typical BSs close to the band edge of an iso-
lated dispersive band [e.g. as in Fig. 1 (c)-(e)]. This is
witnessed by Fig. 3(c) for the Kagomè model, which
shows that the BS localization length gets larger and
larger as the detuning from the FB decreases in con-
trast to the saturation behavior next to an isolated FB
that occurs e.g. in Fig. 1 (f )-(h). Notice that in these
1D examples the dispersive band scales quadratically
in the vicinity of the FB.
FBs falling on the edge of dispersive bands can also
occur in 2D. A paradigmatic instance is the checker-
board lattice in Fig. 5(a). This a bipartite lattice
where nearest-neighbour cavities of the same sublat-
tice are coupled with hopping rate −J, while the hop-
ping rate between nearest-neighbour cavities of differ-
ent sublattices is −Jalong one diagonal and Jalong
the other diagonal. Each cavity has frequency 2J.
The spectrum features a FB and a dispersive band
according to [see also Fig. 5(b)]
ωFB = 0, ωd(kx, ky)=2J[2 −cos(kx)−cos(ky)].
(37)
The two bands touch one another at the Γpoint kx=
ky= 0 (close to this ωd(kx, ky)has a parabolic shape).
Compact
Localised
State
Non-contractible
Loop State
(a)
Non-contractible
Loop State
Flat ba
nd
Energy
(b)
Band touching
Figure 5: (a) Structure of the checkerboard lattice and the
set of CLSs. For completeness, we also display the pair of
non-contractible loop states which must be added to the
CLSs to form a complete basis of the FB subspace (see foot-
note in the main text). (b) Spectrum of the 2D checkerboard
model, showing a band touching (green dot) between the dis-
persive band (blue) and the FB (orange) at zero energy.
A set of non-orthogonal CLSs, one for each unit cell,
is easily identified as [see Fig. 5(b)]
|ϕn⟩=1
2hanx,ny−anx−1,ny+bnx,ny−bnx,ny+1 i,
(38)
where n= (nx, ny)labels unit cells 6. We are thus in
the case U= (2,2) with the overlap between nearest-
neighbour CLSs being given by αx=αy=α= 1/4.
Now, remarkably, we numerically checked that an
atom dispersively coupled to the FB at ωFB = 0 gives
rise to a BS whose features accurately match those
predicted in Section 5.3 for α= 1/4, meaning that lo-
calization length is independent of the detuning and
has the finite value λBS ≃0.567 [recall the discussion
after Eq. (36)]. In other words, we find that the be-
haviour is identical to that of an energetically isolated
FB.
The above case studies can be understood by recall-
ing that a dispersive band that scales quadratically in
the vicinity of a band edge gives a contribution to the
density of states (DOS) which is finite in 2D while it
diverges in 1D (in the latter case a van Hove singu-
larity occurs). In contrast, the DOS of a FB clearly
has δ-like diverging at ωFB. It follows that in 2D the
dispersive band can be neglected compared to the FB,
6Unlike previous examples of overlapping CLSs, the set (38)
is not a complete basis of the FB eigenspace, which can be
shown to be due to the singular behaviour of the present FB
[55]. To get a complete basis, the set (38) must be comple-
mented with a pair of so called non-contractible loop states
[see Fig. 5(b)]. These two states are yet unbound, hence in the
thermodynamic limit we can neglect their contribution to the
BS (this arising from the local atom-field interaction).
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explaining why in this case the properties of the BS
match those predicted for an isolated FB in Section
5.3. In 1D, instead, the contribution of the disper-
sive band can no longer be neglected, explaining why
e.g. in the Kagomè lattice the BS behaviour is differ-
ent from the one in Section 5.2.
7 Giant atoms coupled to a photonic
flat band
In recent years, it has become experimentally possi-
ble fabricating so called giant atoms [73], i.e. quan-
tum emitters which are coupled non-locally to the
field through a discrete set of coupling points (where
a standard atom is retrieved in the special case of only
one coupling point). For a giant atom with Ncou-
pling points the total Hamiltonian (1)is generalized
as
H=ω0σ†σ+HB+N
X
ℓ=1 gℓa†
xℓσ+ H.c.,(39)
where xℓ(coupling point) labels the cavity to which
the giant atom is coupled to and gℓthe corresponding
(generally complex) coupling strength. It is conve-
nient to define the field ladder operator [40]
aχ=N
X
ℓ=1
γ∗
ℓaxℓwith γℓ=gℓ
¯gand ¯g=sX
ℓ|gℓ|2,
(40)
where, due to PN
ℓ=1 |γℓ|2= 1,aχfulfills [aχ, a†
χ] = 1.
