Topological superconductivity from doping a triplet quantum spin liquid in a at band system Manuel Fern andez L opez1Ben J. Powell2and Jaime Merino1

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Topological superconductivity from doping a triplet quantum spin liquid in a ﬂat

band system

Manuel Fern´andez L´opez,1Ben J. Powell,2and Jaime Merino1

1Departamento de F´ısica Te´orica de la Materia Condensada,

Condensed Matter Physics Center (IFIMAC) and Instituto Nicol´as Cabrera,

Universidad Aut´onoma de Madrid, Madrid 28049, Spain

2School of Mathematics and Physics, The University of Queensland, QLD 4072, Australia

(Dated: October 12, 2022)

We explore superconductivity in strongly interacting electrons on a decorated honeycomb lattice

(DHL). An easy-plane ferromagnetic interaction arises from spin-orbit coupling in the Mott insu-

lating phase, which favors a triplet resonance valence bond spin liquid state. Hole doping leads to

partial occupation of a ﬂat band and to triplet superconductivity. The order parameter is highly

sensitive to the doping level and the interaction parameters, with p+ip,fand p+fsupercon-

ductivity found, as the ﬂat band leads to instabilities in multiple channels. Typically, ﬁrst order

transitions separate diﬀerent superconducting phases, but a second order transition separates two

time reversal symmetry breaking p+ip phases with diﬀerent Chern numbers (ν= 0 and 1). The

Majorana edge modes in the topological (ν= 1) superconductor are almost localized due to the

strong electronic correlations in a system with a ﬂat band at the Fermi level. This suggests that

these modes could be useful for topological quantum computing. The ‘hybrid’ p+fstate does not

require two phase transitions as temperature is lowered. This is because the symmetry of the model

is lowered in the p-wave phase, allowing arbitrary admixtures of f-wave basis functions as overtones.

We show that the multiple sites per unit cell of the DHL, and hence multiple bands near the Fermi

energy, lead to very diﬀerent nodal structures in real and reciprocal space. We emphasize that this

should be a generic feature of multi-site/multi-band superconductors.

I. INTRODUCTION

Understanding the mechanism of superconductivity in

strongly correlated materials remains a formidable chal-

lenge. For instance, high-Tccuprates [1–3], organics[4]

and twisted bilayer graphene [5,6] (TBG), display sim-

ilar phase diagrams[7]. In these materials superconduc-

tivity emerges near to a Mott insulating phase, indicat-

ing the important role played by strong correlations on

Cooper pairing. A common ingredient in these systems

is the presence of antiferromagnetic interactions, which

lead to Mott insulators with antiferromagnetic order, as

in the cuprates, or quantum spin liquids, as in the or-

ganic materials κ-(BEDT-TTF)2Cu(CN)3or κ-(BEDT-

TTF)2Ag(CN)3[4,8,9]. Such AF interaction is crucial

to singlet d-wave (or d+id) superconductivity which,

according to Andersons theory of high-Tcsuperconduc-

tivity [10], can emerge when hole doping the resonance

valence bond (RVB) Mott insulator on the square (tri-

angular [11]) lattice [12]. The discovery of superconduc-

tivity in magic angle TBG [6] poses new questions about

strongly correlated superconductivity in ﬂat bands [13].

Deﬁnitive signatures of triplet pairing are rare out-

side the well understood triplet superﬂuidity [14,15] in

3He. At present, there is no unambiguous evidence for,

and increasing evidence against, triplet pairing in (for-

mer) candidate materials such as Sr2RuO4[16–18], the

quasi-one-dimensional (TMTSF)2X Bechgaard salts[19]

or A2Cr3As3[20,21], (A=K, Rb, Cs). A reason for the

scarcity of triplet superconductors is the antiferromag-

netic (AFM) superexchange between spins which often

dominates over ferromagnetic exchange processes, likely

to stabilize triplet superconductivity. However, the su-

perconductivity observed [22] in the ferromagnetic Mott

insulators, CrXTe3with X=Si, Ge has been predicted to

be of the triplet type [23]. Perhaps the most promising

class of materials for observing triplet superconductors

are the uranium based heavy fermion materials [24,25],

where superconductivity is often found near ferromag-

netism.

