RKKY interaction in one-dimensional flat-band lattices Katharina Laubscher1Clara S. Weber2 3Maximilian H unenberger1 Herbert Schoeller4Dante M. Kennes4 5Daniel Loss1and Jelena Klinovaja1

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RKKY interaction in one-dimensional ﬂat-band lattices

Katharina Laubscher,1, ∗Clara S. Weber,2, 3, ∗Maximilian H¨unenberger,1

Herbert Schoeller,4Dante M. Kennes,4, 5 Daniel Loss,1and Jelena Klinovaja1

1Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

2Institut f¨ur Theorie der Statistischen Physik, RWTH Aachen University and JARA

- Fundamentals of Future Information Technology, D-52056 Aachen, Germany

3Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

4Institut f¨ur Theorie der Statistischen Physik, RWTH Aachen, 52056 Aachen,

Germany and JARA - Fundamentals of Future Information Technology

5Max Planck Institute for the Structure and Dynamics of Matter,

Center for Free Electron Laser Science, 22761 Hamburg, Germany

(Dated: November 7, 2023)

We study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two classical mag-

netic impurities in one-dimensional lattice models with ﬂat bands. As two representative examples,

we pick the stub lattice and the diamond lattice at half ﬁlling. We ﬁrst calculate the exact RKKY

interaction numerically and then compare our data to results obtained via diﬀerent analytical tech-

niques. In both our examples, we ﬁnd that the RKKY interaction exhibits peculiar features that

can directly be traced back to the presence of a ﬂat band in the energy spectrum. Importantly,

these features are not captured by the conventional RKKY approximation based on non-degenerate

perturbation theory. Instead, we ﬁnd that degenerate perturbation theory correctly reproduces our

exact results if there is an energy gap between the ﬂat and the dispersive bands, while a nonpertur-

bative approach becomes necessary in the absence of a gap.

I. INTRODUCTION

Magnetic impurities embedded in a host material can

interact indirectly by coupling to the electron spin den-

sity of the host. This so-called Ruderman-Kittel-Kasuya-

Yosida (RKKY) interaction [1–3] can result in a magnetic

ordering of the impurity spins, leading to a wide range

of interesting phenomena with potential applications in

the ﬁelds of spintronics [4, 5], spin-based quantum com-

putation [6–10], or engineered topological superconduc-

tivity [11–21]. The exact form of the RKKY interac-

tion depends on the properties—in particular, the band

structure—of the underlying host material and has been

extensively studied for various types of systems [22–54].

Conventionally, the RKKY interaction is calculated

in second-order perturbation theory assuming that the

exchange coupling between the impurity spins and the

itinerant electrons is small compared to the typical en-

ergy scale of the latter. Recently, however, systems

with so-called ﬂat bands have attracted signiﬁcant at-

tention [55, 56]. The energy of these bands is completely

independent of momentum (see Fig. 1) or, in a weaker

sense, at least approximately constant over a large range

of allowed momenta. While the recent interest in ﬂat-

band systems has mainly been fueled by signiﬁcant the-

oretical and experimental progress on Moir´e materials

such as twisted bilayer graphene [57–62], ﬂat bands can

also emerge as Landau levels in two-dimensional electron

gases subjected to a strong magnetic ﬁeld or in a variety

of artiﬁcial lattice models [63–69], some of which have

successfully been realized in experiments using photonic

lattices or cold-atom setups [70–75].

In the presence of ﬂat bands, the vanishing band width

and the large degeneracy make it questionable whether

FIG. 1. (a,b) Stub lattice. The unit cell (dashed rectan-

gle) consist of three sites (orange dots) labeled A,B, and C.

Nearest-neighbor sites are connected by a hopping term of

strength t(black lines). The ﬂat band is spanned by a set

of CLSs living on three sites each (red and blue dots). The

amplitudes of the unnormalized CLSs are +1 (−1) for the red

(blue) sites. The dispersive bands (green) are separated from

the ﬂat band (cyan) by an energy gap Egap =t. (c,d) Dia-

mond lattice. Here, the CLSs have support on two sites each.

The dispersive bands (green) linearly intersect the ﬂat band

(cyan).

the conventional perturbative approach to the RKKY

interaction is still applicable [76]. This issue was ﬁrst

touched upon in the context of zigzag graphene nanorib-

bons, where exact numerical studies of edge impurities re-

vealed unconventional features of the RKKY interaction

that had not been captured by preceding analytical stud-

ies [33, 77]. Later, Ref. [78] found unconventional ﬁrst-

order contributions to the RKKY interaction in partially

ﬁlled graphene Landau levels via degenerate perturbation

arXiv:2210.10025v3 [cond-mat.mes-hall] 4 Nov 2023

theory. A few more recent studies calculate the standard

second-order contribution to the RKKY interaction in

two-dimensional ﬂat-band lattice models (in particular,

in the Lieb lattice) [79, 80], while Ref. [81] points out

that this does not capture certain ﬂat-band eﬀects in the

Kondo-Lieb model. However, a more general understand-

ing of RKKY eﬀects in ﬂat-band systems—including, in

particular, insights regarding the applicability and lim-

itations of perturbation theory—is still lacking. With

this motivation, we carefully study the RKKY interac-

tion in two simple one-dimensional (1D) ﬂat-band sys-

tems at half ﬁlling, see Fig. 1. We ﬁrst calculate the

exact RKKY interaction numerically and then compare

our data to results obtained via diﬀerent analytical tech-

niques. In both our examples, we ﬁnd that the RKKY

interaction exhibits peculiar features that are not cap-

tured by the conventional RKKY approximation based

on non-degenerate perturbation theory. Instead, we ﬁnd

that degenerate perturbation theory correctly reproduces

our exact results if there is an energy gap between the

ﬂat and the dispersive bands, while a nonperturbative

approach becomes necessary in the absence of a gap.