With this definition, (39)can be arranged formal ly as
the Hamiltonian in the presence of a normal atom
H=HB+ω0σ†σ+ ¯ga†
χσ+ H.c.(41)
(notice however that aχgenerally does not commute
with field operators a†
x, this being a signature of the
non-local nature of atom-photon coupling). Let us
define the single-photon state
|χ⟩=a†
χ|vac⟩,(42)
which for convenience in the remainder we will call site
state since it can be formally thought to be associated
with a fictitious location of the atom.
7.1 Atom-photon bound state and Heff
It can be straightforwardly shown [40] that the the-
ory of atom-photon BSs and effective Hamiltonians,
which we reviewed in Sections 2.2 and 2.3 is naturally
generalized to the case of giant atoms once the atom
location (previously called |x0⟩with only one atom)
is replaced with the site state (42). In particular, the
BS of one atom in the dispersive regime now reads
|ψBS⟩=GB(ω0)|χ⟩[cf. Eqs. (4)and (6)].
Interestingly, in the usual regime where the atom
is dispersively coupled to an isolated FB, the pos-
sibility to engineer the giant atom’s coupling points
and relative strengths allows for the BS to have just
the same shape as a CLS. Indeed, if |ϕn⟩is a CLS,
then clearly GB(ω0)|ϕn⟩=|ϕn⟩/(ω0−ωFB)(since
HB|ϕn⟩=ωFB |ϕn⟩). Accordingly, if we choose the
emitter’s coupling points such that
|χ⟩=|ϕn⟩(43)
(for some given n) then [see Eq. (6)for |x0⟩→|χ⟩]
|ψBS⟩=g
ω0−ωFB |ϕn⟩.(44)
This shows that the BS has just the same wavefunc-
tion as a CLS of the FB. In the case of many gi-
ant atoms labeled by nand each such that |χn⟩=
|ϕn⟩, the photon-mediated interaction strength is then
given by [see Eq.(11)for |xi⟩→|χi⟩]
Knn′=g2
ω0−ωFB ⟨χn|χn′⟩.(45)
An interesting consequence of this is that for nearest-
neighbour CLSs [e.g. in the sawtooth model of
Fig. 1(a) or the stub lattice of Fig. 2(c)] an effec-
tive spin Hamiltonian arises with strictly nearest-
neighbour interactions. In the stub lattice, in par-
ticular, we get [cf. Eq. (23)]
Knn′=g2
ω0−ωFB
1
2+∆,(46)
hence the interaction strength can be in principle
modulated through parameter ∆(provided that this
remains large enough to neglect the effect of the dis-
persive bands).
Taking advantage of the compact nature of CLSs,
this type of dipole-dipole interactions can be imple-
mented in 1D in a relatively straightforward fashion
by using giant atoms with very few coupling points
(only three in the sawtooth and stub lattices).
7.2 General case
The above is immediately generalized to the case that
the site state is an arbitrary superposition of CLSs,
i.e. when Eq. (43)is replaced by
|χ⟩=X
n
cn|ϕn⟩,(47)
which through an analogous argument leads to [see
Eq. (44)]
|ψBS⟩=g
ω0−ωFB X
n
cn|ϕn⟩.(48)
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8 Conclusions
In this work, we carried out a general study of dipole-
dipole interactions between quantum emitters disper-
sively coupled to photonic flat bands (FBs), i.e. bands
with flat dispersion law.