Quantum spin liquids with ferromagnetic interactions

cannot be described through standard singlet RVB the-

ory. However, recent extensions to easy-plane ferromag-

netic triangular lattices predict the existence of triplet

resonance valence bond (tRVB) Mott insulators which

can become unconventional p+ip-wave superconductors

under hole doping[23]. On the other hand, in multior-

bital systems such as the iron pnictides, Hunds coupling

can induce an intra-atomic triplet RVB state [26]. In

these cases, triplet superconductivity may arise under

hole doping. This is allowed by the presence of an even

number of atoms per unit cell, leading to a spatially stag-

gered gap pattern, as proposed by Anderson [27] in the

context of heavy fermion superconductivity.

Triplet pairing induced by ferromagnetic interactions

may arise in certain organic and organometallic materi-

als with unit cells containing many atoms. (EDT-TTF-

CONH2)6[Re6Se8(CN)6] [28], Mo3S7(dmit)3[29], and

Rb3TT·2H2O [30] crystals with layers of decorated hon-

eycomb lattices (DHLs) can potentially host rich physics

arising from the interplay of strong correlations, Dirac

points, quadratic band touching points and ﬂat bands

[31–35]. This lattice also occurs in several metal organic

frameworks and coordination polymers [35]. Indeed, un-

arXiv:2210.05275v1 [cond-mat.supr-con] 11 Oct 2022

conventional singlet f-wave pairing has been found in

an AFM t-Jmodel on the DHL [36]. However, the

spin molecular-orbital coupling (SMOC) present in these

systems,[37–40] can lead to easy-plane ferromagnetic in-

teractions favoring tRVB states. Hence, it is interest-

ing to address the question of whether triplet supercon-

ductivity emerges in a strongly correlated easy-plane fer-

romagnetic model on a DHL. The possibility of ﬁnding

non-trivial topological superconductivity as well as the

role played by the ﬂat bands deserve special attention.

In this paper we study a t-Jmodel on a DHL with XXZ

interactions, which arise due to spin-orbit coupling. Ex-

act diagonalization of small clusters shows that a tRVB

spin liquid, is a competitive ground state of the model

at half-ﬁlling. Hole doping this spin liquid state leads to

partial occupation of a ﬂat band. Based on this we ap-

ply the tRVB approach to search for superconductivity

in the hole doped system. We ﬁnd multiple triplet su-

perconducting phases, including include p+ip,p+f,f.

Presumably, the wide variety of superconducting phases

is a consequence of the ﬂat band leading to instabilities

in multiple Cooper channels.

Interestingly, we ﬁnd both topologically trivial and

topological p+ip superconductivity with Chern numbers,

ν= 0 and 1 respectively. These phases are separated

by a continuous phase transition, where nodes appear in

the otherwise fully gapped p+ip order parameter. In

the topological superconductor phase the single Majo-

rana edge mode expected for ν= 1 is almost localized

since it traverses a tiny gap bounded by nearly ﬂat bands.

We show that the p+fstate does not require a two

superconducting phase transitions as the temperature is

lowered. Rather the once the symmetry of the system is

lowered by going into the p-wave superconducting phase,

the pand fsolutions belong to the same irreducible rep-

resentation of the group describing the symmetry of the

model. Therefore, pand fare overtones and the system

can and does take advantage of this to lower its energy.

The DHL lattice has six sites per unit cell. There-

fore, diagonalizing the Hamiltonian requires a Bogoli-

ubov transformation, a Fourier transformation and a

transformation from the multi-site basis to a multiband

basis. We show that the latter leads to remarkably diﬀer-

ent nodal structures in real and reciprocal space. We em-

phasize that this is a very general phenomenon in multi-

band superconductors.

We show that, at strong coupling (low doping), where

the magnetic exchange is dominant, real space pairing

dominates, leading to order parameters that are fully

gapped or have only isolated nodal points in reciprocal

space. In contrast, at weak coupling (high doping), where

the superconductivity is dominated by the kinetic energy,

the superconducting gap displays nodal points in recip-

rocal space.

The paper is organized as follows: in Sec II we in-

troduce the anisotropic t-Jmodel on a DHL with XXZ

magnetic exchange interactions. In Sec. III we show, us-

ing exact diagonalization of small clusters, that a tRVB

spin liquid, is a competitive ground state of the model at

half-ﬁlling. In Sec. IV we show that a wide variety triplet

superconducting states emerge on hole-doping the tRVB,

including a topological superconductor (TSC), and give

a detailed characterization of these states. In Sec. Vwe

characterize the TSC by calculating the topological in-

variants and Majorana edge modes. Finally, in Sec. VI

we conclude our work by summarizing our main results

and their relevance to actual materials realizing DHLs.