II. MODELS

A unit cell of the stub lattice consists of three sites

labeled by l∈ {A, B, C}, see Fig. 1(a). Neighboring sites

are coupled by a hopping element of strength t > 0, such

that

Hstub =tX

nc†

n,Acn,B +c†

n,Acn,C +c†

n+1,Acn,B + H.c.

(1)

Here, c†

n,l (cn,l) creates (destroys) a spinless electron

on sublattice lin the nth unit cell. Imposing periodic

boundary conditions on a chain with Nunit cells, the

Hamiltonian can be rewritten in momentum space as

Hstub =PkΨ†

kH(k)Ψkwith Ψk= (ck,A, ck,B , ck,C )T

and

H(k) = t



0 1 + eika 1

1 + e−ika 0 0

1 0 0

,(2)

where adenotes the lattice spacing. The corresponding

bulk spectrum consists of two dispersive bands E±(k) =

±tp3 + 2 cos (ka) as well as one completely ﬂat band

E0(k) = 0 that is separated from the dispersive bands

by an energy gap Egap =t, see Fig. 1(b). The ﬂat band

is macroscopically degenerate and is spanned by a set

of Nlinearly independent states. These can be chosen

to have support on only three lattice sites each: |vn⟩=

(|n, C⟩−|n, B⟩+|n+ 1, C⟩)/√3 for n∈ {1, ..., N}and

where we identify N+1 ≡1 to simplify the notation. One

of these so-called compact localized states (CLSs) [56, 64]

is visualized in Fig. 1(a). While the CLSs are chosen

such that they are strictly localized, they are not mutu-

ally orthogonal. In order to construct a set of mutually

orthogonal basis states for the ﬂat band, the strict local-

ization has to be traded in for exponential localization,

e.g., by changing to a basis of maximally localized Wan-

nier states.

A unit cell of the diamond lattice consists of three sites

as well, see Fig. 1(c). The Hamiltonian is given by

Hdia =tX

nc†

n,Acn,B +c†

n,Acn,C

+c†

n+1,Acn,B +c†

n+1,Acn,C + H.c.(3)

In momentum space, this leads to Hdia =PkΨ†

kH(k)Ψk

with

H(k) = t



0 1 + eika 1 + eika

1 + e−ika 0 0

1 + e−ika 0 0 

.(4)

Again, the bulk spectrum consists of two dispersive bands

E±(k) = ±2√2tcos (ka/2) and a ﬂat band E0(k) = 0, see

Fig. 1(d). Importantly, however, there is now no energy

gap separating the ﬂat band from the dispersive bands.

Rather, the two dispersive bands linearly intersect the

ﬂat band at ka =π. The ﬂat band can again be de-

scribed in terms of a set of CLSs having support on two

lattice sites each, see Fig. 1(c). Explicitly, their wave

functions are given by |vn⟩= (|n, C⟩−|n, B⟩)/√2.Both

the stub and the diamond lattice are bipartite lattices

with one sublattice given by all Asites and the other one

by all Band Csites. Furthermore, we note that the ﬂat

band of the stub lattice is topologically trivial, i.e., its 1D

topological invariant (winding number) is zero, while it

is not meaningful to assign a topological invariant to the

ﬂat band of the diamond lattice as it is not energetically

isolated.

III. RKKY INTERACTION

We now consider a system of spinful electrons at zero

temperature with both spin species independently de-

scribed by Hstub or Hdia. Throughout this work, we set

the chemical potential µ= 0 and focus on the case of

a half-ﬁlled ﬂat band. However, we have checked that

our results do not depend on the exact ﬁlling factor as

long as the ﬂat band stays partially ﬁlled. Two mag-

netic impurities are placed in the unit cells n1and n2

at sublattice positions αand β, respectively. The local

exchange coupling between the impurity spins and the

itinerant electrons is described as H(1)

imp +H(2)

imp with

H(i)

imp =¯

σ,σ′

c†

ni,li,σ [Si·σ]σσ′cni,li,σ′,(5)

where we have deﬁned l1=αand l2=β. Compared

to Eqs. (1) and (3), the electronic creation (annihilation)

operators c†

n,l,σ (cn,l,σ) now carry an additional spin label

FIG. 2. Absolute value of the RKKY coupling |Jαβ

RKKY|in the

stub lattice in dependence on the inter-impurity distance R,

calculated via ED and displayed on a logarithmic scale. For all

sublattice conﬁgurations, |Jαβ

RKKY|decays exponentially with

R. Here, J1=J2= 0.2t.

σ∈ {↑,↓}. Furthermore, σis the vector of Pauli matri-

ces, Siare classical impurity spins with Si=|Si| ≫ 1,

and ¯

Ji≥0 denotes the exchange coupling between the

impurity spin and the electron spin density. For simplic-

ity, we also deﬁne Ji=¯

JiSi.