In line with standard theory of dipole-dipole dis-
persive interactions, the spatial shape of such interac-
tions, hence the interaction range, is inherited from
the wavefunction of the atom-photon bound state
(BS) arising when the emitter is off-resonant with the
band. In the case of a FB, such localization length can
saturate to a finite value in a way that the BS extends
even beyond the cell where the emitter sits, thus en-
abling dipole-dipole interactions between atoms cou-
pled to different cells. We showed on a general basis
that this type of atom-photon BS can be connected
with so called compact localized states (CLSs), a key
concept in the theory of FBs. Remarkably, we showed
that for 1D lattices with nearest-neighbour overlap-
ping CLSs the localization length (hence the dipole-
dipole interaction range) monotonically grows with
the overlap between CLSs, according to a universal
law which we derived explicitly, until diverging when
the overlap tends to its maximum value. This sug-
gests that the cross-talk between emitters placed in
different cells is enabled by the CLSs overlap. An
analogous task was carried out in 2D square lattices,
which showed that, unlike 1D, the BS localization now
converges to a finite value as the CLSs overlap ap-
proaches its maximum value. We also considered the
singular situation that a FB, instead of being ener-
getically isolated, touches a dispersive band. In such
case, we showed that the BS behaves like in the pres-
ence of typical dispersive bands in 1D (localization
length diverging with the detuning) while in 2D it be-
haves like in the presence of an isolated FB. Finally,
we considered the effect of replacing the emitter with
a so called giant atom which can couple non-locally to
the lattice to a manifold of distinct cavities, showing
that in this case one can engineer the coupling points
in a way that the BS wavefunction has just the same
shape as a CLS or a linear combination of a few of
these.
Unavoidable occurrence of disorder in real setups
could affect some of the phenomena predicted here,
which is briefly discussed in Appendix F in some case
studies. Yet, it is reasonable to expect that recent
advancements in platforms such as cold atoms [74]
and circuit QED [10] allow for devices that are clean
enough to see FB-induced effects, as recently demon-
strated in Ref. [75].
Within the topical framework of atom-photon in-
teractions in unconventional photonic environments,
which takes advantage of current experimental ca-
pabilities enabling the fabrication of emitters cou-
pled to engineered baths, our work introduces a new
paradigm of dipole-dipole interactions. Additionally,
it establishes a new link with the hot research area
in condensed matter and photonics investigating flat
bands [48,49].
Acknowledgements
We are grateful to D. De Bernardis, L. Leonforte,
P. Scarlino, V. Jouanny and L. Peyruchat for fruit-
ful discussions and A. Miragliotta for reading the
manuscript. E.D.B. acknowledge support from the
Erasmus project during his stay at IFF-CSIC. E.D.B.
and F.C. acknowledge support from European Union
– Next Generation EU through Project Eurostart
2022 “Topological atom-photon interactions for quan-
tum technologies" (MUR D.M. 737/2021) and
through Project PRIN 2022-PNRR no. P202253RLY
“Harnessing topological phases for quantum technolo-
gies”. A.G.T. acknowledges support from the CSIC
Research Platform on Quantum Technologies PTI-
001 and from Spanish projects PID2021-127968NB-
I00 funded by MICIU/AEI/10.13039/501100011033/
and by FEDER Una manera de hacer Eu-
ropa, and TED2021-130552B-C22 funded by MI-
CIU/AEI /10.13039/501100011033 and by the Eu-
ropean Union NextGenerationEU/ PRTR, respec-
tively, and QUANTERA project MOLAR with
reference PCI2024-153449 and funded by MI-
CIU/AEI/10.13039/501100011033 and the European
Union.
A More analytical details on 1D mod-
els with flat bands
In this appendix, we provide more details on 1D
models exhibiting FBs discussed in the main text.
Due their 1D nature, for simplicity we call simply n
the Bravais lattices identifying the unit cells. Based
on the Bloch theorem, one first define traslationally-
invariant ladder operators as
αk,ν =1
√NX
n
e−ikn an,ν (49)
with ν= 1, ..., Q labeling the sublattices and krun-
ning over the first Brillouin zone. In terms of these,
the bare bath Hamiltonian can be arranged in the
form
HB=X
kα†
k,1. . . α†
k,QHk


αk,1
.
.
.
αk,Q


,(50)
where Hkis a Q×QHermitian matrix (Bloch Hamil-
tonian) representing the Hamiltonian in the momen-
tum space. Diagonalizing Hkyields the dispersion
laws of all bands (indexed by m) and the correspond-
ing normal modes.