II. MODEL

Our starting point is a t-Jmodel with easy-plane XXZ

ferromagnetic exchange on the DHL:

H=−tX

αijσ

PGc†

αiσcαj,σ +c†

αjσcαiσ PG

−t0X

hA,Biiσ

PGc†

AiσcBiσ +c†

BiσcAiσPG

−JX

αij Sx

αiSx

αj +Sy

αiSy

αj −Sz

αiSz

αj +1

4nαinαj 

−J0X

hA,BiiSx

AiSx

Bi +Sy

AiSy

Bi −Sz

AiSz

Bi +1

4nAinBi

+µX

αiσ

c†

αiσcαiσ,(1)

where PG= Πi(1 −ni↑ni↓) is the Gutzwiller projec-

tion operator which excludes doubly occupied sites com-

pletely, c(†)

αiσ is the usual annihilation (creation) operator,

and Sr

αi is the rth component of the spin operator. The

α-index runs over the two triangular clusters, while i, j

run over the three sites within each triangular clusters,

with site numbering as illustrated in Fig. 1, and σde-

notes the spin of the electron. Angled brackets indicate

that the sums are restricted to (triangles that contain)

nearest-neighbor sites. We are interested in the prop-

erties of the model close to half-ﬁlling, so we write the

electron density n= 1 −δ, where δis the density of holes

doped into the half-ﬁlled DHL. We ﬁx t= 1 as the en-

ergy scale. The XXZ exchange couplings, J, J0>0, lead

to easy-plane ferromagnetic (FM) interactions and AFM

longitudinal interaction.

A possible microscopic origin of the XXZ exchange in-

teractions of model (1) can be found by considering a sin-

gle s-like orbital with on site (Hubbard) interactions and

SMOC. On transforming to real space SMOC is equiva-

lent to an anisotropic Kane-Mele spin-orbit coupling. If

we consider nearest neighbor (nn) interactions with a z

component of the spin-orbit coupling only we have

H=iλ X

hαiβji

(c†

αi↑cβj↑−c†

αi↓cβj↓) + UX

iα

niα↑niα↓,(2)

where niασ =c†

iασciασ. In the large-Ulimit, the Hubbard

FIG. 1. The t-Jmodel on the decorated honeycomb lattice.

The unit cell consists of a A (red) and a B (blue) triangles

with the sites numbered as shown. The parameters entering

the model are illustrated. A particular triplet covering of the

lattice entering the |tRVBistate of the model is shown.

model maps onto an eﬀective XXZ spin hamiltonian [41]

HXXZ =−4λ2

hαiβjiSx

αiSx

βj +Sy

αiSy

βj −Sz

αiSz

βj .(3)

At half-ﬁlling, the Hubbard model (2) with U |λ|be-

comes an XXZ model with J= 4λ2/U, J0= 4λ02/U,

i.e. Eq. (1), as is corroborated by our exact denation-

alization (ED) calculations of Appendix D. On includ-

ing direct the hopping terms tand t0additional AFM

exchange, Dzyaloshinskii-Moriya, and oﬀ diagonal ex-

change interactions will be introduced [37,42,43]. We

neglect these interactions for simplicity. Thus Eq. (1)

is a toy model for understanding the impact of SMOC

on superconductivity. We further assume that λ∝tand

λ0∝t0so that J0/J = (t0/t)2reducing the number of

independent parameters in our Eq. (1). Other mecha-

nisms for easy-plane ferromagnetic interaction have also

been put forward recently, including interactions arising

from the interplay of Hund’s and Kondo interactions in

heavy fermions.[44]

We emphasize that diﬀerent materials are known to

span a wide range of the parameter space for this

model. For example, in Mo3S7(dmit)3is in the trimer-

ized limit: t0< t and J0< J [38,45], whereas in (EDT-

TTF-CONH2)6[Re6Se8(CN)6] [28] and Rb3TT·2H2O [30]

are in the dimerized limit: t0> t and J0> J.

The physics of these two limits is known to diﬀer

markedly [32,35]. Furthermore, at ambient pressure,

(EDT-TTF-CONH2)6[Re6Se8(CN)6] undergoes a metal-

insulator transition as the temperature is lowered [28],

whereas Mo3S7(dmit)3[29] and Rb3TT·2H2O [30] are

insulating at all temperatures at ambient pressure.