Since there is no spin-orbit interaction in our problem,

the indirect exchange interaction between the two impu-

rity spins is isotropic and can be written as

HRKKY =Jαβ

RKKY ˆ

S1·ˆ

S2(6)

with ˆ

Si=Si/Si. Here, the eﬀective RKKY coupling

constant Jαβ

RKKY ≡Jαβ

RKKY(R) depends on the sublattice

position of the impurities and on the inter-impurity dis-

tance R=r2−r1>0 with ri=nia. The exact RKKY

coupling [76] can be obtained from the exact ground state

energies Eαβ

FM and Eαβ

AFM for the ferromagnetic (FM) and

antiferromagnetic (AFM) conﬁguration of the impurities

with respect to an arbitrarily chosen spin quantization

axis, say, the zaxis, such that Si= (0,0,±Si):

Jαβ

RKKY = (Eαβ

FM −Eαβ

AFM)/2.(7)

The energies Eαβ

FM/AFM can be computed numerically via

exact diagonalization (ED) [33] or, alternatively, via the

exact lattice Green functions using the optimized algo-

rithm presented in Appendix F. This second approach

allows us to study signiﬁcantly larger system sizes while

at the same time improving the numerical accuracy of

our results.

A. Stub lattice

We start by studying the RKKY interaction in the

stub lattice. Our numerical results show that Jαβ

RKKY

decays exponentially with Rfor all sublattice conﬁgura-

tions, see Fig. 2. For the AA conﬁguration, this is not

surprising since the ﬂat-band states do not have support

on the Asublattice. As such, we expect to recover the

FIG. 3. RKKY coupling Jαβ

RKKY in the stub lattice in depen-

dence on J1calculated via ED (black) and lowest-order per-

turbation theory (blue: ﬁrst-order, red: second-order). The

standard second-order approximation [Eq. (8)] gives the cor-

rect lowest-order approximation for (a) JAA

RKKY, (c) JAB

RKKY,

and (d) JBA

RKKY. The approximation in (d) is worse than in

the other cases since, as J1increases, an unconventional third-

order term ∝J2

1J2originating from the ﬂat band becomes

important. (b) JBB

RKKY shows an unconventional behavior due

to a ﬁrst-order contribution originating from the ﬂat band,

see Eq. (9). Here, J2/t = 0.05 and R/a = 5 [85].

usual Bloembergen-Rowland behavior found in conven-

tional insulators [82]. In fact, for all sublattice conﬁgura-

tions involving at least one impurity on the Asublattice,

virtual transitions between the gapped dispersive bands

yield the dominant contribution to the RKKY interac-

tion. For conﬁgurations involving only the Band Csub-

lattice, on the other hand, the ﬂat-band states give an

additional contribution that is responsible for the signiﬁ-

cantly larger absolute value of Jαβ

RKKY in these cases (see

also below). However, the ﬂat-band states are spatially

localized (e.g., they can be constructed as exponentially

localized Wannier states), such that their contribution is

exponentially suppressed with Ras well. Furthermore, in

accordance with the general result for bipartite lattices

at half ﬁlling [27], we ﬁnd that the ground state is FM

(AFM) if the two impurities are located on the same (on

diﬀerent) sublattices of the bipartition.

To gain further insight, we study Jαβ

RKKY in dependence

on one of the exchange coupling constants—say, J1—for

J1,2/t ≪1. We ﬁnd that JAA

RKKY ∝J1, see Fig. 3(a). This

is the functional dependence expected from the standard

expression for the RKKY interaction in second-order per-

turbation theory at zero temperature [83],

Jαβ

RKKY =−J1J2

2πZ0

−∞

dE Im[G(0)

αβ (R, E)G(0)

βα(−R, E)],

(8)

where G(0)

αβ are the retarded single-particle Green func-

tions of the unperturbed system for a single spin

species [84]. Evaluating Eq. (8) by using the analyti-

cal expression for G(0)

AA (see Appendix A), we see that it

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RKKYinteractioninone-dimensionalflat-bandlatticesKatharinaLaubscher,1,∗ClaraS.Weber,2,3,∗MaximilianH¨unenberger,1HerbertSchoeller,4DanteM.Kennes,4,5DanielLoss,1andJelenaKlinovaja11DepartmentofPhysics,UniversityofBasel,Klingelbergstrasse82,CH-4056Basel,Switzerland2Institutf¨urTheoriederStatistischenP...

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RKKY interaction in one-dimensional flat-band lattices Katharina Laubscher1Clara S. Weber2 3Maximilian H unenberger1 Herbert Schoeller4Dante M. Kennes4 5Daniel Loss1and Jelena Klinovaja1

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