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A.1 Sawtooth lattice
The Bloch Hamiltonian is the 2×2matrix
Hk=J0√2(1 + e−ik)
√2(1 + eik)−2 cos (k),(51)
which is readily diagonalised, resulting in the two dis-
persion laws in Eq. (14)and in single-photon eigen-
states (each describing a photon populating a normal
mode)
|ΨFB,k⟩ ∝ h(1 + e−ik )|ak⟩ − √2|bk⟩i,(52)
|Ψd,k⟩ ∝  1 + e−ik
1 + cos (k)|ak⟩+√2|bk⟩(53)
(we omit normalization factors as intended by sym-
bol ∝). Here, |ak⟩and |bk⟩are single-photon states
corresponding to (49), where aand bare the sublat-
tice indexes defined in Fig. 1(a). In particular, notice
that each |ΨFB, k⟩form an orthogonal basis of un-
bound states spanning the FB eigenspace, which is
alternative to the CLS basis of Eq. (21).
A.2 Double-comb lattice
The Bloch Hamiltonian in this case reads
Hk=

ωc0t
0ωct
t t 2Jcos k
,(54)
which is easily diagonalized. The eigenvalues embody
the dispersion laws of the three bands and read
ωFB =ωc,
ω±(k) = ωc
2−Jcos k±rt2
2+ωc
2+Jcos k2.
A.3 Stub lattice
For the three-partite stub lattice [Hamiltonian param-
eters and sublattice indexes are defined in Fig. 2(c)],
the Bloch Hamiltonian can be cast in the form
Hk=J

0√∆ 0
√∆ 0 1 + e−ik
0 1 + eik 0
(55)
which, upon diagonalisation, yields the band disper-
sion laws
ωFB = 0, ω±(k) = ±Jp∆ + 2(1 + cos k),(56)
so that ∆in fact measures the gap between the zero-
energy FB and either dispersive band. In particular,
the FB eigenstates are worked out as
|ΨFB,k⟩ ∝ h(1 + eik )|ak⟩ − √∆|ck⟩i.(57)
and form an orthogonal basis of unbound states alter-
native to the non-orthogonal basis of CLSs (22).
A.4 1D Kagomè lattice
This five-partite lattice represents a sort of 1D coun-
terpart of the Kagomè model [67], where the Hamil-
tonian parameters and sublattice indexes are defined
in Fig. 3(c). The Bloch Hamiltonian Hkis calculated
as the 5×5matrix given by
Hk=





0 1 −e−ik −e−ik 0 0
1−eik 0−1 0 0
−eik −1 0 −1−eik
0 0 −1 0 1 −eik
0 0 −e−ik 1−e−ik 0






,
(58)
giving rise to five bands, one of which is a FB of fre-
quency ωFB = 2Jtouching with the upper edge of a
dispersive band.
B FB projector in the CLS basis
To derive Eq. (27)we need to express projector (15)in
the CLSs basis which requires the expansion of each
FB Bloch eigenstate in the CLS basis
|Ψk,FB⟩=1
ND/2X
n
ak,n |ϕn⟩.(59)
To determine the expansion coefficients ak,n, we ex-
press Eq. (59)in matrix-vector form as
vk=M·ak,(60)
where ak,n is the N-dimensional column vector hav-
ing coefficients ak,n as components while vkand M
are respectively the N-dimensional column vector and
N×Nmatrix having entries [we use Eq. (27)]
vk,n =⟨ϕn|Ψk,FB⟩=pf(k)eik·rn, Mnn′=⟨ϕn|ϕn′⟩.