III. TRIPLET RVB QUANTUM SPIN LIQUID

We analyze the ground state of model (1) by us-

ing tRVB theory[46], which has recently applied to iron

pnictides[26] and transition metal chalcogenides[23]. It is

analogous to Andersons RVB theory of high-Tcsupercon-

ductivity to deal with strongly correlated models contain-

ing ferromagnetic exchange. Triplets rather than singlet

bonds are the building blocks of the theory. The simplest

form of the theory assumes that the ground state of the

ferromagnetic undoped model is a quantum spin liquid

in which spins are paired into triplets.

In general, such triplet RVB state can be expressed as:

|tRVBi=X

AP|Pti,(4)

where |Pti= Πij∈P|ijiand the Sz= 0 spin triplet be-

tween sites i, j is

|iji ≡ |i, ↑i|j, ↓i +|i, ↓i|j, ↑i

√2,(5)

where |iji=−|jiidue to the symmetry of triplets un-

der inversion. Hence, the |tRVBistate is a quantum

superposition of all possible coverings of the lattice into

triplet valence bonds. The simplest version of the |tRVBi

state involves triplets between nn spins only. One possi-

ble snapshot triplet conﬁguration |Ptientering |tRVBiis

shown in Fig. 1.

A. Exact analysis of the triplet RVB in the DHL

We ﬁrst concentrate on model (1) at half-ﬁlling, δ= 0,

which becomes a easy-plane XXZ ferromagnetic model.

Triplet RVB states in such spin model are explored

by performing exact diagonalization on six-site (two-

triangle) clusters. The dependence of the exact level

spectra of the cluster with J0/J is shown in Fig. 2(a).

For any J0/J 6= 0 the four-fold ground state degeneracy

found at J0= 0 is split so that the ground state becomes

a non-degenerate S= 1, Sz= 0 triplet, as expected.

The ﬁrst excitation corresponds to a S= 1, Sz=±1

doublet. The splitting between the S= 1, Sz= 0 and

S= 1, Sz=±1 states is increased by J0/J. The ground

state energy at J0=Jis 0(exact) = −3.541287J, and

the corresponding wavefunction is given in the Appendix.

We now consider a simple nn tRVB ansatz for the

ground state wavefunction of the cluster, which consists

on a linear combination of nn triplet valence bonds (VB)

only:

|tRVBi=|A1B1i|A2A3i|B2B3i+|A1A3i|A2B2i|B1B3i

+|A1A2i|A3B3i|B1B2i − |A1B1i|A2B2i|A3B3i,

(6)

with the corresponding energy

0(nn-tRVB) = hnn-tRVB|HXXZ|nn-tRVBi

hnn-tRVB|nn-tRVBi.(7)

0.0 0.5 1.0 1.5 2.0

-5

-4

-3

-2

-1

J'/J

ϵi

(a)

(b)

<Ψ0(tRVB)Ψ0(exact)>

(ϵ0(tRVB)-ϵ0(exact))/J

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

J'/J

1.0

FIG. 2. Exact triplet RVB analysis of the half-ﬁlled easy-

axis XXZ ferromagnetic model on a six site (two triangle)

cluster. (a) The dependence of the low energy spectrum of the

model (1) on J0/J. The numbers denote the exact energy level

degeneracies. (b) The overlap between the exact ground state

and the tRVB wavefunction for the six site cluster (given in

Eq. (8)), |hΨ0(tRVB)|Ψ0(exact)i|, and the diﬀerence between

the exact ground state energy, 0(exact), and the triplet RVB

energy, 0(tRVB).

For J0=Jthe overlap with the exact ground state is

hΨ0(exact)|nn-tRVBi= 0.9747, its energy is 0(nn-tRVB) =

−3.397058Jwhich is 4.07% higher than the exact ground

state energy. In contrast, an RVB analysis of the Heisen-

berg antiferromagnetic model on the same cluster using nn

singlet VBs would give a much more accurate description of

the exact ground state: hΨHAFM

0(exact)|nn-RVBi= 0.9988

and HAF M

0(nn −RVB) = −3.04412J. The energy of this

RVB state is only 0.16% higher than the exact ground state

energy, HAF M

0(exact) = −3.052775J. Hence, it appears that

the nn RVB gives a much better description of the ground

state of the AFM Heisenberg model than the nn-tRVB of the

easy-plane XXZ magnetic exchange we consider in this work.