(61)
Now, owing to translation symmetry, the overlap ma-
trix Mnn′(which is positive definite) can be written
as
Mnn′=δnn′+
D
X
d=1
α1,d(δn,n′+ˆ
ed+δn,n′−ˆ
ed)+
+
D
X
d=1
α2,d(δn,n′+2ˆ
ed+δn,n′−2ˆ
ed) + . . . ,
(62)
where we sum over nearest-neighboring cavities, next-
to nearest and so on to cavity nalong the dth direction
and accordingly α1,d (α2,d) is the overlap along said
direction. This choice also specifies the shape of func-
tion f, according to the equality between Eqs. (29)
and (62), which holds if
f(k) = 1+2 X
d
α1,d cos(kd)+2 X
d
α2,d cos(2kd)+. . . ,
(63)
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Poles
Poles
(a) (b)
Figure 6: Poles of the integrand of ξnn′in the 1D and 2D
cases [respectively panels (a) and (b)] plotted as a function of
the CLS overlap α(different colours correspond to different
poles). When one of plotted functions takes values within the
region highlighted in red, the corresponding pole falls within
the the unit circle, i.e. inside the integration contour of the
complex integral. In (b), notice that for |α|= 1/4, one of
the two poles stays well inside the unit circle, corresponding
to a BS with finite localization length.
where kddenotes the dth component of the kvector.
We also see that these are basically the eigenvalues of
M(due to translation symmetry).
Inverting Eq. (60)we get ak=M−1·vkand ex-
ploiting again translation symmetry we obtain
|Ψk,FB⟩=1
ND/2
1
pf(k)X
n
eik·rn|ϕn⟩.(64)
By plugging this into Eq. (15)we finally end up with
Eq. (27).
C Explicit form of function ξnn′
Here, we show the derivation of the explicit form of
function ξnn′which was given in the main text.
C.1 1D case
For 1D lattices and CLSs of class U= 2 (only nearest-
neighbour CLSs overlap), the scalar product between
any pair of CLSs has the form ⟨ϕn|ϕn′⟩=δn,n′+
α δn,n′+1 +α δn,n′−1. Hence, function f(k)takes the
form [cf. Eq. (30)]
f(k) = 1 + 2αcos k, (65)
with |α| ≤ 1/2. Replacing this in Eq.(28) for D= 1
yields
ξnn′=1
NX
k
eik(n−n′)
1+2αcos k,(66)
where, since we are working in 1D, we made the re-
placement xn=n. Based on periodic boundary con-
ditions, ktakes Nvalues within the first Brillouin
zone [−π, π]with uniform spacing ∆k= 2π/N. Ac-
cordingly, we can rewrite ξnn′as
ξnn′=1
2πX
k
∆keik(n−n′)
1+2αcos k.(67)
In the thermodynamic limit N→ ∞,kbecomes a
continuous variable and the sum is turned into a con-
tinuous integral
ξnn′=1
2πZπ
−π
dkeik |n−n′|
1+2αcos k.(68)
Notice that in the exponent we replaced n−n′with
its modulus since the integral is invariant under the
substitution n−n′→n′−n.
Applying a standard technique [61,62], (68)can be
expressed as integral on the unit circle on the complex
plane upon the change of variable z=eik (since −π <
k≤π,zlies on the unit circle of the complex plane).
Accordingly, dz=ieikdk=izdkand cos k= (eik +
e−ik)/2=(z+z−1)/2=(z2+ 1)/(2z). This yields
ξnm =1
2πi Idzz|n−n′|
αz2+z+α,(69)
This expression can be computed with the help of the
residue theorem by noting that, since |α|<1/2, the
integrand function always has a pole falling inside the
unit circle at z0=−1
2α1−√1−4α2[see blue curve
in Fig. 6(a)]. Calculating the corresponding residue
we thus end up with
ξnn′=(−sgn α)|n−n′|
√1−4α2e−|n−n′|
λ1D ,(70)
with the localization length λ1D given by Eq. (33).
C.2 2D case
In 2D and for a square geometry, (63)reduces to
f(kx, ky) = 1 + 2αxcos(kx)+2αycos(ky),(71)
with αx=α1,1,αy=α1,2(according to notation
introduced in Eq. (62)) and subject to the constraint
|αx|+|αy|<1/2. Plugging into Eq. (28)and taking
the thermodynamic limit as done previously in the 1D
case produces the double integral
ξnn′=1
(2π)2ZZBZ
dkxdky
eikx(n−n′)eiky(m−m′)
1+2αxcos(kx)+2αycos(ky),
(72)
where BZ stands for the region of integration specified
by −π≤kx≤πand −π≤ky≤π. To evaluate
the integral, we first take m=m′(i.e. we look at the
behaviour of ξnn′along the ˆexdirection) which allows
to write the integral as
ξnn′=1
2πZπ
−π
dkxeikx(n−n′)
×1
2πZπ
−π
dky
1
1+2αxcos(kx)+2αycos(ky),
(73)
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which can be solved by applying the residue theorem
twice, first to the integral over kyand then to that
over kx. There occur four poles given by
z1,±=x+
2±qx2
+−4
2, z2,±=x−
2±qx2
−−4
2(74)
with
x±=−1±2 sgn (αx)|αy|
|αy|.(75)
We henceforth focus on the isotropic case αx=αy=
αwith |α| ≤ 1/4, which ensures that the localization
length calculated along the x-direction will be match
the one along any other direction. In this isotropic
case, the only two poles falling inside the unit circle
are given by
z1,+=2α−1
2|α|+√1−4α
2|α|,
z2,+=−2α+ 1
2|α|+√1+4α
2|α|.
(76)
Calculating the corresponding residues and summing
up as prescribed by the residue theorem we end up
with Eq. (35).
D BS localization length from ξnn′
In this appendix, we show that the BS scales with
distance in the same way as function ξnn′defined in
Eq. (28). For the sake of argument, we will focus on
CLSs of class U= 2 in 1D (the 2D case for = (2,2) is
treated analogously on each of the two directions).
Let |n, ν⟩the state where a single photon lies on
the nth unit cell in the νth sublattice. Since we as-
sume U= 2, the CLS is localized only on two nearest-
neighbor cells, say nand n+1, so that its wavefunction
can be written as
|ϕn⟩=X
ν
c1,ν |n, ν⟩+c2,ν |n−1, ν ⟩,(77)
where ci,ν are the (in general complex) coefficients of
the CLS in front of state |n, ν⟩, which do not depend
on ndue to translational invariance. If the atom is
coupled to cavity (n0, ν0), the weight function wn(x0)
Eq. (32)will take thus the form
wn(x0) = c∗
0,ν0ξnn0+c∗
1,ν0ξnn0+1.(78)
Thus, owing to Eq. (31), the shape of the bound state
will be given by
|ψBS⟩ ∝ X
nν hc0,ν (c∗
0,ν0ξnn0+c∗
1,ν0ξnn0+1)|n, ν ⟩
+c1,ν (c∗
0,ν0ξnn0+c∗
1,ν0ξnn0+1)|n−1, ν ⟩i,
(79)
which, upon shifting n−1→nin the second term
can be turned into
|ψBS⟩ ∝ X
nν h(c0,ν c∗
0,ν0+c1,ν c∗
1,ν0)ξnn0
+c0,ν c∗
1,ν0ξnn0+1 +c1,ν c∗
0,ν0ξnn0−1i|n, ν⟩
=X
nν
Anν |n, ν⟩.
(80)
Now, if ξnn0∝expn−|n−n0|
λo(exponential scaling),
evidently coefficient Anν will also have the same ex-
ponential scaling with n. This shows that in general
that the BS wavefunction scales exponentially with
the same localization length as function ξnn′. An
analogous result holds if ξnn′is the sum of a finite
number of exponentially localised terms.
The proof can be extended to CLSs of arbitrary
class Uas long as the weight function ξnn′scales as the
sum of a finite number of exponentials. Notice that
in the case of a complex pole zin the computation of
ξnn′, the associated localization length will be given
by λ−1= log |z|.
BS population
Cell number
Figure 7: Population of the atom-photon bound state on sub-
lattice aof the lattice in Fig. 3(d) of Ref. [67] (the behavior
is similar on other sublattices). There is a good agreement
between the BS as obtained from exact diagonalization (blue
line) and the one predicted by our theory (orange line).
E Flatbands of class U= 3
In the main text, we on FBs where only nearest-
neighbor CLSs overlap, i.e. U= 2 for 1D lattices, due
to their experimental feasibility in state-of-the-art ex-
perimental platforms. Nonetheless, our formalism can
be naturally extended also to more general cases, such
as U > 2in 1D.