We therefore consider a more general nn tRVB ansatz:

|tRVBi=α(|A1B1i|A2A3i|B2B3i+|A1A3i|A2B2i|B1B3i

+|A1A2i|A3B3i|B1B2i)−β|A1B1i|A2B2i|A3B3i,

(8)

where αand βare determined variationally. For J0=Jwe

ﬁnd an energy of 0(tRVB) = −3.453257Jwhich is only 2.5%

higher than the exact ground state energy. The overlap with

the exact wavefunction is now hΨ0(exact)|tRVBi= 0.98849,

improving our previous result. For comparison a singlet-RVB

version of (8) for the Heisenberg antiferromagnet (HAF) re-

covers its exact ground state, i.e., hRVB|ΨAFM

0(exact)i= 1,

with an energy 0(RVB) = AFM

0(exact) = −5.302775J.

We extend the variational |tRVBianalysis to other J06=J.

The dependence of the overlap and the energy diﬀerence on

J0/J shown in Fig. 2(b) indicates that the |tRVBi(Eq. (8))

becomes a closer description of the exact ground state of the

cluster as J0/J increases since htRVB|Ψ0(exact)i → 1 and

0(tRVB) −0(exact) →0. Furthermore, the large overlap

found htRVB|Ψ0(exact)i ∼ 1 between the tRVB state and

the exact ground state in the whole J0/J > 0 range explored

indicates that the |tRVBiis a good candidate for the ground

state of the model.

The properties of quantum dimer models [47,48] are inde-

pendent of whether the dimers represent singlets or triplets.

This suggest that dimerised states will be common to both

singlet and triplet RVB theories. Therefore, it is interesting

to note that two diﬀerent valence bond solids are found in the

antiferromagnetic Heisenberg model on the DHL: a valance

bond solid with the full symmetry of the lattice for J0&J

and a valence bond solid with broken C3symmetry for J0.J

[35,49].

IV. TRIPLET RVB SUPERCONDUCTIVITY

Having established the tRVB state as a competitive ground

state of the undoped model (1), we now consider the possibil-

ity of superconductivity mediated by the ferromagnetic part

of the XXZ interaction. We ﬁrst describe the pairing states

allowed by the symmetries of the DHL. The most favorable

superconducting states are then determined by a microscopic

calculation. [23,26]

A. Symmetry group analysis of superconducting

states

A general phenomenological formulation of superconduc-

tivity can be achieved using Ginzburg-Landau theory in which

the order parameter is obtained from general symmetry ar-

guments. The full symmetry group of our model (1) is

C6v⊗K⊗U(1)⊗SU(2) where Kis the time-reversal symmetry

group.

On doping the triplets pairs in the tRVB wavefunction be-

come mobile leading to triplet superconductivity. Note that

this condensate contains only opposite spin pairing (OSP),

in contrast to the equal spin pair (ESP) found in, say, the

A-phase of 3He [15]. Thus, one can fully describe the su-

perconductivity by a scalar order parameter ∆(k) instead of

requiring the usual vector order parameter for a triplet su-

perconductor, d(k). However, these are trivially related via

d(k) = ∆(k)ˆz, where ˆzis the unit vector in the longitudinal

direction in spin-space. Thus, all states consider also break

the SU(2) symmetry associated with spin rotation. This is

unsurprising as the XXZ interaction arises from spin-orbit

coupling.

We conveniently [50] express the elements of the C6vgroup

in a 18-dimensional space generated by the pairing ampli-

tudes, ∆αiσ,βjσ0=hcαiσ cβjσ0iconsistent with the transla-

tional symmetry of the lattice. Deﬁning the 18-component

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TopologicalsuperconductivityfromdopingatripletquantumspinliquidinaatbandsystemManuelFernandezLopez,1BenJ.Powell,2andJaimeMerino11DepartamentodeFsicaTeoricadelaMateriaCondensada,CondensedMatterPhysicsCenter(IFIMAC)andInstitutoNicolasCabrera,UniversidadAutonomadeMadrid,Madrid28049,Spain2Schoolo...

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Topological superconductivity from doping a triplet quantum spin liquid in a at band system Manuel Fern andez L opez1Ben J. Powell2and Jaime Merino1

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