To provide evidence of this, in this Appendix we
show how out general theory can be extended to one-
dimensional lattices of class U= 3. In this case,
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function f(k)is written as f(k) = 1 + 2α1,x cos k+
2α2,x cos 2k, where α1,x =⟨ϕn|ϕn±1⟩and α2,x =
⟨ϕn|ϕn±2⟩[see Eq. (62)for notation]. Thus, ξnn′can
be written in the thermodynamic limit as (see Ap-
pendix C)
ξnn′=1
2πZπ
−π
dkeik|n−n′|
1+2α1,x cos k+ 2α2,x cos 2k(81)
which, upon the substitution z=eik can be turned
into a complex integral on the unit circle reading
ξnn′=1
2πi Idzz|n−n′|+1
α2,xz4+α1,x z3+z2+α1,xz+α2,x
.
(82)
Since its denominator is a symmetric quartic poly-
nomial, the poles of the integrand function can be
analytically worked out as
z±
1=−β+±q2α1,xβ+−8α2
2,x −4α2,x
4α2,x
,(83)
z±
2=−β−±q2α1,xβ−−8α2
2,x −4α2,x
4α2,x
,(84)
where β±=α1,x ±qα2
1,x + 8α2
2,x −4α2,x. These are
two pairs of complex and conjugate numbers, meaning
that 0,2or 4poles will exists within the unit circle,
depending on the overlap parameters α1,x and α2,x.
This ensures that (82)is always real.
To test the above, we considered the three-partite
one-dimensional photonic lattice considered in Fig.
3(d) of Ref. [67]. Such lattice consists of three sub-
lattices, namely a, b, c, described by the Hamiltonian
HB/J =X
na†
nbn+b†
ncn+ H.c.
+X
nha†
n(−0.523 an+1 + 0.17 bn+1 + 0.693 cn+1)
+b†
n(−0.627 an+1 −0.115 bn+1 + 0.512 cn+1)
+c†
n(−0.731 an+1 −0.399 bn+1 + 0.332 cn+1)+H.c.i
(85)
where Jis an energy constant. In this lattice, a U= 3
FB appears at energy ωFB =−1.5J, whose CLSs read
|ϕn⟩∝|an⟩−|bn⟩+|cn⟩+ 0.255 |an+1 ⟩
+ 0.286 |bn+1⟩ − 0.599 |cn+1 ⟩+ 0.25 |an+2⟩
−0.5|bn+2⟩+ 0.25 |cn+2 ⟩.
(86)
By coupling an atom dispersively to this FB, we
checked that the atom-photon BS agrees with our the-
ory, which is shown in Fig. 7. Interestingly, such BSs
has a non-monotonic spatial shape.
F Effect of disorder
Inevitable occurrence of disorder in real experimen-
tal setups can affect FBs [76,77], hence it is natu-
ral to ask how the predicted effects in this work are
affected by disorder in the photonic lattice. While
a comprehensive study of the effects of disorder is
beyond the scope of this work, in this Appendix
we discuss briefly some case studies in 1D in the
presence of disorder on the cavity frequencies (di-
agonal disorder) and on the photon hopping rates
(off-diagonal disorder). For diagonal disorder, we
change the photonic bath Hamiltonian as HB→
HB+Pxωx(η)a†
xax, while for off-diagonal disorder
as HB→HB+Px=x′Jxx′(η)(a†
x′ax+ H.c.). Here,
ωx(η)and Jxx′(η)are random values drawn from a
uniform probability distribution in the interval [−η, η]
(in units of J). We will address separately FBs with
orthogonal and non-orthogonal Compact Localised
States (CLSs) in the case studies of the double-comb
lattice and stub lattice, respectively.
In line with the manuscript, as a case study of flat-
bands with orthogonal CLSs we consider the double-
comb lattice (see Fig. 2a-b), whose relevant parame-
ters (without disorder) are the cavity frequencies ωc
and hopping rates Jand t(see also Appendix A.2).
We focus on the representative case J= 1, ωc=
4J, t = 3J,ω0= 3.8J, g = 0.01J. With this choice of
parameters, a FB arises with energy ωFB =ωc, which
is well-separated from the dispersive bands (allowing
us to discern effects due to disorder on the FB from
the standard Anderson localization in the dispersive
bands). Diagonal disorder is simulated by introduc-
ing random cavity frequencies for sublattice b, while
off-diagonal disorder concerns the two hopping rates
Jand t. In the case of an off-diagonal disorder in
J, due to the peculiar shape of the CLSs in this sys-
tem (whose amplitudes vanish on the b-sublattice),
neither the FB nor the shape of the CLSs is affected
by adding disorder on J. As such, dipole-dipole in-
teractions mediated by the FB are unchanged. On
the other hand, in the case of off-diagonal disorder
on t, by calling tn
bc (tn
ac) the local hopping rate be-
tween cavities bn(an) and cn, it is immediately seen
(through the usual destructive interference argument)
that any state |ϕn⟩ ∝ tn
bc |an⟩−tn
ac |cn⟩is a CLS with
energy ωFB =ωc. Therefore, despite the disorder,
the FB still exists at the same frequency where each
CLS now features slightly different amplitudes on the
two sublattices. Accordingly, short-ranged photon-
mediated interactions between emitters still arise in
this case with an interaction strength that will not
depend on the local values of tn
bc and tn
ac. Finally, in
the case of diagonal disorder, as expected, the flat-
ness of the FB is spoiled, which however does not
cause a qualitative significant change of the predicted
effects. In particular, we find that if η≲gthe atom-
atom interaction strength is preserved, while if η > g
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the interaction strength can be enhanced or lowered,
depending on specific realization of disorder. The
interaction range is preserved in both cases. This
striking behaviour somehow is not consistent with the
localization-delocalization transition in FBs [76] due
to disorder, i.e., the fact that FB eigenstates are delo-
calized with weak disorder and localized with strong
disorder (standard Anderson localization is recovered
in the latter case). We conjecture that no significant
effects arise from this transition as long as there exists
a relatively large energy gap between the FB and the
dispersive bands (as is the case here given the consid-
ered choice of parameters). Arguably, an enhanced
interaction range might arise with weak disorder if
the gap were smaller, which however might affect the
validity of the single FB approximation (holding in
the regime g≪δFB ≪δd).
We next address non-orthogonal CLSs in the case
study of the stub lattice [see Fig. 2(c-(d)], which is a
tripartite lattice where cavities bnand cnare coupled
to each other with hopping equal to Jand cavities
bnare side-coupled to cavities anwith strength J√∆
(∆being a dimensionless parameter. Recall that the
energy gap between the FB and each dispersive band
is measured by J√∆. Again, we study the effect of
diagonal and off-diagonal disorder. We recall that in
this lattice the existence of a zero-energy FB is guar-
anteed by chiral symmetry [see Appendix A.3]. For
our simulations, we set J= 1,∆=1.5,ω0= 0.1J
and g= 0.01J. In the case of off-diagonal disorder on
J, where we introduce disorder on both the an-bnand
bn-cnhopping rates, chiral symmetry is not broken,
hence the zero-energy FB is unaffected. However, the
shape of CLSs (which are non-orthogonal) is changed
by the disorder and so are the resulting atom-photon
bound states and mediated interactions. As is rea-
sonable to expect, such changes are weak for η≲g,
hence the localization length and interaction strength
are unaffected in this regime. Instead, in the regime
η≳J√∆, the localization length gets shorter and the
interaction strength weaker, which can be attributed
to the standard effect of Anderson localization. In
the intermediate regime, i.e., when g < η < J √∆we
observe signatures of the localization-delocalization
transition, entailing an increase in the localization
length of the bound states due to delocalization of
the CLSs, which results in the increase of the inter-
action strength between emitters coupled to distant
unit cells. On the other hand, introducing diagonal
disorder on all cavities, chiral symmetry no longer
holds and the FB is no longer flat. Again, we observe
a similar behavior as in the former case (i.e., when
g < η < J√∆). Indeed, in this regime the interaction
strength between distant emitters might be enhanced
for certain disorder realizations. This can be seen as
a signature of the delocalization of the CLSs due to
weak disorder and, as a consequence, an increase of
the localization length of the bound state.
The above discussion provides evidence that there
exist regimes where the essential phenomena pre-
dicted in our work can show up even in the presence
of disorder. A comprehensive study of the effect of
disorder goes beyond the scope of the manuscript and
it will be the subject of a future work.
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